Internal Heat Flux Estimation in Tubes Using the Backward Reciprocity Functional Method: Numerical and Experimental Results

Directly measuring internal heat fluxes in pipes is often impractical, requiring inverse heat conduction problem solutions for a nonintrusive and accurate assessment. This study extends the non-iterative backward reciprocity functional method to estimate time- and space-varying heat fluxes in tubes. To reduce computational cost, the classical integral transform technique is applied to take advantage of the orthogonality between the eigenfunctions and orthonormal basis in the spatial domain. Initially, the method uses synthetic temperature measurements on the exterior surface of a pipe, generated by solving the forward problem for different known heat fluxes on the internal surface of the pipe and adding Gaussian noises to the temperature data. Then, the method is applied to data from pulsating heat pipes experiments, where the fluid inside the tubes shows oscillatory variations over time and space. The pulsating heat pipe operation is strongly correlated with the internal wall heat flux, indicating its working condition, such as the startup and dry-out conditions. The results demonstrate good accuracy in assessing the heat flux with low computational costs using synthetic and experimental data. The total computational time for the estimate using experimental data was 4 s with a code written in MATLAB, which can be reduced to a fraction of a second when using a compiled language, such as C or C++.