Сравнительный анализ оценок коэффициента теплопроводности каркаса пористого твердого тела

Abstract

Heat-insulating porous materials and structural ones having also a porous structure, which are produced by pressing and powder metallurgy methods, are widely used in engineering. One of the important thermophysical characteristics of such materials is a coefficient of thermal conductivity, which affects the choice of specific areas of their application. Along with the experimentally determined coefficient of thermal conductivity of porous materials, there are various approaches to estimate this coefficient. Most of these approaches have an empirical character and are based on various models of the structure of porous solid skelton, which enable us to approximately estimate contribution of this skeleton to the value of effective thermal conductivity of the entire porous body. A reliable estimate of the thermal conductivity of a porous solid skelton can be based on a modification of its structural model through conditional replacement of pores with their surrounding shells of the material by solid particles with an equivalent coefficient of the thermal conductivity. Such a replacement allows us to extend constructibility of computational dependencies, primarily, to obtain the guaranteed two-sided estimates of the effective thermal conductivity of a porous solid, including using the dual variational formulation of the problem of a steady-state heat conductivity in an inhomogeneous solid. The peculiarity of this formulation is that it includes two alternative functionals (minimized and maximized) that reach equal extremal values at the true temperature distribution in an inhomogeneous body. This property of alternative functionals makes it possible, according to their values, calculated at the approximate temperature distributions in this body, to obtain, respectively, the upper and lower bounds of its effective thermal conductivity. However, the use of the initial structural model of the porous solid skelton, provided that there is no thermal energy transfer through the pores, ensures the preservation of the physical sense only for the upper estimate of the effective thermal conductivity of this skeleton, and the lower estimate is either incorrect or corresponds to the zero value. The introduction of an equivalent thermal conductivity coefficient for a conditional solid particle replacing a pore surrounded by a layer of the solid skelton material allows using the variational approach to find correct two-sided estimates of the effective thermal conductivity and simultaneously to estimate the possible greatest error that can arise when using the computational dependencies based on different structural models and assumptions.