blackmetИзвестия высших учебных заведений. Черная МеталлургияIzvestiya. Ferrous Metallurgy0368-07972410-2091National University of Science and Technology "MISIS"10.17073/0368-0797-2022-1-57-65blackmet-2237Research ArticleИНФОРМАЦИОННЫЕ ТЕХНОЛОГИИ И АВТОМАТИЗАЦИЯ В ЧЕРНОЙ МЕТАЛЛУРГИИINFORMATION TECHNOLOGIES AND AUTOMATIC CONTROL IN FERROUS METALLURGYМетод определения температуропроводности и коэффициента теплопроводности по температурам поверхности пластины как полуограниченного телаMethod for determining the thermal diffusivity and thermal conductivity coefficient by temperatures of plate surface as a semi-bounded bodyhttps://orcid.org/0000-0001-5956-567XСоколовА. К.SokolovA. K.

Анатолий Константинович Соколов, д.т.н., профессор кафедры безопасности жизнедеятельности

153003, Иваново, ул. Рабфаковская, 34

Anatolii K. Sokolov, Dr. Sci. (Eng.), Professor of the Chair of Life Safety

34 Rabfakovskaya Str., Ivanovo 153003

sokolov@bjd.ispu.ruИвановский государственный энергетический университет имени В.И. ЛенинаРоссияIvanovo State Power University named after V.I. LeninRussian Federation2022110220226515765Copyright © Соколов А.К., 20222022Соколов А.К.Sokolov A.K.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://fermet.misis.ru/jour/article/view/2237

Проведено исследование численно-аналитической модели полуограниченного тела, которая использовалась для одновременного определения теплофизических характеристик (ТФХ): температуропроводности ат и коэффициента теплопроводности λт материала, по которым легко определить объемную теплоемкость cт . Распределение температур по сечению пластины в конце расчетного интервала времени τ описано степенной функцией, показатель которой n зависел от числа Фурье Fo. Величины ТФХ рассчитывались по динамике изменения температур поверхностей пластины Т(хп = Rп , τ ) и Т(хп = 0, τ) толщиной Rп , нагреваемой при граничных условиях второго рода q = const. По температуре Т(хп = 0, τ) определялся момент времени τк , в который температурное возмущение достигало адиабатной поверхности хп = 0 (Т(Rп , τк ) – Тн(0, τ = 0) = 0,1 К). Вычисления ТФХ (ат и λт ) выполнялись по формулам, параметры которых находились решением нелинейной системы из трех алгебраических уравнений путем подбора числа Фурье, соответствующего τк . Исследование трудоемкости и точности расчета ТФХ выполнено по тестовым (исходным) температурным полям пластины из огнеупорного материала, рассчитанным методом конечных разностей. Зависимости ТФХ от температуры аи(Т ), λи(Т ) и cи(Т ) задавались полиномами. Температуры пластины толщиной Rп = 0,04 м с начальными условиями Тн = Т(хп , τ = 0) = 300, 900, 1200, 1800 К (0 ≤ хп ≤ Rп ) были рассчитаны при удельном потоке теплоты q = 5000 Вт/м2. Время нагрева до τк составляло 105 – 150 с. Среднемассовая температура пластины Tcp, пл за время τк увеличивалась на 5 – 11 К. Значения ТФХ восстанавливались решением обратной задачи теплопроводности для десяти моментов времени τi + 1 = τi + Δτ. Среднеарифметические отклонения ТФК (Tcp, пл ) от исходных значений для расчетов при Тн = 300, 900, 1200, 1800 К составили менее 2,5 %. Установлено, что значения ат и λт , полученные для моментов времени τi , практически постоянны, следовательно возможен упрощенный расчет ат, о и λт, о только по значениям температур Т(Rп , τк ) и Т(0, τк ) в конце нагрева. Значения ат, о и λт, о , которые были рассчитаны сразу для всего времени нагрева, отличались от исходных значений принятых условий теплообмена примерно на 2 %. Параметры простых алгебраических формул для расчета ат, о и λт, о находились решением системы из трех нелинейных уравнений n = n(Fo), ат, о = а(Тн , Т(Rп , τк ), Rп , n, τк ), Fo = Fo(ат, о , Rп , τк ) и выражения для λт, о = λ(Rп , q, n, Тн , Т(Rп , τк)). Предложенный метод значительно упрощает решение обратной задачи теплопроводности.  

The studied numerical and analytical model of a semi-bounded body is used to simultaneously determine the thermophysical characteristics (TFC): thermal diffusivity at and thermal conductivity coefficient λt of the material which make it easy to determine the volumetric heat capacity сt . Temperature distribution over the plate cross-section at the end of the calculated time interval τ is described by a power function, its exponent n depends on the Fourier number Fo. The values of TFC were calculated from the dynamics of changes in surface temperatures T(xp = Rp , τ) and T(xp = 0, τ) of the plate with a thickness Rp heated under boundary conditions of the second kind q = const. The temperature T(xp = 0, τ) was used to determine the time moment τe , at which the temperature perturbation reached the adiabatic surface xp = 0 (T(Rp , τe ) – Tb (0, τe = 0) = 0.1 K). Calculations of TFC (at and λt ) were performed using formulas whose parameters were found by solving a nonlinear system of three algebraic equations by selecting the Fourier number corresponding to τe . The author studied the complexity and accuracy of TFC calculation using the test (initial) temperature fields of a plate made of refractory material by the finite difference method. Dependences of TFC on the temperature ai (T ), λi (T ) and ci (T ) were set by polynomials. Temperatures of the plate with a thickness of Rp = 0.04 m with initial conditions Tb = T(xp , τ = 0) = 300, 900, 1200, 1800 K (0 ≤ xp ≤ Rp ) were calculated for a specific heat flow q = 5000 W/m2. The heating time to τe was 105 – 150 s. The average mass temperature Tm, pl of the plate during the τe increased by 5 – 11 K. The TFC values were restored by solving the inverse thermal diffusivity problem for 10 time points    τi + 1 = τi + Δτ. The arithmetic mean deviations of TFC (Tm, pl ) from the initial values for calculations at Tb = 300, 900, 1200, 1800 K were less than 2.5 %. It was established that the values of at and λt obtained for the time moments ti are practically constant, therefore, a simplified calculation of at, o and λt, o is possible only from the values of temperatures T(Rp , τe ) and T(0, τe ) at the end of heating. The values of at, o and λt, o , which were calculated immediately for the entire heating time, differed from the initial values of the accepted heat exchange conditions by about 2 %. The parameters of simple algebraic formulas for calculating at, o and λt, o were found by solving a system of three nonlinear equations n = n( Fo), at, o = a(Tb , T(Rp , τe ), Rp , n, τe ), Fo = Fo(at, o , Rp , τe ) and expressions for λt, o = λ(Rp , q, n, Tb , T(Rp , τe )). The proposed method significantly simplifies the solution of the inverse problem of thermal conductivity.

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The authors declare that there are no conflicts of interest present.