Fourier’s law of heat conduction, engineering, Other Engineering

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Fourier’s law of heat conduction
For one dimension;
q(x) = - k dT/dx (2.1)
where qx is the heat flux in the x direction (W/m²); k is the thermal conductivity (W/mK, a property of material, see Table 1.1)and dT/dx is the temperature gradient (K/m).
For two dimensions;

Chen et al. (2002) have applied the Laplace transform technique and finite-difference method with a sequential in time concept. The least square scheme is proposed to predict the unknown surface temperature of two sided boundary conditions for a two dimensional inverse heat conduction problem.
Hsu et al. (1990) have used the finite difference method in conjunction with the linear least squares method to estimate the one-sided and two-sided boundary conditions in two-dimensional inverse heat conduction problem. In their work, they suppose that the functional form of the estimated surface temperature is given a priori and then parameterized. However, the effect of the measurement errors on the surface temperature cannot be neglected. Recently, Loulou et al. (2003) used the iterative regularization method in one dimensional inverse heat conduction problem to estimate a combination of two kinds of surface boundary conditions.
In this work, we propose a simultaneous estimation of transient distributions of two boundary heat conditions by using the iterative regularization method and transient temperature histories taken with several sensors inside a two-dimensional specimen.

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