Study of heat and mass transfer from non-isothermal surfaces

THE COLLEGE OF AERONAUTICS

C R A N F I E L D

STUDY O F HEAT AND MASS TRANSFER

FROM NON-ISOTHERMAL SURFACES

by

C O L L E G E O F AERONAUTICS N O T E NO. 135

ASTIA D O C U M E N T NO: A D

C O N T R A C T A F 61(052)-267

TECHNICAL REPORT

THEORETICAL AND EXPERIMENTAL STUDY O F H E A T

A N D MASS TRANSFER F R O M NON-ISOTHERMAL SURFACES

by

A. G. Smith and V. L . Shah

D e p a r t m e n t of A i r c r a f t P r o p u l s i o n College of A e r o n a u t i c s

C r a n f i e l d , England

N o v e m b e r 1962

"The r e s e a r c h r e p o r t e d in t h i s document h a s been s p o n s o r e d by PROPULSION RESEARCH DIVISION, AFOSR, OAR. UNITED STATES AIR F O R C E .

No. AF 61(052)-267, and monitored by the Air Force Office of Scientific Research of the Air Research and Development Command.

The authors are grateful to many individuals in the fulfilment of this r ese ar c h. Professor D. B. Spalding of Imperial College, London, suggested the use of his velocity law in the theoretical investigation. Dr. A. J . M. Spencer, of

Nottingham University, assisted with the discussion of the methods of numerical solution. The Department of Mathematics at The College of Aeronautics gave pernnission to use the Digital Computer, and the Deputy Head of the Department of Aircraft Propulsion, Mr. J. R. Palmer, gave extensive help in the programming for the digital computer.

The authors are also grateful to the Workshop Staff of the Department of Aircraft Propulsion for the manufacture of the test rig. Mr. E. R. Norster

gave his assistance in the use of the Infra-red Gas Analyser, T. McNeill assisted in the running of the test rig, and V. Kaul assisted with the experimental

measurements.

This work was carried out in the Department of Aircraft Propulsion, The College of Aeronautics, Cranfield, England.

ABSTRACT

The numerical solution of the partial differential equation in the in-compressible turbulent boundary layer has been obtained for step

—ZZZ and for Prandtl numbers 0.7, 1 and 7. The Schmidt method of

pu, C VC./2 1 p f

integration was used and the integration was carried out on a Ferranti Pegasus Digital Computer. A method has been developed to apply this numerical solution for obtaining surface and fluid temperature for the case of arbitrary distribution of heat flux at the surface.

Simultaneously, approximate equations for calculating heat transfer over a flat plate with arbitrary heat flux are derived for laminar and turbulent flow.

To verify the theoretical solution, experiments were made in which the concentration profiles of the injected gas (carbon dioxide) at different stations were measured when the pipe had another gas (air) flowing turbulently through in the axial direction, and the gas injected was passed through a porous section of the tube wall.

A b s t r a c t Contents Introduction

1 . 1 . The P r o b l e m 1.2. P r e s e n t Work Review of the P r e v i o u s Work

2 . 1 . Introduction

2 . 2 . Uniform m a i n s t r e a m velocity - l a m i n a r flow 2 . 3 . Uniform m a i n s t r e a m velocity - turbulent flow 2 . 4 . A r b i t r a r y m a i n s t r e a m velocity - l a m i n a r flow 2 . 5 . A r b i t r a r y m a i n s t r e a m velocity - turbulent flow Heat T r a n s f e r over a F l a t P l a t e with A r b i t r a r y Heat Flux -A p p r o x i m a t e -A n a l y s i s

3 . 1 . L a m i n a r Flow 3 . 2 . T u r b u l e n t Flow

3 . 3 . A p p r o x i m a t e A n a l y s i s a s s u m i n g l i n e a r t e m p e r a t u r e profile ( P r = 1)

N u m e r i c a l Solution for A r b i t r a r y Heat Flux 4 . 1 . T h e o r y

4 . 2 . N u m e r i c a l Solution

4 . 3 . F o r m of P r e s e n t a t i o n in the R e s u l t s 4 . 4 . Check by heat b a l a n c e

4 . 5 . Dependence of the Spalding Function and H on the P r a n d t l N u m b e r

4 . 6 . Application of the solutions to the c a s e of a r b i t r a r y heat flux at the wall

D e s c r i p t i o n of A p p a r a t u s 5 . 1 . Introduction 5 . 2 . A i r Section 5 . 3 . Diffusing Section 5 . 4 . T e s t Sections 5 . 5 . CO Section 5 . 6 . M e a s u r e m e n t s

Contents (Continued)

6. Test Procedure - Calculation Procedure 6 . 1 . Introduction

6.2. Procedure of Test 6 . 3 . Preliminary Tests

6.4. Calculation Procedure (Experimental) 6 . 5 . Calculation Procedure (Theoretical) 7. Results and Conclusion

7 . 1 . Results 7.2. Conclusion 8. References 9. Symbols

1. Introduction 1.1. The Problem

The initiation of this investigation was the idea that there might be important advantages in the construction of a solid propellant grain by the assembly of alternate "pineapple rings" of different propellants. This problem was therefore investigated at some length both theoretically and by means of an idealised experiment.

In considering the operation of such a grain, the first problem is that of determining the manner of mixing the gases deriving from the alternate slices.

This problem idealises to the problemi of finding mass transfer coefficient and the concentration in the boundary layer for an arbitrary mass diffusing surface. Heat transfer and mass transfer are similar processes with similar differential equations. Therefore, in the case of heat transfer, this is a problem of finding heat transfer and temperature profile in the boundary layer, for non-isothermial surfaces. The present r e s e a r c h is a contribution to the solution of this problem.

In the uniform surface temperature case, analytical predictions of the local heat transfer, for laminar flow over a flat plate and wedge, have met with considerable success. When the flow in the boundary layer is turbulent, then the momentum and energy equations for the laminar boundary layer apply with the addition of the eddy transport properties to the viscosity and thermal

conductivity. However, these eddy transport properties are not known fundamentally, so that the available solutions are of the analogy type, relating the heat transfer to the friction.

Considerable attention has been directed to the non-uniform wall temperature case, with notable contributions by Lighthill^^', Leveque^23)_ Rubesiml^',

Eckert* ', and Spalding^^^, etc. for the laminar case, and by Seban'-^^', Rubesim^^', etc. for the turbulent case. Reviews of such solutions are given by Tribus and

Klein<^\ Spalding and Pun<22), and Shah<37).

Although there are numerous methods availble for calculating heat transfer over the bodies of a r b i t r a r y shape with arbitrary surface temperature, there is no method available for calculating the temperature within the boundary layer. 1.2. Present Work

In the present investigation, the theoretical approach to this problem has been made in the field of heat transfer, and the experimental approach is made in the field of m a s s transfer.

On the theoretical side, the numerical solution of the partial differential equation in the incompressible turbulent boundary layer has been obtained for

q" w

step ; and for Prandtl numbers 0.7, 1 and 7. The Spalding pu, Cp^^C^2

boundary layer velocity-law was assumed, and the Schmidt method of integration used. Integration was done on a Ferranti Pegasus Digital Computer. A method has been given for applying this solution to the problem of obtaining surface and

2

-fluid temperatures for the case of arbitrary distribution of heat flux at the wall. Simultaneously, approximate equations for calculating heat transfer over a flat plate with arbitrary heat flux are derived for laminar and turbulent flow. These approximate equations are very simple compared with similar equations of Reynolds et al^^^', Rubesim^^) etc. , and still predict fairly accurate heat transfer coefficients.

To verify the theoretical solution, an experimental rig was designed in which the concentration profiles of the injected gas (carbon dioxide) at different sections were measured when the pipe had another gas (air) flowing turbulently through in the axial direction, and the gas injected was passed through a porous section of the tube wall. The concentration of the carbon dioxide was measured by an Infra-red Gas Analyser.

Designing and manufacturing the experimental rig took a long time. After many modifications, the experimental results obtained were reliable and repeatable.

Experimental results showed higher diffusion when compared with the theoretical solution. In consequence, the experimental concentration of carbon dioxide at the surface was lower by 25% of the theoretical concentration. It seems probable that most of the discrepancy is due to the application of two-dimensional theory to the three-dimensional case of pipe flow.

2. Review of the Previous Work 2.1 Introduction

In the period of the last ten years, research in the field of heat transfer has increased at a tremendous rate, and numerous investigations are directed to the problem of "heat transfer from non-isothermal surfaces".

The method for determining heat transfer for non-isothermal surfaces is similar to the methods used in determining the deflection of beams subjected to arbitrary 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 temperature variation can be determined by superimposing a number of "step temperature distributions", 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 rates attributed to all steps upstream of the point in question.

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 mean values can be obtained from the momentum theory and the law of the conservation of energy. These a r e , in turn, derived from the equations of motion 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 approximate methods.

In all cases, if the wall temperature is given, the heat flux is to be calculated from

= ƒ.

q / V = h . ^ , d T .^, ( 2 . 1 ) ^w(x) J <?,x) w(?)

Here the Kernel function h(?,x) is the heat transfer rate at position x due to step temperature rise AT . . at the position ?. If the heat flux is given, the wall temperature is to be calculated from

r

Here the Kernel function g. ^ . i s the wall temperature at x due to unit heat flux at ?. ^^•^'

It should be noted that integral of equations 2.1 and 2. 2 must be taken in the "Stieltjes" sense rather than in the ordinary "Riemann" or "Area" sense.

Various investigators have obtained analytical expressions for the "Kernel" functions, and all these expressions can be divided into four main groups depending on their applicability.

2.2. Uniform Main Stream Velocity - Laminar Flow (23)

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

i 4 _ ^ i _i k(Pr) \K.X) ^ 3(i)i ^9^ du dy _y=oj (x - ?) ( 2 . 3 ) (15)

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 momentum thickness.

He obtained

(?,x) X " " *'"x

0.304k „ i „ 2 T i / ? \ - "• '^

h,„ ..X = "•"""" P r ' Re_; f l - ( V 1 (2.4) (9)

Eckert assumed cubic velocity and temperature profiles and solved the integral energy equation. He obtained

u -_ 0:33k P r ^ R e ^ r 1 . ( 1 ) 1 l (?,x) X ^ L '^ J

(24)

Klein and Tribus have shown a mathematical manipulation procedure by which the value of g, „ , can be obtained from the known function h, „ ,.

•" ^(?,x) (?,x)

Thus the values of the function g,p > obtained from the equations of

(23)

„ , .

(15)

^ „ I

%)

Leveque , Rubesin and Eckert^ ' are

4 -' ( ? , x ) _2_ 3k '(?,x) " 6(i)'. (1)1 1 P r ' 0.304k and r e s p e c t i v e l y . P r =(?.x) ' 6 ( i ) ' . (I)'. 0.33k Re

1 - m

x (^)^ x ( 2 . 6 ) ( 2 . 7 ) ( 2 . 8 )

The p r e s e n t a u t h o r s have a s s u m e d cubic velocity and t e m p e r a t u r e p r o f i l e s and have obtained a v e r y s i m p l e equation for an a r b i t r a r y heat flux p r o b l e m by d i r e c t l y i n t e g r a t i n g the e n e r g y equation for a step heat flux c a s e . Detailed d e r i v a t i o n i s given in C h a p t e r 3.

2 . 3 . Uniform Main S t r e a m Velocity - T u r b u l e n t Flow

Rubesin a s s u m e d velocity and t e m p e r a t u r e profiles a s following a 1/7 p o w e r law, and solved the i n t e g r a l e n e r g y equation. He obtained

-7/39 l i l t / . r ^ n K -r- i i « ( 0.0288k ( ? . x ) P r ' (Re ) x 0.8

(i)

_x (2.9) and ë, _{( r , x ) ( 3 2 / 3 9 ) 1 ( 7 / 3 9 ) ' .} 28/195 _{0.0288k x ^} 1 -0.8 0.8

I:

39/40 ^39/40 x - ? -32/39 (2.10) (12)

Seban *"' followed the s a m e p r o c e d u r e a s Rubesin, except that he a s s u m e d a l i n e a r t e m p e r a t u r e profile in the lanninar s u b l a y e r . He obtained

and 0.0289k (S,x) " x (8/9) Pr^/%e°-« X P r - " » „ - 0 l - ( ^ ) X 9/1Ü •1/9 ( 2 . 1 1 ) ^(r,x) (8/9)'. (1/9)1 0,0289k ,(14) ^ -0.8 0.8 r 9/10 ^9/1 Re X x - C X L 0 •8/9 (2*. 12) M a i s e l and S h e r w o o d ' ' " ' c a r r i e d out e x p e r i m e n t a l m e a s u r e m e n t s on a m a s s

transfer a p p a r a t u s and obtained an e m p i r i c a l equation "best fit" to t h e i r d a t a .

F o r a i r : -(S.x) ,(10) 0.035k „ 0.8 Re X X l - ( i ) X 0.8 _-0.11 (2.13)

Reynolds et al solved the i n t e g r a l e n e r g y equation by a s s u m i n g velocity and t e m p e r a t u r e p r o f i l e s a s following 1/7 power law and turbulent P r a n d t l n u m b e r equal to unity. They obtained

( ? , x ) 0.0296k „ 0.6 „ 0.8 P r Re X x l - ( i ) _X 9/10 •1/9 ( 2 . 1 4 )

and _'(?.x) (9/10) P r - ° - ^ ( R e , ) - " - » r i / 9 r 8 / 9 0.0296k 1 - ( ^ ) X 9/10 •8/9 (2.15)

F o r the s t e p heat flux p r o b l e m , the p r e s e n t a u t h o r s have d e r i v e d a s i m p l e a n a l y t i c a l e x p r e s s i o n by a s s u m i n g a seventh power law for velocity and t e m p e r a t u r e p r o f i l e s and d i r e c t l y i n t e g r a t i n g the e n e r g y equation. The detailed a n a l y s i s i s given in Section 3 .

2 . 4 . A r b i t r a r y Main S t r e a m Velocity - L a m i n a r Flow

Lighthill solved the e n e r g y equation in Von M i s e s f o r m , a s s u m i n g a l i n e a r

1

3

dz (2.16) velocity profile in the t h e r m a l b o u n d a r y l a y e r . He obtained

and

9u- '(z)

2 /jti!\* 1 r r I —

_{dz (2.17)}

(19)

Bond a l s o a s s u m e d a l i n e a r velocity profile and solved the differential equation for wedge flow. F o r the wedge flow (u = Ax"^) he obtained

1 c (C.x) r l+m"| ^ k_

L 2 J (i)'.x

and 2 c ^(C,x) " 9(f)'. k w h e r e b 1+my

i

b ^ Re^ x b " ^ R e - ^ X l - ( i ) X m+l ,_ P r « 6 \o) (2.18) (2.19) ( 2 . 2 0 ) 3/4(1 + m) a«f'-with and ^ o ) f = V = — ) is a d i m e n s i o n l e s s velocity g r a d i e n t m + l m -t- 1 V u^ vx _(2.21) (7)

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 by a s s u m i n g that a r e l a t i o n of the type Nu = A Re (A^)", 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 , and i s a l s o valid for a p r o b l e m 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 assunaption he h a s solved an i n t e g r a l e n e r g y equation. He obtained

Nu Re 0,332 P r * (T w T,) I X . ~ I u (T - T ) dx 1 w 1

[u,x ] ^

]

( 2 . 2 2 )

6

-T h i s equation is 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 heat t r a n s f e r when c o m p a r e d with the exact solution of E c k e r t for wedge flows.

(8)

Spalding h a s improved on L i g h t h i l l ' s method by a c o r r e c t i o n which accounts for the d e p a r t u r e from l i n e a r i t y of the velocity profile within the t h e r m a l boundary l a y e r , and which c o m p r e h e n d s the influence 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 non-coincident s t a r t of velocity 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 m o m e n t u m t h i c k n e s s 5^, of the velocity boundary 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 those of W a l z ' ^ " ' and T h w a i t e s ^ ^ ^ ' . With 6 (x) evaluated 6 (x) i s to be obtained from a table o r graph of 6 / 6

—s. I —! ] . F i n a l l y , the heat t r a n s f e r coefficient to be obtained from

v s . A = 4

tJ

6.410-

^J

dx + a

t)

F dx ( 2 . 2 3 ) 6 A 2 4 du 1 dx w h e r e F i s a g r a p h i c a l l y p r e s e n t e d function of the a r g u m e n t —— (4)

Schuh s e l e c t e d a dependent v a r i a b l e , which i s a function of the r a t i o of heat flow a c r o s s the whole boundary l a y e r to the t e m p e r a t u r e g r a d i e n t at the wall in a suitable d i m e n s i o n l e s s f o r m , and then i n t e g r a t e d the m o m e n t u m and the e n e r g y e q u a t i o n s .

S c h u h ' s method involves s i m i l a r s t e p s to those of Spalding's method, n a m e l y : 1. The d e t e r m i n a t i o n of 6^ t>y the W a l z - T h w a i t e s t e c h n i q u e .

6^ du

2. B and Z , a r e the functions of -^- ——^ and to be r e a d from the g r a p h . V dx

3. Calculation of n from equation 2. 24

d T

n = fr^A-^ \ - ^ (2.24)

T - T

W 1 dx

4. Calculation of functions G and P from equations 2 . 2 5 and 2.27 by an i t e r a t i v e p r o c e d u r e . G = 0.57 03 + 0.205) 0.104 A°-^''^°-°^'^ ( 2 - ^ ) ^ P ^ " ^ ' ' w h e r e A = 1 + (2 - ^) n and v a l u e s of r a r e to be r e a d from h i s t a b l e . ( 2 . 2 5 ) ( 2 . 2 6 ) o .3 1 / R 1

[iP?] ^ |-(T^-T)G(2-^)J^ ^xj

(2-^3)"^ i G ^ / 2 - T / / 2 w < _dx ( 2 . 2 7 )

5. Heat t r a n s f e r coefficient then to be c a l c u l a t e d from

Seban and D r a k e solved the m o m e n t u m equation by a p r o c e d u r e s i m i l a r to that of E c k e r t , and obtained heat t r a n s f e r r a t e by a s s u m i n g that it i s r e l a t e d to the m o m e n t u m t h i c k n e s s in the s a m e m a n n e r a s on the wedge.

T h i s method involves s i m i l a r s t e p s of t h o s e of Spalding and Schuh. 6^ du 1. The d e t e r m i n a t i o n of 6, and the a u x i l i a r y function — ——

2 -^ V dx

2. Evaluation of functions z and j3 from h i s g r a p h .

3. The r a t e of heat t r a n s f e r at the wall for a p a r t i c u l a r c a s e , when wall

t e m p e r a t u r e i s a power function of length, e . g . (T - T^) » A x " , i s given by

q" = - k A x " g' , ^ (2.29) ^w ^(o) 6

o r

^ g^o)

( 2 . 3 0 )

w h e r e g! , i s a t e m p e r a t u r e g r a d i e n t at the wall. g ' . i s a function of ^ , n and P r . Seban h a s suggested the exact solutions of E c k e r t , Levy e t c , to be used to obtain the value of g^ . for different /3, n and P r .

By s u p e r i m p o s i n g , the heat t r a n s f e r coefficient can be d e t e r m i n e d for a v a r i a t i o n of the s u r f a c e t e m p e r a t u r e of the type

n, n , n ,

(T - T ) = A x + A x + A X ( 2 . 3 1 )

W 1 1 2 3

2. 5. A r b i t r a r y Main Stpeam Velocity - T u r b u l e n t Flow

V e r y r e c e n t l y Spalding h a s suggested a single a n a l y t i c a l f o r m u l a for velocity d i s t r i b u t i o n throughout the whole of a " U n i v e r s a l " turbulent b o u n d a r y l a y e r in the f o r m of y"'"(u'*") instead of the usual form u''"(y"'"). With t h i s form of e x p r e s s i o n , it i s p o s s i b l e to obtain n u m e r i c a l solutions of the p a r t i a l differential equation for 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 and for the a r b i t r a r y heat flux c a s e . T h e s e solutions c o v e r both the l a m i n a r and turbulent flows.

Spalding h a s solved the i n t e g r a l form of the e n e r g y equation for s t e p s u r f a c e t e m p e r a t u r e and for unity P r a n d t l n u m b e r . He a s s u m e d a l i n e a r t e m p e r a t u r e profile of the form

„+

u+ < 6 ; e = 1 - ^ -)

^ J (2.32)

(17)

Muralidharan has solved the differential equation for step surface temperature and for P r = 0.7, 1 and 7. He used an analytical formula of Deissler^'^^' for velocity distribution in the boundary layer and obtained the numerical solution on an analogue computer.

(18)

Kestin and Persen used the digital computer to obtain the numerical solution of the partial differential equation for step surface temperature and for P r = 1, They used Spalding's velocity law.

The authors have solved the partial differential equation for arbitrary heat flux problem and for Pr = 0.7, 1 and 7. The detail of the numerical solution is presented in Section 4.

3. Heat Transfer over a Flat Plate with Arbitrary Heat Flux. Approximate Analysis 3 . 1 Laminar Flow q " w rrrTT-TTTTTTT

4

- > X F i g . 3 . 1 )/ii//u" in'

U-l

-> X F i g . 3.2 '^" = m(x-a) w

For a step heat flux (Fig. 3.1), total energy at any section x is:

q ; ( x - a ) = pu^ ^ T , - T ) 6 ƒ ( ^ H _ ^ ( l - e ) d ( j o

For assumed cubic velocity and temperature profiles

V"J

2 \h) 2 \h)

T - T Equation 3.1 becomes (3.1) ( 3 . 2 ) ( 3 . 3 )

q (x - a) = p u, C (T w 1 p w

T , ) 6 _3_

20

(I)"

- ^

(t)'

]

'-'

We a r e t r e a t i n g the c a s e , that the t h e r m a l b o u n d a r y l a y e r i s t h i n n e r than the velocity b o u n d a r y l a y e r , that is ( A / 6 ) < 1. The second t e r m in the right-hand e x p r e s s i o n of equation 3 . 4 i s then s m a l l c o m p a r e d with the f i r s t , and can be n e g l e c t e d . The heat t r a n s f e r at the wall i s given by

q = h(T w w T ) =

L3yJy=o

k ( T w T ) _{( 3 . 5 )}

F o r a l a m i n a r b o u n d a r y l a y e r , velocity b o u n d a r y l a y e r t h i c k n e s s i s given by: 4.64 X

6 =

V ^

( 3 . 6 )

Thus for a s t e p heat flux, frona equations 3 . 4 , 3 . 5 and 3.6 :•

a n d ( T w Nu

/ R i

T ) = 1 2.395 b(Re)^ (Pr)^ p u C ' P

[' -(I)] *

= 0.418 ( P r ) -

^-'1

1-i ( 3 . 7 ) ( 3 . 8 )

W h e r e b i s the magnitude of the s t e p heat flux.

(9) (15)

S i m i l a r equations of E c k e r t and Rubesin for a step heat flux a r e

. T ) = 2.2^.iBj(pJ ^ (, ,/3)

w 1 p u C r a n d ( T , T ) 1 1 i 2.405 b (Re)^ ( P r ) ^ I ^ ( i . 4/3) (3.9) (3.10) r e s p e c t i v e l y . It can be s e e n that equation 3.7 i s r e l a t i v e l y s i m p l e , for it d o e s not contain the i n c o m p l e t e beta function. F o r any a r b i t r a r y heat flux, the wall t e n a p e r a t u r e and the heat t r a n s f e r coefficient can be calcul&ted by s u p e r i m p o s i n g s m a l l s t e p s . T h u s for any a r b i t r a r y heat flux

( T w T ) 1 i 2 2.395 Re^ P r ^ p u C ' P S=x ?=o (1 - C/x) = dK d ? (3.11) a n d N u X 0.418 ( P r ) ^ q " , , = w(x) VRi

r

' C=o (1 - ?/x) 1 d q " , i . > 3 w(g) dt (3.12) d ?

10

It should be noted that the i n t e g r a l of 3.11 and 3.12 m u s t be taken in the " S t i e l t j e s " s e n s e . F o r i n s t a n c e , for a delayed r a m p heat flux ( F i g . 3 . 2 ) :

and 1_ 2 2 / T D „ \ 3 ( T - T ) W 1 1.797 (Re)^ ( P r ) ^ mx p u C ' P

f l - ( a / x ) !

4 / 3 Nu v ^ 1 i r -1 "3 0.557 ( P r ) ^ 1 - ( a / x )

Fl - ( a / x ) l

(3.13) (3.14) 3 . 2 . T u r b u l e n t Boundary L a y e r

F o r a s t e p heat flux, in the turbulent boundary l a y e r , we a s s u m e velocity and t e m p e r a t u r e p r o f i l e s :

-H V (y/6)^/^

and T - T T - T 1 w - = ( y / A ) ^ / ^ Then equation 3.1 b e c o m e s q " (x - a) = p u C ( T - T )6 w 1 p W 1 •_7_ 72 A 8/7

The heat t r a n s f e r m o m e n t u m analogy d e r i v e d by Reynolds et al heat flux i s : -C„ / . - 1 / 7 (10) St q " w (T - T ) p u C w 1 1 p

-/ {I)

(3.15) (3.16) (3.17) for s t e p (3.18)

w h e r e the P r a n d t l n u m b e r dependence factor is applied a f t e r w a r d s . F o r a turbulent b o u n d a r y l a y e r ^ f -0 2 - J = 0.0296 (Re) 6 x 0.37 (Re) •0.2 (3.19) (3.20)

T h u s for a s t e p heat flux, from equations 3 . 1 7 , 3 . 1 8 , 3.19 and 3. 20 : •

and ( T - T ) w 1 Nu (Re) ,0.8 32.95 b - „ ^ 0 . 2 , _ , . 0 . 4 , , , . 1 / 9 —-; (Re) ( P r ) (1 - a/x) p C u P ' 0.03035 ( P r ) ° - ^ (1 - a / x ) " ^ ^ ^ (3.21) ( 3 . 2 2 )

for a s t e p heat flux i s :

-(T - T ) = 32.42 -—^ (Re)°-^ ( P r ) ^ ' * I ( 1 / 9 . 1 0 / 9 ) (3.23) w 1 p C u s

P '

Equation 3 . 2 1 i s r e l a t i v e l y s i m p l e , for it d o e s not contain the i n c o m p l e t e b e t a function. F o r any a r b i t r a r y heat flux, the wall t e m p e r a t u r e and the heat t r a n s f e r coefficient can be calculated by s u p e r i m p o s i n g snnall s t e p s . T h u s for any a r b i t r a r y heat flux:-'

32.95 ( R e ) ° - 2 ( P r ) ° - ^ ; = ' ' „ , , Al9 ^ ^ , ( 0 (T... - T . ) - - ^ ^ - ; " ; ; ' .. " ^ ' / ( l - ? / x ) ^ ' " - — ^ dK w 1 p C u "' d? and ^ . . . (3.24) Nu 0.03035 (Pr)*^'^ 6 " Ï = JUSH {Q 2'j) (Re) 1 /o d q , „ , (1 . 5 / ^ ) 1 / 9 - - ^ ^ i d C

r=o ^^

In e q u a t i o n s 3 . 24 and 3. 25 the i n t e g r a l m u s t be taken in the " S t i e l t j e s " s e n s e . T a k i n g a delayed r a m p heat input ( F i g . 3.2) a s an e x a m p l e :

-^^ _ ^ 29.65 m X ( P r ) ° - ^ R e ) ° : ! (1 - a / x ) ^ ° / « (3.26) and Nu ^ = 0.03373 (Pr)°'® (1 - a / x ) " ^ ^ ^ (3.27) / « \0.8 (Re)

R e s u l t s of the above equations a r e c o m p a r e d with Reynolds et al

E c k e r t ' " ' , and R u b e s i n ' ^ ^ ' , for both lantiinar and t u r b u l e n t boundary l a y e r s , and a r e plotted in F i g s . 3 . 3 and 3 . 4 .

F o r the l a m i n a r c a s e , the r e s u l t s of the above equations a g r e e v e r y well with that of Rubesin' ' , and not with E c k e r t ' ^ ' . The d i s a g r e e m e n t with E c k e r t i s m e r e l y in a n u m e r i c a l f a c t o r . E c k e r t ' s (T - T^) v a l u e s a r e 8% lower than those of equation 3 . 7 .

F o r t

dfio).

i]or the t u r b u l e n t c a s e , the r e s u l t s a g r e e within 1.5% to that of Reynolds et a l ' " " '

12

-3 . -3 . A p p r o x i m a t e A n a l y s i s a s s u m i n g L i n e a r T e m p e r a t u r e P r o f i l e ( P r = 1) F o r a s t e p heat flux, t o t a l e n e r g y at any s e c t i o n x i s :

q " (x - a) w PC u (T - T,) dy P ' ( 3 . 2 8 ) o r St (x - a) u^ = / u 6 dy ( 3 . 2 9 ) With

=^' • { -

P w (x - a) f Z V 'M 2 u y iC, NI p ' f ^' 2 (3.30) (3.31) and w P u 1 X u / r _{1 I ^f} dy-*- ^ du" ( 3 . 3 2 ) ( 3 . 3 3 ) equation 3 . 2 9 s i m p l i f i e s t o : St +' _{u"*- e e+} du"*-A s s u m i n g a l i n e a r t e m p e r a t u r e profile n + 6 = 1 -and knowing that

St ICf 6 u^=0 ( 3 . 3 4 ) ( 3 . 3 5 ) ( 3 . 3 6 ) Equation 3. 34 s i m p l i f i e s to: 6 + Ö / -t-\

- = ƒ u-

( l - O ^" ^"'

MW«<16)

F r o m Spalding's velocity

profile:-with k' = 0.407 and 1 k'u+ 1 - k'u+ E = 0.0991 ' . , - H ^ ' ( k ' u ^ ) ' (k'u-^) 2'. " 3'. ( 3 . 3 7 ) ( 3 . 3 8 )

F r o m e q u a t i o n s 3.37 and 3 . 3 8 ^' (1 - Ji) + ii 6 ^ E ' E k'6 ' 2

(1 - 4 - ) + - (1 + 4-)

k 6 k'^ k'6

.

ILL'

r 1 + 1 _ K'

6 [ 2 20 "^ 6 + 30 (3.39)

Equation 3 . 39 i s c o m p a r e d with p r e v i o u s a p p r o x i m a t e a n a l y s e s for l a m i n a r and t u r b u l e n t flows obtained by a s s u m i n g cubic and seventh power p r o f i l e s r e p s e c t i v e l y . ( E q s . 3 . 4 2 and 3 . 4 5 ) . C o m p a r i s o n i s shown in F i g . 3 . 5 and a g r e e m e n t i s found to be v e r y s a t i s f a c t o r y . F o r P r = 1, equation 3 . 8 of l a m i n a r flow i s : — = 0.418 (1 - 9 / x ) ' ^ ( 3 , 4 0 ) Knowing 'f _ 0.332 VHRC (3.41)

for l a m i n a r flow, equation 3.40 s i m p l i f i e s to:

f-— = 0.603 x"*

F o r P r = 1, equation 3. 22 of turbulent flow i s :

( 3 . 4 2 ) with Nu_ Re Cf 0.8 0.03035 (1 - 9/x) •1/9 0,0296 Re -0.2 ( 3 , 4 3 ) ( 3 . 4 4 ) equation 3 . 4 3 s i m p l i f i e s to: St = 0.1452 X V - l / « ( 3 . 4 5 )

14

4 . N u m e r i c a l Solution for A r b i t r a r y Heat Flux 4 . 1 . T h e o r y

(Ifi)

By use of the Von M i s e s t r a n s f o r m a t i o n , Spalding h a s reduced the e n e r g y equation for a t u r b u l e n t i n c o m p r e s s i b l e boundary l a y e r

aT . „ aT a r aT , aT

ax

to w h e r e

"^ ay ay

|_"

ay ^ ay J

. = J _ i_ r 2 l ar -1 )

e u 9u [_ e au J X IT X w /• u .

ar

ax

x+ = / ^l P dx = / - ^ dx (4.3)

_a^ " T -"a "^

l ^ v

u y

Y+ ='^ 0 y = ( 4 . 4 ) V

u"*- = -^=^ = — and u+ = u+(y"*-) ( 4 . 5 )

IT Ur

' w '

J-

P

e+ = 1 + 1 = E l l (4.6) du+

(this i m p l i e s that s h e a r s t r e s s i s independent of y)

" P r v

F u r t h e r , using a new v a r i a b l e 4 defined by d^ = -—• du"*- (4.8) equation 4 . 2 can be r e d u c e d to a' aT ^ _j_ a' T a x + " u+o-^ 95* with b o u n d a r y conditions ( 4 . 7 ) ( 4 . 9 ) T = 0 at é = ~ ) (4 10) and T = 0 at x+ < 0 ) \ • >

In t h i s s e c t i o n the t e m p e r a t u r e s T and T a r e r e l a t i v e to the fluid t e m p e r a t u r e T , . *

An additional b o u n d a r y condition in the p r e s e n t solution i s

(m

= Constant (4.11)

F r o m equations 3.37 and 3 . 3 8 6 ^ E ' E 'e^^ 2 1 2 / 2 k 6 k 6 1^6 6

-[i

(3.39)

Equation 3 . 39 i s c o m p a r e d with p r e v i o u s a p p r o x i m a t e a n a l y s e s for l a m i n a r and t u r b u l e n t flows obtained by a s s u m i n g cubic and seventh power p r o f i l e s r e p s e c t i v e l y . ( E q s . 3 . 4 2 and 3 . 4 5 ) . C o m p a r i s o n i s shown in F i g . 3 . 5 and a g r e e m e n t i s found to be v e r y s a t i s f a c t o r y . F o r P r = 1, equation 3 . 8 of l a m i n a r flow i s : — = 0 . 4 1 8 ( 1 - 9 / x ) ' ^ / R e ( 3 , 4 0 ) Knowing _0.332

viae

(3.41)

for l a m i n a r flow, equation 3.40 s i m p l i f i e s to:

J

St_ f

T

0.603 X +'

F o r P r = 1, equation 3 . 2 2 of turbulent flow i s :

( 3 . 4 2 ) 0.8 = 0.03035 (1 - 9/x) with Nu Re C f - 0 0 -J- = 0.0296 Re •1/9

equation 3 , 4 3 sim.plifies to: St

0.1452 X + / - 1 / 9

( 3 . 4 3 )

( 3 . 4 4 )

14

-4 . N u m e r i c a l Solution for A r b i t r a r y Heat Flux 4 . 1 . T h e o r y

(16)

By use of the Von M i s e s t r a n s f o r n a a t i o n , Spalding h a s reduced the e n e r g y equation for a turbulent i n c o m p r e s s i b l e b o u n d a r y l a y e r

aT ^ ,^ a T a r aT ^ aT -t . . , . u — + V — = — fl; — + e — ( 4 . 1 ) _{ax ay ay |_" ay ^ ay J} to w h e r e ar a

I = JL i_ r 2l 9T -1

x+ e V au-^ L ^^ ^^'' J X IT" X

x+ = [

^ l ~

dx = / — dx

(4.3)

Ja_ " F - ^a '^ ^w u^ y 0 y = (4.4) V u

+

^z^ = ~ and u+ = u'*-(y+) (4.5)

J>

e+ = 1 + 1 = E l l (4.6) ^ du

(this i m p l i e s that s h e a r s t r e s s i s independent of y) + 1 , ^

'^ = p ; ^

-F u r t h e r , using a new v a r i a b l e ^ defined by d^ = —- du"*" (4.8) equation 4 . 2 can be r e d u c e d to 9T _ _ 1 _ a' T a x + u-^o"*- 3 5 ' with b o u n d a r y conditions ( 4 . 7 ) ( 4 . 9 ) T = 0 at ë = ~ ) (4 10) and T = 0 at x-'-< 0 )

In t h i s s e c t i o n the t e m p e r a t u r e s T and T a r e r e l a t i v e to the fluid t e m p e r a t u r e T , »

An additional b o u n d a r y condition in the p r e s e n t solution i s

= Constant (4.11)

(m _

_ë=o

(1 a\

Spalding h a s d e r i v e d a single e x p r e s s i o n for the d i s t r i b u t i o n of the v e l o c i t y in the u n i v e r s a l turbulent b o u n d a r y l a y e r in the f o r m of y"*" (u"*") i n s t e a d of the usual form u"''(y"*'). It i s :

-+ -+ ^ 1 ) k'u-+ . , , -+ (k'u-^)' ( k V ) ' ( k ' u V "i -, , , , y = u + g j ^ e - 1 - k u - -27 37 4 r - J <4-12) w h e r e k' = 0.407

and 1/E = 0.0991

F r o m equation 4 . 1 2 an e x p r e s s i o n for e"*" can be obtained, using equation 4 . 6 . T h i s gives

,* . , . ^ [ e ^ ' " ^ I - k ' „ - - ! | ; i i t ) " . ( ^ " J ,4.,3)

A f u r t h e r a s s u m p t i o n that the turbulent P r a n d t l n u m b e r is unity (& = e) l e a d s to an e x p r e s s i o n for a^

+ 1_ ^ ^' } k'u+ , , , + (k'u+)' (Jc%i+)' P r

+ X , K j k u , , / + tK u ' ) Vku ; / . ,.>

a = ^ + - j^e - 1 - k u --2T— - -Y— J <4.14)

T h e r e f o r e the quantity (u"*" a ) , for v a r i o u s P r a n d t l n u m b e r s , can be obtained a s a function of S • T h i s r e d u c e s equation 4. 9 to

9 T _ 1 a ' T , . . . .

—•, - 577T "~r" (4.15) The v a l u e s of f(6) for P r = 0.7, 1 and 7 a r e shown in F i g . 4 . 6 .

F o r the uniform s u r f a c e t e m p e r a t u r e c a s e , v a r i o u s i n v e s t i g a t o r s have solved equation 4 . 1 5 . Spalding' ' h a s solved it a p p r o x i m a t e l y for P r = 1, by use of the e n e r g y i n t e g r a l equation and a s s u m i n g a t e m p e r a t u r e profile in the b o u n d a r y l a y e r . Muralidharan* ' ' h a s solved it on an analogue c o m p u t e r for P r = 0,7, 1 and 7. K e s t i n and P e r s e n ' ^ ® ' have solved it on a digital c o m p u t e r for P r = 1.

(

In the p r e s e n t p a p e r , a solution of equation 4 . 1 5 h a s b e e n m a d e for uniform the wall and for P r = 0. 7, 1 and 7.

51) at

3 6 / 36

4 . 2 . N u m e r i c a l Solution

It will be s e e n that equation 4 . 1 5 i s v e r y s i m i l a r to the heat conduction equation, and m a y be solved by the finite difference method of E . Schmidt.

T h e equation 4 , 1 5 can be w r i t t e n in finite-difference form a s Ax +

•(x+-t-Ax+.S) (x+, ë ) f(6) (A6)*

\xt 6 +A6) "^ '^(x+, 5 -Ae)" ^ V + . e ) ]

(4.16)

16

-F r o m equation 4 . 1 6 , T, , , . +> can be calculated from T- +, which

(x+ -I- Ax^) (x^)

in t u r n can be calculated from T, , , +. and so on. The grid for the finite

( x ^ - Ax ) ^

difference s c h e m e i s shown in F i g . 4 . 7 .

F r o m F i g . 4. 7 it can be s e e n that all the vailuss of t e m p e r a t u r e can be c a l c u l a t e d for T, . , ^ +. from T, ., except two end v a l u e s , one at the

(x^ + Ax^) (x^) "^

wall and the second at 4 = ^ . The l a t t e r difficulty was m e t by s e l e c t i n g 6 s o high that T t h e r e w a s z e r o . The value of T at the wall w a s obtained

m a x ^

by adding the wall t e m p e r a t u r e g r a d i e n t x A^ to the value of the t e m p e r a t u r e

at (x+ -I- A x + , A C ) .

To obtain the n u m e r i c a l r e s u l t s in solving equation 4 . 1 5 , the complete c a l c u l a t i o n s w e r e m a d e for an a r b i t r a r y value ( aT) . „ and l a t e r

\ac/4=o " " •

the r e s u l t s w e r e m a d e d i m e n s i o n l e s s . The n u m b e r -40 h a s no p a r t i c u l a r s i g n i f i c a n c e .

It w a s not p o s s i b l e to s t a r t the solution at x"*" = 0 b e c a u s e t e m p e r a t u r e at a l l v a l u e s of 6 at x"*" = 0 w e r e z e r o . T h i s difficulty was o v e r c o m e by s t a r t i n g the solution at a v e r y low value of x"*" called x^ .

T h e s t a r t i n g value of x"J" for

S)

3 4 / -40 w a s s e l e c t e d such that the 4=0

t e m p e r a t u r e at all v a l u e s of ë was z e r o except at the s u r f a c e . T h i s f i r s t

-a s s u m e d v-alue of the s u r f -a c e t e m p e r -a t u r e T w -a s obt-ained from the -a p p r o x i m -a t e (35) *^

a n a l y s i s of Smith and Shah . The final r e s u l t s when checked by i n t e g r a t i n g the heat flux showed that they w e r e not affected by any e r r o r in t h i s a s s u m p t i o n . The s t a r t i n g v a l u e s for v a r i o u s P r a n d t l n u m b e r s a r e tabulated in Table 4 , 1 .

P r 0.7 1.0 7.0 T W 1 19.30 19.41 38.50 Table ^ x+ 1 0.050 0.025 0.004 l . l AS 0.5 0.5 1.0

The solution by the method of E . Schmidt is stable only if the condition

A X '

f ( S ) ( A Ê ) '

(4.17)

i s fulfilled throughout. Since it was decided to solve the equation up to x"*" = 1 0 ° , the i n t e r v a l Ax"*" w a s i n c r e a s e d g r a d u a l l y a s shown in T a b l e s 4 . 2 , 4 . 3 and 4 . 4 . With the i n c r e a s e in the i n t e r v a l Ax"*", it b e c a m e n e c e s s a r y to i n c r e a s e A^ n e a r the s u r f a c e (low v a l u e s of 4 ) to fulfil the above condition.

As the values of f(6) increase with 6 the interval A^ was gradually reduced for the calculations away from the surface. Near the surface, the temperature profile gradually became linear and therefore this increase in A6 near the wall, with the increase in the interval Ax"*" did not impair the accuracy of the final

results. I n t e r v a l x 0 t o 1 1 t o 10 10 t o 10* 10* t o 1 0 ' 10* t o 1 0 ' 1 0 ' t o 1 0 ' AX+ 0 . 0 5 0.1 1 1 50 500 M i n . Aé 0.5 0.5 0.5 0.5 0.5 0.5 1 ~] M a x . A^ 0.5 0 . 5 1 1 4 8

Table 4 . 2 . Steps chosen for P r = 0.7

I n t e r v a l x 0 t o 1 1 t o 10 10 t o 10* 10* t o l O ' 1 0 ' t o 1 0 * 10* t o 10= 10= t o 10* Ax+ 0 . 0 2 5 0.5 1 4 20 25 250 M i n . AS 0.5 1 M a x . AS 0.5 1 1 2 4 4 8

Table 4 . 3 . Steps chosen for P r = 1

I n t e r v a l x 0 t o 1 1 t o 10 10 t o 1 0 * l O ^ t o l O ' l O ' t o 1 0 * 1 0 * t o 1 0 ' Ax-^ 0 . 0 0 5 0 . 0 5 0.5 1 5 2 5 M i n . AS 1 1 1 2 2 2 M a x . AS 1 2 4 6 8 14

18

-4 , 3 . F o r m of P r e s e n t a t i o n in the R e s u l t s

The n u m b e r s r e s u l t i n g from the computations a r e t e m p e r a t u r e differences T in ?, x""" c o - o r d i n a t e s , with ( 9 T | _ Heat t r a n s f e r coefficients m a y

^ 9 | / | = o

be computed conveniently from such r e s u l t s in t e r m s of the "Spalding function" St/ / C „ / 2 , by the r e l a t i o n , deduced from equations 4 . 4 , 4 . 8 and 4 . 1 2 .

Cf ^^«^^5=0 / ^ "2" Thus St/^rcJ2 (x"*") can be p r e s e n t e d . Note, f u r t h e r , that q : (4.19)

J

St Cf ^w p C p U , V C / 2 1 ^ T w

A s s u m i n g that C„/2 , , and u,. . a r e known for a specific e x a m p l e , x^(x) m a y be computed from equation 4 . 3 . Hence from S t / ' C / 2 (x"*"), St(x) m a y be computed. Then knowing the constant value of q" / p u C ' C 7 / 2 , T m a y be found. T r e a t m e n t for a r b i t r a r y q i s given in a l a t e r s e c t i o n .

F o r the t e n n p e r a t u r e within the boundary l a j e r , it will then suffice if 6 ( I , x"^) be p r e s e n t e d .

A p r a c t i c a b l e p r e s e n t a t i o n i s thus of S t / * c T / 2 (x""") and 0(C, x'^).

4 . 4 . Check by Heat Balance

An i n t e g r a l f o r m of the equation 4 . 1 5 m a y be d e r i v e d by i n t e g r a t i n g it along the § c o - o r d i n a t e . T h u s

o

In o u r solution I r-?" ) " Constant and t h e r e f o r e ^ ^ ^ / f = 0

0

ƒ

T u-*- a + d ? = - ( I I ) x+ (4.21) ^^^/f=0

o

T h e r e f o r e from equations 4 . 1 8 and 4 . 21 we have

oa

The temperature profiles obtained in the numerical solution were integrated at various sections and the integrated heat fluxes were compared with the Spalding function and x"*" as shown in equation 4. 22. Comparisons for various Prandtl numbers are tabulated in tables 4 . 5 , 4.6 and 4. 7. The results were found reasonably satisfactory.

4 . 5 . Dependence of the Spalding function and 9 on the Prandtl number The results of the calculations in t e r m s of the temperature profiles are shown in Figs. 4 . 1 , 4 . 2 , and 4 . 3 . The values of the Spalding function for various values of x"*" and for Prandtl number 0.7, 1 and 7, are tabulated in Tables 4 . 8 , 4.9 and 4.10 and plotted in Fig. 4 . 4 .

Having performed these calculations and checks, an effort was made to find the dependence on Prandtl number of the Spalding function and the temperature profile in the form

^* ^* X P r " (4.23)

iCf

5J

Pr=l

Results for P r = 0.7, 1 and 7 showed that in the laminar region (up to x"*" = 1000) the value of n remains constant and is equal to - 2 / 3 , but for higher x"*", the value of n when obtained from the results for P r = 0.7 and 1, changes from -2/3 at x"*- = 1000 to -0.4 at x+ = 10^ and -0.325 at x"*" = 10^, whereas from the results for P r = 7, the value of n decreases from -2/3 at x"*- = 1000 to -0.516 at x"*- = 10^.

In the laminar region (x"*- < 1000) the temperature profile for any Prandtl number can be obtained from

^ 5 ) = ^ ? ' ) ( P r = l ) <^-24> where „,„

f = f ' x P r ' " " ^ (4.25) Temperature profiles for various Prandtl numbers and for x =100 are

shown in Fig. 4 . 5 .

4 . 6 . Application of the Solutions to the Case of Arbitrary Heat Flux at the Wall The working equation by which the numerical solutions given in this paper may be applied is equation 4. 33 below. A full statement of its development is given in Sections (a) and (b) following.

(a) Determination of temperature at the wall.

The numerical solutions given in the previous sections have the boundary condition ^ | T j ^ constant, x"*" > 0. From the definitions of the variables

^ ^ « / 5 = 0

in equations 4. 8, 4 . 7 , 4.6 and e a r l i e r , together with the velocity-law equation 4.12, this boundary condition is the same as q" / p u C v C j 2 = constant. _{•' w 1 p f}

20

-That i s , the heat flux p a r a m e t e r is constant.

T h i s b o u n d a r y condition i s highly p a r t i c u l a r : usually p r o b l e m s will be m e t in the f o r m of a specified d i s t r i b u t i o n of q" , with T (x) the quantity d e s i r e d to be known.

However, the t r e a t m e n t of the p r o b l e m of a r b i t r a r y q" will be m o s t e a s i l y w

p e r c e i v e d after a r e c a p i t u l a t i o n of the method of d e t e r m i n a t i o n of T (x) for w

constant q" / p u. C >^CT2 . ^w ' p f

F i g . 4 . 4 gives v a l u e s of the Spalding function StI ^ l ^ ( P r , x-*"). If S t / V c „ / 2 be known at a given x, t h e n T i s given by

. //

1 / ^ ^

^w = stTT^g > ' U u C p V c 7 l ) <^-2«^

In F i g . 4 . 4 . heat injection s t a r t s at x = a. We set the p r o b l e m , then, of calculation of the t e m p e r a t u r e T . . at x = b , consequent on constant

q* /p u C ^ C „ / 2 between x = a and x = b . w V p f

Define a new v a r i a b l e x"*" by the equation

3.. D

x-^ = / ^ dx ( 4 . 2 7 a , b

Then x"*" , i s e a s i l y c a l c u l a b l e , knowing T , P and v . F r o m F i g . 4 . 4 a . b •' *= w

( S t / / c . / 2 ) m a y be r e a d off. for x"*" = x"^ . T h i s quantity i s the value of

I cL f D c l , D

St/ V c , / 2 at x = b consequent on constant q" / P u C JCJ2 from x = a to f ^ w 1 p f

X = b . Then at x = b , th m a i n s t r e a m i s given by

X = b . Then at x = b , the difference T ,, , between wall t e m p e r a t u r e and w(b)

"w(b) ' / S t

/ a , b

It m u s t be p a r t i c u l a r l y noted that equation 4 . 28 holds for constant

a" I pu C V c „ / 2 between x = a and x = b . However, in the g e n e r a l p r o b l e m , w ' p f

a" I P U.C ^ C , / 2 will be a function of x. Knowing ci" , ,, u,, , and C.. . t h e ^w ' p f ^ ( x ) '(x) f(x) dependence of q " / p u , C ^0/2 on x m a y be computed. The t e m p e r a t u r e at the wall at x = b i s then given by the Stieltjes i n t e g r a l .

f" q "

T ,^, = T7^ : d f ^ - 7 ^ . \ (4.29)

w(b)

Equation 4 . 29 i s the working equation for the deternaination of wall t e m p e r a t u r e . In equation 4. 29, {St/^J2) i s the value of S t / ^ / 2 at b , consequent on

I a , b t ' q" \ constant q" /P u.C ^ C . / 2 between x = a and x = b . d ( ^ „ ) is the

w ' p f \ P u C p V c 7 2 /

i n c r e m e n t of heat flux p a r a m e t e r for the i n c r e m e n t dx containing x = a. It m a y be m o r e convenient for computation if equation 4 . 2 9 i s r e - w r i t t e n .

T 1 d / "^w w(b) J / St,rz:r-,-^\ dx dx (4.30) a ^(b) q " ^ T ,^, =/ ^-^ \ A / ^ \ (4.31)

V ^ / 2 ) ^^ \ Pu,C Vc72

/ a , b \ P

In evaluating equation 4 . 3 0 , h o w e v e r , it m u s t be noted that if t h e r e i s a s t e p in q " / p u . C *C„/2, t h i s step c o n t r i b u t e s an i n c r e m e n t of T . . . given by

w 1 p f w(b) ^ •'

1 q "

-/' _ _ J _ _ \ ^ / w

In equation 4 . 3 1 the s y m b o l A signifies a finite i n c r e m e n t . (b) D e t e r m i n a t i o n of t e m p e r a t u r e within the boundary l a y e r .

T h e p r e v i o u s s e c t i o n showed the method of computing wall t e m p e r a t u r e consequent on an a r b i t r a r y d i s t r i b u t i o n of wall heat flux. A f u r t h e r p r o b l e m i s , h o w e v e r , the d e t e r m i n a t i o n of t e m p e r a t u r e within the b o u n d a r y l a y e r . Again the method will be s e e n m o r e e a s i l y if the computation for s t e p q " / p u , C * C . / 2 i s f i r s t c o n s i d e r e d .

T e m p e r a t u r e s within the boundary l a y e r have been p r e s e n t e d in F i g s . 4 . 1 , 4 . 2 , and 4 . 3 a s 6(x-^, I , P r ) . 6, defined a s in the list of notation, is in fact t e m p e r a t u r e on a s c a l e such that m a i n s t r e a m t e m p e r a t u r e i s z e r o and wall t e m p e r a t u r e i s unity. The t e m p e r a t u r e at a point in the b o u n d a r y l a y e r i s defined if T and 6 be known. The r e l a t i o n i s T = e T .

w w C o n c e n t r a t i n g our attention bn the c a s e of constant ci /pu C VC„/2, the

t e m p e r a t u r e difference between wall and m a i n s t r e a m at x = b , called T ., ., m a y be found by the method of the p r e v i o u s s e c t i o n . The p r o b l e m now i s to find the fluid t e m p e r a t u r e at point c with c o - o r d i n a t e s x , y. Of the p a r a m e t e r s in F i g s , 4 . 1 , 4 . 2 and 4 . 3 , x'^ and P r a r e a l r e a d y known. To d e t e r m i n e 6 and t h u s p e r m i t the d e t e r m i n a t i o n of t e m p e r a t u r e at y, the d i m e n s i o n l e s s c o -o r d i n a t e C m u s t be d e t e r m i n e d f-or the p-oint c. ? i s defined by equati-on 4. 8 and equations 4 . 8 , 4 . 7 , 4 . 6 , 4 . 5 , 4 . 1 2 , 4 . 1 3 and 4 . 1 4 p e r m i t the function 5(y'*-. P r ) to be computed. R e s u l t s of t h i s computation a r e shown in F i g . 4. 8 and Table 4 . 1 1 , Using F i g . 4 . 8 , ? niay t h e r e f o r e be found, after y"'" h a s been d e t e r m i n e d from equation 4 . 4 . Thence 6 m a y b e r e a d off F i g s . 4 . 1 , 4 . 2 o r 4 . 3 . T h e difference of t e m p e r a t u r e between point c and the m a i n s t r e a m is then

( a , b , c ) / % \

T , . , = ^ ^ ° ' " ' , ' ( ^^^— ) (4.32) a , b

22

-In equation 4 . 3 2 , 6, . i s the value of 6 at x"*" = x"*" , and the a p p r o p r i a t e 5. ( a , b , c ) a . b ( S t / V c , / 2 ) , is the value r e a d off F i g . 4 . 4 at x+ = x+ , and q " / p u C V c J l

1 a , b ° a , b w ' p f is the constant value of the heat flux p a r a m e t e r between x = a and x = b .

When t h e r e i s an a r b i t r a r y d i s t r i b u t i o n q" (x) the d i s t r i b u t i o n of the heat flux p a r a m e t e r m a y be computed a s d e s c r i b e d in the p r e v i o u s section of the p a p e r , and the value of the difference in t e m p e r a t u r e between point c and the m a i n s t r e a m conaputed by the Stieltjes i n t e g r a l :

6 c(b) ( a . b . c ) St 7 ^ C ^ w a , b pu C V c / 2 1 p f ( 4 . 3 3 )

Equation 4. 33 i s the g e n e r a l equation by which the r e s u l t s of t h i s p a p e r m a y be applied, s i n c e it c o m p r e h e n d s equation 4 . 2 9 in that at the wall, 6 i s unity. Again it m a y be m o r e convenient for computation if the equation i s r e w r i t t e n

• c ( b )

e

_{( a . b , c )}

i^'t^2y

_d_ dx a . b w p u C VC./2 1 p f dx ( 4 . 3 4 )

In the evaluation of equation 4 . 34 c a r e m u s t be taken in allowing for s t e p s in q* / p u ^ C ^ C , / 2 a s was pointed out after equation 4. 30.

1

""^

10 10* 1 0 ' 10* lOF 10* 1 St + ^ C f / 2 3.810 17.97 97.15 653.6 4952 39660 oo /" e u"^ a"^ df •'o 3.838 18.07 97.21 652.4 4876 39780 % diff. + 0.72 -1- 0.54 + 0.05 - 0.18 - 1 . 5 4 + 0.30

x+ , 10 10* 1 0 ' 10* 10» 10* , ^^ -z+ ^ C f / 2 2.965 14.18 77.28 544.1 4293 35320 OO

ƒ eu+ a+d?

•'o 3.010 14.22 77.48 546.6 4298 35190 %diff, + 1.51 + 0.30 + 0.27 + 0.44 + 0.11 - 0.37

Table 4 . 6 I n t e g r a t e d Check of the R e s u l t s for P r = 1

+ X 10 10» ! 1 0 ' 10* 1 0 ' S ^ x+ ^Cf/2 0.8185 3.844 20.75 171.3 1571 0 0 / e u+ a+ d§ "^o 0.7994 3.856 - 20.57 171.2 1579 % diff. - 2.33 + 0.31 - 0.87 - 0.06 + 0.52

24 -x-^ 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 1.10* 2 3 4 5 6 7 8 9

lo'

St VCj/2 0.801678 0.642647 0.563886 0.513691 0.477754 0.450215 0.428144 0.409884 0.394414 0.381063 0.300255 0.263463 0.240197 0.223636 0.211018 0.200964 0.192695 0.185731 0.179756 0.146030 0.130328 0.120738 0.114091 0.109125 0.105228 0.102059 0.099413 0.097158 i.io' 2 3 4 5 6 7 8 9 1.10* 2 3 4 5 6 7 8 9 1.10= 2 3 4 5 6 7 8 9,

lo'

St VCf/2 0.097158 0.084421 0.078568 0.074916 0.072325 0.070346 0.068760 0.067446 0.066330 0.065363 0.059528 0.056654 0.054776 0.053401 0.052325 0.051448 0.050711 0.050077 0.049523 0.045895 0.044175 0.043020 0.042159 0.041476 0.040913 0.040436 0.040023 0.039660

x+ 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 1.10» 2 3 4 5 8 7 8 9 10' St ^Cf/2 0.61424 0.49137 0.43359 0.39639 0.36957 0.34889 0.33224 0.31841 0.30667 0.29650 0.23723 0.20810 0.18965 0.17709 0.16652 0.15856 0.15201 0.14650 0.14177 0.11513 0.10281 0.09510 0.08998 0.08621 0.08327 0.08091 0.07894 0.07728

x+

1.1 o' 2 3 4 5 6 7 8 9 1.10* 2 3 4 5 6 7 8 9 i.io' 2 3 4 5 6 7 8 9 10' St VCf/2 0.07728 0.067892 0.063685 0.061115 0.059300 0.057914 0.056803 0.055880 0.055095 0.054413 0.050335 0.048244 0.046863 0.045843 0.045042 0.044385 0.043830 0.043351 0.042931 0.040295 0.038927 0.038007 0.037319 0.036774 0.036324 0.035943 0.035612 0.035321

26 -•^+ 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 1.10* 2 3 4 5 6 7 8 9 10' St ^Cf/2 0.172205 0.137977 0.121054 0.110283 0.102576 0.096672 0.091940 0.088025 0.084709 0.081847 0.064768 0.056777 0.051701 0.048080 0.045315 0.043107 0.041289 0.039754 0.038435 0.030841 0.027396 0.025336 0.023950 0.022950 0.022196 0.021607 0.021135 0.020749 .U- + X i.io' 2 3 4 5 6 7 8 9 1.10* 2 3 4 5 6 7 8 9 10' St

/c7/2

0.020749 0.018916 0.018277 0.017932 0.017705 0.017537 0.017406 0.017297 0,017206 0,017127 0.016646 0.016395 0.016225 0.016096 0.015993 0.015907 0.015834 0.015770 0.015713

0 j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0 1 2 3.003 4.013 5.042 6.115 7.274 8.590 10.18 12.23 15.04 19.06 24.98 33.88 47.36 67.88 99.13 146.7 0 0.7 1.4 2.101 2.803 3.509 4.223 4.953 5.702 6.496 7.322 8.189 9.092 10.02 10.98 11.95 12.92 13.91 14.90 0 6.999 13.99 20.88 27.50 33.48 38.48 42.35 45.25 47.44 49.18 50.62 1 51.90 53.07 54.18 55.25 56.29 57.32 58.34 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 219.0 328.6 494.6 745.6 1125 1696 2557 3854 5806 8743 13160 19800 29780 44780 . 67310 101200 152100 228500 343400 15.90 16.89 17.89 18.88 19.87 20.87 21.87 22.87 23.87 24.87 25.87 26.87 27.87 28.87 29.87 30.87 31.87 32.87 33.87 59.35 60.36 61.36 62.37 63.37 64.37 65.38 66.38 67.38 68.38 69.38 70.38 71.38 72.38 73.38 74.38 75.38 76.38 77.38

Table 4 . 1 1 . I (y"^, P r ) and y"*" (u'^)

Note: (a) F o r P r = 1, I = u"*"

(b) F o r y+ > 300 the following r e l a t i o n s hold:

y+ = 0.3255 e x p . 0.4098 ? ( P r = 0.7) y+ = 0.09177 exp. 0 . 4 0 9 3 ? ( P r = 1.0) y+ = 6.317 X 10-9 e x p . 0.40881 ( P r = 7.0)

28

-5. Description of Apparatus 5 . 1 . Introduction

The purpose of this experiment was to obtain an experimental confirmation of the numerical solution that i s obtained for an arbitrary heat flux problem. To obtain this an experimental rig was designed in which carbon dioxide was injected into a pipe from a porous wall, the main stream of the pipe being air. The mixed gases were sampled for concentration of carbon dioxide at various stations, and analysed by means of an infra-red gas analyser; the velocity profile in the pipe, just before the injecting section, was fully developed. This was checked by traversing a probe. (Figs. 5.2 and 5.7).

The general layout of the experimental rig is shown in Fig. 5.1 and 5.4. The complete apparatus can be divided into the following five

groups:-1. Air section 2. Diffusing section 3. Test sections 4. CO2 section 5. Measurements 5.2. Air Section

The test rig was on the suction side of an Allis Chalmers compressor with a suction capacity of 3120 cu.ft/min. Air was sucked from the atmosphere, and the air m a s s flow was controlled by:

1. A control valve on the by-pass line

2. A control valve in the rig after the orifice

Both the control valves were butterfly valves with angular indicators and stop nuts on handle for fixing the valve positions.

The pipe size was made 4.75" internal diameter for the following two r e a s o n s : -1. To facilitate the traverse of the probe for measuring concentration

profile very accurately, it was better to have a fairly large size pipe diameter.

2. The final size 4.75" I.D. was decided by the size of the porous tube available.

To obtain a fully developed and an axisymmetric flow, a bell-mouth entry was chosen. The bell-mouth section was made from wood and polished to obtain a very smooth surface.

There is a possibility of asymmetric flow in the pipe if there is the slightest e r r o r in coincidence of the axis of the bell-mouth section with that of the main pipe. To prevent this, a straightening grid was inserted immediately after the bell-mouth section. The straightening grid was made by inserting 172 stainless steel tubes, 0,323" O.D. , 0.007" thick and 2" long, in a flange 2" thick, 4.75" I.D. (Fig, 5.6).

The gap was filled by a tube separately made to suit the gap. This tube was 0.412" O.D. 0.008" thick and 2" long.

The inlet section was made 22'-6" long and in three pieces, 9 ft., 6ft. and 7ft. - 6 " respectively. This long length of inlet section, approximately 57 times the internal diameter of the tube, was used to ensure that the air flow just before the test section and the diffusing section, was fully developed turbulent. Further, to obtain a very symmetrical flow, the last piece of inlet section, 7ft. - 6 " long, and before the diffusing section, was made from perfectly circular drawn tube of one length and 0.125" thick. The other two pieces were made from 16 SWG. 3ft. wide steel sheet.

Between the test section and orifice, a straight pipe 7ft. long was connected in accordance with the British Standards Specification for orifices. (Fig. 5.8).

5 . 3 . Diffusing Section

The diffusing section mainly consisted of a porous tube and a diffusing chamber. The porous tube was a high grade ceramic tube with uniform porosity.

The aim of the experiment was to measure the concentration profiles of the diffusing substance at different sections for uniform step mass transfer, followed by a step adiabatic wall. Mass transfer through a porous material depends on three

factors:-1. Static pressure difference 2. Thickness of the porous tube 3. Porosity of the material Hence to obtain uniform,

1. p r e s s u r e in the chamber, 2. thickness of the material, and

3. porosity throughout the diffusing length, the following construction was

used:-1. High grade ceramic tube was used to ensure uniform porosity. 2. The outside and inside surfaces were ground to uniform diameters.

During grinding precautions were taken to obtain both the surfaces co-axial and hence of uniform thickness. The manufacturers of the ceramic tube confirmed that grinding does not alter the porosity. 3. The diffusing chamber was made 12" I.D. and 10.25" wide. This

large size of chamber gave enough room for the carbon dioxide to settle before it diffused through the porous tube. This, in other words, ensured uniform pressure in the chamber.

4. Carbon dioxide was admitted into the chamber from 16 places, eight on either side of the chamber, equiplaced on 11.125" P. C D . Two circular ring-mains, made from 0.5" diameter copper tube, supplied carbon dioxide to the chamber at these 16 positions by 0.25" dia. tubes as shown in Fig. 5.5. Carbon dioxide to the ring-mains was supplied from a carbon dioxide cylinder, through the pressure

30

-reducing valve, main valve and the flowmeters.

5. 'O' ring seals were used at all points to prevent any leakages. 5.4, Test Sections

There were two test sections, a small one for measuring the velocity profile and one large one for measuring the concentration profiles. They were made from 4.5" I.D. , 0,281" thick mild steel tubes, and they were bored accurately to

4.75" internal diameter. The flanges of these test sections were made to fit in a circular groove of the adjoining flanges so that they could be rotated around the axis of the pipe. With this arrangement, traversing could be made at different circumferential positions to check the axisymmetry of the flow.

The smaller section was mounted before the diffusing chamber. It was made 8" long with only one boss welded in the middle of the length to take the traversing assembly.

The larger section was mounted after the diffusing chamber. It was made 19" long with four bosses welded at four axial positions and at right angles in a consecutive order (see Fig. 5.5).

During the construction of these test sections, all care and precautions were taken to ensure that

:-1. The axes of these test sections were in line with the axis of the main pipe line.

2. No leakages occurred from side surfaces where they rotate. To ensure this 'O' ring seals were used at thesesurfaces.

3. Traversing axis of the probe was perpendicular to the axis of the pipe and in the same plane.

4. Surfaces of the matching pieces, on which the traversing probe assembly was mounted, were parallel and at a fixed distance from the axis of the pipe.

5. Inside surfaces of the matching pieces on which the traversing probe assembly was mounted, matched the inside curvature of the pipe. To obtain this, test sections were bored accurately to 4.75" I.D. , with matching pieces in their position.

6. Inside surfaces was smooth and straight. 5. 5. CO2 Section

CO2 was obtained from three cylinders (each containing 28 lb of liquid CO2) connected in parallel. From these cylinders, CO2 passed

through:-1. P r e s s u r e regulating valve 2. Settling chamber

3. Control valve

4. Two flowmeters, and 5. Orifice

At the p r e s s u r e reducing valve, the temperature of the CO2 was very low due to its expansion. To bring the temperature of CO2 back to atmospheric temperature at flowmeter and at diffusing sections, the CO2 cylinders and pressure reducing valve were placed at a distance from the test rig and the CO2 cylinders were placed in a water tank. With this arrangement, CO2 passed through the long length of copper tube (exposed to atmosphere) and in consequence obtained atmospheric temperature before reaching the test rig.

5.6. Measurements

(a) Air mass flow: Air mass flow was measured by an orifice at the downstream side of the test section. The orifice plate was of "D and D/2 taps" type, and it was made according to the British Standard Specification B.S. 1042 (1943). To simplify the calculations, the area ratio of orifice to pipe was made 0.5. There were four sets of pressure tappings at four right angle circumferential positions. P r e s s u r e differences across the orifice were measured on vertical water manometers. The mean of these four sets of pressure differences was used to obtain air m a s s flow.

(b) CO2 flow: The flow of CO2 was measured by two Fischer & P o r t e r percentage type flowmeters. They read accurately within - 2% of the maximum flow. Two flowmeters were used in series to cover the wide range of CO2 flow. 100% reading on the small flowmeter represented 0.97 cu.ft. /min. at 14.7 p . s . i . and 60°F. , and 100% reading on large flowmeter represented 3.72 cu.ft./min. at 14.7 p . s . i . and 70°F. For different temperature and pressure conditions, the flow was corrected by the use of curves supplied by the manufacturers.

At first it was found that the amount of CO2 flowing through the pipe and measured by the gas analyser was less than the CO2 measured by the flowmeter. After checking leaks in the CO2 line, the flowmeters were suspected. For an approximate check on flowmeters, an orifice was introduced in the CO2 pipe line, and this confirmed the accuracy of the flownaeters (see Figs. 7. 3 and 7.4). Later on the fault was found in the sampling cell of the infra-red gas analyser. The cell was cracked and, in consequence, air was leaking into the cell resulting in low concentration measurements. The cell was then replaced by a new one.

(c) Concentration of CO2: Concentration of CO2 was measured by an infra-red gas analyser type S . B . I . This gas analyser has two absorbing lengths. One length is a standard cell, whereas the other length has two cells, one for pure air and another for the sample to be tested. Radiation from the nichrome heaters passes through the absorption tubes and thence two two receivers, which are filled with the gas to be detected. These receivers are partitioned off from one another by a thin metal diaphragm, and form an electric condenser. Absorption of infra-red radiation in the sampling cell depends on the concentration of carbon dioxide.

This absorption of radiation causes a pressure difference between the two receiving chambers, with the result that the thin diaphragm deforms and causes a variation of the capacity. The resulting capacity changes are amplified electronically and a final indication is obtained on an output galvanometer.

The sampling length was on the suction side of a Charles Austen Mk. 1 pump with a capacity of 3500 c. c. /min. The amount of suction was adjusted by controlling