Mathematical foundations of fractal heat and mass transfer in the two-phase zone of the metal melt

Abstract

The problem of improving progressive and creating new technologies in metallurgy and foundry production is relevant for obtaining high-quality cast metal and castings. The microscopic and macroscopic properties of casting templates significantly depend on the thermophysical parameters of the casting system and the casting mold, namely, the width of the two-phase zone of melt solidification, the initial temperature of the melt, the cooling rate of the casting, the cooling gradient of the melt, and the temperature on the surface of the casting mold. In order to obtain a fine-grained metal structure. The article presents the results of experimental studies, indicating the fractal nature of structure formation in a two-phase zone of the solidifying metal melt. The thermodynamic statement of the non-stationary problem solidifying of binary systems is considered. Transfer equations are described that are adequate for media with fractal geometry. The mathematical apparatus for describing the curing process from the standpoint of heat and mass transfer in a two-phase zone and diffusion in fractal media is presented. It is shown that the mathematical apparatus of fractional calculation makes it possible to effectively describe the fractal nature of diffuse processes. The analysis of the thermal and mass transfer processes in the melt of the metal, which is in the rare state, and their description using the mathematical apparatus of fractional calculation, have been carried out.