Relativistic moving heat source

The relativistic heat conduction (RHCE) model is particularly important in the analysis of processes involving moving heat sources (MHS) at speeds or frequencies comparable with those of heat propagation in the medium. This paper establishes a unified framework for solving heat conduction problems using the RHCE model. It offers “Fundamental Solutions” in one, two, and three spatial dimensions, for the transient response due to an instantaneous point MHS. Moreover, it presents the transient response due to a continuous point MHS, the quasi-steady response due a periodic point MHS, as well as guidelines for solving the RHCE equation under various loading and boundary conditions.     

The thermodynamic basis of entransy and entransy dissipation

In the present work, the entransy and entransy dissipation are defined from the thermodynamic point of view. It is shown that the entransy is a state variable and can be employed to describe the second law of thermodynamics. For heat conduction, a principle of minimum entransy dissipation is established based on the second law of thermodynamics in terms of entransy dissipation, which leads to the governing equation of the steady Fourier heat conduction without heat source. Furthermore, we derive the expressions of the entransy dissipation in duct flows and heat exchangers from the second law of thermodynamics, which paves the way for applications of the entransy dissipation theory in heat exchanger design.     

Entropy analyses for hyperbolic heat conduction based on the thermomass model

This paper is divided into three major sections with the first one introducing the concept of generalized entropy in extended irreversible thermodynamics briefly, that is, the entropy of a non-equilibrium system depend not only on the classical variables but also on the dissipative fluxes, which makes the hyperbolic equation of heat conduction based on the Cattaneo–Vernotte model compatible with the second law of thermodynamics. The second section deals with the hyperbolic heat conduction based on the thermomass model. According to the Einstein’s mass-energy relation, the phonon gas in dielectrics can be viewed as a kind of weighty compressible fluid, and the momentum equation of the phonon (thermomass) gas in the dielectrics, which consists of the driving force, inertia and resistance of phonon (thermomass) gas, is just the damped thermal wave equation. In the third section our analyses show that the contribution of the kinetic energy of the phonon gas in the expression of extended entropy based on the thermomass model is identical with that of the heat flux in the expression of generalized entropy in extended irreversible thermodynamics. It implies that the hyperbolic heat conduction based on the thermomass model is compatible with the second law of thermodynamics.     

Fractional heat conduction in a space with a source varying harmonically in time and associated thermal stresses

Time-nonlocal generalization of the classical Fourier law with the “long-tail” power kernel can be interpreted in terms of fractional calculus (theory of integrals and derivatives of noninteger order) and leads to the time-fractional heat conduction equation with the Caputo derivative. Fractional heat conduction equation with the harmonic source term under zero initial conditions is studied. Different formulations of the problem for the standard parabolic heat conduction equation and for the hyperbolic wave equation appearing in thermoelasticity without energy dissipation are discussed. The integral transform technique is used. The corresponding thermal stresses are found using the displacement potential.     

ON A NEW C- AND F-PROCESSES HEAT CONDUCTION CONSTITUTIVE MODEL AND THE ASSOCIATED GENERALIZED THEORY OF DYNAMIC THERMOELASTICITY

Emanating from the Boltzmann transport equation, a new C- and F-processes heat conduction constitutive model is derived. The model acknowledges the notion of the simultaneous coexistence of both the slow Cattaneo-type C-processes and fast Fourier-type F-processes in the mechanisms of heat conduction. The C- and F-processes heat conduction constitutive model and the corresponding temperature equation that results from coupling the constitutive model with the energy equation naturally lead to a generalization of the macroscale in space one-temperature theory for heat conduction in solids of the Jeffreys'-type model, Cattaneo model, and the Fourier model for heat conduction in solids. This is unlike the Jeffreys'-type phenomenological model, which cannot reduce to the classical Fourier model (but only to a Fourier-like representation with relaxation) and it cannot explain the underlying physics associated with the C- and F-processes model. Additionally, the microscale in space two-temperature theory for pulse heating of metals is also highlighted via the C- and F-processes heat conduction constitutive model. Emphasis is placed on the development of a new C- and F-processes heat conduction model based on generalized thermoelastic theory to study the dynamic thermoelastic behavior of solids with special features that can lead to and explain the classical and nonclassical dynamic thermoelastic theories. Finally some conceptual pitfalls that appear in the literature are addressed.     

Heat conduction investigation of functionally graded material plates under the exponential heat source load

This paper adopts the hybrid numerical method to research the heat conduction of the functionally graded materials (FGM) plates under the exponential heat source load. Based on the heat balance equation, the hybrid numerical method theory is established through the method of weighted residuals. Then inversely Fourier transformed equation is applied to get the temperature distribution of FGM plates. And the temperature variation law at different positions is obtained in the loading time. The results show that at different positions temperature changing was consistent with heat source and gradually reduced with the distance away from the heat source. The heat transfer efficiency is related to the material performance of the surface.     

On the Theory of Two-Temperature Thermoelasticity with Two Phase-Lags

The present paper is concerned with the theory of two temperature thermoelasticity with two phase-lags in which the theory of heat conduction in deformable bodies depends on two distinct temperatures – the conductive temperature and the thermodynamic temperature. A generalized heat conduction law with dual-phase-lag effects was proposed by Tzou (1995) for the purpose of considering the delayed response in times due to the microstructural interactions in the heat transport mechanism. Recently, Quintanilla (2008) has proposed to combine this constitutive equation with a two temperature heat conduction theory and has proved that a dual-phase-lag theory with two temperatures is a well-posed problem. In the present work we consider the basic equations concerning this dual-phase lag theory of two temperature thermoelasticity and make an attempt to establish some important theorems in this context. A uniqueness theorem has been established for a homogeneous and isotropic body. An alternative characterization of mixed boundary initial value problem is formulated and a variational principle as well as reciprocal principle have been established.     

Fractional phase-lag Green–Naghdi thermoelasticity theories

The constitutive equations are given with a fractional Maxwell–Cattaneo heat conduction law using the Caputo fractional derivative and the fractional order heat transport equation is given. The uniqueness theorem is proved, the reciprocity relation is deduced and the variational characterization of solution is given. Seven thermoelasticity theories result from the given problem as special cases (Coupled, Lord–Shulman, phase-lag Green–Naghdi theories, Green–Naghdi theory without energy dissipation and the Green–Naghdi theory of type III).     

Second law analysis of coupled conduction–radiation heat transfer with phase change

This work considers an exergy-based analysis of two-dimensional solid-liquid phase change processes in a square cavity enclosure. The phase change material (PCM) concerns a semi-transparent absorbing, emitting and anisotropically scattering medium with constant thermodynamic properties. The enthalpy-based energy equation is solved numerically using computational fluid dynamics. Once the energy equation is solved, local exergy loss due to heat conduction and radiative heat transfer during the phase change process is calculated by post processing procedures. In this work, the radiation exergy loss in the medium and at the enclosure boundary is taken into consideration. It is found that radiation exergy loss is significant in the high-temperature phase change process. Parametric investigation is also carried out to study the effects of Stefan number, Biot number, Planck number, single scattering albedo and wall emissivity on exergy loss. The results show that the total exergy loss increases with Biot number, single scattering albedo and wall emissivity. The second law effects of the conduction–radiation coupling in the energy equation are also shown in this work.     

THERMOMECHANICAL ANALYSIS OF FUNCTIONALLY GRADED CYLINDERS AND PLATES

The dynamic thermoelastic response of functionally graded cylinders and plates is studied. Thermomechanical coupling is included in the formulation, and a finite element model of the formulation is developed. The heat conduction and the thermoelastic equations are solved for a functionally graded axisymmetric cylinder subjected to thermal loading. In addition, a thermoelastic boundary value problem using the first-order shear deformation plate theory (FSDT) that accounts for the transverse shear strains and the rotations, coupled with a three-dimensional heat conduction equation, is formulated for a functionally graded plate. Both problems are studied by varying the volume fraction of a ceramic and a metal using a power law distribution.     

An investigation on thermoelastic interactions under an exact heat conduction model with a delay term

The present work is concerned with a very recently proposed heat conduction model: an exact heat conduction model with a single delay term. A generalized thermoelasticity theory was proposed by Roy Choudhuri based on the heat conduction law with three-phase-lag effects for the purpose of considering the delayed response in time due to the microstructural interactions in the heat transport mechanism. However, the model defines an ill-posed problem in Hadamard sense. Quintanilla has recently proposed to reformulate this heat conduction model as an alternative heat conduction theory with a single delay term and subsequently, Leseduarte and Quintanilla investigated the spatial behavior of the solutions for this theory and they extended the results to a thermoelasticity theory by considering the Taylor series approximation of the equation of heat conduction with one delay term. In the present work, we consider the thermoelasticity theory based on this newly proposed heat conduction model and investigate a problem of thermoelastic interactions. State-space approach is used to formulate the problem and the formulation is then applied to a problem of an isotropic elastic half-space with its plane boundary subjected to sudden increase in temperature and zero stress. The integral transform method is applied to obtain the solution of the problem. A detailed analysis of analytical results is provided by finding the short-time approximated solutions of different field variables analytically and comparing the results of the present model with the corresponding results reported for other existing theories. An attempt has also been made to illustrate the problem and numerical values of field variables are obtained for a particular material. Results are analyzed with different graphs. To the best of the author\textquoteright s knowledge, this thermoelastic model is not yet investigated by any researcher in this direction.     

Thermodynamic stability analysis of non-Fourier model-based bio-heat transfer in laser-irradiated cylindrical-shaped multilayered biological tissue

The present research focuses on examining the thermic response of living tissue in the form of a triple-layered cylindrical structure when subjected to laser light and the compatibility analysis of non-Fourier heat transfer with thermodynamics second law. The temperature field in the triple-layered cylindrical living tissue subjected to laser light is determined by numerically solving the transient radiative transfer equation in conjunction with the dual phase lag (DPL) based bioheat equation. Once the temperature field is known, the equilibrium and nonequilibrium entropy production rate (EPR) is calculated based on the hypothesis of classical irreversible thermodynamics and extended irreversible thermodynamics, respectively. The present results are verified against the data from the literature and found a good match between them. A comparative analysis of the Fourier and non-Fourier models is accomplished. The equilibrium and nonequilibrium EPR values for the Fourier model are found to be positive. While the equilibrium EPR is negative for non-Fourier heat conduction and does not satisfy the thermodynamics second law, nonequilibrium EPR is always a positive value for Fourier, DPL, and hyperbolic models and satisfies thermodynamics second law. It has been investigated how thermal relaxation times affect the temperature field and EPRs in tissue are subjected to laser light.     

Numerical analysis of wave-type heat transfer propagating in a thin foil irradiated by short-pulsed laser

This paper presents a numerical simulation of wave-type heat transfer phenomena propagating in an aluminum thin foil irradiated by a pulsed laser using the cubic interpolated propagation method coupled with a thermo-convective model. We did not use the two-step model and dual-phase lag model, which are generally known as the non-Fourier heat conduction law, but wave-type heat transfer phenomena could be observed by our method. The main characteristic of the method is to solve the governing equation including the equation of continuity, the equation of motion, the equation of energy and the equation of state. It is found that when the pulse duration is under the order of picosecond, the pure heat conduction is hardly observed and heat transfer toward the inside of materials occurs only by a thermal shock wave. The heat conduction mode after pulse laser irradiation is strongly dependent upon the value of total incident laser energy density E_in and the threshold value for pure heat conduction is 5.0 × 10⁴ J/m² for an aluminum.     

Green–Naghdi type III viscous fluids

We investigate constitutive theory for a new type of thermoviscous fluid. Equations are derived which reduce to the classical heat conduction equation under Fourier’s law when the new parameters tend to zero. Basic solutions for parallel flows are found and a travelling wave analysis is presented.     

Exponential stability of a thermoviscoelastic mixture with second sound

In this work, we consider a one-dimensional model of a thermoviscoelastic mixture with second sound. Under suitable assumption on the constitutive constants of the system, we prove, using the theory of semigroup of linear operators and results obtained by Prüss (1984) and Borichev and Tomilov (2009), that the damping effect through heat conduction given by the Cattaneo law is strong enough to stabilize the system.     

Exact solution for thermal damping of functionally graded Timoshenko microbeams

This article deals with the thermoelastic damping problem in a functionally graded (FG) Timoshenko microbeam. Thermal and mechanical properties of the microbeam vary in the thickness direction according to the power law relation. Employing Timoshenko beam theory, the governing dynamic equation coupled with thermal effects of the FG microbeam is developed. Afterwards, Using the Taylor series expansion for material properties, the heat conduction equation is solved analytically for temperature in the form of a power series. The free vibration of the FG microbeam is analyzed to achieve the natural frequencies and thermal damping ratio of the FG microbeam. The effect of FG index on the thermoelastic damping ratio is investigated in different aspect ratios. Also comparison studies are made between the results obtained from the models based on the Euler–Bernoulli and Timoshenko beam theories.     

Heat flow choking in carbon nanotubes

Based on Einstein’s mass–energy relation, the equivalent mass of thermal energy or heat is identified and referred to as thermomass. Hence, heat conduction in carbon nanotubes (CNTs) can be regarded as the motion of the weighty phonon gas governed by its mass and momentum conservation equations. The momentum conservation equation of phonon gas is a damped wave equation, which is essentially the general heat conduction law since it reduces to Fourier’s heat conduction law as the heat flux is not very high and the consequent inertial force of phonon gas is negligible. The ratio of the phonon gas velocity to the thermal sound speed (the propagation speed of thermal wave) can be defined as the thermal Mach number. For a CNT electrically heated by high-bias current flows, the phonon gas velocity increases along the heat flow direction, just like the gas flow in a converging nozzle. The heat flow in the CNT is governed by the electrode temperature until the thermal Mach numbers of phonon gas at the tube ends reach unity, and the further reduction of the electrode temperature has no effect on the heat flow in the CNT. Under this condition, the heat flow is said to be choked and temperature jumps will be observed at the tube ends. In this case the predicted temperature profile of the CNT based on Fourier’s law is much lower than that based on the general heat conduction law. The thermal conductivity which is determined by the measured heat flux over the temperature gradient of the CNT will be underestimated, and this thermal conductivity is actually the apparent thermal conductivity. In addition, the heat flow choking should be avoided in engineering situations to prevent the thermal failure of materials.     

Analytical solution of heat conduction problem for a multilayer assembly arising in photothermal reliability testing

An analytical solution of a two-dimensional heat conduction problem is obtained for a multilayer thin coating-substrate assembly irradiated by a laser beam. A method of solution is based on expansion of Laplace and Hankel transforms of temperature distribution in terms of a small parameter which is equal to the ratio of a coating thickness to a laser beam radius. It is shown that the boundary value problem of the second kind reduces to a generalized characteristic equation. It includes as particular cases the known in heat conduction theory characteristic equations which have been obtained for the one-dimensional boundary value problems of the second and third kinds. Also an asymptotic expression for a temperature distribution in a substrate is derived for the case of small values of Fourier number.     

Fractional heat conduction in a thin hollow circular disk and associated thermal deflection

The time nonlocal generalization of the classical Fourier law with the “Long-tail” power kernel can be interpreted in terms of fractional calculus and leads to the time fractional heat conduction equation. The solution to the fractional heat conduction equation under a Dirichlet boundary condition with zero temperature and the physical Neumann boundary condition with zero heat flux are obtained by integral transform. Thermal deflection has been investigated in the context of fractional-order heat conduction by quasi-static approach for a thin hollow circular disk. The numerical results for temperature distribution and thermal deflection using thermal moment are computed and represented graphically for copper material.     

Equivalence Principle for Analyzing Steady-State Heat Conduction Problems

This article presents the concept of equivalence principle in the analysis of steady-state heat conduction problems. A surface integral equation formulation is established for homogeneous thermal media and a volume integral equation formulation for inhomogeneous isotropic/anisotropic thermal media. These formulations are analogous to those commonly used in electromagnetic scattering problems and can be solved with the same numerical algorithms used for electromagnetic analysis. This feature makes the proposed theory useful for simultaneous analysis of electromagnetic fields and heat conduction in an electronic system.