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Journal of Hydraulic Research Vol. 42, No. 4 (2004), pp. 413–425 © 2004 International Association of Hydraulic Engineering and Research Surge damping analysis in pipe systems: modelling and experiments Effet d’atténuation du coup de bélier dans les systèmes de conduits: modelation mathématique et expériences HELENA RAMOS, Department of Civil Engineering, Instituto Superior Técnico (IST), Av. Rovisco Pais, 1049-001 Lisboa, Portugal. E-mail: [email protected]; [email protected] DÍDIA COVAS, Department of Civil Engineering, Instituto Superior Técnico (IST), Av. Rovisco Pais, 1049-001 Lisboa, Portugal. E-mail: [email protected] ALEXANDRE BORGA, Department of Civil Engineering, Instituto Superior de Engenharia de Lisboa (ISEL), Portugal. E-mail: [email protected] DÁLIA LOUREIRO, National Lab of Civil Engineering (LNEC), Av. do Brasil 101, 1799 Lisboa, Portugal. E-mail: [email protected] ABSTRACT The current study focuses on the analysis of pressure surge damping in single pipeline systems generated by a fast change of flow conditions. A dimensionless form of pressurised transient flow equations was developed, presenting the main advantage of being independent of the system characteristics. In lack of flow velocity profiles, the unsteady friction in turbulent regimes is analysed based on two new empirical corrective- coefficients associated with local and convective acceleration terms. A new surge damping approach is also presented taking into account the pressure peak time variation. The observed attenuation effect in the pressure wave for high deformable pipe materials can be described by a combination of the non-elastic behaviour of the pipe-wall with steady and unsteady friction effects. Several simulations and experimental tests have been carried out, in order to analyse the dynamic response of single pipelines with different characteristics, such as pipe materials, diameters, thickness, lengths and transient conditions. RÉSUMÉ Cette étude se concentre sur l’analyse de l’atténuation du coup de bélier dans les systèmes simples des conduites produits par une rapide changement des états d’écoulement. Les équations d’adimensionnel d’écoulement pressurisées en régime transitoire a été développée, présentant l’avantage principal d’être indépendante des caractéristiques de système. Dans le manque de profils de vitesse d’écoulement, l’évaluation du coefficient de résistance dans des régimes turbulents a été analysée avec deux nouveaux affaiblissements coefficients empiriques, qui affectent des paramètres d’accélération locale et convective. Une nouvelle approche des curves atténuant est présentée tenant compte de la variation de pression au long du temps. L’effet d’atténuation peut être une combinaison du comportement non élastique du conduit, de l’effet du facteur résistant dans les régimes permanents et variables. Plusieurs simulations et essais expérimentaux ont été effectués, afin d’examiner la réponse dynamique des pipes simples avec différents matériaux des conduits, diamètres, longueurs et conditions de fonctionnement. Keywords: Surge damping, energy dissipation, elastic or non-elastic behaviour, pressure oscillations. 1 Introduction Waterhammer analysis usually focuses on the estimation of the extreme pressures associated with the worse case scenarios. Valves manoeuvres, pumps trip-off and start-up, turbines stop- page and pipe accidental burst are typical events that generate fluid transients considered in the design of pipe systems. Even though the type of actions which induce stronger effects on the pressure variation are well known, the correct prediction of the pressure wave propagation, in particular, the damping effect, is not always properly accounted for, which will influence the Revision received April 02, 2003 / Open for discussion until November 30, 2004. 413 system re-operation, advanced model calibration and analysis of dynamic behaviour of the system response. The need of a more reliable analysis of the pressure wave prop- agation as well as a better understanding of physical phenomena was the major motivation of the current research work. This study consists of the development of theoretical formulations, based on both experimental tests and computer modelling, to char- acterize the pressure surge damping. According to the type of the system or the accuracy required, different hydraulic transient simulators can be used. However, transient solvers commercially available in the market are not capable of predicting the pressure

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Journal of Hydraulic ResearchVol. 42, No. 4 (2004), pp. 413–425

© 2004 International Association of Hydraulic Engineering and Research

Surge damping analysis in pipe systems: modelling and experiments

Effet d’atténuation du coup de bélier dans les systèmes de conduits:modelation mathématique et expériencesHELENA RAMOS,Department of Civil Engineering, Instituto Superior Técnico (IST), Av. Rovisco Pais,1049-001 Lisboa, Portugal. E-mail: [email protected]; [email protected]

DÍDIA COVAS, Department of Civil Engineering, Instituto Superior Técnico (IST), Av. Rovisco Pais,1049-001 Lisboa, Portugal. E-mail: [email protected]

ALEXANDRE BORGA, Department of Civil Engineering, Instituto Superior de Engenharia de Lisboa (ISEL), Portugal.E-mail: [email protected]

DÁLIA LOUREIRO, National Lab of Civil Engineering (LNEC), Av. do Brasil 101, 1799 Lisboa, Portugal. E-mail: [email protected]

ABSTRACTThe current study focuses on the analysis of pressure surge damping in single pipeline systems generated by a fast change of flow conditions.A dimensionless form of pressurised transient flow equations was developed, presenting the main advantage of being independent of the systemcharacteristics. In lack of flow velocity profiles, the unsteady friction in turbulent regimes is analysed based on two new empirical corrective-coefficients associated with local and convective acceleration terms. A new surge damping approach is also presented taking into account the pressurepeak time variation. The observed attenuation effect in the pressure wave for high deformable pipe materials can be described by a combination ofthe non-elastic behaviour of the pipe-wall with steady and unsteady friction effects. Several simulations and experimental tests have been carriedout,in order to analyse the dynamic response of single pipelines with different characteristics, such as pipe materials, diameters, thickness, lengthsandtransient conditions.

RÉSUMÉCette étude se concentre sur l’analyse de l’atténuation du coup de bélier dans les systèmes simples des conduites produits par une rapide changement desétats d’écoulement. Les équations d’adimensionnel d’écoulement pressurisées en régime transitoire a été développée, présentant l’avantage principald’être indépendante des caractéristiques de système. Dans le manque de profils de vitesse d’écoulement, l’évaluation du coefficient de résistancedans des régimes turbulents a été analysée avec deux nouveaux affaiblissements coefficients empiriques, qui affectent des paramètres d’accélérationlocale et convective. Une nouvelle approche des curves atténuant est présentée tenant compte de la variation de pression au long du temps. L’effetd’atténuation peut être une combinaison du comportement non élastique du conduit, de l’effet du facteur résistant dans les régimes permanents etvariables. Plusieurs simulations et essais expérimentaux ont été effectués, afin d’examiner la réponse dynamique des pipes simples avec différentsmatériaux des conduits, diamètres, longueurs et conditions de fonctionnement.

Keywords: Surge damping, energy dissipation, elastic or non-elastic behaviour, pressure oscillations.

1 Introduction

Waterhammer analysis usually focuses on the estimation of theextreme pressures associated with the worse case scenarios.Valves manoeuvres, pumps trip-off and start-up, turbines stop-page and pipe accidental burst are typical events that generatefluid transients considered in the design of pipe systems. Eventhough the type of actions which induce stronger effects on thepressure variation are well known, the correct prediction of thepressure wave propagation, in particular, the damping effect,is not always properly accounted for, which will influence the

Revision received April 02, 2003 / Open for discussion until November 30, 2004.

413

system re-operation, advanced model calibration and analysis ofdynamic behaviour of the system response.

The need of a more reliable analysis of the pressure wave prop-agation as well as a better understanding of physical phenomenawas the major motivation of the current research work. This studyconsists of the development of theoretical formulations, basedon both experimental tests and computer modelling, to char-acterize the pressure surge damping. According to the type ofthe system or the accuracy required, different hydraulic transientsimulators can be used. However, transient solvers commerciallyavailable in the market are not capable of predicting the pressure

414 Ramos et al.

surge damping observed in real systems. Since water pipe sys-tems are expensive infrastructures, computational modelling iseasily available, even though with important constraints and lim-itations associated with inevitable simplified assumptions. Up tonow, the waterhammer modelling rarely takes into account allthe important waterhammer parameters that significantly influ-ence the system’s response. Whilst, in some cases, these effectscan be neglected with no loss of accuracy, under certain circum-stances and depending on the type of analysis required, the errorcan be unacceptable.

From the practical point of view, engineers must be able toestimate the time pressure variation and understand the phys-ical phenomena associated with transients that occur betweentwo steady state operating conditions, even without advancedand complex models currently developed by researchers. Severalfactors can contribute for pressure surge damping (Covaset al.,2003), such as the non-elastic rheological behaviour of the pipematerial, presence of free gas in the fluid, leaks or ruptures, pipemovement or shear stress in pipe-walls.

2 State-of-the-art

Classical waterhammer models, which normally assume linear-elastic behaviour of the pipe-walls and quasi-steady state frictionlosses, have been widely used and presented in expert literature(Chaudhry, 1987; Almeida and Koelle, 1992; Wylie and Streeter,1993). However, recently research work has changed this ten-dency. Relevant developments in inverse transient analysis havebeen achieved recently, in which the dynamic behaviour of thepipe system during a transient event has been used for modelcalibration and leak detection (Covas and Ramos, 1999, 2001;Covaset al., 2001, 2002c; Covas, 2003). Furthermore, the futureperspective of plastic pipes has recently induced the developmentof more accurate hydraulic transient models taking into accountthe viscoelastic behaviour of these materials (Gallyet al., 1979;Rieutford and Blanchard, 1979; Covaset al., 2002a,b; Pezzinga,2002b; Covas, 2003).

Comparisons between experimental and computational resultsusing the Method of Characteristics (MOC) enhance some effectsthat can be identified as the main sources of surge damping. Threetypes of models have been proposed in literature to describe fasttransient events and to help the identification of dynamic effects:(i) the quasi-steady state 1D modelwith a pseudo-uniform veloc-ity distribution in each cross-section, linear elastic behaviourof the pipe material and pipe constrained from any axial orlateral movements, is well-known for underestimating frictionforces and overestimating pressure oscillations (Chaudhry, 1987;Almeida and Koelle, 1992; Wylie and Streeter, 1993); (ii) anequivalent model to (i), with a modification in friction factor isimplemented by adding the unsteady-friction effect due to thenon-uniformity of velocity profiles that includes both the effectsof local inertia and unsteady wall shear stress on flow, which sev-eral approximate formulas have been presented (Zeilke, 1968;Trikha, 1975; Hinoet al., 1977; Brunoneet al., 1991, 1995;Vardy et al., 1993); (iii) the incorporation of the viscoelastic

behaviour of plastic pipe-walls in the transient-flow equations(Rieutford and Blanchard, 1979; Gallyet al., 1979; Covaset al.,2002a,b). Dynamic effects of type (ii) and (iii) seem to be verydifficult to distinguish as both contribute to the surge dampingand phase shift of the pressure wave.

Fluid flow in a pipeline is essentially of one-dimensional flowtype (1D). Zeilke (1968) proposed a formulation relating the shearstress at the pipe-wall, in transient laminar flow, with the instanta-neous mean velocity and weighted past velocity changes. Trikha(1975) simplified Zeilke’s model using exponential relations forsimulating frequency dependent friction. More recently, Brunoneet al. (1991) proposed a formula to evaluate unsteady frictionlosses with a decay coefficient (k3) related to the velocity dis-tribution in the cross-section. Vitkovskyet al. (2000) proposedchanges to the unsteady-friction model, which produce moredamping and a slight positive shift in the pressure oscillationphase.

In 2D models, the unsteady-friction factor is estimated consid-ering the radial variation of flow characteristics. Developmentspresented by Bratland (1986), Vardy and Hwang (1991), Silva-Araya and Chaudhry (1997), Pezzinga (1999) and Abreu andAlmeida (2000) analysed different approaches based on velocityprofiles for each pipe section and time. Covaset al. (2002b) andPezzinga (2002b) present results of an experimental and theoreti-cal study for a high-density polyethylene (HDPE) additional pipe,allowing the analysis of both linear elastic and Kelvin–Voigt vis-coelastic behaviour of the pipe material. Pezzinga (2002b) alsocompared both one-dimensional (1D) and quasi-2D flow models.The numerical results showed that the viscoelastic model betterdescribes the surge damping in plastic pipes, the elastic modeladequately estimates the maximum and minimum oscillations bythe first peaks, and the difference between 1D and 2D was notsignificant.

3 Flow model

3.1 Basic equations

Considering the flow mainly one-dimensional (1D), hydraulictransients in pressurised pipe systems can be satisfactorilydescribed by a linear elastic model, according to the followingassumptions (Chaudhry, 1987; Pejovicet al., 1987; Almeida andKoelle, 1992; Wylie and Streeter, 1993): (i) the flow is consid-ered with a pseudo-uniform velocity profile in each cross-section;(ii) the fluid is assumed one-phase, homogeneous, compressibleand negligible temperature and density changes; (iii) the rheolog-ical behaviour of the pipe material is linear-elastic; (iv) the pipeis a straight uniform element without lateral inflow or outflow;(v) the pipe is completely constrained from movement.

Any disturbance induced in the flow is propagated with awave speed that will strongly influence the dynamic response inthe pipeline. The elastic wave speed corresponds to the storagecapacity of the fluid compressibility and pipe deformation:

c =√

K

ρ[1 + (K/E)ψ] (1)

Surge damping analysis in pipe systems: modelling and experiments415

in which E = Young′s modulus of elasticity of the pipe (Pa);K = fluid bulk modulus of elasticity (Pa);ρ = fluid density(kg/m3); ψ = dimensionless parameter that takes into accountthe cross-section parameters of the pipe and conduit constraints(Chaudhry, 1987; Wylie and Streeter, 1993).

Pressure transients in pipe systems are usually described bythe well-known waterhammer equations—the continuity andmomentum equations. In most engineering applications, theconvective acceleration terms(Q ∂Q/∂x andQ ∂H/∂x) are neg-ligible and the basic differential equations of unsteady pressurisedflows, can be further simplified to a hyperbolic system of equa-tions (Chaudhry, 1987; Wylie and Streeter, 1993; Ramos, 1995)which can be presented in matrix form as follows (Ramos, 1995;Ramoset al., 2000; Ramos and Almeida, 2001):

∂U

∂t+ ∂F (U)

∂x= D(U) (2)

yielding the following vectors:

U =[H

Q

]F(U) =

[c2

gAQ

gAH

]

D(U) =[

0− JgA

Q2 Q|Q|] (3)

wherex = distance along the pipe axis (m);t = time (s);A = cross-section flow area (m2); Q = discharge (m3/s);H = piezometric head (m);J = hydraulic gradient (–);g = gravitational acceleration (m/s2); c = wave speed (m/s).

This set of equations can be solved by several numerical meth-ods (implicit or explicit finite differences, finite elements, bound-ary elements, or MOC), being the MOC, the finite-differencemethod adopted in this case. For the solution of 1D hydraulictransients, MOC has become extensively used, having proven tobe better than other methods due to easiness programming andefficiency of results. The partial derivatives are replaced by finitedifference approximations. However, the stability conditionsrestrict the time and the space step to Courant–Friedrich–Lewyverification,

�x

�t≥ ±c (4)

that is equivalent to neglecting the kinetic fluid term when com-pared to the wave speed propagation(V � c), leading to straight

Figure 1 Method of characteristics—characteristic lines in plane(x, t) (Chaudhry, 1987).

characteristic lines in time and space that constitute the computa-tional grid as shown in Fig. 1. Ideally, it should be an equality toavoid numerical dispersion and damping, i.e.�x/�t = ±c dueto numerical interpolations. By transforming the set of partialdifferential equations into a set of ordinary differential equationsvalid along the characteristic lines, and integrating in thex − t

plane, the finite difference schemes can be written as follows(Borga, 1986; Ramos, 1995):

C+ : HP − HA + c

gA(QP − QA) + I+ = 0

C− : HP − HB − c

gA(QP − QB) + I− = 0

(5)

Several numerical techniques can be used to integrate the termI±, which represents the friction losses. In this paper, this termwas evaluated by afirst order explicit scheme or a second orderimplicit approximationas follows:

First order explicit scheme

I+ = R|QA |QA

I− = −R|QB|QB

(6)

Second order implicit scheme

I+ = R|QP|QP

I− = −R|QP|QP

(7)

in which R = J�x/Q2 = �H0/NQ2 andN = the number ofpipe elements.

In order to obtain a generic formulation applicable to anysystem characteristics, the use of dimensionless parameters ofrelative head,h, relative head losses,�h0, and relative discharge,q, is considered:

h = H

cQ0/gA= H

�HJ; �h0 = �H0

�HJ; q = Q

Q0. (8)

For a first order integration of head losses, Eq. (5) can betransformed into a set of dimensionless equations:

C+ : hP − hA + qP − qA + �h0

N|qA |qA = 0

C− : hP − hB − qP + qB − �h0

N|qB|qB = 0

(9)

416 Ramos et al.

which is equivalent to the following set of equations:

C+ : hP + qP − c1 = 0

C− : hP − qP − c2 = 0

(10)with

c1 = hA +(

1 − �h0

N|qA |

)qA

c1 = hB −(

1 − �h0

N|qB|

)qB

or in an equivalent form:

hP = c1 + c2

2

qP = c1 − c2

2

orhP = c1 − qP; qP = c1 − hP

hP = c2 + qP; qP = hP − c2

(11)

and for thesecond orderapproximation of head losses, it yields:

C+ : hP − hA + qP − qA + �h0

N|qP|qP = 0

C− : hP − hB − qP + qB − �h0

N|qP|qP = 0

(12)

which is equivalent to,

hP = hA + hB + qA − qB

2

qP = hA − hB + qA + qB

1 + √1 + 2(�h0/N)|hA − hB + qA + qB|

(13)

It is interesting to verify that, forsecond orderapproximation, thehead,hP, is independent of the number of pipe sections (or pipeelements). This dimensionless form of pressurised transient flowequations has the advantage of being independent of the systemcharacteristics.

3.2 Energy dissipation

Several formulations have been proposed to estimate the energydissipation under unsteady conditions in turbulent regime. Anextension of Brunoneet al. (1991) formulation presented byVitkovsky et al. (2000) has been analysed by Loureiro (2002)and Loureiro and Ramos (2003), showing a good agreement ofenergy dissipation for certain system characteristics. This formu-lation can be easily integrated into the MOC, avoiding the use ofcomplex axi-simetric models. Generally, these types of formu-lations are composed of a first component calculated based onquasi-stationary hypothesis,Jqs,

Jqs = f V 2

2gD(14)

with f = Darcy–Weisbach friction factor;V = average velocityof the fluid in the cross-section of the pipe; andD = inner pipe

diameter; and other component which take account the unsteadyeffect,Ju, as follows:

J = Jqs + Ju (15)

According to Vitkovskyet al. (2000), the unsteady componentcan be estimated by the following equation:

Ju = k3

gA

(∂Q

∂t+ c SGN(Q)

∣∣∣∣∂Q

∂x

∣∣∣∣)

(16)

which is based on Brunoneet al. (1991) formula, beingk3 anempirical coefficient which depends on velocity profile, variationbetween steady and unsteady flow conditions, and SGN() thesignal of the instantaneous mean velocity.

In this research work, a novel formulation based on Vitkovskyet al. (2000) formula with two new different empirical corrective-coefficients (Loureiro, 2002; Ramos and Loureiro, 2002) ispresented and analysed:

Ju = 1

gA

(Kv1

∂Q

∂t+ Kv2c SGN(Q)

∣∣∣∣∂Q

∂x

∣∣∣∣)

(17)

This equation can be easily integrated into compatibility equa-tions of MOC. The local and convective acceleration componentscan be solved according to the following numerical schemes(Covas, 2003) in order to avoid numerical instabilities:

Local acceleration

C+ : ∂Q

∂t= θ

Qi,j − Qi,j−1

�t

+ (1 − θ)Qi−1,j−1 − Qi−1,j−2

�t

C− : ∂Q

∂t= θ

Qi,j − Qi,j−1

�t

+ (1 − θ)Qi+1,j−1 − Qi+1,j−2

�t

(18)

with θ = convergence (or relax) coefficient (Ramos, 1986).

Convective acceleration

C+ : ∂Q

∂x= Qi,j−1 − Qi−1,j−1

�x

C− : ∂Q

∂x= Qi,j−1 − Qi+1,j−1

�x

(19)

The term SGN(Q) takes into account the flow direction, whichfor the characteristic lineC+ depends on the signal ofQi−1,j−1

and for theC− depends on the signal ofQi+1,j−1. After the incor-poration of the unsteady friction into the MOC, the characteristiclines are given by:

C+ : Qi,j = C ′′P − C ′

aHi,j

C− : Qi,j = C ′′N + C ′

aHi,j

(20)

Whilst Covas (2003) used asecond order implicitnumericalscheme to calculate the coefficientsC ′′

P, C ′′N andC ′

a and one singledecay coefficientk3, Ramos and Loureiro (2002) and Loureiroand Ramos (2003) used afirst order explicitscheme with twocorrective coefficients,Kv1 and Kv2. The latter approach wasfollowed herein as the use of two coefficients seemed to describebetter the observed surge damping for plastic pipe materials (non-elastic behaviour which induce such greater damping and wave

Surge damping analysis in pipe systems: modelling and experiments417

dispersion). Accordingly, coefficientsC ′′P, C ′′

N andC ′a are defined

as follows:

CP = Qi−1,j−1 + CaHi−1,j−1 − RQi−1,j−1|Qi−1,j−1| (20a)

C ′P = CP + Kv1θQi,j−1 − Kv1(1 − θ)(Qi−1,j−1 − Qi−1,j−2)

− Kv2SGN(Qi−1,j−1)|Qi,j−1 − Qi−1,j−1| (20b)

C ′′P = C ′

P

1 + Kv1θ(20c)

CN = Qi+1,j−1 + CaHi+1,j−1 − RQi+1,j−1|Qi+1,j−1| (20d)

C ′N = CN + Kv1θQi,j−1 − Kv1(1 − θ)(Qi+1,j−1 − Qi+1,j−2)

− Kv2SIGN(Qi+1,j−1)|Qi,j−1 − Qi+1,j−1| (20e)

C ′′N = C ′

N

1 + Kv1θ(20f)

with

C ′a = Ca

1 + Kv1θand Ca = gA

c(20g)

At the boundaries, additional equations must be specified.

3.3 Surge damping curves

A new simplified approach of the surge damping is presented tak-ing into account the pressure peak damping in time. This dampingcan be a combined effect of the non-elastic behaviour of the pipe-wall, the steady and unsteady friction effect. This technique aimsat the characterization of energy dissipation through the variationof the extreme piezometric head in time, in a simple and rapidimplementation.

Generally the mechanical characteristics of pipe materialscan be distinguished in three categories: (1) thermo-plasticbehaviour, which includes HPPE and PVC pipes; (2) thermo-elastic behaviour, which includes PRGF—Polyester reinforcedwith glass fibre, but only available forD > 400 mm, condition-ing the application in lab conditions; (3) elastic behaviour, whichincludes concrete, iron and steel pipes. From a practical point ofview, there are no systems purely dissipative by friction effectsor non-elastic effects, but systems with combined effects.

In the systems analysed in this current research, there are twotypes of pipe materials: (i) a less deformable material with quasi-elastic behaviour such as metal pipes and (ii) a more deformablematerials with viscoelastic behaviour (i.e. plastic), such as HPPEand PVC. Specific formulations were developed and imple-mented in order to carry out the surge damping estimation fordifferent pipe-wall characteristics.

In elastic pipe behaviour, the energy dissipation of the sys-tem in time for a rough turbulent flow, in a dimensionless form,varies withh2 (or q2, due to friction effects). Hence, the extremevalues of head,h = H/(�HJ), can be obtained according to thefollowing equation

h1 − h2 = Kelas�h0h2 (21)

whereh1 − h2 = dimensionless head difference between twosuccessive peaks for different time calculation,Kelas = damping

factor,�h0 = friction effect and�HJ = overhead of Joukowskywhich is given by

�HJ = cQ0

gA(22)

The energy dissipation in time for this type of systems (i.e. lowdeformable pipe material or “rigid” pipes made of concrete ormetal) can be evaluated by the following equation:

dh

dτ= −Kelas�h0h

2 (23)

assuming thatτ = t/(2L/c), and by integration, the time-headvariation is given by

h = 1

1/h0 + Kelas�h0(τ − τ0)(24)

with h0 = dimensionless head atτ0 = t0/(2L/c); andt0 = timefor the first pressure peak where the head is maximum.

According to the same type of analysis, innon-elasticpipebehaviour (e.g. plastic pipes), the pipe-wall retarded behaviouris the main factor for the pressure damping. Thus, in practi-cal, the energy dissipation can be adequately reproduced by aproportionality to the head:

h1 − h2 = Kplas�h0h (25)

and its time-variation can be evaluated by

dh

dτ= −Kplas�h0h (26)

which by integration is equivalent to

h = h0 e−Kplas�h0(τ−τ0) (27)

This equation is in accordance with Pearsall (1965) and withthe typical behaviour of a viscoelastic pipe (Gallyet al., 1979;Ghilardi and Paoletti, 1986; Covaset al., 2002a).

For systems withcombined effects(i.e. elastic and plasticresponse), the surge damping can be evaluated by a combinationof both effects as follow:

h1 − h2 = Kplas�h0h + Kelas�h0h2 (28)

that, in time-variation, is equivalent to

dh

dτ= −Kplas�h0h − Kelas�h0h

2 (29)

and after the integration yields in this final form,

h = 1

(Kelas/Kplas+ 1/h0)eKplas�h0(τ−τ0) − (Kelas/Kplas)(30)

whereKplas andKelas are damping coefficients to take accountthe plastic and elastic effects, respectively.

4 Analysis of results

4.1 Classic waterhammer theory

First of all, a theoretical analysis applied to a simple elastic pipesystem with a reservoir at upstream and a valve at downstreamis discussed in order to better identify the physical phenomenaassociated with pressure head variation, both with instantaneousclosure manoeuvre (i) without head losses,�h = 0, and (ii) with

418 Ramos et al.

h

τ 1 2 3 0 4

ho

Figure 2 Typical elastic pipe-wall behaviour for instantaneous valveclosure and no head losses.

ho

h

τ0 1 2 3 4

h (τ)

Figure 3 Typical elastic pipe-wall behaviour for instantaneous valveclosure and with head losses.

head losses along the pipe flow,�h �= 0:

(i) Instantaneous valve closure without head losses(�h = 0)—Fig. 2. Based on Eq. (24), the extreme dimensionless headvalues can be evaluated by considering�h0 = �H0/�HJ

that, in this case, is equal to 0;h = H/�HJ; h0 =H0/�HJ = 1, sinceH0 = �HJ; τ = t/(2L/c), that resultsh = h0 for extreme positive values, without any type ofdamping.

(ii) Instantaneous valve closure with head losses(�h0 �= 0)—Fig. 3. According to Eq. (24), the extreme dimensionlesshead values along time steps (i.e. whenτ → ∞) tend to be0 (i.e. h → 0). This means, contrary to case (i), that thereis a complete attenuation of the head variation in time at thevalve section, as shown in the damping tendency presentedin Fig. 3.

Based on several types of simulations, for different valuesof discharge,Q, and valve closure time for fast manoeuvres,T c ≤ 2L/c, a pure elastic model based on Eq. (24) fits quite wellto the classic waterhammer model results (Fig. 4), commonlyused for design purposes. This analysis allows the identifica-tion of the damping capacity of commercial software packagesthat generally describe quite accurately the waterhammer inlow deformable pipe systems (e.g. metal pipes), which have apredominant elastic behaviour.

4.2 Experimental procedures

4.2.1 Unsteady friction (UF) estimationThe use of different weight coefficients (Kv1 and Kv2),applied to the convective and the local acceleration terms inEq. (17), to estimate the unsteady friction significantly improvesthe agreement between numerical results and experimental

observations comparatively to a unique coefficient used byBrunoneet al. (1995) and, after, modified by Vitkovskyet al.(2000) (Loureiro, 2002; Ramos and Loureiro, 2002; Loureiro andRamos, 2003). With this novel approach, it is possible to repro-duce reasonably well the experimental observations in terms ofpressure damping and phase (Fig. 5). However, the shape of thispressure wave propagation is much sharper than the real one. Forlong pipelines with high steady state flows, where the packingeffect is very important, the results fit quite well the pressurepeaks at the valve section.

The coefficientsKv1 andKv2 have been estimated based on thefitting of the results of numerical simulations with experimentaldata, for different initial steady state conditions (Loureiro, 2002).This analysis has shown that these parameters are practicallyconstant for different Reynold numbers, defining the limits forKv variation of the formula proposed byVardy and Brown (1995,1996):

Kv = 2

√7.41

Relog(14.3/Re0.05)(31)

Several tests have been carried out in a lab facility System 1 (S1),characterised by an HPPE (high performance polyethylene) pipewith the lengthL1 = 100 m orL2 = 200 m, an internal diameterDi = 0.044 m and a wave speedc = 280 m/s. For this system,it has been verified numerically that the termKv1 ∂Q/∂t affectsthe phase shift of transient pressure waves andKv2 ∂Q/∂x thedamping effect. The numerical pressure wave propagates fasterthan the experimental wave for small values ofKv1 and tends to beslightly delayed than the observed wave for higher values ofKv1.For the experimental procedure of system S1,Kv1 varied between[0.004; 0.0054] andKv2 between [0.033; 0.05], inversely to watercolumn inertia (L = 100; 200 m) (Fig. 6a). Figure 6(b) showsthat the coefficientKv2 is much more dependent on the pipelinehead losses thanKv1, as it attains higher values for smaller headlosses.

Although Eq. (17) allows a good fitting between numeri-cal results and experimental tests in terms of pressure dampingand phase shift, there are still some differences, particularly,regarding the actual shape of the transient pressure wave. Covaset al. (2002a,b) analysed, both theoretically and experimen-tally, waterhammer in a polyethylene pipe. These authors haveobserved that the shape, the amplitude and the dispersion of thetransient pressure wave are mainly caused by the viscoelasticbehaviour of the polyethylene pipe material, rather than by thefrictional shear stress in the pipe-wall.

4.2.2 Damping effect (DE) analysisDifferent case studiesExperimental tests have been performed,in order to analyse the dynamic response of single pipes withdifferent characteristics: pipe materials, diameters, pipe lengthsand thickness and operating conditions. The following five typesof facilities have been analysed (Table 1):

System 1 (S1)—an HPPE (high performance polyethylene) pipewith the lengthL1 = 100 m orL2 = 200 m, an internal diameterDi = 0.044 m and a wave speedc = 280 m/s;

Surge damping analysis in pipe systems: modelling and experiments419

–1.0

–0.8

–0.6

–0.4

–0.2

0.0

0.2

0.4

0.6

0.8

1.0

–1.0

–0.8

–0.6

–0.4

–0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60 70 80 90 100

0.0 0.05 0.2 1.0

h

∆ho

∆ho

τc = 0

τc = 0

0

h

10 20 30 40 50 60 70 80 90 100

0.0 0.05 0.2 1.0

τ

τ

Figure 4 Typical elastic behaviour. Agreement between classic waterhammer simulation and Eq. (24) for differentτc and discharge values.

S1L =100 m; Qo =1.81 l/s; c =280 m/s; Di = 0.044 m

Re = 40933

S1L =100 m; Qo =1.81 l/s; c =280 m/s; Di = 0.044 m

Re = 40933

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10 12 14 16 18 20

H (

m)

measurementsVitkovsky et al. (2000) k = 0.0397modified formulation Kv1 = 0.0048 e kv2 = 0.048

measurementsVitkovsky et al. (2000) k = 0.0397modified formulation Kv1 = 0.0048 e kv2 = 0.048

t (s)

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10 12 14 16 18 20

H (

m)

t (s)

Figure 5 Comparison of piezometric heads: experimental measurements, Vitkovskyet al. (2000) and modified formulation—Eq. (17) at the valvesection and at the middle of the pipe.

System 2 (S2)—an HPPE pipe with the lengthL = 250 m, aninternal diameter Di= 0.050 m and a wave speedc = 330 m/s;System 3 (S3)—a PVC pipe with a lengthL1 = 14 m orL2 =28 m, an internal diameter Di= 0.045 m and a wave speedc = 400 m/s;System 4 (S4)—(based on Doc8664-51-Benchmark Analysis) asteel pipe with a lengthL = 3000 m, an internal diameter Di=0.400 m and a wave speedc = 1000 m/s;System 5 (S5)—(based on Pezzinga and Scandura, 1995) a zincplated steel with a length ofL1 = 78 m andL2 = 144 m,

an internal diameter Di= 0.053 m and a wave speedc =1386 m/s.

Cases studies S1, S2 and S3 are obtained by the authors in lab-oratory system composed by an air-vessel at upstream to keep thepressure constant and at downstream end there are two ball valves,one to discharge control and another to originate the manoeuvre,respectively, discharging to the atmosphere. Systems S4 and S5are case tests available in the literature.

420 Ramos et al.

(a)

(b)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

15000 30000 45000 60000

Re (–)

Kv2 – L = 200 mKv2 – L = 100 mKv (Vardy and Brown)Kv1– L = 200 mKv1– L = 100 m

0.60

0.80

1.00

1.20

1.40

kv2/

kv

0.105

0.110

0.115

0.120

0.125

15000 25000 35000 45000 55000 65000

Re(–)

Re(–)

15000 25000 35000 45000 55000 65000

Re(–)

kv1/

kv

L1 = 100 m

L2 = 200 m

L1 = 100 m

L2 = 200 m

0.0

0.1

0.2

0.3

0.4

15000 20000 25000 30000 35000 40000 45000 50000 55000 60000 65000

∆ho

Kv1 = 0.005; Kv2 = 0.049

Kv1 = 0.0044; Kv2 = 0.033

Figure 6 Values ofKv1 andKv2 for different values of Reynolds number (a) and relative head losses (b).

Table 1 Parameters estimation for damping effect in different system characteristics

System System characteristicstype-pipematerial L Di Q c hf k1 k2(32) k2(33)

(m) (m) (l/s) (ms−1) (–) (–) (–) (–)(1) (2) (3) (4) (5) (6) (7) (8)

S1—HPPE 100 0.044 0.87 280 2.48 1.02 – 0.141.48 1.46 0.162.40 0.90 0.17

200 0.80 2.69 0.261.90 1.13 0.272.75 0.78 0.29

S2—HPPE 250 0.050 1.10 330 2.38 0.99 – 0.231.50 1.75 0.262.00 1.31 0.272.60 1.01 0.30

S3—PVC 14 0.045 1.50 400 0.77 1.02 – 0.0628 1.00 1.20 0.04

1.15 1.08 0.05S4—steel 3000 0.400 57.00 1000 7.06 0.98 0.24 –

98.00 4.11 0.30S5—zinc plated steel 78 0.053 0.40 1386 1.79 1.01 0.03 0.02

0.60 1.20 0.03 0.02144 0.40 1.33 0.03 0.03

Surge damping analysis in pipe systems: modelling and experiments421

Benchmark analysisConsidering Eqs (24) and (27), after ade-quate algebraic transformations based on the final steady statepiezometric headHf , normally associated to each system char-acteristics, or in dimensionless formhf = Hf /�HJ, which allowto better understand the actual head variation,h = H/�HJ.The estimative of the extreme pressure heads during fast changesof flow conditions must include the following terms: (1) thefinal steady state piezometric head,hf , for which the pressurefluctuations tend and for a completely closure, it is equiva-lent to the upstream static head; (2) instantaneous overpressuregiven by Joukowsky formula,K1 Q0, or in dimensionless formk1 ≈ 1 (beingk1 = K1Q0/�HJ); 3) damping effect,K2t , orin dimensionless formk2τ , which depends on total head losses,among other factors (pipe-wall behaviour), that is expressed byan inverse term for elastic behaviour and by an exponential equa-tion for plastic type response. Accordingly, Eqs (24) and (27) canbe replaced by Eqs (32) and (33), after adequate transformationsbased on the tendency to the final steady state piezometric headline, as follows:

Elastic behaviour(i.e. low deformable pipes):

H = Hf ± K1Q0

(1 + K2t)

in dimensionless form:h = hf ± k1

(1 + k2τ)(32)

Plastic behaviour(i.e. viscoelastic materials):

H = Hf ± K1Q0 e−K2t

in dimensionless form:h = hf ± k1 e−k2τ (33)

being the sign± for up-surge and down-surge, respectively.This analysis was carried out for different system character-

istics and for different pipe lengths. For systems S1, S2 and S3composed of plastic pipes (i.e. HPPE and PVC), Eq. (33) fits wellthe observed surge damping (Figs 7–9; Table 1). In case of sys-tems S4 and S5, with an elastic mechanical behaviour as the maincharacteristic of metal materials, Eq. (32) is much more adequate(Figs 10 and 11; Table 1). According to Table 1 for both pipe-wallcases, i.e. plastic and metal pipes,k2 has different values and itis directly proportional to discharge values but inversely to thehead losses associated to each pipe length.

Figure 7 shows the most rapid surge damping in the systemwith smaller length. Maximum and minimum pressure values,attained in the first peaks are almost the same for both length andflow conditions. The packing effect, most notorious in the longercircuit, practically is compensated by the steady state head losses.

Figure 8 presents the surge damping in a longer pipeline (sys-tem S2), with a higher diameter and thickness than system S1,but with equivalent response. As for the former case, the plasticbehaviour obtained by Eq. (33) fits quite well.

In system S3, the damping effect is higher as the discharge andthe smaller is the length (Fig. 9). Definitely, classic waterhammermodels, which follow the elastic type response given by Eq. (32),do not have the capacity to reproduce the real damping inducedby plastic pipe materials.

0

10

20

30

40

50

60

70

80

90

0 2 4 6 8 10 12 14 16 18t (s )

Qo=0.87 ls Qo=1.48 l/s Qo=2.40 l/s

Qo=0.87 l/s Qo=2.40 l/s Qo=1.48 l/s

S1L=100 m; Qo=0.87; 1.48; 2.40 l/s; c=280 m/s; Di=0.044 m

-10

0

10

20

30

40

50

60

70

80

90

100

0 2 4 6 8 10 12 14 16 18t (s )

Qo=0.80 l/s Qo=1.90 l/s Qo=2.75 l/s

Qo=0.80 l/s Qo=1.90 l/s Qo=2.75 l/s

S1L=200 m; Qo=0.80; 1.90; 2.75 l/s; c=280 m/s; Di=0.044 m

Eq. (33)

Eq. (33)

H (

m)

H (

m)

Figure 7 Surge damping of system S1 for different discharge and pipelength.

0

10

20

30

40

50

60

70

80

90

0 2 4 6 8 10 12 14 16 18t (s )

H (

m)

Qo=2.60 l/s Qo=2.00 l/sQo=1.50 l/s Qo=1.10 l/sexp expexp exp

S2

L=250 m; Qo=1.10; 1.50; 2.00; 2.60 l/s; c=330 m/s;Di=0.051 m

Eq. (33)

Figure 8 Surge damping of system S2 for different discharge values.

However, for metal pipes, the behaviour is much more similarto the classic waterhammer models, which are based on a linearelastic rheological behaviour of the pipe-wall (Figs 10 and 11),which the elastic formulation, Eq. (32) describes reasonably wellthis behaviour. Since in system S5 (Fig. 11) the effect of frictionis smaller due to low discharge values, a good agreement for bothformulations is obtained.

System S2 was chosen to analyse the fitting in detail due toits higher thickness, when compared to system S1. Analysis ofcombined effects of elastic and non-elastic type was developed byusing Eq. (30) and presented in Fig. 12. The zero value forK2plas

or K2elas indicates which effect (i.e. plasticity or elasticity) canbe negligible. For this system, it was observed the behaviour ishighly conditioned by a plastic type response (i.e.K2plas = 0.14when compared withK2elas = 0.07), as expected. Figure 12

422 Ramos et al.

S3

0

10

20

30

40

50

60

70

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0t (s)

t (s)

t (s)

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

H (

m)

H (

m)

H (

m)

L = 14 m; Qo = 1.50 l/s; c = 400 m/s; Di = 0.045 m

L =28 m; Qo = 1.00 l/s; c = 400 m/s; Di = 0.045 m

L =28 m; Qo = 1.15 l/s; c = 400 m/s; Di = 0.045 m

Eq. (33)

S3

0

10

20

30

40

50

60

Eq. (33)

S3

0

10

20

30

40

50

60

Eq. (33)

Eq. (32)

Figure 9 Sensitivity analysis to damping effect of S3, for differentsystem characteristics.

220

260

300

340

380

420

0 20 40 60 80 100t (s)

H (

m)

57 l/s 98 l/sEq. (33) exp Eq. (33) expEq. (32) inv Eq. (32) inv

L = 3000 m; Qo = 57;98 l/s; c = 1000 m/s; Di = 0.400 mS4

Figure 10 Selection of the best fit for pressure surge for differentdischarge values of system S4.

shows as well a good fitting for all type of formulations only forthe first time steps, although the viscoelastic damping effect ismost notorious along time pressure wave propagation. A clas-sic formulation of waterhammer (i.e. withK2plas = 0) cannotreproduce in this system the correct damping effect in the timeof simulation.

0

10

20

30

40

50

60

70

80

90

0.0 1.0 2.0 3.0 4.0 5.0 6.0t (s)

t (s)0.0 1.0 2.0 3.0 4.0 5.0 6.0

H (

m)

H (

m)

Qo = 0.4 l/s Qo = 0.6 l/sEq. (33) exp Eq. (33) expEq. (32) inv Eq. (32) inv

S5

0

10

20

30

40

50

60

70S5

L = 78 m; Qo = 0.40 l/s; c = 1386 m/s; Di = 0.053 m

L = 144 m; Qo = 0.40 l/s; c = 1386 m/s; Di = 0.053 m

Qo = 0.4 l/sEq. (33) expEq. (32) inv

Figure 11 Damping analysis for both approaches and for system S5.Best fitting obtained with Eq. (32).

4.2.3 Link between UF and DESince system S1 is composed of a plastic pipe response andit cannot be accurately simulated by the classic waterhammerformulation, it was compared with both approaches presentedherein (i.e. unsteady friction (UF) and damping effects (DE)estimations). These both approaches have different formulationsnot associated, although they have an equivalent response typefor the estimation of surge damping, which is analysed in detail.After identifying that only both termsKv2 from Eq. (17) and K2from Eq. (33) can be related since they reproduce the pressurewave attenuation effects, in Fig. 13, it was observed a constantratio between these parameters (i.e.Kv2/K2 ≈ 0.19), for differ-ent discharge values, independently of the system configuration(i.e. withL = 100 m – diamond marker orL = 200 m – trianglemarker). In Fig. 13, the parameter�H0/L represents the totalhead losses for each pipe length, which is also associated tothe surge damping. This constant ratio means there is a surgedamping effect, which is dependent on the system characteristics.

5 Conclusions

According with this analysis, the following remarks can bemade:

• Generally, all formulations allow the estimation of the max-imum overpressure after a fast manoeuvre. However, thedifferences increase with the simulation time, depending on thecapability of the transient model to describe the most importantparameters of the system.

• Equation (16) provides a good approximate solution forunsteady friction effect; however, it is not enough to reproduce

Surge damping analysis in pipe systems: modelling and experiments423

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

6 8 10 12

t c/(2 L)t c/(2 L)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10 12

h

measurementsk2plas = 0.18; k2elas = 0.00k2plas = 0.14; k2elas = 0.07k2plas = 0.11; K2elas = 0.11k2plas = 0.16; k2elas = 0.03k2plas = 0.00; k2elas = 0.17

measurementsk2plas = 0.18; k2elas = 0.00k2plas = 0.14; k2elas = 0.07k2plas = 0.11; K2elas = 0.11k2plas = 0.16; k2elas = 0.03k2plas = 0.00; k2elas = 0.17

Figure 12 Sensitivity analysis to plastic and elastic response effects when applied to system S2.

Re (–)

�H

o/L

(–)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

15000 30000 45000 60000

Kv2 / K2 = 0.19

Figure 13 Constant correlation between damping coefficients of bothapproaches, for different Reynolds and head losses in the system S1.

accurately some particular experimental tests, specially, inplastic pipe systems.

• Two different corrective coefficients (Kv1 andKv2) associatedto the local and convective acceleration in the unsteady frictionformulation, Eq. (17), have been analysed. It was shown theuse of these two coefficients, instead of a single one, improvesthe fitting between the results of the numerical simulationsand the experimental measurements. The termKv1 (i.e. thewave phase coefficient), which affects the local acceleration, isresponsible for the wave phase and the termKv2 (i.e. dampingeffect coefficient), which affects the convective acceleration,influences the surge damping, being always, for the analysedsystem,Kv1 < Kv2 (i.e.Kv1 ≈ 10%Kv2).

• Classic waterhammer models only describe the elastic effects(using the quasi-stationary condition for friction losses esti-mation), being only reasonably adequate to simulate systemswith elastic behaviour (i.e. low deformable pipe-walls—metalor concrete).

• Being the plastic pipe with a future increasing application, theviscoelastic effect must be considered, either for model cali-bration, leakage detection or in the prediction of operationalconditions (e.g. start-up or trip-off electromechanical equip-ment, valve closure or opening).

• Aiming at the operational system response, the damping effectcan be estimated by Eqs (32) or (33), depending on the mainsystem characteristics (i.e. elastic or plastic type response),beingK1 related to the Joukowsky overpressure (ork1 ≈ 1)

andK2t (ork2τ ) the attenuation factor between two successivewave pressure peaks, which depends on the pipe length, typeof pipe material, pipe roughness or inertial effects (i.e. dQ/dt).Alternatively other equivalent dimensionless formulations canbe applied by using Eqs (24), (27) or (30) for elastic, non-elastic or combined effects, respectively;

• The coefficientK2 depends on each system characteristics(i.e. inverse formulation for low deformable pipe materials—Eq. (32), or exponential formulation for high deformable pipematerials—Eq. (33)). In the inverse formulation,K2 representsthe energy dissipation only due to friction effect, which forwater and turbulent flow can be represented as a function ofQ2.In the exponential formulation,K2 is considered proportionaltoQ, which includes energy dissipation due to the lack of elas-ticity of the system (e.g. fluid, pipe walls, eventual presenceof air pockets) and the effect of inertia effects (i.e. dQ/dt).

Plastic pipes or systems with dissolved air (Pearsall, 1965; Covaset al., 2003), have the predominance of non-elastic effects. Onthe contrary, metal and concrete pipes, with rough material andwithout air mixing, have a typical elastic behaviour.

In practice, the estimation of coefficientK2 (or k2 in dimen-sionless form) can be a difficult task, as this coefficient dependson several factors, such as inertial effects (associated with thepipe length), pipe materials, pipe roughness and inertial forces(dQ/dt).

Acknowledgments

The authors wish to express thanks to SMT4-CT97 2188EU project (Transient Pressures in Pressurised Conduits forMunicipal Water and Sewage Water Transport) for the use ofthe data of system S4, as well as to LHRH of DEC/IST for data

424 Ramos et al.

collection of transient events and FCT by the financial support tothe project POCTI/37798/ECM/2001.

Notation

A = cross-section flow areac = wave speed

Di = internal diameter of the pipeE = Young’s modulus of elasticity of the conduit wallsg = gravitational accelerationh = relative head(h = H/�HJ)

H = piezometric headJ = hydraulic gradient

Jqs = hydraulic gradient due to steady-state conditionsJu = hydraulic gradient due to unsteady conditionsK = the bulk modulus of elasticityK1 = overpressure coefficient from Joukowsky formula

(K1 = c/(gS)) for damping curvesK2 = damping coefficient for damping curves

K2elas= damping coefficient for elastic effectK2plas= damping coefficient for plastic effect

k3 = decay coefficient (Brunoneet al., 1991)Kv1 = wave phase coefficient for unsteady frictionKv2 = damping effect coefficient for unsteady friction

L = pipe lengthN = number of pipe elementsq = relative discharge

Q, Q0 = discharge flow; discharge of the steady-state conditionS = flow cross-sectiont = time

V = fluid velocityx = distance along the pipe axis

�H0 = total head losses�HJ = Joukowsky overpressure(�HJ = cQ/(gS) = K1Q)

�h0 = relative head losses(�h0 = �H0/�HJ)

ρ = fluid densityψ = parameter that depends on the elastic properties of the

conduitτ = relative time(τ = t/(2L/c))

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