Transient heat conduction in non-homogeneous spherical systems

A comprehensive study encompassing a general analytical development and an archival presentation of results is made for transient heat conduction in thermally coupled spherical regions. The system consists of a sphere and its surrounding environment, each region having different thermal properties and different initial temperatures. The general closed form solution, which accommodates an arbitrary inital temperature distribution in the sphere, is specialized to apply to a number of interesting problems. Among these, the situation in which the sphere and the surroundings are initially at different uniform temperatures constitutes a basic problem in the theory of heat conduction. Correspondingly, a comprehensive presentation of transient temperature histories is made for various locations in the sphere and in the surroundings, with relevant thermal property ratios serving as parameters. Characteristics such as the duration of the transient period and the penetration depth of the temperature field into the surroundings are illuminated. Another interesting situation is that in which the thermal conductivity of the sphere is much greater than that of the surroundings, so that the sphere temperature may be regarded as being spatially uniform. In addition to a presentation of temperature histories, the conditions are identified under which the assumed spatial uniformity of the sphere temperature is valid. For the case of a metallic sphere situated in a gaseous environment, it is demonstrated that the transient response can be represented by a single universal curve.