# (410d) Two-Dimensional Heat Transfer through Simple Composite Planar Solids: Effectiveness of the One-Dimensional Thermal Circuit Analysis

Authors:
Purdue University

The concept of “thermal resistance” plays an important role in the analysis of various steady-state heat conduction problems. In particular, the steady-state one-dimensional heat conduction through a given solid can be considered as giving rise to a “thermal circuit”, in which a solid acts like a thermal resistor that provides some amount of resistance to heat transfer (electrical current) that is induced by an imposed temperature (or voltage) difference. This direct analogy to Ohm’s law can be further extended. Solids comprised of more than one material (each with different values of the thermal conductivity) can be treated as if they are comprised of thermal resistors in series or parallel, or combinations of series and parallel resistors that seem to mimic the arrangement of the various materials making up the solid. Thermal circuits are very useful for analyzing the rates of heat transfer through composite solids, as they allow for straightforward back-of-the-envelope calculations (thereby avoiding the need to perform two- or three-dimensional analyses of complicated solids), for understanding how the effective resistances of composite solids depend upon their material properties and for locating those regions in which the heat flux has its highest and lowest values. Despite their convenience, textbooks on heat transfer nevertheless provide little or no discussion on the accuracy of the estimates of the rates of heat transfer obtained using thermal circuits. In addition, a unique thermal circuit cannot be generated for more complicated composite solids. (Multiple circuits can always be drawn to represent such solids.) Again, textbooks offer no insights into whether there possibly exist two particular thermal circuits that can yield upper and lower bounds on the actual rates of heat transfer through composite solids.

We therefore investigate the effectiveness of thermal circuits in describing the rates of heat transfer through various simple two-dimensional composite planar solids. For a solid comprised of two materials with a direct parallel arrangement (upper and lower layers of equal thicknesses in the direction of heat transfer but with different thermal conductivities), the analytical solution of the heat conduction equation indicates that this solid is exactly described by a thermal circuit comprised of two resistors in parallel in the limit of negligible heat losses along both the top and bottom surfaces. While no longer exact, thermal circuits nonetheless provide accurate estimates (within 10% of the results obtained via numerical solution of the heat conduction equation) of the following two other composite planar solids: 1) a solid comprised of three different materials, the upper layer being made of two separate materials in series and the lower layer made of the third material; and 2) a solid comprised of an outer material completely enclosing a different inner material. For these solids, the parallel representation (which includes effective resistors in parallel with each effective resistor being comprised of resistors in series) provides a lower bound on the actual rate of heat transfer, while the series representation (which includes effective resistors in series with each effective resistor being comprised of resistors in parallel) provides an upper bound on the actual rate of heat transfer. In all cases, the averages of these two bounds provide very accurate estimates of the numerically calculated rates of heat transfer. Our analysis suggests that the thermal circuit analogy should provide reliable estimates of the rates of heat transfer through other composite solids.