Turbine Design And Application Vol123

NASA SP-290

/N-.o7

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TIIItI|INI{

I)KSll;l

a,id AI'I'I, II:AI'IIiN

(NASA-SP-290) APPLICATION

TURBINE (NASA)

DESIGN 390

N95-22341

AND

p Unclas

HlI07

NATIONAL

AERONAUTICS

AND

SPACE

0041715

ADMINISTRATION

NASA

SP-290

'lrlllel|lNl IDK Iq ,N annalAIDIDI,Iq;A"I' IqlPm

Edited by Arthur J. Glassman Lewis Research Center

Scientific and Technical Information Program--1994 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Washington,

DC

Available NASA

from the Center

800 Elkridge

for AeroSpace Landing

Linthicum Heights, Price Code: A17 Library

of Congress

Information

Road MD 21090-2934 Catalog

Card Number:

94-67487

PREFACE NASA and

has

space

an interest

turboshaft

spacecraft.

fluids,

interest

for turbine

trains,

etc.)

In view a

Program.

The

1972-73.

Various

entitled

course and

velocity

cooling, The

mechanical notes written

edited

for

are the

U.S.

sets

of units

units

and

after

Such

set

of

will

satisfy

units

consistent symbol

customary

units.

by including defining

a

Research

Application"

was

Graduate again

were

Study

presented

covered

can

serve

means

for

in

including

fundamental

aerodynamic

course,

power.

at Lewis

and

a publication

buses,

turbine design,

blade

revised

and

as a foundation self-study,

or

topics.

used the

and

blade

of

trucks,

In-House

concepts,

losses,

turbine

consistent

Design revised

providing

electrical

efforts

technology

gases,

applications

(cars,

and the

auxiliary

for

Other

and l%o-

inert

design, operation, and performance. and used for the course have been

for selected

given

somewhat of turbine

diagrams,

publication.

commonly

of

fluid-dynamic

introductory

Any

part

using

ground-based

interest as

was

aspects

concepts,

Two

and

aircraft,

studied

land-vehicle

"Turbine

1968-69

thermodynamic

an

power

been

jet

and

engines

spacecraft.

include

for

propulsion

have

for

to aeronautics provide

power

turbine

fluids

power

turbine-system

during

reference

rocket

engines

course

presented

engines

provide

propulsion

of the

primarily

turbine

as auxiliary

metal

electric

related

well

Closed-cycle

and

long-duration

Center,

as

turbines

for

organic

for

Airbreathing

propulsion,

pellant-driven power

in turbines

applications.

as unity

sets

of

definitions. A

single

all constants those

not

,°°

111

the units

These set

of

required required

equations and are

presented.

constant the

SI

equations for the

values units

covers U.S.

and both

customary

for the SI units. ARTHUR J. GLASSMAN

a

i _

,_

CONTENTS CHAPTER

PAGE

PREFACE

°

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • . . . . . . . . . . .

THERMODYNAMIC CONCEPTS

AND

by Arthur

FL UID-D

YNAMI

J. Glassman

C

....................

BASIC CONCEPTS AND RELATIONS ....................... APPLICATION TO FLOW WITH VARYING AREA REFERENCES ...................................... SYMBOLS .........................................

,

BASIC

TURBINE

CONCEPTS

by Arthur

1 1 14 19 20

..............

J. Glassman

....

TURBINE FLOW AND ENERGY TRANSFER .................. DIMENSIONLESS PARAMETERS ......................... REFERENCES ...................................... SYMBOLS ......................................... GLOSSARY .........................................

.

VELOCITY DIAGRAMS by Warren Warner L. Stewart ...................................

J. Whitney

BLADE Arthur

DESIGN

by Warner

J. Glassman

21 45 62 63 65

and

L. Stewart

STUDIES

.....

CHANNEL

FLOW

ANALYSIS

lOl

.

INTRODUCTION by William

102 118 124 125

by Theodore

STREAMAND POTENTIAL-FUNCTION VELOCITY-GRADIENT ANALYSIS REFERENCES ...................................... SYMBOLS .........................................

ANALYSES ........................

TO BOUNDARY-LAYER

D. McNally

70 84 95 96 98

and

..................................

SOLIDITY .......................................... BLADE-PROFILE DESIGN .............................. REFERENCES ...................................... SYMBOLS .........................................

,

21

69

MEAN-SECTION DIAGRAMS ............................ RADIAL VARIATION OF DIAGRAMS ....................... COMPUTER PROGRAMS FOR VELOCITY-DIAGRAM REFERENCES ...................................... SYMBOLS .........................................

,

...

111

Katsanis ............

......

127 130 147 154 155

THEORY

...............................

157

NATURE OF BOUNDARY LAYER ......................... DERIVATION OF BOUNDARY-LAYER EQUATIONS SOLUTION OF BOUNDARY-LAYER EQUATIONS CONCLUDING REMARKS .............................. REFERENCES ...................................... SYMBOLS .........................................

............ ..............

157 160 172 188 188 191

V

iI

PAG'E__

I_T_NTtON_!.Ly BLANg

CHAPTER

o

PAGE

BOUNDARY-LAYER

LOSSES

by Herman

W. Prust,

Jr ....

BOUNDARY-LAYER PARAMETERS ........................ BLADE-ROW LOSS COEFFICIENTS ....................... BLADE-ROW LOSS CHARACTERISTICS .................... REFERENCES ....................................... SYMBOLS .........................................

o

MISCELLANEOUS

LOSSES

by Richard

195 201 217 221 223

J. Roelke

........

225

TIP-CLEARANCE LOSS ................................ DISK-FRICTION LOSS ................................. PARTIAL-ADMISSION LOSSES ........................... INCIDENCE LOSS .................................... REFERENCES ....................................... SYMBOLS ..........................................

o

SUPERSONIC

TURBINES

by Louis

225 231 238 243 246 247

J. Goldman

.........

METHOD OF CHARACTERISTICS ......................... DESIGN OF SUPERSONIC STATOR BLADES ................. DESIGN OF SUPERSONIC ROTOR BLADES .................. OPERATING CHARACTERISTICS OF SUPERSONIC TURBINES REFERENCES ....................................... SYMBOLS .........................................

10.

RADIAL-INFLOW

TURBINES

by Harold

E. Rohlik

OVERALL DESIGN CHARACTERISTICS .................... BLADE DESIGN ..................................... OFF-DESIGN PERFORMANCE ........................... REFERENCES ....................................... SYMBOLS .........................................

11.

TURBINE

COOLING

by Raymond

S. Colladay

EXPERIMENTAL DYNAMIC and Harold

DETERMINATION PERFORMANCE by Edward J. Schum ................................

....

.......

250 263 266 272 277 278

279

...........

307 307 314 328 330 332 340 345 347

OF AEROM. Szanca

TEST FACILITY AND MEASUREMENTS .................... TURBINE PERFORMANCE .............................. REFERENCES ....................................... SYMBOLS .........................................

vi

249

284 295 302 305 306

GENERAL DESCRIPTION .............................. HEAT TRANSFER FROM HOT GAS TO BLADE ................ CONDUCTION WITHIN THE BLADE WALL .................. COOLANT-SIDE CONVECTION ........................... FILM AND TRANSPIRATION COOLING ..................... SIMILARITY ........................................ REFERENCES ....................................... SYMBOLS .........................................

12.

193

351 352 374 387 388

CHAPTER 1

Thermodynamic andFluid-Dynamic Concepts ByArthur J. Glassman This

chapter

cepts are

of the

is intended

thermodynamics concepts

energy-transfer treatment textbooks. the

U.S.

needed

to

processes

purposes

given

and

of this

consistent commonly after

units

by

units

and

of

analyze

including

units

will

all

sets

must

know

generalizations of behavior,

The

flow

more

concerning are

far

how

not

and

complete

of

with

their

covers the

for

U.S.

the

values

are

and

the

units both

sets

of

customary

SI units.

RELATIONS

kind

of calculation

volume,

and

resulted

in

behavior. to

SI

State

any has

presented.

constant the

for

required

AND

gases

referred

and

of equations

pressure,

of

equations

are

required

those

very

study

gases

of units

set

CONCEPTS

get

we

These

the A

the

These

constants

as unity

we can

interrelated.

turbine.

satisfy

Equation

gases,

mechanics.

understand

a

con-

can be found in reference 1 and in many to be steady and one-dimensional for

A single

BASIC

Before

fundamental

fluid

and in

definitions.

units.

defining

compressible

consistent

symbol

customary

of the

chapter. set

used

the

some

occurring

of these subjects Flow is assumed

Any Two

to review

as being

In

temperature certain

discussing

either

involving

ideal

laws these or real.

are and laws The

TURBINE

DESIGN

ideal

is only

the

gas real

The

gas

AND

hypothetical

can

ideal

APPLICATION

only

gas

and

approach

equation

obeys

under

of state

various certain

simplified

laws

that

conditions.

is R*

T

pv=--_-

(1-1)

where P

absolute

pressure,

V

specific

R*

universal gas constant, (lb mole) (°R)

volume,

molecular

lb/ft 2

m3/kg;

weight,

absolute The

N/mS;

fta/lb 8314

kg/(kg

temperature,

quantity

R*/M,_

J/(kg

mole); K;

is often

1545

mole)(K);

lb/(Ib

(ft) (lb)/

mole)

°R used

as

a single

quafftity

such

that

R $

R-----_-_ where

R is the

Density law.

gas

constant,

is often

used

(1-2)

in J/(kg)(K)

instead

or

of specific

(ft)(Ib)/(lb)(°R). volume

in

the

ideal

gas

Thus, 1 p =-

where

p is density,

In

general,

sures

or

within

high

as 50 atmospheres, deviations

of several None

applicable

only

pressure. to use

similarity (ratio

pressure

2

between

above their critical to within 5 percent for

and

gases

may

below

appear

ideal

been the

cannot

their

free

over

have

the as

pressure.

to express

in the the

p-v-T

satisfactory, range

of these

justified

are

tempera-

resulted

a limited

useful

space

molecules

critical

universally

gas

pres-

temperatures, up to pressures

of state

most be

the

low

at 1 atmosphere

behavicr

found

to a single Even

in behavior

and

reduced

behavior.

which

forces

at

unless

and of tem-

equations

a high

are

degree

of

is required.

temperature forms

under

equations

have

behavior

attractive

from

hundred

of these

and

cumbersome The

gases

ideal

conditions the

while

of real

relation.

accuracy

and

which are be accurate

proposal are

(1-3)

approximate

of 2 to 3 percent

Deviations

perature

will

temperatures,

For gases gas law may

most

gas

gas is large

small. ideal tures,

RT = pRT

in kg/m 3 or lb/ft 3.

a real

high

the

V

the

of

of substances

temperature, (ratio

basis

of a relatively

The

method

of

T,

of pressure, simple general

at equal to

critical

p,

method correlation

values

of reduced

temperature,

to critical for

estimating is

to

To)

pressure, real

incorporate

pc) gas a

THERMODYNAMIC

correction law:

term,

called

the

AND

FLUID-DYNAMIC

compressibility

factor,

CONCEPTS

into

the

ideal

gas

p=zpRT where

z is the

The and

compressibility

compressibility

reduced

of the

gas.

Values

temperature other as

figure

with

for

This

may

and

pressures

be

are

The

molal

averages

of

calculations

do

never

quick

not

determination

approximate

error

of

the

associated

we

the

are

use

of the

pres-

using

the

law

ideal

in our

However, is always

factor

can

gas

of

in a large with

region. gas

region

result

concerned

this ideal

by

would

compressibility with

and

is a large

law

within

that

temperatures

components.

there

gas

that

fall

granted

average correla-

approximated

that

ideal

and here

agreement

temperatures

of the

shows

usually

for

an

if pseudocritical are

conditions

from

in rigorous

reduced

properties

of the

texts

compressibility-factor

mixtures

1-1

use

of reduced

in many

is not

The

nature

2 is reproduced

is derived

properties

figure

temperature of the

as a function

presented

and

to calculate

the

take

gas.

critical

where

Fortunately,

should

one

reduced

reference

correlation

to gas

used

are from

of gases

any

of the

conditions

error.

of

pseudocritical

Examination

A

for

extended

sures.

state

type

of

to be independent factor

charts

number

data

tion

a function

pressure

of the

a large

all the

is

is assumed

reduced

One

1-1.

of data

and

of compressibility

and

sources.

factor.

factor

pressure,

(1-4)

we valid.

show

the

I 9,0

F 10.0

law.

Reduced temperature, 1.20 T/Tc_ 1.10 1.00

_

_--

_._

--

1. 1_50

______

N

•90 .80 • 70 t_

.60

--

1.20

.50

-

I \ -1.1o

.30! _-'LO0 1.0

I 2,0

I 3,0

1 4.0

I 5.0

f 6.0

I 7.0

I 8.0

Reduced pressure, P/Pc

FIGURE

1-1.--Effect

of

reduced

pressibility

pressure factor.

(Curves

and

reduced from

ref.

temperature

on

com-

2.)

3

TURBINE

DESIGN

AND

Relation In

a flow

is the and

of

process,

enthalpy

position,

APPLICATION

Energy the

h. For

enthalpy

Change

energy

term

a one-phase

can

be

to

State

Conditions

associated

system

expressed

as

with

work

of constant a

and

chemical

function

of

heat com-

temperature

pressure: h=fcn(T,

where

h is specific

enthalpy

can

enthalpy,

in J/kg

be expressed

partial

derivatives

properties

as follows.

can By

(1-5)

or Btu/lb.

A differential

change

v dT+(00-_h)rp be

expressed

dp in

(1-6) terms

of

determinable

definition, Oh Cv=(-_'_),

where

cp is heat

(lb) (°R).

One

capacity

of the

at

basic

constant

differential

dh=

where s is specific entropy, conversion constant, 1 or derivative

with

determined

respect

from

of the

Maxwell

pressure,

in J/(kg)(K)

equations

of thermodynamics

pressure

or Btu/

vdp

at

is

(1-8)

in J/(kg)(K) or Btu/(lb)(°R), 778 (ft)(lb)/Btu. Therefore,

to

equation

(1-7)

Tds + j

constant

and J is a the partial

temprature

is,

as

(1-8), Oh'_

One

in

as

Oh dh:(-O-T) The

p)

relations

T los\

states

1

that

(1-10)

Substituting

equations

(1-7),

(1-9),

and

(1-10)

j

Ov T (_-_)p_

into

equation

(1-6)

yields dh=c,dT+

Equation change between 4

(1-11)

in

terms

two

states

is the of

the

rigorous state

is calculated

[v--

equation conditions, rigorously

for and as

dp

(1-11)

a differential the

enthalpy

enthalpy change

THERMODYNAMIC

Ts

AND

1

FLUID-DYNAMIC

P*

CONCEPTS

_V (1-12)

If

we

law,

now

assume

we can

that

the

gas

behaves

according

to

the

ideal

gas

set RT v------P

(1-13)

and

( Ov' By

using

these

pressure

on

last

two

enthalpy

equations

change

Empirical

equations

and

in

for most

gases

of equation

Although

one

might

calculations, If and

there

it can

be

/'2, then

not

bT+cT

assumption

gases,

there

of some

average

be within

usually loss,

value

a few

Relation In

is an

there

cv will

of the

true

Conditions

the

loss

be is no

heat

assumed change

in hand-

example,

(1-17)

type it for

of expression computer

between

for hand calculations.

temperatures

T_

T2-- T1) one

of State

a turbine, can

percent

available

If, for

(_--T_)

this

for

variation

for

remains

(1-16)

ca is constant

excellent

is a significant

of

becomes _=c.(

This

there

2

to avoid

that (1-15)

effect

yields

to use

reason

assumed

equation

and

of T are

of interest.

(1-15)

want

is no

the

(1-15)

T2-- T1) +_ b (_--_)+3

Ah=a(

(1-12),

to zero,

for ca as a function

textbooks

integration

equation

frTcz, dT

cp=a-tthen

(1-14)

is reduced

Ah:

books

R

to be

monatomic

in c_ with give

an

gases;

for

T. However,

approximation

other

the

that

use

should

value. for

Constant

is normally adiabatic.

in entropy.

(1-18)

small, For

Therefore,

Entropy

Process

and

flow

the

adiabatic the

flow

process with

no

constant-entropy

(isentropic) process is the ideal process for flow in the various parts of the turbine (inlet manifold, stator, rotor, and exit diffuser) as well as for the overall turbine. Actual conditions within and across the

TURBINE

DESIGN

AND

APPLICATION

turbine are usually determined in coniunction necessary For can

with

to be able a one-phase

some

state

conditions

of constant

as a function

a differential

From

change

equations

Substituting

(1-8)

(1-23)

we get

(1-21)

and

relating

temperature

If we assume into

equation

and

pressure

ideal-gas-law (1-23)

and

but

using

a relation

such

equation

(1-17),

a computer calculation With the additional

not

behavior the

as equation

peratures 6

T_ and

equation

equation

(1-25)

(1-24)

yields

(1-22)

particularly

and

useful,

expression

for an isentropie substitute

integration,

process.

equation

(1-14)

we get

Rln_

(1-24)

(1-16),

integration

yields

(T_--T_)

also

is more

becomes

(1-25)

suitable

than in a hand calculation. assumption that cp is constant

T2, equation

(1-20)

dp

1Rln_=aln_--kb(T2--T_)+2 J p_ Like

(1-20)

and

, _dT----

By

into

conditions

perform

as

(tp

0v (0-T),

ds=0

rigorous,

entropy

pressure (1-19)

(1-10)

dT--j

process,

is the

composition, and

be expressed

and

a constant-entropy

Equation

can

dT÷(_)r

equations

process.

(T, p)

ds----(_),

ds=_

For

chemical

in entropy

(1-7),

It is, therefore,

for an isentropic

of temperature s----fcn

and

isentropic process calculations

efficiency or loss term.

to relate system

be expressed

from

for use

between

in

tem-

THERMODYNAMIC

AND

Jc_ ln T2--R T1--

FLUID-DYNAMIC

CONCEPTS

p2 ln- pl

(1-26)

and p2=('%'_ p, \T1/

Jc,/R (1-27)

But Jcp R where

3" is the

capacity equation

ratio

of heat

at constant (1-27) yields

3' 3"-- 1

capacity

(1-28) at

constant

volume. Substitution of the more familiar form

pressure

equation

to

heat

(1-28)

into

p, _-_----\_]

Where should

specific heat ratio give a reasonable

(1-29)

3" is not constant, approxlmation. Conservation

The

rate

of mass

flow

through

the

use

of an average

value

of Mass an area

A can

be expressed

as

w=pAV

(1-30)

where w A

rate flow

of mass flow, kg/sec; area, m2; ft 2

V

fluid

velocity,

For across across

a steady any any

m/see; flow

lb/sec

ft/see

(and

nonnuclear)

section of the flow other section. That

path is,

process, must

equal

the

rate

of mass

flow

the

rate

of mass

flow

plA, VI = p2A2 V2 This

expresses

(1-31)

the

is referred

principle to as the

of conservation continuity

Newton's

are

All conservation

equations,

consequences

of Newton's

that an unbalanced in the direction force is proportional the body.

Second

Law

Second

product

of mass,

and

equation

equation.

theorems,

force that ac_s of the unbalanced to the

(1-31)

of etc.,

Law

Motion dealing of

with

Motion,

momentum which

states

on a body will cause it to accelerate force in such a manner that the of the

mass

and

acceleration

of

7

TURBINE

DESIGN

AND

APPLICATION

Thus, F= m a g

(1-32)

where F

unbalanced

force,

m

mass,

a

acceleration,

g

conversion

kg;

N;

lbf

lbm m/sec2

; ft/sec

constant,

2

1 ; 32.17

(lbm)

(ft)/(lbf)

(sec 2)

But dV a =--d- 7 where

t is time,

(1-32)

in seconds.

Substituting

the

mass

is constant,

into

equation

g Equation

(1-34b)

is equal mass

(1-34a)

can

to

specifies the

(1-34a)

equation

F=I

Since

(1-33)

dV dt

g

fluid

equation

yields F=m

Since

(1-33)

rate

(1-34a)

the

of

also

be written

as

be

written

time

F =w

as

(1-34b)

unbalanced

of change

increment

also

d(mV__) dt

that

per

can

force

of momentum is the

mass

acting

(mV) flow

on

with

rate,

the

time.

equation

dV

(1-35)

g

be

A useful derived

fluid

as indicated

ligible. of

relation, sometimes called the from second-law considerations.

A fIictional

fluid

forces

in figure resistance

is subjected acting

in

Expanding,

Gravitational (force)

downstream in

+(p+

simplifying,

and direction

the upstream direction is

of are

assumed

can of neg-

element

boundary-surface-pressure and

fluid-pressure Therefore,

(A+dA)--dRt

second-order

dropping

motion, an element

as R s. The

direction.

d-_)dA--(p+dp)

and

forces

is indicated

to fluid-pressure the

friction forces acting force in the downstream

F=pA

1-2.

equation Consider

and the

net

(1-36)

differentials

yields F= 8

-- Adp--

dR I

(1-37)

THERMODYNAMIC

The

mass

of the

element

AND

FLUID-DYNAMIC

is m:pAdx

Substituting

CONCEPTS

equation

(1-38)

into F_

(1-38)

equation

pAdx g

(1-34)

yields

dV dt

(1-39)

Since

(1-40) equation

(1-39)

can

be written

in the

form

F_pAV

dv

(1--41)

g Equating

(1-37)

with

(1-41)

now

yields

F=--Adp--dRs=

pA VdV g

(1-42)

and dp + VdV+

dRs

0

p-F--i-A--

(1-43)

p+dp 2

A+dA f f

% ,%

Flow

p+dp

p dx

V+dV

V

dRf FIGURE

1-2.--Forces

on

an

element

of

fluid.

TURBINE

DESIGN

If we now

AND

APPLICATION

let (1-44)

where

ql is heat

produced

by

friction,

in J/kg

dpp __V_VdgV+Jdq1= For

isentropic

flow,

a steady-flow

a system system

or or

electrical

(and

part

of of that

energy,

etc.,

(1-45)

0

P lvl

equal

If

we still

"Jr- V12

Energy process,

must

system.

kinetic energy work I¥,. Thus,

u,+--j-

of

nannuclear)

a system

part

energy pv, mechanical

we have

dqs----0. Conservation

For

or Btu/lb,

we

have

energy Z,

heat

J-J--

that energy, u, flow q,

and

(1-46)

V2" 2 .4 Z_2_4_W Tr

j-2g

entering

leaving

chemical

internal energy

p2l'2_l__ ,

-j+q=u:+

energy

energy

neglect

potential

_j[_Z1

_gj

the

can

to consider

V2/2g,

the

s

where u

specific

internal

Z

specific

potential

q W,

heat added mechanical For

by

energy,

J/kg;

energy,

to system, work (lone

a gas system,

the

Btu/lb

J/kg;

(ft)

(lbf)/lbm

J/kg; Btu/lb by system, J/kg;

potential

energy

can

Btu/lb be neglected.

In addition,

definition pv

(1-47)

h=u + -j Thus,

equation

(1-46)

reduces

to V

"{72

2

(1-48) gd Equation

(1-48)

as we will be using

is the

basic

g form

The

sum

of the

problems,

steady-flow

energy

balance

it. Total

in flow

of the

enthalpy and

and

Conditions the

kinetic

it is convenient

energy

to use

is always

it as a single

appearing quantity.

Thus, V2 h ' _-h + 2gJ 10

(1-49)

THERMODYNAMIC

where

h' is total

The

enthalpy,

concept

of

temperature. that

corresponds

cept

is most

capacity

to

the

leads can

total

when

us

be

enthalpy.

In

that

to

the

defined The

ideal-gas-law

be assumed.

FLUID-DYNAMIC

CONCEPTS

or Btu/lb.

enthalpy

temperature

useful

can

in J/kg

total

Total

AND

as

the

of

temperature

and

according

con-

constant

heat

to equation

(1-15),

h'--h=cp(T'--T) where

T'

with

is total

equation

temperature,

(1-49)

(1-50)

in K or °R.

Combining

equation

(1-50)

yields V2

T' = T+ 2gJcp The

total

attained

temperature

when

to rest The

total,

pressure

T'

can

at static

these

total

two

brought

equation

of

T and

can

between

as

the

temperature

velocity

is also

used

V is brought

called

stagnation

interchangeably. be regarded

isentropically

relation

(1-29)

are

pressure

to rest the

thought

temperature

terms

or stagnation,

p. Since

be

(1-51)

temperature

Thus,

and

of a fluid use

a gas

adiabatically.

temperature,

total

total-temperature

behavior

case,

concept

from p'

as the

a velocity

and

pressure

V and

p is isentropic,

static we can

to write

(1-52)

where

p'

With

is total

pressure,

regard

to the

in N/m 2 or lb/ft _. above-defined

total

conditions,

certain

points

should its use

be emphasized. The concept of total enthalpy is general, involves no assumptions other than those associated with

energy

balance

be seen,

as we have

is a very

tion,

but

heat

capacity.

useful

it is rigorous For

change,

the

use

pressure,

in addition

ture, involves conditions. Flow Let occurs work. each

us now, with This part

in

neither process

of the

only

for

systems

of an

considered

it.

convenience

total

temperature

to the

assumptions

is

With

terms

of total

conditions,

transfer

(adiabatic

heat

turbine

one

No

that

(including

not

between

Heat

occurs the

rotor,

and

of calculaand

reaction

constant

or

a

recommended.

associated

path

as will

burden

behavior

chemical

Process

is the

temperature, the

ideal-gas-law

involving

isentropic

Total

for easing

with

static

No

Work

examine process) (neglecting at constant

phase Total

total

the

and the

temperaand

total

a process

that

nor

mechanical

heat

losses)

radius,

in

when 11

TURBINE

the

DESIGN

velocities

AND

are

expressed

Substitution

of

rangement

APPLICATION

relative

equation

to the

moving

into

equation

(1-49)

blade). and

(1-48)

rear-

yields h2' -- hi' = q--W_

The

energy

dynamics

balance for

If we set

now

a flow

q and

looks

something

process,

W_

equal

as we

to zero,

(1-53)

like

were

the

first

First

Law

exposed

of Thermo-

to it in college.

we get

h_' =-h,' Therefore,

for

constant. that

adiabatic

Further,

total

flow

from

with

no

equations

temperature

also

(1-54) work,

(1-18)

remains

total

and

enthalpy

(1-50),

remains

it can

be shown

constant.

TJ =T/ Note

that

the

enthalpy Total and

process

and

total

pressure

(1-55)

tions,

not

have

the

matter.

ideal-gas-law

be shown

to be isentropic

to remain

is another

and

it can

does temperature

that

(1-55)

From

and

for

in order

for total

(1-22),

(1-52),

constant. equations

constant-heat-capacity

adiabatic

flow

with

assump-

no work,

eJd' l _= P_' p2' Only

for isentropic

constant.

For

flow

flow

(ds=0),

with

(1-56)

therefore,

loss

(ds>0),

does there

total is

pressure

remain

a decrease

in

total

pressure. Speed An wave

important

of

Sound

and

characteristic

propagation

or,

as

of

otherwise

small-pressure-disturbance

Velocity gases

is

called,

the

Ratios the

speed

speed

of

of sound.

From

a is speed the

ideal

of sound, gas

From

theory

a=_/g(-_p), where

pressure-

law

in m/sec and

(1-57) or ft/sec.

isentropic

process

relations,

this

reduces

to a= The factor called 12

ratio

of fluid

velocity

in determining the flow the Mach number M:

V

_gRT to sound characteristics

(1-58) velocit_y

a is an important

of a gas.

This

ratio

is

THERMODYNAMIC

AND

FLUID-DYNAMIC

CONCEPT_

M= --V

(1-59)

a

Mach

number

is a useful

behavior regimes, but expressions. Consider temperature, (1-59),

given

and

parameter

also the

with

(1-51).

equation

ratio

for

Combining

(1-51)

T'

Another velocity critical velocity

only

identifying

for simplifying and generalizing relation of total temperature

in equation

(1-28)

not

to

equations

certain static (1-58),

yields

l_I_7__

often

flow-

(1-60)

M 2

used

is the

ratio

of fluid

velocity

V V Vc,--acr where

V.

sound

at

is critical critical

is equal

to the

condition (1-60),

at

velocity,

velocity

(1-61)

in m/sec

condition,

in m/sec

of sound

or or

at the

is that condition where the critical condition

to

ft/sec, ft/sec.

critical

M----1.

and The

a.

is speed

critical

condition.

The

Consequently,

of

velocity critical

from

equation

2 T.=3"+ and

substitution

of equation

1 T'

(1-62)

into

(1-62) equation

(1-58)

yields

(1-63)

acr=_/_lgRT' Thus,

in any

no work),

flow process

the

value

of the

for the

entire

process,

as

the

static

temperature

The

ratio

the

critical

because while perature The

the Mach

constant

critical

whilethe

of fluid velocity

with

velocity value

velocity ratio.

to

Its

(Vc,=a,)

of the

critical

speed

velocity

use is often

velocity

ratio

number

is not

(since

the

temperature

(no heat remains

of sound

and

constant (a) changes

changes.

critical

in

total

preferred

is directly there

is sometimes over

proportional

is a square

root

Mach

called number

to velocity, of static

tem-

denominator).

relation

between

critical

velocity

ratio

(1-28),

and

static results

and from

total

temperature

combining

equations

in terms (1-61),

of the (1-63),

(1-51). TT '-1

3,--1(V) 3"-t-1 _

2

(1-64) 13

TURBINE

DESIGN

AND

APPLICATION

APPLICATION The the

equations

flow

(flow the

TO

already

through

in the turbine.

there

to learn

varying-area

passages

We

are

going

change,

ously

to

and

presented

examine

number.

equations

yields

to analyze

are

losses

in a turbine,

we can

about

the

of the

rotor,

are

behavior

and

exit

no

losses

diffuser)

use flow

of the

Regime

relations

a,mon_

Prot)er the

there

completely

that

Flow

the

Much

AREA

provided

(stator, of

VARYING

sufficient

something

Effect

area

are

passages,

Although

process

WITH

presented

turbine

is isentropic). loss-free

FLOW

pressure,

manipulation

following

velocity,

of the

equation

previ-

for isentropic

flo_v:

_(I_M_

Equation

(1-65)

velocity

is opposite

the

changes

whether let

flow),

static

to the

Much

Let

Much

with than

the

various

flow

(M
1. Increasing

flow).

cases

pressure

Decreasing

flow

Increasing Velocity This

This

way

passage

from

equal

to

1

of definition,

in which passage

static

in

equation

which

flow

(M=

Area

change

mum-area

increases

(dA>0).

and

area

decreases

(dA_0).

and

area

(tecreases

area

increases

(dA_0).

diffuser. (dV>0)

supersonic

increasing

:

(dp<0):

increases

Both

(1-65)

area

(dp>0): (dV<0)

pressure

is the

critical,

14

on

(M>I):

supersonic

Decreasing

Sonic

of

depend

(dp<0):

pressure decreases

is the

Velocity C.

directions

flow),

By

in

(dp>0):

pressure

Supersonic

2.

change

in area

is a varying-area

Velocity increases (dV>0) This is the subsonic nozzle.

1.

(2) the

1 (subsonic

Velocity decreases (dV<0) and This is the subsonic diffuser..

B.

amt

the

increases.

Subsonic

2.

numbers

is a varying-area

a diffuser

(1-65)

changes

1 (supersonic

us examine A.

all

dA A

in l)ressure

is less

than

and

for

pressure

a nozzle

decreases

1--M 2alp 7M 2 p

change

number

that

pressure

(1)

and

or greater

us specify

pressure

that

in velocity

the

(sonic

shows

)dV V--

and

(dA>0).

nozzle.

1) : (dp>0) must

condition section

and

equal can

zero

occur

decreasing

(dp<0)

(dA=0). only

of a varying-area

at

the

pressure.

Thus,

the

inlet,

exit,

passage.

sonic,

or

or mini-

THERMODYNAMIC

You

may

dition

also

(M=

must

want

to note

1) going

have

either

AND

that

up

in order

or down

a decreasing-area

FLUID-DYNAMIC

portion

CONCEPTS

to cross

the

in velocity, followed

critical

the

by

an

flow

con-

passage

increasing-area

portion. Flow Since

we

diffuser nozzles.

are

the

nozzle

Convergent nozzle.

primarily

nozzle

the

gas

is subsonic,

nozzle.--Let

This

the

at

T'.

as p, and is designated

mences

and

the

lowered,

flow

At

value

some

velocity,

and

of p,, M=

Mach

number

nozzle.

Therefore,

tion

(M=

the

throat

remains and

lowered

than

the

flow

at

the is

pe is reduced flow

much

the

nozzle

and

then

expansion

with the

the

an

isentropic The fact

pressure ratio

(p'/p_r)

constant the mass

be

throat

the

exhaust

and

the

pressure

to maximum

com-

equal

to pc. to sonic

from

will

than

the

The

a

that condi-

pressure

according

from

mass

to equa-

no

with

shocks

pr6ver_

to the

fact

nozzle.

(which

occur

no

matter

that

this

mathematically.

later),

and

process. for nozzle

critical

rate

on

within

the

a little

flow

effect

to pt=p,

for a fixed value when

constant The

be

has p'

Pt to pe outside

equal

in

(1-66)

for this part of the remains constant or

a

convergent

critical

static

be discussed

nozzle

remains can

seen

in

pressure

expands

not valid condition

is reduced. flow

is

(in the

flow

equal

in the

which

exhaust

gas

and

greater

thereafter

p',

have

attained

remains

Pt to Pe occurs

are throat

that

We

under these conditions. Thus, flow rate reaches a maximum

the

st)on(is

from

means

exit

Pt still

further?

pressure,

further

in entropy

(p'/p,)

a static

pressure

{ 2-_-_ "_/(_-') \_,+ 1/

The

expands

equations that the

ratios

p,,

nozzle.

process

increase

than

with

p_ is lowered.

critical

below

within

less

becomes

throat

P'=P_'=P'

the

and

nozzle

throat

1 cannot at the

how

(1-29)

p'

where

throat.

if pc is now greater

Assume

velocity)

to pc. As p_ is progressively

increase,

at the

interest.

convergent

static

at the

pe is a little

both

than

previously.

pressure

right

velocity velocity

of most

or outside,

pt is equal

the

1) no matter

(1-62)

pressure

rather

simple (zero

total)

pressure

1 at the

happens

(and

When

case

the

a reservoir

exhaust,

static

as pt.

throat

rate

and

What

the

flow

A2 mentioned

from

The

is the

consider

case

gas a static

temperature

throat)

The

nozzle

the discussion to flow in to the case where the flow

this

first

to the with

is maintained

designated

Once

with

since

us

corresponds is supplied

total)

tions

Nozzles

flow in turbines, we will narrow We will further limit the discussion

entering

(and

concerned

in

pressure

also

remains

upstream M becomes to what condition

state, 1 at value corre-

A nozzle

in 15

TURBINE

this

DESIGN

condition

AND

APPLICATION

is said

to be choked.

Convergent-divergent more the

involved nozzle

case p'

sure

ratio

the

exhaust

commences

the

that

to

the

sonic

T'.

consider

Figure will

the

less

lowest

throat

same

p'

decreases

velocity,

Pt,

and

of the lowered and

(curve

the

this (curve

gas

the throat AD

still

in fig.

at

the

passage (curve velocity

1-3).

I /r

Pe

I I I I I

&

.B

A ,ll

-- _r- .....

B

-_'_"_'_'_

.= N Z

K

H

E

Length

FmuR_;

16

1-3.--Nozzle

becomes

flow

processes.

Note

subsonically

Th roat

-

throat

increases.

velocity

/

If 1-3),

is acting AC in fig.

in fig.

I/

of pres-

discussion.

AB

diffuses

_X._I / rpt

assume

maintained plots

occurring

of p_, the

or Mr----1

than

Again,

showing

supplement than

somewhat

reservoir

1-3,

pressure

value

the

nozzle.

the

the divergent section As pe is progressively

particular

higher

from

length,

p, at the

at some

p_ is still

gas

pe is a little

with

now

convergent-divergent with

nozzle

this case, diffuser.

pressure

us

temperature

pressure

Eventually, equal

and

against

(pt_P_). In as a subsonic 1-3),

of the

to be supplied

at pressure

flow

nozzle.--Let

THERMODYNAMIC

in

the

(with

diveruent

section.

pt:pc._pe),

we

see

to

the

critical

required nozzle achieve If

is less than the critical p_

is

condition only

achieve

a_ain

place

nozzle,

and

the

pendently If

the

divergent

flow

is

part

of the

to discharge

area,

servation

of

relations.

This

shows

flow

isentropic The

supersonic

flow

that

Observing

the

optical in the

means flow.

waves.

Shock

changes temperature process weak.

flow

be

considered

across

Weak

Let

a shock

pressure,

with shocks and

of the that

are shock

complete of pressure

If the

exhaust

pressure

isentropic

and

shock the subsonic

small

some called

the

l)ressure diffusion

AD,

either

fig.

AE,

1-3) fig.

1-3). by

changes flow are

and

the

occur shock

fluid

instantaneously. but,

or

conditions

density in the

even

state Total

though

pressure

there

because

the

Shocks may be strong or flow (and are thus called

velocities small

downstream

angle

oblique

with

shocks), but

of the

respect and

the

the

to

the

velocity

Mach

number

of convergent-divergent

between

at some rises

assume

shock.

p, is reduced

occurs

to

be isentropic.

constant

discussion

ratios

which

allow

(curve

supersonic,

of the

AE,

cannot

is a loss in total

con-

process,

curve

p_ that

thickness,

area

the

Pe_Pt

occurring

remains

upstream

region

nozzle,

at

thus

satisfy

nonisentropic

in subsonic

occur

(and

us now

a normal

(curve

these

the

of throat

unreasonable

Pe_pt

remains

mass

inde-

isentropic by

some therefore,

there

result

for the

D,

is

an increase in entropy. occur normal to the

shocks

downstream than

the 1-3 of

as

the

throughout

values

of very

the

behave

ratio

the

under

are

direction

is less

flow

as

to some

place,

waves

shocks)

shock.

take gas

in static occurs Strong

normal

to

with

and

p_ will

reveals that surfaces of abrupt These apparent discontinuities

may

is a rise

well It

between

is the

As long as the nozzle is choked to

given

in figure

expansion

does

as

isentropic

any

continuously.

diffusion

Thus,

pressure as

critical

exist. value. the

to

throat.

for

exhaust

the throat

constant,

and

then

at

continues

the

energy,

falling

(p'/pe)

the

remains

supersonic one

remain

can

nozzle

is represented

subsonic

isentropic

the

nozzle,

is impossible

ratio

us that

its maximum at the throat,

beyond

and

case

critical

a convergent-divergent

must

showed

state

of

be

only

mass

pressure

that

to

is now

pressure

in

condition

throat

conditions

CONCEPTS

condition

nozzle

throat

(1-65)

part

of the

the

the

constant at is maintained

convergent

throat

condition

critical

the

flow must remain critical condition

FLUID-DYNAMIC

pressure ratio (p'/p_:p'/pc,) required in a simple convergent nozzle.

equation the

the

that

lowered,

where

convergent

Since

the nozzle condition

because

AND

points a little

point

from

below

in the

instantaneously

occurs

D and

the

nozzles

E in figure

the

value

divergent to

shock

plane

at point

part

a value

1-3.

such

to the

of the that nozzle

17

TURBINE

DESIGN

exit.

The

flow

with

AK

being

and

LF

the

to point

the

flow

moves H,

path

the

higher

than

nozzle

exit,

When

p_. In with

from

the

in a nonisentropic

It should do

not

occur

abrupt, tions

may

a shock

1-3

exactly

takes

produce

that

over

from

similar

In

order of

books

and

charts ture.

The

carbon and

general

the

nozzle final

outside

listed

air

and

and

factors

are

of

processes,

however,

are

in reference

and

with

also

the

presented

temperaindividual oxygen,

in references

in reference

the

as

4. These

(nitrogen,

presented

5 include

products 3 and

capacity

are

in

3 to 7.

combustion

and

many

published

as references

air

Charts

calculations, and

products

also

although considera-

downstream

flow

in heat

argon)

proceffects

flow

Tables

its

combustion

vapor,

rise,

real-fluid

in references

of

the

shock

1-3.

and

are

and the

constructed

variation

its

compressible-flow

as functions

for

values normal

of Mach

functions number

effect

and

oblique

a listing

of compressible

tions

in terms

of both

Mach

(TIT',

4

5. The

of pressure,

pip',

are presented

of heat-capacity

sents

18

E.

as

as temperature.

for various charts

the

pe occurs

pressure

Also, subsonic

presented

the

and

presented

to

at point

of p_, the

discussion

in figure

properties

5. Compressibility

Isentropic others)

of

are

water

previously,

the

the

been

of these

include

air

dioxide,

properties well

of

and

pressure

occurs

values

Flow-Function

have

thermodynamic

components

is

cannot

a static

actuality,

and

The

properties

tables

In

thermodynamic

of temperature and

shock

shock

lower

previous

shown and

Some

Thermodynamic functions

exit,

as p_ approaches

distance.

isentropic.

charts

reports.

the

make

to those

and

in

pressure

idealized.

that

to facilitate

tables

For

a finite

Thermodynamic-Property

sets

nozzle

a normal

oblique

static

are

effects

different

qualitatively

process

of pe correspond-

at the

as mentioned

instantaneously

place

flow

the

manner.

out

in figure

shock,

further,

the

value

result

weaker

E,

nozzle-exit

be pointed

shown

and

E,

weaker

isentropic.

expansion

esses

the

and

would

becoming

to point

the

H

and

case,

completely

exit, At some

AKLF,

normal

pe is reduced

will be right

points

shock

p_ corresponds

path

the

is AEH.

strong

this

the

the

being

As

nozzle

shock

nozzle

by

KL

as AMNG.

normal

it is too

is again nozzle

the

such

of pe between

because

is illustrated

diffusion.

toward

in the

values

case

expansion,

isentropic

by a path

ing

flow

in this

an isentropic

shock

represented

occur

APPLICATION

process

being

normal

For

AND

ratio. shock flow

number

in references

Also

included

calculations.

function and

p/p',A/Ac,,

and critical

4, 6, and

are

tables

Reference _hock velocity

and

function ratio.

and 6 preequa-

7

THERMODYNAMIC

AND

FLUID-DYNAMIC

CONCEPTS

REFERENCES 1.

KUNKLE,

JOHN

pressed 2.

Gas

NELSON,

L.

S.j

WILSON,

C.;

AND

Compressibility 3.

ENGLISH,

ROBERT

Properties

of

TN

1950.

KEENAN, JOSEPH Inc., 1948.

5.

HILSENRATH,

6.

HOGE,

KIAN,

YERAM

ties

of

Gases

ties

of

Air,

7.

LEwis

Argon,

and 1955.

Rep.

KAYE,

BECKETT, J.;

AND

for

RICHARD

to

col.

Use

61,

the

no.

New

WILLIAM

W.:

Products

from

BENEDICT,

F.;

NUTTALL,

HAROLD

Carbon 564, Tables,

John

Com-

of

Charts

L.;

for

NACA

and

Sons,

S.;

FANO,

TOULOU-

Thermal

Proper-

Transport

Hydrogen, Bureau

203-208.

° R.

WILLIAM

Monoxide,

and

3500

Wiley

and

National

pp.

Thermodynamic

RALPH

Tables

Thermodynamics

Circular

Equations,

Tables.

W.:

of

300 ° to

W.;

JOSEPH

Dioxide,

ED:

. Generalized

1954,

Charts

CHARLES

of

A.; . .

7, July

Gas

MASI,

NBS

COTA,

JOSEPH:

Tables

STAFF: 1135,

How

WOOLLEY,

Carbon

Steam.

LABORATORY

Functions

AND

AND

1969.

Eng.,

Combustion

HAROLD S.;

F.:

WACHTL,

Comprising

RESEARCH

NACA

and

H.;

E.

D.;

SP-3045,

Chem.

AND

JOSEPH;

LILLA;

AMES

OBERT,

E.; Air

4.

Oxygen, Nov. l,

NASA

Charts.

2071,

SAMUEL

Handbook.

PropNitrogen,

of

Standards,

Compressible

Flow.

1953. COMPUTING

Specific-Heat

STAFF: Ratios

from

Tables 1.28

of to

1.38.

Various NACA

Mach TN

Number 3981,

1957.

19

TURBINE

DESIGN

AND

APPLICATION

SYMBOLS A

flow

a

acceleration, m/seC; ft/sec 2 speed of sound, m/sec; ft/sec

aj

b_ c

c_

area,

m2; ft _

general constants for polynomial, heat capacity at constant pressure, unbalanced conversion

F g h

force, N; constant,

lb 1 ; 32.17

eq. (1-16) J/(kg)(K); (lbm)

(ft)/(lbf)(seC)

J

specific enthalpy, J/kg; Btu/lb conversion constant, 1 ; 778 (ft) (lb)/Btu

M

Mach

M_

molecular

weight,

m

mass,

lb

number, kg;

P

absolute

q

heat

added

ql R

heat

produced

defined kg/(kg

pressure,

by

RI R*

universal

8

specific

T

absolute

t

time,

Btu/lb J/kg;

8314

J/(kg)

Btu/lb

J/(kg

(°R)

mole)(K);

1545

(ft)(lbf)/

temperature,

(K) ; Btu/(lb) K;

(°R)

°R

sec internal

energy,

fluid

V

specific

Ws

mechanical

w

mass

x

length,

in; ft

Z

specific

potential

absolute

velocity,

volume, flow

rate,

of heat

J/kg;

Btu/lb

m/sec;

ft/sec

ft3/lb

done

by

kg/sec;

factor,

system,

J/kg; defined

lb/ft 3

Subscripts: c

critical

state

cr

critical

flow

e

exhaust

t

throat

Superscript: '

absolute

total

condition condition

state

Btu/lb

(M:

(ft)(lbf)/lbm by

at constant

volume

kg/m_;

J/kg;

lb/sec

energy,

capacity

at constant density,

m3/kg;

work

compressibility ratio

90

mole)

(°R)

V

P

lb/(lb

(K) ; (ft) (lbf)/(lbm) force, N; lb

constant,

entropy,

specific

Z

(1-59)

lb/ft 2

J/kg;

friction,

gas constant, J/(kg) frictional resistance

(lb mole)

eq. mole);

N/mS;

to system,

gas

by

Btu/(lb)(°R)

1)

eq.

(1-4)

pressure

to heat

capacity

CHAPTER 2

BasicTurbine Concepts ByArthurJ. Glassman This

chapter

introduces

and

performance

efficiency, nitions,

diagrams,

the

blades

the

end

and

and

blading

of this

chapter.

TURBINE

turbine dimensionless geometry

FLOW

An

analysis

of the

requires

some

through

a turning

directed

parallel

ally

through

gentially

to

tangential radial-axial blade eters.

axis rotating

plane

average) For

indicated

radial

many

types

(or blade-to-blade) values.

Such

of

usually the

and

variation made

at some

third

velocity-variation,

constant

coordinate.

within

system.

For

consists

of one

variation

(rather

are

usually

coordinate

coordinate

directed

tan-

are

the

axial,

Analysis

radial, of flow

of the

desired

ignore

values

the and

as made

in the

flow

t)aram-

circumferential just

use

average

analysis.

or radialthan

and

(or blade-to-

an axisymmetric

diagram,

flowing

one

we can

axial-tangential

a tur-

fluid

circumferentially-averaged

value

calculations

TRANSFER

radi-

2-1.

Velocity

at

directed

planes.

of parameter

in the

GLOSSARY,

coordinate

three

is called

to

one

in figure

axial

of defi-

referring

processes

system and

of calculations,

a calculation

Calculations are

the

means

System

These

form

depicts

ENERGY

of rotation, wheel.

coordinates

in the

coordinate

of rotation,

energy-transfer, by

Terms

energy-transfer

a logical axis

the

defined

Coordinate

and

wheel, to the

the

three

are

convenient

directions

These

flow

flow,

primarily parameters.

AND

Analysis

bine

geometric,

characteristics

tangential

for average well

planes

conditions)

as blade-to-blade

in these

planes.

When 21

TURBINE

DESIGN

AND

APPLICATION

2

Vu

FIGURE

flow

2-1.--Velocity

is predominantly

turbine,

the

nantly

radial,

such

as

in

an

for

a generalized

at

the

is used.

axial-flow

rotor.

inlet When

turbine,

to

a radial-flow

flow

the

is predomi-

axial-tangential

is used.

One

of the

most,

be concerned is the

fluid

tions.

To and

For

flow

interest.

velocity

For

relative

other

a rotating of flow

its variation

in and

to the blade

and

across

row

in a stationary

Diagrams

important in the

the

stators,

across

the

blade.

and

different and

that energy

we will transfer

coordinate in depicting

direcblading

diagrams. the rotors,

In

to be discussed

can be analyzed

flow

analyses

velocity-vector

passage.

variables

of turbine

these

rotating

parameters

and

most,

analysis

we use

and

flow

relative

the

in the

us in making

types, in

Vectors

if not

with assist

shapes

_2

as

plane

Velocity

ered

such

radial-tangential

axial,

plane

components

absolute

velocities

velocities

must

terms

of relative

later

in a manner

in this similar

are

of

be consid-

velocities chapter, to the

and flow

analysis

in

BASIC

Velocity-diagram downstream inside

calculations

of the

the

blade

ferential represent

various

rows.

blade

In

velocity

velocities.

diagram

In making

Relative

the

CONCEPTS

upstream

infinitesimal

velocity

note

and

distances

diagrams,

the

circum-

velocity

absolute

diagram,

velocity=Absolute

locations

considered. The of the flow.

both

velocity

at

or at just

the

are not average

shows the

made

rows

making

variations in flow the circumferential

The

are

TURBINE

and

vectors

the

relative

that

velocity--Blade

velocity

(2-1)

or

W=V--U

where W

relative

V

absolute

U

blade

Since

velocity

consider

vector

velocity velocity

blade

vector vector

velocity

the

is always

magnitude,

in the

that

is,

velocity

shows this

the

diagram components

velocity

absolute

in figure

diagram

and

relative

tangential

the

W= The

(2-2)

blade

speed.

we need

So,

we

can

V--U

2-2

represents

absolute

and

to be

drawn

in an

can

only write

(2-3)

of the

velocities

direction,

be

equation

relative

(2-3)

velocities.

axial-tangential expressed

in

terms

and

also

Assuming plane,

the

of

their

Absolute Relative angle

of flow,

angle of flow, cl7 I I I

V V x = Wx

VU

FIGURE

2-2.--Typical of

absolute

velocity-vector and

relative

diagram velocities

having in

the

same

tangential

components

direction.

23

TURBINE

DESIGN

components

AND

in the

APPLICATION

axial

and

tangential

directions

V2:V_2q-V,,

as

2

(2-4)

Wu 2

(2-5)

and W 2=W2+ where V

magnitude

V=

axial

of V, m/see;

Vu

tangential

component

W

magnitude

of W, m/see;

IV=

axial

W_

tangential

component

ft/sec

of absolute

component

velocity,

of absolute

velocity,

velocity,

of relative

figure

2-2,

m/see;

we see that

m/see;

ft/sec

ft/sec

velocity,

If this diagram (fig. 2-2) were drawn the values marked as axial components From

ft/sec

ft/sec

of relative

component

m/see;

we can

m/see;

ft/sec

in the radial-tangential plane, would be radial components. write

W_,=V,,--U

A

sign

convention

tangential are

of

shown gram and in

components the

exact

in figure shown flow

of

same 2-2.

the

directions,

Therefore,

In

will

for

as

the

velocity

and

diagrams

example the

the

diagram

velocity

dia-

tangential

components

relative

velocities

are

it is not

obvious

that

adopt

and

Wu

the

angles

and

t

24

all

example,

instance,

stick

13_

FIGURE 2-3.--Typical absolute

the

not

shape

Relative angle of flow,

._

for

since

have,

this

absolute and

we

established

velocity, could

2-3.

of

be

geometrical We

in figure

angles

opposite

valid.

must

(2-6)

with

the

equation

(2-6)

convention

is that

r-Absolute angle of flow, cI

Wx Wx = = Vx Vx U

velocity-vector ap.d relative

directed

diagram having tangential velocities in opposite directions.

---I components

of

BASIC

all

angles

and

are

in the

direction

in the we

tangential

direction

can

see

in

a larger

Not

all

Some

the

of a rotor

and

then

negative

with

respect

to the

as we are using. you

aware

valid

for

where

above

at

a location and

direction by

convention Energy

The

basic

tively

as applied of

a

and

to a fluid velocity.

by

any

of the

fluid

at

1 and

2 are

at any

sumed.

Further,

regarded being

as

radii

the

working

with

angles

defined

than

the

else,

all

axial

direction

to use

velocity-

make this

sure

that

information.

turbomachines

Figure

the

2 are

rl and

at

Second

0-0

2-1

the

rotor at

any

the

average

velocity

vectors

of

the

2. The

for

through

and

of steady and

and

directions

angle,

inlet

values

a rotor

rotation

1, passes

point

arbitrary

at

of Motion

represents

axis

r2. A condition

vectors

is rein

Law

at point

is discharged

velocity

representing

inlet

mutually

and

of the

to an

change

axial

bearing have

effect

of the

force,

in magnitude

a radial nents

outlet

perpendicular

in magnitude

the

enters

1 and

at locations

points

state the

the

is as-

outlet

are

of

flow

mass

considered.

The

rise

points

for

with

and

cases.

avoids

in generating

a rotor.

Fluid

path,

all

this

occasion

of Newton's

turbomachine,

angular

rotor

work

Vu Wu.

upstream

directions cases

of for

Transfer

a form

traversing

for

someone

relation

is only

generalized

the the

energy-transfer

simple

convention

have

used

value value

rather

should

velocity

immediately

analysts

if you

the

positive

negative

are

convention,

a negative

In many

generated

of the

small

the

many

tangential

a

U yields

positive

Therefore,

information

are

remains

of

if they

if they

(2-6)

use

Also,

negative this

of a rotor.

values.

are

CONCEPTS

positive

With

convention switch

and

are

velocity.

2-3,

downstream

with

velocity

value

analysts

above

immediately

diagram

figure

of velocity

blade

equation

positive

turbine

use

blade to the

that

shown

minus

of the

opposite

now

diagram

components

TURBINE

any

load. effect

of bearing

tangential

in angular

components axial

must

of the

radial

Neither

the

the

components

momentum

be

taken

velocity axial

nor

angular

motion

It is the

change

of velocity

of the

fluid

and

into

previously.

components

which

on

be resolved

discussed

velocity

friction).

can

through by

The the

a thrust

components the

that results

bearing.

The

rotor

rise

to

compo-

(except

in magnitude

in the

change gives

velocity

corresponds

three

rotor gives

radial of the

the

and

for

radius

to a change desired

energy

transfer. Net outlet

rotor products

torque

is equal

of tangential

to

the

force

difference times

r:(F_r)_--(F,,r)2

radius,

between

the

inlet

and

or (2-7) 25

TURBINE

DESIGN

AND

APPLICATION

where net

torque,

F_

tangential

r

radius,

Applying V=O

N-m;

lb-ft

force,

N;

lb

m; ft equation

at t=O

(1-34)

to V=V

in the

at t=t,

tangential

and

direction,

setting

w=m/t

integrating

from

yields

(2-8)

_w_ Vu F,,--g where w

rate

of mass

g

conversion

flow, constant,

Substituting

angular

(rate

lb/sec

1; 32.17

equation

T =--

Power

kg/sec;

(2-8)

g

Yu,

W

lrl

of energy

into

_

(lbm)

(ft)/(lbf)(seC)

(2-7)

then

=-

w V,.2r2

transfer)

yields

(2-9)

g (V_.lrt--V_.2r2)

is equal

to the

product

of torque

and

velocity: p_r_

J -gJ

w

_(rlV_

.

1--r2V_

,

2)

(2-10)

where P

net

power,

angular J

W;

Btu/sec

velocity,

conversion

rad/sec

constant,

1; 778

(ft)(lb)/Btu

Since (2-11)

roo:V

we can

write p=W

gj

(2-12)

(UIV_.,-U2V,.2)

But P=whh' where

h' is total

(2-13)

into

enthalpy,

equation

in J/kg

(2-12)

(2-13) or Btu/lb.

Substituting

equation

yields

(2-14) J

where

Ah' is here

Equation machines

(2-14) and the

fluid

between

the

two

is the and UV_

t

as hi--h2.

is called

between

26

defined

basic the

the

rotor

terms.

work

Euler must The

equation equation.

for All

be accounted way

equation

all

forms

of turbo-

energy

transfer

for by the

difference

the

(2-14)

is stated,

it

BASIC

can

be seen

that

with

the

energy

fluid

is defined

It is useful will

be

done

turbine and

blade outlet.

There inlet sarily

at the

be made From

for

equation

to transform with

the

the aid

section The

the

be positive

balance,

for a turbine. (1-46),

where

This

CONCEPTS

is consistent

work

done

by

the

as positive.

is assumed or

Ah' must

TURBINE

Euler

along

with diagrams

to be no

radial

same

locations,

equations

in

(2-4)

and

(2-5),

diagrams

locations

This

axial-flow

for

the

axial-tangential

following

three-dimensional

form. an

of velocity

these the

another shows

velocity

component

Actually,

into

which

are

although

radius.

a general

2-4,

the

velocity

outlet

equation

of figure

inlet

planes. at either

are

derivation

not

the

necesalso

can

case. we get

Vx *: V 2- V= _

(2-15)

and Wx_=W*--W,, Substituting

equation

(2-6)

into

(2-16)

_

(2-16)

gives

Vx, 1 = Wx, 1

Wu, 1

Ul

of rotation''e*

Direction V_.

_z E'

,__i/_,,"- Vx, 2 = WX,2

u2

\ '- Vu, 2

FIGURE

2-4.--Rotor

section

with

inlet-

and

exit-velocity-vector

diagrams.

27

TURBINE

DESIGN

AND

APPLICATION

W,_=W*-(V,,-U) Since

Vx=

Wz,

combining

equations

V 2- V,,2=W

2

(2-15)

(2-17)

and

_- V,,2+2UV_

(2-17) -

yields (2-18)

U2

Therefore,

UV,,= 1 (V_+U2-W Now,

adding

subscripts

for inlet

U1V.,,

and

(2-19)

2)

outlet

yields

1 (V_+U2

(2-20)

W2 )

U2V _ _:-21 (W+U_2-W_ _) Inserting

these

values

into

the

Euler

(2-21)

equation

(eq.

(2-14))

finally

yields Ah'--2gJ Equation relation. By

(2-22)

1

is an

(V2

V2_+U__U2_+W2_

alternative

form

Wt2 )

of

the

basic

(2-22)

energy-transfer

definition,

,

,

Vl _

V__

Ah'=hl--h2=hl+-_--h2. Therefore, shows that

comparison

of

2gJ

equation

(2-22)

(2-23)

with

equation

(2-23)

1 Ah:h_--h2=_gj Thus,

the

in static

U 2 and

W 2 terms

enthalpy

change

across

in absolute

of terms transfer.

are

(UI2-U22 of equation

the

kinetic

rotor,

energy

sometimes

tum the

of the rotor.

of this energy

28

that

The

to

in

is actually fluid

flows

a centrifugal

in the

discussion

tangential

force

as

change

acts

V 2 terms

the

change

represent

These

the

three

components

of

pairs energy

of energy

tangential from

figure

2-5

concern

and

the

way

momenthe fluid the

to

cause

in which

the

wheel.

curved on

in the

transfer and

to the the

(2-24)

represent

rotor.

the

momentum

transferred through

the the

2)

Loading

it is the

results

following

change

As the blades,

fluid

while

referred

previously,

(2-22)

across

Blade As mentioned

+W22-W_

passage

it in the

between

direction

each of the

pair pressure

of

BASIC

TURBINE

1

Stations

CONCEPTS

2

sur,_ce J \\1

f

/

/- Suction surface

Flow

"_ ] I

Flow

Axial chord

pll

Pl

m

--P2 Suction surface--,'

Axial distance FIGURE

2-5.--Blade

row

with

surface

static-pressure

distribution.

29

TURBINE

DESIGN

AND

APPLICATION

(concave) surface. Since the free to move in the direction must

be established

through the flow

passage surface.

The

resulting

is illustrated distance. At where the tion value. flowing trailing crease

at

centrifugal

pressure (convex)

the

distribution

to

the

edge.

On

the

below

the

exit

the

of the

blade

surfaces

suction

surface,

pressure

curve

blade-loading

The

the

blade

force

diagram. acting

in the

in a rotating

passage

in a stationary

moving similar

by

is relative

examine

what

through

the

to equation

total

happens

rotor.

through

enthalpy,

to relative

For

purely

rotor

flow,

quently,

no change

constant

for the

We enthalpy. 3O

can

also This

the

changes

axial

stagnation

will

back

up

figure

the

the

often

de-

to the

exit

2-5

curves

blade

is called represents

in a manner conditions

similar

relative

total enthalpy enthalpy.

in a manner

J

(2-25)

in J/kg total

or

Btu/lb.

enthalpy

(2-24)

to

to the

as

we substitute

Now the

let

fluid

us

flows

for W 2 according

we get

we see that the

axial point

direction.

relative total

h '2'-- h ','-Therefore,

the

pressure

be analyzed

If in equation

(2-25),

surfaces

toward

in

considering

passage. Let us first define to the definition of absolute

h"

static

the

reaches its stagnapoint for the fluid

between

h"__h+2_ where

blade

at

Conditions

can

passage

normal to the pressure

the

From

illustrated

Relative

fluid

is plotted against there is a stagnation

increase

tangential

not force

the

lowest

decreases

then

area

turn

and on

blade.

the

and

the

Flow

pressure

sides

pressure. The pressure-distribution

flow

surface

zero and the pressure point is the dividing

two

along

and

force is directed surface. Thus,

in figure 2-5, where pressure or near the blade leading edge

pressure

and, therefore, force, a pressure force

pressure

of static

velocity becomes The stagnation

the

the

path. The the suction

is highest

around

point,

to balance

its curved and toward

in the suction

fluid is constrained of the centrifugal

where flow

speed,

enthalpy

of the

fluid

is a change

in the

blade

speed.

in radius

and,

conse-

is no the

(2-26)

change

relative

total

enthalpy

flowing

remains

process.

a temperature

is called

total

if there

there

in blade

rotor define

relative only

U_-- UI_ 2 gJ

the

relative

that

corresponds

total

temperature,

to relative T".

total When

ideal-gas-law we can write

behavior

and

constant

heat

BASIC

TURBINE

capacity

can

CONCEPTS

be

assumed,

h"--h=%(T"--T)

(2-27)

where c_

heat

capacity

T

absolute

at constant

temperature,

Combining

equation

pressure, K;

(2-27)

J/(kg)(K);

Btu/(lb)(°R)

°R with

equation

(2-25)

then

yields

W 2 T" From and

equation relative

(1-51) total

and

-= T _- 2gJc-_

equation

temperatures

(2-28)

(2-28),

are

we see

related

For

the

rotor

flow

process,

T"--

this

with

we can

equation

2

Therefore,

relative

depends flow

only

through

total

on

blade

that

U_-- V12 2gJcp

temperature, and

(2-30)

T_') shows

T_'--

speed

(2-29)

write

(2-26)

T_'--

absolute

-2gJc_

h_'--h_'=cp(T_'-Combining

the

as follows"

V__W

T'--

that

(2-31)

like

relative

remains

total

constant

for

enthalpy, purely

axial

a rotor.

Relative

total

pressure

brought

to rest

pressure

p. Therefore,

can

be

isentropically

defined

from

as

a relative

the

pressure

velocity

of

W and

a fluid a static

]!

where p"

relative

_,

ratio of heat capacity constant volume

From

this

total

pressure,

equation

and

N/m2; at

crease,

the

rotor

flow

or remain

temperature

and

on

the

pressure

(1-52),

p"

(T"y/(-,-')

process,

constant,

constant

equation

p' For

lb/ft 2

we also

to heat

capacity

see that

=\--_-;,T/ relative

depending losses.

For

at

(2-33) total on purely

pressure the

can

change axial

increase,

in relative

flow,

relative

detotal total 31

TURBINE

DESIGN

pressure it must We

AND

will remain decrease. can

define

APPLICATION

constant

only

a relative

Mach

if the

flow

number

M,_=

is isentropic;

Mr_

otherwise,

as

W

(2-34)

a

and

a relative

critical

velocity

as

Wct=a_.,_=_l

gRT"

(2-35)

where

We,

critical

act, ttt

speed

R

gas

Then, and

velocity,

m/sec;

of sound

at relative

constant,

J/(kg)

in a manner (1-64),

critical

condition,

m/sec;

ft/sec

(K) ; (ft) (lb) / (lb) (° R)

similar

we can

ft/sec

to

the

way

we

derived

equations

(1-60)

get T"T --1+

2--1

M_e_

(2-36)

and T,,--1

3,+1

_

(2-37)

Reaction The

fraction

enthalpy)

that

portant

way

energy

total

of the row.

or more

The

of velocity

across

enthalpy

across

enthalpy

across

enthalpy

the

the

we can

write

stage.

exit

kinetic

energy

stage.

the

rotor,

According

ditions 32

R,tc

is stage

upstream

reaction, and

that

and

the

above

the

change

enthalpy

for static total

absolute

total

in absolute

total

of stage

constant reaction,

(2-38)

subscripts

of the

in

in absolute

hl--h_ --hi

the

of

degree

for classifying

remains

detiIlition

way

is the

change in

cha_lge

im-

kinetic

parameter

change

as the

total

to the

is used

as

in

important cases

an important

of the

same

since

downstream

is one

total

is one

change

in both

is defined

Note is the

absolute

enthalpy

The

Reaction

it is also

R_'_--hl' where

used

as a fraction

stage

in

in static

turbine

and

rotor

the

stator.

a change

a

reaction

tile

(change

the reaction.

diagrams,

across

through

transfer

parameter

simply,

correlating losses. Stage reaction.--Stage enthalpy

by

classifying

a blade

of reaction,

energy

is obtained of

as a fraction

classifying types

of

rotor,

1 and

2 refer

respectively.

to con-

BASIC

The

preceding

equation

for

velocities. Substituting (2-38) yields

equations

can

(U_2--U_ Zero

be positive,

2) and

design.

rotor,

and

be

(2-22)

expressed

and

in terms

(2-24)

into

negative,

or zero,

depending

of

equation.

(2-39)

on the

values

of

(W2_--Wl_).

reaction

stage

can

CONCEPTS

(U?-U22) + (W 2-W ?) V22)+ (U2 U22)+ (W22_W2)

R.,,-Reaction

reaction

TURBINE

is one

important

If Rst0=0, all the

value

there

work

done

is no by

that

characterizes

change

the

a particular

in static

stage

enthalpy

is a result

in the

of the

change

in

absolute kinetic energy across the stage. This stage is called an impulse stage. In the general case where the fluid enters and leaves the rotor at

different

of static effect

radii,

an impulse

enthalpy and

an

in one

equal

relative-velocity

stage

direction

change

effect.

other

purely

pressure

people

define

in the

rotor

definition

impulse rather

in terms

of static

used herein. The definitions exactly

difference coincide.

Simple

examples

windmill, from

or the

a stationary

the lawn rotation.

develol)ed

energy

at the

that

blade

kinetic

energy

blade-row stage For

the

blade-row row.

For

the

blade

exit.

These

a

corresponds

reaction

represents

no

are by

the the

in

static

isentropic

child's

This

same

as that flow,

pinwheel,

impingement

example

Thus,

enthalpy. the

For

in static only.

change

in static

the

the

of a fluid

of a reaction

turbine

from

reaction

is

defined

as

the

kinetic

as a

fraction

of

the

kinetic

are

stator

or

to the

change

an effect

the

kinetic

axial-flow

energies

rotor,

in static similar

thus

is

water

row

nozzles,

the

the

enthalpy. to that

causing

relative change

in

Therefore,

represented

by

reaction. a stator

blade

row,

R,,

where

turbines

(U 2) by

change

velocity

of

to losses.

A simple

within

centrifugal

any

basis

operated

ejects

the

a change

contributed

flow,

change

having

is approximately

is due

wheel

that

no

reaction.--Blade-row

energy to

than

by

of relative WI= W2.

the

pressure

nozzle.

sprinkler

Blade-row

on

of impulse paddle

from

direction

axial

enthalpy must be caused by a change an axial-flow impulse stage must have Some

result

contributed

in the

For

may

Rs, is stator

reaction.

reaction

is defined

V1_-- V°2----1 V12 For

a rotor

as

V°2 Vl 2 blade

(2-40) row,

reaction

is defined

as

33

TURBINE

DESIGN

AND

APPLICATION

R

--W22--W1_= to--

where

R,o is rotor

tions

upstream

stream

some

reaction. of the

of the

In

rotor,

The

the of

kinetic

equations

(2-40)

and

(2-41)

rather

than

all adiabatic

velopment when

the

presented this

gas-law shown

work)

and

veniently

temperature entropy

for

down-

later

lines

all

the

velocities

similar

appear

to the

energy

trans-

maximum or

energy

pressure

do it here),

transfer

ratio from

and

Therefore,

variables

pressure.

is obtained

the With

with

Since

previously

and

as can

ideal-

previously by temwe

actual

entropy

the

temperature,

of interest,

pressure,

(de-

we will illustrate

discussion.

(isentropic)

decreasing

to

V2).

can

diagram. against

increases be seen

Constant-entropy-

b.-

E Pl > P2 > P3 E b--

I Entropy, s

FIGURE

34

2-6.--Typical

temperature-entropy

diagram.

con-

expansion

means of a temperature-entropy diagram is a plot of temperature and

of

is

be proven

in this

being

of constant

temperature

terms

assumptions, we have changes can be represented

ideal

by

in

Process

a given can

changes.

the

processes in a turbine The temperature-entropy entropy

for

we will not

a little

represent

increasing

to condiand

definition than

energy)

This

constant-heat-capacity energies and energy and

pressure,

kinetic

is isentropic. (but

the

processes,

of mechanical

graphically

peratures

2 refer stator,

is defined

V rather

Expansion

of

and that

of the

that

(i.e.,

expansion

equations

fact

0, 1, and

This

except

squared

(development process

(2-41)

reaction

energies.

Turbine For

W22

downstream

blade-row

instead

formation

W12

-

respectively.

literature,

power

l

subscripts

stator,

velocities first

W22

with from

BASIC

the

discussion

chapter

1,

example by

entropy,

the

For

the

These The the

coastant-entropy-1)rocess

figure

pressure

purposes four

four

the

A

At

diverge;

is also

diagrams

the

with will

like

process

values

of

therefore,

at

difference

in

looks

the

is repre-

temperature

and

increasing

values

between

any

two

increasing.

of clarity, steps,

diagram

constant-entropy

temperature

CONCEPTS

thermodynamics T-s,

increasing

curves

curves

into

or

2-6.

line.

entropy,

pressure

divided

in

a vertical

constant

given

the

temperature-entropy,

shown

sented of

of a

TURBINE

each

then

be

turbine shown

expansion

process

in a separate

combined

into

T-s

a single

four diagrams represent the stator expansion process relation between absolute and relative conditions

will

be

diagram. diagram.

(fig. 2-7 (a)), at the stator

P()"P'Lid

_,,_Jc-T_= T_

Ti

TO

Tj'

------7

vl p?

T1 Tl, kJ

T1

NJCp 0 (a)

(b)

i.--

p_

P_' T_'

/

/

Ti' : T_'

T_ 2gJcp

P2

>2gJCp

hl--_

T2

CPl

T2 ....

(c)

(d)

l Entropy, s

(a)

Expansion

process

across

stator.

(b)

Relation relative

(c)

Expansion

process

across

rotor.

(d)

Relation absolute

FIGURE

2-7.--Temperature-entropy flow

diagrams turbine.

for

between conditions

absolute at

stator

between conditions

flow-process

and exit.

relative at steps

and

rotor of

exit. an

axial-

35

TURBINE

exit

DESIGN

(fig.

blades

2-7(b)),

(fig. at

Figure

APPLICATION

the

rotor

2-7(c)),

conditions four

AND

and

the

rotor

2-7(a)

shows

before

is represented

point

and

the

the

expansion

indicated state

by

process

by

the

small

by

the

1 with

arrows. actual

It

related

gies

and

The represent If the

static

For

and

were

rotor

exit

process

The

is shown

four

state

as

the

enthalpy (p_,

figure

2-7(c),

it is on

for

the

T2._d), state refers

figure

2-8,

the

entire

stage

36

figure

the

and

by

the

the

which the

ideal

rotor

in

kinetic

in figure

en-

2-7(c)

in

curves

before

and

so

that

would

be that

and

after

the

T;'=

T_'.

indicated

are

is on

the point

ideal 2,id and

absolute

now

are

right state

!

line.

rotor).

The

the

one by

In

figure.

figure

ideal

l_,_t_fing

the

stage

diagram total

the

being, Note

arrows

ignore

the

that

the

line 2-8, the

rotor

alone.

of the

across subscript

in

where

2-7(c),

expansion

the isen-

relative

in figure

across

to the

at

constant-entropy

as in(licated

expansion refers

time

of the

at

into and

indicated

the

ideal

related

energies

total,

an

states

are

combined

For

developed by

total

states

kinetic

expansion on the

energy

developed

These

points.

same

kinetic be

absolute

0 constant-entropy the

relative

absolute

2-7

indicated

stator

relative

assumed, state

would

static,

turbine

subscript

final

2-7(d).

state

to

(both

is

relative

The

the

is not

2,id

by

developed

constant-pressure

flow

the

appropriate

script

that from

figure.

is shown four

pressures

than

of figure

2-8.

rotor

the

that

relative

differences

point

be

proceeds energy

and

total

is less

in

diagrams figure

processes

through

would

If

The actual process proceeds from state 1 to the small arrows, with an increase in entropy.

between

tropically, and the exit are indicated. shown

state

(1-51).

through

in the

The

axial

be noted

it can

process. The relation

static

developed

flow

at each

as indicated

kinetic

absolute

the

isentropic,

Here

actual

the

relative

2,id. by

again

state

process

be

analyze

across

simplicity,

expansion

the

would

conditions.

by the subscript state 2, as indicated by

the

in entropy,

are indicated

process relative

the

expansion.

and

temperatures

of the

than

The total

energy

equation

final

the

stator.

conditions. Figure 2-7(b) shows the relation between relative total states at the stator exit. These states

expansion

terms

with

the

that

we

isentropica]ly, total

absolute

absolute

between

actual

increase

be noted

the

and

kinetic

distance

The

a small

previously,

terms of relative the absolute and

static

The

isentropic,

is less

moving

and

across

the

in accordance

1,id.

can

process

process. As mentioned

are

vertical

were

to the

relative

process

expansion.

point

subscript

0 to state

the

the

state

between

represent

the

relative

2-7(d)).

expansion

after

total

process

relation (fig.

curves and

state

the

the exit

constant-pressure

pressures

expansion

subIn the 2,id

BASIC

TURBINE

CONCEPTS

T_" T_ i1

ii

P2

=__Po

To

Ah'

(h6-

T_? '_ T_' p-

Ahid

\ lh

T1

--

--j,,J

T_ Ca

E

I

T' 2, id

Ahid (h 6 " h2, id )

T2 TZ id

Entropy, FIGURE

is,

2-8.--Temperature-entropy

therefore,

diagram

ambiguous

obvious

from

process

(as represented

be obtained To-- T;. ,d).

figure

but

2-8 an

for

the

work

T o-

ideal

T_)

Since

turbine

used actual blade

for

blade

to express this

rows

is less

purpose

in both from

the

the

work

than

process

operate

performance.

is blade-row

exit kinetic energy row. For the stator,

axial-flow

(as

turbine.

senses. real

It

that

could

represented

by

isentropically, One

efficiency,

divided

by the ideal

common

which exit

we

need

is defined

kinetic

_

is stator

indicated in figure (1-55), we get

efficiency. 2-7(a).

The By

as the

energy

of the

For

the

(2-42)

relation

applying

between

equations

V_ (1-51),

and

where

is

and

(2-43)

rotor

Wi _'°=W_, ,d Vro is rotor

is indicated

2 Vl._a

(1-52),

\pod

L

a

parameter

V_I

where

is

turbine

Efficiency

do not

blade-row

of an

used

obtained

turbine

Blade-Row

parameter

a stage

is commonly

that by

from

s

in figure

efficiency. 2-7(c).

The For

(2-44)

relation

purely

axial

between

W_

and

W_._d

flow,

37

TURBINE

D]_SIGN

AND

APPLICATION

W_,,d=2gJc_,T;' Thus,

with

inlet

it is possible

conditions

and

to calculate

Blade-row

in

as a loss rather

known

for a given

for a specified

terms

than

(2-45)

j

efficiency

exit velocity

performance

expressed

1-\-_,/

of

blade

exit static

kinetic

energy

as an efficiency,

is

sometimes

as

e=l--n where

e is the

kinetic-energy

Blade-row total

differing by dimensionless. actual

also

Several the

(2-46)

loss coefficient.

performance

pressure.

can

be expressed

coefficients

of this

normalizing parameter Inlet total pressure, exit

dynamic

head

have

all

been

in terms

type

used ideal

used

have

this

Axial

I

/I

f

f

!

Y,

Y',

ff

!

between

the

pressure

Y"

Y"o

stated,

and

they

are

total-pressure

can

be derived.

loss

involve

I¢

t!

p2

process

or stage

energy

is isentropic.

Since

parameter that

for expressing

we use

ratio

is the

of actual

This

Overall stage or

stage

to

turbine

we

are

It is the the

ideal

or stage

discussing

process

inlet

transfer condition

aerodynamic

when

simply

which

refers energy

energy

based to

efficiency

the

exit and

transfer. The

definition

are

overall

transferred on

turbine

in the

isentropic pressure. are

as the

efficiency.

above

to the

a

parameter

is defined

or adiabatic the

expansion we need

The

(isentropic)

apply

the

isentropic,

performance.

isentropic

efficiency energy

total-

are not

dependency.

efficiency,

of actual

various

relations

is never

or stage

that we can to follow. ratio

the

Relations

Efficiencies

to ideal

as the

coefficients.

is maximum

or stage

e_ciency.--Overall

process.

the

38

the

transfer

ways sections

Stage

transfer

turbine

is known

several different discussed in the

and

turbine

energy

efficiency

number

(2-47c)

--P2

and

These

(2-47b)

P_

loss

coefficient

a Mach

Turbine Turbine

_f

P2

kinetic-energy

loss coefficients

(2-47a)

,, yT o__P, p_ --P2 --p_

"Po--Pl

and

It

--P2,,

'

• 't--P--_--pl

where

as follows:

PI

Po--p___._ "--P;--Pl

V"

each

the coefficient head, and exit

purpose

YTo-_P-'

Po

v'

used,

in

rotor:

t

po--pl,

of a loss

been

to make dynamic

for

Stator:

y,

row,

pressure.

not,

or

turbine

flow

from

Note

that

at

present,

BASIC

considering seal friction.

mechanical

We will turbine.

define

This

however,

Actual

absolute

total

in figure

2-8.

Now

work

energy

transfer

energy

for conversion conditions are used.

If

plenum, energy

turbine

then

the

could

shaft for

the

work the

in the

down

If we were the

kinetic

loss.

The

are

carried

shaft

to the

exit

ideal

case,

energy kinetic

work.

the

cases

where

pose,

the

entire

from

the

exit

case

is the

high

velocity

leaving

and

effi('iency exit

total

before turbine exit

based by

the exit total kinetic energy)

is always

between

the

two

the

ideal is

situation,

obvious must

as a

wasted,

but

of its exit

to static

of their

serves

of ideal

a useful work

exit pur-

computed

example

of this

be expanded

therefore,

In exit

converted

basis

most and,

not be

basis the

basis

engine, ideal

ideal

enthall)y

2-8.

It

can

must based

higher increasing

than

work

available

is culled

work

called

con(tition than that

efficien('y

are

the

gas

to

the

above

may

the

state

be considered

energy

the

equal

in tile

on

static energy.

would

The

Here

exit

to

exit. kinetic

stages on

converted

a high

to

a

velocity

a waste.

on

on the the

on

the

kinetic

desirable

they

rated

kinetic

('.on(litions

conditions in figure

are

conditions.

based static

the

as in a

wasted

been

would

other

is rated

the

zero

where

stages

leaving

inlet,

is available

be

state

stage

the

turbine.

have

turbine

stage,

stage

is not

the

This

it would

with

last

is rated

state

available

is dissipated,

we use

total

turbine-exit

turbine total

rel)resented in(licated

last

the

efficiency

total

next

other

jet-engine

the

The

the

the

leaving

to the

Thus,

state

a multistage

energies

over

energy At

in

is indicated

energy

wasted.

because

exit

only

the

decrease

this

ideal

energy

a case,

work

the

from

by

l)lus exit kinetic

and

kinetic

if it could

such

static

considering

the

is just

1)ut to use In

done

occasionally,

is the

conditions.

kinetic

of ideal

work

and

At the turbine or stage exit., static and total conditions are sometimes

energy

turbine.

condition, while total conditions. In

been

as bearing

work

or stage,

inlet

CONCEPTS

people;

herein

or total

work. used

shaft

as shaft

to define the

such

most

defined

turbine

exhaust-flow

have

this desirable static state.

the

exit kinetic

computation

expand

as

because

to shaft sometimes

by

is defined

of static

used

as the

used

whether

basis

is ahvays

to items

transfer

across

consider

due

one

transfer

enthalpy

on the

state

energy is the

we must

to do total

actual

definition

actual

energy.

inefficiencies

TURBINE

available tile be

the

between

the

inlet

efficiency.

The

decrease

for each

of these

that

the

be less (as long on the exit static static

efficiency, increasing

ideal

with

inlet The

total

and

conditions

work

as there condition. exit.

the

efficiency.

total seen

with

between

static

cases

are

based

on

is some Thus, the

kinetic

exit total

difference energy. 39

TURBINE

DESIGN

Overall

AND

turbine

APPLICATION

efficiency

_ and

similar equations. The inlet and exit conditions,

subscripts instead

stage.

static

Overall

turbine

stage

efficiency

_tg

are

defined

by

in and ex are used to denote turbine of the subscripts 0 and 2 used for the

efficiency

can

be expressed

as

(2-48a)

For the ideal-gas-law reduces to

and

constant-heat-capacity

assumptions,

this

T',.--T',_ V

/ p,, \(_-l)/_l

Overall

turbine

total

efficiency

(2-48b)

J

T;. is expressed

as

hh' h;.--h',_ _' = .-_-7-, =h' _'

For the ideal-gas-law reduces to

and

constant-heat-capacity

._, _

total

and

appropriate Relation

of turbine

efficiencies

are

(2-49b)

similarly

defined

but

with

the

as

However,

to stage

a measure

it is not

efficiency.--The

of the

a true

overall

indication

overall

turbine

performance

of the

of

efficiency

the

of the

comprising the turbine. There is an inherent thermodynamic hidden in the overall turbine efficiency expression. If equation

(2-48b)

or (2-49b)

a given

stage

which

were

pressure

for a stage

ture

would

gas

entering

2-8,

the

losses

perature stage though

for a stage,

and

stage

the of one

For

stage

appear

the

following

is then

capable

of

delivering

all the

individual

overall

turbine

the

number

of stages.

effect such

can as

stages

efficiency be shown

figure

2-9.

by The

the

a turbine, in the

stage have

del)ends

means solid

to

form

on

the

same the

line

transfer, teml)era-

of a higher

pressure

O-2,id

tem-

following

Therefore, stage

of a temt)erature-entropy verticnl

for

be __een from This

work. the

that

energy

as can

(T_T2,_,).

additional may

still

be seen

is prol)ortional

stage.

entering

it could

efficiency,

be (To--T_),

gas

the

This

written ratio

of the

figure

4O

[1 --\p_--_-/ (P-_= _('-'>"]

e3_ciency

is useful

turbine.

gram,

this

subscripts.

efficiency stages effect

static

assumptions,

T'_.--T',_ T_

Stage

(2-49a)

even

efficiency, ratio

and dia-

represents

BASIC

TURBINE

CONCEPTS

Pk k

_ \\

1

i '

b---

E

\\ A,'i .,tg

\

¢D

E

_\

¢D b--

Ahld:stg

\I:

\\

Ahld, stg

2,id

Entropy, s 2-9.--Temperature-entropy

FIGURE

isentropic

expansion

dashed

line

taking

place

The

0-2

from

obtained

ideal

work

difference

of temperature

increasing

p_),

the

that

represented

greater

by

constant E-F

stage

is greater

ing

the

the

sum

which and

by

virtue

ideal

Thus,

the

than work turbine

the

the

of

actual

work

B-2,id.

With

lines

three

stages,

it can ideal

be seen work

that

the

0-A,

work

where the

increases

stage

(p_

is greater for

this

stage

be greater.

C-D,

and

and

to

than stage

an(l,

is for

Similarly,

E-F

represent-

2; Ahld. stt representing

Z Ah;dstg

represented

second C-D

previous will

_' _'sto.

previously, pressure

the

line

The

efficiency

is ,7_tg Ah_,.stt,

isentropic

for

the

for the

p_.

efficiency

mentioned

of constant

by

inefficiency

efficiency,

of these,

is the

A-B.

of

pressure

stage

stage

Hence,

represented

in a multistage

turbine

same

As

lines

of entropy.

work

each

a stage.

between

values

isentropic

the

effect

exit

of overall

having

from for

reheat

Po to

process

each

work

the

showing

pressure

the

stages,

Ah_d. stg is with

inlet

represents

in three

actual

diagram turbine.

by

is greater the

sum

than of 0-A,

A-h_, A-B,

B-2,id. 41

TURBINE

DESIGN

The

total

to P'2 can these

two

AND

actual

APPLICATION

turbine

be

represented

values

must

work

obtained

by

either

be equal.

from

7'

the

expansion

Ah',d or

_tg

from

Z Ah'_.,t,,

Po and

Thus,

-'Ah'

'

(2-50)

z ah,_.,,, Ah'_

(2-51)

or

, _-_

Since

2_ Ah'_d,s,g

greater This

than effect

be confused between

_Ah_d,

with

the

stages,

The

of

efficiency

turbine

process

which

equation

stages

the

overall

isentropic

efficiency

the stage isentropic efficiencies, or _' _tg. in turbines is called the "reheat" effect. is also

for

constant

of adding called

calculating stage

heat

from

This

an

is

must

external

not

source

"reheat".

overall

pressure

turbine

ratio

efficiency

p_/p_

and

for

several

constant

stage

_t_ is

I--

-,

1--_,t_

J)

1--\_-£]

_1 --

(2-52)

( p i t nt(.y_ l)l_, ]

1 --\_oo/ where n is the number be found in reference The

fact

on the

that

stage

pressure

ratio,

of turbine

efficiencies

not

comparison

a true

higher

pressure

sirable

to be able

In

order

ciency

ratio

of stages. efficiency raises

from

machines

of their

to express

temperature

(T--dT),

by

a true

all reheat

the

consideration.

can

this

a gas where

efficiency

from

is expanded d T is the

we

It

have

ratios

is

as the

one

would

be

of de-

for a turbine. to be the

effi-

stage.

of isentropic

definition,

A comparison

effect. would

depending

pressure

behavior,

reheat

aerodynamic effect,

small

suppose stage

equation

efficiency,

of different

eff_ciency.--Starting

T,

of this

turbine

aerodynamic

is helped

Infinitesimal-stage

tropic-efficiency

derivation

an important

of an infinitesimally

infinitesimal

differs

of two

to eliminate

temperature an

The

1.

pressure

to pressure

increment

efficiency

p

(p--dp)

of temperature

vp. By

using

the

and and for isen-

write

d T = _pT E l --(ppdp

) (_- ' ) /_]

(2-53)

and dTT_ _1o[1--(I 42

__)(_-

z,l_]

(2-54)

BASIC

These

equations

efficiency

the be

the

the

be

than

static

rigorously

authors

the

in accord

ignore

proportional

the

to the

that

stage,

there

temperature

fact

that

the

is used

isentropicactual

in

the

work

differenOther

in kinetic

d T'----d T. However,

that

the

differential.

is no change

so that

with

CONCEPTS

total-temperature

static-temperature

assumption

infinitesimal

quite

Some

should

rather

make

not

definition.

differential tial

are

TURBINE

authors

energy

it ahvays

across seems

to

infinitesimal-efficiency

expression. Using

the

evaluation

series

expansion

of equation

approximation

(2-54)

(1-#x)

n= 1 +nx

dT 7_1 dp T --_ 7 I' Integrating

between

the

for

yields

turbine

inlet

(2-55)

and

exit

yields

Tfn

In _(2-56) z/P--7--

1 In p_" "Y

Equation

(2-56)

The

can

infinitesimal-stage

dynamic

efficiency,

efficiency

is also

from

method

the

constant, called

be written

where

we get

for the

as

efficiency exclusive

known

as the

n is called l)olytropic

the

effect

l)olytropic an

the

process.

_p is SUl)posedly

of

of expressing

a l)olytropic

Pex

of

the

efficiency.

irreversible

polytropic

for

This

process

exl)onent

Substituting

true

aero-

ratio.

This

name

arises

pressure

v from

path

as pv"=

, and

the

process

the

ideal

gas

process

{p,n'_

T,n

(n-l)/n

(2-5s)

\p,,! Equations process relate

(2-57) were

l)olytropic

and

(2-58)

to be expressed efficiency

are as

and

the

n

inlet

and

exit,

very

similar,

a ])olytroi)ic

n--1

If we neglect

is law,

polytrol)ic

and

if

the

process,

then

exponent

as

-),--1

turbine we

couhl

(2-59)

?

kinetic

energies

for

the

overall

turbine 43

TURBINE

DESIGN

process,

we

ciency.

Actual

AND

can

APPLICATION

relate

turbine

temperature

overall

efficiency

could

be expressed

drop

T_ _Te_='_T_,

to

E1 --\_-_,/ (Pe_ y"-l)/_

(_-e_)

1J

_'{(_-

1)/_]

L -\_/ Equating

(2-60)

with

(2-61)

then

(2-60)

k

(2-61)

J

yields

_=1--\_/

This proach

relation

is illustrated

each

other

unity.

However,

ciency

levels,

as at

the

two

in

(2-62)

figure

pressure

higher

effi-

as

or _1

polytropic

2-10.

ratio

and

pressure

efficiencies

can

The

two

efficiency

ratios,

especially

differ

signiticantly.

efficiencies each at

ap-

approach lower

effi-

.9-Turbine pressure

¢-

_.,,_

r_tio /.,///

09 ----

.8

B

b---

.5

1

I

.7 Turbine

FIGURE

2-10.--Relation

between Specific

44

I

,8 poly_ropic efficiency,

turbine heat

overall ratio

I

.9

7,

ffp

and 1.4.

1.0

polytropic

efficiencies.

BASIC

DIMENSIONLESS Dimensionless turbine

number introduced

of

the and

dimensionless

serve

geometry,

and

to correlate

Dimensional

analysis

variables the

or

minimum

more

the

analysis

that some

of the

procedure

variables for

and

obtaining

flow

yields

The

dimensions, term

resultant

ratios

implies

the

actual

insight

that

into

shape

and linear

]'his

based

on

on

is an

various

These viscous

a basic

ideal attributes

include

effects;

an elasticity

number),

which

number, pressing are

the The

matic, similitude.

which real

parameter

effects,

significant concept and

fluid,

ratios

which which for

for gas groups leads conditions

the

Reynolds

and

based

relations.

the

effect

of

surface-tension to the

and Of

analysis

ideal

reduces

effects;

change of the

an

expresses

a gas

than

groups,

expresses

effects. the

of

modify

of

geometrical rather

dimensionless

that

gravitational

quantities operating

physical

is a con trolling

characteristic other

in general,

of dimensionless two

are

(which

parameters

dynamic If

by itself,

compressibility

expresses

fluid

dimensions),

number,

expresses

formal

of fluid

basic

The

of linear

of

1, which

represent

dimension

number,

Weber

of the

of a

pertinent

problem

of velocities.

parameter

There

Reynolds

the

nature

The the

the ratio of the force due to the inertia force due to the motion

of a real

the

forces;

flow

fluid.

general

states

powers

reference

terms

ratios

(as a ratio

of each

factor. Another term expresses of pressure in the fluid to the fluid.

the

dimensionless

of forces,

magnitude

to the

and

form

of

group.

including

analysis

which

from

be

of dimen-

in the

groups

con-

once

basis

product

a dimensionless

the will

at least

7r-Theorem, a

some

each

groups

The

be expressed

in many texts, this discussion.

considerable

relations.

may

groups,

them.

is the

throw

for grouping

variables

dimensionless

of dimensional

are of

use

of variables

they

of such

representing

forming the

variables is presented served as the basis for Application

all the between

equation term

a group

so that

number

1)rocedure

each

allows

of dimensionless The

relation

physical

A

parameters for the

It is a procedure

to include

as a formal terms,

that

to be arranged

number

physical

a complete of

performance.

Analysis

relation.

variables.

necessary

to represent

number

of the

a smaller

two

sional

relation

nature

into

taining

diagrams,

analysis.

is a procedure

a physical

on the

velocity

turbine

is dimensional Dimensional

light

classify

more commonly used dimensionless discussed in this section. The basis

parameters

comprising

CONCEPTS

PARAMETERS

parameters

classify

TURBINE

the

these

Mach Froude

terms

Mach

ex-

numbers

flow. as ratios _o are

the such

of geometric, idea that

of

kine-

similarity all

the

or

dimen45

TURBINE

DESIGN

sionless

terms

of the

separate

obtained.

AND

have

APPLICATION

the

same

variables,

Complete

value,

then

physical

similarity

is ever

be approached

of similarity

the

to the

operation

rather

full-size

at some

Another fluid

design

Turbomachine Application flow

in are rows

the

use

in conjunction

_4th

size,

following variables tant relations : Volume

flow

Head,

to the

rate,

P,

Rotative

speed,

set

to ideal and

Q, m3/sec

or ft3/sec

parameters.

These

and

of the of the

power

fluid.

more

The

impor-

W or Btu/sec N,

rad/sec

or rev/min

Fluid

viscosity,

u, (N) (sec)/m

Fluid

elasticity,

E,

dimension,

variables,

the dimensional five

capacity, 3, which

D, nt or ft

_-or lbm/(ft)

N/m 2 or lbf/ft five

dimensionless

or flow is called

_2_' rate, the

groups

constants groups

can

in order

can

be formed.

to ease

be expressed

p-N3D 5' is exl)ressed capacity

(sec)

2

dimensionless

conversion

_fcn

Q/ND

of fluid

flow rate,

some

p, kg/m 3 or lb/ft 3

The

work),

to demonstrate

density,

the

problem

operational characterin the relation of head

used

Fluid

lation,

of

properties

linear

drop

involves conditions

or (ft)(lbf)/lbm

speed,

these

results

detailed examination of flow within In addition, dimensional analysis

Characteristic

From

46

relates

so that

the

ambient

general

the

are

H, J/kg

Power,

this

scale

with

Parameters

has great utility in the analysis of the overall istics. For any turbomachine, we are interested flow,

use

of similarity

at or near

mentioned

important for the of turbomachines.

(for compressible

it

One

condition.

analysis

previously

linear

the ratios

purposes

utility.

performed

Operational

of dimensional

results

parameters the blade

be

with

the

similar-

the ratios complete

practical

of smaller

can

are

are everywhere the velocity

for most

machine.

severe

(1) geometric

to be of great

of models

of machines

than

but

experiments

values

conditions

which means that is doubtful whether

closely

operation

inexpensive

applicable

attained,

sufficiently

is the

relatively

implies

similarity, same. It

individual

physical

dimension ratios which means that

are the same; and (3) dynamic of the different forces are the can

of the

similar

similarity

ity, which means that the linear same; (2) kinematic similarity,

physical

regardless

exactly

u

the

manipu-

as

(2-63)

' pN2D2/ il_ (limensionless

coefficient.

If we

It

can

form be

further

by

BASIC

rel)resented

TURBINE

CONCEPTS

as Q

VA

VI) 2

V

V

(2-64)

Thus, the capacity coefficient is equivalent to V/U, and a given value of Q/ND 3 implies a l)articular relation of fluid velocity Io blade speed or, in kinematic terms, similar velocity diagrams. The head is expressed in dimensionless form I)v H/N21) ', whi(,h is called the head coefficient. This can be rel)rcscnted as H H N 2D,_a: -_

(2-65)

Thus, a given value of t./N2D 2 iml)lies a particular rclatiol_ of hea_l to rotor kinetic energy, or dynamic similarity. The term P/pN3D 5 is _t power coefficient. It represents the actual power And thus is related to the capacity and hea(t coefficients, as well as to the efficiency. The term pND2/u is the Reynolds number, or viscous effect coefficient. Its effect on overall turbine 1)erformance, while still iml)ortant, can be regarded AS secondary. Ti_e Reynolds number effect will be discussed separately later in this chapter. The term E/aN2D 2 is the compressibility coefficient. Its effect depends on the level of .XIach number. At low NIach number, where the gas is relatively incoml)ressible, the effect is negligible or very secondary. As NIach number increases, the compressibility effect becomes increasingly significant. Velocity-Diagram

Parameters

We have seen that the ratio of fluid velocity to blade velocity and the ratio of fluid energy to blade energy are inq)ortant factors required for achieving similarity in turbomachines. Since completely similar machines shouhl perform similarly, 'these factors become iml)ortanl as a means for correlating performance. Since the fa('tor._ I_/'U aud H/U z are rclatc(I to the velocity (liagrams, factors of this type ar(, refcrrc(I to as velocity-diagram I)aramcters. Several velocity-(liagram parameters are co,mnonly us(,_l ill t_,rt)illc work. NIost of these arc ,ise(l I)rim'trily with rcsl)e('t to axial-ttox_ turbines. One of these parameters is the speed-work parameter X::-

U 2

(2-66

gJ/W 'Fh(_ re(:ipro('al

of the sl)ce(l-work

parameter

is also oflcn

use_l, aml

it 47

TURBINE

DESIGN

is referred

For

AND

to as the

an axial-flow

APPLICATION

loading

turbine,

factor

or loading

1 ¢=X--

g JAb' U2

we can

coefficient

(2-67)

write

(2-68)

Therefore,

equations

(2-66)

and

(2-67)

X=

Another

l)arameter

often

used

1

can

be expressed

as

U

(2-69)

¢--AV_ is the

blade-jet

speed

ratio

U p_

where

V_ is the

or spouting,

jet,

or spouting,

velocity

is defined

ideal

expansion

stage

or turbine.

from

inlet

That

(2-70)

"_fj

velocity, as the

total

in m/see velocity

to exit

or ft/sec.

The

corresponding

st, atie

(:onditions

across

of equation

(2-71)

back

into

(2-71) equation

(2-70)

yields

U _= -7-= _ 2gJAh,d A relation

between

parameter the

static

can

the

be obtained

efficiency

the

is, Vj2= 2gJ,_h,,_

Substitution

jet,

to the

blade-jet by

(2-72)

speed

ratio

use of equations

and

the

(.2-66)

and

speed-work (2-72)

and

definition zXh'

The

resultant

This

shows

must

also

is directh speed

ratio

Another 48

relation

that

if efficiency

be a function related

onh

n =Sh-_

(2-73)

v= _/____ _

(2-74)

is

is a funclion

of the

other.

to the

is related

to the

frequently

used

actual

velocity

of ()ne of these

While

the

veloc)tv diagram

velocity-diagram

parameters

speed-work diagram, and

to the

parameter

it

parameter the

blade-jet

efficiency. Is

the

flow

BASIC

factor,

or flow

TURBINE

CONCEPTS

coefficient

(2-75) The

flow

coefficient

can

be related

to the

loading

coefficient

as follows:

v, / v, By

using

equation

(2-69)

and

the

velocity-diagram

_=#

cot

(V,,. ,'_ al kAVu /

The term Vu. I/AV,, cannot be completely specific types of velocity diagrams, such next

chapter,

(a different each can

this

term

function

of the

becomes

for each

different

be expressed

types

of the

(2-77)

of loading

of velocity

of velocity

in terms

we get

generalized. However, for as will be discussed in the

a function type

geometry,

coefficient

diagram).

diagrams,

Therefore,

the

loadingcoefficient

alone

flow

and

for

coefficient

the

stator

exit

angle. It

is thus

related

seen

to each This

next

and

chapter. one

Where

of

the

efficiency. or the two flow

We

related

that

we have

with

constant

stage

of this this

at

of

For

are other

type

use

only loss, therefore, definition is

stage is

the

in figure exit

kinetic

of equation

(2-68)

into

in the

for

only

correlating parameter

efficiency

correlation,

ease

how static speed

impulse Further

(total energy.

equation

the

(W_:

Assume W2)

stage

diagram assume

efficiency The

efficiency

ratio.

A velocity

2-11.

be

Parameters

blade-jet U_)

the next

is specified,

static

h'o-h'2 Ah' ,7-- h£--h2. ,_ --_h,_ Substitution

in

One parameter must the loading coefficient.

(U_=

is isentropic

to these

speed-work

general

specific the

are

case

case

diagram

(V_._=V_,2).

is shown

turbine

real

Velocity-Diagram

to

axial-flow

be related specific

is required

a more

for an idealized

velocity

can

of velocity

required. is usually to

parameters

idealized general

generally

ratio.

a single

an

more

mathematically axial

efficiency

parameters

Efficiency

show

velocity-diagram

for

type

Lewis

parameters and the

will now

through

addition, shown

a somewhat

speed

of these coefficient,

be

for

In be

a particular

blade-jet

We

will

four

velocity-diagram

Relation

can

these

other.

parameters. section

that

for

that _'=

a

flow

1).

The

efficiency

(2-78) (2-78)

yichts 49

TURBINE

DESIGN

AND

APPLICATION

a1

Vu,1 j/ V2

Wu, 2

FIGURE

2-11.--Velocity-vector

diagram

for

an

axial-flow,

impulse

stage.

UAV,,

(2-79)

= gJAh,.,t The

change

in fluid

tangential

velocity

is (2-80)

zxV,,=V_,,-V_,z From

the

assumptions

convention

(Wl---- W2)

and

(W,._=

and

W,.2)

the

we adopted, (2-81)

W_. 2= --W_. 1 From

sign

equations V,.2=W,.5+

(2-6),

(2-81),

and

(2-80),

U=-W,._+U=-(V,.

we get 1- U) + U=-

V_.I+2U (2-82)

and AV.=V..,I--V.,2=V..1--(--V..,+2U)--2V..I--2U From

the

velocity-diagram

(2-83)

geometry (2-84)

V_, 1= V1 sin m Since

flow

is isentropic

and

the

turbine

stage

is of the

impluse

type

(h2, _d=h_=hl), Vl=_ Substitution 50

of equations

(2-84)

2gJAh_e and

(2-85) (2-85)

into

equation

(2-83)

BASIC

TURBINE

CONCEPTS

yields AVe=2 Substitution

of

sin al_/2gJAhia-2U

equation

(2-86)

back

into

4U sin al

definition

Equation stator only. an

The

(2-88) angle, with

is reached speed

tion

(2-88)

shows

ratio and

that

static

variation

example

0.88 jet

exit

speed

for this and

mathematically

ratio

derivative

of axial-flow,

case

and

of 0.47.

The

by

for of

optimum

bladeequa-

(2-89)

I

I

I

.6

.8

stage.

2-12

to zero:

.4

speed

speed ratio,

ratio Stator

on

ratio

efficiency

differentiating

I

impulse

constant

speed

in figure

.2

blade-jet

any

of blade-jet

sin al 2

Blade-jet

2-12.--Effect

(2-72) (2-88)

equal

Vo_t--

FIGURE

equation

of 70 °. A maximum

speed

the

from

is illustrated

angle

can

setting

ratio

particular

at a blade-jet be found

(2-87)

is a function

is parabolic exit

yields

sin a,--4_3

efficiency

a stator

(2-79)

2gJAh_

of blade-jet n=4z,

equation

4U _

_--4'2gJAh_a Now using the finally yields

(2-86)

static exit

1.0

v

efficiency angle,

of

an

isentropic,

70 °.

51

TURBINE

Since the

DESIGN

the

AND

stator

sine

APPLICATION

exit

of the

angle

angle

is normally

does

not

in the

vary

greatly,

speed ratio for most cases of interest be in the range of 0.4 to 0.5. Equation specific.

(2-88) While

ideal

case,

indeed, very

the

the

figure

2-12

levels

and

values

basic

it does. good

and

We

parabolic find

correlating

Likewise,

for

other

tion

(2-63).

This

does

would

of all sizes.

Such groups.

apply

would

however,

to

apply

be

parameter is found

exit

used

for

or turbine

blade-jet static

would

idealized the

from

same;

speed

and

and

ratio

total

Ds and

parameter

efficiency.

on the variables relating to parameters shown in equathe

number

A parameter not because values of

case,

by

having of the

geometrically not

this

of dimension-

rotative

values

of

at D

similar

having

two

turbospeed

the

all rotative

combining

excludes

the linear remaining

Q

remaining speeds.

of the

is known

\l/2/N2D2\3/4

the

volume

excludes

as

previous

the

specific

NQI/2

flow

rate

is taken

at the

the

Commonly, are 52

quoted

stage

N

(2-91)

is known

as the

specific

diameter

as ( H D_=\-_VD2]

With

N

Thus,

that

is found

is a

parameters.

NE),/2

The

the and,

as

a turbine, exit.

type

will differ

case,

exhaust

found

that

case remain

to a turbomachine

can

very

should both

range in

/

When

of course,

a parameter

because,

parameters N8 and

a

Also,

desirable

The

speed

of this

blade-iet

Parameters

are possible. be desirable

machines variables

not,

that would

variables be

a turbine

of dimensional analysis led to the dimensionless

less parameters dimension D

would

optimum

for a real

for

of 60 ° to 80 °, where

the

velocity-diagram Design

The operation turbomachines

are,

a real

parameter

so are the

with

trend

that

range

volume

but with

flow

not rotative

rate

,,_1/4 (V)

1/2-

taken

at

D,--

DH1/4 Qi/_x

exclusively,

the

speed

N

the

in

DH1/, Q1/2

stage

exit

(2-92) or

turbine

exit, (2-93)

values revolutions

for

these per

parameters minute,

exit

BASIC

volume flow rate H in foot-pounds of units,

Qe_ in cubic per pound,

specific

less because

speed

the

to be the

and

units

are

total-to-total

it is specified Specific

as the

speed

presented

not

diameter

consistent. (5h;_),

total-to-static diagram

parameters.

dimensional

The

ratio

of total

differing eters. all

efficiency

definitions Some

cases, The

thus

eliminating

parameter

speed-work (2-96)

to the

parameter

by

to the

blade

speed

previously

is

or 60 sec/min).

The

(n)-_

(2-95)

and

(2-72)

efficiency

used

use

the can

appears the work

ideal

ratio

from

expressed equation

equations

because

in defining

be

substituting

with

same

efficiency

interrelation

taken

convenience,

(2-94)

(2-93),

work

prefer

for

(27r rad/rev

to static

of ideal

authors

H is usually

be related

The

constant

equations (2-91), (2-95) yields

dimension-

(hh_).

can

H=JAh_d=J_d Combining (2-94) and

head

truly

or head, this set

_ND K

U-where K is the head is

not

sometimes,

value

diameter

are

The but

and specific

velocity

CONCEPTS

feet per second, ideal work, and diameter D in feet. With

specific

value

TURBINE

of the

various

param-

definition

equation in

(2-74)

(2-96).

terms

of

into

equation

NsD8 ' :-- K -v/g_'X -

speed The

and specific exit volume

diameter flow rate

can is

also

be related

to the

Qe_=A_V_ where (2-91),

Since diagram

A_x

is the

(2-93),

specific

flow

area,

(2-94),

and

(2-75)

with

NsD,

3=-- KD2 _'_A_

parameters,

and

specific

which

can

in m 2 or

diameter be used

flow

(2-98)

exit

speed

the

(2-97)

"/l"

Specific coefficient.

in

ft 2. Combining

equation

(2-98)

equations yields

(2-99)

are to

related correlate

to the

velocity-

efficiency,

then 53

TURBINE

DESIGN

specific

AND

speed

and

APPLICATION

specific

diameter

can

also

be

used

to

correlate

efficiency. Specific

speed

and

velocity-diagram flow rate, and (2-99),

diameter

are

sometimes

often

dictate

diameter

that

imply

sometimes referred the

shape. to

type

of design

these for

are

the

equal

applied

thus

overall

following

are

entire

can

that

be

might the

we

have

turbine.

and the

been

When represent

used

some

the

and volume appearing in

speed since

the

specific They

are

shape

will

Parameters

turbine,

the type of design that rapid means for estimating

that

parameters.

parameters,

parameters

and

specific shape

that

or to the

similarity

values to the

The

variables

to be selected.

parameters

to a stage

as

to as design

dimensionless

be applied

Thus,

referred

Overall The

contain

parameters do not. These are diameter their use leads to terms, such as D2/Aex

equation also

specific

most

to a stage,

similar

conditions

efficiency.

parameters

be most appropriate number of required commonly

can

applied

to correlate

of these

discussing

help

and stages.

When identify

serve

encountered

as

a

overall

parameters: Overall

specific

speed __

No1/2

N.----Overall

specific

(2-100)

D"_H1/'

(2-101)

diameter "_

Overall

___e, H3/4

speed-work

parameter (2-102) gj_'_

Overall

blade-jet

speed

ratio

;=

The

subscript

script

(--)

Of nificant

av refers refers

these

overall

considerations erally for that

its on the

specific

contribute

U..

some

value

the

nature speed

to the

value

for

perhaps

determined the

the

super-

evolved

geometry. to show specific

speed.

is most by

other

be restated of overall

and

turbine.

speed

always

Q,_=wv,_ 54

condition,

entire

values

of the can

the

specific

is almost

while

(2-103)

average

for

parameters, value

only,

depend overall

to

to the

because

,

parameters Equation the

sig-

application gen(2-100)

considerations Let (2-104)

BASIC

where

ve_ is specific

flow

rate

volume

be expressed

at exit,

in m3/sec

TURBINE

or ft3/lb.

CONCEPTS

Also,

let

mass

as JP w----_

(2-105)

_'H

Then, (2-100)

substitution yields

of

equations

-_,=1_.7 Thus, three

the

overall

terms.

The

be reasonably

estimated.

gas

thermodynamic

and

the

useful choice

for evaluating is available)

by the

application.

in other

cases,

of N and The

P,

second

both

product

by

in which

as the only

conditions.

the

and

than

of

which

can

on the specified

This

speed

rather

product

performance,

depends

rotative the

equation

oo)

be expressed

term

N_/J__,

into

,=\

expected

cycle

is established

manner

(2-105)

effects that different fluids on the turbine. The third

Often, the

can

reflects

The

the have

vex kll21

speed

term

and

t

/

specific first

(2-104)

second

term

is

(in cases where term is dictated

power

the

are

specified

individual

;

values

application.

overall

specific

speed

influences

the

tur-

axial flow

axial flow One-stage, _'_

A

flow

[

I

I

I

I

I

I

I

I

I

I

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.1

1.2

I 1.3

Specific speed, Ns, dimensionless

I

I

I

I

I

20

30

40

50

60

I 70

I

I

I

I

I

I

I

I

I

80

90

100

110

120

DO

140

150

160

112)(ibf3/4) Specific speed, Ns, ( ft3/4 )(Ibm314)/(min)(sec t FIGURE

2-13.--Effect

of specific

speed

on turbine-blade

a

shape.

55

TURBINE

DESIGN

AND

bine

passage

shape

bine

and

one-

ratio

for

of hub

speed.

For

decreases

is illustrated and

radius the

APPLICATION

axial-flow,

tip

decreases

to

radius

radius turbines,

ratio.

of the

Dividing

equation

for stage

specific

number (2-100)

speed

for

the

a radial-flow

tur-

turbines.

The

example with

increasing

Thus,

application indicates the type The values of some of the proximation

2-13

two-stage,

axial-flow

the

in figure

increasing

the

overall

specific

or types of design overall parameters

specific

number

of stages

speed

for

any

that will be required. give us a rapid ap-

of stages

required

for

for overall

specific

speed

a given

application.

by equation

(2-91)

yields N_

Qe_

1/2 (2-107)

If we neglect change

per

the

reheat

stage,

effect,

we can

which

is small,

and

assume

equal

H=nH Further,

if the

expansion

compressibility these

effect

last

two

head

write

ratio

and

is not

assume

conditions

(2-108)

into

too

that

large,

we

can

Qe_:-Q_:._t_.

equation

(2-107)

neglect

the

Substitution

and

of

rearrangement

yields n:(NsY/3

(2-109)

\NJ Since

stage

specific

experience assume

can

number

type

overall

speed-work Often,

basis

of stress

suming

2, where

by

requires

value

of blade

equal

Dividing

equation work

Or,

equation

(2-102) blade

per

blade

speed

on

this

can

speed

for

the

for is

is presented.

for

obtained the

of be

the

for

for stage (U2=U_v),

the

overall blade on the

varied

_peed-work

turbine

the

estimate

be selected

may

(2-66) overall

from

value

a value

to

stage

estimate

is often

a knowledge

for

para-

speed-work parameter, and

as-

stage,

ah' =n h' 56

an

an assumed

speed

speed with

known us

correction

of

however,

Thus,

of stages and

Knowledge

efficiency,

specific

speed gives

number

parameter

considerations.

a constant

specific

of compressibility

parameter.

if desired.

of stage

(2-109)

for

for

of efficiency.

a compressibility

speed-work

a reasonable

parameter assuming

effect

parameter,

speed.

overall

of estimate

speed-work

level

equation

The

in reference

metrically

and

parameter

value

a given

assumed

A similar the

a reasonable

requirement, of stages.

discussed

stage

us

is a correlating

to achieve

speed

application

from

tell

in order

specific

speed

(2-110)

BASIC

TURBINE

CONCEPTS

yields n==-

Equations studies

(2-109)

and

associated

(2-111)

with

are particularly

preliminary

Performance The

turbomachinery

perfectly

correct

variables, rate

Q because

considerably Change

for

the

ideal

is the

work

ratio,

depends

we include

number

power work

on the on

as

D

gas #. For

Now,

operating

still R,

simplicity with

heat

ratio

.

some complicated, and speed terms. Let

us operate

we can by using tionality

get

out

but on

of them.

f

not

of the

The

ratio

initial

and to

of

Since

pressure

Since

Mach

temperature

N and

fluid

is

a character-

properties

a molecular heat

H,

express

Ah'.

as on the

variable. speed

specific

T'_,,

are

weight,

ratio

inand

3' is assumed

Ah'

ND

D, R, #)

(2-112)

p,..

w

, -_ ) _/ RT,,,'_T"--7 'P_

been

assumed terms

above

mass-flow

equation,

N,

following:

second-order,

some

the continuity AocD _, so that

the

-- Icn_R-_n'

had

to

variables

produces

-, P; D_

specific

implies the

p'_,, p_,,

w_/RT,.

If the

which

constant.

term

as well

flow

preferred

enthalpy

introduction

here, the

preferred

The

of

Q changes

remains is

in total

Rotative

volume

expansion, w

as another

of interest.

the

pressure

temperature

elasticity.

are

analysis

the

temperature,

w=fcn(_h', dimensional

P,

are

choice

nondimensional to

ratio

on both

temperature

constant

viscosity constant.

while

or drop

initial

of

(2-63)

Another

expressing

pressure

depends

of

to introducing

dimension

turbine,

specific initial

depends

equivalent eluded

flow

Instead

work

for degree

as

parametric

in equation

machines.

w is preferred

significant

expressed

for compressible

actual

presented

rate

for

Parameters

flow

preferred

flow

any

pressure

temperature.

istic

mass

throughout

of

which

is often

The

useful

analyses.

Specification

compressible

however,

performance.

system

parameters for

(2-111)

),

the

(2-113)

constant,

there

modifying

terms parameter ideal-gas

to see

the what

may law,

would

flow,

be

work,

significance

be transformed and

RT .

the

propor-

(2-114) 57

TURBINE

DESIGN

Substitution (2-113)

of this

APPLICATION

relation

into

the

mass

flow

parameter

of equation

yields

Thus,

the

mass

of actual the

AND

critical, The

flow

mass speed

flow

rate rate

is represented to the

or sonic,

velocity.

parameter

may

nondimensionally

mass

flow rate

when

be transformed

by

the

the

velocity

ratio equals

as

ND

U U oc ,-7-_,_, oc-_]RT't,, _RT,_ ac, Thus,

the

rotative

of rotor-blade Mach

speed

velocity

number.

parameter fluid

have

certain

with

V/U,

Mach

number,

as for

temperature

a given

(2-113)

with

can

in order

be

the

for

rotor

critical the flow,

the

effect

only

velocity.

For speed

becomes

of varying

speed

be inlet

presented

The

must

also

must

parameters

of rotor the

must

but

ratio

similarity.

rotative

All variables

dimensionless

expressed

not

to the

fluid.

by

of

the

the

is a kind

parameter

condition but

incompressible

by

similarity,

therefore,

of the

gas,

for

respect

dimensions,

dimensionless form to be correlated. For

kinematic

a certain velocity

which

mass-flow

is that

variable the

the

nondimensionally velocity,

the

analysis

of fixed

singular

of

of this

fixed

machine

to critical

Division

gives

implication

is represented

(2-116)

the

have

a

a given is not

a

associated expressed

in

temperature as equation

as

(2-117) p_---_

For

a given

gas

-_-Icn

in a given

_T-'

turbine,

_=fcn

Depending tions

on

the

(2-113),

particular

(2-117),

or

the

_-r,

case,

--7'

the

(2-118)

parameters

further

be

to

(2-118)

-r-,

parameters

can

reduce

presented

used

to

in equa-

express

turbine

performance. Equivalent It

is very

useful

of temperature and 58

specific

to report

and heat

pressure

ratio.

This

Conditions

performance and

under

sometimes

is done

in order

standard

of fluid that

conditions

molecular results

weight

obtained

at

BASIC

different easily

conditions used

following sure,

may

are

101

the

325

as

of these Let constant.

With

subscript can then

are

parameters

the

known

with

basis

diameter

conditions similarity

and

the

conditions

(2-119)

!

Ah'eq l_,d

N

(2-120)

N_q

/

these

The

the

Pst_

ah' RT_.--R_,.

of

air. on

but

the

or are

conditions.

standard

P

P_.

Rearrangement conditions

standard

expressed

(2-113)

conditions,

K

1.4. These

as equivalent

std denoting

equivalent as

pres-

288.2

ratio,

NACA

speed

also The

atmospheric

heat

or and

and

we desire.

temperature,

of equation

subscript

eq denoting be expressed

used:

specific

work,

CONCEPTS

compared

condition

psia;

and

conditions,

of flow,

conditions

the

readily

usually

14.696 29.0;

standard

variables

us use

or

weight,

standard

and

for any

conditions

N/m _ abs

NACA

performance

directly

performance

standard

518.7 ° R; molecular known

be

to determine

TURBINE

,

(2-121)

equations

then

yields

for

the

equivalent

p',. w,q.=w

taT' ta

,

Ah_q-----,_h

(2-122)

, RT',_

(2-123)

N_q= N 4R--_ta / T:,a ,

by

As you may recall, assuming constant

always and

the

case,

fluid.

Let

since

The

used

yield

do not

(sonic)

velocity.

specific

heat

and

have

specific add

similarity and

a very

commonly used are expressed as

heat

ratio

change

corrections

under the

Mach small

can

a specific-heat-ratio

specific-heat-ratio However,

ratio

only

we started off the discussion of these specific heat ratio for all conditions.

us now

parameters.

(2-124)

all

terms number,

effect

specific-heat-ratio

conditions, that are

are

the

temperature

effect

into

the

that

are

commonly

but

only

out

depend

cumbersome

on equivalent terms,

left

with

parameters This is not

at

critical on both

to work

conditions. equivalent

above

with,

With

the

conditions

59

TURBINE

DESIGN

AND

APPLICATION

(2-125)

V_.v_,,.J \p,.:

8q--

----

t

(2-126)

_:,=t,h' ,,,_7 ) N_q_

Nj(V_'d.)

'

(2-127)

where

.>,,S. _2 V,;(,,,:,) "\%,,+1/

(2-128)

.d__Ly<.,-,,

_-

'\,y+ 1/ and,

as you

recall, 2 27 V<,--_-_l

Therefore, 127)

for

reduce

constant

specific

to equations

Finally,

heat

(2-122)

(2-129)

gRT' ratio,

equations

(2-125)

to

(2-

to (2-124).

we define 0=(

Vc, kYcr.

)2

(2-130)

8ldl

and #

.--_--P-_",

(2-131)

Pstd

The

equivalent

conditions

are

then

expressed

w,q=w

_

as

_

_kh

(2-132)

#

0

_'_--

(2-133)

N Neq=_ One

point

operation of both reduce effect

that

can

be seen

at temperatures actual the

mass

output

is the reduction

greater

flow of

and

The

6O

be

effect one

in takeoff

of viscosity of

the

these than

in the

dimensionless

similarity

standard

equivalent

equations will cause

speed.

a powerplant.

Reynolds

to

from

(2-134)

Both

A well-known

performance Number form

of jet

a reduction

of these example

aircraft

on hot

factors of this days.

Effect

of Reynolds

parameters

is that

number

affecting

was

shown

turbomachine

BASIC

performance.

While

its

effect

effect of Reynolds number the following manner: Expressing

efficiency

it is still

efficiency

CONCEPTS

important.

is usually

The

correlated

in

as 7'

we can

is secondary,

on turbine

TURBINE

A '

'

hh'_

Ah' 5h'

(2-135)

ld

write _h

,

,

• -7 If we assume

that

the

only

loss

J

,o,

=

(2-136)

is friction

loss,

(2-137)

where length.

f

is the friction factor, For turbulent flow,

and

L is the

characteristic

flow-path

1 fOCR--_0._ where and

Re

is the

Reynolds

number.

into

equation

(2-136)

(2-137)

1--_'

Adding

subscripts

dividing yield

Since

the

for

equation

for geometric

V_/Ah_.,_-=V22/Ah

This the

range

all

the

attributable another

is an for

1 and

(2-140)

correlation. type

of 0.1 to 0.2, losses

are

not

to viscous suggested

type

the

(2-139) for

reduces

depending

on the losses,

mmilarity

varies

of correlation

(2-141)

0.2,

machine. and

between

2

to

it has

is not

and

condition

and for dynamic

Actually,

viscous

equation

equation

(Re_'_ °'2 \Re1/

of correlation

loss

2 to

1 by

L_/DI--:L_/D2

'_,_d, equation

this

(2-138)

(2-139)

condition

similarity

ideal

equations

1

1--7; 1--72

exponent

Substituting yields

conditions for

(2-138)

the

been but

found

This

of In

the

varies

occurs

fraction

machines.

that

usually

in

because total

view

loss

of this,

is 61

TURBINE

DESIGN

AND

APPLICATION

1--v_--A

where the

A and

exponent

B are

such

is maintained

and

the

loss

is viscous

coefficients

discussion

fractions

loss.

A

to 0.4 for A and

be

a good

that

and in

to represent

turbine

tests

for

here

viscous the

values

correlating

fact

that

of 0.7

Reynolds

(2-142)

loss

at Lewis,

1, indicate

corresponding

compromise

1. In equation the

B serve

reference

(2-142)

A+B=

at 0.2 to reflect

Recent

presented

0.3

{Re2"_ °_

exponent,

that

not

as well

values

of

to 0.6 for number

all

as the about

B seem effects.

REFERENCES 1.

SHEPHERD,

D.

2.

GLASSMAN,

ARTHUR

eters Ratio

62

for

G.:

Principles J.;

Examination

Axial-Flow

of

AND

of Turbines.

Turbomachincry.

STED/ART,

WARNER

Geometry NASA

Macmillan L.:

Use

Characteristics TN

D-4248,

of of

1967.

Co., Similarity

1956. Param-

tligh-Expansion-

to

BASIC

TURBINE

CONCEPTS

SYMBOLS A

flow area, Reynolds

a

speed

B

Reynolds

C_

heat

D

diameter,

D_ E

specific

e

kinetic

F

force, N ; lb friction factor

]

m2; ft 2 number correlation

of sound,

m/sec;

number

capacity

(2-142)

coefficient

in eq.

(2-142)

pressure,

J/(kg)(K);

dimensionless;

of elasticity, energy

N/m2;

loss

(sec I/2) (lbfl/4) / (ft I/4) (lbm 1/4) lb/ft 2

coefficient,

constant,

defined

g H

conversion

h J

specific enthalpy, J/kg; Btu/lb conversion constant, 1; 778 (ft)(lb)/Btu

K

conversion

L M

characteristic Mach number

N

rotative

N,

specific

?%

number polytropic

P

power,

P

absolute

Q

volume

R

gas constant, reaction

Re

Reynolds

r

radius,

T

absolute

U

blade

head,

J/kg;

constant,

eq.

(2-46)

(Ibm)(ft)/(lbf)(sec

2_r rad/rev;

length,

speed,

2)

60 sec/min

m; ft

rad/sec;

speed,

rev/min

dimensionless;

(ft 3/4) (lbm 3/4)/ (rain)

(sec 1/_) (lbf 3/4)

of stages exponent W;

Btu/sec

pressure, flow

rate,

N/m2;

lb/ft 2

m3/sec;

ft3/sec

J/(kg)

(K) ; (ft) (lbf)/(lbm)

(°R)

number m; ft temperature,

speed,

velocity,

Vj

ideal

V

specific

volume,

W

relative

velocity,

20

mass

jet speed

flow

K;

m/sec;

absolute

rate,

!:1

total-pressure fluid

1 ; 32.17

by

(ft) (lbf)/lbm

V

OL

Btu/(lb)(°R)

ft

diameter,

modulus

in eq.

ft/sec

correlation

at constant

m;

coefficient

absolute

deg fluid relative

°R

ft/sec m/sec;

(defined

by

mZ/kg;

loss

(2-71)),

m/sec;

ft/sec

ft/sec lb/sec

coefficient,

angle angle

eq.

ft3/lb

m/sec; kg/sec;

ft/sec

defined

by

eqs.

(2-47)

measured

from

axial

or radial

direction,

measured

from

axial

or radial

direction,

deg 63

TURBINE

DESIGN

ratio

7

AND

of heat

constant ratio

APPLICATION

capacity

at constant

pressure

to heat

capacity

at

volume

of inlet

function

total

pressure

of specific

heat

to NACA ratio,

standard

defined

by

eq.

pressure (2-128)

efficiency 0

squared

ratio

perature

of critical to

velocity

critical

based

velocity

on

based

on

temperature speed-work viscosity, y

blade-jet

p

density,

T

torque, flow

¢

parameter, (N)(sec)/m_; speed

ratio,

kg/mS; N-m;

defined by lb/(ft) (sec) defined

defined

coefficient,

angular

velocity,

by

defined rad/sec

Subscripts: av cr

average critical

eq ex

equivalent exit

/d

ideal

in

inlet

loss

loss

opt

optimum

p

polytropic

r rel

radial component relative

ro

rotor

st

stator

std

NACA

stg u

stage tangential

x 0 1

axial component at stator inlet at stator exit or rotor

2

at rotor

condition

(M---- 1)

standard

condition

component

exit

Superscripts:

'

absolute

"

relative

64

(2-72)

lb/ft a

coefficient,

vector overall

eq.

(2-66)

lb-ft

loading

--_ --

by

eq.

quantity turbine total total

state state

inlet

eq. (2-75) by

eq.

(2-67)

turbine NACA

inlet

tem-

standard

BASIC

TURBINE

CONCEPTS

GLOSSARY The

terms

defined

herein

are

illustrated

in figure

2-14.

Tip-___

FLow_

I Blad_

_Hu b

I/-Suction

surface

Leading edge-/" Pressure surface J/

Camber line-_ /-Trailing

edge

Axis-/ /

_ Opening, or th roat

/ /

Tangent to camber / line at leading edge 4.

Chord

\ \

l Spacing. or pitch

Blade inlet angle 7 \...\ /

Axis-,,

/

Flow inlet angle-/ FIo_ / I

Incidence angle J /

/_-Tangent to camber line at trailing edge Flow exit angle

Axial chord Deviation ant

FIOURF

2-14.--Blade

-Blade exit angle

terminology.

65

TURBINE

DESIGN

aspect

ratio.

axial

chord.

AND

The

ratio

The

length

turbine, onto of the blade. axial

solidity.

blade

exit

at the

The

height. inlet

parallel

The

angle

The

and

radius

angle.

tangents equal

The

to the to the

chord.

The

line

of the

at the

length

the

chord

between chord line. up on where

axial

length

to the

camber

line

direction. the

radius

at the

tangent

by

the

leading

formed

to the

hub. camber

line

direction.

formed

angles

in the

spacing.

tangent

axial

mean

line

of the

intersection

and by

blade

edge to the trailing edge, and the suction surface.

The

onto

camber

to the

the

turbine

as set

axis. It is the

tip minus

angle

blade,

axial

between

external

chord.

of the

trailing

the

of the

edges.

chord

line

It

is

and

the

from

the

tangents.

line.

leading surface

the

turbine

and the blade.

sum

camber-line camber

between

angle

at the leading edge bucket. Same as rotor camber

chord

the

to the

turbine

axial

at the

The

height

projection to the

of the

edge

angle.

blade

of the

ratio

angle.

blade

of the

a line

trailing

blade

APPLICATION

of the line.

profile.

halfway

perpendicular

a flat surface, the front and

extends

between

projection

It is approximately

the leading edge and If a two-dimensional

It

equal

of the to the

the trailing edge. blade section were

the chord line the rear of the

is the blade

the

pressure

blade

linear laid

profile distance

convex

line between section would

side

the points touch the

surface.

deflection.

The

ference

total

between

deviation

the

angle.

The

flow exit angle. exit and the

The

hub-tip hubleading mean nozzle

section. blade.

pitch.

The

points

66

The The

ratio.

and

ratio

inlet

in the

adjacent are

flow

the

blade

flow

angle.

exit

direction flow

dif-

angle. at the

direction

blade at

the

ratio.

of the

minus

of the

direction

hub

radius

to the

the

blade

between

the

inlet

tip radius. angle.

blade.

of rotatioll

hub

between

and the

tip.

corresponding

blades.

The

concave

surface

of the

highest.

Same

exit

to the

blade.

angle

or nose,

It is equal

the

fluid

to tip-radius

The

flow

front,

fluid.

minus

The blade section halfway Same as stator blade.

surface.

pressures radius

angle

angle

of the

as hubratio.

distance on

pressure

of the

between the fluid axial direction.

section

Same

angle. edge.

inlet exit

The angle the turbine

to tip-radius

incidence

flow flow

innermost ratio.

angle

The angle between the turbine axial direction.

flow inlet angle. blade inlet and hub.

turning

as hub-

to tip-radius

ratio.

blade.

Along

this

surface,

BASIC

root.

Same

rotor

blade. The

spacing.

Same

A rotating

stagger angle. direction. stator

tip.

blade.

ratio

blade.

of the

chord

angle

between

A stationary

surface.

edge.

to the

spacing.

as pitch. The

The

The

rear,

the

chord

line

and

the

turbine

axial

blade.

convex

pressures are lowest. The outermost section

trailing

CONCEPTS

as hub.

solidity.

suction

TURBINE

surface of the

or tail,

of the

blade.

Along

this

surface,

blade.

of the

blade.

67

CHAPTER 3

Velocity Diagrams By Warren J. Whitneyand WarnerL.Stewart As

indicated

in chapter

be considered the

fluid

in the

velocities

universally overall

design

these

blading

next

cussed

the

to the

in chapter velocity

efficiency

is devoted

entirely

part

of this

Usually, second

the part

diagrams and

chapter

representative

from

turbines

of this

that the are

variation considered

rotative

expected

of the

also

concerns at

illustrated

blade

balance

blade

in this

speed

conditions

mean to the of forces with

that

radius radial in the radius.

the

diagrams turbine.

turbine

and

were

dis-

associated

with

2,

and

their

case.

This

an idealized

diagram

flow

in

diagrams

chapter

of velocity

a single

is devoted the

by

subject

average the

in

are

important

the

parameters

presented

are the

diagrams.

velocity

from

capacity

row Once

velocities

the

flow

was

in

and

of and

speed

velocity is very

angles

level

to the

from

and

addition,

dimensionless

chapter

result

work,

of the

velocity

blade

diagrams.

geometry

flow In

of the

and

of the

conditions

speed

to

absolute

of velocity

velocity

were

to turbine

The

constructing

2. Various

diagram

is the

next.

for

work

chapter sidered

the

efficiency

relation first

blading

variables

of turbines

evolution

produce.

methods

them

flow,

is the

specify to

affect

general

of

use

important

to the

to the

the

required

diagrams

most

row

relation

step

to the

significantly relating

their

blade

through

is required

The

one

requirements the

relation

that

the

and

of the

or analysis

from

described

established, Their

design

as it passes

relative

2, one

diagrams. that

can for

are

The be

the used.

variations

constage. The

in the

radial

direction

Only

axial-flow

chapter. 69

TURBINE

DESIGN

AND

APPLICATION

MEAN-SECTION In

this

section,

(halfway

the

between

conditions grams,

hub

review, the

clature.

to 3-1

vector

are

the

assumed

turbine.

stage

at the

mean

section

the

average

to represent

The

efficiency,

shows

no

(2-14)

an

relations

Assuming

equation

tip)

occurring

different

and

of dia-

selection

when

their

types

are discussed.

figure

indicating

diagrams

by

relation

is required

In

and

encountered their

staging

velocity

DIAGRAMS

illustrative

described in

be written

as

can

change

stage

velocity

in chapter

mean

2 and

radius

diagram the

through

nomen-

the

stage,

ah ' =--UA V_' gj

(3-1)

where h'

total

enthalpy, speed,

J/kg;

Btu/lb

U

blade

V,

tangential

component

m/see;

g

conversion

constant,

1 ; 32.17

J

conversion

constant,

1 ; 778

relates

ft/sec of velocity,

This

equation

the

The

axial

component

of

rate,

state

conditions,

and

m/see;

(lbm)

(ft)/(lbf)

specific

work

velocity

the

(sec 2)

(ft) (lb)/Btu

stage the

ftlsec

area

to the

vector

by

the

velocity

is related

diagram. to

the

flow

relation

_0 Vz

=

(3-2)

P-_an

where Vx

axial

component

w

mass

flow

o Aan

density, annulus

Flow only

are

the

axial

the

absolute

also

affect

kg/see;

key

the studies

because

shape.

Such

such

are

to

and

in

parameter

parameter,

7O

the

swirl

velocity)

geometry. can

discussed can

they

not

component

association

which

be

but

In addition, with

in

of

velocity-

related chapter

be expressed

to the 2 and

in several

as U2 U _=gJAh'--AV,,--

The

as

values

were

because

tangential

blading

used

parameters

speed-work

(the

referred

efficiency the

parameters

velocities

is often

parameters

diagram ways,

swirl

expected

diagram

the

ft/sec

velocity-diagram

and

velocity

dimensionless

include

m/see;

lb/sec

kg/m3; lb/ft 3 area, mS; ft 2

angles

link

of velocity,

rate,

speed-work

parameter

is used

in

g JAb' AV_ 2 this

chapter

(3-3) because

diagram

VELOCITY

DIAORAMS

FI0w

VX,0 Station 0

Vx,

VI

al-'"

i" i'll

,,

I

w z v/Ivu, Vx, 2

../

u

!

_/Vu .2

I

WU,2 3-1.--Velocity-vector

FIGURE

types

are

related

normalize

the

to

the

diagram

diagrams

swirl

and nomenclature.

distribution

velocities

by

and

Velocity-Diagram After

the

diagrams

can

shapes

design on the

Diagram

exit

(V_._) and diagrams in this

type

refers

the

rotor their

the and

exit

(V_._).

reaction

Zero-exit-swirl

diagram

(V_._---- 0)

(2)

Rotor-impulse

diagram

(W, = W2)

(3)

Symmetrical

diagram

three diagrams in figure 3-2.

Zero-exit-swirl velocity efficiency.

head The

for

(V_--W2 several

diagram.--In or

the

swirl

zero-exit-swirl

value

reaction

(1)

These shown

the

to

values many

component diagram,

Vu., _ 1 hVu--

of the

values

swirl

and

V2:W1)

either

and

speed-work imposed between

three

the

common

characteristics

of speed-work

thereof

sizes

of performancesplit

following

and

cases,

velocity

constraint

swirl

The

the

different

physical

determines

as stage

and section:

established, have

and

to some

shape

such

are

diagrams

diagram

type

parameters,

types of discussed

Velocity

is convenient

Types

requirements

evolved.

Diagram

diagram.

related stator

be

depending

parameter. on the

overall

it

AV,.

are

parameter the

represents

are

entire

exit

a loss

in

where

(3-4a)

and 71

TURBINE

DESIGN

AND

APPLICATION

Speedwork

Diagram type

parameter

Zero exit swirl

Impulse

Symmetrical

O.25

0.5

1.O

FIGURE

3-2.--Effects

of

speed-work stage

parameter

be

used

For

an

to reduce axial-flow

(Vz.1=Vz.2), (2-39)

such

the

reduces

diagram

type

on

shape

of

diagram.

V_.2_ AV,-can

and

velocity-vector

0 (3--4b)

loss.

rotor

(UI=

definition

U2)

of

to

W2 R.,--

having

stage

constant

reaction

axial

presented

in

velocity equation

W2 u, 1 2+W ,2-W .t

(3-5)

_, 2--

1-

where Rst_

stage

W_

tangential

By

using

(3-5)

reaction component

equation

can

be

(2-6)

expressed

of relative and

equations

0.5, is

equation which

zero,

reaction --0.5, velocity 72

is plotted

indicates which

and

indicates as can increase

and

ft/sec

(3-4),

equation

1 1--_-_

in

figure

a conservative an

is encountered. which,

(3-3)

m/sec;

as Rs,g=

This

velocity,

be

shown,

in static

3-3(a).

At

diagram.

impulse

For

(3-6)

represents pressure

At

rotor.

example,

at

_---1,

the

k=0.5,

Below

the

_=0.5,

_,----0.33,

the

a substantial across

the

reaction

rotor.

is

reaction negative

reaction

is

decrease

in

Because

of

VELOCITY

DIAGRAMS

1.0--

Symmetrical_ \

.o_

ZeroS,

_- Impulse

(a)

I

/

I

1

I

.5-::3

N

Zero exit swirl-, 0

,m

(b)

I 0

I

.25

I

1

.50 .75 Speed-workparameter, _,

1.0

(a) (b) FIGURE 3-3.--Effects

potentially avoided;

high Figure

losses,

Impulse

such

presents

negative

reduces

and

are

zero-exit-swirl negative-reaction

this

case,

WI=W2

axial

equation velocity,

(2-6), the

rotor

are

type

usually

seldom

used

for

diagrams

for

the

cases. and

the

equation

for

to Rst_=O

From

diagram

reactions

diagrams the

impulse,

diagram.--For

reaction

high

zero-exit-swirl

3-2

positive-reaction, stage

Reaction. Exit swirl.

of speed-work parameter and velocity-vector on reaction and exit swirl.

therefore,

X<0.5.

.

equation inlet

(3-3), and

exit

(3-7) and swirl

the

assumption

velocities

can

of constant be expressed 73

TURBINE

DESIGN

AND

APPLICATION

as

V_, _ AVu--

X+0.5

(3-8a)

_,--0.5

(3-8b)

and Vu

_-_= The

exit

swirls

swirl

are

characteristics

encountered

at

swirls are obtained and zero-exit-swirl figure

3-2.

positive

seldom,

if ever,

Symmetrical one

in

specified

_ values

swirl swirl

used

when

the

to have

the

same

0.5,

Positive

and

negative

leaving

is

a

turbine

work, than

type and

shape.

than

3-3(b).

the impulse illustrated in

_ is greater third

figure

than 0.5. At _=0.5, These effects are

stage

stator-exit-

in

greater

velocity

decreases

diagram.--A

which

shown

at _ values less cases coincide.

Because

because

are

impulse

and are

0.5.

of diagram

commonly

rotor-exit-velocity

In terms

a loss

diagrams used

is

triangles

are

of velocities,

V1 =W2

(3-9a)

V2=W,

(3-9b)

and

Under

this

condition,

the

equation

for stage

reaction

reduces

to

1 R,tg=_ From

equation

axial

(2-6),

velocity,

the

equation

swirl

(3-10)

(3-3),

velocity

and

the

components

Vu. 1 AV_

assumption can

of constant

be expressed

as

X+ 1 2

(3-11 a)

and Vu. 2 h_l A-V_= 2-These

reaction

typical is the

same

decreases, at 0.5. this (e.g.,

and

diagrams as the the This

type the

swirl

swirl

of diagram and

attractive middle

74

aspect

but

is conducive

The at

the

of a turbine

in figure

symmetrical _=

1. As

reaction

to high

for stages stages

Stage A significant

3-2.

diagram

increases,

reaction

are shown

in figure

zero-exit-swirl

exit

good

front

characteristics

illustrated

(3-11b)

where

the

value

swirl

of

constant

efficiency,

exit

of a multistage

with

diagram

remains

total

3-3,

making

is not

a loss

turbine).

Efficiency design

is the

expected

efficiency.

VELOCITY

The

efficiency

type blade on

is an

important

of velocity diagram surface. Therefore, the

efficiency

basic

relations

sented

and

effects.

References

function

between

in

of

diagram

herein

to

1 and

As presented written as

among

used and the pressure the diagram selection

requirements

used

of,

the

2 are

chapter

2,

up

of

used

as a basis

turbine

stage

the

on the dependent

application.

and

some

things,

distribution is greatly

intended

parameters

point

other

DIAGRAMS

efficiency

are

more

important

the for

Some

this

static

pre-

development.

efficiency

can

be

hh' _--Ah_d

(3-12)

where stage

static

efficiency

Ah'

stage

work,

J/kg;

Btu/lb

hh_

stage

ideal

work

based

static Expressing

pressure, ideal

on

J/kg;

work

ratio

of inlet

total

pressure

to exit

Btu/lb

in terms

of actual

work

plus

losses

yields

ah' ,hh' +Lot'-b

Lro+

V22

(3-13)

where Lst

stator

Lro

rotor

loss,

y /2gJ

stage

leaving

The

equation

nation into

of the equation

loss,

for

J/kg;

Btu/lb

J/kg;

Btu/lb

loss,

total

stage

J/kg;

efficiency

leaving

(3-13)

it

was

kinetic

relating

the

assumed energy

stator that

across

n' is the

loss,

V]/2gJ.

same

except

for

Substituting

the

equation

elimi(3-3)

yields

,1--

In

Btu/lb

gJ(L,,TLro)

and the

the

rotor losses

blade

losses were

rows.

L,,:K_,

1

V2 2

to the proportional

That

(3-14)

diagram to

parameters, the

average

is,

V°2-b V_2 2gJ

(3-15a)

and L_o=K_o

where/4

is constant

Wl2+W2_ 2gJ

(3-15b)

of proportionality. 75

TURBINE

DESIGN

Equations

AND

(3-14)

efficiency.

The

and

exact

in references

efficiency

is as follows:

(1)

velocities

The

(3-15)

nature

be found

axial

APPLICATION

of

1 and are

serve the

(2) The

tangential

(2-6)

expressed

related

according

and

(3) The

the

in

eq.

axial

(3-4),

the

(3-8),

components

mass-flow

are

to

or

are

for

and

of

estimating

equations

procedure

terms

for

their

assumption

or

can

estimating

tangential

and

by

an angle

assumption.

values

for

constant

basis (5)

of previous

Efficiency

work

parameter

The

total-

erence

2 by

presented that

for

various

determined

the

the

reaction

The For

total

each

efficiency

parameter, higher

},, value than

the

efficiency _-_0.5, other the

at two

for

efficiency

decreases High rather

swirls

this

chapter,

The

static

symmetrical

ref-

curves diagram

diagram,

as

diagram,

and

of those

curves for the zero-exit-swirl less than 0.5 because of the

for

diagram undesirable

region. the

are

highest

presented

efficiency

in figure

occurs

s)_nmetrical-diagram

equal

is equal to

not

the

shown, less

are rather

than fiat.

at

for

all

to the

0.5.

is slightly

values

of

_. The

symmetrical-diagram

impulse-diagram becomes

3-4(a).

a speed-work

efficiency

efficiency

less

efficiency than

Between

either

_1

As h is reduced

still

are

greater

than

consider

the

is the aspects

three-dimensional

before

a diagram

efficiency exit

therefore,

efficiency

efficiency" the

the

This

representative

_ values

static, and

static

because

for

must

exit

for

efficiency. the

efficiencies,

types

designer

actually

The

and

below

0.5,

at

of the },_0.5, efficiency

rapidly.

total than

from

3-4.

are

}, values

curves

more

diagram

is

on

of speed-

as obtained

characteristics

efficiency

although

selected

a range

in figure

are

total

impulse-diagram

}_--1,

and,

tangential

are

over

presented

diagram

of 1. The

zero-exit-swirl-diagram

the

types.

characteristics

type,

to

2, approximates

in that

diagram

speed-

considered

of an application-

them

characteristics are

a symmetrical diagram. The were not obtained for )_ values negative

relating

diagram

maximum

efficiency

means

be generated

symmetrical

in reference

associated

then

method

)4elds

by

of proportionality

static-efficiency above

of the

being

experience. can

and the for

test

curves

analytically

the

the

in terms type

(3-11)). by

(4)

The

expressed diagram

evaluated

components

76

basis

assumptions

2. Briefly,

components

parameter

(eq.

The

the

components.

work

the

as

type

is substantially head

about

criterion such

with

0.5. of

as the

effects,

characteristics

velocity

achievable

any

Even

of these

where

merit,

total,

however,

the

previously

discussed

to be discussed

later

is selected. are

presented

lower

than

represents

a loss.

in figure the The

total

3-4(b). efficiency

highest

static

in

VELOCITY

efficiency diagram, diagram. _=0.5,

for

h values

less

than

0.5

and

for

X values

greater

For

the

impulse

diagram,

where

there

is no

is

obtained

than

exit

0.5,

the

swirl.

with

with

efficiency

For

the

DIAGRAMS

the

the

impulse

zero-exit-swirl

is a maximum

symmetrical

at

diagram,

1.0

.8

f,,_.._=--_'---u-

.6 e-

-% _6

Diagram type

.4

Zero exit swirl Impulse Symmetrical .2

(a)

I

/

/

I

///

I

I

I

/ /

// /

I

(b) 0

.2

.4

Speed-work parameter,

FIGURE

3-4.--Effects

(a)

Total

efficiency.

(b)

Static

efficiency.

of speed-work on

efficiency.

I .8

.6

parameter (Curves

and from

I 1.0

;_

velocity-vector ref.

diagram

type

2.)

77

TURBINE

DESIGN

AND

APPLICATION

.90 -Total efficiency, 7'

.80-t--

L_

• 70

iency, "q

6O

1

I

I

J

65

70

75

8O

Stator exit angle, % deg 3-5.--Effect

FIGURE

of stator exit eter X, 0.5.

the

efficiency

The

zero-exit-swirl-diagram

very

little

is a maximum

which

type

but

obtained shows

from the

and

that

maximized.

If

should

stator

exit

of

angle of

through-flow

stress

influence

It

has

large

been

exit

are

One

means

use

of downstream

back

to axial. presented

stream shows 78

the

total

which

total

efficiency

does

swirl.

decreases

not

parameter component

of this

is taken

from

upon

which

is

desired,

and,

by the

exit

the

is to be

stator

complete

exit

affects

the

annulus

area

a

freedom

it

the

annulus

1,

angle.

is desired,

since

therefore,

be

reference

efficiency exist

can

efficiency

However,

always

V=,

effect

of stator

static

and

area.

and,

hence,

selection.

that

at

low

values with

of increasing

stators,

are

example

depends

encountered,

The

param-

exit

at _ = 1, but

as functions

is also influenced

which

efficiency

in figure

staters that

3-5,

of velocity

shown

no

speed-work

75 ° is indicated.

angle

swirls

efficiency.

are

level

is

through-flow

An

angle

angle

the

velocity

angles.

60 °. If maximum

of about this

the

efficiencies

best

about

there

is highest by

1. Figure

component

rotor

could

the

flow

static

the

only

by

maximum

be

selection

The

also

reference

It is evident angle

not

to the

total

where

Speed-work

to 0.5.

is affected

is related

X= 1,

at

efficiency

as _, is reduced

Efficiency diagram

angle on stage efficiency. (Curves from ref. I.)

3-6.

referred

of

associated

the remove

static the

characteristics In

this

to

as

efficiencies

of

speed-work

1X-stage

the

diffuse

flow

(ref.

with Figure

turbines

the

the

turbines

turbines

turbines.

l_-stage

in static

is through

and

of such

figure, the

reductions

efficienc:_ swirl

parameter,

3)

down3-6(a)

are

lower

VELOCITY

than

those

are

due

Because no

gain

of the

1-stage

turbines.

to the

additional

friction

of this

additional

friction

in static

efficiency

value

of X is below

below

about

through

impulse

0.35,

use

over

losses loss, that

approximately substantial

lower

of the

total

efficiencies

downstream

stators.

the

l_-stage

of the

1-stage

0.35

gains

of downstream

These

(fig.

in static

DIAGRAMS

turbine turbine

3-6(b)).

For

efficiency

can

achieves until

the

X values

be achieved

stators.

1. O0

Diagram type

impulse

1-Stage ll-St_e

impulse

ll-Stage

symmetrical

l_-Stage

impulse (two

downstream

o

I

I

stators}

I

I

.60

¢.-

.40

0

FmuR_;

3-6.--Effect

.i

of

.2

.3

Speed-work

parameter,

(a)

Total

(b)

Static

downstream

.4

.5

h

efficiency. efficiency. stator

on

efficiency.

(Curves

from

ref.

3.)

79

TURBINE

DESIGN

AND

APPLICATION

Multistage

Turbine

Efficiency

When the turbine requirements are such that the speed-work parameter is quite low and high efficiencies are still desired, multistage turbines are used, and the required work is split amongst the various stages.

t-

stages Turbine

o_ ro

2 2

ta_

.!

/

7° I

o--

_ N

I

I

I

1

I

.f/..i

/ ,/,/"

50

//,,," /,

•40

tb) 0

FIGURE 3-7.--Comparison

8O

I • 10

1

I

.20 .30 Overall speed-work parameter,

] .ZlO

(a) Total efficiency• (b) Static efficiency. of efficiencies of 1-, l_/r -, and 2-stage from ref. 4.)

I .50

turbines.

(Curves

VELOCITY

Two-stage turbine the

$urbines.--The

results

in about

reduction

stage

addition

of

doubling

stage

work.

_, is accompanied

with

two

the

stages

exit

swirls

A study sented (from

4. The

4)

are

total the

the

2- and

efficiency

1-stage 1-stage and

difference

between

than

for

24

the

total

1-stage

The

2-stage

turbine

work

good

symmetrical

maintained by

increased

the

first

stages

diagram

The

as well

is illustrated

in figure

velocity

compounded. (or

increase)

Figure

3-8(b)

two-stage stator. exit

with

The swirl

second-stage

diagram and

characteristics ciencies obtained

the

with

of this

positive

the

exit

3-7

speeds

with

swirl

of turbine

upon

the

and

no

50:50

each

stage.

exit

swirl

of the

work

work impulse

split

has

first

and

This

a type

in which

type

of

of turbine

all expansion

stator

and

and

the

equal. was

for without

another

subse-

first-stage type

X=0.125

case of the

with

in reference

the

first

zero

efficiency 5. Effi-

two-stage turbines blade row. Because

leaving

of

a second-stage

A study made

all

in velocity. condition is

decreasing

diagram

swirl

the swirl

a

zero

stator.

first

swirl

for

are exit

with for

the

a velocity-compounded

turbine

is again

blade

for

output.

and

exit

represents

the

velocity

turbine

work

split

total

smaller

loss

fraction

turbine in

counterrotating

depends

The

optimum

general,

for

2-stage

total

features

higher than those for conventional because of the elimination of one work

points

the flow with no change the velocity-compounded

the

type

between

diagrams

and

In

increasing

shown

efficiency

figure

second-stage

is achieved

both

static difference

is maximized

three-stage)

illustrates

turbine,

of no

3-8 (a)

quent blade rows merely turn As k is reduced below 0.125, maintained, but work fraction.

work

diagram

impulse

is a two-stage velocity

in

stage

an increasing

associated

as

(fluid

presented

_=0.125,

turbines

speed-work

leaving

is achieved

and

At

turbine

and

is pre-

2-stage

the

of the

efficiency

kno_:a

split

turbines

overall

the

the

fraction

efficiency

as an

in

addition,

work

efficiency.

because

criteria

stage

stage.

increase

a 2-percentage-point-

for

zero-exit-swirl

second

to 75:25.

second

an

static

varying

_=0.5,

maximum

in the

produced

smaller

diagram

and

As _ is reduced,

stage

has

to 0.15,

for

occurs

by

At

split

At

efficiencies

efficiencies

reaction.

through

In

to 5 percentage

points

obtained

imposing

3-7. turbine

static

turbine

is a much

while

figure

increases

and

turbine

negative

in

percentage

values

the

of 2-stage

As _ is reduced

2-stage maximum

an

efficiency.

of 1-, 1Y2-, and

2-stage

efficiencies

efficiency

previously,

to adjust

a 1-stage

X value

and a 9-percentage-point-higher

turbine.

to

efficiency.

efficiencies

the

stage

stage

in stage

characteristics

compared

_, of 0.50,

shown

possible

in reference

than

As

average

an increase

it becomes efficiency

ref.

the

so as to maximize

of the

parameter, higher

by

of a second

DIAGRAMS

stage,

were the the 81

TURBINE

DESIGN

AND

APPLICATION

(a)

fb)

(a) Velocity-compounded turbine. FIGURE 3-8.--Velocity-vector diagrams Overall speed-work

second

stage,

75:25 are

for

the

also

related

for nuclear

(for

_<0.5)

in reference

6. Equal

assumed.

(derived

cr

Overall

as eq.

Such

split

work of

n

is the

(stator-inlet

number

velocity

to the in such aircraft. of

in which

general

stage

is equal

to stage

exit

were

then

efficiencies static 82

efficiency,

(intermediate

applications

include

production,

and

turbines stages

work

and

composed (for

constant

speed-work

h>0.5) stage

neglecting

Total

and

efficiency

total

and

stage,

from the

are blade

parameters

are

(3-16)

were

obtained

of

as

or last

velocity)

work

consider-

rockets.

stage

of stages.

is axial)

high

due utilized

_,=_ where

is

splits

their

V/STOL

power

zero-exit-swirl stage

turbines.

combination

multistage

and

(2-111))

for

for

hydrogen

and

being

the

used of

(work

of turbines

required.

turbines

characteristics

stages were

are

use

stage

compactness are

applications

stages vapor

turbines efficiency

examined speed

two

their engines

the

turbine. 2-stage

Because

turbines

direct-lift

dictates

turbopump impulse

as many

than

and

of

efficiencies ratio.

), levels

requirements turbines,

The

blade-speed

turbines.--In

more

fan-drive

a low-work

counterrotating

applications

speed

The

the

at low

row,

be

diagram).

of

potential

n-stage ably

would

illustrated

of a blade

advanced and

general,

functions

efficiency lack

in

(b) Counterrotating for special types parameter _, 0.125.

reheat

where

obtained the

static

stage

effect

for

stator

inlet

as functions

stage for

velocity

of X. Overall

efficiencies. discussed

a first

efficiencies

For

in chapter

overall 2,

a

VELOCITY

DIAGRAMS

w

n_'

o+

(3-17)

,

where first-stage

ideal

exit

total

ideal

to exit

total

This

equation

small.

By

J/kg;

Btu/lb

work

the

based

of inlet

on ratio

J/kg;

Btu/lb

work

based

on

pressure,

J/kg;

ideal static

neglects

using

on ratio

pressure,

general-stage to exit

based

pressure,

general-stage Ah_, ,

work

the

reheat

pressure

to

of inlet

total

pressure

of inlet

total

pressure

Btu/lb

effect,

stage-efficiency

ratio

total

which

reference

definition,

6 shows

equation

(3-17)

to be

becomes

Tb

_--

Overall on

total

the

efficiency

basis

1,.4_n

differs

of stage

total

2+1_

only

(3-18)

in that

efficiency.

the

last

stage

is evaluated

Therefore, n

_'--

The are

multistage

efficiency

presented

Figure

3-9(a)

efficiency This function

this

of

ratio, vary

ations

This

large

increases

total

efficiencies

because

of the

Another in terms 2 (eq. ing ratio

with

leaving

commonly

blade-jet

to the across

the

described

the

of stages must

be

(0.1

are

required

on

the

is and

The

vari-

the

concern either

to achieve

high

of stages

static

although

a

blade

or less),

number The

trends,

k----1.

shape

value.

however,

expected.

similar

number,

indicated _, values

at

herein,

to diagram

are,

low

limiting

are

Reynolds

diagrams at

stages

6.

is

efficiencies

at lower

levels

loss. method

ratio. ratio

energy tubine.

all

restriction

show used

speed

kinetic

from

of presenting

parameters

as the

when

addition

manner

reference

of _. The

those

angle,

in

that

if some

3-9(b)

all

(stator

varying

efficiencies

of diagram

(2-72))

as

this

from

as a function

etc.)

number

or,

lower in figure

factors

illustrates in the

shown

well

in

obtained

is reached

as

or downward

figure

obtained

was

efficiency

solidity,

in efficiency

here.

which

0.88)

other

blade upward

imposed,

total

case,

(3-19)

characteristics

3-9,

efficiency,

of many

aspect

overall

figure

shows

(in

level

may

in

1, f n--1,

is to plot This

of the

parameter blade

associated Blade-jet

was

speed with

speed

turbine

efficiency

performance

as a function

described

to a velocity the

ratio

in chapter correspond-

total-to-static is related

of

pressure

to speed-work 83

TURBINE

DESIGN

AND

APPLICATION

11

'_

.S .6_

.....

(_

Number of stages

I

_.... I z l Illl

Limiting efficiency

I

l

I

L

I J I llJl

.8 i_-

.5

•_

.4

32 I/

of stages/ 1

.z( b_

J I = I= I=1

t

.01

.02

.04

(a) Total efficiency. (b) Static efficiency. efficiency characteristics.

FIGURE 3-9.--Overall

parameter

and

From of the effect

the

efficiency

discussions

number on the

specific

(actual

or ideal)

imposed

the

final

selection

among

such

requirements),

compactness,

assumed In 84

the

first

half

to represent

a turbine

of the

design

and blade

goals

structural

type

are very

turbine

from

I

6.)

that

the

selections

have

an important

dependent speed

diagrams

upon

utilized. must

as performance

integrity

ref.

.8

(related

the In

an

represent

(dictated to

by

component

and weight.

RADIAL In

and

.6

(2-74).

it is clear diagram

level

design, cycle

life),

velocity

efficiency

a compromise the

and

section,

.4

(Curves

to equation

expected

work

actual

according in this

of stages

l = 1= I=1

.06 ,08 .I .2 Overallspeed-work parameter,

having

VARIATION of

this

OF

chapter,

average

conditions

a relatively

high

DIAGRAMS

a single over hub-

velocity the

entire

to tip-radius

diagram blade ratio

was span. (about

VELOCITY

0.85

or

lower

greater),

hub-

velocity

such

an

to tip-radius

diagrams

may

or may

blade

span.

ratios,

are

not The

must

important

in

radial

and

be applied final

variations

and

are

conditions

in the

radial exist in section

become will

effect

entire

to the

section

and their

the

must mean

which

This

of

diagrams for

due

end regions,

selection.

case

mean-section conditions

diagrams

to the

the

variations

balance of forces that were described for the

diagram

in flow

the flow

in

the that

In

substantial

average

variations

diagrams

the

however,

the

radial

in blade speed The considerations also

is reasonable.

encountered,

represent

variation the flow.

the

assumption

DIAGRAMS

on

very

consider

the

velocity

diagrams. Radial Consider

an element

3-10(a).

When

sulting

there

force. acting

on the

path.

When

the

force

must

fluid

be accounted

The

is termed

radial

The

radial

The

and

3-10(b))

to

maintain

for

as part

the

to fluid

the

of forces

net

as in figure

be

the

moving

along

along

the

maintained

balance is curved

flow

of the

field,

of velocity,

(streamline)

flow must have radial direction

balance

flow

must

serves

to keep path

by

its curved

curved

path

Any

linear

force.

an associated pressure force, if the streamline is inclined

required

to account

a

centrifugal

(fig. 3-10(c)),

this

pressure

re-

for these

part from

factors

equilibrium. equilibrium

pressure

turbine component

force

through-flow

required

horizontal.

(fig.

pressure

acceleration of the of which is in the

in the

is a tangential flow

The

force the

of fluid

circumferential

pressure

Equilibrium

forces

figure

3-10(b).

in the

x direction,

will

acting

Fluid

weight

the

now

on

net

an

be

formulated

element

is neglected. pressure

force

Fp.,et:(p+dp)(r+dr)dO--prdO--2

mathematically.

of fluid If

unit

are

indicated

length

(directed

radially

(p+?)

dr

in

is assumed inward)

is

sin dO 2 (3-20a)

where F_,

n_

g

net

inward

pressure

P 0

static

pressure,

angle

of rotation,

r

radius

Neglecting setting

of rotation, higher-order

sin

(d0/2)=d0/2

force,

N'm

N;

lb

2; lb/ft _

rad m; terms

ft (product

of

three

differentials)

yields (3-20b)

F_. ,,, =rdpdO The

mass

m of the

fluid

and

being

acted

on by

the

pressure

force

is 85

TURBINE

DESIGN

AND

APPLICATION

!_!ii2iiii!_i!_::: dr

J (a)

p+dp + dpl2-_ P

____

dBI2

.-

Vu e

(-/dr_

_

or_

_

/

/ p+ dp/2

\ \

!p/

rVm.

._____Vx

ame .I vr

(b)

(c) (a)

(b)

Rotation

plane

Element

of

fluid

(c)

FIGURE

3-10.--Radial

reduces

flow

field.

Meridional

equilibrium

m=p[r(r+ which

in turbine

(r-e).

dr)_--wr

plane

(r-x).

factors.

_] d_f0 2r

(3-21a)

to m=--prdrdO

The

net

previously. ferential

pressure To

flow,

force

balance the

the

radial

g

from

centrifugal

pressure

F_,c--m

86

results

Vu _ r

(3-21b)

force

prdrd0 g

the force

three

factors

associated

mentioned with

circum-

is

Vu 2_p V2drd r g

0

(3-22)

VELOCITY

The

radial

component

centrifugal is

force

of the

associated

FT, 0-----m

pressure

with

V_,

cos rm,

g

force

flow

required

along

a,_=

the

prdrdO g

DIAGRAMS

to

balance

meridional

V*. r_

cos

the

streamline

am_

(3-23)

where V,_

velocity

along

r,_

radius

a,_e

angle

of inclination

The

positive

directions

of curvature

are as indicated indicates case. the

the

radial

linear

various

the

plete line

(3-25)

deg

3-10(c).

For

The

of the the

(3-22),

V_ r

Thus,

radial

flow

approximation

as "simple"

those

order

that

(eq.

(3-20(b))

and

(3-24))

convenient flow),

the

angles

(3-24)

equal

to

the

yields

(3-25)

and

includes

to use

in its

meridional

(am,)

all com-

stream-

are

both

side of equation term and can

V_ r

quite

(3-25) often be

(3-26)

equation

Variations

the

nature

there

definition

will

is no (eq.

in

of the

second

assumptions

to be zero,

enthalpy

is

dV,,_ dt - sin a,_

on the right (rotational)

by

are usually

simplifying

assumed total

to illustrate

effects

other

to produce

(3-26)

has

become

known

equilib.rium. Radial

In

required

in this

write

represented

radial

(3-23)

outward

equation

not

inclination

gdp pdr-The

in equation

streamline

equilibrium

(or near-axial

and

we can

angle

V_, dVm_ cos a_e---sin a,_e r_, dt

It is, however,

(1/r,_)

sign

prdrdO g

(3-23),

and inclination

force

meridional

force

ft

is directed

pressure

small. Therefore, the last two terms are small as compared to the first neglected.

minus force

pressure

is the axial

curvatures

curvature

pressure

(eqs.

factors.

form.

streamline,

radial

_gdp p dr--

contributing

of meridional

dVme dt " sin _----

components

Equation

m;

along

net

ft/sec

streamline,

component

Fv • Z=--g

m/sec;

of meridional

balancing

acceleration

Setting

streamline,

for streamline

in figure

that

The

meridional

radial

(1-49))

order be

Velocity radial

variations

will be neglected, made.

If

component can

in velocity,

be written

•

and certain

streamline

slope

of velocity,

and

is the

as

87

TURBINE

DESIGN

AND

APPLICATION

" -- 2gJ-Differentiating

with

substitute

for

dh

respect

(and

dh' ds dr --T-_r-_ If

the

flow

enthalpy

entering

at the

to

since

radius

and

using

equation

(1-8)

to

p= 1/v) yields

1 jp

the

(3-z7)

2gJ

dp dr

1 _ 2gJ

turbine

first-stator

d(V_ 2) 1 dr -_ 2gJ

is radially

exit

d(Vx 2) dr

uniform,

is radially

(3-28)

then

constant.

the

total

Further,

if the

stator loss is radially constant, then the entropy at the first-stator exit is also radially constant. The rotor, as will be discussed later this

chapter,

enthalpy) loss. At

may

extraction any place

enthalpy

and

gradients

imposed

due

to radial

For

or

are

tion

(3-26),

into

equation

and in the

entropy

not probably turbine,

depend

by the

have

radially

constant

does not have therefore, radial

on

the

various

uniformity

blade

rows,

work

radially gradients

of the

in

(total

constant in total

inlet

and the gradient

flow,

the

damping

mixing.

simplicity,

entropy

may

it is here

radially the

assumed

constant.

With

"simple"

(3-28),

that

the

these

radial

total

enthalpy

assumptions

equilibrium

and

and with

expression,

the

equa-

substituted

we get

v. 2 , 1 d(V. 2) 1 -_ 2 In

order

to

specify

a relation

often,

a variation

or, in terms

solve

this

between of swirl

dr

--I-_

equation,

dr

(3-29)

it is necessary

V_ or V_ and

to

r or between

velocity

with

V,,:

Kr N

of mean-section

--0

radius

independently

V,

has

and V_. Most

been

specified (3-30a)

conditions,

v.

(3-30b)

vZ:=,,E: Substituting (3-29)

equation

and

then

V,.,_-V, where

a_

(3-31)

is not

88

is the valid

(3-30b)

integrating

{1--tan' absolute for

the

as

and

its

between

am flow special

differential the

limits

( i)rc,r angle case

at

form

I_\_/

--

the

mean

of N=0

into

of r,_ and

,l

_I}

(3-31)

radius.

(constant

equation

r yields

V_).

Equation For

this

VELOCITY

special

case,

integration

of equation

v=V_=[1--2 A case the

of interest

absolute

and

flow

equation

not

angle

(3-29)

The

radial

is radially

integrates

variations

for

from

a mean

velocity

flow

E

_

_.._>= m_

flow

equation

constant.

V=(r'_

(3-32)

(3-30b) In this

case,

is that

where

V,-=-V=

tan

-sin'a

velocity,

above

angle

In _.j

angle are

of 60 °. largely

(3-33)

axial

velocity,

are

presented

equations,

The

radial

dependent

_

and flow angle, in figure

variations on

the

in

specified

3-11 axial swirl

___"'Exp°nent'N1 0

1.0

]

.5

I

I

- J

2.0 V u = Kr N Q .m

"N

_-2

L.

_ E "- :>_

Vu = KVx 1.C

-1

x

........

B e--

O

60[

30 .7

-1

< .8

I

I

I

.9

1.0

1.1

Ratio of radius to mean radius, FIGURE

3-11.--Radial

a,

to

in swirl

the

radius

and

by

yields r 71/2

tan _ _

covered

Vu

as computed

(3-29)

DIAGRAMS

variations

of flow

angle

velocity a,,,

and

I ""-z 1.2

I 1.3

r/r m

flow

angle.

Mean-section

60 °.

89

TURBINE

DESIGN

velocity increases and

AND

APPLICATION

variation (value of N). As the swirl distribution exponent N or decreases from a value of --1, the changes in axial velocity

flow

angle

with

changing

seen, the axial velocities of N cannot be obtained can

be used

shorter

for design

(values

variations

purposes

of rh/r,,

and

illustrated

discussed

radius

become

and flow angles with all blade becomes

rt/rm

in figure

in subsequent

When

a value

stage

of this

Free-Vortex of -- 1 is used

as the

to 1). The on

pronounced.

As

with certain values The range of N that

larger

closer 3-11

sections

more

associated lengths.

blades

effects

become

of the

velocity

radial

diagrams

are

chapter.

Diagrams

for the

exponent

N in equation

(3-30a),

then rVu=K This

is the

for

such

condition a swirl

in a free

distribution

or a free-vortex The turbines

for flow

to

and as

a turbine

designed

a free-vortex

design,

turbine.

free-vortex in which

products

Thus, valid

vortex,

is referred

design is used radial variation

this condition is specified at there is no radial variation UVu

(3-34)

both

entering

in the of the

both the stator and rotor in specific work, _x(UV_),

and

leaving

the specific work computed for the entire flow. Further,

axial

velocity

mass from

flow per unit area the mean-section

Vx

entire

flow

design

simplicity

is radially

within

vortex

designs

An

example

rotor

from the if N=--I

constant.

accuracy

of

of the

main

is one

for axial-flow of

the

outlets, because

are radially

constant.

mean-section in equation

Thus,

the

then the

diagram (3-31),

radial

is the

variation

in

(pV_) is small, and the mass flow rate obtained velocity diagram can be used to represent the

an

set

vast majority of axial-flow diagram is accounted for. If

0.1

percent

reasons

in

for

most

the

cases.

wide

use

This of free-

turbines.

velocity

diagrams

for

a

free-vortex

design

is

shown in figure 3-12 for the hub, mean, and with a radius ratio of 0.6. The radial variation

tip sections of a blade in the diagram shape

is

for

considerable.

The

symmetrical )_m of

zero-exit-swirl

1. The

(_h=0.56), reaction hub

(lowest blades, 9O

while

the

(_t----1.56). is the

efficiency). the in order

diagram

diagram

associated

section

selecting

mean-section hub tip

diagram

Thus, critical

for

to ensure

special diagram,

satisfactory

an

swirl

care

is diagram

with

high

distribution,

must for at the

a

parameter

impulse

aerodynamic

especially diagrams

an

conservative,

free-vortex from

example

a speed-work

is nearly is very

a

section

Therefore,

mean-section

having

diagram

this

the

standpoint

be

taken

when

low-radius-ratio hub

section.

A

VELOCITY

DIAGRAMS

45. 30....

Radius ratio, rh/r t = O.6

Hu_ section

rm/r t ° O.8

Mean section

rt/r t = 1.0

Tip section 3-12.--Radial variation flow. Stator mean-section exit eter Xm, 1.

of velocity-vector diagrams for free-vortex angle a,,, 60°; mean-section speed-work param-

FIGURE

very

high

increases Another

reaction

to

diagram

leakage across the potential problem

considerable trated

tip

radial

in figure

--38 ° at the

an overhanging bending stresses. blade is illustrated

tip,

the

in rotor rotor

a variation

tip section,

also

be

troublesome

because

blade tip clearance space. is that of rotor-blade twist.

variation

3-12,

can

thus

The positioning in figure 3-13.

inlet

inlet angle

of 83 °. This causing of the

some hub

angle.

For

varies

from

results tip

case

hub

having

problems sections

is a illus-

45 ° at the

in a blade

fabrication and

the

There

it

and

of such

a

91

TURBINE

DESIGN

AND

APPLICATION

.-/*-

_ ip section

\\

n

FmURE

3-13.--Relative

positioning

of hub

and

Non-Free-Vortex Free-vortex are often

designs

classified

non-free-vortex potential trated

the

vortex design

1.111, the

r/r,_

sections,

to the

3-11,

3-14

no real

At

any

swirl

having The

For

a blade

of 0.889

mean

section.

for those

exit

blade

particular

exist

ratio, swirl

constant-flow-angle

There

the

rotor

cases superwheel-

the

constar_t-flow-angle

--1)

design.

The

r/r,_ of 1, are

radius

ratios

mean-

the

same

r/r,_ of 0.75,

to tip-radius to the

ratio

hub

and

to tip-radius

ratio

to the

and

hub

ratio

decreases,

to a blade

section

relatively

no

for which,

Gf tip

to tip-radius

of course,

of tip

a hub-

cases

for axial

Illus-

for the

correspond

hubare,

design.

the

correspond

with

The of the

The

a hub-

of r/rm corresponds

values

radius

at

arLd 1.111

As the

value

1.25

some

3-11.

(N:

with

vortex.

to alleviate

design,

and

diagrams

of non-free

(N=0)

ratio

a blade

designs

in figure

free-vortex

and

turbine.

other

in diagrams

at a radius

of 0.75

distributions. zero

are For

respectively.

in figure

92

values

particular

the

design,

all

free-vortex

constant-swirl

are

shown 1.25.

r/r,_ values

sections,

the

Also and

respectively.

the

closer

with which

the

variations

illustrated

the

of freeovorgex

that

heading with

(N=I)

compared

used

in an attempt

radial

variations

solid-rotation,

all cases.

any

the

design,

diagrams,

0.889,

common

used

associated are

velocity

are

section

0.8,

3-14

(N-------2) or

0.6,

the

sections

Diagrams

so commonly

are

disadvantages

having

for

under

designs

in figure

flow,

are

tip

diagrams

to show

as shown

in figure

velocity. exit

diagrams

This is due to the selected

are

the

same

mean-section

for

all

diagram

(az._:0). diagrams

are

quite

similar

to

the

free-

VELOCITY

DIAGRAMS

Radial swirl distribution

Ratio of radius to mean radius,

Super vortex (N = -2)

Free vortex (N = -l)

Constant swirl (N = O)

Wheel flow {N = 1)

(a)

(a)

Constant flow angle

rlr m

1. 250

(a)

1.111

1.flOf

O. 889

O.150

(a)

aNo real value for axial velocity. FIGURE

3-14.--Radial

vortex

diagrams

rotor-blade the

variation

and,

twist

and

therefore, low

constant-flow-angle

stator

has

The

a small

sort.

The

case.

The

radial

present

hub

amount

of twist

(N= blade

has

--2)

twist

variation

The

constant-swirl

alleviate

the

design.

is more

large

and

below

cannot

about

0.85

for

and

relative

than

for

turbines, those

blade-twist

the

0.70

be sustained

velocities these

too

for the

wheel-flow

the

and

here

free-vortex higher

of a free-vortex

severe

(N= at

the

flow

blades

with

hub-

stator

exit For

velocities

are

could

velocity about

absolute

these Mach

higher

is

ratios

below hub

for

high

do

free-vortex

axial

the

higher cause

diagrams

and

addition,

can-

to tip-

to tip-radius

design

relatively

and

hub-

of the in

of

free-vortex

with

(N--l)

(N=O) In

the is large

problems

1) design.

design.

for

variation

on blades

is that free-vortex

no advantage

velocity

wheel-flow

constant-swirl

the

to have

on

radial

of high

advantage

than

axial

hub-reaction the

swirl

12°).

imaginary) and

various

problems

while

appear

of stator-exit

(N=0)

However,

twist,

(about

for

same

A possible no

diagrams

not be sustained (Vx becomes radius ratios much below 0.8.

diagrams

the

reaction.

stator

super-vortex

any

of velocity-vector distributions.

designs number

losses

than

design. 93

TURBINE

DESIGN

AND

APPLICATION

M - 0.58

M/_ =o.sr - 0.77/ 4z" M

-8

Tip section, rlrt = 1

Sectionat r/rt - O.68 (a)

16. 6°--__

_r \

_ "_'_ Mr "0./_,_

M,.oW Mr" 1"01_'_

_45.

3o

_'" &'i_

-\

I1__, "7°

/M-

0.74

Sectionat rlr t - O.67

Tip section, rlr t- 1.0 (b)

FmURE

94

3-15.--Comparison twisted

(a) Free-vortex turbine. (b) Nontwisted turbine. of velocity-vector diagrams of free-vortex turbines. (Diagrams from ref. 9.)

add

non-

VELOCITY

A design termed

procedure

for

a "nontwisted"

design

completely

should

be easy

results

exit tions

_. Although

rotor

10 ° for

free-vortex

design.

At

the

at the

tip.

The

for the

same

with

vaIiation

7.

exit,

the

relative

stator

designs.

swirl

at

design

Mach

(0.85)

than

has

stator

from

and

the

hub

free-vortex

of

nontwisted

hub

at

value

increased the

at the

The

condi-

(N)

number

for the

the

stator-exit

30 ° for

is negative

blade-inlet

a

9 are shown

velocity

twist

than

Such

therefore,

of reference The

angle,

experimental

nontwisted

in axial

to more

exit

which,

9 contain

design.

is eliminated, design

rotor

rotor,

study

nontwisted

and

in reference

the 8 and

the

inlet

to a swirl-distribution-exponent

twist

nontwisted

However, over that

for

radial

in this closely

in

designs

used

A large

correspond the

is presented twist

free-vortex

present

of constant

References

diagrams

3-15.

is also

design,

to fabricate.

velocity

in figure

blades

eliminates

comparing

design

positive is higher

design

(0.72).

the reaction at the hub of the nontwisted design is improved of the free-vortex turbine. The two turbines have about the

efficiency.

The

non-free-vortex

work

and,

tion

because

in mass

tions

may

error

may

section by

rotor

DIAGRAMS

occur

if such

As

seen

from

free-vortex Mach

conditions

and

flow

rate.

this

It

designs

been be

from

used

increased

has

cannot

deviations

results

to obtain

than

design the

in

use

other

stator

sustained

basis and

of the

mean-

be designed

tip

in

order

to

of a non-free-vortex the

design

procedures,

of a free-

however,

of m_n-free-vortex

problems

this

over

all

to with

higher design

from

spans.

as reported

turbine

as

increased

deviations blade

designs associated

such

and

large

designs,

PROGRAMS

problems

twist,

that

improved

COMPUTER

design

considerable

should

hub

hub-reaction

shown

free-vortex

on the

proper

variacondi-

disadvantage.

discussion, and

and

turbine

complex

computerized

rotor-twist

numbers,

The more

is no real

designs

plexity.

been

much

radial

mean-section

conditions,

between

in specific

velocity,

the

is designed

flow

With

the

Thus,

average

a turbine

variation

in axial

area.

the

complexity

alleviate

radial

A non-free-vortex

therefore,

turbine.

additional

feature

gradient

unit true

conditions.

work

all

radial per

represent

is,

vortex

rate

not

integrating

turbine

of the

flow

flow

compute

designs

hub com-

free-vortex

However,

in reference

small 10, have

performance.

FOR

VELOCITY-DIAGRAM

STUDIES This diagram

chapter

has

selection,

staging,

and

the

diagrams

best

radial

presented including variations.

and number

some diagram

of the

basic

aspects

types,

their

relation

It is evident

that

the

of stages

for a given

of velocityto efficiency,

determination

application

of

requires 95

TURBINE

many

DESIGN

tion. such

then

One such The program radial

is

computer

computer includes

allows

program

uses

reflects

equation

and

real

exit

dependent

the

in reference

(swirl

14,

ef swirl

proven

very

wherein

to

any

an internal a basis.

rotor

work

However,

of

velocity)

loss This (which

for

either

a

with

The

it not

13 as

(Vme=V_/cos

because

12. in and

addition,

includes

inputs.

perform

11 and effects

specifications,

velocity).

many there

a,,_)

large

is

or

the

variation

small

in

variations

existence

of

these

in condi-

3-11. in

the and

a program

radial

is used

successful

with

as

calcula-

in enthalpy In

and

velocity

resulted

velocity

also

input

(meridional

has

instead

variation

evolved

reference

distribution

solution

by figure

problem

generated

swirl

meridional

variable

is indicated

radial of hand

gradients

but

distribution

find

variable

independent

and

in flow.

from

of these

for

cannot

radial

as an input

swirl)

solution

non-free-vortex

realm

been

variations

information

combinations

computer

This

loss

the

exit

effects, of the

have

and

radial

stator

rotor

values

include

program is described in references consideration of streamline-curvature

for blade using

to

are out

programs

in determining

correlation

desired

analyses

equilibrium

entropy

tions

it

curvature

such

Therefore, tasks.

only

If

meridional-streamline

in efficiency,

no

APPLICATION

considerations.

designs,

the

AND

variation

as input.

shows

reasonable

modification,

that

in

The

meridional

modified

valid

variation

as reported velocity

program

turbine

designs

in meridional

has can

be

velocity.

REFERENCES 1. STEWART,

2.

WARNER

teristics

in

ASME,

Dec.

STEWART,

L.:

Terms

A Study

of

of Axial-Flow

Velocity

Diagram

L.:

Analytical

4.

WILLIAM T.;

WINTUCKY,

Characteristics

of

Terms

and

of Work

ments. 5.

NACA

WINTUCKY, ments.

RM

Paper

Turbine RM

Charac61-WA-37,

WARNER Turbine

with

WILLIAM

Requirements.

Analysis

of Efficiency

Downstream T.:

RM

Analysis

of Work

Stators

E56J19, and

in

1957.

of Two-StageSpeed

Require-

1957.

Efficiencies

E57L05,

Speed

L.:

NACA

in Terms

Single-Stage-Turbine

and

ANY STEWART, WARNER in Terms

L. : Analysis of Work

and

.of Two-Stage Speed

Require-

1958.

L. : Analytical 1958.

of

Work

Requirements.

E57F12, T.;

Characteristics E57K22b,

of

STEWART,

Single-Stage Speed

RM

6. STEWART, WARNER ciency

Terms

Characteristics

WILLIA_a

NACA

AND

Investigation

L.; AND WINTUCKY,

Efficiency

Counterrotating

96

a

STEWART, WARNER Turbine

Efficiency

1961.

WARNER

Efficiency Characteristics in NACA RM E56G31, 1956. 3.

Turbine

Parameters.

in Terms

Investigation of Work

and

of Multistage-Turbine Speed

Requirements.

EffiNACA

VELOCITY WILLIAM R.; AND SILVE_RN, DAVID H.: Analytical Aerodynamic Characteristics of Turbines with Nontwisted NACA TN 2365, 1951.

DIAGRAMS

7.

SLIVKA,

8.

HEATON, THOMAS R.; SLIVKA, WILLIAM R.; AND WESTRA, LEONARD F.: Cold-Air Investigation of a Turbine with Nontwisted Rotor Blades Suitable for Air Cooling. NACA RM E52A25, 1952.

9. WHITNEY,

WARREN

J.; STEWART,

WARNER

L.;

AND

Evaluation of Rotor Blades.

DANIEL

MONROE,

E.:

Investigation of Turbines for Driving Supersonic Compressors. V-Design and Performance of Third Configuration with Nontwisted Rotor Blades. NACA RM E53G27, 1953. I0. DORMAN, T. E. ; WELNA, H. ; AND LINDLAUF, R. W. : The Application of Controlled-Vortex Aerodynamics to Advanced Axial Flow Turbines. J. Eng. Power, vol. 90, no. 3, July 1968, pp. 245-250. 11.

CARTER,

A. F.; PLATT, M.; AND LENHERR, F. K.: Analysis of Geometry and Design Point Performance of Axial Flow Turbines. I-Development of the Analysis Method and the Loss Coefficient Correlation. NASA CR-1181, 1968.

12.

PLATT,

M.;

formance 1968. 13.

SMITH,

vol. 14.

AND

CARTER,

of Axial

Flow

A. F.:

Analysis

Turbines.

S. F. : A Simple Correlation 69, no. 655, July 1965, pp.

of Geometry

II-Computer of Turbine 467-470.

and

Program. Efficiency.

Design

Point

NASA J. Roy.

CARTER, A. F.; AND LENHERR, F. I_.: Analysis of Geometry Point Performance of Axial-Flow Turbines Using Specified Velocity Gradients. NASA CR-1456, 1969.

Per-

CR-1187, Aeron.

Soc.,

and DesignMeridional

97

TURBINE

DESIGN

AND

APPLICATION

SYMBOLS A

flow

area,

m_; ft 2

pressure conversion

g h

force, N; lb constant, 1 ; 32.17

(lbm)

J

specific enthalpy, J/kg; Btu/lb conversion constant, 1;778 (ft)(lb)/Btu

K

proportionality

L

loss,

m

mass,

N

swirl

(ft)/(lbf)

constant

J/kg;

Btuflb

kg;

lb

distribution

exponent

number

of stages

P R

pressure, reaction

N/m_;

r

radius,

8

T

specific entropy, temperature, K;

m;

lb/ft 2

ft

U

blade

V

absolute

speed,

t)

specific

volume,

W

relative

velocity,

W

mass

flow

o_

fluid

absolute

J/(kg) °R

m/sec;

velocity,

(K) ; Btu/(lb)

ft/sec m/sec;

mS/kg; m/sec;

rate,

(°R)

kg/sec; flow

angle,

ft/sec ft3/lb ft/sec lb/sec deg

efficiency 8

angle

k

speed-work

of rotation,

deg

parameter

density,

kg/ma;

lb/ft 3

Subscripts: a

first stage annulus

c

component hub

h i /d

general ideal

1

component

m

mean

me

meridional

net

net

r

radial

ro

rotor

8

component

st

stator stage

98

due

to circumferential

due

to linear

flow

stage acceleration

section

component

due

to streamline

curvature

(sec 2)

VELOCITY

t

tip tangential

_g

DIAGRAMS

component

0

axial component at stator inlet

1

at stator

2

at rotor

exit

or rotor

inlet

exit

Superscripts: --

overall

'

absolute

turbine total

state

99

CHAPTER 4

BladeDesign By Warner L.Stewartand ArthurJ. Glassman The

design

of a turbine

determination are usually the

overall

established

by

evolution

and/or

of velocity

number

is the design required This The

shape,

height

of the state

fluid

usually

The

assure

chord

many

chapter.

chord

and

by the

or axial profile

as well as the of this

used

third

and

step

velocities of

with

to allow

operation.

flow

mechanical

con-

selection

of blade (ratio

chord

to spacing), first

blade

exit

is then which profile

part

and

inlet

discussed is the design,

is

fabrication

in the

profiles, the

chord

as solidity

theory,

to accomplish

speed, dictates

blade

accurate

The

of axial

includes

which

The

consistent

design.

of flow,

diagram,

turbine.

enough

(ratio

of blade

requirements

will be discussed

surface

Channel

is

the determination

nondimensionally

which

connecting

chapter.

procedures

value

solidity

design,

step efficiency

3. The

aspects

velocity the

during

that

is the These

second

desired

flow angles

important overall

be long

be expressed

the

involves

first

speed.

blades.

throughout must

the

in chapter

step

the selected

integrity

can

part

next

of the

to be a minimum

tries

in the

This

The and

The

with

will produce

of the more

is set

considerations

Blade

analytical

that

some

blade

to spacing)

volves

discussed

was

steps. work,

application.

This

spacing

structural which

particular

consistent

conditions

selected

spacing,

the

major of flow,

diagrams

diagrams.

conditions

state

siderations. and

and covers

of three

requirements

of the blading

chapter

and inlet the

of stages.

by the velocity

the size,

consists

of the

inof this

geome-

in the

basis

of

last

for

is discussed

chapter.

_A_}l___

i_._f,9_i.(L

the

Y _,. A_

TURBINE

DESIGN

AND

APPLICATION

SOLIDITY One

of the

important

of the blade

solidity,

A minimum

value

weight,

cooling

mechanical

flow,

the

aerodynamic

will include

description suppress

between

separation

Figure

4-1

as the

static-pressure

figure

are

pertains equations

and

rows rather used

must

inlet

tions,

and

If one height

rows

stages,

in previous are

and

taken

The

as negative

negative

if in the

considers

the

same

blades,

from

blade

in this

bla(t(_ rows. absolute

When

will differ

The

as well in this chapter referring

velocities

concerned

inlet

components

in the

with

blade

slightly

from

component

values

are positive

ar_ in opposite

direc-

direction.

then inlet

velocities

tang(_ntial-velocity

values.

two-dimensional

two

solidity.

The

convention

tangential-velocity

for use to

discussion

we are

exit

dis-

will be a

studied

ai_d (,xit diagrams

than

chapter

The

on solidity

Solidity

a blade.

rather

in

will concern

Also included

on

by

results

selection.

permissible

The

the angle

chapters.

exit

as it flows

around

is limited

requirements

inlet

velocities.

to spacing. of reducing

section

are being

as well as to stator

with

than

of blade

in this

This

solidity.

the

selection

eventually

solidity

Diagrams

relative

Since

between

fluid

reduce

figures.

flow angle

if the

thereby

use

flow.

that

chord

reduction

spacing

and

concepts

is the

standpoint

diagram

loading

distribution

blade

we

or axial

the

affecting

blading

set

design

chord

of velocity

as absolute

to rotor

to a rotor,

factors

a typical

shown

from

increased

of Velocity

shows

of chord

However,

blade

and

blading

to separated

the effect

Effect

and

cost. due

of advanced

ratio

desired and

efficiency

and the relation

that

is the

and

blade

with

cussion

which

of turbine

is usually

considerations,

decreased itself

aspects

flow the

through

tangential

(subscript

a passage

force

1) to exit

exerted (subscript

of unit by

the

2) is

1 F,, =-

smV.,2(

V,, ,_- V,_ ,_)

(4-1)

g

where Fu

tangential

force,

conversion 8

blade

P

density, axial

Vu

tangential

This due

lb/ft

component

forc_

(Ibm)

(ft)/(lbf)

(see 2)

s

of velocity,

component

lower

m/see

of velocity,

exerted

to the static-pressure 2. The

1; $2.17

m ; ft

kg/m'_;

tangential

in chapter 102

constant,

spacing,

V_

N; lb

by

the

distribution part

of figure

; ft/s(,c m/see;

fluid

must

around 4-1

ft/sec b(, _h(_ sam(, t h(_ blade,

shows

a typical

_s the

as was

force

discussed

static-pressure

BLADE Stations

DESIGN

1

surface-'" "_ /- Such'on'_--r-

Vx, 1

--L y2_Vu,

.,_-_...._

2

C X --------_

pl

P

Pl

13-

P2. "'" rs, mln Axial distance

FIOUP, E

4-1.--Typical

distribution between

around the

flow in the

two

tangential

blade-row

the blade curves

velocity diagrams distribution.

row as a function

represents

direction.

the

and

of axial

total

blade

surface

static-pressure

distance. force

acting

The

area

on

the

Thus,

F,, = c_

L'

(pp-

p_) d

(4-2)

where C_

axial

Pv

pressure-surface

P,

suction-surface

X

axial

The

axial

chord,

distance,

solidity,

m; ft static static

pressure,

N/m2;

pressure,

N/m2;

lb/ft lb/ft

2 2

m; ft ¢z, is

{T x

_-

--

(4-3)

8

103

TURBINE

DESIGN

Substituting

AND

APPLICATION

equations

(4-1)

and

(4-2)

into

equation

(4-3)

then

yields

a_ =

(4-4)

/0

g At this point, been

used

first

is the

to relate

and

pressure

blade

surface

loading

that

pressure.

In equation

t'

equal

blade

Zweifel (l)

have

loading. (ref.

This

pressure

total

to be constant

The

1).

the static

to the inlet

surface

that

pressure and

equal

form,

(pp--

_, =

by

assumes

and

oil the suction

coefficients

to an ideal

introduced

to be constant

pressure

loading

loading

coefficient

on an ideal

static

exit static

two tangential

actual

used

is based

(2) the

to the

the

widely

coefficient on the

we introduce

d

p.)

d (4-5)

pl' -- p_

where Zweifel

_Z

loading

coefficient

pl r

inlet

total

pressure,

N/m2;

lb/ft 2

P_

exit static

pressure,

N/m2;

lb/ft

The

second

constant

coefficient

static

is similarly

pressure

on the

value

of static

pressure

(see

cient

can never

exceed

a value

ahvays

be less than

ceed a value fined as

2

1. The

defined

suction

fig. 4-1)

of 1. In equation

on that

of 1, and

Zweifel

surface.

(pp--

loading

the

assumed

to the

This

minimum

loading

purposes,

on the other

this second

/o

that

is equal

for all practical

coefficient,

form,

except

surface

coeffiit must

hand,

coefficient

can

ex-

_b is de-

p_) d (4-6)

pl p-

where p,,m,, is the N/m 2 or lb/ft2_ _. The pressed

velocity

minimum

components

static in terms

ps

,min

pressure

on

of veloerity

the

suction

and

flow

surface

angle

are

in ex-

as V_ = V sin a

(4-7)

V_ = V cos a

(4-8)

and

104

BLADE

DESIGN

where V

fluid velocity,

ot

fluid flow angle,

Substituting (4-4)

m/see; deg

equations

and

using

the

ft/sec

(4-5)

or

(4-6),

trigonometric

(4-7),

relation

mVQ

K is the

inlet

to that

1) sin 2a2

of tangential at the

flow

and

with

Bernoulli's

blade

equation

1) sin 2a2

component

(Vu.1)

relations.--Relations are

(4--9)

at the

blade

With

usually

this

involving

evolved

by

assumption,

solidity,

assuming

density

incom-

p is constant,

equation 1 P-k--z pV 2 zg

P'=

can be used.

into

exit.

loading

no loss.

(K-

velocity

of incompressible-flow

diagrams,

pressible and

ratio

(V,.2)

Derivation velocity

(4-8)

VQ (K-

where

and

sin 2a = 2 sin a cos a yields

Substituting

equation (K-1)

(4-10)

sin2a2

-

(4-10)

into

equation

(K-1)

(4-9)

yields

sin2a_

(4-11)

¢ ,,W,, where

V_

Let

is the

us now

velocity

define

on the

suction

a suction-surface

surface

diffusion

where

p = p_._,.

parameter

D, as

D_ =

(4-12) V_ 2

Many parameters the deceleration indication this

of this type have been of the flow on the suction

of the

definition

susceptibility

(eq.

(4-12)) (K-

of the flow on the in equation

Equation

(4-13) for each

coefficient can be seen

_, which that

shows particular cannot

decreasing

(4-11)

1) sin 2a2

_ =

constant

used to represent a measure of surface. This deceleration is an

(K-

blade

to separate.

1) sin 2a2

=

that

the

solidity

velocity-diagram exceed

a value

solidity

results

Using

yields

(4-13)

parameter

a,_D,

requirement. of 1, does primarily

not

or

Since vary

in increased

a,_,

is

loading

greatly,

it

suction105

TURBINE

DESIGN

surface

diffusion

later

in this

gential

AND

(higher

chapter.

velocity

4-2(a).

of K=-

represents and

exit

solidity sents

K

for

an

in the

there

is no turning K

is to be avoided,

from

value

of K, a xnaximum

exit angle

reaction

same

solidity

must

of solidity

(4-13)

to yield

the

can

be

equation

modified

For brevity,

derived

shows

that

the

angles

COS

o_2

_z

COS

O/1

of the

several

values

exit

angle,

solidity

inlet

which

of most

angle

inlet

encountered (4-13), repre-

parameter

suction-surface velocity

diagrams

for

any

is obtained

given

with

was defined

can

inlet

interest

an

with

and

exit

(4-14)

coefficient

can

angle

_,.

Equation

be expressed

is plotted

in terms

against

in figure

with

solidity

exit

4-2(b).

increasing

0 °, a2 <-45°),

decreasing

be evolved,

in chapter

_

flow

inlet

1.

of the

increases

(m>

of the

a2)

parameter

parameter

decreases

relation

in terms

parameter

Solidity

for

region

only

solidity

angle

A third

are

that

parameter

sin (m--

only.

given each

1

solidity

as the

be seen

in reference

2

this is expressed

flow

In the

of K <-

for K = 1. This

to a function

ax-

(4-14)

a

equation

if excessive

can

inlet,

of 45 ° .

Equation

of the

and

angles

increase

in figure

axial

a value

As seen from

It

tan-

of K represent

direction

Thus,

impulse.

value

and

the

angle

with

of the flow. The

values.

toward

flow

values

to zero for all exit

decreasing

move

blade,

blades.

against

blade

Positive

of rotor

is equal

of exit

a reaction

impulse

will be discussed

is plotted

values

blade.

velocities

the case where

diffusion

several

reaction

of which

parameter

1 represents

parameter

angles

solidity

in the tip sections

with

consequence

represents

tangential

increases

the

of K=0

a negative

primarily

D,),

The

ratio

A value

value

APPLICATION

flow For

inlet

a

angle.

parameter

for

exit. angle.

this

one

in terms

of blade

reaction

R,

2 as Wl 2

R--l----

(4-15) V= 2

Substituting

equation

(4-8)

into

R=

equation

1-(

the

two-dimensional,

Substitution

of equation

back

(4-16)

al/

incompressible-flow (4-16)

yields

c°s a2_ 2 \cos

for

(4-15)

into

case, equation

where (4-14)

V,a=Vx,2. then

yields

2 _=-wkere 106

Aa is m--a2.

-x/l-Z- R sin Aa

(4-17)

BLADE

DESIGN

Exit-flow angle, m

% deg

x O

-45 __ X tD t..

E

-75 or -15 -60 or -30

1--

(a) O

°.5

3F

I

I

I

J

0 -.5 -1.0 Tangential velocity ratio, K

_lnlet-flow

-1.5

angle,

Y, I -30

-45

-60

-75

Exit-flow angle, % deg

3 [--

Turn ing angle,

!

z_ deg

, g

,_

2__--_...._

0 -.25

0

.25 .50 Reaction, R

or 120

.75

I.O0

(a) Effect

of tangential-velocity ratio and exit-flow angle. (b) Effect of exit- and inlet-flow angles. (c) Effect of reaction and turning angle. FIGURE 4-2.--Effect of velocity diagrams on solidity.

107

TURBINE

DESIGN

Equation

(4-17)

reaction

and

reaction that,

AND

expresses

turning

values The

of 90 ° and

turnings Radial

parameter

in terms

of blade

parametor

is plotted

against

of turning solidity

varies

angle

the

in figure

solidity

little

with

4-2(c).

parameter

parameter

variation.--Chapter that

radial

varying

must

It can be seen

decreases

is a maximum

turning

unless

hub-to-tip-radius

with

for

very

solidity

nature

high

or very

low

swirls),

turbine

ratio

of 0.7, an impulse free-vortex

hub

and

from

tip

and

equation

the

hub

speed

vary

in the

axial

with

with desired

inlet

and

For

this

exit

ease,

shown

in the

following

table:

Rotor

Exit

Solidity

Inlet

Exit

angle,

angle,

parameter,

angle,

angle,

deg

deg

ffx_z

deg

(leg

ffz_

Hub

0

- 70

0.64

54

--54

1.90

Tip

0

- 62

--2

--63

again

what

different

that

negative.

constant

the

assumption

convention

that

Assume

radially.

.83

angle

from

of previous

that This

enables

the is

us

the

solidity-parameter

Inlet

Note

exit

a constant

a stator-hub

distribution. are

to

velocities,

corresponding

(4-14)

blade

will be illustrated

having axial

rotor

swirl

shown variation

variation

constant

in velocity

varying

Stator

are

in-

a turning

variations the

was

of this radial

a single-stage

°, and

radial both

will be a radial

exit

at the

computed

axial

and

of -70

angles

The

the

to satisfy

there

Consider

inlet

flow angle

Since

solidity.

(zero

values

in order

diagrams,

by an example. flows

3 discussed

occur

equilibrium.

velocity of axial

flow

solidity solidity

are used.

diagrams

value

the The

previously,

reaction.

angle

and

angle.

for several as indicated

creasing

APPLICATION

to

a

being

used

chapters.

loading

in this

Herein,

coefficient

reasonably

proportion

z

.79

chapter stator

is someexit

angles

_b_ is to be maintained

desirable solidity

Solidity parameter,

condition,

directly

to

and the

the

solidity

parameter. Let us now determine shown solidity radius 108

in the

preceding

variation (because

how table

in any blade

the

hub

and

can bc made

blade

spacing

row

must

is directly

tip values

of solidity

physically

consistent.

be

inversely

proportional

parameter The

proportional to

radius)

axial to and

directly proportional to parameter

at the

corresponding

hub

tip

0.64X0.7=0.45, a considerable

axial

chord

chord.

is 0.64.

value

which

fore,

axial

If axial

of

the

can increase

with

chord

half

taper

the

axial

is almost

axial

For

radius

were

tip

axial

desired

solidity

constant,

parameter

to hub

and yield

the

held

solidity

of the

from

stator,

value

is often

the be

of 0.83. used

the higher

then would

There-

so that

solidities

the

desired

at the tip. In the

case

If axial

of the

chord

axial

solidity

than

the

often

used

radius

rotor,

were

held

desired yield

the is not

cally

from

desirable

fabrication axial this

there

solidity radial

where turbine

variation

the

axial

solidities

especially

a radial in loading

blading

is not

axial

taper tip.

hub

with

Taper

desirable,

but

blade

for smaller

turbines,

axial velocity

cases,

is

hub

to

To simplify

mean-section have

to tip increasing

is also mechani-

stress. coefficient.

will not

of the larger

from

in loading in many

loaded,

is 1.90.

is still

from

of reducing

of the

hub

tip value

which

can decrease at the

cocfficient

highly

1.33,

chord

variation

on the basis

at the

corresponding

desired

standpoint

cases,

results

selected

the

1.90X0.7=

only aerodynamically the

in many

and

parameter

Therefore,

so that

lower

tip in the rotor

be

of 0.79.

blades

solidity then

would

value

in rotor

axial

constant,

parameter

and

used,

the

taper

is not

With

the

diagrams,

especially

a severe

those effect

on

performance.

Effect

of compressibility.--Thc

term

pl r-

in equation

(4-9)

shown

by equation

loading

coefficient

by equation

reduces

to 1/D,

(4-13).

For

p*,min

for incompressible

a compressible

¢ as for incompressible

(4-13)

flow

conditions,

flow case having

flow,

division

of equation

as

the

same (4-9)

yields

1 2g t_ V_D' O"x

(4-18) _*,inc

where

ax,,,c

is the

incompressible

equation

such

between

critical

velocity

(1-61),

(1-63),

and

equation

(4-18)

plt--

as (4-13),

(4-14), ratio,

(1-64))

is modified

Ps,,nin

flow

value

or (4-17). density,

and

using

as

determined

By introducing

and

pressure

the

definition

(eqs.

from

an

the

relations

(1-3),

(1-52),

of D, (eq.

(4-12)),

to 109

TURBINE

DESIGN

AND

APPLICATION

0"2:

O'X

(4-19)

, inC

7'+1

V-c, 2

where 7

ratio of specific heat constant volume

Vc.

critical

velocity,

m/sec;

by using

binomial

expansion

Then, equation

(4-19)

at

constant

pressure

to

specific

heat

at

ft/sec and

can be approximated

by neglecting

the

secondary

terms,

as

O" x

- 1

e

a_,,._ The (V

approximation

Vcr)_ values

ratio

diffusion

in figure

as D, either

4-3.

effect

required

for any

solidity

1.50

equation

solidity

value

decreases

effect

from

(4-20)

ratio

for several

compressibility

or decreases

(4-20) 2(7+1)

by

1. The

parameter

The

increases

compressibility 2, the

represented

up to about

suction-surface

_'+1

a value

of ( V with

is quite

values

of critical

becomes

more

values

1.2 .__1. 25 O

1.0 .6 0

110

1

I

2 3 Suction-surface diffusionparameter,Os

4-3.--Effect

of compressibility

on

axial

solidity.

is no

of less than of

Exitcritical-velocity ratio, (V/Vcr)2

FIGURE

velocity

pronounced

V,r) 2. For D, values increasing

for

against

of 2. At D, = 2, there

--

.50

good

_x/¢=._,_ is plotted

(V/Vc_)_.

BLADE

For

Do values

because

of more

it is beyond

than the

2, a region

limits

of good

increases with increasing (V/V_r)2. should be maintained below about Relation It is well recognized pressor

blade

is an

Correlation described This

that loss

in reference

parameter

reflecting

and

a

terms,

solidity.

energy

is defined

neglect

that

the

and

pressure

(Vp.,,i,

pressure

=0),

then

the

overall

D-

V,_.,-

the

or of a comreaction.

parameter

was

compressor

field.

and the second

diffusion 3, where

sum of the

surfaces

surface

ratio

D, values

and

reaction

analogous

of the

blade

within

in reference

ratio

solidity

solidity

diffusion

widely

An

as the

on the suction

is assumed

of both

one reflecting

for the case of the turbine

parameter

the

of a turbine

compressor

is used

two

practice,

interest

to Solidity

loading

with

of academic

Experience has shown that 2 to avoid excessive losses.

function

2 and

includes

turning

evolved

the

is only

design

of Loss

important

of blade

that

DESIGN

parameter

an overall

decelerations

to the exit

in kinetic

kinetic

minimum

velocity

diffusion

parameter

was

diffusion

energy.

is low

If it

enough

is defined

to as

2

With

the

use

equation

of the

(4-21)

definitions

reduces

V22+ V12 V_ 2

of D,

(eq.

(4-21

(4-12))

and

R

(eq.

(4-15)),

to D=D,-R

As seen

from

equation

(4-22)

(4-13), _ =_D_

Substitution

)

of equations

(4-23)

and

(4-23) (4-14)

into

equation

(4-22)

then

yields 2

COS

_2

.

D -

sm Aa--R _z_

This

relation

reaction

is like

and

Attempts overall

(ref.

loss

different not

4)

with

give the

figure

4-4(a).

been and

made

compressors,

with

diffusion of reaction

same first

to correlate

suction-surface

of increasing correlation

values

Consider

for

(4-24)

_1

the

two

terms

blade

loss

involving

solidity. have

definite trend but complete blade

that

COS

turbine

(ref.

5)

diffusion

loss with could not

increasing diffusion be obtained. Such

parameter

alone

and

solidity

would giving

not the

with

parameters.

both A

was established, a correlation of be expected,

same

value

since of D do

loss. the

effect

As reaction

of reaction

on loss,

is reduced

from

as shown

a relatively

qualitatively high

value

in near 111

TURBINE unity,

DESIGN AND APPLICATION

there

reaction tion

occurs

to negative

in loss varies

regime, of the

frictional on the

other

the increased these

are

desired

in many

surface

opposing

the

in chapter

applications,

per

the

solidity.

loss per

unit

surface

diffusion

required. The

value

is used.

4-4(b).

increases,

loss occurs

o .-I

(b)

Axialsolidity,ox

112

the

is increasing suction-surface

and solidity.

of the because

A minimum

As solidity

Reaction,R

FIGURE 4-4.--Loss

varia-

reaction

blading

A minimum

(a) Reaction. (b) Solidity. trend with reaction

nature

negative avoided

area

of the

as the

is usually

As solidity is increasing.

This

in

in boundary-layer

The

in figure

flow

factors.

6)

conventional

unit

reductions

rapidly.

change

to diffusing.

when

optimum

hand,

by

on loss is indicated

area

loss. Further

loss to increase

discussed

accelerating

at some surface

the

highly

of solidity

in blade

is caused

loss encountered

effect

loss occurs

of

(which from

high

cause

reaction

although

The

increase

values

with

characteristics flow

a gradual

amount

of

is reduced, because

of

as a result diffusion

BLADE

parameter

corresponding

factors

such

tribution,

to the

as Reynolds

and

rate

not exceeding

number,

of turning.

about

analytical

optimum tion

(4-14),

the

blade-row

locus

experimental

inlet range

of points

In order

_b, is equal

axial

and

solidity

of angles.

The

for impulse

have

velocity

previously,

been

1, minimum

to 0.8. can

exit flow angles,

disvalues

to identify

loss occurs

By using

this

be determined

and

dashed

made

when

value

the

in equa-

as a function

this is plotted

(long-short)

in figure

curve

of

4-5 (a)

represents

the

blading.

to determine

it is necessary

surface

of many

Solidity

attempts

to reference

coefficient optimum

for a wide

of suction as mentioned

of Optimum

According

loading

shape

is a function

2.0 are used.

and

solidity.

Zweifel

solidity

In gencral,

Selection Both

optimum

DESIGN

the

to determine

optimum the

stagger

values

in terms

of actual

angle

as, because

solidity,

_X

_-

(4-25) COS

An analytical to the

blade

flow angles

solidity

were

figure

and

The

determined

with

solidity

is seen

Loss coefficients function

blades

in relative various

exit

solidity

from

the

angle

exit

gets

nounced

and

optimum

the

the

loss

spacing.

cannot be correlation

Thus,

the

optimized is somewhat

obtained blade for each clouded.

any

shape

and

solidity,

a figure

the

such

exit

of selecting

some increase

in loss.

becomes

more

a given resultant and

as solidity

that

the

curves

blade

shape

velocity significance

the

deviation

significant severe

for

of exit angle,

loss region more

for

solidity

importance

loss, and

8 as a

angle coefficients,

against

values

as

efficiency

in reference and

4-7

be recognized by using

4-6.

(al = - a_). These

of minimum

become

was

in figure

maximum

presented

negative)

cause

It must

6 from

in

solidity

efficiency

here

solidity)

indicate

minimum

penalties

value.

4-7 are usually

not the

as shown

7, where

in figure

(more

region

does

smaller,

are

blades

curves

angles,

angle

of actual

an optimum

yielding

of

here

larger

in the

optimum

from

the

flat

the

data

impulse

These For

to that

stagger

values

as shown

in reference

(inverse

replotted

exit

of reference

close

ratio

and

solidities,

on cascade

are

angles.

are rather

of figure

to be quite

optimum

6 compared

data

rotor

(al = 0) and

solidity.

the

6 to relate

Thus,

of inlet

as optimum

based

terms,

optimum curves

with

of pitch/chord

reaction

solidity.

of reference

different

determined

figure 4-5(b) for this case.

in reference

as a function

way

four

used

axial

authors

in this

measured

was

the

obtained

4-5(b).

The

model

as

such

in As pro-

departs as those

and varying distribution of such

a

113

TURBINE

DESIGN

AND

APPLICATION

Inlet-flow

6--

\

\

\

angle,

\

\ \

_4

Typeof

\ -2

_ "" _. _ _ _ _

"_3-

_1_

bladerow

"_

40

-----Impulse -- ---- Decelerating

\ \ \\\\\

\\

g' 1 (a)

o1.1_

I

m

I

I

I

\

\

\ \

% \ deg

\80

\

\

\70

\

\ \

I

Inlet-flow angle,

\ \

I

\

\ \

\

\ \

\

\ \

\

\ X

\ \

\ \

(b)

o -20

FIGURE 4-5.--Effect

114

I -30

I

I

-40 -50 -60 Exit-flowangle,o2, deg

(a) Axial solidity. (b) Actual solidity. of inlet and exit angles on optimum coefficient _= =0.8.

I

I

-70

-80

solidity.

Zweifel loading

BLADE

•9O _

Optimumsolidity from

DESIGN

Number blades

•_

. 88 •

-

o

24

o

of 64

" I 1.2

.86 .8

FmURE 4-6.--Variation

I [ 1.6 2.0 Solidity, o

of efficiency

10,--

with solidity

I 2.8

for four turbines

Reactionblades(ol = O) Impulseblades(al = -a2)

------

\

I 2.4

\

i

_I

6'_

_ .... %

I

""*-

_

Ol

The

are

with

those

that

agreement

not

good

obtained

analytical and

the

solidity time

curves impulse with

is that

mental

results

factors

that

obtained

the

exit-angle do cross

exit

angle

pertain act

3.0

and values. other

blading,

are just results

to one

to determine

angle

the

3.5

shown

for both

optimum

and

blade solidity

It is obvious results

experimental

and

can and

the

in optimum be said

assumptions,

profile,

is

(c_ = 0) blading

variations

All that

in

compared

analytical

the reaction

many

shown

are

4-5(b).

the the

indicated

similar.

involve

particular

and

coeffi(.ient.

results

4-8

in figure

Although the

not

cascade

in figure

experimental

each

analytical

from

exit

analytically

(al = -a_)

the

,,--7o

2.0 2.5 Solidity, o

against

between

for most

"_-40

of solidity and exit angle on blade-loss

solidities plotted

-60

80

1.5

optimum 4-7

_

>"

_

FIGURE 4-7.---Effe(:t

figure

I

__

4

-70

i" _

7.

Exit-flow angle, a2, deg

\ 8--

of reference

the there

in a manner

are that

at

this

experimany we do 115

TURBINE

DESIGN

AND

APPLICATION

Basedon analyticalresultsof fig. 4-_b) Basedon experimental resultsoffig. 4-7

------

Inlet-flow angle, al, deg - 2

-a 2

1-0 E -(]2

o

I

-30

FIGURE

not

yet

are

more

fully

mental

results, ¢,

such

recommended

of

Analytical

used

to

in reference

which

In the

past,

some

the

on the

modification

is suppressed The

to reduced

boundary

layer

increasing

the

the

Certain

Two

are

tandem

the

applying

alternate

blade

in references test

references The

15 and

practice

than

the

0.8

have

boundary-layer to stator

blades

blades

low-solidity

and

are presented tandem

removing

blades

jet-flap

on

marginal

better

potential

in figure as well

4--9. as the

are summarized

of low-solidity

in references

and

or

with

concepts tests

the

of turbulators

illustrated

rotor

Cascade

is one

by blowing,

perhaps,

are

separation

layer explored

treatment

10, respectively.

that

solidities,

include

by use

been have,

separation

lower

of separation

could layer

that

been

such

boundary

boundary

which

plain,

11 to 14. Turbine

rotors

are

presented

in

16, respectively. operates

of suction-surface

diffusion

minated

the

116

the

blades,

blade

at about

4-5,

experi-

occur.

region

treatments

energizing of the

do not

in the

concepts

jet-flap

tandem

Such

})e utilized

blade

concepts

with

layer

has

To achieve

losses

jet-flap

the

9 and

and

results

must

high

concepts

and

design

higher

in solidity blade.

of these

alternate

Studies

tandem,

solidity.

by suction, turbulence

success.

concept

of the boundary

approach

blade.

of the

the associated

treatment

Current

are

Blading

to reductions

surface

in blade

and

4-7.

of figure

than

1.

limitation suction

as those solidity

is slightly

Ultralow-Solidity

occurring

I -80

such

in figure

1.0,

I -10

solidities.

optimum

shown

of 0.9

optimum

results,

to determine

as those

values

I

-50 -_ Exit-flow angle,(]p deg

4-8.--Comparison

understand.

frequently

is to use

I

-40

point

on the principle is utilized of separation.

that,

(perhaps The

2),

although the

remaining

a high value

front

foil is ter-

diffusion

then

Tandemblades

FIGURE

takes

place

on the

20 to 30 percent The trailing the own

lift.

foil with

blading

a clean

mainstream

point

around

In addition,

momentum.

the

the

Figure

edge,

delivers

4-10

layer

through

perhaps

and thc

slot.

a secondary air stream main stream. This jet

trailing

jet

with

concepts.

boundary

air going

jet-flap blade operates with edge perpendicular to the

stagnation

DESIGN

Jet-flapblades

4-9.--Low-solidity

rear

of the

BLADE

thereby

some

shows

force

jetting moves

substantially to the

experimental

blade

velocity

out the the rear increasing

througl_

its

distributions

1.2 (-1 Jeton (4 percentflow) 0 Jetoff .-._- -O..[:r

1.0

_,

Suctionsurface IJ

_e'-

6I I°

4

2

Pressure

FIGURE

f_

I

__ ,

0

surface

I 20

4-10.---Jet-flap

I

I

40 60 Axialchord,percent

experimental

velocity

I

I

80

100

distributions.

117

TURBINE

DESIGN

AND

around

one such

longer

a requirement

APPLICATION

blade

with

to be equal

at the blade

a rectangular

shape,

unity.

Also,

thus

the

for solidity

sidered

only

other

load

on the

tendency

reductions. such

The

loading

coefficient

suction

the

blade

the The

determined This

from

involves

jet

connecting

flap,

to

channel

and

exit must

thc

provide

has

The

stream.

the

concepts

however,

been

reduced,

offer

the

poten-

will probably air

flow

and

the blade and

and

be con-

is required

parts

profiles

flow turning

blade

for

must

be designed.

of the

and

blade

between the

minimum

the

must the

connecting

with

spacing

geometries

transition

surface

the

itself exit

exit

efficient

The

required

selected

inlet

inlet

a smooth,

free

approaching

DESIGN

of the

profiles.

provide

now approaches

closely

is substantially

a secondary

considerations,

determination

surface

designed

is no

surfaces

cooling.

length

solidity

pressure

diagram

tandem-blade

where

as blade

chord

jet on, there

and

_ more

surface

BLADE-PROFILE After

the

to separate.

and

for applications

purposes,

on. With

on the suction

edge.

the

the jet-flap-blade

off and

velocities

trailing

with

diffusion

suppressing

Both tial

the

the jet

for the

be

blade

inlet

and

edge,

the

loss.

Exit Consideration throat,

and

the

Trailing trailing

of the suction

edge.--In edge

blade surface

17, an increase loss.

This

discussion

in chapter

significant

effect

4-11,

which

of the shows

new

exit-velocity

just

within

trailing-edge station The

example diagram

the

blade

blockage

2, which

that

at 2a include

sections

have

region.

in a higher just

been

conservation

beyond used

with

continuity

in

turbine-loss also

has

with

the

2a, which

reduced

velocity

use of figure

area

at station

blade this

trailing-edge

used. due

to the

2a than

(4-26)

_)_

at

region.

"within-the-blade"

momentum:

(p

A

is located

:

or2

a

exit region. nomenclature

of tangential

S COS

118

an increase

of the

the

obtain

smallest in

at station The

edge.

shown

thickness

blade

V,,.2, = V,,,2 and

the As

causes

as part

the

to

trailing

considerations.

will be made

is constructed

results

the

trailing-edge

effect

blade

and

in the

trailing-edge

is located

equations

diagram

addition,

blockage

throat

further

on the flow blockage

Consideration

trailing

thickness

is discussed

7. In

the

it is wise to utilize

mechanical

in trailing-edge

effect

includes the

of turbines,

with

the

section

between

the design

consistent

reference blade

exit

(4-27)

BLADE

DESIGN

Station 1

0

$

FIQU_E 4-11.--Blade

where

t is the

trailing-edge

is determined between

from

stations

are usually

thickness,

equations

2a and

small)

Mach

preceding

at

and

flow 3Iach

blockage

can cause

station

design

whether flow rate

Throat.--Since,

the

throat

design

assumes

no

change

inside a turbine

use of the

up to the throat, One

also

number

high

in general, o (see

fig. 4-11)

technique

flow

(since

at the

blade

conditions

region,

used

area, a rather

successfully velocity and

exit

a straight

the large

(station

2)

trailing-edge important

will occur

row operates

or minimum becomes

from

as is often the

an

as at station

It is, therefore, row

flow

changes

to have

angle

angle

asa

the the

be designed

the

blade

blade

flow angle

assuming

determined

choked. the

The

be

subsonic

"inside-the-trailing-edge" in

by

must

Because

be obtained.

procedure.

can

2a to become

choking

opening

sion makes

blade

cannot

the flow accelerating of the

in the

(4-27)

incompressible

2a

the

to be

the

station assumptions.

specified

or feet.

a velocity-diagram

and

is often

to determine

and

The

to produce

number

(65 ° or greater)

(4-26)

or isentropic.

equations

in meters

2 to bc either

exit angle of a2a in order 2 outside the blade row. The

section and nomenclature.

such

that

as a nozzle,

with

the determination critical

aspect

to give this diagram. suction

of

dimenIf one surface 119

TURBINE

DESIGN

between

the

obtained

from

AND

APPLICATION

throat

and

the

station

velocity

2a,

then

diagram

the

throat

at station

dimension

2a by

using

can

the

be

following

equation:

o( =

where

o is the

throat

If it is assumed throat

and

the

1

opening, that

cos a:_ 8

cos

in meters

the

velocity

"free-stream"

(4-28)

(T2a

or feet.

and

station

loss do not

change

between

the

2, then

0 -= 8

When the

this

method

angle

(4-28)

of the

and

solidities

(4-29)

applies

flow the

sonic

exit.

For

exit

(throat) perhaps, achieved additional the

case

predicted at exit

blade

condition

at the

flow

back

the

channel

occurs

channel

exit

downstream dimension

condition

i\[ach

o would

then

for expansion choking

such that

from

(4-28)

velocity,

1.3, the

low supersonic

the

be

across

flow is subsonic.

to account

supersonic

about

could

occur

at the section

a convergent-

numbers

been found that satisfactory performance is still located at the exit of the channel,

expansion

required

For

to the than

within

exit

This 60 ° and

equation

to a supersonic

be modified

throat

would

from

blade-row

than

deviation

that

(cqs.

8 compares (4-29).

greater

determined

changes

methods

by equation angles

to 35 ° . This

expands

greater

Both Reference

gradients

the

must

is obtained.

1.3), it has if the throat

as

row

numbers

be located

passage

down

wherc

dimension

5[ach

must

divergent

the

throat

from

following

those

thickness

its length.

dimensions.

agreement

dimension to the

within

this computed

not

as well as larger

throat-opening

If the

but

with

close

(4-29)

of trailing-edge

throat

of up to 5 ° for exit angles

due to lower the throat. The

angles

q2

effect

position

indicates

deviations

the

give similar

exit-flow

comparison

case,

throat

(4-29))

measured

or

is used,

COS

throat.

be computed

(up

to,

can be and the In by

this the

equation:

(4-30) o = o_ \A _} where

058

throat

opening

supersonic

computed velocity,

Act

flow area

for sonic

A,,

flow area

for supersonic

120

flow,

from

equation

m; ft m2; ft 2 flow,

me; fC"

(4-28)

or

(4-29)

for

BLADE

DESIGN

1.0

¢3

.9--

1

Rt

I

1.1 Mach

FIGUttE 4-12.--Variati(m

This

area

exit,

correction,

is shown

Suction surface

surface

be made edge

the

from

throat

such

region,

assumed

isentropic

from

throat.--The

and

trailing

considerations

5[ach

suction-surface

number

flow Mach munl_er.

flow between

throat

and

4-12.

downstream

between

1

1.4

ill flow area with supersonic

with

in figure

I

number

diffusion

on the

as structural

level (D_),

selection

edge

and

and

in the

losses,

surface

area

type

surface

integrity

associated

blade

of the

suction

of

must

trailing-

desired

level

resulting

of

from

the

design. A "straight unity)

back"

are specified

or transonic the

uses tail

problems solidity ture ably at

this

to the

curw'd

and

ref. 8), if the is small. 5[ach

surface

suction-surface is improved of remaining

and integrity

are

great. 5Iach

regions.

between

the

velocity

distribution.

curvature

region.

This

permits

some

numbers design type

and

it adds

by introducing is used,

t)y figure

(greater

0.8, than

curvatures

trailing throat

edge the

effect

(which

the

curvature

0.8),

the

has an effect

to trailing

is

effect

be lower

selected

velocity

angle of this

4-13

should

of curvature

dif-

consider-

a wedge the

is less than

In general, from

flimsy. of curva-

number

decreases

structurally

low-

amount

blading

and

Principal

preclude

some

As indicated

throat

next

utilizes

blade

The

low.

low D, values

tail of the blade,

Therefore,

number

if the

can become

trailing-edge

(,xit Mach

severe.

edge

subsonic

in the

flow acceleration

losses

the

()f the

High

to prevent

associated

blading

on the

(approximately

discussion

that

loaded

exit-flow

At higher

on loss can become suction

the

on loss is not

effect

higher

and

If eonv(,ntionally

surface

by the

in order

keep

of D,

are permissible.

of surface

h)ading

structural

exit.

edges

long trailing

throat

low values

be indicated

surface

the

when

trailing

gas-turbine

the

from

the

blade

a straight

and additional

the

long

type

conventional

between

fusion

is used

as would

of the

with designs

5lost

and

blading,

paragraph, on

design

in

for the on the

distribution edge

instead

constant. 121

TURBINE

DESIGN

AND

APPLICATION

Mach number 1.0

__o .-

j

m

_\\ L°7

.2 .4 .6 .8 Ratio ofbladespacingto surfaceradiusofcurvature

FIauR_

4-13.--Variation

of

profile

between

h)ss

throat

with and

Math

exit

numl_er

(from

ref.

and

surface

curvature

_).

Inlet The

leading-edge

than

the

exit-region

leading-edge erally

geometry

diffusion

and

numbers,

inlet

the

must blade

for thc

lead

that

inlet.

the

.\lach

through

the

blade

blading

area

With

5[aeh

arc

usually

lead5 [ach high

high

inlet

5[ach

is not so severe

and

to det('rmin('

and

The high

of suction-surface

contraction (4-26)

row.

is gen-

excessively

values

losses.

large

number and

binding,

to high

Equations

can also be used

a relatively

the

i:_creased

flow angle

circular

could

edges

can

limit

leading the

region. result

pressure-surface ellipses,

(4-27),

which

as

were

a blade-inlet

opening

to check

for blade-

number

freedom The

of

velocity

peaks

leading

edge.

permit

of the

variations or eliminate

the

leading-

remaining

and

is t:o join

the velocity

them

with

is arbitrary

selection

associated

in curvature

trailing-edge

this

velocity-distribution

curvatures

Blade-Surface Once

specified,

large

portions

which

edges

in undesirable

be used to minimize

122

inlet,

less critical

choking.

leading-edge

task

blade because

of low-reaction can

be taken exit,

row is usually

for low-reaction

toward

at the

bIade

case

region

"within-the-blade"

Although and

the

inlet

a tendency

care

to choke and

In

in the

increases

concern

blade

the

be used,

and then

a serious

blading.

velocities

At

earl usually

low at the inlet

number

used

geometry.

radius

ing edge bccomes

of a turbine

with

on both Other

around

in

circular the

leading

suction-

geometries, the leading

the

such edge,

and as can

peaks.

Profile geometri('s a profih_ that,

have

been

yields

the

selected, required

the flow

BLADE

turning

and

desigu

a satisfactory

procedure

must

to an accuracy Two

sufficient

of the

major

Velocity

gradients

pressure

surface

turn

the

flow

occur

as a result

of these

factors

influence

The

channel

and

of the

theory

from

Pressure

_ Suction

surface7

_su rface

The rows

in the

figure

difference and,

considerations.

serves

to

the

required

to

therefore, Since

distribution,

as the basis

available

4-14.

suction

position

velocity

limits).

both

the design

of a quasi-three-dimensional

that

ncxt

blade.

the blade diffusion

illustrated

channel

programs

in the

(e.g.,

static-pressure

least

the

through

in streamlinc

be at

computer

are discussed

arc

the

of radial-equilibrium

should the

around

controls

the blade-surface

flow analysis

procedures putations

across

variations

velocity

used

design

considerations

occur

Radial

distribution

the flow conditions

to impose

as a result

flow.

procedures

velocity

describe

DESIGN

nature.

for these

to perform

the

design com-

chapter.

..-Pressure surface

Suction surface_..

Cross-channel

distance

(a)

__

, 'lEi.:_.i:!_._.:._::..::::::_i_!| Tip : _i_ Rotorf_ii!i|

-'Nl ,

"'- Tip

_

_I

°--

NN

¢";1

,-Hub

_:: /

Velocity (b)

(a)

Cross-channel (b)

FIGURE

4-14.--Turbine

Radial

variation. variation.

blade-row

velocity

variations.

123

TURBINE

DESIGN

AND

APPLICATION

REFERENCES 1.

O.:

ZWEIFEL,

Angular 2.

LIEBLEIN, sion

The

STEWART,

Blade

Paper

Effective

RM

E56B29, Y.;

for

a Series

Four 6.

HELLER,

E56F21, A.;

Investigation

8.

AINLEY,

l).

G.;

BETTNER,

RM

Loaded

10.

H.

cepts

Designed

2, Apr. 11.

STABE,

ROY

NOSEK, tion

13.

NOSEK,

STABS,

U.;

AND

Gt.

E55BOS,

of

Turbine-Blade-

WARREN

WHITNEY,

by

Parameter

Cascade

and

for

1955.

Changes

in

J.:

Blade

RICHARD

Turbine

at

R.:

An

Analysis

Geometry.

H.:

Experimental

Four

Rotor-Blade

Examination

of

Turbines.

Rep.

Axial-Flow Britain,

the

Flow

R&M

and 2891,

1955. M.

:

Summary

of Tests

in Three-Dimensional

R.

J.:

Turbine

Some

on

Cascade

Experimental

Blade

Two

Highly

Sector.

Paper

Results

Loading.

J. Eng.

Ratio

Cascade

to Spacing

KLINE,

JOHN

Blade

Blade. and

with

Two-Dimensional Chord

of Two

Power,

NASA

JOHN NASA

of 0.5.

F.:

Chord

of

TM

vol.

Con92,

Turbine

no.

X-1836,

X-2183,

Stator

X-1991,

Cascade TM

1970. Investiga-

1969.

Two-l)imensional

TM

Two-Dimensional of Axial

Test

NASA

F.:Two-Dimensional

Design.

AND KLINE,

Rotor

l)_ign

Blade

of

1969.

and

Tandem M.;

ROY G.:

Use

Losses.

Diffusion

Two-Dimensional

AND

STANLEY

Nov.

of Axial

Turbine

Stator

C.

of

Concepts

])esign

Ratio

STANLEY

W.:

198-206.

of a Turbine

Jet-Flap 14.

pp.

STANLEY

JAMES

Rotor-Blade

with

AND CAVlCCHI,

G.

AND ROELKE,

G.:

with

Ex-

Diffusion

1952.

Rows

to Increase

1970,

Blade 12.

G.;

R.:

Using

Correlation

Designed

NOSEK,

ASME,

LUEDZRS,

L.;

L.;

Council,

Blade

69-WA/GT-5,

ROSE

Blade

; AND

Turbine

in

Affected

MATHIESON,

in

L.

L.:

RM

WARNER

E52C17,

Research

JAMES

MICHAEL

MISER,

Turbine

Thickness

NACA

Conservatively

AND

Aeronautical 9.

a

Losses

Diffu-

1956.

NACA

Pressure

AND

])escribing

Blades

Losses

WHITNEY,

of

Solidities.

L.:

in Axial-Flow-

1967.

J.;

WARNER

Rotors.

Viscous

RM

Large

436-444.

1953.

AND VANCO,

Nov.

Momentum

Turbine

STEWART,

JACK

pp.

ROBERT

Loadings

Characteristics

ASME,

on Wake

Turbine W.;

with

1945,

1956.

of Subsonic

JAMES

J.;

WARREN

AND STEWART,

Turbomachine

NACA 7.

Blade-Lolling

in

Blade

E531)O1,

Turbine

WHITNEY,

Based

Transonic

MISER,

of

Losses

RM

Especially 12, Dec.

AND BEODERICK,

ARTHUR

Thickness

ROBERT

Element

NACA

32, no.

Limiting

GLASSMAN,

Momentum

NACA WosG,

L.;

C.;

and

67-WA/GT-8,

WARNER

vol.

FRANCIS

Elements. L.;

Blading,

Rev.,

Losses

of Axial-Flow

Parameters.

5.

SCHWENK,

WARNER

STEWART,

Turbo-Machine

Boveri

for Estimating

amination

4.

of

Brown

SEYMOUR;

Factor

Compressor 3.

Spacing

Deflection.

Cascade

Test

of a

1971.

Cascade

Test

to Spacing

of a Jet-Flap

of 0.5.

NASA

TM

Turbine X-2426,

1971. 15.

16.

BETTNER,

JA_ES

Solidity

Tandem

BETTNER,

JAMES

Solidity, 17.

PRUST,

Jet

ing.

124

NASA

and TN

l)esign Rotor.

L.:

Flap

HERMAN

Geometry

L.:

l)esign Rotor.

W.,

Jm;

and

Experimental

NASA and

on

1)-6637.

1972.

Results

CR-1968,

AND the

HELON,

of a Highly

Loaded,

Low

of a Highly

Loaded,

Low

1971.

Experimental

NASA

Thickness

Results

CR-1803,

1972. RONALD

Performance

M.:

of Certain

Effect

of

Turbine

Trailing-Edge Stator

Bind-

BLADE

DESIGN

SYMBOLS A

flow area,

C

chord,

D

diffusion

F

force, N; lb conversion constant,

g K

ratio

m_; ft 2

m; ft parameter

of inlet

o

throat

P R

absolute reaction

8

blade

t

trailing-edge

V

absolute

X

axial

o_8

1; 32.17

to exit tangential

opening,

(Ibm)

(ft)/(lbf)

components

(sec 2) of velocity

m; ft

pressure, spacing,

N/m2;

lb/ft

_

m; ft thickness,

velocity,

distance,

m; ft

m/sec;

ft/scc

m; ft

fluid absolute

angle

from

axial

direction,

deg

blade

angle

from

axial direction,

deg

stagger

ratio of specific heat constant volume density,

( V_ ._/V,._)

kg/m'_;

lb/ft

at

constant

pressure

to specific

heat

at

a

solidity

_bz

loading

coefficient

defined

by equation

(4-6)

loading

coefficient

defined

by equation

(4-5)

Subscripts" cr

critical

inc max

incompressible maximum value

rain

minimum

opt

optimum

p s

pressure suction

ss u

supersonic tangential

x 1

axial component blade row inlet

2

blade

2a

within

value

surface surface

row

component

exit

trailing

edge

of blade

row

Superscript" '

absolute

total

state

125

CHAPTER 5

Channel FlowAnalysis ByTheodore Katsanis

The

design

of a proper

blade

profile,

chapter

4, requires

calculation

determine analysis

the velocities on the blade theory for several methods

of the

discusses associated computer Lewis Research Center. The

actual

cannot

velocity

be calculated viscous,

passages.

To calculate

simplifying simplified surfaces Similar mean

distribution

surfaces. This used for this

to flow

on or through

hub-to-shroud shown

directly,

but

stream in figure

provides

surface

extreme

to

at

the also

NASA

flow

field

complexity

geometrically therefore,

two-dimensional

of

complex certain

three-dimensional

5-1 (a),

(commonly

does

information

There

are

tribution

two

over

formulation

one

of the

mathematical we

velocity-gradient

parts

will

to a method

of these

flow

surfaces.

and

problem. discuss

the For

stream-

(stream-filament)

yield

The second

is

Such

and

is the

is the

mathematical The

surface which

yield

a velocity

dis-

mathematical

numerical

solution

formulation

potential-function

methods.

velocities

solutions,

part

part

meridional

blade-to-blade

to obtain

first

the

blade-surface

5-1 (c))

of analysis

the

called

for the

(fig.

surfaces.

problem,

not

required

(fig. 5-1 (b)) and orthogonal surface the desired blade-surface velocities.

problem,

in order

are illustrated in figure 5-1 for the case of a radial-inflow turbine. surfaces are used for an axial-flo(v turbine. A flow solution on the

surface),

of the

The

various

of

developed

distribution,

made.

field

section

a blade-row

of the

velocity be

last

chapter presents calculation and

were

flow through

a theoretical must

that

because

three-dimensional

assumptions

flow

throughout

time

in the

blade-row

programs

at this

nonsteady,

as indicated

stream-

of the

methods and

and

potential-

TURBINE

DESIGN

AND

APPLICATION

Blade-_,-blade

Hub-to-shroud stream surface

-_ |

surface] I

(a)

0_)

,- Ortho_nm

(c) (a) Hub-to-shroud

stream surface. (b) Blade-to-blade (c) Orthogonal surface across flow passage. FIOURE 5--1.--Surfaces used for velocity-distribution calculations.

128

surface.

CHANNEL

function

methods

will be described

solution.

A similar

type

The

of analysis

velocity-gradient

for solutions The

equation

on any of the

following

of analysis (1)

The

surface

flow

Thus,

if the

fixed

coordinate

(2)

The

blade-to-blade

surface

for the meridional is general

surface.

and can be used

are

made

in deriving

the

various

This

means

methods

herein:

at any

blade

can be made

ANALYSIS

surfaces.

is steady

velocity

to the

to be presented

assumptions

discussed

relative

FLOW

relative

given

point

is rotating,

the

to

the

blade.

on the blade flow

would

does not

not

that

vary

be steady

the

with

time.

relative

to a

system.

fluid obeys

the

ideal-gas

law

p=pRT

(5-1)

where p

absolute

pressure,

p

density,

kg/m_;

R

gas constant,

T

absolute

The

lb/ft

lb/ft

2

S

J/(kg)

(K);

temperature,

or is incompressible (3)

N/m_;

(ft) (lbf)/(lbm)

K; °R

(p = constant).

fluid is nonviscous.

A nonviscous

The blade-surface velocity is calculated, extends to the blade surface. (4)

The

fluid

(5)

The

flow is isentropic.

(6) inlet.

The

total

(7)

For

the

assumption

has

a constant

heat

temperature stream-

is made

and

and

that

Y is the

absolute

do

particles

total

vector.

not

change

their

absolute

change.

For

example,

particle

at

times

t and

t-t-At.

particle

changes

blade,

rotation the

is zero. particle

has rotated,

In

and

Of course,

layer.

free

stream

are

uniform

analyses,

across

the

irrotational.

the

shape

additional

because

(5-2)

Intuitively,

orientation figure

this with

5-2

means

time,

shows

absolute

frame instant

of time,

of reference

relative

the

frame

that

although

a hypothetical

at a later

in a frame

the

Therefore,

Y = VX V =0

may

net

no boundary as if the

pressure

flow is absolutely

velocity

its location

has

capacity.

shape

their

fluid therefore,

potential-function

the

curl where

(°R)

of reference,

of reference

the

but

the

to the

has rotated.

129

TURBINE

DESIGN

AND

APPLICATION

Direction of rotation

Time = t

r-l LJ

Time-t+At

I

I

(Absolute frame of reference)

Time=t + At (Relativeframe ofreference)

FIGURE

Some also

numerical

techniques

be discussed.

techniques excellent given

However,

for solving theoretical

in Chapter

STREAM-

5--2.--Absolutely

irrotational

for solving it must

the

simplest cascade 130

stream

IV of reference

AND

be emphasized

equations

that

there

can

be

of streamlines.

in figure

5-3.

only

will

are many a few. cascades

An is

1.

POTENTIAL-FUNCTION

function

is in terms as shown

mathematical

these equations, and we will discuss discussion of flow in two-dimensional

Stream-Function The

flow.

ANALYSES

Method

defined

several

Suppose

It is assumed

ways,

but

we consider that

there

two

perhaps blades

is two-dimensional

the of a

CHANNEL

FLOW

ANALYSIS

1 .8 Mass

.6

flow fractbn

.4

0

\\\\

FIGURE

axial

flow here,

5-3.--Streamlines

so that

the

there is no variation of the rotation about the centerline. Shown the

in figure

blades

passing line.

Thus,

the lower streamlines

the

radius

r from

flow

in the

5-3 are a number

is w. The

between

for a stator

number

the upper upper

surface have

surface

centerline

radial

streamline

of the lower

(which

The

is a streamline)

It will be recalled

There

mass

indicates blade

and has

and

may

be

flow between

the fraction the given the value

of w

stream0, and

of the upper blade has the value of 1, while the remaining values between 0 and 1. Note that a value can be asso-

ciated with any point in the passage. This value function value and can be used to define the stream (or uniform)

is constant

direction.

of streamlines.

by each surface

the

cascade.

that

mass

flow can be calculated

is called function.

the

stream-

for a one-dimensional

flow by w= pVA

(5-3) 131

TURBINE

DESIGN

AND

APPLICATION

where W

rate of mass

V

fluid absolute

A

flow area

This

flow,

kg/sec;

velocity,

normal

can be extended

lb/sec m/sec;

to the

ft/sec

direction

to a varying

of the velocity

flow by using

V, m2; ft _

an integral

expression:

w= f a pV dA Since

this stream-function

cascades relative

(blade velocity

analysis

rows),

the

W, which

velocity V. We will assume the mass flow wl._ between fig. 5-4)

can be calculated

fluid

applies velocity

(5-4) to both will

be

stationary

and

expressed

in terms

reduces

blade

to absolute

that any

has a uniform height b. Then, QI and Q2 in the passage (see

by QI

wl._ --/Q

132

5-4.--Arbitrary

(5-5)

_ pW,,b dq

-'_Q1

FIGURE

of

for a stationary our cascade two points

row

rotating

•

curve

joining

two

points

in

flow

passage.

CHANNEL

where

Wn is the

hand

normal

relative

of the line going

that

wl.t will be negative

The

integral

dependent With

velocity

is a line of path

from

for steady

function

in the direction

QI to Qt. This

if Qs is below integral

the use of equation

for the stream

component

the

flow relative (5-5),

points

of the right-

passing

means

through

Q_ and

Q2 and

Q1. is in-

to the cascade.

an analytical

u at a point

ANALYSIS

sign convention

a streamline

between

FLOW

expression

can be written

(x, y) :

/Q(_'Y) pWnb dq o

u(x,

(5-6)

y) = W

where integral in figure Since tively calculate is still

Qo is any

point

on the

is taken

along

any curve

upper

surface

between

of the

Qo and

lower

(x, y).

blade, This

and

the

is indicated

5--5. the

easy

integral

in equation

to calculate Ou/Ox

at the

the

partial

point

in the flow passage,

(5-6)

derivatives

(x, y).

as shown

Let

of path,

of u. For

Xo<X such

in figure

/c_ pWnb u(x,

is independent

5-6.

that

it is rela-

example, the

we will

point

(x0, y)

Then

dq-}- fc ffipW_b

dq (5-7)

y) W

FIGURE

5-5.--Curve

joining

(x,y)

with

a point

on the

upper

surface

of the lower

blade.

133

TURBINE

DESIGN

AND

APPLICATION

(Xo.Y)ff c2 (x,y}

FIGURE 5--6.--Curve

joining

where

C_ is an arbi'_rary

zontal

line

depend

between

on x. Along

horizontal line through (x,y) surface of the lower blade.

curve

(Xo, y)

between

and

Qo and

(x, y).

C2, we have

The

W, = -W_

Ou (x, y) = --

a point

on the upper

(x0, y),

and

C2 is a hori-

integral and

pW_b o

Ox

with

along

C_ does

not

dq = dx. Hence,

(5-8)

dx

w

or

In a similar

manner,

Ou

aW,,b

Ox

w

we can calculate Ou_ Oy

Now From

we will make the

definition

curlV=fOV.

wherei, 134

j, and

\or

(5-9)

use of the fact of the

curl

OV=_ i

k are the

unit

pW_b w that

operator [OV=

vectors

the and

OV._

in the

(5-10)

flow is absolutely the

above

irrotational.

assumption,

+(OV_

OV,_'_k=O

(5-11)

x, y, and

z directions,

respec-

\ox

ov/

CHANNEL

tively,

and

m/sec

Vx,

V_, and

or ft/sec)

considering

in the

V,

are

the

x, y, and

two-dimensional

absolute

FLOW

velocity

z directions,

ANALYSIS

components

respectively.

Since

(in we are

flow only, V.=O

(5-12)

and OVa_ Oz Hence,

equation

(5-11)

requires

OV:,=O Oz

(5-13)

only that OV_

OVx

Ox

Oy

(5-14)

Since V_= W_

(5-15)

Vu-- W_+_r

(5-16)

and

where

_ is the

equation

Actually, in this

angular

(5-14)

the

speed

(in

can be expressed

case.

Now,

and

in terms

of relative

OW3,

OW_

Ox

Oy

flow is irrotational

particular

rad/sec)

with

from

radius

r is constant,

velocities

as

(5-17)

respect

equations w

the

to the (5-10)

moving and

coordinates

(5.9),

0_t

W_ -

(5.18) pb cOy w Ou

= Substituting

equations

(5-18)

and

0 (1 0u\ cox; since

w and

b are both

For incompressible

(5.19)

pb Ox (5-19)

into

; oy/

0 (10u_=

equation

(_-17)

0

yields

(5-20)

constant. flow,

p is constant, 0_U

and 02U

(5-91) which

is Laplace's

equation.

called

a harmonic

function.

Any There

function

satisfying

is a great

deal

Laplace's of theory

equation

is

concerning 135

TURBINE

DESIGN

harmonic

functions

complex

AND

that

important

thing

number

of functions

solution

that

equation

specifying

two

the

The

entire

first

in every figure

5-8.

and

function boundary along

that

HG, HG

satisfy

of analytic

we along

functions

that

the

angle

we can

ABCDEFGHA. CD,

and FE,

AH

part

will (2)

be

a

to either by

condition

Since

region

far

upstream

_0,_ is known. BC,

u--0;

where

than

that

flow

From

boundary

the

distance

flow

way

the

AB and CD.

cascade.

stream entire of u Along

direction

x

infinite

along

value

Wx

5-7.--Two-dimensional

the

u= 1. Along

is, the in the

in

angle

on the

FG,

that

it is along

_ is the

so that the

conditions

exists;

same

as shown

is uniform

and along

condition

1 greater

and the

A typical

the flow is the

solution

boundary that

region.

yl

FIGUI_E

find

determined

a boundary

solution

is sufficiently

specify

Along

is known,

5-7.

a finite

of the

a periodic

and FE is exactly

is the

in figure

it is assumed

flow

we must

region.

be specified consider

Similarly,

AH and DE, Ou/O_ the outer normal.

136

of

a tremendous

The solution

(5-20) and

are and

conditions.

region,

is shown can

there

(5-21),

equation

of the

this

defined,

equation

a finite

must

cascade

is that

boundary or

(1)

It is assumed

was

know

certain

that

passage,

is known.

AB,

here

boundary

thing

is uniform

DE,

to

(5-21)

things:

two-dimensional

/_

to the theory

that

satisfies

Laplace's

flow

is related

variables.

The

along

APPLICATION

of

CHANNELFLOWANALYSIS

H

6/

"_'. E

Uniform flow

Uniform flow

n

L A

y

W

B P

_-_x

_ C FIGURE

Consider

the

differential

5-8.--Finite

solution

of u in the direction

du = OxO'-u dx+_ The

differential

streamline, Along

is 0 because

and

the

region.

velocity

the

of the velocity

W:

dy -- 0

stream

(5-22)

function

vector

must

Ou

8u

Or -

8x

be

is constant

tangent

to

along

a

a streamline.

AH,

and substitution

from

equation

(5-22)

(5-23)

yields

Ou_ 07

Ou dy Oy dx

(5-24)

du -"-dx

tan _

(5-25)

However,

Further, uniform

Ou/Oy there.

is constant

along

AH,

since

it is assumed

(5-25)

s is the and

the

flow is

Therefore, Ou [u(H)-u(A)'] .... 8y s

where

that

blade

(5-26)

spacing in equation

in the (5-24)

1

(5-26)

s

y direction. gives

along

Substituting

equations

AH

(5-27) ia

8

137

TURBINE

DESIGN

Similarly,

AND

along

DE,

APPLICATION

one can calculate

o,,, = We now have

a boundary

condition

shown

in figure

5-8.

unique

solution

to Laplace's

(5-20)),

a unique

subsonic

throughout

There After

This

problem

is always

can

for solving

will be discussed velocity

as the later.

If lines

by

The

as the stream

determine flow

flow

a (eq.

is strictly

differentiation

indirect,

function,

If the potential function be defined so that

¢ exists

blade

surface

stream

of solving

problem and

distribution.

a

potential

is to

from

this

This

will

drawn,

they

function

properties

(i.e.,

can

be

will be orthogonal

will not be defined

but the main

of the

A method

Method

are

function

or (5-21). and velocities

or inverse,

this velocity

flow,

potential

potential

(5-20)

velocities

problem.

on the

will give

irrotational

of equal

The

if the

equation

direct

distribution that

two-dimensional

streamlines.

of the region

compressible

blade-surface

Potential-Function

defined.

For

determined

be obtained

is known

determine a blade shape not be discussed here.

For

boundary will always

(5-21).

is obtained,

passage

a desired

the entire

conditions

equation

techniques

is what

(5-28)

region.

function

the

function. specify

the

along

boundary

solution

are numerous

the stream

throughout this

These

s

in the

and relations

same

to detail

will be given.

the flow is irrotational),

then

it can

-- = Vx Ox

(5-29)

- Vv

(5-30)

and

Oy We will refer tional

relative

assumption system for pure

to absolute to

the

of absolute

does not rotate. axial

flow,

since

velocities coordinate irrotational This the

here,

since

system flow,

does not exclude rotation

we must

have

This,

coupled

used.

implies

the

continuity

relationship

for steady

o(pvx) +o(pv,) =o Ox 138

Oy

the

with

if there respect

the

coordinate

use of the potential

has no effect

radius; that is, the flow is actually irrotational with as we saw in the discussion of the stream function. From

that

flow irrota-

function

is no change to the

blades,

flow, (5-31)

in

CHANNEL

Substituting

equations

(5-29)

and

(5-30)

FLOW

in equation

ANALYSIS

(5-31)

yields

0(oo,o() If the

flow is incompressible,

p is constant, a_

and

a2_

=0 So,

the

potential

pressible,

irrotational

function

satisfy

difference We

function flow,

the

same

lies in the

can consider

specify BC and

Laplace's

both

the

stream

differential

boundary the

boundary

satisfies

same

equation

(5-33) equation.

Thus,

function

and

(Laplace's

for incom-

the

potential

equation).

The

conditions. solution

conditions

over

region

the

entire

shown

in figure

boundary

5-8.

as follows:

We can Along

FG, -

V, = 0

(5-34)

0_ where

V, is the

velocity

normal

to the blade

surface.

Along

AH, (5-35)

and

along

DE,

oue

The

inlet

and

outlet

axial

velocities

are given

by the

equations

W

(v_) ,. -

(5-37)

p_nbs

and W

(V_) o_t-

Along

AB,

uniform

GH,

along

CD,

and

EF,

a periodic

(5-38)

po_tbs condition

exists.

Since

the flow is

AH,

(0yy_),.,=

[_(H)-

_(A)Is

- (V_) _,

(5-39)

Substituting V_ = V: tan/_

(5-40) 139

TURBINE

into

DESIGN

equation

AND

APPLICATION

(5-39)

yields _(H)

Because than

of the

along

=¢(A)q-s(Vx)_,

periodicity,

AB.

¢ is exactly

Similarly,

at the

O(E) Equation This

(5-42)

gives

completes

The

boundary

but

only

the

+s(Vx)o,,,

difference

boundary

point,

_,

greater

along

HG

these

the

lines

for equation

do not

FE

and

(5-32)

determine

additive

boundary

(5-33)

(5-42)

#o_,

in ¢ along

an arbitrary

to equation

tan

tan

conditions

however,

within

at one

solution

s(Vz)_,_

(5-41)

_,,

outlet,

= ¢(D)

conditions,

a solution

¢ is specified unique

the

tan

a unique

constant.

solution,

If the

conditions

for incompressible

CD.

or (5-33). value

of

will determine flow,

a

or to equation

(5-32), for strictly subsonic compressible As for the stream function, there are

flow throughout the region. numerous methods for solving

equation

preceding

ary the

(5-32)

or (5-33)

subject

conditions. A method velocity distribution

references

2 and

to the

for solving to determine

of Stream-

choice (the flow)

of the

three

is not

of stream

function

The

existence

of the

equation. For a line between

the flow be either

tion

is necessary flow was

relative

This of

blade

could function

can easily mass

may

be

function

flow

or steady.

irrotational, the

restricted

is proven

flow to be unique.

for

or incomas to the

function.

incompressible

for the

absolutely

to the

assumptions the stream

we

irrotational,

the stream function to be defined, two points must be independent

that the

(steady,

then

or potential stream

there is little to function. In this

of ease of solution for the boundary is the same: Laplace's equation).

assumptions

applicable,

choice

Method

and incompressible, and the potential

is made on the basis differential equation

if any

pressible

problem of specifying shape is described in

or Potential-Function

If the flow is steady, irrotational, choose between the stream function

However,

the inverse the blade

bound-

3.

Choice

case, the conditions

or equivalent

used

turned

case

the

the mass of path. Some

We

which

axial-flow

from

continuity

flow This

crossing requires

additional the

out

assump-

assumption

that

to be irrotational

considered.

However,

other

be made for other problems. Another restriction is that it can be defined only for two-dimensional

be seen

since

between

two

the stream points,

function and

this

is defined

on flow.

as a percentage

is meaningless

in

three

dimensions. The

existence

irrotational 140

of the

relative

potential

to the

given

function coordinate

can

be shown system.

This

if the

flow

is necessary

is

CHANNEL

because

we must

have

equality

of mixed

second

FL_OW

partial

ANALYSIS

derivatives;

that

is, if 02_

02& (5--43)

m

OxOy

OyOx

then

0 V ----_ Ox _-Oy and

the

flow

must

be

dimensional

flow;

that

irrotational

with

respect

an

assumption

done

is, the

must

by using

the

irrotational.

continuity

Finite-Difference As

stated

posed

by

detail

flow.

rate

In this case,

region

first

step

shown at

typical 5-10. points

The

mesh

point

point

hi, and

in figure tively.

each

5-10. With

equation

mesh

points.

When

this

2ul h_(h_Th2)

finite

is done,

the

2u_ 4-h2(h_+h2)

can

be

can

grid

stream

neighboring

mesh

is labeled

series

distance

following 2Uo]

of this

[

but

2U3

to the

bound-

The

method

with

a lower

points

in figure (eq.

5-9.

(5-21))

function

in the Then

a

can be

is unknown.

points

is shown

the

four

between

A

in figure

neighboring

points

0 to 4 are labeled

expression

for the

irrotational

are h2, h3, and

expansion

be approximated

explanation

problem

of mesh

0, and

distances

of u at points

consider

solution.

equation

The

problems

We will

similar,

is shown

the other

various

function.

grid the

Method

subject

to Laplace's

1 to 4, as shown.

(5-21)

equation

is quite

where

four

similarly,

(Further

Finally,

This

incompressible,

on the stream

A typical

point

value

flow is

used.

of the direct

difference

in consideration

The

being

theory.

a rectangular

the use of a Taylor

tions,

Laplace's

function

5-8.

with

in three-

if the

solution.

of solving

solution

solve

for the

are labeled

denoted

ways

case of steady,

approximation

mesh

only

Stream-Function

in the section

potential

in figure

system a unique

for many

is to establish

finite-difference written

are

we must

for the

exists

exists

or potential-function

discussed

of convergence

The

there

for the simplest

ary conditions of solution

coordinate

the finite-difference

function

function

to assure

Solution

before,

situation

equation.

stream-function

in further stream

potential

made

(5-44)

A similar

to the

be

V.

1 and

0 is

h4 as indicated u0 to u4, respec-

for u in the x- and y-direcby

using

is given

only

values

in ch.

of u at

6 of ref.

4.)

is obtained: 2U4

+h_(h_+h4)

2Uo ] _j=O (5-45) 141

TURBINE

DESIGN

AND

APPLICATION

N I ] Ii-'_ "_ _- .."

. !

_

"

!

\

1

li, ii ¸ I I

_i

fl II, IJ.-"_

--'-..I

[I \_, i\ I .

N

_

ilII i, I

1

_\! " '

1

"

!

i

;

I

r A-

•

_

e

1

,l

"F

'_\{

j

E Jl

il ,II II

i

: !

.,

Ill LI] Lil

!

\\ :

I

I

I

I)

FIGURE

5-9.--Mesh

used

for

a finite_lifference

solution.

h2

h3

0

D

h4

,4

1

FIGURE 5-10.--Notation

142

for adjacent

mesh points and mesh spaces.

CHANNEL

Solving

equation

(5-45)

for u0 yields

the

FLOW

ANALYSIS

expression

4 UO=

E i-1

aiui

(5--46)

where h 3+ h4 al= a0h--_

(5-47)

h3--_- h4

= --

(5-48)

aoh_

hi+h2 a3 --

(5-49)

aoh3

hl+h_ a4 -

(5--50)

aoh4

1 1 ao=(h3Th4)(_T_)+(h.+h,)(_+-_,)

Equation

(5-46)

boring

points

used.

At other

but

the

these

points

boundary

points.

Ou/O_

holds

along

can be used along

by equation

if point

the

the

periodic

boundary

that

1 in the

of the

neigh-

point

can be

(5-46)

to obtain

alternate

boundary

cannot AH

0 is on line AH,

be used,

equations

at

in figure

5-9,

then,

a finite

gives

(5-52)

(tan-Bin)

0 is on line DE,

along

boundary between

it is known point

points

If one

of u at that

equation

If point

Uo= u3-

For

point.

value

the upstream

(5-27).

Uo= u4+h4 Similarly,

the

the boundary,

conditions

approximation

mesh

then

1 (5-51)

interior

surface,

For example,

is given

difference

at every

is on a blade

1

AB

CD,

condition.

A and ul=ul.s-1, y-direction,

and

h3 (tans-_°"'

If

B, the point where as shown

(5-53)

)

equations

can be derived

the

0

point

I is outside the

point in figure

(fig.

5-11)

the boundary.

by

is on

the

However,

1,s is a distance 5-11.

using

Substituting

s above this

143

TURBINE

DESIGN

AND

APPLICATION

H

2 31

I

G.

I

u2 "u2,-s

A

2,-s

FIGURE 5-11.--Mesh condition

+1

in equation

(5-46)

point on line AB.

gives 4

uo=alui..-I-

_,

aiui--al

(5-54)

i--2

This

equation

The

points

greater

than

along

CD

along

HG

need

the

mesh

line below

HG.

In

below

holds

this

HG,

this

therefore,

equation

One mesh

point

interest knowns.

These

tions

144

values

so forth The

the

where

there

first

mesh to

stream

up to un at the

last

at a typical

point. point,

distance

in figure

s

5-12.

(fig. 5-8).

be applied

to each

is unknown

in the

region

equations

as there

are

points.

can be numbered

be ui at the

2 is on line

is a

FE

can

function

mesh

1

(5-55)

line below

of linear

are n unknown points

point

2,-s

just

gives

(5-55)

points.

The

since

are

for the first

aaus _ a4u4 -{- as

function

the

they

equation

as indicated

mesh

of u will then

equation

-_-

(5-52)

number

The

point

(5-46)

,_.

stream

as unknown

in n unknowns.

n. The apply.

that

or

same

points

to simply

Suppose

to the

(5-46) the

the

y-direction,

a_Na

since

AB.

be modified,

in equation

applies

for which

to give

referred

and

also

of equations

considered,

where

negative

condition

also.

along

must

Zto = alltl-_

This

be

point

us=u_._._-l,

2 in the

Substituting

not

corresponding

case,

point

(fig. 5-8)

At each

We then consecutively

first point, i, could

is unknown

will

have

point,

one

be

n equafrom

us at the second

be written

of un-

equation

I to

point, will

CHANNEL

1,2

H

FLOW

ANALYSIS

G

/

Ul "Ul, s - 1

FIGURE 5-12.--Mesh

point on first line below HG.

_"_aijuj

The

values

(5-55).

of the

All but

value

aij are determined five,

around

the

singular;

hence

there

of linear

These

of this

unknowns,

but

small,

roundoff

iteration,

few an

are type;

terms

initial

estimate

The

simplest

iterative

the

estimated

value

the

The

procedure

ever,

the

time the

equation

is simple

change When

The

w= 1, the

underrelaxation)

It

it always

iteration procedure is proven

is convergent occurs

when

After

this

at every

change

in the

for this

so that _, called

is straight

relaxation,

in reference

1 < _ < 2. In fact,

4 that

However, there

point

the

of

point, values

the of u. How-

computer

by increasing overrelaxation and

when

_:> 1,

overrelaxation the

is

so as to

problem.

greatly

start

consists

exci_ssive

by a factor

if 0 <:_ <2.

This

are To

mesh

is done

can be accelerated

of

requirements

in succession

converges slow,

systems number

methods.

unknown

point

non-

by iterative

in solving

is relaxation.

each

points

is always

are a large

iterative

every

The

= --1.

uj.

Storage

with

aii

be obtained

there

is negligible

is extremely

convergence

in u at each

it is overrelaxation. convergence

and rate

for the

equation.

procedure

there

the

through

unknown

valuable

is, where

of u at

and

aij matrix

can

of u at

point.

until

convergence

is required.

factor.

for that

is repeated

the

solution

is minimized

changing procedure

that

particularly that

(5-47)

outermost

(5-56)

in each

error

zero,

the

a unique

required. satisfy

for

to equation

techniques

equations and

is always

of equations

aij are

except

It can be shown

solution

techniques.

by one

of the

zero,

boundary.

A numerical

the

at most,

of ki is always

(5--56)

_-ki

greatest

is an optimum

(or rate

value

of of 145

TURBINE

DESIGN

between

AND

1 and

2 which

overrelaxation To give

APPLICATION

factor

gives

an explicit

use a superscript estimates

are

the

most

expression

on the denoted

ul. That

of ui°=0

calculate

ui re+l, for i = 1, 2,...,

--

overrelaxation

be any Then,

a solution

method), tions

for

Z

aijui

and

of ui. The example,

initial

an initial

for all i, we can

aij_J

m'gf-]gi-

(5-57)

uim

j-i-i-1

by

to calculate

(5-10)

For

we will

by

re+l-

u is obtained

it is necessary

(5-9)

value.

n in succession

optimum 4.

procedure,

if ui m is known

j-I

After

This

in reference

is, ui m is the m th iterate may

is satisfactory.

uim-_-0J

convergence.

as explained

for the

ui ° and

estimate

uim+l-_-_

rapid

can be calculated

the

overrelaxation velocities

(or

with

the

any

other

use of equa-

as

w(OZ) (5-58)

Yz-

pb and

o W_ =

The

partial

calculated

discrete

differences, the

values

or by

points.

The

at unknown from

fitting

mesh

As can

be seen,

of velocities puter.

which mixed

solution

analysis In

shown

in reference aqcordance

stream-function

method,

solution

The

146

region.

from the

the

been

flow must

TSONIC

program,

by finite

curve,

through

two

components is calculated

Analyses

The

and

the

which

is best

written

at the

through

5, can be used

the

the

either

velocity

equation

of these

the

from

angle.

procedure

in fig. 5-9). with

done,

Stream-Function

of flow

Most

estimated

as a spline

surface,

tangent

have

analysis

methods.

is described flow.

programs

(region

such

of Laplace's

calculation

for the

stream-function

blade

for

be

can be readily

blade

blade

Programs

computer

Center

the

must

is calculated

On the

and

the

au/Oy curve,

velocity

is a lengthy

Several

Research

a smooth

points.

one component

and

of ui. This

resultant

Computer

by

au/ax

derivatives

(5-59)

pb

are

program to analyze

be subsonic described

done

by com-

NASA

turbomachine

programs

constraints

calculation

for

called axial,

associated throughout in reference

Lewis blading

blade-toTURBLE, radial, with

or the

the entire 6, super-

CHANNEL

sedes

TURBLE

addition, flow

extends

the

problems.

gradient

called

analyze

8, obtains blade and

a detailed slot

MAGNFY

Flow indicated

by

fig.

MAGNFY

solution

in the

leading-

region

of tandem are or

in references

9 and

in reference

indicated

methods

of analysis

within give

possible sonic,

to use

called

indicated

the

intersect

method

depends

blades.

For

such

region,

defined.

On

the

in figure

guided

region,

must

other 5-9, and

of the suction

hand, less

surface.

alone

In this latter

can

than

is often equation

position.

suction

by

surface row,

surface

be computed

only

the stream-function

of the suction-surface

the

small

distribution blade

within

velocity

a

streamline

usefulness

and/or

suction

case,

other

of all

provided

to

is also sub-

give solutions the

velocity

and

to obtain

and

Therefore,

of the

It

analysis

ends

of the

subsonic

regime.

both

a low-solidity

half

by

subsonic

velocity-gradient

solidity

surface

velocities

definition

be obtained

assumptions

can only

where

(high

most

is

can be extended

flow

of flow guidance

for

than

which

equation

curvature

boundary.

4-11,

surface

be used if better

the

of analysis

associated

mixed-flow

entirely

velocity-gradient

because

passage

in figure the

solution

without

The

a passage

degree

as

potential-function

are

of analysis

solutions

a solid

on the

and

shown half

is,

a well-guided

as shown

guided

that

or stream-filament,

that

or

a preliminary

transonic

method

method

passage;

orthogonals

the

analysis

streamline,

A velocity-gradient guided

in

earlier.

a stream-filament

involves

surface,

can

and

the subsonic

solution

or supersonic

ones

flow

ANALYSIS

to solutions

a velocity-gradient

transonic, basic

TANDEM

By use of a velocity-gradient

however,

approximate

of any

The

_IERIDL,

solutions

stream-function

limited region.

assumptions,

an

the

are

the computation

additional

the

previously,

with

in reference

axial-

called

to extend

rows

flow.

of any

a program

VELOCITY-GRADIENT As

A

regions

blades.

hub-to-shroud

equation

a

solution.

7, can be used

described

to subsonic

10. Transonic

the use of a velocity-gradient stream-function solution.

to extend

or blade

and

5-1(a)),

by

a velocity-

section

rows

in

velocities)

using

or trailing-edge

(mean

fig.

be analyzed

by

or slotted

restricted

plane

supersonic

next

blade

called

4-14(b)

can

slotted

and,

stream-function

is described

or

meridional

turbomachine described

which

programs

in the

in the

subsonic

ANALYSIS

calculations

obtained

described

in tandem

or in the

same (local

are

flow rate)

program,

the

to transonic

type

mass

Another

all

solutions

TANDEM,

flow

splitters.

performs

solution

of the

(lower

program

it

Transonic

equation

preliminary to

in that

FLOW

of this turbine angles),

is within

the

can be well such

as that

is within on the

the front

analysis distribution

is required. 147

TURBINE

DESIGN

AND

APPLICATION

Method The

idea

sidering

of a velocity-gradient

a simple

case.

method

Suppose

narrow

passage

passage velocity

to be b, and the can be calculated

as shown

can

we have

in figure

be demonstrated

two-dimensional

5-13.

width d. If the approximately

We

by

con-

flow through

assume

the

height

mass flow is known, from continuity by

the

a

of the average

W

W.,o -

However, and

is a variation

in turbomachinery

With the

there

pressure

gradient

equilibrium,

where radius

in velocity

it is this

a force-equilibrium

it can be shown

across

velocity

equation, as was

(5-60)

pbd width

difference

by balancing

done

the

in chapter

of the

we are

passage,

interested

centrifugal

force

3 for consideration

in.

against of radial

that

q is the distance from of curvature for the

dW

W

dq

ro

the suction streamline.

(5-61)

(convex) surface, and The sign convention

re is the for rc is

important; rc is positive cave downward. For the

if it is concave upward, and negative if it is consimple case shown in figure 5-13, equation (5-61)

can be integrated

a radial

curvature

along

to be equal

for integration

from

line by assuming

in magnitude the

inner

radius W Wo

to the

passage

to any

point

the

streamline radius.

in the

148

results,

passage,

ro r

(5-62)

Row

FIGURE 5-13.--Flow

There

radius

through a curved passage.

of

CHANNEL

FLOW

ANALYSIS

where Wa

relative

r.

radius

of inner,

r

radius

of passage,

The

mass

velocity

on inner,

or suction,

or suction,

surface,

surface,

m/sec;

ft/sec

m; ft

m; ft

flow through

the

passage

w---

is expressed

rm+d

as

pWb dr

(5-63)

rl

and

substitution

constant

of equation

density

assumed,

(5-62)

into

(5-63)

and

integration,

with

yields w

w. =

In a similar puted

manner,

the

outer,

(5-64)

or pressure,

surface

velocity

can be com-

as W

-

Thus, by

using

not

an estimate

of the

equation

blade-surface

(5-62),

necessarily

(5-65)

which

restricted

to

velocities

is a velocity-gradient

two-dimensional

variation of velocity in the height of the be calculated in that direction also. We will now we are

consider

interested system

5-14.

indicated

Also

meridional

is a plane

angle

between

meridional

with

the

containing W_

plane.

and The

the the

x axis,

following Ws=

0, and

Also shown

and/_,

relations

the hold

W sin/_

W,_ = W cos/_

equation.

Since

angle

cylindrical

5-14

between

for the

in figure

W_, and

and W_. The in figure

are

some could

x, as shown W,,

of W,

We were

gradient

a rotating

axis

components,

resultant

x axis.

there

a velocity

we will use

velocity

W_ is the

If

velocity-gradient

r, angle

simply

equation.

flow.

passage,

general

radius

are

component

plane

a very

in turbomachinery,

coordinate

can be obtained

We. The

meridional are a, the W and

the

components: (5--66) (5-67)

Wr = W_, sin a

(5-68)

W_ = W_

(5-69)

cos a

149

TURBINE

DESIGN

AND

APPLICATION

W

FIGURE

In

5-14.--Cylindrical

addition

to the

r-, 0-, and

m-coordinate.

The

line,

in figure

as shown

line distance of the

The

radius

positive We

in

the

of curvature if the streamline

want

the

distance

along

constant

angular

velocity this

The

_0.

m-distance

The

of the of the

plane;

For

along

the

streamline that

is,

the

0-coordinate

streamline.

The

where sign

is r_ is

of rc is

upward. along

an arbitrary

case of constant

(rV_,)

d--q=a

stream-

is the projection

meridional

dW

an

stream-

true

is 1/rc,

the

to use

a meridional

is less than

streamline

gradient

curve.

components.

meridional

is concave

momentum

velocity

it is convenient

meridional

meridional

curvature

and

is the distance

5-15.

angle

system

x-coordinate,

m-coordinate

if the

a streamline

neglected.

coordinate

at the

curve. total

Let

temperature

q be the and

inlet,

dr

dx

dO

jq+b

dq +c--dq

(5-70)

where W

COS

ol COS 2

W sin _

arc

150

r

/_+sin

dW,n

a cos _ -----2_ dm

sin _

(5-71)

CHANNEL

FLOW

ANALYSIS

Meridional streamline7

_' rc

Q

m

r

J

_x Axis

5-15.--The

FIGURE

b = - W sin a cos 2 _+cos re

m-coordinate.

a cos/_

dW_______ dm

c=Wsinasin[3c°s[3+rc°s_(ddWWm These

equations

In using ential

any

A great

number

(5-73).

blades, tial

channel, For

as shown

(0) direction

distance We=0,

normal then

are

these

not

dO/dn=O

cases suppose

in figure to the

known

5-16, page). and

have

an

and

no velocity

We

can

l_=0.

from annular component

calculate q=n

Further,

dW/dn, in equation

from

figure

11.

a differ-

such

in advance.

can be obtained

Let

of reference to solve

parameters,

can be estimated

we

streamline.

(B14)

precisely

parameters

of special example, (into

and

it is necessary

streamline-geometry

8. These

(5-73)

+2_sina

(B13)

equation,

involving

a, and

for a guided

as equations

velocity-gradient

equation

vature,

to

are derived

).

(5-72)

as cur-

However,

reasonably equations

well. (5-70)

passage

with

no

in the tangenwhere

n is the

(5-70). 5-16,

Since

it can

be

Outerwall--, Meridional streamline-,

Llnner wall

FIGURE

5--16.--Annular

passage

with

no

blades.

151

TURBINE

DESIGN

seen

dr/dn

to

that

AND

APPLICATION

= cos a and dx/dn

= - sin a. Then,

from

equations

(5-70)

(5-73), dW_ dn

Thus,

for this

tion

case,

equation

W rc

(5-70)

reduces

Several

computer

machine

blading

NASA

Lewis

CTTD

program,

and easier

programs

by

This

CHANEL

flow

turbines

program

program

velocity

streamline

orthogonals

surface, rate.

form

of equa-

through

turbo-

as illustrated

(choking)

mass

medium-

definition

than blading,

sections

of the

The

to

mean-,

a specified surfaces

program

program

used

meridionaland

for an orthogonal

the

gives

indicated

only

mass

flow

along

the

maximum good

results

previously,

may

be needed

be obtained

more for low-

for fully

guided

15 for backward-swept compressors.

use

of

in this chapter,

for

the

not

for

known

the

which

a blade-to-blade

velocity-gradient a subsonic

turbine equation,

use

of the called

impeller vaned

as

stream-function

are

and

in

diffusers

plane

impeller

it

were analysis

turbine and

in

streamline

in advance,

on

lines,

impellers

for a radial-inflow

is to extend

is presented along

meridional-plane

or radial for

for a meridi-

program

straight

a radial-inflow

A program

meridional-

method

impellers

are

a computer fixed

to obtain

equation

lengths

programs 11

basic

centrifugal

along

Such

in reference

used

The

velocity-gradient

to base

equation

also been

solutions.

orthogonal

that uses quasi-orthogonals in reference 16.

viously

hub-,

to compute

As

can

have

the

the

quasi-orthogonals.

A further

channel.

mixed-flow

uses

convenient

centrifugal

along

13.

or mixedare

along

of these

be used

by this

plane

for

Since

velocity-gradient

reference

a number

blading.

methods

analysis

presented

for the

radial-,

tip

satisfies

general

passage.

14, which

more

to blade

which

more

equations to

is the to axial-

in reference

axial-,

hub

at the

years

by the

is described

in a flow solution

also

solutions

and blade-to-blade

orthogonals.

for can

high-solidity

Velocity-gradient

reference

5-17,

made

because

from

written

for many

superseded

to analyze

blade

been

12 and is limited

which

results

can be provided

solidity

onal-plane

from

program

flow rate to

used

Velocity-gradient

This

are

have

was

now been

be used

in figure

This

that

both

and

orthogonals.

passage.

plane

can

of flow

in reference

program,

variations

Computations

blade

has

analysis methods

One

compressors.

determine tip-streamline

the

is described

CHANEL

or

for

Center.

which

to use

The

152

simple

Programs

velocity-gradient

Research

flow turbines.

was

to the

(5--61). Computer

for

(5-74)

of

analysis

is described

mentioned solution

preto

obtain the

local flow

gradient

supersonic

angles

and

equation.

velocities. streamline Programs

for

FLOW

ANALYSIS

solution

is used

to obtain

subsonic

curvatures

method are presented in references in reference 6 for a blade-to-blade

Orthogonal

The

CHANNEL

required

transonic-flow

for

solutions

the

velocity-

based

9 and 10 for a meridional solution.

on

solution

this and

j,f

su Hace _/ /

su rface

.-Suction Tip

I surface

orthogonal

f Midchannel stream line--, i

Mean

Parallel to axis of rotation-_

i Hub

T

FIGURE

5-17.--Turbine

blades

with flow

three-dimensional

orthogonal

surface

across

passage.

153

TURBINE

DESIGN

AND

APPLICATION

REFERENCES 1. JOHNSEN,

IRVING

Axial-Flow 2.

A.;

AND BULLOCK,

Compressors.

COSTELLO,

GEORGE

Velocity

NASA

R.:

Method

Distributions

in

ROBERT SP-36,

of

O.,

EDS.:

Aerodynamic

Design

of

1965.

Designing

Compressible

Cascade

Blades

Potential

Flows.

with

Prescribed

NACA

Rep.

978.

1950. 3.

COSTELLO,

GEORGE

Detailed scribed

Velocity

1060, 4.

VARGA,

Procedure

Distributions

RICHARD

gram

S. : Matrix

THEODORE; for

for

in

L.;

AND

Design

JOHN

SINNETTE,

of

Compressible

Surface

Iterative

AND

Calculating

Cascade

Potential

Analysis.

MCNALLY,

Velocities

of a Turbomachine.

KATSANIS, on

ROBERT

CUMMINGS,

Blades Flows.

T.,

JR.:

with

Pre-

NACA

Rep.

1952.

5. KATSANIS,

6.

R.;

Computational

THEODORE:

and

NASA

a Blade-to-Blade

Surface

Inc.,

Revised

on

X-1764,

Program

Stream

I).:

Streamlines

TM

FORTRAN

Prentice-Hall,

WILLIAM

1962.

FORTRAN

Pro-

a Blade-to-Blade

Stream

1969.

for

Culculating

Transonic

of a Turbomachine.

NASA

Velocities TN

D-5427,

1969. 7.

KATSANIS,

THEODORE;

Calculating a Tandem

Blade

8. KATSANIS, face

NASA

and or

O.:

TN

D-5044,

WILLIAM

TN

Mixed-Flow

for

Surface

of

FORTRAN

Program

a Blade-to-Blade

Stream

l).:

Program

for Sur-

1969.

WILLIAM on

Program Stream

1969.

]).:

on

D-5091,

Streamlines

FORTRAN

a Blade-to-Blade

Region

the

FORTRAN

Hub-Shroud

Turbomachine.

for

Mid-Channel

I--User's

Flow

Manual.

NASA

1973.

THEODORE;

AND

MCNALLY,

Velocities

and

Streamlines

Calculating Surface

McNALLY,

NASA

Axial-

D-7343,

KATSANIS,

NASA

AND McNALLY,

Velocities of an

on

in a Magnified

THEODORE;

Surface TN

AND

of a Turbomachine.

KATSANIS,

WILLIAM

Streamlines

Turbomachine.

Velocities

Calculating

10.

and

THEODORE;

Calculating

9.

AND McNALLY,

Velocities

of an TN

Axial-

on

or Mixed-Flow

D-7344,

WILLIAM

1).:

the

FORTRAN

Hub-Shroud

Turbomachine.

Program

Mid-Channel

for Flow

H--Programmer's

Manual.

1974.

11.

KATSANIS, THEODORE: Distribution in the

12.

KATSANIS,

Use of Arbitrary Meridional Plane

Quasi-Orthogonals of a Turbomachine.

for Calculating NASA TN

Flow D-2546,

1964. THEODORE;

Method Blade. 13.

for NASA

KATSANIS,

NASA

TM

TN

of Analysis

VANCO,

Meridional D-6701, 16.

KATSANIS,

154

GINSBURG,

Design.

Choking

for Flow

AMBROSE;

Flow

NACA

R.:

Plane

of

for

an

Axial

Flow

Turbine

Quasi-Three-Dimensional for Turbomachine

CalBlade

Rows.

Rep.

AND

Through 1082,

FORTRAN a Turbomachine.

OSBORNE,

WALTER

Mixed-Flow

M.:

Centrifugal

Method

Impellers

1952.

Program

for

CMculating

I--Centrifugal

Velocities

Compressor.

in NASA

the TN

1972. THEODORE:

Distribution D-2809,

Quasi-Three-Dimensional

A

1971.

T.;

MICHAEL

T.:

Velocities

Program and

for Compressible

of Arbitrary 15.

Velocities

D-6177,

JOSEPH

LOIS

1967.

FORTRAN

of Surface

HAMRICK,

DELLNER,

Blade-Surface

X-1394,

THEODORE:

culation

14.

AND

Calculating

on 1965.

a

Use

of

Arbitrary

Blade-to-Blade

Quasi-Orthogonals Surface

in

a

for Turbomachine.

Calculating NASA

Flow TN

CHANNEL

FLOW

ANALYSIS

SYMBOLS A

flow area, mS; ft 2

ai

coefficients

b

cascade

height,

m; ft

d

passage

width,

m; ft

h

distance

between

ki

constant

in equation

m ?t

distance distance

along meridional streamline, normal to streamline, m ; ft

P

absolute

pressure,

q R

distance

along

r

radius,

8

blade

T

absolute

t u

time, sec stream function

V

absolute

W

relative

W

mass

flow rate,

fluid

absolute

for equation

(5-46)

mesh

points, (5-56)

N/m2;

J/(kg)

(K);

m; ft

2

curve,

m; ft

(ft)

(lbf)/(lbm)

(°R)

m; ft spacing,

m; ft

temperature,

velocity,

K; °R

m/see;

velocity,

m/see; kg/sec; angle

(in the tangential distance in direction 0

angular

distance

P

density,

kg/m3;

potential

ft/sec ft/sec lb/sec

of inclination

meridional plane, deg fluid flow angle, relative

o3

lb/ft

arbitrary

gas constant,

m; ft

to blades,

direction), deg of outer normal

in direction lb/ft

from

direction

in the

of the

meridional

plane

to cascade

boundary,

out

of rotation,

axial

m; ft

rad

a

function

angular velocity, tad/see verrelaxation factor

Subscripts: c

curvature

in

inlet

m

meridional

component

n out

component outlet

normal

p

pressure

r s

radial component suction surface

x

axial

y

component

in y-direction

z

component

in z-direction

to streamline

surface

component

155

TURBINE

8

DESIGN

AND

tangential

APPLICATION

component

o 1 mesh-point 1, 2, 3, 4 I

156

designations

CHAPTER 6

Introductionto BoundaryLayerTheory By WilliamD. McNally As shown

in chapter

certain

amount

ducing

work.

work

of ideal The

causes. on the

the

next

chapter.

the and

layer

hand,

This

that

more

of the

(windage),

chapter

provides

turbine

is not

a

for

pro-

converted

to

important

and

difficult

to understand

their

losses.

losses.

_Iethods

fluid

as air)

the

influence

there

velocity

agree

of the

on the

outer with flow

fluid

edge those

basic

are

viscous

presented

a turbine

blade

flow is confined of the

of this

blade.

layer

loss in the

At

is zero in all directions

layer

is called

flow is frictionless,

with the

at normal

to a relatively

This

the

calculated

assumptions.

to estimate

LAYER

past

neighborhood

At the

nonviscous)

flows

the

partial-

to boundary-

needed

losses

BOUNDARY

of viscosity

immediate

layer.

mixing

and

introduction

the parameters

and

a real in the

an

incidence,

for determining

trailing-edge

(such

flow

gives

to calculate

OF

conditions thc

friction

is used

boundary

(frictionless,

to the

energy

of the

a turbine

it is necessary

NATURE

velocities, thin

disk

associated

When

One

predicted,

which

(friction)

and

is available

ideal

is the prediction

be

operation.

theory,

viscous

can

flows,

admission

that

of the

across

primary cause of losses is the boundary layer that develops and end-wall surfaces. Other losses occur because of shocks,

tip-clearance layer

design

losses

The blade

ratio

to be a loss.

of turbine

Before

pressure

energy

portion

is considered

aspects

2, the

the

wall, (no-slip

use on

of ideal

the

other

condition). 157

TURBINE

DESIGN

AND

It is the frictional, velocity

from

point

layer

layer

at the

and

or viscous,

slide

over damped

ness of the

overall

or changes

in some

Most are

flows

to a turbine, The

on the

bulent

boundary

layer.

amplified,

as in turbulent fashion Figure layer.

6-1

a boundary The

a mean also

Separation layer

manner

stream a turbine

blade,

the

shows separates, this

static

Laminar boundary

the along

pressure _- Transition _ region

for any

great

dis-

becomes

disturbances

a turin the

in velocity

boundary

oscillates

layer,

in a random

moves

away

rear

portion

turbulent

boundary

boundary from

the

layer. blade

in figure

6-3.

of the

suction

increases.

This

Separated -,,_,,a, lull

FIGURE 6-1.--Boundary

_'

_Z////'_

layer

\

on blade.

When surface.

As the free-

r-Turbulent boundary layer

__

158

a the

layer --....

_,

have flow,

and

in the

laminar

correspondingly

/

a combustor,

fluctuations

is illustrated the

time

overall

turbulent

point

region in the

fluid

happens

with

6-2 (b) indicates.

a separated occur

are

smooth-

of velocity

weak

random

any

steady

laminar

In the

at

as figure

can likewise

decreases

flow.

blade

in velocity

this

region

region,

to the

velocity

value,

in which

velocity

leads

to the

it from

With

remain

transition

of turbulent

flow,

about

flow.

suction

indicates.

components

of

The

is always

on the

is either

a transition

In the this

influence

or entering

cannot

through

and

are characteristic

fluctuations

6-2(a)

the

layer

parallel

local

as figure

type

blades

passes

layers

6-1.

stagnation

both

negligible

fuctuating

in this

along

fluid

wall.

at the

boundary

at a point

ducted

It usually

that

velocity way,

tance. flow are

Any minute have

grows

the

in figure

thickness

fluid

reduce

to zero at the

of the

layer,

smooth

in nature. layer

and

portion

they

flow. The

influence

boundary

other.

that

is illustrated finite

blade

boundary so that

being

turbulent

significant

blade

initial

layer

value

a small

of the

The

each

sufficiently

frictionless from

edge

surfaces.

In a laminar

surface

in this thin

on a turbine

develops

leading

pressure

laminar.

forces

its free-stream,

A boundary boundary

APPLICATION

region-t

surface positive

of

INTRODUCTION

TO

BOUNDARY-LAYER

THEORY

Steady Steady t

t

I

Unsteady

t (a)

Laminar

t flow.

FxovaE

(b)

6-2.--V_riation

FIoUrtE

pressure

gradient

boundary

layer

layer

in a direction point. can

The also

turbulent

laminar

layers

boundary can

be

Mach

laminar

equati()ns

for

and the

layer This

be noted

either

the

immediately

layer.

it should

lew_'l ()f the sent

and

that

Just

turbulent incompressible

closc

flow passing

the

itself

at the

leading

edge

reattach both

itself in figure

laminar

and

or compressible, as there

are

boundary-layer and

different flow,

compressible

flow

in the

to the

reverses

incompressible

number.

the

flow

flow

is illustrated that

The

very

of the mean

which

boundary

separatc

Finally,

at

flow.

at a point.

retards

energy.

a degree

to that

point

gradient)

it to lose

to such

Turbulent

time

separation.

pressure

causes

opposite The

with

6-3.--Boundary-layer

(adverse and

can be retarded

separation.

of velocity

in

wall it moves blade.

is the

This

the

is

separation

of a turbine

to

the

boundary

blade

surface

as

a

6-4. turbulent

boundary

depending equations there

are

variations

on

the

to repredifferent of each. 159

TURBINE

DESIGN

AND

APPLICATION Turbulent boundary Separat ion

Laminar

bubble;,

boundary layer_

layer 7 _

,Z__...1-f_.rrt_

II I I __I __' ' .....

c Stagnation point

FIGURE

6-4.--Laminar

separation

Boundary layers should be considered relative Mach number exceeds values equations

for these

discussed

in this

various

The

general

OF

equations

equations.

equations, equations

In

compressible of 0.3 to 0.4.

are derived

of motion of the

be derived

from

be repeated

here.

of viscous

fluids

systems,

coordinate

directions.

the

tion

assumptions

are

represents

methods

the

for a compressible

during

Navier-Stokes fluid with

du dt

1 and

their

The

are

The

the

equations, The

combined

such

Navier-

the law of conis lengthy, and

have

derivation.

Navier-

three

boundary-layer

equations.

2 both

equations

constant

the

are

by applying This exercise

different forms. of the Navier-Stokes

made

are called there

Navier-Stokes

References

derivation, in two somewhat There are various forms what

solution

EQUATIONS

coordinate

Stokes equations themselves can be derived servation of momentum to a fluid element. will not

and

if the free-stream The boundary-layer

BOUNDARY-LAYER

normal

one for each can

reattachment.

chapter.

DERIVATION

Stokes

cases

and

complete

depending following into

vector

on equaform

viscosity:

gf_g__ Vp+_U V2u+ p p

__

p

V(V.u)

(6-1)

where u

general

t

time,

g f

conversion

160

general

velocity

vector,

m/sec;

ft/sec

sec constant, body

force

1;32.17 acting

(lbm)

on a unit

(ft)/(lbf) mass

(sec _)

of fluid,

N/kg;

lbf/lbm

INTRODUCTION

P

density,

P

static

/z

dynamic

In this

kg/mS; pressure,

s

N/m2;

lbf/ft

viscosity,

equation,

u, v, and

lbm/ft

three

BOUNDARY-LAYER

THEORY

2

(N) (see)/m

u represents

w in the

TO

2; Ibm/(ft)

a general

coordinate

(see)

velocity

directions

vector

with

x, y, and

components

z, respectively.

u=uiTvj+wk where The

i, j, and total,

k are

the

or substantial,

unit

vectors

derivative

(6-2) in the

three

of u is du/dt.

coordinate In any

directions.

of the

coordinate

directions, d

0

0

0

0

Oz In equation

(6-1),

rather

than

vector

quantities, du _ -_=g'--p

operator If the

(6-1)

V2 is applied

term

_

which

be more

[VX(VXu)-]+_

in terms

familiar

to the

p-f-_- grad(div P

vector

into

u

simple

1

VP+U-o V(V.u)--p V operator

to the

V2u is expanded

becomes

g

the

du gf-'q grad d--/= p

Laplacian

function.

equation

Expressing may

the

to a scalar

(6-3)

u) -_-

of gradients, reader,

p

curls,

and

equation

(6-4)

u)

1 _l P grad

curl(curl

(6-4)

p- V(V.u)

+_

divergences, becomes

(div

u)

(6-5) In

order

to derive

the

to be expanded

into three

directions.

three

Ou

Ou

The Ou

+U x+V y+W

boundary-layer scalar

resulting

equations,

equations, equations

Ov

Ov

Ou Op __u /O_u _z-z=g f . .... gpox p t0_-t-

Ov

g Op __u_/O2v

of the

(6-1)

has

coordinate

are 02u 02u\ _y2+0-_)

l u 0 [Ou

Ov

equation

one for each

020

Ov

Ow\

Ov

Ow\

02v\

-_-t- u -_x -k-v -_y + W oz = gf u ....

l u 0 [Ou

161

TURBINE

DESIGN

aw

aw

AND

_

APPLICATION

g Op.

+ w aw

tt [CO2w. O_w.

COho'_

1 tt cO[cOu

cOy

cow\

+5-_ t,_+_+_)(6-s) P

where

f,, fv, and f, are the

In

order

Laminar

Incompressible

to

Prandtl's

incompressible (1)

derive flow,

Viscosity

writing

of the

(2)

the following

assumptions has

the final terms

in equations

Flow

and

Layer equations

for

laminar

will be made:

already

been

assumed

in the

Since

for

incompressible

flow

the

con-

is

V. u=div

(3)

f.

equations.

/cOu

as well

cOy

cOw\

u=t_x+_yy+_z (6-6)

to (6-8)

is two-dimensional.

consideration,

force

Boundary

This

is incompressible.

equation

of the body

boundary-layer

is a constant. preceding

Flow

tinuity

components

This

as all terms

)=0

(6-9)

can be eliminated.

eliminates

involving

equation

(6-8)

from

w or O/cOz in equations

(6-6)

(6-7). (4)

Flow

is steady.

(5)

Body

forces

Thus,

f, andf_

With

these

following

This

are

eliminates

negligible

can be discarded assumptions,

two equations

cOOr terms.

in relation from

the for the

to inertia

equations

and viscous

(6-6)

Navier-Stokes

and

equations

(6-7). reduce

COy

the

continuity

(6-11)

. _ . t,_+_)

equation

becomes cOu

cOy

_x+_yy=0 In order

to make

boundary-layer and

flow,

the

some

velocities

and

check

are negligible coordinate

(6-10)

equations

an order-of-magnitude

show that

162

equations

the

(6-_0>

g__ cOp + __ / cO_v cO2v\

u_+__: Likewise,

to

x- and y-directions:

COu cOu /O_u CO'u\ u_+v _= 00p __ ___,,_,_+_)

COy

forces.

with

directions

(6-12/

to (6-12)

suitable

are traditionally is performed respect pertinent

for the analysis made

on

to others. to the

the

of

dimensionless,

various

Figure boundary

6-5

terms

to

shows

the

layer.

INTRODUCTION

U =u uO_

TO

u_5 _

BOUNDARY-LAYER

full

THEORY

Trailing

y

x

L

FmURE 6-5.--Boundary-layer

The

following

dimensionless

velocities

parameters

and dimensions.

are defined:

X X

(6-13a)

=-

L

(6-13b)

u

(6-13c)

U0

(6-13d) Uo

(6-13e)

Re = o.L

Uo (6-13f)

where X

dimensionless

x-coordinate

L

characteristic

length

Y

dimensionless

y-coordinate

U

dimensionless

velocity

u0

free-stream

V

dimensionless

velocity velocity

(in this case,

the blade

chord),

m; ft

in x-direction upstream

of blade,

m/see;

ft/sec

in y-direction 163

TURBINE

DESIGN

AND

P

dimensionless

Re

Reynolds From

And order

figure

since

APPLICATION

pressure

number 6-5,

we see that

y is proportional

$r,m/L

the y-direction x-direction. In order quantities,

to the

= _, a quantity

U0, U = u Uo is of order in the

to put (6-12)

equations

are

much

equations

are

(6-10) and

is multiplied

(6-11)

1 /O_U

02U\

OV

OV

OP

1 [02V

O2V\

are

of order

of magnitude with

each

by

resulting

OP

those

of the

L/Uo _, and

(6-15)

OV

various Since

in the

dimensionless

.....

other.

in

of dimensionless

multiplied

The

OU

_--_+_-_ order

are

L/Uo.

Y is of

velocities

than

in terms

1.

u is of order

_, since

smaller

to (6-12)

by

OU

be compared

much

_/,u,

since

OU

u +v

The

thickness

1. Likewise,

V = v Uo is of order layer

(6-10)

to L, X is of order

boundary-layer less than

1. And

boundary

equations

equation

since x is proportional

= 0 terms

X and

(6-16) in these

equations

U are of order

1, and

can

now

Y and

V

e, OU 1 -- =-= OX 1 OU ----OY

1

(6-17a)

1 (6-175)

oV OX

- = _ 1

(6-17c)

1

(6-17d)

OV --=-= OY

O_U 1 -=-= 1 OX 2 1.1

164

O_U

1

1

0 y2

e- e

e2

(6-17e)

(6-17f)

INTRODUCTION

TO

BOUNDARY-LAYER

THEORY

a2V

e ----- E 0X _ 1.1 02V

_

OY _ Furthermore, magnitude order 1. Relating to (6-16)

the

change

as the

change

these

--

of

orders

1 -

_.e

in P with

(6--17g)

(6-17h)

e

respect

U with

to X is of the

respect

of magnitude

same

to X,

so that

terms

in equations

to the

order

OP/OX

of is of

(6-14)

yields OU

OU

v

OP

1 /02U

v o-T=

(6-18)

i3-X+

(1) (1) + (e) (!)

= -- 1 + (_)

OV

OP

OV

v

02U\

(1+_)

1 /02V

+ v oT= or

+

(1) (_) + (E) (1) = -_+

OU

02V\

)

(6-19)

(t2) (_+!)

OV

(6-20) 1+1 By examining can be reached: (1)

In

terms

(6-18),

1Re

02U/OX

2 and

Reynolds (2)

is to dominate, smaller fore, allows

than P=P(X)

(6-20),

must

be must

02V/OX

the

_, the

it too must OP/OX,

and that

two

since terms

with terms

this

the

conclusions

viscous

terms

of magnitude

to be true 2 is much

as the

in equation larger

than

Therefore,

the

large.

are

and

that order

02U/OY

of

of order

order

and

e. Therefore, a function

OP/OX=dP/dX

pressure

_

e or less. Therefore,

P can be considered the

following

in parentheses.

l/Re

be of order

or p=p(x),

us to assume

d,

be relatively

(6-19),

same For

of order

dominates

the

it is assumed

2) -] are of the

U(OU/OX)+V(OU/OY).

equation

dominating

to

theory,

2) + (02U/OY

number In

(6-18)

boundary-layer

1/Re[(O2U/OX inertia

equations

across

with unless

2.

OP/OY

OP/O Y is much of X alone.

or Op/Ox=dp/dx. the

02V/OY

boundary

layer

ThereThis in the 165

TURBINE

DESIGN

y-direction

the

_, the

(4)

In

small

constant.

existing

Since

order

APPLICATION

is essentially

flow pressure (3)

AND

at the

first

equation

second

equation

equation

(6-18),

in comparison

It can be assumed

outside

of the

is of order

1, and

can be neglected 02U/OX

with

02U/OY

_ can

2. This

equal

boundary

to the potential

layer.

the

second

equation

is of

completely. be

neglected

leaves

the

because

following

it is so dimension-

less equations: OU OU dP 1 02U U _ T V O----Y = - d--X -{-Re 0 Y_ OU

(6-21)

OV

OX +-_=O These

are

The the

Prandtl's

boundary-layer

boundary-layer

influence

for different

From

number

equation

magnitude, the viscous-force smaller. The boundary-layer decreasing

layer

by

The

Uo/L.

useful

in determining

of the

boundary

we see that

layer

as Re increases

be

(6-21)

equations Ou u --+v Ox

increasing

viscosity

can

equation

resulting

are size

rule,

laminar,

Prandtl's

decreases.

put by

Ou

g dp

-

Oy

_t

p dx

in terms Uo2/L

of

and

dimensional

equation

(6-22)

_ 02u

(6-23)

p Oy 2

Ov

incompressible

flow.

and equations (6-23) It should be noted the

presence

of large

boundary

layer

shock Mach

viscosity

waves

(i.e.,

where

occur).

on mainly

the

about

the

interaction

two-dimensional,

arc assumed surface, unknowns

for their equations

instantaneous Just

as

flow

Reynolds

of shock

waves

constant

dp/dx, is also are u and v,

calculation. arc not valid adverse

in

pressure

phenomena

number,

wave depend on primarily the Mach number. number is not included in the boundary-layer

us nothing 166

and

for

along the blade The remaining

(6-24) are sufficient the boundary-layer

magnitude

depend

equations

Density

gradient solution.

and that

of shock

gradients

(6-24)

boundary-layer

and known. The pressure known from an ideal-flow

to

the thickness

--+--=0 Ox Oy are

and So,

are

Ou

These

in

Re corresponds

So, as a general

as the

equations

by multiplying

form

Furthermore,

decreases

variables

form.

on the

(6--21)

if pL Uo is constant.

boundary-layer

in dimensionless

terms (1Re) (02U/O y2) will get smaller thickness will correspondingly decrease.

_i_u decreases.

viscosity

of the boundary The

in this

Reynolds

fluids.

as Re increases,

equations

equations

of the

(6-22)

in the

conditions

in a

Since the influence of equations, they tell and

boundary

layers.

INTRODUCTION

The

boundary-layer

is approached. velocity

v is much

boundary

layer

Nonetheless, right

significant

than

and

up

to the

separation and

calculations

The

Navier-Stokes

However, in the

close

v begins

to the

error these

since

used

is quite

question flow

over

can

(i.e.,

the

y-axis

be derived

reference of curvature

terms

r at a position of the

as was

previously.

thickness for the the

such

x along

individual With

is small

compared

with

no large

variations

same

boundary-layer the walls

as would

as well, occur

near

radius result

sharp

there

for

surface.

of the curved equations are

given

in

radius

relative

orders

in the same

manner

the

boundary-layer

of curvature

of the

occur,

no large

for

system

on the

The

that

as were

The

change

coordinate

so that

obtained

equations are

negligible). would

equations

in curvature

boundary-layer

provided

be used development

dependent

assumption

equations

fiat-plate

are

These

arc very

the

of

for an orthogonal system of each of the coordinate

can be estimated

the

V is

point

set of Navier-Stokes

the blade

terms

not

is in the direction

a system.

equations

case where

Therefore, curved

in such

where

flow region.

orthogonal

the x-axis

as u. in cal-

of the

equations

to it, a new

in the

region

in the

effects

If a curvilinear

wherein

of magnitude done

curvature

boundary-layer

is normal

for flow

1. The

the

wall.

is introduced

and

where

as to how

a curved

(fig. 6-6) wall

large

arises

the

order used

should

of the boundary-layer equations were derived of coordinates in which the radius of curvature axes

the

the

point,

same

of a separated to (6-8)

is that

separation

location

equations

(6-6)

as separation

generally

in the

neighborhood

equations

are

THEORY

derivation

to be of the

point,

little

reliable

in their

equations

small,

is incurred.

detailed

BOUNDARY-LAYER

completely used

u. Very

rapidly,

boundary-layer

is very

separation

are net

assumptions

smaller

grows the

culations

equations

One of the

TO

may variations

wall,

and

dr/dx

._ l,

for fiat bc

walls.

applied

to

in curvature,

edges.

y x

x

Y

FIGURE

6-6.--Curvilinear

coordinate

system

on

a blade.

167

TURBINE

DESIGN

AND

APPLICATION

Laminar An

Compressible

order-of-magnitude

equations

and

neglected, case,

analysis

for a compressible

viscosity

density

and

can

assumed

energy

temperature,

the equation

of state

temperature,

and,

is required.

nonisothermal, can

be

thermal

In the

not

used.

For

boundary-layer These

are

compressible of

and density

to

for

three

viscosity,

were

a function

form

equations

flow will involve

to temperature.

the

pressure some

case,

variations

is considered

to relate

the

incompressible

temperature

viscosity is used

to derive

of the energy compressible,

parameters specific

which heat,

and

conductivity.

There The

The

was

Layer performed

is not isothermal,

variable-viscosity

related

layer.

equation constant,

if the process

be

constant,

is no longer

equation

also

boundary

were

the

density

Boundary

are

most

several

common

relations

for viscosity

is probably

Sutherland's

as a function relation

of temperature.

(rot.

1)

z [T_3/2To+S _=\-_o] T÷S

(6-25)

where _o

dynamic

viscosity

T

absolute

To

reference

S

a constant,

Ibm/(ft)

A less

power

reference

temperature,

temperature,

K; °R

K; °R (for air, but

also

S=

by

heat

and

least-squares

perature the

less accurate,

and

for

viscosity, 168

one

equations

for

be related

the

variables

relating

component

of the

to temperature

particular related

gas

problem

these

variables

momentun_

and

tem-

to temperature, reduce

will be the equation,

the

to u, constate

equation. analysis

equations equations

can

fits these

0.65.

compressible-boundary-layer

the energy

incompressible the

conductivity With

order-of-magnitude

(Navier-Stokes) that

in the

cquation,

The

thermal

T. The four

equation,

relation

(6-26)

For air, _0 is approximately

involved.

unknowns

tinuity

temperature-viscosity

0.5
polynomial-curve

range

v, p, and

110 K or 198 ° R)

law

_ is a constant.

Specific

To, (N) (sec)/m_;

K; °R

_o= \_o/ where

temperature

(sec) static

complicated,

is the

at the

for flow.

of the

continuity

compr,.ssible For

analogous

flow

con,pressible to _()-10)

and

momentum

is almost flow

to (6-12)

with

identical nonconstant

are the

following:

to

INTRODUCTION

Ou

Ou

Op,

TO

BOUNDARY-LAYER

THEORY

0 [

L2" ou ---- 2

pu _+ pv oy--= -g _-v_

Ov\l O [ IOu Ov\l

[Ou

(6-27) Ov

Ov

pu _xx+P,

g Op.

Oy

0 [

Ov

/Ou

2

Ov\l

a [.lay

au\l (6-28)

O(pu)

+O(pv) Oy

Ox If an order-of-magnitude to that layer

for

the

analysis

on these

equations,

the

equations

following

similar

boundary-

result:

pu -_x-{-pv --=--g Oy

o(pu)

-d-x-t--_y

equation

(6-30)

-0

Ox

boundary-layer

_

o(pv) +

The

(6-29)

is performed

incompressible-flow

equations

=0

(6-31)

Oy

of state

is also

required

flow. The

state

equation

for the

solution

of compressible

is

p=pRT where

R is the

The

final

gas

constant,

equation equation,

The

energy

equation

for

the

energy

equation

for

magnitude

check.

two-dimensional

pc,

in J/(kg)

required

momentum

and

The

the

(K)

besides

a perfect

=_xx-i-_Oyy.Oxx

Cp

specific

at constant

J

conversion

k

thermal

continuity

energy gas,

written

the

equation.

is derived

of another

equation

k_-x)T_yy

energy

layer

means

(°R). equation,

is the

boundary by

flow of a perfect

U_xxTV_yy

(ft) (lbf)/(lbm)

of state,

gas

is the

or the

equation

a compressible

following

steady

(6-32)

from

order-of-

for compressible, in full:

k_yy)+_j_

(6-33)

where heat

constant, conductivity,

pressure,

J/(kg)

(K);

Btu/(lbm)

(°R)

1 ; 778 (ft) (lbf)/Btu W/(m)

(K);

Btu/(sec)

(ft) (°R)

and

j

(6-34)

169

TURBINE

DESIGN

AND

APPLICATION

If an order-of-magnitude

check

following

energy

boundary-layer

is performed equation

on the

above

equations,

the

results:

(6-35)

Equations layer

(6-30),

equations

(6-31),

thc

ideal

It is desirable blades.

Turbulent

motions)

superimposed

present. the

are due

magnitude flow.

are

The

sional,

flow

first

those

since

stretching ever, handle

motion

the

to represent

per-

(mixing

or

eddy

fig. 6-2). are

important,

since

of velocity

are

of turbulent

of the

time-dependent,

is a prime

These

not feasible

at

the

in

stresses

often

of greater

boundary-layer three-dimen-

three-dimensional could

mechanism

available of these

fluctuating

their

motion.

The

solutions

the

will

in

(see

solutions

portion

separation

decrease

motion

calculations

computers

the major

turbulent,

solution

solution

which

three-dimensional

mesh

fluid

mean

equations.

over

fluctuations

is very

to the

exact

largest

is not

components

two-dimensional

of eddies,

even

flow of

Methods

layer

closed-form

due to the

Navier-Stokes

required,

main

that

approaches

is the

boundary-

compressible

a resulting

irregular

on the

mixing

two

with

has

to fluctuating

than

There

layer

blades,

so complex

Yet,, the

fluid

the laminar

Solution

boundary

boundary

the

formance. fluctuations

a turbulent

If the on

are

two-dimensional,

Boundary-Layer

to have

occur

(6-35)

gas law.

Turbulent

probably

and

for nonisothermal,

a gas obeying

of turbine

(6-32),

never

equations

arc

represent

the

of turbulent

at

the

present

equations

flow.

How-

time

cannot

on a small

enough

components

of velocity

the equations

of continuity,

of turbulent

flOW.

The and

second

energy

approach

density,

temperature,

of the

u component

velocity temperature

is to write

in terms

of mean and

and

velocity.

of velocity,

of fluctuation are written

by

fluctuating

u'.

In for

So the

170

fluctuations

approach,

example,

the

is denoted

velocities,

of pressure,

density,

time by

average

_ and

pressure,

the and

as follows: u = _+ u'

(6-36a)

v=_+v'

(6-36b)

p = _+

The

this

momentum,

components

in viscosity,

(6-36c)

p'

p = p + p'

(6-36d)

T= T+

(6-36e)

thermal

T' conductivity,

and

specific

heat

are

INTRODUCTION

negligible

and

as functions If the

are not considered.

of the

listed

momentum,

and

continuity, compressible These They the

flow,

are called are

unknowns are

the

turbulent where

u'v' is the

terms

in the

problem

available.

the

stresses,

For

substituted

this

for the

equations

relations and

following

over

time

of

additional

additional

equations

expressions

stress

terms

or

before

th_

can be solved. Boundary (6-36)

performing

equations

and stresses.

add

empirical

Reynolds

of equations

then

average

the

equations.

or Reynolds

for which

Incompressible

(6-12),

in thees

equations

reason,

into

incompressible

arises

new

Substituting

flow:

for

terms

pu'v',

Turbulent

layer

equations

of stress

v'. These

boundary-layer

yields

are substituted

pu '_ and

are

and

(6-36)

form

approximations

(6-11),

energy

boundary-layer

are calculated

of temperature.

in equations

set

THEORY

parameters

of u r and

presently

turbulent

three

"apparent"

to the

not

a new

value

BOUNDARY-LAYER

the

of the

product

So these

time-averaged

flow properties

TO

for

Layer into

equations

(6-10),

an order-of-magnitude

turbulent,

analysis

incompressible,

boundary-

(6-37)

(6-a8) These flow.

equations Notice,

momentum (_ and

are analogous however,

equation.

_), thereby

the This

(6-28), magnitude pressible,

the

(6-29),

presence adds

making

Turbulent Substituting

to equations of the

a new

three

unknowns

(6-32),

analysis boundary-layer

and the

(6-33)

and

following

term

in the

two

two

equations.

Layer

(6-36) then

for laminar

to the original

only

Boundary

(6-24) stress

(u'v')

with

of equations

yields

and

Reynolds

unknown

Compressible relations

(6-23)

into

equations

performing

equations

for

all turbulent,

(6-27), order-ofcom-

flow:

o(_)

ox

. o(_,)

, o(p'v')

+

-o

(6-39) (6-40)

/5 =_RT

(6-41) 171

TURBINE

DESIGN

AND

APPLICATION

1

where

T, is the

absolute

0

_

total

k

0_

0

temperature,

(6-42)

in K or °R, and

is defined

T,= + We

have

now

dimensional,

laminar

boundary-layer the

point

equations

for

SOLUTION velocity

parameters Included

and as

are

presently exist circumstances.

After

basic

far

profiles

defined,

the

basis

tions the

time

that

for the

many,

boundary-layer

are

discussed of

flat-plate,

the

for

two-

and

compressible

this

is really

solutions

are

many

only

concerned.

methods

solutions

BOUNDARY-LAYER

some

under

which various

EQUATIONS

and

the

solution

important

methods

incompressible

boundary-layer will

solution,

be

discussed.

as well

as com-

boundary-layer

solu-

methods.

of the

principal

is a description blade

surface

results of the

(fig. 6-7).

FIGURE

172

equations

incompressible at this

boundary-layer

Velocity One

boundary-layer

note

as

only

OF

(6--43)

turbulent,

obtaining

will be the

pressible

the

flow. We should

starting

These

derived

as

Profiles

obtained velocity

The

profile

velocity

6-7.--Boundary-layer

from

most in the

profile

velocity

boundary

describes

profiles.

layer

along

mathematically

INTRODUCTION

TO

BOUNDARY-LAYER

THEORY

Outeredgeof boundarylayer_ /

!

°e:J

|

u J

la)

/ (b)

IlL rll//llllTII/I/,

U/1/:frillill/I1

(a) Laminar profile, FIauaz 6-8.--Laminar

the dimensionless y/5:,,,

from

at a distance stream

velocity

the blade. y from

velocity

_],m,

from

from

the

velocity,

the

at

a

the

surface.

blade

where

surface,

u is the and

distance the

the

equal

Alternately,

profiles

of the velocity velocity

to _:_,

velocity

for laminar

shape,

while

those

monly

used

mathematical

for turbulent

by Pohlhausen

u-a The

as a function

velocity

the by

distance

in the boundary u, is the defined

1 percent

free-

thickness,

as that from

layer

external

boundary-layer

is often

differs

dimensionless

the

distance external

u,.

Velocity

originated

u/ue

The

(b) Turbulent profile. and turbulent velocity profiles.

constants

_-}-b

a, b, c, and

flow

(fig. 6-8(a))

flow are

expression

for

tend

to be parabolic

blunted

(fig.

u/ue

in laminar

6-8(b)). flow

in

A comis that

(sec ref. 1) :

( ,_y+c ( ."__ Y+ ( :,y d are defined

in terms

of a dimensionless

(6-44) shape

parameter

u

dx

(6-45)

wherc }, a = 2 +-6

(6-46a)

k b = -- 2

(6-46b)

X

c=

(6-46c) 173

TURBINE

DESIGN

AND

APPLICATION

Shapeparameter,

;:!-

o. 1

==

0

FIGURE 6-9.--Laminar

veh)city

d = 1 ---

Velocity

profiles

Velocity law

for various

profiles

values

for turbulent

Pipe-flow

experiments

Reynolds

number appropriate

n can

be related

to other _fand

Definitions Solutions

In

of the

order

thickness to

define

thickness

of the

thickness

is rather

boundary

layer

little

distance 174

a value from

exponent 4 up

5, the

however, which the

wall.

since because

is very

close

It is possible

0, and

the

from

place

the

the

the

are

most dis-

factor

H.

define

the

of boundary-layer velocity

in the

external

to define

dis-

are the

form

asymptotically.

velocity

to the

the

are described

to first

definition

transition

it takes

namely

These

it is necessary The

exponent

equations

parameters.

thickness

8/,,n.

The

Parameters

boundary-layer

layer,

of n= 7

0, which

Boundary-Layer

important

of the

value

plate.

parameters, thickness

momentum

outside

power

function

10. The

flow on a fiat

parameters,

arbitrary, to that

by the

n is a mild

to about

boundary-layer

of three

these

represented

(6-47)

the momentum

boundary

importance,

attains

the

from

two-dimensional

in terms

often

6-9.

_

of Important

obtained

placement

that

varies

shc, wn in figure

kS/,,H/

for boundary-layer

placement thickness in the next section.

often

show and

(6-46d)

flow are

Ue

profiles.

6

of _ are

-- =

is most

1.0

.2 .4 .6 .8 Fraction of boundary-layer height, y/Sfull

inside This

boundary

velocity boundary-layer

at

the is of layer

a small thick-

INTRODUCTION

TO

BOUNDARY-LAYER

THEORY

ue Ue

,-(u e - u) ue

r/Ill,

rllllilllllllll

(a) Actual

velocity

FzGva_

ness as that from

the

The

distance

profile.

external

from

the the

pe is the

outward

5, for

compressible

the help of figure 6-10. flow within the boundary

density, layer.

5, the

distance

velocity

differs

by I percent

boundary-layer

flow,

As seen from figure 6-10(a), layer due to the influence of

by

boundary

distance

the

for

u,.

thickness

_[ass

where

Equivalent profile equal mass flow. of a boundary layer.

thickness

where

'llll[lll//

'lllll/ll/llfll

(b)

the blade

velocity

displacement

is given

llill,

6-10.--])isplacement

can be defined with the decrease in mass friction

(b)

(a)

II!_1111/_

defect

in kg/m This

which

3 or lbm/ft

external

as a consequence

of the

on)

defect

decrease

free

stream

outside

can be represented

as shown potential

(6-48)

dy

3, in the

mass

thickness,

the

(p,u.-

t/=81u.ll

integrated

displacement by

=

in figure field

6-10(b).

of flow

in velocity

of by a It is

is displaced

in the

boundary

layer. As figure

6-10

shows,

the

distance

_ can

be defined

by the

equation

[.y=Sfutt

(p.U.--pU)

dy

(6-49)

y--O

Solving

for _ gives

= -- 1

[_-,s.,,

(p.u.--

peue _u-O

The

displacement

thickness

The

for incompressible

friction

is given

dy=

_ y--O

loss of momentum

dy

(6-50)

0

_=-- 1 f_,=_, `n (u.--u) Ue

pu) dy = IJ_:z"':"u (Pp-_u_) 1 -flow reduces

f.=',,<,,

(_)1--

to dy

(6-51)

_ _0

in the

boundary

layer

due

to the

presence

of

by 175

TURBINE

DESIGN

AND

APPLICATION

y= I/ull

l"

Momentum

defect

= I

pu( u,--

d

This

momentum

defect

can be represented

from

the

by a distance

momentum

0, defined

p,u,20 =

u) dy

(6-52)

y=0

of purely

by the

pu (u,--

Y=PJfull

potential

flow

equation

u) dg

(6-53)

" y=0

Solving

for 0 in this equation

ness for compressible

boundary

0=-LIe2

_

momentum

as the

1--

ratio

y=0

dy=

-y=0

(6--54)

to

1--

dy

(6-55)

l"/e

compressible

of displacement

dg

flow reduces

] "

both

thick-

pc_le

for incompressible

H for

momentum

as

y=0

factor

of the

dg=

u(u,--u) /_e2

defined

layers

_

thickness

form

definition

y=0

0=-The

the

pu(u_--u)

Pe

The

gives

and

thickness

incompressible

to momentum

flow

is

thickness:

5 H =There

are many

other

two-dimensional, These

boundary-layer

and

especially

three,

however,

are

boundary-layer

studies. Physical

When retarded toward

of flow

fluid in the

boundary

exists

along

penetrate small

a surface,

too

kinetic

far

and

moves

behind

the point

the

principal

from

a region

stream.

follow

neighborhood

of the

wall.

the

occurs,

176

6-11

illustrates

separation

general

particles

layer

cannot,

pressure

pressure

the

the

gradient

of the surface

gradient in general,

because

is deflected

In general,

stream. The and reverse

some

pressure

away fluid and

of their from

the

particles move

in a

point of separation is flow in the layer in the

At separation,

_yy/_=0=0 Figure

H for layers.

in

from

an adverse

fluid

main

used

away

of increased

opposite to the external as the limit between forward

immediate

or a casing with

retarded

of separation

boundary

parameters

is transported

the boundary

the

_, 0, and

of Separation

a blade

layer

region

Thus, into

besides

for three-dimensional,

the

When the

into

energy.

surface direction defined

stream.

parameters

Interpretation

separation the main

(6-56)

0

occurring

(6-57) along

a surface.

INTRODUCTION

TO

(;),.o >o

,_\\\',, .....

FIOURE

By relation the

Prandtl's

between

it is possible presence

dp/dx>O.

being

- -- _x_

as flow

boundary-layer

pressure

gradient

to infer

that

of an

adverse

From

surface

gradients

separation

equation

undergoes

and

flow

gradient with

the

and

velocity

in a steady

pressure (6-23),

separation.

equations

dp/dx

TItEOR_/

-o

LSeperation point

6-11.--Velocity

examining

BOUNDARY-LAYER

(i.e., boundary

considering

distribution

u(y),

will occur

only

decelerated

can

dp/dx

conditions

at

relate

through

velocity

equation

velocity

\Oy_/__o

= g dx

profiles

to

(6-58).

neighborhood

02u/Oy 2, depends of the

the

u = v = 0, we have

now

immediate

in

flow),

dp We

the

of the

only

on the

profile

at

The wall,

the

pressure

the

(6-58)

Ou/Oy,

O_u/Oy _, and

cquation

indicates

curvature

of the

gradient,

wall

changes

dp/dx, its

sign

that velocity

and

the

with

the

finally

to

in

the

profile, curvature pressure

gradient. Figure layer

6-12(a)

subjected

shows

a velocity

to a decreasing

profile

pressure.

that

would

For such

exist

a profile,

in a boundary figure

6-12 (b)

y

(c)

(a) Velocity profile, FIGURE 6-12.--Velocity

(b) Velocity gradient, distribution

in a boundary

(c) Velocityprofile curvature. layer with pressure

decrease. 177

TURBINE

DESIGN

indicates

that

Ou/Oy

Furthermore, Ou/Oy,

AND

figure

is negative

profiles

is positive 6-12(c)

are

not

all

y and

that

dp/dx.

pressure

(6-58),

as y increases. slope

that

negative

a boundary

dp/dx)

of impending

is the

we know

Consequently,

(negative

indicative

decreases

02u/Oy _, which

equation

to negative

to a decreasing which

for

indicates

for all y. From

02u/Oy _ corresponds subjected

APPLICATION

will

layer

have

separation

of

velocity

(the

form

of

fig. 6-12(a)). Figure with

6-13(a)

gradient). near

shows

decelerated the

flow

Here, blade

corresponds be less than

a profile due

figure

6-13 (b)

surface;

which

to an

that

would

increasing

indicates

for which

layer

velocity

02u/Oy_= profile.

O. This

It follows

in a boundary (adverse

Ou/Oy

is, O_u/Oy _ is positive

to positive dp/dx. However, zero at some distance from

point

that

exist pressure

is a point that

has

a positive

slope

(fig.

6-13(c)).

This

since in all the surface,

cases there

of inflection

in a region

layer pressure

O_u/Oy 2 must must exist a

of the

of retarded

boundary-

potential

flow

• j Y

:::_/-Point of inflection

(a|

(b)

TRY-

FIOURE

_y2 .

(a) Velocity profile, 6-13.--Velocity

distribution

(b) Velocity gradient, in a boundary

(c) Velocityprofile curvature. layer with pressure

Stagnation

Sudion -_'_"_Adve

Fmuug

178

6--14.--Pressure

distribution

rse gradient

on a turbine

blade.

increase.

INTRODUCTION

(positive

dp/dx),

the

point

of inflection.

(with

8u/Oy

that,

with

can

adverse Figure

6-14

when

the

used

the

indicates The

It was

a typical danger

This

(ref.

On a fiat plate

3).

point

will have of separation

of inflection,

equation

a

it follows

(6-23), (i.e.,

distribution

on the

separation in regions

of

first

was

steady

separation

Layer

of a is

the major

in 1928

fiat-plate

as an

Therefore, therefore, 8u

8u

82u

u _+v

o_

oy=

viscosity

,/p,

0U

in m'/sec

of

(ref.

Prandtl's in 1908.

4).

the velocity

is constant

reduce

Technical

of Blasius

by NACA p(x)

Plate

NACA

solution

solution

translated

a Flat

in 1904 in Germany.

flow at zero incidence,

equations,

where

on

mathematical

the

is constant.

surface

is concerned,

surface,

first reported

published

The

kinematic

as

place.

was

was also later

The boundary-layer

v is the

THEORY

layer

is retarded

Boundary

and

with

solution

where

a point

of the suction

theory

to be published work

as far

is taking

translated

German

potential

pressure

portion

diffusion

Memorandum equations

at the

flow

zone,

boundary-layer

later

profile

in deriving

Incompressible

Prandtl's

boundary

have

potential

seen to be the rear of the blade

Laminar

in the

velocity must

BOUNDARY-LAYER

gradient).

blade.

readily

profile

surface)

assumptions

only

pressure

turbine part

Since

= 0 at the the

occur

velocity

TO

and

from dp/dx

the ffiO.

to

(6-59) or ft'/see,

and

¢9V

(6-e0) The

With

following

the

differential

are

use

the boundary u=v=O

at

y=O

U=Ue

at

y=

of a stream

equation

conditions:

function

(6-59)

into

(6--61) oo

_b, Blasius the

following

transformed ordinary

the

partial

differential

equation: f d2f+2 dy 2 where

f is a normalized

stream

daf=0 dy 3

(6-62)

function

f(,)179

TURBINE

which

DESIGN

depends

AND

APPLICATION

on the dimensionless

y-coordinate, Y

=

This

equation

has

the

following

n, where

(6--64)

boundary

f=_--fy=O

conditions:

at

n=O

(6-65) df _=1 dy Equation mate

(6-62)

solution

cannot

in the

asymptotic

expansion

able

More

point.

(6-62)

with

df/dy, the

and

degree

profile at the

wall

series

expansion

two

solutions

(ref.

of accuracy,

turns

oo

Blasius

Howarth

of figure and

the

_=

exactly.

of a power

for ,1 = _,

recently,

a high

be solved

d2f/dy _ as functions

velocity

curvature

form

at

6-15.

This

profile

rather

abruptly

.4

.6

about

71= 0 and

at a suit-

the

Blasius

equation

tabular

= u/u_,

the

possesses further

values

gives

a very

small

from

it in order

E .2

.8

1.0

Boundary-layervelocity ratio, u/ue Fmuaz 180

6-15.--Blasius-Howarth

velocity

for f,

solution

!_ s-. 0

an

joined

provided df/dy

an approxi-

being

5) solved

and

of 7. Since

obtained

profile for flow on a flat plate.

INTRODUCTION

to reach

the

inflection,

asymptotic

since

From

the

value.

At the

BOUNDARY-LAYER

wall itself,

the

curve

has

THEORY

a point

of

for y = O, O_u/Oy _= O.

order-of-magnitude

boundary-layer

TO

equations,

analysis we had

the

performed

to obtain

Prandtl's

relation

(6-66) For a semi-infinite

flat plate,

the

Reynolds Rez-

number

can be expressed

u,x

as (6-67)

p

In order

to make

equation

(6-66)

dimensionally

x2

correct,

we can say

(6-68)

Rez

or

_is.u ¢c

The

constant

solution

of proportionality

and is equal

in laminar

flow,

(6-69)

can be obtained

to 5. So, for a semi-infinite

we

obtain

the

useful

from Howarth's flat plate

relation

for

numerical

at zero incidence

the

boundary-layer

thickness

(_s_zz=5.0 With

the

ing relations

use of Howarth's for other

flow on a flat plate

solution

important

v/_

to the Blasius

boundary-layer

(6-70) equations, parameters

the

follow-

for laminar

can also be obtained:

= 1.72

0=0.664

gr,, pUe 2

= 0.332

v_

(6-71)

v]_

(6-72)

0.332

_/"-:-_U_g

(6-73) =

1.328 D = _ b _¢/-_plu2 g

(6-74)

181

TURBIN]_

DESIGN

AND

APPLICATION

1.328

(6-75)

CI = 1.328 _-_ "uJ where I"W

shear

D

total

b

width

of fiat plate,

l

length

of fiat plate,

Cf

dimensionless

Re_

Reynolds

It should flow;

stress

on the surface,

drag

on both

equations plate

then

will

are valid

Rez < 106, a value

will probably

larger

occur,

for laminar

that

and the

is indicative expressions

the leading

to turbulent than

only

For Re_ > 106, transition

only from

If transition be

for fiat plate

l

of the plate.

will be valid

point.

drag

length

relations

length

layer

to (6-75)

the

N; lbf

coefficient

on plate

only where

the entire

to the transition

occur,

based

all of these

boundary (6-71)

_

m; ft

are valid

flow over

to turbulent

of fiat plate,

drag or skin friction

that

that is, they

lbf/ft

m; ft

number

be noted

of laminar

sides

N/m_;

boundary

that

calculated

Solving

the

in

edge of the layer by

does

equation

(6-74). Integral

M_thods

for

Laminar-Boundary-Layer The tions

two are

principal by

integral

means provide cumbersome. Integral

means

methods

of solving

methods

approximate are

and

based

Von

K_rm_n's

original

was

later

translated

by NACA

was

not

necessary Instead, in the

boundary layer,

only

are

over are

the

integrated thickness

introduced,

pressible

over the

the

differential

y=O

(6-50))

following

of fluid

equation

equation

(eq.

thickness. to y=_/,,n, and

momentum

equations

realized

result.

close

in the

(6-23)

the

a mean

(6-30)) (6-23)

definitions

thickness For

to the

boundary

Such or

it

fluid

by satisfying

is satisfied.

if the

that

for every

If equations and

integral

in 1912 in Germany

Khrm_n equations

region

Both

extremely

momentum

flow is approached

remaining

are

equations

boundary-layer

(eq.

laminar,

by and of dis-

(6-54)) incom-

flow,

dO+ (20+,_) u2 dx

182

6). Von

equa-

methods.

solutions

Khrm_n's

boundary-layer

boundary-layer from

exact

was published

(ref.

the

momentum

(eq. the

yon

external

In the

a mean from

integration placement

where

finite-difference

since

work

the

he satisfied

conditions.

is obtained (6-30)

to satisfy region

by

on

and

wall and

the laminar-boundary-layer

solutions,

formula.

particle.

Equations

u. du. = gr._.._. d'---x p

(6-76)

INTRODUCTION

For laminar,

TO

THEORY

compressible flow, uJ d_ + (20+_-MJ0)

where

BOUNDARY-LAYER

the subscript

e denotes

u, du, d--_ = pT._2_ p,

conditions

at the outer

(6-77)

edge of the boundary

layer. Equation for the

(6-76)

boundary-layer

sumed

for the

placement

at the wall, a solution

published

in

Pohlhausen

families

1921

7 and earlier

Pohlhausen's

distributions pressible

form

profile. and

exact

and

the

profile

shearing to

work

was

assumed

by

"Velocity

the simplest,

As a result, by

dis-

(6-76)

His

under

method

with

the

use

that

does the

Pro-

it is known

various

authors

assuming

solutions

not require

different was

solutions

velocity

for

laminar

incom-

solution

of ordinary at the wall,

a type

quantities

for the

relation

gradients.

of

known

its derivative

of these

that

all

specifying

universal

pressure

the

wall shear, forms

of exact

Pohlhausen's

compared

without

a nearly

favorable

and

approximate

method

nondimensional

It developed for

velocity

followed

collected

to one another

To do this,

quantities

layers.

chapter,

his

which

He relates

factor

evaluated

layer.

those

Thwaites'

equations.

the

The

pressure.

extend

Thwaites

from

differential and

among

8).

flow.

0, and

is asthe

distributions.

work (ref.

thickness,

is probably

of rising

and

form

first to use equation

in this

solution

in regions

a suitable

us to calculate

boundary

1).

equation

allows

was the

(refs.

differential

that

This

momentum

discussed

of velocity

Thwaites

provided

incompressible

to improve

A famous

to an ordinary

u/u,.

T_. Pohlhausen

to give poor results tried

6, the for

was

Although

have

profile,

thickness,

obtain

leads

thickness,

velocity

stress

files."

or (6-77)

were

laminar

existed

For

of velocity defined

boundary among

adverse

these

gradients,

Thwaites selected a single representative relation. A unique correlation was chosen that reduced the solution of an incompressible problem to the evaluation

of a single

pressible heat

fluids

transfer

formation compressible One Their

best

and

by

the

method

applies or

pressure

Prandtl

symmetric

distribution

and

extended

recognized is equal

could

be

and

to date Reshotko

when

to 1, a transused

for the (refs.

or incompressible surfaces.

performs

to comthat to

relate

solutions.

to appear

of Cohen

to compressible

axially

10)

boundary-layer

is that

was

number

(ref.

methods

layers

method

(ref. 9). They

Stewartson

integral

boundary

stream

Thwaites'

Crabtree

to incompressible

of the

dimensional

and

is negligible, proposed

laminar

integral.

by Rott

It

flow

handles

well in areas

solution 11 and over

of 12). two-

arbitrary

free-

of adverse

pres-

sure gradient. A surface temperature level may be specified, and heat transfer is calculated. Cohen and Reshotko's method is based on Thwaites' correlation

concept.

Stewartson's

transformation

(ref.

10)

is first

applied

183

TURBINE

to

DESIGN

Prandtl's

equations the

equations.

wall shear,

the

Then

parameters by

The

surface

relations,

methods

In

to have

values

and and

for

Laminar-Boundary-Layer Finite-difference digital

come

computers.

work

with

and

this relatively

Clutter

short and

Turbulent referencing

boundary-layer length" used

flow,

should in many

stresses matical

form

which,

and the

pressure.

first

worked

of viscosity

14 and 16).

of

on the

computer.

for

recent give

very

in

solving

viscosity" to relate mean

into

the

only

transformed

of the

governing values

been

Reynolds of velocity a matheequations, of density,

equations

mean

in 1877.

law for laminar

the

values

differential

"mixing

have

are given

mean

turbulent

and

concepts

stresses

problem

of

Flow

to the

calculation

Another

methods

to date

containing

on this

in Stokes'

15).

of

amount

Concepts

"eddy

substitution

equations

development

These

approximation

Reynolds

for the

1.

the

the

methods

motion

the

a

which

from

a considerable

times

current

mixing

These

point

running

developed

means,

but

of

Mixing-Length

equations

starting

Boussinesq coefficient

the

published

boundary-layer

done

(ref.

These

upon

to differential

velocity, stitute flow.

this

have

concepts

methods

by

By

the

discussed.

of the

produced

components. leads

be

the

(refs.

of the

Young different

Boundary-Layer

any

important

Reshotko's,

Solving

because

of Krause

Eddy-Viscosity

Before

solving

technique

is that

carried resulting

Equations

prominence

Smith

of interest

results

for

into

in developing

reference good

methods

recently

the

of all the

slightly

Methods

of these

is then

11. With

Luxton

as Cohen

to

free-stream

quantities

calculation

1960,

is as general

Finite-Difference

have

transformed

of these the

related

interdependence

of reference

for

differential

parameters

and the

evaluation

number

first-order

of a unique

derived

13) which

Prandtl

transfer,

solutions

parameters.

(ref. the

The exact

are

boundary-layer method

heat

nonlinear,

of dimensionless

concept

is assumed. the

resulting

in terms

Thwaites'

utilizing

allows

APPLICATION

are expressed

velocity. out

AND

con-

boundary-layer

In analogy

with

the

flow

t_ Ou (6-78)

r,g Oy where mixing

rz is the coefficient,

laminar A,,

shear

stress,

in N/m s or lbf/ft

for the Reynolds

stress

in turbulent

2, he introduced flow by putting

A, 0_ r_ -

184

g 0y

(6-79)

a

INTRODUCTION

where

rt is the

introduced

turbulent

shear

concept

of eddy,

the

stress,

TO

BOUNDARY-LAYER

in N/m _ or lbf/ft

or virtual,

THEORY

_. In 1880, Reynolds

viscosity,

_, where

AY

, =--

(6--80)

p

Thus,

the

eddy

Turbulent

viscosity

stress

is analogous

to the

be expressed

as

can then

paa g Oy With

the use of this concept,

can be written

A similar

concept

eddy-viscosity

In

mean

1925,

free theory

itself

concept

good

deal

with

deals

the

with

Deriving

(6-82) than

and (6-40)

such

equation

where

as

an eddy,

difficulty

with

_ depend

on velocity.

between

applying

these

l is the mixing (6-81)),

the It is,

coefficients

1. His

Prandtl's

_ of the length

is generally is equation

more (6-81).

--P-

}-_y dy

mean kinetic

whereas

of large

clusters

of

stress

requires

a

shear

of turbulent

expression

to the is that

of particles,

motion

model

p l 2 Ida da --=

flow,

all of which

is (6-82)

U'V'

g

in m or ft. expression

that

little

first expression l of the

analogous

for

for

mixing-length

difference

motion

expression final

approximation

Prandtl's

main

macroscopic

of his physical

it appears

The

microscopic

the

length,

different

is called is somewhat

Prandtl's

in reference

On comparing

mixing

The hence

of gases.

g

viscosity

energy

relations

length

theory

rt =-

(eq.

and

argument

mixing

of discussion

is contained

where

(6-37)

a completely

His

the

concerns particles.

A,

empirical

introduced

kinetic

Prandtl's

is that

stresses,

in the

fluid

to the

can be defined.

to find

Prandtl since

path

(6--81)

velocity.

Reynolds

hypothesis,

p= u/p.

-Pu'v' g

in equations

be applied

method necessary

the

the

can

conductivity

therefore,

viscosity

as

or a virtual,

and

terms

kinematic

second

(eq. has

(6-82)) been

with that

gained.

has merely

been

expression.

However,

suitable Turbulent

for the dra_:

The

replaced

calculation is roughly

of Boussinesq unknown

by the Prandtl's

of turbulent proportional

eddy

unknown equation motion to the 185

TURBINE

DESIGN

AND

square

of velocity,

mixing

length

So, mixing

length

superiority

the

same

fluid.

local

It is far

1 than

about

of Prandtl's

integral

with

the

methods

since

was

the

first

for an incompressible in 1931.

improvements

to the

many

in 1949 and

empirical

was

relation

equation.

This

and

relation

transformations

10)

Maskell, pressible

in 1951

and

thus

obtained Tillmann pressure

to

the

equations

was

published

of them

empirical was

in 1950

in many

layers.

the

current

used

published

(ref.

in the

making

data

by

in Ger-

17), proposed

momentum

methods.

an

integral

Stewartson's

used

in many

methods

for solving

an

improved

method

for incom-

He replaced

approximation momentum

gradient.

determine

Truckenbrodt, translated

whose

by NACA

turbulent

and,

like

tion.

the momentum

which

thickness.

It

applies

Because

brodt's

equation

is directly A profile

method

is still

use

20).

Prior

been

utilized law,

of integral

to their with and

and

use

the

integrable parameter

is

the

of several

layers by

K_irm_tn

turbulent

Reshotko

and

momentum

boundary-layer empirical

first

equa-

symmetrical results,

Trucken-

boundary treated

integral

skin-friction

is simple

integral

layers.

adequately

Tucker

velocity

was

laminar

method

rotationally

were

adverse

for both

The

accurate

for incompressible

with

in 1952 and

momentum

and

relatively

boundary

an assumed one

not

flows

solutions flows.

two-dimensional

methods

work,

19), proposed

does

for

in Germany

boundary-layer

used

turbulent

point

published

(ref.

method, to both

was

of its simplicity

Compressible the

work

in I955

Maskell's

with

a separation

incompressible

flows.

186

turbulent solutions,

from an empirical auxiliary differential equation. The Ludwiegskin-friction formula is used to calculate the skin-friction disand

power

work

term

work

most

whose

proposed

determined

determines

tribution

and

18),

boundary

by an empirically

both

equations.

(ref.

turbulent

followed, and

likewise

the turbulent-boundary-layer

His

technique by NACA

are

the

for solving

layer.

skin-friction

is still used

(ref.

the

are

approximate

a method

Tillmann,

the

there

for solving

provide

of works

translated

for

the

the

equations,

boundary

calculational

Ludwieg

about

constitutes

Equations

methods

to propose

A rash

it is a

flow are now impossible.

turbulent

in Germany Gruschwitz.

Solving

of these

for turbulent

say

of Boussinesq.

Boundary-Layer

Both

this

if the

of velocity.

assumptions

_, and

that for

finite-difference

solutions

Gruschwitz

over

(6-82)

we cannot

to make viscosity

Methods

equations.

exact

although

laminar-boundary-layer

and

boundary-layer

eddy

from

of the magnitude

function,

expression

Turbulent as

is obtained

simpler

the

Integral

Just

result

to be independent

is a purely

of the

mixing

and

is assumed

length

property

APPLICATION

in 1957

(ref.

equation

profile,

had

usually

relations.

the

When

INTRODUCTION

pressure

gradient

of-momentum tiplying The

equation,

the

normal

was present,

integrand

to the

and

Tucker's

and

pressure

transfer

form

and

(ref.

10)

usually

used.

equation

is obtained

are and

momentum

transfer

(ref.

21).

by

by

auxiliary

mul-

a distance

to that

distance.)

equation

were

then

to compressible

flow

with

with

of Maskell

results is used

through

years

boundary

the

One

of

the

best

integral

an extension

the

same

and

moment-of-momentum

analysis,

simultaneously tributions

of the

Reshotko-Tucker

boundary-layer

boundary-layer

11 and 12)

source

analysis

is the

work

Finite-Difference

Finit_difference equations

portion Atwell

(refs.

28

Patankar

recently

begun

of this work have

29)

and

Spalding

is going

to date

based

on have

boundary-layer on in this

of Herring

(refs.

the

use

developed equations

field

at the

the

It

momentum are solved

shear-stress is better

dis-

turbulent than

of adverse program (ref.

compressible

Solving

is

somewhat

equations

and Mellor

for

that of pressure based on

22)

tech-

turbulent (ref.

24).

the

Equations

solving

the

turbulent

Cebeci

25, 26, and

methods

22).

uses

Sasman-Cresci on

today.

compressible (ref.

It

in regions a computer

to appear.

also developed

and

turbulent

for

turbulent

for

Cresci

analysis

and

until

of equilibrium

Boundary-Layer

methods

have

results

Methods

Turbulent

today

These

of information

where

programs

boundary-layer

numerical

point

method,

compressible

computer

to uncouple

Sasman-Cresci

(refs.

additional

as the This

method.

at predicting separation (ref. 23) has developed

Cohen-Reshotko An

The

the

and

is made of

recent

to simplify

for

equations.

introduction

analysis.

Reshotko-Tucker gradient. McNaUy niques.

no attempt

from

are used zero.

of Sasman

integral

after

the

available

is that

but

obtained

methods

through

is located

in many

concept

distribution

profile

available

layers

flow with

reference-enthalpy

becomes

used

Ludwieg-Tillmann

for compressible

Separation

best

moment-

transformation

The

shear-stress velocity

It is still widely

boundary

simply

the

18).

suitable

extrapolated,

and

in incompressible

of Stewartson's

of Eckert's

power-law

was

expressed

(ref.

in a form

equation.

ago,

are

use

for the

when

layers.

turbulent

the

application

and

friction,

several

also uses the momentum These

uncoupled

layer

skin

work

respect

applicable

the

moment-of-momentum

the

the

the moment-

equation

with

and

gradient,

An approximation

boundary

and

integral

integrating

equations.

relation

heat

large

(This

method,

integral

skin-friction

the

equation,

equation

and

of-momentum

the

THEORY

an auxiliary

then

integral

BOUNDARY-LAYER

simultaneously.

Reshotko heat

of the

surface

momentum

solved

was

TO

boundary-layer

and

Smith

27).

Bradshaw,

for the turbulent of the still (refs.

present

turbulent another

time,

energy 31).

and

done

a

Ferriss,

boundary

method

30 and

have

layer

equation. for handling

A great

no method

deal

of

is yet 187

TURBINE

clearly

DESIGN

AND

superior

(refs.

32

integral

and

to 33)

and

finite

APPLICATION

any

of the

compare

others.

many

difference,

for

Two

of

the

The

selection

of a method

problem

able. that of

This

can

have

been

solution

solution of

requires

the

the

be achieved mentioned

has techniques,

whole

been

herein.

the

variety

boundary-layer

methods,

turbulent

with

studying

intended

publications

prominent

suitable

familiarity

by

recent

boundary

both layer.

REMARKS

of solution

some

most

solving

CONCLUDING

layer

relatively

some

The to

various

of the

more

present

show

the

of methods problem,

to a particular the

recent

discussion

avail-

references

of the

historical

available, especially

boundarymethods

methods

development and

where

the

of

complexity

turbulent

flows

are involved.

REFERENCES 1. 2. 3. 4. 5. 6. 7.

HERMAN_ (J. KESTIN, TRANS.)" Boundary Layer Theory. McGrawHill Book Co., Inc., any edition. BIRD, R. BYRON; STEWART, WARREN E.; AND LIGHTFOOT, EDWIN N.: Transport Phenomena. John Wiley & Sons, Inc., 1960. PRAN_rL, L. : Motion of Fluids with Very Little Viscosity. NACA TM 452, 1928. BLASIUS, H.: The Boundary Layers in Fluids with Little Friction. NACA TM 1256, 1950. HOWARTH, L." On the Solution of the Laminar Boundary Layer Equations. Proc. Roy. Soc. (London), Set. A, vol. 164, no. 919, Feb. 18, 1938, pp. 547-579. YON K_RM_N, TH: On Laminar and Turbulent Friction. NACA TM 1092, 1946. POHLHAUSEN, K. : Approximate Integration of the Differential Equation of the Limit Surface of Laminar Motion. Zeit. f. Math. Mech., vol. 1, Aug. 1921, pp. 252-268. SCHLICHTING,

8. THWAITES, B.: Approximate

Calculation

of the Laminar

Boundary

Layer.

Aero-

naut. Quart., vol. 1, Nov. 1949, pp. 245-280. 9. ROTT, NICHOLAS; AND CRABTREE, L. F.: Simplified Laminar Boundary-Layer Calculations for Bodies of Revolution and for Yawed Wings. J. Aeron. Sci., vol. 19, no. 8, Aug. J,952, pp. 553-565. 10. STEWARTSON, K. : Correlated Incompressible and Compressible Boundary Layers. Proc. Roy. Soc. (London), Ser. A, vol. 200, no. 1060, Dec. 22, 1949, pp. 84-100. 11. CO_EN, CLAaENCS B.; AND RESHOTXO, ELI: Similar So|utions for the Compressible Laminar Boundary Layer with Heat Transfer and Pressure Gradient. NACA TR 1293, 1956. 12. COHEN, CLARENCE B. ; ANY RESHOTKO, ELI: The Compressible Laminar Boundary Layer with Heat Transfer and Arbitrary Pressure Gradient. NACA TR 1294, 1956. 13.

188

LVXTOS, R. E.; AND YOUNG, A. D. : Generalized Methods for the Calculation of the Laminar Compressible Boundary-Layer Characteristics with Heat Transfer and Non-Uniform Pressure Distribution. R&M-3233, Aeronautical Research Council, Gt. Britain, 1962.

INTRODUCTION

TO

BOUNDARY-LAYER

THEORY

14. SMITH, A. M. 0.; AND CLUTTER, DARWIN W." Solution of the Incompressible Laminar Boundary-Layer Equations. AIAA J., vol. 1, no. 9, Sept. 1963, pp. 2062-2071. 15. 16.

SMITH, A. M. O.; ANY CLUTTER, DARWIN W.: Machine Calculation of Compressible Laminar Boundary Layers. AIA_ J., vol. 3, no. 4, Apr. 1965, pp. 639-647. KRAUSE, EGON: Numerical Solution of the Boundary-Layer Equations. AIAA J., vol. 5, no. 7, July 1967, pp. 1231-1237.

17.

LUDWIEG, HUBERT; Stress in Turbulent

AND TILLMANN, W.: Investigations Boundary Layers. NACA TM 1285,

18.

MASKI':LL, E. C. : Approximate Calculation Two-Dimensional Incompressible Flow. Establishment, Nov. 1951.

19.

TRUCKENBROIYr,

E.:

A Method

and Turbulent Boundary Flow. NACA TM 1379, 20.

21.

for Calculation

of the

in Case of Plane and Rotationally

Layer in Aircraft Laminar

Symmetrical

ELI; AND TUCKER, MAURICE: Approximate Calculation of the Compressible Turbulent Boundary Layer with Heat Transfer and Arbitrary Pressure Gradient. NACA TN 4154, 1957. ECKERT, E. R. G. : Engineering Relations for Friction and Heat Transfer to SurRESHOTKO,

faces 587. 22.

Wall-Shearing

of the Turbulent Boundary Rep. AERO 2443, Royal

of Quadrature

Layer 1955.

of the 1950.

in High

Velocity

Flow.

J. Aeron.

SASMAN, PHILIP K.; AND CRESCI, Layer with Pressure Gradient 1966, pp. 19-25.

Sci., vol. 22, no. 8, Aug.

ROBERT

and

Compressible Transfer. AIAA

J.:

Heat

1955,

pp. 585-

Turbulent Boundary J., vol. 4, no. 1, Jan.

23.

MCNALLY, WILLIAM D.: FORTRAN Laminar and Turbulent Boundary NASA TN D-5681, 1970.

24. 25.

A Method of Calculating Compressible CR-1144, 1968. CEBECI, T.; SMITH, A. M. 0.; AND MOSINSKIS, G.: Solution of the Incompressible Turbulent Boundary-Layer Equations with Heat Transfer. J. Heat Transfer, vol. 92, no. 1, Feb. 1970, pp. 133-143.

26.

SMITH,

HERRING,

H.

Turbulent

A.

Layer 1967.

M.

JAMES;

AND

Boundary

0.;

AND

Equations.

MELLOR,

Layers.

G.

Program for Calculating Compressible Layers in Arbitrary Pressure Gradients. L.:

NASA

T. : Numerical Solution of the Turbulent-BoundaryDAC-33735, Douglas Aircraft Co. (AD-656430), May

CEBECI,

Rep.

27.

CERECl, T.; ANn SMITH, A. M. O.: A Finite-Difference Method for Calculating Compressible Laminar and Turbulent Boundary Layers. J. Basic Eng., vol. 92, no. 3, Sept. 1970, pp. 523-535.

28.

P.; FERRISS, D. H.; AND ATWELL, N. P.: Calculation of BoundaryLayer Development Using the Turbulent Energy Equation. J. Fluid Mech., vol. 28, pt. 3, May 26, 1967, pp. 593-616. BRADSHAW, P.: Calculation of Boundary-Layer Development Using the Turbu-

29.

BRADSHAW,

lent Energy Equation. Lab., Jan. 30, 1969.

IX: Summary.

Rep.

NPL-Aero-1287,

National

Physical

30.

PATANKAR, S. V.; ANY SPALDING, D. B. : A Finite-Difference Procedure for Solving the Equations of the Two-Dimensional Boundary Layer. Int. J. Heat Mass Transfer, vol. 10, no. 10, Oct. 1967, pp. 1389-1411.

31.

PATANKAR,

Layers. 32.

S. V.; AND C.R.C. Press,

SPALDING,

D. B." Heat

and

Mass

Transfer

in Boundary

1967.

COLES, D. E.; ANn HIRST, E. A., ED8.: Proceedings, Boundary Layers--1968, AFOSR-IFP-Stanford Press, 1969.

Computation of Turbulent Conference. Stanford Univ.

189

TURBINE

33.

DESIGN

BERTRAM, SP-216,

190

AND

MITCHEL 1969.

APPLICATION

H.,

ED.:

Compressible

Turbulent

Boundary

Layers.

NASA

INTRODUCTION

TO

BOUNDARY-LAYER

THEORY

SYMBOLS

A_r

turbulent

a

constant

b

flow mixing in eq.

coefficient,

(N)

(sec)/m_;

lbm/(ft)

(6-44)

m; fwidth of flat plate, ft constant in eq. (6-44)

Cf

skin-friction

e,

constant

cp

specific

D

total

d

constant

f

Blasius

dimensionless

f

general

body

coefficient in eq.

heat drag

for a flat

plate

(6-44)

at constant

pressure,

on flat plate, in eq.

J/(kg)(K);

Btu/(lbm)(°R)

N; lbf

(6-44) stream

force

vector,

function N/kg;

defined

by eq. (6--63)

lbf/lbm

component

of body

force

f in x-direction,

N/kg;

lbf/lbm

f, f.

component

of body

force

f in y-direction,

N/kg;

lbf/lbm

component

of body

force

f in z-direction,

N/kg;

g H

conversion

constant,

1; 32.17

(Ibm)

i

form factor, unit vector

defined by eq. (6-56) in the x-direction

J

conversion

constant,

J

unit

in the y-direction

k

vector

1 ; 778

(ft)/(lbf)

k L

characteristic

1

Prandtl mixing length, m; ft length of flat plate, m; ft Mach number external to the boundary

length

on the

(e.g.,

(K);

Btu/(sec)

the blade

turbulent

chord),

m; ft

layer

exponent

P

dimensionless

p R

static

Re

Reynolds

number

based

Rez

Reynolds

number

based

on l, as defined

in eq.

Re_

Reynolds

number

based

on x, as defined

by eq.

r

radius

pressure,

gas constant,

defined

N/m2; J/(kg)

velocity

(ft) (°R)

n

pressure,

lbf/ft (K)

of curvature

constant

in eq.

T

absolute

static

T,

absolute

total

To

reference

t

time,

U

dimensionless

Uo

free-stream

velocity

U

component

of general

(6-25),

eq.

(6-47)

(6-13e)

(lbf) / (lbm)

on L and

(°R)

U0, as defined

surface,

by eq.

(6-13f)

(6-75) (6-67)

m; ft

K; °R

temperature, temperature,

temperature

profile,

by eq.

2

; (ft)

of blade

S

lbf/lbm (sec _)

(ft) (lbf)/Btu

thermal conductivity, W/(m) unit vector in the z-direction

M.

(sec)

K; °R K; °R

used

in eq.

(6-25),

K; °R

sec velocity

in x-direction, upstream velocity

of blade, vector

defined m/sec; u in the

by

eq.

(6--13c)

ft/sec x-direction,

m/sec;

ft/sec 191

TURBINE

DESIGN

AND

APPLICATION

U

general

velocity

vector,

U.

free-stream

V

m/sec; ft/sec dimensionless velocity

p

component

m/sec;

velocity

at the

ft/sec outer

edge

in y-direction,

of the

defined

boundary

by eq.

layer,

(6-13d)

of general

velocity

vector

u in the

y-direction,

m/sec;

of general

velocity

vector

u in the

z-direction,

m/sec;

ft/sec tO

component

X

ft/sec dimensionless

Y

x-coordinate, m; ft coordinate parallel to boundary surface, m; ft dimensionless y-coordinate, defined by eq. (6-13b)

Y Z

x-coordinate,

y-coordinate, m; ft coordinate perpendicular z-coordinate, m; ft displacement

_yuu

eddy

surface,

m; ft

m; ft

thickness,

viscosity

by eq. (6-13a)

to boundary

thickness,

boundary-layer

defined

defined

m; ft

by eq.

(6-80),

m2/sec;

ft2/sec

¢

a dimensionless quantity much less than 1 Blasius transformed y-coordinate defined by 0

momentum

thickness,

dimensionless

viscosity,

dynamic

viscosity

Ibm/(ft)

parameter (N)

P P_

T!

lbm/ft laminar

Tt

turbulent

Tw

shear

viscosity,

a shear

time

at the wall,

defined by eq. stream function, in eq.

(6-26)

192

average

fluctuating

N/m2;

stress,

Superscripts"

t

m2/sec;

stress,

shear

stress

constant

2; lbm/(ft)

at reference

density, kg/m _; lbm/ft 3 free-stream density external

60

by eq.

(6-45)

(sec)

temperature

To,

(N)(sec)/m_;

(sec)

kinematic

function Blasius

defined

(sec)/m

p

_0

(6-64)

m; ft

shape

dynamic

eq.

component

ft2/sec to

the

lbf/ft

N/mS;

2

lbf/ft

N/m_; (6-34) m2/sec;

lbf/ft

_ 2

ft_/sec

boundary

layer,

kg/mS;

CHAPTER 7

Boundary-Layer Losses By Herman W.Prust,Jr. The

primary

builds are

up

the

friction

surfaces, blade the

cause

on the the

theory,

fluid. by

Chapter

means

analytically for

associated

the

A fundamental loss resulting

objective from

expressions

the

loss coefficients. as a fractional the

blade

subtracting

these

the

blade-row

efficiency

pressure 7-1.

proceeding

loss

coefficients, and

These

velocity pressure

the

coefficients

the

ideal based from used

discussion

blade-row

distributions and

The

theory

velocity

the

the

buildup and

presented

herein

refers

layers. losses

row.

the

Methods from

and

in chapter

of the

energy this

twoenergy the

of kinetickinetic

actual

flow

can be obtained is consistent

with

2. locations

will be introduced distributions

the

for

Therefore,

of boundary-layer station

the

loss in fluid

energy

on kinetic

be

losses

are in terms

express

can

experimental mixing

blade

kinetic unity,

high-velocity

is to minimize

chapter

the from

and

end-wall design

resulting

the

to boundary-layer

boundary

in this

definition

past

blades

analytical

of fluid through

coefficients

flow of fluid

trailing-edge,

in blade-row

of the

over

boundary-layer

blade plus discussed.

Efficiency

with

the

that losses

fluid

fluid with

covers

layer.

layer these

viscous

of the

blade-section

These part

row.

by

Before

surface

for loss developed

energy through

flow

from

friction,

boundary

energy

and

the

three-dimensional results are also

of the

an introduction

chapter

to two-dimensional

obtaining dimensional

final

the

This

determining

with

primarily

flow

boundary-layer 6 presented

boundary

In particular,

loss downstream

of which

described.

methods

the

of the low-velocity

free-stream

the

loss resulting

and

is the

surfaces.

from

pressure-drag edge,

in a turbine

and end-wall

loss resulting

trailing mixing

of losses

blade

with and

parameters and

the

associated

the aid of figure associated

dis193

TURBINE

cussion layer,

DESIGN

refer

AND

APPLICATION

to an attached

with its associated

higher Figure

loss,

and

7-1 (a)

cannot indicates

boundary

reversal

only.

of flow at the

be analyzed the

layer in the

four

station

A separated

surface,

same

is thicker,

manner,

locations

boundary

that

yields

if at all. will be referred

Station

,_---- $ --.--_

0

r la r"

'-1

(a)

.... --

Station 0

Total pressure Static pressure Velocity

I'--'--'I

_-r]

rq

v,,,,, .1| !,/ Station la

Station i

Station 2

(b)

FIovr_

194

7-1.--Station

(a) Station locations. (b) Pressure and velocity distributions. locations and associated press_Ire and

velocity

distributions.

a

BOUNDARY-LAYER

to in this

chapter.

Station

station,

a uniform

total

7-1 (b).

Station

boundary pressure stream

la

developed

profiles

as shown

value through

from

the

surfaces.

region value

pressure

Station

This

and flow angle la and

are

l, the

sufficiently associated profiles In

the

order been

inlet

conditions

both

free

stances,

constant

tests

but

and

seldom

the

exists

stream

and

boundary

a real

in the

the region

velocity _:,u. the these

to

downstream

mixing

downstream

state

fluid flows and

adjacent

over

the

boundaryfree stream profiles

static

pressure

Between

stations

at a distance

and

with

the

total-pressure

applications. do vary

at stations variation

of the

flow

in the last chapter

specifically used for obtaining be introduced and defined.

of

Experiments

somewhat and

across

1. In some

in-

can be accounted does

take

place,

a hypothetical

a loss results

and friction

for.

a com-

convenience.

surface. V:,

As shown varies at

the

layer,

certain

momentum desired

the

by

from full

zero

friction

layers

figure

7-2,

of fluid the

velocity

and

energy

thickness, here;

kinetic-energy

form

the

height

resulting

are used.

fluid

on

boundary-layer

parameters

and will be reviewed the

due to both

between

in flow, momentum,

thickness,

la

this

region

boundary

Uniformity

can be approached

PARAMETERS

velocity

the losses

of the

of variables

stations. usually

in actual

a surface,

to the

a number

various

is merely

surface

boundary-layer

free-stream

(displacement

introduced

as is

occurred.

mixing,

velocity

no

blade

la,

station-1

2 is located

that

layer

identified,

To describe presence

the the

too,

complete

and flow angle

will be later

fluid

in the

surface

the

station.

that The

BOUNDARY-LAYER When

with

discussion,

across

pressure

uniform

between

where

the

convenience

static

some

pletely

pl_ at the loss has

Here

freevaries

station

edge,

Station row

place.

analysis

that

which

Although

taken

is a universal

in component shown

has

blade

pressure

mixing by

the

is, of course,

across

region.

across

from

uniform.

assumed

have

constant

and

pressure

1 (b)

wake

loss occurs.

to simplify

have

little

The

Total

friction

trailing

in figure entire

of the

loss,

There

edge.

constant

where

blade.

in velocity

varies

surfaces.

At this in figure

of the

result

Velocity

trailing

blade

but

the

trailing-edge

mixing

edge

la, only the surface

assumed

are again

blade

row.

as indicated

surfaces

7-1 (b).

solid

is indicated

downstream

to the blade

trailing

is assumed

the void,

flow throughout

inlet

P']8.1,,= po' to the static

beyond

has filled

occurred.

showing

blade

a_o. At station

1 is just

fluid

the

on the at the

This

static

within

of the

free-stream

the

po' is assumed,

in figure

V:,.I,_ to zero the

the flow angle

has

is just

layers

flow

layer

0 represents pressure

LOSSES

from Some

factor)

in addition, coefficients

of

were others will

195

TURBINE

DESIGN

AND

APPLICATION

Free-streem velocit L Vfs

=i

,- Full boundary

W)odty, v

=;/

layer height.

¢////////.4,

_full

_'//////////_ Surface

FmuRz

The

7-2.--Typical

displacement

flow, is defined

boundary-layer

thickness

5, which

velocity

profile.

is indicative

of the

loss in mass

by

(pV) f. dY-

(0V)s,=L

_lul!

(pV)

dY

(7-1)

_0

where $

displacement

thickness,

boundary-layer V

fluid velocity,

P

fluid

Y

distance

()f.

free-stream

ary

is equal

layer

lb/fP normal

(ideal)

states

the loss in mass ideal

to the

The

momentum

the

196

layer,

m; ft

thickness

dy_

flow of the

which

would

pass

thickness.

fluid in the through

Solving

for

bounda length

_ yields

dY fo_:=u ...... pV (pV):.

O, which

(7-2)

is indicative

of the

momentum

by

O(pV_):.=Jo where

flow

displacement

['/"u =-o

loss, is defined

to boundary

conditions

that

to the

equal

m; ft

ft/sec

kg/m3;

in direction

(7-1)

(or an area)

m/sec;

density,

Equation

m; ft

thickness,

0 is the momentum

loss in momentum

fs:.u

(pVV/,)

thickness, of the

fluid

dy_

fo_,,u

pV 2 dY

in m or ft. Equation in the

boundary

layer

(7-3)

(7-3)

states

is equal

that to the

BOUNDARY-LAYER

ideal

momentum

an area)

of the

equal

to the 8=

The energy

loss

fa'"' -o

in kinetic

thickness

ideal

where

ff is the

energy

loss in kinetic

ideal

kinetic

(or an area)

equal

dY-

can

through

dY

(pV )i.

similarly

a length

(or

for 0 yields

PV_

fo s_'"

be

pass

Solving

(7-4)

expressed

in terms

of an

by

(pVV_,)

_0

thickness,

energy

energy

would

thickness.

pV (pV)t.

energy

defined

_ _b(pV_)/,=_

the

flow which

momentum

LOSSES

fluid in the

ideal

to the

energy

[6/.,, =-o

,V

flow

which

dy_

Ratios of the aforementioned

Equation

boundary

thickness.

(pV 3) dY

_0

in m or ft.

of the

of the

dY-

would Solving

fo sf"'z

(7-5) layer

pass

(7-5) states

is equal

through

that to the

a length

for ff yields

,V 3 dY (pV3)s.

(7--fi)

thickness terms are also used as basic

boundary-layer parameters. The form factorH isdefined as H=

Substituting dimensionless

equations

(7-2)

distance

y as

and

-_

(7-4)

y-

(7-7)

into equation

(7-7)

and defining

Y

a

(7-8)

_futt yields 1

1

(pV)I, H =

(7-9) 1

(or)f, An energy

factor

E is defined

(or'),,,

as E= _8

Substituting

equations

(7-6),

(7-4),

(7-10)

and

(7-8)

into

equation

(7-10)

yields

197

TURBINE

DESIGN

AND

APPLICATION

ff

pV (pV)/,

fo 1 -- pV 3 dy (pV3)I.

dy-

E=

(7-11) fo

Velocity profile

profiles

of the

(pV)j, pV

dy-- f[

for turbulent

flow

pV2 dy (pV_)1.

are

often

represented

by

a power

type V _yn

where

the

Note

that

profile

this

as

1In

the

numerical

exponent

power 6 (eq.

is here

(6-47)) as n,

that

to be used

often

between

expressed

is expressed

with

expressed equations

value

n is most

profile

is consistent

exponent

wherein the

of the

in chapter

pressed The

value

(7-12)

general

as yn, while as yl/n. The

derived.

for n will depend

with

exusage.

the

form

same

reference

Therefore,

on the

the

theory

is consistent

are

0.25.

exponent

boundary-layer

however,

follow

0.1 and

1,

specific

being

used

for

exponent. With

in series expressed ratio

this velocity form,

and

in terms

V/Vcr.

The

profile, the

form

of the resulting 1 --4 n+l

equations and

(7-9)

energy

exponent

and

factors

n and the

equations 3A f,

{

3n+l

) can be integrated

free-stream

derived 5A_.

(7-11

for turbulent

flow can be

critical

in reference

velocity

1 are

F'--

5n+1

H=

(7-13) 1

A/,

+

(n+l)(2n+l)

A_,

+

(3n+l)(4n+l)

-_--Jl*

(5n+1)(6n+l)

and

2

(n_l_l)(3n_q_l)--I-

(3n+l)(5n+l)+(5n+l)(7n+l)

E.__

1 (n+l)(2n+l)

t-

Af,

A_,

+

(3n+l)(4n+l)

-_+-

•

I

o

(5n+l)(6n+l) (7-14)

where

As._"/-

198

_+1

1 (_-)

2 _

(7-15)

BOUNDARY-LAYER

and

_, is the

constant

ratio

of specific

wlume,

critical

and

(Mach

approaches

heat

at constant

Vc, is the

1) flow

fluid

condition.

zero, equations

(7-13)

pressure

velocity,

to specific

in m/sec

For

incompressible

and

(7-14)

LOSSES

flow,

reduce

heat

or ft/sec,

at

at the

where

V/V,r

to

H_.c=2n+l

(7-16)

and 2(2n+1) E_._ =

Values

of the

form

and

energy

(7-17)

3n+l

factors

for turbulent

are shown in figure 7-3 for V/V_r varying from 0 to 1.5. It can be seen that the form

from factor

does

exponent

the

energy

is almost

layer

aerodynamic directly the

simpler ideal

from more

the

so

parameters"

n, the

type

presented

of body.

instance,

momentum to the

the

physical

could are

defined

pass on

They

n varying more than

energy

the

the

factor

work,

blade

row.

The

of zero

in

be ob-

however, row, it is

as a fractional

"dimensionless

can

useful can

of the blade

losses the

and

are directly

In turbine

termed basis

general

of a body

boundaries

through

herein

are

drag

thickness.

to express

that are

just

For

expressed

and

0 to 1.4 and varies much

flow

V¢r.

meaningful

quantities

parameters

constant

on any

work.

flow is confined

and

any

parameters

to a boundary

where the

of V

boundary-layer

certain tained

For

independent

The refer

factor.

compressible

part

of

thickness thickness

trailing-edge

thickness.

Free-stream criticalvelocity

ratio, (VNcr)fs f-0 //-0.6 2.2

-V/21.o ,,,,

hJ

_ 1.4-7

Power n usedin velocity equation VNfs =yn

,-0

L=

1.8 E"

-\\\

\ rL00 1

1.4

I 2

FIGURE 7-3.--Effect

r].25 ' _LS0

I 3

of compressibility (Data

4 Form factor. H

5

on variation of energy from ref. 1.)

6

factor

7

with form

factor.

199

TURBINE

DESIGN

AND

APPLICATION

"%..\ Station

_

_'_'_

t 8

____-_

_ J

Ys

_cos

ala

FIGURE 7-4.--Nomenclature

These

dimensionless

suction-

and

With the

the

in flow, corresponding

row

channel.

are

composed

the

thicknesses

parameters and

ideal total

of the

subscripts

pressure-surface

defined

flow conditions

thickness

The

must

region.

represent

the

sum

of the

are the

same,

thicknesses.

that

momentum,

the

where

parameters

pressure-surface assumption

dimensionless

losses

and

thickness

for trailing-edge

value,

boundary-layer

could

loss plus

by

the

channel

through

as indicated the

dividing

blade-row

pass

channel,

one

blade-

in figure

pressure-surface

by 7-4,

loss,

Stot=$,A-_p

(7-18)

Otot-- O,-.FOp

(7-19)

q/tot= _,A-_

(7-20)

p denote

total

respectively.

thicknesses,

are expressed

obtained

for a single

that

for one

suction-surface

tot, s, and

are

energy

quantities losses

in all channels

value,

Thus, the

suction-surface

in terms

dimensionless

of the

value, previously

boundary-layer

as

($*=

s cos a(pV)I, e,o,(pV2)i.

0* =

S COS

ot(pU2)$,

-

(7-21)

s cos a

-

O,o, 8 cos

(7-22) c_

_/ tot

_* ....

(7-23) 8 COS 8 COS

200

0t (_)(pVa)l.

a

or

BOUNDARY-LAYER LOSSES where

0*

dimensionless dimensionless

displacement momentum

dimensionless

energy

8

blade

O_

fluid flow angle

Equations tum,

(7-21),

and

energy,

ties for the These the

spacing,

equations

trailing

thickness

m; ft from

axial

(7-22), row

(7-23)

be subscripted

or station

As mentioned friction, the

kinetic-energy

to apply the

LOSS

previously,

the

and

energy

are

losses

of the

ideal

kinetic

in

terms

of

coefficient

_11a8

7-4

for the

Since

coefficient

edge,

in the

COS

dimensionless

thickness

of the

in terms

the

of the

O_la(pWa)

of

in kinetic

blade-row

actual

dimensionless

fs,la

(pV3)i,

thickness, of the

aa

(7-24)

in m or ft. trailing

la, just

surface-friction

is expressed

loss

boundary-layer

to station

only

them

as the

energy

the

region

is referenced

it represents

in terms

for evaluating

expressing

_la, defined

trailing-edge

nomenclature

this

trailing

blade-row

within

Losses

as a fraction

t is the

la,

expressed

methods

_1_= (s cos oq_--_*_s cos axe--t) where

station

will be presented.

loss

be expressed as

to be zero.

edge.

to be

and

kinetic-energy

flow, can thicknesses

quanti-

COEFFICIENTS

losses

loss coefficients

momen-

ideal

is assumed

at either

In this section,

mixing

in flow,

respective

trailing

Surface-Friction The

the losses

of their thickness

1, beyond

loss coefficients.

trailing-edge,

express

trailing-edge

BLADE-ROW

kinetic-energy

deg

as fractions

if the

can

edge,

direction,

and

respectively,

blade

thickness thickness

edge within loss.

(Refer of the

to fig. blade.)

the blade-row

If a trailing-edge

as t

t* -

(7-25) S COS

equation

(7-24)

reduces

to

ela

In order necessary

to evaluate to know

the

_la

the

--

1 -- _*_-- t*

loss coefficient

values

of the

_

(7-26) from

dimensionless

equation energy

(7-26), thickness

it is _b_* 201

TURBINE

and

DESIGN

the

either

AND

APPLICATION

dimensionless

displacement

experimentally

Experimental data

are

pressure

functions.

The

the

pressure

of (see fig. 7-1)

pressure

density

and

p,,, and

the

can be evaluated

as will be discussed

the

taken,

$1". These

determining

to measure

pressure

static

or analytically,

determination.--In

it is impractical

loss consist

thickness

experimental

and

density

data

velocity and

required

upstream

loss values,

directly.

velocity

are

pressure

loss survey

Instead, related

for computing

total

the total-pressure

herein.

the

po', the data

p0'-

friction

blade-exit P'lo for one

blade space. Since the dimensionless boundary-layer thicknesses the losses of the blade row as a fractional part of the ideal quantities could ness

pass

through

the

can be expressed

blade

row,

in terms

the

dimensionless

of the

flow

across

displacement one

blade

to

pitch

express which thickas

8

s cos ,._.(pV)i.,1o-t*s cos a,a(pV)s,,,_- cos a_of0 (pV)_odu _i*.--where (7-27)

u is the

distance

simplifies

in the

In a similar

tangential

direction,

in m or ft. Equation

to

_l_a

nesses

(7-27)

s cos _o(pV)_,,_.

manner,

=

I--t*--

the

can be expressed

dimensionless

,.

d

(7-28)

momentum

and

energy

thick-

as

01*=

(pV2)1o,1,, 1

= fo

pV

fo

V

pV

[1-(V-_f,),,]

(_),d

(7-29)

(u)

and

¢l_a --

(pf.V}.),.

pVd Assuming 202

that

the

total

temperature

T' and

the

(7-30)

static

pressure

p_, in

BOUNDARY-LAYER

the

boundary

(p/p/,)

layer

are the

l, can be related

isentropic

same

to the

as in the free

pressure

ratio

P'ljPo'

the

density

as follows:

From

the

(p_V.T

I.,1=ps,.l==po' '

and

T'I,=T_o.la=To'),

plJp/.a==plJpo ' '

division

\p--7 1_,.,.,

'

'

of equation

(7-32)

(from (7-31)

the by

ideal

gas

equation

law,

The

velocity

ratio

1,, and pl=/po'

isentropic write

relation,

(V/Vs,)la

can

with

(7-32)

-P_s,/,. - \po' ]

(p/p')

ratio

relation, p

Since

stream,

LOSSES

yields

(7-33)

be

related

to

as follows:

From

equations

(1-51)

V_ 2gJcj, T'-

T fp_C.y-_>/v 1--_-_ = 1-\_/

the

pressure

the total-temperature and

(1-52)

ratios

definition

of chapter

and

1, we can

(7-34)

where g J

conversion

constant,

1;32.17

conversion

constant,

1; 778 (ft)

Cp

specific

Subscripting values

at constant

equation

at

recalling

heat

la, that

(7-34)

dividing

the

(lbm)

(ft)/(lbf)

(lb)/Btu

pressure, once

first

(sec 2)

J/(kg)(K);

for station of these

Btu/(lb)(°R)

la and again

equations

by

for free-stream the

second,

and

P,_oa,,= Po' and T_°aa = T_= yields

pin _ (v-l)/v

1-- ---7" \Pla]

V

(7-35)

With

the

pressures

density

and velocity

by equations

equations

(7-28),

boundary-layer

(7-29),

and and

thicknesses.

can be computed The

ratios

(7-33)

kinetic-energy

from

equation

expressed (7-35),

(7-30)

Then,

the

in terms

it is now and

evaluate

kinetic-energy

of the

possible

measured

to integrate

the

dimensionless

loss

coefficient

$1=

(7-26).

loss coefficient

thus

determined

is a two-dimensional 203

TURBINE

DESIGN

coefficient; cascade order

that or from

cascade

APPLICATION

is, it is based a constant

can be, and to obtain

are taken and

AND

the

often

on data

radius is, the

full stator

a three-dimensional

at a number as shown

ary-layer

thicknesses

or rotor

radius.

obtained

by

(The

from

annular

a turbine.)

for a blade cover

boundary-layer

for each

are then

a two-dimensional

cascade.

to adequately

dimensionless

previously

from

loss coefficient

of radii sufficient

two-dimensional

calculated

either

of an annular

row,

the annulus,

thicknesses

Three-dimensional radial

In data

integration

are

boundfrom

hub

to tip:

ff'

$*_(pV)I.,I,,

cos alo r dr

h

(7-36) "' (pV)/°jo

cos al. r dr

k

cos. , r dr Ol*.a_ -

(7-37) " (pV2)f.,1,, h

cos alo r dr

d/_,,(pV 3)i.,h cos al. r dr (7-38)

_la,|D

"* (pV3)s..I.

cos al_ r dr

k

In terms

of the

measured

pressures,

these

integrals

L:i,_o(pia)i/'y[l__(P"_(_-')i'_l'l' 5"Ia,SD

expressed

as

COSalo r dr (7-39)

-_"

S,:' <,,->"'[ \p-_41

204

j

are

j

cos al. r dr

BOUNDARY-LAYER

O*a(Px_}l/v [

\po']

I_SSES

j cosaz_rdr

(7-4O) \p-_0' ]

j cos a,_ r dr

\p0']

j

cos ax, r dr

&:_,_ =

(7-41) \-_

The

three-dimensional

in a manner

similar

/

kinetic-energy to equation

j

loss

(7-26)

cos a,a r dr

coefficient

is then

obtained

: _/la,SD

_1_,3_ = 1 -- _*o.aD-- t_* where

t_* is the

trailing-edge

and is used to represent Analytical

values

lytical

methods

discussed

Center another An

average

the

for

not

methods currently

include one (ref. 2) based based on the finite difference equation

used

boundary-layer

in the

momentum

study thickness

coefficient

determined consuming

in use

values,

at the

are

solutions

are

solution. NASA

4 to

Ana-

parameters

Lewis

BoundaryResearch

on an integral method solution method of reference 3. of reference

ana-

to obtain.

boundary-layer computer

_1_ can

boundary-layer

6. The boundary-layer require

radius

row.

loss

as experimental

time

the basic

in chapter

programs

and

at the mean

for the blade

as reliable

costly

calculating

and the better

computer

value

use of analytically

less

referenced

thickness

kinetic-energy

While

are much

and

simple,

layer

with

parameters.

lytical

not

the

determination.--The

also be evaluated thickness

dimensionless

(7--42)

compute

and

turbulent

was

0.231 01a

_---

,

×

\_ccrlfa,

la

(7-43)

205

TURBINE

DESIGN

AND

APPLICATION

where parameter

defined

X

distance

l

blade-surface

along

by equation

blade

(7-15)

surface

distance

from

from

forward

forward

stagnation

to

rear

point,

m; ft

stagnation

point,

m; ft viscosity, The

(N)

development

sumed

that

reference

(sec)/m_;

of this

the

equation

boundary

4, the

lb/(ft)

is presented

layer

exponent

has

n was

,

(7-43)

surfaces

of the

equktions

blade.

(7-43)

blade-surface channel

and

factor

from

for

the

parameters from stator from

thickness

for the

ably

close

tained from

blade

the computer

programs

from

parameters

would

sufficient have

to be

reference

shown

In (1) by

206

the The

the

losses good

over

is commonly with

of reference

used.

as

were

reason-

results

ob-

obtained

be

calculated thickness

at a number length

and surfaces.

simplified from

obtained

of radii

would

also

Such method

two-dimensional by this

method

results.

5, the

following blade

thickness

values, momentum

could

losses

loss for the

momentum

the

3.

end-wall

Results

be

For

as those

so the

experimental

known

$_ can

however,

blade

the

can be

chapter.

two-dimensional

effort,

of the

boundary-layer

analytical

determined

three-dimensional

momentum

dimensionless

The the

surface

individually

2 and

over

considerable

agreement

method average

variation somehow,

require

5 for predicting

mean-section have

determined,

would

the

parameters

to (7-38).

and

coefficient

In general,

to be analytically the

various

be as accurate

of references

of the

any

boundary-layer

surfaces

not

to the

by (7-43)

in this

4,

for

5.

loss

the

required

01a, Hla, and E_

earlier

two

will

(7-36)

have

to establish

procedure

the

pressure

adjacent

la for each

surfaces,

boundary-layer

equations

values

With

values.

(7--43)

densities

(7-14).

of

for the

experimental

equation

Three-dimensional directly

and

and

in equation

in reference (7-43),

and

obtained

kinetic-energy

studied

suction

in chapter

presented

equation

to the

from

the

equations

blade

calculated

pressure

the

be

Ela at station and

In

equation

(7-44)

free-stream can

is as-

profile.

referenced

1

both

discussed

4. It

velocity

the

velocities

those

factor

and

the

for

These

(7-13) and

xj

H as required

energy

suction

evaluated turbine

factor

equations

thickness

are

layers. techniques

Hla and

obtained

evaluated

(7-44)

form

from

L\-_-Iy.

free-stream

boundary of the

both

be

The

flow analysis

Values form

must

in reference

a power-law

obtained

n Equation

(sec)

assumptions surface

at

the

are

made:

can be represented blade

mean

section;

a of

BOUNDARY-LAYER

LOSSES-

Approximate area o{one end wall

sccosas)7

t

Approximate

F/:iiii::iilfi:,i::iiiiiii::::i::ili::f|iiiiii::iiiil direction [ ]!{i]_[i] []]]_]]]_][:i]i[]i][][ ]]]]]]]][]i]i ][_ _]]i]i]_]![i]i]i !1

s_de of blade

__:_;_i_;;_i_!_i_ _I_i_::_]_;_ :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::×

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::i:::::::::::::::::::::::::::: :':':"::_i_''"' •

" "::::_!:i:i:_:

============================= .....

_%%_,_. •

::::::::::::::::::::::::::::

•

_''"

_

I------ s-----

FIGURE

7-5.--Schematic

diagram the effect

late

(2)

the

momentum

the same and

(3)

stant

the

blade

cross

section

loss per unit

as the average

equivalent

of equivalent of end-wall

area

momentum

configuration

can blade,

section,

and

spacing,

given

(see fig. 7-5)

blade.

on the

inner

loss per unit

two-dimensional of the

two-dimensional area on blade loss.

stagger

outer

end

7-5,

equal

to those

area

to

of one

calcu-

walls

is

surface;

approximated

in figure

angle

surface

used

on the blade

be satisfactorily as shown

The

and

area

blade

by an

having at the

equivalent

a conmean blade

is Ab = 2ch

(7-45)

where Ab

total

surface

areas) C

blade

chord,

h

blade

height,

The

inner

and

area

(sum

of one blade,

outer

of suction-surface

and

pressure-surface

m_; ft 2

m; ft m; ft end-wall

area

for one passage

Aw=2sc

is

cos a,

(7-46)

where Aw

total

surface

end-wall blade Now,

taking

stagger

area

of passage

areas), angle,

the average

end

walls

(sum

of inner

and

outer

m2; ft _ deg

momentum

loss O'a.,,, over

the blade

radial

length

207

TURBINE

and

DESIGN

modifying

AND

APPLICATION

it to include

the

01"3D=0" The

calculated

losses

yields

fAb-t-Aw_-O*

three-dimensional

are then

end-wall

energy

and

(1_

sc°sa')

displacement

(7-47)

thickness

parameters

as * _la,3D

_

* ,mOla,3

Ela

(7-48)

D

and 5"la,3D = Hla , m01*o , 3D Mean-section the

values

energy

and form

boundary-layer

factors

used

dimensional

for the

were

thicknesses,

satisfactorily tion

are used

energy

as is done

and

originally

(7-48)

coefficient

factors.

in terms

in reference

in equations loss

form

defined

it is indicated

kinetic-energy

(7-49)

of individual

4 that

and

is then

Although they

(7-49).

obtained

can be

The

three-

from

equa-

(7-42). Trailing-Edge

The with

kinetic-energy flow

past

mentally

loss coefficient

the

blade

_te that

trailing

edge

represents

can

be

the loss associated

determined

either

experi-

or analytically.

Experimental

determination.--Experimental

edge loss coefficient _te are obtained two-dimensional loss coefficients loss and blade

Loss

trailing-edge

loss, and

surface-friction

values

from differences _1, which include

loss coefficients

of blade

trailing-

between experimental both surface-friction

_1_, which

include

only

loss. Thus,

e,e= :1- e_. Loss

coefficients

_1_, which

tained as described surface-friction

loss and

on data

where

the

the

obtained

which

measurements

blade

include must

little

Analytical

mixing,

:1, which

include

obboth are were

not

determined

are

the

the

are

loss,

in exactly

blade

yet

trailing

occurred.

surface-friction

be made

at

to station

edge

To

determine

loss and

a location

1 in figure

just

at station the

trailing-edge downstream

7-1, where

la, loss loss,

of the

the trailing-edge

has occurred.

determination.--In

are

presented

for a large

the

reference

are

208

coefficients loss,

within

both

surface-friction

total-pressure loss and static pressure The surface-friction loss coefficients

loss has

row, corresponding

loss, but

trailing-edge

just

(7-50)

only

Loss

that the locations.

trailing-edge

coefficients

include

previously.

same manner except measured at different based

the

reference number

experimental

6, experimental

of surface data

for

drag

discontinuities.

sheet-metal

joints

coefficients Included of different

in

BOUNDARY-LAYER

LOSSES

T ¥I

_u11

FIGURE

geometry, edges.

bolt

7-6.--Schematic

and

rivet

It is indicated

behaves

similarly

Therefore,

the

lytically

diagram

heads

that

of different

the

regardless loss due

of the

and to the

a trailing

to a body

layer.

loss due

flow direction

past

due

in boundary

geometry,

pressure-drag

to flow

as if the loss were

of body

over edge

placed

the

airfoil

trailing

discontinuity discontinuity.

will be treated

in the

path

ana-

of a boundary

layer. As indicated

in reference

to or less

than

boundary

layer,

effective height

the

6, the

drag

full boundary-layer

as shown

dynamic

pressure

of the body.

Thus,

in figure of the

of a small height

7-6, part

body

6y=u, placed

corresponds of the

of height

h, equal

in a turbulent

approximately

to the

layer

to the

boundary

equal

D = q_IihCD

(7-51)

where D

drag

h

height

on body,

drag and

the

of body,

N/m;

lb/ft

m; ft

coefficient

effective

dynamic

pressure

qell=h

Drag

is related

to momentum

q_f_ is expressed

J o

thickness

2g

as (7-52)

dY

as

D= O(pV_)I° g Therefore,

a

trailing-edge with

a properly

dimensionless loss is obtained subscripted

momentum by form

combining of equation

(7-53)

thickness

e*,

equations (7-22)

representing (7-51)

and

the (7-53)

: 209

TURBINE

DESIGN

AND

APPLICATION

q,LrhCo

8",-

(7-54) (pV2)fs 8 COS 0_1 --

g The

flow angle

(4-26)

and

Before sure the

a, is related

to the

angle

ala as discussed

must

(7-54)

be determined.

free-stream

dynamic

can

be evaluated,

The

ratio

pressure

the

of the

is equal

turbulent with

Combining

flow,

the

the

use

equations

effective

effective

dynamic

dynamic

pressure

that

the

layer

and

free

stream

dr

variation

of velocity

of the (7-8)

tion

(1-64)

total

simple

and

(7-55)

in the

power

(7-12)

boundary

profile

layer

presented

of chapter

the

can be

previously.

yields

(7-56)

temperature

are

to

h

V/. Assuming

pres-

to

-

expressed

4 (eqs.

(4-27)).

equation

ql, For

in chapter

and

same

and

static

using

pressure the

ideal

in the

boundary

gas law and

equa-

1 gives

To'

7+ I

_

I, (7-57)

P/*

To' Substituting the parameter

equations

(7-56)

A _, defined

"y+ 1 and

1-

Performing qy!=(l_A/, qy,

a binomial

(7-57)

by equation

expansion

and

in equation

(7-15)

A

fs

and

using

\_u/l/

integrating

then

gives

) [( h ) 2" 1 ( h _4'_ A,. L\ (i/,, ,z/ 2 n -l---_l-I- k _. zt/ (4n+l)

(h_y + \(_f.../ 210

(7-55)

yields

_ (6n+l) A_

+.--

]

(7-59)

BOUNDARY-LAYER

Substituting

equation

thickness

t in place

(7-59) of body

tCD (l_As,) 2s cos a_

0"-

in equation height

(7-54)

h finally

L\_-_-!

_,

(4n+

+ The

boundary-layer

be the

thickness

sum of the

Equation

suction-

(7-60)

n is not

well

incompressible

the

flow

and

n=l/_

(7-60)

(7-60)

should

values.

flow.

following

+""

in equation

pressure-surface

is for compressible

known,

trailing-edge

1)

(6n+l)

_f_u to be used

and

using

yields

+

2n+l

and

LOSSES

In many

simplified

cases,

equation,

(commonly

used

at least which

for

when

assumes

turbulent

flow),

is adequate: 0",=0.375_The information be set

equal

trailing verted edge

in reference to 0.16

edge.

The

0.22

basis

for

will be available.

ted S COS oq

6 indicates

a rounded

corresponding

to the same and

for

t _futZ

trailing values

as equation

a square In such

that

a case,

the

drag

edge

and

reported

(7-61)

trailing

(7-61) coefficient 0.22

C9 can

for

in reference

a square

7 and

are 0.14 for a rounded

edge.

Frequently,

for incompressible

con-

trailing

5to, instead

of _i/_,

flow, (7-62)

and

for compressible

6full

flow, (7-63)

--

1-

Equations momentum

(7-60) due

kinetic-energy

the

(7-61) blade

loss coefficient,

evaluated

flow and from used to obtain

and

to

in flow and kinetic factors,

(1 -- Ay.)

energy. from

equations

1 , (n--_-t-_

give

trailing

the edge.

it is necessary

As a simple equations (7-16)

A/.

A_, t 5-_-t-

(7-17)

5*, = He*,

, • • ")

fractional To

find

to find

approximation, (7-13)

and

,

and

loss the the

in

corresponding fractional

the form (7-14)

blade-row

for

for incompressible

losses

and

energy

compressible flow,

are

(7-64)

and d/*, = EO*,

( 7-65 ) 211

TURBINE

DESIGN

At station flowed

AND

1, which

into

the

APPLICATION

is just

downstream

area behind

due to trailing-edge is obtained as

the

blockage.

of the

trailing

blade

edge

Therefore,

trailing

and there

edge,

fluid has

is no longer

a kinetic-energy

loss coefficient

* This

loss coefficient

ideal

kinetic

were

the

energy

only values

thickness.

This

energy the

The

surface-friction

would

flow

associated

loss

is approximately, coefficient.

and trailing-edge loss in terms cussed in the next section.

as a fraction

if the

trailing-edge

on

loss

exist

of trailing-edge

(7-66).

energy

kinetic-energy

against

is based

in equation

loss coefficient

that

trailing-edge

of the ratio figure

(7-61).

included

flow

flow is plotted

for several

not

The

(7-66)

the loss in kinetic

of the

loss.

incompressible

equation

expresses

the

a void

trailing-edge loss

thickness with

Therefore, but

as expressed

blade-surface

not rigorously,

Expression

7-7

of the

by

friction

trailing-edge

of a kinetic-energy

for

in figure

to boundary-layer loss

this

loss

coefficient

thickness

momentum

of the

additive

combined

is

kineticwith friction

loss coefficient

is dis-

Ratio of trailing-edge thickness to boundary-layer height,

. O2O

U_ul; 1.O i

.015 .5

_

.olo .!

_

.005

0

.05

.10

.D

Dimensionlesstrailing-edge thickness, t*

FIGURE 7-7.--Effect of trailing-edge blockage on kinetic-energy factor H--1.3; energy factor E-1.8; drag coefficient

212

loss coefficient. Cv--0.16.

Form

BOUNDARY-LAYER

Combined As stated

in the

Friction

discussion

thickness

parameters

combined

friction

and and

and

of the

Trailing-Edge

trailing-edge

a kinetic-energy

trailing-edge at

a location

which

downstream

of the

trailing

experimental versions obtained

values

of &*, 01", and

of equations as

(7-28),

Analytically, Before

adding

the friction

can be added, The

must

in terms

ever,

there

with

blockage

there

is no blockage.

must

parameters are flow as follows:

adjusted

to

the

subscripted of _1 is then

la

trailing-edge la,

ideal

friction-loss for

at station

the

sCOS c°s Ola--t al"

ideal 01*a,

flow.

_l*a) are

blockage. the

(with

How-

ideal

at station

boundary-layer true

loss.

parameters

(61",

where

flow

1 are

trailing-edge

thickness

at station to the

to account

1,

we obtain

of the same

at station

flow without

the

station

way,

parameters

on the basis

blockage

Therefore,

the

The value

boundary-layer

be comparable

by making

appropriately

loss

thicknesses

of an ideal

the

(7-67)

thickness

be expressed

is a trailing-edge

expressing to

In this

(7-30).

surface-friction

dimensionless

boundary-layer

1--61"

and trailing-edge

they

friction-loss

expressed

the

edge.

and

the boundary-layer

by

the

corresponding

_1" from

(7-29),

-

obtained

loss,

loss can be obtained

measurements

is just

Los

loss coefficient

experimental

LOSSES

flow

1, where thickness

blockage)

)

ideal

_*1,/_

_*(s la

(7-68)

0*

0* (s sc°sal"

(7-69)

, = _bx_ ,( s scosa_ _bl'! cos al_-- t )

(7-70)

and

where

the

friction parameters

subscript

and

f refers

trailing-edge at station

to the loss due

to surface

loss parameters

then

friction.

yields

the

Adding combined

the loss

1: 6x* = _I.IA-_ t.

(7-71)

01" = Ox,/A-O t,

(7-72)

and _ll*

And

the value

of _1 is then

obtained

--

IY l,f

from

(7-73)

'fire

equation

(7-67). 213

TURBINE

DESIGN

AND

APPLICATION

After-Mix The

after-mix

loss is the total

the trailing-edge $2 is determined is obtained

loss, and as described

pressure plete

the

the

has

includes

previously

after-mix

measurements mixing

loss that

the

surface-friction

the mixing loss. The after-mix in this section, and the mixing

by subtracting

To determine

Loss

determined

loss experimentally

be made

occurred.

downstream

This

loss coefficient loss, if desired,

_1 from would

of the

is impractical

$_.

require

blading

for

loss,

that

where

several

mixed, that

length,

thus

values

of after-mix

the

these use

leading

possibility

reasons, of

error;

po'-p2'

would

of measurement

error

values

either

(station

to possible

of after-mix

experimentally

would

analytically

and

axial

momentum

flow rate

before

in the mixing

conservation

f0 from

(pV2)l

these

in the

small

enough

large.

analytically

are

For

with

the

before-mix

in the

and

could flow

were

before-mix

station

is not

station

direction

a2 cos a_(pV2)_ axial

direction

(7-75) we get

_ a2(oV2)_

for two-dimensional at station

evaluated conservation

1, it would

mass

we get

flow by integrating

These

the

2) yields

(7-74)

= gp_+cos

available

be directly angle.

Equating (station

a2(pV)2

tangential

subscripted

data

mixing. mixing

written for any before-mix location at which data used to evaluate the after-mix loss coefficient.

214

flow had

after-mix conditions are in the tangential direction,

=cos

=sin

to three-dimensional survey

equations

pressure

d

(oV 2) cos 2 al d

equations

also be applied above

o1

of momentum

+

If experimental static

cos

after

sin al cos al d

conservation

'

the

and

determined

the

during

1) and

of momentum

g fo p_ d Although

direction

(station

o(pV)1

can

the

1) loss parameters. for determining of mass, momentum

and

(1)

(2) the for the

after

be relatively

obtained

The basic equations those for conservation

From

(3)

be constant

loss are or

and

com-

reasons:

The length for complete mixing, while quite long, is unknown; after-mix loss would have to be corrected for side-wall friction mixing

the

even

(7-76) flow,

they

radially. 1, the

with

integrals variations

equations

could

were available, In the case

not be possible

in in be

and then where the

to determine

BOUNDARY-LAYER

the

mixing

ever,

loss completely

it is only the

ments

are usually

where

angle

possible used

to express

herein

differs

used

farther

that

for

station

l,

to (7-76) 1 only

where

there

(pV)ld

in that la

is no

measure-

trailing station

for station

of the

previously

1. The

analysis

the

before-mix

station

herein.

Equation

(7-28)

trailing-edge

I and

edge, 1, it is

void,

=(1--Sl*)(pV)l.a

(7-29)

can

be

(7-77)

combining

it with

equation

yields

(pV2)l Substituting (7-76)

d

equations

yields

momentum mined:

the

= (1-8,*-01")

(7-77)

and

following

in terms

gpl+cos

(pV2)ioa=cos

equations,

along

with

the

ideal

equation

(TI'=

T_'),

can

be

1 to

for both

cos

(pV2)s0a

energy

obtain

_,

compressible

and

incompressible

flow,

the

gas

after-mix solution

(7-74) of

_ a2(pV2)_Wgp_ and

and deter-

(7-80) (7-81)

the

conservation-of-

simultaneously

as shown

kinetic-energy

to

(7-79)

a2(pV)2

law

mass

previously

= sin a_ cos a_(pV2)_

solved

incompressible the

equations

parameters

sin al cos al(1 --81"--01") 2 al(1-81"-01")

(7-78)

conservation

(pV)f,,1--

These reference

into

for

boundary-layer

oq(1--81")

(pY_)l..l

(7-78)

equations

of the COS

For

across in terms

how-

out.

in reference

to station

1

equation

of the

constant

cases,

survey

damped

as was done

of reference

and

somewhat

are

(7-74)

In most

downstream

have

angle

1 corresponds

fo

(7-77)

is desired,

parameters,

from

Subscripting

flow

equations

in reference

subscripted written as

loss that

variations

and

boundary-layer

means.

a little

pressure

pressure

experimental

after-mix

made

and

If static

final

by

LOSSES

loss

in

coefficient,

flow. for _2 is

)+co 2

sin 2 al (1 - _l*-- 01"\ _ _2=1--

(7-82) 1+2

For steps

compressible are required

flow,

COS 20t1[-(1--_1*)

no explicit

to obtain

solution

2-

(]--81*--01*)_

was

found,

and

the

following

_:

215

TURBINE

(1)

DESIGN

AND

The parameters

APPLICATION

C and D are computed

?+1 --_-7 +cos_

(1--As.a)

from

(V)' _

at(X--*t*-01*)

y.,t

C-

(7-83)

D

(2)

The quantity

V =(V-f_,)t,,tsinal\

(V,/Vc,)2

The

density

ratio

is obtained

(o/p'):

The total

pressure

from

is obtained

(_)---_1 (4)

(7-84)

/

(7-s5)

.yC X/(.yC )'_l+(.y-1)D, ,+1 - 7-4-i / 7-;il

(___,.)_ (3)

(l-_x*-0,*_ ]_-_1"

('),-I'_[-

ratio

i0_I

from

p2'/po'

P

c,

(7-86)

,V_, \:]'I''('-')

is obtained

from

j.,_.-7_., cos _x(1- _,*) (7-87)

,o, \O

(5)

The

pressure

ratio

(p/p')_

is obtained p

(6)

Finally,

_2 is obtained

gc,l_

from

p

v

(7-8s)

from

(7-89) (p,'_(',-"/, -\pi

Values the

216

blade

of _ include row,

the

all the blade-row trailing-edge

1

/

loss,

loss; and

that the

is, the frictional

mixing

loss.

Values

loss of of $1

BOUNDARY-LAYER

include

all the

blade-row

losses

except

mixing

LOSSES

loss. Therefore,

_miz = &-- _ where

_m;z is the

fractional

loss in available

BLADE-ROW In

this

the

section,

various

effect

LOSS

types

will

geometry

7-8,

analytically

angle

senting

at the the

losses. edge,

(arithmetic

edge,

section;

annulus

The

including

the

stator

the

and

experimentally loss coefficients

and

at

three

the

the the

friction

end-wall

between good.

_1._,

at three

at

repre-

within

friction,

mean

obtained

just

the

total

tra_ling-edge

experimental

the

the

loss plus trailing-edge

_2.3D represents the

just

loss

coefficient

and

stations

$1,.m, obtained

surface-friction

coefficient and

mixing. In general, agreement loss coefficients is reasonably

0

compares

of kinetic-energy

represents

blade

.04 __

8,

section;

and

compared,

of

of Losses

loss coefficient

radius)

the trailing

Comparison

a given

represents

mean

mean

for

and

losses

will be discussed

reference

values

settings

different trailing

beyond

from

determined

be presented

and

determined

different blade

taken

due to mixing.

analytically

on losses

Distribution Figure

and

considered

energy

CHARACTERISTICS

experimentally

of blade-row

(7-90)

and

loss loss for

drag,

and

analytical

Experimental results

t'3 Analyticalresults O_"

e2.3D"

.0:

!

Mixing and end-v/all losses

Trailing-

edgeloss g_ ¢-

ela. m

?

t_

Mean-section blade suf/ace friction loss

.0: ¢,m v

o 70

I

I

100

Do

Percentstatorareasetting

FIOURE 7-8.--Comparison

of experimental stator

area

settings.

and analytical (Data

from

loss coefficients for different ref. 8.)

217

TURBINE

DESIGN

AND

APPLICATION

A el,30 .05

0 el, m 0 ela, m

,o ,Mixinq loss End-wall loss

¢,.

Bladesurface friction loss

.01 --

o

I

I

.5

.6

.7

I ....

I

.8

.9

Mean-section ideal after-mix critical velocity, (VNcr)i, m, 2 FIGURE 7-9.--Variation

Figure row,

but

7-8

gives

does

of loss coefficients

some

not

from

reference

locity.

Loss coefficient

In this particular loss was

about

loss

will

The

end-wall

case,

the

vary

is seen

The

case,

The

one-quarter with

the

friction of the

was

the

design,

does

will,

total

total

stator

loss.

as was

15 percent

of course,

vary

indicate

that

each

velocity.

losses,

as well

of the

trailing-edge

the

in figure

total on radius

the

stator

losses

stator

trailing-edge

10 percent

of the

ve-

2 percent

shown

with

7-9,

loss. The

primarily

remaining

Figure

increasing

of the

up the

blade

with

In general,

blockage depending

in a stator

end-wall

loss was about

ref. 9.)

coefficient

with

and

from

losses.

in loss

mixing

about

loss made

loss breakdown

end-wall

slightly

of the

trailing-edge

with

and

the

(Data

of losses

variation

to decrease

which

mixing

comparison

mixing

one-half

velocity.

distribution

the

separately losses.

about

loss,

will vary

spacing. loss.

and

the

9, shows

This figure also shows as the other blade-row energy

of the

separate

taken

ideal

idea

with

may

7-7.

loss for this ratio

and

of the

total

design, be

of

but con-

sequence. Effect A study layer 218

of the

of Blade-Row

effect

loss is presented

of turbine in reference

Geometry geometry 10. In that

on Losses

on turbulent-flow study,

the

boundary-

assumption

was

BOUNDARY-LAYER

made

that

the momentum

of the chord

Reynolds

loss per unit number

blade

surface

LOSSES

varies

as the

inverse

to the m power:

--_Re7

_

(7-91)

C

where

Rec

equation into

is

a Reynolds

(7-91)

number

based

by multiplying

equation

(7-22),

three-dimensional

and

effect

and

then

0,*¢¢ (_)m

chord

c. Expanding

by like terms,

equation

an equation

_1 +cos

blade

dividing

using

yielded

on

(7-47)

substituting to express

the

of the form

a._ (O,ot_

(7-92)

(_)1-_Re__,,,

fC.),(",,) where the

Reh is a Reynolds

number

three-dimensional

as

a function

blade given

c/s,

(there values

minimum

from

reference

each

figure

change

analysis. of the

values

of the

results

from

number

nature

shows the the

and

be varied

loss.

Comparison

a wide

optimum

causes

two

shape

chord

Reynolds

The

7-10,

was

and

thickin

of the other With the loss around

of the

7-12.

variation

ex-

obtained

in terms

results

7-11,

geometry variation little

counteracting area.

considerably of the

sensitive

curve

momentum

variable

determined.

in figures of the

that

end-wall

may

loss is more

then

The

a

analysis

Also shown

associated

with

in the

variable.

7-10

around

were

value

loss for

of solidity.

variables

of each

h/s,

reference

minimum

dimensionless

geometric

value

10 are shown is the

value

be expressed ratio

Reh. The on the

a function

derivative to each

can

is no minimum for height Reynolds number). known, the relative variations in momentum

in each

Figure

number

becomes

to find the minimum-loss

variables optimum the

respect

h. As indicated,

parameter

9, is based

to _/_ in the

10, the

08* with

Reynolds

therefore,

m is set equal

order

height

height

variables--height-to-spacing

in reference

and,

In reference hess

and

on blade

thickness

geometric

as explained

solidity

ponent

momentum

of the

solidity

of O,,,/c,

based

of figure number

7-11

from

to solidity 7-11 and

increase

effects

Figure

results

(50

optimum than

reflects end-wall

with

to the also

that

7-10 the

or more)

in momentum

of changes

shows

in figures

percent

loss.

This

in chord

Reynolds

solidity

of a blade

but

excessive,

the

some, and

in h/s

7-11

height-to-spacing counteracting

not shows

that ratio.

influences

the The of

area. 219

TURBINE

DESIGN

AND

APPLICATION

/_

_/////_/////// I

! I I

i

I

/ H n

,;7 .=L4-

J/,4

E._I 1.21-_'--,==P

I \

==o-

¢/s and Reh are constant

IX, .2

.6

LO

L4

2.2

1.8

Z6

Height-to-spacing ratio relative to optimum ratio, (htslllhlslopt FIGURE 7-10.--Variation

of momentum-thickness ratio with spacing ratio. (Data from ref. 10.)

variation

in he_ht-t_

_<'//////////(/ I I I I

Y/'//////

///i

_ I

,/

b;'/ _:,,, j', J" _i/I-

= .=_ i. 4o . \ _'_Z_ _

his and Reh are co

Lo

I

t"--_ I _

.4

.6

.8

LO

1.2

I

J

1.4

1.6

Solidity ratio, (cls)/(c/s)opt

FIGURE 7-11.--Variation

220

of momentum-thickness from ref. 10.)

ratio

with

solidity

ratio.

(Data

BOUNDARY-LAYER

LOSSEF

t //.

/////Z I ! I

1

_Ls -

1.6

_

1.4

/

.,

4 //[

his and c/s areconstant

.s

I ._

o

I .4

I .6

I .s

I Lo

I _2

I

I

].4

L6

I--7 L8

2.0

HeightReynolds numberratio, RehlReh,ref FIGURE 7-12.--Variation

Figure height

7-12

shows

Reynolds

Reynolds

could

the

variation

number

number

number

of momentum-thickness ratio with height Reynolds number ratio. (Data from ref. 10.)

due

ratio.

While

to change

also result

from

in height

Reynolds

figure

indicates the

in inlet

ratio

results

change

is sometimes

used

The

curve to the an in-

performance.

in correlating

the

in

in Reynolds

flow conditions.

in improved

with

a change

loss being inversely proportional power. These results show that

number

height Reynolds number of different turbomachines.

the

in geometry,

change

shape, then, results from the Reynolds number to the m=_ crease

of momentum-thickness

The

performance

REFERENCES 1.

STEWART,

L.:

WARNER,

Characteristics Boundary-Layer 2.

McNALLY, NASA

3.

PATANKAR, Layers.

4.

WHITNEY, mental acteristics

of of

and TN

l).:

S. V.; CRC

TN

3515,

Program

Boundary

AND SPALDING,

WARREN

Compressible-Flow Blade

Layers

Loss

in Terms

of

Basic

1955. for

in

Rows

Calculating

Arbitrary

Compressible

Pressure

Gradients.

1970.

Press,

Investigation and

NACA

FORTRAN

Turbulent

D-5681,

Two-Dimensional

Turbomachine

Characteristics.

WZLLIAM

Laminar

Analysis

Downstream

D.

B.:

Heat

and

Mass

Transfer

in

Boundary

1967. J.;

STEWART, of

WARNER

Turbine

a Comparison

with

L.;

AND

MISER,

Stator-Blade-Outlet Theoretical

JAMES

W.:

Boundary-Layer Results.

NACA

RM

ExperiCharE55K24,

1956.

221

TURBINE

5.

6.

DESIGN

STEWART,

WARNER

WHITNEY,

Turbine

Losses.

Stator

HOERNER,

SIGHARD

HERMAN and

NASA

Cooling.

TM

X-1696,

Variable

the

AND WONG, in

E,55L12a,

Y.:

Use

of

Three-Dimensional

1956.

Drag.

Midland

RONALD

Performance

ROBERT

Predicting

Park,

M. : Effect of

Certain

N.J.,

1965.

of Trailing-Edge Turbine

Geom-

Stator

Bla_ling.

MOFFITT, on

THOMAS

Performance Detailed

P.; of

Losses

AND

BIDER,

a Single-Stage with

70-Percent

BERNARD: Turbine Design

Effect Suitable

Area.

of for

NASA

1968. P.; PRUST,

Stator

Area

on

II--Stator

X-1635,

HERMAN

W., JR.; AND

Performance Detailed

of

Losses

BIDER,

a Single-Stage with

130-Percent

BERNARD: Turbine

Effect of

Suitable

Design

Area.

WARREN

J.:

for NASA

1968.

JAMES

of

Turbomachine RM

RM

AND HELON,

V--Stator

MISER, NACA

J.;

1972. Area

THOMAS

Air Cooling.

NACA

on

W.;

Stator

WARREN Parameters

Fluid-Dynamic

JR.;

D-6637,

Air

TM

W.,

HERMAN

9. MOFFITT,

F.:

Thickness

TN

PRUST,

Variable

222

L.;

Boundary-Layer

etry

10.

APPLICATION

Mean-Section

7. PRUST,

8.

AND

W.;

STEWART,

E56F21,

Viscous 1956.

WARNER Losses

L.] Affected

AND WHITNEY, by

Changes

in

Blade

Analysis

Geometry.

BOUNDARY-LAYER

LOSSES

SYMBOLS

Ab

surface

area of one blade,

,41,

parameter

A. C

surface

defined

area

parameter drag

by equation

of end

walls

defined

(7-15)

for one passage,

by equation

m2; ft _

(7-83)

coefficient

C

blade

Cp

specific

D

drag, N/m; lb/ft parameter defined energy factor

E

m2; ft 2

chord,

m; ft

heat

at constant

kinetic-energy

pressure,

J/(kg)

by equation

(K) ; Btu/(lb)

(°R)

(7-84)

loss coefficient

g H

conversion form factor

constant,

h J

blade height, m; ft height of body placed in boundary layer, conversion constant, 1 ; 778 (ft) (lb)/Btu

l

blade

m

exponent

n

turbulent

P

absolute

pressure,

N/m2;

lb/ft _

q Re_

dynamic

pressure,

N/m_;

lb/ft

Reh

height

r

radius,

m; ft

8

blade

spacing,

T

absolute

t

trailing-edge

U

distance

V

fluid

X

distance

Y

distance

Y

distance

surface

1; 32.17

distance

(lbm)

from

(ft)/(lbf)

forward

(sec _)

m; ft to rear

stagnation

point,

m; ft

chord

in equation

(7-91)

boundary-layer

Reynolds

velocity

profile

exponent

2

number

Reynolds

number m; ft

temperature,

K; °R

thickness, in tangential

velocity,

direction,

m/sec;

along

m; ft m; ft

ft/sec

blade

surface

from

forward

stagnation

point,

m; ft from

surface

from

fraction

normal

surface

to boundary

normal

of boundary-layer

(x

fluid flow angle

o_m

blade

stagger

3_

ratio

of specific

constant

from angle

heat

boundary

m; ft

layer

expressed

as

thickness

axial from

to

layer,

direction, axial

at

deg

direction,

constant

deg

pressure

to

specific

heat

at

volume

boundary-layer

displacement

boundary-layer

thickness,

thickness,

m; ft

m; ft

223

TURBINE

0

DESIGN

AND

boundary-layer viscosity,

P

APPLICATION

density,

momentum

(N)

(see)/m2;

kg/m3;

thickness, lb/(ft)

(sec)

lb/ft 3

boundary-layer

energy

thickness,

m; ft

Subscripts: cr

critical

flow conditions

eft

effective

f

friction

fs h

free hub

i

ideal

inc m

incompressible mean section

min

minimum

mix

mixing

opt

optimum

stream

p

pressure

ref s

reference suction surface

surface

t

tip

te tot

trailing total

x 0

axial component blade-row inlet

1

just

la 2

just within downstream

3D

three

edge

beyond

trailing

Superscripts: absolute

*

dimensionless

224

of blade

trailing edge of blade uniform state

dimensional

t

edge

total

state value

row row

m ; ft

CHAPTER 8

Miscellaneous Losses By Richard J. Roelke In the

last

process

chapter,

in the

design-point include

these

losses

neglected;

of

in other

part

instances,

comprises

losses

turbine

these

losses

turbine

all the

losses

turbine.

turbine

considered,

there

is being The

blade

partial-admission

inactive

passages

blade

as they

a loss that

occurs

at off-design

loss, which

will also be covered

pass

of the the

a turbine

rotor

tips,

blades

thus

amount

shrouds. the

reaction)

first

causes

since more

casing,

nature by

geometry, a large

some

some

point.

be

mag-

The

sum in the

a partiallosses

considered

the

that

are the

filling-and-emptying

admission turbine

clearance

fraction

in turbine

clearance, tip

be of such

additional

the

may

arc. Finally,

is the

incidence

LOSS

with

a reduction

a given loss,

operate

the

and

herein.

of all, by the

of radial

For

leakage

and

causing

loss is affected, the

must

instances,

however,

usually

of any

flow

overall

are considered

are and

TIP-CLEARANCE Because

can design

If,

through

operation

In some output

that

losses channels

the

the

also be considered;

loss.

of the

of the

with

determine

must

disk-friction

axial-flow

loss in the

loss in the

other

the selection

normally

associated To

a full-admission

be included.

pumping

small

losses

discussed.

loss and

a very

however,

admission must

represent

losses

design

were

of a turbine,

tip-clearance

as to influence

of these

boundary-layer

channel

efficiency

these

nitude

the

blade

pressure

higher-kinetic-energy

of the

work

of the

recesses the

between

tip in

amount difference

the

fluid

output.

leaks This

geometry;

across

flow to leak

and

reaction the through

tips

across leakage

that

casing,

of blade

the

is, by by

tip

affects tip

(high the

tip 225

TURBINE

DESIGN

gap

the

from

shrouded

but

problem. developed,

flow

not

side of the

only

causes

rather

of the

in turbine

drop

difficult

of tests

to determine

the

and tests

loss. Figure

have

effect

reaction helps 8-1

and been

made

of tip

because

shows

the

60

and

that

caused

complex

none

flow

of some

at the blade

on of the

of the exit

is entirely

Research

geometry

understanding

traces

things space

own in the

efficiency

Lewis

tip

An examination

angle

stage turbine (ref. 2). Two that the flow in the clearance

to its

primarily

of the

states

at the NASA

a better

an un-

clearance loss have been in reference 1; however,

author

clearance

turbines.

to obtain

the

With

a loss due

of the blade,

is inherently

complicated,

blade.

an unloading

evaluation

leakage

satisfactory. A number

these

suction

Several empirical expressions for and some of these are summarized

are

impulse

leakage

also causes

Analytical

by tip-clearance

they

side to the

this

work,

region.

APPLICATION

pressure

blade,

reduced tip

AND

Center axial-flow results

of

tip-clearance

of a 5-inch

single-

to be noted from the angle traces are and near the tip was not fully turned,

Tip clearance, percent of passage height

4_

2'

2O

0

1 2

1:3

•

5.0

o

8.0

Axial

,,

direction =

-6 4--P

-2O

-4C .5

FIGURE

8-1.--Variation

lTr +,,I .6

of exit

I

.7 .8 Ratioof hubradiusto tip radius

flow

angle

(Data

226

I

with from

radius ref.

2.)

ratio

Oo,+rw.,, I,

.9

for four

1.0

rotor

tip clearances.

MISCELLANEOUS

LOSSES

Turbine 0 C] Z_

1-Stage,reactionIref. 2) 2-Stage,reaction(ref. 3) l-Stage, impulse(ref. 4)

--Estimate (ref. 5) for ref. 2 turbine m m Estimate (ref. 5)for ref. 4 turbine 1.00

m

.96 _

\\

.80

I

.76' 0

FIGURE

even

at the

increased way

down

to the

hub.

in figure The (ref.

the

2) and

results

in figure

I

I ,12

on efficiency.

tested,

and that

underturning

clearance,

and

effect

in lower turbine,

4).

this

of the

flow

of the flow

occurred unloads

turbine

output

as well

as for two

of reaction the same

lines

and

Extrapolation

of the

satisfactory

in the

5 (as

8-2

all the the

blade

efficiency.

The

others,

and from

is shown

about

double

2 and data

that

loss

importance evident the

from here

in

losses

that

for

tip

for

the curves

as figure

8-2 shows small

from

losses

for the impulse

efficiency

4) as obtained

of figure

of tip-leakage

a single-stage

height,

of the

and reproduced

single-stage

The

loss is clearly to blade

estimates

fig. 1.6)

experimental

estimates

are

from

unshrouded.

clearance

were

(refs.

results

turbines were

of tip clearance

turbines

turbines

in reference

test

3) reaction

plays ratio

in figure

single-stage

published

represent

All turbines

for the reaction dashed

8-2

(ref.

(ref.

For

efficiency turbine. the two

clearance

underturning

for this

two-stage

the level figure.

gives

lines

turbine

The

I,

8-2. solid

impulse that

tip This

and

in efficiency

I

of tip

clearance

increasing

aerodynamically decrease

8-2.--Effect

smallest

with

I

.02 .04 .06 .08 .10 Tipclearance,fractionofpassageheight

figure

8-3. 8-3

clearances. 227

TURBINE

DESIGN

AND

APPLICATION

1.00

•99

.98 U O

.97

r-

8

•95

.95--

.1 .2 .3

¢-

.o

•94

.4

_J r'-

•93

•92 -b-

.91--

I

.90 0

FIGURE 8-3.--Tip-clearance

Reviewing loss

the results

in efficiency

large

ratios

creasing

the

schemes

can

be used

impulse

turbine

the

tip

blade

A clearer

both

8-4

loss mechanisms

flow,

of the are

blade-height consist (3)

and three

blade These

with

The blade

leakage

at

to

flow

with

in-

The

single-

ratios

recessed

are shown

of tip

casing

and tested

in figure

8-5.

is possible

affecting

as compared loading

tip-

configurations

characteristics factors

the

tip while loss-reduction

several

the

general results

configuration of the

tested

performance

considered.

for reducing the

or in combination.

the

that the

and for moderate

a tip shroud.

without

of (1) reduced

mixing

above

4 was

shows

it is apparent

methods

casing

individually

of reference

from ref. 5.)

the loss is appreciable•

clearance,

adding

blades• (Data

and 8-3,

height,

the

.03

reaction,

4, and the turbine-performance

configuration

228

and

either

height,

Figure

tip

recessing

understanding

for the reduced leakage

the

8-2

increasing

to blade

height,

to blade

in reference the

include

shroud•

I

for unshrouded

in figures with

to reducing

losses

clearance

shown

increases

leakage

stage

correlation

of tip clearance

In addition

I

.01 .OZ Tipclearance,fractionofpassageheight

turbine

work

to a zero-clearance

area,

if

(2)

clearance-gap

channel

throughflow,

MISCELLANEOUS

LOSSES

R_s_casi_

Flow -_.

E

"- Rotor blade

(b)

(a)

Tip shroud-_

_',-Rotor blade

Flow_

(c) (a) Reduced blade zero-clearance

height (relative to blade height). (c) Shrouded rotor. 8-4.--Tip-clearance configurations investigated

FmvaE

and

(4) blade

to the

suction

extended clearance fore, was

the

side).

to the

to the blade,

With

the blade Note

the

outer

changed blade

because

reduced.

(as a result

passage

gap was reduced

reduced

further

unloading

of the from

area

indirect

unloading

casing.

for impulse

turbine

of flow going

recessed-casing radius

by varying

loading

(b) Recessed

was

the

amount

of constant

was eliminated, path.

was eliminated, figure

8-5,

the

however,

and With

the height

of casing the the

blade There-

leakage shroud

side as the

recess.

and the leakage that

4).

pressure

configuration,

and

leakage

from

(ref.

flow added

flow was

at tip-clearance 229

TURBINE

DESIGN

AND

APPLICATION

"O

R

38

I

1

.04 .06 .08 .10 Rotortipclearance,fractionofpassageheight

i

.12

36

FIGURE 8-5.--Effect

fractions longer

of tip-clearance

below

some

provides

an

The to other the

turbines.

leakage

difference

but

extend In

the

by The

diameter

(larger

loss increases

clearance as the

and

diameter) gap

to an

different

With

gap

blade-tip not

apply

blade,

since

and

pressure

respect

the blade

to

should

section

will

the not

just

be

losses.

a complicated

easily

no

clearance

span

that

flow

predicted

for a turbine

of clearance

attributed

shrouded

used.

from

shroud

and may

overlapping

is not

for larger ratio

of the

be noted

presents

required

for

clearance

additional

loss

be

the

as the

design

the

the

creating

gap

8-5

of seals

If it does,

factors

clearance

on

can

casing

true

only

efficiency. (Data

instance,

This

particular

it should

fluid and

many

in this

in figure

that

number

tip-clearance

accuracy.

230

the

recess.

stagnant

summary,

influenced

the

not

on turbine

and

is particularly

configuration,

into

churning

on

shroud

shown

depends

also

recessed-casing

0.035

in efficiency.

upon

This

flow

about

results

dependent

I

configurations ref. 4.)

loss between

comparative are

value, increase

increasing friction is decreased. geometries

I

with

depends and,

to blade

problem consistent

primarily

as seen height

previously, increases.

on

MISCELLANEOUS

For

any

given

increasing maintain

for

to

and

hence

therefore, larger

the

case,

leakage

small the

ratio

blade

tip-clearance

It

becomes

of clearance

height,

If tip

it might

the

ratio.

loss is more

turbines.

particular

therefore,

tip-radius

a desired

turbine, ratio,

diameter,

hub-

severe

leakage

for small

disk-friction

height

as the

a given

radius

and

to carry

out tests

be worthwhile

some

a small rotor

disk.

near

the

engine

tive

nature

throughflow

turbines

steady

the

timating

loss

less severe

a problem

in a

to evaluate

the

the

cooling flow

of cooling the associated

gas

outward patterns gas

with

shown

skin

the

base

aircraft

disks

in figure

The

without

8-6.

cools from

qualitaand

Equations

In

engines, and

surface

of the blades.

and

casing.

bathes

rotor-disk

rotor

friction

stationary

gas that

along

to the

to the

and the

for example

are presented

with for

es-

layer

of

herein.

Throughflow

no throughflow,

fluid close to the rotating

flows

around

are

losses No

the case

is due

disk

of lower-temperature

centerline

of the

loss)

rotating

for hot applications,

stream

This

LOSS

(or windage

of fluid between

addition,

For

turbines

to

effects.

circulation have

For

to be

with

diffacult

to blade

smaller.

is considered

DISK-FRICTION The

loss increases

increasingly

gap

becomes

LOSSES

surface

as in figure is thrown

8-6(a),

outward

by

the

thin

centrifugal

action

7//A//////////_

(b)

(a_

(a)

Without FmuaE

throughflow. 8-6.--

(t)) Flow

patterns

for

With rotating

throughflow. disks.

231

TURBINE

and

DESIGN

returns

AND

via

the

up a continuous side of the disk

APPLICATION

stationary

wall

circulatory

to the

effect.

inner

Consider

radius,

an

thereby

element

building

of area

on one

dA = 2,rr dr where

A is the area,

in m or ft,

of the

(8-1)

in m 2 or ft 2, of one side area

element

lb/ft _, acting over this the disk rotation of

area

dA.

at the

The

of the fluid

radius

disk,

shear

and

r is the radius,

stress

r produces

r, in N/m

a resisting

2 or

torque

to

dMo - r2rr 2 dr

2 where

Mo is the

in the

case

resisting

torque,

of no throughflow.

in N-m The

(8--2)

or lb-ft,

shear

for both

stress

can

sides

of a disk

be expressed

as

C/

(8-3) where C!

fluid shear-stress

9

conversion

P

density,

v.

tangential

At the

disk surface,

coefficient

constant, kg/ma;

1;32.17

lb/ft

(Ibm)

(ft)/(lbf)

(sec 2)

_

component

of fluid absolute

the fluid tangential

velocity,

velocity

m/sec,

ft/sec

is

V_ = too where into

w is the angular (8-3),

the

total

velocity, torque

(8-4)

in rad/sec. for both

By substituting

sides

of the

disk

equation

(8-4)

can be written

Mo = fo a 2__#Cip_2r 4 dr where

a is the

disk rim radius,

(8-5)

in m or ft. Performing

the

integration

yields

w2a 5 --

Mo=CM,op

as

(8-'6)

2g where friction

CM,o is a torque loss

expressed

coefficient

for the

as power

ease of no throughflow.

is then

the

torque

times

The the

disk-

angular

velocity: Mow Pd$ --

pwaa 5 -- CM,o

J where 232

Pd/is

the

disk-friction

power

---

(8-7)

2gJ loss,

in W or Btu/sec,

and

J

is a

MISCELLANEOUS

conversion

constant

equation

(8-7)

(equal

that

to

is found

1,

or

in most

778

(ft)(Ib)/Btu).

handbooks

is

Pdl = KdlpNaDr

LOSSES

The

form

5

of

(8--8)

where Kd!

disk-friction

N

rotative speed disk rim diameter

Dr

A number equation made

(8-8)

assortment

doubt,

due

model the

is that

a smaller

diameter

four

picture

modes

casing the

effect

flow

speed,

has been

several

rotating

and

the

and

Flow,

casing across

are the

in the

in the gap.

The

empirically,

is

theoretically

and

radial

and

that

space

on the

chamber

coefficient in each for the

so that

torque

a

tangential

general,

between

the and

evaluated of

coefficient

are as

layers

on the

variation

indicates

components

equation

In

a

A description

continuous

8-7(a)

7) to deterto present

dimensions

Boundary

s. Figure

best

the (8-7)

by having

CM.o was

regime.

Clearance.

and

6 and

can exist in the axial

gap

of the variations

(refs. friction

exist.

merged,

axial

the

between

equation

may

equations

at a given

radius

of flow

torque

Small

from

and

loss is obtained

on disk

experimentally

associated

I: Laminar exists

The

somewhat

speed.

depending

number. and

lower

The

this loss is, no the

in the space

be noted

have

data.

is derived,

conducted

modes

disk,

can occur

rotative

proportions

Reynolds

disk

velocity

blade

a higher

(8-7)

can

of chamber of the

the

each regime follows:

rotor

that

investigation

theoretically

Regime

thing

to predict

Kds in

others

available

geometry,

equation

of flow, or flow regimes,

and

both

and

used

constant

while

fit the

test-apparatus

of flow that

One

of the

circumstances, to better

which

for a given

An extensive clearer

types

casing.

values

equations

of the from

of different

the

exponents

of semiempirical

or (8-8)

mine

published

for different

to the

to variations

oversimplified and

have

to be used

changes

wide

rotor

coefficient

of investigators

small

existence

power-loss

for torque

the

in

nature

of fluid

velocity

coefficient,

both

211-

CM.owhere the

s is the Reynolds

axial

distance,

number

(s/a)i

_

in m or ft, between

defined

(8-9) disk

and

casing,

and

R is

as wa 2p

R = --

where

u is the

dynamic

viscosity,

in (N)

(8-10)

(sec)/m

2 or lb/(ft)

(see). 233

TURBINE

DESIGN

AND

APPLICATION

Radial component ofvelocity

Tangential component of velocity

0 Disk r_

"_}/////////////////////A_ (a)

Radial component ofvelocity

Tangential component ofvelocity

0

0

r(d I

(b)

(a) Flow regimes I and Ill. (t)) Flow regimes II and IV. patterns around rotating (links without

FIGURE 8-7.--Velocity

Regime

II:

Laminar

ness of the boundary

Flow, layers

axial gap, and between ing fluid in which no the

variations

case. arc

The

best

in the

Large on the

Ch,arance. rotor

and

The

theoretical

and and

tangential

velocity

empirical

combined

on the casing

these boundary layers there change ill veh)city occurs. radial

throughflow.

equations

exists Figure

thick-

is less than

the

a core of rotat8-7(i)) shows

components for torque

for

this

coefficient

CII

CM.o--

(8-11) 1_

where 234

CII is a function

of (s/a),

112

as shown

in figure

8-8(a),

and

MISCELLANEOUS

LOSSES

3.3_

3.1

m

2.9 2.7

I

2.5

I

I

I

I

I

I

1

I

I

.10 .15 .20 .25 Ratioofaxialgaptorim radius, s/a

.30

.09

.oz

I

0

.05

(a) Flow regime II. (b) Flow regime IV. FIGURE 8-8.--Evaluation of torque coefficients. (Data from ref. 6.)

CM.o --

3.70 (s/a)

1/1o

(8--12)

R 1/2

respectively. Regime counterpart for torque

III:

Turbulent

of Regime coefficient

Flow, I. The

best

Small

Clearance.

theoretical

and

The empirical

turbulent equations

are 0.0622 CM,o--

(s/a)

V4RV4

(8-13)

and 0.080 CM,o-- (s/a)1/6R1/4

(8-14)

respectively. 235

TURBINE

DESIGN

Regime

IV:

counterpart tions

AND

APPLICATION

Turbulent

Flow,

of Regime

for torque

II.

coefficient

Large

The

best

Clearance.

theoretical

The and

turbulent

empirical

equa-

are Civ

C_,O-RI/5 where

Civ is a function

of (s/a),

(8-15)

as shown

in figure

O.102(s/a) C_,o -

8-8(b),

and

1/l° (8-16)

RII5

respectively. The

particular

determined

by plotting

equations several of figure the

flow regime

(8-9), values 8-9

torque

(8-11), of s/a.

at any

coefficient and from

determined

against (changes

by matching

_'lOI---

_ IV

lines

with

flow; separate

Ratioofaxialgap to disk rim radius,

- --00, %_

- _

:_I I I I I 1o2 lo3 I@ lo 5 lo 6 ld Reynolds number,

236

lines

boundarylayers

_\

,L

FmuRE 8-9.--Theoretical

Turbulent

_

'_ \\

"_

for

boundarylayers _

\ \

_,\_,, I "\\

of the

Laminarflow; merged boundarylayers Laminarflow; separate boundarylayers Turbulentflow;merged

III \

slopes

in the

I

_'_ \

the

from 8-9

In this figure,

Description

II

•,

in slope)

to another.

can be

number in figure

Flow regime

'

-2

number

Reynolds as shown

one regime

Slopeofcurve

_10

Reynolds

(8-15),

discontinuities

transition

are

exists

(8-13),

The

indicate

flow regimes

that

,05

I [ ["1 I@ I@ lol° lon R

variation of torque coefficient with Reynolds number for no throughflow. (Data from ref. 6.)

MISCELLANEOUS

those

shown

determined water

in the

insert

in the

experimentally

and

figure.

with

oil for several

values

Torque-coefficient

a 50.8-centimeter of s/a verify

values

(20-in.)

the

LOSSES

(ref.

6)

disk rotated

in

theory.

Throughflow For the the

case of the

friction

torque

rotating

disk

increases

with

with

velocity

The

symbol

core

of gas

AM,

over

and

leaves

Ko represents to the that

momentum

the

without

of the

at the

angular

ratio

where

Q is the

with

some

of the angular of the

throughflow

disk.

is the

through

AM=2p

volumetric

throughflow of the

M = Mo+AM

value

An

assessment

friction

torque

throughflow

of the of the

velocity

velocity

of the

The

K_a. rotating

in torque, of angular

rate,

disk.

The

2

loss

or fP/sec,

torque

p QKowa

for the

in the through-

(8-18)

2

g

can

throughflow

(8-17)

in m$/sec total

0.45 for s/a ratios

power

increase

of change

Q K_a g

- CM'°pw2aS+2

of Ko is approximately

has been

angular

rate

2g The

problem

the system:

Q- (gowa)a=2p g

clearance space on one side flow case is then

This

8--6(b),

with regime-IV flow. In this case, chamber near the centerline with no

rim

velocity

fluid flowing

as in figure

the throughflow.

analyzed for low values of throughflow it is assumed that the fluid enters the angular

throughflow,

from

be obtained

case

0.025

by

compared

to

to 0.12.

calculating that

of

the

the

no-

case: M --=It Mo

2pQK_a

_

4Ko

-1+

CM

1

Q ,o wa

(8-19) 3

-_ C M,op_a 5 Substituting

equation

(8-16)

M

for CM,o yields

KoR 1/5

--_-o= 1 + o.0255is/a)l/l where

T is a dimensionless

Q ° we _

throughflow

V= _

= 1 +39.2

number

Ko (s/a)m defined

o

T

(8-20)

as

Rm

(8-21)

wa

According that

are

to the

somewhat

data high;

of reference moreover,

7, equation the

effect

(8-20) of s/a

predicts is not

values

accurately 237

TURBINE

DESIGN

AND

APPLICATION

Ratio of axial gap to disk rim radius, sla

f

/o.o,

_

1.

--

1.

--

.12

1.2--

g

10 0

.01

.02

.03

Throughflow

FIGURE

8-10.--Empirical

variation

of torque ref.

given

by

(s/a)

+ 5 percent

m°.

.04

Empirically,

.05

number. T

the

with

throughflow

number.

(Data

test

data

are

represented

to within

by the relation M

T - 1 + 13.9Ko

(8-22)

Mo Equation

(8-22)

is plotted

(s/a) in figure

8-10

Full-admission turbine

axial-flow

unusual may

be a better

so small

that

a normal

heights,

then

it may

due

to partial

with

freedom

In addition, of larger

use of partial 238

for a given diameter

design

admission

may

would

blades

may

rotative higher

speed,

be a convenient

way

mass-flow very-small

admission.

be less

The

than

the

turbine

partial

blade-jet

applications;

a partial-admission

give

of the full-admission

and

values.

most

the design

to use partial

long

losses

for

for which

If, for example,

full-admission

admission

used

arise

be advantageous

and low Reynolds-number blades.

are

sometimes

choice.

s/a

LOSSES

turbines

conditions

1/8

for several

PARTIAL-ADMISSION

however,

from

7.)

ratios.

to reduce

is

blade losses leakage

having

admission speed

rate

short

allows

the

Also,

the

power

output

MISCELLANEOUS

of an existing passages). output

turbine (physically

full-admission

In general,

partial-admission

and low volumetric-flow

As mentioned are the emptying loss.

The mechanisms but

referred

they do result

turbine

The pumping

loss is that

loss caused

several showed power

that

the

effects

loss are quite terms.

rows,

open

casing

sides

empirical

blade

loss

perhaps

in to,

for the

are

disk-friction

in the

accounted

Therefore,

for

pumping-

in the exponents

of obstructions

of such

were

of

investigations

on the

by variations

These

results

equations

these

diameter

and location

the

The

from

and

loss.

8, where

summarized.

resulted

height

most

and efficiency

full admission. in form

(adjacent

vicinity

of the

three

for only by differences

it

for pumping-power

one equation

power

or fully

rotating

or lack

coefficient.

expression

clearly

similar

as evidenced

etc.)

with

sector. or sector

blades

that

channel

are not

to reference

the nature

wall,

of the

applicable The

uncertain,

Further,

active

by the inactive

back

of blade

the

losses

filling-and-

scavenging,

in output

expression

loss

the

for it are somewhat

investigations

pumping-power

on these blade

the

to trace

experimental

estimating

the

with,

all seem

and

through

operating

and expressions

combined

expressions

partial-admission

losses

in a decrease

to the same

and often

pass

of partial-admission

casing,

of the stator

high specific-work

channels

to as expansion,

compared

a fluid-filled

have

the

blade

as the blades

loss has been

understood,

chapter,

inactive

loss encountered

latter

when

in this

loss in the

This

turbines

some

rates.

previously

pumping

block

LOSSES

appears

that

in

a generally

loss is yet to be found.

often

used

is

(8-23)

Pp = KpoU,,,al l'sD,,, ( 1 - _) where Pp

pumping-power

loss, W;

K_

pumping-power

loss coefficient,

u.

blade

mean-section

l

blade

height,

D,,

blade active

mean-section diameter, fraction of stator-exit

E

The

value

to the for

units

an

values the

used

friction

coefficient herein

by

turbine

loss estimated

losses

of reference

equation

(8-23),

(sec _)/(Ibm)

(lbf)

(seO)/(Ibm)

(ft 3n)

ft/sec

For and

same

coefficient

(ft3n).

This

rotors

of the

pumping were

by equation if the

in reference

l/m _/2, or 0.0105 the

housing

9 and the

m; ft area

to one-half

disk-friction the

1/m_/_;

m/sec;

Kp as reported

is 3.63

rotor.

one-quarter

combined

speed,

(lb)/sec

m; ft

unenclosed were

enclosed

(lbf)

of the

(ft)

(8-7)

(lbf)

losses

from

to be 5.92 higher

recently,

a single-stage

loss is converted

Kp is found is significantly

(fts/2), coefficient

More

in reference

is subtracted

remaining

the

values. for

converted

(sec2)/(lbm)

enclosed,

above

reported

8 and

than

rotor

9. If a diskthe

combined

to the

form

of

]/m _/_, or 0.0171 the

coefficients 239

TURBINE

DESIGN

reported lack

other

Imagine filled

8, and

momentum

fluid

less

passes

to the

leaving

out

of the

fluid passing

of the

fluid.

where

by

sector,

into,

cause

The

effect

multiplying

pitch,

sector

loss

on

the associated velocity diagram work of an axial-flow turbine as

and

has

the

decrease

(2-14),

geometry,

we

can

energy

in momenby

nozzle

a loss

active

arc

that

ac-

coefficient

efficiency

and

as it flows

in the momentum

f is the

Wu ,2) = _Um ( Wl sin 01-

( W_ a-

less

velocity

turbine

(2-6)

arc,

momentum

in m or ft, and

As the

the available

this

blade

fluid

diffused

decreasing

until

the

this

decrease

K, is a rotor

As

active

Since

rotor-exit

It is high-

loss occurs.

nozzle

10 that

the

the use of equations

Ah' = g_

thus

loss.

sector.

sector.

it is rapidly

in reference

rotor-blade

of the

With

the

sector

out by the

sector

an overall

the rotor,

in m or ft. Effectively, for the sector loss.

follows.

the

will continue

active

channel.

the

active

scavenging the

off from

to flow losses

the

be pushed

a second

the

reported

be found

This

enters

through

It was

p is the

length, counts

area

These

must

within

is cut

fluid

channel

of the

to primarily

be called

to enter

that

nozzle.

active

channel

the rotor.

tum may coefficient

the

herein

starts

fluid

is completely

blade

blade

through

as it just

stagnant

high-momentum

entire

is attributed

loss shall

channel

channel

channel

difference

loss model.

relatively

blade

the

partial-admission

a blade

with

inlet

APPLICATION

in reference

of an adequate

The

the

AND

is determined from

volume

express

the

as 1, and

specific

(s-25)

W2 sin 02)

where Ah'

turbine

Wu W

tangential

0

fluid relative

The

relative

subscripts

an impulse

specific

work,

J/kg;

component velocity,

1 and turbine

of relative

m/sec;

angle

velocity,

m/sec;

ft/sec

ft/sec

measured

2 refer (which

Btu/lb

to the most

from rotor

axial inlet

direction, and

partial-admission

exit,

deg respectively.

turbines

are),

For where

01= -02, Um Ah' = _-_ W, sin Ox(1 +K,_) where turbine. 240

K_ is the For

rotor the

relative-velocity

partial-admission

ratio turbine,

W2/W_

(8-26) for the

applying

full-admission the

sector

loss

,

MISCELLANEOUS

coefficient

LOSSES

yields W_ = KwK,WI

So, for the partial-admission

(8-27)

turbine,

Ah'r_=-_

(8-28)

Wt sin _I(1+K_K,) qJ

Since

efficiency

is hh' (8-29)

- Ah_a where

Abed is the

turbine

ideal

specific

work,

in J/kg

turbine

with

respect

ficiency of the partial-admission admission turbine is

nm

or Btu/lb, to that

the

of the

Ahrro

effull-

(8-30)

Ah' Substituting

equations

(8-26)

and

(8-28)

into

equation

(8--30)

then

yields I+K,,K, 1 -I-K_

_=n The

efficiency

sector

penalty

loss only;

efficiency turbine

the

further. rotor

should

as more

increase.

Also,

is not

admission

study

was

determined

total

loss due

between and

the

closely of the be done

a range

losses

were

to leakage

admission in figure with stant

losses 8--11.

decreasing over

Predicted

from

the

of reference The arc

fraction,

the range

of arcs

cfficiencies

from

pumping while

the

sector

loss;

loss will

on the

pumping

of a partial-

was

taken

separately

to the

other

turbine

as the The

and

The

difference

blade were

pumping subtracted

was called other partialsector loss and any loss

inactive

against and

axial-flow

12 to 100 percent.

efficiencies.

loss to give what losses include the sector

blades

of a small

operation

9 are plotted

combined

profile

at present.

efficiency

measured

active

the

blade

optimization

of admissions

to partial-admission

the overall

a partial-admission

the

of rotor

analytically

for the

to reduce

rotor,

complete

accounts

will reduce

that

blades

the full- and the partial-admission

disk-friction

earlier

to the

9, the

from the total partial-admission admission losses. These other due

spaced

the

of reference

(8--31)

indicates

number

Therefore,

over

equation

loss discussed

are added

effect cannot

by

(8-24)

have

blades

known. design

In the

pumping Equation

however, loss

expressed

(8-31)

sector.

partial-

admission-arc

disk-friction losses

The

fraction

loss

remained

increased

nearly

con-

tested.

(from

ref.

10) are plotted

against

blade-jet

speed 241

TURBINE

DESIGN

.6

AND

APPLICATION

T

-- - -- Estimated pumpinganddisk-frictionlosses

i.5

C3 0

I I I

K.4

Pumping anddisk-frictionlosses Otherpartial-admission losses

I

I "I"

'Is

I I t

C

% .|

-

gJ .1

0

Fmtmm

.2

8-11.--Variation

l

I

I

I

.3 .4 .5 .6 .7 .8 Activefraction of statorexitarea, •

.9

1.0

of partial-+_cimission losses with active area. (l)ata from ref+ 9.)

fraction of stator

mission

o ,a U

qJ

$

.1

I .1

.2

.3 .4 .5 .6 Blade-jetspeedratio, Urn/_

Fmuam 8-12.--Design-point

ratio

(see discussion

operating admission. admission reduction Aerodynamic 242

with The

performance of partial(Data from ref. 10.)

in vol.

1, ch. 2) in figure

full admission expected

is seen.

The

in optimum efficiency

and

reduction important blade-jet

with

three

in peak thing speed

is a maximum

.7 .8 Ahtd

8-12

turbines.

for a particular

different

to note

1.0

and full-admission

efficiency

ratio

.9

amounts with

from

at a blade-jet

of partial

reduced

this

as admission

turbine

arc

speed

figure

arc

of

is the

is reduced. ratio

of 0.5,

MISCELLANEOUS

irrespective

of admission

Blade-pumping blade

and

speed, arc

maximum

net

and

design

decreases.

speed

into ratio

which

The

decrease

as

admission

arc

power

minus

at lower

turbine, before

blade with

(aerodynamic

speed.

decreasing

aerodynamic

blade

power

is reduced,

as the

blade-pumping

speeds.

Thus,

the partial-admission

an optimum

for the

losses

or near-optimum

must

be

blade-jet

can be selected.

INCIDENCE

row

decreasing

gross

is obtained

design

with

of the

Therefore,

power) the

part

power

of a partial-admission

factored

decreases losses,

a larger

output

disk-friction

and

disk-friction

become

admission

arc,

LOSSES

incidence (either

loss is that

stator

angle.

Flow

since,

theoretically

design

condition.

shown

in figure

is the camber is defined as

loss which

or rotor)

at some

would

normally

incidence

at least, The

occurs angle

The

and

all gas

and line

the

i

blade

used

dashed

defines

a

--

the gas enters than

the

angles

when

are

speaking through

inlet

angle.

The

a blade

optimum

at off-design

running

blade

=

when other

only occur

nomenclature

8-13. line

LOSS

flow

conditions,

matched

at the

of incidence the

is

blade

profile

incidence

angle (8-32)

ab

where i

incidence

angle,

fluid flow angle ab

blade

The

inlet

for

indicated because angle

rotors.

from

angle

fluid flow angle

angle

deg from

must

The

axial

direction,

axial

be the

incidence

deg

direction,

absolute angle

deg angle

may

for stators be

positive

in figure 8-13. The sign of the incidence cascade tests have shown that the variation

is different

Axial direction

for positive

and

negative

and

the relative

or negative,

as

angle is important of loss with incidence

angles.

Vp /

IB,

i -a - ob

FIOuRE

8-13.--Blade

incidence

nomenclature.

243

TURBINE

DESIGN

AND

APPLICATION

Low-reaction _/_ blades-_. o.

n •*,._/2___

_,_i_

U.j'_''_

Incidence angle,

The

8-14.--Characteristics

FiauaE

8-15.--L(ical

general

is shown

The

incidence.

figure

at large

may

positive

whereas

low-reaction

range. Another not

nation

thing occur

incidence.

This streamlines

cidence

is larger

be due

blade

and

the inlet

have

of incidence

to some

have

to be noted at zero may

from

incidence,

be explained for

other angle.

two

figure

small

tests

the

and

angle test

zero incidence

incidence

in figure

higher

flow

incidence of cascade

than

angle,

for

negative

on the suction

surface

8-15,

and

the

lack,

value of negative incidence. of the gas flow is large over

losses 8-14

at some

by the

inlet

at some Both

about

of incidence

but

surface.

loss with

local separation

range

loss.

a summary

positive

as indicated

blades

incidence

on blade

represents for

a wide

streamline curves upward and the true zero incidence 244

which

incidence,

blades)

the

separation

area, of separation at the same in which the mean acceleration

reaction

does

flow

is not symmetrical

a loss that This

i

of blade

of the variation 8-14,

loss curve

shows

smaller blades

nature

by

results. but

FIGUaE

b de

for

negative theory

as the flow impacts occurs when there

the

is that small

sketch

angles

which

the

8-16. one

incidence

with

that

loss

of negative

shown;

show

incidence

minimum

of figure are

loss is low,

same

amount

or

Also, (high-

the

The at

stag-

zero

in-

respect

to

stagnation

on the blade leading edge, is some negative incidence

MISCELLANEOUS

LOSSES

a
ab

F[aua_

8-16.--Curvature

relative

to the

free-stream

usually

-4 ° to -8 °. Because

blades because

with a small amount of the small difference

The

magnitude

off-design An

component any

that entry

recovered

incidence

here

to the

into

blade

parallel

loss and energy

the

kinetic-energy

and

to account

negative

loss due

on

be

others

importance

their do not

when

the

predicted.

A method

for

data

is described

in reference

11.

with

the

aid

a component direction passes

normal

of figure V.

8-13.

normal

The

to, and

a

(camber

line at inlet).

through

the

component

(V,_ \-_J

2

V_' =2_

blade

is entirely

c°s_ i

to incidence

is

V12 -2_ (1--cos

2i)

for the

incidence,

loss not occurring to

while

loss is

design

row lost.,

is

Li-

In order

incidence, takes

inlet the

at minimum

designers

component

that

angle

turbine

must

on test

V_ 2 VI' 2gJ-2gd and

some

loss

is described

the

kinetic

incidence

of a turbine

Vp parallel

stre,_mline at blade inlet.

of negative involved.

V_ can be resolved

If it is assumed the

The

of this,

loss based

method

velocity

without

of the

incidence

analytical

inlet

flow.

performance

determining

of stagnation

differences

the effect

(8-34)

in loss variation

of blade-row

at zero incidence,

(8-33)

reaction,

equation

(8-34)

and

with

positive

the minimum

has been

generalized

V12

L,=_

I-l--cos"

(8-35)

(i-io7,_)"1

zg,I where equation

ion, is the has

optimum

proved

(minimum-loss)

satisfactory

when

incidence used

angle.

in off-design

This

type

of

performance 245

TURBINE

prediction loss

data

positive

DESIGN

AND APPLICATION

methods are

such

lacking,

incidence

have

as that

values been

of reference

of n = 2 for used

12. Where negative

specific

incidence

incidenceand

n = 3 for

satisfactorily.

REFERENCES 1. HORLOCK, JOHN H.: Axial Flow Turbines. Butterworth Inc., 1966. 2. HOLESKI, DONALD E.; AND FUTRAL, SAMUEL M., JR.:Effect of Rotor Tip Clearance on the Performance of a 5-Inch Single-StageAxial-Flow Turbine. NASA TM X-1757, 1969. 3. KOFSKEY, MILTON G.; AND NUSnAUM, WXLLIAM J.:Performance Evaluation of a Two-Stage Axial-Flow Turbine for Two Values of Tip Clearance. NASA TN I)-4388, 1968. 4. KOFSKEY, MILTON G.: Experimental Investigationof Three Tip-Clearance ConfigurationsOver a Range of Tip Clearance Using a Single-StageTurbine of High Hub- to Tip-Radius Ratio. NASA TM X-472, 1961. 5. HONO, YONO S.; AND GROH, F. G.: Axial Turbine Loss Analysis and Efficiency Prediction Method. Rep. D4-3220, Boeing Co., Mar. 11, 1966. 6. DAILY, J. W.; AND NECE, R. E.: Chamber l)imension Effects on Induced Flow and Frictional Resistance of Enclosed Rotating Disks. J. Basic Eng., voi. 82, no. 1, Mar. 1960, pp. 217-232. 7. DAXLY, J. W.; ERNST, W. I).; AND ASnEDIAN, V. V.: Enclosed Rotating Disks with Superposed Throughflow: Mean Study and Periodic Unsteady Characteristics of the Induced Flow. Rep. R-_4-16, Massachusetts Inst. Tech. (ARO1)-2500-2, AD-443060), Apr. 1964. 8. STODOLA, A. (Louis C. LOEWENSTEIN, TRANS.): Steam and Gas Turbines. Vol. I. McGraw-Hill Book Co., Inc., 1927. Reprinted by Peter Smith, 1945, pp. 200201. 9. KLASSEN, HUOH A.: Cold-Air Investigation of Effects of Partial Admission on Performance of 3.75-Inch Mean-l)iameter Single-Stage Axial-Flow Turbine. NASA TN D-4700, 1968. 10. STENNING, ALAN H.: Design of Turbines for High-Energy-Euel Low-PowerOutput Applications. Rep. 79, Dynamic Analysis and Control Lab., Massachusetts Inst. Tech., Sept. 30, 1953. ll. AINLEY, D. G.; ^ND M^THIESON, G. C. R.: An Examination of the Flow and Pressure Losses in Blade Rows of Axial-Flow Turbines. R&M-2891, Aeronautical Research Council, Gt. Britain, 1955. 12. FLAGO, E. E.: Analytical Procedure and Computer Program for Determining the Off-De._ign Performance of Axial-Flow Turbines. NASA CR-710, 1967.

246

MISCELLANEOUS

LOSSES

SYMBOLS A

area

¢t

disk rim radius,

CII Civ

coefficient

C/ CM

,o

on one side

of rotor

disk,

m2; ft 2

m; ft

used

to evaluate

Ci.o

in regime

I I by equation

(8-11)

coefficient used fluid shear-stress

to evaluate coefficient

Ci,o

in regime

IV by equation

(8-15)

torque

coefficient

with

no throughflow

D

diameter,

f

nozzle

9 Ah'

conversion turbine

specific

_h_d

turbine

ideal

i

incidence

J

conversion

Kdl

disk-friction

Ko

ratio

Kp K.

pumping power loss coefficient, sector loss coefficient

K,o

rotor

Li l

incidence

M

frictional lb-ft

N

rotative

n

exponcnt

m; ft active

arc length, constant, work,

rotor-blade volumetric

1/mY2;

angular

velocity

(lbf)(seC)/(lbm)(ft

for full-admission

torque

tad/see;

impulse

'v2)

turbine

Reynolds radius,

8

axial

distance

U

blade

speed,

sides

of rotor

disk,

N-m;

h)ss, W ; Btu/sec Btu/sec

m; ft

throughflow

r

both

rev/min

loss, W;

pitch,

for

(8-35)

power power

R

rate,

m_/sec;

ft'_/scc

number m; ft

V

absolute

W

relative

between m/see;

velocity,

blade

inlet

rotor

disk and

from

angle

dynamic density,

static

ft/sec

axial

from

axial

direction,

deg

direction,

deg axial

direction,

deg

efficiency

viscosity, kg/m'_;

m; ft

m/sec/ft/sec

fluid relative angle measured from active fraction of stator exit area turbine

casing,

ft/sec m/see;

velocity,

fluid flow angle

P

to disk

m; ft

disk-friction P

velocity

Btu/lb

in equation

Q

pressure

coefficient

coefficient

speed,

pumping

of inlet-total

Btu/lb

angular

resistance

Pp

on ratio

1; 778 (ft) (lb)/Btu

loss, J/kg;

height,

based

J/kg;

power-loss

of rotating-core

blade

(seC)

deg

constant,

velocity

(ft)/(lbf)

Btu/Ib

work

pressure,

angle,

(Ibm)

J/kg;

specific

to exit-static

m; ft

1;32.17

(N)

(sec)/m2;

lb/(ft)

(see)

lb/ft 3

247

TURBINE

DESIGN

AND

T

fluid shear

T

throughflow angular

APPLICATION

stress,

N/m_;

number

velocity,

lb/ft _

defined

by equation

rad/sec

Subscripts: m

mean

n

component

o

no throughflow

opt

optimum

p

component

pa r

partial admission disk rim

u

tangential

1 2

248

section normal

parallel

component

_rotor inlet ],blade-row inlet rotor exit

to blade

inlet

direction

to blade

inlet

direction

(8-21)

CHAPTER 9

Supersonic Turbines ByLouisJ. Goldman A supersonic supersonic have

turbine

relative

potential

having

weights

used

They

have

been

have

been

studied

and/or

used

for space. Supersonic

turbines

puts

because

of the

type

of turbine

would

require

small

number

of stages.

It

atively

simple.

turbines than high

As

ratios are

a minimum

consumption, could

the

more

than

supersonic To

the

proper rotors

chapter, the

ideal

efficiency

are

losses.

For

systems

available

lower

the

optimum

designed

by

supersonic

turbine

following

headings:

must the design

design

(1) method

and (often

low

relless

blade-jet because

primary

high

a

of

design

minimum

fluid

pressure

ratio

and

may

result

in a

the

highest

possible

supersonic

stators

choice.

used.

and

the

this and

supersonic

1),

with

the

turbines

method

level,

ratios

where

out-

fluid

primarily

efficiency

be

power

(vol.

along from

turbine

of supersonic

methods

3

of stages

work

work

however,

efficiencies,

and

systems

specific

speed

(those

available.

be light-weight

2 and

static

the

being

design

chapters

are

of driving

velocities,

a

expansion systems

a given

blade-jet

to low number

offset

turbine

keep

level,

under

in

For amount

jet

low

ratios

large

therefore,

high at

indicated

correspond

ratio.

would,

fluids high

turbopump

for

a small

with turbines

auxiliary-power

potential

pressure

of

pressure

open-cycle

the

operate

exit-kinetic-energy

criteria

and

high

Because

generally 0.2).

speed

have

high-energy

in rocket

in

operates

Supersonic

consequently,

high

NASA use

that

rotor.

where

and',

where

by

for

as one the

in systems

molecular are

is defined entering

application

low

velocities)

stage

velocity

at Both

of

characteristics.

performance of characteristics,

In are

this

discussed (2) design 249

TURBINE

DESIGN

AND

APPI.,I_ATION

of supersonic

stator

(4) operating

characteristics

blades,

(3) design

METHOD The

method

certain tions

of

type

are

chapter. certain

OF

characteristics

is a

of partial

for

two-dimensional

the

of this

type.

Other

types

Only

this

flows)

method

blades,

of flow flow

can

be

for

equation.

supersonic

type

also

general

differential

of supersonic

non-steady

rotor

and

turbines.

CHARACTERISTICS

(hyperbolic)

of motion

gas

of supersonic

of supersonic

flow

will

(i.e.,

solving The

of a perfect

be discussed

axially

handled

by

a

equain this

symmetric this

and

method

(see

ref.1). The by

method

formal

of characteristics

mathematical

siderations.

The

derivation

processes

involved.

ematical

derivation

equations.

Both

It

based

on only

is useful

motion can

than weak

be considered

uniform

through

the

and

compression

wave

subsequently,

one

the

field

weak

flow)

waves an

are

expansion

the

here.

method

to other

is the

flow

waves, shock zero. shown wave

the

waves.

The

Examples figure

is

produced

"// V//////

_UX/, :/;//_Wa II

9-1.

wave

V

" V__ctV

X .....

7"/////i /

(a)

(a) FIGURE

250

(b)

Expansion. 9-1.--Weak

(b) expansion

and

compression

Compression. waves.

waves, entropy

of weak

in

of

a single

Mach

Mach

dV

similar

equations

through

ye

V

math-

2.

called

Mach wa

physical

The

1 and

satisfies

oblique

is essentially

stresses

(1) con-

Wall

that

These

ways:

dynamical

presented

in references

a Single

parallel wave.

to be very

change shown

flow

standing

pansion be

Along

in two

simple

dynamics

are given

supersonic

(other

vanishingly

developed (2) by

in extending

developments

simplest

be

and

will be the

Flow The

can

methods

ex-

As

will

when

the

SUPBRS_NIC

wall bends away from the flow, and a compression when the wall bends toward the flow. The

bend

(of

angular

as a disturbance

which

is to follow

the

a boundary

condition

with

uniform

solution may The the

wall. flow

The

bend

flow

along

a curved

through

a weak

Consider

the

of initial

of mass

requires

expansion standing

wave Mach

velocity

of it.

can,

The

wave

if the flow

be considered

any

as wave

of this

curved

of straight be

The

Mach

importance

that

therefore,

waves.

surface sections.

approximated

dynamics

as

of the

flow

be discussed.

included

V as shown

also

number

will now

can be considered

is a standing

it is realized

of Mach

is produced

is required

may

up of a finite

surface

a series

the wall

wall

sides

wave

which

one solution

when

to be made

through

in the

on both

can be appreciated

flow

dO) in the wave,

to which fields

be considered

rection

magnitude produces

TURBINES

at an angle

in figure

9-2.

The

_ to the

di-

conservation

that A=

p V,, = (p+dp)

( V. +dV.)

= constant

(9-1)

where w

mass

flow

A

flow

area

p

density,

kg/m3;

V.

velocity

component

Neglecting

rate,

kg/sec;

along

Mach

lb/sec wave,

m_; ft 2

lb/ft a

second-order

normal terms

to Mach (i.e.,

wave,

dp dV.)

m/sec;

ft/sec

gives

V.+V dp=o Conservation

of momentum pV.V

in the

(9-2) tangential

,= (p k- do) ('[7. + dV.)

direction

gives

(V ,+ dV ,)

(9-3)

t

Ma¢,// wave/

Z_Z_ n

p

p +dp + dV

dO

U//////////////////_ •. _ /////,

FIGURE 9-2.--Flow

through

a weak

expansion

wave,

and associated

nomenclature.

251

TURBINE

_, I_E,SIGN

where

Vt

is the

m/sec

or

ft/sec.

AND

APPLICATION

velocity

component

Substituting

tangent

equation

to

(9-1)

the

Mach

into

wave,

equation

in

(9-3)

gives

y,,V,=

pV,,(V,+dVt)

(9-4)

or

dV,=o This

means

stant dV

that

the

flow

crosses

to dV,

and

as the is equal

Conservation

tangential the

component wave.

of velocity

Consequently,

is directed

normal

of momentum gp--b pV,,Z=g(p

(9-2)

in the

remains

the

velocity

to tile Mach

normal

change

wave.

direction

+ dp) --b (p--b dp) (V,,--b dV,)

con-

gives _

(9-6)

where g

conversion

p

absolute

constant,

1 ; 32.17

pressure,

Substituting

N/m2;

equation

(9-I)

dV,_ by

using

(ft)/(lbf)

(sec 2)

lb/ft 2 into

O=g Eliminating

(lbm)

equation

dp.-}-pVn

equation

(9-6)

and

expanding

dVn (9-2)

(9-7) results

in

V.--g Equation

(1-57)

of chapter

where

a is speed

process (9-8)

being into

of sound,

considered

equation

1) states

(_)s

in m/sec here

(9-9)

(9-8)

1 (vol.

a---- _/g

(9-9)

or

ft/sec.

is isentropic,

shows

Since

the

substitution

differential of equation

that V,,:a

Therefore,

the

be equal

to the

component speed

yields

(9-10)

of velocity of sound.

normal

Noting

Vn=V

to the

from

figure

Mach 9-2

wave

must

that

sin tt

(9-11)

gives V, sin _V-V where and 252

M has

is the

Mach

number.

meaning

only

for M>

The 1.

a

angle

1 M ft is called

(9-12) the

Mach

angle

SUPF__SONrC

TURBINE,S

Mach wa ve

V

_" _

/ FIGURE

The dV

can

9-3.--Velocity

be 9-3.

du

713

v+dv

relation

figure

"-- -....

diagram

between

for

change

found

from

the

In the

limit

(dO-*0),

flow

through

in flow

velocity

angle

relations

a weak

expansion

wave.

dO and

velocity

change

shown

geometrically

in

du=dV

(9-13)

dv=Vdo

(9-14)

du dV B= _-_= _

(9-15)

and tan where du

component

of dV

parallel

to initial

velocity

dv

component

of dV

normal

to initial

velocity

Since,

as can

be determined

from

equation

V, m/sec;

ft/sec

V, m/sec;

ft/sec

(9-12),

1 tan/_=_/_-1_ equation

(9-15)

(9-16)

1

becomes dV

do

V -- 4_-1 It

is more

convenient

if dV/V

velocity

ratio

velocity

Vcr is equal

to

(M=I)

and

evaluated

(vol.

1). The

M*=V/VcT can relation

be

rather the

between

is expressed than

speed

Mach

of sound from

M*

(9-17)

and

in terms number at

equation M is given

the

of the M.

The

critical

(1-63) by the

of

critical critical

condition chapter

1

equation 253

TURBINE

l)E_IGN

AND

APPLICATION

/ M=

2

/

7 is the

heat

at

ratio

constant

temperature

of specific

volume.

_

_+i

V where

M,

1

(9-18)

7--1

heat

Since

M,

_

at constant

Vc,

is

pressure

constant

to specific

(because

the

total

is constant), dV V

Substituting

equations

(9-18)

dM* M*

and

(9-19)

(9-19)

into

equation

(9-17)

gives,

finally, dO:

1 M7--1.2-1 M, 7+1

This

is the

velocity relation

could

wave,

bends,

by equation to

type

each

(9-20)

changes

solution

9-4.

been

flow

Assume

that

producing

the

The

surface

through

each

motion

for

Mach will,

flow,

indicated

wave

provided

therefore, values

or simple

wave

.._- _lach waves //

I

\

//

V1

---,.-

/

/

/""

'_

\

, ve

; /

I/ /

",

"-,

>'

, //

d83 FmURE

254

9-4.--Representation

of

flow

along

a

convex

as

of a number

infinitesimal

Prandtl-Meyer

a

surface,

relation

field

and similar

compression

(convex) The

flow

A

sign.

is composed

wave.

combined

of

weak

a minus

a curved

a Mach

will be satisfied

is called

have

angle

wave.

a single

would along

in flow

expansion

for

(9-20)

equations

of flow

a change

weak

obtained

the

in 8 are small. the

between

a single

equation

consider

in figure

of small

This

that

us now

shown

relation

through have

except

Let

the

differential

change

(9-20)

_ dM* M*

wall.

be a of

dO.

flow.

SUPERSONIC

If the

number

continuous.

of segments

approaches

Integration

•0= _1 _I/4/_--4-1 arc_

infinity,

of equation

sin [(3,-- 1 )M*_--

(9-20)

the

TURBINES

flow

field

becomes

gives

_¢]+ 1 arc sin t/'_ _,_-_ -4-1 -- 3,)\ + constan (9-21)

If

the

angle

constant

given

and

it

Meyer

Mach

symbol

such

equation

is tabulated angle

from

is chosen

by

in

is the 1 to

(9-21) many

angle the

that

0=0

when

is called

the

references

through

required

(e.g.,

which Mach

the

M*=I(M=I),

the

Prandtl-Meyer

angle,

ref. flow

number

1).

The

Prandtl-

must

turn

in going

and

is often

the

_ (or u). Therefore,

/,-45

1

given

I_ _r _A- arc sin [(_--1

sin

J"_ --_ 1E2-arc

)M*_--_]

(9-22) Note

that

given

by

the the

change change

in flow in the

direction

respective

Oz-- 01: 50: The waves,

derivation there

the

velocity

sion

wave.

along

has

would

be

decreases This

a concave

means wall,

been

for

a minus (M

(50)

expansion sign

shown

the

from

V1 to V2 is

angles.

That

_2-- _I waves.

for Mach

in figure

is,

(9-23) For

in equation

decreases)

that

in going

Prandtl-Meyer

(9-17).

flow

through

angle 9-5.

compression Therefore, a compres-

_ increases

The

Mach

for lines,

flow there-

hock

Mach waves ,4/,

_\\\\\\\\\\_\\\\\

\"r ....

_o2

d61

FIGURE

9-5.--Representation

of

flow

along

a

concave

wall.

255

TURBINE

fore,

converge

relations, the

DESIGN

entropy

AND

and

of course,

a shock

would

as shown

be invalid

in the

in the

figure.

shock

region

increase.

FIGURE

256

form

APPLICATION

9-6.--Hodograph

characteristic

curves.

The

derived

because

of

SUPERSONIC

The

relation

ratio

M*

(eq.

in figure curves

between

The

9-6.

This

erate

these

An

The

on

the

/)2

must

Mach (shown

in

the

plane.

the

line

wave

The

may

plot.

The

flow

any

two-

has

around

characteristics.

been

M*=

with

critical

hodograph

varied

to gen-

1 at 0----0 represent velocity

ratio

Point

OP2 the

as

After the

P_ is located

parallel

preceding

the

procedure

_

Math

is

wall

by

drawing

diagram

by

Note P1.

the

segment

the the

The This

physical additional

to use.

if it is recalled

that

normal

P_P2.

in

cumbersome

into (line

S_).

through

be

V1

through

direction

continued

at best,

to

segment curve

to

example.

PI is located

hodograph

wall

wave

numerical

field

is divided

point

characteristic

Mach

is,

entirely

in the

V2 is found

process

be made

the

flow

a simple

initial

characteristic to the

the by

the

corresponding

corresponding

to Ur2 (or

V1 and to

9-7(b))

is that

the

allows

explained

wall.

(fig.

figure)

parallel

to

(fig. 9-7(a)),

expansion

separating

This

is best

a curved

characteristics

parallel

plane.

curve

as N1 in the is

as shown

a hodograph

through

are

and

along

to V1).

segments. The graphical cedure

of the

of segments

lie on

velocity

diagram,

hodograph

(9-21)

angle

physical

characteristic

the

called

passing

characteristics

flow

number

direction

are

the

critical

(9-22).

graphically

OP_ parallel drawing

curves

property the

wave

Consider

and

of equation

of Prandtl-Meyer

constructed a finite

of

surface

the

a polar

is called

curves.

to

on

of diagram

constant

important

Mach

be plotted

0 and

characteristic

by equation

normals

angle

are

of the

variation

expressed

can

flow

type

plot

convex

value

the

(9-21))

of this

dimensional

the

TURBINES

The

that

the

lo _-,___.._/

/

prodirec-

waves

-15,-J \

/

\,

Cb_

(a)

Phy,_ical

plane. FIGURE

(b) 9-7.--Flow

along

convex

Hodograph

plane.

wall.

257

TURBINE

I)E_SIGN

AND

/

APPLICATION

'_,]dC h waves

0 cai

(a) FIaURE

(hi

General

9-8.--Flow

case.

along

(b)

convex

wall

with

Math

Limiting waves

case.

intersecting

at

one

point.

tion

of

the

Mach

changes

occur

assumed

that

average

speed

direction

of

may

now

waves

A special shaped in figure

9-8(a).

point

0.

The

seen

that

This

type

corner,

Now

imagine

limiting

of flow

and

case

large

sonic nozzles for this case.

with

The method of solution generalized to handle the uniform figure

parallel 9-9.

amount. As finite and

The

before, number S'_,

represented represents 258

flow

the

that will

$3 and by

the

the

The

a wall

hodograph

point,

approaches

Two

or

to the

Equation

flow both

where

flow design

(9-21)

are

about

constructed initial

P_ in the

by

two

deflected the by

around

a

of super-

is still

valid

hodograph of the

as shown

outward the

here flow

the

of the

dividing

denoted

wall can be the initially

walls

centerline

parallel

magnitude

it is bends.

Walls

bounded

walls

common

of small

flow,

is so

as shown

9-8(b),

a number

corner-type throats.

figure

wall

the

used for the flow along a single flow between two walls. Consider

and

average

past

if the

by

is important

The

usually to the

to the

occurs

wall

later,

Between

be

finite

is

flow

a common

replaced

line segments, S'3.

point

direction

the

called

is symmetric

flow

of straight

and

has

supersonic

Suppose

The

wall

through

that

sharp-edged

Flow

relative

numerically.

is represented

is often

it

corresponding

points.

a single

pass

bend

as will be seen

_. Since

for visualization.

along

lines

angle

two

completely

of flow

angle problems,

measured

the

useful

Math

Mach

two points,

Mach

a single

the

practical

the

between

is still

the

of

at

constructed

case

that

lie

the

flow

though,

by

solution

between the

is given

the

the

be

diagram,

wave

in

diagram. velocity

same

channel.

walls

into

by S_ and field

a

S'_, S_

in region The

in

line

V_. The

1 is OP_ flow

S_PF__

V [

,F 2

.... /!:

>_,""'/L_

.... s3__ i ,////' , si .,,,,_>;_L_

s

S_NIC

TURBINES

\

15v--...

u

I C'I

,

/

c2

]

// 0

P7

_1-5/ J _ "" _///////_ k- V k_V 2

6 _ 3 _///////'_ ,..,,/////.

(a)

(a)

(b)

Physical

plane. FIOUR_

in regions

2 and

be determined, The

lines

The two

2' (points

problem

for the

OP'2

now

Mach

between

what

intersect.

a weak V'2,

fields

diagram)

can

expansion

wave.

respectively.

happens

Flow

plane.

hodograph

through to V2 and

is to determine

Hodograph

walls.

in the

flow

are parallel

waves

two

P2 and P'2

as before,

OP2 and

initial

(b) 9-9.--Flow

to the 2 and

flow

after

the

2' must

be

sep-

---4

arated

by

another

flow

field,

direction. Consider that continuation, in modified A jump

from

region

of motion

only

is,

C'2

C1 or

To

both

satisfy

ing the The the

point

makes

little

physical

makes

sense

direction given in

by

the

flow,

of the the

the

construct across

velocity the

the

P,

flow

channel

are

can

end

point

end

plane.

strikes

are

the

M'2

being

of the P'2P3

of

the

until wall.

equations P2; that

graphically). C'L or C2.

jump

this

further

would

represent-

and

one

P2P3,

A new

type

hodograph The

waves

are

respectively,

symmetry

of the

which

expansion. Mach

procedures

that

waves,

flow initial

plane

mean

P3 in the

assumed

to V_. These

piecemeal

the

(9-21) of the

same

2 and 2' by a M, and M'I.

through

compression

point

segments

Because

V3 is parallel field

and

the

characteristic

because

it represents

to the

satisfy

point

out

sense.

The

in

P3 or P1 in the hodograph

waves

M_

not

from waves

eq.

lie on

the

be ruled

because

can

represent

expansion

normals

V'2

wave

2' must

be either

extensions

hodograph

curves

region

3 must of the

and

lies on a characteristic

of conditions,

being

extensions

plane

these

sets

any

point

from

flow field

end

end

(since

a iump

V2

the flow field 3 is separated form, of the initial Mach

2 through

if the

Similarly,

since

can

of

the

be used

to

waves

advancing

of solution

is

now

required. 259

TURBINE

DES,IGN

Consider Sa.

The

wall. P_

the flow

5 cannot The

because

in field wall,

in field

hodograph the

flows

According

one

that

wave

4 and

As

seen

from

the

Mach

waves.

across

the

used

in the

wave

stream

and,

the point

line

OP4,

Also,

P_

is, C_ or C'3.

these

conditions

diagram,

wave.

In

where

general,

an

as an expansion of the channel

net.

flow for

the

summarized

are

shown

along

(c) show

properties

are found

found

known

at the are,

method" in reference

design

of supersonic 9-10.

the

for

most

will 1.

not

Flow

each

waves,

and

the

because of the higher the reflected wave.

Mach

9-10(d)

boundary.

in the

same

produced without

shows A bend

direction by

and

the

wave.

any

additional

cancellation

at the

proper

of the

The

same

points,

situations, further.

as others blade

that

9-10(a), expansion of an

associated

location

(b),

and the

expansion

magnitude

boundary

conditions

(reflected)

waves.

as the are,

the reincident the

wave

top

wall

flow

is

wave,

with

expansion

in the

are

situation

In figure 9-10(c), angle than is the

of an

are

sections,

physical

reflection

number

the

or

procedure,

discussed

Figures

and

wave across

Figure

rotor

of a weak

respectively. smaller Mach

this

or lattice

the

cases

from a solid boundary, wave is at a slightly

regions,

practical be

"field

Solutions

case,

wave flected

through-

"lattice-point

In

as well

solution.

discussed

the

intersections,

stator

For

as

the form

the

in small

problems.

discussed,

hodograph

of expansion

finite

satisfied

flow

of

are

in

is called

previously

figure

quadrilaterals satisfied

may

in each

of procedure

of Elementary

previously

intersection

are

channel

regions

approximately

solutions

with

the

are

methods

"lattice-point are described

in

of the

procedure,

Summary The

sides

in supersonic

Both

in the

type

This

calculation

properties

Mach

The

flow

quadrilateral

of motion

stream

used

the

of small

therefore,

field.

the

identical. The Both methods

260

before,

hodograph

discussion,

equations

is often

the

foregoing

flow

Another

method,"

fied

of

that

a solid boundary reflects of the flow in the interior

is constant.

The

since

fields.

solid

field

therefore,

direction.

Ps,

is an expansion

by a number

entire

method,"

used

and,

same

segment

it from

extension

in the

to those

wall

Therefore,

separate

wall the

through

as shown P6

wall.

6 must

on

6 are

similar

P5 and

velocity

each the

the

lie

to the

the

as before.

which

of

field

characteristics

P8 be located

be approximated

out

to

to the

must

in fields of the

expansion wave striking wave. The construction proceeds

a new

6 is parallel plane

between

is parallel

parallel

and

to arguments

require

4, which

5 is not

the

flow

lie on

the

flow

APPLICATION

in field

extend

in the

must

AND

flow

at

is made deflection

therefore,

satis-

a

SfUPE./_SONI_C

Figure section point istic

9-10(e)

shows

the

of an expansion representing curves

continues

for the

and

a compression

wave

region

passing

solution D must

through

unchanged

lie on

B and

in type

flow field

the

the

beyond

wave.

The

intersection

C, as shown.

beyond

TURBINES

the interhodograph

of character-

Therefore,

each

wave

intersection.

6

(al 6

OD

BA

/

b

?//'//_}i"///d

;:P; ;,, .¢;

; ;4,:,_/_

_c_

(a) (b) (c)

Reflection FlovRs

Weak

expansion

Intersection of an

expansion

9-10.--Elementary

of

wave.

expansion wave

waves.

from

a solid

flow

solutions.

boundary.

261

TURBINE,

DESIGN

AND

APPLI_ATrON

6 It//l.I//l/l_'//.ft/lllll_ A

t

"V/I/i/////

0 B : 04 - 6

L

6_ (dt

Z////////21//7_

_-B

/---_1'--Compression wave

O0 : 0A + 25 "////Z ////I/I_

(e)

P : Constant

A

wave

_on

(fl

(f)

(d)

Cancellation

of expansion

wave

(e)

Intersection

of

and

Reflection

of

expansion

expansion wave

FIGURE

A case sections,

that but

This

is the

free

boundary.

262

is not may

be

from

a

at

boundary

the

design

in

interest,

of an boundary

waves. (constant

pressure).

9-10.--Concluded.

of general

The

boundary.

free

encountered

reflection

solid

compression

expansion condition

of supersonic

is shown

wave

from

requires

in figure

blade 9-10(f).

a constant-pressure that

the

pressure

be

SUPERSONIC

constant

along

velocity

the outside

magnitude

be equal.

Therefore,

In general,

field

wave

as a compression

DESIGN One

of

the

most

at

dimensional has

speeds.

to have

uniform

design

of a stator

based

Nozzles A supersonic in figure sonic,

9-11.

the

(DE), Point

that

(M----1).

Because

line

may,

and

Therefore, The

the for

flow

on this

that first

AD

generates

The

calculation

the

the

flow

plane. (constant

to produce of

of nozzle

Uniform

design

This

type

blades, the

will

uniform,

outward

(AD)

at

the

nozzle

throat

purposes, of the

and

ABCDA

is called

expansion

the

waves

is the

the here.

curve

super-

in again

to the initial flow. slope. It is usually parallel,

by

is shown and

and

sonic

axis is a streama solid

boundary.

be designed. expansion

which

same

flow

then

the nozzle

need

it is

Only

be parallel

be replaced

nozzle

since

Flow

is uniform,

symmetry,

two-

of nozzle

rotor.

parallel

flow

parallel of

be discussed

Parallel

the

uniform,

the

stator entering

that

of characteristics

tunnels.

flow type

BLADES

the wall is again parallel the wall has its maximum

design

procedure

the must

hodograph

method

basis

wind

produces

curve

one half

region

wall

is

parallel

it is required

of the

only

of

of a channel This

so that at the exit, D is the point where

assumed

uses

Producing

nozzle Since

wall must

in the

a free boundary

to supersonic-turbine

desired

is isentropic,

the boundary

STATOR

for supersonic

application

from

flow

along

be as shown

SUPERSONIC

to the design

nozzles

the

lying

reflects

important

supersonic

Since

fields

wave.

OF

is its application

also

C must

an expansion

pressure)

flow

streamline.

of all the flow

TURBINES

as was

reflect

zone. off

discussed

The

the

curved

centerline.

in the

section

[

B

FIGURE

9-11.--Supersonic

¢

nozzle

producing

uniform,

parallel

flow.

263

TURBINE'

"Flow

DE,SIGN

Between

ing section, in the

flow

past

for A

with

form

points

A

(corner

flow),

parallel that

one

at

of the half

wall

bounding _,

A computer with

Only

the

The

input

number output 3 does

flow

waves

that

waves

Mach

A

may

and

2 is set

long

form

of

been throats

ratio

coordinates for

any

_

the

of the

flow, the

to the

Prandtl-Meyer angle

of

the

working The

shows

corner

3) to design

includes

uniform

Therefore,

method

nozzle.

centerline.

to obtain

is designed

7 of the

of the flow

(ref.

edge

waves.

equal

nozzle

program

specific-heat

account

by

of the

the

Prandtl-Meyer

written

where

sharp

(fig. 9-12(b))

as a result reflected

It is a

9-11),

the by

used

9-12(a),

length.

(fig.

diagram

exit

figure

around

reflected

at an angle design

in

possible

is again

occurs

computer

shown

are

of the

portion

A,D

_.

supersonic

characteristics. by

the

desired fluid.

program

program. exit

The

of reference

losses.

[

_,_

3

/ _,5" ] 181

Physical FIGURE

I1);

plane. 9-12.--Nozzle

Mach

program

I

264

D.

be too

a limiting

previously

expansion

has

to the

(a)

was The

final

between

cases,

shortest

hodograph

program

the

these

incoming

Solutions."

type

expands

waves

The

sharp-edged

gives not

of this

described

of the

the

the

The

occurs

throat, in the

The

is half

supersonic and

Flow

supersonic.

In

as a result

region

which

nozzles

exit.

half

the

nozzle

reflected

the

other

and angle

the

coincide.

of the

flow

and

a nozzle

flow

producing

Cancellation

so that

straighten-

of cancelling

expansion

sharp-edged

parallel

D

method

applications.

a

of

and

is curved

the

is used.

uniform,

limiting

much

numbers,

turbine

nozzle

The

is called

of Elementary

parallel,

how

DCED

section

cancelled.

Mach

of nozzle

produces

in this

region

"Summary on

exit

type

The

is uniform,

supersonic

this

are

depends

large

wall

section CE

number

APPLICATION

Walls."

the

waves

seen

For

Two

and

expansion

AND

(b) Hodograph with

sharp-edged

throat.

plane.

SUPgRSONIC

The for

the

tional

sharp-edged-throat design

Stator

Nozzles

nozzle

just

of minimum-length

considerations

discussed

(chord)

for a stator

serves

supersonic

as compared

to the

TURBINES

as

the

stators. nozzle

basis Addi-

previously

Ideal nozzle -_ t

/ / /

Displacementth ickness-,

/

\ \ \

/

/

Straight sectior_ angle /_ozzle

/ Diverging section-'-.

/ /

/ /

Tangential direction

t_

Converging section-_ \

Axial _Trection

FIow_

FIGURE

9-13.--Design

of supersonic

stator

nozzle

with

sharp-edged

throat.

265

TURBINE

D_IGN

discussed

are

energy

AND

the

APPL_CATrON

flow

turning

A supersonic-turbine discussed

a stator

flow.

the

section

The

stream

suction

determined

sonic

by the

nozzles,

of the

type

the

nates

is first

are

then

local

rotors

OF

methods are

that

discussed

characteristics.

The

caused

entering

by

the

the

rotor

by

entering surface

of the

of

cancel

the

incoming

zontal

flow

(0=0)

corner-type

flow

The

expansion,

with

the

by

ideal

methods

obtained nozzle

by

coordi-

is obtained 7 (vol.

from 2).

BLADES

the

design

methods

use

of supersonic the

any

compression

method

shock

waves.

to be uniform

rotor

type

blades

is shown

(region

I)

along

the

and

of

forma-

The

flow

parallel.

waves. This

waves

concave surface In

parallel being

is given

in figure undergoes

(suction)

compression is obtained.

momentum

in chapter

for

in

dashed

Method

flow upper

by the

profile

to prevent

supersonic

from

is

is presented

efficiency

design

is assumed

parallel

blade.

to

proposed

convergence

resulting

length

super-

is then

ROTOR

is designed

of this

the

of characteristics.

ideal

profile

The nozzle

Both

blade

uniform,

compression,

the

as described

herein.

of designing

1). A typical

method

thicknesses

Corner-Flow One method

freesection

its

thickness,

final

channel passage

the

by

straight and

the

of

desired

indicated

the

for

The

been

the

for losses,

SUPERSONIC

have

by

losses,

of sharp-edged-throat

(displacement

9-13.

designed

is designed The

a

section,

turning

to

profile,

profile,

designed

parameters

DESIGN

flow

a correction

computed

in figure

of the

section

design

2).

(1)

converging

to minimize

nozzle

nozzle

displacement

as indicated

be

can

to

angle.

the

6 (vol.

boundary-layer

Two

for

sections:

and

discussed.

the

nozzle

(no loss)

in chapter

adding

266

and

be referred

The

the

as previously

parameters

etc.)

discussed

(ref.

flow

(supersonic)

all

This

will

surface.

order

produce

exit.

section

of three

speed In

to

including

9-13,

Boundary-layer thickness,

2).

the

required

4. An ideal

in figure

sonic

completes

program

stator

reference

suction

accelerates

at

surface

A computer

a diverging

to

section

of characteristics

the

tion

include

and

the

5 (vol.

number

channel

(2)

is designed

diverging

Mach

method

the

to

9-13

consists

on

flow

and

nozzle

section,

of chapter

converging

lines

desire

in figure

stator

section

accelerates

methods

on

The

a straight

section

blade

is shown

(subsonic)

(3)

the

stator

herein

nozzle.

converging and

the

losses.

being as

and

region flow

cancelled

by Shapiro 9-14(a).

The

a comer-type lower

(pressure)

is curved 2, then by

so as to

parallel

hori-

undergoes the

concave

a

SUPERSONIC

surface,

until

at the

blade

to the

inlet

For the tion

uniform exit. and

needs

of comer

since

only

diagram 9--14(c) quite

flow

blade

of one

kind

because

the

drawback

This loading

zero

upper

number

in any region.

one

in the

middle

half

of

The

specifica-

easy

to design,

The

hodograph

figure 9-14(b). Shown velocity distribution,

is that

parallel

profile.

only

particularly

distribution

occurs

surface,

the blade

9-14(a)),

of blade

of velocity method

Mach

it is symmetrical.

are present

becomes

of this design

in fig.

is shown in blade-surface

type

on the

since type

desired

complete

shown

this

for this blade is the theoretical unusual.

segments

(as

makes

of the

directions,

to be designed,

flow

waves

flow

Straight-line outlet

an impulse

blade

parallel

TURBINES

in figure which is

is not

very

of the

blade.

Another

inlet

Mach

number,

for a given

desirable,

rParallel flow I

t t

Parall

,_

Parallel flow -_ (a)

M*

Inlet

Outlet Distance alon9 chord

_b)

(c) (a)

(b) Hodograph

Blade

and

passage.

diagram.

FIGURE 9-14.--Supersonic

(c) Blade rotor

design

by the

loading

corner-flow

diagram method.

267

TURBINE

the

I_ESIGN

amount

sonic

(Mach

is limited.

1) or higher.

equal

to the

reasonable

turning

APPLICATION

of flow turning

therefore, For

AND

The

sum

Mach

inlet

levels

(120 ° to 150 °) would

method 5. This

the passage.

In a vortex-flow

critical out

shown

ratio

field.

A typical

in figure

The

(1.5

arcs

passage

inlet

the

by

field sect.

begins where The concentric

The

outlet

cular

arcs.

by

are

any

bution

to be quite

outlet,

program

input and

The

output

the

blade

shape.

without

locations

includes using

circular

of turning.

the

inlet

the

the

the

of on

blade

the

to the

hodo-

amount arcs

velocity

figure

the

9-15(a). the

circular

surface

especially

rotor and

can

distri-

9-15(c).

This

as compared

of

of this

to the

type

6. The

outlet

angles,

blade

method

blades

in reference and

rotor

method

figure BF

figure

with

9-14.

numbers,

approximate

and

cir-

parallel

in

shown, arcs

IK

the

and

is no limit The

by profile.

in

interflow.

uniform

is shown

diagram,

loaded,

into

blade

are

because

blade-loading

vortex-flow

surface

design

there

waves gener-

waves first the vortex

circular

rotor,

in figure

the

arcs

of design,

highly

The

indicated the

tranat the

waves

flow

upper

of

inlet flow

generated

surfaces

is presented

Mach An

the

for designing

includes

surface

gas.

268

shown

of characteristics

program

is

transition

compression

vortex waves

type

in the

the

The

expansion

complete

along

degree

on

design

A computer method

throughmethod

parallel

9-15(a)).

the

blade

the

type

arcs.

and expansion turn and maintain

this

the by

this

fig.

on

the

flows

necessary

is seen

corner-flow

(see

inlet

the

the

remaining

for to

of

and

directions

obtainable

is shown

blade

arcs

flow

along

In

turning

provide

within

therefore,

this

(1)

uniform

means

reconvert the

represented

transition

arc

segments

outlet

flows

diagram.

of flow

by

parts:

the

by arc

arcs

constant-velocity

graph

flow

(also,

is a constant

designed

three

outlet

the compression circular arcs

diagram

The

surfaces

radius

convert

flow

Straight-line and

of

transition

corresponding

The

of flow

is described

vortex

of velocity

passage

transition

cancelling

A hodograph letters

and

(3)

vortex

transition

to the inlet

9-15(b).

and upper)

upper

flow

amounts

blades

on establishing

the product

essentially

lower

ated

parallel

rotor

and streamline blade

and

into

by the

large

is,

angles.

Method

is based

arcs,

(lower

generated

turning

Prandtl-Meyer

to 3.0),

supersonic

field,

M*)

consist

(2) circular

sition

of flow

exit

be

9-15(a).

blades

arcs,

method

velocity

the

amount

2 must

be impossible.

of designing

in reference

in region

and

Vortex-Flow Another

velocity

maximum

of the

number

The

the

flow

specific-heat

by

computer the ratio

inlet, of the

coordinates

and

a plot

for obtaining

the

transition

characteristics

is

the

described

of in

SUPERSONIC

TURBI_,ES

references 7 and 8. In this procedure, the vortex flow is established by making the curvature of the transition arcs one-half the curvature of the circular arcs. For very small curvatures, this method is correct. In this blade design, the lower- and upper-surface Mach numbers are specified. This permits blades of various shapes to be designed for

Region Vortex flow transition

AB and FG BC and EF HI and KL CDEand1JK

Straight lines Upper transition arcs Lower transition arcs Circular arcs

(a)

M*..- _H,B

C

E

L,G

A,H

X'x// )

\

/ I

K

Inlet

Outlet Distance along chord

mh_7,F (c)

Ib) (a) (b)

Hodograph FIGURE

Blade

and

passage.

diagram. 9-15.--Supersonic

(c) rotor

design

Blade by

the

loading vortex-flow

diagram. method.

269

TURBINE

I_E,SIGN

AND

APPLICATrON AA AB BD CC CD

k

Circular arc Uppertransition arc Straight line Circular arc Lowertransition arc

\ D/

(b!

ta)

A A

A

B

B

Lower-surface

Prandtl-Meyer

an-

gle, 0° (M = 1) ; upper-surface Prandtl-Meyer angle, 59 ° (M = 3.5) ; total flow turning angle, 130 °. (c) Lower-surface Prandtl-Meyer angle, 18 ° (M---- 1.7) ; upper-surface Prandtl-Meyer angle, 59 ° (M = 3.5) ; total flow turning angle, 130 ° . (e)

Lower-surface Prandtl-Meyer angle, 21 ° (M-- 1.8) ; upper-surface Prandtl-Meyer angle, 59 ° (M---- 3.5) ; total

flow

turning

FIGURE 9-16.--Turbine Meyer 270

B

A B

A

D

D

(a)

B

angle,

120 ° .

(b)

B

Lower-surface Prandtl-Meyer angle, 12 ° (M=l.5); upper-surface Prandtl-Meyer angle, 59 ° (M = 3.5) ; total flow turning angle, 130 °.

(d)

Lower-surface Prandtl-Meyer angle, 18 ° (M----1.7) ; upper-surface Prandtl-Meyer angle, 104 ° (M----10.7) ; total flow turning angle, 130 °. (f) Lower-surface Prandtl-Meyer angle, 21 ° (M---- 1.8); upper-surface Prandtl-Meyer angle, 59 ° (M = 3.5) ; total flow turning angle, 140 ° .

blade shapes at inlet Mach angle of 39 °) and specific-heat

number of 2.5 ratio of 1.4.

(inlet

Prandtl-

SUPEJRSONIC

a given

inlet

program

Mach

of reference

figure

9-16.

From

number

(cf. figs.

whereas

the

(c))

number.

and

effects.

the

9-16(c)

(cf.

in

will

of which

the

flow

for

program

for the design

including

a correction

(no loss) eters ing

passage

designed are then

in figure

the

of

later

case

computed,

indicated

final

Rotor

loss

profile

parameters

.... _--

shape, (b),

and

significant

is obtained

starting

by

problems,

characteristics" flow.

dashed

lines

design A computer

rotor

in reference

to the

is then ideal

in figure

_

._ --'__/'x

"

/__/

_,.

i/

9-17,

is

by

add-

as indicated

are determined 7 (vol.

ideal

param-

obtained profile

in chapter

sections,

9. The

Boundary-layer

profile

as described

blade

have

vortex-flow

coefBcients

in

chapter.

of characteristics.

and the

(f))

(isentropie)

by the

the

Mach

9-16(a),

design

of

is presented

thicknesses

boundary-layer

in this

of ideal

displacement

9-17.

and

of supersonic-turbine

the method

on the

figs.

supersonic

"method

by

upper-surface

effect

a blade

and

for losses,

profile,

by

the local

selection

designed

of 2.5 axe shown

the

(cf.

9-16(e)

separation

discussed

is only

number

has little

figs.

blades

that

number

be discussed

previously

Mach

(d))

flow

both

first

and

turning

of

it is seen

Mach

of

procedure

figure,

lower-surface

Guidance

number

6 for an inlet the

consideration The

A

_URBINES

from

the

2).

Displacement thickness

_

,--Loss-free

passage

\

Z

\

\

FIOURE

9-17.--Design

of supersonic

rotor

blade

section.

271

TURBINE

As

DESIGN

seen

from

transition

arcs

results

severe

in large

of the

upper-

9-15,

and

OPERATING

the flow

exist

along

passages.

The

of whether

places

Mach

blade

separation.

it would

criterion

lower-surface

gradients

rotor

an indication

to cause

If it is possible,

separation

and

pressure

within give

enough

losses.

The

adverse

EF

calculations

are

separation.

APPLICATION

figure HI

boundary-layer gradients

AND

the

Flow

be desirable limitations

pressure

separation to prevent

on

the

choice

numbers.

CHARACTERISTICS

OF

SUPERSONIC

TURBINES Supersonic

the

Problems diffuser

Starting

occur in the starting must be able to swallow

of supersonic the shock that

diffusers because forms at the inlet Lowe r-s u rface PrandtI-Meyer angle,

140

o)/, de3 ,/_---

120

120 ll5 llO 105 9O 85 8O 75 7O 65 6O 55 5O 45 40 3O 75 2O 15 10

___------------

20

0

t

1

20

40

1 60

Upper-surface

FIGURE

9-18.--Maximum

272

1

I

I

l O0

120

140

starting.

Specific-

Prandtl%_eyer

Prandtl-Meyer heat

t 80

angle ratio,

1.4.

an(jl¢,

for

'%,

supersonic

deg

SUPE_

during

startup.

in shape,

Since

similar

of supersonic

turbines.

a normal

shock

ing.

permissible

The

condition, wave

Mach

flow

can

this rotor

blades

angles.

(often

angles

w_ and

(or inlet

established. value

the

must

the then

wz), there

starting

instant

passage

enough

that

of start-

is set

to permit

by

the

velocity

circular-arc of

this shock

diagram,

and

the

order

to

determining the

maxi-

of vortex-flow

blade-surface inlet

of the

supersonic

9-18,

starting

the

in

value

for

6. In figure

of the

of

corresponding

for which

procedure

problem,

determined

segments

the

a maximum

angle)

for supersonic

as a function design

be

exists

in reference

angle

the terms

calculation

is given

usual

from

in

Prandtl-Meyer

The

is plotted

In

at the

blade

along

expressed

Prandtl-Meyer

is known angles

numbers

the

be large

in the

it is assumed

passage

of

Mach

to occur

approximation,

blade

must

flow

maximum inlet

a first rotor

contraction

number

mum

is convergent-divergent

be expected

passage

surfaces

be

passage

TURBINES

through.

Prandtl-Meyer inlet

blade

would

the

the

specified

blade

rotor As

spans

since

to pass

For the

the

problems

SONI_

Prandtl-Meyer

Prandtl-Meyer

angle

surface

Prandtl-Meyer

obtain

the

rotor

blade

8O

-

k

Lower-surface Prandtl-Meyer angle,

E

_

_o

&

39

r_

•_

I

4o

g &

c 5

__e_,

o,io, S9 0

F_,...'_, ,//

,

7" ,'_7,_97._/'/77"7"//'77/'I

,'

,

,

,/

'

,

/,

20

E

I

I

r_

o

l_°l

i_oo

I lJ

1

_

50

30

l 70

L

I

,,

90

I 1 110

Upper-surface PrandtI-Mever angle, wu, deg FIGURE 9-19.--Supersonic Mach number 2.5;

inlet

starting criterion Prandtl-Meyer

applied to example angle, 39°; inlet flow

turbine. Inlet angle, 65 °.

273

TURBINE

and

DESIGN

passage

AND

profiles.

conditions,

the

in figure

APPLICATION

To

point

9-18

must

assure

representing

the

lie on or above

the

inlet

Prandtl-Meyer

angle.

tion

places

restriction

surface

a severe

Mach

Suppose

numbers. a

(1) _=1.4;

blade

is

noted

that

39 ° to 104 °. The

must

remain the

at

inlet

shown.

be

would starting limit

than

indicated

the

limit

the

ported

in

efficiency

not

be

in the

velocity ratio)

efficiency rapidly

as

to that efficiencies

pressure (circles

for

14.

at ratio

in fig.

a subsonic at

curve.

the The

ratio

(blade

angle

Prandtl-

this

example.

design

purposes

are

region

shown because

consideration lower values

If

For

pressure

ratios

divided

were

the

data

ideal

for

given

ratio

and in

have

the

speed, falls

off

maximum

ratio

a subsonic

in supersonic-turbine

by

any

speed

would

re-

exit-static-pressure

variation

blade-jet

are

supersonic-turbine

presents

The

this

turbines

speed

14.

design

with

pressure

in

to

which

decreased.

9-20)

decrease

flow

the

inlet-total-

reference

turbine.

lower

variation

about is

fact

inlet

°, and

supersonic

The

9-20, of

the

Performance for

turbine

turbine

is maximum

efficiency

274

to

flow

previously

Flow separation value of w_ to much

data

in figure

for

vary

the

figure.

speed

corresponding is illustrated

of _,

for

to

inlet

discussed

_,=39

Turbine

to

blade-jet

partial-admission

envelope

10

the

°.

_, can that

maximum

_,

permissible

that fact

w, is due

exceed the

conditions:

(3)0,.----0,_=65

to the on

of

an example.

following and

condivalues

from

0 ° to 39 °, and

9-19,

considerations. the maximum

design

seen

the

_z is due

w_ and

represents

performance

with

for

to the

starting

permissible

as a function

line

references

corresponding

is best

cannot

of

Supersonic Experimental

angles

supersonic

104 ° limit

In figure

bounds

of supersonic will, in general,

the

from on

is plotted

dashed

crosshatched

vary

turning

the

Prandtl-Meyer

ordinate the

designed

sonic;

w_,. ,,_

The

can

104°).

only

given inlet

(_,.=_,x=39°);

transition

angle clarity,

surface

restriction

0 ° limit

least

of 65 ° (39°+65°= For

to

_

from

Meyer

on

This

for the

starting

In general,

(2) M,,=Mex=2.5

It is first

that

satisfactory

is similar turbine,

the

fallen

on

the

efficiency

at

the

8UPF_R_)NIC

TURBII_S

Ratio of turbine-inlet total pressure to turbine-exit _

a

static pressure 150

o o

120 63 (design)

_

"_

_

I

.30

cCD

.2O _3 t_

.10

....... 0

F,GuR_.

L__ ......... 2___ .04

9--20.--Static

.08

efficiency

The

1 .16

of turbine as function for conseant speeds.

lower pressure ratios is due expanded

A_ .12 Blade-jet speedratio

to

J

l

.20

of blade-jet

.24

speed

ratio

the shocks occurring in the under-

stator nozzles.

variation in static pressure throughout

the turbine of reference 14 is shown of the shock waves

the stator nozzle of

in figure 9-21. The

in the underexpanded

formation

nozzle is readily apparent.

It can also be seen from this figure that at pressure ratios near design, the divergent section of the nozzle performed pressure did not remain

as expected, but the

constant in the straight section. There

some overexpansion followed by some compression. havior was found in the data of reference 10.

This

same

was be-

275

TURBINE

I_E_IGN

AND

APPLICATION

• 10-__

Divergent -section

Straight .... section

["-

I

•09 --

.0_

•07 -a

c

9

•05 -L_

V o

•03 --

A •02 -[] Theoretical .01 -Throat I

Exhaust

Ii

i

.2

.4

It .6

L

_

.8

1.0

Fraction of axial distance FIGURE for

276

9-21.--Variation constant

of ratios

nozzle

of nozzle

pressure exit

static

ratio pressure

with to

axial inlet

distance total

in

pressure.

nozzle

SUPE_tSONIC

TURBINES

REFERENCES 1. 2. 3.

4.

5.

6.

7.

8. 9.

10.

11.

12.

13.

14.

ASCHER H. : The Dynamics and Thermodynamics of Compressible Fluid Flow. vol. 1. Ronald Press Co., 1953. LIEPMANN, HANS WOLFGANG; AND PUCKETT, ALLEN E.: Introduction to Aerodynamics of a Compressible Fluid. John Wiley & Sons, 1947. VANCO, MICHAEL R. ; AND GOLDMAN, LOUIS J. ; Computer Program for Design of Two-Dimensional Supersonic Nozzle with Sharp-Edged Throat. NASA TM X-1502, 1968. GOLDMAN, Louis J.; AND VANCO, MICHAEL R.; Computer Program for Design of Two-Dimensional Sharp-Edged-Th_roat Supersonic Nozzle with Boundary-Layer Correction. NASA TM X-2343, 1971. BOXER, EMANUEL; STERRETT, JAMES R. ;AND WLODARSKI, JOHN; Application of Supersonic Vortex-Flow Theory to the Design of Supersonic Impulse Compressoror Turbine-Blade Sections. NACA RM L52B06, 1952. GOLDMAN, LOUIS J. ; AND SCULLIN, VINCENT J. : Analytical Investigation of Supersonic Turbomachinery Blading. I--Computer Program for Blading Design. NASA TN D-4421, 1968. STRATFORD, B. S. ; AND SANSOME, G. E. : Theory and Tunnel Tests of Rotor Blades for Supersonic Turbines. R&M 3275, Aero. Res. Council, 1962. HORLOCK, J. H. : Axial Flow Turbines: Fluid Mechanics and Thermodynamics. Butterworths, 1966. GOLDMAN, LOUIS J.; AND SCULLIN, VINCENT J.: Computer Program for Design of Two-Dimensional Supersonic Turbine Rotor Blades with Boundary-Layer Correction. NASA TM X-2434, 1971. MOFFITT, THOMAS P.: Design and Experimental Investigation of a SingleStage Turbine with a Rotor Entering Relative Mach Number of 2. NACA RM E58F20a, 1958. STABE, ROY G.; KLINE, JOHN F.; AND GIBBS, EDWARD H.; Cold-Air Per-

SHAPIRO,

formance Evaluation of a Scale-Model Fuel Pump Turbine for the M-1 Hydrogen-Oxygen Rocket Engine. NASA TN D-3819, 1967. MOFFITT, THOMAS P. ; AND KLAG, FREDERICK W., JR. : Experimental Investigation of Partialand Full-Admission Characteristics of a Two-Stage Velocity-Compounded Turbine. NASA TM X-410, 1960. JOHNSON, I. H.; AND DRANSFIELD, D. C.: The Test Performance of Highly Loaded Turbine Stages Designed for High Pressure Ratio. R&M 3242, Aero. Res. Council, 1962. GOLDMAN, LOUIS J.: Experimental Investigation of a Low Reynolds Number Partial-Admission Single-Stage Supersonic Turbine. NASA TM X-2382, 1971.

277

TURBINE

I)E_IGN

AND

APPLICATIO_

SYMBOLS A

flow

a

speed

g M

conversion

M*

critical

P

absolute

area

along

of sound,

Mach

wave,

m/sec;

mS; ft _

ft/sec

constant,

1;32.17

(lbm)(ft)/(lbf)(sec

_)

number velocity

ratio

pressure,

component

of

m/sec; V

Mach

lb/ft _

velocity

parallel

to

initial

flow

direction,

initial

flow

direction,

ft/sec

velocity, critical

V

component

Y)

mass

m/sec;

ft/sec

velocity

(M--

of

m/sec;

1), m/sec;

velocity

ft/sec

normal

to

ft/sec

flow

Mach

(V/Vc,)

N/m_;

rate,

angle,

kg/sec;

lb/sec

deg

ratio of specific heat constant volume

at constant

small

direction,

change

0

flow

p

Prandtl-Meyer

P

density,

in flow

angle,

pressure

to specific

deg

deg angle,

deg

kg/m 3; lb/ft 3

Prandtl-Meyer

angle,

deg

Subscripts e_

rotor

exit

in

rotor

inlet

l

lower

surface

ITI4IZ

maximum normal

direction

7"

relative

8

isentropic

t

tangential upper

278

of blade with

direction

surface

respect

with

of blade

to Mach

respect

wave

to Mach

wave

heat

at

CHAPTER 10

Radial-lnflow Turbines ByHarold E.Rohlik Radial-inflow space

power

turbines systems,

are suitable

and other

for many

systems

applications

where

compact

in aircraft, power

sources

are required. Turbines of this type have a number of desirable characteristics such as high efficiency, ease of manufacture, sturdy construction,

and

reliability.

radial-inflow

There

turbines

in nature

and cover

most

machines.

In this

chapter,

its

features

addition, design

flow

turning long 8.

Radial

with

a section

the flow

those

stator takes

In

through

radially place

axial

height

the varies

rotor

and from

stator

which

is a doughnut-shaped is a spiral

inlet.

torus

is fed inlet

axial-flow

and

turbine.

design,

In

and

off-

by pipe.

flow

the

rotor

passage, blade blades,

This

is relatively

ratio,

about on

turbine.

axially.

which

aspect

from

plenum,

passage,

a radial In

radial-inflow

1 to

the

which

is

as much

other

hand,

the

inlet case

or a volute

usually pipe,

surrounds while

of a volute,

the

(shown the

in

stator

volute

is fed

a prewhirl

(tan-

of velocity) is imparted to the gas before it enters row. This results in stator blades with little or no

It can be seen

radial-inflow

of these

0.1 to 0.5.

which

camber.

on

is described,

blade

leaves

turbines,

ratios

gential component the stator blade

an

a typical

and

A torus, The

of

to chord,

aspect

a tangential

performance

turbine

in the rotor

fig. 10--2), by

of information

1 to 6 are general

and

performance,

turbine

have

design

radial-inflow

and

of blade

generally

of the

the

amount References

are discussed.

narrow.

ratio

areas

geometry

shows

enters of the

and

the as

10-1

literature.

compared

performance

Figure The

are design

is a substantial

in the

turbine

from

figure

is considerably

10-2

that

larger

the than

overall the

diameter rotor

of a

diameter. 279

TURBINE

DE, SIGN

AND

APPLICATION

Station :,

0

Stator blade -

1 2

Rotor blaue

FIGURE

At little

the

rotor

or

no

usually

10-1.--Schematic

inlet,

generally

is rather

varies

where the

velocity

has

Figure shows solidity generally turbine blades 280

to

or no

shows

the

prewhirl

(ratio

of

used shown in the

of

turn

radius.

the

blading

the

stators

They

section

angular

blade

rV,

of absolute

V,,=U_r,

the

that

the

The

stator

(where

velocity)

W_=O, At

has are

rotor

momentum

(Since so

rotor blades

of the

rV,,a:r2.)

flow,

to the rotor

rotor

exit

exit,

absolute

whirl.

to

has

the

component

more

is developed

in

rotor.

since

turbine.

relative

straight

Therefore,

chord

here

velocity

tangential

square

of radial-inflow

(W_=O),

This

speed.

curved

little

10-3 that

the blade

are

flow

loaded,

V_ is the

with

U is the blades

radial.

highly

and

the

section

component

and

r is the radius, here

where

tangential

straight

cross

clearly.

in the

inlet

volute.

and

low

aspect

spacing)

of radial

splitter,

or

are

in the

used

turbines

partial, radial

blades part

blade Also, ratio

can

be

between of the

shape the

low

that

are

seen. the

The full

flow passage

RADIAL-INFLOW

TURBINES

Stator

/- Rotor /

Volute

C-72323

FIGURE

to reduce the the "BLADE The in

The

relative

radius,

an

axial

in a radial because

temperature

discussed

discussion).

turbine

because

overall

expansion.

diagram

process turbine

total

as was

associated

This

in figure

and

the

_hown

for

tile

p_'

line

fig.

2-8

only

of ch.

to rotor

2), losses.

to both

the

If

be only

this

the the

relative

were

an

below

losses

from and

the the

(2-31)

for

the

change

and radial

a given

the

turbine

p_'

line p['

rotor

temperature

pressure

as shown

p't' line

for

through total

axial

between

turbine,

decreasing

eq. level

total

the

rotor.

temperature-entropy

expansion

in

from

the

with

velocity

in relative

difference radial

removed rotor

the

change

slightly

because For

shows

in

1 (see

the

in

appreciably

advantage

from

further

change

decrease

of a lower seen

change

expansion.

would

use

Tile

corresponding

the

pressure

a distinct

be

which

turbine.

the p;' line is farther is due

10--4,

the

discussed

differs

radius

2 of volume

is

can

are

turbine

and

This

blades

of the

in chapter

it permits

of a radial-inflow T"

turbine.

blade loading. Splitter DESIGN" section.

expansion

that

10-2.--Radial-inflow

p" (T['=T_'),

(as and

shown p_"

in figure

because in radius.

are

the

in

is due 10-4,

difference Therefore, 281

TURBINR

I)F__SIGN AND

APPLICATION

Rotor splitter blade--

Stator blade--.

Rotor full blade

/

/

/ t

C-?1863 -

FIGURE 10-3.--Turbine

expansion

from

exit the

pressure exit in

static rotor

vertical

distance

friction

losses

the relative is clear. A

the

same

gas

U_ and the

increase

velocity,

the

velocity

approximately turbine. 282

Us is very relative three

total

assembly.

pressure

a higher than in

diagram inlet 0.5.

times

to

is shown

the

level

in

W= at (larger

figure

For

a typical

as high

W] if

zero-exit-whirl

leaving

U2 equaled

the UI,

rotor

fluid

square

of

of velocity 10-5

volute and a mean diameter The difference between the

energy

same

Since

with

of a lower

the

velocity turbine

turbine.)

approximately

advantage

evident.

kinetic

p'_'

relative a radial

p_" and p= for an axial

a rotor

turbine with prewhirl in the (exit-mean to inlet) of about diagram,

and rotor

inlet

p_ would require an axial turbine

radial-turbine

speeds

stator

rotor

between in

. _gak-

for

a

ratio blade

velocity would

as in an

be axial

RADIAL-INFLOW

TURBINES

i

T1

tl

2gJCp

TI

E

2gJcp

F--

Entropy FIGURE 10-4.--Temperature-entropy

diagram

for a radial-inflow

turbine

rotor.

V0

W]

W2 /

p2-.

U2 FIGURE 10-5.--Velocity

diagram.

283

TURBINE

DE'SIGN

AND

APPLICATION

OVERALL

Since

the

shown

blades

in figure

angle

that

provides

This

angle

blade.

This

in a centrifugal

of the

blade

Before

ferentially gradient the

that

the

point

when

separate sive cally

from

loss. and

The

so that pattern

is an

there

If

suction

relation

experimentally

properly

"optimum"

10-6.

angle

Note this

that were

surface

between in

FIGURE

the

near

the

10-6.--Streamline

with

the

unloading

flow

blades,

_1. This the

U1 has

compressors

toward

flow condition

inlet

stagnation is shown

stagnation

flow

would

edge,

causing

been

studied

and

turbines.

inlet.

rotor

it is circum-

pattern

/

flow at rotor

in the

shift

the

flow at the

"slip"

static-pressure

of this

locates

so,

;uction urface

to the

is a streamline

leading

V_._ and both

leading

a large

the

incidence a radial

rotor

not

fll

rotor-blade

of mass

produces

angle

as 40 ° with

is analogous

flow analyses

Pressure surface

284

by then

flow

is some

is associated

distribution

loading

passage,

radial. the

the

inlet

at the as high

condition and

the

There

conditions

Stream-function

in figure

is approximately

angle.

is influenced

streamline

inlet,

sometimes

and

Blade

the

there

schematically

tip flow

surface.

show

flow

a value

the

rotor

incidence

incidence

the

across

at the

compressor,

uniform.

suction

Incidence

is an

optimum

factor passage.

Optimum

optimum has

near

CHARACTERISTICS

are radial 10-5

edge.

DESIGN

point tend

to

excesanalytiIt

has

RADIAIr-INFLOW

been

determined

optimum

that

ratio

number

there

depends

and is often

is an optimum

on

blade

expressed

as

loading

V_"=1---2 where

n is the

number

of

ratio and,

TURBINES

of V,. 1 to

U_. This

consequently,

blade

(10-1)

U,

n

blades

(total

of full

blades

plus

splitter

blades). Effect The vol.

of Specific

specific

Speed

speed

1) is given

by

on Design

parameter the

Geometry

N_ (derived

and

and

Performance

discussed

in ch. 2 of

equation NQ21/2

N_--

(10-2)

H314

where N

Q2

rotative volume

H

ideal

speed, rad/sec; rev/min flow rate at turbine exit, work,

or head,

based

m3/sec;

on inlet

ft3/sec

and

exit

total

pressures,

J/kg;

(ft) (lbf)/lbm In

its

most

units), size

commonly

it is not and

be

and

Analytical examined

truly

may

geometric

by

used

form

(with

dimensionless.

considered

Specific

as

velocity-diagram

the

a

shape

stated

U.S.

speed

customary

is independent

parameter

that

of

expresses

similarity.

study.--The

effect

substituting

for N,

of specific Q2, and

N

=

speed

on efficiency

may

KUI

(10-3)

Q2=_-D2h2V2 H=Vj2

be

H as follows:

(10-4)

(hh'_

(10-5)

where K

dimensional

D1

rotor

D2

rotor-exit

mean-section

h2

rotor-exit

passage

V2

rotor-exit

fluid

m/sec;

ft/sec

v,

inlet

ideal

jet

ratio, g

constant,

conversion

(tip)

speed, m/sec;

2_ rad/rev;

diameter,

m;

velocity based

ft

diameter,

height,

60 sec/min m;

ft

m; ft (assumed on

to be in axial

inlet-total

to

direction),

exit-static

pressure

ft/sec

constant,

1 ; 32.17

(lbm)

(ft)/(lbf)

(sec 2) 285

TURBINE,

DESIGN

Ah_

ideal

AND

work

APPLICATION

based

on inlet-total

and

exit-total

pressures,

J/kg;

based

on inlet-total

and exit-static

pressures,

J/kg;

Btu/lb Ahid

ideal

work

Btu/lb These

substitutions

expression

for

and

specific

some

manipulation

result

in

the

following

speed:

y. N,=(Constant)\_-7]u The

terms

teristics by an

\-_j/

of equation

(10-6)

are

\U],

related

\D,]

of these

combinations

7 to determine

were

optimum

(10-6)

to velocity-diagram

and overall geometry. Any specific speed infinite number of combinations of these

number

\D.]2

examined

combinations

value ratio

charac-

can be achieved terms. A large

analytically

over

a wide

in reference

range

of specific

speed. The

analysis

of reference

properties,

neglecting

were

caused

those

shroud

clearance,

kinetic

energy.

exit

flow

the

stator

windage

on

number

to provide

This

was

the

tion

(10-1)

included

the

total

the

to establish

a maximum

limit

of

the

rotor, varied

blade-to-

and

the

with

(10-7) would

avoid

splitter)

angle. zero

and

exit

stator-

equation

plus

incidence

Dt.2/D_,

considered

layers,

was

that

(full

(W_:2W_),

of 0.7 for

flow

12

number

rotor

losses

the

to the

of blades

reaction

The

blades

(a_--57)_T

minimum

mean-diameter

boundary

back

according

number

a favorable

rotor

of rotor

al (in degrees)

to

variations. and

n:0.03 in order

losses

hub-to-shroud by

The

angle

7 related

separation.

used

Other exit

in equa-

assumptions

whirl

a minimum

(V_,2:0),

limit

of 0.4

for

(nh/D,)2. The

effects

examined number

of

height Vc,)_. fell

The

to be

region

for

set

by

geometric large values. 286

mentioned

combinations

of

stator-exit

flow

D,.2/D_

static

the

any

in static

angle.

is the

value

angle

The

against

as much

ratios

specific

falls

boundaries and speed, the

speed. points

flow

angle

into

a small

of each by

the there

as 45 to 50 points of all

(U/

calculated

Stator-exit

variables

a large

to rotor-inlet

velocity

which

were

for

stator-blade-

rotor-exit

of specific

envelope

a_,

all of the 10-7.

of input

given

efficiency,

curve

plotted

study,

of efficiency,

flow values

For

and

losses

critical

in figure

determinant

extreme

dashed

then

in the

shown

stator-exit

hdD_,

rotor-tip

was

used

areas

a prime each

three

efficiency

limits. The

ratio at

of values

shaded

variation

characteristics

previously

range

in the

velocity-diagram

the

ratio

the

seen are

and

calculating

to rotor-inlet-diameter

diameter For

of geometry

by

computed

is

region assumed can

be a

for some static

RADIAL-INFLOW 1.0

Total efficiency corresponding to curve of maximum static efficiency

Stator-exit flow angle, aI , deg

.9 _

TURBINE_

of maximum static efficiency

8

\ \

.7

\

8

\ \

.5

.4

.3

._ 0

.2

[

I

[

.4

.6

.8

t

I

I

1.0

1.2

1.4

Specific speed, Ns, dimensionless

l

I

I

I

I

I

l

I

I

I

0

20

40

60

80

lO0

120

140

160

180

Specific speed, Ns, (ft314)',lbm314)/(min)(secll2}(Ibf3t4) FIGURE

10-7.--Effect

efficiencies, total

of

and

the

efficiencies.

achievable

associated

with

primary mum

Most the ure

specific height creases

of that

study,

velocity

curves. of specific

10-8

specific

model

geometric

envelope

functions

shows

The

that and

speed.

Figure increasing

efficiency.

corresponding

do not

are

many

study was

necessarily assumptions

of reference

to determine

shows

specific

values

vary

7. The the

opti-

to a larger

flow flow

that

the optimum

is small

at low

speed

continuously

of some in figures

stator-exit

(opens

diameter

ratios

presented

optimum

10-9

there in the

velocity

are

decreases

to rotor-inlet

the

of efficiency

however,

optimum

the

it represents

values used

design-point

ratios. and

speed

speed

with

above because

loss

and

of the

curve

computed values,

the

concern geometry

solid

The

represent

specific speed on computed (Data from ref. 7.)

until

of 10-8 angle

area) ratio

specific

a maximum

these

along ratios

to 10-11. is large with

as Fig-

at low

increasing

of stator-blade speed

and

in-

is reached

at 287

TURBINE

DESIGN

AND

APPLICATI.ON

90-XD

_" 8O

¢D

_o 70

60

I

5O .2

.4

.6

.8

1.0

1.2

1.4

Specific speed, Ns, dimensionless

l

1

I

I

I

I

t

1

I

20

40

60

80

i00

120

la3

160

180

Specific speed, Ns, (ft3J41(Ibrn3/4)/IminJlseclJ2jIIbf3/4_ FIGUaE

10-8.--Effect

of specific speed on optimum (Data from ref. 7.)

stator-exit

angle.

Blade critical velocity ratio at rotor inlet, • 20 --

(UtVcr)l 0.20

O

oA XZ

L_"

_z u_

C

0

0

a:

.08

.04

0

I .4

.2

I .6

1 .8

[ 1.0

I

l '1.2

l.a

Specific speed, Ns, dimensionless

I

I

I

I

I

I

J

I

I

20

40

60

80

l O0

120

140

160

180

Specific speed, Ns, (ft3/d_tlbm314t/Imin!lseclIZJ_lbf3i4_ FIGURE

288

10-9.--Effect of specific speed blade height to rotor-inlet

and blade diameter.

speed on optimum (Data from rcf.

ratio 7.)

of stator-

RADIAL-INFLOW

TURBINES

.l E 7-

.6 O

.4

--

¢_

.o f-f

I

.2 0

.2

.4

.6 Specific

I,

I

0

20

I 40

I

I

I

I

.8

1.0

1.2

1.4

speed, Ns, dimensionless

I

1

60

80

I

I

I

I

I

100

!20

140

160

180

Specific speed, Ns, (ftJld}(Ibm3/41/(min_secll2)ilbf314_

FIGURE

10-10.--Effect diameter

14

specific

speed

on

diameter.

optimum (Data

ratio from

ref.

of

rotor-exit

tip

7.)

--

L

I

0

FIGURE

of

to rotor-inlet

20

10--I

1.--Effect

of specific (Data

speed

on optimum

from

ref.

blade-jet

speed

ratio.

7.)

289

TURBINE.

some

value

in the mum

DESIGN

AND

APPLICATION

of specific

speed

turbine.

The

only

geometric

and

velocity

it is seen blade

that

height

Figure

10-10

higher to

depending

effect

shows

that

ratios

velocity the

level

any

ratio

of velocity on

in figure

in smaller

at

optimum

has

shown

result

diameter

that

overall

compressibility is also

levels

rotor-inlet

on the

the

10-9,

ratios

given

optiwhere

of stator-

specific

of rotor-exit

speed. tip

diam-

[]

qb_

(a)

Specific

speed,

0.23;

30

(ft

3/_)(lbm

flow (b)

Specific

speed,

0.54;

70

(ft

3/_)(lbm flow

(c)

Specific

speed,

1.16;

150

290

10-12.--Sections

of

_tZ)(lbf

3/_). Stator-exit

l!Z)(lbf

air).

Stator-exit

l/2)(lbf

3/4).

Stator-exit

81 °. 3/_)/(min)(sec

angle,

(ft 3/_)(lbm flow

FmURE

3z_)/(min)(sec

angle,

angle,

radial

turbines

(Data

from

75 ° . 3/4)/(min)(sec 60 ° . of ref.

7.)

maximum

static

efficiency.

RADIAL-INFLOW

eter

to

rotor-inlet

creases

rapidly

diameter with

of D,2/DI_-0.7 optimum

is

is reached.

blade-iet

to the

variation

The

optimum

values

shown

design

of radial-inflow

geometries These

are

shown

sections

that

varies

of static

10-12

figure with

10-8

and

imposed

10-11

inlimit

that

specific

to 10-11

Sections

specific

speed

the

the

speed

in a

efficiency.

in figures

in figure

show

specific until

from

UI/Vj

turbines.

low

speed

is seen

ratio

similar

at

specific It

speed

manner the

small

increasing

TURBINE_

for three speed

can

of turbines values

be used

with

of specific

is largely

an

for

optimum

index

speed. of

flow

capacity. The

design

different

study

losses

shown

in figure

values

of specific

.2

1.0

of reference

along

the

10-13

7 also

curve

for

the

speed,

the

indicated

of maximum range

stator

the static

efficiency.

of specific

speed

and

viscous

rotor

variation

in the This

covered.

For

losses

are

is low

very

--

.9-p

¢x3

to

¥ Loss Stator

c <19

Rotor Clearance

:>

Windage Exit velocity

c_

I

.4

,

.4 Specific

1

_ Specific

10-13.--Loss

.6

.8

I 1

2

speed, Ns, dimensionless

n

_0

FIGURE

1 , 1,1,1

.2

J 50

I

1

J

80

100

200

speed. Ns, l[t3_'4Jllhm3/d_/fi:lin_secll2_ltbf3'4_

distribution

along (Data

from

curve ref.

of

maximum

static

efficiency.

7.)

291

TURBINE,

large

DESIGN

because

of

clearance

loss

relatively

large

which large

the

APPI_CATION

high

is large on specific

speed increases, loss all decrease kinetic-energy

of

primarily speed

the stator because loss

ratio

of wall

because

fraction

depends at low

AND

the

passage

area.

the

diameter

and

of the

The

the

flow

is

windage

rotative

low

Also,

clearance

height.

and rotor losses, of the increased

becomes

to flow

blade-to-shroud

the because

area

speed, rate.

As

a

loss, is also specific

clearance loss, and windage flow and area. The exit

predominant

at

high

values

of

specific

speed.

C-/I-159

C-69q816 {a)

ib_

C-?0-3533 to)

(a)

Design

rotor.

(b) (c)

FIGURE

292

10-14.--Rotor

configurations

Cut-back used

Rotor

with

exducer

extension.

rotor. in

specific-speed

study

of reference

8.

RADIAL-INFLOW

Experimental

study.--In

effect

of specific

(ref.

8) to accept

of

blades

and

extension stator

the

turbine

rotor

back.

Details

calculations Performance

from a large with

the

was

area

area over

and

range the

rotor

rotor

was

was

also

with

to

results,

the

area

and

allowed

the

This

and

in-

vary

throat

speed.

an

for

Figure

extension,

10-14 and

internal

cut

velocity

8. experimentally

area.

numbers

back

used

design

of specific

modified

fitted cut

were of the

the

was

different

reduced-area

test

in reference determined

with

53 to 137 percent.

geometry,

given

The and

modifications

as designed, of

are

angles.

experimentally a turbine

rows

20 to 144 percent

throat

to be operated the

blade

operation

These

from

rotor

shows

of stator

blade

determine

efficiency,

of stator

area

operation. area

to

turbine

a series

reduced

area throat

to vary

on

different

for

creased

order

speed

TURBINE_

Figure

1 0-15

for shows

13

the

combinations

envelopes

of the

1.00--

• 9O

.8o .2

¢_

_

WithConfigu design ration rotor With rotor extension -With cutback rotor

/,_/'" . 7O

I-,-

[

.6O

1

[

[

L

L

• 9C

.8_ t'oJ o

• 7{3

1 .2

.3

l

l

.4

.5

1

1

.6

.7

I

I

.g

.9

Specific speed, Ns, dirnensi_nless

FIGURE

t

t

I

I

20

30

40

.50

l

I

I

1

I

60 70 80 90 lO0 t3/4 3/4 t/2 Specific speed, Ns, _f )_lbm )/trnin)[sec )(Ibf 3/4)

10-15.--Experimental

variation of efficiency from ref. 8.)

with

I 1lO

specific

speed.

(Data

293

TURBINE

DE,SIGN

AND

APPLICATION

design-speed efficiency as well as the overall and-rotor ratio

combination

at

design

measured

for

design

ratio.

The

speed

and,

used

even off

turbine

might

stators.

In

the

from

overall

efficiencies

0.37

to

efficiencies

of about

of

be

considerably

of nearly

total

varying

over

0.80

(48

of about

0.4

to 0.5

pressure 0.90

to

were

103

(ft 3/4)

were obtained area was near the

0.90

were

(51

to 65

that

a

measured

(ft 3/4) (lbm31_)/

(lbf3/4)).

could

speeds)

by

Maximum efficiencies area to rotor throat

static range

with each rotor configuration, Specific speed for each stator-

simply

that

speeds

investigation

design

varied

a)(lbf3/4)). of stator throat

Maximum

(seC/2)

obtained curve.

Note

specific

specific

(min)

was

speed.

(lbm314)/(min)(seO when the ratio in the

curves envelope

parallel

though

the still

distribution

endwalls

volume

efficiency

of the

internal

efficiency.

in applications

the

total

flow

rate

remaining

stator

blade

specific

velocities

is

Further,

a radial

requiring

variable

varied

over

row

basic

(different

of

high

to advantage

with

particular

of applications

yield

investigation,

three,

8 showed

a variety

design,

be used

this

reference

for

by

0.90.

minimize

a factor

In addition, the

potential

for leakage. Effect Clearance avoid

of Blade-to-Shroud

between

contact

minimized

the

during to

blade

speed

avoid

loss

and

and

previously

discussed

figure 10-13, as determined

the

The

effects

rotor

must

transients, to

flow

fraction

Since

it

design

For

clearance

radial-inflow study

in figure exit

of the is the

flow

turbine

the

With

of passage

height),

294

percent

be

it must the

blades,

efficiency loss included in the losses

10-16,

which

at

the

rotor

efficiency

of reference

clearance.

flow

that

turbine

turning

clearance.

each

but

The losses

9. The shows

Increasing

were

results the

shown

in

clearance rotor-exit

increase

stator

can

equal

is fully that

be achieved inlet

there

was

and

exit

about

in clearance.

turned produces even

with

clearances a 1-percent

to

and

at

determined studies

of both

clearance

inlet

causes

a

does a comparable that determines

the the

inlet

of these

effects

exit

significantly greater loss in turbine efficiency than increase in inlet clearance. It is the exit clearance the

to

bypassing

unloading. one of the analysis.

be adequate

ratios.

in the and

shroud

clearance loss was based on an average from constant values of rotor-inlet and

on

presented

clearance

blade was

of blade-to-shroud

exit

experimentally are

due

specific-speed

clearance-to-diameter

the

the

thermal

of work

generation of turbulence, and due to blade-to-shroud clearance

Clearance

exit

blade

angle.

rotor

inlet

whirl,

a relatively (in loss

terms

large

inlet

of percent

in efficiency

for

RADIAL-INFLOW Exit clearance, percent of passageheight v 0.25 .92 o 3

4

"_E

0

TURBINE6

\ __

D

A

7

\ .E "-4

_"'-b_-. \

/

"<

1- Equal percent, exit and inlet clearance

1'-

-8

FIGURE

An

axial-flow turbine

of inlet

turbine would

passage

height)

rotor-exit

diameter

absolute area).

and This

energy level radial-inflow

the

have

a larger

than

would

a smaller

]

the

on

flow

relative

] 28

total

clearance

(percent

would

clearance height

(in order

of the

reasons,

along

of rotorThe

result

is largely

passage

(Data

as a radial-

turbine.

turbine

discussed, over a small

efficiency.

conditions

radial-inflow

required

be one

previously turbine

same

axial-flow

(since

may

and exit clearances from ref. 9.)

with

of the

clearance

diameter)

_ I

L.

4 8 12 16 20 24 Inlet clearance, percent of passageheight

10-16.--Effects

inflow exit

i 0

a function

to have with

larger

in a larger

the

for the efficiency axial-flow turbine

of

same

annulus

lower

kinetic-

advantage for the

of a same

application. BLADE The

curves

velocity

ratios

for

a particular

bine

size

part

of any

and

(figs.

to

10-11)

to specific

speed

turbine

problem.

shape,

as well

design

problem

in order

to determine

methods

and

used

10-8

the

computer

are

relating useful

They

can

as the

design the

best

stator

programs

turbine

geometry

in preliminary

involves

be used velocity

and

rotor

studies

to determine diagram.

examination

discussed

and

design

tur-

The

of internal blade

in chapter

flow

profiles. 5 (vol.

next The 2) are

for this purpose. Internal

Stator.--Stator forward. chords, large

DESIGN

blade

Typically, and

blade

parallel profiles

the

Flow

Analysis

aerodynamic

design

blades

relatively

endwalls. are easier

have

A long

chord

to machine

and

is

is usually because

relatively little

straight-

camber,

long

specified

because

the

number

small

295

TURBINE,

I_E_IGN

of blades lower serve

(long

cost.

AND

chord

Also,

long

as structural

associated

with

APPLICATION

means

large

chords

supports the

added

spacing

for

are desirable

a given

because

for the

shroud.

The

endwall

area

(over

solidity) the

stator

aerodynamic that

of

penalty short-chord

Pressure

.9--

.... Free-stream velocity

Suction

Pressure su rface

I

t

.2 .4 .6 .8 1.0 Meridional distance, dimensionless {b_

FIGURE

296

l()-17.--Stator

(a)

Blades

(b)

Surface

blades

and

passage.

velocities. with

surface-velocity

means blades

distributions.

blades)

is small

because

boundary-layer

of the high

parallel

flow

in reference

solutions.

Input

rate,

properties

fluid

complete

surface and

solidity,

and

number,

shown

in

figure

surfaces.

used

to determine If at the

angle if the

trailing

Rotor.--The than

that the

encountered The

for

screening

suitable contour,

This

onal

plane.

approach, uses

which

the

is shown

of velocities calculated

equation

the

10-18. in the

irrotational blades.

various

accelerations

11 is particularly

avoiding

severe

hub

and

blade

for radial-

with

several

integration

straight lines in the meridiof

these

meridional

Blade-surface

flow

and

quasisolution

velocities

of reference

blade-surface primarily

contour,

specifically

program

absolute These

geometries and

three-dimensionality

A complete

approximately

difficult gradients

distribution,

with

flow

provide

more pressure

method,

section

exit will

fixed arbitrarily located intersect all streamlines

is obtained.

on

blades

the

developed

design remain

Conversely,

of shroud

velocity-gradient along that

in figure

between

to evaluate

was

be

angle.

adverse

thickness

the on

the

free-stream

the

of

no are

may

curves

provide.

of reference

streamlines

based

distribution

smooth

and

the

combinations blade

edge

is appreciably of

a

has at

accomplish

flow

because

A meridional-plane

orthogonals then

blading

various

of directional derivatives (called quasi-orthogonals)

input

for

stator

trailing

can

edge,

the

program

of blades,

turbines,

trailing

and

profiles

velocity

for

and

continuously

the

blades

until

pressure

deceleration

can

value

because

computer

number

curvature. inflow

blading

design.

row

varying

velocities

accelerates near

the

by

of rotor

stator

(decelerations) of

the

the

the

small

magnitude

with

passage

surface

input

is specified

design of

a

and blade-

curvature

for

and

flow

blade

than

blade

entering

for

pressure

the

before

than

of

angles,

and

are made

obtained

flow

calculated

pressure-surface

the

the

and

method transonic

includes

acceleration

flow

because

and flow

The

trials

velocities

turning

cross

turning

edge,

edge,

more

curves

more

the

calculated

suction

specifies

outlet

and

blade

Except

whether

the

inlet

suction-and

10-17(b).

The

program

are

stator

subsonic

geometry.

stator

the

stream-function

distribution

which

leading

both

open

in

for

computer

for smooth

the

calculated

The

Successive

shows

pressure-surface

turning.

blade

and

turbine The

the

distributions

10-17(a)

radial-inflow

of

are examined

blade

prewhirl.

favorable

the resultant

satisfactory

conditions,

decelerations.

satisfactory velocity suction surfaces. Figure

and

be used

endwalls. for

specification of local

may

10 provides

information

velocities

rate

analysis

or near-parallel

described

a

TURBINE6

conditions.

A two-dimensional of the

reaction

RADIAL-INFLOW

a

are

11 with

linear

an

velocity

velocities

are

basis

of obtaining

on the

used

decelerations. 297

TURBINE,

DE:SIGN

AND

APPLICATION

1.8_ 1.6 Shroud contour

Quasi orthogonal oa

cfi [:Z

,,,,_

ela CE

1 .2

FIGURE

"In figure

I

1 l.O

10-18.--Meridional

Blade-surface ional-plane sections of

I

.4 .6 .8 Axial distance, dimensionless

velocity

section

distributions,

are

shown

the

blading, but as calculated by 10. The surface velocities

between

linear

method

does

velocity not

these regions. surface velocities 298

solutions variation reflect

the

radial-inflow

as calculated

turbine.

the

merid-

the hub, mean, and are shown in figure

shroud 10-19.

velocity

from

distributions

the stream-function calculated from

solution agree fairly well with over most of the blade. It can The

through

solution of reference 11, at a radial-inflow turbine rotor 10-20

difference

1

the

for

used blade

The stream-function in a more rigorous

at in

the the

unloading

leading

and

solution appreciable

trailing

meridional-plane that

same

method of reference meridional-plane

those of the stream-function be seen, however, that an

occurs

the

actually

method determines manner. However, the

edges. solution occurs

in

the blade meridional-

RADIAIrINFLOW

TURBINES

1.0 .8 _

_

Suction surface Free-stream velocity

.6 .4 .2 O--

.... _

I

t

I

I

.... 1

t

1

I

I

(al ,... 1.0

w

.8 o"

.6 .4

'_

•"_--

h

.2

-.2

F

-.4

I

1

I

I

fb) ].0

.6

.2_

....

____

0 -.2

-.4 0

.2

.4 .6 .8 l.O Meridional distance, dimensionless

1.2

t 1.4

(c)

(b)

Mean

FIGURE 10-19.--Rotor-blade

(a) Shroud section. (50-percent streamline) (c) Hub section, surface-velocity solution.

section.

distributions.from

merEdional-plane

plane program (ref. 11) is easier and quicker to use than the streamfunction program (ref. 10) and, thus, provides a better means for rapid screening of the many design variables. A lesser difference occurs in the intermediate portion of the blade passage. In the meridional299

TURBINE

I_EISIGN

AND

APPD£CATION

1.0 I

Suction

surface

.8 _Mean

.6 /

4

_

t- tee-stream

_--

""_/

velocity

.2

.... 1_1

0

t

1

J

__I__

I

_3b

1.0 .8

d 3

.6 .4

1

.2

% _2

0 q)J

C)

1.0 .8 .6 .4 .2 0

__----

1

()

.2

.4

.6

Meridional

L

._q

distance,

I _1

].0

1.2

1,4

dif]Te!_%lc,rqle%

_c_

(a) (b)

Mean

Shroud

(c) FZGURE

section.

(50-percent

10-20.--Rotor-blade

Hub

streamline)

section.

section.

surface-velocity

distributions

from

stream-function

solution.

plane

analysis,

the

flow

and the mean stream mean blade surface. blade

variations

blades

that

of blade Figures along

the

than

generally 300

and

velocities

flow path.

elsewhere the

from

then,

10-19

surface

Also,

in the flow

loading,

considered

and

defines

the

mean somewhat

10-20

illustrate

and

the

blade of the

is a region the

most

the

circumferentially

a mean

the loading,

is more

heavily

solidity

of high critical

stream

surface.

surface The

between

distribution

different.

blade lower

uniform,

blades follows the prescribed solution considers blade-to-

blade

is also

The

because

shroud

to be

surface between The stream-function

deviates

in the

is assumed

flow. region

hub-to-shroud

variations

as well

as the

variations

loaded

along

and

the

Therefore, and

shorter the

the

shroud

flow path. shroud

is examined

most

is

RADIAL-INFLOW

carefully

for

loading

favorable

near

the

siderably

higher

is nearly

axial.

previously 50 next

the

The

section.

percent

high

loading.

This

decrease

in blade

particular

rear

loading use

part

results

at

change

the

The

principally

indicated

inlet,

where

the

and/or by

large

using

the

the

Such

reduced

loading

of surface the must on

be made,

The

are

the

therefore,

this

effect

in the was

blades

splitters

shown

were

hub

almost

large

increase

study

the

in

trailing-edge

the

and

to

the

rotor surface

be

reduced

radial

10-3

part

and

used,

the

per

unit

losses surface area.

of

are com-

are

unit

use

turbine

area

of

A judgment

of splitter

blades

performance

The

splitter

calculated

will,

examined

designed

blades

with

then

were

in the upstream for both cases.

negative

indicated

half When

velocities

on

a reverse-flow

eddy

extending

50-percent

streamline,

what

been

meridional upstream

was

12. A turbine

tested.

the

loading data

showed the

when

of

of blade-shroud flow

conditions

velocity

and

favorable

12)

little

The

were

area.

had

result

clearance rotor margin

the

and

the

and

a

splitter

in

and

between

the

due

was

apparently

offset

the

previously

discussed

leading

tolerance

of speed

increase

to the

an insensitivity

rotor

reaction

a range

in efficiency

loss

indicate the of

over

removed

This near

taken

difference

cases.

splitters

surface

appreciable

(ref.

very

no-splitter the

to poor an

this

is excessive,

additional

the

at

blades

loss per

of reference and

blade

performance ratio

l l_e reduced

vide

great In

location.

Turbine

effect

the

pressure

boundary-layer

is now

on

removed,

of

the

increase

blade

75 percent.

it can

in figure splitter

reduced

blades

built

side

from

splitter

about

in the

removed, thereby doubling the blade loading of the rotor. Channel velocities were calculated

pressure

low

and

loading

blades

as to whether

of splitter

splitter

pressure

by the

be beneficial.

experimentally

the

r _)

in

section.

for the surface,

full

When

there

to offset

path hub

is highest

If

decreased

However,

blades

balance,

in

loading

suction the

blades.

results

area.

splitter

the

blades

splitter

_

be reduced a very

by

calculated

between

partial

called

flow

at the

to the (rVu

discussed

shows

long

inward.

on

are

flow

Blades

velocities

blades

primarily

could

decreased

blade

is radially

decelerations

the

to exit

the

momentum

surface

is con-

where

is due

inlet

The

radial,

exit,

which

hub

spacing

the

negative

partial

rotor.

monly

flow

by

inlet

near

inlet

blade

previously,

as indicated

the

rotor

angular

Splitter As

the

distributions.

is nearly

in

from

from

the

flow

of splitters, of the

spacing

turbine,

the near

loading

rapid

This

velocity

where

loading

high

through

section.

inlet,

than

discussed

in that

blade-surface

rotor

TURBINE6

edge.

by

of efficiency The

a radial-inflow toward

loading

low

turbine such

inlet pro-

conditions. 301

TURBINE,

I_E_IGN

AND

APPLICATION

OFF-DESIGN

PERFORMANCE

The performance characteristics different from those of axial-flow all

rotor

speeds,

pressure

ratio

inflow in

figure

The for

is only

rotation,

must

be

with with

variation

the

no flow

(see fig.

for turbine.

with The

Prediction studies

any

design to

in

modifications

design

transients

where

the

use

of

calculation studies.

system

methods In

the

is somewhat off-design

used

The

be

from the

4

E Z

.2

/

FIGURE

302

[

I

1,2

1.4

Inlet-total-

to e×it_staticq)res,_Jre

10-21.--Radial-inflow

1 1,6

turbine

flow

[

I

1.8

2.0

ralio characteristics.

select

matched in

these

that

used

geometry

o

1,O

help

not

Zero speeO

0

in

conditions to

.6

N

in

in system

approach

different

speed rapid

valuable

e_

¥

to that

more

operating

calculations,

.8

illustrated

are

may

geometry.

force.

be useful

be

components

variable

pressure

as blade-jet

can

also

across

similar

is slightly

various

can

ratio

ratio,

performance and

They

fluid

directed

centrifugal

is very

data

the

zero-flow

decrease

off-design

is built.

This speed

point

a radial-

on

pressure

turbine

efficiency

In

gradient

increasing

performance

start

hardware

study

off-design

for

Estimated

to examine

before or

techniques

situations.

small

turbine

as illustrated

force

blade-jet

ratio varies from the peak-efficiency the case of the radial-inflow turbine. many

is one.

10-21).

a radial-inflow

the

speed,

a pressure

of the

are slightly turbine at

when

centrifugal

is some

because

of efficiency

axial-flow

only

at zero

by

there

speed

section,

true

balanced

Therefore,

even

in this an

this

With

inward.

zero

to exit-static-pressure)

however,

increases

later

becomes

rotor

turbine

ratio

rate

10-21. the

radially

flow

(inlet-total-

turbine,

within the

the

of radial-inflow turbines turbines. In an axial-flow

is

RADIAL-INFLOW

fixed,

the

are blade rotor

working-fluid speed

depend

inlet

and pressure

conditions

on loss

ratio.

are fixed,

Losses

coefficients

selected

Joss and

the

A radial-inflow developed 13,

reference from

turbine

at

erence

the

and 14.

NASA the

mation

this

static in figure

10--22

for the agreement

10-22

variation

efficiencies

of mass plotted

10-23.

program

10-23

The

an accurate

flow

rate

against

calculated

is

and

presented obtained presenting

pressure

ratio

mass

pressure speed

efficiencies

and

flow

representation

with

in

by

The

blade-jet

in ref-

results

program

of speed

method

is described

illustrate

computer performance.

shows

and

between

calculation

computer

a range

stator

loss. Center

experimental

variables

the design operating flows are the rotor

performance

this

the

to force

Research and

of over

with

in figure

perimental

off-design Lewis

version

performance

comparing

kinetic-energy

associated

Figures

a modified

calculated

exit

and

calculated

calculated and experimental or design values at point. Additional losses considered for subsonic incidence

TURBINE6

esti-

of the

ratio. ratio

are generally

Total are within

exand

shown 1 per-

Percentof designspeed

Ii"

Experimental

120

70 90 100 110

Calculated -."-

¢-

_"-110

30 50

B

100 "- 110 90 Design 8O 1.4

1.5

I

I

I

I

I

I

1,6

1.7

1,8

1.9

2,0

2.1

Ratio of inlet total pressure to exit static pressure FIGURE

10-22.--Comparison

of calculated and design operation.

experimental

flow

rates

for

off-

303

TURBINE

cent

and

lations

DE'SIGN

AND

at most

2 percent

are

sufficiently

examination testing

of

the

accurate

to

of overall

system

various

components.

of the

1.00

APPLICATION

experimental provide

values. a

performance

valuable

prior

to fabrication

.__I__

the and

110

•80 --

90 _ _-. vu,, Percent of design speed

.70 --

5O

¢1)

"5

calcuin

--

• 90 --

c o_

The tool

Experimental

•60 --

70 90 100 ,_

• 50 --

llO

Calculated

Design ! •30

I

1

I

I

1

JJ

I

•90 --

,- 110

• 80 --

100 _'

.70 -o_

,60 -

•50 --

,40--

• 30 ,, .2

_ 30

I

I

I

.3

.4

.5

Desiqn

] ..... I .6

.7

I .8

J .9

Blade-jet speed ratio, U]"V i FIGURE

304

10-23.--Comparison

of calculated and design operation.

experimental

efficiencies

for

off-

RAD_AL-I"NF_A)W

TURBI_S

REFERENCES 1.

SAWYER,

JOHN W., ed.: Gas Publications, Inc., 1966.

2. 3.

SHEPHERD, D. G.: Principles RODOERS, C. : Efficiency and Paper 660754, SAE, 1966.

4.

LAGNEAU,

5.

bines. Int. Note 38, yon WOOD, HOMER J.: Current

J. P.:

Turbine

Engineering

Handbook.

Gas

Turbine

of Turbomachinery. Macmillan Co., 1956. Performance Characteristics of Radial Turbines.

Contribution

to the

Study

of Advanced

Karman Institute of Fluid Technology of Radial-Inflow

Small

Radial

Tur-

Mar. for

1970. Com-

Dynamics, Turbines

7.

pressible Fluids. J. Eng. Power, vol. 85, no. 1, Jan. 1963, pp. 72-83. HIETT, G. F. ; AND JOHNSON, I. H.: Experiments Concerning the Aerodynamic Performance of Inward Flow Radial Turbines. Paper 13 presented at the Thermodynamics and Fluid Mechanics Convention, Inst. Mech. Eng., London, Apr. 1964. ROHLIK, HAROLD E.: Analytical Determination of Radial Inflow Turbine

8.

KOFSKEY,

6.

Design

Geometry

for

MILTON

on Experimental D-6605, 1972.

G.;

9.

10.

SAMUEL M., JR.; AND of Varying the Blade-Shroud Turbine. NASA TN D-5513, KATSANIS, THEODORE: Fortran

HOLESKI,

FUTRAL,

KATSANIS,

THEODORE:

13.

14.

Use

in

the

SAMUEL M., JR.; Performance Evaluation Without Splitter Blades.

FUTRAL,

a

Surface

DONALD

in

CARROLL

to Estimate NASA TN

E.: Experimental Results 6.02-Inch Radial-Inflow

Calculating

Arbitrary Meridional

Transonic

WASSERBAUER,

CHARLES

of a 4.59-Inch Radial-Inflow NASA TN D-7015, 1970.

A.; AND FUTRAL, the Off-Design D-5059, 1969.

NASA

Velocities TN

D-5427,

Quasi-Orthogonals for Calculating Plane of a Turbomachine. NASA

FUTRAL, SAMUEL M., JR.; AND WASSERBAUER, Performance Prediction with Experimental Inflow Turbine. NASA TN D-2621, 1965. TODD,

a

TN D-4384, 1968. Effects of Specific Speed Turbine. NASA TN

of a Turbomachine.

of

AND

J.:

Radial-Inflow

Clearance 1970. Program for

Stream

NASA

WILLIAM

of

Flow Distribution TN D-2546, 1964. 12.

Efficiency.

NUSBAUM,

Performance

on a Blade-to-Blade 1969. 1 1.

Maximum AND

SAMUEL

Performance

CHARLES

Verification M.,

JR.:

of

A.: Experimental Turbine With and

A.: for

A Fortran Radial-Inflow

Off-Design a RadialIV

Program Turbines.

305

TURBINE

DESIGN

AND

APPLICATION

SYMBOLS Cp

D g H

specific

heat

diameter, conversion ideal

at constant

m; ft constant,

work,

J/kg;

pressure, 1; 32.17

or head,

based

J/(kg)

(lbm)

on inlet

(K) ; Btu/(lb)

(ft)/(lbf)

(see _)

and

total

exit

(°R)

pressures,

(ft) (lbf)/lbm

passage height, m; ft ideal work based on

inlet-total

J/kg; Btu/lb ideal work based

inlet-total

on

J

J/kg; Btu/lb conversion constant,

1 ; 778

K

conversion

2r

N

rotative

speed,

N,

specific

speed,

constant,

and

pressures,

exit-total

pressures,

(ft) (lb)/Btu

rad/rev;

rad/sec;

and

exit-static

60 sec/min

rev/min

dimensionless;

(ft a/4) (lbma/4)/(min)

(sec 1/_)

(lbP '4) total

number

P

absolute

Q

volume

r

radius,

T

absolute

U

blade

V

absolute

Yj

ideal

flow m;

relative

Ot

fluid

0

(full

partial)

rate,

mS/see;

ftS/sec

ft

speed, jet

plus

N/m 2 ; lb/ft _

temperature,

ratio, W

of blades

pressure,

K;

m/see;

ft/sec

velocity, speed,

m/see; based

m/see;

deg fluid relative

ft/sec on inlet-total-

m/see; flow flow

ft/sec

angle angle

measured

from

meridional

plane,

measured

from

meridional

plane,

deg Subscripts: flow

b'r

critical

h

hub

t

tip

U

0

tangential at stator

component inlet

1

at stator

exit

2

at rotor

condition

or rotor

exit

Superscripts" '

absolute

"

relative

306

total total

to exit-static-pressure

ft/sec

velocity, absolute

°R

state state

(sonic

inlet

velocity)

CHAPTER 11

Turbine Cooling, ByRaymondS.Colladay The inlet

trend

towards

temperatures

necessity

of cooling

requirements. foil,

In

while

bustor

the

frequently

term

of the

bled

the

airfoils

for

discrete

of

K

and

then

the

turbine

and

very

of air are

effective

]n

any

turbine on

the

configuration analysis prediction the

an

airfoil,

flow, profile

be

of the

requires the

cooling blade

which can

location

potential-flow

(pattern

factor)

the

hot

com-

temperatures

to

preserve

the

environment,

air

passages

main

gas

cycle

of the

stream

at

results

in

inevitably

overall cooling

one

vane,

meets flux

or

must

end

a given

up

thermodynamic

schemes

make

wall)

which

utilize

to the of

of

the

velocity of the

blade

the

gas

a complete

to

metal

arrive

into from

the

three hot

boundary-layer

transition

from

distribution, leaving

at

temperature

conceptually

understanding the

order

This

life air-

DESCRIPTION

design,

(or

broken heat

to meet The

internal

into

to the

to a stator

at peak

hostile

the

turbineled

required.

GENERAL

balance

in this

in the

walls

airfoil.

In

or vane.

and has

refers

row

through

blade

end

rotor

vane

is dumped

the

and

(2500 ° F).

is routed

Consequently,

a minimum

first

ratios

efficiency

"vane"

to the

the

components

around

across

vanes, term

refers

1644

turbine

cooling

efficiency.

cycle

and

the

enter

compressor

locations both

pressure

thrust blades,

"blade"

excess

integrity

compressor

chapter,

gases

in

from

turbine

this

discharge

losses,

higher to increase

the

energy a cooling

limit. parts:

gas

stream.

development laminar and

combustor

the

to

The

(1)

The This over

turbulent

temperature (or other

heat 307

TURBINE,

source).

(2)

provide tions.

I_E,SIGN

AND

APPI._CATION

A steady-state

a detailed And,

(3)

map the

blade, to

the

transient

of metal

prediction

for convection-cooling balance,

or

temperatures

entire

heat

transfer

through

coolant--must

To

be

the

treated

(see fig. 11-1).

The

heat

pressed

gas

adiabatic

would

reach

let

the

or recovery if there

adiabatic

were wall

the

flux

heat-transfer gas and the

is the

wall,

predic-

coolant

maintain

closure

flow on the

from

and

to

hot

convection

from

paths energy gas

to

blade

simultaneously.

product of a hot-gas-side difference between the as an effective

internal

process--convection blade

Let us for a moment oversimplify dimensional model of a turbine-blade surface

analysis

for blade-stress

of complex

calculations.

conduction

heat-conduction

problem wall on

to the

by considering a onethe suction or pressure

blade

can be expressed

as a

coefficient and the temperature wall. The gas temperature is ex-

temperature,

which

temperature

(the

no cooling).

for convection temperature

For

purposes

be

the

temperature

of this

total

gas

cooling the

surface

illustration, temperature.

Therefore, q=he(Tg'--Tw,

o)

(11-1)

where q

heat

flux,

hz

heat-transfer

W/m_;

Btu/(hr)(ft

coefficient

_) of hot

gas,

W/(m

2) (K);

(°R) Tg!

total temperature

T_.o

temperature

of hot gas, K; °R

of wall outer surface,K; °R

TW,

0

Tw, i

FIOURE 11-1.--Simplified

308

one-dimensional

model.

Btu/(hr)(ft

_)

TURBINE

The

heat

removed

from

the

wall,

expressed

in the

same

CIOOLING

manner,

q=h_(T_._--TJ)

is

(11-2)

where heat-transfer

he

coefficient

of coolant,

W/(m

_) (K);

Btu/(hr)(ft

2)

(OR) w,

temperature

i

T' C

of wall inner surface,K; °R

total temperature The

temperature

drop

of coolant, K; °R through k

q=-

the

wall

is given

dT k_ -dy=-[ (T_. ,--T_.

by (11-3)

,)

where kw

thermal

y

coordinate

1

wall

The

holds

is frequently For

number

a

Nu

only

done

he be

of wall, to wall

W/(m)(K);

surface,

Btu/(hr)(ft)(°R)

m; ft

m; ft

equality

coefficient plate.

normal

thickness,

second

As

conductivity

for constant

in a first-order

approximated

turbulent is given

by

thermal design,

conductivity. let

a correlation

boundary

layer,

the

for

the

heat-transfer

flow

fiat-plate

over

local

a fiat Nusselt

by

Nu,=

__gx__ 0.0296Re_"

(11-4)

8Prl/3

where distance Re, Pr The

along

Reynolds Prandtl Reynolds

number number number

surface

from

based

leading

on distance

is defined

edge

of flat

plate,

m;

ft

z

as

Re_ = pugx

(11-5)

la

where P

density,

_tg

component

_t

viscosity,

and

the

Prandtl

kg/ma;

lb/ft 3

of hot-gas

velocity

(N) (sec)/mS;

lb/(ft)

number

is defined

in x direction,

m/sec;

ft/sec

(sec) as (11-6)

where 309

TURBINE,

])E_IGN

AND

K

dimensional

C_,

specific

For

an

constant, heat

ideal

(11-4)

APPLICATION

gas,

at

1; 3600

constant

equation

sec/hr

pressure, can

(11-5)

J/(kg)(K);

be

Btu/(lb)(°R)

substituted

into

equation

to yield h =_

(0.0296)Pr

'/3 FPz'

" x

-/

k

Mx

_'g,

yW ,

#

I

T s

T--1

j

2\(v+l)/2('_-Dm

(11-7) where

p',

total pressure of hot ratio of specific heat constant volume

T

conversion

g R

constant

gas constant, Mach number

M On the

gas, N/m2; at constant

coolant

1 ; 32.17

J/(kg)

side,

lb/ft 2 pressure

(lbm)

(ft)/(lbf)

(K) ; (ft) (lbf)/(lbm)

a number

to specific

of cooling

heat

at

(sec 2)

(°R)

schemes

can

be used,

but

in

general, h_=CRe/,"Pr"=C(

w'_"

"]"Pr"

(11-8)

where C

constant

Re f

Reynolds

We

coolant

J

characteristic

dependent mass

turbulent

cooling

flow

Now, depicted the

hot

the

m2;

cooling,

m-----0.8

laminar

flow

pressure

are increased

and

the

wall

from

(11-1),

0.8 power

pressure

ratio),

for

convection

so the

cooling

otherwise

the

outer

is sharply

the outer

wall

11-2). gas

temperature

From

The wall

(i.e.,

At

the

same

(Tw._-Tc') flux

will increase.

to

the

(higher

heat

of

equations pressure

temperature.

T,_.o).

The

as

Tw.o is kept

through

difference

wall, Tg'

with

increases

reduced.

blade

temperature

temperature

air temperature temperature

the

increases

drop

efficient

be avoided.

temperature

increasing

temperature

wall

bleed

outer

Since

should

and

blade

raises

compressor

p_'

2 in fig.

with

n-----l3.

through

1 to state

flux

a fixed

profile

flux to the

it increases

T_._ for

sor

heat

and

decreases the

state

the

and

(m=0.5)

the

(going

temperature

m; ft

ft 2

when

heat

310

passage,

11-2,

and

removed,

area,

length___

lb/sec

in figure

increased

time,

kg/sec;

the

gas

geometry

characteristic

for coolant

flow

internal

on

consider

constant (11-7)

rate,

length

convection

is desired,

coolant-passage

based

coolant-passage For

on

number

compresavailable q must Therefore,

be

WPURBINE

COOLING

i

Tg

hg(T_ - Tw,o)

_ kw / _- (Tw,o - Tw, i I

1 / C

FIGURE 11-2.--Gas

h, must

be increased

case,

as seen

ture

and

and

an

from

infinite

by

figure

coolant

impossible

11-2,

coolant

reference

1 shows

required

for convection limit K

taining

in the

the or

we. The

inside

course,

through

limiting

wall

temperabe infinite,

this

condition

exceed

is

savings

cooling

only.

components

stream

Figure

cooled

cooling and

conditions

11-4

by

one

is about

while

the

use

cooling

illustrates

the

l l-4(a)

to (e)),

or more

of these

20

maincooling

cooling.

air with

air

about

temperatures,

convection (figs.

cooling

Figure

11-3

of transpiration as

compared

basic

to

methods

and it also cooling

for shows

methods

to (i)).

cooling

a protective,

and

to

11-3

increase.

convection

transpiration

in cooling

film

turbine of blades

or

in

temperature gas

the

a limit Figure

increase

blade-metal

film

on

pressure,

and temperature

these

operating

combined

l l-4(f)

as pressure of advanced

air-cooling examples

highly

restriction

is apparent.

turbine-inlet

To

the

and on its supply nonlinear

convection

gas

drop

he must

and

the

incorporate

potential

size cooling

hot-spot

reasonable

Film

the

Of

convection

application

pressure.

must

cooling

(figs.

flow

therefore,

passage

cooling

(2500 ° F)

atmospheres

shows

3, where

is required.

air available

of plain

designs

on temperature

coolant

are equal;

internal

of cooling

The

effect

the

is state

flow

capabilities

1644

pressure the wall.

increasing

temperature

of limited

quantity from

and

to achieve.

Because the

temperature

is an effective

by directing cool

film

way

cooling along

the

to protect

air into

the

surface.

The

the

surface

boundary effective

from

layer gas

the

hot

to provide temperature 311

TURBINE,

I_E_IGN

AND

APPLICATION

Convection cooling Film and convection

pressure, Pg, in, atm Turbine-inlet 3

4O

1400

1600

I

--

--

cooling Transpiration

cooling

lO

20

I

[

I

1800

2O00

22O0

Turbine inlet temperature,

2OOO

--

T_, in, K

I

I

t

25OO

3O0O

35OO

I

Turbine inlet temperature,

FIou]_.

l l-3.--Effect

of

Tg, in, oF

turbine-inlet flow

in equation (11-1) becomes flux to the blade is then

pressure

the

local

film

q=hg(T's_.,-where

T_,z_

frequently is the same The

tion

objectives 312

losses

which

designs

which

temperature

of film

pressures

transfer

total

yet

tend

and must

ensures

air into

minimizes

the

to reduce

temperatures. be integrated

blade

metal the

temperature

on

coolant

temperature,

and

the

T,,,,.) of the

assumed that the heat-transfer as in the non-film-coo|ed case.

injection

dynamic higher

is the

and

requirements.

(11-9) gas

film,

coefficient

boundary

layer

some

of the

The

loss in turbine

in K or °R. in

causes

this

turbine

efficiency.

aeroof using

and

an optimum consistent

It

equation

advantages

aerodynamic

to achieve temperatures

heat

heat-

configurawith

long-life

is

q_UR,BINE

Transpiration cooling

cooling

scheme

available,

currently

limit

heat-flux

conditions.

should and

because To

its

be small,

foreign

use

which

piration

cooling

A typical coverage

leads

figure l l-4(h), transpiration

the

however, less

cooling

from

is an cooling

attempt without

wall

is the operating

of blockage air

air

due

losses

into

the

blade

is shown

an array

of discrete

extreme the

be severe

boundary

other

trans-

schemes.

11-4(i).

Full-

as illustrated

in

to draw on some of the advantages paying the penalties mentioned.

of

Cooling air

Zn

holes,

layer.

that

cooling

in figure

pores

to oxidation can

be recognized

than

airwhich

under

cooling,

aerodynamic it must

efficient

drawbacks

transpiration

cooling

most

significant

of cooling

point,

transpiration-cooled film

has

to problems Also,

requires

blade designs

efficient

injection

latter

it

to advanced

contaminates. this

but

For

of normal

offset

of a porous

COOLING

_

Xn

(aJ

__

(bt

o

c:_

c>xE//Ej

(d)

J

JJ (e)

(a) (c)

Convection cooling. (b) Impingement cooling. Film cooling. (d) Full-coverage film cooling. (e) Transpiration cooling. FmuaE 11-4.--Methods for turbine blade cooling.

313

TURBINE

I_E,SIGN

AND

APPLICATION

Radial outward airflow into chamber1, Film cooled--_ \ Convection \_

i_f// 1/// /.////

L Impingement cooled

. /

f-Convection

inlet airflow

(fl

/_\lmpingement

cooled

Convection cooled

(g)

(hl

_Transpiration i,

cooled

//

Wire-form porous sheet

(il

(f)

Convection-, (g)

impingement-,

Convection(h)

Full-coverage (i)

HEAT

and

and

TRANSFER

film-cooled

FROM

General

314

equations.--The

boundary-layer

region

transfer very

blade

HOT

configuration.

configuration.

configuration. configuration.

GAS

TO

BLADE

Equations of heat

near

blade blade

Transpiration-cooled blade FIGURE I I-4.--Concluded.

Boundary-Layer

the

film-cooled

impingement-cooled

the

to the surface,

blade where

is confined large

velocity

to

'tURBINE

and

temperature

gradients

heat-transfer

are

process,

introduced

in chapter

Conservation

present.

the

Consequently,

following

6 (vol.

2), must

C(K)LING

to describe

boundary-layer

the

equations,

be solved:

of mass

o 0%(pu)

O (pv+

(11-10)

where time-average m/sec;

()' ()

value

velocity

component

in

y

direction,

ft/sec

fluctuating

component

time-averaged

Conservation

of

quantity

of momentum Ou . pu 6-_+(pv+p

_-7=7..,_ Ou v ) -_=--g

shear

N/m2;

dp -_+g

0 -_

(11-11)

r-t-gpB_

where T

local

B.

component

Conservation

stress, of body

lb/ft 2

force

in the

x direction,

N/kg;

lbf/lbm

of energy OH

.

_

OH

0 / 1

ur\

(11-12)

where H

total

J

conversion

Q

heat-generation

The u,

enthalpy,

dependent v, and

The for

the

H,

as denoted

solution

of

of heat

understanding tions but

bears has

turbulent for

eddy

little

persisted transport diffusivity

heat flux are contributions"

(ft)(lb)/Btu

W/m3;

in ch. and

and

6),

heat

layers.

H are with

flux

The

the

flow

overbar the

requires

the

resemblance of its and

momentum. sum

of

the

and molecular

our

of various such

structure

and

is Prandtl's the

to but

One

_,

expressions

hydrodynamic

use

to the

simplicity

(i.e.,

understood.

contribution

counterpart.

processes,

being

appropriate

through

because

as

the

values

is straightforward,

turbulent

heat

time-average

laminar

physical

of

(ft z)

requires

momentum

expressed

Btu/(see)

equations

of turbulent

in describing

which

1; 778

p, u, v, and

these

stress

boundary

diffusivity

Btu/lb

term,

variables

shear

thermal

J/kg;

constant,

limited assump-

assumption, of turbulence

success

in predicting

mixing-length

hypothesis

The laminar

shear and

stress

and

turbulent

315

TURBINE

DESIGN

AND

APPLICATION

p/

Ou

_--_..,\

(11-13)

and { aL_jj0/_

q=Ko[

_)

(11-14)

where yr,

laminar

c()mponent

ity), aL

laminar

,4

static

The

m2/sec;

diffusivity

(kinematic

viscos-

ft2/sec

component enthalpy,

turbulent

tional

of momentum of heat J/kg;

shear

stress

to tile respective

diffusivity,

m2/see

ft2/sec

Btu/lb u'v---; and

gradients

heat

flux _

in tile

mean

are flow

assumed

propor-

variable;

that

'tt'v' = -- vr _-ff

is,

(11-15)

and O,4 v-'_g_ ' = -- O_r _-_ where and

the subscript heat

T denotes

the

(11-16)

turbulent

(:omponent

of momentum

diffusivity.

Equations

(11-13)

and

(11-14)

can

p

then

0u

be written

p

as

0u

(t1-17)

and 0_'. _-_-- Koa

q=-Ko(aL+ar) The

preceding

boundary-layer

properties

and

flow

where

both

heat

cp is neglected

equation

,r

reduces

ar

and

there

mental

The

variable section.

properties

results

from

it

are These

an

is

are

taken

equations.--As

equation,

equations 316

final

under

then

integral

of

variation

generation,

assumed

must

laminar

in specitic the energy

approach

to

be

and

isothermal

to account will

of the expericondi-

for temperature-

considered

6 (vol.

2) with

solve

the

in term_

onset,

simultaneously.

'tssumed,

approximately

in chapter

convenient

at the

be solved

freq_)ently

corrected

corrections we saw

often

If the heat

arc

equations

usually

properties.

Integral tum

are

zero).

(inclusive

(6-42). properties

constant data

flows

is no intern,d

to equation

all boundary-layer

However, tions.

approach

(ll-lS)

as_u me t e lnperature-variable

turbulent

and

If temperature-variable analysis,

equations

compressible,

O/

in the

a later momen-

boundary-layer

of integral

parameters

TURBINE

such

as momentum

and

of discrete

velocity

thicknesses

derive

tion,

enthalpy

so the

displacement

profiles. their

Just

as the

meaning

from

thickness

eter for the integral defined as follows:

thicknesses

rather

displacement the

is a significant

energy

equation.

than

and

integral

O0_LING

in terms

momentum

momentum

equa-

boundary-layer

The

enthalpy

param-

thickness

/_= fo¢* pu( H-- H,)dY p,u,(H,,,.o--H,)

z_ is

Note the

that

the

subscript

subscript

g refers

e in chapter

to the

6. For

(11-19)

free-stream

low-velocity,

value

denoted

constant-property

A=fo_*u(T'--T',)dy u,(T_,.o--T,') The

enthalpy

ment

thickness

caused

The

by

integral

equation across

the

is a measure

boundary

energy

(11-3)

or

the boundary

equation

boundary

layers

the

resulting

integral

energy

with

that

if we make

pressure at

the

gradient, wall,

_, then equation

derived

either

the

equation

see for

and mass

by

transport

containing

details

decre-

integrating of

energy

the hydrodynamic

ref.

2).

In either

compressible

transfer

and

_ __d__ 1 du, q_ (H,_.°--Ht)dx 1 restrictive

at the

case,

flow

with

wall is

low-speed (11-21)

heat-transfer

flow

of constant (incompressible),

temperature reduces

coefficient

h,,x

difference

to its simplest

q c_,(T_,, o--T,')=

(11-21)

d (H'*' o--11,)]

assumptions

constant

Kpu, If a local

energy

-F PgU,c

= dh ____t_A [ (l_Mg)

flux

be

volume (for

properties

Kp,u,(H,.

zero

eonveeted

balancing

of a control

thermal

Note

of the

can by

and

q

(11-20)

layer.

directly

temperature-variable

by flow,

dA d--x

is defined

properties, no

mass

(T,,.o--Tz')

form, (11-22)

as

q h'.'=(T..,--T,')

(11-23)

h,., da Stz Kpu, c_,=d-x=

(11-24)

then,

317

TURBII_E

I_E@IGN

AND

APPLICATION

The group of variables on the left side is dimensionless the local Stanton number Sty, which is also equal to S

Notice

from

tions,

the

equations

integral

is called

Nu, t_--R_p r

(6-72)

and

momentum

and

(11-25)

(6-75)

equation

that

with

resulted

similar

assump-

in

CI'_--dO 2 dx For

compressible

thermal

flow,

energy

by

characterized as shown

by

wall

temperature uncooled

the

boundary

the

recovery

viscous 11-5.

within

in the

The

T,_,o, and

is dissipation

shear

an increase

in figure

were

there

is the

layer.

tz is the

flow,

the

dissipation by

turbulent

boundary

ing

the

that

temperature. sponsible allowing that

for

should

Prandtl

the

for energy

number

following

to

free-stream a high

kinetic

FZOUR_

318

l l-5.--Temperature

distribution

on

in

is related

to

°R.

For

laminar

Pr 11_, while the

adiabatic

of the

for

(re-

(mechanism

This

a high

temperature,

would

Prandtl and

suggest number

vice

versa.

Ii Tg,e

//

in

_

Iq " 0

a

a

wall

viscosity

diffusivity

layer).

energy,

r 2g-_j Cp---I

_

if it

Pr z_. It is not surpris-

ratio

thermal

u2

_'_

by

effect

wall

tg /

Thermal

an

boundary

adiabatic

reach heating

(11-27)

to equal

to the the

would

in K or

is the

is wall

or adiabatic

viscous energy

This

equation:

approximated

has

from

wall

into

the

ug2 2gJcp

number

dissipation)

to escape

a given lead

be

Tt.,,

of the

of kinetic

the

r is assumed

Prandtl

near

temperature,

can

layer,

The heat

static

factor

layer.

a measure

This

hot-gas

recovery

boundary

temperature

r defined

energy

temperature

gas temperature

Tg,=T,o.=t,+r ' ' where

of kinetic

the

static

effective

is, therefore,

factor

(11-26)

high-velocity

boundary

layer.

The

heat

(either

flux to the

effective

As we

have

terms ture

or static

seen,

The

gas problem

is to design wall

a cooling

temperature.

never

the

refined

The

effect

in

a

side

of the for

expression

gradient

wall

heat

flux

which

that

the

must

in

tempera-

blade

be

the

is to find

a

objective

a constant

surface

be

flux

always

hg.:`. The

varying

can

heat

temperature.

will yield

actual

layer

the

to the

the

(11-28)

gas-to-wall

coefficient

however,

boundary

to

outer-

temperature

surface

is

temperature

accounted

Boundary-Layer

blade

for

in

more

the In

surprisingly because the

blade

pertains,

flow,

proximation.

by

a

are

often

sense,

accurate the

to

for

the

fiat-

zero-pressure-

a first-order

correlation

close to those of more sophisticated Stanton number St is relatively

a cylinder

Though

only

enough

fiat-plate

correlation

around

region.

strict

to the solution the suction or

a fiat-plate

distribution

leading-edge in

results fact,

simplest approach coefficient on

coefficient

the

Equations

is approximated

a heat-transfer

crossflow

plate

the

approximation.--The that the heat-transfer

into

wall:

stages.

First-order is to assume faired

the case

heat-transfer

reality,

Solutions

pressure

and

in this adiabatic

configuration

thermal

design

the

In

constant. on

or the

for

gradient

to express

h,.,

temperature

in determining

expression

temperature

at the

it is convenient

temperature,

O00LING

OT, _ h -_ v=0---- g.:`(T,.e--T,_.o)

coefficient

gas

to the

temperature)

already

heat-transfer

The

gas

Ot Nlv=0 =-k_

of the

suitable

is proportional

q=--]¢'

difference.

effective

has

blade

TURBINE

yields

analyses, insensitive

apresults

primarily to pressure

gradient. For

laminar

dynamic energy

equation

similarity With

flow

over

boundary

the

wall

plate

both

be

solved

can

solution

a fiat

layers discussed

temperature

with

the

beginning

in

directly chapter

assumed h_.,=0.332

thermal at

by

the

and

means

6 for

the

leading

the

to be constant,

of

the

velocity the

hydro-

edge,

result

_ Re:,'12pr'z3

the

Blasius profile. is (11-29)

X

The

turbulent

counterpart

is given hz.,=0.0296_Re

The

local

For the

the

following

velocity

u_. x is used

heat-transfer correlation

in the

coefficient is frequently

by °" Spr'/a Reynolds he. ,, in

(11-30) number.

the

leading-edge

region,

assumed: 319

TURBINE

DE,SIGN

h,,'_=a

AND

APPLI_CATION

--80°_¢_80

E1" ]14k" _\ (p'u' _ _D'_'/_pro.,(l__)]

° (11-31)

where a

augmentation

D

diameter

The

factor of leading-edge

velocity

of gas

approaching

angular

distance

from

bracketed

diameter

term

D

(see

is the

fig.

11-6)

FIGURE

The

term

a vane from

highly

blade to

uniquely

leading

1.8,

have

leading-edge

factor

edge.

layer.

the

Kestin

with

yet,

turbulent (ref.

no general

Reynolds

number

the

boundary

ent,

it can

Reynolds stream 320

layer generally number

turbulence

studied

laminar

free

becomes

be assumed of 200 surface

of

amplification

velocities this

of

stream.

coefficient

approaching the

factor

of heat

heat

within

flow high

a fiat

plate

that

transition

000

to 500

roughness.

the

vortex thereby boundary

in detail,

transfer

with

will

to

occur

allow

with

zero will

000,

However,

a,

flux 'is flows.

stretches of flow,

phenomenon

turbulent

For

the

flow

stagnation region in the direction

sufficiently

to grow.

to adjust

magnitudes

of stagnation

is available. laminar to

and

a cylinder

favorable-pressure-gradient

fluctuating

3) has

range

deg

geometry.

used

This

large,

correlation

scale and intensity Transition from

ft/sec point,

for

mainstream

Various

used.

The highly accelerated flow at the filaments oriented with their axes increasing

m/sec;

stagnation coefficient

turbulent

been

associated

edge,

a cross-flowing,

11-6.--Blade

augmentation

or 1.2

leading

x/

for

the

ft

leading-edge

in

a is an

account

m;

heat-transfer

Ug,_ ==

to

circle,

but

turbulence when

instabilities pressure take

as

a Reynolds

in gradi-

place

depending

the

in the on

freehum-

TURBINE

ber

based

origin)

on

parameter. as the then

distance,

the

independent

state. comes

the

of x. A value

number

corresponding

for pipe

flow.

turbulence, An

For

from

the

blade

functional

layer

(see

momentum is given by

be derived (6-76))

2).

local

integrating

velocity

x is the

surface

meters

or feet.

The

tJo

distance

U,Sdx)

measured

momentum

of diameter

Fx

momentum assumptions

through

the

(6-76),

as to laminar velocity

+0,,_.

(11-32)

the

with

equaboundary

the

free-stream

stagnation

8,t_s at

D in a crossflow

variation

integral

\o. 5

2000

be assumed.

suitable

from

thickness

ReD=

free-stream

thickness

profile

given

Reynolds

high

can

of a variable

5 /

is, it is

to the

and

very

equation

as a function

where

the

making

that

is

layer never beof 0 but not with

plate

with

point

critical

momentum

from by

Upon

OL__O.67vO. Ug 3

of a cylinder

the

of the

thickness

layer;

of Ree.cr_t=200

for

can

form ref.

blade

0,

determining

in getting

000 for a flat

value

6 (eq.

for

a given

the boundary with the use

a local

thickness,

at

is a "universal"

a turbine

expression

chapter

boundary

developed

to Rex=300

a conservative

it is not

number

of the layer

(boundary-layer

momentum

number

history

over

the

Reynolds

accelerated flow, fact is consistent

flow

edge because

Reynolds

of Reoccur=360

approximate

on a turbine tion

in the

boundary

For a strongly turbulent. This

use

to use

critical

of the

how

leading

criterion,

convenient length

because

the

transition

is more

characteristic

immaterial

x, from

a practical

It

transition,

the

the

is not

O00LIN_

the an

point,

stagnation approach

in

point velocity

ug_ is 0.1D

(11-33)

°.'..- ?..o,5 ¥ Turbulent such as

or

that

transitional

flow,

(ouz0/u)_200.

The

2_

then,

value

exists

when

of x where

the

this

value

occurs

of 0 is

is denoted

T,cr_ t.

The

turbulent

the

equation

by

momentum

0T--

4.

L

This

assumes

Integral tum

and

accurate discussed. complexity

I1

method.--

The

solution the

than

the

penalty

of the

_g

transition

to obtain

approach The

,,,..

_/,g

an abrupt

energy

thickness

heat

_x-r-t,L._ti,----_e

in a similar

manner

]

(11-34)

from

laminar

to turbulent

of the

integral

equations

flux

to a blade

"fiat-plate

for more

computation.

is obtained

accuracy In many

of momen-

is a more

approximation" is, of course, cases,

the

more

flow.

refined

and

previously the

increased

sophisticated 321

TURBINE

DE,SIGN

methods

are not

method

accounts

and

effect

the

warranted

integral

Consider

in the

early

for free-stream

However,

the

specific

APPLLCATION

the

heat

more

temperature

assumptions

must

The

realistically,

on

still

integral

h=._ can

be made

also

in order

equations.

no

_p,u,cp

of design.

variation

surface

some

integral

and

stages

velocity

of a nonconstant

be included. to solve

AND

energy

mass

ax

equation

transfer

_ _

(eq.

across

u,

dx

(11-21))

the

wall

with

constant

boundary.

_- (T_.o--T,')

dx (T=.o--T,') (11-35)

Ordinarily,

the

solved

in

(1i-20).

to

Ambrok

equation

by

Stanton

making

that

the

length,

proposed independently

Stanton

and

that

fact

that

weak

this

can

approach of

show

If ] is independent give

For and

of pressure

us the

as a function

turbulent

(11-25)

flow

He of a

as the

character-

of pressure

gradient. ( 11-36

gradient,

functional

the

gradient.

Stx----f(Re_)

should

be

momentum

data

thickness

is independent

to

equation whereby

the

of pressure

be written

on enthalpy

function

an

have

5 in

experimental

function

number

based

first

thickness

solved

of the

number

would

enthalpy

however,

be

use

equation

the

to be a very

Reynolds

istic

4),

could

number

proposed

momentum

evaluate (ref.

(11-35)

equation

local

integral

order

then

the

flat-plate

solution

form.

over

a flat

plate,

combining

equations

(11-4)

yields St_=0.0296

Recalling

from

equation

Re-_ °2 Pr - 2/3

(11-24)

that

(11-37)

for a flat, plate dA

Stx=-_ the

local

thickness

Stanton by

number

combining

can

be

equations

St_ = (0.0296 Hence, (11-39) free-steam flows.) integrating

the for

function

f

from

turbulent velocity

Substituting

flow

expressed (11-37)

equation

enthalpy

so as to obtain

Pr-2/3) 1"25(0.8 Rea) - 0.2_

(11-39)

equation

(11-36)

by

(The

(11

of the

38)

and_

variation.

in terms and

is given

assumption, same

(11-39)

for

argument into

by any

holds

equation

equation arbitrary

for laminar (11-35)

and

yields Sty-

rT hg'_ l"kpe'_=C p

322

(11-38)

--0.0296

Pr-2/3(

T '

T

_o.z_/-o._

(11-40)

)

TURBINE

COOLING

where I =-: .f]_*.... p?I_(T_'--T.,o)' u

25 •

J,=,....

. ['0.SRe_(T/--

T_ o)71"25

(11-41) The

integration

is performed

enthalpy-thickness

Reynolds

laminar-boundary-layer

heat

to

a

equations

simultaneously

are

good

several

was

by

and

of

(eq.

kinetic

W.

In

addition

0z

to do

(o,y+ o

-_.v=p_

\0v/

Ov

of these

procedure

equations

of

for

(11-11)),

that

the and

of turbulent

with

ix given

There

One

(eq.

equation,

energy

of calculating

this.

to the

2.

boundary-layer

numerical

simultaneously

kinetic

°z/+lo,.,+¢,.'t

the

momentum

solved

the

approach.

the

conservation

also

of turbulent

available

(11-10)),

a fourth

all

critical

see reference

finite-(lifference uses

the

from

metho(1

solve

an
5).

(eq.

.._is

Conservation

l(ays (ref.

mass

(11-12)),

energy

a

details,

accurate

is to

programs

M.

P'ttankar

censervation energy

by

numerical

developed

Spalding

blade

with

ewduated

further

most

turbine

hg._,

being

For

solution.--The

flux

for

mnnber

equation.

IGnite-difference the

numerically

the

others.

by

az" -9

P(_+_)

0.v

(11-42) wheref/r The

is a turbulent

turbulent

kinetic

dissipation energy

is defined

.j//_

in

w'

the

is the

fluctuating

direction

All with lion wall

properties

are

no restrictive of surface

11

bound'n'v integration

Also,

in m/sec

plane.

mixing

the

locally

an(I

x-y

By length

effects

ix

the

l)rotile..Mass film

turbu-

boundary

made

assumptions

on

layer

the

varia-

transfer

cooling

the

calculated

of fi'ee-stream

through

local

or ft/sec,

including

are

at

the

also

handled

plots

showing

manner. example

the in

must

started

results

mmwrical

high-pressure

is given l,Jyer

the

m' velocity

cooling)

of

temperature,

a) (sec).

(1 1-43)

of velocity, the

or approximating

7 presents

tlcxihilitv

protile

layer. for.

temperature

(transpir'ttion

I;igtll'e the

to

evaluated

in a slraightforwar(I

Btu/(ft

as

component equation,

locally in the boundqry lence can be accounted

W/m s or

(u'2+v'2+w'2)

perpendicular

turbulent-kinetic-energy

in

1

2gd where

term,

figure

turbine l l-7(a).

be supplied (fig.

ll-7(b)),

from

apI)roa<:h wine. The

computer for The

initial

as a boundary but,

from

then

the

case

of

free-stream profiles

high-

velocity through

condition on,

a

the

to get, the

profiles

can

be 323

TURBINE

IYE,SIGN

AND

APPLICATION

2500-700 6O0

150(]

m

E

.

200 500 10C_ 0

_

(a)

I 0

I

I

.Ol

.02

I

I

O

.04

I

I

I

.03 .04 .05 Surfacedistance, x, m

I

I

[

I

.06

.07

[

.08 .12 .16 Surfacedistance, x.It

I

I

.20

.24

Total enthalpy (b)

I

(a) (b)

Initial

profiles

Free-stream reference FIGURE

for

reference enthalpy,

Surface

1.0447)<

I

106

J/(kg)(K)

turbine

} 1.O

boundary-layer

m/see

development

I .9

profile.

numerical 30.87

pressure

324

I

velocity

finite-difference velocity,

l l-7.--Boundary-layer

I

.2 .3 .4 .5 .6 .l .8 Dimensionless boundary-layer variable

.1

or

or

101.28

249.7 over

vane.

ft/sec;

program. free-stream

Btu/(lb)(°R). a

high-t_mperature,

high-

TURBINE

O00LIN(_

calculated through the boundary layer at discrete x locations. The boundary-layer thickness, momentum thickness, momentum-thickness Reynolds number, and heat-transfer coefficient are shown in figures 11-7(c) to 11-7(f), respectively. Notice that just upstream of the

• 16 --

.OO4

(c) ]

I

0

I

1

3.2_E

_, 2.4--

o8° I 60

E

(.3

¢-

E =

1.6

E

i°

*

>"

.8

b20

r-

I

[)--

20

(c) (d)

Pressure-side

Pressure-side

4O 60 Percentsurfacedistance boundary-layer

boundary-bayer FIGUaE

momentum

80

100

thickness. thickness.

11-7.--Continued.

325

TURBINE

DESIGN

20-percent

surface

thickness

(fig.

increase

rapidly

before

AND

APPLI.CATION

distance

11-7(c))

on

the

momentum

and

and

continuing

location

then

to

decrease

increase.

the

slightly

This

"blip"

vane,

the

boundary-layer

thickness over

(fig. a short

is caused

by

ll-7(d)) distance the

rapid

2400--

le)

I

1

I

I

I

80

1_

1600--

% m

t-

v

8.6_

_

_

7.0

ii '°

._

5.4

1000

8OO

"T"

3.8

-7-

600

2.2

4OO

I

0

(e)

Pressure-side (f)

20

momentum-thickness Pressure-side FIGURE

326

40 60 Perceni surface distance

heat-transfer 11-7.--Concluded.

Reynolds coefficient.

number.

_URBINE

O0_LIN_

48--

4[--

•8I

24--

.6

16-E E

E

>:.

>_

I .2bg

fl

(al

c

0l

(l--x=

g, o

48 -- E 8 r_

m

E o

E

e_ c21

32--

.8i--

24--

.6i--

16--

.2i--

8--

(b)

"--Total enthalpy

I

0--

_

I

I

I

.2 .4 .6 .8 Dimensionlessboundary-layer variable

(a)

Initial

profiles

ft/sec; Btu/(lb) (b)

1151.75 FIGURE

slot.

Free-stream

reference

reference

enthalpy,

velocity,

4.8189X108

609.6

m/sec

J/(kg)(K)

or

or

2000

1151.75

(°R).

Three m/sec

at

free-stream

1.0

slot-widths or

2004 Btu/(lb)

downstream

ft/sec;

free-stream

of

slot.

Free-stream

reference

enthalpy,

reference 4.8189X

velocity,

610.8

106 J/(kg)(K)

or

(°R).

11-8.--Boundary-layer

profiles

along

adiabatic

wall

with

film

cooling.

327

TURBINE

DE,SIGN

deceleration the

AND

and

APPL]:CATION

acceleration

of the

adverse-pressure-gradient

pressure

side.

about

the

Initial film

Transition

from

velocity the

have

to the

and

relations

involving

St, discussed

c_, which these

in earlier

all vary

transport

perature

with (and,

to results variations

temperature

are

Properties

or the are

analytical

corrected

common the ture

data

to account

use

for

the

In

the

latter

for

method,

(for

all transport

temperature-ratio

subscript

free-stream m=0.12. influence

CP static

For than

method

refers

to

coolant

the

local

side

are

heat-conduction up 328

into

a number

large

at

what

layer,

constantmethod)

Two

differences schemes

are

results;

the

in

namely,

reference-tempera-

are evaluated

at

the

flow, flow.

tg+0.22

(11-44)

(T_e'_"{ t_ "_" _] \T-_.J

constant

(11-45)

properties

For n=0.4

WITHIN

heat-transfer

Tg,_

assumes

temperature.

turbulent in laminar

CONDUCTION Once

and

tem-

Since

temperature

properties

Tw.o+0.28

Nu St N---_ce--Z-_c_=\ The

and

Usually,

property

gases)

of

coefficient)

constant.

variation.

Nu,

p, k, u, and

finite-difference

small

of constant

Pr,

dependence

boundary

in the

11-8 (b).

Re,

velocity

evaluated?

with

property

the

of

down-

: Tre.,':O.5

The

(except

method

temperature

the be

obtained

correction

temperature-ratio method.

reference

to

is

example

heat-transfer were

at

layer

gas properties in

occurs

in figure

temperature

the

across

properties solutions

experimental

in

the

widths

parameters

if properties

occur

the

an

3 slot

Fluid

a change

on

11-7(f).

shown

The

causes

layer

shapes

contain

therefore,

obtained

temperature property

sections,

edge

illustrating

dimensionless

temperature.

properties

profiles

compared

the

from

boundary

figure

About

Temperature-Dependent The

leading

of the

from

11-8(a).

changed

resulting

boundary

profiles

in figure

flow

of the

little

be seen

enthalpy

given

profiles

Very

as can

and

are

aft

a laminar

location.

state,

cooling

stream,

region

10-percent

in a transitional

mainstream

evaluated

laminar and

THE

coefficients

flow,

n=0.08

m:=0.6,

a

BLADE

WALL

on

the

at

much

hot-gas

the and

greater

side

and

known,

the

heat-flux

boundary

conditions

for

problem

are

available.

The

or vane

is broken

of finite

elements,

as shown,

blade for

example,

the

in figure

TURBINE

FIGURE

1 l-9.--Typical

node

breakdown

for

a turbine-blade

COOLING

conduction

analysis.

oj+4 _

I \ I \

_

f

-

iI

_

I

_

I

\

I

ii

J+

_

II

i

\\

I

II

il il ......

_11

oj+$ FIGURE

11-9,

and

a system equations then

be

computer. temperature thermal-stress Consider

l l-10.--Typical

an energy

boundary

balance

of algebraic equal solved Once

element

is written

finite-difference

to the

total

number

simultaneously such

a conduction

distribution calculations. a typical

by

throughout

boundary

element

for heat-conduction

for each

element.

equations,

with

of elements. means

of

analysis the

The the

equations

is available

figure

result

11-10.

is

number

high-speed,

is completed,

blade from

All a

analysis.

of must

digital a detailed for

use

in

Accounting 329

TURBINE

DESIGN

AND

for

all the

energy

and

those

transfer

adjacent

boundary,

APPLICATION

leads

between

to it

to the

+

(elements

following

)m+ k_Aj

dc Ac 1 (Tj--Tc'

the

T

(Tj--

• " • +_s

given

element

(the

jth element)

j-l-1

to j+5),

including

algebraic

equation

(see fig. 11-10).

_

j+l)

k_42 +--_--

5 (Tj--T_+5)"--

the

fluid

(Tj--Tj+2)"

pc_,Vj A(time)

(11-46)

(T']+I--TT')

where surface

A_

area

noted by

between

an explicit

the

may

equations

infinitesimal

transient

and

step

element

for every

the

n or n+

scheme

de-

or boundary

denoted

element

must

or steady-state,

If the

energy

l, depending

on whether

is used.

volume

transient

structured.

size,

heat-conduction

time

be either

are

or boundary

m 3 ; ft s

ra denotes

equation

calculation

element

i, m; ft

or implicit

A similar

and

i, m s ; ft _

ofj th element,

superscript

element

jth element

subscript

volume The

jth

by subscript

distance

44

between

element

balance

at

be written. depending

is allowed a point

The on how

to reduce

yields

the

to an familiar

equation 0T

(11-47)

Pcp0(time) where

x, y, and

z are

the coordinate

direction_.

COOLANT-SIDE There transfer

can by

impossible the local

problem

be many

CONVECTION

internal

flow

convection

to the

coolant,

to

each

convection-cooling

discuss

is to determine

coolant

the

temperature,

is not,

however,

can be very complex, must be known before An internal equations 330

flow network that

describe

as simple and the

is established, internal

used

for that

to promote

reason,

shown

he, and

the

equation

T'c)

(11-2) The

flow and coefficient and

be

Essentially,

coefficient,

previously

heat

it would

scheme•

as it sounds.

the internal heat-transfer

the

and

heat-transfer

T_, in the q----hc(T,_._--

This

geometries

coolant

path

pressure distribution can be determined.

conservation

pressure

flow

distribution

of momentum are

solved

_URBINE

to determine there

the

discussed the

flow

is interaction (convection

surface,

between

and

the

for

Fins

lators"

to keep

They

also help

most

effective

11-4(b)),

be

the by

highly

as seen

from

blade

and

the

convection

distance

Zn

D

is

boundary

11-4(b)

and

The

hole power

both

diameter,

m

on

functions

Reynolds

number.

thin.

One

of the

cooling

(fig.

cooling

the

inside

representative gives

\0.091

(11-48)

number wall,

based

on hole

diameter

m; ft

m; ft

the

of

and

as "turbu-

toward

(g). One

heat

(k_)

Nusselt dimension hole

deterscheme

layers

area.

impingement

irap-_-_lq_2ReDraprl/S

between

been

to act

surface

6 for impingement

impingement-cooling as characteristic

has

of

convection

air are directed

in figures

percentage

convection

passages

/,._ NUD,

the

coolant-side

mixed

of cooling

reference

After

cycle

he.

methods

jets

through

an iterative

particular

cooling

the

Since steps

conduction

of the

the

to the

increasing

small

of the blade,

correlation

flow

surface,

region

blade.

heat-transfer

to coolant),

to enhance

added

of the

three

be made.

for

used

parts

the

to

must

convection

where

gas

to determine

are

can

of

surface

a given

be used

methods

transfer.

hot from

correlations

must

Various

from

cooling

various

each

calculations

empirical

considered

between

convection

three

air available mined,

wall

split between

COOLIN_

Reynolds

the

number

and

impingement-hole

the

array

A least-squares-curve

coefficient geometry

fit of the

data

_1 and

are the

in reference

6

gives Xn

2

m----a_(-_)

Xn

+b,(,)+c,

(11-49)

and =expEa2 where holes

x,

in the

in table factor from

is the

11-I to

center-to-center

direction

for

rows

2

and

the

of ReD.

crossflow

in meters

coefficients

The

coefficient

caused

of impingement

(11-50)

x.

distance,

of flow,

as functions

account

multiple

xn

or feet,

a, b, and _o2 is an

by

the

accumulation

It

can

be expressed

jets.

between

c are given attenuation of fluid as

1 _o2 where

aa and

ba are

given

in

1 +a3_b_3

table

11-I,

(11-51) and

_b for

the

i th row

of 331

TURBINE

I_E,SIGN

TABLE

AND

APPLICATION

ll--I.--IMPINGEMENT-CooLING

Coefficient

CORRELATION

COEFFICIENTS

Reynolds number range, 3 000 to 30 000

Reynolds number range, 300 to 3 000

--O. 0025 • 0685 .5070

-- 0.0015 .0428 •5165

al

bl ¢1

a2

b2 c2 a3

b3

impingement

holes

is defined

0. 0126 --.5106 --.2057

0.0260 --. 8259 • 3985

0.4215 .580

0.4696 •965

as Go!

Zn

(11-52)

where Gcr

crossflow

Gh

impingement-hole

mass

FILM As

that

reduce

blade

in fig.

11-3).

cooling

metal

and

combined

and

cooling-air

are

in the

cooling

yields

film

convection

or

temperature coo]ing,

but

332

then

of combining

and region

cooling

cooling

the

cooling

air

same

film

only, all

lower alone.

film

cooling

only

the

wall

temperature

wall

also

that

the

same

and

of holes

full-coverage

much

only, hot-gas rate.

combined does

average

as for

either wall

convection

higher

or slots

surface

flow the

the

shown

cooling

than

Notice

to

convection

coolant

is about

arc

(as

same

hole,

temperature

of the protective film. film cooling from rows cooling

the

ap-

cooling

blade

film

for

percent

gradients

and

Here,

of the film injection

for

transpiration

film

11-11.

cooling,

it becomes by

both

in figure

convection

a significantly

increase,

conserve

for convection

(ft 2)

COOLING

be augmented

and

is shown

and

immediate

of the rapid decay First, localized cussed,

given

(ft 2)

(m 2) ; lb/(hr)

pressure

must

importance

conditions

Except

and

cooling

design

fihn

kg/(sec)

TRANSPIRATION

temperatures

The

(m 2) ; lb/(hr)

flux,

temperature

convection

in a given

temperatures

kg/(sec) mass

AND

turbine-inlet

parent

flux,

because

will

be dis-

discrete-hole

film

TURBINE

1800L-

_6

_

2600--v

/

:

I

J

_ ,A,'_|

f__'-

2200

COOLING

Hot9as

_3_1_"

Film coolingonly

_Convection _

coolingonly

Hotgas

1200 ---..Combinedfilm and convection cooling

_

_

1000

:_-

_ooI 0

FIGURE

l l-ll.--Effect

cooling.

To

i

of

l

combining coolant

successfully

film-cooled"

analysis

expression

temperature

I

I

the

becomes

and

on the

heat

the

film

must

be

preceding

flux

film

convection

model

coefficient

builds

for

film and flow rate.

analyze

heat-transfer

film-cooling ing

I

.2 .4 .6 .8 l.O Dimensionless distancedownstreamof slot

to

cooling.

cooling,

surface,

temperature

the

known.

"non-

Hence,

discussion.

the

Constant

In the the

follow-

effective

h,.x

is the

heat-transfer

Very

near

the

point

by

the

However,

the

effect

assumed

to be

adiabatic

wall

somewhat

from

of injection, injection

between

data

under

of an uncooled it and

correlated

the

hot

gas

in dimensionless

outer

T_, o is the wall).

The

injected film

out

film having

(see

fig.

form

by

effectiveness

and

wall

film

film

called (i.e.,

layer

film

it is the

of cool

decays

a'r

temperature

effectiveness

is

nl, z_:

T.,.--T_.,,, Tg..-- T', o temperature

the

it is obtained

conditions The

for this.

so _ is frequently

because

a buffer

11-12)). the

is altered

to account

is sometimes

cooling,

adiabatic

film

cooling,

coefficient

rapidly,

temperature

wall

_'"'"= where

heat-transfer

damped with

film

(11-54)

and _ is included

film

temperature

experimental

temperature

The

without

(he, x)I,,,_ hg, x the

itself,

is usually

unity.

(11-53)

coefficient

e --

gas

T),_m:

q=_hg._(T'.,,_=--Tw.o) where

the

(11-55) (coolant from

temperature

a value

of 1, at

at the 333

TURBINE

DE,SIGN

AND

APPIAI;C_TION Boundary

layer -7 /

Y I

_.,,_"_

_

I r_

T,

/

//

_'mm /

I

-J ....

'(

Coolant film

I Coolingair FIGURE

ll-12.--Experimental

determination

•+1:-___

+

deg

o++ 90 _'__----

.1 ---

+:_:

--%

08 --

._

.oz

I

10

_ I ,1,1,1

20

40

60

I

FIGURE

to zero,

of film

downstream.

of

(x--x_)

for

investigators.

is normalized

between

the is with

film

the

air and

the

respect

90 ° is perpendicular). increasing The film 334

As

injection following

cooling

to the

I 1000

hot-gas surface the

for

slots

s and

stream.

slots.

values

as determined the

The

(0 ° is parallel film

4000

experimental

downstream

width

, I

2000

xs#gs

gives

from

distance slot

seen,

(x

11-13

injection

The by

600

effectiveness

Figure

film

400

distance,

ll-13.--Film-cooling

effectiveness

number

angle

far

, 1,1,1,!

200

100

Dimensionless

slot,

temperature.

Injection angle,

1_ .8 _-"_

¥

of film

effectiveness

by

from

the

mass-flux indicated to the

a

slot

ratio

F

injection surface,

decreases

and with

angle. expressions

reasonably

well:

(from

ref.

7) correlate

turbine-blade

slot

TURBINE

Ik

_I'z'_=exp

for small

values

-0. 2

--2.9

k,p-_2]

of (z--z,),

OOOLIN(_ O.

_s

\p_u_s/

and

(11-57)

for large edge

values

of the

Values

slot,

of (z--x,),

where

z8 is the

location

in meters

or feet,

measured

from

for the coefficient

for a 30 ° injection angle. Film

effectiveness

distances ref.

from

the

8) for film

small

lateral

effectiveness

C and

angle,

and

as

cooling

a single

hole

downstream

and n----0.21

and

lateral

11-14

(from

of holes.

in this

distance,

case,

as previously

For film

shown

Dimensionless lateral distance from injection hole, hole diameters

o °

.40--

o

A

"_ E

30i_ xji_ ' "-_.,-'="_ " _ ."__

u:.

20--

,

.50

1.00

Plain s,,ymbolsdenote single hole at 35v injection angle Tailed symbolsdenote single row

_

I

FIGURE

a row

diameter

• 50 --

O

in figure

and from

1 hole

point.

for a 15 ° injection

downstream

is presented

up to about with

n are C----2.7

the

downstream

the stagnation

and n----0.155 of

hole

from

decreases

exponent 1.95

a function

injection

distances,

the

G=

of the

of holes

l

I

I

I

i

I

I

10 20 30 40 50 60 70 80 Dimensionless distance downstream from injection hole, hole diameters l l-14.--Film-cooling

effectiveness

as function

of

stream and lateral distances from injection holes. Mass-flux hole diameter, 1.18 cm or 0.464 in.; gas velocity, 30.5 Reynolds number, 0.22X 10 s.

dimensionless ratio, m/sec

] 9O

down-

0.5; injectionor 100 ft]sec;

335

TURBINE

DE,SIGN

for slots,

and

AND

the

APPI.,ICJ_TION

same

values

of holes. For larger lateral with downstream distance flow,

and

the

values

hole

because

that

_mm is not

of hot

of the

gases

surface.

for

the

row

interaction

unity

data

for single

of holes

are

are

larger

from

as for a row

jet

This

as

the

available

than

adjacent

hole.

film

holes

effectiveness initially increases of the spreading of the injected

injection

the

limited

obtained

of flows

at the

underneath

Very

are

distances, as a result

for

for the

holes.

is due jet

single

Notice

also

to entrainment

separates

from

a staggered

row

of film-

cooling holes. Frequently, the slot data are used for this case, an effective slot width s defined such that the total area of the equals

the

area

Figure single the

11-15

hole

of the

slot.

(from

ref.

at various

direction

of the

diameters,

methods

of cooling

boundary

through the this method extremely tion or

small and, contaminants

standpoint,

is one

mass

transfer

With

stream In

essentially order

cooling from

a large

surface.

This

In

amount

of heat

through

the

wall

maze of interconnected effectiveness. Convection

336

film yet

angle

of

persist

as

effective wall

into

convection

flow

still

is

as it passes in applying tend to be

v_o,v

obtain

cooling, cooling,

into

the

gas

some

of

the

full-coverage

the

is

a

of the

the

trans-

air

internal

flowing

flow

pas-

straight-through holes, or it may consist of a

a relatively

ability

in

pure

cooling

of the

term

issues

a continuous mass the other end. The

to tile

tortuosity

film

air holes

between

with essentially film cooling on

with

cooling

discrete

spectrum

convection

the

is a measure

the

efficient

air is injected

closely-spaced,

flow passages,

and

from

2 hole

to blockage due to oxidafrom an aerodynamic-loss

be constructed of simple, convection effectiveness,

effectiveness theory

the

lies in the

on

most

in

boundary.

the one end, and localized

depends

sages. The wall may with a low resultant

subject Also,

film

by

not

of the

in counter

problems

of small,

transferred

does

with

of transpiration

of cooling

piration cooling on flux over the surface,

exchanger

these

than

there are problems blades. The pores

are air. since

less

compound

film

a

angle

a very effective heat exchanger, wall from the hot gas stream

coolant

full-coverage

number type

the

the

cooling

However, to turbine

advantages

is used.

as the

to the

alleviate

characteristic

serves into

is paid,

normal

to

film

therefore, in the

a penalty

A

wall

to

small pores. of cooling

coverage.

of a porous

it combines

transferred

film spreads

with holes from

a 35 ° injection

but

cooling. The porous wall where the heat conducted continuously

the

film

of a film layer

For

coverage,

available.

layer,

spreading

stream,

local

lateral

cooling

the

of injection.

gas

very

more

Transpiration the

main

giving

injection gives far downstream.

8) shows

angles

the

borrowed of the

high

convection from

wall

heat-

(or blade

TURBINE

O00LING

Fi Im-cooling effectiveness, "r/film tO. 10

-2 -_

0--@

E

2

/

\

o r-

I

d

\

.25

--

1

'

6

_. 20

_. 15

1

I

I

I

(a)

r-

.g- -2 -.E

0 --

E

/

•30-I g

L. 25

X13

--

I

6

I

lO

L. 15

L.20

4 --

r.

I

I

I

I

tb) u_

_

-2 --

r.

25

"_,

E

_

2

\

4 -6

,-- .20

'--.30

0

I

1

I

I

I

I

5

10

15

20

25

30

Dimensionless distance downstream from injection hole, hole diameters (c} (a) Injection (b) Injection (c) Injection FIGURE

ll-15.--Lines

angle, angle, angle,

35°; lateral 90°; lateral 90°; lateral

injection, injection, injection,

of constant film-cooling injection. Mass-flux ratio,

acting as a heat exchanger) convection.

to transfer f

90 °. 35% 15 °.

effectiveness 0.5.

for

single-hole

heat to the cooling

air by

!

T c.o--T c.,.

(11-58)

Since an optimum design utilizes as much of the heat sink available in the cooling air as possible for convection cooling, ,7_o,, values approaching the limit of 1 are desirable. However, the convection 337

TURBINE I)E,SIGNAND effectiveness

is

available.

turbine

a

solid

ref.

perature

wall

by

matrix

The

resulting the

the

so does

the

cooling-air

pressure

model

in figure

metal

9).

through

wall,

limited

one-dimensional

blade

the

(see

usually

As Vco,v increases,

Consider on

APPLICATION

11-16. and

of

An

on the

wall,

Tw,

the

energy

and

local

be written

through

for

wall.

perforated

can

flow

coolant

the

or

balance

equations

pressure

through

porous

cooling-air

differential

supply

drop

local

the

wall

metal

tem-

temperature

in the

T'c, are d3Tw hv d2T,_ dy 3 _ Go% dy _

' hv dTw kw.e dy

0

(11-59)

and T'_=Tw

kw'ed_T'_ hr. dy 2

(11-60)

where effective

thermal

conductivity

Btu/(hr) (ft) (°R) internal volumetric

hv

of the

heat-transfer

porous

coefficient,

wall,

W/(m)(K)

;

W/(m

3) (K) ; Btu/

(hr) (ft 3) (°R) The

boundary

conditions

are e dTw @-_=o

(11-61)

_,)=kw

e dTw " dy _=o

(11-62)

11-16,

G_ is the

h_(Tw.,--T'e,_.)=kw, and

G_cp(T_ In this case, as seen of surface area. An the

overall

heat

energy

flux

from

ba]ance

to the

1,1-16.

wall

derivative, and matrix The side

and

They

heat

are

,--T"

figure

gives,

338

as a third

coolant both

temperature

nonlinear

heat-transfer

is somewhat

flux

boundary

to the

wall

can

profiles

with

which is a consequence heat transfer. flux

mass

o-- T_,_.)=Gc%_co._,(Tw,o--

per

condition

unit

for

opposite

of the

T:,_.) are

shown

signs

interaction

also be written

(11-63) in

figure

in

the

second

of

the

coolant

in terms

of a hot-gas-

coefficient: q=h,,_(

This

'

wall,

q=Gccp(T' Typical

'

different

from

Tz,e-the

Tw.o) heat

flux

(11-64) expression

with

local

TUR,BINE

OODLIN(_

Tw,o

1"1

C,O ]w,J

q

Gc

Tc, i

T__J

\

'\.,

_

ay--_t

c, in

hvA .,Xy(T w - Tc) '_

\

dl-w + dy d (k w, eA -_-y) ClTw_Ay kw,eA dy

\ \ '_

Typical element FmuRE

film

cooling

duced" than

in

11-16.--Porous-wall

that

the

heat-transfer the

acual

temperature

recovery

coefficient

film temperature

and

ht._ the

profile

gas

due

model.

temperature

to blowing

solid-blade

and are

used

heat-transfer

a "rerather

coefficient

hg,Z"

Consistent wall, not

with

which too

the

incidentally

large,

we can

local

one-dimensional

gives

good

results

model if the

pressure

of

the

blade

gradient

is

write F ht_ Stt_ _:;--_=eF/__

where

the

as shown mass

flux

correction in figure (surface

factor] 11-17

(from

averaged)

1J

is a function ref. to the

.

Str_

10), and hot-zas

F -(pu)_

(pu)

(11-65)

of convection Fis mass

the ratio

effectiveness of the

coolant

flux: (11-66)

339

TURBINE

I_E_IGN

AND

APPLICATION

Convection effectiveness, 11cony

1. E

0.9

o

J

f

J

J

f

• 7j

J

J

o

k_

.7

.6 0

1

2 Blowing

FIGURE

l l-17.--Correction

to

3

parameter,

equation

4

FtStcj '

(11-65)

for

wall

convection

effectiveness.

the

heat-transfer

SIMILARITY It

is

often

of

performance actual

of

engine

to evaluate size

turbine

the the

co_lditions.

conditions To

which are formance

and at

application.

a cooled test

answer

at

Generally,

hardware

actual

to

components

heat-transfer

to whether at

necessity

environment.

prototype

than

economic

this

other

tests

temperatures

practice

raises

design

similarly

question,

the

with

valid

various

actualpressures

question

as

specifications

under

actual

similarity

important in relating test performance of an actual-size film-convection-cooled

the

conducted

and a

meeting

behave

than

are

performance

gas

configuration will

conditions initial

aerodynamic lower This

blade

evaluate

engine

parameters

to engine turbine

perblade

are discussed. The number engine 340

Mach

number

distribution and

test

distribution around

conditions.

and the

vane

Similarity

momentum-thickness must in

be these

the two

Reynolds same

between

parameters

is

TUITBINE

essential

to

ensure

the

coefficient

and

adiabatic

transition

from

laminar

Let

superscript

same wall to

(t) refer

conditions.

To

tion

does

not

between

flow

must

be the

same

temperature

ensure

in both

F is an approximate

for the

variation

the

two

p;,.

and

local

point

superscript

Mach

/(RT')_,r(.

the

distribu-

equivalent

the

remain

local

_

(11-67)

eqs.

(2-128)

temperature

and

given

(2-129))

by

\(_+t)/2(_-t)

between

Reynolds

(t)

and

(e)

number

must

also

conditions,

o;,,.:.,

/z

#

(11-68)

momentum-thickness

unchanged

mass

,,

F-- _f_ _--_) Since

of

(e) refer

number

ry'

(from

with

/

same

Therefore,

'e' _(-R-_)g("

heat

the

heat-transfer

layer.

conditions,

correction

of specific

and

conditions

cases.

wy'--p'_

of

boundary

that the

w(.,,

where

distributions

turbulent

to test

to engine

change

relative

O00LIN_]

/(,r.);.,r,,, s

(p_),',-o;., .;') _"- ,".;',p';"

__

_,.,-1

(11-69)

\--_/z

If

the

engine

local and

(ou)J(Cu)z

,

(or density

conditions,

the

coolant

is

the to

0Jo_),

ratio

hardware

effectiveness

test ratio

hole-diameter size

film

and

to

coolant

hot-gas

the

OJD must

same

(11-69)

between

mass-flux ratio

ratio

(pu_)¢/(pu_)=

to film-ejection-

in both

cases.

Since

actual-

then

(_)(,, equation

hot-gas

momentum

--

and

to

unchanged

momentum-thickness be the

is presumed,

remain

--1

(11-70)

o;,,

becomes

_':' _","___,,"'-'"'e_I(_T_)',', r;" #g(t) w_" Equation with

the

pressure number Parametric for air.

(11-71) viscosity and and

,(,,p,(,,

shows

that

and

gives

temperature Mach curves

the the

which

number of

_/(R-_)z(,, gas

flow

(11-71)

(11-71)

rate

functional will

distributions

equation

Fg(,,--1

provide

must

vary

relation the

directly

between same

for test

and

engine

are

shown

in

gas

Reynolds conditions.

figure

11-18

341

TURBINE

I_E_IGN

AND

APPIAC_TION

4O

E

30

_ l-

20

eD

a_

i0

J

0 250

500

150

1000

1250

1500

1750

2000

2250

25OO

I

I

I

3000

3500

i

Temperature,Tq, K

FIGURE thickness

I

I

I

I

I

0

500

1000

1500

2000

ll-18.--Similarity Reynolds number

The

cooling-air

coolant-to-gas

curves of distributions

flow

rate

mass-flux

1

2500 , Temperature,lg, oF

constant around

and

ratio

Mach number a turbine vane

temperature

and

are

momentum

and momentumfor air properties.

then

ratio.

set

by

the

Requiring (11-72)

(7 ,J - L(--#u-),j implies

(11-73) Wg/

and

it is necessary

to ensure

equality

Neglecting in

the

that

of test

conduction

direction

T_,o is related

normal to the

and

engine

in the

plane

of the

to

wall,

the

supply

the

coolant

(He, o--He, (Hc.o--H_, where

the

(11-71)). hot-gas-side 342

viscosity

ratio

Satisfying heat-transfer

momentum

here

equations

ratios, wall

film

compared

ejection

temperature

since

(11-71)

coefficient

the to

that

by

mass (11-75)

distribution

to

temperature

,,)(t>_q(,) _,(_) ,n) (_) q(C) u_"

represents

pc.o=p_.

(11-75) flow

ratio

ensures around

the

(see

eq.

that

the

vane

will

TURBINE

have the same shape for both test transfer coefficient in dimensionless St(,'):St(, Since

the

Prandtl

similarity

number

conditions

number

ratio

equation

and engine conditions. Stanton number form

cannot

unity

The is

be

are

set

met,

depends

on

(11-76) independently

the

if all

departure

the

of the

Prandtl

the

Nusselt

coolant

side,

number

form

the

heat-transfer

is given

Stanton

number

ratio

where

m is the

power

tion. The viscosity coolant Reynolds factor

cannot

on the

in

dimensionless

/ _'* _'*' °_

Reynolds

factor number

be set

coefficient

(*'

number

in equation ratio. As

independently,

is small. In fact, if the viscosity t ct)¢.o could be approximated by

(11-77)

for coolant-side

(11-77) with the although

is the Prandtl

its

convec-

test-to-engine number, this

departure

from

over the full temperature a power law

then,

by equation

ratio

(based

perature)

(11-74),

tg(') to

(11-78)

test-to-engine

on

the

film-cooling

bole

would

be

1. The

coolant

simulated

same

conditions

through the

the

the

cooled

in the

engine,

which

remains

there

most

convenient

cooling

must

are

known,

same

test

wall

and

the

for

namely

the

for

same

the

two

during

wall

pressure as it does

temperature

conditions.

includes coolant

only

supply

Tf._.

Hence,

or some

similar

Strict

equality

actual

hardware,

scaled

properly

and

grouping

as a measure

pressure.

because

the

proportionately

of the

of these cooling

in _ for test since

the

because It

greater

temperature

of the

is easier

driving

three engine

lower

to cool

potential than

temperatures,

the for

at high

dimen-

is commonly

of a given is, however,

drop heat

those

(11-79)

performance

and

The

temperathe

T_.,--T_.o

used

temactual

conditions.

a test

engine

temperature

ture T_, _, and the effective gas temperature sionless wall temperature _, defined as

ejection

the

outer

and

number

number

ensure

normalized

between

Reynolds

Reynolds to

the

be some

dimensionless which

diameter

air passages

is to perform

invariant

temperatures

coolant

is important

internal

blade

unity

range

cc t_

If

in

by

Nuc (') =Nuc

drop

other

(11-76).

On

and

heat-

*) {'Pr(')Y/a

discussed

from

C_OLING

flux

blade

temperature

impossible

through

the

at reduced

at the

convection

blade

and

with is not

temperature

reduced

cooling

wall

design.

conditions, (T_._--T/)

pressure.

is How343

TURBINE

ever,

I)E,SIGN

for

and

AND

properly

APPI.JCATION

scaled

_(e) is well

within

test the

conditions, range

the

difference

of experimental

between

accuracy

_(t)

in most

cases. An

example

(11-71)

to

pressure,

of

(11-77) were

For

condition

a test

temperature

under directly

conveniently test conditions, in equation

rather using

_(_) is

than

high-temperature

344

than

and

equations

11-II

for

a high-

environment. a given the

what of the

air.

ratio

gas.

total

heat

radiation

and heat-flux

wall

be

conditions

Since

at the low-temperature be accounted for in the

fuel-air

Air

dimensionless it would

high-pressure with

solving

in

flux

the to a

and cannot

ll--II.--SIMILARITY

temperature

Takeoff

Gas total pressure, atm.

STATES

condition

Coolant

temperature

oF

_3500 200 400 600 800

1700 1811 1922 2033 2144

2600 2800 3000 3200 3400

905 1000 1200 1400 1600 1800 2000 2200 2400

condition.

' 33. 7 4.3 6.0 7.7 9.4 10. 3 11.1 12. 9 14.6 16.4 18. 2 19. 9 21.7 23. 5 25. 3 27. 1 28. 9 30. 9 32. 8

145 188 23O 273 294 315 357 399 442 485 528 571 613 656 699 743 786 828

is be

low-pressure ratio q(')/q(e)

(11-75).

_2200 367 478 589 700 758 811 922 1033 1144 1255 1367 1478 1589

' Reference

in table

air,

component

by film cooling

simulated it must

for

cooling

higher

be a significant

(a)

K

those

ambient

1 percent

affected

is given

by

gas-turbine-engine

TABLE

Gas total

generated

simultaneously

used

actual engine. Radiation can not

states

high-temperature

properties

blade

similarity

--199 --122 --45 31 70 107 182 259 335 413 490 568 644 721 799 878 955 1030

1.04 1.03 1.02 1.01 1.01 1.01 1.00 1.00 1.00 1.00 I. 00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

TURBINE TABLE

11-II.--Concluded

(b) Cruise

Gas

total K

12200 367 478 589 700 799 811 922 1033 1144

condition

Coolant

' 3500 2O0 4OO 6OO 8OO 978 1000 1200 1400 1600 1800 2000 2200 2400 2600 28OO 3000 3200 3400

temperature _( t)/ _o(e)

!

oF

1255 1367 1478 1589 1700 1811 1922 2033 2144

I Reference

Gas total pressure, arm.

temperature

(X_LING

K

'13.8 1.7

°F

1801

1983

139 180 220 259 294 299 338 378

2.5 3.2 3.9 4.5 4.6 5.3 6.0 6.7 7.4 8.2 8.9 9.6 10.4 11.1 11.8 12.7 13.4

--209 -- 136 --64 7 70 78 148 220

417 458 498 539 579 619

!

660 702 743

,

729 804 878

782

!

948

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1. O0 1. O0

ALBERT:

An Analysis

!

291 364 437 510 582 655

1.03 1.02 1.01 1.01 1.01 1.01 1.00 1.00 1.00

condition.

REFERENCES 1. EsQAB,

2. 3.

4.

5. 6.

JACK B.:

COLLADAY,

RAYMOND

S.;

KAUFMAN,

of the Capabilities and Limitations of Turbine Air Cooling Methods. NASA TN D-5992, 1970. KAYS, W. M.: Convective Heat and Mass Transfer. McGraw-Hill Book Co., Inc., 1966. KESTIN, J.: The Effect of Free-Stream Turbulence on Heat Transfer Rates. Advances in Heat Transfer. Vol. 3. T. F. Irvine, Jr.; and J.P. Hartnett, cds., Academic Press, 1966, pp. 1-32. AMBRpK, G. S. : Approximate Solution of Equations for the Thermal Boundary Layer With Variations in Boundary Layer Structure. Soviet Phys.-Tech. Phys., vol. 2, no. 9, 1957, pp. 1979-1986. SPALDING, D. B.; AND PATANKAR, S. V.: Heat and Mass Transfer in Boundary Layers. Chemical Rubber Co., 1968. KERCHFR, D. M.; AND TABAKOFF, W. : Heat Transfer by a Square Array of Round Effect

Air Jets of Spent

Impinging Perpendicular Air. J. Eng. Power, vol.

to a Flat Surface Including the 92, no. 1, Jan. 1970, pp. 73-82. 345

TURBINE 7.

DESIGN

AND

APPLICATION

ARTT, D. W.; BROWN, A.; AND MILLER, P. P.: An Experimental Investigation Into Film Cooling With Particular Application to Cooled Turbine Blades. Heat Transfer 1970. Vol. 2. Ulrich Grigull and Erich Hahne, eds.,

Elsevier Publ. Co., 1970, pp. FC1. 7.1-FC1. 7.10. 8. GOLDSTEIN, R. J.; ECKERT, E. R. G.; ERIKSEN, V. L.; AND RAMSEY, J. W.: Film Cooling Following Injection Through Inclined Circular Tubes. Rep. HTL-TR-91, Minnesota Univ. (NASA CR-72612), Nov. 1969. 9. COLLADAY, RAYMOND S. ; AND STEPKA, FRANCIS S. : Examination of Boundary Conditions for Heat Transfer Through a Porous Wall. NASA TN D-6405, 1971. 10. L'ECUYER, MEL R.; AND COLLADAY, RAYMOND S. : Influence of Porous-Wall Thermal Effectiveness on Turbulent-Boundary-Layer Heat Transfer. NASA

346

TN

D-6837,

1972.

TURBINE

CODLING

SYMBOLS A

surface

area

of one

coolant-passage

flow

augmentation

g

side

coefficients

in eqs.

body-force

component

bl, b2, b3 C

coefficients

in eqs.

cto,

aa

constants

coefficient.,

Cp

specific

heat

el,

coefficients diameter

d

dissipation term distance between

F

ratio

.[

hole,

correction

factor

mass

kg/(see)

g H

conversion

h

heat-transfer

hv

internal

/; I

static

and

J

conversion

K

dimensional

flux

thermal

hole,

or

film-

(m _) ; lb/(hr) 1 ; 32.17

J/kg;

mass

flux

(ft 2) (lbm)

(ft)/lbf)

(see 2)

Btu/lb W/(m

2) (K) ; Btu/(hr)

heat-transfer

(ft 2) (°R)

coefficien't,

W/(m3)(K);

fit s) (°R)

defined

turbulent

impingement

to hot-gas

coefficient,

enthalpy,

k

; Btu/(lb)(°R)

in eq. (11-65)

volumetric

,W

J/(kg)(K) (11-50)

circle,

used

constant,

Btu/(hr)

J/kg

; Btu/lb

by eq. (11-41) constant,

1 ; 778

constant, kinetic

energy, characteristic m; ft

exponents Nusselt

used number

in eqs.

exponents Prandtl

used number

in eqs.

pressure,

(2

heat-generation

q 1l

heat

flux,

gas

constant,

Re

Reynolds

r

recovery

factor,

St

Stanton

number

J/kg;

W/(m)

wall thickness, Xlach number

P

(ft) (lb)/Btu

1 ; 3600

(:onductivity,

eo()l'mt-passage M

pressure,

(11-49)

mass

enthalpy,

(11-57)

m ; ft

G

I

and

in eq. (11-42), W/m3 ; Btu/(ft 3) (see) volume elements (see fig. 11-10), m ; ft

of coolant flux,

lbf/lbm

(11-26)

of leading-edge

injection

term

(11-8)

N/kg;

to (11-51)

at constant

D

total

to (11-51)

in x direction,

eq.

m2; ft 2

(11-31)

(11-49)

in eqs.

element,

m2; ft z

eq. (11-49)

in eqs.

friction

C2

area,

factor,

B_

al,

of volume

N/me;

(11-8),

sec/hr Btu/lb (K) ; Btu/(hr) length,

(11-45),

m;

(ft) (°R) ft

and

(ll-4S)

(ll-S),

(11-45),

(11-56),

W/ma;

Btu/(sec)(ft

a)

and

(11-57)

lb/ft" term,

W/m_;

Btu/(hr)(fC)

J/(kg)

(K) ; (ft) (lbf)/(lbm)

(°R)

number eq.

(11-27)

347

TURBINE

DE,SIGN

AND

8

slot

T

temperature,

t

gas

%

component

APPLICATION

width,

m;

K;

static

°R

temperature, of

direction), volume V

ft K;

gas

m/sec; of gas

(y-direction), 'tO

component mass flow boundary

X

distance

Xn

center-to-center

Y

coordinate

Zn

distance

direction

along

surface

direction

normal

velocity

in

to

surface

ft/see

of gas velocity in direction perpendicular rate, kg/sec; lb/sec layer plane (x-y plane), m/sec; ft/sec surface

from

distance

direction

(z-

m3; ft 3

m/sec;

along

the

in

ft/sec

of jth element,

component

°R

velocity

of flow,

distance

edge,

m; ft

impingement

holes

in

m; ft

normal

between

leading between

to the

to surface,

impingement

m; ft

holes

and

blade

inner

wall,

m; ft heat F

diffusivity,

specific

heat

m2/sec; ratio

correction

ratio of specific heat constant volume ,h

enthalpy ratio

of heat

transfer

coefficient

coefficient

without

cooling

0

momentum

thickness,

tt

viscosity,

(N) (see)/m2;

l,t

momentum

T

(11-68) to specific

heat

at

with

film

cooling

to heat

film cooling

effectiveness m ; ft lb/(ft)

diffusivity

angular

distance

dimensionless coefficients

in eq.

defined

exponent

from wall

(sec)

(kinematic

density, kg/m3; lb/ft 3 local shear stress, N/mS;

term

eq. pressure

m; ft

7/

P

factor,

at constant

thickness,

transfer

ft2/sec

viscosity),

m2/sec;

stagnation

point,

ft2/,_ec

lb/ft _ leading-edge

deg

temperature (11-48)

by eq.

(11-52)

in eq. (11-78)

Subscripts: a

adiabatic

CP

constant

C

coolant

property

CFOSSflOW

COlbY

convection

crit

critical,

348

referring

to transition

from

laminar

to turbulent

flow

TURBINE

])

with

hole

e

effective

film

fihn

g

hot gas hole

h i

(li'uneter

at

free-stream

impingement inlet

J

jth clement laminar

J

with_._,

a,s characteristic

leading outer

edge

0

re]

reference

8

downstream

stag T

stagnation turbulent

t

transpiration wall

le

W X

local with

edge

length

value x as charttcteristic

with

A as characteristic

8

with

0 as characteristic

Sul)ersc,

condition

of slot

A

oO

dimension

inner

imp in L

as characteristic

C'(YOLING

apl)roaching

leading

length length length edge

ril)ts"

(e)

engine

(t)

test,

,

] total

condition ('mid ition stale

/ flu(:tuating

(referring component

to T anti (referring

p) to p, v,/,

u, and

w)

349

CHAPTER 12

Experimental Determination of Aerodynamic Performance ByEdward M.SzancaandHarold J. Schum The

preceding

theoretical deal

chapters

aspects

relating

of consideration

entire

etc.)

the

turbine

dynamic this

on

basis

it must

design

goals

have

turbine

shafting,

of mechanical

criteria.

be determined met.

design.

by

A great since,

hai'dware

bearings, Once

whether

Only

the

blading,

Associated

disks,

been

primarily

turbine

turbine.

(rotor

built,

and

with

toward

is the

and

concerned

directed

blading

the

been turbines

been

assembly

is designed

is designed

to

has

aerodynamically, the

have

casings

the

or not

testing

the

for

turbine the

aero-

turbine

can

be determined.

In

addition

of the The

to

separate stator

ments, obtained

overall

losses

loss

as

the

contributing

can

be

discussed easily

indirectly

chapter

direct

the

of a turbine,

to the

obtained

in

from

from

performance

overall

readily

from

7

2).

(vol.

stator

loss

loss

they

and

the

is often

experimental Rotor

measurements;

a breakdown

are

measure-

losses

cannot

usually

overall

desired.

be

obtained

performance

measurements. In

developing

determine ing.

the

which

The

obtain

the

test

of the

desired

the

test

stator

in which are

facility

performance

nature

manner

conditions

a

and parameters

facility,

and/or these

subject

program,

researcher

he is interested

the

instrumentation

overall

performance

parameters of this

the

vary

with

must

in evaluatnecessary

parameters, turbine

to and

operating

chapter. 351

PAGE _-_--'_

INTENTIONALLY 8LAtIK

TURBINE

DE,SIGN

AND

TEST The

manner

is discussed herein.

FACILITY

in which

parameters pressure

bine.

The

sures,

mass

flow

rate,

directly

in the

calculated

from

be

work,

and

ranges

performance

speed,

discussed overall efficiency.

of rotative inlet

speed

of the

tur-

exit

pres-

and

determine

facility.

computed

turbine

the to

test

not

for

rotative (needed

turbine

will

and

define

specific

define

torque,

measured

the

torque,

to fully

temperature

and to

determined

in order

inlet

2)

used

rate,

are usually

ratio

and

flow

is expressed

7 (vol.

generally

are mass

and

MEASUREMENTS

performance

in chapter

parameters

performance

AND

stator

in detail

The

These

APPLICATION

efficiency)

Specific

work

are is then

equation rN -dw

Ah'-_K

(12-1)

where ,_h'

turbine

K

conversion

specific

F

torque,

N

rotative

J

conversion

w

mass

N-m;

rate,

(2-49b)

depends

is based

on

the

1; 778 (ft) (lb)/Btu

ratio

based

on

of the

outlet

across

The on

outlet

outlet

the

outlet

static

velocity

accordingly,

of the

commonly

used

In the will

this types

of devices

be discussed.

discussed manometers automatic 352

section,

in

as are

the

generally

test

used

acquisition

this

chapter. gages data

and

all

efficiency

is

where

all

on

a pressure

component where

Rating

static

efficiency

of outlet

only

efficiency

efficiencies,

the

axial

is not which

as

were

1).

a representative

Data

axial

work,

where

meaningful is based

meaningful is useful.

Total

Static

is most

the

ideal

meaningful

efficiency

of only

total

2 (vol.

and electroaic

and

velocity

desired.

equation is always

define

recoverable.

Rating

is most

outlet

in chapter

or

by

pressure

to

is most

by the ideal temperature

shown

inlet

used

efficiency and

pressure

recovery

as

1). The

pressure

useful

is lost.

to the

component

2 (vol.

pressure is

velocity

turbine,

particular

velocity

and,

the

outlet

total

corresponding

discussed

lb/sec

of chapter

pressure.

however, of

(rad)(min)/(rev)(sec)

is obtained by dividing the actual work hh' work is a function of the turbine inlet total

or

total

Btu/lb

rev/min

kg/sec;

pressure

(2-48b) the

rad/sec;

constant,

flow

the

J/kg; 1; _/30

lb-ft

speed,

Efficiency work. Ideal and

work,

constant,

These with

to make

will

the

reduction

can

vary

from

computations with

be described,

required

and

slide-rule

acquisition

facility

on-line

and

measurements

systems visual to computer

will not

be

reading

of

completely process-

EXPERI1VIENTAL

DETERMINATION

OF

AERODYNAMIC

PERFORMANCE

ing. A general discussion of data measurement, acquisition, mission, and recording systems can be found in texts such erence 1. Description of Test Facility

transas ref-

A turbine test facility consists of the research turbine, a gas supply, an exhaust system, associated piping with control valves, a power absorber, and the instrumentation needed to make the desired measurements. A schematic diagram of a turbine test facility at the NASA Lewis Research Center is shown in figure 12-1. This facility, a photograph of which is shown in figure 12-2, is used to test single-stage or multistage turbines of about 76 centimeters (30 in.) in diameter. It is generally representative of most turbine test facilities and is used here as an example for this discussion. In such a facility, removal of the turbine rotor gives the room necessary to place survey instrumentation behind the stator and, thereby, transforms the rotating rig into a stator annular cascade. Most turbine component testing is conducted with temperature or slightly heated. This is commonly

Laboratory air system

Venturi

acilit

isolation

air at ambient called cold-air

Burst-disc

Vent to

safety valve -.

roof

Sonic valve

", ,_i_

•

Isolation,'

valve

valves

• _

/kl I-

Gas supply .f-

control

joint

-

_

valve -,

--I

Bypass control valve -,

Expansion Main exhaust

....

LBurner_j_,,,

L

-=

"-_ Main control.-

/

"@

valve

valve @ _,- Facility

isolation f

Laboratory system

FIGURE

12-1.--Flow

exhaust

schematic

of a turbine

test

facility.

353

TURBINE

DE,SIGN

AND

APPLICATION

ductim

:FIGURE

testing.

Performance

values

based

N/cm was

2 or

and

thereby are

to and

however, on

turbine

smaller

turbines,

turbine

inlet

the

for

the

larger

performance

where

can

to figure

12-1,

(40-psig)

air

be

the

model are

number, found

number

to

effects

to

the

this

effect be

the

applivelocity

same can

as those present

a

of Reynolds

negligible.

are more

obtain

in lower

actual

turbine

then,

This

conditions

results

in an

turbines,

varied

(10.133

test

values

numbers

was

Reynolds

pressure

Yet,

Mach

pressure

K or 518.7 ° R).

standard

Reynolds

of equivalent

turbine-inlet

be encountered

testing.

Only

of

(288.2

the

would

in terms

conditions

1). Using

near

than

facility.

reported

temperature

2 (vol.

turbine.

dissimilarity;

are

facilitates

similar

actual

number

and

levels

test

sea-level

temperature

power

diagrams of the

psia)

in chapter and

and

cation

standard

14.696

of pressure flow

parameters

on

discussed

12-2.--Turbine

proper

For

important, Reynolds

number. Referring N/cm2-gage venturi

meter

(16-in.)

air-supply

354

is located line

air for the system

of

in a straight for

the

purpose

turbine the

is supplied laboratory.

section

A

of the

of metering

by the

the

27.6-

ca}ibrated

40.6-centimeter air flow.

This

EX_PERIME,NTAL

DF_TERMINATION

piping

is sized

exceed

61 meters

(as was

device

in

the

per

all piping) second

pressure

required

for

and

a range

of

flows.

the

turbine

vides

a high

operator.

pressure

inlet-pressure protection turbine-inlet (30

main

pressure

in.)

of Hg

valve

permits

diverts

into

of lower

fine

be seen

valve

the

control

used

example

The

to

sonic

valve

and

turbine,

turbine-inlet (20-in.)

turbine

desired

was

76.2

bypass

pressure. lines

entry

line provide is the 50.8the

this (6-in.)

air

dual

(These

cm

control

The

to provide

plenum.

pro-

automatic

establish

15.2-centimeter

of the

to the

controllable

safety valve and vent Further downstream

control

devices

air-metering

valve

operation

(for

air

of

(12-in.)

50.8-centimeter

velocity

discussion

not

constant

of metering

section. is an isolation

burner

will

air-metering

a relatively

number

to facilitate

absolute).

two

velocities of the

A 30.5-centimeter

drop

(20-in.)

air

affords

the

A further

control. A burst-disc from excessive pressure.

centimeter

the

PERFORMANCE

Location

line

minimize

devices is presented in a subsequent Downstream of the venturi meter by

that

ft/sec).

supply

may

AERODYNAMIC

such

(200

high-pressure

upstream

OF

then entry

lines

can

in fig. 12-2.)

After

passing

altitude

exhaust

(48-in.)

main

control

control

valve.

These

pressure

through

the

system

ratio

turbine,

of the

and

valves

air

laboratory

valve

acrossthe

the

a

permit

discharged

through

(16-in.)

turbine

while

the

to

the

a 121.9-centimeter

40.6-centimeter the

turbine

is

operator

bypass

to vary

turbine-inlet

the

pressure

is

maintained constant by automatic control. This type of pressureratio control has been most successful with small turbines. With large

turbines,

large

pipe

volume

A burner its use inlet In

air.

controlling

422 In

avoid gas,

the

burner

inlet

at

to heat

lb/sec)

the

from

and

the

valve. because

the

turbine-

turbine can,

exit.

modified

high-pressure

is then

mixed

temperature air

of

control

burner of the

air

of bypassed capability

inlet

is to elevate

Some

heated

because

in phantom,

problems

is used.

(52

the

12-1)

jet-engine

turbine

second

and fig.

icing

the

amount

has per

(in

response

air is with

the

is maintained the

fuel

a maximum ambient

by

flow. air

This

flow

of

temperature

to

(300 ° F). general,

burners

a relatively

simple

combustion

the

turbine of the

and

desired the

burner

kilograms K

natural

The

a slower

commercial

burner

both

particular 23.6

so as to with

remaining

the

purpose

a single

to the

is

is shown

The

facility

for operation

heat,

between

temperature

bypassed

there

installation

is optional.

this

for

however,

in

products

the but

cost

and

performance are

and

generally complexity

using

gasoline,

inexpensive are

added

to the

calculations. used

only

of large

jet

fuel,

means air,

and

Electrical where

or natural

of heating

flow

these

the must

heaters rates

are

gas

provide

air. However, be accounted provide low,

because

clean of

installations. 355

TURBINE

DEISIGN

All turbines The

and

test

are against

following

be guarded

AND

A.PPLICATION

facilities

must

be designed

some

of the

potentially

by

constant

unsafe

Low

supply

(2)

Low

pressures

(3)

High

temperature

in bearings

(4)

High

temperature

of dynamometer

(5)

Low

(6)

High

temperature

(7)

High

(or low)

temperature

(8)

High

pressure

of turbine

inlet

(9)

High

pressure

of turbine

exit

(10)

Overspeed

(11) (12)

Reduced Excessive

(13)

Excessive

Interlocks starting. air

rotor

rotation

of turbine

line

and

some

of

water inlet

was

previously,

for

designed

velocity

with

gas

and

upstream

minimum

of

circumferential has

an

pressure

approximate

of 2 dynamic A short, inlet inlet low,

straight,

section

entering

and

measuring

turbine

356

the

from

two

in the

quickly

stops

and

facility.

sides;

A screen

to the

used

view also

to provide

further area,

in

this

of the shown.

test As

the

plenum

for minimum

is shown

ensure

turbine

effective

the

row

and

upstream

insertion

located

a symmetrical

blades.

giving

This

screen

a pressure

drop

of blades

for

temperature

(station of the

of probes

is provided

does

the

the

purpose

12-3)

blades. not

converging of installing

measurement

0 of fig. stator

between

devices.

is located

Since

the

significantly

The

about

1/_

inlet

velocity

disturb

the

is flow

blading. annular

blades 2,

to

passage

first

station

chords

A straight, station

valves

turbines

An enlarged indicated is

distortion.

distribution

annular

pressure

the

turbine

during

heads.

turbine-inlet blade

to the

as feasible

section

50 percent

the This

research

a plenum

pressure

shut

system.

turbine

alarm

Turbine

volume

a converging

prevent

audible

to rapidly

of the

air enters

casing

an

damage

of one

as much

gas

systems

only

heater

to prevent

the

exit

and

test facility is presented in figure 12-3. section with instrumentation stations stated

supply

gas

monitoring

a signal

diagram

water

gas rotor

Research A schematic

oil

gas

of turbine

the

as in the

in order

oil

]ubricating

outlet

provide

provide

must

rotor

orbit

as well

that

lubricating

dynamometer

of turbine

monitors

features.

conditions

dynamometer

of dynamometer

shaft

Others

inlet

and

of turbine

on

operation.

of turbine

clearance between rig vibration

Some

safety

monitoring:

(1)

pressure

with

fig.

to 12-3).

flow measure

passage the

Measurements

is also

provided

turbine-outlet are

made

downstream air about

state

of the

(measuring 2 blade-chord

EXPERIMENTAL

DETERMINATION

OF

AERODYNAMIC

PERFORMANCE

f Station 0

1

I

2 ]

1

I

I

Stator blade-t-

'LJ :" Rotor blade

Plenum

]'0 dynamometer

i

\

i! \

_

i

i_\ ¸,

\ CD-11683-11

[ -

FIGURe:

]engths bilize(1

downstream and uniform

Both

the

idealized The

inlet as

latter

tail

cone

axial

outlet

compared

to

a burner

vary.

diflicult,

passages

the aml

in the

rotor, sections an

for

the

this test

actual

positions

flow-property indicated

last

sake

reason

test

section.

air

is relatively

turbine

are

turbine the

row

weight

inlet,,

and

a

Flow-passage minimum

state

conditiolls

the

in varying

area use

somewhat

s_,ving,

measuring for the

sta-

installation.

turbine

of blades.

of engine

blading

passages

of straight,

is exannular

turbine. Measurements

measuring in

test

preceding

Flow-Property The

exit

jet-engine

the

is the

the

of the

Accurately

turbine

of turbine

where

immediately

Also,

after

diagram

following

is required.

aml

tremely

and

has

length

before

of the

immediately

diameters

flow

12-3.--Schematic

figure

stations 12-3.

There

are are

loe,_ted numerous

in

the types

uxia! and

357

TURBINE

DE_TGN

AND

APPLICATION x o • E_

Measuring stations

1

0

FIGURE 12-4.--Schematic

variations of probes will discuss primarily turbine and

to obtain

flow

outlet, In size

all

size

of the

turbine

mentation probes

can with

In small ated

respect the

could

limits

Static 12-4,

have

on the

90 ° apart,

are on

values

number

and each

four both

the

on the

taps

manifolded

are often

served

differ,

pressures. As part termine As

calculate 358

reading

of the the

explained the

blade

effect

on

being size

can the

inner

of research

stator a

station,

outer

associprocess,

consideration

as

walls.

This

section,

along

of the

blade taps

the

ob-

individual

desired

the

pro-

individual

and

information

Static-pressure

and

In order multiple

If the

it is often

this

figure

multiplicity

occurs

average

test,

distribution velocities.

reading.

true

in

opposed

distribution. and data,

circulation

be the

shown

diametrically

a single

flow

performance

subsequent surface

not

their

expansion This

of the

negligible.

with

turbine

The instru-

size

be considered

measured.

taps,

to provide may

relative

probes

of

data.

of probes.

and

some

and

irrespective overall

the

various

measuring

static-pressure in

passages of the

inlet

12-4.

same

the circumferential pressure amount of instrumentation however,

pressure

the

turbines,

static-pressure

vides a check to minimize pressures

an

the

pressure.--At there

flow

in figure

duplication

large

presence

temperature,

turbine

investigations,

require

whether For

at the

is shown

experimental

to the

turbines,

therefore,

usually

exit,

determines

of pressure,

located

of stages,

be afforded.

blockage

and,

turbine

instrumentation.

the desired measurements. We used in the example research

measurements

stator

or number

2

of turbine

instrumentation

as at the

general,

turbine

desired

The

as well

diagram

available to make the instruments

the

angle.

Temperature rake ]-otal-pressure probe Static-pressure tap Angle probe

to desurface.

is are

used installed

to

EXPERIMENTAL

along

the

tubes

that

the

hub,

mean,

can

blade,

problem.

0.0254

centimeter

so as not Inlet are

at the to

is shown pressure

The

unshielded

used

for

the

in.)

is limited

be

in

If

divided

among

being

discussed,

installing

the

taps

used

is

size

diameter.

A

however,

too

area

center

small

(a)

The and

are relatively

probe

shown

in

measurement.

station the

or more blades no

is desirable results

in a

which that

insensitive figure This

to yaw is

probe

has

I

Shielded

However, to define on

is about

0.48

for

is such some

also

40 ° .

commonly

an insensitivity

tb_ (b)

total-pressure pressure

to establish

used

such

Flow

probe.

turbine-inlet

sure.

and

One

in length,

12-5(b)

90 °

12-4)

passage.

shielding, twice

probes, fig.

20 ° .

circumferential

based

of

size

presented

(see

flow

FZ(_URE 12-5.--Total-pressure

value

the

approximately

a hole

ia_

testing

of

the

hole

small

number two

total-pressure of

12-5(a).

readings

about

shielded measuring

Flow I

The

the by

hole

in diameter

total-pressure for

thus,

inlet

in figure

(0.19

that

to yaw

in.) flow;

blade

turbine

pressure.--Four

all immersed

centimeter

can

pressure-tap the

blade.

time.

are located

probe

taps

PERFORMANCE

of the

one

research

(0.010

AERODYNAMIC

sections

large;

The

total

apart,

fairly

OF

in any

the

to disturb

response

tip

desired

For

and

major

long

the

blades.

hollow

and

be installed

then

adjacent are

D_TERMINATION

distribution

and

maintain

the

turbine-inlet

pressure

ratio

experimental

readings and a constant total-pressure

when

Unshielded

probe.

probes.

reporting

measurements

serve are

as a check used

on

during

turbine-inlet value

the

turbine total

that

turbine

performance

of mass

flow

rate,

is

presoften is a static 359

TURBINE

DESIGN

pressure, tion,

and

with

AND

total

the

APPLICATION

temperature

flow

angle

and

obtained

a assumed

by

the

to be zero

[-i,

at

following

equa-

turbine

inlet:

the

V &s%jT'"-"

(12-2)

where p'

total

P

static

pressure,

N/m2;

5'

ratio of specific heat constant volume

g

conversion

Aan

annulus

R

gas

T'

total

pressure,

lb/ft 2 at

constant, area,

1; 32.17

J/(kg)

K;

measured

This

calculated

more

rel)resentative

l)ressure

to

specific

(lbm)(ft)/(lbf)(sec

(K) ; (ft) (lbf)/(lbm)

temperature, angle

constant

heat

at

2)

m2; ft 2

constant,

flow

lb/ft 2

N/mg;

value

°R from

axial

direction,

of turbine-inlet of the

(°R)

true

(teg

total

aver'_ge

pressure

value

is thought

than

is the

to be

experimental

value. are

Inlet temperature.---Two located at the turbine

These

rakes,

contain

areas.

The

to determine and

the

Provisions

are

all the

the

wall

of

made

the

operation,

where

a constant

matic

regulation

Measurement for

the

was

latter

to the

to heat research

encountered

testing

conventional good shown 170.

360

length

accuracy This

inlet

modified

being

Reynolds

12-6(b) probe

shown

of

small

has gives

in

for

an

average

burner

in

Auto-

tmrl)ose. can did

example

present not

exist

herein,

but

as discussed

in ref-

wire must be exposed error negligible. The 12-6(a)

number.

The

has

a ratio

is inadequate

modified

length-to-diameter results.

this

number

turbines

figure

the

the

problem

12, whictl

excellent

12-6(a) area-mean

is maintained.

This as

radii

turbine.

as for with

of about a wire

the

a radial-inflow a.s well

Reynolds

used

12-6(a),

center

in figure

center,

is I)rovided effects.

shown

area

testing

at low

some

in figure

temI)erature

burner

to diameter

at low

in figure

duct

shown ,_1 the

the

of bare thermocouple make the conduction

thermocouple

wire

at

facilita.tes

turbine

in

rake

readings

conduction

erence 2. A large amount to the flow in order to exposed

outlet

of temperature

due large

particular

for individual The

a problem

to that situated

temperatures

readings.

fuel

siinilar

of thermocouples

annular

used

radius, of

are of a type

a number

of equal was

which

thermoeouple rakes, spaced 180 ° apart, inlet measuring station (see fig. 12-4).

of for

thermocouple ratio

of about

EXPERIMENTAL

DIhTERMINATION

OF

AERODYNAMIC

PERFORMANCE

! / /

/

/

I /

/

/

/

/ /

Thermocoul)le junction

Therrnocouplejunction

(a) (a)

Stator

stator probe

probe

is the

and with

blockage

testing,

the

rotor

Although

circumferential conventional

on the

rake

previously

testing,

is installed.

radial

associated

turbine

outlet

performance

survey outlet

type. 12-6.--Thermocouple

outlet.--During

at the stator

Conventional FIGURE

(b)

value

the

only

mentioned

wedge

probes,

measured,

also

For

a total-pressure to obtain

surveys, especially make

made

pressure.

and

it is desirable

type.

measurement static

is removed,

static-pressure being

(b) Modified configurations.

such

stator-

the

problems

the

effect

of

measurements

361

TURBINE

DE,SIGN

FIGURE

unreliable. from probe flow

angle.

The actuator driven

Note

angle

that

the

previously

the

test

hub

has

to be

sensing the

equipment

is shown of the

figure

does

probe regarding

not

This

have

the

same

that

wa_

average these

outer

in figure and

circumferential

effects

total-

12-7. the

and

probe,

(fig. 12-7) the

The

elements; inner

movement for

taps.

determined two

facility.

by interpolation

wall

survey

shown

Considerations

tip

test

in figure

at both

provides

in

is shown

for radial saddle

in this

and

previously probe

installed

are obtained

facility

measurements

provides shown

362

an

probe

pressures

at

in the

total-pressure

as the facility.

at

to obtain

probe

static

measured used

outer-wall

survey

required

values probe

is fixed

required

APPLICATION

12-7.--Total-pressure

The

the

pressure

AND

are walls.

12-8. the

movement. stem used

of stem

The

motorThe

configuration in the

blockage,

example sensing

•EXPERIMENTAL

DI_TERMINATION

OF

AERODYNAMIC

PERFORMANCE

r Outer wall saddleassembly

r Actuator t /

cm 0

5

LLLLU

C-06-2250 FIGURE 12-8.--Total-pressure

element diameter, edge are discussed Although example use

not

and measurement in reference 3. used

herein,

for

velocimeter,

velocity probes.

measurements

shown

in figure

12-9,

angle

the

of five

number

tional design,

with

the

side

tubes

and

are exposed

and the making connected

use

the

total

A laser allows

type

of sensing

pressure, Figure

general,

head

temperature,

are distributed circumferentially is located at the area center In

the

of flow-disturbing

probes probe areas.

exit.

4,

turbine

angle

probe center

direction.

into

12-4

turbine

shows

size

that at radius

influences

permissible.

of flow

self-balancing

with

discussion come

reference the

trailing

at the

annular

of probes

Measurement

without

as the

and

in

blade

recently

magnitude

of measuring

the

serving have

described

probes,

capable

are located

equal

facility

to be made

five of these combination measuring station 2. Each of one

as

outlet.--Combination

and flow

behind

techniques

velocity

such

equipment.

distance

particular

laser

measuring

Doppler

Turbine

in the

optical

directly

survey

tube

are symmetrically to a pressure

is accomplished

system.

The

used

to measure

located that

with ranges

by

probe

means

shown total

respect between

of a convenis of

pressure. to

the

the

a 3-tube The

center

total

two tube

pressure

static pressure. The openings in the side tubes are in planes an angle of 45 ° with the center tube. These side tubes are to

the

two

sides

of a diaphragm

in a balance

capsule.

A 363

TURBINE

I)E,SIGN

AND

FIGURE

differential so that tubes the

error

is generated

Exit

probe

not

used

are

than

does

true

if the

in flow

measurement occur

a self-balancing mined

is usually pressure 364

found

that more

the

integrated

range exit

operates

order

to

for

check of

for

and have

specific

use work

is especially

large

variations

conditions.

wouhl

gross

and

specific

This

position

the

are

efficiency,

of torque

values

temperature.

passage

reduce

measuremertts

of operating

in order

to

side

flow.

measurement

total

in the

of turbine

reliable

is in a fixed

diaphragm,

pressures

the

in

the

Even

to be work

with

surveyed

to be

deter-

of ideal

work,

accurately.

Exit is small,

into

values

of exit

probe,

results

probe

determirtation

the

by

the

total-pressure

probe over

when

and

been

temperature

is actuated

primary

provides

head.

A servo-system

the

to determine

(12-1)

angle

the

for the

It has

equation

capsule

unequal.

pointing

used

discrepancies.

and

are

by

total-temperature

they

of

signal

to zero

generally

sensing

in the

of the

but

12-9.--Combination-probe

transformer an

error

KPPLIGATI,0N

total

pressure,

which

determined use than

from

of equation does

direct

is used equation

(12-2)

yields

measurement.

for

the

(12-2). more

calculation When reliable

When

the

exit

values the

exit

flow

angle

of exit

total

flow

angle

is

EXPERIMENTAL

large,

the

become

DETERMINATION

radial

variations

large.

In

this

case,

(12-2),

ment

pressure

total

unless the

Flows on

pipe.

The

orifice

are

usually

the

pressure

primary

is not

placed

secondary pressure and on

in

the

recording

the All

the

computation

Each

of these

selection

obtained measure-

meters

rate

of flow:

fluid

meters

have

defluid

nozzle,

is flowing.

or The

or an intricate

has

certain

particular

advantages

meter

particular the

which in the

as a venturi,

the

of the

head

W=

be

meters,

manometer

of any

constraints

of the

passage

and

a constriction

such

which

and

variable

by

U-tube

the

exit

can

calculation

variable-head

caused

through

and

requirements of these

with

be a simple

device.

disadvantages,

values

between

is a restriction

pipe

may

in the

clear-cut.

differential

element

angle

PERFORMANCE

Measurement

measured element

and

integrated

choice

Mass-Flow

pend

AERODYNAMIC

in pressure

for use in equation of exit

OF

depends

application.

same

basic

equation

for

AtMCEY_/2gp.,(p.,--pt)

(12-3)

where A,

flow

area

of meter

throat,

m2; ft _

M

approach

velocity

C

discharge

coefficient

E

thermal

Y

compressibility

p_n

density

p_.

static

pressure

at meter

inlet,

Pt

static

pressure

at meter

throat,

The

approach

factor

expansion

factor factor

at meter

inlet,

velocity

kg/m3;

factor

lb/ft 3 N/m2;

lb/ft _

N/mS;

lb/ft 2

is 1

M:-

(12-4) 1

where The

D is diameter,

coefficient

actual

flow

rate

and

ferent

for

each

type

the

throat

area

C accounts

for

theoretical

of

dis<.harge coefficients bration of the meter expansion

4

in m or ft.

discharge

thermal

D,

meter.

flow

the

rate

Although

difference and

good

is significantly approximations

can be made from published data, should be made to assure accurate factor

is usually

E

accounts

determined

room temperature, which usually fluid flowing through the meter.

for

from is not

the

fact

measurements

equal

between

to the

a direct results.

that

the obtained

temperature

the diffor caliThe meter at of the

365

TURBINE

DESIGN

The

AND

APPLICATION

compressibility

factor

for nozzles

and

venturi

meters

is

(12-5) The

derivation

function

of equation

of pJpt_

and

concentric

orifices

determined

from

(12-5),

D#D,,

with

is presented

having the

along

the

empirical

equation

Ventur4, verging

which

section,

tubes

are

fuser the

tube.--Figure tube,

a cylindrical

usually

section object

12-10 consists

cast

usually

shows

important entrance

is made

section,

with

of accomplishing

and

machined

features

of

section, section.

The

The

of about

dif-

7 ° with energy venturipressure

in that

inlet

and

it is bulky,

reproducibility), a long,

the

difficult more

straight

throat.

run

The

venturi

to construct

expensive

than

tube

(particularly other

head

has

a

a con-

of kinetic

the

recovery

5:

(12-6)

surfaces.

angle

be

while minimizing friction loss. The total pressure loss from the tube inlet to exit is from 10 to 20 percent of the differential between

a maximum

most

can

reference

a diffuser

internal

an included

Y

')

the

have

of

from

of a cylindrical throat

and

Y as a

5. For

value

y --_ l -- [ O.41q- O.3 5 ( _-_--_/. )' ] ! P2_-_/

venturi

showing

in reference

(Ap/p_,)<0.3,

following

curves

disadvantages so as to provide

meters,

and

requires

of piping.

F low

"v_"

FIGURE

Flow

nozzle.--Figure

flow

nozzle.

tube

without

with

the

366

The the

venturi

flow

Pressure

12-10.--Venturi

12-11 nozzle

diffuser is thus

shows

tube.

the

approaches,

section. lost,

taps

and

The the

shape to some

high nozzle

pressure has

of a commonly extent, recovery a pressure

the

used venturi obtained

loss

of 30

EXPERIMENTAL

DETERMINATION

OF

Static-pressure

laps

AERODYNAMIC

PERFORMANCE

,

Flow Din

Dl

I

i

J

\,\

/// :: ii\\ FIGURE

percent

or more,

between approach 0.98.

the

depending

inlet

factor

M

and

on

the

and

streamlines

on the

0.65. vena

upstream

published cording

to make,

coefficients

ditions

is possible.

length

of the

The

cylindrical

of the flow

coefficient

the and

of the

differential

(7) for

be a high

upstream portion

(product

a nozzle

of

is about

is probably the most of the inward flow of

plate,

the

minimum

stream

orifice edge. This minimum area is it is at this area that the minimum

has a pressure flow coefficient

in most

pressure

coefficient

loss somewhat for an orifice

to the effective minimum at the orifice itself.

may

to specifications,

ratio, The

side

The orifice nozzle. The

This low value is due contracta rather than

It is possible

nozzle.

orifice (fig. 12-12) head meters. Because

area occurs downstream from known as the vena contracta, pressure is obtained. than that for a flow

area

throat.

discharge

Orifice.--The sharp-edged widely used of the various the

12-11.--Flow

machine

used.

edge must

being

shops,

an orifice

hole

is carefully

If the

degree

area

of reproducibility must not

be

exceed

sharp,

greater is about

with

which

made

ac-

of flow

con-

and

5 percent

at the

the of the

axial in367

TURBINE

DESIGN

AND

APPL]:CATION

L f low Din

Dt

- Static pressure FIGURE

side diameter more orifices the

orifice

12-12.--Sharp-edged

of the pipe. Because which will have the

is extensively

In

turbine-component by

the

performance. torque

dynamometer This

heat,

serves turbine,

converts useful

and

(1)

(3) electric methods other

than

brakes,

as the

generators

to

since

the

dynamometers

fluid

in figure

generally

used

in

speed

controller. in

as brakes;

dynamometer.--These is almost

12-13,

which

dynamometer

where

provides

(2) and

absorption into

this

for

section

types

the

include

electromagnetic

and

heat.

it generally

a load

(4) airbrakes.

force

turbine

the

turbine

discussed

dynamometer

absorption

is shown

be

the

surroundings,

brakes;

used

by

torque turbine

to determine Simply,

dynamometer turbine

of

in evaluating

used

supplied

to the

The

fluid-friction,

for measuring

Hydraulic

368

is dissipated

or

measurement

commonly

energy

dynamometers

hydraulic,

accurate consideration

dynamometers.

the

purpose.

this is used

Absorption

mos_

absorption

in turn,

no

an

is of prime

devices

cradled

it is possible to construct two or same coefficients when calibrated,

Measurement

tests,

turbine

The

are

orifice.

used. Torque

produced

taps

brakes; In

addition,

of torque

meters

are discussed. units always shows

are water. clearly

installations.

frequently A typical the The

cradle shaft

called

water

water

brake

mounting is

coupled

EXPERIMENTAL

DETERMINATION

OF

AE:RODYNAMIC

PERFORMANCE

Disk

Water in let -_ Shaft bearing\

/- Water in let

Packing gland ,, \

,/

/- Packing gland Shaft / /- bearing

Shaft

Pedestal bearing

FIGVRE

12-13.--Hydraulic

directly

to the

housing

through on the

it is free

housing

tends

housing

permits

absorbed

may periphery are

the or

is increased;

pacity

of the to

housing

rotate,

of Murray

the the

it.

A scale

of the

As

torque

rotates,

to

arm

moment,

fluid

bearings

shaft an

Co.)

to the

and

pedestal

the

attached

turning

Works

glands,

in the

limits.

Iron

developed

packing

is supported

_thin

with

disk

of the

closed

and

transmits

bearings,

determination

fed into

disk

The to

(Courtesy

and

the on

the

the

power

be computed.

valves

cient

disk.

and

shaft

to rotate

to the

the

turbine the

so that

Water

dynamometer.

test

friction

depth

Pedestal bearing

the

prevent

disk,

the

this

and

The the

where

inlet results

housing,

brake.

compartment

it forms

valves in greater

of water of

steam

by

a ring.

opened

frictional

the the

resistance

increased circulated at

centrifugal As

further,

a consequently

amount

formation

is thrown

any

force

discharge water

between

absorption should point,

ring ca-

be suffisince

such 369

TURBINE

DE,SIGN

action

would

level, the

the

AND

cause

cube

of the

outer

can

increase

Care

must

periphery.

supported

ends ing

on

device

vicinity

the

pedestal

the being

of these

the stator.

with and

desirable

the

the

erosion

Another that

12-14.

In

so that

stator

and

a small lines

teeth,

rotor.

this any

On

so that of force

can to

using

occur,

increase

more

the

torque

than

the

enter

stator

may

be

is

trans-

the scale. The in the starer in with

rotor

are

teeth

them

and

the

the

rotor

rotor

caused

is

by

energized

as the are

disk

device,

between

of force

and

dynamometer

when

air gap

near

friction,

way

is, by

arm and measured which is supported which,

disk

of magnitude.

of the

eddy-current

a coil,

lines

in the

increase

holes.

bearings

The

of the

tested,

particularly

as an order

staging;

of figure

to produce ends

a_

of torque

through-holes

as much

is by

carries

magnetizes of the

water almost

all fluid-friction

them

because

of the torque on the shaft,

starer

machined

constant varies

increase of

to further by

however,

views

The

through

tends

two

by means is mounted

face

This

with

dynamometer.--The

in the

current,

provided

capability

Eddy.current

(i.e.,

makes

a

brake

is typical

and

absorption

in the

power absorption one disk.

bearings.

versa)

are

be exercised,

particularly

With this

engines.

brakes power

of

characteristic

vice

nongoverned water

shown

This

and

dynamometers

for testing the

unloading.

capacity

speed.

of speed,

electrical-type

mitted rotor

momentary

power-absorbing

increase

Some

APPLICATION

with oppos-

principally

is moved

to sweep

direct

by

the

through

the

/- Stator I I

itator

Water passages-,,

c_:Q(_-_I /_

Shaft

___

bearing-_

_k_--7-_:_-i'---'_;

/'//

.... 'K

OCilile /-Shaft

', /

-:--_:

'

It

Scale

L ],

o

i

,lorquearmI

FIGURE

12-14.--Eddy-current

dynamometer. mometer

370

and

Engineering

(Courtesy Co.)

of

Mid-West

Dyna-

EXPERIMENTAL

iron

of the

stator force

stator.

causes sweeping

The

current

dynamometer

cylindrical stator

coolant

connected dc

cradled, to measure

provides

a compressor)

(a turbine

and

shaft,

windage

and

torque in order

the

added

they

power

can

absorber It

about

19 kW

mometer

that

is used (25

an

inlet

airbrake which

in

an it

direction

of

momentum power

from

straighteners

The

casing

and

axial

the

After and

the

of speed. For

is cradled loads.

diiection,

the

The

mover unit

acts

bearing, the

frictional

turbine

tests

smaller

turbines,

percentage

of

turbines, torque

the

these output,

the

air

and,

leaving

on the

thereby, the

iotor

arm

the

is attached

to

removes

entry of the stator, to the

research

passes

torque

wheel

tangential turbine

through

in an axial

to the

which

the

opposite

the

air

airbrake

is equal

bearings,

through

rotor the

of a throttle

to ensure axial inlet collector

absorbs

rotor,

from

air

The

than dyna-

a paddle

a direction

of

Research (less

consists

the

type

airbrake

either

is accelerated

rotor.

Lewis turbines

of the

with

in

is a

NASA

airbrake

a rotor

it

of

A torque

be

can

a pump

This the

the

small view

momentum

in

It

driving

dynamometer

testing

12-15.

is discharged

torque

rotor,

appreciable

at

for

a stator,

tangential

rotation

output.

Therefore,

developed

in figure

axial

or

as a generator.

during

blading, and flow straighteners of the air. After the air enters

gives

frame

to determine

turbine

torque.

A cross-sectional

collector,

or airfoil type and discharge

the

airbrake

was

hp)).

capability.

the

acts

be

be neglected.

extensively

is shown

valve,

often

in the

of a prime

unit

a

can

(testing

the larger jet-engine type when compared to the total

dynamometer.hThe

Center.

the

an

has

passages

it is driving,

measured

represent

type

with

output

as a function

turbine

wet-gap

absorption

researcher

removing

true

may

type, eddy-

a device

the

energy,

torque

a dry-gap

dynamometer.

When

the

torque

turbine power. For are generally small hence,

absorb

stator.

A wet-gap

from

power

the

The lines of this energy

dynamometers

to drive

to the the

losses

Airbrake

ment

by

flows

versatile

engine).

losses

to obtain

friction

and

as to

permits

measuring

is then

total losses

most

it is absorbing

capability

The

and

in the

motor-generator,

required

as well

when

driving

seal,

the

power

used.

required

electric

or a reciprocating

as a motor; The

dc

rotor.

Eddy-current

any

rotor. therein;

is called the

water

rotor

passages

12-14

contact

the

PERFORMANCE

the

with the currents

commonly

rotor.

to give

the

not

and

the

dynaraometer.--A

used

the

rotor,

in series

between

tluough

in figure does

is also

onto

stator or

water

iron-core directly

flowing

shown

the

attraction

water

dynamometer

AE.RODYNAMIC

to try to turn the stator induce

to cooling

because

OF

magnetic

the stator through

is dissipated The

DETERMINATION

direction.

on the

are

designed

the

casing

for for

flow

casing. radial

measure-

of torque. 371

TURBINE

DE,SIGN

AND

APPL£CATION

Thrust Journal Flow tubes_,,I\

il

-_-

t_-

I '

1,',-I Y_::_._:.._.__--"--?i

i_:_-

valve J

air

bearings

bearings-r-...

....

Throttle

air

_-z-.-X.Jl Inlet collector

1

,

-/

straighteners

x

i

[ \

/_

stator

i

_ _t_tteor r

I

. -J _.J

CD-10167-09 FIGURE

It

can

be noted

sections,

an

for

blade

each

extremely a stator

from

outer

momentum other

case,

the be

driving the

stator

opposite rotor used

capability,

Measurement Such

can

a scale

absorber forces relatively The readily and 372

torque acting

the

are

vertically,

for

regardless

small. of the

observable scales

may

not

of the

have

the

rotor. the

in this

aiIbrake

(It dynamometer,

This permits

losses. dynamometer by

Because

of

these

should

magnitude are

force

a spring-balance

displacements

of the that force

scale. the

scales

and

telescopes proper

rotor

For

turbine.

scale

(low-power

as the

the

the

impart

momentum Thus,

arm

of a spring-balance

can

design.

simply

torque

example,

to drive

frictional

the

row

of rotation

force.--The

because very

or

seal

most

the

blade

two

to measure For

tangential

type

of

valving

adapted

of rotation

case

and

used

arm

direction

power

consists

independent

pressures.

one

imparts

dynamometer be

horizontal,

available

turbine

inlet

where

direction

be obtained

disadvantages remotely

row

stator

is well

low

used same

as was of

can

system at

blade

of bearing

measurement

with

paddle-wheel

absorb

measurement

the

row,

in the to the

is of the to

that

blade

stator be

dynamometer.

12-15

outputs can

while

a direction

inner

This

turbine

tangential

can

an

configuration the

figure

and

row.

small

12-15.--Airbrake

indicate

does force

the have range.

power remain

involved.

reading been

is not used),

EXPERIMENTAL

DETERMINATION

Hydrostatic (e.g.,

air)

devices as the

of operation with

the

the

fluid,

produce

be

Most

particularly it

is

sorber,

signal

which

operates

strain

are

on

fine

the

through

the

The

the

a gas

principle

on the

greater

the

fluid.

fluid,

force

This

or force

has

on

fluid

required

to

slip rings, where this

type

occurrence with

system

torque

torquemeter

optical

projects each

separated

by

illuminated

reflecting

Experience

has

bearings surfaces.

kept

and/or

facilitate the

twist

intensity accurate

load.

the these

gearbox

strain-gage under

twist

highly

because

A higher

possibly and

be

difficult,

the

tile null-balance twist. that

reflecting

The

by successive onto

surfaces maintain usually

such The

must have

as

optical

accuracy. close

tends

to the to cloud

a laser

design

provide been

position

of the

to

systems

illumrepositions

are

torquemeters

photocells

Photocell

source,

end.

optical

unbalanced

oil mist

of

reflection

two

thereupon

readings.

the

at each

surfaces

light

and consists

polished and

equip-

measurements.

shaft.

produces

a

encountered

surfaces

of the shaft,

nearly

transmitted

life

the

condition.

bearings torque

Both

are

is

absorber.

basically

of a slit, on

Shaft

power

brush

A servomechanism

indicated

must be

twist

surface

is very

with

box.

gage

electronic

short

reflecting

the

image

gap.

to restore of shaft

fiat

surfaces

two photocells.

the photocells is a measure

torquemeter

and

shaft strain

Problems

torquemeter

measures

a hairline

on the

may

parallel unit

the

of the

optical

the

Readings

interfere

ab-

gear

and

to appropriate

that

the

torquemeter,

resistance

indicated.

for

wherein

of

wire

and

include

voltages

polished

torque

its

and

arise

strain-gage

applied. is

required

capability

A bonded

that

brushes,

as

of an intermediate

turbine

is

component

may

the

shaft

used. the

onto of

use

that be

strain

torquemeter.--An

A stationary

the

can signal

recording.

situation

than

a high-speed,

shaft

This

equipment

The

is higher

property

the

of induced

Optical a shaft

the

the

an

electronics,

voltmeter. data

calibrated provide

in turbomachinery

between

the

with devices

appropriate

a digital

cradle

can

of

equipped These

digital

principle

shaft,

function

ment,

with

necessitate

on

wire

unique

are

torque. on

problem,

proportional,

mounted

facilities

to

speed

this

torsional

load

measurement.

rotative would

optical

on

of the

which,

impractical

which

these

pressure

for automatic

circumvent

may

The

the

readings

torque

turbine

turbine

force.

or

is impressed

space.

torquemeter.--Sometimes,

conventional

This

mercury)

to measure

terms

test

suitable

testing,

ination

in

to measure

torque

Strain-gage

from

becomes

PERFORMANCE

(e.g.,

to be measured

cells

output direct

with

used

turbine

load

provide

The

a liquid

in a confined

calibrated

current

electrical

To

AERODYNAMIC

it.

strain-gage

the

been

force

held

greater

can

either

have

the

being

the

pressure

fluid

is that fluid

with

OF

beam,

of both for

operated

the

adequate in con373

TURBINE

DE,SIGN

junction

with

types

of

could

be

AND

APPL]_CATION

dynamometers,

torquemeters used

for

with that

turbine

good

are

One

of the

simplest

is that

of rotative

to give

a continuous

manent shaft

the

the

voltage

usually

and

accurate The

output

of the

in

tachometer

can

Adc

a rotating

generator

generator,

armature,

is to be measured.

indicator

measurements

electric

of speed.

and

of which

remote

most

speed.

field

speed

are

other

that

also

Measurement

indication

magnetic

There available

testing.

Rotative-Speed

testing

correlation.

commercially

is

Since

the

with

a per-

driven

by

the

field is constant,

is proportional

is a voltmeter

turbine be used

to its

graduated

to

speed.

read

The

rotative

speed. For

greater

revolution and

accuracy

counter

should

disengaging

called

in speed be used.

they

yield

are

an

with

is provided

a timer.

available.

average

a positively

A means

it simultaneously

chronotachometers,

because

measurements,

These

rotative

for engaging

Commercial

units

speed

driven

are

units,

advantageous

for a given

time

(usually

1 minute). The the

currently

most

accepted

use of an electromagnetic

larly

suited

sprocket An

for

with

electronic

given

time

Rotative supply

there to

number accurately

displays

speed

the

count

Since

faster.

accelerations

control

system.

It

provide

greater

accuracy

by one

performance

means

The

absorber

and

decelerations when

the

conditions contours

of speed

of equivalent inlet

conditions

of equivalent 374

maps. flow

and

of efficiency. conditions

the

a

air

power

pressure

to correct

are tends

for this,

the

and

accuracy

a steady

of

air supply

data.

and

of turbines Such

flow,

so that are

the and

discussed

usually

presented

map

as functions

of the

ratio. work,

are

a performance

work

pressure

The

of temperature conditions

for

PERFORMANCE

characteristics turbine

therefore,

to have

taking

shaft.

when

in supply within

a

speed.

and,

tends

ideal

turbine

tests

rate

method,

(or impulses)

turbine

an increase

is, therefore,

of performance

figure,

flow

this

to the

teeth

is through

It is particu-

as rotative

during

mass

TURBINE The

the

directly

to pressure,

For

is secured

counts

speed

counter.

machines.

somewhat

varies.

turbine

result

pulse

of teeth

the

varies

proportional

to drive the

a given

pressure

directly

of measuring

high-rotative-speed

pickup and

method or electronic

Also and

map

shown

speed can

pressure.

on

are

The

concept 2 (vol.

on

operating

the

shown

be readily

in chapter

shows, map

are

in terms

used and 1).

for

any

nature

EXPERIMENTAL

In

brief

DETERMINATION

review,

the

OF

equivalent

AERODYNAMIC

conditions

w.q=w

PERFORMANCE

are

--_ e

(12-7)

_h'

ah;q=

0

(12-8)

and N

Ne,= where rection

the subscript eq refers factors are defined as

to

o=(

(12-9)

the

equivalent

condition.

The

V_,,o ) _ \Vc_. ,,a/

cor-

(12-10)

_=p,'0 Ps,a

(12-11)

and ( e--%'a

2

"_%,_/(,.ta-_)

\_-_-t_+ 1/

(12-12)

( 2 y,(,-,, where

the

square

of the

critical

velocity

V_r=

The

subscript

pressure

An

specific

flow

and

the out

follow,

there

variations

may

ratio Also,

in this

performance

a

great

the speed.

or no

speed.

deal a better

if some as functions

curves

type

to be

of

are

included

of pressure

be

of the

performance ratio in the

mass

flow

to

rate

With

ratio

(total

presented

on

the

for completeness. can

understanding

presented

are

Equiva-

discussion

pressure

speed

information

of the

mass

of constant

or

conveniently

in the

in the

K

equivalent

product

shown

of

(1.4).

12-16.

of the

This be

constant

ratio

in figure

variation

Lines and

conditions (288.2

heat

product

as will

air

temperature specific

is presented

rotative

independently of this

and

of efficiency

map,

obtained

psia),

(29.0),

because,

case)

sea-level

14.696

against

be little

contours

Although be

data

(12-13)

standard

map

is plotted

in rotative

pressure

_ or

equivalent the

2----Z-_ gRT' 3,-t-1

the

weight

performance work

spreads

to

N/cm

molecular

example

lent

can

refers

(10.133

518.7 ° R),

map.

std

Vc, is

obtained turbine

parameters for

a range

following

from

the

performance are

plotted

of speed. sections

The

are not 375

TURBINE

DESIGN

AND

APPLICATION

Percent of design equivalent speed, N/%/B Ratio of inlet-total to exit-total pressure, pb/@ Efficiency

t

Design equivalent speedand work output

90xlO3 39

IO0

37

85

35

105 084.4

_- Y 3.6 / _;3.4 _-:_-_1 4.0 i-/3. 2

80

_ 5-<-t-7-h-q701:?_'_S:_937 _5_ _V

31 70-

3. o z.8

29 33

i

6oh

O

25

li

i

-

i

7.6

--. d-- Yd- -t--A=z_

2.4

-4

2.z

I

_._;c.;-t-'.-i'l

55[ _'

f/

--

23 .SO

21

l,'J-5" 7..I.. k Ob / 4 / ._'_ / ,_..-'T

"_

85

_' " / ?1 /"7 _ i" F;.. ,. ;]¢.. >.4:-I-

42.o

_'-.. 70;_.--80-U7 -1/-

"E 19

2

-

P x...t[../,c,(-/_2_._-'1.. /'IV-" /I/--1--/7

40

17

/

i 1.8

I..i.I

u -C,q'-_/- e,--7 ,

15

"i"l

13

30

11

2=. 3

LI -'>

;j-1

___L

7

I

9 4 5 6 7 8 Product of equivalent mass flow and equivalent rotative speed, _wN/6, (kg)iradl/sec 2

lOxlO3

_8

211xl04

9I

I 10

I 11

_ 12

I 13

l 14

_L.... 16 I 15

I 17

i 19

I 18

I 20

Product of equivalent mass flow and equivalent rotative speed, ¢wN/8, (Ibl(rpm)/sec

FIGURE

for

the

same

selected

turbine

to illustrate

12-16.--Turbine

whose

map

certain

performance

is shown

are

shown

ated

with

in mass in figures two

stator

having

figure

12-18

376

flow 12-17

different a large

was

rate and stators.

stagger

obtained

in figure

12-16.

but

were

points. Mass

Variations

map.

with

Flow

with

turbine

12-18

for

Figure angle a stator

pressure

ratio

a single-stage 12-17

(small

was

speed

turbine obtained

stator-throat

having

and

a small

operwith

area), stagger

a and

angle

EXPERIMENTAL

(large

DETERMINATION

stator-throat

identical, In

and

both

rate

area).

the

figures

increases

as

value is reached. increase in mass that

either

mum this

value

the

rate

with In

12-17

would

32

for

the

speed,

by

]arge

choked

speed; maximum indicates the

maxi-

rotational speed. in the rotor inlet

In some

losses

which rotor,

maxi-

causes

cases,

This rela-

however,

a decrease

the

in maxi-

speed.

discussion

a multistage indicate

speed.

the

speed,

of the

the

rotational

12-18,

rotational

incidence

flow

maximum

area,

the

In figure

case

decreasing

mass

some

stator-throat

increases with decreasing and is due to an increase

decreasing

were

choked.

small

the

the

until

is unaffected by

blades

in pressure ratio produces no this maximum in mass flow is

is choked.

For

stator

a given

increases

has

the

rate

the

has

been

turbine,

the

a first-stage

for

the

flow

stator

case

of

variation choke.

a

single-stage

shown A flow

in figure

variation

of

14.5_

31 30

3

rotor

stator

with

foregoing

turbine.

that ratio

is for

flow the

of very

flow

The

which

is choked.

pressure

occurrence mum

or the

flow rate behavior

total

cases,

PERFORMANCE

used.

seen

is influenced

rotor

both

was

pressure

stator

that

mum mass is the usual tive

be

of mass

flow

that

it can

12-17,

indicates

mass

rotor

AERODYNAMIC

A further increase flow. The reason for

the

In figure

For

same the

OF

_

i

E

_:

!

•_ 24

_

14.0_ 13.5_

_

f_

.%_

.... _---__

_spee_

12.o}- /F

o

4o

II.5-

o

60

I0.5

o

II0 I l_i

._

23!

l I, __

22 t 21

t

i

1.4

1.6

I

_

i

I_

1.8 2.0

2.2

2.4

I

1

I

_

3.0

3.2

3.4

3.6

i0.0!

9.5 ' 1.0 1.2

2.6

2.8

3.8

Ratio o1'inlet-total to exit-total pressure, pb/p_ FIGURE

12-17.--Variation of for turbine

equivalent with small

mass flow stator-throat

with area.

total-pressure

ratio

377

TURBINE

I_EISIGN

AND

&PPLZCATION

49 48 47

45

=

44

"E 43 "5

42 41

4O 39 17.5L

1.2

1.3

I

1.4

I

1.5

I

1.6

t

1.7

i

1.8

I

1.9

I

2.0

I

2.1

I

2.2

2.3

Ratio of inlet-total to exit-total pressure, p_/p_ FIGURE

12-18.--Variation turbine

the

type

shown

downstream exactly between

row

stator

at

378

figure

row

row

3.7. flow

the

the

static

about

3.2.

second

rotor

then

It is, of course, the rate for the turbine.

by

For As

ratio static

while

the in

the

particular the

turbine

chokes first

each

the

constant first

are

pressure for

pressure

is indicated

of

some

determine

shown

turbine

to decrease.

for

measurements Such data

is

occurs

in

To

in hub

remaining

choking

ratio

stator.

speed)

blade

continues

to increase,

ratio of about the maximum

variation As

12-19,

a

ratio

choking

static-pressure to be obtained.

turbine. blade

pressure

or

constunt

given

that

with total-pressure area.

indicate

rotor

the

(at

pressure in

a turbine

continues

pressure tablishes

any

of

static

illustrated

would a

where

ratio

in

upstream

downstream

either

of a two-stage

choking

pressure

12-18

12-19,

pressure exit

mass flow stator-throat

choking occurred, rows would have

in figure

turbine

increases,

ratio

row,

where this the blade

blade

case

figure

blade

illustrated with

in

of equivalent with large

at choke

second pressure

a turbine that

es-

EXPERIMENTAL

DETERMINATION

OF

AERODYNAMIC

PERFORMA.__TCE

First stator exit

First rotor exit Second stator chokes Second rotor chokes I Second stator exit

I I

I

I

3.0

3.5

.i 2.0

1.5

2.5

I I I

Second rotor exit

i

l

l

4.0

4.5

Ratio of inlet-total to exit-total pressure, pb/p_ FIGURE

12-19.--Effect

of turbine total-pressure ratio two-stage tu_rbine.

on hub

static

pressure

in a

Torque As

indicated

should

vary

exit

for

any

12-20.

creases

the

AVu

(absolute) with

constant

For torque

tuining

due

in the

rotor.

rotor

increases)

flow

speed,

(exit

ratio

pressure to flow decrease

with

angle

pressure and the

becomes flow

varies

higher

the

in

ratio

in-

values turning

torque in

in inlet

is shown

increased

in mass

change

torque

speed

rate

ratio,

torque

rotor

and

a decrease

the

the

between

tbe

flow

1),

in which and

velocities

is due

absolute

and

(hV_)

manner

mass

high

a possible

rate

2 (vol.

increasing

a given

This

and

The

pressure

the At

speed.

chapter

velocity

to a higher

from

in the

mass

radius.

turbine a given

resulting

increasing

as speed

the

of

of absolute

with

figure

(2-9)

with

component

experimentally

of

equation

directly

tangential and

by

decreases amount

more

of

positive

rate. 379

TURBINE

I_E,SIGN

4000 --

AND

APPLICATI,0N

5500 50O0 /- Design point

4500 32OO ,

E

2800

,.o" 2400 --

g

4OOO

z

Percent of

_o- 3500 C

design equivalent

_" 3000 E o

_ __

1500

g

1290

20 40

0

6o

1500'

A _7 I_.

--

1000'

,4 17

100 110

--

500'

I>

120 130

2500

_ •_ _

--

800 4OO

speed 0 []

o-

o 1.4

1.8

2.2

I

I

I

I

I

2.6

3.0

3.4

3.8

4.2

Ratio of inlet-total FIGURE

12-20.--Variation

of

I 4.6

to exit-static pressure,

equivalent

torque

with

I

I

I

5.0

5.4

5.8

70 80 90

p_IP2

turbine

pressure

ratio

and

speed.

Figure speed,

12-20 the

shows

torque

that

tends

as the

pressure

ratio

off

reach

to level

Above this limit, additional torque.

any further increase This phenomenon

and

indicated

on

ratio

converging

is

pressure specific

work

for each

approached

but

the

area

annulus

axial

Mach

The

flow data.

performance map ratio

efficiency

the

to

not

map

yield

12-16,

reached.

turbine

by

exit

for

value.

ratio results in no "limiting loading"

the

lines

of

value limiting

Limiting is choked;

a given

a maximum

a maximuin

In figure

been

increases

in pressure is termed

performance

speed.

at the

of

constant

equivalent

loading

loading that

is being

occurs

is, when

when the

exit

is unity. and

torque

These

map.

is to select for

has

number

mass

measured

a

and

The

the

are then

usual

mass

various

curves

curves

flow

speeds.

calculated,

just

are then procedure and

torque

Specific and

the

discussed used

can

be plotted

to construct

in constructing at even work,

from turbine

a performance

increments

ideal

performance

the

of pressure

specific map

work,

and

can

be drawn.

of presenting

turbine

Efficiency Another 380

convenient

and

widely

used

method

EXPERIMENTAL

performance _, which

DETERMINATION

is to plot is given

by

OF

efficiency

the

AERODYNAMIC

as a function

PERFORMANCE

of blade-jet

speed

ratio

equation U (12-14) _/2gJ,_h_

where U

blade

mean-section

speed,

m/sec;

Ah_

ideal

specific

based

on

pressure, This

was

J/kg;

discussed for

to vary

speed

an

2 (vol.

idealized

parabolically

Experimentally jet

in figure

of

inlet-total

static

12-21

1), where

case.

with

obtained

ratio

ratio

to

exit-static

Btu/lb

in chapter

mathematically shown

work

ft/sec

For

blade-jet

a correlation that

case,

speed

ratio.

was

shown

efficiency

was

efficiencies

are plotted

against

blade-

for a two-stage

axial-flow

turbine

over

___

<_

50

_

30

a

_q

/ / /

/ /

Design value

.1

I .1

1 .2

I

I

I

I

I

I

.3

.4

.5

.6

.7

.8

Blade-jet

FIGURE

12-21.--V_riation

speed ratio,

of efficiency

with

v

blade-jet

speed

ratio.

381

TURBINE

wide

I)E,SIGN

range

turbine, only jet

of speed

because very

and

AND

and

of the

slightly

are,

therefore,

speed

ratio

generalized

not serves for

be noted,

as for

the

where

limiting

somewhat,

turbine

parameters,

we

The

total

and

very

low

stages

higher

presented.

Figure

very

to

well

a real

turbine

represented

by

this

figure.

is approached, at the

angles should

the blade-jet

Flow

Angles

not

understand

as they

the

ideal always

operating lines

speed

were blade-

efficiency an

this

efficiencies

that

is not For

considered how

static

as for

speed

lower

are

the shows

correlation

for

velocities,

turbine

as well

the

efficiencies exit

than 12-21

correlate

that

loading

flow

two

ratio.

however,

especially

Although

pressure

(1 or 2 percent)

manner

It should

A.PPLI.CATION

tend

in

turbine. as good

conditions to separate

ratios.

turbine

vary

over

performance the

turbine

Percent of design equ ivalent speed

3O

2O --

80

100

10

0 "0

120

_'_ -10 g

2- -2o -30

-40

-50

-6O

1

I

I

I

I

I

1.4

1.6

1.8

2.0

2.2

2.4

Ratio of inlet-total to exit-total pressure, P_P_t FIOURE

382

12-22.--Variation

of rotor

incidence and speed.

angle

with

turbine

pressure

ratio

a

EXPERIMENTAL

operating row to

DETERMINATION

conditions.

determines

the

off-design

incidence inlet

angle,

a range

turbine, angles

the

resultant

herein

defined

of the

velocity

velocity.

The

following

this

figure:

tential

4

which in

entering

each

is an important

chapter

as the

8

contributor

(vol.

difference

blade

2).

The

rotor

the

rotor-

between

the rotor

blade

inlet

angle,

and

pressure

ratio

for

a typical

single-stage

in figure

12-22.

values

are presented

as being

positive

is in the

generalized variation

range

flow

PERFORMANCE

and

vector

(1) a large

operating

loss,

is defined

angle

ponent

of the

discussed

of speed

and are

as

which

flow

direction

incidence

losses,

relative

over

The

OF AERODYNAMIC

when same

of a turbine,

(2) the

com-

the

blade

made

from

be

occurs

change

Flow

as

can

angle

calculated

tangential

direction

observations

in incidence

the

was

over

the

in incidence

po-

angle

Percent of design equivalent speed

m

16-_" A _3 ©

8--

0--

4O 5O 6O 7O 8O 90 100

[] ©

-16 --

o Q_

-2_--

L.

-32 --

-40 --

-48 --

-56 2

1

I

I

I

I

I

t

1.3

1.4

1.5

1.6

1.7

1.8

1.9

I 2.0

] 2.1

Ratio of inlet-total to exit-total pressure, p_/p_, FIGURE

12-23.--Variation

of

outlet

flow speed.

angle

with

turbine

pressure

ratio

and

383

TURBINE

with

I)E,SIGN

pressure

rotor

AND

ratio

incidence

becomes

angle

increases and The turbine

APPLICATION

speed outlet

becomes

angle

12-23

for

graph.

The

angle the

is plotted

over

single-stage

trends

also

to the

in outlet

direction

of speed

Stator

loss

outlet

flow

angle

to that

is

shown

total-pressure trailing of

directly

loss

edge

the

loss

occurs

circumferential such

as shown

trated

near

affected with

by

in figure

the

hub

the

end-wall

increasing

boundary-layer 7 (vol.

2). the

the

_=_

--

1.5

--

in and

12-8.

at

wake

one

region.

The

majority where

boundary

ratio

be noted.

and

loss

of

para-

the incidence being

speed

Once

Suction

the

behind

the seen

stator that

of many

such

total-pressure

of the

traces

ratio

were

concen-

were

greatly

The

pressure

increased flow)

and

total-pressure the

the

end-wall data

as described area

turbine

of one and

--

stator

passage perform-

surface

_c_

__ressure

0

384

12-24.--Typical

total-pressure

loss

survey

data

at

can

in chapter full

One blade pitch

FIGURE

loss

loss

/

su rfac/

all

measurements

,'

o

by

as that

circumferential

composite stator

(and

over

pressure

- (=_ ca. "--

that

is in the

such

be plainly

coefficients

losses

total

just

can

The

.1.0-r,--

in figure last

difference

A typical

of

layers.

velocity

stator.

for

equipment

A

tip regions,

of the

only

radius It

12-25.

can

ratio

to in the

terms

contours

loss for the

2.0

design

angle.

survey

12-24.

to kinetic-energy

total

ratio

to the

pressure

ratio

and

yields

critical

Integration

it)

taken

and

buildup

be converted gives

in

traces

the

Loss

12-7

in figure

the

incidence

probe

survey

is shown

(3)

pressure

respect

made

pressure

measurable

in figures

and

angle,

with

for rotor

of a total-pressure

previously

and

the

referred

generalizations

Stator

means

as with

turbine

and

flow

opposite

positive

is important

a range

observed

apply

change

increases,

may be downstream of the turbine or to the can be obtained from the outlet flow. Outlet

the

same

as speed

more

decreases. flow angle

of whatever component amount of thrust that flow

greater

blade

exit.

EXPERIMENTAL

DETERMINATION

Total-press ure ratio (blade exit to blade inlet_

OF

AERODYNAMIC

PERFORMANCE

Total-pressure ratio (bladeex_t 1o Nade inlet)

v,

>0.98 0 90 to 098 >0.80 to 0.90

E::3 >0 98 090 Io 098 0.80 to 0.90 >070 to 0.80

Pressure surface side Suction surface ,, side -

(b) Total-pressure ratio (blade exit to blade inlet)

Total-pressure ratio (blade exit to blade inlet} r_>

r---I >098 _zza 0,90 to 0.98 i_ 0.80 to 090 I_ 0.70 to 080 _lm >0.60 to 070

0.98 0 90 0.80 0.70 >060

to to to to

0.98 0,90 0.80 070

I I 4

L_j (d_

(c) (a)

Ideal ity

(c)

Ideal ity

FIGURE

after-mix

ratio,

critical

veloc-

critical

veloc-

(d)

0.823. of

obtained

total-pressure

Surface

profiles

Ideal

part

of the

that

yield

ratio

experimentally,

blading favorable

after-mix

ratio,

ity

12-25.--Contours

An important

Ideal ity

after-mix

ratio,

ance have been can be made.

surface

(b)

0.512.

a

critical

veloc-

critical

veloc-

0.671. after-mix

ratio,

0.859.

from

stator

turbine

annular

loss

surveys.

breakdown

Velocity design

is the

surface

selection velocity

of the

blade

distributions. 385

TURBINE

DE'SIGN

Analytical

AND

methods

chapter

5 (vol.

mine

achieved.

During

the

To

the

the

velocity

ments.

With

the

known,

the

velocity

static

pressure

can

v-f_'+i[-1

L_--1L-\p_-/

tribution

shown

Acceleration

on

smooth,

and

no

flow

large

(force

the

the

12-26(a)

suction

maximum is well

a blade

surface,

the

along

surfaces

the

in the

measureblade

from

the

surfaces relation

to

(12-15)

determined surface similar conditions.

is considered

surface

distributed

blade

jj

maximum

along

1.4

the

surface.

one.

velocity

is

There

are

subsonic.

on either

other hand, Flow on

velocity The dis-

to be a desirable

the

is maintained

(diffusions)

12-26(b), on the to be undesirable.

along

( P']'"-"'"7"lr"

velocity

decelerations

on blade)

Figure considered

in figure

to deteractually

be determined

Figure 12-26 shows the experimentally distributions for two stators tested under

in

were

on static-pressure

distribution

distribution

Vc,

along

section

discussed

velocities

distribution

in the

were

it is of interest

surface

are made

previously

velocities

program,

for"

measurements

discussed

surface test

"designed

obtain

static-pressure manner

for calculating

2).

whether

APPLICATION

The

loading

blade.

shows a velocity the suction surface

distribution accelerates

B

1.2

._

>_1.0

r- Suction surface

¢''_\ _ Suction surface

--

_

_.

i,/.....E,/ '

/

•2 _

.Z/

"0 f_).

0.. 3_. _

surface

i

.

I 0

.2

.4

surface

__Pressure .6

I

I_

I

'_ Pressure

I

I

.8 1.0 0 .2 Fraction of blade surface length

I

.6

(a)

(a)

Desirable FIGURE

386

I .8

(b)

distribution. 12-26.--Experimental

I

.4

(b) surface

Undesirable velocity

distribution.

distributions.

I 1.0

EXPERIMENTAL

to

a supersonic

deceleration thickening and face.

could

DETERMINATION

velocity

back

to a subsonic boundary

possibly

lead

A deceleration

result

in reattachment

distributions blades

with

are being

because sharp

and

velocity. layer

observed

of the

flow

of

any

the by

separated and

valleys

causes

increase

off the

pressure an

a rapid

a deceleration

associated

on

PERFORMANCE

undergoes

an

it is followed peaks

then

Such

with

to separation

is also

as critical,

AERODYNAIVIIC

(V/Vcr=l.2)

of the

is not

OF

suction

surface,

acceleration

flow. should

In

sur-

but that

general,

be avoided

a

in loss this

would velocity

when

the

designed.

REFERENCES 1.

DOEBELIN,

ERNEST 0.: Measurement Systems: Application and Design. McGraw-HiU Book Co., 1966. 2. PUTRAL, SAMUEL M.; KOFSKEY, MILTON; AND ROHLIK, HAROLD E.: Instrumentation Used to Define Performance of Small Size, Low Power Gas Turbines. Paper 69-GT-104, ASME, Mar. 1969. 3. MOFFITT, THOMAS P.; PRUST, HERMAN W.; AND SCHUM, HAROLD J.: Some Measurement Problems Encountered When Determining the Performance of Certain Turbine Stator Blades from Total Pressure Surveys. Paper 69-GT103, ASME, Mar. 1969. 4. WISLER, D. C.j AND MOSSEY, P. W.: Gas Velocity Measurements Within a

5.

Compressor Rotor Passage Using 72-WA]GT-2, ASME, Nov. 1972. ASME RESEARCH COMMITTEE ON Theory and Application, 5th ed., Engineers, 1959.

the

Laser

Doppler

FLUXD METERS: The American

Velocimeter.

Fluid Society

Paper

Meters, Their of Mechanical

387

TURBINE

DE,SIGN

AND

APPLICATION

SYMBOLS A

area,

C

discharge

D

diameter,

E

thermal

g hh_d

conversion ideal

m2; ft 2 coefficient m;

ft

expansion constant,

specific

pressure, turbine

J

conversion

K

conversion

M

approach

N

rotative

speed,

P R

absolute

pressure,

absolute

V

absolute change

AVu

work,

absolute

and exit,

kg/sec; angle,

torque,

5'

ratio of specific heat constant volume ratio

of

defined

ratio

from

to

kg/m

annulus

cr

critical

eq in

equivalent meter inlet

std

standard

t

meter

0

measuring

388

be-

or (12-6)

direction,

pressure

pressure

velocity

deg

critical

ratio, 8; lb/ft

defined

condition

by

sea-level

Mach

eq.

1)

condition

throat station

at turbine

standard

inlet

heat sea-level

by eq. (12-12) on

based

by eqs.

3

(at

to

velocity

defined

to specific

based

Subscripts" an

axial

defined

critical

temperature, speed

density,

velocity

ft/sec

at constant total

of

temperature blade-jet

absolute

by eq. (12-5)

pressure function of specific-heatratio,

sea-level

of

lb-ft

turbine-inlet

squared

ft/sec

m/sec;

measured

F

N-m;

(o R)

lb/sec

factor,

flow

m/sec;

m/sec; ft/sec component

inlet

flow rate,

(sec)

(12-4)

K; °R speed,

gas velocity, in tangential

compressibility

(min)/(rev) by eq.

rev/min

temperature,

mass

(rad)

defined

(K) ; (ft) (lbf)/(lbm)

mean-section

Y

to exit-static

N/m S; lb/ft 2

J/(kg)

W

of inlet-total

Btuflb

1 ; 9/30 factor,

rad/sec;

rotor

2)

1; 778 (ft)(lb)/Btu

constant, velocity

tween

on ratio

J/kg;

constant,

gas constant, blade

based

(lbm)(ft)/(lbf)(sec

Btuflb

specific

U

1;32.17

work J/kg;

hh'

T

factor

(12-10)

(12-14)

turbine-inlet on and

standard (12-13)

at

EXPERIMENTALDETERMINATIONOF 1

measuring

station

at stator

2

measuring

station

at turbine

AERODYNAMIC

PERFORMANCE

outlet outlet

Superscript: '

absolute

total

state

389

REPORT r_,_t?,.¢,r_

_r-_

, oll_ct4_f_ i;,,l_,_

,t t_l_;l_._,

,_._r,t,_,r_+,_

i T_,,

_r_f,_r_r,,_,,'_,r_. ,.

r.

_,j*_,_

!2C,t

]

_TI

_,,c,r_'._ ,,v!,r_

DOCUMENTATION

"_e'_. ,_,,_;

;_c,

_,

,in_ _,'_tq,_r,_

, :,

:c, rmpl_llnq _,.,r

r_ducln]

22202.,1_02,

_ncl

_6

_t,,_

r_.._,,._n,:

'_...

;,,_t_,_n

_r._,

:_f,,.,,

PAGE _h_ _'_

_f

2;

_h,rTrOn J'_h_n_on

_.t_n_,_ment

Of

,nl,_rmat_on

_teadquar ind

oMs _o. o7o4-o188 Form Approved S_'nd qr%

Budeet

Ser_

cP._.

` Paperwork

3. REPORT

I" AGENCY USEONLY(Leaveb/ank) 12" REPORT 1994 DATEjune 4. TITLE AND

I

comment_

regarding

Directorate

_ot

Reducteon

TYPE

SDecial

the%

Design

eslrmate

AND

DATES

and

Wa%hington.

.Jny

other

asDect

_epott_, DC

1215

Ot

thi%

]._fferson

2050].

COVERED

Publication 5. FUNDING

and

or

ODeratlon%

(0104-018B),

SUBTITLE

Turbine

burden

Information

Prolect

NUMBERS

Application

6. AUTHOR(S)

A.

J. Glassman

7. PERFORMINGORGANIZATION NAME(S) AND ADDRESS(ES) Lewis

Research

Cleveland,

OH

8. PERFORMING ORGANIZATION REPORT NUMBER

Center 44135 R-5666

9. SPONSORING/MONITORINGAGENCYNAME(S) AND AODRESS(ES) National Aeronautics and Washington, DC 20546

Space

10. SPONSORING / MONITORING AGENCY REPORT NUMBER

Administration NASA-SP-290

11. SUPPLEMENTARY NOTES Responsible 433-5890. 12a.

DISTRIBUTION

/ AVAILABILITY

Unclassified Subject

13, ABqTRACT

C.

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organization

code

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STATEMENT

2760,

(216)

Ol S T"iB U TI--'0-_0

D E'

- Unlimited

Category

(Maxmnum

NASA turbine driven engines electric trucks,

Kestutis

person,

- 7

200 words)

has an interest in turbines relatedprJrnarily to aeronautics and space applications. Airbreathing engines provide jet and turboshaft propulsion, as well as auxiliary power for aircraft. Propellantturbines provide rocket propulsion and auxiliary power for space craft. Closed-cycle turbine using inert gases, organic fluids, and metal fluids have been studied forproviding long-duration power for spacecraft. Other applications of interest for turbine engines include land-vehicle (cars, buses, trains, etc.) propulsion power and ground-based electrical power.

In view of the turbine-s_stem interest and efforts at Lewis Research Center, a course entitled "Turbine Design and Application was presented during ]968-69 as part of the ]n-house Graduate Study Program. The course was somewhat rewsed and again presented in 1_72-73. Various aspects of turbine technology were covered including thermodynamic and fluid-dynamic concepts, fundamental turbine concepts, velocity diagrams, losses, blade aerodynamic design, blade cooling, mechanical design, operation, and performance. The notes written and used for the course have been revised and edited for publication. Such apublicaLion can serve as a foundation for an introductory turbine course, a means f-or self-study, or a reference for selected topics. Any consistent set of units will satisfy the equations presented. Two commonly used cons!stent sets of units and constant values are given after the symbol definitions. These are the ST units and the U.S. customary units. A single set of equations covers both sets of units by including all constants required for the U.S. customary units and defimn 8 as unity tho_o not required for the ST units. 14.

SUBJECT

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15. NUMBER

engine design, turbine engines, cooling 17.

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