Physical Understanding of Sudden Pressurization of PipeSystems with Entrapped Air: Energy Auditing Approach

A. Malekpour, A.M.ASCE1; B. W. Karney, M.ASCE2; and J. Nault, S.M.ASCE3

Abstract: A global energy-auditing approach is used to describe the complex transient flow associated with sudden pressurization of aconfined pipe system containing entrapped air pockets. The key concept is that the greatest pressures following pressurization are directlyrelated to the amount of energy absorbed by the air pockets. The energy approach leads to an insightful understanding of the transientpressurization event, one that clarifies the underlying physics and that better explains several previously published observations. Moreover,the energy formulation leads to analytical algebraic expressions that can be numerically confirmed. DOI: 10.1061/(ASCE)HY.1943-7900.0001067. © 2015 American Society of Civil Engineers.

Author keywords: Pipeline; Rapid pressurization; Entrapped air; Transient flow; Energy approach.

Introduction

Rapid pressurization of liquid pipe systems naturally induces tran-sient pressures. However, if the system contains liquid only (and noair pockets), the magnitude of the resulting transient pressures maynot be dramatic. In fact, Abreu et al. (1992) show that the maximumtransient pressure in an all-liquid pipe is usually limited to approx-imately two times the driving head under which the system is pres-surized, but if the pipe system does contain entrapped air, themagnitude of the induced transient pressure may be significantlyhigher (Wylie and Streeter 1993). Of course, the essential originof the transient flow is similar in both cases. That is, when the boun-dary pressure of the system suddenly increases, the induced flow inthe pipe suddenly becomes arrested when it encounters the deadend, thus generating a water hammer response at that location.If the system contains air, however, the greater system compress-ibility permits a momentary but much higher flow velocity to enterthe system, which can then produce significantly greater transientpressures when it is arrested.

Sudden air compression has been identified as the cause of ex-treme transient pressure in some pipe systems (Martin 1976; Wylieand Steeter 1993). For example, the rapid compression of an airpocket trapped between the check valve of a pumping line andthe returning liquid column following a power failure can leadto severe overpressures (Jönson 1985). The sequence of eventsin a dry lake tap can suddenly compress the air in the associatedtunnel and thus produce extreme overpressures (Chaudhry andReddy 2011). Under wet-weather flow conditions, a hydraulic boremoving along a sewer pipe may compress an air pocket at a dropshaft and induce extreme pressures; subsequent reflection of the

induced overpressures may then result in negative pressures(Wright et al. 2012; Vasconcelos and Leite 2012).

The magnitude of the resulting transient pressures dependson many factors, including the system’s driving head, amount ofentrapped air, pipe friction, liquid and pipe elasticity, and pipeslope. How these parameters interact to create the resulting tran-sient pressure has motivated considerable research. A number ofauthors (Martin 1976; Cabrera et al. 1992; Abreu et al. 1992;Izquierdo et al. 1999) used a rigid water column model to describethe overpressure associated with a rapid air compression event.Some (Abreu et al. 1992; Lee and Martin 1999; Chaiko andBrinckman 2002; Epstein 2008; Liu et al. 2011; Zhou et al.2011a) utilized a water hammer model. Zhou et al. (2011b) appliedboth two-dimensional (2D) and three-dimensional (3D) computa-tional fluid dynamic (CFD) models to capture the transientconditions caused by rapid pressurization of a simple pipe systemand thereby confirmed the occurrence of extreme pressure rises.

Several experimental studies have also been conducted. BothLee (2005) and Zhou et al. (2011a) experimented on a simplepipe-reservoir system, confirming the occurrence of the extremepressure rises predicted by numerical models. Jönson (1985) ex-plored the onset of extreme overpressures caused by compressionof an air pocket in the discharge line of a pump due to liquid col-umn reversal following a power failure. Nakamura and Tomita(1999) showed that the rapid compression of air pockets due toquick start-up of a pumping line can produce an extreme pressurerise. Wright et al. (2012) and Vasconcelos and Leite (2012) exper-imented with both partial and full flow stoppage in a pipe-reservoirsystem with a downstream air pocket, and like others, they reportedthe occurrence of extreme positive and negative pressures.

The studies just mentioned illustrate the implications of en-trapped air that need to be considered in design and operation.In particular, they show how the resulting transient conditionsare sensitive to the characteristics of the pipe systems. Nevertheless,the complex nature of entrapped air has often hindered a more com-plete understanding of these conditions.

As part of his study, Martin (1976) employed a rigid columnmodel to study rapid pressurization of a simple pipe system com-prising an upstream constant-water-level reservoir, a horizontalfrictionless pipe, and an entrapped air pocket at the downstreamend. This study showed that, for a specific reservoir level, themaximum pressure following rapid pressurization of the system

1Hydraulic Specialist, HydraTek and Associates, 216 Chrislea Rd.,Suite 204, Vaughan, ON, Canada L4L 8S5 (corresponding author). E-mail:a.malekpour@hydratek.com

2Professor, Univ. of Toronto, 35 St. George St., Toronto, ON, CanadaM5S 1A4. E-mail: karney@ecf.utoronto.ca

3M.A.Sc. Candidate, Univ. of Toronto, 35 St. George St., Toronto, ON,Canada M5S 1A4. E-mail: Johnathan.nault@mail.utoronto.ca

Note. This manuscript was submitted on July 6, 2014; approved on June16, 2015; published online on August 13, 2015. Discussion period openuntil January 13, 2016; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Hydraulic Engineer-ing, © ASCE, ISSN 0733-9429/04015044(11)/$25.00.

© ASCE 04015044-1 J. Hydraul. Eng.

J. Hydraul. Eng.

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TY

OF

TO

RO

NT

O o

n 08

/14/

15. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

is independent of the size of the air pocket. This response is counter-intuitive because larger air pockets would seem to lead to a greatergain in momentum that should presumably lead to greater pressurerises. Yet this effect appears exactly countered by the increased cush-ioning effect of the larger air pocket. Such a fortuitous balance ofeffects is not convincingly explained in the literature.

Cabrera et al. (1992) studied the pressurization of a pipe withentrapped air by applying both complete and simplified rigid col-umn models while ignoring length-related terms in the governingequation. This study showed that the simplified model can replicatethe peak pressure predicted by the complete model if the velocityhead term is removed from the governing equation of the simplifiedmodel. These results led to a discussion of the underlying physics(Ghidaoui and Karney 1994), which in turn led to a further clari-fication of the findings of Cabrera et al. (1994). Nevertheless thearguments presented by Cabrera et al. (1994) were made numeri-cally, not through a physical argument.

Many other studies have identified, either numerically or exper-imentally, some peculiar transient conditions following suddenpressurization of systems with entrapped air; however, none provideda comprehensive physical explanation. What perhaps has hinderedresearchers is the fact that experimental and numerical approachesprovideonly a local and case-specific perspective. Even in simple pipesystems, the local transient response may be the result of the interac-tion of variouswaves that are affected bydifferent systemcomponents(i.e., boundary conditions, friction, the elastic characteristics of pipeand water, leaky joints). Although other studies have been beneficialto a point, a more holistic view would be helpful.

Karney (1990) proposed an energy approach to studying andunderstanding transient conditions. This approach balances thenet energy flux entering the system with the time rate of changein different energy components in the system, including the kinetic,elastic, and energy dissipation terms. These researchers showedthat the partitioning of the energy components in such a mannerleads to an integrated understanding of both a system’s transientresponse and the influence of specific factors.

The current paper uses an energy audit approach to better under-stand the underlying physics of transient conditions following therapid pressurization of pipe systems containing entrapped air.Karney’s (1990) approach is first adapted to the context of this study.Several published case studies are then revisited in light of the re-fined energy approach. Because many experimental and numericalstudies have shown that compressibility effects are seldom signifi-cant, only published rigid water column results are considered here.

Theoretical Background

Numerical Model

Although this paper does not propose a new numerical approach, arigid water column model is developed both to reproduce thenumerical results provided in the literature and to confirm the pro-posed energy approach. In addition, in more complex cases forwhich the energy approach cannot provide quantitative results, re-sults of the numerical model are used to support qualitative physicalinterpretations. For completeness this model is summarized here.

The governing equation of the rigid water column model is asfollows (Abreu et al. 1992):

ðL0 þ xÞg

dVdt

¼ Hd − ðL0 þ xÞ sin θ

−�1þ Kl þ

fðL0 þ xÞD

�VjVj2g

− Px

ρgð1Þ

V ¼ dxdt

ð2Þ

Px ¼ ðP0 þ PatmÞ�

L − L0

L − L0 − x

�γ − Patm ð3Þ

where V = liquid column velocity; g = acceleration due to gravity;Kl = local loss coefficient;D = pipe diameter; ρ = density of liquid;Hd = driving head; f = Darcy-Weisbach friction factor; Px = airpressure at a distance; P0 = initial air pressure; Patm = atmosphericpressure; γ = polytropic exponent; θ = pipe angle relative to thehorizontal (positive values are for adverse slopes); x = increasein liquid column length; L = pipe length; L0 = initial liquid columnlength, A = pipe cross-sectional area; and t = time.

The following assumptions are made: (1) the liquid-pipe systemis incompressible; (2) the pipe is either completely full or emptywith a vertical filling front; and (3) the head-loss relationshipfor steady flow conditions is valid. The MATLAB functionODE45, which uses a Runge-Kutta method, is used to numericallyintegrate the governing equations.

Energy Relations in Pipe Systems with Entrapped Air

By manipulating the one-dimensional (1D) continuity and momen-tum equations, Karney (1990) derived the following equation:

ρA2

�ga

�2 ddt

ZH2dxþ ρA

2

ddt

ZV2dxþ fρA

2D

ZjVj3dx

þ ρgAVðL; tÞHðL; tÞ − ρgAVð0; tÞHð0; tÞ ¼ 0 ð4Þ

where H = piezometric head; and a = acoustic wave velocity.The collective terms in Eq. (4) have units of power. In order

these are the rates of change of elastic energy of both the liquidand the pipe, of kinetic energy of the liquid, of mechanical energydissipation, and of the energy fluxes leaving and entering the pipe-line, respectively. A more compact form is

dUdt

þ dTdt

þD 0 þW 0 ¼ 0 ð5Þ

where U = elastic energy stored in both the pipe and the liquid; T =kinetic energy of the liquid column;D 0 = rate of energy dissipation;and W = net energy or work flux entering the pipe.

To use the energy approach for systems containing entrappedair, several additional terms are needed. Consider Fig. 1, which il-lustrates the motion of a liquid column in a system containing adownstream air pocket. As the liquid column elongates, the airpocket is compressed as it stores a portion of the liquid’s kineticenergy. If the liquid column moves in a sloping pipe, it also gainsor loses gravitational potential energy. Thus, Eq. (5) can be ex-tended as follows:

Fig. 1. Typical pipe with entrapped air

© ASCE 04015044-2 J. Hydraul. Eng.

J. Hydraul. Eng.

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TY

OF

TO

RO

NT

O o

n 08

/14/

15. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

dUdt

þ dUa

dtþ dEp

dtþ dT

dtþD 0 þW 0 ¼ 0 ð6Þ

where Ua = elastic energy stored in the air pocket; and Ep =gravitational potential energy of the liquid inside the pipe. If elasticeffects in the pipe and liquid are ignored, the nature of the otherterms is easily summarized.

Air Pocket EnergyThe rate of change in the elastic energy accumulated in the airpocket over a time period Δt can be calculated as follows:

dUa

dt¼ 1

Δt

0B@ZX

0

APxdx

1CA ð7Þ

Substituting Px from Eq. (3) into Eq. (7), the following equationis obtained after integration:

dUa

dt¼ AðL − L0Þ

Δt

�ðP0 þ PatmÞγ − 1

½ð1 − X�Þ1−γ − 1� − PatmX��ð8Þ

where X� ¼ X=ðL − L0Þ is the air pocket compression rate.

Potential EnergyThe rate of change in the potential energy of the liquid column overthe same time period can be calculated as

dEP

dt¼ 1

Δt½ðEPÞtþΔt − ðEPÞt� ð9Þ

The potential energy of the liquid column at the beginning andend of the time period can be calculated as

ðEPÞt ¼ 0.5ρgA sin θL20 ð10Þ

ðEPÞtþΔt ¼ZX0

ρgAðxþ L0Þ sin θdx ¼ 0.5ρgA sin θðX þ L0Þ2

ð11Þ

sin θ ¼ Z1 − Z2

Lð12Þ

By substituting Eqs. (10)–(12) into Eq. (9), the rate of change inthe gravitational potential energy of the liquid column can be cal-culated as

dEP

dt¼ 0.5

ΔtρgA sin θXðX þ 2L0Þ ð13Þ

In the previous equations, X = increase in length of the liquidcolumn; and Z1 and Z2 = elevations of the upstream and down-stream ends of the pipe, respectively.

Kinetic EnergyThe rate of change in the kinetic energy of the liquid column can becalculated as

dTdt

¼ 1

ΔtðTtþΔt − TtÞ ¼

1

Δt½0.5ρAðL0 þ XÞV2 − 0� ð14Þ

Boundary Energy FluxThe net energy flux (i.e., flow work) entering the system dependson whether the pipe system is pressurized by a constant-water-levelreservoir or by use of a pump. For constant-water-level reservoirs

W 0 ¼ − 1

ΔtρgAXHr ð15Þ

For a pump

W 0 ¼ − 1

ΔtρgA

ZX0

Hpdx ð16Þ

where Hp = pump head.Assuming that the head and discharge of the pump are approxi-

mated by a parabolic curve, the energy flux provided by the pumpcan be written as a function of the liquid column velocity:

W 0 ¼ − 1

ΔtρgA

Zx0

ðCp1 − Cp2A2V2Þdx

¼ − 1

ΔtρgACp1xþ

1

ΔtρgCp2A3

Zx0

V2dx ð17Þ

where Cp1 = pump shutoff head; and Cp2 = constant defining thepump’s characteristic curve shape.

Frictional Dissipation

Using the Darcy-Weisbach equation, the rate of mechanical energyloss over a period of time can be calculated as follows:

D 0 ¼ 1

Δt

ZX0

ρAfðL0 þ xÞ

DV2

2dx ð18Þ

Numerical Exploration

Given the background just described, some historical studies withimportant contributions can now be revisited using the energy ap-proach to provide a more complete understanding of the underlyingtransient flow physics. Since most previous work used either a res-ervoir-pipe system or a pumped pipeline, both types of system areconsidered in this paper.

Reservoir-Pipe Systems

Considering the typical pipe-reservoir system shown in Fig. 2 andneglecting pipe and liquid elastic effects, the energy equation canbe developed by plugging different components of energy fromEqs. (7)–(18) into Eq. (6) as follows:

AðL − L0Þ�P0 þ Patm

γ − 1½ð1 − X�Þ1−γ − 1� − PatmX�

�

þZX0

ρAfðL0 þ xÞ

DV2

2dxþ 1

2ρAðL0 þ XÞV2 − ρgAXHr

þ 0.5ρgA sin θXðX þ 2L0Þ ¼ 0 ð19Þ

This quite general equation can be employed to examine theeffects of reservoir head, slope, and friction on maximum air

© ASCE 04015044-3 J. Hydraul. Eng.

J. Hydraul. Eng.

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TY

OF

TO

RO

NT

O o

n 08

/14/

15. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

pressure. However, to better understand the effects of individualparameters, the following exploration starts from the simplestsystems and progressively adds complicating factors.

Frictionless-Horizontal Reservoir-Pipe SystemAs part of his research, Martin (1976) studied the rapid pressuriza-tion of a pipe system comprising a frictionless-horizontal pipe and anupstream constant-head reservoir with a downstream entrapped airpocket. His results unexpectedly showed that, for a specific reservoirwater level, the resultingmaximumpressure is independent of the airpocket size and the initial length of the liquid column.

By neglecting the terms associated with both friction and slopein Eq. (19), it can be simply concluded that, as the liquid columncompresses the air pocket, the energy entering the system throughthe upstream reservoir is partitioned between that accumulated inthe air pocket and that in the kinetic energy of the liquid column.After the liquid column velocity reaches its peak value, the partiallycompressed air pocket begins to decelerate the liquid column byabsorbing its kinetic energy.

By rearranging Eq. (19) and neglecting the terms associatedwith both the pipe slope and the friction, the liquid column velocitycan be calculated as a function of the compression rate of the airpocket, X�, as follows:

V ¼�

2g

X� þ ð L0

L−L0Þ

�X�ðHr þHatmÞ

−Hatm þH0

γ − 1½ð1 − X�Þ1−γ − 1�

��0.5

ð20Þ

Once the liquid column comes to rest, the pressure of the airpocket reaches its maximum, having absorbed all of the column’skinetic energy. By setting the velocity in Eq. (20) to zero, the maxi-mum air compression rate can be calculated as

ð1 − X�maxÞ1−γ − 1

ðγ − 1ÞX�max

¼ Hr þHatm

H0 þHatmð21Þ

Alternatively, Eq. (20) can be expressed in terms of the pressurehead: �

1

1 − γ

�"1 − ð HatmþH0

HmaxþHatmÞð1−γÞ=γ

1 − ð HatmþH0

HmaxþHatmÞ1=γ

#¼ Hr þHatm

H0 þHatmð22Þ

Both Eqs. (21) and (22) contain only one unknown (X�max

and Hmax, respectively), which can be obtained via iteration.Eq. (20) shows that the initial air void fraction, ∀f ¼L − L0=L0, is an important factor that affects the velocity of theliquid column during pressurization. As the initial air void fractionincreases, the liquid column’s potential to accelerate also increases.However, Eqs. (21) and (22) show that the maximum air compres-sion rate and the maximum pressure are unaffected by the initial air

void fraction, depending only on the reservoir level and the poly-tropic exponent. In other words, regardless of the initial length ofthe liquid column and the initial volume of the air pocket, boththe maximum air compression rate and the maximum pressureare fixed. This confirms the observations of Martin (1976).

To validate the energy approach, the results obtained fromEqs. (20) and (22) can be compared with those of a numericalanalysis for a specific pipe system (Fig. 2). Hereafter, the “fictitiouspipe system” refers to the system in Fig. 2, which has dimensionsof, L0 ¼ 5 m, Hr ¼ 20 m, Hatm ¼ 10.44 m, and D ¼ 0.2 m.

Fig. 3 depicts liquid column velocity and air pressure head asfunctions of the air compression rate for initial air void fractions of1.0 and 10% (corresponding with L values of 5.05 and 5.55 m,respectively). As shown in Fig. 3, the results obtained from theenergy approach exactly coincide with those of the numericalmodel. Also, they confirm that, while the maximum air pressureis independent of the initial air void fraction, the resulting maxi-mum liquid column velocity is highly sensitive to the initial air voidfraction. As expected, the greater the initial air void fraction, thegreater the liquid column velocity.

Although Eq. (22) mathematically explains the hypothesispresented by Martin (1976), further interpretation is possible. Toobtain a better understanding, Eq. (19) can be rearranged to obtainthe maximum energy stored in the air pocket per unit volume:

ðP0 þ PatmÞγ − 1

½ð1 − X�maxÞ1−γ − 1� − PatmX�

max ¼½Ua�max

AðL − L0Þð23Þ

Fig. 2. Hypothetical pipe-reservoir system with entrapped air

Fig. 3. Liquid column velocity and air pressure versus air void fractionfor a frictionless-horizontal pipe

© ASCE 04015044-4 J. Hydraul. Eng.

J. Hydraul. Eng.

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TY

OF

TO

RO

NT

O o

n 08

/14/

15. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Eq. (23) shows that if the air pocket receives a constant supplyof energy per unit volume, the maximum air compression rate andmaximum pressure remain constant regardless of the size of the airpocket. As was the case in Martin (1976), because each part of theflow entering the pipe system from the reservoir has an identicalenergy per unit volume of ρgHr, the reservoir can supply constantenergy per unit volume of the air pocket regardless of the air pock-et’s size . However, this hypothesis is invalidated if pipe slope,friction, or both are considered.

Effect of Pipe SlopeAbreu et al. (1992) showed numerically that pipe slope has a sig-nificant effect on the resulting maximum pressure, particularly atgreater initial air void fractions. They observed that pipe systemswith adverse slopes experience a decrease in maximum pressure asthe initial air void fraction increases, whereas pipe systems withfavorable slopes experience the opposite (i.e., the resulting maxi-mum pressure increases as the initial air void fraction increases).

Eq. (19) shows that, in the absence of friction, when the liquidcolumn is arrested, the energy in the compressed air pocket doesnot simply balance that released by the reservoir. Depending onwhether the slope is adverse or favorable, the total energy em-bedded in the air pocket can be either less or greater than that in-troduced by the reservoir. For adverse slopes, as the liquid columnadvances in the pipe, a portion of the energy provided by the res-ervoir is expended in the gain in gravitational potential energy. Thetotal amount of gravitational potential energy eventually accumu-lated in the liquid column depends on the final length of the watercolumn and increases as the air pocket size increases. The impli-cation is obvious: the larger the air pocket, the lower the maximumpressure.

In favorable slopes, the relationship is naturally reversed. As theliquid column becomes longer, it gains potential energy; thus, thelarger the air pocket, the higher the maximum pressure.

By manipulating Eq. (19) and neglecting friction, the velocity ofthe liquid column can be expressed as

V ¼��

2g

X� þ ð L0

L−L0Þ

��X�ðHr þHatmÞ

−�H0 þHatm

γ − 1

�½ð1 − X�Þ1−γ − 1�

− 0.5 sinðθÞX�ðX þ 2L0Þ��

2

ð24Þ

When the liquid column eventually comes to rest, there is zerokinetic energy and Eq. (24) can be simplified as follows:

ð1 − X�maxÞ1−γ − 1

ðγ − 1ÞX�max

¼ Hr þHatm − 0.5ðXmax þ 2L0Þ sinðθÞH0 þHatm

ð25Þ

Alternatively, Eq. (26) can be expressed in terms of pressurehead:

�1

1 − γ

�241 − ð HatmþH0

HmaxþHatmÞ1−γ=ðγÞ

1 − ð HatmþH0

HmaxþHatmÞ1=γ

35

¼ Hr þHatm − 0.5ðXmax þ 2L0Þ sinðθÞH0 þHatm

ð26Þ

From these equations, it can be seen that the pipe slope effect isincluded through the last term in the numerator on the right-handside of the expressions. This term affects the driving head ofthe system, justifying the driving-head approach employedby Abreu et al. (1992) to explain the effect of pipe slope on maxi-mum pressure. Eq. (26) makes explicit that the slope term,−0.5ðXmax þ 2L0Þ sinðθÞ, can increase or decrease the driving headof the reservoir depending on whether the pipe is adverse or fa-vorable.

To validate this relation, analytical and numerical results can becompared in the context of the aforementioned hypothetical pipesystem. Fig. 4 shows the results obtained from Eq. (24) along withthose of the analytical and numerical solutions. It can be seen thatthe analytical and numerical results are identical. Fig. 5 illustratesthe maximum air pressures obtained from Eq. (26) along with thoseof the numerical model. Again, the analytical results coincide withthe numerical results.

Figs. 6(a and b) show different components of energy, includingthe energy supplied by the reservoir, the potential energy associatedwith the pipe slope, and the energy accumulated in the air pocketfor two pipe slopes, 10 and −10°, and different air pocket lengths.As can be seen, with a favorable slope, increases in the size of theair pocket increase the released potential energy and thus maximumair pressure increases. However, with an adverse slope, the energyabsorbed by the air pocket decreases with an increase in air pocketsize because a greater portion of the reservoir energy is stored in thepipe as gravitational potential energy.

Effect of FrictionBy setting the velocity of the liquid column to zero in Eq. (19)and manipulating the expression, the following equation isobtained:

�1

1 − γ

�241 − ð HatmþH0

HmaxþHatmÞð1−γÞ=γ

1 − ð HatmþH0

HmaxþHatmÞ1=γ

35 ¼

Hr þHatm − 0.5ðXmax þ 2L0Þ sinðθÞ − f2DgXmax

R Xmax0 ðL0 þ XÞV2dx

H0 þHatmð27Þ

Eq. (27) is nearly identical to that obtained for the frictionlesspipe system [Eq. (26)] except for the extra term on the right-handside, which incorporates the effect of friction. The negative sign ofthis additional term implies that the driving head of the system (Hr)is reduced by an amount equal to the magnitude of the friction term;that is, the greater the magnitude of this additional term, the lowerthe effective driving head and maximum pressure. The magnitudeof the friction term varies with the initial air void fraction. A largerfraction allows the liquid column to attain a greater velocity during

pressurization, which in turn increases the effect of friction andtherefore the magnitude of the dissipation term.

The friction term complicates the solution of Eq. (27). For thehypothetical system considered in the previous section and a pipeslope = −10°, Fig. 7 compares the maximum pressure headsinduced for f ¼ 0.01, 0.02, and 0.03. As can be seen, friction sig-nificantly reduces the effect of pipe slope and causes the maximumpressure head to dramatically decrease compared with the friction-less system. The figure also reveals that in the frictional system,

© ASCE 04015044-5 J. Hydraul. Eng.

J. Hydraul. Eng.

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TY

OF

TO

RO

NT

O o

n 08

/14/

15. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

below the air pocket length of approximately 80 m, the maximumpressure head decreases as the air pocket size increases, but beyondthis point the maximum air pressure grows as the size of the airpocket increases.

To explain the appearance of this extremum, consider theenergy components for f ¼ 0.01, 0.02, and 0.03 summarized inFigs. 8(a–c). The figures show that, below the air pocket lengthof 80 m, as the length increases, the rate of increase in energy dis-sipation per unit volume of the air pocket becomes greater than thesum of the rates at which the recovered potential energy and theenergy injected by the reservoir per unit volume of the air pocketincreases. Thus, the energy received per unit volume of the airpocket increases as the air pocket size increases. However, beyondthis size the trend reverses and the air pocket receives more energyper unit volume its size increases, causing the maximum air pres-sure to grow. The appearance of such an extermum point had notbeen previously identified.

Effect of Liquid Length Variation

Cabrera et al. (1992) numerically explored the rapid pressurizationof a pipe system containing entrapped air at the downstream end.They employed both full and simplified versions of the rigid watercolumn model in which the variation in liquid column length wasignored. The governing equations associated with the full and sim-plified models are

gðHr þHatmÞ − 1

2V2 ¼ ðL0 þ XÞ dV

dtþ f

ðL0 þ XÞD

VjVj2

þ gðH0 þHatmÞ�

L − L0

L − L0 − X

�γ

ð28Þ

gðHr þHatmÞ − 1

2V2z}|{¼0

¼ L0

dVdt

þ fL0

DVjVj2

þ gðH0 þHatmÞ�

L − L0

L − L0 − X

�γ

ð29Þ

Cabrera et al. demonstrated that the maximum air pressureobtained from the simplified model is the same as that from thecomplete model if the velocity head is ignored in the simplifiedmodel. Such a counter-intuitive result raised a discussion between

Fig. 4. Liquid column velocity versus air pocket compression rate fordifferent pipe slopes (solid line: numerical; signs: analytical)

Fig. 5. Maximum air pressure head versus air pocket length for dif-ferent pipe slopes (solid line: numerical; signs: analytical)

Fig. 6. Energy components in the frictionless reservoir-pipe system: (a) slope ¼ −10°; (b) slope ¼ 10°

© ASCE 04015044-6 J. Hydraul. Eng.

J. Hydraul. Eng.

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TY

OF

TO

RO

NT

O o

n 08

/14/

15. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Ghidaoui and Karney (1994) and Cabrera et al. (1992) in whichdetails of the underlying behavior were clarified. The discussionconcluded that all of the results presented in the original paper heldtrue (Cabrera et al. 1994), but the underlying physics were notyet understood. Lee (2005) attempted a better explanation of the

problem by providing a closed-form solution for both models.Although the analytical solutions presented by Lee confirmedthe numerical results obtained by Cabrera et al. (1992), his analyti-cal solutions provided little additional insight.

The energy approach can be used to further elucidate this issue,first for a frictionless pipe system. The energy equation in this casesimply balances the energy entering the system through the reser-voir with the kinetic energy of the liquid column and the energyaccumulated in the air pocket. The energy equations for the com-plete and simplified models are

AðL − L0Þ�ðP0 þ PatmÞ

γ − 1½ð1 − X�

CÞ1−γ − 1� − PatmX�C

�

þ 1

2ρAðL0 þ XCÞV2

C − ρgXCAHr ¼ 0 ð30Þ

AðL − L0Þ�P0 þ Patm

γ − 1½ð1 − X�

SÞ1−γ − 1� − PatmX�S

�

þ 1

2ρAL0V2

S − ρgXSAHr ¼ 0 ð31Þ

The subscripts C and S refer to the complete and simplifiedmodels, respectively. For a given liquid column advancementlength (XC ¼ XS and X�

C ¼ X�S), the energy entering the system

and the energy accumulated in the air pocket are identical for both

Fig. 7. Pressure rise in the reservoir-pipe system with different frictionlevels and slope ¼ −10°

Fig. 8. Different components of energy in the reservoir pipe system with friction: (a) f ¼ 0.01; (b) f ¼ 0.02; (c) f ¼ 0.03

© ASCE 04015044-7 J. Hydraul. Eng.

J. Hydraul. Eng.

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TY

OF

TO

RO

NT

O o

n 08

/14/

15. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

models. Consequently, the kinetic energy of the liquid column mustbe equal in both models. The same amount of kinetic energy givesrise to a higher velocity in the simplified model because the liquidcolumn in the simplified model has less mass. By equating the ki-netic energy terms, the following relationship between the liquidcolumn velocities of the two models can be obtained:

VS ¼�1þ x

L0

�0.5VC ð32Þ

This is the same result reached algebraically by Cabreraet al. (1992).

Because the velocity in the simplified model always exceedsthat in the so-called complete model, the conservation-of-mass re-lation implies that filling is faster in the simplified method. There-fore, as was also recognized by Cabrera et al. (1992), the simplifiedmethod produces erroneous temporal results. Nevertheless, whenthe liquid column finally comes to a rest against the compressedair pocket, the resulting maximum air pressure in both cases isthe same because the air pockets in both models are compressedby the same amount of energy.

By taking the derivative of Eq. (31) and, through manipulation,the equation presented byCabrera et al. (1992) [Eq. (29)] is obtained.This reconfirms that, for the air pocket in the simplified model to becompressed to the same degree as in the full model, the velocity headmust be ignored in the governing equations of the simplified model.

To explore the role of the velocity head, consider the casein which the initial liquid column is very short. In this case,Eq. (32) implies that velocity can increase without limit in the sim-plified model as the initial liquid column length tends to zero. Suchan instantaneously high-velocity flow requires an energy sourcethat can provide instantaneously high mass and energy flux. How-ever, the limited head of the reservoir does not physically permitsuch fluxes to occur.

Considering that inclusion of the velocity head in the boundaryequations physically harnesses the amount of the mass and energyreleased through the reservoir, in order to allow the reservoir boun-dary to release any unphysical amounts of mass and energy, thevelocity head must be removed from the governing equation.Whereas the instantaneous release of energy and mass fluxes inthe simplified model is greater than that in the complete model,the time period of these unrealistic fluxes in the simplified modelis proportionally enforced by the continuity equation such that whenthe liquid columns are eventually brought to rest, the total mass andenergy released by the reservoir are the same for both models. Thisclearly explains that it is only by violating the real physics thatthe simplified model can correctly calculate the maximum airpressure.

Cabrera et al. (1992) also showed that for a frictional system thesame hypothesis holds true. In this case, the energy equations cor-responding to the complete and simplified models can be written asfollows:

AðL−L0Þ�ðP0 þPatmÞ

γ − 1½ð1−X�

CÞ1−γ − 1�−PatmX�C

�

þ 0.5ρAðL0 þXCÞV2C þ

ZXC

0

ρAfðL0 þ xÞV2

C

2Ddx− ρgXCAHr ¼ 0

ð33Þ

AðL − L0Þ�ðP0 þ PatmÞ

γ − 1½ð1 − X�

SÞ1−γ − 1� − PatmX�S

�

þ 0.5ρAL0V2S þ

ZXS

0

ρAfxV2

S

2Ddx − ρgXSAHr ¼ 0 ð34Þ

For an arbitrary liquid column length variation (XC ¼ XS),the following equation can be concluded by comparing Eqs. (33)and (34): ZX

0

ρAfðL0 þ xÞV2

C

2Ddxþ 0.5ρAðL0 þ XÞV2

C

¼ZX0

ρAfxV2

S

2Ddxþ 0.5ρAL0V2

S ð35Þ

Equation (35) implies that, for a given liquid column length, thesums of the kinetic energy of the liquid column and the friction lossare the same for both models. Eqs. (33) and (34) also show that, whenthe liquidcolumnsare finallybrought to rest against the compressedairpockets, the resultingmaximumairpressurescapturedbythe twomod-els are the same if the total energy losses in both models are the same.Equating the twoenergy loss terms results in a relationshipbetween theliquid column velocities that is equivalent to Eq. (32). Because thisrelation satisfies Eq. (35), it can be concluded that, for a given liquidcolumn advancement length, both the energy loss and the kinetic en-ergy terms in the completemodel are equal to their counterpart terms inthe simplified model. Thus, when the liquid columns are finallybrought to rest, the total energy loss in both models is equal and sothe air pockets are compressed with the same amount of energy. Thisconfirms that both models calculate the same maximum air pressure.

By taking the derivative of Eq. (34), it can be shown that the re-sulting equation also lacks a velocity head term. This confirms that forthe simplifiedmodel to simulate the correctmaximumair pressure, thevelocity head must be removed from the respective governing equa-tion. The physics of theproblem in this case can be also explainedwiththe same argument presented for a frictionless pipe system.

Pumping Systems

Izquierdo et al. (1999) studied rapid pressurization in pumpinglines containing multiple air pockets and concluded that compres-sion of the first air pocket is significant. They also found thatsmaller air pockets produce higher peak pressures. Fuertes et al.(1999) expanded this work and studied the influence of several rel-evant parameters, including pump characteristic curve shape, pipeslope, and friction, on the resulting peak air pressure. This resultedin the creation of some useful nondimensional design charts;however, a better physical understanding can be achieved by apply-ing the proposed energy approach.

The energy equation [Eq. (19)] developed for pipe systems sup-plied by a constant-water-level reservoir can be easily adjusted toaccount for the effect of pumps. This can be accomplished byreplacing the boundary energy flux in that equation, W 0, withthe one given in Eq. (17).

When the liquid column eventually comes to rest, the total en-ergy released by the pump is entirely stored in the air pocket. In thiscase, Eq. (19) can be rearranged as

�1

1 − γ

�241 − ð HatmþH0

HmaxþHatmÞð1−γÞ=γ

1 − ð HatmþH0

HmaxþHatmÞ1=γ

35 ¼

Cp1 þHatm − Cp2A2

Xmax

R Xmax0 V2dx − 0.5ðXmax þ 2L0Þ sinðθÞ − f

2DgXmax

R Xmax0 ðL0 þ XÞV2dx

H0 þHatmð36Þ

© ASCE 04015044-8 J. Hydraul. Eng.

J. Hydraul. Eng.

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TY

OF

TO

RO

NT

O o

n 08

/14/

15. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Note that Eq. (36) is remarkably similar to Eq. (27) for a pipe-reservoir system. The pump shut-off head (Cp1) plays the samerole as the reservoir head, and the Cp2 includes a term very similarto the friction term. This means that, even if both friction and slopeare neglected, this term ensures that the pumps can never provideconstant energy per unit volume of the air pocket.

Eq. (36) further implies that the pump characteristic curve shape isalso an important factor that affects the amount of energy that a pumpprovides per unit volume of the air pocket. Steeper pump curves thathave greater Cp2 values (Fig. 9) correspond to a steeper head dropduring pressurization. Thus, for a given air pocket size, pumps withgreater Cp2 values tend to compress the air pocket with a lower en-ergy per unit volume, which in turn results in lower maximum airpressures. The effect of the pump characteristic curve shape on themaximumair pressurewas first explainedbyAbreu et al. (1992) usingthe driving head approach discussed earlier in this paper.

To justify the preceding qualitative arguments, a numericalanalysis is conducted using a modified version of the hypotheticalpipe system in which the reservoir is replaced with a pump; bothpipe slope and friction are also neglected. Fig. 10 depicts both theresulting maximum air pressures and the energy released by thepump per unit volume of the air pockets for two pumps withthe same shut-off head of Cp1 ¼ 20 m but different Cp2 valuesof 3,000 and 5,000 m−6=s2. It confirms the preceding reasoningand justifies the following:

1. The maximum air pressure and energy released per unit vo-lume of the air pocket decrease as the initial air void fractionincreases;

2. For a given air void fraction, a steeper pump characteristiccurve provides less energy per unit volume of the air pocketand results in a lower maximum pressure;

3. For very small initial air pocket sizes, the resulting maximumpressure is similar to that produced by a reservoir with a headequal to the shut-off head of the pump; and

4. In the absence of the pipe slope, the fact that smaller air pock-ets produce higher maximum pressures holds true for pipe-reservoir systems only when the systems contain friction;for pumping systems, this is true regardless of friction.

To examine how pipe friction and slope affect response, theaforementioned example is revisited by assuming that friction fac-tor, pipe slope, and Cp2 are 0.018, −10, and 3,000, respectively (inconsistent SI units). Fig. 11 shows that, unlike the frictionless-horizontal pipe system, the maximum air pressure in this case doesnot occur at smaller air pocket sizes but rather at an extermumpoint. The energy flow of the system shown in Fig. 11 reveals thatincreasing the air pocket size decreases the energy unit per unitvolume of the air pocket and then increases it. In larger air pockets,the role of the pump in energizing the air pocket is rendered un-important and the energy recovered because of the action of theslope supplies a significant portion of the energy received bythe air pocket. Also, in this case, even if the friction is neglected,the maximum pressure still occurs at an extremum point.

The obtained results are apparently not in full agreement withthose obtained by Izquierdo et al. (1999) and Fuertes et al. (1999),both of which concluded that maximum pressure occurs withsmaller air pocket sizes. The pumping systems considered by theseresearchers contain several air pockets separated by a series ofblocking water columns. In fact, the following example can be seenas a special case of the pumping systems considered in thosestudies in which just one air pocket is compressed against an in-finite blocking water column, not moving during the air pocketpressurization.

Additional Remarks

To this point, the elastic effects of the fluid and the pipe have beenneglected. Although both experimental and numerical results show

Fig. 9. Example pump characteristic curves

Fig. 10.Maximum air pressure and energy released per unit volume ofthe air pocket in the -frictionless-horizontal pumping line

Fig. 11. Energy components and pressure rise for the pumping linewith both friction and slope

© ASCE 04015044-9 J. Hydraul. Eng.

J. Hydraul. Eng.

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TY

OF

TO

RO

NT

O o

n 08

/14/

15. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

that in a few cases the elastic effect may significantly affect tran-sient responses (Lee 2005; Zhou et al. 2011a; Nakamura andTomita 1999; Abreu et al. 1991, 1992), these studies confirm that,unless the air pocket is very small, the elastic effect is insignificant.Even under such conditions, inclusion of the elastic effect reducesthe maximum pressures obtained from rigid column theory.

The proposed energy approach simply concludes that ignoringthe elastic effect causes the air pocket to receive additional energythat could have otherwise been accumulated in the liquid and pipebody as strain energy. This clearly explains why the rigid columnmodel overestimates pressures. Nonetheless, except for very smallair pockets, the energy storage capability of the air pocket is sig-nificantly greater than that of the liquid and pipe body. Therefore,this additional energy can significantly affect the air pocket’s en-ergy (and consequently its maximum pressure) only when thepocket is very small. Further to this short qualitative explanation,the energy approach has also been successfully employed to ex-plain some aspects of transient flow that arise because of the elasticnature of pipe systems (Malekpour and Karney 2014).

Finally, experimental studies show that existing 1D models areunable to accurately resolve the dampening of the pressure oscilla-tions that follow rapid pressurization of pipe systemswith entrappedair pockets. Lee (2005) numerically showed that considering thethermal effect of the air pocket allows the numerical result to bettermimic reality, but in some of the examples provided the thermal ef-fects cannot produce as strong an energy dissipation as those ob-served experimentally. This leads to the conclusion that thethermal effect might not to be the sole cause of the problem.

The proposed energy approach confirms that any sink of energysuch as air pocket thermal effects can reduce the amount of energymechanically received by the air pocket and consequently reducethe maximum air pocket pressure. However, inspection of the ex-perimental results of Vasconcelos and Leite (2012) shows that 1Dmodels resolve the first pressure peak much better than the sub-sequent peaks. This may imply that the energy dissipation afterthe first peak becomes more intense.

A hypothesis that can physically explain this behavior may beestablished by considering air pocket expansion after a peak pres-sure. CFD results (Zhou et al. 2011b) show that strong air-pocketbreak-up forms as the pocket expands. This break-up may signifi-cantly increase local head loss and intensify dampening. The val-idity of such a hypothesis needs to be justified through a detailedstudy. The authors’ future plan is to test this hypothesis in a CFD-based study.

Summary and Conclusion

An energy approach was applied to published cases to better under-stand the complex transient responses following rapid pressuriza-tion of confined pipe systems with entrapped air pockets. Thisapproach was employed to explain the underlying physics of someimportant aspects of transient flow that were not well understood inthe literature. Because most contributions were made using rigidwater column theory, this paper did not consider the elastic effectsof liquid and pipe.

The essential conclusion drawn from the energy approach issimple: because the resulting maximum pressure rise directly de-pends on the amount of energy received by the air pocket, auditingthe partition of energy attributed to air pocket compression revealsthe effect of each pipe system parameter (i.e., reservoir head, pumpcharacteristics, pipe slope, friction) during the pressurization event.

The results show that the energy approach can sometimes leadto an analytical solution. In a comparison of this approach and

corresponding numerical solutions, both give identical results. Evenwhen analytical solutions are not available, energy methods remainpowerful conceptual tools for explaining the underlying physics.

In addition to providing a better understanding of different as-pects of rapid pressurization in pipe systems, the energy approachhelps to identify one important aspect of this complex phenomenonthat had not been previously identified. The energy approach showsthat the interaction of the pipe slope with the friction effect in res-ervoir pipe systems and with the friction and/or pump shape effectin pumping lines may cause the maximum air pressure to occur at aspecific air pocket size. This counters the widespread belief that therigid column model always produces higher pressures with smallerair pockets.

Notation

The following symbols are used in this paper:A = pipe cross-sectional area;a = acoustic wave velocity;

Cp1 = pump shut-off head;Cp2 = constant defining the shape of the pump characteristic

curves;D = pipe diameter;D 0 = rate of energy dissipation;Ep = potential energy of the liquid column;f = Darcy-Weisbach friction factor;g = gravitational acceleration;H = piezometric head;

Hatm = atmospheric pressure head;Hd = driving head;

Hmax = maximum pressure head;H0 = initial air pressure head;Kl = local (or minor) loss coefficient;L = pipe length;La = air pocket length;L0 = initial liquid column length;

Patm = atmospheric pressure;Px = air pressure at a specific liquid column advancement

length;P0 = initial air pressure;T = kinetic energy of the liquid column;t = time independent variable;U = elastic energy stored in both the pipe body and the liquid;Ua = elastic energy stored in the air pocket;V = liquid column velocity;V0 = initial liquid column velocity;W 0 = net energy flux entering a pipe system;x = increase in the liquid column length;

X� = air pocket compression rate;X�max = maximum air pocket compression rate;Z = pipe elevation;γ = polytropic exponent;

∀f = air void fraction;θ = pipe angle to the horizontal; andρ = liquid density.

References

Abreu, J., Cabrera, E., García-Serra, J., Iglesias, P., and Izquierdo, J.(1991). “Boundary between elastic and inelastic models in hydraulictransients analysis with entrapped air pocket.” Proc., Int. Conf. Hy-draulic Transients with Water Column Separation, International Asso-ciation for Hydro-Environment, Madrid, Spain.

© ASCE 04015044-10 J. Hydraul. Eng.

J. Hydraul. Eng.

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TY

OF

TO

RO

NT

O o

n 08

/14/

15. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Abreu, J., Cabrera, E., García-Serra, J., Iglesias, P., and Izquierdo, J.(1992). “Influence of trapped air pockets in pumping systems.” 4thInt. Conf. Hydraulic Engineering Software: HYDROSOFT 92, Compu-tational Mechanics, Boston.

Cabrera, E., Abreu, J., Perez, R., and Vela, A. (1992). “Influence of liquidlength variation in hydraulic transients.” J. Hydraul. Eng., 10.1061/(ASCE)0733-9429(1992)118:12(1639), 1639–1650.

Cabrera, E., Abreu, J., Perez, R., and Vela, A. (1994). “Influence of liquidlength variation in hydraulic transients.” J. Hydraul. Eng., 10.1061/(ASCE)0733-9429(1994)120:5(663), 663–666.

Chaiko, M. A., and Brinckman, K. W. (2002). “Models for analysis ofwater hammer in piping with entrapped air.” J. Fluid Eng. Trans.,124(1), 194–204.

Chaudhry, M. H., and Reddy, H. P. (2011). “Mathematical modeling of laketap flows.” J. Hydraul. Eng., 10.1061/(ASCE)HY.1943-7900.0000339,611–614.

Epstein, M. (2008). “A simple approach to the prediction of waterhammertransients in a pipe line with entrapped air.” J. Nucl. Eng. Des., 238(9),2182–2188.

Fuertes, V., Cabrera, E., Izquierdo, J., and Iglesias, P. (1999). “Peak pres-sure evaluation in pipelines with entrapped air pockets.” Proc.,3rdASME/JSME Joint Fluids Engineering Conf., ASCE, New York.

Ghidaoui, M. S., and Karney, B. W. (1994). “Influence of liquid lengthvariation in hydraulic transients.” J. Hydraul. Eng., 10.1061/(ASCE)0733-9429(1994)120:5(661), 661–663.

Izquierdo, J., Fuertes, V., Cabrera, E., Iglesias, P., andGarcía Serra, J. (1999).“Pipeline start-up with entrapped air.” J. Hydraul. Res., 37(5), 579–590.

Jönson, L. (1985). “Maximum transient pressure in a conduit with checkvalve and air entrapment.” Proc., Int. Conf. Hydraulics of PumpingStations, BHRA, Cranfield, U.K., 55–76.

Karney, B. W. (1990). “Energy relations in transient closed-conduit flow.”J. Hydraul. Eng., 10.1061/(ASCE)0733-9429(1990)116:10(1180),1180–1196.

Lee, N. H. (2005). “Effect of pressurization and explosion of entrapped airin pipelines.” Ph.D. thesis, Georgia Institute of Technology, Atlanta.

Lee, N. H., and Martin, C. S. (1999). “Experimental and analytical inves-tigation of entrapped air in a horizontal pipe.” Proc., 3rdASME/JSMEJoint Fluids Engineering Conf., ASCE, New York.

Liu, D., Zhou, L., Karney, B., Zhang, Q., and Ou, C. (2011). “Rigid-plugelastic-water model for transient pipe flow with entrapped air pocket.”J. Hydraul. Res., 49(6), 799–803.

Malekpour, A., and Karney, B. W. (2014). “Understanding of the risks ofhigh pressures following rapid pressurization in pipelines containingentrapped air pockets: A novel energy auditing approach.” Proc.,10th Int. Pipeline Conf., ASME, Calgary, AB, Canada.

Martin, C. S. (1976). “Entrapped air in pipelines.” Proc., 2nd Int. Conf. onPressure Surges, BHRA Fluid Engineering, Cranfield, U.K.

Nakamura, K., and Tomita, T. (1999). “Water hammer caused by air in apump started with the discharge valve closed.” Proc., 3rd ASME/JSMEJoint Fluids Engineering Conf., ASCE, New York.

Vasconcelos, J. G., and Leite, G. M. (2012). “Pressure surges followingsudden air pocket entrapment in storm-water tunnels.” J. Hydraul.Eng., 10.1061/(ASCE)HY.1943-7900.0000616, 1081–1089.

Wright, S. J., Determan, K. V., and Vargas, S. M. (2012). “Pressure tran-sients due to compression of trapped air in rapidly filling sewer storagetunnel.”On modeling urban water systems—Monograph 20, CHI Press,Guelph, ON, 2–18.

Wylie, E. B., and Streeter, V. L. (1993). Fluid transients in systems,Prentice-Hall, Upper Saddle River, NJ.

Zhou, L., Liu, D., Karney, B., Zhang, Q., and Ou, C. (2011a). “Influence ofentrapped air pockets on hydraulic transients in water pipelines.” J. Hy-draul. Eng. 10.1061/(ASCE)HY.1943-7900.0000460, 1686–1692.

Zhou, L., Liu, D., and Ou, C. (2011b). “Simulation of flow transients inwater filling pipe containing entrapped air pocket with VOF model.”Eng. Appl. Comput. Fluid Mech., 5(1), 127–140.

© ASCE 04015044-11 J. Hydraul. Eng.

J. Hydraul. Eng.

Dow

nloa

ded

from

asc

elib

rary

.org

by

UN

IVE

RSI

TY

OF

TO

RO

NT

O o

n 08

/14/

15. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.