Regularization of inverse heat conduction by combination of rate sensor analysis and analytic continuation

Abstract

This paper describes some recent observations associated with (1) clarifying and expanding upon the integral relationship between temperature and heat flux in a half-space; (2) offering an analytic-continuation approach for estimating the surface temperature and heat flux in a one-dimensional geometry based on embedded measurements; and, (3) offering a novel digital filter that supports the use of analytic continuation based on a minimal number of embedded sensors. Key to future inverse analysis must be the proper understanding and generation of rate data associated with both the temperature and heat flux at the embedded location. For this paper, some results are presented that are theoretrically motivated but presently adapted to implement digital filtering. A pulsed surface heat flux is reconstructed by way of a single thermocouple sensor located at a well-defined embedded location in a half space. The proposed low-pass, Gaussian digital filter requires the specification of a cut-off frequency that is obtained by viewing the power spectra of the temperature signal as generated by the Discrete Fourier Transform (DFT). With this in hand, and through the use of an integral relationship between the local temperature and heat flux at the embedded location, the embedded heat flux can be accurately estimated. The time derivatives of the filtered temperature and heat flux are approximated by a simple finite-difference method to provide a sufficient number of terms required by the Taylor series for estimating (i.e., the projection) the surface temperature and heat flux. A numerical example demonstrates the accuracy of the proposed scheme. A series of appendices are offered that describe the mathematical details omitted in the body for ease of reading. These appendices contain important and subtle details germane to future studies.