Majid Keyhani

A calibration integral equation method is proposed for estimating the surface temperature in the context of a nonlinear inverse heat conduction problem. The temperature-dependent thermophysical properties and probe positioning are implicitly accounted in the integral equation formulation through calibration tests. A first kind Chebyshev expansion is applied to represent the temperature-dependent property transform function. The undetermined expansion coefficients associated with the Chebyshev expansion are then estimated through two calibration tests. Regularization of the ill-posed problem is achieved by the future-time method. The optimal regularization parameter is estimated using a phase plane and cross-correlation phase plane analyses. Numerical simulation for stainless steel yields highly favorable surface temperature prediction.

This paper presents experimental data from a two-layer test sample made up of a copper layer and an AISI type 316 stainless steel (SS) layer that was heated with a laser power source. Experiments were conducted to generate high-temperature benchmark data that ranged from room temperature to 820 °C. The concept of time rescaling was employed to account for the dependence of thermal diffusivity on temperature in order to utilize the calibration integral equation method (CIEM). The future time regularization method was used to obtain a stable prediction for the surface temperature. An estimate for the future time regularization parameter was acquired through analysis of the in-depth calibration test thermocouple (TC) response. Results for three test cases consisting of selected pairs of calibration data and reconstruction data (to be predicted) are presented and discussed. Four different values of the future time regularization parameter were employed in the three test cases. The proposed nonlinear (NL) formulation improved the prediction accuracy when compared to the constant properties formulation of the CIEM. It should be emphasized that no knowledge of TC probe depth or TC response properties is required.

A novel thermophysical property estimation method is proposed, which incorporates both calibration and rescaling principles for estimating both unknown thermal diffusivity and thermal conductivity of materials. In this process, temperature and heat flux calibration equations are developed, which account for temperature-dependent thermophysical property combinations. This approach utilizes a single in-depth temperature measurement and a known set of boundary conditions. To acquire both thermal diffusivity and thermal conductivity, two distinct stages are proposed for extracting these properties. The first stage uses a temperature calibration equation for estimating the unknown thermal diffusivity. This process determines the thermal diffusivity by minimizing the residual of the temperature calibration equation with respect to the thermal diffusivity. The second stage uses the estimated thermal diffusivity and a heat flux calibration equation for estimating the unknown thermal conductivity. This stage produces the desired thermal conductivity by minimizing the residual of the heat flux calibration equation with respect to the thermal conductivity. Results verify that the proposed estimation process works well even in the presence of significant measurement noise for the chosen two representative materials. The relative error between the exact properties and the estimated values is shown to be small. For both test materials (stainless steel 304 and a representative carbon composite), the maximum relative prediction error is approximately 2–3%. Finally, as an added benefit, this method does not require explicit knowledge of the slab thickness or sensor position which further reduces systematic errors.

A novel surface heat flux calibration method applicable to nonlinear inverse heat conduction problems and a new strategy for obtaining an optimal regularization parameter independent of the regularization technique are presented herein. For the proposed nonlinear calibration integral equation, quasi-linearization of the nonlinear heat equation is achieved through both time domain and heat flux magnitude rescaling based on the local temperature measurement. The inverse heat conduction problem is then resolved in terms of rescaled variables through a calibration framework. The proposed calibration equation has a form of the Volterra equation of the first kind that relates the rescaled net unknown heat flux to the rescaled net calibration heat flux and their corresponding rescaled in-depth temperature measurements during both calibration and reconstruction tests. Because the functional form of a Volterra integral equation of the first kind is ill-posed, three different regularization methods are considered and compared. The three methods consist of 1) local future time method, 2) singular-value-decomposition-based regularization, and 3) Tikhonov regularization. A new strategy is proposed to determine the corresponding optimal regularization parameters. This strategy uses a Gaussian filter for estimating the variance in a group of predictions. The best regularization parameters are obtained by balancing the weighted bias and variance. Encouraging numerical results are observed independent of applied regularization approaches in presence of a significant noise, verifying the nonlinear calibration method and the merit of the new regularization parameter strategy.