Numerical Simulating Nonlinear Effects of Ultrasonic Propagation on High-speed Ultrasonic Gas Flow Measurement

Transcription

1 Appl. Math. Inf. Si. 7, No. 5, (2013) 1963 Applied Mathematis & Information Sienes An International Journal Numerial Simulating Nonlinear Effets of Ultrasoni Propagation on High-speed Ultrasoni Gas Flow Measurement i Yue-zhong 1,2,, Wu Jiang-tao 1 and Hu Kai-ming 2 1 Shool of Energy and Power Engineering, Xi an Jiaotong University, Xi an , China 2 Shool of Mehanial and Eletroni Engineering, East China Institute of Tehnology, Nanhang , China Reeived: 28 Jan. 2013, Revised: 29 May. 2013, Aepted: 30 May Published online: 1 Sep Abstrat: The effets of nonlinear ultrasoni propagation on high-speed ultrasoni gas flow measurement are analyzed based on the sound line equation derived from Snell s geometri aousti law. A mathematial model for the ultrasoni propagation path in ultrasoni flowmeter pipe is built, and the relationship between x and r is alulated by MATAB programming and ode45 simulating. The ultrasoni propagation path at the flow rate of 3 30m/s is simulated in the assumed boundary onditions of the pipeline, transduer installation and fluid state. It shows that the nonlinear propagation harateristis ause the large deviations of the position of reeived ultrasoni waves in onditions of different flow rates, whih strongly affets the stability and auray of flow measurement. The simulated offset data of the reeiving position are useful for the high-speed gas ultrasoni flowmeter installation and dry alibration on lamp-on ultrasoni flowmeters. Keywords: Numerial Simulating, Ultrasoni Propagation, Ultrasoni Gas 1. Introdution As a non-intrusive flow measurement devie, the ultrasoni flowmeter has a unique advantage in flammable and explosive gas flow measurement. In the large-diameter pipe natural gas transmission industry, the pipeline flow rate in medium-pressure gas transmission is mostly restrited to 10 m/s or so. In order to improve gas transport effiieny, the natural gas transmission speed is inreased to 15m/s 20m/s under 0.2MPa 0.4MPa, the steam transmission speed is mostly ontrolled in the range of 30m/s 40m/s, and the superheated steam flow rate is more than 60m/s. Hydrogenation in the hydrogen stations generally uses small diameter pipe(15mm 18mm) at the pressure of 30MPa or more, and the hydrogen flow rate is more than 65m/s. Reently, the hydrogenation pressure is inreased to 70MPa or 75MPa, and the alulated hydrogen flow rate is 120m/s or more. However, due to the effets of the flow rate on nonlinear ultrasoni propagation, most of the ultrasoni gas flowmeters measurement range is less than 30m/s. This paper mainly onsiders the appliation of the ultrasoni flowmeter for high-speed gas flow measurement. At first, the ultrasoni propagation harateristis in the high-speed gas medium are analyzed, then, the effets of the flow rate on nonlinear ultrasoni propagation are simulated to provide the theoretial and methodologial bases to develop the high-speed gas ultrasoni flowmeter. Some researhers have studied the nonlinear harateristis of ultrasoni propagation. Vanhille et al. [1] simulated the nonlinear ultrasound propagation properties in the fluid with the agrangian equation, and they mainly studied the nonlinear relationship between the sound pressure and the propagation distane. Willatzen et al. [2, 3] studied the non-linear relationship between the propagation time and the flow rate based on the fluid dynami equations and the basi formula for measuring the flow rate. Q. Ji et al. [4 9] studied the ultrasoni propagation harateristis and model in aluminum, olloids and other materials. Franeso anza di Salea et al. [10] observed and measured the ultrasoni propagation harateristis through establishing the physial experimental apparatus and mainly studied the relationship between the re- Corresponding author li-yuezhong@sohu.om

2 1964. Yue-zhong et al: Numerial Simulating Nonlinear Effets of... eived eho signal and the propagation harateristis. The Amerian Gas Assoiation (AGA) and the China s Administration of Quality Supervision, Inspetion and Quarantine (AQSIQ) [11, 12] have established suessively the relevant standards of gas and natural gas flow measurement for ultrasoni flowmeters, in whih the ultrasound propagates in the pipeline from the emitting transduer to the reeiving by the S-shaped path, and the ultrasoni propagation equation determined by Snell s law is presented roughly. However, a few papers have been reported about the relationship between the nonlinear ultrasoni propagation path and the flow rate. So, this work mainly deals with the effets of flow rates on ultrasoni propagation path based on the ultrasoni propagation equation determined by Snell s law and the wave propagation trajetory equation derived by Boone and Vermaas. 2. Ultrasoni propagation model in the ultrasoni flowmeter pipe The struture of the typial lamp-on ultrasoni flowmeter investigated in this work is shown in Fig.1 [13], and it onsists of lamp holder, upstream and downstream transduers, preamplifier, and ultrasoni flowmeter. It is the typial struture of the small diameter lamp-on ultrasoni flowmeters in Amerian GE, German Flexim, Japan s Fuji Corp et. Aording to GB/TI and AGA Report No.9, the average axial flow veloity formula for transittime single path ultrasoni flowmeter is: V = K C 2osϕ 0 t U t D t D t U, (1) The related parameters are shown in Fig.2, where, is the straight line length between the ultrasoni sensors A and B, and φ 0 is the initial transmission angle for the ultrasoni path. In equation (1), t D is the ultrasoni downstream propagation time, t U is the ultrasoni upstream propagation time, and K is the veloity distribution oeffiient. As the flow rate along the pipe ross-setion is not onstant, the ultrasoni wave propagation path in the pipeline is not linear. Aording to veloity distribution funtion V(r) and Snell s geometrial aousti law, the ultrasoni transmission line equation is determined as C +V(r)= Const, (2) osϕ(r) Where, C is the ultrasoni propagation veloity in air. Aording to Boone and Vergas s model, the aousti line equation is written as: dx = C osϕ(r)+v(r) dt dr = C sinϕ(r), (3) dt dϕ(r) dt = os 2 ϕ(r) dv(r) dr Where, x is the ultrasoni propagation horizontal displaement from the transmitting sensor as a starting point, r is the ultrasoni propagation radial displaement, and φ(r) is the angle between the partile motion tangent and the axis in position r of ultrasoni propagation path. Aording to the empirial formula of Prandtl veloity distribution, and onsidering the fully developed turbulent flow, the veloity distribution is: V(r)= V m (1 r R ) 1 n, (4) Where, V m is the surfae veloity along the axial line, and R is the radius of the pipeline. Considering only the smooth pipe, the index n is: n=2log 10 ( R e ) 0.8, (5) n Where, Re is Reynolds number of fluid in the pipelinewhih is defined as R e = V ν D, and ν is the motion visosity. The mean flow veloity V z on the vertial enter parallel pipeline setion and the pipeline mean flow veloity V both are defined respetively as V Z = 1 V(r)d, (6) V = 1 V(r)dS, (7) S S Where, is the ultrasoni path length; S is the pipeline inner setional area. So the relation of V z and V is: V = 2n (2n+1) V Z, (8) Eq.(8) is ompared with Eq.(1), and there are definitions as follows K C = 2n (2n+1), (9) V Z = 2osϕ 0 t U t D t D t U, (10) From Eq.(3) and Eq.(4), aording to the downstream and upstream situations a model is established as follows: (1) Ultrasoni propagation in the downstream diretion The relationship of φ and r an be derived as: d(ϕ) = nrsinϕ v m os 2 ϕ(1, (11) d(ϕ) = nrsinϕ v m os 2 ϕ(1+ And the relationship of x and r is obtained as: sinϕ = d(x) v m (1 R r) n 1 + osϕ, (12) sinϕ = d(x) v m (1+ R r) n 1 + osϕ

3 Appl. Math. Inf. Si. 7, No. 5, (2013) / (2) Ultrasoni propagation in the upstream diretion The relationship of φ and r an be derived as: d(ϕ) = nrsinϕ v m os 2 ϕ(1+, (13) d(ϕ) = nrsinϕ v m os 2 ϕ(1 And the relationship of x and r is obtained as sinϕ = d(x) v m (1+ R r) n 1 + osϕ, (14) sinϕ = d(x) v m (1 R r) n 1 + osϕ 3. Simulation and analysis of models 3.1. Method Aording to Eq. (11) and Eq. (13), the results an be derived as follows: (1) Ultrasoni propagation in the downstream diretion: osϕ = seϕ 0 Rv m (1, (15) osϕ = seϕ 0 Rv m (1+ (2) Ultrasoni propagation in the upstream diretion: osϕ = seϕ 0 + Rv m (1+, (16) osϕ = seϕ 0 + Rv m (1 When Eq. (15) is put into (12), and (16) into (14), a omplex equation an be derived, and it is diffiult to solve the integral diretly. With the numerial solution method of ordinary differential equation MATAB ode45, the relationship of x and r an be solved. Using the Runge-Kutta algorithm, ode45 is a variable step size solver. Ode45 has fourth-order and fifth-order Runge-Kutta single-step algorithm. Its trunation error is ( x) 3. It an be mainly used to solve Nonstiff (non-rigid) of ordinary differential equations, and it is the preferred method to solve the problem of numerial solution Disussion The measurement pipeline ondition is assumed as R = 100mm and = 340m/s, and the initial ultrasound launh angle is φ 0 = 30. Figure 1 lamp-on ultrasoni flowmeter. If Re=4e6, n= The ultrasoni propagation trajetory under V ={12,15,18,24,27,30}(m/s) is alulated and depited by alulating and simulating with MATAB and ode45. Fig.3 and Fig.4 show respetively ultrasound downstream and upstream transmission paths at higher flow rates. In order to show the level migration of ultrasoni wave reahing the end of the pipeline wall at the different flow rates more learly, Fig. 3 and 4 are respetively enlarged partly as shown in Fig. 5 and 6. In Fig.3 Fig.6, the solid line is the ultrasoni propagation path at zero fluid veloity. The others are the ultrasoni propagation paths from the left to the right at V = {12,15,18,24,27,30}(m/s). As an be seen from Fig. 3 and Fig. 5, the deviation from the zero veloity of ultrasoni downstream propagation paths is larger with the higher flow rate. As shown in Fig. 4 and Fig. 6, the deviation from the zero veloity of ultrasoni upstream propagation paths is larger too with the higher flow rate. However, as presented in Fig. 5 and Fig. 6, the downstream offset is larger learly than the upstream one, and is about 2 times that of the upstream. In order to ompare the offsets of ultrasoni propagation path to reah the end of the pipeline wall under the lower flow rates, Fig. 7 and Fig. 8,show propagation paths of the ultrasoni wave respetively alulated by MAT- AB and ode45 at V = 3 12m/s in downstream and upstream. Fig. 3 8 shows that the maximum offsets of the pipeline end of ultrasoni propagation path under the high speed fluid are 2 times or more than under the zero speed fluid. Through alulating by MATAB programming, when V = 30m/s, the displaement of ultrasoni downstream propagation in x diretion is obtained, i.e. X 30 = 0.363m. When V = 0m/s, the displaement is obtained, i.e. X 0 = 0.346m and then the offset of ultrasoni downstream propagation is x=x 30 X 0 = 0.015m. The same method an be used to alulate the offset of ultrasoni upstream propagation, that is, x U = 0.007m. By this way, the above onlusion is verified.

4 1966. Yue-zhong et al: Numerial Simulating Nonlinear Effets of... Figure 2 ultrasoni propagation in the pipeline. Figure 5 the enlargement diagram of ultrasoni downstream propagation path in High-veloity fluids. Figure 3 ultrasoni downstream propagation path in Highveloity fluids. Figure 6 the enlargement diagram of ultrasoni upstream propagation path in High-veloity fluids. Figure 4 ultrasoni upstream propagation path in High-veloity fluids. Figure 7 the enlargement diagram of ultrasoni downstream propagation path in ow-veloity fluids.

5 Appl. Math. Inf. Si. 7, No. 5, (2013) / Figure 8 the enlargement diagram of ultrasoni upstream propagation path in ow-veloity fluids. 4. Conlusion Theoretially, with analysis on nonlinear harater of the ultrasoni propagation in gas flow measurement, we have drawn the onlusion that the nonlinear ultrasoni propagation auses a deviation of ultrasoni wave reahing the end of the pipeline wall from the reeiving ultrasoni sensor. Sine the position of reeived ultrasoni wave hanges at the flow rate, the position will deviate from the enter of the reeiving ultrasoni sensors effetive area (usually with the diameter of 7 12mm), espeially at the higher flow rate. The propagation time also hanges with the flow rate, whih affets the auray of the ultrasoni flowmeter. Aording to the above simulation results, the maximum offset is 15mm at the flow rate V=30m/s, and will be even greater for the higher speed fluid. So, the large-diameter insert-type ultrasoni gas flowmeter an be usually used at the oasions of the relatively steady flow field and flow veloity. And for the lamp-on ultrasoni gas flowmeter, two movable ultrasoni sensors an be designed to improve the flow measurement stability and auray. The above simulating results are very useful for the high-speed ultrasoni gas flowmeter installation and the dry alibration on lamp-on ultrasoni flowmeters. Referenes [1] C. Vanhille, C. Campos-Pozuelo. A numerial formulation for nonlinear ultrasoni waves propagation in fluids. Ultrasonis 2004, 42, (2004) [2] M. Willatzen, H. Kamath. Nonlinearities in ultrasoni flow measurement. Flow Measurement and Instrumentation, 19, (2008). [3] H. M. Habib, E. R. El-Zahar. A New Algorithm for Solving Nonlinear Stationary Shok Problems with Mehanization Applied Mathematis & Information Sienes, 1, (2010). [4] Q. Ji,. H. e,. J. Filipow, et. Ultrasoni wave propagation in water-saturated aluminum foams. Ultrasonis, 36, (1998). [5] J. S. Tebbutt, R. E. Challis. Ultrasoni wave propagation in olloidal suspensions and emulsions: a omparison of four models. Ultrasonis, 34, (1996). [6] B. C. ee, M. Palaz, M. Krawzuk, et.wave propagation in a sensor/atuator diffusion bond model. JOURNA OF SOUND AND VIBRATION, 276, (2004). [7] P. P. Delsanto, Sigrun Hirsekorn, V. Agostini, et. Modeling the propagation of ultrasoni waves in the interfae region between two bonded elements. Ultrasonis, 40, (2002). [8] Zhang Rongxin, Qin Guoliang, Xu ina. High-auray numerial simulation of aousti propagation problem in pipe, Journal of Vibration and Shok, 29, (2010). [9] Franeso anza di Salea, Robert E. Green Jr. Experimental observation of the intrusive effet of a ontat transduer on ultrasound propagation. Ultrasonis, 37, (1997). [10] Yuto Inoue, Hiroshige Kikura, Hideki Murakawa et. A study of ultrasoni propagation for ultrasoni flow rate measurement. Flow Measurement and Instrumentation, 19, (2008). [11] Amerian Gas Assoiation.A.G.A.Report No.9: Measurement of Gas by Multipath ultrasoni Meters, (1998). [12] Chinese General Administration of Quality Supervision, Inspetion and Quarantine. Ultrasoni gas flowmeter measuring gas flow (GB/TI ). Beijing: Standards Press of China, (2001). [13] GE Sensing & Inspetion Tehnologies. Ultrasoni flowmeter. i Yuezhong reeived the PhD. in Detetion tehnology and automation equipment from Huazhong University of Siene and Tehnology in 2010, and he is as a postdotoral researh fellow in Xi an Jiaotong University. He is urrently an assoiate professor in East China Institute of Tehnology. His researh interest is in the areas of ultrasoni flow measurement, and intelligent instrumentation. Wu Jiangtao is a professor and dotoral tutor in Xi an Jiaotong University. He is a Fellow of IUPAC (International Union of Pure and Applied Chemistry)and assoiate Editor of Journal of Chemial and Engineering Dataand a member of Advisory Board of the Journal of Chemial Thermodynamis. He does researh on thermophysial properties Moleular simulation of fluidsdatabase of thermophysial properties based on the Web and alternative refrigerants and lean fuels.