Thermoacoustics of multi-burner combustors with plenum and chamber cross-talk

Abstract

This paper presents a thermoacoustic model in the frequency domain for multi-burner combustors composed of multiple plenum-nozzle-chamber modules with cross-talk at both plenums and chambers. The thermoacoustic model is given as a multi-input-multi-output transfer function matrix from heat fluctuation input vector to velocity fluctuation output vector. The discrete rotational symmetry of a multi-burner combustor is exploited to obtain a simple, diagonal transfer function matrix. The resonance equation, whose roots are resonances, is given as a set of scalar equations. The explicit form of the resonance equation reveals how combustor and operating parameters influence the resonances, and which parameters are significant for the longitudinal-azimuthal coupling and the plenum-chamber coupling. The diagonalization also allows us to show constructively that every mode shape is a Bloch wave and those mode shapes can characterize all resonances in terms of longitudinal and azimuthal mode numbers. At a low frequency, the resonance equation can be approximated as biquadratic and quadratic equations for a typical combustor geometry and operating conditions. A bunch of low-frequency resonances, peculiar to multi-burner combustors with cross-talk, are interpreted as azimuthal Helmholtz modes in which plenums/chambers and cross-talk ducts serve as the cavities and necks, respectively. Our combustor model is combined with a diagonal flame transfer function matrix for combustion instability analysis and we develop a set of scalar stability equations whose roots determine combustion instability. This stability equation reveals how combustion instability depends on the combustor geometry and operating parameters. Several numerical examples are presented to validate our thermoacoustic model and theoretical findings.