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Macro-scale conjugate heat transfer in periodically developed flow through solid structures

Published online by Cambridge University Press:  09 September 2016

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 treats the macro-scale description of the periodically developed conjugate heat transfer regime, in which heat transfer takes place between an incompressible viscous flow and spatially periodic solid structures through a spatially periodic interfacial heat flux. The macro-scale temperature of the fluid and the solid structures are defined through a spatial averaging operator with a specific weighting function. It is shown that a double volume average is necessary in order to have a linearly changing macro-scale temperature in response to a constant macro-scale heat flux. Furthermore, with the aid of a double volume average, the thermal dispersion source, the thermal tortuosity and the interfacial heat transfer coefficient all become spatially constant in the developed regime. That way, these closure terms of the macro-scale temperature equations can be exactly determined from the periodic temperature part on a unit cell of the solid structures without taking the spatial moments of the solid into account. The theoretical derivations of this paper are illustrated for a case study describing the heat transfer between a fluid flow and an array of solid squares with a uniform volumetric heat source.

JFM classification

Convection: Convection Convection: Convection in porous media Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 804 , 10 October 2016 , pp. 298 - 322
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Copyright
© 2016 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.CrossRefGoogle Scholar
Baveye, Ph. & 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. 2015a Macro-scale heat transfer in periodically developed flow through isothermal solids. J. Fluid Mech. 780, 274–298.CrossRefGoogle Scholar
Buckinx, G. & Baelmans, M. 2015b Multi-scale modelling of flow in periodic solid structures through spatial averaging. J. Comput. Phys. 291, 34–51.CrossRefGoogle Scholar
Catton, I. 2006 Transport phenomena in heterogeneous media based on volume averaging theory. Heat Mass Transfer 42 (6), 537–551.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.CrossRefGoogle Scholar
DeGroot, C. T. & Straatman, A. G. 2012 Numerical results for the effective flow and thermal properties of idealized graphite foam. Trans. ASME J. Heat Transfer 134 (4), 042603.Google Scholar
Gagnon, R. J. 1970 Distribution theory of vector fields. Am. J. Phys. 38 (7), 879–891.CrossRefGoogle Scholar
Ghaddar, C. K. 1995 On the permeability of unidirectional fibrous media: a parallel computational approach. Phys. Fluids 7 (11), 2563–2586.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.Google 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.CrossRefGoogle Scholar
Hsu, C. T. & Cheng, P. 1990 Thermal dispersion in a porous medium. Intl J. Heat Mass Transfer 33 (8), 1587–1597.CrossRefGoogle Scholar
Kuwahara, F. & Nakayama, A. 1999 Numerical determination of thermal dispersion coefficients using a periodic porous structure. Trans. ASME J. Heat Transfer 121, 160–163.Google Scholar
Kuwahara, F., Nakayama, A. & Koyama, H. 1996 A numerical study of thermal dispersion in porous media. Trans. ASME J. Heat Transfer 118, 756–761.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
Lopez Penha, D. J., Stolz, S., Kuerten, J. G. M., Nordlund, M., Kuczaj, A. K. & Geurts, B. J. 2012 Fully-developed conjugate heat transfer in porous media with uniform heating. Intl J. Heat Fluid Flow 38, 94–106.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. Trans. ASME 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. Trans. ASME J. Heat Transfer 99 (2), 180–186.CrossRefGoogle 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
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. Trans. Porous Med. 14 (2), 163–177.Google Scholar
Quintard, M. & Whitaker, S. 1994b Transport in ordered and disordered porous media. ii: generalized volume averaging. Trans. Porous Med. 14 (2), 179–206.Google Scholar
Quintard, M. & Whitaker, S. 1994c Transport in ordered and disordered porous media. iv: computer generated porous media for three-dimensional systems. Trans. Porous Med. 15 (1), 51–70.CrossRefGoogle 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
Schroeder, C., Stomakhin, A., Howes, R. & Teran, J. M. 2014 A second order virtual node algorithm for Navier–Stokes flow problems with interfacial forces and discontinuous material properties. J. Comput. Phys. 265, 221–245.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 Aata. Academic.Google 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. 1996 The Forchheimer equation: a theoretical development. Trans. Porous Med. 25 (1), 27–61.Google Scholar
Whitaker, S. 1999 The Method of Volume Averaging (Theory and Applications of Transport in Porous Media). Kluwer.Google Scholar
Wood, B. D. 2013 Technical note: revisiting the geometric theorems for volume averaging. Adv. Water Resour. 62, 340–352.CrossRefGoogle 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. Int. 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). Trans. ASME J. Heat Transfer 133 (11), 111802.Google Scholar