ANALYSIS OF SURGE EFFECTS IN PIPE SYSTEMS BY AIR

RELEASE / VENTING

Helena Ramos Professor of Instituto Superior Técnico, APRH member nº 822, Portugal, hr@civil.ist.utl.pt A. Borga Professor of ISEL,Portugal, aborga@sapo.pt Anton Bergant Chief Research Engineer, Litostroj E.I. d.o.o., Slovenia, anton.bergant@litostroj-ei.si Dídia Covas Assistant Professor of Instituto Superior Técnico, APRH member nº 1387 , Portugal, d.covas@civil.ist.utl.pt A. Betâmio de Almeida Professor of Instituto Superior Técnico, APRH member nº 80, Portugal, aba@civil.ist.utl.pt

ABSTRACT - A transient event in a pressurized pipe system corresponds to an intermediate

flow state between two steady state regimes. The control of dynamic effects normally

associated to the occurrence of hydraulic transients must be considered during design and

exploitation phase to guarantee the best solution in terms of safety and reliability. Extreme

pressures, even with small duration, can reach excessive values with desired safe and

operational conditions: the maximum pressure values can cause ruptures in pipes and fittings,

while low values can induce the collapse of the pipe-wall, air admission and release or the

formation of vapour cavities. The analysis of surge effects due to cavitation occurrence in two

pipelines with different pipe materials (i.e., metal and plastic), including comparisons with

experiments as well as the modelling of an air-valve behaviour for different pipe profiles, type

of discharge closure manoeuvres and quantity of retained air, are performed. Mathematical

models are increasingly more used in design, as well as in the daily management and control

of hydraulic systems, in particular for the prediction of unstable dynamic conditions. RESUMO - Um transitório hidráulico quando ocorre num sistema em pressão corresponde a

um estado intermédio do escoamento entre dois regimes permanentes. O controlo de efeitos

dinâmicos normalmente associados à ocorrência de transitórios hidráulicos deve ser

considerado desde as fases de projecto até à fase de exploração para que seja garantida a

melhor solução em termos de segurança e funcionalidade de cada sistema. Pressões extremas,

mesmo de curta duração, podem atingir valores incompatíveis com os desejados em termos de

segurança e operacionalidade: valores de pressão máxima podem causar rupturas em condutas

e acessórios, enquanto que valores baixos podem induzir o colapso da parede da conduta,

entrada de ar, libertação de ar dissolvido no escoamento ou a formação de cavidades de vapor.

A análise dos efeitos dinâmicos devido à ocorrência de cavitação em duas condutas de

material diferente (i.e., metal e plástico), mediante a comparação com resultados

experimentais, assim como a modelação do comportamento de uma ventosa para diferentes

perfis da conduta, tipo de manobra e quantidade de ar que poderá ficar retido no sistema, são

aspectos abordados neste estudo. A utilização de modelos matemáticos faz, cada vez mais,

parte das rotinas de projecto assim como na gestão e controlo do sistema hidráulico, em

particular na previsão de condições instáveis de funcionamento.

Keywords: dissipative effects, cavitation, air-valves, air release, air venting, pressure surges

1. INTRODUCTION

In transient flows with cavitation or water column separation, the presence of free air or

vapour modifies the elastic wave celerity and, consequently, the dissipative effects of the

bubbly flow. Several transient tests have been carried out to calibrate numerical models

developed by using the Method of Characteristics with localised gas cavities and variable

wave speed to describe the presence of gas. The developed analysis aims at alert the existence

of special dynamic effects induced during different transient conditions. This paper explains

the distinction between different types of system responses and shows associated phenomena

with air release/venting.

The use of air valves at high points along the pipeline profile allows air admission and release,

thus avoiding the occurrence of cavitation at those points and maintaining practically constant

pressure equal to the atmospheric one. When the piezometric line drops below the

atmospheric pressure the air valve opens to allow air admission. In same way when the

pressure raises above the atmospheric pressure the air valve allows air release into the

atmosphere.

2. CAVITATION

In water, there is always a small quantity of air, which flows with the liquid either as free air

(bubbly flow) or/and dissolved. In numeric modelling, the fluid is considered as pseudo

(homogeneous) liquid. Special attention must be given to pressure-dependent wave celerity.

The free gas is considered as isothermal and ideal inducing a variation in the wave speed of

about 40 to 70% (Almeida, 1981, Borga et al., 2004) of the initial value of the wave speed

(Figure 1).

Figure 1 –Typical variation of the wave speed in time

The presence of free air or vapour cavities modifies the actual elastic wave celerity and

induces higher dissipative effects. However, the system response is significantly different in a

metal and a plastic pipe. Given the lack of complete mathematical models suitable to analyse

the mixed flow and the pipe material rheological behaviour in a satisfactory way, a new

model has been developed based on special characteristic parameters. The model allows

identifying and distinguishing different dynamic effects as well as assessing the importance of

intervenient factors, and their parameterisation. This model, that has been developed based on

the Method of Characteristics (MOC), is calibrated and tested using transient data collected in

different type of systems (e.g. metal and plastic pipes).

270

290

310

330

350

370

390

410

0 5 10 15 20time (s)

c (m

/s)

Numerical results are presented in terms of piezometric head variation in time, allowing the

characterisation and analysis of relevant issues such as (i) maximum and minimum observed

pressures, (ii) time period oscillation and (iii) damping effect.

Mathematical formulation

The conventional vapour-liquid model assumes that discrete vapour cavities are formed at

fixed pipe sections and considers a constant wave speed in pipe reaches between the cavities

(Wylie and Streeter, 1993; Chaudhry, 1987). This model is particularly adequate when the

vapour tends to cumulate in particular pipe sections, forming pockets or, when, due to the

pipe profile, only part of the system is under vapour pressure. In this formulation, the MOC is

solved with a fixed grid and, whenever a particular section of the pipeline has a pressure

below vaporization pressure, it is calculated as an internal boundary condition.

Numerical results presented in this paper are obtained based on the traditional vapour-liquid

model, with a second order approximation for the calculation of quasi-steady friction losses,

introducing adequate changes in different characteristic parameters (e.g. wave celerity, head

losses) in order to better simulate observed dissipation and dispersion effects due to

mechanical, frictional and inertial effects. The following changes have been incorporated: (i)

modification of Courant number; (ii) modification of friction loss coefficient (or headloss);

(iii) modification of wave speed by an exponential law in time but uniform along the pipe

axis; and (iv) modification of coefficients of the characteristic equations which affect the

transformation of kinetic energy into elastic and vice-versa.

The simulated systems are basically composed of a uniform horizontal transmission pipeline

with a reservoir and a sectioning valve at the upstream end, and a reservoir at the downstream

end. The numerical gridline is chosen in order to satisfy the Courant-Friedrich-Lewy (CFL)

stability condition

Cr = c ∆t/∆x ≤ 1 (1)

where Cr = Courant number; c = wave speed; ∆t = numerical time increment; ∆x = numerical

space increment. Cr is generally equal to unity, but in some simulations was adopted Cr < 1.

For modelling purposes, the effect of air release is neglected and discrete vapour cavities can

open at all pipe sections. Hence macrocavitation (large cavities) can be characterized by the

existence of a vapour cavity volume ∀ i,j at the pipe section i and for the time j as

( ) 2/tQQQQ 1j,iLj,iL1j,iRj,iR1j,ij,i ∆−−++∀=∀ −−− (2)

where QR and QL = discharge at the right and the left of the cavity, respectively.

This condition is imposed when the absolute pressure drops to the liquid vapour pressure

(vaporous cavitation inception) HIC (ca.-10 m at room temperature) and maintains this value

as long as the cavity volume is positive. The piezometric head within the cavity is given by

γ+=∀

atmIC

pHH (3)

The application of the MOC to the typical “reservoir-pipe- valve” gives the following

equations (Ramos, 2004):

C+: ( ) 0QQRBQQHH LLLPLP =+−+− (4)

C-: ( ) 0QQRBQQHH RRRPRP =−−−− (5)

with gScB = . For a reservoir with constant water level (Z):

ZHP = (6)

when solved simultaneously with the characteristic line C- allows the calculation of QP (by

using a first order integration),

B

CZQ 2P

−= (7)

For an intermediate pipe section, characteristic lines C+ and C- must be used and if the air

volume inside the pipe is null (i.e., 0=∀ ) , then

( ) ( ) 0QQQQRQQBBQ2HH RRLLLRPLR =+++−+− (8)

yielding for QP,

B2CCQ 21

P−

= (9)

and

2CCH 21

P+

= (10)

When ∀< HHP then ∀= HHP and the volume of the cavity is calculated according to

equation (4), then solving for

BQQRHHBQ

Q LLPLLPL

−−+= (11)

and the same procedure for the point at the right side of the cavity

BQQRHHBQ

Q RRPRRPR

++−= (12)

being the volume of the cavity given by

( ) tQQ LR12 ∆−+∀=∀ (13)

in which the subscript 1 and 2 refers to the previous and current time step.

If ∀<0 then ∀=0 is imposed and the procedure is repeated.

At the valve section, with a quasi-instantaneous discharge reduction and if the cavity volume

is∀= 0 then QP = 0 and based on C+

( ) LLLP QQRBHH −+= (14)

but if HP< ∀H then

BHC

Q 1P

∀−=

(15)

and

( ) tQQ LP12 ∆−+∀=∀ (16)

In the calculus if ∀<0 then is considered∀=0 and the procedures is repeated.

The modification of the headloss coefficient is obtained by a multiply constant factor, KR. In

the simulation of the variable celerity (Borga et al, 2004) due to air releasing during cavitation

occurrence, an exponential variation along time is considered, uniform along the entire pipe,

according to the following equation:

( ) CTt

ecfcocfc−

−+= (17)

where co, cf and c = wave speed values at initial, final and calculation time, respectively; CT

is the time decay coefficient of the wave speed.

For the description of non-elastic behaviour due to fluid and pipe material, two parameters

(KH and KQ) were included in the MOC equations:

IQgScKHH −∆=∆ (18)

)gS/(cIHKQQ −∆

=∆ (19)

where I = headloss term; g = gravitational acceleration; S = pipe cross-sectional area; ∆H and

∆Q = head and discharge variation, respectively.

Parameter KH stands for the head decay induced by a discharge variation due to the non-

elastic behaviour of the fluid (i.e., with the presence of free gas) and the pipe viscoelasticity.

KQ is a raised coefficient in the discharge value caused by a head variation, due to a non-

elastic response in the recuperation phase of the deformation.

Analysis of Results

The following analysis includes the parameterisation of special dynamic effects induced by

cavitational effects (Borga et al., 2004; Ramos, 2004). The values of input data are in SI units

and appear in left column of each figure with the following meaning: Q – discharge; L – pipe

length; D – pipe diameter; HM – upstream head; HJ – downstream head; N – number of

reaches in pipeline; Co – initial wave speed; CF – final wave speed; CT – parameter of wave

speed reduction; KR – multiplicative head loss coefficient; KT – coefficient of second order

term in the integration of head loss; TV – maximum vapour pressure; Cr – Courant number;

KH - gives a reduction in the head variation; KQ - a reduction coefficient in the discharge

value; TF – time closure of the valve; TT – total time for simulation.

Figures 1, 2, 3, 4 and 5 refer to an experimental set-up with a horizontal cast iron pipe with

110 m of length and an inner diameter of 0.067 m. This pipe is connected, at the extremities,

with two reservoirs of constant water level of 12 m at the upstream end and 1.8 m at the

downstream end, respectively. A pneumatically actuated ball valve was installed close to the

upstream reservoir, which allows a quasi-instantaneous change of flow conditions. The elastic

wave speed in the pipe is c = 1300 m/s that was experimentally estimated based on the

measurement of small amplitude of the elastic oscillating period. The initial flow conditions

are: ∆Ho = 12 m and Qo = 0.007 m3/s. The time for ball valve closure is approximately 0.1 s.

The successive incorporation of different effects changes the transient response of the system.

In Figure 2 the incorporation of localised vapour cavities is not sufficient because the

dissipative effects and the wave period are not correctly modelled.

Figure 2 – MOC with localised cavities (dark line - experiments; grey line - model).

Figures 3 and 4 show that an additional head loss and a wave speed reduction improve the

results.

Figure 3 – MOC with localised cavities and larger value of headloss (KR) (dark line - exp; grey line - mod).

Figure 4 - MOC without cavitation and with local. cav., larger value of headloss (KR) and

wave speed decreasing (dark line - exp; grey line - mod).

Q= 0.007L= 110D= 0.067

HM= 12HJ= 1.8N= 10

C0= 1300CF= 1CT= 5KR= 1KT= 0.5TV= -8.2Cr= 1

KH= 1KQ= 1TF= 0.1TT= 20

Q= 0.007L= 110D= 0.067

HM= 12HJ= 1.8N= 10

C0= 1300CF= 1CT= 5KR= 2.3KT= 0.5TV= -8.2Cr= 1

KH= 1KQ= 1TF= 0.1TT= 20

Q= 0.007L= 110D= 0.067

HM= 12HJ= 1.8N= 10

C0= 1300CF= 1CT= 6KR= 4KT= 0.5TV= -8.2Cr= 0.84

KH= 1KQ= 1TF= 0.1TT= 20

middle pipe section

-50.0

0.0

50.0

100.0

150.0

200.0

0 2 4 6 8 10 12 14 16 18 20-50.0

0.0

50.0

100.0

150.0

200.0

0 2 4 6 8 10 12 14 16 18 20

An equivalent effect can be obtained by a numerical manipulation based on a fixed Courant

number (Cr) lower than unity and combining this with an increasing of parameter KR (Borga

et al., 2004). The attempt with Cr different than 1 is not recommended in engineering practice

(numerical damping) at current stage of the research. Further research is needed to investigate

physical source of this effect.

Figure 5 - MOC with local. cav., wave speed variation and special dynamic parameters

(KH<1 e KQ>1) (dark line - exp; grey line - mod).

Figure 5 shows how the use of specific parameters can reproduce reasonably well the flow

energy dissipation as well as the non-elastic effects (i.e. induced by the expansibility and

compressibility of gas bubbles inside the flow).

In the set-up with a plastic pipeline, the type of response is completely different. This set-up is

composed of a single transmission pipeline connected to an air vessel at the upstream end and

with free discharge to a constant water level at the downstream end. The pipe material is high-

density polyethylene (HDPE), with a total length of 200 m, an inner diameter of 0.043 m and

an elastic wave speed of 280 m/s. A ball valve is installed followed the air vessel and it is

used to interrupt the flow in a fast closing manoeuvre. The incorporation of localised vapour

cavity model is also considered (Figure 6), as well as the headloss increase (Figure 7).

Figure 6 - MOC with localised cavities (dark line - exp; grey line - mod).

Q= 0.007L= 110D= 0.067

HM= 12HJ= 1.8

N= 10C0= 1300CF= 0.0001CT= 7KR= 1KT= 0.5TV= -8.2Cr= 1

KH= 0.75KQ= 1.2TF= 0.1TT= 20

middle pipe section

upstream pipe section (1)

-10

0

10

20

30

40

50

0 5 10 15 20 25 30 35 40time (s)

H (m)Q= 0.004L= 200D= 0.045

HM= 29HJ= 0.5N= 10

C0= 300CF= 1CT= 5KR= 1KT= 0.5TV= -8.2Cr= 1

KH= 1KQ= 1TF= 0.2TT= 40

middle pipe section

-10

0

10

20

30

40

50

0 5 10 15 20 25 30 35 40

time (s)

H (m)

Figure 7 - MOC with localised cavities and larger value of headloss (KR) (dark line - exp; grey line - mod).

The presence of vapour cavities (Bergant and Simpson, 1994; Bughazem and Anderson,

2000) and other dynamic effects, such as additional dissipative effects and plastic pipe

rheological behaviour (Covas et al, 2004a,b; Covas et al., 2005), increase the pressure

damping. Pressure fluctuations in the mathematical model are due to the collapse of vapour

cavities in different pipe sections, which induces some numeric disturbances (i.e. for low N

these effects are almost neglected).

When pressure decreases, gaseous cavitation is formed in the flow, which consequently

decreases the wave speed (Qiu and Burrows, 1996; Burrows and Qiu, 1995; Lee, 1999). This

is visible in the transient pressure wave period shown in Figure 8.

Figure 8 - MOC with local. cav., larger value of headloss (KR) and wave speed decreasing (dark line - exp; grey line - mod).

However, there is a numerical effect which does not have an immediate or evident physical

correspondence but that allows to obtain the same type of behaviour: when the Courant

number decreases the cavity duration increases, as well when total energy dissipation (KR)

increases the cavity duration decreases but both parameters will increase the attenuation

(Figure 9).

upstream pipe section (1)

-10

-5

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35 40

time (s)

H (m)

middle pipe section

-10

-5

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35 40

time (s)

H (m)

Q= 0.004L= 200D= 0.045

HM= 29HJ= 0.5N= 10

C0= 300CF= 1CT= 5KR= 2.3KT= 0.5TV= -8.2Cr= 1

KH= 1KQ= 1TF= 0.2TT= 40

upstream pipe section (1)

-10

-5

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35 40

time (s)

H (m)

middle pipe section

-10

-5

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35 40

time (s)

H (m)Q= 0.004L= 200D= 0.045

HM= 29HJ= 0.5N= 10

C0= 300CF= 0.6CT= 5KR= 2.3KT= 0.5TV= -8.2Cr= 1

KH= 1KQ= 1TF= 0.2TT= 40

Figure 9 - MOC with local. cav., greater KR and lower Cr (dark line - exp; grey line - mod).

The use of special dynamic parameters (Figure 10) can reproduce the pressure flow variation,

producing equivalent results as the Courant encapsulation effect.

Figure 10 - MOC without cavitation and with local. cav., wave speed variation and special

dynamic parameters (dark line - exp; grey line - mod).

Real systems have frequently inlet / outlet to the exterior by using air-valves or existing leaks.

This inlet / outlet is not currently considered in the modelling, but it modifies significantly the

system behaviour.

upstream pipe section (1)

-10

-5

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35 40

time (s)

H (m)

middle pipe section

-10

-5

0

5

10

15

20

0 5 10 15 20 25 30 35 40

time (s)

H (m)

Q= 0.004L= 200D= 0.045

HM= 29HJ= 0.5N= 10

C0= 300CF= 1CT= 5KR= 8KT= 0.5TV= -8.2Cr= 0.6

KH= 1KQ= 1TF= 0.2TT= 40

upstream pipe section (1)

-10

-5

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35 40

time (s)

H (m)

Q= 0.004L= 200D= 0.045

HM= 29HJ= 0.5N= 10

C0= 300CF= 0.8CT= 5KR= 1KT= 0.5TV= -8.2Cr= 1

KH= 0.4KQ= 1.4TF= 0.2TT= 40

midle pipe section

-10

-5

0

5

10

15

20

0 5 10 15 20 25 30 35 40

time (s)

H (m)

middle pipe section

upstream pipe section (1)

-10

-5

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35 40

time (s)

H (m)

Figure 11 - MOC with localised cav., variable wave speed and specific dynamic effects for set-up 2 with a leak at upstream (dark line - exp; grey line - mod).

It is well known that free air in liquid decreases the wave speed, which is, in some cases, a

beneficial effect. However, sometimes (similar to air-valves problem) the presence of air

inside the pipes induces non-elastic effects that can provoke unexpected, non-controlled and

violent pressure peaks, as shown in the first isolated and sudden peak in Figure 11.

3. AIR-VALVES

The presence of air can generate extreme pressure surges during normal operating conditions

as well as a reduction in system transport capacity caused by the reduction of the flow section

(De Martino et al, 2003). The use of air-valves ensures, sometimes, sufficient air venting.

This phenomenon prevents the formation of large air pockets subjected to high-pressure

values during the pressure compression procedures. However, the operation of these devices

may trigger pressure surges, in some cases of considerable importance due to a sudden

deceleration of water column following expulsion of the air.

Mathematical Formulation

For modelling purposes an air valve is an interior boundary condition and it is considered that

the water level inside the pipe at the air valve section is constant (i.e., it is supposed the inlet

air volume is relatively small when compared to the pipe length).

The methodology is quite similar to the case with cavitation, being different due to the

possibility to consider a fraction of air retained inside the fluid by dragging flow to neighbour

sections of the air-valve during transients (Ramos, 2004). In the section with air-valve pipe

the following equations should be considered:

Continuity equation for the air

upstream pipe section (1)

-10

-5

0

5

10

15

20

25

30

0 5 10 15 20 25 30 35 40

time (s)

H (m)

middle pipe section

-10

-5

0

5

10

15

20

0 5 10 15 20 25 30 35 40

time (s)

H (m)Q= 0.004L= 200D= 0.045

HM= 29HJ= 0.5N= 10

C0= 300CF= 0.6CT= 5KR= 1KT= 0.5TV= -8.2Cr= 1

KH= 0.35KQ= 1.45TF= 0.2TT= 40

( )PRPLairair QQt −∆−∀=∀ (20)

with air∀ the air volume inside the pipe.

When the air-valve is opened then

VP ZH = (21)

But if the air valve is closed with air inside

ARVP

K310ZH∀

=+− ∀. (22)

It is considered that the percentage of air that can be retained in the flow has an isothermal

behaviour ( cteH nair =∀ with n=1).

However if the air-valve is closed but without air inside the pipe then

PJPM QQ = (23)

The following analysis shows the variation of the system response for an air-valve located at

the high point of the pipe profile with different closure times and percentages of retained air

inside the pipe due to air dragging by transient conditions with forwards and backwards

pressure waves.

Analysis of Results

The following analysis is for a pumping system with a pump at the upstream, a check valve

immediately downstream the pump (which can have different closure times) and no surge

protection device installed. It shows the influence of the air-valve located at high sections of

the pipe profile (see Figure 12 – profile) by the piezometric head variation at the check valve

and at the air-valve sections, and the maximum and minimum values along the pipeline.

Figure 12 – Effect of an air-valve in the high point of the profile of a pumping system with a

slow check valve closure at upstream (Z(1)=60; TF=4 s).

There is a marked difference on the system behaviour when a fraction of air (20% and 60%) is

retained inside the flow (Figure 13).

Figure 13 – Effect of an air-valve with TF=4 s and a percentage of retention air (FV) inside

the flow of 20% and 60% (Z(1)=60).

Hence a new case is analysed with an upstream water level Z(1)=75 higher than the initial

value Z(1)=60. It was verified that the maximum value of piezometric head is greater

(Hmax=159) than initial value Z(1) = 60 (Hmax=145) – Figure 14. Figure 15 shows the effect

of the pipe profile modification.

40 60 80

100 120 140 160 180 200

0 50 100 150 200 250 300 350 400 450 500

Max. envelopMin. envelopprofile

L=500 m; Lair-valve=290 m; c=1000 m/s; TF=4s; FV=0.2; HM=100 m; HJ=90 m ; Qo=0.01 m3/s

H (m)

L (m)

40

60

80

100

120

140

160

180

200

0 50 100 150 200 250 300 350 400 450 500

Max. envelopMin. envelopprofile

L=500 m; Lair-valve=290 m; c=1000 m/s; TF=4s; FV=0.6; HM=100 m; HJ=90 m ; Qo=0.01 m3/s

H (m)

L (m)

40 60 80

100 120 140 160 180 200

0 10 20 30 40 50 60 70 80

Check valve Z valve = 60air-valve Z air-valve = 80

H (m)

t (s)

L=500 m; Lair-valve=290 m; c=1000 m/s; TF=4s; FV=0.0; HM=100 m; HJ=90 m; Qo=0.01 m3/s

40

60

80

100

120

140

160

180

200

0 50 100 150 200 250 300 350 400 450 500

Max. envelopMin. envelopprofile

L=500 m; Lair-valve=290 m; c=1000 m/s; TF=4s; FV=0.0; HM=100 m; HJ=90 m ; Qo=0.01 m3/s

H (m)

L (m)

Figure 14 - Effect of an air-valve in the high point of the profile of a pumping system with a

slow check valve closure at upstream (TF=10 s) and with increasing of the profile level.

Figure 15 – Effect of an air-valve in the high point of the profile of a pumping system with a

slow check valve closure at upstream (TF=10 s), with increasing of the profile level and a

60% of retained air inside the flow.

In practice, air-valves are often not sufficient to prevent occurrence of column separation

(formation of large vapour cavities) and the use of air vessels or one-way tank is not only

advisable but necessary. The lack of data on the dynamic performance of air-valves is a

concern of designers.

4. FINAL REMARKS

The main conclusions of this research work can be summarized in the following points:

40 60 80

100 120 140 160 180 200

0 10 20 30 40 50 60 70 80

Check valve Z valv = 75Air-valve Z air-valve = 86

H (m)

t (s)

L=500 m; Lair-valvet=290 m; c=1000 m/s; TF=10s; FV=0.6; HM=100 m; HJ=90 m; Qo=0.01 m3/s

40

60

80

100

120

140

160

180

200

0 50 100 150 200 250 300 350 400 450 500

Max. envelopMin. envelopprofile

L=500 m; Lair-valve=290 m; c=1000 m/s; TF=10s; FV=0.6; HM=100 m; HJ=90 m ; Qo=0.01 m3/s

H (m)

L (m)

40

60

80

100

120

140

160

180

200

0 50 100 150 200 250 300 350 400 450 500

Max. envelopMin. envelopprofile

L=500 m; Lvent=290 m; c=1000 m/s; TF=10s; FV=0.0; HM=100 m; HJ=90 m ; Qo=0.01 m3/s; Z1=75

H (m)

L (m)

40 60 80

100 120 140 160 180 200

0 50 100 150 200 250 300 350 400 450 500

M ax. envelopM i n.

envelop

profile

L=500 m; Lvent=290 m; c=1000 m/s; TF=10s; FV=0.0; HM=100 m; HJ=90 m ; Qo=0.01 m3/s ; Z1=60

H (m)

L (m)

(i) a model only based on the basic equations of MOC considering the stationary headloss

(∆Ho) to simulate the dissipative effects cannot describe the system response when it is

subjected to cavitation or air-venting conditions;

(ii) a model based on MOC which incorporates the modelling of localised cavities can give

reasonable results that can be used with some confidence for design purposes;

(iii) once it was experimentally verified for both experimental set-ups (i.e., with metal and

plastic pipes), the gas-release induces the increase of dissipative effects, as well as the

modification of the pressure wave period, it advises one to incorporate in model (ii) an

additional headloss (KR . ∆Ho, with KR>1) and a decrease of the wave speed value in

time given by equation (3) (i.e. cfinal is from 40 to 70 % of co, being co the stationary

wave speed value);

(iv) an equivalent numerical effect can be obtained by the combination of parameters Cr and

KR which cannot be correlated with the facilities in an evident way, however, the

attempt with Cr different from 1 is not recommended in engineering practice (because it

induces numerical damping) at current stage of the research - further research is needed

to investigate physical origin of this effect.;

(v) the use of air valves at high pipe sections along the pipe profile is inevitable to avoid the

occurrence of cavitation with severe pressure surges;

(vi) the retention of air inside the flow by air-valve operation modifies the transient

behaviour, inducing a greater damping and dispersion of the pressure waves.

ACKNOWLEDGEMENTS The authors wish to thank the Surge-Net project (WP3 - Leak Detection) which is supported

by funding under the European Commission's Fifth Framework 'Growth' Programme via the

Thematic Network "Surge-Net" contract reference: G1RT-CT-2002-05069 (see

http://www.surge-net.info), as well as to the Hydraulic Laboratory of Civil Engineering

Department, Instituto Superior Técnico (Lisbon, Portugal) and LNEC, where the experimental

programme was run. The authors wish also to thank to FCT through the projects

POCTI/ECM/37798/2001 and POCTI/ECM/58375/2004, which are partly supported by

funding under the European Commission – FEDER.

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SIMBOLOGY

c = wave speed;

co = initial wave speed;

cf = wave speed value at final time;

Cr = Courant number;

CT = time decay coefficient of wave speed reduction;

C1, C2 = known parameters in water hammer compatibility equations;

D = pipe diameter;

g = the gravitational acceleration;

H = piezometric head;

HIC = liquid vapour pressure (or initial of cavitation);

HJ = downstream head;

HM = upstream head;

HP = piezometric head at calculus pipe section;

Hv = piezometric head inside the cavity;

I = the headloss term;

KH = reduction in the head variation by non-elastic fluid and pipe deformation;

KQ = reduction in the discharge value caused by a head variation, due to a non-elastic

response in the recuperation phase of the occurred deformation;

KR = multiplicative head loss coefficient;

KT = coefficient of second order term in the integration of head loss;

L = pipe length;

N = number of pipe reaches;

patm = atmospheric absolute pressure;

Q = discharge;

QL = discharge values at the left;

QR = discharge values at the right;

R = head loss coefficient;

S = the pipe cross-sectional area;

TF = time closure of the valve;

TT = total time for simulation;

TV = maximum vapour pressure (depends on the type of the fluid);

air∀ = air volume;

∀ i,j = vapour cavity volume at the pipe section i and for the time j;

Z = water level in the reservoir;

Zv = air-valve topographic level;

∆H = head variation;

∆Q = discharge variation;

∆t = numerical time increment;

∆x = numerical space increment;

γ = liquid specific weight.