Transient radiative cooling of an absorbing and scattering cylinder - A separable solution

A cylindrical region filled with absorbing-emitting material is cooled by radiation to surroundings at a much lower temperature. A solution is found showing that, for each set of parameters, the transient radial temperature distribution reaches a fixed shape, although the temperatures are decreasing with time. This 'fully developed' transient region is characterized by having a constant emittance based on instantaneous values of the cylinder heat loss and mean temperature. This emittance depends only on the optical radius of the cylinder and the scattering albedo. The emittance is lower than that for a cylinder at uniform temperature. This arises from the larger local cooling and, hence, reduced temperatures of the outer layers of the cylinder. An examination of this transient emittance provides the ranges of parameters within which the simplification can be made that the cylinder has uniform radial temperature distribution throughout the cooling process.