Mathematical model of the two-phase zone motion during heating of the two-component cylindrical object

If due to heating or cooling of an object, a phase transition occurs, the propagation of heat significantly changes. Unlike one-component metal, upon cooling of the multicomponent alloy, when passing through the liquidus temperature, enthalpy, density and thermal conductivity are continuous, but their derivatives and, in particular, the heat capacity become discontinuous. By heating the surface of the solid metal cylindrical object with two-component composition above the liquidus temperature, the two-phase zone occurs at the surface, and moves to the cylinder axis. We propose a method of introducing “effective” specific heat that allows calculating the speed of this zone movement, as well as the temperature of the object at any point and at any time. The ordinary equation of heat conductivity with variable coefficients which, in this case, depend on temperature was used. Liquid, solid and transitional zones of a sample were counted by the same equation, its coefficients were counted by the same method. For each point of the sample for each respective temperature the proportion of solid and liquid phases was calculated. In this paper, for simplicity, the solution was considered as ideal. Specific volume, thermal conductivity and specific enthalpy were calculated as the weighted average of the corresponding values for the liquid and solid states. The specific heat was calculated as the derivative of the enthalpy with respect to temperature. The resulting system of differential equations was reduced to finite-difference equations. A computer program was developed to solve the resulting system of difference equations. Results of one of such calculations are given in the article. The developed method allows to calculate the velocity of the phase boundary, as well as the object temperature at any point and at any time. The results may be interesting not only for metallurgists, but for metrology to develop self-testing temperature sensor theory.