Research Article Hydraulic Transients Induced by Pigging Operation in Pipeline with a Long Slope

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1 Applied Mthemtics Volume 203, Article ID 23260, 9 pges Reserch Article Hydrulic Trnsients Induced by Pigging Opertion in Pipeline with Long Slope To Deng, Jing Gong, Hiho Wu, Yu Zhng, 2 Siqi Zhng, Qi Lin, nd Huishu Liu Beijing Key Lbortory of Urbn Oil nd Gs Distribution Technology, Chin University of Petroleum, Beijing 02249, Chin 2 South Est Asi Pipeline Compny Limited, Chin Ntionl Petroleum Corportion, Beijing 00028, Chin Correspondence should be ddressed to Jing Gong; ydgj@cup.edu.cn Received 4 July 203; Revised 22 August 203; Accepted 9 September 203 AcdemicEditor:BoYu Copyright 203 To Deng et l. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Pigging in pipelines bsiclly performs opertions for five resons including clening the pipe interior, btching or seprting dissimilr products, displcement, mesurement, nd internl inspection. A model hs been proposed for the dynmic simultion of the pigging process fter wter pressure testing in long slope pipeline. In this study, n ttempt hs been mde to nlyze two serious ccidents during pigging opertion in 200 by the model which is developed by the method of chrcteristic (MOC) by Wylie et l. (993) nd the two-phse homogeneous equilibrium vporous cvittion model deveoped by Shu (2003) for vporous cvittion. Moreover, simultion results of the third opertion show good greement with field dt from the previous field tril. After investigtion, it ws showed tht the impulse pressures produced during collpse of vpor cvity result in severe dmge of tubes.. Introduction Modern hydrulic systems re widely pplied to vrious industril fields. Routinely, for pressure testing of segmentl pigging, pigs re usully employed to remove liquids nd deposits fter the pressure testing. Although there re lrgenumberofvritionsndspecilpplictions,pigs re bsiclly utilized in pipelines to perform opertions for five resons including clening the pipe interior, btching or seprting dissimilr products, displcement, mesurement, nd internl inspection. Pressure testing, which includes strength tests nd tightness tests, is n importnt prt of the gurntee for the sfe opertion of oil nd gs pipelines. As, ir is ccessible nd inexpensive, it ws utilized s medium nd the pressure is incresed grdully to vlues tht the pressure testing requires. However, opertion stffs hve difficulty to find qulity defects for smll spills. For long pipeline, once pipeline lekge tkes plce, pressure would not decrese drmticlly due to the compressibility of ir. Moreover, there is lso the risk of getting pipes burst by pressurized ir. Thus, ir pressure testing hs been out of use since the 970s. Fortuntely, wter pressure testing cn cover up those problems. The pipelines of thousnds of kilometers in length re divided into segments vrying in length ccording to elevtion profiles nd mximum llowble differences. B 3.8 Code Committee nd Americn Gs Assocition jointly mde gret contribution to promote wter pressure testing for decdes. Pressure testing grdully took dvntge of wter for its sfety nd stbility. Generlly, pigging process is sfe under low pressure becuse fter pressure testing, opertion stffs will decrese pressure by dischrging the wter through the vlves t the end of pipes.however,duetovryingelevtionprofiles,therere lwys some segments with long slope, especilly in mountinous re nd pigging opertion, tht should del with new problems. More specificlly, for full-flow pipes with longslope,vlvesopeningndmovingpigsmycuseliquid trnsients in pipeline. Pigging process is subject to rpid pressure trnsients, resulting in wter-hmmer events. In system, ny chnge in flow velocity cuses chnge of pressure instntneously. Lrge pressure vritions nd distributed cvittion (bubble flow) my be involved due to the sudden shutdown of pump or closure of vlve. Column seprtion therefore my occur nd my hve significnt impct on subsequent trnsients in the system [].

2 2 Applied Mthemtics During the course of trnsient, usully t high point, the pressure in pipeline flls to the vpor pressure of wter, resulting in loclized liquid column seprtion [2 5] or vporous cvittion. Streeter nd Wylie, in their textbooks [4, 6] (Wylie nd Streeter 978), summrized previous work on column seprtion in detil. Beuthe [7] provided n extensive generl review with emphsis on stem condenstion. For vporous cvittion, vpor is distributed long portion of pipeline, rther thn being concentrted t one loction sforloclizedliquidcolumnseprtion.however,the formtion nd collpse of vpor cvity in pipeline my led to unexpectedly high pressure rises in the form of short-durtion pressure pulses. Angus [8], emphsized tht the dmge done to pipes s result of wter hmmer is so serious tht no engineer cn fford to neglect it in the design of long pipes, prticulrly those under low heds. Most of the erly efforts in liquid trnsients in pipelines emphsized prticulrly on the prediction of the mximum pressure rise due to closure of vlve [9 ] orshutdownof hydropower turbines [2]. Joukowsky s simple wter hmmer formul is used to determine the mximum hed rise due to the instntneous closure of vlve without considertion of liquid column seprtion or linepck during trnsient event. When locl column seprtion forms, the pressure mgnitude produced by the collpse of the cvity is greter thn the Joukowsky rise which is referred to s shortdurtion pressure pulse. Previous nlyticl studies [8, 3 5] predicted tht the pressure rise, due to the collpse of vpor cvity, my exceed the Joukowsky pressure rise. Much previous experimentl dt showed strong ttenution of the mximum pressure rise upon the first collpse of cvity t vlve[6 8]. Jeger et l. [9], Mrtin [20], nd Thorley [2] were devoted to comprehensive bibliogrphies of the historicl development of mny spects of wter hmmer including column seprtion. As previously mentioned, column seprtion is common pproch in trnsmission line modeling. However, such n pproch is oversimplistic nd cnledtounrelisticresults.shu[22] proposedthetwophse homogeneous equilibrium vporous cvittion model which could void unrelisticlly high pressure spikes with considertion of frequency-dependent friction. A pipe is divided in to two prts including the upstrem ir section nd the downstrem liquid section by pig. Becuse wter directly dischrges into tmospheric environment, the liquid flow will be under depressurized condition. Additionlly, compressed ir t upstrem is to be t low pressure. In order to ensure tht the pig is moving forwrd, the pressure is just greter thn the resisting force cting on the pig. The pig slowly moves nd hs gret influence on the hydrodynmic pressure of the downstrem flow. Accordingly, the pressure downstrem my drop to or below vpor pressure nd it would result in lrge cvity of vpor usully t high points. In previous reserches [23, 24], cvities were ssumed to form only t high points, t chnges in pipeline slope (convex up), nd t system boundries. Moreover, the gs nd liquid two-phse flow my pper in the sloping pipe where the hydrulic grde line is found to be t or below the elevtion of vpor pressure hed. Ultimtely, sever ccident my occur when column seprtion collpses () (2) (3) Figure : Schemtic digrm of collpse of cvity by pig. by the hevy pig t high speed. Due to the fctors such s the terrins, lrge number of bubbles will be produced, nd the flow pttern is the bubbly flow with the intense pressure oscilltionsttheendofthepipeline.thelrgecvityofvpor nd ir between the pig nd downstrem liquid-filled pipe will be compressed t the downhill when the pig runs fst fter the high point. The compression is n isotherml process, nd the gs would undergo the extrusion process t the end of the outlet. Eventully the gs is dispersed into wter in the form ofbubblessshowninfigure. 2. Mthemticl Methods 2.. Conventionl MOC. One-dimensionl continuity nd momentum equtions re pplied to nlyze the hydrulic trnsients. A number of pproches hve been introduced for the simultion of the pipeline trnsients including the method of chrcteristics (MOC), wve chrcteristics method (WCM), finite volume method (FVM) [25], finite element method (FEM), nd finite difference method (FDM). Among these methods, MOC is extensively used due to its simplicity. It is n explicit method nd powerful tool to nlyze hydrulic trnsients in pipeline flow. With the method of chrcteristics (MOC), the prtil differentil equtions cn be converted into ordinry differentil equtions.

3 Applied Mthemtics 3 The method of chrcteristics is technique tht tkes dvntgeoftheknownphysiclinformtiontechpointof the regulr rectngulr grid. Therefore, the results of physicl chrcteristics cn be quickly clculted in every time step. For fixed cross-sectionl re, the mss conservtion equtioncnbewrittensfollows: t t 0 +Δt C + P C by: H t +V H x + ( 2 ga ) Q =0. () x The momentum conservtion eqution cn be expressed t 0 t 0 Δt A F E B G x where ga ( Q t +V Q x )+ H x +fq Q m =0, (2) f = ]m. (3) D5 m The MOC pproch trnsforms the bove prtil differentil equtions into the ordinry differentil equtions long the chrcteristic lines nd is defined s C + C dx dt =V+ ga dq + dh + fq Q m dt = 0, dx dt =V ga dq dh + fq Q m dt = 0. Due to the fct tht V,thetermofV cn be omitted from the equtions. Flow rte Q is unknown nd difficult to determine between time steps t nd t+δt,becuseitvries with time nd spce. Therefore, it is impossible to identify the vlue of Q P.However,thetermof P B fq Q m dx cn be delt with pproximtely by the method of Streeter nd Wylie. So we ssume tht Q Q m =Q P Q A m, Q Q m =Q P Q B m. Due to the bove simplifiction, these equtions re integrted on the chrcteristic lines between time steps t nd t+δt,sshowninfigure 2, nd solved by the known vribles: Δx C + Δt =+ : ga (Q P Q A )+(H P H A )+fq P Q A m Δt = 0, Δx C Δt = : ga (Q P Q B ) (H P H B )+fq P Q B m Δt = 0. (6) (4) (5) Figure 2: Chrcteristic lines in x-t pln. Theseequtionscnbewritteninthesimplifiedformss follows: where C + :H P =R A S A Q P, C :H P =R B +S B Q P, R A =H A +C W Q A, R B =H B C W Q B, S A =C W +f Q A m Δt, S B =C W +f Q B m Δt, C W = ga, where the Q P is unknown flow t point P t time t+δt, H P is the unknown hydrulic hed t point P t time t+δt, Q A nd Q B re flows t neighboring sections of P t the previous time t,ndh A nd H B re heds t neighboring sections of P t the previous time t The Two-Phse Homogeneous Equilibrium Vporous Cvittion. When vporous cvities re loclly incipient, the locl pressures my be less thn or greter thn the vpor pressure of the liquid. However, for modeling purposes in engineering, it is ssumed tht the locl pressures re equl to the vpor pressure when vporous cvittion is occurring. During the pigging process, vporiztion occurs nd vpor cvities my be physiclly dispersed homogeneously in the form of bubbles. Due to the fctors such s the terrins, lrge number of bubbles will be produced, nd the flow pttern is the bubbly flow with the intense pressure oscilltions long the pipeline. The behvior of the flow should be described by the two-phse flow theory. Otherwise, wter hmmer equtionssolvedbythemoccnbedoptedforthesinglephse flow. (7) (8)

4 4 Applied Mthemtics The bsic equtions for the unstedy homogeneous equilibrium flow model in tube re ρ m πr0 2 P 2 t +(ρ l ρ V ) α t + ρ m πr0 2 x (Q )=0, (9) α t (Q α )+ P x +F 0 ( Q α,α)+ρ mg sin θ 0 =0. (0) In terms of the volumetricfrctionα of liquid nd vpor phse density ρ V, the men density ρ m cnbewrittens ρ m =αρ l + ( α) ρ V. () Intheboveequtions, thesecondtermin(9) describes the interfcil mss trnsfer rte nd the term Q/α in (9), nd () shows the flow rte differences between the liquid nd the vpor phse. Using the Drcy-Weisbch friction fctor f,the term F 0 (Q/α, α) cnbeexpressedsfollows: F 0 ( Q α,α)= fρ mq Q 4π 2 α 2 r0 5. (2) The method of chrcteristics is used to trnsform the bove equtions to four ordinry differentil equtions: C + : C : πr 2 o d dt (Q α )+ ρ l fq Q + 4π 2 α 2 r0 5 +gsin θ 0 =0 dx dt =, πr 2 o d dt (Q α ) ρ l fq Q + 4π 2 α 2 r0 5 +gsin θ 0 =0 d dt (P p V)+ t (ln ρ m ) ρ l d dt (P p V) t (ln ρ m ) ρ l dx dt =. (3) The equtions needed to solve the vribles t ech time step re C + : C : πr 2 0 πr 2 0 Q p α p + ρ l (P p P V )+ 2 ln ρ mp ρ l =C A, (4) Q p α p ρ l (P p P V ) 2 ln ρ mp ρ l =C B, (5) where Q p, P p,ndα D =(ρ md ρ V )/(ρ l ρ V ) cn be solved by the bove equtions, C A nd C B re constnt: C A = πr0 2 ( Q A )+ α A ρ l (P A P V )+ 2 ln ρ meρ mf ρ l ρ ma fδxq A Q A 4π 2 αa 2 r5 0 gδx sin θ 0, C B = πr0 2 ( Q B )+ α B ρ l (P B P V )+ 2 ln ρ meρ mg ρ l ρ mb fδxq B Q B 4π 2 αb 2r5 0 gδx sin θ 0, (6) The two-phse homogeneous equilibrium vporous cvittion model hs no conflict between negtive cvity sizes nd pressures below the vpor pressure. If C A C B,thenα p =, P p = ρ l 2 (C A C B )+P V. (7) If C A <C B,P p =P V,then α p = ρ l exp ((C A C B ) /) ρ V. (8) ρ l ρ V In either cse, Q p = πr2 0 α p (C 2 A +C B ). (9) Numericl results show tht the volume rte rnge is from 0 to 0.95 m 3 /s. According to the present study, ρ l = 000 kg/m 3, ρ V =.293 kg/m 3, = 00 m/s, A =.099 m 2, θ 0 0, f = , P V = 2340 P, nd Δx = 0 mre given; then we ssume tht the volumetric frction of liquid α vries between nd 0.5. Therefore, it is possible to identify the vlues of C A nd C B by our progrm, nd the rnge is from m/s to in the present study Boundry Conditions. Boundries include the inlet of pipeline, the outlet of pipeline, the til of the pig, nd nose of the pig. In order to solve the flow dynmic equtions, boundries conditions must be given. Boundries t the pipeline inlet nd outlet re constnt flow rte nd constnt pressure, respectively. In ddition, it is ssumed tht pressure ndflowrtetthetilofthepigrethesmestheyre in the upstrem fluid, close to the pig. We ssume tht the pig is moving boundry with no thickness compred with the length of the pipeline, but its weight would be considered. Bsedonthebovessumptions,thebehviorofthepigis tken into ccount to solve the flow dynmics equtions. The behvior of the pig in the pipeline is determined by blnceofforcesctingonthepigsshowninfigure 3.The pig will move forwrd if the drg force is less thn the driving force,butitwillstopwhenthedrgforceisdominnt: P =P 2 +ΔP s + A M g sin θ 0 + M A dv dt, (20)

5 Applied Mthemtics 5 ΔP s P 2 P Pig ΔP s Figure 3: Forces cting on the pig. Tble : Summry of bsic prmeters for field opertions. Prmeters Vlues The length of pipeline 6.93 Km Pipe size Φ mm The mximum elevtion difference 78.5 m Wll equivlent roughness 0.0 mm Mss of pig.0 kg Frictionl resistnce between pig nd pipe wll 0.03 MP Sttic friction resistnce between pig nd pipe wll 0.04 MP Type of compressor XHP070 Rted operting pressure 2.2 MP Air displcement 30.0 Nm 3 /min where V, M, P, P, ndθ 0 re the pig velocity, pig mss, the pressure on the upstrem, downstrem fces of the pig, nd ngle between xis nd horizontl direction. Term ΔP s represents the xil contct force between the pig nd the pipe wll, cting in opposition to the pig motion, in the direction of the pig xis. The xil contct force between the pig nd the pipe wll is obtined from the contct force eqution. 3. Results nd Discussion For the convenience of nlysis, the min stges (lbeled with roughness of bck line in Figure 4)ofthepiggingprocessre defined s follows, ccording to the pig position during the process: () the pig flt-segment movement stg, (2) the pig gully-segment movement stge, (3) the pig downhill-segment movement stge, (4) the pig ner outlet-segment movement nd overpressure stge. As shown in Figure 4, the pipeline with long slope is nerly 7 kilometers long nd 29 millimeters in dimeter. Air compressors nd vlve were instlled t the high point nd the low point, respectively. A pig ws put into pipeline from the high point before ir compressors strted to work to mke the pig move nd drin wy wter t the low point. Tble summrizes some importnt informtion of the pigging process in prctice including pipeline, pig, nd ir compressor. θ 0 Tble 2: Summry of two ccidents. The first ccident The second ccident Number of compressor 2 Totl time of pigging process 20Hours 39.5 Hours The length of dringe pipe 50.0 m 50.0 m The dringe pipe size Φ59.0 Φ mm 8.7 mm The length of rupture 2.6 m 2.5 m Mximumofirpressure.0 MP Bursting pressure of tube 20.83MP 20.83MP 3.. Numericl Simultion of Two Accidents 3... The First Accident Description. On July 9, couple of XHP070 ir compressors nd DN50 vlve were instlled for dewtering fter wter pressure testing. All the preprtions for the pigging process were completed nd the pig ws put in the pipeline. At nine o clock in the morning, the compressorsstrtedtoworkndthevlvewssimultneously opened. At five o clock in the next morning, n eruption of the mixture of wter nd gs occurred t the outlet of the pipeline, nd the pigging opertion hd tken nerly twenty hours. Finlly, frcture of 2.6 m in length ws found t the lst piece of steel tube The Second Accident Description. On September 2, n ir compressor nd DN50 vlve were instlled for dewtering fter wter pressure testing. At three o clock in the morning, pig ws put in the pipe before the compressor strtedtowork,whilethevlvewsopened.athlfpstsix in the next fternoon, the second ccident hppened, nd frctureof2.5minlengthwsbout5mwyfromthe firstfrcture.thepiggingopertionhdtotllytkennerly thirty nine hours nd hlf n hour. During this period of time, ccording to the records, the mximum of ir pressure ws bout.0 MP. After investigtion, however, the tubes hve no qulity defult, nd the rted operting pressure of the ir compressor is 2.2 MP so tht the compressed ir unlikely cused the dmges. Additionlly, bursting pressure is determined from Brlow s eqution tht instntneous pressure for severe rupture of the tube should be over MP s shown in Tble 2. A mthemticl model hs been suggested for simulting the pigging process, bsed on our own previous work. Menwhile, these results were obtined from simulting the process by our own progrm. The simulted time of the first pigging opertion ws 9.6 hours. During the pig downhill-segment movement stge, some mount of wter ws gthered in the prt of low elevtion ner the outlet due to the grvity. When the pigwsclosetotheendofthepipe,somemountofgs downstrem would undergo extrusion process nd eventully ws dispersed into wter s shown in Figure.Consequently, the gs cvity collpsed resulting in n impulse pressure up

6 6 Applied Mthemtics Elevtion (m) Elevtion (m) Distnce (m) () () Elevtion (m) Elevtion (m) Distnce (m) (2) (b) Distnce (m) (3) (c) Distnce (m) (4) (d) Figure 4: Four stges for pigging process. Pressure (log 0 MP) 00 0 Elevtion (m) m Position (m) Pressure (log 0 MP) 00 0 Elevtion (m) 6930 m Position (m) Elevtion Pressure (MP) Elevtion Pressure (MP) Figure 5: Numericl result of pressure t the end of pipe for the first ccident. Figure 6: Numericl result of pressure t the end of pipe for the second ccident. to MP s shown in Figure 5. Thevlueoftheimpulse pressure is lrger thn the bursting pressure of the tube, nd therefore it cused the rupture. Thereislrgeslopettheendofpipesothtpressure vried with the liquid level in the pipe. In other words, pressure would decrese when the liquid level dropped, nd pressure would increse when the liquid level rose. According to the results, the liquid level quickly dropped ner 9 h nd 28 h, nd the liquid level rose ner h becuse the flow rte of outlet nd top ws vrying during the pigging process. The flow rte t the outlet ws much lrger thn tht t the top when the liquid level quickly dropped. On the contrry, the flowrteoftopwsmuchlrgerthnthtttheoutletwhen theliquidlevelquicklyrose. The simulted time of the first pigging opertion ws 9.6 hours. During this period of time, the mximum pressure of compressed ir ws s much s.03 MP s shown in Figure 7. Finlly, the gs cvity collpsed resulting in n impulse pressure up to MP s shown in Figure 6. The vlue of the impulse pressure is lso lrger thn the bursting pressure of the tube, nd the second ccident occurred s shown in Tble 3.

7 Applied Mthemtics 7 Tble 3: Summry of numericl simultion of the two ccidents. The first ccident The second ccident Number of compressor 2 Totl time of pigging process 9.6 Hours 36.7 Hours Mximum of ir pressure.03mp Mximum of outlet pressure MP MP Bursting pressure 20.83MP 20.83MP Tble 4: Summry of the field tril nd numericl simultion. The field tril Numericl simultion Number of compressor Totl time of pigging process 36 Hours 35.5 Hours Vlue of the impulse pressure.576 MP.5 MP Bursting pressure 20.83MP 20.83MP Pressure (MP) Figure 7: Numericl result of ir pressure for the second ccident Comprison of Field Results to Simulting Predictions. To study the vrition of outlet pressure due to the pigging process, field tril ws conducted in 20 by Luo [26] from Chin Ntionl Petroleum Corportion (CNPC). After two ccidents, the third opertion ws conducted nd trnsient pressures were recorded by two pressure trnsducers which were instlled t the third nd fourth tubes from the end, respectively. In this experiment, the NS-B pressure trnsducers re 50 MP in mesurement rnge nd 0.3% in precision instlled with n ngle of 90 degrees to ssure the results vlidity. While the dringe tube is 59 mm in dimeter for the first two pigging processes, the lst tube of pipe which is 29 mm in dimeter hs four bores of mm in dimeter for dringe, nd the totl re of the cross sections of the four bores will be equl to the re of the cross section of the tube. As Figure 8 nd Figure 9 show, the simulted time of the first pigging opertion ws 9.6 hours, nd the mximum pressure ws s much s.5 MP. However, the pressure t the end ws constntly kept in lower level for most of the time. Becuse the pressure t the loction of cvittion is usully below the sturted vpor pressure, the vpor pressure oftheliquidisdoptedsthecvittioninceptionpressure in this mthemticl model for trnsient cvittion. When the pressure is under the sturted vpor pressure, the wter wouldchngeintovpor.furthermore,sitwsbelowthe vpor pressure for long period of time, the ir dissolved in the wter is relesed, nd thus there would be lrge cvity of the wter vpor nd ir. When the pig ws close to the end of the pipe, some mount of gs downstrem would undergo extrusion process, nd eventully n eruption of the mixture of wter nd gs ppered t the outlet ccording to field records. With comprison between model prediction nd field dt,itwsfoundthtthemplitudesoftheimpulsepressures inducedbythecollpseofthecvitywerenerlythesme,nd their totl time ws 36 hours, nd 35.5 hours (see Figure 0) respectively s shown in Tble 4. Pressure (MP) Figure 8: Numericl result of pressure t the end of pipe for the field tril. 4. Conclusions The model proposed for the pigging process ws employed to understndtheflowdynmicsinthepipelinendtoobtin trnsient pressures for the two ccidents nd the field tril. The lrge cvity of wter vpor nd ir ws in the downslopepipefollowingthepekndslcklineflowthtexited for long period of time. When the pig ws close to the outlet, the cvity ws compressed, nd gs underwent the extrusion process nd eventully ws dispersed into wter in the form of bubbles. Finlly, the cvity collpsed by the pig, nd the serious collision resulted in considerble impulse pressures. The results of the simultion illustrte tht the impulse pressures cused the severe dmges during the pigging process. Additionlly, our model predictions for the third opertion showed good greement with field dt, nd the dimeter of dringe hd significnt effect on impulse

8 8 Applied Mthemtics Pressure (MP) ( ) Figure 9: Numericl result of the impulse pressure for the field tril. Pressure (MP) Figure 0: Result of the impulse pressure for the field tril. pressures. Generlly, the terrin of pipeline is key fctor for the liquid-fill flow behvior, nd slck line flow my pper in long-slope pipeline. Then, column seprtion would occur s common phenomenon for hilly pipeline, nd it cn cuse devstting effects such s severe dmges. Therefore, preventive mesures re of criticl significnce for prcticl resons. Nomenclture f: Friction fctor x: Distnce long the pipeline (m) g: Grvity ccelertion (m s 2 ) t: Time(s) A: Cross-section re of pipeline (m 2 ) m: The index number of Drcy formul H: Wter hed of fluid (m) : Acoustic speed of fluid (m/s) Q: Volume rte of fluid (m 3 /s) Q A,Q B : Volume rtes for given points (m 3 /s) H A,H B : Wter heds for given points (m) Q P : Volume rte for unknown point (m 3 /s) H P : Volume rte for unknown point (m) V: Velocity(m/s) D: Dimeter of the pipeline (mm) ]: Kinemtic viscosity of liquid ((m 2 /s)) P V : Sturted vpor pressure t liquid temperture (P) ρ: Density of liquid (kg/m 3 ) M: Pig mss (kg) P : Thepressureontheupstremfcesofthepig (P) P 2 : The pressure on the downstrem fces of the pig (P) β: A ngle between xis nd horizontl direction (rd) ΔP s : The xil contct force (P) α: The volumetric frction of liquid ρ V : Vpor phse density (kg/m 3 ) ρ l : Liquid phse density (kg/m 3 ) ρ m : The men density (kg/m 3 ). Abbrevitions MOC: The method of chrcteristic IAHR: Interntionl Assocition For Hydrulic Reserch HC: Hydrodynmic cvittion. Acknowledgments The uthors thnk the Ntionl Science & Technology Specific Project (Grnt no. 20ZX ), nd the Key Ntionl Science nd Technology Specific Project (20ZX ), the Ntionl Nturl Science Foundtion of Chin (50467) for their finncil support. References [] A. Bergnt, A. R. Simpson, nd A. S. Tijsseling, Wter hmmer with column seprtion: historicl review, Fluids nd Structures,vol.22,no.2,pp.35 7,2006. [2] R. A. Bltzer, A study of column seprtion ccompnying trnsient flow of liquid in pipes [Ph.D. thesis], The University of Michign,AnnArbor,Mich,USA,967. [3] R. A. Bltzer, Column seprtion ccompnying liquid trnsients in pipes, ASME Bsic Engineering D, vol. 89, pp , 967. [4] V.L.StreeterndE.B.Wylie,Hydrulic Trnsients, McGrw- Hill Book Compny, New York, NY, USA, 967. [5] J. A. Swffield, A study of column seprtion following vlve closure in pipeline crrying vition kerosine, Proceedings of the Institution of Mechnicl Engineers,vol.84,no.7,pp.57 64, 969. [6] E.B.Wylie,V.L.Streeter,ndL.SuoJones,Fluid Trnsients in Systems, chpter 3, Prentice Hll, Enlewood Cliffs, NJ, USA, 993.

9 Applied Mthemtics 9 [7] T. G. Beuthe, Review of two-phse wter hmmer, in Proceedings of the 8th Cndin Nucler Society Conference,p.20, Toronto, Cnd, 997. [8] R. W. Angus, Simple grphicl solution for pressure rise in pipes nd dischrge lines, Engineering Institute of Cnd,vol.8,no.2,pp.72 8,935. [9] N. E. Joukowsky, Wter Hmmer, Proceedings of the Americn Wter Works Assocition,vol.24,pp ,904. [0] L. Allievi, Generl Theory of Perturbed Flow of Wter in Pressure Conduits, Annli dell Societ degli Ingegneri ed rchitetti itlini, Milno, Itly, 903. [] L. Allievi, Theory of Wter Hmmer. Notes I to V, ASME,New York, NY, USA, 93. [2] N. R. Gibson, Pressure in penstocks cused by grdul closing of turbine gte, Trnsctions of the Americn Society of Civil Engineers,vol.83,pp ,99. [3] R. W. Angus, Wter hmmer in pipes, includin those supplied by centrifugl pumps: grphicl tretment, Proceedings of the InstitutionofMechniclEngineers,vol.36,pp ,937. [4] I. C. O Neill, WterHmmerinSimplePipeSystems[M.S.thesis], Deprtment of civil Engineering, University of Melbourne, Victori, Austrli, 959. [5] A. R. Simpson nd E. B. Wylie, Problems encountered in modeling vpor column seprtion, in Proceedings of the Symposium on Fluid Trnsients in Fluid-Structure Interction, pp.03 07,ASME,MimiBech,Fl,USA,November985. [6] J. P. Klkwijk nd C. Krnenburg, Cvittion in horizontl pipelines due to wter hmmer, ASCEJournloftheHydrulics Divsion,vol.97,no.0,pp ,97. [7] C. Krnenburg, Trnsient cvittion in pipelines, Report 73-2, Lbortory of Fluid Mechnics, Communictions on Hydrulics, Deprtment of Civil Engineering, Delft University of Technology, 973. [8] C. Krnenburg, Gs relese during trnsient cvittion in pipes, ASCE Journl of the Hydrulic Division, vol. 00, no. 0, pp , 974. [9] C.Jeger,L.S.Kerr,ndE.B.Wylie, Selectedbibliogrphy, in Proceedings of the Interntionl Symposium on Wter Hmmer in Pumped Storge Projects, ASME Winter Annul Meeting, pp , Chicgo, ILL, USA, 965. [20] C. S. Mrtin, Sttus of fluid trnsients in W estern Europe nd the United Kingdom: report on lbortory visits by Freemn scholr, ASME Fluids Engineering, vol.95,pp.30 38, 973. [2] A. R. D. Thorley, A Survey of Investigtions into Pressure Surge Phenomen, The City University, Deprtment of Mechnicl Engineering, London, UK, 976. [22] J.-J. Shu, Modelling vporous cvittion on fluid trnsients, Interntionl Pressure Vessels nd Piping, vol.80,no. 3, pp , [23] C. Krnenburg, The effect of gs relese on column seprtion, Report 74-3, Delft University of Technology, Deprtment of Civil Engineering, 974, Communiction on Hydrulics. [24] V. L. Streeter, Trnsient cvittion pipe flow, ASCE Hydrulic Engineering,vol.09,no.,pp ,983. [25] M. H. Afshr nd M. Rohni, Wter hmmer simultion by implicit method of chrcteristic, Interntionl Pressure Vessels nd Piping,vol.85,no.2,pp ,2008. [26] J. Luo, An nlysis of the problems of frcture t new gret drop line pipe, Oil nd Gs Storge nd Trnsporttion, pp , 20.

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