Response function formulation for inverse heat conduction: concept

Abstract

The Calibration Integral Equation Method (CIEM) represents an alternative view for resolving inverse heat-conduction problems based on a closely integrated mathematical and calibratory framework. The development of the measurement equation is constructed in the frequency domain and then inverted to the physical time domain through a convolution theorem. Hence, a first-kind Volterra integral equation arises from which the surface thermal condition can be resolved. This paper develops a "solution" formulation for the surface thermal condition initiated by CIEM principles. As with all inverse problems, stability in the prediction is key to the success of the methodology. All inverse problems require regularization by some technique for producing a stable approximate solution. The solution approach is formulated in the frequency domain in terms of a ratio of Laplace-transformed in-depth temperature measurements multiplied by the transformed calibration surface condition. The Laplace-transformed response function contains the calibration and reconstruction information from the in-depth sensor. The reconstructed surface condition is obtained in two steps. First, the inversion of this response function to the time domain is meticulously illustrated. Second, this temporal representation of the response function is then convolved with the measured calibration surface condition for predicting the surface condition. Fundamental to this paper is the mathematical framework required for inverting the Laplace-transformed response function composed of noisy in-depth data. The methodology is demonstrated by investigating a transient, one-dimensional linear heat equation in a semi-infinite geometry as associated with null-point calorimetry. This preliminary study relates the relevant details for the inversion of the response function based on Fourier analysis and presents results highlighting the merit of the concept.

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