1. Nondegenerate eigenvalues in annular gas turbines
Driek Rouwenhorst
6th TANGO Meeting
Ansaldo, Genoa
September 14th 2015

2. Overview Low-order TA model for annular combustion systems Results
Monitoring strategy for annular gas turbines
Example: Analysis of simulated data
Potential work for Tango2

3. Traveling wave solution through an annulus
Black box: Unknown effects can alter 𝝎 and 𝜶 𝑭 𝜽,𝒕 = 𝑭 𝟎 𝒆 𝒊( 𝝎 𝑭 𝒕−𝜽/𝒎)− 𝜶 𝑭 𝒕 𝑮 𝜽,𝒕 = 𝑮 𝟎 𝒆 𝒊( 𝝎 𝑮 𝒕+𝜽/𝒎)− 𝜶 𝑮 𝒕 𝒑 𝜽,𝒕 =𝑭+𝑮
Different frequencies and decay rates
Azimuthal bulk flow
Temperature gradients
Reflections
Swirl
etc.
Difference in damping possibly caused by dispersion (for example velocity gradient in annulus)

4. Heat release couples the traveling waves
Heat release: continuous 𝒏−𝝉 model 𝑸 𝒕,𝜽 ∝ 𝑵 𝜽 𝒑 𝒕−𝝉,𝜽 Thermoacoustic coupling: Rayleigh 𝑫𝑬 𝑫𝒕 = 𝜸−𝟏 𝑸 𝒕,𝜽 𝒑 𝒕,𝜽
Simplification: 𝑬=𝒑 𝒑 ⋆ → Feedback only on acoustic pressure, 𝒖(𝒑)
𝑫𝑭 𝑫𝒕 =− 𝜶 𝑭 𝑭+ 𝑵 𝒅𝒑 𝒅𝑭 ⋆ 𝒑(𝒕−𝝉,𝜽) 𝑫𝑮 𝑫𝒕 =− 𝜶 𝑮 𝑭+ 𝑵 𝒅𝒑 𝒅𝑮 ⋆ 𝒑(𝒕−𝝉,𝜽)

5. Results very similar to simplified network model approach of Bauerheim et al.
𝒅 𝒅𝒕 𝑭 𝑮 =𝐌 𝑭 𝑮
𝐌(𝝎, 𝜶,𝒎, 𝐍 ,𝝉)

6. Azimuthally traveling wave spectra observed in gas turbine data
Stochastic differential equation:
𝑭 𝑮 𝒊+𝟏 = 𝐈+𝐌𝚫𝒕 𝑭 𝑮 𝒊 + 𝝈 𝟐 𝚫𝒕 𝒘 𝒊
Sensor readings:
𝒔 𝟏 𝒔 𝟐 … 𝒔 𝒌 𝒊 =𝑨 𝑭 𝑮 𝒊

7. Better damping estimation and more physical information after decomposition

8. Conclusions Low-order TA-model for annular combustion systems
Takes into account azimuthal effects
Azimuthal modes split
Traveling, standing and mixed modes
Decomposition in azimuthal modes
Improved stability margin determination

9. Future challenges? Measurement options in industrial gas turbines
Heat release, inlet velocity
Excitation possibilities
Better identification
Passive control strategies
Specific action to stabilize identified system