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Non-Homogeneous Dual-Phase-Lag Hyperbolic Heat Conduction Problem: Analytical Solution and Select Case Studies

[+] Author and Article Information
Julius Simon

Technion - Israel Institute of Technology, Turbomachinery and Heat Transfer Laboratory, Faculty of Aerospace Engineering, Technion-IIT, Haifa, 3200003, Israel
shimonjulius@tx.technion.ac.il

Leizeronok Boris

Technion - Israel Institute of Technology, Turbomachinery and Heat Transfer Laboratory, Faculty of Aerospace Engineering, Technion-IIT, Haifa, 3200003, Israel
borisl@technion.ac.il

Cukurel Beni

Technion - Israel Institute of Technology, Turbomachinery and Heat Transfer Laboratory, Faculty of Aerospace Engineering, Technion-IIT, Haifa, 3200003, Israel
beni@cukurel.org

1Corresponding author.

ASME doi:10.1115/1.4037775 History: Received October 27, 2016; Revised July 16, 2017

Abstract

Finite integral transform techniques are applied to solve the one-dimensional dual-phase hyperbolic heat conduction problem, and a comprehensive analysis is provided for general time-dependent heat generation and arbitrary combinations of various boundary conditions (Dirichlet, Neumann & Robin). Through the dependence on the relative differences in heat flux and temperature relaxation times, this analytical solution effectively models both parabolic and hyperbolic heat conduction. In order to demonstrate several exemplary physical phenomena, four distinct cases that illustrate the hyperbolic heat conduction behavior are presented. In the first model, following an initial temperature spike in a slab, the thermal evolution portrays immediate dissipation in parabolic systems, whereas the hyperbolic solution depicts wavelike temperature propagation - the intensity of which depends on the relaxation times. Next, the analysis of periodic surface heat flux at the slab boundaries provides evidence of interference patterns formed by temperature waves. In following, the study of Joule-heating driven periodic generation inside the slab demonstrates that the steady-periodic parabolic temperature response depends on the ratio of pulsatile electrical excitation and the electrical resistivity of the slab. As for the hyperbolic model, thermal resonance conditions are observed at distinct excitation frequencies. Building on findings of the other models, the case of moving constant-amplitude heat generation is considered, and the occurrences of thermal shock and thermal expansion waves are demonstrated at particular conditions.

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