Impact of fuel supply impedance and fuel staging on gas turbine ...

More documents

Recommendations

Info

Acoustics velocity fluctuations. The loudspeaker has the same characteristics as a closed end boundary condition (Eqn. (3.59)), where, in addition, a fluctuation <strong>of</strong> velocity ǫ is imposed: [ 1+ Mi M i − 1 ][ f i g i ] = ǫ. (3.62) 3.3.2 Numerical solutions <strong>of</strong> the acoustic network model To analyze the acoustic behavior <strong>of</strong> a network model representing a practical premixed combustion system two different mathematical techniques are applied in the present work: The calculation <strong>of</strong> the response <strong>of</strong> the inhomogeneous system, which is excited over a certain range <strong>of</strong> frequencies or the determination <strong>of</strong> the eigenfrequencies <strong>of</strong> the unexcited homogeneous system. The individual elements <strong>of</strong> the entire acoustic network model are combined in one matrix equation with the quadratic system matrix S, which consists <strong>of</strong> the matrix coefficients, the system vector with the Riemann Invariants <strong>and</strong> the source term vector on the right h<strong>and</strong> side ⎡ ⎢ ⎣ T 11 (ω)···T 1(2n) (ω) . . T (2n)1 (ω)···T (2n)(2n) (ω) ⎡ ⎤ ⎥ ⎦ ⎢ ⎣ ⎤ f 1 ⎡ g 1 . ⎢ = ⎣ ⎥ f n ⎦ g n ⎤ ǫ 1 . ǫ 2n ⎥ ⎦. (3.63) Here, n is the number <strong>of</strong> interfaces <strong>of</strong> the system which implies 2×n Riemann Invariants. In case <strong>of</strong> a homogeneous system the source term vector [ǫ 1 ...ǫ 2n ] is zero. If the system is excited at one or more boundaries the corresponding ǫ refers to the amplitude <strong>of</strong> the excitation which leads to an inhomogeneous system. In the latter case the frequency response <strong>of</strong> the linear inhomogeneous system can be calculated for a desired, fixed range <strong>of</strong> frequencies using a simple algorithm based on LU decomposition with partial pivoting. 64
3.3 Acoustic network model The eigenvalues <strong>of</strong> the homogeneous system are determined by calculating the roots <strong>of</strong> the determinant <strong>of</strong> the system matrix S: ⎛⎡ ⎜⎢ det(S)=det⎝⎣ T 11 (ω)···T 1(2n) (ω) . . T (2n)1 (ω)···T (2n)(2n) (ω) ⎤⎞ ⎥⎟ ⎦⎠=0. (3.64) If the system considers damping effects or acoustic sources like the flame, the determinant <strong>of</strong> the system matrix becomes zero for complex eigenfrequencies (ω=ω real +i ω im ). Inserting the complex eigenfrequency into the assumed harmonic time dependence, the time dependence can be written as: exp(i ωt )=exp(i (ω real + i ω im )t = exp(i ω real t )exp(−ω im t ). (3.65) If the imaginary part <strong>of</strong> the eigenfrequency has a negative value (ω im < 0) the oscillation will grow exponentially in time. On the other h<strong>and</strong>, a positive imaginary part (ω im > 0) will dampen possible oscillations. While the real part <strong>of</strong> the eigenfrequency determines the frequency <strong>of</strong> the eigenmode, the imaginary part is thus an indication for the stability <strong>of</strong> the system at this frequency. The stability can also be described by the cycle increment <strong>of</strong> the eigenmode, which is defined by the imaginary <strong>and</strong> real part as: ( C I = exp −2π ω ) im . (3.66) ω real The cycle increment can be considered as a growth rate per period <strong>of</strong> the oscillation. It is usually reduced by one to shift the stability border from one to zero. A cycle increment <strong>of</strong> C I < 1 or (C I−1) 1 ((C I − 1) > 0) indicates an unstable eigenmode <strong>and</strong> would thus amplify an oscillation. In the following the term C I − 1 is used. The routine to find frequencies ω, where |det(S(ω real ,ω im ))| → 0, is based on the Nelder-Mead simplex method [63], which is a direct search method <strong>and</strong> finds the minimum <strong>of</strong> a scalar function <strong>of</strong> several variables. Especially as the determinant exhibits large changes in its value over small frequency steps, the 65
Page 1:
Technische Universität München In
Page 4 and 5:
und ihre Geduld besonders für die
Page 7 and 8:
Contents 1 Introduction 1 1.1 Techn
Page 9 and 10:
CONTENTS 7.2.2 Impact</stro
Page 11 and 12:
Nomenclature G stretch factor [-] h
Page 13 and 14:
Nomenclature λ air-fuel</s
Page 15 and 16:
1 Introduction 1.1 Technology backg
Page 17 and 18:
1.2 Thermo-acoustic instabilities 1
Page 19 and 20:
1.2 Thermo-acoustic instabilities d
Page 21 and 22:
1.3 Control of the
Page 23 and 24:
1.3 Control of the
Page 25 and 26:
1.4 Intention and
Page 27 and 28: 1.5 Terminology and</strong
Page 29 and 30: 1.5 Terminology and</strong
Page 31 and 32: 2 Numerical analysis of</st
Page 33 and 34: 2.1 Overview of me
Page 39 and 40: 2.2 Flame dynamics of</stro
Page 47 and 48: 2.3 Identification of</stro
Page 57 and 58: 3 Acoustics Before analyzing the ac
Page 59 and 60: 3.1 Basic acoustic equations as a f
Page 61 and 62: 3.2 Linear acoustic equations Here,
Page 63 and 64: 3.2 Linear acoustic equations p ′
Page 65 and 66: 3.3 Acoustic network model p ′ (x
Page 67 and 68: 3.3 Acoustic network model matrix e
Page 69 and 70: 3.3 Acoustic network model The <str
Page 71 and 72: 3.3 Acoustic network model where A
Page 73 and 74: 3.3 Acoustic network model coming c
Page 75 and 76: 3.3 Acoustic network model Z i Air
Page 77: 3.3 Acoustic network model pressure
Page 81 and 82: 3.3 Acoustic network model tine. To
Page 83 and 84: 4 Modeling of turb
Page 85 and 86: 4.1 Theory and num
Page 87 and 88: 4.1 Theory and num
Page 89 and 90: 4.2 Theoretical and</strong
Page 95 and 96: 5 System Identification Flame trans
Page 97 and 98: 5.1 Model structures and</s
Page 99 and 100: 5.2 System Identification 5.2 Syste
Page 101 and 102: 5.3 Excitation signals the zero mea
Page 103 and 104: 5.4 A posteriori validation <strong
Page 105 and 106: 5.4 A posteriori validation <strong
Page 107 and 108: 5.5 Proof
Page 119 and 120: 6 Numerical simulation MISO model i
Page 121 and 122: 6.1 Practical premixed combustor co
Page 129 and 130:
6.2 Steady-state simulations <stron
Page 131 and 132:
6.2 Steady-state simulations <stron
Page 133 and 134:
6.3 Identifiction of</stron
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
6.4 Identification of</stro
Page 151 and 152:
6.4 Identification of</stro
Page 153 and 154:
7 Acoustic analysis Once the acoust
Page 155 and 156:
7.1 Setup of the a
Page 157 and 158:
7.2 Acoustic network model results
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
8 Summary and Conc
Page 183 and 184:
The flame element of</stron
Page 185 and 186:
Bibliography [1] Verband</s
Page 187 and 188:
BIBLIOGRAPHY [18] L. Crocco. Theore
Page 189 and 190:
BIBLIOGRAPHY [36] A. Giauque, L. Se
Page 191 and 192:
BIBLIOGRAPHY [54] J.J. Keller, W. E
Page 193 and 194:
BIBLIOGRAPHY [72] T. Lieuwen <stron
Page 195 and 196:
BIBLIOGRAPHY Number 98-GT-269 in In
Page 197 and 198:
BIBLIOGRAPHY [112] T. Sattelmayer.
Page 199 and 200:
BIBLIOGRAPHY [130] A. Widenhorn. pr
Page 201 and 202:
A.2 Estimation of
Page 203 and 204:
A.3 Main flow parameters an
show all

Impact of fuel supply impedance and fuel staging on gas turbine ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?