Statistical Characteristics of Pressure Oscillations in an Unstable Gas Turbine Combustor

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Statistical Characteristics of Pressure
Oscillations in an Unstable Gas Turbine Combustor

• Tim Lieuwen
• Assistant Professor
• School of Aerospace Engineering
• Georgia Institute of Technology
• Atlanta, GA
• __________________________________________________
  __________________

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Combustion Instabilities in Industrial Gas
Turbines

• Gas turbines have become the preferred power
  generation technology in the U.S. and abroad
• Development of these systems hindered by the
  occurrence of self excited, combustion driven
  oscillations
• Reduces combustor life
• Constrains regions of operability

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Overview

• A number of experimental and theoretical
  investigations have investigated the mechanisms
  of instability
• Anderson and Morford, ASME 98-GT-568,
• Straub and Richards, ASME Paper 98-GT-492
• Lieuwen and Zinn, 27th Intl Symposium on
  Combustion
• Broda et al., 27th Intl Symposium on Combustion
• Processes controlling stochastic and nonlinear
  characteristics have received less attention
• Some theoretical work reported (Culick and
  co-workers)
• Few comparisons of theory and data

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Motivation and Objective

• Understanding of stochastic, nonlinear processes
  in combustors needed
• Nonlinear processes predict instability
  amplitude, controller performance
• Stochastic processes quantify uncertainties in
  dynamic performance, robust active control
  methodologies
• Study objective Characterize cyclic variability
  in combustor oscillations and compare results
  with model predictions

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Examples of Cyclic VariabilityEvolution of state
space trajectories with increasing amplitude
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Cyclic Variations in Pressure Oscillations

• Our past studies strongly suggest variability due
  to random processes in combustor and do not arise
  from low dimensional dynamics (e.g., chaos)
• Can the statistical characteristics of these
  cyclic variations be predicted by assuming that
  they arise from stochastic processes?

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Investigated Model
Energy Addition
Random Excitation
System Nonlinearities

• Similar approach used previously by Culick and
  co-workers in assessing the role of noise in
  combustors with gas-dynamic nonlinearities

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Treatment of Nonlinear Terms in Oscillator
Equation

• No specific form for nonlinearities assumed
• Expand nonlinear terms in Taylor series,
  truncated at fourth order

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Pressure Amplitude and Phase

• Write pressure as
• p(t) A(t)cos (wts(t))
• Because system disturbed by random noise, A(t)
  and s(t) are random variables
• Objective of analysis is to determine their
  statistical characteristics

Amplitude
Phase
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Fokker-Planck Equation (FPE) for PDFs of A(t)
and s(t)

• In general, FPE is a nonlinear partial
  differential equation
• W - Probability Density Function
• Di Drift Coefficients
• Dij Diffusion Coefficients
• Specific form of coefficients given in paper

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Stationary Solution of Fokker-Planck Equation
Phase PDF
Amplitude PDF
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Predicted Pressure Amplitude PDF, W(A)
a ltlt 0
a gtgt 0
Increasing Growth Rate, a
a ? 0
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Schematic of Facility
Air
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Combustor Section-Front View
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Studied Parameter Space

• Equivalence Ratio ?0.65-1
• Combustor Pressure 1-10 atm.
• Inlet Velocity 10-60 m/s
• Inlet Length 104 164 cm
• Mass Flow Rate 6.1-21.1 g/s

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Time Evolution of Pressure and Flame Structure
(Flame visualized with CH radical
chemiluminescence)
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Procedure to Determine Amplitude and Phase PDFs
from Experimental Data

• (1) Determine reference signal based upon the
  average frequency of oscillations
• (2) Divide data record and reference signal into
  N pieces

• (3) Determine amplitude, An, and phase, sn, from
  Fourier transform
• (4) Determine W(A) and W(s) from calculated Anand
  sn

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Comparison of Measurements and Model, Amplitude
PDF
Measured
Modeled
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Comparison of Measurements and Model, Phase PDF
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Conclusions

• Statistical characteristics of cycle-to-cycle
  variations in limit cycle pressure oscillations
  captured qualitatively by stochastic model
• Amplitude transitions from Rayleigh (stable) to
  Gaussian (unstable) PDF
• Phase Uniform
• i.e., given enough cycles, oscillatory phase
  achieves all values with uniform probability
• Future Work Use statistical characteristics for
  nonlinear system identification