Inverse determination of thermal conductivity for one-dimensional problems

Abstract

Two finite difference procedures are presented for the inverse determination of the thermal conductivity in a one-dimensional heat conduction domain. The thermal conductivity is reconstructed from the inverse analysis based on the assumption that the temperature measurements are either available continuously over the entire domain or at discrete grid points. The convergence and stability of the computational algorithms are investigated. It is concluded that both procedures are first-order accurate methods. A comparison of the exact thermal conductivity with the one estimated was made to confirm the validity of the numerical procedures. The close agreement between the two results confirms that the proposed finite difference techniques are effective procedures for the inverse determination of thermal conductivity in a one-dimensional heat conduction domain. The methods are applicable for linear and nonlinear spatially- as well as temperature-d ependent thermal conductivities. Additionally, the special feature of the present techniques is that a priori knowledge of the functional form for the thermal conductivity is not mandatory. Nomenclature C = arbitrary constant F = function defined by Eq. (10) / = function defined at the boundary G = function defined by Eq. (10) g — heat generation, W/nr k = thermal conductivity, W/m-°C kM = initial thermal conductivity calculated at dT/dx — 0 in the discrete formulation, W/m-°C k(} = initial thermal conductivity calculated at dT/dx = 0 in the continuous formulation, W/m-°C q = heat flux, W/m2 T = temperature, °C t = time, s / = selected time in the continuous formulation, s /,- — selected time in the discrete formulation, s A' = spatial coordinate, m XM = spatial coordinate where dT/dx = 0 in the discrete formulation, m A',, = spatial coordinate where dT/dx = 0 in the continuous formulation, m Subscripts /, /, k = indices Superscript = approximated value