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Heat Transfer Research
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Heat Transfer Research

DOI: 10.1615/HeatTransRes.2014006434
pages 447-464

ANALYTICAL SOLUTION OF THE PROBLEM OF NON-FOURIER HEAT CONDUCTION IN A SLAB USING THE SOLUTION STRUCTURE THEOREMS

Mohammad Akbari
Department of Mechanical Engineering, Semnan University, Semnan, Iran
Seyfolah Saedodin
Department of Mechanical Engineering, Semnan University, Semnan, Iran
Davood Semiromi Toghraie
Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran
Farshad Kowsary
Department of Mechanical Engineering, University College of Engineering, University of Tehran, Tehran 515-14395, Iran

ABSTRACT

This paper studies an analytical method which combines the superposition technique along with the solution structure theorem such that a closed-form solution of the hyperbolic heat conduction equation can be obtained by using the fundamental mathematics. In this paper, the non-Fourier heat conduction in a slab at whose a left boundary there is a constant heat flux and at the right boundary, a constant temperature Ts = 15, has been investigated. The complicated problem is split into multiple simpler problems that in turn can be combined to obtain a solution to the original problem. The original problem is divided into five subproblems by setting the heat generation term, the initial conditions, and the boundary conditions for different values in each subproblem. All the solutions given in this paper can be easily proven by substituting them into the governing equation. The results show that the temperature will start retreating at approximately t = 2 and for t = 2 the temperature at the left boundary decreases leading to a decrease in the temperature in the domain. Also, the shape of the profiles remains nearly the same after t = 4. The solution presented in this study can be used as benchmark problems for validation of future numerical methods.


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