Gas Dynamics-compressible Flow

ME 1303 GAS DYNAMICS AND JET PROPULSION

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1-D Flow: Revisited Stagnation Quantities 2 U T0 γ −1 2 C PT + = C pT0 or = 1+ ( )M 2 T 2 Furthermore for Isentropic process

P0  γ −1 2  = 1 + ( )M  P  2 

γ γ −1

ρ0  γ −1 2  = 1 + ( )M  ρ  2 

1 γ −1

Relationship with Critical quantities γRT U 2 γRT * U *2 + = + γ −1 2 γ −1 2

a 2 U 2 a *2 a *2 γ + 1 *2 + = + = a γ −1 2 γ −1 2 2(γ − 1) P  2  =   P0  γ + 1  *

2

 a*  T * 2   = = T0 γ + 1  a0 

 2  ρ =   ρ 0  γ + 1  *

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γ γ −1

1 γ −1

a02 γ + 1 *2 = a γ − 1 2(γ − 1)

Relationship between T0,T & M The acoustic speed & Mach number For a compressible flow, the speed of propagation of small disturbances, called the acoustic speed and the ratio of the flow velocity to the acoustic speed, called the Mach number .

The magnitude of K will depend on the process in which the compression is executed. For isentropic flow process,

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Cont.. Mach number

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Cont..

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Basic characteristics of air

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Why is the speed of sound so important?? n

n

n

Fluid particles send signals in form of acoustic (pressure) waves. If signals reach faster than the object itself, fluid particles will “hear” and “clear out” (Subsonic case) If the object is traveling faster than these acoustic waves (speed given by speed of sound), then there is “shock”. (Supersonic case)

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1

2

a + da p + dp ρ + dρ T + dT

a p ρ T

A sound wave, by definition, ie: weak wave ( Implies that the irreversible, dissipative conduction are negligible)

Wave front

uContinuity equation ρa = ( ρ + dρ )( a + da ) = ρa + adρ + ρda + dρda a = −ρ

da dρ

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Speed of Sound in different medium For Liquids

For Solids

B,E- Bulk modulus and Young’s modulus

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Effect of Mach number on Compressibility From Bernoulli eqn w.k.t, compressibility factor is unity for incompressible fluid.

For compressible flow this value deviates from unity; the magnitude of this deviation increases with the mach number of the flow.

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Compressibility Factor Comparison

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Wave Propagation Q.

An airplane is traveling while you are observing from the ground. How will you know whether it is subsonic or supersonic? Moving disturbance Point disturbance is at rest M=0 ( M = u/a = 0.5)

Always stays inside the family of circular sound waves PDF created with pdfFactory trial version www.pdffactory.com

Wave fronts from Sonic disturbance Ø

Ø

All the wave fronts coalesce on the left side and move along with the disturbance. No region upstream is forewarned of the disturbance as the disturbance arrives at the same time as the wave front. Zone of Silence

Zone of Silence

Zone of action

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Wave fronts from Supersonic disturbance n

n

The wave fronts have coalesced to form a cone with the disturbance at the apex. The half angle at the apex is called Mach angle ( µ )

Always stays outside the family of circular sound waves PDF created with pdfFactory trial version www.pdffactory.com

Disturbance Propagation

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n

In Subsonic flow, both Raj and Lisa can hear Joy talking, since sound waves travel from Joy’s mouth in all directions.

n

In Supersonic flow, sound waves (and other disturbances in the flow) travel only in the downstream direction; thus, while Lisa can hear Joy talking, Raj can’t. Disturbances can not travel upstream in a supersonic flow

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Problems

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Cont..

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Isentropic flow with variable area passages a) Nozzle b) Diffuser

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OBJECTIVES OF THIS LECTURE 1.

To examine different scenarios of nozzle & diffuser flows

2. Investigate the relation of flow velocity & pressure in subsonic & supersonic flow regimes 3. To understand how mass flow rate through a nozzle will change with the exit pressure 4. Determine the implications of choking

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Distinction Between True 1-D Flow and Quasi 1-D Flow • In

“true” 1-D flow Cross sectional area is strictly constant • In quasi-1-D flow, cross section varies as a Function of the longitudinal coordinate, x • Flow Properties are assumed constant across any cross-section • Analytical simplification very useful for evaluating Flow properties in Nozzles, tubes, ducts, and diffusers Where the cross sectional area is large when compared to length

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Nozzle - Function ü

From an energy view point : Nozzle is a device that converts static enthalpy into kinetic energy

Expansion Process in nozzle

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Diffuser - Function ü

From an energy view point : Diffuser is a device that converts kinetic energy into static enthalpy

Compression Process in diffuser

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One--Dimensional Isentropic One Flow n

For flow through nozzles, diffusers, and turbine blade passages, flow quantities vary primarily in the flow direction n

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Can be approximated as 1D isentropic flow

Applications Ram-jet engine

nozzle

Diffuser(compressor) combustion chamber

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Applications – Cont..

Space Shuttle

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One-Dimensional Isentropic Flow OneVariation of Fluid Velocity with Flow Area n n

n

Relationship between V , ρ, and A are complex Derive relationship using continuity, energy, speed of sound equations Continuity

n

Differentiate and divide by mass flow rate (ρ (ρAV)

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One--Dimensional Isentropic Flow One

Variation of Fluid Velocity with Flow Area n

n

n

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Derived relation (on image at left) is the differential form of Bernoulli’s equation. Combining this with result from continuity gives

Using thermodynamic relations and rearranging

One--Dimensional Isentropic Flow One Variation of Fluid Velocity with Flow Area

n

This is an important relationship n

For Ma < 1, (1 - Ma2) is positive ⇒ dA and dP have the same sign. n

n

Pressure of fluid must increase as the flow area of the duct increases, and must decrease as the flow area decreases

For Ma > 1, (1 - Ma2) is negative ⇒ dA and dP have opposite signs. n

Pressure must increase as the flow area decreases, and must decrease as the area increases

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Cont..

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One--Dimensional Isentropic Flow One Variation of Fluid Velocity with Flow Area Comparison of flow properties in subsonic and supersonic nozzles and diffusers

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Subsonic Vs Supersonic flow

Nozzle

Diffuser

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Diffuser

Nozzle

Property variation with area change At low Mach no. density variations is less and the velocity changes compensate for area changes.

At M = 1.0 , we reach a situation where density changes and velocity changes compensate for one another and thus dA = 0

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Sonic Properties Ø Ø

Let [ * ] denote a property at the sonic state M = 1 Then giving M =1 in stagnation state set eqns, @ γ = 1.4

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Sonic properties – Cont..

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Remarks on Isentropic Nozzle Design n

n

n

n

Length of the nozzle is immaterial for an isentropic nozzle. Strength requirements of nozzle material may decide the nozzle length. Either Mach number variation or Area variation or Pressure variation is specified as a function or arbitrary length unit. Nozzle design attains maximum capacity when the exit Mach number is unity.

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