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

Abstract

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