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

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Acoustics The term D Dt = ∂ + u·∇ is the material derivative. ρ denotes the density, t the ∂t time, u the velocity, p the pressure <strong>and</strong> e is the internal energy per unit <strong>of</strong> mass 1 . In the energy equation (3.3) 1/2|u| 2 is the kinetic energy where|u| denotes the absolute value <strong>of</strong> the velocity vector u. q represents the heat flux <strong>and</strong> ˙Q V the heat production per unit <strong>of</strong> volume which can be caused by chemical reactions or electrical heating. The energy equation can be replaced by the equation for the entropy s, which is derived by a combination <strong>of</strong> the energy equation with the second law <strong>of</strong> thermodynamics. If the heat transfer <strong>and</strong> dissipation are negligible the entropy equation can be written as ρT Ds Dt = ˙Q V , (3.4) where T denotes the temperature. In the following it is assumed that the flow is homentropic except in the region where the flame is located. Homentropy implies, that the flow is adiabatic, reversible ( Ds Dt = 0) <strong>and</strong> uniform (∇s = 0). The combustion process adds heat to the system, which does not result in a uniform temperature distribution – especially if equivalence ratio fluctuations are present. The flame is therefore treated as a discontinuity. If the flow is uniform, entropy waves, which are caused by equivalence ratio fluctuations at the flame, propagate independently with the mean flow speed <strong>and</strong> may also be coupled to the acoustics <strong>of</strong> the system at boundaries, at which the flow faces strong velocity gradients 2 . As these boundaries, e.g. nozzles or choked cross sections, are not considered in the present work, entropy waves are neglected <strong>and</strong> the assumption <strong>of</strong> homentropy for the flow field upstream <strong>and</strong> downstream the flame is valid. The conservation laws (3.1), (3.2), (3.3) do not form a complete set <strong>of</strong> equations. Additional information is required which is provided by the constitutive equations. Assuming that the flow is homogeneous <strong>and</strong> in a state <strong>of</strong> local thermodynamic equilibrium, two intrinsic state variables are sufficient to specify the thermodynamic state <strong>of</strong> the fluid. It is common to use the density ρ <strong>and</strong> the entropy s as the basic state variables. The pressure can then be expressed 1 The internal energy is also called the specific internal energy or simply the energy. 2 Here, a temperature fluctuation results in a volume flux fluctuation <strong>and</strong> can generate expansion waves propagating upstream. 44
3.1 Basic acoustic equations as a function <strong>of</strong> the state variables p = p(ρ, s) (equation <strong>of</strong> state). In differential form this equation yields where ( ) ( ∂p ∂p d p = dρ+ ∂ρ s ∂s ( ) ∂p ∂ρ s ) p d s, (3.5) ≡ c 2 . (3.6) Equation (3.6) is a definition <strong>of</strong> the thermodynamic variable c = c(ρ, s) <strong>and</strong> is a measure for the speed <strong>of</strong> sound. In the present work, air at atmospheric pressure is considered as the main fluid, which behaves like an ideal gas. The equation <strong>of</strong> state can therefore be simplified to p = ρRT, (3.7) where R denotes the specific gas constant, which is defined by the Boltzmann constant k b <strong>and</strong> the Avogadro number N A divided by the molar mass M <strong>of</strong> the gas (R = k b N A /M). For an ideal gas R can be expressed also as R = c p − c v , where c p <strong>and</strong> c v are the specific heat at constant pressure <strong>and</strong> volume, respectively. The ratio <strong>of</strong> the specific heats is defined by γ≡c p /c v . A further useful relation is the combining <strong>of</strong> the first <strong>and</strong> second law <strong>of</strong> thermodynamics 3 : T d s= d e+ pd 1 ρ . (3.8) Using this equation, the equation <strong>of</strong> state (3.7) for an ideal gas <strong>and</strong> assuming that e= c v T , which is a valid assumption for most ideal gases over a wide range <strong>of</strong> temperatures, one obtains a further formulation <strong>of</strong> the equation <strong>of</strong> state relating pressure <strong>and</strong> density: 3 which is also called the fundamental equation <strong>of</strong> thermodynamics ( ) p ρ γ ( ) s− s0 = exp . (3.9) p 0 ρ 0 c v 45
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Technische Universität München In
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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: 3 Acoustics Before analyzing the ac
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 and 78: 3.3 Acoustic network model pressure
Page 79 and 80: 3.3 Acoustic network model The eige
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
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5.5 Proof
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5.5 Proof
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5.5 Proof
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5.5 Proof
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5.5 Proof
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6 Numerical simulation MISO model i
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6.1 Practical premixed combustor co
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6.2 Steady-state simulations <stron
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6.2 Steady-state simulations <stron
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6.3 Identifiction of</stron
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6.4 Identification of</stro
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6.4 Identification of</stro
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7 Acoustic analysis Once the acoust
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7.1 Setup of the a
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7.2 Acoustic network model results
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8 Summary and Conc
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The flame element of</stron
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Bibliography [1] Verband</s
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BIBLIOGRAPHY [18] L. Crocco. Theore
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BIBLIOGRAPHY [36] A. Giauque, L. Se
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BIBLIOGRAPHY [54] J.J. Keller, W. E
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BIBLIOGRAPHY [72] T. Lieuwen <stron
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BIBLIOGRAPHY Number 98-GT-269 in In
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BIBLIOGRAPHY [112] T. Sattelmayer.
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BIBLIOGRAPHY [130] A. Widenhorn. pr
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A.2 Estimation of
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A.3 Main flow parameters an
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