Transient Temperature Rise in a Semi-Infinite Solid Due to a Uniform Disk Source

Abstract
A transient solution is presented for the temperature rise in a semi-infinite solid due to a circular-disk source with a uniform and constant rate of heat flux. The solution is obtained only along the axis of the disk. It is shown that the temperature-time relation at any point along this axis is obtainable from the corresponding well-known one-dimensional solution for a semi-infinite rod by the relation [ θ ( z , t ) ] 3-dimensional = [ θ ( z , t ) - θ ( z 2 + a 2 , t ) ] 1-dimensional where θ, z, a, and t are the temperature rise, distance from the surface, radius of the heat source, and time, respectively.