Non-stationary Temperature Distribution Research Articles

MATHEMATICAL MODELING OF THE NON-STATIONARY THERMAL STATE OF A COMPOSITE COATING OF CLOSE TO EQUIMOLAR COMPOSITION DURING LASER REMELTING

Based on assumptions and limitations, a mathematical model of the non-stationary thermal state of a composite coating during laser remelting on a metal substrate to obtain an alloy of equimolar or close to equimolar composition has been developed. The model includes the boundary value problem of calculating the nonstationary thermal state of the temperature distribution in the solid, two-phase and liquid regions, considering the heat release of crystallization according to the theory of a quasi-equilibrium two-phase zone in the presence of porosity of a mixture of metal powders. Based on the mathematical model, the computer program “Composite coating thermal state” was created, which allows to determine the non-stationary temperature distribution in the formed coating and substrate, the depth of penetration of the high-entropy coating layer, to evaluate the shrinkage of the molten coating under different technological conditions: laser power, the diameter of its focal spot, as well as the speed of movement along the surface of the processed material. The created program also allows to determine the dynamics of the position of two-dimensional liquidus and solidus lines for each metal included in the mixture. Using the created program, a computer simulation of the process of unsteady heating of a mixture of three metal powders in the same mass fractions: nickel, chromium and iron was carried out. Non-stationary temperature fields in the composite coating are obtained, considering its penetration and heating of the substrate. The depth of penetration of all metals, including the most low-melting one, was determined during laser remelting on a metal substrate in the production of a high-entropy alloy. The practical significance of the developed computer model lies in its use to predict the penetrationdepth of a composite coating under specified technological modes of laser remelting. At the same time, its adaptation is necessary, based on experimental data on the porosity and thickness of the coating before and after melting of simple metal systems with similar thermophysical properties.

THE EQUATION OF THE NON-STATIONARY THERMAL CONDUCTIVITY PROBLEM OF THE THERMOMETRICAL DEVICE CONSTRUCTIVE ELEMENT OF WEAPON SYSTEMS AND MILITARY EQUIPMENT

Means of contact thermometry are widely used in various complexes of devices of weapons systems and military equipment. Such means basically involve the use of mechanical contact, in particular, bimetallic temperature transmitters, which are the most widely used thermometer variants in practice. To ensure the proper level of metrological and operational characteristics of bimetallic thermometers, reliable and adequate methods of calculating thermodynamic parameters (distributions of temperature, deformations, stresses caused by force and temperature loads) of the bimetallic package are necessary. The presented work describes an approach to the calculation of non-stationary temperature distribution in a solid-state element of a complex shape, which simulates the operation of a bimetallic thermometer sensor using the contact method. The calculated dependences obtained are the basis for modeling bimetallic thermometers of various structural forms and standard sizes. In order to constructively improve the thermometric system, it is proposed to make bimetallic plates non-one-pieces, but with additional structural inclusions, in the presence of which the temperature distribution and, accordingly, the stress state will approach to uniaxial one. Keywords: differential equations, integration constants, displays of primary heat exchangers, calculation dependencies, information transfer, metrological, operational characteristics, temperature, density, thermal conductivity of the medium.

Nonstationary Temperature Distribution in a Thermally Insulated Concentric Cylindrical Channel with Biomass Moving Under the Influence of Rotation of an Electrically Heated Helix

Nonstationary Temperature Distribution in a Thermally Insulated Concentric Cylindrical Channel with Biomass Moving Under the Influence of Rotation of an Electrically Heated Helix

Study of the Influence of Thermal Convection on Temperature Measurement in Thermal Control Boreholes during Artificial Ground Freezing

This paper considers the problem of thermal convection of a calcium chloride solution in a vertical borehole. A non-uniform temperature distribution with a given vertical gradient is set at the walls of the borehole. The non-stationary temperature distribution along the borehole axis was analyzed, and its deviations from the temperature at the walls were investigated. From a practical point of view, this problem is important for estimating the error in distributed temperature measurements over the depth of thermal control boreholes during artificial ground freezing. In this study, an area near the bottom of the borehole was identified where the fluid temperature at the borehole axis deviates significantly from the temperature at the wall. The maximum deviations of the fluid temperature from the temperature at the walls, as well as the length of the temperature deviation sections, were determined.

Analytical Model of Heating an Isotropic Half-Space by a Moving Laser Source with a Gaussian Distribution

This study presents the solution of the transient spatial problem of the impact of a moving source of heat flux induced by laser radiation on the surface of a half-space using the superposition principle and the method of transient functions. The solution is based on the Green’s function method, according to which the influence function of a surface-concentrated heat source is found at the first stage. The influence function has axial symmetry and the problem of finding the influence function is axisymmetric. To find the Green’s function, Laplace and Fourier integral transforms are used. The novelty of the obtained analytical solution is that the heat transfer at the free surface of the half-space is taken into account. The Green’s function that was obtained is used to construct an analytical solution to the moving heat-source problem in the integral form. The kernel of the advising integral operator is the constructed Green’s function. The Gaussian distribution is used to calculate integrals on spatial variables analytically. Gaussian law models the distribution of heat flux in the laser beam. As a result, the corresponding integrals on the spatial variables can be calculated analytically. A convenient formula that allows one to study the non-stationary temperature distribution when the heat source moves along arbitrary trajectories is obtained. A numerical, analytical algorithm has been developed and implemented that allows one to determine temperature distribution both on the surface and on the depth of a half-space. For verification purposes, the results were compared with the solution obtained using FEM.

INFLUENCE OF UNSTEADY CONDITIONS ON HEAT EXCHANGE DURING A SHARPY TRANSITION TO FILM BOILING

Boiling is one of the main physical processes, which that take place in heat exchange equipment designed for various purposes. The problem of removing large thermal loads from the heated surface is important for nuclear energy, chemical industry, metallurgy, electronics and other areas where intense heat is released. Boiling processes in process equipment perform important protective functions and can control its effectiveness.
 According to the boiling curve, with increasing temperature power, the flow passes through five regions, starting from the single-phase region of free convection and ending with the region of developed film boiling.
 The purpose of this article is an analytical study of heat transfer at spontaneous transition to the film boiling (explosive type of boiling), taking into account the unsteady nature of this process.
 In order to achieve the aim of this research, two analytical approaches were used, namely, the symmetry method and the Laplace method. As a result of mathematical transformations, expressions for the nonstationary temperature distribution and the Nusselt number are obtained. The obtained expressions make it possible to analyze the dynamics of non-stationary heat exchange processes. The results of analytical and numerical modeling were also compared. It was found that the results of the self-similar solution have a better comparison with numerical data compared to the results according to the Laplace method.

Перенос и аккумуляция газов, растворенных в воде, в неизотермическом массиве пористой среды с учетом зон замерзания

A one-dimensional problem of gas transport through a bubbly medium under the influence of a temperature wave is investigated taking into account freezing of the near-surface layer. Such freezing means the presence of a phase transition under the action of a temperature wave on the massif of the porous medium when the temperature goes below the freezing point of water. In this case, the Stefan problem is to be solved in order to obtain a nonstationary temperature distribution. With the knowledge of the temperature distribution in the massif at each moment of time, it is possible to take into consideration the influence of the temperature wave on the gas transport process. This process is described within the framework of the bubbly medium approximation. The calculation of the solubility of gas in water is based on the scaled particle theory. It is shown that a temperature wave causes the rise of an average flow of gases generated in the porous massif toward the surface, which leads to the effect of gas accumulation in the near-surface layer. The distribution of the gas concentration in the near-surface layer is obtained and the depth of this layer is estimated. In addition, the influence of the parameters of the problem on the accumulation effect is examined. The gas accumulation rate is found to be constant if the surface of the massif is at a temperature below the freezing point all year round and to decrease with an increase in the average surface temperature. At the same time, the accumulation rate increases with the intensification of gas generation in the porous massif.

A methodology for computing the relative icing degrees of logs stored in an open warehouse at ambient air temperature in winter

ABSTRACT This paper describes a methodology for mathematical modeling, computing, and research of two mutually connected problems: 2D non-stationary temperature distribution in logs stored for a long time in an open warehouse at periodically changing ambient air temperature in winter and change in the icing degrees of the logs during this time. Mathematical descriptions of the periodically changing ambient air temperature and of three types of relative icing degree of the logs that result under the influence of that temperature have been presented. These descriptions are introduced in coupled 2D non-linear mathematical models of the heat distribution in logs during their freezing and defrosting. The paper presents solutions of the models with explicit form of the finite-difference method. Results from a simulative investigation of the 2D non-stationary temperature distribution, average mass temperature, and three types of icing degree of beech logs with industrial dimensions (diameter of 0.4 m and length of 0.8 m), moisture content of 0.6 kg·kg−1, and initial temperature of 0°C during their 5 days and nights alternating freezing and defrosting in an open warehouse at sinusoidal change of the ambient air temperature with various initial values below −5°C and different amplitudes are graphically presented and analyzed.

Non-stationary temperature distribution in a thermally insulated concentric cylindrical channel with biomass moving under the influence of rotation of electrically heated helix

Non-stationary temperature distribution in a thermally insulated concentric cylindrical channel with biomass moving under the influence of rotation of electrically heated helix

CFD MODELING OF HEAT TRANSFER AND HYDRODYNAMICS PROCESSES IN A HEAT STORAGE TANK

Background. Today, heat storage tanks are an integral part of heating systems. However, modern designs of capacitive storage tanks are characterized by the phenomenon of thermocline and high thermal inertia. To minimize the aforementioned disadvantages, it is proposed to use a “thermal core”, for the formation of which it is necessary to select a substance with a high value of heat capacity.Objective. The purpose of the presented study is to simulate the process of heat and mass transfer in a storage tank with a “thermal core”, in the form of the binary tube located on the central axis, the inter tube space of which is filled with paraffin (a mixture of saturated hydrocarbons with melting point – from 45 °C to 65 °C; dense – 0.880–0.915 g/cm3 at 15 °C).Methods. Fluent software determines the temperature distribution in the heat storage tank under free convection conditions. The resulting data were then converted into a calculation module of the Transient Thermal software complex ANSYS, where further calculations of the non-stationary temperature distribution of the “thermal core” were carried out.Results. It is determined that the heat storage tank with a capacity of 1400 liters, heated for one hour by a heat transfer agent with a temperature of 115 °C, is cooled to 50 °C in 4 hours. The analysis of the hydrodynamic structure of the flow based on the distribution of the trajectories of movement of free-convective flows in the water column of the storage tank indicates the need to improve the design of the tank. It is determined that the use of the “thermal core”, regardless of the type of paraffin used for its formation, helps to reduce the stratification of temperature by tank height. The type of paraffin used to form the “heat core” has no significant effect on the cooling time of the heat storage tank as a whole. However, when using “ceresin” as a filler for the “heat core”, the average tank temperature is generally about 0.5 °C higher than for other paraffin types studied. Thus, it is “ceresin” that should be used as a “thermal core”.Conclusions. The result of the calculation of the inhomogeneous temperature field of all elements of the heat storage tank was used to determine the time of its complete cooling. The conducted research allows automating the process of calculation of the storage tanks as well as carrying out their modernization to increase the efficiency of use.

Mathematical modeling of the electron-beam wire deposition additive manufacturing by the smoothed particle hydrodynamics method

BackgroundThe actual problem for calculating a shape of free surface of the melt when analyzing the processes of wire-based electron-beam surfacing on the substrate, being introduced into additive manufacturing, is the development of adequate mathematical models of heat and mass transfer. The paper proposed a formulation of the problem of melt motion in the framework of the Lagrangian description. The mathematical statement includes the balance equations for mass, momentum and energy, and physical equations for describing heat and mass transfer.MethodsThe smoothed particle hydrodynamics method was used for numerical simulation of the process of wire-based electron-beam surfacing on the substrate made from same materials (titanium or steel). A finite-difference analog of the equations is given and the algorithm for solving the problem is implemented. To integrate the discretized equations Verlet method was utilized. Algorithms are implemented in the open software package LAMMPS.ResultsThe numerical simulation results allow the estimation of non-stationary volume temperature distributions, melt flow velocities and pressures, and characteristics of process.ConclusionThe possibility of applying the developed mathematical model to describe additive production is shown. The comparison of numerical calculations with experimental studies showed good agreement.

Modeling Latent Heat Fluxes of Water in Logs during their Freezing

This paper suggests a methodology for mathematical modelling and research of two interconnected problems: 2D non-stationary temperature distribution in logs subjected to freezing and change in the latent heat fluxes of the free and bound water in logs during the freezing process. For the purpose of this methodology, a 2-dimensional mathematical model has been created, solved, and verified for the transient non-linear heat conduction in logs during their freezing at convective boundary conditions. The model includes a mathematical description of the specific latent heat fluxes, qLHv-fw and qLHv-bw, formed by the freezing of the free and bound water in the logs, respectively. The paper presents solutions of the model with explicit form of the finite-difference method in the calculation environment of Visual Fortran Professional and its verification in accordance with our own experimental studies. The paper presents the results of simulation analysis of 2D non-stationary temperature distribution in the longitudinal section of pine log with a diameter of 0.24 m, length of 0.48 m, and moisture content above the hygroscopic range during its 30-hour freezing in a freezer at the temperature of the processing air medium of approximately –30 °C. The change in the latent heat fluxes qLHv-fw and qLHv-bw during the log freezing is presented, visualized, and analyzed.

Influence of Brazing Process Parameters on the Strength of Liquid Rocket Engine Brazed Structures

The article addresses the issues of brazing parameters optimization: mediums; pressure; rate of temperature change; transition from one rate of temperature change to another one; the excess pressure in the cavity of the installation with respect to the medium pressure in the brazing cavity. Impact of the parameters is demonstrated by the examples of the engine gimbal flange, booster pump unit rotor and guide vanes of the turbo-pump unit pump. Improvement in the quality of the soldered joints of the flange can be achieved by introducing homogenizing annealing after brazing. A theoretical calculation of the nonstationary temperature distribution in the structural elements during soldering was preliminarily carried out to solve the problem of soldering the guide apparatus. The calculation resulted in that a reduction in the rate of heating and cooling, and temperature exclusions were excluded. The problem of an impregnated rotor of the booster aggregate was solved by analyzing the thermal interaction of the brazed components, which determined the need for a slow rate of the initial stage of heating the rotor and introducing a soak in temperature.

Designing of the digital casting process for the gas turbine engine blades with a single-crystal structure

The work represents the approaches for designing the digital casting process of turbine blades. The development of the mathematical model for determining the non-stationary temperature distribution in the “furnace-moulding-flask-filler-metal-stopper” process system has appeared to be the core of the digital blade manufacturing process considering particular structural, geometry and thermal-physical features of this system. The approbation of the developed mathematical model as well as the computer designing methods under the pilot production conditions through manufacturing of the GTE turbine blades has demonstrated the decreased amount of defects in the macrostructure of blanks of turbine blades with the longitudinally-oriented and single-crystal structure by 30% in total.

A modified approach for the thermoelastic large deflection in the elliptical plate

The present paper deals with an investigation into the thermoelastic effect on the elliptical plate during large deflection while heating with non-stationary temperature distribution. The governing equation for the deflection is formulated with modification within the existing methodology devised by Berger. Thermally induced deflection results and its associated stresses are obtained in terms of Mathieu function of the first kind of order 2n. Furthermore, aforementioned problems can be degenerated into the problems of the circular region by applying limiting conditions. Some results which are derived by means of computational tools are illustrated numerically and depicted graphically.

Modeling of the Energy Consumption for Warming up of Furniture Elements during their Unilateral Convective Heating before Lacquering

Abstract A mathematical model and a numerical approach for the computation of the specific energy consumption, which is needed for warming up of flat furniture elements before their lacquering, have been suggested. The approach is based on the integration of the solutions of a non-linear model for the calculation of the nonstationary 1D temperature distribution along the thickness of subjected to unilateral convective heating furniture elements. With the help of a self-prepared software program, computations have been carried out for the determination of the change in the specific energy, which is consumed by oak furniture elements with an initial temperature of 20 °C, moisture content of 8 %, thickness of 16 mm, and length of 0.6 m, 1.2 m, and 1.8 m, during their 10 min unilateral convective heating by hot air with temperature of 100 °C and velocity of 5 m·s−1.

Numerical Approach for Computation of the Heat and Heat Flux for Covering of the Emission in the Surrounding Air of Subjected to Unilateral Heating Flat Wood Details before their Bending

A numerical approach for the computation of the specific (for 1 m2) energy consumption, qe, and the specific heat flux, dqe/dτ, needed for covering of the emission in the surrounding environment of the subjected to unilateral heating flat wood details aimed at their plasticizing and following bending has been suggested. The approach is based on the integration and differentiation of the solutions of a linear model for the calculation of the non-stationary 1D temperature distribution along the thickness of subjected to unilateral heating flat wood details, suggested by the authors earlier.For the numerical solution of the model aimed at the determination of qe and dqe/dτ software program has been prepared, which was input in the calculation environment of Visual Fortran. Using the program, computations have been carried out for the determination of the change in the energy qe and in the flux dqe/dτ, which are consumed by spruce details with an initial temperature of 20 °C, moisture content of 0.15 kg·kg-1, and thicknesses of 6 mm, 8 mm, and 10 mm during their 10 min unilateral heating at temperatures of the heating metal band of 100 °C, 120 °C, and 140 °C and of the surrounding air of 20 °C. The obtained results are graphically presented and analyzed.

Mathematical model of non-stationary temperature distribution in the metal body produced by induction heating process

Indukcni ohřev tělesa může být popsan parabolickou diferencialni rovnici, ktera ma jako parametr Joulovske tepelne ztraty. Výpocet Joulovských ztrat může být proveden Fredholmovou integralni rovnici druheho druhu pro viřive proudy se singularnim jadrem. Fredholmovou integralni rovnici rovnici lze řesit Nystomovou metodou se zhlazenim singularity. Tento postup znacně zkrati výpocetni cas modelu. clanek popisuje aplikaci teto metody a obsahuje ilustrativni přiklady.

Method of Heat Balances for Calculating Heat Transfer in Flat Multilayer Nanostructures

We suggest method of calculation of one-dimensional temperature field in multilayer nanostructures. Our method allows obtaining non-stationary temperature distribution in the periodical and non-periodical spatial structures with a different degree of periodicity. Comparison temperature distributions in the multilayer nanostructures and equivalent continuous samples are performed in this study. In addition we suggest experimental application of our method for estimation an average value of thermal Kapitza resistance.

Peculiarities of Non-Stationary Temperature Distribution of Optical Non-Transparent Anisotropic Thermoelement at Impulse Ray Excitement

The solutions of the non-stationary equation of thermoconductivity for the optical non-transparent thermoelement at impulse ray excitement are presented in the paper for two-time ranges: when the ray current falls (0<t<tau) and when it stops (t<tau). It is shown that the behavior of temperature distributions essentially depends on the relationship between the impulse duration (tau) and relaxation time (tau0) of temperature shifts over the whole volume of anisotropic thermoelement, arising at the impulse ray excitement. The non-stationary temperature distributions are studied for the long, short and middle impulse excitement.