Inverse Method for Estimating Thermal Conductivity in One-Dimensional Heat Conduction Problems

Abstract

An inverse analysis is provided to determine the spatial- and temperature-dependent thermal conductivities in several one-dimensional heat conduction problems. A e nite difference method is used to discretize the governing equations, and then a linear inverse model is constructed to identify the undetermined thermal conductivities. The present approach is to rearrange the matrix forms of the differential governing equations so that the unknown thermal conductivity can be represented explicitly. Then, the linear least-squares-error method is adopted to e nd the solutions. The results show that only a few measuring points at discrete grid points are needed to estimate the unknown quantitiesofthethermalconductivities, evenwhenmeasurementerrorsareconsidered.Incontrasttothe traditional approach, the advantages of this method arethatno prior information is needed on thefunctional form oftheunknown quantities,no initialguessesarerequired,andno iterations in thecalculating processarenecessary and that the inverse problem can be solved in a linear domain. Furthermore, the existence and uniqueness of the solutions can be easily identie ed. Nomenclature A = coefe cient matrix of vector T B = coefe cient matrix of vector C C = vector constructed from the unknown thermal conductivities D = vector constructed from the functions of the unknown thermal conductivities E = productof A i1 and B F = error function g = heat generation, W/m 3 k = thermal conductivity, W/m ¢ ± C q = heat e ux, W/m 2 R = reverse matrix T = temperature, ± C T = temperature vector t = time, s x = spatial coordinate, m 1t = increment of time domain, s 1x = increment of spatial coordinate, m ae = standard deviation ! = random variable