Abstract
A mathematical description of a material’s thermal diffusivity ae in a semi-bounded body is proposed with a relatively simple algorithm for its numerical and analytical calculation 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 Fourier number Fo ≤ Foe (Foe = 0.04–0.06). It is assumed that the temperature distribution over a cross-section of the heated layer of the plate with thickness R is sufficiently described by a power-like function whose exponent depends linearly on the Fourier number. A simple algebraic expression is obtained for calculating ahc in time interval Δτ from the dynamics of temperature change T(Rp, τ) of a plate surface with thickness Rp heated at boundary conditions of the second kind. Temperature T(0, τ) of the second surface of the plate is used only to determine end time τe of the experiment. Moment of time τe, at which the temperature perturbation reaches adiabatic surface x = 0, can be set by the condition T(Rp, τe) – T(0, τ = 0) = 0.1 K. An approximate method of calculating the dynamics of changes in depth of heated layer R by values of Rp, τe, and τ is proposed. The calculation of ahc for time interval Δτ 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 calculation of ahc at radiation-convective heating was performed using the initial temperature field of the refractory plate with thickness Rp = 0.05 m, calculated by the finite difference method under initial condition T(x, τ = 0) = 300 (0 ≤ x ≤ Rp). The heating time was 260 s. Calculation of ahc, i has been performed for ten time moments τi + 1 = τi + Δτ, Δτ = 26 s. Average mass temperature of the heated layer for the entire time was τe T = 302 K. The arithmetic-mean absolute deviation of ae (T = 302 K) from the initial value at the same temperature was 2.8%. Application of the method will simplify conducting and processing experiments to determine 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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