Composite lumped model and algebraic solutions of unsteady temperatures and accumulated heat transfer

Calculations of temperature histories and accumulated heat transfer in simp le bodies (long plates, long cylinders, and spheres) for applications in th ermal engineering have been traditionally obtained from the standard Heisle r/Grober charts (Heisler, M. P., "Temperature Charts for Induction and Cons tant Temperature Heating,: Transactions of the American Society of Mechanic al Engineers, Vol. 69, 1947, pp. 227-236 and Grober, H., Einfurhrung in die Lehre von der Warmeiibertragung, Springer, Berlin, 1926). The deterrent of this tandem of charts is that they are exclusively applicable to dimension less times tau greater than 0.2. From a mathematical perspective, this impo sing restriction is associated with the utilization of truncated one term s eries for the construction of these charts. For practical purposes, the pai r of Heisler charts for each of the three simple bodies is difficult to rea d, necessitating skillful visual interpolation. The time variations of mean and surface temperatures, as well as the total heat transfer of the three basic bodies via a composite lumped model, are predicted. Then the time var iations of the center temperatures in the three bodies are determined from a variant of the integral balance method. This unique alliance turns out to be very accessible and leads to a handful of easily evaluated algebraic ex pressions at any time, even at arduous dimensionless times less than 0.2, w here the standard Heisler/Grober charts falter.