Natural convection in a thermally stratified fluid saturated porous medium

Natural convection in a thermally stratified fluid saturated porous medium

O&?O-7225192 $5.00+ 0.00 PergamonPressLtd ant.1. EngngSci. Vol. 30, NO.8, PP. 1008-1~7, 1992 Printedin GreatBritain NATURAL CONVECTION IN A THERMALL...

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O&?O-7225192 $5.00+ 0.00 PergamonPressLtd

ant.1. EngngSci. Vol. 30, NO.8, PP. 1008-1~7, 1992 Printedin GreatBritain

NATURAL CONVECTION IN A THERMALLY STRATIFIED FLUID SATURATED POROUS MEDIUM KALPANA

TEWARI

and PUNYATMA

SINGH

Department of Mathematics, Indian Institute of Technology, Kanpur, Kanpur-208016, India Abstract-Natural convection heat transfer results are presented for a vertical plate of finite length immersed in a fluid saturated porous medium. The ambient vertical temperature gradient is positive but the plate temperature is everywhere higher than that of the surrounding fluid. Solutions are obtained by series expansion about the leading edge of the plate. The effects of stratification parameter on local and overall heat transfer coefficients, velocity and temperature fields are studied.

1.

INTRODUCTION

The problem of natural convection heat transfer from bodies submersed in thermally stratified fluid saturated porous medium arises in many important applications [l, 21. An important engineering application can be found in the problem of natural convection in an enclosed rectangular cell filled with some fluid saturated porous medium with one wall heated and the other cooled. Heated fluid rising from the hot wall overlays the top of the cell and the cool fluid falling from the cold wall lies along the bottom thereby creating a stratification inside the cell. Recently, Bejan [3] and Singh and Sharma [5] have studied the boundary layer free convection along a vertical plate immersed in a stable, thermally stratified saturated porous medium. Takhar and Pop [4] have obtained similar solutions when a vertical plate of uniform surface heat flux is embedded in a thermally stratified Darcian fluid. The present work deals with heat transfer from a vertical plate at uniform temperature, immersed in a fluid saturated porous medium whose temperature increases linearly with height. The leading edge of the plate is at a temperature above that of the surrounding fluid at the same elevation. It must be presumed that the plate is of finite length or else its temperature above some elevation would fall below that of the surrounding fluid and the dominant flow direction would be towards the leading edge. The problem is solved by series expansion about the leading edge of the plate; the first term in this series corresponds to the standard result for the case when there is a fixed temperature difference between the plate and the external environment. The solutions for three terms in the series expansion are presented and the effect of stratification parameter on heat transfer, temperature and flow fields are studied.

2. MATHEMATICAL

ANALYSIS

Consider, steady free convection on a vertical flat plate embedded in a stable, thermally stratified saturated porous medium. The ambient temperature T,(x), of the fluid is assumed T,(x) = T,, + Bx

(2.1)

where T,, is the ambient temperature at x = 0 and B = dT,ld.x, the slope of the ambient temperature profile with vertical distance. If we assume that (1) the temperature of the plate is uniform throughout its length H and surrounding temperature T, is proportional to n with T, > T,, (2) the convective fluid and the porous medium are everywhere in local thermodynamic equilibrium, (3) the properties of the fluid and the porous medium are constant and 1003

1004

(4) the Boussinesq

K. TEWARI

approximation

and P. SINGH

is employed,

the governing equations are [4],

(2.2) (2.3) (2.4) where U, v are the Darcian velocities in the (x, y) directions. p and Y are thermal expansion coefficient and kinematic viscosity of the fluid, g is the acceleration due to gravity, K is the permeability of the porous medium, CEis the thermal diffusivity of the porous medium and T denotes the temperature inside the boundary layer. The boundary conditions at the wall are T = Tw

(2.5)

T + T,(x).

(2.6)

v = 0, and at infinity are U-+0, Let us introduce the following variables

r = tY/~)(~~)l~

(2.7)

(2.8) T = tl(q, i)0,,

+ T,

(2.9)

where R, = g/3K( T, - Tw)xlva. In terms of new variables it is easy to show that the governing equations (2.1)-(2.4) boundary conditions (2.5)-(2.6) become

(2.10)

f, = 6 @,, + WJ2

-X(f,

rl =o,

f =O,

+!&

-fXj)

=0

e=[l-f]

fq = 0,

n-,03,

with the

(2.11) (2.12)

0 = 0.

(2.13)

To solve equations (2.10) and (2.11) a series expansion about the leading edge is used, Thus f and 8 may be assumed as

f (?I,3 = $

(-lY~ml;,(rl)

(2.14) (2.15)

@(rj, 2) = i: (-l)“F&(rj). m=o Inserting (2.14) and (2.15) in equations differential equations:

(2.10)-(2.13)

for R”;

F;-Ho=0

we get the fo~owing sets of ordinary (2.16) (2.17)

Hg + (&E&)/2 = 0 MO) = 0,

Ho(O) = I,

11;1o(@J) =0

F,’ - H, = 0

for i’:

(2.19) (2.20)

Ii’;+(FoH;+3F,H&)/2+F;--l&F;=0 F,(O) = 0,

H,(O) = 1,

(2.18)

HJ=)

= 0

(2.21)

Natural convection in a thermally stratified fluid

(2.22)

F;-Hz=0

for f2;

1005

H; + (F,H; + 3&H; + 5F,H&)/2 + F; - H,F; - 2HzF; = 0

(2.23)

F,(O) = Hz(O) = H,(m) = 0

(2.24)

where the primes refer to differentiation with respect to rl. Of course, the set of ordinary differential equations go on ad infinitum, and their complexity also increases. The local rate of heat transfer from the surface of the plate to the medium can be computed from 4 = -k(aTlay),,=o

(2.25)

which in terms of the Nusselt number may be written as

-$ =(-H&(O) + _W;(O) x

.f’H;(O) + * * *).

(2.26)

The overall surface heat transfer rate from the plate with a length H can be computed from the relation

Q=$&W The final expression for the non-dimensional -NO-H

(2.27)

overall heat transfer is obtained as

= -2( H&(O)- ; bH;(O) + 5 b2H;(0) f * * +.

R’”H

(2.28)

Equations (2.16)-(2.24) are integrated nume~cally on a digital computer. A shooting technique is employed to solve equations (2.16)-(2.21), after which equations (2.16)-(2.24) were solved simultaneously.

3. RESULTS

AND DISCUSSION

The variation of local Nusselt number NxlRia with P is shown in Table 1. The results obtained by the integral method [5] are also presented in the table. It is found that the local Nusselt number decreases as R increases. This is expected because T, - T, is constant and the local temperature diierence between wall and the fluid decreases as B increases. For R = 0, the local Nusselt number is 0.444 which is same as obtained by Cheng and ~inko~~ [2f. The variation of N~-H/~~ with strat~~tion parameter b is computed and shown in Table 2. As expected it is found that the value of No..+/R~ decreases as b increases that is the coefficient in Table 1. The variation of local Nusselt number with the local stratification parameter f

WR:”

N,/R:a

2

(present solution)

(integral method [5])

x.1 0.2 0.3 0.4 0.5

x*zi 0:330 0.277 0.227 0.176

0.427 0.368 0.311 0.257 0.207 0.160

1006

K. TEWARI

and P. SINGH

Table 2. The variation of No-*/RF with the stratification parameter b b

No-JR;’ (present solution)

No.+,/R~ (integral method [5])

0.5 0.8 1.0

0.888 0.811 0.702 0.601 0.538

0.854 0.775 0.665 0.552 0.512

the NO--H- RF decreases monotonically as b increases. The reason for this expectation is that the effective temperature difference between wall and fluid decreases as B increases, When b = 0 the present solution gives No_H/R$2 = 0.888 which is same as obtained by Cheng and Minkowycz [2]. The functions Hi are shown in Fig. 1. Figure 2 shows the temperature profiles for different values of 3. The second order corrections to 8 are small and for the most part should be negligible.

1

t I

i 2

I

L

t

1

I

3

4

5

6

7

rlFig, 1. Temperature

Fig. 2. Temperature

functions, Hi for i = 0, 1 and 2.

profiles for various values of local stratification parameter.

Natural convection in a thermally stratified fluid

1007

Acknowledgements-The authors are grateful to the anonymous referee for his useful comments. Financial assistance from CUR, New Delhi is gratefully acknowledged.

REFERENCES [l] [2] [3] [4] [5]

P. CHENG, Adu. Heat Transfer 14, 1 (1978). P. CHENG and W. J. MINCKOWYCZ, J. Geophys. Res. 82, 2040 (1977). A. BEJAN, Convection Heat Transfer. Wiley, New York (1984). H. S. TAKHAR and I. POP, Me&. Res. Commun. 14,81 (1987). P. SINGH and K. SHARMA, Acta Me& 83, 157 (1990). (Revision received 9 July 1991; accepted 18 December 1991)