Correlations Spanning the Entire Biot Number Range to Predict Near Steady State Condition in Transient 1-D Heat Conduction Problems

Abstract

Steady state criteria for vanishing small values of Biot number (lumped case) is well known and is reported in every undergraduate text on heat transfer. The heat conduction time scale for pure thermal diffusion problems (extremely large values of Biot number) is also a very well established fact and common knowledge to all well-schooled thermal engineers. However, to the best of the authors’ knowledge no attempt has been made so far to develop a generalized criterion encompassing the entire Biot number range. Hence, the objective of this paper is to construct a simple, but accurate, correlation to predict the onset of steady state for the three basic configurations (plane layer, cylinder, and sphere) for the complete range of Biot number from the high (Bi → ∞) to the low (Bi →0) Bi value limits, while spanning all values in between. Correlations are developed and reported in this paper such that they predict the transient time duration very close to those obtained from the theoretical solution to the problem. Moreover these proposed correlations are extremely simple in form and, as such, are ideal to be used by practicing thermal engineers in need of a quick estimate for the required time period to achieve steady state for problems that can be modeled from these basic geometries. A more accurate correlation, for the case of slab has also been proposed (containing two additional terms) which can be used if higher accuracy in the intermediate Biot number range were to be desired.