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1 Journal of Porous Media 7(3), (2004) )OXLG)ORZDQG+HDW7UDQVIHUDURXQG&LUFXODU &\OLQGHUVLQWKH3UHVHQFHDQG1R3UHVHQFH RI 3RURXV 0HGLD Mohammad Layeghi and Ali Nouri-Borujerdi School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran ABSTRACT Steady-state laminar and incompressible fluid flow and forced-convection heat transfer from a circular cylinder and an array of circular cylinders in the presence and no-presence of porous media are investigated. Various mathematical and numerical models are compared and the effects of porous media on heat transfer enhancement are studied. Navier Stokes equations are used for the analysis of laminar fluid flow and heat transfer. However, the Darcy and extended Darcy Brinkman models are used for the analysis of fluid flow and heat transfer in porous media. The cylinders are at constant temperature and the analysis is restricted to the low- and intermediate-peclet-number regimes (Pr = 1, Re 40 for a single cylinder and Re 300 for an array of cylinders), and high Darcy numbers (10 4 /4 Da 10 2 ). The governing equations are discretized using the finite-volume approach based on collocated and staggered grids. The first-order upwind and power-law schemes and also the QUICK scheme are used in the numerical solutions. Parametric studies have been done for better understanding of the porous-medium effects on the Nusselt-number distribution, and flow, temperature, and pressure fields around a single cylinder and an array of cylinders. The results are compared with available numerical and experimental data in the literature and show good agreement in all compared cases. Received February 17, 2003; Accepted July 24, 2003 Copyright 2004 Begell House, Inc. 239

2 240 Layeghi and Nouri-Borujerdi 1. INTRODUCTION Analysis of fluid flow and heat transfer around a circular cylinder or an array of cylinders embedded in a porous medium has great practical importance in many engineering applications. These applications include the utilization of geothermal energy, the control of pollutant spread in groundwater, as well as nuclear engineering (reactor safety analysis) and combustion (Golombok et al., 1990), compact heat exchangers (Zukauskas, 1987), solar power collectors, high-performance insulation for buildings, and many more. Moreover, the study of flows in the interstitial spaces of a periodic array of cylinders can provide important clues for the modeling of convective transport in idealized, two-dimensional, fully saturated packed beds (Wang and Georgiadis, 1996). In many of these applications, such as in compact heat exchangers, heat transfer enhancement from a cylinder or an array of cylinders is a very important subject. The enhancement of heat transfer from a solid substrate can be achieved by covering its surface with an appropriate porous material. Previous experience has shown that all porous materials are not appropriate for this purpose (Koh and Colony, 1974; Koh and Stevens, 1975; Vafai and Kim, 1990; Nield and Bejan, 1999). In general, the efficiency of a porous medium in heat transfer enhancement from a solid substrate depends on the porous medium structure, thermophysical properties, and fluid flow conditions. Fibrous materials and metal foams (Phanikumar and Mahajan, 2002) saturated with a fluid can serve as an effective medium for augmenting forced-convection heat transfer. There are a number of effective methods for enhancement of heat transfer from an array of cylinders. Ohadi et al. (1994) studied the electrostatic enhancement of heat transfer in a tube-bundle gas-to-gas heat exchanger. Watterson et al. (1999) discussed the effects of turbulence in a tube bundle. However, there are a number of other methods, such as utilizing vortex generators, oval tubes, pulsating flows, etc., which can be used to enhance the heat transfer from a tube bundle. Many different theoretical and experimental models have been used for the analysis of fluid flow and heat transfer in a porous medium. The Darcy model proposed by Henri Darcy (1856) is one of the basic models which have been used extensively in the literature. Following Wooding (1957), many early authors on convection in porous media used various types of extended Darcy models. Neild and Bejan (1999) published many of these extended models. The most important studies were those of Sano (1980), Cheng (1982), Bejan (1984), Kimura (1988), and Kimura (1989). The incompressible fluid flow and heat transfer around a circular cylinder have also been studied by many researchers. In many of these studies, the fluid properties are assumed constant and the flow field is decoupled from the temperature field. As a consequence, the flow field can be solved independently. The most important studies were those of White (1974), Fornberg (1980), Dennis et al. (1968), Dennis and Chang (1970), Chang and Finlyson, (1987), and Chang et al. (1987). The predicted results have also been compared with available experimental data in some cases. There is a growing body of literature concerning hydrothermal fields over isothermal, in-line arrays of cylinders immersed in crossflow. The numerical study of the fluid flow past tube bundles can be traced back to Thom and Apelt (1961). They used the conformal mapping technique to transform the boundaries of the staggered tubes to coincide with equivalued surfaces of one of the independent variables. The most important later works were those of Le Feuvre (1973), Launder and Massey (1978), Ho and Chen (1983), Antonopoulos (1985), Dhaubhadel et al. (1986), Wung and Chen (Parts I and II, 1989), and Li and Chen (1990). In these numerical studies, it was frequently assumed that the flow was steady and symmetric with respect to the centerline of the cylinder in the same row, at least for small to intermediate Reynolds numbers. Other experimental and theoretical works concerning forced-convection heat transfer past an array of circular cylinders in the presence and no-presence of porous media were carried out by Jubran et al. (1993), Fand et al. (1993), Bejan (1995), Massarotti et al. (1998), Manzari (1999), Wang (1999), and Wang (2001). In this article, the classical problem of fluid flow and heat transfer around a single circular cylinder and an array of circular cylinders at low and intermediate Peclet numbers is investigated. The main objective of this work is the investigation of the effects of porous media on heat transfer enhancement from a circular cylinder and an array of circular cylinders. The numerical results are compared with available numerical and experimental data in the literature for various ranges of Peclet and Darcy numbers. The differences between various models are also discussed.

3 Fluid Flow and Heat Transfer around Circular Cylinders GOVERNING EQUATIONS Two sets of governing equations are used in the numerical analysis. The first set includes the momentum and energy equations, which are two-dimensional Navier Stokes equations, and the extended Darcy Brinkman equations, in a Cartesian coordinate system. The second set of governing equations includes the Darcy model and the two-dimensional Navier Stokes equations in polar coordinates. The main assumptions in the present work are summarized as follows: 1. The flow is steady and incompressible. 2. The properties of the porous medium and the fluid are isotropic and homogeneous. 3. The thermophysical properties of the fluid and the porous matrix are assumed to be constant. 4. The effects of viscous dissipation and thermal dispersion are neglected. 5. Near-wall porosity variation is neglected. 6. Local thermal equilibrium (LTE) condition is assumed (Amiri and Vafai, 1994; Kim and Jung, 2002). 7. The inertia effect (Forchheimer term) is neglected (Nield and Bejan, 1999) Laminar-Flow Model (Cartesian) The steady, laminar motion of an incompressible fluid around a circular cylinder or an array of cylinders in presence and no-presence of a porous medium in a Cartesian coordinate system is described by the following equations (Fig. 1). Continuity: V i x i = 0 (1) Momentum: (ρ f V i V j ) = p + τ ij + S i (2) x j x i x j Energy: (ρ f V i T) = k f T x i C pf x i x (3) i where the stress tensor is given by τ ij = µ V i x j + V j x i (4) and ρ f and µ are the fluid medium density and viscosity, respectively, V i is the velocity vector (V x, V y ), T is the temperature, p is the static pressure, and k f and C pf are the fluid medium thermal conductivity and heat capacity, respectively. For the case of laminar flow around a circular cylinder or an array of cylinders in the absence of porous material, the source term in Eq. (2) is zero Extended Darcy Brinkman Model (Cartesian) For a circular cylinder or an array of cylinders embedded in a porous medium, the formal shape of the momentum equations remains the same as the above equations except that the following source term should be added to the momentum equation: Figure 1. Schematic of the computational domains and coordinate system: (a) a single cylinder; (b) an array of cylinders.

4 242 Layeghi and Nouri-Borujerdi S i = µ K V i i = x, y (5) where V x and V y in a porous medium are in fact volumeaveraged velocity components (Bejan, 1984) and K is the permeability of the porous medium. These source terms can be used in low-peclet-number regimes and need to be modified for high-peclet-number regimes. This model is called the extended Darcy Brinkman model. The energy equation for this model is slightly different from Eq. (3) and can be written as (ρ f V i T) = k eff T x i C pf x i x + S T (6) i where the source term S T is zero for a flow with negligible viscous dissipation, k eff is the effective thermal conductivity in the porous medium and is the volume average of the fluid-medium and solid-medium conductivities: k eff = φk f + (1 φ)k s (7) where φ is the porosity of the porous medium and k s is the solid-medium thermal conductivity Darcy Model (Polar) In this classical porous-medium model, the steady-state momentum and energy equations in a polar coordinate system can be written as (Fig. 2) 1 R R (Ru) + 1 R u = K P Re 2 D R v ω = 0 (8) (9a) v = K D 2 Re 1 R P ω (9b) 1 J R R R + 1 J ω R ω = 0 (10) where the nondimensional parameters in Eqs. (8) (10) are defined as R = r D θ = T T T w T v = V ω p P = U 2 ρ f U u = V r U The overall radial and circumferential fluxes are J R = R uθ 1 θ Pe R J ω = R vθ 1 1 θ Pe R ω (11a) (11b) In the above equations, r and ω are the radial and angular coordinates, and V r and V ω are the radial and circumferential velocities, respectively; T is temperature, p is pressure, R is the nondimensional radial coordinate, D is the cylinder diameter, and u and v are nondimensional radial and circumferential velocities, respectively; θ and P are the nondimensional temperature and pressure, and T and U are the free-stream temperature and velocity, respectively; T w is the surface temperature of the cylinder, Pe = U D α is the Peclet number, Re = U D ν f is the Reynolds number, and ν f is the kinematic viscosity of the fluid in the porous medium. The effective thermal diffusivity, α, is defined as α = k eff ρ f C pf (12) Figure 2. Fluid flow around a cylinder embedded in a porous medium, together with coordinate system. In this model, the effects of porous media on the fluid flow and heat transfer around a circular cylinder are investigated. The results predicted by this model are compared with the laminar-flow model results. The most important parameters which, along with the Peclet number, are used to describe the results, are the Darcy number, Da = K D 2, and porosity, φ (which is the ratio of pore volume to the total volume of a porous medium).

5 Fluid Flow and Heat Transfer around Circular Cylinders Laminar-Flow Model (Polar) The conservative form of the continuity, momentum, and energy equations in the polar coordinate system can be written as (Fig. 3) 1 r r rj Φ,r + 1 r ω rj Φ,ω = S Φ Φ = V r, V ω, T J Φ,r = ρv r Φ Γ Φ Φ r J Φ,ω = ρv ω Φ Γ Φ 1 r Φ ω (13a) (13b) (13c) Figure 3. Fluid flow around a cylinder, together with coordinate system. The general variables in the above equations are listed in Table 1. In the above equations, r and ω are the radial and angular coordinates, and V r and V ω are the radial and circumferential velocities, respectively; T is temperature, and p is pressure of the fluid. 3. TEST CASES The numerical analysis consists of two distinct parts. In the first part, four test cases are used for the analysis of fluid Table 1 General variables for various conservation equations Equation Φ Γ Φ J Φ,r JΦ,ω S Φ Continuity 1 0 ρ f V r ρ f V ω 0 r Momentum V r µ ρ f V r V r µ V r r ω Momentum V ω µ ρ f V r V ω µ V ω r Energy T k f C pf ρ f V r T k f C pf T r ρ f V ω V r µ 1 r ρ f V ω V ω µ 1 r V r ω V ω ω ρ f V ω T k f 1 T C pf r ω ρ f r V ω 2 µ r 2 V r 2 r 2 µ V ω ω p r ρ f r V ωv r µ r 2 V ω + 2 r 2 µ V r ω 1 r 0 p ω Table 2 Specifications for the test cases used in the numerical solution Test case no. Configuration Coordinate system Medium surrounding cylinder/cylinders Model name 1 A single cylinder Cartesian Nonporous (fluid) Laminar flow 2 A single cylinder Cartesian Porous Extended Darcy Brinkman 3 A single cylinder Polar Porous Darcy 4 A single cylinder Polar Nonporous (fluid) Laminar flow 5 Array of cylinders Cartesian Nonporous (fluid) Laminar flow 6 Array of cylinders Cartesian Porous Extended Darcy Brinkman

6 244 Layeghi and Nouri-Borujerdi Figure 4. Nusselt-number distributions along a single cylinder surface predicted by various models. flow and heat transfer around a single cylinder in the presence and no-presence of a porous medium. In the second part, two test cases are used for the analysis of fluid flow and heat transfer around an array of cylinders in the presence and no-presence of a porous medium. These test cases are listed in Table Method of Solution The governing equations for the test cases in Table 2 together with appropriate boundary conditions are solved using the finite-volume approach based on collocated and staggered grids in Cartesian and polar coordinate systems. A segregated solver based on the SIMPLE algorithm (Patankar, 1980) is used in the numerical analysis. The details of the numerical methods can be found in the literature (Ferziger and Peric, 1999) Description of Results The effects of porous media on heat transfer enhancement around a circular cylinder and an array of cylinders have been investigated. The incompressible flow model, Darcy, and extended Darcy Brinkman models have been used in the numerical solutions. The fluid flow and heat transfer around a circular cylinder have been studied at small to intermediate Peclet numbers (Pr = 1, Re 40). As shown in Fig. 4, the effect of a porous medium on enhancement of heat transfer around a circular cylinder depends on both the Darcy and Reynolds numbers. For a single cylinder at high Darcy numbers (Da > 10 3 ), the porous medium can cause a significant increase in Nusselt number (over 80%). At low Darcy numbers (Da < 10 3 ), the porous-medium models predicted different results. The extended Darcy Brinkman model showed a decrease in Nusselt number at low Darcy numbers, 10 3 and 10 4, in some regions on the cylinder surface in comparison with high Darcy numbers. However, the Darcy model always predicts an increase in Nusselt number when porosity decreases. For the cases studied in this article, a porous medium with φ = 0.6 and K = 10 4 m 2 (which implies that Da = 10 4 for a single cylinder with D = 1 m), could be very effective for heat transfer augmentation in the range of low to intermediate Reynolds numbers (Re 40). The fluid flow and heat transfer around an array of cylinders have also been studied at small to intermediate Peclet numbers (Pr = 1, Re 300). As shown in Fig. 5, the effect of the porous medium on enhancement of heat transfer around an array of cylinders depends on both the Darcy and Reynolds numbers, much like a single cylinder. However, the effects of the porous medium on heat transfer from an array of cylinders could be more complex than those for a single cylinder embedded in a porous medium. It can be concluded that the presence of an appropriate porous medium increases the heat transfer rate from the first and second rows in comparison with the laminar-flow case. However, the porous medium increases the heat transfer from the first row more than from the second row.

7 Fluid Flow and Heat Transfer around Circular Cylinders 245 Figure 5. Nusselt-number distributions along the first- and second-row cylinder surfaces of an array of cylinders predicted by various models.

8 246 Layeghi and Nouri-Borujerdi For the cases studied in this article, a porous medium with φ = 0.6 and K = 10 4 m 2 (which implies that Da = for an array of cylinders with d = 2 m), could be very effective for heat transfer augmentation in the range of intermediate Reynolds numbers (100 Re 300), for both the first- and second-row cylinders. However, at Re 20, the use of a porous medium introduced here is not very effective for heat transfer augmentation from the first row except near the front stagnation region. It also has a negative effect on the heat transfer from the second row. The pressure drop always increases in the presence of a porous medium, which may be unfavorable in designing heat exchangers. Therefore, both pressure drop and heat transfer augmentation effects of porous media should be considered in designing heat exchangers. Numerical simulations have also shown that the efficiency of a porous material on enhancement of heat transfer from a circular cylinder or an array of cylinders depends on its structure, the thermophysical properties of the fluid and the solid medium, and the fluid flow characteristic Reynolds number. However, further numerical and experimental studies are needed to analyze the effects of porous media on enhancement or decrement of heat transfer from a single cylinder or an array of cylinders in wide ranges of Darcy and Peclet numbers. ACKNOWLEDGEMENTS The first author would like to thank Professor Kambiz Vafai at the University of California, Riverside for his efficient comments during the preparation of this article. REFERENCES Amiri, A. and Vafai, K., Analysis of dispersion effects and non-thermal equilibrium, non-darcian, variable porosity incompressible flow through porous media, Int. J. Heat Mass Transfer, vol. 37, pp , Antonopoulos, K. A., Heat transfer in tube banks under conditions of turbulent inclined flow, Int. J. Heat Mass Transfer, vol. 28, pp , Bejan, A., Convective Heat Transfer, Wiley, New York, NY, Bejan, A., The optimal spacing for cylinders in cross flow forced convection, J. Heat Transfer, vol. 117, pp , Chang, M. W. and Finlyson, B. A., Heat transfer in flow past cylinders at Re < 150-Part I. Calculations for constant fluid properties, Numer. Heat Transfer, vol. 12, pp , Chang, M. W., Finlyson, B. A., and Sleicher, C. A., Heat transfer in flow past cylinders at Re < 150 Part II. Experiments and theory for variable fluid properties, Numer. Heat Transfer, vol. 12, pp , Chen, C. H. and Weng, F. B., Heat transfer for incompressible and compressible fluid flows over a heated cylinder, Numer. Heat Transfer A, vol. 18, pp , Chen, C. J. and Wung, T. S., Finite analytic solution of convective heat transfer from tube arrays in cross flow II. Heat transfer analysis, J. Heat Transfer, vol. 111, pp , Cheng, P., Mixed convection about a horizontal cylinder and a sphere in a fluid saturated porous medium, Int. J. Heat Mass Transfer, vol. 25, pp , Darcy, H. P. G., Les fontaines publiques de la ville de Dijon, Victor Dalmont, Paris, Dennis, S. C. R. and Chang, G. Z., Numerical solution for steady flow past a circular cylinder at Reynolds numbers up to 100, J. Fluid Mech., vol. 42, pp , Dennis, S. C. R., Hudson, J. D., and Smith, N., Steady laminar forced convection from a circular cylinder at low Reynolds numbers, Phys. Fluids, vol. 11, pp , Dhaubhadel, M. N., Reddy, J. N., and Telionis, D. P., Penalty-finite element analysis of coupled fluid flow and heat transfer for in-line bundle of cylinders in crossflow, Int. J. Nonlinear Mech., vol. 21, pp , Fand, R. M., Varahasamy, M., and Greer, L. S., Empirical correlation equations for heat transfer by forced convection from cylinders embedded in porous media that account for the wall effect and dispersion, Int. J. Heat Mass Transfer, vol. 36, pp , Ferziger, J. H. and Peric, M., Computational Methods for Fluid Dynamics, Springer-Verlag, Berlin, Heidelberg, Germany, Fornberg, B. A., Numerical study of steady viscous flow past a circular cylinder, J. Fluid Mech., vol. 98, part 4, pp , Golombok, M., Jariwala, J., and Shirvill, L. C., Gas-solid heat exchange in a fiberous metallic material measured by a heat regenerator technique, Int. J. Heat Mass Transfer, vol. 33, pp , Ho, K. S. and Chen, C. J., Finite analytical solution of laminar convective heat transfer in tube bundles, ASME Paper 83- WA/HT-23, Huang, P. C. and Vafai, K., Analysis of flow and heat transfer over an external boundary covered with a porous substrate, ASME J. Heat Transfer, vol. 116, pp , Jubran, B. A., Hamdan, M. A., and Abdualh, R. M., Enhanced heat transfer, missing pin, and optimization of cylindrical pin fin arrays, ASME J. Heat Transfer, vol. 115, pp , Kim, S. J. and Jang, S. P., Effects of the Darcy number, Prandtl number, and the Reynolds number on local thermal non-equilibrium, Int. J. Heat Mass Transfer, vol. 45, pp , 2002.

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