Heat transfer through an incompressible laminar boundary layer

CoA. Note No. 130

T H E C O L L E G E O F A E R O N A U T I C S

C R A N F I E L D

HEAT TRANSFER THROUGH AN INCOMPRESSIBLE

LAMINAR BOUNDARY LAYER

by

V, L. Shah

TKHNISCHE HOGESCHOOl D E F T

VLItGTUlGBOUWKUNDE

BiBLlOTMEEK

5 $ nnY 5 f

THE C O L L E G E OF A E R O N A U T I C S

CRANFIELD

Heat Transfer through an Incompressible Laminar Boundary Layer

b y

-V. L. Shah, B . E . ( M e c h . ) , D . I . C . , M . S c . ( E n g . ) .

SUMMARY

The study of the boundary layer theory has been done by reviewing briefly the boundary layer concept, the boundary layer equations, and the exact solutions for wedge flow.

All the methods for the calculations of the heat transfer coefficient are surveyed briefly, describing their procedure and their limitations.

CONTENTS

Page Summary

List of Symbols

Introduction 1 Boundary Layer Theory 2

2 . 1 . Introduction 2 2 . 2 . Boundary Layer Assumption 2

2 . 3 . Dimensional analysis 3 2 . 4 . Boundary Layer Equations for two-dimensional flow 4

2 . 5 . Axisymmetric flow 7

Wedge Flow 8 3 . 1 . Velocity Boundary Layer 8

3 . 2 . Thermal Boundary Layer 11

Description of Methods 15 4 . 1 . Introduction 15 4. 2. Arbitrary main stream velocity and

uniform surface temperature 15 4 . 3 . Uniform main stream velocity with

non-isothermal surface 21 4 . 4 . Arbitrary main stream velocity and a r b i t r a r y

surface temperature 22

References 28 Figures

b o u n d a r y l a y e r , ft. X D i s t a n c e w h e r e the h e a t i n g s t a r t s , ft. y D i s t a n c e n o r m a l t o the w a l l . ft. r R a d i u s of the body. ft. ? V a r i a b l e of the i n t e g r a t i o n ^ S t r e a m function 6 B o u n d a r y l a y e r t h i c k n e s s , ft. 6 B o u n d a r y l a y e r d i s p l a c e m e n t t h i c k n e s s , ft. 6jj M o m e n t u m t h i c k n e s s , ft. A T h e r m a l b o u n d a r y l a y e r t h i c k n e s s , ft. A , T h e r m a l d i s p l a c e m e n t t h i c k n e s s , ft. A^ E n t h a l p y flux t h i c k n e s s , ft. A H e a t flux t h i c k n e s s , ft. 4 u Velocity in the b o u n d a r y l a y e r p a r a l l e l to w a l l , f t / s e c . u Main s t r e a m v e l o c i t y p a r a l l e l to the w a l l . F t / s e c , v Velocity n o r m a l to the w a l l , f t / s e c . T T e m p e r a t u r e in the b o u n d a r y l a y e r . F . T Main s t r e a m t e m p e r a t u r e . F . T T e m p e r a t u r e of the s u r f a c e . F . w T - T Ö w T - T 1 w H V i s c o s i t y , l b / f t . s e c . 2 V K i n e m a t i c v i s c o s i t y , ft / s e c . 2 h Heat t r a n s f e r coefficient. B . T . U . / s e c . f t . F . k T h e r m a l conductivity. B . T . U . / s e c . f t . F .

List of Symbols (Continued) p H C P a T

K

-Cf Nu X P r Re X Density. Ib/ft^. Enthalpy Specific heat Thermal diffusivity. k / p C 3 2 Shear s t r e s s at the surface, lb/ft / s e c .

Heat transfer rate at the wall per unit area and unit time Represents axisjrmmetric flow.

Friction factor, T / i p u , Nusselt Number, hx/k Prandtl Number. C ^x/k

P Reynolds Number. U,x/i/

1. Introduction

The calculation of the heat transfer to the laminar boundary layer is of importance in many engineering applications. This type of flow occurs in the aerodynamic heating of bodies in flight, in the cooling of gas turbine blades, along aerofoil surfaces, and along rocket motor nozzles. In such instances, the variation of the wall temperature and the variation of the main stream velocity profile

have an important influence upon the rate of heat transfer, and a quantitative knowledge of heat transfer coefficient is necessary if the surfaces are deliberately heated or cooled.

Theoretically, the problem of heat transfer in a laminar boundary layer is to find, mathematically, a simultaneous solution to the three basic boundary layer equations for a given wall condition, longitudinal p r e s s u r e distribution, and a gas with specified thermodynannic properties.

Analytical predictions of the local heat transfer, for flow over a flat plate and wedge, have met with considerable success. An explicit solution for more general problemis of flow over a body of a r b i t r a r y shape, in which the velocity at the edge of the boundary layer varies arbitrarily, is either impossible, or in naany cases so cumbersome and time-consuming, that it cannot be carried out in practice. It i s , therefore, desirable to possess at least approximate methods of solution to be applied in cases where an exact solution is impossible, or cannot be obtained with the reasonable amount of work, even if the accuracy is only limited.

A considerable amount of r e s e a r c h has been done in this field and numerous methods are available for calculating heat transfer over the bodies of a r b i t r a r y shape with a r b i t r a r y surface temperature. But there is not a single report available which covers all these methods describing their procedure and their limitations. This presents great difficulty to any beginner entering this field and also to an industrial designer. The purpose of this report is to survey all the literature pertaining to this field and to present their conclusions in an orderly naanner.

In selecting suitable material from the extensive available literature, the basic objective has been to provide an introduction to the field of heat transfer through an incompressible laminar boundary layer. Thus the exact solution for w edge flow has been briefly reviewed after describing the basic boundary layer theory. All the approxinaate methods are then surveyed describing their procedure and the limitations.

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Recently Spalding and Pun have published a similar work in which they have surveyed all the methods very briefly and presented their conclusions in a tabular form.

2

-2. Boundary Layer Theory 2 . 1 . Introduction

In 1904, L. Prandtl proved that at a moderate Reynolds Number, the flow about a solid body can be divided into two regions.

(1) A very thin layer in the neighbourhood of the body called the boundary layer, where the vsicous t e r m s and the conduction t e r m s in the momentum and the energy equation play an essential part.

(2) The potential flow outside the boundary layer, where the viscous s t r e s s e s are negligible compared to the inertia s t r e s s e s .

This theory of Prandtl, with the few boundary layer assumptions, enabled the three basic equations of the boundary layer to be derived:

(1) Equation of Continuity (2) Equation of Motion (3) Energy Equation

Calculations of the heat transfer and the viscous s t r e s s e s can be made by solving these three basic equations. The subject of the boundary layer theory can be divided into two main c l a s s e s , depending on the type of flow:

1. Laminar 2. Turbulent

In laminar flow, the individual particles of fluid flow in a straight line parallel to the stream line without appreciable ti'ansverse to and fro motion, whereas in the turbulent flow innumerable eddies or vortices are present.

The laminar boundary layer can be further divided into three groups, depending on the velocity of the fluid:

1. High speed flow

2. Flow with moderate velocity 3. Natural convection

The present report is restricted to incompressible laminar flow with moderate velocity.

For incompressible flow, the presence of a boundary layer on a body Influences the potential flow only in a secondary way, (unless r e v e r s e flow or separation occurs) through an alteration in the effective boundaries of the potential flow by the amount of the boundary layer displacement thickness. The potential flow, on the other hand, establishes the longitudinal p r e s s u r e distribution for the boundary layer and thereby plays a controlling role in the behaviour and the formation of the boundary layer. 2. 2. Boundary Layer Assumption

In the boundary layer theory, the velocity and temperature gradients parallel to the wall are assumed to be smaller than those in the direction nornaal to the wall.

conduction in the direction normal to the wall.

2. The component of velocity normal to the wall is very small compared to the component parallel to the wall, and so the boundary layer flow is almost parallel to the wall.

3. The pressure within the boundary layer varies only in the direction parallel to the wall and, therefore, the p r e s s u r e within the boundary layer is

established by the potential flow outside the boundary layer.

These boundary layer assumptions are true when the boundary layer thickness is small compared to the radius of the curved body, and the distance from the stagnation point.

In the case of flow past a flat plate, the laminar boundary layer thickness is given by the equation

^ . i : ^ (2.2.1) x

Therefore, boundary layer assumptions do not hold good for low Reynolds Number and at the leading edge.

2 . 3 . Dimensional analysis

An analytical solution of the boundary layer is very cotnplex, due to the large number of t e r m s involved. The method of dimensional analysis can be applied to simplify the solution. By this method the fundamental equations can be so arranged that the quantities enter the equations through certain combinations that are dimensionless. The forming of such equations is independent of the size of the units involved in the various t e r m s in the equations. This method indicates the logical grouping of factors into dimensionless combinations. This is very helpful in interpreting data where two or more factors have been varied in different experiments.

The present report is restricted to the forced convection in a laminar in-compressible flow with moderate velocities, where the heat due to friction and compression need not be taken into account. For such flows the dimensional analysis leads to the conclusion that the solution of the above system of equations, for the velocity field, temperature field, and local coefficient of heat transfer, depend upon the two dimensionless groups.

1. Reynolds Number, which is a ratio of the inertia force to the friction force. 2. Prandtl Number, which is a ratio of the kinematic diffusion to thermal

diffusion.

Vectorial dim.ensional analysis further leads to a relationship:

Nn

— = f ( P r , Geometry) (2.3.1) VRi

- 4

There is a third dimensionless group called Grashof Number which becomes important only at very small velocities of flow, particularly if the motion is carried out by the buoyancy forces and not by the pressure differences. In such cases the flow becomes independent of Reynolds Nunaber and the process is called natural convection.

With moderate velocities, the buoyancy forces caused by temperature differences are small compared with the inertia and frictional forces. In such cases the problem ceases to depend on Grashof Number, and Nusselt Number depends only on Reynolds Number and Prandtl Number.

2.4. Boundary Layer Equations for two-dimensional flow 2 . 4 . 1 . Differential equations

Three differential equations form the basis of the boundary layer theory and they are reviewed below.

(a) Continuity Equation

In the case of the steady flow, the law of the conservation of mass asserts that the mass flow entering the control volume is equal to the mass flow leaving the control volume.

(2.4.1) and for incompressible flow

| i i + 1 ^ = 0 (2.4.1) 8x ay

(b) Monaentum Equation

For steady flow, Newton's 2nd Law of Motion states that the increase in the momentum per unit time of all the particles passing through the control volume is equivalent to the inertia force, and must be in equilibrium, with the external forces acting on the surface and within the control volume. The increase of the momentum of all the particles passing through the control volume can be expressed as the

difference between the momentum leaving the volunae per unit time, and the naonaentum entering through the surface per unit time. Thus

- ^ + ^ < M | ^ = ^ ( p u ^ + - ^ ( p v u ) (2.4.2) dx 9y 9y 8x '^ 9y

Pressure force + Viscous force » Change in naonaentuna

Expanding with the knowledge of the continuity equation, the equation of naomentuna simplifies to du. „2 8u 8u ' , 8 u in A r,\ pur— + pv r— = pu, -T— + M— (2.4.3) •^ 9x '^ By "^ ' dx „ a J" 8 y

o r 2 8u , 9u 1 dp 8 u In A ^\

u r — + v r — = - - - f ^ + V ( 2 . 4 . 4 ) 8x 8y p dx z

^^\ 1 dp

w h e r e u, —T— = - — -*- (frona the potential flow) ' dx p dx

(c) E n e r g y Equation

T h e law of the c o n s e r v a t i o n of e n e r g y for the s t e a d y flow t h r o u g h a c o n t r o l volume s t a t e s that the net efflux of enthalpy plus the k i n e t i c e n e r g y i s equal to the net r a t e of heat t r a n s f e r into the c o n t r o l v o l u m e , p l u s the net r a t e of s h e a r w o r k into the c o n t r o l v o l u m e .

F o r m o d e r a t e v e l o c i t i e s the r a t e of s h e a r work can be n e g l e c t e d . T h e r e f o r e the e n e r g y equation i s

8x

' - ' [ H ^ ^ " ] ^ » T ' - ' [ - ^ 1 - » T

[^a-f]

(2.4,5)

E x p a n s i o n of t h e s e t e r m s , with the known continuity e q u a t i o n , s i m p l i f i e s equation 2 . 4 . 5 to

9 FTT U* + V S 8 r „ U* + v^'-l 8 / , 9 T \ ,„ . „>

A s s u m p t i o n that the enthalpy i s a function only of t e m p e r a t u r e and the use of g a s l a w s E • RT l e a d s to the therm.odynamic r e l a t i o n s h i p P dH . C d T and | i ï - C | ^ p 8x p 8x T h e e n e r g y equation m a y now be w r i t t e n a s r „ 9T ^ 8 f n + v ' n ^ r ^ 8T ^ 8 f Ü' + v' \ "I 8 / . 8 T \ (2.4.7)

and, for m o d e r a t e v e l o c i t i e s , the k i n e t i c e n e r g y is n e g l i g i b l e . A s s u m i n g k : c o n s t a n t , the e n e r g y equation s i m p l i f i e s to

9T 8 T 8 ' T -_ . „.

u r - + v - r - = a • ( 2 . 4 . 8 ) 8x 8y dy

M a t h e m a t i c a l l y , the p r o b l e m of the i n c o n a p r e s s i b l e l a m i n a r b o u n d a r y l a y e r with m o d e r a t e velocity naay be sunanaarised a s that of finding the sinaultaneous solution to

(1) i r - 1^ = o (2.4.1)

Q o d u - 2 / o \ 8u , 8u 1 9 u ,„ . .V (2) u S~ + "^ Sr~ "^ 1*1 T~ + V r ( 2 . 4 . 4 ) 8x 8 y ' dx gy2 (3) u | ^ + v | ^ = a ^ ( 2 . 4 . 8 )

for a given w a l l condition, longitudinal p r e s s u r e d i s t r i b u t i o n , and a g a s with specified t h e r m o d y n a m i c p r o p e r t i e s ,

2 . 4 . 2 . I n t e g r a l E q u a t i o n s

I n t e g r a l e q u a t i o n s of m o t i o n and e n e r g y a r e obtained by i n t e g r a t i n g the

d i f f e r e n t i a l e q u a t i o n s of the b o u n d a r y l a y e r flow o v e r the b o u n d a r y l a y e r t h i c k n e s s . T h e s e e q u a t i o n s , r a t h e r than satisfying the b o u n d a r y conditions for e v e r y individual fluid p a r t i c l e , s a t i s f y the b o u n d a r y l a y e r flow only in the a v e r a g e .

T h e s e e q u a t i o n s with a s s u m e d v e l o c i t y and t e m p e r a t u r e p r o f i l e s , f o r m the b a s i s of a l l the a p p r o x i m a t e m e t h o d s .

1. Monaentum equation

— " T - <u! ^ ) + 6 , u , - T ^ ( 2 . 4 , 9 )

p dx ' 2 1 1 dx

w h e r e 6 and 6 a r e the d i s p l a c e m e n t t h i c k n e s s and the monaentum t h i c k n e s s r e s p e c t i v e l y .

and

2. Heat flux equation

w dx and the d i n a e n s i o n l e s s f o r m / p C u ( T - T,) dy ( 2 . 5 . 0 ) o St . A. j H. (1 _ e) dy ( 2 . 5 . 0 ) dx / u o

T h e d i f f e r e n t i a l e q u a t i o n s for the v e l o c i t y (Equation of motion) and the t h e r m a l b o u n d a r y l a y e r ( E n e r g y equation) a r e v e r y s i m i l a r in s t r u c t u r e . E x c e p t for the n a t u r a l convection, the v e l o c i t y field d o e s not depend on the t e n a p e r a t u r e field, although the c o n v e r s e i s t r u e .

At m o d e r a t e v e l o c i t i e s when the h e a t due to friction and c o n a p r e s s i o n m a y b e n e g l e c t e d , the d e p e n d e n c e of the t e m p e r a t u r e field on the velocity field i s governed

s o l e l y b y the P r a n d t l N u m b e r . To e a c h single velocity field t h e r e c o r r e s p o n d s a single infinite fanaily of t e m p e r a t u r e d i s t r i b u t i o n , with the P r a n d t l N u m b e r a s i t s p a r a m e t e r .

E x a c t i n v e s t i g a t i o n for wedge flow h a s been achieved with s u c c e s s and the r e s u l t s of wedge flow f o r m t h e b a s i s of c o m p a r i s o n f o r a l l t h e a p p r o x i m a t e naethods. In t h e next c h a p t e r t h e solution for t h e wedge i s b r i e f l y r e v i e w e d .

2 , 5 , Axisynrmaetric flow

T h e b o u n d a r y l a y e r , which e x i s t s on a c y l i n d r i c a l body (two d i m e n s i o n a l ) ,

d e p e n d s only on t h e p o t e n t i a l flows around t h e c y l i n d e r . T h e s h a p e of the c y l i n d r i c a l body d o e s not e n t e r t h e c a l c u l a t i o n e x p l i c i t l y , but i n d i r e c t l y i n f l u e n c e s t h e p o t e n t i a l flow. On t h e o t h e r hand, in the a x i a l l y s y n a m e t r i c flow, t h e b o u n d a r y l a y e r d e p e n d s d i r e c t l y on t h e s h a p e of the body, b e c a u s e the r a d i u s of the c r o s s s e c t i o n a p p e a r s e x p l i c i t l y in t h e d i f f e r e n t i a l e q u a t i o n s , a p a r t from influencing t h e p o t e n t i a l v e l o c i t y d i s t r i b u t i o n ,

B o u n d a r y l a y e r e q u a t i o n s for a fluid p a s t an a x i s y m m e t r i c body can be d e r i v e d in t h e s a m e way a s f o r t h e two dinaensional flow.

T h e y a r e

1. Mill + 8iZr) = 0 (2,5,1)

8x 8y (Continuity equation) 2 . _ 8ÏÏ ^ _ 8Ü 1 dp ^ d'xi ,_ - _v u -= + V -= = - — -rt: + V ( 2 , 5 , 2 ) 8x 8y p dx Q-a (Equation of motion) _ 8T _ 8T 8 ' T . „ c _. 3 . u - = + V rt=: = a ( 2 . 5 . 3 ) 8x 8y 8 y 2 ( E n e r g y equation)

I n t e g r a l e q u a t i o n s of naotion and e n e r g y a r e obtained b y i n t e g r a t i n g e q u a t i o n s 2 . 5 . 2 and 2 . 5 . 3 r e s p e c t i v e l y , o v e r t h e b o u n d a r y l a y e r t h i c k n e s s . T h u s T J du 5" . _{<^ d ,— z — » •«- — ' - 2 2 d r / _ r- A\} — =« T= ( u . 3^,) + F u :T= + u. — -7= ( 2 , 5 . 4 ) p dx 1 2' "i 1 dx 1 r dx (Equation of motion)

7 i / ' » T d J . i | ' ü T a 7 . - . (|)__^ (2.5.5,

( E n e r g y equation)

8

-(8)

W, Mangier has shown that, by using the transformations presented below, it is possible to transform, these three boundary layer differential equations for an axisynametr ic flow into the three differential equations for the two-dimensional case. Hence, it pernaits the use of the solutions for the two dimensional case to derive the solutions for axisymmetric flow.

According to Mangier, the equations which transform the co-ordinates, velocities, and thermal properties of the axisymmetrical problem to those of the equivalent two-dimensional are as follows

1 ^ 1 . X • - 5 - / r*(5c) dx o , r (3D _ , L r _ ^ r ' - - - ] 3 . u = Tr 6. h = — h r 4. u, = Ü, 7. A = £ 5

With the above-m.entioned transformations it is possible to apply a solution of two-dimensional case to an axisymmetrical case. Hence, in the present report, attention is given to the two-dinaensional flow and wherever necessary a conaparison with an axisymmetrical case is made.

3. Wedge Flow

3 . 1 . Velocity Boundary Layer 3 . 1 . 1 . Momentuna equation

U

In the case of the flow past a wedge, the velocity of the potential flow is proportional to a power of the length co-ordinate from the stagnation point. i . e . u, = c x"^

wher^ 1. m JL

2 - ^

and 2, ir ^ : angle between the two planes of the wedge.

This condition reduces the partial differential equation of m^otion to an ordinary differential equation. Equation of motion is

8u 8u du, 8*u /o , n \

u r - + V ^ - - u, —r- + V • ( 3 . 1 . 1 . ) 9x 8y ' dx Q y2

with the transformations

the equation of motion reduces to

f + f f ^ ^ [ l - f ' * ] = 0 (3.1.3) where the boundary conditions are

f = 0 ; f « 0 , f ' = 0 "^

I = 0= . f'= 1 J (3.1.4)

It is interesting to note that for the case ^ : i , m : j , the equation 3 . 1 . 3 becomes

f"+ f f"+ 1 ( 1 - f " ) " 0 (3.1.5) The differential equation is identical to the differential equation of

rotationally symmetrical flow with stagnation point. Therefore, the solution for a two-dimensional wedge flow with ^ i ^ is also a solution for the axially symmetrical flow with stagnation point.

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Equation 3 . 1 . 3 was first deduced by Falkner and Skan and its solution was later investigated in detail by Hartree*^^'. The solution in t e r m s of the velocity profile is represented in Fig. 3 . 1 .

3 . 1 . 2 . Flat Plates

For a flat plate, where m : 0, the equation 3.3 simplifies to

f + ff" " 0 (3.1.6) This equation is the Blasius non-linear differential equation and of the third

order. The three boundary conditions a r e , therefore, sufficient to determine the solution conapletely.

10

-(32) (14) B l a s i u s solved t h i s equation in the f o r m of a power s e r i e s but Howarth

h a s solved t h i s equation with a h i g h e r d e g r e e of a c c u r a c y . Velocity profile obtained by Howarth i s shown in F i g . 3 . 2 .

(33)

L a t e r , P i e r c y and P r e s t o n pointed out a n o t h e r method which r e s u l t s in a s i m p l e solution b y a s u c c e s s i v e a p p r o x i m a t i o n in the equation.

f' o

TTT

f dC d€ ( 3 , 1 . 7 ) f df df 3 . 1 . 3 . S h e a r S t r e s s

S h e a r s t r e s s r on the wall at the position x i s given by 8u 9y _y=>0 ( 3 . 1 . 8 ) i . e .

H'

Mc-o

_{( 3 . 1 . 9 )} p u , V a l u e s of [ f ^ e « for s o m e v a l u e s of m a r e r e p r o d u c e d in T a b l e 3 . 1 and the v e l o c i t y d i s t r i b u t i o n i s shown in F i g . 3 . 1 . na - 0.091 - 0.0476 0 0.111 0.333 1 oo ^ - 0.199 - 0.10 0 0.2 0.5 1 2

Ff'l

0 0.319 0.470 0.687 0.928 1,233 1,687 C, x VSe _{f x} 0 0.44 0.664 1.024 1.516 2.466 OO TABLE 3 . 1 .

3.2, Thernaal Boundary Layer 3 . 2 . 1 . Energy Equation

For incompressible flow, in which the dissipation is neglected, the energy equation is

8T ^ 8T 9^T .„ - , .

u S~ + V r— = a—=- (3.2.1) 9x 8y 8y2

Equation 3,2.1 is linear in tenaperature, and consequently any suna of solutions of equation 3 . 2 . 1 . is also a solution of equation 3 . 2 . 1 . Equation 3.2.1 is transformed by utilising the variables of the Hartree equation (eq. 3.1.3) and a tenaperature function (T - T ) = B'X'^. For the particular case of this power function wall temperature variation, an ordinary differential equation is obtained. This is

0"+ f P r 0' - P r ( j | : ; i ) r f'0 = 0 (3.2.2) where the boundary conditions are

0 = 0 for § = 0 "A

and ƒ (3.2.3)

0 = 1 for § = CO J

3 . 2 . 2 . r : 0

( 3 . 2 . 4 ) For the isothernaal case,

0 ' d the solution is 0 '+ f P r 0' = O QO

i =

O r I 0, the equation 3, 0 f d l d€ f d § df 2 2 simplifies t o (3.2.5) (13)

Pohlhausen has solved this equation for a flat plate and a few values of Prandtl Number. Eckert has solved the equation for a Prandtl Number of 0.7 and for a few values of m.

It can be seen from equations 3. 2. 5 and 3.1.2 that for the flow over a flat plate with Prandtl Number equal to unitys

1. The velocity distribution and the temperature distribution in the boundary layer are identical.

2. The thickness of the boundary l a y e r s , both the thermal and the hydrodynamic, are equal.

12

-3 . St =

PC u,

P ' Pn' 2

( 3 . 2 . 6 )

T e m p e r a t u r e d i s t r i b u t i o n for the flat plate in the b o u n d a r y l a y e r , c a l c u l a t e d b y P o h l h a u s e n , i s shown in F i g . 3 . 3 . A s a l r e a d y m e n t i o n e d , the c u r v e for P r a n d t l N u m b e r : 1 g i v e s a l s o the velocity d i s t r i b u t i o n in the b o u n d a r y l a y e r .

3 . 2. 3 , Heat t r a n s f e r coefficient H e a t t r a n s f e r coefficient i s given b y

- - ( I f )

o r w h e r e Nu ( 0 ' ) y=0

i ^ (6')

1 = 0 = f ( P r ) 1 5=0 - P r / f dg e "' d? ( 3 . 2 . 7 ) ( 3 . 2 . 8 ) ( 3 . 2 . 9 )

R e s u l t s for s o m e v a l u e s of m and P r a n d t l Nunaber a r e r e p r o d u c e d in T a b l e s 3 . 2 and 3 . 3 . P r f(Pr) P r f(Pr) P r f(Pr) 0.6 0.7 0.8 0.276 0.293 0,307 0,9 1 1.1 0.320 0.332 0.344 7 10 15 0.645 0.730 0.835 TABLE 3 , 2 . m : 0 na 0 0.111 0.333 1 4

b%-o

0,414 0.444 0.471 0.496 0.514 Nu VRe 0.2925 0.331 0.384 0.496 0.813 TABLE 3 . 3 , P r : 0.7

A good i n t e r p o l a t i o n fornaula for m : 1 (Stagnation flow) i s

^ = 0,57 P r ° - ^ ( 3 . 3 , 0 ) VRe

and for m : 0 (flat plate) Nu

-= 0,332 P r ^ ( 3 . 3 . 1 )

F o r a fluid with a v e r y l a r g e P r a n d t l Nunaber, flowing o v e r a flat p l a t e , assunaption of l i n e a r v e l o c i t y profile l e a d s to

^ X 0.3387 P r ' ( 3 , 3 . 2 ) VRe

and for a fluid with a v e r y low P r a n d t l N u m b e r , flowing o v e r a flat p l a t e , a s s u m p t i o n of uniform v e l o c i t y profile l e a d s to

^ « 0.565 P r ^ ( 3 . 3 . 3 ) VRe

3 . 2 . 3 . (2 - / ? ) r : - 1

F o r t h i s p a r t i c u l a r c a s e , the equation 3 . 2 . 2 i s e x a c t and the solution by q u a d r a t u r e i s p o s s i b l e . S o l u t i o n i s , p ^ j ^ g

r

e = e ° ( 3 . 3 . 4 ) T h e t e m p e r a t u r e g r a d i e n t at the wall ( ^ ) = B'(e') X -^==^ ( 3 . 3 . 5 ) _{ö y / „ _ n €=0 ^ rjT} y=o X , , ,

i s s e e n t o be z e r o for a l l v a l u e s , except the indeternainate point at x : 0, At that point, the t e m p e r a t u r e of the fluid adjacent to w a l l i s infinite and the heat i s i n t r o d u c e d i m p u l s i v e l y at t h i s point,

In the p h y s i c a l a s p e c t , t h i s i s a c a s e of an a d i a b a t i c wall when the heat i s i n t r o d u c e d i m p u l s i v e l y at the s t a g n a t i o n point.

It can be s e e n f r o m equation 3 . 3 . 5 that for r ( : ) > - 1 the heat flow Vm+1 y / 2 \ at the s u r f a c e i s f r o m the s u r f a c e to the b o u n d a r y l a y e r , and for r I ——: ) < " 1 the h e a t flow at the s u r f a c e i s frona the b o u n d a r y l a y e r to the s u r f a c e .

14

-3 . 2 . 4 . Other values of r

For other values of r , the numerical solution of equation 3.2,2 is necessary. Chapnaan and Rubesin'^^', Schuh(22) and Levy'23) have carried out numerical solutions for certain values of r , P r , and m. Levy has covered a wider range of r , P r and na,

The results for P r : 0.7 are plotted in Fig. 3.4.

F r o m his r e s u l t s . Levy found that the local heat transfer coefficient can, with the exception of large negative r values, be expressed within 5 per cent a s :

. — ' B (m, r) (Pr)^

/ R e ( 3 . 3 . 6 )

where the function B(m, r) can be approximated by the equation

B(m, r) = -^-1 + 1 _m+1

0.37 + ( ° : i ^ )

m

_{+ 1 r / 2,r, \ °-l°^'}

X 0.57 { 7 ^ 1 + 0.205) (3,3.7)

and the exponent X of the Prandtl Number varies from 0. 254 to 0,367 for - 0.904 < m < 4. The values of X a r e given in Table 3.4.

^ X 1.6 0.367 1 0.353 0 0.327 - 0.199 0.254 TABLE 3.4

H. Schuh , Leveque and Lighthill have found a solution for the equation 3.2.2 by assuming that the thermal boundary layer is very small compared to the hydrodynanaic boundary layer and hence by replacing velocity profile by a tangent at the wall f' = b é • Thus an asymptotic solution of equation 3.2.2 is 6 = 2"^ r ( | ) L ( " B i ( L ' ) - B^ ( L ' ) 1 where 3 '

rPrb(2-/?)l [r^el*

( 3 . 3 . 8 ) (3.3.9)

and B the modified Bessel function of the first kind and order c, as defined, '^ (31)

for instance, by McLachlan and V the Gamma function. The heat transfer coefficient becomes, Nu >'Rë = 0.72895 ( P r b r)^ (2 - ^ ) ' ' ( 3 . 4 . 0 )

4. Description of Methods 4 . 1 . Introduction

All the approximate methods, instead of satisfying the differential equation for every fluid particle, satisfy boundary conditions near the wall and the region of transition to the external flow, together with certain compatibility conditions. In the remaining region of the fluid in the boundary layer, only an average value over the differential equation is satisfied, the average being taken over the whole thickness of the boundary layer. Such naean values can be obtained frona the monaentum theory and the law of the conservation of energy. These a r e , in turn, derived frona the equations of naotion and energy by integrating over the boundary layer thickness. These integral equations of motion and heat flux, with assumed velocity and temperature profiles in the boundary layer, form the basis of the approxinaate naethods.

All methods can be divided into three groups, depending on their applicability. 4. 2. Arbitrary main stream velocity and uniform surface temperature

(a) Frossling (29)

Frossling solved the energy equation in a very similar manner to that of Pohlhausen'^3' by assuming a power s e r i e s for the main s t r e a m velocity, velocity in the boundary layer, and the temperature distribution in the boundary layer. This method is very cumbersome, particularly for slender body forms, where a large number of t e r m s in the power expansion are required.

(b) Eckert

Eckert solved the differential equation for the thickness of the thermal boundary layer by assuming that the temperature profile and the gradient of the thermal boundary layer thickness are the same, as on the wedge with the same velocity gradient outside the boundary layer. This naethod, although l e s s than Frossling, involves lengthy and tedious calculations,

(c) Allen and Look

Allen and Look have used Reynolds' analogy to obtain the heat transfer from the wall shear s t r e s s . The shear s t r e s s is obtained by assuming a velocity

profile in the boundary layer similar to that over the flat plate. Frick and McCullough^^^' extended this naethod for any Prandtl Number by suggesting a simple multiplier.

This method, though very simple, gives very high values of heat transfer when compared to E c k e r t ' s exact solution for wedge flow.

Procedure for the calculation of heat transfer can be summarised as: 1. The calculation of the boundary layer thickness from

/X

5.31"/ u " • d x vl/ u , 8.1^ •I o ' (u,)

16

-The boundary layer thickness at the stagnation point is obtained from

R2 = ^ (4 2)

^ Stag 5u, ^^-^^ where F is the radius of curvature at the stagnation point.

2. The calculation of heat transfer coefficient from

^ = X ( | ) ( P r ) » (4.3) where 5t is the shape factor and is equal to 0.765 for the Blasius velocity profile.

X is assumed to be uniforna up to the nainimuna pressure point and then reducing linearly to zero at the separation.

(d) Squire (4)

Squire assumed a Blasius velocity profile in the boundary layer and allowed the displacement thickness to vary with x in a manner appropriate to the assunaed main stream velocity. He then solved the energy equation by assuming a tenaperature profile in the boundary layer sinailar to the velocity profile. This naethod contains more calculations in comparison to that of Snaith and Spalding^^', Allen and Look^^^' and Merk'3«), The method of Squire requires the following four steps.

Calculation of I -r-j (j> i T~ j from equation 4.4

X

(^)K^)

0.3861 P r X u, dx dx ( 4 . 4 ) ( ^ \

2. Deternaination of the values of I g—j frona the graph or table given by Squire. They are reproduced in Fig, 4,1 and Table 4.1 respectively.

3. Calculation of 6 from equation 4. 5

. 1.721

6 a

1 Ï " / x

u, dx (4.5)

4. Heat transfer from 2, 3 and equation 4.6 h 1 0,5715

k A A <^-«>

0,5 0.625 0.667 0.833 1.0 1.25 1,429 1.667 1.818 2,0 0.2075 ' 0.257 0.272 0.332 0.3«61 0.4563 0.4988 0.5478 0.5750 0.5994

(t)Xt)

0.052 0.100 1 0.121 0.230 0.386 0.713 I 1.018 1.522 1.901 2.398 T A B L E 4 . 1

(e) Snaith and Spalding

Smith and Spalding used the s a m e p r o c e d u r e a s t h a t of E c k e r t but r e p l a c e d E c k e r t ' s g r a p h i c a l i n t e g r a t i o n by a s i m p l e q u a d r a t u r e . F u r t h e r , an e r r o r function i s included to account for the deviation of the s i m p l e q u a d r a t u r e f r o m t h e e x a c t wedge s o l u t i o n . T h i s m e t h o d i s v e r y s i m p l e and s t i l l g i v e s the r e s u l t s that a r e in a g r e e m e n t with E c k e r t . The equation d e r i v e d by Smith and Spalding can be used only for fluids with P r a n d t l N u m b e r : 0.7. Smith and S h a h ' 3 ^ ' * 2 ' have extended t h i s m e t h o d s o that the heat t r a n s f e r coefficient can b e c a l c u l a t e d for fluids with P r a n d t l N u m b e r in the r a n g e of 0.5 < P r < 20,000. T h i s method r e d u c e s to the e v a l u a t i o n of A^ f r o m equation 4 . 7.

A* _ ± = V A_ B /" B - 1 , 1 / B - 1 j u, dx + -g j u^ E dx 4 (4.7)

and at the front s t a g n a t i o n point A / B

d(u,) dx

(4.8)

w h e r e A and B a r e the c o n s t a n t s depending on the P r a n d t l N u m b e r . A^ i s r e l a t e d t o the h e a t t r a n s f e r coefficient by the equation:

k/ A ( 4 . 9 )

T h e v a l u e s of A and B for different P r a n d t l N u m b e r s a r e p r e s e n t e d in T a b l e 4 . 2 and F i g . 4 . 2 .

18 -P r 0 . 5 0.7 0 . 8 1.0 1.4 2 3 5 10 50 1 0 0 5 0 0 1000 5000 10000 20000 A ( P r ) ^ 9.369 9.204 9.146 9.069 8.975 8.902 8.840 8.791 8.754 8.724 8.712 8.717 8.716 8.716 8.716 8.716 F • B 2.791 2.869 2.900 2.952 3,026 3.102 3.181 3.273 3,380 3.558 3.606 3.692 3.716 3753 3.764 3,773 ^ ! < - > ' 3.357 3.207 3,154 3.072 2.966 2.870 2.779 2.686 2.590 2.452 2.416 2.361 2.346 2.322 2,316 2.310 TABLE 4.2

E4, which is a function of

du

dx - , is given in Ref. 39 for 0.7 < P r < 10 and they a r e reproduced in Table 4 . 3 , For P r values greater than 10 it will often be satisfactory, when it is desired to include the e r r o r term in the calculation, to assume sinailarity of the E4 values with those for P r : 10. By this it is meant that

^ - [ ( ^

^)l^]

(4.10)

P r P r P r P r P r : 0.7 0.8 1.0 5 10 < V E4 A' V E4 A» 4 V E . A ' V E . A* 4-V E4 du, dx d u , dx du, dx du, dx d u , dx 6.05 1.042 5.427 1.027 4.541 0,919 1.327 0.343 0.798 0.238 4,07 0 3.663 0.011 3.074 -0.002 0.919 0 0,554 0

r

2.26 -0.68 2.039 -0,618 1,721 -0.551 0.531 -0.202 0.325 -0,126 1.01 -0.67 0,917 -0.614 0.779 0,543 0.248 -0.208 0.154 -0.132 0 0 0 0 0 0 0 0 0 0 -1.02 2.06 -0.937 1.935 -0.809 1.716 -0.287 0.740 -0.185 0.491 TABLE 4 . 3

T h e e r r o r t e r m E^ i s s m a l l and m a y be ignored when high a c c u r a c y i s not d e m a n d e d . If the e r r o r t e r m i s not excluded in equation 4 . 7, then it can be solved by an i t e r a t i o n p r o c e d u r e .

(41) (37) The e x p e r i m e n t a l r e s u l t s of Shah and Sogin e t . a l have c o n f i r m e d that t h i s method p r e d i c t s the h e a t t r a n s f e r coefficient within 5%, except n e a r s e p a r a t i o n , w h e r e it p r e d i c t s r a t h e r high v a l u e s .

(19)

Smith and Spalding have a l s o s u g g e s t e d a n o t h e r method in which t h e y give an equation for mixed t e m p e r a t u r e t h i c k n e s s .

na

V

19.78

u,dx ( 4 . 1 1 )

w h e r e A i s r e l a t e d to the heat t r a n s f e r coefficient b y

m •'

k C

m _(4.12)

and C d e p e n d s on m

r ^ du,"i

_{The g r a p h and t a b l e for t h i s r e l a t i o n s h i p a r e}

20 -^ m du, V d *

c

m 9.89 1.56 4.88 1.47 1.96 1.39 0 1.30 1 -1.48 1.20 TABLE 4 . 4

T h i s method h a s not been extended for a r a n g e of P r a n d t l N u m b e r s . The equation 4 , 1 1 i s , t h e r e f o r e , valid only for fluid with P r a n d t l N u m b e r : 0.7, (f) M e r k

M e r k h a s used the e x a c t solution of E c k e r t in d e r i v i n g h i s a p p r o x i m a t e m e t h o d , w h e r e he u s e s the wedge solutions of the d y n a m i c p r o b l e m in a

m a t h e m a t i c a l a r g u m e n t l e a d i n g to a s e r i e s expansion in which only the f i r s t t e r m i s r e t a i n e d . T h i s method i s a l s o s i m p l e and r a p i d a s the method of Smith and Spalding. The heat t r a n s f e r coefficient can be calculated f r o m

i

A 4 2

f.

dx ( 4 . 1 3 ) u (E . ' o

w h e r e A i s r e l a t e d to the heat t r a n s f e r coefficient by k

h ( 4 . 1 4 )

The quantity E i s a function of the P r a n d t l N u m b e r and the wedge v a r i a b l e A , which p l a y s a r o l e quite analogous to /9 and, indeed, i s i d e n t i c a l l y /3 for a p u r e wedge flow. The v a l u e s of E , a s a function of A for s o m e v a l u e s of the P r a n d t l N u m b e r , a r e r e p r o d u c e d in F i g . 4 . 4 .

u, dx d u ,

dx~ ( 4 . 1 5 )

The two d i m e n s i o n a l s t a g n a t i o n value c o r r e s p o n d s to A : 1, and the s e p a r a t i o n v a l u e s to A : - 0 . 1 9 8 8 .

4. 3. Uniform main stream velocity with non-isothermal surface 4 . 3 . 1 . Arbitrary surface temperature

The method for determining heat transfer for non-isothermal surfaces is similar to the methods used in deternaining the deflection of beams subjected to a r b i t r a r y load distributions. The energy equation of the boundary layer is linear in the fluid temperature, if the fluid properties are assumed to be constant. This allows the super-position technique to be employed. Rubesin has shown that the heat transfer rate for an arbitrary tenaperature variation can be determined

by superimposing a number of "step tenaperature distribution", so that the summation of the steps yields the actual variable temperature distribution, and the heat

transfer at any point is equal to the sum of the heat transfer r a t e s attributed to all steps upstream of the point in question. This results in heat transfer r a t e s from the non-isothernaal given by the following integral expression.

^ ( x ) " / '^^^-^^'^MQ <'-^«^ Here the Kernel function h(5x) is the heat transfer rate at position x due to

step tem.perature rise dT .p. at the position ë . It should be noted that the integral of equation 4.16 must be taken in the "Stieltjes" sense rather than in the ordinary "Rienaann" or " a r e a " sense. Various investigators have obtained an equation of heat transfer rate for step rise in the surface temperature case, by assuming velocity and tenaperature profiles and then solving the energy equation.

(38)

Leveque assumed a linear velocity profile, independent of x, and solved the differential energy equation. He obtained

, k P r ' , _P_ >3 ^ è , x ) ' 3(I)J~" \ n ' I r- , . - J. y=0J ( x - ê ) ' ^ (4.17) (18)

Rubesin assumed linear velocity and temperature profiles and solved the integral energy equation by further assuming that the thermal boundary layer varies proportionally to the monaentum thickness. He obtained

_ l

h / . ^ - ^^^^ P ^ ' Re* [l-<iU)^'] ' (4.18) ( é . x ) X X |_ J

velocitj integral energy equation. He obtained

(17)

Eckert . assunaed cubic velocity and temperature profiles and solved the

( ^ . x )

22

-4 . 3 , 2 . Arbitrary surface heat flux

Similar to a problem of finding heat transfer for an a r b i t r a r y surface temperature case, it is also an equally important problem to find surface tenap-erature when the surface heat flux varies arbitrarily. Again, in a similar way, the wall temperature can be determined frona

ë=x

(T - T J w

C=o

g ( 4 . x ) q ^ ( ^ ) d 5 (4.20)

where g(4,x) is the wall temperature at x due to a unit heat flux at 4 . Klein and Tribus have shown a mathenaatical manipulation procedure by which the value of function g(S>x) can be obtained from the known function h(4.x).

Thus the values of the function g(5,x) obtained from the equations of Leveque Rubesin(^^) and E c k e r t ^ ? ) are

(38) g(S.x) - 3j^ g(4.x) P r

(ï)i

6(i)'. (1)1 and g(4.x) respectively. 6(i)'. (1)1 0.33k (40) 1 Pr"» 0,304k

Pr-i

[(^4=0 J

Re"^

[l

-X _ i r-3 (x (5/x 3

-a-«

2 _ 2

1 '

(4.21) (4,22) (4.23)

Smith and Shah assunaed the cubic velocity and temperature profiles and obtained a very simple equation for an a r b i t r a r y heat flux problem by directly integrating the energy equation for a step heat flux case. They obtained

i 2 2 / r > „ \ 3 (T - T ) 2.395 (Re )^ (Pr) X (1 - £/x)» d q

% (a

(4.24)

4 . 4 . Arbitrary naain streana velocity and a r b i t r a r y surface temperature (a) Lighthill

(6)

Lighthill solved the energy equation in Von Mises form, assuming a linear velocity profile in the thermal boundary layer. He obtained

hU.x) k

(in

Pr^ f-^f/r, ,

VT , , dz (j(z) 1

and t h e r e f o r e for an a r b i t r a r y s u r f a c e t e m p e r a t u r e c a s e 4 — ^ r ^"^ n"» Nu 0.486 P r * | x ^ 1 ; — / / i— , ,rp (4.26) F o r wedge flow

r.-r^ UMl'-f) l'ö]

( 4 . 2 7 )

and the heat t r a n s f e r coefficient for wedge flow i s given by C=x

Nu

^ ^ ( T - T , )

0.3935 „ i , ^ , .

P r (na -^ 1)

^ f('o) ƒ [ l-(Ux)*<™^'>l dT..,,, (4.28)

_w(5) 4»o

(30)

U s i n g a sinailar a p p r o a c h to that of Klein and T r i b u s , the equation for a n a r b i t r a r y h e a t flux beconaes: 1 , 2 \ ï ' ( 5 , x )

2 / i i f l V 1

^(ITTk'' \pPrJ -JZ

^-1

X •- o T , . dz w(z) ( 4 . 2 9 ) T h i s method i s expected to give good r e s u l t s for the fluid with high P r a n d t l N u m b e r , a s the a p p r o x i m a t i o n of a t a n g e n t i a l velocity profile at the wall i s t r u e only for the c a s e when the t e m p e r a t u r e b o u n d a r y l a y e r i s a snaall f r a c t i o n of the v e l o c i t y b o u n d a r y l a y e r . H o w e v e r , it i s found to give f a i r l y good r e s u l t s even when the P r a n d t l N u m b e r i s a s low a s 0,7. T h i s method n e e d s an i n i t i a l knowledge of the d i s t r i b u t i o n of the s h e a r s t r e s s o v e r the s u r f a c e ,

(b) Bond (43)

Bond a l s o a s s u m e d a l i n e a r v e l o c i t y profile and solved the d i f f e r e n t i a l equation for wedge flow. F o r the wedge flow be obtained

and w h e r e and

^.x)"[^]^^x ^-< i^-^^i-n

-i

2c / 2 Y ^ - i „ - i / X g(4,x) = 9-(|)..k \ l - ^ ) ^ « ^ b : P r / 6 f''(o) c : 1(1 -I- m) m_+l 2 ( 4 . 3 0 ) ( 4 . 3 2 )

24

(c) Anabrok

A m b r o k h a s d e r i v e d an a p p r o x i m a t e method for calculating the h e a t t r a n s f e r coefficient b y a s s u m i n g that a r e l a t i o n of the type Nu : A(Re \)^, which i s valid for a flat plate with a uniform s u r f a c e t e m p e r a t u r e , i s a l s o valid for a problena w h e r e m a i n s t r e a m , v e l o c i t y and s u r f a c e t e m p e r a t u r e v a r y a r b i t r a r i l y . With t h i s a s s u m p t i o n h e h a s solved an i n t e g r a l e n e r g y e q u a t i o n . He obtained Nu ^^ÏÏê 0,332 P r ^ (T - T ) w 1 — /

u,(T

- T,

f

dx u,x J ^ w ' x=x (4.33)

w h e r e h e a t i n g s t a r t s at x : x . T h i s method i s v e r y s i m p l e but it i s found to give low v a l u e s of h e a t t r a n s i e r when conapared to t h o s e of E c k e r t for wedge flow.

(d) Spalding (12)

Spalding h a s inaproved on L i g h t h i l l ' s naethod by a c o r r e c t i o n which a c c o u n t s vfor the d e p a r t u r e f r o m l i n e a r i t y of the v e l o c i t y profile within the t h e r n a a l b o u n d a r y

l a y e r , and which c o m p r e h e n d s the influences of P r a n d t l N u m b e r , p r e s s u r e g r a d i e n t , body f o r c e s and n o n - c o i n c i d e n t s t a r t of v e l o c i t y and t h e r m a l l a y e r s .

In t h i s m e t h o d , the momentuna t h i c k n e s s , 6^ , of the velocity b o u n d a r y l a y e r i s to be evaluated f i r s t by a p r o c e d u r e s i m i l a r to t h o s e of Walz(35) and T h w a i t e s . T h i s involves the evaluation of

.X 6» _0.4418 5.17 4.17 dx u , 5.17 4.17 Egdx (4.34) 52 du w h e r e E_ i s a t a b u l a t e d function of — -r-' ' V dx 6* du _{2 1} V dx -0.0682 0.0266 0 0.0333 0.0611 0.0855 E 2 -0.026 -0,0058 0 0,0033 0,0019 0.0

ikJ

00 1 36.9 20.5 12.75 9.50 7.70 TABLE 4 . 5

The integral containing E^ is once again a snaall correction term which can be onaitted for most purposes.

With 6 . . given by equation 4.34, 5 . is obtained by interpolation from a table of the ratio 6^/6^ with argument F _2_ £^i | • ^ is the "shear

L I' dx J thickness".

Thereafter, the heat transfer coefficient can be obtained by the evolution of the equation 4, 35

(4.35)

\ CtJ '•''"/, (y''^'^""My ^^^

L o o

6 A du where F is a graphically presented function of the argument —^— -r-(Fig. 4,5), and x is the value of x where the heating s t a r t s .

This time the correction term F is not always snaall. Since A^ appears in Its argunaent, sonae iteration is necessary.

(e) Schuh (5)

Schuh selected a dependent variable, which is a function of the ratio of heat flow across the whole boundary layer to the tenaperature gradient at the wall in a suitable dimensionless form, and then integrated the momentum and the energy equations.

Schuh's method involves similar steps to those of Spalding's method,nanaely: 1. The deternaination of 6 by the Walz-Thwaites technique.

2 2 &! du,

2. Calculation of the auxiliary function -— — frona 1. and the known velocity gradient.

6* du

3. iS and Z , a r e the functions of ~ — and to be read from the graph ( F i g . 4 . 6 ) . 4. Calculation of n from equation 4. 36

dT

^

= (

T " ^

^ w 1 w dx

(4.36)

5. Calculation of functions P and G from equations 4. 37 and 4, 38 by an iterative procedure.

26-w' ^'

J T ^

-

T ]

G (2 - #)

-j*

[u,xf

/ O 1_ ( 2 i 9 ) x 4 3/ ;y _, t Q / 2 (,J. _ , j , )7a dx and w h e r e G : 0.57 (^ . 0.205)^-^°^ ^ 0 . 3 7 . 0 , 0 6 ^ ^^_^^^, ^l-Sr A : 1 -H(2 -/3)n and v a l u e s of r a r e given in T a b l e 4 . 6 . (4,37) (4.38) (4.39)

1 ^

T 1.6 0.367 1.0 0.355 0 0.327 -0.199 0.254 TABLE 4 . 6

E q u a t i o n 4 . 38 i s to be used for n < 4 and 0.3 < P < 1.3 but for n > 4, -0.14 < /S < 1,0 and P > 0.2, the value of function G to be obtained f r o m equation 4 . 4 0 and T a b l e 4 . 7

/

^l

du \ i

6. Heat t r a n s f e r coefficient then to be c a l c u l a t e d frona equation 4 . 4 1 ^ , P G M u. (4.40) (4,41) ^ -0.14 0 0.2 0.4 0,6 0.8 1.0 V d x -0.041 0 0.033 0.053 0.068 0.078 0.0854

fö" du 1

c —s —1

L" dx J

0.583 0.714 0.782 0.809 0.814 0.805 0.781 T A B L E 4 . 7

(f) Seban and Drake

(2) (3)

Seban and Drake solved the momentum equation by a procedure similar to that of Eckert, and obtained heat transfer rate by assuming that it is related to the monaentum thickness in the sanae manner as on the wedge.

This method also involves sinailar steps to those of Spalding and Schuh. 1. The determination of 6^ and the auxiliary function — — .

2. Evaluation of function Z, and /9 from Fig. 4 . 6 .

3. The rate of heat transfer at the wall for a particular case, when wall temperature is a power function of length, e.g. (T - T ) : AX"^, is given by n Z, q^ - - k A x g'(o) y (4.42) or ^ 1 -*- - — (4.43) ^2 gto) Z

Where g'(o) is a function of /3, n and P r , The value of g'(o) is relatively. insensitive to /S but depends critically upon the exponent n of the wall tenaperature function Ax . The function g'(o) for Prandtl Number : 0.7 is shown in Fig. 4 . 7 .

By superimposing, the heat transfer coefficient can be determined for a variation of the surface temperature of the type

(T - T ) » A x ' ^ + Ax"^-(- A x " ' (4.44) W < 1 2 3

28 -References 1. Eckert, E . R . C . 2. Seban, R.A. 3. Drake, R.M. 4. Squire, H . B . 5. Schuh, H. 6. Lighthill, M.I. 7. Tribus, M. , Klein. J . 8. Mangier, W, 9. Smith, A.G. , Spalding, D . B . 10. Ambrok, G.S. 1 1 . H a r t r e e , D.R. 12. Spalding, D . B . 13. Pohlhausen, E, 14. Howarth, L. 15. Schlichting, H. 16. Shapiro, A.H. 17. Eckert, E . R . G . , Drake, R.M. 18. Rubesin, M.W. 19. Hartnett, J . P . , e t . a l . 20. Allen, H . J . , Look, B . C . 21. Frick, C . W . , McCullough, G . B . V.D.I. Forschungsheft 416, 1942. University of California, Institute of

Engineering, Research Report 2-12, 1950. J . A e r o . S c i . , Vol.20, p.309, 1953.

R. & M. 1966, 1942,

K . T . H . A e r o T.N. 33, Stockholm, 1953.

P r o c . of Royal Society, A, Vol.202, p.359, 1950. J . A e r o . S c i . Vol.22, p.62, 1955.

Z . A . M . M . , Vol.28, 1948.

Jnl. Royal Aero. Soc, , Tech. Notes, Vol, 62 , 1958, Sov.Phys.Tech.Phys. 2,1979, 1957.

P r o c . Camb.Phil.Soc. , 33.223, 1937. J . Fluid Mech. Vol.4, 1958.

Z . A . M . M . , Vol.1, p.115, 1921.

P r o c . Roy. Society, A, Vol.154, p 364, 1936. Boundary Layer Theory, Pergamon P r e s s , 1955. The Dynamic and Thermodynamics of Compressible Fluid Flow.

Heat and Mass Transfer. McGraw-Hill. 1959.

M.S. Thesis, University of California, Berkeley, 1945.

W . A . D . C . Tech. Report 56-373, 1958, N . A . C . A . Report 784, 1943.

References (Continued) 22. Schuh, H. 23. Levy, S. 24. Falkner, V,M. , Skan, S,W. 25. Livingood, J , N , B. , Donoughe, P . L. 26. Rubesin, M.W. 27. Baxter, D . C . , Reynolds, W.C. 28. Drake, R.M. 29. Frossling. 30. Klein, J . , Tribus, M. 31. McLachlan, N.W. 32. 3 3 . 34. 3 5 . 36. 37. 3 8 . 39. 4 0 . 4 1 . 4 2 . 4 3 . 4 4 . Blasius, H.

Piercy and Preston Chapman, D . R . , Rubesin, M.W, A. Walz Merk, H . J . Sogin, H.H. , et. al Leveque, M,A, Smith, A,G. , Shah, V . L . Smith, A . G , , Shah, V , L . Shah, V . L . Smith, A.G. , Shah, V . L . Bond, R. Spalding, D . B . , Pun, W.M.

J . Aero.Sci. , VoL20, No. 2, p. 146, 1953. J . Aero.Sci. Vol.19, No. 5, 1952.

R.M.1314, British A , R . C . 1930.

N . A . C . A , TN,3588,

Paper No. 48-A-43 (Preprint) A.S.M.E., 1948.

Jnl.Aero/Space Sci. Vol.25, No.6, 1958.

Jnl. Aero. Sci. , Vol.20, No. 5, 1953.

Lunds Universitets Arsskrift, N . F . A v d , 2 , 36 No.4, 1940.

Eng, Res. Inst. , Univ. Michigan, 1952.

Bessel Function for Engineers, Oxford University P r e s s , 1934, Z . Math, U , P h y s , , 56, 1, 1908. Phil.Mag. (7), 21, 1936. J . A e r o . S c i . , Vol.16, No. 19, 1949, Lilienthal-Bericht, 141. 8. 1941. J , F l u i d Mech, Vol,5, P a r t 3, 1959. A . F . O . S . R . TR-60-78, 1960. Ann, Mines, 13, s 283. 1928,

Int. J . Heat Mass Transfer, Vol.3, p . l 2 6 , 1961.

J . Aero/Space Sci. , Vol.28, p 738, 1961,

M,Sc.(Eng.), Thesis. University of London, 1961. Int. J.Heat Mass Transfer, Vol.5, p 697, 1962.

University of California, March 15, 1950. Int. J.Heat Mass Transfer, Vol.5, p 239, 1962.

FIG. 3 1 . VELOCITY DISTRIBUTION IN THE LAMINAR BOUNDARY LAYER I N . THE FLOW PAST A WEDGE.

FIG.3-3. TEMPERATURE DISTRIBUTION OVER HEATED PLATE FOR VARIOUS PRANDTL NUMBERS.

O B 0 6 O 2 /

,y

r

y

/ y yf' ^ ^<!^ ^ =s— o O S I 7 - \ 57

FIG. 3-2. VELOCITY DISTRIBUTION IN THE BOUNDARY LAYER ALONG A FLAT PLATE.

-R" t 2 / / / / / / / /

//l

s - i 1 ^ 6 ^ ^ ^ 0 o 199

FIG.3-4. LOCAL HEAT TRANSFER COEFFICIENTS FOR WEDGE FLOWS WITH VARIABLE WALL TEMPERATURE

(^)*Gf)

y

'^

0^

L(3:

^

^y

/

, / ^

/

A

*(-^) 0 4 O 6 O B 1 8 2 0

C^)

FIG.4L GRAPH OF THE FUNCTIONS ( T ^ ) * ( J ^ ) AND * ( j * )

-2 9 2 7 2 5 , > */

/ ^

^x

A f f

ii)J_

~

FIG.4 2. GRAPHS OF DEPENDENCE OF NUMBERS A AND 8 ON a.

Cr = PRANDTL NUMBER I i

y

— t »

y

1 6 15 ! 4Ï 1 2 1 ^

r^

^^

^ .

u_-^

—1

y / . ^

r

^psMS--rTi

1 O B -•^^ 0 7

-—H

o 2 0 4 O fc 0 1 FIG. 4.3. C „ AS A FUNCTION OF _{d x}

FIG. 4-4. Eo AS A FUNCTION OF THE WEDGE VARIABLE A, FOR SOME VALUES OF THE

" 6 4 "'•' ' ' f . \ ^ » / / ' "

, . i

/ r / A . 6 . du. ~ \ FIG. 4-5. GRAPH OF THE FUNCTION F ( -^—^ -—!• )•

V, V dx /

FlG.4-6. HARTREE PARAMETER ^ AND Z, AS A FUNCTION OF

PARAMETER

4 4^ =

r

o o e ó / V . — - o P,= o 70 O I 2 3 4