Analytical Solution for Temperature Distribution in a Multilayer Body With Spatially Varying Convective Heat Transfer Boundary Conditions on Both Ends

Analytical modeling of thermal conduction in a multilayer body is of practical importance in several engineering applications such as microelectronics cooling, building insulation, and micro-electromechanical systems. A number of analytical methods have been used in past work to determine multilayer temperature distribution for various boundary conditions. However, there is a lack of work on solving the multilayer thermal conduction problem in the presence of spatially varying convective heat transfer boundary condition. This paper derives the steady-state temperature distribution in a multilayer body with spatially varying convective heat transfer coefficients on both ends of the body. Internal heat generation within each layer and thermal contact resistance between layers are both accounted for. The solution is presented in the form of an eigenfunction series, the coefficients of which are shown to be governed by a set of linear, algebraic equations that can be easily solved. Results are shown to be in good agreement with numerical simulation and with a standard solution for a special case. The model is used to analyze heat transfer for two specific problems of interest involving spatially varying convective heat transfer representative of jet impingement and laminar flow past a flat plate. In addition to enhancing the theoretical understanding of multilayer heat transfer, this work also contributes toward design and optimization of practical engineering systems comprising multilayer bodies.