JET  21
ANALYTICAL AND NUMERICAL 
ANALYSIS OF TRAPPED AIR POCKET 
DYNAMIC RESPONSE DUE TO 
PRESSURE CHANGE IN LIQUID-FILLED 
PIPELINES
ANALITIČNA IN NUMERIČNA ANALIZA 
DINAMIČNEGA ODZIVA ZRAČNEGA 
MEHURJA NA TLAČNO MOTNJO V 
CEVOVODIH, NAPOLNJENIH Z VODO
Jošt Pekolj
1
, Anton Bergant
1,2ℜ
, Matjaž Perpar
1
Keywords: pipelines, air pocket, transient pressures, analytical solution, method of characteris-
tics
Abstract
This paper serves as an overview of the different approaches to modelling the filling and empty-
ing of liquid-filled pipelines. Specifically, it compares an analytical and numerical analysis of the 
start-up of a liquid column in a one-dimensional pipeline with a trapped air pocket. The analytical 
analysis is based on the premise that there are no pressure waves travelling in the water column. 
For this reason, it is not suitable for pipeline systems with very small initial trapped air pockets 
JET Volume 15 (2022) p.p. 21-30
Issue 2, 2022
Type of article: 1.01 
www.fe.um.si/si/jet.html
 
ℜ Corresponding author: Anton Bergant, PhD, Full time: Litostroj Power d.o.o., Litostrojska 50, 1000 Ljubljana, Slo-
venia, anton.bergant@litostrojpower.eu; Part time: Faculty of Mechanical Engineering, University of Ljubljana, Ašk-
erčeva 6, 1000 Ljubljana, Slovenia
1
 Faculty of Mechanical Engineering, University of Ljubljana, Aškerčeva 6, 1000 Ljubljana, Slovenia
2 
Litostroj Power d.o.o., Litostrojska 50, 1000 Ljubljana, Slovenia 
Analytical and numerical analysis of trapped air pocket dynamic response due to pressure change in liquid-filled pipe-
lines
Analitična in numerična analiza dinamičnega odziva zračnega mehurja na tlačno motnjo v cevovodih, napolnjenih z 
vodo
Jošt Pekolj, Anton Bergant, Matjaž Perpar

22 JET 22 JET
Jošt Pekolj, Anton Bergant, Matjaž Perpar JET Vol. 15 (2022)
Issue 2
or very high reservoir pressures. In the pipelines with parameters, as shown in this paper, the 
results of the analytical analysis correlate sufficiently with those of the numerical analysis.
Povzetek
Prispevek predstavlja pregled dveh različnih pristopov k modeliranju polnjenja in praznjenja 
cevovodov s kapljevino. Primerja analitično in numerično analizo polnjenja v enodimenzijskem 
cevovodu z ujetim zračnim mehurjem na enem koncu. Analitična analiza temelji na predpostav-
ki, da po vodnem stolpcu ne potujejo tlačni valovi. Zaradi tega ta pristop ni primeren za cevne 
sisteme z zelo majhnimi začetnimi zračnimi mehurji ali zelo visokimi tlaki rezervoarja. Za predsta-
vljene primere v tem delu se analitični in numerični rezultati ujemajo.
1 INTRODUCTION
During filling or emptying of liquid-filled pipelines an air pocket, or pockets, of various sizes can 
get trapped at distinct positions (high point, pipe end). The presence of an air pocket influences 
the system’s response to a pressure disturbance. Pressure in an air pocket could increase by 
multiples of the original pressure change. The response of an air pocket can be simulated using 
different computational models. In this paper, the authors examine the one-dimensional analyt-
ical and numerical models of air pocket dynamic response. Air pockets become trapped at the 
downstream end of the gravitational type of pipeline. The basis of the analytical analysis is to 
describe the behaviour of the system based on differential equations. The method of character-
istics will be used for the numerical analysis, as it is convenient for tracking pressure waves in the 
water filled region of the pipe. The analytical and numerical results will be compared for different 
parameters of the hydraulic pipeline system during partial filling, as shown in Figure 1. At the 
leftmost part of the pipeline there is a reservoir with a pressure head H
rez
, which is attached to 
the pipeline with a water column of length L(t) and an air pocket of length L
gas
(t). The water and 
air regions are initially separated by a valve, which is located at x
1
. After the rapid valve opening, 
the water column moves towards the air pocket at velocity v(t). The response of an air pocket in 
similar pipelines has also been studied experimentally in [1], [2] and [3]. When comparing the 
two types of analysis, the numerical one shows better agreement with the experimental results, 
because it also includes the interaction of pressure waves and unsteady friction [2].
Figure 1: A hydraulic pipeline system with a trapped air pocket

JET  23
Analytical and numerical analysis of trapped air pocket dynamic response due to pressure change in liquid-filled pipelines
2 THEORETICAL MODELLING
The following subchapters describe the most important equations used in analytical and numer-
ical analysis. The equations are valid for the system shown in Figure 1.
2.1 Water region
The water column in the water region of the pipe can be modelled as rigid or elastic. The one-di-
mensional motion of the rigid water column is governed by the momentum conservation, as 
shown in equation (2.1):
dv
dt
( t ) =
g ∙ ( H
a , r ez
− H
a , gas
( t ) )
L ( t )
+ g ∙ sin θ − ¿ 
− λ
2 D
∙ v ( t ) | v ( t ) | −
K
v
2 L ( t )
∙ v ( t ) | v ( t ) | −
K
e
+ 1
2 L ( t )
∙ v
2
( t ) ∙ H ( v ( t ) ) . (2.1)
dv
dt
( t ) =
g ∙ ( H
a , r ez
− H
a , gas
( t ) )
L ( t )
+ g ∙ sin θ − ¿ 
− λ
2 D
∙ v ( t ) | v ( t ) | −
K
v
2 L ( t )
∙ v ( t ) | v ( t ) | −
K
e
+ 1
2 L ( t )
∙ v
2
( t ) ∙ H ( v ( t ) ) . (2.1)
dv
dt
( t ) =
g ∙ ( H
a , r ez
− H
a , gas
( t ) )
L ( t )
+ g ∙ sin θ − ¿ 
− λ
2 D
∙ v ( t ) | v ( t ) | −
K
v
2 L ( t )
∙ v ( t ) | v ( t ) | −
K
e
+ 1
2 L ( t )
∙ v
2
( t ) ∙ H ( v ( t ) ) . (2.1)
dv
dt
( t ) =
g ∙ ( H
a , r ez
− H
a , gas
( t ) )
L ( t )
+ g ∙ sin θ − ¿ 
− λ
2 D
∙ v ( t ) | v ( t ) | −
K
v
2 L ( t )
∙ v ( t ) | v ( t ) | −
K
e
+ 1
2 L ( t )
∙ v
2
( t ) ∙ H ( v ( t ) ) . (2.1)
(2.1)
Similar equations can be seen in [4] and [5]. Equation (2.1) is simplified by neglecting some of 
the terms; namely, friction factor λ, coefficients of valve head losses K
v
 and reservoir exit losses 
K
e
, while the slope term of the pipe will be set to zero (horizontal pipe). For the elastic water 
region, it is necessary to consider the motion of pressure waves in the water. This is based on the 
assumption that the waves are one-dimensional, which is justified, since the diameter of the pipe 
is substantially smaller than its length. The motion of the water is governed by the continuity and 
momentum conservation, as shown in equations (2.2) and (2.3) [6]:
∂ H
∂ t
+ v ∙
∂ H
∂ x
− v ∙ s in θ +
a
2
g
∂ v
∂ x
= 0, ( 2. 2)
g ∙
∂ H
∂ x
+
∂ v
∂ t
+ v ∙
∂ v
∂ x
+
λ ∙ v ∙ | v |
2 D
= 0 . ( 2. 3)
H
i
=
C
p
B
m
+ C
m
B
p
B
p
+ B
m
, ( 2. 4)
Q
i
=
C
p
− C
m
B
p
+ B
m
. ( 2. 5)
H
a , gas
( t ) ∙ L
gas
n
( t ) = C
A
. ( 2. 6)
(2.2)
∂ H
∂ t
+ v ∙
∂ H
∂ x
− v ∙ s in θ +
a
2
g
∂ v
∂ x
= 0, ( 2. 2)
g ∙
∂ H
∂ x
+
∂ v
∂ t
+ v ∙
∂ v
∂ x
+
λ ∙ v ∙| v |
2 D
= 0 . ( 2. 3)
H
i
=
C
p
B
m
+ C
m
B
p
B
p
+ B
m
, ( 2. 4)
Q
i
=
C
p
− C
m
B
p
+ B
m
. ( 2. 5)
H
a , gas
( t ) ∙ L
gas
n
( t ) = C
A
. ( 2. 6)
(2.3)
It is assumed that the velocity of pressure waves a is constant. Thereafter, the method of charac-
teristics is used to transform equations (2.2) and (2.3) into four ordinary differential equations, 
which are solved using the finite difference method [6]. They are combined to derive a solution 
for the piezometric head H
i
 and volume flow rate Q
i
 at a node in a characteristic grid, as shown in 
equations (2.4) and (2.5), where the friction head losses are neglected:
∂ H
∂ t
+ v ∙
∂ H
∂ x
− v ∙ s in θ +
a
2
g
∂ v
∂ x
= 0, ( 2. 2)
g ∙
∂ H
∂ x
+
∂ v
∂ t
+ v ∙
∂ v
∂ x
+
λ ∙ v ∙ | v |
2 D
= 0 . ( 2. 3)
H
i
=
C
p
B
m
+ C
m
B
p
B
p
+ B
m
, ( 2. 4)
Q
i
=
C
p
− C
m
B
p
+ B
m
. ( 2. 5)
H
a , gas
( t ) ∙ L
gas
n
( t ) = C
A
. ( 2. 6)
(2.4)
∂ H
∂ t
+ v ∙
∂ H
∂ x
− v ∙ s in θ +
a
2
g
∂ v
∂ x
= 0, ( 2. 2)
g ∙
∂ H
∂ x
+
∂ v
∂ t
+ v ∙
∂ v
∂ x
+
λ ∙ v ∙ | v |
2 D
= 0 . ( 2. 3)
H
i
=
C
p
B
m
+ C
m
B
p
B
p
+ B
m
, ( 2. 4)
Q
i
=
C
p
− C
m
B
p
+ B
m
. ( 2. 5)
H
a , gas
( t ) ∙ L
gas
n
( t ) = C
A
. ( 2. 6)
(2.5)
In order to calculate variables H
i
 and Q
i
 at the boundary, a device-specific equation, or equations, 
replaces one of the water hammer compatibility equations (the reservoir and air pocket in this 
case) [2], [6].

24 JET 24 JET
Jošt Pekolj, Anton Bergant, Matjaž Perpar JET Vol. 15 (2022)
Issue 2
2.2 Air region
The air pocket is modelled as a lumped body. Therefore, the presence of pressure waves is not 
included in the air region in the model. A similar description is illustrated in [6]. During the filling 
of the pipeline, the water column compresses the air pocket, which undergoes a polytropic pro-
cess. The changes in pressure and volume of the air pocket are described using equation (2.6):
∂ H
∂ t
+ v ∙
∂ H
∂ x
− v ∙ s in θ +
a
2
g
∂ v
∂ x
= 0, ( 2. 2)
g ∙
∂ H
∂ x
+
∂ v
∂ t
+ v ∙
∂ v
∂ x
+
λ ∙ v ∙ | v |
2 D
= 0 . ( 2. 3)
H
i
=
C
p
B
m
+ C
m
B
p
B
p
+ B
m
, ( 2. 4)
Q
i
=
C
p
− C
m
B
p
+ B
m
. ( 2. 5)
H
a , gas
( t ) ∙ L
gas
n
( t ) = C
A
. ( 2. 6)
(2.6)
The gas constant CA is calculated by using the known initial conditions (pressure head and air 
pocket length) in the equation (2.6). From previous experimental investigations, such as [1-3], it 
was determined that the numerical model most closely matches the experimental data when the 
polytropic coefficient for the process in air equals 1.4.
2.3 Gas-water interface
When using the rigid water column approach, equations (2.1) and (2.6) are coupled for the air 
and water regions. The result is a linear first-order ordinary differential equation. Equation (2.7) 
describes the motion of a gas-water interface [5]:
d v
2
dL
+
(
K + 1
L
+
λ
D
)
v
2
= 2 g
(
H
a , r ez
L
−
C
A
(
x
L
− L
)
n
+ s in θ
)
.
( 2. 7)
v = L
−
(
K + 1
2
)
∙ √ 2 g ∙
√
H
a , r ez ∫
L 0
L
L
¿ K
∙ d L
¿
− C
A ∫
L 0
L
L
¿ K
( x
L
− L
¿
)
n
∙ d L
¿
+ ¿ + sin θ ∫
L 0
L
L
¿ K + 1
∙ d L
¿
.
( 2. 8)
(2.7)
It is assumed that the gas-water interface is always vertical (both in the rigid and elastic water 
approach). Equation (2.7) can be solved analytically (in this case) or numerically [5]. By using the 
elastic water column approach, the interface changes with the volume of the air pocket.
3 ANALYTICAL AND NUMERICAL SOLUTIONS
3.1 Analytical solution
The analytical method solves the differential equation for the gas-water interface (2.7) [5]. This 
equation was solved by using an integration factor. The solution is an expression for the speed of 
the water column in terms of its length (2.8):
 
Analytical and numerical analysis of trapped air pocket dynamic response due 
to pressure change in liquid-filled pipelines 
5 
   
---------- 
 𝑣𝑣 = 𝐿𝐿 � �
���
�
�
∙ � 2𝑔𝑔 ∙ �
𝐻𝐻 �,���
∫ 𝐿𝐿 ∗
�
∙ 𝑑𝑑𝐿𝐿
∗
�
�
�
− 𝐶𝐶 �
∫
�
∗
�
(�
�
��
∗
)
�
∙ 𝑑𝑑𝐿𝐿
∗
�
�
�
+
+ sin 𝜃𝜃 ∫ 𝐿𝐿 ∗
���
∙ 𝑑𝑑𝐿𝐿
∗
�
�
�
  . (2.8) 
 
With the help of equation (2.6) for the air region, it was also possible to calculate how the 
pressure in the air pocket changes over time.  
 
3.2 Numerical solution 
For the numerical analysis, the numerical model based on the method of characteristics [6] was 
used. The nodes in the numerical grid are fixed and are distributed over the length of the initial 
water column – from the leftmost node at the reservoir to the rightmost node at the valve. For 
the interior nodes of the water region characteristic grid, the piezometric head and volume flow 
rate are calculated using (2.4) and (2.5). At the leftmost node, at the boundary with the reservoir, 
the modified characteristic equations (2.4) and (2.5) are coupled with the boundary equation of 
the constant head reservoir [6]. For the node at the gas-water interface, the air region equation 
coupled with the modified characteristic equations was used. When the water-air interface 
moves, it is assumed that the variables Hi and Qi at the interface and at the fixed node on the 
valve are the same. 
 
4 COMPARISONS OF ANALYTICAL AND NUMERICAL RESULTS 
The analytical and numerical solutions to the equations were calculated using different 
parameters of the hydraulic pipeline system, as depicted in Figure 1. The parameters for the 
different pipelines are shown in Tables 1, 2 and 3. For all of the cases illustrated, the pressure loss 
coefficient K and slope of the pipe θ are equal to zero. 
Table 1: Pipeline system parameters for Case 1 [4], [5] 
Hrez [m] Ha, gas0 [m] L0 [m] Lgas0 [m] D [m] n [-] ρv [kg/m
3
] a [m/s] 
31 10.3 100 15 0.3 1.4 1000 1000 
Table 2: Pipeline system parameters for Case 2 [1] 
Hrez [m] Ha, gas0 [m] L0 [m] Lgas0 [m] D [m] n [-] ρv [kg/m
3
] a [m/s] 
8.15 10.32 5.57 3.25 0.04 1.4 1000 400 
Table 3: Pipeline system parameters for Case 3 [1] 
Hrez [m] Ha, gas0 [m] L0 [m] Lgas0 [m] D [m] n [-] ρv [kg/m
3
] a [m/s] 
12.23 10.28 5.57 3.25 0.04 1.4 1000 400 
 
 
(2.8)
With the help of equation (2.6) for the air region, it was also possible to calculate how the pres-
sure in the air pocket changes over time.

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Analytical and numerical analysis of trapped air pocket dynamic response due to pressure change in liquid-filled pipelines
3.2 Numerical solution
For the numerical analysis, the numerical model based on the method of characteristics [6] was 
used. The nodes in the numerical grid are fixed and are distributed over the length of the initial 
water column – from the leftmost node at the reservoir to the rightmost node at the valve. For 
the interior nodes of the water region characteristic grid, the piezometric head and volume flow 
rate are calculated using (2.4) and (2.5). At the leftmost node, at the boundary with the reservoir, 
the modified characteristic equations (2.4) and (2.5) are coupled with the boundary equation 
of the constant head reservoir [6]. For the node at the gas-water interface, the air region equa-
tion coupled with the modified characteristic equations was used. When the water-air interface 
moves, it is assumed that the variables H
i
 and Q
i
 at the interface and at the fixed node on the 
valve are the same.
4 COMPARISONS OF ANALYTICAL AND NUMERICAL RESULTS
The analytical and numerical solutions to the equations were calculated using different param-
eters of the hydraulic pipeline system, as depicted in Figure 1. The parameters for the different 
pipelines are shown in Tables 1, 2 and 3. For all of the cases illustrated, the pressure loss coeffi-
cient K and slope of the pipe θ are equal to zero.
Table 1: Pipeline system parameters for Case 1 [4], [5]
H
rez
 [m] H
a, gas0
 [m] L
0
 [m] L
gas0
 [m] D [m] n [-] ρ
v
 [kg/m
3
] a [m/s]
31 10.3 100 15 0.3 1.4 1000 1000
Table 2: Pipeline system parameters for Case 2 [1]
H
rez
 [m] H
a, gas0
 [m] L
0
 [m] L
gas0
 [m] D [m] n [-] ρ
v
 [kg/m
3
] a [m/s]
8.15 10.32 5.57 3.25 0.04 1.4 1000 400
Table 3: Pipeline system parameters for Case 3 [1]
H
rez
 [m] H
a, gas0
 [m] L
0
 [m] L
gas0
 [m] D [m] n [-] ρ
v
 [kg/m
3
] a [m/s]
12.23 10.28 5.57 3.25 0.04 1.4 1000 400
Figures 2 to 4 show the analytical solutions for the velocity of the water column in terms of its 
length for all the three cases, as illustrated in Tables 1, 2 and 3. The results shown in Figure 2 
agree with those presented by Tijsseling et al. [5].

26 JET 26 JET
Jošt Pekolj, Anton Bergant, Matjaž Perpar JET Vol. 15 (2022)
Issue 2
Figure 2: Water column velocity in terms of column length for Case 1
Figure 3: Water column velocity in terms of column length for Case 2

JET  27
Analytical and numerical analysis of trapped air pocket dynamic response due to pressure change in liquid-filled pipelines
Figure 4: Water column velocity in terms of column length for Case 3
Figures 5 to 7 show the comparison of calculated pressure in the air pocket by using the analyt-
ical and numerical methods (Chapter 3) for all three cases in Tables 1, 2 and 3. As can be seen 
from the figures, the air pocket pressure response for the discussed cases is similar to that of the 
analytical and numerical analysis.
Figure 5: A comparison of analytical and numerical air pocket pressure for Case 1

28 JET 28 JET
Jošt Pekolj, Anton Bergant, Matjaž Perpar JET Vol. 15 (2022)
Issue 2
Figure 6: A comparison of analytical and numerical air pocket pressure for Case 2
Figure 7: A comparison of analytical and numerical air pocket pressure for Case 3

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Analytical and numerical analysis of trapped air pocket dynamic response due to pressure change in liquid-filled pipelines
5 CONCLUSION
The analytical and numerical results match quite well, however there are still some differences. 
The reason for this is that the physical models are based on slightly different approaches, e.g. 
the rigid and elastic water column, and a different consideration of the water region between 
the valve and air pocket. The models match for those cases with large initial air pockets and low 
reservoir pressures. Since the analytical analysis does not include pressure waves in the water 
column and unsteady friction [2], it is not suitable for cases with small initial air pockets and high 
reservoir pressures. The numerical method could be improved by providing a better description 
of the gas-water interface. The part of the water column that extends over the valve will be in-
cluded in the elastic water region in future research.
References
[1] L. Zhou, D. Liu, B. Karney, P . Wang: Phenomenon of White Mist in Pipelines Rapidly Fill-
ing with Water with Entrapped Air Pockets, Journal of Hydraulic Engineering, Vol.139, 
Iss. 10, p.p.1041-1051, 2013
[2] A. Bergant, A. Tijsseling, Y. Kim, U. Karadžić, L. Zhou, M.F. Lambert, A.R. Simpson: Un-
steady Pressures Influenced by Trapped Air Pockets in Water-Filled Pipelines, Strojniški 
vestnik – Journal of Mechanical Engineering, Vol.64, Iss.6, pp.501-512, 2018
[3] L. Zhou, Y. Cao, B. Karney, A. Bergant, A. S. Tijsseling, D. Liu, P.Wang: Expulsion of 
Entrapped Air in a Rapidly Filling Horizontal Pipe, Journal of Hydraulic Engineering, 
Vol.146, Iss.7, p.p.1-16, 2020
[4] E. Cabrera, J. Abreu, R. Perez, A. Vela: Influence of Liquid Length Variation in Hydraulic 
Transients, Journal of Hydraulic Engineering, Vol.118, Iss.12, p.p.1639-1650, 1992
[5] A. S. Tijsseling, Q. Hou, Z. Bozkuş: Analytical expressions for liquid-column velocities 
in pipelines with entrapped air, In: Proceedings of the ASME 2015 Pressure Vessels & 
Piping Division Conference, ASME, Paper PVP2015-45184, 2015
[6] F. B. Wylie, V. L. Streeter: Fluid Transients in Systems, Prentice Hall, 1993
Nomenclature
A Area
a Velocity of pressure waves
B
m
Proportionality constant
B
p
Proportionality constant
C
A
Gas constant
C
m
Proportionality constant
C
p
Proportionality constant
D Internal diameter of the pipe

30 JET 30 JET
Jošt Pekolj, Anton Bergant, Matjaž Perpar JET Vol. 15 (2022)
Issue 2
g Gravitational acceleration (g = 9.81 m/s
2
)
H Heaviside step function
H
a,gas
Air pocket absolute pressure head
H
a,gas0
Air pocket initial absolute pressure head
H
a,rez
Reservoir absolute pressure head
H
i
Pressure head at a node
H
rez
Reservoir pressure head
K Combined pressure head loss coefficient
K
e
Pressure head loss coefficient for reservoir exit
K
v
Pressure head loss coefficient for the valve
L Water column length
L
*
Dummy length in integral
L
0
Initial water column length
L
gas
Air pocket length
L
gas0
Initial air pocket length
n Polytropic coefficient
Q
i
Volumetric flow rate at a node
t Time
v Water column velocity
x Space coordinate in direction of the pipeline
x
1
Location of the valve
x
L
Full length of the pipeline
λ Friction factor
θ Slope of the pipe
ρ
v
Water density Acknowledgments
Acknowledgments 
The authors gratefully acknowledge the support of the Slovenian Research Agency (ARRS) con-
ducted through research project L2-1825.