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Homogenisation applied to thermal radiation in porous media

Part of: Partial differential equations Basic methods

Published online by Cambridge University Press:  03 December 2020

C. M. ROONEY Open the ORCID record for C. M. ROONEY [Opens in a new window] ,
C. P. PLEASE  and
S. D. HOWISON
C. M. ROONEY
Affiliation:
Mathematical Institute, 24–29 St Giles, Oxford OX1 3LB, UK email: caoimhe.rooney@nasa.gov
C. P. PLEASE
Affiliation:
Mathematical Institute, 24–29 St Giles, Oxford OX1 3LB, UK email: caoimhe.rooney@nasa.gov
S. D. HOWISON
Affiliation:
Mathematical Institute, 24–29 St Giles, Oxford OX1 3LB, UK email: caoimhe.rooney@nasa.gov
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Abstract

Heat transport in granular and porous media occurs through conduction in the solid and radiation through the voids. By exploiting the separation of length scales between the small typical particles or voids and the large size of whole region, the method of multiple scales can be applied. For a purely diffusive system, this yields a problem on the macroscale with an effective conductivity, deduced by solving a ‘cell problem’ on the microscale. Here, we apply the method when radiation and conduction are both present; however, care must be taken to correctly handle the integral nature of the radiative boundary condition. Again, an effective conductivity is found by solving a ‘cell problem’ which, because of the non-linearity of radiative transfer, to be solved for each temperature value. We also incorporate modifications to the basic theory of multiple scales in order to deal with the non-local nature of the radiative boundary condition. We derive the multiple scales formulation of the problem and report on numerical comparisons between the homogenised problem and direct solution of the problem. We also compare the effective conductivity to that derived using Maxwell models and effective medium theory.

Keywords

Asymptotic homogenisationradiative heat transferintegral constraints

MSC classification

Primary: 35B27: Homogenization; equations in media with periodic structure 80M40: Homogenization
Secondary: 80M35: Asymptotic analysis
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 5: 30th Anniversary Special Issue , October 2021 , pp. 784 - 805
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Allaire, G. & El Ganaoui, K. (2009) Homogenization of a conductive and radiative heat transfer problem. Multiscale Model. Simul. 7(3), 1148–1170.CrossRefGoogle Scholar
Allaire, G. & Habibi, Z. (2013) Homogenization of a conductive, convective, and radiative heat transfer problem in a heterogeneous domain. SIAM J. Math. Anal. 45(3), 1136–1178.CrossRefGoogle Scholar
Allaire, G. & Habibi, Z. (2013) Second order corrector in the homogenization of a conductive-radiative heat transfer problem. Discrete Continuous Dyn. Syst. Ser. B 18(1), 1–36.CrossRefGoogle Scholar
Chapman, S. J. & McBurnie, S. E. (2015) Integral constraints in multiple-scales problems. Eur. J. Appl. Math. 26(5), 595–614.CrossRefGoogle Scholar
Chapman, S. J., Shipley, R. J. & Jawad, R. (2008) Multiscale modeling of fluid transport in tumors. Bull. Math. Biol. 70(8), 2334.CrossRefGoogle ScholarPubMed
Dalwadi, M. P., Griffiths, I. M. & Bruna, M. (2015) Understanding how porosity gradients can make a better filter using homogenization theory. Proc. R. Soc. A Math. Phys. Eng. Sci. 471(2182) 20150464.CrossRefGoogle Scholar
Francl, J. & Kingery, W. D. (1954) Thermal conductivity: Ix, experimental investigation of effect of porosity on thermal conductivity. J. Am. Ceramic Soc. 37(2), 99–107.CrossRefGoogle Scholar
Howell, J. R., Menguc, M. P. & Siegel, R. (2015) Thermal Radiation Heat Transfer. CRC Press, Abingdon.CrossRefGoogle Scholar
Jones, E., Oliphant, T., Peterson, P. (2001) SciPy: open source scientific tools for Python.Google Scholar
Kiradjiev, K. B., Halvorsen, S. A., Van Gorder, R. A. & Howison, S. D. (2019) Maxwell-type models for the effective thermal conductivity of a porous material with radiative transfer in the voids. Int. J. Therm. Sci. 145, 106009.CrossRefGoogle Scholar
Luikov, A. V., Shashkov, A. G., Vasiliev, L. L. & Fraiman, Y. E. (1968) Thermal conductivity of porous systems. Int. J. Heat Mass Transfer 11(2), 117–140.CrossRefGoogle Scholar
McBurnie, S. E. (2010) Sound Propagation through Bubbly Liquids. PhD thesis, Oxford University.Google Scholar
Modest, M. F. (2013) Radiative Heat Transfer. Academic Press, Oxford.CrossRefGoogle Scholar
Ockendon, J. R., Howison, S., Lacey, A. & Movchan, A. (2003) Applied Partial Differential Equations. Oxford University Press, Oxford.Google Scholar
Palencia, E. S. (1980) Non-Homogeneous Media and Vibration Theory . Lecture Notes in Physics, vol. 127, Springer, Berlin.Google Scholar
Pasternak, E. & Mühlhaus, H. B. (2005) Generalised homogenisation procedures for granular materials. In: Mathematics and Mechanics of Granular Materials, Springer, Dordrecht, pp. 199–229.CrossRefGoogle Scholar
Pavliotis, G. & Stuart, A. (2008) Multiscale Methods: Averaging and Homogenization. Springer Science and Business Media, New York.Google Scholar
Rathgeber, F., Ham, D. A., Mitchell, L., Lange, M., Luporini, F., McRae, A. T. T., Bercea, G.-T., Markall, G. R. & Kelly, P. H. J. (2017) Firedrake: automating the finite element method by composing abstractions. ACM Trans. Math. Software (TOMS) 43(3), 24.CrossRefGoogle Scholar
Rooney, C. M. (2019) Homogenisation of the Electrothermal Behaviour of Granular Material. PhD thesis, University of Oxford.Google Scholar
Rosseland, S. (1936) Theoretical Astrophysics; Atomic Theory and the Analysis of Stellar AtMospheres and Envelopes, Clarendon Press, Oxford.Google Scholar
Schatz, J. F. & Simmons, G. (1972) Thermal conductivity of Earth materials at high temperatures. J. Geophys. Res. 77(35), 6966–6983.CrossRefGoogle Scholar
Spinnler, M., Winter, E. R. & Viskanta, R. (2004) Studies on high-temperature multilayer thermal insulations. Int. J. Heat Mass Transfer 47(6–7), 1305–1312.CrossRefGoogle Scholar
Tiihonen, T. (1997) Stefan–Boltzmann radiation on non-convex surfaces. Math. Methods Appl. Sci. 20(1), 47–57.3.0.CO;2-B>CrossRefGoogle Scholar
Zhao, C. Y., Lu, T. J., Hodson, H. P. & Jackson, J. D. (2004) The temperature dependence of effective thermal conductivity of open-celled steel alloy foams. Mater. Sci. Eng. A 367(1–2), 123–131.CrossRefGoogle Scholar
Zhou, Q., Zhang, H. W. & Zheng, Y. G. (2012) A homogenization technique for heat transfer in periodic granular materials. Adv. Powder Tech. 23(1), 104–114.CrossRefGoogle Scholar