Maxwell-type models for the effective thermal conductivity of a porous material with radiative transfer in the voids

Abstract

There are several models for the effective thermal conductivity of two-phase composite materials in terms of the conductivity of the solid and the disperse material. In this paper, we generalise three models of Maxwell type (namely, the classical Maxwell model and two generalisations of it obtained from effective medium theory and differential effective medium theory) so that the resulting effective thermal conductivity accounts for radiative heat transfer within gas voids. In the high-temperature regime, radiative transfer within voids strongly influences the thermal conductivity of the bulk material. Indeed, the utility of these models over classical Maxwell-type models is seen in the high-temperature regime, where they predict that the effective thermal conductivity of the composite material levels off to a constant value (as a function of temperature) at very high temperatures, provided that the material is not too porous, in agreement with experiments. This behaviour is in contrast to models which neglect radiative transfer within the pores, or lumped parameter models, as such models do not resolve the radiative transfer independently from other physical phenomena. Our results may be of particular use for industrial and scientific applications involving heat transfer within porous composite materials taking place in the high-temperature regime.Common models for calculating the effective thermal conductivity of a solid composite material include the Maxwell model (based on the pioneering work of Maxwell [3], where an expression for the effective thermal conductivity was derived via far-field perturbations to solutions of the steady heat equation), as well as variants thereof, including the effective medium theory (EMT) model and the differential effective medium theory (DEMT) model. EMT uses a similar approach to that used in deriving the Maxwell model, and can be applied to many other physical properties (see [4], for instance, for the electrical resistance problem). In [5,6], there is a comparison between the Maxwell model and EMT, and those works outline how certain bounds on the thermal conductivities can be obtained. The multipole expansion method [7] gives similar results. In applying the DEMT, one incrementally adds one of the materials to the composite, and considers the effect of an infinitesimal change in the composite material composition on the effective thermal conductivity, obtaining a differential equation for the effective thermal conductivity in terms of the volume fraction of inclusions; see [8,9]. Reviews of many current models and methodologies, including those outlined above, can be found in [10,11,12].Recent work has involved the application of Maxwell-like models to the study of composite materials [13], including polymer composites [14,15]. Such models have recently proven useful in understanding nanoflake thermal annealing [16], and in the understanding of effective thermal conductivity in a variety of materials, such as for a wood cell modelled as a constituent element of briquette c...

Melting and dripping of a heated material with temperature-dependent viscosity in a thin vertical tube

“…However, the results are only really relevant to lower temperature applications because at high furnace temperatures radiation between particles plays a strong role in the diffusive thermal mechanism. One option is to consider the thermal conductivity to be a cubic function of the temperature, although for the high temperature regime this functional form should likely be generalised (Kiradjiev et al 2019).…”

“…One option is to consider the thermal conductivity to be a cubic function of the temperature, although for the high temperature regime this functional form should likely be generalised (Kiradjiev et al. 2019).…”

Моделирование Окисления Расплава Активной Зоны Ядерного Реактора При Наличии Оксидной Корки На Поверхности Расплава

“…Отметим, что использование для определения теплопроводности корки экспериментальных данных, полученных в близких для опытов по окислению расплава условиях по температуре, материалу, пористости и размеру пор корки, позволяет отказаться от расчетного определения эффективной теплопроводности, учитывающего распространение теплового излучения в пористой полупрозрачной среде [33][34][35][36].…”

Homogenisation applied to thermal radiation in porous media

“…We are interested in how the void radius affects the effective conductivity tensor. Kiradjiev et al [10] investigated this relationship for effective conductivities derived using Maxwell models, effective medium theory (EMT) and differential effective medium theory (DEMT). The theory and formulation of these models are well explained in [10] .…”

“…Tiihonen [23] studied radiation on non-convex surfaces and proved the existence and uniqueness of the conductive-radiative problem. Kiradjiev et al [10] generalised Maxwell models to account for radiative transfer for a dilute void distribution. They derived a relationship between the effective thermal conductivity and temperature and concluded that the effective thermal conductivity saturates with increasing temperature, or rather the rate of increase of the conductivity slows down as the temperature increases for the dilute distribution.…”