Thermal Conductivity Equation Research Articles

Non-axisymmetric coupled non-stationary problem of thermoelectroelasticity for a long piezoceramic cylinder

A new closed solution to the non-axisymmetric coupled non-stationary problem of thermoelectroelasticity was constructed for a long piezoceramic cylinder for the case of satisfaction of the first and the third kind boundary conditions. Cylindrical surfaces were made as electrodes and connected to a measurement device with large input resistance. Limitation of a temperature change “load” rate made it possible to include equations of statics, electrostatics and thermal conductivity in the initial formula. The finite biorthogonal transforms are applying to explore a non-selfadjoint system of differential equations and to develop a closed solution. The obtained relations made it possible to determine the temperature and electric fields, and the stress-strain state in the piezoceramic cylinder, as well as the potential difference between cylindrical surfaces (electrodes) under non-stationary non-axisymmetric temperature impact.

Differential Equation of Thermal Conductivity and Convective Heat Transfer in a Cylindrical Coordinate System

Differential Equation of Thermal Conductivity and Convective Heat Transfer in a Cylindrical Coordinate System

RESEARCH OF THE DESTRUCTION CONDITIONS OF GLAZING OF ENCLOSING STRUCTURES UNDER THE CONDITIONS OF THERMAL INFLUENCE OF FIRE

The article presents the main results of mathematical modelling of the destruction of glazing of enclosing building structures with translucent elements under fire conditions. The authors analyse literature data on calculating the fire resistance of enclosing building structures with glazing. The results show that improving methods for predicting the fire resistance of glazing is relevant. Mathematical models for the numerical study of the behaviour of glazing under the thermal influence of the standard fire temperature regime are substantiated. Mathematical models of the heat transfer process involve using a non-stationary differential equation of heat conduction with boundary conditions of the third kind, accounting for the convective and radiant heat exchange between the air in the room with the fire and the glazing. As a criterion for the destruction of glazing under the influence of fire, the authors establish the temperature resistance parameter, which is determined based on the mechanical characteristics of the glass. The article highlights the relevant data on the study of temperature indicators and the mechanism of glazing failure, using mathematical modelling with well-founded mathematical models. It obtains the results of the thermal calculation by applying the finite difference method for solving the non-stationary differential equation of thermal conductivity. The methodology for determining the time of glazing failure under the influence of the standard temperature regime of a fire using the calculation parameters obtained based on well-founded mathematical models is substantiated. An analysis of the adequacy of the obtained calculation data was carried out by comparing them with the results of experimental studies using statistical criteria. As a result of the analysis, the authors confirmed the validity of the calculation data. A complete factorial experiment was conducted based on well-founded mathematical models, which revealed the regularities of the dependence of the parameters of thermomechanical processes in single-layer glass on its structural characteristics. The revealed regularities have the form of linear regression dependences of the destruction time on the thickness of the glazing and the maximum length of the glass panel without bars. Using the regularities of the dependence of the parameters of thermomechanical processes in single-layer glazing on its structural characteristics, the authors constructed reference tables for evaluating the fire resistance of enclosing structures with glazing. The research obtained new scientific data on the study of the behaviour of glazing of enclosing building structures under the thermal influence of fire through mathematical modelling, which is the basis for improving the methods of calculating the fire resistance of enclosing structures with translucent elements. Keywords: enclosing building structures, glazing, fire resistance limit, calculation method, glass heat resistance.

MATHEMATICAL MODELS OF THE THERMAL PROTECTION COATING OF THE GAS GENERATOR OF THE HYDROGEN STORAGE AND SUPPLY SYSTEM

The article describes the properties of the thermal protection coating of the gas generator of the hydrogen storage and supply system by two transfer functions with Hurwitz polynomials of the third order. Obtaining such transfer functions is based on the solution of the non-stationary heat conduction equation, represented in operator form using the integral Laplace transform, and in which approximating spline functions in the form of second-order polynomials are used. The article gives the solution of the non-stationary thermal conductivity equation, which describes the thermal processes in the heat-protective coating of the gas generator of the hydrogen storage and supply system under the thermal effect of a fire. This solution comes in the operator form for the surface temperature of the heat-protective coating on the side of the gas generator wall. The peculiarity of this decision is the presence of hyperbolic functions of an irrational argument in its composition. The structural and dynamic scheme of the thermodynamic system ‘gas generator wall – heat-protective coating’ is presented, the feature of which is the presence of two entrances. The signal at the scheme’s first input reflects the thermal effect of the fire, and the signal at the second input reflects the thermal state of the gas generator cavity. An equivalent mathematical transition to the description of thermal processes in the heat-protective coating of the gas generator of the hydrogen storage and supply system was carried out. This transition happened due to the use of spline functions, which approximate the hyperbolic functions of the irrational argument and are polynomials of the second order. The article gives a verbal interpretation of the algorithm for determining the transfer functions of the heat-protective coating of the gas generator of the hydrogen storage and supply system and also an example of its implementation. Furthermore, it shows that for the given conditions of functioning of the gas generator heat-protective coating, the relative errors in approximation of hyperbolic functions by second-order polynomials do not exceed 1.7 %, and the average relative error when equivalent replacement of transfer functions does not exceed 3.8 %. Keywords: gas generator, hydrogen storage and supply system, thermal protection coating.

Algorithm for the synthesis of equations of thermal conductivity of lithium-ion accumulator for finite volumes during division

Algorithm for the synthesis of equations of thermal conductivity of lithium-ion accumulator for finite volumes during division

Development of a transient analysis code for trans-critical simulation of SCWR

Supercritical water-cooled reactors (SCWR) experience pressure reductions below the critical point during a loss of coolant accident (LOCA). During the reactor depressurization, boiling crises may occur, leading to pressure fluctuations and significant increases in wall temperatures, posing a severe threat to the cladding. In this study, a one-dimensional transient analysis code has been developed to incorporate one-dimensional steady-state flow equations in the fluid domain and transient thermal conductivity equations in the solid domain. The code also includes a wall heat transfer model and a model for the velocity of the moving quench front to simulate transcritical depressurization transient processes. The simulation successfully predicts the typical experimental phenomena. It is reasonable to use the critical temperature as the demarcation point between the dry and wet conditions of the axial wall of the rod bundle under transcritical conditions. By combining the experimental data with the program calculations, the critical interface for the occurrence of boiling crisis under transcritical conditions is determined using mass flow rate, heat flow density and fluid temperature as inputs. The criterion for the occurrence of CHF under transcritical conditions is obtained, which provides a reference to the safety analysis of SCWRs.

Solution of the plane thermoelasticity problems based on a new set of equations

The interaction of thermal and mechanical fields in modern technologies for the synthesis and processing of materials attracts more and more attention. This is due to the need to predict the behavior and properties of new materials and products depending on the conditions of their creation. In the present work, attention is focused on the comparison of the solutions to the thermoelasticity plane problems with and without accounting for the temperature dependence of the material properties. It is assumed that an external heat source traveling over the surface with a specific velocity and following a given trajectory is responsible for a significant temperature increase. It may correspond to a laser source, for example. The new way of problem formulation in stresses is used. The temperature dependence of the characteristics leads to nonlinear thermoelasticity equations when the equilibrium equations, the requirement of strain compatibility, and the Duhamel-Neuman relation for the plane stress state case are combined. To determine the temperature field, a nonlinear thermal conductivity equation is used. To solve the problem numerically, the use of an implicit finite-difference scheme and coordinate splitting for the thermal conductivity equation and the finite-difference method of successive over-relaxation with Chebyshev acceleration for the equations for stresses was made. As a result, the difference in the solution of linear and nonlinear problems is demonstrated for the considered cases. Using the example of a mixture of titanium and aluminum powders, numerical modeling demonstrated that some stress components can reach values up to 3.5 times higher than those obtained in the case of temperature-independent material properties when there is a 500-degree local temperature increase.

Mathematical modeling of insulating coating of thermal conductivity including body’s own radiation and non-local spatial effects

Abstract In this paper thermal conductivity analysis of insulating coating is discussed. The general partial differential thermal conductivity equation is used with consideration to nonlocal boundary effects. We relying on the Eringen’s approach involves heat-flow vector transformation using nonlocal influence function and weight coefficients. Also presented model includes own surface radiation of the coating.

Comparative calculations of trapped field and trapped magnetic flux with different multi-pulse magnetization methods for a bulk high-temperature superconductor

Bulk high-temperature superconductors (HTSs) can trap high magnetic field and are potentially useful for a variety of applications as pseudo-permanent magnets. The pulsed field magnetization (PFM) for bulk HTSs is cost effective and flexible in application compared with quasi-static field cooling and zero field cooling techniques. Many PFM methods have been proposed in many studies to achieve the excellent magnetization performances such as high trapped field and large trapped magnetic flux. In order to clarify the magnetization characteristics of bulk HTSs using different typical PFM methods, we comparatively analyze several typical PFM methods using a simulation model based on the H-formulation combining the thermal conductivity equation. The electromagnetic and thermal behaviors during the magnetization of a bulk HTS with different PFM methods are numerically achieved using the solenoid-type coil to magnetize the bulk. The calculations show that multi-pulse magnetization methods can effectively enhance the trapped field and trapped magnetic flux of the bulk, and different multi-pulse magnetization methods have different efficiencies enhancing these performances. Among all considered PFM methods in the study, the combination method of modified multi-pulse technique with step-wise cooling and iteratively magnetizing pulsed-field method with reducing amplitude has the largest improvement for the trapped magnetic field and the trapped flux simultaneously.

Calculation of the heat stabilizer effective parameters depending on its location relative to the pile

Maintaining the soil in a frozen state in a cryolithozone is necessary for preventing the collapse of structures. The most effective technical solution for this is the use of two-phase passive heat stabilizers. Monitoring the state of permafrost soil allows to accept technical decisions that prevent soil thawing. Calculation the effective parameters of such decisions requires mathematical modeling of thermal and hydrodynamic processes. An approach is proposed to effectively compensate the heat flux from the pile on the ground, assuming the location of the heat stabilizer in the same well with the pile or at a slight distance from it. The aim of the work is to calculate the effective parameters of the heat stabilizer depending on its location relative to the pile. The developed mathematical model assumes the five tasks: 1) blowing the above-ground part of the heat stabilizer with air; 2) movement of liquid refrigerant to the bottom of the heat stabilizer; 3) cooling of the casing of the heat stabilizer with an two-phase flow of refrigerant; 4) heat exchange in the system refrigerant — the casing — frozen soil; 5) compensation of heat flow from the pile into the ground with heat flow from the ground into the heat stabilizer. The first problem is solved on the basis of an empirical criterion equation. The second and third tasks are solved using the laws of conservation of mass, momentum and energy. The fourth and fifth tasks involve solving the equation of thermal conductivity. As a result of calculations, the effective parameters of the heat stabilizer were obtained. The minimum lateral dimensions of the heat stabilizer casing have been established to compensate the heat flow from the pile into the ground. The time of freezing front reaching the pile from the heat stabilizer located at a distance of 0.5 m from one of its corners has been determined.

Research on effective thermal conductivity of hollow glass microspheres under vacuum at low temperature

As insulation materials with excellent thermal insulation properties, high robustness, and low maintenance costs, hollow glass microspheres (HGM) are gradually being applied in large liquid hydrogen storage tanks. However, the low-temperature insulation mechanism of HGMs is still unknown, and the insulation effect of HGMs of different types under various conditions has been neglected in recent decades. In this study, an existing theoretical model was improved by considering the gas thermal conductivity equations under various pressure intervals, and an experimental platform was built based on the steady vertical heat flux method to measure the effective thermal conductivities of three commercially available HGM models. The effective thermal conductivities of HGM models K1, T15, and HL20 at 0.1 Pa were 10, 8.9, and 7.6 mW/(m·K), respectively. The values decreased to 2.5, 2.9, and 2.0 mW/(m·K), respectively, when the cold boundary temperature was 90 K. To analyse the effect of the HGM physical parameters and environment on the HGM insulation, the thermal conductivities were calculated using the derived theoretical model. It was found that continuing to reduce the pressure after the ambient pressure reached 10 Pa hardly improved the insulation performances of the HGMs. The radiative heat transfer was very sensitive to temperature changes and jointly dominated the heat transfer with integrated thermal conductivity in the low-temperature region. In summary, HGMs are advanced vacuum-insensitive insulation materials, which have great potential in the field of cryogenic liquid storage and transportation.

Study on the structure, elasticity, and thermal conductivity of orthocarbonate Sr2CO4

Orthocarbonate Sr2CO4 is a recently discovered Sr-carbonate that plays a crucial role in understanding the global long-term carbon cycle. In this work, the structure, equation of state, elasticity, and thermal conductivity of Sr2CO4 were studied by first-principles calculations. The results show that the surfaces of the SrO9 and SrO11 polyhedrons, which make up Sr2CO4, are composed of triangles and quadrangles. By comparison with other alkaline earth metal orthocarbonates, SrCO3 polymorphs, and the Preliminary Reference Earth Model (PREM), Sr2CO4 is found to be a carbonate with high density, low wave velocity, low anisotropy, and low thermal conductivity in the entire Earth’s mantle. This study may help to understand the high-pressure behavior of alkaline earth metal orthocarbonates in the Earth’s interior.

Влияние режима пожара на огнестойкость несущих строительных конструкций машинных залов ГРЭС

Machine rooms of a state district power plant (GRES) are sizeable. Heat impact on the structures of the building in the case of a real fire can significantly differ from the impact in the case of a standard fire. Therefore, the calculation of fire resistance of building structures should be conducted not only in conditions of a standard fire but of a real fire as well. The objective of the study is the theoretical evaluation of the actual limits of fire resistance of load-bearing building structures of the machine room of the main building of the Nizhneturinskaya GRES in conditions of standard and real fire regimes. Numerical experiments to calculate the heating of load-bearing building structures in the case of a standard fire and the case of the most hazardous scenario of the real fire development have been conducted in the framework of the study. A field 3-dimensional model to calculate fire hazardous factors and a non-stationary thermal conductivity equation have been applied. The obtained results have been analyzed. The field model adapted to the space-planning solutions of the machine room has been described. The most hazardous scenario of the development of real fire in case of combustion of the turbine oil spill in the sector of the machine room with the lowest height of ceiling has been chosen. The temperature fields in the gaseous environment of the machine room at different moments are presented. It has been detected that the heating of reinforced concrete structures of the machine room ceiling is significantly uneven due to the 3-dimensional gas dynamic pattern of fire. Dependencies of temperatures of the frame of reinforced concrete load-bearing structures of the ceiling on the time of standard and real fires in the most heated section of the ceiling have been obtained. It has been determined that the maximum temperature of the frame in the case of the standard fire is lower than in the case of real fire. Due to the features of the 3-dimensional thermal gas dynamic pattern of fire in machine rooms of the GRES, the fire resistance of load-bearing building structures in these premises must be determined in both conditions of standard fire and real fire, considering the most hazardous scenario of its development.

Инженерная методика определения температурного состояния металла при его нагреве

Currently, the determination of the temperature profile of the furnace required to ensure the specified heating parameters of the metal is carried out using the temperature measurement method or mathematical models. They are most based on solution of the differential equation of non-stationary thermal conductivity. Previously, the thermal diagram method of I.D. Semikin was widely used. It allows us, under given heating conditions, to determine heating duration. In practice, it is important to develop a simple engineering method that would allow the temperature field of the metal to be determined with sufficient accuracy under given heating conditions. The development is based on the thermal diagram method of I.D. Semikina. But at the same time, the final equation describing the heat balance of the metal within the furnace zone is solved relative to the enthalpy of the metal at the exit from the zone. The specified heating conditions (productivity, zone temperature) are used to determine the quantities of the right-hand side of the equation (time heating, average heat flux density). The authors have proposed the method for continuous furnaces of various types (push-type and with a mechanized hearth) that allows us for a given productivity and temperature profile of the furnace to calculate the change in the temperature state of the metal for all heating zones, from the moment of loading to the release of metal. The adequacy of the proposed methodology has been tested for two types of furnaces. For methodical push-type furnaces, the results obtained have been compared to the results of modeling the temperature state of the metal using a numerical method. The discrepancy between the results after passing the inertial heating period does not exceed 1 %. For a walking-beam furnace the calculated value of the metal surface temperature after the furnace has been compared with the results of operational measurements. The discrepancy is less than 2 %. The totality of the results obtained ensures the achievement of the research goal. The proposed method is based on the heat balance equation and allows us to consider it to be correct. It is an engineering method and quite simple to implement. The method is recommended to be used to design furnaces and to calculate the temperature profile of furnaces by engineering departments of metallurgical enterprises.

Математическое моделирование пиролиза резиновых отходов в горизонтальном цилиндрическом реакторе

Recycling of cushioned rubber products is an important environmental and technical-economic issue. Among the various methods of processing such waste, one of the most effective is the pyrolysis process, since it allows us to obtain fuel and energy, and provides the possibility of secondary use of carbon black and metal. Mathematical modeling of this process is necessary to solve problems of optimization and automated control. Thus, the purpose of this study is to simulate the pyrolysis process in a horizontal cylindrical batch reactor, which is currently becoming increasingly widespread. To mathematically describe the process, a model with distributed parameters is used in the form of a two-dimensional thermal conductivity equation and a system of equations of chemical kinetics of polymer thermal destruction reactions. The problem is solved numerically using the finite element method. As a result of the numerical solution of the equations of the mathematical model, non-stationary distributions of temperature and degree of conversion over the cross section of the reactor with non-uniform filling have been obtained. The authors have studied the kinetics of the reaction of polymer thermal destruction in reactors of different diameters. The results obtained can be used to predict the kinetics of material destruction under given reactor heating conditions, which can be useful when we design reactors and automate pyrolysis process control.

Solution one-dimensional problems of heat conduction with convection using Poisson integral

There is the relevant problem of developing high-viscosity oil in the oil and gas industry. One of the ways increasing its production is the thermal method. It is necessary to combine hydrodynamic equations with the thermal conductivity equation for simulation of filtration processes. The resulting mathematical models are solved by using different numerical methods. The accuracy and convergence of such algorithms is rarely tested. One way to carry out this check is to simulate problems that can be solved analytically in particular cases. For example, Fourier series are used for simple boundary conditions. Another method is usage the Poisson integral. This method is more convenient for comparing results than the Fourier method, because with the same accuracy requires significantly less calculations. But the Poisson method requires knowledge of the initial condition over the entire space, and in real problems it is usually given in a limited area. In this paper we propose an algorithm for extending the initial conditions to the entire space. In this way, two heat conduction problems are solved taking into account convection using the Poisson integral. It is shown that the accuracy is comparable to the Fourier method, but with fewer calculations.

Моделирование и анализ распределений температуры и термомеханических напряжений в многокристальном электронном модуле

A mathematical thermomechanical model of a multi-chip electronic module (MEM) containing three silicon dies of high-power transistors fixed with a conductive adhesive on a copper plate placed on a radiator is considered in the form of a system of equations of thermal conductivity and thermoelasticity with specified boundary conditions. As a result of computational studies of the model in the COMSOL Multiphysics software environment, distributions of temperature and thermomechanical stresses in the MEM structural elements were obtained depending on the heating power, the thickness of the adhesive and the size of the model defect in the contact connection of one of the MEM crystals with a copper plate in the form of voids in the adhesive layer. It is shown that voids in the adhesive layer lead to a sharply non-uniform temperature distribution over the crystal area and thermomechanical shear stresses in the contact layer that exceed the maximum permissible values for the adhesive. It has been established that thermomechanical stresses decrease with increasing thickness of the adhesive layer and increase with increasing size of the defect(voids) in this layer. The simulation results are in good agreement with the results of measuring the thermal impedance of power transistor crystals using the modulation method, which indicates the correctness and adequacy of the developed model.

Numerical-analytical approach to solving problems of non-stationary thermal conductivity of a non-thin annular plate

This paper considers the first stage of calculating the initial boundary value problem of non-stationary thermal conductivity of cylindrical bodies using a modified method of lines, namely dimension reduction of the original differential equations, initial and boundary conditions. The original equations of thermal conductivity are defined in a cylindrical coordinate system in a spatial setting. An object is a cylindrical body with commensurate dimensions. This area of research is relevant, because when calculating the load bearing elements of structures to thermal effects, the first step is to determine the distribution of temperature fields. Boundary conditions are considered as conditions of convective heat transfer, which by means of boundary transition are transformed into boundary conditions of the first and second types. Dimension reduction with respect to spatial coordinate is performed by the Bubnov-Galorkin-Petrov projection method using local basis functions. These functions are called "cover" functions, which are related to the lines drawn on the domain of the task. Normalized trigonometric series are used to reduce the dimensionality of the equations with respect to circular coordinate. All transformations are performed in index form. In addition to differential equations, the projection method reduces the dimensionality of the initial and boundary conditions. In this paper, the most optimal form of writing reduced equations is determined, which provides the ease of reducing the dimensionality of the original differential equations. Initial and boundary conditions take into account the impact of the environment. All this makes it possible to set a reduced initial boundary value problem for further calculation by numerical finite difference methods.

Расчет времени охлаждения надземного водовода с теплоизоляцией в условиях отрицательных температур

Objective: transport infrastructure facilities include a variety of water supply systems. In the event of an emergency, after stopping the movement of water in the pipeline, it is cooled first, and then there is a risk of freezing and destruction of water pipelines. This work is devoted to the calculation of the cooling time of water in the above-ground water conduit with thermal insulation from the specified temperature value in the initial state to the freezing temperature. Methods: when building a mathematical model of the water cooling process, an approach is used based on averaging the equations of hydrodynamics by the volume of water in the pipeline and averaging the equations of thermal conductivity in the wall of the pipeline and in the layer of the heat insulator by the polar angle. To obtain a quasi-stationary form of equations, a comparative analysis of the rates of thermal processes in different layers of the water conduit is used. Results: a new mathematical model for cooling the water pipeline is formulated — a model of average temperatures. The applicability of the quasi-stationary form of the equations of the model is justified and its analytical solution is found. Explicit formulas are obtained for cooling time of water conduit as a function of its parameters. Cooling time was calculated in a wide range of parameters. The results of model calculations are compared with calculations according to traditional semi-empirical formulas. Practical importance: the formulas obtained in the work can be used to estimate the cooling time of water in an above-ground water pipeline with heat insulation to the freezing temperature in the case when the ambient temperature drops to negative values.

Investigation of thermal stress effects on subthreshold conduction in nanoscale p-FinFET from Multiphysics perspective

The rising temperature due to a self-heating or thermal environment not only degrades the subthreshold performance but also intensifies thermal stress, posing a severe challenge to device performance and reliability design. The thermal stress effects on the ON-state performance of the p-type fin field-effect transistor were previously studied. However, as far as we know, how thermal stress affects its subthreshold conduction remains unclear, which is studied in this manuscript. The impact of thermal stress due to the self-heating of adjacent devices on subthreshold conduction is investigated by solving the quantum transport, thermal conduction, and force balance equations for ballistic transport and dissipative transport with phonon scattering. Then, the thermal stress effects at different ambient temperatures are further discussed and analyzed. The simulation results show that the OFF-state leakage current can be reduced by thermal stress, even up to 9.28% for the (110)/[001] device operating at an ambient temperature of 550 K, and its reduction is the comprehensive result of the thermal stress effects on the band structure, potential profile, carrier distribution, and source-to-drain tunneling. In addition, the thermal stress has no significant effects on subthreshold swing although it can change the magnitude of the subthreshold current. Moreover, the effect of thermal stress on subthreshold conduction is highly dependent on the thermal environment of the device and the crystal orientation of the channel semiconductor material.