1、31 6 2010 M6N NVChineseJournalofScientificInstrumentVol.31No.6Jun.2010l :2009-10 ReceivedDate:2009-10 “:SE8639(2008AA042207) “ 2 2 9sZE# s胡鹤鸣,王 池, 孟 涛(S9 S 100013)K 1: 2 9 T 9, 2 2 91 ? , “d1il。s 2 9 , 2Gauss 2 Z,w 1 2 “ 。GausssZ i, L= 2 EM T Vr。GausssZ ( ide F (,7 4 Ws 2 “ , s A F (,O 2 ,s 。1oM: 2
2、9; ; s;ms |:TK313 DS M :A SES Ss :470.2010Integrationmethodofmulti-chordultrasonicflowmeteranditsaccuracyanalysisHuHeming, WangChi, MengTao(NationalInstituteofMetrology, Beijing100013, China)Abstract:Ultrasonicflowmeterisavelocity-typeflowmeter.Themathematicmodelitadoptsisveryimportanttoitsmeasureme
3、ntaccuracy.Integrationtheoryoftheflowmeterisintroducedinthispaper;twosensorarangementschemesareanalyzedandthepairingsensorpositionsandtheirweightcoeficientsarecalculated.Thecauseoftheintegrationerroristhattheactualdistributionofthemeasuredvelocitycannotbeexpressedbyalgebraicpolynomialswithcorrespond
4、ingorder.Gaussintegrationschemedoesnotjustsimplyuseareaweightingaverageastheaverageflowrate, butincreasestheweightsofthemiddleacousticlines.Itsintegrationaccuracyisobviouslybeterthantheareaweightingmethod;andthemorethenumberofacousticlinesis, thebetertheaccuracyis.Keywords:ultrasonicflowmeter;mathem
5、aticmodel;Gaussintegration;truncationerror1 “ ? oC,B 1“S9 3v S ?Z, , w g、 2 。t “cBt g v5, _F5,? 9E a。 M 7? 2 2 9, z % v g / 45, 9/ s5 gK, 2 Va s,# 2 9Xv g KD/ 41。 2 9 T 9,YV 2o8 I .l HW 9 2 L ( ,iYV H 2 F (9 ( 。 2 91 ?K ( , “d1il。A , 2 , s , 2 9Fvv9F 9, 4 2 、 “ Es1。 2 9 VV4 HW#+ 6 : 2 2 9sZE# s1219
6、4 9 +Y “0/ #|) / f ?Z, 2 2-3,sv4 。4 2 、8 2 2 9,s 9 1 ; 2 f ,s9 9 1Yy 。 P 9 -5 1 p, s9G i;7 KE H,s i。2 s 2 9 2o 8.l HWis+, B ? ( m1 U), 2o I .l HWtd,i、tu,i,M 2 (_ 4(e 2 ):vi= Li2cos i1tu,i-1td,i (1)T:Li 2 , i 2 。 2 9, ( 2 i+1“, s s Y。 4 9 , H 2 , 2 VV M H = ( , m2 U,iG H “ i,F pZE9 ,Q=RNi=1i vili (2)
7、T:R5, li=Lisin i。m1 2 Fig.1 MeasurementoflineaveragevelocityT(2)F p9 ZE L= s ,YVK 2 “9 l(z) v(z), / uW-R,R s:Q=R-Rl(z) v(z)dz=R1-1l(tR) v(tR)dtRNi=1il(tiR) v(tiR) (3)m2 UimFig.2 SketchoftheflowintegrationT:z=tR 2 , tM 2 , l(z) 2 z。 T(3)| sM -1, 1 uW ,ZL H9 。 2 zl(z)=2 R2 -z2 =2R 1 -t2 ,5 :Q=2R21-1(t
8、) v(tR)dt2R2Ni=1i v(tiR)(ti)(4)T:(t)= 1 -t2 。M0 T、 Ts1 p“% . 7,GaussZE5 “B、14K/sKBZE5 。5 2 9BGauss-JaccobisE 2 KtiM “ i,IEC416#PTC187?X 2 N H 2 “ ,BGN#“ 2hi9 。3 2 “ w 3.1 Gauss-JaccobiZ Gausss ,M 2 ti (t) TPN(t)。 5, T(4) V A f (t)= 1 -t2 ,zJaccobi T f (1+t)(1 -t)+ yf , = =0.5,#Jaccobi T V T(5)w8:Pj+
9、1 =Pj(2j+2+1)x-Pj-1(j+)(j+1)(j+1)(j+2+1) (5) SP-1 =0,P0 =1。B V9 PNNti,M 2 。7 5 , f(t)=1,5Legendre T5。N5RGaus-JaccobiZ、Z5RGauss-LegendreZ 。1220 N N V 3 1 “ i VYV/ s:i= 1(ti)1-1(t)Nk=0,kit-tkti-tkdt (6)Gammaf (x)=0e-ttx-1dt, T(6) V 9 T:i= 1(ti)2(+N)(+N)22+1(N+1)(2+N+1)PN-1(ti)PN(ti) (7)T:PN-1(ti)N-1Ja
10、ccobi T, PN(ti)NJaccobi T 。i , “ Gauss-Jaccobis “ Y, L= 2 z V ? s In,|(ti)V “ 1“ ,7s H L(ti)=Lisin i/2 9 , T(4) U。V1 4 2 #9 2 HM 2 tiM “ i。Gaus-JaccobiZ=0.5,V wM 2 “ e9 T9 :ti=cosiN+1, i=1, 2,Ni= 1(ti)N+1sin2 iN+1 (8)T:N 2 。3.2 OWICSZ 2 s v(z) VM TVr f ,Gaus-JaccobiZi; L= 2 s X TVr TWi vs,+Y E8C5C)
11、,B +, sT ,7 2 , 9 t 。I n s?Z5 L= 2 s (1-t2)1/10 s, V| V 2 v(z)4 | , P v(z)1,V7 v(z) T Vr。B| f (t)=1-t2:v(z)(t)= v(z)(1 -t2)1/10(t)=v(z)(1-t2)0.6 = v(z)(t) (9)v f (t)=(1 -t2)0.6 ,= =0.6, V9 2 N HM 2 ti “ i, 4 2 #9 2 TnV1。 EKD (OWICS)Z, L=9 TGausssZ10。3.3 “ (Z1 GausssZ s $, 2Hq/, 2 H M1,id11“。9 M 2 ti
12、,9 2 H ,sW09 :Si=12(xi+xi+1)zi+iR-siniR2 (10)T:i= arcsinti-arcsinti+1 , xixi+1 2 H / ?,?z (sM # 2 W ,?xi V H M7 p。 2 H = ( ,5 2 H Si VT ( 9 H “ , - GausssZ1, “ ) , s,i=Si/(ti)。1GausssZ (Z “ ?C, Y, V1 U。GausssZide F p, “ F (M1,W 2 v7H 2 l, Gausss W 2 ( D, V ?9GausssZ 2 s a1 (Zy 。V1 s# p“ Table1 Theab
13、scissasandweightsoftheflowintegrationNGauss-JaccobiZOWICSZM 2 ti “ i i-s,iS,i(%)M 2 ti “ i i-s,iS,i(%)40.309 016 994 0.597 566 433 12.20 0.303 783 158 0.58 822 759 11.810.809 016 994 0.369 316 366 -22.16 0.799 639 363 0.371 884 135 -21.410.000 000 000 0.314 159 265 3.33 0.000 000 000 0.311 215 811 3
14、.270.309 016 994 0.298 783 216 2.97 0.306 221 533 0.296 281 302 2.929 0.587 785 252 0.254 160 185 1.55 0.583 053 275 0.252 910 792 1.560.809 016 994 0.184 658 183 -2.80 0.803 925 148 0.185 264 674 -2.540.951 056 516 0.097 080 552 -30.66 0.947 630 511 0.099 815 408 -29.646 : 2 2 9sZE# s1221 4 s s4.1
15、2 Hs 2 9 K 2 s f ,A7n 2 , s 。 5 ,5 s?Z sY V s , n v MY1“,/ n=9 Hv=(1-r)1/9 (0r1) (11) ssE=Q-QiQi100% (12)Yy 。 T, QZ H sT,Qi s T。m3 3Z 2 Hs。V A,Gauss-JaccobiZs (v , O “ 2 N9F,s hl,N=6sX0.05%/, O?9F 2 vM, e ,V 2 4 9 ilv。 F (Z,s9 “ 2 N9F7hl,hl . ,36 2 H 0.05%Ps。 1 V?C,GausssZ11 (Z 。KD Z,s , /,“ 2 9F,Sv
16、hl。 2 N=2 9,OWICSZs1Gaus-JaccobiZ1 l, - L; 2 v H,OWICSZ X1 。m3 5 Hq/sFig.3 Integrationerrorinthedevelopedpipeflowcondition B)3sZs, T11 :u(r,)=(1-r)1/n+mr(1-r)1/kf() (13)sBs。 T, n、k、m、f()V , 0,2 =。m4 3Zs9 T, T(13) BF /s (,7 L5V U Svl。 ? p5 f , GausssrT11 F (z, -s ? “ 2 9F l ,7 Oo91 2 F (Zl。 6B ,OWICS
17、Z 18C 2 H, 2 sK,OWICSZD。m4 Hq/sFig.4 Integrationerrorinthedisturbedflowcondition4.2 s Gauss sV 2N-1 , L= 2 s V2N-1Q TVr,5 V ,。 L= 2 s wL sHHqY,/ E2N-1Q T Vr, 9 sV| B, vl L= 2 s wL1“,B ,sv。 T(11) s , s 。 s15 VYV sQ=2.678 620。m5 2 wL f s, s+,Wu ,H , Gaus-Jac-cobiZ |N=5“, sTQ=2.681 950,M0.12%。m5 6 VF“
18、TB(Q v9, 3H H , ), BF wLs , 2 ssTQ=2.681 950, sT M,9 Gauss 2N-1222 N N V 3 1 1 。 Y L BC L= 2 wLz T f ,#sV9 is。m5 2 s UimFig.5 SketchofdistributionofthelineaveragevelocityGauss-JaccobisEOWICSsE (y 2 s V T VrL !$ ,s 2 s a T Vr12, s f ,A.Nichtaw-itz13 PCHIP T(s8: + T) Vr s1 a,HC f /, 4 2 PCHIPsE9 2 Gau
19、s-JaccobisEv Ms。 6914-15 9C d、 V,7 2i * ) , w2 9 。5 s 2 9 s ,i s ,w 2 9 2 Z(Gaus-JaccobiZ#OWICSZ) 2 “ 。YV1GausssZE F (ZE 2 M H “ Y,?CGaussZEF jWs 2 ( D, V ? 1y。 6s GausssZ 3y,i1 2 sY。s L= 2 E T Vr; 2 ,s ,V 2 4 9 il9v。 ID 1 .v 2 2 9C. S “, 2006:158-166.HUODZH.Largeflow-ratemeasurementandmulti-chordsu
20、ltrasonicflowmeterC.InProceedingoftheflow-ratemeasurementconference, 2006:158-166. 2 d, ,fo,. 3N 2o 9 !9# LCJ.N NV,2008, 29(10):2024-2028.NIUYH, PENGLH, ZHANGBF, etal.Designandimplementationofdigitalinstrumentbasedplatformforul-trasonicflow-meterJ.ChineseJournalofScientificIn-strument, 2008, 29(10):
21、2024-2028. 3 g b.los 2or HW MYJ.0 N , 2004, 18(9 1):78-83.LVQH.Firstzero-crossingpointdetectionofultrasonicsignalbasedonwaveletanalysisJ.JournalofElectron-icMeasurementandInstrument, 2004, 18(Suppl.1):78-83. 4 JJG1030 2007. 2 9, SSE9 _?S.SE S_f9, 2007.JJG1030 2007.Ultrasonicflowmeter, nationalverifi
22、ca-tionregulationS.GeneralAdministrationofQualitySu-pervision, InspectionandQuarantineofChina, 2007. 5 ,?, X. ZEM.=: , 1992.WANGXM, YANGTX, LISHQ.AppliednumericalmethodsM.Changchun:JinlinEducationPress, 1992. 6 IEC41 1991.InternationalcodeforthefieldacceptancetestofhydraulicturbinesS.InternationalEl
23、ectro-tech-nicalCommission, 1991. 7 PTC18 2002.Hydraulicturbinesandpump-turbinesperformancetestcodesS.AmericanSocietyofMe-chanicalEngineers, 2002. 8 PRESSWH, TEUKOLSKYSA.Numericalrecipes, theartofscientificcomputingM.CambridgeUniversityPress, 2007. 9 k),S.5 2 2o #9 ZEJ.q , 1991,24(1):84-92.CHENXX, F
24、ANHX.Mathematicalmodelsforacousticflowmeasurementincircularpenstockandtheircalcula-tionmethodsJ.JournalofWuhanUniversityofHy-draulicandElectricEngineering, 1991, 24(1):84-92. 10 TRESCHT, GRUBERP, STAUBLIT.ComparisonofintegrationmethodsformultipathacousticdischargemeasurementsC.Proceedingof6thInterna
25、tionalCon-ferenceonIGHEM, Portland, OregonUSA, 2006. 11 MOOREPI, BROWNG.J, STIMPSONBP.Ultrasonic6 : 2 2 9sZE# s1223 transit-timeflowmetersmodelledwiththeoreticalvelocityprofiles J.MesurementScienceandTechnology,2000, 11(12):1802-1811. 12 RYCHAGOVMN, TERESHCHENKOSA.Ultrasonicflowmeasurementsbymultipa
26、thmeasuringspoolpiecesquadratureintegrationandtomographicreconstructionC.Proceedingof5thISFFM, 2002. 13 NICHTAWITZA, GRAFENBERGERP, TORMANNJ,etal.Discussiononacousticflowmeasure-mentmethod C.Proceedingof5thInternationalConferenceonIGHEM, Lucerne, Switzer-land, 2004. 14 LUNTTAE, HALTTUNENJ.Neuralnetw
27、orkapproachtoultrasonicflowmeasurementsJ.FlowMeasurementandInstrumentation, 1999, 10, 35-43. 15 , ,. * 22 8 9J.N NV, 2007, 28(12):2280-2284.LIYZH, LICHX, HUAZHB.Researchonmulti-pathultrasonicgasflowmeterbasedonneuralnetwork J .ChineseJournalofScientificInstrument, 2007, 28(12):2280-2284.Te , 2008 M
28、bvp V,CS9 S ,1Z_ 9 。E-mail:HuHemingreceivedPh.D.degreein2008fromTsinghuaUniversity, andnowheisanassistantresearcherinNationalInstituteofMetrology.Hiscurrentresearchinterestisflow-ratemeasurement., 1989 M bv V,CS9 S ,1Z_ 9 。E-mail:WangChireceivedmasterdegreein1989fromTsinghuaUniversity, andnowsheisthedirec
