Finite Speed Of Heat Propagation Research Articles

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.

Математичне моделювання полів температури антенних рефлекторів космічних апаратів за допомогою нового інтегрального перетворення

Prediction of the mechanical properties of the reflector, and above all, the deviation of the highly accurate shape of the reflecting surface (RSS) from the given one is the main goal of designing spacecraft antennas. Distortion of the RSS is determined by the stress-deformed state of the elements of the reflector structures under the conditions of orbital operation. At the same time, the main factor determining the distortion of reflectors with RSS in open space is temperature deformation due to the uneven distribution of solar heat fluxes among structural elements. Therefore, the development of methods and models for calculating temperature fields in reflectors during heat flows on the surface is relevant. In the article, for the first time, a new finite integral transformation for the Laplace equation in a cylindrical coordinate system is constructed for a region bounded by several closed piecewise smooth contours. The inverse transformation formula is given. In the article, for the first time, a mathematical model for the calculation of temperature fields in a paraboloid rotating with a constant angular velocity is constructed, taking into account the finite speed of heat propagation in the form of a boundary value problem of mathematical physics for the hyperbolic equation of heat conduction with Dirichlet boundary conditions. With the help of the developed integral transformation, the temperature fields in the paraboloid were found in the form of convergent series according to the Fourier functions. The found solution to the generalized boundary value problem of the heat transfer of the paraboloid of rotation can find application in modulating the temperature fields that arise in the antenna reflectors of space vehicles. The developed integral transformation makes it possible to obtain solutions to complex boundary value problems of mathematical physics.

Determination of the thermo-physical properties of multi-layered biological tissues

Abnormal biological tissues possess unknown properties that must be estimated in order to decide the appropriate method of treatment in terms of amount of dosage and appropriate place of inoculation. The current work numerically investigates, for the first time, the simultaneous estimation of several thermo-physical properties of multi-layered tissues from internal temperature measurements. The heat propagation within the tissues is modelled by the thermal-wave model of bio-heat transfer, accounting for the finite speed of heat propagation. 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 tissues a Dirichlet boundary temperature condition is prescribed. Accurate and stable reconstruction of the piecewise constant properties given by the thermal conductivity, heat capacity and blood perfusion rate is successfully achieved for two physical examples concerning three- and four-layered, one-dimensional tissues, subjected to externally induced aggression.

On a strain gradient theory of porous thermoelastic solids

The interest in the gradient theory of elasticity is stimulated by the fact that this theory is adequate to investigate important problems related to size effects and nanotechnology. In this paper, we use the theory of microstretch solids and the Green-Naghdi theory of thermomechanics of continua to derive a linear strain gradient theory of porous thermoelastic bodies, which is capable of predicting a finite speed of heat propagation and leads to a symmetric conductivity tensor. A uniqueness theorem for the mixed problem is presented. In the case of isotropic solids we establish the continuous dependence of solutions upon initial data and body supplies. The problem of a concentrated heat source is also investigated.

Numerical analysis of non-Fourier thermal response of lung tissue based on experimental data with application in laser therapy

Background and objectiveThe thermal therapy is a minimally invasive technique used as an alternative approach to conventional cancer treatments. There is an increasing concern about the accuracy of the thermal simulation during the process of tumor ablation. This study is aimed at investigating the effect of finite speed of heat propagation in the biological lung tissue, experimentally and numerically. MethodsIn the experimental study, a boundary heat flux is applied to the lung tissue specimens and the temperature variation is measured during a transient heat transfer procedure. In the numerical study, a code is developed based on the finite volume method to solve the classical bio-heat transfer, the Cattaneo and Vernotte, and the Dual-phase-lag (DPL) equations. The thermal response of tissue during the experiments is compared with the predictions of the three heat transfer models. ResultsIt is found that the trend of temperature variation by the DPL model resembles the experimental results. The experimental observation in parallel with the numerical results reveals that the accumulated thermal energy diffuses to the surrounding tissue with a slower rate in comparison with the conventional bio-heat transfer model. The DPL model is implemented to study the temperature elevation in the laser irradiation to lung tissue in the presence of gold nanoparticles (GNPs). It is concluded that the extent of the necrotic tumoral region and the area of the damaged healthy tissue are reduced, when the non-Fourier heat transfer is taken into account. ConclusionsResults show that considering the phase lags is crucial in planning for an effective thermal treatment, in which the cancerous tissue is ablated and the surrounding tissues are preserved from irreversible thermal damage.

Modelling ultra-fast nanoparticle melting with the Maxwell–Cattaneo equation

• New models of nanoparticle melting based on the Maxwell–Cattaneo equation are proposed. • Consider temperature profiles that are continuous or have a jump across the solid–liquid interface. • The jump condition for the temperature is derived by taking a sharp-interface limit of a phase-field model. • The jump condition is the most suitable condition to use in phase-change models based on the Maxwell–Cattaneo equation. • Thermal relaxation is shown to substantially slow the melting process. The role of thermal relaxation in nanoparticle melting is studied using a mathematical model based on the Maxwell–Cattaneo equation for heat conduction . The model is formulated in terms of a two-phase Stefan problem . We consider the cases of the temperature profile being continuous or having a jump across the solid–liquid interface. The jump conditions are derived from the sharp-interface limit of a phase-field model that accounts for variations in the thermal properties between the solid and liquid. The Stefan problem is solved using asymptotic and numerical methods. The analysis reveals that the Fourier-based solution can be recovered from the classical limit of zero relaxation time when either boundary condition is used. However, only the jump condition avoids the onset of unphysical “supersonic” melting, where the speed of the melt front exceeds the finite speed of heat propagation. These results conclusively demonstrate that the jump condition, not the continuity condition , is the most suitable for use in models of phase change based on the Maxwell–Cattaneo equation. Numerical investigations show that thermal relaxation can increase the time required to melt a nanoparticle by more than a factor of ten. Thus, thermal relaxation is an important process to include in models of nanoparticle melting and is expected to be relevant in other rapid phase-change processes.

Theoretical estimation of quench occurrence and propagation based on generalized thermoelasticity for LTS/HTS tapes triggered by a spot heater

The present study deals with the thermal characteristics and mechanical behaviors of low/high temperature superconducting (LTS/HTS) composite tapes during quench processes triggered by a spot heater. Based on the generalized thermoelastic theory, a dynamic thermoelastic model with a relaxation time is developed which takes into account the temperature dependence and finite speed of heat propagation for the superconducting tapes under cryogenic condition. The analyses were performed using the finite element method to solve the coupled differential equations of dynamic heat conduction and elastic equilibrium. The results show that the thermoelastic behaviors exhibit a strong relevance to quench characteristics of the superconductors. As a quench occurs, the thermoelastic strain-rate has an obvious jumping variation with the instant of time of its peak being fortunately coincident with the time at which the critical temperature is reached. Such a jumping change of strain-rate could be a way of estimation and detection of quench occurrence, and the theoretical predictions coincide with the existing experimental observations on thermoelastic strain-rate in LTS magnets. For a HTS tape, the thermoelastic strain-rate or temperature-rate variation and a small jump also are illustrated as the quench occurrence is determined. Additionally, the normal zone propagation velocities for the LTS/HTS tapes are predicted by the critical temperature and thermoelastic strain-rate to show quite good agreements with the results evaluated by Wilson’s formula for a LTS tape or the experimental measurements for a HTS tape. The influences of the relaxation time of heat conduction and thermoelastic coupling on the thermal distribution and strain profile are also discussed in details.

A novel hybrid Bézier based multi-step and differential quadrature method for analysis of rotating FG conical shells under thermal shock

Abstract Transient response of rotating multi-layered FG truncated conical shells subjected to thermal shock with temperature dependent properties is studied. In order to consider the finite speed of heat propagation, the non-Fourier heat transfer equation is employed. Based on the linear theory of elasticity, the initial dynamic equilibrium equations are derived. The equations of motion of each layer around this equilibrium state and also the related boundary and compatibility conditions are obtained using Hamilton's principle. A layerwise-differential quadrature method in conjunction with the mapping technique is applied to discretize the resulting equations in the spatial domain. This converts the governing partial differential equations to a system of ordinary differential equations in the time domain. Then, to obtain solution of the resulting system of differential equations, a new general multi-step method based on the Bezier curves is implemented. Verification of the formulation and the method of solution are shown using benchmark solutions available for particular problems in the literature. Eventually, the influence of different parameters on the transient analysis of the shells subjected to thermal loading is also examined.

Strain gradient theory of chiral Cosserat thermoelasticity without energy dissipation

In this paper, we use the Green–Naghdi theory of thermomechanics of continua to derive a linear strain gradient theory of Cosserat thermoelastic bodies. The theory is capable of predicting a finite speed of heat propagation and leads to a symmetric conductivity tensor. The constitutive equations for isotropic chiral thermoelastic materials are presented. In this case, in contrast with the classical Cosserat thermoelasticity, a thermal field produces a microrotation of the particles. The thermal field is influenced by the displacement and microrotation fields even in the equilibrium theory. Existence and uniqueness results are established. The theory is used to study the effects of a concentrated heat source in an unbounded homogeneous and isotropic chiral solid.

Numerical solution of integrodifferential heat conduction equation for a nonlocal medium

A heat conduction model is proposed based on the relations of rational thermodynamics of irreversible processes, taking into account the nonlocal nature of the medium and the finite speed of heat propagation. In the one-dimensional case, the numerical solution of the integral-differential equation of heat conduction is obtained by the finite-element method.

Transport processes: Perturbations propagation, the local time concept and experimental data for perturbations propagation in plasma

Classical equations for transport processes predict infinite speed of the perturbations propagation. From molecular point of view, this feature is rather doubtful, because, for instance in gas or plasma, any perturbation propagates as a result of molecules or charged particles (ions, electrons) interaction. In the present work, exact statistical description of the perturbations propagation in gas and plasma is presented based on the Boltzmann equation written in the most general form. As a result of statistical analysis, for transport processes in the systems with zero or small hydrodynamic velocity, the local time concept and the eikonal type equation for 3D propagation of the perturbations with varying speed are formulated. The local time concept shows how classical transport equations predicting infinite speed of the perturbations propagation, must be modified in order to describe transport processes with finite speed of the perturbations propagation. I also determine conditions, when solutions of classical transport governing equations are accurate enough in the case of finite speed of the perturbations propagation. Three dimensional heat conduction governing equation with finite speed of heat propagation is derived, and illustrative examples of its analytical solutions are obtained and analyzed. Three dimensional governing equation for non-compressible viscous fluid with finite speed of the perturbations propagation is formulated, and its particular analytical solution is derived and studied. Experimental data for thermal perturbations propagation in magnetically confined fusion plasma is compared with the theoretical predictions.

Relativistic heat conduction

The hyperbolic heat conduction equation (HHCE), which acknowledges the finite speed of heat propagation, is based on microscopic evidence from the kinetic theory and statistical mechanics. However, it was argued that the HHCE could violate the second law of thermodynamics. This paper shows that a HHCE-like equation (RHCE) can be derived directly from the theory of relativity, as a direct consequence of space–time duality, without any consideration of the microstructure of the heat-conducting medium. This approach results in an alternative expression for the heat flux vector that is more compatible with the second law. Therefore, the RHCE brings the classical field theory of heat conduction into agreement with other branches of modern physics.

Thermal Control in Solids With Rapid Relaxation

Feedback control of temperature in solids with a finite speed of heat propagation is investigated in this work. The emphases are placed on the effect of rapid thermal relaxation in high-rate heating and the possible delay during the short-time operations. Fundamental characteristics and dominating parameters are identified for effective feedback control in proportional heating. Transient instability, including the intrinsic transition from the desirable stability, neutral stability, to the ultimate unstable response are investigated in a wide spectrum of heating rates. Delayed heating is found to have a destabilizing effect during the short-time transient.

Thermodynamically consistent description of heat conduction with finite speed of heat propagation

In this work, governing equations for heat conduction with finite speed of heat propagation are derived directly from classical thermodynamics. For a one-dimensional flow of heat, the developed governing equation is linear and of parabolic type. In a three dimensional case, the system of nonlinear equations is formulated. Analytical solutions of the equations for one-dimensional flow of heat are obtained, and their analysis shows characteristic features of heat propagation with finite speed, being fully consistent with classical thermodynamics.

The impact of a thermoelastic rod against a rigid heated barrier

The impact of a thermoelastic rod with a heat-insulated lateral surface against a rigid heated barrier is considered. The heat exchange between the rod and the wall occurs at one of its ends contacting with the wall, while the other end is heat-insulated and free from external forces. The behaviour of the rod during the impact process is described by the Green-Naghdy theory which allows one to take finite speed of heat propagation into account, neglecting therewith thermal relaxation. The Laplace integral transform with the subsequent expansion of the found images in terms of the natural functions of the problem is used as a method of solution, which is found in explicit exact closed form. The analytical time-dependence of displacements, stresses, and temperature at each rod particle is obtained. The emphasis is on the analysis of the contact stress, the temperature of the colliding bodies during their contact interaction, and on the detection of the duration of contact of the rod with the rigid wall. It is shown that the contact time essentially depends on the relationship between the mechanical and thermal values.

Nonstationary Rayleigh waves on the thermally-insulated surfaces of some thermoelastic bodies of revolution

The influence of heat conduction and thermal relaxation on the propagation of the surface waves polarized in the sagittal plane along the heat-insulated surfaces of the following thermoelastic bodies of revolution: a cylinder, a sphere, a torus, and a cone is investigated. The modified Maxwell law is used as the law of heat conduction, which allows one to take a finite speed of heat propagation into account. The nonstationary surface waves are interpreted as lines (a straight line or a diverging or converging circumference) on which the temperature and the components of the stress and strain tensors experience a discontinuity. Each of the discontinuity lines propagates with a constant normal velocity across the free from stresses and thermally-insulated surface of the body of revolution along the corresponding lines of curvature and is obtained by coming onto the body's surface of the three strong discontinuity complex wave surfaces which intersect along this line: quasi-thermal, quasi-longitudinal and quasi-transverse volume waves. By applying the theory of discontinuities, the velocities and the intensities of the surface waves have been found. It has been shown that the attenuation of the surface wave intensity is determined by the two factors: the coupling between the related strain and temperature fields and the change in curvature of the surface wave with time if the wave is a curvilinear one.

Importance of a finite speed of heat propagation in metals irradiated by femtosecond laser pulses.

We study theoretically the propagation of heat in a metal, due to irradiation with an ultrashort laser pulse. The target is treated in an extended two-fluid model for electrons and phonons, which accounts for a finite speed of heat propagation in the electron gas. As a result, the absorbed laser energy is more localized in the electronic system yielding an enhanced peak electron temperature. \textcopyright{} 1996 The American Physical Society.

Relaxation equation of heat conduction and generation — an analytical solution by Laplace transforms method

The relaxation equation of heat conduction and generation permits the relaxation of heat flux (a finite speed of heat propagation) as well as the relaxation of heat source capacity. The parabolic and hyperbolic heat conduction equations can be treated as special cases of the relaxation equation. A one-dimensional case of the relaxation equation, in which the relaxation of heat flux is neglected, is solved analytically by the Laplace transforms method to investigate the effect of the inertia of the heat source on the temperature field. The results of sample calculations show that as the relaxation time of heat source capacity increases from zero to infinity the temperature profile for a given time moves from the parabolic solution with heat generation towards the parabolic solution without heat generation. It is also demonstrated that differences between relaxation solutions and the related parabolic solutions do not vanish with time.

The resonance phenomenon in thermal waves

The thermal resonance of temperature wave propagating in a solid with finite speed of heat propagation is studied in this work. The emphasis is placed on the externally applied exciting frequency, in relation to the modal frequency of the thermal waves, at which the amplitude of the thermal waves reaches a maximum. The amplitude of thermal waves thus excited will be related to the exciting frequencies as well as the modal frequencies of the modes. It has been shown that the thermal resonance is a high-mode phenomenon which exists only for the mode with its eigenmode number exceeding a critical value. This critical value depends on the relaxation distance, an intrinsic thermal property extended from the relaxation time, of the solid medium. Such an unusual phenomenon is attributed to the distributions of the over- and the under-damped modes of thermal waves in propagation.

Three dimensional structures of the thermal shock waves around a rapidly moving heat source

This paper investigates the thermal shock formation in a three-dimensional solid due to energy accumulation in preferential direction around a rapidly moving heat source. An analytical method is employed to yield closed form solutions for the temperature field which not only provides an exact means for studying the shock wave structures but also suitable forms readily extendable for the study of the resulting thermal stresses. The temperature field is characterized by a thermal Mach number ( M) defined as the ratio of the speed of the moving heat source (ν) to the finite speed of heat propagation ( C) in the solid. Mathematically, M = ν/ C. In the subsonic range with M < 1, a detailed comparison with the classical diffusion theory is made to reveal the importance of considering the wave nature when the response time is short. At the transonic and in the supersonic range with M ≥ 1, the emphasis is placed on the physical mechanism of the thermal shock formation. All the special features in the two-dimensional theory for the thermal shock formation are preserved in the present three-dimensional analysis. The temperature level, additionally, is found to increase with the distance away from the heat source in the direction perpendicular to the plane containing the moving heat source.