AbstractâA technique for processing interferometric signals intended for estimation of the amount of deposit in pipes is proposed, which is based on a 2D ...
Russian Journal of Nondestructive Testing, Vol. 41, No. 7, 2005, pp. 430–435. Translated from Defektoskopiya, Vol. 41, No. 7, 2005, pp. 23–30. Original Russian Text Copyright © 2005 by Sukatskas, Volkovas.
ACOUSTIC METHODS
A Technique of Signal Processing for Interferometric Estimation of the Amount of Deposit in Pipes V. Sukatskas and V. Volkovas Kaunas University of Technology, Kaunas, Lithuania Received February 25, 2005
Abstract—A technique for processing interferometric signals intended for estimation of the amount of deposit in pipes is proposed, which is based on a 2D Fourier transform of the signals considered as functions of two variables (frequency and time). The proposed processing technique is shown to enable determination of the rate with which peaks of signal images diminish. This rate, being a measure of wave absorption in pipe walls, is a reliable characteristic of the deposit amount in a pipe. Application of this technique reduces the time needed for measurements.
Recently, we have proposed an interferometric method intended for measuring the thickness of deposits or coke on the interior surface of pipes, which is based on the usage of antisymmetrical Lamb waves excited on the outer surface of a pipe [1]. The following scheme proved to be the most efficient: a pointlike probe excites continuous waves with a gradually varying frequency that are recorded at another point. In other words, this scheme involves measurement of a response characteristic (RC) and its subsequent analysis [2]. Theoretically, each RC is a set of resonance peaks corresponding to equidistant frequencies that emerge whenever an integer number of waves fits the pipe circumference. Initially, we had proposed to estimate the ratio of the RC maximum and average values, which is related to absorption of waves, i.e., to the thickness of deposits [3]. However, an undistorted RC is observed only in practically clean pipes where measurements of deposits are not needed. Distortions are caused, in particular, by differences in the thickness and/or composition of the deposit layer. As a result, adjacent segments resonate at somewhat different frequencies that are characterized at the same time by approximately the same quality factor. This circumstance substantiates the determination of the RC autocorrelation function and its maximum steepness [4, 5]. The autocorrelation function (ACF) can be represented in a discrete form as follows: k
F( f i) =
k
∑ ∑ U( f
m )U (
f m + i ),
(1)
i = 0m = 1
where F( fi) is the ith value of the ACF, fi is the ith value of frequency (ACF argument), U( fm) and U( fm + i) are the ACF values at discrete frequencies fm and fm + i, and k is the number of points where the discrete ACF is defined. If m + i > k, U( f ) = 0. The last technique is characterized by a number of serious disadvantages, some of which are listed below: (i) The measurement time is large since resonance peaks are created by the waves that have traveled around the pipe many times. This factor limits the rate with which the frequency can be tuned. For a 150-mm-dia steel pipe with 8-mm thick walls, the time needed to measure the ACF in the frequency range 100–150 kHz is over one minute. (ii) The results are affected by reflections from inhomogeneities located within the “reach” of the waves. We have used longitudinal wave probes pressed against a pipe through a coupling liquid. Since the probe’s diameter is smaller than the length of a Lamb wave, the probe’s characteristics are close to those of a pointlike omnidirectional probe. (iii) Calculation of the ACF is not supported by the standard software supplied with measuring equipment and needs a special code to be developed. The first two disadvantages may be partly removed by exciting a piezoelectric probe with radio pulses that maintain simultaneously the frequency band of the system at a comparatively narrow value. The duration of the interference process between the waves that have traveled around a pipe many times and the number of these waves are limited in this case by the radio-pulse duration. As was shown in [3, 4, 6], this results 1061-8309/05/4107-0430 © 2005 MAIK “Nauka /Interperiodica”
A TECHNIQUE OF SIGNAL PROCESSING
in broadening and distortions of resonance peaks. We have estimated the error in the RC’s ACF related to these factors in [7]. The evolution of the recorded pulse was also studied, albeit for only some particular cases [5, 8]. The amplitudes of the waves forming the received radio pulse depend on the absorption of waves in the pipe wall. Therefore, the pulse envelope contains information about the deposit thickness. However, the possibility of extraction of this information for the specific conditions of the considered problem has not been studied yet.
(‡)
431 (b)
3
1
2
FORMATION OF THE ENVELOPE OF A RECEIVED PULSE Fig. 1. Layout of the experiment (a); formation of a
The layout of a system for excitation and detecreceived pulse (b). tion of waves is shown in Fig. 1a. Piezoemitter 1 and piezoreceiver 2 are located diametrically opposite to each other in the same section of pipe 3. The waves arrive at piezoreceiver 2, having traveled many times around the pipe in the clockwise and counterclockwise directions. If a transient process after the arrival of a wave is shorter than the time interval until the arrival of the next wave, the complex-valued amplitude of a summed wave, u, represented as a vector is described by the expression N
∑u ,
u = u0 +
(2)
i
i=1
where u0 and ui are the complex-valued amplitudes (vectors) of the first and ith detected wave and N is the number of detected waves. The wave amplitudes decrease owing to divergence and absorption according to u i = u s exp ( – αr i )/ r i ,
(3)
where us is the amplitude of an emitted signal, ri is the path passed by the wave, and α is the absorption coefficient. The set of waves can be represented most conveniently as a chain of vectors (Fig. 1b), where the sum of these vectors is a vector connecting the beginning and end of the chain. Each vector is shown as a pair of collinear line segments conventionally representing the waves that arrive from the opposite sides. The chain degenerates into a straight line when the difference between the wave phases is 2π (resonance-peak zone) and curls otherwise. The reference frame in Fig. 1b is related to wave u0, and chains correspond to different cases near a resonance. When analyzing the envelope of the detected pulse u = u( f, t), where f is the filling frequency and t is time, appearance of a periodic component along the time axis can be expected since the waves arrive after the equal time intervals τ = c/2πR, where c is the wave velocity and R is the average pipe radius. Each arriving wave changes amplitude u with the exception of those rare cases when u and ui are perpendicular to each other. However, the character of these changes (increase, decrease, or periodic variations) depends on f. A periodic behavior may be expected as well along the f axis. We have shown in [6] that, when a ring-shaped probe is used, N – 1 secondary maxima appear between two resonance peaks in the ACF after the arrival of N waves due to interference phenomena. The larger the value of N, the smaller the amplitude of the maxima. To conduct experimental studies of the envelope, we have used a 150-mm-dia steel pipe with 8-mm-thick walls containing a 0-, 10-, and 20-mm-thick coke layer. Measurements were made in a frequency range of 115–146 kHz. The envelopes of two-dimensional function u( f, t) describing the received signal are shown in Figs. 2 and 3. The isolines f = const plotted for each kilohertz of the filling frequency and t = const in the middle of an interval between the arrival of N and N + 1 waves (N = 0–5) are also shown in the plots. Formation of resoRUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING
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SUKATSKAS, VOLKOVAS 1
urel
0 120
130
140
f, kHz 0
0.5
t, ns
Fig. 2. Envelope of received pulse u(f, t) in an empty pipe.
nance peaks after approximately every 7 kHz is clearly seen. The decrease of the amplitude with increasing frequency (Fig. 3) is related to frequency properties of the pair of probes. A modification of the envelope along f = const is seen in both plots; the isolines t = const show secondary maxima for N = 2 and 3 that are more pronounced in an empty pipe. More detailed conclusions about their properties may be drawn by applying a two-dimensional Fourier transform. A TWO-DIMENSIONAL FOURIER TRANSFORM OF FUNCTIONS U( F, T) A two-dimensional Fourier transform (2D FFT) was successfully applied in ultrasonic testing for spatial filtering [9], analysis of a complex multimode signal [10], etc. A 1D Fourier transform is widely used and packages of the respective software are supplied with many analyzers. A 2D transform can be obtained using the same algorithm applied first to rows and then to columns (or vice versa) of a 2D array [11]. In our study, we have used a freeware package [12]. Function u( f, t) was recorded as an array with dimensions 32 × 256 where 32 columns correspond to the filling frequency of an exciting pulse with an interval of 1 kHz, and 256 rows correspond to values of u(f, t) at a fixed frequency after every 3 µs. The 2D FFT yields array U(f, t) with dimensions 32 × 512 where information about the absolute values of the Fourier components is contained in elements from U(0,0) to U(16, 128); other elements up to U(32, 256) are their mirror reflections. The first index corresponds to the row number. Similarly, elements U(0, 257) to U(32, 512) contain information about the phase. Transform results (elements U(0, 0) to U(16, 16)) are shown in Figs. 4–6. The axis of rows (t in Figs. 4–6) yields the value 1/f = t where f is the filling frequency of a radio pulse. Although this quantity has the dimension of time, it should be understood as the number of periodic changes of function u(f, t) (for given t in the original) per unit interval of the filling frequencies. The other horizontal axis yields the frequency of periodic changes (their number per unit time) of function u(f, t) at f = const (in the original). The scale of images is established by the ratio of the respective scales of the image and the original: F = 1/dt and T = 1/df, where F and T are the maximum values of the respective scales (along rows and columns, receptively) of the original and dt and df are the intervals along the respective axes of the image [12]. For F = 32 kHz, interval dt = 0.03125 1/kHz. Since T = 768 µs, interval df = 1.302 kHz. RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING
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urel 1 0 120
130
140 f, kHz 0
0.5
t, ns
Fig. 3. Envelope of received pulse u(f, t) in a pipe containing a 20-mm-thick coke deposit.
4
13
2
.30
Fig. 4. Two-dimensional Fourier transform of function u(f, t) for an empty pipe.
9
×1
1
5
Hz
02
1.3
5 3 9 7 1.25 11 3 15 13 t, µs ×
1
f, k
× Hz
7 10 13 16
0 –10 –20 –30 –40 –50 –60 –70 –80 –90 –100
U(t, f), dB
U(t, f), dB
1
f, k
0 –10 –20 –30 –40 –50 –60 –70 –80 –90 –100
3 9 7 5 11 13 5 15 t, µs × 31.2
1
Fig. 5. Two-dimensional Fourier transform of function u(f, t) for a pipe containing a 10-mm-thick coke layer.
RESULTS AND DISCUSSION Studying two-dimensional Fourier transform ν(t, f) that envelopes received pulse u(f, t), we may conclude that periodicity of the image (elements U(0, 0) to U(0, 128)) along axis f is virtually absent. However, columns can be found in the original (see Figs. 2, 3) that are characterized by significant periodic variations along f = const. The differences in the character of variations along adjacent isolines f = const seem to result in their averaging in the image. Peaks of function U(t, 0) are clearly seen at t = 0, 5, 9, and 12. Interval dt being taken into account, this means that function u(f, t) changes after every 7, 3.5, and 2.6 kHz of the frequency of filling. The first change corresponds to resonance peaks, and the two subsequent changes correRUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING
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0 –10 –20 –30 –40 –50 –60 –70 –80 –90 –100
U(t, f), dB
434
1 f, 5 kH 9 z× 1. 13 30 2
15 13
11
5 7 9 5 .2 31 × t, µs
3
1
Fig. 6. Two-dimensional Fourier transform of function u(f, t) for a pipe containing a 20-mm-thick coke layer.
U(t, 0) 0.20
0 cm 1 cm 2 cm
0.16 1.0 0.12
0.8 0.6
0.08
1
0.4 4
0.2
7 10
0 0 cm 2 cm
13
0 cm 1 cm
0.04
2 cm
0
16
Fig. 7. First rows of the normalized plots displayed in Figs. 4–6.
3
6
9
12
15 1/f
Fig. 8. First rows (elements U(0, 3) to U(17, 3)) of the plots displayed in Figs. 4–6.
spond to the aforementioned secondary maxima at N = 2 and 3. Figure 7 shows elements U(0, 0) – U(16, 0) of the 2D FFT of images in Figs. 4–6 on a linear scale normalized to the value of U(0, 0). The decrease of the peak amplitudes of function U(t, f) is the smallest in an empty pipe and the largest in a pipe with a 20-mm-thick coke layer. This finding is a direct consequence of the fact that absorption of the second and third (as well as subsequent) waves participating in the interference becomes stronger as the coke layer becomes thicker. Therefore, application of, e.g., the regression analysis allows determination of the rate with which peaks characterizing the presence of coke in a pipe and its amount decrease. Element U(0, 0) is an invariable component (an average value of the array) and depends on the beginning (the arrival of the first wave or the moment when the signal attains a certain level) and end of the procedure that records the envelope of received pulse u(f, t). If these values are not recorded, the zero element can be excluded from consideration by comparing peaks of function U(t, n) alone where n > 0. Figure 8 shows rows U(0, 3) to U(17, 3) normalized with respect to the value of the first maximum. By investigating the plot, a reliable conclusion can be drawn that the second peak is largest for an empty pipe and smallest for a pipe containing a 20-mm-thick coke layer. The third maximum is diffuse. To make this peak more pronounced, it is necessary to increase the range of axis t and simultaneously decrease dt. Owing to the relationship between the scale ranges and scale intervals for the image and the original [12], the first requirement may RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING
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be met by decreasing the interval when the original is recorded and the second requirement may be met by increasing the range of radio-pulse filling. Therefore, the rate with which the peaks of row U(t, 0) or U(t, n) of the image decrease may be used as a measure of absorption in the pipe wall and, thus, as an indicator of the presence and amount of deposits. CONCLUSIONS The method for estimation of the thickness of deposit layers in pipes using excitation of Lamb waves by radio pulses and a two-dimensional Fourier transform of the envelope of the received pulse may have certain advantages As compared to the method based on determination of the RC’s ACF: The time needed for recording information is significantly reduced (in the described case 32 × 1 m as compared to 1 min needed for determination of the RC’s ACF). The information obtained is more reliable owing to its more comprehensive utilization (dynamics of a process is recorded instead of a steady-state process, as in the technique using the RC’s ACF). The effects of closely located inhomogeneities can be filtered out. The proposed technique for processing signals is implemented in virtual and real time; therefore, it can be widely used not only as a means to improve a nondestructive testing technique but also as an effective instrument for monitoring creation of deposits on pipe walls. ACKNOWLEDGMENTS The authors are grateful to the German Academic Exchange Service (DAAD) for the equipment that was provided as part of a support program for former trainees, thus enabling this study to be conducted. REFERENCES 1. Volkov, V., Sukatskas, V., and Potapenko, V., Patent of Lithuanian Republic no. 1P567, Byull. Izobret., 1994, no. 12. 2. Volkov, V. and Sukatskas, V., An Interferometric Method of Evaluating Engineering Pipeline State, Rus. J. of Nondestructive Testing, 1995, vol. 31, no. 8, pp. 604–607. 3. Volkov, V. and Sukatskas, V., Technical Diagnostics of Pipe-Lines by the Approach of Wave Interference, Proc. of IMEKO XIV World Congress, Tampere, Finland, Tampere: Finnish Soc. Automation, 1997, vol. 7, pp. 67–70. 4. Sukatskas, V., Giedraitiene, V., Ramanauskas, R., Nondestructive Testing of the Pipe Inner Cavity, Proc. of IMEKO XVI World Congress, 2000, Vienna, Austria, Vienna: Austrian Soc. Measur. Automation, 2000, vol. 6, pp. 273–277. 5. Sukatskas, V. and Volkovas, V., On the Control of Wave Interference in Complex Acoustic Systems, Tekhn. Diagn. Nerazr. Kontr., 1998, no. 1, pp. 22–26. 6. Sukatskas, V. and Volkovas, V., Investigation of the Lamb Wave Interference in a Pipeline with Sediments on the Inner Surface, Defektoskopiya, 2003, no. 6, pp. 39–47 [Rus. J. of Nondestructive Testing (Engl. Transl.), 2003, vol. 39, no. 6, p. 445]. 7. Sukatskas, V. and Ramanauskas, R.A., PC Simulation of the Propagation of Lamb Waves in Pipes, Proceedings of the 3rd Int. Conference on Modeling and Experimental Measurements in Acoustics, Cadiz, Spain, Boston, WIT Press Southampton, 2003, pp. 391–398. 8. Shanaurin, A.M., Komlev, D.G., Kravchenko, G.I., and Kuz’min, S.Yu., New Approach to Testing Wheel Disks of Railroad Cars, Defektoskopiya, 2002, no. 9, pp. 90–95 [Rus. J. of Nondestructive Testing (Engl. Transl.), 2002, vol. 38, no. 9, p. 709]. 9. Alleyne, D. and Cawley, P., A Two-Dimensional Fourier Transform Method for the Measurement of the Propagating Multimode Signals, J. Acoust. Soc. Am., 1991, vol. 89, no. 3, pp. 1159–1168. 10. John C. Ross, The Image Processing Handbook, Florida, USA: CRC Press Inc., 1992. 11. Multi Precision Floating Point Computing for EXCEL: http://digilander.libero.it/foxes/ XNUMBERS 4.1. RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING
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