fluid transients in complex systems with air … transients in complex systems with air entrainment...
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FLUID TRANSIENTS IN COMPLEX SYSTEMS
WITH AIR ENTRAINMENT
NGUYEN DINH TAM (B.Eng., HCMUT)
A THESIS SUBMITTED
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2009
i
ACKNOWLEDGEMENTS
I am enormously grateful to my supervisors at National University of
Singapore: Associate Professor Lee Thong See, and Associate Professor Low
Hong Tong, for their personal support and encouragement as well their
guidance in this study. Their advice and support played an important role in
the success of this thesis. I wish to thank my former supervisors at Ho Chi
Minh City University of Technology: Associate Professor Nguyen Thien
Tong, and Associate Professor Le Thi Minh Nghia, for their encouragement.
I wish to especially acknowledge Miss Koh Jie Ying, and Mr. Neo
Wei Rong, Avan for their cooperation in the experiment study.
I am very grateful acknowledges the financial support of the National
University of Singapore. I would like to thank the Fluid mechanics group
members and graduate students for their invaluable assistance and friendship
during this study.
I especially thank my flat-mates Nguyen Khang, The Cuong and Khoi
Khoa for helping me overcome difficulties in my daily life during my PhD
study.
Gratitude is also extended to Associate Professor Loh Wai Lam for his
help, and support.
I wish to dedicate this thesis to my lovely wife Lien Minh and my son
Huu Loc. I would also like to dedicate this work to my family, especially my
mum and dad. I will always be thankful to them for their huge support,
encouragement and love.
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CONTENTS
ACKNOWLEDGEMENTS .............................................................................i
CONTENTS .....................................................................................................ii
SUMMARY ......................................................................................................v
LIST OF TABLES.........................................................................................vii
LIST OF FIGURES.......................................................................................vii
LIST OF SYMBOLS .....................................................................................ix
LIST OF ABBREVIATIONS ..................................................................... xii
CHAPTER 1 INTRODUCTION....................................................................1
1.1. BACKGROUND ..............................................................................1
1.2. SCOPE AND OBJECTIVES...........................................................11
1.3. ORGANISATION OF THESIS.......................................................12
CHAPTER 2 LITERATURE REVIEW......................................................13
2.1. INTRODUCTION ..........................................................................13
2.2. WATER HAMMER THEORY AND PRACTICE .........................13
2.2.1. Numerical solutions for 1-D water hammer equations ....................17
2.2.2. Quasi-two-dimensional water hammer simulation ..........................20
2.2.3. Practical and research needs in water hammer ................................22
2.3. FLUID TRANSIENT WITH AIR ENTRAINMENT .....................25
2.4. FLUID TRANSIENT WITH VAPOROUS CAVITATION AND
COLUMN SEPARATION ..............................................................31
2.4.1. Single vapor cavity numerical models .............................................32
2.4.2. Discrete multiple cavity models.......................................................33
iii
2.4.3. Shallow water flow or separated flow models.................................37
2.4.4. Two phase or distributed vaporous cavitation models.....................38
2.4.5. Combined models / interface models...............................................41
2.4.6. A comparison of models ..................................................................42
2.4.7. State of the art - the recommended models......................................43
2.4.8. Fluid structure interaction (FSI).......................................................46
2.5. SUMMARY.....................................................................................47
CHAPTER 3 FLUID TRANSIENT ANALYSIS METHOD............... 50
3.1. INTRODUCTION ..........................................................................50
3.2. GOVERNING EQUATIONS FOR TRANSIENT FLOW..............50
3.3. VARIABLE WAVE SPEED MODEL............................................51
3.4. FRICTION FACTOR CALCULATION.........................................57
3.5. NUMERICAL METHOD................................................................60
3.6. BOUNDARY CONDITIONS .........................................................64
3.7. COMPUTATION OF PUMP RUN-DOWN CHARACTERISTICS
.........................................................................................................66
CHAPTER 4 VALIDATION OF THE NUMERICAL MODEL ......... 71
4.1. INTRODUCTION ..........................................................................71
4.2. COMPARISON BETWEEN EXPERIMENTAL AND
NUMERICAL RESULT..................................................................71
4.2.1. Test rig and instrumentation ...........................................................71
4.2.2. Results and discussion .....................................................................73
4.3. COMPARISON BETWEEN THE RESULTS FROM VARIABLE
WAVE SPEED MODEL AND PUBLISHED RESULTS..............77
4.4. SUMMARY.....................................................................................80
CHAPTER 5 NUMERICAL MODELLING AND COMPUTATION OF
FLUID TRANSIENT IN COMPEX SYSTEM WITH AIR
ENTRAINMENT ..........................................................................................82
5.1. INTRODUCTION ..........................................................................82
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5.2. GRID INDEPENDENCE TEST......................................................83
5.3. WATER HAMMER WITH AIR ENTRAINMENT ......................87
5.4. FLUID TRANSIENT WITH GASEOUS CAVITATION ..............94
5.5. SUMMARY...................................................................................101
CHAPTER 6 EXPERIMENTAL STUDY OF CHECK VALVE
PERPORMANCES IN FLUID TRANSIENT WITH AIR
ENTRAINMENT ..................................................................103
6.1. INTRODUCTION ........................................................................103
6.2. TEST RIG, INSTRUMENTATION AND TEST METHOD.......105
6.3. RESULTS AND DISCUSSION....................................................110
6.3.1. Pressure surge analysis ..................................................................110
6.3.2. Dynamic characteristics .................................................................117
6.3.3. Dimensionless dynamic characteristics .........................................120
6.4. SUMMARY...................................................................................118
CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS .........124
7.1. CONCLUSIONS ...........................................................................124
7.2. RECOMMENDATIONS FOR FUTURE WORK .......................125
REFERENCES.............................................................................................128
PUBLICATIONS .........................................................................................145
APPENDICES ..............................................................................................146
Appendix A: Experimental setup specifications............................................146
Appendix B: Chaudhry et al. (1990) experimental setup specifications. ......147
Appendix C: Technical data for the simulation of pumping systems............148
v
SUMMARY
Fluid transient analysis is commonly based on the assumption of no air in the
liquid. In fact, air entrainment, trapped air pockets, free gas, and dissolved
gases frequently present in the pipeline. The effects of entrapped or entrained
air on pressure transient in pipeline systems can be either beneficial or
detrimental; the outcome highly depends on the characteristics of the pipeline
concerned and the nature and cause of the transient. This thesis presents a
variable wave speed model which can improve the computational and
modeling of fluid transients in pipelines with air entrainment. By using the
variable wave speed model, wave speed is calculated depending on the local
pressure and the local air void fraction at any local point along the pipeline.
Therefore, wave speed was no longer constant as in the constant wave speed
model, it varied along the pipeline and varied in time. Free gas in fluid and
released/absorbed gas from gaseous cavitation is modeled. The variable wave
speed model is validated by comparison the numerical results with
experimental results and published results.
The variable wave speed model was then applied to investigate the
fluid transient with air entrainment in the pumping system. The numerical
results showed that entrained, entrapped or released gases amplified the first
pressure peak, increased surge damping and produced asymmetric pressure
surges with respect to the static head. These results are consistent with the
experimental and field data observed by other investigators. The findings
show that even with a very small amount of air entrainment in the liquid; the
pressure transients are considerably different from the case of pure liquid.
vi
Hence the inclusion of the effects of air entrainment can improve the accuracy
of fluid transient analysis.
In addition, we also study the mechanisms of the effects of air
entrainment on the pressure transient. To explain the increase in peak pressure,
the study suggested that the higher pressure peak is caused by the lapping of
the effects of two factors: the delay wave reflection at reservoir and the change
of wave speed. We also study experimentally the check valve performances in
fluid transients with air entrainment. The experimental study presents the
comparison of the dynamic behaviour of difference types of check valve under
pressure transient condition, and three useful methods to evaluate the pressure
transient characteristics of check valves.
In this thesis, the investigation of pressure transient was restricted to
the complex system without the installation of pressure surge protection
devices such as air vessels, air valves, surge tanks etc. In practical systems,
these devices are used to protect the system under excessive pressure transient
conditions. The ability of these hydraulic components in pressure surge
suppressions should be affected by air entrainment. The variable wave speed
model can be applied to carry out these further investigations.
Keywords: Pressure transient, Air entrainment, Variable wave speed,
Check valve
vii
LIST OF TABLES
Table. 5.1. Grid independence test result .......................................................86
LIST OF FIGURES
Fig. 3.1. Angle between horizontal direction and fluid velocity direction ....51
Fig. 3.2. Computational grid .........................................................................62
Fig. 3.3. Schematic diagram of typical pumping system ..............................64
Fig. 3.4. Boundary condition at pump ...........................................................65
Fig. 3.5. Boundary condition at reservoir.......................................................66
Fig. 4.1. Hydraulic schematic of the pumping system ..................................73
Fig. 4.2. Transient pressure measured from the experiment at check valve 74
Fig. 4.3. Transient pressures predicted from present method at check valve 74
Fig. 4.4. Effects of air content on maximum and minimum pressure head ..75
Fig. 4.5. Comparison between experimental resutls and numerical results...77
Fig. 4.6. Schematic of experiment by Chaudhry et al. (1990) ........................78
Fig. 4.7. Comparison of computed and experimental results at Station 1 .......79
Fig. 4.8. Comparison of computed and experimental results at Station 2 .......79
Fig. 5.1. Pumping station pipeline profile ....................................................84
Fig. 5.2. Pressure transient at check valve using different grid resolution ...84
Fig. 5.3. The change of pressure value with grid resolution ........................85
Fig. 5.4. Pipeline contour for pumping station ..............................................86
Fig. 5.5. Pressure head downstream of pump ..............................................88
Fig. 5.6. Max. and min. pressure head along pipeline ...................................89
Fig. 5.7. Wave speed with different initial air void fractions .......................90
Fig. 5.8. Air void fraction at check valve.......................................................91
Fig. 5.9. Pressure head with different initial air void fractions ....................91
Fig. 5.10. Max. and min. pressure head along pipeline ...................................92
Fig. 5.11. Effects of air content on max. and min. transient pressure head ...93
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Fig. 5.12. Pressure head downstream of pump ...............................................97
Fig. 5.13. Pressure head with different initial air void fraction .....................97
Fig. 5.14. Pressure head of first pressure peak with initial air void fraction ...98
Fig. 5.15. Variation of air void fraction with initial value ε0 = 0.001 ............98
Fig. 5.16. Variation of wave speed with initial air void fraction ε0 = 0.001 ..99
Fig. 5.17. Pressure transient without the effects of gas release (ε0 = 0.001) .100
Fig. 5.18. Pressure transient with the effects of gas release (ε0 = 0.001) .....100
Fig. 5.19. Maximum and minimum pressure head long pipeline ..................101
Fig. 6.1. Hydraulic schematic of the pumping system ...............................106
Fig. 6.2. Experimental sequence flowchart .................................................107
Fig. 6.3. Check valves used in the test and test section ..............................108
Fig. 6.4. Pressure transient in horizontal orientation of ball check valve ....110
Fig. 6.5. Pressure transient in horizontal orientation of swing check valve 111
Fig. 6.6. Pressure transient in horizontal orientation of piston check valve 111
Fig. 6.7. Pressure transient in horizontal orientation of nozzle check valve112
Fig. 6.8. Pressure transient in horizontal orientation of double flap check
valve ..............................................................................................112
Fig. 6.9. Pressure transient in vertical orientation of ball check valve ......113
Fig. 6.10. Pressure transient in vertical orientation of swing check valve .....113
Fig. 6.11. Pressure transient in vertical orientation of piston check valve ...114
Fig. 6.12. Pressure transient in vertical orientation of nozzle check valve ..114
Fig. 6.13. Pressure transient in vertical orientation of double flap check valve
.......................................................................................................115
Fig. 6.14. Dynamic characteristics chart in horizontal orientation ...............117
Fig. 6.15. Dynamic characteristics chart in vertical orientation.....................118
Fig. 6.16. Dimensionless dynamic characteristics in horizontal orientation 120
Fig. 6.17. Dimensionless dynamic characteristics in vertical orientation .....121
ix
LIST OF SYMBOLS
a wave speed
A cross-sectional area of pipe
A1, A2, A3 constants for pump H-Q curve
B1, B2, B3 constants for pump T-Q curve
cl parameter describing pipe constraint
C1, C2, C3 constants for pump η-Q curve
D Inner diameter of pipe
E modulus of elasticity
e local pipe wall thickness
ƒ friction factor
ƒl local equivalent loss factor
g gravitational acceleration
H gauge piezometric pressure head
i node along the pipeline
I pump set moment of inertia
k time level
K bulk modulus of elasticity
Kloss total local loss factor
Km local loss factor due to pipe features
Kf local loss factor due to nature of flow
Ka, Kr time delay factors
L pipeline length
n the polytropic index
np number of pumps operating in parallel
N total number of node points
Nik pump speed in rpm
Np number of transient periods
P pressure inside the pipe
x
Pg saturation pressure of the liquid
Po reference absolute pressure
Pv vapour pressure of the liquid
Q fluid flow rate
R C+ line intercept on x-axis
Re Reynold number
S C- line intercept on x-axis
T pump torque
t time
v local radial velocity
V cross-sectional average velocity
VR reverse velocity
Z elevation of the pipe centerline
x distance along pipeline
Greek symbols
αga gas absorbed fractional papameter
αgr gas released fractional parameter
αvr gas released fractional parameter at vapour pressure
∆t time step
∆tk time step at k
th time level
∆x node point distance along pipeline
ε fraction of gas in liquid
εo initial air void fraction
εg fraction of dissolved gas in liquid
εv fraction of released gas at vapour pressure
η pump efficiency
θ angle between horizontal direction and fluid velocity direction
ρ density of fluid
ρg density of gas
ρl density of liquid
xi
τ shear stress
τw shear stress at the pipe wall
ν kinematic viscosity
Ψg volume of gas
Ψl volume of liquid
Ψt total volume of gas and liquid
Subscripts
e equivalent
g gas
i node point
l liquid
o initial
t total
T temporary value
xii
LIST OF ABBREVIATIONS
CFL Courant-Friedrichs-Lewy
DGCM discrete gas cavity model
DVCM discrete vapor cavity model
FD finite difference
FSI fluid structure interaction
FV finite volume
GIVCM generalized interface vapor cavity model
MOC method of characteristics
NPSH net positive suction head
TVD total variation diminishing
1
CHAPTER 1
INTRODUCTION
1.1. BACKGROUND
During the operations of complex fluid systems, such as pumping installation,
oil-gas pipeline system, and nuclear power plant, unsteady and transient flow
conditions will be inevitably encountered. Pressure transients in pipeline
systems are caused by fluid flow interruption from operational changes such
as starting/stopping of pumps, changes to valve setting, changes in power
demand, etc. Consequently, there are unexpectedly high pressure surges
occurring in the pipeline, these pressure surges may cause the damage/collapse
of the pipeline and hydraulic components, devices in the system. One typical
case of the fluid transient accident is the burst pipe of the Oigawa power
station in 1950 in Japan (Bonin, 1960) in which three workers died. The plant
was designed in the early 20th century. A fast valve-closure due to the
draining of an oil control system during maintenance caused an extremely
high-pressure water hammer wave that split the penstock open. The resultant
release of water generated a low-pressure wave resulting in substantial column
separation that caused crushing (pipe collapse) of a significant portion of the
upstream pipeline. Many more severe accidents caused by fluid transient is
reported and investigated by Jaeger (1948), Parmakian (1985), Kottmann
(1989), De Almeida (1991) and Ivetic (2004). Careful considerations are thus
required in the system design stages to make sure that the unsteady fluid
system operations do not give rise to unacceptable flow and/or excessive
2
pressure transient conditions. Suitable methods for system control must be
designed to avoid such severe flow situation.
The principal use of transient analysis, both historically and present
day, is the prediction of peak positive and negative pressures in pipe systems
to aid in the selection of appropriate strength pipe materials and
appurtenances, and to design effective transient pressure control systems.
Therefore, computational and modelling fluid transient in complex systems
has been attracted research efforts in recent years (Borga et al., 2004; Covas et
al., 2003; Lauchlan et al., 2005; Wahba, 2006; 2008; Li et al., 2008; Afshar
and Rohani, 2008; Liu, 2009).
Fluid transient analysis is commonly based on the assumption that
there is no air in the liquid. In fact, air entrainment, trapped air pockets, free
gas, and dissolved gases frequently present in the pipeline. Air bubbles will
be evolved from the liquid during the passage of low-pressure transients.
When the liquid is subject to high transient pressure, the free gas will be
compressed and some may be dissolved into the liquid. The process is highly
time- and pressure- dependent. The effects of entrapped or entrained air on
pressure transient in pipeline systems can be either beneficial or detrimental;
the outcome highly depends on the characteristics of the pipeline concerned
and the nature and cause of the transient. The previous studies show that
reasonable predictions of initial pressure surges are obtained by including gas
release. However, the existence of entrained air bubbles within the fluid,
together with the presence of pockets of air complicates the analysis of the
transient pressures and makes it increasingly difficult to predict the true effects
on surge pressures.
3
Fluid transient is unsteady flow in pipe which is followed by the
change in the flow rate condition. Unsteady or transient flows may be initiated
by the system operator, be imposed by an external event, be caused by a badly
selected component or develop insidiously as a result of poor maintenance.
The causes of unsteady and transient flows in fluid systems can be
summarized as follow:
• Uncontrolled pump trip, often resulting from a power failure. The
magnitude of transient pressure caused by a sudden pump stop can be
significant for low-pressure pipelines whose initial section goes uphill
for a certain extent.
• Check valve slam.
• Rapidly closure of pump delivery valves.
• Valves and similar flow control devices anywhere in the system can
initiate unwelcome fluctuations in pressure and flow.
• The most serious pump-start problem is in system in which borehole
and submerged deep well pumps with check valves mounted at ground
level.
• Pipeline supports are a matter of compromise.
• The potential for resonance to occur should also be considered.
• Changing elevation of reservoir.
• Waves on a “reservoir” or surge tank.
• Vibration of impellers or guide vanes in pumps.
• Suction instability due to vortexing.
4
• Unstable pump characteristics.
If fluid transient happens in complex systems, unacceptable conditions
or failure can be created. Some of these fluid transient events can be predicted
and controlled by designer and plant operator, but other events, such as power
failure or self-excited resonance can be unplanned and possibly unexpected.
Even though, designer should still assess the risks for any unacceptable
conditions that may arise. Some of unacceptable conditions may be listed
below:
• Pressure too high – leading to permanent deformation or rupture of the
pipeline and components; damage to joints, seals and anchor blocks;
leakage out of the pipeline, causing wastage, environment
contamination and fire hazard.
• Pressures too low – may cause collapse of the pipeline; leakage into
the line at joints and seals under sub-atmospheric conditions;
contamination of the fluid being pumped; fire hazard with some fluids
if air is sucked in.
• Reverse flow – causing damage to pump seals and brush gear on
motors; draining of storage tanks and reservoirs.
• Pipeline movement and vibration; overstressing and failure of
supports. Leading to failure of the pipe; mechanical damage to
adjacent equipment and structures.
• Low flow velocity – mainly a problem in slurry lines, causing
settlement of entrained solids and line blockage.
5
Surge pressure is defined as the rapid change in pressure as
consequence of fluid transient in a pipeline. The surge pressure can be
dangerously high if the change of flow rate is too rapid. The excessive
pressure surges may cause the collapse of the pipeline or the damage of the
hydraulic equipments in the system. Therefore, in order to protect complex
systems from severe accidents, the transient or unsteady behaviour of the
systems needs to be analyzed, and suitable surge protection devices and
operation process need to be proposed to control the fluid transient conditions
at the design stage. In short, there are three very important reasons to carry out
an analysis of the fluid transient in complex systems:
• To protect the pipe network against abnormal or faulty conditions that
can provoke too high or too low pressures which can eventual cause
pipe ruptures with fluid leakage or contamination and indirect hazards.
• To verify the hydraulic behaviour of both the overall network and of its
each component for different conditions, including the transient
regimes (e.g. pump start-up or trip-off and valve or gate manouevers)
due to operational needs.
• To implement advanced operational control techniques for the pipe net
work, both off-line and on-line, in order to minimize energy and fluid
losses or to improve the system capacity and the system water quality.
The tasks of a transient analysis usually include:
• Evaluate and modify the pipeline wall thickness distribution
determined by the steady state hydraulic design.
• Determine the pressure class of station piping components.
6
• Provide the peak pressure at key locations for anchor and piping
support designs.
• Determine the pump ramp time and valve travel time.
• Predict system performance under upset conditions.
• Identify the worst transient scenarios.
• Determine the locations and set points of safety (pressure relief)
devices.
• Simulate pipeline operations.
• Optimize the pipeline shutdown and restart sequences.
There is a wealth of literature available addressing the study of fluid
transients or ‘water hammer’, the most notable work is of Wylie and Streeter
(1978). Many hydraulics textbooks provide a useful elementary overview of
the background theory (e.g. Nalluri and Featherstone, 2001) for non-specialist
civil engineers. The works of Thorley (2004) provide, in case of the former,
guidelines for computational formulations, and in the latter, a broader
descriptive background with practical case studies.
The transient flow in a pipeline can be divided into three phases: water
hammer, cavitation and column separation. In the water hammer phase the
release of dissolved gas is small and the wave speed depends on the void
fraction, which in turn depends on the local pressure. In the cavitation phase,
gas bubbles are dispersed throughout the liquid owing to the reduction of the
local transient pressures to the vapour pressure of the liquid. The liquid boils
at that pressure and the local pressure will not drop further. The liquid in this
7
phase behaves like a gas-liquid mixture. Depending upon the pipeline
geometry and velocity gradient, the gas bubbles may become as large as to fill
the entire cross-section of the pipe. This is the column separation phase.
Fluid transient analysis is commonly based on the assumption of no air
in the water. However, air entrainment, trapped air pockets, free gas, and
dissolved gases frequently present in the pipeline. Air in pressurized system
comes from three primary sources. The first source of air is trapped air pockets
at the top of the pipe cross-section at high points along the pipe profile. Prior
to start-up, pipeline is full of air. As the line fills, much of this air will be
removed through hydrants, faucets, etc. However, a large amount of air will
still be trapped at high points since air is lighter than water. This air will
continuously be added due to the progressive upward migration of pockets of
air as the system operation continues. The second source of air is free gas,
dissolved gases in the flow. For example, water contains approximately 2%
dissolved air by volume (Fox, 1977). During system operation, the entrained
air can be evolved from the liquid or compressed, dissolved into the liquid due
to the pressure transient. The third source of air is that which enters through
mechanical equipments. This air may be forced into the system as a result of:
falling jets of fluid from the inlet into the sump; attached vortex formation;
and the adverse flow path towards the operating pump. Air may also be
admitted through packing, valves, air vessel, etc. under vacuum conditions. In
short, air always presents in a pressurized pipeline.
The pockets of air accumulating at a high point can result in a line
restriction which increases head loss, extends pumping cycles and increases
energy consumption. As the air pockets grow, the fluid velocity will be
8
increased and one of the following two phenomena will occur. The first
possibility is a total flow stoppage. As the flow decreases in a pipeline due to
the present of air-entrainment, the pumps are forced to work harder and are
less efficient; this could result in a total system blockage. The second
possibility is that all or part of the pocket would suddenly dislodge and be
pushed downstream. The sudden and rapid change in fluid velocity when the
pocket dislodges and is then stopped by another high point, can lead to a high
pressure surge. Under low pressures, the phenomenon of gas release, or
cavitation, creates vapour cavities which, when swept with the flow to
locations of higher pressure or subject to the high pressures of a transient
pressure wave, can be collapsed suddenly and creating further ‘impact’
pressure rise, thus potentially causing severe damage to the pipeline. In normal
pipeline design, cavitation risk is to be avoided as far as is possible or
practicable. The work of Burrows and Qui (1995) highlighted that the
presence of air pockets can be further detrimental to pipelines subject to un-
suppressed pressure transients and localized caviation, such that substantial
underestimation of the peak pressures might result.
Generally, fluid transient with air entrainment are considerably
different from those computed according to models with no air. Numerous
practical and numerical experiments show several distinct characteristic
differences of fluid transients with and without air entrainment. In general, the
first pressure peak with entrained air is found to be higher than that predicted
by models with no air. The pressure periods are longer when air entrainment is
considered. The pressure surges are asymmetric with respect to the static head,
while the pressure surges are symmetric with respect to the static head for
9
models with no air presented in the flow. The pressure transient damping with
air entrainment is faster than the damping with no air entrainment.
Computational and modeling of fluid transient with air entrainment has
been carried out by many researchers together with practical experiments and
field measurements. Most fluid transient studies based on single fluid models
use the method of characteristics to solve the resulting finite difference
equations which are derived from the continuity equation and momentum
equation of one dimension fluid flow. The governing equations of motion
(continuity and momentum) are expressed in terms of changes over finite
intervals in space along the pipeline (∆x) and time (∆t). The resulting finite
difference equations can then be solved by the method of characteristics
(MOC), derivations being widely available (Wiley and Streeter, 1978;
Thorley, 2004). For the single fluid problem, this approach is normally
acceptable for predictive design. Many researches introduce refinements to the
single fluid models to improve the fluid transient prediction in terms of shape
of the pressure peaks, the frequency of the oscillations and the rate of decay.
These refinements include making better allowance of energy dissipation
(non-steady friction) in the mathematical formulation (Abreu and Almeida,
2000; Prado and Larreteguy, 2002), and non-elastic behaviour (Borga et al.,
2004 and Covas et al., 2003). Further refinement is called for to account for
the cavitation process explicitly, whereby vapour filled voids will grow and
callapse as the pressure changes.
To consider the effect of air entrainment, a variety of approaches like
one-fluid model and two-fluid model coupled with numerous numerical
schemes have been introduced, for example, the concentrate vaporous cavity
10
model (Brown, 1968 amd Provoost, 1976), the air release model (Fox, 1972
and Wylie, 1980), and homogeneous gas-liquid model (Chaudhry, 1990).
Fluid transient result from these models shows reasonable prediction of
pressure transient behaviours in pipeline systems. When air is entrained such
that the gas void fraction is significant and two phase motion occurs, it
become necessary to introduce multi-phase modeling (Huygens et al., 1998;
Fujii and Akagawa, 2000 and Lee et al., 2004). Lauchlan et al. (2005) showed
that the predictions from above models may be regarded as “fit for purpose” in
the sense they indicate that unacceptable fluid transient conditions will occur.
However, the occurrence of discrepancies between the computational
predicted results and reality points to the need for further development of two-
phase transient flow models.
The studies of the increase in the first peak pressure during the
pressure transient with air entrainment also have the attribution of many
researchers. Dawson and Fox (1983) explained that the accumulation of
relatively minor changes in flow during the period of the transient had a
significant effect upon the peak pressures causing them to rise, while Jonsson
(1985) attributes the results to the compression of “an isolated air cushion” in
the flow field after valve closure. More recent studies (Kapelan et al., 2003;
Covas et al., 2003) have also identified peak pressure enhancement and
transient distortions from suspected air pocket formation.
Air entrainment has substantial effects on fluid transients. However the
existence of vapour cavity, trapped air pockets and entrained free gas bubbles
greatly complicates fluid transient analysis by making transient wave speed a
function of transient pressure. In practice, the analysis is also more difficult
11
due to the lack of information such as the location and size of trapped air
pockets in the pipelines, the amount of free air bubbles distributed throughout
the liquid, and the rate of gas release and absorption in the liquid as a function
of pressure and time. With few exceptions, proper phasing and attenuation of
subsequent predicted peaks remained a question. The process of gaseous
diffusion in a closed conduit subjected to unsteady flow is not still fully
understandable. The difficulty of the analysis is also due to the random nature
of bubble nucleation, coalescence and growth in turbulent flow fields.
1.2. SCOPE AND OBJECTIVES
According to Bergant et al., (2006), the inability of pressure waves to
propagate through a vapour bubble zone is a major feature distinguishing the
flow with vaporous cavitation from the flow with gaseous cavitation. This
distinction makes the development of a numerical model which can solve fluid
transient problem in all circumstances become very challenging. In this thesis,
we focus on study fluid transient in complex systems with air entrainment and
released gas. The objectives of this study are:
i. To develop a variable wave speed model for analyzing the fluid
transient in complex systems with air entrainment. The proposed
model includes the effects of free gas in the liquid and released gas
on the pressure transient in the pipeline. This model is solved
numerically by using the method of characteristics.
ii. To validate the proposed variable wave speed model by
experimental and published results.
12
iii. To evaluation of the effects of free entrained air and released gas in
the fluid transient in typical pumping systems due to pump trip
using the variable wave speed model.
iv. To study check valve performances in fluid transient with air
entrainment by experiments.
The main targets are systems with very little air as these situations,
which in most circumstances, will result in high transient pressures.
1.3. ORGANISATION OF THESIS
The importance and necessity of the study, as well as the general background
of the study are discussed in Chapter 1. In Chapter 2, a detailed review of
literature of fluid transient in pipeline systems is presented. Based on the
literature review, the scope of the present study is outlined. In Chapter 3, a
variable wave speed model for computational and modelling fluid transient in
complex systems with air entrainment and released gas is introduced. The
numerical scheme adopted for the developing variable wave speed model is
also presented in this chapter. In Chapter 4, numerical result from the variable
wave speed model is compared with experimental and published results to
validate the model. In Chapter 5, analysis of fluid transient in typical pumping
systems with air entrainment is presented. In Chapter 6, experiment study of
the check valve performances in fluid transient with air entrainment is
provided. Finally, in Chapter 7, some conclusions from this study, together
with some suggestions for future works are drawn.
13
CHAPTER 2
LITERATURE REVIEW
2.1. INTRODUCTION
Fluid transient has been an important and attractive research topic since the
turn of the nineteenth century. Considering liquid transients in a pipeline
system, there are two different types of flow regimes. The first is referred to as
the water hammer regime (or “no cavitation” case) where the pressure is above
the vapor pressure of the liquid. The second is the cavitation regime where the
pressure is equal to the liquid vapor pressure. In the cavitation regime,
cavitation can occur in three situations: gaseous cavitation, vaporous
cavitation and column separation. This literature review is written with close
reference to the reviews by Ghidaoui et al. (2005), Lauchlan et al. (2005) and
Bergant et al. (2006). Firstly, a review of water hammer theory and practice is
presented. A review of fluid transient with air entrainment, which also covers
gaseous cavitation due to a close association between gaseous cavitation and
air entrainment, is provided. Finally, a review of fluid transient with vaporous
cavitation and column separation is presented.
2.2. WATER HAMMER THEORY AND PRACTICE
According to Ghidaoui et al. (2005), the problem of water hammer was first
studied by Menabrea (1885). The following researchers like Weston (1885),
Carpenter (1893) and Frizell (1898) attempted to develop expressions relating
pressure and velocity changes in a pipe. Frizell (1898) was successful in this
14
endeavor. However, similar work by Joukowsky (1898) and Allievi (1903,
1913) attracted greater attention. Joukowsky produced the best known
equation in transient flow theory called the ‘‘fundamental equation of water
hammer.’’
Joukowsky’s fundamental equation of water hammer is given as
follows:
VaP ∆±=∆ ρ or g
VaH
∆±=∆ (2.1)
where a is the acoustic (waterhammer) wave speed, P = ρg(H-Z) is the
piezometric pressure, Z is the elevation of the pipe centerline from a given
datum, H is the piezometric head, ρ is the fluid density, V is the cross-sectional
average velocity, and g is the gravitational acceleration. The positive sign in
Eq. (2.1) is applicable for a water-hammer wave moving downstream while
the negative sign is applicable for a water-hammer wave moving upstream.
The combined efforts of Allievi (1903, 1913), Jaeger (1933, 1956),
Parmakian (1955), Streeter and Lai (1963), and Streeter and Wylie (1967)
have resulted in the following classical mass and momentum equations for
one-dimensional water-hammer flows
02
=∂
∂+
∂
∂
t
H
x
V
g
a (2.2)
04
=+∂
∂+
∂
∂w
Dx
Hg
t
Vτ
ρ (2.3)
in which τw is the shear stress at the pipe wall, D = pipe diameter. Equations
(2.2) and (2.3) are the fundamental equations for 1-D water hammer problems
and are capable to model wave propagation in complex pipe systems
15
physically. The research of Mitra and Rouleau (1985) for the laminar water
hammer case and of Vardy and Hwang (1991) for turbulent water-hammer
supports the validity of the unidirectional approach when studying water-
hammer problems in pipe systems.
The water hammer wave speed is (Joukowski, 1898; Chaudhry, 1987,
Wylie et al., 1993)
dP
dA
AdP
d
a
ρρ+=
2
1 (2.4)
where A is the cross-sectional area of the pipe.
The first term on the right hand side of Eq. (2.4) represents the effect
of fluid compressibility on the wave speed and the second term represents the
effect of pipe flexibility on the wave speed. Korteweg (1878) related the right
hand side of Eq. (2.4) to the material properties of the fluid and to the material
and geometrical properties of the pipe. As a result, Korteweg (1878)
developed a formula to estimate the wave speed:
( )( )eDEK
Ka
//1
/
+=
ρ (2.5)
where K is the bulk modulus, ρ is the mass density, E is the Young’s modulus
of the pipe wall material, D is the inner diameter of the pipe, and e is the wall
thickness.
The modeling of wall friction is essential for practical applications that
warrant transient simulation well beyond the first wave cycle. Many wall shear
stress models have been introduced in transient analysis. In quasi-steady wall
shear stress models, it is assumed that phenomenological expressions relating
16
wall shear to cross-sectionally averaged velocity in steady-state flows remain
valid under unsteady conditions. That is, wall shear stress expressions, such as
the Darcy-Weisbach and Hazen- Williams formulas, are assumed to always
hold during a transient. Indeed, discrepancies between numerical results and
experimental and field data are found whenever a steady-state based shear
stress equation is used to model wall shear in water hammer problems (e.g.,
Vardy and Hwang, 1991; Axworthy et al., 2000).
Various empirical-based corrections to quasi-steady wall shear models
have been introduced by Daily et al. (1956), Brunone et al. (1991), Vardy and
Brown (1996), and Axworthy et al. (2000). Although both the Darcy-
Weisbach formular and Brunoe et al. (1991) model cannot produce enough
energy dissipation in the pressure head traces, the model by Brunone et al.
(1991) is quite successful in producing the necessary damping features of
pressure peaks. Vardy and Brown (1996) rely on steady-state-based turbulence
models to adequately represent unsteady turbulence. However, modeling
turbulent pipe transients is currently not well understood. Therefore, the
reliability of the model by Vardy and Brown (1996) is limited.
The mechanism that accounts for the dissipation of the pressure head is
addressed by Ghidaoui et al. (2002) who found that the additional dissipation
associated with the instantaneous acceleration based unsteady friction model
occurs only at the boundary due to the wave reflection.
Physically based wall shear models are a class of unsteady wall shear
stress model, based on the analytical solution of the unidirectional flow
equation. The analytical approach of Zielke (1968) is applied for laminar
flows, and later is extended for turbulent flows by Vardy and Brown (1996).
17
The results from the proposed approximate models are in good agreement with
both laboratory and numerical experiments over a wide Reynolds numbers and
wave frequencies range.
2.2.1. Numerical solutions for 1-D water hammer equations
In general, the equations governing 1D water hammer can not be solved
analytically. Therefore, numerical techniques are used to find approximated
solution. The method of characteristics (MOC), which has the desirable
attributes of accuracy, simplicity, numerical efficiency, and programming
simplicity is the most popular numerical method. Other techniques that have
also been applied to solve water hammer equation include the wave plan,
finite difference (FD), and finite volume (FV) methods (Ghidaoui et al. 2002).
A significant development in the numerical solution of hyperbolic
equations was introduced by Lister (1960). Lister study shows that the fixed-
grid MOC was an easier approach since it provides full control over the grid
selection and enabling the computation of both the pressure and velocity fields
in space at constants time. Fixed-grid MOC has since been used with great
success to calculate transient conditions in pipe systems and networks. The
fixed-grid MOC requires that a common time step (∆t) be used for the solution
of the governing equations in all pipes. However, pipes in the system tend to
have different lengths and sometimes different wave speeds, making the
Courant condition (Courant number Cr = a∆t/∆x ≤ 1) impossible to satisfy
exactly if a common time step ∆t is used. This discretization problem can be
addressed by interpolation techniques, or artificial adjustment of the wave
speed or a hybrid of both.
18
Various interpolation techniques have been introduced to deal with this
discretization problem. Lister (1960) used linear space-line interpolation to
approximate heads and flows at the foot of each characteristic line. Trikha
(1975) suggested using different time step for each pipe in order to use large
time steps, resulting in shorter execution time and the avoidance of spatial
interpolation error. However, Trikha’s approach requires interpolation at the
boundaries, which can be inaccurate for complex, rapidly changing control
actions. Wiggert and Sundquist (1977) derived a single scheme that combines
the classical space-line interpolation with reach-out in space interpolation.
However, this scheme generates more grid points and requires longer
computational time and computer storage. Furthermore, an alternative scheme
must be used to carry out the boundary computations. The reach-back time-
line interpolation scheme, developed by Goldberg and Wylie (1983), uses the
solution from m previously calculated time levels. Lai (1989) combined the
implicit, temporal reach-back, spatial reach-back, spatial reachout, and the
classical time and space-line interpolations into one technique called the
multimode scheme. The multimode scheme gives the user the flexibility to
select the interpolation scheme that provides the best performance for a
particular problem. Sibetheros et al. (1991) showed that the spline technique is
well suited to predicting transient conditions in simple pipelines subject to
simple disturbances when the nature of the transient behaviour of the systems
is known in advance. The most serious problem with the spline interpolation is
the specification of the spline boundary conditions. Karney and Ghidaoui
(1997) developed ‘‘hybrid’’ interpolation approaches that include
interpolation along a secondary characteristic line, ‘‘minimum-point’’
19
interpolation, and a method of ‘‘wave path adjustment’’ that distorts the path
of propagation but does not directly change the wave speed. The resulting
composite algorithm can be implemented as a preprocessor step and thus uses
memory efficiently, executes quickly, and provides a flexible tool for
investigating the importance of discretization errors in pipeline systems.
Afshar and Rohani (2008) proposed an implicit MOC where an
element-wise definition is used for all the devices that may be used in a
pipeline system and the corresponding equations are derived in an element-
wise manner. The proposed method allows for any arbitrary combination of
devices in the pipeline system. The authors claimed that the proposed implicit
MOC is a remedy to the shortcomings and limitations of the conventional
MOC.
Beside MOC based schemes, other schemes have been developed to
solve the water hammer equation. The wave plan method approximated flow
disturbances by a series of instantaneous changes in flow condition. The by-
piecewise-constant approximation to disturbance functions implies that the
accuracy of the scheme is of first order in both space and time. Therefore, fine
discretization is required for achieving accurate solutions to water hammer
problems. Wylie and Streeter (1970) proposed the implicit central difference
method in order to allow larger time steps. However, implicit schemes
increase both the computational time and the storage requirement, moreover, it
requires a relatively quality solver. Furthermore, mathematically, implicit
methods are not suitable for wave propagation problems because they entirely
distort the path of propagation of information, thereby misrepresenting the
mathematical model. Chaudhry and Hussaini (1985) applied the MacCormack,
20
Lambda and Gabutti schemes, which are explicit, second-order finite
difference schemes, to the water hammer equations. However, spurious
numerical oscillations are observed in the wave profile. Hwang and Chung
(2002) tried to develop a scheme using the Finite Volume (FV) method for
water hammer. The application of such a scheme in practice would require a
state equation relating density to head. However, at present, no such equation
of state exists for water. Application of this method would be further
complicated at the boundaries where incompressible conditions are generally
assumed to apply.
Several approaches have been developed to deal with the
quantification of numerical dissipation and dispersion such as Von Neumann
method by O’Brian et al. (1951), L1 and L2 norms method by Chaudhry and
Hussaini (1985), three parameters approach by Sibetheros et al. (1991), and
energy approach by Ghidaoui et al. (1998).
2.2.2. Quasi-two-dimensional water hammer simulation
Quasi-two-dimensional water hammer simulation using turbulence models can
enhance the current state of understanding of energy dissipation in transient
pipe flow, provide detailed information about transport and turbulent mixing,
and provide data needed to assess the validity of 1-D water hammer models.
Examples of turbulence models for water hammer problems, their
applicability, and their limitations can be found in Vardy and Hwang (1991),
Silva-Araya and Chaudhry (2001), Pezzinga (1999) and Ghidaoui et al.
(2002). The governing equations for quasi-two-dimensional modeling can be
expressed as the following pair of continuity and momentum equations:
21
01
2=
∂
∂+
∂
∂+
∂
∂+
∂
∂
r
rv
rx
u
x
Hu
t
H
a
g (2.6)
r
r
rx
Hg
r
uv
x
uu
t
u
∂
∂+
∂
∂−=
∂
∂+
∂
∂+
∂
∂ τ
ρ
1 (2.7)
respectively, where v is the local radial velocity and τ is the shear stress. In this
set of equations, compressibility is only considered in the continuity equation.
Radial momentum is neglected therefore theses equations are only quasi-two
dimensional.
The 2-D governing equations are a system of hyperbolic-parabolic
partial differential equations. The numerical solution of Vardy and Hwang
(1991) solves the hyperbolic part of governing equations by MOC method and
the parabolic part by using finite difference method. This hybrid solution
approach has several merits. First, the solution method is consistent with the
physics of the flow since it uses MOC for the wave part and central
differencing for the diffusion part. Second, the use of MOC allows modelers to
take advantage of the wealth of strategies, methods, and analysis developed in
conjunction with 1-D MOC water hammer models. Third, although the radial
mass flux is often small, its inclusion in the continuity equation by Vardy and
Hwang is more correct and accurate physically. A major drawback of the
numerical model of Vardy and Hwang, however, is that it is computationally
demanding. The numerical solution by Pezzinga (1999) solves for pressure
head using explicit FD from the continuity equation. Once the pressure head
has been obtained, the momentum equation is solved by implicit FD for
velocity profiles. This velocity distribution is then integrated across the pipe
section to calculate the total discharge. The Pezzinga scheme is fast since it
22
decouples the continuity and momentum equations and adopts the tri-diagonal
coefficient matrix for the momentum equation. However, the scheme has a
difficulty in the numerical integration step because the approximated
integration leads to spurious oscillations in the solution for pressure.
Wahba (2006, 2008) used Runge–Kutta schemes to simulate unsteady
flow in elastic pipes due to sudden valve closure. The spatial derivatives are
discretized using a central difference scheme. Second-order dissipative terms
are added in regions of high gradients while they are switched off in smooth
flow regions using a total variation diminishing (TVD) switch. The method is
applied to both one and two dimensions water hammer formulations. Both
laminar and turbulent flow cases are simulated. Different turbulence models
are tested including the Baldwin–Lomax and Cebeci–Smith models. The
results reported in good agreement with analytical results and with
experimental data available in the literature. The two-dimensional model is
shown to predict more accurately the frictional damping of the pressure
transient. Moreover, through order of magnitude and dimensional analysis, a
non-dimensional parameter is identified to control the damping of pressure
transients in elastic pipes.
2.2.3. Practical and research needs in water hammer
Existing transient pipe flow models are derived under the premise that no
helical type vortices emerge (i.e., the flow remains stable and axisymmetric
during a transient event). Recent experimental and theoretical works indicate
that flow instabilities, in the form of helical vortices, can be developed in
transient flows. These instabilities lead to the breakdown of flow symmetry
with respect to the pipe axis. However, the conditions under which helical
23
vortices emerge in transient flows, and the influence of these vortices on the
velocity, pressure, and shear stress fields, are currently not well understood
and, thus, are not incorporated in transient flow models. Ghidaoui et al. (2005)
suggested that future research is required to accomplish the following:
• Understand the physical mechanisms responsible for the emergence of
helical type vortices in transient pipe flows.
• Determine the range in the parameter space, defined by Reynolds
number and dimensionless transient time scale over which helical
vortices develop.
• Investigate flow structure together with pressure, velocity, and shear
stress fields at subcritical, critical, and supercritical values of Reynolds
number and dimensionless time scale.
Understanding the helical vortices in transient pipe flows, and
incorporating this new phenomenon in practical unsteady flow models would
significantly reduce discrepancies in the observed and predicted behavior of
energy dissipation beyond the first wave cycle.
Current physically based 1-D and 2-D water hammer models assume
that turbulence in a pipe is either quasi-steady or quasi-laminar; and the
turbulent relations that have been derived and tested in steady flows remain
valid in unsteady pipe flows. However, the lack in-depth understanding of the
changes in turbulence during transient flow conditions lead to a difficulty for
establishing the domain of applicability of models that utilize these turbulence
assumptions and for seeking appropriate alternative models where existing
model fail. Therefore, more researches are needed to develop an understanding
24
of the turbulence behavior and energy dissipation in unsteady pipe flows.
These researches need to accomplish the following:
• Improve the ability to quantify changes in turbulent strength and
structure in transient events at different Reynolds numbers and time
scales.
• Use the understanding gained to determine the range of applicability of
existing models and to seek more appropriate alternative models to
replace those failed models.
The development of inverse water hammer techniques is another
important future research area. Inverse models have the potential to utilize
field measurements of transient events to accurately and inexpensively
calibrate a wide range of hydraulic parameters, including pipe friction factors,
system demands, and leakage.
The practical significance of the above research goals is considerable.
An improved understanding of transient flow behavior gained from such
research would permit development of transient models able to accurately
predict flows and pressures beyond the first wave cycle. Water hammer
models are becoming more widely used for the design, analysis, and safe
operation of complex pipeline systems and their protective devices; for the
assessment and mitigation of transient-induced water quality problems; and
for the identification of system leakage, closed or partially closed valves, and
hydraulic parameters such as friction factors and wave speeds. Understanding
the governing equations for water hammer research and practice and their
limitations is essential to interpret the results of the numerical models that are
25
based on these equations, for judging the reliability of the data obtained from
these models, and for minimizing misuse of water hammer models.
2.3. FLUID TRANSIENT WITH AIR ENTRAINMENT
The effects of air entrainment on the pressure transient in pumping systems
were firstly studied by Whiteman and Pearsall (1959, 1962) in their pump
shut-down tests. The study showed that even a small amount of air
entrainment in the flow could produce significant effects to the fluid transient.
Until the 1960s, a comprehensive investigation of fluid transient with air
entrainment in pipelines was not possible due to the unavailability of
computers. Until around 1960, most studies used graphical and arithmetic
procedures originally set forth Parmakian (1955). The first computer-oriented
procedures for the treatment and analysis of water hammer include work by
Lai (1961), Streeter and Lai (1963), and Van De Riet (1964).
The water hammer equations are applied to calculate unsteady pipe
liquid flow when the pressure is greater than the vapor pressure. They
comprise the continuity equation and the equation of motion. Research by
Streeter and Wylie (1967) led the world to the direct use of the method of
characteristics as a numerical method on a digital computer to provide
solutions to the water hammer equations. The method of characteristics has
been the standard solution method for solving water hammer in pipeline
systems for the last 40 years. The work of Chaiko and Brinckman (2002),
developed upon the experimental work of Lee and Martin (1999), presented a
range of applicability for the models under evaluation for differing proportions
of air to liquid. The finding is that standard MOC methods are likely to be
acceptable when the liquid fraction in the system exceeds 90%.
26
Many papers, starting in the 1970s and early 1980s, have addressed the
effects of dissolved gas and gas release on transients in pipelines (Enever,
1972; Kranenburg, 1974; Wiggert and Sundquist, 1979; Wylie, 1980; Hadj-
Taieb and Lili, 1998; Kessal and Amaouche, 2001). One of the main features
of liquids is their capability of absorbing a certain amount of gas when they
contact free surface. In contrast to vapor release, which takes only a few
microseconds, the time for gas release is in the order of several seconds. Gas
absorption is slower than gas release (Zielke et al., 1989). Gas release occurs
in several types of hydraulic systems (cooling water systems, long pipelines
with high points, oil pipelines, etc.). Dissolved gas is an important
consideration in sewage water lines and aviation fuel lines. Gases come out of
solution when the pressure drops in the pipeline. If a cavity forms, it may be
assumed that released gas stays in the cavity and does not immediately
redissolve following a rise in pressure. Pearsall (1965) showed that the
presence of entrained air or free gas reduces the wave propagation velocity
and accordingly the transient pressure variations. A significant limitation in
the numerical models proposed in each of the above studies was required,
rather arbitrary, assumptions regarding to the rate of release of gas. Dijkman
and Vreugdenhil (1969) investigated theoretically the effect of dissolved gas
on wave dispersion and pressure rise following column separation.
To consider the effect of air entrainment, the concentrate vaporous
cavity model (Brown 1968, Provoost 1976) and the air release model (Fox,
1972; Wylie, 1980) shows reasonable prediction of pressure transient
behaviours in pipeline systems. The vaporous concentrated model (Provoost
1976), which confines the vapor cavities to fixed computing sections and uses
27
a constant wave speed of the length between the cavities, produced
satisfactory results in slow transients but unstable solutions for rapidly
changing transients such as pump shutdowns. The second type is the air
release model (Fox, 1983) which assumes the evolved and free gas to be
distributed homogeneously throughout the pipeline, thereby requiring variable
wave speeds. In air release models, wave speed varies along the pipeline and
is depended on the air content and local pressure at particular point. The air
release model produced satisfactory results in pump shutdown cases but was
susceptible to numerical damping (Ewing, 1980; Jonsson, 1985).
When air is entrained such that the gas void fraction is significant and
two phase motion occurs between the water and air in bubbles, pockets and/or
voids, it become necessary to introduce multi-phase modelling. This can be
introduced at different levels (Falk and Gudmundsson, 2000; Fuji and
Akagawa, 2000; Huygens et al., 1998) ranging from a two-fluid (two
component) model which satisfies the equations of motion (conservation
equations) in each fluid concurrently, to a homogeneous flow model
(Chaudhry et al., 1990), which assumes the same velocities in each phase,
effectively requiring input of mean parameters (i.e. density and pressure wave
speed) into the normal formulation. Falk and Gudmundsson (2000) reports
that the modified MOC gives a good picture of the pressure waves but is
unable to predict void waves, a proposition also concluded by Huygens.
Lauchlan et al. (2005) showed that the predictions from above models
may be regarded as “fit for purpose” in the sense that they indicate that
unacceptable fluid transient conditions will occur. However, the occurrence of
28
discrepancies between the computational predicted results and reality points to
the need for further development of fluid transient models.
In a very recent research, Epstein (2008) introduced a convenient
integral method, which takes full account of air/water interface movement and
liquid compressibility, to calculate theoretically the pressure histories within
entrapped air bubbles in a pipeline during waterhammer transient. In this
method, the governing partial differential waterhammer equations and initial
conditions are replaced by an approximate first order system of ordinary
differential equation using a variant of the integral (moment) method. The
method is shown to be a computationally simple and efficient way of assessing
the impact of liquid compressibility on pressure rise when multiple water
columns and air pockets are present in a pipeline.
The studies of the increase in the first peak pressure during the
pressure transient with air entrainment also have the attribution of many other
researchers. Dawson and Fox (1983) reasoned that the accumulation of
relatively minor changes in flow during the period of the transient had a
significant effect upon the peak pressures causing them to rise. Jonsson (1985)
attributes the results to compression of “an isolated air cushion” in the flow
field after valve closure. Jonsson (1985) justified this by application of a
standard (constant wave speed, elastic theory) model and concluded that there
would be a lower limit of the volume of air to which the descriptor ‘air-
cushion’ is still valid. Burrows and Qiu (1995) had taken Jonsson’s finding as
an early independent validation for use of ‘air pockets’ to better explain
discrepancies between observed transients and model results. They further
suggested that a combination of the ‘variable wave speed’ and ‘discrete air
29
pocket’ approaches might provide a more rigorous model but verification
requires high quality field or laboratory data. More recent laboratory and pilot
scale studies (Kapelan et al., 2003; Covas et al., 2003) have also identified
peak pressure enhancement and transient distortions from suspected air pocket
formation. Independently, Lai et al. (2000) investigate water hammer in
presence on non-condensable gas voids (i.e. air) together with vapor cavities
and found that whilst the presence of air is generally beneficial in reducing
water hammer loads, it can result in an increase in the ‘longer term’ transient
(i.e. not the first positive pressure peak).
The experimental measurements by Van de Sande and Belde (1981)
presented pressure peak values higher than those calculated by the Joukowsky
formula. In referring to this experimental result, De Almeida (1983)
commented that though old, this apparently very simple problem had not been
resolved yet. De Almeida (1983) cited five possible reasons for obtaining large
pressures due to cavity collapse including: non-uniform velocity distribution,
unsteady friction effects, “column elastic effect”, local and point effects. The
“column elastic effect” was the result of the time of existence of the cavity not
being an integer value of 2L/a. He presented an expression for estimating the
upper bound of the overpressure in a frictionless system.
Hadj-Taieb and Lili (2000) analyzed transient flow of homogeneous
gas liquid mixture in pipelines by taking into account the pipe elasticity effect
on the pressure wave propagation. The developed models used the gas-fluid
mass ratio which is assumed to be constant and not depend on the pressure. A
conservative finite difference scheme was used in computing pressure
evolution at different equidistant sections of the pipe. The results showed that
30
including liquid compressibility and pipe wall elasticity in the theory has no
significant influence in case the gas-fluid mass ratio large enough. However,
the influence is considerable when the gas-fluid mass ratio is relatively small
or at the limit near to zero.
Chaiko and Brinckman (2002) analyzed water hammer transients in a
pipe line with an entrapped air pocket by using three different one-
dimensional models of varying complexity. The simplest model neglects the
influence of gas-liquid interface movement on wave propagation through the
liquid region and assumes uniform compression of the entrapped non-
condensable gas. In the most complex model, the full two-region wave
propagation problem is solved for adjoining gas and liquid regions with time
varying domains. An intermediate model which allows for time variation of
the liquid domain, but assumes uniform gas compression, is also considered.
Results show that time variation of the liquid domain and non-uniform gas
compression can be neglected for initial air volumes comprising 5% or less of
the initial pipe volume. The uniform compression model with time-varying
liquid domain captures all of the essential features predicted by the full two-
region model for the entire range of pressure and initial gas volume considered
in the study, and it is the recommended model for analysis of water hammer in
pipe lines with entrapped air.
Wang et al. (2003) introduced a computational model that combines
the method of characteristics and the shock wave theory to simulate the
propagation of pressure surges with the formation of an air pocket in pipelines.
The study found that the air pressure changes greatly during the early stage of
formation of an air pocket. For the case of an air pocket trapped between two
31
positive interfaces, an open surge front may be emerged from the upstream
interface and eventually reverses the upstream surge to propagate upstream as
a negative wave.
Cannizzaro and Pezzinga (2005) examined the effect of gaseous
cavitation on thermic exchange between the gas bubbles and the surrounding
liquid by means of a 2-D model. They used continuity equations for gas,
continuity equation for mixture, energy and momentum equations for the
solution. The two dimensional model of constant temperature and mass was
able to predict the experimental data only at the first set of oscillations. They
found that incorporation of thermic exchange between the gas bubbles and the
surrounding liquid into the model improved the performance of the model.
2.4. FLUID TRANSIENT WITH VAPOROUS CAVITATION AND
COLUMN SEPARATION
There have been considerable studies of column separation during water
hammer or transient events. Bergant et al. (2006) wrote an excellent and detail
literature review of waterhammer with column separation. This part of our
review presents briefly summary of Bergant’s review to show all the
significant research about fluid transient with vaporous cavitation and column
separation that has been carried out. The occurrence of low pressures and
associated column separation during water hammer events has been a concern
for much of the twentieth century in the design of pipe and water distribution
systems. The closure of a valve or shutdown of a pump may cause low
pressures during transient events. The collapse of vapor cavities and rejoinder
of water columns can generate extremely large pressure that may cause
significant damage or ultimately failure of the pipe system. Two types of
32
cavitation in pipelines are now distinguished: (i) discrete vapor cavity or local
liquid column separation (large α) and (ii) distributed vaporous cavitation or
bubbly flow (small α).
As early as 1900, Joukowsky had identified the physical occurrence of
column separation. The 1930s produced the first mathematical models of
vapor cavity formation and collapse based on the graphical method. The
identification of the various physical attributes of column separation occurred
in the mid-20th century (distributed or vaporous cavitation in the 1930s;
intermediate vapor cavities in the 1950s). These both led to a better physical
understanding of the process of column separation and ultimately laid out the
groundwork for the development of computer based numerical models. The
late 1960s saw the development of the first computer models of column
separation within the framework of the method of characteristics solution of
the water hammer equations. A variety of alternative numerical models were
developed. The most significant models that have been developed include: the
discrete vapor cavity model (DVCM), the discrete gas cavity model (DGCM)
and the generalized interface vapor cavity model (GIVCM). The first two
models are the easiest to implement. The DVCM is the most popular model
used in currently available commercial computer codes for water hammer
analysis.
2.4.1. Single vapor cavity numerical models
Single vapor cavity numerical models deal with discrete vapor cavity or local
column separations. A single cavity is used either at a boundary, at a high
point in the pipeline, or at a change in pipe slope. Most graphical solutions of
water hammer problems employed this modeling approach. Rigid column
33
theory has also been used to compute the behavior of systems with single
cavities. Streeter and Wylie (1967) presented a computer model describing
vaporous cavitation using only a single vapor cavity in a pipeline. A single
cavity was assumed to form at the point in the pipeline that first dropped to the
vapor pressure of the liquid. Weyler (1969) used a single vapor cavity at the
valve to study liquid column separation. Cavities were not permitted to form at
the other computational sections. A semi-empirical “bubble shear stress” was
proposed to predict the increased momentum loss observed under liquid
column separation conditions. A single spherical bubble was examined and
compressive dissipation was concluded to be large compared with the viscous
dissipation for a single spherical bubble. De-aerated liquid was found to
undergo much more violent opening and closing of cavities, a behavior
characteristic of distributed vaporous cavitation. Safwat (1972) also
considered the wave attenuation problem and introduced an equivalent shear
stress concept. However, Kranenburg (1974) disagreed with Weyler and
Safwat and contended that the thermodynamic behavior was essentially
isothermal (because of the small size of the bubbles) and that no dissipation
would occur due to this bubble shear stress mechanism.
2.4.2. Discrete multiple cavity models
Discrete Multiple Cavity Models includes the discrete vapor cavity models
(DVCM) and the discrete free gas cavity model (DGCM). Liquid column
separations and regions of vaporous cavitation are both modeled using discrete
cavities at all computational sections. Liquid is assumed to be in between all
computational sections and the method of characteristics is applied throughout
the pipeline, even in vaporous cavitation regions. The discrete free gas cavity
34
model (DGCM) (Wylie 1984) is similar to the discrete vapor cavity model,
with a quantity of free air assumed to be concentrated at each computational
section.
Wylie and Streeter (1993) described the DVCM in detail. Cavities are
allowed to form at any of the computational sections if the pressure drops
below the vapor pressure of the liquid. The DVCM does not specifically
differentiate between localized vapor cavities and distributed vaporous
cavitation (Simpson and Wylie 1989, Bergant and Simpson 1999). Vapor
cavities are thus confined to computational sections, and a constant pressure
wave speed is assumed for the liquid between computational sections. Upon
formation of a cavity, a computational section is treated as a fixed internal
boundary condition. The pressure is set equal to the vapor pressure of the
liquid until the cavity collapses. Both upstream and downstream discharges for
the computational section are computed, using the C+ and C– compatibility
relationships for each of the positive and negative characteristics within the
method of characteristics (MOC) solution. Wylie and Streeter (1993) showed
a comparison of the results of the DVCM and the experimental results by Li
and Walsh (1964) of an isolated cavity formation at the downstream side of a
valve. The magnitudes of the pressures were reasonably predicted, whereas the
timing of existence of column separation was not well predicted. Evans and
Sage (1983) had confidence in the DVCM and used it for the water-hammer
analysis of a practical situation. Bergant and Simpson (1999) incorporated
cavitation inception with “negative” pressure into the discrete vapor cavity
(DVCM) numerical model and found that the local "negative" pressure spike
at cavitation inception did not significantly affect the column separation
35
phenomena. Kojima et al. (1984) found that the consideration of unsteady
friction in DVCM improved the numerical results. Similar approaches using
unsteady friction model have been used by Bughazem and Anderson (2000)
and Bergant and Tijsseling (2001).
In referring to discrete multiple cavity models, the second type of
models is the discrete gas cavity model (DGCM). Brown (1968) presented the
first attempt at describing liquid column separation with the effects of
entrained air. Entrained air was assumed to be evenly distributed in
concentrated pockets at equal distances along the pipeline. The presence of air
decreased the overall pressure wave celerity. The presence of entrained air was
neglected above a certain head, where the solution reverted back to normal
water hammer computations. The DGCM has been used for modeling column
separation over the last 30 years but not as widely as DVCM. In fact, the
discrete free gas cavity model is a modification of the discrete vapor cavity
model by utilizes free gas volumes to simulate distributed free gas (Wylie
1992). This model was developed by Brown (1968), and followed by Provoost
(1976), Provoost and Wylie (1981) and Wylie (1984). Provoost (1976) and
Wylie (1984) presented a detailed description of a DGCM, in which cavities
were concentrated at “grid points” (or computational sections). Pure liquid
was assumed to remain in each computational reach. A quantity of free gas
was introduced at each computational section. Wylie (1984) and Wylie and
Streeter (1993) described the discrete free gas model for simulating vaporous
and gaseous cavitation. Gas volumes at each computational section expanded
and contracted as the pressure varied and were assumed to behave as an
isothermal perfect gas. This model exhibited dispersion of the wave front
36
during rarefaction waves and steepening of the wave front for compressive
waves. Distributed cavitation in pipelines may be successfully simulated by
using very small quantities of free gas.
Wylie and Streeter (1993) compared the performance of the DGCM
against experimental results where only one vapor cavity formed adjacent to
the valve with no distributed cavitation. The results clearly show the
occurrence of short-duration pressure pulses of different relative magnitudes
and widths. Wylie (1992) and Wylie and Streeter (1993, pp. 202-205)
compared results from the DGCM with analytical and experimental data in a
low void-fraction system during rapid transient events. Favorable comparisons
were obtained although highly non-linear behavior was observed. The wave
speed variation with pressure in a system with free gas was demonstrated.
Wylie and Streeter (1993, pp. 188-192) tested the DGCM for a bubbly flow
case in which air was dispersed in the continuous liquid phase. The
experimental results were taken from Akagawa and Fujii (1987). Wylie and
Streeter (1993) considered three different void fraction distributions in
applying the DGCM. The comparison of numerical and experimental results
for two of the three cases was quite favorable despite the results being
extremely sensitive to the amount of free gas in the system. In the third case,
with a uniform void fraction in one section and a single phase upstream, the
DGCM produced oscillations that were not present in the experimental results.
Bergant and Simpson (1999) validated the DGCM (including DVCM
and generalized interface vaporous cavitation model (GIVCM) results against
experimental results for the rapid closure of a downstream valve. In addition,
the authors presented a global comparison of DGCM and experimental results
37
for a number of flow regimes in downward and upward sloping pipes (30
cases). A comparison was made for the maximum head at the valve and the
duration of maximum cavity volume at the valve. The agreement between the
computed and measured results was acceptable. Barbero and Ciaponi (1991)
performed a similar global comparison analysis between DGCM simulations
and measurements.
2.4.3. Shallow water flow or separated flow models
This type of numerical model describes liquid column separation or cavitation
regions with shallow water (open channel) theory. The water hammer (liquid)
regions are calculated by the method of characteristics. They are separated
from the shallow water regions by moving boundaries. Shallow water flow
modeling of regions of vapor pressure provided the first real attempt at a more
realistic description of transient cavitation. Vapor bubbles were assumed to
form, rise quickly and agglomerate to form a single long thin cavity when the
pressure reached the vapor pressure (Provoost, 1976). Baltzer (1967)
developed a shallow water flow numerical model for column separation at a
valve while the water hammer equations were applied in the remaining part of
the pipe. The shape, movement, and collapse of a vapor cavity formed at the
upstream side of a valve were considered. Once the vapor cavity formed, it
usually expanded and propagated in the direction of the flow as a bubble.
Siemons (1967) at Delft Hydraulics Laboratory also developed a shallow
water flow model of liquid column separation to describe cavitating flow. This
model was referred to as a “separated flow” model by Vreugdenhil et al.
(1972). Kalkwijk and Kranenburg (1971) noted that Siemons' results did not
maintain a mass balance at the boundary of the cavity, and therefore they
38
questioned the validity of the conclusion concerning the generation of high
pressures. The transition from the water hammer region to the cavitating
region was one of the major problems in the shallow water flow approach. For
this reason, Kranenburg (1974) concluded that the description of cavitating
flow and liquid column separation by shallow water flow theory did not seem
attractive due to the appearance of gravity waves. Furthermore, the model was
proved to be physically incorrect for vertical pipes.
2.4.4. Two phase or distributed vaporous cavitation models
This type of numerical model distinguishes between water hammer regions
(with pure liquid) and distributed vaporous cavitation regions (with a
homogeneous liquid-vapor mixture). Velocity and vapor void fraction are
computed in the mixture region. Kalkwijk and Kranenburg (1971) presented a
theory to describe the occurrence of distributed vaporous cavitation in a
horizontal pipeline. They referred to this as the “bubble model”. Their
approach ignored free gas content in the liquid and the diffusion of gas
towards cavities. An upstream end pump failure was considered. Wylie and
Streeter (1978) presented a similar analytical development of a model for
vaporous cavitation in a horizontal pipeline. An example was considered that
involved the rupture of a pipeline at the upstream reservoir.
Kalkwijk and Kranenburg (1971) presented two approaches to the
theory. The first approach was based on the dynamic behavior of nuclei or gas
bubbles. However, the method failed at the point where the radius of the
bubbles exceeded a critical value. At this size the bubbles became unstable
and the characteristics became imaginary. The second approach separate out
regions with and without cavitation. Different systems of equations held for
39
the water hammer and the vaporous cavitation region. The wave celerity in the
cavitation region, with respect to the fluid particles, is reduced to zero.
Kalkwijk and Kranenburg (1971) assumed that the liquid in a cavitation region
had a pressure equal to the vapor pressure. Analytical expressions were
developed for the velocity and void fraction for the vaporous zone in a
horizontal pipe. When a cavitation region stopped growing, a shock formed at
the transition from the water hammer to the vaporous region, which penetrated
into the cavitation region. This interface was described using the laws of
conservation of mass and momentum, which resulted in “shock equations”
analogous to the equations for a moving hydraulic jump. A shock-fitting
technique was used (Kranenburg 1972). Use of analytical methods for the
treatment of the cavitation region and the explicit shock calculation was
implemented to avoid numerical distortions (Vreugdenhil et al. 1972). A
number of systematic computations for a simple horizontal pump discharge
line showed that the maximum pressure after cavitation did not exceed the
steady state operating pressure for the pump.
Kranenburg (1974) presented an extensive work on the effect of free
gas on cavitation in pipelines. A detailed description was given of the
phenomenon of transient cavitation in pipelines. A simplified one-dimensional
mathematical model, referred to as the “simplified bubble flow” model, was
presented (Kranenburg 1972, 1974). Continuity and momentum equations in
conservation form were presented for the vaporous regions. The slope of the
pipe and the influence of gas release were both considered. Kranenburg (1974)
found that there was considerable difficulty in using the method of
characteristics due to the pressure dependence of the wave celerity because of
40
the presence of free gas. He asserted that discontinuities or shocks between the
water hammer and vaporous region should be fitted in the continuous solution
only for simple cases. The bubble flow or vaporous cavitation regime was
assumed for the whole pipe, even for the water hammer regions to simplify the
model. A modified surface tension term was used to achieve this
simplification. As a result, this model did not show explicit transitions
between the water hammer and vaporous cavitation regions.
Kranenburg (1972) used a Lax-Wendroff two-step scheme despite the
occurrence of shock waves. A numerical viscosity was used to suppress the
non-linear instability resulting in the spreading of the developing shock wave
over a number of “mesh points”. In addition, a smoothing operator was
introduced to reduce oscillations and instability of computations caused by the
pressure dropping to vapor pressure. Liquid column separation was explicitly
taken into account at mesh points where it may be expected to occur.
Kalkwijk and Kranenburg (1971) also presented experimental results.
Dispersion of the negative wave was observed for pressures below
atmospheric pressure. This was attributed to the growth of nuclei. Some gas
bubbles were observed to remain after the passage of the shock wave that
collapsed the vaporous region, and this suggested that gas content played a
certain role. Computations did not exhibit the dispersive effect observed for
the experimental results. In conclusion, the authors stated “this method gives a
reasonable description of the overall behavior of the process”.
Fanelli (2000) also addressed methods for solving ordinary and partial
differential equations including shock-fitting and shock-capturing procedures.
41
Wylie and Streeter (1978, 1993) developed a case-specific model
involving vaporous cavitation and referred to it as an “analytic model”, while
Streeter (1983) referred to a more general model as a “consolidation model”.
A distributed vaporous cavitation region (zone) is described by the two-phase
flow equations for a homogeneous mixture of liquid and liquid-vapor bubbles
(liquid-vapor mixture). A homogeneous mixture of liquid and liquid-vapor
bubbles in pressurized pipe flow is assumed to occur when a negative pressure
wave traveling into a region in which the pressure along the pipe drops to the
liquid vapor pressure over an extended length of the pipe. Pressure waves do
not propagate through an established distributed vaporous cavitation zone.
2.4.5. Combined models / interface models
This type of model allows for distributed vaporous cavitation in the same way
as the previous type, however, the formation of local column separations at
any point in the pipeline is taken into account (Kranenburg 1974, Streeter
1983). Flow regions with different characteristics (that is – water hammer,
distributed vaporous cavitation, end cavities and intermediate cavities) are
modeled separately, while the region interfaces are tracked.
Streeter (1983) was the first to develop a combined analysis for
modeling local liquid column separations at high points and a number of
distributed vaporous cavitation regions, while retaining the shock-fitting
approach to explicitly compute the locations of transitions between water
hammer and vaporous cavitation regions. Gas release was not considered,
thereby removing the problem associated with the variable wave speed due to
the presence of free gas. This model was referred to as an “analytical
approach” or the “consolidation model”. The model was applicable to pipes at
42
any angle with the horizontal. Many separate distributed vaporous cavitation
zones could be modeled, as well as the collapse and reforming of vaporous
regions. All water-hammer regions were described by the method of
characteristics. The equations developed for the vaporous region were for
various combinations of slope and initial (inception) velocity. A computational
section became vaporous once a pressure less than the vapor pressure had been
computed from the method of characteristics. A time-line interpolation scheme
was used to find the first occurrence of vapor at a computational section.
Wylie and Streeter (1993, pp. 196-207) give a detailed presentation of
the so-called interface model for modeling distributed cavitation. Although
interface models give reliable results (Bergant and Simpson 1999), they are
quite complicated for general use. More accurate treatment of distributed
vaporous cavitation zones, shock waves and various types of discrete cavities
contribute to improved accuracy of the pipe column separation model. The
drawback of this type of model in comparison with discrete cavity models is
the complex structure of the algorithm and the longer computation times.
2.4.6. A comparison of models
Provoost (1976) compared the results of the “separated flow” (open channel
flow) model and Kranenburg's (1974) “simplified bubble flow” model for a
horizontal pipeline and a pipeline with high points. Provoost (1976) concluded
that the “separated flow” model did not reproduce the field measurements for
the pipeline system with two high points. The “simplified bubble model” was
not suited to describe the local liquid column separations at the high points.
This model assumed that the equations for the cavitation region were applied
to the entire pipeline. Explicit transitions were not shown between water-
43
hammer regions and vaporous cavitation zones. A filtering procedure was
required to suppress a slowly developing instability in the cavitation regions.
As a result, the discrete free gas model was developed by Provoost (1976) in
order to deal with local liquid column separations at high points.
Bergant and Simpson (1999) compared numerical results from discrete
vapor (DVCM), discrete gas (DGCM) and generalized interface vaporous
cavitation models (GIVCM) with results of measurements performed in a 37.2
m long sloping pipeline of 22 mm diameter. The principle source of
discrepancies between the computed and measured column separation results
was found to originate from the method of physical description of vaporous
cavitation zones and resulting phenomena along the pipeline.
Dudlik et al. (2000) compared the DVCM with a three-phase model
that allowed for the calculation of sudden changes of gas content in the liquid.
Shu (2003) compared the DVCM with a two-phase model and with
experimental data.
2.4.7. State of the art - the recommended models
The discrete vapor cavity model (DVCM) gives acceptable results when
clearly defined isolated cavity positions occur rather than distributed
cavitation. The DVCM and its variations like DGCM involve a relatively
simple numerical algorithm in comparison to the interface models. The
discrepancies between measured data and DVCM, DGCM and GIVCM
predictions found by temporal and global comparisons (Bergant and Simpson
1999) may be attributed to approximate modeling of column separation along
the pipeline (distributed vaporous cavitation region, actual number and
44
position of intermediate cavities) resulting in slightly different timing of cavity
collapse and different superposition of waves. In addition, discrepancies may
also originate from discretization in the numerical models, the unsteady
friction term being approximated as a quasi-steady friction term, and
uncertainties and stochastic behavior in the measurement. At the present stage,
the GIVCM is used as a research tool whereas the DVCM and DGCM models
are used in most commercial software packages for water-hammer analysis.
When the absolute pressure reaches the vapor pressure, cavities or
bubbles will develop in the liquid. In the DVCM these cavities are
concentrated, or lumped, at the grid points. Between the grid points, pure
liquid is assumed for which the basic water hammer equations remain valid.
This means that the pressure wave speed is maintained (and convective terms
neglected) between grid points in distributed cavitation regions. However, in
bubble flow the pressure wave speed is very low and pressure-dependent.
These matters are implicit in the model (Liou, 2000). Pressure waves actually
do not propagate through an established distributed cavitation region, since
this is at an assumed constant vapor pressure. The annihilation of a distributed
cavitation region by a pressure wave causes a delay in propagation, which
must be regarded as a reduction of the wave speed.
As stated, the discrete cavity model is a relatively simple model, which
is able to cover the essential phenomena in transient cavitation. It fits in with
the standard MOC approach, so that it can be used in general water-hammer
computer-codes. Its main deficiency is in the numerical oscillations and
unrealistic spikes appearing in the calculated pressure histories, when regions
of distributed cavitation occur (Bergant and Simpson 1999). One way of partly
45
suppressing the oscillations and spikes is to assume a small amount of initial
free gas in the grid points (Provoost 1976; Wylie 1984; Zielke and Perko
1985; Barbero and Ciaponi 1991). The overall conclusion is that discrete
cavity models are adequate, but improvements concerning the numerical
oscillations are welcome, because engineers tend to take the highest pressure,
which might be an unrealistic peak, as a measure for design and operation.
Unsteady friction models help to predict more accurately the time intervals
between successive column separations.
Keller and Zielke (1976) measured free gas variations subsequent to a
rapid drop in pressure in a 32 m long plastic pipe with a diameter of 125 mm
that was connected to a cavitation tunnel. Wiggert and Sundquist (1979)
conducted experiments using a 129 m and a 295 m long coiled copper-tube
apparatus with a diameter of 25 mm. They investigated gas release during
transients at different initial gas concentrations. The effects of gas release,
cavitation nuclei and turbulence were studied. Martin (1981) used a Plexiglas
pipe apparatus (length 32 m, diameter 26 mm) where the water was saturated
with injected air.
Shinada (1994) studied experimentally and theoretically column
separation with gas release. In a 2.5 m long, 19 mm diameter pipe he tested
with saturated and deaerated oil. He measured air content, surface tension and
diffusion rate. On the basis of experimental results, the proposed bubble-
diffusion model allowed for gas release only at the column separation. Gas
release has a significant effect on column separation in saturated oil.
46
2.4.8. Fluid structure interaction (FSI)
The repeated collapse of column separations, and the almost instantaneous
pressure rises associated with them, forms a severe load for pipelines and their
supporting structures. Structural vibration is likely to occur. Fluid-induced
structural motion, structure-induced fluid motion and the underlying coupling
mechanisms are commonly referred to as FSI (fluid-structure interaction).
Most of the researchers mentioned in this review paper prevented unwanted
FSI effects by rigidly anchoring their pipes. Fan and Tijsseling (1992),
however, focused on the simultaneous occurrence of cavitation and FSI. They
performed experiments in a closed pipe, the vibrating ends of which interacted
with transient column separations. They observed distributed vaporous
cavitation caused by a stress wave in the pipe wall. More information on
combined cavitation/FSI models and on FSI in general can be found in review
papers by Wiggert and Tijsseling (2001).
In short, the GIVCM handles a number of pipeline configurations
(sloping and horizontal pipe) and various interactions between water hammer
regions, distributed vaporous cavitation zones, intermediate cavities (along the
pipeline) and cavities at boundaries (valve, high point). More accurate
treatment of distributed vaporous cavitation zones, shock waves and various
types of discrete cavities, contribute to improved performance of the pipe
column separation model. Although the interface model gives reliable results,
it is quite complicated for general use. The drawback of this model in
comparison to the discrete cavity models is the complex structure of the
algorithm and longer computational time. At the present stage, the GIVCM is
useful as a research tool but not in commercial codes.
47
Numerous experimental studies have been carried out over the last 40
years. From all the validation tests presented in the research literature it may
be concluded that, despite its simplicity, the discrete vapor cavity model
(DVCM) reproduces the essential features of transient cavitation. The
versatility of the model has been demonstrated by the variety of pipe systems
used in the tests. The major deficiency of the model is the appearance of non-
physical oscillations in the results. The DGCM is recommended in developing
and revising industrial engineering water hammer computer codes.
In the last 10 years, the amount of research effort into column
separation has slowed. There is still room for further research in a number of
areas. Laboratory testing of the impact of various cases of distributed
cavitation (or vaporous cavitation) in conjunction with the testing of the
performance of the three most commonly used models (DVCM, DGCM and
GIVCM) needs to be undertaken. Over the last 10 years a number of advances
in unsteady friction have occurred. The impact of these new models on the
behavior of column separation modeling also needs to be investigated. Finally,
further work could also be carried out on the GIVCM to explore whether
simplifications could be made to the complexity of the approach so that it
becomes viable to introduce this approach into commercial water hammer
codes.
2.5. SUMMARY
A critical review of literature pertaining to transient flow in pipelines systems
is presented in this chapter. From the review, transient flow in pipelines
systems can be divided into water hammer, fluid transient (water hammer)
with air entrainment, and fluid transient with vaporous cavitation and/or
48
column separation. The method of computational and modeling is different for
each case. The existing models for water hammer problem give a good
prediction of the first pressure peak. However, there are discrepancies in the
observed and predicted behavior of energy dissipation beyond the first wave
cycle. Hence, understanding the helical vortices in transient pipe flows, and
incorporating this new phenomenon in practical unsteady flow models, as well
as understanding of the turbulence behavior and energy dissipation in unsteady
pipe flows would lead to significantly discrepancy reduction in the observed
and predicted behavior of energy dissipation beyond the first wave cycle. For
fluid transient with column separation, the overall conclusion is that numerical
models are adequate, but improvements concerning the numerical oscillations
are welcome. More accurate treatment of distributed vaporous cavitation
zones, shock waves and various types of discrete cavities, contribute to
improved performance of the pipe column separation model. The review
shows that air entrainment complicates the computational and modeling of
fluid transient in pipeline systems. A significant limitation in the numerical
models proposed in dealing with free entrained air and released gas was the
need to make rather arbitrary assumptions regarding the rate of release of gas.
As mentioned above, transient flow in pipelines systems can be
divided into water hammer, fluid transient with air entrainment, and fluid
transient with vaporous cavitation and/or column separation. The applicable
methods of computational and modeling for these three cases are so different
that the development of a special method to solve all fluid transient problems
is a very challenging objective. Therefore, researchers should focus on a
segmentation of the study of transient flow.
49
In this thesis, the fluid transient in complex systems with air
entrainment is the chosen research segmentation because of its possible
application in practical engineering. A well-designed pipeline system should
have avoided all vaporous cavitations and/or column separation to happen.
However, that pipeline system is always subjected to fluid transient with air
entrainment due to its unavoidable operational changes such as
starting/stopping of pumps, changes to valve setting, changes in power
demand, etc. From the literature review, the method of characteristics has the
desirable attributes of accuracy, simplicity, numerical efficiency, and
programming simplicity; and the standard MOC methods are likely to be
acceptable when the air void fraction in the fluid is small. Therefore, in this
thesis, MOC method is used to develop a new numerical model to study fluid
transients in complex systems with air entrainment.
50
CHAPTER 3
FLUID TRANSIENT ANALYSIS METHOD
3.1. INTRODUCTION
This chapter presents a numerical approach for computational and modelling
of fluid transient in complex system with air entrainment. We first present the
governing equations for 1D transient flow. We then introduce a variable wave
speed model to count on the effects of free entrained air and released gas from
gaseous cavitation. Finally, an improved method of characteristics with
variable time steps was employed to solve numerically the governing
equations.
3.2. GOVERNING EQUATIONS FOR TRANSIENT FLOW
The transient flow of a fluid closed conduit is described by the momentum and
continuity equations. Since the flow velocity and pressure developed during
transient flows are both functions of position and time, the momentum and
continuity equations are a set of partial differential equations. The following
simplifying assumptions are made in the derivation of the equations:
(i) The flow in the conduit is one-dimensional.
(ii) The conduit wall and the fluid are linearly elastic.
(iii) The conduit remains full of fluid at a pressure exceeding the vapour
pressure of the liquid or the gas release pressure.
The governing equations for the transient fluid flow in pipeline are:
51
Equation of Motion:
( ) 02
1 =+++= VVD
fVVVgHL xtx (3.1)
Continuity Equation:
( ) 0sin2
2 =+++= θVVHHVg
aL xtx (3.2)
where
H = Pressure head,
V = Velocity of fluid flow,
f = The Darcy-Weisbach friction factor,
a = Wave speed,
D = Internal diameter of pipeline, and
θ = Angle between horizontal direction and fluid velocity direction,
Fig. 3.1. Angle between horizontal direction and fluid velocity direction
3.3. VARIABLE WAVE SPEED MODEL
In order to predict the pressure changes occurring in a fluid system under
unsteady fluid operating condition, it is always necessary to model the wave
speed a correctly. Detail development of the wave speed equations is available
52
in major text books (Wylie and Streeter 1993, Chaudhry 1987). In general, the
wave speed for a pipe system is given by:
( )[ ]lceDEKaV
Ka
)/)(/(1/1
/2
++=
ρ (3.3)
where K is the Bulk modulus of elasticity of fluid, ρ is the density of the fluid,
E is the modulus of elasticity of the pipe, e is the pipe-wall thickness, and cl is
the dimensionless parameter that describes the effect of system pipe-constraint
condition on wave speed.
Transient can generate very different pressure at different parts of a
pipeline at the same instant of time. Thus, it is possible to have a wave speed
of 1000 m/s at some point of a pipeline, and another at wave speed as low as
10 m/s due to fluid property variations. Therefore, to ignore the variation in
wave speed can lead to error in predicting the real transient in practice. The
above wave speed calculation (Eq. (3.3)) did not take into consideration the
local variation of the wave speed due to local air content ε and its effects due
to extreme pressure transients and the absorption and release of gases during
the extreme pressure transients. We introduce a variable wave speed model to
take the above effects into consideration.
Consider a control mass of gas-liquid mixture containing a fractional
volume ε of gas in free bubble form (Pearsall 1965), where Ψt is the total
volume of the gas plus liquid,
Ψl = (1 - ε) Ψt (3.4)
is the volume of liquid, and
Ψg = ε Ψt (3.5)
53
is the free gas volume.
Assuming a pressure increment ∆P to the liquid. The liquid volume will
change to
( )( ) tt KP Ψ−∆−=Ψ ε11
* (3.6)
The gas volume is assumed to be distributed in small bubble form and to have
a polytrophic change in properties due to partial heat being transfer to the
water.
The gas volume change will then be related through
( ) ( )( )ng
n
g PPP*
Ψ∆+=Ψ (3.7)
where Ψg* is the fractional gas volume at pressure P + ∆P.
The volume of gas/liquid mixture at pressure P + ∆P will then become
( )( ) tt
n
t KP
PP
PΨ−∆−+Ψ
∆+=Ψ εε 11
1
* (3.8)
Expanding ( )
∆−≈∆+−
PP
nPP n 1
111
and discard all the smaller terms, we
will arrive at
∆+
∆−=
Ψ
Ψ
K
P
P
P
nt
t ε1
*
(3.9)
Hence, the effective bulk modulus of the fluid with air fraction content of ε
and at a pressure P is given by:
+
=
Ψ
Ψ−
∆=
nPK
PK
t
tε1
1
1
*
* (3.10)
54
The effective bulk modulus of the gas/liquid mixture KT including the pipe
distensability effects and pipe constraint condition cl is thus given by:
eE
Dc
nPKK
l
T
++=ε11
(3.11)
Thus, the wave speed of the fluid system with free air bubble fraction of ε is
given by:
21
1−
++=
eE
Dc
nPKa l
e
ερ (3.12)
where ρe is the equivalent mass density of the gas liquid mixture and is
approximated by neglecting the weight of the free gas:
( ) ( )ερερερρ −≈+−= 11 lgle (3.13)
For a fluid system, at the time t equal to k∆t and at the location along the
pipeline x equal to i∆x, the local wave speed aik at the local absolute pressure
Pik and air volumetric void fraction of εi
k is given by:
( )2
1
11
−
++−=
eE
Dc
nPKa l
k
i
k
ik
il
k
i
εερ (3.14)
For this model of variable wave speed the initial free air volumetric
void fraction εo and dissolved gas relative volumetric void fraction εg at a
reference absolute pressure Po must be specified. The initial variable wave
speed along a pipeline (at node points i = 0, 1, …, N) is then computed
through the absolute pressure distribution along the pipeline from Eq. (3.14) at
k = 0 (steady state). The transient computation of the fractional air content
55
along the pipeline depends on the transient local pressure and the transient
local air fraction content.
The computation of the above transient local air volumetric void
fraction in Eq. (3.14) along the pipeline is given by
k
i
n
k
i
k
ik
TP
Pεε
/1
1
1
=
+
+ and o
n
k
i
ok
oP
Pεε
/1
1
1
=
+
+ (3.15a)
For g
k
i PP ≥+1 and ggr
k
o
k
T εαεε +≤ ++ 11
11 ++ = k
T
k
i εε (3.15b)
For g
k
i PP ≥+1 and ggr
k
o
k
T εαεε +> ++ 11 and with a time delay of Ka∆t
( )gga
k
i
n
k
i
k
ik
iP
Pεαεε −
=
+
+
/1
1
1 (3.15c)
For g
k
i PP <+1 but > Pv and with a time delay of Kr∆t
( )ggr
k
i
n
g
k
ik
iP
Pεαεε +
=+
/1
1 (3.15d)
For v
k
i PP <+1 ; Pik+1
is assumed equal to Pv and instantaneously:
( )v
k
i
n
v
k
ik
iP
Pεεε +
=+
/1
1 (3.15e)
The model proposed here considers the presence of an initial free
entrained air of volumetric void fraction εo and dissolved gas content of
relative volumetric void fraction εg in the liquid at atmospheric pressure and
ambient temperature. We also assume that: (i) the gas-liquid mixture is
56
homogeneous; (ii) the free gas bubbles in the liquid follow a polytrophic
compression law with n = 1.2-1.3 (due to the occurrence of some heat transfer
from the heated fluid in the pipeline to the surrounding soil); and (iii) the
pressure and temperature within the air bubbles during the transient process is
in equilibrium with the local fluid pressure and temperature. The above model
considers that when the local pressure falls below the vapour pressure Pv, an
instant cavitation at Pv under transient conditions occurs. The local transient
pressure remains at Pv until the transient pressure recovers to a value above Pv.
When the computed local transient pressure falls below the gas release
pressure Pg, it is assumed that there is a release of dissolved gas of αgrεg with a
time delay of Kr∆t. The local pressure remains constant and is equal to the
vapour pressure. When the computed transient pressure recovers to a value
above Pg, it is assumed that an equivalent amount of αgaεg gas is absorbed into
the fluid with a time delay of Ka∆t. When compared with the field data
obtained so far, both Ka and Kr are slightly larger than 1.000 and Ka > Kr (Lee
& Cheong 1998). Typical values used for Ka and Kr are 1.005 and 1.001
respectively. Typical free air content in the industrial water intake at
atmospheric pressure is about 0.1% by volume and the free gas content
evolved at gas release head is about 2% of the gas content at atmospheric
pressure head. The fractional parameter of gas absorption is αga ≈ 0.3 and the
fractional parameter of gas release is αgr ≈ 0.6 (Pearsall (1965, 1966),
Kranenburg (1974), Provoost (1976)). For the comparative study of constant
wave speed cases, the present study assumed ε = 0.000 (fully de-aerated
water) in the fluid system even when the transient pressure falls below the Pg
or Pv, i.e. the system is assumed completely free from the influence of
57
entrapped air or water vapour. Whenever the transient pressure falls below Pv,
it is assumed that minimum pressure remained at Pv for the duration of this
transient. Hence, the values of the variable speed will have a lower bound,
which is consistent with the field data observed.
The model expressions in Eq. (3.15) are very different from that given
by Pearsall (1965; 1966), Fox (1984), Chaudhry et al. (1990), and Wylie and
Streeter (1993) in which the evolution of dissolved gases and the re-absorption
of released gases below and above Pg, respectively, were not taken into
account in the transported fluid under transient flow conditions. The air
entrainment models of Pearsall and Provoost assumed that the air volumetric
void fraction ε is constant throughout the pipeline without local variation due
to the transient pressure changes. Proper phasing of the transient pressure
variation was difficult to obtain with the existing models especially when air
entrainment in the fluid system exists. Equation 3.15 is only valid for small
values of ε where the assumption of homogeneous of the gas liquid mixture is
still applicable. When values of ε are large fractions of unity, the flow
becomes frothing flow which may separate into open channel flow with a gas
flow over the top of it, or it may become a slug flow in which large bubbles
are interspersed with liquid.
3.4. FRICTION FACTOR CALCULATION
There is experimental evidence which demonstrates that losses due to friction,
fittings, etc. are affected by unsteadiness in the flow. The unsteadiness in the
flows affects the turbulence, boundary layer velocity profiles and hence the
losses. Kita et al. (1980) showed experimentally that in very slow transients
the quasi-steady approximation of losses due to friction can be sued with
58
satisfactory accuracy. Recent research evidence has shown that the unsteady
flow friction factor is greater than the steady flow friction factor in accelerated
flows for more rapid transients, and the converse for decelerated flows. For
such flows, an appropriate representation of the unsteady loss factor and
frictional effects is necessary. A review of the available literature indicates
that satisfactory theoretical loss models have been developed for unsteady
laminar and single-phase flow, while no satisfactory model is available for
unsteady turbulent and two-phase flow.
Thus, a modified variable-loss-factor model is proposed here for the
study of unsteady pressure transient flow with air entrainment and gas release
problems. The present model takes into account the losses due to the two-
phase nature of the flow and the features of the pipelines. This model was
tested for the case of pump trips due to power failures and valve closure in
sewage pumping stations. For the variable-loss-factor model proposed here the
results show that the maximum pressures transient is marginally higher and
the minimum pressure transient is marginally lower than for the corresponding
model with a constant steady stare loss factors.
The loss factor model proposed for use in conjunction with the method
of characteristics with air entrainment and gas release in a pipeline system can
be evaluated at a local point i as follows:
(a) Rei = ViDi/νi (3.16)
where
Rei is local effective Reynolds number,
Vi is local cross-section average velocity,
59
Di is inner diameter of pipe and,
νi = (1 – εi)νl + εiνg is the effective kinematic viscosity of the air-liquid
mixture.
(b) Assuming that the total local loss factor (Kloss)i at node point i includes the
losses due to the two-phase nature of the flow and the features of the pipelines.
(Kloss)i = Km(Rei) + Kf(Rei) (3.17)
where
Km(Rei) = local loss factor due to pipe features,
Kf(Rei) = (∆xi/Di)f(Rei) is local loss factor due to nature of flow.
The total losses at a node point are thus represented by
( )
∑
g
VVK
ii
iiloss
2
(c) Depending on the value local effective Reynolds number calculated by Eq.
(3.16), the value of f(Rei) is calculated from one of the following three
situations
If Rei ≤ 1, f(Rei) = 64
If 1 ≤ Rei ≤ 2000, f(Rei) = 64/Rei
If Rei > 2000, (fi)0 = f (Moody’s formula (Moody 1947))
The value of f(Rei) is obtained through iteration of the corresponding
Colebrook-White equation with the appropriate roughness factor for the pipe
at the node point i (Streeter and Wylie, 1978).
60
(d) For [ ] 001.0)(Re/)()(Re 0 ≥− iii fff , set )(Re)( 0
ii ff = and repeat
iteration.
(e) Then the total local loss factor is calculated from Eq. (3.17). Hence a local
equivalent loss factor ilf )( is then defined as
)/()()( iiilossil xDKf ∆= (3.18)
The steady state overall loss factor at the operating point of a system
can be determined from the pump characteristic curve and the system curve.
This value is used as a check against the value obtained from Eq. (3.18) for the
summation of i = 0, 1, …, N at the steady state Reynolds number.
3.5. NUMERICAL METHOD
There are many methods have been used to solve one-dimensional fluid
transient flow. Among these methods, the method of characteristics (MOC) is
the most popular computational technique for solving the basic water hammer
equations since:
• The MOC concept and basis follows the true physical nature of the
pressure unsteady flow. Numerical waves propagate along the pipes,
carrying pressure or velocity discontinuities, as the elastic waves do.
• Accuracy of results as well as small terms is retained.
• There is proper inclusion of friction.
• It affords ease of handling boundary conditions.
• A computer program for an explicit MOC scheme is easy to write and
to implement. Most of the programming errors can be detected by
61
comparing the numerical results with the true physical behaviour such
as propagation, attenuation, reflection, etc…
• No need for large capacity in computer.
• Detail results are completely tabulated.
• Since the sixties until now, most of the general water hammer
computer programs were developed and improved by the users for
several years. The numerical scheme of a useful and reliable long life
code needs to be easy to remember and to understand for young
members of the team, as it is the MOC.
The method of characteristics applied to the equation of motion (Eq.
(3.1)) and the continuity equation (Eq. (3.2)) can be described by the
respective C+ and C
- characteristic equations (Fox 1984).
The C+ characteristic equations are:
02
sin =+++D
VVfV
a
g
dt
dV
dt
dH
a
g lθ (3.19a)
aVdt
dx+= (3.19b)
And, C- characteristic equations are
02
sin =+−+−D
VVfV
a
g
dt
dV
dt
dH
a
g lθ (3.20a)
aVdt
dx−= (3.20b)
With air entrainment, the wave speed a is calculated by using variable
wave speed model (Eq. (3.14)). The transient computation of the fraction of
62
air content ε along the pipeline depends on the local pressure and local air
volume and is given by Eq. (3.15). The equivalent loss factor ƒl used in
conjunction with the method of characteristics with air entrainment and gas
released in a pipeline system is evaluated at the local point i using the
characteristics of the flow at that point (Eq. (3.18)). The steady state overall
loss factor at the operating point of a system can be determined from the pump
characteristic curve and the system curve.
Fig. 3.2. Computational grid
With reference to the irregular t-grid and regular x-grid notation used
in Fig. 3.2, i denotes the regular x-mesh point value at location x = i∆x and k
denotes the irregular time level corresponding to the time at ( )kk tt ∆Σ= . The
value of the time step ∆tk at each time level is determined by the CFL criterion
( )
+
∆=∆
ii
i
k
aV
xkt min for i = 0, 1, …, N (3.21)
where ki is a constant less than 1.0.
63
The characteristics C+ and C
- equations specified by Eqs. (3.19) and (3.20) can
thus be approximated by the finite difference expressions.
02
||sin
11
=++∆
−+
∆
−++
D
VVfV
a
g
t
VV
t
HH
a
g RR
k
l
iRk
i
k
R
k
i
k
R
k
i
k
i
iθ (3.22a)
k
iRk
Ri aVt
xx+=
∆
− (3.22b)
And
02
||sin
11
=+−∆
−+
∆
−−
++
D
VVfV
a
g
t
VV
t
HH
a
g SS
k
l
iSk
i
k
S
k
i
k
S
k
i
k
i
iθ (3.23a)
k
iSk
Si aVt
xx−=
∆
− (3.23b)
where R is the point of interception of the C+ characteristic line on the x-axis
between node points (i-1) and i at the kth
time level and S is the point of
interception of the C- characteristic line on the x-axis between node points i
and i + 1 as shown in Fig. 3.2. With conditions knows at points (i – 1), i and (i
+ 1) (A, B and C) at the kth
time level, the conditions at R and S can be
evaluated by a linear interpolation procedure.
( ) ( )( )( )AC
k
AC
k
i
k
C
R
VVx
t
VVax
tVV
−∆
∆+
−∆
∆−=
1 (3.24)
( ) ( )( )( )BC
k
BC
k
i
k
C
S
VVx
t
VVax
tVV
−∆
∆−
−∆
∆−=
1 (3.25)
( )( )( )AC
k
iR
k
CR HHaVx
tHH −+∆
∆−= (3.26)
( )( )( )BC
k
iS
k
CS HHaVx
tHH −−∆
∆+= (3.27)
64
The conditions at R and S are substituted into Eqs. (3.22) and (3.23).
Solutions at the (k + 1)th
time level at point i are obtained for i = 0, 1, …, N.
( ) ( )
( )
+∆
−−∆−−++
=+
SSRR
kk
l
SRi
k
k
i
SRk
i
SR
k
i
VVVVD
tf
VVta
gHH
a
gVV
V
i
2
sin
2
11
θ
(3.28)
( ) ( )
( )
−∆
−+∆−−++
=+
SSRR
kk
lk
i
SRi
k
SR
k
iSR
k
i
VVVVD
tf
g
a
VVtVVg
aHH
H
i
2
sin
2
11
θ
(3.29)
3.6. BOUNDARY CONDITIONS
A common flow arrangement in fluid engineering consists of a lower
reservoir, a group of pumps with a check valve in each branch, and a pipeline
discharging into an upper reservoir (water tower, gravity conduit, aeration
well, etc.).
Fig. 3.3. Schematic diagram of typical pumping system
Boundary condition at the pump is shown in Fig. 3.4 and the response
of the pump (pump set) operating at constant speed may be included in
analysis by defining the pump characteristic curve.
65
Fig. 3.4. Boundary condition at pump
The pump characteristic gives,
H0P = A1 + A2 V0P + A3 V0P2 (3.30)
where A1, A2 and A3 are constants of pump characteristics.
From C- characteristic line:
( ) 02
.sin00 =∆
+∆−−−−D
VVtfVt
a
gHH
a
gVV
SS
SSPSP θ (3.31)
From Eq. (3.30) and (3.31) above, V0P can be obtained from a 2nd
order
equation, and then H0P can be obtained from Eq. (3.30).
When reverse flow is encountered in the pump, the check valve is
assumed closed. At this instant, V0Pk+1
is assumed to be zero for the C-
characteristic line (Eq. (3.31)) at i = 0 for all subsequent time levels.
66
Fig. 3.5. Boundary condition at reservoir
Boundary condition at the downstream reservoir is shown in Fig. 3.5.
At large reservoir, the elevation of free surface level can be considered as
constants.
At point i = N of the pipeline:
HNP = HHR (3.32)
C+ characteristic line gives
( ) 02
sin =∆
+∆+−+−D
VVtfVt
a
gHH
a
gVV
RR
RRNPRHP θ (3.33)
3.7. COMPUTATION OF PUMP RUN-DOWN CHARACTERISTICS
Pump stoppage is an operational case which has to be investigated and
which often gives rise to maximum and minimum pressures. The most severe
case occurs when all the pumps in a station fail simultaneously due to power
failure. In this case the flow in the pipeline rapidly diminishes to zero and then
reverses. The pump also rapidly loses its forward rotation and reverses shortly
after each pump. When the flow reverses, the check valve is activated and
67
closed. A large pressure transient occurs in the pipeline when the flow
reverses, and the check valves of the pumps close rapidly.
The followings were the assumptions made in the numerical simulation of the
transients caused by a pump trip:
i. The effects of minor losses and partial reflection of surge waves at
bends along the pipelines are ignored.
ii. The pump net positive suction head (NPSH) remains positive.
iii. The difference in pumps and sump water level is assumed to be
negligible.
iv. The check valves close instantaneously when reverse flow takes place
(except for the study of check valve).
v. The pipe diameter and wall thickness remain constant throughout the
transient period.
vi. The pipe remains full of water except at the peak (air valve) location.
vii. The losses at junctions of air vessel and air valve with the main
pipeline are ignored.
viii. The efficiency curve (η) remains unchanged as the speed changes
during pump run-down.
The analysis presented in this thesis does not include fluid-structure
interaction therefore we are only interested in the pressure behaviour during
the time period when the effects of the deformations and vibration of the pipe
structure on the fluid are of less importance.
Characteristics of pump at steady state are modeled by:
68
2
32
2
1 ooooo QAQNANAH ++= (3.34)
2
32
2
1 ooooo QBQNBNBT ++= (3.35)
( ) ( )2
321 // ooooo NQCNQCC ++=η (3.36)
When power to the pump is interrupted (during pump run-down;
pump-trip), the unbalance torque T is applied to the rotating parts of the pump
and motor. The basic equation governing the pump speed change is:
dt
dNIT
60
2π−= (3.37)
where I is the moment of inertia of the rotating parts of the motor and pump
including the water in the pump impellers, and N is the pump speed in RPM.
The pump speed Nk+1
during the pump run-down is thus given by:
I
tTTNN
kkkk
π2
.60
2
11 ∆
+−=
++
(3.38)
At the pump run-down speed of Nk+1
, the pump characteristics can be
described by:
( ) ( )21
3
11
2
21
1
1 +++++ ++= kkkkkQAQNANAH (3.39)
( ) ( )21
3
11
2
21
1
1 +++++ ++= kkkkkQBQNBNBT (3.40)
( ) ( )211
3
11
21
1//
+++++ ++= kkkkkNQCNQCCη (3.41)
In this thesis, the computation of pump run-down characteristics (using
pump homologous relationships) is followed by the below procedure.
69
At time tk = k∆t, (N
k, Q
k, H
k, T
k, η
k) are assumed known. Hence the pump
characteristics are known:
( ) ( )2
32
2
1
kkkkkQAQNANAH ++= (3.42)
( ) ( )2
32
2
1
kkkkkQBQNBNBT ++= (3.43)
( ) ( )2
321 //kkkkk
NQCNQCC ++=η (3.44)
For time tk+1
= (k+1)∆t, the computation for pump run-down characteristics is
given by:
1) Estimated a new pump speed
I
tTNN
kkk
π2
601
0
∆−=+
(3.45)
2) Solve for Hk+1
, Qk+1
by solving Eq. (3.39) together with the C-
characteristics line Eq. (3.31).
3) Obtain Tk+1
and ηk+1
from Eq. (3.40), (3.41)
4) Obtain improved estimation of pump run-down speed:
I
tTTNN
kkkk
π2
.60
2
11
1
∆
+−=
++
(3.46)
5) If ε>−++ 1
0
1
1
kkNN (0.1RPM), set N0
k+1 = N1
k+1 and repeat steps 2) to 5)
6) Else, 1
1
1 ++ =kk NN and obtain (Q
k+1, H
k+1, T
k+1, ηk+1
) which are defined at
tk+1
= (k+1)∆t.
In the case, there is a group of np number of pumps operate in parallel.
The equivalent pump H-Q characteristic is given by:
70
( ) ( )( ) ( )( )212
3
11
2
21
1
1//
+++++++=
k
ep
k
e
k
p
kk
e QnAQNnANAH (3.47)
where for pumps in parallel operations, the pump pressure head is similar for
all operating pumps
1k
n
1k
2
1k
1
1k
e pH H H H
++++=…=== (3.48)
Flow rate Qe that the groups produce is the sum of that produced by the
individual pump
Qn p=+…++=pn21e Q Q Q Q (3.49)
Torque characteristics can be modeled as:
( ) ( ) ( )( )21
3
11
2
21
1
1/
+++++++=
k
eP
k
e
kk
P
k
e QnBQNBNnBT (3.50)
The pump efficiency can be modeled as:
( )( ) ( )( )2112
3
11
21
1////
+++++++= kk
eP
kk
eP
k
e NQnCNQnCCη (3.51)
The basic equation governing the pump speed change is:
dt
dNIT e
e60
2π−= (3.52)
where Ie = nPI, ω = 2πN, (A1, A2, A3, B1, B2, B3, and C1, C2, C3) are single
pump constants.
71
CHAPTER 4
VALIDATION OF THE NUMERICAL MODEL
4.1. INTRODUCTION
In chapter 3, a numerical approach with variable wave speed model to solve
fluid transient flow in complex system with air entrainment was introduced. In
this chapter, the validation of this numerical model is presented. The
computational results from this numerical model were compared with the
experimental results from an experiment at Fluid Mechanics Laboratory,
National University of Singapore, to verify the variable wave speed model. In
addition, the model was verified with published data from experimental and
computational study of Chaudhry et al. (1990).
4.2. COMPARISON BETWEEN EXPERIMENTAL RESULTS AND
NUMERICAL RESULTS (*)
In this section, experimental and computational result of the pressure transient
generated by pump trip was compared using a pumping system as shown in
Fig. 4.1. The value of variables at the check valve position is used in this
comparison.
4.2.1. Test rig and instrumentation
The test rig for this experiment is shown in Fig. 4.1. The test facility consisted
of a 2 HP single pump with speed 1450 rpm, a head tank with overflow
located 5.5 m above sump level, a nominal 89 mm diameter PVC piping of
(*) Part of this work has been published in:
Lee, T. S., Nguyen, D. T., Low, H. T., and Koh, J. Y., (2008). Investigation of fluid transients in
pipelines with air entrainment. Advanced and Applications in Fluid Mechanics, Vol. 4, Issue 2, pp. 117-
133.
72
15.75 m length, test section, a control valve and an air compressor for air
entrainment studies. The pump supplies water at temperature about 25oC from
the basement tank to the head tank. The water level in the head tank is
constant by means of an overflow at a height of about 5.5 m above sump level.
Excess of water flows back in to the supply reservoir. For air entrainment
studies, the compressor is used to supply external air through a tube connected
to the suction bell-mouth at the lower reservoir. The air supplied from the
compressor is being controlled by a release valve and the amount of air
introduced can be read through a rotameter. Check valves are installed at test
section 5.75 m downstream of the pump. The flow through the 1.2 m long test
section has fully recovered at the measuring points to reduce the irregular fluid
distribution inside the tube itself. Pressure measurements are made using a
Kistler 4603B10 piezoresistive amplifier with frequency 1 kHz. To avoid
vorticity effects due to local turbulence, the pressure transducer is flush with
the tube surface so that there is no protraction into the tube. An
electromagnetic flow-meter is used to measure the flow velocity in the pipe.
The steady flow conditions are controlled by a control valve. The hydraulic
pressure transients are initiated by pump trip due to power failure. The key
measurement location is at the check valve.
73
Fig. 4.1. Hydraulic schematic of the pumping system
4.2.2. Results and discussion
Typical pressure transients at the check valve are shown in Fig. 4.2 and 4.3.
Following the stop of the pump due to power failure, the flow rapidly
diminishes to zero and then reverses. When the flow reverses, positive
pressure waves propagate downstream of the pump towards the reservoir, and
negative pressure waves propagate upstream of the pipe towards the suction of
the pump. The pump also rapidly losses its forward rotation, and reverses. To
prevent the reverse flow through the pump, the check valve is activated and
closed. A pressure peak was created as the check valve closed. Many pressure
surges followed the first pressure peak as the result of the pressure wave
reflections from the check valve and the upper reservoir. The pressure surge
magnitudes were progressively reduced with time.
74
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Time (s)
Pre
ssu
re h
ea
d (b
arg
)
void fraction: 0.002
void fraction: 0.006
void fraction: 0.01
Fig. 4.2. Transient pressure measured from the experiment at check valve
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Time (s)
Pre
ss
ure
he
ad
(b
arg
)
void fraction: 0.002
void fraction: 0.006
void fraction: 0.01
Fig. 4.3. Transient pressure predicted from present method at check valve
75
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
Air void fraction
Pre
ssu
re h
ead
(b
arg
)
experimental
present method
Maximum pressure
Minimum pressure
Fig. 4.4. Effects of air content on maximum and minimum pressure head
Figs. 4.2, 4.3, and 4.4 show that the pressure transient behaviour varies
significantly with the initial amount of air within the pumping system. Fig. 4.2
shows the results of decreasing measured pressure magnitude with higher air
content. The numerical computation permitting the investigation with very
small amount of air content shows that when the initial air void fraction was
increased, the pressure head of the first pressure peak grossly increased to a
maximum value then slightly decreased (Fig. 4.4).
Fig. 4.5 shows the comparison between the pressure transients from the
instrumented test on the actual system and the computed data from numerical
computation. Although the variable wave speed model can provide a closely
transient pressure prediction for the first pressure peak in the comparison with
experimental value, cumulative discrepancies still developed with time. It is
observed that there was a difference in the damping rate of the pressure surges
between the experimental results and computational results. Numerical results
captured the increase in the damping effect for increased air content. However,
76
the damping effect predicted by the numerical computation is less than the
damping effect presented by the experimental results. In the literature, the
damping of the pressure surges with air entrainment possibly caused by
increased bulk viscosity of the fluid due to the air entrainment (Taylor, 1954),
slipping between air bubbles and water (Van Wijngaarden, 1976), and thermal
exchange between gas bubbles and surrounding liquid (Ewing, 1980). The
variable wave speed model can include the damping caused by increased bulk
viscosity of the fluid but not includes the damping caused by slipping between
air bubbles and water, and thermal exchange between gas bubbles and
surrounding liquid. These mechanisms may help explaining the inaccurate
calculation of the damping of the pressure surges in the variable wave speed
model with the presence of air.
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Time (s)
Pre
ssu
re h
ead
(b
arg
)
experimental
present method
a) Air void fraction ε = 0.002
77
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Time (s)
Pre
ss
ure
he
ad
(b
arg
)
experimental
present method
b) Air void fraction ε = 0.006
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Time (s)
Pre
ss
ure
he
ad
(b
arg
)
experimental
present method
c) Air void fraction ε = 0.010
Fig. 4.5. Comparison between experimental results and numerical results
4.3. COMPARISON BETWEEN THE RESULTS FROM VARIABLE
WAVE SPEED MODEL AND PUBLISHED RESULTS
To further verify the validity of the variable wave speed model, numerical
results from the variable wave speed model were compared with the
78
experimental results and numerical results from MacCormack Scheme
published by Chaudhry et al. (1990).
Fig. 4.6. Schematic of experiment by Chaudhry et al. (1990)
A schematic of experiment by Chaudhry et al. (1990) is shown in Fig.
4.6. The length of the pipe is to 30.6 m and its diameter is 0.026 m. The test
procedure was as follows: A steady-state flow of an air-water mixture was
established in the test pipe. Then, the downstream valve was rapidly closed
and the pressures at three locations (1, 2, and 3) were continuously recorded.
The three stations are located at x equal to 8.0 m, 21.1 m and 30.6 m,
respectively, from the upstream end. The test conditions was as follows: The
constant upstream reservoir pressure was 18.46 m of water absolute and the
steady flow velocity was 2.42 m/s. Steady air mass flow rate was equal to 4.1 x
10-6
kg/s with a downstream void ratio of 0.0023. Steady flow friction factor
was equal to 0.0205.
79
0
20
40
60
80
100
0 0.4 0.8 1.2 1.6 2 2.4
Time (s)
Pre
ss
ure
(m
of
wa
ter)
Experiment
MacCormack method
Variable wave speed model
Fig. 4.7. Comparison of computed and experimental results at Station 1
0
20
40
60
80
100
0 0.4 0.8 1.2 1.6 2 2.4
Time (s)
Pre
ss
ure
(m
of
wa
ter)
Experiment
MacCormack method
Variable wave speed model
Fig. 4.8. Comparison of computed and experimental results at Station 2
The variable wave speed model was used to compute the transient
pressures in the above pipeline. We applied the same boundary condition with
Chaudhry et al. (1990) simulation using MacCormack Scheme. The upstream
boundary was a constant-level reservoir while the downstream boundary was
the known pressure history at pressure transducer no. 3. Typical comparisons
80
between the numerical results from the variable wave speed model and the
results of Chaudhry et al. (1990) were shown in Fig. 4.7 and Fig. 4.8 for the
transient pressures at location 1 and 2. It is clear from these figures that
transient pressures are satisfactorily simulated by the variable wave speed
model. In addition, it can be seen the same finding with previous comparison
in the section 4.2 that the damping effect predicted by the numerical
computations is less than the damping effect presented by the experimental
results.
4.4. SUMMARY
Fluid transient modeling was improved by using the variable wave speed
model. The wave speed is calculated depending on the local pressure and local
air void fraction. Therefore, the wave speed was no longer constant as in the
constant wave speed model, it varied along the pipeline and varied in time.
Free gas in the fluid and released/absorbed gas from gaseous cavitation at the
fluid vapor pressure was also modeled. To consider the gaseous cavitation
problem, there is an instantaneous release of dissolved gas content is assumed
when the computed local transient pressure falls below the fluid vapor
pressure; and an equivalent amount of released gas content is assumed to have
re-dissolved into the liquid when the computed transient pressure recovers to a
value above the fluid vapor pressure. The above refinements allow the
pressure transient simulation closely captures the actual complex physical
phenomena such as the effect of air entrainment and the effect of gaseous
cavitation.
The computed results compared satisfactorily with the experimental
results demonstrating the validity of the variable wave speed model. The
81
variable wave speed model can generally predict an abrupt and sharp pressure
rise for the pressure surge shape, predict detail of the characteristics of the
pressure surge period, as presented by the test measurement; conservatively
predict the maximum and minimum pressure transients; and capture the effects
of air entrainment on fluid transients. Similar to other models, the variable
wave speed model produces some discrepancies in the predicted behavior of
energy dissipation beyond the first wave cycle. Unfortunately, this limitation
will not prevent the applicable of the variable wave speed model because in
the fluid transient analysis, premiere pressure surges are the most important
pressure surges to be considered. In the next chapter, the variable wave speed
model was applied to analyze the effects of air entrainment and released gas
for fluid transient in typical pumping systems.
82
CHAPTER 5
NUMERICAL MODELLING AND COMPUTATION
OF FLUID TRANSIENT IN COMPLEX SYSTEM
WITH AIR ENTRAINMENT (*)
5.1. INTRODUCTION
In pumping installation, fluid transient computation is necessary to predict
excessive transient pressures which may cause collapse of pipelines, and
damage of hydraulic components. In these systems, air content and air
entrainment always exist and affect the pressure transient. This chapter
presents an investigation of the effects of air entrainment on pressure transient
in typical pumping systems. The systems consist of a lower reservoir, a group
of three pumps operating in parallel which has a check valve in each branch,
and a pipeline system discharging water from the lower reservoir into an upper
reservoir. The most dangerous case of pressure transient is the stoppage of all
three pumps in the station due to a power failure. In this case, the following
events take place: (i) the flow rapidly diminishes to zero and then reverses, (ii)
when the flow reverses, positive pressure waves propagate downstream of the
pump towards the reservoir, and negative pressure waves propagate upstream
of the pipe towards the suction of the pump, (iii) the pump rapidly losses its
forward rotation, and reverses, (iv) when the flow reverses, to prevent reverse
(*) Part of this work has been published in:
Lee, T. S., Low, H. T., and Nguyen, D. T. (2007). Effects of air entrainment on fluid transients in
pumping systems. Journal of Applied Fluid Mechanics, Vol. 1, No. 1, pp55-61.
83
flow through the pump, the check valve is activated and closed. A large
pressure transient occurs in the pipeline. The investigation considered two
cases of studies. The first case was for water hammer with air entrainment,
where the transient pressure is always above the saturation pressure. The
second case was for fluid transient with gaseous cavitation, where the transient
pressure fluctuates above and below the saturation pressure. The numerical
method with the variable wave speed model is applied for both cases.
5.2. GRID INDEPENDENCE TEST
The purpose of grid independence test is to determine minimum grid
resolution required to generate a solution that is independent of the grid used.
Grid resolution relates directly to the time step ∆t by the CFL
criterion ( )aVxkt +∆≤∆ / , a small ∆x requires a vey small time step ∆t. The
using of a too fine grid may cost longer computational time and computer
storage. Starting with a coarse grid (N = 101) the number of nodes was
increased until the solution from each grid was almost unchanged for
successive grid refinements. The grid independence test was performed on a
pipeline profile as shown in Fig. 5.1 (Lee and Leow 1999). The equivalent
steady flow rate is 0.89m3/s with the initial air void fraction 0.01. The grid
resolution was based on the number of nodes used to model the pipeline.
Seven grid refinements were performed. The results of the grid independence
test are presented in Fig. 5.2 comparing the pressure transient at the
downstream of check valve.
84
90
95
100
105
110
115
0 1000 2000 3000 4000 5000
Chainage (m)
Ele
vati
on
(m
)
Sump level = 92.8m
Peak level = 107.5m
Inlet level = 106.0m
Resevoir level = 112.5m
Check valve
Fig. 5.1. Pumping station pipeline profile
0
5
10
15
20
25
30
35
40
0 50 100 150 200 250 300
Time (s)
Pre
ss
ure
he
ad
(m
wa
ter)
N = 101N = 201N = 501N = 1001N = 2001N = 3001N = 4001
Fig. 5.2. Pressure transient at check valve using different grid resolution
85
30
31
32
33
34
35
36
37
38
100 102 104 106 108 110 112 114
Time (s)
Press
ure h
ea
d (
m w
ate
r)
N = 101
N = 201
N = 501
N = 1001
N = 2001
N = 3001
N = 4001
(a) Peak pressure P1
10
11
12
13
14
15
155 160 165 170 175
Time (s)
Press
ure h
ead
(m
wate
r)
N = 101
N = 201
N = 501
N = 1001
N = 2001
N = 3001
N = 4001
(b) Lowest pressure P2
Fig. 5.3. The change of pressure value with grid resolution
86
Table 5.1. Grid independence test result
N P1
(m water) ∆P
(m water) % error
P2 (m water)
∆P (m water)
% error
101 37.37 12.13
201 33.83 -3.54 9.47 12.62 0.49 4.04
501 34.48 0.66 1.94 12.15 -0.48 3.77
1001 34.85 0.36 1.06 11.84 -0.31 2.51
2001 35.25 0.40 1.16 11.58 -0.26 2.20
3001 35.42 0.17 0.49 11.48 -0.10 0.85
4001 35.57 0.15 0.41 11.43 -0.05 0.47
Some change in the predicted pressure surges was found with each
successive grid refinement. Fig. 5.3 shows the change in the peak pressure P1
and the lowest pressure after the first peak P2. By increasing the number of
node N of the computational grid, the error of the predicted pressure P1 and P2
between two successive grid is reduced as shown in Table 5.1. By considering
an expected error less than 1%, the difference between the N = 2001 and N =
3001 predictions was 0.85% and smaller than the expected error. Therefore,
any further grid refinement would not improve the current pressure transient
solution. The grid resolution N = 3001 was used for the simulation of pumping
system with pipeline profile shown in Fig. 5.1.
90
95
100
105
110
115
0 200 400 600 800 1000 1200 1400 1600 1800
Chainage (m)
Ele
vati
on
(m
)
Sump level = 92.8m
Inlet level = 106.0m
Resevoir level = 112.5m
Check valve
Fig. 5.4. Pipeline contour for pumping station
87
The same grid independence test was carried out for the simulation of
pumping system with pipeline profile shown in Fig. 5.4. We chose the grid
resolution N = 1001.
5.3. WATER HAMMER WITH AIR ENTRAINMENT
The effects of air entrainment on water hammer generated by simultaneous
pump trip at pumping station were studied using a pipeline contour in Fig. 5.4.
The pumping station consists of three parallel centrifugal pumps to supply 0.5
m3/s water to an upper reservoir through a 0.985m diameter main of 1550 m
length. The material of pipeline is mild steel. The value of variables
downstream of pump (check valve position) is specially noticed. In this case,
the value of transient pressure due to pump trip is above the saturation
pressure, therefore the dissolved gas in the liquid is not considered. The effects
free entrained air in the liquid is investigated.
Fig. 5.5 and 5.6 show the general effects of air entrainment on pressure
transient in pumping system after pump trip. The transient pressure with
presence of an initial free entrained air of volumetric void fraction ε0 = 0.001
(at ambient temperature and pressure) in the liquid is compared with the
transient pressure with no air content. The figures show distinct characteristic
differences between two transient pressures:
(i) The first pressure peak predicted by the variable wave speed model
is higher than the first pressure peak which was predicted by the
constant wave speed model, and both the time to get the first peak
and the time to complete a pressure surge period is longer.
88
(ii) The damping of the pressure surges is noticeably faster in
comparison with the damping in constant wave speed model.
(iii) The pressure surges are asymmetric with respect to the static head,
while the pressure surges for the constant wave speed model are
symmetric with respect to the static head.
This pressure transient behaviour of the system with air entrainment
was also observed by other researchers (Whiteman and Pearsall 1959, 1966,
Dawson and Fox 1983, Jonsson 1985, Burrows and Qiu 1995, Kapelan et al.
2003; Covas et al. 2003, Malanca et al. 2006).
0
10
20
30
40
50
60
0 10 20 30 40 50 60 70 80
Time (seconds)
Pre
ssu
re h
ead
(m
ete
rs)
Constant wave speed
Variable wave speed
Fig. 5.5. Pressure head downstream of pump
89
Fig. 5.6. Max. and min. pressure head along pipeline
Many studies have also given attempts to explain the above effects of
air entrainment on pressure transient. For the increase in peak pressure,
Jonsson (1985) attributed the increase in peak pressure to the compression of
‘an isolated air pocket’ in the flow. Dawson and Fox (1983) proposed the
‘cumulative effect of minor flow changes during the transient’. In this
investigation, before proposing an explanation of the pressure transient
characteristics with air entrainment, more numerical experiment has been
carried out by changing the initial air void fraction.
90
0
200
400
600
800
1000
1200
1400
0 20 40 60 80
Time (seconds)
Wave s
peed
(m
/s)
eps = 0.0001
eps = 0.001
eps = 0.01
Fig. 5.7. Wave speed with different initial air void fractions
Fig. 5.7 shows the variation of the wave speed at the point after the
check valve with different initial air void fractions. In the variable wave speed
model, wave speed at every point in the system depends on the local pressure
and local air void fraction at that point. Local pressure and local air void
fraction varied along the pipeline and varied on time (Fig. 5.8). Therefore, the
wave speed was no longer a constant as in the constant wave speed model but
varied along the pipeline and varied on time. In addition, by increasing the
value of the initial air void fraction, the value of wave speed was greatly
reduced. Consequently, the pressure surge periods were increased
proportionally with air void fraction due to the greatly reduce of wave speed.
Meanwhile, the dependence of wave speed on the initial air void fraction
implies that the effects of air entrainment on pressure transient may be more
significant under low pressure conditions, where its volume is greater than
under high-pressure conditions. This also explains the asymmetric and
different periods of down-surge and upsurge. This result is consistent with the
91
experimental of observation of various earlier investigators such as Brown
(1968) and Evans (1983).
0.00E+00
1.00E-04
2.00E-04
3.00E-04
4.00E-04
5.00E-04
6.00E-04
7.00E-04
8.00E-04
9.00E-04
1.00E-03
0 10 20 30 40 50 60 70 80
Time (seconds)
Air
vo
id f
rac
tio
n
eps = 0.001
Fig. 5.8. Air void fraction at check valve
0
10
20
30
40
50
60
0 10 20 30 40 50 60 70 80
Time (seconds)
Pre
ssu
re h
ead
(m
ete
rs)
eps = 0.0001eps = 0.001eps = 0.01
Fig. 5.9. Pressure head with different initial air void fractions
92
-10.00
0.00
10.00
20.00
30.00
40.00
50.00
0 200 400 600 800 1000 1200 1400
Chainage (meters)
Pre
ss
ure
he
ad
(m
ete
rs)
eps = 0.0
eps = 0.0001
eps = 0.001
eps = 0.01
Fig. 5.10. Max. and min. pressure head along pipeline
The comparison of the transient pressures with different amout of air
void fraction was shown in Fig. 5.9 and 5.10. The numerical result shows that
when the initial air void fraction was increased, the pressure head of the first
pressure peak grossly increased to a maximum value then slightly decreased
(Fig. 5.11). The increase in the peak pressure is now explained by the lapping
of the effects of two factors: (i) the delay wave reflection at reservoir, and (ii)
the change of wave speed. Free air in the liquid increases the effective bulk
modulus and thus lower the average wave speed. As a result, the wave
reflection at reservoir is delayed. Therefore, a more complex variation in
pressure interaction occurs in the system, culminating in a peak at a specific
transient interval. Meanwhile, the reduction of the wave speed of the mixture
directly causes changes in the strength of pressure oscillations. When the
positive effect from the delayed wave reflection couldn’t compensate for or
exceed the negative effect from the reduction of wave speed, a suppressed
93
pressure peak happens. This explains why the first peak pressure increases
initially and then decreases with the increase of the initial air void fraction.
The initial air void fraction which gives the highest first pressure peak depends
on the characteristics of the pumping systems and could be found out by doing
this numerical simulation with changing of initial air void fraction. For this
pumping system, the initial air void fraction ε ≈ 0.003 gives the highest
pressure peak (Fig. 5.11).
0
10
20
30
40
50
60
70
0 0.005 0.01 0.015 0.02 0.025
Initial Air void fraction
Ma
x.
an
d M
in.
Pre
ssu
re
Maximum pressure
Minimum pressure
Fig. 5.11. Effects of air content on max. and min. transient pressure head
The damping of the pressure surge is fast for the variable wave speed
model. This is due to the fact that air content is an important source of
damping beside friction and minor losses. Generally, the numerical
experiments show that the damping produced by the loss factor is much
smaller than that produced by the gas content in the fluid during the pressure
transient process. Damping effect of air content is suggested in the forms of (i)
direct damping owing to the increased effective bulk viscosity of the fluid-gas
mixture, (ii) losses due to slip between air bubbles and water, (iii)
94
thermodynamic losses, and (iv) indirect damping due to partial wave
reflection. This may explain the fast damping of the pressure surge in the
variable wave speed model in comparison with the constant wave speed
model. However, the precise physical cause of large surge damping in
transient flow with air content is still a subject of current researches (Wolters
and Muller, 2004; Cannizzaro and Pezzinga, 2005).
In short, this investigation shows that in comparison with the standard
constant wave speed model, the variable wave speed model, which counted on
the effects of air entrainment, can improves the simulation of experimental
observations in terms of the shape of the pressure peaks, the frequency of the
oscillations and the rate of decay.
5.4. FLUID TRANSIENT WITH GASEOUS CAVITATION
In transient conditions, when the pressure in a pipe drops, free gas and vapour
cavities in liquid flow may develop (Bergant and Simpson, 1999; Cannizzaro
and Pezzinga, 2003). In general, two cases can be distinguished (Zielke et al.
1990): either the pressure drops below the saturation pressure but still keeps
above the vapour pressure, or the pressure drops to the vapour pressure of the
liquid. In the former case, gaseous cavitation takes place, characterized by the
presence of a large number of gas nuclei (Cannizzaro and Pezzinga, 2005).
When the pressure drops suddenly, a significant gas release may occur. In the
later case, vapourous caviatation takes place, and when the fluid pressure
drops to its vopour pressure, a sudden growth of the nuclei containing vapour
occurs (Wiggert and Sundquist, 1979).
95
In this section, the pressure transient with gaseous cavitation generated
by simultaneous pump trip at pumping station was studied using a pipeline
contour in Fig. 5.1. The pumping station consists of three parallel centrifugal
pumps to supply 0.89 m3/s water to an upper reservoir through a 0.985 m
diameter main of 4725 m length. The material of pipeline is mild steel. The
value of variables at check valve position and at the peak level position is
specially noticed.
The model proposed here considers the presence of an initial free
entrained air of volumetric void fraction εo and dissolved gas content of
relative volumetric void fraction εg in the liquid at atmospheric pressure and
ambient temperature. We also assume that: (i) the gas-liquid mixture is
homogeneous; (ii) the free gas bubbles in the liquid follow a polytrophic
compression law with n = 1.2-1.3 (due to the occurrence of some heat transfer
from the heated fluid in the pipeline to the surrounding soil); and (iii) the
pressure and temperature within the air bubbles during the transient process is
in equilibrium with the local fluid pressure and temperature. The above model
considers that when the local pressure falls below the vapour pressure Pv, an
instant cavitation at Pv under transient conditions occurs. The local transient
pressure remains at Pv until the transient pressure recovers to a value above Pv..
When the computed local transient pressure falls below the gas release
pressure Pg, it is assumed that there is a release of dissolved gas of αgrεg with a
time delay of Kr∆t. The local pressure remains constant and is equal to the
vapour pressure. When the computed transient pressure recovers to a value
above Pg, it is assumed that an equivalent amount of αgaεg gas is absorbed into
the fluid with a time delay of Ka∆t. For the comparative study of constant
96
wave speed cases, the present study assumed ε = 0.000 (fully de-aerated
water) in the fluid system even when the transient pressure falls below the Pg
or Pv, i.e. the system is assumed completely free from the influence of
entrapped air or water vapour. Whenever the transient pressure falls below Pv,
it is assumed that minimum pressure remained at Pv for the duration of this
transient. Hence, the values of the variable speed will have a lower bound,
which is consistent with the field data observed.
The transient pressures with εg = 0.01, αgr = 0.6, αga = 0.3, Ka = 1.005,
Kr = 1.001,εv =0.021 and different initial air void fraction ε0 were shown in
Fig. 5.12, Fig. 5.13 and Fig. 5.14. With this pumping system, the transient
pressure was reach the saturation pressure and vapor pressure at some region
along the pipeline. If there is evolution and subsequent absorption of gas in the
liquid along the pipeline, the free air entrainment and the release of gas at the
saturation pressure reduce the local wave speed considerably and produce a
complicated phenomenon of reflection of pressure waves off these ‘cavities’.
The lower local wave speed also increases the duration of the pressure down-
surge as compared with the duration of the pressure up-surge. In general, the
pressure transient with gaseous cavitation showed the same characteristics
with the water hammer with air entrainment.
97
0
5
10
15
20
25
30
35
40
0 50 100 150 200 250
Time (seconds)
Pre
ssu
re h
ea
d (
me
ters
) Constant wave speed
Variable wave speed
Fig. 5.12. Pressure head downstream of pump
0
5
10
15
20
25
30
35
40
0 50 100 150 200 250Time (seconds)
Pre
ssu
re h
ea
d (
me
ters
) eps = 0.0001
eps = 0.001
eps = 0.01
Fig. 5.13. Pressure head with different initial air void fraction
98
25
27
29
31
33
35
37
39
0 0.005 0.01 0.015 0.02Initial air void fraction
Pre
ss
ure
he
ad
of
firs
t p
ea
k (
m)
Fig. 5.14. Pressure head of first pressure peak with initial air void fraction
0.0E+00
2.0E-04
4.0E-04
6.0E-04
8.0E-04
1.0E-03
1.2E-03
1.4E-03
1.6E-03
0 50 100 150 200 250
Time (seconds)
Air
vo
id f
rac
tio
n
Check valve position
Peak level position
Fig. 5.15. Variation of air void fraction with initial value ε0 = 0.001
99
0
100
200
300
400
500
600
700
800
900
1000
0 50 100 150 200 250
Time (seconds)
Wa
ve
sp
ee
d (
m/s
)
Check valve positionPeak level position
Fig. 5.16. Variation of wave speed with initial air void fraction ε0 = 0.001
The effects of gaseous cavition was investigated by comparing
between the pressure transient with free entrained air only and the pressure
transient with free entrained air and gas release at saturation pressure. This
simulation was done with the initial air void fraction ε0 = 0.001. At check
valve position, the transient pressure was above the saturation pressure. At the
peak level position, when only free entrained air was considered, the transient
pressure reached the vapor pressure of the liquid. Theoretically, vaporous
caviation and column separation may occur at the peak level position since
transient pressure reduces to vapor pressure. However, when gaseous
cavitation was considered at the peak level position, the effects of gas release
increased the transient pressure to exceed the vapor pressure of the liquid.
Hence, by counting on the effects of gas release, the pressure transient at the
peak level position could avoid the happening of unexpected vaporous
cavitation and/or column separation. By considering the effects of gas release
along the pipeline, the vapour pressure region had been reduced. Generally, a
100
small amount of free entrained air and released gas could increase the first
pressure peak leading to a more severe transient; while a large amount of free
entrained air and released gas could help the system to avoid vaporous
cavitation and/or column separation by increasing the value of the down-surge
pressure to above the vaporous pressure of the liquid as shown in Fig. 5.19.
-5
0
5
10
15
20
25
30
35
40
0 50 100 150 200 250
Time (seconds)
Pre
ss
ure
he
ad
(m
ete
rs) Check valve position
Peak level position
Fig. 5.17. Pressure transient without the effects of gas release (ε0 = 0.001)
-5
0
5
10
15
20
25
30
35
40
0 50 100 150 200 250
Time (seconds)
Pre
ss
ure
he
ad
(m
ete
rs) Check valve position
Peak level position
Fig. 5.18. Pressure transient with the effects of gas release (ε0 = 0.001)
101
-5
0
5
10
15
20
25
30
35
40
0 1000 2000 3000 4000
Chainage (meters)
Pre
ss
ure
he
ad
(m
ete
rs)
eps = 0.001
eps = 0.02
Fig. 5.19. Maximum and minimum pressure head long pipeline
5.5. SUMMARY
This chapter presents the application of the variable wave speed model to
study the effects of air entrainment on the pressure transients in pumping
systems. Free gas in the liquid and gaseous cavitation at the saturation
pressure was modeled. The numerical experiments were carrired out for two
case of study: water hammer with air entrainment and fluid transient with
gaseous cavitation. Numerical experiments show that where a small amount of
gas content and air entrainment exist, the first peak pressure was amplified;
the damping effect was fast; and pressure surges were asymmetric with respect
to static head. The pressure transient showed a longer period for down-surge
and a shorter period for up-surge. The increase in the peak pressure was
proposed an explanation by the lapping of the effects of two factors: the delay
wave reflection at reservoir and the change of wave speed. By increasing the
air void fraction, the pressure head of the first pressure peak grossly increased
102
to a maximum value then slightly decreased. The down-surge pressure value
was increased with the increasing of the air void fraction. A large enough
amount of free entrained air and released gas could increase the transient
pressure value to above the vaporous pressure of the liquid and could help
avoid the vaporous cavitation and/or column separation.
103
CHAPTER 6
EXPERIMENTAL STUDY OF
CHECK VALVE PERFORMANCES IN
FLUID TRANSIENT WITH AIR ENTRAINMENT (*)
6.1. INTRODUCTION
A common flow system arrangement in a pumping station consists of a lower
reservoir, a group of pumps with a check valve in each branch and a pipeline
discharging into an upper reservoir. Check valves are fitted to pipelines in
order to prevent the lines from draining backwards when the pumps stop and
sometimes also to prevent the downstream reservoirs from emptying. In
addition, they prevent reverse rotation of pumps to protect pumps and others
devices such as seals and brush gear (Thorley, 1989). An ideal check valve
closes at the instant of flow reversal. In practice, check valves seldom close
precisely at zero reverse flow velocity. The most dangerous case of pressure
transients in pumping system is the stoppage of pumps due to a power failure.
To prevent reversal of flow through the pump, when the flow reverses the
check valve is activated and closed. However, limited flow reversal will still
occur due to the inertia and friction of the check valve components. The
sudden closure of the check valve at a reverse velocity can cause large
(*) Part of this work has been published in:
Lee, T. S., Low, H. T., Nguyen, D. T., and Neo, W. R. A., (2009). Experimental study of check valves
performances in fluid transient. Proc. IMechE., Journal of Process Mechanical Engineering. Vol. 223,
No. 2, pp. 61-69.
Lee, T. S., Low, H. T., Nguyen, D. T., and Neo, W. R. A., (2008). Experimental Study of Check Valves
in Pumping System with Air Entrainment. International Journal of Fluid Machinery and Systems. Vol. 1,
No. 1, pp. 140-147.
104
pressure surges downstream of the check valve and negative pressures
upstream of the check valve. These pressure transients may cause a collapse of
the pipeline or damage to the hydraulic equipments in the system. Therefore,
in order to minimize the pressure surge, the maximum reverse velocity Vr
which occurs virtually at the instant of closure, should be kept as close to zero
as possible. This can only be achieved by careful design and selection of check
valves.
According to Ballun (2007), the solution to preventing check valve
slam is not to find the fastest-closing check valve and make it the “standard”
but to match the non-slam characteristics of the check valve to the pumping
system. In his paper, Thorley (1989) suggested a few criteria to follow when
selecting a check valve in order to avoid valve slamming. According to these
suggestions, check valves should have low inertia of moving parts, small
travel distance/angle and motion assisted by springs. To predict the
performances of check valves, data of the valves under dynamic conditions
have to be known in order to aid in analysis. The extend to which actual valves
possess these features can be illustrated graphically in the form of Pressure
Surge Analysis, Dynamic Performance Characteristics (Provoost 1982) and
Dimensionless Dynamic characteristics (Koetzier et al., 1986). Even if the
valve has the best performances, other factors such as cost and head loss also
need to be considered. In many cases, no valve satisfies all criterions; hence,
compromises have to be made in the final choice of valves.
Some studies have been done on the effect of air entrainment in a
pipeline system and its subsequent effect on check valve performances.
Through the numerical studies, Lee (1995; 2001) reported that the transient
105
flow velocities near the check valve of a fluid system should also depend on
the characteristics of air being entrained in the fluid system. Consequently,
different amount of air entrainment were also found to affect the pressure
peaks. The maximum and minimum pressure peak need not necessary occurs
at the highest or lowest amount of air entrainment but can also occur anywhere
in between the range of air level.
The present experimental study aims to investigate the performance of
five different self-actuating check valves in the event of a pump-trip through a
set of various comparison methods and analyze how the methods measure up
to each other. In addition, the effects of air entrainment on the pressure
transient induced by check valve closure are also being analyzed. The focus
would be mainly on the downstream pressure transient since in real-life
pumping station, it is the water hammer effect downstream of the check valves
rather than upstream which poses a greater danger to the whole piping system.
6.2. TEST RIG, INSTRUMENTATION AND TEST METHOD
The test rig for check valves is shown in Fig. 6.1. The test facility consisted of
a 2 hp single pump with speed 1450 rpm, a upper reservoir with overflow
located 5.5 m above sump level, a nominal 89 mm diameter PVC piping of
15.75 m length, a control valve and an air compressor. The pump supplies
water at temperature of about 25oC from the basement tank to the head tank.
The water level in the head tank is constant by means of an overflow at a
height of about 5.5 m above sump level. Excess of water flows back in to the
supply reservoir. The compressor is used to supply external air through a tube
connected to the suction bell-mouth at the lower reservoir. The air supplied
from the compressor is being controlled by a release valve and the amount of
106
air introduced can be read through a rotameter. Check valves are installed at
test section 5.75 m downstream of the pump. The flow through the 1.2 m long
test section has fully recovered at the measuring points to reduce the irregular
fluid distribution inside the tube itself. A Kistler 4603B10 piezoresistive
amplifier with frequency 1 kHz is used for pressure measurement. The
pressure transducer is flush with the tube surface so that there is no protraction
into the tube to avoid vorticity effects due to local turbulence. An
electromagnetic flow-meter is used to measure the flow velocity in the pipe.
The steady flow conditions are controlled by a control valve. The hydraulic
pressure transients are initiated by pump trip due to power failure. The key
measurement location is at the check valve.
Fig. 6.1. Hydraulic schematic of the pumping system
Experimental variables were the type of check valves used, setup
orientation, flow rates and the air entrainment levels. Each check valve was
tested at 5 different air entrainment levels ranging from 0.0% to 0.6%, with a
0.15% interval. Under each air entrainment level, each check valve was further
subjected to different flow rates ranging from 2.5 l/s to 5 l/s, with a 0.5 l/s
interval. Each experiment is repeated three times. The experimental procedure
107
flowchart is shown in Fig. 6.2. The five different types of check valve being
used in the test as shown in Fig. 6.3 are swing check valve, ball check valve,
piston check valve, double-flap check valve and nozzle check valve.
Fig. 6.2. Experimental procedure flowchart
108
a) Ball Check Valve b) Swing Check Valve
c) Piston Check Valve d) Nozzle Check Valve
e) Double-Flap Check Valve f) Test Section
Fig. 6.3. Check valves used in the test and test section
The reverse flow velocity at the check valve is not measured; it is
calculated from the measured pressure surge by use of the water hammer
equation.
a
HgVR
∆= (6.1)
where VR is the reverse velocity in m/s, ∆H is the transient pressure in meters
of water, g is the gravitation constant 9.81 m/s2 and a is the wave speed in m/s.
109
Wave speed is a function of a number of different variables such as air
content, pipe material, pipe connections, etc. Along the pipeline, the fraction
of air content depends on the local pressure and local air volume. Therefore,
wave speed is not constant and is calculated for each point i which has local
pressure Pi and air fraction content εi by Eq. (3.14):
( )2
1
11
−
++−=
eE
Dc
nPKa l
k
i
k
ik
il
k
i
εερ (6.2)
In practice, it is difficult to identify the local air content at every point.
Therefore in this paper, the overall pressure wave speed a is determined from
the measured pressure transient period and overall time.
a
LNT p
2=∆ (6.3)
where T is the overall time, 2L/a is pressure transient period and Np is the
number of transient periods.
Hence
T
LNa p
∆=
2
(6.4)
Provoost (1982) proposed a graphical illustration of check valve
characteristics in the form of Dynamic Performance Characteristics. The
vertical axis is the maximum reverse velocity that occurs during valve seated.
The base line is the mean deceleration of the liquid column as the flow is
brought to rest. The deceleration of flow is calculated from Eq. (6.5)
ta
Hg
t
V
t
VV
dt
dV ORO
∆
∆+
∆=
∆
+≈ (6.5)
110
where the pressure head change ∆H is measured from the transient plots, and
VO is the measured initial velocity in m/s.
Furthermore, the dimensionless dynamic characteristic for check valve
can be shown by a plot of VR/VO vs (D/VO2).(dV/dt) (Koetzier, 1986).
6.3. RESULTS AND DISCUSSION
6.3.1. Pressure surge analysis
Ball check valve (horizontal), εεεε = 0.0%, Q = 4.5 l/s
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Ball check valve (horizontal), εεεε = 0.15%, Q = 4.5 l/s
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Fig. 6.4. Pressure transient in horizontal orientation of ball check valve
111
Swing check valve (horizontal), εεεε = 0.0%, Q = 4.5 l/s
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Swing check valve (horizontal), εεεε = 0.15%, Q = 4.5 l/s
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Fig. 6.5. Pressure transient in horizontal orientation of swing check valve
Piston check valve (horizontal), εεεε = 0.0%, Q = 4.5 l/s
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Piston check valve (horizontal), εεεε = 0.15%, Q = 4.5 l/s
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Fig. 6.6. Pressure transient in horizontal orientation of piston check valve
112
Nozzle check valve (horizontal), εεεε = 0.0%, Q = 4.5 l/s
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Nozzle check valve (horizontal), εεεε = 0.15%, Q = 4.5 l/s
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Fig. 6.7. Pressure transient in horizontal orientation of nozzle check valve
Double flap check valve (horizontal), εεεε = 0.0%, Q = 4.5 l/s
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Double Flap Check Valve (horizontal), εεεε = 0.15%, Q = 4.5 l/s
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Fig. 6.8. Pressure transient in horizontal orientation of double flap check
valve
113
Ball check valve (vertical), εεεε = 0.0%, Q = 4.5 l/s
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Ball check valve (vertical), εεεε = 0.15%, Q = 4.5 l/s
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Fig. 6.9. Pressure transient in vertical orientation of ball check valve
Swing check valve (vertical), εεεε = 0.0%, Q = 4.5 l/s
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Swing check valve (vertical), εεεε = 0.15%, Q = 4.5 l/s
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Fig. 6.10. Pressure transient in vertical orientation of swing check valve
114
Piston check valve (vertical), ε ε ε ε = 0.0%, Q = 4.5 l/s
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Piston check valve (vertical), εεεε = 0.15%, Q = 4.5 l/s
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Fig. 6.11. Pressure transient in vertical orientation of piston check valve
Nozzle check valve (vertical), εεεε = 0.0%, Q = 4.5 l/s
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Nozzle check valve (vertical), εεεε = 0.15%, Q = 4.5 l/s
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Fig. 6.12. Pressure transient in vertical orientation of nozzle check valve
115
Double flap check valve (vertical), εεεε = 0.0%, Q = 4.5 l/s
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Double flap check valve (vertical), εεεε = 0.15%, Q = 4.5 l/s
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0 1 2 3 4 5
Time (s)
Pre
ssu
re h
ead
(b
arg
)
Fig. 6.13. Pressure transient in vertical orientation of double flap check
valve
Fig. 6.4 to Fig. 6.13 show the pressure transients at a flow rate of 4.5
l/s for the different valves. The sample transient graphs are for initial air
fraction ε equal to 0% and 0.15%. From Fig. 6.4 to Fig. 6.8, it can be seen that
in horizontal orientation, the valves can be separate into two groups based on
their surge responses analysis, swing check valve and ball check valve in one
group while double flap, piston and nozzle check valve form the other group.
Both swing check valve and ball check valve displayed similar pressure surge
responses in the horizontal orientation. Both will cause a high pressure
transient at low air entrainment level which decreases with the increase of air
entrainment levels. On the other hand, double flap check valves, piston check
valves and nozzle check valves displayed similar response too. Their pressure
surges are smaller than the pressure surges formed by using the check valves
116
in the first group. Even at higher air entrainment levels, the surge responses of
these three check valves are still smaller than the other two. Generally, these
finding indicates that the valves in the second group are better pressure
protection devices than the valves in the first group.
In the vertical orientation, the check valves were installed in a vertical
test section with the up flow. The pressure transient results in the vertical
orientation are shown in Fig. 6.9 to Fig. 6.13. The swing check valve was
found to be the only valve to have a poorer response as compared to the other
valves at all air entrainment level. The surge responses of ball check valve is
reduced and in the range of the other three valves responses. Similar
characteristics of surge responses by all the check valves at the various flow
rates being tested can be observed, with swing and ball check valves
displaying poorer responses in the horizontal orientation while only the swing
check valves carry on showing poor responses in the vertical orientation. The
good performance of ball check valve in the vertical orientation compared
with its performance in the horizontal orientation shows the strong effect of
gravity on the ball check valve. Therefore, the preferred installation of a ball
check valve is in the vertical position. This will insure that gravity will
position the ball check valve properly.
117
6.3.2. Dynamic characteristics
Dynamic characteristics (horizontal), εεεε = 0.0%
Ball
Swing
DF (S)
DF (W)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.0 1.0 2.0 3.0 4.0
dv/dt (m/s2)
Vr
(m/s
)
DF
Swing
Piston
Nozzle
Ball
Thorley's experiment
Present experiment
(a) for ε = 0%
Dynamic characteristics (horizontal), εεεε = 0.15%
Ball
Swing
DF (S)
DF (W)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.0 1.0 2.0 3.0 4.0dv/dt (m/s
2)
Vr
(m/s
)
DF
Swing
Piston
Nozzle
Ball
Thorley's experiment
Present experiment
(b) for ε = 0.15%
Fig. 6.14. Dynamic characteristics chart in horizontal orientation
118
Dynamic characteristics (vertical), εεεε = 0.0%
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1.0 1.5 2.0 2.5 3.0dv/dt (m/s
2)
Vr
(m/s
)
DF
Swing
Piston
Nozzle
Ball
(a) for ε = 0%
Dynamic characteristics (vertical), εεεε = 0.15%
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1.0 1.5 2.0 2.5 3.0dv/dt (m/s
2)
Vr
(m/s
)
DF
Swing
Piston
Nozzle
Ball
(b) for ε = 0.15%
Fig. 6.15. Dynamic characteristics chart in vertical orientation
119
Provoost (1982) suggested that for each type of check valve, a response curve
of the dynamic characteristics should be generated to show the relationship
between the deceleration of the liquid column and the maximum reverse
velocity through the check valve. From this experiment study, the dynamic
characteristics of five types of check valves are shown in Fig. 6.14 and 6.15.
The dotted-lines in Fig. 6.14 and 6.15 represent the data obtained by Thorley
(1989). It can be seen that the results obtained are comparable to those
observed by Thorley in terms of valve performances. A good check valve
should have the value of VR minimized and have a small gradient in terms of
dv/dt. In other words, the maximum reverse velocity when the check valve
close should always be kept at a minimal at all deceleration values. From Fig.
6.14, the swing check valve has the steepest gradient, followed by the ball
check valve. Piston, nozzle and double flap check valve displayed similar
trend in terms of dynamic characteristics.
In the vertical orientation (Fig. 6.15), swing check valve is the only
valve to have a much steeper gradient as compared to the other four valves.
From the use of these dynamic characteristic curves to determine valve
performances, it was found that swing check valve consistently displayed a
poorer response as compared to piston, nozzle and double flap check valve in
both orientations. Beside, the responses of ball check valve is greatly
dependent on the orientation of the valve. The dynamic characteristics of
check valves were found to be similar when the valves were tested at different
air entrainment level. Pressure surge analysis and dynamics characteristics of
check valve both give the same indication to predict the potential valves which
have the great possibility to slam pressure transient condition. However, the
120
dynamics characteristics can show clearer and more readable information. It is
why the dynamics characteristics of check valve is applied in practice.
6.3.3. Dimensionless dynamic characteristics
Dimensionless dynamic characteristics (horizontal), εεεε = 0.0%
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.2 0.4 0.6 0.8
(dv/dt)(D/Vo2)
Vr/
Vo
Ball
Swing
DF
Piston
Nozzle
(a) for ε = 0.0%
Dimensionless dynamic characteristics (horizontal), εεεε = 0.15%
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.2 0.4 0.6 0.8
(dv/dt)(D/Vo2)
Vr/
Vo
Swing
DF
Piston
Nozzle
Ball
(b) for ε = 0.15%
Fig. 6.16. Dimensionless dynamic characteristics in horizontal orientation
121
Dimensionless dynamic characteristics (vertical), εεεε = 0.0%
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.2 0.4 0.6 0.8
(dv/dt)(D/Vo2)
Vr/
Vo
Swing
DF
piston
Nozzle
Ball
(a) for ε = 0.0%
Dimensionless dynamic characteristics (vertical), εεεε = 0.15%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0 0.2 0.4 0.6 0.8
(dv/dt)(D/Vo2)
Vr/
Vo
Swing
Ball
DF
Piston
Nozzle
(b) for ε = 0.15%
Fig. 6.17. Dimensionless dynamic characteristics in vertical orientation
122
By normalizing the dynamic chart with the steady state velocity, it was found
that the dimensionless chart (Fig. 6.16 and 6.17) displayed similar
performances of the check valves despite the slight change in gradient. Swing
check valves shows poorer performances in both orientation while nozzle,
piston and double flap check valve consistently displayed a better response.
Thus, dimensionless chart may be preferred in place of a dynamic chart as
performances of valves will not be distorted. In addition, the dimensionless
chart helps reduce the complexity of analysis and allows for fewer tests to be
carried out. Results of dimensionless chart at other air entrainment levels were
observed to display consistent trends as shown by both the pressure surge
analysis and dynamic characteristics chart.
6.4. SUMMARY
An experiment setup was introduced to study dynamic behaviour of difference
types of check valves under pressure transient condition. The experiment data
of the check valves were compared by using pressure surge analysis, check
valve dynamic characteristic and check valve dimensionless dynamic
characteristic respectively. All three comparisons give same findings that in
the horizontal orientation, the check valves with low inertia, assisted by
springs or small traveling distance/angle such as piston check valve, nozzle
check valve and double flap check valve gave better performance under
pressure transient condition than check valves without these features such as
swing check valve and ball check valve. Meanwhile in the vertical orientation,
only the swing check valves carry on showing poor responses. The better
performance of ball check valve in the vertical orientation in comparison with
123
its performance in the horizontal orientation shows the strong effect of gravity
on the ball check valve.
Although different amount of air entrainment was found to affect the
experimental readings, the general characteristics of each check valves
remains the same when compared among check valves. Pressure surge
analysis, check valve dynamic characteristic or check valve dimensionless
dynamic characteristic can offer the designer the tools necessary to evaluate
the pressure transient characteristics of various check valves. This information
combined with other valve characteristics such as head-loss, laying length, and
cost will provide the designer necessary tools to help choosing suitable check
valves for a particular pumping system.
124
CHAPTER 7
CONCLUSIONS AND RECOMMENDATIONS
7.1. CONCLUSIONS
In this thesis, a variable wave speed model was developed for computational
and modeling of fluid transient in complex systems with air entrainment. An
experiment setup was build to validate the proposed variable wave speed
model. The variable wave speed model was then used to investigate the fluid
transient with air entrainment in the pumping system. In addition, this thesis
presented an experimental study of check valve performances in fluid
transients with air entrainment. The following conclusions were derived from
the present study:
i. The proposed variable wave speed model can predict an abrupt and
sharp pressure rise for the pressure surge shape, predict detail of
the characteristics of the pressure surge period, as presented by the
test measurement; conservatively predict the maximum and
minimum pressure transients; and capture the effects of air
entrainment on fluid transients.
ii. In comparison with the standard constant wave speed model, the
variable wave speed model can improve the simulation of
experimental observations in terms of shape of the pressure peaks,
the frequency of the oscillations and the rate of decay.
125
iii. Entrained, entrapped or released gases amplified the first pressure
peak, increased surge damping and produced asymmetric pressure
surges with respect to the static head.
iv. To explain the increase in peak pressure, the study suggested that
the higher pressure peak is caused by the lapping of the effects of
two factors: the delay wave reflection at reservoir and the change
of wave speed.
v. By increasing the air void fraction, the pressure head of the first
pressure peak grossly increased to a maximum value then slightly
decreased. The down-surge pressure value was increased with the
increasing of the air void fraction.
vi. A large enough amount of free entrained air and released gas in the
liquid could increase the transient pressure value to be greater the
vaporous pressure of the liquid and could help to avoid the
vaporous cavitation and/or column separation.
vii. This study indicated that the exclusion of the effects of air
entrainment may cause the transient pressure calculation to be
inaccurate.
viii. Although different amount of air entrainment was found to affect
the experimental readings, the general characteristics of each check
valves remains the same when compared among check valves.
7.2. RECOMMENDATIONS FOR FUTURE WORK
Although the variable wave speed model can provide a closely transient
pressure prediction for the first pressure peak in comparison with experimental
126
value, this model could not capture accurately the damping rate of pressure
surges. The reason may come from the fact that the variable wave speed model
can only count the damping caused by increased bulk viscosity of the fluid but
ignores the damping caused by slipping between air bubbles and water, and
the thermal exchange between gas bubbles and surrounding liquid. In addition
this model only provides 1-D simulation; the characteristic of the flow at every
section is represented by the average value at the centre point. Therefore, the
simulation may miss out some effects of the real flow such as wall effects,
liquid-structure interaction, and turbulent flow. For future work, it is necessary
to refine the variable wave speed models so that it can improve the accuracy of
the results especially in the damping prediction. These refinements should
consider the thermal exchange between gas bubbles and the surrounding liquid
as well as develop a 2-D model.
The proposed variable wave speed model is based on the assumption
of homogeneous two phase flow. Therefore the model is only applicable for a
small amount of air content in the liquid. When air is entrained such that the
gas void fraction is significant and two phase motion occurs between the water
and air in bubbles, pockets and/or voids, the fluid transient in the flow must be
considered by other model. On the other hand, the variable wave speed model
does not include a shock-fitting technique/shock-capturing procedure to
handle discontinuities or shocks between the water hammer and vaporous
region. Therefore this variable wave speed model could not model the
vaporous cavitation and column separation phenomena.
In this thesis, the investigation of pressure transient was restricted to
the complex system without the installation of pressure surge protection
127
devices such as air vessels, air valves, surge tanks etc. In practical systems,
these devices are used properly to protect the system under excessive pressure
transient conditions. The ability of these hydraulic components in pressure
surge suppressions should be affected by air entrainment. The variable wave
speed model can be applied to carry out these further investigations.
128
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PUBLICATIONS
1. Lee, T. S., Low, H. T., and Nguyen, D. T. (2006). Effects of air
entrainment on fluid transients in pumping systems. Proceedings of
the Eleventh Asian Congress of Fluid Mechanics，Kuala Lumpur,
Malaysia 22nd
-25th
May 2006
2. Lee, T. S., Low, H. T., and Nguyen, D. T. (2007). Effects of air
entrainment on fluid transients in pumping systems. Journal of
Applied Fluid Mechanics, Vol. 1, No. 1, pp55-61.
3. Lee, T. S., Nguyen, D. T., Low, H. T., and Koh, J. Y., (2008).
Investigation of fluid transients in pipelines with air entrainment.
Advanced and Applications in Fluid Mechanics, Vol. 4, Issue 2, pp.
117-133.
4. Lee, T. S., Low, H. T., Nguyen, D. T., and Neo, W. R. A., (2008).
Experimental Study of Check Valves in Pumping System with Air
Entrainment. International Journal of Fluid Machinery and Systems.
Vol. 1, No. 1, pp. 140-147.
5. Lee, T. S., Low, H. T., and Nguyen, D. T., (2008). The Effects of air
entrainment on the pressure damping in pipeline transient flows.
Proceedings of the 12th Asian Congress of Fluid
Mechanics，Daejeon, Korea 18-21 August 2008.
6. Lee, T. S., Low, H. T., Nguyen, D. T., and Neo, W. R. A., (2009).
Experimental study of check valves performances in fluid transient.
Proc. IMechE., Journal of Process Mechanical Engineering. Vol.
223, No. 2, pp. 61-69.
146
APPENDICES
Appendix A: Experimental setup specifications
Pipe Length 15.75 m
Pipe Material Perspex, PVC
Modulus Elasticity of Pipe, E 2.835 GN/m3
Poisson’s Ratio 0.38
Pipe External Diameter, De 89 mm
Pipe Internal Diameter, Di 77 mm
Pipe-wall Thickness, e 6 mm
Pump Specifications
Speed, N 1450 rpm
Total Head, H 5.12 m
Capacity, Q 60 m3/h
Motor 2 HP
147
Appendix B: Chaudhry et at. (1990) experimental setup specifications
Pipe Length, L 30.6 m
Constant wave speed 715 m/s
Constant upstream reservoir pressure 18.46 m
Steady flow velocity 2.42 m/s
Steady air mass flow rate 4.1 x 10-6
kg/s
Pipe Diameter 0.026 m
Downstream void ratio 0.0023
Steady flow friction factor 0.0205
148
Appendix C: Technical data for the simulation of pumping systems
Pipeline longitudinal profile As given in Fig. 5.1 As given in Fig.
5.4
Sump Level 92.8 m 92.8 m
Delivery Exit Level 112.5 m 112.5 m
Distance from pumping station to
delivery exit
4725 m 1550 m
Pipe Material Steel Steel
Modulus Elasticity of Pipe, E 205 x 109 N/m
2 205 x 10
9 N/m
2
Poisson’s Ratio 0.28 0.28
Estimated Factor of Friction f 0.0088 0.0088
Pipe External Diameter 1.05 m 1.05 m
Pipe Internal Diameter 0.985 m 0.985 m
Wall Thickness (mm) 20 mm 20 mm
Pump Specifications
Pump Nos. 3 3
Speed, N 730 RPM 730 RPM
Moment of Inertial of Pumpset 33.30 kg.m2 33.30 kg.m
2
Diameter of Impeller (mm) 655 mm 655 mm