Simulation of Water Hammer

5/28/2018 Simulation of Water Hammer

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SIMULATION OF WATER HAMMER

IN VISCOELASTIC PIPES

Hassan Mansour, Mohamed Safwat, Berge Djebedjian, and Mohamed Tawfik

Mechanical Power Engineering Department, Faculty of Engineering,Mansoura University, El-Mansoura 35516, Egypt

Emails: [email protected], [email protected], [email protected],[email protected]

(KelvinVoigt model)

Method of Characteristics fluid-structure interaction unsteady friction

(Courant number)

ABSTRACTThe mechanical behavior of the pipe material affects the pressure response of a fluid system during

water hammer. In viscoelastic pipes, the pressure fluctuations are rapidly attenuated and the pressure wave isdelayed in time. This is due to the retarded deformation of the pipe-wall. In this work, a mathematical model tosimulate water hammer in viscoelastic pipes taking into account the viscoelastic behavior of pipe walls, applyingthe KelvinVoigt model, has been developed. The developed model was solved using the Method ofCharacteristics (MOC), neglecting fluid-structure interactions (FSI) and unsteady friction effects. The modelresults were tested against the experimental results obtained by Covas et al. [1], which carried out on a highdensity polyethylene (HDPE) pipe-rig at Imperial College, London. The effects of time step and wave speed werestudied. The pressure fluctuation obtained with the proposed viscoelastic model showed a good agreement withthe experimental results. Conversely, the pressure obtained by the elastic model solution showed a largediscrepancy with the experimental and numerical data. The time step affected the pressure-head wave amplitudeand frequency, while wave speed affected only the wave frequency. The best results were obtained at higher timestep values, corresponding to a Courant number ranging from 0.983 to unity, as the average amplitude andfrequency of the numerical solution are 96% and 95.1% of their values for the experimentalresults, respectively. The best wave speed was at about 388.7 m/s.

Keywords:Water Hammer Viscoelasticity Methods of Characteristics Courant Number Wave Speed

1. INTRODUCTION

In recent years, the application ofplastic pipes, such as polyethylene (PE),

polyvinyl chloride (PVC) and

polypropylene (PP) have been increasinglyused in water supply systems due to theirhigh mechanical, chemical and temperature


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resistant properties, light weight and easyand fast installation. Hydraulic transient inpipes is important for the design of waterpipeline systems. Analysis of transient flowhelps in selecting pipe materials, pressure

classes and specifying of surge protectiondevices to withstand additional loadsresulting from water hammer phenomenon.

Classic water hammer theory based onthe assumptions of linear elastic behaviorof pipe walls and quasi-steady-state frictionlosses (Chaudhry [2], Wylie and Streeter[3,4]). This approach is relatively accurateto predict hydraulic transients in metal orconcrete pipes, but for plastic pipes it isconsiderably imprecise. This is because

polymers, in general, exhibit a viscoelasticmechanical behavior (Ferry [5], Riande etal. [6]) which influences the pressureresponse of the pipe system duringtransient events. The viscoelastic behaviorattenuates the pressure fluctuations rapidly,delays the pressure wave in time andincreases the dispersion of the travellingwave. This effect has been experimentallyobserved by several researchers (Covas etal. [1], Fox and Stepnewski [7], MeiBnerand Franke [8], Williams [9], Mitosek andRoszkowski [10], Brunone et al. [11],Kodura and Weinerowska [12], Bergant etal. [13] and Bergant et al. [14]). Manyother authors proposed mathematicalmodels to simulate fluid transients inviscoelastic pipes taking viscoelasticity ofpipe-wall into account (Bergant et al. [14],Gally et al. [15], Rieutford and Blanchard[16], Rieutford [17], Franke and Seyler

[18], Suo and Wylie [19], Covas [20],Covas et al. [21], Duan [22], Bergant et al.[23] and Keramat et al. [24]).

Although viscoelasticity has asignificant effect on simulation of fluidtransients in viscoelastic pipes, FSI andunsteady friction affect also simulationresults. FSI effect was investigated bymany authors (Keramat et al. [24], Lavooijand Tijsseling [25], Tijsseling [26],Heinsbroek [27], Wiggert and Tijsseling

[28], Neuhaus and Dudlik [29] andAchouyab and Bahrar [30]). They found

that the FSI has no significant effect if thepipe was fixed rigidly and constrained fromany axial movement.

Several researchers took the effect ofunsteady friction into account while

simulating fluid transients in viscoelasticpipes (Duan [22], Zielke [31], Brunone etal. [32], Bergant et al. [33],Ghidaoui et al.[34], Adamkowski and Lewandowski [35],Pezzinga [36], Brunone and Berni [37] andStorli and Nielsen [38]). They found thatthe viscoelastic effect becomes more andmore dominant with respect to unsteadyfriction, as time progresses (Meniconi et al.[39]).

In this paper, a mathematical model

which simulates water hammer inviscoelastic pipes was developed andsolved using the MOC. In the proposedmodel, only viscoelasticity was taken intoaccount, neglecting FSI and unsteadyfriction effects. Viscoelasticity wassimulated applying a generalized KelvinVoigt model. The model was verified bycomparing the model numerical resultswith the experimental results observed byCovas et al. [1].

2. MATHEMATICAL MODEL

2.1. Viscoelasticity

Viscoelastic materials exhibit bothviscous and elastic characteristics whenundergoing deformation due to itsmolecular nature. When viscoelasticmaterials subjected to a certaininstantaneous stress 0, they do not respond

according to Hookes law. The viscosity ofthe viscoelastic material gives thesubstance a strain rate which is dependenton time. These materials have aninstantaneous elastic response and aretarded viscous response, as shown inFig. 1(a), so that the total strain can bedecomposed into an instantaneous-elasticstrain, e, and a retarded strain, ras:

)(tre += (1)

For small strains, applyingBoltzmann superposition principle, a


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combination of stresses that actindependently results in strains that can beadded linearly. This is shown in Fig. 1(b)for the particular case of two stresses. So,the total strain generated by a continuous

application of stress (t) is (Bergant et al.[23]):

+=

t

tJtJt

)(*)()( 0 (2)

where J0 is the instantaneous creep-compliance and "*" denotes convolution.For linearly elastic materials, the constantcreep-complianceJ0is equal to the inverseof Youngs modulus of elasticity, i.e., J0=

1/E0.

r(t)e

t

t

(t)0

(t)

0

LoadingPhase UnloadingPhase

1(t)

t

t

(t)

1(t)

0,1

1+ 22

t1

t1

0,20,2

2(t)

1(t)+ 2(t)

(a) (b)

Figure 1 (a) Stress and strain for an

instantaneous constant load

(b) Boltzmann superposition principle for

two stresses applied sequentially

E0E1 E2 KVNE

1 2KVN

Figure 2 Generalized KelvinVoigt modelIn order to determine stress-strain

interactions and temporal dependencies ofviscoelastic materials, many viscoelasticmodels have been developed. For smalldeformations, it is usually applicable toapply linear viscoelastic models, forexample a generalized KelvinVoigt modelconsisting of NKV parallel spring and

dashpot elements in series with oneadditional spring, as shown in Fig. 2. In

this mechanical model, the elastic behaviorof viscoelastic materials is modeled usingsprings, while dashpots are used to modelthe viscous behavior. It is used to describethe creep function (Aklonis et al. [40]) as

follows:

( )kKV

tN

k

k eJJtJ /

10 1)(

=

+= (3)

whereJk is defined by Jk= 1/Ek,Jkand Ekare the creep-compliance and the modulusof elasticity of the spring of the KelvinVoigt k-element, and k= k/Ek. k and kare the retardation time and the viscosity ofthe dashpots of k-element. The parameters

Jk and k of the viscoelastic mechanicalmodel are adjusted to the creep-complianceexperimental data. The creep-compliancefunction for a material is dependent ontemperature, stress, age, and orientation asa result of the manufacturing process (Laiand Bakker [41]). These effects are notincluded in Eq. (3).

The total strain indicated by Eq. (2)consists of an elastic and a viscoelasticpart. The viscoelastic part is a function of

the whole loading history. In the case ofwater hammer, this loading comes from thefluid pressure head within the pipe. Thesteady head H0 accounts for the staticsituation as one can assume that a longtime has been passed after the

establishment ofH0. The dynamic head Hrepresents the difference between the fluidpressure head H and static head H0 asfollows:

0HHH = (4)

If the dynamic head is substituted for stress in the integrand of Eq. (2), a functionwhich represents up to a constant factor theretarded response to water hammer is givenby (Keramat et al. [24]):

=

=

KV

k

N

k

t

s

k

k

HdsestH

JtI

1 0

/)()(

==

KVN

kkH

tI1

)( (5)


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2.2. Governing Equations

In this section, the governing equationsfor water hammer in a viscoelastic pipe arepresented (Keramat et al. [24]). In the

presented model, the relevant assumptionsare: the piping system consists of thin-walled, linearly viscoelastic pipes with nobuckling and no large deformations andneglecting fluid-structure interactions(FSI), unsteady friction and convectiveterms. Another important assumption madein this derivation is that the pipe is totallyrestrained from axial movements. Applyingthe Kelvin-Voigt model to simulateviscoelasticity, then the continuity and

momentum equations can be written in thefollowing forms:

022

=

+

+

tt

H

a

g

s

V (6)

0||2

1=++

+

VV

D

f

ds

dzg

s

p

t

V

(7)

where,

( )( )dJ*1 2 = (8)

The thin-walled pipe assumption alsoallows the dynamic circumferential hoopstress to be calculated according to:

e

HgD

2

= (9)

Using Eqs. (8) and (9) in (6), andeliminating the convolution operator,Eq. (6) can be rewritten in the pressure

form as follows:

01

012 =+

+

C

t

p

s

Va

(10)

where

( )

+

=

t

I

e

gDa

s

z

t

sgC H

2201 1

2.3. MOC Implementation

From the previous section, thecontinuity and momentum equations,which govern unsteady fluid flow in

pipelines, are found as partial differentialequations. The standard procedure to solvethese equations is the method ofcharacteristics (MOC). This procedureyields water-hammer compatibilityequations that are valid along characteristiclines in the distance (s)time (t) plane.

2.3.1 Development of the characteristic

equations

Multiplying Eq. (7) by Lagrangemultiplier, a constant linear scale factor(), and adding the result to Eq. (10), get:

+

+

+

+

s

p

t

p

s

Va

t

V

12

0||2

01=++ CVVD

f

ds

dzg

(11)

The first group can be replaced by

dtdV , then dtdsa =2 , while the

second group can be replaced bydtdp1 , then dtds= . To satisfy

these two requirements for dtds , then22

a= . This leads to:

a= (12)

Then Eq. (11) after replacing groups can berewritten as follows:

++

+

ds

dzg

dt

dp

dt

dV

1

0||2

01=+CVVD

f (13)

Replacing Lagrange multiplier with itsvalues from Eq. (12), then Eq. (13) can berewritten in the head form, as it is easier tovisualize the propagation of pressurewaves, as follows:


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0||2

01 =+

a

CVV

D

f

dt

dH

a

g

dt

dV

only when

= a

dt

ds (14)

Eq. (14) represents the characteristicequations. These equations describe afamily of straight lines of slope (1/a) onthe s-tplan. Figure 3 shows the C+and C-characteristic lines on the s-t plane for areservoir-pipe-valve system.

Figure 3 The MOC grid for a reservoir-pipe-valve system

2.3.2 The finite difference equations

representation

In order to get the numerical solutionof Eq. (14), they have to be written in finitedifference form. It will become:

( ) ( )++ LepLep HHa

gVV

0||2

01 =

+

a

tCVV

D

tfLeLe (15)

( ) ( )+RipRip

HHa

gVV

0||2

01 =

a

tCVV

D

tfRiRi (16)

The term

a

tC01 can be rewritten in the

following form:

+

=

t

IC

s

ztg

a

tCH

0701

only when

= a

dt

ds (17)

where, ( )e

gDatC

.1 207 = .

By substitution in Eqs. (15) and (16), get:

( ) ( ) ||2

LeLeLepLep VVD

tfHH

a

gVV

++

( ) 0sin. 07 =

++

t

ICtg H (18)

( ) ( ) ||2 RiRiRipRipVV

DtfHH

agVV +

( ) 0sin. 07 =

+

t

ICtg H (19)

where sin = (z/s) is positive for pipessloping upward in the downstreamdirection. Keramat et al. [24] evaluated theterm tI

H / using the following linear

relation for the unknown head at the

current time step:

211

)( atHat

I

t

Ip

N

k

kHHKV

+=

=

=

(20)

where,

011

1 1 Sumet

Ja

KV

k

N

k

t

k

=

=

(21)

( ) = 032 HttHaa p

( ) 41 attHa p (22)

021

3 SumeJ

aKV

k

N

k

t

k

k

=

=

(23)

( ) 031

4 SumttIe

aKV kN

kkH

k

t

==

(24)

Then, Eqs. (18) and (19) can be rewrittenas follows:

i-1 i+1i


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( ) +

++ LepLep H

a

gHC

a

gVV 08

( ) 0sin.||2

09=++

CtgVVD

tfLeLe (25)

( ) +

+

+ RipRip H

a

gHC

a

gVV 08

( ) 0sin.||2

09=+

CtgVVD

tfRiRi (26)

where ( )10708 *aCC = and ( )20709 *aCC = .Equations (25) and (26) represent the

C+ and C-characteristic lines equations.

3. MODEL VERIFICATIONTo verify the proposed mathematical

model and its solutions, a case studypresented experimentally by Covas et al.[1] was investigated.

Covas et al. [1] performed anexperiment on a high density polyethylene(HDPE) pipe-rig, which is rigidly fixed to awall with the specifications given inTable 1. It is a reservoir-pipeline-valvesystem where the length between the vesseland the downstream globe valve is 277 m.

Covas et al. [21] performed creep teststo determine the creep function of HDPE.They also estimated the order of magnitudeof this creep function based on datacollected from the pipe-rig and in thecalibration of a mathematical modeldeveloped by Covas et al. [1]. The creepfunction provided by Covas et al. [21] isused (Table 2).

The developed mathematical modelresults for the heads at location 5,corresponding to a distance of 197 m fromthe upstream end are compared with theexperimental results of Covas et al. [1] inFigure 4.

It can be observed that viscoelasticityhas been modelled and implemented hereincorrectly, The effect of FSI was notsignificant in the experimental resultsbecause the test-rig pipe sections were

rigidly fixed and assumed to be constrainedfrom any axial movement.

Table 1. Specifications of the reservoir-

pipeline-valve experiment performed

by Covas et al. [1]

Length 277 mInner diameter 50.6 mmWall thickness 6.3 mmPoisson ratio 0.46Steady state discharge 1.01 l/sReservoir head 45 mValve closure time 0.09 sPressure wave speed 385.0 m/sStress wave speed 630.0 m/s

Table 2. Calibrated creep coefficientsJk

Covas et al. [21]

Sample size 20 sNumber of K-Velements

5

1= 0.05 J1= 1.0572= 0.50 J2= 1.0543= 1.50 J3= 0.90514= 5.0 J4= 0.2617

Retardationtimes k(s) andcreepcoefficients Jk(10-10Pa-1) 5= 10.0 J5= 0.7456

Figure 4 Comparison of the mathematical

model results against the experimental

results of Covas et al. [1] at location (5)

The viscoelastic effect becomes moredominant with respect to unsteady friction,as time progresses. So, acceptable

discrepancy between the numerical andexperimental results with time progress canbe observed, due to neglecting the unsteady


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friction effect in the present model. For theviscoelastic model, the figure also showsthat the pressure wave damps faster thandamping in the elastic model, which agreeswith the experimental results. Therefore, in

case of studying viscoelastic pipes theviscoelastic model should be applied topredict pressure head fluctuations.

4. PARAMETERS AFFECTING

VISCOELASTIC MODEL

From Eqs. (25) and (26), it is clear thatthe time step, t, and the wave speed, a,appear in more than one term. Therefore,these parameters may lead to a significant

effect on viscoelastic results. Therefore,their effect on the viscoelastic modelresults will be investigated herein.

4.1 Time Step Effect

In this section, the time step effect onthe viscoelastic model is investigated byvarying the time step value. The stabilitycriterion for explicit time steppingdeveloped by Courant et al. [42] must be

satisfied. This criterion requires that theCourant number, C, which is defined byEq. (27), must not exceed the unity.

s

VamaxtC

+= 1 (27)

Therefore, the time step value can bechanged within the following limit:

Vamax

st

+ (28)

For a wave speed of 385 m/s, theeffect of varying the time step andconsequently Courant number values isshown in Figure 5. It reveals that changingtime step and Courant number causesamplitude and frequency distortions. Theamplitude distortion is illustrated inFigure 6, whereas the frequency distortionis shown in Figure 7.

Figure 6 represents a comparison

between the average head of Covas et al.[1] data and those of the numerical results

at different time steps. The average headratio is the ratio of the numerical resultsaverage head to their corresponding valuesof the experimental data obtained by Covaset al. [1] at different time steps. It shows

that the average amplitude of the numericalsolution at Courant number of 1 is 96 % ofits value for the experimental results.

Figure 5 Time-step and Courant number

effects on head fluctuations

Figure 6 Time step and Courant number

effects on average head value

Figure 7 represents a comparisonbetween the frequency of fluctuating headof Covas et al. [1] data and those of thenumerical results at different time steps.The average frequency ratio is the ratio of


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the numerical results average frequency totheir corresponding values of theexperimental data obtained by Covas et al.[1] at different time steps. It shows that theaverage frequency of the numerical

solution at Courant number of 1 is 95.1 %of its value for the experimental results.

From Figures 6 and 7, it is clear thatthe higher time step and Courant numbergives the best match. The best amplitudematch can be obtained at Courant numberequals unity, while the best frequencymatch is achieved at Courant number of0.983. At this Courant number, theamplitude distortion will be quiet large.While at Courant number equals to unity,

the frequency distortion will be quiet small.So, it is recommended to choose Courantnumber equals to unity to get the optimummatch.

Figure 7 Time step and Courant number

effects on average frequency offluctuating head

4.2 Wave Speed Effect

Wave speed for a pipe anchoredthroughout against axial movement waspresented by Larock et al. [43] as follows:

( )

+

=211

/

e

D

E

K

Ka (29)

While the HDPE modulus of elasticityvaries from 0.8 to 1.43 GPa [44,45], thenthe wave speed value can vary from345 m/s to 452 m/s, respectively. Figure 8illustrates the effect of wave speed change

within the previous range at Courantnumber of unity.

Figure 8 shows that changing wavespeed has no significant effect on pressure-head wave amplitude, while it affects thewave frequency.

Figure 8 Wave speed effect on head

fluctuations

For a certain wave speed, it can beobserved that the wave frequency changeswith time. This effect can be observed fromFigure 9. It shows the change in wavefrequency as a function of time and thecomparison with experimental results.

From Figure 9, it is clear that for all

wave speeds the pressure-head wavefrequency decreases with time, while forthe experimental results it tends to decreasefirstly and there is a slight increase whentime progress.

Figure 10 represents the relationbetween average pressure-head wavefrequency as a function of wave speed. Itshows that the higher wave speed, thehigher average pressure-head wavefrequency.


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Figure 9 Wave speed effect on wave

frequency

Figure 10 Wave speed effect on average

wave frequency

5. CONCLUSIONS

A mathematical model to calculatewater hammer in viscoelastic pipes takinginto account the viscoelastic behavior ofpipe walls has been developed in thecurrent paper. Viscoelastic behavior wassimulated by applying the Kelvin-Voigtmodel. Governing equations for waterhammer in a viscoelastic pipe were solvedusing MOC neglecting FSI and unsteadyfriction. To verify the proposed

mathematical model and its numericalimplementation, its results were compared

with the experimental data. The numericalresults obtained by the elastic and theviscoelastic models were compared withexperimental results carried out on a HDPEpipe-rig by Covas et al. [1]. The effects of

time-step, Courant number and wave speedwere studied.

The pressure fluctuation obtained withthe proposed viscoelastic model agreed,within 96%, with the experimental data.Conversely, the pressure obtained by theelastic model solution showed a largediscrepancy with the experimental andtheoretical data. So, taking viscoelasticityinto account was very important whilesimulating transient events in a viscoelastic

pipe. However, neglecting FSI in thecurrent model did not affect the resultssignificantly, as the test rig pipe sectionswere rigidly fixed and assumed to beconstrained from any axial movement. Onother hand, neglecting the unsteady frictionaffected the model results slightly withtime.

The time step and Courant numberaffects the pressure-head wave amplitudeand frequency. The higher time step andCourant number, the best match betweenthe numerical and experimental results.Best results can be obtained at a Courantnumber ranging from 0.983 to unity.

Wave speed affects the pressure-headwave frequency, as increasing wave speedincreases the average pressure-head wavefrequency. The deviation between thenumerical and experimental results may bedue to the wave speed value, a, which

depends on the fixation type and the valueof the modulus of elasticity that variesfrom 0.8 to 1.43 GPa.

NOMENCLATURE

a Wave speed, m/sC Courant number

D Pipe internal diameter, me Pipe wall thickness, m

E Pipe modulus of elasticity, Pa

f Friction coefficientg Gravitational acceleration, m/s2


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H Head, mH0 Head at steady state condition, m

H Dynamic head (HH0), mHsump Sump level, mhP Head increase across a pump, m

HI a function representing the retarded

response to water hammerJ Creep compliance function, Pa-1K Fluid bulk modulus, Pa

L Pipe length, mNKV Number of Kelvin-Voigt elementsp Pressure, PaQ Discharge, m3/sQP Discharge of a pump, m

3/ss Axial coordinate

t Time, stc Valve closure time, sV Fluid velocity, m/sV0 Fluid velocity at steady state

condition, m/sz Elevation, m Difference Strain Circumferential strain Pipe inclination angle, degree Lagrange multiplier

Poissons ratio Fluid density, kg/m3 Circumferential stress, Pa Retardation time, s

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Simulation of Water Hammer

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