Ultrasonic pressure measurement in pressure vessels Yao Bi, Hongliang Zhou, Zhiyao Huang, Hanhua Zhou, and Xianglong Yang Citation: Review of Scientific Instruments 85, 125002 (2014); doi: 10.1063/1.4902338 View online: http://dx.doi.org/10.1063/1.4902338 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/85/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Double threshold ultrasonic distance measurement technique and its application Rev. Sci. Instrum. 85, 044905 (2014); 10.1063/1.4871993 Ultrasonic measurement of condensate film thickness J. Acoust. Soc. Am. 124, EL196 (2008); 10.1121/1.2968297 Investigation of Noninvasive Approaches for Pressure Measurement AIP Conf. Proc. 894, 1653 (2007); 10.1063/1.2718163 Application of the acousto-optic effect to pressure measurements in ultrasound fields in water using a laser vibrometer Rev. Sci. Instrum. 75, 3203 (2004); 10.1063/1.1790556 Ultrasonic waveguide techniques for the measurement of material properties AIP Conf. Proc. 615, 1742 (2002); 10.1063/1.1473003
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 125002 (2014)
Ultrasonic pressure measurement in pressure vessels Yao Bi,1 Hongliang Zhou,1,a) Zhiyao Huang,1 Hanhua Zhou,1 and Xianglong Yang2 1 State Key Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou, 310027, People’s Republic of China 2 School of Biosystems Engineering and Food Science, Zhejiang University, Hangzhou, 310058, People’s Republic of China
(Received 31 December 2013; accepted 9 November 2014; published online 2 December 2014) Based on the reflected longitudinal wave, a new non-intrusive method for pressure measurement is proposed. The acoustoelastic theory and the thin-shell theory are introduced to develop the pressure measurement model in cylindrical pressure vessels. And a pressure measurement system is constructed to evaluate the effectiveness of the proposed method. The pressure measurement is implemented by measuring the travel-time change between two received ultrasonic sensors. The experimental results verify the feasibility and effectiveness of this new non-intrusive method. Compared with the non-intrusive pressure measurement method based on the critically refracted longitudinal wave (LCR wave), the proposed non-intrusive pressure measurement method has the advantages of higher sensitivity and higher signal-to-noise ratio. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4902338] I. INTRODUCTION
Pressure vessels, as special equipment, are widely used in many industrial fields, such as chemical, energy, and metallurgical engineering, etc. Overpressure of pressure vessel is one of the major reasons for the safety accidents which cause severe disasters and economic loss. The effective measurement and control of the vessel pressure is an important approach to prevent accidents.1 The traditional methods for pressure measurement usually require drilling holes in the vessel wall, which will cause local stress concentration and may make the wall prone to crack. There is a practical requirement for non-intrusive pressure measurement. Unfortunately, up to date, the effective non-intrusive pressure measurement methods or instruments are very limited, although some non-intrusive pressure measurement methods have been proposed, such as the strain gauge method,2 the capacitor method,3 and the ultrasonic method. So, more research works should be undertaken in this area. Among the above non-intrusive pressure measurement methods, the non-intrusive pressure measurement method based on ultrasonic wave technique is an attractive and potential method. It has the advantages of low cost and simple construction. To implement the non-intrusive pressure measurement by ultrasonic technique, many research groups have made their efforts. Diodati found there was a linear dependence between the amplitude of an acoustic pulse propagating inside the vessel wall and the pressure of the internal fluid, and further implemented the pressure measurement by Lamb wave.4 However, the linear dependence will be affected by different liquid in the container. Yu et al. found that acoustic velocity in oil increased with the rise of pressure and established the pressure measurement model for the hydraulic a) Author to whom correspondence should be addressed. Electronic mail:
[email protected]
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system.5 But the acoustic velocity was also affected by the medium in the vessel. Rosenkrantz et al. developed an acoustic method which was on basis of the ultrasonic wave velocity of the longitudinal wave, and realized the pressure measurement in the upper plenum of a standard light water reactor (LWR) fuel rod.6 Unfortunately, the amplitude of the acoustic signals was relatively weak, which meant lower measurement accuracy for the ultrasonic wave velocity. Ling et al. developed a pressure measurement model which was based on the travel-time of ultrasonic wave, and implemented the pressure measurement by utilizing the LCR wave and Rayleigh wave.7 But the amplitude of the LCR wave was also weak, which will cause more calculation error for the travel-time. In summary, the conventional non-intrusive pressure measurement methods based on ultrasonic wave technique still have some drawbacks. The measurement performances (measurement sensitivity and accuracy) cannot meet the requirements of academic research and industrial application. In this work, we will introduce a new non-intrusive pressure measurement method based on the reflected longitudinal wave. As shown in Fig. 1, when a beam of longitudinal wave is incident from medium I to medium II, waveform conversion will take place at the interface according to the Snell’s law. A portion of energy is reflected to medium I and converts to the reflected longitudinal wave and the reflected shear wave. The other portion of energy is refracted to medium II and converts to the refracted longitudinal wave and the refracted shear wave. Further, if a longitudinal wave is incident into the interface (between the transmitting probe and the vessel outer wall) with the first critical angle, a LCR wave and a refracted shear wave will be generated, as shown in Fig. 2. The LCR wave will travel along the outer wall of the vessel and will be received by the receiving probe. The refracted shear wave will be reflected on the inner wall of the pressure vessel, which will generate reflected longitudinal wave-I1 and
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FIG. 1. Reflection and refraction of the ultrasonic wave.
reflected shear wave-1 (as shown in Fig. 2). Based on the Snell’s Law, because the reflection angle of reflected longitudinal-I1 wave is 90◦ , the reflected longitudinal wave-I1 will travel along the inner wall of the vessel. But, the reflected shear wave-1 will continue to propagate and will be reflected on the outer wall of the vessel. That will generate reflected longitudinal wave-1 (which will travel along the outer wall of the vessel and will be finally received by the receiving probe) and reflected shear wave-2 (which will continue to propagate and generate succeeding reflected longitudinal wave-I2 and reflected shear wave-3). And so on. Thus, the receiving probe will receive one LCR wave and several reflected longitudinal waves, respectively (reflected longitudinal wave-1, reflected longitudinal wave-2, and so on), as shown in Fig. 3. Research works have verified that the change of pressure of the cylindrical pressure vessel will cause the change of the stress of the cylindrical pressure vessel.8 Research works also indicated that the change of the stress of the cylindrical pressure vessel will cause the change of ultrasonic wave velocity in the wall.9 Thus, the obtained reflected longitudinal waves contain the information of the pressure of the cylindrical pressure vessel. That means it is possible to use reflected longitudinal waves to implement the pressure measurement. Meanwhile, our experimental tests have indicated that the reflected
FIG. 3. Received ultrasonic signal observed by the digital oscilloscope.
longitudinal waves have higher amplitude and signal-to-noise ratio (SNR) compared with the LCR wave, as shown in Fig. 3. However, up to date, the research works concerning the application of reflected longitudinal wave to the non-intrusive pressure measurement are very limited. Few research works have been reported. This work attempts to introduce the reflected longitudinal waves into the research field of pressure measurement and hence to propose a new non-intrusive pressure measurement method which is on the basis of reflected longitudinal waves. Fig. 4 shows the study scheme of the new non-intrusive pressure measurement method. The whole research works include six parts: (1) The acoustoelastic theory is introduced to obtain the relationship between the stress of the pressure vessel and the ultrasonic wave velocity.8 (2) The thin-shell theory is introduced to obtain the relationship between the pressure of the cylindrical pressure vessel and the stress.9 (3) Based on
FIG. 2. Propagation of ultrasonic wave inside the vessel wall.
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described as 2 = λ + 2μ + (2l + λ)(εA + εR + εC ) ρ0 VAA
+ (4m + 4λ + 10μ)εA ,
(1)
where VAA denotes the longitudinal wave velocity in the pressure vessel wall along the axial direction. εA , εR , and εC are the stresses in the pressure vessel wall along the axial, radial, and circumferential directions, respectively. ρ 0 is the initial density of the pressure vessel. λ and μ are second-order elastic constants, while l, m, and n are third-order elastic constants. Based on the thin-shell theory,9 the stresses are related to the internal pressure. The stress field in the wall of the cylindrical pressure vessel is two-dimensional, including the stresses in the axial direction and in the circumferential direction, and can be described as σA =
FIG. 4. Study scheme of the new non-intrusive pressure measurement method.
the acoustoelastic theory and thin-shell theory, the relationship between the pressure and ultrasonic wave velocity can be developed. (4) The pressure measurement model, which describes the relationship between the pressure and the variation of travel-time of the reflected longitudinal wave, is derived. (5) With one transmitting probe and two receiving probes, a pressure measurement system is constructed. (6) The variation of travel-time between two receiving probes is obtained by experimental tests and the pressure measurement is implemented by the pressure measurement model. Besides, because the temperature is an important factor for ultrasonic measurement technique,10 in this work, the influence of temperature on the pressure measurement will also be investigated. II. DEVELOPMENT OF THE PRESSURE MEASUREMENT MODEL A. The relationship between pressure and ultrasonic wave velocity
The ultrasonic method realizes the non-intrusive pressure measurement by monitoring the variation of the ultrasonic wave velocity. It is necessary to obtain the relationship between the pressure and ultrasonic wave velocity. Unfortunately, the direct relationship between the pressure and ultrasonic wave velocity has not been established. Therefore, in this work, we first develop the relationship between ultrasonic wave velocity and stress. Then, we establish the relationship between the pressure and stress. Finally, the relationship between the pressure and ultrasonic wave velocity can be developed. As derived by Hughes and Kelly,8 the relationship between the ultrasonic wave velocities and the strain can be
pR , 2d
(2) pR σC = , d where σ A and σ C are the stresses in the pressure vessel wall along the axial and the circumferential directions, respectively, p represents the internal pressure, R denotes the average radius of the cylindrical pressure vessel, and d is the thickness of the wall. Based on the Hooke’s Law,11 the strain components in the pressure vessel wall under the state of two-dimensional stress are given in the following equations: 1 (σ − νσC ), E A ν (3) εR = − (σA + σC ), E 1 εC = (σC − νσA ), E where εA , εR , and εC represent the strains in the pressure vessel wall along the axial, radial, and circumferential directions, respectively, E is the elasticity modulus of the vessel material, and ν is the Poisson’s ratio of the vessel material. Thus, with Eqs. (1)–(3), the relationship between pressure and the longitudinal wave velocity can be obtained, as given in the following equation: εA =
E d 2 ρ0 VAA − λ − 2μ , (4) LR where L represents the constant, which is related to the second-order and third-order elastic constants 6l + 7λ + 4m + 10μ − 12lν − 14λν − 8mν − 20μν . L= 2 (5) p=
B. Pressure measurement model based on the reflected longitudinal wave
For ultrasonic measurement system, the velocity measurement of the ultrasonic wave is usually implemented by the travel-time measurement.12
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Under the free pressure state, the stresses in the pressure vessel wall along the axial, radial, and circumferential directions could be considered as free (εA = εR = εC = 0). With Eq. (1), the relationship between ρ 0 and VL0 , which is the longitudinal wave velocity under the free pressure state, can be described as 0 2 λ + 2μ VL = . (6) ρ0 Besides, the VL0 can also be defined as VL0
D = 0, tL
p = −
E d tL . L1 R tL0
Meanwhile, the relationship between the shear wave velocity and the strain is also summarized by Hughes and Kelly,8 which can be described as
(7)
where represents the reference travel-time of the longitudinal wave under the free pressure state, D represents the propagation distance of the longitudinal wave. With Eqs. (4)–(7), the relationship between the internal pressure and the travel-time of the longitudinal wave can be described as 2 tL0 Ed (λ + 2μ) −1 , (8) p= LR tL where tL is the travel-time of the longitudinal wave under the measured pressure state. With Eq. (8), the acoustoelastic model for the longitudinal wave traveling along the axial direction of the pressure vessel can be developed as 3 tL E d tL0 , (9) p = − L1 R t L tL0 Where p is the observed change in pressure, tL is the travel-time changes for the longitudinal wave (tL = tL − tL0 ), and L1 is the acoustoelastic constant for the longitudinal wave, which is related to the second-order and third-order elastic constants 1 6l +7λ+4m + 10μ−12lν −14λν −8mν −20μν . L1 = 4 λ + 2μ (10)
1 + 2μεR − nεC , (12) 3 where VAR represents the shear wave velocity in the pressure vessel wall along the axial direction. However, the shear wave is reflected several times inside the wall. There is an angle (90◦ − β S ) between its propagation direction and the axial direction of the vessel wall, as shown in Fig. 2. Hence, the strain should be decomposed in its propagation direction and the direction of particles movement. Finally, the acoustoelastic model for the reflected shear waves inside vessel wall can be developed as p = −
E d tS , L2 R tS0
The reflected longitudinal waves are generated as the result of the reflection of the shear wave inside the vessel wall. Thus, the travel-time change of the reflected longitudinal wave (t) should be the sum of two parts. One part is the travel-time change of the shear wave reflected inside the vessel wall and the other part is the travel-time change of the longitudinal wave travelling along the outer wall of the pressure vessel, as given in the following equation: p R 0 L1 tL + L2 tS0 . E d
(15)
Equation (15) shows that the travel-time change of the reflected longitudinal wave is proportional to the pressure change. With Eq. (15), the pressure measurement mode can
(13)
where tS0 is the reference travel-time of the shear wave under the free pressure state, tS is the travel-time of the shear wave under the measured pressure state, and tS is the travel-time changes for the shear wave (tS = tS − tS0 ). Since the change in the wave speed of the shear wave induced by the pressure or temperature is very small, the refraction angle β S can be approximately considered as a constant. And L2 can be regarded as an acoustoelastic constant for the shear wave, which is also related to the second-order, third-order elastic constants and β S
1 9λ + 9m − 18λν − 18mν − 45μν + nν + 9μ − 2n − 3μ(1 + ν) cos 2βS . 12 μ
t = tS + tL = −
(11)
2 = μ + (m + λ)(εA + εR + εC ) + 4μεA ρ0 VAR
tL0
L2 =
In practical measurement, the travel-time change induced by pressure (tL ) is much smaller than tL0 . That means tL0 is approximate to tL , and Eq. (9) can be simplified as
(14)
be developed, as shown in Eq. (16). And this is the model based on the reflected longitudinal wave we proposed in this paper d E t. p = − 0 0 R L 1 t L + L2 t S
(16)
III. NON-INTRUSIVE PRESSURE MEASUREMENT SCHEME
In the previous works, temperature is identified as the main influence variable in the pressure measurement using ultrasonic waves.10 In this work, the dual-receiver mode is in-
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FIG. 5. Measurement system.
troduced to minimize the temperature influence. That means three probes are arranged in the axial direction of the vessel, including one transmitting probe (T) and two receiving probes (R1, R2), as shown in Fig. 5. For the single-receiver mode, which is one transmitting probe (T) and one receiving probe (R1), the total travel-time is the sum of the propagation time in the vessel wall, the wedge, and the couplant, as shown in Eq. (17) tT →R1 = tT∗ →R1 + twT + twR1 + tCT + tCR1 ,
(17)
where tT→R1 represents the total travel-time from transmitter T to receiver R1, tT∗ →R1 is the propagation time in the T R1 , tW are the propagation vessel wall between T and R1, tW time in the wedges of T and R1, respectively, and tCT , tCR1 denote the propagation time in the couplant of T and R1, respectively. So the travel-time for the single-receiver mode is also affected by the wedge and couplant, which are significantly influenced by the temperature. Moreover, Bray validated that the temperature influence for the ultrasonic velocity in polymethyl methacrylate (PMMA) wedge is greater than that in the steel.13 Hence, the single-receiver mode will aggravate the temperature influence arising from the wedge and couplant. For the dual-receiver mode, the travel-time can be defined as the differential propagation time between probe R1 and probe R2 and is shown in Eq. (18) tR1→R2 = tT →R2 − tT →R1 = tT∗ →R2 + twT + twR2 + tCT + tCR2 − tT∗ →R1 + twT + twR1 + tCT + tCR1 = tT∗ →R2 − tT∗ →R1 + twR2 − twR1 + tCR2 − tCR1 ∗ ≈ tR1→R2 ,
(18)
where tR1→R2 is defined as the travel-time for the dual-receiver mode, tT→R2 represents the total travel-time from transmitter T to receiver R2, tT→R1 is the total travel-time from T to R1, tT∗ →R2 is the propagation time in the vessel wall between T and R2, tT∗ →R1 is the propagation time in the vessel wall be-
∗ is the propagation time in the tween T and R1, and tR1→R2 vessel wall between R1 and R2. Because of the uniformity of the two receivers and couplant (twR2 ≈ twR1 , tCR2 ≈ tCR1 ), the travel-time for the dualreceiver mode tR1→R2 is approximately equal to the propa∗ . gation time in the vessel wall between R1 and R2 tR1→R2 Therefore, dual-receiver mode is adopted in the measurement system to minimize the influence of temperature.
IV. MEASUREMENT SYSTEM
The measurement system mainly includes six parts: pressure vessel, pressure test pump, ultrasonic probes, ultrasonic exciting circuit, digital acquisition system, and the computer, as shown in Fig. 5. The internal pressure of the cylindrical pressure vessel is applied by the pressure test pump. The ultrasonic exciting circuit is used to excite the ultrasonic signal for the transmitter (T), while the digital acquisition system is used for acquiring the ultrasonic signal from two receivers (R1 and R2). And the computer is utilized to calculate the travel-time between R1 and R2. The pressure is measured by a precise manometer with the full scale of 16 MPa and the maximum fiducial error of 0.4%. In order to ensure consistency of the contact characteristics of probes, they are mounted tightly on the vessel wall by three clamping fixtures of the same kind. The spacing between ultrasonic probes is described as T-R1, T-R2, and R1R2, respectively. To test the feasibility of the measurement method, two pressure vessels with different material and size are used in this work, labeled as pressure vessel A and pressure vessel B, respectively. The properties of the measurement system are shown in Table I. The inclined angle of the PMMA wedge is about 27◦ which is the first critical incidence angle. And the frequency of the three ultrasonic probes is set to 5 MHz. The sampling frequency of the digital acquisition system is 5 GHz with the time resolution of 0.2 ns. Besides, in order to investigate the temperature influence on the measurement, a digital biochemical incubator is used to change the surface temperature of the pressure vessel, as shown in Fig. 6. Thermometer TES-1315
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TABLE I. The properties of the measurement system. Pressure vessel no.
Pressure vessel A
Pressure vessel B
Component Outer radius (mm) Inner radius (mm) Thickness (mm) Height (mm) T-R1 (cm) T-R2 (cm) R1-R2 (cm)
30 CrMo 139.5 131.9 7.6 1360 7.5 13 5.5
37 Mn 114.5 108.8 5.7 1140 7.5 13 5.5
is utilized to measure the temperature with the resolution of 0.1 ◦ C and the measurement accuracy of ±0.5 ◦ C. Fig. 7 shows the flowchart of the travel-time calculation. First, the signals from R1 and R2 are received and filtered by the FIR digital filter. Then, the travel-time between R1 and R2 can be obtained by utilizing the cross-correlation algorithm. Besides, in order to measure the travel-time change between R1 and R2, the travel-time under the free pressure state and the measured pressure state are obtained, respectively. V. EXPERIMENTAL RESULTS AND DISCUSSION A. Comparison between the reflected longitudinal wave and the LCR wave
According to previous analysis, the interval time ti between two adjacent longitudinal waves (LCR wave, reflected longitudinal wave-1, reflected longitudinal wave-2, and so on) is identical, as shown in Fig. 3. And it can be described as ti =
2d 2d tan βS − , VS cos βS VL
(19)
where VL and VS represent the velocity of the longitudinal wave and shear wave in the wall of pressure vessel, respectively. Because the propagation is shortest and the velocity of the longitudinal wave is faster than that of the shear wave, the
FIG. 7. The flowchart of the travel-time calculation.
LCR wave is the first wave reaching the receiving probe. Then the reflected longitudinal waves can be easily recognized. The amplitude of reflected longitudinal waves is the joint effect of many factors, including the initial energy of the refracted shear wave, the propagation distance, the reflecting times, and the attenuation characteristics in the vessel wall and etc. From Fig. 3, we find that reflected longitudinal wave-4 has the highest amplitude and SNR, which could improve the accuracy for the travel-time. So reflected longitudinal wave-4 is utilized to implement the non-intrusive pressure measure in this work. Fig. 8 shows the travel-time changes under different pressures for the LCR wave and reflected longitudinal wave-4 on pressure vessel B. The experiment is conducted at the temperature of 18.8 ◦ C. The reference travel-time for the LCR wave and reflected longitudinal wave-4 are 9.2488 μs and 9.3090 μs, respectively. Obviously, the variation rate of the travel-time with the pressure for reflected longitudinal wave-4 is larger than the LCR wave. The sensitivity of reflected longitudinal wave-4 is increased by 40.1% compared with the LCR wave. B. Feasibility of the pressure measurement method based on the reflected longitudinal wave
Fig. 9 shows the pressure changes with the travel-time in pressure vessel A and pressure vessel B, with experimental temperatures of 21 ◦ C and 18.8 ◦ C, respectively. In the dualreceiver mode, the reference travel-time of pressure vessel A and pressure vessel B are 9.0190 μs and 9.3090 μs, respectively. Obviously, the pressure change is linearly proportional to the travel-time of reflected longitudinal wave-4 in both pressure vessels, which validates the effectiveness of pressure measurement model in Eq. (16). C. Influence of temperature on the travel-time
FIG. 6. Experimental setup.
As is well known, temperature is an important factor affecting the wave speed of the ultrasonic wave. To investigate the influence of temperature on the travel-time, temperature experiments are carried out with pressure vessel B. In Sec. III, we have discussed the merit of dual-receiver mode over single-receiver mode. Experiments are conducted at zero pressure state from 10.3 ◦ C to 30.2 ◦ C, and 10.3 ◦ C is selected as the reference temperature. The probe spacing for the single-receiver mode is T-R1 (7.5 cm) and that for the dual-receiver mode is R1-R2 (5.5 cm). The experiment results are shown in Fig. 10. It can be concluded that the travel-time change linearly increased with the temperature for both the
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FIG. 8. Travel-time changes with pressure for the LCR wave and reflected longitudinal wave-4 on pressure vessel B.
dual-received mode and the single-received mode. Moreover, comparing with the single-receiver mode, the dual-receiver mode can minimize the influence of temperature effectively. Therefore, in the following paragraph, we just discuss the situation in dual-receiver mode. More experiments are carried out to investigate the effect of temperature. Experiment results are shown in Fig. 11. The temperature ranges from 10.3 ◦ C to 30.2 ◦ C and temper-
ature of 10.3 ◦ C is regarded as the reference temperature. The reference travel-time under different pressure states (0 Mpa, 1 Mpa, 4 Mpa, and 7 Mpa) are 9.3048 μs, 9.3056 μs, 9.3098 μs, and 9.3140 μs, respectively. From Fig. 11, it is found that the influence of temperature on the measurement results is significant and there is a linear relationship between the travel-time and the temperature. The travel-time increases with the temperature and the
FIG. 9. Pressure changes with the travel-time in pressure vessel A and B.
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FIG. 10. Travel-time change with the temperature at zero pressure.
average sensitivity is about 0.45 ns/◦ C. Thus, to obtain better measurement results, it is necessary to seek an effective approach to overcome the influence of temperature. In this work, a calibration coefficient KT is introduced, as shown in Eq. (20). t (T ) = t (0,T ) − t (0,T0 ) = KT T ,
(20)
where t(T) is the temperature-induced travel-time change, t(0, T) represents the propagation time at free pressure and tem-
perature T, t (0,T0 ) denotes the propagation time at free pressure and reference temperature T0 , T is the temperature change (T = T − T0 ), and KT denotes the variation rate of the traveltime change with the temperature. In addition, the relationship between the travel-time change and the temperature change remains appropriately invariable under different pressure states. That means the temperature-induced travel-time change could be considered independent of the pressure-induced travel-time change.
FIG. 11. Travel-time changes with the temperature under different pressure.
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FIG. 12. Measured travel-time changes with pressure and temperature.
Therefore, the travel-time change considering the pressure and the influence of temperature could be described as t (p,T ) = t (p,T ) − t (0,T0 ) = Kp p + KT T ,
(21)
where t(p, T) is the measured travel-time at pressure p and temperature T, t(p, T) is the measured travel-time change in practical measurement, and Kp is the variation rate of the traveltime with the pressure. With Eq. (21), pressure measurement model with temperature compensation can be developed as p=
1 (t (p,T ) − KT T ). Kp
(22)
D. Experimental results of pressure measurement with temperature compensation
In order to determine the coefficients Kp and KT in Eq. (22), multiple linear regression analysis is applied.14 Fig. 12 shows the analysis sets obtained in pressure vessel B, which shows the measured travel-time change (t(p, T) ) with pressure and temperature. The reference temperature and the reference travel-time are 10.3 ◦ C and 9.3048 μs, respectively. Finally, the pressure measurement model with temperature compensation is shown in Eq. (23),with the determination coefficient (R2 ) of 0.9916 p = 0.7911t (p,T ) − 0.3348T − 0.5210.
pressure vessel. A better fit of probes that ensured the uniform coupling conditions would improve the performance.
(23)
With Eq. (23), the pressure measurement could be achieved. And pressure measurement results with temperature compensation are shown in Table II. The results have shown the feasibility and efficiency of the new approach, which can realize the pressure measurement with the maximum relative error of 9%. The errors of pressure measurement might be caused by the coupling condition between the probes and the
VI. SUMMARY AND CONCLUSIONS
This paper presents a new non-intrusive method for pressure measurement, which is based on the reflected longitudinal wave. By acoustoelastic theory and thin-shell theory, the relationship between pressure and ultrasonic wave velocity is derived. The pressure measurement model, which describes the relationship between the pressure change and the traveltime change, is also developed. With one transmitting ultrasonic sensor and two receiving ultrasonic sensors, the measurement system is constructed. To validate the feasibility of the new non-intrusive method, experiments are carried out under different pressure and temperature. According to the theoretical analysis and experimental results, the conclusions can be summarized as: 1. The pressure in pressure vessels is linearly proportional to the travel-time change of the reflected longitudinal wave in pressure vessel wall. TABLE II. Pressure measurement results with temperature compensation.
Temperature (◦ C) 16.2 26.4 18.8 21.9 16.2 26.4 18.8
Measured travel-time change (ns)
Measured pressure (MPa)
Practical pressure (MPa)
Absolute error (MPa)
Relative error (%)
4.4 10.0 8.0 10.2 9.8 14.8 13.2
0.98 1.99 2.96 3.66 5.26 5.80 7.08
1 2 3 4 5 6 7
0.02 0.01 0.04 0.34 0.26 0.2 0.08
2 0.05 1.4 8.5 5.2 3.3 1.1
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2. Comparing with the LCR wave, the reflected longitudinal waves have the advantages of higher sensitivity and SNR. In this work, the sensitivity is increased by 40.1% for pressure measurement when using reflected longitudinal wave-4. 3. Temperature is an important factor affecting the pressure measurement based on ultrasonic wave. Comparing with the single-receiver mode, the dual-receiver mode has the advantage of minimizing the influence of temperature significantly. And there is a linear dependence between the travel-time change and the temperature with a slope of about 0.45 ns/◦ C. 4. The travel-time change induced by temperature could be considered as independent as that induced by the pressure within a wide temperature. 5. A pressure measurement model with temperature compensation is proposed and achieved the accuracy with the maximum relative error of 9%. As a preliminary study, the research work verified that the reflected longitudinal wave could be applied to achieve the non-intrusive pressure measurement in pressure vessels. In order to take the technique into the industrial application, more research works should be done in this area, for example, the optimization of probe mounting and the determination of measurement models in a wider temperature range.
Rev. Sci. Instrum. 85, 125002 (2014)
ACKNOWLEDGMENTS
This work was supported by Zhejiang Provincial Natural Science Foundation of China (Grant No. LY12F03022) and the Fundamental Research Funds for the Central Universities (No. 1A5000*172210111). 1 G.
T. Shen and W. L. Zhang, Nondestruct. Test. 26(1), 37–40 (2004). Ji, H. P. Zhang, X. B. Zhang, and Y. Q. Shun, Mach. Tool Hydraul. 36(9), 203 (2008). 3 J. Y. Huang, H. W. Yuan, Y. Cui, and Z. Q. Zheng, IEEE Trans. Instrum. Meas. 59(11), 2914–2920 (2010). 4 P. Diodati, Rev. Sci. Instrum. 57(2), 293–295 (1986). 5 F. Yu, N. Gupta, and J. Hoy, Insight: Non-destruct. Test. Cond. Monit. 47(5), 285–288 (2005). 6 E. Rosenkrantz, J. Y. Ferrandis, G. Leveque, and D. Baron, Nucl. Instrum. Methods Phys. Res. A 603(3), 504–509 (2009). 7 Z. W. Ling, H. L. Zhou, and H. J. Zhang, IEEE Trans. Instrum. Meas. 58, 1578–1584 (2009). 8 D. S. Hughes and J. L. Kelly, Phys. Rev. 92, 1145–1149 (1953). 9 H. W. Liu, J. X. Lin, and M. L. Cao, Shell Theory (Zhejiang University Press, Hangzhou, China, 1987), p. 297. 10 R. S. Fraga, A. A. Santos, and M. H. Andrino, in Proceedings of the ASME International Mechanical Engineering Congress and Exposition, Boston, MA, USA, 2008. 11 G. T. Yang, Introduction to Elasticity (Tsinghua University Press, Beijing, China, 1998). 12 S. Chaki, G. Corneloup, I. Lillamand, and H. Walaszek, J. Press. Vessel Technol. ASME 129, 383 (2007). 13 D. E. Bray, U.S. patent 20020078759A1[P/OL] (27 June 2007). 14 J. L. Wang, X. J. Liang, and B. H. Li, Applied Linear Regression (China Statistics Press, Beijing, China, 1998). 2 A.
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REVIEW OF SCIENTIFIC INSTRUMENTS 85, 125002 (2014)
Ultrasonic pressure measurement in pressure vessels Yao Bi,1 Hongliang Zhou,1,a) Zhiyao Huang,1 Hanhua Zhou,1 and Xianglong Yang2 1 State Key Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou, 310027, People’s Republic of China 2 School of Biosystems Engineering and Food Science, Zhejiang University, Hangzhou, 310058, People’s Republic of China
(Received 31 December 2013; accepted 9 November 2014; published online 2 December 2014) Based on the reflected longitudinal wave, a new non-intrusive method for pressure measurement is proposed. The acoustoelastic theory and the thin-shell theory are introduced to develop the pressure measurement model in cylindrical pressure vessels. And a pressure measurement system is constructed to evaluate the effectiveness of the proposed method. The pressure measurement is implemented by measuring the travel-time change between two received ultrasonic sensors. The experimental results verify the feasibility and effectiveness of this new non-intrusive method. Compared with the non-intrusive pressure measurement method based on the critically refracted longitudinal wave (LCR wave), the proposed non-intrusive pressure measurement method has the advantages of higher sensitivity and higher signal-to-noise ratio. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4902338] I. INTRODUCTION
Pressure vessels, as special equipment, are widely used in many industrial fields, such as chemical, energy, and metallurgical engineering, etc. Overpressure of pressure vessel is one of the major reasons for the safety accidents which cause severe disasters and economic loss. The effective measurement and control of the vessel pressure is an important approach to prevent accidents.1 The traditional methods for pressure measurement usually require drilling holes in the vessel wall, which will cause local stress concentration and may make the wall prone to crack. There is a practical requirement for non-intrusive pressure measurement. Unfortunately, up to date, the effective non-intrusive pressure measurement methods or instruments are very limited, although some non-intrusive pressure measurement methods have been proposed, such as the strain gauge method,2 the capacitor method,3 and the ultrasonic method. So, more research works should be undertaken in this area. Among the above non-intrusive pressure measurement methods, the non-intrusive pressure measurement method based on ultrasonic wave technique is an attractive and potential method. It has the advantages of low cost and simple construction. To implement the non-intrusive pressure measurement by ultrasonic technique, many research groups have made their efforts. Diodati found there was a linear dependence between the amplitude of an acoustic pulse propagating inside the vessel wall and the pressure of the internal fluid, and further implemented the pressure measurement by Lamb wave.4 However, the linear dependence will be affected by different liquid in the container. Yu et al. found that acoustic velocity in oil increased with the rise of pressure and established the pressure measurement model for the hydraulic a) Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0034-6748/2014/85(12)/125002/10/$30.00
system.5 But the acoustic velocity was also affected by the medium in the vessel. Rosenkrantz et al. developed an acoustic method which was on basis of the ultrasonic wave velocity of the longitudinal wave, and realized the pressure measurement in the upper plenum of a standard light water reactor (LWR) fuel rod.6 Unfortunately, the amplitude of the acoustic signals was relatively weak, which meant lower measurement accuracy for the ultrasonic wave velocity. Ling et al. developed a pressure measurement model which was based on the travel-time of ultrasonic wave, and implemented the pressure measurement by utilizing the LCR wave and Rayleigh wave.7 But the amplitude of the LCR wave was also weak, which will cause more calculation error for the travel-time. In summary, the conventional non-intrusive pressure measurement methods based on ultrasonic wave technique still have some drawbacks. The measurement performances (measurement sensitivity and accuracy) cannot meet the requirements of academic research and industrial application. In this work, we will introduce a new non-intrusive pressure measurement method based on the reflected longitudinal wave. As shown in Fig. 1, when a beam of longitudinal wave is incident from medium I to medium II, waveform conversion will take place at the interface according to the Snell’s law. A portion of energy is reflected to medium I and converts to the reflected longitudinal wave and the reflected shear wave. The other portion of energy is refracted to medium II and converts to the refracted longitudinal wave and the refracted shear wave. Further, if a longitudinal wave is incident into the interface (between the transmitting probe and the vessel outer wall) with the first critical angle, a LCR wave and a refracted shear wave will be generated, as shown in Fig. 2. The LCR wave will travel along the outer wall of the vessel and will be received by the receiving probe. The refracted shear wave will be reflected on the inner wall of the pressure vessel, which will generate reflected longitudinal wave-I1 and
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© 2014 AIP Publishing LLC
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FIG. 1. Reflection and refraction of the ultrasonic wave.
reflected shear wave-1 (as shown in Fig. 2). Based on the Snell’s Law, because the reflection angle of reflected longitudinal-I1 wave is 90◦ , the reflected longitudinal wave-I1 will travel along the inner wall of the vessel. But, the reflected shear wave-1 will continue to propagate and will be reflected on the outer wall of the vessel. That will generate reflected longitudinal wave-1 (which will travel along the outer wall of the vessel and will be finally received by the receiving probe) and reflected shear wave-2 (which will continue to propagate and generate succeeding reflected longitudinal wave-I2 and reflected shear wave-3). And so on. Thus, the receiving probe will receive one LCR wave and several reflected longitudinal waves, respectively (reflected longitudinal wave-1, reflected longitudinal wave-2, and so on), as shown in Fig. 3. Research works have verified that the change of pressure of the cylindrical pressure vessel will cause the change of the stress of the cylindrical pressure vessel.8 Research works also indicated that the change of the stress of the cylindrical pressure vessel will cause the change of ultrasonic wave velocity in the wall.9 Thus, the obtained reflected longitudinal waves contain the information of the pressure of the cylindrical pressure vessel. That means it is possible to use reflected longitudinal waves to implement the pressure measurement. Meanwhile, our experimental tests have indicated that the reflected
FIG. 3. Received ultrasonic signal observed by the digital oscilloscope.
longitudinal waves have higher amplitude and signal-to-noise ratio (SNR) compared with the LCR wave, as shown in Fig. 3. However, up to date, the research works concerning the application of reflected longitudinal wave to the non-intrusive pressure measurement are very limited. Few research works have been reported. This work attempts to introduce the reflected longitudinal waves into the research field of pressure measurement and hence to propose a new non-intrusive pressure measurement method which is on the basis of reflected longitudinal waves. Fig. 4 shows the study scheme of the new non-intrusive pressure measurement method. The whole research works include six parts: (1) The acoustoelastic theory is introduced to obtain the relationship between the stress of the pressure vessel and the ultrasonic wave velocity.8 (2) The thin-shell theory is introduced to obtain the relationship between the pressure of the cylindrical pressure vessel and the stress.9 (3) Based on
FIG. 2. Propagation of ultrasonic wave inside the vessel wall.
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described as 2 = λ + 2μ + (2l + λ)(εA + εR + εC ) ρ0 VAA
+ (4m + 4λ + 10μ)εA ,
(1)
where VAA denotes the longitudinal wave velocity in the pressure vessel wall along the axial direction. εA , εR , and εC are the stresses in the pressure vessel wall along the axial, radial, and circumferential directions, respectively. ρ 0 is the initial density of the pressure vessel. λ and μ are second-order elastic constants, while l, m, and n are third-order elastic constants. Based on the thin-shell theory,9 the stresses are related to the internal pressure. The stress field in the wall of the cylindrical pressure vessel is two-dimensional, including the stresses in the axial direction and in the circumferential direction, and can be described as σA =
FIG. 4. Study scheme of the new non-intrusive pressure measurement method.
the acoustoelastic theory and thin-shell theory, the relationship between the pressure and ultrasonic wave velocity can be developed. (4) The pressure measurement model, which describes the relationship between the pressure and the variation of travel-time of the reflected longitudinal wave, is derived. (5) With one transmitting probe and two receiving probes, a pressure measurement system is constructed. (6) The variation of travel-time between two receiving probes is obtained by experimental tests and the pressure measurement is implemented by the pressure measurement model. Besides, because the temperature is an important factor for ultrasonic measurement technique,10 in this work, the influence of temperature on the pressure measurement will also be investigated. II. DEVELOPMENT OF THE PRESSURE MEASUREMENT MODEL A. The relationship between pressure and ultrasonic wave velocity
The ultrasonic method realizes the non-intrusive pressure measurement by monitoring the variation of the ultrasonic wave velocity. It is necessary to obtain the relationship between the pressure and ultrasonic wave velocity. Unfortunately, the direct relationship between the pressure and ultrasonic wave velocity has not been established. Therefore, in this work, we first develop the relationship between ultrasonic wave velocity and stress. Then, we establish the relationship between the pressure and stress. Finally, the relationship between the pressure and ultrasonic wave velocity can be developed. As derived by Hughes and Kelly,8 the relationship between the ultrasonic wave velocities and the strain can be
pR , 2d
(2) pR σC = , d where σ A and σ C are the stresses in the pressure vessel wall along the axial and the circumferential directions, respectively, p represents the internal pressure, R denotes the average radius of the cylindrical pressure vessel, and d is the thickness of the wall. Based on the Hooke’s Law,11 the strain components in the pressure vessel wall under the state of two-dimensional stress are given in the following equations: 1 (σ − νσC ), E A ν (3) εR = − (σA + σC ), E 1 εC = (σC − νσA ), E where εA , εR , and εC represent the strains in the pressure vessel wall along the axial, radial, and circumferential directions, respectively, E is the elasticity modulus of the vessel material, and ν is the Poisson’s ratio of the vessel material. Thus, with Eqs. (1)–(3), the relationship between pressure and the longitudinal wave velocity can be obtained, as given in the following equation: εA =
E d 2 ρ0 VAA − λ − 2μ , (4) LR where L represents the constant, which is related to the second-order and third-order elastic constants 6l + 7λ + 4m + 10μ − 12lν − 14λν − 8mν − 20μν . L= 2 (5) p=
B. Pressure measurement model based on the reflected longitudinal wave
For ultrasonic measurement system, the velocity measurement of the ultrasonic wave is usually implemented by the travel-time measurement.12
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Under the free pressure state, the stresses in the pressure vessel wall along the axial, radial, and circumferential directions could be considered as free (εA = εR = εC = 0). With Eq. (1), the relationship between ρ 0 and VL0 , which is the longitudinal wave velocity under the free pressure state, can be described as 0 2 λ + 2μ VL = . (6) ρ0 Besides, the VL0 can also be defined as VL0
D = 0, tL
p = −
E d tL . L1 R tL0
Meanwhile, the relationship between the shear wave velocity and the strain is also summarized by Hughes and Kelly,8 which can be described as
(7)
where represents the reference travel-time of the longitudinal wave under the free pressure state, D represents the propagation distance of the longitudinal wave. With Eqs. (4)–(7), the relationship between the internal pressure and the travel-time of the longitudinal wave can be described as 2 tL0 Ed (λ + 2μ) −1 , (8) p= LR tL where tL is the travel-time of the longitudinal wave under the measured pressure state. With Eq. (8), the acoustoelastic model for the longitudinal wave traveling along the axial direction of the pressure vessel can be developed as 3 tL E d tL0 , (9) p = − L1 R t L tL0 Where p is the observed change in pressure, tL is the travel-time changes for the longitudinal wave (tL = tL − tL0 ), and L1 is the acoustoelastic constant for the longitudinal wave, which is related to the second-order and third-order elastic constants 1 6l +7λ+4m + 10μ−12lν −14λν −8mν −20μν . L1 = 4 λ + 2μ (10)
1 + 2μεR − nεC , (12) 3 where VAR represents the shear wave velocity in the pressure vessel wall along the axial direction. However, the shear wave is reflected several times inside the wall. There is an angle (90◦ − β S ) between its propagation direction and the axial direction of the vessel wall, as shown in Fig. 2. Hence, the strain should be decomposed in its propagation direction and the direction of particles movement. Finally, the acoustoelastic model for the reflected shear waves inside vessel wall can be developed as p = −
E d tS , L2 R tS0
The reflected longitudinal waves are generated as the result of the reflection of the shear wave inside the vessel wall. Thus, the travel-time change of the reflected longitudinal wave (t) should be the sum of two parts. One part is the travel-time change of the shear wave reflected inside the vessel wall and the other part is the travel-time change of the longitudinal wave travelling along the outer wall of the pressure vessel, as given in the following equation: p R 0 L1 tL + L2 tS0 . E d
(15)
Equation (15) shows that the travel-time change of the reflected longitudinal wave is proportional to the pressure change. With Eq. (15), the pressure measurement mode can
(13)
where tS0 is the reference travel-time of the shear wave under the free pressure state, tS is the travel-time of the shear wave under the measured pressure state, and tS is the travel-time changes for the shear wave (tS = tS − tS0 ). Since the change in the wave speed of the shear wave induced by the pressure or temperature is very small, the refraction angle β S can be approximately considered as a constant. And L2 can be regarded as an acoustoelastic constant for the shear wave, which is also related to the second-order, third-order elastic constants and β S
1 9λ + 9m − 18λν − 18mν − 45μν + nν + 9μ − 2n − 3μ(1 + ν) cos 2βS . 12 μ
t = tS + tL = −
(11)
2 = μ + (m + λ)(εA + εR + εC ) + 4μεA ρ0 VAR
tL0
L2 =
In practical measurement, the travel-time change induced by pressure (tL ) is much smaller than tL0 . That means tL0 is approximate to tL , and Eq. (9) can be simplified as
(14)
be developed, as shown in Eq. (16). And this is the model based on the reflected longitudinal wave we proposed in this paper d E t. p = − 0 0 R L 1 t L + L2 t S
(16)
III. NON-INTRUSIVE PRESSURE MEASUREMENT SCHEME
In the previous works, temperature is identified as the main influence variable in the pressure measurement using ultrasonic waves.10 In this work, the dual-receiver mode is in-
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FIG. 5. Measurement system.
troduced to minimize the temperature influence. That means three probes are arranged in the axial direction of the vessel, including one transmitting probe (T) and two receiving probes (R1, R2), as shown in Fig. 5. For the single-receiver mode, which is one transmitting probe (T) and one receiving probe (R1), the total travel-time is the sum of the propagation time in the vessel wall, the wedge, and the couplant, as shown in Eq. (17) tT →R1 = tT∗ →R1 + twT + twR1 + tCT + tCR1 ,
(17)
where tT→R1 represents the total travel-time from transmitter T to receiver R1, tT∗ →R1 is the propagation time in the T R1 , tW are the propagation vessel wall between T and R1, tW time in the wedges of T and R1, respectively, and tCT , tCR1 denote the propagation time in the couplant of T and R1, respectively. So the travel-time for the single-receiver mode is also affected by the wedge and couplant, which are significantly influenced by the temperature. Moreover, Bray validated that the temperature influence for the ultrasonic velocity in polymethyl methacrylate (PMMA) wedge is greater than that in the steel.13 Hence, the single-receiver mode will aggravate the temperature influence arising from the wedge and couplant. For the dual-receiver mode, the travel-time can be defined as the differential propagation time between probe R1 and probe R2 and is shown in Eq. (18) tR1→R2 = tT →R2 − tT →R1 = tT∗ →R2 + twT + twR2 + tCT + tCR2 − tT∗ →R1 + twT + twR1 + tCT + tCR1 = tT∗ →R2 − tT∗ →R1 + twR2 − twR1 + tCR2 − tCR1 ∗ ≈ tR1→R2 ,
(18)
where tR1→R2 is defined as the travel-time for the dual-receiver mode, tT→R2 represents the total travel-time from transmitter T to receiver R2, tT→R1 is the total travel-time from T to R1, tT∗ →R2 is the propagation time in the vessel wall between T and R2, tT∗ →R1 is the propagation time in the vessel wall be-
∗ is the propagation time in the tween T and R1, and tR1→R2 vessel wall between R1 and R2. Because of the uniformity of the two receivers and couplant (twR2 ≈ twR1 , tCR2 ≈ tCR1 ), the travel-time for the dualreceiver mode tR1→R2 is approximately equal to the propa∗ . gation time in the vessel wall between R1 and R2 tR1→R2 Therefore, dual-receiver mode is adopted in the measurement system to minimize the influence of temperature.
IV. MEASUREMENT SYSTEM
The measurement system mainly includes six parts: pressure vessel, pressure test pump, ultrasonic probes, ultrasonic exciting circuit, digital acquisition system, and the computer, as shown in Fig. 5. The internal pressure of the cylindrical pressure vessel is applied by the pressure test pump. The ultrasonic exciting circuit is used to excite the ultrasonic signal for the transmitter (T), while the digital acquisition system is used for acquiring the ultrasonic signal from two receivers (R1 and R2). And the computer is utilized to calculate the travel-time between R1 and R2. The pressure is measured by a precise manometer with the full scale of 16 MPa and the maximum fiducial error of 0.4%. In order to ensure consistency of the contact characteristics of probes, they are mounted tightly on the vessel wall by three clamping fixtures of the same kind. The spacing between ultrasonic probes is described as T-R1, T-R2, and R1R2, respectively. To test the feasibility of the measurement method, two pressure vessels with different material and size are used in this work, labeled as pressure vessel A and pressure vessel B, respectively. The properties of the measurement system are shown in Table I. The inclined angle of the PMMA wedge is about 27◦ which is the first critical incidence angle. And the frequency of the three ultrasonic probes is set to 5 MHz. The sampling frequency of the digital acquisition system is 5 GHz with the time resolution of 0.2 ns. Besides, in order to investigate the temperature influence on the measurement, a digital biochemical incubator is used to change the surface temperature of the pressure vessel, as shown in Fig. 6. Thermometer TES-1315
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TABLE I. The properties of the measurement system. Pressure vessel no.
Pressure vessel A
Pressure vessel B
Component Outer radius (mm) Inner radius (mm) Thickness (mm) Height (mm) T-R1 (cm) T-R2 (cm) R1-R2 (cm)
30 CrMo 139.5 131.9 7.6 1360 7.5 13 5.5
37 Mn 114.5 108.8 5.7 1140 7.5 13 5.5
is utilized to measure the temperature with the resolution of 0.1 ◦ C and the measurement accuracy of ±0.5 ◦ C. Fig. 7 shows the flowchart of the travel-time calculation. First, the signals from R1 and R2 are received and filtered by the FIR digital filter. Then, the travel-time between R1 and R2 can be obtained by utilizing the cross-correlation algorithm. Besides, in order to measure the travel-time change between R1 and R2, the travel-time under the free pressure state and the measured pressure state are obtained, respectively. V. EXPERIMENTAL RESULTS AND DISCUSSION A. Comparison between the reflected longitudinal wave and the LCR wave
According to previous analysis, the interval time ti between two adjacent longitudinal waves (LCR wave, reflected longitudinal wave-1, reflected longitudinal wave-2, and so on) is identical, as shown in Fig. 3. And it can be described as ti =
2d 2d tan βS − , VS cos βS VL
(19)
where VL and VS represent the velocity of the longitudinal wave and shear wave in the wall of pressure vessel, respectively. Because the propagation is shortest and the velocity of the longitudinal wave is faster than that of the shear wave, the
FIG. 7. The flowchart of the travel-time calculation.
LCR wave is the first wave reaching the receiving probe. Then the reflected longitudinal waves can be easily recognized. The amplitude of reflected longitudinal waves is the joint effect of many factors, including the initial energy of the refracted shear wave, the propagation distance, the reflecting times, and the attenuation characteristics in the vessel wall and etc. From Fig. 3, we find that reflected longitudinal wave-4 has the highest amplitude and SNR, which could improve the accuracy for the travel-time. So reflected longitudinal wave-4 is utilized to implement the non-intrusive pressure measure in this work. Fig. 8 shows the travel-time changes under different pressures for the LCR wave and reflected longitudinal wave-4 on pressure vessel B. The experiment is conducted at the temperature of 18.8 ◦ C. The reference travel-time for the LCR wave and reflected longitudinal wave-4 are 9.2488 μs and 9.3090 μs, respectively. Obviously, the variation rate of the travel-time with the pressure for reflected longitudinal wave-4 is larger than the LCR wave. The sensitivity of reflected longitudinal wave-4 is increased by 40.1% compared with the LCR wave. B. Feasibility of the pressure measurement method based on the reflected longitudinal wave
Fig. 9 shows the pressure changes with the travel-time in pressure vessel A and pressure vessel B, with experimental temperatures of 21 ◦ C and 18.8 ◦ C, respectively. In the dualreceiver mode, the reference travel-time of pressure vessel A and pressure vessel B are 9.0190 μs and 9.3090 μs, respectively. Obviously, the pressure change is linearly proportional to the travel-time of reflected longitudinal wave-4 in both pressure vessels, which validates the effectiveness of pressure measurement model in Eq. (16). C. Influence of temperature on the travel-time
FIG. 6. Experimental setup.
As is well known, temperature is an important factor affecting the wave speed of the ultrasonic wave. To investigate the influence of temperature on the travel-time, temperature experiments are carried out with pressure vessel B. In Sec. III, we have discussed the merit of dual-receiver mode over single-receiver mode. Experiments are conducted at zero pressure state from 10.3 ◦ C to 30.2 ◦ C, and 10.3 ◦ C is selected as the reference temperature. The probe spacing for the single-receiver mode is T-R1 (7.5 cm) and that for the dual-receiver mode is R1-R2 (5.5 cm). The experiment results are shown in Fig. 10. It can be concluded that the travel-time change linearly increased with the temperature for both the
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FIG. 8. Travel-time changes with pressure for the LCR wave and reflected longitudinal wave-4 on pressure vessel B.
dual-received mode and the single-received mode. Moreover, comparing with the single-receiver mode, the dual-receiver mode can minimize the influence of temperature effectively. Therefore, in the following paragraph, we just discuss the situation in dual-receiver mode. More experiments are carried out to investigate the effect of temperature. Experiment results are shown in Fig. 11. The temperature ranges from 10.3 ◦ C to 30.2 ◦ C and temper-
ature of 10.3 ◦ C is regarded as the reference temperature. The reference travel-time under different pressure states (0 Mpa, 1 Mpa, 4 Mpa, and 7 Mpa) are 9.3048 μs, 9.3056 μs, 9.3098 μs, and 9.3140 μs, respectively. From Fig. 11, it is found that the influence of temperature on the measurement results is significant and there is a linear relationship between the travel-time and the temperature. The travel-time increases with the temperature and the
FIG. 9. Pressure changes with the travel-time in pressure vessel A and B.
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FIG. 10. Travel-time change with the temperature at zero pressure.
average sensitivity is about 0.45 ns/◦ C. Thus, to obtain better measurement results, it is necessary to seek an effective approach to overcome the influence of temperature. In this work, a calibration coefficient KT is introduced, as shown in Eq. (20). t (T ) = t (0,T ) − t (0,T0 ) = KT T ,
(20)
where t(T) is the temperature-induced travel-time change, t(0, T) represents the propagation time at free pressure and tem-
perature T, t (0,T0 ) denotes the propagation time at free pressure and reference temperature T0 , T is the temperature change (T = T − T0 ), and KT denotes the variation rate of the traveltime change with the temperature. In addition, the relationship between the travel-time change and the temperature change remains appropriately invariable under different pressure states. That means the temperature-induced travel-time change could be considered independent of the pressure-induced travel-time change.
FIG. 11. Travel-time changes with the temperature under different pressure.
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FIG. 12. Measured travel-time changes with pressure and temperature.
Therefore, the travel-time change considering the pressure and the influence of temperature could be described as t (p,T ) = t (p,T ) − t (0,T0 ) = Kp p + KT T ,
(21)
where t(p, T) is the measured travel-time at pressure p and temperature T, t(p, T) is the measured travel-time change in practical measurement, and Kp is the variation rate of the traveltime with the pressure. With Eq. (21), pressure measurement model with temperature compensation can be developed as p=
1 (t (p,T ) − KT T ). Kp
(22)
D. Experimental results of pressure measurement with temperature compensation
In order to determine the coefficients Kp and KT in Eq. (22), multiple linear regression analysis is applied.14 Fig. 12 shows the analysis sets obtained in pressure vessel B, which shows the measured travel-time change (t(p, T) ) with pressure and temperature. The reference temperature and the reference travel-time are 10.3 ◦ C and 9.3048 μs, respectively. Finally, the pressure measurement model with temperature compensation is shown in Eq. (23),with the determination coefficient (R2 ) of 0.9916 p = 0.7911t (p,T ) − 0.3348T − 0.5210.
pressure vessel. A better fit of probes that ensured the uniform coupling conditions would improve the performance.
(23)
With Eq. (23), the pressure measurement could be achieved. And pressure measurement results with temperature compensation are shown in Table II. The results have shown the feasibility and efficiency of the new approach, which can realize the pressure measurement with the maximum relative error of 9%. The errors of pressure measurement might be caused by the coupling condition between the probes and the
VI. SUMMARY AND CONCLUSIONS
This paper presents a new non-intrusive method for pressure measurement, which is based on the reflected longitudinal wave. By acoustoelastic theory and thin-shell theory, the relationship between pressure and ultrasonic wave velocity is derived. The pressure measurement model, which describes the relationship between the pressure change and the traveltime change, is also developed. With one transmitting ultrasonic sensor and two receiving ultrasonic sensors, the measurement system is constructed. To validate the feasibility of the new non-intrusive method, experiments are carried out under different pressure and temperature. According to the theoretical analysis and experimental results, the conclusions can be summarized as: 1. The pressure in pressure vessels is linearly proportional to the travel-time change of the reflected longitudinal wave in pressure vessel wall. TABLE II. Pressure measurement results with temperature compensation.
Temperature (◦ C) 16.2 26.4 18.8 21.9 16.2 26.4 18.8
Measured travel-time change (ns)
Measured pressure (MPa)
Practical pressure (MPa)
Absolute error (MPa)
Relative error (%)
4.4 10.0 8.0 10.2 9.8 14.8 13.2
0.98 1.99 2.96 3.66 5.26 5.80 7.08
1 2 3 4 5 6 7
0.02 0.01 0.04 0.34 0.26 0.2 0.08
2 0.05 1.4 8.5 5.2 3.3 1.1
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2. Comparing with the LCR wave, the reflected longitudinal waves have the advantages of higher sensitivity and SNR. In this work, the sensitivity is increased by 40.1% for pressure measurement when using reflected longitudinal wave-4. 3. Temperature is an important factor affecting the pressure measurement based on ultrasonic wave. Comparing with the single-receiver mode, the dual-receiver mode has the advantage of minimizing the influence of temperature significantly. And there is a linear dependence between the travel-time change and the temperature with a slope of about 0.45 ns/◦ C. 4. The travel-time change induced by temperature could be considered as independent as that induced by the pressure within a wide temperature. 5. A pressure measurement model with temperature compensation is proposed and achieved the accuracy with the maximum relative error of 9%. As a preliminary study, the research work verified that the reflected longitudinal wave could be applied to achieve the non-intrusive pressure measurement in pressure vessels. In order to take the technique into the industrial application, more research works should be done in this area, for example, the optimization of probe mounting and the determination of measurement models in a wider temperature range.
Rev. Sci. Instrum. 85, 125002 (2014)
ACKNOWLEDGMENTS
This work was supported by Zhejiang Provincial Natural Science Foundation of China (Grant No. LY12F03022) and the Fundamental Research Funds for the Central Universities (No. 1A5000*172210111). 1 G.
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