Problem Of Nonstationary Heat Conduction Research Articles

Termostressed state of a three-layer rectangular plate under non-stationary convective heating conditions

The study considers a rectangular isotropic plate with a layered irregular structure. It is convectively non-stationarily heated by an external environment. The initial relationships of the non-stationary heat conduction and thermoelasticity problem are formulated using a five-mode mathematical model based on the shear deformation theory of thermoelasticity. Using the methods of Fourier and Laplace integral transforms, general solutions have been obtained for the non-stationary heat conduction problem and the quasi-static thermoelasticity problem for a hinge-supported plate along its edges. A numerical analysis of the temperature field, radial deflections, normal forces, bending moments, and normal stresses, depending on geometric parameters and the Bi criterion, has been performed for a three-layer plate. The materials of its layers are made of ceramics and metal. The temperature and mechanical parameters have been analyzed for the layering configuration of the plate: metal-ceramic-metal.

DETERMINATION AND ANALYSIS OF THE TEMPERATURE FIELD IN A THREE-LAYER ISOTROPIC PLATE WITH A GIVENINITIAL TEMPERATURE DISTRIBUTION

An approximate system of two-dimensional heat conduction equations for a multilayer isotropic plate is written down. Boundary conditions for a rectangular plate of finite dimensions are formulated. A general solution of the non-stationary heat conduction problem for this plate was found using integral Fourier transforms in spatial variables and Laplace transform in time.On the basis of the obtained general solutions, the solution of the thermal conductivity problem for a three-layer plate, which at the initial moment of time is heated by a temperature field linear in its thickness, which is uniformly distributed over the surface of the plate in a rectangular region, was analyzed.The numerical analysis of the temperature field was performed for a three-layer plate, the middle layer of which is made of metal, and the outer layers are made of ceramics.

Thermoelastic State of a Thermosensitive Half Space and a Thermosensitive Layer in Contact Under the Conditions of Complex Heat Exchange

We solve a nonstationary problem of heat conduction for a thermosensitive half space in contact with a thermosensitive layer under the conditions of complex heat exchange with the medium of constant temperature. To construct the solution, we use a numerical-analytic approach characterized by the application of a version of the method of successive approximations, linearizing parameters, the Laplace integral transformation, and its numerical inversion with the help of the Prudnikov formula adapted to the problems of heat conduction. The thermoelastic state of the outlined thermosensitive piecewise homogeneous structure is analyzed by applying the proposed algorithm.

Nonstationary Problems of Heat Conduction for a Thermosensitive Plate with Nonlinear Boundary Condition on One Surface

We propose a method for the determination of temperature fields formed in a plate with regard for the thermal radiation, temperature dependence of thermal characteristics, and densities of surface and volumetric heat sources for a nonuniform distribution of the initial temperature. By using the Kirchhoff transformation, Green’s function, generalized functions, and linear splines, we reduce the problems of heat conduction to the solution of a recurrence nonlinear algebraic equation for the values of the Kirchhoff variable at the spline nodes on the corresponding boundary surface. The results of numerical analysis are presented.

Numerical Determination of the Nonsteady Thermal State of a Three-Layer Hollow Thermosensitive Cylinder Under the Conditions of Complex Heat Exchange

By using the straight-line method and conservative difference schemes, we reduce the nonlinear nonstationary boundary-value problem of heat conduction for a three-layer hollow cylinder to a Cauchy problem for a system of ordinary differential equations, which is solved numerically with the help of the formulas of backward differentiation. The conservative discretization of the heat conduction equation and the boundary conditions with respect to the space variable is performed by the integro-interpolation method. We also analyze the influence of temperature dependence of the thermal characteristics of the chosen materials on the temperature field formed in the layers of a three-layer cylinder.

Analytical Method for Calculating Eigenvalues in the Problem of Nonstationary Heat Conduction of a Spherical Body

Analytical Method for Calculating Eigenvalues in the Problem of Nonstationary Heat Conduction of a Spherical Body

Конечно-элементное моделирование нестационарных задач теплопроводности при сложном теплообмене

The article presents a numerical simulation of nonstationary heat conduction problems under complex heat transfer, which includes such heat transfer mechanisms as heat conduction, convection, and radiation. The Stefan-Boltzmann law describes the resulting heat transfer by radiation between two bodies, where the heat transfer coefficient is a function of the body surface temperature. An algorithm and software for solving the heat conduction problem using the finite element method were developed, and the influence of external impacts on the temperature field distribution in the vicinity of an insulated circular hole in the center of the body was studied. The temperature fields were investigated for various boundary conditions in the hole of the plate and the corresponding isotherms were given.

Numerical calculation of heat transfer in a multilayer composite structure with honeycomb filler during autoclave molding at the heating stage

The production of any multilayer structure begins with the development of technological documentation, which describes, among other things, the temperature regimes during their manufacture. Conventionally, the temperature process of polymerization can be divided into three stages: preheating, temperature stabilization and cooling. The paper presents the results of a numerical calculation of temperature fields in a multilayer composite structure with a honeycomb filler during its manufacture by autoclave molding at the preheating stage. This method of manufacturing composite structures makes it possible to mold parts of varying complexity and dimensions, the demand for which is growing in such industries as mechanical engineering, aircraft building, and shipbuilding. The requirements for the quality of such products are increasing, which is greatly influenced by compliance with the temperature regime during molding. Conducting direct experiments requires large energy costs, therefore, to solve the problem of controlling heat exchange processes inside the structure, mathematical models were developed that describe these processes. A non-stationary heat conduction problem is formulated for a multilayer unbounded plate with a constant initial distribution, boundary conditions of the third kind on the outer boundaries, and boundary conditions of the fourth kind on the contact surfaces of the layers. Using the finite element method, the problem is reduced to three-point difference equations, the solution of which is found by the sweep method. The determination of the sweep coefficients is shown taking into account the thermal characteristics of the layers. The results of a numerical calculation of the temperature distribution for a nine-layer composite structure with a honeycomb core are presented. The numerical calculation was carried out using the developed program in the BorlandDelphi 7.0 object-oriented programming environment. The results obtained are presented in the form of graphic dependences of the temperature over the thickness of the sample at different points in time, as well as the dependence of temperature on time at various nodes of the sample in comparison with the theoretical curve. An analysis of these dependences was carried out, which showed that the heating of the sample occurs unevenly over its thickness. The deviation from the theoretical temperature values is observed in the layers located closer to the honeycomb layer. This can adversely affect the course of the polymerization step, which is characterized by the conversion of the binder into a polymer, and occurs at certain temperatures. Therefore, achieving the desired temperature values at the stage of heating the structure is important for the manufacture of reliable and durable structures that can withstand extreme operating conditions. The obtained temperature distributions make it possible to correct the technological process of manufacturing various multilayer structures at the stage of its development, which will reduce the economic costs of production.

Solution of Inverse Problem of Non-Stationary Heat Conduction Using a Laplace Transform

This paper presents some solutions to the one-dimensional Cauchy-type transient inverse heat conduction problem. Based on known courses of temperature and the heat flux on one of the boundaries of the region, courses of temperature and heat flux on the opposite boundary of the region were determined with the use of the Laplace’s transform. Solution of the inverse problem does not require determination of temperature distribution on subsequent time moments inside the region. Stability of the solution to the inverse problem of the Cauchy type was investigated by means of determining the minimal time step ensuring the stability of the solution. Two examples of one-dimensional transient inverse problems were solved. The first one is related to the reconstruction of shock temperature change on the boundary of the region based on known courses of temperature and heat flux on the opposite boundary. The second one concerned the reconstruction of the heat flux acting on one of the boundaries through a period of time. In both considered problems the reconstructed boundary conditions were discontinuous. Stable results were obtained for both investigated examples, however, the stability of the inverse problem depended on the length of time step.

SOFTWARE FOR SIMULATION OF NON-STATIONARY HEAT TRANSFER IN ANISOTROPIC SOLIDS

The software «AnisotropicHeatTransfer3D» for the numerical solution of three-dimensional non-stationary heat conduction problems in anisotropic solids of complex domain by meshless method [1] was developed. The numerical solution of the non-stationary anisotropic heat conduction problem in the software «AnisotropicHeatTransfer3D» is based on a combination of the dual reciprocity method [2] with anisotropic radial basis functions [3] and the method of fundamental solutions [4]. The dual reciprocity method with anisotropic radial basis functions is used to obtain particular solution, and the method of fundamental solutions is used to obtain a homogeneous solution of boundary-value problem.

Simulation of Heat Transfer in Single-Crystal Lithium Niobate in Interaction with Continuous-Wave Laser Radiation

The paper presents the simulation results of heat transfer in single-crystal lithium niobate (LiNbO3) in the form of cylinder of diameter mm and height mm in interaction with continuous-wave laser radiation with the output power of W and the wavelength of nm. The density of the LiNbO3 crystal is kg/m3; the thermal conductivity along the [001] direction is W/(m×K); the thermal conductivity in the (001) plane is W/(m×K); the specific heat at constant pressure is J/(kg×K); the absorption coefficient is %/cm @ 1064 nm. The laser beam propagates along the optical axis of the crystal. The laser beam intensity profile is represented as a Gaussian function, and the absorption of laser radiation of the single-crystal lithium niobate is described by Beer-Lambert’s law. The numerical solution of the non-stationary heat conduction problem is obtained by meshless scheme using anisotropic radial basis functions. The time interval of the non-stationary boundary-value problem is 2 h 30 min. The results of numerical calculations of the temperature distribution inside and on the surface of the single-crystal lithium niobate at times s are presented. The time required to achieve the steady-state heating mode of the LiNbO3 crystal, as well as its temperature range over the entire time interval, have been determined. The accuracy of the approximate solution of the boundary-value problem at the n-th iteration is estimated by the value of the norm of relative residual . The results of the numerical solution of the non-stationary heat conduction problem obtained by meshless method show its high efficiency even at a small number of interpolation nodes.

Numerical-Analytical Method for Reducing Problems of Non-Stationary Heat Conduction with Boundary Conditions of the III Kind to Problems with Conditions of the I Kind

Numerical-Analytical Method for Reducing Problems of Non-Stationary Heat Conduction with Boundary Conditions of the III Kind to Problems with Conditions of the I Kind

Family of the Atomic Radial Basis Functions of Three Independent Variables Generated by Helmholtz-Type Operator

The paper presents an algorithm for constructing the family of the atomic radial basis functions of three independent variables generated by Helmholtz-type operator, which may be used as basis functions for the implementation of meshless methods for solving boundary-value problems in anisotropic solids. Helmholtz-type equations play a significant role in mathematical physics because of the applications in which they arise. In particular, the heat equation in anisotropic solids in the process of numerical solution is reduced to the equation that contains the differential operator of the special form (Helmholtz-type operator), which includes components of the tensor of the second rank, which determines the anisotropy of the material. The family of functions is infinitely differentiable and finite (compactly supported) solutions of the functional-differential equation of the special form. The choice of compactly supported functions as basis functions makes it possible to consider boundary-value problems on domains with complex geometric shapes. Functions include the shape parameter , which allows varying the size of the support and may be adjusted in the process of solving the boundary-value problem. Explicit formulas for calculating the considered functions and their Fourier transform are obtained. Visualizations of the atomic functions and their first derivatives with respect to the variables and at the fixed value of the variable for isotropic and anisotropic cases are presented. The efficiency of using atomic functions as basis functions is demonstrated by the solution of the non-stationary heat conduction problem with the moving heat source. This work contains the results of the numerical solution of the considered boundary-value problem, as well as average relative error, average absolute error and maximum error are calculated using atomic radial basis functions and multiquadric radial basis functions.

An Extended Analytical Solution of the Non-Stationary Heat Conduction Problem in Multi-Track Thick-Walled Products during the Additive Manufacturing Process.

An analytical model has been developed for calculating three-dimensional transient temperature fields arising in the direct deposition process to study the thermal behavior of multi-track walls with various configurations. The model allows the calculation of all characteristics of the temperature fields (thermal cycles, cooling rates, temperature gradients) in the wall during the direct deposition process at any time. The solution of the non-stationary heat conduction equation for a moving heat source is used to determine the temperature field in the deposited wall, taking into account heat transfer to the environment. The method considers the size of the wall and the substrate, the change in power from layer to layer, the change in the cladding speed, the interpass dwell time (pause time), and the heat source trajectory. Experiments on the deposition of multi-track block samples are carried out, as a result of which the values of the temperatures are obtained at fixed points. The proposed model makes it possible to reproduce temperature fields at various values of the technological process parameters. It is confirmed by comparisons with experimental thermocouple data. The relative difference in the interlayer temperature does not exceed 15%.

Верифікація математичної моделі для розв’язання задачі Стефана в рамках методу “mushy layer”

The use of solar energy has limitations due to its periodic availability: solar plants do not operate at night and are ineffective in dull weather. The solution of this problem involves the introduction of energy storage and duplication systems into the conversion loop. Among the energy storage systems, solid–liquid phase transition modules have significant energy, ecologic, and cost advantages. Physical processes in modules of this type are described by a system of non-stationary nonlinear partial differential equations with specific boundary conditions at the phase interface. The verification of a method for solving the Stefan problem for a heat-storage material is presented in this paper. The use of the mushy layer method made it possible to simplify the classical mathematical model of the Stefan problem by reducing it to a nonstationary heat conduction problem with an implicit heat source that takes into account the latent heat of transition. The phase transition is considered to occur in an intermediate zone determined by the solidus and liquidus temperatures rather than in in infinite region. To develop a Python code, use was made of an implicit computational scheme in which the solidus and liquidus temperatures remain constant and are determined in the course of numerical experiments. The physical model chosen for computer simulation and algorithm verification is the process of ice layer formation on a water surface at a constant ambient temperature. The numerical results obtained allow one to determine the temperature fields in the solid and the liquid phase and the position of the phase interface and calculate its advance speed. The algorithm developed was verified by analyzing the classical analytical solution of the Stefan problem for the one-dimensional case at a constant advance speed of the phase interface. The value of the verification coefficient was determined from a numerical solution of a nonlinear equation with the use of special built-in Python functions. Substituting the data for the physical model under consideration into the analytical solution and comparing them with the numerical simulation data obtained with the use of the mushy layer method shows that the results are in close agreement, thus demonstrating the correctness of the computer algorithm developed. These studies will allow one to adapt the Python code developed on the basis of the mushy layer method to the calculation of heat storage systems with a solid-liquid phase transition with account for the features of their geometry, the temperature level, and actual boundary conditions.

Parallel technology for solving nonstationary heat conduction problems in axisymmetric domains

The paper develops a parallel algorithm and program for solving nonstationary heat conduction and diffusion problems in axisymmetric domains with axisymmetric boundary conditions. The numerical solution is based on the boundary element method. In order to optimize and enhance the effectiveness of the computer implementation of the algorithm, the computations are parallelized and the OpenMP application program interface is used. The program is tested by comparing the calculation results with the data of known exact solutions. The calculations confirm the correctness of the numerical solutions and the possibility of full scaling at different numbers of boundary elements according to the number of cores/processors available. The program is applicable to solving axisymmetric heat conduction and diffusion problems and, as a component of a software system, to solving nonlinear problems.

Numerical modelling in problems of thermal control for three-layer structures with defects

This article describes solutions to the direct and inverse problems of the three-dimensional non-stationary heat conduction problem in a three-layer structure, using the finite element method for the direct problem and the gradient descent method for the inverse problem. A comparison of the FEM-solution and the analytical solution for a solid with a simple geometry is presented. Here are presented solutions of the direct and inverse three-dimensional non-stationary heat conductivity problem for a free three-stage turbine. The accuracy of the found and exact solutions is compared.

Analytical Solution of the Non-Stationary Heat Conduction Problem in Thin-Walled Products during the Additive Manufacturing Process.

The work is devoted to the development of a model for calculating transient quasiperiodic temperature fields arising in the direct deposition process of thin walls with various configurations. The model allows calculating the temperature field, thermal cycles, temperature gradients, and the cooling rate in the wall during the direct deposition process at any time. The temperature field in the deposited wall is determined based on the analytical solution of the non-stationary heat conduction equation for a moving heat source, taking into account heat transfer to the environment. Heat accumulation and temperature change are calculated based on the superposition principle of transient temperature fields resulting from the heat source action at each pass. The proposed method for calculating temperature fields describes the heat-transfer process and heat accumulation in the wall with satisfactory accuracy. This was confirmed by comparisons with experimental thermocouple data. It takes into account the size of the wall and the substrate, the change in power from layer to layer, the pause time between passes, and the heat-source trajectory. In addition, this calculation method is easy to adapt to various additive manufacturing processes that use both laser and arc heat sources.

Thermoelastic State of Contacting Thermally Sensitive Half Space and Thermally Sensitive Layer

By using the numerical inverse Laplace transform based on the modified Prudnikov formula, we construct the solution of a nonstationary problem of heat conduction for a thermally sensitive half space operating in contact with a thermally sensitive layer under the conditions of convective heat exchange with a medium of constant temperature. The thermoelastic state of this thermally sensitive piecewise homogeneous structure is investigated for the case where a constant temperature is kept on the surface.

A Meshless Method of Solving Three-Dimensional Nonstationary Heat Conduction Problems in Anisotropic Materials

A Meshless Method of Solving Three-Dimensional Nonstationary Heat Conduction Problems in Anisotropic Materials