Gorter-Mellink pulsed-source problem in cylindrical and spherical geometry

Abstract

An exact solution to the Gorter-Mellink pulsed-source problem is known in plane geometry [Dresner, L. Advances in Cryogenic Engineering (1984) 29 323]. According to this solution, the central temperature (i.e., the temperature at the source location) falls as t − 3 2 , where t is the elapsed time after an instantaneous, plane pulse. No such exact solutions are known to the pulsed-source problem in cylindrical and spherical geometry. But in cylindrical geometry, it can be shown that if the initial condition is an instantaneous temperature rise ΔT inside a cylinder of radius R, the central temperature is bounded from above by a decreasing exponential function of time. The relaxation time of this exponential is related to ΔT and R. In spherical geometry, it can be shown that if the initial condition is an instantaneous temperature rise ΔT inside a sphere of radius R, the central temperature is bounded from above by a function proportional to ( t 0-t) 9 2 . The extinction time t 0 is related to ΔT and R. These predictions have not been tested by experiment, and the author recommends such experiments.