Mathematical model of the transition zone motion during heating of the three-component cylindrical object

If due to heating or cooling of an object, a phase transition occurs, the propagation of heat significantly changes. Behavior of the enthalpy, heat capacity, density, coefficient of the heat-conductivity as a function of temperature, changes in a complicated manner. For example, when a ternary eutectic alloy is cooling, one solid phase is formed first, then the second, and finally the entire alloy crystallizes. By heating the surface of the solid metal cylindrical object with three-component composition above the liquidus temperature, the two- or three-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 was used. For each point of a sample for each corresponding temperature, the fractions of liquid and all solid phases were calculated. 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 is reduced to finite-difference equations. A compu¬ter 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 presented method may be useful for metallurgists (calculation of heating of parts in heat treatment, calculation of heating of pieces of charge in the steel smelting furnace), for metrologists (self-testing temperature sensors) and others.