A Closed Form Solution of Dual-Phase Lag Heat Conduction Problem With Time Periodic Boundary Conditions

Abstract

The Laplace transform (LT) is a widely used methodology for analytical solutions of dual phase lag (DPL) heat conduction problems with consistent DPL boundary conditions (BCs). However, the inversion of LT requires a series summation with large number of terms for reasonably converged solution, thereby, increasing computational cost. In this work, an alternative approach is proposed for this inversion which is valid only for time-periodic BCs. In this approach, an approximate convolution integral is used to get an analytical closed-form solution for sinusoidal BCs (which is obviously free of numerical inversion or series summation). The ease of implementation and simplicity of the proposed alternative LT approach is demonstrated through illustrative examples for different kind of sinusoidal BCs. It is noted that the solution has very small error only during the very short initial transient and is (almost) exact for longer time. Moreover, it is seen from the illustrative examples that for high frequency periodic BCs the Fourier and DPL model give quite different results; however, for low frequency BCs the results are almost identical. For nonsinusoidal periodic function as BCs, Fourier series expansion of the function in time can be obtained and then present approach can be used for each term of the series. An illustrative example with a triangular periodic wave as one of the BC is solved and the error with different number of terms in the expansion is shown. It is observed that quite accurate solutions can be obtained with a fewer number of terms.