Exact analytical solution for thermal conduction in a Cartesian body with heat-generating regions of arbitrary shapes and thermal properties

Abstract

Despite considerable advances in numerical simulations, the development of analytical tools for exact solutions of thermal conduction problems remains to be of much importance. While multiple techniques are available for analyzing thermal conduction in a homogeneous body, it is a lot more challenging to derive exact solutions for problems containing multiple materials, particularly when the geometry may be irregular. For example, solving the problem of circular heat-generating regions of different thermal properties inside a Cartesian body – such as in a Li-ion battery pack – presents considerable theoretical difficulties. This work presents an exact analytical technique for solving thermal conduction problems containing multiple heat-generating regions of arbitrary non-Cartesian shapes and distinct thermal properties within a Cartesian body. The spatial distribution of heat generation and thermal properties of the non-Cartesian regions is represented mathematically using Heaviside step functions. Closed-form expressions for the coefficients of a series solution for the transient and steady state temperature fields are derived using the differential and integral properties of Heaviside functions. In limiting conditions, these expressions are shown to correctly reduce to well-known results for homogeneous bodies. Good agreement with numerical simulations, and with past work for a specific two-layer problem is also demonstrated. The general technique developed here is used to solve a variety of geometrically complex problems that are not solvable using traditional analytical methods, such as one with four heat-generating sources of different shapes and materials, transient thermal conduction due to a heart-shaped heater and a thermal decay problem. While such problems may be solved using numerical simulations, analytical techniques such as the one developed here advance the theoretical understanding of thermal conduction, and are often more practical to implement in real-life engineering scenarios, such as battery thermal management and composites manufacturing.