Small and Large Time Solutions for Surface Temperature, Surface Heat Flux, and Energy Input in Transient, One-Dimensional Conduction

This paper addresses one-dimensional transient conduction in simple geometries. It is well known that the transient thermal responses of various objects, or of an infinite medium surrounding such objects, collapse to the same behavior as a semi-infinite solid at small dimensionless time. At large dimensionless time, the temperature reaches a steady state (for a constant surface temperature boundary condition) or increases linearly with time (for a constant heat flux boundary condition). The objectives of this paper are to bring together existing small and large time solutions for transient conduction in simple geometries, put them into forms that will promote their usage, and quantify the errors associated with the approximations. Approximate solutions in the form of simple algebraic expressions are derived (or compiled from existing solutions) for use at both small and large times. In particular, approximate solutions, which are accurate for Fo <0.2 and which bridge the gap between the large Fo (single-term) approximation and the semi-infinite solid solution (valid only at very small Fo), are presented. Solutions are provided for the surface temperature when there is a constant surface heat flux boundary condition, or for the surface heat flux when there is a constant surface temperature boundary condition. These results are provided in terms of a dimensionless heat transfer rate. In addition, the dimensionless energy input is given for the constant surface temperature cases. The approximate expressions may be used with good accuracy over the entire Fourier number range to rapidly estimate important features of the transient thermal response. With the use of the approximations, it is now a trivial matter to calculate the dimensionless heat transfer rate and dimensionless energy input, using simple closedform expressions.