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

/lp/multidisciplinary-digital-publishing-institute/effects-of-velocity-profiles-on-measuring-accuracy-of-transit-time-IfG7blGhlN

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- Multidisciplinary Digital Publishing Institute
- Copyright
- © 1996-2019 MDPI (Basel, Switzerland) unless otherwise stated
- ISSN
- 2076-3417
- DOI
- 10.3390/app9081648
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- See Article on Publisher Site

applied sciences Article Eects 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 eects 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 dierential 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 dierence 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 dierences [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 eect 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 eects 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 eects 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 dierence 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 eects 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 eect of velocity proﬁle on the ﬂowrate measured. If the eect 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 eect 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 eect 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 eect 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 dierence 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 eect 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. 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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/).

Applied Sciences – Multidisciplinary Digital Publishing Institute

**Published: ** Apr 20, 2019

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