Nonlinear Heat Conduction Equation Research Articles

Active control of thermally induced vibrations of temperature-dependent FGM circular plate with piezoelectric sensor/actuator layers

This study investigates the novel application of active control methods in mitigating thermally induced vibrations of functionally graded material (FGM) circular plates subjected to thermal shock. The top surface of the FGM circular plate is exposed to rapid surface heating, resulting in thermally induced vibrations. To counteract these deflections, two piezoelectric layers, one functioning as a sensor and the other as an actuator, are strategically integrated into the plate opposite sides. This layered configuration effectively distances the direct rapid surface heating from piezoelectric layers. All thermos-electro-mechanical properties of the used materials are considered temperature-dependent. The research employs the Generalized differential quadrature element (GDQE) method and the Crank-Nicolson time marching procedure to solve the nonlinear Fourier-type heat conduction equations for the layered structure. After obtaining the transient temperature profile, the study calculates thermal moments, forces, and pyroelectric terms. These thermally induced effects are then integrated into the motion and electrostatic equations, which inherently exhibit nonlinearity due to the von Kármán large deformations theory. The plate displacements are approximated using the first-order shear deformation theory (FSDT) along the plate thickness, while a quadrature distribution is considered for approximating the electric potential within the piezoelectric layers. The highly coupled electro-mechanical equations are solved using the GDQ method and the Newmark approach, with the Newton-Raphson iterative scheme utilized to address nonlinearity. A significant contribution of this research lies in the exploration of three distinct active control strategies: proportional control, derivative control, and proportional-derivative control. The study thoroughly analyzes the impact of each control type on the plate response, considering various boundary conditions.

Математичні моделі локального нагрівання елементів електронних пристроїв

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in isotropic spatial heat-active media sub-jected to internal and external local heat load, have been developed. In the case of nonlinear boundary-value problems, the Kirchhoff transformation was applied, using which the original nonlinear heat conduction equations and nonlinear boundary conditions were linearized, and as a result, linearized second-order differential equations with partial derivatives and a discontinu-ous right-hand side and partially linearized boundary conditions were obtained. For the final linearization of the partially linearized differential equation and boundary conditions, the ap-proximation of the temperature according to one of the spatial coordinates on the boundary sur-faces of the inclusion was performed by piecewise constant functions. To solve linear bounda-ry-value problems, as well as obtained linearized boundary-value problems with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which analytical solutions of these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, analytical relations for determining the temperature distribution in this envi-ronment were obtained. Numerical analysis of temperature behavior as a function of spatial co-ordinates for given values of geometric and thermophysical parameters was performed. The in-fluence of the power of internal heat sources and environmental materials on the temperature distribution was studied. To determine the numerical values of the temperature in the given structure, as well as to analyze the heat exchange processes in the middle of these structures, caused by the internal and external heat load, software tools were developed, using which a ge-ometric image of the temperature distribution depending on the spatial coordinates was made.

Dynamic behaviors of the non-linear local fractional heat conduction equation on the cantor sets

Dynamic behaviors of the non-linear local fractional heat conduction equation on the cantor sets

Mathematical models for analysis of temperature regimes in vehicle braking systems

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in the braking systems of vehicles, which are geometrically depicted as isotropic spatial heat-active media that are subject to internal local thermal heating, have been developed. With the use of classical methods, it is not possible to obtain analytical solutions of linear and nonlinear boundary value problems of mathematical physics in a closed form. This is especially the case when the right-hand sides of differential equations with partial derivatives and boundary conditions are discontinuous functions. The given approach is based on the application of the apparatus of generalized functions to describe the local concentration of thermal influence. This made it possible to apply the integral transformation and, on this basis, to obtain analytical solutions of both linear and nonlinear boundary value problems. In the case of a nonlinear boundary value problem, the Kirchhoff transformation was applied, using which the original nonlinear heat conduction equation and nonlinear boundary conditions were linearized, and as a result, a linearized second-order differential equation with partial derivatives and boundary conditions with a discontinuous right-hand side were obtained. To solve the linear boundary value problem, as well as the obtained linearized boundary value problem with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which analytical solutions of these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, an analytical relationship was obtained for determining the temperature distribution in this medium. On the basis of the developed mathematical models, a computational algorithm was created and on this basis, software tools were created, using which the heat exchange processes in the middle of the brake structures for the selected materials of the brake pads were analyzed in terms of their effectiveness, as well as the determination of the optimal temperature values for the effective operation of the braking system of vehicles. The developed linear and nonlinear mathematical models for determining the temperature field in spatial heat-active media with internal heating make it possible to analyze their thermal stability. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and individual elements, but also the entire structure.

Physical and mathematical model of friction welding

The physical and mathematical model of the friction welding process in the form of a system of differential nonlinear heat conduction equation and a variational problem to determine the pressure distribution and shortening of the rods is proposed. The computer model of the friction welding process of solid cross-section rods on its basis is created. Computer simulation reproduces the main effects observed in the friction welding process: equalization of heat generation intensity in the joint plane; stabilization of the joint temperature when the plastic state of the metal is reached; characteristic sequence of increase, decrease and stabilization of torque during welding.

A new solution to a weakly non-linear heat conduction equation in a spherical droplet: Basic idea and applications

A new analytical solution to a non-linear heat transfer equation in a spherically-symmetric droplet is suggested. All thermophysical properties inside the droplet are considered to be close to their average values. This allows us to consider the non-linearity of this equation as weak. The solution is presented as T=T0+T1, where T0 is the solution to a linear heat conduction equation, and T1≪T0. The equation for T1 is presented as a linear heat conduction equation with a source term depending on the distribution of T0 and its spatial derivatives inside the droplet. The latter equation is solved analytically alongside the linear equation for T0, and the final solution is presented as T=T0+T1. The predictions of the numerical code in which this solution was implemented are verified based on a comparison of those predictions with the predictions of COMSOL Multiphysics code using input parameter values that are typical for nanofluid (water and SiO2 nanoparticles) droplet evaporation in atmospheric conditions. It is demonstrated that for these experiments T1≪T0 which justifies the applicability of the linear heat conduction equation used for the analysis of this process. Small differences in the temperatures predicted by both non-linear and linear models lead to a much more noticeable difference in integral characteristics such as time before the start of the formation of the cenosphere when the mass fraction of nanoparticles at the droplet surface reaches about 40%.

Thermoelastic Behaviors of Temperature-Dependent Multilayer Arches under Thermomechanical Loadings

This work presents analytical solutions for thermoelastic behaviors of multilayer arches with temperature-dependent (TD) thermomechanical properties under thermomechanical loadings. The temperature is varied across the thickness of the arch. Firstly, an arched-slice model is developed, which divides every layer of the arch into numerous hypothetical arched slices with uniform thermomechanical properties. Based on the model, the nonlinear heat conduction equations across the thickness of the arch are solved using the iteration approach, and then the thermoelastic equations obtained from the two-dimensional thermoelasticity theory are solved using the state-space approach and transfer-matrix approach. The present solutions are compared with those obtained using the finite element method and the Euler–Bernoulli theory (EBT). It is found that the error of the EBT increases when the angle of the arch increases or the length-to-thickness ratio decreases. Finally, numerical examples are conducted to analyze the effects of surface temperature and TD thermomechanical properties on the temperature, displacement, and stress distributions of a sandwich arch. The results show that the temperature dependency of thermomechanical properties is a key parameter in predicting the thermoelastic behaviors of the arch in a high-temperature environment.

THE CONTRIBUTION OF THE RELAXATION OF THE HEAT FLOW TO THE CHARACTERISTICS OF THE FUNDAMENTAL HARMONICS OF A NONLINEAR PHOTOACOUSTIC SIGNAL IN CONDENSED MEDIA

Based on the system of nonlinear heat conduction equations, which takes into account the relaxation of the heat flux for the gas layer, the sample, and the substrate, a system of equations is obtained for temperature fluctuations at the fundamental harmonic (FH) of a nonlinear photoacoustic (PA) signal. It is believed that all three layers in the PA-chamber have their own relaxation times, and the irradiated photon flux is modulated according to the harmonic law. The temperature dependence of thermophysical and optical parameters is represented by thermal coefficients. By solving the boundary problem, general expressions were obtained for the oscillatory component of the temperature perturbation in the exhaust gas, and then, according to a well-known scheme, for pressure fluctuations in the gas layer. The obtained expressions are analyzed. The frequency dependence of the amplitude and phase of this signal is numerically calculated for two possible substrates: the substrate is a thermal insulator and the substrate is an ideal conductor. It has been established that the frequency dependence of the PA-signal amplitude for these cases is antiphase. While, this dependence for the phase of the PA-signal is consistent in phases moreover, the values of the maximum and minimum coincide. The only significant difference is that in the region of zero modulation frequencies, the phase of the signal for a substrate made of a thermal insulator is positive, while for a substrate made of an ideal conductor is negative.

Математичні моделі температурного поля у елементах електронних пристроїв із чужорідним включенням

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in isotropic media with semi-transparent foreign inclusions that are also subject to external heat load, have been developed. To solve the nonlinear boundary-value problem, a linearizing function was introduced, using which the original nonlinear heat conduction equation and nonlinear boundary conditions were linearized, and as a result, a partially linearized second-order differential equation with partial derivatives and discontinuous and singular coefficients relative to the linearizing function and partially linearized boundary conditions was obtained. For the final linearization of the partially linearized differential equation and boundary conditions, the approximation of the temperature according to one of the spatial coordinates on the boundary surfaces of the inclusion was performed by piecewise constant functions. To solve the obtained linear boundary value problem, the Hankel integral transformation method was used, as a result of which an analytical solution was obtained, which determines the introduced linearizing function. For the numerical analysis of temperature behavior and heat exchange processes caused by external heat load, software tools have been developed, which are used to create a geometric image of temperature distribution depending on spatial coordinates. The obtained results indicate the correspondence of the developed mathematical models of the analysis of heat exchange processes in heterogeneous media with external heating to a real physical process. The obtained results indicate the correspondence of the developed mathematical models of the analysis of heat exchange processes in heterogeneous media with external heating to a real physical process. The developed models make it possible to analyze environments with foreign elements under external thermal loads regarding their thermal resistance. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual elements, but also the entire structure.

Causal explicit algorithm for heat conduction in a plasma

Hyperbolic heat conduction extends standard Spitzer-Harm heat conduction by including a term proportional to the time derivative of the heat flux. The new term arises from a kinetic derivation of the heat flux that includes higher order corrections. We present a causal explicit numerical algorithm for solving the nonlinear hyperbolic heat conduction equation in an unmagnetized plasma. The maximum stable timestep for the causal explicit algorithm scales linearly with the cell size, owing to the hyperbolic nature of the problem. This is in contrast to the quadratic scaling of the maximum stable timestep with the cell size for the parabolic forward time centered space algorithm. The favorable scaling of the timestep with the cell size enables a practical explicit implementation of heat conduction in high-performance massively parallel plasma codes. In particular, we have implemented the causal explicit algorithm in the laser plasma interaction code pF3D. We verify the CE algorithm and analyze its convergence rate by simulating a harmonic mode, which has an analytic solution within the context of the HHC model. We also compare simulations using the CE algorithm to those using the forward time centered space algorithm on a pair of test problems: evolution in time of a Gaussian temperature perturbation in a uniform plasma and heat transport in the presence of inverse bremsstrahlung heating by a Gaussian laser speckle.

The Analytical View of Solution of the First Boundary Value Problem for the Nonlinear Equation of Heat Conduction with Deviation of the Argument

The Analytical View of Solution of the First Boundary Value Problem for the Nonlinear Equation of Heat Conduction with Deviation of the Argument

Analytical Solution of Transient Heat Conduction through a Hollow Cylindrical Thermal Insulation Material of a Temperature Dependant Thermal Conductivity

The one-dimensional, cylindrical coordinate, non-linear partial differential equation of transient heat conduction through a hollow cylindrical thermal insulation material of a thermal conductivity temperature dependent property proposed by an available empirical function is solved analytically using Kirchhoff’s transformation. It is assumed that this insulating material is initially at a uniform temperature. Then, it is suddenly subjected at its inner radius with a step change in temperature. Four thermal insulation materials were selected. An identical analytical solution was achieved when comparing the results of temperature distribution with available analytical solution for the same four case studies that assume a constant thermal conductivity. It is found that the characteristics of the thermal insulation material and the pressure value between its particles have a major effect on the rate of heat transfer and temperature profile.

An Explicit Difference Scheme for a Nonlinear Heat Conduction Equation

An Explicit Difference Scheme for a Nonlinear Heat Conduction Equation

Study of the model of the phase transition envelope taking into account the process of thermal storage under natural draft and by air injection

The study proposed mathematical modeling of the enclosing structure of the building, which includes an energy-active panel accumulating solar radiation due to the phase transition of the heat-accumulating material. The mathematical model is based on a two-dimensional non-stationary nonlinear heat conduction equation describing the process of heat transfer in the bearing layer of the enclosing structure and the energy-active panel, taking into account the process of heat accumulation under natural and forced traction by air injection. In comparison of the calculation results, it was found. that the efficiency of the use of forced convection relative to natural was 33–44% in the question of the selection of useful heat, depending on the time (second or first day), and the value of heat accumulation showed almost the same results. Taking into account the research carried out by the authors, it is shown by calculation that enclosing structures in which a heat-accumulating material with a phase transition is used to accumulate solar energy are more energy efficient than simple heat accumulation in the carrier layer. The introduction of highly conductive inclusions into the energy-active panel and the use of forced convection instead of natural traction during heat extraction also significantly increase the energy efficiency of the structure, which can subsequently be used as an option for external fencing under appropriate climatic conditions.

Analytical Solution of Transient Heat Conduction through a Hollow Spherical Thermal Insulation Material of a Temperature Dependant Thermal Conductivity

The one-dimensional, spherical coordinate, non-linear partial differential equation of transient heat conduction through a hollow spherical thermal insulation material of a thermal conductivity temperature dependent property proposed by an available empirical function is solved analytically using Kirchhoff’s transformation. It is assumed that this insulating material is initially at a uniform temperature. Then, it is suddenly subjected at its inner radius with a step change in temperature. Four thermal insulation materials were selected. An identical analytical solution was achieved when comparing the results of temperature distribution with available analytical solution for the same four case studies that assume a constant thermal conductivity. It is found that the characteristics of the thermal insulation material and the pressure value between its particles have a major effect on the rate of heat transfer and temperature profile.

МАТЕМАТИЧНІ МОДЕЛІ ВИЗНАЧЕННЯ ТЕМПЕРАТУРНИХ ПОЛІВ У НЕОДНОРІДНИХ ЕЛЕМЕНТАХ ЦИФРОВИХ ПРИСТРОЇВ ІЗ УРАХУВАННЯМ ТЕРМОЧУТЛИВОСТІ Primary tabs View(active tab) Edit Translate

Linear and nonlinear mathematical models for determining the temperature field and subsequently analyzing temperature regimes in isotropic spatial media with semi-through foreign inclusions subjected to internal and external thermal loads are developed. For this purpose, the heat transfer coefficient for such structures is described as a single unit using asymmetric unit functions, which makes it possible to consider boundary value problems of heat transfer with one linear and nonlinear differential equations of heat transfer with discontinuous and singular coefficients and linear and nonlinear boundary conditions on the boundary surfaces of the media. In the case of a nonlinear boundary value problem, the introduced linearizing function is used to linearize the original nonlinear heat conduction equation and nonlinear boundary conditions, and as a result, a partially linearized second-order differential equation with partial derivatives and discontinuous and singular coefficients is obtained relative to the linearizing function with partially linearized boundary conditions. For the final linearization of the partially linearized differential equation and boundary conditions, the temperature is approximated by one of the spatial coordinates on the boundary surfaces of the inclusion by piecewise linear functions, as a result of which both the differential equation and boundary conditions become fully linearized. To solve the resulting linear boundary value problem, the Hankel integral transformation method is used, which results in an analytical solution that determines the introduced linearizing function. As an example, the linear dependence of the thermal conductivity coefficient of structural materials of a structure on temperature, which is often used in many practical problems, is chosen. As a result, analytical relations in the form of quadratic equations were obtained to determine the temperature distribution in a thermally sensitive layer with a foreign semi-through inclusion under external heating in the form of a heat flux. A numerical analysis of the temperature behavior as a function of spatial coordinates for given values of geometric and thermophysical parameters is performed. The influence of a foreign inclusion on the temperature distribution is investigated if the VK94-I ceramic is chosen as the material of the medium and the inclusion is silver. To determine the numerical values of temperature in the above structures, as well as to analyze heat transfer processes inside these structures caused by internal and external thermal loads, software tools have been developed that have been used to perform a geometric image of the temperature distribution depending on spatial coordinates. The obtained numerical temperature values indicate that the developed mathematical models for analyzing heat transfer processes in spatially heterogeneous environments with internal and external heating correspond to a real physical process. The software also makes it possible to analyze such environments subjected to internal and external thermal loads in terms of their thermal resistance. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual elements but also the entire structure.

Математичні моделі визначення температурних полів у теплоактивних елементах цифрових пристроїв з локальним внутрішнім нагріванням та із урахуванням термочутливості

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in isotropic spatial heat-active media subjected to internal local heat load, have been developed. In the case of a nonlinear boundary-value problem, the Kirchhoff transformation is applied, using which the original nonlinear heat conduction equation and nonlinear boundary conditions are linearized, and as a result, a linearized second-order differential equation with partial derivatives and a discontinuous right-hand side and partially linearized boundary conditions is obtained. For the final linearization of the partially linearized boundary conditions, the approximation of the temperature by the radial spatial coordinate on the boundary surface of the thermosensitive medium was performed by a piecewise constant function, as a result of which the boundary value problem was obtained completely linearized. To solve the linear boundary value problem, as well as the obtained linearized boundary value problem with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which analytical solutions of these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, an analytical relationship was obtained for determining the temperature distribution in this medium. Numerical analysis of temperature behavior as a function of spatial coordinates for given values of geometric and thermophysical parameters was performed. The influence of the power of internal heat sources and environmental materials on the temperature distribution was studied. To determine the numerical values of the temperature in the given structure, as well as to analyze the heat exchange processes in the middle of these structures, caused by the internal heat load, software tools were developed, using which a geometric image of the temperature distribution depending on the spatial coordinates was made. The developed linear and nonlinear mathematical models for determining the temperature field in spatial heat-active environments with internal heating testify to their adequacy to a real physical process. They make it possible to analyze such environments for their thermal stability. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and their elements, but also the entire structure.

Efficient utilization of PCM in building envelope in a hot environment condition

The thermal performance of the building envelope must be adjusted to promote sustainability, reduce energy consumption, and reduce greenhouse gas emissions. In this case, the elements' thermal resistance and heat storage capacity are critical. Due to their high energy density and ability to absorb/release heat under nearly isothermal conditions, phase change materials are extremely competitive and have a wide range of applications. The effectiveness of using PCM in the building envelope is studied in this investigation under exceptionally hot climatic conditions. A thermal model is developed based on the nonlinear transient heat conduction equation to predict the time-dependent thermal response of a multilayer building envelope wall. The enthalpy formulation is used to account for the phase change process, and the mathematical model is solved numerically using the finite element method. A typical exterior multilayer wall's thermal performance is studied and utilized as a benchmark. Then, at various points throughout the wall, a PCM layer is incorporated. The effect of the PCM on wall temperature distribution, heat movement into the interior area, and the fraction of molten material in the PCM layer is investigated. Optimization analysis is used to establish the suitable PCM melting temperature interval, with the heat flow at the internal surface of the wall as the objective function to be minimized. The thermal energy flow into the internal space can be reduced by approximately 50% when the PCM settings are improved. The findings of this investigation demonstrate that the suggested computational model is effective in calculating the optimal PCM, resulting in a significant improvement in the external wall's thermal performance. Integrating PCM into the building envelope is a viable technique for the region, as it has the potential to reduce the amount of heat entering the building, hence lowering the air conditioning load and lowering energy consumption.

Barycentric Rational Collocation Method for Nonlinear Heat Conduction Equation

Nonlinear heat equation solved by the barycentric rational collocation method (BRCM) is presented. Direct linearization method and Newton linearization method are presented to transform the nonlinear heat conduction equation into linear equations. The matrix form of nonlinear heat conduction equation is also obtained. Several numerical examples are provided to valid our schemes.

Unified modeling and kinetic analysis of the near-cathode region and hot cathode in atmospheric-pressure arc discharges

The near-cathode region plays a crucial role in exploring the transport characteristics of the transition from arc column to the hot cathode in atmospheric-pressure arc discharges because of the existing non-equilibrium phenomena. A one-dimensional unified model, including the near-cathode region and the cathode body, is developed for an argon arc discharge with the tungsten cathode at atmospheric pressure in this paper. The electrostatic model coupled with an external circuit in the near-cathode region is solved based on the implicit particle-in-cell coupled Monte Carlo collision method without any assumptions of thermal or ionization equilibrium or quasi-neutrality. A detailed description of the arc plasma–cathode and cathode–gas interactions is obtained by calculating the nonlinear heat conduction equation in the cathode. It is shown that the space-charge sheath strongly affects particle transport in the near-cathode region and energy transport from arc plasma to the thermionic cathode. The total current density has significant effects on the kinetic characteristics of arc plasma by feedback-like mechanisms. The Joule heating by the external circuit and charged particles deposited into the cathode are dominating mechanisms of energy transfer from the near-cathode region to the cathode, while energy loss by radiation is more significant compared with natural convection.