Transport in Porous Media 24: 157-166, 1996. 9 1996 KluwerAcademic Publishers, Printed in the Netherlands.
157
Combined Convection in Power-Law Fluids Along a Nonisothermal Vertical Plate in a Porous Medium MAHESH KUMARI 1 and RAMA SUBBA REDDY GORLA 2
1Department of Mathematics, Indian Institute of Science, Bangalore, 560 012, India 2Mechanical Engineering Department, Cleveland State University, Cleveland, OH, 44115, U.S.A. (Received 12 June 1995; in final form: 8 February 1996) Abstract. The problem of combined convection from vertical surfaces in a porous medium saturated with a power-law type non-Newtonian fluid is investigated. The transformed conservation laws are solved numerically for the case of variable surface heat flux conditions. Results for the details of the velocity and temperature fields as well as the Nusselt number have been presented. The viscosity index ranged from 0.5 to 2.0. Key words: non-Newtonianfluids, combined convection, heat transfer
Nomenclatures d f 9 h k [( m n Nu Pe q~ Ra* T u, v z, y c~ /3 7/ O p e X r
particle diameter (m) dimensionless stream function acceleration due to gravity (m/s2) heat transfer coefficient (W/m2K) thermal conductivity (W/mK) permeability coefficient of the porous medium (m l+n) consistency index for power-law fluid (pa sn) viscosity index Nusselt number P6clet number wall heat flux (W/m2) modified Rayleigh number temperature (K) velocity components in z and y directions (m/s) axial and normal coordinates (m) thermal diffusivity (toO/s) coefficient of thermal expansion (l/K) dimensionless distance dimensionless temperature density (kg/m3) porosity combined convection nonsimilar parameter stream function
Subscripts w c~
surface conditions conditions far away from the surface
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MAHESH KUMARI AND RAMA SUBBA REDDY GORLA
1. Introduction
The study of combined convection boundary-layer flow along a vertical surface embedded in fluid-saturated porous media has received considerable interest recently. The interest in such studies was motivated by numerous thermal engineering applications in several areas such as geothermal engineering, thermal insulation systems, petroleum recovery, filtration processes, packed bed reactors, sensible heat storage beds, ceramic processing, and ground water pollution. Cheng and Minkowycz [1] presented similarity solutions for free convective heat transfer from a vertical plate in a fluid-saturated porous media. Gorla and coworkers [2,3] developed a procedure for the corresponding nonsimilar boundary layer problem with an arbitrarily varying surface temperature or heat flux. The problem of combined convection from surfaces embedded in porous media was studied by Minkowycz et al. [4] as well as Ranganathan and Viskanta [5]. A general transformation for similarity solutions in combined convection in a porous medium was obtained by Nakayama and Koyama [6]. Nakayama and Pop [7] proposed similarity transformations for the free, forced and combined convection. Hsieh [8] et al. presented nonsimilar solutions for combined convection in porous media. All these studies were concerned with Newtonian fluid flows. A number of industrially important fluids including fossil fuels which may saturate underground beds display non-Newtonian fluid behavior. Non-Newtonian fluids exhibit a nonlinear relationship between shear stress and shear rate. Chert and Chen [9] and Mehta and Rao [10] presented similarity solutions for free convection of non-Newtonian fluids over vertical surfaces in porous media. Nakayama and Koyama [11] studied the natural convection over a nonisothermal body of arbitrary geometry placed in a porous medium. A similarity solution was derived by Mehta and Rao [12] for the natural convective boundary layer flow of a non-Newtonian fluid over a nonisothermal horizontal plate immersed in a porous medium. The problem of combined convection from vertical surfaces in porous media saturated with non-Newtonian fluids has not been investigated. The present work has been undertaken in order to analyze the problem of combined convection from a vertical nonisothermal flat plate embedded in nonNewtonian fluid-saturated porous media. The boundary condition of variable surface heat flux is treated in this paper. The power law model of Ostwald-de-Waele which is adequate for many non-Newtonian fluids will be considered here. The transformed boundary layer equations are solved using a finite difference method. The numerical results for the velocity and temperature fields are obtained.
2. Analysis Let us consider the combined convection flow in a porous medium saturated with a non-Newtonian fluid beside a heated vertical impermeable flat plate. The coordinate system and flow model are shown in Figure 1. The z-coordinate is measured along
159
COMBINED CONVECTION IN POWER-LAW FLUIDS
X~ U
g q w(x)
, y,v,T..
o
Figure I. Coordinatesystem and flow model.
the plate and the y-coordinate normal to it. The gravitational acceleration g is in a direction opposite to x-direction. The flow velocity and the pores of the porous medium are assumed to be small so that Darcy's model can be used. The governing equations under Boussinesq and boundary-layer approximations may be written as (see [10, 12]):
Ou Ox +
Ov = o,
(1)
nun_iOn ( ~ ) O---y= OT
OT
+ v-b-- v =
OT Oy ' 02T 2
(2)
(3)
In the above equations, u and v are the Darcian velocity components in x and y directions, respectively; T the temperature; n the viscosity index; p the density; m the consistency of power-law fluids; fl the volumetric coefficient of thermal expansion; K the permeability of the porous medium and ~ the equivalent thermal diffusivity of the porous medium. Non-Newtonian fluids generally exhibit a nonlinear relationship between shear stress and shear rate. These fluids may be classified as inelastic and viscoelastic. The inelastic fluids may be subdivided as time-dependent and time-independent fluids. The time-dependent fluids, in turn, are subdivided into two groups: thixotropic and rheopectic. The time-independent fluids can be subdivided into four groups: (1) pseudoplastic, (2) dilatant, (3) Bingham plastic and (4) pseudoplastic with yield stress.
MAHESHKUMARIAND RAMASUBBAREDDYGORLA
160
Inelastic time-independent non-Newtonian fluids have recieved the greatest attention from rheologists which has resulted in the development of a number of equations or models proposed to represent their flow behavior. The Ostwaldde Waele power law model represents several inelastic time-independent nonNewtonian fluids of practical interest and therefore has been used in this paper. When n < 1, the model describes pseudoplastic behavior whereas n > 1 represents dilatant behavior. Christopher and Middleman [13] were the first to propose the form of Darcy law applicable to power-law fluids. In essence, the modified Darcy law as obtained by [13] can be written in vector notation:
( Ttzlv]n-1 ~ Vp=pg-\
K
(4)
iv,
where ra is the consistency of the power-law fluid and K is the modified permeability. For the power law model of Ostwald-de-Waele, Christopher and Middleman [13] and Dharmadhikari and Kale [14] proposed, respectively, the following relationships for the permeability:
nE
] n+l
t(-- 72LS( ~)]n+l(10n6n-+----13] ~ \75](16~ 3(lOn-3)/(lOnTll) In the above equation, d is the particle diameter and e the porosity. The boundary conditions are given by u = O : v = O, qw = a x A , y ~ ec : u = u~,T = T~,
(5)
where a and A are constants. We note that A = 0 corresponds to uniform heat flux wall conditions. In practical applications, the constant temperature or constant heat flux boundary conditions represent limiting extreme conditions. Therefore, the power law variatoin with distance for the wall heat flux is chosen to be a more realistic boundary condition. The continuity equation is automatically satisfied by defining a stream function r y) such that
u =
and v -
O~b Ox"
(6)
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COMBINEDCONVECTIONIN POWER-LAWFLUIDS
Proceeding with the analysis, we define the following transformations: ~] =
rex X
,
X -1 = 1 + [Ra;/Pe(~2n+l)/2]U(2n+l), 1/2 -1
~b = a P e .
X f(x, rl), (7)
0 = (Y - Too)Pe~1/2 X- 1
(w-) qwx
Pex Ra*-
?-too g
oz
o~n+1 [
mk
J"
Substituting the expressions in (7) into Equations (2), (3) and (5), the transformed governing equations may be written as
n(f')n-lf"
= (1 - x)2n+10 ',
(8)
0" + ~ 1 1 1 + ( ~ - ~ +[ ] ) f 0 '1-
2a+
= 2(2n + 1) X(1
+( ~ )
- X) [o'Of [ OX
2nlf ' (Ol - x+) ]l
_f, o0]]
(9)
~xJ]'
1 + 2n+2)~ + (1 - X) f(x,O) + -----~X(X2n+ - 1)
(X,0) = 0,
(10)
e'(x,O) = - 1 , f ' ( x , oo) = )C2, e(x , oo) = 0. Primes in the above equations denote partial differentiation with respect to z]. We note that X = 0 and 1 correspond to pure free and forced convection cases, respectively. Between these limits, it represents combines free and forced convection case. For practical applications, it is usually the velocity components, friction factor and Nusselt number that are of interest. These are given by
u= u~x - 2 f i (x, rl)
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MAI-IESH KUMARI AND RAMA SUBBA REDDY GORLA
V
--
a r~''/2 e~ [X2 f + (1_~) x
2A+l 2(2n + 1) I + ( X -
., 2A + 1 Of l)2r2n[ + I ) 0 x -
X-1 2 .N u x
pex1 / X2-1
(11)
r
3. Numerical Scheme
The numerical scheme to solve Equations (8) and (9) adopted here is based on a combination of the following concepts: (a) The boundary conditions for r / = ~ are replaced by f ' ( x , ?']max) : )s
O(X~ r/max) : 0
where r/maxis a sufficiently large value of r/where the boundary condition (12) for velocity is satisfied. ?]maxvaries with the value of n. We have set r/max 25 in the present work. The two-dimensional domain of interest, (X, r/) is discretized with an equispaced mesh in the X direction and another equispaced mesh in the r/direction. The partial derivatives with respect to X and r/are all evaluated by the central difference approximations. The central difference approximation for the partial derivatives with respect to X vanish when X = 0 and X = 1 which correspond to the first and the last mesh points on the X axis, or free and forced convection, respectively. Two iteration loops based on the successive substitution are used because of the nonlinearity of the equations. In each inner iteration loop, the value of X is fixed while each of Equations (8) and (9) is solved as a linear second-order boundary-value problem of ODE on the r/domain. The inner iteration is continued until the nonlinear solution converges for the fixed value of X. In the outer iteration loop, the value of y is advanced from 0 to 1. The derivatives with respect to X are updated after every outer iteration step. =
(b) (c)
(d) (e)
(f)
More details of the numerical solution scheme are explained in reference [15]. The numerical results are affected by the number of mesh points in both directions. To obtain accurate results, a mesh sensitivity study was performed. In the r/direction, after the results for the mesh points of 51, 100, 200, and 800 were compared it was found that 200 points give the same results as 800. In the X direction, only 11 mesh points were found to give as accurate results as with 21 points.
163
C O M B I N E D CONVECTION IN POWER-LAW FLUIDS 2.0
1.5 n=0.5
-- ~=0.0 --X=I.0 -- Z=2.0
1,5 . ~
~\
~, I.o
n = 0.5
1.0
,~Z
0.5
"~\ \ ~ L o
X= 1.0
'~\~ ~"
0.5
0.5
0.0
0
1
0.0
2
0
~
g=O.O
-"
Z=I.O
=0-0
1
2
3
rl
Figure 2. Velocity
a n d t e m p e r a t u r e profiles ( n = 0.5).
1.5 n=l.O
1.0
---X = 1.0
~k
2.0 X=O.O
n=l.O
---
X=l.0 ~=2.0 "
Z=0.0 ~,= 1.0
Z=0.0
t=
~, 1.o
,.~
0.5
0.5 .
X
0.5 "" 0.0
0
1
0.0 2
3
4
0.0 0
I
2
3
4
Figure 3. Velocityand temperatureprofiles (n = 1).
Therefore, the remainder of the computations were performed with 200 times 11 mesh points. 4. R e s u l t s a n d D i s c u s s i o n
Numerical solution for the governing equations is presented for X ranging from 0-1. We have chosen A and n as prescribable parameters. To assess the accuracy of the present results, we have shown a comparison of our results with those of Hsieh et al. [8] for the case of Newtonian fluid, namely, n = 1. It may be noted that the agreement between our results and the literature values is within 0.5%. For the Newtonian fluid (n = 1), the literature value for Nux Pe~ U2 in the case of pure forced convection is 0.8862 whereas our results indicate this value is 0.8863. For pure natural convection case the value of Nu~: Ra~-1/3 from the literature and present data are 0.7715 and 0.77647, respectively. We therefore conclude that our results are very accurate.
164
MAHESH KUMARI AND RAMA SUBBA REDDY GORLA 1.5
n=1.5 X~
2.0~
- - 7,=0.0 - - Z=I.O -- ~,= 2.0
Z= 1,0
n=l.5
~
g=O,O
--
g=l.O
1.0
"-,\ 0_5
0.0
1
0
2
3
4
0
rl
0.9
'~
X=O.O
0.8
' N
+
0.7
~
1.5 0.2
0..
0.4
0.6
0.8
1.0
7(
Figure 5. Local Nusselt number versus X(A = 0.0). 1.2
~=0.5
~ 1:0
+ o.8
15
0"-0.0
2
)i
Figure 4. Velocity and temperature profiles (n = 1.5).
~,
1
t
t
|
!
0.2
0.4
0.6
0.8
1.0
Z
Figure 6. Local Nusselt number versus x(A = 0.5).
3
COMBINED CONVECTION IN POWER-LAW FLUIDS
,_~
1.4
165
x= LO
' ..aj % +
1.0
~.
0.8
1-~~5[[~ 111~
d "~
z
2.0 fi 0"-0.0
I
I
!
!
0.2
0.4
0.6
0.8
1.0
g
Figure 7. Local Nusseltnumberversus x(A = 1).
1.6
h=2.0
t'q
"-~
1.4
~XS
~
+
1.2
~'N 1.0
1.0
z 0.8
0.0
I
I
I
i
0.2
0A
0.6
0.8
1.0
X
Figure 8. Local Nusselt number versus X(A = 2.0).
Figures 2 through 4 display results for the velocity and temperature profiles. We have treated the viscosity index n, combined convection parameter X and the temperature power law exponent A as parameters. We note that X = 0 and 1 represent pure natural convection and forced convection, respectively. As X increases, we note that the momentum boundary layer thickness increases. As A increases, the momentum and thermal boundary layer thicknesses decrease. We note that A = 0 corresponds to uniform surface heat flux boundary condition. The streamwise velocity at the porous wall decreases with A and n. The surface temperature of the porous wall decreases with A but increases with n. The surface heat transfer rate increases with A and decreases with n. Figures 5 through 8 display the variation of the local heat transfer rate parameter {Nux/[Pelz/2 + Rax*U(Zn+l)]} with X. As X varies from 0 to 1, the heat transfer rate decreases initially, reaches a minimum at about X = 0.5 and then increases as X approaches 1.
166
MAHESH KUMARI AND RAMA SUBBA REDDY GORLA
5. Concluding Remarks I n t h i s p a p e r , w e h a v e p r e s e n t e d an a n a l y s i s f o r t h e p r o b l e m o f c o m b i n e d c o n v e c t i o n f r o m a h o r i z o n t a l s u r f a c e w i t h v a r i a b l e w a l l h e a t flux a n d e m b e d d e d in a p o r o u s m e d i u m s a t u r a t e d w i t h O s t w a l d - d e - W a e l e t y p e n o n - N e w t o n i a n fluid. T h e n o n s i m i l a r p a r a m e t e r X is i n t r o d u c e d a n d as X v a r i e s f r o m 0 to 1, t h e e n t i r e r e g i m e o f t h e m i x e d c o n v e c t i o n c a s e is d e s c r i b e d . T h e n o n s i m i l a r b o u n d a r y l a y e r e q u a t i o n s a r e s o l v e d n u m e r i c a l l y b y m e a n s o f a finite d i f f e r e n c e s c h e m e . V e l o c i t y a n d t e m p e r a t u r e p r o f i l e s are p r e s e n t e d f o r t h e e n t i r e m i x e d c o n v e c t i o n r e g i m e ( 0 ~< X ~< 1). The local Nusselt numbers for the entire mixed convection regime for a range of v a l u e s o f n a n d A are p r e s e n t e d .
Acknowledgement T h e a u t h o r s a r e g r a t e f u l to t h e r e v i e w e r s f o r t h e i r h e l p f u l c o m m e n t s .
References 1. Cheng, P. and Minkowycz, W. J.: 1977, Free convection about a vertical plate embedded in a porous medium with application to heat tranfer from a dike, J. Geophys. Res. 82, 2040-2044. 2. Gorla, R. S. R. and Zinalabedini, A. H.: 1987, Free convection from a vertical plate with nonuniform surface temperature and embedded in a porous medium, Trans. ASME, J. Energy Resourc. Technol. 109, 26-30. 3. Gorla, R. S. R. and Tornabene, R.: 1988, Free convection from a vertical plate with nonuniform surface heat flux and embedded in a porous medium, Transport in Porous Media 3, 95-106. 4. Minkowycz, W. J., Cheng, E and Chang, C. H.: 1985, Mixed convection about a nonisothermal cylinder and sphere in a porous medium, Numerical Heat Transfer 8, 349-359. 5. Ranganathan, E and Viskanta, R.: 1984, mixed convection boundary layer flow along a vertical surface in a porous medium, Numerical Heat Transfer 7, 305-317. 6. Nakayama, A. and Koyama, H.: 1987, A general similarity transformation for combined free and forced convection flows within a fluid-saturated porous medium, J. Heat Transfer 109, 1041-1045. 7. Nakayama, A. and Pop, I.: 1985, A unified similarity transformation for free, forced and mixed convection in Darcy and non-Darcy porous media, Int. J. Heat Mass Transfer 28, 683-697. 8. Hsieh, J. C., Chen, T. S. and Armaly, B. E: 1993, Nonsimilarity solutions for mixed convection from vertical surfaces in porous media, Int. J. HeatMass Transfer36, 1485-1493. 9. Chen, H.T. andChen, C. K.: 1987,Naturalconvectionofnon-Newtonianfluids about a horizontal surface in a porous medium, Trans. ASME, J. Energy Resour. TechnoL 109, 119-123. 10. Mehta, K. N. and Rao, K. N.: 1994, Buoyancy-inducedflow ofnon-Newtonian fluids in a porous medium past a vertical plate with nonuniform surface heat flux, Int. J. Eng. Sci. 32, 297-302. 11. Nakayama, A. and Koyama, H.: 1991, Buoyancy-induced flow of non-Newtonian fluids over a non-isothermal body of arbitrary shape in a fluid-saturated porous medium, AppL Sci. Res. 48, 55-70. 12. Mehta, K. N. and Rao, K. N.: 1994, Buoyancy-induced flow of non-Newtonian fluids over a non-isothermal horizontal plate embedded in a porous medium, Int. J. Eng. Sci. 32, 521-525. 13. Christopher, R. H. and Middleman, S.: 1965: Power-law flow through a packed tube, I & EC Fundamentals 4, 422-426. 14. Dharmadhikari, R. V. and Kale, D. D.: 1985, Flow of non-Newtonian fluids through porous media, Chem. Eng. Sci. 40, 527-529. 15. Gorla, R. S. R., Lee, J. K., Nakamura, S. and Pop, I.: 1933, Effects on transverse magnetic field on mixed convection in wall plume of power law fluids, Int. J. Eng. Sci. 31, 1035-1045.
