Abstract
A mathematical description of the material thermal diffusivity aт in a semi-bounded body is proposed with a relatively simple algorithm for its numerical and analytical by solving the inverse problem of thermal conductivity. To solve the problem, it is necessary to obtain the temperature values of the unbounded plate as a result of a thermophysical experiment. A plate can be conditionally considered as a semi-bounded body as long as the Fourier number Fo ≤ Foк (Foк ≈ 0.04–0.06). It is assumed that the temperature distribution over cross-section of the heated layer of the plate R is sufficiently described by a power function whose exponent depends linearly on the Fourier number. A simple algebraic expression is obtained for calculating aт in the time interval ∆τ from the dynamics of temperature change T(Rп , τ) of a plate surface with thickness Rп heated under boundary conditions of the second kind. Temperature of the second surface of the plate T(0, τ) is used only to determine the time of the end of experiment τк. The moment of time τк, in which the temperature perturbation reaches the adiabatic surface x = 0, can be set by the condition T(Rп , τк) – T(0, τ = 0) = 0,1 K. The method of approximate calculation of dynamics of changes in depth of the heated layer R by the values of Rп , τк , and τ is proposed. Calculation of a т for the time interval ∆τ is reduced to an iterative solution of a system of three algebraic equations by matching the Fourier number, for example, using a standard Microsoft Excel procedure. Estimation of the accuracy of a т calculation was made by the test (initial) temperature field of the refractory plate with the thickness Rп = 0.05 m, calculated by the finite difference method under the initial condition T(x, τ = 0) = 300 (0 ≤ x ≤ Rп) at radiation-convective heating. The heating time was 260 s. Calculation of aт, i was performed for 10 time moments τi + 1 = τi + Δτ, τ = 26 s. Average mass temperature of the heated layer for the whole time was T = 302 K. The arithmetic-mean absolute deviation of aт(T = 302 K) from the initial value at the same temperature was 2.8 %. Application of the method will simplify the conduct and processing of experiments to determine the thermal diffusivity of materials.
Highlights
Поскольку толщина пластины Rп известна, а момент времени τк определится по условию Т(0, τк ) – Тн = ΔТ (ΔТ ≈ 0,1 К), то по формуле (18) можно рассчитать толщину прогретого слоя Ri для каждого момента времени τi, в который выполнялось измерение температуры Т1(τi )
A method to determine the thermal conductivity from measured temperature profiles // Int
A plate can be conditionally considered as a semi-bounded body as long as the Fourier number Fo ≤ Foк (Foк ≈ 0.04–0.06)
Summary
Методы определения теплофизических характерис тик материалов в основном основаны на решениях обратных задач теплопроводности по параметрам температурных полей, полученным в результате теплофизического эксперимента. В работах [21 – 25] для определения величин теплофизических характеристик по известному температурному полю предложено применять довольно простой (инженерный) метод численно-аналитического моделирования процессов теплопроводности, описанный в [23 – 25]. В этом методе используются аналитические решения дифференциального уравнения теплопроводности в виде алгебраических выражений, полученные для расчетного интервала времени ∆τ. При описании распределения температур по толщине прогретого слоя в конце расчетного интервала времени Δτi + 1 = τi + 1 – τi функцией (1) [21,22,23,24] граничное условие (5) запишется в виде (6). Составим уравнение баланса теплоты прогретого слоя пластины (0 ≤ x ≤ R, 0 ≤ X ≤ 1) для расчетного i-го интервала времени Δτ:. Где τк – момент времени, в который температура Т0 (τк ) превысит Тн на ΔТ
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