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

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 876 ◽  
Author(s):  
Stanisław Kukla ◽  
Urszula Siedlecka
Keyword(s):  
Heat Conduction ◽  
Hollow Cylinder ◽  
Single Phase ◽  
Heat Conduction Problem ◽  
Expansion Method ◽  
Maximum Temperature ◽  
Phase Lag ◽  
Conduction Model ◽  
Laplace Transform Technique ◽  
Finite Medium

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.

scholarly journals Nonlinear Solution to a Non-Fourier Heat Conduction Problem in a Slab Heated by Laser Source

2016 ◽  
Vol 63 (1) ◽  
pp. 129-144
Author(s):  
Mohammad Javad Noroozi ◽  
Seyfolah Saedodin ◽  
Davood Domiri Ganji
Keyword(s):  
Heat Conduction ◽  
Heat Conduction Problem ◽  
Adomian Decomposition Method ◽  
Laser Source ◽  
Temperature Dependent ◽  
Conduction Model ◽  
Non Linear Equation ◽  
Adomian Decomposition ◽  
Non Linear ◽  
Fourier Heat Conduction

Abstract The effect of laser, as a heat source, on a one-dimensional finite body was studied in this paper. The Cattaneo-Vernotte non-Fourier heat conduction model was used for thermal analysis. The thermal conductivity was assumed temperature-dependent which resulted in a non-linear equation. The obtained equations were solved using the approximate-analytical Adomian Decomposition Method (ADM). It was concluded that the non-linear analysis is important in non-Fourier heat conduction problems. Significant differences were observed between the Fourier and non-Fourier solutions which stresses the importance of non-Fourier solutions in the similar problems.

Nonhomogeneous Dual-Phase-Lag Heat Conduction Problem: Analytical Solution and Select Case Studies

2017 ◽  
Vol 140 (3) ◽  
Author(s):  
Simon Julius ◽  
Boris Leizeronok ◽  
Beni Cukurel
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Analytical Solution ◽  
Heat Generation ◽  
Integral Transform ◽  
Heat Conduction Problem ◽  
Relaxation Times ◽  
Hyperbolic Heat Conduction ◽  
Phase Lag ◽  
Dual Phase

Finite integral transform techniques are applied to solve the one-dimensional (1D) dual-phase heat conduction problem, and a comprehensive analysis is provided for general time-dependent heat generation and arbitrary combinations of various boundary conditions (Dirichlet, Neumann, and 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 wavelike 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 dual-phase 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 dual-phase 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.

Nonlinear analysis of a non-Fourier heat conduction problem in a living tissue heated by laser source

2017 ◽  
Vol 10 (08) ◽  
pp. 1750107 ◽  
Author(s):  
Mohammad Javad Noroozi ◽  
Majid Goodarzi
Keyword(s):  
Heat Conduction ◽  
Nonlinear Analysis ◽  
Heat Conduction Problem ◽  
Adomian Decomposition Method ◽  
Experimental Result ◽  
Laser Source ◽  
Living Tissue ◽  
Conduction Model ◽  
Adomian Decomposition ◽  
Fourier Heat Conduction

The effect of laser, as a heat source, on a one-dimensional finite living tissue was studied in this paper. The dual phase lagging (DPL) non-Fourier heat conduction model was used for thermal analysis. The thermal conductivity was assumed temperature-dependent, resulting in a nonlinear equation. The obtained equations were solved using the approximate-analytical Adomian decomposition method (ADM). It was concluded that the nonlinear analysis was important in non-Fourier heat conduction problems. Moreover, a good agreement between the present nonlinear model and experimental result was obtained.

Some theorems on linear theory of thermoelasticity for an anisotropic medium under an exact heat conduction model with a delay

2016 ◽  
Vol 22 (5) ◽  
pp. 1177-1189 ◽  
Author(s):  
Bharti Kumari ◽  
Santwana Mukhopadhyay
Keyword(s):  
Heat Conduction ◽  
Anisotropic Body ◽  
Spatial Behavior ◽  
Inhomogeneous Material ◽  
Phase Lag ◽  
Conduction Model ◽  
Delay Term ◽  
Heat Conduction Model ◽  
Thermoelasticity Theory ◽  
Equation Of Heat Conduction

The present work is concerned with a very recently proposed heat conduction model—an exact heat conduction model with a delay term for an anisotropic and inhomogeneous material—and some important theorems within this theory. A generalized thermoelasticity theory was proposed based on the heat conduction law with three phase-lag effects for the purpose of considering the delayed responses in time due to the micro-structural interactions in the heat transport mechanism. However, the model defines an ill-posed problem in the Hadamard sense. Subsequently, a proposal was made to reformulate this constitutive equation of heat conduction theory with a single delay term and the spatial behavior of the solutions for this theory have been investigated. A Phragmen–Lindelof type alternative was obtained and it has been shown that the solutions either decay in an exponential way or blow-up at infinity in an exponential way. The obtained results are extended to a thermoelasticity theory by considering the Taylor series approximation of the equation of heat conduction to the delay term and a Phragmen–Lindelof type alternative was obtained for the forward and backward in time equations. In the present work, we consider the basic equations concerning this new theory of thermoelasticity for an anisotropic and inhomogeneous material and make an attempt to establish some important theorems in this context. A uniqueness theorem has been established for an anisotropic body. An alternative characterization of the mixed initial-boundary value problem is formulated and a variational principle as well as a reciprocity principle is established.

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