Ch 2 Heat Transfer

Chapter 2 Heat Transfer Calculations Learning Objectives At the end of this chapter students will: •

Be familiar with the mechanisms of heat transfer which are encountered in heat exchangers and other engineering heat transfer problems

•

Be able to calculate an overall heat transfer coefficient (or resistance) from knowledge of individual heat transfer coefficients (or resistances)

•

Understand the concept of fin efficiency and be able to determine overall heat transfer coefficients in geometries involving extended surfaces. Be able to apply published correlations to determine heat transfer coefficients in single-phase, boiling and condensing flows. Heat transfer specialists may be able to quote some correlations for convective heat transfer from memory, but will generally check their own notes or a published source to find the most appropriate for a particular application. Students should NOT attempt to learn the correlations given in this section.

•

Have a knowledge of fouling and fouling mechanisms

•

Be able to calculate radiative heat transfer for simple geometries

2.1 One-Dimensional Conduction The governing equation for one dimensional conduction is:

dt Q& = −kA dx

(2.1)

which, for constant area, for example through a plane wall, may be integrated: Q& = − kA(t 2 − t1 )

(2.2)

At this stage it is also convenient to introduce an equation relating heat transfer between a fluid and a solid boundary: Q& = −αA∆t

(2.3)

2.1

where α is the convective heat transfer coefficient. We will discuss the evaluation of heat transfer coefficients in Chapter 4 In both cases the negative sign signifies heat transfer is from the higher temperature to the lower temperature. We shall look firstly at how these equations may be combined to give us a relationship between heat transferred from one fluid stream to another, the appropriate heat transfer coefficients, and the geometry of the barrier between the fluids.

2.2

Heat Transfer between Two Fluids

In many applications two fluids are separated by a solid boundary, the two simplest situations are shown in figures 2.1(a) and 2.1(b), representing fluids flowing on either side of a flat plate and on the inside and outside of a tube, respectively.

Th ro Tw1

ri

To

Tw2 Tc

Two

Twi

t

Ti

Hot fluid outside a) Plane wall

Ti Two

Twi

Two

Hot fluid inside b) Tube

Figure 2.1 Heat Transfer through plate and tube

2.2

There are several resistances to heat transfer between the bulk of the hot fluid and the bulk of the cool fluid as shown in Fig 2.1. For steady state conditions, the rate of heat transfer from the hot fluid to the wall, the rate of heat flow through the wall, and the rate of heat transfer from the wall to the cool fluid must be equal, defining this as Q& , the surface area of the wall as A, and the heat flux by q& = Q& A ,we can write:

Q& = −αa A(Th − Tw1 )

or

q& = −αa ( Th − Tw1 )

kA Q& = − (Tw1 − Tw 2 ) t

or

q& = −

Q& = −αb A(Tw 2 − Tc )

or

q& = −αb (Tw 2 − Tc )

k (T − T ) t w1 w 2

(2.4a) (2.4b) (2.4c)

(The negative signs are included to indicate that energy flows from hot to cold) Rearranging equations 2.4 gives

q&

αa

= −( Th − Tw1 )

& qt = −(Tw1 − Tw 2 ) k q&

αb

(2.5)

= −( Tw 2 − Tc )

Adding these equations gives: ⎛ 1 t 1 ⎞ ⎜⎜ + + ⎟⎟q& = −(Th − Tc ) ⎝ α a k αb ⎠

(2.6)

which is equivalent to: q& = −U (Th − Tc )

(2.7)

or Q& = −UA(Th − Tc )

(2.8)

with

1 ⎛ 1 t 1⎞ =⎜ + + ⎟ U ⎝ α a k αb ⎠

(2.9)

For a multi-layered wall this analysis may easily be extended to give:

2.3

1 ⎛ 1 i =n t i 1⎞ =⎜ +∑ + ⎟ U ⎝ α a i =1 k i α b ⎠

(2.10)

Examples of this situation encountered in practice, include be the wall of a building comprising structural decorative and insulating layers and heat exchanger plates. By analogy to the analysis of electrical circuits, we may consider the heat flux q& as analogous to the current I flowing through a series of resistances, r, and the driving temperature ∆T difference as equivalent to a voltage drop, V,. With reference to the diagram below: V1

V2 r1

I=

V1 − V2 R

r2

r3

rn

where R = r1 + r2 + ............rn

The heat transfer through a wall of unit area may be expressed:

q& = − R= =

Th − Tc R

1 U 1

αa

+

(2.11)

1 t1 t t +......+ i +.............+ n + k1 ki k n αb

(2.12)

= ra + r1 +..........+ ri +.................. rn + rb

Where the individual rs represent the heat transfer resistance for each boundary. Often the values of these resistances differ by an order of magnitude or more and it is permissible to neglect the smaller resistances. In building design, for example, it is common to encounter structures having multiple layers, some primarily structural or aesthetic, others providing thermal insulation.

2.4

There are also situations when intermediate temperatures are known, or must be calculated. For example if the internal surface temperature of the wall illustrated in Fig. 2.1(a) is known then equations 2.4(b) and (c) may be used to give: ⎛t 1 ⎞ ⎟⎟q& = −(Tw1 − Tc ) ⎜⎜ + k α b ⎠ ⎝

(2.6(a))

Obviously it is undesirable to have a high thermal resistance in a heat exchanger, so it would be unusual to have a composite wall separating the fluids by design. Fouling, however, may lead to the deposition of a layer of material with poor thermal conductivity on one or both sides of the wall. Hence, we must often consider the situation where an additional fouling resistance must be included in equation 2.9.

q& = −

1 (T − Tc ) t 1 1 h w + r fa + + r fb + kt αa αb

(2.13)

The terms rfa and rfb representing the fouling resistances, or fouling factors on each side of the wall. Even for this simplest of geometries we can write this expression incorporating fouling resistances in a number of ways, all of which are equally valid, for example:

Q& = −U f A(Th − Tc )

(2.14)

1 ⎛1 ⎞ = ⎜ + r fa + r fb ⎟ ⎠ U f ⎝U

(2.15)

or 1 1 tw 1 = + + U f αaf k t αbf

(2.16)

with the appropriate heat transfer coefficient, α f , for the fouled surface being calculated from: 1

αf

=

1

α

+ rf

(2.17)

2.5

To determine the heat transfer through a tubular element we follow a similar line of reasoning. As for the plane wall, during steady state conditions the rate of heat flow into a section of wall must equal the heat flow out at the other side. Furthermore, if the tube is long and longitudinal temperature gradients are small, we can assume that heat flow is one dimensional and in the radial direction. Firstly consider the heat flow and temperature distribution within the tube wall: For an element radius r and thickness δ r ,

δT Q& = − kA δr

(2.18)

or, in the limit:

dT Q& = − kA dr

(2.18a)

For a length l of tube: A = 2π rl Therefore

dT Q& = − k 2π rl dr

(2.19)

This may be integrated between the limits ri and ro to give:

Q& = −

2π kl(To − Ti )

(2.20)

⎛r ⎞ ln⎜ o ⎟ ⎝ ri ⎠

Now if we consider the internal and external thermal resistances we can write a set of equations: Q& = −2π lroα o ( To − Two )

Q& = −

(2.21a)

2π kl (T − Ti ) ⎛ ro ⎞ o ln⎜ ⎟ ⎝ ri ⎠

(2.21b)

Q& = −2π lri αi ( Twi − Ti )

(2.21c)

Which may be combined to give:

2.6

1 ⎛r ⎞ 1 ⎞ Q& ⎛ 1 ⎜ ⎟ = −(To − Ti ) + ln⎜ o ⎟ + 2π ⎝ roα o l kl ⎝ ri ⎠ ri αi l ⎠

(2.22)

As for the wall we may define an overall heat transfer coefficient or “U-value” such that: Q& = −UA(To − Ti )

(2.23)

and ⎛r ⎞ 1 1 1 1 = + ln⎜ o ⎟ + UA 2π roαo l 2π kl ⎝ ri ⎠ 2π riαi l

(2.24)

This implies that, with reference to the internal surface, having area Ai = 2π ri l ,

1 r r r 1 = i + i ln o + U i roαo k ri αi

(2.25)

and, with reference to the external surface, having area Ao = 2π rol ,

r 1 1 r r = i + o ln o + o U o αo k ri riαi

(2.26)

To avoid confusion regarding the area used , it is also possible to work in terms of unit length:

⎛r ⎞ Q& ⎛ 1 1 1 ⎞ ⎜ ⎟ = −(To − Ti ) + ln⎜ o ⎟ + l ⎝ 2π roα o 2π k ⎝ ri ⎠ 2π ri αi ⎠

(2.27)

leading to Q& = −U ′l (To − Ti )

(2.28)

where

⎛r ⎞ 1 ⎛ 1 1 1 ⎞ ⎟ ln⎜ o ⎟ + =⎜ + U ′ ⎝ 2π roαo 2π k ⎝ ri ⎠ 2π riαi ⎠

(2.29)

Before considering how we might incorporate a fouling factor into a tubular geometry it is worthwhile to look at the above expressions more closely, and, in particular, examine what happens as ro approaches ri, i.e. the tube wall becomes thin compared to the radius of the tube.

2.7

For a thin walled tube the mean radius may be calculated as rm = (ro + ri ) 2 , the area is then given by Ai ≈ Ao ≈ A = 2π rml and the tube wall may be considered as a flat wall, thickness t = ro − ri . The overall heat transfer coefficient and rate of heat transfer may thus be calculated: 1 ⎛ 1 t 1⎞ =⎜ + + ⎟ U ⎝ αo k αi ⎠

(2.30)

Which is identical to equation 2.6. If the tube is thin walled and the thermal resistance, t/k, is small compared with the two film resistances: 1 ⎛ 1 1⎞ =⎜ + ⎟ U ⎝ αo αi ⎠

(2.31)

and Q& = −U 2π rm l(To − Ti )

(2.32)

The engineer must make a judgment as to whether these

approximations are

reasonable for a given situation. If fouling resistances are to be included, we have already seen that these may be incorporated in the expression for the overall heat transfer coefficient through a plane wall by simply adding the resistances. For the thin walled tube this may be expressed:

⎞ 1 ⎛ 1 t 1 = ⎜ + rfo + + + ri ⎟ U f ⎝ αo k αi ⎠

(2.33)

or: 1 ⎛ 1 t 1⎞ =⎜ + + ⎟ U f ⎝ αo′ k αi′⎠

(2.34)

where

1 1 = + rf α′ α

(2.35)

2.8

It is convenient to use this modified heat transfer coefficient when dealing with thick walled tubes, and as we shall see later, with all geometries where the heat transfer areas differ for each stream. For the thick walled tube we can, for example, write:

r 1 1 r r = i + o ln o + o U o′ αo′ k ri riαi′

(2.36)

EXTENDED SURFACES - FINS We have seen that the rate of heat transfer from a plane surface is proportional to the surface area, an obvious technique for increasing heat transfer is to increase the surface area available. This may involve using more or longer tubes in a heat exchanger or by adding to the surface area using fins. In applications where the geometry is fixed – for example the top of a microprocessor or the cylinder of an air-cooled engine, the use of an extended surface is the only option.

Figure 2.2 Typical Fin Types

he surface area on one or both sides of a heat exchanger may be increased by the use of extended surfaces or fins. There is a wide range of geometries employed in extending the surface in contact with a fluid. Surfaces which are separated from the

2.9

other fluid only by the thin layer of material through which conduction occurs (e.g. the plane wall or tube discussed above) are referred to as the primary surface of a heat exchanger. Additional surface which is in contact with one fluid but from which there is a tortuous conduction path to the other fluid is known as the secondary surface. We will first analyse the simplest fin type and then discuss how the results may be used in more complex geometries. We shall consider a rectangular fin on a plain surface as shown in Figure 2.3.

Figure 2.3 Diagram of heat flow in rectangular fin

If the length, l, and breadth, L, of the fin are large compared to the thickness, b, we can assume that conduction through the fin is approximately one dimensional. The heat flow into the element dx at some position, x, from the root of the fin is given by:

⎛ dT ⎞ ⎛ dT ⎞ Q& x = − kA⎜ ⎟ = − kLb⎜ ⎟ ⎝ dx ⎠ ⎝ dx ⎠

(2.37)

and the heat flow out of the element is:

2.10

⎛ dQ& ⎞ ⎛ dT ⎞ d ⎛ ⎛ dT ⎞ ⎞ Q& ( x + dx ) = Q& x + ⎜ ⎟ dx = − kLb⎜ ⎟ − ⎜ kLb⎜ ⎟ ⎟ dx ⎝ dx ⎠ dx ⎝ ⎝ dx ⎠ ⎠ ⎝ dx ⎠ ⎛ d 2T ⎞ ⎛ dT ⎞ = − kLb⎜ ⎟ − kLb⎜ 2 ⎟ dx ⎝ dx ⎠ ⎝ dx ⎠

(2.38)

The difference between the inflow and outflow by conduction must be equal to the net outflow of heat from the element to the surroundings:

⎛ d T⎞ dQ& = Q& x − Q& ( x + dx ) = kLb⎜ 2 ⎟ dx ⎝ dx ⎠ 2

(2.39)

and, for surroundings at Ts this is also given by:

dQ& = α A(T − Ts ) = α A(2(L + b )dx )(T − Ts ) ≈ α A2 Ldx(T − Ts ) since L >> b (2.40)

defining θ = (T − Ts )

d 2 T d 2θ so that = 2 , since Ts is constant, and equating the two dx 2 dx

expressions for heat loss from the element: d 2θ 2α = θ dx 2 kb

(2.41)

This differential equation has a general solution of the form:

θ = Memx + Ne − mx

(2.43)

where m=

2α kb

The values of the constants M and N are then determined with reference to appropriate boundary conditions: At the root of the fin x=0 and the fin temperature is equal to the root temperature, To.

θ0 = T0 − Ts = M + N

(2.44)

The second boundary condition is less obvious, but if the fin is very slender so that the heat loss from the tip can be neglected, or if the fin is insulated, then, at x=l,

2.11

⎛ dθ ⎞ Ql = − kLb⎜ ⎟ = 0, ⎝ dx ⎠ l

⎛ dθ ⎞ or ⎜ ⎟ = 0 ⎝ dx ⎠ l

(2.45)

Differentiating equation 2.43 and putting x=l gives: Mme ml − Nme − ml = 0

(2.46)

Combining equations 2.44 and 2.46 gives values for the constants M and N: M = θ0

e − ml eml + e − ml

and N = θ0

eml eml + e − ml

(2.47)

Substituting these values in equation 2.43 gives: ⎛ e m( l − x ) + e − m ( l − x ) ⎞ ⎛ cosh( m( l − x ) ) ⎞ ⎟ = θ0 ⎜ ⎟ θ = θ0 ⎜ ml − ml ⎝ cosh( ml ) ⎠ ⎠ ⎝ e +e

(2.48)

The heat flow from the surface of the fin is equal to the heat flow through the base of the fin, therefore at xo, using equation 2.45, gives:

⎛ sinh( m( l − x ) ) ⎞ ⎛ dt ⎞ ⎟ Q& 0 = − kLb⎜ ⎟ = mkLbθ0 ⎜ ⎝ dx ⎠ 0 ⎝ cosh( ml ) ⎠ = mkLbθ0 tanh( ml )

x =0

(2.49)

Now an ideal fin would have infinite thermal conductivity, hence the entire fin would have a surface temperature equal to the temperature of the root, and the rate of heat transfer from the fin would be given by: Q& ideal = α Aθ0 = α 2 Llθ0

(2.50)

Defining the fin efficiency as:

ηfin =

Rate of heat transfer from fin (2.51) Rate of heat transfer from ideal fin of the same geometry

gives:

2.12

ηfin =

mkLbθ0 tanh( ml ) α 2 Llθ0

Remembering that m =

(2.52)

2α , we can write equation 2.52 as: kb

ηfin =

tanh( ml ) ml

(2.53)

If fouling occurs the fouling resistance should be taken into account when evaluating the heat transfer coefficient used in determining m. i.e.

1 1 = + rf α′ α

(2.54)

It is implicit in the above analysis that the tip of the fin is adiabatic. This approximation holds if the tip of the fin is not insulated or if it butts on to the tip of an adjacent fin. However, if there is heat transfer from the tip of the fin then this may be taken into account by correcting the length of the fin by adding 1/2b, i.e. lc=l+1/2b. The corrected length may then be used in both the evaluation of fin efficiency and fin area. Expressions, often presented in graphical form, are available for the fin efficiency of many shapes of fin. Examples are given in Figures 2.4 and 2.5. In order to use an expression of the form: Q& = UA∆T

to determine the rate of heat transfer across a boundary which includes extended surfaces we can derive an appropriate equation, following the example of equations 2.4 and 2.21;

(

)i

Q& = −α′i m fin ,c& x fin area + unfinned area (Ti − Twi ) T − Two Q& = − wi rw Q& = −α′o m fin ,o x fin area + unfinned area

(

(2.55)

)o (Two − To )

2.13

which may be rearranged to give:

−

(Ti − To ) Q&

=

(

α ′i η fin

1 1 + rw + x fin area + unfinned area α ′o η fin x fin area + unfinned area

)i

(

(2.56)

)o

or 1

Q& =

(

α ′i η fin

1 1 + rw + x fin area + unfinned area α ′o η fin x fin area + unfinned area

)i

(

(2.57)

(Ti − To )

)o

so ⎛ 1 1 UA ≡ ⎜ + rw + ⎜ α ′i η fin x fin area + unfinned area α ′o η fin x fin area + unfinned area i ⎝

(

)

(

(2.58)

As we shall see later, the product UA is an important parameter in heat exchanger design. In applications where fins are employed, it is likely that the purpose is to increase heat transfer (or reduce temperature difference) It is therefore desirable to make each term on the right hand side of equation 2.58 small. Normally rw is the smallest of the terms and, for calculation purposes, may often be neglected. It is more effective to reduce the larger of the two remaining resistances, especially if they differ significantly. Therefore, if finning is to be applied to one surface only, it should be applied to the side with the lower heat transfer coefficient (unless fouling, corrosion or other considerations render this impracticable). This is clearly seen in liquid to gas heat exchangers where it is common for the liquid (or evaporating or condensing fluid) flows inside tubes which are externally finned.

2.14

)o

⎞ ⎟ ⎟ ⎠

−1

Figure 2.4 Efficiency of Radial Fins - constant cross section (SI units with l,b in meters)

Figure 2.5 Efficiency of Axial Fins (SI units with l,b meters)

2.15

2.3

Convective heat transfer

We have already applied the relationship:

(

Q& = −αA∆t ≡ −αA T fluid − Twall

)

(2.3)

to determine the rate of heat transfer from a surface of area A and at temperature Twall to a fluid at temperature Tfluid. Representative values of the heat transfer coefficient, α, may be used in preliminary designs, but it is important that we should be able to estimate values of α for a particular geometry and flow conditions with some accuracy if we are to have confidence in a heat transfer calculation, whether it be a heat exchanger design or calculation of a component temperature. There are few conditions for which analytical solutions for α are available, and these are not often encountered in engineering situations. It is therefore usually necessary to rely upon empirical correlations in the determination of convective heat transfer coefficients. Convection may be forced or free, free convection is also known as natural convection. In forced convection an external driving force (e.g. a pump or fan) causes he movement of fluid over the heat transfer surface. In free or natural convection the fluid movement is induced by the heat transfer and the resulting density change within the fluid. If the fluid movement induced by density changes is significant compared to forced fluid movement heat transfer is said to be by mixed convection. The majority of heat exchangers operate with forced convection. The exceptions being most condensers and some boilers

We shall look at single-phase forced

convection, boiling and condensation in some detail. While the importance of natural convection in many applications (e.g. electronics cooling and space heating) should not be underestimated it is rarely encountered in heat exchangers. Empirical correlations are based upon experimental observations. While the form of the correlations may have some theoretical or conceptual justification, their accuracy relies upon the reliability of the experimental observations and the similarity of the experimental conditions and those to which the correlation is to be applied. It is obvious that correlations are geometry dependent (e.g. applying to flow inside tubes, crossflow

outside tubes or over flat plates) There are other, less obvious

2.16

transitions, the most important being between laminar and turbulent flow and the occurrence or otherwise of a phase change. From the designer’s point of view it is essential that an appropriate correlation is applied, use of a correlation outside the range of conditions for which it has been experimentally verified is very dangerous.

2.17

2.3.1 Dimensionless groups and units Quantity

Symbol(s)

Dimensions

SI Units

Length

L,x

L

m

Time

t

t

s

Mass

M

M

kg

Temperature

T,t,θ

T

K

Absolute Viscosity

µ

ML-1t-1

kg/ms

Acceleration

a

Lt-2

m/s2

Coefficient of Expansion

β

T-1

K-1

Density

ρ

ML-3

kg/m3

Enthalpy

H

ML2t-2

J

-2

Force

F

MLt

N

Heat

Q

ML2t-2

J

-2

Heat Flux

q&

Mt

W/m2

Heat Transfer Coefficient

h,α

Mt-3T

W/m2K

Internal Energy

U

ML2t-2

J

Kinematic Viscosity

ν=µ/ρ

2 -1

Lt

m2/s

Mass Flow Rate

& m

Mt-1

kg/s

Mass Flux

G

Mt-1L-2

kg/m2s

Power

W&

ML2t-3

W

Pressure

p

Mt-2L-1

N/m2

Shear Stress

τ

Mt-2L-1

N/m2

Specific Enthalpy

h

L2t-2

J/kg

2 -2 -1

Specific Heat Capacity

c,cp,,cv

Lt T

J/kgK

Surface Tension

σ

Mt-2

N/m

Thermal Conductivity

k.λ

MLt-3T-1

W/mK

Thermal Diffusivity

a,α

Lt

m2/s

Thermal Resistance

r,R

Tt3M-1

m2K/W

Velocity

V,U,v

Lt-1

m/s

3

2 -1

Volume

V

L

m3

Work

W

ML2t-2

J

Table 2.1 Quantities, units and dimensions

Table 2.1 lists various quantities and their dimensions commonly encountered in heat transfer calculations. Heat transfer data and correlations are frequently presented in the form of dimensionless groups. Some of the dimensionless groups which we will deal with are listed below. Other groups will be introduced when we consider two2.18

phase heat transfer. The student should verify that the groups below are indeed dimensionless. It is possible to devise other groupings by combining the common groups. Dimensionless Groups Darcy friction factor

f =

8τ o = 4c f ρV 2 cf =

Fanning skin friction coefficient Colburn j factor

jh = St Pr

(2.59(a))

1 2

τo f 2 = 4 ρV

2 3

(2.59(c))

gβ∆TL3

Grashof Number

Gr =

Nusselt Number

Nu =

Prandtl Number

Pr =

Rayleigh Number

Ra = GrPr

Reynolds Number

Re =

Stanton Number

St =

(2.59(b))

(2.59(d))

ν2 αd e

(2.59(e))

k fluid

µcp

(2.59(f))

k fluid

(2.59(g))

ρVd e Vde Gd e = = µ ν µ Nu α q& = = ρVc p ∆T ρVc p RePr

(2.59(h))

(2.59(i))

In evaluating the above groups there is often some ambiguity in the choice of values which must be resolved. The physical dimension for flow over plates is generally taken as the distance along the plate, for flow in or around ducts it is the hydraulic or equivalent diameter defined:

de =

4 x Cross sectional area Wetted Perimeter

(2.60)

As expected, for a circular duct or pipe, diameter d, this is given by: de =

4π d 2 4 =d πd

(2.61(a))

For a square duct, side length x,

2.19

4x2 de = =x 4x

(2.61(b))

and for a rectangular duct, width a and depth b:

de =

4ab 2( a + b)

(2.61(c))

if a >> b , for example closely spaced plates, this becomes: de =

4 ab 4 ab ≈ = 2b 2(a + b ) 2 a

(2.61(d))

i.e. the equivalent diameter is equal to twice the plate spacing. For flow through an annulus having inner and outer diameters d1 and d2, respectively the hydraulic diameter may be calculated:

de =

4π (d 22 − d12 ) 4

π ( d 2 + d1 )

=

4π (d 2 − d1 )(d 2 + d1 ) 4

π ( d 2 + d1 )

= ( d 2 − d1 )

(2.61(d))

which is equal to twice the thickness of the annular gap. The hydraulic radius, re,, is defined:

re =

Cross sectional area d e = Wetted Perimeter 4

(2.62)

Thermophysical fluid properties (density, viscosity etc.) vary with temperature. It is important, particularly if the temperature difference between the wall and the fluid is large, that the appropriate temperature is chosen. Normally this is the fluid bulk temperature, but some correlations require properties to be evaluated at the mean film temperature: T film =

T fluid + Twall

(2.63)

2

2.3.2 Single-phase convection. As stated above convection may be free or forced. While free convenction is important in many applications, few heat exchangers rely on single-phase free

2.20

convection. We will therefore concentrate on forced convection, however, for completeness some correlations which may be used to determine free convection heat transfer coefficients are briefly discussed. Whether the flow is induced by natural convection or forced it can be described as either laminar or turbulent. Laminar and turbulent flow If one imagines a deck of playing cards or a sheaf of paper, initially stacked to produce a rectangle, to be sheared as shown in Fig. 2.6, it can be seen that the individual cards, or lamina, slide over each other. There is no movement of material perpendicular to the shear direction.

Figure 2.6 Shear applied to parallel sheets

Similarly, in laminar fluid flow there is no mixing of the fluid and the fluid can be regarded as a series of layers sliding past each other. If the flow is laminar a thin filament of dye inserted in the fluid will remain as a thin filament as it follows the flow. Consideration of a simple laminar flow allows us to define viscosity. Fig. 2.7 illustrates the velocity profile for a laminar flow of a fluid over a flat plate:

2.21

Free stream velocity

y Plate v Figure 2.7 Velocity profile in laminar flow over a flat plate

The absolute or dynamic viscosity of a fluid, µ, is defined by:

τ=µ

dv dy

(2.64)

where τ is the shear stress. At the wall, the velocity of the fluid must be zero, and the wall shear stress is given by: ⎛ dv ⎞ ⎟ ⎝ dy ⎠ w

τ w = µ⎜

(2.65)

The kinematic viscosity of a fluid is defined:

ν=

µ ρ

(2.66)

(Be careful not to confuse ν and v!! ) In practice, laminar flow is observed at low speeds, in small tubes or channels, with highly viscous fluids and very close to solid walls. If the fluid layers seen in laminar flow break up and fluid mixes between the layers then the flow is said to be turbulent. The turbulent mixing of fluid perpendicular to the flow direction leads to a more effective transfer of momentum and internal energy between the wall and the bulk of the fluid. Turbulent flow is the more common regime for bulk flow in most heat transfer equipment, but laminar flow is

2.22

encountered in highly compact heat exchangers and those handling very viscous fluids. Even when the bulk of the flow is turbulent a very thin laminar layer exists close to the wall, this is important when considering processes close to the wall. It should come as no surprise to the student that the heat transfer characteristics of laminar and turbulent flows are very different. In forced convection the magnitude of the Reynolds number provides an indication of whether the flow is likely to be laminar or turbulent: For flow over a flat plate, as shown in Fig 2.8 we may determine whether the flow in the boundary layer is likely to be laminar or turbulent by applying the following conditions:

⎛ ρV ∞ x ⎞ 5 Re x ⎜ = ⎟ < 10 Laminar flow µ ⎠ ⎝ (2.67)

⎛ ρV ∞ x ⎞ 6 Re x ⎜ = ⎟ > 10 Turbulent flow µ ⎠ ⎝ where x is the distance from the leading edge of the plate. V∞ x

Laminar

Transition

Turbulent

Laminar sublayer

Figure 2.8 Development of the boundary layer over a flat plate

For values of Reynolds number between 105 and 106 the situation is complicated by two factors. Firstly, the transition is not sharp, it occurs over a finite length of plate. In the transition region the flow may intermittently take on turbulent and laminar characteristics. Secondly, the position of the transition zone depends not only upon the Reynolds number, it is also influenced by the nature of the flow in the free

2.23

stream and the nature of the surface. Surface roughness or protuberances on the surface tend to trip the boundary layer from laminar to turbulent. For flow in pipes, channels or ducts the situation is similar to that for a flat plate in the entry region, but in long channels the boundary layers from all walls meet and fully developed temperature and velocity profiles are established. For fully developed flow in pipes or channels the transition from laminar to turbulent flow occurs at a Reynolds number, based on the channel hydraulic diameter of approximately 2000. As with the boundary layer on a flat plate, the transition may occur at higher or lower values of Red. If the flow at entry to the channel contains no turbulence and the channel is very smooth, laminar flow may be sustained up to Reynolds numbers of 5-10000. Turbulence may occur at values of Red as low as 1000 but at low Reynolds numbers may decay if induced by, for example, sharp corners. As we shall see, heat transfer coefficients are generally higher in turbulent flow than in laminar flow, and higher in the entry region than in the fully developed region. Heat exchanger designers may therefore incorporate features which either promote turbulence or lead to a geometry which approximates to many short channels. The velocity distribution and variation in local heat transfer coefficient observed at entry to a tube at Red>>2000 is illustrated in Fig. 2.9.

2.24

Figure 2.9 Velocity distribution and variation of local heat transfer coefficient for turbulent flow near the entrance of a uniformly heated tube

Laminar forced convection in ducts1 Examination of the velocity and thermal boundary layers permits the development of analytic solutions for heat transfer between a wall and a fluid in laminar flow, at least in simple geometries. It is beyond the scope of these notes to derive the solutions, however some of the more useful are presented in Table 2.2. Three values of Nusselt number are given for each geometry, the appropriate value depending upon the boundary conditions. These are: •

Nu H 1 =Average Nusselt no. for uniform heat flux in flow direction and

uniform wall temperature around perimeter at any cross section •

Nu H 2 =Average Nusselt no. for uniform heat flux both axially and

around the perimeter •

NuT = Average Nusselt no. for uniform wall temperature.

also tabulated are values of the product fRed.

1

The term ducts here encompasses tubes and channels

2.25

Table 2.2 Nusselt number and friction factor for fully developed flow in ducts

2.26

Table 2.3 gives values of the Nusselt number for heat transfer to or from laminar flow in an annulus with one wall insulated (adiabatic) and the other is maintained at constant temperature.

Table 2.3 Nusselt number and friction factor for fully developed flow in annuli

Particularly in compact heat exchangers and heat sinks on microelectronic systems, the effective duct length may be quite short and entry effects must be taken into consideration. Analytic and empirical solutions for the variation of heat transfer in the entry region are available. Typically these are presented as either:

Nu Nu fully developed

⎛ x dh ⎞ ⎟⎟ = f ⎜⎜ ⎝ Re dh Pr ⎠

or Nu x Nu fully developed

⎛ x dh ⎞ ⎟⎟ = f ⎜⎜ ⎝ Re dh Pr ⎠

where Nu is the mean Nusselt number from duct entry to a position x along the duct and Nux is the local Nusselt number at a distance x from the entry. Nufully developed is the Nusselt number for fully developed flow for the corresponding

boundary conditions. An example of entrance length effect is given in 2.10.

2.27

Figure 2.10 Ratio of mean Nusselt number from entry to x to fully developed Nusselt number ( constant temperature wall)

The thermal entry lengths, Le, for the simultaneously developing hydroynamic and thermal profiles in laminar flow are given by:

Le ≈ 0.037 Redh Pr (Uniform surface temperature) dh Le ≈ 0.053 Redh Pr (Uniform heat flux) dh

(2.67)

Turbulent forced convection in ducts As we have already seen, the Reynolds Number is a particularly important group when dealing with forced convection. The value of the Reynolds Number may be used to determine whether the flow is laminar or turbulent. The Reynolds number is also included in most turbulent flow heat transfer correlations, many of which are expressed in the form: Nu=f(Re,Pr, fluid property correction)

For example, the heat transfer coefficient in single-phase turbulent flow is commonly determined from the Dittus-Boelter equation:

2.28

Nu = 0.023 Re 0.8 Pr n

(2.68)

n = 0.4 for heating Tw > Tf n = 0.3 for cooling Tw < Tf

with properties evaluated at the bulk or mean bulk fluid temperature. This gives results for Nu within 20% for uniform wall temperature and uniform wall thickness conditions within the following ranges. 0.5 < Pr < 120 6000 < Red < 107 60 < L / De

A modification to equation 2.68, taking into account the change in viscosity with temperature in the thermal boundary layer is:

⎛ µ⎞ Nu = 0.027 Re Pr ⎜ ⎟ ⎝ µs ⎠ 0.8

0.14

n

(2.69)

If evaluating local Nusselt number then the bulk fluid temperature is equal to the local bulk temperature, i.e. the temperature which would be measured if the fluid at that station were to be fully mixed. If evaluating the mean Nusselt number over a length of tube the mean bulk temperature is given by:

Tm =

Tm,in + Tm,out

(2.70)

2

Reynolds Analogy Let us consider a turbulent flow past a wall as shown in Fig. 2.9

m

Free stream at V, Tf

Wall at Tw 2.29

Assume that in unit time, over an area A, a mass m, of fluid moves from the free stream to the wall and a corresponding mass moves away from the wall. Assuming that the fluid at the wall is stationary and reaches thermal equilibrium with the wall we can say: Transfer of momentum from the fluid to the wall in flow direction = m(V − 0)

(

Transfer of heat from the wall to the fluid = − mcp Tf − Tw

)

Remembering that force = rate of change of momentum and the shear stress on the wall is equal to the force exerted by the fluid in the flow direction per unit area:

(

c p Tf − Tw Rate of heat transfer / A q& = =− Rate of momentum transfer / A τ w V

(

Rearranging and putting ∆T = Tw − Tf

α=

)

(2.71)

)

c pτ w q& = V ∆T

(2.72)

showing that the heat transfer coefficient and wall shear stress are closely related. In dimensionless form we can write equation 2.72 as:

α ρVc p

=

1 τw 2 21 ρV 2

(2.72)

Using the groups defined in equation 2.59:

St =

Nu 1 = cf RePr 2

(2.73(a))

or

1 Nu = c f RePr 2

(2.73(b))

2.30

We will discuss the pressure drop through ducts pipes and fittings in Section 4, but at this stage it is worth noting that a force balance on a length, L, of circular pipe as shown in Fig. 2.12 gives:

(p

1

− p2 ) Ac = τ w As

(2.74)

L Ac

p1 Vm

p2 Vm

τw

Figure 2.12. Nomenclature used in equation 3.74

Substituting expressions for the cross sectional area, Ac, the surface area, As, and the wall shear stress from equation 2.59(b)

π D2

1 ρ Vm2 c f π DL 4 2 (2.75) 2 2 ρ Vm c f L ∆p = D Thus, if the heat transfer coefficient and wall shear stress are related, the pressure ∆p

=

drop and heat transfer coefficient are also closely linked. For turbulent flow in smooth pipes with Reynolds Number up to 105 the Blasius equation, equation 2.76, gives reasonable results for friction factor, cf, gives: c f = 0.079 Re0.25

(2.76)

Reynolds analogy (as presented in equation 2.73(b)) thus suggests: Nu = 0.0395 Re0.75 Pr

(2.77)

which, for Pr=1 gives very similar results to the Dittus Boelter equation. This is illustrated in Fig.2.11 which shows calculated values of Nusselt number using Dittus Boelter equation and the combination of Reynolds analogy and the Blasius equation as presented in equation 2.77. The Nusselt number for laminar flow with constant heat flux is also included on Fig. 2.13. It is clear that, for a given Reynolds number the

2.31

value of estimated Nusselt number differs significantly depending whether the flow is assumed to be turbulent or laminar. This confirms the importance of ensuring that the appropriate flow type is identified. It also shows that heat transfer enhancement may be achieved for Reynolds numbers from c1000 to the transition region by “tripping” the flow to induce turbulence. The Prandtl Number is the ratio of molecular momentum diffusivity to the thermal diffusivity of a fluid. It is therefore to be expected that Reynolds Analogy is only valid for Pr ≈1, since the derivation implied equal momentum and thermal diffusivity. Alternative correlations While the Dittus Boelter correlation is widely used, its accuracy is limited. A more complex (and thus more awkward to use!) correlation is that due to Gnielinsky, this is regarded as having an accuracy within 6%. With all properties evaluated at the mean bulk temperature: For 02300

( f 8)( Re − 1000) Pr ⎡⎢ ⎛ d ⎞ 1+ ⎜ ⎟ Nu = ⎝ L⎠ ⎢ 1 + 12.7( f 8) ( Pr − 1) ⎣ . ) f = (0.79 ln Re − 164 dh

h

dh

1 2

2 3

2 3

⎤ ⎥ ⎥⎦

(2.78)

−2

dh

approximations to equation 2.78 may be used over the appropriate ranges: For 0.5
Nu dh = 0.0214( Re

0.8 dh

− 100) Pr

0. 4

(2.79(b))

For 1.5
0.87 dh

− 280) Pr

⎡ ⎛d ⎞ 3⎤ ⎢1 + ⎜ h ⎟ ⎥ ⎢⎣ ⎝ L ⎠ ⎥⎦ 2

0 .4

(2.79(c))

Equations 2.78 and 2.79 give mean Nusselt numbers over the length of the tube. If applied to fully developed conditions local Nusselt numbers may be obtained by 2.32

setting

dh to zero. Gnielinsky recommended that a further correction may be L

included to take into account property variations due to temperature. The Nusselt no calculated using equation 2.78 or 2.79 above should be multiplied by:

⎛ Tb ⎞ ⎜ ⎟ ⎝ Ts ⎠

0.45

⎛ Pr ⎞ for gases or ⎜ b ⎟ ⎝ Prs ⎠

0.11

for liquids.

The Gnielinsky correlation was derived for uniform wall temperature, but gives good results for uniform heat flux conditions. An alternative correlation, based on data for uniform heat flux was proposed by Pethukov for fully developed flow:

( f 8) Re Pr Nu = 107 . + 12.7( f 8) ( Pr f = (0.79 ln Re − 164 . ) dh

1 2

dh

⎛ µ⎞ ⎜ ⎟ 2 3 − 1 ⎝ µs ⎠

n

)

(2.80)

−2

dh

Pethukov’s correlation applies when all properties (except µs ) are evaluated at the bulk temperature and values of the constants used are: n = 0.11

(liquids, heating)

n = 0.25

(liquids, cooling)

n=0

(gases)

104
6% uncertainty

200
2.33

Forced convection over cylinders, rods and tube banks The boundary layer observed when a fluid flows over a cylinder or rod cannot be uniquely described as laminar or turbulent. The boundary layer itself is laminar at the front of the cylinder and, depending upon the Reynolds number, may become turbulent. Additionally, at all but the lowest Reynolds numbers, the boundary layer separates from the surface of the cylinder at some point and a wake is formed. The wake may be laminar or turbulent. Data for air (Pr=approximately 0.7) has been correlated in a form: Nud = c Remd Pr n

(2.81)

The value of n in equation 2.81 is 0.33. Properties are evaluated at the mean film temperature Values of c and m are tabulated in Table 2.4 Red

c

m

4-35

0.895

0.384

35-5000

0.657

0.471

5000-50,000

0.167

0.633

50,000-230,000

0.0234

0.814

Table 2.4 Values of c and m for use in equation 2.81

It is recommended that the use of equation 2.81 and the constants of Table 2.4 be restricted to the range 0.50.2 and with properties evaluated at the mean film temperature is: Red>400.000

2.34

4

5 0.62 Red Pr ⎡ ⎛ Red ⎞ 8 ⎤ Nu d = 0.3 + ⎟ ⎥ 1 ⎢1 + ⎜ 2 ⎝ 282000⎠ ⎥ 4 ⎢ ⎡ 3⎤ ⎣ ⎦ ⎢⎣1 + ( 0.4 Pr ) ⎥⎦ 1 2

1 3

5

(2.82(a))

20,000
(2.82(b))

Red<20,000 1

Nu d = 0.3 +

1

0.62 Red2 Pr 3

(2.82(c))

1

⎡1 + (0.4 Pr ) 3 ⎤ 4 ⎥⎦ ⎣⎢ 2

Heat exchangers rarely comprise single tubes in crossflow, they usually incorporate tube banks and the flow differs from that around a single tube in two ways: • The velocity between the tubes is greater than the free stream velocity • The flow field on a tube row is influenced by the presence of upstream row(s)

Tube banks may be in-line or staggered, as shown in Fig.2.14 (a) and (b) respectively. Zukauskas recommends that a correlation of the form:

⎛ Pr ⎞ Nud = c Re d Pr ⎜ ⎟ ⎝ Prs ⎠ m

0.25

n

(2.83)

should be applied to determine the mean Nusselt no (and hence heat transfer coefficient) tube banks having more than 16 rows. The velocity used in evaluating the Reynolds number is the maximum fluid velocity in the bank. All properties are evaluated at the mean bulk temperature,

with the exception of Prs which is

evaluated at the surface temperature of the tubes. Values of c, m and n for use in equation 2.83 are given in Table 2.6.

2.35

Column V∞,T∞ d ST

Row

SL (a) in line

Column V∞,T∞ X

Y d SD

ST

Row

SL (b) staggered

Figure 2.14 Tube bank arrangements

2.36

Red

c

m

n

10-100

0.9

0.4

0.36

100-1000

0.52

0.5

0.36

1000-200,000

0.27

0.63

0.36

200,000-2,000,000

0.033

0.8

0.4

(a) in line arrangement

Red

c

m

n

10-500

1.04

0.4

0.36

500-1000

0.71

0.5

0.36

1000-200,000

0.35

0.63

0.36

200,000-2,000,000

0.031

0.8

0.36

(b)staggered arrangement

Table 2.6 Constants for use in equation 2.83

For in line tube banks the maximum velocity may be calculated by considering conservation of mass, assuming incompressible flow:

Vmax =

V∞ ST ST − d

(2.84a)

For staggered tube banks the maximum velocity may occur either between adjacent tubes in a row or between one tube and a neighbouring tube in the succeeding row, i.e. through the planes X or Y marked on fig. (b). Conservation of mass, assuming incompressible flow, gives: V∞ ST = VX ( ST − d ) = 2VY ( S D − d )

(2.84b)

and the maximum velocity is the larger of VX and VY. Equation 2.83 is valid for 16 or more tube rows. For N rows, where N is less than 16, the mean Nusselt number should be reduced by a factor c1. NuN = c1 Nu16

(2.85)

2.37

where c1 is given in Fig . 2.15.

Figure 2.15 Correction factor c1 for use in equation 3.85

Complex Geometries For complex geometries it is unlikely that an appropriate correlation is available. Experimental data for a number of configurations typical of those used in compact heat exchangers has been published by Kays and London2. If new geometries are to be developed it is likely that experimental measurements will be required to produce a correlation. Manufacturers may publish such data, or they may be proprietary. Sample figures from Kays and London are given as Figs 2.16(a)-(f) - unfortunately, the data are in American units.

2

Kays W.M.and London A.L.,Compact heat exchangers, McGraw-Hill, 2nd Edition, 1964

2.38

f/4 f/4

Figure 2.16 (a) Heat transfer and friction factor for Plain platefin surface 9.03 (h=heat transfer coefficient)

Figure 2.16 (b) Heat transfer and friction factor for Plain platefin surface 11.1 (h=heat transfer coefficient)

2.39

f/4 f/4

Figure 2.16 (c) Heat transfer and friction factor for Plain platefin surface 5.3 (h=heat transfer coefficient)

Figure 2.16 (d) Heat transfer and friction factor for Plain platefin surface 6.2 (h=heat transfer coefficient)

2.40

f/4

Figure 2.16 (e) Heat transfer and friction factor for finned circular tubes, surface CF-7.34 (h=heat transfer coefficient)

2.41

2.4

Boiling and Evaporation

2.4.1 Introduction. Many heat transfer applications involve the evaporation of a liquid. Boiling of a single substance is a vital part of vapour power and refrigeration cycles. If we are to design boilers or evaporators we must be able to determine the relationship between the rate of boiling heat transfer, operating conditions and wall temperature for our heat exchanger. In this introductory study. This introductory note is limited to the case of boiling single fluids, but it should be remembered that evaporation occurs frequently as part of a separation process, in which case the vapour formed has a different composition from the boiling liquid While the terms "boiling" and "evaporation" are used loosely to describe the action of converting a liquid to a vapour by the transfer of energy to the liquid at its saturation temperature it is necessary to be more precise when describing the mechanisms involved. Boiling is the addition of heat causing liquid to evaporate and the vapour to flow away from the heated surface. Evaporation is the conversion of liquid to vapour which occurs at the liquid vapour interface. Boiling is categorised according to the geometric situation and according to the mechanism occurring. The geometric situations commonly encountered are: Pool Boiling - this is defined as boiling from a heated surface submerged in a stagnant pool of liquid. The only movement of the liquid being that induced by the boiling process. Flow Boiling - This is defined as boiling of a liquid as it is pumped through a heated channel. These are analogous to free and forced convection. Boiling outside tube bundles, for example in a fire-tube boiler, combines elements of both situations- a recirculating flow is induced through the bundle due to the vapour generation. The three mechanisms of boiling which are observed are:

2.42

Nucleate Boiling- This involves the formation and growth of bubbles, usually on the heated surface, the bubbles then leave the heated surface and rise to the surface of the liquid. Fig.1 illustrates nucleate and film boiling. Convective Boiling - This mechanism, sometimes referred to as evaporation, involves transfer of heat from the heated surface through a thin layer of liquid and evaporation of liquid at the liquid vapour interface. Film Boiling- This mechanism occurs when the heated surface is blanketed by a film of vapour, heat transfer is then by conduction through the vapour layer and evaporation occurs from the liquid in contact with this liquid film. Fig 2.17 illustrates nucleate and film boiling.

Film Boiling

Nucleate Boiling

Figure 2.17 Schematic representation of film and nucleate boiling

The mechanisms involved in boiling are complex and the relationships used in design and analysis are almost all empirical or semi-empirical, however, in formulating and using empirical correlations it is necessary to have an understanding of the underlying processes. 2.4.2 Pool Boiling In 1934 Nukiyama performed a pool boiling experiment, passing an electric current through a platinum wire immersed in water. The apparatus is shown schematically in Fig. 2.18. The heat flux was controlled by the current through and voltage across the wire and the temperature of the wire was determined from its resistance. Nukiyama then proposed a boiling curve of the form shown in Fig. 2.19

2.43

Condenser

Vapour

Liquid Heated Cylinder (or flat surface)

Figure 2.18 - Simple Pool Boiling Experiment

Since we have a liquid and vapour coexisting in the cylinder both must be at (or during boiling, very close to,) the saturation temperature of the fluid at the pressure in the container. If we measure the surface temperature of the heater, T, the temperature of the fluid, Tsat, the rate of energy supply to the heater, Q& , and the heater surface area, A, we may carry out a series of tests and plot a graph of log Q& , or more usually log q& = log(Q& A) against log ∆Tsat , where ∆Tsat = (T − Tsat ) , often referred to as the wall superheat. As the heat flux, q& , is increased while keeping the temperature of the fluid constant, we would expect the temperature of the rod to increase. The designer of heat transfer apparatus must be able to determine the relationship between heat flux and temperature difference. The relationship between heat flux and wall superheat for a typical fluid is shown schematically in fig. 2.19.

2.44

H

log( q& )

E

F

D

G* C A

G B

B* log( ∆Tsat )

Figure 2.19 Schematic representation of boiling curve

For the case of controlled heat flux (for example, electric heating) the various regimes may be described: For increasing heat flux, in the region 'A'-'B' heat transfer from the heater surface is purely by single-phase natural convection. Superheated liquid rises to the surface of the reservoir and evaporation takes place at this surface. As the heat flux is increased beyond the value at 'B' bubbles begin to form on the surface of the heater, depart from the heater surface and rise through the liquid this process is referred to as nucleate boiling. At this stage a reduction of heater surface temperature to 'C' may be observed. Reducing the heat flux would now result in the heat flux temperature difference relationship following the curve 'C'-'B*'. This type of phenomenon, for which the relationship between a dependent and independent variable is different for increasing and decreasing values of the independent variable, is known as hysterisis. After the commencement of nucleate boiling further increase in heat flux leads to increased heater surface temperature to point 'D'. Further increase beyond the value

2.45

at 'D' leads to vapour generation at such a rate that it impedes the flow of liquid back to the surface and transition boiling occurs between 'D' and 'E'. At 'E' a stable vapour film forms over the surface of the heater and this has the effect of an insulating layer on the heater resulting in a rapid increase in temperature from: 'E' to 'F'. The heat flux at 'E' is known as the critical heat flux. The large temperature increase which occurs if an attempt is made to maintain the heat flux above the level of the critical heat flux is frequently referred to as burn-out. However, if physical burn out does not occur it is possible to maintain boiling at point 'F' and then adjust the heat flux, the heat flux temperature difference relationship will then follow the line 'G'-'H'. This region on the boiling curve corresponds to the stable film boiling regime. Reduction of the heat flux below the value at 'G' causes a return to the nucleate boiling regime at 'G*'. The factors which influence the shape of the boiling curve for a particular fluid include: Fluid properties, heated surface characteristics and physical dimensions and orientation of the heater. The previous history of the system also influences the behaviour, particularly at low heat flux. Clearly several relationships, defining both the extent of each region and the appropriate shape of the curve for that region, would be required to describe the entire curve. It is the nucleate boiling region, 'C'-'D' which is of greatest importance in most engineering applications. However, it is clearly important that the designer ensures that the critical heat flux is not inadvertantly exceeded, and there are some systems which operate in the film boiling regime. Many correlations describing each region of the boiling curve have been published. Additionally, the temperature difference at which nucleation first occurs, i.e. the temperature at 'C' influences the boiling regime during flow boiling and the hysterisis. If the temperature of the heater, rather than the heat flux, was to be controlled then increasing temperature above that corresponding to the critical heat flux would result in a decrease in heat flux with increasing temperature from ‘E’ to ‘G’, followed by an increase along the line G-H. The point ‘G’ is sometimes referred to as the Liedenfrost Point. Temperature controlled heating of a surface is found in many heat exchangers and boilers - the temperature of the wall being necessarily below the

2.46

temperature of the other fluid in the heat exchanger. Experimentally, it is difficult to maintain surface temperatures over a wide range with the corresponding range of heat fluxes. To obtain boiling curves for varying ∆Tsat it is usual to plunge an ingot of high conductivity material into a bath of the relevant fluid. The surface temperature is measured directly and the heat flux can then be calculated from the geometry of the ingot and the rate of change of temperature. The explanation for the importance of surface finish lies in the mechanism of bubble formation. Observation of boiling is difficult because of the vigour with which the process occurs, high speed photographic or video techniques are necessary to get anything more than an approximate qualitative overview. However, even this can give us some insight into the process. Observation of the formation of bubbles in a carbonated drink in a glass can also be instructive, the following experiment works best with carbonated mineral water but other drinks can be used. Pour the drink onto a glass and observe the bubbles. You will note that, once any initial “froth” has dispersed: i.

Bubbles are formed at the surface of the glass3

ii.

Bubbles rise in a chain originating from the same point on the surface.

iii.

If the glass is emptied and refilled many of the sites where bubbles form will

correspond to those observed during the first attempt. This suggests that some feature of the surface encourages bubble nucleation. It has been observed that nucleation occurs in cavities within the surface, these cavities contain minute bubbles of trapped gas or vapour which act as starting points for bubble growth. This is illustrated schematically in Fig. 2.20. When the bubble leaves the site a small bubble remains in the cavity which acts as the start for the next bubble.

3

Any bubbles which arise from a point within the bulk of the liquid almost certainly originate at a solid impurity, for example dust or a particle of organic matter.

2.47

Trapped bubbles of gas or vapour Liquid

Surface

Figure 2.20 Schematic representation of surface showing nucleation sites.

Consideration of idealised nucleation sites allows some indication of their necessary size if they are to play a part in boiling. With reference to an idealised conical cavity as shown in Fig. 2.21.

Liquid

Growing Bubble

2R

Figure 2.21 Idealised cavity acting as a nucleation site

The pressure, pB, inside a bubble is somewhat higher than the pressure in the surrounding liquid: pB = p +

2σ r

(2.86)

Where p is the liquid pressure, r is the radius of curvature of the bubble and σ is the surface tension of the liquid. The radius of curvature is a maximum when the bubble forms a hemispherical cap over the cavity, i.e. r=R, the radius of the mouth of the

2.48

cavity. This is the condition for pB to be a maximum. If the bubble is to grow then the wall temperature must be sufficiently high to vapourise the liquid at a pressure pB. In order for the bubble to grow:

TW > Tsat +

dT ( pB − p ) dp

(2.87)

The Clausius-Clapeyron Equation states that the slope of the vapour pressure curve is given by: h fg dp = dT vg − v f Tsat

(

(2.88)

)

if vg is very much greater than vf we can simplify this: dT vg Tsat = dp h fg

(2.89)

Hence, for the bubble to grow: TW > Tsat +

2σ vg Tsat Rp h fg

(2.90)

The radius of the cavity and the superheat, ∆T sat , at which nucleation from the cavity starts can be related: R=

2σvg Tsat

(2.91)

h fg ∆T sat

For water boiling at 1bar ∆T sat is commonly of the order of 5K. Substitution of values for the properties of water gives a value for the smallest active cavity to be approximately 6.5 x 10-6m radius. This demonstrates that typical active cavities are of the order of 1-10µm. Clearly, real surfaces have a range of cavities of varying size and shape. Surfaces which are designed to improve boiling heat transfer (enhanced surfaces) are made to have large numbers of suitable cavities.

2.49

Some useful pool boiling correlations The symbols used are:

α

Heat transfer coefficient W/m2K

g

acceleration due to gravity m/s2

σ

Surface tension N/m

Gr

Grashof Number

ρf

Liquid density kg/m3

hfg

Latent heat J/kg

µf

Liquid viscosity Ns/m2

kf

Liquid thermal conductivity W/mK

ρg

Vapour density kg/m3

kg

Vapour thermal conductivity W/mK

∆Tsat

Temperature difference K

Nuf

Nusselt Number (liquid conductivity)

cpf

Specific heat capacity of liquid J/kgK

Nug

Nusselt Number (vapour conductivity)

cpg

Specific heat capacity of vapour J/kgK

Pr

Prandtl Number

q

Heat flux W/m2

It can be argued that the heat transfer coefficient, defined by α = q& ∆Tsat , is of limited use when it is not constant, but varies with heat flux (or temperature difference). However many correlations are given in terms of heat transfer coefficient. Natural Convection Region: Typically: Nu = C Gr m Pr m

(2.92)

Where C and m depend on the geometry and whether the induced flow is laminar or turbulent Nucleate Boiling Region: There are a wide number of correlations which have been applied to nucleate pool boiling. Some of the more commonly used are given below: Rohsenow (1952) This is essentially an empirical correlation, but it is instructive to see the way in which it was derived. It is evident that it will be difficult, if not impossible, to produce a theoretical model of boiling which can be used to predict heat transfer coefficients. The situation is 2.50

complicated by the dependence of the heat transfer on the condition and history of the surface. It has already been noted that experimental results for nucleate boiling may be represented by an equation of the form: q& = a (Twall − Tsat )m

or

(2.93(a))

q& = a∆Tsat m

This may be rearranged in terms of a heat transfer coefficient, α,

α=

q& = a∆Tsatm −1 ∆Tsat

α = bq&

m −1 m

(3.93 (b)) (3.93 (c))

≡ bq& n

the value of m is generally in the range 3 - 2.33, corresponding to n being in the range 0.67- 0.7. An early nucleate boiling correlation is that due to Rohsenow, following the example of turbulent forced convective heat transfer correlations Rohsenow argued that: Nu= f(Re,Pr)

Nu = Re = Pr =

αL k ρUL

µ

(U = Velocity)

µc p k

If the fluid properties are all those for the liquid this still left the problem of choosing a suitable velocity and representative length, L. The velocity may be taken as the velocity with which the liquid flows towards the surface to replace that which has been vapourised: U =

q& ρ f h fg

(2.94)

and the representative length is given by: ⎡ ⎤ σ L=⎢ ⎥ ⎣⎢ g (ρ f − ρ g )⎦⎥

0.5

(2.95)

2.51

The correlation thus produced was:

Nu =

1 Re 1− x Pr − y Csf

(2.96)

Which is frequently presented in the form: c pf ∆Tsat h fg

⎡ q& = Csf ⎢ ⎢ µ f h fg ⎣

⎞ ⎛ σ ⎟ ⎜ ⎜ g (ρ − ρ ) ⎟ f g ⎠ ⎝

0.5

⎤ ⎥ ⎥ ⎦

x

⎡ µ f c pf ⎢ ⎢⎣ k f

⎤ ⎥ ⎥⎦

1+ y

(2.97(a))

For most fluids the recommended values of the exponents were: x=0.33, y=0.7. This correlation then corresponds to:

q = [Constant depending upon fluid properties and surface] x ∆T 3 It may also be written: c pf ∆Tsat h fg

⎡ q& = Csf ⎢ ⎢ µ f h fg ⎣

⎞ ⎛ σ ⎟ ⎜ ⎜ g (ρ − ρ ) ⎟ f g ⎠ ⎝

0.5 0.333

⎤ ⎥ ⎥ ⎦

Pr n

(2.97(b))

or

(

⎛g ρ −ρ f g q& = µ f h fg ⎜ ⎜ σ ⎝

) ⎞⎟

0.5

⎛ c pf ∆Tsat ⎞ ⎜ ⎟ ⎟ ⎜⎝ Csf h fg Pr n ⎟⎠ ⎠

3

(2.97(c))

The value of the constant Csf depends upon the fluid and the surface and typical values range between 0.0025 and 0.015. Since, for a given value of ∆Tsat the heat flux is proportional to Csf3 the correlation is very sensitive to selection of the correct value. It is arguable that the complexity of the correlation is not warranted because of the need for this factor.

2.52

Some values of Csf for use in Equation 2.97 and are given in the table below. Fluid

Surface

Csf

Water

Nickel

0.006

Water

Platinum

0.013

Water

Copper

0.013

Water

Brass

0.006

Carbon Tetrachloride

Copper

0.013

Benzene

Chromium

0.010

n-Penthane

Chromium

0.015

Ethanol

Chromium

0.0027

Isopropanol

Copper

0.0025

n-Butanol

Copper

0.0030

Forster and Zuber (1955) ⎛ k 0f .79 c 0pf.45 ρ f0.49 ⎞ 0.25 0.75 α = 0.0122⎜⎜ 0.5 0.29 0.24 0.24 ⎟⎟ ∆Tsat ∆psat ⎝ σ µ f h fg ρg ⎠ ∆p sat =

(

h fg ∆Tsat

Tsat 1 ρ g − 1 ρ f

)

(2.98)

(2.99)

Mostinski (1963)

α = 0106 . pcr0.69 (18 . pr0.17 + 4 pr1.2 + 10 pr10 )q& 0.7

(2.100)

Cooper (1980)

α = 55 pr( 0.12 − 0.2 log ε ) ( − log pr )

−0.55

α = 55 pr0.12 ( − log pr )

−0.55

M − 0.5 q& 0.67

(2.101(a))

M −0.5 q& 0.67

(2.101(b))

ε is the surface roughness in microns. Typically a value of 1 may be used, thus

simplifying the equation. Mostinski and Cooper are both dimensional equations, therefore the units must be consistent with the constants given. For the forms quoted here pressures are in bar and heat flux in W/m2, giving heat transfer coefficients in W/m2K.

2.53

Critical Heat Flux: Kutateladze (1963) & Lienhard et al (1970,1973)

( (

CHF = q& max = Cρ g0.5 h fg σg ρ f − ρ g For Plates

))

0.25

C is in the range 0.13 to 0.18 depending on geometry

For Cylinders

(

(

C = 0.13 0.89 + 2.27 exp − 3.44 R Lb ⎛ ⎞ σ ⎟ Lb = ⎜ ⎜ g (ρ − ρ ) ⎟ f v ⎠ ⎝

))

0.5

Film Boiling Region For spheres and cylinders

(

)

⎡ ρ ρ − ρ gh * d 3 ⎤ v f g fg ⎥ Nuv = c ⎢ ⎢ ⎥ k g µ g ( ∆Tsat ) ⎣ ⎦

0.25

(2.103)

c=0.62 for cylinders and 0.67 for spheres. Liquid properties are evaluated at the saturation temperature and the vapour properties at the average of the surface temperature, Ts, and the liquid saturation temperature, Tsat. The corrected enthalpy of vapourisation is calculated from: h *fg = h fg + 0.4c pg ∆Tsat

(2.104)

This correction accounts for the sensible heating of the vapour. For large horizontal surfaces the expression:

(

)

3 ⎤ ⎡ ρv ρ f − ρg gh *fg d rep ⎥ ⎢ Nuv = 0.425 ⎥ ⎢ k g µ g ( ∆Tsat ) ⎦ ⎣

0.25

(2.105)

may be used, where the Nusselt Number is based on the representative dimension drep, defined by: d rep

⎡ σ =⎢ ⎢g ρ − ρ f g ⎣

(

)

⎤ ⎥ ⎥ ⎦

0.5

(2.106)

2.54

The situation is further complicated in film boiling because of the high surface temperatures which may be involved. The radiative heat transferred can be calculated from:

(

4 q& rad = σ SB ε r Ts4 − Tsat

)

(2.107)

where σSB is the Stefan Boltzman constant = 56.7 × 10−9 W / m 2 k 4 and ε r is the emmissivity of the surface. The convective component is calculated separately. To account for the interaction between the two mechanisms they should be combined:

q& = q con + 0.75q& rad

(2.108)

Film boiling is rarely encountered in heat exchangers, the designer usually wants to ensure that there is no risk of exceeding the internal hear flux within the exchanger. This is particularly important when reduction in the heat transfer results in an increase in temperature of the heating medium. Exceeding the critical heat flux in these circumstances results in a rapid temperature increase, or burnout.

2.4.3 Flow Boiling The prediction of heat transfer coefficients in flow boiling is even more difficult (and often less reliable!) than in pool boiling. In addition to the influence of heat flux (or temperature difference), fluid and surface properties and geometry we must also consider the flow velocity and the quality of the fluid.

2.55

Fig 2.22 Regions of convective boiling

We shall firstly discuss boiling in vertical tubes, similar considerations apply to horizontal tubes. Let us first consider the flow patterns and boiling regimes in a vertical tube heated uniformly along its length. This is illustrated in Fig.2.22. As we proceed up the tube we would observe: • Region A single phase-convection • Region B Sub-cooled boiling- as the fluid approaches its saturation temperature the

wall and the fluid adjacent to the wall will exceed the saturation temperature and nucleation may commence. Bubbles will form and then collapse as they move into the cooler bulk fluid. (This phenomenon can occur during pool boiling and is responsible for the “singing” of a kettle prior to boiling) • Regions C and D are the saturated nucleate boiling regions • Regions E and F are the convective boiling regions • Region G is the liquid deficient region

2.56

• Region F Involves single-phase convection to the vapour.

The flow patterns which one observes may be described: • Bubbly flow The gas phase is present as discrete bubbles dispersed throughout the

liquid phase. • Slug flow Gas bubbles approaching the diameter of the pipe move up the pipe,

separated from the wall by a descending liquid film. The gas bubbles have approximately spherical caps (in round tubes). The bubbles are separated by slugs of liquid, which may contain entrained gas bubbles. • Churn Flow Long bubbles formed as slug flow develops become unstable and the

gas bubbles and liquid slugs become intermingled. The liquid tends to be displaced towards the tube wall but intermittent, irregularly shaped liquid bridges pass up the tube. • Wispy Annular At high mass velocities the majority of the liquid flow is attached to

the duct walls but "fingers" of liquid flow in the gas core. • Annular Flow The liquid phase flows principally as a film on the pipe wall while the

gas flows up the central core. Waves forming on the film may break up causing liquid to be entrained in the gas core as discrete droplets. • Drop Flow Since the presence of the heated wall causes the liquid film to evaporate

the wall will dry out prior to the thermodynamic quality of the fluid reaching unity. Drops, entrained in the vapour during annular flow remain in the vapour stream, only evaporating when the bulk vapour temperature is increased to a value slightly above the local saturation temperature. Flow in horizontal channels yields similar patterns, but the effects of gravity result in stratification, particularly at low velocities. The resulting patterns are shown schematically in Figure 2.23.

2.57

Figure 2.23 Flow patterns during boiling in a horixontal tube

As you would expect relationships are required for each of the flow regimes and heat transfer regions. We will deal only with the annular flow regime which occurs for vapour quality in excess of a few percent and is therefore the most prevalent regime in practice. For example, in refrigeration evaporators which receive a vapour liquid mixture from the expansion valve the flow will be entirely annular.

2.58

Figure 2.24 Variation in heat transfer coefficient Figure with quality, heat flux and mass flux

In

fact the correlations which we will examine can be used with reasonable accuracy for the complete range of saturated boiling.

All flow boiling correlations are

empirical, but are based upon observations of the mechanisms involved as well as heat transfer data. Heat transfer in flow boiling can be regarded as being due to one or both of two mechanisms: namely nucleate boiling and convective boiling.

In

general low quality and high heat flux favour nucleate boiling while high quality and low heat flux lead to convective boiling. High mass flux is conducive to convective boiling. Figure 2.24 illustrates the nucleate and convective boiling regions. .

The way in which correlations account for the two mechanisms differ, some add the contributions for each mechanism, some take only one contribution, and some combine the contributions so that the effect of the larger is dominant. The general form of these correlations is illustrated below and in figure 2.25. It must be remembered that very high heat fluxes can lead to formation of a vapour film, analogous to that encountered in pool boiling. The analysis presented here assumes that the heat fluxes encountered are molecular.

2.59

Figure 2.25

Schematic illustration of Correlations

Additive or superposition (e.g. Chen, (1963)): α = α nb + α conv α nb = Sα nbp

(2.109)

α conv = Fα L

Where the subscripts have the following meanings nb

nucleate boiling contribution

nbp

predicted for pool boiling at the same temperature difference from Forster

and Zuber according to Chen) conv

convective boiling contribution

l

predicted for the single phase flow of liquid (either all fluid flowing as liquid

or based on the liquid component only) (for Chen this is from Dittus Boelter with liquid only Reynolds No.) S and F are factors which are correlated against flow parameters

Enhancement or Substitution (e.g. Shah (1976) )

α = Eα l E is an enhancement factor, the value of which is given by one of several expressions

depending upon the flow parameters and heat flux.

2.60

Asymtotic (e.g. Liu and Winterton (1988)) 2 α 2 = αnb2 + αconv αnb = Sαnbp

(2.110 )

αconv = Fαl The factors in the Liu-Winterton correlation may be determined from: ⎡ ⎛ρ ⎞⎤ F = ⎢1 + xPrl ⎜⎜ l − 1⎟⎟ ⎥ ⎝ ρg ⎠ ⎥⎦ ⎢⎣ 1 S= 1 + 0.055F 0.1 ReL0.16

0.35

(3.110)

αnbp = 55 pr0.12 ( − log pr ) α L = 0.023 Re L =

Gd

µL

−0.55

M − 0.5q& 0.67

k L 0.8 0.4 Re L PrL d

,

G=

m& total Flow Area

Examination of Figs 2.22 and 2.25 and the form of typical boiling correlations shows that the heat transfer coefficient during flow boiling varies significantly as the quality goes from 0 (pure liquid) to 1 (dry vapour). If the vapour is then superheated there will be a step change in heat transfer coefficient as the wall dries out. This means that a stepwise approach must be taken in the design of flow boilers: The local film heat transfer coefficients and overall heat transfer coefficient must be evaluated at entry to the channel, the heat transferred over a short length of channel evaluated thus permitting calculation of the increase in quality over the short length. This process must be repeated over the length of the tube to determine the total heat transferred. Clearly this is a very time consuming process and best carried out using a computer package. Correlations are available which can be used to give estimates of the mean heat transfer over a range of vapour qualities, for example the Pierre (1964) correlation:

2.61

k α =C l d

⎛ 2 ( x out − xin )∆h fg ⎜⎜ Re L Tube Length ⎝

⎞ ⎟⎟ ⎠

n

(2.111)

where C=0.0009 and n=0.5 for exit vapour quality up to 0.9 and C=0.0082 and n=0.4 for higher vapour qualities and exit superheat of up to 6K. It should also be noted that the concepts of mean temperature difference and effectiveness covered in Section 5 rely upon an assumption that the heat transfer coefficient is constant over the entire heat exchanger area. In at least the preliminary stages of thermal design it may be permissible to use an average heat transfer coefficient either for the whole heat exchanger, or for particular sections. For example, if subcooled liquid enters and this is fully evaporated and then superheated the heat exchanger may be considered in three sections - the economiser, the boiling section and a superheater. Finally, in many applications involving boiling, for example fired boilers, the boiling side heat transfer coefficient is likely to be very much higher than the heat transfer coefficient from the heating medium to the wall, hence variations in the boiling side heat transfer coefficient have little influence on the overall heat transfer coefficient.

2.62

2.4.4 Condensation Condensation involves the formation of a liquid from a vapour due to heat transfer from the fluid or a change in pressure of the fluid. The various modes of condensation which may be observed are illustrated in Figure 2.26.

Figure 2.26 Modes of Condensation

2.63

Modes of condensation • Filmwise condensation: The condensate forms a continuous film on the cooled

surface. This is the most important mode of condensation occurring in industrial equipment and is discussed further below. • Homogeneous condensation: The vapour condenses out as droplets suspended in

the gas phase, thus forming a fog. A necessary condition for this to occur is that the vapour is below saturation temperature, which may be achieved (as illustrated) by increasing the pressure as the vapour flows through a smooth expansion in flow area. In condensers, however, it usually occurs when condensing high-molecularweight vapours in the presence of noncondensable gas. Fogs may also form when cold gas is mixed with vapour, for example, during the mixing warm, humid air with cold air. • Dropwise condensation: This occurs when the condensate is formed as droplets

on a cooled surface instead of as a continuous film. High heat transfer coefficients can be obtained with dropwise condensation, but this is difficult to maintain continuously In heat exchangers. • Direct contact condensation: This occurs where vapour is brought directly into

contact with a cold liquid. • Condensation of vapour mixtures forming immiscible liquids: A typical example of

this is when a steam-hydrocarbon mixture is condensed. The pattern; formed by the liquid phases are complicated and varied Filmwise Condensation: Filmwise condensation occurs when the condensate vapour forms a film on the surface which runs down the surface, as shown in Fig. 2.27. The film will be laminar (and amenable to analysis) at the top of the surface, as the film becomes thicker the laminar flow is not stable and waves form in the film, lower down the surface the film becomes turbulent. In many heat exchanger applications, it is satisfactory to assume laminar flow. This gives a conservative estimate of the heat transfer, since both waves and turbulence lead to an increase in the heat transfer coefficient.

2.64

Fig. 2.27 Schematic representation of filmwise condensation on a vertical plate

Rogers and Mayhew (1980) present the analysis of filmwise condensation in the laminar non-wavy region originally derived by Nusselt. This analysis is summarised below. The heat ransfer coefficient a distance x from the top of the plate may be calculated from:

⎛ h'fg ρ 2f gx 3 Nu x = ⎜ ⎜ 4µ f k f ∆Tsat ⎝

⎞ ⎟ ⎟ ⎠

0.25

(2.112)

The mean heat transfer coefficient from the top of the plate to some point l below the top may be calculated from: ' 2 3 4 ⎛ h fg ρ f gl ⎞ ⎟ ⎜ Nu = ⎜ 3 ⎝ 4 µ f k f ∆Tsat ⎟⎠

0.25

2.65

and, for a horizontal tube, diameter d: ⎛ h fg' ρ f2 gd 3 ⎞ ⎟⎟ Nud = 103 . ⎜⎜ ⎝ 4 µ f k f ∆Tsat ⎠

0.25

( ∆Tsat = (Tsat − Tw ) in the above equations) Properties are evaluated at the arithmetic mean film temperature, with the exception of the latent heat which should be calculated from: hfg' = hfg + 0.68cpf ∆Tsat

with hfg calculated at the saturation temperature. Corrections are available to take account of: Waves and Turbulence Shear between the liquid and vapour In the absence of the above, the Nusselt Equation for horizontal tubes may also be used for tubes in the bundles of heat exchangers. However, condensate will drain from the upper tubes to the lower tubes thus increasing the film thickness on all but the top tube. If the liquid flows uniformly then the mean heat transfer coefficient for a bank N rows deep is given by:

αN = N − 0.25 α1 and for a tube on the Nth row the heat transfer coefficient is given by:

αN 0. 75 = N 0.75 − ( N − 1) α1

2.66

In fact, in most practical situations the liquid flows to the lower tubes in rivulets as shown in (b), and the reduction in heat transfer coefficient is not as marked as predicted by the above equations. Filmwise condensation - Nusselt analysis Filmwise condensation on a vertical surface is one of the few aspects of convective heat transfer, which yields to an analytical solution. Nusselt derived a solution based on the following assumptions: • The shear force between the vapour and the condensate film is negligible • Inertia ad hydrostatic forces in the film may be neglected • The flow of liquid in he film is laminar • The resistance to heat and mass

transfer at the liquid vapour interface is

negligible With reference to figure2.28 If the thermal conductivity of the film, thickness δ, is constant then:

t −t dQ& = − k w s = −α x dx(t w − t s ) = −α x dx∆t s k

(2.117)

where:

αx =

k

δ

Defining the Nusselt Number, a distance x from the top of the film as:

Nu x =

αx x k

=

x

δ

suggests that to find the heat transfer coefficient we must first find the film thickness

δ at a distance x from the top of the film.

2.67

(a)

(b)

Figure 2.28: Schematic representation of filmwise condensation

Now, if we consider an element of the film, length dx and an element of the fluid shaded in figure 2.26 , we can equate the shear force and gravity acting on the element:

⎛ dU ⎞ ⎟ dx = (δ − y) dxρ g ⎝ dy ⎠

τdx = µ ⎜

(2.118)

or

dU =

ρg (δ − y)dy µ

(2.119)

This can be integrated to find the velocity distribution through the film: y2 ⎞ ρg ⎛ ⎜ δy − ⎟ U= 2⎠ µ ⎝

(2.120)

The mass of liquid flowing through the film at some distance x from the top is then given by: δ

m& = ∫ ρUdy = 0

ρ2 g µ

δ

2⎞ 2 3 ⎛ ⎜ δy − y ⎟dy = ρ gδ ∫⎜ 2 ⎟⎠ 3µ 0⎝

2.68

(2.121)

At a distance dx below, when the film thickness has increased by dδ, we can write:

( m& + dm& ) − m& = dm& =

ρ 2 gδ 2 dδ µ

(2.122)

Now the heat transferred may be related to the latent heat given up by the condensing vapour:

ρ 2 gδ 2 & & dQ = h fg dm = h fg dδ

(2.123)

µ

Giving: ∆t ρ 2 gδ 2 dQ& = kdx s = h fg dδ

δ

(2.124)

µ

Integrating between x=0, where δ= 0, and x, the film thickness at x may be determined:

δ4 =

4 kµ ∆t s x h fg ρ 2 g

(2.125)

or: 0.25

⎛ h'fg ρ 2f gx 3 ⎞ ⎟ Nu x = ⎜ ⎜ 4µ f k f ∆Tsat ⎟ ⎝ ⎠

(2.112)

If the plate is inclined to the vertical at an angle β to the vertical then we may substitute g cos β for g. The mean heat transfer coefficient from x=0 to x=l is given by: l

1 4 α = ∫ α x dx = α l l0 3

4 or N u = Nu l . 3

(2.126)

For a cylinder, diameter d, the effective length or equivalent plate height is l=2.85d. Any aspect of the geometry or fluid flow which causes the film to break up or become wavy tends to enhance condensation heat transfer, as does shear due to high velocity vapour.

2.69

The presence of non-condensable gases, even in small quantities, in a condenser can have highly detrimental effects on the condenser performance. The non-condensable gas (usually air) becomes concentrated adjacent to the liquid film, thus forming a layer through which vapour must diffuse. The partial pressure of the vapour, and hence its condensing temperature, is reduced by the presence of non-condensables therefore the temperature at the surface of the film, and consequently the temperature difference across the film, is reduced.

2.70

2.5

Fouling of Heat Exchangers

Fouling of Mechanisms The deposition of foreign matter on a heat transfer surface is known as fouling. The presence of a foulant on a surface introduces an additional thermal resistance between the surface and the heat transfer fluid. As the layer becomes thicker this effect becomes more marked and, since the foulant occupies space within the flow passage, effective diameter of the flow passage decreases with a consequential increase in pressure drop (or reduction in flow rate). Both of these effects are undesirable and therefore heat transfer equipment and process conditions must be designed to minimise the effects of fouling. Measures to mitigate the effects of fouling may be preventative (eg treatment of cooling water, high fluid flow velocities), or remedial (eg regular cleaning of the affected surfaces). Additionally, it is usual to allow for a thermal resistance due to fouling when specifying or designing a heat exchanger. Unfortunately, the complex mechanisms involved in fouling are not fully understood and there is only a limited theoretical background to permit the fouling propensity of new designs or applications to be predicted. In practice a designer must rely upon the TEMA fouling factors4 which is additional thermal resistance’s which should be incorporated into the determination of the overall heat transfer coefficient when designing a shell-and-tube heat exchanger. For convenience fouling is generally classified under one of six headings depending upon the mechanism causing the deposition eg5: a)

Crystallisation or precipitation fouling occurs when a solute in

the fluid stream is precipitates and crystals are formed either on the heat transfer surface or in the fluid and subsequently deposited on the heat transfer surface. When the fluid concerned is water and calcium or magnesium, salts are deposited. This mechanism is frequently referred to as scaling. Tubular Exchanger Manufacturers Association Bott T.R., General Fouling problems, Fouling Science and Technology, NATO ASI Series, Ed., Melo L.F., Bott T.R. and Bernado C.A., Kluwer Academic Publishers 1988.

4 5

2.71

b)

Particulate fouling (silting) occurs when solid particles from the

fluid stream are deposited on the heat transfer surface. Most streams contain some particulate matter originating from a variety of sources. c)

Biological fouling is caused by the deposition and growth of

organisms on the heat transfer surface. d)

Corrosion fouling is the result of a chemical reaction involving the

heat transfer surface leading to a build up of corrosion products on the surface. e)

Chemical reaction fouling occurs when a reaction involving one

or more constituents in the process fluid results in the formation of a solid layer on the heat transfer surface. The surface itself is not involved in the chemical reaction. f)

Freezing or solidification fouling occurs when the temperature of

the process fluid is reduced sufficiently to cause freezing at the heat transfer surface. The above definitions are commonly used. However it must be noted that other classification are also found in the literature and in specialist publications.

For

example, defines scale, microbiological contamination and corrosion corresponding broadly with (a), (c) and (d) above but reserves the term fouling for deposition of particulate matter, as in (b) above. Fouling of Open Cooling Water Systems In open cooling water systems neither chemical reaction nor freezing (e) nor (f) above is likely to occur (freezing of water in the cooling tower pond or connecting pipe work during cold weather is a separate problem). System design, materials selection and water treatment combine to mitigate the effect of scaling, corrosion and particulate and biological fouling, however one or more of these mechanisms causes some degree of fouling in most practical open cooling systems. It should also be emphasised that the mechanisms described do not operate independently of each other but usually occur concurrently and can interact. Use of a biocide together with maintenance of the cooling tower to prevent the establishment and build up of any biological growth can minimise, if not entirely eliminate, biological fouling.

Biocide treatment of the water in cooling tower

systems is essential to eliminate the build up of bacteria which may be harmful to

2.72

health, the best known of these being the legionella bacteria.

The materials of

construction of heat exchangers used in cooling water circuits should be chosen so that corrosion is acceptable, remembering that any corrosion products may act as a foulant where formed or break away and contribute to particulate fouling elsewhere. Calcium and magnesium compounds (carbonates, sulphates and phosphates) are inverse solubility salts, that is there solubility decreases with increasing temperature. These salts are the principal components of scale in open water systems. Water treatment must be employed to prevent (or at least minimise) scaling. An adequate purge rate should prevent unacceptable concentrations of the salts likely to crystalise, while chemical additives increase the solubility of the common hardness salts. Lowering the pH of the cooling water increases the solubility of the scale forming constituents, but tends to raise the potential for corrosion. Dispersants are chemicals which impart electrical charges to the heat transfer surfaces and particles so as to keep the particles in suspension. Cooling water treatment is a specialised field and in designing or operating cooling water plant it is usual to consult a chemical supplier for advice on the use of additives. An additional problem associated with compact heat exchanges and related to, but not normally classified as, fouling must be considered: It is inevitable that the small flow passages inherent in most forms of compact heat exchanger will be susceptible to blockage or plugging by large or fibrous particles. Therefore process fluids for use in PCHE’s must be filtered to ensure that particles of dimensions comparable to or larger than the passage cross-section do not reach the heat exchanger. This may require the use of special filtration equipment and/or more rigorous maintenance then would be normal for shell-and-tube units. For the purposes of this report the term ‘blockage’ is used to describe the obstruction of flow passages by relatively large particles.

2.73

Fouling Rate It is beyond the scope of this course to evaluate the various models which have been proposed to facilitate the prediction of fouling behaviour under various conditions but it is necessary to enumerate some of the factors which influence fouling. Fouling involves the deposition of material onto the heat transfer surface occurring concurrently with removal of material previously deposited. A simple model, due to Kern and Seaton, expresses this:

dx f = a1cu − a2 τW x f dt

(2.127)

Where the first term on the right hand side represents the rate of deposition on the surface and the second term represents the removal rate. Integration of equation (2.127) gives:

xF = x*f

(1 −

exp(− Bt

))

(2.128)

Implying that the fouling thickness approaches a value x *f asymptotically. The values of x *f and B are given by:

a1c u a2 τ w

(2.129)

B = a2 τ w

(2.130)

x*f =

Equation 2.129 may be derived from examination of equation 2.127: x *f is the value of the thickness of the fouling layer at which the rate of deposition onto the surface is equal to the rate of removal.

2.74

Figure 2.29 Typical fouling curves

This model is not universally applicable. Several curves of fouling resistance (or thickness) against time have been observed and some typical shapes are shown in Fig. 2.29, the Kern-Seaton model applies only to curve B, representing an asymptotic deposit with no induction period.

However, this simple model is adequate as a qualitative

indication of the importance of various parameters in determining the rate and severity of fouling. If the fouling follows a curve of the form B or D then the heat exchanger can be designed for continuous operation with a fouling factor corresponding to the resistance of the layer at the asymptotic thickness. If the foulant continues to build up then a permissible resistance should be included in the design and cleaning scheduled to take place before this level is reached. The existence of an induction period in many situations may lull the operator into a false sense of security, fouling

2.75

is not immediately apparent but appears after an extended period of operation. There are two possible explanations for this phenomenon. •

Foulant may not initially adhere to the heat transfer surface and the

layer does not build up until the surface has become conditioned in some way. •

Alternatively, if the layer thickness is inferred from heat transfer

measurements then the existence of a fouling resistance may be masked by an enhancement of the heat transfer coefficient by roughening of the surface. Indeed, a net increase in heat transfer (corresponding to an apparently negative heat transfer coefficient) is sometimes observed during the early stages of a heat exchangers life. Remembering that the wall shear stress, τ w , increases with mean velocity, u, to a power greater that l, we can see from equations 2-4 that the fouling rate and final thickness of the fouling layer can be expected to increase with decreasing velocity. For this reason it is essential that heat exchanger designers avoid regions of low velocity in their designs. Designers must also be wary of including too conservative a fouling factor - it may be self fulfilling. If the incorporation of additional heat transfer area is accompanied by an increase in flow area and corresponding reduction in fluid velocity (which in itself will reduce the film heat transfer coefficient), then the propensity to foul will be greater. In general the higher the temperature of a surface the greater its propensity to foul. This is clearly the case for deposition of inverse solubility salts or the products of decomposition. There are obvious exceptions, for example freezing occurs at low temperatures as does the condensation of liquids or tars from a gas stream (eg combustion products).

The designer should attempt to ensure a uniform

temperature where possible in a heat exchanger. Tema Fouling Factors It is often the case that the best that a designer can do is to incorporate TEMA fouling factors into the evaluation of heat exchanger overall heat transfer coefficient. These fouling factors have many shortcomings: they take little account of fluid

2.76

velocity or temperature, they apply only to tubular exchangers and to a limited range of fluids. Typical values of fouling factors are given in Table 2.7.

2.77

Table 2.7 Typical film transfer coefficients for shell -and-tube heat exchangers (taken from Handbook of Heat Exchangers Design by G.F. Hewitt)

2.78

Table 1.2 Typical film transfer coefficients for shell -and-tube heat exchangers (taken from Handbook of Heat Exchangers Design by G.F. Hewitt)

2.79

Table 2.7 continued. Typical film transfer coefficients for shell -and-tube heat exchangers (taken from Handbook of Heat Exchangers Design by G.F. Hewitt)

2.80

2.5.1 Example Showing Effect of Fouling a) What features of gasketed plate heat exchangers make them attractive for use for processing foodstuffs. b) The figure below shows a pasteurisation system treating 7600 l/hour of milk. It incorporates two gasketed plate heat exchangers. The regenerator has 51 thin plates clamped between end plates. The channels between the plates (including those between the end plates and heat exchanger plates) may be regarded as rectangular, having width 300mm and the spacing between the plates is 1mm. The pasteurisation process requires that the milk leaving the heater and returning to the regenerator is always at 77oC. When the plates of the regenerator are clean the milk enters the heater at 65oC. Calculate the height of the plates in the regenerator. After a period of operation, fouling of the plates occurs and a fouling resistance of 0.0001m2K/W is applied to each surface. Estimate the percentage increase in the rate of energy supplied to the heater to maintain the milk peak temperature at 77oC. Calculate the temperature of the milk leaving the plant.

For the plate heat exchanger the heat transfer coefficient may be calculated from: Nu = 0.2536 Re0.65 Pr 0.4

Properties of milk: Density 1030kg/m3 Specific heat capacity 3.92kJ/kgK Dynamic viscosity 1100 x 10-6kg/ms Thermal conductivity 0.565W/mK

2.81

Milk from storage at 4oC

Pasteurised milk Regenerator

77oC

Heater

Solution Showing Effect of Fouling Plate Heat Exchanger 51 plates + ends therefore 52 channels 26 channels per side. Flow area/channel

dh =

4 × 300 × 2mm 602

Flow = 7600l / hour =

G =

m& m& = A 300 × 10 − 6 × 26

7600 l / sec = 2.111l / sec 3600

m& = 2.111 × 1.03 = 2.174 kg / sec 2.174 ⎛ ⎞ −3 ⎜ ⎟ × 2 × 10 − 6 Gdh ⎝ 300 × 10 × 26 ⎠ Re = = 506 = µ 1100 × 10 −6

Pr =

µ Cp

k

1100 × 10 −6 × 3920 = = 7.63 0.565

Nu = 0.2536 Re 0.65 Pr 0.4 = 32.8

α=

Nu × k 32.8 × 0.565 = = 9254W / m 2 K −3 dn 2 × 10

Since properties are constant and the same flow in each side: α

1

=α

2

= α for thin plate

t ≈0 k

2.82

1 1 1 = = ∴U = 4627W / m 2 K U α1 α2

(m& c p )c (Tc ,out − Tc ,in ) = (m& c p )(Th ,in − Th ,out ) Tc ,out = 65 0 C ,

Tmin = 77 0 C

Tc ,in = 4 0 C ,

(m& c p )c = (m& c p )n ∴ Tn ,out = 77 − ( 65 − 4 ) = 16 0 C

∆T1 = 16 − 4 = 12 0 C

∆T2 = 77 − 65 = 12 0 C

Q = m& c p (Th ,in − Th ,out ) = 2.174 × 392 × 61 = 520 kW Q = UA∆Tm

520 × 10 3 A= = 9.36 m 2 4627 × 12

After Fouling 1 Uf

= Uf

1 + rf 1 + rf 2 U

=

=

1 + 0.0001 + 0.0001 4627 2403W / m 2 k

Q = U f A∆Tm

(T c ,out

− T c ,in

) = (T h ,in

Heat Transfer

− T n ,out

Q = m& c p (Tc ,out − Tc ,in )

∆Tm = (Th ,out − Tc ,in ) = (Tmin − Tc ,out )

)

Heat Balance

(1)

[ Equal m& C p ]

2.83

∴ ∆Tm = 12 0 C

Th ,out = Th ,in − (Tc ,out − Tc ,in )

(2)

Q = U f A(Th ,out − Tc ,in )

(3)

Combine (1), (2) and (3)

m& c p UA

m& c p UfA

=

=

Th ,out − Tc ,in Tc ,out − Tc ,in

Th ,in − Tc ,in Tc ,out − Tc ,in

=

Th ,in − (Tc ,out − Tc ,in ) − Tc ,in Tc ,out − Tc ,in

−1

Substitute values for

Th ,in = 77 0 C

Tc ,in = 4 0

(m& c p ) = 2.174 × 3.92kW / 0 C U f A = 2.403 × 9.36 kW / 0 C Tc ,out = 57 0 C Original heater power Fouled heater power

Percentage waste

m& c p ( 77 − 65 ) m& c p ( 77 − 57 )

( 77 − 57 ) − ( 77 − 65 ) 77 − 65 = 67% in ncrease

= 100 ×

Tn ,out = 77 − ( 57 − 4 ) = 24 0 C

2.84

2.6

Heat Transfer by Radiation

Unless at a temperature of absolute zero (i.e 0.0K or -273.15oC, a situation never encountered in practice) all matter emits electromagnetic radiation. The higher the temperature of the body the greater is the rate of energy emission. Bodies also absorb at least a proportion of the thermal radiation which is incident upon them. Therefore if two bodies which are at different temperatures are placed so that each intercepts radiation from the other then there will be a net interchange of energy from the hotter to the cooler body. This is commonly referred to as heat transfer by radiation or radiative heat transfer. Electromagnetic radiation requires no medium for its propogation and will therefore pass through a vacuum. Electromagnetic radiation at the frequencies which are of interest for heat transfer (thermal radiation) will also pass through most gases. For most applications it can be assumed that gases are transparent to thermal radiation and do not emit thermal radiation. There are, however, some important exceptions: the influence of the so called “greenhouse gases” in the atmosphere being one and radiation from flames and combustion products being another. Before considering the transfer of energy it is necessary to remind ourselves of the nature of the radiation involved.

2.6.1 The spectrum of electromagnetic radiation. We do not need to study the physics of electromagnetic waves in any detail, it is sufficient to know that they are characterised by their frequency or wavelength. Wavelength is inversely proportional to frequency:

λ = cv Where: λ = wavelength c = the velocity of light v = frequency Frequency is expressed in Hertz (1/s) and velocity in m/s therefore, for consistency of units wavelength is given in metres. However, in descriptive work the wavelength may ne quoted in cm, mm or µm. Fig shows the electromagnetic spectrum and the names associated wih various wavelengths of radiation. The frequency and hence wavelength depend upon the nature of the source. Radiation in the wavelength range 0.1-1000 µm ( 10-5-10-1cm) will heat any body on which it is incident and is known as thermal radiation. Thermal radiation encompasses ultra-violet radiation, visible light and

2.85

infer-red radiation. The visible spectrum falls within the wavelength band 0.38-0.76 µm. The quantity and frequency of the radiation emitted by a body depends upon the temperature of the body, we cannot see thermal radiation emitted from a body at a temperature below about 500oC.

Thermal Radiation

2.6.2 Black Body Radiation When radiation is incident upon a body it may be reflected, transmitted or absorbed, or a combination of two or three of these. Defining the following terms:

Reflected Energy =ρ Incident Energy Absorbed Energy Absorbtivity = =α Incident Energy Transmitted Energy Transmissivity = =τ Incident Energy Reflectivity =

(Note symbols – these are widely used in the literature but are also used in other areas of heat transfer) Since the entire incident radiation must be reflected, transmitted or absorbed:

ρ + α +τ = 1

(2.131)

For most engineering applications solids are opaque to thermal radiation, i.e. τ=0. Even optically transparent substances are opaque to all but a narrow range of wavelengths. If τ=0

ρ +α =1

2.86

(2.131(a))

If all the radiation of all frequencies incident on an object is absorbed then the object is known as a Black Body. (Note: the terms object or body and surface are almost interchangeable in this context, almost all absorption occurs within a few microns of the surface of an object)

For a black body

α = 1 and ρ = 0 . The visual appearance of an object or material is not always a

good guide to its “blackness”. For example snow is almost black to thermal radiation outside the visible range. It can be demonstrated that a black body is, for a given size and temperature, the best possible emitter of radiation. Consider a small object in a large enclosure:

Fig. 2.30 Object in large surroundings If the body and the enclosure are at the same temperature then there can be no net exchange of energy by radiation (or by conduction or convection). This is a consequence of the Second Law of Thermodynamics, but hopefully it is intuitively obvious. First assume that the object is a black body. All the energy incident on the body is absorbed. For equilibrium an equal amount of energy must be radiated by the body. Let this amount be E b A , where A is the surface area of the body. If the black body is then removed and replaced by a body of the same shape and size but having a surface such that

α < 1 then some of the incident radiation must be

reflected from the surface of the body. The amount of radiation incident on the body will be unchanged since this depends only on the temperature of the surroundings and the dimensions of the object. The rate of energy incident upon the body is still E b A and the rate of energy absorption is

αEb A . For thermal equilibrium these must be equal to the rate at which energy is radiated from the body, EA.

EA = αAE b

α = EE

2.87

b

(2.132)

Since α must be less than one, then E is less than Eb. E is known as the emissive power of the body and is equal to the energy radiated per unit time per unit surface area of the body. The ratio of the emissive power of a body to the emissive power of a black body having the same dimensions is known as the emissivity, ε.

ε = EE

(2.133) b

It can be deduced from the above discussion that the emissivity of a body is equal to its absorbtivity at a given temperature. The emissivity of a body radiating energy at a temperature T is equal to its absorptivity at the same temperature , T. The rate at which energy is radiated from a black body may be determined from the Stefan-Boltzmann law:

Eb = AσT 4

(2.134)

Where: Eb = the emissive power W σ = the Stefan-Boltzmann constant W/m2K4 T = the absolute temperature K A = the area of the body m2 The Stefan-Boltzman constant has a value 5.67 x 10-8W/m2K4 In practice, no surface is absolutely black to thermal radiation but many surfaces approach the ideal having emissivities in excess of 0.95. The surroundings may frequently be considered to behave like a black body. Reference to figure 2.31(a) shows that very little radiation leaving a small object in large surroundings will be reflected back to the object, therefore the surroundings appear to be black. Similarly a small hole leading to a relatively large chamber, as shown in figure 2.31(b), will appear black since radiation entering the hole will not be reflected out. Even if the surfaces in question have high reflectivity, radiation will be absorbed during multiple reflections.

(a) Small object in surroundings

(b) Small hole in chamber

Fig. 2.31Approximations to black body

2.88

When a relatively small object radiates heat to large surroundings at uniform temperature, the net rate of heat transfer to the body is given by:

Q& = − Aσ (ε T14 − α Ts4 )

(2.135(a))

which, if ε and α are independent of temperature, can be written:

Q& = − Aεσ (T14 − Ts4 )

(2.135(b))

The range of wavelengths of the radiation emitted from a black body, as well as the rate of energy emission depends upon its temperature. Figure 2.32 shows the variation with temperature of wavelength in terms of the emissive power/micron of wavelength for a black body.

Fig. 2.32 Spectral distribution from a black body The distribution is given by the equation:

E bλ = and the wavelength,

⎤ 2πhc 2 ⎡ 1 ⎥ ⎢ 5 λ ⎣ exp(hc λK oT ) − 1⎦

(2.136)

λ max at which the emissive power is a maximum is at a given temperature can be

determined from:

λ max T = c where the symbols have the following meanings and values.

2.89

(2.137)

A body or surface which emits less than a black body but has the same shape spectral distribution, as shown in figure 2.33 is said to be grey. For a grey surface ε does not vary with temperature, however in many instances the error introduced by assuming constant emissivity is acceptable.

Black body distribution Grey body distribution

Eλ

Real surface approximating grey body distribution

λ Fig. 2.33 Spectral distribution for grey body The emissivities of various surfaces are given in Table 2.8. Note that for real surfaces the emissivity may vary significantly with temperature. If the body is at T1 and the surroundings are at Ts then when calculating the energy exchange the emmisivity at T1 and the absorbtivity at Ts should be used. i.e. the net rate of heat transfer to the body is given by:

Q& = − Aσ (ε T 1T14 − α Ts Ts4 )

(2.138)

Selective surfaces are those which have very different values of emmissivity ans absorptivity at different temperatures are known as selective surfaces. They are particularly useful in solar energy applications. For solar collectors it is desirable to have a surface with high absorptivity for radiation emanating from a high temperature source and low emissivity at low temperature (the nature of solar radiation at the earth’s surface is such that the sun may be approximated as a black body having temperature ~6000K while the collector surface is at ~350K ) .

2.90

Table 2.8 Emisivities of various surfaces

2.6.3 Practical heat transfer calculations Body in black surroundings We have already seen that the heat transfer between a body and relatively large surroundings is given by:

Q& = − Aσ (ε T 1T14 − α Ts Ts4 )

(2.138)

Radiation exchange between two black surfaces In general, for any two objects in space, a given object 1 radiates to object 2, and object 2 radiates to object 1 and both radiate to space. This is illustrated for the general case in figure 2.34

2.91

Radiation to space Radiative exchange Radiation to space

Fig. 2.34 Radiation between two bodies

A1, T1

Surface 1

Heat transfer

A2, T2

Surface 2

Fig. 2.35 Radiation between two arbitrary surfaces

In order to calculate the energy interchange between the two surfaces at different temperatures it is necessary to calculate both the total quantity of radiation leaving each surface and the proportion of the radiation which reaches the other surface. The radiation leaving a black surface is given by equation 2.134. The proportion of the radiation which is incident on the other surface is given by the radiation shape factor or view factor, F,. With reference to figure 2.35:, F1-2 = fraction of energy leaving 1 which reaches 2 F2-1 = fraction of energy leaving 2 which reaches 1 F1-2 and F2-1 are functions of geometry only.

2.92

For body 1, we know that is the emissive power of a black body, so the energy leaving body 1 is σA1T14 . The energy leaving body 1 and arriving (and being absorbed since, by definition, α = 1 for a black body) at body 2 is σA1T14 F1− 2 . The energy leaving body 2 and being absorbed at body 1 is σA2T24 F2−1 . The net rate of energy interchange from body 1 to body 2 is:

σA1T14 F1− 2 − σA2T24 F2−1 = Q& 1− 2

(2.139)

Note that if the two bodies are at the same temperature T1 = T2 = T then there can be no heat transfer between them:

σA1T 4 F1− 2 − σA2T 4 F2−1 = 0 hence:

A1 F1− 2 = A2 F2−1

2.140

Equation 2.140 is a useful relationship in determining view factors. View factors may be obtained analytically for simple shapes and resulting relationships are given in figure 2.36 or graphically as shown in figure 2.37.

2.93

Fig. 2.36 View factors for various geometries (From Fundamentals of heat transfer, F.P. Incropera and D.P. DeWitt, John Wiley and Sons)

2.94

Fig. 2.37 Graphical representation of view factors

Radiation exchange between two grey surfaces When dealing with finite grey surfaces it is necessary to consider both the view factor and the radiation reflected from one body which is returned to the original 2.95

source. The mathematical manipulation becomes rather complex. We will therefore limit our considerations to the relatively simple case of infinite parallel plates ( or long concentric cylinders with a small gap between them) this is shown schematically in figure 2.40. A,α1 , ε1 , ρ1 , T1 Multiple reflection of radiation

A, α 2 , ε 2 , ρ 2 , T2

Fig. 2.40 schematic representation of radiation between infinite grey surfaces The radiation emitted from surface 1 is given by Aε1σT14 . Surface 2 absorbs: Aε1α 2σT14 ≡ Aε1ε 2σT14

and reflects: Aε1ρ 2σT14

Surface 1 absorbs a portion of this radiation: Aα1ε1 ρ 2σT14 while reflecting Aρ1ε 1ρ 2σT14 This series can be developed over the multiple reflections, and the net energy leaving surface 1 and being absorbed by surface 2 is:

(

)

2 3 Q&1 = ε1ε 2σT14 A 1 + (ρ1ρ 2 ) + (ρ1ρ 2 ) + (ρ1ρ 2 ) ................

Similar logic gives:

(

)

2 3 Q& 2 = ε1ε 2σT24 A 1 + (ρ1ρ 2 ) + (ρ1ρ 2 ) + (ρ1ρ 2 ) ................

and the net rate of heat transfer is:

(

)

)(

(

)

2 3 Q& = Q& 2 − Q&1 = ε1ε 2σ T24 − T14 A 1 + (ρ1ρ 2 ) + (ρ1ρ 2 ) + (ρ1ρ 2 ) .........

The series

1 + (ρ1ρ 2 ) + (ρ1ρ 2 ) + (ρ1ρ 2 ) ........ = 2

Q& =

1

3

1 − ρ1ρ 2

ε1ε 2 σ (T24 − T14 )A 1 − ρ1ρ 2

which may be rearranged with the substitution

ρ = (1 − α ) = (1 − ε )

2.96

Q& =

1 1

⎛1 ⎞ ⎜⎜ + − 1⎟⎟ ⎝ ε1 ε 2 ⎠

σ (T24 − T14 )A

(2.141)

This reduces to equation 2.135(a) if one surface is black , and 2.135(b) if both surfaces are black.

Summary Points •

In order to design or analyse the performance of a heat exchanger or evaluate the heat transfer performance of a system it is necessary to be able to relate the rate of heat transfer to the temperature difference between the two fluid streams or between surfaces and the surrounding fluid.

•

Heat transfer between two streams occurs by convection from the hot stream to the wall, by conduction through the wall and then by convection from the wall to the cool stream.

•

The rate of heat transfer is generally expressed:

Q& = UA(Th − Tc ) •

The overall heat transfer coefficient, U, and the appropriate area, A, may be calculated from a knowledge of the heat exchanger geometry and the fluid flow characteristics.

•

Convective heat transfer coefficients are frequently empirical or semiempirical and it is essential that an appropriate correlations is used.

•

Heat transfer may be adversely influenced by fouling which must be considered at the design stage.

•

Techniques for calculating radiative heat transfer are available

2.97