Inferring convective and radiative heating loads from transient surface temperature measurements in the half-space

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Abstract
The classical, one-dimensional, transient half-space heat conduction problem in an opaque material is revisited in order to highlight several important observations that can impact a variety of ill-posed problems. First, the differential statement is recast into an equivalent Abel integral equation that describes the penetrating (conductive) heat flux in terms of the surface temperature. In many aerospace studies, the surface temperature is measured and the surface heat flux is calculated. Though intuitively simple, using unfiltered, noisy temperature data leads to unstable heat flux predictions as the sample density increases. This article presents a clear mathematical proof using the Discrete Fourier Transform Method verifying that the root-mean square error of the surface heat flux grows as the square root of the sample size (i.e., it is ill-posed when based on surface temperature measurements containing white noise). Second, a digital filtering method is proposed to reduce the instability problem while permitting an accurate depiction of the surface heat flux. Third, the study indicates that it is possible to predict the radiative and convective heat loads based on surface temperature measurements without a priori specification of the heat transfer coefficient or emissivity. That is, the Abel formulation effectively uncouples the interior (conductive region) from the surface (convective and radiative regions). Finally, it is demonstrated that the average convective heat transfer coefficient and average emissivity can be determined through the decoupled formulation using a simple least-squares approach. Further, the effect of data filtering is illustrated on the predictions of both the convective and radiative heat loads. The proposed Gauss filter is well suited to this problem owing to (i) its inherent behavior as a low-pass filter in the frequency domain and (ii) maintaining smooth, analytic support in the time domain.

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