Parabolic Heat Conduction Equation Research Articles

Neural network method for solving nonlocal two-temperature nanoscale heat conduction in gold films exposed to ultrashort-pulsed lasers

Recently, we have presented an artificial neural network (ANN) method for solving the parabolic two-temperature heat conduction equations (PTTM) in double-layered thin films exposed to ultrashort-pulsed lasers. In this article, we extend this study and develop an ANN method for the nonlocal (or weakly nonlocal) two-temperature model (NTTM), which takes into account both time and space nonlocal effects when the relaxation time becomes more effective such as in low temperature and when the characteristic length scale becomes comparable with the mean free path of heat carriers such as in nano systems under femtosecond laser irradiation where the PTTM may not be suitable. We analyze the well-posedness of the NTTM and the convergence of the ANN solution to the analytical solution. The ANN method is finally used to predict the electron and lattice temperatures in nanoscale gold films when exposed to ultrashort-pulsed lasers. As compared with the experimental data, the NTTM-based ANN method gives more accurate solutions than the PTTM-based ANN method.

Neural network method for solving parabolic two-temperature microscale heat conduction in double-layered thin films exposed to ultrashort-pulsed lasers

Simulation of the micro/nanoscale heat conduction induced by ultrashort-pulsed laser heating has been attracting great attention. Additionally, machine and deep learning techniques are becoming an important tool in engineering and science research. This article presents an artificial neural network (ANN) method for solving the parabolic two-temperature heat conduction equations in double-layered thin films exposed to ultrashort-pulsed lasers. Convergence of the ANN solution to the analytical solution is theoretically analyzed. Finally, the ANN method is used to predict the electron and lattice temperatures in a gold film padding on a chromium film when exposed to ultrashort-pulsed lasers, which is based on the parabolic two-temperature heat conduction model.

Parabolic two-step model and accurate numerical scheme for nanoscale heat conduction induced by ultrashort-pulsed laser heating

Simulation of nanoscale heat transfer phenomena has attracted great attention, particularly the nano-scale heat conduction induced by ultrashort-pulsed laser heating. In this article, we propose a nanoscale parabolic two-step model and an accurate numerical scheme for thermal analysis of the nanoscale heat conduction induced by ultrashort-pulsed laser heating. To this end, we first introduce the Knudsen number (Kn) into the original parabolic two-step heat conduction equations and couple them with the Kn-dependent and temperature-jump boundary condition. We then develop a fourth-order accurate compact finite difference method for solving the nanoscale model. Stability and convergence of the obtained numerical scheme are analyzed theoretically. We finally test the accuracy and applicability of the nanoscale model and the obtained numerical scheme in three examples. By choosing various values of the Kn and the parameter α in the boundary condition, the simulation could be a tool for analyzing the nanoscale heat conduction induced by ultrashort-pulsed laser heating.

A hyperbolic lattice Boltzmann method for simulating non-Fourier heat conduction

Classical lattice Boltzmann method (LBM) can recover Fourier's law. Since Fourier's law results in a parabolic heat conduction equation, the classical LBM is called the parabolic Boltzmann method (PLBM). But the Fourier’s law is based on the absurd assumption that the heat transfer rate is infinite, so whether the classical LBM can be used to observe the non-Fourier heat transfer problem remains controversial. In this paper, a hyperbolic lattice Boltzmann method (HLBM), which can be used to analyze the non-Fourier effect, was derived based on the Cattaneo-Vernotte (CV) model. To verify the accuracy of the HLBM, the process of a gold film irradiated by the laser was simulated using the HLBM and the PLBM and the results were compared with the experimental data. Because of the non-equilibrium heat transfer between electron and lattice during the laser irradiation, the two-step HLBM/PLBM models were proposed according to the two-temperature model. The results show that the electron temperatures simulated by the two models are not much different from each other, and both of them coincide with the experimental values. However, the thresholds obtained by the two models are different, and the results of HLBM are closer to the experimental values.

Hyperbolic heat conduction at a microscopic sliding contact with account of adhesion-deformational heat generation and wear

Different non-Fourier models were proposed to simulate temperatures in materials subjected to extremely fast thermal disturbances, when the speed of heat propagation should be concerned. The present study investigated temperature and heat balance at a microscopic sliding contact during a single frictional interaction based on the Cattaneo-Vernotte hyperbolic heat conduction equation. Two fundamental features of friction, namely, adhesion-deformational heat generation and wear, were taken into account. By applying the Laplace transform approach, non-stationary temperature expressions were derived for the hyperbolic and classical parabolic heat conduction equations. Parametric analysis was then done for parameter ranges typical of brake materials. It was found that the hyperbolic heat conduction generally results in a higher temperature at the sliding surface compared to the parabolic heat conduction. The influence of the heat propagation speed can be significant for thermal relaxation time of the order above microsecond. It becomes stronger with an increase in the contribution of the adhesive heat generation. Another finding obtained is that a considerable fraction of heat is removed from the contact zone along with wear debris, resulting in a lower temperature. This fraction is larger for the hyperbolic heat conduction.

Fourier Variant Homogenization of the Heat Transfer Processes in Periodic Composites

Abstract The well-known parabolic Heat Transfer Equation is a simplest recognized description of phenomena related to the heat conductivity in solids with microstructure. However, it is a tool difficult to use due to the discontinuity of coefficients appearing here. The purpose of the paper is to reformulate this equation to the form that allows to represent solutions in the form of Fourier’s expansions. This equivalent re-formulation has the form of infinite number of equations with Fourier coefficients in expansion of the temperature field as the basic unknowns. The first term in Fourier representation, being an average temperature field, should satisfy the well-known parabolic heat conduction equation with Fourier coefficients as fields controlling average temperature behavior. The proposed description takes into account changes of the composite periodicity accompanying changes in the variable perpendicular to the surfaces separating components, concerning FGM - type materials and can be treated as the asymptotic version of Heat Transfer Equation obtained as a result of a certain limit passage where the cell size remains unchanged.

Wave Heat Transfer in the Orthotropic Half-Space Under the Action of a Nonstationary Point Source of Thermal Energy

A new analytical solution to the problem of wave heat transfer in the orthotropic half-space under the action of a time-dependent point heat flux is obtained and studied. The heat transfer is described by a hyperbolic heat conduction wave equation, in which the directions of the thermal conductivity coincide with the Cartesian coordinate system axes (the orthotropic solid). The obtained analytical solution has allowed us to trace the behavior of the point temperature profile in the vicinity of the initial time moment during a number of relaxation times, which is impossible to do when the classical parabolic heat conduction equation is used.

Mesoscopic Moment Equations for Heat Conduction: Characteristic Features and Slow-Fast Mode Decomposition.

In this work, we derive different systems of mesoscopic moment equations for the heat-conduction problem and analyze the basic features that they must hold. We discuss two- and three-equation systems, showing that the resulting mesoscopic equation from two-equation systems is of the telegraphist’s type and complies with the Cattaneo equation in the Extended Irreversible Thermodynamics Framework. The solution of the proposed systems is analyzed, and it is shown that it accounts for two modes: a slow diffusive mode, and a fast advective mode. This latter additional mode makes them suitable for heat transfer phenomena on fast time-scales, such as high-frequency pulses and heat transfer in small-scale devices. We finally show that, if proper initial conditions are provided, the advective mode disappears, and the solution of the system tends asymptotically to the transient solution of the classical parabolic heat-conduction equation.

Additional boundary conditions in unsteady-state heat conduction problems

Using some additional sought function and boundary conditions, a precise analytical solution of the heat conduction problem for an infinite plate was obtained using the integral heat balance method with symmetric first-order boundary conditions. The additional sought function represents the variation of temperature with time at the center of a plate and, due to an infinite heat propagation velocity described with a parabolic heat conduction equation, changes immediately after application of a first-order boundary condition. Hence, the range of its time and temperature variation completely incorporates the ranges of unsteadystate process times and temperature changes. The additional boundary conditions are such that their fulfilment is equivalent the fulfilment of a differential equation at boundary points. It has been shown that the fulfilment of an equation at boundary points leads to its fulfilment inside the region. The consideration of an additional sought function in the integral heat balance method provide a possibility to confine the solution of an equation in partial derivatives to the integration of an ordinary differential equation, so this method can be applied to the solution of equations, which do not admit the separation of variables (nonlinear, with variable physical properties of a medium, etc.).

Discrete space-time model for heat conduction: Application to size-dependent thermal conductivity in nano-films

An analytical discrete-variable model has been developed to describe heat conduction in nano-sized systems. The model assumes that the system consists of a homogeneous array of cells with characteristic size h; each cell interacts with the nearest neighbors in discrete time step τ and all the cells compute their new state simultaneously. In the continuum limit h→0 and τ→0, the model reduces to classical heat diffusion equation of parabolic type or heat conduction equation of hyperbolic type, depending on the choice of scaling invariant. The model is applied to heat conduction in nano-films with emphasis on the transition from the diffusive to ballistic heat transport, which occurs with decreasing film thickness. This model provides a simple method for predicting in a self consistent manner the effective cross-plane thermal conductivities, the temperature jump at the boundaries, the heat flux across the film, and the temperature gradient within the film as functions of the film thickness. The results are in good agreement with molecular dynamic and Monte Carlo simulations.

MATHEMATICAL MODELING OF THERMAL PROCESSES AT SURFACE TREATMENT OF METAL PRODUCTS WITH HIGHLY CONCENTRATED ENERGY FLOWS

The modeling tasks of thermal effect of power impulse action on a surface of the plate of VК10 (КS) alloy were considered and solved. As modeling tasks for homogeneous equations of parabolic and hyperbolic heat conductions, a wave equation in a cylindrical solid of final sizes with boundary conditions of III kind were chosen. The action of power impulse from an exterior radiant was modeled by sudden appearance of initial high temperature, which spreads on a plate body under the laws expressed by various heat conduction equations, on one of the surface ends of a cylinder. Approaches of temperature fields were received in the form of a series segment of functions from eigenvalues of tasks, gradients of fields were defined. Simultaneous presence in the equation of heat conductivity of private derivatives on time of the first and second usages (the hyperbolic equation), statement of the task for it with boundary conditions of III kind and the entry condition at a cylinder end surface provides two ways (modes) of the problem’s decision, both of diffusion type. For the value of the relaxation time of the heat flux of 10–11 s, the complete cooling of the cylindrical sample (tungsten carbide) in the first mode is minutes, in the second – 10–10 s. It can be concluded that the modes for solving the problem for the hyperbolic heat equation do not correspond to the actual pattern of heat propagation. However, the linear combination of these modes as a solution of the problem preserves the possibility of obtaining a diffusion dynamics adequate to the actual process. Gradients of the temperature field in the solutions of the problems for the parabolic heat conduction equation and the wave equation are in the same order of values. The temperature field of the moving thermal wave for several of its first reflections in experimental samples should be taken into account when evaluating phase transformations and temperature stresses. The results of the theoretical analysis are compared with changes in the microstructure of the near-surface layer of a plate of alloy VK10 (KS), subjected to electric explosive loading by plasma of a titanium foil.

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

ABSTRACTTime-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.

Generalized variational principles for heat conduction models based on Laplace transforms

The classical variational principle does not exist for parabolic and hyperbolic heat conduction equations, which has led to the demand for special variational methods for heat conduction. O’Toole (1967) first used Laplace transforms for the variational principle only for Fourier’s law with the first type of boundary condition. In this paper, the Laplace transform strategy is extended to other parabolic and hyperbolic heat conduction models and other types of boundary conditions. Generalized variational principles are given for heat conduction models including Fourier’s law, the Cattaneo–Vernotte (CV) model, the Jeffrey model, the two-temperature model and the Guyer–Krumhansl (GK) model, based on Laplace transforms. The Laplace transform method transforms the heat conduction equations of these models into linear variational equations whose variational principles are already known. For the three standard types of boundary conditions, these generalized variational principles are strictly equivalent to the heat conduction equations for these models. The Laplace transform method has stronger convergence in infinite temporal domain problems. In physics, the Laplace transform method is understood as replacing the time dimension with the frequency of the temperature change and the rate of the entropy change.

On measurement of the thermal relaxation time in transient thermal processes of solids

Experimental transient processes are compared with theoretical processes described by unsteady boundary value problems of the third kind with parabolic and hyperbolic heat conduction equations of a solid. Physical meaning of transfer rate is specified in the thermal relaxation formula. Thermal relaxation time of the solid is measured.

Thermal Responses of a Thin Plate under Periodic Heat Flux Oscillation

Thermal behaviors of a thin plate under a periodic surface thermal disturbance are investigated. The temperature and the displacement responses are predicted using the parabolic heat conduction model. The solution of temperature response is obtained by separation of variables. Firstly, an analytic expression of heat flux filed within the plate is obtained by using the parabolic heat conduction equation which is described by heat flux vector, then, based on the conservation equation of energy, the temperature response is given. The displacement response of the plate is solved analytically using the Nowachi’s and the Navier’s approachs. The thermal behaviors of a plate with various relatively parameters is calculated numerically and the results shown graphically.

Investigation Into Pulse Laser Heating of Nanoscale Au Film Using Dual-Phase-Lag Model

In this study the thermal field is presented for pulse laser processing of nanoscale Au films. Fourier law is inadequate for describing the heat conduction in nanoscale process due to the boundary scattering and the finite relaxation time of heat carriers. In the regime where the particle description of electrons and phonons is valid, the Boltzmann equation is the most accurate option to model heat transfer in such problems. However, solving the Boltzmann equation is generally difficult due to involving three spatial, three momentums and one time. Dual-phase-lag (DPL) model is averaged over the momentum space and thus involves only spatial coordinates plus time, as in the Fourier equation. Therefore this paper utilizes the dual-phase-lag (DPL) model with scattering boundary condition to study the temperature field for laser processing of nanometer-sized thin films instead of Boltzmann equation. The results obtained from the dual-phase-lag heat conduction model, hyperbolic and parabolic heat conduction equations were compared with the available experimental data to validate the compatibility of the thermal models for analyzing the heat transfer in nanoscale thin film irradiated by laser. The temperature history at different locations of the thin film and the effects of boundary phonon scattering on the normalized temperature were also discussed.

Influence of microstructure on thermoelastic wave propagation

Numerical simulations of the thermoelastic response of a microstructured material on a thermal loading are performed in the one-dimensional setting to examine the influence of temperature gradient effects at the microstructure level predicted by the thermoelastic description of microstructured solids (Berezovski et al. in J. Therm. Stress. 34:413–430, 2011). The system of equations consisting of a hyperbolic equation of motion, a parabolic macroscopic heat conduction equation, and a hyperbolic evolution equation for the microtemperature is solved by a finite-volume numerical scheme. Effects of microtemperature gradients exhibit themselves on the macrolevel due to the coupling of equations of the macromotion and evolution equations for macro- and microtemperatures.

Differential-difference heat-conduction and diffusion models and equations with a finite relaxation time

Differential-difference heat conduction and diffusion models and equations with a finite relaxation time are described that give a finite disturbance propagation rate. The modified Biot-Fourier law with delay is used for the heat flux, which leads to the differential-difference heat-conduction equation $$\left. {\frac{{\partial T}} {{\partial t}}} \right|_{t + \tau } = a\Delta T, $$ where the left-hand side is calculated at t + τ (τ is the relaxation time) and the right-hand side is calculated, as usual, at t (without a time shift). At τ = 0, the differential-difference heat-conduction equation turns into the classical parabolic heat-conduction equation; if the left-hand side is expanded into a series in τ and two main terms of expansion are retained, we obtain the Cattaneo-Vernotte hyperbolic heat-conduction equation. An exact solution to the differential-difference heat-conduction equation is derived for a one-dimensional problem without the initial conditions with an arbitrary periodic boundary condition. The approximate solution is constructed to the general three-dimensional initial-boundary value problem of heat propagation with a finite relaxation time in a bounded domain with arbitrary initial heat distribution and the boundary condition of the third kind. It is shown that the differential-difference model makes it possible to derive the Oldroyd-type differential heat-conduction model.

A moving least square reproducing polynomial meshless method

Interest in meshless methods has grown rapidly in recent years in solving boundary value problems (BVPs) arising in science and engineering. In this paper, we present the moving least square radial reproducing polynomial (MLSRRP) meshless method as a generalization of the moving least square reproducing kernel particle method (MLSRKPM). The proposed method is established upon the extension of the MLSRKPM basis by using the radial basis functions. Some important properties of the shape functions are discussed. An interpolation error estimate is given to assess the convergence rate of the approximation. Also, for some class of time-dependent partial differential equations, the error estimate is acquired. The efficiency of the present method is examined by several test problems. The studied method is applied to the parabolic two-dimensional transient heat conduction equation and the hyperbolic two-dimensional sine-Gordon equation which are discretized by the aid of the meshless local Petrov–Galerkin (MLPG) method.

Laser heating of a finite silver selenideslab

The problem of heating a finite homogeneous silver selenide slab induced by time dependent laser irradiance is studied. The Fourier Series Expansion Technique and the Parabolic Heat Conduction Equation(PHCE) are considered to obtain the required temperature field within the solid target. The critical time required to initiate melting at the front surface and the time interval required to initiate phase transition for the considered slab are computed as an illustrative example.