Modeling of heat flow and effective thermal conductivity of fractured media: Analytical and numerical methods

Abstract

International audienceThe present work aims to modeling the thermal conductivity of fractured materials using homogenization-based analytical and pattern-based numerical methods. These materials are considered as a network of cracks distributed inside a solid matrix. Heat flow through such media is perturbed by the crack system. The problem of heat flow across a single crack is firstly investigated. The classical Eshelby's solution, extended to the thermal conduction problem of an ellipsoidal inclusion embedding in an infinite homogeneous matrix, gives an analytical solution of temperature discontinuity across a non-conducting penny-shaped crack. This solution is then validated by the numerical simulation based on the finite elements method. The numerical simulation allows analyzing the effect of crack conductivity. The problem of a single crack is then extended to a medium containing multiple cracks. Analytical estimations for effective thermal conductivity, that take into account the interaction between cracks and their spatial distribution, are developed for the case of non-conducting cracks. Pattern-based numerical method is then employed for both cases non-conducting and conducting cracks. In the case of non-conducting cracks, numerical and analytical methods, both account for the spatial distribution of the cracks, fit perfectly. In the case of conducting cracks, the numerical analyzing of crack conductivity effect shows that highly conducting cracks weakly affect heat flow and the effective thermal conductivity of fractured media

Thermal conductivity of crack-containing media: A numerical study

“…Nguyen et al 27 with the Mori-Tanaka 45 homogenization scheme (MT),and Nguyen et al 27 with the Pont Castaneda and Willis 46 scheme (PC&W),…”

“…[18][19][20] The theoretical framework has found extensive applications to composite systems, and is able to deliver high-fidelity results across a wide range of geometric features of the heterogeneity encompassing fibers and porosity. [21][22][23][24] As for cracks as a specific form of defects, there have been theoretical estimates of steadystate thermal conductivity of solids with various crack configurations, such as oriented cracks at various angles in two dimensions, 25 random penny-shaped cracks, 26,27 oriented elliptical cracks, 28 randomly distributed elliptical cracks, 29 cracks of arbitrary non-flat shape, 30 uniformly oriented penny-shaped cracks, 14,27,31 and ribbonshaped cracks. 32 While the analytical expressions offer useful closed forms and shed light on the general trend, they are less capable of providing sufficient insight into how local arrangement of cracks dictates the overall behavior.…”

Steady-state heat transfer in microcracked media

“…While several studies account for the arbitrary value of matrix/inclusion conductivity and arbitrary crack's orientation or shape, most of the existing papers generally provide thermal conductivity of microcracked media in the non-interacting case. Nguyen et al [19] give closedform expression for different schemes but consider only one orientation of the crack. Nevertheless, that is not the only challenge.…”