A Fractional Single-Phase-Lag Model of Heat Conduction for Describing Propagation of the Maximum Temperature in a Finite Medium.

Stanisław Kukla,Urszula Siedlecka

doi:10.3390/e20110876

Abstract

In this paper, an investigation of the maximum temperature propagation in a finite medium is presented. The heat conduction in the medium was modelled by using a single-phase-lag equation with fractional Caputo derivatives. The formulation and solution of the problem concern the heat conduction in a slab, a hollow cylinder, and a hollow sphere, which are subjected to a heat source represented by the Robotnov function and a harmonically varying ambient temperature. The problem with time-dependent Robin and homogenous Neumann boundary conditions has been solved by using an eigenfunction expansion method and the Laplace transform technique. The solution of the heat conduction problem was used for determination of the maximum temperature trajectories. The trajectories and propagation speeds of the temperature maxima in the medium depend on the order of fractional derivatives occurring in the heat conduction model. These dependencies for the heat conduction in the hollow cylinder have been numerically investigated.

Highlights

The classical Fourier’s law of the heat conduction establishes the relationship between the heat flux vector and the gradient of the temperature [1]q(r, t) = −k∇ T (r, t) (1)where q is the heat flux vector, r is the point in the considered region, t is the time, k is the thermal conductivity of the material, ∇ is the gradient operator and T is the temperature
There are no works devoted to the propagation of the maximum temperature in a medium based on the heat conduction model in which the non-local and phase-lag properties are considered
The solution to the heat conduction problem based on the fractional single-phase-lag model

Summary

Introduction

The classical Fourier’s law of the heat conduction establishes the relationship between the heat flux vector and the gradient of the temperature [1]. Where q is the heat flux vector, r is the point in the considered region, t is the time, k is the thermal conductivity of the material, ∇ is the gradient operator and T is the temperature This relationship implies a nonphysical infinite speed of a thermal signal in the medium. A generalization of the heat conduction model can be obtained by replacing the time derivative in Equation (4) with the fractional derivative of order β. There are no works devoted to the propagation of the maximum temperature in a medium based on the heat conduction model in which the non-local and phase-lag properties are considered. A solution to the heat conduction problem according to the time-fractional single-phase-lag model is presented. The presented numerical results concern the hollow cylinder with the Robin–Neumann boundary conditions, which is subjected to a variable ambient temperature or impulsive heat source

Formulation of the Problem

Solution to the Problem

Numerical Analysis and Discussion

The indicate an important of the orders of fractional derivatives

Conclusions for the fractional derivative order