Proceedings of the ASME 2017 Fluids Engineering Division Summer Meeting FEDSM2017 July 30-August 3, 2017, Waikoloa, Hawaii, USA FEDSM2017-69582 NUMERICAL STUDY OF HEAT TRANSFER IN TURBULENT CROSS-FLOW OVER POROUS-COATED CYLINDER N. Rahmati Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran Z. Mansoori Energy Research Center, Amirkabir University of Technology, Tehran, Iran M. Saffar-Avval Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran G. Ahmadi Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY, USA engineering applications [1,2]. The properties of porous media such as high thermal conductivity, high effective surface area, and increased fluid mixing has led to the use of porous wrapping as an effective tool to enhance heat transfer [3]. In recent years, several researchers have studied porouscoated cylinders for enhancing heat transfer. However, these efforts have been limited mostly to laminar flows over porous layer. For example, Odabaee et al. [4] studied forced convection heat transfer from a cylinder embedded inside a metal foam layer numerically, where the laminar flow over and inside the porous medium was assumed. They evaluated the optimum porous layer thickness for maximum heat transfer enhancement, and showed that the porous wrapped cylinder has higher heat transfer rate compared to the fin-tubed heat exchanger. Saada et al. [5] investigated the natural convection heat transfer around a horizontal porous-covered cylinder and assumed that the fluid flow over and within the porous layer is laminar. Bhattacharyya and Singh [1] conducted a numerical study of the laminar flow around a porous layer wrapped cylinder, where the mixed convection is studied for different values of flow parameters. Al-Sumaily et al. [3] investigate numerically the forced convection heat transfer from a porous wrapped cylinder. In many cases the cross-flow over porous-covered cylinder is in turbulent regime and the fluid flow in the porous layer could be laminar or turbulent depending on porosity properties. In [6,7] the turbulent fluid flow in porous layer was modeled and the corresponding flow configuration and heat transfer ABSTRACT In the present paper, a numerical study has been conducted to investigate the heat transfer from a constant temperature cylinder covered with metal foam. The cylinder is placed horizontally and is subjected to a constant mean cross-flow in turbulent regime. The Reynolds Averaged Navier-Stokes (RANS) and Darcy-Brinkman-Forchheimer equations are combined and used for flow analysis. The energy equation used assumes local thermal equilibrium between fluid and solid phases in porous media. The k-ω SST turbulence model is used to evaluate the eddy viscosity that is implemented in the momentum and energy equations. The flow in the metal foam (porous media) is in laminar regime. Governing equations are solved using the finite volume SIMPLEC algorithm. The effect of thermophysical properties of metal foam such as porosity and permeability on the Nusselt number is investigated. The results showed that using a metal porous layer with low porosity and high Darcy number in high Reynolds number turbulent flows markedly increases heat transfer rates. The corresponding increase in the Nusselt number is as high as 10 times that of a bare tube without the metal foam. INTRODUCTION Heat transfer from a horizontal cylinder covered with a porous layer in cross flows has received considerable attention over the last several decades. Design of compact heat exchanger, effective cooling of electronic systems, geothermal energy system, catalytic reactors, and refrigeration are a few 1 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use Reynolds number, 2 f U Rs f Re t Time, (s) Temperature, (K) T u Radial velocity component U Free stream velocity, (m/s) v Tangential velocity component Greek symbols Porosity were investigated. In most situations, a composite porous-fluid system is used where the flow in the clear fluid region is turbulent while in the porous layer the flow remains in laminar regime. For example, Sobera et al. [8] conducted a numerical study for heat and mass transfer in turbulent flow from a circular cylinder sheathed by a porous layer with applications in human clothing. Kuznetsov [9] investigated the turbulent flow in a composite porous-fluid duct and reported that the roughness of the porous-fluid interface significantly impacts the turbulent flow regime in the clear fluid region as well as the overall heat transfer in the duct. In this analysis, a roughness related to the pore diameters of the porous layer was applied at the interface. In the present study, a computational model was used and the convective heat transfer under turbulent flow regime from a porous-coated cylinder was studied. Here it is assumed that the flow in the porous region remains laminar, and the surface roughness at the porous-fluid interface was accounted for in the boundary condition of the clear fluid turbulent flow equations. The SST k-ω turbulence model, which is suitable for predicting the separation of flow over the porous-fluid interface, was used in the simulations. The simulations results showed that the porous surface wrapping significantly increase the heat transfer. p Porous layer thickness f Dynamic viscosity, (kg/ms) t t , n Turbulent viscosity, (kg/ms) Dimensionless turbulent viscosity, (kg/ms) c p eff 1 Density, (kg/m3) Effective thermal capacity of the porous 1 c ,(J/m3K) Specific heat ratio, c c medium, c p s f p eff p f Tangential coordinate Tangential angle measured from the forward stagnation point Subscripts f p s NOMENCLATURE c CF cp Forchheimer coefficient Cp Pressure coefficient Da e F kf Darcy number Binary parameter for porosity Binary parameter Thermal conductivity of fluid, (W/mK) Effective thermal conductivity of porous material, k f 1 k s , (W/mK) keff K k Nu Nuave Nu0 P Pr R Solid specific heat, J/kg K Free stream Fluid Porous medium Solid Specific heat, (J/kgK) GOVERNING EQUATIONS In this section, the governing equations for turbulent flow over a porous-coated cylinder are described. Fig. 1 shows the computational domain for the flow over a porous-coated cylinder that is is placed horizontally in a uniform cross flow. The outer boundary condition is set at R∞=30Rp. The tube surface temperature is kept constant at a temperature that is higher than the ambient temperature (T∞). It is assumed that the porous media is isotropic, and the fluid and solid phases in the porous media are locally at thermal equilibrium. Permeability of the metal foam, (m2) Turbulent kinetic energy, (m2/s2) Local Nusselt number Total average Nusselt number Total average Nusselt number of a solid cylinder pressure, (N/m2) Prandtl number, c p f k f Rs Radial coordinate Radius of cylinder, (m) Rc Thermal conductivity ratio, keff k f Fig. 1 A schematic of computational domain and fluid flow in the cylindrical coordinate. 2 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use In this paper, we use the single-domain approach to model the composite system of the clear fluid zone and porous medium. Hence, a single set of equations is used in the computations to model the fluid flow in the porous medium as well as the fluid region. The momentum equations are generated by combining the Reynolds Averaged Navier-Stokes (RANS) and Darcy-Brinkman-Forchheimer equations [10]. Also, other equations are obtained by combining the governing equations in the porous medium and fluid region. The final non-dimensional equations can be expressed as: ru r v 0 T 1 ruT 1 vT t r r r T 1 T Pr Pr Rc t ,n (4) r Rc t ,n t t 2 1 r 1 r r r RePr r where t =0.85. The non-dimensional parameters in equations (1)-(4) are: 2 f U Rs cp f K Re , Da 2 , Pr , f kf Rs (5) c eff keff t ,R , c p c k f t ,n f (1) ru 2 uv u 1 e 1 e v2 t r r r er f The switching parameters in these equations are given in Table1. The SST k-ω turbulence model developed by Menter [11] is used for evaluation of turbulence properties including the nondimensional eddy viscosity in the fluid region. The transport equations for k and ω are not shown here for the sake of brevity. 1 t ,n u r 1 u t n , r P 2 1 r 1 r Re r r r v ( ) 1 u 1 r r 1 t ,n r r r 1 t ,n r 2 r r e Re 2 1 t ,n 1 v u r r r C e 2e F u F u 2 v2 u Da Re Da Table 1. The switching parameters in clear fluid region and fluid flow in porous medium Fluid flow in porous Fluid region F 1 0 1 e (2) ruv v2 v 1 e 1 e uv t r r r er ( cp )eff c p f 1 cs c keff k f 1 ks kf t 0 t p f NUMERICAL METHOD The set of governing equations is discretized on the staggered grid by using finite volume method. A pressure correction-based iterative algorithm, SIMPLEC (Van Doormal and Raithby) is used for solving the governing equations. The convective and diffusion terms are discretized through a power law differencing scheme. For convergence criteria, the maximum relative errors on the dependent variables are set to 10-5. A non-uniform grid distribution along the radial direction and uniform grid distribution along the tangential direction is generated. The grid independence study is examined by considering four sets of non-uniform grids, namely, 197×100 (grid 1), 259×140 (grid 2), 293×200 (grid 3), and 350×220 (grid 4). Here the first and second numbers being the number of grid points, respectively, in r- and θ-directions. The minimum cell size in each grid in r-direction is 0.3mm, 0.1mm, v v 1 r 1 t ,n 1 t ,n 1 P 2 1 1 r r r r Re r r u v r 1 t ,n 1 t ,n v r r 1 1 2u r r r r 2 v e Re 1 t ,n r u r r r r 2e C e F v F (u 2 v2 v Da Re Da Parameter (3) 3 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use ratio for the cylinder with metal foams is in the order of 4-10, which represents significant augmentation of the heat transfer by using a porous layer over the cylinder. 0.05 mm and 0.025 mm. The difference between the average Nusselt number in grid 3 and grid 4 is less than 0.3%. Hence, grid 3 with the minimum cell size of 0.05 mm in r-direction is selected and used for the computation. To check the validity of the numerical methods, the predicted local Nusselt number for a bare tube for laminar flow at Reynolds number of Re=20 are compared with the earlier results of Denis and Chang [12] in Fig. 2. This figure shows that good agreement of the present predictions of the local Nusselt number with the earlier results. The predicted average Nusselt number of a bare tube for different Reynolds numbers in turbulent flow regime are compared with the empirical correlations presented by Incropera et al. [13] in Table 2. This table shows that the model predictions are in reasonable agreement with the data. Fig. 3 Pressure coefficient distributions at Re=106. Fig. 2 Local Nusselt numbers for a bare cylinder with those at Re=20. Fig. 4 Average Nusselt number ratio (Nu/Nu0) versus Re number. Table 2. Comparison of the predicted average Nusselt numbers for a bare cylinder with the empirical models of Incropera et al. [13], at different Reynolds number. Re N u a v e (our results) N u a v e [1 3 ] 104 105 5×105 Relative error (%) 58.35 229.1 670.8 53.84 216.19 701.88 8.38 5.97 4.43 The effect of variations of Darcy number on the average Nusselt number are also studied and the results for Re=104 are shown in Fig. 5. It is seen that the Nusselt number increases with increasing the Darcy number. Foams with low Darcy numbers behave the same as a solid layer because low Darcy number represents low permeability. For Re=104, Fig. 5 shows that there no change in the heat transfer by reducing the Darcy number from 10-6 to 10-8. One key parameters is the porosity of porous layer. The porosity affects the Darcy resistance in the momentum equations and also the effective thermal conductivity of porous material (keff). The effect of porosity on the velocity is small, but the effective thermal conductivity significantly changes with the porosity. Fig. 6 shows that increasing the porosity reduces the effective thermal conductivity, and decreasing the porosity enhances the heat transfer. The predicted pressure coefficient distributions for Re=106 are plotted in Fig. 3 and are compared with the earlier numerical simulations of Ong et al. [14]. It is seen that the present results are in conformity with the earlier works. RESULTS AND DISCUSSION In this section, the effects of key parameters of fluid and metal foam layer such as Reynolds number, Darcy number, and porosity of porous layer on the heat transfer are evaluated and discussed. Fig. 4 shows the variations of the average Nusselt number ratio (Nu/Nu0) with flow Reynolds number for a porous wrapped cylinder with Da=10-4. Here metal foam with the following properties is used: porosity=0.9, dimensionless thickness=1.0, CF=0.167. It is seen that the Nusselt number CONCLUSIONS In this study, we investigated the heat transfer from a horizontal cylinder covered with the metal foam layer under 4 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use turbulent flow regime with constant temperature. The RANS equations with the SST k-ω turbulence model in conjunction with the single-domain approach were used in the simulations. It was assumed that the fluid flow in porous medium remained in laminar regime. The finite volume method with SIMPLEC algorithm was used for numerical integration of the governing equations, and the computation model was validated by comparison of the model predictions with the earlier results and empirical equations. It was shown that the presence of porous wrapping significantly augmented the heat transfer from the cylinder. The corresponding Nusselt number increase to almost 10 times that of a bare tube. Effects of thermophysical properties of foam layer including the Darcy number and porosity on the Nusselt number were also studied. [3] [4] [5] [6] [7] [8] Fig. 5 Average Nusselt number versus log(Da) at Re=104. [9] [10] [11] [12] [13] Fig. 6 Average Nusselt number versus porosity at Re=104, Da=10-4. [14] REFERENCES Bhattacharyya, [1] [2] Brinkman model for the mixed convection boundary layer flow past a horizontal circular cylinder in a porous medium. International Journal of Heat and Mass Transfer, 46(17), pp. 3167-3178. Al-Sumaily, G.F., Nakayama, A., Sheridan, J. and Thompson, M.C., 2012. The effect of porous media particle size on forced convection from a circular cylinder without assuming local thermal equilibrium between phases. International Journal of Heat and Mass Transfer, 55(13), pp. 3366-3378. Odabaee, M., Hooman, K. and Gurgenci, H., 2011. Metal foam heat exchangers for heat transfer augmentation from a cylinder in cross-flow. Transport in Porous Media, 86(3), pp. 911-923. Saada, M.A., Chikh, S. and Campo, A., 2007. Natural convection around a horizontal solid cylinder wrapped with a layer of fibrous or porous material. International journal of heat and fluid flow, 28(3), pp. 483-495. Pedras, M.H. and de Lemos, M.J., 2001. Macroscopic turbulence modeling for incompressible flow through undeformable porous media, International Journal of Heat and Mass Transfer, 44(6), pp. 1081-1093. Silva, R.A. and de Lemos, M.J., 2003. Turbulent flow in a channel occupied by a porous layer considering the stress jump at the interface. International Journal of Heat and Mass Transfer, 46(26), pp. 5113-5121. Sobera, M.P., Kleijn, C.R., Van den Akker, H.E. and Brasser, P., 2003. Convective heat and mass transfer to a cylinder sheathed by a porous layer. AIChE Journal, 49(12), pp. 3018-3028. Kuznetsov, A.V., 2004. Numerical modeling of turbulent flow in a composite porous/fluid duct utilizing a two-layer k–ε model to account for interface roughness. International journal of thermal sciences, 43(11), pp. 1047-1056. Nield, D.A. and Bejan, A., 2006. Convection in porous media. Springer Science & Business Media. Menter, F.R., 1994. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA journal, 32(8), pp. 1598-1605. Dennis, S. C. R., and Chang, G.Z., 1970. Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100. Journal of Fluid Mechanics, 42(03). pp. 471-489. Incropera, F.P., DeWitt, D.P., Bergman, T.L. and Lavine, A.S., 1985. Introduction to conduction. Introduction to Heat Transfer, C. Robichaud, Ed. John Wiley & Sons Inc, New York, pp.43-72 Ong, M.C., Utnes, T., Holmedal, L.E., Myrhaug, D. and Pettersen, B., 2009. Numerical simulation of flow around a smooth circular cylinder at very high Reynolds numbers. Marine Structures, 22(2), pp. 142-153. S. and Singh, A.K., 2009. Augmentation of heat transfer from a solid cylinder wrapped with a porous layer. International Journal of Heat and Mass Transfer, 52(7), pp. 1991-2001. Nazar, R., Amin, N., Filip, D. and Pop, I., 2003. The 5 Copyright © 2017 ASME Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 10/25/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use

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