Solving two-dimensional Cartesian unsteady heat conduction problems for small values of the time

This paper is intended to provide very accurate analytical solutions modeling transient heat conduction processes in 2D Cartesian finite bodies for small values of the time. Analysis of diffusion of thermal deviation effects indicates that, when the space and time coordinates satisfy a criterion developed in the paper, the simple transient 1D semi-infinite solutions may be “used” for generating extremely accurate values for temperature and heat flux at any point of a finite rectangle. Also, they may be “used” with excellent accuracy as short-time solutions when the time-partitioning method is applied (so avoiding the usually difficult integration of the short-cotime Green's functions). A complex 2D semi-infinite problem is solved explicitly and evaluated numerically as part of the analysis. The proposed criterion is based on an accuracy of one part in 10n (n = 1,2,...,10,...), where n = 2 is for engineering insight and visual comparison while n = 10 is for verification purposes of large numerical codes.

► An exact solution for heat conduction in a quarter-infinity region is derived.
► Two-dimensional deviation effects in transient heat conduction are evaluated.
► The deviation time is nearly insensitive to the Biot number at the heated boundary.
► The use of the 1D semi-infinite solutions for solving 2D transient problems is shown.