Linear time-continuous state-space realization of flame transfer functions by means of a propagation equation.

Abstract

Low-order network models, commonly used to assess the thermo-acoustic stability of combustors, can be cast in a linear, time-continuous state-space representation. A standard linear eigenvalue problem for the system modes results, which can be solved in a robust and efficient manner. To represent the linear dynamics of any time-invariant flame in the state-space framework, this study presents an approximation of the distributed time-delayed flame response to acoustic velocity perturbations based on a spatially discretized propagation equation (PE). We derive the rational flame transfer function of a first-order-upwind-PE state-space model and discuss its relation to the Tustin approximation of flame transfer functions. For an exemplary discrete finite impulse response of a flame, a third-order-upwind-PE state-space model is shown to match the discrete flame frequency response with comparable accuracy as a rational approximation found by non-linear optimization. The numerical dissipation introduced by discretization of the PE ensures zero gain above the Nyquist frequency of the underlying discrete flame impulse response. Finally, we apply the PE state-space flame model to a generic Rijke tube and show that the predicted thermo-acoustic modes agree well with results obtained from a classical non-linearly optimized rational approximation of the frequency response function of the flame.