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Numerical Simulation of Hydraulic Shock in a Water Pumping System
Protected by Air
ANCA CONSTANTIN , CLAUDIU STEFAN NITESCU
Hydraulic Engineering Department , Faculty of Civil Engineering
“Ovidius” University
Constanta, 22B Unirii Str.
ROMANIA
aconstantina@univ-ovidius.ro, http://www.univ-ovidius.ro/faculties/civil_eng
Abstract: - Air may be efficiently used in water pumping system protection from hydraulic shock, due to its
elasticity. The paper presents the results regarding the extreme pressures in the discharge duct of a pumping
installation, obtained by numerical simulation of the water hammer phenomenon in two cases of protection
with air: using air chamber and using free air dispersed throughout the pumped water. The air chamber
increases the pipe wall elasticity in the particular section where the chamber is mounted. The air bubbles
transform single-phase water into a biphasic fluid with greater compressibility. Specific parameters are
related to these methods, in order to establish criteria for best protection solution choice in water supply
engineering design.
Key-Words: - Hydraulic Shock, Pumping Installation, Air Chamber, Biphasic Flow, Dissolution.
1 Introduction
The engineering design of a pumping installation aims to
achieve the requested discharge and head at an optimal
energetic efficiency, and also to prevent operation
instabilities and damaging phenomena such as cavitation
and water hammer. Our study focuses on a water
pumping installation and two of the ways of protection
from hydraulic shock: by air chamber and by free air
dispersed in pumped water. The air chamber locally
offers elasticity to the conduit due to air compressibility
[7], [14]. The presence of small amounts of free air in
the water results in a smaller celerity, due to the change
of density of the biphasic fluid. In the engineering
design, the choice of one or the other of these protection
solutions must be theoretically studied. There must be
known their efficiency and limitation.
Numerical simulation, using specially written or
commercial computer programs, is the most reliable and
low cost method of comparative study.
2 Pumping Installation Protection from
Water Hammer
The hydraulic shock is one of the most damaging
phenomena in a water supply system. Therefore, the
specialists in hydraulics and mathematicians have been
paying their attention to the rigorous study of the water
transient movement in closed conduits. The hydraulic
shock may occur in a pumping installation equipped
with N pumps mounted in parallel, in three main cases:
when the first pump is turned on, when the last pump is
turned off and, the most dangerous case, when all the
pumps accidentally stop due to a power failure.
There have been conceived different methods to solve
the hydraulic shock problems, such as: arithmetic water
hammer, graphical water hammer, algebraic method,
characteristics method, miscellaneous methods based on
the linearization of the differential equations [7], [9].
[14].
We may also mention a series of devices conceived
and designed to protect tubes from the threat of water
hammer: surge tank, air chamber, water chamber,
asymmetrical flow resistance device etc. [10].
Air chamber is frequently used in practice. In
engineering design, the dimensioning of such a device is
carried out according to the recommendation that the
ratio air volume to total chamber volume has to be
approximately 0,3. There are no references with respect
to the air volume decrease by dissolution in water inside
the air chamber. There were reported some accidents in
pumping installations equipped with air chamber
mounted on the discharge duct, when the air volume
hadn’t been monitored for a long period of time [5]. In
these cases, the pressure values reached during water
hammer were greater than the ones recorded in the
absence of the air chamber.
Free air dispersed into the pumped water is a more
recently conceived method. The biphasic flow inside the
pipeline is rather complex, therefore this method has
been studied assuming simplifying hypotheses [2]. The
free air may be deliberately introduced in the discharge
WSEAS TRANSACTIONS on SYSTEMS
Anca Constantin, Claudiu Stefan Nitescu
ISSN: 1109-2777
1009
Issue 10, Volume 9, October 2010
duct using a compressor, which allows a rigorous
control on the air flow rate. The influence of air on the
wave propagation speed is depicted in [2], [5], [10]. We
are interested in further information regarding extreme
pressure during hydraulic shock and the range of air
volume fraction in which this method is efficient.
3 Case Study
3.1 Hydraulic Shock Equations
The installation we focused on is equipped with a double
flux centrifugal hydraulic pump, type 12NDS. The
operation point is given by the discharge
at the head . The steel made
discharge duct is horizontally mounted. It has in
length and a diameter of . The layout of the
installation is shown in Fig.1. Water is pumped at a
geodetic height of 9m. The discharge duct is equipped
with a check valve that prevents water from flowing
back in an accidentally pump stoppage situation. In fact,
this type of stoppage is taken into account as the source
of perturbation in our discussion.
Fig.1. Pumping installation with horizontal
discharge duct
Our purpose is to determine the best solution of
protection from hydraulic shock for this installation,
considering the air chamber method versus the free air
method. Consequently, there were determined, by
numerical simulation, the extreme pressures in the
discharge duct, in the section next to the pump, during
water hammer, in three hydraulic shock situations: when
the pipeline is not protected by any devices (as the
reference case), when the discharge duct is protected by
an air chamber mounted next to the pump and when free
air is homogenously spread into the pumped water by
the help of a compressor. Further more, we envisage
finding out the range of efficient protection for each
method, considering the following parameters:
-the ratio of air volume to chamber volume, β;
-the volume fraction of free air in the biphasic
mixture, α.
The pressure variation in the discharge duct was
simulated by the program Hammer. The mathematical
model used for the water hammer phenomenon assumes
that the water flows in a single direction (along the
longitudinal axis of the pipe) and the head losses may be
calculated using the same formulas as in the steady
regime. The main equations are the momentum
conservation equation (1):
(1)
and the continuity equation (2):
(2)
where: v-velocity, [m/s];
H-head, [m];
c-celerity, [m/s];
-pipe’s diameter, ;
-Darcy’s coefficient;
g-gravitational acceleration, .
The term in equation (1) stands for the
head losses on pipe’s length unit, calculated with
Darcy’s formula [5]. This term may be neglected only
for the first water oscillation. For the subsequent
oscillations, it contributes to the water oscillation
damping.
The characteristics method is the most used in numerical
simulation of the hydraulic shock, due to its precise
results, even in solving complex system problems. It
allows a relative easy implementation of various
boundary conditions.
The characteristics method transforms the partial
differential equations (1) and (2) into four total
differential equations, knowing that :
(3)
(4)
The sign “+” stands for the direct wave and the sign
“–“ for the indirect wave. In current technical problems
the celerity is constant for a given duct, consequently the
relations (3) represent straight lines along which the
relations (4) are compatible. The method is clearly
explained in detail in[10], [14]. The equation system
can be solved by a finite –difference technique. Written
in finite differences the main equations governing the
hydraulic shock become the velocity equation (5):
(5)
and the head equation (6):
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Anca Constantin, Claudiu Stefan Nitescu
ISSN: 1109-2777
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Issue 10, Volume 9, October 2010
(6)
where: subscript i accounts for time,
subscript j accounts for node,
-step of time, [s].
The studied duct is divided into sections and the
equations (5) and (6) allow the calculation of speed and
head in each node, knowing the initial regime
conditions.
Specific relationships are added to these equations
according to the envisaged type of water hammer
protection of the discharge pipe.
3.2 Protection with Air Chamber
3.2.1 Specific Hydraulic Shock Equations
Specific boundary conditions are imposed for the
upstream and downstream end of the duct and for the
node where the air chamber is mounted. These
conditions result in additional equations written also in
finite differences.
Fig.2. Upstream end of the discharge duct.
Node P with check valve
At the upstream end of the duct, Fig.2, the boundary
conditions are imposed by the check valve clothing. The
closing law allows the determination of . The head
results as [10]:
(7)
At the downstream end of the duct, there is an open
reservoir with constant head, Fig. 3. Therefore:
(8)
Fig.3. Downstream end of the discharge duct.
Node N with constant head reservoir
(9)
Fig.4. Node with air chamber
It was assumed that the discharge duct is equipped
with an air chamber of total volume as it may
be seen in Fig.4. The equations (10)-(12) are specific for
a calculus node with an air chamber [10], assuming a
polytropic process for air.
(10)
(11)
(12)
where -air pressure, in the air layer inside the
chamber, ;
- volume of air in the chamber, ;
-ascensional velocity of water surface level
in the air chamber, [m/s];
n-polytropic exponent;
y-elevation of water surface in the air chamber,
[m];
A-air-water interface area, [m2].
The celerity in a water pipeline depends both on
water compressibility and pipe wall elasticity [3], [5],
[9]. The relation (13) stands for celerity in a single liquid
phase, taking into account the elastic behaviour of the
pipe wall:
y
C
+
x
C
-
Air-water
interface, A
p
N
C
+
x
x
P
C
-
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Anca Constantin, Claudiu Stefan Nitescu
ISSN: 1109-2777
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Issue 10, Volume 9, October 2010
(13)
where -water density,[ ];
-modulus of elasticity of water, ;
-modulus of elasticity of the pipe wall,
;
e-pipe-wall thickness, ;
k-coefficient depending on the pipe’s
emplacement.
There was assumed a constant celerity along the
pipeline.
3.2.2 Model for Air Dissolution in Water
A problem encountered in pumping stations
exploitation, in the case the air chamber isn’t monitored
for a long time, is the decrease of the compressed air
volume by the air dissolution in water. The phenomenon
is more intense in the systems with large aria air-water
interface and high air pressure.
Dissolution of compressed air into the water, inside
the chamber may change the ratio of the volume of air to
the total volume of the reservoir:
(14)
where - total volume of chamber, .
An evaluation of the air volume rate absorbed in the
water is possible considering the transport equation of
the gas. We assumed air to be single sort of perfect gas.
In the absence of any chemical reaction between the two
fluids, the transport equation is [2], [15]:
(15)
where -diffusion coefficient of air in
water, ;
- velocity, .
C- concentration of air dissolved in water,
;
Both fluids are at rest inside the chamber, so the
convective term in equation (15) equals zero. The
equation (15) becomes:
(16)
We considered that pressure p in the air layer inside
the chamber remains constant, which allowed us to
impose a constant concentration at the air water
interface, determined by the use of Henry’s law [13]:
(17)
where - air pressure in the chamber, [ ];
--Henry’s constant, ;
We also assumed a constant concentration in the
inferior section of the chamber, where water in the
chamber is adjacent to water in the discharge duct.
Considering that concentration of air in water varies
only on the y axis of the chamber, Fig.4, we obtained by
integration, with Dirichlet boundary conditions, the
concentration variation as showed in Fig.5.
The normal diffusive flux of air (expressed on unit
of length) through the inferior section of the chamber
towards the protected discharge duct is represented in
Fig.6. It exponentially varies in time. It may be noticed
that the water in the chamber becomes saturated after a
week. Thus, after this period we may assume a steady
state for the mass transfer.
Fig.5. Variation of air concentration in water,
inside the chamber, for different exposure times
Taking into account the dimensions of the section,
the flux tends to a constant value,
.
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Anca Constantin, Claudiu Stefan Nitescu
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Fig.6. Air diffusive flux on unit of length, at the
inferior section of the chamber, towards the
water in the discharge duct
The mass flow rate transferred between the two
phases, in steady state, is depicted by the relationship:
(18)
where -mass flow rate transferred
between air and water, [ ];
M-equivalent molecular weight of air,
[ ].
The equivalent molecular weight of air, M, the
equivalent Henry’s constant, X, and the diffusivity
coefficient, , were calculated as weighted averages
of the oxygen and nitrogen correspondent properties,
assuming air to be a perfect gas composed only of the
two mentioned gases [2].
The mass flow rate value determined with the
relation (18) allows us to determine the volume flow rate
of air, at constant temperature and pressure.
(19)
where -air constant, ;
-absolute temperature, ;
3.3 Protection with Free Air
The model used to simulate the hydraulic shock in the
case of free air protection assumed a homogenous air-
water mixture, where water was the continuous phase
and the air the dispersed one.
The mathematical model is the same used for the air
chamber protection (except for the specific equations
referring to the calculus node with chamber), but celerity
is calculated with a modified relationship [3], as follows.
We assumed the flow of the biphasic fluid was
homogenous, that means the velocity for gas is equal to
the water velocity [2], [11]. Both phases are subjected to
the same pressure field.
Taking into account a control volume filled with
biphasic fluid, the volume fraction of air , is defined by
the ratio:
(20)
A similar volume fraction might be written for water :
(21)
Thus, the relationship between the two volume fractions
becomes [2]:
(22)
The presence of air bubbles modifies the parameters
of fluid [8]. So, instead of considering the water density,
the density of the biphasic fluid, , was taken into
account. The biphasic fluid density may be expressed
using the volume fractions [2]:
(23)
The term related to the air density may be neglected.
Consequently, the mixture density will be:
(24)
The new expression of the density leads to the following
form for the celerity relationship: [5]
(25)
where- -air mass on biphasic unit volume,
[ ].
Assuming the air mass on biphasic unit volume to be
constant, that means the air dissolution and liberation are
neglected (no mass transfer between bulk water and air
bubbles), the term has an important weight on
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Anca Constantin, Claudiu Stefan Nitescu
ISSN: 1109-2777
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Issue 10, Volume 9, October 2010
celerity at small pressures. This term may be neglected
at pressures exceeding 40 bar [5].
4 Results
The numerical simulation of the hydraulic shock in the
specified pumping discharge duct was achieved by the
use of a special computer program, based on the
mathematical model of the water hammer phenomenon
written in finite differences and solved by the method of
characteristics.
The numerical simulation for the hydraulic shock in
the given installation was carried out in the following
circumstances:
a. the pipeline is not provided with any protection
device;
b. the pipeline is equipped with an air chamber and
the ratio drops from 0,75 to 0,025;
c. the pipeline hasn’t any protection device, but
free air is continuously introduced by the help of a
compressor, right downstream the pump; the air volume
fraction is rigorously varied from 0,5 to10 %.
In order to compare the extreme pressures obtained
in the above mentioned cases during hydraulic shock,
the variation of pressure is graphically represented for
the same cross section of the discharge duct, next to the
pump.
4.1 Pipeline without Protection from Water
Hammer
In the case of no protection, the extreme values of
pressure may be seen in Fig. 7. Maximal pressure
reaches 132 mwc, value that might be dangerous for an
installation conceived to resist at maximum 10 bar.
Fig.7. Pressure variation in the unprotected conduit
Cavitation occurs during the first 20s of the unsteady
movement.
4.2 Pipeline Protected by Air Chamber
In the second case, when the duct is protected by an air
chamber, the maximal pressure in the same node
decreases very much for , Fig.8. The
pressure rises only to 28mwc.
Fig.8 Pressure variation in the case of the discharge duct
protected by air chamber; large air volume in the
chamber
When the ratio range is the air
chamber still works as a protection device, but is less
efficient. After the first 5 seconds, time for the check
valve to close, negative pressure inside the duct
disappears. Fig.9 shows that a decrease with 5% of the
ratio (from 0,15 to 0,1) results in a decrease of the
maximal pressure with about 20 mwc.
Fig.9 Pressure variation in the case of the discharge duct
protected by air chamber; medium air volume in the
chamber
The maximal pressure rises up to larger values as
continues to decrease.
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Issue 10, Volume 9, October 2010
Finally, if the air volume isn’t monitored and reaches
the value , the air chamber becomes a threat
to the duct, because the maximal pressure reaches
160mwc, greater than the correspondent value in the
case when there are no protection means for the duct.
Pressure variation in the case of such small air volumes
in the chamber is graphically represented in Fig. 10.
Fig.10. Pressure variation in the case of the discharge
duct protected by air chamber; small air volume inside
the air chamber
The decrease in time of ratio , due to air
dissolution in water, was represented in Fig.11. The
volume flow rate of absorbed air was determined from
the mass flow rate given by the relationship (12),
at a temperature of 15 oC.
Fig.11. Variation in time of the ratio air volume to
reservoir volume, , due to the air dissolution in water
It may be seen that an exposure of pressurized air to
water, during a month, may decrease the value of
from 0,35 to 0,12, in the specific air chamber considered
above.
4.3 Pipeline protected by Free Air
4.3.1 .Extreme pressures. Method efficiency range
In the third case, when the discharge duct is protected
from water hammer by free air deliberately introduced in
the pumped water, there was considered an air volume
fraction variation between 0,5% and 10 %. These values
for the air volume fraction are expressed with respect to
the normal atmospheric pressure (101,3 kPa ).
Figures 12-13 present the pressure variation during
hydraulic shock, in the case of the discharge duct
protected by free air spread into water at different
percent values for . As the volume fraction of air
increases, extreme values of pressure attenuate.
Extreme pressure variation for small values of is
shown in Fig.12. Comparing with the case of
unprotected discharge duct, the maximal pressure
decreases from 132mwc to 67mwc and the minimal
pressure value increases with about 3 mwc for
. Cavitation disappears. This method takes
effect even at small amounts of dispersed air.
It may also be noticed that the frequency of pressure
oscillation is smaller as the volume fraction of gas is
greater. This difference is enhanced at small values, as
it may be observed in Fig. 12.
Fig.12. Pressure variation in the case of the discharge
duct protected by free air present in water at small
values for .
As the volume fraction increases, the extreme
pressure values attenuate, which means pressure varies
in a narrower range.
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Anca Constantin, Claudiu Stefan Nitescu
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Issue 10, Volume 9, October 2010
Fig.13. Pressure variation in the case of the discharge
duct protected by free air present in water at large values
for .
If rises above 5%, the attenuation of the extreme
pressures is slower. The pressure variation curves are
very much alike for , as it may be seen
in Fig.13.
We may conclude that there is no reason to increase
the volume fraction of gas above 5 %.
The simulation of hydraulic shock in a pipeline
protected by free air spread homogenously into the
pumped water neglected the tendency of air to
accumulate in the upper side of the conduit.
4.3.2 .Air Accumulation. Free Air Protection Method
Limitations
The free air may form pockets of gas, according to the
inner geometry of the duct and of its fittings. Therefore,
the use of free air must be done cautiously, to prevent
such accumulations.
The investigation on the air tendency to accumulate
in the upper side of the pipeline was carried out on a
duct section of 2,2 m in length, Fig.14. This section of
duct is equipped with a check valve and an electro valve,
both of butterfly type. The inner geometry was
simplified, both obturators being considered of
ellipsoidal shape.
The study was computationally made using the
program Comsol, on the above presented section of the
discharge duct. We assumed a non homogeneous flow of
the two phases inside the horizontal duct, in the presence
of the gravitational field [15]. The relative velocity
between phases was determined on the basis of the
balance of the forces acting on the air bubbles: the drag
and the buoyancy force [2], [11].
In Fig.14, the surface colour indicates the air volume
fraction field. The regions of light grey represent air
accumulations. The arrows indicate the air bubbles
velocity at a moment of unsteady flow. The check valve
is completely open, but the electro vale is only partially
open.
Fig.14. Biphasic (water-air) non homogeneous flow.
Surface colour: air volume fraction field. Arrows: air
velocity
Fig.15. Variation of the air volume fraction. Rotation
angle with respect to the duct axis: 30o for both
obturators
Fig.16 Variation of the air volume fraction. Rotation
angle with respect to the duct axis: 0o for the check
valve (completely open) and 30o for the electro valve
obturator
We computed the variation of the volume fraction of
air for different rotation angles between the main axis of
each ellipse and the duct axis. Some of these graphs are
presented in Fig. 15-18.
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Anca Constantin, Claudiu Stefan Nitescu
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Fig.17. Variation of the air volume fraction. Rotation
angle with respect to the duct axis: 0o for both obturators
(both valves completely open)
The values of the air volume fraction in these figures
are expressed at local pressure.
This sequence of rotation angles illustrates moments
during the opening of the electro valve. The biphasic
discharge is taken in accordance with the pump’s
characteristic [3],[4].
Fig.18. Variation of the air volume fraction. Rotation
angle with respect to the duct axis: 0o for the check
valve (completely open) and 45o for the electro valve
obturator
The maximal value for the air volume fraction, for
each of the three cases is recorded immediately
downstream the second obturator, in the cross section
situated at 1,5-1,6m. The maximum value is reached in
the case both obturators are partially open, Fig.15, when
the turbulence is most intense.
When both valves are completely open, Fig. 17, air
bubbles are entrained by water and the air volume
fraction field tends to be uniform.
Modifications in the field of air volume fraction may
occur at any manoeuvre of the electro valve. For
example, when the obturator of the electro valve is
partially closed at an angle of 45o, Fig.18, the volume
fraction of air records the largest value in the considered
duct, about 0,027.
In reality, the inner geometry is complex and air is
trapped in the upper side of the pipeline, forming
pockets.
5 Conclusion
Analysing the pressure variation obtained by numerical
simulation results the best options to protect the
horizontal discharge duct are:
-by the use of air chamber, at ;
-by the use of free air at
-5
0
5
10
15
20
25
30
35
40
45
0 20 40 60 80
100
Time [s]
Pressure [mwa]
air chamber b=0.35
5% free air
Fig.19.The best variants for protecting the installation by
the use of air
The two protection solutions offer almost the same
values of maximal pressure, as may be seen in Fig.19.
The use of air chamber provides a good protection of the
discharge duct, but only if . Thus, if accidentally
the volume of air in the chamber decreases such as
, the presence of this chamber becomes a threat
to the duct by increasing the maximal pressure during
the water hammer. Continuous surveillance of the air
volume in the chamber must be imposed as a caution
measure.
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Free air method seems to be very efficient, but there
are a few inconvenient aspects. The air bubbles must be
deliberately introduced into the hydraulic circuit, in
order to control the volume fraction of gas. The
considered circuit is open; therefore a compressor might
continuously supply the free air, which results in an
increase of electrical power consumption. The use of
free air is recommended only in installations where the
configuration doesn’t allow air to accumulate in
different sections of the discharge duct.
We will attempt to extend our study on free air
protection method in the case of non homogeneous
biphasic movement. Even if the hypothesis of
homogeneous flow is enough accurate for the pressure
variation determination which interests from the pipe
mechanical resistance point of view, the consequences
of biphasic movement are complex.
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Anca Constantin, Claudiu Stefan Nitescu
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