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Homogenisation applied to thermal radiation in porous media

Part of: Basic methods Partial differential equations

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: [email protected]
C. P. PLEASE
Affiliation:
Mathematical Institute, 24–29 St Giles, Oxford OX1 3LB, UK email: [email protected]
S. D. HOWISON
Affiliation:
Mathematical Institute, 24–29 St Giles, Oxford OX1 3LB, UK email: [email protected]
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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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