Thermal Profile Produced by a Gaussian-In-Time Heat Source on a Semi-Infinite Solid

This work presents an analysis of transient conduction heat transfer in a semi-infinite solid with uniform surface heating of intensity varying as Gaussian, without a physically meaningful start. Practical and accurate explicit, closed-form expressions for the characteristic values of the temperature profile are presented. Exact solutions were obtained for the surface temperature and its heating or cooling rate, while approximate solutions were obtained for the sub-surface temperature profile. Explicit expressions are presented for surface temperature, surface heating or cooling rate, maximum surface temperature and the time at which it is attained, maximum distribution parameter of the heat source which guarantees the attainment of a target temperature and its asymptotic behaviour at early and late times, maximum heating and cooling rates and the times and temperatures at which they happen, asymptotic behaviour of surface heating and cooling rate at early, late, and intermediate times, maximum penetration of a target temperature and their associated times and a function for intermediate cases of maximum temperature penetration, temperature profile for the entire time domain. The expressions presented do not require numerical methods or specialized software, and are applicable as practical engineering tools to fields such as welding, additive manufacturing, heat treatment, or flash heating.