Thermal Wave Speed Research Articles

Introducing electromagnetic effects in Soledge3X

AbstractIn the pedestal region, electromagnetic effects affect the evolution of micro‐instabilities and plasma turbulence. The transport code Soledge3X developed by the CEA offers an efficient framework for turbulent 3D simulation on an electrostatic model with a fixed magnetic field. The physical accuracy of the model is improved with electromagnetic induction, driven by the local value of the parallel component of the electromagnetic vector potential , known from Ampère's law. It is solved implicitly in a coupled system with the vorticity equation on the electric potential . The consequence is a basic electromagnetic behavior in the form of shear Alfvén waves. A finite electron mass prevents unphysical speeds but requires solving for the time evolution of the parallel current density in the generalized Ohm's law. This term can be analytically included with little computational overhead in the system on and and improves its numerical condition, facilitating the iterative solving procedure. Simulations on a periodic slab case let us observe the predicted bifurcation of the wave propagation speed between the Alfvén wave and the electron thermal wave speeds for varying perpendicular wavenumbers. The first results on a circular geometry with a limiter attest to the feasibility of turbulent electromagnetic scenarios.

Thermal shock fracture of honeycomb-based porous thermoelectric materials with non-Fourier heat conduction

Compared with dense thermoelectric materials (TEMs), the auxetic honeycomb porous TEMs can enhance mechanical performance and energy conversion efficiency. The Fourier heat conduction law becomes unavailable due to the thermal wave speed in the porous TEMs is limited. Considering the non-Fourier law, this paper analyzes the thermal shock fracture characteristics of the porous TEMs bonded to an elastic substrate. A theoretical model for the effects of non-Fourier heat conduction, thickness of TME, shape and dimensions of honeycomb cell on strength failure and fracture failure is proposed. The analysis demonstrates that the maximum thermal stresses in the porous TEMs are underestimated without considering the non-Fourier heat conduction. The stress level in the auxetic honeycomb-based TEMs is much smaller than that in the traditional honeycomb TEMs with the positive Poisson's ratio architecture. It is found that a thicker TEM, a bigger relative density and a larger length ratio of the horizontal to the inclined strut can result in a smaller thermal stress. Based on the strength and fracture criterions, the envelope of safety angle between the horizontal and the inclined strut is identified. The simplified expressions of the safety angle envelope versus the applied thermal load are given.

Numerical modelling of hyperbolic phase change problems: Application to continuous casting

Heat diffusion processes are generally modeled based on Fourier’s law to estimate how the temperature propagates inside a body. This type of modeling leads to a parabolic partial differential equation, which predicts an infinite thermal wave speed of propagation. However, experimental evidence shows that diffusive processes occur with a finite velocity of thermal propagation in many applications. In this paper, we develop a mathematical formulation to predict the finite speed of heat propagation in multidimensional phase change problems. The model generalizes the enthalpy formulation by adding a hyperbolic term. The governing equations are simulated by the finite element method. The proposed model is first verified by comparing numerical and experimental results illustrating the difference between the infinite and finite propagation velocity for heat inside biological tissues. Then, the results of the two and three-dimensional numerical solution of the continuous steel casting process are presented. We will illustrate that the effects of the initial conditions vanish faster when using the parabolic equation, while they persist in the hyperbolic modeling approach. The results demonstrate significant differences in the initial thermal dynamics and at the solid-liquid interface position when adding the hyperbolic term. The changes are more noticeable in the regions of the steel beam where rapid heat loss and, consequently, faster phase change occur.

Transient Finite-Speed Heat Transfer Influence on Deformation of a Nanoplate with Ultrafast Circular Ring Heating

The present study provides a theoretical estimate for the thermal stress distribution and the displacement vector inside a nano-thick infinite plate due to an exponentially temporal decaying boundary heating on the front surface of the elastic plate. The surface heating is in the form of a circular ring; therefore, the axisymmetric formulation is adopted. Three different hyperbolic models of thermal transport are considered: the Maxwell-Cattaneo-Vernotte (MCV), hyperbolic Dual-Phase-Lag (HDPL) and modified hyperbolic Dual-Phase-Lag (MHDPL), which coincides with the two-step model under certain constraints. A focus is directed to the main features of the corresponding hyperbolic thermoelastic models, e.g., finite-speed thermal waves, singular surfaces (wave fronts) and wave reflection on the rear surface of the plate. Explicit expressions for the thermal and mechanical wave speeds are derived and discussed. Exact solution for the temperature in the short-time domain is derived when the thermalization time on the front surface is very long. The temperature, hydrostatic stress and displacement vector are represented in the space-time domain, with concentrating attention on the thermal reflection phenomenon on the thermally insulated rear surface. We find that the mechanical wave speeds are approximately equal for the considered models, while the thermal wave speeds are entirely different such that the modified hyperbolic dual-phase-lag thermoelasticity has the faster thermal wave speed and the Lord-Shulman thermoelasticity has the slower thermal wave speed.

Dynamic crack analysis in anisotropic media under wave-like thermal loading

In this paper, the eXtended finite element method (XFEM) is used to study a cracked orthotropic medium under a non-classic thermal shock considering the Green-Naghdi (GN) theory of thermoelasticity (type II). A brief critical review of the generalized thermoelasticity in anisotropic materials is presented, and outstanding features of the Green-Naghdi theory in the description of the thermal wave traveling in an anisotropic medium are discussed. Tip enrichment functions are derived for the temperature according to the governing equations of the Green-Naghdi type II model. In addition, new simple and efficient tip enrichment functions are proposed, which are identical for both displacement and temperature fields. The effect of the thermal wave speed in the material directions on the thermal wave traveling and its reflection is studied. In particular, the time variation of the Stress Intensity Factors (SIFs) and the temperature distribution based on the GN theory are numerically compared with the classic thermoelasticity (CTE) theory.

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.

Identification of the thermo-physical properties of a stratified tissue. Adiabatic hypodermic wall

Biological living tissues possess thermophysical properties that are difficult to measure directly and their estimation is very important in order to decide the appropriate treatment procedure (e.g. place of inoculation and appropriate amount of drug dosage) in case an abnormality is detected. In the present paper, we numerically investigate, for the first time, the determination of several thermo-physical blood-tissue properties of a one-dimensional, multi-layered biological skin tissue subjected to external heating. In contrast to the usual parabolic model that assumes infinitely fast propagation of the heat signal, in the present paper, the bio-heat transfer in such a biological body is mathematically modelled by a hyperbolic partial differential equation, which takes into account that the thermal wave speed is finite. On the skin surface a convective boundary condition holds taking into account the heat exchange with the environment, whilst on the most inner wall of the tissue an adiabatic boundary condition applies. Then, in this framework, the piecewise constant thermo-physical properties of interest given by the thermal conductivity, heat capacity and blood perfusion rate are accurately and stably reconstructed from non-intrusive temperature measurements on the inner and outer boundaries of the multi-layered tissue by minimizing a weighted nonlinear least-squares objective function.

Effects of non-Fourier heat conduction and surface heating rate on thermoelastic waves in semi-infinite ceramics subject to thermal shock

Non-Fourier heat conduction has been observed in many fast heating experiments. This paper studies the thermoelastic fields of semi-infinite ceramics subjected to surface heating shock based on non-Fourier C–V model, which may have its root application for future thermal protection system of space vehicles. To simulate the surface heating process, a family of exponential functions is presented. Furthermore, the ratio of heating duration to thermal relaxation time, and the ratio of thermal wave speed to elastic wave speed are introduced to study the coupling effects of heating rate, non-Fourier heat conduction and inertia. Firstly, the time horizon and spatial scale where the non-Fourier effect is significant has been delimited. Secondly, it is demonstrated that the thermal stress component along depth increases and converges as the depth increasing. Thirdly, the domain where the maximum thermal stresses influenced by non-Fourier heat conduction are drawn by the critical lines in the plane of dimensionless heating duration vs thermoelastic wave speed. This study may be useful for designs of ceramic heat insulations.

Non-contact, non-destructive mapping of thermal diffusivity and surface acoustic wave speed using transient grating spectroscopy

We present new developments of the laser-induced transient grating spectroscopy (TGS) technique that enable the measurement of large area 2D maps of thermal diffusivity and surface acoustic wave speed. Additional capabilities include targeted measurements and the ability to accommodate samples with increased surface roughness. These new capabilities are demonstrated by recording large TGS maps of deuterium implanted tungsten, linear friction welded aerospace alloys, and high entropy alloys with a range of grain sizes. The results illustrate the ability to view the grain microstructure in elastically anisotropic samples and to detect anomalies in samples, for example, due to irradiation and previous measurements. They also point to the possibility of using TGS to quantify grain size at the surface of polycrystalline materials.

Experimental study of propane/air premixed flame dynamics with hydrogen addition in a meso-scale quartz tube

The combustion process of propane/air premixed flame in meso-scale quartz tubes with different hydrogen additions was investigated experimentally to explain the flame-wall interaction mechanism. The ranges of different flame regimes were obtained by changing the flow rates of propane and hydrogen. The effects of hydrogen addition, inlet velocity and equivalence ratio were analyzed. The results show that the hydrogen addition broadens the operation ranges of fast flame regime and slow flame regime significantly. The flame propagation speed is in the same order of the thermal wave speed in solid wall for the slow flames. In fast flame regime, the flame propagation speed has an inverse correlation with the inlet flow velocity irrespective of the equivalence ratio. With the increase of the equivalence ratio, the maximum flame speed in fast flame regime decreases gradually, while the maximum flame speed in slow flame regime increases continually. It indicates that rich fuel condition suppresses the fast flame and promotes the slow flame. In slow flame regime, the output thermal efficiency is dominated by the inlet velocity and equivalence ratio.

Dynamic model of fractional thermoelasticity due to ramp-type heating with two relaxation times

This is an attempt to design a fractional heat conduction model in a bounded cylindrical region exposed to axisymmetric ramp-type heat flux and discuss its thermal behaviour. This model is drafted using the classical theory of fractional thermoelasticity due to the involvement of two relaxation times to the heat conduction equation and the equations of motion. In order to achieve finite thermal wave speed, the heat conduction equation is derived from the viewpoint of Maxwell–Cattaneo law in the context of fractional derivative. Analytical results for the distribution of temperature, displacement and thermal stresses are obtained using integral transforms in the Laplace domain. The Gaver–Stehfest method has been used to invert the results of Laplace domain. The convergence of infinite series solutions has been discussed. As a specific case this model has been applied to a thick circular plate subjected to the axisymmetric ramp-type heat flux. The results obtained for the thermal variations are validated by comparing to coupled and generalized theory of thermoelasticity. These comparisons are shown graphically.

Transient grating spectroscopy of thermal diffusivity degradation in deuterium implanted tungsten

Using transient grating spectroscopy, we measure thermal diffusivity and surface acoustic wave speed in tungsten exposed to different fluences of deuterium plasma. Scanning electron microscopy (SEM) shows the formation of surface blisters that have similar morphology for all fluences considered. A significant reduction in thermal diffusivity and surface acoustic wave speed occurs as a result of plasma exposure. A saturation of the thermal diffusivity reduction with fluence is seen. Deuterium ion flux density appears to play a more important role in thermal diffusivity reduction than exposure time. These observations have important implications for plasma facing components in future fusion reactors.

Numerical Technique for Resolving the Dual Phase Lag Heat Conduction in Thin Film Metal

In this article, a three-time level finite difference scheme is used to resolve the dual phase lag’s (DPL) heat conduction in a micro scale gold film subjected to spontaneous temperature boundary conditions without knowing the heat flux. Finite difference analog of DPL equation on applying to the intermediate grid points of the computational domain results into a system of linear, algebraic equations which can be solved using Thomas’ algorithm to finally obtain the transient temperature solution distributions in the film. The solution predicted by the DPL model is compared with that obtained by the single-phase Cattaneo–Vernotte’s model. Further, the way in which non-Fourier’s temperature distributions affected by the diffusion due to the increase in Heat Conduction Model numbers agree with the predecessor’s published results. The results by both the models revealed a finite thermal wave speed in the film contrasting the infinite speed of heat propagation as stated by the classical Fourier’s thermal model. Low spatial step and higher order finite difference schemes are recommended for better accurate numerical results of the non-Fourier’s temperature distributions occurring in the very short transient period between the instants of the suddenly applied spatial temperature gradient and the reaching of the steady state conditions.

Thermal Stress Analysis of an Object with a Focus Considering Finite Thermal Wave Speed

Thermal Stress Analysis of an Object with a Focus Considering Finite Thermal Wave Speed

Dual-phase-lag thermoelastic problem in finite cylindrical domain with relaxation time

Purpose The purpose of this paper is to frame a dual-phase-lag model using the fractional theory of thermoelasticity with relaxation time. The generalized Fourier law of heat conduction based upon Tzou model that includes temperature gradient, the thermal displacement and two different translations of heat flux vector and temperature gradient has been used to formulate the heat conduction model. The microstructural interactions and corresponding thermal changes have been studied due to the involvement of relaxation time and delay time translations. This results in achieving the finite speed of thermal wave. Classical coupled and generalized thermoelasticity theories are recovered by considering the various special cases for different order of fractional derivatives and two different translations under consideration. Design/methodology/approach The work presented in this manuscript proposes a dual-phase-lag mathematical model of a thick circular plate in a finite cylindrical domain subjected to axis-symmetric heat flux. The model has been designed in the context of fractional thermoelasticity by considering two successive terms in Taylor’s series expansion of fractional Fourier law of heat conduction in the two different translations of heat flux vector and temperature gradient. The analytical results have been obtained in Laplace transform domain by transforming the original problem into eigenvalue problem using Hankel and Laplace transforms. The numerical inversions of Laplace transforms have been achieved using the Gaver−Stehfast algorithm, and convergence criterion has been discussed. For illustrative purpose, the dual-phase-lag model proposed in this manuscript has been applied to a periodically varying heat source. The numerical results have been depicted graphically and compared with classical, fractional and generalized thermoelasticity for various fractional orders under consideration. Findings The microstructural interactions and corresponding thermal changes have been studied due to the involvement of relaxation time and delay time translations. This results in achieving the finite speed of thermal wave. Classical coupled and generalized thermoelasticity theories are recovered by considering the various special cases for different order of fractional derivatives and two different translations under consideration. This model has been applied to study the thermal effects in a thick circular plate subjected to a periodically varying heat source. Practical implications A dual-phase-lag model can effectively be incorporated to study the transient heat conduction problems for an exponentially decaying pulse boundary heat flux and/or for a short-pulse boundary heat flux in long solid tubes and cylinders. This model is also applicable to study the various effects of the thermal lag ratio and the shift time. These dual-phase-lag models are also practically applicable in the problems of modeling of nanoscale heat transport problems of semiconductor devices and accordingly semiconductors can be classified as per their ability of heat conduction. Originality/value To the authors’ knowledge, no one has discussed fractional thermoelastic dual-phase-lag problem associated with relaxation time in a finite cylindrical domain for a thick circular plate subjected to an axis-symmetric heat source. This is the latest and novel contribution to the field of thermal mechanics.

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.

Non-Fourier heat conduction/convection in moving medium

Non-Fourier one-dimensional unsteady heat conduction in a moving medium is investigated by using the Cattaneo-Vernotte- Christov-Jordan (CVCJ) heat flux model for medium speeds less than (sub-critical), equal to (critical), and greater than (super-critical) the thermal wave speed. Coupled partial differential equations are solved simultaneously by a finite volume numerical method. Temperature and heat flux distributions for sub-critical, critical, and super-critical flow conditions are presented for two example problems. The importance of boundary conditions on the thermal wave propagation in both sub-critical and super-critical cases is discussed. Approximate analytical solutions are presented which qualitatively substantiate the numerical results.

A modified Green–Lindsay thermoelasticity with strain rate to eliminate the discontinuity

Probing the mechanism of ultrafast thermoelastic processes is becoming increasingly important with the development of laser-assisted micro/nano machining. Although thermoelastic models containing temperature rate have been historically proposed, the strain rate has not been considered yet. In this work, a generalized thermoelastic model is theoretically established by introducing the strain rate in Green–Lindsay (GL) thermoelastic model with the aid of extended thermodynamics. Numerically, a semi-infinite one-dimensional problem is considered with traction free at one end and subjected to a temperature rise. The problem is solved using the Laplace transform method, and the transient responses, i.e. displacement, temperature and stresses are graphically depicted. Interestingly, it is found that the strain rate may eliminate the discontinuity of the displacement at the elastic and thermal wave front. Also, the present model is compared with Green–Naghdi (GN) models. It is found that the thermal wave speed of the present model is faster than GN model without energy dissipation, and slower than GN model with energy dissipation. In addition, the thermoelastic responses from the present model are the largest. The present model based upon GL model is free of the jump of GL model in the displacement distribution, and is safer in engineering practices than GN model. The present work will benefit the theoretical modeling and numerical prediction of thermoelastic process, especially for those under extreme fast heating.

Reflection and refraction of a thermal wave at an ideal interface

Thermal waves are of great significance as non-Fourier effects arise with ultrafast heating rates and small system length. This study analytically and numerically investigated the behavior of thermal waves based on the Cattaneo-Vernotte model at an ideal interface. A stable, fast algorithm based on the alternative direction implicit method is introduced to solve the two-dimensional heat conduction problem. When thermal waves meet with an ideal interface, some energy is reflected back while the rest is conveyed across the interface, which are called the reflection and refraction of thermal waves. The changes of the profile and direction and the energy distribution between the reflection and refraction of the thermal waves are studied both analytically and numerically. Regardless of the boundary conditions imposed on the interface, the reflection angle is always identical to the incident angle, and the ratio of the sine of the refraction angle of the thermal waves to that of the incident angle is equal to the ratio of the thermal wave speeds in the two material layers. A theoretical equation to describe the relationships between the energy distribution and the material thermal properties shows that the thermal wave speeds in the materials, the specific heat and the incident angle determine the thermal energy transmittance ratio. Total reflection can occur for some conditions, and the nature of the energy conveyed by thermal waves is interesting and instructive.

Coupling effects of inertia and dual-phase-lag heat conduction on thermal shock fracture of a cracked piezoelectric layer

This paper studies the thermal shock fracture mechanics of a piezoelectric material thin layer with an internal crack. The typical model I and model II cracking problems are separately considered. The analyses are based on the dual-phase-lag, non-Fourier heat conduction theory with consideration of material inertia. Laplace transform and dual integral equation technique are applied to solve the problems. Some numerical results of the stress intensity factor and energy release rate are obtained and drawn in figures. It is demonstrated that the effect of the inertia on the fracture behavior depends on the ratio of the thermal wave speed to the stress wave velocity. In addition, the large ratio of the temperature gradient lag to the thermal flux lag will greatly enhance the mode I and mode II stress intensity factors.