Thermal Conductivity Equation Research Articles

Modeling of temperature processes in orthotropic boards of radio-electronic devices

The design of electronic equipment that is resistant to external influences is a topical problem. The arrangement of individual circuit parts is modeled in the form of an orthotropic plate. The paper describes the solutions of homogeneous and inhomogeneous thermal conductivity equation for an orthotropic rectangular plate, obtained by the recurrence-operator method. Numerical results are given for two cases of temperature distribution, satisfying zero boundary conditions, during cooling and heating and given initial conditions.

Локальное лазерное окисление фотонных кристаллов на основе пористого кремния

The process of thermal oxidation of porous silicon photonic crystals by laser treatment has been studied experimentally and theoretically. The temperature of laser heating was calculated by solving of the non-stationary thermal conductivity equation. The degree of silicon oxidation was estimated experimentally by measuring the spectral shift of the reflection curve of photonic crystals. We demonstrate the controllable shift of the photonic band gap of porous silicon photonic crystals made by means of laser irradiation.

Construction of a model for an enclosing structure with a heat-accumulating material with phase transition taking into account the process of solar energy accumulation

This paper proposes a mathematical model and a procedure for calculating the thermal state of the enclosing structure of the building, which includes an energy-active panel that accumulates solar radiation due to the phase transition of the heat-accumulating material. The mathematical model is based on a two-dimensional non-stationary nonlinear equation of thermal conductivity, which describes the process of heat transfer in the bearing layer of the enclosing structure and the energy-active panel. The model also includes equations describing radiant heat transfer between opaque and translucent bodies. To correctly describe solar insolation, the ASHRAE 2009 model was used in conjunction with the daily change in the position of the Sun in the sky. To solve the system of equations that make up the mathematical model, an iterative procedure has been developed, which involves alternating solution at each time step of the two-dimensional equation of thermal conductivity and a set of algebraic equations of convective and radiant heat transfer. The study’s result established that the amount of accumulated energy in the heat-accumulating material of the phase transition during daylight hours increases significantly, from 15 to 35 %. At night, the surface temperature of the heat-accumulating element in structures using a material with a phase transition is greater than in the case of heat accumulation only in the bearing layer. As a result, it is possible to select from 70 to 120 % more accumulated heat while the presence of high-thermal partitions in a heat-accumulating material with a phase transition contributes to an increase in accumulated heat and usable heat

Особенности расчета нестационарного теплообмена в стенах вращающихся печей

A refined calculation of the non-stationary temperature of the walls of a rotary kiln is proposed, taking into account the non-stationary thermal conductivity in the enclosing walls and the accumulation of heat in them, intended for use in numerical simulation of combustion and heat transfer processes. The influence on heat transfer of the alternating contact of the inner surface of the walls of a rotary kiln with a gaseous medium and a layer of technological material is considered. The adequacy of the method was verified by numerical solution of the differential equation of non-stationary thermal conductivity of walls in the conditions of a computational experiment.

Theoretical Analysis of Self-Heating and Cooling Processes in the Pulsed Field Emission From ZnO Nanowire for Achieving High Emission Current

Quasi-one-dimensional (Q1D) nanowire field emitter arrays (FEAs) have important applications in vacuum microelectronics. Investigation on its pulsed field emission characteristics is important for not only avoiding self-heating-induced vacuum breakdown but also achieving high emission current. In this work, a model for quantitative calculation of the transient emission current and temperature distribution of individual nanowire during pulsed field emission has been established by considering the time-dependent thermal conduction equation. Taking ZnO nanowire as an example, the influences of the pulsewidth, the nanowire length, the nanowire radius, and the electrical properties of nanowire on its maximum transient emission current have been investigated. Besides, the cooling process is found to be mainly related on the nanowire length when the nanowire temperature is below 1000 K. By calculating the average emission current under different pulsewidths and duty ratios, it is found that its maximum is determined by the maximum emission current in the steady state that the nanowire is self-heating steadily to the critical temperature below the melting point. Finally, discussions for improving the emission current in different potential applications of FEAs are given. The results provide guidelines for designing Q1D nanowire FEAs.

Analysis of the Influence of the Gas Infrared Heater and Equipment Element Relative Positions on Industrial Premises Thermal Conditions

The creation of local heated areas in large-sized premises using systems based on gas infrared heaters has recently become the most rational alternative in terms of energy efficiency. However, the lack of information about the thermal characteristics in such areas limits the effective application of these systems. To determine the main thermal parameters characterizing the scheduled thermal conditions in heated local working areas of industrial premises, experimental and mathematical modeling of heat transfer processes in a closed area with the presence of equipment in it was carried out. The experimental area was equipped with a gas infrared heater and a model of the equipment (a horizontally oriented panel). The system of equations of thermal conductivity, radiant heat transfer, as well as energy and Navier–Stokes was solved by the finite element method. A significant influence of the equipment position on the temperature field and the air movement hydrodynamics in the local working area has been established. The equipment presence in the room intensifies the air movement due to thermal convection and, as a result, a more uniform temperature distribution over the local working area was obtained. Analysis of the obtained results shows the possibility to control the temperature fields’ formation in local working areas during the gas infrared heater operation by varying the position and configuration of the equipment in the room.

Thermomagnetic behavior of a semiconductor material heated by pulsed excitation based on the fourth-order MGT photothermal model

This article proposes a photothermal model to reveal the thermo-magneto-mechanical properties of semiconductor materials, including coupled diffusion equations for thermal conductivity, elasticity, and excess carrier density. The proposed model is developed to account for the optical heating that occurs through the semiconductor medium. The Moore–Gibson–Thompson (MGT) equation of the fourth-order serves as the theoretical framework to establish the photothermal model. It is well-known that the optical and heat transfer properties of such materials behave as random functions of photoexcited-carrier density; therefore, the current model is remarkably more reliable compared to the earlier closed-form theories which are limited to a single form. The constructed theoretical framework is able to investigate the magneto-photo-thermoelastic problems in a semiconductor medium due to laser pulse excitation as a case study. Some parametric studies are used to exhibit the impact of thermal parameters, electromagnetic fields, laser pulses and thermoelectric coupling factors on the thermomagnetic behavior of physical variables. Finally, several numerical examples have been presented to draw the distributions of the examined field variables.

Determination of thermophysical characteristics in a nonlinear inverse heat transfer problem

This paper presents a detailed and developed method for finding soil nonlinear thermophysical characteristics. Two-layer container complexes are constructed. The side faces of which are thermally insulated, so that one can use the one dimensional heat equation. In order not to solve the boundary value problem with a contact discontinuity and not to lose the accuracy of the solution method, a temperature sensor was placed at the junction of the two media, and a mixed boundary value problem is solved in each region of the container. In order to provide the initial data for the inverse coefficient problem, two temperature sensors are used: one sensor was placed on the open border of the container and recorded the temperature of the soil at this border, and the second sensor was placed at a short distance from the border, which recorded the air temperature. The measurements were carried out in the time interval (0,4tmax). At first, the initial-boundary value problem of nonlinear thermal conductivity equation with temperature dependent coefficients of thermal conductivity, heat capacity, heat transfer and material density is investigated numerically. The nonlinear initial-boundary value problem is solved by the finite difference method. Two types of difference schemes are constructed: linearized and nonlinear. The linearized difference scheme is implemented numerically by the scalar Thomas’ method, and the nonlinear difference problem is solved by the Newton’s method. Assuming the constancy of all thermophysical parameters, except for the thermal conductivity of the material, on the segment (0,tmax), the quadratic convergence of Newton’s method is proved and the thermal conductivity is calculated, while the solution of the linearized difference problem was taken as the initial approximation for the Newton’s method. By freezing all other thermophysical parameters, and using the measured temperatures in the segment (tmax,2tmax), the coefficient of specific heat of the material is found. Proceeding in a similar way, using the initial data on the segment (2tmax,3tmax), the density of the soil is determined. Also, based on the measured initial data on the segment (3tmax,4tmax), the heat transfer coefficient is determined.Based on the experimentally measured data, at each time interval, the corresponding functional is minimized using the gradient descent method. The differentiation of a nonlinear difference problem with respect to the desired parameter method is used to find the gradient of the functional. This method allows you to find the functional gradient and the damping coefficient of the gradient descent in an explicit form. In this case, the compilation and solution of the conjugate problem, which is a frequently used method for solving the inverse problems, is not needed.The performed numerical calculations show that for small time intervals the solutions of the linearized difference problem differ little from the solution of the nonlinear difference problem (1 - 3%). And for long periods of time, tens of days or months, the solutions of the two methods differ significantly, sometimes exceeding 20%. In addition, all thermophysical characteristics (8 coefficients) were found for a two-layer container with sand and black soil. Also the dependence of the temperature difference at the boundary between the environment and the soil for the heat transfer coefficient was shown.

An Artificial Neural Network Model for Electro-Thermal Effect Affected Hot Carrier Injection Reliability in 14-nm FinFETs

FinFETs have a bright future for advanced analog/RF and digital circuit design in 5G/6G technology. As the integration density increases, the self-heating effect (SHE) becomes more severe, and the hot carrier injection (HCI) reliability of FinFETs further deteriorates since it is highly dependent on temperature. In this article, an artificial neural network (ANN) model for HCI prediction is proposed for 14-nm FinFET with consideration of SHE under various voltage stresses and environment temperature. The HCI numerical simulation can be mainly divided into two parts: 1) the temperature responses for the device under different stresses are obtained by numerically solving time-dependent thermal conduction equation with heat generation in the nanoscale FinFET and 2) based on the transient temperature response to different stress voltages, HCI-induced threshold voltage shift (TVS) as a function of time is captured numerically. Our HCI-induced TVS simulation results agree well with others’ experiment results, which verifies the simulation methods. An ANN model is then developed to predict the TVS induced by HCI with the training data obtained from numerical simulations. The ANN model has good accuracy with an average relative error of 0.36% in predicting TVS and greatly reduces simulation cost once the ANN model is built for the HCI prediction of FinFETs, which has great potential for reliability design in circuits and systems.

Anomalous high thermal conductivity in heavy element compounds with van der Waals interaction

It is conventionally believed that lattice thermal conductivity (κ) decreases with increasing atomic mass (negative atomic-mass correlation), and the high κ can only occur in crystals composed of strongly bonded light elements. By solving the fundamental thermal conductivity equation using first-principles calculations, here we reveal the anomalous κ departing from the long-held concept, that is, a positive atomic-mass correlation and high κ with heavy elements and weakly bonded interaction. We demonstrate this anomalous phenomenon by performing calculations of the cross-plane κ of the layered compounds, i.e., the h-BX family with X = N, P, and As. We find that the anomalous increase in the cross-plane κ with X going from N to As results in the cross-plane/in-plane conductivity ratio, generally expected to be much smaller than 1 in layered compounds, reaching as large as 2.6 at low temperatures. We also find that the unusually high cross-plane κ (660 W m−1 K−1), which is comparable to the bulk silicon with strong covalent bonding interactions, can be generated by a weak van der Waals interaction. Our analysis shows that the anomalous κ arises from one-dimensional-like phonons propagating in the cross-plane direction, which is due to the extremely large phonon anisotropy induced by the combined effect of atomic-mass difference and structural anisotropy. This discovery paves an avenue to realize thermally conductive materials that have weakly bonded structures, which can be potentially applied in the design of high-performance nanoelectronic devices.

Morphological structure of shell mould in investment casting

The stress state (SS) of multilayer shell mould (SM) is significantly influenced by its morphological structure, its variants, and the connections between the contacting layers. The purpose of this work is to establish the effect of sliding without friction of one of the layers of SS relative to the neighboring contacting layer on SS of the multilayer SM. The process of determining the stress-strain state (SSS) in the SM is modeled when it is filled with steel, in which there is a surface between the layers, where one layer of the SM is completely sliding relative to the adjacent neighboring layer. The task is to find the geometric location of a given surface by thickness of SM from the condition of the formulated objective function with the corresponding system of constraints. The construction of the mathematical model is based on equations of the linear theory of elasticity, the equations of thermal conductivity, solution of which is carried out by a proven numerical method. In the axisymmetric formulation, a complex three-component system is considered: liquid metal, solid metal, ceramic SM. Solid metal and SM are considered to be isotropic. Crack resistance of the SM is estimated by the stresses arising in it. On contact with the support filler (SF), it is possible to move the SM surface from the SF during the liquid metal cooling. In this case, the contact problem is solved – reassignment of the boundary conditions. An algorithm for solving the problem is compiled. The results of numerical modeling are displayed graphically in the form of plots. A detailed analysis of the obtained results was carried out. The inconsistency of manufacturing such a multilayer mould is shown. The results of the research can be useful in calculations of other functional multilayer shell systems.

Mathematical model of the thermal mode of overhead power lines considering temperature variations along the line length

A mathematical model representing the temperature mode of an overhead power line wire and taking into account the axial heat transfer was developed. Processes in overhead power lines were analyzed using analytical and numerical methods for solving differential equations, including the finite difference method. The equation of thermal conductivity for AS-240/32 and SIP-2 3х95+1х95 wires was solved for the case of current variations along the line length. An analytical solution to the equation of thermal conductivity was proposed for the steady-state operation of an overhead wire under the same current in all sections of the line, taking into account the temperature dependence of active resistance. The results obtained by the analytical method agree well with those obtained by the method of finite differences. The boundary conditions at the beginning and at the end of the line were established to affect the line temperature only within a few meters. At the same time, despite the slight increase in the degree of this effect at an increase in the current due to the temperature dependence of heat emission, it remains small up to emergency level currents. Therefore, the calculations of the line thermal mode require no high accuracy in setting boundary conditions. A line with a uniformly distributed load demonstrates differing results at large current variations along the wire length. Thus, the absolute error of the analytical solution (compared to the finite difference method) for the maximum temperature equals 77.9°C, while the relative error for losses equals 10%. The same errors in temperature calculations for an infinitely long wire in terms of the length function comprise 2.5°C and 0.1%, respectively. Therefore, despite the high thermal conductivity of a metal, a model with a zero thermal conductivity along the wire axis gives more accurate results as compared to a model with an infinitely high thermal conductivity. The obtained results are applicable when clarifying the total loss of active power, as well as for estimating the line capacity according to the maximum permissible temperature, which depends on the type of wires and comprises 70 and 90°C for uninsulated steel-aluminum and self-supporting insulated wires, respectively.

Energy harvesting from thermally induced vibrations of antenna panels

• A novel piezoelectric energy harvester excited by thermally induced vibration is proposed. • A fully coupled model considering thermal-mechanical-electrical interactions is established. • A new computational framework with high accuracy is developed for the multi-physics problem. • The feasibility of the designed energy harvester on the antenna panel has been proven. Though vibration energy harvesting technology has been extensively explored in the past decades, harnessing energy from thermally induced vibration has been rarely investigated. This study, for the first time, proposes a piezoelectric energy harvester (PEH) excited by time-varying thermal loading in outer space to power wireless electronics in antenna panels of satellites. A novel thermal-mechanical-electrical coupling model is developed to accurately predict the dynamic response of the system. Firstly, based on the comprehensive analysis of spatial heat fluxes, the transient thermal conduction equations are derived via the variational principle. Subsequently, different from conventional incremental finite elements, the thermoelasticity of the panel is characterized by the absolute nodal formulation. Taking advantage of invariant matrices, an enhanced mathematical model is constructed to improve the computational efficiency of the thermoelastic forces and their Jacobian matrices. Furthermore, an electromechanically-coupled analytical model is put forward for the PEH installed on the antenna panel. Finally, an integrated computational framework is established to iteratively solve the multi-physics coupled problem with second-order accuracy. A corresponding finite element model is also built for verification. The effectiveness and efficiency of the developed multi-physics model are demonstrated through a comparison with the simulation results. In particular, the proposed analytical model not only considers the bidirectional interaction between the elastic deformation and heat absorption, but also incorporates the coupling relationship between the piezoelectric effect and structural vibration. Moreover, the investigation results provide pivotal insights into the design of the energy harvesting system excited by thermally induced vibration.

Effects of Variable Properties on the Convective Flow of Water near Its Density Extremum in an Inclined Enclosure with Entropy Generation

The aim of the current numerical simulation was to understand the effects of the temperature-dependent properties of viscosity and thermal conductivity on the buoyant convection of cold water around its density extremum in a tilting chamber. The equations for thermal conductivity and viscosity were derived based on the reference temperature parameter. The entropy generation and cup mixing temperature were also calculated. The governing mathematical model was solved by the finite-volume-based iterative technique. The obtained results were deliberated for several values of inclination angles and for the density reversal parameter for variable viscosity and thermal conductivity cases. It was detected that density inversion has a strong effect on stream and thermal patterns through the development of a bi-cellular arrangement due to density inversion.

Conductive Heat Transfer in Materials under Intense Heat Flows

The paper presents the solution of the spatial transient problem of the impact of a moving heat flux source induced by the laser radiation on the surface of a half-space using the superposition principle and the method of transient functions. The hyperbolic equation of transient thermal conductivity accounting for the relaxation time is used to model the laser heating process. It is assumed that the heat flux is distributed symmetrically with respect to the center of the heating spot. The combined numerical and analytical algorithm has been developed and implemented, which allows one to determine the temperature distribution both on the surface and on the depth of the half-space. In this case, the principle of superposition is used with the use of a special symmetric Gaussian distribution to describe the model of a source of high-intensity heat flux. The use of such a symmetric distribution made it possible to calculate the integrals over the spatial variables analytically. The results of the work could be used to estimate the contribution of the conductive component in the overall heat transfer of materials exposed to intense heat flows (laser surface treatment, laser additive technologies, streamlining and heating of materials by high-enthalpy gases, etc.).

Nonlinear inverse problem of finding thermophysical characteristics

In this paper, a method has been developed for determining the nonlinear heat-conducting characteristics of the soil. Two-layer container complexes were created, the side faces of which are thermally insulated, so the 1D thermal conductivity equation is used. The temperature sensor is placed at the junction of two media, and a mixed boundary value problem is solved in each region. In order to provide the inverse coefficient problem with initial data, two temperature sensors are used: one sensor was placed at the open boundary of the container and recorded the soil temperature at this boundary, and the second sensor was placed a short distance from the boundary, which recorded the air temperature. The measurements were carried out in the time interval (0,4tmax). First, the initial-boundary problem of heat conduction with nonlinear coefficients is studied by the finite difference method. Two types of difference schemes are constructed: linearized and nonlinear. The linearized difference scheme is implemented numerically by the scalar Thomas method, and the nonlinear difference problem is solved by the Newton method. The solution of a linearized difference problem was taken as the initial approximation of Newton’s method. To find the thermophysical parameters, the corresponding functional is minimized using the gradient descent method. In addition, all thermophysical characteristics (8 coefficients) were found for a two-layer container with sand and chernozem.

Computation of conductive thermal distribution using non-homogenous graph theory for real-time applications in metal PBF process

The Powder Bed Fusion (PBF) process is inherently a thermal process with complex thermal interactions between different printed zones as well as different layers. There exist only a few methods such as finite element analysis (FEA), finite element differences (FDM), graph theory (GT), Goldak’s FEA, and Rosenthal equation, which are able to predict thermal temperature distribution throughout a printed layer (2D) or part (3D). All these approaches suffer from inherent limitations including the applied boundary conditions and computational time. A rapid and reliable method to compute thermal distribution throughout a printed part is pivotal to supporting real-time closed-loop monitoring and control, enabling thermal simulation software with rapid and precise prediction, and advancing current research on thermal-related abnormalities such as residual stress and distortion. The literature shows that the conventional graph theory is the fastest approach that generates relatively precise results in a fraction of the computational time of other techniques; however, the lack of a solution to the non-homogeneous governing thermal equation through GT has hampered this method in terms of thermal load resolution, accuracy in highly rapid process such as PBF, and scope of application. In this paper, we describe the characteristics that make GT a superior approach for real-time computation of thermal field compared to other similar approaches such as FDM. Also, we develop a solution to the non-homogeneous term of the thermal conduction equation by using GT. This solution represents a breakthrough for the development of precise real-time closed-loop monitoring and control systems by providing a precise numerical solution to the thermal conduction equation in a fraction of time compared with previous traditional methods such as FEA and FDM. Ongoing work includes the development of an intelligent monitoring and control system that leverages this solution in order to optimize scan strategy real-time in metal PBF.

Волны деформации в двух соосных, физически нелинейных оболочках с конструкционным демпфированием, взаимодействующих с окружающей средой и заполненных жидкостью

The paper investigates longitudinal strain waves in physically nonlinear coaxial elastic shells containing a viscous incompressible fluid situated both between the shells and within the inner shell. The study takes into account the effects that the following factors have upon the magnitude and velocity of the wave: viscosity, fluid motion inertia, the medium surrounding the outer shell, and structural damping in the shell material. As it is impossible to evaluate the strain wave models proposed via qualitative analysis methods, using numerical methods is necessary. We numerically investigated the model presented using a specific difference scheme to represent its equations, one similar to the Crank --- Nicolson scheme for the thermal conductivity equation. We show that the velocities and amplitudes of strain waves in the shells do not change when disregarding the following factors: the fluid inside the shell, structural damping in the shell material, and the influence of the surrounding elastic medium. The waves evolve in the positive direction of the abscissa axis. At the same time, taking nonlinearity into account leads to increased wave velocity as compared to the linear case, when the wave propagation velocity equals the speed of sound, which means the waves become supersonic. We performed a numerical study of this case and obtained results matching the exact solution. If we take into account the effect that the fluid motion inertia within the inner shell has upon the wave velocity, then a decrease in the strain wave velocity is observed. Taking into account the presence of an elastic medium surrounding the outer shell leads to an increase in velocity. Considering the viscous properties of the fluid within the inner shell and the damping properties of the shell material results in decreased strain wave magnitudes. The models proposed can form the basis for designing modern non-destructive testing devices

Dust control efficiency improvement

High dustiness of operating space in coal mines and active methane emission detains faster driving of coal production and weakens competitive ability of coal mines. From the physical research of inertia movement of rotation liquid drops, the mathematical model of the circulation in the gas environment is constructed to develop more effective dust catching techniques. It is proved that the equation of fluid whirl diffusion in motion of a liquid drop along a spiral line is identical to the equation of thermal conductivity with dispersion coefficient of liquid drop rotation energy. It is confirmed that circulation of liquid drops, both in super Stokes and Stokes flows, enlarges the relaxation time owing to reduction in aerodynamic drag factor of the gas environment, conditioned by the increase in the effective value of the Reynolds number with the increased rotational velocity of liquid drops. The averaging of the aerodynamic drag factors of liquid drop movement allows using the obtained formulas for calculating  hydraulic vortex coagulation in a wide range of the Reynolds number, 1 < Re < 104. For enhancing energy of high-head hydraulic dedusting, the mathematical model and technology of hydro-vortex dust collection are proposed. The similarity criteria and indicators of the hydro-vortex dust collection efficiency are obtained. The experiments prove that dedusting efficiency depends not on the fluid flow rate but on the energy of dynamic head of liquid drops and on their dispersion degree. The hydro-vortex dust collection technique allows enhancing the dust control efficiency up to 99 % owing to the increased rotational velocity of liquid drops.

Transient simulation of dielectric breakdown during high-power microwave propagation based on the SETD method

This article is mainly devoted to the research of dielectric breakdown phenomenon during high power microwave propagation. Under the force of the high-intensity fields, bound charges break free and a part of an insulator becomes electrically conductive, which may lead to electrical breakdown and thermal breakdown in high power microwave (HPM) devices or on the output window. A three-dimensional (3D) multi-physics electromagnetic-thermal model is developed and solved by spectral-element time-domain (SETD) method. The nonlinear conductivity caused by HPM is introduced into Maxwell's equations, which makes its nonlinear. The heat flux from transient power loss is utilized to construct the heat source for solving the thermal conduction equation. In the SETD process, the Galerkin's method and central difference scheme are employed for the space discretization and temporal discretization, respectively. The numerical results show that physical mechanism of electrical breakdown, thermal effect and nonlinear effect can be captured accurately during high power microwave propagation. Once the strength of the electric field exceeds the critical value experienced by the dielectric, the current increases rapidly, along with the temperature change is very severe. The validity and accuracy of the proposed method are demonstrated, and our research is beneficial for the HPM devices protection.