Analytical solution of unsteady heat conduction in multilayer internal combustion engine walls

Abstract

An analytical solution of the unsteady heat conduction problem in multilayer walls was developed and applied to study thermal barrier coatings for reciprocating internal combustion engines. The mathematical solution was derived using the matrix method and complex analysis/residue-calculus Laplace transform inversion techniques. The one-dimensional domain includes a time-varying heat flux on the combustion chamber surface, and a time-varying temperature on the backside coolant surface. These boundary conditions are simulated as the superposition of two adjacent triangular, unit-magnitude pulses, which gives a piece-wise linear representation of the applied flux or temperature history. The temperature at any location of interest is found as the convolution of the transfer function, which results from the inversion, with the discrete-time heat flux or backside temperature history. The transfer functions describe the exact response and only depend on multilayer architecture, therefore, they can be computed a priori. The triangular pulse method provides a more than two order of magnitude reduction in computation time compared to a finite difference solution at the same level of accuracy. A 104-fold speed increase relative to a finite difference solution was realized when using the Overlap-add convolution technique for a full drive cycle simulation, i.e., a long time record. The surface response or transfer function acts like a finite impulse response and can be viewed in the frequency domain to reveal complementary information about coating performance.