Open AccessThe problem under consideration is to estimate the effective coefficient of heat conductivity of a material with included balls of zero heat conductivity, being either in a cubic lattice order or chaotically.
The solution of heat conductivity equation can be obtained using a Wiener process. In this mathematical model, the process of heat conduction is represented by random motion of \heat particles", although these \particles" do not exist in a physical sense: they are special formal objects, they represent a sample of a distribution the density of which is proportional to the density of heat energy in each time moment. If one has a solid without heat exchange on its surface, the trajectories of randomly moving particles must reflect from the surface.
Consider a non-bounded flat layer of a composite with its effective heat conductivity to be evaluated. As a criterion of heat conductivity, consider the probability P, which may be that a heat particle, starting from one side of the layer reaches its other side for the time less than T. For a homogeneous isotropic material, this probability is calculated analytically.
Having performed a series of computing experiments simulating heat conductivity through the layer of a composite (the source of heat is applied to its surface, and on the opposite surface is heat absorbing) and processed the experiments' results statistically, one obtains confidence intervals for P, wherefrom appear the confidence intervals for the effective temperature conductivity (under what temperature conductivity a homogeneous material yields the same value of P). Finally, the effective coefficient of heat conductivity is calculated by multiplying the effective coefficient of temperature conductivity with the average volume heat capacity.
Various ratios of the inclusion radius to the cube lattice period (or the corresponding space densities of chaotic inclusions) were considered. For series of 4,300 randomly moving particles, the obtained results appear to agree with those obtained by analytical methods.
Our method allows finding the effective coefficient of heat conductivity for composites with inclusions of arbitrary shapes.