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Effects of Velocity Profiles on Measuring Accuracy of Transit-Time Ultrasonic Flowmeter

Effects of Velocity Profiles on Measuring Accuracy of Transit-Time Ultrasonic Flowmeter applied sciences Article E ects of Velocity Profiles on Measuring Accuracy of Transit-Time Ultrasonic Flowmeter 1 , 2 1 , 1 Hui Zhang , Chuwen Guo * and Jie Lin School of Electrical and Power Engineering, China University of Mining and Technology, Xuzhou 221116, China; cumthui@126.com (H.Z.); TS17130003A3@cumt.edu.cn (J.L.) Xuhai College, China University of Mining and Technology, Xuzhou 221008, China * Correspondence: cwguo@cumt.edu.cn; Tel.: +86-516-8359-2200 Received: 13 March 2019; Accepted: 18 April 2019; Published: 20 April 2019 Featured Application: The research results of this paper could be used to improve the measuring accuracy of the transit-time ultrasonic flowmeter. Abstract: Ultrasonic wave carries the information for flowing velocity when it is propagating in flowing fluids. Flowrate can be obtained by measuring the propagation time of ultrasonic wave. The principle of transit-time ultrasonic flowmeters used today was based on that the velocity is uniform along the propagation path of the ultrasonic wave. However, it is well known that the velocity profiles in a pipe are not uniform both in laminar flow and turbulent flow. Emphasis on the e ects of velocity profiles across the pipe on the propagation time of ultrasonic wave, theoretical flowrate correction factors considering the real velocity profile were proposed for laminar and turbulent flow to obtain higher accuracy. Experiment data of ultrasonic flowmeter and weighting method are compared to verify the proposed theoretical correction factors. The average relative error of proposed correction factor is determined to be 0.976% for laminar flow and 0.25% for turbulent flow. Keywords: velocity profile; transit-time; ultrasonic flowmeter; accuracy; correction factor 1. Introduction Flow measurement is necessary in engineering field for fluid metrology and process control [1,2]. A variety of flow measurement techniques were invented, such as di erential pressure flowmeter [3], float flowmeter [4], volumetric flowmeter [5], electromagnetic flowmeter [6], and vortex shedding flowmeter [7]. However, the common disadvantage of these flow measurement techniques is that the fluid properties are tightly restricted due to direct contact with the flowmeters. In order to overcome the above disadvantage, non-contact ultrasonic flowmeter was developed, which is immune to temperature, causticity, and conductivity of measured fluid [8]. There is a great variety of ultrasonic flowmeters for measurement of liquid and gas flow [9–12]. Today, ultrasonic flowmeters utilize clamp-on and wetted transducers, single and multiple paths, paths on and o the diameter, passive and active principles, contra-propagating transmission, reflection (Doppler), tag correlation, vortex shedding, liquid level sensing of open channel flow or flow in partially-full conduits, and other interactions. Due to the simplicity of the measurement principle, the ultrasonic transit-time method is one of the most common techniques in industrial applications. Flowrate measurement using the ultrasonic transit-time method is based on the apparent di erence of the sound velocity in the flow direction and in the opposite direction [13,14]. It was pointed out in reference [15] that the transit-time ultrasonic flowmeter has a relatively high uncertainty, approximately 5%. The uncertainty is mainly caused by three factors, the first is the installation method of transducers; the second is the measurement of transit time, especially for pipe with a small inner diameter; the last is Appl. Sci. 2019, 9, 1648; doi:10.3390/app9081648 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 1648 2 of 11 Appl. Sci. 2019, 9, x FOR PEER REVIEW 2 of 11 velocity distribution across the pipe [16]. A large amount of studies have been carried out to reduce the studies have been carried out to reduce the uncertainty caused by these three reasons [17–20]. For the uncertainty caused by these three reasons [17–20]. For the first reason, Mahadeva et al. [21,22] improved first reason, Mahadeva et al. [21,22] improved measurement accuracy by changing the distance measurement accuracy by changing the distance between transducers. Rajita et al. [23] indicated that between transducers. Rajita et al. [23] indicated that multipath ultrasonic flowmeters can provide multipath ultrasonic flowmeters can provide more accurate flow velocities than a single path. For the more accurate flow velocities than a single path. For the second reason, several research focused on second reason, several research focused on developing new algorithms to obtain more accurate time developing new algorithms to obtain more accurate time differences [24]. And high time resolution di erences [24]. And high time resolution electronic components have significant progress in recent electronic components have significant progress in recent years to achieve nanosecond measurement years to achieve nanosecond measurement of transit time. However, there is a limited amount of of transit time. However, there is a limited amount of information available on the effect of velocity information available on the e ect of velocity profile on the measurement uncertainty. Looss et al. [13] profile on the measurement uncertainty. Looss et al. [13] have proved that the flow rate is have proved that the flow rate is overestimated by the e ects of the assumption of uniform velocity overestimated by the effects of the assumption of uniform velocity distribution, and a correction distribution, and a correction factor is obtained empirically based on numerical simulation results factor is obtained empirically based on numerical simulation results for fully developed turbulence. for fully developed turbulence. Zheng et al. [25] gave a correction factor for the transition region Zheng et al. [25] gave a correction factor for the transition region with Reynolds number in the range with Reynolds number in the range of 2000–20000 based on experiment results by Particle Image of 2000–20000 based on experiment results by Particle Image Velocimetry (PIV) measurement. Little Velocimetry (PIV) measurement. Little attention was paid to obtain correction factor for transit time attention was paid to obtain correction factor for transit time ultrasonic flowmeter through theoretical ultrasonic flowmeter through theoretical analysis. analysis. In this paper, we put emphasis on the e ects of velocity profiles across the pipe on the time of In this paper, we put emphasis on the effects of velocity profiles across the pipe on the time of ultrasonic wave propagation. Theoretical correction factors for transit-time ultrasonic flowmeters ultrasonic wave propagation. Theoretical correction factors for transit-time ultrasonic flowmeters were proposed for laminar and turbulent flow to improve measurement accuracy. Flow measurement were proposed for laminar and turbulent flow to improve measurement accuracy. Flow experiments were performed for laminar and turbulent flow, respectively, and experiment data measurement experiments were performed for laminar and turbulent flow, respectively, and of ultrasonic flowmeter and weighting method are compared to verify the proposed theoretical experiment data of ultrasonic flowmeter and weighting method are compared to verify the proposed correction factors. theoretical correction factors. 2. Methodology 2. Methodology In transit-time type ultrasonic flowmeter, the two transducers can be arranged in W-type, V-type In transit-time type ultrasonic flowmeter, the two transducers can be arranged in W-type, V- or Z-type, as shown in Figure 1. The di erence between any of the two types is the propagation type or Z-type, as shown in Figure 1. The difference between any of the two types is the propagation distance of ultrasonic wave. The W-type and V-type arrangement have longer distance, which will distance of ultrasonic wave. The W-type and V-type arrangement have longer distance, which will result in a longer time to be detected. So that the measurement accuracy could be improved. However, result in a longer time to be detected. So that the measurement accuracy could be improved. their measuring principle is the same, that is, one transducer works as transmitter and the other as a However, their measuring principle is the same, that is, one transducer works as transmitter and the receiver, and the transmit time from one transducer to the other will reflect the flowing velocity of the other as a receiver, and the transmit time from one transducer to the other will reflect the flowing fluid. Therefore, in order to make it easier for understanding, the Z-type is discussed in this paper. velocity of the fluid. Therefore, in order to make it easier for understanding, the Z-type is discussed Additionally, the final results can be used for W-type and V-type. in this paper. Additionally, the final results can be used for W-type and V-type. (a) (b) (c) Figure 1. Arrangement of ultrasonic transducers. (a) W-type; (b) V-type; and (c) Z-type. Figure 1. Arrangement of ultrasonic transducers. (a) W-type; (b) V-type; and (c) Z-type. The basic principle of the transit-time ultrasonic flowmeter is shown in Figure 2, two ultrasonic The basic principle of the transit-time ultrasonic flowmeter is shown in Figure 2, two ultrasonic transducers, upstream P1 and downstream P2, send and detect a short sound pulse with an oblique transducers, upstream P and downstream P , send and detect a short sound pulse with an oblique 1 2 propagation direction (angle θ with the pipe axis). propagation direction (angle  with the pipe axis). Appl. Sci. 2019, 9, 1648 3 of 11 Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 11 Figure 2. Principle of the transit-time ultrasonic flowmeter. Figure 2. Principle of the transit-time ultrasonic flowmeter. When the ultrasonic wave propagates downstream from transducer P , its propagation speed When the ultrasonic wave propagates downstream from transducer P1, its propagation speed will be coupled with the velocity projection of fluid in the direction of propagation. Therefore, the will be coupled with the velocity projection of fluid in the direction of propagation. Therefore, the propagation time for downstream signal from P to P will be 1 2 propagation time for downstream signal from P1 to P2 will be T = (1) 𝑇 = (1) c + ucos 𝑐 +𝑢𝑐𝑜𝑠𝜃 wherwhere e L is the L is pr topagation he propagat distance ion dist of ance o ultrasonic f ultrawave sonic w from avetransducer from transP ducer P to P , 1 c to P is the 2, c is t sound he so speed und 1 2 speed in the fluid, in the ufluid is the , u is the velocity velocity of the fluid, of tand he fluid,  is the and angle θ is the an between gle between the dire the directions of fluid ctions o flowing f fluid and wave flowing and wav propagation. e propagation. Similarly, wh Similarly, when en ultrasonic ultrasonic wave wave prop propagates agates upstr upstre eam am from from t transducer ransducer P P 2, , it its s p pr rop opagation agation sp speed eed will will subtra subtractct the vel the velocity ocipr ty projecti ojection ofon fluid of fl inui the d dir in the direction ection of propagation. of propagat Therefor ion. There e, the propagation fore, the p time ropagat for upstr ion tieam me fo signal r upst fr ream om P sign to al P fr will om P be 2 to P1 will be 2 1 (2) 𝑇 = T = (2) 2 𝑐 −𝑢𝑐𝑜𝑠𝜃 c ucos Eliminating the propagation speed of ultrasonic wave from Equation (1) and (2), the velocity of Eliminating the propagation speed of ultrasonic wave from Equations (1) and (2), the velocity of fluid can be expressed as fluid can be expressed as L T T 2 1 L T −T 2 1 u =  (3) u = ⋅ (3) 2 cos T  T 1 2 2cosθ T ⋅T 1 2 If the flow is axially uniform, the flowrate in the pipe can be estimated by measuring the transit If the flow is axially uniform, the flowrate in the pipe can be estimated by measuring the transit times of downstream signal T and upstream signal T , expressed as 1 2 times of downstream signal T1 and upstream signal T2, expressed as L L DT = T T = (4) 2 1 c u cos c + u cos LL Δ=TT −T = − (4) However, the velocity distribution21 across the pipe is not uniform. The variation in velocity will cu−+ cosθθ c ucos have e ects on the propagation of ultrasonic wave. Thus, Equation (4) can be used only within a very However, the velocity distribution across the pipe is not uniform. The variation in velocity will thin layer of fluid. Then, have effects on the propagation of ultrasonic wave. dL Thus, Equa dL tion (4) can be used only within a very dT dT = (5) 2 1 thin layer of fluid. Then, c u cos c + u cos where dL is the propagation distance of ultrasonic wave across the thin layer. dL dL dT − dT = − (5) According to geometric relation, we have 2 1 c − ucosθ c + ucosθ 2dr where dL is the propagation distance of ultrasonic wave across the thin layer. dL = (6) sin According to geometric relation, we have where, dr is the thickness of the thin layer. 2dr dL = (6) sin θ Appl. Sci. 2019, 9, 1648 4 of 11 Substituting Equation (6) into Equation (5), the transit times must be expressed as 2 1 1 DT = T T = [ ]dr (7) 2 1 sin c u cos c + u cos where R is the radius of the pipe. Introducing a dimensionless radius s = , then 2R 1 1 DT = [ ]ds (8) sin c u cos c + u cos Re-written as 2R 1 1 DT = [ ]ds (9) u sin cos K u/u K + u/u m m m where c is the sound speed; K = . u cos 2.1. Laminar Flow For laminar flow in a pipe, the velocity profile will be u = u (1 ) = u (1 s ) (10) m m Substitute into Equation (9), we get 2R 1 1 DT = [ ]ds (11) 2 2 u sin cos K (1 s ) K + (1 s ) then 2R 2 2s DT = ds (12) u sin cos 2 2 0 K (1 s ) 2 2 Because K >> (1 s ) , then 2R 2 2s DT = ds (13) u sin cos m 0 Finally, we have 8Ru cos DT = (14) 3c sin 2 m For a laminar flow, the flowrate can be expressed as Q = R . Then, it is easy to get the flowrate for laminar flow from Equation (14) 3Rc tan Q =  DT (15) Equation (15) clearly indicates the e ect of velocity profile on the flowrate measured. If the e ect of velocity distribution is neglected, the velocity u in Equation (4) is considered to be a constant, then L L 2Lu cos DT = = (16) 2 2 2 c u cos c + u cos c u cos 2R 2 2 2 Since c >> u cos  and L = , then sin 2Lu cos 4Ru cos DT = = (17) 2 2 c c sin Appl. Sci. 2019, 9, 1648 5 of 11 Therefore, the flowrate will be Rc Q = R u = tan  DT (18) Comparing Equation (15) with (18), it is found that Q = Q . That is to say, the flowrate Q 0 0 measured with an ultrasonic flowmeter without considering the e ect of velocity profile in the pipe should be multiplied by 3/4, in order to get the real flowrate. 2.2. Turbulent Flow For the turbulent flow in a pipe, the velocity profile will be 1/n 1/n u = u (1 ) = u (1 s) (19) m m where, the exponent n depends on Reynolds number, as shown in Table 1 [19,26]. Table 1. Relationship between Re and n. 3 4 5 5 5 5 5 Re 4  10 2.56  10 1.05  10 2.06  10 3.2  10 3.84  10 4.28  10 n 6.0 7.0 7.3 8.0 8.3 8.5 8.6 Substitute Equation (19) into (9), we get 1/n 2(1 s) 2R DT = ds (20) 2/n u sin cos 2 0 K (1 s) 2/n Because K >> (1 s) , we finally obtain n 4Ru cos DT = (21) n + 1 c sin The flowrate for turbulent flow is nRc tan Q =  DT (22) 4n + 2 If the e ect of velocity distribution is neglected, the velocity u in Equation (9) is considered to be a constant, u = u , then 2R 1 1 2R 2 DT = [ ]ds =  (23) u sin cos K 1 K + 1 u sin cos K 1 m m Because K >> 1, we finally obtain 4R 4Ru cos DT = = (24) 2 2 K u sin cos c sin Therefore, the flowrate will be Rc 2 2 Q = R u = R u = tan  DT (25) 0 m Comparing Equation (22) with (25), it is found that 2n Q =  Q (26) 2n + 1 Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 11 π Rc QR== ππu Ru = tanθ⋅ΔT (25) 0 m Appl. Sci. 2019, 9, 1648 6 of 11 Comparing Equation (22) with (25), it is found that Since the exponent n depends on Reynolds number, the e ect of turbulent velocity profile will 2n QQ =⋅ (26) also depends on the Reynolds number. 21 n + Since the exponent n depends on Reynolds number, the effect of turbulent velocity profile will 2.3. Correction Factors for Measurement with Ultrasonic Flowmeter also depends on the Reynolds number. For a laminar flow, it is easy to see from Equations (15) and (18) that the laminar correction factor will be 0.75. That is to say, the real flowrate is the result of multiplying the reading flowrate by the 2.3. Correction Factors for Measurement with Ultrasonic Flowmeter correction factor. For a laminar flow, it is easy to see from equation (15) and (18) that the laminar correction factor However, it is not so easy to get the final result for a turbulent flow because the correction factor is will be 0.75. That is to say, the real flowrate is the result of multiplying the reading flowrate by the not a constant in this case. From Equation (26), the correction factor for a turbulent flow will be correction factor. However, it is not so easy to get the final result for a turbulent flow because the correction factor 2n k = (27) is not a constant in this case. From Equation (2 t 6), the correction factor for a turbulent flow will be 2n + 1 2n The relationship between the exponent n and k = the Reynolds number in Table 1 can be expressed (27) as 2n +1 n = f(Re) (28) The relationship between the exponent n and the Reynolds number in table 1 can be expressed as n = f (Re) (28) The Reynolds number depends on the real flowrate Q, as follows The Reynolds number depends on the real flowrate Q, as follows 2Q Re = (29) 2 ρQ Re = (29) πμ R The relationship between real flowrate and the reading flowrate Q is The relationship between real flowrate and the reading flowrate Q0 is 2n 2n Q = k Q = Q (30) Q = tk Q 0 = Q 0 (30) t 0 0 2n + 1 2n +1 The real flowrate can be obtained from the simultaneous solution of the above Equations (28) to The real flowrate can be obtained from the simultaneous solution of the above Equations (28) to (30). (30 Figur ). Figu e r 3e gives 3 givethe s the corr cor ection rection factor factor for for a turbulent flow a turbulent flow.. Figure 3. Correction factor for a turbulent flow. Figure 3. Correction factor for a turbulent flow. 3. Experiment In order to verify the above analytical results, experiments were carried out at two testing systems, as shown in Figures 4 and 5. Figure 4 is a Reynolds’ apparatus, with which accurate results can be obtained when the Reynolds number is low. In order to keep the head of the upper tank be a constant, a stabilizing plate and an Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 11 Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 11 3. Experiment 3. Experiment Appl. Sci. 2019, 9, 1648 7 of 11 In order to verify the above analytical results, experiments were carried out at two testing In order to verify the above analytical results, experiments were carried out at two testing systems, as shown in Figures 4 and 5. systems, as shown in Figures 4 and 5. Figure 4 is a Reynolds’ apparatus, with which accurate results can be obtained when the Reynolds Figure 4 is a Reynolds’ apparatus, with which accurate results can be obtained when the Reynolds overflow plate are set. The head is 0.4 m. The experimental tube is made of organic glass with an number is low. In order to keep the head of the upper tank be a constant, a stabilizing plate and an number is low. In order to keep the head of the upper tank be a constant, a stabilizing plate and an inner diameter of 14mm. Therefore, the maximum Reynolds number with this apparatus will be about overflow plate are set. The head is 0.4 m. The experimental tube is made of organic glass with an inner overfl5ow plate are set. The head is 0.4 m. The experimental tube is made of organic glass with an inner 1.7  10 . diameter of 14mm. Therefore, the maximum Reynolds number with this apparatus will be about 1.7 × 10 . diameter of 14mm. Therefore, the maximum Reynolds number with this apparatus will be about 1.7 × 10 . Figure 4. The Reynolds’ apparatus. 1-water tank with pump; 2-tank with constant head; 3-stabilizing Figure 4. The Reynolds’ apparatus. 1-water tank with pump; 2-tank with constant head; 3-stabilizing Figure 4. The Reynolds’ apparatus. 1-water tank with pump; 2-tank with constant head; 3-stabilizing plate; 4-overflow plate; 5-regulating valve; 6-ultrasonic transducers; 7-experimental tube; 8-small plate; 4-overflow plate; 5-regulating valve; 6-ultrasonic transducers; 7-experimental tube; 8-small plate; 4-overflow plate; 5-regulating valve; 6-ultrasonic transducers; 7-experimental tube; 8-small container; and 9-return tube. container; and 9-return tube. container; and 9-return tube. Since the Reynolds’ apparatus could not reach a higher Reynolds number, a performance testing Since the Reynolds’ apparatus could not reach a higher Reynolds number, a performance testing Since the Reynolds’ apparatus could not reach a higher Reynolds number, a performance testing system for centrifugal pump is used, as shown in Figure 5. The rated flowrate of the centrifugal pump system for centrifugal pump is used, as shown in Figure 5. The rated flowrate of the centrifugal pump system for centrifugal pump is used, as shown in Figure 5. The rated flowrate of the centrifugal pump in the system is 65 m /h, and the inner diameter of the experimental pipe is 50.8 mm. The maximum in the system is 65 m /h, and the inner diameter of the experimental pipe is 50.8 mm. The maximum in the system is 65 m /h, and the inner diameter of the experimental pipe is 50.8 mm. The maximum Reynolds number with this system can be as high as 4.24  10 5 . Reynolds number with this system can be as high as 4.24 × 10 . Reynolds number with this system can be as high as 4.24 × 10 . Figure 5. The performance testing system for centrifugal pumps. 1-centrifugal pump; 2-regulating valve; 3-ultrasonic transducers; 4-water tank; 5-electronic scale; 6-water tank. The Reynolds number can be easily adjusted by the regulating valves in both systems. Flowrate was measured by a TDS-100H portable ultrasonic flowmeter and by the weight of water flowing out of the pipe within a certain time interval. Appl. Sci. 2019, 9, 1648 8 of 11 4. Results 4.1. Verification Table 2 gives the results of flowrate measurement at the Reynolds’ apparatus. The flowrate Q is measured by the ultrasonic flowmeter, while Q is measured by the weight discharged within a certain period of time. The flowrate ratio Q :Q is the measured correction factor. The relative error 2 1 ( ) defined in Equation (31), is used to evaluate the di erence between theoretical correction factor and measured data, where k indicates the theoretical correction factor. =  100% (31) Table 2. Results of flowrate measured at the Reynolds’ apparatus. Ultrasonic Flowmeter Weighing Method Flowrate No. Flowrate Flowrate Ratio  (%) DT  10 Time Mass Velocity 5 5 = Q :Q Q  10 Q  10 Re 2 1 1 2 (s) (kg) (m/s) (s) 3 3 (m /s) (m /s) 1 0.378 0.970 60 0.430 0.717 0.047 652 0.739 1.443 2 0.462 1.185 60 0.524 0.873 0.057 795 0.737 1.772 3 0.632 1.622 60 0.721 1.202 0.078 1094 0.741 1.192 4 1.002 2.572 50 0.965 1.930 0.125 1757 0.751 0.052 5 1.121 2.877 50 1.073 2.147 0.140 1954 0.746 0.498 6 1.310 3.363 40 1.005 2.512 0.163 2286 0.747 0.406 7 1.317 3.380 40 1.019 2.548 0.166 2318 0.754 0.513 8 1.553 3.987 40 1.219 3.048 0.198 2774 0.764 1.931 From Equations (15) and (18), it is clearly indicated that the theoretical correction factor for laminar flow is 0.75. It can be seen from Table 2 that the flowrate ratios ( ) are in range of 0.737 to 0.764, the average value of is 0.747. The average relative error is 0.976%, and the maximum relative error is 1.931%, which is in good agreement with the experiment result. The last line in Table 2 is in the transition zone from laminar flow to turbulent flow, and the correction factor is a little bit larger than 0.75, which is caused by the gradual changing of velocity profile from laminar flow to turbulent flow. Table 3 gives the results of flowrate measured at the performance testing system for centrifugal pumps. The Reynolds numbers are from 4311 to 422102. The theoretical turbulent flow correction factor (k ) is calculated by Equation (27). The average relative error is 0.25% and maximum error is 1.178%. Therefore, the theoretical correction factor is reliable. Table 3. Results of flowrate measurement at the performance testing system for centrifugal pumps. Ultrasonic Flowmeter Weighing Method Flowrate No. Flowrate Flowrate Ratio k  (%) t 2 DT  10 Time Mass Velocity 5 5 Q  10 Q  10 Re n = Q :Q 1 2 2 1 (s) (s) (kg) (m/s) 3 3 (m /s) (m /s) 1 7.270 0.186 60 10.314 0.172 0.085 4311 6 0.925 0.923 0.216 2 17.714 0.454 40 16.939 0.424 0.209 10619 6 0.934 0.923 1.178 3 36.956 0.948 30 26.438 0.881 0.435 22099 6 0.929 0.923 0.646 4 43.889 1.125 20 20.984 1.049 0.518 26310 7 0.932 0.933 0.107 5 76.663 1.966 15 27.566 1.838 0.907 46084 7 0.935 0.933 0.214 6 112.156 2.876 15 40.387 2.693 1.329 67518 7 0.936 0.933 0.321 7 143.424 3.677 10 34.384 3.438 1.697 86223 7 0.935 0.933 0.214 8 171.690 4.402 10 41.125 4.113 2.030 103127 7.3 0.934 0.936 0.214 9 244.767 6.276 10 58.683 5.868 2.897 147156 7.3 0.935 0.936 0.110 10 312.871 8.022 10 75.124 7.513 3.708 188385 7.3 0.937 0.936 0.107 11 341.571 8.758 8 66.056 8.257 4.076 207057 8 0.943 0.941 0.212 12 440.921 11.305 8 85.452 10.681 5.273 267854 8 0.945 0.941 0.423 13 532.958 13.664 7 90.218 12.888 6.362 323193 8.3 0.943 0.943 0.000 14 597.541 15.320 5 72.367 14.473 7.145 362942 8.3 0.945 0.943 0.212 Appl. Sci. 2019, 9, 1648 9 of 11 Table 3. Cont. Ultrasonic Flowmeter Weighing Method Flowrate No. Flowrate Flowrate Ratio k  (%) t 2 DT  10 Time Mass Velocity 5 5 Q  10 Q  10 Re n = Q :Q 1 2 2 1 (s) (s) (kg) (m/s) 3 3 (m /s) (m /s) 15 629.177 16.131 5 76.218 15.244 7.525 382256 8.5 0.945 0.944 0.106 16 662.113 16.976 5 80.136 16.027 7.912 401906 8.5 0.944 0.944 0.000 17 694.919 17.817 5 84.163 16.833 8.309 422102 8.6 0.945 0.945 0.000 4.2. Uncertainty Analysis The flowrate obtained by the weighing method is regarded as the standard to evaluate the measurement results of the ultrasonic flowmeter. Therefore, the experiment uncertainty of weighting method is analyzed and results for laminar and turbulent flow are shown in Tables 4 and 5, respectively. Collected time is controlled by an electronic timer, the uncertainty of time is 10ms. Collected mass is weighed by an electronic scale, the uncertainty of mass is 1 g. The standard relative uncertainty of time is from 0.017% to 0.200%; from 0.001% to 0.233% of mass. The combined uncertainty is from 0.019% to 0.233%. The expanded uncertainty has been reported at k = 2, which from 0.039% to 0.466%. Table 4. Experiment uncertainty for laminar flow. Collected Mass Collected Time Combined Expanded No. Estimate Uncertainty Estimate Uncertainty Uncertainty Uncertainty (M)(kg) ( )(%) (Dt)(s) ( )(%) ( )(%) (U)(%) M c Dt 1 0.43 0.233 60 0.017 0.233 0.466 2 0.524 0.191 60 0.017 0.192 0.383 3 0.721 0.139 60 0.017 0.140 0.279 4 0.965 0.104 50 0.020 0.106 0.211 5 1.073 0.093 50 0.020 0.095 0.191 6 1.005 0.100 40 0.025 0.103 0.205 7 1.019 0.098 40 0.025 0.101 0.203 8 1.219 0.082 40 0.025 0.086 0.172 Table 5. Experiment uncertainty for turbulent flow. Collected Mass Collected Time Combined Expanded No. Estimate Uncertainty Estimate Uncertainty Uncertainty Uncertainty (M)(kg) ( )(%) (Dt)(s) ( )(%) ( )(%) (U)(%) M Dt c 1 10.314 0.010 60 0.017 0.019 0.039 2 16.939 0.006 40 0.025 0.026 0.051 3 26.438 0.004 30 0.033 0.034 0.067 4 20.984 0.005 20 0.050 0.050 0.100 5 27.566 0.004 15 0.067 0.067 0.134 6 40.387 0.002 15 0.067 0.067 0.133 7 34.384 0.003 10 0.100 0.100 0.200 8 41.125 0.002 10 0.100 0.100 0.200 9 58.683 0.002 10 0.100 0.100 0.200 10 75.124 0.001 10 0.100 0.100 0.200 11 66.056 0.002 8 0.125 0.125 0.250 12 85.452 0.001 8 0.125 0.125 0.250 13 90.218 0.001 7 0.143 0.143 0.286 14 72.367 0.001 5 0.200 0.200 0.400 15 76.218 0.001 5 0.200 0.200 0.400 16 80.136 0.001 5 0.200 0.200 0.400 17 84.163 0.001 5 0.200 0.200 0.400 Appl. Sci. 2019, 9, 1648 10 of 11 5. Conclusions The e ect of velocity profiles across the pipe on the propagation time of ultrasonic wave is considered for transit-time ultrasonic flowmeters. Theoretical correction factors were proposed to improve measurement accuracy. For laminar flow, the correction factor is a constant of 0.75. For turbulent flow, the correction factor varies with Reynolds number, show as Equation (27). Flow measurement experiments were performed for laminar and turbulent flow respectively, and experiment results showed that the proposed correction factors agree well with measured correction factors. The average relative error is determined to be 0.976% for laminar flow and 0.25% for turbulent flow. Author Contributions: Conceptualization, C.G.; Data curation, H.Z.; Formal analysis, H.Z.; Funding acquisition, C.G.; Investigation, H.Z.; Methodology, C.G.; Resources, J.L.; Software, J.L.; Writing—original draft, H.Z.; Writing—review & editing, C.G. Funding: This research was funded by the Fundamental Research Funds for the Central Universities, grant number 2017XKZD02. Acknowledgments: The authors are very grateful to Dr. Wang Fengchao for his help in the experiments. Conflicts of Interest: The authors declare no conflict of interest. References 1. Ferrari, A.; Pizzo, P.; Rundo, M. Modelling and experimental studies on a proportional valve using an innovative dynamic flow-rate measurement in fluid power systems. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 2018, 232, 2404–2418. [CrossRef] 2. Sun, Y.; Zhang, T.; Zheng, D. New Analysis Scheme of Flow-Acoustic Coupling for Gas Ultrasonic Flowmeter with Vortex near the Transducer. Sensors 2018, 18, 1151. 3. Grände, P.O.; Borgström, P. An electromic di erential pressure flowmeter and a resistance meter for continuous measurement of vascular resistance. Acta Physiol. Scand. 2010, 102, 224–230. [CrossRef] [PubMed] 4. Bückle, U.; Durst, F.; Howe, B.; Melling, A. Investigation of a floating element flowmeter. Flow Meas. Instrum. 1992, 3, 215–225. [CrossRef] 5. Gibson, W.G.; Cobbold, R.S.; Johnston, K.W. Principles and design feasibility of a Doppler ultrasound intravascular volumetric flowmeter. IEEE Trans. Bio.-Med. Eng. 1994, 41, 898. [CrossRef] [PubMed] 6. Bevir, M.K. The theory of induced voltage electromagnetic flowmeter. J. Fluid Mech. 2006, 43, 577–590. [CrossRef] 7. Pasquale, G.D.; Graziani, S.; Pollicino, A.; Strazzeri, S. A vortex-shedding flowmeter based on IPMCs. Smart Mater. Struct. 2016, 25, 015011. [CrossRef] 8. Lynnworth, L.C.; Liu, Y. Ultrasonic flowmeters: Half-century progress report, 1955–2005. Ultrasonics 2006, 44, E1371–E1378. [CrossRef] 9. Chhantyal, K.; Jondahl, M.H.; Viumdal, H.; Mylvaganam, S. Upstream Ultrasonic Level Based Soft Sensing of Volumetric Flow of Non-Newtonian Fluids in Open Venturi Channels. IEEE Sens. J. 2018, 18, 5002–5013. [CrossRef] 10. Jiang, Y.D.; Wang, B.L.; Li, X.; Liu, D.D.; Wang, Y.Q.; Huang, Z.Y. A Model-Based Hybrid Ultrasonic Gas Flowmeter. IEEE Sens. J. 2018, 18, 4443–4452. [CrossRef] 11. Liu, B.; Xu, K.J.; Mu, L.B.; Tian, L. Echo energy integral based signal processing method for ultrasonic gas flow meter. Sens. Actuat. A-Phys. 2018, 277, 181–189. [CrossRef] 12. Raine, A.B.; Aslam, N.; Underwood, C.P.; Danaher, S. Development of an Ultrasonic Airflow Measurement Device for Ducted Air. Sensors 2015, 15, 10705–10722. [CrossRef] [PubMed] 13. Iooss, B.; Lhuillier, C.; Jeanneau, H. Numerical simulation of transit-time ultrasonic flowmeters: Uncertainties due to flow profile and fluid turbulence. Ultrasonics 2002, 40, 1009–1015. [CrossRef] 14. Mandard, E.; Kouame, D.; Battault, R.; Remenieras, J.P.; Patat, F. Transit time ultrasonic flowmeter: Velocity profile estimation. Ultrason 2005, 2, 763–766. 15. Sanderson, M.L.; Yeung, H. Guidelines for the use of ultrasonic non-invasive metering techniques. Flow Meas. Instrum. 2002, 13, 125–142. [CrossRef] Appl. Sci. 2019, 9, 1648 11 of 11 16. Jaiswal, S.K.; Yadav, S.; Agarwal, R. Multiple Weighing Based Method for Realizing Flow. Mapan 2015, 30, 119–123. [CrossRef] 17. Willatzen, M. Perturbation theory applied to sound propagation in flowing media confined by a cylindrical waveguide. J. Acoust. Soc. Am. 2001, 109, 102–107. [CrossRef] [PubMed] 18. Willatzen, M.; Kamath, H. Nonlinearities in ultrasonic flow measurement. Flow Meas. Instrum. 2008, 19, 79–84. [CrossRef] 19. Willatzen, M. Ultrasonic flowmeters: Temperature gradients and transducer geometry e ects. Ultrasonics 2003, 41, 105–114. [CrossRef] 20. Dadashnialehi, A.; Moshiri, B. Online monitoring of transit-time ultrasonic flowmeters based on fusion of optical observation. Measurement 2011, 44, 1028–1037. [CrossRef] 21. Mahadeva, D.V.; Baker, R.C.; Woodhouse, J. Further Studies of the Accuracy of Clamp-on Transit-Time Ultrasonic Flowmeters for Liquids. IEEE Trans. Instrum. Meas. 2009, 58, 1602–1609. [CrossRef] 22. Mahadeva, D.V.; Baker, R.C.; Woodhouse, J. Studies of the Accuracy of Clamp-on Transit Time Ultrasonic Flowmeters. In Proceedings of the 2008 IEEE Instrumentation and Measurement Technology Conference, Victoria, Canada, 12–15 May 2008; pp. 969–973. 23. Rajita, G.; Mandal, N. Review on transit time ultrasonic flowmeter. In Proceedings of the International Conference on Control, Kolkata, India, 28–30 January 2016. 24. Wang, X.F.; Tang, Z.A. Note: Ultrasonic gas flowmeter based on optimized time-of-flight algorithms. Rev. Sci. Instrum. 2011, 82, 1371–1584. [CrossRef] [PubMed] 25. Liu, Z.G.; Du, G.S.; Shao, Z.F.; He, Q.R.; Zhou, C.L. Measurement of transitional flow in pipes using ultrasonic flowmeters. Fluid Dyn. Res. 2014, 46, 055501. 26. Xiong, Y.T.; Su, Z.D.; Lin, J.D.; Zhang, Y.C. Calibration of large diameter gas flowmeter by the corresponding velocity distribution equation. J. China Univ. Metrol. 2011, 22, 25–29. © 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Sciences Multidisciplinary Digital Publishing Institute

Effects of Velocity Profiles on Measuring Accuracy of Transit-Time Ultrasonic Flowmeter

Applied Sciences , Volume 9 (8) – Apr 20, 2019

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Abstract

applied sciences Article E ects of Velocity Profiles on Measuring Accuracy of Transit-Time Ultrasonic Flowmeter 1 , 2 1 , 1 Hui Zhang , Chuwen Guo * and Jie Lin School of Electrical and Power Engineering, China University of Mining and Technology, Xuzhou 221116, China; cumthui@126.com (H.Z.); TS17130003A3@cumt.edu.cn (J.L.) Xuhai College, China University of Mining and Technology, Xuzhou 221008, China * Correspondence: cwguo@cumt.edu.cn; Tel.: +86-516-8359-2200 Received: 13 March 2019; Accepted: 18 April 2019; Published: 20 April 2019 Featured Application: The research results of this paper could be used to improve the measuring accuracy of the transit-time ultrasonic flowmeter. Abstract: Ultrasonic wave carries the information for flowing velocity when it is propagating in flowing fluids. Flowrate can be obtained by measuring the propagation time of ultrasonic wave. The principle of transit-time ultrasonic flowmeters used today was based on that the velocity is uniform along the propagation path of the ultrasonic wave. However, it is well known that the velocity profiles in a pipe are not uniform both in laminar flow and turbulent flow. Emphasis on the e ects of velocity profiles across the pipe on the propagation time of ultrasonic wave, theoretical flowrate correction factors considering the real velocity profile were proposed for laminar and turbulent flow to obtain higher accuracy. Experiment data of ultrasonic flowmeter and weighting method are compared to verify the proposed theoretical correction factors. The average relative error of proposed correction factor is determined to be 0.976% for laminar flow and 0.25% for turbulent flow. Keywords: velocity profile; transit-time; ultrasonic flowmeter; accuracy; correction factor 1. Introduction Flow measurement is necessary in engineering field for fluid metrology and process control [1,2]. A variety of flow measurement techniques were invented, such as di erential pressure flowmeter [3], float flowmeter [4], volumetric flowmeter [5], electromagnetic flowmeter [6], and vortex shedding flowmeter [7]. However, the common disadvantage of these flow measurement techniques is that the fluid properties are tightly restricted due to direct contact with the flowmeters. In order to overcome the above disadvantage, non-contact ultrasonic flowmeter was developed, which is immune to temperature, causticity, and conductivity of measured fluid [8]. There is a great variety of ultrasonic flowmeters for measurement of liquid and gas flow [9–12]. Today, ultrasonic flowmeters utilize clamp-on and wetted transducers, single and multiple paths, paths on and o the diameter, passive and active principles, contra-propagating transmission, reflection (Doppler), tag correlation, vortex shedding, liquid level sensing of open channel flow or flow in partially-full conduits, and other interactions. Due to the simplicity of the measurement principle, the ultrasonic transit-time method is one of the most common techniques in industrial applications. Flowrate measurement using the ultrasonic transit-time method is based on the apparent di erence of the sound velocity in the flow direction and in the opposite direction [13,14]. It was pointed out in reference [15] that the transit-time ultrasonic flowmeter has a relatively high uncertainty, approximately 5%. The uncertainty is mainly caused by three factors, the first is the installation method of transducers; the second is the measurement of transit time, especially for pipe with a small inner diameter; the last is Appl. Sci. 2019, 9, 1648; doi:10.3390/app9081648 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 1648 2 of 11 Appl. Sci. 2019, 9, x FOR PEER REVIEW 2 of 11 velocity distribution across the pipe [16]. A large amount of studies have been carried out to reduce the studies have been carried out to reduce the uncertainty caused by these three reasons [17–20]. For the uncertainty caused by these three reasons [17–20]. For the first reason, Mahadeva et al. [21,22] improved first reason, Mahadeva et al. [21,22] improved measurement accuracy by changing the distance measurement accuracy by changing the distance between transducers. Rajita et al. [23] indicated that between transducers. Rajita et al. [23] indicated that multipath ultrasonic flowmeters can provide multipath ultrasonic flowmeters can provide more accurate flow velocities than a single path. For the more accurate flow velocities than a single path. For the second reason, several research focused on second reason, several research focused on developing new algorithms to obtain more accurate time developing new algorithms to obtain more accurate time differences [24]. And high time resolution di erences [24]. And high time resolution electronic components have significant progress in recent electronic components have significant progress in recent years to achieve nanosecond measurement years to achieve nanosecond measurement of transit time. However, there is a limited amount of of transit time. However, there is a limited amount of information available on the effect of velocity information available on the e ect of velocity profile on the measurement uncertainty. Looss et al. [13] profile on the measurement uncertainty. Looss et al. [13] have proved that the flow rate is have proved that the flow rate is overestimated by the e ects of the assumption of uniform velocity overestimated by the effects of the assumption of uniform velocity distribution, and a correction distribution, and a correction factor is obtained empirically based on numerical simulation results factor is obtained empirically based on numerical simulation results for fully developed turbulence. for fully developed turbulence. Zheng et al. [25] gave a correction factor for the transition region Zheng et al. [25] gave a correction factor for the transition region with Reynolds number in the range with Reynolds number in the range of 2000–20000 based on experiment results by Particle Image of 2000–20000 based on experiment results by Particle Image Velocimetry (PIV) measurement. Little Velocimetry (PIV) measurement. Little attention was paid to obtain correction factor for transit time attention was paid to obtain correction factor for transit time ultrasonic flowmeter through theoretical ultrasonic flowmeter through theoretical analysis. analysis. In this paper, we put emphasis on the e ects of velocity profiles across the pipe on the time of In this paper, we put emphasis on the effects of velocity profiles across the pipe on the time of ultrasonic wave propagation. Theoretical correction factors for transit-time ultrasonic flowmeters ultrasonic wave propagation. Theoretical correction factors for transit-time ultrasonic flowmeters were proposed for laminar and turbulent flow to improve measurement accuracy. Flow measurement were proposed for laminar and turbulent flow to improve measurement accuracy. Flow experiments were performed for laminar and turbulent flow, respectively, and experiment data measurement experiments were performed for laminar and turbulent flow, respectively, and of ultrasonic flowmeter and weighting method are compared to verify the proposed theoretical experiment data of ultrasonic flowmeter and weighting method are compared to verify the proposed correction factors. theoretical correction factors. 2. Methodology 2. Methodology In transit-time type ultrasonic flowmeter, the two transducers can be arranged in W-type, V-type In transit-time type ultrasonic flowmeter, the two transducers can be arranged in W-type, V- or Z-type, as shown in Figure 1. The di erence between any of the two types is the propagation type or Z-type, as shown in Figure 1. The difference between any of the two types is the propagation distance of ultrasonic wave. The W-type and V-type arrangement have longer distance, which will distance of ultrasonic wave. The W-type and V-type arrangement have longer distance, which will result in a longer time to be detected. So that the measurement accuracy could be improved. However, result in a longer time to be detected. So that the measurement accuracy could be improved. their measuring principle is the same, that is, one transducer works as transmitter and the other as a However, their measuring principle is the same, that is, one transducer works as transmitter and the receiver, and the transmit time from one transducer to the other will reflect the flowing velocity of the other as a receiver, and the transmit time from one transducer to the other will reflect the flowing fluid. Therefore, in order to make it easier for understanding, the Z-type is discussed in this paper. velocity of the fluid. Therefore, in order to make it easier for understanding, the Z-type is discussed Additionally, the final results can be used for W-type and V-type. in this paper. Additionally, the final results can be used for W-type and V-type. (a) (b) (c) Figure 1. Arrangement of ultrasonic transducers. (a) W-type; (b) V-type; and (c) Z-type. Figure 1. Arrangement of ultrasonic transducers. (a) W-type; (b) V-type; and (c) Z-type. The basic principle of the transit-time ultrasonic flowmeter is shown in Figure 2, two ultrasonic The basic principle of the transit-time ultrasonic flowmeter is shown in Figure 2, two ultrasonic transducers, upstream P1 and downstream P2, send and detect a short sound pulse with an oblique transducers, upstream P and downstream P , send and detect a short sound pulse with an oblique 1 2 propagation direction (angle θ with the pipe axis). propagation direction (angle  with the pipe axis). Appl. Sci. 2019, 9, 1648 3 of 11 Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 11 Figure 2. Principle of the transit-time ultrasonic flowmeter. Figure 2. Principle of the transit-time ultrasonic flowmeter. When the ultrasonic wave propagates downstream from transducer P , its propagation speed When the ultrasonic wave propagates downstream from transducer P1, its propagation speed will be coupled with the velocity projection of fluid in the direction of propagation. Therefore, the will be coupled with the velocity projection of fluid in the direction of propagation. Therefore, the propagation time for downstream signal from P to P will be 1 2 propagation time for downstream signal from P1 to P2 will be T = (1) 𝑇 = (1) c + ucos 𝑐 +𝑢𝑐𝑜𝑠𝜃 wherwhere e L is the L is pr topagation he propagat distance ion dist of ance o ultrasonic f ultrawave sonic w from avetransducer from transP ducer P to P , 1 c to P is the 2, c is t sound he so speed und 1 2 speed in the fluid, in the ufluid is the , u is the velocity velocity of the fluid, of tand he fluid,  is the and angle θ is the an between gle between the dire the directions of fluid ctions o flowing f fluid and wave flowing and wav propagation. e propagation. Similarly, wh Similarly, when en ultrasonic ultrasonic wave wave prop propagates agates upstr upstre eam am from from t transducer ransducer P P 2, , it its s p pr rop opagation agation sp speed eed will will subtra subtractct the vel the velocity ocipr ty projecti ojection ofon fluid of fl inui the d dir in the direction ection of propagation. of propagat Therefor ion. There e, the propagation fore, the p time ropagat for upstr ion tieam me fo signal r upst fr ream om P sign to al P fr will om P be 2 to P1 will be 2 1 (2) 𝑇 = T = (2) 2 𝑐 −𝑢𝑐𝑜𝑠𝜃 c ucos Eliminating the propagation speed of ultrasonic wave from Equation (1) and (2), the velocity of Eliminating the propagation speed of ultrasonic wave from Equations (1) and (2), the velocity of fluid can be expressed as fluid can be expressed as L T T 2 1 L T −T 2 1 u =  (3) u = ⋅ (3) 2 cos T  T 1 2 2cosθ T ⋅T 1 2 If the flow is axially uniform, the flowrate in the pipe can be estimated by measuring the transit If the flow is axially uniform, the flowrate in the pipe can be estimated by measuring the transit times of downstream signal T and upstream signal T , expressed as 1 2 times of downstream signal T1 and upstream signal T2, expressed as L L DT = T T = (4) 2 1 c u cos c + u cos LL Δ=TT −T = − (4) However, the velocity distribution21 across the pipe is not uniform. The variation in velocity will cu−+ cosθθ c ucos have e ects on the propagation of ultrasonic wave. Thus, Equation (4) can be used only within a very However, the velocity distribution across the pipe is not uniform. The variation in velocity will thin layer of fluid. Then, have effects on the propagation of ultrasonic wave. dL Thus, Equa dL tion (4) can be used only within a very dT dT = (5) 2 1 thin layer of fluid. Then, c u cos c + u cos where dL is the propagation distance of ultrasonic wave across the thin layer. dL dL dT − dT = − (5) According to geometric relation, we have 2 1 c − ucosθ c + ucosθ 2dr where dL is the propagation distance of ultrasonic wave across the thin layer. dL = (6) sin According to geometric relation, we have where, dr is the thickness of the thin layer. 2dr dL = (6) sin θ Appl. Sci. 2019, 9, 1648 4 of 11 Substituting Equation (6) into Equation (5), the transit times must be expressed as 2 1 1 DT = T T = [ ]dr (7) 2 1 sin c u cos c + u cos where R is the radius of the pipe. Introducing a dimensionless radius s = , then 2R 1 1 DT = [ ]ds (8) sin c u cos c + u cos Re-written as 2R 1 1 DT = [ ]ds (9) u sin cos K u/u K + u/u m m m where c is the sound speed; K = . u cos 2.1. Laminar Flow For laminar flow in a pipe, the velocity profile will be u = u (1 ) = u (1 s ) (10) m m Substitute into Equation (9), we get 2R 1 1 DT = [ ]ds (11) 2 2 u sin cos K (1 s ) K + (1 s ) then 2R 2 2s DT = ds (12) u sin cos 2 2 0 K (1 s ) 2 2 Because K >> (1 s ) , then 2R 2 2s DT = ds (13) u sin cos m 0 Finally, we have 8Ru cos DT = (14) 3c sin 2 m For a laminar flow, the flowrate can be expressed as Q = R . Then, it is easy to get the flowrate for laminar flow from Equation (14) 3Rc tan Q =  DT (15) Equation (15) clearly indicates the e ect of velocity profile on the flowrate measured. If the e ect of velocity distribution is neglected, the velocity u in Equation (4) is considered to be a constant, then L L 2Lu cos DT = = (16) 2 2 2 c u cos c + u cos c u cos 2R 2 2 2 Since c >> u cos  and L = , then sin 2Lu cos 4Ru cos DT = = (17) 2 2 c c sin Appl. Sci. 2019, 9, 1648 5 of 11 Therefore, the flowrate will be Rc Q = R u = tan  DT (18) Comparing Equation (15) with (18), it is found that Q = Q . That is to say, the flowrate Q 0 0 measured with an ultrasonic flowmeter without considering the e ect of velocity profile in the pipe should be multiplied by 3/4, in order to get the real flowrate. 2.2. Turbulent Flow For the turbulent flow in a pipe, the velocity profile will be 1/n 1/n u = u (1 ) = u (1 s) (19) m m where, the exponent n depends on Reynolds number, as shown in Table 1 [19,26]. Table 1. Relationship between Re and n. 3 4 5 5 5 5 5 Re 4  10 2.56  10 1.05  10 2.06  10 3.2  10 3.84  10 4.28  10 n 6.0 7.0 7.3 8.0 8.3 8.5 8.6 Substitute Equation (19) into (9), we get 1/n 2(1 s) 2R DT = ds (20) 2/n u sin cos 2 0 K (1 s) 2/n Because K >> (1 s) , we finally obtain n 4Ru cos DT = (21) n + 1 c sin The flowrate for turbulent flow is nRc tan Q =  DT (22) 4n + 2 If the e ect of velocity distribution is neglected, the velocity u in Equation (9) is considered to be a constant, u = u , then 2R 1 1 2R 2 DT = [ ]ds =  (23) u sin cos K 1 K + 1 u sin cos K 1 m m Because K >> 1, we finally obtain 4R 4Ru cos DT = = (24) 2 2 K u sin cos c sin Therefore, the flowrate will be Rc 2 2 Q = R u = R u = tan  DT (25) 0 m Comparing Equation (22) with (25), it is found that 2n Q =  Q (26) 2n + 1 Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 11 π Rc QR== ππu Ru = tanθ⋅ΔT (25) 0 m Appl. Sci. 2019, 9, 1648 6 of 11 Comparing Equation (22) with (25), it is found that Since the exponent n depends on Reynolds number, the e ect of turbulent velocity profile will 2n QQ =⋅ (26) also depends on the Reynolds number. 21 n + Since the exponent n depends on Reynolds number, the effect of turbulent velocity profile will 2.3. Correction Factors for Measurement with Ultrasonic Flowmeter also depends on the Reynolds number. For a laminar flow, it is easy to see from Equations (15) and (18) that the laminar correction factor will be 0.75. That is to say, the real flowrate is the result of multiplying the reading flowrate by the 2.3. Correction Factors for Measurement with Ultrasonic Flowmeter correction factor. For a laminar flow, it is easy to see from equation (15) and (18) that the laminar correction factor However, it is not so easy to get the final result for a turbulent flow because the correction factor is will be 0.75. That is to say, the real flowrate is the result of multiplying the reading flowrate by the not a constant in this case. From Equation (26), the correction factor for a turbulent flow will be correction factor. However, it is not so easy to get the final result for a turbulent flow because the correction factor 2n k = (27) is not a constant in this case. From Equation (2 t 6), the correction factor for a turbulent flow will be 2n + 1 2n The relationship between the exponent n and k = the Reynolds number in Table 1 can be expressed (27) as 2n +1 n = f(Re) (28) The relationship between the exponent n and the Reynolds number in table 1 can be expressed as n = f (Re) (28) The Reynolds number depends on the real flowrate Q, as follows The Reynolds number depends on the real flowrate Q, as follows 2Q Re = (29) 2 ρQ Re = (29) πμ R The relationship between real flowrate and the reading flowrate Q is The relationship between real flowrate and the reading flowrate Q0 is 2n 2n Q = k Q = Q (30) Q = tk Q 0 = Q 0 (30) t 0 0 2n + 1 2n +1 The real flowrate can be obtained from the simultaneous solution of the above Equations (28) to The real flowrate can be obtained from the simultaneous solution of the above Equations (28) to (30). (30 Figur ). Figu e r 3e gives 3 givethe s the corr cor ection rection factor factor for for a turbulent flow a turbulent flow.. Figure 3. Correction factor for a turbulent flow. Figure 3. Correction factor for a turbulent flow. 3. Experiment In order to verify the above analytical results, experiments were carried out at two testing systems, as shown in Figures 4 and 5. Figure 4 is a Reynolds’ apparatus, with which accurate results can be obtained when the Reynolds number is low. In order to keep the head of the upper tank be a constant, a stabilizing plate and an Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 11 Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 11 3. Experiment 3. Experiment Appl. Sci. 2019, 9, 1648 7 of 11 In order to verify the above analytical results, experiments were carried out at two testing In order to verify the above analytical results, experiments were carried out at two testing systems, as shown in Figures 4 and 5. systems, as shown in Figures 4 and 5. Figure 4 is a Reynolds’ apparatus, with which accurate results can be obtained when the Reynolds Figure 4 is a Reynolds’ apparatus, with which accurate results can be obtained when the Reynolds overflow plate are set. The head is 0.4 m. The experimental tube is made of organic glass with an number is low. In order to keep the head of the upper tank be a constant, a stabilizing plate and an number is low. In order to keep the head of the upper tank be a constant, a stabilizing plate and an inner diameter of 14mm. Therefore, the maximum Reynolds number with this apparatus will be about overflow plate are set. The head is 0.4 m. The experimental tube is made of organic glass with an inner overfl5ow plate are set. The head is 0.4 m. The experimental tube is made of organic glass with an inner 1.7  10 . diameter of 14mm. Therefore, the maximum Reynolds number with this apparatus will be about 1.7 × 10 . diameter of 14mm. Therefore, the maximum Reynolds number with this apparatus will be about 1.7 × 10 . Figure 4. The Reynolds’ apparatus. 1-water tank with pump; 2-tank with constant head; 3-stabilizing Figure 4. The Reynolds’ apparatus. 1-water tank with pump; 2-tank with constant head; 3-stabilizing Figure 4. The Reynolds’ apparatus. 1-water tank with pump; 2-tank with constant head; 3-stabilizing plate; 4-overflow plate; 5-regulating valve; 6-ultrasonic transducers; 7-experimental tube; 8-small plate; 4-overflow plate; 5-regulating valve; 6-ultrasonic transducers; 7-experimental tube; 8-small plate; 4-overflow plate; 5-regulating valve; 6-ultrasonic transducers; 7-experimental tube; 8-small container; and 9-return tube. container; and 9-return tube. container; and 9-return tube. Since the Reynolds’ apparatus could not reach a higher Reynolds number, a performance testing Since the Reynolds’ apparatus could not reach a higher Reynolds number, a performance testing Since the Reynolds’ apparatus could not reach a higher Reynolds number, a performance testing system for centrifugal pump is used, as shown in Figure 5. The rated flowrate of the centrifugal pump system for centrifugal pump is used, as shown in Figure 5. The rated flowrate of the centrifugal pump system for centrifugal pump is used, as shown in Figure 5. The rated flowrate of the centrifugal pump in the system is 65 m /h, and the inner diameter of the experimental pipe is 50.8 mm. The maximum in the system is 65 m /h, and the inner diameter of the experimental pipe is 50.8 mm. The maximum in the system is 65 m /h, and the inner diameter of the experimental pipe is 50.8 mm. The maximum Reynolds number with this system can be as high as 4.24  10 5 . Reynolds number with this system can be as high as 4.24 × 10 . Reynolds number with this system can be as high as 4.24 × 10 . Figure 5. The performance testing system for centrifugal pumps. 1-centrifugal pump; 2-regulating valve; 3-ultrasonic transducers; 4-water tank; 5-electronic scale; 6-water tank. The Reynolds number can be easily adjusted by the regulating valves in both systems. Flowrate was measured by a TDS-100H portable ultrasonic flowmeter and by the weight of water flowing out of the pipe within a certain time interval. Appl. Sci. 2019, 9, 1648 8 of 11 4. Results 4.1. Verification Table 2 gives the results of flowrate measurement at the Reynolds’ apparatus. The flowrate Q is measured by the ultrasonic flowmeter, while Q is measured by the weight discharged within a certain period of time. The flowrate ratio Q :Q is the measured correction factor. The relative error 2 1 ( ) defined in Equation (31), is used to evaluate the di erence between theoretical correction factor and measured data, where k indicates the theoretical correction factor. =  100% (31) Table 2. Results of flowrate measured at the Reynolds’ apparatus. Ultrasonic Flowmeter Weighing Method Flowrate No. Flowrate Flowrate Ratio  (%) DT  10 Time Mass Velocity 5 5 = Q :Q Q  10 Q  10 Re 2 1 1 2 (s) (kg) (m/s) (s) 3 3 (m /s) (m /s) 1 0.378 0.970 60 0.430 0.717 0.047 652 0.739 1.443 2 0.462 1.185 60 0.524 0.873 0.057 795 0.737 1.772 3 0.632 1.622 60 0.721 1.202 0.078 1094 0.741 1.192 4 1.002 2.572 50 0.965 1.930 0.125 1757 0.751 0.052 5 1.121 2.877 50 1.073 2.147 0.140 1954 0.746 0.498 6 1.310 3.363 40 1.005 2.512 0.163 2286 0.747 0.406 7 1.317 3.380 40 1.019 2.548 0.166 2318 0.754 0.513 8 1.553 3.987 40 1.219 3.048 0.198 2774 0.764 1.931 From Equations (15) and (18), it is clearly indicated that the theoretical correction factor for laminar flow is 0.75. It can be seen from Table 2 that the flowrate ratios ( ) are in range of 0.737 to 0.764, the average value of is 0.747. The average relative error is 0.976%, and the maximum relative error is 1.931%, which is in good agreement with the experiment result. The last line in Table 2 is in the transition zone from laminar flow to turbulent flow, and the correction factor is a little bit larger than 0.75, which is caused by the gradual changing of velocity profile from laminar flow to turbulent flow. Table 3 gives the results of flowrate measured at the performance testing system for centrifugal pumps. The Reynolds numbers are from 4311 to 422102. The theoretical turbulent flow correction factor (k ) is calculated by Equation (27). The average relative error is 0.25% and maximum error is 1.178%. Therefore, the theoretical correction factor is reliable. Table 3. Results of flowrate measurement at the performance testing system for centrifugal pumps. Ultrasonic Flowmeter Weighing Method Flowrate No. Flowrate Flowrate Ratio k  (%) t 2 DT  10 Time Mass Velocity 5 5 Q  10 Q  10 Re n = Q :Q 1 2 2 1 (s) (s) (kg) (m/s) 3 3 (m /s) (m /s) 1 7.270 0.186 60 10.314 0.172 0.085 4311 6 0.925 0.923 0.216 2 17.714 0.454 40 16.939 0.424 0.209 10619 6 0.934 0.923 1.178 3 36.956 0.948 30 26.438 0.881 0.435 22099 6 0.929 0.923 0.646 4 43.889 1.125 20 20.984 1.049 0.518 26310 7 0.932 0.933 0.107 5 76.663 1.966 15 27.566 1.838 0.907 46084 7 0.935 0.933 0.214 6 112.156 2.876 15 40.387 2.693 1.329 67518 7 0.936 0.933 0.321 7 143.424 3.677 10 34.384 3.438 1.697 86223 7 0.935 0.933 0.214 8 171.690 4.402 10 41.125 4.113 2.030 103127 7.3 0.934 0.936 0.214 9 244.767 6.276 10 58.683 5.868 2.897 147156 7.3 0.935 0.936 0.110 10 312.871 8.022 10 75.124 7.513 3.708 188385 7.3 0.937 0.936 0.107 11 341.571 8.758 8 66.056 8.257 4.076 207057 8 0.943 0.941 0.212 12 440.921 11.305 8 85.452 10.681 5.273 267854 8 0.945 0.941 0.423 13 532.958 13.664 7 90.218 12.888 6.362 323193 8.3 0.943 0.943 0.000 14 597.541 15.320 5 72.367 14.473 7.145 362942 8.3 0.945 0.943 0.212 Appl. Sci. 2019, 9, 1648 9 of 11 Table 3. Cont. Ultrasonic Flowmeter Weighing Method Flowrate No. Flowrate Flowrate Ratio k  (%) t 2 DT  10 Time Mass Velocity 5 5 Q  10 Q  10 Re n = Q :Q 1 2 2 1 (s) (s) (kg) (m/s) 3 3 (m /s) (m /s) 15 629.177 16.131 5 76.218 15.244 7.525 382256 8.5 0.945 0.944 0.106 16 662.113 16.976 5 80.136 16.027 7.912 401906 8.5 0.944 0.944 0.000 17 694.919 17.817 5 84.163 16.833 8.309 422102 8.6 0.945 0.945 0.000 4.2. Uncertainty Analysis The flowrate obtained by the weighing method is regarded as the standard to evaluate the measurement results of the ultrasonic flowmeter. Therefore, the experiment uncertainty of weighting method is analyzed and results for laminar and turbulent flow are shown in Tables 4 and 5, respectively. Collected time is controlled by an electronic timer, the uncertainty of time is 10ms. Collected mass is weighed by an electronic scale, the uncertainty of mass is 1 g. The standard relative uncertainty of time is from 0.017% to 0.200%; from 0.001% to 0.233% of mass. The combined uncertainty is from 0.019% to 0.233%. The expanded uncertainty has been reported at k = 2, which from 0.039% to 0.466%. Table 4. Experiment uncertainty for laminar flow. Collected Mass Collected Time Combined Expanded No. Estimate Uncertainty Estimate Uncertainty Uncertainty Uncertainty (M)(kg) ( )(%) (Dt)(s) ( )(%) ( )(%) (U)(%) M c Dt 1 0.43 0.233 60 0.017 0.233 0.466 2 0.524 0.191 60 0.017 0.192 0.383 3 0.721 0.139 60 0.017 0.140 0.279 4 0.965 0.104 50 0.020 0.106 0.211 5 1.073 0.093 50 0.020 0.095 0.191 6 1.005 0.100 40 0.025 0.103 0.205 7 1.019 0.098 40 0.025 0.101 0.203 8 1.219 0.082 40 0.025 0.086 0.172 Table 5. Experiment uncertainty for turbulent flow. Collected Mass Collected Time Combined Expanded No. Estimate Uncertainty Estimate Uncertainty Uncertainty Uncertainty (M)(kg) ( )(%) (Dt)(s) ( )(%) ( )(%) (U)(%) M Dt c 1 10.314 0.010 60 0.017 0.019 0.039 2 16.939 0.006 40 0.025 0.026 0.051 3 26.438 0.004 30 0.033 0.034 0.067 4 20.984 0.005 20 0.050 0.050 0.100 5 27.566 0.004 15 0.067 0.067 0.134 6 40.387 0.002 15 0.067 0.067 0.133 7 34.384 0.003 10 0.100 0.100 0.200 8 41.125 0.002 10 0.100 0.100 0.200 9 58.683 0.002 10 0.100 0.100 0.200 10 75.124 0.001 10 0.100 0.100 0.200 11 66.056 0.002 8 0.125 0.125 0.250 12 85.452 0.001 8 0.125 0.125 0.250 13 90.218 0.001 7 0.143 0.143 0.286 14 72.367 0.001 5 0.200 0.200 0.400 15 76.218 0.001 5 0.200 0.200 0.400 16 80.136 0.001 5 0.200 0.200 0.400 17 84.163 0.001 5 0.200 0.200 0.400 Appl. Sci. 2019, 9, 1648 10 of 11 5. Conclusions The e ect of velocity profiles across the pipe on the propagation time of ultrasonic wave is considered for transit-time ultrasonic flowmeters. Theoretical correction factors were proposed to improve measurement accuracy. For laminar flow, the correction factor is a constant of 0.75. For turbulent flow, the correction factor varies with Reynolds number, show as Equation (27). Flow measurement experiments were performed for laminar and turbulent flow respectively, and experiment results showed that the proposed correction factors agree well with measured correction factors. The average relative error is determined to be 0.976% for laminar flow and 0.25% for turbulent flow. Author Contributions: Conceptualization, C.G.; Data curation, H.Z.; Formal analysis, H.Z.; Funding acquisition, C.G.; Investigation, H.Z.; Methodology, C.G.; Resources, J.L.; Software, J.L.; Writing—original draft, H.Z.; Writing—review & editing, C.G. Funding: This research was funded by the Fundamental Research Funds for the Central Universities, grant number 2017XKZD02. Acknowledgments: The authors are very grateful to Dr. Wang Fengchao for his help in the experiments. Conflicts of Interest: The authors declare no conflict of interest. References 1. Ferrari, A.; Pizzo, P.; Rundo, M. Modelling and experimental studies on a proportional valve using an innovative dynamic flow-rate measurement in fluid power systems. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 2018, 232, 2404–2418. [CrossRef] 2. Sun, Y.; Zhang, T.; Zheng, D. New Analysis Scheme of Flow-Acoustic Coupling for Gas Ultrasonic Flowmeter with Vortex near the Transducer. Sensors 2018, 18, 1151. 3. Grände, P.O.; Borgström, P. An electromic di erential pressure flowmeter and a resistance meter for continuous measurement of vascular resistance. Acta Physiol. Scand. 2010, 102, 224–230. [CrossRef] [PubMed] 4. Bückle, U.; Durst, F.; Howe, B.; Melling, A. Investigation of a floating element flowmeter. Flow Meas. Instrum. 1992, 3, 215–225. [CrossRef] 5. Gibson, W.G.; Cobbold, R.S.; Johnston, K.W. Principles and design feasibility of a Doppler ultrasound intravascular volumetric flowmeter. IEEE Trans. Bio.-Med. Eng. 1994, 41, 898. [CrossRef] [PubMed] 6. Bevir, M.K. The theory of induced voltage electromagnetic flowmeter. J. Fluid Mech. 2006, 43, 577–590. [CrossRef] 7. Pasquale, G.D.; Graziani, S.; Pollicino, A.; Strazzeri, S. A vortex-shedding flowmeter based on IPMCs. Smart Mater. Struct. 2016, 25, 015011. [CrossRef] 8. Lynnworth, L.C.; Liu, Y. Ultrasonic flowmeters: Half-century progress report, 1955–2005. Ultrasonics 2006, 44, E1371–E1378. [CrossRef] 9. Chhantyal, K.; Jondahl, M.H.; Viumdal, H.; Mylvaganam, S. Upstream Ultrasonic Level Based Soft Sensing of Volumetric Flow of Non-Newtonian Fluids in Open Venturi Channels. IEEE Sens. J. 2018, 18, 5002–5013. [CrossRef] 10. Jiang, Y.D.; Wang, B.L.; Li, X.; Liu, D.D.; Wang, Y.Q.; Huang, Z.Y. A Model-Based Hybrid Ultrasonic Gas Flowmeter. IEEE Sens. J. 2018, 18, 4443–4452. [CrossRef] 11. Liu, B.; Xu, K.J.; Mu, L.B.; Tian, L. Echo energy integral based signal processing method for ultrasonic gas flow meter. Sens. Actuat. A-Phys. 2018, 277, 181–189. [CrossRef] 12. Raine, A.B.; Aslam, N.; Underwood, C.P.; Danaher, S. Development of an Ultrasonic Airflow Measurement Device for Ducted Air. Sensors 2015, 15, 10705–10722. [CrossRef] [PubMed] 13. Iooss, B.; Lhuillier, C.; Jeanneau, H. Numerical simulation of transit-time ultrasonic flowmeters: Uncertainties due to flow profile and fluid turbulence. Ultrasonics 2002, 40, 1009–1015. [CrossRef] 14. Mandard, E.; Kouame, D.; Battault, R.; Remenieras, J.P.; Patat, F. Transit time ultrasonic flowmeter: Velocity profile estimation. Ultrason 2005, 2, 763–766. 15. Sanderson, M.L.; Yeung, H. Guidelines for the use of ultrasonic non-invasive metering techniques. Flow Meas. Instrum. 2002, 13, 125–142. [CrossRef] Appl. Sci. 2019, 9, 1648 11 of 11 16. Jaiswal, S.K.; Yadav, S.; Agarwal, R. Multiple Weighing Based Method for Realizing Flow. Mapan 2015, 30, 119–123. [CrossRef] 17. Willatzen, M. Perturbation theory applied to sound propagation in flowing media confined by a cylindrical waveguide. J. Acoust. Soc. Am. 2001, 109, 102–107. [CrossRef] [PubMed] 18. Willatzen, M.; Kamath, H. Nonlinearities in ultrasonic flow measurement. Flow Meas. Instrum. 2008, 19, 79–84. [CrossRef] 19. Willatzen, M. Ultrasonic flowmeters: Temperature gradients and transducer geometry e ects. Ultrasonics 2003, 41, 105–114. [CrossRef] 20. Dadashnialehi, A.; Moshiri, B. Online monitoring of transit-time ultrasonic flowmeters based on fusion of optical observation. Measurement 2011, 44, 1028–1037. [CrossRef] 21. Mahadeva, D.V.; Baker, R.C.; Woodhouse, J. Further Studies of the Accuracy of Clamp-on Transit-Time Ultrasonic Flowmeters for Liquids. IEEE Trans. Instrum. Meas. 2009, 58, 1602–1609. [CrossRef] 22. Mahadeva, D.V.; Baker, R.C.; Woodhouse, J. Studies of the Accuracy of Clamp-on Transit Time Ultrasonic Flowmeters. In Proceedings of the 2008 IEEE Instrumentation and Measurement Technology Conference, Victoria, Canada, 12–15 May 2008; pp. 969–973. 23. Rajita, G.; Mandal, N. Review on transit time ultrasonic flowmeter. In Proceedings of the International Conference on Control, Kolkata, India, 28–30 January 2016. 24. Wang, X.F.; Tang, Z.A. Note: Ultrasonic gas flowmeter based on optimized time-of-flight algorithms. Rev. Sci. Instrum. 2011, 82, 1371–1584. [CrossRef] [PubMed] 25. Liu, Z.G.; Du, G.S.; Shao, Z.F.; He, Q.R.; Zhou, C.L. Measurement of transitional flow in pipes using ultrasonic flowmeters. Fluid Dyn. Res. 2014, 46, 055501. 26. Xiong, Y.T.; Su, Z.D.; Lin, J.D.; Zhang, Y.C. Calibration of large diameter gas flowmeter by the corresponding velocity distribution equation. J. China Univ. Metrol. 2011, 22, 25–29. © 2019 by the authors. Licensee MDPI, Basel, Switzerland. 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Published: Apr 20, 2019

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