Research Article Shock Wave Speed and Transient Response ...€¦ · Improved Wave Speed Formula...

Research ArticleShock Wave Speed and Transient Response of PE Pipe withSteel-Mesh Reinforcement

Wuyi Wan1 and Xinwei Mao2

1Department of Hydraulic Engineering College of Civil Engineering and Architecture Zhejiang University Hangzhou 310058 China2Research and Teaching Labs for Civil and Hydraulic Engineering College of Civil Engineering and ArchitectureZhejiang University Hangzhou 310058 China

Correspondence should be addressed to Wuyi Wan wanwuyizjueducn

Received 7 December 2015 Revised 1 May 2016 Accepted 5 May 2016

Academic Editor Tai Thai

Copyright copy 2016 W Wan and X MaoThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A steel mesh can improve the tensile strength and stability of a polyethylene (PE) pipe in a water supply pipeline system Howeverit can also cause more severe water hammer hazard due to increasing wave speed In order to analyze the influence of the steelmesh on the shock wave speed and transient response processes an improved wave speed formula is proposed by incorporatingthe equivalent elastic modulus A field measurement validates the wave speed formula Moreover the transient wave propagationand extreme pressures are simulated and compared by the method of characteristics (MOC) for reinforced PE pipes with varioussteel-mesh densities Results show that a steel mesh can significantly increase the shock wave speed in a PE pipe and thus can causesevere peak pressure and hydraulic surges in a water supply pipeline systemThe proposed wave speed formula canmore reasonablyevaluate the wave speed and improve the transient simulation of steel-mesh-reinforced PE pipes

1 Introduction

Water hammer events can cause noise vibration distortionand fracture in a water supply pipeline system due to suddenincrease or decrease of pressure [1 2] Numerical simulationand prediction are very important to protect the pipelinefrom undesired damage due to the water hammer effect [3]It is the fundamental to control the peak transient pressureby optimal operation [4ndash6] as well as select and designprotection devices inwater supply pipelines [7] Furthermoreit is also applicable to some other pipeline transportationareas such as oil pipelines [8 9] and natural gas pipelines [1011] Shock wave speed is a primary factor in various transientsimulation models [12ndash16] It represents the basic propertiesof the fluid and pipeline system in transient flow simulations[17] Usually wave speed can greatly change the frequencyand amplitude of the water hammer waves as well as theextreme transient pressure distributions along the pipeline[18] Therefore the reliability of the result depends heavilyon the wave speed in the numerical simulation of the waterhammer Generally the wave speed is subjected to manyfactors such as the density and elastic modulus of the fluid

the material [19 20] and shape of the pipe [21ndash23] and themeans of fixation of the pipe [24]Moreover the temperaturepressure and gas content can also affect the wave speedin a pipe system [25 26] In fact it is difficult to estimateaccurately the shock wave speed Since the presentation ofseveral basic formulas by Wylie and Streeter [18] for theshock wave speeds they are widely used in common pipesand conduits However it is inapplicable for some specificfluids and composite material pipes and so some modifiedapproaches have been developed to estimate the shock wavespeed for various areas Sun et al [24] presented the waterhammer speed of fiber-reinforced plastic composite pipesbased on three different fixed means Hachem and Schleiss[27] studied the influence of local wall stiffness decreaseon the shock wave speed by experiments and the resultshowed the transient pressure change with the wave speedHan et al [28] and Zhou et al [29] studied the shock wavespeed of a slurry flow carrying solid particles Hadj-Taıeband Lili [30] and Ando et al [31] studied the shock wavepropagation through a deformable tube Lee and Pejovic [26]studied the influence of air on the similarity of hydraulictransients and vibrations Soares et al [22] and Apollonio

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 8705031 10 pageshttpdxdoiorg10115520168705031

2 Shock and Vibration

et al [23] investigated separately the hydraulic transientproperties of a pipeline made of plastic material Mitosek[32] measured the shock wave speed of PVC pipes Especiallyfor polyethylene pipe Covas et al [19 33 34] developeda viscoelastic model to take account of the influences ofviscoelasticity on transient processes Evangelista et al [20]simulated complex plastic pipes system by viscoelastic modeland showed that the viscoelasticity needs to be consideredin the transient response of a plastic pipe These previousachievements are significant but more accurate simulationsare required to prevent the water hammer However con-sidering the influence of complicated steel-mesh structuresit is presently still difficult to reasonably estimate the wavespeed of a large PE pipe with steel-mesh reinforcement Inorder to improve the transient simulation of a PE pipe withsteel-mesh reinforcement we propose an improved wavespeed formula for reinforced PE pipes by incorporating theequivalent modulus Also we conduct a field test to validatethe approach Based on the proposed approach the transientresponses are simulated and compared for PE pipes withvarious steel-mesh reinforcement ratios Then the influenceof the steel mesh on the transient response is evaluated anddiscussed The result shows that the steel mesh can increasethe shock wave speed and cause more serve hydraulic surgesin the pipeline The proposed formula is significant to revisethe wave speed and improve the transient simulation of steel-mesh-reinforced PE pipe

2 Basic Shock Wave Speed Formula

Wave speed can primarily affect the wave propagation andtransient pressure and it is an important parameter forwater hammer simulations as shown in the simplified closedexpression equations for water hammer simulation [1]

120597V120597119905

+ V120597V120597119909

+ 119892120597ℎ

120597119909+119891V |V|2119863

= 0 (1)

120597ℎ

120597119905+ V

120597ℎ

120597119909minus V sin120573 +

1198862

0

119892

120597V120597119909

= 0(2)

For PE pipe considering the additional viscoelastic term[20 34] the improved continuity equation can be written as

120597ℎ

120597119905+ V

120597ℎ

120597119909minus V sin120573 +

1198862

0

119892

120597V120597119909

+

2 1198862

0

119892

119889120576119903

119889119905= 0

(3)

Generally the shock wave speed can be determined byYoungrsquos modulus of the fluid and the wall material for someregular water supply pipes Based on basic water hammerequation and mass conservation equations [1] provided abasic shock wave speed formula

1198862

0= (

119860

120588)(

119889119901

119889119860) (4)

Steel mesh PE pipe wall

Figure 1 Composition of a PE pipe with steel-mesh reinforcement

SteelPER

120575

B

Figure 2 Schematic of a steel-mesh-reinforced PE pipe

For general elastic pipes the shock wave speed formulacan be expressed as [1]

1198860=

radic119870120588

radic1 + 120572 (2119877120575) (119870119864) (5)

This formula is widely applicable to circular pipes How-ever for some complex pipe materials this formula mayneed to be improved As seen in this formula the modulusof elasticity is an important parameter that determines thewave speed A reinforced PE pipe has a composite structurewhere the steel mesh and polyethylene play different rolesAccordingly the formula cannot directly determine the wavespeed in reinforced PE Based on the conventional formulatherefore we will analyze the wave speed for reinforced PEpipe with steel-mesh reinforcement

3 Wave Speed Analysis in Reinforced PE Pipes

31 Composition of Steel-Mesh-Reinforced PE Pipe A generalPE pipe is made of polyethylene In order to improvethe strength steel meshes are embedded in the pipe wallCompared to the common PE pipe reinforced PE pipes havemore strength and are inflexible Figures 1 and 2 illustrate thebasic structure of a PE pipe with steel-mesh reinforcementThe pipe wall consists of PE and steel meshesThemodulus ofthemesh is much larger than that of PEThus the shock wavespeed changes greatly with the steel-mesh reinforcementIt is necessary to revise the shock wave speed due to the

Shock and Vibration 3

CP

120579

120579

minus120579

minus120579

B

Clockwise sp

iral

Anticlockwise spiral

Figure 3 Schematic of unfolded pipe wall structure

steel meshes in order to obtain a more reasonable transientsimulation result

32 Force Analysis of the Pipe Wall under Small DeformationIn order to determine the shock wave speed of a reinforcedPE pipe an equivalent modulus is proposed for the com-posite reinforced PE pipe wall Based on the conventionalwave speed formula the equivalent modulus is used as themodifiedmodulus of pipe wall In fact stress and strain occursimultaneously when a pipe is subjected to a water hammerwave Thus the transient wave propagation represents thevariations of fluid and pipe in pressure and deformation

Before deriving the equivalentmodulus wewillmake twoassumptions (1)Thedeformation is sufficiently small and canbe considered as elasticity in instantaneous time so that thestress and strain can satisfy Hookersquos law for both steel meshesand PE (2) For a finite deformation there is no relativedisplacement between the steel thread and the PE materialIn other words steel and PE have the same deformation tobear the extra pressure due to the water hammer pressure

We select a unit pipe of specific length around which asteel thread is wound As shown in Figure 3 the axial lengthof the pipe is calculated as

119861 = 119862119875cot 120579 (6)

119862119875= 2120587 (119877 + 05120575) (7)

Figure 4 shows the unfolded pipe wall When an internalhydraulic pressure 119901 acts on the pipe wall the tensile force isequal to the total horizontal component

119875 = 119863119861119901 (8)

In the specific length a spiral stirrup is considered asan equivalent hoop For a differential deformation the totalcircular force in steel mesh can be expressed as

119879s = 2119899119860 s120576119864s sin 120579 (9)

Simultaneously the circular force in the PE material canbe expressed as

119879p = 2 (119861120575 minus119899119860 ssin 120579

)119864p120576 (10)

33 Equivalent Modulus of PE Pipe with Steel-Mesh Reinforce-ment Based on the above assumptions the force of pipe wallcan be expressed as

119879 = 2 [119899119860 s119864s120576 sin 120579 + (119861120575 minus119899119860 ssin 120579

)119864p120576] (11)

If we define an equivalentmodulus119864e then the total forcecan be written as

119865e = 2119861120575119864e120576 (12)

Equations (10) and (11) represent the equal force in mag-nitude to (7) for an equilibrium state analysis Accordingly

2119861120575120576119864e = 2 [119899119860 s119864s120576 sin 120579 + (119861120575 minus119899119860 ssin 120579

)119864p120576] (13)

Then the equivalent modulus of the pipe wall can bedetermined as

119864e =119899119860 s119864s sin 120579 + (119861120575 minus 119899119860 s sin 120579) 119864p

119861120575 (14)

We define the pipe wall reinforcement ratio as

119901s =119899119860 s119861120575

(15)

The reinforcement ratio represents the ratio between thesteel areas and the wall section areas Substituting it into (13)the equivalent elastic modulus can be expressed as

119864e = 119901s sin 120579119864s + (1 minus 119901s csc 120579) (16)

34 Improved Wave Speed Formula for a PE Pipe with Steel-Mesh Reinforcement Equation (15) provides the equivalentYoungrsquos modulus in terms of the reinforcement ratio of thepipe wall That is to say we can deal with the reinforced PEpipe wall as a composite material with an equivalent elasticmodulus Accordingly the wave speed can be expressed as

1198860

=radic119870120588

radic1 + 120572 (2119877120575) (119870 (119901s sin 120579119864s + (1 minus 119901s csc 120579) 119864p))

(17)

Provided we know the number and winding angle of thespiral steel wire in pipe cross section the improved formulacan calculate the shock wave speed for a PE pipe with steel-mesh reinforcement


Table 1 Comparison of test and calculation

119877 (mm) 120575 (mm) 119901s () 120579 119864p (GPa) 119864s (GPa) 119864e (GPa) 119886n (ms) 119886m (ms) 120576119886

232 18 148 4226∘ 143 207 343 3850 3798 137

B120575120590

B120575120590

Steel mesh

PE pipe wall

Figure 4 Force analysis of the unfolded pipe wall

35 Validation of the Wave Speed Formula In fact it isdifficult to conduct an experiment to obtain the shock wavespeed for a large PE pipe with steel-mesh reinforcement inthe field Fortunately a practical project supports the field tovalidate the proposed formula As a result the experiment issubject to the practical scale and layout of the project Figure 5shows the schematic of pipeline and stations in the field testwhere 119904

1is a 2016 km PE pipe 119904

2is a 04 km steel pipe and

1199043is a 01 km steel pipe Moreover the PE pipe is 0232m in

internal radius and 0018m in thicknessIn the experiment the control valve will give rise to a neg-

ative pressure wave by sudden opening and discharging waterfrom the main pipe at initial time 119905

0 and then the negative

wave propagates along the pipe to the stations Stations 1 and2 detect the wave at times 119905

1and 1199052 respectively According

to the time and wave speed the following equations can bewritten

1199041

119886p+1199042

119886s= 1199051minus 1199050 (18)

1199041

119886p+1199042

119886s+1199043

119886s= 1199052minus 1199050 (19)

According to 1199050 1199051 and 119905

2 the experimental wave speed

can be obtained as follows

119886p =11990411199043

(1199051minus 1199050) (1199042+ 1199043) minus (1199052minus 1199050) 1199042

(20)

Figure 6 shows the pressure response at station 1 andstation 2 respectively The beginning time of the impulse 119905

0

is 2961 s and the response times 1199051and 1199052are respectively

83109 and 83214 s at station 1 and station 2 According to (19)the experimental wave speed can be obtained

As shown in Table 1 the experimental wave speed is3798ms According to the steel-mesh density and windingangle of the spiral steel wire the proposed theoretical formula

yields a collective wave speed 3850ms The error is about137 The comparison shows that the computational resultagrees well with themeasurementThe proposed formula canprovide a reasonable result for the PE pipe with steel-meshreinforcement

36 Influence of Reinforcement Ratio on Wave Speeds of a PEPipe As is well known the elastic modulus can greatly affectthe shock wave speed Since the reinforcement can primarilyaffect the elastic modulus it can also change the shock wavespeed Next the influence of the steel-mesh density on thewave speed is studied including the reinforcement ratios andwinding angles Based on the proposed formula Figure 7shows the influence of the reinforcement ratio on shock wavespeed for three different PE pipe walls For a given pipediameter wall thickness and winding angle of the steel wirethe wave speed increases with the reinforcement ratio Ifwe define reinforcement ratio as 0 it will indicate actuallythe original PE material pipe without steel meshes and thewave speed is only about 250ms Obviously the wave speedincreases with the reinforcement ratio Moreover for a givendiameter wall thickness and reinforcement ratio the wavespeed also increases with the winding angle of steel wire asshown in Figure 8 In fact for a specified reinforcement ratioin the section the density of the steel mesh increases withthe winding angle along the axial direction It shows thatthe wave speed changes because of the steel mesh enhancingthe elastic modulus of the PE pipe In practice the range ofthe reinforcement ratio is 1-2 and the winding angle ofa rhombic metal mesh is about 4226∘ accordingly the wavespeed in the PE pipe may increase to 300ndash400ms

4 Effects of Steel Mesh on TransientResponses of the PE Pipe

41 Selection of Transient Simulation Model Consideringthe viscoelasticity of PE materials the viscoelastic model isneeded to simulate the hydraulic transient processes Covaset al [34] developed a conventional viscoelastic model andthemodel is also well verified in complex plastic pipes system[20] Given the wave speed (1) and (3) form a closed systemfor thewater hammer simulations Considering the viscoelas-tic model [34] and coupling the method of characteristics [1]the modified MOC equation systems can be expressed as

119892

1198860

119889ℎ

119889119905+119889V119889119905

+119891V |V|2119863

+ 21198860(120597120576119903

120597119905) = 0

10038161003816100381610038161003816100381610038161003816119889119909119889119905=+1198860

minus119892

1198860

119889ℎ

119889119905+119889V119889119905

+119891V |V|2119863

+ 21198860(120597120576119903

120597119905) = 0

10038161003816100381610038161003816100381610038161003816119889119909119889119905=minus1198860

(21)


PE pipe with reinforcement mesh

Impulse generator (valve) Water

Steel pipe Steel pipe

Station 1 Station 2

s1 s2 s3

Figure 5 Schematic of wave speed measurement in the field

0 30 60 90 120 150 180061

062

063

064

065

066

Station 1

0 30 60 90 120 150 180061

062

063

064

065

066

p-t

Station 2

p(M

Pa)

t (s)

p(M

Pa)

t (s)

t0 = 2961 s t1 = 83109 s t0 = 2961 s t1 = 83214 s

p-t

Figure 6 Pressure surge in the stations

Integral along the characteristic line the system of equa-tions can be converted as

ℎ119894119905+Δ119905

+1198860

119892V119894119905+Δ119905

+ 21198862

0Δ119905

119892(120597120576119903

120597119905) minus 119862

119865= 0

ℎ119894119905+Δ119905

minus1198860

119892V119894119905+Δ119905

+ 21198862

0Δ119905

119892(120597120576119903

120597119905) minus 119862

119877= 0

(22)

In the equations119862119865= ℎ119894minus1119905

+V119894minus1119905

1198860119892minussgn (V)119891Δ119909V2

119894minus1119905

(2119892119863) and 119862119877= ℎ119894+1119905

minus V119894+1119905

1198860119892 + sgn (V)119891Δ119909V2

119894+1119905(2119892119863)

About the term 120597120576119903120597119905 Covas et al [34] have proposed

an applied model According to the model the term can beexpressed as

120597120576119903(119894 119905 + Δ119905)

120597119905= 119862119878ℎ119894119905+Δ119905

+ 119862119879 (23)

where 119862119878= (120572120588119886

2

0119863120575)sum

119899

119896=1119869119896(1 minus 119890

119896) 119890119896= 119890minusΔ119905120591119896 119862

119879=

2(1198862

0Δ119905119892)sum

119899

119896=1(119869119896119890119896120591119896)(1205721198632120575)(ℎ

119894119905minus ℎ1198940) minus (119869

119896(1 minus

119890119896)Δ119905)(1205721198632120575)120588119892ℎ

119894119905minus 119890119896119903119896(119894119905)

119865119894119905

= (1205721198632120575)120588119892[ℎ119894119905minus

ℎ1198940] and

119903119896(119894119905)= 119869119896119865119894119905minus 119869119896119890119896119865119894119905minusΔ119905

minus (119869119896120591119896(1 minus 119890119896)Δ119905)(119865

119894119905minus

119865119894119905minusΔ119905

) + 119890119896119903119896(119894119905minusΔ119905)

For a present step all values at initial time 119905 are known Asystem of closed equations is obtained Then the solution ofthe simultaneous equations can be written as

ℎ119894119905+Δ119905

=05 (119862

119865+ 119862119877+ 2119862119879)

(1 + 119862119878)

V119894119905+Δ119905

=(ℎ119894119905+Δ119905

+ 119862119878+ 119862119879minus 119862119877) 119892

1198860

(24)

To simulate the transient process with viscoelastic prop-erties a six-elementKelvin-Voigtmodel and basic creep func-tions of polyethylene [34] are adopted In order to analyzethe influence of the elastic steel mesh an extra spring isadded Figure 9 shows the improved six-element Kelvin-Voigtmodel In the model the instantaneous elastic is modified tothe compound of the PE instantaneous elastic and the steelinstantaneous elastic according to their contribution factorsIn the figure 119864es = 119901s sin 120579119864s and119864ep = (1minus119901s csc 120579)119864p asymp 119864p

For viscoelastic solid the basic Kelvin-Voigt creep func-tion can be written as [35]

119869 (119905) = 1198690+

119873

sum

119896=1

119869119896(1 minus 119890

minus119905120591119896) (25)


0 2 4 6 8 100

200

400

600

800

a(m

s)

ps ()

120575 = 00180m120575 = 00155m120575 = 00125m

Figure 7 Influence of reinforcement ratio on the wave speed

0 15 30 45 60 75 900

200

400

600

a(m

s)

120579 (∘)

ps = 037

ps = 074

ps = 148

Figure 8 Influence of spiral winding angle on the wave speed

Covas et al [34] have fixed several creep functions fortypical polyethylenematerials In order to describe the behav-ior of the polyethylene especially the basic creep coefficientis approximated by the six-element Kelvin-Voigt model [34]which has the corresponding instantaneous creep compliance1198690

= 07 times 10minus9 Paminus1 For PE pipe with steel mesh the

improved model is modified to 1198690= 1(119864ep + 119864es) as shown

in Figure 9 Considering the influence of the steel meshFigure 10 shows the modified creep functions for differentreinforcement ratios

42 Pipe and Material Parameters As shown in the previoussection the steel mesh can greatly increase the shock wavespeed meanwhile it can also cause more severe transient

Ees

Eep

E1 E2 E3 E4 E5

1205831 1205832 1205833 1205834 1205835

Figure 9 Six-element Kelvin-Voigt viscoelastic model with steelmesh

0 2 4 6 8 1000

02

04

06

08

10

12

14

Covas et al [34]

J

T (s)

ps = 037

ps = 074

ps = 148

(times10

minus9

Paminus

1 )

Figure 10 Modified creep function for PE pipe with steel mesh

Reinforced PE pipe

Booster pump

Control valveVender pool

Objective reservoir

Figure 11 Basic model of a water supply PE pipe system

pressure In order to analyze the influence of the steel meshon the transient response various reinforcement ratios areconsidered in the same scale pipeline system As shown inFigure 11 the system is composed of a booster pump a controlvalve the main pipe and the downstream reservoir In theexample the pipe is 05m in external diameter 0018m inwallthickness and 2500m in length Four types of reinforcementsare simulated to analyze the influence of the steel mesh on thetransient responseThe reinforcement ratios are 0 (the purePE materials) 037 074 and 148 separately Table 2shows the equivalent elastic modulus and shock wave speedparameters for next transient simulation according to theproposed method

43 Influence of Steel-Mesh Reinforcement on TransientResponse of PE Pipe In the example the hydraulic transientsof pump failure were numerically simulated and comparedfor the water supply system Figure 12 compares the transient


Table 2 Wave speeds of PE pipes with various reinforcement ratios

PE R (mm) 120575 (mm) 119901s () 120579 (∘) 119864p (GPa) 119864s (GPa) 119864e (GPa) 1198860(ms)

Φ500 232 18 000 mdash 143 207 143 25304Φ500 232 18 037 4226 143 207 194 29282Φ500 232 18 074 4226 143 207 244 32710Φ500 232 18 148 4226 143 207 345 38495

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70

Without steel meshClassic modelViscoelastic model

Classic modelViscoelastic model

F

F

h(m

)

h(m

)

t (s) t (s)

h(m

)

h(m

)

t (s) t (s)

ps = 037



ps = 074 ps = 148

Figure 12 Comparison of pressure waves by different transient models

response by the classic model and the viscoelastic modelThe result shows that viscoelasticity has a great effect ontransient process it needs to take account of viscoelas-ticity in transient simulation of PE pipe with steel meshFigure 13 shows the transient pressure waves of the samescale PE pipe with different reinforcement ratios As seenin the figure the negative pressure is the largest whenthe PE pipe has the reinforcement ratio of 148 Con-versely the negative pressure is the least for the PE pipewithout the steel mesh Obviously the transient intensitiesincrease with the reinforcement ratio as well as the wavefrequency and amplitude Accordingly Figure 14 shows thedistribution of extreme transient pressure along the pipelineCompared with the original extreme pressures Table 3shows that the steel-mesh reinforcement has increasedthe amplitudes of the transient pressure along the entirepipeline

Table 3 Extreme pressure with various reinforcement ratios (vis-coelastic model)

119901s () 119886 (ms) ℎ0(m) ℎmin (m) ℎmax (m) 119860m (m)

000 25304 4995 3784 5675 1211037 29282 4995 3688 5691 1307074 32710 4995 3616 5692 1379148 38495 4995 3520 5678 1475

Figure 15 shows the changes of the extreme pressuresand amplitudes with the reinforcement ratio As seen in thisfigure the pressure surge increases with the reinforcementratio The result shows that steel-mesh reinforcement cancause more severe transient response due to increase in theshock wave speed of the PE pipes Therefore the modifiedwave speed formula and viscoelastic model can improve


20

30

40

50

60

34

36

38

40

Without mesh

10 20 30 40

t (s)

t (s)

h(m

)

h(m

)

Am = 1211mAm = 1307m

Am = 1379mAm = 1475m

ps = 037ps = 074ps = 148

0 100 200 300 400 500

Figure 13 Pressure waves for various reinforcement ratios

00 05 10 15 20 2530

40

50

60

Without mesh

Steady hydraulic slope

h(m

)

x (km)

ps = 037

ps = 074

ps = 148

Figure 14 Extreme pressure distributions along the pipeline

the transient simulation of the PE pipe with steel-meshreinforcement for a water supply system

5 Discussion

For the composite structure of a PE pipe with steel-meshreinforcement an equivalent instantaneous elastic modulusis used to determine the water hammer properties of thereinforced system Then an improved shock wave speedformula is proposed for the reinforced PE pipe based onthe reinforcement ratios and winding angles of spiral steelthreads Accordingly the formula is validated by a fieldmeasurement in a submarine PE pipe with steel-mesh rein-forcement Finally transient simulation is applied to analyzethe influence of the steel mesh on the hydraulic transients ofthe reinforced PE pipe Compared to the original uniform PEmaterial the steel-mesh-reinforced PE material has a higher

00 05 10 1511

12

13

14

15

Am

(m)

ps ()

Am

Figure 15 Transient intensions for various reinforcement ratios

elastic modulus and a larger shock wave speed Thereforeit can cause more severe transient pressure For examplethe negative pressure wave amplitudes and frequenciesgreatly increase with the steel-mesh reinforcement ratioViscoelastic model especially is needed to take account ofthe viscoelasticity of the PE pipe Obviously a higher wavespeed increases the water hammer risks due to the steel-mesh reinforcement in a PE pipe It is necessary to evaluatethe water hammer pressure more seriously because of theeffects of the steel mesh in PE pipe system The viscoelasticmodel and the proposed shock wave speed can yield a morereasonable transient simulation in the PE pipe with steel-mesh reinforcement

6 Conclusion

A steel mesh can greatly affect the transient intensity in aPE pipe water supply system because it increases the shockwave speed in the PE pipe Generally a steel mesh can causemore serious water hammer peak pressure and hydraulicfluctuation thus more consideration on the reinforcementmesh is necessary to protect the pipeline from water hammerdamage It is worth reevaluating the shock wave speedaccording to the density and winding angle of the steel meshThe equivalent elastic modulus is presented to indicate thematerial properties of steel-mesh-reinforced PE pipe wallAccordingly an improved wave speed formula is proposedto evaluate reasonably the water hammer speed for thereinforced PE pipe It agrees well with a field measurement ina submarine reinforced PE pipeline system Given the rein-forcement ratio and winding angle of the spiral steel threadsthe improved formula can conveniently be applied to thecalculation of water hammer speed Moreover viscoelasticityhas a great effect on transient process it needs to take accountof viscoelasticity in transient simulation of PE pipe with steelmeshThe comparison of various reinforcement ratios showsthat the steel mesh can increase the transient pressure as


well as the wave frequency and amplitude Consequentlyit is significant to evaluate reasonably the wave speed bythe proposed formula and improve the transient simulationby viscoelastic model for water hammer prediction andprevention in a reinforced PE pipe system

Nomenclature

119886m Measured water hammer speed (ms)119886n Calculated water hammer speed (ms)119886p Wave speed of PE pipe in field test (ms)119886s Wave speed of steel pipe in field test (ms)1198860 Wave speed of water hammer (ms)

119860 Internal section area of pipe (m2)119860m Maximum amplitude of pressure surge (m)119860 s Section area of the steel wire (m2)119861 Length of the unit pipe (m)119862119875 Mean perimeter of pipe wall (m)

119862119865 Specified aggregative variable




119863 Internal diameter of pipe (m)119890119896 Specified dependent variable

119864 Youngrsquos modulus of pipe materials (Pa)119864e Equivalent modulus of pipe wall (Pa)119864ep Instantaneous bulk modulus of PE

materials (Pa)119864es Equivalent bulk modulus of steel (Pa)119864p Youngrsquos modulus of PE materials (Pa)119864s Modulus of steel (Pa)119891 Darcy-Weisbach friction factor119865e Equivalent circumferential force in per

unit length pipe wall (N)119892 Acceleration of gravity (ms2)ℎ Pressure head (m)ℎmax Maximum water hammer pressure head

(m)ℎmin Minimum water hammer pressure head

(m)119894 Serial number of nodes (s)119869119896 Creep of the springs in Kelvin-Voigt model

(Paminus1)119870 Youngrsquos modulus of fluid (Pa)119899 Number of steel lines in specific length

pipe wall119873 Number of elements in Kelvin-Voigt model119901 Pressure in internal side of the pipe (Pa)119901s Reinforcement ratio of pipe wall119875 Horizontal force in unit length (N)119903s Radius of the steel wire (m)119877 Internal radius of the pipe (m)1199041 Length of PE pipe (m)

1199042 Length of steel pipe (m)

1199043 Distance between stations (m)

119905 Time as subscript to denote time (s)1199050 Beginning time of impulse (s)

1199051 Initial response time in the first station (s)

1199052 Initial response time in the second station (s)

119879p Circular tensile on PE pipe wall (N)119879 Circumferential tensile force in unit length

pipe wall (N)119879s Circular tensile on steel wire (N)V Flow velocity (ms)119909 Distance from inlet (m)120572 Dimensionless constant of pipe constraint

conditions120573 Pipe slope (rad)120579 Winding angle of spiral steel wire (rad)120575 Thickness of the pipe wall (m)120576 Circumferential strain of pipe wall (mm)120576119886 Error between calculation and measurement

120576119903 Retarded strain (mm)

119903119896 Strain in initial time (mm)

120588 Density of fluid (kgm3)120590 Stress in pipe wall (Pa)120591119896 Retardation time of dashpots (s)

120583119896 Viscosity of the dashpots (kgsm)

Δ119905 Time step (s)Δ119909 Step of segment (m)

Acronyms

MOC Method of characteristicsPE Polyethylene

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no 51279175) the ZhejiangProvincial Natural Science Foundation of China (Grant noLZ16E090001) and the Open Foundation of the State KeyLaboratory of Hydraulic Engineering Simulation and SafetyTianjin University (HESS-1505)

References

[1] E BWylie V L Streeter and L Suo Fluid Transients in SystemsPrentice Hall Englewood Cliffs NJ USA 1993

[2] M Chaudhry Applied Hydraulic Transients Van NostranaReinhold New York NY USA 1987

[3] D J Wood ldquoWaterhammer analysis-essential and easy (andefficient)rdquo Journal of Environmental Engineering (ASCE) vol131 no 8 pp 1123ndash1131 2005

[4] M R Bazargan-Lari R Kerachian H Afshar and S N Bashi-Azghadi ldquoDeveloping an optimal valve closing rule curve forreal-time pressure control in pipesrdquo Journal of MechanicalScience and Technology vol 27 no 1 pp 215ndash225 2013


[5] W Wan and W Huang ldquoInvestigation of fluid transients incentrifugal pump integrated system with multi-channel pres-sure vesselrdquo Transactions of the ASME Journal of Pressure VesselTechnology vol 135 no 6 Article ID 061301 2013

[6] W Y Wan W R Huang and C Li ldquoSensitivity analysis forthe resistance on the performance of a pressure vessel for waterhammer protectionrdquo Journal of Pressure Vessel Technology-Transactions of the ASME vol 136 no 1 Article ID 011303 2014

[7] S-H Kim ldquoDesign of surge tank for water supply systems usingthe impulse responsemethodwith theGAalgorithmrdquo Journal ofMechanical Science and Technology vol 24 no 2 pp 629ndash6362010

[8] F Esmaeilzadeh D Mowla and M Asemani ldquoMathematicalmodeling and simulation of pigging operation in gas and liquidpipelinesrdquo Journal of Petroleum Science and Engineering vol 69no 1-2 pp 100ndash106 2009

[9] M Behbahani-Nejad and A Bagheri ldquoThe accuracy andefficiency of a MATLAB-Simulink library for transient flowsimulation of gas pipelines and networksrdquo Journal of PetroleumScience and Engineering vol 70 no 3-4 pp 256ndash265 2010

[10] R Alamian M Behbahani-Nejad and A Ghanbarzadeh ldquoAstate space model for transient flow simulation in natural gaspipelinesrdquo Journal of Natural Gas Science and Engineering vol9 pp 51ndash59 2012

[11] M Abbaspour and K S Chapman ldquoNonisothermal transientflow in natural gas pipelinerdquo Journal of Applied Mechanics-Transactions ASME vol 75 no 3 2008

[12] S-S Deng S-Q Zhou Z-F Liao Z-Y Qiu and S-P ZengldquoTheoretical analysis on hydraulic transient resulted by suddenincrease of inlet pressure for laminar pipeline flowrdquo AppliedMathematics and Mechanics vol 25 no 6 pp 672ndash678 2004

[13] W-Y Wan S Zhu and Y-J Hu ldquoAttenuation analysis ofhydraulic transients with laminar-turbulent flow alternationsrdquoApplied Mathematics and MechanicsmdashEnglish Edition vol 31no 10 pp 1209ndash1216 2010

[14] J-S Lee B-K KimW-R Lee and K-Y Oh ldquoAnalysis of waterhammer in pipelines by partial fraction expansion of transferfunction in frequency domainrdquo Journal of Mechanical Scienceand Technology vol 24 no 10 pp 1975ndash1980 2010

[15] E Yao G Kember and D Hansen ldquoAnalysis of water hammerattenuation in applications with varying valve closure timesrdquoJournal of Engineering Mechanics vol 141 no 1 Article ID04014107 2015

[16] X Yu J Zhang and D Miao ldquoInnovative closure law forpump-turbines and field test verificationrdquo Journal of HydraulicEngineering vol 141 no 3 2015

[17] M S Ghidaoui M Zhao D A McInnis and D H AxworthyldquoA review of water hammer theory and practicerdquo AppliedMechanics Reviews vol 58 no 1ndash6 pp 49ndash75 2005

[18] E B Wylie and V L Streeter Fluid Transients McGraw-HillNew York NY USA 1978

[19] D Covas I Stoianov J F Mano H Ramos N Graham andC Maksimovic ldquoThe dynamic effect of pipe-wall viscoelasticityin hydraulic transients Part Imdashexperimental analysis and creepcharacterizationrdquo Journal of Hydraulic Research vol 42 no 5pp 516ndash530 2004

[20] S Evangelista A Leopardi R Pignatelli and G de MarinisldquoHydraulic transients in viscoelastic branched pipelinesrdquo Jour-nal of Hydraulic Engineering vol 141 no 8 2015

[21] H Ramos S Tamminen and D Covas ldquoWater supply systemperformance for different pipe materials part II sensitivity

analysis to pressure variationrdquo Water Resources Managementvol 23 no 2 pp 367ndash393 2009

[22] A K Soares D I C Covas and L F R Reis ldquoAnalysis ofPVC pipe-wall viscoelasticity during water hammerrdquo Journalof Hydraulic Engineering (ASCE) vol 134 no 9 pp 1389ndash13942008

[23] C Apollonio D I C Covas G de Marinis A Leopardi andH M Ramos ldquoCreep functions for transients in HDPE pipesrdquoUrban Water Journal vol 11 no 2 pp 160ndash166 2014

[24] C Sun S S Pang Y Zhao andM A Stubblefield Estimation ofWater Hammer Speed in Composite Pipeline Composite Mate-rials Design and Analysis American Society of MechanicalEngineers Petroleum Division Publication 1998

[25] I S Pearsall ldquoThe velocity of water hammer wavesrdquo in Proceed-ings of the Institution of Mechanical Engineers vol 180 no 5 pp12ndash20 SAGE 1965

[26] T S Lee and S Pejovic ldquoAir influence on similarity of hydraulictransients and vibrationsrdquo Journal of Fluids Engineering vol 118no 4 pp 706ndash709 1996

[27] F E Hachem and A J Schleiss ldquoEffect of drop in pipe wallstiffness on water-hammer speed and attenuationrdquo Journal ofHydraulic Research vol 50 no 2 pp 218ndash227 2012

[28] WHan ZDong andHChai ldquoWater hammer in pipelineswithhyperconcentrated slurry flows carrying solid particlesrdquo Sciencein China Series E Technological Sciences vol 41 no 4 pp 337ndash347 1998

[29] Y-L Zhou B Sun X-N Duan W-P Hong and L ZhangldquoThe calculation of slurry water hammer on liquid-solid two-phase flow in complex pipeline systemsrdquo Journal of EngineeringThermophysics vol 25 no 2 pp 251ndash254 2004

[30] E Hadj-Taıeb and T Lili ldquoValidation of hyperbolic modelfor water-hammer in deformable pipesrdquo Journal of FluidsEngineering vol 122 no 1 pp 57ndash64 2000

[31] K Ando T Sanada K Inaba et al ldquoShock propagationthrough a bubbly liquid in a deformable tuberdquo Journal of FluidMechanics vol 671 pp 339ndash363 2011

[32] M Mitosek ldquoStudy of transient vapor cavitation in series pipesystemsrdquo Journal of Hydraulic Engineering (ASCE) vol 126 no12 pp 904ndash911 2000

[33] D Covas I Stoianov H Ramos N Graham C Maksimovicand D Butler ldquoWater hammer in pressurized polyethylenepipes conceptual model and experimental analysisrdquo UrbanWater Journal vol 1 no 2 pp 177ndash197 2004

[34] D Covas I Stoianov J F Mano H Ramos N Graham and CMaksimovic ldquoThe dynamic effect of pipe-wall viscoelasticity inhydraulic transients Part IImdashmodel development calibrationand verificationrdquo Journal of Hydraulic Research vol 43 no 1pp 56ndash70 2005

[35] J H Aklonis W J MacKnight M Shen and W P MasonIntroduction to PolymerViscoelasticityWiley-Interscience NewYork NY USA 1972

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



et al [23] investigated separately the hydraulic transientproperties of a pipeline made of plastic material Mitosek[32] measured the shock wave speed of PVC pipes Especiallyfor polyethylene pipe Covas et al [19 33 34] developeda viscoelastic model to take account of the influences ofviscoelasticity on transient processes Evangelista et al [20]simulated complex plastic pipes system by viscoelastic modeland showed that the viscoelasticity needs to be consideredin the transient response of a plastic pipe These previousachievements are significant but more accurate simulationsare required to prevent the water hammer However con-sidering the influence of complicated steel-mesh structuresit is presently still difficult to reasonably estimate the wavespeed of a large PE pipe with steel-mesh reinforcement Inorder to improve the transient simulation of a PE pipe withsteel-mesh reinforcement we propose an improved wavespeed formula for reinforced PE pipes by incorporating theequivalent modulus Also we conduct a field test to validatethe approach Based on the proposed approach the transientresponses are simulated and compared for PE pipes withvarious steel-mesh reinforcement ratios Then the influenceof the steel mesh on the transient response is evaluated anddiscussed The result shows that the steel mesh can increasethe shock wave speed and cause more serve hydraulic surgesin the pipeline The proposed formula is significant to revisethe wave speed and improve the transient simulation of steel-mesh-reinforced PE pipe

2 Basic Shock Wave Speed Formula

Wave speed can primarily affect the wave propagation andtransient pressure and it is an important parameter forwater hammer simulations as shown in the simplified closedexpression equations for water hammer simulation [1]

120597V120597119905

+ V120597V120597119909

+ 119892120597ℎ

120597119909+119891V |V|2119863

= 0 (1)

120597ℎ

120597119905+ V

120597ℎ

120597119909minus V sin120573 +

1198862

0

119892

120597V120597119909

= 0(2)

For PE pipe considering the additional viscoelastic term[20 34] the improved continuity equation can be written as

120597ℎ

120597119905+ V

120597ℎ

120597119909minus V sin120573 +

1198862

0

119892

120597V120597119909

+

2 1198862

0

119892

119889120576119903

119889119905= 0

(3)

Generally the shock wave speed can be determined byYoungrsquos modulus of the fluid and the wall material for someregular water supply pipes Based on basic water hammerequation and mass conservation equations [1] provided abasic shock wave speed formula

1198862

0= (

119860

120588)(

119889119901

119889119860) (4)

Steel mesh PE pipe wall

Figure 1 Composition of a PE pipe with steel-mesh reinforcement

SteelPER

120575

B

Figure 2 Schematic of a steel-mesh-reinforced PE pipe

For general elastic pipes the shock wave speed formulacan be expressed as [1]

1198860=

radic119870120588

radic1 + 120572 (2119877120575) (119870119864) (5)

This formula is widely applicable to circular pipes How-ever for some complex pipe materials this formula mayneed to be improved As seen in this formula the modulusof elasticity is an important parameter that determines thewave speed A reinforced PE pipe has a composite structurewhere the steel mesh and polyethylene play different rolesAccordingly the formula cannot directly determine the wavespeed in reinforced PE Based on the conventional formulatherefore we will analyze the wave speed for reinforced PEpipe with steel-mesh reinforcement

3 Wave Speed Analysis in Reinforced PE Pipes

31 Composition of Steel-Mesh-Reinforced PE Pipe A generalPE pipe is made of polyethylene In order to improvethe strength steel meshes are embedded in the pipe wallCompared to the common PE pipe reinforced PE pipes havemore strength and are inflexible Figures 1 and 2 illustrate thebasic structure of a PE pipe with steel-mesh reinforcementThe pipe wall consists of PE and steel meshesThemodulus ofthemesh is much larger than that of PEThus the shock wavespeed changes greatly with the steel-mesh reinforcementIt is necessary to revise the shock wave speed due to the


CP

120579

120579

minus120579

minus120579

B

Clockwise sp

iral







119861 = 119862119875cot 120579 (6)

119862119875= 2120587 (119877 + 05120575) (7)


119875 = 119863119861119901 (8)


119879s = 2119899119860 s120576119864s sin 120579 (9)


119879p = 2 (119861120575 minus119899119860 ssin 120579

)119864p120576 (10)


119879 = 2 [119899119860 s119864s120576 sin 120579 + (119861120575 minus119899119860 ssin 120579

)119864p120576] (11)


119865e = 2119861120575119864e120576 (12)



)119864p120576] (13)



119861120575 (14)


119901s =119899119860 s119861120575

(15)




1198860

=radic119870120588


(17)





232 18 148 4226∘ 143 207 343 3850 3798 137

B120575120590

B120575120590

Steel mesh

PE pipe wall



1is a 2016 km PE pipe 119904









1199041

119886p+1199042

119886s= 1199051minus 1199050 (18)

1199041

119886p+1199042

119886s+1199043

119886s= 1199052minus 1199050 (19)

According to 1199050 1199051 and 119905



119886p =11990411199043


(20)


0








119892

1198860

119889ℎ

119889119905+119889V119889119905

+119891V |V|2119863

+ 21198860(120597120576119903

120597119905) = 0

10038161003816100381610038161003816100381610038161003816119889119909119889119905=+1198860

minus119892

1198860

119889ℎ

119889119905+119889V119889119905

+119891V |V|2119863

+ 21198860(120597120576119903

120597119905) = 0

10038161003816100381610038161003816100381610038161003816119889119909119889119905=minus1198860

(21)





Station 1 Station 2

s1 s2 s3


0 30 60 90 120 150 180061

062

063

064

065

066

Station 1

0 30 60 90 120 150 180061

062

063

064

065

066

p-t

Station 2

p(M

Pa)

t (s)

p(M

Pa)

t (s)

t0 = 2961 s t1 = 83109 s t0 = 2961 s t1 = 83214 s

p-t



ℎ119894119905+Δ119905

+1198860

119892V119894119905+Δ119905

+ 21198862

0Δ119905

119892(120597120576119903

120597119905) minus 119862

119865= 0

ℎ119894119905+Δ119905

minus1198860

119892V119894119905+Δ119905

+ 21198862

0Δ119905

119892(120597120576119903

120597119905) minus 119862

119877= 0

(22)


+V119894minus1119905

1198860119892minussgn (V)119891Δ119909V2

119894minus1119905

(2119892119863) and 119862119877= ℎ119894+1119905

minus V119894+1119905

1198860119892 + sgn (V)119891Δ119909V2

119894+1119905(2119892119863)



120597120576119903(119894 119905 + Δ119905)

120597119905= 119862119878ℎ119894119905+Δ119905

+ 119862119879 (23)

where 119862119878= (120572120588119886

2

0119863120575)sum

119899

119896=1119869119896(1 minus 119890

119896) 119890119896= 119890minusΔ119905120591119896 119862

119879=

2(1198862

0Δ119905119892)sum

119899

119896=1(119869119896119890119896120591119896)(1205721198632120575)(ℎ

119894119905minus ℎ1198940) minus (119869

119896(1 minus

119890119896)Δ119905)(1205721198632120575)120588119892ℎ

119894119905minus 119890119896119903119896(119894119905)

119865119894119905

= (1205721198632120575)120588119892[ℎ119894119905minus

ℎ1198940] and

119903119896(119894119905)= 119869119896119865119894119905minus 119869119896119890119896119865119894119905minusΔ119905

minus (119869119896120591119896(1 minus 119890119896)Δ119905)(119865

119894119905minus

119865119894119905minusΔ119905

) + 119890119896119903119896(119894119905minusΔ119905)


ℎ119894119905+Δ119905

=05 (119862

119865+ 119862119877+ 2119862119879)

(1 + 119862119878)

V119894119905+Δ119905

=(ℎ119894119905+Δ119905

+ 119862119878+ 119862119879minus 119862119877) 119892

1198860

(24)



119869 (119905) = 1198690+

119873

sum

119896=1

119869119896(1 minus 119890

minus119905120591119896) (25)


0 2 4 6 8 100

200

400

600

800

a(m

s)

ps ()

120575 = 00180m120575 = 00155m120575 = 00125m


0 15 30 45 60 75 900

200

400

600

a(m

s)

120579 (∘)

ps = 037

ps = 074

ps = 148







Ees

Eep

E1 E2 E3 E4 E5

1205831 1205832 1205833 1205834 1205835


0 2 4 6 8 1000

02

04

06

08

10

12

14

Covas et al [34]

J

T (s)

ps = 037

ps = 074

ps = 148

(times10

minus9

Paminus

1 )


Reinforced PE pipe

Booster pump


Objective reservoir







Φ500 232 18 000 mdash 143 207 143 25304Φ500 232 18 037 4226 143 207 194 29282Φ500 232 18 074 4226 143 207 244 32710Φ500 232 18 148 4226 143 207 345 38495

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70



F

F

h(m

)

h(m

)

t (s) t (s)

h(m

)

h(m

)

t (s) t (s)

ps = 037



ps = 074 ps = 148





000 25304 4995 3784 5675 1211037 29282 4995 3688 5691 1307074 32710 4995 3616 5692 1379148 38495 4995 3520 5678 1475



20

30

40

50

60

34

36

38

40

Without mesh

10 20 30 40

t (s)

t (s)

h(m

)

h(m

)

Am = 1211mAm = 1307m

Am = 1379mAm = 1475m

ps = 037ps = 074ps = 148

0 100 200 300 400 500


00 05 10 15 20 2530

40

50

60

Without mesh


h(m

)

x (km)

ps = 037

ps = 074

ps = 148



5 Discussion


00 05 10 1511

12

13

14

15

Am

(m)

ps ()

Am



6 Conclusion




Nomenclature




























Acronyms


Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










CP

120579

120579

minus120579

minus120579

B

Clockwise sp

iral







119861 = 119862119875cot 120579 (6)

119862119875= 2120587 (119877 + 05120575) (7)


119875 = 119863119861119901 (8)


119879s = 2119899119860 s120576119864s sin 120579 (9)


119879p = 2 (119861120575 minus119899119860 ssin 120579

)119864p120576 (10)


119879 = 2 [119899119860 s119864s120576 sin 120579 + (119861120575 minus119899119860 ssin 120579

)119864p120576] (11)


119865e = 2119861120575119864e120576 (12)



)119864p120576] (13)



119861120575 (14)


119901s =119899119860 s119861120575

(15)




1198860

=radic119870120588


(17)





232 18 148 4226∘ 143 207 343 3850 3798 137

B120575120590

B120575120590

Steel mesh

PE pipe wall



1is a 2016 km PE pipe 119904









1199041

119886p+1199042

119886s= 1199051minus 1199050 (18)

1199041

119886p+1199042

119886s+1199043

119886s= 1199052minus 1199050 (19)

According to 1199050 1199051 and 119905



119886p =11990411199043


(20)


0








119892

1198860

119889ℎ

119889119905+119889V119889119905

+119891V |V|2119863

+ 21198860(120597120576119903

120597119905) = 0

10038161003816100381610038161003816100381610038161003816119889119909119889119905=+1198860

minus119892

1198860

119889ℎ

119889119905+119889V119889119905

+119891V |V|2119863

+ 21198860(120597120576119903

120597119905) = 0

10038161003816100381610038161003816100381610038161003816119889119909119889119905=minus1198860

(21)





Station 1 Station 2

s1 s2 s3


0 30 60 90 120 150 180061

062

063

064

065

066

Station 1

0 30 60 90 120 150 180061

062

063

064

065

066

p-t

Station 2

p(M

Pa)

t (s)

p(M

Pa)

t (s)

t0 = 2961 s t1 = 83109 s t0 = 2961 s t1 = 83214 s

p-t



ℎ119894119905+Δ119905

+1198860

119892V119894119905+Δ119905

+ 21198862

0Δ119905

119892(120597120576119903

120597119905) minus 119862

119865= 0

ℎ119894119905+Δ119905

minus1198860

119892V119894119905+Δ119905

+ 21198862

0Δ119905

119892(120597120576119903

120597119905) minus 119862

119877= 0

(22)


+V119894minus1119905

1198860119892minussgn (V)119891Δ119909V2

119894minus1119905

(2119892119863) and 119862119877= ℎ119894+1119905

minus V119894+1119905

1198860119892 + sgn (V)119891Δ119909V2

119894+1119905(2119892119863)



120597120576119903(119894 119905 + Δ119905)

120597119905= 119862119878ℎ119894119905+Δ119905

+ 119862119879 (23)

where 119862119878= (120572120588119886

2

0119863120575)sum

119899

119896=1119869119896(1 minus 119890

119896) 119890119896= 119890minusΔ119905120591119896 119862

119879=

2(1198862

0Δ119905119892)sum

119899

119896=1(119869119896119890119896120591119896)(1205721198632120575)(ℎ

119894119905minus ℎ1198940) minus (119869

119896(1 minus

119890119896)Δ119905)(1205721198632120575)120588119892ℎ

119894119905minus 119890119896119903119896(119894119905)

119865119894119905

= (1205721198632120575)120588119892[ℎ119894119905minus

ℎ1198940] and

119903119896(119894119905)= 119869119896119865119894119905minus 119869119896119890119896119865119894119905minusΔ119905

minus (119869119896120591119896(1 minus 119890119896)Δ119905)(119865

119894119905minus

119865119894119905minusΔ119905

) + 119890119896119903119896(119894119905minusΔ119905)


ℎ119894119905+Δ119905

=05 (119862

119865+ 119862119877+ 2119862119879)

(1 + 119862119878)

V119894119905+Δ119905

=(ℎ119894119905+Δ119905

+ 119862119878+ 119862119879minus 119862119877) 119892

1198860

(24)



119869 (119905) = 1198690+

119873

sum

119896=1

119869119896(1 minus 119890

minus119905120591119896) (25)


0 2 4 6 8 100

200

400

600

800

a(m

s)

ps ()

120575 = 00180m120575 = 00155m120575 = 00125m


0 15 30 45 60 75 900

200

400

600

a(m

s)

120579 (∘)

ps = 037

ps = 074

ps = 148







Ees

Eep

E1 E2 E3 E4 E5

1205831 1205832 1205833 1205834 1205835


0 2 4 6 8 1000

02

04

06

08

10

12

14

Covas et al [34]

J

T (s)

ps = 037

ps = 074

ps = 148

(times10

minus9

Paminus

1 )


Reinforced PE pipe

Booster pump


Objective reservoir







Φ500 232 18 000 mdash 143 207 143 25304Φ500 232 18 037 4226 143 207 194 29282Φ500 232 18 074 4226 143 207 244 32710Φ500 232 18 148 4226 143 207 345 38495

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70



F

F

h(m

)

h(m

)

t (s) t (s)

h(m

)

h(m

)

t (s) t (s)

ps = 037



ps = 074 ps = 148





000 25304 4995 3784 5675 1211037 29282 4995 3688 5691 1307074 32710 4995 3616 5692 1379148 38495 4995 3520 5678 1475



20

30

40

50

60

34

36

38

40

Without mesh

10 20 30 40

t (s)

t (s)

h(m

)

h(m

)

Am = 1211mAm = 1307m

Am = 1379mAm = 1475m

ps = 037ps = 074ps = 148

0 100 200 300 400 500


00 05 10 15 20 2530

40

50

60

Without mesh


h(m

)

x (km)

ps = 037

ps = 074

ps = 148



5 Discussion


00 05 10 1511

12

13

14

15

Am

(m)

ps ()

Am



6 Conclusion




Nomenclature




























Acronyms


Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












232 18 148 4226∘ 143 207 343 3850 3798 137

B120575120590

B120575120590

Steel mesh

PE pipe wall



1is a 2016 km PE pipe 119904









1199041

119886p+1199042

119886s= 1199051minus 1199050 (18)

1199041

119886p+1199042

119886s+1199043

119886s= 1199052minus 1199050 (19)

According to 1199050 1199051 and 119905



119886p =11990411199043


(20)


0








119892

1198860

119889ℎ

119889119905+119889V119889119905

+119891V |V|2119863

+ 21198860(120597120576119903

120597119905) = 0

10038161003816100381610038161003816100381610038161003816119889119909119889119905=+1198860

minus119892

1198860

119889ℎ

119889119905+119889V119889119905

+119891V |V|2119863

+ 21198860(120597120576119903

120597119905) = 0

10038161003816100381610038161003816100381610038161003816119889119909119889119905=minus1198860

(21)





Station 1 Station 2

s1 s2 s3


0 30 60 90 120 150 180061

062

063

064

065

066

Station 1

0 30 60 90 120 150 180061

062

063

064

065

066

p-t

Station 2

p(M

Pa)

t (s)

p(M

Pa)

t (s)

t0 = 2961 s t1 = 83109 s t0 = 2961 s t1 = 83214 s

p-t



ℎ119894119905+Δ119905

+1198860

119892V119894119905+Δ119905

+ 21198862

0Δ119905

119892(120597120576119903

120597119905) minus 119862

119865= 0

ℎ119894119905+Δ119905

minus1198860

119892V119894119905+Δ119905

+ 21198862

0Δ119905

119892(120597120576119903

120597119905) minus 119862

119877= 0

(22)


+V119894minus1119905

1198860119892minussgn (V)119891Δ119909V2

119894minus1119905

(2119892119863) and 119862119877= ℎ119894+1119905

minus V119894+1119905

1198860119892 + sgn (V)119891Δ119909V2

119894+1119905(2119892119863)



120597120576119903(119894 119905 + Δ119905)

120597119905= 119862119878ℎ119894119905+Δ119905

+ 119862119879 (23)

where 119862119878= (120572120588119886

2

0119863120575)sum

119899

119896=1119869119896(1 minus 119890

119896) 119890119896= 119890minusΔ119905120591119896 119862

119879=

2(1198862

0Δ119905119892)sum

119899

119896=1(119869119896119890119896120591119896)(1205721198632120575)(ℎ

119894119905minus ℎ1198940) minus (119869

119896(1 minus

119890119896)Δ119905)(1205721198632120575)120588119892ℎ

119894119905minus 119890119896119903119896(119894119905)

119865119894119905

= (1205721198632120575)120588119892[ℎ119894119905minus

ℎ1198940] and

119903119896(119894119905)= 119869119896119865119894119905minus 119869119896119890119896119865119894119905minusΔ119905

minus (119869119896120591119896(1 minus 119890119896)Δ119905)(119865

119894119905minus

119865119894119905minusΔ119905

) + 119890119896119903119896(119894119905minusΔ119905)


ℎ119894119905+Δ119905

=05 (119862

119865+ 119862119877+ 2119862119879)

(1 + 119862119878)

V119894119905+Δ119905

=(ℎ119894119905+Δ119905

+ 119862119878+ 119862119879minus 119862119877) 119892

1198860

(24)



119869 (119905) = 1198690+

119873

sum

119896=1

119869119896(1 minus 119890

minus119905120591119896) (25)


0 2 4 6 8 100

200

400

600

800

a(m

s)

ps ()

120575 = 00180m120575 = 00155m120575 = 00125m


0 15 30 45 60 75 900

200

400

600

a(m

s)

120579 (∘)

ps = 037

ps = 074

ps = 148







Ees

Eep

E1 E2 E3 E4 E5

1205831 1205832 1205833 1205834 1205835


0 2 4 6 8 1000

02

04

06

08

10

12

14

Covas et al [34]

J

T (s)

ps = 037

ps = 074

ps = 148

(times10

minus9

Paminus

1 )


Reinforced PE pipe

Booster pump


Objective reservoir







Φ500 232 18 000 mdash 143 207 143 25304Φ500 232 18 037 4226 143 207 194 29282Φ500 232 18 074 4226 143 207 244 32710Φ500 232 18 148 4226 143 207 345 38495

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70



F

F

h(m

)

h(m

)

t (s) t (s)

h(m

)

h(m

)

t (s) t (s)

ps = 037



ps = 074 ps = 148





000 25304 4995 3784 5675 1211037 29282 4995 3688 5691 1307074 32710 4995 3616 5692 1379148 38495 4995 3520 5678 1475



20

30

40

50

60

34

36

38

40

Without mesh

10 20 30 40

t (s)

t (s)

h(m

)

h(m

)

Am = 1211mAm = 1307m

Am = 1379mAm = 1475m

ps = 037ps = 074ps = 148

0 100 200 300 400 500


00 05 10 15 20 2530

40

50

60

Without mesh


h(m

)

x (km)

ps = 037

ps = 074

ps = 148



5 Discussion


00 05 10 1511

12

13

14

15

Am

(m)

ps ()

Am



6 Conclusion




Nomenclature




























Acronyms


Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Station 1 Station 2

s1 s2 s3


0 30 60 90 120 150 180061

062

063

064

065

066

Station 1

0 30 60 90 120 150 180061

062

063

064

065

066

p-t

Station 2

p(M

Pa)

t (s)

p(M

Pa)

t (s)

t0 = 2961 s t1 = 83109 s t0 = 2961 s t1 = 83214 s

p-t



ℎ119894119905+Δ119905

+1198860

119892V119894119905+Δ119905

+ 21198862

0Δ119905

119892(120597120576119903

120597119905) minus 119862

119865= 0

ℎ119894119905+Δ119905

minus1198860

119892V119894119905+Δ119905

+ 21198862

0Δ119905

119892(120597120576119903

120597119905) minus 119862

119877= 0

(22)


+V119894minus1119905

1198860119892minussgn (V)119891Δ119909V2

119894minus1119905

(2119892119863) and 119862119877= ℎ119894+1119905

minus V119894+1119905

1198860119892 + sgn (V)119891Δ119909V2

119894+1119905(2119892119863)



120597120576119903(119894 119905 + Δ119905)

120597119905= 119862119878ℎ119894119905+Δ119905

+ 119862119879 (23)

where 119862119878= (120572120588119886

2

0119863120575)sum

119899

119896=1119869119896(1 minus 119890

119896) 119890119896= 119890minusΔ119905120591119896 119862

119879=

2(1198862

0Δ119905119892)sum

119899

119896=1(119869119896119890119896120591119896)(1205721198632120575)(ℎ

119894119905minus ℎ1198940) minus (119869

119896(1 minus

119890119896)Δ119905)(1205721198632120575)120588119892ℎ

119894119905minus 119890119896119903119896(119894119905)

119865119894119905

= (1205721198632120575)120588119892[ℎ119894119905minus

ℎ1198940] and

119903119896(119894119905)= 119869119896119865119894119905minus 119869119896119890119896119865119894119905minusΔ119905

minus (119869119896120591119896(1 minus 119890119896)Δ119905)(119865

119894119905minus

119865119894119905minusΔ119905

) + 119890119896119903119896(119894119905minusΔ119905)


ℎ119894119905+Δ119905

=05 (119862

119865+ 119862119877+ 2119862119879)

(1 + 119862119878)

V119894119905+Δ119905

=(ℎ119894119905+Δ119905

+ 119862119878+ 119862119879minus 119862119877) 119892

1198860

(24)



119869 (119905) = 1198690+

119873

sum

119896=1

119869119896(1 minus 119890

minus119905120591119896) (25)


0 2 4 6 8 100

200

400

600

800

a(m

s)

ps ()

120575 = 00180m120575 = 00155m120575 = 00125m


0 15 30 45 60 75 900

200

400

600

a(m

s)

120579 (∘)

ps = 037

ps = 074

ps = 148







Ees

Eep

E1 E2 E3 E4 E5

1205831 1205832 1205833 1205834 1205835


0 2 4 6 8 1000

02

04

06

08

10

12

14

Covas et al [34]

J

T (s)

ps = 037

ps = 074

ps = 148

(times10

minus9

Paminus

1 )


Reinforced PE pipe

Booster pump


Objective reservoir







Φ500 232 18 000 mdash 143 207 143 25304Φ500 232 18 037 4226 143 207 194 29282Φ500 232 18 074 4226 143 207 244 32710Φ500 232 18 148 4226 143 207 345 38495

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70



F

F

h(m

)

h(m

)

t (s) t (s)

h(m

)

h(m

)

t (s) t (s)

ps = 037



ps = 074 ps = 148





000 25304 4995 3784 5675 1211037 29282 4995 3688 5691 1307074 32710 4995 3616 5692 1379148 38495 4995 3520 5678 1475



20

30

40

50

60

34

36

38

40

Without mesh

10 20 30 40

t (s)

t (s)

h(m

)

h(m

)

Am = 1211mAm = 1307m

Am = 1379mAm = 1475m

ps = 037ps = 074ps = 148

0 100 200 300 400 500


00 05 10 15 20 2530

40

50

60

Without mesh


h(m

)

x (km)

ps = 037

ps = 074

ps = 148



5 Discussion


00 05 10 1511

12

13

14

15

Am

(m)

ps ()

Am



6 Conclusion




Nomenclature




























Acronyms


Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 2 4 6 8 100

200

400

600

800

a(m

s)

ps ()

120575 = 00180m120575 = 00155m120575 = 00125m


0 15 30 45 60 75 900

200

400

600

a(m

s)

120579 (∘)

ps = 037

ps = 074

ps = 148







Ees

Eep

E1 E2 E3 E4 E5

1205831 1205832 1205833 1205834 1205835


0 2 4 6 8 1000

02

04

06

08

10

12

14

Covas et al [34]

J

T (s)

ps = 037

ps = 074

ps = 148

(times10

minus9

Paminus

1 )


Reinforced PE pipe

Booster pump


Objective reservoir







Φ500 232 18 000 mdash 143 207 143 25304Φ500 232 18 037 4226 143 207 194 29282Φ500 232 18 074 4226 143 207 244 32710Φ500 232 18 148 4226 143 207 345 38495

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70



F

F

h(m

)

h(m

)

t (s) t (s)

h(m

)

h(m

)

t (s) t (s)

ps = 037



ps = 074 ps = 148





000 25304 4995 3784 5675 1211037 29282 4995 3688 5691 1307074 32710 4995 3616 5692 1379148 38495 4995 3520 5678 1475



20

30

40

50

60

34

36

38

40

Without mesh

10 20 30 40

t (s)

t (s)

h(m

)

h(m

)

Am = 1211mAm = 1307m

Am = 1379mAm = 1475m

ps = 037ps = 074ps = 148

0 100 200 300 400 500


00 05 10 15 20 2530

40

50

60

Without mesh


h(m

)

x (km)

ps = 037

ps = 074

ps = 148



5 Discussion


00 05 10 1511

12

13

14

15

Am

(m)

ps ()

Am



6 Conclusion




Nomenclature




























Acronyms


Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Φ500 232 18 000 mdash 143 207 143 25304Φ500 232 18 037 4226 143 207 194 29282Φ500 232 18 074 4226 143 207 244 32710Φ500 232 18 148 4226 143 207 345 38495

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70

0 200 400 600 80020

30

40

50

60

70



F

F

h(m

)

h(m

)

t (s) t (s)

h(m

)

h(m

)

t (s) t (s)

ps = 037



ps = 074 ps = 148





000 25304 4995 3784 5675 1211037 29282 4995 3688 5691 1307074 32710 4995 3616 5692 1379148 38495 4995 3520 5678 1475



20

30

40

50

60

34

36

38

40

Without mesh

10 20 30 40

t (s)

t (s)

h(m

)

h(m

)

Am = 1211mAm = 1307m

Am = 1379mAm = 1475m

ps = 037ps = 074ps = 148

0 100 200 300 400 500


00 05 10 15 20 2530

40

50

60

Without mesh


h(m

)

x (km)

ps = 037

ps = 074

ps = 148



5 Discussion


00 05 10 1511

12

13

14

15

Am

(m)

ps ()

Am



6 Conclusion




Nomenclature




























Acronyms


Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










20

30

40

50

60

34

36

38

40

Without mesh

10 20 30 40

t (s)

t (s)

h(m

)

h(m

)

Am = 1211mAm = 1307m

Am = 1379mAm = 1475m

ps = 037ps = 074ps = 148

0 100 200 300 400 500


00 05 10 15 20 2530

40

50

60

Without mesh


h(m

)

x (km)

ps = 037

ps = 074

ps = 148



5 Discussion


00 05 10 1511

12

13

14

15

Am

(m)

ps ()

Am



6 Conclusion




Nomenclature




























Acronyms


Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Nomenclature




























Acronyms


Competing Interests


Acknowledgments


References








































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Shock Wave Speed and Transient Response ...€¦ · Improved Wave Speed Formula...

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Transcript of Research Article Shock Wave Speed and Transient Response ...€¦ · Improved Wave Speed Formula...