1

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: tneuhaus@tuev-nord.de

Andreas Dudlik

Fraunhofer UMSICHT, Oberhausen, Germany Phone: +49 (0) 208 / 8598-1125, Fax: +49 (0) 208 / 8598-1425, E-Mail: du@umsicht.fhg.de

Arris S. Tijsseling

Eindhoven University of Technology, Eindhoven, The Netherlands Phone: +31 (0) 40 / 247-2755, Fax: +31 (0) 40 / 244-2489, E-Mail: a.s.tijsseling@tue.nl

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

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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.

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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

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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.

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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. 10 sec

Yes / no -

Initial system pressure 1 bar ≤ p ≤ 20 bar 3 static pressure levels

Repetition of experiments At least 2 / experiment -

Table 1: Matrix of experiments

Experiment number

Fluid velocity [m/sec]

Flow rate [m3/h]

Temperature [oC]

Pressure B1 [bar]

124 1.02 33.7 19.9 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 VΓ and AΓ are used for the vaporisation on the one hand and for air

release on the other hand:

( ) ( ) ( ) AVLL

L

L

Lz

w

zw

z

p

a

w

tt

p

aΓ−Γ−=

∂∂−+

∂∂−

∂∂−+

∂∂−

∂∂− ρααρααρα

111

22 (1)

AVGG

G

G

Gz

w

zw

z

p

a

w

tt

p

aΓ+Γ=

∂∂+

∂∂+

∂∂+

∂∂+

∂∂ αραρααρα

22 (2)

βρ

τρ

sin41

gdz

p

z

ww

t

w

mm

−−=∂∂+

∂∂+

∂∂

(3)

( ) GLm αρραρ +−= 1 , (4)

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where p pressure w fluid velocity α void fraction

La ( Ga ) wave propagation speed in the liquid (vapour/gas) phase

Lρ ( Gρ ) density of the liquid (vapour/gas) phase

mρ density of the liquid-vapour/gas mixture

VΓ mass transfer due to vaporisation/condensation

AΓ 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 Gρ is calculated using

the ideal gas law, while the liquid density Lρ is assumed constant herein. The mass transfer

terms VΓ and AΓ are eliminated during the source term integration in establishing a direct

relation between p and α . The wave speeds La and Ga 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:

( )5.0

22

11

−

⋅⋅−+=

sE

d

ca

M

i

i

i

ρµ , (5)

with

ia wave propagation speed of the respective phase

ic speed of sound in unconfined fluid of the respective phase

iρ density of the respective phase

µ Poisson’s ratio of the pipe material

ME elasticity modulus of the pipe material s wall thickness of the pipe. The speeds of sound in the unconfined fluid ic are assumed constant herein.

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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 1k and 2k . Thus the effects of

damping ( 2k ) and change of frequency of pressure oscillations ( 1k ) 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:

( )

∂∂⋅+

∂∂+=

z

wwsignak

t

wk

dww L

L21424

ρρλτ , (6)

with λ steady Darcy-Weisbach friction coefficient

1k unsteady friction coefficient accounting for change of pressure wave frequency

2k 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 La is chosen to be 0.8, so that t∆ is between 0.13 and 0.17 msec. Otherwise, different values are mentioned.

Step 1: 0=∂∂+

∂∂

zB

tA

ψψ, (7)

Step 2: St

A =∂

∂ψ, (8)

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 split-coefficient matrix scheme of Chakravarthy (see Toro 1999), an explicit first-order upwind scheme, after that the source terms are calculated.

Step 1: ( ) ( )( )

∂∂∆−= −

zBAt

nnj

nj

njj

ψψψψψ 1* , (9)

Step 2: ( ) ( )( )nj

njj

nj SAt ψψψψ 1*1 −+ ∆+= . (10)

The superscripts n and 1+n denote time levels, * denotes the intermediate time level and t∆ represents the time step. The subscript j denotes the position.

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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 VΓ and AΓ are eliminated in adding

the source terms of the equations (1) and (2). Then the following equation can be obtained:

( ) ( )

tt

pp

aaj

nj

GLj

nj

G

nj

L

nj

∆−

−=∆

−

+

− ++ *1*1

22

1 ααρρ

αα, (11)

with the two unknown variables 1+njp and 1+n

jα . The proposed model does not use an explicit

approach for the mass transfer terms VΓ and AΓ , 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 1+n

jp and 1+njα . 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: Spp > , 1210−=α , (12)

2. Pure vapour phase: Spp < , 12101 −−=α , (13)

3. Two-phase mixture: Spp = , 1212 10110 −− −<< α , (14)

where Sp is the saturation pressure at the given temperature. The value 1210− is an arbitrary

small value, such that α does not become zero or one. With the equations (11) – (14) the variables 1+n

jp and 1+njα 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: AV ppp += , (15)

with

Vp partial pressure of the vapour

Ap partial pressure of air in the vapour/gas phase. The void fraction is the sum of the volume fractions of air and vapour: ααα =+ VA , (16)

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with

Vα volume fraction of vapour

Aα 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):

ppVA

AA αα

α+

= . (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: SV pp = . (18)

So the pressure can be calculated:

SV

VA ppα

αα += , (19)

or

SA

ppαα

α−

= . (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 Am is known. The following equation can be applied (ideal gas law):

p

TRm AAA =α , (21)

with

Am mass of free air per volume (obtained from Fig. 15)

AR gas constant of air T temperature. With the equations (11), (20) and (21) the variables 1+n

jp and 1+njα as well as Aα 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 Ga are calculated as follows:

VV

AA

G ρα

αρααρ += (22)

TR

p

AA =ρ (23)

TR

p

VV =ρ (24)

+

+

=

VV

AA

VV

V

AA

A

G

aa

a

ρααρ

αα

αρα

αρα

22

1, (25)

with

Vρ density of vapour

10

Aρ density of air

Va wave propagation speed in pure vapour

Aa wave propagation speed in pure air

VR 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 La , Va , Aa , Lρ and Sp 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. RFpAcxxbxm +∆=++ &&& (26)

LrrR wwLd

F ρπλ8

= , (27)

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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

RF friction between fluid and pipe segment L length of the pipe segment

rw 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.

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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:

t

xx

t

ww nn

m

nFD

nSD

m ∆−=

∆− +++

&&111

ρρ , (28)

with

1+nFDw fluid velocity after the fluid dynamic calculation

1+nSDw 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

FSI are generated within one pipe segment. Just where t

xx nn

∆−+&&

1

differs spatially, that is at a

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:

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• 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 att = 0 s

Relevant pipe

50 m50 m 80 bar80 bar

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.

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0

40

80

120

160

0 0.1 0.2 0.3Time [sec]

Pre

ssu

re [

bar

]

MOCNumerical

Figure 7. Consistency check of the applied model and numerical method by means of the WAHALoads Benchmark Test 1.1a

30

60

90

120

150

0.0 0.1 0.2 0.3Time [sec]

Pre

ssu

re [

bar

]

MOC with DVC modelNumerical

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

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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 01 =k and 18.02 =k . 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.

0

10

20

30

40

0 1 2 3Time [sec]

Pre

ssu

re [

bar

]

ExperimentSimulation without FSI and with quasisteady frictionSimulation without FSI and with unsteady friction, k2 = 0.18

p03

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.

0

10

20

30

40

0 1 2 3Time [sec]

Pre

ssu

re [

bar

]

ExperimentSimulation with FSI and unsteady friction, k2 = 0.18

p03

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

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≈=L

af L

42.1 Hz, is close to the frequency of the first eigenmode of the second pipe bridge.

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.

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 1 2 3Time [sec]

Dis

pla

cem

ent

[m]

ExperimentSimulationw18x

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

-0.002

0

0.002

0.004

0 1 2 3Time [sec]

Dis

pla

cem

ent

[m]

ExperimentSimulation

w06z

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

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vaporous cavitation. The damping and the frequency of the pressure surges after 7 sec cannot be reproduced by the model.

0

10

20

30

0 5 10 15 20Time [sec]

Pre

ssu

re [

bar

]

Experiment

Simulation with FSI and quasisteadyfriction and without air release

p03

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 18.02 =k and 01 =k 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.

0

10

20

30

0 5 10 15 20Time [sec]

Pre

ssu

re [

bar

]

Experiment

Simulation with FSI and unsteadyfriction and without air release

p03

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.

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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.

0

10

20

30

0 5 10 15 20Time [sec]

Pre

ssu

re [

bar

]

Experiment

Simulation with FSI, quasisteadyfriction and air release

p03

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.

19

0

10

20

30

0 5 10 15 20Time [sec]

Pre

ssu

re [

bar

]

Experiment

Simulation with FSI, quasisteadyfriction and air release

p09

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.

0

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ExperimentSimulation

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.

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w18x

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.

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w06z

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).

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p03

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.

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p03

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.

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p06

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.

22

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 01 =k

and 18.02 =k 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 ( 1210−>α ) occurs, the calculated void fraction is smaller than using the quasisteady friction approach.

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p03

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.

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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.

23

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.

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Simulation with FSI but without air release and unsteady frictionp03

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 18.02 =k and 01 =k , 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.

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Simulation with FSI and unsteady friction but without air releasep03

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

24

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.

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Simulation with FSI, unsteady friction and air releasep03

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.

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ExperimentSimulation with FSI, unsteady friction and air release

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.

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w18x

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 fluid-structure-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.

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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

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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.

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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.

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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.

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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.