Thermal Conductivity Equation Research Articles

Mathematical Model of Fuse Effect Initiation in Fiber Core

This work focuses on the methods of creating in-fiber devices, such as sensors, filters, and scatterers, using the fiber fuse effect. The effect allows for the creation of structures in a fiber core. However, it is necessary to know exactly how this process works, when the plasma spark occurs, what size it reaches, and how it depends on external parameters such as power and wavelength of radiation. Thus, this present study aims to create the possibility of predicting the consequences of optical breakdown. This paper describes a mathematical model of the optical breakdown initiation in a fiber core based on the thermal conductivity equation. The breakdown generates a plasma spark, which subsequently moves along the fiber. The problem is solved in the axisymmetric formulation. The computational domain consists of four elements with different thermophysical properties at the boundaries of which conjugation conditions are fulfilled. The term describing the heat source in the model is determined by the wavelength of radiation and the refractive indices of the core and the shell and also includes the radiation absorption on the released electrons during the thermal ionization of the quartz glass. The temperature field distributions in the optical fiber are obtained. Based on the calculations, it is possible to estimate the occurrence times of various phase states inside the fiber, in particular, the plasma spark occurrence time.

On a Method of Solving the Thermal Conductivity Equation of an Inhomogeneous Isotropic Body

Key words: coefficient of internal heat conductivity, classical problems, Cauchy problem, small parameter method, second-order partial differential equation, inhomogeneity function Classical problems in various formulations have been solved by many authors (Cauchy, Dirichle, Neumann and others) for isotropic homogeneous bodies in the theory of elasticity. Such boundary value problems have been reduced to solving differential equations with constant coefficients [1-3; 5]. The paper considers the problem of thermal conduction of an inhomogeneous isotropic body, which leads to the solution of a second-order partial differential equation with variable coefficients (1.3). With sufficient smoothness of the variable coefficients, through transformations, as well as the characteristic equation, this second-order differential equation in a given region can be reduced to one of the following forms; elliptic, hyperbolic and parabolic. Here, the boundary value problem is reduced to solving a second order differential equation with variable parabolic coefficients. A small physical parameter is introduced, and a recurrent sequence of boundary value problems for the Cauchy boundary value problem is obtained in the article. When solving these boundary value problems, the general solution is represented as a series by formula (2.2) and under certain conditions depending on δ and σ the convergence of the series is proved and the solution is reduced to (5.4). In case of varying the inhomogeneity function f (x,y,z), a small physical parameter δ and a constant σ, a set of inhomogeneous isotropic bodies is obtained, the results of which will be compared with the results of homogeneous isotrpic bodies.

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.

Influence of heterogeneous catalysis on aerothermodynamics at hypersonic speeds based on gas-interface-solid coupling simulation

Heterogeneous catalysis plays an important role in causing the aerodynamic heating surge in the hypersonic nonequilibrium flow/thermal protection material interactions for high-speed aircrafts. To model the flow/material interaction problem, a gas-interface-solid coupling method for heterogeneous catalysis was established by introducing interface balance relation into the coupling framework of nonequilibrium Navier-Stokes equations and thermal conduction equation. The influence of coating heterogeneous catalysis on heat transfer into the heat shield of a blunt body was then numerically investigated to reveal the catalysis-induced aerothermodynamics. The coupling simulation results reveal that the codominant role of material mediated catalysis and gaseous reactions on boundary layer and their induced near-wall evolutions are the main features of the chemically reacting boundary layer that differ from the conventional boundary layer theory. It was found that catalytically inert coatings can weaken the local aerodynamic heating due to the low activity of local catalysis and enhance the downstream catalytic heating through driving atomic species to the downstream region. Appropriate layouts of thermal protection coatings can make full use of the nature of catalytic exothermicity to effectively manage hypersonic flow and catalysis-induced heat-transfer characteristics for the lightweight and low-redundancy thermal protection designs of future spacecraft.

Terrestrial heat flow measurement and numerical simulation of lithospheric thermal structure in western Sichuan, China

The western Sichuan in the eastern Qinghai-Tibet plateau has significant geothermal potential. Heat flow is one of the most important parameters in geothermic, its measurement can help obtain a better understanding of the regional lithospheric thermal structure. In this study, we reported 4 heat flow data based on the continuous steady-state temperature logging and rock thermal conductivity test, including two class-A and two class-D data. Furthermore, we calculated the lithospheric thermal structure by a 2D steady-state thermal conduction equation in COMSOL, analyzing the lateral differences of lithospheric thermal structure in the eastern Qinghai-Tibet plateau. Our study has determined that the heat flow values in the western Sichuan range from 94.7 to 116.2 mW/m2, with a corresponding geothermal gradient of 32.1–37.3 °C/km. Moreover, numerical simulation results indicate significant differences in lithospheric thermal structure between western Sichuan and Sichuan Basin, where the Longmen Shan fault zone is a crucial regional tectonic boundary and a significant temperature gradient zone. Specifically, the Moho temperature in the western Sichuan region ranges from 1000 to 1200 °C with an average temperature of 1060 °C, while the Sichuan Basin ranges from 600 to 700 °C. Besides, the thermal lithosphere thickness in the western Sichuan and the Sichuan Basin is 88–97 km and 105–110 km, respectively. These observations suggest that the western Sichuan region is characterized by high thermal anomaly and provide theoretical evidence for the geodynamic and geothermal development in this area. Finally, we suggest that the high thermal anomaly in western Sichuan is the result of a combination of factors, such as thickened crust, high radioactive heat production, and shear frictional heating from deep fault and so on.

The approximate solution of nonlinear heat equation

AbstractNowadays there are many approximate methods for thermal conductivity calculation, that lead to satisfactory outcomes in engineering practice. With the new approach, the solution of the nonlinear thermal conductivity equation was investigated. In addition to this, we reviewed the approximate solution of the nonlinear equation of thermal conductivity with cubic nonlinearity. To solve the problems of mathematical analysis, differential and integral equations, and boundary value problems of mathematical physics, difference, and interpolation are applied. Thus, for thinking of the effectiveness and reasonableness of these approaches, it is crucial for their theoretical investigation. The solution to these questions was found for each class of equation and each of its methods in their way and was often represented as significant difficulty, and in many cases was an obstacle for the current time. A natural approach to solving this issue is the use of the ideas of functional analysis. The variational principle initially was considered as a variational approach for solving linear functional equations and finding eigenvalues of linear operators. As in any variational approach, the problem of solving an equation will be brought to finding the extremum of the certain function of a special type, given over the entire space. It was found that the approach is useful in a way of minimizing functions of the more general type.

Experimental and DEM simulations of the mechanical properties of rock under freeze–thaw cycles

The investigation of the physical-mechanical properties of rock after freeze–thaw (F-T) cycles has important theoretical significance for understanding freeze–thaw disaster mechanisms, disaster prediction, and tunnel protection system design in cold regions. Physical-mechanical tests are carried out to quantify the deterioration of the physical-mechanical properties of sandstone subjected to long-term F-T cycles. The damage mechanism of sandstone subject to F-T cycles is discussed based on scanning electron microscopy (SEM) results. A new three-dimensional discrete element method (DEM) model for simulating freeze–thaw damage of water-saturated rocks is established based on the volume expansion theory. In the new method, a one-dimensional thermal conduction equation is introduced to realistically simulate the distribution of the temperature field in a cylindrical sample during thermal conduction, and the evolution equation of ice volume expansion is introduced to control the volume expansion of ice particles in the sample as the freeze point. The crack development mechanism during the F-T cycle and the correlation between the cracks generated by the F-T cycle and subsequent loading are also analyzed. The results show that approximately 80% of the frost-heave cracks are distributed in the annular column area 10–25 mm from the axis of the sample. In the radial direction, there is a significant correlation between the formatting of cracks during F-T cycles and loading. In the axial direction, due to the relatively uniform distribution of F-T cracks, the formation of cracks under subsequent loading is mainly controlled by stress concentration.

Development of a Thermal Model of Welding by the Finite Element Method in Software "Bazis"

The paper presents a model and an algorithm for calculating the operational properties of a metal and a structure after its assembly-welding. The model is based on the equations of non-stationary thermal conductivity and mechanical equilibrium, as well as the analysis of structural-phase transformations using thermokinetic diagrams. It is implemented in the original BAZIS software package. Validation of software was carried out when calculating thermal fields and deformations as applied to arc welding, arc deposition, and laser welding of standard joints.

Theoretical foundations of the construction of the operation of heat flow devices

Numerous studies show that non-destructive testing methods satisfy most of the requirements of technical diagnostics of heating networks and technological facilities. Methods of non-destructive testing are based on the observation and automated registration of the temperature state of processes. The developed device is designed to analyze the state of thermal insulation of underground pipelines. The development and research of devices for measuring heat flow requires mandatory consideration of the temperature field of the sensing element, i.e. solutions of the differential equation of thermal conductivity for a body of a certain shape under given boundary conditions. In general, the sensing elements are multi-layered: black coating, calorimetric load, heat-sensitive elements, alternating lacquer and adhesive layers, i.e., the sensing elements are heterogeneous, both in the direction perpendicular to the irradiated surface and in parallel. The heterogeneity in the first case is due to the multi-layering of the sensing element. The article describes solutions to the thermal conductivity equation describing the temperature field of a sensitive element in the form of a hemisphere and a spherical zone, due to the nonequivalence of heat losses during irradiation and calibration by electric current. Taking into account this systematic error makes it possible to increase the accuracy of measuring the energy parameters of radiation. These solutions of the equations formed the basis of the design of the device for measuring heat flow.

The nonlocal thermo‐hydro‐mechanical model for cracking behaviors of quasi‐brittle materials in the framework of advanced general particle dynamics

AbstractA novel nonlocal thermo‐hydro‐mechanical coupling model is proposed to investigate cracking behaviors of quasi‐brittle materials in the framework of advanced general particle dynamics (GPD). The nonlocal thermal conduction equation and the nonlocal flow diffusion equation are derived based on nonlocal vector calculus and the microscopic constitutive equation. Moreover, the coupling influences are considered based on the nonlocal equation and the coupling principle. The governing equation of the coupling system is established in the framework of GPD. The correctness of the proposed model is verified through comparing with the corresponding experimental data, and the numerical results are in good agreement with experimental results. Finally, the proposed model is applied to analyze the deformation and cracking behaviors of rock mass around tunnels, and the stress redistribution is revealed under thermo‐hydro‐mechanical coupling condition.

Thermal Conductivity Calculation Model for Refrigerant Mixtures in the Vapor Phase

Thermal conductivity measurements of ten refrigerant mixtures (R-404A, R-406A, R-407C, R-409A, R-410A, R-415A, R-507A, R-227ea/R-134a 61.5/38.5, R-227ea/R-134a 88.8/11.2 and R-227ea/R-134a 45/55) in the gas phase are analyzed. The thermal conductivity was studied with the same experimental setup, which improved the reliability of the results, excluding thereby systematic errors caused by using different methods to measure thermal conductivity. The reported experimental data have 1.5 % to 2.5 % uncertainty at 0.95 confidence level with a coverage factor of k = 2. Equations for calculating thermal conductivity depending on temperature and pressure are given for each mixture. The equations for thermal conductivity on the dew line and in the ideal gas state are obtained. A model for predicting thermal conductivity on a wide range of state parameters is proposed based on obtained experimental data and the theory of thermodynamic similarity. A comparison of the experimental data with the calculation model gives the standard deviation at 0.4 % to 2.1 %, which does not exceed the measurement error.

The Theoretical and Experimental Study on High Power Quasi Single Mode Tm-Doped All-Fiber Laser at 1908 Nm

Thulium-doped all-fiber lasers at wavelength of 1908 nm have developed rapidly because of more applications. In this paper, a quasi single mode thulium-doped quasi single mode all-fiber laser is reported, which can generate laser powers of 30.8 W in oscillator and 83.5 W in amplifier at wavelength of 1908 nm. The slope efficiencies are 43.93% and 55.65%, respectively. Besides, the effect of active fiber length on the conversion efficiency are studied theoretically and experimentally. The results show that optimal fiber length could improve the cross-relaxation process and pump utilization. Based on the thermal conduction equations, experiments theoretically study the cooling temperature and the coefficient of convective heat transfer impacted on thermal management to obtain high laser power output.

SOLUTION OF THE THERMAL CONDUCTIVITY EQUATION OF A RODS WITH A SQUARE SECTION BY THE DIFFERENCE METHOD

The purpose of this work is to study the thermophysical state of a rod of constant cross section and limited length.This work is devoted to the automation of the study of the thermophysical state of a rod of constant cross-section and limited length. The process of automating research is based on the laws of conservation of energy.A three-dimensional body is considered, the constant cross section of which has the shape of a square. It is assumed that the left end of the rod coincides with the origin of coordinates and the heat transfer coefficient is assumed to be constant over the entire ВЕСТНИК КазНПУ им. Абая, серия «Физико-математические науки», No1(77), 2022 г.34surface of the rod. It is also assumed that the rod is subject to point temperature and surface heat transfer. The stated problem is solved by the difference method, i.e. the heat equation is approximated by a difference scheme. A program has been developed for finding the temperature distribution along the rod, which places the results of numerical calculations in several files. The results of numerical calculations in dynamics (over time) are displayed in the form of one-dimensional and two-dimensional graphs Бұл жұмыстың мақсаты –ұзындығы шектеулі тұрақты қималы сырықтың жылуфизикалық күйін зерттеу болып. Бұл жұмыс тұрақты көлденең қиманың және ұзындығы шектеулі сырықтың жылуфизикалық күйін зерттеуді автоматтандыруға арналған. Зерттеуді автоматтандыру процесі энергияның сақталу заңдарына негізделген. Тұрақты көлденең қимасы квадрат түрінде болатын үшөлшемді дене қарастырылады. Қиманың сол жақ шеті координатаның басталуымен сәйкес келеді және жылу беру коэффициенті сырықтың бүкіл бетінде тұрақты болып саналады. Сондай-ақ, сырық нүктелік температура мен беттік жылу алмасудың әсерінен болады деп болжанады. Қойылған есеп айырмашылық әдісімен шешіледі, яғни жылу өткізгіштік теңдеуі айырмашылық схемасымен жуықталады. Сандық есептеулердің нәтижелерін бірнеше файлдарға орналастыратын, сырықтағы температураның таралуын табу бағдарламасы жасалды. Динамикадағысандық есептеулердің нәтижелері (уақыт бойынша) бір өлшемді және екі өлшемді графиктер түрінде көрсетіледі. Целью данной работы является исследование теплофизического состояния стержня постоянного сечения и ограниченной длины. Данная работа посвящена к автоматизации исследования теплофизического состояния стержня постоянного сечения и ограниченной длины. Процесс автоматизации исследования опирается на законы сохранения энергии. Рассматривается трехмерное тело, постоянное поперечное сечение которого имеет форму квадрата. Предполагается, что левый конец стержня совпадает с началом координат и коэффициент теплообмена считается постоянным по всей поверхности стержня. Также предполагается, что стержень находится под воздействием точечной температуры и поверхностного теплообмена. Поставленная задача решается разностным методом, т.е. уравнение теплопроводности аппроксимируется разностной схемой. Разработана программа нахождения распространения температуры по стрежню, которая помещает результаты численных расчетов в несколько файлов. Результаты численных расчетов в динамике (по времени) отображаются в виде одномерных и двумерных графиков.

Mathematical Modeling of the Wave Dynamics of an Encapsulated Perfluorocarbon Droplet in a Viscoelastic Liquid

A mathematical model has been developed and a numerical study of vapor bubble growth as a result of acoustic evaporation of an encapsulated perfluorocarbon droplet in a viscoelastic liquid is presented. The viscoelasticity of the droplet shell and the carrier liquid is taken into account according to the Kelvin–Voigt rheological model. The problem is reduced to solving a system of ordinary differential equations for the radius and temperature of the bubble, the radius of the droplet and the shell together with the thermal conductivity equation for the internal liquid. Spatial discretization of the thermal conductivity equation is carried out using an implicit finite difference scheme. ODEs are solved by the fifth order Runge–Kutta method with an adaptive computational step. To check the correctness of the numerical calculation in a particular case, the theory has been compared with known experimental data. The influence of the shear modulus of the shell and the carrier liquid, and the shell thickness on the radial dynamics of a vapor bubble inside an encapsulated droplet in an external viscoelastic liquid is demonstrated.

SIMULATION OF TEMPERATURE EFFECTS IN MASSIVE BODIES USING THE MODIFIED METHOD OF LINES

This article presents the main ideas and possibilities of the modified straight line method for modeling temperature effects in massive bodies. The calculation model of a rectangular beam with a defined thermal state is adopted. The procedure for reducing the dimensionality of the original equations using the modified method of straight lines and the Bubnov-Galyorkin-Petrov method for boundary conditions is applied. The reduced equation of thermal conductivity and the reduced equation of heat balance are given. Conclusions are made regarding the addition of reduced second-order equations with components from the boundary conditions on the lateral surfaces, and finally reduced second-order equations in spatial variables are given.
 We are considering a beam of rectangular cross-section, the three dimensions of which are of the same order. The thermal state of such an object is three-dimensional and is therefore described systematically using three-dimensional heat conduction equations. When applying the modified method of straight lines to reduce the dimensionality of the original equations, it is convenient to use them in the form of a system of partial differential equations of the first order in spatial coordinates.
 According to the modified method of straight lines, two systems of straight lines parallel to the coordinate axes Oy and Oz are applied to the beam (a decrease in dimensionality in terms of spatial variables y and z is assumed). The choice of such straight lines is due to the fact that the reduced equations will depend on the variables x and t, that is, they will resemble the one-dimensional equations of the one-dimensional non-stationary problem of thermal conductivity, for which efficient and convenient numerical methods have been developed in the literature of mathematical physics at the moment, to which it is easy to adapt the reduced equation.

Computer Simulation of Coke Sediments Burning from the Whole Cylindrical Catalyst Grain

The article is devoted to the development of the mathematical model of oxidative regeneration of the cylindrical catalyst grain. The model is constructed using a diffusion approach to modeling catalytic processes. The model is based on the equations of material and thermal balance. Mass transfer in the catalyst grain is carried out due to diffusion and the Stefan flow resulting from a decrease in the reaction volume during sorption processes. Chemical transformations of substances are taken into account as a source term in the equation. The thermal balance of the catalyst grain is described by a thermal conductivity equation, with an inhomogeneous term responsible for heating the grain during exothermic chemical reactions. The effective coefficients of heat capacity and thermal conductivity of the catalyst grain, which are determined taking into account the porosity of the grain depending on temperature, were used to calculate the thermal balance of the catalyst grain. The dependencies are approximated using the method of least squares based on experimental data. Different boundary conditions for the developed model allow calculating the main characteristics of the oxidative regeneration process for a whole catalyst grain under different conditions. The mathematical model of oxidative regeneration of a cylindrical catalyst grain is described by a stiff system of differential equations. Splitting by physical processes is applied to avoid computational difficulties. The calculation of flows is carried out sequentially: first, chemical problems are solved using the Radau method, then the diffusion and thermal conductivity equations are solved by the finite volume method. The result of the algorithm implemented in C++ is a picture of the distribution of substances and temperature along the cylindrical grain of the catalyst.

Accounting for heat release in small volumes of matter on the example of the growth of ZnO micro-rods: search for a modeling technique

Using examples of an exothermic chemical reaction and self-heating of the region of a conducting filament of a memristor, heat-induced phase transitions, disadvantages of applying the classical Fourier approach on the nanoscale, and advantages of the molecular mechanics method at modeling the temperature factor are discussed. The correction for Arrhenius relationship, taking into account that the temperature becomes a random variable is proposed. Based on the introduced concepts (elementary act of heat release, distance and region of thermal impact) method for taking into account the thermal factor, is proposed.The correction is based on splitting the entire pool of particles into several, each of which corresponds to a fixed temperature value taken from a certain range. Although continuous and discrete correction options are given both, but the discrete option is more preferable. This is due to the fact that the methodology focuses on the application of methods of molecular mechanics, and, intentionally, in the most primitive version. The role of amorphization is noted as an example of the structural restructuring of matter in nano-volumes. It is indicated that the phonon spectra themselves, which determine heat transfer, depend on temperature. The technique is consistent with the ideology of multiscale modeling. The integral temperature increase is calculated outside the region of thermal exposure, where nonequilibrium effects are significant, by solving the standard equation of thermal conductivity.

Mathematical modeling of the temperature field of the arc of a conductive channel taking into account the temperature dependence of thermal conductivity

The article is devoted to mathematical modelling of the plasma temperature field taking into account the temperature dependence of thermal conductivity. The channel model was taken as a basis. Neglecting a very small fraction of the energy received by the ions during their acceleration in the longitudinal field, it can be assumed that all the energy taken by the arc discharge from an external source in the arc column passes directly to the plasma electrons. To find the temperature field by the integration method, the equation of thermal conductivity was solved. At the same time, the arc column is considered as a cylindrical continuous conductive rod in which all the supplied electrical energy is diverted due to thermal conductivity to the cooled walls of the discharge tube. The arc itself is represented by two regions: conductive and non-conductive. Based on mathematical modelling, the law of temperature field change in the arc cross-section was obtained. A formula was also derived to determine the effective radius of an electrically conductive channel.

ИССЛЕДОВАНИЕ ПРИМЕНЕНИЯ ИНТЕРВАЛЬНОЙ АРИФМЕТИКИ ПРИ РЕШЕНИИ ЗАДАЧИ РАСПРЕДЕЛЕНИЯ ТЕМПЕРАТУРЫ ПОЛОСЫ НА ПРОМЕЖУТОЧНОМ РОЛЬГАНГЕ СТАНА ГОРЯЧЕЙ ПРОКАТКИ

The article shows the modeling of the temperature in the strip on the intermediate roller of а hot rolling mill using the thermal conductivity equation with interval parameters. The resulting model is presented, the behavior of the model is investigated when several interval parameters are added to it. The results of modeling the interval model were compared with the results of a comparable real model.

Assessment of compliance of foreign geotechnical complexes for calculations of freezing-thawing of soils with Russian legal normatives

The design of construction objects in the conditions of the far north relates to changes in the temperature regime of the soil mass. In this regard, to choose the right design, it is necessary to be able to predict the change in the temperature field over time. Taking into account that heat release or absorption occurs when passing through the freezing onset temperature, the thermal conductivity equation, even for the onedimensional case, becomes non-linear. Several software packages have been developed to solve heat conduction problems. Meanwhile, theoretical assumptions concerning the assignment of heat capacity and heat conductivity coefficients may differ. It is of interest to assess the compliance of the results of calculations of freezing and thawing issues with Russian norms.