Research Papers: Gas Turbines: Combustion, Fuels, and Emissions

Methods for the Calculation of Thermoacoustic Stability Boundaries and Monte Carlo-Free Uncertainty Quantification

[+] Author and Article Information
Georg A. Mensah

Institut für Strömungsmechanik
und Technische Akustik,
Technische Universität Berlin,
Berlin 10623, Germany
e-mail: georg.a.mensah@tu-berlin.de

Luca Magri

Engineering Department,
University of Cambridge,
Cambridge CB2 1PZ, UK

Jonas P. Moeck

Institut für Strömungsmechanik
und Technische Akustik,
Technische Universität Berlin,
Berlin 10623, Germany

1Corresponding author.

Contributed by the Combustion and Fuels Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received July 18, 2017; final manuscript received August 10, 2017; published online January 17, 2018. Editor: David Wisler.

J. Eng. Gas Turbines Power 140(6), 061501 (Jan 17, 2018) (10 pages) Paper No: GTP-17-1375; doi: 10.1115/1.4038156 History: Received July 18, 2017; Revised August 10, 2017

Thermoacoustic instabilities are a major threat for modern gas turbines. Frequency-domain-based stability methods, such as network models and Helmholtz solvers, are common design tools because they are fast compared to compressible flow computations. They result in an eigenvalue problem, which is nonlinear with respect to the eigenvalue. Thus, the influence of the relevant parameters on mode stability is only given implicitly. Small changes in some model parameters, may have a great impact on stability. The assessment of how parameter uncertainties propagate to system stability is therefore crucial for safe gas turbine operation. This question is addressed by uncertainty quantification. A common strategy for uncertainty quantification in thermoacoustics is risk factor analysis. One general challenge regarding uncertainty quantification is the sheer number of uncertain parameter combinations to be quantified. For instance, uncertain parameters in an annular combustor might be the equivalence ratio, convection times, geometrical parameters, boundary impedances, flame response model parameters, etc. A new and fast way to obtain algebraic parameter models in order to tackle the implicit nature of the problem is using adjoint perturbation theory. This paper aims to further utilize adjoint methods for the quantification of uncertainties. This analytical method avoids the usual random Monte Carlo (MC) simulations, making it particularly attractive for industrial purposes. Using network models and the open-source Helmholtz solver PyHoltz, it is also discussed how to apply the method with standard modeling techniques. The theory is exemplified based on a simple ducted flame and a combustor of EM2C laboratory for which experimental data are available.

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Grahic Jump Location
Fig. 1

Schematic illustration of risk factor evaluation in the parameter space using a linear eigenfrequency model. A uniform parameter distribution is assumed for both parameters. Thus, the parameters may take values within the rectangle with same probability. The linear model divides this area into two parts. The part lying in the unstable region is highlighted by the hatching. The ratio between the hatched area and the total area is the risk factor.

Grahic Jump Location
Fig. 2

The marginally stable solution under consideration. This result of the Rijke tube model was computed with PyHoltz.

Grahic Jump Location
Fig. 3

Evaluation of 121 uniformly distributed samples for the Rijke tube in the parameter space (top) with the Helmholtz solver (bottom left) and the network model (bottom right). There is a good agreement between the two methods. About 50% of the samples lie in the unstable regime; thus, the risk factor is about 0.5. Though more points were used in the present study to estimate the risk factor, the low number of samples already gives a good estimate. This would not be possible by Monte Carlo sampling.

Grahic Jump Location
Fig. 4

Comparison of the prediction of the risk factor by use of uniformly distributed Monte Carlo samples (solid black) and uniformly distributed samples (dashed cyan) for the Rijke tube network model. Uniform sampling requires fewer samples (N) to converge than MC sampling.

Grahic Jump Location
Fig. 5

Mapping of the parameters to the resulting frequencies modeled by second-order two-parameter perturbation theory. For the Rijke tube case, this results in a hyperbola in parameter space separating the stable from the unstable regime. The small orange shaded rectangle encloses all parameters of 2.5% maximum deviation while the large cyan shaded rectangle corresponds to a maximum deviation of 10%.

Grahic Jump Location
Fig. 6

Turbulent swirled combustor configuration under investigation. Reproduced with permission from Silva et al. [28]. (Copyright 2017 by American Society of Mechanical Engineers.)

Grahic Jump Location
Fig. 7

Real part of the axial mode computed with PyHoltz

Grahic Jump Location
Fig. 8

Stability boundary in parameter space for the two modeling approaches computed by sample interpolation (solid black), first (dashed cyan)- and second (dashed orange)-order adjoint theory

Grahic Jump Location
Fig. 9

Wave-based thermoacoustic model. The flame located at x = b divides the duct into two segments. The waves' reflection coefficients at each end are R1 and R2.




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