JORDI DILMÉ. Supervisor: Department JUNE 20111

Transcription

1 Modelling of Thermoacoustic Instabilities with a Graphical Interface (Simulink) JORDI DILMÉ Supervisor: Dr Aimeee S. Morgans IMPERIALL COLLEGEE LONDON Report submitted to the Department of Aeronautics, Imperial College London for thee Master ss Thesis JUNE 2111

2 To my mother, because she is always with me. To my father and to Toni, because they are always with me, too.

3 ABSTRACT Combustion units frequently experience thermoacoustic instabilities, also known as combustion oscillations, which are consequence of the internal coupling between acoustic waves and unsteady heat release. Deterioration in system performance, starting by increased emissions or higher levels of fuel consumption, may occur due to this largeamplitude flow oscillations and their associated pressure fluctuations and, under some circumstances, these can be intense enough to cause structural damage on the installation. After introducing its physical background, this acoustic phenomenon is modelled in the time domain using a one dimensional linearized analytical approach and then implemented into Simulink; thus providing a pioneering tool which models combustion instabilities in an interactive and customizable environment. A first model reproduces the behaviour of an acoustically excited pipe without combustion, whereas a second one simulates the performance of a generic combustor with unsteady heat release without mean flow. The validity of the conceived models is checked through comparison with acoustics theory and earlier research developed in the Laplace domain. This graphical modelling aspires to become a faster, more visual alternative to the complex current approaches to the analysis of combustion oscillations, especially suitable for shorter projects thanks to its simplicity of use. As an initial approach to the interruption of combustion oscillations, feedback control is applied to the modelled combustor. A fixed parameter controller is designed in the time domain using Nyquist and Bode techniques and then implanted into the Simulink model. Finally, the robustness of the controller to slight changes in the heat release time delay is assessed. I

4 ACKNOWLEDGEMENTS I would like to express my sincere thanks my supervisor, Dr Aimee S. Morgans, for her attention and advice throughout the course of this Master s project. Her guidance was crucial and made possible the eventual success of this work. I would like to extend my gratitude to German Gambon and Damián Álvarez, who provided priceless help during the first stages of the project to get familiar with the software involved. Finally, I am very grateful to my mother, my family and my friends. Without their encouragement and love I would not have overcome this challenge. JORDI DILMÉ London, June 211 II

5 Contents ABSTRACT.... I ACKNOWLEDGEMENTS. II 1. Introduction Aim of the project Energy and combustion oscillations Physical fundamentals Acoustic analysis Acoustically excited tube without combustion General description of the model and reason for modelling Description of the numerical model Pressure modelling Results and model checking Model combustor General description of the model and reason for modelling Description of the numerical model Pressure modelling Results and model checking 2 6. Practical approach to feedback control Fixed parameter control applied to the combustor modelled with Simulink Controller design in Laplace domain Practical implementation of the controller into the Simulink model Analysis of control robustness Future work Conclusions References Additional bibliography.. 4 Appendices: A. Simulink block diagram of the acoustically excited tube without combustion B. Simulink block diagram of the model combustor C. Simulink block diagram of the model combustor with masked subsystems D. Simulink block diagram of the model combustor with feedback control E. Simulink block diagram of the model combustor with feedback control and masked subsystems 1

6 List of figures Fig. 1 Control volume of perfect gas within a combustor... 5 Fig. 2 Schematic of the acoustically excited tube without combustion... 8 Fig. 3 Simulink block diagram of the model (acoustically excited tube) 11 Fig. 4 Pressure amplitude response to variation of excitation frequency without mean flow.. 13 Fig. 5 Pressure amplitude response to variation of excitation frequency with mean flow.. 13 Fig. 6 Mode shapes of pressure amplitude for the first three resonant frequencies.. 14 Fig. 7 Diagram of the combustor model.. 16 Fig. 8 Simulink block diagram of the combustor model... 2 Fig. 9 Stability regions of the first mode of G(s) as a function of the heat release time delay. 22 Fig. 1 Time evolution of pressure measurements for two different values of.. 22 Fig. 11 Stability regions of the time response of the Simulink model (max. step size of 1 5 s.). 23 Fig. 12 Stability regions of the i first modes (n = 1 i) of G(s).. 23 Fig. 13 (a) Stability regions of the ten first modes of G(s) obtained from Bode plot analysis.. 24 (b) Stability regions of the time response of the Simulink model (max. step size of 1 5 s.). 24 Fig. 14 Stability regions of the time response of the Simulink model (max. step size of 1 4 s.). 24 Fig. 15 (a) Stability regions of the four first modes of G(s) obtained from Bode plot analysis. 25 (b) Stability regions of the time response of the Simulink model (max. step size of 1 4 s.). 25 Fig. 16 Gain and phase shift checking for k = (maximum step size = 1 5 s.). 26 Fig. 17 Gain and phase shift checking for k = 2 (maximum step size = 1 5 s.). 26 Fig. 18 Gain and phase shift checking for k = 1.3 (maximum step size = 1 5 s.) 27 Fig. 19 Gain and phase shift checking for k = 3.3 (maximum step size = 1 5 s.) 27 Fig. 2 Generic arrangement for feedback control of combustion oscillations.. 28 Fig. 21 Structure of the negative feedback closed loop control system.. 29 Fig. 22 Bode plot of the open loop transfer function of G(jω) for k = Fig. 23 Nyquist diagram of the OLTF from the loudspeaker input to the filtered pressure 31 Fig. 24 Bode diagram of a generic phase lag compensator.. 31 Fig. 25 Nyquist diagram of the controlled system with two anticlockwise encirclements of 1 point. 32 Fig. 26 Simulink block diagram of the controlled system 33 Fig. 27 Comparison between pressure measurements with control OFF and ON, respectively. 34 Fig. 28 Pressure time response under varying values of k. 35 Fig. 29 Pressure measurement with control ON for k =

7 1. INTRODUCTION Combustion units, from gas turbine combustors to rocket motors, frequently experience thermoacoustic instabilities, also known as combustion oscillations or instabilities. These are consequence of the internal coupling between acoustic waves and the combustion process itself: unsteady heat release generates acoustic waves, these propagate along the combustor and reflect from boundaries to get back to the combustion zone, where they generate more unsteady heat release, for example through hydrodynamic instabilities [1,2] or local changes in the fuel air ratio [3]. Depending on the phase relationship of this unsteady heat release response, the energy associated to the acoustic waves may increase rapidly resulting in the occurrence of large amplitude instability. When this instability appears, oscillation amplitudes start to increase exponentially; at some point, some nonlinearity in the system limits these amplitudes, leading to self excited oscillations [4, 5, 6]. Deterioration in the system performance may occur due to these almost always unwanted largeamplitude flow oscillations and their associated pressure fluctuations and, under some circumstances, these can be intense enough to cause structural damage on the installation. Study and control of combustion instabilities is currently a subject of great importance, inasmuch as their occurrence in the new generation gas turbines, in which reduced emissions of NO x are a priority, is, at least, frequent [4]: both industrial land based gas turbines [7, 8, 9] and aero engines [1, 11] are particularly susceptible to them. Because these gas turbine combustors are operated under lean premixed conditions, NO x emissions are reduced, but, at the same time, this also makes combustors especially prone to experience combustion instabilities [12, 13]. This problematic, however, is not limited to gas turbine combustors: aeroengine afterburners [14, 15, 16], rocket motors [17], ramjets [18], boilers or furnaces [19] are other systems particularly susceptible to it. On a laboratory scale, the most common device used to reproduce and analyse this phenomenon is a simple open ended vertical tube with a heat source in its lower half, known as Rijke tube [2]. The elimination of thermoacoustic instabilities is achieved by interrupting the coupling between the acoustic waves and the unsteady heat release. Existing methods used to reach this interruption may be classified into passive control methods, active control methods or, the most recent alternative, tuned passive control methods. The first ones [19, 21] are based on permanent changes and aim to achieve one of these two objectives: either reduce the susceptibility of the combustion process to acoustic excitation through hardware design changes, such as modifying the combustor geometry [12, 22, 23] or the fuel injection system, or remove energy from sound waves using acoustic dampers, such as Helmholtz resonators [24] or quarter mode tubes [25]. The main inconvenient associated to these mechanisms is that they might be ineffective at the low frequencies at which some of the most damaging oscillations occur and, furthermore, required changes of design are usually expensive and time consuming [4]. 3

8 On the other hand, active control principle is based on introducing one or more inputs to the system which are actively varied using an actuator [4, 5, 6]. Active control may be subdivided into openloop, where the mentioned input is independent from measurements on the system, and closed loop (i.e. with feedback), where an actuator modifies a parameter of the system in response to a measured signal. The aim of closed loop active control is to design the control relationship between de sensor and actuator signal such that the closed loop system is stable. The main advantage of active control, especially closed loop, lies in its power to interrupt oscillations over a range of operating settings. However, if the controller design is incorrect, it may make the instabilities even more damaging. As the third option appears the tuned passive control, which has focused recent interest. It consists of designing the damping devices, such as Helmholtz resonators [24], with a variable geometry and/or variable forced flow throw them. The fundamental is simple: these dampers offer peak damping performance at a certain frequency depending on the previously mentioned parameters, so, by tuning them, high damping and even instability suppression may be achieved across a wider range of frequency. In comparison to active control, bandwidth requirements are lower, as geometry/flow rate actuation is only required when operating conditions are modified. 2. AIM OF THE PROJECT Given the susceptibility of the listed devices to experience these undesired combustion instabilities, the aim of this project is to model the appearance and behaviour of these large amplitude flow oscillations in the time domain, in certain different circumstances and settings, using Simulink 1 (Matlab). As well as that, the application of closed loop control to interrupt the coupling between acoustic waves and unsteady heat release that gives rise to combustion oscillations is introduced and a fixed parameter controller is designed and applied to the main combustor model. The idea is to provide a useful but easy to use tool to predict, analyse and, consequently, avoid the occurrence of thermoacoustic instabilities. The numerical background of the developed models is based on previous modelling approaches: Morgans, Evesque and Zhao have developed several Fortran codes in time domain that simulate different combustor settings, under specific assumptions, whereas Stow and Dowling have implemented more complicated codes, again in Fortran, combining both frequency and time domains. This project represents a step forward in combustion oscillations analysis as it seeks to provide the first tool which models this thermoacoustic phenomenon in an interactive and customizable graphical environment. This implementation is intended to offer a faster, more visual alternative to the complex current modelling approaches as none of them offers graphical user interface (GUI). As a result of this, it is thought to be more suitable for short future projects and researches thanks to the simplicity of use and the easiness with which it can be customized to satisfy users specific requirements. 1 Simulink is an environment for multidomain simulation and Model Based Design for dynamic and embedded systems. 4

9 3. ENERGY AND COMBUSTION OSCILLATIONS 3.1. Physical fundamentals [4] As introduced in Section 1, combustion instabilities are large amplitude flow oscillations that arise due to feedback between sound wavess and unsteady heat release. John William Strutt, 3 rd Baron Rayleigh, studied the physics of acoustics in depth and was the first to provide somee physical approach to combustion instabilities in the late nineteenth century [ 2]. In 1878, he formulatedd a criterion, known as Rayleigh s criterion, which states that an acoustic wave gains energy when heatt is added in phase with pressure, but loses energyy when heat iss added out of phase withh pressure. This principle may be quickly extrapolated to the physics of combustion oscillations to explain the energy y exchange between acoustic waves and heat releasee on which most of these oscillations are based. In 1964, B. T. Chu expanded Rayleigh s criterion through thee incorporation of the effect of boundary conditions to it [26],, thus providing a useful insight into thermoacoustic instabilities and establishing the fundamentals to develop methods for controlling them. If we follow Chu s analysis [26] and reproduce the simplification presented by Dowling and Morgans [4], a perfect gas burning within a combustor of volume V, bounded d by a surface S is considered (Fig. 1). The gas is assumed to be linearly disturbed from thee rest with noo mean heat release r (further analysis accounting for mean flow and mean heat release have been published [27]) and viscous v forces are neglected. The pressure, density, heat release rate per unit volume, particle velocity, speed of sound and ratio of specific heatt capacities are denoted by p, ρ, q, u, c and γ, respectively. An overbar denotes a mean value and a prime denotes a fluctuating value. Pressure fluctuations and changes in entropy lead to the density change that arisess from unsteady heat release. The relationship between these three terms results in: combustor (volume V) surface, S acoustic waves unsteady heat release, Fig. 1 Control volumee of perfect gas within a combustorr [4] 5

10 By combining this equation (3.1) with the linearized continuity and momentum equations, and integrating over the volume V, an acoustic energy equation is obtained [26]: S 3.2 The left hand side term represents the rate of change of the sum of the kinetic and potential energies within volume V. The first term on the right hand side represents the exchange of energy between the combustion and sound waves. Following Rayleigh, when the pressure, p, and the heat release, q, have a component which is in phase (in other words, when the phase difference lies between 9 and +9 ), the acoustic energy tends to be increased. Finally, the second term on the right hand side, the surface term, captures energy losses across the bounding surface, S, because the fluid within this surface does work on the surroundings [4]. Furthermore, it can be observed that acoustic disturbances will grow in magnitude if their gain from combustion is larger than the energy losses across boundaries. This can be formulated in the following inequality, which is the generalized form of the Rayleigh s criterion: 1.S 3.3 where the overbar denotes an average over one period of the acoustic oscillation. If the inequality is satisfied, acoustic waves amplitude will increase until non linear effects limit its growth. If these nonlinearities appear primarily in the heat release rate with the sound waves remaining linear and it is assumed that acoustic waves are initially growing, heat release saturation or phase change effects may tend to equalize the terms in equation (3.3) at a certain pressure amplitude. At this amplitude, limit cycle oscillations occur [28, 29, 3, 31]. Equation (3.3) does not only explain why combustion instabilities occur and why their size is limited by non linear effects, but also shows that these oscillations may be eliminated by either decreasing the energy source term,, or increasing the surface loss term,.s. This is what control methods mentioned in Section 1, both passive and active, pursue Acoustic analysis The models presented in this project simulate and analyse the behaviour of acoustic waves travelling along an open ended pipe with a constant cross section, A, and a total length of L, under specific assumptions in each case: different sources and locations of the tube excitation, with or without mean flow, absence or presence of combustion. Denoting the distance along the tube by x, the combustion zone (or the harmonically forcing in the absence of combustion) is located at x =, with the upstream and downstream open ends at x = x u and x = x d, respectively. Similarly, upstream region is noted as Region 1 and downstream region as Region 2. 6

11 When analysing the system in any of the cases provided, it is assumed that the frequencies of interest are low enough to ensure that all nonplanar modes (both radial and transversal) are well cut off, so only one dimensional disturbances are important [32], and for the combustion zone to be short compared to the wavelength [28]. Contributions from entropy waves are neglected, acoustic waves are assumed to behave linearly to the mean flow [24] and both the mean density,, and the speed of sound,, are assumed constant all along the tube [4]. Wave strengths (vid Fig. 2 and Fig. 7) are denoted as: L 1 (t) for left travelling waves in Region 1 R 1 (t) for right travelling waves in Region 1 L 2 (t) for left travelling waves in Region 2 R 2 (t) for right travelling waves in Region 2 As pressure obeys the linear wave equation, pressure and velocity upstream the flame, x u < x <, can be written as a linear combination of the waves L 1 and R 1 :,, Similarly, downstream the combustion zone, > x > x d, pressure and velocity can be expressed as a linear combination of the waves L 2 and R 2 :,, where an overbar denotes a mean value. The boundary conditions of the combustor are characterized by upstream and downstream pressure reflection coefficients R u and R d, respectively. Therefore, the reflected waves R 1 and L 2 are easily obtained as a function of L 1 and R 2 using the following relationships: And hence:

12 where τ u and τ d are, respectively, the upstream and downstream propagation time delays, whose numerical developments are: where / is the mean flow Mach number. 4. ACOUSTICALLY EXCITED TUBE WITHOUT COMBUSTION 4.1. General description of the model and reason for modelling A first approach to the study of combustion instabilities consists of modelling the acoustically excited tube shown in Figure 1 and analysing the behaviour of acoustic waves, initially injected by a loudspeaker located at x =, travelling along the setup in the absence of combustion, both with and without mean flow. The reason why the study firstly models a pipe excited by a loudspeaker instead of directly introducing a flame as an excitation source is no other than the complexity this second option involves. An acoustically excited tube is easier to model numerically and implementing this setting is a necessary stage before proceeding to the sophisticated modelling of a thermoacoustically excited pipe, where the interaction of combustion and acoustic waves is by no means trivial to describe (vid. Section 5). In the current model (Fig. 2), two sensors are implemented to measure the pressure fluctuations along the pipe, each one located at an arbitrary point of each region, these locations denoted as x = x 1 for the upstream sensor and x = x 2 for the downstream sensor. Values for the main parameters used in this first approach are summarized in Table 1. Loudspeaker V ls t V sinwt p 1 p 2 L 1 t R 2 t R 1 t L 2 t u d Region 1 Region 2 Fig. 2 Schematic of the acoustically excited tube without combustion 8

13 4.2. Description of the numerical model The device simulated consists of an axial pipe which incorporates the previously mentioned loudspeaker to generate acoustic waves which propagate along the tube. In this initial analysis, two different settings or cases regarding the mean flow (ū) are modelled: a) Assuming negligible mean flow (ū = ) b) Assuming steady and one dimensional mean flow. The rate of unsteady volume injection, V ls (t), is implemented as a sinusoidal wave: sin 4.1 where V denotes the amplitude of the wave and ω the frequency. This wave acts as the acoustic source which gets the physical system started and our aim is to study and analyse the behaviour of the derived waves resulting from its interaction with the tube s ends and the mean flow at different frequencies over the range of interest. To do so, a matrix system expressing these derived waves (L 1 (t), R 1 (t), L 2 (t) and R 2 (t)) as a function of V ls (t) and, by extension, as a function of frequency ω, is required. The equations of conservation of mass and pressure continuity across the loudspeaker at x = are given by:,, Substitution from (3.4) and (3.5) into (4.2a) and (4.2b) leads to: Parameter Value Effective length or the pipe, L, m 1 Pipe cross-sectional area, A, m x 1-2 Axial distance from the loudspeaker to the upstream end, x u, m.2 Axial distance from the loudspeaker to the downstream end, x d, m.8 Axial distance from the loudspeaker to the upstream pressure sensor, x 1, m.1 Axial distance from the loudspeaker to the downstream pressure sensor, x 2, m.4 Reflection coefficient at upstream end, R u, / -.95 Reflection coefficient at downstream end, R d, / -.95 Mean density,, kg m Mean speed of sound,, m s Loudspeaker sine wave amplitude, V, m 3 s -1 1 Table 1 Geometrical and acoustic parameters 9

14 Making use of the pipe end boundary conditions specified in (3.7), enough information is provided to solve for each of the four wave strengths in Figure 2. The resulting matrix equation is given by: 4.4 where X and Y are the following coefficient matrices: ; 4.5 Having solved the matrix equation, the expressions for the time evolution of the outgoing waves L 1 (t) and R 2 (t) as a function of ω are obtained: Pressure modelling In order to obtain the time evolution expressions of the pressure fluctuation, theoretically measured by the pressure sensors at x = x 1 and x = x 2, these points have to be substituted in equations (3.4a) and (3.5a), respectively, as follows:,, Application of the open end boundary conditions and the correspondent time delays leads to: Combining these two expressions with (4.6a) and (4.6b), the graphical block modelling of this case is implemented with Simulink so that the pressure fluctuation s behaviour at the pressure sensors, located at x = x 1 and x = x 2, can be analysed when varying the input frequency ω of V ls (t). The arrangement of blocks in the definitive Simulink model is reproduced in Figure 3 (a larger view of the model is shown in Appendix A). 1

15 4.4. Results and modell checking The developed model, by adjusting the customizable value of some of its parameters, can c be used to study the pressure evolution along the pipe in two different cases or circumstances, depending on the assumptions over the mean flow (ū): a) Assuming negligible mean floww (ū=) b) Assuming steady and one dimensional mean flow. In case a), it is obviouss that neglecting the mean flow implies that ū =, and, as a consequence,. direct simulations and it is assumed that ū =.1, so the mean flow Mach number is to both settings a) and b), remain as summarized in Table 1. These are logical valuess useful to compare the two cases, but may be easily modified inn Matlab s Command Window to fit user s requirements in further analysis. when In case b), the mean flow, regarded as steady and one dimensional, is noww considered in the The numerical value given to the rest of geometrical and flow parameters of the model, common Among all the information that this model can provide, the evolution of the pressure amplitude the frequency of the acoustic excitation is varied and the main m mode shapes at resonant frequencies are plotted, for both cases a) and b), in order to carryy out a double checking of the agreement between the results derived from this numerical model and the previous experimental research as well as the theory based on planee wave description [32]...1. Fig. 3 Simulink block diagram of the model 11

16 First checking: Pressure amplitude behaviour around resonance If we let λ denote the wave length and L has been previously defined as the total length of the tube, the acoustic system is in resonance when L is a multiple of half the wavelength, λ/2. That is: 2 ; 4.9 As a result of this, if, resonant frequencies are those that fulfill the following condition: ; 4.1 For example, the first resonant frequency (n = 1), also known as first mode frequency or fundamental frequency, is and the second resonant frequency (n = 2) is In order to illustrate the specific behaviour of the pressure s amplitude around resonance, the graphs plotting the results of this initial study represent the factor n [ ] from equation (4.9) on the horizontal axis and pressure amplitude [Pa] on the vertical axis. Not only two different settings are simulated, without mean flow (Fig. 4) and with mean flow (Fig. 5), but in each of these cases the results are calculated for two different values of the reflection coefficients R u and R d. As detailed in Table 1, the model is simulated and plotted for R u = R d =.98 and R u = R d =.95. These values, especially the second one, are slightly less (in magnitude) than the theoretical value of 1 for an open end [32]. Through this assumption, the numerical model is considering the acoustic energy loss that occurs at both ends of the tube. The lower (in magnitude) the reflection coefficient is, a greater loss of energy is assumed. Graphs presented below confirm the expected results: the plotting of pressure amplitude as a function of frequency presents peak values around resonance (i.e. when index n is an integer; vid. 4.1), regardless the presence or the absence of mean flow. However, whereas these peaks exactly coincide with resonant frequencies when no mean flow is considered (Fig. 4[a,b]), they appear slightly before resonance in both regions of the tube in the second case (Fig. 5[a,b]) due to the interaction between the unsteady volume injected and the mean flow. At the same time, it can be observed that the amplitude s peak values are lower when the reflection coefficients are lower (in magnitude) (Fig. 4[a] and Fig. 5[a]), and higher when these are closer to the theoretical value of 1 for an open end (Fig. 4[b] and Fig 5[b]). The explanation is simple: as the reflection coefficients approach to their ideal value, the loss of energy at open ends is reduced and, consequently, the amplitude of pressure fluctuations remains higher. Finally, it can be observed that in the four figures, the peak of amplitude for the first mode is higher in Region 2. However, this tendency changes in the third peak thanks to the damping properties derived from the length of each region.. 12

17 The matching between the theoretical results and those obtained from the developed model in the time domain constitutes a solid validation for this first numerical development and its graphical implementation with Simulink (Fig. 3). Amplitude [Pa] 6 x [ Ru = Rd = -.95 ] p1 p n (L=n*/2) [ ] Ampltiude [Pa] 15 x [ Ru = Rd = -.98 ] p1 p n (L=n*/2) [ ] Fig. 4[a,b] Pressure amplitude response to variation of excitation frequency without mean flow 6 x 15 [ Ru = Rd = -.95 ] Amplitude [Pa] n (L=n*/2) [ ] 15 x 15 [ Ru = Rd = -.98 ] Amplitude [Pa] n (L=n*/2) [ ] Fig. 5[a,b] Pressure amplitude response to variation of excitation frequency with mean flow ( =.1) 13

18 Second checking: Mode shapes of the resonant frequencies The second validation consists of plotting the pressure amplitude along the horizontal axis, from the upstream end at x = x u to the downstream end at x = x d, for the first three resonant frequencies (ni=i1,i2,i3) in order to check the agreement between the information this model provides and the theoretical mode shapes at this frequencies. Analysing the figures below, it can rapidly be affirmed that the mode shapes perfectly fit the theory regarding standing waves in air columns in an open ended tube. Precisely, given that the pipe is open at both ends, the pressure at the ends would theoretically have to be atmospheric, p(t) =, for any resonant frequency [32]. However, as reflection coefficients are assumed to present values slightly lower in magnitude than the theoretical values of 1, as possible energy losses across pipe ends are considered, pressure nodes (i.e. points of zero amplitude) suffer a slight shift from their ideal position outwards the pipe (Fig. 6). 2 x 16 n = 1 2 x 16 n = 1 Amplitude [Pa] 1 Amplitude [Pa] x [m] x [m] 2 x 16 n = 2 2 x 16 n = 2 Amplitude [Pa] 1 Amplitude [Pa] x [m] x [m] 2 x 16 n = 3 2 x 16 n = 3 Amplitude [Pa] 1 Amplitude [Pa] x [m] x [m] (a) Without mean flow (b) With mean flow Fig. 6 Mode shapes of pressure amplitude for the first three resonant frequencies (n = 1, 2, 3) with R u = R d =

19 Furthermore, acoustics theory raise that pressure nodes do not only occur at the extremes of an open ended tube, but at every position of x where: 2 ; 4.11 (q is a multiplying factor and n is the mode number). At the same time, an antinode of pressure, corresponding to a point of maximum pressure, is due to appear at every position of x where: ; 4.12 Remembering that for a system in resonance, conditions (4.11) and (4.12) clearly match with the pressure amplitude behaviour shown in Fig. 6 for the first three (n=1,2,3) resonant frequencies. It must be pointed out that, although it cannot be easily appreciated in the presented figures, pressure amplitude at pressure nodes all along the tube is not exactly zero. The explanation of this phenomenon is based on the energy losses which had been previously mentioned when discussing about the pressure nodes at the extremes of the pipe. To sum up, the analysis of mode shapes appears to corroborate the validity of this model, as the agreement between the physical behaviour of the pressure amplitude, the theory of acoustics and the results plotted is satisfactory. 5. MODEL COMBUSTOR 5.1. General description of the model and reason for modelling The next step in our way to model combustion instabilities lies in developing a Simulink model which includes a source of unsteady heat release. To be precise, the following system consists of a horizontal tube without mean flow, open at both ends, with a heat source contained in its left half and a loudspeaker in its upstream end which applies a white noise input to get the system started. Figure 7 shows a schematic diagram of the geometry and devices included. The analysis that follows transfers to the time domain the one that was firstly conceived in the Laplace domain by Evesque [33], aiming to find the open loop transfer function from the loudspeaker input to the sensor pressure in the absence of a control heat input. This study in the Laplace domain was later expanded by Dowling and Morgans, so their corresponding paper [4] is taken as reference in this project. This modelling approach offers a graphical interface to identify the stability regions of the uncontrolled system in a specific setting. At the same time, it constitutes the base model for which a fixed parameter controller will be designed in the next step of the current project (vid. Section 6). 15

20 5.2. Description of the numerical model Following the assumptions specified in Section 3, equations of conservation of mass, momentum and energy across the flame at x = can be written in the form [28]: We firstly seek to obtain a matrix equation that relates the upstream and downstream acoustic waves to the instantaneous rate of heat release, Q(t), and the loudspeaker signal, i(t). This can be achieved by combining (5.1), (5.2), (5.3) and the perfect gas equation, and making use of the specific boundary conditions for this setting. The first equation needed is obtained directly from substitution of (5.1) intro (5.2), giving: 5.4 form: To obtain the second equation, the left hand side of the energy equation (5.3) is expanded in the Use of the perfect gas equation to rewrite Tρ as p/r (where R is the gas constant) and substitution of (5.1) into (5.5) simplify the previous expression to: Recalling the relationships between specific heat capacities, C p and C v, and their ratio, γ, we have: Loudspeaker input V c white noise Heat release Q t P ref Pressure sensor L 1 t R 2 t R 1 t L 2 t u d Region 1 Region 2 Fig. 7 Diagram of the combustor model 16

21 By introducing this equivalence into (5.6), the second equation of the matrix system is obtained: where Q is the instantaneous rate of heat release, γ is the ratio of specific heat capacities and A comb is the combustor cross sectional area. Once these two equations, (5.4) and (5.8), are deduced, substituting their flow variables (p 1, p 2, u 1, u 2 ) for their expressions, detailed in (3.4a,b) and (3.5a,b), assuming, making use of the isentropic condition / ) and the corresponding boundary conditions and linearizing in the flow perturbations give a matrix system which expresses the time evolution of the outgoing waves L 1 (t) and R 2 (t) generated by the unsteady heat release Q(t) (Q(t) = Q (t), as and, then, ). However, it is necessary to highlight that, since the mean flow is negligible and due to the presence of the loudspeaker output, reflected waves R 1 (t) and L 2 (t) may be here expressed as a function of L 1 (t), R 2 (t) and the loudspeaker signal, i(t), using the boundary conditions as follows: where i(t) is the white noise injected by the loudspeaker at x = x u and 2 / and 2 / are, respectively, the upstream and downstream propagation time delays in the absence of mean flow. Ultimately, the resulting matrix expression of the model shows that: Similarly to McManus formulation for unsteady heat release, known as the (n τ) model [5], heat release is expressed as a function of the fluid velocity just upstream the combustion zone (u 1 (x=,t)). Specifically, Dowling and Morgans [4] use the following expression in the Laplace domain for the instantaneous rate of heat release per unit area, q: 5.11 where H(s) is the flame transfer function. If we transfer this expression to the time domain and develop the velocity term as a function of the waves in Region 1 by using the expression (3.4b), then: Application of the upstream boundary condition (5.9a) gives:

22 At his point, substitution of (5.12) intro (5.1) and grouping of terms, leads to: If we then isolate the time evolution of the outgoing waves L 1 (t) and R 2 (t), we have: Finally, having solved the matrix equation, the expressions for the time evolution of the outgoing waves L 1 (t) and R 2 (t) as a function of the instantaneous rate of heat release expressed through Ht) and the acoustic loudspeaker signal, i(t), are obtained: However, the ultimate aim of the model is to relate pressure measured at the sensor located downstream, at x = x ref, with the loudspeaker input, V c (t). The first step to reach this relationship is describing the loudspeaker dynamics through its transfer function in the Laplace domain, W ac (s): 5.16 where i(s) is the loudspeaker output or signal and V c (s) is the loudspeaker white noise input. Nevertheless, the loudspeaker dynamics is assumed to be flat over the low frequencies of interest [4], consequently, W ac (s) constant, both in the Laplace and the time domain. In parallel, once more, Dowling and Morgan s work [4] is followed when it comes to choose a specific flame model, H(s): the simplest type, based on a time lag concept, is assumed

23 The inclusion of these two last considerations into expressions (5.15a) and (5.15b) results in the definite expressions of the outgoing waves, L 1 (t) and R 2 (t), in the time domain: Pressure modelling The last step before drawing the block diagram of this model consists of deducing the expression of P ref, the pressure measured at the sensor located at x = x ref, as a function of the travelling waves. This is obtained by substituting this specific location into equation (3.5a) assuming negligible mean flow:, 5.19 Application of the downstream boundary condition (3.7b) leads to:, 5.2 where 2 / since negligible mean flow is assumed. So, in the end, we get to the equivalent expression of sensor pressure as the one previously achieved by Dowling and Morgans [4], but placed in the time domain:, When connecting expressions (5.18a) and (5.18b) with (5.21), enough information is provided to draw a Simulink block diagram of the model described, which relates the pressure measured at the sensor, P ref, with the loudspeaker input, V c (t). However, before designing the final graphical model, one last improvement should be introduced: Dowling & Morgans analysis in the Laplace domain of this model shows that the stability of the model strongly depends on the value of the flame model time delay, (5.17); in order to check that our Simulink model presents the same behaviour (Section 5.4), this time delay is normalized in the same way it is done in the reference study [4], so the results obtained with our design in the time domain can be easily compared to the ones presented by Dowling and Morgans. As a result of this, blocks corresponding to the flame model time delay are implemented as follows:

24 wheree ω is the theoretical firstt mode frequency (n = 1) correspondingg to a wavelength of double the tube length (vid. 4..1) and k is the t normalizing parameterr used in the following f section to illustrate the stability map of the model. Taking into account all these t considerations, this is the Simulink block diagram of the model combustor with unsteady heat release and without mean flow (Fig. 8): Fig. 8 Simulink block diagram of the model Appendix B includes a larger view of the Simulink block diagram, whereas Appendix C reproduces an equivalent model structured in subsystems that mask smaller subdivisions of the diagram and offer a more aesthetic view of the graphical model presented above Results and modell checking The checking of the developed Simulink model ( Fig. 9) basically arises from the comparison between the results derived from our model and the results derived from the analysis firstly presented by Evesque [33] and later studied in depth byy Dowling and Morgans [4]. Therefore, as it was previously mentioned, their paper is used as a main reference to assure the validity of o our model. Essentially, our model ( Fig. 8), basedd on a numerical development in the time domain, is a graphical equivalence to the open loop transfer function, G(s), from the loudspeaker input, V c, to the microphone pressure, P ref, presented in section of Dowling and Morgans report of their study. Transferring it to the notation used in our model, G(s) would have the following form:

25 where: W ac (s) is the loudspeaker transfer function: 1 H(s) is flame transfer function, assuming a flame model based on a time lag concept: J is the determinant of a matrix derived from the numerical development in the Laplace domain: Geometry and flow parameters defining our model are simulated assuming the same values that are used in [4] to evaluate the open loop transfer function, G(s), so the results derived from each method may be fairly compared. These parameters and their value are summarized in Table 2. Dowling and Morgans prove in their paper that the stability of the detailed system, expressed through the transfer function G(s), depends strongly on the value of the flame model time delay,. Our next aim is, therefore, to confirm that the stability of our Simulink system depends on this parameter in the same way. In the reference paper, each mode is considered to be caused by a second order transfer function with the form: where is the system s natural frequency and ξ is the system s damping ratio. Consequently, at mode peaks, where, we have: Parameter Value Effective length or the pipe, L, m.75 Axial distance from the heat release to the upstream end, x u, m.25 Axial distance from the heat release to the downstream end, x d, m.5 Axial distance from the heat release to the pressure sensor, x ref, m.9 Reflection coefficient at upstream and downstream end, R u and R d, Loudspeaker transfer function, W ac (s), - 1 Ratio of specific heat capacities, γ, Mean speed of sound,, m s -1 4 Table 2 Geometrical and flow parameters 21

26 This means that a positive ξ implies a phase decrease of 18 and, consequently, a stable mode; and a negative ξ results in a phase increase of 18 and, therefore, an unstable mode. As a result of this, we can deduce the stability of each mode analyzing the change of phase across the mode peaks in the Bode plots: a phase decrease of 18 indicates a stable conjugate pair of poles, whereas a phase increase of 18 indicates an unstable conjugate pair [4]. If we do trace the Bode plots of G(jω), the open loop transfer function from V c to P ref defined by Dowling and Morgans (vid. 5.22) for successive values of the flame model time delay,, normalized as a function of the multiplying factor k (vid. 5.22) and analyze the phase change across the first mode peak, we obtain the following stability map: Stability of first mode of G(jω) with flame model time delay Stable Unstable k = τ H ω /Π Fig. 9 Stability regions of the first mode of G(s), presented by Dowling and Morgans in the Laplace domain, as a function of the heat-release time delay, τ H. Using the Simulink block diagram (Fig. 8), when analyzing the time evolution of the pressure measurements in the pressure sensor for different values of the heat release time delay,, we also face two different responses depending on the value of the normalizing factor k: a stable response or an unstable response. As an example, Figure 1a shows the stable response for k = (which is unsurprising given that = means that there is no heat release and, by extension, no coupling mechanism to cause instability), whereas Figure 1b shows the unstable response for k =.5, where the coupling between acoustic waves and unsteady heat release is responsible for the occurrence of combustion instabilities. 1 (a) k = (b) k =.5 4 Amplitude [Pa] 5-5 Amplitude [Pa] Time [s] Time [s] Fig. 1 Time evolution of pressure measurements for two different values of the flame model time delay, τ H = kπ/ω 22

27 If we simulate (imposing an initially arbitrary maximum step size along each simulation of 1 5 s.) and analyze the time response of the Simulink model for the same range of values for the parameter k as in Figure 9, in order to classify them between unstable and stable following the criterion exemplified with Figures 1a and 1b, the following stability regions arise: Stability map obtained with Simulink model (max step size = 1 5 s) Stable Unstable k = τ H ω /Π Fig. 11 Stability regions of the time response of the Simulink model as a function of the heat-release time delay, τ H. (imposing a maximum step size of 1-5 s. in the simulation) It may be quickly observed that stability regions derived from Simulink simulations clearly differ from the stability map presented in Figure 9. However, further research in the interpretation of each analysis seems to indicate that this difference lies in the fact that higher order modes are not considered in the first case (Fig. 9). Taking this into account, following the steps that led to the stability map for the first mode, the changes of phase across higher order mode peaks are studied and the coupling, in terms of stability, of an increasing number of modes are captured in the following figure: Stability of i first modes (n = 1 i) with flame model time delay Stable Unstable n = 1 n = 1 2 n = 1 4 n = 1 6 n = 1 8 n = k = τ H ω /Π Fig. 12 Stability regions of the i first modes (n = 1 i) of G(s), as a function of the heat-release time delay, τ H, obtained from Bode plot analysis 23

28 Considering the ten first modes, the stability regions which can be obtained analyzing the Bode plots of the system s transfer function, G(s) (vid. 5.23), presented by Dowling and Morgans [4], are nearly the same as the one resulting from the Simulink model simulations (Fig. 11). Stability evolution in Figure 12 clearly shows that the higher the number of modes considered to draw the stability maps above is, the closer they will get to the one obtained through our Simulink model imposing a maximum step size of 1 5 s. in the simulations. Our Simulink model is then a useful tool to study the system s real behaviour over a determinate range of frequencies of interest. The range of frequency considered by the model will depend on the maximum step size fixed, so this parameter must be carefully chosen. Comparison of stability maps (I) Stable Unstable (a) (b) k = τ H ω /Π Fig. 13 (a) Stability regions of the ten first modes (n=1-1) of G(s) obtained from Bode plot analysis (b) Stability regions of the time response of the Simulink model (maximum step size of 1-5 s.) Therefore, having permormed our simulations in Simulink imposing such a small maximum step size, 1 5 s., a wide range of frequencies are captured in the time response and, therefore, up to ten modes have to be considered to obtain a similar stability map through Bode plot analysis (Fig. 13). This statement is verified when observing how the stability map provided by our Simulink model changes if we impose a significantly wider maximum step size: 1 4 s. Stability map obtained with Simulink model (max step size = 1 4 s) Stable Unstable k = τ H ω /Π Fig. 14 Stability regions of the time response of the Simulink model as a function of the heat-release time delay, τ H. (imposing a maximum step size of 1-4 s. in the simulation) 24

29 Figure 14 shows that, after changing the maximum step size of the simulation, the new stability map obtained by Simulink presents similar stability regions to the one included in Figure 12 in which only the first six modes were considered (n = 1 6). These two stability maps are reproduced below for easier comparison (Fig. 15): Comparison of stability maps (II) Stable Unstable (a) (b) k = τ H ω /Π Fig. 15 (a) Stability regions of the four first modes (n=1-6) of G(s) obtained from Bode plot analysis (b) Stability regions of the time response of the Simulink model (maximum step size of 1-4 s.) Therefore, by running the simulation with wider step sizes, higher order modes are well cut off. As a result of this, this tool may be used as a trial and error low pass filter. Considering this, the selection of the maximum step size of the simulations is a powerful tool that lets the user restrict his acoustic study to a determinate range of frequency. In the study of combustion oscillations, the attention is focused on the low frequencies at which some of the most damaging instabilities occur, so the second option, a maximum step size of 1 4 s., appears to be an adequate choice. This hypothesis will be further discussed in Section 6 when conceiving the feedback control. At this point, although a clear agreement between the results included in Dowling and Morgans paper and the results derived from our Simulink model has been proved, this is yet restricted to stability regions. Therefore, a deeper checking strategy is performed to ensure total agreement. To run this ultimate checking, a prior change has to be made in our Simulink model: the white noise input is substituted by a sine wave block, so the loudspeaker becomes harmonically forced and, as a result of this, both the input and output waves of the system are sinusoidal. Thanks to this slight modification, gain and phase shift can be measured at any given frequency for a specific value of parameter k (used as a normalization of the heat release time delay, ; vid. 5.22). Obviously, this methodology is restricted to those values of k that stabilize our system, as gain and phase shift can only be measured in the temporal response when this response is stable. Having repeated this procedure for a range of spot check frequencies and different values of parameter k, results are scattered on the Bode diagrams that arise from the Laplace analysis, described in equation (5.23) by Dowling and Morgans [4], for each specific value of parameter k. 25

30 It was above demonstrated (Fig. 11), that Simulink model s response is only stable for k = (without heat release) and for k = 2 (when a maximum step size of 1 5 s if fixed). So, using these values, the previously detailed checking method leads to the following diagrams: 4 Gain and phase shift checking (k=) Gain [db] [rad/s] Phase [deg] -5-1 Dowling & Morgans Simulink Model [rad/s] Fig. 16 Gain and phase shift checking for k = (maximum step size = 1-5 s.) 4 Gain and phase shift checking (k=2) Gain [db] [rad/s] Phase [deg] -5-1 Dowling & Morgans Simulink model [rad/s] Fig. 17 Gain and phase shift checking for k = 2 (maximum step size = 1-5 s.) The matching between the Bode plots and the scattered points, both for gain and phase, obtained with our Simulink model with k = and k = 2 appears to be total. 26

31 However, in order to go a step further, not only checking the validity of the Simulink model, but also the explanation given to Figure 15, the graphs are now plotted for k = 1.3 and k = 3.3, values that also lead to a stable response in the time domain when the maximum step size imposed is 1 4 s. 4 Open Loop Transfer Function (k=1.3) Gain [db] [rad/s] Phase [deg] -5-1 Dowling & Morgans Simulink model [rad/s] Fig. 18 Gain and phase shift checking for k = 1.3 (maximum step size = 1-4 s.) 4 Open Loop Transfer Function (k=3.3) Gain [db] [rad/s] Phase [deg] -5-1 Dowling & Morgans Simulink Model [rad/s] Fig. 19 Gain and phase shift checking for k = 3.3 (maximum step size = 1-4 s.) Once more, plots show the expected matching between the two methods. In conclusion, the reliability of the developed Simulink model (Fig. 8) that relates the pressure measured at a given downstream location, P ref, to the white noise loudspeaker input, V c, in presence of heat release and without mean flow is ensured. 27

32 6. PRACTICAL APPROACH TO FEEDBACK CONTROL Feedback control, also known as closed loop control, involves modifying some input to an unstable combustion system (using a controller and an actuator) in response to an output measurement (obtained using a sensor). To be precise, feedback control seeks to design the control relationship between the sensor signal and the actuator signal such that the closed loop comprising combustion system, the sensor, the controller and the actuator is stable. Figure 2 shows the feedback layout for a typical combustion system. A sensor measures a timevarying output from the combustor, S(t), and feeds it to a controller. This controller produces a signal, V(t), and this signal drives an actuator to produce a time varying input to the combustion system, A(t). As specified above, the aim of controller design is to choose the relationship between the sensor signal, S(t), and the actuator signal, A(t), that stabilizes the entire closed loop combustion system [4]. + Combustion system Sensor Actuator signal A(t) Actuator Controller signal V(t) Controller Sensor signal S(t) Fig. 2 Generic arrangement for feedback control of combustion oscillations The choice of sensor and actuator is one of the most delicate points when implementing feedback control on practical combustion systems. In spite of this, another main complexity of controller design for such applications basically lies in the large number of features which have to be taken into account simultaneously [8, 34]. In 25, Dowling and Morgans reviewed and summarized all the systematic approaches to controller design that had been applied to combustion instabilities to date [4]. Schuermans had presented a similar table in 23 which also included developments in open loop control [35] Fixed parameter control applied to the combustor modelled with Simulink Controller design in Laplace domain As a first stage, previous to much more complex systematic controller design techniques, the objective of this section is to design a fixed parameter controller for the combustor which was modelled with Simulink in Section 5 (Fig. 8) and implement it adding the feedback control for a given configuration of this model in the time domain. 28

33 Because the modelled system is single input single output (SISO), it is possible to design a robust controller using straightforward Nyquist and Bode criterions [4]. Once the controller has been designed in the Laplace domain, it will be added to the Simulink model, which operates in the time domain, in order to check its effectiveness to interrupt combustion instabilities. Once more, the open loop transfer function (OLTF) of interest, G(s), is the transfer function from the loudspeaker input, V c, to the pressure measurement, P ref. We now seek to conceive a controller, K(s), such that the closed loop system illustrated in Figure 21 is stable. + G(s) K(s) Fig. 21 Structure of the negative feedback closed-loop control system The first step consists of the selection of the specific setting of the model for which the controller will be designed. Geometry and flow parameters keep the same values that were used in Section 5 and summarized in Table 2. Choosing a specific heat release time delay,, is, however, much more challenging because the controller design analysis that follows is valid for systems which present only one unstable mode over the low frequencies of interest. Taking this into account, Bode plots of G(jω) are drawn for a wide range of values of the parameter k, taken as the previously detailed normalization of the flame model time delay, τh (vid. 5.22). Considering the phase change across the resonant peaks of G(jω), system s behaviour for k =.5 seems to be susceptible of stabilization using the fixed parameter control technique, because only the first mode (n = 1) is unstable over low frequencies and this instability corresponds to an unstable conjugate pair of poles (Fig. 22; the blue line remarks the phase change across the first resonant peak). Gain in db Phase in degrees in rad/s Fig. 22 Bode plot of the open-loop transfer function of G(jω) for k =.5 29

34 The Nyquist Stability Criterion [36] can be expressed as Z = N + P where Z = number of zeros of 1 + K(s)G(s) in the right half s plane N = number of clockwise encirclements of the 1 + j point P = number of poles of K(s)G(s) in the right half s plane Our modelled system with k =.5 has two open loop unstable poles over the low frequencies of interest, which means K(s)G(s) presents two poles in the right half s plane and, consequently, P = 2. As a result of this, for a stable control system, we must have Z =, or N = P, therefore we must have P anticlockwise encirclements of 1 point. In other words, the Nyquist plot for K(jω)G(jω) needs to encircle the 1 point in an anticlockwise direction twice in order to obtain a stable closed loop system. Before drawing the Nyquist plot, however, a remark on the stability of higher order modes must be made: despite Figure 22 clearly shows that in our model only the first mode (n = 1) is unstable over the low frequencies of interest, the eleventh mode (n = 11), whose frequency (ω 11 = 11Π /L rad s 1 ) is assumed to be out of the range of interest, is again unstable and could distort the time response. Considering this, a second order low pass filter, G f (s), is added to the open loop transfer function of our system, G(s), so it attenuates high frequencies and the Nyquist plot for G f (jω)g(jω) is restricted to the low frequencies of interest, thus highlighting the first mode (n = 1) which causes the instability. A generic second order low pass filter, G f (s), has the following structure: where = natural frequency of the system ξ = damping ration ( < ξ < 1) = gain of the low pass filter Because we want the filter to attenuate the frequency response just above the unstable mode, which from the Bode plot in Figure 22 is at a frequency of ω = 1675 rad s 1, a good choice of value for the low pass filter s natural frequency is = 2 rad s 1. The importance of the damping ratio s exact value is lower, so an arbitrary but realistic value of ξ =.5 is chosen. Finally, a unity gain is fixed for the filter, K f = 1, because it is thought that if the filter does not modify the response magnitude, the influence of the controller, K(s), will be later easier to understand and interpret. After adding the filter, the Nyquist plot for G f (jω)g(jω) is shown in Figure 23. The anticlockwise loops correspond to the unstable mode, for positive and negative frequencies, respectively. By considering the unit circle plotted over the Nyquist diagram, it is clear that there are no encirclements of the 1 + j point. To achieve these two anticlockwise encirclements, the controller K(s) has to introduce additional phase lag near the frequency of the unstable mode. 3

35 6 Nyquist plot for G f (j)g(j) 4 Imaginary Part Real part Fig. 23 Nyquist diagram of the OLTF from the loudspeaker input to the filtered pressure ( G f (jω) G(jω) for positive ω, --- G f (jω) G(jω) for positive ω, unit circle) By analyzing the figure above, an interesting choice of controller structure to achieve this phase lag is that of a phase lag compensator with the following form: The value of the compensator s should be such that the maximum lag provided by the compensator appears close to the location of the unstable mode we seek to stabilize, which, as previously detailed, is at a frequency of ω = 1675 rad s 1 (Fig. 22). The corner frequencies of the phase lag compensator described in (6.2) are at ω = a and ω = βa rad s 1, so the maximum phase lag occurs at an approximate frequency of rad s 1 [4, 36] (Fig. 24). -5 Diagram of a phase lag compensator Bode Diagram of a phase-lag compensator Phase (deg) Magnitude (db) Phase [deg] Gain [db] ω [rad/s] 1 5 Frequency (rad/sec) Fig. 24 Bode diagram of a generic phase-lag compensator 31

36 The exact choice of values for the controller s parameters responds to the compromise between maximising the phase and gain margins. Thus, having calculated the phase margin and the gain margin through Nyquist diagram analysis for several different combinations of a and β that satisfy the condition 1675 rad s 1, the finally selected values for the controller are a = and β = The value of K c is then chosen to be.1 to achieve the two required encirclements of the 1 point and a sufficiently wide gain margin. Considering these values, the ultimate transfer function of the phase lag compensator, K(s), is: The resulting Nyquist plot for K(jω)G f (jω)g(jω) is shown in Figure Nyquist plot for K(j)G f (j)g(j) Imaginary part Real part Fig. 25 Nyquist diagram of the controlled system with two anticlockwise encirclements of-1 point ( G f (jω) G(jω) for positive ω, --- G f (jω) G(jω) for positive ω, unit circle) As the Nyquist Stability Criterion required (vid. 6.1), there are two anticlockwise encirclements of the 1 +j point. The gain margin is 4.47 db and the phase margin is 21.5 ; consequently, the closed loop system should be stable once the controller is implemented and, considering the value of these margins, the controlled system should be robust to slight plant uncertainties or changes. 32

37 Practical implementation of thee designed controller into the Simulink model Once the controller aimed to stabilize the Simulink model developed in Section 5 hass been designed using Nyquist and Bode techniques, the final step consists of implementing it into the model as a negative feedback loop and analyse its robustness to slight changes in the heat release time delay,. If we implement the phase lag compensator as the transfer function specified in (6.3) into the Simulink model (Fig. 8), the ultimate graphical block diagram for the closed loop controlled system is obtained (Fig. 26). Larger views of the model can be found in Appendices D and E. However, before simulating this system to prove the compensator s power to stabilize the system for a given flame model time delay, (normalized through parameter k; in the current analysis, for k =.5), as it happened when this controller was designed, a mention regardingg the elimination of higher order modes needs to be made. If in the controller design we had h to introduce a low pass filter to analyse the Nyquist diagram (Fig. 23) so that the analysis was focused on the low frequencies of interest and the only unstable mode (n = 1) ) over this range of frequency was highlighted in the plot (higher order unstable modes the t first one at n = 11, were filtered because they were out of the range of interest), this filtering will be here achievedd by fixing a specific maximum step size in the simulations. Fig. 26 Simulink block diagram of the t controlled system 33

38 In Section 5, we traced the stability maps of our model combustor derived from two different configurations of the simulations: the first one imposing a maximum step size of 1 5 s and the second one limiting this simulation parameter to 1 4 s. As it was then explained, the first setting provided similar results to ones obtained through Bode plot analysis when the ten first modes were considered, whereas the second setting led to a stability map really close to the one which could be traced considering only the first six modes. Because we now need to restrict our analysis to low frequencies, so that we face a system arrangement with just one unstable mode over the frequencies of study and, thus, higher order unstable modes have to be well cut off, we will perform the following simulations aimed to check the validity of the designed controller fixing a maximum step size of 1 4 s. In order words, we are using this parameter setting as a trial and error low pass filter to make sure that out of interest high frequencies do not affect the time response of the system, an application which had been previously suggested in Section 5. Having done this remark, the Simulink model is simulated without and with the negative feedback loop so the pressure time response at x = x ref for the same settings, specially the value of factor k (k =.5), can be compared with the control OFF and ON, respectively (Fig. 27). 4 Pressure measurement with control OFF Amplitude Pressure [Pa] Pressure Amplitude [Pa] Time [s] Pressure measurement with control ON Time [s] Fig. 27 Comparison between pressure measurements with control OFF and ON, respectively 34

39 Analysis of control robustness The last step of the implementation of feedback control to the modelled combustor focuses on assessing the robustness of the designed controller when the system faces slight changes in the flame model time delay,. The aim of this analysis is to determine a margin of the parameter k for which the first frequency mode and, consequently, the whole system, remain stable without modifying the values of the compensator K(s). Knowing that /, the study simply consists of analyzing the pressure time response of the system under varying values of the k, starting for the central value k =.5 for which the controller has been specifically designed, and classify it between stable or unstable for each case. k =.39 k =.4 k =.45 k =.5 k =.52 k =.53 Amplitude Pressure [Pa] [Pa] Amplitude Pressure [Pa] Amplitude Pressure [Pa] Amplitude Pressure [Pa] Amplitude Pressure [Pa] Amplitude Pressure [Pa] 1 x x Time [s] [s] Fig. 28 Pressure time response under varying values of k (normalization of heat-release time delay) 35

40 By observing Figure 28, it is easy to deduce the interval of k around the face value k =.5 for which the modelled system is stabilized by the designed fixed parameter controller. This interval, which informs about the robustness of the controller to changes in the heat release time delay, may be expressed as: If the flame model time delay suffers any change such that the value of its normalization, expressed as k, is moved out of this interval, in principle a new controller will have to be designed to stabilize the modeled combustor (some exceptions to this statement will be later explained). If we examine it in depth, the information provided by Figure 28 does not end here: interval (6.4) does not only inform about the robustness of the controller but also confirms the usefulness as a lowpass filter of a correct setting of the maximum step size of the simulation. In section 6.1.2, it was explained that, when imposing a maximum simulation step size of 1 4 s., approximately only the first six modes (n = 1 6) of frequency were captured in the time response because higher order modes were filtered and therefore had a negligible influence on the system s plotted behavior. This assumption is again corroborated in Figure 28. Bode plot analysis show that for values from k =.15 to k = 1, the first mode of the system is unstable, but this instability is meant to be interrupted by the designed controller. Then, once the controller has been added to the model to stabilize this first mode, the new responsible for desstabilizing the system is the next unstable mode, which will appear at higher frequencies. However, we only want this second, and higher, unstable modes to affect the system s response if they are over the low frequencies of interest and that is why we use the step size of the simulation as a low pass filter. Starting with the face value of k =.5 and moving down, for k =.45 the second unstable mode is the eighth (n = 8), but this is filtered when a maximum step size of 1 4 is imposed and so the system remains stable. The same happens for k =.4: the second unstable mode is the seventh (n = 7), which is filtered too and consequently a stable response arises. Nevertheless, for k =.39, the sixth mode (n = 6) is unstable, so it is not filtered by our simulation settings and the time response is plotted as unstable. A similar casuistry is observed if the value of k is moved up. For k =.52, the second mode (n = 2) is right on the limit between stability and instability and this is reflected on the time response: peak values are higher but the response is still stable. For k =.53, however, the second (n = 2) and the fourth (n = 4) modes are clearly unstable and, as these modes are included in the six first modes that are considered when fixing a maximum step size of 1 4 s, the response of the simulation becomes unstable. Figure 29 shows the ultimate example and constitutes one of the exceptions previously announced. When analyzing the system for k = 2.45, the situation is quite similar to the one derived from k =.5: unstable modes are n = 1, 8, 1 As a result of this, the system response is expected to be stable because the first unstable mode (n = 1) is stabilized by the designed phase lag compensator and higher order unstable modes (n = 8, 1 ) are filtered by the simulation s step size restriction. 36

41 6 Pressure measurement with control ON for k= Amplitude [Pa] Time [s] Fig. 29 Pressure measurement with control ON for k = 2.45 In conclusion, the designed controller can stabilize any configuration of the model such that only the first mode is unstable over the low frequencies of interest, in this case, limited up to sixth mode. Thus, by fixing a determinate maximum step size of the simulation, the user can choose how low the range of frequency to which he/she wants to restrict the research is. In the current analysis, it was decided to focus it on, approximately, the six first modes of frequency. 7. FUTURE WORK At this point, it is clear that research aimed to model combustion instabilities with Simulink presents several lines of future development, as different stages of the work done up to date may be improved in order to obtain models that present an acoustic behaviour closer to the performance by real combustors. Firstly, some of the assumptions and simplifications specified in Section 3.2 can be modified, starting for including the contribution of entropy waves in future models, following the analysis by Morgans, Annaswamy and Chee Su Goh [37,38], or accounting for changes in flow parameters across the flame axial plane (temperature, density, speed of sound, Mach number, etc.). Focusing on the model combustor develop in Section 5, the logical next step would imply adding mean flow to the numerical development of the model. As well as that, future research could select a more complex flame model (vid. 5.22) aiming to emphasize the effect of combustion oscillations at low frequencies and, on parallel, accounting for important factors such as combustor geometry, laminar or turbulent flows, diffusion or premixed flames or combustion efficiency [38]. Finally, regarding the feedback control of the model, it is obvious that the controller designed in this project is just an initial approach to control techniques and constitutes the base for further applications of closed loop control to Simulink model. Consequently, the complexity of the implemented feedback control may be widely expanded, for example, through the application of parameters that are varied in time to respond to measurements on the system. 37

42 8. CONCLUSIONS At this point, it seems clear that the goals that were settled at the beginning of the project have been achieved. Although a first model simulating the behaviour of an acoustically excited tube has been designed and the derived results perfectly match the theory of acoustics, this has been conceived as a prior but necessary step to face the true aim of the study: the modeling of thermoacoustic instabilities using a graphical interface. Therefore, a generic combustor experiencing combustion oscillations has been successfully developed in the time domain using a one dimensional linearized approach and then modelled using Simulink 2. Results obtained with this model have been doubled checked with the ones provided by a similar study carried out by Dowling and Morgans in the Laplace domain [4]: analysis of stability and the comparison of the gain and phase shifts provided by these two methods for different values of the heat release time delay are fully satisfactory. This model constitutes a first version of the easy to use tool to model thermoacoustic instabilities in a graphical and customizable environment we aimed to provide when this project was initiated. Although several improvements may be applied to this first model (vid. Section 7), it represents an important and firm step to reach the ultimate ambitious objective. Finally, as an initial approach to the interruption of combustion instabilities, fixed parameter feedback control has been added to the system and its potential to stabilize the system for a specific given setting with only one unstable mode over the low frequencies of interest has been demonstrated. This last section of the project has been completed with an analysis of the controller s robustness to slight changes in the flame model time delay. Here arises the importance of an adequate configuration of the simulations in Simulink: the maximum step size setting has been proved as a powerful tool to restrict the scope of the study to the low frequencies at which combustion oscillations tend to occur. Over and above the objectives and results attained in this project, it may be affirmed that the modelling of thermoacoustic instabilities with Simulink presents an important room for improvement. In this sense, moving towards more accurate and complex numerical developments but preserving the fundamental idea of providing easily understandable visual models represents an important but involving challenge for the near future. 2 Simulink is an environment for multidomain simulation and Model Based Design for dynamic and embedded systems. 38

43 9. REFERENCES [1] Poinsot, T., Trouve, A., Veynante and D., Candel, S., J. of Fluid Mech. 177 (1987) [2] Renard, P. H., Thevenin, D., Rolon, J. C. and Candel, S., Prog. Energy Combust. Sci. 26 (3) (2) [3] Richards, G. A. and Janus, M. C., ASME J. of Eng. For Gas Turbines and Power 12 (1998) [4] Dowling, A. P., and Morgans, A. S., Feedback control of combustion oscillations, Annu. Rev. Fluid Mech. 37 (25) [5] McManus, K. R., Poinsot, T. and Candel, S. M., A review of active control of combustion instabilities, Prog. Energy Combust. Sci 19 (1993) 1 29 [6] Ducruix, S., Schuller, T., Durox, D. and Candel, S., J. of Prop. and Power 19 (5) (23) [7] Seume, J., Vortmeyer, N., Krause, W., Hermann, J., Hantschk, C., Zangl, P., Gleiss, S., Vortmeyer, D. and Orthmann, A., J. of Eng. For Gas Turbines and Power 12 (1998) [8] Mongia, H. C., Held, T. J., Hsiao, G. C. and Pandalai, R. P., J of Prop. and Power 19 (5) (23) [9] Schadow, K. and Yang, V., AGARD Workshop Report (1996) [1] Zhu, M., Dowling, A. P and Bray, K. N., J. of Eng. for Gas Turbines and Power Trans. of the ASME 123 (4) (21) [11] Giuliani, F., Gajan, P., Diers, O. and Ledoux, M., Proc. of the Combustion Inst. 29 (23) [12] Richards, G. A. and Janus, M. C., ASME J. of Eng. for Gas Turbines and Power 12 (1998) [13] Correa, S. M., Proc. of the 27 th Int. Symp. On Combustion, Vol. 2 (1998), pp [14] Bloxsidge, J. G., Dowling, A. P. and Langhorne, P. J., J. of Fluid Mech. 193 (1988) [15] Bloxsidge, J. G., Dowling, A. P. and Langhorne, P. J., J. of Fluid Mech. 193 (1988) [16] Schadow, K., Wilson, R. and Gutmark, E., AAIA J. 25 (1988) [17] Crocco, L. and Cheng, S. I., Theory of combustion instability in liquid propellant rocket motors, Butterworths, London, [18] Hedge, U., Reuter, D., Zinn, B. and Daniel, B., AAIA (1987) [19] Putnam, A. A., Combustion Driven Oscillations in Industry, American Elsevier, [2] Rayleigh, J. W. S., The Theory of Sound, Vol. II, Dover Publications, [21] Culick, F., AGARD Conf. on Combustion Instabilities in Liquid Fueled Propulsion Systems (1988) [22] Richards, G. A., Straub, D. L. and Robey, E. H., J. of Prop and Power 19 (5) (23) [23] Steele, R. C., Cowell, L. H., Cannon, S. M. and Smith, C. E., J. of Eng. for Gas Turbines and Power 122 (3) (2) [24] Zhao, D., Barrow, C. A., Morgans, A. S., and Carrotte, J., Acoustic damping of a Helmoltz resonator with an oscillating volume, AAIA Journal, Vol. 47, No. 7 (29) [25] Joshi, N., Epstein, M., Durlak, S., Marakovits, S. and Sabla, P., 94 GT 253 (1994) [26] Chu, B. T., Acta Mechanica 1 (1964)

44 [27] Bloxsidge, J. G., Dowling, A. P., Hooper, N. and Langhorne, P. J., AAIA J. 26 (1988) [28] Dowling, A. P., Nonlinear self excited oscillations of a ducted flame, J. of Fluid Mech. 346 (1997) [29] Peracchio, A. A. and Proscia, W. M., ASME J. of Eng. for Gas Turbines and Power 121 (3) (1999) [3] Stow, S. R. and Dowling, A. P., Proc. of the ASME Turbo Expo, GT (24) [31] Noiray, N., Durox, D., Schuller, T. and Candel, S., J. of Fluid Mech. 615 (28) [32] Dowling, A. P., and Ffowcs Williams, J. E., Sound and sources of sound, Department of Engineering, Univ. of Cambridge, Ellis Horwood Publishers, [33] Evesque, S., Adaptive control of combustion oscillations, Department of Engineering, Univ. of Cambridge, 2. [34] Cohen, J. M. and Banaszuk, A., Factors affecting the control of unstable combustors, J. Prop. Power 19(5) (23) [35] Schuermans, B., Modeling and control of thermoacoustic instabilities, PhD Thesis, École Polytechnique Fédérale de Lausanne (23). [36] Ogata, K., Modern Control Engineering, Pearson Education International (22). [37] Morgans, A. S. and Annaswamy A. M., Adaptive control of combustion instabilities for combustion systems with right half plane zeros, Combust. Sci. and Tech, 18 (28) [38] Su Goh, C. and Morgans, A. S., The effect of entropy wave dissipation and dispersion on thermoacoustic instability in a model combustor. 1. COMPLEMENTARY BIBLIOGRAPHY Illingworth, S. J. and Morgans, A. S., Advances in feedback control of the Rijke tube thermoacoustic instability, International Journal of Flow Control (211) Morgans, A. S. and Stow, S. R., Model based control of combustion instabilities in annular combustors, Combustion and Flame, 15 (27) Dowling, A. P. and Stow, S. R., Acoustic analysis of gas turbine combustors, Journal of Propulsion and Power, 19 (23) Dowling, A. P., The Calculation of Thermoacoustic Oscillations, Journal of Sound and Vibration, 18 (1995) Wang, C. H. and Dowling, A. P., The absorption of axial acoustic waves by a Helmholtz resonator, Journal of Sound and Vibration (Draft paper) 4

45 Appendices A. Simulink block diagram of the acoustically excited tube without combustion B. Simulink block diagram of the model combustor C. Simulink block diagram of the model combustor with masked subsystems (aesthetic arrangement) D. Simulink block diagram of the model combustor with feedback control E. Simulink block diagram of the model combustor with feedback control and masked subsystems

46 Appendix A. Simulink block diagram of the acoustically excited tube without combustion (Fig. 3)

47 Appendix B. Simulink block diagram of the model combustor (Fig. 8)

48 Appendix C. Simulink block diagram of the model combustor with masked subsystems (aesthetic arrangement)

49 Appendix D. Simulink block diagram of the model combustor with feedback control

50 Appendix E. Simulink block diagram of the model combustor with feedback control and masked subsystems