Estimating sensor number and spacing for inverse calculation of thermal boundary conditions using the conjugate gradient method

Abstract

The conjugate gradient method is a popular tool for solving inverse heat transfer problems which arise for example from the estimation of unknown thermal boundary conditions or temperature dependent material properties. Depending on the desired accuracy of the results, input data of numerous temperature sensors need to be considered. However, due to limited access and available space many systems offer only a few temperature measurement spots. Therefore, this paper focuses on the question how changes in the number of temperature measurements affect the inversely estimated boundary condition. This behavior is studied by two numerical test cases with different boundary conditions, thermal properties and geometries, investigating also varying resolutions of the boundary condition and the effect of measurement errors. The results of both test cases show, that for undisturbed measurement data as well as superimposed measurement errors, fewer temperature readings than unknowns are sufficient to accurately estimate the boundary condition. Also, after exceeding a threshold in sensor number, only little improvement of inverse estimated results can be observed in both test cases. To transfer these findings on further inverse heat transfer scenarios, the key heat conduction parameters are summarized in a characteristic parameter, the Fourier number. This number supports the estimation of necessary sensor count in future inverse investigations with varying thermal parameters, geometries and investigation time.