Determination of Thermal Diffusivity of Material by the Numerical-Analytical Model of a Semi-Bounded Body

• Published in: Steel in translation 2020-06, Vol.50 (6), p.391-396
• Main Author: Sokolov, A. K.
• Format: Article
• Language: English
• Subjects: Algebra; Algorithms; Approximation; Boundary conditions; Chemistry and Materials Science; Diffusivity; Finite difference method; Heating; Inverse problems; Iterative methods; Iterative solution; Materials Science; Perturbation; Temperature distribution; Thermal conductivity; Thermal diffusivity; Thickness
• ISSN: 0967-0912; 1935-0988
• DOI: 10.3103/S096709122006008X

A mathematical description of a material’s thermal diffusivity a e 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 ≤ Fo e (Fo e = 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 a hc in time interval Δτ from the dynamics of temperature change T ( R p , τ) of a plate surface with thickness R p 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 τ e of the experiment. Moment of time τ e , at which the temperature perturbation reaches adiabatic surface x = 0, can be set by the condition T ( R p , τ e ) – 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 R p , τ e , and τ is proposed. The calculation of a hc 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 a hc at radiation-convective heating was performed using the initial temperature field of the refractory plate with thickness R p = 0.05 m, calculated by the finite difference method under initial condition T ( x , τ = 0) = 300 (0 ≤ x ≤ R p ). The heating time was 260 s. Calculation of a hc, i has been performed for ten time moments τ i + 1 = τ i + Δτ, Δτ = 26 s. Average mass temperature of the heated layer for the entire time was τ e T = 302 K. The arithmetic-mean absolute deviation of a e ( 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.