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 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 nd the solutions. The results show that only a few measuring points at discrete grid points are needed to estimate the unknown quantities of the thermal conductivities, even when measurement errors are considered. In contrast to the traditional approach, the advantages of this method are that no prior information is needed on the functional form of the unknown quantities, no initial guesses are required, and no iterations in the calculating process are necessary and that the inverse problem can be solved in a linear domain. Furthermore, the existence and uniqueness of the solutions can be easily identi ed. NomenclatureA = coef cient matrix of vector T B = coef 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 = product of A ¡1 and B F = error function g = heat generation, W/m 3 k = thermal conductivity, W/m ¢ ± C q = heat 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 ¾ = standard deviation ! = random variable Subscripts esti = estimated data exact = exact data i = index of spatial coordinate j = index of time domain meas = measured data n = index at the boundary when x is equal to 1, m

An LGSM to identify nonhomogeneous heat conductivity functions by an extra measurement of temperature

“…In order to overcome this problem, there have been many studies, for example, Yeung and Lam [1], Keung and Zou [2], Lin et al [3], Chang and Chang [4], Engl and Zou [5], Ben-yu and Zou [6], Jia and Wang [7], and references therein. Most of the studies applied the least squares method to estimate the heat conductivity in inverse heat conduction problems.…”

Inverse determination of thermal conductivity using semi-discretization method

“…Lam and Yeung [12] used a first order finite difference and Yeung and Lam [13] further applied a second-order finite difference technique to estimate the thermal conductivity by solving the discretized linear equations. Lin et al [14] employed finite difference method to discretize the heat conduction equation and then the linear least-squares-error was adopted to solve IHCP. Kim et al [7] used an integral approach to solve the temperaturedependent thermal conductivity in one-dimensional domain.…”

Investigation of the effect of the measurement with the abrupt noise and the number of approximated terms in conductional boundary estimation problem

“…The method is used to determine the thermal conditions by Yang [15,16], Lin et al [17], and Hsu et al [18], to estimate the thermophysics properties by Yang [19], Lin et al [20], and Chang et al [21], to solve the dynamics by Yang [22], Shaw [23], and Kau and Yang [24], and to deal with the manufacturing problems by Hong and Lo [25] and Lee et al [26]. However, there are two problems in the reverse matrix approach.…”

“…Many researches [15][16][17][18][19][20][21][22][23][24][25][26] in various domains have used the reverse matrix method to solve the inverse problems. The method is used to determine the thermal conditions by Yang [15,16], Lin et al [17], and Hsu et al [18], to estimate the thermophysics properties by Yang [19], Lin et al [20], and Chang et al [21], to solve the dynamics by Yang [22], Shaw [23], and Kau and Yang [24], and to deal with the manufacturing problems by Hong and Lo [25] and Lee et al [26].…”