Radiation Effects on Mixed Convection Flow and Viscous Heating in a Vertical Channel Partially Filled with a Porous Medium

Transcription

1 Tamkang Journal of Science and Engineering, Vol. 14, No. 2, pp (2011) 97 Radiation Effects on Mixed Convection Flow and Viscous Heating in a Vertical Channel Partially Filled with a Porous Medium Dileep Singh Chauhan* and Vikas Kumar Department of Mathematics, University of Rajasthan, Jaipur , India Abstract A theoretical analysis is made for a fully developed mixed convection viscous fluid flow between two infinite vertical parallel plane walls, where a porous substrate of finite thickness is attached to the left vertical wall, in the presence of radiation and viscous dissipation effects. It is assumed that the viscous fluid is gray, absorbing-emitting radiation but a non-scattering medium. The Boussinesq approximation and Rosseland approximation are employed. The analytic expressions for temperature and velocity profiles are obtained and the effects of the permeability of the porous substrate, Grashof number, conduction-radiation parameter (stark number), and perturbation parameter on the flow and temperature fields and in the Nusselt number have been discussed. Key Words: Radiation, Permeability, Mixed Convection, Vertical Channel, Viscous Dissipation 1. Introduction Buoyancy is of considerable importance in technological applications, such as in heated rooms or reactor configurations, where temperature differences give rise to complicated flow patterns. In fact, it is known that heat exchangers technology often involves convective flows in vertical channels, where these flows in most cases imply thermal conditions of uniform heating of channel walls either by isothermal or iso-heat flux boundary conditions. The most of the interest in such study is therefore, due to its applications in heat exchangers technology, for example, in the design of cooling and solar energy collection systems, etc. Several papers have been published that deal with the study of the velocity and temperature profiles for the fully developed vertical parallel-flow regime. A theoretical study of fully developed, mixed convection in vertical channel was conducted by Aung and Worku [1] and Cheng et al. [2] including flow reversal. *Corresponding author. dileepschauhan@gmail.com Zanchini [3] and Barletta [4 6] investigated the effects of viscous dissipation on mixed convection flow in vertical channels by taking thermal boundary conditions as prescribed uniform heat fluxes on both walls and the case of prescribed uniform temperatures on both walls, or isothermal-isoflux boundary conditions. In these studies the effect of buoyancy is accounted for by writing the temperature and the velocity as a power series in a dimensionless parameter which is proportional to the Brinkman number or in a mixed convection parameter which is the ratio of the Grashof number and the Reynolds number. Boulama and Galanis [7] discussed an analytical solution for mixed convection in vertical channel with heat and mass transfer. Barletta et al. [8] discussed dual mixed convection flow problem in vertical parallel-plate channel. In many environmental and scientific processes radiative convective flows are frequently encountered, for example, in aeronautics, fire research, heating and cooling of channels, etc. It is observed that radiative transport is often comparable and hence associated with that of convective heat transfer in several practical applications. Therefore it is of great significance to the researchers to

2 98 Dileep Singh Chauhan and Vikas Kumar study combined radiative and convective flow and heat transfer aspects. Many authors have investigated such problems in non-porous and porous medium, e.g. Siegel and Howell [9], Chamkha [10], Raptis [11,12], Bakier [13], Raptis and Perdikis [14], Bég et al. [15], Ghosh and Bég [16]. The study of heat transfer and flow of viscous fluids through and across a porous medium has wide ranging applications in various fields of science and engineering. As a result of its technological import to geothermal and reservoir engineering, and cooling of nuclear reactors, etc. several researchers have studied such problems in channels, composed of porous materials. An excellent review of the literature on this subject is given in the monograph by Nield and Bejan [17]. Lai et al. [18] investigated two-dimensional mixed convection in a vertical porous layer with a finite isothermal heat source on one vertical wall, which is otherwise adiabatic and the other wall is isothermally cooled. Ingham et al. [19] studied effects of viscous dissipation on mixed convection in a porous medium between two vertical plates. Al- Hadhrami et al. [20] investigated the mixed convection in a fully developed fluid flow by taking viscous dissipation effects into consideration, in a vertical channel filled with a porous material. Forced convection is studied in a channel filled by a porous material with viscous dissipation and flow work by Nield et al. [21]. The fully developed mixed convection flow with viscous dissipation is investigated by Barletta et al. [22] and Umavathi et al. [23,24] in a vertical parallel-plate channel filled with a porous medium subject to isoflux-isothermal, and isothermal-isothermal boundary conditions at the walls. Chauhan and Kumar [25] studied forced convection and entropy generation in a circular channel filled by a highly porous medium with velocity and temperature slip conditions and uniform heat flux at the wall. Enhancement of heat transfer in channels by attaching a porous substrate to the inner wall has been the subject of many investigations. Convection effects are investigated by Chauhan and Soni [26] in an inclined channel partially filled with two permeable layers attached to the inner walls. Chang and Chang [27] studied mixed convection in a vertical channel partially filled with highly porous medium. Chauhan and Gupta [28] investigated heat transfer in couette compressible fluid flow through a channel with highly permeable layer at the bottom. Alkam et al. [29] numerically simulated the forced convection in a parallel-plate channel partially filled with two porous substrates deposited at the inner walls. Transient free convection viscous fluid flow in domains partially filled with porous substrates is studied by Al- Nimr and Khadrawi [30]. This paper addresses the situation in a vertical composite channel in which a constant pressure gradient drives a fully developed flow modified by the buoyancy force caused due to temperature differences of channel walls with viscous dissipation and radiation effects. This, in turn generates a non-linear and coupled system of differential equations. Thus the evaluation of the temperature and velocity profiles is performed using a perturbation series method. A dimensionless parameter proportional to the Brinkman number is taken as perturbation parameter. The results on flow and heat transfer for a range of values of the pertinent parameters have been reported and discussed. 2. Formulation of the Problem A steady, laminar, viscous fluid flow in an infinite vertical parallel-plate channel of width d, is considered, where a porous substrate of thickness h isper- fectly attached to the left channel wall. The x-axis is taken parallel to the gravitational field but in the opposite direction and the y-axis is normal to the plates. A schematic diagram of the system is shown in Figure 1. The impermeable channel walls at y = 0 and y = d, kept at constant but different temperatures T 1 and T 2 respectively, with T 2 > T 1. Following Barletta and Zanchini [31], we assume the characteristic temperature T 0 as T 0 = (T 1 + T 2 ) / 2. The flow in the channel is assumed to be fully developed, thus the only non-vanishing component of the velocity is in the x-direction and both velocity and temperature of the fluid depend only on y. In fact, the vertical channel is infinite in length. The steady-state mass and momentum conservation equations are simplified greatly for the flow in the fully developed region. It is that section of the channel flow that is situated far enough from the entrance such that the scale of transverse velocity is negligible. Therefore in the fully developed flow limit, we have v = 0 and from the mass continuity equation u 0. In fact, in the en- x

3 Radiation Effects on Mixed Convection Flow and Viscous Heating in a Vertical Channel Partially Filled with a Porous Medium 99 Further the fluid considered here is optically thick which is gray, emitting-absorbing radiation but not scattering and that the channel walls are gray and diffuse. We also limit to the case in which radiation along the x-direction is negligible as compared to its transverse value, which is a good assumption for channel locations that are of small plate spacing d removed from the inlet. Thus the radiation heat flux is considered negligible in this paper in the x-direction, in comparison with that in the y-direction (Sparrow and Cess [32]). The fluid and the porous medium are assumed to be in local thermal equilibrium. The Rosseland radiation flux model for an optically dense viscous fluid that flows through the channel, is employed to simulate radiative heat transfer, which following Siegel and Howell [9] takes the form, as follows: Figure 1. Schematic diagram. (2) trance region, the scale of y is very small and not fixed, therefore neither v nor u is negligible. But in a fully x developed region the scale of y is d (width of the channel and fixed). Thus y-momentum equation reduces to simply which indicates that p is a function of x only and dp dx = constant (let dp dx = A). Here P = p + 0 gx is the hydrodynamic pressure, g, the acceleration due to gravity, 0, the fluid density at the reference temperature T 0. For fully developed temperature profiles, we assume that the flow is hydrodynamically fully developed and the effect of thermal diffusion has had enough time to reach the centerline of the channel stream. Thus it is the profile in the region situated far from the entrance region such that both u and t are developed. Further, Boussinesq approximation is assumed and the fluid density is taken to depend on the temperature t by where, is the thermal expansion coefficient. (1) where, q r is the radiative heat flux; and, k are, respectively, the Stefan-Boltzmann constant and mean absorption coefficient for thermal radiation. Following Raptis [11], the temperature function t 4 in (2) can be written as a linear function by expanding it in a Taylor series about T 0, and neglecting higher-order terms, as follows: (3) The channel is divided into two regions. Region-I is porous medium region (0 y h), where momentum equation is taken as the Brinkman equation; and Region-II is clear fluid region (h y d), where Navier-Stokes equations hold. Let u, t and u, t, denote dimensional velocity components in x-direction and temperature in these regions, respectively. The equations, governing the radiative fluid flow and temperature distribution in the channel for this problem, are given by Porous region - I (4)

4 100 Dileep Singh Chauhan and Vikas Kumar (5) (11) Clear fluid region - II (6) Using equations (2), (3) and above non-dimensional quantities, equations (4-7) are reduced to the following: (12) (7) where, m, m, n = (m / r 0 ), k, k, C p and K0 denote viscosity of the clear fluid, effective viscosity of the fluid in porous medium, kinematic viscosity, thermal conductivity, effective thermal conductivity in the porous medium, specific heat at constant pressure and permeability of the porous medium, respectively. The no-slip boundary conditions on the velocity components are taken on the channel impermeable boundaries, y = 0 and y = d: (8) (13) (14) (15) where, U 0 = gb(t2 - T1 ) d 3 m Ad 2 d k, f 1 =, f 2 =, Gr =, s= k 48m m K0, Re = r 0nC p U 0d mu 02,, Pr =, Br = k n k (T2 - T1 ) Constant temperatures are prescribed on these boundaries: n2 kk and asterisks are dropped for convenience. N = 4s T03 t (0) = T1 and t ( d ) = T2. Eliminating t from (12) and (13), we obtain Using equations (4), (6), (8) and above boundary conditions on temperature, we obtain (16) Similarly, eliminating t from (14) and (15), we obtain (9) (17) The corresponding boundary conditions (8-10) reduce to Some suitable matching conditions are prescribed at the fluid-porous medium interface (10) Let us introduce the following non-dimensional quantities: (18)

5 Radiation Effects on Mixed Convection Flow and Viscous Heating in a Vertical Channel Partially Filled with a Porous Medium Method of Solution (25) Since equations (16) and (17) are highly non-linear, these are solved analytically using Perturbation series æ Gr ö method by taking perturbation parameter e = Brç << 1. è Re ø Thus we assume where, (19) (20) Substituting (19) and (20) into equations (16-18) and comparing the coefficients of like powers of e on both sides, we obtain the boundary value problem for n = 0 and n = 1 (neglecting the terms for n ³ 2): First order For first order solution, we have Zeroth order For zeroth order solution, we have (26) (21) (27) (22) The corresponding boundary conditions are given by The corresponding boundary conditions are given by (28) (23) where, s 1 = s f1 where, A1 =. On solving the above boundary value problem (2123) for n = 0, we obtain (24) 3N 3N, A2 =. 3 Nf N + 4 On solving the above boundary value problem (2628) for n = 1, we obtain (29)

6 102 Dileep Singh Chauhan and Vikas Kumar (30) We neglect the higher order solutions (n ³ 2). Thus, we obtain the expressions of velocity distributions for both - porous and clear fluid regions, as follows: (31) (32) where, u0, u0, u1, u1 are given in (24), (25), (29), (30), respectively. The non-dimensional temperature fields, for both regions are obtained using (12), (14), (31) and (32), as (33) (34) The Nusselt numbers at the left and right channel walls are respectively, given by free and forced convection flow in a vertical channel with a porous substrate attached to the left wall is conducted in the presence of radiation and viscous effects. The flow feature encountered in the channel is a result of the buoyancy force and driving pressure gradient on the one hand, and the non-linear effects of heat generation because of viscous dissipation with radiation on the other hand. For asymmetric heating, it is assumed that Boussinesq approximation holds. The flow and temperature fields depend on the perturbation parameter e = Br(Gr / Re), where Brinkman number Br accounts for the relevance of the viscous dissipation and (Gr / Re) is the mixed convection parameter. The situations in which viscous dissipation is important are those involving flows of relatively large velocities which lie either in the clear fluid model or the Brinkman model. Hence we used the Brinkman model, so that the effects of the viscous dissipation can be included in the study. Figure 2 illustrates the effects of different parameters, e.g., the perturbation parameter e, viscosity ratio parameter f1, Grashof number Gr, Stark number N, and permeability parameter K0 (or s) on the velocity profiles in the channel. In the case, Gr / Re > 0 (buoyancy-assisted flow) the fluid rises in the channel because of the temperature gradient in the system. It is observed that the velocity in the channel increases with the increase in the value of the perturbation parameter e for positive Gr. Further, we see that the presence of porous substrate in the channel produces flow resistance. In addition, the effect due to an increase in the value of the viscosity parameter f1, adds on this resistance mechanism which (35) (36) 4. Discussion In this paper, the study of fully developed combined Figure 2. Velocity profiles, u vs. y for f2 = 1.67, N = 1, Gr = 1, Re = 1, a = 0.2.

7 Radiation Effects on Mixed Convection Flow and Viscous Heating in a Vertical Channel Partially Filled with a Porous Medium 103 further reduces the flow in the channel, however the permeability of the porous substrate increases the flow in the channel. It also increases with the increase in the value of the Stark number N. Figure 3 compares the temperature profiles in the composite channel with that when there is no porous substrate is attached to the left vertical wall. It is observed that with the introduction of the porous layer at the wall the temperature in the channel reduces. The perturbation parameter is taken a dimensionless parameter proportional to the Brinkman number (Br), and it is well known that it accounts for the relevance of viscous heating. So the temperature profile is linear, when the heat transfer is purely by conduction in the absence of viscous dissipation. While, we observe from the figure that the temperature field increases with the increase in the value of, because for 0 the convection regime dominates in the channel, and viscous dissipation forces generates greater energy to yield greater fluid temperature in the channel. We further observe that even when 0, the temperature profile in the mid part of the channel is almost linear indicating that the viscous effects dominate only in the boundary layer region. It can be deduced from Figures 2 and 3 that the magnitudes of the velocity and temperature profiles increase as the viscous dissipation parameter,, increases. These results can be explained physically since in the case p when A 0 then during the flow the pressure gradient above its hydrostatic value is supported by both the x buoyancy force and the heat generated from the viscous dissipation. Hence, both the velocity and temperature profiles are positive and assume their largest value in this situation. The influence of thermal radiation can be interpreted through the Stark number N which defines the relative contribution of heat transfer due to thermal conduction to thermal radiation. Observing the energy conservation equations (13, 15), supplemented by the thermal radiation flux term, it is seen that in the limit when N thermal radiation flux vanishes and temperature distribution is due to thermal conduction with viscous dissipation, while for N = 0, it is only due to thermal radiation. Figure 4 shows that the temperature in the channel increases with the increase in N values. However it reduces by increasing the Grashof number Gr or the viscosity ratio parameter 1. Figures 5 and 6 illustrate the variations of the Nusselt number Nu 0 at the left cold wall and Nu 1 at the right hot wall for different values of the parameters, Gr,, 1 and N. It is noticed in Figure 5 that the dimensionless rate of heat transfer, Nu 0 increases at the left wall where porous substrate is attached with the increase in the permeability parameter K 0 (or with decrease in ). It also increases by the Stark number N or the perturbation parameter. However Grashof number Gr or viscosity ratio parameter 1 decreases Nu 0. Figure 6 shows that Nu 1 decreases with the increase in the perturbation parameterand becomes zero at certain value ( * ), then changes sign and increases numerically in magnitude further. It is observed that the critical value ( * ) where Nu 1 changes sign increases with the increase in the Grashof number Gr or the viscosity ratio parameter 1, whereas it decreases with the increases in the permeability K 0 (or de- Figure 3. Temperature profiles, t vs. y for 1 = 1.25, 2 = 1.67, N =1,Gr =1,Re =1,a = 0.2. Figure 4. Temperature profiles, t vs. y for 2 = 1.67, = 0.1, =1,Re =1,a = 0.2.

8 104 Dileep Singh Chauhan and Vikas Kumar Figure 5. Nusselt number at left wall, Nu0 vs. e for a = 0.2, f2 = 1.67, Re = 1. Figure 6. Nusselt number at right wall, Nu1 vs. e for a = 0.2, f2 = 1.67, Re = 1. crease in s). The conduction-radiation parameter N decreases this critical value of e. Some particular cases have been plotted in Figures 7 and 8 for the flow and thermal field in vertical channel without radiation, when (i) the channel is free from porous material with no viscous dissipation effects (a = 0, s = 0, e = 0); (ii) channel is free from porous material with viscous dissipation effects (a = 0, s = 0, e = 0.1); (iii) channel is partially filled with porous medium and viscous dissipation effects are taken into account (a = 0.2, s = 1, e = 0.1); (iv) channel is fully composed of porous material and viscous dissipation effects are taken into account (a = 1, s = 1, e = 0.1). Results displayed for these limiting cases in Figures 7 and 8 depict that in a vertical channel, free from porous material and with no viscous dissipation and radiation effects, the temperature distribution in the fluid is a linear function of the transverse coordinate y in the case of asymmetric heating. It indicates that in the case of asymmetric heating, heat transfer between channel plates occurs by pure conduction; however the buoyancy force influences the velocity profiles. Further we see that the effect of the viscous dissipation is to increase the flow and temperature profiles in the channels free from porous material, which are reduced in the channel partially filled with a porous medium. These are further suppressed in the channel fully filled with a porous medium. These cases are examined by Aung and Worku [1], Barletta [4], Umavathi et al. [23,24] and others for various thermal boundary conditions. The results obtained in this study in absence of radiation effects are similar to Figure 7. Velocity profiles, u vs. y for Re = 1. Figure 8. Temperature profiles, t vs. y for Gr = 1, Re = 1.

9 Radiation Effects on Mixed Convection Flow and Viscous Heating in a Vertical Channel Partially Filled with a Porous Medium 105 those reported in the existing literature. 5. Conclusion The present study of flow and heat transfer in a composite vertical channel concludes that for asymmetric heating, viscous dissipation enhances the flow through the channel in the upward direction for buoyancy assisted flow. It further rises in the upward direction by increasing the permeability (measure of ease to flow in porous medium) of the porous substrate attached to the left wall of the channel, or by increasing the value of the conduction-radiation parameter. However viscosity ratio parameter 1 reduces the flow in the channel because of increase in the viscous resistance in the porous substrate. Temperature in the channel raises due to viscous dissipation effects. It increases also by increasing the permeability of the porous substrate or the value of the conduction-radiation parameter. The rate of heat transfer at the left wall increases by viscous dissipation effects or by increasing value of the conduction-radiation parameter or by increasing the permeability of the porous substrate. On the other hand the rate of heat transfer at the right wall is a decreasing function of, so the viscous dissipation effect reduces the rate of heat transfer at the right wall. It becomes zero at certain critical value of the perturbation parameter, and for greater values of the parameter it becomes negative. This change in sign of the rate of heat transfer is due to the change of direction of the heat flux at the hot wall when viscous dissipation is sufficiently large. It is well known that heat exchangers technology involves convective viscous fluid flows in vertical channels. In such cases, mostly these flows imply conditions of uniform heating of the channel, modeled by uniform wall temperature (isothermal) thermal boundary conditions. The results of this study may therefore find applications in such engineering technology. Acknowledgements The authors are thankful to the referees for their valuable comments and suggestions. The support provided by Council of Scientific and Industrial Research through Junior Research Fellowship to one of the authors Vikas Kumar is gratefully acknowledged. References [1] Aung, W. and Worku, G., Theory of Fully Developed, Combined Convection Including Flow Reversal, ASME J of Heat Transfer, Vol. 108, pp (1986). [2] Cheng, C. H., Kou, H. S. and Huang, W. H., Analysis on the Flow Reversal and Heat Transfer of Fully Developed Mixed Convection in Vertical Channels, AIAA Journal of Thermophysics and Heat Transfer, Vol. 4, pp (1990). [3] Zanchini, E., Effect of Viscous Dissipation on Mixed Convection in a Vertical Channel with Boundary Conditions of the Third Kind, Int J Heat Mass Transfer, Vol. 41, pp (1998). [4] Barletta, A., Laminar Mixed Convection with Viscous Dissipation in a Vertical Channel, Int J Heat Mass Transfer, Vol. 41, pp (1998). [5] Barletta, A., Heat Transfer by Fully Developed Flow and Viscous Heating in a Vertical Channel with Prescribed Wall Heat Fluxes, Int J Heat Mass Transfer, Vol. 42, pp (1999a). [6] Barletta, A., Analysis of Combined Forced and Free Flow in a Vertical Channel with Viscous Dissipation and Isothermal-Isoflux Boundary Conditions, ASME J Heat Transfer, Vol. 121, pp (1999b). [7] Boulama, K. and Galanis, N., Analytical Solution for Fully Developed Mixed Convection between Parallel Vertical Plates with Heat and Mass Transfer, J Heat Transfer, Vol. 126, pp (2004). [8] Barletta, A., Magyari, E. and Keller, B., Dual Mixed Convection Flows in a Vertical Channel, Int J Heat Mass Transfer, Vol. 48, pp (2005). [9] Siegel, R. and Howell, J. R., Thermal Radiation Heat Transfer, International Student Edition, New York: McGraw-Hill (1972). [10] Chamkha, A. J., Solar Radiation Assisted Natural Convection in a Uniform Porous Medium Supported by a Vertical Flat Plate, ASME J Heat Transfer, Vol. 119, pp (1997). [11] Raptis, A., Radiation and Free Convection Flow through a Porous Medium, Int Commun. Heat Mass Transfer, Vol. 25, pp (1998). [12] Raptis, A., Radiation and Flow through a Porous Medium, J Porous Media, Vol. 4, pp (2001). [13] Bakier, A. Y., Thermal Radiation Effects on Mixed

10 106 Dileep Singh Chauhan and Vikas Kumar Convection from Vertical Surfaces in Saturated Porous Media, Int Comm Heat and Mass Transfer, Vol. 28, pp (2001). [14] Raptis, A. and Perdikis, C., Unsteady Flow through a Highly Porous Medium in the Presence of Radiation, Transport Porous Media, Vol. 57, pp (2004). [15] Bég, O. A., Zeuco, J., Takhar, H. S. and Bég, T. A., Network Numerical Simulation of Impulsively-Started Transient Radiation-Convection Heat and Mass Transfer in a Saturated Darcy-Forchheimer Porous Medium, Non Linear Analysis: Modelling and Control, Vol. 13, pp (2008). [16] Ghosh, S. K. and Bég, O. A., Theoretical Analysis of Radiative Effects on Transient Free Convection Heat Transfer Past a Hot Vertical Surface in Porous Media, Non Linear Analysis: Modelling and Control, Vol. 13, pp (2008). [17] Nield, D. A. and Bejan, A., Convection in Porous Media, New York: Springer (2006). [18] Lai, F. C., Prasad, V. and Kulacki, F. A., Aiding and Opposing Mixed Convection in a Vertical Porous Layer with a Finite Wall Heat Source, Int J Heat Mass Transfer, Vol. 31, pp (1988). [19] Ingham, D. B., Pop, I. and Cheng, P., Combined Free and Forced Convection in a Porous Medium between Two Vertical Walls with Viscous Dissipation, Transport Porous Media, Vol. 5, pp (1990). [20] Al-Hadhrami, A. K., Elliott, L. and Ingham, D. B., Combined Free and Forced Convection in Vertical Channels of Porous Media, Transport Porous Media, Vol. 49, pp (2002). [21] Nield, D. A., Kuznetsov, A. V. and Xiong, M., Effects of Viscous Dissipation and Flow Work on Forced Convection in a Channel Filled by a Saturated Porous Medium, Transport in Porous Media, Vol. 56, pp (2004). [22] Barletta, A., Magyari, E., Pop, I. and Storesletten, L., Mixed Convection with Viscous Dissipation in a Vertical Channels Filled with a Porous Medium, Acta Mech., Vol. 194, pp (2007). [23] Umavathi, J. C., Kumar, J. P., Chamkha, A. J. and Pop, I., Mixed Convection in a Vertical Porous Channel, Transport Porous Media, Vol. 61, pp (2005). [24] Umavathi, J. C., Kumar, J. P., Chamkha, A. J. and Pop, I., Mixed Convection in a Vertical Porous Channel, Transport Porous Media, Vol. 75, pp (2008). [25] Chauhan, D. S. and Kumar, V., Effects of Slip Conditions on Forced Convection and Entropy Generation in a Circular Channel Occupied by a Highly Porous Medium: Darcy Extended Brinkman-Forchheimer Model, Turkish J Eng Env Sci., Vol. 33, pp (2009). [26] Chauhan, D. S. and Soni, V., Convection Effects in an Inclined Channel with Highly Permeable Layers, Arch Mech., Vol. 46, pp (1994). [27] Chang, W. J. and Chang, W. L., Mixed Convection in a Vertical Parallel Plate Channel Partially Filled with Porous Media of High Permeability, Int J Heat Mass Transfer, Vol. 39, pp (1996). [28] Chauhan, D. S. and Gupta, S., Heat Transfer in Couette Flow of a Compressible Newtonian Fluid through a Channel with Highly Permeable Layer at the Bottom, Modelling, Measurement and Control: B, Vol. 67, pp (1999). [29] Alkam, M. K., Al-Nimr, M. A. and Hamdan, M. O., On Forced Convection in Channels Partially Filled with Porous Substrates, Heat and Mass Transfer, Vol. 38, pp (2002). [30] Al-Nimr, M. A. and Khadrawi, A. F., Transient Free Convection Fluid Flow in Domains Partially Filled with Porous Media, Transport Porous Media, Vol. 51, pp (2003). [31] Barletta, A. and Zanchini, E., On the Choice of the Reference Temperature for Fully Developed Mixed Convection in a Vertical Channel, Int. J. Heat Mass Transfer, Vol. 42, pp (1999). [32] Sparrow, E. M. and Cess, R. D., Radiation Heat Transfer, Augmented ed. Washington, D.C.: Hemisphere Publication (1978). Manuscript Received: Dec. 1, 2009 Accepted: Jul. 7, 2010