- Email: [email protected]
Experimental analysis of thermo-acoustic combustion instability
Applied Energy 70 (2001) 179–191 www.elsevier.com/locate/apenergy
Experimental analysis of thermo-acoustic combustion instability A. Ficheraa,*, C. Losennob, A. Paganoa a
Istituto di Fisica Tecnica, Universita` di Catania, Viale A. Doria 6, 95125 Catania, Italy b School of Mechanical Engineering, University of Edinburgh, The King’s Buildings, Mayfield Road, Edinburgh EH9 3JL, Scotland, UK Received 26 February 2001; accepted 24 March 2001
Abstract Thermo-acoustic instabilities are dynamic phenomena that represent a major threat for most modern combustion systems. Many studies, mainly undertaking a linear analysis of experimental data, have been carried out to provide a deeper understanding of the underlying phenomena. However, linear analysis may lead to an oversimplified view of the problem, which involves many complex non-linear interactions, and a more detailed non-linear analysis may be necessary. This paper presents both linear and non-linear analyses of experimental measurements observed in a methane-fuelled laboratory combustor. The linear analysis aims to verify the existence of thermo-acoustic instabilities and consists of the elaboration of both power-spectral density distribution and Rayleigh Index of the experimental time series. Nonlinear analysis aims to investigate dynamic behaviours by means of the deterministic chaos theory. Results of the analyses show that combustion instabilities occur in all the experimental operating conditions. Moreover, the existence of a chaotic source in the combustion system under study is demonstrated. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: Combustion instability; Experimental nonlinar dynamics; Chaos
1. Introduction Thermo-acoustic instability represents one of the major problems affecting modern high-performance combustion chambers. It consists of the coupling and self-sustenance * Corresponding author. Tel.: +39-95-738-2450; fax: +39-95-337994. E-mail addresses: afi[email protected] (A. Fichera), [email protected] (C. Losenno), [email protected] (A. Pagano). 0306-2619/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0306-2619(01)00020-4
180
A. Fichera et al. / Applied Energy 70 (2001) 179–191
of large-amplitude and low-frequency pressure and heat release oscillations [1]. In fact, the coupling between acoustic waves and heat sources modulates the instantaneous heat-release in phase with pressure oscillations [2], exciting the resonant acoustic modes of the combustion chamber. Many harmful effects arise due to combustion instability: non-uniform exhausted gas thermal distribution, combustion efficiency reduction, thermal-NOx growth and wall thermal stress (both caused by local temperature peaks), vibrational phenomena, all contributing to compromise the performances and the structural integrity of the burner. Many studies have been performed to provide a deeper understanding of the underlying phenomena. Common approaches to the analysis of combustion instabilities are generally based on linear tools for experimental data analysis. Indeed, linear approaches are beneficial in revealing the unstable behaviour of combustion processes. However, combustion instabilities are in general very complex due to the coupled interactions and the inherent non-linearities of the involved phenomena, such as rapid chemical reactions, turbulent flow and acoustic phenomena. A few years ago, Culick [3] pointed out that ‘‘measured time histories of the pressure in a combustion chamber will always show aspects of randomness as well as contributions from well-defined oscillations’’. In these cases, power-spectral densities distribution of the experimental time series consists of sharp peaks rising from a broadband background [4,5], traditionally explained as superposition of noise and acoustic oscillations. On the other hand, modern dynamical system theory proposes as different explanation an intrinsic constraint of FFT analysis: the accuracy of the FFT representation depends on the basic assumption of a system behaving linearly (or not strongly influenced by non-linear effects). When this assumption does not hold, the representation in the frequency domain is typically characterised by the broadband power-spectral density distribution. These results may often be due to the intrinsic complexity of the system dynamics and the noisy spectrum simply indicates the inadequacy of the representation in the frequency-amplitude space. The first part of this study is devoted to demonstrate the existence of thermoacoustic combustion instability through the classical linear approaches to the analysis of experimental data. The second part of the paper aims to verify, by means of non-linear time series analysis techniques, whether or not unstable combustion is governed by chaos.
2. Experimental set-up The experimental facilities consisted of a laboratory combustor and an optical system. Fig. 1 shows the experimental set-up. The laboratory-scale combustor was 1:4 scale model of a Dry Low NOx combustor devoted to 5–10 MWe cogeneration plants. In this system, no water injection is required for the reduction of NOx. This is obtained by using high values of the flame stoichiometric ratio in order to reduce the temperature below the threshold at which thermal NOx is formed. The burner was tested at atmospheric pressure (without the turbine) and with inlet air at ambient temperature, considering a thermal load of 250 kW. The fuel used
A. Fichera et al. / Applied Energy 70 (2001) 179–191
181
Fig. 1. Experimental apparatus (dimensions are reported in mm).
during the tests was natural gas with 93.5% of CH4, 2.5% of N2, 2% of C2H6 and 2% of other minor compounds, which result in a higher heating value HHV=51 MJ/kg and in a lower heating value LHV= 49 MJ/kg. Fig. 2 shows the premixing zone. The main percentage of the total gas, namely primary gas, fed the premixed flame. A pilot diffusive flame was used to burn a percentage ranging from 0 to 15% of the total gas, which was injected by 120.9 mm diameter orifices directly into the primary combustion zone. The combustor was also provided with 12 gas injectors for the premixed flame, each consisting of six orifices of 1 mm internal diameter. Moreover, the burner was equipped with 12 swirling blades, placed downstream the premixing fuel injectors; the position of the blades, in the experimental tests was set to zero so that no swirl was introduced into the flow. A reduction of the diameter of
Fig. 2. Combustor premixing zone.
182
A. Fichera et al. / Applied Energy 70 (2001) 179–191
the inlet duct from 64.5 to 33.3 mm, corresponding to a reduction factor of the duct section of 3.75, accelerated the mixture so that flashback did not pose a threat. The combustion chamber consisted of a 1030 mm long steel cylinder of 350 mm diameter. The sides of the combustion duct were fitted with quartz windows allowing optical access. Dilution ports were not present. Measurements of the heat release fluctuations were carried out using an optical system sensing the CH intensity emissions (wavelength l=430 nm, l=10 nm) [6–8]. Fig. 3 shows the optical system; it consisted of a sapphire fibre (diameter 1.1 mm, length 305 mm) that detected the emissions signal from the flame. The light beam was sent through a convex lens to a filter and through another convex lens to a photodiode. A water-cooled jacket contained the lenses and the filter in order to protect them from the high temperature [9]. The experimental time series were recorded with a sampling time step Ts=0.1 ms and sampling frequency fc=10 kHz [10]. The data-acquisition system consisted of a data logger (Yokogawa) for the CH emissions, pressure transducer and microphone. Signals coming from the sensors were collected at a frequency of 10 kHz over a sampling interval of 12.8 s for each operating condition. Experimental tests were performed by varying the stoichiometric ratio l (ratio of air rate over stochiometric air rate) and the pilot fuel percentage PFP (percentage of methane fuelling the diffusive stabilising flames). Two different values of l (1.3–1.5) and two values of PFP (0, 7) were considered. Table 1 reports the experimental conditions.
3. Linear analysis Common methods of studying thermo-acoustic instabilities include a linear analysis performed by evaluating the Rayleigh Index [11] and the power spectra of pressure and heat release oscillations [12,13]. The Rayleigh Index is defined by Eq. (1) as the mean value of the product of pressure oscillations p’(x,t) and heat release fluctuations q0 (x,t):
Fig. 3. Optical system.
183
A. Fichera et al. / Applied Energy 70 (2001) 179–191 Table 1 Experimental conditions Operating conditions
l
PFP
C130 C137 C150 C157
1.3 1.3 1.5 1.5
0 7 0 7
G ð xÞ
1 T
ð
q0 ðx; tÞp0 ðx; tÞdt
ð1Þ
T
where T is the period of one oscillation cycle. If G(x)>0, the combustion is unstable as coupling takes place and local amplification of fluctuations occurs. If G(x) < 0, the combustion is stable because of the local damping of fluctuation. The Rayleigh Index was calculated using the internal pressure oscillations and heat release fluctuations and Table 2 shows the results of the calculation. All the operating conditions are characterised by positive values of the Rayleigh Index, confirming that the process is unstable. Also, power spectrum analysis of pressure and heat release fluctuations can be used to verify the excitation of unstable modes of the combustion system. Fig. 4 shows power spectral density distributions of time series detected by the microphone, the pressure transducer and the optical sensor in the experimental tests C130 and C137. The analysis of the plots evidences that all of the operating conditions manifest unstable behaviours. Indeed, the power spectra of pressure oscillations present the same unstable modes of heat release fluctuations. This is consistent with the definition of thermo-acoustic instability stated by the Rayleigh criterion. The spectra show a dominant frequency around 120 Hz. Moreover, the combustion process is affected by the excitation of several different unstable modes, in general hardly distinguishable, in a broad band of frequencies. In these cases, excited frequency bands exist instead of specific frequencies. In other words, the results reported so far are indeed helpful in revealing the unstable behaviour of the combustion process. On the other hand, the excitation of broad frequency-bands does not allow reaching a full insight of its dynamical behaviour. In order to achieve a better understanding of unstable combustion phenomena, a non-linear approach is needed. Table 2 Rayleigh Index of the experimental operating conditions Rayleigh Index C130 C137 C150 C157
0.3515 0.0044 0.0212 0.0155
184
A. Fichera et al. / Applied Energy 70 (2001) 179–191
Fig. 4. Power spectral distributions of signals detected by the optical sensor (a), the pressure transducer (b) and microphone (c) during the experimental operating conditions C130 (l=1.3 and PFP=0) and C137 (l=1.3 and PFP=7).
4. Nonlinear analysis The application of chaos theory to investigate non-linear processes may reveal important aspects of the system dynamics, such as the existence of complex behaviours. The dynamic behaviour of any system must be depicted in a representation space having adequate dimensions. Complex systems typically require the representation space as having more than two dimensions. Generally speaking, the adoption of a n-D Phase Space or State Space is necessary, i.e. a mathematical space with n orthogonal co-ordinate directions corresponding to the variables needed to specify the instantaneous state of the system [4,5]. If a system is deterministic (i.e. not stochastic) and not diverging, the dimension of the phase space needed to give a correct representation is finite and sufficiently low and the representation of its regime solution lies on a bounded region of the phase space, which is therefore called as attractor. When a low dimensional representation of the system dynamics can be drawn but the power spectrum is of the broad-band type, the system is not affected by noise and is more correctly referred to as chaotic. In such cases, the
A. Fichera et al. / Applied Energy 70 (2001) 179–191
185
existence of a regular structure is a primary aspect that is often very important to exploit. This is usually done by characterising the attractor on a geometrical stand point through the dynamical invariants or simply invariants. Some of the most meaningful invariants are Fractal Dimension [5], and Lyapunov Exponents [5,14]. The knowledge of these quantities, for example, allows assessing whether the globally stable behaviour of a deterministic system (i.e. not stochastic) is chaotic or not. This is the reason why the above mentioned invariants have been considered in this paper. 4.1. Fractal dimension dA A typical property of a chaotic behaviour relies on its connection to fractal sets, which describe the geometrical distribution of the chaotic attractor. This connection is so strong that a practical definition of chaotic motion takes a chaotic attractor to be an attracting set with fractal properties [14]. The main features characterising fractal structures are self-similarity and lack of smoothness. The former describes that, after repeated amplification, fractal structures usually maintain the same geometrical appearance, which corresponds to the existence of inherent scale invariance. The latter refers to the jagged or disconnected distributions shown by the groups of trajectories constituting a chaotic attractor. Analytically, these properties are measured by the Fractal or System Dimension dA, which is not necessarily integer and must be greater than two for a chaotic system [5,14]. Despite its importance, there is not a unique analytical definition of fractal dimension. The most common definitions are those of capacity dimension, information dimension and correlation dimension [14]. A major problem of these definitions is that the calculations are computationally intense and are sensitive to noise, which is always present in experimental time-series. For these reasons an indirect approximate evaluation of fractal dimension, based on calculation of the Embedding Dimension and on Takens’ Embedding Theorem [15], is often considered. In fact, the dynamics of a generalized non-linear system can be described using some observable output states, i.e. observable time series that measure specific variables of the system. The Embedding Dimension can be defined as the number of state variables necessary to correctly represent the system dynamics and its value must be chosen suitably. This problem, usually addressed as Phase Space Reconstruction, is often solved using Takens’ method of delays [15]. This method allows evaluating either the minimum dimension of the state space in which the attractor is correctly embedded or the variables that can be used to define the state space. Takens’ method is based on embedding a single scalar time series, measuring a state variable of a dynamical dA-dimensional system, into a dE-dimensional vector. The Embedding theorem ensure that if: dE52dA þ 1
ð2Þ
the creation of the dE-dimensional vector results in the reconstruction of a state space containing a smooth manifold for the dA-dimensional system. In the following, dA will indicate the system (fractal) dimension whereas dE will indicate the
186
A. Fichera et al. / Applied Energy 70 (2001) 179–191
Embedding Dimension, which is not the dimension of the system itself but the number of phase space co-ordinates necessary to correctly embed the attractor in phase space. Eq. (2) can be used to evaluate an indirect approximation of the fractal dimension, dA. If the aim of the evaluation is limited to detect the existence of a possible chaotic behaviour, it is sufficient to verify that dE is greater than 5, i.e. dA greater than 2. Finally, the fractal dimension can be directly evaluated once the Lyapunov exponents have been calculated. 4.2. Lyapunov exponents and Lyapunov dimension The concept of global or local embedding dimension is connected to the existence of a corresponding number of directions along which the system dynamics evolve in phase space. Chaos theory associates to these directions a measure of the exponential tendency of two contiguous (or indistinguishable) trajectories to converge or diverge. Considering for simplicity a 1-D system, which correspond to one direction, if 0 is the distance between two trajectories at time t0, the distance (t) at time t will be (t)=0eht. The exponent h is called the Lyapunov exponent and strongly characterises the dynamic behaviour of the system [14]. The sign of this exponent is of primary importance as it describes if the trajectories diverge (h>0), converge (h< 0) or maintain the same distance (h=0). For a generic system embedded in a dL-dimensional space, the knowledge of the dL Lyapunov exponents can be used to give a direct proof of the existence of chaos [5,14]. From a physical point of view, positive and negative Lyapunov exponents correspond to the existence of directions along which expansion and contraction of the attractor occurs, respectively. A chaotic system must have at least one positive Lyapunov exponent, i.e. one direction in which expansion occurs. Moreover, as it is necessary a net contraction effect for the system to be globally stable, the sum of all Lyapunov exponents must be negative. As noted previously, a direct way of defining the fractal dimension dA is based on Lyapunov exponents and the corresponding fractal dimension is called the Lyapunov dimension, d-Lyap, which is defined by the following expression [14]: k P
hi d Lyap ¼ k il hkþl
ð3Þ
In Eq. (3), hi denotes the ith Lyapunov exponent (ordered for decreasing value, from positive to negative) and k labels the last exponents for which h1+h2+. . .+hk50.
5. Results This section describes the results of the phase-space analysis performed on the experimentally-detected combustion chamber dynamics.
A. Fichera et al. / Applied Energy 70 (2001) 179–191
187
5.1. Attractors Figs. 5 and 6 show the 2-D attractors of the time series detected during the experimental tests. In these plots os(t), tr(t) and mic(t) indicate the time series detected, respectively, by the optical sensor, the pressure transducer and the microphone, and os(t+tau), tr(t+tau) and mic(t+tau) the corresponding delayed copies. As the plots show, the attractors describing different operating conditions of the combustor present sensible shape variations. This represents the first important point of the phase-space representation as it may allow one to characterise the various operating conditions in terms of the morphological differences shown by their attractors. In particular, the existence of such variations indicates a strong phase-space nonuniformity of the combustion process at different values of the stoichiometric ratio. Moreover, the attractors depicted in Figs. 5 and 6 are morphologically very similar to those shown by other experimental systems. In particular, the attractors of the acoustical signals [i.e. mic(t) and tr(t)] are characterised by a phase-space distribution similar to that of the ‘‘air noise system’’ described by Mackenzie and Sandler [16].
Fig. 5. Attractors of the time series detected by the optical sensor (a), the pressure transducer (b) and the microphone (c), during the experimental operating conditions C130 (l=1.3 and PFP=0) and C137 (l=1.3 and PFP=7).
188
A. Fichera et al. / Applied Energy 70 (2001) 179–191
Fig. 6. Attractors of the time series detected by the optical sensor (a), the pressure transducer (b) and the microphone (c), during the experimental operating conditions C150 (l=1.5 and PFP=0) and C157 (l=1.5 and PFP=7).
Far more interesting is the attractor of heat release during operating condition l=1.3 PFP=0 [left-hand side of Fig. 5(a)]. This attractor, at least in its 2-D representation, seems to be characterised by the same canonical structure, namely foldedband hyper-chaotic, displayed by Ro˜ssler’s 4-D non linear circuit [17]. The existence of similarities between the attractors of physically different systems represents a very important feature of chaos analysis. Such similarities may reflect the existence of an analogous (if not identical) mathematical model for the similarly behaving systems, though it is clear that the variable of the model itself refers to physically different quantities. In such cases, it may be possible to define the model of a unknown system simply on the basis of the morphological analogy between its attractors and those of a system for which the mathematical model is known. 5.2. Embedding dimensions and Lyapunov dimension Table 3 reports the values of global and local embedding dimensions, dE and dL, and of Lyapunov dimension d-Lyap.
189
A. Fichera et al. / Applied Energy 70 (2001) 179–191 Table 3 Characteristic dimensions of the time series of the external and internal pressure and of heat release Heat release
C130 C137 C150 C157
Internal pressure
External pressure
dE
d-Lyap
dE
d-Lyap
dE
d-Lyap
6 5 6 6
4.574 4.457 5.359 5.075
6 6 5 5
5.353 4.474 4.570 4.570
5 5 6 5
4.426 4.393 5.189 4.438
The results reported in Table 3 show that the external and internal acoustic fields and the heat release of the flame front are always characterised by a global embedding dimension not less than five. The local dimension behaves in the same way. Moreover, the Lyapunov dimension is always fractional and greater than 2. These results demonstrate the existence of a chaotic source of the dynamics in the combustion system. 5.3. Lyapunov exponents The values of the Lyapunov exponents are reported in Tables 4–6, which refer respectively to the heat release from the flame front, to the internal pressure field and to the external pressure field. The Lyapunov spectra of the three variables herein examined give clear evidence of the existence of chaos. In fact, for all of the time series: Table 4 Lyapunov exponents for the flame-front heat release Heat release
C130 C137 C150 C157
h6
Sum
0.804 0.613 0.283 0.341
0.872 0.925
0.296 0.521 0.445 1.107
h1
h2
h3
h4
h5
0.381 0.218 0.369 0.189
0.241 0.116 0.241 0.082
0.068 0.060 0.122 0.002
0.182 0.182 0.022 0.114
Table 5 Lyapunov exponents for the internal pressure field Internal pressure
C130 C137 C150 C157
h1
h2
h3
h4
h5
h6
Sum
0.243 0.292 0.341 0.382
0.142 0.167 0.189 0.225
0.041 0.025 0.020 0.080
0.075 0.172 0.202 0.161
0.241 0.622 0.784 0.698
0.725
0.615 0.310 0.436 0.172
190
A. Fichera et al. / Applied Energy 70 (2001) 179–191
Table 6 Lyapunov exponents for the external pressure field External pressure
C130 C137 C150 C157
h1
h2
h3
h4
h5
0.252 0.301 0.221 0.301
0.139 0.199 0.161 0.191
0.022 0.062 0.063 0.042
0.218 0.142 0.032 0.196
0.659 0.598 0.164 0.694
h6
0.516
Sum 0.508 0.178 0.267 0.356
a. There are at least two positive Lyapunov exponents, i.e. there are two directions along which expansion of the attractor occurs; hence, the system must be addressed as hyper-chaotic, confirming from an analytical standpoint the morphological similarity encountered with the folded-band attractor, displayed by Rossler’s 4-D non linear circuit [17]. b. There are at least two negative Lyapunov exponents, which correspond to two directions along which contraction of the attractor occurs c. One of the Lyapunov exponents is close to zero, confirming that the system is autonomous [4], i.e. the system dynamics are governed by an internal source and are not produce by an external forcing term d. The sum of the whole set of Lyapunov exponents is negative, i.e. a net contraction effect dominates the system dynamics and ensures that the system is globally stable.
6. Conclusions This paper has presented a study of thermo-acoustic combustion instabilities performed with both a traditional linear approach and a novel non-linear analysis. The linear analysis of experimental data was used in order to verify the existence of unstable combustion conditions but did not permit achieving a full insight of the process, due to the excitation of various unstable modes. The use of non-linear analysis tools allowed characterising the dynamical behaviour of the system. The main result of this work consisted in showing the existence of chaos in the system under study. This was achieved firstly by the morphological similarities existing between the attractors of the system herein examined and other chaotic system (folded-band Ro˜ssler’s attractor and Mackenzie and Sandler air noise systems). Secondly, chaos existence was analytically proved by means of the calculation of global and local embedding dimensions, Lyapunov spectra and Lyapunov dimensions for the time series constituting the whole set of experimental tests. The results point out the opportunity to define a model of the system that takes into account the non-linearities affecting the process, which would represent an important step towards the definition of an effective control system. A non-linear model-based control system might be able to ensure a reduction of combustion instabilities stronger than the one obtained by controllers based on linear approaches.
A. Fichera et al. / Applied Energy 70 (2001) 179–191
191
References [1] Rayleigh JW. The explanation of certain acoustical phenomena. In: Royal Institute Proceeding, vol. 3, London, 1878. p. 536–42. [2] Rayleigh JW. The theory of sound — vol. II. New York: Dover Publications, 1945. [3] Culick FEC. Some recent results for non-linear acoustics in combustion chambers. AIAA Journal (Paper 90-3927) 1996;32(1):146–69. [4] Thompson JMT, Stewart HB. Non-linear dynamics and chaos. New York: J. Wiley & Sons, 1986. [5] Abarbanel HDI. Analysis of observed chaotic data. San Diego: The Institute for Non-Linear Science, 1995. [6] Gutmark E, Parr TP, Hanson-Parr DP, Schadow KC. Use of chemiluminescence and neural networks in active combustion control. In: Proceedings of the 23rd Symposium (International) on Combustion, The Combustion Institute, Pittsburg, 1990. p. 1101–6. [7] John RR, Summerfield M. Studies of mechanism of flame stabilization by a spectral intensity method. Jet Propulsion 1955;27:169–79. [8] John RR, Summerfield M. Effect of turbulence on radiation intensity from propane-air flames. Jet Propulsion 1955;25:535–42. [9] Losenno C, Presaghi M, Romanello F, Sica M, Sidoti G. Sensore Ottico per la Misura del Rilascio Termico e della Stabilita` di Fiamma nella Combustione. ENEA Patent RM99A000623, 12 October, 1999. [10] Cammarata, G., Fichera, A., Losenno, C., Romanello, F, An experimental study of combustion instabilities in premixed combustor. In: Proc. 5th International Conference on Technologies and Combustion for a Clean Environment, vol. 1. Lisbon, Portugal, 1, 1999 p. 501–5. [11] Poinsot T, Trouve´ A, Veynante D, Candel S, Esposito E. Vortex-driven acoustically coupled combustion instabilities. Journal of Fluid Mechanics 1987;177:265–92. [12] McManus KR, Vandsburger U, Bowman CT. Combustor Performance enhancement through direct shear layer excitation. Combustion and Flame 1990;82:75–92. [13] Shih WP, Lee JG, Santavicca DA. Stability and emissions characteristics of a lean premixed gas turbine combustor. In: Proceedings of the 26th Symposium (International) on Combustion, The Combustion Institute, Pittsburg, 1996. p. 2771–8. [14] Rasband SN. Chaotic dynamics of nonlinear systems. New York: J. Wiley & Sons, 1990. [15] Takens F, Detecting strange attractors in fluid turbulence. In: Rand DA, Young LS, editors. Lecture Notes in Mathematics, Dynamical System and Turbulence, 1981, p. 366–81. [16] Mackenzie J, Sandler M. Modelling sound with chaos. In: Proc. IEEE ISCAS94, London, 1994. p. 6.93–6. [17] Hudson JL, Ro´ssler OE, Killory HC. Chaos in a four-variable piecewise-linear system of differential equations. IEEE Transaction on Circuits and Systems 1998;35(7):902–8.
Experimental analysis of thermo-acoustic combustion instability A. Ficheraa,*, C. Losennob, A. Paganoa a
Istituto di Fisica Tecnica, Universita` di Catania, Viale A. Doria 6, 95125 Catania, Italy b School of Mechanical Engineering, University of Edinburgh, The King’s Buildings, Mayfield Road, Edinburgh EH9 3JL, Scotland, UK Received 26 February 2001; accepted 24 March 2001
Abstract Thermo-acoustic instabilities are dynamic phenomena that represent a major threat for most modern combustion systems. Many studies, mainly undertaking a linear analysis of experimental data, have been carried out to provide a deeper understanding of the underlying phenomena. However, linear analysis may lead to an oversimplified view of the problem, which involves many complex non-linear interactions, and a more detailed non-linear analysis may be necessary. This paper presents both linear and non-linear analyses of experimental measurements observed in a methane-fuelled laboratory combustor. The linear analysis aims to verify the existence of thermo-acoustic instabilities and consists of the elaboration of both power-spectral density distribution and Rayleigh Index of the experimental time series. Nonlinear analysis aims to investigate dynamic behaviours by means of the deterministic chaos theory. Results of the analyses show that combustion instabilities occur in all the experimental operating conditions. Moreover, the existence of a chaotic source in the combustion system under study is demonstrated. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: Combustion instability; Experimental nonlinar dynamics; Chaos
1. Introduction Thermo-acoustic instability represents one of the major problems affecting modern high-performance combustion chambers. It consists of the coupling and self-sustenance * Corresponding author. Tel.: +39-95-738-2450; fax: +39-95-337994. E-mail addresses: afi[email protected] (A. Fichera), [email protected] (C. Losenno), [email protected] (A. Pagano). 0306-2619/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0306-2619(01)00020-4
180
A. Fichera et al. / Applied Energy 70 (2001) 179–191
of large-amplitude and low-frequency pressure and heat release oscillations [1]. In fact, the coupling between acoustic waves and heat sources modulates the instantaneous heat-release in phase with pressure oscillations [2], exciting the resonant acoustic modes of the combustion chamber. Many harmful effects arise due to combustion instability: non-uniform exhausted gas thermal distribution, combustion efficiency reduction, thermal-NOx growth and wall thermal stress (both caused by local temperature peaks), vibrational phenomena, all contributing to compromise the performances and the structural integrity of the burner. Many studies have been performed to provide a deeper understanding of the underlying phenomena. Common approaches to the analysis of combustion instabilities are generally based on linear tools for experimental data analysis. Indeed, linear approaches are beneficial in revealing the unstable behaviour of combustion processes. However, combustion instabilities are in general very complex due to the coupled interactions and the inherent non-linearities of the involved phenomena, such as rapid chemical reactions, turbulent flow and acoustic phenomena. A few years ago, Culick [3] pointed out that ‘‘measured time histories of the pressure in a combustion chamber will always show aspects of randomness as well as contributions from well-defined oscillations’’. In these cases, power-spectral densities distribution of the experimental time series consists of sharp peaks rising from a broadband background [4,5], traditionally explained as superposition of noise and acoustic oscillations. On the other hand, modern dynamical system theory proposes as different explanation an intrinsic constraint of FFT analysis: the accuracy of the FFT representation depends on the basic assumption of a system behaving linearly (or not strongly influenced by non-linear effects). When this assumption does not hold, the representation in the frequency domain is typically characterised by the broadband power-spectral density distribution. These results may often be due to the intrinsic complexity of the system dynamics and the noisy spectrum simply indicates the inadequacy of the representation in the frequency-amplitude space. The first part of this study is devoted to demonstrate the existence of thermoacoustic combustion instability through the classical linear approaches to the analysis of experimental data. The second part of the paper aims to verify, by means of non-linear time series analysis techniques, whether or not unstable combustion is governed by chaos.
2. Experimental set-up The experimental facilities consisted of a laboratory combustor and an optical system. Fig. 1 shows the experimental set-up. The laboratory-scale combustor was 1:4 scale model of a Dry Low NOx combustor devoted to 5–10 MWe cogeneration plants. In this system, no water injection is required for the reduction of NOx. This is obtained by using high values of the flame stoichiometric ratio in order to reduce the temperature below the threshold at which thermal NOx is formed. The burner was tested at atmospheric pressure (without the turbine) and with inlet air at ambient temperature, considering a thermal load of 250 kW. The fuel used
A. Fichera et al. / Applied Energy 70 (2001) 179–191
181
Fig. 1. Experimental apparatus (dimensions are reported in mm).
during the tests was natural gas with 93.5% of CH4, 2.5% of N2, 2% of C2H6 and 2% of other minor compounds, which result in a higher heating value HHV=51 MJ/kg and in a lower heating value LHV= 49 MJ/kg. Fig. 2 shows the premixing zone. The main percentage of the total gas, namely primary gas, fed the premixed flame. A pilot diffusive flame was used to burn a percentage ranging from 0 to 15% of the total gas, which was injected by 120.9 mm diameter orifices directly into the primary combustion zone. The combustor was also provided with 12 gas injectors for the premixed flame, each consisting of six orifices of 1 mm internal diameter. Moreover, the burner was equipped with 12 swirling blades, placed downstream the premixing fuel injectors; the position of the blades, in the experimental tests was set to zero so that no swirl was introduced into the flow. A reduction of the diameter of
Fig. 2. Combustor premixing zone.
182
A. Fichera et al. / Applied Energy 70 (2001) 179–191
the inlet duct from 64.5 to 33.3 mm, corresponding to a reduction factor of the duct section of 3.75, accelerated the mixture so that flashback did not pose a threat. The combustion chamber consisted of a 1030 mm long steel cylinder of 350 mm diameter. The sides of the combustion duct were fitted with quartz windows allowing optical access. Dilution ports were not present. Measurements of the heat release fluctuations were carried out using an optical system sensing the CH intensity emissions (wavelength l=430 nm, l=10 nm) [6–8]. Fig. 3 shows the optical system; it consisted of a sapphire fibre (diameter 1.1 mm, length 305 mm) that detected the emissions signal from the flame. The light beam was sent through a convex lens to a filter and through another convex lens to a photodiode. A water-cooled jacket contained the lenses and the filter in order to protect them from the high temperature [9]. The experimental time series were recorded with a sampling time step Ts=0.1 ms and sampling frequency fc=10 kHz [10]. The data-acquisition system consisted of a data logger (Yokogawa) for the CH emissions, pressure transducer and microphone. Signals coming from the sensors were collected at a frequency of 10 kHz over a sampling interval of 12.8 s for each operating condition. Experimental tests were performed by varying the stoichiometric ratio l (ratio of air rate over stochiometric air rate) and the pilot fuel percentage PFP (percentage of methane fuelling the diffusive stabilising flames). Two different values of l (1.3–1.5) and two values of PFP (0, 7) were considered. Table 1 reports the experimental conditions.
3. Linear analysis Common methods of studying thermo-acoustic instabilities include a linear analysis performed by evaluating the Rayleigh Index [11] and the power spectra of pressure and heat release oscillations [12,13]. The Rayleigh Index is defined by Eq. (1) as the mean value of the product of pressure oscillations p’(x,t) and heat release fluctuations q0 (x,t):
Fig. 3. Optical system.
183
A. Fichera et al. / Applied Energy 70 (2001) 179–191 Table 1 Experimental conditions Operating conditions
l
PFP
C130 C137 C150 C157
1.3 1.3 1.5 1.5
0 7 0 7
G ð xÞ
1 T
ð
q0 ðx; tÞp0 ðx; tÞdt
ð1Þ
T
where T is the period of one oscillation cycle. If G(x)>0, the combustion is unstable as coupling takes place and local amplification of fluctuations occurs. If G(x) < 0, the combustion is stable because of the local damping of fluctuation. The Rayleigh Index was calculated using the internal pressure oscillations and heat release fluctuations and Table 2 shows the results of the calculation. All the operating conditions are characterised by positive values of the Rayleigh Index, confirming that the process is unstable. Also, power spectrum analysis of pressure and heat release fluctuations can be used to verify the excitation of unstable modes of the combustion system. Fig. 4 shows power spectral density distributions of time series detected by the microphone, the pressure transducer and the optical sensor in the experimental tests C130 and C137. The analysis of the plots evidences that all of the operating conditions manifest unstable behaviours. Indeed, the power spectra of pressure oscillations present the same unstable modes of heat release fluctuations. This is consistent with the definition of thermo-acoustic instability stated by the Rayleigh criterion. The spectra show a dominant frequency around 120 Hz. Moreover, the combustion process is affected by the excitation of several different unstable modes, in general hardly distinguishable, in a broad band of frequencies. In these cases, excited frequency bands exist instead of specific frequencies. In other words, the results reported so far are indeed helpful in revealing the unstable behaviour of the combustion process. On the other hand, the excitation of broad frequency-bands does not allow reaching a full insight of its dynamical behaviour. In order to achieve a better understanding of unstable combustion phenomena, a non-linear approach is needed. Table 2 Rayleigh Index of the experimental operating conditions Rayleigh Index C130 C137 C150 C157
0.3515 0.0044 0.0212 0.0155
184
A. Fichera et al. / Applied Energy 70 (2001) 179–191
Fig. 4. Power spectral distributions of signals detected by the optical sensor (a), the pressure transducer (b) and microphone (c) during the experimental operating conditions C130 (l=1.3 and PFP=0) and C137 (l=1.3 and PFP=7).
4. Nonlinear analysis The application of chaos theory to investigate non-linear processes may reveal important aspects of the system dynamics, such as the existence of complex behaviours. The dynamic behaviour of any system must be depicted in a representation space having adequate dimensions. Complex systems typically require the representation space as having more than two dimensions. Generally speaking, the adoption of a n-D Phase Space or State Space is necessary, i.e. a mathematical space with n orthogonal co-ordinate directions corresponding to the variables needed to specify the instantaneous state of the system [4,5]. If a system is deterministic (i.e. not stochastic) and not diverging, the dimension of the phase space needed to give a correct representation is finite and sufficiently low and the representation of its regime solution lies on a bounded region of the phase space, which is therefore called as attractor. When a low dimensional representation of the system dynamics can be drawn but the power spectrum is of the broad-band type, the system is not affected by noise and is more correctly referred to as chaotic. In such cases, the
A. Fichera et al. / Applied Energy 70 (2001) 179–191
185
existence of a regular structure is a primary aspect that is often very important to exploit. This is usually done by characterising the attractor on a geometrical stand point through the dynamical invariants or simply invariants. Some of the most meaningful invariants are Fractal Dimension [5], and Lyapunov Exponents [5,14]. The knowledge of these quantities, for example, allows assessing whether the globally stable behaviour of a deterministic system (i.e. not stochastic) is chaotic or not. This is the reason why the above mentioned invariants have been considered in this paper. 4.1. Fractal dimension dA A typical property of a chaotic behaviour relies on its connection to fractal sets, which describe the geometrical distribution of the chaotic attractor. This connection is so strong that a practical definition of chaotic motion takes a chaotic attractor to be an attracting set with fractal properties [14]. The main features characterising fractal structures are self-similarity and lack of smoothness. The former describes that, after repeated amplification, fractal structures usually maintain the same geometrical appearance, which corresponds to the existence of inherent scale invariance. The latter refers to the jagged or disconnected distributions shown by the groups of trajectories constituting a chaotic attractor. Analytically, these properties are measured by the Fractal or System Dimension dA, which is not necessarily integer and must be greater than two for a chaotic system [5,14]. Despite its importance, there is not a unique analytical definition of fractal dimension. The most common definitions are those of capacity dimension, information dimension and correlation dimension [14]. A major problem of these definitions is that the calculations are computationally intense and are sensitive to noise, which is always present in experimental time-series. For these reasons an indirect approximate evaluation of fractal dimension, based on calculation of the Embedding Dimension and on Takens’ Embedding Theorem [15], is often considered. In fact, the dynamics of a generalized non-linear system can be described using some observable output states, i.e. observable time series that measure specific variables of the system. The Embedding Dimension can be defined as the number of state variables necessary to correctly represent the system dynamics and its value must be chosen suitably. This problem, usually addressed as Phase Space Reconstruction, is often solved using Takens’ method of delays [15]. This method allows evaluating either the minimum dimension of the state space in which the attractor is correctly embedded or the variables that can be used to define the state space. Takens’ method is based on embedding a single scalar time series, measuring a state variable of a dynamical dA-dimensional system, into a dE-dimensional vector. The Embedding theorem ensure that if: dE52dA þ 1
ð2Þ
the creation of the dE-dimensional vector results in the reconstruction of a state space containing a smooth manifold for the dA-dimensional system. In the following, dA will indicate the system (fractal) dimension whereas dE will indicate the
186
A. Fichera et al. / Applied Energy 70 (2001) 179–191
Embedding Dimension, which is not the dimension of the system itself but the number of phase space co-ordinates necessary to correctly embed the attractor in phase space. Eq. (2) can be used to evaluate an indirect approximation of the fractal dimension, dA. If the aim of the evaluation is limited to detect the existence of a possible chaotic behaviour, it is sufficient to verify that dE is greater than 5, i.e. dA greater than 2. Finally, the fractal dimension can be directly evaluated once the Lyapunov exponents have been calculated. 4.2. Lyapunov exponents and Lyapunov dimension The concept of global or local embedding dimension is connected to the existence of a corresponding number of directions along which the system dynamics evolve in phase space. Chaos theory associates to these directions a measure of the exponential tendency of two contiguous (or indistinguishable) trajectories to converge or diverge. Considering for simplicity a 1-D system, which correspond to one direction, if 0 is the distance between two trajectories at time t0, the distance (t) at time t will be (t)=0eht. The exponent h is called the Lyapunov exponent and strongly characterises the dynamic behaviour of the system [14]. The sign of this exponent is of primary importance as it describes if the trajectories diverge (h>0), converge (h< 0) or maintain the same distance (h=0). For a generic system embedded in a dL-dimensional space, the knowledge of the dL Lyapunov exponents can be used to give a direct proof of the existence of chaos [5,14]. From a physical point of view, positive and negative Lyapunov exponents correspond to the existence of directions along which expansion and contraction of the attractor occurs, respectively. A chaotic system must have at least one positive Lyapunov exponent, i.e. one direction in which expansion occurs. Moreover, as it is necessary a net contraction effect for the system to be globally stable, the sum of all Lyapunov exponents must be negative. As noted previously, a direct way of defining the fractal dimension dA is based on Lyapunov exponents and the corresponding fractal dimension is called the Lyapunov dimension, d-Lyap, which is defined by the following expression [14]: k P
hi d Lyap ¼ k il hkþl
ð3Þ
In Eq. (3), hi denotes the ith Lyapunov exponent (ordered for decreasing value, from positive to negative) and k labels the last exponents for which h1+h2+. . .+hk50.
5. Results This section describes the results of the phase-space analysis performed on the experimentally-detected combustion chamber dynamics.
A. Fichera et al. / Applied Energy 70 (2001) 179–191
187
5.1. Attractors Figs. 5 and 6 show the 2-D attractors of the time series detected during the experimental tests. In these plots os(t), tr(t) and mic(t) indicate the time series detected, respectively, by the optical sensor, the pressure transducer and the microphone, and os(t+tau), tr(t+tau) and mic(t+tau) the corresponding delayed copies. As the plots show, the attractors describing different operating conditions of the combustor present sensible shape variations. This represents the first important point of the phase-space representation as it may allow one to characterise the various operating conditions in terms of the morphological differences shown by their attractors. In particular, the existence of such variations indicates a strong phase-space nonuniformity of the combustion process at different values of the stoichiometric ratio. Moreover, the attractors depicted in Figs. 5 and 6 are morphologically very similar to those shown by other experimental systems. In particular, the attractors of the acoustical signals [i.e. mic(t) and tr(t)] are characterised by a phase-space distribution similar to that of the ‘‘air noise system’’ described by Mackenzie and Sandler [16].
Fig. 5. Attractors of the time series detected by the optical sensor (a), the pressure transducer (b) and the microphone (c), during the experimental operating conditions C130 (l=1.3 and PFP=0) and C137 (l=1.3 and PFP=7).
188
A. Fichera et al. / Applied Energy 70 (2001) 179–191
Fig. 6. Attractors of the time series detected by the optical sensor (a), the pressure transducer (b) and the microphone (c), during the experimental operating conditions C150 (l=1.5 and PFP=0) and C157 (l=1.5 and PFP=7).
Far more interesting is the attractor of heat release during operating condition l=1.3 PFP=0 [left-hand side of Fig. 5(a)]. This attractor, at least in its 2-D representation, seems to be characterised by the same canonical structure, namely foldedband hyper-chaotic, displayed by Ro˜ssler’s 4-D non linear circuit [17]. The existence of similarities between the attractors of physically different systems represents a very important feature of chaos analysis. Such similarities may reflect the existence of an analogous (if not identical) mathematical model for the similarly behaving systems, though it is clear that the variable of the model itself refers to physically different quantities. In such cases, it may be possible to define the model of a unknown system simply on the basis of the morphological analogy between its attractors and those of a system for which the mathematical model is known. 5.2. Embedding dimensions and Lyapunov dimension Table 3 reports the values of global and local embedding dimensions, dE and dL, and of Lyapunov dimension d-Lyap.
189
A. Fichera et al. / Applied Energy 70 (2001) 179–191 Table 3 Characteristic dimensions of the time series of the external and internal pressure and of heat release Heat release
C130 C137 C150 C157
Internal pressure
External pressure
dE
d-Lyap
dE
d-Lyap
dE
d-Lyap
6 5 6 6
4.574 4.457 5.359 5.075
6 6 5 5
5.353 4.474 4.570 4.570
5 5 6 5
4.426 4.393 5.189 4.438
The results reported in Table 3 show that the external and internal acoustic fields and the heat release of the flame front are always characterised by a global embedding dimension not less than five. The local dimension behaves in the same way. Moreover, the Lyapunov dimension is always fractional and greater than 2. These results demonstrate the existence of a chaotic source of the dynamics in the combustion system. 5.3. Lyapunov exponents The values of the Lyapunov exponents are reported in Tables 4–6, which refer respectively to the heat release from the flame front, to the internal pressure field and to the external pressure field. The Lyapunov spectra of the three variables herein examined give clear evidence of the existence of chaos. In fact, for all of the time series: Table 4 Lyapunov exponents for the flame-front heat release Heat release
C130 C137 C150 C157
h6
Sum
0.804 0.613 0.283 0.341
0.872 0.925
0.296 0.521 0.445 1.107
h1
h2
h3
h4
h5
0.381 0.218 0.369 0.189
0.241 0.116 0.241 0.082
0.068 0.060 0.122 0.002
0.182 0.182 0.022 0.114
Table 5 Lyapunov exponents for the internal pressure field Internal pressure
C130 C137 C150 C157
h1
h2
h3
h4
h5
h6
Sum
0.243 0.292 0.341 0.382
0.142 0.167 0.189 0.225
0.041 0.025 0.020 0.080
0.075 0.172 0.202 0.161
0.241 0.622 0.784 0.698
0.725
0.615 0.310 0.436 0.172
190
A. Fichera et al. / Applied Energy 70 (2001) 179–191
Table 6 Lyapunov exponents for the external pressure field External pressure
C130 C137 C150 C157
h1
h2
h3
h4
h5
0.252 0.301 0.221 0.301
0.139 0.199 0.161 0.191
0.022 0.062 0.063 0.042
0.218 0.142 0.032 0.196
0.659 0.598 0.164 0.694
h6
0.516
Sum 0.508 0.178 0.267 0.356
a. There are at least two positive Lyapunov exponents, i.e. there are two directions along which expansion of the attractor occurs; hence, the system must be addressed as hyper-chaotic, confirming from an analytical standpoint the morphological similarity encountered with the folded-band attractor, displayed by Rossler’s 4-D non linear circuit [17]. b. There are at least two negative Lyapunov exponents, which correspond to two directions along which contraction of the attractor occurs c. One of the Lyapunov exponents is close to zero, confirming that the system is autonomous [4], i.e. the system dynamics are governed by an internal source and are not produce by an external forcing term d. The sum of the whole set of Lyapunov exponents is negative, i.e. a net contraction effect dominates the system dynamics and ensures that the system is globally stable.
6. Conclusions This paper has presented a study of thermo-acoustic combustion instabilities performed with both a traditional linear approach and a novel non-linear analysis. The linear analysis of experimental data was used in order to verify the existence of unstable combustion conditions but did not permit achieving a full insight of the process, due to the excitation of various unstable modes. The use of non-linear analysis tools allowed characterising the dynamical behaviour of the system. The main result of this work consisted in showing the existence of chaos in the system under study. This was achieved firstly by the morphological similarities existing between the attractors of the system herein examined and other chaotic system (folded-band Ro˜ssler’s attractor and Mackenzie and Sandler air noise systems). Secondly, chaos existence was analytically proved by means of the calculation of global and local embedding dimensions, Lyapunov spectra and Lyapunov dimensions for the time series constituting the whole set of experimental tests. The results point out the opportunity to define a model of the system that takes into account the non-linearities affecting the process, which would represent an important step towards the definition of an effective control system. A non-linear model-based control system might be able to ensure a reduction of combustion instabilities stronger than the one obtained by controllers based on linear approaches.
A. Fichera et al. / Applied Energy 70 (2001) 179–191
191
References [1] Rayleigh JW. The explanation of certain acoustical phenomena. In: Royal Institute Proceeding, vol. 3, London, 1878. p. 536–42. [2] Rayleigh JW. The theory of sound — vol. II. New York: Dover Publications, 1945. [3] Culick FEC. Some recent results for non-linear acoustics in combustion chambers. AIAA Journal (Paper 90-3927) 1996;32(1):146–69. [4] Thompson JMT, Stewart HB. Non-linear dynamics and chaos. New York: J. Wiley & Sons, 1986. [5] Abarbanel HDI. Analysis of observed chaotic data. San Diego: The Institute for Non-Linear Science, 1995. [6] Gutmark E, Parr TP, Hanson-Parr DP, Schadow KC. Use of chemiluminescence and neural networks in active combustion control. In: Proceedings of the 23rd Symposium (International) on Combustion, The Combustion Institute, Pittsburg, 1990. p. 1101–6. [7] John RR, Summerfield M. Studies of mechanism of flame stabilization by a spectral intensity method. Jet Propulsion 1955;27:169–79. [8] John RR, Summerfield M. Effect of turbulence on radiation intensity from propane-air flames. Jet Propulsion 1955;25:535–42. [9] Losenno C, Presaghi M, Romanello F, Sica M, Sidoti G. Sensore Ottico per la Misura del Rilascio Termico e della Stabilita` di Fiamma nella Combustione. ENEA Patent RM99A000623, 12 October, 1999. [10] Cammarata, G., Fichera, A., Losenno, C., Romanello, F, An experimental study of combustion instabilities in premixed combustor. In: Proc. 5th International Conference on Technologies and Combustion for a Clean Environment, vol. 1. Lisbon, Portugal, 1, 1999 p. 501–5. [11] Poinsot T, Trouve´ A, Veynante D, Candel S, Esposito E. Vortex-driven acoustically coupled combustion instabilities. Journal of Fluid Mechanics 1987;177:265–92. [12] McManus KR, Vandsburger U, Bowman CT. Combustor Performance enhancement through direct shear layer excitation. Combustion and Flame 1990;82:75–92. [13] Shih WP, Lee JG, Santavicca DA. Stability and emissions characteristics of a lean premixed gas turbine combustor. In: Proceedings of the 26th Symposium (International) on Combustion, The Combustion Institute, Pittsburg, 1996. p. 2771–8. [14] Rasband SN. Chaotic dynamics of nonlinear systems. New York: J. Wiley & Sons, 1990. [15] Takens F, Detecting strange attractors in fluid turbulence. In: Rand DA, Young LS, editors. Lecture Notes in Mathematics, Dynamical System and Turbulence, 1981, p. 366–81. [16] Mackenzie J, Sandler M. Modelling sound with chaos. In: Proc. IEEE ISCAS94, London, 1994. p. 6.93–6. [17] Hudson JL, Ro´ssler OE, Killory HC. Chaos in a four-variable piecewise-linear system of differential equations. IEEE Transaction on Circuits and Systems 1998;35(7):902–8.