System Of Heat Conduction Equations Research Articles

An innovative subdivision collocation algorithm for heat conduction equation with non-uniform thermal diffusivity.

This paper presents a subdivision collocation algorithm for numerically solving the heat conduction equation with non-uniform thermal diffusivity, considering both initial and boundary conditions. The algorithm involves transforming the differential form of the heat conduction equation into a system of equations and discretizing the time variable using the finite difference formula. The numerical solution of the system of heat conduction equations is then obtained. The feasibility of the algorithm is verified through theoretical and numerical analyses. Additionally, numerical and graphical representations of the obtained numerical solutions are provided, along with a comparison to existing methods. The results demonstrate that our proposed method outperforms the existing methods in terms of accuracy.

An implicit thermo-mechanical coupling model for heat conduction and thermal cracking in brittle materials

In this paper, an implicit thermo-mechanical coupling model is developed to simulate heat conduction and thermal cracking in brittle materials. In the thermal analysis, an implicit particle heat conduction model (PHCM) is proposed. A linear algebraic system of heat conduction equations is derived based on the temperatures of the particle system and the heat flux among adjacent particles. An analytical expression is introduced for the involved correction coefficient, with which the macroscopic thermal conductivity coefficient can be directly adopted without calibration. Furthermore, a thermal conductivity reduction coefficient is introduced to account for the thermal resistance of the cracks. In the mechanical analysis, particle discontinuous deformation analysis (PDDA), which is an implicit variant of the discrete element method, is employed. The PHCM is first verified against analytical and finite element solutions for transient heat conduction problems. The results show that the PHCM is an effective and accurate heat conduction model that readily reflects the thermal resistance of cracks. Then, the thermo-mechanical coupling model is verified by the expansion of heated specimens under different conditions. Finally, the successful application of the coupling model in the simulation of thermal shock tests demonstrates its effectiveness for thermal cracking problems.

Численное моделирование максимального перепада температур в термоэлектрической ветви из градиентно-варизонного сплава висмут-сурьма в области температур 70–120 К

Inhomogeneous bismuth-antimony alloys are the most efficient thermoelectrics at 100 K. This article describes a numerical simulation by means of COMSOL Multiphysics of the temperature field as a function of the current in a thermoelectric branch made of a gradient-inhomogeneous Bi100-x-Sbx al-loy within the semiconductor region of the interval 11 at. %< x<19 at. %. The model takes into account that the parameters of the substance affecting the thermoelectric figure of merit, namely ther-mal conductivity, thermoelectric power, and electrical conductivity, are variable along the branch. In this case, the alloy inhomogeneity and temperature gradients lie in the cleavage plane of the sample. To simulate this problem, the authors numerically solved a system of heat conduction equations and the generalized Ohm's law equations, taking into account continuously distributed heat sources, for which the Joule heat, the Thomson effect, and the Peltier effect are responsible. It can be seen from numerical calculations that in the region of 70–120 K, there is an increase in the maximum tempera-ture difference in the presence of opposite gradients of the antimony concentration and temperature in the sample, compared to unidirectional gradients or a homogeneous alloy. The results obtained were compared with experimental data for a thermal branch with the same inhomogeneity composi-tion (11 at. %<x<19 at. % Sb).

Photothermal Transformation of Bessel Light Beams in Periodically Polarized Nonlinear Crystals

A mechanism of the formation of a photodeflection response in a periodically polarized nonlinear crystal illuminated by a Bessel light beam (BLB) is considered. The thermal-field distribution is found, and the exact solution of the system of heat-conduction equations is obtained using the Fourier–Bessel and Laplace integral transforms for a three-layer system consisting of a nonlinear crystal, a substrate, and environment. A method for controlling the amplitude of the photodeflection signal is proposed based on the application of axicons with an adjustable taper angle or optical schemes providing a change in the conicity of BLBs.

Mathematical simulation of the temperature profile of metal films under pulsed photon annealing

The exact solution to the system of heat conduction equations for the electron and phonon subsystem is obtained using the thermal peak model; the temperature of the film subjected to incoherent optical irradiation is determined. The effects of thermal contact with surroundings and electromagnetic radiation damping on the spatial temperature distribution in an aluminum film are analyzed. Recommendations are given for the choice of the optimal regime of the photon annealing.

Calculation of the thermal mode in semiconductor devices in conditions of their operation in semiconductor apparatuses

The study of the temperature field of power semiconductor devices, operating in semiconductor apparatuses, either non-contact or hybrid was carried out in the paper.It was also shown that the main mode of the current load of power semiconductor devices, operating in semiconductor apparatuses is a pulse mode.Analytical method for calculating the values of the temperature rise in the structure of power semiconductor devices when subjected to a current pulse of arbitrary shape based on a model that adequately reflects the design of semiconductor devices was used during the studies. To calculate the thermal mode of power semiconductor devices, a technique, based on using the simplified thermal models, where the solder layers are taken into account due to expansion of thermal-resistance equivalent tungsten layers is also used. However, unlike it, the method, proposed in the paper provides a higher accuracy in calculating the temperature rise in the structure of power semiconductor devices since all layers of these devices are taken into account. Also, calculation by this method takes into account the dependence of physical properties of materials of the thyristor components on the change in their temperature. Solving the system of heat conduction equations, set up for the model that adequately reflects the thyristor design, was performed by finite difference method using the implicit scheme.When calculating the transient thermal resistance using the real model for different types of thyristors, difference in its value reaches 5 - 16% compared with the calculation using the simplified models. Thus, calculation of the temperature rise in the structure of power semiconductor devices is appropriate to carry out using the proposed method based on a model that adequately reflects the design of these devices.The described calculation method can be used for the rational choice of a power semiconductor device as a basic element of hybrid and non-contact semiconductor apparatuses in systems of their computer-aided design.The paper presents examples of calculating the thermal mode of semiconductor devices for the most common types of semiconductor apparatuses.

Light and thermal fields in multilayer skin tissue exposed to laser irradiation

A simple model for calculating the light fields created by a laser beam in a multilayer biological tissue is proposed. On the basis of this model, the coefficient of diffuse reflection of skin and the penetration depth of light are found for the wavelength range 400–850 nm and for a wide range of variation in the structural and optical characteristics of the medium. The effect of localized light absorption by blood vessels is taken into account. The results obtained are shown to agree reasonably with published theoretical and experimental data. Using the found depth distributions of the absorbed energy as source functions for the thermal problem, the analytical solution for the spatiotemporal structure of the temperature field in the cutaneous covering irradiated by a laser beam was obtained. The thermal fields determined in terms of single-and two-layer models are compared, and the corresponding differences are discussed. Examples of the temperature depth profiles obtained at different instants after the beginning of irradiation are presented. The results obtained in terms of the analytical method proposed are demonstrated to agree satisfactorily with those yielded by the numerical finite difference scheme of the solution of the system of heat conduction equations.

Analytic description of the laser-beam heating of multicomponent biological tissues

An analytic method is developed for solving a system of heat conduction equations in two-component biological tissues irradiated by a narrow laser beam. The axial and radial heat transfer in a medium, heat exchange between capillaries and the surrounding tissue, and heat transfer to an external medium are taken into account. Simple expressions describing the spatiotemporal distributions of the tissue temperature are obtained, which in fact do not require any software for calculations. The method was used for calculating temperature fields in a medium at different irradiation wavelengths in the region between 400 and 700 nm for a broad range of geometrical, optical, and thermal parameters of the problem. The time intervals are estimated in which a simpler approximation of an infinitely wide beam can be used to analyse the thermal regime of a tissue irradiated by a narrow light beam. The analytic representation of the Green function for the problem with a narrow beam obtained in the paper shows that the applicability of this approximation is determined not by the condition of a substantial excess of the beam width over the light penetration depth into the medium, but by the beam radius, the thermal diffusivity, and the instant of time after the onset of laser beam action.

Determination of thermoelastic parameters of a layered medium by the photodeflection spectroscopy method

For determining the thermal stresses produced in the film-substrate system under the action of modulated laser radiation, the system of heat conduction equations is solved and the distribution of the potential of thermoelastic forces acting in the media under study is found. Elastic strains and the corresponding stresses are determined, taking into account the initial and boundary conditions, by using the known thermoelastic potentials. The numerical and graphical analysis of thermal stresses is made, which takes into account gyrotropic properties of the film. It is shown that the optical gyrotropy and multibeam interference of eigenwaves in a sample can substantially change the magnitude of photodeflection response of the system. A method for efficient control of the photodeflection signal is proposed, which uses changes in energy, phase, and polarization characteristics of electromagnetic radiation acting on the layered system under study.

Nonlocal ansatze and solutions of a nonlinear system of heat-conduction equations

By a nonlocal substitution, a nonlinear system of heat-conduction equations is reduced to a scalar nonlinear heat-conduction equation. The Lie and conditional invariance of the scalar equation is used to find nonlocal ansatze which reduce the original system to systems of ordinary differential equations.

A computational model of laser-induced melting and solidification with density change

A finite-element model of laser-induced thermal processes including density changes due to thermal expansion and phase change is presented. In contrast to the classical Stefan formulation of the models of phase change, the local thermodynamic equilibrium condition is replaced by a relation between the solid/liquid interface velocity and its temperature, which follows from the Jackson-Chalmers theory. Moreover, the model accounts for the thermal expansion of both the phases and for the motion of the sample surface due to the density jump at the phase transition. This approach leads to a more accurate determination of the position of the phase interface. To solve the resulting system of heat conduction equations with moving boundaries, standard finite-element procedure is employed. Since the model is nonlinear an iterative procedure has to be used after space and time discretization. A detailed description of the algorithm is given. As an example, results of numerical experiments with XeCl- and ArF-excimer laser processing of mono-Si in ns regime are presented and discussed.

Solution of inverse problems for a system of quasilinear equations of heat conduction in a self-similar regime

Explicit solutions are obtained for inverse problems for a system of heat-conduction equations in a self-similar regime. The thermophysical characteristic being sought depend on the temperature distribution.