Analysis of signal characteristics of swirlmeter in oscillatory flow based on Hilbert–Huang Transform (HHT)

Measurement 45 (2012) 1765–1781

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Analysis of signal characteristics of swirlmeter in oscillatory flow based on Hilbert–Huang Transform (HHT) Jiegang Peng ⇑, Gang Zhang School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 610054, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 18 July 2011 Received in revised form 9 April 2012 Accepted 12 April 2012 Available online 28 April 2012 Keywords: Hilbert–Huang Transform (HHT) Swirlmeter Oscillatory flow Marginal spectrum

a b s t r a c t In the paper, firstly, on an experimental facility, we investigated the measurement characteristics of a diameter 50 mm swirlmeter in uniform flow and oscillatory flow. At the same time, the interference characteristics of oscillatory flow were studied. Then, the signal characteristics of swirlmeter in oscillatory flow were analyzed by Hilbert–Huang Transform (HHT) method. Results show that the response characteristics of swirlmeter in oscillatory flow are addition of that of swirlmeter in uniform flow and the interference characteristics of oscillatory flow. They further prove the conclusions which suppose that the correlation between the velocity pressures of fluid disturbs wave and that of vortex precession in swirlmeter is linear in the literature, and a new method for the oscillatory flow swirlmeter noise removal on HHT was provided. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction and background A swirlmeter (vortex precession flowmeter) is basically a velocity/pressure-sensitive device that measures the volumetric flow rate of liquids, gases and steam. It is gaining more and more acceptance in the field of fluid flow measurement [1–3]. The simplified sketch of a swirlmeter is shown in Fig. 1. The frequency f of these fluctuations is directly proportional to the volumetric flow rate Q, through the relationship below:

Q¼

f K

ð1Þ

where K is the so-called K-factor, with the unit of pulse per unit volume (pulse/m3). The flowmeter of this type has many advantages in that the K-factor is independent of the fluid properties (gas, liquid or gas–liquid two-phase flow), no moving parts, and wide rangeability [4–9]. ⇑ Corresponding author. E-mail address: [email protected] (J. Peng). 0263-2241/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.measurement.2012.04.007

However, in the range of small flow rates, the amplitude of the tiny pressure fluctuations becomes very small. In this way, the limited sensitivity of the sensor restricts the measuring range of the swirlmeter. Further pulsations of the static system pressure, vibrations of conduit and oscillatory flow can interfere with the true signal. In order to avoid the above drawbacks, Heinrichs [6] developed a measuring tube with two pressure ports placed face to face to each other, using a differential pressure sensor to measure the pure alternating pressure that is wanted from the pressure signals with a 180° phase difference in steady fluid flow. The measuring tube can increase the limited sensitivity of the sensor restricts the measuring range of the swirlmeter in steady fluid flow. Xin and Huayong [7] studied dynamic characteristics of swirlmeter by using CFD. Then, he suggests that a method of symmetry velocity pressure signals difference would overcome interference from vibrations of conduit. In recent years, Peng et al. [8] investigated fluid oscillation characteristics of swirlmeter in oscillatory flow by experiment. In their work, the Fourier transform (FT) has been applied to analyze the signal of swirlmeter in oscillatory flow. However, it is difficult to localize an event precisely in both time and frequency domains concurrently

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Nomenclature K fd Q f HHT LF HF MF fssf1 Assf1 fpof fpof1 Apof1 fpofL fpofH ApofL

the so-called K-factor, with the unit of pulses per volume (pulse/m3) the imposed velocity frequency of oscillatory flow the experiment volume flow rate the frequency of swirlmeter fluctuations Hilbert–Huang Transform the low-frequency part of main frequency the high-frequency part of main frequency the middle-frequency part of main frequency main frequency for the swirlmeter in steady flow amplitude of main frequency for the swirlmeter in steady flow main frequency of a pipe to oscillatory flow main frequency of Type I signal for a pipe to oscillatory flow amplitude of main frequency of Type I signal for a pipe to oscillatory flow low frequency main frequency of Type II signal for a pipe to oscillatory flow high frequency main frequency of Type II signal for a pipe to oscillatory flow the amplitude of low frequency main frequency of Type II signal for a pipe to oscillatory flow

ApofH

fsof fsof1 Asof1 fsofL fsofH

AsofL

AsofH

fsofM AsofM

the amplitude of high frequency main frequency of Type II signal for a pipe to oscillatory flow main frequency of a swirlmeter in oscillatory flow main frequency of Type I signal for a swirlmeter in oscillatory flow amplitude of main frequency of Type I signal for a swirlmeter in oscillatory flow low frequency main frequency of Type II and Type III signal for a swirlmeter in oscillatory flow high frequency main frequency of Type II and Type III signal for a swirlmeter in oscillatory flow the amplitude of low frequency main frequency of Type II and Type III signal for a swirlmeter in oscillatory flow the amplitude of high frequency main frequency of Type II and Type III signal for a swirlmeter in oscillatory flow middle frequency main frequency of Type II and Type III signal for a swirlmeter in oscillatory flow the amplitude of middle frequency main frequency of Type II and Type III signal for a swirlmeter in oscillatory flow

because the signal is nonlinear and non-stationary in oscillatory flow. In this paper, they suppose that the correlation between the velocity pressures of fluid disturbs wave and that of vortex precession in swirlmeter is linear. The studies of fluid oscillation characteristics for swirlmeter in oscillatory flow have been promoted by the above work. However, the fluid oscillation characteristics of swirlmeter in oscillatory flow are still an unresolved problem, which deserves further investigation. The data of swirlmeter in oscillatory flow are inherently non-stationary because the recordings are the result of propagation of various type waves with different amplitude, frequency, and wave speed in fluid that are likely nonlinear. It should be noted that swirlmeter in oscillatory flow measurement are often viewed as non-stationary, nonlinear data. Hilbert–Huang Transform (HHT) is a novel analysis method for nonlinear and non-stationary data, which was developed by Huang et al. [11] in 1998. The primary advantages

of the HHT over other methods are that it can deal with the nonlinear problem objectively [13]. The HHT has been widely applied in recent years in the fields of meteorology, ocean engineering, earthquake studies, two-phase flow measurement etc. [10–14]. Useful research results have been achieved to date, but very limited work has been reported on the method for the monitoring and characterization of swirlmeter in oscillatory flow. In this paper, the signal characteristics of swirlmeter in oscillatory flow were analyzed by HHT method. Those dates were decomposed by Empirical Mode Decomposition (EMD). Hilbert spectral analysis was conducted the various components of the decomposed IMF. Finally, the swirlmeter response characteristics in oscillatory flow were studied by comparison of the relevant main frequency. Results show that the response characteristics of swirlmeter in oscillatory flow are addition of the response characteristics of swirlmeter response characteristics in uniform flow

Fig. 1. Simplified sketch of a swirlmeter.

Fig. 2. Flowmeter experiment instrument.

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and the response characteristics of the interference characteristics of oscillatory flow.

Mode Decomposition (EMD), and (2) Hilbert Spectral Analysis [10–12]. 3.1. Empirical Mode Decomposition

2. Experimental apparatus and procedure The schematic diagram of the experimental apparatus is shown in Fig. 2. The experimental apparatus is composed of five parts: Fluid oscillation imitate device, test swirlmeter, standard flow device, vacuum pump and computer measurement system (V). The fluid oscillation imitate device imitate oscillatory flow in laboratory, shown in Fig. 3. Basically it consists of (i) DC electromotor; (ii) throttle round plate. In order to obtain an oscillatory flow in the test section, the flow is throttled by throttle round plate before reaching the test swirlmeter. The throttle round plate is derived by a DC electromotor. When the electromotor rotates a circuit, the flow is throttled for 10 times by the throttle round plate throttle. The standard speed of the electromotor is 3000 rpm. The maximum value of imposed velocity frequency of oscillatory flow (fd) is 500 Hz. The imposed velocity frequency of oscillatory flow (fd) can be adjusted by varying speed of rotation of the electromotor. The speed regulation system of the electromotor is closed loop control system of voltage feedback. The range of infinitely variable speed of the electromotor is from 0 to 3000 rpm by using this speed regulation system. The vortex precession frequency is obtained based on measured fluctuating pressure by piezoelectric press sensor. A straight circular pipe having inner diameter D = 50 mm and 5 m length is used and is located on 100D downstream from the inlet of the pipe. Flow is conditioned by this upstream of test section. Flow through the test section has a fully developed turbulent flow. A mean uniformity of the flow is less than ±3% and a free stream turbulence intensity of that is less than 0.7% over volume flow rate range of 5.5 m3/h to 220.5 m3/h. The diameter of the pipe and test swirlmeter is 50 mm. 3. Hilbert–Huang Transform Hilbert–Huang Transform (HHT) was proposed by Huang et al. [10,11]. It consists of two parts: (1) Empirical

With EMD, any complicated data set can be decomposed into a finite and often less number of intrinsic mode functions (IMFs). An IMF is defined as a function satisfying the following conditions [10–12]: 1. Over the entire data set, the number of extrema and the number of zero-crossings must be equal or differ at most by one; and 2. At any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero. An IMF represents a simple oscillatory mode similar to a component in the Fourier-based simple harmonic function, but more general. One can decompose any waveform as follows. First, identify all the local extrema. Connect all the local maxima by a cubic spline to produce the upper envelope, and repeat the procedure for the local minima to produce the lower envelope. The upper and lower envelopes should encompass all the data between them. The mean of these two envelopes is designated as m1, and the difference between the data X and m1 is the first component h1; i.e.

h1 ðtÞ ¼ XðtÞ  m1 ðtÞ

ð2Þ

Ideally, h1 should be an IMF, since the construction of h1 described above should have made it satisfy all the conditions set in the definition of an IMF. Yet, in practice, all the conditions of an IMF cannot be achieved until the previous process, called the sifting process, is repeated. In the subsequent sifting process, h1 is treated as the data, then

h11 ðtÞ ¼ h1 ðtÞ  m11 ðtÞ

ð3Þ

where m11 = mean of the upper and lower envelopes of h1. After repeated sifting, up to k times which is usually less than 10, h1k, given by

h1k ðtÞ ¼ h1ðkÞ ðtÞ  m1k ðtÞ

is designated as the first IMF component c1 from the data, or

throttle round plate

c1 ðtÞ ¼ h1k ðtÞ

D.C. electromotor

ð4Þ

flow out flow in

ð5Þ

Typically, c1 will contain the finest-scale or the shortest-period component of the signal. One then removes c1 from the rest of the data to obtain the residue

r 1 ðtÞ ¼ XðtÞ  C 1 ðtÞ

ð6Þ

The residue r1, which contains longer-period components, is treated as the new data and subjected to the same sifting process as described above. This procedure can be repeated to obtain all the subsequent rj functions as follows: Fig. 3. Fluid oscillation imitate device.

r j1 ðtÞ  cj ðtÞ ¼ rj ðtÞ;

j ¼ 2; 3; . . . ; n

ð7Þ

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The sifting process can be terminated by either of the following predetermined criteria: (1) either the component cn or the residue rn becomes so small that it is less than a predetermined value of consequence; and (2) the residue rn becomes a monotonic function, from which no more IMF can be extracted. If the data have a trend, the final residue will be that trend. The original data are thus the sum of the IMF components plus the final residue

xðtÞ ¼

n X cj ðtÞ þ r n ðtÞ

ð8Þ

the IMF can be considered as a generalized Fourier expansion. The time-dependent amplitude and instantaneous frequency in Eq. (14) might not only improve the flexibility of the expansion, but also enable the expansion to accommodate non-stationary data. The frequency-time distribution of the amplitude is designated as the Hilbert amplitude spectrum, H(x, s), or simply Hilbert spectrum, defined as

Hðx; tÞ ¼

Thus, the data are decomposed into n IMF components and a residue rn that can be either the mean trend or a constant.

hðxÞ ¼ For given data, C(t), the Hilbert transform, Y(t), is defined as

p

p

Z

cðt 0 Þ 0 dt t  t0

ð9Þ

where P denotes the Cauchy principal value. With this definition, C(t) and Y(t) can be combined to form the analytical signal Z(t), given by

ZðtÞ ¼ CðtÞ þ iYðtÞ ¼ aðtÞeihðtÞ

ð10Þ

where time-dependent amplitude a(t) and phase u(t) are found as

aðtÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 ðtÞ þ Y 2 ðtÞ

hðtÞ ¼ arctan

YðtÞ CðtÞ

ð11Þ ð12Þ

From the polar coordinate expression of Eq. (9), the instantaneous frequency can be defined as

xðtÞ ¼

n X aj ðtÞ

dhðtÞ dt

ð13Þ

Applying the Hilbert transform to the n IMF components of X(t) in Eq. (8), the data X(t) can be written as

R n X XðtÞ ¼ b aj ðtÞei xj ðtÞdt

ð14Þ

j¼1

where b real part of the value to be calculated and aj – the analytic signal associated with the jth IMF. The residue rn is not included because of its monotonic property Huang et al. [10]. Eq. (14) is written in terms of amplitude and instantaneous frequency associated with each component as functions of time, which differ from the time-independent amplitude and phase in the Fourier series representation of n X XðtÞ ¼ b Aj eiXj t

n n Z X X ~ ðxÞ  h j j¼1

j¼1

T

aj ðtÞdt

j¼1

where Aj = Fourier transform of X(t), a function of frequency Xj. A comparison of the two representations in Eqs. (14) and (15) suggests that the Hilbert transform of

ð17Þ

0

Provides a measure of total amplitude contribution from each frequency value, in which T denotes the time duration of data. It should be noted that the Hilbert transform described in Eqs. (9)-(13) is not new. However, the incorporation of the Hilbert transform into the IMF components and thus the HHT representation of data in Eq. (14) are entirely novel. Huang et al. [11] shows that the instantaneous frequency has physical meaning only through its definition on each IMF component; by contrast, the instantaneous frequency defined through the Hilbert transform of the original data might be less directly related to frequency content because of the violation of the monocomponent condition on the Hilbert transform. 3.3. Summary The EMD is based on the local characteristic time scale of the data and is thus an adaptive and efficient characterization of non-stationary data. The HSA of IMF defines the instantaneous or time-dependent frequency of the data, a generalized version of Fourier-based fixed or time-independent frequency. These two unique properties enable HHT analysis to reveal the possible enhanced interpretive value of decomposed components (i.e., IMF or its grouped components) and Hilbert spectra, alternative to Fourier components and spectra. In addition, the HHT data representation makes it possible to capture the low-frequency components without requiring long data length, and nonlinear waveform distortions without resorting to the spurious harmonics. The details of the method can be seen in Huang et al. [10,11]. To expose the relation between the main frequency and the period of a signal, let us observe a simple example firstly. Let

xðtÞ ¼ cos 20pt þ 1:5 cos 100pt ð15Þ

ð16Þ

j¼1

~ j ¼ jth component where  denotes ‘‘by definition’’ and H of the total Hilbert spectrum H. The square of H reveals the evolutionary energy distribution or energy density. The marginal spectrum, h(x), defined as

3.2. Hilbert Spectral Analysis

YðtÞ ¼

~ j ðx; tÞ  H

j¼1

j¼1

1

n X

ð18Þ

Their waveforms which are distorted from the sine waveform can be treated as the results caused by nonlinearity. As shown in Fig. 4, in which are the waveforms correspond to f = 10 Hz and 50 Hz respectively from top to bottom. This example implies that the nonlinearity, which

J. Peng, G. Zhang / Measurement 45 (2012) 1765–1781

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Fig. 4. IMF of simple example signal.

brings plenty of frequency components in signal, will cause main frequency shift (MFS). In other words, it will be unreliable to estimate signal period according to its main frequency. Most similar experiments conducted by us support the observation. Its marginal spectrum is shown in Fig. 5. One has no difficult in seeing the main frequency of x(t) is 10 Hz and 50 Hz, which is precisely the reciprocal of signal period. On the other hand, it is easy to see that the signal energy is mainly concentrated on the main frequency. It is worth noting that the main frequency energy of such a special signal should theoretically be equal to its total energy because it does not contain any other frequency components. The difference between theory and reality is completely caused by computation. However, we think the difference will not cause the vital effect on analysis result.

4. Hilbert–Huang Transform analysis of signal characteristics of swirlmeter in oscillatory flow In order to investigate on signal characteristics of swirlmeter in oscillatory flow, two experiments were designed. One is the response of a pipe to oscillatory flow, and the other is the response of a swirlmeter to steady flow or oscillatory flow. In this section, the HHT method was applied to analyze the characteristics of swirlmeter signal in oscillatory flow, using MATLAB as the main software tool.

4.1. Typical signal analysis In order to describe the analysis clearly, some important concepts of analysis, such as main frequency and

Fig. 5. Marginal spectrum of simple example signal.

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Fig. 6. Typical Time–Frequency–Amplitude distribution for the swirlmeter in steady flow.

amplitude of main frequency, are explained in this section. In this paper, the main frequency was defined as the apparent frequency components except that the power line interference frequency in marginal spectrum. 4.1.1. Typical signal of the swirlmeter in steady flow The sketch map of the experiment of response of the swirlmeter in steady flow is shown in Fig. 2. The Time–Frequency–Amplitude distribution of the typical signal of swirlmeter in steady flow is shown in Fig. 6. In Fig. 6, there are two apparent frequency components within 1 s of sampling time. The marginal spectrum of it is shown in Fig. 7. In the Figure, there are two larger amplitude frequencies (45 and 255 Hz). In this paper, the frequency components of about 60 Hz are considered the frequencies of power line interference. So, the 45 Hz frequency component is the frequency of power line interference. In this case, the main frequency (fssf1) is 255 Hz. The amplitude of the main frequency (Assf1) is 6.4  103. 4.1.2. Typical signal of a pipe to oscillatory flow The sketch map of the experiment of the response of a pipe to oscillatory flow is shown in Fig. 8. In the experiment, the response of a pipe to oscillatory flow is obtained based on the fluctuating pressure measured by the piezoelectric press sensor. The experimental volume flow rate is ranged from 5.5 to 220.5 m3/h. The imposed velocity frequency of oscillatory flow is ranged from 0 to 300 Hz. For the response of a pipe to oscillatory flow, the marginal spectrums have two types.

Fig. 8. Sketch map of response of a pipe to oscillatory.

For Type I signal of it, the marginal spectrums includes a frequency component of about 60 Hz and a high frequency component (>60 Hz). The Time–Frequency–Amplitude distribution of it is shown in Fig. 9. In Fig. 9, there are two apparent frequency components within 1 s of sampling time. The marginal spectrum of it is shown in Fig. 10. There are two larger amplitude frequencies (75 Hz and 615 Hz) in Fig. 10. In this case, the main frequency (fpof1) is 615 Hz. The amplitude of main frequency (Apof1) is 7  103. For Type II signal of a pipe to oscillatory flow, the marginal spectrums include a frequency component of about 60 Hz and two high frequency components (>60 Hz). The Time–Frequency–Amplitude distribution of it is shown in Fig. 11. In the Fig. 11, there are three apparent frequency components within 1 s of sampling time. The marginal spectrum of it is shown in Fig. 12. There are three larger amplitude frequency (45, 105 and 795 Hz) in Fig. 12. In this case, a low frequency (LF) main frequency and a high frequency (HF) main frequency were adopted. So, a LF main frequency (fpofL) is 105 Hz and a HF main frequency (fpofH) is 795 Hz. The amplitude of fL (ApofL) is 8.1  103 and that of fH (ApofH) is 1.765  102.

Fig. 7. Typical marginal spectrum for the swirlmeter in steady flow.

J. Peng, G. Zhang / Measurement 45 (2012) 1765–1781

Fig. 9. Typical Time–Frequency–Amplitude distribution for Type I signal for a pipe to oscillatory flow.

Fig. 10. Typical marginal spectrum for Type I signal for a pipe to oscillatory flow.

Fig. 11. Typical Time–Frequency–Amplitude distribution for Type II signal for a pipe to oscillatory flow.

Fig. 12. Typical marginal spectrum for Type II signal for a pipe to oscillatory flow.

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Fig. 13. Typical Time–Frequency–Amplitude distribution for Type I signal for a swirlmeter in oscillatory flow.

Fig. 14. Typical marginal spectrum for Type I signal for a swirlmeter in oscillatory flow.

Fig. 15. Typical Time–Frequency–Amplitude distribution for Type II signal for a swirlmeter in oscillatory flow.

Fig. 16. Typical marginal spectrum for Type II signal for a swirlmeter in oscillatory flow.

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Fig. 17. Typical Time–Frequency–Amplitude distribution for Type III signal for a swirlmeter in oscillatory flow.

Fig. 18. Typical marginal spectrum for Type III signal for a swirlmeter in oscillatory flow.

Table 1 The data between fsof, fssf1, fpof and Q in different typical fd = 68 Hz. fd = 68 Hz Flow rate (m3/h)

Frequency swirlmeter oscillatory flow fsof (Hz)

Frequency swirlmeter steady flow fssf1 (Hz)

Frequency pipe oscillatory flow fpof (Hz)

fsof1 or fsofL

fsofM

fsofH

fssf1

fpof1 or fpofL

fpofH

36.7 66.6 103.3 148.4 178.3 200.4 215

195 345 315 255 435 285 315

No No No No No No No

375 No 555 825 975 1095 1125

255 375 585 825 975 1095 1125

315 315 315 285 405 255 285

No No No No No No No

Frequency swirlmeter oscillatory flow fsof (Hz)

Frequency swirlmeter steady flow fssf1 (Hz)

Frequency pipe oscillatory flow fpof (Hz)

fsof1 or fsofL

fsofM

fsofH

fssf1

fpof1 or fpofL

fpofH

255 405 165 615 675 675

No No 555 No No No

No 615 645 No 1095 1185

255 375 585 645 1095 1125

255 645 165 615 645 615

No No 645 No No No

Table 2 The data between fsof, fssf1, fpof and Q in different typical fd = 136 Hz. fd = 136 Hz Flow rate (m3/h)

36.7 66.6 103.3 111.7 200.4 215

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Table 3 The data between fsof, fssf1, fpof and Q in different typical fd = 181 Hz. fd = 181 Hz Flow rate (m3/h)

Frequency swirlmeter oscillatory flow fsof (Hz)

36.7 66.6 103.3 148.4 178.3 200.4 215

195 165 135 195 935 765 945

fsof1

or fsofL

Frequency swirlmeter steady flow fssf1 (Hz)

Frequency pipe oscillatory flow fpof (Hz)

fsofM

fsofH

fssf1

fpof1 or fpofL

fpofH

No 405 585 825 No No No

No 855 855 855 1105 1125 1125

255 375 585 825 975 1095 1125

135 165 105 195 735 795 945

No 825 795 795 No No No

Frequency swirlmeter steady flow fssf1 (Hz)

Frequency pipe oscillatory flow fpof (Hz)

Table 4 The data between fsof, fssf1, fpof and Q in different typical fd = 227 Hz. fd = 227 Hz flow rate (m3/h)

36.7 66.6 103.3 148.4 178.3 200.4 215

Frequency swirlmeter oscillatory flow fsof (Hz) fsof1 or fsofL

fsofM

fsofH

fssf1

fpof1 or fpofL

fpofH

135 405 195 165 1005 885 225

No No 555 855 No No 1005

195 975 885 975 No 1065 1125

255 375 585 825 975 1095 1125

135 375 165 135 945 885 225

No 1005 915 945 No No 1005

Frequency swirlmeter steady flow fssf1 (Hz)

Frequency pipe oscillatory flow fpof (Hz)

Table 5 The data between fsof, fssf1, fpof and Q in different typical fd = 295 Hz. fd = 295 Hz Flow rate (m3/h)

Frequency swirlmeter oscillatory flow fsof (Hz) fsof1 or fsofL

fsofM

fsofH

fssf1

fpof1 or fpofL

fpofH

36.7 66.6 103.3 111.7 148.4 178.3 200.4 215

195 405 165 135 105 135 165 105

No No 585 645 825 975 1065 No

255 1005 1125 1065 1035 1215 1155 1185

255 375 585 645 825 975 1095 1125

135 135 135 195 135 105 195 105

No 1125 1095 1095 1095 1125 1125 1125

4.1.3. Typical signal of the swirlmeter to oscillatory flow The sketch map of the experiment of response of the swirlmeter in oscillatory flow is shown in Fig. 2. For the response of a swirlmeter in oscillatory flow, the marginal spectrums have three types. For Type I signal of a swirlmeter in oscillatory flow, the marginal spectrums include a frequency component of about 60 Hz and a high frequency component (>60 Hz). In this case, the Time–Frequency-Amplitude distribution of it is shown in Fig. 13 and the marginal spectrum of it is shown in Fig. 14. In the Fig. 13, there are two apparent frequency components within 1 s of sampling time. There are two larger amplitude frequencies (75 Hz and 195 Hz)

in Fig. 14. In this case, the main frequency (fsof1) is 195 Hz. The amplitude of main frequency (Asof1) is 5.2287  103. For Type II signal of a swirlmeter in oscillatory flow, the marginal spectrums include a frequency component of about 60 Hz and two high frequency components (>60 Hz). The Time–Frequency–Amplitude distribution of it is shown in Fig. 15. In Fig. 15, there are three apparent frequency components within 1 s of sampling time. The marginal spectrum of it is shown in Fig. 16. There are three larger amplitude frequencies (75, 375, and 615 Hz) in Fig. 16. In this case, a low frequency (LF) main frequency and a high frequency (HF) main frequency were adopted.

J. Peng, G. Zhang / Measurement 45 (2012) 1765–1781

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Fig. 19. The relation between fsof, fssf1, fpof and Q in typical fd = 68 Hz.

Fig. 20. The relation between fsof, fssf1, fpof and Q in typical fd = 136 Hz.

So, LF main frequency (fsofL) is 375 Hz and (HF) main frequency (fsofH) is 615 Hz. The amplitude of fL (AsofL) is 8.4  103 and that of fH (AsofH) is 2.2  102. For Type III signal of a swirlmeter in oscillatory flow, the marginal spectrums include a frequency component of about 60 Hz and three high frequency components (>60 Hz). The Time–Frequency–Amplitude distribution of it is shown in Fig. 17. In the Fig. 17, there are four apparent

frequency components within 1 s of sampling time. The marginal spectrum of it is shown in Fig. 18. There are four larger amplitude frequency (45, 165, 315, and 555 Hz) in Fig. 18. In this case, a low frequency (LF) main frequency, a middle frequency (MF) main frequency and a high frequency (HF) main frequency were adopted. So, the LF main frequency (fsofL) is 165 Hz, the MF main frequency (fsofM) is 315 Hz and the HF main frequency (fsofH) is 555 Hz. The

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Fig. 21. The relation between fsof, fssf1, fpof and Q in typical fd = 181 Hz.

Fig. 22. The relation between fsof, fssf1, fpof and Q in typical fd = 227 Hz.

amplitude of fL (AsofL) is 1.126  102, the amplitude of fM (AsofM) is 1.21  102 and that of fH (AsofH) is 1.805  102. 4.2. Experimental results and discussion In order to describe the experiment results more clearly, the relationship between fsof, fssf1 fpof and Q, and

the relationship between fsof, fssf1, fpof and fd were investigated. The data between fsof, fssf1, fpof and Q in different typical fd(68, 136, 181, 227, and 295 Hz) were shown in Tables 1– 5. The relationship between fsof, fssf1 fpof and Q was shown in Figs. 19–23. Based on Tables 1–5 and Figs. 19–23, we can make the following conclusions:

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Fig. 23. The relation between fsof, fssf1, fpof and Q in typical fd = 295 Hz.

Table 6 The data between fsof, fssf1, fpof and fd in different typical Q = 36.7 m3/h. Q = 36.7 m3/h fd (Hz)

68 136 181 227 295

Frequency swirlmeter oscillatory flow fsof (Hz) fsof1

or

fsofL

195 255 195 135 195

Frequency swirlmeter steady flow fssf1 (Hz)

Frequency pipe oscillatory flow fpof (Hz)

fsofM

fsofH

fssf1

fpof1 or fpofL

fpofH

No No No No No

375 No No 195 225

255

315 255 135 135 135

No 585 No No No

Table 7 The data between fsof, fssf1, fpof and fd in different typical Q = 148.4 m3/h. Q = 148.4 m3/h fd (Hz)

Frequency swirlmeter oscillatory flow fsof (Hz) fsof1 or fsofL

fsofM

fsofH

fssf1

fpof1 or fpofL

fpofH

68 136 181 227 295

255 345 195 165 105

No 525 No 855 825

825 825 825 975 1035

825

285 285 195 135 135

No 585 795 945 1095

Frequency swirlmeter steady flow fssf1 (Hz)

1. With the increase of fd, the fpofH frequency component of fpof appears. This phenomenon may be attributed to fluid–structure interaction between the pipe and oscillatory flow. 2. With whatever flow rate, there is one main frequency (fssf1) of swirlmeter in steady flow, and the fssf1 is directly proportional to Q. 3. With fpofH frequency component appear, the middle frequency component fsofM appears.

Frequency pipe oscillatory flow fpof (Hz)

4. For the fsof, there must be a frequency component (fsofL or fsofM or fsofH) close to fssf1. Apart from the frequency component near fssf1, for fsof, other frequency components of fsof must be close to the fpof. Therefore, the fsof is the addition of fpof and fssf1. The reason of above phenomenon can be assumed as fellow: The fluid oscillation of swirlmeter in oscillatory flow is the coupled vibrations between the vibration of a

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Table 8 The data between fsof, fssf1, fpof and fd in different typical Q = 215 m3/h. Q = 215 m3/h fd (Hz)

68 136 181 227 295

Frequency swirlmeter oscillatory flow fsof (Hz)

Frequency swirlmeter steady flow fssf1 (Hz)

Frequency pipe oscillatory flow fpof (Hz)

fsof1 or fsofL

fsofM

fsofH

fssf1

fpof1 or fpofL

fpofH

315 675 945 255 105

No 525 No 1005 No

1155 1185 1125 1125 1185

1125

285 615 945 225 105

No No No 1005 1155

Fig. 24. The relation between fsof, fssf1, fpof and fd in typical Q = 36.7 m3/h.

pipe to oscillatory flow and the vibration of the swirlmeter in steady flow. The coupled relation can be decoupled by HHT. From what has been discussed above, A new method which is based on HHT spectral analysis can be used for the oscillatory flow swirlmeter noise removal. The functional block diagram of this method is shown in Fig. 28. The data between fsof, fssf1, fpof and fd in different typical Q (36 m3/h, 148 m3/h, 215 m3/h) were shown in Tables 6– 8. The relationship between fsof, fssf1, fpof and fd in different typical Q (36 m3/h, 148 m3/h, 215 m3/h) were shown in Figs. 24–26. Based on Tables 6–8 and Figs. 24–26, we can make the following conclusions. In the lower flow rate (Q = 36 m3/h), the frequency component of fsof near fssf1 drifts dramatically to low frequency direction. In the higher flow rate (Q = 148 m3/h and Q = 215 m3/h), the frequency component of fsof near fssf1 drifts lightly to high frequency direction. In order to describe the experiment results more clearly, the bias of the frequency component of fsof near fssf1 (d) is defined as:

d¼

the frequency component of f sdf near f ssf 1  fssf 1  100% fssf 1

The relationship between d and fd for different flow rate was shown in Fig. 27. For the lower flow rate (Q = 36 m3/h), the value of d is between 10% and 25%. For the higher flow rate (Q = 148 m3/h and Q = 215 m3/h), the value of d is between 0 and 5%. The reason of above phenomenon can be assumed as fellow: In the lower flow rate, the fluid vibration of swirlmeter in steady flow is very weak. The vibration of a pipe to oscillatory flow can dramatically affect the vibration of the swirlmeter in steady flow. In the higher flow rate, the fluid vibration of swirlmeter in steady flow is very strong. The vibration of a pipe to oscillatory flow cannot dramatically affect the vibration of the swirlmeter in steady flow.

5. Conclusion The signal characteristics of swirlmeter in oscillatory flow were studied by HHT method. The results led to the following conclusions: First, the response characteristics of swirlmeter in oscillatory flow are addition of the response characteristics of swirlmeter response characteristics in steady flow and

J. Peng, G. Zhang / Measurement 45 (2012) 1765–1781

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Fig. 25. The relation between fsof, fssf1, fpof and fd in typical Q = 148.4 m3/h.

Fig. 26. The relation between fsof, fssf1, fpof and fd in typical Q = 215 m3/h.

the response characteristics of the interference characteristics of oscillatory flow. Based on above conclusion, a new method which is based on HHT spectral analysis for the oscillatory flow swirlmeter noise removal was advanced.

Second, the vortex precession effect of swirlmeter is affected by oscillatory flow. In the lower flow rate, the vibration of a pipe to oscillatory flow can dramatically affect the vibration of the swirlmeter in steady flow. In the higher flow rate, the vibration of a pipe to oscillatory flow cannot

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Fig. 27. The bias of the frequency component of fsof near fssf1 (d).

Fig. 28. The functional block diagram of the method based on HHT spectral analysis.

dramatically affect the vibration of the swirlmeter in steady flow. Acknowledgment This project is supported by the National Natural Science Foundation of China (Grant No. 61074182).

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