Variabilities and their upper and lower bounds of the equivalent thermal conductivity and resistance defined by the entransy dissipation rate

The prediction of equivalent thermal conductivity (ETC) and resistance (ETR) is an important topic in the composite material field. Many expressions of ETCs and ETRs have been proposed using different homogenization methods in last several decades. Especially, expressions defined by the entransy dissipation rate, keff and Reff, excel for their wide range of application. Usually, it was agreed that keff and Reff were determined by the thermal conductivity distribution inside. However, we find that they are also vary with the boundary temperature gradient distribution (TGD). In this paper, we simulate and mathematically prove the influence of boundary TGD on keff. The results indicate that keff with uniform boundary TGD is higher than that with non-uniform boundary TGD. In addition, the mathematical proof after a little adjustment is applied to confirm the minimum thermal resistance principle based on the uniform TGD inside. The results show that keff is the maximum when TGD are uniform in all regions including boundaries, and keff reaches the minimum when the non-uniformity of TGD extends to the limit. Upper and lower bounds of keff and Reff found in this paper are greatly valuable for optimizing the heat transfer ability and heat insulation ability of materials. And the methodology as well as conclusion in this paper can be reproduced in some generalized irreversible transport systems.