Unrestraining the Biot Number for Systems With Internal Heat Generation

Thermal runaway from hot spots in systems such as solar energy storage poses a safety concern. Tools for rapid analysis of these systems would be exceedingly useful in their development and maintenance. The “lumped capacitance” (LC) assumption is one of these analysis tools and is limited to Biot numbers less than about 0.1. However, for systems like energy storage batteries with internal heat generation, there is no such tool. A numerical solution was, therefore, used to compute the spatiotemporal temperature of cooling spheres with varying thermal conductivity, characteristic length scale, and internal heat generation rate to determine the effects that internal heat generation has on LC accuracy. Increasing the heating time or decreasing the thermal conductivity hinders LC accuracy, while increasing the internal heat generation rate or characteristic length scale improves it. This means that larger volumes improve the accuracy of LC, completely inverting its previous relationship. The Buckingham–Pi theorem was then used to create a new nondimensional group, the Yonkist number, in order to provide an analogous Biot number for systems with heat generation. Ultimately, it was found that LC can be utilized for systems with unlimited Biot numbers, as long as the internal heat generation rate is sufficiently large or the heating time is sufficiently small to make the Yonkist number less than the Biot number. The use of the new Yonkist number removes the upper boundary from the range of Biot numbers to which the LC assumption can be applied and allows expedient heat transfer analyses for thermal runaway problems.