Method for determining the thermal diffusivity and thermal conductivity coefficient by temperatures of plate surface as a semi-bounded body

The studied numerical and analytical model of a semi-bounded body is used to simultaneously determine the thermophysical characteristics (TFC): thermal diffusivity a_t and thermal conductivity coefficient λ_t of the material which make it easy to determine the volumetric heat capacity с_t . 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(x_p = R_p , τ) and T(x_p = 0, τ) of the plate with a thickness R_p heated under boundary conditions of the second kind q = const. The temperature T(x_p = 0, τ) was used to determine the time moment τ_e , at which the temperature perturbation reached the adiabatic surface x_p = 0 (T(R_p , τ_e ) – T_b (0, τ_e = 0) = 0.1 K). Calculations of TFC (a_t and λ_t ) were performed using formulas whose parameters were found by solving a nonlinear system of three algebraic equations by selecting the Fourier number corresponding to τ_e . 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 a_i (T ), λ_i (T ) and c_i (T ) were set by polynomials. Temperatures of the plate with a thickness of R_p = 0.04 m with initial conditions T_b = T(x_p , τ = 0) = 300, 900, 1200, 1800 K (0 ≤ x_p ≤ R_p ) were calculated for a specific heat flow q = 5000 W/m². The heating time to τ_e was 105 – 150 s. The average mass temperature T_{m, pl} of the plate during the τ_e increased by 5 – 11 K. The TFC values were restored by solving the inverse thermal diffusivity problem for 10 time points    τ_{i + 1} = τ_i + Δτ. The arithmetic mean deviations of TFC (T_{m, pl} ) from the initial values for calculations at T_b = 300, 900, 1200, 1800 K were less than 2.5 %. It was established that the values of a_t and λ_t obtained for the time moments t_i are practically constant, therefore, a simplified calculation of a_{t, o} and λ_{t, o} is possible only from the values of temperatures T(R_p , τ_e ) and T(0, τ_e ) at the end of heating. The values of a_{t, o} and λ_{t, 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 a_{t, o} and λ_{t, o} were found by solving a system of three nonlinear equations n = n( Fo), a_{t, o} = a(T_b , T(R_p , τ_e ), R_p , n, τ_e ), Fo = Fo(a_{t, o} , R_p , τ_e ) and expressions for λ_{t, o} = λ(R_p , q, n, T_b , T(R_p , τ_e )). The proposed method significantly simplifies the solution of the inverse problem of thermal conductivity.

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