Thermoacoustic Damping Rate Determination From Combustion Noise Using Bayesian Statistics

Abstract

In this paper, we present a method to determine the quantitative stability level of a lean-premixed combustor from dynamic pressure data. Specifically, we make use of the autocorrelation function of the dynamic pressure signal acquired in a combustor where a turbulent flame acts as a thermoacoustic driver. In the proposed approach, the unfiltered pressure signal including several modes is analyzed by an algorithm based on Bayesian statistics. For this purpose, a Gibbs sampler is used to calculate parameters like damping rates and eigenfrequencies in the form of probability density functions (PDF) by a Markov-chain Monte Carlo (MCMC) method. The method provides a robust solution algorithm for fitting problems without requiring initial values. A further advantage lies in the nature of the statistical approach since the results can be assessed regarding its quality by means of the PDF and its standard deviation for each of the obtained parameters. First, a simulation of a stochastically forced van-der-Pol oscillator with preset input values is carried out to demonstrate accuracy and robustness of the method. In this context, it is shown that, despite a large amount of uncorrelated background noise, the identified damping rates are in a good agreement with the simulated parameters. Second, this technique is applied to measured pressure data. By doing so, the combustor is initially operated under stable conditions before the thermal power is gradually increased by adjusting the fuel mass flow rate until a limit-cycle oscillation is established. It is found that the obtained damping rates are qualitatively in line with the amplitude levels observed during operation of the combustor.

Comparison of Center Nozzle Staging to Outer Nozzle Staging in a Multi-Flame Combustor

“…In addition to understanding the timescale of stability transitions between the staging configurations, we also compare the relative stability margins using the combustor damping rates [18,19]. The physical effect of damping is to de-correlate fluctuations at one time with those at a later time.…”

“…This oscillation correlation time is proportional to the inverse of the damping rate, and it is this proportionality that allows the total system damping to be related to the system damping rates. Combustor damping rates are calculated using the method described by Stadlmaier et al [19], and is only briefly summarized here. This method, which is based on the method proposed by Lieuwen [18], assumes that the acoustic pressure inside of a combustor is due to the superposition of I nonlinearly-interacting oscillators [18][19][20].…”

“…Combustor damping rates are calculated using the method described by Stadlmaier et al [19], and is only briefly summarized here. This method, which is based on the method proposed by Lieuwen [18], assumes that the acoustic pressure inside of a combustor is due to the superposition of I nonlinearly-interacting oscillators [18][19][20]. The motion of these oscillators is governed by a second order system in η, which is written as Eq.…”

“…The motion of these oscillators is governed by a second order system in η, which is written as Eq. 5 for a linearly stable system [19].…”

“…5, it can be shown that the autocorrelation of the pressure is related to the damping rate as shown in Eq. 6 [19,21].…”