Natural Convection in Vertical Channels with Porous Media and Adiabatic Extensions

Natural Convection in Vertical Channels with Porous Media and Adiabatic Extensions Assunta Andreozzi 1,a, Bernardo Buonomo 2,b, Oronzio Manca 2,c and Sergio Nardini 2,d 1 DETEC, Università degli Studi Federico II, Piazzale Tecchio 80, 80125 Napoli, Italy 2 DIAM, Seconda Università degli Studi di Napoli, Via Roma 29, 81031 Aversa, Italy a asandreo@unina.it, b bernardo.buonomo@unina2.it, c oronzio.manca@unina2.it, d sergio.nardini@unina2.it Keywords: Natural convection, Porous media, Chimney effect, Vertical Channel, Wall Heat Flux, electronic cooling, thermal insulation. Abstract. A numerical investigation on natural convection in air in a vertical heated channel, partially filled with porous medium, with adiabatic extensions downward and collinear the heated plates is accomplished. The fluid flow is assumed two-dimensional, laminar, steady state and incompressible. The porous material is considered as homogeneous and isotropic and the Brinkman-Forchheimer-extended Darcy model is considered. A finite-extension computational domain is employed to simulate the free-stream condition and allows to account for the diffusive effects and the numerical results are obtained using the finite volume method by FLUENT. Results in terms of wall temperature profiles are presented to evaluate the effects of the main thermal and geometrical parameters. The adiabatic extensions determine a wall temperature decrease and wall temperature decreases increasing Darcy number. In full filled heated channels wall temperature presents a significant increase for Darcy number decrease. Introduction Natural convection in fluid-saturated porous media has received, during the past decade, and receives considerable attention due to its applications in many technological and geophysical systems [1-4]. The natural convection heat transfer in enclosures or cavities filled or partially filled with porous medium has been the subject of various studies [5,6]. Open-ended vertical channels fully or partially filled with porous media have been accomplished in various studies [7-15]. An analytical study on natural convection in porous channels was carried out in [7]. Analytical solutions for fully developed natural convection in an open-ended vertical totally and partially filled porous channel were presented in [8], respectively. Expressions for the transient fully developed volumetric flow rate, the mixing cup temperature and the local Nusselt number were evaluated for four fundamental boundary conditions. Natural convection fluid flow in open-ended vertical parallel-plate channels partially filled with porous material were studied in [9]. The role of the local macroscopic inertial term in the porous domain momentum equation was investigated. A numerical study on local thermal equilibrium assumption in the transient natural convection channel flow, partially filled with porous medium, was carried out in [10]. The Darcy-Brinkman-Forchheimer model was used to model the flow. The role of the local macroscopic inertial term in the porous domain momentum equation was studied. A numerical study of coupled fluid flow and heat transfer by transient natural convection and thermal radiation in a vertical channel opened at both ends and filled with a saturated porous medium was achieved in [11]. The Darcy law and of the local thermal equilibrium were assumed. The results indicated that the controlling parameters of the problems had significant effects on the

flow and thermal fields and on the transient process of heating or cooling of the medium. The extension to anisotropic, in both thermal conductivity and permeability, fluid-saturated porous media was accomplished in [12]. The developing hydrodynamic and thermal behaviors of natural convection gas flow in a vertical open-ended parallel-plate microchannel filled with porous media were investigated numerically in [13]. The extended Darcy-Brinkman-Forchheimer model was employed to model the flow in porous medium and the solid and fluid media were not assumed in local thermal equilibrium. The slip flow regime was considered for the microflow regime. A numerical simulation of the steady-state, laminar, two-dimensional, natural convection heat transfer in an open-ended channel partially filled with an isotropic porous medium was investigated in [14]. The Darcy-Brinkman-Forchheimer model and the Boussinesq approximation were assumed to describe the fluid flow in the porous region. The results indicated that air gap presence may reduce the average flow in the porous substrate to zero. This led to the presence of an optimum average Nusselt number at low and high values of the effective thermal conductivity ratios. The optimal configuration of a stacking of porous medium structure with the objective to minimize the hot spot temperature in natural convection was investigated in [15]. As far as the present authors knowledge is concerned, natural convection in air in a vertical channel filled with porous media and adiabatic extensions has not been dealt with. Then, in this paper, reference is made to natural convection in air in a vertical channel partially filled with an isotropic porous medium and the two principal heated flat plates at uniform heat flux with adiabatic extensions. The numerical analysis is accomplished in steady laminar and two dimensional regime. The working fluid is air. The study is carried out employing the Brinkman-Forchheimer-extended Darcy model. The numerical results are obtained using the commercial code FLUENT, which is based on the finite volume method. A finite-extension computational domain is employed to simulate the free-stream condition and allows to account for the diffusive effects which are peculiar to the elliptic model. Problem Description and Governing Equations The physical system and geometry under investigation are shown in Figure 1a. It consists of two parallel plates that form a vertical channel partially filled with a fluid-saturated porous medium and two parallel adiabatic extensions which are placed downward and collinear to the heated plates. Both heated plates are at uniform heat flux. The porous material is considered as homogeneous and isotropic. It is assumed that the steady state fluid flow in the channel is two-dimensional, laminar and incompressible. Viscous dissipation, heat generation and pressure work are all assumed to have negligible effect on the velocity and temperature fields and are therefore neglected. The working fluid is air, Pr=0.71. All the thermophysical properties of the fluid and the solid matrix of the porous medium are assumed constant except for the variation in density with temperature (Boussinesq approximation) giving rise to the buoyancy forces. The thermophysical properties of the fluid and the solid matrix of the porous medium are evaluated at the ambient temperature, T 0, which is equal to 300 K in all cases. With the above assumptions, the governing equations in the fluid region and in dimensionless variables are: U V + = 0. Y (1)

clear fluid porous medium L L h q w q w x b t y L y L x Figure 1. Geometric configuration: (a) physical domain; (b)computational domain. U U P U U U + V = + + Grθ. Y + Y V V P U V V V + = + + Y Y Y. 1 U θ V θ θ θ + = + Y Pr Y. (2) (3) (4) In the porous medium region, the generalized flow model, known as the Brinkman-Forchheimerextended Darcy model, is used in the governing equations. The equations for mass, momentum and energy in the porous medium region are: U V + = 0. Y U U 2 P U U 2 1 C 2 U + V = ε + ε + ε U V U ε Grθ. Y + + + Y Da Da V V 2 P V V 2 1 C U + V = ε + ε + ε U V V. Y Y + + Y Da Da 1 U θ V θ θ θ + = +. Y Pr e Y (5) (6) (7) (8) The employed dimensionless variables are:

( ) 2 p p0 b T T0 θ 2 ν / ν / 2 ρν / f x y tν u v =, Y =, τ =, U =, V =, P=, =, b b b b b qb k 4 L / b 2 0 w w w gβqb 1 b 1 ν Gr =, Ra = Pr Gr, Nu( ) =, Nu = d, Pr =, k fν θ ( ) L θ ( ) α ν K k Pr e =, Da =, =. α with: e α 2 e e b ρfcpf ( ε ) w (9) k = 1 k + ε k. (10) e s f where ε is the porosity coefficient or porosity. The permeability coefficient K and inertia coefficient C of porous medium are based on Ergun's experimental investigation [17] and it is expressed by Vafai [18] as follows: K 2 3 d b ε 1.75 =, C =. 2 1.5 175 1 150ε ( ε ) (11) Numerical Model Since the vertical channel is in an infinite medium, from a numerical point of view the problem is solved with reference to a computational domain of finite extension, as shown in Figure 1b. Following the approach given in [19] the finite-extension computational domain is employed to simulate the free-stream condition and allows to account for the diffusive effects which are peculiar to the elliptic model. The choice of rectangular reservoirs with respect to other geometric configurations, such as circular, is related only to a simpler discretization of the extended computational domain. The imposed boundary conditions are the following: in the inlet permeable surfaces of the lower reservoir, the free stream conditions are considered and the air temperature is assumed equal to the ambient temperature. On the impermeable surfaces, in the lower and upper reservoir, the no slip condition are assumed and these surfaces are adiabatic. At permeable surfaces of the upper reservoir, free stream conditions are considered, whereas thermal conditions are to assign an adiabatic condition if the motion is outgoing, or the fluid temperature at ambient value if the motion is incoming. On the solid wall in the channel, no slip conditions and uniform heat flux are assumed. The commercial CFD code Fluent [20] is employed to solve the governing equations. The SIMPLE scheme is chosen to couple pressure and velocity. The porous medium model is active in the porous region. The convergence criteria of 10-4 for the residual of continuity equation and velocity components and 10-8 for the residuals of energy are assumed. A grid dependence test is accomplished to realize the most convenient grid size by monitoring the average Nusselt number, for Ra=10 4, an aspect ratio equal to 10 and for a case without porous medium and with porous medium (Da=1 and ε=). Three different grids were tested with 43x19, 85x38 and 171x77 nodes in the channel. The variations of the average Nusselt number values when the number of nodes were 85x38 in the channel, with respect to the reference values, obtained by the Richardson extrapolation, was 1.3%. The mesh size 85x38 was employed in this investigation because it ensures a good compromise between computational time and accuracy requirements.

An analogous analysis is carried out to set the optimal reservoirs dimensions, L x and L y. A reservoir vertical dimension L x, equal to the plate height L, and a reservoir horizontal dimension L y, equal to 11 times the channel gap, b, were chosen since the velocity and temperature in the channel, for larger dimensions of the reservoirs, present very small variation, as indicated in [20]. Results and Discussions The results are presented for an aspect ratio equal to 10, the ratio between the total channel height and the heated channel height is and 1.5 and a Rayleigh number equal to 10 4. The ratio between the channel gap and the porous medium thickness is equal to 0.1, 0.3 and 05. The porous material in has values of Da=10-1 -10-4 and ε=, and. The fluid is air and the solid matrix has a thermal conductivity from 1 to 10 times the one for the air. The heat capacity per unit volume, ρ c p, for fluid and porous medium are assumed to have the same value. A comparison between the wall temperature profiles for the simple channel, without adiabatic extensions, and the channel with L/L h =1.5 is reported in Fig. 2, for t/b=0.1 and 0.5. As expected the wall temperature along the heated channel increases increasing the axial coordinate and it attains a maximum value for <L h /b due to the end effects. Wall temperature decreases passing from the simple channel (L/L h =) to the channel with the adiabatic extensions. The decrease is higher for t/b=0.5 than for t/b=0.1 and this indicates that the presence of the adiabatic extension determines an increase of the chimney effect and consequently of the mass flow rate inside the channel. It is the same tendency presents in the channel-chimney system without porous material [21]. The effect of porosity, ε=, and, for L/L h =1.5, t/b=0.5 and Da=10-2 and 10-4 are given in Fig. 3. It is shown that for the highest considered Da value (10-4 ), in Fig. 3b, the effect of ε is negligible whereas for Da=10-2 some evident differences are noted at lowest considered porosity, ε=. In Fig. 4, for t/=0.5, completely filled heated channel, and e= wall temperature profiles present increasing values as the Da value decreases. The increase is greater as greater Da is and a very high increase is noted from Da=10-4 to 10-6 ; in fact, maximum wall temperature is equal to about 2.0 for Da=10-4 and about 15 for Da=10-6. The increase in the ratio between the thermal conductivities (γ=k s /k f ), in Fig. 5, determines the decrement of wall temperature. The differences between the wall temperatures decreases increasing the value of k s /k f. Wall temperature profiles for porosity equal to and t/b=0, 0.1 and 0.3 for different Da values are shown in Fig. 6. The effect of t/b at lowest considered Da value, in Figure 6a, is very weak at low t/b; in fact, the differences between the profile at t/b= and t/b=0.1 are negligible respect to the same differences between t/b = 0 and t/b = 0.3. The presence of porous medium on the heated walls determines, for the examined cases with k s /k f =1, an increase of wall temperature. For lower Da also at low t/b values the differences between the wall temperatures increase and they are higher as lower Da is. Moreover, decreasing Da a very significant increase of the difference between wall temperature profiles at t/b=0.1 and 0.3 is observed. This allow to employ this configurations as a thermal insulation system. Summary Natural convection in air in a vertical heated channel, partially filled with porous medium, with adiabatic extensions downward and collinear the heated plates was studied numerically. Results in terms of wall temperature profiles were carried out in order to evaluate the effects of the main thermal and geometrical parameters. It was found that the adiabatic extensions determined a wall

1.2 Ra=10 4, ε=, Da=10-2 L/b=10 L h /b=10 L/L h =1 L/L h =1.5 (a) t/b=0.1 (b) t/b=0.5 Figure 2. Wall temperature profiles for Ra=10 4, Da=10-2, ε= and (a) t/b=0.1, (b) t/b=0.5. 1.2 6.0 Ra=10 4, L/b=10, L/L h =1.5, full porous 5.0 4.0 ε= ε= ε= 3.0 (a) Da=10-2 2.0 (b) Da=10-4 0 25 5 75 0.1 0 25 5 75 0.1 Figure 3 Wall temperature profiles for Ra=10 4, for full filled heated channel, different porosities, L h /L=1.5 and (a) Da=10-2, Da=10-4. 2 1.2 15.0 Da=10-1 Da=10-2 Da=10-3 Da=10-4 Da=10-5 Da=10-6 t/b=0.5 Ra=10 4, ε=, Da=10-4 L/b=10, L/L h =1.5, t/b=0.3 L h /b=10 1 5.0 (b) ε= γ=1 γ=2 γ=5 γ=10 Figure 4. Wall temperature profiles for Ra=10 4, for t/b=0.5, ε= and different Da values. Figure 5. Wall temperature profiles for Ra=10 4, for t/b=0.3, ε=, L h /L=1.5, Da=10-2 and different thermal conductivity ratio.

Ra=10 4, ε=, L/b=10, L/L h =1.5 L h /b=10 t/b=0 t/b=0.1 t/b=0.3 (a) Da=10-1 (b) Da=10-2 (c) Da=10-3 (d) Da=10-4 Figure 6. Wall temperature profiles for Ra=10 4, for t/b=0, 0.1, 0.3, ε= and different Da values: (a) Da=10-1, (b) Da=10-2, (c) Da=10-3 and (d) Da=10-4. temperature decrease respect to the simple heated channel, without adiabatic extensions. Wall temperature decreased increasing Darcy number and in full filled heated channels wall temperature attained a significant increase for Darcy number decrease. Nomenclature Symbol Quantity SI Unit Symbol Quantity SI Unit b channel spacing m p pressure Pa C inertia coefficient eq. (11) P dimensionless pressure eq. (9) c p specific heat at constant Jkg -1 K -1 Pr Prandtl number eq. (9) pressure Da Darcy number eq. (9) q heat flux Wm -2 g gravitational acceleration ms -2 Ra Rayleigh number eq (9) Gr Grashof number eq. (9) t porous medium thickness m k thermal conductivity Wm -1 K -1 T temperature K K permeability m 2 u,v velocity components ms -1 L total height of the channel m U,V dimensionless velocity eq. (9) components L h heated channel length m x,y coordinate distance m Nu() local Nusselt number eq. (9),Y dimensionless coordinate eq. (9) Nu average Nusselt number eq. (9)

Greek symbols Subscripts α thermal diffusivity m 2 s -1 a ambient β volumetric expansion K -1 e average coefficient δ convergence angle f fluid ε porosity coefficient s solid θ dimensionless temperature eq. (9) w wall ν kinematic viscosity m 2 s -1 0 initial or ambient ρ density kg/m 3 τ dimensionless time eq. (9) Acknowledgments This work was supported by MIUR with Art. 12 D. M. 593/2000 Grandi Laboratori EliosLab. References [1] D.A. Nield and A. Bejan: Convection in Porous Media, third ed. (Springer, New York 2006). [2] D.B. Ingham and I. Pop: Transport Phenomena in Porous Media (Pergamon, Oxford 2002). [3] K.Vafai: Handbook of Porous Media, vol. II (Marcel Dekker, New York 2005). [4] A. Bejan, I. Dincer, S. Lorente, A. F. Miguel and A. H. Reis: Porous and Complex Flow Structures in Modern Technologies (Springer, New York 2004). [5] S. Hsiao, C. Chen and P. Cheng, P.: Int. J. Heat Mass Transfer Vol. 37 (1994), p. 2193. [6] A. Merrikh and A. Mohamad: Int. J. Heat Mass Transfer Vol. 45 (2002), p. 4305. [7] T. Nilsen and L. Storesletten: J. Heat Transfer Vol. 112 (1990),pp. 396. [8] M. Al-Nimr and O. Haddad: J. Porous Media Vol. 2 (1999), p. 179. [9] A. F. Khadrawi and M. A. Al-Nimr: ASME, Fluids Engineering Division, Vol. FED-257 (2002), p. 1293. [10] A. F. Khadrawi and M.A. Al-Nimr: J. Porous Media Vol. 6 (2003), p. 59. [11] K. Slimi, L. Zili-Ghdira, S. Ben Nasrallah and A. A. Mohamad: Num. Heat Transfer Part A Vol. 45 (2004), p. 451. [12] O. M. Haddad, M. M. Abuzaid and M. A. Al-Nimr: Num. Heat Transfer Part A Vol. 48 (2005), p. 693. [13] K. Slimi, A. Mhimid, M. Ben Salah, S. Ben Nasrallah, A.A. Mohamad and L. Storesletten: Num. Heat Transfer Part A Vol. 48 (2005), p. 763. [14] S. Kiwanand and M. Khodier: Heat Transfer Eng. Vol. 29 (1) (2008), p. 67. [15] A. Bejan: Int. J. Heat Mass Transfer Vol. 47 (2004), p. 3073. [16] K.,Vafai and C. L. Tien: Int. J. Heat Mass Transfer Vol. 24 (1981), p. 195. [17] S. Ergun: Chem. Engng. Proc. Vol. 48 (1952), p. 89. [18] K. Vafai: J. Fluid Mech. Vol. 147 (1984), p. 233. [19] A. Andreozzi and O. Manca: Int. J. Heat Fluid Flow Vol. 22 (2001), p. 424. [20] Fluent Inc., Fluent 6.3, User Manual (Fluent Inc., Lebanon NH 2006). [21] A. Andreozzi, B. Buonomo and O. Manca: Num. Heat Transfer Part A Vol. 47 (2005), p. 741.