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A hybrid algorithm of extended Kalman filtering and recursive least-square estimator (EKF-RLSE) is developed to solve the nonlinear inverse heat conduction-radiation problems in participating medium. By measuring the time-resolved temperature signals from a thermocouple installed on the other boundary of the participating medium, the proposed EKF-RLSE algorithm can realize real-time retrieval of the time-varying heat flux on the medium surface. The influence factors of the sampling interval, measurement and process covariance, forgetting factor, and medium thickness on the reconstruction accuracy and stability of the EKF-RLSE are investigated by numerical experiments. All the estimated results show that the present algorithm is robust and accurate in real-time estimating transient heat flux. If the nonlinear coupled conduction and radiation heat transfer problems is regard as the linear problems, the unknown time-dependent heat flux predicted by the KF-RLSE are also illustrated in this work.

The Kalman smoothing (KS) technique, which is consisted of the Kalman filtering and Rauch-Tung-Striebel smoothing technique, is introduced to resolve the inverse radiation-conduction heat transfer problem by using the future temperature measurement information. For the forward problem, the discrete ordinate method is employed to solve the radiative transfer equation and the energy equation is resolved by using the finite volume method. The boundary time-dependent heat flux and internal temperature filed in participating media are retrieved in near real time from measurement temperature on the right surface. Different forms of time-dependent heat flux are imposed on the left surface to examine the performance of the proposed algorithm. All the reconstruction results indicate that the KS technique is effective and robust for resolving the retrieval of the boundary time-dependent heat flux and internal temperature fields in near real time. The effect of different parameters on the accuracy and stability of the reconstruction results, including the future temperature information, sampling interval, measurement noise covariance, initial state error, and initial state error covariance, are analyzed in detail. Compared with the KF technique, the reconstruction accuracy of KS technique can be improved obviously. Meanwhile, the time delay and oscillation of reconstructed heat flux are reduced significantly and the deviation of the retrieval temperature is also decreased greatly.

The non-intrusive inverse heat transfer technique based on the Kalman filtering (KF) method is proposed for on-line retrieving the time-dependent boundary heat flux, internal temperature distribution, and liquid-phase interface of the participating medium simultaneously. To obtain the measured temperature signals, the nonlinear conduction-radiation heat transfer with phase change in the participating medium is solved by the enthalpy method combined with discrete ordinate method. Three different types of retrieval algorithms have been employed: the original Kalman filter (KF), the extended KF (EKF), and the unscented KF (UKF). The ideal participating media, which is assumed to be anisotropic scattering, and opaque and diffuse gray boundary, is employed to verify the reliability and validity of the proposed algorithm. Comparing with the EKF and UKF, the original KF cannot be applied to solve the nonlinear problems. Comparing with the UKF, the EKF can only be employed to solve the weak nonlinear problem. For the inverse coupled conduction-radiation problem with phase change, the UKF is proved to be more robust to estimate the time-dependent heat flux, internal temperature distribution, and liquid-phase interface simultaneously in real time.