Investigation on critical equilibrium of trapped air pocket in water supply pipeline system *

Abstract

A trapped air pocket can cause a partial air lock in the top of a hump pipe zone. It increases the resistance and decreases the hydraulic cross section, as well as the capacity of the water supply pipeline. A hydraulic model experiment is conducted 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 reference 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 empirical 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.

Full text

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

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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

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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.
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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