Diffusion Penetration Time for Transient Heat Conduction

The time duration for processes involving transient thermal diffusion can be a critical piece of information related to thermal processes in engineering applications. Analytical solutions must be used to calculate these types of time durations because the boundary conditions in such cases can be effectively like semi-infinite conditions. This research involves an investigation into analytical solutions for six geometries, including one-dimensional cases for Cartesian, cylindrical, and spherical coordinates. The fifth case involves a heated surface on the inside of a hole bored through an infinite body, which is a one-dimensional problem in cylindrical coordinates. The sixth case involves two-dimensional conduction from a point heat source on the surface of a slab subjected to insulated boundary conditions elsewhere. The mathematical modeling for this case is done in cylindrical coordinates. For each geometric configuration, a relationship is developed to determine the time required for a temperature rise to occur at a nonheated point in the body in response to a sudden change at a heated boundary. A range of time values is computed for each configuration based on the amount of temperature rise used as a criterion. Plots are given for each case, showing the relationships between the temperature rise at the point of interest and the amount of time required to reach that temperature. It was found that a dimensionless parameter, defined herein as dimensionless penetration time, remains reasonably constant between the various geometries studied. The definition of this term is the dimensionless time required to bring about a desired temperature rise, with the characteristic length being the distance between the point of heating and the point of interest.