Research Article Hydraulic Transients Induced by Pigging Operation in Pipeline … · 2018. 12....

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 231260 9 pageshttpdxdoiorg1011552013231260

Research ArticleHydraulic Transients Induced by Pigging Operationin Pipeline with a Long Slope

Tao Deng1 Jing Gong1 Haihao Wu1 Yu Zhang2 Siqi Zhang1 Qi Lin1 and Huishu Liu1

1 Beijing Key Laboratory of Urban Oil and Gas Distribution Technology China University of Petroleum Beijing 102249 China2 South East Asia Pipeline Company Limited China National Petroleum Corporation Beijing 100028 China

Correspondence should be addressed to Jing Gong ydgjcupeducn

Received 14 July 2013 Revised 22 August 2013 Accepted 9 September 2013

Academic Editor Bo Yu

Copyright copy 2013 Tao Deng et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Pigging in pipelines basically performs operations for five reasons including cleaning the pipe interior batching or separatingdissimilar products displacement measurement and internal inspection A model has been proposed for the dynamic simulationof the pigging process after water pressure testing in a long slope pipeline In this study an attempt has been made to analyze twoserious accidents during pigging operation in 2010 by the model which is developed by the method of characteristic (MOC) byWylie et al (1993) and the two-phase homogeneous equilibrium vaporous cavitation model deveoped by Shu (2003) for vaporouscavitation Moreover simulation results of the third operation show good agreement with field data from the previous field trialAfter investigation it was showed that the impulse pressures produced during collapse of a vapor cavity result in severe damage oftubes

1 Introduction

Modern hydraulic systems are widely applied to variousindustrial fields Routinely for pressure testing of segmentalpigging pigs are usually employed to remove liquids anddeposits after the pressure testing Although there are alarge number of variations and special applications pigsare basically utilized in pipelines to perform operations forfive reasons including cleaning the pipe interior batching orseparating dissimilar products displacement measurementand internal inspection Pressure testing which includesstrength tests and tightness tests is an important part ofthe guarantee for the safe operation of oil and gas pipelinesAs air is accessible and inexpensive it was utilized as amedium and the pressure is increased gradually to valuesthat the pressure testing requires However operation staffshave difficulty to find quality defects for small spills For along pipeline once a pipeline leakage takes place pressurewould not decrease dramatically due to the compressibilityof air Moreover there is also the risk of getting pipes burstby pressurized air Thus air pressure testing has been outof use since the 1970s Fortunately water pressure testingcan cover up those problems The pipelines of thousands

of kilometers in length are divided into segments varyingin length according to elevation profiles and maximumallowable differences B 318 Code Committee and AmericanGas Association jointly made great contribution to promotewater pressure testing for decades Pressure testing graduallytook advantage of water for its safety and stability Generallypigging process is safe under low pressure because afterpressure testing operation staffs will decrease pressure bydischarging the water through the valves at the end ofpipes However due to varying elevation profiles there arealways some segments with a long slope especially in amountainous area and pigging operation that should dealwith new problems More specifically for full-flow pipes witha long slope valves opening andmoving pigsmay cause liquidtransients in a pipeline Pigging process is subject to rapidpressure transients resulting in water-hammer events

In a system any change in flow velocity causes a changeof pressure instantaneously Large pressure variations anddistributed cavitation (bubble flow) may be involved dueto the sudden shutdown of a pump or closure of a valveColumn separation therefore may occur and may have asignificant impact on subsequent transients in the system [1]

2 Journal of Applied Mathematics

During the course of a transient usually at high point thepressure in a pipeline falls to the vapor pressure of waterresulting in a localized liquid column separation [2ndash5] orvaporous cavitation Streeter andWylie in their textbooks [46] (Wylie and Streeter 1978a) summarized previous work oncolumn separation in detail Beuthe [7] provided an extensivegeneral review with emphasis on steam condensation Forvaporous cavitation vapor is distributed along a portion ofa pipeline rather than being concentrated at one locationas for a localized liquid column separation However theformation and collapse of a vapor cavity in a pipeline maylead to unexpectedly high pressure rises in the form ofshort-duration pressure pulses Angus [8] emphasized thatthe damage done to pipes as a result of water hammer isso serious that no engineer can afford to neglect it in thedesign of long pipes particularly those under low headsMost of the early efforts in liquid transients in pipelinesemphasized particularly on the prediction of the maximumpressure rise due to closure of a valve [9ndash11] or shutdown ofhydropower turbines [12] Joukowskyrsquos simple water hammerformula is used to determine the maximum head rise due tothe instantaneous closure of a valve without considerationof liquid column separation or linepack during a transientevent When a local column separation forms the pressuremagnitude produced by the collapse of the cavity is greaterthan the Joukowsky rise which is referred to as a short-duration pressure pulse Previous analytical studies [8 13ndash15] predicted that the pressure rise due to the collapse of avapor cavity may exceed the Joukowsky pressure rise Muchprevious experimental data showed strong attenuation of themaximum pressure rise upon the first collapse of a cavity ata valve [16ndash18] Jaeger et al [19] Martin [20] and Thorley[21] were devoted to comprehensive bibliographies of thehistorical development of many aspects of water hammerincluding column separation As previously mentioned col-umn separation is a common approach in transmission linemodeling However such an approach is oversimplistic andcan lead to unrealistic results Shu [22] proposed the two-phase homogeneous equilibrium vaporous cavitation modelwhich could avoid unrealistically high pressure spikes withconsideration of frequency-dependent friction

A pipe is divided in to two parts including the upstreamair section and the downstream liquid section by a pigBecause water directly discharges into atmospheric environ-ment the liquid flow will be under depressurized conditionAdditionally compressed air at upstream is to be at lowpressure In order to ensure that the pig is moving forwardthe pressure is just greater than the resisting force actingon the pig The pig slowly moves and has great influenceon the hydrodynamic pressure of the downstream flowAccordingly the pressure downstream may drop to or belowvapor pressure and it would result in a large cavity ofvapor usually at high points In previous researches [23 24]cavities were assumed to form only at high points at changesin pipeline slope (convex up) and at system boundariesMoreover the gas and liquid two-phase flow may appear inthe sloping pipe where the hydraulic grade line is found to beat or below the elevation of vapor pressure head Ultimatelysever accident may occur when column separation collapses

(1)

(2)

(3)

Figure 1 Schematic diagram of collapse of a cavity by a pig

by the heavy pig at high speed Due to the factors such asthe terrains a large number of bubbles will be produced andthe flow pattern is the bubbly flow with the intense pressureoscillations at the end of the pipelineThe large cavity of vaporand air between the pig and downstream liquid-filled pipewill be compressed at the downhill when the pig runs fast afterthe high pointThe compression is an isothermal process andthe gas would undergo the extrusion process at the end of theoutlet Eventually the gas is dispersed into water in the formof bubbles as shown in Figure 1

2 Mathematical Methods

21 Conventional MOC One-dimensional continuity andmomentum equations are applied to analyze the hydraulictransients A number of approaches have been introducedfor the simulation of the pipeline transients includingthe method of characteristics (MOC) wave characteristicsmethod (WCM) finite volume method (FVM) [25] finiteelementmethod (FEM) and finite differencemethod (FDM)Among these methods MOC is extensively used due to itssimplicity It is an explicit method and a powerful tool to ana-lyze hydraulic transients in pipeline flow With the methodof characteristics (MOC) the partial differential equa-tions can be converted into ordinary differential equations

Journal of Applied Mathematics 3

The method of characteristics is a technique that takesadvantage of the known physical information at each point ofthe regular rectangular gridTherefore the results of physicalcharacteristics can be quickly calculated in every time step

For a fixed cross-sectional area the mass conservationequation can be written as follows

120597119867

120597119905+ 119881

120597119867

120597119909+ (

1198862

119892119860)120597119876

120597119909= 0 (1)

The momentum conservation equation can be expressedby

1

119892119860(120597119876

120597119905+ 119881

120597119876

120597119909) +

120597119867

120597119909+ 119891119876|119876|

1minus119898= 0 (2)

where

119891 = 00246]119898

1198635minus119898 (3)

The MOC approach transforms the above partial differ-ential equations into the ordinary differential equations alongthe characteristic lines and is defined as

119862+

119889119909

119889119905= 119881 + 119886

119886

119892119860119889119876 + 119889119867 + 119891119876|119876|

1minus119898119886119889119905 = 0

119862minus

119889119909

119889119905= 119881 minus 119886

119886

119892119860119889119876 minus 119889119867 + 119891119876|119876|

1minus119898119886119889119905 = 0

(4)

Due to the fact that 119881 ≪ 119886 the term of 119881 can be omittedfrom the equations Flow rate 119876 is unknown and difficult todetermine between time steps 119905 and 119905 + Δ119905 because it varieswith time and space Therefore it is impossible to identifythe value of119876

119875 However the term of int119875

119861119891119876|119876|

1minus119898119889119909 can be

dealt with approximately by themethod of Streeter andWylieSo we assume that

119876|119876|1minus119898

= 119876119875

100381610038161003816100381611987611986010038161003816100381610038161minus119898

119876|119876|1minus119898

= 119876119875

100381610038161003816100381611987611986110038161003816100381610038161minus119898

(5)

Due to the above simplification these equations areintegrated on the characteristic lines between time steps 119905and 119905 + Δ119905 as shown in Figure 2 and solved by the knownvariables

119862+

Δ119909

Δ119905= +119886

119886

119892119860(119876119875minus 119876119860) + (119867

119875minus 119867119860) +119891119876

119875

100381610038161003816100381611987611986010038161003816100381610038161minus119898

119886Δ119905 = 0

119862minus

Δ119909

Δ119905= minus119886

119886

119892119860(119876119875minus 119876119861) minus (119867

119875minus 119867119861) + 119891119876

119875

100381610038161003816100381611987611986110038161003816100381610038161minus119898

119886Δ119905 = 0

(6)

t0 minus Δt

t0

t0 + Δt

t

x

A

F G

E

P

B

Cminus

C+

Figure 2 Characteristic lines in 119909-119905 plan

These equations can be written in the simplified forms asfollows

119862+ 119867119875= 119877119860minus 119878119860119876119875

119862minus 119867119875= 119877119861+ 119878119861119876119875

(7)

where

119877119860= 119867119860+ 119862119882119876119860

119877119861= 119867119861minus 119862119882119876119861

119878119860= 119862119882+ 119891

100381610038161003816100381611987611986010038161003816100381610038161minus119898

119886Δ119905

119878119861= 119862119882+ 119891

100381610038161003816100381611987611986110038161003816100381610038161minus119898

119886Δ119905

119862119882=

119886

119892119860

(8)

where the 119876119875is unknown flow at point 119875 at time 119905 + Δ119905 119867

119875

is the unknown hydraulic head at point 119875 at time 119905 + Δ119905 119876119860

and 119876119861are flows at neighboring sections of 119875 at the previous

time 119905 and119867119860and119867

119861are heads at neighboring sections of 119875

at the previous time 119905

22The Two-Phase Homogeneous Equilibrium Vaporous Cav-itation When vaporous cavities are locally incipient thelocal pressures may be less than or greater than the vaporpressure of the liquid However for modeling purposes inengineering it is assumed that the local pressures are equalto the vapor pressure when vaporous cavitation is occurringDuring the pigging process vaporization occurs and vaporcavities may be physically dispersed homogeneously in theformof bubbles Due to the factors such as the terrains a largenumber of bubbles will be produced and the flow pattern isthe bubbly flow with the intense pressure oscillations alongthe pipeline The behavior of the flow should be describedby the two-phase flow theory Otherwise water hammerequations solved by the MOC can be adopted for the single-phase flow


The basic equations for the unsteady homogeneous equi-librium flow model in a tube are

1

1198862

120597119875

120597119905+ (120588119897minus 120588V)

120597120572

120597119905+

120588119898

12058711990320

120597

120597119909(119876

120572) = 0 (9)

120588119898

12058711990320

120597

120597119905(119876

120572) +

120597119875

120597119909+ 1198650(119876

120572 120572) + 120588

119898119892 sin 120579

0= 0 (10)

In terms of the volumetric fraction 120572 of liquid and vaporphase density 120588V the mean density 120588

119898can be written as

120588119898= 120572120588119897+ (1 minus 120572) 120588V (11)

In the above equations the second term in (9) describesthe interfacial mass transfer rate and the term119876120572 in (9) and(11) shows the flow rate differences between the liquid and thevapor phase Using the Darcy-Weisbach friction factor 119891 theterm 119865

0(119876120572 120572) can be expressed as follows

1198650(119876

120572 120572) =

119891120588119898119876 |119876|

41205872120572211990350

(12)

The method of characteristics is used to transform theabove equations to four ordinary differential equations

119862+

1

1205871199032119900

119889

119889119905(119876

120572) +

1

120588119897119886

119889

119889119905(119875 minus 119901V) + 119886

120597

120597119905(ln

120588119898

120588119897

)

+119891119876 |119876|

41205872120572211990350

+ 119892 sin 1205790= 0

119889119909

119889119905= 119886

119862minus

1

1205871199032119900

119889

119889119905(119876

120572) minus

1

120588119897119886

119889

119889119905(119875 minus 119901V) minus 119886

120597

120597119905(ln

120588119898

120588119897

)

+119891119876 |119876|

41205872120572211990350

+ 119892 sin 1205790= 0

119889119909

119889119905= minus119886

(13)

The equations needed to solve the variables at each timestep are

119862+

1

12058711990320

119876119901

120572119901

+1

120588119897119886(119875119901minus 119875119881) +

119886

2ln

120588119898119901

120588119897

= 119862119860 (14)

119862minus

1

12058711990320

119876119901

120572119901

minus1

120588119897119886(119875119901minus 119875119881) minus

119886

2ln

120588119898119901

120588119897

= 119862119861 (15)

where119876119901 119875119901 and 120572

119863= (120588119898119863

minus120588119881)(120588119897minus120588V) can be solved by

the above equations 119862119860and 119862

119861are constant

119862119860=

1

12058711990320

(119876119860

120572119860

) +1

120588119897119886(119875119860minus 119875119881) +

119886

2ln

120588119898119864120588119898119865

120588119897120588119898119860

minus119891Δ119909119876

119860

10038161003816100381610038161198761198601003816100381610038161003816

41205872120572211986011990350119886

minus119892Δ119909 sin 120579

0

119886

119862119861=

1

12058711990320

(119876119861

120572119861

) +1

120588119897119886(119875119861minus 119875119881) +

119886

2ln

120588119898119864120588119898119866

120588119897120588119898119861

minus119891Δ119909119876

119861

10038161003816100381610038161198761198611003816100381610038161003816

41205872120572211986111990350119886

minus119892Δ119909 sin 120579

0

119886

(16)

The two-phase homogeneous equilibrium vaporous cavi-tationmodel has no conflict between negative cavity sizes andpressures below the vapor pressure

If 119862119860ge 119862119861 then 120572

119901= 1

119875119901=120588119897119886

2(119862119860minus 119862119861) + 119875119881 (17)

If 119862119860lt 119862119861119875119901= 119875119881 then

120572119901=120588119897exp ((119862

119860minus 119862119861) 119886) minus 120588V

120588119897minus 120588119881

(18)

In either case

119876119901=1205871199032

0120572119901

2(119862119860+ 119862119861) (19)

Numerical results show that the volume rate range isfrom 0 to 095m3s According to the present study 120588

119897=

1000 kgm3 120588V = 1293 kgm3 119886 = 1100ms 119860 = 1099m21205790asymp 0 119891 = 000035 119875

119881= 2340Pa and Δ119909 = 110m are

given then we assume that the volumetric fraction of liquid120572 varies between 0999 and 05 Therefore it is possible toidentify the values of 119862

119860and 119862

119861by our program and the

range is from minus380837ms to 22864 in the present study

23 Boundary Conditions Boundaries include the inlet ofpipeline the outlet of pipeline the tail of the pig and noseof the pig In order to solve the flow dynamic equationsboundaries conditions must be given Boundaries at thepipeline inlet and outlet are constant flow rate and constantpressure respectively In addition it is assumed that pressureand flow rate at the tail of the pig are the same as they arein the upstream fluid close to the pig We assume that thepig is a moving boundary with no thickness compared withthe length of the pipeline but its weight would be consideredBased on the above assumptions the behavior of the pig istaken into account to solve the flow dynamics equations

The behavior of the pig in the pipeline is determined by abalance of forces acting on the pig as shown in Figure 3 Thepig will move forward if the drag force is less than the drivingforce but it will stop when the drag force is dominant

1198751= 1198752+ Δ119875119904+1

119860119872 sdot 119892 sdot sin 120579

0+119872

119860

119889119881

119889119905 (20)


PigP1

P2

ΔPs

ΔPs

1205790

Figure 3 Forces acting on the pig

Table 1 Summary of basic parameters for field operations

Parameters ValuesThe length of pipeline 693 KmPipe size Φ1219 times 184mmThe maximum elevation difference 1785mWall equivalent roughness 001mmMass of pig 7000 kgFrictional resistance between pig and pipe wall 003MPaStatic friction resistance between pig and pipewall 004MPa

Type of compressor XHP1070Rated operating pressure 22MPaAir displacement 300Nm3min

where 119881 119872 1198751 119875 and 120579

0are the pig velocity pig mass

the pressure on the upstream downstream faces of the pigand angle between axis and horizontal direction Term Δ119875

119904

represents the axial contact force between the pig and the pipewall acting in opposition to the pig motion in the directionof the pig axisThe axial contact force between the pig and thepipe wall is obtained from the contact force equation

3 Results and Discussion

For the convenience of analysis the main stages (labeled withroughness of back line in Figure 4) of the pigging process aredefined as follows according to the pig position during theprocess

(1) the pig flat-segment movement stag

(2) the pig gully-segment movement stage

(3) the pig downhill-segment movement stage

(4) the pig near outlet-segment movement and overpres-sure stage

As shown in Figure 4 the pipeline with a long slope isnearly 7 kilometers long and 1219millimeters in diameter Aircompressors and a valve were installed at the high point andthe low point respectively A pig was put into pipeline fromthe high point before air compressors started to work tomakethe pig move and drain away water at the low point

Table 1 summarizes some important information of thepigging process in practice including pipeline pig and aircompressor

Table 2 Summary of two accidents

The firstaccident

The secondaccident

Number of compressor 2 1Total time of pigging process 20 Hours 395 HoursThe length of drainage pipe 1500m 1500m

The drainage pipe size Φ1590 times87mm

Φ1590 times87mm

The length of rupture 26m 25mMaximum of air pressure 101MPaBursting pressure of tube 2083MPa 2083MPa

31 Numerical Simulation of Two Accidents

311 The First Accident Description On July 19 a couple ofXHP1070 air compressors and a DN150 valve were installedfor dewatering after water pressure testing All the prepa-rations for the pigging process were completed and the pigwas put in the pipeline At nine orsquoclock in the morning thecompressors started towork and the valvewas simultaneouslyopened At five orsquoclock in the next morning an eruption ofthe mixture of water and gas occurred at the outlet of thepipeline and the pigging operation had taken nearly twentyhours Finally a fracture of 26m in length was found at thelast piece of steel tube

312 The Second Accident Description On September 21an air compressor and a DN150 valve were installed fordewatering after water pressure testing At three orsquoclock inthe morning a pig was put in the pipe before the compressorstarted to work while the valve was opened At half past sixin the next afternoon the second accident happened anda fracture of 25m in length was about 5m away from thefirst fracture The pigging operation had totally taken nearlythirty nine hours and half an hour During this period of timeaccording to the records the maximum of air pressure wasabout 101MPa

After investigation however the tubes have no qualitydefault and the rated operating pressure of the air compressoris 22MPa so that the compressed air unlikely caused thedamages Additionally bursting pressure is determined fromBarlowrsquos equation that instantaneous pressure for severerupture of the tube should be over 2083MPa as shown inTable 2

A mathematical model has been suggested for simulatingthe pigging process based on our own previous workMeanwhile these results were obtained from simulating theprocess by our own program

The simulated time of the first pigging operation was196 hours During the pig downhill-segment movementstage some amount of water was gathered in the part oflow elevation near the outlet due to the gravity When thepig was close to the end of the pipe some amount of gasdownstreamwould undergo extrusion process and eventuallywas dispersed into water as shown in Figure 1 Consequentlythe gas cavity collapsed resulting in an impulse pressure up


0

750

700

650

600

5501000 2000 3000 4000 5000 6000 7000

Distance (m) (1)

Elev

atio

n (m

)

(a)

750

700

650

600

550

Elev

atio

n (m

)

Distance (m) (2)0 1000 2000 3000 4000 5000 6000 7000

(b)

750

700

650

600

5500 1000 2000 3000 4000

Elev

atio

n (m

)

Distance (m) (3)5000 6000 7000

(c)

750

700

650

600

550El

evat

ion

(m)

Distance (m) (4)0 1000 2000 3000 4000 5000 6000 7000

(d)

Figure 4 Four stages for pigging process

0

0

2 4 6 8 10Time (h)

Position (m)

12 14 16 18 20

01

1

10

100

80007000600050004000300020001000550

600

650

700

750

Elevation

Elev

atio

n (m

)

Pressure (MPa)

Pres

sure

(log

10 M

Pa)

6930 m

Figure 5 Numerical result of pressure at the end of pipe for the firstaccident

to 3050MPa as shown in Figure 5 The value of the impulsepressure is larger than the bursting pressure of the tube andtherefore it caused the rupture

There is a large slope at the end of pipe so that pressurevaried with the liquid level in the pipe In other wordspressure would decrease when the liquid level dropped andpressure would increase when the liquid level rose Accordingto the results the liquid level quickly dropped near 9 h and28 h and the liquid level rose near 11 h because the flow rateof outlet and top was varying during the pigging processTheflow rate at the outlet was much larger than that at the top

Time (h)

01

1

10

100

0 4 8 12 16 20 24 28 32 36

Pres

sure

(log

10 M

Pa)

0 80007000600050004000300020001000550

600

650

700

750

Elev

atio

n (m

)

6930 m

ElevationPressure (MPa)

Position (m)

Figure 6 Numerical result of pressure at the end of pipe for thesecond accident

when the liquid level quickly dropped On the contrary theflow rate of top was much larger than that at the outlet whenthe liquid level quickly rose

The simulated time of the first pigging operation was 196hours During this period of time the maximum pressureof compressed air was as much as 103MPa as shown inFigure 7 Finally the gas cavity collapsed resulting in animpulse pressure up to 3703MPa as shown in Figure 6 Thevalue of the impulse pressure is also larger than the burstingpressure of the tube and the second accident occurred asshown in Table 3


Table 3 Summary of numerical simulation of the two accidents

The firstaccident

The secondaccident

Number of compressor 2 1Total time of pigging process 196 Hours 367 HoursMaximum of air pressure 103MPaMaximum of outlet pressure 3050MPa 3703MPaBursting pressure 2083MPa 2083MPa

Table 4 Summary of the field trial and numerical simulation

The field trial Numericalsimulation

Number of compressor 1 1Total time of pigging process 36 Hours 355 HoursValue of the impulse pressure 1576MPa 151MPaBursting pressure 2083MPa 2083MPa

32 Comparison of Field Results to Simulating PredictionsTo study the variation of outlet pressure due to the piggingprocess a field trial was conducted in 2011 by Luo [26] fromChina National Petroleum Corporation (CNPC) After twoaccidents the third operation was conducted and transientpressures were recorded by two pressure transducers whichwere installed at the third and fourth tubes from the endrespectively In this experiment the NS-B pressure transduc-ers are 50MPa in measurement range and 03 in precisioninstalled with an angle of 90 degrees to assure the resultsvalidity While the drainage tube is 159mm in diameter forthe first two pigging processes the last tube of pipe which is1219mm in diameter has four bores of 600mm in diameterfor drainage and the total area of the cross sections of thefour bores will be equal to the area of the cross section of thetube

As Figure 8 and Figure 9 show the simulated time of thefirst pigging operation was 196 hours and the maximumpressure was as much as 151MPa However the pressure atthe end was constantly kept in lower level for most of thetime Because the pressure at the location of cavitation isusually below the saturated vapor pressure the vapor pressureof the liquid is adopted as the cavitation inception pressurein this mathematical model for transient cavitation Whenthe pressure is under the saturated vapor pressure the waterwould change into vapor Furthermore as it was below thevapor pressure for a long period of time the air dissolved inthe water is released and thus there would be a large cavityof the water vapor and air When the pig was close to the endof the pipe some amount of gas downstream would undergoextrusion process and eventually an eruption of the mixtureof water and gas appeared at the outlet according to fieldrecords

With comparison between model prediction and fielddata it was found that the amplitudes of the impulse pressuresinduced by the collapse of the cavitywere nearly the same andtheir total time was 36 hours and 355 hours (see Figure 10)respectively as shown in Table 4

00010203040506070809101112

Pres

sure

(MPa

)

12 16 20 24 28 32 364 8Time (h)

Figure 7 Numerical result of air pressure for the second accident

0 12 16 20 24 28 32 36

05

0

10

15

20Pr

essu

re (M

Pa)

Time (h)4 8

Figure 8 Numerical result of pressure at the end of pipe for the fieldtrial

4 Conclusions

Themodel proposed for the pigging process was employed tounderstand the flow dynamics in the pipeline and to obtaintransient pressures for the two accidents and the field trialThe large cavity of water vapor and air was in the down-slope pipe following the peak and slack line flow that exitedfor a long period of time When the pig was close to theoutlet the cavity was compressed and gas underwent theextrusion process and eventually was dispersed into waterin the form of bubbles Finally the cavity collapsed by thepig and the serious collision resulted in considerable impulsepressures The results of the simulation illustrate that theimpulse pressures caused the severe damages during thepigging process Additionally our model predictions for thethird operation showed good agreement with field data andthe diameter of drainage had a significant effect on impulse


3450 3475 3500 3525 3550 3575 3600

00

05

10

15

20

Pres

sure

(MPa

)

Time (h)

6930 (560633)

Figure 9 Numerical result of the impulse pressure for the field trial

00

10

05

15

20

Pres

sure

(MPa

)

2510 2515 2520 2525Time (h)

Figure 10 Result of the impulse pressure for the field trial

pressures Generally the terrain of pipeline is a key factor forthe liquid-fill flow behavior and slack line flowmay appear ina long-slope pipeline Then column separation would occuras a common phenomenon for a hilly pipeline and it cancause devastating effects such as severe damages Thereforepreventive measures are of a critical significance for practicalreasons

Nomenclature

119891 Friction factor119909 Distance along the pipeline (m)119892 Gravity acceleration (m sdot sminus2)119905 Time (s)119860 Cross-section area of pipeline (m2)119898 The index number of Darcy formula119867 Water head of fluid (m)119886 Acoustic speed of fluid (ms)

119876 Volume rate of fluid (m3s)119876119860119876119861 Volume rates for given points (m3s)

119867119860119867119861 Water heads for given points (m)

119876119875 Volume rate for unknown point (m3s)

119867119875 Volume rate for unknown point (m)

119881 Velocity (ms)119863 Diameter of the pipeline (mm)] Kinematic viscosity of liquid ((m2s))119875119881 Saturated vapor pressure at liquid

temperature (Pa)120588 Density of liquid (kgm3)119872 Pig mass (kg)1198751 The pressure on the upstream faces of the pig

(Pa)1198752 The pressure on the downstream faces of the

pig (Pa)120573 A angle between axis and horizontal

direction (rad)Δ119875119904 The axial contact force (Pa)

120572 The volumetric fraction of liquid120588V Vapor phase density (kgm3)120588119897 Liquid phase density (kgm3)

120588119898 The mean density (kgm3)

Abbreviations

MOC The method of characteristicIAHR International Association For Hydraulic ResearchHC Hydrodynamic cavitation

Acknowledgments

The authors thank the National Science amp TechnologySpecific Project (Grant no 2011ZX05039-002) and theKey National Science and Technology Specific Project(2011ZX05026-004-03) the National Natural Science Foun-dation of China (51104167) for their financial support

References

[1] A Bergant A R Simpson and A S Tijsseling ldquoWater hammerwith column separation a historical reviewrdquo Journal of Fluidsand Structures vol 22 no 2 pp 135ndash171 2006

[2] R A Baltzer A study of column separation accompanyingtransient flow of liquid in pipes [PhD thesis] The University ofMichigan Ann Arbor Mich USA 1967

[3] R A Baltzer ldquoColumn separation accompanying liquid tran-sients in pipesrdquo ASME Journal of Basic Engineering D vol 89pp 837ndash846 1967

[4] V L Streeter and E B Wylie Hydraulic Transients McGraw-Hill Book Company New York NY USA 1967

[5] J A Swaffield ldquoA study of column separation following valveclosure in a pipeline carrying aviation kerosinerdquo Proceedings ofthe Institution of Mechanical Engineers vol 184 no 7 pp 57ndash641969

[6] E B Wylie V L Streeter and L SuoJones Fluid Transients inSystems chapter 3 Prentice Hall Enalewood Cliffs NJ USA1993


[7] T G Beuthe ldquoReview of two-phase water hammerrdquo in Pro-ceedings of the 18th Canadian Nuclear Society Conference p 20Toronto Canada 1997

[8] R W Angus ldquoSimple graphical solution for pressure rise inpipes and discharge linesrdquo Journal of Engineering Institute ofCanada vol 18 no 2 pp 72ndash81 1935

[9] N E Joukowsky ldquoWater Hammerrdquo Proceedings of the AmericanWater Works Association vol 24 pp 314ndash424 1904

[10] L AllieviGeneralTheory of Perturbed Flow ofWater in PressureConduits Annali della Societa degli Ingegneri ed architettiitaliani Milano Italy 1903

[11] L Allievi Theory of Water Hammer Notes I to V ASME NewYork NY USA 1913

[12] N R Gibson ldquoPressure in penstocks caused by gradual closingof turbine gaterdquo Transactions of the American Society of CivilEngineers vol 83 pp 707ndash775 1919

[13] R W Angus ldquoWater hammer in pipes includin those suppliedby centrifugal pumps graphical treatmentrdquo Proceedings of theInstitution of Mechanical Engineers vol 136 pp 245ndash331 1937

[14] I COrsquoNeillWaterHammer in Simple Pipe Systems [MS thesis]Department of civil Engineering University of MelbourneVictoria Australia 1959

[15] A R Simpson and E B Wylie ldquoProblems encountered inmodeling vapor column separationrdquo in Proceedings of theSymposium on Fluid Transients in Fluid-Structure Interactionpp 103ndash107 ASME Miami Beach Fla USA November 1985

[16] J P Kalkwijk and C Kranenburg ldquoCavitation in horizontalpipelines due to water hammerrdquoASCE Journal of the HydraulicsDivsion vol 97 no 10 pp 1585ndash1605 1971

[17] C Kranenburg ldquoTransient cavitation in pipelinesrdquo Report73-2 Laboratory of Fluid Mechanics Communications onHydraulics Department of Civil Engineering Delft Universityof Technology 1973

[18] C Kranenburg ldquoGas release during transient cavitation inpipesrdquo ASCE Jouranl of the Hydraulic Division vol 100 no 10pp 1383ndash1398 1974

[19] C Jaeger L S Kerr and E B Wylie ldquoSelected bibliographyrdquo inProceedings of the International Symposium on Water Hammerin Pumped Storage Projects ASMEWinter Annual Meeting pp233ndash241 Chicago ILL USA 1965

[20] C S Martin ldquoStatus of fluid transients in W estern Europe andthe United Kingdom report on laboratory visits by Freemanscholarrdquo ASME Journal of Fluids Engineering vol 95 pp 301ndash318 1973

[21] A R D Thorley A Survey of Investigations into Pressure SurgePhenomena The City University Department of MechanicalEngineering London UK 1976

[22] J-J Shu ldquoModelling vaporous cavitation on fluid transientsrdquoInternational Journal of Pressure Vessels and Piping vol 80 no3 pp 187ndash195 2003

[23] CKranenburg ldquoThe effect of gas release on column separationrdquoReport 74-3 Delft University of Technology Department ofCivil Engineering 1974 Communication on Hydraulics

[24] V L Streeter ldquoTransient cavitation pipe flowrdquo ASCE Journal ofHydraulic Engineering vol 109 no 11 pp 1407ndash1423 1983

[25] M H Afshar and M Rohani ldquoWater hammer simulationby implicit method of characteristicrdquo International Journal ofPressure Vessels and Piping vol 85 no 12 pp 851ndash859 2008

[26] J Luo ldquoAn analysis of the problems of fracture at new great dropline piperdquo Oil and Gas Storage and Transportation pp 1000ndash8241 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


During the course of a transient usually at high point thepressure in a pipeline falls to the vapor pressure of waterresulting in a localized liquid column separation [2ndash5] orvaporous cavitation Streeter andWylie in their textbooks [46] (Wylie and Streeter 1978a) summarized previous work oncolumn separation in detail Beuthe [7] provided an extensivegeneral review with emphasis on steam condensation Forvaporous cavitation vapor is distributed along a portion ofa pipeline rather than being concentrated at one locationas for a localized liquid column separation However theformation and collapse of a vapor cavity in a pipeline maylead to unexpectedly high pressure rises in the form ofshort-duration pressure pulses Angus [8] emphasized thatthe damage done to pipes as a result of water hammer isso serious that no engineer can afford to neglect it in thedesign of long pipes particularly those under low headsMost of the early efforts in liquid transients in pipelinesemphasized particularly on the prediction of the maximumpressure rise due to closure of a valve [9ndash11] or shutdown ofhydropower turbines [12] Joukowskyrsquos simple water hammerformula is used to determine the maximum head rise due tothe instantaneous closure of a valve without considerationof liquid column separation or linepack during a transientevent When a local column separation forms the pressuremagnitude produced by the collapse of the cavity is greaterthan the Joukowsky rise which is referred to as a short-duration pressure pulse Previous analytical studies [8 13ndash15] predicted that the pressure rise due to the collapse of avapor cavity may exceed the Joukowsky pressure rise Muchprevious experimental data showed strong attenuation of themaximum pressure rise upon the first collapse of a cavity ata valve [16ndash18] Jaeger et al [19] Martin [20] and Thorley[21] were devoted to comprehensive bibliographies of thehistorical development of many aspects of water hammerincluding column separation As previously mentioned col-umn separation is a common approach in transmission linemodeling However such an approach is oversimplistic andcan lead to unrealistic results Shu [22] proposed the two-phase homogeneous equilibrium vaporous cavitation modelwhich could avoid unrealistically high pressure spikes withconsideration of frequency-dependent friction

A pipe is divided in to two parts including the upstreamair section and the downstream liquid section by a pigBecause water directly discharges into atmospheric environ-ment the liquid flow will be under depressurized conditionAdditionally compressed air at upstream is to be at lowpressure In order to ensure that the pig is moving forwardthe pressure is just greater than the resisting force actingon the pig The pig slowly moves and has great influenceon the hydrodynamic pressure of the downstream flowAccordingly the pressure downstream may drop to or belowvapor pressure and it would result in a large cavity ofvapor usually at high points In previous researches [23 24]cavities were assumed to form only at high points at changesin pipeline slope (convex up) and at system boundariesMoreover the gas and liquid two-phase flow may appear inthe sloping pipe where the hydraulic grade line is found to beat or below the elevation of vapor pressure head Ultimatelysever accident may occur when column separation collapses

(1)

(2)

(3)

Figure 1 Schematic diagram of collapse of a cavity by a pig

by the heavy pig at high speed Due to the factors such asthe terrains a large number of bubbles will be produced andthe flow pattern is the bubbly flow with the intense pressureoscillations at the end of the pipelineThe large cavity of vaporand air between the pig and downstream liquid-filled pipewill be compressed at the downhill when the pig runs fast afterthe high pointThe compression is an isothermal process andthe gas would undergo the extrusion process at the end of theoutlet Eventually the gas is dispersed into water in the formof bubbles as shown in Figure 1

2 Mathematical Methods

21 Conventional MOC One-dimensional continuity andmomentum equations are applied to analyze the hydraulictransients A number of approaches have been introducedfor the simulation of the pipeline transients includingthe method of characteristics (MOC) wave characteristicsmethod (WCM) finite volume method (FVM) [25] finiteelementmethod (FEM) and finite differencemethod (FDM)Among these methods MOC is extensively used due to itssimplicity It is an explicit method and a powerful tool to ana-lyze hydraulic transients in pipeline flow With the methodof characteristics (MOC) the partial differential equa-tions can be converted into ordinary differential equations




120597119867

120597119905+ 119881

120597119867

120597119909+ (

1198862

119892119860)120597119876

120597119909= 0 (1)


1

119892119860(120597119876

120597119905+ 119881

120597119876

120597119909) +

120597119867

120597119909+ 119891119876|119876|

1minus119898= 0 (2)

where

119891 = 00246]119898

1198635minus119898 (3)


119862+

119889119909

119889119905= 119881 + 119886

119886

119892119860119889119876 + 119889119867 + 119891119876|119876|

1minus119898119886119889119905 = 0

119862minus

119889119909

119889119905= 119881 minus 119886

119886

119892119860119889119876 minus 119889119867 + 119891119876|119876|

1minus119898119886119889119905 = 0

(4)



119861119891119876|119876|

1minus119898119889119909 can be


119876|119876|1minus119898

= 119876119875

100381610038161003816100381611987611986010038161003816100381610038161minus119898

119876|119876|1minus119898

= 119876119875

100381610038161003816100381611987611986110038161003816100381610038161minus119898

(5)


119862+

Δ119909

Δ119905= +119886

119886

119892119860(119876119875minus 119876119860) + (119867

119875minus 119867119860) +119891119876

119875

100381610038161003816100381611987611986010038161003816100381610038161minus119898

119886Δ119905 = 0

119862minus

Δ119909

Δ119905= minus119886

119886

119892119860(119876119875minus 119876119861) minus (119867

119875minus 119867119861) + 119891119876

119875

100381610038161003816100381611987611986110038161003816100381610038161minus119898

119886Δ119905 = 0

(6)

t0 minus Δt

t0

t0 + Δt

t

x

A

F G

E

P

B

Cminus

C+



119862+ 119867119875= 119877119860minus 119878119860119876119875

119862minus 119867119875= 119877119861+ 119878119861119876119875

(7)

where

119877119860= 119867119860+ 119862119882119876119860

119877119861= 119867119861minus 119862119882119876119861

119878119860= 119862119882+ 119891

100381610038161003816100381611987611986010038161003816100381610038161minus119898

119886Δ119905

119878119861= 119862119882+ 119891

100381610038161003816100381611987611986110038161003816100381610038161minus119898

119886Δ119905

119862119882=

119886

119892119860

(8)


119875



time 119905 and119867119860and119867






1

1198862

120597119875

120597119905+ (120588119897minus 120588V)

120597120572

120597119905+

120588119898

12058711990320

120597

120597119909(119876

120572) = 0 (9)

120588119898

12058711990320

120597

120597119905(119876

120572) +

120597119875

120597119909+ 1198650(119876

120572 120572) + 120588

119898119892 sin 120579

0= 0 (10)



120588119898= 120572120588119897+ (1 minus 120572) 120588V (11)



1198650(119876

120572 120572) =

119891120588119898119876 |119876|

41205872120572211990350

(12)


119862+

1

1205871199032119900

119889

119889119905(119876

120572) +

1

120588119897119886

119889

119889119905(119875 minus 119901V) + 119886

120597

120597119905(ln

120588119898

120588119897

)

+119891119876 |119876|

41205872120572211990350

+ 119892 sin 1205790= 0

119889119909

119889119905= 119886

119862minus

1

1205871199032119900

119889

119889119905(119876

120572) minus

1

120588119897119886

119889

119889119905(119875 minus 119901V) minus 119886

120597

120597119905(ln

120588119898

120588119897

)

+119891119876 |119876|

41205872120572211990350

+ 119892 sin 1205790= 0

119889119909

119889119905= minus119886

(13)


119862+

1

12058711990320

119876119901

120572119901

+1

120588119897119886(119875119901minus 119875119881) +

119886

2ln

120588119898119901

120588119897

= 119862119860 (14)

119862minus

1

12058711990320

119876119901

120572119901

minus1

120588119897119886(119875119901minus 119875119881) minus

119886

2ln

120588119898119901

120588119897

= 119862119861 (15)

where119876119901 119875119901 and 120572

119863= (120588119898119863



119861are constant

119862119860=

1

12058711990320

(119876119860

120572119860

) +1

120588119897119886(119875119860minus 119875119881) +

119886

2ln

120588119898119864120588119898119865

120588119897120588119898119860

minus119891Δ119909119876

119860

10038161003816100381610038161198761198601003816100381610038161003816

41205872120572211986011990350119886

minus119892Δ119909 sin 120579

0

119886

119862119861=

1

12058711990320

(119876119861

120572119861

) +1

120588119897119886(119875119861minus 119875119881) +

119886

2ln

120588119898119864120588119898119866

120588119897120588119898119861

minus119891Δ119909119876

119861

10038161003816100381610038161198761198611003816100381610038161003816

41205872120572211986111990350119886

minus119892Δ119909 sin 120579

0

119886

(16)


If 119862119860ge 119862119861 then 120572

119901= 1

119875119901=120588119897119886

2(119862119860minus 119862119861) + 119875119881 (17)

If 119862119860lt 119862119861119875119901= 119875119881 then

120572119901=120588119897exp ((119862

119860minus 119862119861) 119886) minus 120588V

120588119897minus 120588119881

(18)

In either case

119876119901=1205871199032

0120572119901

2(119862119860+ 119862119861) (19)


119897=


119881= 2340Pa and Δ119909 = 110m are


119860and 119862





1198751= 1198752+ Δ119875119904+1

119860119872 sdot 119892 sdot sin 120579

0+119872

119860

119889119881

119889119905 (20)


PigP1

P2

ΔPs

ΔPs

1205790





where 119881 119872 1198751 119875 and 120579



119904











The firstaccident

The secondaccident



Φ1590 times87mm









0

750

700

650

600

5501000 2000 3000 4000 5000 6000 7000

Distance (m) (1)

Elev

atio

n (m

)

(a)

750

700

650

600

550

Elev

atio

n (m

)

Distance (m) (2)0 1000 2000 3000 4000 5000 6000 7000

(b)

750

700

650

600

5500 1000 2000 3000 4000

Elev

atio

n (m

)

Distance (m) (3)5000 6000 7000

(c)

750

700

650

600

550El

evat

ion

(m)

Distance (m) (4)0 1000 2000 3000 4000 5000 6000 7000

(d)


0

0

2 4 6 8 10Time (h)

Position (m)

12 14 16 18 20

01

1

10

100

80007000600050004000300020001000550

600

650

700

750

Elevation

Elev

atio

n (m

)

Pressure (MPa)

Pres

sure

(log

10 M

Pa)

6930 m




Time (h)

01

1

10

100

0 4 8 12 16 20 24 28 32 36

Pres

sure

(log

10 M

Pa)

0 80007000600050004000300020001000550

600

650

700

750

Elev

atio

n (m

)

6930 m


Position (m)






The firstaccident

The secondaccident








00010203040506070809101112

Pres

sure

(MPa

)

12 16 20 24 28 32 364 8Time (h)


0 12 16 20 24 28 32 36

05

0

10

15

20Pr

essu

re (M

Pa)

Time (h)4 8


4 Conclusions



3450 3475 3500 3525 3550 3575 3600

00

05

10

15

20

Pres

sure

(MPa

)

Time (h)

6930 (560633)


00

10

05

15

20

Pres

sure

(MPa

)

2510 2515 2520 2525Time (h)



Nomenclature













Abbreviations


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120597119867

120597119905+ 119881

120597119867

120597119909+ (

1198862

119892119860)120597119876

120597119909= 0 (1)


1

119892119860(120597119876

120597119905+ 119881

120597119876

120597119909) +

120597119867

120597119909+ 119891119876|119876|

1minus119898= 0 (2)

where

119891 = 00246]119898

1198635minus119898 (3)


119862+

119889119909

119889119905= 119881 + 119886

119886

119892119860119889119876 + 119889119867 + 119891119876|119876|

1minus119898119886119889119905 = 0

119862minus

119889119909

119889119905= 119881 minus 119886

119886

119892119860119889119876 minus 119889119867 + 119891119876|119876|

1minus119898119886119889119905 = 0

(4)



119861119891119876|119876|

1minus119898119889119909 can be


119876|119876|1minus119898

= 119876119875

100381610038161003816100381611987611986010038161003816100381610038161minus119898

119876|119876|1minus119898

= 119876119875

100381610038161003816100381611987611986110038161003816100381610038161minus119898

(5)


119862+

Δ119909

Δ119905= +119886

119886

119892119860(119876119875minus 119876119860) + (119867

119875minus 119867119860) +119891119876

119875

100381610038161003816100381611987611986010038161003816100381610038161minus119898

119886Δ119905 = 0

119862minus

Δ119909

Δ119905= minus119886

119886

119892119860(119876119875minus 119876119861) minus (119867

119875minus 119867119861) + 119891119876

119875

100381610038161003816100381611987611986110038161003816100381610038161minus119898

119886Δ119905 = 0

(6)

t0 minus Δt

t0

t0 + Δt

t

x

A

F G

E

P

B

Cminus

C+



119862+ 119867119875= 119877119860minus 119878119860119876119875

119862minus 119867119875= 119877119861+ 119878119861119876119875

(7)

where

119877119860= 119867119860+ 119862119882119876119860

119877119861= 119867119861minus 119862119882119876119861

119878119860= 119862119882+ 119891

100381610038161003816100381611987611986010038161003816100381610038161minus119898

119886Δ119905

119878119861= 119862119882+ 119891

100381610038161003816100381611987611986110038161003816100381610038161minus119898

119886Δ119905

119862119882=

119886

119892119860

(8)


119875



time 119905 and119867119860and119867






1

1198862

120597119875

120597119905+ (120588119897minus 120588V)

120597120572

120597119905+

120588119898

12058711990320

120597

120597119909(119876

120572) = 0 (9)

120588119898

12058711990320

120597

120597119905(119876

120572) +

120597119875

120597119909+ 1198650(119876

120572 120572) + 120588

119898119892 sin 120579

0= 0 (10)



120588119898= 120572120588119897+ (1 minus 120572) 120588V (11)



1198650(119876

120572 120572) =

119891120588119898119876 |119876|

41205872120572211990350

(12)


119862+

1

1205871199032119900

119889

119889119905(119876

120572) +

1

120588119897119886

119889

119889119905(119875 minus 119901V) + 119886

120597

120597119905(ln

120588119898

120588119897

)

+119891119876 |119876|

41205872120572211990350

+ 119892 sin 1205790= 0

119889119909

119889119905= 119886

119862minus

1

1205871199032119900

119889

119889119905(119876

120572) minus

1

120588119897119886

119889

119889119905(119875 minus 119901V) minus 119886

120597

120597119905(ln

120588119898

120588119897

)

+119891119876 |119876|

41205872120572211990350

+ 119892 sin 1205790= 0

119889119909

119889119905= minus119886

(13)


119862+

1

12058711990320

119876119901

120572119901

+1

120588119897119886(119875119901minus 119875119881) +

119886

2ln

120588119898119901

120588119897

= 119862119860 (14)

119862minus

1

12058711990320

119876119901

120572119901

minus1

120588119897119886(119875119901minus 119875119881) minus

119886

2ln

120588119898119901

120588119897

= 119862119861 (15)

where119876119901 119875119901 and 120572

119863= (120588119898119863



119861are constant

119862119860=

1

12058711990320

(119876119860

120572119860

) +1

120588119897119886(119875119860minus 119875119881) +

119886

2ln

120588119898119864120588119898119865

120588119897120588119898119860

minus119891Δ119909119876

119860

10038161003816100381610038161198761198601003816100381610038161003816

41205872120572211986011990350119886

minus119892Δ119909 sin 120579

0

119886

119862119861=

1

12058711990320

(119876119861

120572119861

) +1

120588119897119886(119875119861minus 119875119881) +

119886

2ln

120588119898119864120588119898119866

120588119897120588119898119861

minus119891Δ119909119876

119861

10038161003816100381610038161198761198611003816100381610038161003816

41205872120572211986111990350119886

minus119892Δ119909 sin 120579

0

119886

(16)


If 119862119860ge 119862119861 then 120572

119901= 1

119875119901=120588119897119886

2(119862119860minus 119862119861) + 119875119881 (17)

If 119862119860lt 119862119861119875119901= 119875119881 then

120572119901=120588119897exp ((119862

119860minus 119862119861) 119886) minus 120588V

120588119897minus 120588119881

(18)

In either case

119876119901=1205871199032

0120572119901

2(119862119860+ 119862119861) (19)


119897=


119881= 2340Pa and Δ119909 = 110m are


119860and 119862





1198751= 1198752+ Δ119875119904+1

119860119872 sdot 119892 sdot sin 120579

0+119872

119860

119889119881

119889119905 (20)


PigP1

P2

ΔPs

ΔPs

1205790





where 119881 119872 1198751 119875 and 120579



119904











The firstaccident

The secondaccident



Φ1590 times87mm









0

750

700

650

600

5501000 2000 3000 4000 5000 6000 7000

Distance (m) (1)

Elev

atio

n (m

)

(a)

750

700

650

600

550

Elev

atio

n (m

)

Distance (m) (2)0 1000 2000 3000 4000 5000 6000 7000

(b)

750

700

650

600

5500 1000 2000 3000 4000

Elev

atio

n (m

)

Distance (m) (3)5000 6000 7000

(c)

750

700

650

600

550El

evat

ion

(m)

Distance (m) (4)0 1000 2000 3000 4000 5000 6000 7000

(d)


0

0

2 4 6 8 10Time (h)

Position (m)

12 14 16 18 20

01

1

10

100

80007000600050004000300020001000550

600

650

700

750

Elevation

Elev

atio

n (m

)

Pressure (MPa)

Pres

sure

(log

10 M

Pa)

6930 m




Time (h)

01

1

10

100

0 4 8 12 16 20 24 28 32 36

Pres

sure

(log

10 M

Pa)

0 80007000600050004000300020001000550

600

650

700

750

Elev

atio

n (m

)

6930 m


Position (m)






The firstaccident

The secondaccident








00010203040506070809101112

Pres

sure

(MPa

)

12 16 20 24 28 32 364 8Time (h)


0 12 16 20 24 28 32 36

05

0

10

15

20Pr

essu

re (M

Pa)

Time (h)4 8


4 Conclusions



3450 3475 3500 3525 3550 3575 3600

00

05

10

15

20

Pres

sure

(MPa

)

Time (h)

6930 (560633)


00

10

05

15

20

Pres

sure

(MPa

)

2510 2515 2520 2525Time (h)



Nomenclature













Abbreviations


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1

1198862

120597119875

120597119905+ (120588119897minus 120588V)

120597120572

120597119905+

120588119898

12058711990320

120597

120597119909(119876

120572) = 0 (9)

120588119898

12058711990320

120597

120597119905(119876

120572) +

120597119875

120597119909+ 1198650(119876

120572 120572) + 120588

119898119892 sin 120579

0= 0 (10)



120588119898= 120572120588119897+ (1 minus 120572) 120588V (11)



1198650(119876

120572 120572) =

119891120588119898119876 |119876|

41205872120572211990350

(12)


119862+

1

1205871199032119900

119889

119889119905(119876

120572) +

1

120588119897119886

119889

119889119905(119875 minus 119901V) + 119886

120597

120597119905(ln

120588119898

120588119897

)

+119891119876 |119876|

41205872120572211990350

+ 119892 sin 1205790= 0

119889119909

119889119905= 119886

119862minus

1

1205871199032119900

119889

119889119905(119876

120572) minus

1

120588119897119886

119889

119889119905(119875 minus 119901V) minus 119886

120597

120597119905(ln

120588119898

120588119897

)

+119891119876 |119876|

41205872120572211990350

+ 119892 sin 1205790= 0

119889119909

119889119905= minus119886

(13)


119862+

1

12058711990320

119876119901

120572119901

+1

120588119897119886(119875119901minus 119875119881) +

119886

2ln

120588119898119901

120588119897

= 119862119860 (14)

119862minus

1

12058711990320

119876119901

120572119901

minus1

120588119897119886(119875119901minus 119875119881) minus

119886

2ln

120588119898119901

120588119897

= 119862119861 (15)

where119876119901 119875119901 and 120572

119863= (120588119898119863



119861are constant

119862119860=

1

12058711990320

(119876119860

120572119860

) +1

120588119897119886(119875119860minus 119875119881) +

119886

2ln

120588119898119864120588119898119865

120588119897120588119898119860

minus119891Δ119909119876

119860

10038161003816100381610038161198761198601003816100381610038161003816

41205872120572211986011990350119886

minus119892Δ119909 sin 120579

0

119886

119862119861=

1

12058711990320

(119876119861

120572119861

) +1

120588119897119886(119875119861minus 119875119881) +

119886

2ln

120588119898119864120588119898119866

120588119897120588119898119861

minus119891Δ119909119876

119861

10038161003816100381610038161198761198611003816100381610038161003816

41205872120572211986111990350119886

minus119892Δ119909 sin 120579

0

119886

(16)


If 119862119860ge 119862119861 then 120572

119901= 1

119875119901=120588119897119886

2(119862119860minus 119862119861) + 119875119881 (17)

If 119862119860lt 119862119861119875119901= 119875119881 then

120572119901=120588119897exp ((119862

119860minus 119862119861) 119886) minus 120588V

120588119897minus 120588119881

(18)

In either case

119876119901=1205871199032

0120572119901

2(119862119860+ 119862119861) (19)


119897=


119881= 2340Pa and Δ119909 = 110m are


119860and 119862





1198751= 1198752+ Δ119875119904+1

119860119872 sdot 119892 sdot sin 120579

0+119872

119860

119889119881

119889119905 (20)


PigP1

P2

ΔPs

ΔPs

1205790





where 119881 119872 1198751 119875 and 120579



119904











The firstaccident

The secondaccident



Φ1590 times87mm









0

750

700

650

600

5501000 2000 3000 4000 5000 6000 7000

Distance (m) (1)

Elev

atio

n (m

)

(a)

750

700

650

600

550

Elev

atio

n (m

)

Distance (m) (2)0 1000 2000 3000 4000 5000 6000 7000

(b)

750

700

650

600

5500 1000 2000 3000 4000

Elev

atio

n (m

)

Distance (m) (3)5000 6000 7000

(c)

750

700

650

600

550El

evat

ion

(m)

Distance (m) (4)0 1000 2000 3000 4000 5000 6000 7000

(d)


0

0

2 4 6 8 10Time (h)

Position (m)

12 14 16 18 20

01

1

10

100

80007000600050004000300020001000550

600

650

700

750

Elevation

Elev

atio

n (m

)

Pressure (MPa)

Pres

sure

(log

10 M

Pa)

6930 m




Time (h)

01

1

10

100

0 4 8 12 16 20 24 28 32 36

Pres

sure

(log

10 M

Pa)

0 80007000600050004000300020001000550

600

650

700

750

Elev

atio

n (m

)

6930 m


Position (m)






The firstaccident

The secondaccident








00010203040506070809101112

Pres

sure

(MPa

)

12 16 20 24 28 32 364 8Time (h)


0 12 16 20 24 28 32 36

05

0

10

15

20Pr

essu

re (M

Pa)

Time (h)4 8


4 Conclusions



3450 3475 3500 3525 3550 3575 3600

00

05

10

15

20

Pres

sure

(MPa

)

Time (h)

6930 (560633)


00

10

05

15

20

Pres

sure

(MPa

)

2510 2515 2520 2525Time (h)



Nomenclature













Abbreviations


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










PigP1

P2

ΔPs

ΔPs

1205790





where 119881 119872 1198751 119875 and 120579



119904











The firstaccident

The secondaccident



Φ1590 times87mm









0

750

700

650

600

5501000 2000 3000 4000 5000 6000 7000

Distance (m) (1)

Elev

atio

n (m

)

(a)

750

700

650

600

550

Elev

atio

n (m

)

Distance (m) (2)0 1000 2000 3000 4000 5000 6000 7000

(b)

750

700

650

600

5500 1000 2000 3000 4000

Elev

atio

n (m

)

Distance (m) (3)5000 6000 7000

(c)

750

700

650

600

550El

evat

ion

(m)

Distance (m) (4)0 1000 2000 3000 4000 5000 6000 7000

(d)


0

0

2 4 6 8 10Time (h)

Position (m)

12 14 16 18 20

01

1

10

100

80007000600050004000300020001000550

600

650

700

750

Elevation

Elev

atio

n (m

)

Pressure (MPa)

Pres

sure

(log

10 M

Pa)

6930 m




Time (h)

01

1

10

100

0 4 8 12 16 20 24 28 32 36

Pres

sure

(log

10 M

Pa)

0 80007000600050004000300020001000550

600

650

700

750

Elev

atio

n (m

)

6930 m


Position (m)






The firstaccident

The secondaccident








00010203040506070809101112

Pres

sure

(MPa

)

12 16 20 24 28 32 364 8Time (h)


0 12 16 20 24 28 32 36

05

0

10

15

20Pr

essu

re (M

Pa)

Time (h)4 8


4 Conclusions



3450 3475 3500 3525 3550 3575 3600

00

05

10

15

20

Pres

sure

(MPa

)

Time (h)

6930 (560633)


00

10

05

15

20

Pres

sure

(MPa

)

2510 2515 2520 2525Time (h)



Nomenclature













Abbreviations


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

750

700

650

600

5501000 2000 3000 4000 5000 6000 7000

Distance (m) (1)

Elev

atio

n (m

)

(a)

750

700

650

600

550

Elev

atio

n (m

)

Distance (m) (2)0 1000 2000 3000 4000 5000 6000 7000

(b)

750

700

650

600

5500 1000 2000 3000 4000

Elev

atio

n (m

)

Distance (m) (3)5000 6000 7000

(c)

750

700

650

600

550El

evat

ion

(m)

Distance (m) (4)0 1000 2000 3000 4000 5000 6000 7000

(d)


0

0

2 4 6 8 10Time (h)

Position (m)

12 14 16 18 20

01

1

10

100

80007000600050004000300020001000550

600

650

700

750

Elevation

Elev

atio

n (m

)

Pressure (MPa)

Pres

sure

(log

10 M

Pa)

6930 m




Time (h)

01

1

10

100

0 4 8 12 16 20 24 28 32 36

Pres

sure

(log

10 M

Pa)

0 80007000600050004000300020001000550

600

650

700

750

Elev

atio

n (m

)

6930 m


Position (m)






The firstaccident

The secondaccident








00010203040506070809101112

Pres

sure

(MPa

)

12 16 20 24 28 32 364 8Time (h)


0 12 16 20 24 28 32 36

05

0

10

15

20Pr

essu

re (M

Pa)

Time (h)4 8


4 Conclusions



3450 3475 3500 3525 3550 3575 3600

00

05

10

15

20

Pres

sure

(MPa

)

Time (h)

6930 (560633)


00

10

05

15

20

Pres

sure

(MPa

)

2510 2515 2520 2525Time (h)



Nomenclature













Abbreviations


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











The firstaccident

The secondaccident








00010203040506070809101112

Pres

sure

(MPa

)

12 16 20 24 28 32 364 8Time (h)


0 12 16 20 24 28 32 36

05

0

10

15

20Pr

essu

re (M

Pa)

Time (h)4 8


4 Conclusions



3450 3475 3500 3525 3550 3575 3600

00

05

10

15

20

Pres

sure

(MPa

)

Time (h)

6930 (560633)


00

10

05

15

20

Pres

sure

(MPa

)

2510 2515 2520 2525Time (h)



Nomenclature













Abbreviations


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










3450 3475 3500 3525 3550 3575 3600

00

05

10

15

20

Pres

sure

(MPa

)

Time (h)

6930 (560633)


00

10

05

15

20

Pres

sure

(MPa

)

2510 2515 2520 2525Time (h)



Nomenclature













Abbreviations


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Hydraulic Transients Induced by Pigging Operation in Pipeline … · 2018. 12....

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Transcript of Research Article Hydraulic Transients Induced by Pigging Operation in Pipeline … · 2018. 12....