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MEASUREMENT AND SPECTRAL ANALYSIS OF PRESSURE–TIME VARIATION IN A VALVELESS PULSED COMBUSTOR
Mechanical Systems and Signal Processing (1997) 11(2), 307–322
CASE HISTORY MEASUREMENT AND SPECTRAL ANALYSIS OF PRESSURE–TIME VARIATION IN A VALVELESS PULSED COMBUSTOR J. A. O Department of Mechanical Engineering, University of Ilorin, Ilorin, Nigeria
M. J. E. S Electrical Engineering Department, King Saud University, Riyadh, Saudi Arabia (Received July 1995, accepted July 1996) Pressure–time variations were measured at six axial locations in a propane-fuelled SNECMA-Lockwood valveless pulsed combustor. Two techniques were used to acquire the data. The collected data were preprocessed and then analysed by using the fast Fourier transform (FFT) and parametric modeling techniques so as to examine the frequency distribution of the pressure waves at each location. A spectral matching technique is proposed for analysis to ensure accurate frequency estimation. The estimated power spectra of the collected data were found to have peaks close to 220 Hz and at harmonically related frequencies in all the considered locations. Since the presence of higher frequency harmonics in the pressure spectra at the ports in the tailpipe shows that an appreciable proportion of the fuel is burned there, the definition of combustion intensity for pulsed combustors should be based on the whole volume of the combustor rather than the volume of the combustion chamber only. The enhancement of convective heat transfer in the tailpipe due to combustion-driven oscillations can be attributed to the presence of higher frequency harmonics there. Consequently, the combustor is expected to generate broadband acoustical signals having spectral peaks at 220 Hz and its harmonics.
7 1997 Academic Press Limited
1. INTRODUCTION
Pulsed combustors are variously called pulse jets, pulse pots, resonant-type jet propulsion motors, harmonic burners, and pulse reactors. They are burners in which pressure waves are generated by intermittent combustion. The intermittent combustion is induced by the use of special geometries. The three major parts of a pulsed combustor are: inlet, combustion chamber, and tailpipe. The inlet may be equipped with valves or it may be valveless, in which case the inlet is designed to function as an aerodynamic valve. These combustors have been used in domestic heating units, as propulsive devices and as hot gas generators in drying applications. Work is being done to develop this type of combustor for use in gas turbine plants [1]. Pulsed combustors of so many geometries have been built by various inventors. A description of various types of pulsed combustor can be found elsewhere [2]. 0888–3270/97/020322 + 16 $25.00/0/pg960059
7 1997 Academic Press Limited
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The exploitation of the phenomenon of pulsating combustion is not without its problems. These combustors tend to generate excessive noise due to the rapidity of their combustion process and the high amplitude of pressure waves in the inlet and tailpipe. By the very nature of their operation, pulsed combustors can also cause strong mechanical vibrations in nearby structures and machine components. The sound power radiated from the open ends of the pulsed combustor and the induced vibrations depend on the pressure–time variations in the combustor. The oscillating stresses in the wall of the pulsed combustor caused by pressure–time variations can cause fatigue cracking of the oxide film on the wall leading to accelerated corrosion and premature failure of the combustor. Therefore, it is important to study pressure–time variation in the pulsed combustor in order to properly address the problems mentioned above which are concomitant with its operation. Dhar et al. [3] measured pressure–time variations in the combustion chamber and exhaust chamber of a Helmholtz-type pulsed combustor fuelled with natural gas. An FFT spectrum analyser was used to obtain the pressure spectra. With the combustor operating at 3.7 kW heating capacity, it was found that the pressure amplitude in the combustion chamber was 2.4 kPa and was an almost pure 65 Hz sine wave. However, the amplitudes of the two harmonics at 130 and 195 Hz were negligible as compared to that of the fundamental frequency. The sound pressure level spectrum measured at 0.61 m (2 ft) from the exit of the combustor exhibited peaks at 65, 130, 195, 260, 325 and 390 Hz. The lower amplitude peaks at higher frequencies is believed to be caused by either structural vibrations or sidelobes (leakages) due to the effect of window on the data. The work reported here is an extension of these previous studies to propane-fuelled valveless SNECMA-Lockwood pulsed combustor having the geometry shown in Fig. 1. In these studies, an FFT technique was used to analyse the collected data. This procedure is computationally efficient and is capable of producing some reasonable results, depending on the length and quality of the data. However, it has some inherent performance limitations such as producing poor resolution and/or high variance spectral estimates. These drawbacks are undesirable in this study since we are interested in accurate harmonic analysis of the data so as to evaluate the performance of the pulsed combustor. This paper considers the use of autoregressive moving average (ARMA) modelling technique for the processing of the collected data. The parameters of the ARMA model
Figure 1. Proportions of a SNECMA-Lockwood valveless pulsed combustor.
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50.8 mm 12.7 mm
Radiation shield
19.05 mm
Pressure tap tubing Combustor wall
25.4 mm
12.7 mm
Pressure transducer boss
2.29 mm Figure 2. Pressure transducer adapter.
are determined by singular value decomposition (SVD) algorithm which is known to be computationally efficient and stable. A spectral matching procedure is also advocated in this analysis to ensure accurate frequency estimation. 2. EXPERIMENTAL APPARATUS
The fuel flow rate was measured using a choked nozzle meter. The orifices used in the flow meter were of the standard Amal jet type 187/001, No. 300 or No. 500 depending upon operating conditions. The varying static pressure was measured at six ports called port 1, 2, 3, 4, 5 and 6 which were set at distances 0.98d, 2.26d, 3.72d, 8.42d, 12.25d, and 15.73d respectively (where combustion chamber diameter, d = 73 mm) from the left end of the combustor. Kistler (model 601B1) water-cooled quartz pressure transducers were used for pressure measurements. The high natural frequency of the transducer element of about 120 kHz makes the transducer suitable for the dynamic measurements in the present work since it can respond well to the pressure variations at the operating frequency and several harmonics of the fundamental frequency. The maximum allowable temperature of the transducer element is about 260°C [4]. Therefore, each transducer was mounted in an adaptor to protect it from the high temperature of the combustor. The adaptor is shown in Fig. 2. Assuming the air column in the adaptor passage vibrated as that in a quarter-wavelength tube, and taking the mean temperature of the gas column to be 240°C, the fundamental resonant frequency for the adaptor passage was estimated to be 4.25 kHz. This is much higher than the fundamental frequency of the combustor and, indeed, of several harmonics contained therein. The output of each of the three transducers was fed to a model 504A Kistler charge amplifier through a low-noise cable supplied with the transducer. Two different arrangements were used to record the output signal from the charge amplifiers. In the first arrangement shown in Fig. 3, the output was recorded with a model 1508B Honeywell Visicorder. The calibration time-base signal supplied to the viscorder was
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Figure 3. Pressure measurement system.
obtained from HP 3310B function generator. Kodak linagraph paper (direct print type 2022 extra thin 80 × 150’ Spec. 2) was used for recording the pressure–time variation. In the second set-up, the output of the three charge amplifiers were fed to front-end analog amplifiers of three of the channels of a data acquisition system. A diagram of the data acquisition sytem is shown in Fig. 4. As the combustor ran at a fuel flow rate of 7.24 kg/h, signals were sampled at intervals of 125 ms with synchronous sample and hold capability. The three channels were sampled simultaneously. The data was multiplexed onto digital magnetic tape. 3. DATA ANALYSIS TECHNIQUE
This section discusses some technique for analysing the recorded data. The pressure– time variation of the data is shown in Fig. 5. These data were obtained from the visicoder and can be seen to be of a weakly random nature. Since the mean value of the pressure
Figure 4. Block diagram of the data acquisition system. 1, Front-end amplifier (one per channel); 2, analog–digital converter (ADC); 3, ADC control; 4, dual channel port controller no. 1; 5, dual channel port controller no. 2; 6, pressure–time buffer; 7, Hewlett-Packard computer; 8, program; 9, magnetic tape unit; 10, gate; 11, input/display terminal; 12, printer.
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Figure 5. Pressure–time trace recorded at the six ports with combustor operating at a fuel flow rate of 7.24 kg/h.
at each port is a constant component of the total signal, it is subtracted out resulting in a zero mean stationary signal. As observed from Fig. 5, each signal from the six ports consists of a repetitive complex waveform which needs to be broken down into its sinusoidal and/or cosinusoidal components, termed harmonics. This Fourier series representation would indicate the relative amplitude of the different frequency components contained within a complete waveform. Unfortunately, time domain method of analysis such as the correlation analysis would not be able to provide all the desirable information hence the need to carry out spectral analysis on the data. Spectral analysis is a transformation procedure whereby frequency contents of a complex data or waveform can be obtained. Techniques for spectral analysis are many but they can be broadly classified into two groups, namely, parametric and non-parametric. Parametric techniques are based on a finite set of parameters which define a closed form mathematical model for the spectrum of a given data. On the other hand, non-parametric methods compute the power spectrum from either the data directly or the estimated autocorrelation lags. The two most widely used examples of the non-parametric methods are the periodogram and Blackman–Tukey techniques [5, 6]. Both methods of spectral analysis are attractive because they can easily be implemented by an FFT algorithm. However, they often produce spectral estimates with poor resolution and high variance, especially for short data records. Several procedures [5, 7] have been successfully used to reduce the variance of spectral estimates, but at the expense of poorer frequency resolution. Parametric techniques such as autoregressive (AR), moving average (MA), and ARMA models are often used so as to overcome the above limitations of the nonparameteric methods. The AR spectral estimator is the most popular because many computationally efficient algorithms exist for computing its model parameters. However,
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it produces poor results for noisy data. The MA spectral estimator is not popular because of the difficulty in estimating its model parameters. Moreover, it produces poor spectral estimates for narrowband processes. An ARMA spectral estimator, being a generalisation of AR and MA models, often produces better spectral estimates than any of the above methods, especially when the nature of the data under investigation is unknown. Gutowski et al. [8] have highlighted some of the problems that can arise when incorrect models are used for power spectrum estimation. That is, a spectral estimator that approximates the data model as accurately as possible must always be selected. Consequently, the ARMA modeling technique is proposed here for the data analysis since this is well known to produce high resolution estimates especially for noisy and short length data. 3.1. Denoting the collected data at any port by x[n] so that its autocorrelation lag is given by Rx [n] = E[x[n + m]x[m]] Consider an ARMA( p, q) model, with an input excitation o[n] and output x[n], which satisfies a linear difference equation p
q
k=1
k=0
x[n] + s ak x[n − k] = s bk $ [n − k],
(1)
where p, {ak , k = 1, 2, . . . p} and q, {bk , k = 0, 1, 2, . . . q} respectively represent the AR and MA parameters; o[n] is assumed to be zero mean, unit variance and uncorrelated noise sequence. The estimated power spectrum of x[n] is obtained from Sx (v) =
b
b0 + b1 e−jv + . . . + bq e−jqv 1 + a1 e−jv + . . . + ap e−jpv
b
2
(2)
It is desirable to determine the ARMA parameters for use in this equation by optimal techniques. However, such approaches pose many numerical problems as well as being computationally inefficient. A less arduous task is to use suboptimal procedures whereby the AR and MA parameters are separately determined from a set of Yule–Walker equations [9] q F p G− s am Rx [k − m] + s bm h*[m − k], 0 E k E q, m=k Rx [k] = g m =p 1 G am Rx [k − m], k e q + 1, f−ms =1
(3)
where h[n] is the unit impulse response of the ARMA model described by equation (1). A set of high-order Yule–Walker (HOYW) equations p
Rx [k] + s ai Rx [k − i] = 0
(4)
i=1
is obtained from equation (3) for k e q + 1 and this describes a linear relationship between ak and Rx [k] from which the AR parameters can be obtained. Using a minimum number of HOYW equations in solving for ak can cause parameter hypersensitivity [10]. This is avoided if a set of t(qp) equations is formed from (4), and in which the
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estimated values R x [k], p' and q' are used since their true values are unknown. The resulting overdetermined equation is expressed in matrix form as R1 a = e
(5)
where the elements of R1 are obtained as r1 (i, j) = R x (q' + 1 + i − j), 1 E i E t, 1 E j E p' + 1. R1 is a t × ( p' + 1) matrix whilst a and e are respectively ( p' + 1) × 1 and (t × 1) vectors. Several procedures [5, 9, 10, 11] are available for computing a in equation (5). Here, however, the SVD approach is used because it: (i) provides a low-rank matrix with improved SNR from R1; (ii) alleviates the ill-conditioned nature of R1; and (iii) provides a minimum norm least squares solution. Decomposing R1 into the product of three matrices [11]: R1 = UDV H where U and V are respectively t × t and ( p' + 1) × ( p' + 1) unitary orthogonal matrices; D is a diagonal matrix of the same size as R1 but with elements (termed the singular values) d1 q d2 q · · · q dp + 1 ; and H denotes complex conjugate transpose of a matrix. A unique t × ( p' + 1) matrix RQ that best approximates R1 in the least squares sense is obtained as [10, 11] RQ = UDQ V H = [h : B] where DQ is obtained from D by setting all singular values smaller than dQ to zero; h is the left most t × 1 column vector while B is a t × p' matrix corresponding to the p' rightmost columns of RQ . A minimum norm solution for the AR coefficients is obtained from a = −B(h where B( is the pseudo inverse of the matrix B. Once the AR coefficients are obtainable then the MA spectrum can be derived from severval techniques [12, 13]. A particularly efficient procedure is to compute the coefficient Ck from [12] p'
p'
Ck = s s ai aj R x (k − i + j)
(6)
i=0 j=0
for −q E k E q so that the spectral estimates for x[n] are determined from Sx (e jv) =
C(e jv) vA(e jv)v2
(7)
where C(z) is the Z-transform of Ck evaluated on the unit circle. Usually an appropriate window is applied on Ck to obtian positive definite spectral estimates. For this analysis, an exponential function Wk = a vkv, 0 Q a Q 1, =k Q q is found to generate good results. 3.2. Spectral matching is a procedure whereby the model and signal (true) spectra are compared to ensure accurate frequency estimation. That is, both techniques should produce approximately the same result, thereby guaranteeing the correctness of the estimated frequency. Unfortunately, the true spectrum is unknown, making spectral matching a difficult task. Here, the FFT spectral estimate is used as a reference since the
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FFT algorithm would always produce the correct spectral shape of the signal spectrum. However, it may be biased with high variance. The transfer function of the ARMA ( p, q) model described in (1) is H(z) = (B(z)/A(z)) where B(z) and A(z) represent respectively the Z-transform of the bk and ak coefficients. An error sequence, e[n] is generated when x[n] is passed through an inverse filter A(z)/B(z). The matching error, E between the signal spectrum, P(v) and the model spectrum, Sx (v) is given by E=
1 2p
g
p
P(v) dv, S x (v) −p
(8)
where P(v) =
b
b
1 N s x[n] exp{−jvn} N n=1
2
is the periodogram of x[n]. Thus, the objective of the spectral matching is to select the ARMA parameters so as to minimise E. Unfortunately, minimisation of E results in a set of non-linear equations which must be solved iteratively in order to compute the above parameters [14]. This approach is computationally inefficient as it may have convergence problems especially if the initial (or starting) values are poorly selected. A non-iterative procedure for achieving spectral matching is employed in this paper. The performance of this approach relies on the proper selection of p and q so as to obtain good estimates of the ak and bk coefficients for use in equation (2). The main problem is how to establish a criterion for selecting the smallest singular value of R1 for providing good estimates of p and ak . The spread of the singular values is influenced by many factors amongst which are the data SNR as well as the number of distinct components, amplitude distribution, and frequency spacings of the various components in the data. In this analysis, p is selected as the value of dp for which d1 d 1. dp + 1 dp Using small values of dp in the estimation of p and ak lead to high variance with spurious peaks whilst too large values produce smoothed but poorly resolved harmonics. Selection of p using the above criterion and spectral matching often produce good spectral estimates. This aspect is further illustrated in Section 4. To obtain good spectral matching, the parameters of the MA model should be computed, and this is done by equating expressions (2) and (6), that is vB(ejv) =B(e jv )= 2 = C(e jv)
(9)
The MA parameters q, bk are then computed by spectral factorisation. There are many spectral factorisation techniques [15], but cepstral deconvolution procedure is used here since spectral factorisation is effectively reduced to DFT processing of C(ejv) via the FFT algorithm. Solving equation (9) by cepstral analysis [13] yields a recursive equation K
(k + 1)bk + 1 = s [k + 1 − m]b[k + 1 − m]bm ; m=0
0 E k E N' − 1
315
for computing the minimum phase bk coefficients, where b[k] is the cepstral coefficient of B(e jv) and N' is the number of FFT points. The cepstrum decays, at least, as 1/k [6], hence q is selected when vbq /b0 v is less than a threshold values (usually Q1/N') to produce good spectral matching. The contributions to the total error in equation (8) are determined by the integrated ratio of the signal and model spectra. Thus, using the ‘‘best’’ values of p, ak , q, bk in estimating the model spectrum ensures that the approximation of P(v) by Sx (v) is far superior at the harmonics peaks than at the spectral valleys. That is a combined use of the SVD and cepstral analysis algorithms guarantees that the true spectral peaks of Sx (v) are matched to those of P(v) in such a way as to minimise E. Consequently, Sx (v) is a smoothed version of P(v) with greatly reduced variance and enhanced harmonics peaks, thereby producing correct frequency estimates of the harmonics in x[n]. 4. RESULTS
Figure 5 shows the pressure–time trace obtained from the visicorder recording at the six ports. Due to the DC drift superimposed on the output signals from the pressure transducers, it was not possible to obtain absolute values of the pressure at the peak and trough of the pressure–time trace of the measurements. Correlation lags are computed for each set of the collected data and these are used for generating the extended higher order correlation matrix. The AR parameters are then determined from this matrix using the SVD algorithm. Table 1 shows the distribution of the singular values of the correlation matrix for the collected data at the various ports. The spread and size of the singular values provide useful information about the number and relative strength of harmonics present in the collected data. For example using the ratio test for port 1 data in Table 1, the rank of the matrix is 4. However, this test fails if the singular values are poorly distributed as for ports 2, 5 and 6. In this case, matrix truncation is achieved through the spectral matching procedure. Thus, p was determined to be 5 for port 2, 10 and 14 for ports 5 and 6 respectively. However, p was clearly obtained as 4 for ports 1, 3 and 4. Because p is related to the AR model, it also gives an indication of the number of harmonics (usually it is equal to p/2) present in the data. Figures 6–11 show the estimated power spectra for the pressure waves at the six ports as a function of frequency. In each figure the model spectrum is compared to the periodogram so as to ensure accurate frequency estimation. The folding frequency is 1/(2Dt) which corresponds to 4 kHz since Dt = 125 ms. Thus, for port 1 the estimated power spectrum contains two peaks located at 219 Hz and 448 Hz. These values are close to the expected values of 220 Hz and 440 Hz respectively. Identical results are obtained for the data collected at ports 2 and 3, as depicted in Fig. 7 and 8. For port 4, the two peaks are located at 227 Hz and 672 Hz, whereas the data collected at port 5 indicate two main peaks at 220 Hz and 447 Hz and a rather weak amplitude harmonic at 1352 Hz. The data collected at port 6 contain many spectral peaks located at 219, 448, 672, 891, 1117, 1352 and 1563 Hz. The summary of these results and the magnitudes of the average pressure range at each port are shown in Table 2. 5. DISCUSSION
From Fig. 5, it can be seen that not only does pressure–time variation differ from one port to another, the pressure–time trace is not perfectly repeated from cycle to cycle at a given location. As a result of this variation from cycle to cycle, the uncertainties in the
. . . . .
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T 1 Singular values of the extended correlation matrix for port data 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
Port 1
Port 2
Port 3
Port 4
Port 5
Port 6
11935.20898 8373.05273 1104.16736 583.53149 23.83881 18.95292 14.28537 9.94217 9.15664 8.51047 5.12876 4.76556 3.24489 2.00536 1.94456 1.33980
5085.25879 4989.51465 1115.09241 1020.54126 311.52737 229.59258 169.44687 161.73494 156.28532 142.57089 89.55776 85.29453 69.92095 66.89046 45.68165 42.26416 40.37534 34.37437 22.73998 14.39294 8.70326 5.66671 4.17971
11470.74316 11005.49023 1836.83765 1417.28674 57.99976 54.67427 33.78836 26.16177 24.62028 20.35104 18.31054 14.66772 13.43668 10.94784 9.60498 8.19890 7.49910 6.12841 5.86137 4.23738 3.91467
5957.75488 5172.35938 1701.20667 1582.59290 181.19153 155.56604 136.50002 99.88354 86.48573 81.38601 78.79207 76.31448 69.15928 60.84648 56.97273 43.34847 37.03294 27.71640 26.40346 23.91868 20.12556
2918.63232 2455.25513 766.70813 692.08545 255.77934 194.94090 158.30460 156.28021 69.90155 29.64744 14.51829 11.46196 7.23857 5.01444 4.25434 3.89600 3.75496 3.22454 2.63990 2.31287 1.80716
1320.27319 1203.08325 951.1898 887.44623 736.45691 693.30414 279.71078 273.67728 152.18881 142.63162 134.61488 129.50581 83.47752 80.27852 21.97401 20.95443 17.53509 15.90479 14.34618 12.72540 11.12245 10.30945 8.30926 7.88265 5.15213 3.81884 3.50838 2.37429 2.23894 1.93310 1.64057 1.35085 1.04064 1.00923 0.39223 0.37475 0.31916 0.22161 0.17855 0.02606 0.00063
pressure range values at the ports were estimated to vary between 28 and 218%. Such a lack of cyclic repeatability has been reported for cylinder pressure–time trace of internal combustion engines by Karim and Sarpal [16]. The factors that contribute to non-repeatability of the cyclic process in the pulsed combustor are: (1) random fluctuation from cycle to cycle in magnitude of any variable that may influence the combustion process (such as the mass of fuel burned during the cycle, delay period, the amount of precompression, and the wave processes); (2) susceptibility of flame development to turbulent flow fluctuations which are inherently random.
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Figure 6. FFT (*) and ARMA model (+) spectral estimates of data from port 1.
The latter point also appears in the work of Hansel [17], who reported that the cycle-to-cycle variation in peak cylinder pressure of spark ignition engines is a function of air/fuel ratio and the random variation in turbulence intensity. The cyclic variations in pulsed combustors are worse than in reciprocating internal combustion engines because pulsed combustors have nothing to act as flywheels which can minimise cyclic variation in performance. The presence of high-frequency harmonics in the tailpipe is due to pockets of fuel which do not burn with the main mass in the combustion chamber, but have delayed combustion and so form a combustible mixture in the tailpipe thereby causing an instantaneous pressure rise because when they eventually burn, they burn rapidly. It is well known that transient signals with short duration contain high-frequency components whereas those with longer duration do not contain such components. Where the amplitudes of these high-frequency harmonics are small, it indicates that the mass of the pockets of combustible mixtures formed are small. Boundary condition effects due to
T 2 Average pressure range and harmonics at the six ports with combustor operating at a fuel flow rate of 7.24 kg/h Harmonics* Port
Pressure Range (kPa)
1 2 3 4 5
106 115 105 111 77
6
60
Normalised
Absolute (Hz)
0.0274, 0.0560 0.0274, 0.0560 0.0274, 0.0560 0.0284, 0.0840 0.0274, 0.0560, 0.084, 0.1690 0.0274, 0.0560, 0.0840, 0.1113, 0.1396, 0.1690, 0.1953
219,448 219,448 219,448 227,672 219,448,672,1352
* Absolute harmonic frequency = 8000 × normalised frequency.
219,448,672,891 1117,1352,1563
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Figure 7. FFT (*) and ARMA model (+) spectral estimates of data from port 2.
reflections of waves and inflow of cold air from the atmosphere also affect the pressuretime trace at ports 1 and 6. Results of numerical simulation of the pulsed combustor show that at a fuel flow rate of 7.24 kg/h, the percentage of fuel burned in the combustion chamber is about 76%, the rest being largely burned in the tailpipe [2]. Results of the simulation also showed that the back flow of ambient air from the tailpipe exit into the combustor only travels a distance of about 81 mm from the tailpipe exit (port 6 is at a distance of 76 mm from tailpipe exit) [2]. This explains why higher frequency harmonics are more present in port 6 since fresh air is made available there to mix with the gas and complete the combustion process. The combustion intensity of the pulsed combustor is usually specified with reference to the combustion chamber volume. In view of the fact that an appreciable proportion of
Figure 8. FFT (*) and ARMA model (+) spectral estimates of data from port 3.
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Figure 9. FFT (*) and ARMA model (+) spectral estimates of data from port 4.
the fuel is burned in the tailpipe, it is more meaningful to specify combustion intensity of valveless pulsed combustors with reference to the whole combustor volume or at least the combustion chamber plus tailpipe volume. The sound power radiated from the open end of the combustor is proportional to the square of the pressure amplitude there [18]. It has been found experimentally that both the spectra of pressure in the pulsed combustor and the noise field outside the combustor have peaks at the firing frequency and its harmonics [3]. Also, there is efficient conversion of thermal energy into acoustic energy in a ducted combustion system like the pulsed combustor [19]. Therefore, from the spectra of pressure at the six ports, one can expect sound pressure levels of the combustor to have a multi-peak broadband structure with peaks at the firing frequency of 220 Hz and its harmonics up to at least 1540 Hz, when the combustor is operated at a fuel flow rate of 7.24 kg/h.
Figure 10. FFT (*) and ARMA model (+) spectral estimates of data from port 5.
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Figure 11. FFT (*) and ARMA model (+) spectral estimates of data from port 6.
Since a pulsed combustor causes strong mechanical vibrations in nearby structures and machine components, the excitation frequencies of the vibrations will be the firing frequency and its harmonics mentioned above. If any of these frequencies happen to coincide with one of the many resonance frequencies of the structures, an amplification will occur. High pulsation frequencies and amplitudes have been found to increase convective heat transfer in unsteady flow when compared with the corresponding average, steady flow [20]. It was found that the ratio of convective heat transfer coefficient in the pulsed combustor to that of corresponding steady flow (heat transfer coefficient improvement ratio) were 1.2, 1.9, and 2.8 at ports 4, 5 and 6 respectively, at a fuel flow rate of 7.24 kg/h. Whereas, at ports 2 and 3 in the combustion chamber where these higher frequency harmonics are not found, the values of heat transfer coefficient improvement ratio were 0.9 and 1.0 respectively [2]. The increase in convective heat transfer coefficient improvement ratio as one moves towards the tailpipe exit is due to the higher frequency harmonics present in the pressure spectra as one moves from ports 4 to 6. An important use of measured pressure has been the computation of combustion or heat release rate in computer simulations of engines. For combustion-driven oscillation to be sustained, Rayleigh’s criterion requires that combustion must take place intermittently in the combustion chamber and it should occur when the pressure is rising towards or just falling from the peak. Hence, the heat release is transient in each cycle of operation of the combustor. The measurement and spectral analysis of pressure data presented in this paper gives useful information for modeling the heat release spectrum more realistically. The heat release rate can then be obtained as the inverse Fourier transform of the heat release spectrum. In their acoustic modeling of a cylindrical pulsed combustor that is nearly closed at one end except for fuel and air inlet, Chiu and Summerfield [19] modeled the combustion heat release using a first half-cycle sinusoidal heat release spectrum. As in reciprocating internal combustion engines research, where an important use of engine cyclinder pressures is a comparison of measured pressures with pressures computed using a mathematical engine simulation [21], the results presented in this work can serve as standard for assessing how accurately a mathematical model of the valveless pulsed combustor can predict pressures at various axial locations.
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6. CONCLUSIONS
From these results, one can expect broadband acoustical signals to have spectral peaks at 220 Hz and its harmonics when the combustor is operated at a fuel flow rate of 7.24 kg/h. The presence of higher frequency harmonics in the pressure spectra of the tailpipe shows that an appreciable proportion of the fuel is burned there. Therefore, a more meaningful definition of combustion intensity of valveless pulsed combustor should be based on the whole combustor volume or at least on the volume of the combustion chamber and the tailpipe rather than on the volume of combustion chamber only, as is presently done. The results presented can be used in modeling heat release spectra more realistically in valveless pulsed combustors. The results may also be used to assess the accuracy of pressure predicted using mathematical models of the combustor. ACKNOWLEDGMENTS
The authors are grateful for the advice of J. A. C. Kentfield and the assistance of William Anson when the experiments were done. The provisions of facilities for this experimental work at the University of Calgary is also gratefully acknowledged. Finally, the authors are also grateful to the King Saud University for providing the facilities for analysing the collected data. REFERENCES 1. J. A. C. K and L. C. V. F 1989 American Society of Mechanical Engineers paper No. 89-GT-277, Improvements to the performance of a prototype pulse, pressure gain, gas turbine combustor. 2. J. A. O 1985 Numerical simulation and experimental studies of highly-loaded valveless pulsed combustors. Ph.D. dissertation, University of Calgary. 3. B. D, W. S and R. J. S, 1979 Argonne National Laboratory, Report ANL/EES-TM-87 , Proceedings of the Symposium on Pulse-combustion Water Heater. 105–128. Measurements and interpretation of pressure and sound spectra of a pulse combustion water heater. 4. H. & L. K I C 1967 Instrumentation Manual: Kistler Quartz Pressure Transducers Model 601. 5. D. C 1978 Modern Spectrum Analysis. New York: IEEE Press. 6. A. V. O and R. W. S 1975 Digital Signal Processing, Englewood Cliffs New Jersey: Prentice-Hall. 7. P. D. W 1967 IEEE Transactions on Audio Electroacoustics AU-15, 70–73. The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms. 8. P. R. G, E. A. R and S. T 1978 IEEE Transactions on Geoscience and Electronics GE-16, 80–84. Spectral Estimation: Fact or Fiction. 9. S. M. K and S. L. M 1981 Proceedings of the IEEE 69, 1390–1419. Spectrum analysis: a modern perspective. 10. J. A. C 1982 Proceedings of the IEEE 70, 907–939. Spectral estimation: an overdetermined rational model equation approach. 11. D. W. T, R. K and I. K 1982 Proceedings of the IEEE 70, 684–685. Data adaptive signal estimation by singular value decomposition of a data matrix. 12. R. L. M and A. A. B 1986 IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP-34, 1668–1670. A comparison of Numerator Estimators for ARMA Spectra. 13. M. J. E. S 1985 Ph.D. Dissertation, The University of Calgary ARMA models in multicomponent signal analysis. 14. J. M 1975 Proceedings of the IEEE 63, 561–580. Linear prediction: a tutorial review. 15. J. F. C 1976 Fundamental of Geophysical Data Processing. New York: McGraw-Hill.
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16. G. A. K and G. S. S 1976 Transactions of CSME 4, 234–236. A statistical analysis of engine cyclic pressure variations using a data acquisition system. 17. J. G. H 1970 Combustion Science and Technology 2, 223–225. A turbulent combustion model of the cycle-to-cycle combustion variations in spark-ignition engines. 18. V. I. H 1971 Journal of the Institute of Fuel 595–599. Basic considerations on the operation of a simple pulse combustor. 19. H. H. C and M. S 1974 Acta Astronautica 1, 967–984. Theory of combustion noise. 20. A. E. B 1964 Journal of Heat Transfer, Transactions of ASME 559–560. The influence of flow vibrations on forced convection heat transfer. 21. D. R. L, R. B. K and J. H. L 1975 SAE Transactions 84, Paper 750026. Measurement and analysis of engine pressure data.
CASE HISTORY MEASUREMENT AND SPECTRAL ANALYSIS OF PRESSURE–TIME VARIATION IN A VALVELESS PULSED COMBUSTOR J. A. O Department of Mechanical Engineering, University of Ilorin, Ilorin, Nigeria
M. J. E. S Electrical Engineering Department, King Saud University, Riyadh, Saudi Arabia (Received July 1995, accepted July 1996) Pressure–time variations were measured at six axial locations in a propane-fuelled SNECMA-Lockwood valveless pulsed combustor. Two techniques were used to acquire the data. The collected data were preprocessed and then analysed by using the fast Fourier transform (FFT) and parametric modeling techniques so as to examine the frequency distribution of the pressure waves at each location. A spectral matching technique is proposed for analysis to ensure accurate frequency estimation. The estimated power spectra of the collected data were found to have peaks close to 220 Hz and at harmonically related frequencies in all the considered locations. Since the presence of higher frequency harmonics in the pressure spectra at the ports in the tailpipe shows that an appreciable proportion of the fuel is burned there, the definition of combustion intensity for pulsed combustors should be based on the whole volume of the combustor rather than the volume of the combustion chamber only. The enhancement of convective heat transfer in the tailpipe due to combustion-driven oscillations can be attributed to the presence of higher frequency harmonics there. Consequently, the combustor is expected to generate broadband acoustical signals having spectral peaks at 220 Hz and its harmonics.
7 1997 Academic Press Limited
1. INTRODUCTION
Pulsed combustors are variously called pulse jets, pulse pots, resonant-type jet propulsion motors, harmonic burners, and pulse reactors. They are burners in which pressure waves are generated by intermittent combustion. The intermittent combustion is induced by the use of special geometries. The three major parts of a pulsed combustor are: inlet, combustion chamber, and tailpipe. The inlet may be equipped with valves or it may be valveless, in which case the inlet is designed to function as an aerodynamic valve. These combustors have been used in domestic heating units, as propulsive devices and as hot gas generators in drying applications. Work is being done to develop this type of combustor for use in gas turbine plants [1]. Pulsed combustors of so many geometries have been built by various inventors. A description of various types of pulsed combustor can be found elsewhere [2]. 0888–3270/97/020322 + 16 $25.00/0/pg960059
7 1997 Academic Press Limited
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The exploitation of the phenomenon of pulsating combustion is not without its problems. These combustors tend to generate excessive noise due to the rapidity of their combustion process and the high amplitude of pressure waves in the inlet and tailpipe. By the very nature of their operation, pulsed combustors can also cause strong mechanical vibrations in nearby structures and machine components. The sound power radiated from the open ends of the pulsed combustor and the induced vibrations depend on the pressure–time variations in the combustor. The oscillating stresses in the wall of the pulsed combustor caused by pressure–time variations can cause fatigue cracking of the oxide film on the wall leading to accelerated corrosion and premature failure of the combustor. Therefore, it is important to study pressure–time variation in the pulsed combustor in order to properly address the problems mentioned above which are concomitant with its operation. Dhar et al. [3] measured pressure–time variations in the combustion chamber and exhaust chamber of a Helmholtz-type pulsed combustor fuelled with natural gas. An FFT spectrum analyser was used to obtain the pressure spectra. With the combustor operating at 3.7 kW heating capacity, it was found that the pressure amplitude in the combustion chamber was 2.4 kPa and was an almost pure 65 Hz sine wave. However, the amplitudes of the two harmonics at 130 and 195 Hz were negligible as compared to that of the fundamental frequency. The sound pressure level spectrum measured at 0.61 m (2 ft) from the exit of the combustor exhibited peaks at 65, 130, 195, 260, 325 and 390 Hz. The lower amplitude peaks at higher frequencies is believed to be caused by either structural vibrations or sidelobes (leakages) due to the effect of window on the data. The work reported here is an extension of these previous studies to propane-fuelled valveless SNECMA-Lockwood pulsed combustor having the geometry shown in Fig. 1. In these studies, an FFT technique was used to analyse the collected data. This procedure is computationally efficient and is capable of producing some reasonable results, depending on the length and quality of the data. However, it has some inherent performance limitations such as producing poor resolution and/or high variance spectral estimates. These drawbacks are undesirable in this study since we are interested in accurate harmonic analysis of the data so as to evaluate the performance of the pulsed combustor. This paper considers the use of autoregressive moving average (ARMA) modelling technique for the processing of the collected data. The parameters of the ARMA model
Figure 1. Proportions of a SNECMA-Lockwood valveless pulsed combustor.
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50.8 mm 12.7 mm
Radiation shield
19.05 mm
Pressure tap tubing Combustor wall
25.4 mm
12.7 mm
Pressure transducer boss
2.29 mm Figure 2. Pressure transducer adapter.
are determined by singular value decomposition (SVD) algorithm which is known to be computationally efficient and stable. A spectral matching procedure is also advocated in this analysis to ensure accurate frequency estimation. 2. EXPERIMENTAL APPARATUS
The fuel flow rate was measured using a choked nozzle meter. The orifices used in the flow meter were of the standard Amal jet type 187/001, No. 300 or No. 500 depending upon operating conditions. The varying static pressure was measured at six ports called port 1, 2, 3, 4, 5 and 6 which were set at distances 0.98d, 2.26d, 3.72d, 8.42d, 12.25d, and 15.73d respectively (where combustion chamber diameter, d = 73 mm) from the left end of the combustor. Kistler (model 601B1) water-cooled quartz pressure transducers were used for pressure measurements. The high natural frequency of the transducer element of about 120 kHz makes the transducer suitable for the dynamic measurements in the present work since it can respond well to the pressure variations at the operating frequency and several harmonics of the fundamental frequency. The maximum allowable temperature of the transducer element is about 260°C [4]. Therefore, each transducer was mounted in an adaptor to protect it from the high temperature of the combustor. The adaptor is shown in Fig. 2. Assuming the air column in the adaptor passage vibrated as that in a quarter-wavelength tube, and taking the mean temperature of the gas column to be 240°C, the fundamental resonant frequency for the adaptor passage was estimated to be 4.25 kHz. This is much higher than the fundamental frequency of the combustor and, indeed, of several harmonics contained therein. The output of each of the three transducers was fed to a model 504A Kistler charge amplifier through a low-noise cable supplied with the transducer. Two different arrangements were used to record the output signal from the charge amplifiers. In the first arrangement shown in Fig. 3, the output was recorded with a model 1508B Honeywell Visicorder. The calibration time-base signal supplied to the viscorder was
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. . . . .
Figure 3. Pressure measurement system.
obtained from HP 3310B function generator. Kodak linagraph paper (direct print type 2022 extra thin 80 × 150’ Spec. 2) was used for recording the pressure–time variation. In the second set-up, the output of the three charge amplifiers were fed to front-end analog amplifiers of three of the channels of a data acquisition system. A diagram of the data acquisition sytem is shown in Fig. 4. As the combustor ran at a fuel flow rate of 7.24 kg/h, signals were sampled at intervals of 125 ms with synchronous sample and hold capability. The three channels were sampled simultaneously. The data was multiplexed onto digital magnetic tape. 3. DATA ANALYSIS TECHNIQUE
This section discusses some technique for analysing the recorded data. The pressure– time variation of the data is shown in Fig. 5. These data were obtained from the visicoder and can be seen to be of a weakly random nature. Since the mean value of the pressure
Figure 4. Block diagram of the data acquisition system. 1, Front-end amplifier (one per channel); 2, analog–digital converter (ADC); 3, ADC control; 4, dual channel port controller no. 1; 5, dual channel port controller no. 2; 6, pressure–time buffer; 7, Hewlett-Packard computer; 8, program; 9, magnetic tape unit; 10, gate; 11, input/display terminal; 12, printer.
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Figure 5. Pressure–time trace recorded at the six ports with combustor operating at a fuel flow rate of 7.24 kg/h.
at each port is a constant component of the total signal, it is subtracted out resulting in a zero mean stationary signal. As observed from Fig. 5, each signal from the six ports consists of a repetitive complex waveform which needs to be broken down into its sinusoidal and/or cosinusoidal components, termed harmonics. This Fourier series representation would indicate the relative amplitude of the different frequency components contained within a complete waveform. Unfortunately, time domain method of analysis such as the correlation analysis would not be able to provide all the desirable information hence the need to carry out spectral analysis on the data. Spectral analysis is a transformation procedure whereby frequency contents of a complex data or waveform can be obtained. Techniques for spectral analysis are many but they can be broadly classified into two groups, namely, parametric and non-parametric. Parametric techniques are based on a finite set of parameters which define a closed form mathematical model for the spectrum of a given data. On the other hand, non-parametric methods compute the power spectrum from either the data directly or the estimated autocorrelation lags. The two most widely used examples of the non-parametric methods are the periodogram and Blackman–Tukey techniques [5, 6]. Both methods of spectral analysis are attractive because they can easily be implemented by an FFT algorithm. However, they often produce spectral estimates with poor resolution and high variance, especially for short data records. Several procedures [5, 7] have been successfully used to reduce the variance of spectral estimates, but at the expense of poorer frequency resolution. Parametric techniques such as autoregressive (AR), moving average (MA), and ARMA models are often used so as to overcome the above limitations of the nonparameteric methods. The AR spectral estimator is the most popular because many computationally efficient algorithms exist for computing its model parameters. However,
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it produces poor results for noisy data. The MA spectral estimator is not popular because of the difficulty in estimating its model parameters. Moreover, it produces poor spectral estimates for narrowband processes. An ARMA spectral estimator, being a generalisation of AR and MA models, often produces better spectral estimates than any of the above methods, especially when the nature of the data under investigation is unknown. Gutowski et al. [8] have highlighted some of the problems that can arise when incorrect models are used for power spectrum estimation. That is, a spectral estimator that approximates the data model as accurately as possible must always be selected. Consequently, the ARMA modeling technique is proposed here for the data analysis since this is well known to produce high resolution estimates especially for noisy and short length data. 3.1. Denoting the collected data at any port by x[n] so that its autocorrelation lag is given by Rx [n] = E[x[n + m]x[m]] Consider an ARMA( p, q) model, with an input excitation o[n] and output x[n], which satisfies a linear difference equation p
q
k=1
k=0
x[n] + s ak x[n − k] = s bk $ [n − k],
(1)
where p, {ak , k = 1, 2, . . . p} and q, {bk , k = 0, 1, 2, . . . q} respectively represent the AR and MA parameters; o[n] is assumed to be zero mean, unit variance and uncorrelated noise sequence. The estimated power spectrum of x[n] is obtained from Sx (v) =
b
b0 + b1 e−jv + . . . + bq e−jqv 1 + a1 e−jv + . . . + ap e−jpv
b
2
(2)
It is desirable to determine the ARMA parameters for use in this equation by optimal techniques. However, such approaches pose many numerical problems as well as being computationally inefficient. A less arduous task is to use suboptimal procedures whereby the AR and MA parameters are separately determined from a set of Yule–Walker equations [9] q F p G− s am Rx [k − m] + s bm h*[m − k], 0 E k E q, m=k Rx [k] = g m =p 1 G am Rx [k − m], k e q + 1, f−ms =1
(3)
where h[n] is the unit impulse response of the ARMA model described by equation (1). A set of high-order Yule–Walker (HOYW) equations p
Rx [k] + s ai Rx [k − i] = 0
(4)
i=1
is obtained from equation (3) for k e q + 1 and this describes a linear relationship between ak and Rx [k] from which the AR parameters can be obtained. Using a minimum number of HOYW equations in solving for ak can cause parameter hypersensitivity [10]. This is avoided if a set of t(qp) equations is formed from (4), and in which the
313
estimated values R x [k], p' and q' are used since their true values are unknown. The resulting overdetermined equation is expressed in matrix form as R1 a = e
(5)
where the elements of R1 are obtained as r1 (i, j) = R x (q' + 1 + i − j), 1 E i E t, 1 E j E p' + 1. R1 is a t × ( p' + 1) matrix whilst a and e are respectively ( p' + 1) × 1 and (t × 1) vectors. Several procedures [5, 9, 10, 11] are available for computing a in equation (5). Here, however, the SVD approach is used because it: (i) provides a low-rank matrix with improved SNR from R1; (ii) alleviates the ill-conditioned nature of R1; and (iii) provides a minimum norm least squares solution. Decomposing R1 into the product of three matrices [11]: R1 = UDV H where U and V are respectively t × t and ( p' + 1) × ( p' + 1) unitary orthogonal matrices; D is a diagonal matrix of the same size as R1 but with elements (termed the singular values) d1 q d2 q · · · q dp + 1 ; and H denotes complex conjugate transpose of a matrix. A unique t × ( p' + 1) matrix RQ that best approximates R1 in the least squares sense is obtained as [10, 11] RQ = UDQ V H = [h : B] where DQ is obtained from D by setting all singular values smaller than dQ to zero; h is the left most t × 1 column vector while B is a t × p' matrix corresponding to the p' rightmost columns of RQ . A minimum norm solution for the AR coefficients is obtained from a = −B(h where B( is the pseudo inverse of the matrix B. Once the AR coefficients are obtainable then the MA spectrum can be derived from severval techniques [12, 13]. A particularly efficient procedure is to compute the coefficient Ck from [12] p'
p'
Ck = s s ai aj R x (k − i + j)
(6)
i=0 j=0
for −q E k E q so that the spectral estimates for x[n] are determined from Sx (e jv) =
C(e jv) vA(e jv)v2
(7)
where C(z) is the Z-transform of Ck evaluated on the unit circle. Usually an appropriate window is applied on Ck to obtian positive definite spectral estimates. For this analysis, an exponential function Wk = a vkv, 0 Q a Q 1, =k Q q is found to generate good results. 3.2. Spectral matching is a procedure whereby the model and signal (true) spectra are compared to ensure accurate frequency estimation. That is, both techniques should produce approximately the same result, thereby guaranteeing the correctness of the estimated frequency. Unfortunately, the true spectrum is unknown, making spectral matching a difficult task. Here, the FFT spectral estimate is used as a reference since the
314
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FFT algorithm would always produce the correct spectral shape of the signal spectrum. However, it may be biased with high variance. The transfer function of the ARMA ( p, q) model described in (1) is H(z) = (B(z)/A(z)) where B(z) and A(z) represent respectively the Z-transform of the bk and ak coefficients. An error sequence, e[n] is generated when x[n] is passed through an inverse filter A(z)/B(z). The matching error, E between the signal spectrum, P(v) and the model spectrum, Sx (v) is given by E=
1 2p
g
p
P(v) dv, S x (v) −p
(8)
where P(v) =
b
b
1 N s x[n] exp{−jvn} N n=1
2
is the periodogram of x[n]. Thus, the objective of the spectral matching is to select the ARMA parameters so as to minimise E. Unfortunately, minimisation of E results in a set of non-linear equations which must be solved iteratively in order to compute the above parameters [14]. This approach is computationally inefficient as it may have convergence problems especially if the initial (or starting) values are poorly selected. A non-iterative procedure for achieving spectral matching is employed in this paper. The performance of this approach relies on the proper selection of p and q so as to obtain good estimates of the ak and bk coefficients for use in equation (2). The main problem is how to establish a criterion for selecting the smallest singular value of R1 for providing good estimates of p and ak . The spread of the singular values is influenced by many factors amongst which are the data SNR as well as the number of distinct components, amplitude distribution, and frequency spacings of the various components in the data. In this analysis, p is selected as the value of dp for which d1 d 1. dp + 1 dp Using small values of dp in the estimation of p and ak lead to high variance with spurious peaks whilst too large values produce smoothed but poorly resolved harmonics. Selection of p using the above criterion and spectral matching often produce good spectral estimates. This aspect is further illustrated in Section 4. To obtain good spectral matching, the parameters of the MA model should be computed, and this is done by equating expressions (2) and (6), that is vB(ejv) =B(e jv )= 2 = C(e jv)
(9)
The MA parameters q, bk are then computed by spectral factorisation. There are many spectral factorisation techniques [15], but cepstral deconvolution procedure is used here since spectral factorisation is effectively reduced to DFT processing of C(ejv) via the FFT algorithm. Solving equation (9) by cepstral analysis [13] yields a recursive equation K
(k + 1)bk + 1 = s [k + 1 − m]b[k + 1 − m]bm ; m=0
0 E k E N' − 1
315
for computing the minimum phase bk coefficients, where b[k] is the cepstral coefficient of B(e jv) and N' is the number of FFT points. The cepstrum decays, at least, as 1/k [6], hence q is selected when vbq /b0 v is less than a threshold values (usually Q1/N') to produce good spectral matching. The contributions to the total error in equation (8) are determined by the integrated ratio of the signal and model spectra. Thus, using the ‘‘best’’ values of p, ak , q, bk in estimating the model spectrum ensures that the approximation of P(v) by Sx (v) is far superior at the harmonics peaks than at the spectral valleys. That is a combined use of the SVD and cepstral analysis algorithms guarantees that the true spectral peaks of Sx (v) are matched to those of P(v) in such a way as to minimise E. Consequently, Sx (v) is a smoothed version of P(v) with greatly reduced variance and enhanced harmonics peaks, thereby producing correct frequency estimates of the harmonics in x[n]. 4. RESULTS
Figure 5 shows the pressure–time trace obtained from the visicorder recording at the six ports. Due to the DC drift superimposed on the output signals from the pressure transducers, it was not possible to obtain absolute values of the pressure at the peak and trough of the pressure–time trace of the measurements. Correlation lags are computed for each set of the collected data and these are used for generating the extended higher order correlation matrix. The AR parameters are then determined from this matrix using the SVD algorithm. Table 1 shows the distribution of the singular values of the correlation matrix for the collected data at the various ports. The spread and size of the singular values provide useful information about the number and relative strength of harmonics present in the collected data. For example using the ratio test for port 1 data in Table 1, the rank of the matrix is 4. However, this test fails if the singular values are poorly distributed as for ports 2, 5 and 6. In this case, matrix truncation is achieved through the spectral matching procedure. Thus, p was determined to be 5 for port 2, 10 and 14 for ports 5 and 6 respectively. However, p was clearly obtained as 4 for ports 1, 3 and 4. Because p is related to the AR model, it also gives an indication of the number of harmonics (usually it is equal to p/2) present in the data. Figures 6–11 show the estimated power spectra for the pressure waves at the six ports as a function of frequency. In each figure the model spectrum is compared to the periodogram so as to ensure accurate frequency estimation. The folding frequency is 1/(2Dt) which corresponds to 4 kHz since Dt = 125 ms. Thus, for port 1 the estimated power spectrum contains two peaks located at 219 Hz and 448 Hz. These values are close to the expected values of 220 Hz and 440 Hz respectively. Identical results are obtained for the data collected at ports 2 and 3, as depicted in Fig. 7 and 8. For port 4, the two peaks are located at 227 Hz and 672 Hz, whereas the data collected at port 5 indicate two main peaks at 220 Hz and 447 Hz and a rather weak amplitude harmonic at 1352 Hz. The data collected at port 6 contain many spectral peaks located at 219, 448, 672, 891, 1117, 1352 and 1563 Hz. The summary of these results and the magnitudes of the average pressure range at each port are shown in Table 2. 5. DISCUSSION
From Fig. 5, it can be seen that not only does pressure–time variation differ from one port to another, the pressure–time trace is not perfectly repeated from cycle to cycle at a given location. As a result of this variation from cycle to cycle, the uncertainties in the
. . . . .
316
T 1 Singular values of the extended correlation matrix for port data 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
Port 1
Port 2
Port 3
Port 4
Port 5
Port 6
11935.20898 8373.05273 1104.16736 583.53149 23.83881 18.95292 14.28537 9.94217 9.15664 8.51047 5.12876 4.76556 3.24489 2.00536 1.94456 1.33980
5085.25879 4989.51465 1115.09241 1020.54126 311.52737 229.59258 169.44687 161.73494 156.28532 142.57089 89.55776 85.29453 69.92095 66.89046 45.68165 42.26416 40.37534 34.37437 22.73998 14.39294 8.70326 5.66671 4.17971
11470.74316 11005.49023 1836.83765 1417.28674 57.99976 54.67427 33.78836 26.16177 24.62028 20.35104 18.31054 14.66772 13.43668 10.94784 9.60498 8.19890 7.49910 6.12841 5.86137 4.23738 3.91467
5957.75488 5172.35938 1701.20667 1582.59290 181.19153 155.56604 136.50002 99.88354 86.48573 81.38601 78.79207 76.31448 69.15928 60.84648 56.97273 43.34847 37.03294 27.71640 26.40346 23.91868 20.12556
2918.63232 2455.25513 766.70813 692.08545 255.77934 194.94090 158.30460 156.28021 69.90155 29.64744 14.51829 11.46196 7.23857 5.01444 4.25434 3.89600 3.75496 3.22454 2.63990 2.31287 1.80716
1320.27319 1203.08325 951.1898 887.44623 736.45691 693.30414 279.71078 273.67728 152.18881 142.63162 134.61488 129.50581 83.47752 80.27852 21.97401 20.95443 17.53509 15.90479 14.34618 12.72540 11.12245 10.30945 8.30926 7.88265 5.15213 3.81884 3.50838 2.37429 2.23894 1.93310 1.64057 1.35085 1.04064 1.00923 0.39223 0.37475 0.31916 0.22161 0.17855 0.02606 0.00063
pressure range values at the ports were estimated to vary between 28 and 218%. Such a lack of cyclic repeatability has been reported for cylinder pressure–time trace of internal combustion engines by Karim and Sarpal [16]. The factors that contribute to non-repeatability of the cyclic process in the pulsed combustor are: (1) random fluctuation from cycle to cycle in magnitude of any variable that may influence the combustion process (such as the mass of fuel burned during the cycle, delay period, the amount of precompression, and the wave processes); (2) susceptibility of flame development to turbulent flow fluctuations which are inherently random.
317
Figure 6. FFT (*) and ARMA model (+) spectral estimates of data from port 1.
The latter point also appears in the work of Hansel [17], who reported that the cycle-to-cycle variation in peak cylinder pressure of spark ignition engines is a function of air/fuel ratio and the random variation in turbulence intensity. The cyclic variations in pulsed combustors are worse than in reciprocating internal combustion engines because pulsed combustors have nothing to act as flywheels which can minimise cyclic variation in performance. The presence of high-frequency harmonics in the tailpipe is due to pockets of fuel which do not burn with the main mass in the combustion chamber, but have delayed combustion and so form a combustible mixture in the tailpipe thereby causing an instantaneous pressure rise because when they eventually burn, they burn rapidly. It is well known that transient signals with short duration contain high-frequency components whereas those with longer duration do not contain such components. Where the amplitudes of these high-frequency harmonics are small, it indicates that the mass of the pockets of combustible mixtures formed are small. Boundary condition effects due to
T 2 Average pressure range and harmonics at the six ports with combustor operating at a fuel flow rate of 7.24 kg/h Harmonics* Port
Pressure Range (kPa)
1 2 3 4 5
106 115 105 111 77
6
60
Normalised
Absolute (Hz)
0.0274, 0.0560 0.0274, 0.0560 0.0274, 0.0560 0.0284, 0.0840 0.0274, 0.0560, 0.084, 0.1690 0.0274, 0.0560, 0.0840, 0.1113, 0.1396, 0.1690, 0.1953
219,448 219,448 219,448 227,672 219,448,672,1352
* Absolute harmonic frequency = 8000 × normalised frequency.
219,448,672,891 1117,1352,1563
318
. . . . .
Figure 7. FFT (*) and ARMA model (+) spectral estimates of data from port 2.
reflections of waves and inflow of cold air from the atmosphere also affect the pressuretime trace at ports 1 and 6. Results of numerical simulation of the pulsed combustor show that at a fuel flow rate of 7.24 kg/h, the percentage of fuel burned in the combustion chamber is about 76%, the rest being largely burned in the tailpipe [2]. Results of the simulation also showed that the back flow of ambient air from the tailpipe exit into the combustor only travels a distance of about 81 mm from the tailpipe exit (port 6 is at a distance of 76 mm from tailpipe exit) [2]. This explains why higher frequency harmonics are more present in port 6 since fresh air is made available there to mix with the gas and complete the combustion process. The combustion intensity of the pulsed combustor is usually specified with reference to the combustion chamber volume. In view of the fact that an appreciable proportion of
Figure 8. FFT (*) and ARMA model (+) spectral estimates of data from port 3.
319
Figure 9. FFT (*) and ARMA model (+) spectral estimates of data from port 4.
the fuel is burned in the tailpipe, it is more meaningful to specify combustion intensity of valveless pulsed combustors with reference to the whole combustor volume or at least the combustion chamber plus tailpipe volume. The sound power radiated from the open end of the combustor is proportional to the square of the pressure amplitude there [18]. It has been found experimentally that both the spectra of pressure in the pulsed combustor and the noise field outside the combustor have peaks at the firing frequency and its harmonics [3]. Also, there is efficient conversion of thermal energy into acoustic energy in a ducted combustion system like the pulsed combustor [19]. Therefore, from the spectra of pressure at the six ports, one can expect sound pressure levels of the combustor to have a multi-peak broadband structure with peaks at the firing frequency of 220 Hz and its harmonics up to at least 1540 Hz, when the combustor is operated at a fuel flow rate of 7.24 kg/h.
Figure 10. FFT (*) and ARMA model (+) spectral estimates of data from port 5.
320
. . . . .
Figure 11. FFT (*) and ARMA model (+) spectral estimates of data from port 6.
Since a pulsed combustor causes strong mechanical vibrations in nearby structures and machine components, the excitation frequencies of the vibrations will be the firing frequency and its harmonics mentioned above. If any of these frequencies happen to coincide with one of the many resonance frequencies of the structures, an amplification will occur. High pulsation frequencies and amplitudes have been found to increase convective heat transfer in unsteady flow when compared with the corresponding average, steady flow [20]. It was found that the ratio of convective heat transfer coefficient in the pulsed combustor to that of corresponding steady flow (heat transfer coefficient improvement ratio) were 1.2, 1.9, and 2.8 at ports 4, 5 and 6 respectively, at a fuel flow rate of 7.24 kg/h. Whereas, at ports 2 and 3 in the combustion chamber where these higher frequency harmonics are not found, the values of heat transfer coefficient improvement ratio were 0.9 and 1.0 respectively [2]. The increase in convective heat transfer coefficient improvement ratio as one moves towards the tailpipe exit is due to the higher frequency harmonics present in the pressure spectra as one moves from ports 4 to 6. An important use of measured pressure has been the computation of combustion or heat release rate in computer simulations of engines. For combustion-driven oscillation to be sustained, Rayleigh’s criterion requires that combustion must take place intermittently in the combustion chamber and it should occur when the pressure is rising towards or just falling from the peak. Hence, the heat release is transient in each cycle of operation of the combustor. The measurement and spectral analysis of pressure data presented in this paper gives useful information for modeling the heat release spectrum more realistically. The heat release rate can then be obtained as the inverse Fourier transform of the heat release spectrum. In their acoustic modeling of a cylindrical pulsed combustor that is nearly closed at one end except for fuel and air inlet, Chiu and Summerfield [19] modeled the combustion heat release using a first half-cycle sinusoidal heat release spectrum. As in reciprocating internal combustion engines research, where an important use of engine cyclinder pressures is a comparison of measured pressures with pressures computed using a mathematical engine simulation [21], the results presented in this work can serve as standard for assessing how accurately a mathematical model of the valveless pulsed combustor can predict pressures at various axial locations.
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6. CONCLUSIONS
From these results, one can expect broadband acoustical signals to have spectral peaks at 220 Hz and its harmonics when the combustor is operated at a fuel flow rate of 7.24 kg/h. The presence of higher frequency harmonics in the pressure spectra of the tailpipe shows that an appreciable proportion of the fuel is burned there. Therefore, a more meaningful definition of combustion intensity of valveless pulsed combustor should be based on the whole combustor volume or at least on the volume of the combustion chamber and the tailpipe rather than on the volume of combustion chamber only, as is presently done. The results presented can be used in modeling heat release spectra more realistically in valveless pulsed combustors. The results may also be used to assess the accuracy of pressure predicted using mathematical models of the combustor. ACKNOWLEDGMENTS
The authors are grateful for the advice of J. A. C. Kentfield and the assistance of William Anson when the experiments were done. The provisions of facilities for this experimental work at the University of Calgary is also gratefully acknowledged. Finally, the authors are also grateful to the King Saud University for providing the facilities for analysing the collected data. REFERENCES 1. J. A. C. K and L. C. V. F 1989 American Society of Mechanical Engineers paper No. 89-GT-277, Improvements to the performance of a prototype pulse, pressure gain, gas turbine combustor. 2. J. A. O 1985 Numerical simulation and experimental studies of highly-loaded valveless pulsed combustors. Ph.D. dissertation, University of Calgary. 3. B. D, W. S and R. J. S, 1979 Argonne National Laboratory, Report ANL/EES-TM-87 , Proceedings of the Symposium on Pulse-combustion Water Heater. 105–128. Measurements and interpretation of pressure and sound spectra of a pulse combustion water heater. 4. H. & L. K I C 1967 Instrumentation Manual: Kistler Quartz Pressure Transducers Model 601. 5. D. C 1978 Modern Spectrum Analysis. New York: IEEE Press. 6. A. V. O and R. W. S 1975 Digital Signal Processing, Englewood Cliffs New Jersey: Prentice-Hall. 7. P. D. W 1967 IEEE Transactions on Audio Electroacoustics AU-15, 70–73. The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms. 8. P. R. G, E. A. R and S. T 1978 IEEE Transactions on Geoscience and Electronics GE-16, 80–84. Spectral Estimation: Fact or Fiction. 9. S. M. K and S. L. M 1981 Proceedings of the IEEE 69, 1390–1419. Spectrum analysis: a modern perspective. 10. J. A. C 1982 Proceedings of the IEEE 70, 907–939. Spectral estimation: an overdetermined rational model equation approach. 11. D. W. T, R. K and I. K 1982 Proceedings of the IEEE 70, 684–685. Data adaptive signal estimation by singular value decomposition of a data matrix. 12. R. L. M and A. A. B 1986 IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP-34, 1668–1670. A comparison of Numerator Estimators for ARMA Spectra. 13. M. J. E. S 1985 Ph.D. Dissertation, The University of Calgary ARMA models in multicomponent signal analysis. 14. J. M 1975 Proceedings of the IEEE 63, 561–580. Linear prediction: a tutorial review. 15. J. F. C 1976 Fundamental of Geophysical Data Processing. New York: McGraw-Hill.
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