Differential Equation Of Heat Conduction Research Articles

A method to calculate the effective thermal conductivity of spherical particle-laden composite

Accurate calculation of the effective thermal conductivity of composite is of great importance. Based on the heat conduction differential equation, the effective thermal conductivity of spherical particle-laden composite was derived in this paper. To validate the calculation method, comparisons with the Maxwell model, Karayacoubian model, Lewis-Nielsen model, Woodside-Messmer model, and the experimental data were conducted. It was found that the results of the present method are closer to the experimental data. Further validation was conducted with comparison to the finite-element simulations. The effective thermal conductivity of a matrix cube with a spherical particle was simulated and calculated. The results obtained by these two methods were consistent with the relative error within 7%.

Simplified Grinding Temperature Model Study

A study of a simplified mathematical model for determining the grinding temperature is performed. According to the obtained results, the equations of this model differ slightly from the corresponding more exact solution of the one-dimensional differential equation of heat conduction under the boundary conditions of the second kind. The model under study is represented by a system of two equations that describe the grinding temperature at the heating and cooling stages without the use of forced cooling. The scope of the studied model corresponds to the modern technological operations of grinding on CNC machines for conditions where the numerical value of the Peclet number is more than 4. This, in turn, corresponds to the Jaeger criterion for the so-called fast-moving heat source, for which the operation parameter of the workpiece velocity may be equivalently (in temperature) replaced by the action time of the heat source. This makes it possible to use a simpler solution of the one-dimensional differential equation of heat conduction at the boundary conditions of the second kind (one-dimensional analytical model) instead of a similar solution of the two-dimensional one with a slight deviation of the grinding temperature calculation result. It is established that the proposed simplified mathematical expression for determining the grinding temperature differs from the more accurate one-dimensional analytical solution by no more than 11 % and 15 % at the stages of heating and cooling, respectively. Comparison of the data on the grinding temperature change according to the conventional and developed equations has shown that these equations are close and have two points of coincidence: on the surface and at the depth of approximately threefold decrease in temperature. It is also established that the nature of the ratio between the scales of change of the Peclet number 0.09 and 9 and the grinding temperature depth 1 and 10 is of 100 to 10. Additionally, another unusual mechanism is revealed for both compared equations: a higher temperature at the surface is accompanied by a lower temperature at the depth. Keywords: grinding temperature, heating stage, cooling stage, dimensionless temperature, temperature model.

The theoretical rationale for the use of contact heating when drying seeds in a vacuum

Purpose. The rationale for heating seeds by contact in a vacuum by analyzing the solutions of the differential equation of unsteady heat conduction. Methods. The differential heat equation, the Fourier method, the Sturm-Livull problem and the Dirac δ-function are used. Results. Based on the Fourier method and the Sturm-Livull problem, a differential heat equation is obtained, which makes it possible to determine the kinetics of seed heating depending on the thickness of the layer at a known temperature of the heating surface. A graphical analysis of the solution of the differential heat equation showed that for uniform drying of the seeds in vacuum it is necessary to apply mixing during drying. Conclusions. The analysis of the contact heating technological process, which was carried out on the basis of solving the differential equation of unsteady heat conduction, showed that the heating rate of seeds nonlinearly decreases with increasing layer thickness. This leads to uneven drying of the seeds in a vacuum. Therefore, the use of only one contact surface for heating seeds in a vacuum is impractical. Keywords: seeds, contact heating, heating temperature, layer thickness, vacuum, drying, heating.

The infrared thermal wave imaging detection of micro-crack defects in Ti-Al alloy plate

The Ti-Al alloy has good physical, chemical and mechanical properties, and it is the preferred material in aerospace and other special fields. The laser spots array is used for thermal excitation of the Ti-Al alloy specimen. Based on the fractional differential equation and Fourier heat conduction equation, the fractional heat transfer model of Ti-Al alloy plate specimen ex-cited by short pulse laser spots array is established, and the infrared thermal imaging simulation analysis is carried out by finite element method. The effects of crack width, crack depth and the distance between the crack and its nearest laser spot center on temperature abrupt jump is analyzed. With the increase of crack width and depth, the temperature abrupt jump increases, but the trend gradually slows down. The distance between the crack and its nearest laser spot center has a significant effect on the temperature abrupt jump. With the increase of the distance, the temperature abrupt jump first increases and then decreases. When the distance equals the laser spot radius, that is, the crack is tangent to the spot, the temperature abrupt jump reaches its maximum.

Creation of Model and Calculation of Thermal Processes in the Ring Graphite Electrode of the Plasmatron

This paper presents the theoretical and experimental material obtained in the study of the erosion and thermal state of the ring graphite electrode for a plasmatron. Thermal processes in graphite electrodes of plasmatrons are quite complex and multifaceted. A mathematical model of thermal processes that occur at the ring electrodes of plasmatrones has been developed. The mathematical model is based on the differential heat conduction equation for a ring electrode in cylindrical coordinates. With the use of this mathematical model, the inverse problem of heat conduction is solved: determination of the regularities of the heat exchange process by the temperatures of individual points on a solid surface. An experimental study of the temperature distribution at the end of the electrode and along the length of the electrode was carried out. Experiments have shown that the temperature on the side surfaces drops sharply towards the cold end of the electrode. When reducing the length of the electrode, the maximum temperature at the end decreases, and the temperature on the inner and outer edges of the electrode increases slightly. The most significant factors determining the temperature field at the end of the ring electrode are the power and size of the heat source. Comparison of the results of experimental studies and mathematical modeling showed a match with an acceptable degree of accuracy.

A new look at the integral methods for solving heat and mass transfer problems

A new concept of construction of integral relations for approximate solving problems on heat and mass transfer is proposed. This concept is based on the introduction of local functions for the heat flow or the temperature that are determined directly from the differential equation of heat conduction with boundary conditions at the temperature disturbance front. This made it possible to obtain a number of new efficient integral relations, mainly an improved integral relation for the temperature momentum and integrals of the square-law heat flow and the square-law temperature function. New schemes for optimization of the exponent n in the parabolic description of the temperature field with the use of the error norms E1 and introduced for the first time are proposed. In comparison with the Langford norm EL (the scheme of T. Myers), the effectiveness of determining optimum parabolic solutions has been substantially increased. On the basis of the integral relations proposed in combination with the new schemes for minimizing an error, optimum solutions of simple parabolic form have been obtained with an error of 1.23% and an integral error E1 = 0.0301. The solutions obtained are much better in approximation representation and error than the solutions obtained by known integral methods.

Parabolic Profile in Heat-Conduction Problems. 1. Semi-Bounded Space with a Surface of Constant Temperature

A new approach to the construction of constitutive integral relations, involving the introduction into consideration of local functions for a heat flow or a temperature determined directly from the differential heat-conduction equation, is proposed. With the use of this approach, new integral relations: the temperature-momentum integral of single modification, the temperature-momentum integral of double modification, the heat-flow integral, and the temperature-function integral, have been obtained. On the basis of these integrals as well as the integral method of heat balance and the refined integral method, different variants of the hybrid integral method realized with the use of manifold integral relations were investigated. In addition to the Langford norm, new error norms were introduced into consideration. The parabolic solutions obtained possess much better qualities compared to those of the analogous known solutions.

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.