Mathematical modeling of thermal stabilization systems based on phase transitions

Abstract

At present, first of all, due to success of biotechnologies, tasks have arisen in which it is required to ensure the thermal regime of the protected product within strictly defined and fairly close temperature limits. Most traditional thermal protection, cooling and heating systems have proven to be ill-suited to such tasks. Among the principles of thermal protection used in modern practice, active thermal protection technologies based on the latent heat of phase transitions are considered to be the most effective. It seems to be quite natural to use the same principle in new thermal stabilization systems in a narrow temperature range. To do this, sufficiently small inclusions of a different phase with a phase transition temperature close to one of the limiting temperatures of the required range should be introduced into the heat-shielding layer. As the geometric dimensions of the inclusions of a different phase are significantly smaller than the dimensions of the object as a whole, a multiscale problem is generated with all the difficulties inherent in solving such problems. To overcome the difficulties associated with the multiscale nature of the problem, the thermal effects of inclusions of a different phase were modeled by point heat sources. The thermophysical properties of the heat-shielding coating material were assumed to be constant due to the narrowness of the considered temperature range. Boundary conditions for the heat equation at the outer boundary of the heat-shielding layer were set of the first, second, or third kind. On the inner surface of the heat-shielding layer, conditions were set for conjugation with the temperature field of the protected body. In this case, two limiting models of the protected body were considered: a solid body with some effective thermal conductivity and an integral heat capacity with a temperature constant over the volume. In the first case, the conditions of the fourth kind were set, and in the second, the condition of the first kind. To calculate the temperature field in the system under consideration, the boundary element method was used, which easily takes into account the presence of point heat sources. Thus, a mathematical model and the corresponding calculation scheme of the thermal stabilization system based on the latent heat of the phase transition were constructed in the work. The obtained results of numerical calculations could not be confirmed either by comparison with analytical results, or with experimental studies, or with the works of other authors, since the latter were not found. With an increase in the number of inclusions, the material of the heat-shielding layer begins to resemble a heterogeneous medium, which inspires hope for the possibility of comparison with asymptotic results. Prospects for further research are related to the optimization of thermal stabilization systems based on the latent heat of phase transitions. The illustrative calculations carried out confirm the conclusion about the high efficiency of this approach.