Propagation Speed Of Thermal Wave Research Articles

Thermoelastic responses of biological tissue under thermal shock based on three phase lag model

The paper dealt with the thermoelastic responses of biological tissues subjected to an abrupt input of temperature load in the context of generalized thermoelasticity with three phase lag times. A modified thermoelastic model with three phase lag times was proposed to resolve the discrepancy between the three-phase-lag thermoelastic model and existing models in propagation of thermal wave. The Laplace transform and numerical inverse transform methods were applied to acquire the distributions of temperature, displacement and stresses. The effects of rate of thermal conductivity, phase lag times and rate of blood perfusion on thermoelastic behaviors were discussed specifically. The rate of thermal conductivity plays a crucial role in determining the propagation speed of thermal wave in the framework of Three-Phase-Lag (TPL) or Green-Naghdi (G-N) thermoelasticity.

Non‐Fourier fractional thermoelastic two‐dimensional model of a hollow sphere

Assuming non‐Fourier thermal effects, Tzou's dual‐phase‐lag model has been applied to introduce the governing heat conduction equation in the presented mathematical model. Moreover, in order to design a well‐posed dual‐phase‐lag model, the governing time fractional dual‐phase‐lag heat equation has been established by introducing conductive temperature and thermodynamical temperature, satisfying the two‐temperature theory. Due to the application of phase lags τΩ, τΘ satisfying τΩ > τΘ, the finite speed of thermal wave propagation has been achieved. The corresponding governing equations of motion and stresses have been considered in two‐dimensional bounded spherical domain. The spherical boundaries are assumed to be traction free. The Laplace and the Legendre integral transforms have been applied to obtain the analytical solutions of conductive and thermodynamical temperatures, displacement components, and thermal stresses. The Gaver–Stehfest algorithm has been employed to achieve the time domain inversions of Laplace transforms numerically, satisfying the Kuznetsov convergence criteria. Classical, fractional and generalized thermoelasticity theories has been recovered theoretically and numerically as well for various fractional orders and phase‐lags.

A numerical approach to non-Fourier heat transfer in liver tumor during laser irradiation

Thermal therapy is a type of cancer treatment that uses heat to kill cancer cells, but it also may harm healthy tissue. Numerical simulations can help to accurately analyze the thermal damage of the tissue during heat exposure. The target of this study is to investigate the effect of time lags on the thermal response of the biological tissue during laser irradiation to the tumoral tissue. The classical Fourier, single phase lag (SPL) and dual phase lag (DPL) models of bio-heat transfer are implemented and compared. The numerical solution based on the finite volume method (FVM) is applied to solve the bio-heat transfer equations. Beer-Lambert’s law is applied to determine the heat source distribution caused by the laser irradiation. The thermal damage caused by the laser exposure for the three models is discussed. Results show that the DPL model predicts a significantly different thermal damage from the classical Fourier and the SPL models. It is observed that the DPL model predicts the maximum temperature 4.1 ˚C and 5.7 ˚C less than the Fourier and the SPL models, respectively. The deviation between the maximum temperatures obtained by the three models can be attributed to the finite speed of thermal wave propagation in the non-Fourier models.

Two temperature fractional order thermoelasticity theory in a spherical domain

This article is an application of fractional thermoelasticity in association with two-temperature theory. The fractional heat conduction model has been proposed to investigate the thermal variations within the bounded spherical region. The corresponding heat conduction equation has been derived in the context of the generalized two-temperature theory of fractional thermoelasticity. The analytical solutions of thermal variations have been obtained in the Laplace domain, which are inverted using the Gaver–Stehfest algorithm in the time domain. Kuznetsov convergence criterion has been discussed for the bounded variations and stability of the problem. The delay time translations used in the heat flux vector and the temperature gradient result in the finite speed of thermal wave propagation. As a special case of time fractional derivative, the classical and generalized thermoelasticity theories have been recovered.

Numerical studies on the radiation uniformity of Z-pinch dynamic hohlraum

Radiation uniformity is important for Z-pinch dynamic hohlraum driven fusion. In order to understand the radiation uniformity of Z-pinch dynamic hohlraum, the code MULTI-2D with a new developed magnetic field package is employed to investigate the related physical processes on Julong-I facility with drive current about 7–8 MA. Numerical simulations suggest that Z-pinch dynamic hohlraum with radiation temperature more than 100 eV can be created on Julong-I facility. Although some X-rays can escape out of the hohlraum from Z-pinch plasma and electrodes, the radiation field near the foam center is quite uniform after a transition time. For the load parameters used in this paper, the transition time for the thermal wave transports from r = 1 mm to r = 0 mm is about 2.0 ns. Implosion of a testing pellet driven by cylindrical dynamic hohlraum shows that symmetrical implosion is hard to achieve due to the relatively slow propagation speed of thermal wave and the compression of cylindrical shock in the foam. With the help of quasi-spherical implosion, the hohlraum radiation uniformity and corresponding pellet implosion symmetry can be significantly improved thanks to the shape modulation of thermal wave front and shock wave front.

One-dimensional numerical investigation on the formation of Z-pinch dynamic hohlraum using the code MULTI

Z-pinch dynamic hohlraum can effectively convert Z-pinch plasma kinetic energy into radiation field energy, which has a potential to implode a pellet filled with deuterium-tritium fuel to fusion conditions when the drive current is sufficiently large. To understand the formation process of Z-pinch dynamic hohlraum on JULONG-I facility with a typical drive current of 8-10 MA, a new radiation magneto-hydrodynamics code is developed based on the program MULTI-IFE. MULTI-IFE is a one-dimensional, two-temperature, multi-group, open-source radiation hydrodynamic code, which is initially designed for laser and heavy ion driven fusion. The original program is upgraded to simulate Z-pinch related experiments by introducing Lorentz force, Joule heating and the evolution of magnetic field into the code. Numerical results suggest that a shock wave and a thermal wave will be launched when the high speed plasma impacts onto the foam converter. The thermal wave propagates much faster than shock wave, making the foam become hot prior to the arrival of shock wave. For the load parameters and drive current of shot 0180, the calculated propagation speed of thermal wave and shock wave are about 36.1 cm/s and 17.6 cm/s, respectively. The shock wave will be reflected when it arrives at the foam center and the speed of reflected shock wave is about 12.9 cm/s. Calculations also indicate that the plastic foam will expand obviously due to the high temperature radiation environment (~30 eV) around it before the collision between tungsten plasma and foam converter. The evolution of radial radiation temperature profile shows that a pair of bright strips pointing to the foam center can be observed by an on-axis streak camera and the radiation temperature in the foam center achieves its highest value when the shock arrives at the axis. A bright emission ring moving towards the foam center can also be observed by an on-axis X-ray frame camera. The best time to capture the bright strips and bright emission rings is before the thermal wave reaches the foam center. Even though some amount of X-ray radiation in the foam is expected to escape from the hohlraum via radiation transport process, simulation results suggest that the tungsten plasma can serve as a good hohlraum wall. The radiation temperature is about 80 eV when the dynamic hohlraum is created and can rise more than 100 eV before the shock arrives at the foam center. Most of the X-rays emitted by the wire-array plasma surface have energies below 1000 eV. In this paper, the physical model of the code MULTI-IFE and the simulation results of array implosions on Saturn facility are presented as well.

A GDQ approach to thermally nonlinear generalized thermoelasticity of disks

ABSTRACTGeneralized thermoelasticity response of an annular disk subjected to thermal shock on its inner surface is analyzed in this research. The Lord–Shulman theory, which accounts for one relaxation time in the conventional Fourier law, is used to avoid the infinite speed of thermal wave propagation. Unlike the other available works in which the first law of thermodynamics is linearized, the nonlinearity arising from the temperature change is taken into consideration. The first law of thermodynamics in this case becomes nonlinear and the analysis under such formulation is called thermally nonlinear. Two coupled equations, i.e., the radial displacement wave equation and temperature wave propagation equation are obtained. These equations and the associated boundary conditions are discreted through the generalized differential quadrature method. Solution of the time-dependent system of equations is obtained using the Newmark time marching scheme and the successive Picard method. Numerical results are provided for both thermally linear and thermally nonlinear temperature and radial displacement wave propagations. Parametric studies reveal that at higher temperature levels, thermally nonlinear first law of thermodynamics should be considered instead of thermally linear one. Furthermore, the higher the coupling parameter and/or relaxation time, the higher the divergence of thermally nonlinear-/linear-based results.

Finite element model for femtosecond laser pulse heating using dual phase lag effect

The ultrashort pulse laser in material processing of micro/nanoscale has emerged as a useful application due to its capability of high precision control on heating and heat affected zone. The non-Fourier effect in heat conduction becomes more predominant as the duration of heating becomes extremely small which is the case of ultrashort pulse laser. To analyze the heat transfer process by an ultrashort pulse laser, a three dimensional finite element based numerical model is developed assuming finite speed of thermal wave propagation with two phase lags. The present work also describes the finite element formulation using phase lag boundary conditions. The numerical model is developed indigenously using an Intel fortran compiler. The transient heating and cooling phenomena are analyzed for the effect of two relaxation times, variation of pulse width, and multiple pulses. The model results are validated with ultrashort pulse laser heating of a thin gold film. The accuracy of the computed results shows promising application of the developed numerical model for differential influence of process parameters in micro/nano scale heat transport by ultrashort pulse laser processing.

On the phase- lag Green–Naghdi thermoelasticity theories

In this work, we proposed three models of generalized thermoelasticity: a single –phase - lag Green–Naghdi theory of type III, a dual-phase–lag Green–Naghdi theory of type II and of type III. The solid is assumed to be linear anisotropic inhomogeneous. A unified heat conduction law and heat transport equation are given, which consolidate the three theories and also the Lord–Shulman theory and the Green–Naghdi theory of type II. Dissipative inequality is derived, constitutive equations and thermodynamic restrictions are given. The speed of thermal wave propagation is given, for each of the three theories in case of isotropic solid. A uniqueness theorem is proved for the considered theories. Variational characterization of solution is established. An application is given for isotropic thermoelastic solid and the results are presented graphically.

Analysis of the dual phase lag bio-heat transfer equation with constant and time-dependent heat flux conditions on skin surface

This article focuses on temperature response of skin tissue due to time-dependent surface heat fluxes. Analytical solution is constructed for DPL bio-heat transfer equation with constant, periodic and pulse train heat flux conditions on skin surface. Separation of variables and Duhamel?s theorem for a skin tissue as a finite domain are employed. The transient temperature responses for constant and time-dependent boundary conditions are obtained and discussed. The results show that there is major discrepancy between the predicted temperature of parabolic (Pennes bio-heat transfer), hyperbolic (thermal wave) and DPL bio-heat transfer models when high heat flux accidents on the skin surface with a short duration or propagation speed of thermal wave is finite. The results illustrate that the DPL model reduces to the hyperbolic model when ?T approaches zero and the classic Fourier model when both thermal relaxations approach zero. However for ?q = ?T the DPL model anticipates different temperature distribution with that predicted by the Pennes model. Such discrepancy is due to the blood perfusion term in energy equation. It is in contrast to results from the literature for pure conduction material, where the DPL model approaches the Fourier heat conduction model when ?q = ?T . The burn injury is also investigated.

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.

Numerical analysis of dual-phase-lag heat transfer in a layered cylinder with nonlinear interface boundary conditions

To accommodate the micro-structural effect, this work analyzes the heat transfer induced by a pulsed surface heating in a two-layered solid cylinder with the dual-phase-lag model. The interface thermal resistance is specified with the radiation boundary condition model. Due to the difference in the relaxation times between two dissimilar materials, the strong nonlinearity of the interfacial boundary condition, and the singularity at the center axis, it introduces the complexity and causes some mathematical difficulties to analyze the present problem. Thus a numerical scheme is developed. Results show the lagging thermal behavior is affected with the geometry effect and the interface thermal resistance and depends on the magnitude of the relaxation times more than the ratio value between the relaxation times. The microstructure effect would destroy the structure of thermal wave. The propagation speed of thermal wave is independent of the interface thermal resistance and the microstructure effect.

On shear band nucleation and the finite propagation speed of thermal disturbances

This paper examines the onset of shear flow localization in thermal viscoplastic materials at high rates of loading while accounting for the finite speed of thermal wave propagation. It is well known that the Fourier's classical law of heat conduction implies that thermal disturbances travel with infinite speed. We introduce a modification to the classical Fourier law which renders a finite speed for thermal wave propagation. We then utilize the energy viewpoint of dynamic localization introduced in Shawki (Shawki, T. G. (1994). An energy criterion for the onset of shear localization in thermal viscoplastic materials, part I: Necessary and sufficient initiation conditions. ASME J. Appl. Mech.61, 530–537) and (Shawki, T. G. (1994). An energy criterion for the onset of shear localization in thermal viscoplastic materials, part II: Applications and implications. ASME J. Appl. Mech.61, 538–547) as well as a linear stability analysis to determine the necessary conditions for the onset of localization. An exact linear solution is obtained for adiabatic deformations of thermal viscoplastic materials with no strain dependence. A matched asymptotic expansion is obtained for the general case involving non-adiabatic deformations of the foregoing class of materials. The foregoing solutions illustrate the effect of finite speed of thermal wave propagation on localization initiation.

Novel model for evolution of normal zones in composite superconductors

This paper presents a numerical analysis of the evolution of normal zones in a composite superconductor based on a novel heat conduction equation which takes into account the finite speed of thermal wave propagation and the inertia of the internal heat source. The results are compared with those obtained from the classical parabolic heat conduction equation. It is shown that appreciable differences can occur between the two models, in qualitative as well as quantitative terms. For large times these differences do not disappear because of the effect of internal heat generation.

Effect of relaxation of internal heat source capacity on the temperature field in the semi-infinite body

The novel heat conduction equation [1,2] which takes into account both the heat flux relaxation (the finite speed of thermal wave propagation) and the relaxation of internal heat source capacity is solved numerically by the MacCormack predictor-corrector method [3]. The temperature field in the body due to a step change in heat flux at the surface is analysed for various ratios of relaxation times. It is shown that the relaxation of internal heat source capacity can have an appreciable influence on the temperature distribution in the body.

SPECIALLY TAILORED TRANS FINITE-ELEMENT FORMULATIONS FOR HYPERBOLIC HEAT CONDUCTION INVOLVING NON-FOURIER EFFECTS

The phenomenon of hyperbolic heat conduction in contrast to the classical (parabolic) form of Fourier heat conduction involves thermal energy transport that propagates only at finite speeds as opposed to an infinite speed of thermal energy transport. To accommodate the finite speed of thermal wave propagation, a more precise form of heat flux law is involved, thereby modifying the heat flux originally postulated in the classical theory of heat conduction. As a consequence, for hyperbolic heat conduction problems, the thermal energy propagates with very sharp discontinuities at the wave front. The primary purpose of the present paper is to provide accurate solutions to a class of one-dimensional hyperbolic heat conduction problems involving non-Fourier effects that can precisely help understand the true response and furthermore can be used effectively for representative benchmark tests and for validating alternate schemes. As a consequence, the present paper purposely describes modeling/analysis formulations via specially tailored hybrid computations for accurately modeling the sharp discontinuities of the propagating thermal wave front. Comparative numerical test models are presented for various hyperbolic heat conduction models involving non-Fourier effects to demonstrate the present formulations.