Transport #I Porous Media 6: 159-171, 1991. @ 1991 Kluwer Academic Publishers. Printed in the Netherlands.

159

Horizontal Boundary-Layer Natural Convection in a Porous Medium Saturated with a Gas I O A N POP

Facuhy of Mathematics, UniversiO, of Cluj, Cluj, Romania and RAMA SUBBA REDDY GORLA

Department of Mechanical Engineering, Cleveland State University, Cleveland, OH 44115, U.S.A. (Received: 8 August 1989; in final form: 7 June 1990) Abstract. Boundary-layer analysis is performed for free convection flow over a hot horizontal surface embedded in a porous medium saturated with a gas of variable properties. The variable gas properties are accounted for via the assumption that thermal conductivity and dynamic viscosity are proportional to temperature. A similarity solution is shown to exist for the case of constant surface temperature. Numerical results for the stream function, horizontal velocity, and temperature profiles within the boundary layer as well as for the mass of entrained gas, surface slip velocity, and heat transfer rate at different values of the wall-temperature parameter are presented. Asymptotic solutions for large heating are also available to support the numerical work. Key words. Convection over a hot horizontal surface, boundary layers in porous media saturated with a gas of variable properties.

Nomenclature Ch

f(q) g gO1) k K L nj P q Ra /2,

heat transfer coefficient, specific heat, similarity stream function, similarity stream function (large A formulation), acceleration due to gravity, similarity temperature function, similarity temperature function (large A formulation), thermal conductivity, permeability of the porous medium, characteristic length of the plate, unit vector in the direction of gravitational force, pressure, heat transfer rate, Modified Rayleigh number, velocity components in x and y directions,

160

U,V x, y

X,Y Z

IOAN POP A N D R A M A SUBBA REDDY GORLA

boundary-layer velocity components in X and Y directions, Cartesian coordinates, boundary-layer coordinates, Howarth-Dorodnitsyn coordinate

Greek Symbols o~ thermal diffusivity, heating or cooling parameter, A small parameter, ~,~ similarity variables, temperature, 0 dynamic viscosity, density, P stream function. 4, Superscript ' differentiation with respect to r/or 4Subscripts o condition in the undisturbed state, W condition at the plate.

1. Introduction Thermal convection problems in porous media occur in a broad spectrum of disciplines ranging from chemical engineering to geophysics. Some problems of technological interest, whose performance depends on a better understanding of natural convection in porous media, include heat insulation by fibrous materials, nuclear reactors, thermal insulations, spreading of pollutants, underground heat exchangers for energy storage and recovery, and flows in water-percolated soils. The literature on free and forced convection boundary-layer flows adjacent to surfaces embedded in porous media is extensive. As far as the basic equations are concerned, there is some variety in the form of these equations. For our purpose, it is sufficient to refer to the review articles by Cheng (1978, 1985) and Bejan (1987). From the literature, we know that almost all studies of convective flows in porous media were based on the Boussinesq form of the boundary-layer equations. Thus, variations of density, viscosity, and thermal conductivity (collectively referred to here as variable properties effects) were ignored except for the essential inclusion of the density variation in the relevant buoyancy term. However, this approximation is reasonable if the temperature and pressure gradients that enter into the problem are not too large. In contrast with the forced convection problems, the density has to be considered as temperature-dependent in free convection problems, because temperature and density gradients are generators of motion. For that reason, it

HORIZONTAL BOUNDARY-LAYERNATURAL CONVECTION

161

could be generally expected that the variable properties have a bigger effect on free than on forced convection flow and this can be an important consideration in practical problms. A systematic study on the effect of variable properties on free and forced convection boundary-layer problems in a viscous fluid has been carried out recently by Herwig (1985). For a porous medium, there are also several theoretical studies which consider fluids of variable properties. Thus, Saatdjian (1980) has considered the natural convection in a porous layer bounded by two horizontal planes saturated with a compressible ideal gas. Blythe and Simpkins (1981) have examined the thermal convection in a rectangular enclosure containing a porous medium for the case in which the viscosity depends strongly on the local temperature. Further, Mey and Merker (1987) gave an analysis of free-convection flow in a porous shallow cavity with variable properties using the method of matched asymptotic expansions as described by Herwig (1985). For a body of arbitrary shape immersed in a porous medium saturated with a fluid of variable viscosity, we mention the paper by Taunton and Lightfoot (1970). When the wall temperature of a semi-infinite horizontal upward facing plate is kept higher than that of a surrounding porous medium, a vertical density gradient is generated within the thermal boundary layer over the plate, which then will create a pressure gradient along the plate. If the pressure force is greater than the buoyance force, the fluid moves along the horizontal plate so as to relax its pressure. The problem has many important applications such as the convective flows above the heated bedrock and below the cooled caprock in a liquid dominant geothermal reservoir. The present paper is concerned with steady, free convection boundary-layer flows in a porous medium saturated with a gas subjected to severe heating. The particular geometry of interest corresponds to a porous medium adjacent to a hot horizontal impermeable surface. Use of Howarth Dorodnitksyn transformation makes it possible to dispense with the usual Boussinesq approximation, and variable gas properties are accounted for via the assumption that dynamic viscosity and thermal conductivity are proportional to temperature. A similarity solution is shown to exist for the case of constant surface temperature. Although the resultant coupled nonlinear ordinary differential equations are in functional form, the numerical method used encounters no difficulties, and an asymptotic solution for very large heating is also available to support the numerical work. As a check on the numerical method used, on the other hand, the known results of other authors are reproduced. Comparisons with other available solutions show excellent agreement. The main finding of the present paper, however, is concerned not with the accuracy of the technique, but with the remarkable conclusion that there is a substantial variation in the transport rates as compared to results assuming constant fluid properties. This result should be of great interest to experimentalists in various fields.

162

IOAN POP AND RAMA SUBBA REDDY GORLA

2. Basic Equations Consider a semi-infinite horizontal impermeable surface lying beneath a porous medium. Assuming that the porous medium is saturated with a gas, the set of equations which describes natural convection of a variable-density flow can be expressed in dimensionless form as (pu,) = o,

@

/~uj = - O~j + (p -

(1)

1)nj,

(2)

pO'= 1

(4)

along with the law of Charles and Gay-Lussac (Clarke and Riley 1976). Equations (1) through (3) are written in Cartesian tensor notation and are rendered dimensionless with the following reference parameters: L for the space vector xj; pogK/l~o for the velocity vector uj; pogL for which is the pressure; To for the absolute temperature O" so that it has the value 1 in the undisturbed state; Po, Ilo and ko for the density p, dynamic viscosity/1, and thermal conductivity k, respectively; nj is the unit vector in the direction of gravitational force and Ra is the modified Rayleigh number for a porous medium defined as Ra = pogKL/l~o~o, where % = ko/C~po. For the further analysis, it proves convenient to define 0 = 0' - 1 so tht 0 = 0 in the undisturbed state; accordingly Equation (4) becomes p(l+0)=l.

(5)

We are now in a position to make a study of the present problem under the particular parameter limit Ra ~ oo, or the boundary layer approximation. The geometrical configurations that we investigate here is sketched in Figure 1, which exhibits the relevant boundary conditions together with the notation that will be adopted from now on. However, it is of interest to note at this stage that since 0 -~ 0 in the outer region of the boundary layer, Equation (5) implied p ~ 1 to a first order there. The velocity vector then satisfies an incompressible form of the continuity equation, from (1), and using (2) it can be easily shown that the outer fluid is potential. We shall further derive the boundary-layer equations for the present problem. Following the formalism used by Cheng and Chang (1976) for treating this type of problem, we introduce the boundary layer variables

X=x,

Y=y/e,

p = eP,

0= O

u =eU,

V=~2V, (6)

where e = ( R a ) -1/3 is a small parameter. Substituting (6) into Equations (1)

HORIZONTAL BOUNDARY-LAYER NATURAL CONVECTION g ~1;

163

O, p. u ~ 0

as y ~

eo

x2-Y

l

U2_= V

g

Darcy's velocity components

T

n i (0,-1) /

Ul-

U

/

I

I

/////////////~/////// / /

0

Ow(x
= To

O (x > 0 , 0 )

= O w (x) > 0

v (x>O,O)

= 0

Xl--~ X

Fig. 1. Physicalmodel and coordinate system.

through (3) and retaining the terms of order e only gives the boundary layer equations: ( p U ) x + ( p V ) r = 0,

(7)

~tU = --P~,

(8)

Pv = l - p = pO,

(9)

o u o x + pVOy = (kOy)~,.

(lO)

Now, in order to simplify these equations and bring them close to an incompressible form, we must necessarily make use of several similarity transformations. Thus, we define a stream function ~ such that

pu=oy,

p v = -~,x

(11)

and continuity Equation (7) is then satisfied. Next, independent variables are changed from (x, Y) to (X, Z) by using the Howarth-Dorodnitsyn transformation (Stewartson, 1964).

X =X,

Z =

~0Yp(x,

s) ds =

~0Y{1 -

P, (x, s)} ds = Y - P(x, Y ) + P ( x , o).

(12) Defined thus, Z, fits conveniently within the canon of compressible boundary-layer

IOAN POP AND RAMA SUBBA REDDY GORLA

164

theory. Thus, from the definition of the new variable Z, it follows that a(o,

4,)

~(o,

r

(13)

u = ~,z, ~(x, ]9 - p a(x, z )

and

Px(x, y) = Px(X, Z) + Pz(X, Z) { - P x ( X , Y) + Px(X, o)}.

(14)

The later relation (14) is required for the transformation of the term in Equation (8). Hence, Equations (8) through (10) become (~tp)r

(15)

= - p~ - oPt.o,

Yz=O,

(16)

~ z O x -- ~ x O z = (pkOz)z.

(17)

Note that Pxo in Equation (15) is written for Px(X, o) and the appearance of this term makes the boundary-layer equations a set of functional differential equations. Before looking for similarity solutions of these equations, it is worth examining properties of the solution when some simplifying assumptions are made about p/~ or pk. The simplest of these, and in many ways the most useful and relevant, is that the porous medium is saturated with a fluid in which l~, k ~ (9 so that

(18)

Pl~ = l = pk. For this fluid, Equation (15) through (17) reduce to [(1 + O ) 2 0 z ] z = --Ox + Oz

OzdZ,

~gzOx - ~ x O z = Ozz.

(19) (20)

after P is eliminated. The boundary conditions of these equations are r

o) = o,

o(x,

~p(X,Z~,zo)~O,

o) = o,,,(x),

|

Z ~ m) --,0.

(21)

If the imposed surface temperature 0w(X) is given by O,,(X) = AX a,

(22)

where A and 2 are constants, the Boussinesq version of Equations (19) and (20), subject to (21), have similarity solutions of the form (Cheng and Chang, 1976). q(X, Z) = A'/3X (~+ ,)/3f(q),

(23)

H(X, Z) = AX~G(q)

(24)

/1 = AI"3ZX(A- 2)/3,

(25)

and

HORIZONTAL BOUNDARY-LAYER NATURAL CONVECTION

165

where f ( q ) is the similarity stream function, g(q) is the similarity temperature, q is the similarity variable. Substitution of (23) and (24) into Equations (19) (20) shows that the variable-density problem has a similarity solution for only special value of the temperature index 2, namely zero. Hence, we obtain following set of ordinary differential equations.

and and one the

[l + Ag)2gq

= ( 2 / 3 ) ~ g ' - (2/3)

Ag'

f[

,Ig" d~,

g" + ( 1/3)fg' = O,

(26/

(27)

subject to the boundary conditions f(O) = 0,

g(0) = 1,

f ' ( m ) = g(oo) = 0,

(28)

where the primes denote differentiation with respect to q. In this respect, the present analysis deals with a situation that is quite different from that of fluids with constant properties where boundary-layer similarity solutions exist for a wall temperature being a power function of distance from the origin. One point to note about Equation (26) is that there is only the wall temperature parameter A left in the problem and it plays a central role in the physical interpretation of the solution. For the plate to be hotter than the ambient gas, it is necessary to have A > 0. It is relevant to note here that setting A = 0, Equation (26) reduces to that found by Chang and Cheng (1983) or Riley and Rees (1985) for natural convection from heated surfaces in porous media saturated with fluids of constant properties. This means that all physical properties of the fluid-saturated porous medium are taken as constant except the density in the buoyancy force term that drives the natural convection (Boussinesq approximation). The assumption that A = 0 does not conflict with the physics of the problem, since in this case the ambient temperature To is not appropriate as a reference temperature. However, Equation (26) includes the effect of variable fluid properties which have not been considered in previous work of convective flow from a horizontal surface immersed in a porous medium.

3. Numerical Method The numerical procedure used here solves the two-point boundary-value problems for a system of N ordinary differential equations in the range (X, Xt). The system is written as

~

,%

=f(x,y,,y2

.....

YN),

i = 1 , 2 . . . . . N,

and the derivatives f are evaluated by a procedure that evaluates the derivatives of Yl, Y2 YN at a general point X. Initially, N boundary values of the variable y,. must be specified, some of which will be specified at X and some at X1. The . . . .

,

166

IOAN POP AND RAMA SUBBA REDDY GORLA

remaining N boundary values are guessed and the procedure corrects them by a form of Newton iteration. Starting from the known and guessed values of y; at X, the procedure integrates the equations forward to a matching point R, using Merson's method. Similarly starting from Xl, it integrates backwards to R. The difference between the forward and backward values of y; at R should be zero for a true solution. The procedure uses a generalized Newton method to reduce these differences to zero, by calculating corrections to the estimated boundary values. This procedure is repeated iteratively until convergence is obtained to a specified accuracy. The tests for convergence, and the perturbation of the boundary conditions are carried out in a mixed form, e.g. if the error estimate for y; is E R R O R , we test whether ABS(ERRORi) < ERRORi • (1 + ABSy~). Essentially, this makes the test absolute for Yi <~ 1 and relative for 3,~ >> 1. Note that convergence is not guaranteed, particularly from a poor starting approximation. However, the algorithm is computationally simple, numerically stable, and suitable for obtaining physically meaningful results.

4. Results and Discussion Equations (26) through (28) have been solved numerically by the method described in the previous section for arbitrary values of the heating parameter (A > 0). The results from the reduced stream function ~,/(Al"3x j/3) =f01), horizontal velocity U / ( A 2 / 3 x - 1,3) = f , ( q ) and reduced temperature profiles gO1) are displayed in Figures 2 to 4. These figures also show results for the constant fluid properties case A = 0, which are in excellent agreement with the numerical solution reported by Riley and Rees (1985). It should, however, be noted that the results for A = 0 were calculated by making A zero in Equation (26) but not elsewhere (Boussinesq approximation). From Figures 2 and 3 it may be observed that as A increases the dimensionless stream function assumes an S-shape, while the horizontal velocity profiles exhibit quite different characteristics when compared to the case A = 0; the curves o f f ' ( q ) versus q are concave towards the ordinate in the wall region. Note that both the stream function and velocity profiles decrease substantially as A increases. Moreover, the velocity profile within the boundary has a maximum value away from the surface and the location of the maximum velocity moves further away from the surface as A increases. Figure 3 also shows that the momentum boundary-layer thickness becomes thicker in terms of r/. Further, the dimensionless temperature distribution is depicted in Figure 4. We see that as A increases, the temperature profiles become broader and the thermal boundary-layer thickness increases. However, it is important to note that actual physical boundary-layer thickness vary with A in quite a complicated way which must be traced back through Equations (9), (12), (24) and (25), respectively. There is no doubt that the Boussinesq approximation will prove quite inadequate whenever the difference between the plate temperature and the ambient temperature is comparable with the absolute ambient temperature.

HORIZONTAL BOUNDARY-LAYER NATURAL CONVECTION

./

2 I

/

f

~__~...----

167

--~"

40.

f

Present results

0

I

I

I

I

5

10

15

20

Fig. 2. Stream function profiles divided by A~3X ~/3 for selected values of A. The curve labeled A = 0 was calculated by making A zero in Equation (26) but not elsewhere (Boussinesq approximation).

=

9

1.0

Riley and Rees (1985) Present results

8

O

5

10

15

20

q Fig. 3.

Horizontal velocity profiles divided by A2/3X- U3 for selected values of A.

168

IOAN POP AND RAMA SUBBA REDDY GORLA 1.0

~~~

9

Riley and Rees (1985) Present results

I \\\\\\\

0.6

0.2

o

5

10

15

20

q Fig. 4. Temperature profiles divided by A for selected values of A.

An appropriate heat transfer coefficient is defined by

Ch =

-- Ra '/3A4/3X

- -

2/3G'(0),

(29)

where Ch = qwL/To and qw = - ( T o / L ) ( k O0"/O>')y= o being the surface heat transfer. Variations o f f ( o o ) , f ' ( 0 ) and g'(0) are also plotted in Figures 5 - 7 versus A; the quantities are relevant, respectively to the mass of entrained gas, surface slip velocity and heat transfer rate. For A = 0 , Chang and Cheng (1983) found that f ( o o ) =2.813, f ' ( 0 ) = 1.053 and g ' ( 0 ) = - 0 . 4 2 9 9 while the present numerical method gives f ( o o ) = 2 . 8 1 3 1 , f ' ( 0 ) = 1.0529 and g ' ( 0 ) = 0 . 4 2 9 9 . Note that the results predicted by the present theory agree well with those of Chang and Cheng (1983). Finally we consider asymptotic solutions of Equations (26) through (28) to supplement the numerical solutions just obtained. Thus, we examine the case of a very hot wall for which A >> 1. An inspection of these equations show that when A is large, the new variables can be represented by F(~) = Al/3f,

G(~) = g

(30)

and

(31)

ll = A l / 3 ~ 9

Upon using these variables in (26) and (27), we then obtain (G2F') ' = --~G'

t

c~9

~G' d~,

(32)

do G" + } F C ' = 0,

(33)

HORIZONTAL BOUNDARY-LAYER NATURAL CONVECTION

169

0.5

0.4 <~ X

o

0.3

I

I

I

10

20

30

0.2 0

m

40

&

Fig. 5. Variation with A of mass of entrained gas F(oc,); full line represents computation of Equations (26) through (28); dashed line is from asymptotic large A results in Equations (35).

0.38

0.36

<1 X

0.34

o

0.32

0.30 0

i

i

10

20

L G'

30

& Fig. 6. Variation with A of surface slip velocity f'(O).

(0) 40

170

IOAN POP AND RAMA SUBBA REDDY GORLA

8

F(~) 7

2 <~

6

X

5

4

3 0

I

I

I

10

20

30

40

A Fig. 7.

Variation with A of heat transfer rate g'(O).

with the boundary conditions F(0) = 0,

G(0) = l,

F'(oo) = G(oc,) = 0,

(34)

where now the primes denote differentiation with respect to r Equations (32) through (34) are now integrated numerically and we deduce that f(r

= A-l/3F(oo) 4- o(A-I).

f ' ( O ) = A 2/3F'(0) + 0(A-4/3),

(35)

g'(0) = A-1i3G'(0) + 0(A-4/3). These asymptotic expansions are compared with results obtained from numerical solutions of Equations (26) through (28) for finite A in Figures 5-7. These figures indicate that the asymptotic expansions (35), shown by dashed lines in Figure 5-7, agree quite well with the numerical solution obtained from the boundary-layer Equations (26) through (28).

Acknowledgement The authors appreciate very much the constructive suggestions that were made by the two reviewers of the original manuscript.

HORIZONTAL BOUNDARY-LAYER NATURAL CONVECTION

171

References Bejan, A., 1987~ Convective heat transfer in porous media, in S. Kakac et al. (eds), Handbook of Single-Phase Convective Heat Transfer, Wiley, New York, pp. 16.1 16.34. Blythe, P. A. and Simpkins, P. G., 1981, Convection in a porous layer for a temperature dependent viscosity, Int. J. IIeat Mass Transfer 24, 497-506. Chang, I. D. and Cheng, P., 1983, Matched asymptotic expansions for free convection about an impermeable horizontal surface in a porous medium, Int. J. Heat Mass Transfer 26, 163-174. Cheng, P. and Chang, I. D., 1976, Buoyancy induced flows in a saturated porous medium adjacent to impermeable horizontal surfaces, bit. J. Heat Mass Transfer 19, 1267-1272. Cheng, P., 1978, Heat transfer in geothermal systems, Adv Heat Transfer 14, 1-105. Cheng, P., 1985, Heat transfer in porous media: External flows, in S. Kakac et al. (eds), Natural Convection: Fundamentals and Applications, Hemisphere, Washington, D. C. Clarke, J. F. and Riley, N., 1976, Natural convection induced in a gas by the presence of a hot porous horizontal surface, Q. J. Mech. Appl. Math. 28, 373 396. Herwig, H., 1985, Asymptotische Theorie zer Erfassung des Einflusses Variabler Stoffwerte auf Impulse und Warmeubertragung, Fortschr.-Ber, VDL Reihe 7" Stromungstechnik Hr. 93. Dusseldorf. May. S. T. and Merker, G. P., 1987, Free convection in a shallow cavity with variable properties - 2. Porous media, Int. J. Heat Mass Transfer 30, 1833-1837. Riley, D. S. and Rees, D. A. S., 1985, Non-Darcy natural convection from arbitrarily inclined heated surfaces in saturated porous media, Q. J. Mech. Appl. Math. 38, 277-295. Saatdjian, E., 1980, Natural convection in a porous layer saturated with a compressible ideal gas, Int. J. Heat Mass Transfer 23, 1681-1683. Stewartson, K., 1964, The Theory of Laminar Boundary Layers in Compressible Fhdds, Oxford University Press. Taunton, J. W. and Lightfoot, E. N., 1970, Free convection heat or mass transfer in porous media, Chem. Engng. Sci. 25, 1939-1945.