Математичні моделі процесу теплопровідності для кусково-однорідних структур із урахуванням термочутливості

Abstract

Дисертація присвячена створенню лінійних та нелінійних математичних моделей процесу теплопровідності для шаруватих, однорідних і шаруватих із чужорідним включенням структур. Розвинено теорію методів лінеаризації нелінійних крайових задач теплопровідності для термочутливих кусково-однорідних структур завдяки запровадженню нових перетворень, які для однорідних тіл стають перетворенням Кірхгофа, що дають змогу частково лінеаризувати вихідні нелінійні диференціальні рівняння, подальшу лінеаризацію яких і крайових умов виконано за допомогою кусково-лінійної апроксимації температури із використанням узагальнених функцій на поверхнях спряження чужорідних конструкційних елементів. Інтегральними перетвореннями Фур'є і Ганкеля отримано аналітично-числові розв'язки частково лінеаризованих задач для визначення запроваджених функцій, завдяки яким одержано співвідношення, що виражають розподіл температури за просторовими координатами у наведених структурах із конкретними залежностями коефіцієнта теплопровідності конструкційних матеріалів від температури. На основі отриманих аналітично-числових розв’язків лінійних і нелінійних крайових задач теплопровідності розроблено алгоритми та програмні засоби їхньої числової реалізації для аналізу температурних режимів у кусково-однорідних структурах. Диссертация посвящена созданию линейных и нелинейных математических моделей процесса теплопроводности для слоистых, однородных и слоистых с инородным включением структур. Развита теория методов линеаризации нелинейных граничных задач теплопроводности для термочувствительных кусочно-однородных структур с помощью введенных новых преобразований, которые для однородных тел стают преобразованием Кирхгофа, и позволяют частично линеаризовать исходные нелинейные дифференциальные уравнения, дальшейшая линеаризация которых и граничных условий проведена кусочно-линейной аппроксимацией температуры с использованием обобщенных функций на поверхностях сопряжения инородных конструкционных элементов. Интегральными преобразованиями Фурье и Ханкеля получены аналитически-числовые решения частично линеаризованных задач для определения введенных функций, с помощью которых получены соотношения, что определяют распределение температуры в рассматриваемых структурах с конкретными зависимостями коэффициента теплопроводности конструкционных материалов от температуры. На основе полученных аналитически-числовых решений линейных и нелинейных граничных задач теплопроводности разработаны алгоритмы и програмные средства их численной реализации для анализа температурных режимов в кусочно-однородных структурах. The thesis is dedicated to creation of linear and nonlinear mathematical models of the process of heat conduction for layered homogeneous and layered with foreign inclusions structures. Due to introduction of new transformations which for homogeneous bodies become Kirchhoff’s transformations which enable us to partially linearize initial differential equations, a theory of methods of linearization of non-linear boundary value problems of heat conduction for thermo-sensitive piecewise homogeneous structures has been developed; further linearization of them and of the boundary conditions has been carried out with a help of piecewise linear approximation of temperature with a use of generalized functions at the mated surfaces of foreign elements of structure. By Fourier and Henkel’s integral transformations, analyticalnumerical solutions of partially linearized problems for determination of introduced functions due to which there are obtained relations which express temperature fields in the aforesaid structures with concrete dependences of heat conductivity of structure materials on temperature are obtained. On the basis of obtained analytical-numerical solutions of linear and non-linear boundary value problems of heat conduction, algorithms and software of their numerical implementation for the analysis of temperature regimes in piecewise homogeneous structures are obtained. In the first section, the problem of modeling of heat transfer for homogeneous bodies and homogeneous bodies of different dimensionality and geometric shape is presented; a review of main literature sources in the aspect of creation of linear and nonlinear mathematical models of the process of heat conduction and in the aspect of development of solving the boundary value problems and initial-boundary value problems has been made. This has shown that the process of heat propagation in twoand three dimensional piecewise-homogeneous structures remains to be poorly investigated. In the second section, there are presented linear mathematical models of the process of heat conduction for an infinite plate and for the multi-layer infinite plate with heat insulted face surfaces and with parallelepipedon-shaped foreign inclusions, at one of the boundary surfaces of these layers there is heat flow concentrated, on the other convective heat exchange with the outer medium according to Newton’s law; on the mated surfaces of the foreign bodies conditions of ideal heat are set. With help of Fourier integral transformation, due to piecewise linear approximation of temperature on mated surfaces of the inclusion, analytical-numerical solutions of corresponding boundary value problems are obtained. For these problems, in case when a foreign inclusion is heatactive and through, second-type boundary conditions are observed is also considered. In the third section, there are presented linear mathematical models of the process of heat conduction for a layer and piecewise-homogeneous layer with an heat-active cylindrical foreign inclusion, at the boundary surfaces of these layers conductive heat exchange with a outer medium proceedes and at mated surfaces of the foreign inclusions conditions of ideal heat contact are set. With a help of Henkel’s integral transformation, due to piecewise linear approximation of temperature on mated surfaces of inclusions, analytical-numerical solutions of corresponding boundary value problems are obtained. For the suggested structures, in case when a foreign inclusion is through and at boundary surfaces second-type the boundary conditions are observed is also considered. In the fourth section, there are given linear and non-linear mathematical models of the process of heat conduction for a layer with a small and thin inclusion, at boundary surfaces of the layer there proceed convective heat exchange with outer medium and second-kind boundary conditions are set; and at the surfaces of the inclusion, ideal heat contact takes place. Due to introduction of reduced thermal parameters for the case of a small and thin inclusion, the initial differential equations are considerably simplified, this enables us to apply Fourier integral transformation and, as result, to obtain analyticalnumerical solutions of corresponding linear boundary problems. The solution of nonlinear boundary problem is obtained with a help of Kirchhoff’s transformation and Fourier transformation with concrete dependences of heat conductivity on the temperature. In the fifth section, there are given non-linear mathematical models of the process of heat conduction for a layered structures which are heated locally by a concentrated internal source of heat on the boundary surfaces of which convective heat exchange with the outer medium is set; and at the mated surfaces of different substances ideal heat contact takes place. With a help of suggested new transformations which for homogeneous solids become Kirchhoff’s transformations, the initial differential equations are partially linearized. Due to piecewise linear approximations of temperature at boundary surfaces of structures and at the mated surfaces of different elements, problems are perfectly linearized; this enables us to apply method of integral transformation and to obtain solutions for determination of the suggested functions, concrete dependences of conductivity on the temperature are considered for constructional materials of three-layer structures; relations for finding the temperature fields are obtained. In the sixth section, there are presented mathematical models of the process of heat conduction for an infinite plate and for a multi-layered infinite plate with heat-insulted face surfaces containing foreign heat-active parallelepipedon-shaped inclusions at the boundary surfaces of which convectional heat exchange with outer medium takes place and at the mated surfaces of different substances ideal heat contact is set. With a help of suggested new transformations, piecewise linear approximation of temperature at boundary surfaces and at mated surfaces of different substances and method of integral transformations, formulae for determination of the temperature fields with concrete dependencies of the heat conductivity on the temperature are obtained for constructional materials of homogeneous and three-layer with foreign inclusions structures. In the seventh section, there are given mathematical models of the process of heat conductions for a layer and a piecewise homogeneous layer containing a foreign heatactive cylindrical inclusion. Analogically, as in the sixth section, formulae for determination of the temperature fields with concrete dependencies of heat conductivity on temperature are obtained for constructional materials.