Thermoacoustics and combustion instability analysis for multi-burner combustors

Abstract

This paper presents a low-order thermoacoustic model for multi-burner combustors and proposes a control theoretic approach to linear combustion instability analysis in frequency domain, taking acoustic and flame interaction effects among burners into account under a non-zero mean flow condition. Our thermoacoustic model can deal with can-annular and annular combustors in a unified way and is given as a multi-input multi-output transfer function matrix from heat fluctuation input vector to velocity fluctuation output vector. We also propose a flame transfer function matrix description where off-diagonal components describe the effects of a local acoustic field of one burner location on the flame response at another burner location. The discrete rotational symmetry of burner arrangement in a combustor makes it possible to diagonalize both the aforementioned thermoacoustic and flame transfer function matrices analytically from a spatial discrete-time Fourier transformation. The decoupled thermoacoustic transfer function matrix provides a characterization of acoustic resonances, particularly the longitudinal-azimuthal coupled modes unique to multi-burner combustors. In addition, from the Bloch wave interpretation, we find general patterns in the mode shape of a multi-burner combustor and its relations to the mode shape of a single-burner combustor. Furthermore, from a diagonalized closed loop transfer function matrix, a simple combustion instability condition for a multi-burner combustor is developed. This stability condition shows how a single-burner combustor and a multi-burner combustor are different quantitatively, when it comes to combustion instability. Numerical examples are presented to validate our thermoacoustic model against three-dimensional FEM results, and to illustrate our developments and findings.