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.

A closed‐form solution of DPL bioheat transfer problem with time‐periodic boundary conditions

“…h is identical to the approach discussed in Biswas et al 54 for homogeneous materials. The above sinusoidal BFs are presented in the general complex form:…”

“…One can see the detailed formation of DPL consistent BCs in Biswas et al 54 The above form is also capable of taking nonzero heat flux into account.…”

“…The BCs are consistent with the DPL model and have been used earlier. 31,33,54 A general form of these BCs is given as follows 54 :…”

“…The alternative LT approach assumes that the boundary functions (BFs) are time periodic in nature. The approach developed here is based on that employed in the work of Lu et al [50][51][52][53] for Fourier heat conduction problems, which was extended by Biswas et al 54 for the closed-form solutions of the DPL heat conduction with consistent BCs for nonliving bodies. It is to be noted that one gets an analytical expression of thermal wave impedance as part of the solution of the problem.…”

Thermal Step Response of N-Layer Composite Walls—Accurate Approximative Formulas

“…The time scale also impacts the accuracy of the approximation method. For heat flows at relatively smaller time scales, the Laplace transform can be used effectively [14], whereas Fourier series are more suitable for larger time scales [15].…”