P. S. Grinchuk

Mathematical Model for Calculating the Heat-Protection Properties of the Composite Coating “Ceramic Microspheres–Binder”

A mathematical model which makes it possible to calculate the heat-protection properties of the composite coating “ceramic microspheres–binder” is proposed. The model takes account of the optical and thermophysical properties of the components and the optical properties of a substrate. It is shown that the decrease in the heat loss due to suppression of its radiative component by the coating in question can reach more than 50%. Under certain conditions, the heat-insulation effect can be absent or even inverted, i.e., the presence of the coating is capable of resulting in an increase in the heat flux from the surface being protected. Comparison of the data of the performed full-scale experiment and the calculations based on the proposed model showed that the latter adequately reflects the actual process of heat transfer and can be used for calculating the heat-protection properties of coatings representing a compound of a binder and ceramic microspheres.

Extinction Threshold in Random Heterogeneous Condensed Mixtures with an Extremely Small Amount of a Combustible

A mathematical model describing the process of combustion of heterogeneous condensed mixtures with account for the randomness of the distribution of combustible particles over the system is proposed. The step dependence of the reaction rate on the temperature is used as a model of chemical kinetics. The case where the concentration of the particles is extremely low but the heat energy released in their combustion is high has been investigated. It is shown that for an adiabatic system there exists a concentration limit of combustion. The relation of this combustion limit to the geometric phase transition occurring in a heterogeneous condensed mixture with rare combustion inclusions has been demonstrated. An analytical approach to the evaluation of the threshold concentration of the combustible in such systems, based on the continual theory of percolation, is proposed.

System of Differential Equations for the Lattice Problems of the Percolation Theory

A model of cluster formation in the percolation system is proposed. On its basis a system of differential equations for finite clusters and a differential equation for and infinite cluster are obtained. The solutions of these equations for several limiting cases are investigated. A method of approximate closure of the equation for an infinite cluster using a system of equations for a number of finite clusters is developed, and an expression for the percolation probability in the entire range of change of the part of the conducting bonds in the system is obtained by means of this method.