Localization of unsteady heat source in a tube from pressure measurements with the inverse method

Abstract

This paper presents work to identify the position of an unsteady heat source in a one-dimensional tube from acoustic pressure measurements with the inverse method. The relationship between the oscillation heat release rate and the pressure can be represented as a Volterra integral equation of the first kind. The discretization method was applied to transform the integral equation into matrix form. To stabilize the solution of the matrix equation, the Tikhonov regularization method was proposed. Experiments were performed to validate the inverse method. A semi-infinite probe system was used to measure the pressure perturbations in the tube, to avoid the high temperature damaging the microphone. Before the pressure measurements were taken, calibration was performed for the semi-infinite probe system to obtain accurate pressure data in the tube. The experiments were performed in three steps. First, the localization of a pure sound source in the tube at ambient temperature was studied. Second, localization of a pure sound source at hot conditions was considered. Third, the pure sound source was replaced with an unsteady heat source, and pressure data were used to determine the position of the unsteady heat source. Results show that calibration and the regularization are both necessary for the determination of the sound source position in the tube. Meanwhile, at the hot and heat release rate conditions, with the consideration of the temperature distribution in the thermoacoustic model, the position of the sound source and the unsteady heat release source can be determined successfully.