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K u r z e Mitteilungen - Brief R e p o r t s - C o m m u n i c a t i o n s br&ves

L a m i n a r F l o w Heat T r a n s f e r f r o m a Flat Plate w i t h Variable T h e r m a l Conductivity B y P. C. SINHA,Dept. of Mathematics, I n d i a n I n s t i t u t e of Technology, Kharagpur, I n d i a

Introduction A simple solution for the case of h e a t transfer from a plate or a pipe for fully developed viscous b o u n d a r y layer with c o n s t a n t physical properties was given by Ls [11 ~). R e c e n t l y a n u m b e r of workers [3-5] h a v e studied the effect of the v a r i a t i o n of physical properties on h e a t transfer. I n t h e present note the effect of linear v a r i a t i o n of t h e r m a l c o n d u c t i v i t y with t e m p e r a t u r e on h e a t transfer from a flat plate is studied.

Analysis The energy e q u a t i o n governing the present p r o b l e m [6], neglecting viscous dissipation, is

~

~I

~

'

where u = c y (vide [6]). The b o u n d a r y conditions for the p r o b l e m are T=

T~(const)

for y =

0 ( x > 0),

T=

Too(const)

for

y+

~.

W i t h 0 = ( T -- Too)/(To~ -- T~) and k = k 0 (1 + /~ 0), t h e E q u a t i o n (1) reduces to

a0

~ (k

00~,

where k o = t h e r m a l c o n d u c t i v i t y at T = Tc~, e = ko/~ Cp and fl = constant. I n t r o d u c i n g the similarity variable [ c ~1/3 7= Y \b~-/ ' E q u a t i o n (2) can be w r i t t e n as d20

dO

-drl2 + 3 v]e ~ q + ~

[ d20 [0--+

{ gO ? ]

=0

(3)

r/--> oo.

(3a)

w i t h the b o u n d a r y conditions 0 = 1 for

r]~ 0

and

0 ~ 0 for

E q u a t i o n (3) is solved by the m e t h o d of successive approximation. The first approxim a t i o n corresponding to fi = 0 is given b y exp (-- ~a) d~

00(~l) = 1-

0 oo f e x p (-- ,)3) drl 0

1) Numbers in brackets refer to References, page 902.

,

(4)

Vol. 18, 1967

Kurze Mitteilungen - Brief Reports - Communications br~ves

901

w h e r e a s t h e second a p p r o x i m a t i o n is f o u n d to b e 01(~)

1+ CII+

~[2(I oxff

o-

I) ~ e x p ( - -

r~a ) + 4 E - -

2II 0-

310~],

(5)

where 00

[ =fexp

(-- rfl) d, 7 ,

Io

=fe• (-- r

0 t~

E=

;

0 00

f~iexp(--

2~fl) drl,

J

0

E o= f~]exp(-J

2~7 a) d~ ;

0

a n d C 1 is a c o n s t a n t of i n t e g r a t i o n , g i v e n b y Cl=fl

1 6I o

Io"

3 ~

T h e t e m p e r a t u r e profiles are c a l c u l a t e d for/~ = 0.5, 0 a n d -- 0.5, t a k i n g t h e v a l u e s of I f r o m [7] while t h e v a l u e s of E are e v a l u a t e d u s i n g W e d d l e ' s rule. E v e n t h o u g h t h e v a l u e s selected for fl m a y n o t b e realistic, y e t one c a n h a v e a n idea of q u a l i t a t i v e n a t u r e of t h e effect of t h e v a r i a t i o n of t h e r m a l c o n d u c t i v i t y w i t h t e m p e r a t u r e on h e a t t r a n s f e r , I n t h e Figure, t h e t e m p e r a t u r e profiles are d r a w n for d i f f e r e n t v a l u e s of ft. I t is o b s e r v e d t h a t a l i n e a r v a r i a t i o n of t h e r m a l c o n d u c t i v i t y w i t h t e m p e r a t u r e is h a v i n g a l i n e a r v a r i a t i o n in t h e t e m p e r a t u r e d i s t r i b u t i o n . f8

8.6 =

8.2

0

0.4

8.8

Z2

i.5

2.0

rl ,,-Effect of varying thermal conductivity on temperature distribution.

T h e local heat transfer coefficient and the local Nusselt n u m b e r are found to be

h--

k

r~o-- T00

~Ty o by

= =--k~

( c ~11~ dot \9o:x!

~

~=o

'

A t a n y g i v e n x, i t is seen t h a t t h e s e q u a n t i t i e s are d i r e c t l y p r o p o r t i o n a l t o dO1/d~ l~- o 9 V a l u e s of -- dO1/dl 7 [~=0 for d i f f e r e n t fl are g i v e n in t h e Table. I t is clear f r o m t h e t a b l e t h a t t h e r a t e of h e a t t r a n s f e r increases as fl decreases.

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Kurze Mitteilungen - Brief Reports - Communications braves

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Table fl

+0.5

0

-0.5

- dO~/drl [ ~ = 0

0.7864

1.1198

1.4533

I n conclusion I wish to express m y sincere t h a n k s to Dr. A. K. RAo for his help in t h e p r e p a r a t i o n of t h i s n o t e a n d t o Dr. K. S. SASTRI for his suggestions. REFERENCES [1] M. A. L~V~QUE, Ann. m i n e s 73, 201, 305, 381 (1928). [2] K. T. YANO, J. appl. Mech., Trans. A S M E 25, 146 (1958). [3] E. M. SPARROW a n d J. L. GReGG, Trans. A S M E 80, 879 (1958). [4] L. B. KoPPXL a n d J. M. SmTH, J. H e a t T r a n s f e r , T r a n s . A S M E 84, 2, 157 (1962). [5] G. POOTS a n d M. H. ROGERS, I n t . J. H e a t Mass T r a n s f e r 8, 12, 1515 (1965). [6] J. G. KNUDS~N a n d D. L. KATZ, Fluid Dynamics and Heat Trans/er (McGraw-Hill B o o k Co., Inc.), I n t e r n a t i o n a l S t u d e n t E d i t i o n , 363 (1958). [7] M. ABRAMOWITZ, J. M a t h . P h y s . 30, 162 (1951).

Zusammen/assung E s wird der E i n f l u s s einer m i t d e r T e m p e r a t u r l i n e a r v e r g n d e r l i c h e n L e i t f g h i g k e i t a u f die W g r m e i i b e r t r a g u n g a n einer e b e n e n P l a t t e u n t e r s u c h t . (Received: May 16, 1967.)

A b o u t Malkus' T r a n s i t i o n s in T h e r m a l T u r b u l e n c e B y LIONEL RINTEL, G e o p h y s i c a l Fluid D y n a m i c s L a b o r a t o r y , E S S A , W a s h i n g t o n , D.C., U S A l) T h e first a t t e m p t s to r e p r e s e n t t h e t r a n s i t i o n to t u r b u l e n c e as a succession of i n s t a b i l i t i e s were m a d e b y Bose a n d S o r k a u (see NOETHER [112)). L a t e r LANDAIJ [2] d r e w a n i m a g i n a t i v e p i c t u r e of h o w s e c o n d a r y i n s t a b i l i t i e s increase t h e n u m b e r of degrees of f r e e d o m a n d t h u s , b y b e c o m i n g m o r e a n d m o r e i n t r i c a t e , t h e m o t i o n a p p r o a c h e s o u r i n t u i t i v e n o t i o n of t u r b u l e n c e . I t was f o u n d t h a t o t h e r c h a r a c t e r i s t i c s of t u r b u l e n c e , such as t u r b u l e n t spots, c a n r e s u l t f r o m i n t e r a c t i n g i n s t a b i l i t i e s [31 . H o w e v e r , d o u b t s still exist t h a t n o t all aspects of t u r b u l e n c e r e s u l t f r o m i n s t a b i l i t i e s a l o n e [41. More r e c e n t l y MALKUS [5] p e r f o r m e d e x p e r i m e n t s o n h e a t t r a n s f e r in a fluid b e t w e e n t w o h o r i z o n t a l p l a t e s a n d i n t e r p r e t e d t h e r e s u l t s in t h e s p i r i t of Bose a n d S o r k a u ' s w o r k s as d i f f e r e n t regimes of t u r b u l e n t c o n v e c t i o n . T h e b o u n d a r i e s of t h e regimes were d e t e r m i n e d b y slope d i s c o n t i n u i t i e s o n t h e p l o t of h e a t t r a n s f e r versus t e m p e r a t u r e , a n d for R a y I e i g h n u m b e r s below 106 , six s u c h t r a n s i t i o n s h a v e b e e n claimed. T h e o r e t i c a l l y , for e a c h v a l u e of t h e w a v e n u m b e r k t h e n e u t r a l s t a b i l i t y e q u a t i o n yields s e v e r a l v a l u e s for t h e R a y l e i g h n u m b e r Ra. T h e m i n i m a l v a l u e s of R a for t h e v a r i o u s b r a n c h e s of t h e n e u t r a l s t a b i l i t y c u r v e so o b t a i n e d are t h e critical R a y l e i g h n u m b e r s R a t r, a n d t h e v a l u e s of t h e w a v e n u m b e r c o r r e s p o n d i n g to t h e m - t h e critical w a v e n u m b e r s k~r. B y assumillg a l i n e a r t e m p e r a t u r e profile b e t w e e n t h e p l a t e s a n d u s i n g some s i m p l i f y i n g a p p r o x i m a t i o n s of t h e n e u t r a l s t a b i l i t y curves, M a l k u s o b t a i n e d n u m e r i c a l e s t i m a t e s of t h e critical R a y l e i g h 1) Now at the College of William and Mary, Williamsburg, Virginia, USA. 2) Numbers in brackets refer to References, page 904.