K.D. Edwards, C.E.A. Finney, K. Nguyen. University of Tennessee. Knoxville TN 37996-2210. C.S. Daw. Oak Ridge National Laboratory. Oak Ridge TN 37831- ...
Use of Symbol Statistics to Characterize Combustion in a Pulse Combustor Operating Near the Fuel-Lean Limit K.D. Edwards, C.E.A. Finney, K. Nguyen University of Tennessee Knoxville TN 37996-2210 C.S. Daw Oak Ridge National Laboratory Oak Ridge TN 37831-8087
Abstract
Experimental apparatus and methods
We apply symbol-sequence analysis to pulse-combustor oscillations to characterize the onset of instabilities near the fuel-lean limit. Using data from a laboratory combustor, we demonstrate that such techniques can be very useful for finding and classifying complex pressure patterns which can be precursors to misfires and flameout. On-line diagnosis of these instabilities could facilitate active control for modulating the pressure pulses or extending the lean limit.
The data for this study consists of pressure time-series information collected from a laboratory-scale thermal pulse combustor operating over a global equivalence-ratio range from near-stoichiometric to flameout in the fuel-lean condition.
Introduction Pulsating combustion has many potential applications, such as industrial boilers and dryers where the continual disruption of thermal boundary layers by the resulting pressure waves can increase convective heat transfer. In other circumstances, pulsating combustion may occur spontaneously and unintentionally, such as when gas turbines are operated in the lean premix mode. In both intentional and unintentional pulse combustion, operation at very lean conditions tends to increase the complexity of the oscillations, thereby making them extremely difficult to understand and predict. It is highly desirable to be able to rapidly characterize these unstable patterns so that active control strategies can be implemented. In the following discussion, we describe a method which is especially useful for characterizing the complex nonlinear pressure patterns in lean pulse combustion. Our technique is based on the idea of symbolic time-series analysis, which transforms the initial measurements into a sequence of discrete symbols which can be processed very rapidly and efficiently. By using these methods, we specifically intend to detect precursors to misfires and flameout. Our eventual goal is to develop a simple and efficient scheme for real-time detection of so-called “mediating trajectories” [1] which signal the onset of misfire and flameout.
fuel spark plug
1 0 0 1 0 1 0 1 0 1 ceramic 0 1 flame-holder 0 1 0 1
exhaust
cooling water air pressure transducer damping volume
Figure 1: Schematic of pulse combustor. The combustor, shown schematically in Fig. 1, consists of a 0.295-L combustion chamber coupled with a 0.91-m long tailpipe. A ceramic flameholder is installed in the combustion chamber to help stabilize the flame. Air and fuel (propane) enter the combustor at a constant flow rate through separate choked-flow orifices. A motorcycle spark plug is used to ignite the mixture initially. Once ignited, the combustion reaction is self-sustaining and the spark plug can be deactivated. A fraction of the exhaust energy leaving the tailpipe is reflected at the boundary creating a standing compression wave at the acoustic frequency of the tailpipe. This acoustic wave propagates back into the combustion chamber compressing the air-fuel mixture which then ignites. Expansion of the exhaust gases through the tailpipe completes the cycle. A piezoelectric pressure transducer (Kistler 206) was used to measure the pressure fluctuations inside the combustion chamber. A damping volume attached to the pres-
Corresponding author Proceedings of the 1998 Technical Meeting of the Central States Section of the Combustion Institute
sure tap prevents acoustic coupling between the tap and the combustion chamber. The transducer signal was conditioned using a dual-mode amplifier and bandpass filtered from 0.1 to 2000 Hz. The data collection rate was 5000 Hz. The global equivalence ratio of the air-fuel mixture is controlled by varying the pressure drop across the chokedflow orifices. By monitoring this pressure drop and measuring the orifice flow area, the global equivalence ratio is calculated using compressible-flow relations.
(a) = 0.93
(b) = 0.72
(c) = 0.55
(d) = 0.29 (e) = 0.27
(f) = 0.25
(g) = 0.25
0.0
0.1
0.2 0.3 Time [sec]
0.4
0.5
Figure 2: Representative pressure traces over a range of global equivalence ratios. Data are plotted on the same scale. As the stoichiometry approaches the fuel-lean flammability limit, combustion becomes increasingly complex and unstable, until flameout occurs (at the end of segment (g)). Figure 2 shows example time series of the pressure data collected over a range of global equivalence ratios. At nearstoichiometric conditions (a) the pressure trace appears to be somewhat unstable and to transition between a highamplitude and a low-amplitude behavior. As the global equivalence ratio is decreased, the behavior initially becomes more regular (b) with fewer transitions to the lowamplitude condition. Further reductions in global equivalence ratio (c) leads to increased cyclic variation. Low-
frequency oscillations begin to appear which are reminiscent of beating of incommensurate frequencies. At very lean global equivalence ratios (d), combustion instabilities occur which result in misfiring of the combustor. The misfires are often followed by a characteristic multi-peak sequence which becomes more frequent as global equivalence ratio is decreased (e–f). (Examples of this sequence are visible at 0.2, 0.3 and 0.4 sec on segment (f).) Due to frequent misfiring, the ceramic flameholder and combustor walls begin to cool leading to more misfirings (f). Eventually, the combustor becomes too cool to sustain combustion and unrecoverable flameout occurs (g). It should be noted that the equivalence ratios cited are global values based on the flow rates of air and fuel through the choked-flow orifices. A well-mixed air-propane mixture of this ratio normally would not be expected to react and indeed, when introduced to a cold combustor, the spark plug will not ignite it. During operation of the combustor, the ceramic flameholder is heated to temperatures in excess of 2000 K. Also, since the fuel and air are introduced separately, the mixture is not well-mixed, resulting in regions with a local equivalence ratio far different from the global value. We believe the high-temperature environment and mixing effects allow the combustor to be operated at such low global equivalence ratios. The fact that the combustor will not reignite once flameout has occurred supports this hypothesis. As the air-fuel mixture is made more fuel-lean, the exhaust-gas temperature and flame speed both decrease resulting in a lower tailpipe acoustic frequency and a slower reaction rate. We conjecture that as equivalence ratio is reduced, a phase difference develops between combustion and acoustics which leads to the combustion instabilities. Specifically, as the mixture is made more fuel-lean, the combustion timescale increases faster than the acoustic timescale and, consequently, combustion is not complete before the beginning of the next acoustic oscillation. Initially, this timescale difference produces a quasiperiodicity [2] due to the fact that the acoustic and combustion frequencies are incommensurate. As the level of residual fuel increases, the nonlinear coupling apparently leads to deterministic chaos. A hallmark of chaotic dynamical systems is divergence of initially close state points over time. Figure 3 shows two pressure-trace segments recorded within a 6-sec interval. At around 0.05 sec, the pressure traces are nearly identical, but within a dozen combustion cycles, the oscillation timescales differ, seen by the phase shifting of the traces after 0.1 sec. This irregular behavior is indicative of an unstable periodicity. In theory, chaotic oscillations involve an infinite number of unstable periodicities, and it is possible to to use active control to stabilize the system on any selected periodicity. It is also theoretically possible to design active controls which avoid certain unstable periodici-
0.0
0.1 Time [sec]
0.2
Figure 3: Illustration of the divergence of initially similar pressure traces along an unstable periodic orbit. ties. In our work with the pulse combustor, we would like specifically to avoid unstable periodicities leading to flameout. We believe symbolic time-series analysis may provide a key tool for accomplishing such control. Symbolization methods The data-symbolization methods we describe here are based on the general approach suggested by Tang et al. [3] with some modifications to address time irreversibility. For more detailed information on the theoretical basis and development of symbolization as a data-analysis tool, the reader is referred to [4,5]. The key step in applying symbolization to time-series measurements involves transforming the original values into a stream of discretized symbols. We accomplish this by partitioning the range of the observed values into a finite number of regions (usually less than 10) and then assigning a symbol to each measurement based on which region it falls into. For example, the simplest scheme is to assign values of 0 or 1 to each observation depending on whether it is above or below some critical value (a so-called binary partition). More partitions yield larger “alphabets” of symbols, one symbol for each discrete region of the data range. Typically, we define discretization partitions such that 1) the occurrence frequency of any particular symbol is equiprobable with all others, or 2) the measurement range covered by each region is equal. Once a time series is symbolized, the relative frequencies of all possible sequences of length L in the data provide a fast and efficient way to characterize temporal patterns. A simple way to create symbol-sequence histograms is to assign a unique number that identifies each possible sequence, thus making it possible to plot frequency versus sequence number to depict the relative frequency for each pattern. We assign a sequence “code” to each pattern by using the equivalent base-10 value of each base-n sequence, where n is the number of symbols. For example, a sequence of 1 0 1 for a binary partition would have a sequence code of 5. With our equiprobable partitioning convention, the relative frequency of each sequence for random data will be equal (subject to sampling fluctuations). Thus any significant deviation from equiprobable sequences is indicative of time correlation and determinism.
From the above discussion, it can be seen that symbol sequences can be used to represent any possible variation over time, depending on the number of symbols used and the sequence lengths. This is a very powerful property because it does not make any assumption about the nature of the patterns (e.g., it works equally well for linear and nonlinear phenomena). A detailed explanation of the “optimal” choice of number of symbols and sequence length is beyond the scope of this paper but is addressed in [4]. For practical purposes, we have empirically found that partitions of 2–8 symbols and sequence lengths of 2 to 6 are most useful for depicting the observed patterns in pulsecombustor data. For continuous data, such as discussed here, there is also a sampling-rate issue. Specifically, the pulse-combustor signals could be digitized at any selected rate up to several thousand samples per second. At excessive sampling rates, a problem arises because we simply produce long sequences of the same repeated symbol because adjacent measurements do not differ significantly in value. Thus it is important not to oversample the data. In the examples given below, we present a specific approach for selecting a meaningful sample interval. In fact, we show that this approach can be used to generate diagnostics about relevant timescales in the data. Various statistics can be used to characterize symbolsequence histograms. For example, we define a modified Shannon entropy as:
HS (L) = ? log 1N
X Pi;L log Pi;L ;
seq i
(1)
where Nseq is the total number of sequences with non-zero frequency, i is a symbol-sequence index of sequence vector length L, and Pi;L is the relative frequency of symbol sequence i. This choice of Nseq reflects the fact that many possible sequences may not be realized because of finite data-set length. The result is to bias HS upward when the number of possible sequences becomes large relative to the available data. For random data HS should equal 1, whereas for nonrandom data it should be between 0 and 1. To compare two symbol-sequence histograms A and B , we use the Euclidean norm as a difference statistic:
s X(Ai;L ? Bi;L) AB (L) = i
2
;
(2)
where i is indexed over all possible sequence codes. Computing is useful for comparing SSHs of different operating conditions or for testing for time reversibility by comparing the SSH for a time series with that of its time reverse. For more details about data symbolization for measurements of engineering systems, see [5,4,6]. Another quantity readily determined from symbolsequence frequency distributions is time irreversibility. Briefly, time irreversibility is present when the relative frequency of some symbol sequences is significantly different than their time-reverse sequences. For example, if the symbol sequence 0 5 8 is much more frequent than the frequency of the sequence 8 5 0, there is clearly a timedirection bias in the process. The presence of time irreversibility implies that the observed measurements cannot be explained as a Gaussian linear random process [7] (i.e., it cannot be generated by Gaussian noise exciting a linear system). While time irreversibility is not an absolute test for nonlinearity, its presence is a significant indicator of nonlinear structure. The reader is referred to [7,8,9] for further discussion. As noted above, when dealing with time series which vary smoothly with time, such as the continuous, fast-timesampled analog signals used in this study, some data treatment is necessary in order to obtain a good symbolic transformation of the data. In the coarse-grained sense, the measurement signal is oversampled, meaning that many consecutive data points would be symbolized in a long string of the same symbols. To obtain information about how the data change meaningfully in time, it is necessary to choose a symbolization interval which defines the number of actual data points between successive symbols (the other data points within that interval are ignored). The appropriate symbolization interval may be chosen on the basis of how symbol statistics vary as a function of that interval. Figure 4 shows how HS and time vary with symbolization interval for 4 symbols and sequence length 3 for time-series data collected while the combustor was operating at a global equivalence ratio of 0.27. Here, time refers to the difference statistic of SSHs computed from the (forward-time) time series and with the reverse-ordered (backward-time) time series. As seen in (a), at short symbolization intervals, HS is low because the data are effectively oversampled, and the predominant symbol patterns are long sequences of the same repeated symbols and the transitions between symbols. As the symbolization interval increases, the data irregularity increases HS . At an interval near the mean oscillation period of 54 timesteps, HS decreases. At very long symbolization intervals, the effects of aperiodicity and noise in the data reduce apparent deterministic structure, and HS approaches unity. In Fig. 4(b), the measured irreversibility shows shifts in the apparent effective degree of nonlinearity. The irreversibility measure time distinguishes different
1
(a) Modified Shannon entropy HS
0.9 0.8 0.7 (b) Time irreversibility ∆time 0.06 0.04 0.02 0 0
10
20 30 40 50 Symbolization interval
60
Figure 4: Variation of modified Shannon entropy (a) and time irreversibility (b) with symbolization interval for data from a fuel-lean operating condition. timescales than entropy. Although both statistics experience a minimum at the mean oscillation period of 54 timesteps, the half-period timescale range is viewed differently by time than by HS . Specifically, near a symbolization interval of 30 timesteps, the maximal degree of irreversibility is observed. To observe important features such as those described below, this interval might be more appropriate than is indicated by HS . Once the appropriate symbolization interval is chosen, the SSHs can be formed to show the relative frequency of occurrence of each symbol pattern. Due to the nature of the combustor behavior, the dominant peaks on the SSH are often long strings of the same repeated symbol. By using the sequence code, we can decide which of the histogram peaks correspond to symbol patterns which represent important dynamical behavior.
Results Figure 5 illustrates two examples of unstable periodicities visible in the pulse combustor pressure measurements for a global equivalence ratio of 0.27. These periodicities were located by identifying symbol sequences that appear to be associated with or precursors to extreme variations in pressure. Once these symbol sequences were defined, the converted symbol stream was scanned to identify the temporal location of each segment in the data series. Figure 5(a) shows six occurrences of the symbol sequence 2 1 1 2 us-
2
Symbol
5
1
1
2
(a)
4 3 2 1 0 0.00
0.01
0.02
0.03
0.04
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0.04
0.05
Time [sec]
Symbol
0 9 8 7 6 5 4 3 2 1 0
9
(b)
0.00
0.01
0.02 Time [sec]
Figure 5: Time-series segments containing two symbol sequences corresponding to extreme combustion events. Sequence 2 1 1 2 (a) was determined from a 6-symbol discretization with a 10-timestep symbolization interval, and sequence 0 9 (b) was determined from a 10-symbol discretization with a 25-timestep symbolization interval. ing an equiprobable 6-partition with an intersymbol time interval of 0.002 sec. This unstable oscillation appears to coincide with a misfire. Figure 5(b) shows seven occurrences of the symbol sequence 0 9 using an equiprobable 10-partition with an intersymbol time interval of 0.005 sec (approximately half of the average oscillation period). This particular unstable periodicity represents the largest pressure variations seen and correspond to much larger than normal combustion events following a misfire. We conjecture that misfires leave unburned fuel in the combustion chamber which exaggerates the combustion in successive cycles. It is noteworthy that the time-reverse sequence 9 0 never occurs in our data. The six segments containing the symbol sequence 2 1 1 2 have fairly similar trajectories for a period of approximately 0.015 sec before they visibly diverge, and the seven segments containing the symbol sequence 0 9 follow similar trajectories for almost 0.010 sec. Thus it would seem our method is successful in locating segments of the time-series data which visit the same unstable periodic orbits.
Summary Data symbolization shows promise as a new approach to analyzing pulse-combustor data. Symbol statistics can provide new information about relevant timescales or data patterns using a low-precision approach which is less adversely affected by the presence of noise and which does not depend significantly on the nature of the data (linear or nonlinear). The computational simplicity of symbolic data analysis suggests that it could be useful in real-time monitoring of pulse-combustor measurement signals. We have shown that our method is capable of detecting the presence of frequently visited unstable periodic orbits. By careful selection of the target symbol sequence, we have demonstrated that misfire events can be identified. By applying this method to real-time monitoring, we hope to predict the occurrence of extreme events so that active control may be applied to prevent misfire and flameout [1].
References 1. In V., Spano M.L., Neff J.D., Ditto W.L., Daw C.S., Edwards K.D., Nguyen K. (1997). “Maintenance of chaos in a computational model of a thermal pulse combustor”, Chaos 7:4, 605–613. 2. Moon F.C. Chaotic and Fractal Dynamics, WileyInterscience, ISBN 0-471-54571-6, 1992. 3. Tang X.Z., Tracy E.R., Boozer A.D , deBrauw A., Brown R. (1995). “Symbol sequence statistics in noisy chaotic signal reconstruction”, Physical Review E 51:5, 3871–3889. 4. Finney C.E.A., Green J.B. Jr., Daw C.S. (1998). “Symbolic time-series analysis of engine combustion measurements”, SAE Paper No. 980624. 5. Daw C.S., Kennel M.B., Finney C.E.A., Connolly F.T. (1998). “Observing and modeling nonlinear dynamics in an internal combustion engine”, Physical Review E 57:3, 2811– 2819. 6. Daw C.S. (1998). “Symbolization: A new approach for analyzing multiphase flow data”, to be presented at the 1998 ASME International Congress & Exposition (Anaheim, California USA; 1998 November 15-20). 7. Diks C., Houwelingen J.C. van, Takens F., DeGoede J. (1995). “Reversibility as a criterion for discriminating time series”, Physics Letters A 201, 221–228. 8. Heyden M.J. van der, Diks C., Pijn J.P.M., Velis D.N. (1996). “Time reversibility of intracranial human EEG recordings in mesial temporal lobe epilepsy”, Physics Letters A 216, 283– 288. 9. Green J.B. Jr., Daw C.S., Armfield J.S., Finney C.E.A., Durbetaki P. (1998). “Time irreversibility of cycle-by-cycle engine combustion variations”, to be presented at the 1998 Spring Technical Meeting of the Central States Section of the Combustion Institute (Lexington, Kentucky, May 31 - June 2).
