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

A Finite Element Method for Three-Dimensional Analysis of Thermo-acoustic Combustion Instability

[+] Author and Article Information
S. M. Camporeale1

Dipartimento DIMeG, Sezione Macchine ed Energetica, Politecnico di Bari, Via Re David 200, 70125 Bari, Italycamporeale@poliba.it

B. Fortunato, G. Campa

Dipartimento DIMeG, Sezione Macchine ed Energetica, Politecnico di Bari, Via Re David 200, 70125 Bari, Italy

1

Corresponding author.

J. Eng. Gas Turbines Power 133(1), 011506 (Sep 24, 2010) (13 pages) doi:10.1115/1.4000606 History: Received May 25, 2009; Revised October 12, 2009; Published September 24, 2010; Online September 24, 2010

A method for predicting the onset of acoustically driven combustion instabilities in gas turbine combustor is examined. The basic idea is that the governing equations of the acoustic waves can be coupled with a flame heat release model and solved in the frequency domain. The paper shows that a complex eigenvalue problem is obtained that can be solved numerically by implementing the governing equations in a finite element code. This procedure allows one to identify the frequencies at which thermo-acoustic instabilities are expected and the growth rate of the pressure oscillations, at the onset of instability, when the hypothesis of linear behavior of the acoustic waves can be applied. The method can be applied virtually to any three-dimensional geometry, provided the necessary computational resources that are, anyway, much less than those required by computational fluid dynamics methods proposed for analyzing the combustion chamber under instability condition. Furthermore, in comparison with the “lumped” approach that characterizes popular acoustics networks, the proposed method allows one for much more flexibility in defining the geometry of the combustion chamber. The paper shows that different types of heat release laws, for instance, heat release concentrated in a flame sheet, as well as distributed in a larger domain, can be adopted. Moreover, experimentally or numerically determined flame transfer functions, giving the response of heat release to acoustic velocity fluctuations, can be incorporated in the model. To establish proof of concept, the method is validated at the beginning against simple test cases taken from literature. Over the frequency range considered, frequencies and growth rates both of stable and unstable eigenmodes are accurately evaluated. Then the method is applied to a much more complex annular combustor geometry in order to evaluate frequencies and growth rates of the unstable modes and to show how the variation in the parameters of the heat release law can influence the transition to instability.

Copyright © 2011 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Simplified scheme of flame location in a straight duct with uniform cross section

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Figure 2

Variation with τ of the eigenvalues of the duct with heat release fluctuation given by Eq. 10. Symbols represent the results of the FEM code. Lines are obtained from the solution of Eq. 11.

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Figure 3

Variation with β of the normalized frequency for the first mode of the duct with heat release fluctuation concentrated in a flame sheet. Time delay τ=0 and b=l/10. Symbols represent the results from the acoustic code. Line is obtained from the solution of Eq. 16.

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Figure 4

Variation with τ of the growth rate for the first mode of the duct with heat release fluctuation concentrated in a flame sheet. Case b=l/10. Symbols represent the results from the acoustic code. Lines are obtained from the solution of Eq. 16.

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Figure 5

Variation in the normalized frequency (a) and growth rate (b) with the temperature ratio T2/T1 for the first mode. β=0.6, τ=0.005 s, and b=l/10.

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Figure 6

Simplified scheme of flame location and boundary conditions, for benchmark tests on straight duct with variation in section

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Figure 7

Mesh visualization of the quasi-one-dimensional combustion system

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Figure 8

Resonant mode of a simple combustor. Crosses represent the results from the acoustic code. Circles represent the results from Ref. 3.

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Figure 9

Some modeshapes for the simple combustor

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Figure 10

Geometry of the annular combustion chamber

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Figure 11

Computational grid of the annular combustion chamber

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Figure 12

Variation with k of the normalized frequencies of the annular combustion chamber. Time delay τ=0.

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Figure 13

First four acoustic eigenmodes of the combustor

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Figure 14

Variation with the time delay of the normalized frequency (a) and the growth rate (b) of the first axial mode

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Figure 15

Variation with the time delay of the normalized frequency (a) and of the growth rate (b) of the first circumferential mode of the plenum

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Figure 16

Variation with the time delay of the normalized frequency (a) and of the growth rate (b) of the circumferential mode of the combustion chamber

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Figure 17

Variation with the time delay of the normalized frequency (a) and of the growth rate (b) of the second circumferential mode of the plenum. Symbols represent the results from the acoustic code for different values of β.

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Figure 18

Three examined geometries for the flame zone: (a) whole ring divided in 12 parts; (b) 12 cylinders; and (c) 12 hollow cones

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Figure 19

Variation with the time delay of the normalized frequency (a) and of the growth rate (b) of the first circumferential mode of the plenum. Symbols represent the results of the FEM code for different geometries of the heat release zone.

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