Appl. Math. Inf. Sci.7, No. 5, 1963-1967 (2013) 1963

Applied Mathematics & Information SciencesAn International Journal

http://dx.doi.org/10.12785/amis/070535

Numerical Simulating Nonlinear Effects of UltrasonicPropagation on High-speed Ultrasonic Gas FlowMeasurementLi Yue-zhong1,2,∗, Wu Jiang-tao1 and Hu Kai-ming2

1 School of Energy and Power Engineering, Xi ’an Jiaotong University, Xi ’an 710049, China2 School of Mechanical and Electronic Engineering, East China Institute of Technology, Nanchang 330013, China

Received: 28 Jan. 2013, Revised: 29 May. 2013, Accepted: 30 May. 2013Published online: 1 Sep. 2013

Abstract: The effects of nonlinear ultrasonic propagation on high-speed ultrasonic gas flow measurement are analyzed based onthe sound line equation derived from Snell’s geometric acoustic law. A mathematical model for the ultrasonic propagation path inultrasonic flowmeter pipe is built, and the relationship between x and r is calculated by MATLAB programming and ode45 simulating.The ultrasonic propagation path at the flow rate of 3∼30m/s is simulated in the assumed boundary conditions of the pipeline, transducerinstallation and fluid state. It shows that the nonlinear propagation characteristics cause the large deviations of the position of receivedultrasonic waves in conditions of different flow rates, which strongly affects the stability and accuracy of flow measurement. Thesimulated offset data of the receiving position are useful for the high-speed gas ultrasonic flowmeter installation and dry calibration onclamp-on ultrasonic flowmeters.

Keywords: Numerical Simulating, Ultrasonic Propagation, Ultrasonic Gas

1. Introduction

As a non-intrusive flow measurement device, the ultra-sonic flowmeter has a unique advantage in flammable andexplosive gas flow measurement. In the large-diameter pipenatural gas transmission industry, the pipeline flow rate inmedium-pressure gas transmission is mostly restricted to10 m/s or so. In order to improve gas transport efficiency,the natural gas transmission speed is increased to 15m/s∼ 20m/s under 0.2MPa∼ 0.4MPa, the steam transmis-sion speed is mostly controlled in the range of 30m/s∼40m/s, and the superheated steam flow rate is more than60m/s. Hydrogenation in the hydrogen stations generallyuses small diameter pipe(15mm∼ 18mm) at the pressureof 30MPa or more, and the hydrogen flow rate is morethan 65m/s. Recently, the hydrogenation pressure is in-creased to 70MPa or 75MPa, and the calculated hydrogenflow rate is 120m/s or more. However, due to the effects ofthe flow rate on nonlinear ultrasonic propagation, most ofthe ultrasonic gas flowmeters’ measurement range is lessthan 30m/s. This paper mainly considers the applicationof the ultrasonic flowmeter for high-speed gas flow mea-

surement. At first, the ultrasonic propagation characteris-tics in the high-speed gas medium are analyzed, then, theeffects of the flow rate on nonlinear ultrasonic propagationare simulated to provide the theoretical and methodologi-cal bases to develop the high-speed gas ultrasonic flowme-ter.

Some researchers have studied the nonlinear charac-teristics of ultrasonic propagation. Vanhille et al. [1] sim-ulated the nonlinear ultrasound propagation properties inthe fluid with the Lagrangian equation, and they mainlystudied the nonlinear relationship between the sound pres-sure and the propagation distance. Willatzen et al. [2, 3]studied the non-linear relationship between the propaga-tion time and the flow rate based on the fluid dynamicequations and the basic formula for measuring the flowrate. Q. Ji et al. [4–9] studied the ultrasonic propagationcharacteristics and model in aluminum, colloids and othermaterials. Francesco Lanza di Scalea et al. [10] observedand measured the ultrasonic propagation characteristics

through establishing the physical experimental appa-ratus and mainly studied the relationship between the re-

∗ Corresponding author e-mail:li-yuezhong@sohu.com

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1964 L. Yue-zhong et al: Numerical Simulating Nonlinear Effects of...

ceived echo signal and the propagation characteristics. TheAmerican Gas Association (AGA) and the China’s Admin-istration of Quality Supervision, Inspection and Quaran-tine (AQSIQ) [11, 12] have established successively therelevant standards of gas and natural gas flow measure-ment for ultrasonic flowmeters, in which the ultrasoundpropagates in the pipeline from the emitting transducerto the receiving by the S-shaped path, and the ultrasonicpropagation equation determined by Snell’s law is presentedroughly. However, a few papers have been reported aboutthe relationship between the nonlinear ultrasonic propaga-tion path and the flow rate. So, this work mainly deals withthe effects of flow rates on ultrasonic propagation pathbased on the ultrasonic propagation equation determinedby Snell’s law and the wave propagation trajectory equa-tion derived by Boone and Vermaas.

2. Ultrasonic propagation model in theultrasonic flowmeter pipe

The structure of the typical clamp-on ultrasonic flowmeterinvestigated in this work is shown in Fig.1 [13], and it con-sists of clamp holder, upstream and downstream transduc-ers, preamplifier, and ultrasonic flowmeter. It is the typ-ical structure of the small diameter clamp-on ultrasonicflowmeters in American GE, German Flexim, Japan’s FujiCorp etc. According to GB/TI8604-2001 and AGA ReportNo.9, the average axial flow velocity formula for transit-time single path ultrasonic flowmeter is:

V = KC •L

2cosϕ0•

tU − tDtDtU

, (1)

The related parameters are shown in Fig.2, where,Lis the straight line length between the ultrasonic sensorsAandB, andφ0 is the initial transmission angle for the ultra-sonic path. In equation (1),tD is the ultrasonic downstreampropagation time,tU is the ultrasonic upstream propaga-tion time, andKc is the velocity distribution coefficient.

As the flow rate along the pipe cross-section is not con-stant, the ultrasonic wave propagation path in the pipelineis not linear. According to velocity distribution functionV (r) and Snell’s geometrical acoustic law, the ultrasonictransmission line equation is determined as

Ccosϕ(r)

+V (r) =Const, (2)

Where,C is the ultrasonic propagation velocity in air.According to Boone and Vergas’s model, the acoustic

line equation is written as:

dxdt

=C×cosϕ(r)+V (r)

drdt

=C×sinϕ(r)

dϕ(r)dt

=−cos2ϕ(r)dV (r)

dr

, (3)

Where,x is the ultrasonic propagation horizontal displace-ment from the transmitting sensor as a starting point,r isthe ultrasonic propagation radial displacement, andφ(r) isthe angle between the particle motion tangent and the axisin positionr of ultrasonic propagation path.

According to the empirical formula of Prandtl velocitydistribution, and considering the fully developed turbulentflow, the velocity distribution is:

V (r) =Vm(1−|r|R)

1n , (4)

Where,Vm is the surface velocity along the axial line, andR is the radius of the pipeline. Considering only the smoothpipe, the indexn is:

n = 2log10(Re

n)−0.8, (5)

Where,Re is Reynolds number of fluid in the pipelinewhichis defined asRe =

V Dν , andν is the motion viscosity.

The mean flow velocityVz on the vertical center paral-lel pipeline section and the pipeline mean flow velocityVboth are defined respectively as

VZ =1L

∫

LV (r)dL, (6)

V =1S

∫∫

SV (r)dS, (7)

Where,L is the ultrasonic path length;S is the pipelineinner sectional area. So the relation ofVz andV is:

V =2n

(2n+1)VZ , (8)

Eq.(8) is compared with Eq.(1), and there are defini-tions as follows

KC =2n

(2n+1), (9)

VZ =L

2cosϕ0•

tU − tDtDtU

, (10)

From Eq.(3) and Eq.(4), according to the downstreamand upstream situations a model is established as follows:

(1) Ultrasonic propagation in the downstream directionThe relationship ofφ andr can be derived as:

d (r)d (ϕ)

=−nRcsinϕ

vmcos2ϕ(1− rR )

1n−1

06 r 6 R

d (r)d (ϕ)

=−nRcsinϕ

vmcos2ϕ(1+ rR )

1n−1

−R 6 r 6 0, (11)

And the relationship ofx andr is obtained as:

d (r) =−csinϕ

vm(1− rR )

1n + ccosϕ

d (x) 06 r 6 R

d (r) =csinϕ

vm(1+ rR )

1n + ccosϕ

d (x) −R 6 r 6 0, (12)

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Appl. Math. Inf. Sci.7, No. 5, 1963-1967 (2013) /www.naturalspublishing.com/Journals.asp 1965

(2) Ultrasonic propagation in the upstream directionThe relationship ofφ andr can be derived as:

d (r)d (ϕ)

=nRcsinϕ

vmcos2ϕ(1+ rR )

1n−1

−R 6 r 6 0

d (r)d (ϕ)

=nRcsinϕ

vmcos2ϕ(1− rR )

1n−1

06 r 6 R, (13)

And the relationship ofx andr is obtained as

d (r) =−csinϕ

vm(1+ rR )

1n + ccosϕ

d (x) −R 6 r 6 0

d (r) =csinϕ

vm(1− rR )

1n + ccosϕ

d (x) 06 r 6 R, (14)

3. Simulation and analysis of models

3.1. Method

According to Eq. (11) and Eq. (13), the results can be de-rived as follows:

(1) Ultrasonic propagation in the downstream direc-tion:

cosϕ =c

csecϕ0−Rvm(1− rR )

1n

06 r 6 R

cosϕ =c

csecϕ0−Rvm(1+ rR )

1n

−R 6 r 6 0, (15)

(2) Ultrasonic propagation in the upstream direction:

cosϕ =c

csecϕ0+Rvm(1+ rR )

1n

−R 6 r 6 0

cosϕ =c

csecϕ0+Rvm(1− rR )

1n

06 r 6 R, (16)

When Eq. (15) is put into (12), and (16) into (14), acomplex equation can be derived, and it is difficult to solvethe integral directly. With the numerical solution methodof ordinary differential equation MATLAB ode45, the re-lationship ofx andr can be solved. Using the Runge-Kuttaalgorithm, ode45 is a variable step size solver. Ode45 hasfourth-order and fifth-order Runge-Kutta single-step algo-rithm. Its truncation error is (∆x) 3. It can be mainly usedto solve Nonstiff (non-rigid) of ordinary differential equa-tions, and it is the preferred method to solve the problemof numerical solution.

3.2. Discussion

The measurement pipeline condition is assumed asR =100mm andc = 340m/s, and the initial ultrasound launchangle isφ0 = 30◦ .

Figure 1 clamp-on ultrasonic flowmeter.

If Re = 4e6, n = 10.3724.The ultrasonic propagation trajectory underV = {12,15,18,24,27,30}(m/s)is calculated and depicted by calculating and simulat-

ing with MATLAB and ode45.Fig.3 and Fig.4 show respectively ultrasound down-

stream and upstream transmission paths at higher flow rates.In order to show the level migration of ultrasonic wavereaching the end of the pipeline wall at the different flowrates more clearly, Fig. 3 and 4 are respectively enlargedpartly as shown in Fig. 5 and 6.

In Fig.3∼ Fig.6, the solid line is the ultrasonic propa-gation path at zero fluid velocity. The others are the ultra-sonic propagation paths from the left to the right atV ={12,15,18,24,27,30}(m/s) . As can be seen from Fig.3 and Fig. 5, the deviation from the zero velocity of ul-trasonic downstream propagation paths is larger with thehigher flow rate. As shown in Fig. 4 and Fig. 6, the devia-tion from the zero velocity of ultrasonic upstream propaga-tion paths is larger too with the higher flow rate. However,as presented in Fig. 5 and Fig. 6, the downstream offset islarger clearly than the upstream one, and is about 2 timesthat of the upstream.

In order to compare the offsets of ultrasonic propaga-tion path to reach the end of the pipeline wall under thelower flow rates, Fig. 7 and Fig. 8,show propagation pathsof the ultrasonic wave respectively calculated by MAT-LAB and ode45 atV = 3 ∼ 12m/s in downstream andupstream. Fig. 3∼ 8 shows that the maximum offsets ofthe pipeline end of ultrasonic propagation path under thehigh speed fluid are 2 times or more than under the zerospeed fluid. Through calculating by MATLAB program-ming, whenV = 30m/s , the displacement of ultrasonicdownstream propagation inx direction is obtained, i.e.X30=0.363m. WhenV = 0m/s , the displacement is obtained,i.e. X0 = 0.346m and then the offset of ultrasonic down-stream propagation is∆x = X30−X0 = 0.015m . The samemethod can be used to calculate the offset of ultrasonic up-stream propagation, that is,∆xU = 0.007m . By this way,the above conclusion is verified.

c© 2013 NSPNatural Sciences Publishing Cor.

1966 L. Yue-zhong et al: Numerical Simulating Nonlinear Effects of...

Figure 2 ultrasonic propagation in the pipeline.

Figure 3 ultrasonic downstream propagation path in High-velocity fluids.

Figure 4 ultrasonic upstream propagation path in High-velocityfluids.

Figure 5 the enlargement diagram of ultrasonic downstreampropagation path in High-velocity fluids.

Figure 6 the enlargement diagram of ultrasonic upstream prop-agation path in High-velocity fluids.

Figure 7 the enlargement diagram of ultrasonic downstreampropagation path in Low-velocity fluids.

c© 2013 NSPNatural Sciences Publishing Cor.

Appl. Math. Inf. Sci.7, No. 5, 1963-1967 (2013) /www.naturalspublishing.com/Journals.asp 1967

Figure 8 the enlargement diagram of ultrasonic upstream prop-agation path in Low-velocity fluids.

4. Conclusion

Theoretically, with analysis on nonlinear character of theultrasonic propagation in gas flow measurement, we havedrawn the conclusion that the nonlinear ultrasonic propa-gation causes a deviation of ultrasonic wave reaching theend of the pipeline wall from the receiving ultrasonic sen-sor. Since the position of received ultrasonic wave changesat the flow rate, the position will deviate from the cen-ter of the receiving ultrasonic sensors’ effective area (usu-ally with the diameter of 7∼ 12mm), especially at thehigher flow rate. The propagation time also changes withthe flow rate, which affects the accuracy of the ultrasonicflowmeter. According to the above simulation results, themaximum offset is 15mm at the flow rate V=30m/s, andwill be even greater for the higher speed fluid. So, thelarge-diameter insert-type ultrasonic gas flowmeter can beusually used at the occasions of the relatively steady flowfield and flow velocity. And for the clamp-on ultrasonicgas flowmeter, two movable ultrasonic sensors can be de-signed to improve the flow measurement stability and ac-curacy. The above simulating results are very useful forthe high-speed ultrasonic gas flowmeter installation andthe dry calibration on clamp-on ultrasonic flowmeters.

References

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Li Yuezhongreceived the PhD. in Detectiontechnology and automationequipment from Huazhong Uni-versity of Science and Tech-nology in 2010, and he is asa postdoctoral research fellowin Xi ’an Jiaotong University.He is currently an associate pro-fessor in East China Institute

of Technology. His research interest is in the areas of ultra-sonic flow measurement, and intelligent instrumentation.

Wu Jiangtaois a professor and doctoral tu-tor in Xi ’an Jiaotong Univer-sity. He is a Fellow of IUPAC(International Union of Pure andApplied Chemistry)and asso-ciate Editor of Journal of Chem-ical and Engineering Dataanda member of Advisory Boardof the Journal of Chemical Ther-modynamics. He does research

on thermophysical properties Molecular simulation of flu-idsdatabase of thermophysical properties based on the Weband alternative refrigerants and clean fuels.

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