Thermoacoustic Instabilities in a Gas Turbine Combustor

Transcription

1 Thermoacoustic Instabilities in a Gas Turbine Combustor The Royal Institute of Technology School of Technological Sciences Department of Vehicle and Aeronautical Engineering The Marcus Wallenberg Laboratory for Sound and Vibration Research Karl Bengtson, karlbeng@kth.se

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3 Acknowledgements This report summarizes my study of thermoacoustic instabilities in a gas turbine combustor performed at Siemens Industrial Turbomachinery AB during the autumn of The work was performed as the final part of the Master s program in engineering mechanics at the Royal Institute of Technology, Stockholm, Sweden. I would like to thank my supervisor Dr. Jan Pettersson who has encouraged and guided me through the work at Siemens. I have learned a lot even though there are still much to learn and investigate within the complex world of thermoacoustics. Many thanks to all other colleagues at Siemens for support and especially to Joachim Nordin and Anders Häggmark who gave me the opportunity to perform this work at Siemens. I would also like to send my appreciation to Prof. Mats Åbom at the Royal Institute of Technology for being my supervisor at the university. Last but not least I would like to thank my fiancée Madeleine for support, patience and understanding throughout my master studies. Karl Bengtson Finspång I

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5 Abstract Stationary gas turbines are widely used today for power generation and mechanical drive applications. The introduction of new regulations on emissions in the last decades have led to extensive development and new technologies used within modern gas turbines. The majority of the gas turbines sold today have a so called DLE (Dry Low Emission) combustion system that mainly operates in the leanpremixed combustion regime. The lean-premixed regime is characterized by low emission capabilities but are more likely to exhibit stability issues compared to traditional non-premixed combustion systems. Thermoacoustic instabilities are a highly unwanted phenomena characterized by an interaction between an acoustic field and a combustion process. This interaction may lead to self-sustained large amplitude oscillations which can cause severe structural damage to the gas turbine if it couples with a structural mode. However, since a coupled phenomena, prediction of thermoacoustic stability is a complex topic still under research. In this work, the mechanisms responsible for thermoacoustic instabilities are described and a 1- dimensional stability modelling approach is applied to the Siemens SGT-750 combustion system. The complete combustor is modelled by so called acoustic two-port elements in which a 1-dimensional flame model is incorporated. The simulations is done using a generalized network code developed by Siemens. The SGT-750 shows today excellent stability and combustion performance but a deeper knowledge in the thermoacoustic behaviour is highly valued for future development. In addition, measurement data from an engine test is evaluated, post-processed and compared with the results from the 1-dimensional network model. The results are found to be in good agreement and the thermoacoustic response of the SGT-750 is found to be dominated by both global modes including all cans as well as local modes within the individual cans. II

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7 Contents Acknowledgements Abstract Contents Nomenclature I II III V 1 Introduction Gas Turbines Siemens Gas Turbines The SGT Thermoacoustic Modeling Thesis Objectives Thesis Motivation Theory and Fundamental Concepts Combustion Fundamentals Classifications of Flames Equivalence Ratio Adiabatic Flame Temperature Combustion Instabilities Rayleigh s Criterion The Rijke Tube Driving Mechanisms for Thermoacoustic Instabilities Eigenmodes in Gas Turbine Combustors Non-linear Effects Acoustic Theory The Linearized Wave Equation The Convective Wave Equation Solutions to the Wave Equation Impedance and Reflection Coefficient Damping of Acoustic Waves The Heat Release Source Term Acoustic Network Modelling The Straight Duct Element The Area Discontinuity Element Thermoacoustic Stability Analysis Flame Models Acoustic Jump Conditions III

8 2.6.2 The n τ Model An Extended Flame Model Fuel Injection Time Lag τ i Distributed Flame Models Boundary Conditions at Combustor In- and Outlet Method and Numerical Tool Network Modelling Tool Sample Cases Eigenfrequencies for a Simple Duct Wave Transmission through an Expansion Chamber Rijke Tube Simple Burner Featuring a Sudden Expansion Application to the SGT Evaluation of Measurement Data Stability Analysis by the Network Modelling Approach Establishing the Acoustic Network Model The Influence of a Mean Flow on the Acoustics Introducing the Flame - Perfectly Premixed Case The Outlet Acoustic Boundary Condition Fuel Line Impedance Equivalence Ratio Fluctuations Utilizing the Full Flame Model Measures to Improve Stability Change Combustor length Including Helmholtz Resonators C-stage Conclusions from the SGT-750 Study Discussion 66 6 Recommended Future Work 67 APPENDIX A The Two Microphone Method IV

9 Nomenclature Lower case letters c - Speed of sound f - Frequency h f - Specific reaction enthalpy i - Imaginary unit (i = 1) k - Wave number (k = ω/c) k + - Wave number for waves propagating in the positive direction k - Wave number for waves propagating in the negative direction m - Mass ṁ - Mass flow rate p - Pressure q - Heat release density s - Entropy t - Time u - Velocity magnitude u - Velocity vector x - Spatial coordinate (1-D) x - Spatial coordinate vector (3-D) y f - Mass fraction of fuel z - Characteristic impedance (z = ρc) Upper case letters A - Area C + - Amplitude of wave propagating in positive direction C - Amplitude of wave propagating in negative direction C xy - Coherence between signal x and y D - Diameter F - Flame transfer function H xy - Complex transfer function (H xy = x(ω)/y(ω)) L - Length M - Mach number (M = u/c) P xx - Auto power spectrum of signal x P xy - Cross spectrum between signal x and y Q - Heat release R - Wave amplitude reflection coefficient T - Period of oscillation (T = 1/f) T ij - Transfer element ij T - Transfer matrix V

10 V - Volume V - Volume flow rate Z - Specific impedance Greek letters γ - Ratio of specific heats (γ = C p /C v ) φ - Equivalence ratio ρ - Density ω - Angular frequency (ω = 2πf) τ - Time lag τ b - Time lag related to volume flow fluctuations τ i - Time lag related to equivalence ratio fluctuations - Nabla operator, = ( / x, / y, / z) Notations p - Complex conjugate p - Mean value of quantity p - Fluctuating part of quantity (Acoustic part) ˆp - Complex amplitude - Frequency domain representation VI

11 1 Introduction Gas turbines are used all around the world for power generation and as power sources for pumps and compressors. The rapid development and construction of renewable energy sources such as wind and solar energy has created a new demand for flexible power sources that in combination constitutes a reliable energy system. Gas turbines have the ability to start up quickly when the sun is not shining or the wind is not blowing. For this reason, gas turbines will be a central part within the power generation business for many years to come. The development of the next generation gas turbines primarily focuses on improved efficiency and reduced NOx (nitrous oxides) emissions. The majority of the stationary gas turbines produced today have a DLE (Dry Low Emission) combustion system to meet emission regulations. DLE combustion systems operate primarily in the premixed combustion regime and are more likely to experience combustion instabilities than conventional combustors. There are several measures used today to reduce NOx emissions in gas turbines. The most efficient option is to lower the combustion temperature. However, this is in general counter-productive when it comes to engine efficiency and the general trend within thermal machinery is instead to increase temperatures in order to improve engine efficiency. Another way to reduce emissions often utilized is different measures to improve combustion stability allowing for reduction of the stabilizing diffusion pilot and hence lower NOx emissions. To improve combustion stability, understanding and prediction of thermoacoustic instabilities is an important key. Thermoacoustic instabilities are characterized by an interaction process between a combustion process and an acoustic field which may lead to self-sustained oscillations. This oscillations may grow in amplitude and can cause wear and severe structural damage if not kept below acceptable limits. Since thermoacoustic instabilities is formed from the coupling between combustion and acoustics, accurate modelling need to include both phenomena which makes thermoacoustic modelling complex and difficult to deal with. However, for a successful combustion system, thermoacoustic instabilities need to be understood and predicted. This will be even more important in the future with tougher regulations on emissions as well as new demands for flexible operation. A frequently used modelling approach for thermoacoustics is by using so called 1-dimensional loworder models. This work is one attempt to model and predict thermoacoustic instabilities for a gas turbine combustor using a low-order network approach. 1.1 Gas Turbines Gas turbine engines work according to the Brayton cycle and can be divided in three main parts, compressor, combustor and turbine. Compressed air from the compressor enters the combustor where fuel is introduced and the mixture is burnt. The combustion gases are expanded through the turbine 1

12 which drives the compressor as well as giving a net power output that can be used to drive a generator or a pump. The main fuel used in modern gas turbines are natural gas but many engines have the capability to operate on liquid fuel as well as alternative gas mixtures. Only a brief description of the gas turbine working principle is given here, more details can be found in e.g. the gas turbine handbook by Boyce [4]. Two different types of gas turbine combustors are commonly used in modern gas turbines. These are can and annular type combustors. In a can combustor system, each burner has its own combustion chamber while in an annular system all burners goes into one large annular combustion chamber. Annular combustors in general have less area to cool while the can combustors have advantages when it comes to service and maintenance. Also, in annular combustors the burner-to-burner interaction is in general more apparent than for can combustors. 1.2 Siemens Gas Turbines Siemens is a well known manufacturer of industrial gas turbines with a portfolio including gas turbines in the range from 4 to 400MW,[26]. The Siemens gas turbines are sold to customers all around the world while development and manufacturing are at present concentrated mainly to Europe and North America. Siemens Industrial Turbomachinery AB in Finspång, Sweden is responsible for the industrial mid-size engines. The turbine production in Finspång started in 1913 and since then more than 800 gas turbines have been manufactured and delivered. The Finspång site has today about 2700 employees and the site is responsible for development of both core and package, production, sales, commissioning and after-market/service of their engines, [26]. 1.3 The SGT-750 The SGT-750 is one of the latest industrial gas turbines from Siemens with a power output of around 40MW and world class efficiency. The engine has a twin-shaft configuration meaning the compressor and power turbine are disconnected from each other, this is especially suitable for mechanical drive applications. The SGT-750 core engine is shown in figure 1. 2

13 Figure 1: The SGT-750 core engine, [26]. The SGT-750 combustor features eight individual cans which provides single digit NOx capabilities in a wide range of operation conditions. One of the eight combustor cans is shown in figure 2. Each of the combustor cans are fed by compressor discharge air through a common annular casing. Convective and impingement cooling techniques are utilized for cooling of the combustion chamber walls. The burner comprises two separate main fuel lines (main 1 and main 2) for further improved tuning flexibilities. An optimized aerodynamic design ensures a well-defined recirculation zone for stabilizing the flame. In addition, a pilot and a RPL (Rich Pilot Lean) burner are used for central stabilization of the main flame. After combustion, the combustion gases are led through a transition duct to the inlet of the first turbine stage. Figure 2: The SGT-750 combustion system, [13]. 3

14 1.4 Thermoacoustic Modeling Many studies with different modelling approaches to thermoacoustics in gas turbines and jet-engines can be found in the literature. Those includes forced response analysis, complex eigenvalue analysis, URANS (Unsteady Reynolds Averaged Navier Stokes) and LES (Large Eddy Simulations). Even though thermoacoustics is a coupled phenomenon, many studies focus more on either combustion or acoustics. However, the necessity of including the coupling and e.g. complementing LES with acoustic analysis methods is pointed out by Poinsot, [22]. A common used approach to study stability in a coupled manner is to use a so called low-order network model. Such model assumes 1-dimensional acoustics and uses elements known as two-ports to describe the combustor geometry and the flame. Together with up- and downstream boundary conditions a stability criterion can be formulated and analysed. This is the approach used in this work. 1.5 Thesis Objectives The objectives with this work were to: Perform a literature study on thermoacoustic stability analysis and low-order modelling. Evaluate and understand a general network code developed by Siemens AG. Create a thermoacoustic network model of the SGT-750 combustion system. Evaluate and post-process combustion dynamic measurement data from full engine tests. Compare the results and give recommendation of future work. 1.6 Thesis Motivation The main objective with thermoacoustic stability analysis is to predict the systems dynamic behaviour early in the design phase. Anyhow, due to the complex coupled phenomena, this has been found to be easier said than done. Using different modelling techniques is one way to gain understanding in the field and to make justified design changes to improve stability. In this thesis work, one approach to model thermoacoustic stability is utilized. The SGT-750 shows today excellent combustion stability and performance. However, a better understanding of the dynamic behaviour is beneficial for further development and possible future upgrades of the engine. 4

15 2 Theory and Fundamental Concepts In this section, concepts and relevant theory needed to understand combustion instabilities are presented. Since the combustion process itself is the driver for the instabilities a review of some fundamental combustion concepts is given first. 2.1 Combustion Fundamentals Combustion is an exothermal reaction process characterized by conversion of chemical spices and heat release. A combustion process involves fuel and an oxidizer which for gas turbines commonly are hydrocarbons and air. The chemical reaction when methane is burnt in air on a global level, not considering intermediate species can be expressed as CH 4 + 2O 2 CO 2 + 2H 2 O + Heat. (1) The heat release within a combustion process appears as a flame which propagates through the unburned mixture with a certain burning velocity. The flame can be defined as a thin layer with rapid chemical changes and a steep temperature gradient. On a macroscopic level, the flame is often seen as the interface that divides the burned mixture from the unburned mixture. In the science of combustion, both physics and chemistry play important roles, [17]. To describe heat release in thermoacoustic studies the physics and thermodynamics is often sufficient while a detailed chemistry description is more vital when e.g. emissions are to be predicted Classifications of Flames Flames in general can be classified as either premixed or non-premixed flames. In addition, flames can also be categorized as laminar or turbulent depending on the initial state of the reactants, [28]. Both premixed and non-premixed turbulent flames are utilized in modern gas turbines. For premixed flames the fuel and oxidizer are mixed before entering the flame zone while for nonpremixed flames the fuel and oxidizer are mixed by diffusion within the flame zone. Premixed flames are in general more sensitive to disturbances and stability issues than non-premixed flames. The main reason is that non-premixed flames stabilizes in the intermediate mixing region between the fuel and the oxidizer which constitutes steep concentration gradients, [28]. On the other hand, for a perfectly premixed mixture, no such concentration gradients exist and hence no obvious location for the flame to stabilize. This makes the flame location very sensitive to disturbances. Combustors operating in the premixed regime generally features a sudden expansion where the combustible mixture enters the combustion chamber. The velocity before the expansion is high enough to prevent the flame from propagating upstream. This will create a defined location for the flame to stabilize. A schematic picture of general combustor featuring a sudden expansion is shown in figure 3. 5

16 Figure 3: Schematic picture of a simple burner featuring a sudden expansion. Gas turbine combustors featuring DLE combustion systems utilize in general both premixed and nonpremixed flames. The largest portion of the fuel is introduced in the main flame which is a lean premixed flame that produces low levels of NOx emissions. Ideally the main combustible mixture is homogeneous but this is difficult to realize in practice and mixing performance is a topic of continuously development and research. In addition to main fuel, some smaller portion of the fuel is introduced as a pilot which gives a fuel-rich often close to non-premixed flame. The pilot flame is used to stabilize and maintain the main flame but has the drawback to produce higher levels of NOx emissions. With improved combustion stability the PFR (Pilot Flow Rate) can be reduced and hence NOx emission levels are reduced Equivalence Ratio Equivalence ratio describes the fuel-to-air ratio and is an important parameter within the science of combustion. Equivalence ratio is defined as The index st φ = m fuel/m ox (m fuel /m ox ) st. (2) denotes stoichiometric conditions which is the situation when the amount of air is exactly what is needed to completely burn the fuel. Hence, stoichiometric conditions correspond to an equivalence ratio equal to 1. An equivalence ratio higher than 1 means there is an excess of fuel to the available amount of oxidizer, this is called fuel-rich and will give unburnt fuel as a rest product. If the equivalence ratio is less than 1 there is an excess of air to the amount of fuel to be burnt. This condition is called fuel-lean, [28] Adiabatic Flame Temperature Flame temperature has a strong influence on the chemical reaction taking place within a flame and are strongly coupled to both emissions and heat release. The maximum flame temperature is achieved at stoichiometric conditions. Adiabatic flame temperature is a quantity that can be calculated from the thermodynamic properties of the reactants. It is defined as the temperature that the net energy 6

17 released in the flame would give to the combustion products under adiabatic conditions. A real combustion process is not adiabatic and the actual flame temperature is always lower than the adiabatic flame temperature. The adiabatic flame temperature is mainly affected by equivalence ratio, initial temperature and pressure, [28]. 2.2 Combustion Instabilities Combustion instabilities can be divided in two categories, Combustion noise and Thermoacoustic instabilities. Those are both driven by the combustion process but the characteristics and physical phenomena is different. Combustion Noise The flow in gas turbine combustors is inherently turbulent. This turbulence creates flow variations that affects the combustion process and results in combustion noise. This noise is sometimes called combustion roar and is of a broadband character with relatively low amplitude, [9]. Combustion noise is not that critical in modern gas turbines and has not been treated in this work. Thermoacoustic Instabilities Thermoacoustic instabilities on the other hand commonly appears as large amplitude oscillations at one of the systems natural frequencies. Those instabilities are spontaneously excited and the oscillations are maintained by a feedback loop between the combustion and the acoustic field. The principle for thermoacoustic instabilities is illustrated in figure 4 and 5. The unsteady heat release in the flame generates acoustic waves which are reflected at the system boundaries and standing waves are formed. The acoustic fluctuations give rise to flow and mixture perturbations which in turn affects the flame with a fluctuation of the heat release as the result, the loop is closed, [19]. Figure 4: Acoustic waves are created by the flame and reflected at the system boundaries. The oscillations will be amplified or damped depending on the phase between the heat release and the pressure. In contrast to combustion noise, thermoacoustic instabilities are characterized by high amplitude oscillations at distinct frequencies. Those large oscillations in velocity and pressure are highly unwanted and can cause severe wear and structural damage to the gas turbine. 7

18 Figure 5: The feedback loop responsible for thermoacoustic instabilities, [19]. Combustion processes can create acoustic waves in at least two different ways. First, acoustic waves will be created directly by the volume fluctuations resulting from the unsteady heat release. Additionally, unsteady heat release gives rise to temperature fluctuations often refereed to as entropy waves which convect downstream with the mean flow. Those entropy waves are not associated with any acoustic fluctuation of pressure and velocity, hence no noise. However, when an entropy wave is accelerated as happens at the combustor exit, acoustic waves will be indirectly generated. Entropy waves and indirect noise has become an important topic for aero-engines to reduce the overall noise level, [9]. The effect of entropy waves and indirect noise on thermoacoustic instabilities is an ongoing discussion and have been discussed by e.g. Goh and Morgans, [10] and Sattelmayer, [6] Rayleigh s Criterion The situation under which thermoacoustic instabilities occurs was first described by Lord Rayleigh in the 1880s. His criteria is important for the understanding of thermoacoustic instabilities and can be formulated as, [19], ˆ ˆ p (x, t)q (x, t)dtdv > 0. (3) V T Energy is transferred to the acoustic field if the phase difference between the unsteady pressure and heat release is less than 90 leading to amplified oscillations. Maximum energy input to the acoustic field is archived if pressure and heat release are perfectly in phase. If on the other hand the heat release and the pressure is out-of-phase, energy is removed from the acoustic field and the oscillations are damped. Rayleigh s criteria as formulated in equation 3 is strictly valid for undamped systems. For real systems, some portion of the acoustic energy propagates out through the boundaries or gets dissipated by friction and viscous effects. For a system that includes damping, the Rayleigh integral need to render a larger value than the energy dissipated to get amplified oscillations, [19]. 8

19 2.2.2 The Rijke Tube A classical experiment that has highly contributed to the understanding of thermoacoustic instabilities is the Rijke tube. In its most basic form, a Rijke tube constitutes an open-ended pipe with a heat source which under certain conditions will generate a strong tonal sound. For sound to be generated a mean flow is required which can be created by convection if placing the tube vertically. The heat source is commonly an electrically heated grid or a combustion flame, e.g. a Bunsen burner. Figure 6 shows a schematic representation of a Rijke tube. Figure 6: Schematic representation of a Rijke tube The phenomena generating the sound can be understood from the Rayleigh s criteria (equation 3). The first mode of an open-ended tube has pressure nodes at the ends and a velocity node in the middle as shown in figure 6. As can be seen the pressure has the same sign (positive) in the whole tube while the velocity changes sign in the middle of the tube. Considering an electrically heated grid, the heat release from the source to the medium in the tube will be due to convection. The convection process is influenced by the velocity at the grid which will be the mean velocity superimposed by the acoustic fluctuations. The unsteady heat release by the source is obviously related to the acoustic velocity fluctuations and the same is true for a combustion heat source. Placing the heat source in the lower half of the tube will render a positive value of the Rayleigh s integral and the acoustic oscillations will be sustained and amplified. In contrary, placing the heat source in the upper part of the tube the 9

20 Rayleigh s integral will give a negative value and the acoustic oscillations will instead be attenuated, [2]. For an acoustic field without a source, pressure and velocity fluctuations are 90 degrees out-of-phase. This means, if the heat release is perfectly in-phase with the velocity, no energy would be transferred to the acoustic field according to Rayleigh s criteria. This gives a second criteria for instability. It must be a delay between the heat release and the acoustic velocity oscillations. Referring to the electrical heater the heat release is due to convection which is a process that takes time. The same is true for a combustion process were some time is required for the chemical reaction to take place. The tone generated by a Rijke tube is normally the fundamental tone, with a wavelength corresponding to twice the length of the tube. However, the frequency is not easy to predict in practice. First, the phase of the heat release at the heat source location will influence the acoustic impedance which will influence the eigenfrequencis, this has e.g. been studied by Mcintosh, [20]. In addition, due to the heat source, there will be a complex temperature distribution in the tube affecting the speed of sound and hence the eigenfrequencies. The influence of the temperature field in a Rijke tube has been studied by L.Nord, [21] Driving Mechanisms for Thermoacoustic Instabilities There are several mechanisms responsible for driving of thermoacoustic instabilities in gas turbines. This section gives a description of some of the most important mechanisms. Equivalence Ratio Fluctuations Equivalence ratio fluctuations are an important source to combustion instabilities in premixed combustors operating at fuel-lean conditions. Acoustic perturbations within the premixing section may influence the air and/or fuel supply leading to periodic equivalence ratio oscillations. Those fluctuations are convected by the mean flow to the flame front resulting in an unsteady heat release. Equivalence ratio fluctuations due to acoustic coupling is strongly affected by the pressure drop over fuel injectors. In general, a larger pressure drop makes the system less sensitive to acoustic disturbances, [9]. Coupling Acoustic-Fuel Feed Line Fuel feed line-acoustic coupling is a special case of equivalence ratio fluctuations. The mechanism refers to pressure drop fluctuations over non-chocked nozzles which makes the fuel injection rate to be modulated. The origin of the pressure drop fluctuations can be both on the fuel line or due to the acoustic field in the combustor, [9]. The later case is illustrated in figure 7. The flame generates acoustic waves leading to pressure fluctuations over the fuel nozzles located upstream. This imply the amount of fuel injected will vary periodically in time and the resulting mixture is convected to the flame front. The time required for the disturbance to convect to the flame front is here denoted, 10

21 τ i. Depending on the phase between the unsteady heat release and the pressure at the flame front, the oscillation will be amplified or damped. It can be concluded that convective times are important parameters for control of thermocoustic instabilities. Figure 7: Coupling acoustic-fuel feed line. Flame Area Variation The heat release at the flame front is proportional to the flame surface area. There are several reasons the flame area may vary in gas turbine combustors leading to a fluctuating heat release. The most important mechanisms are: Acoustic velocity oscillations within the combustor will affect the flame area. Flame-vortex interaction which refers to periodic separations created by e.g. sudden expansions, flame holders or other obstacles in the flow path. The vorticity generated convects with the flow and stretches the flame when passing through leading to periodic variations of flame area. The frequency at which this occur does not need to be the natural shedding frequency. If the amplitude is high enough, vortex separation can be forced to occur by an external excitation. Vortex interaction with boundaries. Vortexes generated interact with a wall which periodically introduce fresh unburned mixture into the flame zone. The fresh mixture ignites after some time delay and creating a fluctuating heat release. Thermal losses when a flame impinges on a cold wall can affect the chemical reaction and make the flame area to vary. Interaction between flames may affect the flame surface area. This could be between pilot and main flame as well as between different burners in annular combustion systems Eigenmodes in Gas Turbine Combustors Thermoacoustic instabilities is normally associated with one or more acoustic eigenmodes of the system. Gas turbine combustors generally features cylindrical geometries with hard metal walls. Possible mode shapes are illustrated in figure 8. 11

22 Figure 8: Modes in combustor geometries. a) Longitudinal mode. b) Transverse Azimuthal Mode. c) Transverse Radial Mode. The longitudinal modes are to a large extent controlled by the boundary conditions at the combustor inlet and outlet while the transverse modes are more defined due to the hard metal walls. It is assumed the interaction with the walls is negligible. No transverse modes can exist bellow the cut-on frequency for the first higher order mode. The cut-on frequency for the first transverse azimuthal mode in a circular geometry is given by f 1,cut on = c πd. (4) In the low frequency range, below the cut-on frequency for the first higher order mode, the acoustic field is more or less one-dimensional. Typical temperatures in gas turbine combustors are 750K upstream the flame (compressor discharge air temperature) and 1750K downstream of the flame which gives the speed of sound to be about 550m/s and 830m/s respectively Non-linear Effects According to Rayleigh s integral, energy will be added to the acoustic field as long as the integral is larger than the dissipation. The amplitude of the oscillations will grow exponentially in the beginning but this cannot continue forever. Non-linear effects will cause the amplitude of the oscillations to saturate at some finite limit-cycle amplitude. To determine the limit-cycle amplitude, non-linear effects cannot be neglected and hence, the limit-cycle amplitude cannot be determined by linear models, [19]. Linear models however are able to predict potential critical frequencies. In general the acoustic quantities are small enough to be treated as linear while the heat release includes non-linear phenomena. Taking advantage of this have led to hybrid models where linear acoustics is coupled to non-linear heat release models, this approach has been used by e.g. Graham and Dowling, [11]. Only linear models will be used through out this work. 12

23 2.3 Acoustic Theory The Linearized Wave Equation The most fundamental equation within acoustics is the linearized wave equation. The derivation starts from the well known conservation equations from fluid mechanics. These are conservation of mass, momentum and energy which in differential form may be written as, [25], ρ + (ρu) = m, (5) t [ ] u ρ t + u u + p = f, (6) ρt [ ] s t + u s = q. (7) Where m, f, q are source terms representing mass sources, external forces and heat sources respectively. Viscous effects are neglected. Considering acoustic wave propagation in a homogeneous media without any sources. Zero mean flow is assumed at this point. In most cases the acoustic perturbations can be assumed to be small in comparison to the mean value. The acoustic field variables are described by a steady mean value plus a small fluctuating part as p(x, t) =p + p (x, t) ρ(x, t) =ρ + ρ (x, t) u(x, t) =0 + u (x, t). (8) The overline denotes the mean value and the prime denotes the fluctuating (acoustic) part. Substitution of equation 8 into the conservation equations for mass and momentum and neglecting products of primed quantities gives the linearized acoustic conservation equations as ρ t + ρ u = 0, (9) ρ u t + p = 0. (10) Since the acoustic disturbance being small a frequently used assumption is a sound wave being isentropic and reversible. The relation between acoustic pressure and density can be found from the equation of state, p = p(ρ, s) which may be expressed as 13

24 p t = ( ) p ρ ρ s t + ( ) p s s ρ t. (11) For an isentropic process the entropy is constant which makes the second term in equation 11 to vanish. By using the definition of the speed of sound, c 2 = ( p ρ ) s, the relation between the acoustic pressure and density after linearization is found to be p = ρ c 2. (12) This relation is used to eliminate the density in the conservation equation for acoustic continuity, the result is p t + ρc2 u = 0. (13) Now, taking the time derivative of Equation 13, the spatial derivative of equation 10 and subtracting the two yields the linearized wave equation in 3-dimensions 2 p t 2 c2 2 p = 0. (14) Through out this work, wave propagation is assumed to be 1-dimensional only and the wave equation in 1-D is given below for reference. 2 p t 2 c2 2 p x 2 = 0 (15) It should be mentioned that the assumptions made in the derivation of the wave equation may be appropriate for sound propagation in ambient temperature and pressure conditions. However, the assumptions may not always be suitable for the conditions in a gas turbine combustor The Convective Wave Equation The convective wave equation describes wave propagation when a mean flow is present. The effects of a mean flow to acoustic wave propagation can be included in the previously derived wave equation by a change of reference system. Derivation is straight forward and the result is found be replacing the time derivative in equation 15 by the convective derivative, the result is ( t + u ) 2 p c 2 2 p = 0. (16) x x2 14

25 2.3.3 Solutions to the Wave Equation The solution to the wave equation in 1-dimension is a linear combination of two waves, one propagating in the positive direction and one in the negative direction. If a mean flow is present, this positive travelling wave propagates with a speed c + u while the negative travelling wave propagates with a speed c u. The well known d Alembert s solution can be written as p (x, t) = f(x (c + u)t) + g(x + (c u)t), (17) with f and g being two arbitrary functions. A convenient description often used in acoustics is obtained by assuming a harmonic time dependence for the acoustic quantities. The relation between acoustic pressure and velocity for a plane wave is given by the characteristic impedance z = ±ρc, for the positive (+) and the negative (-) travelling wave respectively. Using this, the solution with separated time and space dependence can be written as p (x, t) = C + e i(ωt k+x) + C e i(ωt+k x) u (x, t) = C + ρc ei(ωt k+x) C ρc ei(ωt+k x). (18) The wave number in the positive respective negative direction is given by k + = ω c(1 + M), k ω = c(1 M). (19) If the mean flow being zero the wave number in both directions are reduced to k = ω/c. Acoustic analysis is commonly performed in the frequency domain, the frequency domain solution to the wave equation is obtained by a Fourier transformation which yields ˆp(x) = ˆp + e ik+x + ˆp e ik x û(x) = 1 (ˆp+ e ik+x ˆp e ik x). ρc (20) Where ˆp + and ˆp are the complex amplitude of the waves propagating in the positive and negative direction respectively Impedance and Reflection Coefficient Since the acoustic theory described here is linear the ratio between the acoustic pressure and acoustic velocity at any point will be independent of the sources. They will be related by the specific impedance which in 1-D is defined as Z = ˆp(x, ω) û(x, ω). (21) 15

26 For a plane wave in the direction of propagation, the specific impedance equals the characteristic impedance (z = ρc). Impedance can also be used to describe transmissions and reflections at a given section. The sign of the impedance depends if one are looking in the upstream or downstream direction and careful use is required. A more intuitive way to describe wave reflections is by a reflection coefficient defined as the ratio of the reflected wave amplitude to the incident wave amplitude. For an upstream respective downstream boundary in 1-D, the reflection coefficient relates to impedance as R upstream = Z + ρc Z ρc, R downstream = Z ρc Z + ρc. (22) Where Z is a prescribed impedance at the boundary. Three important cases can be identified from equation 22. For an acoustically hard wall (u = 0) which results in R=1 and for a soft wall such as an open pipe (p = 0) and hence R=-1. For a non-reflecting boundary, impedance matching is required (Z = ρc) and the refection coefficient is R= Damping of Acoustic Waves Even though gas turbine combustors in general are lightly damped, acoustic damping due to viscous effects in the flow will be important at higher frequencies. For low frequencies the viscous effects are small and may be neglected. Acoustic energy dissipation in gas turbine combustors due to viscous effects can be divided in three categories, [7]. 1, Acoustic energy dissipation in boundary layers which becomes important for narrow tubes and pipes. 2, Flow induced damping which refers to the dissipation of acoustics energy in regions with strong vorticity generation such as area discontinuities. Dissipation occurs due to acoustic energy is converted to turbulence within the vortexes. 3, Acoustic energy dissipation in the free field, this damping is in general small compared to the other two damping mechanisms. Damping of a propagating acoustic waves is normally included in models by a complex wave number which accounts for the damping. Different corrections for damping can be found in the literature, one way of including the damping in boundary layers can be found in [1]. Low frequencies are the topic of this work and damping effects due to viscous effects will be neglected The Heat Release Source Term When a fluctuating heat source is present the isentropic assumption used in the derivation of the wave equation without sources is not longer valid. This imply the second term in equation 11 do not longer vanish and a new relation between acoustic pressure and density need to be established. The second term in equation 11 can be rewritten by using the general gas law and the definition of entropy. The equation of state can be expressed as, [25], 16

27 p ρ = c2 + (γ 1) q t t t. (23) The heat release is assumed to consist of a steady and fluctuating part as q(x, t) = q + q (x, t). Linearization of equation 23 gives the relation between acoustic pressure and density when an unsteady heat release source is present. The result is p t = c2 ρ t + (γ 1) q t. (24) This result is used to eliminate the acoustic density in equation 9 and 10. The acoustic conservation equations in 1-D become p t + ρc2 u x = (γ 1) q t, (25) ρ u t + p = 0. (26) x The wave equation is then derived in the same way as before, by taking the time derivative of equation 25, the spatial derivative of equation 26 and subtracting the two. The result is 2 p t 2 c2 2 p x 2 q = (γ 1) t. (27) Comparing the result with the wave equation without any sources (equation 15), the source term representing unsteady heat release is found to be (γ 1) q t. 2.5 Acoustic Network Modelling For duct like systems such as gas turbine combustors, only plane waves can exist at sufficiently low frequencies when the wave length is much longer than the geometrical dimensions of the cross-sections. If only plane waves exist and the effect of coupled wall vibrations is negligible the system can be described by so called acoustic two-port theory. Acoustic two-port theory is based on 1-dimensional acoustics for which the acoustic field is fully determined by two field variables. Different formalisms is used for two-ports in the literature but the principle is the same. In this work the acoustic field is described by the acoustic pressure p and the acoustic velocity u. A two-port element relates the acoustic state at the inlet to the outlet by either analytical expressions or measured transfer functions. In figure 9, a two-port element with inlet (a) and outlet (b) is illustrated. The relation between the acoustic states, up- and downstream of the element is described by a 2x2 transfer matrix (T) as given in equation 28. The strength in the two-port theory is that complex systems can be modelled as a network or cascade of two-port elements. 17

28 Figure 9: Two-port element. The inlet (a) is related to the outlet (b) by the transfer matrix T. ) ( ) ) (ˆpb T11 T 12 (ˆpa = û b T 21 T 22 û a (28) The relation of the inlet to the outlet of a cascade of elements on the form as given in equation 28 can be obtained as a total transfer matrix as N T tot = T n, (29) n=1 where N is the total number of elements. The two-port theory can be extended into a multi-port formulation which can be used for modeling two-dimensional waves. E.g. 2-D multi-port elements for wave propagation in annular cavities are described by [25]. The two most basic and used two-port elements in the modelling of gas turbine combustors are the straight duct and area discontinuity elements, the derivation of the analytical transfer functions is given below The Straight Duct Element The transfer matrix for a straight duct element is derived from the 1-D solution to the convective wave equation. A duct oriented along the x-direction with length L as shown in figure 10 is considered. Figure 10: The straight duct element. 18

29 The acoustic field inside the duct constitutes a left and a right travelling wave which can be described as ˆp(x) = ˆp + e ik+x + ˆp e ik x û(x) = 1 (ˆp+ e ik+x ˆp e ik x). ρc (30) The acoustic state at the inlet respective outlet are found by evaluating equation 30 at x = 0 and x = L. This gives ˆp a = ˆp(x = 0) = ˆp + + ˆp û a = û(x = 0) = 1 ρc (ˆp + ˆp ), (31) ˆp b = ˆp(x = L) = ˆp + e ik+l + ˆp e ik L û b = û(x = L) = 1 (ˆp+ e ik+l ˆp e ik L). ρc (32) Putting these equations together and eliminating ˆp + and ˆp gives two equations relating the inlet state to the outlet state. Formulation in the form of a transfer matrix yields T = 1 2 ( e ik +L + e ik L ρc ( e ik+l e ik L) ) ( 1 ρc e ik +L e ik L) e ik+l + e ik L. (33) The Area Discontinuity Element The most simple way to model an area discontinuity is by using continuity relations. This means the pressure must be the same on both sides of the discontinuity. Correspondingly the oscillating mass flow must be conserved which gives the relation for acoustic velocity. For the linearized case the unsteady part of the mass flow between the up- and downstream sections can be written as ρ 1 A 1 u 1 = ρ 2 A 2 u 2. Formulating this in the form of a transfer matrix yields T = ( A 1ρ 1 A 2ρ 2. ) (34) Thermoacoustic Stability Analysis To analyse thermoacoustic stability of a system, the transfer matrix modelling approach is frequently used. As before, the acoustic quantities are assumed to have a harmonic time dependence, that is p (t) = ˆp(x)e iωt, u (t) = û(x)e iωt. (35) 19

30 For stability analysis the angular frequency ω is allowed to be complex on the form ω = Re(ω) + iim(ω). (36) From the assumed time dependence e iωt it is seen that the real part of the angular frequency represents the oscillation while the imaginary part determines the rate of growth in time. If the imaginary part is positive the amplitude of the acoustic quantities will decay in time and the oscillation is damped. In contrary, if the imaginary part is negative the amplitude will grow in time and the system may become unstable. Consider a system consisting of 3 elements and with known impedance at the inlet and outlet as outlined in figure 11. The system has 4 nodes with an acoustic state described by ˆp i and û i at each node, hence 8 degrees of freedom. By using the definition of impedance and the transfer matrix of respective element, 8 equations can be formulated and assembled into a global system matrix as shown in equation 37. Figure 11: A simple network model for stability analysis. 1 Z in T (1) 11 T (1) T (1) 21 T (1) T (2) 11 T (2) T (2) 21 T (2) T (3) 11 T (3) T (3) 21 T (3) Z out ˆp 1 0 û 1 0 ˆp 2 0 û 2 ˆp = û 3 0 ˆp 4 0 û 4 0 (37) In equation 37, the transfer matrix elements are functions of complex frequency. By using a complex eigenvalue solver the eigenfrequencies of the system can be determined numerically. For a system without any sources or damping the imaginary part of the eigenvalues will be zero. If some small amount of damping is present but still no source, any oscillations will decay in time and the imaginary part of the eigenfrequency will be positive. However, by introducing a flame described by e.g. a time 20

31 lag model, an abrupt change of phase across the flame is introduced which constitutes a feedback loop and unstable modes may appear. Furthermore, the corresponding eigenvectors represents the mode shapes which can be used for further investigation of a systems dynamic behaviour. 2.6 Flame Models Flames in network models are commonly represented by analytical expressions or measured transfer functions. A large number of different flame models can be found in the literature. Within this work, an analytical flame model is utilized. In recent years the rapid development of computational performance has open for studying flames in reactive CFD models which has become an important reference for tuning of analytical models, [1]. The most basic analytical models assumes the flame being acoustically compact. This assumption is generally good in the low frequency range where the acoustic wavelength is much longer than the flame region. For an acoustically compact flame the heat release distribution within the flame are less important and can be neglected. However, in a temporal perspective the flame cannot be assumed as compact as will be described later. Within a flame the mechanisms going on involves different time lags. This can be understood by the fact that a flame within a gas mixture propagates with a certain speed. Hence, a burning combustible mixture need some time to react. In addition, the flame in premixed combustors is located some distance downstream of where the fuel is injected and the fuel burnt in the flame at any instant of time was injected at an earlier time. Obviously, disturbances created upstream the burner will reach the flame after a certain time delay. The analytical models usually involve these time lags and it is well known from control theory that systems involving time lags are inherently unstable Acoustic Jump Conditions The so called jump conditions constitute the basis in many analytical flame models. The jump conditions relates the acoustic pressure and velocity downstream of a compact flame to the upstream state. For the derivation of the jump conditions, a thin flame with thickness x f and volume V f as outlined in figure 12 is considered, and 21

32 Figure 12: Schematic illustration of a thin flame the linearized acoustic conservation equations are integrated over the flame volume. Starting with acoustic momentum, equation 26 gives V f ) ˆ (ρ u t + p dv = A x x f ) ˆ (ρ u dx + A t dp = 0 (38) In the limit of an infinitely thin flame, x f approaches zero. Since the integrand in the first integral being finite this integral will vanish and the result obtained is that the acoustic pressure is unchanged over a thin flame. p downstream(t) = p upstream(t) (39) The same procedure is applied to equation 25, integration over the flame volume yields V f ( p t ) dv + V f ) (ρc 2 u dv = ((γ 1) q ) dv. (40) x V f In the limit of an infinitely thin flame the first integral vanishes and the equation simplifies to ˆ A du = V f ( ) (γ 1) ρc 2 q dv. (41) Since the total unsteady heat release rate is given by Q = V f q dv, the relation for acoustic velocity across the flame is obtained as u downstream(t) = u upstream(t) + 1 γ 1 Q A ρc 2 (t). (42) These results show the acoustic velocity features a jump across the flame in contrast to the acoustic pressure which stays the same. The compact flame acts as a an acoustic monopole source creating a 22

33 fluctuating volume flow. To complete the formulation of a flame model, an expression for the unsteady heat release rate ( Q (t)) is needed The n τ Model The classical way of modelling the unsteady heat rate release is the so called n τ formulation originally developed by L. Crocco in the 1950s for rocket engines. The n τ model can be expressed as, [23], 1 γ 1 Q A ρc 2 (t) = n u upstream(t τ). (43) The unsteady heat release rate is assumed to be related to the velocity fluctuations upstream of the thin flame. Furthermore, the heat release rate lags the velocity by the time lag τ and n is a proportionality constant (often referred to as the interaction index) determining the degree of coupling of the flame response to the velocity fluctuation. In the literature the model is often called the sensitive time lag model due to its characteristics being very sensitive to the time lag value. The frequency domain formulation of the n τ model is obtained by a Fourier transformation as 1 γ 1 ˆ Q A ρc 2 = n û upstream e iωτ. (44) In the above formulation, the acoustic velocity upstream of the flame is used as the reference for the heat release. However, in the literature it is possible to find formulations were the heat release is related to the acoustic velocity in some other location. How to estimate the time lag parameter will be addressed in succeeding sections. Combining equation 44 with the frequency domain version of equation 39 and 42 gives the acoustic jump condition for a thin flame to be ˆp downstream = ˆp upstream û downstream = û upstream (1 + n e iωτ ). (45) For stability analysis using complex eigenvalues, the above result is rewritten in the form of a transfer matrix. The transfer matrix for a thin flame following the n τ model reads T = ( ne iωτ ). (46) This is a passive flame model which means the acoustic state downstream of the flame does only depend on the upstream state without any active source inside the element. 23

34 2.6.3 An Extended Flame Model The classical n τ model assumes the heat release fluctuations being related to the acoustic velocity at a reference position. This is however not always sufficient, e.g. equivalence ratio fluctuations originating from the pressure and velocity fluctuations at the fuel nozzle location will also influence the flame response. In this section a more detailed flame model is given following a formulation developed by Siemens and which is further described in [14] and [15]. The total heat release rate for a combustion process is given by Q(t) = y f ρ V h f, (47) where y f is the mass fraction of fuel, V is the volume flow and hf is the specific reaction enthalpy. By a series expansion around the mean and linearization, the expression for the unsteady heat release rate is obtained as Q (t) = h f V yf ρ (t) + h f ρy f V (t) + h f ρ V y f (t). (48) The first two terms in equation 48 relates heat release to volume flow and density fluctuations just upstream of the flame while the last term including y f accounts for equivalence ratio fluctuations. A simple burner featuring a sudden expansion as depicted in figure 13 is now considered. Fuel is injected in the upstream part of the premixing section. Figure 13: Schematic illustration of burner with fuel injection in the premixing passage. The volume flow fluctuation (second term in equation 48) is assumed to be related to the acoustic velocity fluctuation at a reference position with a delay (τ b ). This reference position is commonly set to the burner exit plane and the time lag is given by the convective time from the burner outlet to the flame. The volume flow fluctuation can then be expressed as, [14], 24

35 V = u b(t τ b )A b, (49) where the subscript b denotes the burner outlet. The volume flow fluctuations characterized by the burner time lag is discussed by i.e. [19] and [23]. The underlying mechanism highlighted is the vortex created at the burner outlet by the induced velocity fluctuations. This vortex is convected with the mean flow through the flame region which makes the heat release to fluctuate. The approach to relate volume flow fluctuations to the velocity at the burner outlet plane has been confirmed successful by many studies on laboratory scale burners described by e.g. [5] and [19]. For gas turbine burners featuring turbulent and swirling flows, determination of the time lag is a challenge on its own. Several different approaches has been suggested in the literature to determine the time lag, i.e. [1] and [25] determined the burner time lag by fitting an analytical flame model to measured flame response functions. W. Krebs et.al. [14] suggests a CFD approach to determine the time lag. An expression for the fluctuating mass fraction of fuel y f can be obtained by studying the fuel mass flow through a fuel nozzle. Assuming incompressible flow, the mass flow through the nozzle is given by ṁ fuel = A nozzle 2ρ fuel (p fuelline p premix ). (50) where A nozzle is the effective area of the nozzle and p fuelline is the pressure of the fuel feed. The pressure in the premixing section is given by p premix and hence p nozzle = (p fuelline p premix ) constitutes the pressure difference over the nozzle. The mass fraction of fuel can then be expressed as y f = ṁ fuel ṁ air + ṁ fuel ṁfuel ṁ air = A nozzle 2ρfuel (p fuelline p premix ). (51) ρ premix u premix A premix Moreover, the fuel line feed pressure, p fuelline is assumed to be constant while the pressure, velocity and density in the premixing passage are acoustic quantities. By a series expansion around the mean and linearization, the fluctuating mass fraction of fuel can be expressed as, [14], y f (t) p premix = (t τ i) 2(p fuelline p premix ) u premix (t τ i) ρ premix (t τ i). (52) u premix ρ premix y f Hence, the fluctuating mass fraction of fuel at the flame is dependent on pressure, velocity and density fluctuations in the position of fuel injection. Disturbances generated are convected to the flame with the mean flow. Furthermore, the fluctuating mass fraction of fuel at the flame front will be delayed by the fuel time lag (τ i ). Upstream the flame, the isentropic relation holds and is now used to rewrite 25

36 the last term in equation 52 in terms of acoustic pressure instead of density. Putting it all together, the unsteady heat release rate is found to be Q p (t) =h f V flame (t) yf c 2 + h f ρ b y f u b(t τ b )A b flame p + h f ρ premix V yf ( premix (t τ i) 2(p fuelline p premix ) u premix (t τ i) p premix (t τ ) (53) i) u premix ρ premix c 2 premix It should be mentioned here that in the premixing section, the strength of the equivalence ratio fluctuations will reduce due to the mixing. This phenomena is not included in this modelling approach and the equivalence ratio fluctuations are assumed to have the same strength when reaching the flame as when they were produced Fuel Injection Time Lag τ i Fluctuations in equivalence ratio will be produced due fluctuations of the pressure drop over nonchoked nozzles which makes the fuel supply rate to oscillate. In addition, fluctuations of equivalence ratio will also be created due to fluctuations of the air supply. Referring to figure 13 and considering a pressure fluctuation at the flame. The events following due to this pressure fluctuation is illustrated in figure 14. Figure 14: Qualitative description of the contributing parts to the fuel injection time lag, from [18]. The time lag characterizing the unsteady heat release due to equivalence ratio fluctuations can be divided in 3 parts. First part, there will be a phase difference between the pressure oscillation at 26

37 the nozzle and the flame due to the distance. The time lag associated with this phase difference is denoted τ 1. In the low frequency region and with a distance between fuel nozzles and the flame being in the range of 100mm, this phase difference will be small and can in most cases be neglected. Second part, the equivalence ratio fluctuations generated at one instant will be convected by the mean flow reaching the flame at a later instant. The delay associated with this transport time is denoted τ conv. For simple burner configurations this τ conv can be estimated by the mean air velocity and the length between the fuel injector and the flame. However, burners used in gas turbines often comprises a swirler generating turbulence which complicates the estimation of the convective time lag. In such case, a detailed CFD analysis of the turbulent flow field can be used to determine the convective time lag as suggested by [24]. Third part, when the equivalence ratio fluctuations reaches the flame the heat release will occur some time later due to the delay related to the chemical reaction, here denoted τ chem. Moreover, the total time lag for equivalence ratio fluctuations is given by τ tot = τ 1 + τ conv + τ chem when the acoutic pressure at the flame is used as reference. Following the definition used in equation 53, the fuel time lag is referenced to the acoustic perturbations at the fuel injection position and hence the time lag will be given by τ i = τ conv + τ chem. At low frequencies τ chem /T will be small and the time lag associated with the chemical reaction can in most cases be neglected, [19] Distributed Flame Models The thin flame models described in preceding sections are based on the assumption of acoustic compactness. This assumption does not generally hold in practice. In the low frequency range the wavelengths will be much longer than the spatial distribution of the flame. Hence, in a spatial perspective the flame may be considered as compact and the compact flame approach holds, [25]. This is however not valid in a temporal perspective which can be illustrated as follows. It has been shown that the time lags involved in stability analysis are dominated by convective transport times. Typical mean flow velocities in gas turbine combustors are around 10 80m/s. Assuming the flame is distributed over an axial distance of 100mm and the mean flow velocity being 50m/s gives the convective time to pass the flame to be 2ms. At 200Hz the period of oscillation is 5ms and hence τ conv /T is not negligible and the flame cannot be assumed as compact in a temporal perspective. A more suitable model should therefore include a distribution of the heat release as depicted in figure 15. The time lag is here a function of x to describe the distribution within the flame. 27

38 Figure 15: Schematic illustration of burner with a distributed conical flame. Many different flame models can be found in the literature featuring distributed heat release and time lags. E.g. a flame model with distrubuted heat release is descibed by [29]. The local heat release rate is then expressed as q (x, t) = n(x) q u u (x ref, t τ(x)) (54) where τ(x) is the time lag distribution, x ref is the reference location for velocity perturbations and n(x) is an interaction index describing the spatial distribution of the heat release within the flame zone. For acoustic characterization of a flame, another common used approach is to study the flame transfer function (also denoted flame response function) defined as F (ω) = ˆ Q(ω)/ Q û ref (ω)/u ref. (55) The flame transfer function relates the unsteady heat release rate to the acoustic velocity fluctuations at a reference position. For easy comparisons, the reference position is normally selected at a location where the acoustic velocity is easily measured. The heat release is commonly measured by OH* chemiluminescence techniques. Measured flame response functions are commonly used to calibrate an analytical flame model. This has e.g. been done by [25] who tuned an analytical flame model in which a probability density function was used for description of the time lag distribution. Distributed flame models is not the topic of this work. 2.7 Boundary Conditions at Combustor In- and Outlet Determination of the up- and downstream acoustic boundary conditions are important for accurate modelling of thermoacoustics. Poinsot, [16], showed that even with a detailed model of the combustor, the result is still very dependent on the boundary conditions. However, these acoustic boundary conditions are not easily determined for a gas turbine combustor at operation conditions. A detailed 28

39 review of possible boundary conditions for acoustic eigenmode calculations can be found in, [16]. A gas turbine compressor is used to increase the static pressure. The stationary guide vane stages in an axial compressor therefore feature diverging flow channels to reduce velocity and increase static pressure. For an acoustic model of a gas turbine combustor the upstream boundary is commonly set to the outlet of the last compressor guide vane stage. In contrary, the flow out from the combustor flows through the turbine guide vane nozzles which feature converging flow channels in order to increase velocity. For acoustic modelling of a gas turbine combustor, the inlet to the first turbine guide vane stage is commonly specified as the downstream boundary. For low frequencies the acoustic boundaries can be approximated from the theory of converging and diverging compact nozzles. Under this assumption some analytical expressions for the acoustic boundary can be found in the literature. In the derivation of such expressions it is assumed that the characteristic length is small compared to the acoustic wavelength and hence geometrical details can be neglected. Consider the flow in the first turbine guide vanes at the combustor outlet to be choked converging nozzles. This chocked nozzle assumption implies acoustic waves can only travel in one direction through the nozzle and hence no acoustic waves from the turbine section can travel upstream to the combustion chamber. For a choked converging nozzle with the upstream Mach number (M 1 ) being low, the reflection coefficient can be expressed as, [16], R 1 = (γ 1)M (γ 1)M. (56) 1 Where the index 1 denotes the reflection coefficient for acoustic waves incident on the nozzle from upstream. It should be noted that for low upstream Mach numbers, M 1 0 and R 1, acoustically the boundary acts as a hard wall. Similarly, for a choked diverging nozzle the reflection coefficient can be expressed as a function of the downstream Mach number (M 2 ) as, [16], Here, index 2 R 2 = 1 γm 2 + (γ 1)M γm 2 + (γ 1)M2 2. (57) denotes the reflection coefficient for waves indent from downstream on the nozzle. Moreover, if the downstream Mach number being low (M 1 0), the diverging nozzle will also act an acoustically hard wall (R 1). For non-choked nozzles, acoustic waves can travel in both directions. Analytical expressions of the reflection coefficient for non-choked compact nozzles can be established for both diverging and converging geometries, [16]. However, those requires information of the acoustic boundary at the troat of the nozzle to be known. Analytical expressions for non-choked nozzles and suggestions for the troat 29

40 boundary conditon are given in [16]. Commonly for the compressor outlet vane row, constant flow velocity rate is assumed at the inlet to the guide vanes which means u = 0 and R vane inlet = 1. For turbine nozzles, constant pressure may be assumed at the outlet of the guide vanes and hence p = 0 and R vane outlet = 1. For higher frequencies the detailed geometry of the inlet and outlet vanes cannot longer be neglected. The reflection coefficient can then be solved for in a quasi-one-dimensional manner by linearizing the Euler equations. The result will be a frequency dependent boundary conditon. This approach is further described by Poinsot, [16]. Another extensive study of the downstream boundary condition using LES can be found in [8]. Some authors have also tried measuring the reflection coefficients for the inlet respective outlet, i.e. [27] measured the reflection coefficient at the combustor inlet and outlet for a model gas turbine. 30

41 3 Method and Numerical Tool 3.1 Network Modelling Tool In the remainder of the report, a generalized network code developed by Siemens AG is used to study thermoacoustic stability as outlined in the theory section. The network code solves stationary flow, kinetics, heat transfer, thermodynamics and acoustics in a coupled manner. This means that an acoustic model must also include relevant flow features and fuel spices to capture heat release, temperatures etc. which are of importance for thermoacoustics. A built in library with predefined elements is available and the code features a graphical interface through Simulink in which the elements and interconnection of elements are defined. MATLAB is used as the primary solver for the underlying equations and the open source package Cantera has been integrated for solving kinetics and chemical reactions. Two types of acoustic analysis is available, these are forced response analysis and complex eigenfrequency analysis. The thermoacoustic module has previously been validated by Siemens AG against an academic burner with successful results. 3.2 Sample Cases Several sample cases were created and investigated to learn the code and understand the behaviour of the available acoustic elements. Some of those sample cases are described in this section Eigenfrequencies for a Simple Duct As a first study, the eigenfrequencies for a simple duct with hard walls at both ends were studied and the results were compared to well known analytical expressions. The available complex eigenvalue solver was used and the network model for the simple duct is shown in figure 16. The working procedure for the solver is as follows. First, the flow and chemical reactions are solved and when finished the acoustic solver is initiated. Input parameters needed for the acoustic study such as speed o sound, Mach number etc. for each node in the model are automatically imported from the flow simulation. Figure 16: Network model of a simple duct. Air was used as the medium and a parameter study was performed with different mass flows to capture the effect of a mean flow on the acoustics. The analysis was done for a duct with length 1m and a total temperature of the inlet air being 293K. The eigenfrequencies determined by the network code as a function of the Mach number is shown in figure 17 along with the analytical eigenfrequency for the n:th mode given by 31

42 f n = nc 2L (1 M 2 ). (58) The results from the complex eigenvalue solver correspond well to the analytically obtained eigenfrequencies. Slight differences are seen at higher Mach numbers which is a result of the coupled modelling approach utilized in the network code. The analytic eigenfrequencies are here calculated by assuming a constant speed of sound while for the network model, total conditions are specified and the static conditions and thereby the speed of sound is recalculated depending on the mass flow rate. imaginary part of the eigenfrequencies was found to be almost zero as it should for a system without sources and damping. The Figure 17: Eigenfrequencies for a simple duct as a function of the mean flow Wave Transmission through an Expansion Chamber In the next study, wave transmission through an expansion chamber was studied using the forced response analysis option. Within the code, the forced response analysis option splits the network at a selected node for excitation. At this node a predefined acoustic velocity (û) is then enforced and the acoustics is solved for each frequency in a specified range. The dimensions of the expansion chamber used for this study is shown in figure 18. Air with a temperature of 293K and without any mean flow was used as medium. The upstream boundary was specified as the excitation node and the downstream boundary was specified as a reflection free termination. The network model is shown in figure

43 Figure 18: Illustration with dimensions of the expansion chamber. Figure 19: Network model of the expansion chamber. The transmission of waves incident from upstream was evaluated as the ratio of the incident pressure amplitude (ˆp i ) to the transmitted amplitude (ˆp t ). The results was then compared to an analytical expression found in [3]. Due to the modelling approach utilized for the forced response analysis, decomposition of the acoustic field upstream of the expansion chamber was required to distinguish the incident from the reflected wave. For decomposition the measurement method known as the two microphone method was applied to the numerical results, the method is further described in Appendix A. The results from the network model was found to correspond well to the analytic results as shown in figure

44 Figure 20: Incident to transmitted amplitude ratio for the expansion chamber Rijke Tube As a first model including a flame, a network model of a simple Rijke tube was created and is shown in figure 21. Up- and downstream boundary conditions was set to R = 1, i.e. open to atmosphere and constant pressure. Within the network code, the acoustic flame model is merged into the available reactor elements. A combustible air-methane mixture with temperature 300K, equivalence ratio Φ = 0.67 and a mass flow of 1g/s was specified at the inlet. This mass flow gives a maximum Mach number of 0.02 in the tube and the mean flow effects will be negligible. The flame divides the tube into a low temperature and a high temperature region. The temperature upstream of the flame is given by the temperature of the inlet mixture while the temperature downstream is calculated by the reactor for the given combustible mixture. Figure 21: Model for thermoacoustic analysis of a Rijke tube. Several types of chemical reactors are available in the network element library and the choice of reactor is strongly connected to the purpose of the model. For thermoacoustic analysis, all the reactor elements use the same thin flame model including volume flow fluctuations and eqvivlence ratio fluc- 34

45 tuation and which is futher described in [15]. The coupling between the chemical reaction and the acoustics is the heat release in the flame. An equilibrium reactor assumes infinite reaction time inside the reactor and the temperature downstream will then be the adiabatic flame temperature. The equilibrium reactor was selected to be used through out this work. For the Rijke tube, no equivalence ratio fluctuations were included as a homogeneous air-fuel mixture was specified at the inlet. The flame model will therefore have the form of a simple time lag model. Results from the complex eigenfrequency solver is shown as a stability plot in figure 22. Positive values indicates unstable modes that may grow in time. Figure 22: Thermoacoustic stability plot for the Rijke tube. Complex eigenfrequencies were calculated for three cases. First case with only a tube with two different temperature regions and no flame (i.e. no jump in acoustic velocity at the interface of the two temperature regions). The second and the third case comprise a flame with a time lag being τ = 0 and τ = 0.1ms. Furthermore, the Rijke tube has no sudden expansion as in the derivation of the flame model and volume flow fluctuations were related to the immediate upstream side of the flame. Interpretation of the time lag for a Rijke tube where the flow could be assumed to be close to laminar is therefore a bit tricky. Since no obstacles generating disturbances or such thing, the convective time lag assumption cannot be applied and the time lag will be more connected to the chemical reaction. Anyhow, the actual value of the time lag was not analysed in depth for the Rijke tube but will be further investigated in succeeding sections. It was the behaviour of the flame that was of interest in this study. Conclusions Zero time lag (τ = 0) implies an acoustic velocity jump at the flame but no change of phase. This does not create any unstable modes but a significant change of the eigenfrequencies are seen compared to the no flame case (no velocity jump). The reason to the change in eigenfrequencies 35

46 is due to the changed impedance at the interface between the hot and cold temperature regions. The system will still be stable due to if no time delay the acoustic pressure and unsteady heat release will be 90 degrees out-of-phase and the Rayleigh s integral will render zero. Introducing a time delay will make some modes to become unstable and some to be damped depending on the phase between the acoustic pressure and the unsteady heat release. For this Rijke tube configuration, unstable modes are predicted at around 190Hz and 680Hz. Positioning the flame in the downstream half of the tube was found to not create any unstable modes as expected (stability plot for this study is not shown here) Simple Burner Featuring a Sudden Expansion As a next sample study, a network model of a simple burner featuring a sudden expansion with area ratio=2 (A 2 /A 1 ) was created. The model with the baseline dimensions and input parameters is shown in figure 23. Dimensions were arbitrarily selected without any further reference. Acoustically the inlet boundary was specified as a hard wall while outlet was assumed to be an open end. As for the Rijke tube an equilibrium reactor was used to include the flame. As a first case, a perfectly premixed combustible air-methane mixture with equivalence ratio Φ = 0.67 and a mass flow of 0.1kg/s was specified at the inlet. The mass flow of 0.1kg/s gives a maximum Mach number of 0.17 at the outlet. Figure 23: Sample burner model. Baseline case. The heat release fluctuations at the flame are assumed to be related to volume flow fluctuations at the burner exit plane only (as introduced in equation 49). The burner time lag (τ b ) is dominated by the convective time a fluid particle need to travel from the sudden expansion to the flame. It is obliviously crucial to determine the position of the flame in order to determine the burner time lag. A parameter study was performed varying the input parameters and geometrical dimensions up and 36

47 down from the baseline case. Only one parameter was varied at a time. Stability plots for different input parameters and geometrical dimensions are shown in figure 24. The stability measure on the y-axis is here presented as the growth rate defined as ( Growth rate = exp 2π Im(f) ) 1. (59) Re(f) The growth rate is a measure of how much the amplitude is changed over one cycle. For a positive growth rate the oscillation grows and for a negative growth rate the oscillation is damped. A growth rate of unity means the amplitude will increase by 100% (doubling) over one period of oscillation. For a high growth rate it is easily understood the amplitude may grow very fast and give rise to instabilities. However, since this is a linear code the growth rate cannot be used to determined the final limit-cycle amplitude which is controlled by non-linear phenomena. 37

48 Figure 24: Stability plots. Sensitivity study to different input parameters and geometrical dimensions. Conclusions Geometrical changes such as varying L1 and L3 will change the acoustic response of the system and the geometry itself is hence strongly connected to stability. Making L1 shorter seems to be better for this configuration and it is possible to find a value of L1 for which no unstable modes at all is predicted. For the second mode, the length of L1 has a significant influence on the frequency as well. Also increasing the length of the outlet pipe (L3) is predicted to act 38

49 stabilizing for this burner configuration. The results shown here indicate very high sensitivity to geometrical changes. However, it should be said that the geometrical changes investigated here have been quite dramatic compared to the size of the model. The position of the flame (L2) has a minor influence on stability given a fixed value of the time lag. This follows from the frequently good assumption of the flame being acoustically compact in a spatial perspective. At higher frequency it can be seen some change in frequencies due to the change of flame position is larger with respect to the wavelength. Equivalence ratio influences the growth rate slightly but not so much the frequencies. This is a bit counter intuitive since a higher equivalence ratio (still below 1) will give a higher downstream temperature and hence the eigenfrequencies should go up. However, the unsteady heat release will also go up and those phenomena together will determine the response. The time lag and hence the phase change of velocity across the flame has a large influence on the frequencies but not so much the growth rate. Hence, the flow velocity controls the convective time lag and will be very important even though the influence on the acoustic field is small for low Mach numbers. The burner was now modified to include fuel injection nozzles to study stability due to equivalence ratio fluctuations. In the flame model implemented in the network code, the expression for equivalence ratio fluctuations introduced in equation 52 is slightly rewritten in terms of fuel line impedance, this is futher described in [15]. The fuel line impedance characterizes the fuel line and is defined as the ratio of acoustic pressure to acoustic velocity at the outlet of the fuel injection holes. Since the rate of fuel injection is determined by the velocity through the nozzles, a high fuel line impedance value means the velocity fluctuations will be small and the rate of fuel injected becomes more or less constant. Anyway, with high fuel line impedance values implying a constant fuel supply, equivalence ratio fluctuations will still be present due to fluctuations of the air in the pre-mixer. Two fuel injection locations were specified as outlined in figure 25. In the network code, this is done in a merger element where the air and fuel are mixed. Within the merger element it has to be specified that equivalence ratio fluctuations are to be included in the thermoacoustic study. This makes the acoustic perturbations in the location of fuel injection to be used as reference for equivalence ratio fluctuations. The same amount of fuel as in before was specified in order to give the same overall equivalence ratio at the flame and hence the same downstream temperature. Within this study a very high value of the fuel line impedance was used which was justified by a large area difference between the fuel injection nozzles and the burner premixing section. The influence due to volume flow fluctuations at the burner exit plane was excluded in this study in order to investigate the influence of equivalence ratio fluctuations only. 39

50 Figure 25: Sample burner model, with two locations for fuel injection added. As a first case, the importance of the location of the fuel injector was studied by letting all the fuel to be injected through the ṁ fuel,1 port. Stability plots for different injector locations as well as different fuel time lags are shown in figure 26. For equivalence ratio fluctuations the fuel time lag (τ i ) is dominated by the time required for a fluid particle to travel from the fuel injection location to the flame. The results confirm the location of the injector has a small influence on the stability for a fixed fuel time lag. This is due to the long wavelength and hence the phase difference between the different injection locations is very small. The value of the fuel time lag on the other hand has a large influence on both stability and frequencies which is in line with previously observed results for volume flow fluctuations (controlled by the burner time lag). Figure 26: Stability plots. Influence due to equivalence ratio fluctuations. The fuel was now split equally between the two fuel injection locations as shown in figure 25 in order to investigate the interaction between two fuel injectors. The fuel time lag for the injector at location 1 was kept constant while the fuel time lag for the injector in location 2 was varied. 40

51 Figure 27: Stability plot. Interaction of two fuel injection positions. Its clearly seen for the first mode that increasing the fuel time lag for one of the fuel nozzles is predicted to act stabilizing. This is however only true to a certain limit which can be understood from the period of oscillation. The criteria for instability is given by the Rayleigh s integral and includes the phase of the pressure and the unsteady heat release. More in detail, the integral will change sign if the time lag is lager than T/2 (but still less than T ). It is often useful to think of time lags in relation to the period of oscillation in order to get an idea of the expected influence to a certain change of time lag. For this case, the first mode has a period of oscillation of T = 5ms (1/200). Hence, a time lag difference of 2ms as is the maximum change investigated here is significant and a large impact is expected on stability. For the higher mode at around 400Hz the period of oscillation is 2.5ms and half of that is 1.25ms. Hence a time lag difference of 2ms are expected to have a large impact on stability but may also flip the results around totally. For this configuration τ i,2 = 4ms seems to be better for stability than τ i,2 = 5ms. Since there are several different terms included in the full flame model, the response to a certain change of time lag is anyhow difficult to predict which is why the model becomes very useful. However, the results suggest that a careful selection of the time lags could be an efficient way to improve stability. In this section, some sample cases have been studied and the results have been discussed. It has been shown that even for simple models the stability behaviour becomes complicated and difficult to predict before hand. Parameter studies was found to be very useful to investigate trends even though too many cases may be more confusing than helpful. In succeeding sections, the network modelling approach will be applied to a commercial gas turbine combustor. 41

52 4 Application to the SGT-750 The Siemens SGT-750 has a combustion system consisting of eight identical cans. At present, the combustion system provides excellent combustion stability and performance. However, a deeper knowledge in its thermoacoustic response will be useful for possible future upgrades and further reduction of emission levels. The network modelling approach as outlined in the previous sections was therefore applied to the SGT-750 and the results were compared to measurement data. Considering low frequencies and a single can, no transverse modes can exist due to the relatively small cross-sections. With the largest diameter, the cut-on frequency for the first transverse mode is of the order of 1000Hz for the SGT-750 combustor at operating conditions. In a full engine though, longitudinal modes are not the only ones that can exist at low frequencies. Exhaust gases from the combustion chamber are led through a transition duct to the turbine inlet. Due to thermal expansion reasons the transition duct is not tightly sealed to the turbine guide vanes and a gap between the transition duct and the guide vanes is required. This gap provides a connection for the different cans to acoustically interact at the hot side. Additionally, the air feed from the compressor diffuser to each can is through a common casing which creates the possibility for the cans to acoustically interact at the cold side as well. By this said, the conclusion is that there is a possibility of can-to-can interaction which may be present at low frequencies together with longitudinal oscillations. The thermoacoustic stability study for the SGT-750 was started with an investigation of available measurement data after which the network modelling approach was applied and the results compared. 4.1 Evaluation of Measurement Data Available measurement data from a prototype engine test was analysed. For the SGT-750, combustion dynamic levels are measured in the upstream part of each burner using transducers with high temperature resistance. Two different full load operation conditions were investigated, SGT-750 standard operation conditions and conditions when instability was provoked. The measurement data was available as time signals which were processed using the MATLAB signal processing package. Auto power spectrum for the two conditions are shown in figure 28, the frequency axis has been normalized and the same normalization is used from now on for easy comparisons. As expected, the dynamic levels are much higher for all cans when instability is provoked even though there are quite large differences between the individual cans. It can be observed the dynamic response is dominated by two distinct frequencies for both operation conditions. Those two peaks are from now on referred to as the first respective second peak where the first occurs at a lower frequency than the second. 42

53 Figure 28: Auto power spectrum. Left figure: Standard operation. Right figure: Provoked instability. To investigate if the signals from the individual cans were correlated, the coherence measure was calculated. The coherence between two signals x and y is defined as C xy = P xyp xy P xx P yy, (60) where P xx and P yy is the auto spectrum of the respective signal and P xy the cross spectrum between the two. Coherence was calculated with each of the cans as the reference for the provoked instability condition. As an example, the coherence with burner/can no.5 as the reference is shown to the left in figure 29. Figure 29: Left figure: Coherence for each can with can no.5 as reference. Right figure: Transfer function phase between each can and can no.5. In general, low coherence values were achieved which indicate the signals are not coherent or high 43

54 levels of noise. However, at the frequency of the first peak seen in the auto power spectrum, the coherence shows a peak as well indicating it could be some correlation. For each can as reference, the distinct peak in coherence was observed for the other cans except two. In figure 29, with can no.5 as reference it is seen the coherence is low for can no.3 and no.7 at the frequency of the first peak. At the frequency of the second peak though, the coherence is about zero and hence no correlation at all between the individual cans. Furthermore, if two signals are coherent it is of interest to determine the phase difference between them. The phase difference between each can and the reference was investigated by studying the phase of the cross spectrum since it is the same as the transfer function which can be estimated as H xy = P xy P xx. (61) The phase of the transfer function between each can and can no.5 is shown to the right in figure 29 for the interesting frequency range. A rectangular window was used in the data processing in order to retain the phase as accurate as possible and enough data was available to perform a large number of averages. Referring to figure 29, the phase shown is the phase difference to the reference can no.5. The phase for can no.3 and no.7 should not be considered since the coherence indicated those to be completely uncorrelated to the reference. For the other cans, it can be seen that the phase difference for can no.4 and no.6 is close to zero indicating they are in-phase. The remaining cans has a phase difference of about ±π which indicates they are completely out-of-phase to the reference. This means the reference can is in-phase with the closest cans while being out-of-phase with the cans on the opposite side of the engine. The two cans with low coherence was found to form a line in between the in-phase and out-of-phase side. That explains the low coherence which is probably due to a zero crossing and thereby low signal levels. Similar results where obtained independent on the selection of reference can. The results are visualized in figure

55 Figure 30: Phase relations at the frequency of the first peak. The results show the first peak seems to a global mode including all of the cans. This global mode has also been found in previous FEM-calculations of the full SGT-750 combustion system. By this finding, it is apparent that this instability mode cannot be predicted if only longitudinal modes in one combustor can are studied. Conclusions There are two distinct frequencies that dominate the thermoacoustic response at low frequencies. Coherence indicates the first peak (lower frequency) most likely is coherent between the cans and hence characterized by a global mode including all of the cans. Further study of the transfer function phase showed this probably is a mode with nodal diameter 1 where each can is in-phase with the two closest cans and out-of-phase with the three cans on the opposite side of the engine. Low coherence at the frequency of the second peak indicates no correlation between cans. This frequency is due to a local phenomena in the individual cans. 4.2 Stability Analysis by the Network Modelling Approach Establishing the Acoustic Network Model An acoustic network model of a 45 degree sector was developed to be used for stability analysis of the second peak. The model comprises one of the identical cans and the compressor diffuser and central 45

56 casing featuring annular cavities was divided accordingly. The objective of the model was to capture the low frequency behaviour and the modelling strategy was therefore to: Capture lengths and over all volumes. Capture effective area and flow velocities in the fuel injection passages (velocities in premixing section are important in the study of equivalence ratio fluctuations) The geometry of the combustor system was divided into duct, area discontinuity and variable area section elements. Dead volumes of significant size was modelled as side branches to capture quarter wave resonator effects. Some significant assumptions had to be made for modelling of the common casing since the geometry was difficult to translate into a 1-D model. The main focus for the casing was to capture the volume and cross-sectional area. This since the casing volume is significantly larger than for the swirlers and will be important for correct acoustic reflections at the interfaces. The split into different elements is shown in figure 31. Figure 31: Schematic illustration of the network model. Division into acoustic network elements. The SGT-750 burner comprises two radial swirlers in where the main fuel is introduced. Due to low Mach numbers, the radial swirlers were assumed to be acoustically transparent meaning any flow effects generated by the swirlers were neglected in the acoustic model. Each of the passages in the swirler were modelled by duct elements with specified effective area in order to get the air split between the two swirlers correct. The final version of the network model is shown in figure 32. Sub-models for the swirlers and flame region were created which will be further described in succeeding sections. 46

57 Figure 32: The acoustic network model. The geometrical dimension of the model were specified directly in each element while the inlet parameters such as compressor mass flow, inlet pressure and temperature were specified in an excel sheet for easy adjustments to different operation conditions. The total mass flow from the compressor was divided by eight to get the mass flow through one of the cans. Cooling air consumed by downstream parts of the combustor, mainly the transition duct and the interface between combustor and turbine was extracted from the casing. This was required to get the correct amount of air in the combustion zone and since a coupled solver, the amount of air in the combustion zone together with the amount of fuel will determine the temperature in the flame. Boundaries at both the inlet and the outlet were specified as acoustically hard walls for now. This assumption will be further investigated in the next section. Referring to figure 31, a stream wise coordinate from the combustor inlet to the outlet of the model was defined. This stream wise coordinate is used for plots of acoustic mode shapes in succeeding sections. The scale of the stream wise coordinate is element wise and hence not actual lengths which should be kept in mind. Anyway, this representation of the mode shape is very useful to understand the systems behaviour. Following Rayleigh, the location of the flame within a mode is of high interest and the location of the flame is therefore indicated in the mode shape plots. The eigenfrequencies and corresponding mode shapes from the network model were compared against available FEM calculations for the case with a uniform temperature and pressure corresponding to the compressor discharge air at full load. The mode shapes from the network model are shown in figure 33 and the longitudinal mode shapes from FEM calculations are shown in figure 34. It can be seen the mode shapes corresponds well. However, the frequencies was found to differ by up to 10%. 47

58 Figure 33: Acoustic mode shapes from the network model. Figure 34: Acoustic mode shapes from FEM calculations The Influence of a Mean Flow on the Acoustics The presence of a mean flow makes the speed of acoustic wave propagation in the upstream respective downstream direction to be different. For gas turbine combustors in general, the Mach numbers are low due to the high temperatures and hence high sound speed. The maximum Mach number within the SGT-750 combustor is about 0.2. Higher Mach number values are obtained outside the acoustic domain in the turbine guide vanes which constitutes the acoustic boundary at the combustor outlet. The influence of the mean flow on the acoustics was investigated by comparing the eigenfrequencies obtained by the network model with and without a mean flow. The resulting eigenfrequencies is presented in table 1. Full load conditions No mean flow Table 1: Normalized acoustic eigenfrequenices with and without a mean flow. As can be seen the differences in eigenfrequencies due to the mean flow was found to be very small. 48

59 A slight difference is seen for higher frequencies while the lower is barely changed. The conclusion is that a correct mean flow is not crucial for the acoustic analysis. Anyway, the mass flow and flow velocities will be of importance later on when convective time lags are to be estimated Introducing the Flame - Perfectly Premixed Case In this section, the flame is introduced and the stability due to volume flow fluctuations at the flame is studied. No equivalence ratio fluctuations are included yet. Results from a previously performed CFD calculation was used to estimate the position of the flame. The actual flame has a conical shape with significant axial distribution while an infinitely thin flame is assumed in the flame model available. To tackle this mismatch, the axial position that best represents the heat release was estimated to be in the middle of the actual flame. This position was found to be just before the combustion chamber expansion section. The thin flame together with the actual conical flame are illustrated in 35. Figure 35: Schematic illustration of the network model. Including the flame. The flame was introduced as an equilibrium reactor element further described in the study of the Rijke tube. The sub-model for the flame region is shown in figure 36. Figure 36: Network sub-model of the flame region. 49