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Heat and Mass Transfer 33 (1998) 515±519 Ó Springer-Verlag 1998

Free-convection between vertical walls partially filled with porous medium T. Paul, B. K. Jha, A. K. Singh

Abstract This paper presents an analytical study of laminar fully developed free-convection ¯ow between two vertical walls partially ®lled with porous matrix and partially with a clear ¯uid having interface vertically. The momentum transfer in porous medium is described by the Brinkman-extended Darcy model and the two regions are coupled by equating the velocity and shear stress at the interface. The governing equations having non-linear nature have been solved by using perturbation method. It has been found that effect of Brinkman term is in entire porous domain for large values of Darcy number while its effect is con®ned nearer to interface and wall for small values of Darcy number. List of symbols Da Darcy Number d0 distance of interface from the wall y0 ˆ 0 d d0 =H g acceleration due to gravity H distance between vertical walls k0 permeability of the porous matrix k thermal conductivity N Buoyancy parameter T0 temperature of the ¯uid temperature of the hot wall Th0 Tc0 temperature of the cold wall u0 velocity U velocity in non-dimensional form y0 coordinate y non-dimensional coordinate Greek b l m q h

symbols coef®cient of thermal expansion dynamic viscosity kinematic viscosity density temperature in non-dimensional form

Received on 19 March 1997

T. Paul, B. K. Jha, A. K. Singh Department of Mathematics Banaras Hindu University Varanasi-221 005 INDIA

Subscripts f ¯uid layer p porous layer h hot wall c cold wall

1 Introduction The existence of a ¯uid-layer adjacent to a layer of ¯uidsaturated porous medium is a common occurrence in both geophysical and industrial environments, including engineering applications such as thermal-energy storage system, a solar collector with a porous absorber and porous journal-bearings. Furthermore convection through a porous medium may be found in ®bre and granular insulation, including structures for high power density electric machines, and cores of nuclear reactors. Beavers and Joseph [1] ®rst time studied the ¯ow mechanism at porous/¯uid interface. Comprehensive literature survey concerning with this subject is given in monographs [2, 3]. Rudraiah and Nagaraj [4] studied the fully developed free-convection ¯ow of a viscous ¯uid through a porous medium bounded by two heated vertical plates. Sacheti and Singh [5] extended the same problem in a rotating system. Recently Kuznetsov [6] presented an analytical solution of the ¯uid ¯ow in the interface region between a porous medium and a clear ¯uid in channels partially ®lled with a porous medium. The purpose of the present work is to present an analytical solution of free-convective ¯ow in the interface region between a porous medium and a clear ¯uid bounded by two vertical walls. In most of the works [7± 11] while using Brinkman model it has been assumed that viscosity of the ¯uid is equal to effective viscosity of the porous matrix. However, a recent experimental investigation has demonstrated that for some cases it is important to distinguish these two coef®cients. But, here we have also preferred to take them equal in order to see the effect of other physical parameters. Furthermore, thermal conductivities of the ¯uid and porous medium are also assumed to be the same.

2 Basic equations and description of the problem The second author (B. K. Jha) is thankful to C.S.I.R., New Delhi Steady fully developed free-convective ¯ow between two vertical parallel walls partially ®lled with porous medium for ®nancial assistance. and partially ®lled with a clear ¯uid has been considered when one wall is heated and other is cooled as shown in Correspondence to: A. K. Singh

515

Da ˆ Uf ˆ hf

k0 ; H2

yˆ u0f m

y0 ; H

dˆ

;

Up ˆ

gbH 2 …Th0 ÿ Tc0 † …Tf0 ÿ Tc0 † ; ˆ 0 …Th ÿ Tc0 †

d0 ; H

u0p m

gbH 2 …Th0 ÿ Tc0 † …Tp0 ÿ Tc0 † hp ˆ 0 ; …Th ÿ Tc0 †

;

g 2 b2 H 4 …Th0 ÿ Tc0 † : …6† km The physical quantities used in above equations are de®ned in list of symbols. It should be noted that Eq. (3) describing momentum transfer in porous domain is based on Brinkman-extended Darcy model [12]. Furthermore in (5) matching conditions for velocity are due to continuity of velocity and shear stress at the interface. The continuity of temperature and heat ¯ux at the interface has been considered as matching conditions for temperature. The constants A and B Fig. 1. Physical con®guration of the system appeared in boundary conditions given by (5) are due to consideration of temperature of walls at y ˆ 0 and y ˆ 1 as TC0 ‡ A…Th0 ÿ TC0 † and TC0 ‡ B…Th0 ÿ TC0 † respectively. The Fig. 1. The x0 -axis is taken along one of the wall and y0 -axis combination A ˆ 1 and B ˆ 0 corresponds when wall at y ˆ 0 is heated and y ˆ 1 is cooled while combination normal to it. It has been found that under usual BousA ˆ 0 and B ˆ 1 gives its vice versa. sinesq approximation, the ¯ow in ¯uid and porous regions is governed by the following equations in non-dimensional 3 form Method of solution Fluid region: It can be observed that problem is non-linear due to vis2 cous and Darcy dissipation terms. This problem can be d Uf ; …1† ‡ h ˆ 0 tackled by using a perturbation method as N is small in f dy2 most of the practical problems. Accordingly we assume,   for small N, the expansions: d2 hf dUf 2 ; …2† ‡ N ˆ 0 dy2 dy Uf ˆ U0f ‡ NU1f ‡ O…N 2 †; N ˆ

516

Up ˆ U0p ‡ NU1p ‡ O…N 2 †;

Porous region:

d2 Up Up ‡ hp ˆ 0 ; ÿ dy2 Da   d2 hp dUP 2 N 2 ‡N ‡ Up ˆ 0 : dy2 dy Da

…3† …4†

The boundary and matching conditions in dimensionless form are:

y ˆ 0:

Uf ˆ 0;

hf ˆ A;

y ˆ 1:

Up ˆ 0;

hp ˆ B;

y ˆ d:

Uf ˆ Up ;

dUf dUp ˆ ; dy dy

hf ˆ hp ;

dhf dhp ˆ : dy dy

…5†

Above equations as well as boundary conditions in nondimensional form have been obtained by use of the following non-dimensional quantities:

hf ˆ h0f ‡ Nh1f ‡ O…N 2 †; hp ˆ h0p ‡ Nh1p ‡ O…N 2 † :

…7†

The quantities U0f , U0p , h0f and h0p are the solutions for N equal to zero i.e., when the viscous and Darcy dissipations are neglected whereas U1f , U1p , h1f , h1p are perturbed quantities relative to U0f ; U0p ; h0f and h0p respectively when the viscous and Darcy dissipations are taken into account. Use of Eqs. (7) in Eqs. (1)±(4) and boundary conditions (5) has yielded the following solutions for U0f ; U0p ; h0f and h0p .

 …A ÿ B† 3 A 2 …B ÿ A† 2 U0f ˆ y ÿ y ‡ d ‡ …B ÿ A†Da 6 2 6       d Da …B ÿ A† 2 d Da ‡ A‡ m2 d ‡ ÿ A ‡ 2 d 3 2 d    1 ÿ exp f2m…1 ÿ d†g  …8† ‡ 2mm2 BDa exp fm…1 ÿ d†g y ;



 …B ÿ A† 3 A 2 d ‡ d ÿ A Da 3 2

medium for Da ˆ 10ÿ3 for all the three values of d in both cases. In which ®rst portion lies in the momentum boundary layer near the interface y ˆ d while third portion lies in the momentum boundary layer near the wall …y ˆ 1†. The momentum boundary layer near the wall …y ˆ 1† is more visible when the wall at y ˆ 1 is heated and y ˆ 0 is cooled. Between these two layers the second portion indicates that velocity is constant and effect of Brinkman term is almost negligible and ¯ow is only characterised by classical Darcy law [13]. This result also …10† con®rms the ®ndings of Singh and Thorpe [14] and Kuh0f ˆ h0p ˆ A ‡ …B ÿ A†y ; znetsov [6] that Brinkman term has its effect only nearer where  ÿ1 to impermeable wall and interface when Darcy number is …1 ‡ md† small. When Da ˆ 10ÿ1 and 10ÿ2 , there is no any portion m ˆ Daÿ1=2 ; m1 ˆ 1 ÿ exp …2m…1 ÿ d†† ; having a constant velocity and momentum boundary layer …1 ÿ md† is throughout the porous medium. Thus the ¯ow is charm1 : …11† acterised by Brinkman-extended Darcy model in the entire m2 ˆ …1 ÿ md† porous matrix. A comparative study of these ®gures sugWhen viscous and Darcy dissipations are zero, velocity gests that intensity of free convection currents can be pro®les between vertical walls have been illustrated in suppressed by imposing porous matrix along the heated Figs. 2 and 3 for the cases A ˆ 1, B ˆ 0 and A ˆ 0, B ˆ 1 wall. respectively. A clear indication from these ®gures is that The next task is to obtain the solution corresponding to velocity is divided into three portions in the porous ®rst order to see the effect of viscous and Darcy dissipa-

U0p ˆ m2   exp fm…y ÿ d†g ÿ exp fm…2 ÿ d ÿ y†g   1 ‡ md exp fm…1 ÿ 2d ‡ y†g ‡ m1 B Da 1 ÿ md  ÿ expfm…1 ÿ y†g ‡ ‰…B ÿ A† y ‡ AŠ Da ; …9†

Fig. 2. Velocity pro®les when dissipations are neglected for A = 1 Fig. 3. Velocity pro®les when dissipations are neglected for and B = 0 A = 0.0, B = 1.0

517

tion terms. Corresponding to this case, the obtained differential equations are

d2 U1f ‡ h1f ˆ 0 ; dy2   d2 h1f dU0f 2 ‡ ˆ0 ; dy2 dy

518

…13† …14†

d2 U1p U1p ‡ h1p ˆ 0 ; ÿ dy2 Da   d2 h1p dU0p 2 1 2 ‡ ‡ U0p ˆ 0 ; dy2 dy Da

…15† …16†

with the boundary and matching conditions

y ˆ 0: y ˆ 1:

U1f ˆ 0; U1p ˆ 0;

y ˆ d:

U1f ˆ U1p ; h1f ˆ h1p ;

h1f ˆ 0; h1p ˆ 0; dU1f dU1p ˆ ; dy dy dh1f dh1p ˆ : dy dy

Fig. 5. Temperature correction pro®les when A=0, B=1

…17†

The analytical solution of above equations are obtained and due to long expressions, they are not given here to curb the length of the paper. The numerical values corresponding to these solutions are given in the Figs. 4 and 5 showing the effect of dissipation terms. Order of magnitude analysis indicates that in¯uence of dissipation terms on both temperature and velocity ®elds are very small while consideration of these terms makes the problem non-linear. So these terms can be neglected in a problem of complex nature. Another quantity of physical interest is skin-friction on both walls. It has been calculated from the gradient of

Fig. 6. Velocity correction pro®les

velocity ®eld. In Table 1, s1 and s2 represent the numerical values of skin-friction on the walls y ˆ 0 and y ˆ 1 respectively. A clear view from this table is that skin-friction increases on the both walls as width of ¯uid layer increases. The effect of dissipation terms are negligible on it. Effect of Darcy number is to increase the skin-friction on both walls.

Fig. 4. Temperature correction pro®les when A=1, B=0

4 Conclusions Analytical solution of fully developed free-convective ¯ow between two-vertical walls partially ®lled with porous matrix is obtained. Brinkman term has its in¯uence in entire porous matrix for higher Darcy number and its effect is con®ned nearer to the interface and wall for small

Table 1. Numerical values of skin-friction Da )1

10

N 0.0 0.1

10)2

0.0 0.1

d 0.3 0.5 0.7 0.3 0.5 0.7 0.3 0.5 0.7 0.3 0.5 0.7

A = 1.0, B = 0.0

A = 0.0, B = 1.0

s1

s2

s1

s2

0.2601317 0.2960826 0.3221576 0.2601831 0.2961280 0.3222126 0.1800000 0.2427759 0.2940734 0.1800058 0.2427946 0.2941132

0.0772881 0.0916772 0.1221194 0.0773226 0.0917072 0.1221665 0.0101185 0.0116470 0.0249912 0.0101196 0.0116497 0.0249999

0.0947285 0.1223570 0.1504263 0.0948044 0.1224479 0.1505186 0.0324571 0.0653299 0.1107526 0.0324571 0.0653386 0.1107787

0.2196207 0.2310641 0.2647968 0.2199626 0.2314134 0.2651384 0.0900409 0.0909297 0.1038333 0.0900432 0.0909328 0.1038413

values of Darcy number. The effect of dissipation terms is negligible.

7. Neale G; Nader W (1974) Canad J Chem Engng 52: 475±478 8. Beckerman C; Ramadhyani S; Viskanta R (1987) ASME J Heat Transfer 109: 367±370 9. Beckerman C; Viskanta R; Ramadhyani S (1988) J Fluid Mech References 186: 257±286 1. Beavers GS; Joseph DD (1967) J Fluid Mech 30: 197±207 10. Sathe S B; Lin W Q; Tong T W (1988) Int J Heat Fluid Flow 9: 2. Nield D A; Bejan A (1992) Convection in porous media. New 389±395 York, Springer-Verlag 11. Singh A K; Leonardi E; Thorpe G R (1993) ASME J Heat Trans 3. Kaviany M (1991) Principles of heat transfer in porous media. 115: 631±638 New York, Springer-Verlag 12. Brinkman H C (1947) Appl Sci Res A1: 27±34 4. Rudraiah N; Nagraj S T (1977) Int J Engng Sci 15: 589±600 13. Darcy H P G (1830) Victor Dalamount 5. Sacheti N C; Singh A K (1992) Int Comm Heat Mass Trans 19: 14. Singh A K; Thorpe G R (1995) Ind J Pure Appl Maths 26: 81± 423±433 95 6. Kuznetsov A V (1996) Appl Sci Res 56: 53±67

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