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Macro-scale heat transfer in periodically developed flow through isothermal solids

Published online by Cambridge University Press:  04 September 2015

G. Buckinx  and
M. Baelmans
G. Buckinx*
Affiliation:
KU Leuven, Department of Mechanical Engineering, Celestijnenlaan 300A, 3001 Leuven, Belgium
M. Baelmans
Affiliation:
KU Leuven, Department of Mechanical Engineering, Celestijnenlaan 300A, 3001 Leuven, Belgium
*
†Email address for correspondence: geert.buckinx@kuleuven.be
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Abstract

This paper presents spatially averaged temperature equations for modelling macro-scale heat transfer in periodic solid structures such as fin and tube arrays. The governing equations for the periodically developed heat transfer regime in isothermal solids are derived. It is shown that the appropriate macro-scale temperature in the periodically developed heat transfer regime is obtained by averaging the temperature with a specific weighting function which is adapted to the temperature decay rate. This matched weighting function allows the representation of the macro-scale interfacial heat transfer and thermal dispersion source by means of a spatially constant interfacial heat transfer coefficient and thermal dispersion vector, which both can be calculated from the periodic rescaled temperature on a unit cell of the solid structures. Moreover, it is proved that for small temperature decay rates, the matched weighting function yields the same macro-scale description as repeated volume averaging. The theoretical framework of this paper is applied to a case study, describing the heat transfer between a fluid and an array of solid cylinders at constant temperature.

JFM classification

Convection: Convection in porous media Mathematical Foundations: General fluid mechanics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 780 , 10 October 2015 , pp. 274 - 298
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Copyright
© 2015 Cambridge University Press 

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References

Alshare, A. A., Strykowski, P. J. & Simon, T. W. 2010a Modeling of unsteady and steady fluid flow, heat transfer and dispersion in porous media using unit cell scale. Intl J. Heat Mass Transfer 53 (9–10), 2294–2310.Google Scholar
Alshare, A. A., Strykowski, P. J. & Simon, T. W. 2010b Simulations of flow and heat transfer in a serpentine heat exchanger having dispersed resistance with porous-continuum and continuum models. Intl J. Heat Mass Transfer 53 (5–6), 1088–1099.Google Scholar
Baveye, P. & Sposito, G. 1984 The operational significance of the continuum hypothesis in the theory of water movement through soils and aquifers. Water Resour. Res. 20 (5), 521–530.Google Scholar
Buckinx, G. & Baelmans, M. 2015 Multi-scale modelling of flow in periodic solid structures through spatial averaging. J. Comput. Phys. 291, 34–51.Google Scholar
Catton, I. 2006 Transport phenomena in heterogeneous media based on volume averaging theory. Heat Mass Transfer 42 (6), 537–551.Google Scholar
Davit, Y., Debenest, G., Wood, B. D. & Quintard, M. 2010 Modeling non-equilibrium mass transport in biologically reactive porous media. Adv. Water Resour. 33 (9), 1075–1093.Google Scholar
DeGroot, C. T. & Straatman, A. G. 2011 Closure of non-equilibrium volume-averaged energy equations in high-conductivity porous media. Intl J. Heat Mass Transfer 54 (23–24), 5039–5048.Google Scholar
Gagnon, R. J. 1970 Distribution theory of vector fields. Am. J. Phys. 38 (7), 879–891.Google Scholar
Haselbacher, A. & Blazek, J. 2000 Accurate and efficient discretisation of Navier–Stokes equations on mixed grids. AIAA J. 38 (11), 2094–2102.Google Scholar
Hassanizadeh, M. & Gray, W. G. 1979 General conservation equations for multi-phase systems: 1. Averaging procedure. Adv. Water Resour. 2 (C), 131–144.CrossRefGoogle Scholar
Horvat, A. & Catton, I. 2003 Numerical technique for modeling conjugate heat transfer in an electronic device heat sink. Intl J. Heat Mass Transfer 46 (12), 2155–2168.Google Scholar
Horvat, A. & Mavko, B. 2005 Hierarchic modeling of heat transfer processes in heat exchangers. Intl J. Heat Mass Transfer 48 (2), 361–371.Google Scholar
Hsu, C. T. & Cheng, P. 1990 Thermal dispersion in a porous medium. Intl J. Heat Mass Transfer 33 (8), 1587–1597.Google Scholar
Koch, D. L. & Ladd, A. J. C. 1997 Moderate Reynolds number flows through periodic and random arrays of aligned cylinders. J. Fluid Mech. 349, 31–66.Google Scholar
Kuwahara, F., Shirota, M. & Nakayama, A 2001 A numerical study of interfacial convective heat transfer coefficient in two-energy equation model for convection in porous media. Intl J. Heat Mass Transfer 44, 1153–1159.Google Scholar
Leroy, V., Goyeau, B. & Taine, J. 2013 Coupled upscaling approaches for conduction, convection, and radiation in porous media: theoretical developments. Transp. Porous Med. 98, 323–347.Google Scholar
Marle, C. M. 1965 Application de la méthode de la thermodynamique des processus irréversible à l’écoulement d’un fluide à travers un milieux poreux. Bull. RILEM 29, 1066–1071.Google Scholar
Marle, C. M. 1967 Ecoulements monophasiques en milieu poreux. Rev. Inst. Fr. Pét. 22 (10), 1471–1509.Google Scholar
Nakayama, A., Kuwahara, F. & Hayashi, T. 2004 Numerical modelling for three-dimensional heat and fluid flow through a bank of cylinders in yaw. J. Fluid Mech. 498, 139–159.Google Scholar
Nakayama, A., Kuwahara, F. & Kodama, Y. 2006 An equation for thermal dispersion flux transport and its mathematical modelling for heat and fluid flow in a porous medium. J. Fluid Mech. 563, 81–96.Google Scholar
Nakayama, A., Kuwahara, F., Umemoto, T. & Hayashi, T. 2002 Heat and fluid flow within an anisotropic porous medium. J. Heat Transfer 124 (4), 746–753.Google Scholar
Patankar, S. V., Liu, C. H. & Sparrow, E. M. 1977 Fully developed flow and heat transfer in ducts having streamwise-periodic variations of cross-sectional area. J. Heat Transfer 99 (2), 180–186.Google Scholar
Pedras, M. H. J. & de Lemos, M. J. S. 2008 Thermal dispersion in porous media as a function of the solid–fluid conductivity ratio. Intl J. Heat Mass Transfer 51 (21–22), 5359–5367.Google Scholar
Pride, S. R. & Berryman, J. G. 1998 Connecting theory to experiment in poroelasticity. J. Mech. Phys. Solids 46 (4), 719–747.CrossRefGoogle Scholar
Quintard, M., Kaviany, M. & Whitaker, S. 1997 Two-medium treatment of heat transfer in porous media: numerical results for effective properties. Adv. Water Resour. 20 (2–3), 77–94.Google Scholar
Quintard, M. & Whitaker, S. 1994a Transport in ordered and disordered porous media I: the cellular average and the use of weighting functions. Transp. Porous Med. 14 (2), 163–177.Google Scholar
Quintard, M. & Whitaker, S. 1994b Transport in ordered and disordered porous media II: generalized volume averaging. Transp. Porous Med. 14 (2), 179–206.Google Scholar
Quintard, M. & Whitaker, S. 2000 Theoretical analysis of transport in porous media. In Handbook of Heat Transfer in Porous Media (ed. Vadai, K. & Hadim, H. A.). Marcel Dekker.Google Scholar
Sahraoui, M. & Kaviany, M. 1994 Slip and no-slip temperature boundary conditions at the interface of porous, plain media: convection. Intl J. Heat Mass Transfer 37 (6), 1029–1044.Google Scholar
Schwartz, L. 1978 Théorie des Distributions. Hermann.Google Scholar
Shah, R. K. & London, A. L. 1978 Advances in Heat Transfer: Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data. Academic.Google Scholar
Stalio, E. & Piller, M. 2006 Direct numerical simulation of heat transfer in converging–diverging wavy channels. J. Heat Transfer 129 (7), 769–777.CrossRefGoogle Scholar
Teruel, F. E. & Díaz, L. 2013 Calculation of the interfacial heat transfer coefficient in porous media employing numerical simulations. Intl J. Heat Mass Transfer 60, 406–412.Google Scholar
Traoré, Ph., Ahipo, Y. M. & Louste, Ch. 2009 A robust and efficient finite volume scheme for the discretization of diffusive flux on extremely skewed meshes in complex geometries. J. Comput. Phys. 228, 5148–5159.CrossRefGoogle Scholar
Vafai, K. & Tien, C. L. 1981 Boundary and inertia effects on flow and heat transfer in porous media. Intl J. Heat Mass Transfer 24 (2), 195–203.Google Scholar
Whitaker, S. 1999 The Method of Volume Averaging (Theory and Applications of Transport in Porous Media). Kluwer Academic.Google Scholar
Zhou, F. & Catton, I. 2013 Obtaining closure for a plane fin heat sink with elliptic scale-roughened surfaces for Volume Averaging Theory (VAT) based modeling. Intl J. Therm. Sci. 71, 264–273.Google Scholar
Zhou, F., Hansen, N. E., Geb, D. J. & Catton, I. 2011 Obtaining closure for fin-and-tube heat exchanger modeling based on volume averaging theory (VAT). J. Heat Transfer 133 (11), 111802.Google Scholar