Fractional Heat Conduction Models and Thermal Diffusivity Determination

Monika Žecová,Ján Terpák

doi:10.1155/2015/753936

Monika Žecová, Ján Terpák

Open Access

https://doi.org/10.1155/2015/753936

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Abstract

The contribution deals with the fractional heat conduction models and their use for determining thermal diffusivity. A brief historical overview of the authors who have dealt with the heat conduction equation is described in the introduction of the paper. The one-dimensional heat conduction models with using integer- and fractional-order derivatives are listed. Analytical and numerical methods of solution of the heat conduction models with using integer- and fractional-order derivatives are described. Individual methods have been implemented in MATLAB and the examples of simulations are listed. The proposal and experimental verification of the methods for determining thermal diffusivity using half-order derivative of temperature by time are listed at the conclusion of the paper.

Highlights

Unsteady heat conduction process, described by partial differential equation, was first formulated by Jean Baptiste Joseph Fourier (1768–1830)
Nonstationary models are described by Fourier heat conduction equation, where the temperature T (K) is a function of spatial coordinate x (m) and time τ (s)
Implementation of the one-dimensional heat conduction model was realized in the programming environment MATLAB

Summary

Introduction

Unsteady heat conduction process, described by partial differential equation, was first formulated by Jean Baptiste Joseph Fourier (1768–1830). The various analytical and numerical methods are used to solve the Fourier heat conduction equation (FHCE) [10, 11]. In the case of heat conduction in materials with nonstandard structure, such as polymers, granular and porous materials, and composite materials, a standard description is insufficient and required the creation of more adequate models with using derivatives of fractional-order [12,13,14,15]. The more adequate models of processes subsequently require new methods to determine the parameters of these models. The issue of research and development methods and tools for processes modeling with using fractional-order derivatives is very actual, since it means a qualitatively new level of modeling. For the Crank-Nicolson scheme, the literature describes the use of Grunwald-Letnikov definition only for a spatial derivative [73, 76,77,78]

Models of Heat Conduction Processes

Solutions

Simulations

Proposal Method for Thermal Diffusivity Determination

Experimental Verification

Conclusion