Transport in Porous Media 30: 113–124, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.
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Mixed Convection–Radiation Interaction in Power-Law Fluids Along a Nonisothermal Wedge Embedded in a Porous Medium M. A. MANSOUR1 and RAMA SUBBA REDDY GORLA2
1 Mathematics Department, Faculty of Science, Assiut University, Assiut, Egypt. 2 Mechanical Engineering Department, Cleveland State University, OH 44115, U.S.A.
(Received: 17 January 1997; in final form: 10 September 1997) Abstract. A boundary layer analysis has been presented for the interaction of mixed convection with thermal radiation in laminar boundary flow from a vertical wedge in a porous medium saturated with a power-law type non-Newtonian fluid. The fluid considered is a gray medium, and the Rosseland approximation is used to describe the radiative heat flux in the energy equation. The transformed conservation laws are solved numerically for the case of variable surface temperature conditions. Results for the details of the velocity and temperature fields as well as the Nusselt number have been presented. Key words: mixed convection, convection–radiation interaction, power law fluids.
Nomenclature d f g h k K ke σ R CT m n Nux Pex qr
particle diameter (m) dimensionless stream function acceleration due to gravity (m/s2 ) heat transfer coefficient (W/m2 K) thermal conductivity (W/mK) Permeability coefficient of the porous medium (mn+1 ) mean absorption coefficient (1/m) Stefan–Boltzman const radiation parameter temperature ratio wedge angle parameter viscosity index Nusselt number P´eclet number radiative flux (Wm−2 )
qw Ra T u, v x, y α β η θ ρ ε χ 9 µ
wall heat flux (Wm−2 ) Rayleigh number temperature velocity components in x and y direction (m/s) axial and normal coordinates (m) thermal diffusivity (m2 /s) cofficient of thermal expansion (1/K) dimensionless distance dimensionless temperature density (kg/m3 ) porosity combined convection nonsimilar parameter stream function consistency index for power-law fluid (pa sn )
Subscripts w ∞
surface conditions conditions far away from surface
1. Introduction The study of mixed convection boundary layer flow along surfaces embedded in fluid saturated porous media has received considerable interest recently. Interest in such
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M. A. MANSOUR AND RAMA SUBBA REDDY GORLA
studies was inspired by energy applications such as geothermal energy technology, petroleum recovery, filtration processes, sensible heat storage beds, packed bed reactors and underground disposal of chemical and nuclear waste. The Problem of free convection in non-Newtonian fluids over vertical plates in porous media was studied by Chen and Chen (1988) and Mehta and Rao (1994) based upon the similarity solution approach. Nakayama and Koyama (1987) presented similarity solutions for the natural convection in non-Newtonian power-law fluids over bodies of arbitrary geometry placed in a porous medium. On the other hand, heat transfer by simultaneous natural convection and thermal radiation in a participating fluid has not received as much attention. This is unfortunate because thermal radiation will play a significant role in the overall surface heat transfer in situations where convective heat transfer coefficients are small. Gorla (1988, 1993) has investigated the effects of radiation on mixed convection flow over vertical cylinders. In the present study, the focus is on the effect of mixed convection and radiation from a vertical non-isothermal wedge embedded in non-Newtonian fluid saturated porous media. The boundary condition of variable surface temperature is treated in this paper. The power-law model of Ostwald–de Waele which is adequate for many non-Newtonian fluids is considered here. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The transformed nonsimilar boundary-layer equations were solved using the finite difference method. 2. Mathematical Formulation Let us consider the combined convection flow in a porous medium saturated with a non-Newtonian fluid beside a heated vertical impermeable wedge of a small wedge angle. The coordinate system and flow model are shown in Figure 1. The x coordinate is measured along the plate and 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 the model of Darcy can be
Figure 1. Coordinate system and flow model.
MIXED CONVECTION–RADIATION INTERACTION IN POWER-LAW FLUIDS
115
used. The radiative heat flux in the x direction is considered negligible in comparison with that in the y direction. Thus, the equations governing the conservation of mass, momentum and energy can be written within the boundary-layer approximation (see Mehta and Rao (1994), Nakayama and Koyama (1987)) as ∂u ∂v + = 0, ∂x ∂v n−1 ∂u
nu
∂y
=
ρKgβ µ
(1)
∂T , ∂y
(2) !
∂T ∂ 2T 1 ∂qr ∂T +v =α − , u ∂x ∂y k ∂y ∂y 2
(3)
the appropriate boundary conditions are y = 0; v = 0; T = Tw = T∞ + ax λ , y = ∞; u = u∞ ; T = T∞ ,
(4)
where u, v are the Darcian velocity components in x and y direction, respectively; T the temperature; n the viscosity index; ρ the density; µ the consistency of power-law fluids; β the volumetric coefficient of thermal expansion; K the permeability of the porous medium; α the equivalent thermal diffusivity of the porous medium; k the effective thermal conductivity. The quantity qr on the right-hand side of Equation (3) represents the radiative heat flux in the y direction. The radiative heat flux term is simplified by using the Rosseland approximation as qr = −
4σ ∂T 4 , 3ke ∂y
(5)
where σ and ke are the Stefan–Boltzman constant and the mean absorption coefficient, respectively. Non-Newtonian fluids generally exhibit a nonlinear relation between shear stress and shear rate. These fluids may be classified as inelastic and viscoelastic. The inelastic fluids may be subdivided as time-dependent fluids 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: pseudoplastic, dilatant, Bingham plastic and pseudoplastic with yield stress. Inelastic time-independent non-Newtonian fluids have received 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 Ostwald–de Waele powerlaw model represents several inelastic time-independent non-Newtonian 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.
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M. A. MANSOUR AND RAMA SUBBA REDDY GORLA
Christopher and Middleman (1965) were the first to propose the form of Darcy law applicable to power-law fluids. In essence, the modified Darcy law as obtained by them can be written in vector notation: !
µ|v|n−1 v, ∇p = ρg − K
(6)
where µ is the consistency of the power-law fluid and K is the modified permeability. For the power law of Ostwald–de Waele, Christopher and Middleman (1965) and Dharmadhikari and Kale (1985) proposed the following relationships for the permeability:
K=
n n+1 6 nε εd 3(1 − ε) 25 3n + 1 " 2
ε
dε2 8(1 − ε)
#n+1
6n + 1 10n − 3
16 75
3(10n−3)/(10n+11)
.
In the above equation, d is the particle diameter and ε the porosity. The continuity equation is automatically satisfied by defining a stream function 9 such that u = ∂9/∂y
and
v = −∂9/∂x.
Proceeding with the analysis, we define the following transformation:
y −1 Pe1/2 x χ , x
η=
χ −1 = 1 +
Rax Pex
1/2
,
−1 9 = αPe1/2 x χ f (χ , η),
θ=
(7)
T − T∞ , Tw − T ∞
Pex =
u∞ x , α
Rax =
x ρKgβ∇Tw α µ
1/n
,
u∞ = cx m and m = γ /(2π − γ ), where γ is the half-wedge angle.
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MIXED CONVECTION–RADIATION INTERACTION IN POWER-LAW FLUIDS
Substituting the expressions (7) into Equations (2), (3) and (4), the transformed governing equations may be written as n(f 0 )n−1 f 00 = (1 − χ )2n θ 0 ,
(8)
λ − mn 4R[(θ + Cr )3 θ 0 ]0 1 θ + + (m + 1) + (1 − χ) f θ 0 − λf 0 θ 3 2 n λ − mn 0 ∂f 0 ∂θ = χ (1 − χ ) θ −f (9) 2n ∂χ ∂χ 00
with transformed boundary conditions
m + 1 (γ − mn)(1 − χ ) λ − mn ∂f + f (χ, 0) + χ(1 − χ) (χ , 0) = 0, 2 2n 2n ∂χ f 0 (χ, ∞) = χ 2 ,
θ(χ , 0) = 1,
θ (χ , ∞) = 0.
(10)
Primes in the above equations denote partial differentiation with respect to η. We note that χ = 0 and 1 correspond to pure free and forced convection cases, respectively. Between these limits, it represents combined free and forced convection cases. For practical applications, it is usually the velocity components, friction factor and Nusselt number are of interest. These are given by u = u∞ f 0 (χ, η) 1/2
m+1 λ − mn 1−χ αPex χ −1 f + f + v = − x 2 2n χ ∂f λ − mn m−1 λ − mn (1 − χ) 0 + +η (χ − 1) + f , 2n ∂χ 2 2n χ
Nux = −Pex χ −1 1 +
4R (θ + Cr )3 θ 0 (χ , 0). 3
(11)
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 η = ∞ are replaced by f 0 (χ, ηmax ) = χ 2 , θ(χ, ηmax ) = 0, where ηmax is a sufficiently large value of η where the boundary condition (10) for velocity is satisfied. We have set ηmax = 25 in the present work.
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M. A. MANSOUR AND RAMA SUBBA REDDY GORLA
(b) The two-dimensional domain of interest, (χ, η ) is discretized with an equispaced mesh in the χ direction and another equispaced mesh in the η direction. (c) The partial derivatives with respect to χ and η are all evaluated by the central difference approximations. The central difference approximation for the partial derivatives with respect to χ vanish when χ = 0 and χ = 1. (d) Two iteration loops based on the successive substitution are used because of the nonlinearity of the equations . (e) In each inner iteration loop, the value of χ is fixed while each of Equations (8) and (9) is solved as a linear second-order boundary-value problem of ODE on the η domain. The inner integration is continued until the nonlinear solution converges for the fixed value of χ . (f) In the outer iteration loop, the value of χ is advanced from 0 to 1. The derivatives with respect to χ are updated after every outer iteration step. More details of the numerical solution scheme are explained by Gorla (1993). The numerical results are affected by the number of mesh in both directions. To obtain accurate results, a mesh sensitivity study was performed. In the χ 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 χ direction, only 11 mesh points were found to give as accurate results as with 21 points. Therefore, the remainder of the computations were performed with 200 times 11 mesh points. 4. Results and Discussion Numerical solution for the governing equations is presented an Tables I–IV for χ ranging from 0 to 1. We have chosen λ, m, R 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. (1993) for the case of a vertical plate in a Newtonian fluid (m = 0, R = 0, n = 1). It may be noted that the agreement between our results and the literature values is within 0.1%. Figures 2 and 3 display results for the velocity and temperature profiles. We have treated the viscosity index n, the wedge angle parameter m, combined convection parameter χ , the radiation parameter R and the temperature power-law exponent λ as parameters. We note χ = 0 and 1 represent pure natural convection and forced convection, respectively. As R increases, we note that the boundary-layer thickness increases for all cases. We note that λ = 0 corresponds to uniform surface temperature boundary condition. Figures 4 and 5 display the variation of the temperature gradient. For dilatant fluids (n > 1 ), as χ varies from 0 to 1, the temperature gradient decreases initially, reaches a minumum at about χ = 0.5 and then increases as χ approaches 1. For pseudoplastic fluids (n< 1), the temperature gradient varies approximately linearly with χ. As the radiation parameter increases, the temperature gradient decreases.
MIXED CONVECTION–RADIATION INTERACTION IN POWER-LAW FLUIDS
Table I. Comparison of values −θ 0 (χ , 0) for n = 0, n = 1 and R = 0. −θ 0 (χ, 0) Present results
Hsieh et al.(1993)
χ
λ = 0.5
λ = 1.0
λ = 0.5
λ = 1.0
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
0.88512 0.80058 0.72530 0.66243 0.61569 0.58877 0.58423 0.60260 0.64209 0.69926 0.77026
1.12697 1.01951 0.92422 0.84523 0.78735 0.75527 0.75215 0.77839 0.83160 0.90726 1.00007
0.8862 0.8014 0.7259 0.6629 0.6160 0.5890 0.5844 0.6026 0.6419 0.6991 0.7704
1.1284 1.0206 0.9250 0.8457 0.7877 0.7555 0.7522 0.7783 0.8314 0.9071 1.0000
Table II. Values of −θ 0 (χ, 0) for λ = 0.5. n
R
χ
m=0
m = 1/3
m = 0.5
m=1
0.5
0
0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0
0.71051 0.78312 0.88623 0.48048 0.51616 0.57218 0.77026 0.58877 0.88512 0.51197 0.38400 0.57218 0.80054 0.53549 0.88623 0.52701 0.34841 0.57218
0.71072 0.81165 0.94198 0.48039 0.53496 0.60784 0.77153 0.61025 0.94198 0.51197 0.39819 0.60784 0.78266 0.55588 0.94198 0.48895 0.36140 0.60784
0.71079 0.82563 0.96889 0.48016 0.54412 0.62507 0.77142 0.62118 0.96889 0.51191 0.40519 0.62507 0.80036 0.56601 0.96889 0.52625 0.36783 0.62507
0.72852 0.86644 1.04605 0.48021 0.57066 0.67453 0.77142 0.65343 1.04605 0.51191 0.42575 0.67453 0.80036 0.59601 1.04605 0.52701 0.38682 0.67453
0.5
1.0
0
0.5
1.5
0
0.5
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M. A. MANSOUR AND RAMA SUBBA REDDY GORLA
Table III. Values of −θ 0 (χ, 0) for λ = 1.0. n
R
χ
m=0
m = 1/3
m = 0.5
m=1
0.5
0
0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0
0.93184 1.01590 1.12838 0.63000 0.66974 0.72952 1.00007 0.75527 1.12697 0.66449 0.49438 0.72952 1.03365 0.68601 1.12838 0.68116 0.44674 0.72952
0.93093 1.03588 1.17083 0.62999 0.68309 0.75654 1.00048 0.77212 1.17083 0.74330 0.50443 0.75654 1.00855 0.70051 1.17083 0.67494 0.45606 0.75654
0.93096 1.04596 1.19177 0.62998 0.68978 0.76989 1.00048 0.78001 1.19177 0.66449 0.50950 0.76989 1.01182 0.70784 1.19177 0.67180 0.46076 0.76989
0.93176 1.07631 1.25331 0.63010 0.70975 0.80918 1.00048 0.80396 1.25331 0.74330 0.52484 0.80918 1.03354 0.77032 1.25331 0.68107 0.47498 0.80918
0.5
1.0
0
0.5
1.5
0
0.5
Table IV. Values of −θ 0 (χ, 0) for λ = 2.0. n
R
χ
m=0
m = 1/3
m = 0.5
m=1
0.5
0
0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0
1.26229 1.37261 1.50451 0.85593 0.90506 0.97384 1.34859 1.01679 1.504514 0.89625 0.66488 0.97384 1.38968 0.91955 1.50451 0.91657 0.59942 0.97384
1.26266 1.38573 1.53499 0.88349 0.91391 0.99312 1.34859 1.02697 1.53499 0.89625 0.67149 0.99312 1.38968 0.92913 1.53499 0.91657 0.60555 0.99312
1.26336 1.39240 1.55023 0.85480 0.91839 1.00278 1.34859 1.03219 1.55023 0.89625 0.67487 1.00278 1.38968 0.93405 1.55023 0.91657 0.60869 1.00278
1.26230 1.41286 1.59577 0.85468 0.93206 1.03169 1.34859 1.04830 1.59577 0.89625 0.68524 1.03169 1.38968 0.94925 1.59577 0.91657 0.61835 1.03169
0.5
1.0
0
0.5
1.5
0
0.5
MIXED CONVECTION–RADIATION INTERACTION IN POWER-LAW FLUIDS
Figure 2. Velocity profiles (λ = 0.5).
Figure 3. Temperature profiles (λ = 0.5).
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M. A. MANSOUR AND RAMA SUBBA REDDY GORLA
Figure 4. Nusselt number results (λ = 1.0).
MIXED CONVECTION–RADIATION INTERACTION IN POWER-LAW FLUIDS
Figure 5. Nusselt number results (λ = 2.0).
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References Chen, H. T. and Chen, C. K.: 1988, Free convection of non-Newtonian fluids along a vertical plate embedded in a porous medium, Trans. ASME, J. Heat Transfer 110, 257–260. Mehta, K. N. and Rao, K. N.: 1994, Buoyancy-induced flow of non-Newtonian fluids in a porous medium past a vertical plate with nonuniform surface heat flux, Int. J. Eng. Sci. 32, 297–302. 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. 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. Gorla, R. S. R., Lee, J. K., Nakumara, S. and Pop, I.: 1993, Effects of transverse magnetic field on mixed convection in a wall plume of power-law fluids, Int. J. Eng. Sci. 31, 1035–1045. Sparrow, E. M. and Cess, R. D.: 1978, Radiation Heat Transfer, Hemisphere Publishing Corp., Washington, DC. Christopher, R. H. and Middleman, S.: 1965, Power-law flow through a packed tube, I & EC Fundamentals 4, 422–426. Dharmadhikari, R. V. and Kale, D. D.: 1985, Flow of non-Newtonian fluids through porous media, Chemical Eng. Sci. 40, 527–529. Hsieh, J. C., Chen, T. S. and Armaly, B. F.: 1993, Nonsimilar solutions for mixed convection from vertical surfaces in porous media, Int. J. Heat Mass Transfer 36, 1485–1493.
