Content uploaded by Wuyi Wan

Author content

All content in this area was uploaded by Wuyi Wan on Nov 24, 2017

Content may be subject to copyright.

Wan et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2017 18(3):167-178 167

Investigation on critical equilibrium of trapped air pocket in

water supply pipeline system*

Wu-yi WAN†1,2, Chen-yu LI1, Yun-qi YU1

(1Department of Hydraulic Engineering, College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China)

(2State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 310072, China)

†E-mail: wanwuyi@zju.edu.cn

Received Apr. 20, 2016; Revision accepted Aug. 27, 2016; Crosschecked Feb. 7, 2017

Abstract: A trapped air pocket can cause a partial air lock in the top of a hump pipe zone. It increases the resistance and de-

creases the hydraulic cross section, as well as the capacity of the water supply pipeline. A hydraulic model experiment is con-

ducted to observe the deflection and movement of the trapped air pocket in the hump pipe zone. For various pipe flow velocities

and air volumes, the head losses and the equilibrium slope angles are measured. The extra head losses are also obtained by ref-

erence to the original flow without the trapped air pocket. Accordingly, the equivalent sphere model is proposed to simplify the

drag coefficients and estimate the critical slope angles. To predict the possibility and reduce the risk of a hump air lock, an em-

pirical criterion is established using dimensional analysis and experimental fitting. Results show that the extra head losses increase

with the increase of the flow velocity and air volume. Meanwhile, the central angle changes significantly with the flow velocity but

only slightly with the air volume. An air lock in a hump zone can be prevented and removed by increasing the pipe flow velocity or

decreasing the maximum slope of the pipe.

Key words: Hump pipe; Pipe flow; Trapped air pocket; Hydraulic experiment; Water supply pipeline

http://dx.doi.org/10.1631/jzus.A1600325 CLC number: TV131.2

1 Introduction

In an irregular submarine water supply pipeline

system, air can remain and accumulate at the top of a

hump zone when the pipe flow velocity is not large

enough to remove it. The trapped air pocket in the

hump can obstruct flow and reduce the conveying

capacity of the pipe (Pozos et al., 2010). Conse-

quently, this kind of local hydraulic phenomenon is

also called an air lock (Greenshields and Leevers,

1995; Reynolds and Yitayew, 1995; Brown, 2006). It

is a problem in irrigation and drainage systems (Zhou

et al., 2004; Burch and Locke, 2012; Pozos-Estrada

et al., 2015), hydraulic spillway conduits (Liu and

Yang, 2013), and water pipeline systems (Burrows

and Qiu, 1995; Carlos et al., 2011; Pozos-Estrada et

al., 2012), since it increases the head losses, decreases

the cross section, and causes pipe burst failures. Ac-

cording to the scale of the air lock, it can be classified

as an entire air lock or a partial air lock (Brown,

2006). Generally, an entire air lock occurs in small-

scale pipes. It increases significantly the head losses

in the pipe flow, and even partially or entirely blocks

the pipe flow in low-pressure gravity flow pipe sys-

tems (Reynolds and Yitayew, 1995). For storm sewer

and water supply pipe systems, the trapped air usually

forms a partial air lock even if it does not entirely

block the hydraulic cross section of the closed pipe

(Yu, 2015). However, the trapped air mass can cause

pressure oscillations (Vasconcelos and Leite, 2012),

increase the transient pressure peak (Burrows and

Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering)

ISSN 1673-565X (Print); ISSN 1862-1775 (Online)

www.zju.edu.cn/jzus; www.springerlink.com

E-mail: jzus@zju.edu.cn

* Project supported by the National Natural Science Foundation of

China (No. 51279175), the Zhejiang Provincial Natural Science

Foundation of China (No. LZ16E090001), and the Open Foundation

of State Key Laboratory of Hydraulic Engineering Simulation and

Safety, Tianjin University, China (No. HESS-1505)

ORCID: Wu-yi WAN, http://orcid.org/0000-0002-8740-749X

© Zhejiang University and Springer-Verlag Berlin Heidelberg 2017

Wan et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2017 18(3):167-178

168

Qiu, 1995; Chaiko and Brinckman, 2002), and cause

burst failures (Zhou et al., 2004; Pozos-Estrada et al.,

2015) in some closed pipeline systems.

To avoid the hazard of air locks, various air lock

categories and their hydraulic properties have been

investigated in the last few years (Pothof and Clem-

ens, 2011; Pozos-Estrada et al., 2012). Reynolds and

Yitayew (1995) studied the air lock phenomenon of a

low head irrigation pipeline, with small internal di-

ameters of 6, 8, 10, and 13 mm, and the results show

that an entire air lock can form a partial or full

blockage in a small-scale pipe. Pozos et al. (2010)

studied the equilibrium and movement of air pockets

in gravity and pumping pipeline systems by hydraulic

experiments. A useful evaluation criterion was estab-

lished to predict the motion direction of an air pocket

in a straight downward sloping pipe. In the meantime,

Pothof and Clemens (2010) provided two important

clearing velocity criteria according to energy consid-

erations and momentum balance for elongated air

pockets in straight downward sloping pipes.

Izquierdo et al. (1999) studied the influence of an air

pocket in a pipe start-up. Escarameia (2007) and Lin

et al. (2015) measured air pocket movement in a

pressurized conduit pipe system. In addition, the in-

fluence of air pockets on water hammer has been

widely considered by experiments and numerical

simulations (Epstein, 2008; Zhou et al., 2013; Ferreri

et al., 2014).

Previous research shows that air locks can cause

undesired obstructions and pressure fluctuations in

closed irrigation and pipe systems. These are signif-

icant for pipe design and water hammer protection by

avoiding air pocket hazards in straight sloping pipes.

However, the air lock is more complicated in the

hump pipe zone than in a straight downward sloping

pipe, considering that the pipe fluctuates with the

irregular submarine profile. Moreover, air valves

cannot be used in submarine conditions to eliminate

air pockets. To prevent partial air locks in an undu-

lating submarine water supply pipeline system, in this

study the physical equilibrium and extra losses are

investigated with a uniform annular circular pipe. The

partial air locks have been observed by experiments

and the equilibrium equations are established by di-

mensional analysis and force equilibrium. Consider-

ing the complexity of the trapped air pocket shape, the

equivalent sphere model (ESM) is proposed to

simplify and establish a critical equilibrium rela-

tionship. Finally, an empirical criterion coefficient is

proposed to evaluate the possibility of an air lock and

to prevent the air lock. It provides guidance for the

design of pipe slope and flow velocity to prevent and

remove partial air locks in irregular submarine

pipelines.

2 Basic profiles and force equilibrium of an

air lock in a hump pipe zone

In closed pipe flow, bubbles always move to the

top of a pipe because air is far less dense than liquid.

If the drag force of a flow is not large enough to re-

move these bubbles, they will accumulate gradually

and cause an air lock in the top of the hump pipe zone.

Unfortunately, an air lock can increase the resistance

and reduce the flow capacity of a closed pipe system.

Sometimes, in some low-pressure pipe systems, it can

even entirely block the water supply. Fig. 1 shows an

entire air lock, where the air occupies the complete

flow cross section. An entire air lock can greatly af-

fect the flow capacity of a pipe. If the flow velocity

and pressure difference are large enough, the air

pocket will move in the same direction as the pipe

flow. Conversely, it can partially or entirely block the

pipe flow. Usually, this kind of phenomenon occurs

only in small-scale low-pressure pipe systems as in

the examples in Reynolds and Yitayew (1995) and

Brown (2006). Our experiment also shows that it is

difficult to observe an entire air lock in a large-scale

water supply pipe with a 90-mm internal diameter.

Unlike the entire air lock, a partial air lock often

occurs in large-diameter pipe systems. As shown in

Fig. 1 Schematic of an entire air lock in a small-diameter

pipe system (Zu: upstream water level; Zd: downstream

water level)

Zu

Zd

Upstream

Downstream

Entire air lock

Wan et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2017 18(3):167-178 169

Fig. 2, a partial air lock usually partially fills the pipe

cross section near the top of the hump zone in a

submarine water supply pipeline. The deflection and

dissipation of the partial air lock are more compli-

cated in a hump pipe than in a straight slope pipe.

Therefore, we focus on a partial air lock in the hump

zone, and conduct an experiment to observe and

measure that partial air lock. Moreover, the critical

equilibrium conditions are established by force anal-

ysis and dimensional analysis, as well as empirical

fitting according to the experimental results.

In general, the shape of an air lock varies with

the pipe diameter and flow velocity, as well as the

hydraulic pressure (Liu and Yang, 2013; Lin et al.,

2015). As a typical air lock, Fig. 3 shows the basic

profile observed in our experiment. The upper shape

is profiled by the internal surface of the pipe wall, and

the lower shape is approximately a bent flow inter-

face. The profile of the air pocket changes with the

flow velocity and pressure; therefore, it is difficult to

describe accurately the shape by a regular model. To

simplify the air lock, ESM is proposed in the next

section to describe the trapped air pocket.

Considering the air lock as the control volume

for a critical equilibrium condition, there are primarily

four kinds of forces acting on the trapped air pocket,

i.e., gravity, its buoyancy, the drag force of the flow,

and the normal force of the pipe wall (Fig. 4). In this

figure, the effect of the friction between air and pipe

wall is neglected, since the viscosity of air is negli-

gible at ordinary temperatures and pressures. For an

air mass at rest, the sum of the force vectors equals

zero. It can be written as follows:

F+G+D+N=0, (1)

where D is the drag force vector, F the buoyancy

vector, G the gravity vector, and N the normal force of

the pipe wall. In vector analysis, through vector de-

composition along the radial and tangential directions

in the 2D plane, the equilibrium of the tangential

component can be expressed as follows:

TT

sin cos ,

DN FG FG (2)

where θ is the central depression angle of the pocket.

Therefore, the location depression angle of the air

lock is

arcsin .

D

FG (3)

Referring to the air lock at rest, the buoyancy force

can be calculated simply by Archimedes’ principle:

F=ρwgV, (4)

Fig. 2 Schematic of a partial air lock in a submarine

water supply system (Zu: upstream water level;

Zd: downstream water level)

Zu

Zd

Upstream

Downstream

Sea Partial air lock

Fig. 3 Typical profile of a partial air lock in a hump

Pipe wall

Interface

Fig. 4 Force analysis of a partial air lock in a hump

F: buoyancy vector; G: gravity vector; D: drag force vector;

N: normal force; α: left depression angle; β: right depres-

sion angle; θ: central depression angle

F

F

G

β

D

G D N

N

θ

θ

α

θ

Trapped air lock

Pipe wall

Vector analysis

Wan et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2017 18(3):167-178

170

where F is the gravity, ρw the water density, g the

acceleration of gravity, and V the volume of the

trapped air pocket. In fact, buoyancy is also the only

original force driving the air lock to the top. It always

provides a component force opposite to the direc-

tional flow.

The typical drag equation, originally established

by Lord Rayleigh, can be expressed as

2

wD

1,

2

D

vC A

(5)

where D is the drag force, ν the mean flow velocity in

the pipe, CD the drag coefficient, and A the flow

direction-projected area. Thus, combining Eqs. (4)

and (5) with Eq. (3), the depression angle of the air

pocket can be expressed as

2

w

D

wa

arcsin ,

2

CA v

Vg

(6)

where ρa is the air density. Practically, it is very dif-

ficult to determine the drag force of the flow acting on

the air lock if the original shape of the trapped air

pocket is considered as shown in Fig. 3. Here, an

ESM is proposed to simplify the model of the trapped

air pocket. In the model, the suppositional ESM is

defined as: (1) a sphere equals the air pocket in

volume; (2) the sphere is subjected to the same forces

as the original air pocket. Then the relevant equiva-

lent radius is

30.75 π,rV

(7)

where r* is the equivalent radius of the air pocket. The

projected area of the equivalent sphere in the flow

section is

2

π,Ar

(8)

where A* is the equivalent section area. Based on the

ESM, Eq. (6) is converted to

2

w

D

wa

3

arcsin ,

42

Cv

rg

(9)

where D

C

is the equivalent drag coefficient in the

ESM. Moreover, to describe the relative scale of the

air pocket to the pipe section, the dimensionless ra-

dius is defined as

0

/,rrr

(10)

where r

is the dimensionless radius of the air pocket

and r0 is the internal radius of the pipe. Then Eq. (9) is

written as

2

w

D

0w a

3

arcsin .

42

Cv

rr g

(11)

Eqs. (9) and (11) are not subject to the effects of

the air pocket shape and they give an approach for

establishing the drag coefficient of the equivalent

sphere in the following experimental research and

data analysis.

3 Experimental layout and measure principle

Generally, the critical equilibrium of the air lock

in the hump is complicated, because of deformation

and separation. To study the critical state of the air

lock and its influence on the pipe flow, an experiment

was conducted. Fig. 5 shows the schematic of the

experimental principle and Fig. 6 shows the experi-

mental facilities in the field. The experiment consists

mainly of a hump pipe, with upstream and down-

stream pools, a slope differential manometer, an air

supply measuring cylinder, and a flow control valve.

The hump pipe is 0.090 m in internal diameter. The

hump is 0.450 m in external radius and the angle of

the hump is /2.

In the experiment, the measurements include the

volumes and the location depression angle of the air

lock, the pressure, the flow velocity, and the hydraulic

head losses, under various flow and air lock condi-

tions. The volumes of the air lock are 40, 80, and

160 cm3, and the pipe flow velocity varies from 0 to

0.7 m/s.

The head losses are very small and difficult to

measure. To improve the measurement precision of

the pressure difference between station 1 and station 2,

a differential manometer was fixed aslant with a

Wan et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2017 18(3):167-178 171

specific slope. The gradient is 1:4.899; in other

words, the angle is αg=arcsin(1/5). The minimum

interval of the ruler is 1 mm and the corresponding

measuring precision is 0.2 mm in head losses. As

shown in Fig. 5, the head losses are determined as

follows:

12 s1 s2 g s1 s2 g 0

()sin()sin,

v

hhh hh

(12)

where h12 is the head difference between stations 1

and 2, αg the slope angle of the pressure gauge, hs1 the

relative head at station 1, and hs2 the relative head at

station 2. The air supply measuring the cylinder can

provide a desired air quantity under atmospheric

conditions. For example, if an air volume V0 is

needed, the equipment is operated as follows: (1)

close valve #1 and open valve #2; (2) lower the re-

movable cylinder until the air in the fixed cylinder is

V0; (3) close valve #2 and raise the removable cylin-

der to a reasonable level, and then open valve #1 until

all air is injected into the hump pipe.

In the experiment, the upstream water level can

be set by the backflow control valve, and the flow

discharge is set by the flow control valve. A tapeline

adhered to the external surface of the hump pipe

marks the external arc length. Accordingly, the cen-

tral angle of the air pocket is

ud

b

0.5 ( ) ,llR

(13)

where lu is the left arc length of the air pocket, ld the

right arc length of the air pocket, and Rb the external

radius of the hump.

4 Experimental analysis and empirical fitting

Fig. 7 shows a typical air lock observed in the

hump pipe zone in the experiment. These air pockets

are separately 40, 80, and 160 cm3, the corresponding

equivalent radii are respectively 0.021, 0.027, and

0.034 m, and the flow velocity varies from 0 to

0.6 m/s.

As seen in these pictures, the partial air lock can

occur near the top of the hump. With the increase of

the pipe flow velocity, the air pockets go downstream

and spill from the outlet. In other words, if the drag

force is large enough to move the air pocket at the

maximum depression angle, it can entirely remove the

air pocket from the hump pipe zone.

To analyze the influence of the air lock on the

flow capacity of the pipe, extra head losses caused by

the air lock are analyzed based on the measured

v=0 m/s, V=40 cm3

v=0.2 m/s, V=40 cm3

v=0.4 m/s, V=40 cm3

v=0.6 m/s, V=40 cm3v=0.6 m/s, V=160 cm3

v=0.6 m/s, V=80 cm3

v=0 m/s, V=80 cm3

v=0.2 m/s, V=80 cm3

v=0.4 m/s, V=80 cm3

v=0 m/s, V=160 cm3

v=0.2 m/s, V=160 cm3

v=0.4 m/s, V=160 cm3

Fig. 7 Typical air lock patterns in a hump pipe zone

Fig. 6 Experimental facilities in the field

Fig. 5 Schematic of the experimental principle

Wan et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2017 18(3):167-178

172

results. Various air volumes are used to investigate

the resistance properties of the trapped air pocket in

the hump pipe. Considering a dimensionless format, a

maximum equivalent sphere ( 1)r

is chosen as a

reference. The corresponding length of the air column

is h0=4r0/3. Considered as the velocity head

2

00

(2 ),hv g the corresponding velocity is v0=

(2gh0)1/2. Fig. 8 shows the dimensionless head losses

between station 1 and station 2 for four kinds of

conditions, where the air pocket volumes are sepa-

rately 0 (without trapped air pocket), 40, 80, and

160 cm3. The dimensionless scales of the air pockets

for the last three kinds of conditions are separately

0.472,r

0.594, and 0.748.

In fact, it is difficult to measure directly the head

losses caused by the air pocket, since they are always

coupled with the friction and minor losses of the

hump pipe. Usually, the extra head losses can be

determined by the subtraction of the original head

losses without air pockets from the total head losses

with air pockets. As seen in Fig. 8, all the measured

data are presented as scatters. Considering the alter-

nation of laminar and turbulent flows during the in-

crease of flow velocity, the general head losses can be

expressed as follows:

2

12

ft

64 ,,

2

lv

hf

vd d g

(14)

where f is a function symbol, d the internal diameter

of the pipe,

the kinematic viscosity, l12 the pipe

length between stations 1 and 2, and λt the Darcy

friction factor in turbulent flow which can be deter-

mined by the Colebrook formula (Finnemore and

Franzini, 2002):

tt

12.51

2log ,

3.7

e

dRe

in which e is the roughness coefficient and Re the

Reynolds number. Define dimensionless head losses

ff0

,hhh

dimensionless velocity 0,vvv

lami-

nar head loss coefficient 12

11

0

64 ,

l

KC

vd d

and tur-

bulent head loss coefficient 12

2t 2

,

l

KC

d

where C1

and C2 are constant coefficients. For the original head

losses without the air lock, a polynomial function

through the origin of coordinates can be expressed as

2

f1 2

.hKvKv

(15)

Fig. 8 Hydraulic head losses between station 1 and

station 2

(a) Air pocket volume is 0, without trapped air pocket;

(b) Air pocket volume is 40 cm3, 0.472;r

(c) Air pocket

volume is 80 cm3, 0.594;r

(d) Air pocket volume is

160 cm3, 0.748r

Wan et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2017 18(3):167-178 173

The head losses induced by the air pocket are con-

sidered as a minor head loss 2

aa

,hKv

where Ka is

the head loss coefficient of an air lock. Then the di-

mensionless entire head losses can be written as

22

12 a

.hKvKv Kv

(16)

Based on the experimental data of the original head

losses without air pocket, coefficients K1 and K2 can

be determined by the method of least squares in

Eq. (15). Then according to Eq. (16), coefficient Ka of

extra head losses can be determined for various scale-

trapped air pockets. The coefficients are presented in

Table 1.

In Fig. 8, the solid lines show the fitting curves

of the flow velocities with head losses without air

locks, which are consistent with Eq. (15). In this

study, they are considered as the references for extra

head losses of the air lock. The dash lines are the

fitting curves of the head losses with air pockets. As

shown in the figure, these curves are steeper than the

reference line. In other words, head losses with air

locks are generally larger than those without air locks.

Considering the same conditions, a larger trapped air

pocket always brings more extra head losses. In fact,

the extra head losses due to air locks are only a part of

the minor losses. To analyze the extra head losses, the

fitting head loss line without an air lock is selected as

the reference.

According to these coefficients, it is shown that

the head losses increase with the increase of the

trapped air volume. The lines in Fig. 9 show sepa-

rately the extra head losses caused by various air

locks. These results show that the air lock can in-

crease the resistance to the flow in the hump pipe

zone. The resistance increases with the increase of the

flow velocity and the volume of the air lock. Ac-

cording to dimensional analysis, it is proportional to

the second power of the pipe flow velocity, and the

coefficient increases with the increase of the volume

of the air lock. In summary, the above analysis shows

that air locks can cause extra head losses in closed

pipe flow. Under the same conditions, the extra head

losses increase with the increase of air volumes and

pipe flow velocities.

In fact, the trapped air pocket deforms and splits

with the pipe flow patterns. Therefore, the experiment

refers particularly to a macro-scale trapped air pocket.

As shown in Fig. 4, the location of the air lock is

defined by three angles, i.e., α, β, and θ, where θ=

(α+β)/2. As the hump radius is very large in compar-

ison with the air lock, the three angles are similar in

magnitude. Thus, the mean angle θ is chosen to rep-

resent approximately the location of the air lock. The

experimental scatters in Fig. 10 show the location

angles of the air lock with various pipe flow velocities

and volumes. As seen in the figure, the location slope

angle increases significantly with the increase of the

flow velocities and decreases slightly with the in-

crease of the volumes.

Table 1 Extra head loss coefficients for various volumes

K1 K2 Ka

0.472r

0.594r

0.748r

0.175 0.849 0.094 0.141 0.247

Fig. 9 Extra head losses caused by air locks

0.0 0.2 0.4 0.6 0.8

0.00

0.05

0.10

0.15

0.20

v

~

r=0.748

~

r=0.594

~

r=0.472

~

Fig. 10 Influence of the flow velocity on the location of

an air lock

Wan et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2017 18(3):167-178

174

According to Eq. (9), defining dimensionless

density ratio ww a

(),

and substituting

variable θ=sinθ, the general location of the critical

equilibrium conditions can be expressed as the di-

mensionless format:

2

D.

v

Cr

(17)

Accordingly, the coefficients can be determined

based on the experiments for a constant volume:

*

D

22

*

D

1

,

n

ii

i

Ci

vv

MC

rr

(18)

*

D

**

DD 0,

C

M

CC

(19)

where *

D

C

M

is a squared residual, subscript i the ex-

perimental serial number, and n the total number of

the experimental data. Considering *

D

C as a constant

coefficient model (CCM), Eq. (19) is consistent with

the least squares method. However, it is very difficult

to solve directly. Here, a clamp trial method is used to

solve coefficient *

D

C by a computer program. Table 2

shows the coefficients for various air volumes by the

CCM.

These fittings show, for an equivalent sphere,

that the drag coefficients are about 0.4. The corre-

sponding curves can be drawn in Fig. 10. As seen in

the figure, these curves meet the dimensional princi-

ple; however, they obviously distort the experimental

results. In fact, the fitting curves can be greatly im-

proved if the influence of the flow patterns is con-

sidered by coupling the Reynolds number. To still

comply with the above dimension concordant princi-

ple, the Reynolds number is considered as an inde-

pendent factor coupled with the drag coefficient.

Considering *

D

C as a variable coefficient model

(VCM), which depends on the Reynolds number, the

proposed drag coefficient is written as

*

Dd

,

k

CKRe (20)

where Kd and k are constant coefficients, and

.Re vd

Then

2

d.

kv

KRe r

(21)

To obtain the optimal sets of k and Kd, consid-

ering θ and Re as the data sets in Eq. (21), for a spe-

cific k, a matched Kd and the corresponding squared

residual M can be determined. Fig. 11 shows that the

squared residual varies with the sets of k and Kd. The

optimal set is determined at the point with the mini-

mum squared residual Mmin.

Based on the optimal sets of k and Kd in Table 3,

the final empirical formula can be written as

Table 2 Coefficients for various air volumes

r

*

D

C

0.472 0.372

0.594 0.382

0.748 0.440

Table 3 Optimal coefficient sets and the squared residual

k Kd M

1.98 1.44×10−10 0.112

Fig. 11 Squared residuals and the sets of k and Kd

Wan et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2017 18(3):167-178 175

2

d

arcsin .

kv

KRe r

(22)

Accordingly, the revised curves are shown in

Fig. 10. Obviously, after considering the influence of

the flow patterns, these fitting curves are better

approximations to the experimental results. As shown

in the figure, the revised curve can represent the dis-

tribution principle of the location slope angles with

the pipe flow velocities and air volumes. The equi-

librium angle increases significantly with the increase

of the flow velocity, but it decreases slightly with the

increase of the air volume. The above result shows

that it is more difficult to remove trapped air from a

large-slope angle pipe than from a gentle incline pipe.

In other words, a greater flow velocity and a smaller

pipe depression angle are needed to avoid an unde-

sired air lock in the hump zone of a water supply

pipeline system.

5 Prevention and elimination of an air lock in

a hump pipe

As is known, an air lock can bring undesired

head losses and failure risk for closed pipe flow.

Therefore, it is important to design so as to prevent

potential air locks in a long-distance water supply

pipeline system. The above research shows that the

flow velocity, the pipe slope, and the volume of the

trapped air pocket, can greatly affect the critical

equilibrium of an air lock in a hump pipe zone.

In the experiment, with the increase of the pipe

flow velocity, the trapped air moves to a new equi-

librium location with a larger slope angle to resist the

drag force. Practically, a trapped air pocket will move

downstream if the pipe velocity increases and exceeds

the critical velocity of the maximum slope angle in

the hump zone. As shown in Fig. 10, the equilibrium

angle increases with the increase of the pipe flow

velocity. The trapped air is entirely removed when the

flow exceeds the critical flow velocity for the maxi-

mum slope angle.

In the design stage, it is important for the de-

signer to choose the pipe slope and pipe flow velocity.

According to Eq. (22), there are two approaches to

prevent an air lock in an irregularly undulating sub-

marine water supply pipeline system. One is in-

creasing the pipe flow velocity, and the other is de-

creasing the maximum slope angle of the pipeline.

For a specified velocity, the allowable critical slope

angle can be determined by

2

max d c

arcsin / .

k

KRe v r

(23)

To avoid an air lock, the slope angle at any point

should be smaller than the critical slope.

As shown in the above research, besides the pipe

flow velocity, the air volume can slightly affect the

equilibrium location. For example, to prevent a

160-cm3 air pocket in the experimental pipe system,

the allowable maximum slope angles are determined

by Eq. (23) for various specified flow velocities.

Fig. 12 shows the different operating zones. The solid

line is the critical equilibrium line, where Kc is a cri-

terion coefficient. The upper operating zone may

cause a potential air lock and the lower operating zone

can prevent an air lock. In other words, the maximum

slope angle should be smaller than the critical slope

angle to prevent an air lock.

Considering the effects of the trapped air vol-

umes, Fig. 13 shows the effect of the trapped air

volume on the critical slope line. As shown in this

figure, the volume can affect the equilibrium slope

angle, which increases with the increase of the vol-

ume. To prevent a larger air pocket, a greater velocity

or a smaller slope angle is needed.

An existing air lock can bring extra head losses

and potential hazards in a water supply pipeline. It is

important to remove the existing air lock during op-

eration. As presented in the above analysis, for a

given slope angle, the pipe flow velocity can greatly

Fig. 12 Operating zones for the prevention of trapped

air dissipation

2

c

d

= arcsin

k

v

θKReρ

r

Wan et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2017 18(3):167-178

176

affect the air lock equilibrium. The trapped air pocket

can move downstream if the pipe flow velocity is

large enough. For a given pipe system, according to

Eq. (23), the dimensionless critical flow velocity c

v

for removing the air pocket can be expressed as

cmax

d

sin ,

k

r

vKRe

(24)

where θmax is the allowable maximum depression

angle.

Practically, the existing air pocket can be re-

moved by running a higher pipe flow velocity tem-

porarily. Fig. 14 shows the critical flow velocity line

for a specific air volume. As shown in the figure, the

critical velocity increases with the increase of the

slope angle. For a known pipeline, the critical velocity

depends on the maximum slope angle. If an air lock

occurs in a practical project, it can be removed by

operating at a greater velocity than the critical one for

a short period.

Analogously, Fig. 15 shows the effect of the

trapped air volume on the critical velocity line. As

shown in this figure, the critical velocity increases

with the increase of the air volume, which shows that

a greater velocity is needed to remove a larger air

pocket.

To predict or evaluate the possibility of an air

lock in an irregular undulating submarine water sup-

ply pipeline system, a simple evaluation criterion is

proposed. According to the critical equilibrium of the

trapped air pocket in the hump pipe zone, the criterion

coefficient can be expressed as follows:

2

dc

c

max

.

sin

k

KRe v

Kr

(25)

The coefficient represents the relationship

among the operating flow velocity, the maximum

slope angle, and the possible air pocket scale in the

hump pipe zone. The pipe system will not trap an air

pocket in the hump zone when the coefficient is

greater than 1.0. Conversely, the hump zone can trap

potential air and accumulate a partial air lock. Obvi-

ously, a greater flow velocity and a less depression

angle are advantageous in avoiding air locks, since

they both can increase the criterion coefficient.

6 Analysis and discussion

As an undesired phenomenon in water supply

pipeline systems, air locks can restrict flow and even

Fig. 13 Effect of trapped air volume on the critical

slope lines

2

c

d

=arcsin

k

v

θKReρ

r

Fig. 15 Effect of the trapped air volume on the critical

velocity line

0.8

cmax

d

sin

k

r

vKRe

1.0

0.6

0.4

0.2

0.0

0.0 0.5 1.0 1.5

θ

θmax

v

c

r

=1.0

r

=0.8

r

=0.6

r

=0.4

r

=0.2

k

r

vθ

KRe ρ

cmax

d

=sin

v

~

Fig. 14 Critical flow velocity to remove trapped air pocket

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5

Flow velocity to remove air lock

critical control line

potential air lock zone

without air lock zone

vc

~

Kc>1.0,

Kc=1.0,

θmax θ

Kc<1.0,

cmax

d

=sin

k

r

vθ

KRe ρ

Wan et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2017 18(3):167-178 177

cause pipe burst. Thus, it is important to protect pipes

against air lock hazards. In an annular pipe, the ex-

periment simulates the movement of the trapped air

pocket in the hump zone of a submarine water supply

pipeline system. A hump air lock can decrease the

flow section and capacity of a water supply pipeline

system. The volume of the trapped air pocket plays an

important role in hydraulic head losses. Generally, a

larger trapped air pocket can bring greater extra head

losses, and it is more difficult to remove the air pocket

from the hump zone. Based on the experimental

measurement and the ESM, the extra head loss is

analyzed and it can be considered as a minor head

loss. For the critical equilibrium, the pipe flow ve-

locity can greatly affect the critical slope angle of the

air lock in the hump zone, but the air volume only

slightly affects the critical angle of the air lock. To

avoid air lock hazards in submarine water supply

pipelines, some designers may seek to employ an

excess velocity and a small slope angle. In fact, it is

essential and easy to avoid air locks in hump zones by

using the proposed criterion coefficient. The investi-

gation analyzes only the macroscopic movement of

the air pocket as an entire mass. In fact, some small

bubbles may separate from the trapped air mass and

move downstream with the pipe flow before the

trapped air pocket reaches the critical flow velocity.

7 Conclusions

An air lock can cause extra head losses and de-

crease the local flow section; consequently, it de-

creases the flow capacity of a water supply pipeline.

ESM is proposed to simplify the air lock patterns. The

head losses increase with the increase of the pipe flow

velocity and the volume of the air lock, and can be

considered as minor losses. The pipe flow velocity

has a great influence on the critical equilibrium of the

trapped air pocket in the hump pipe zone. The critical

equilibrium angle increases and the air lock moves

downstream to a new equilibrium location with an

increasing pipe flow velocity. There are two ap-

proaches to prevent potential air locks. One is in-

creasing the pipe flow velocity, and the other is de-

creasing the maximum slope angle of the pipe. A

criterion coefficient is proposed to evaluate the pos-

sibility of an air lock. According to this criterion, a

greater flow velocity and a less depression angle are

advantageous to avoid the air lock hazard in a sub-

marine water supply pipeline system. It can be a ref-

erence for preventing or estimating the partial hump

air lock in the design and operation of submarine

water supply pipelines.

References

Brown, L., 2006. Understanding Gravity-Flow Pipelines Wa-

ter Flow, Air Locks and Siphons. http://www.itacanet.

org/understanding-gravity-flow-pipelines-water-flow-air-

locks-and-siphons/

Burch, T.M., Locke, A.Q., 2012. Air lock and embolism upon

attempted initiation of cardiopulmonary bypass while

using vacuum-assisted venous drainage. Journal of Car-

diothoracic and Vascular Anesthesia, 26(3):468-470.

http://dx.doi.org/10.1053/j.jvca.2011.01.019

Burrows, R., Qiu, D.Q., 1995. Effect of air pockets on pipeline

surge pressure. Proceedings of the Institution of Civil

Engineers-Water, Maritime and Energy, 112(4):349-361.

http://dx.doi.org/10.1680/iwtme.1995.28115

Carlos, M., Arregui, F.J., Cabrera, E., et al., 2011. Under-

standing air release through air valves. Journal of Hy-

draulic Engineering, 137(4):461-469.

Chaiko, M.A., Brinckman, K.W., 2002. Models for analysis of

water hammer in piping with entrapped air. Journal of

Fluids Engineering-Transactions of the ASME, 124(1):

194-204.

http://dx.doi.org/10.1115/1.1430668

Epstein, M., 2008. A simple approach to the prediction of

waterhammer transients in a pipe line with entrapped air.

Nuclear Engineering and Design, 238(9):2182-2188.

http://dx.doi.org/10.1016/j.nucengdes.2008.02.023

Escarameia, M., 2007. Investigating hydraulic removal of air

from water pipelines. Proceedings of the Institution of

Civil Engineers-Water Management, 160(1):25-34.

http://dx.doi.org/10.1680/wama.2007.160.1.25

Ferreri, G.B., Ciraolo, G., Lo Re, C., 2014. Storm sewer

pressurization transient―an experimental investigation.

Journal of Hydraulic Research, 52(5):666-675.

http://dx.doi.org/10.1080/00221686.2014.917726

Finnemore, E., Franzini, J., 2002. Fluid Mechanics with En-

gineering Applications (10th Edition). McGraw-Hill

Education, Boston, USA, p.261-284.

Greenshields, C.J., Leevers, P.S., 1995. The effect of air

pockets on rapid crack propagation in PVC and polyeth-

ylene water pipe. Plastics, Rubber and Composites Pro-

cessing and Applications, 24(1):7-12.

Izquierdo, J., Fuertes, V.S., Cabrera, E., et al., 1999. Pipeline

start-up with entrapped air. Journal of Hydraulic Re-

search, 37(5):579-590.

http://dx.doi.org/10.1080/00221689909498518

Lin, C., Liu, T., Yang, J., et al., 2015. Visualizing conduit

flows around solitary air pockets by FVT and HSPIV.

Journal of Engineering Mechanics, 141(5):04014156

Wan et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2017 18(3):167-178

178

http://dx.doi.org/10.1061/(asce)em.1943-7889.0000867

Liu, T., Yang, J., 2013. Experimental studies of air pocket

movement in a pressurized spillway conduit. Journal of

Hydraulic Research, 51(3):265-272.

http://dx.doi.org/10.1080/00221686.2013.777371

Pothof, I., Clemens, F., 2010. On elongated air pockets in

downward sloping pipes. Journal of Hydraulic Research,

48(4):499-503.

http://dx.doi.org/10.1080/00221686.2010.491651

Pothof, I., Clemens, F., 2011. Experimental study of air-water

flow in downward sloping pipes. International Journal of

Multiphase Flow, 37(3):278-292.

http://dx.doi.org/10.1016/j.ijmultiphaseflow.2010.10.006

Pozos, O., Gonzalez, C.A., Giesecke, J., et al., 2010. Air en-

trapped in gravity pipeline systems. Journal of Hydraulic

Research, 48(3):338-347.

http://dx.doi.org/10.1080/00221686.2010.481839

Pozos-Estrada, O., Fuentes-Mariles, O.A., Pozos-Estrada, A.,

2012. Gas pockets in a wastewater rising main: a case

study. Water Science and Technology, 66(10):2265-2274.

http://dx.doi.org/10.2166/wst.2012.462

Pozos-Estrada, O., Pothof, I., Fuentes-Mariles, O.A., et al.,

2015. Failure of a drainage tunnel caused by an entrapped

air pocket. Urban Water Journal, 12(6):446-454.

http://dx.doi.org/10.1080/1573062x.2015.1041990

Reynolds, C., Yitayew, M., 1995. Low-head bubbler irrigation

systems. Part II: Air lock problems. Agricultural Water

Management, 29(1):25-35.

http://dx.doi.org/10.1016/0378-3774(95)01189-7

Vasconcelos, J.G., Leite, G.M., 2012. Pressure surges fol-

lowing sudden air pocket entrapment in storm-water

tunnels. Journal of Hydraulic Engineering, 138(12):

1081-1089.

Yu, Y., 2015. Study of Critical Characteristic of Hump Air

Resistor in Submarine Water Supply Pipeline. MS Thesis,

Zhejiang University, Hangzhou, China (in Chinese).

Zhou, F., Hicks, F., Steffler, P., 2004. Analysis of effects of air

pocket on hydraulic failure of urban drainage infrastruc-

ture. Canadian Journal of Civil Engineering, 31(1):86-94.

http://dx.doi.org/10.1139/l03-077

Zhou, L., Liu, D.Y., Karney, B., 2013. Investigation of hy-

draulic transients of two entrapped air pockets in a water

pipeline. Journal of Hydraulic Engineering, 139(9):949-

959.

http://dx.doi.org/10.1061/(asce)hy.1943-7900.0000750

中文概要

题目：供水管路中驼峰气阻的临界平衡实验研究

目的：输水管道的驼峰气阻是指由于管路高峰位置的滞

气作用使气体不断聚积在峰顶附近、产生的气体

阻碍水流的局部水力现象。它能够导致管路过水

断面减小、输水能耗增加、输送效率降低和管路

压力振荡等后果，严重威胁海底管道输水的稳定

性和安全性。本文旨在分析滞留气团在供水管道

中的力学平衡、能量损失、移动和溢出机理，研

究水流流速对气团的推移特性，提出预测和消除

驼峰气阻的方法，使输水管道免受驼峰气阻的危

害，提高输水管道的供水效率。

创新点：1. 设计了具有连续坡角变化的圆弧形驼峰管道实

验，该实验可以定量模拟气团体积和平衡角度；

2. 建立了驼峰气阻的水头损失经验公式和恒定

流情况下驼峰气阻的管道坡角和流速的对应关

系式，可用于预测和消除驼峰气阻的危害。

方法：1. 通过驼峰气团的受力特性分析，获得满足量纲

和谐的力学平衡方程；2. 采用试验观察和测试获

得有无气泡情况下的水头损失和平衡状态下的

坡角，通过等价球体方法对测试数据进行无量纲

拟合，获得气阻的水头损失方程系数，并通过流

速和平衡坡角建立恒定流情况下的临界平衡方

程；3. 基于试验拟合获得临界平衡方程，建立预

测和评估气阻的准则系数，并提出消除气阻的水

流临界流速。

结论：1. 当管路流速较小时，供水管路的驼峰顶端可能

滞留和聚集气体，形成驼峰气阻；气体体积越大

对水流阻碍越明显，可能造成的水头损失也越

大；2. 利用等价球体法可以极大地简化驼峰气阻

的形状，并良好地模拟气阻的平衡特性和阻力特

性；3. 管道流速是影响驼峰气阻临界平衡位置的

最重要因素，通过减小管道起伏的坡角或增加水

流流速可以防止和消除驼峰气阻的危害。

关键词：驼峰管道；管流；滞留气团；水力试验；供水

管道