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 τeT = 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.

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用半有界体的数值解析模型确定材料的热扩散率

通过求解热导率的逆问题，提出了一种材料在半有界体中的热扩散率ae的数学描述，其数值和解析计算相对简单。为了解决这个问题，需要通过热物理实验获得无界板的温度值。只要傅立叶数 Fo ≤ Foe (Foe = 0.04–0.06)，就可以有条件地将板视为半有界体。假设厚度为 R 的板的加热层横截面的温度分布可以通过幂函数充分描述，该函数的指数与傅立叶数呈线性关系。从温度变化动态 T(Rp, τ) 厚度为 Rp 的板表面在第二类边界条件下加热。板第二个表面的温度 T(0, τ) 仅用于确定实验的结束时间 τe。温度扰动达到绝热面 x = 0 的时刻 τe 可由条件 T(Rp, τe) – T(0, τ = 0) = 0.1 K 设置。 计算动力学的近似方法建议通过 Rp、τe 和 τ 的值来改变加热层 R 的深度。通过匹配傅立叶数，例如使用标准 Microsoft Excel 程序，将时间间隔 Δτ 的 ahc 计算简化为三个代数方程组的迭代解。使用厚度Rp = 0.05 m的耐火板的初始温度场，在初始条件T(x, τ = 0) = 300下通过有限差分法计算，对辐射-对流加热计算ahc的精度进行估计(0 ≤ x ≤ Rp)。加热时间为260秒。ahc,i 的计算已经执行了十个时刻τi + 1 = τi + Δτ, Δτ = 26 s。整个时间内加热层的平均质量温度为 τeT = 302 K。ae (T = 302 K) 在相同温度下与初始值的算术平均绝对偏差为 2.8%。该方法的应用将简化确定材料热扩散率的传导和加工实验。加热时间为260秒。ahc,i 的计算已经执行了十个时刻τi + 1 = τi + Δτ, Δτ = 26 s。整个时间内加热层的平均质量温度为 τeT = 302 K。在相同温度下，ae (T = 302 K) 与初始值的算术平均绝对偏差为 2.8%。该方法的应用将简化确定材料热扩散率的传导和加工实验。加热时间为260秒。ahc,i 的计算已经执行了十个时刻τi + 1 = τi + Δτ, Δτ = 26 s。整个时间内加热层的平均质量温度为 τeT = 302 K。在相同温度下，ae (T = 302 K) 与初始值的算术平均绝对偏差为 2.8%。该方法的应用将简化确定材料热扩散率的传导和加工实验。