Taiwan 300, R.O.C. Published online: 03 Mar 2011.

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1 This article was downloaded by: [National Chiao Tung University 國立交通大學 ] On: 27 Aril 2014, At: 23:02 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Journal of the Chinese Institute of Engineers Publication details, including instructions for authors and subscrition information: htt:// Heat transfer of orous medium with random orosity model in a laminar channel flow Wu Shung Fu a, Ke Nan Wang a & Wen Wang Ke a a Deartment of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan 300, R.O.C. Published online: 03 Mar To cite this article: Wu Shung Fu, Ke Nan Wang & Wen Wang Ke (2001) Heat transfer of orous medium with random orosity model in a laminar channel flow, Journal of the Chinese Institute of Engineers, 24:4, To link to this article: htt://dx.doi.org/ / PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the ublications on our latform. However, Taylor & Francis, our agents, and our licensors make no reresentations or warranties whatsoever as to the accuracy, comleteness, or suitability for any urose of the Content. Any oinions and views exressed in this ublication are the oinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied uon and should be indeendently verified with rimary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, roceedings, demands, costs, exenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and rivate study uroses. Any substantial or systematic reroduction, redistribution, reselling, loan, sub-licensing, systematic suly, or distribution in any form to anyone is exressly forbidden. Terms & Conditions of access and use can be found at htt://

2 Journal of the Chinese Institute of Engineers, Vol. 24, No. 4, (2001) 431 HEAT TRANSFER OF POROUS MEDIUM WITH RANDOM POROSITY MODEL IN A LAMINAR CHANNEL FLOW Wu-Shung Fu*, Ke-Nan Wang and Wen-Wang Ke Deartment of Mechanical Engineering National Chiao Tung University Hsinchu, Taiwan 300, R.O.C. Key Words: random orosity, orous medium, channel flow. ABSTRACT In this aer, force convection heat transfer of a orous medium in a laminar channel flow is numerically investigated. The orous medium with random orosities is used to enhance heat transfer and the random orosities are derived by the Kinderman-Ramage rocedure. The results show that when the mean orosity is larger than 0.5, the average Nusselt numbers are enhanced and better than the solid block case. Therefore, a orous medium with larger orosity and the roer bead diameter could rovide more heat dissiation. For the random orosity distribution, the rofiles of velocities are disordered and the channeling effect is not aarent near the late region. I. INTRODUCTION Recently, numerous investigations of heat and mass transort henomena in orous media which can simulate many industrial alications such as heat exchangers, acked-shere beds, electronic cooling, chemical catalytic reactors, drying rocesses, and heat ie technology have been studied numerically and exerimentally. The orosity is one of the main arameters for studying orous media. Two major models are used to define the distribution of orosity in orous media for conveniently investigating the henomena of thermal and flow fields. The one in which the orosity distributed in the orous medium is uniform is called the constant orosity model. Vafai and Tien (1981) used local volume averaging analysis to derive the governing equations of fluid flow and heat transfer for a orous medium. However, Roblee et al. (1958) and Benenati and Brosilow (1962) observed that orosity varied significantly in the near wall region. Schwartz et al. (1952) conducted exerimental studies and measured the maximum velocity, which is normally called the channeling effect, in the near wall region. These henomena directly validated the idea that orosity regarded as a variable was more realistic. Then Hsu and Cheng (1990) utilized the volume averaging method to derive the governing equations for orous media. The effects of inertial term, solid boundary and variable orosity were considered. Cheng et al. (1991) ointed out that orosity was simulated as a damed oscillatory function of the distance from the wall and the damed oscillatory henomenon was insignificant as the distance was larger than five bead diameters for acked beds. Therefore, concerning both ractical use and a convenient theoretic model, the variation of the orosity is assumed to be an exonential function of the distance from the solid wall and is called a variable orosity model. Besides, Georgiadis et al. (1987, 1988, 1991) studied unidirectional flow and heat transort he- *Corresondence addressee

3 432 Journal of the Chinese Institute of Engineers, Vol. 24, No. 4 (2001) nomena in a random orous medium. The results indicated the mean flow rate U(ε) based on random orosity was larger than that U( ε ) based on constant orosity when the Forchheimer model of flow was held. Saito et al. (1995) studied effects of the orosity and void distributions on the ermeability by the Direct Simulation Monte Carlo method and found that the ermeability deended, strongly, not only on the orosity but also on the void distribution. These facts indicated that excet for secial screen rocesses the sizes of the beads are extremely difficult to kee uniform. In fact, the orosity distributions of orous media seldom agree with the distributions of the constant or variable orosity model mentioned above. That the orosity in orous media is irregular or random is more realistic. This orosity distribution is conveniently named as a random orosity model. Although the effect of random orosity distribution on the flow field has been discussed in the ast, most discussions only concentrated on secial henomena. Fu and Huang (1999) studied the flow and thermal fields of a orous medium with random orosity distribution under a laminar slot iminging jet. The results showed that the orosity near the heat wall should be smaller for enhancing the heat transfer rate of an iminging jet flow. This numerical simulation intends to adot the random orosity model to simulate the heat transfer of a orous medium which is mounted on a horizontal hot late in a laminar channel flow. Differet mean orosities and standard deviations are taken into consideration. The results show that the distributions of velocities U and V are chaotic and disordered in a random orosity distribution. The channelling effect is not aarently near the hot late in the random orosity model. A orous medium with small mean orosity, ε, or small bead diameter, D, could result in reduction of thermal erformance. II. PHYSICAL DESCRI IO The hysical model is shown in Fig. 1. A orous medium is mounted on a hot late in a channel. The height of the channel is h. The length l and height h of the orous medium are same and equal to h/2. The inlet velocity distribution u(y) of the fluid is fully develoed and the inlet temerature T o of the fluid is constant. The hot late is mounted with the orous medium and the other regions are insulated. The length of the hot late is l (=h/2), and the temerature of this region is T w which is higher than T o. The whole comutation domain is large enough for fully develoed distributions of the velocity and temerature at the outlet, (l1=8h and l2=60h). In order to facilitate the analysis, the following assumtions are made. (1) The material of orous medium is coer bead. (2) The flow field is steady, two-dimensional, single hase, laminar and incomressible. (3) The fluid roerties are constant based on T ref =T o (T w T o ) and the effect of gravity is neglected. (4) The effective viscosity of the orous medium equals to the viscosity of the external fluid. The ermeability K and inertia factor F of orous medium roosed by Vafai (1984) are defined as follows. K = ε 2 d 2 150(1 ε) 2 (1) F = ε 1.5 (2) The effective thermal conductivity k e of a orous medium is a combination of the solid conductivity k s and the fluid conductivity k f validated by Shonnard and Whitaker (1989) for a high ratio of k s /k f. k e = k f (4 ln( k s k f ) 11) (3) Based on the above assumtions, the governing equations, boundary conditions and geometric dimensions are normalized in tension forms as follows: 1. Governing equations of the external flow field. U i f =0 (4) U j f U i f Fig. 1 Physical model = P f + X 1 2 f U i (5) i Re 2 U j f θ f = 1 Re Pr f 2 θ f 2 (6) 2. Governing equations of the internal flow field roosed by Cheng (1987) for the orous medium. U i =0 (7)

4 W.S. Fu et al.: Heat Transfer of Porous Medium with Random Porosity Model in a Laminar Channel Flow 433 U j ( U i X ε j )= P + X 1 2 U i i Re 2 F U Da εu i 1 Re Da εu i (8) Table 1 The main arameters H H L D Re ε σ ε Pr f Pr % % where U j θ = ( 1 Re Pr θ ) (9) U f = U, V f = V, P f = P, U f Y = U Y, V f Y = V Y, X i =x i +h, U i =u i /u o, P=/ρ u 2 o, θ=(t T o )/(T w T o ), Pr=ν/α, Da=K/H 2, Re=u o h/ν f, U = U 2 + V 2. (10) u o : the maximum inlet velocity. 3. Boundary conditions U f =0, V f =0, θ f Y =0 on the surfaces AH, BC, and FG. (11) U f = 6(Y 2 Y), V f =0, θ f =0 on the surface AB. (12) U f X =0, V f X =0, θ f X =0 on the surface HG. (13) U =0, V =0, θ =1 on the surface CF. (14) The interfacial conditions at the fluid/orous medium interfaces are automatically satisfied to the Brinkman extension in the governing equations recommended by Hadim (1994). 4. Interfacial conditions on the surfaces CD and EF U f = U, V f = V, P f = P, U f X = U X, V f X θ f = θ, k θ f f X on the surface DE = V X, = k e θ X. (15) θ f = θ, k f θ f Y = k e θ Y. (16) III. NUMERICAL SCHEME The SIMPLEC algorithm with TDMA solver is used to solve the governing Eqs. (4)-(9) for the flow and thermal fields. Staggered and non-uniform meshes are used. The finer meshes are set in the front and rear regions of the orous medium and at the inlet and outlet regions. Non-uniform meshes with a scale ratio of 1.1 are adoted. The uniform meshes are laced in the orous medium. The harmonic mean formulation of thermohysical roerties is used to avoid the effects of abrut change of these roerties across the interfacial regions of the orous medium and the external flow field for accuracy of comutation. The under-relaxation factors are 0.3~ 0.5 for both the fields of velocity and temerature. The conservation residues of the equations of momentum, energy and continuity and the relative errors of each variable are used to examine the convergence criteria and defined as follows: Σ Residue of Φ equation 2 C.V. 1/2 10 4, Φ=U, V, and θ. (17) max Φ n +1 Φ n max Φ n , Φ=U, V, P and θ (18) In this aer, the arameters which include the Reynolds number, Re, mean orosity, ε, standard deviation, σ ε, mean diameter of beads, D, Prandtl number, Pr, height of the orous medium, H, length, L, inlet temerature, θ o, and the hot late temerature, θ w, are tabulated in Table 1. The results of grid tests are listed in Table 2, meshes in the domain and meshes in the orous medium are chosen. The distribution of orosity follows the random

5 434 Journal of the Chinese Institute of Engineers, Vol. 24, No. 4 (2001) Table 2 Grid tests for the case of Re=500, H= 0.5, L=0.5, ε =0.5, σ ε =10% D=0.1 and Pr=0.7. Meshes of X Meshes of Y (A+B+C) (D+E) Nu 150 ( ) 32 (16+16) ( ) 40 (20+20) ( ) 60 (30+30) ( ) 80 (40+40) ( ) 92 (46+46) ( ) 100 (50+50) orosity model. According to Fu and Huang (1999), the theoretical form of the orosity distribution of the random orosity model is generated from the Kinderman-Ramage rocedure, which is used to generate a random variable, ξ, of the standard normal distribution, N(0,1), first. The random variable, ξ, is transformed to gain a general random variable, ε, corresonding to a general normal distribution, N( ε, σ ε2 ), of which the mean, ε, and standard deviation, σ ε, are equal to designed constants, resectively. Therefore, the distribution of the general random variable, ε, is regarded as the orosity distribution of the random orosity model in this study. Since infinite random atterns can be generated with a given mean orosity ε and standard deviation σ ε, it is difficult to solve all atterns. Therefore, only a few atterns with the mean orosity ε (=0.5 and 0.7) and standard deviation σ ε (=5% and 10%) are resented to investigate the heat transfer erformance of the orous medium with the random orosity model. IV. RESULTS AND DISCUSSION In Figs. 2(a)-(c), the global orosity distribution mas of the three selected cases of ε =0.5, 0.7 and σ ε =5%, 10% are shown. In the global orosity distribution mas, the total area of each case is aroximately divided into several different orosity regions with different grey scales and the darker scale reresents the smaller orosity. The mas show that the higher the standard deviation, σ ε, is, the larger the fluctuation of the orosity distribution becomes. The higher the mean orosity, ε, is, the larger local ore size is. Fig. 2 Global random orosity distribution mas To illustrate the flow and thermal fields more clearly, only the henomena near the orous medium are resented. The region shown in the figure of the flow field is 1h ustream and 2.5h downstream from the orous medium. As shown in Fig. 3, there are streamlines for different ε and D cases. The

6 W.S. Fu et al.: Heat Transfer of Porous Medium with Random Porosity Model in a Laminar Channel Flow 435 Fig. 3 Streamlines of different cases with standard deviation σ ε =100% dimensionless stream function, ψ, is defined as and U = ψ and V = ψ Y Y (19) In Figs. 3(a) and (b), two recirculation zones are found in both the ustream and downstream regions from the orous medium. Since the flow enetrates into the orous medium, the downstream recirculation zone is hardly neighboring to the orous medium. In Fig. 3(b), the coer bead is smaller and the resistance becomes larger which results in the fluid hardly flowing through the orous medium. This situation is disadvantageous to the heat transfer rate of the hot late. When D becomes larger ( ε =0.7) as shown in Fig. 3(c), the streamline of ψ=0.02 is closer to the hot late. The results mean more fluids flow through the near wall region easily, which is advantageous to the heat transfer rate of the hot late. The recirculation zone forms further downstream, and in the ustream region the recirculation zone is hardly observed. As shown in Fig. 4(a), the variations of the velocity distributions of U at the middle osition (X= 0.25) of the orous medium are illustrated. Due to Fig. 4 (a) The rofiles of velocity U of central orous medium at X=0.25. (b) The rofiles of velocity V of central orous medium at Y=0.25 the drastic variations of orosity, the variations of local velocity U resent jagged rofiles and the channelling effect does not aear. For the different mean orosities ε (=0.5, 0.7) and the same standard deviation σ ε (=10%) cases, the distributions of local velocity U are different. When the mean orosity, ε, is larger, more fluids may flow through the orous medium, and the distribution of local velocity U becomes larger. In Fig. 4(b), the values of local velocity V are negative near the front edge of the orous medium and ositive near the back edge of the orous medium. This means most fluids flow downward and flow uward through out the orous medium, which is good for the heat transfer rate at the front of the hot late. The isotherms accomanying Figs. 3(a)-(c) are shown in Figs. 5(a)-(c). In larger D cases the fluid enetrates into the orous medium, the temerature

7 436 Journal of the Chinese Institute of Engineers, Vol. 24, No. 4 (2001) Table 3 The results of the different mean orosity cases based on the random orosity model under Re=500, σ ε =10%, D=0.1 and Pr=0.7 ε Nu 0.0 (Solid block) (W. P. M.) W.P.M. : Without orous medium gradient is larger and the isotherms extend farther downstream from the orous medium. On the other hand, the isotherms diffuse ustream in the (b) case with small D, and this would reduce the convective heat transfer caacity remarkably. The local Nusselt number, Nu, and the average Nusselt number, Nu, on the hot late are defined as follows, resectively. Nu = h xl x k f Fig. 5 Isotherms of different cases = k e k f θ Y Y =0, h x = q x (T w T 0 ) y =0 (20) Nu = 1 L 0 L Nu dx (21) As shown in Fig. 6(a), based on the larger D= 0.1, due to the large contact surface between the fluids and the orous medium, the heat transfer rates are aarently enhanced for both ε =0.5 and 0.7 cases comared with the cases of a solid block without orosity. The higher velocity U is attained for the ε = 0.7 case (Fig. 4(a)). The values of Nu for ε = 0.7 cases are then larger than those for the ε =0.5 cases. For the same mean orosity, ε, in Fig. 6(b), the larger D is, the larger local Nusselt number is Fig. 6 The distributions of local Nusselt number Nu for different conditions (a) Comarison of solid block and without orous medium (b) Comarison of different bead diameters D attained. For orous medium cases, the results show that the maximum Nusselt number aears at the front of the orous medium. For a solid block case,

8 W.S. Fu et al.: Heat Transfer of Porous Medium with Random Porosity Model in a Laminar Channel Flow 437 because of the high conductivity of coer and the existence of flow searation in the front region, the distribution of local Nusselt numbers is flatter and more uniform than those of the orous medium cases. In Table 3, the average Nusselt number Nu of the solid block case is larger than those of ε 0.4 cases. The result indicates that the small mean orosity orous medium does not imrove heat transfer. As the mean orosity ε is larger than 0.5, the values of Nu are larger than those of the solid block case. V. CONCLUSIONS The force convection heat transfer rates of a random-ore orous medium mounted on a hot late in a laminar channel flow is numerically investigated. The main results can be summarized as follows: 1. For the random orosity model, the distributions of velocities U and V are chaotic and disordered. The channeling effect is not aarent near the late region. 2. A orous medium with a larger orosity can rovide more heat dissiation than a solid block. 3. A orous medium with small mean orosity, ε, or small bead diameter, D, could result in reduction of thermal erformance. D d F h H NOMENCLATURE Dimensionless mean bead diameter, D=d/ h Mean bead diameter (m) Inertial factor Height of the channel Dimensionless height of the orous medium, H=h/h h x Local heat transfer coefficient (Wm 2 K 1 ) K Permeability (m 2 ) k Thermal conductivity (Wm 1 K 1 ) L Dimensionless length N(a,b 2 ) Normal distribution with mean a and standard deviation b Nu Nusselt number Pr Prandtl number, Pr=ν/α Pressure (Nm 2 ) q x Local heat flux (Wm 2 ) Re Reynolds number, Re=ρu o h/µ T Temerature (K) U, V Dimensionless velocity in the X- and Y- direction, resectively u o Maximum inlet velocity (msec 1 ) X, Y Dimensionless Cartesian coordinates Greek Symbols α Thermal diffusivity (m 2 sec 1 ) ε Mean orosity ξ Random variable ν Kinematic viscosity (m 2 sec 1 ) θ Dimensionless temerature ρ Fluid density (kgm 3 ) σ ε Standard deviation of orosity Φ Comutational variable ψ Dimensionless stream function Suerscrits n Subscrits e f i s w The nth iteration index Mean value Velocity vector Effective value External flow field Index; inlet Porous medium Solid block Hot wall ACKNOWLEDGMENT The suort of this work by National Science Council, Taiwan, R.O.C. under contract NSC E is gratefully acknowledgment. REFERENCES 1. Benenati, R. F., and Brosilow, C. B., 1962, Void Fraction Distribution in Packed Beds, American Institute of Chemical Engineers Journal, Vol. 8, Catton, I., Georgiads, J. G., and Adnani, P., 1988, The Imact of Nonlinear Convective Processes on Transort Phenomena in Porous Media, Proceedings of 1988 National Heat Transfer Conference, Vol. 1, ASME HTD-96, Houston, Cheng, P., Chowdhury, A., and Hsu, C. T., 1991, Forced Convection in Packed Tubes and Channels with Variable Porosity and Thermal Disersion Effects, In Convective Heat Mass Transfer in Porous Media, ed. S. Kakac et al., Fu, W. S., and Huang, H. C., 1999, Effects of a Random Porosity Model on Heat Transfer Performance of Porous Media, International Journal of Heat and Mass Transfer, Vol. 42, Georgiads, J. G., and Catton, I., 1987, Stochastic Modeling of Unidirectional Fluid Transort in Uniform and Random Packed Beds, The

9 438 Journal of the Chinese Institute of Engineers, Vol. 24, No. 4 (2001) Physics of Fluids, Vol. 30, Hadim, A., 1994, Forced Convection in a Porous Channel with Localized Heat Sources, Journal of Heat Transfer, Vol. 116, Hsu, C. T., and Cheng, P., 1990, Thermal Dissiation in a Porous Medium, International Journal of Heat and Mass Transfer, Vol. 33, Georgiads, J. G., 1991, Effect of Randomness on Heat and Mass Transfer in Porous Media, In Convective Heat Mass Transfer in Porous Media, ed. S. Kakac et al., Roblee, L. H. S., Baird, R. M., and Tiern, J. W., 1958, Radial Porosity Variations in Packed Beds, American Institute of Chemical Engineers Journal, Vol. 4, Saito, A., Okawa, S., Suzuki, T., and Maeda, H., 1995, Calculation of Permeability of Porous Media Using Direct Simulation Monte Carlo Method (Effect of Porosity and Void Distribution on Permeability), Proceedings of ASME/JSME Thermal Engineering Joint Conference, Vol. 3, Maui, Schwartz, C. E., and Smith, J. M., 1952, Flow Distribution in Packed Beds, Industrial and Engineering Chemistry, Vol. 45, Shonnard, D. R., and Whitaker, S., 1989, The Effective Thermal Conductivity for Point Contact Porous Medium: An Exerimental Study, International Journal of Heat and Mass Transfer, Vol. 23, Vafai, K., 1984, Convective Flow and Heat Transfer in Variable-Porosity Media, Journal of Fluid Mechanics, Vol. 147, Vafai, K., and Tien, C. L., 1981, Boundary and Inertia Effects on Flow and Heat Transfer in Porous Medium, International Journal of Heat and Mass Transfer, Vol. 24, Kinderman-Ramage 0.5 Manuscrit Received: Jul, 17, 2000 Revision Received: Nov. 24, 2000 and Acceted: Dec. 04, 2000