Adv. Theor. Appl. Mech., Vol. 5, 2012, no. 5, 237-245 Numerical Study of Natural Convection in an Inclined L-shaped Porous Enclosure S. M. Moghimi 1 *, G. Domairry 2, H. Bararnia 2, Soheil Soleimani 2 and E. Ghasemi 3 1 Department of Mechanical Engineering, Islamic Azad University, Qaemshahr Branch, Qaemshahr, Iran 2 Babol Noshirvani University of Technology, Department of Mechanical Engineering, Babol, P. O. Box 484, Iran 3 Department of Mechanical Engineering, University of Idaho, 177 Science Center Dr, Idaho Falls, ID 83402, USA Abstract Brinkman Forchheimer extended Darcy model of 2D natural convection heat transfer in an inclined L-shaped porous cavity is studied numerically by finite difference method via Marker And Cell (MAC) approach. The numerical results are reported with streamlines, isotherms, and mean Nusselt numbers for Rayleigh 3 2 number Ra = 10 10, Darcy number, Da = 10 10, Porosity of porous region ε = 0.5, and different inclination angle varying from 30 o to 90 o. It is concluded that decrement of Darcy number decreases the heat transfer and inclination angle has an important effect on isotherms and streamlines in high Rayleigh numbers. Keywords: Natural convection; Porous media; Inclined cavity; Numerical study * Corresponding Author: Email: m.moghimi@qaemshahriau.ac.ir, moghimi4999@yahoo.com
238 S. M. Moghimi et al 1. Introduction Natural convection heat transfer inside irregular and complex shaped enclosures has a wide variety of technological applications. Between the various geometries L-shaped enclosures have been receiving a growing interest recently due to its potential engineering applications in electronic packages, electrical equipment, building corners, etc.heat transfer inside L-shaped enclosures governed by several factors. Interwall spacing, enclosure height, sharpness of the corners, multicellular flow pattern at high Rayleigh numbers are some of the common influential factors The works of Angirasa and Mahajan [1], Angirasa et al. [2], and Chinnakotla et al. [3] are devoted to study the flow and thermal fields' behavior near an L-shaped corner at different boundary conditions. Mahmud [4] reported buoyancy induced flow pattern and heat transfer characteristics inside an L-shaped enclosure for three different aspect ratios. Existence of multicellular flow is also reported for lower aspect ratio which affects both local and global heat transfer rates. The research of natural convection in porous media has been conducted widely in recent years, which involves post-accidental heat removal in nuclear reactors, cooling of radioactive waste containers, heat exchangers, solar power collectors, grain storage, food processing, energy efficient drying processes, to name of a few. Nield and Bejan [5] and Ingham and Pop [] contributed to a wide overview of this important area in heat transfer of porous media. There are many published studies related to natural convection in rectangular porous enclosures. Moya et al. [7], Bejan [8], Parasad and Kulacki [9], Baytas and Pop [10], Beckerman et al. [11], Gross et al. [12], Lai and Kulacki [13], Monale and Lage [14], and Walker and Homsy [15] have donated many important results for this problem. Caltagirone and Bories [1] studied the stability criteria of free convective flow in an inclined porous layer. In this study natural convection heat transfer in an inclined L-shaped porous cavity is studied numerically for various values of controlling parameters implying finite difference method. 2. Problem definition and governing equations The schematic of the present problem with related boundary conditions is depicted in figure 1.
Numerical study of natural convection 239 Fig. 1. Schematic of the problem and related boundary condition. No slip boundary condition is implied for all solid boundaries of the cavity. The left wall is kept at constant high temperature, the right wall is thermally insulated, and the other walls are cold. The dimensionless form of the governing equations can now be obtained as given below [17]: Continuity equation: u v + = 0 (1) x y x-momentum equation : 1 u 1 u u + u + v = ε t ε x ε y ε (2) ( ) 1 2 2 1 Pr 1/75 u + v 2 u Pr 2 u 2 u ( εpf ) u + 3 + 2 2 ε x Da 150 Da ε x y 2 ε y-momentum equation : 1 v 1 v v + u + v = ε t ε x ε y ε ( ) 1 2 2 u v 2 v 2 v 2 v 3 2 2 Da ε 2 1 Pr 1/75 + Pr ( εpf ) v + + + Pr RaT ε y Da 150 x y ε Energy equation: 2 2 σ T T T T T + u + v = + 2 2 t x y x y (3) (4)
240 S. M. Moghimi et al where the following scales have been used for nondimensionalization x y u v t P x =, y =, u =, v =, t =, P = * * * * * * 2 2 2 Lref Lref α / Lref α / Lref α / L ref ρα / L ref 3 * T Ti gbδt Lref v k = = = = 2 Tw Ti vα α Lref, T, Ra,Pr, Da and ε σ = ( PC p) + ( 1 ε)( PC p) f ( ρc ) D f s The asterisks have been omitted from the dimensionless governing equations (1)-(4) for the sake of convenience. 3. Results and Discussion The continuity and momentum equations (1) - (3) are solved using the MAC method [18]. To prevent oscillations in pressure values staggered arrangement is used. Energy equations (4) should be computed after finding the solution of the momentum equation in each time step, since the momentum and energy equations are coupled in free convection. The equations were presented in an implicit scheme and were solved using an iterative technique in each time step. The steady state solutions are calculated by marching in time and satisfying the following condition. i max j max n+ 1 n 1 Ti, j T i, j 7 10 (5) imax j max i= 1 j= 1 Δt To test and assess grid independence of the solution scheme, numerical experiments are performed for 1 1, 81 81, 101 101 and 121 121. These experiments show that a grid mesh of 101 101 is adequate to describe the flow and heat and mass transfer processes correctly. The code validation was checked by comparison of the results in porous cavity with the results of Nithiarasu [17] in Table 1.
Numerical study of natural convection 241 Da Ra Table. 1. Comparison of Nusselt number of present study with [17]. Ref[17] ε = 0.4 Present study Ref [17] ε = 0. Present study Ref [17] ε = 0.9 Present study 10 7 10 1.079 1.078 1.079 1.079 1.08 1.08 8 10 2.97 3.09 2.997 3.08 3.00 3.095 9 10 11.4 12.01 11.79 12.21 12.01 12.41 2 10 3 10 1.01 1.008 1.015 1.013 1.023 1.017 4 10 1.408 1.35 1.530 1.483 1.40 1.2 5 10 2.983 3.055 3.555 3.474 3.91 3.928 The local Nusselt number and space-averaged Nusselt numbers are evaluated as follows: Nu loc θ =, n wall 1 L loc Nu = Nu dx L 0 () where n denotes the external unit normal to the hot wall surface. Fig.4 shows the effects of inclination angle and Darcy number on isotherms and 4 streamlines for Ra = 10 and ε = 0.5. As shown for angles range from θ = 30 o to θ = 30 o there is no noticeable effect on isotherms and streamlines; for θ = 0 o two vortex are formed at the two corners of the cavity with the same rotating direction with each other but opposite with the main vortex; for θ = 90 o two secondary vortex are formed at the center of the cavity with the same rotating direction with the main vortex. As can be seen from the isotherms, variation in inclination angles has a negligible effect on them and that is also expected that for low Rayleigh numbers, Nusselt number does not change noticeably.
242 S. M. Moghimi et al θ 30 o Da=10-2 Da=10-30 o 0 o 90 o Fig. 4. Effects of inclination angle and Darcy number on isotherms and 4 streamlines for ε = 0.5 and Ra = 10 Fig. 5 displays the effect of inclination angles and Darcy numbers for Ra = 10 and ε = 0.5 which clearly shows the effect of various inclination angles on isotherms and streamlines in high Rayleigh numbers. Moreover decrease in Darcy number from Da=10-2 to Da=10 - cause isotherms to be stratified within the cavity which prevent increment of heat transfer and therefore this decrement also decrease the Nusselt number.
Numerical study of natural convection 243 θ 30 o Da=10-2 Da=10-30 o 0 o 90 o Fig. 5. Effects of inclination angle and Darcy number on isotherms and streamlines for ε = 0.5 and Ra = 10 Influence of inclination angle and Rayleigh number on Nusselt number is depicted in Fig. (a). As seen the maximum Nusselt occurs on hot wall atθ = 0 o. Besides, for θ = 0 o and Ra = 10 the Nusselt number has a minimum value which is not observed for lower Rayleigh numbers.
244 S. M. Moghimi et al Fig.(a). Effect of inclination angle and Rayleigh number on Nusselt number 2 forε = 0.5, Da = 10, (b) Effect of inclination angle and Darcy number on Nusselt number forε = 0.5, Ra = 10. Fig..(b) shows the effect of inclination angle and Darcy number on Nusselt number for ε = 0.5 and Ra = 10 ; as mentioned before decrease of Darcy number from Da=10-2 to Da=10 - decreases the heat transfer resulting in Nusselt number decrement. 4. Conclusion Natural convection heat transfer in an L-shaped porous cavity is studied numerically for different values of governing parameters by finite difference method using the marker and cell method (MAC). It is revealed that inclination angle can be a noticeable parameter in investigation of fluid flow behavior and heat transfer and has an important effect on the vortex formation inside the L-shape cavity. In addition, decreasing of Darcy number results in Nusselt number decrement. References [1] D. Angirasa, R.L. Mahajan, Natural convection from L-shaped corners with adiabatic and cold isothermal horizontal walls, J. Heat. Transfer 115 (1993), 149 157. [2] D. Angirasa, R.B. Chinnakotla, R.L. Mahajan, Buoyancy induced convection from isothermal L-shaped corners with symmetrically heated surfaces, Int. J. Heat. Mass.Transfer 37 (1994), 2439 243.
Numerical study of natural convection 245 [3] R.B. Chinnakotla, D. Angirasa, R.L. Mahajan, Parametric study of buoyancy induced flow and heat transfer from L-shaped corners with asymmetrically heated surface, Int. J. Heat. Mass.Transfer 39 (199), 851 855. [4] S. Mahmud, Free convection inside an L-shaped enclosure, Int. Communicat. Heat. Mass Transfer 29 (2002), 1005 1013. [5] D.A. Nield, A. Bejan, Convection in Porous Media, Springer, New York, 1998. [] D.B. Ingham, I. Pop (Eds.), Transport. Phenomen. Porous. Media, Elsevier, Amsterdam, 1998. [7] S.L. Moya, E. Ramos, M. Sen, Numerical study of natural convection in a tilted rectangular porous material, Int. J. Heat Mass Transfer 30 (1987),741 75. [8] A. Bejan, On the boundary layer regime in a vertical enclosure filled with a porous medium, Lett. Heat. Mass. Transfer (1979), 93 102. [9] V. Prasad, F.A. Kulacki, Convective heat transfer in a rectangular porous cavity effect of aspect ratio on flow structure and heat transfer, J. Heat. Transfer 10 (1984), 158 15. [10] A.C. Baytas, I. Pop, Free convection in oblique enclosures filled with a porous medium, Int. J. Heat. Mass. transfer 42 (1999), 1047 1057. [11] C. Beckermann, R. Viskanta, S. Ramadhyani, A numerical study of non-darcian natural convection in a vertical enclosure filled with a porous medium, Num. Heat. Transfer 10 (198), 557 570. [12] R.J. Gross, M.R. Bear, C.E. Hickox, The application of flux-corrected transport (FCT) to high Rayleigh number natural convection in a porous medium, in: Proc. 8th Int. Heat Transfer Conf., San Francisco, CA, 198. [13] F.C. Lai, F.A. Kulacki, Natural convection across a vertical layered porous cavity, Int. J. Heat Mass. Transfer 31 (1988), 1247 120. [14] D.M. Manole, J.L. Lage, Numerical Benchmark results for natural convection in a porous medium cavity, in: ASME Conference, Heat and Mass Transfer in Porous Media. 21(1992), 55 0. [15] K.L. Walker, G.M. Homsy, Convection in a porous cavity, J. Fluid. Mech 87 (1978), 449 474. [1] J.-P. Caltagirone, S. Bories, Solutions and stability criteria of natural convective flow in an inclined porous layer, J. Fluid. Mech. 155 (1985), 27 287. [17] P. Nithiarasu, K. N. Seetharamu, T. Sundarajan, Natural convective heat transfer in a fluid saturated variable porosity medium, Int. J. Heat Man Transfer. 40 (1997), 3955-397. [18] Harlow FH, Welch JE. Numerical calculations of time-dependent viscous incompressible flow of fluid with free surface. Phys Fluids, 8 12 (195) 2182 9. Received: May, 2012