Differential Equation Of Heat Conduction Research Articles

Calculation of the kinetics of heat transfer using the experimental data of moisture exchange in the process of convective drying of thin flat materiаls

In the paper, the authors analyzed the solution of the differential equation of non-stationary heat conduction for an unbounded plate during the heat exchange of plate surfaces with the surrounding medium according to Newton’s law at a constant temperature of the medium. To use the results of solving the equations in the drying of thin flat materials, the dependence of the heat transfer coefficients on temperature and moisture content was studied. As a result of studying and analyzing a number of literature sources, the regularities of the change in the heat transfer coefficients during drying are established with high reliability. Studies of drying of thin wet plates of white and red clays with known heat transfer coefficients have shown that for small values of the heat transfer criterion of the Bio and small temperature gradients over the section of a thin material, application of the results of solutions of the heat transfer equations gives completely satisfactory agreement between the calculated and experimental values of the temperatures and the duration of drying. It is established that for small Bio numbers, the main factor is the external heat and mass transfer of the surface of the material with the surrounding medium and the rate of drying depends little on internal mass transfer. It is shown that the use of numerical methods for solving differential equations is possible with varying degrees of approximation only for accurate and reliable dependences of heat and mass transfer coefficients on moisture content and temperature. For a number of materials with known heat transfer coefficients, the use of analytical methods in calculations is of considerable interest and brings the theory closer to the practice of drying.

A thermo-mechanical model for SFRC beams or slabs at elevated temperatures

The bearing capacity of steel fibre reinforced concrete (SFRC) at elevated temperatures is the subject of significant ongoing research, as the effect of steel fibres on concrete performance at higher temperatures is poorly understood. On one hand, steel fibres increase the average thermal conductivity of the concrete cross section and lead to greater heating within a concrete structural member during fire exposure, and on the other, fibres reduce crack widths and prevent excessive spalling. The former effect negatively impacts SFRC performance at high temperature, whereas the latter effect protects the inner structure from direct fire exposure. Additionally, the decreasing strength of steel at higher temperatures can result in the sudden failure of fibre or traditionally reinforced concrete. Within this contribution, a coupled thermo-mechanical model is developed in order to investigate the influence of steel fibres on the thermal loading of concrete. The effect of fibres on heat transmission within concrete, the length of time for which concrete can sustain thermal loads, and on the bending stiffness of reinforced concrete beams or slabs is investigated. The heat transfer process is modelled using Fourier’s partial differential equation of transient heat conduction. A modified plastic hinge model and moment–curvature relations are used to describe stress-dependent deformations. Thus, two alternative approaches are used to adequately track the localisation of damage for single cracks and for distributed and multiple cracking. Thermo-mechanical coupling is achieved by means of temperature-dependent stress–strain relations. These are derived for SFRC based on experimental data from the literature. Experiments are performed in which concrete slabs reinforced with variable amounts of fibres and rebar are exposed to combined thermal and mechanical loadings. The results of these experiments are used to validate the proposed model. The measured and predicted results agree well and indicate that steel fibres have a positive effect on the fire resistance of structures, assuming additional rebar is provided to prevent crack localisation. Additionally, it is shown that temperature fields within concrete remain almost unaffected by variations in fibre content.

Method of Measuring Thermal Relaxation in the Solid State

The differential equations of heat conduction of a solid body, which are a consequence of the Fourier, Cattaneo–Vernotte, and Lykov’s equations, are considered. A mathematical model of the transient, three-period process in a circular plate is constructed in the form of a solution to the hyperbolic boundary value problem of heat conduction with boundary conditions of the third kind. The method to determine the Bio numbers in each period of the transition process and the time of thermal relaxation is described by the results of experimental and theoretical studies of transient thermal processes in the center of round plates of different thicknesses made of polymethylmethacrylate upon their sudden immersion in hot water.

МАТЕМАТИЧНІ МОДЕЛІ АНАЛІЗУ ТЕМПЕРАТУРНИХ РЕЖИМІВ У 3D СТРУКТУРАХ ІЗ ТОНКИМИ ЧУЖОРІДНИМИ ВКЛЮЧЕННЯМИ


 
 
 
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 Інформація про авторів:
 Гавриш Василь Іванович, д-р техн. наук, професор кафедри програмного забезпечення. Email: gavryshvasyl@gmail.com
 Лоїк Василь Богданович, канд. техн. наук, доцент кафедри пожежної тактики та аварійно-рятувальних робіт. Email: v.loik1984@gmail.com
 Синельніков Олександр Дмитрович, канд. техн. наук, доцент кафедри пожежної тактики та аварійно-рятувальних робіт. Email: o.synelnikov@gmail.com
 Бойко Тарас Володимирович, канд. техн. наук, доцент, заступник начальника інституту. Email: boykotaras@gmail.com
 Шкраб Роман Романович, асистент кафедри програмного забезпечення. Email: ikni.pz@gmail.com
 Цитування за ДСТУ: Гавриш В. І., Лоїк В. Б., Синельніков О. Д., Бойко Т. В., Шкраб Р. Р. Математичні моделі аналізу температур­них режимів у 3D структурах із тонкими чужорідними включеннями. Науковий вісник НЛТУ України. 2018, т. 28, № 2. С. 144–149.
 Citation APA: Havrysh, V. I., Loik, V. B., Synelnikov, O. D., Bojko, T. V., & Shkrab, R. R. (2018). Mathematical Models of the Analysis of Temperature Regimes in 3D Structures with Thin Foreign Inclusions. Scientific Bulletin of UNFU, 28(2), 144–149. https://doi.org/10.15421/40280227
 
 
 
 
 Нерівномірне нагрівання − один із факторів, що спричиняють деформації та напруження у пружних конструкціях. Якщо з підвищенням температури ніщо не перешкоджає розширенню структури, то вона деформуватиметься і жодних напружень не виникатиме. Однак, якщо в конструкції температура зростає нерівномірно і воно неоднорідне, то внаслідок розширення формуються температурні напруження. Першим і незалежним кроком для дослідження температурних напружень є визначення температурного поля, що становить основну задачу аналітичної теорії теплопровідності. В окремих випадках визначення температурних полів є самостійною технічною задачею, розв'язання якої допомагає визначити температурні напруження. Тому розроблено лінійні математичні моделі визначення температурних режимів у 3D (просторових) середовищах із локально зосередженими тонкими теплоактивними чужорідними включеннями. Класичні методи не дають змоги розв'язувати крайові задачі математичної фізики, що відповідають таким моделям, у замкнутому вигляді. З огляду на це описано спосіб, який полягає в тому, що теплофізичні параметри для неоднорідних середовищ описують за допомогою асиметричних одиничних функцій як єдине ціле для всієї системи. Внаслідок цього отримують одне диференціальне рівняння теплопровідності з узагальненими похідними і крайовими умовами тільки на межових поверхнях цих середовищ. У класичному випадку такий процес описують системою диференціальних рівнянь теплопровідності для кожного з елементів неоднорідного середовища з умовами ідеального теплового контакту на поверхнях спряження та крайовими умовами на межових поверхнях. Враховуючи зазначене вище, запропоновано спосіб, який полягає в тому, що температуру, як функцію однієї з просторових координат, на боковій поверхні включення апроксимовано кусково-лінійною функцією. Це дало змогу застосувати інтегральне перетворення Фур'є до перетвореного диференціального рівняння теплопровідності із узагальненими похідними та крайових умов. Внаслідок отримано аналітичний розв'язок для визначення температурного поля в наведених просторових середовищах з внутрішнім та наскрізним включеннями. Із використанням отриманих аналітичних розв'язків крайових задач створено обчислювальні програми, що дають змогу отримати розподіл температури та аналізувати конструкції щодо термостійкості. Як наслідок, стає можливим її підвищити і цим самим захистити від перегрівання, яке може спричинити руйнування як окремих елементів, так і конструкцій загалом.

Performance modeling of thermoelectric devices by perturbation method

A nonlinear differential heat conduction equation for a leg of a thermoelement with temperature-dependent thermoelectric material properties is reduced to an integral equation. Its solution is obtained by a perturbation method in the form of a series in terms of powers of parameters proportional to Thomson and Joule heat. The first six coefficients of the series and the heat balance equations, i.e. the dependence of heat fluxes at the junctions on its temperatures and the electric current are found. The calculation results of main energy characteristics of thermoelectric devices (thermoelectric generators and thermoelectric coolers) is compared by five methods: exact solution, solution at constant thermoelectric properties taken at a mean temperature of hot and cold junctions, method of average parameters (thermoelectric material properties averaged over the relevant temperature range by integration), linear and quadratic approximations of the perturbation theory. It is shown that the accuracy acceptable in real-world applications (error less than 1%) is obtained for thermoelectric generators by the method of average parameters and for thermoelectric coolers - by quadratic approximation of the perturbation method.

Insulation Failure Prediction Model of Power Cable in Fire

With input parameters being determined, simplified physical model and heat conduction equation were adopted to establish a power cable insulation failure prediction model in fire, and the validation test was carried out. The internal structure of the cable was simplified to a one-dimensional physical model with three layers, to derive the calculation method of internal structure parameters of cable physical model. Differential equation of heat conduction, which could reflect the internal temperature of cable, was constructed based on basic assumptions, to establish a cable insulation cable failure prediction model with mixed layer heat diffusion coefficient, environment temperature change and insulation cable failure temperature as input parameters. The method to determine model input parameters was also proposed. ZR-YJV and SDR-1 cable thermal radiation furnace were selected in the experiments to obtain insulation failure temperature and time, as well as environmental temperature change of the cable. The experimental results showed that the constructed model had relative small error in predicting cable insulation failure time.

Computer Simulation of the Solidification Process Including Air Gap Formation

Abstract The paper presents an approach of numerical modelling of alloy solidification in permanent mold and transient heat transport between the casting and the mold in two-dimensional space. The gap of time-dependent width called "air gap", filled with heat conducting gaseous medium is included in the model. The coefficient of thermal conductivity of the gas filling the space between the casting and the mold is small enough to introduce significant thermal resistance into the heat transport process. The mathematical model of heat transport is based on the partial differential equation of heat conduction written independently for the solidifying region and the mold. Appropriate solidification model based on the latent heat of solidification is also included in the mathematical description. These equations are supplemented by appropriate initial and boundary conditions. The formation process of air gap depends on the thermal deformations of the mold and the casting. The numerical model is based on the finite element method (FEM) with independent spatial discretization of interacting regions. It results in multi-mesh problem because the considered regions are disconnected.

MATHEMATICAL MODELS AND METHODS OF SOLVING SPATIAL GENERALIZED BOUNDARY VALUE PROBLEM HEAT ROTATING HOLLOW PIECEWISE HOMOGENEOUS CYLINDER

Context. The phenomenological theory of heat conduction speed of heat propagation is assumed infinitely large. However, the high intensity of the observed transient processes such as explosions, supersonic flow, high speeds of rotation of the use of this assumption leads to errors, so it is necessary to take into account that the distribution of heat takes place at a finite rate.Objective. Development of a new generalized mathematical model of the temperature distribution in the hollow piecewise uniform cylinder in the form of a boundary value problem of mathematical physics for the heat equation and the solution of the resulting boundary value problem.Method. The use of known integral Laplace transforms, Fourier series, and developed a new integral transformation for piecewise homogeneous space.Results. Found Polga transient temperature field of a circular cylinder in a cylindrical coordinate system, a piecewise homogeneous polar radius direction, which rotates at a constant angular velocity about the axis OZ, with the ultimate heat propagation speed. Thermal properties of each layer does not depend on the temperature at an ideal thermal contact between the layers, and there are no internal sources of heat.Conclusions. For the first time developed a mathematical model of the temperature distribution in the empty piecewise uniform cylinder, which rotates at a constant angular velocity about the axis OZ, taking into account the finite speed of propagation of heat in the form of mathematical physics boundary value problem for hyperbolic partial differential equations of heat conduction with boundary conditions of the first kind. Thermal properties are in each layer does not depend on the temperature at an ideal thermal contact between the layers, and there are no internal sources of heat.Created a new integral transform of a piecewise-homogeneous space, with which found the temperature field of the hollow piecewise homogeneous circular cylinder in the form of convergent orthogonal series of Bessel functions and of Fourier. The obtained analytical solution of a generalized boundary value problem of heat transfer cylinder, which rotates, given a finite amount of heat propagation velocity can be used for modeling of temperature fields, which occur in many technical systems (satellites, forming rolls, turbines, etc.).

Investigation of Contact Thermal Resistance in a Finite Cylinder with an Internal Source by the Fast Expansion Method and the Problem of Consistency of Boundary Conditions

By the fast expansion method, the authors have obtained the approximate solution of the problem in analytical form, which holds true at all points of the cylinder up to the boundary. From an analysis of the solution, it follows that the broken thermal contact between annular regions gives rise to a weak thermal resistance. Beginning with four annular regions, the thermal resistance remains constant, in practice. This result was obtained for the first time. When no more than three terms in a Fourier series are used, the maximum residual of the differential heat-conduction equation is 10–2.

The influence of thermal relaxation and thermal damping on transient processes with cyclic boundary conditions

Variants of the differential equation of heat conduction in a solid body, which follow from the Fourier and Cattaneo–Vernotte hypotheses and the Lykov equation, are considered. A boundary value problem describing temperature fields in a body (cylinder) upon cyclic heat transfer with cold and hot media is formulated. An analytical solution to the boundary value problem with a hyperbolic differential equation of heat conduction with allowance for thermal relaxation and temperature damping with cyclic boundary conditions of the third kind is given. The thermal transient processes calculated by the classical heat conductance equation and hyperbolic equation of heat conduction on the axis of the cylinder at different values of factors such as the ratio of the thermal damping time to the thermal relaxation time, the duration of cyclic periods, the Fourier relaxation number, and the Biot number are compared. A conclusion is made that the theory of regenerative air heater should be improved by taking into account thermal relaxation and thermal damping in the nozzle and measurements of the thermal relaxation and thermal damping times of the corresponding materials.

Extrapolating POD reduced-order model based on temperature gradient for unsteady heat conduction

A new proper orthogonal decomposition (POD) reduced-order model based on temperature gradient is established for heat conduction differential equation. In the most of present studies of heat conduction POD reduced-order model, the POD basis functions are extracted from temperature fields by adopting proper orthogonal decomposition or singular value decomposition (SVD). Different from conventional POD reduced-order method, this paper optimize the gradient of POD solutions by extracting the optimal orthogonal basis from temperature gradient. After getting POD basis, the POD reduced-order model is solved by using Galerkin projection. POD-Galerkin reduced-order model of two dimensional, constant property and non-steady heat conduction differential equations is established, and the accuracy of the reduced-order model under different boundary conditions is further studied. The results suggests that the relative error of POD solution is less than 0.1% while the average computing time reduces from 1.64 s to 0.02 s.

A revised approach for an exact analytical solution for thermal response in biological tissues significant in therapeutic treatments

The genesis of the present research paper is to develop a revised exact analytical solution of thermal profile of 1-D Pennes’ bioheat equation (PBHE) for living tissues influenced in thermal therapeutic treatments. In order to illustrate the temperature distribution in living tissue both Fourier and non-Fourier model of 1-D PBHE has been solved by ‘Separation of variables’ technique. Till date most of the research works have been carried out with the constant initial steady temperature of tissue which is not at all relevant for the biological body due to its nonhomogeneous living cells. There should be a temperature variation in the body before the therapeutic treatment. Therefore, a coupled heat transfer in skin surface before therapeutic heating must be taken account for establishment of exact temperature propagation. This approach has not yet been considered in any research work. In this work, an initial condition for solving governing differential equation of heat conduction in biological tissues has been represented as a function of spatial coordinate. In a few research work, initial temperature distribution with PBHE has been coupled in such a way that it eliminates metabolic heat generation. The study has been devoted to establish the comparison of thermal profile between present approach and published theoretical approach for particular initial and boundary conditions inflicted in this investigation. It has been studied that maximum temperature difference of existing approach for Fourier temperature distribution is 19.6% while in case of non-Fourier, it is 52.8%. We have validated our present analysis with experimental results and it has been observed that the temperature response based on the spatial dependent variable initial condition matches more accurately than other approaches.

The Discrete Model for the System of the Myocardium and Coronary Vessels

Background. The numerical heat transfer model for a system of myocardium coronary vessels is considered. Objective. The goal is to develop a discrete model for the physical system of myocardium and coronary vessels that would make it possible to explore the process of hypo- and hyperthermia with cardiopulmonary bypass. Methods. To solve the differential equation of heat conduction in the MSC Sinda thermal system the network method (TNM – Thermal Network Method) is used, in which system of heat equations is presented in the form of cellular-centered nodes and resistances between the nodes using the finite difference method. In constructing the model of myocardial in the MSC Sinda system the thermal contact between three-dimensional bodies is implemented – the myocardium, coronary arteries, a liquid cooling of heart. Results. Implementation of the model of heat exchange in the MSC Sinda system for infarction cooling process gives on the final process step in establishing the heat balance the temperature difference at the boundary between the myocardium and coronary vessels not more than 0,5 °C. However, in the areas of the myocardium that are removed from the coronary vessels the temperature difference exceeds 1,0 °C. The use of additional cooling for hearts allows for the cooling of myocardium with using of ice surface, that provides the unevenness reduction of the heart temperature during its cooling with cardiopulmonary bypass. This result allows exploring the dynamics of the process of hypo- and hyperthermia with cardiopulmonary bypass. Conclusions. The discrete 3D-model of heat transfer in the layer structure of the myocardium and coronary vessels allows us to investigate the process of hypo- and hyperthermia with cardiopulmonary bypass. The simulation results also make it possible to perform the analysis of the temperature distribution on the surface of the myocardium provided free convection of heat between the layers.

Theoretical estimation of quench occurrence and propagation based on generalized thermoelasticity for LTS/HTS tapes triggered by a spot heater

The present study deals with the thermal characteristics and mechanical behaviors of low/high temperature superconducting (LTS/HTS) composite tapes during quench processes triggered by a spot heater. Based on the generalized thermoelastic theory, a dynamic thermoelastic model with a relaxation time is developed which takes into account the temperature dependence and finite speed of heat propagation for the superconducting tapes under cryogenic condition. The analyses were performed using the finite element method to solve the coupled differential equations of dynamic heat conduction and elastic equilibrium. The results show that the thermoelastic behaviors exhibit a strong relevance to quench characteristics of the superconductors. As a quench occurs, the thermoelastic strain-rate has an obvious jumping variation with the instant of time of its peak being fortunately coincident with the time at which the critical temperature is reached. Such a jumping change of strain-rate could be a way of estimation and detection of quench occurrence, and the theoretical predictions coincide with the existing experimental observations on thermoelastic strain-rate in LTS magnets. For a HTS tape, the thermoelastic strain-rate or temperature-rate variation and a small jump also are illustrated as the quench occurrence is determined. Additionally, the normal zone propagation velocities for the LTS/HTS tapes are predicted by the critical temperature and thermoelastic strain-rate to show quite good agreements with the results evaluated by Wilson’s formula for a LTS tape or the experimental measurements for a HTS tape. The influences of the relaxation time of heat conduction and thermoelastic coupling on the thermal distribution and strain profile are also discussed in details.

Simulation Management of Industrial Projects Based on the Calculated Coupling: The Case for Anchorage Temperature Field with Passenger Rope-bridge in Mengshan-Tianmeng

This paper carries out numerical simulation analysis on temperature field of massive concrete anchorage; integrate the control of thermal analysis differential equations and integrating heat-conduction differential equations of thermodynamics. According to actual of Wanghailou anchorage and Yuhuangding anchorage, determine reasonably the relevant parameters and construction conditions and launch the numerical simulation of the subparagraph layering and pouring. According to variations in temperature and stress distribution in Anchorage pouring process, develop a program of mass concrete pouring, and simulation results are compared with the results of traditional norms. Research for large volume of hydration heat and crack analysis is of great guiding significance.

TEMPERATURE DISTRIBUTION IN MULTILAYER METAL-CERAMIC COATINGS UNDER NONSTATIONARY THERMAL EFFECTS

Progress in the aircraft engine construction is determined by the increase of operation parameters of gas turbine engines, which is inevitably accompanied by an increase of operating temperatures and load for the vital elements of the turbine hot ducts. Furthermore, the requirements for reliability of the engine in general are also increasing. Achievement of these requirements is determined by the performance of the materials turbine blades are made of and is made possible by the application of high-heat Nickel alloys in combination with combined heat-shielding coatings. This article dwells on the problem of assessing the impact of non-stationary thermal effects on the temperature distribution in a multilayer heatproof coating. With the aim of assessing the working capacity of heatproof coatings we propose a method of calculating the temperature field for the blade profile and the coating depth, based on the solution of the basic one-dimensional differential equation of heat conduction. This method allows us to assess the performance of heatproof coating and also gives us an opportunity to choose a combination of heatproof coating layers for the specific operating conditions of a gas turbine engine’s blades. In addition, using the proposed method it is possible to evaluate the effect of non-stationary heat flux on the structure of high-temperature alloy of the engine’s turbine blades and, therefore, to evaluate the capacity with the given coating. At temperatures of 1150–1200 °C and higher in heat-resistant Nickel alloys there starts a coagulation process of the main reinforcing coherent particle phase on the basis of the intermetallic compound, long plates with wavy shapes are formed instead of the cuboids, a formation of topologically close-packed phases which are needle-like compositions happens. These processes lead to a significant deterioration of the strength characteristics of heat-resistant alloys. Making calculations according to the proposed method it is possible to predict the performance of turbine blades, having the information about the intensity of phase transformations in the alloy and the temperature transitions in the process of gas turbine engine functioning (using the data of the on-board parameters monitoring and recording systems).

A Finite Element Multi-mesh Approach for Heat Transport Between Disconnected Regions

The paper presents a method of numerical modeling of transient heat transport between separated areas in two-dimensional space. The gap of variable width filled with heat conducting material separates considered regions wherein the case of ideal contact is also possible. The mathematical model is based on the partial differential equation of heat conduction supplemented by appropriate initial and boundary conditions of I-IV kind. The numerical model uses finite element method (FEM) with independent spatial discretization of interacting regions resulting in multi-mesh problem. The movement of spatially discretized regions relative to each other is also possible. The results of numerical simulations for the two and three areas that were in thermal interaction are presented and discussed.

Calculation of Concrete Strength in Winter Conditions Based on Heat Exchange Processes Modeling

The article deals with the problem of using a mathematical model of the three-dimensional non-stationary temperature field to calculate the strength of concrete, the hardening in winter conditions. There is data on the developed mathematical model describing the processes of heat exchange based on the differential equation of heat conduction. The boundary conditions of the III - kind: the heat exchange of a solid with the ambient air, and the IV - kind: the condition of the thermal junction of two solids have been modeled. The realization of the model with the numerical method of heat balances has been proposed. The analysis of advantages and disadvantages of the classical method based on the regular cooling has been conducted. The comparison of the data produced as a result of the calculation through the classical method and the data obtained by the calculation based on the mathematical modeling has been carried out. The conclusion on the effectiveness of using the mathematical modeling to predict the strength of concrete incubating in winter conditions has been made.

Prediction of the temperature of a drill in drilling lunar rock simulant in a vacuum

In this article, the temperature of a sampling drill in drilling lunar rock simulant in a high-vacuum environment was studied. The thermal problem was viewed as a 1-D transient heat transfer problem in a semi-infinite object. The simplified drill was modeled using heat conduction differential equation and a fast numerical calculation method is proposed on this basis, with time and the drill discretized. The model was modified to consider the effects of radiation, drill bit configuration, and non-constant heat source. A thermal analysis was conducted using ANSYS Workbench to determine the value of the equivalent correction coefficient proposed in this paper. Using fiber Bragg grating temperature measurement method, drilling experiments were conducted in a vacuum, and the results were compared to the model. The agreement between model and experiment was very good.

The Thermophysical Bases of Monitoring of the Fireproof Lining Wear in the Blast Furnace Hearth

This paper presents a two-dimensional description of the temperature field in refractory lining of the hearth in the blast furnace. The mathematical model is based on Fourier differential heat conduction equation. Different solutions of this equation are presented and the optimal quantity and location of thermosensors in the hearth is proposed. This paper also presents a methodology to obtain the heat conductivity of refractory materials.