experiments and corresponding calculations on thermohydraulic

EXPERIMENTS AND CORRESPONDING CALCULATIONS ON
THERMOHYDRAULIC PRESSURE SURGES IN PIPES
Thorsten Neuhaus1
Fraunhofer UMSICHT, Oberhausen, Germany (since 02/2005: TUEV Nord SysTec GmbH &
Co. KG, Hamburg, Germany) Phone: +49 (0) 40 / 8557-2642, Fax: +49 (0) 40 / 8557-2429,
E-Mail: [email protected]
Andreas Dudlik
Fraunhofer UMSICHT, Oberhausen, Germany
Phone: +49 (0) 208 / 8598-1125, Fax: +49 (0) 208 / 8598-1425, E-Mail: [email protected]
Arris S. Tijsseling
Eindhoven University of Technology, Eindhoven, The Netherlands
Phone: +31 (0) 40 / 247-2755, Fax: +31 (0) 40 / 244-2489, E-Mail: [email protected]
ABSTRACT
Modelling of pressure surges in power plant pipe systems means to carefully judge and
classify the relevant phenomena. A decision must be made whether the numerous effects that
can occur during a transient event should be modelled. One main point is to determine if
fluid-structure interaction (FSI) is important for the special situation / scenario. Also unsteady
friction and degassing of dissolved gases can occur. But with each effect more sources of
errors are introduced into the mathematical model, like:
•
•
•
•
•
•
uncertainty about bearing clearance within the calculation of structural dynamics
determination of parameters of the support characteristics like stiffness / spring
constants / mass
determination of friction coefficients and damping parameters
uncertainty about the amount of non-condensables in the liquid
dynamics of degassing processes
unsteady friction coefficients.
Another complicated problem is the definition of boundary conditions, including the
specifications of pumps, valves, fittings, etc. It is obvious that a combination of all
1
Corresponding author
1
uncertainties may lead to quite inaccurate predictions. Therefore they must be taken carefully
into account.
Within this paper experimental data are presented that were obtained at the test facility
PPP at Fraunhofer UMSICHT, a test facility of industrial size with clearly defined boundary
conditions and latest measuring techniques for pressures, void fractions, forces and
displacements. The valve closure tests that have been conducted with tap water at
temperatures between 20 °C and 120 °C and pressures between 1 bar and 20 bar comprise the
effects of vaporous and gaseous cavitation, fluid-structure interaction and unsteady friction.
In order to simulate the experiments a new three-equation two-phase flow model has been
developed including unsteady friction and degassing. It is coupled with a structural model of
the pipe system to incorporate FSI. The proposed model is validated using five valve-closing
experiments with different initial fluid velocities, temperatures and system pressures.
Dependent on the effects that have been taken into account in the model, a good or less good
agreement between experimental and simulated data could be achieved. From the numerical
simulations it could be assessed, which effects occurred during the diverse experiments.
2
1. INTRODUCTION
Water hammer and cavitational hammer appear in several sectors of industry. They are
caused by different processes like contact condensation between vapour and water (power
plants), sudden change of the liquid’s flow velocity, e.g. due to valve closure or pump trip
(water supply, chemical industry, etc.), plug flow of the liquid (oil and gas supply) as well as
bad design or incorrect / critical plant operation, e.g. during start-up or shutdown procedures.
Fluid transient flow processes can generate fluid-structure interaction (FSI) between the
transported medium and the pipe and its supports. If pressures or forces exceed the design
criteria of pipes and supports, system components may be damaged (e.g. pipe deformation,
pipe leakage, erosion of pumps and fittings, noise and damage of pipe supports) and the
operation of the whole pipe system may be disturbed.
For the simulation of pressure surges in pipe systems usually a one-dimensional approach
is applied. For single-phase liquid flow often the method of characteristics is used with the
discrete vapour cavity model for the calculation of small locally fixed vapour bubbles. For
two-phase flow the equations for the conservation of momentum, mass and energy are solved
using FEM, FDM or FVM methods. Here the conservation equations are sometimes
established for each phase and sometimes for the mixture. The two-phase flow model consists
mostly of a hyperbolic system of partial differential equations.
For the simulation of FSI the equations for structural dynamics must be solved. A rigid
pipe model may be applied for the calculation of the structural movement, which can
reproduce junction-coupling effects. In this case a system of ordinary differential equations
must be solved. On the other hand, for the simulation of wave-propagation in the pipe wall,
e.g. for the calculation of precursor waves in the fluid, another hyperbolic system of partial
differential equations must be adopted.
To get information about the quality of software calculations using different models,
boundary conditions and numerical schemes, an experimental set-up with very precisely
defined boundary conditions is required for code validation.
Within the European project WAHALoads and in cooperation with European partners,
Fraunhofer UMSICHT investigates the mechanism and the prediction of water and
cavitational hammer. Several experimental test series have been performed that can be used
for example for code validation.
2. EXPERIMENTAL INVESTIGATIONS AT FRAUNHOFER UMSICHT
The set-up for the experiments used for code validation in this paper, including the main
measuring positions, is given in Figures 1 and 2.
Demineralised tap water is pumped from the pressurised vessel B1 into the test system
made of 110 mm inner diameter steel pipes and having a total length of about 170 m. At
t = 0 sec the valve located between P02 and P03 closes rapidly while the pump remains
running. During the first phase of the transient process, a pressure wave travels upstream the
valve towards B1, where it is partially reflected at the pump. Simultaneously a rarefaction
wave travels downstream the valve also towards B1. In the cases where the saturation
3
pressure is reached, cavitation occurs downstream the valve and also in the upper part of the
10 m high pipe bridge, so that vapour bubbles are formed. The generated rarefaction/pressure
waves oscillate in the pipe system until the cavities condense, inducing cavitational hammers.
The pressure waves are measured by fast pressure transducers (P01 – P23). Forces on
pipe supports are determined (FP1 – FP3) as well as displacements (W1 – 18) in horizontal
(x) or vertical (z) direction. The measuring frequency is 2 kHz. Phase and temperature
distribution are measured with a newly developed wire mesh sensor (GS) and local void
probes.
Figure 1. Experimental set-up of the PPP test rig
Figure 2. Measuring positions in the PPP test rig
FSI effects occur mainly in the 10 m high pipe bridge as well as in the newly constructed
2 m high, 3 m wide pipe bridge 2 (measuring position W18), where pipe supports are quite
elastic on the basis of typical pipe support conditions in power plants.
4
The experimental parameters are given in Table 1 while the performed experiments
considered for the validation of the proposed model are given in Table 2. For more
information about set-up and performance of experiments, please refer to Dudlik (1999,
2004) and Neuhaus (2005).
Initial steady state velocity
1 m/s ≤ w ≤ 5 m/s
5 velocities
Liquid temperature
20 °C ≤ ϑ ≤ 150 °C
9 temperature levels
Re-opening of valve after approx. Yes / no
10 sec
-
Initial system pressure
1 bar ≤ p ≤ 20 bar
Repetition of experiments
At least 2 / experiment -
3 static pressure levels
Table 1: Matrix of experiments
Experiment
number
124
Fluid velocity
[m/sec]
1.02
Flow rate
[m3/h]
33.7
Temperature
[oC]
19.9
Pressure B1
[bar]
1.14
132
2.97
98.4
20.3
1.14
307
3.99
132.2
119.7
9.92
347
1.01
33.4
20.3
9.83
415
1.00
33.2
21.9
19.65
Table 2: Experiments selected for code validation
For code validation only the pipe section downstream the valve up to vessel B1 is
modelled. This section is 149.4 m long.
3. MODELLING
The following set of equations is used to model the phenomena occurring in the experiments.
The equations (1) and (2) represent the mass balances of the liquid and the gas/vapour phases,
respectively, whereas equation (3) is the combined momentum balance for both phases. The
temperature is assumed to be constant, so energy balances are not applied. Two different
mass transfer terms ΓV and ΓA are used for the vaporisation on the one hand and for air
release on the other hand:
(1 − α ) ∂p − ρ ∂α + (1 − α )w ∂p − ρ w ∂α + (1 − α )ρ ∂w = −Γ − Γ
(1)
L
L
L
V
A
2
2
∂t
∂z
∂z
∂z
a L ∂t
aL
α ∂p
aG ∂t
2
∂α αw ∂p
∂α
∂w
= ΓV + ΓA
+ 2
+ ρG w
+ αρ G
∂z
∂t aG ∂z
∂z
∂w
∂w 1 ∂p
4τ
+w
+
=−
− g sin β
∂t
∂z ρ m ∂z
ρmd
ρ m = (1 − α )ρ L + αρ G ,
+ ρG
5
(2)
(3)
(4)
where
p pressure
w fluid velocity
α void fraction
a L ( aG ) wave propagation speed in the liquid (vapour/gas) phase
ρ L ( ρ G ) density of the liquid (vapour/gas) phase
ρ m density of the liquid-vapour/gas mixture
ΓV mass transfer due to vaporisation/condensation
ΓA mass transfer due to air release
τ shear force due to skin friction
d inner pipe diameter
g gravitational acceleration
β angle of the pipe slope with the horizontal
t time
z distance.
The equations (1), (2) and (3) form a system of partial differential equations with the
dependent variables p, w and α . The density of the gas/vapour phase ρ G is calculated using
the ideal gas law, while the liquid density ρ L is assumed constant herein. The mass transfer
terms ΓV and ΓA are eliminated during the source term integration in establishing a direct
relation between p and α . The wave speeds a L and a G as well as the friction force τ are
defined below. Flow regimes where the phases are moving separately cannot be covered,
since only one momentum balance is applied.
Wave propagation speeds
The following equation is used to determine the wave propagation speeds in the single
phases, thereby taking into account the radial expansion of the axially restraint pipe due to a
pressure rise:
 1
ρ ⋅ d 
ai =  2 + 1 − µ 2 i
E M ⋅ s 
 ci
(
)
−0.5
,
with
ai wave propagation speed of the respective phase
ci speed of sound in unconfined fluid of the respective phase
ρ i density of the respective phase
µ Poisson’s ratio of the pipe material
EM elasticity modulus of the pipe material
s wall thickness of the pipe.
The speeds of sound in the unconfined fluid ci are assumed constant herein.
6
(5)
Friction force
The applied friction law is a modified approach of Vítkovský et al. (2000). The friction force
consists of a quasisteady friction part modelled by the approach of Darcy-Weisbach and an
unsteady friction part. The steady friction coefficient for turbulent flow is calculated using the
formulation of Colebrook-White, for laminar flow Hagen-Poiseuille’s law is applied. An
explicit friction law for two-phase flow is not used. In contrast to the approach of Vítkovský,
the unsteady friction coefficient is split into two coefficients k1 and k 2 . Thus the effects of
damping ( k 2 ) and change of frequency of pressure oscillations ( k1 ) can be separated. The
unsteady friction part is only used for the pure liquid phase. For the gas/vapour phase and for
a two-phase mixture only the Darcy-Weisbach equation is applied. Thus:
ρ d  ∂w
∂w 
λρ
w w + L  k1
+ k 2 a L ⋅ sign(w)
(6)
τ=
,
4 2
4  ∂t
∂z 
with
λ steady Darcy-Weisbach friction coefficient
k1 unsteady friction coefficient accounting for change of pressure wave frequency
k 2 unsteady friction coefficient accounting for pressure wave damping.
Numerical solution
To solve the system of equations a first-order finite-volume method (FVM) is chosen with
operator splitting, i.e. convection and source terms are treated separately. In most cases the
spatial increment ∆z is set to 0.25 m and the Courant number based on a L is chosen to be
0.8, so that ∆t is between 0.13 and 0.17 msec. Otherwise, different values are mentioned.
∂ψ
∂ψ
Step 1:
A
+B
=0,
(7)
∂t
∂z
∂ψ
Step 2:
A
=S,
(8)
∂t
with
ψ vector of dependent variables
A coefficient matrix
B coefficient matrix
S source vector.
One time step includes two substeps. First the convective part is solved using the splitcoefficient matrix scheme of Chakravarthy (see Toro 1999), an explicit first-order upwind
scheme, after that the source terms are calculated.
 −1
∂ψ n 
,
ψ *j = ψ nj − ∆t  A ψ nj B ψ nj
Step 1:
(9)
∂z 

Step 2:
ψ nj +1 = ψ *j + ∆t A−1 ψ nj S ψ nj .
(10)
( ( ) ( ))
( ( ) ( ))
The superscripts n and n + 1 denote time levels, * denotes the intermediate time level
and ∆t represents the time step. The subscript j denotes the position.
7
Interphase exchange terms
In the applied model a combined momentum balance is adopted, so the exchange of the
momentum between the phases is presumed infinitely fast and the phase velocities are always
identical.
The mass exchange between the phases is calculated within the source term integration
(Step 2 of the numerical scheme, eq. (10)). The variables ΓV and ΓA are eliminated in adding
the source terms of the equations (1) and (2). Then the following equation can be obtained:
 1 − α nj
α nj  p nj +1 − p *j
α nj +1 − α *j

+ 2
= (ρ L − ρ G )
,
(11)
 a 2

∆
t
∆
t
a
L
G


n +1
with the two unknown variables p j and α nj+1 . The proposed model does not use an explicit
(
)
approach for the mass transfer terms ΓV and ΓA , so the equations (1) and (2) are not further
used within the source term integration step. However, another equation is needed to solve
the mathematical system. The closure relation describes another correlation between the
variables p nj+1 and α nj+1 . It depends on the nature of the cavitation (vaporous/gaseous) and is
described below.
Vaporous cavitation
In a one-component thermodynamic equilibrium model it is assumed that the following three
conditions can occur:
1. Pure liquid phase:
p > p S , α = 10 −12 ,
(12)
p < p S , α = 1 − 10 −12 ,
(13)
2. Pure vapour phase:
3. Two-phase mixture:
p = p S , 10 −12 < α < 1 − 10 −12 ,
(14)
−12
where p S is the saturation pressure at the given temperature. The value 10 is an arbitrary
small value, such that α does not become zero or one. With the equations (11) – (14) the
variables p nj+1 and α nj+1 can be calculated during the source term integration step.
Gaseous cavitation (Air release)
In a two-component model, Dalton’s law indicates that the total pressure in the gas/vapour
phase is the sum of the partial pressures of the components. For a water-air mixture the
following equation can be adopted:
p = pV + p A ,
(15)
with
pV partial pressure of the vapour
p A partial pressure of air in the vapour/gas phase.
The void fraction is the sum of the volume fractions of air and vapour:
α A + αV = α ,
8
(16)
with
α V volume fraction of vapour
α A volume fraction of free air.
The partial pressure of air is the mole fraction or volume fraction multiplied with the total
pressure (ideal gas law):
pA =
αA
p.
α A + αV
(17)
With the assumption of a relative humidity of 100% in the bubbles and thermodynamic
equilibrium, the partial pressure of the vapour is the saturation pressure:
pV = p S .
(18)
So the pressure can be calculated:
α + αV
p= A
pS ,
(19)
αV
or
p=
α
pS .
α −α A
(20)
For the specification of the air release process experimental data from Perko (1985) are
chosen. He determined the free air mass as a function of time after a rarefaction wave passed
through water in a pipe at 20 °C and at initial pressure of 2 bar. So the free gas mass per
volume m A is known. The following equation can be applied (ideal gas law):
m RT
αA = A A ,
(21)
p
with
m A mass of free air per volume (obtained from Fig. 15)
R A gas constant of air
T temperature.
With the equations (11), (20) and (21) the variables p nj+1 and α nj+1 as well as α A can be
calculated during the source term integration step.
In the last step of the fluid dynamic calculation the density of the vapour/gas mixture and
the pressure wave propagation speed aG are calculated as follows:
ρG =
α
αA
ρ A + V ρV
α
α
p
R AT
p
ρV =
RV T
1
(22)
ρA =
aG =
 αA
α V  α A
αV


ρ
+
+
ρV 

A
2
2
 αρ a

α

 A A αρ V aV  α
with
ρV density of vapour
9
(23)
(24)
,
(25)
ρ A density of air
aV wave propagation speed in pure vapour
a A wave propagation speed in pure air
RV gas constant of vapour.
The gas/vapour phase is assumed to be a homogenous mixture of air and vapour, so
equation (25) is used that constitutes the velocity of a compression wave in a homogenous
mixture.
The parameters a L , aV , a A , ρ L and p S are dependent on the initial pressure and
temperature and remain constant during the transient calculation. Nevertheless in the model
the wave propagation speeds change with time and distance, because they strongly depend on
the calculated void fraction and gas/vapour density. The wave speeds are the calculated
eigenvalues of the set of partial differential equations.
Boundary conditions
The geometrical pipe model ranges from the valve to vessel B1 (see Figure 2), thus two
boundaries have to be modelled. The closure of the valve is simulated in linearly decreasing
the fluid velocity from the steady-state value to zero within several milliseconds. The vessel
at the end of the pipe is modelled using a constant-pressure condition. In the proposed model,
special boundary conditions for column separation that occurs in some experiments are not
needed.
Fluid-structure interaction
Concerning fluid-structure-interaction (FSI) a rigid pipe model is applied, this means that the
deformation of a pipe is not possible but solely its rigid-body movement. Only the motion of
the first 10 m high pipe bridge and the second 2 m high, 3 m wide pipe bridge have been
taken into account.
To determine the eigenmodes of the first 10 m high pipe bridge a structural analysis has
been performed. For one eigenmode the three pipe segments always move together
(Figures 3-4). If the displacement of one pipe segment is known, also the displacements of
the other two pipe segments are known, because the motion is coupled. The ratio of the
displacements of the pipe segments is a result of the structural analysis.
Although the structural analysis evaluates also the deformation of the pipe like bending
(Figures 3-4), for the structural model only the axial rigid-body movement has been taken
into account. Therefore one spring-mass system with viscous friction is used to model one
pipe segment (eq. (26) - (27)). For each eigenmode the displacements of the pipe segments
are coupled by applying the ratios of displacement determined within the structural analysis.
The displacements of the single eigenmodes are finally superposed. Only the eigenmodes
with the lowest frequencies have been taken into account, because they imply the biggest
displacements.
m&x& + bx& + cx = ∆pA + FR
λLdπ
FR =
wr wr ρ L ,
8
10
(26)
(27)
with
x axial displacement of the pipe segment
x& axial velocity of the pipe segment
&x& axial acceleration of the pipe segment
m adapted mass of the pipe segment including water (constant during calculation)
b adapted attenuation constant
c spring constant determined in the structural analysis
A cross sectional area of the pipe
∆p pressure difference between the junctions of the pipe segment
FR friction between fluid and pipe segment
L length of the pipe segment
wr relative axial velocity between fluid and pipe.
Figure 3. Model for the structural analysis of the first pipe bridge: First eigenmode at 4.1 Hz.
Figure 4. Model for the structural analysis of the first pipe bridge: Second eigenmode at 7.1 Hz.
11
To determine the frequency of the major eigenmode of the second pipe bridge a 3D-FEM
structural analysis has been performed (Figure 5). A frequency of 2.5 Hz has been identified.
The structural model comprises only the central leg, because it performs the most significant
movements. Here just one spring-mass system with internal viscous friction is applied.
Figure 5. Model for the structural analysis of the second pipe bridge: First eigenmode at 2.5 Hz
The coordinate system of the fluid mechanics calculation is assumed to oscillate together
with the coordinate system of the structure, so that the fluid is accelerated in axial direction at
each point of one moving pipe segment. The effect of the pipe motion on the fluid dynamics
can be taken into account by the following equation:
n +1
w n +1 − wFD
x& n +1 − x& n
ρ m SD
= ρm
,
(28)
∆t
∆t
with
n +1
wFD
fluid velocity after the fluid dynamic calculation
n +1
wSD
fluid velocity after the structural dynamic calculation.
Within one pipe segment each finite volume of the fluid dynamic calculation experiences
the same acceleration due to the structural movement, so that no pressure surges caused by
x& n +1 − x& n
differs spatially, that is at a
FSI are generated within one pipe segment. Just where
∆t
transition from one pipe segment to another or at a closed end, adjacent finite volumes
experience different accelerations and pressure surges are generated. The fact that in reality
pressure surges caused by junction coupling are generated only at elbows or closed ends can
therefore be reproduced.
The advantage of this approach is that the equations of the fluid dynamic process do not
need to be modified by introducing an FSI model. Thus the calculation of one time step
consists of the following three substeps:
12
•
•
•
fluid dynamic step (eq. (1)-(25))
calculation of pipe movement (eq. (26)-(27))
influence of the pipe motion on the fluid dynamics (eq. (28)).
4. CONSISTENCY CHECK
To check the consistency of the fluid dynamic equations and the numerical method, two
theoretical test cases are applied that are also used for benchmark tests in the WAHALoads
project (see Bibliography). The tests are performed without friction, FSI, air release, radial
expansion of the pipe diameter, geodetic terms and thermodynamic non-equilibria. The
model geometry and the initial conditions are depicted in Figure 6. Water flows with an
initial velocity of 5 m/sec through a 100 m long pipe at 80 bar. In the middle of the pipe a
valve is positioned that closes infinitely fast at the beginning of the tests, so that a pressure
wave is generated upstream the valve. BM 1.1a is performed at 20 °C, BM 1.2a at 250 °C, so
that in the latter case a vapour bubble is generated after about 0.9 sec.
Initial velocity: 5 m/s
Valve closes at
t=0s
80 bar
50 m
50 m
80 bar
Relevant pipe
Water,
Temperature: 20 °C / 250 °C
Figure 6. Theoretical test cases BM 1.1a and BM 1.2a of the WAHALoads project used for a consistency check
of the applied model and numerical method.
For the consistency check the discretisation is chosen very fine. In Figure 7 the pressure
history of benchmark test 1.1a is shown 40 m upstream of the valve with a finite volume
length of 1 cm. The theoretical solution is almost obtained. In Figure 8 the pressure history of
benchmark test 1.2a is shown 10 m upstream the valve with a finite volume length of
0.28 cm. The simulated pressure curve nearly matches the theoretical curve.
13
Pressure [bar]
160
120
80
40
MOC
Numerical
0
0
0.1
0.2
0.3
Time [sec]
Figure 7. Consistency check of the applied model and numerical method by means of the WAHALoads
Benchmark Test 1.1a
Pressure [bar]
150
MOC with DVC model
Numerical
120
90
60
30
0.0
0.1
Time [sec]
0.2
0.3
Figure 8. Consistency check of the applied model and numerical method by means of the WAHALoads
Benchmark Test 1.2a
5. MODEL VALIDATION
Five experiments are chosen for model validation (see also Table 2):
• experiment 415, with an initial pressure of 19.65 bar in B1, a temperature of 21.9 °C
and an initial fluid velocity of 1.00 m/sec
• experiment 132, with an initial pressure of 1.14 bar in B1, a temperature of 20.3 °C
and an initial fluid velocity of 2.97 m/sec
• experiment 124, with an initial pressure of 1.14 bar in B1, a temperature of 19.9 °C
and an initial fluid velocity of 1.02 m/sec
• experiment 347, with an initial pressure of 9.83 bar in B1, a temperature of 20.3 °C
and an initial fluid velocity of 1.01 m/sec.
• experiment 307, with an initial pressure of 9.92 bar in B1, a temperature of 119.7 °C
and an initial fluid velocity of 3.99 m/sec.
Experiment 415
Experiment 415 is the first used for validation, because vaporisation and air release
processes did not occur. So the main focus can be set on the effects of FSI and unsteady
14
friction. In Figure 9 the measured and calculated pressures at P03 for experiment 415 are
compared. The model does not include FSI and air release, but unsteady respectively
quasisteady friction is incorporated. The quasisteady friction model calculates a slightly
rising pressure within the pressure plateaus at 33 bar. This effect is called line packing, but it
is not relevant for experiment 415. More important is the pressure damping that is captured
well by the unsteady friction model with k1 = 0 and k 2 = 0.18 . These values are tuned to
experiment 415. Nevertheless, the shape of the measured pressure curve cannot be
reproduced, because it includes secondary high-frequency oscillations. A more triangular
shape of the experimental pressure curve appears in contrast to the rectangular shape of the
simulated curves.
40
p03
Pressure [bar]
30
20
10
Experiment
Simulation without FSI and with quasisteady friction
Simulation without FSI and with unsteady friction, k2 = 0.18
0
0
1
Time [sec]
2
3
Figure 9: Comparison of measured and calculated pressure at P03 for experiment 415. The model does not
include FSI and air release, but quasisteady respectively unsteady friction.
Figure 10 shows that the introduction of the FSI-model strongly improves the simulation
results.
40
p03
Experiment
Simulation with FSI and unsteady friction, k2 = 0.18
Pressure [bar]
30
20
10
0
0
1
Time [sec]
2
3
Figure 10: Comparison of measured and calculated pressure at P03 for experiment 415. The model does not
include air release, but FSI and unsteady friction.
In Figure 11 it can be seen that the amplitude of the oscillations of the second pipe bridge
rises during the first 1.5 sec, since the frequency of the pressure oscillations, which is
15
aL
≈ 2.1 Hz, is close to the frequency of the first eigenmode of the second pipe bridge.
4L
The movement can be captured well by a spring-mass-system with viscous friction. In
Figure 12 the movement of the first pipe bridge is shown. There are several superposing
eigenmodes in the experimental data. The model can roughly catch the amplitude, but the
superposition is not correctly reproduced.
f =
0.06
Experiment
Simulation
w18x
Displacement [m]
0.04
0.02
0
-0.02
-0.04
-0.06
0
1
Time [sec]
2
3
Figure 11: Comparison of measured and calculated displacement at W18X for experiment 415. The model does
not include air release, but FSI and unsteady friction.
0.004
Experiment
Simulation
Displacement [m]
w06z
0.002
0
-0.002
-0.004
0
1
Time [sec]
2
3
Figure 12: Comparison of measured and calculated displacement at W06Z for experiment 415. The model does
not include air release, but FSI and unsteady friction.
In the following the FSI model is always applied.
Experiment 132
During experiment 132 large cavitational bubbles arise downstream the closed valve and in
the upper part of the first pipe bridge. The vapour bubble in the pipe bridge collapses for the
first time after 14 sec.
In Figure 13 the measured and calculated pressures at P03 for experiment 132 are
compared. The model does not include unsteady friction and air release, but FSI and
16
vaporous cavitation. The damping and the frequency of the pressure surges after 7 sec cannot
be reproduced by the model.
Experiment
p03
Pressure [bar]
30
Simulation with FSI and quasisteady
friction and without air release
20
10
0
0
5
10
Time [sec]
15
20
Figure 13: Comparison of measured and calculated pressure at P03 for experiment 132. The model does not
include unsteady friction and air release, but FSI and vaporous cavitation.
Choosing an unsteady friction approach with k 2 = 0.18 and k1 = 0 as determined in
experiment 415 leads to the results shown in Figure 14. Again the frequency of the pressure
oscillations cannot be reproduced and the damping is too high in the simulation.
Experiment
p03
Pressure [bar]
30
Simulation with FSI and unsteady
friction and without air release
20
10
0
0
5
10
Time [sec]
15
20
Figure 14: Comparison of measured and calculated pressure at P03 for experiment 132. The model does not
include air release, but FSI, vaporous cavitation and unsteady friction.
Perko (1985) showed that an air release process following a rarefaction wave in water
takes place in three stages. After an incubation time a constant growth of free gas mass
occurs until a constant value is reached (Figure 15). For the simulation of experiment 132 the
air release curve of Figure 15 is chosen, because the conditions of the UMSICHT-experiment
and Perko-experiment are nearly the same (pressure, temperature, pressure decrease, initial
air content). It is assumed that at each point in the pipe air release occurs, since the
rarefaction wave travels through the whole system.
17
Figure 15: Air release measured by Perko (1985) in water at 20 °C, an initial pressure of 2 bar, fluid velocity of
1.6 m/sec and initially dissolved air of 38 g/m³.
The simulated pressure at P03 using the air release model with the quasisteady friction
approach is shown in Figure 16. The frequency of the pressure oscillations at 14 sec can be
captured well because the wave propagation speed, which drops due to the increasing
compressibility of the water-air-mixture to about 250 m/sec, is calculated correctly. After
12 sec the shape of the experimental and simulated pressure curves look like a row of several
‘U’s. This effect can be reproduced by the model due to the dependency of the wave speeds
on the void fraction and the gas/vapour density. The air release leads also to higher damping
of the pressure waves.
Experiment
p03
30
Pressure [bar]
Simulation with FSI, quasisteady
friction and air release
20
10
0
0
5
10
Time [sec]
15
20
Figure 16: Comparison of measured and calculated pressure at P03 for experiment 132. The model does not
include unsteady friction, but air release, vaporous cavitation and FSI.
In Figure 17 the evolution of the pressure at P09 is depicted. The pressure oscillations at
P03 during the first 14 sec do not appear at P09, because the vapour bubble in the pipe bridge
separates the system hydraulically (column separation). Only the pressure rise caused by the
collapse of the vapour bubble in the pipe bridge can be seen at P09.
18
30
Experiment
Pressure [bar]
p09
Simulation with FSI, quasisteady
friction and air release
20
10
0
0
5
10
Time [sec]
15
20
Figure 17: Comparison of measured and calculated pressure at P09 for experiment 132. The model does not
include unsteady friction, but air release, vaporous cavitation and FSI.
In Figure 18 the evolution of the void fraction behind the fast-closing valve is depicted.
The experimental curve is recorded with wire-mesh sensor technology (Dudlik 2004). It is
confirmed by the simulated curve.
1
Experiment
Simulation
Void fraction [-]
0.8
0.6
0.4
0.2
0
0
5
Time [sec]
10
15
Figure 18: Comparison of measured and calculated void fraction at GS1 for experiment 132. The model does
not include unsteady friction, but air release, vaporous cavitation and FSI.
In Figure 19 the movement of the second pipe bridge is shown. It does not move
considerably during the first 13 sec of the event since the pressure surges that occur directly
downstream the valve are reflected at the vapour bubble in the first, 10 m high pipe bridge.
The mass-spring model with viscous friction can reproduce the real process.
In Figure 20 the movement of the first pipe bridge is shown. In contrast to experiment
415, where the excitation of the structure was generated due to a nearly harmonic pressure
wave oscillation, in experiment 132 the structure is incited to oscillate with each pressure
surge. Therefore the model leads here to more satisfactory results than during the validation
of experiment 415.
19
0.02
Experiment
Simulation
w18x
Displacement [m]
0.01
0
-0.01
-0.02
-0.03
0
5
10
Time [sec]
15
20
Figure 19: Comparison of measured and calculated displacement at W18X for experiment 132. The model does
not include unsteady friction, but air release, vaporous cavitation and FSI.
0.006
Experiment
Simulation
w06z
Displacement [m]
0.004
0.002
0
-0.002
-0.004
-0.006
-0.008
0
2
4
6
Time [sec]
8
10
Figure 20: Comparison of measured and calculated displacement at W06Z for experiment 132. The model does
not include unsteady friction, but air release, vaporous cavitation and FSI.
Experiment 124
Experiment 124 is similar to experiment 132 but the initial fluid velocity is about 1 m/sec,
so the expansion of the vapour bubbles behind the fast-acting valve and in the pipe bridge is
smaller. In the simulation the length of a finite volume is chosen to be 0.44 m, because a
better agreement to experimental data is achieved than using a finite volume length of 0.25m.
In Figure 21 the measured and calculated pressures at P03 for experiment 124 are
compared. The model does not include unsteady friction and air release, but FSI and
vaporous cavitation. After 5 sec the frequency of the experimental and simulated pressure
oscillations differ from each other. Better results can be obtained using the air release model
(Figure 22). The FSI effect in the pressure history can be seen well at P06 (Figure 23).
20
15
Experiment
Simulation without air release
Pressure [bar]
p03
10
5
0
0
2
4
Time [sec]
6
8
10
Figure 21: Comparison of measured and calculated pressure at P03 for experiment 124. The model does not
include unsteady friction and air release, but FSI and vaporous cavitation.
15
Experiment
p03
Pressure [bar]
Simulation with air release
10
5
0
0
2
4
Time [sec]
6
8
10
Figure 22: Comparison of measured and calculated pressure at P03 for experiment 124. The model does not
include unsteady friction, but FSI, vaporous cavitation and air release.
15
Experiment
Simulation
Pressure [bar]
p06
10
5
0
0
2
4
6
8
10
Time [sec]
Figure 23: Comparison of measured and calculated pressure at P06 for experiment 124. The model does not
include unsteady friction, but FSI, vaporous cavitation and air release.
21
Experiment 347
Until now the unsteady friction approach has only been useful in experiment 415 which has a
single-phase pressure oscillation without vaporisation. In the following it is shown that the
unsteady friction approach may also be needed, when vapour bubbles occur.
Because experiment 347 is quite short (5 sec) an air release model is not applied. In
Figure 24 the measured and calculated pressures at P03 for experiment 347 are compared.
The pressure damping can be reproduced using the unsteady friction approach with k1 = 0
and k 2 = 0.18 as determined in experiment 415. The vapour bubble downstream the fast
acting valve remains rather small (Figure 25). Though the unsteady friction approach is not
applied in the finite volumes where two-phase flow ( α > 10 −12 ) occurs, the calculated void
fraction is smaller than using the quasisteady friction approach.
40
Quasisteady friction
Unsteady friction
Experiment
p03
Pressure [bar]
30
20
10
0
0
1
2
Time [sec]
3
4
5
Figure 24: Comparison of measured and calculated pressure at P03 for experiment 347. The model does not
include air release, but FSI, vaporous cavitation and unsteady respectively quasisteady friction.
0.07
Quasisteady friction
Unsteady friction
Experiment
Void fraction [-]
0.06
0.05
0.04
0.03
0.02
0.01
0
0
1
Time [sec]
2
3
Figure 25: Comparison of measured and calculated void fraction at GS1 for experiment 347. The model does
not include air release, but FSI, vaporous cavitation and unsteady respectively quasisteady friction.
22
Experiment 307
The previous four experiments have all been conducted at an ambient temperature of around
20 °C. Experiment 307 is performed at a much higher temperature of about 120 °C.
In Figure 26 the measured and calculated pressures at P03 for experiment 307 are
compared. The model does not include unsteady friction and air release, but FSI and
vaporous cavitation. The frequency and especially the damping of the pressure surges cannot
be reproduced by the model.
70
Experiment
p03
Simulation with FSI but without air release and unsteady friction
Pressure [bar]
60
50
40
30
20
10
0
0
2
4
Time [sec]
6
8
Figure 26: Comparison of measured and calculated pressure at P03 for experiment 307. The model does not
include unsteady friction and air release but FSI.
Choosing an unsteady friction approach with k 2 = 0.18 and k1 = 0 , as determined in the
experiments 415 and 347, leads to results shown in Figure 27. After 3 sec the frequency of
the pressure oscillations cannot be reproduced, but the damping is captured reasonably well.
70
Experiment
p03
Simulation with FSI and unsteady friction but without air release
Pressure [bar]
60
50
40
30
20
10
0
0
2
4
Time[sec]
6
8
Figure 27: Comparison of measured and calculated pressure at P03 for experiment 307. The model does not
include air release but unsteady friction and FSI.
Good simulation results can be obtained choosing the air release model together with the
unsteady friction model (Figure 28). The frequency of the pressure oscillations and the
23
damping is predicted well. Because the conditions of experiment 307 are very different to the
conditions of the Perko (1985) experiments, the air release curve has been adapted. The
incubation time has been chosen to be about 2 sec and the free air content at the end of the
experiment can be indirectly adjusted using the pressure oscillation frequency.
70
Experiment
p03
Simulation with FSI, unsteady friction and air release
Pressure [bar]
60
50
40
30
20
10
0
0
2
4
Time [sec]
6
8
Figure 28: Comparison of measured and calculated pressure at P03 for experiment 307. The model includes air
release, vaporous cavitation, unsteady friction and FSI.
In Figure 29 the evolution of the void fraction behind the fast-closing valve is depicted.
The development of the first cavitation can be simulated quite well.
1
Experiment
Simulation with FSI, unsteady friction and air release
Void fraction [-]
0.8
0.6
0.4
0.2
0
0
1
2
3
Time [sec]
4
5
6
Figure 29: Comparison of measured and calculated void fraction at GS1 for experiment 307. The model
includes air release, vaporous cavitation, unsteady friction and FSI.
In Figure 30 the movement of the second pipe bridge is shown. The motion can nearly be
reproduced by the model.
24
0.2
w18x
Displacement [m]
0.15
Experiment
Simulation with FSI, unsteady friction and air release
0.1
0.05
0
-0.05
-0.1
-0.15
0
2
4
Time [sec]
6
8
Figure 30: Comparison of measured and calculated displacement at W18X for experiment 307. The model
includes air release, unsteady friction and FSI.
6. CONCLUSIONS
About 350 pressure surge experiments with a tap water pipe system have been conducted in a
new test rig of industrial size and precisely described experimental data have been obtained.
The tests may represent processes occurring in different industries, especially in water
supply, but also in gas / oil / energy supply and chemical / power industry. Varying operation
conditions like temperature, pressure and fluid velocity have a considerable impact on the
transient processes.
The presented three-equation model comprising two mass balances and one momentum
balance, numerically solved with the split coefficient matrix scheme, is quite fast and stable.
For liquid flow simulation with local cavities it is therefore an alternative to the well-known
method of characteristics in combination with the discrete vapour-cavity model (DVC), since
the latter implies several simplifications that may lead to unsatisfactory results. Furthermore
the three-equation model is able to simulate isothermal two-phase flow processes, where the
phases form a continuum as in bubbly flow. Flow regimes where the phases are moving
separately cannot be covered, since only one momentum balance is applied. Additionally a
two-component-system can easily be described by our three-equation model.
Usually the simulation of two-phase flow processes, where the densities of the phases
differ strongly, is quite difficult. In a water-steam mixture at 20 °C the liquid density is about
1000 kg/m³ and the vapour density is about 0.02 kg/m³. Due to the thermodynamic and
mechanic equilibrium approaches the applied model did not show any problems during the
calculation processes.
This paper shows how models describing different physical phenomena, like fluidstructure-interaction (FSI), degassing of non-condensables or unsteady friction may influence
the quality of prediction of fluid transients in pipes. It is shown that the use of FSI and
degassing models can highly improve the predictions. A case-dependent unsteady friction
model led to satisfactory results, but sometimes its application might not be necessary or even
erroneous.
25
ACKNOWLEDGEMENTS
We would like to thank the European Union for the promotion within the scope of the
WAHALoads and SurgeNet projects (see Bibliography). Also we like to thank all
coordinators, participants and contractors of the projects for the scientific discussions and
encouragements. The development of this paper was also enabled by the Marie Curie
Fellowship program promoted by the EU.
BIBLIOGRAPHY
Brunone, B., Golia, U. M. and Greco, M. 1991. Some Remarks on the Momentum Equation
for Fast Transients. Int. Meeting on Hydraulic Transients with Column Separation, 9th
Round Table, IAHR, Valencia, Spain, pp. 140-148.
Dudlik, A. 1999. Vergleichende Untersuchungen zur Beschreibung von transienten
Strömungsvorgängen in Rohrleitungen. UMSICHT-Schriftenreihe Band 20, Germany.
Dudlik, A., Schönfeld, S. B. H., Hagemann, O., Carl, H. and Prasser, H.-M. 2004. Water
Hammer and Condensation Hammer Scenarios in Power Plants Using New
Measurement System. The Practical Application of Surge Analysis for Design and
Operation, 9th International Conference on Pressure Surges, Chester, UK, 24-26 March
2004, Vol. 1, pp. 151-165.
Horlacher, H.-B. and Lüdecke, H.-J. 1992. Strömungsberechnung für Rohrsysteme. Expert
Verlag, Ehningen bei Böblingen, Germany.
Lamb, H. 1945. Hydrodynamics. 6th Edition, Dover Publications, New York.
Mahaffy, J. H. 1993. Numerics of Codes: Stability, Diffusion, and Convergence. Nuclear
Engineering and Design 145, pp. 131-145.
Neuhaus, T. 2005. Mathematische Modellierung und vergleichende Untersuchungen zur
Beschreibung von transienten Ein- und Mehrphasenströmungen in Rohrleitungen. PhD
Thesis, University of Dortmund, Department of Biochemical and Chemical Engineering,
will be published in 2005.
Perko, H.-D. 1985. Gasausscheidung in instationärer Rohrströmung. Institut für
Strömungsmechanik und Elektron. Rechnen im Bauwesen der Universität Hannover,
Germany, Bericht Nr. 16/1985, ISSN 0177-9028.
Stewart, H. B. and Wendroff, B. 1984. Two-Phase Flow: Models and Methods. Journal of
Computational Physics 56, pp. 363-409.
SurgeNet, research project within the 5th Framework Program of the European Union,
contract reference G1RT-CT-2002-05069, Thematic Network of European organizations
involved in the prediction and analysis of fluid transients in pipe systems for the purpose
of improved design and operation of such systems.
Thorley, A. R. D. and Tiley, C. H. 1987. Unsteady and Transient Flow of Compressible
Fluids in Pipelines – A Review of Theoretical and some Experimental Studies. Heat and
Fluid Flow, Vol. 8, No. 1, pp. 3-15.
26
Tijsseling, A. S. 1996. Fluid-Structure Interaction in Liquid-Filled Pipe Systems: A Review.
Journal of Fluids and Structures 10, pp. 109-146.
Tiselj, I. and Petelin, S. 1997. Modelling of Two-Phase Flow with Second-Order Accurate
Scheme. Journal of Computational Physics 136, pp. 503-521.
Toro, E. E. 1999. Riemann Solvers and Numerical Methods for Fluid Dynamics. 2nd Edition,
Springer Verlag.
Vítkovský, J., Lambert, M. F., Simpson, A. R. and Bergant, A. 2000. Advances in Unsteady
Friction Modelling in Transient Pipe Flow. Pressure Surges, Safe Design and Operation
of Industrial Pipe Systems, A. Anderson, ed., Professional Engineering Publishing Ltd.,
Bury St. Edmunds, England, pp. 471-482.
Wallis, G. B. 1969. One-dimensional Two-phase Flow. McGraw-Hill Book Company.
WAHALoads, research project within the 5th Framework Program of the European Union,
contract reference FIS5-1999-00114/-00341, WAHALoads - „Two-Phase Flow Water
Hammer Transients and Induced Loads on Materials and Structures of Nuclear Power
Plants“.
Wylie, E. B., Streeter, V. L. and Suo, L. 1993. Fluid Transients in Systems. Prentice Hall,
Englewood Cliffs, USA.
27