A p p l . sci. Res.

Section A, Vol. 14

HEAT OVER

TRANSFER

A FLAT

PLATE

CONSTANT

IN THE WITH

HEAT

FLOW

SUCTION

AND

SOURCES

by K. S. SASTRI Department of Mathematics, Indian Institute of Technology, Nharagpur, India

Summary An analytical study has been made to estimate the effect of suction Oll heat transfer in the laminar boundary layer Oll a flat plate when the fluid is flowing with heat sources in it. Results have been obtained for Prandtl number equal to unity when the suction parameter is greater than 3.

§ 1. Introduclion. In recent years, considerable interest has been devoted to the study of the effect of heat generation on heat transfer l-a). Analytical studies have been carried out for several different forms of the heat generation 1-4). In 4) numerical solutions have been obtained for the energy equation and the distributions of heat transfer coefficient have been analyse& In this paper the effect of suction on the laminar flow heat transfer in the presence of heat generation is estimated for values of the suction parameter greater than 3. It is observed that the trend of the change of the temperatures in the boundary layer becomes steeper near the wall as suction increases. However the heat transfer parameter increases steadily with the non-dimensional longitudinal coordinate to an asymptotic value.

§ 2. Formulation o/ the problem. The boundary layer equations of motion, continuity and energy of an incompressible viscous liquid flowing steadily over a flat plate in the presence of heat sources, are ~u ~u ~2u u ~x + v ~Y v -0y2 (2.1) ' --

126

--

H E A T T R A N S F E R IN F L O W OVER A FLAT P L A T E

au - -

+

ax

av --

=

ay

aT aT v u -[- v - - -Ox Oy Pr

o,

127

(2.2)

a2T

h -1---, Oy2 pCv

(2.3)

where h is t h e heat generated per unit volume in unit time, v the kinematic viscosity, cv the specific heat. The term due to viscous dissipation is considered to be negligible in comparison to the h e a t transfer across the plate. The b o u n d a r y conditions are

u=O, v= vo(x) u=ul:constant

at

y=0,

at

y:oo,

T=

at

y=0,

at

y = oo.

(2.4)

and Tw=constant

T = Tl(X)

(2.5)

Setting

= (ulh,xP y and (2.6)

v,(x, y) = ( ~ u l x p / ( ~ ) , we obtain the velocity components (u, v) as

?V u

--

Oy

--

Ul/',

(2.7) v

-

ax

-

2

VI -

~/'],

where primes denote differentiation with regard to ~. Substituting (2.6) and (2.7) we observe t h a t (2.2) is identically satisfied and t h a t (2.1) takes the form

2/" + 1/" = O.

(2.8)

The b o u n d a r y conditions (2.4) become

]'=0,

/=

K=

constant

I':1

at

~=0,

at

~=oo,

(2»)

where vo(x) is assumed to be of the form

vo(x) = ½Kul/(ulx/v) ~.

(2.10)

128

K.S. SASTRI

Introducing the variable ~ by the relation

(2.11) equations (2.8) and (2.9) become (2.12)

2 K / ' + iß'---- 0 and at at

/=K f=0, f=(1/K)

~=0, ~=oo,

(2.13)

where f-

dl d¢'

)/_

d2I d$ 2 '

]._

aal d« a

Taking the Blasius function/(~) as oo

1

1(«) = K + 2 ~ - # i ~ ( ¢ ) , we obtain the values of/1, /3 as

la = 5

1 1 - = - - 2 + $ + 2 e -~«, e-~¢ [½¢2 + 2« + 6] + e-C,

(2.14)

(terms containing 1/K 2, 1/K 4.... can be shown to vanish) The Blasius function/(~) m a y be approximated by 1

/ = K -}- -K-il + ~

1

la.

(2.15)

This approximation is seen to lead to inconsistent results for K ~< 3 and hence is valid only for K > 3. Using (2.15), equation (2.3) is to be solved for the temperature subject to the conditions (2.5). § 3. Solution o/ the energy equation. The motion being one-dimensional outside the boundary layer, the temperature T1 m a y be • obtained from dT1 h ul - (2.1) dx pcv

129

HEAT T R A N S F E R IN FLOW OVER A FLAT PLATE

in the form T1 -- Tlo --

h

(3.2)

X,

pCv«l

where Tlo = Tl(0).

(3.3)

T -- Tw Tlo -- Tw

(3.4)

Substituting t=.

and using (2.7), the energy equation (2.3) m a y be rewritten as 1

~2t -+½i

Pr @2

~t

~~

-xl'

@

hx

+

~xx

pCv~~l(Tlo- Tw)

=0.

(3.5)

The b o u n d a r y conditions (2.5) become t=0

~ =0,

at

hx

t=l+

at

pcvul(Tlo -- Tw)

~=~.

(3.6)

Assuming the non-dimensional t e m p e r a t u r e t(x, ~) as

(3.7)

t(x, ~) = lo(~) + rtl(~), where

hx

y =

pCvUl(TlO -- Tw)

can be t a k e n as the non-dimensional longitudinal coordinate, we get from (3.5) and (3.6) 1

tt

Pr to + ½/tó = O, 1

tt

Pr tl + ½/ti --

l'tl

(3.8) + 1= 0

(3.9)

and to =

0,

tl =

to =

1,

tl= I

0

~t

~/ =

O,

at

~=oo.

(3.10)

130

K. S. SASTRI

Using (2.11) and assuming to, tl in the form co 1 to ---- ~~--0~

ton(~),

(3.12)

1

oo

n=0 K

we obtain the values of to, tl from equations (3.8) to (3.10) in the form 1 1 to = too + ~ to2 -[- - - ~ to4,

1 1 tl ---- tlo + - ~ - tl2 -[- ~ ~ - tl4, where, with P r = 1 too ---- 1 -- e -ic = tlo, to2 ---- e -½¢ (1¢2 _}_ 1) -- e-C,

to4 = e-~«(~«4 - I¢~ + ¢ + ?) + + e - « ( _ ~ ~ 2 _ 2~ -

i o) -

~e-%

tlü = e-~¢ (}~2 + 4¢ -- 1) + e-C,

t~~ = --e-~* (1«~ + «~ + ~:~ + ~ « _ -- e-« (½¢2 + 6¢ -]- 24) + ½e-~¢.

_~)_ (3.13)

The non-dimensional t e m p e r a t u r e t has been e v a l u a t e d for P r = 1 and K resp. 4 and 10. The profiles are d r a w n in figures 1 and 2 for various values of r.

§4.

Heat trans/er parameter.

T h e net heat flow across the wall

for a n y x m a y be w r i t t e n as q=

\ ~y /y=o

= k(Tlo

(4.1)

- - T w ) ( U l / X ~ ) t t'(O),

k being the t h e r m a l c o n d u c t i v i t y . The Nusselt n u m b e r for a n y x, Nu-

q

x

T1 - - T w

k

H E A T T R A N S F E R IN F L O W O V E R

r=20

22

2O lE

;~!

"

i

131

A FLAT PLATE

22 20 I 18 16

r=20

rs

2

o

I

2

3~1

o

Fig. 1. Nondimension~l temperature profile (t vs. ~1) f o r K = 4.

~2

I

I

~.4,1o~~8

i/0

Fig. 2. Nondimensional temperature profile (t vs. ~) for K = 10.

reduces to

Nu = K(Re)~ E i0(0)-/ril(0) ~ l+r

(4.2)

Re being the Reynolds number equal to (ulx[v). The behaviour of N = Nu. (Re)-~ with respect to r is shown in fig. 3. § 5. Discussion o~ the results. From figs. (1) and (X) it is seen that the temperature increases continuously to an asymptotic value. As the suction parameter increases, the temperatures become steady nearer the wall itself as observed from a comparative study of figs. (1) and (2). The striking difference between the case discussed in 4) and the present case where the fiow is affected by suction, lies in that there exists in the former, a zone where the temperatures in the boundary layer are greater than those outside, for the same longitudinal coordinate, hut in the latter, there is no such zone for any longitudinal coordinate.

132

I-IEAT T R A N S F E R IN F L O W OVER A F L A T P L A T E

5.~97

52

K=IO

J

Z II Z

5.1

r

10 ~ r

25

i

15

20

25

2A69 f

--24

;7|1

f

K-~

Z

23

22

F i g . 3. H e a t t r a n s f e r p a r a m e t e r

lO---,.rlS

2'O

N r s . t h e n o n d i m e n s i o l l a l l o n g i t u d i n a l coo r d i n a t e «.

In the presence of heat sonrces when the flow is affected b y suction, the heat transfer parameter (see fig. 3) steadily increases to an asymptotic value - a result in agreement with the Olle obtained in 4). A c k l l o w l e d g e m e n t s . I am indebted to Prof. M. K. J a i n for his suggesting the problem and for his inspiring guidance. My thanks are also due to the Council of Scientific and Industrial Research, New Delhi, for having sanctioned me a research fellowship. Received 30th December, 1963. REFERENCES 1) 2) 3) 4)

F a y , J. A. and F. R. R i d d e l l , J. Aero. Sci. 25 (1958) 73. L e e s , L., Jet Propulsion 26 (1956) 259. C h a m b r e , P. L., Appl. sci. Res. A 6 (1957) 393. J o j i Y a m a g a , Proe. 9th J a p a n National Congress for Appl. Mech. (1959) 305.