Evaluation of Effective Heat Conductivity of Composite Materials by the Moments Method

The purpose of the research was to elaborate a new method of finding the effective heat conductivity coefficient of a composite material. As an example, we consider a matrix with ball inclusions of a material having another heat conductivity, the heat contact being ideal. The process of heat conduction is modeled via random motion of virtual heat particles. The speed of diffusion in each material is proportional to its temperature conductivity coefficient. When a particle is passing from one material to another, one having smaller heat conductivity, it is reflected from the frontier with a certain probability. We formulate an estimate of heat conductivity via moment of heat energy, this value is exactly known for a homogeneous material, and this estimate is statistically evaluated for composite materials. A computing experiment models the process of heat conduction through a layer of a composite material, having heated one side of the layer at the start. For a layer of a composite, we perform a multiple computational experiment modeling heat conduction, and, having processed the experiment results statistically, we obtain confidence intervals for the effective temperature conductivity and heat conductivity coefficients. We have considered inclusions of materials with heat conductivity coefficients differing from those of the matrix in 3 times up or down, and with zero heat conductivity. Ball inclusions of equal size were situated in a cubic order or chaotically. In series of 4300 randomly moving particles, in all cases considered, the difference between the effective heat conductivity coefficients and those calculated by other methods does not exceed a statistical error. The method elaborated makes it possible to obtain effective heat conductivity coefficients for composites with inclusions of any size and shape; it can be applied also in a case of inclusions of several materials. The results obtained are reliable, their exactness is limited only by the power of computers.

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