Monitoring internal heat fluxes on Pulsating Heat Pipes using Kalman filter – Numerical and experimental results

The present study proposes an approach to monitor the wall-to-fluid internal heat flux on Pulsating Heat Pipes (PHP), which provides meaningful information about the efficiency and limiting operation conditions of such devices. In this sense, the current work adopts the Kalman Filter (KF) to estimate the heat flux between the working fluid and the internal tube wall along the adiabatic section of a PHP device by solving an inverse heat conduction problem. The KF is a well-known tracking technique, and therefore the current study is attractive due to the possibility of real-time evaluation of internal heat fluxes, a prospect that is extremely difficult to realize by standard whole domain inverse problem techniques. A high-speed and high-resolution infrared camera performs the temperature acquisition on the external wall of the PHP device. Two mathematical heat conduction models are considered: complete and thin-wall models. The complete model determines the temperature and heat flux evolution over time, and the thin-wall model is used to correct the random walk parameter on the KF formulation. Since the time step needed to the KF evolution model is smaller than the measurements acquisition interval, a matrix manipulation is used during the prediction step, reducing the number of mathematical operations. Hence, instead of predicting the states until reaching the measurement times, a proper mathematical manipulation reduces several mathematical operations to a single one. Then, the KF estimation capability is evaluated by considering synthetic data with different noise levels and different heat flux values. Subsequently, real experimental data are employed, and the results are compared with the ones obtained via a standard whole domain technique. The results obtained with the KF presented good visual agreement with the exact solution. Once the KF presents a delayed response, which is a characteristic of some filtering techniques, the errors with synthetic data were calculated offsetting the delay of the estimation in time in order to make a fair comparison, showing a maximum estimation error lower than 36 % considering the highest noise level. The synthetic data considered 20 s of time experiment, and the KF was able to estimate the PHP internal heat flux in less than 12 s, including the smoothing temperature step, showing fast and accurate results. For the real experimental data, the agreement between the KF and a whole domain technique is highly commendable, demonstrating a deviation of 25.5 % between both techniques. Moreover, the KF required significantly less computational time. The proposed methodology for heat flux estimation on PHPs is promising to evaluate their thermal behavior in real-time due to a low CPU time and computational memory required, as well as its simplicity to be implemented, possibly on embarked devices. Such information may therefore provide a thermal indicator of the correct functioning of PHPs in real time applications.