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Single- and Two-phase Flow through a Globe Valve: experiments and numerical simulations

M.H.M.Lemmens Master Internship Project, reportnumber: WPC 2007.03 Roma, 1 september – 18 december 2006 Supervisors: Università di Roma “La Sapienza”: Prof. C. Alimonti Eindhoven University of Technology: Dr. C.W.M. van der Geld Numerical Simulations: Prof. A. Corsini, Bruno Perugini

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Table of Contents

Table of Contents .....................................................................................................................................1 Nomenclature ...........................................................................................................................................3 1. Introduction ..........................................................................................................................................5 2. Valve and set-up design and practicalities............................................................................................6

2.1 Valve design ...................................................................................................................................6 2.2 Set-up design ..................................................................................................................................7 2.3 Use of the set-up.............................................................................................................................8

2.3.1 Start set-up...............................................................................................................................8 2.3.2 Measurements..........................................................................................................................8

3. Experiment strategy..............................................................................................................................9 3.1 Operation points .............................................................................................................................9 3.2 Data processing ..............................................................................................................................9

4. Single-phase Experiment Results ....................................................................................................... 10 4.1 Pressure in the set-up.................................................................................................................... 10 4.2. Pressure drop over the valve........................................................................................................ 11

4.2.1 DpValve................................................................................................................................. 11 4.2.2 DpASK92 .............................................................................................................................. 11

4.2 Pitot pressure drop........................................................................................................................ 12 4.3 Fluid velocity in the globe valve .................................................................................................. 13

5. Two-phase flow .................................................................................................................................. 14 5.1 Introduction .................................................................................................................................. 14 5.2 Theory .......................................................................................................................................... 14

5.2.1 Pressure drop ......................................................................................................................... 14 5.2.2 Choking ................................................................................................................................. 15

5.3 Experiment Strategy ......................................................................................................................... 16 5.3.1 Operation points ........................................................................................................................ 16 5.3.2 Tylan calibration........................................................................................................................ 17

6. Two-phase Experiment Results .......................................................................................................... 18 6.1 Pressure drop over the valve......................................................................................................... 18

6.2 DpValve.................................................................................................................................... 18 6.3 DpASK92 ................................................................................................................................. 19

6.4 Pitot pressure drop........................................................................................................................ 19 6.5 Fluid velocity in the globe valve .................................................................................................. 21

7. Two-phase Multiplicator .................................................................................................................... 23 7.1 Theory .......................................................................................................................................... 23 7.2 Experiments .................................................................................................................................. 23

8. Numerical analysis ............................................................................................................................. 25 8.1 XENIOS ....................................................................................................................................... 25 8.2 Grid generation............................................................................................................................. 26 8.3 Boundary conditions..................................................................................................................... 27

8.3.1 Velocity ................................................................................................................................. 27 8.3.2 Turbulent kinetic energy, viscous dissipation rate................................................................. 28

8.4 Pressure and Velocity normalization ............................................................................................ 29 8.3 Flow rate and Valve closings........................................................................................................ 29 8.4 Results .......................................................................................................................................... 29

8.4.1 Valve Pressure drop and Pitot pressure drop ......................................................................... 30 8.4.2 Flow field............................................................................................................................... 30 8.4.3 Pressure contour in valve....................................................................................................... 31 8.4.4 Velocity contour in valve ...................................................................................................... 33

9. Conclusions ........................................................................................................................................ 35 10. Recommendations ............................................................................................................................ 36 11. Bibliography..................................................................................................................................... 37 12. Appendix .......................................................................................................................................... 38

12.1. Single-phase old measurements................................................................................................. 38 12.1.1. Pa ........................................................................................................................................ 38 12.1.2. DpValve.............................................................................................................................. 38 12.1.3. DpPitot................................................................................................................................ 39

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12.2 Two-phase old measurements..................................................................................................... 40 12.2.1 DpValve............................................................................................................................... 40 12.2.2 Pitot Pressure ....................................................................................................................... 40 12.2.3 Two-phase multiplicator...................................................................................................... 40

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Nomenclature

Letter Quantity Unit (SI)

Chapter 4

p pressure N/m2

v velocity m/s

Chapter 5

A flow area m2

cfric friction factor -

Dt tube diameter m

D0 momentum flux density -

G mass flux kg/ (m2s)

GVF homogeneous void fraction -

i index -

Mtp two-phase Mach-number -

Q flow rate m3/s

u fluid velocity m/s

x mass fraction -

Chapter 7

i index -

j index -

Chapter 8

C XENIOS numerical simulation

constant

-

cµ model constant -

Cε1 model constant -

Cε2 model constant -

fε1 model constant -

fε2 model constant -

k turbulent kinetic energy m2/s

2

l*

turbulent length scale m

Lref reference length m

m velocity profile parameter -

r radial coordinate m

R radius inlet tube m

Re Reynolds number -

Sij mean strain rate tensor s-1

u*

turbulent velocity scale m/s

Symbol Quantity Unit (SI)

Chapter 5

α void fraction -

ΦL liquid multiplication factor -

ρ density kg/m3

µ dynamic viscosity kg/(ms)

4

Chapter 7

∆p pressure drop N/m2

ΦLo liquid only multiplication factor -

Chapter 8

ε viscous dissipation rate m2/s3

ε~ modified viscous dissipation rate m2/s

3

σk effective Prandtl number for diffusion m

σε effective Prandtl number for dissipation m

νt eddy viscosity m2/s

θ tangential coordinate -

τ turbulent time scale s

Subscript Quantity

Chapter 4

dyn dynamic

pitot pitot tube

stat static

tot total

Chapter 5

a air

acc acceleration

g gas

grav gravitation

fric friction

m mixture

w water

0 reference position 0

Chapter 6

mix mixture of water and air

tot Total of water flow rate and air flow rate

Chapter 7

exp experiment

s single-phase

t two-phase

Chapter 8

max maximum

mean mean

XENIOS as measured in XENIOS

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1. Introduction

Two-phase flows have a wide range of applications in both science and engineering. The problem that

is often faced when studying two-phase flow is the lack of knowledge of its behavior. Especially in

process technical equipment, in contrary to single-phase flow, predicting pressure drop in two-phase

flow is very complicated. Most relations that exist for predicting two-phase flow pressure drop are

empirical. To get familiar with the behavior of two-phase flows in process technical equipment,

experiments are thus necessary. The topic of this report is the behavior of a two-phase flow in a Globe

Valve designed by the Dutch company ‘Mokveld’. The final goal is the development of a model of the

valve for two-phase flow. This valve is e.g. used in the oil- and gas industry for pressure and flow

control. Two-phase flow, during this study, consists out of water and air. The two-phase flow through

the valve will be studied under several flow conditions. Variable conditions are the flow rate, the

closing of the valve and the void fraction. The experimental data enables to develop a model for two-

phase flow through such a globe valve.

In accordance with the before said, experiments have to be done to get insight in the behavior of the

two-phase flow through the globe valve. The experiments are carried out in the laboratory of the

engineering faculty of ‘Università "La Sapienza" di Roma’. An experimental set-up is build around the

globe valve to be able to do measurements on the valve. Before a two-phase flow is applied to the test

set-up, experiments are done with only one phase: water. This is done to test the sensors and to get

insight in the behavior of the test set-up as a whole. Practicalities of these experiments are discussed in

chapter 2 and 3; the results are presented in chapter 4. For different flow conditions, different

measurements are done. Experiments with almost the same set-up are already done before. Only a

small difference in the set-up is made. Since the change made in the set-up is very small, the

differences between the measurements are expected to be negligible. The old measurements are thus a

good reference for the new ones.

After the single-phase flow experiments are done, experiments with the same water flow rates and

valve closings are done with two-phase flow. Air is added to the water flow. Theory and practicalities

involved with the two-phase flow experiments are discussed in chapter 5. Thereafter, in chapter 6, the

results of these two-phase flow experiments are presented and analyzed. Here extra attention will be

paid to the influence of the void fraction, since it is the most important parameter in two-phase flows.

The results from the experiments are also compared to those of the old set-up. Moreover single-phase

flow experiments and two-phase experimental results will be compared (chapter 7). The comparison of

the two-phase flow data with the single-phase flow data is a first start for the development of a model

for the valve.

The last topic treated in this report is a numerical model (chapter 8) of the globe valve created in

XENIOS. A finite element model is made of the valve. With this model the flow field in the valve can

be studied for different flow conditions. In this study only a model is made for the single-phase flow.

The flow field can give insight in the experimental results. The experimental data functions as a

reference for the data obtained by the numerical model. The experimental data together with the

numerical model is a starting point for the final development of a model for the valve in two-phase

flow.

First the design of the valve and the set-up will be discussed in chapter 2.

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2. Valve and set-up design and practicalities

2.1 Valve design

The valve is of the RZD-RQX type, a Globe Valve made by Mokveld. ‘Globe’ refers to the spherical

shape of the valve. This valve is used for pressure and flow control for liquid, gas and two-phase flows.

Therefore also is referred to the valve as ‘Control Valve’. The typical design of the closing enables to

set the pressure drop and thus the pressure in the system. The inlet of the valve has a diameter of 3”

(76.2 mm). In figure 2.1, two cross-sections of the valve can be seen. The axial axis will be indicated as

the x-axis, the vertical direction as the y-axis. The direction perpendicular to the x-y plane is the z-axis.

The postion (0,0,0) is at the center of the valve , at the intersection point of the horizontal and vertical

dashed lines. The left picture in figure 2.1 shows a cross-section at the x-y plane through the point

(0,0,0). The right picture shows a cross-section at the y-z plane through the point (0,0,0).

Figure 2.1 Cross Section RZD-RQX Mokveld Valve

The flow enters the valve at the left and flows around the body in the center. After the flow has passed

the body it flows through multiple holes (indicated in figure 2.1 as ‘standard cage’). The flow exits the

the valve on the right. By turning the vertical shaft on top of the valve, the cylinder in the middle of the

body can be moved in horizontal direction. When moving to the right, it closes the holes of the cage.

When the cylinder is at the most right position, the flow is blocked. When the cylinder is at the most

left position all holes are opened. The number of closed holes determines the pressure drop over the

valve. The pressure drop over the valve can be set by its closing. Evidentially the fluid and the flow

rate also determine the pressure drop.

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2.2 Set-up design

To be able to make measurements on the valve, a set-up is build around the valve. A scheme of the set-

up is given in figure 2.2. This figure gives only the connections between the different elements in the

set-up. The black lines represent the connections. The drawing is not on scale and the locations are not

like in reality. The flow directions are 1-2-3-4-6-11-10-9-8-5-7-1 and 10-9-11-10. In table 2.1 the

different elements of figure 2.2 are explained.

Figure 2.2 Lay-out of the set-up

Number Explanation

1 Diverging tube to bridge the flow from the 2” to the 3” part.

2 Straight tube to develop the flow after the diverging section.

3 RZD-RQX Mokveld Globe Valve.

4 Converging tube to bridge the flow from the 3” to the 2” part.

5 Pressure tap to measure the static pressure at the pressure side of the pump in the set-up.

6 Separator to separate the air from the two-phase flow.

7 Valve to control the flow rate through the part of the set-up with the Globe Valve.

8 Electronic flow sensor.

9 Valve to divide the flow between the part of the set-up with the Globe Valve and the start-

up cycle (10-9-11-10).

10 Rotary Pump.

11 Water tank. The pump takes the water from the bottom of the tank. At the top of the tank

water flows in, coming from valve 9 and the separator.

12 The air supply which delivers the gas for the two-phase flow.

13 Pressure tap at the inlet of the Globe Valve.

14 Pressure tap at the outlet of the Globe Valve.

15 Pitot pressure tap, to determine dynamic pressure.

Table 2.1 Explanation figure 2.2

In the sequel will be referred to the elements in figure 2.2 by their numbers.

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2.3 Use of the set-up

2.3.1 Start set-up

To be able to start the pump, a small start-up cycle (10-9-11-10) is made in the set-up. At first tank 11

has to be filled with water. When starting the pump, valve 7 is closed and valve 9 is totally opened. The

water sequentially flows from the pump through valve 9, the tank 11 and back to the pump.

2.3.2 Measurements

To let the water flow through the globe valve, after the start-up, valve 7 has to be opened entirely.

Sensors 5, 8, 13, 14 and 15 start to give signals. The signals are send to the computer and can be read

with the program Labview. With Labview the experimental data can be saved.

The flow rate through the globe valve can be increased by, partially, closing valve 9. By closing this

valve, the flow through the start-up loop decreases and thus the flow through the part with the globe

valve increases. The flow through the valve is measured with flow meter 8. The flow is increased from

100 l/min to 250 l/min. Increasing the flow-rate is done in different experiments with different closings

of the globe valve (3). This will be explained extensively in the sequel of the report. The closing of the

valve can be set with the shaft on top (chapter 2.1).

The air supply 12 is controlled by a pump that is connected to the computer. With Labview the air flow

which flows into the set-up can be set. Herewith the homogeneous void fraction is set.

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3. Experiment strategy

3.1 Operation points

The set-up has different sensors with which process parameters can be measured. Sensors 1 and 2 in

figure 2.1 respectively measure the inlet – and outlet pressure of the valve. The sensor at point 3, point

of local zero velocity, measures the total pressure of the flow. Sensor 4 at last measures the static

pressure. The experiments are done for a range of different flow rates and closings of the Globe Valve.

Each combination of flow rate and valve closing will in the sequel be denoted as ‘operation point’. The

flow rate in the set-up ranges during the experiments from 100 l/min to 250 l/min with steps of 25

l/min. The closing of the globe valve is varied from 0 mm to 25 mm with measurements at 0, 10, 15,

20, 22.5 and 25 mm. For safety reasons the measurements are only carried out if the pressure

(measured by sensor 5 in figure 2.2) in the loop doesn’t exceed 1.3 bars and the water temperature is

less than 30°C. The operation points are the same for both the single- and two-phase flow

measurements. If different it will be denoted.

The different outputs of the sensors which will be presented in the sequel of this report are: the pressure

drop over the globe valve; pitot pressure drop in the globe valve and a single point pressure

measurement (sensor 5 in figure 2.2) in the loop.

3.2 Data processing

At each different operation point, more measurements are done. Every single measurement in the

graphs in the sequel is represented by a red ‘+’. From all the measurements on one operation point,

mean values are calculated, these are represented by a dot. In some graphs, ‘+’ and dots are left out for

a better visibility of the lines. Through the mean values, polynomial functions are fitted with use of a

MATLAB routine. The polynomials are in general 2nd order; if different it will be denoted. The

polynomials are represented in the graphs by the lines. In the legends can be seen which valve closing

is represented by a certain polynomial.

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4. Single-phase Experiment Results

4.1 Pressure in the set-up

To check the pressure in the loop a single point pressure tap is placed. The pressure tap is placed on the

pressure side of the pump, before the valve. The actual pressure measured with this pressure tap is

represented by Pa. In the graph below, the measurements on Pa are presented.

0 50 100 150 200 2500

0.5

1

1.5

2

2.5

025mm

Qw [l/min]

Pa [

bar]

Pa single phase

open

10mm

15mm

20mm

22.5mm

Figure 4.1 Pa for single-phase flow

In figure 4.1, on the horizontal axis the water flow rate, in l/min, can be seen and on the vertical axis

the pressure Pa in bars. Like stated in chapter two, measurements are only allowed if Pa < 1.3 bars. As

can be seen in figure 4.1, most measurement points satisfy the restriction of Pa < 1.3 bars. The

measurement points that don’t satisfy the restriction exceed 1.3 bars only slightly. This won’t harm the

set-up. The operation points specified in chapter 3.1 can thus safely be applied to the set-up. The only

exception is a valve closing of 25mm. For this closing only measurements are done with Qw = 100

l/min, because for this lowest flow rate the Pa is already much too high. For increasing flow rate Pa

would only increase more.

In figure 4.1 can be seen, as expected, that Pa increases with increasing closing of the valve and

increasing flow rate. Namely, the resistance for the flow increases with increasing fluid velocity and a

smaller orifice through which it flows. Thus when the flow rate is increased and the valve is closed, Pa

also has to increase to maintain the flow rate at the higher value. Pa can be understood as the driving

pressure for the flow through the whole set-up.

Pa is relative to the atmospheric pressure, it can have values smaller than 1 bar. This measurement

shows good accordance with the measurement done on the old set-up (Appendix 12.1.1).

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4.2. Pressure drop over the valve

The pressure drop over the valve is measured with two different methods. One method measures the

inlet and outlet pressure of the valve and subtracts the outlet from the inlet pressure. The other method

directly measures the difference between the inlet and outlet pressure of the valve. The sensors are

placed at position 1 and 2 in figure 2.1. In the figure of the set-up figure 2.2 the sensors can be seen at

position 13 and 14. In the sequel these different methods will be denoted by ‘DpValve’ and

‘DpASK92’ (the name of this output is equal to the name of the sensor) respectively.

4.2.1 DpValve The measurements on DpValve for a single-phase flow are represented in figure 4.2. On the horizontal

axis the water flow rate, in l/min, can be seen and on the vertical axis the DpValve pressure drop in

mbar.

0 50 100 150 200 2500

200

400

600

800

1000

1200

Qw [l/min]

DpV

alv

e [

mbar]

DpValve single phase

open

10mm

15mm

20mm

22.5mm

25mm

Figure 4.2 DpValve, single-phase flow

DpValve increases with increasing flow rate and increasing closing of the valve. This can be seen

clearly in figure 4.2. An increasing flow rate and increasing closing of the valve result in increasing

resistance to the flow. An increasing resistance results in a higher pressure drop over the valve. The

measurements for a closing of the valve of 25mm are done for smaller flow rates than the other

measurements. For measurements at flow rates higher than 100 l/min (valve closing 25mm), Pa would

immediately become larger than the maximum allowed value of 1.3 bars. The results in Figure 4.2 are

in good accordance to the measurements for DpValve at the old set-up (see Appendix 12.1.2).

4.2.2 DpASK92

The measurements on DpASK92 are shown in figure 4.3. On the horizontal axis the water flow rate, in

l/min, can be seen and on the vertical axis the pressure drop DpASK92 in mbar. The maximum

pressure difference which can be measured by DpASK92, with good accuracy, is about 200 mbar.

Therefore only measurements could be done on valve closings up to 15mm. For higher valve closings,

at the considered flow rate range, the pressure drop over the valve exceeds the scope of DpASK92. The

values found for DpASK92 are in accordance to the values found for DpValve in the scope of

DpASK92. DpValve measured for the new and the old set-up are in good accordance like shown

before. The measurements for DpASK92 thus show accordance to the measurements on the old set-up.

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0 50 100 150 200 2500

20

40

60

80

100

120

140

160

180

200

Qa [l/min]

DpA

SK

92 [

mbar]

DpASK92 single phase

open

10mm

15mm

Figure 4.3 DpASK92, single-phase flow

4.2 Pitot pressure drop

Pitot pressure drop is measured with use of the body in the valve (see figure 2.1). In order to measure

this Pitot pressure drop, one pressure tap is placed at the head of the body (3, figure 2.1). The other

pressure tap is placed at the wall of the valve (4, figure 2.1). The pitot pressure is thus measured at the

wall. Fluid velocity which will be calculated from the pitot pressure is thus the velocity at the wall. The

static pressure, pstat, on the wall of the valve is subtracted from the total pressure, ptot, which is

measured at the head of the body (where the velocity is locally equal to zero). With this method, the

dynamic pressure, pdyn, can be measured and thus the velocity around the body be determined. In

formulas this can be represented by:

dynstattot ppp += (4.1)

2

2

1vdpppp pitotdynstattot ρ===− (4.2)

It is adopted that the stationary incompressible Bernoulli equation is valid. The density of the fluid is

represented by ρ and the velocity of the fluid by ν. The velocity of the fluid in the valve can now

directly be related to the measured pitot pressure drop (which is equal to the dynamic pressure):

ρpitotdp

v2

= (4.3)

This velocity is thus only valid at the position where the dynamic pressure is measured.

The measurements on DpPitot are represented in figure 4.4. On the horizontal axis the water flow rate,

in l/min, can be seen and on the vertical axis the pressure drop DpPitot in mbar. For valve closings of

22.5mm and 25mm, to less measurements are available to make polynomials. The tendency of the

graphs in figure 4.4 is like expected. Pitot pressure drop increases with increasing flow rate and

increasing closing of the valve. Namely the velocity of the fluid increases with increasing flow rate. In

figure 4.4 also can be seen that, when you leave out the measurements for zero closing, the pitot

pressure drop is almost independent of the valve closing. This is caused by the design of this typical

valve. The closing of the valve only results in a locally, in the holes in the cage, velocity change. At the

location where the pitot pressure drop of figure 4.4 is determined, the flow area is constant and so is the

pitot pressure drop only dependent of the flow rate. The same can later be seen in figure 4.5 for the

fluid velocity. When compared to the old measurements on DpPitot (see Appendix 12.1.3), there is a

large discrepancy. Both measurements show the same tendencies, but the values for DpPitot in the new

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experiments have significant higher values than the old measured values. Until now this large

discrepancy can’t be declared. Only for a fully opened valve, measurements on DpPitot on the old set-

up are available.

0 50 100 150 200 250

0

5

10

15

20

25

Qw [l/min]

DpP

itot

[mbar]

DpPitot single phase

open

10mm

15mm

20mm

22.5mm

25mm

Figure 4.4 DpPitot, single-phase flow

4.3 Fluid velocity in the globe valve

With use of formula 4.3, the fluid velocity can easily be determined from the measured DpPitot. The

results are shown in figure 4.5. For the same reason as in figure 4.4 no polynomials are made for the

measurements on valve closings of 22.5mm and 25mm. The velocity shows linear dependency on the

flow rate. Since the fluid in these experiments is incompressible, the conservation of mass relation

indeed gives this linear dependency. The flow area at the place where the velocity is measured is

constant; it’s not affected by the closing of the valve. Therefore the lines for the different closings are

almost the same (same as seen in figure 4.4). The kink at a zero flow rate is because of numerical

reasons and of no importance for the discussion here. It’s trivial that the fluid velocity in the new

experiments has just as DpPitot much higher values than in the old experiments.

0 50 100 150 200 2500

0.5

1

1.5

2

2.5

Qw [l/min]

velo

city [

m/s

]

Fluid velocity around body in valve single phase

open

10mm

15mm

20mm

22.5mm

25mm

Figure 4.5 measured fluid velocity around body in valve

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5. Two-phase flow

5.1 Introduction

The main topic of this study is two-phase flows. The same experiments as with the single-phase flow

are done for two-phase flow. Results of these experiments are presented in chapter 6. Two-phase flows

studied here, consist out of a liquid phase (water) and a gaseous phase (air) flow. The most important

parameter for two-phase flow is the void fraction. It describes the gas to (gas and liquid) ratio. The void

fraction is a locally determined quantity defined as [3]:

A

Ai

ig∑=

,

α (5.1)

When considering for example a tube with a two-phase flow, A is the flow area of the tube and Ag is the

area occupied by the gas in the cross section of that tube. The summation sign is because of the

existence of more bubbles in one cross section. The main difference of two-phase flow to single-phase

flow is its compressibility. The two-phase flow consisting of water and air is compressible; the single-

phase flow consisting solely of water is incompressible. A feature of compressible flows is choking.

This phenomenon, applied to the globe valve, will be discussed in chapter 5.2.2. First the influence of

the void fraction on the pressure drop will be discussed in chapter 5.2.1.

5.2 Theory

5.2.1 Pressure drop

The content of this chapter is derived from v.d Geld [3].

Strongly idealized the Globe Valve can be seen as a tube. This simplification can be used to gain

insight in the relation between the pressure drop over the valve and the void fraction. The pressure drop

gradient in axial direction for a two-phase flow in a tube can be written as:

gravaccfric dz

dp

dz

dp

dz

dp

dz

dp++= (5.2)

The total pressure drop consists of three parts: friction, acceleration and gravitation. In this analysis

gravitation will be left out, because the set-up is horizontal. The pressure drop due to friction can be

written as:

w

fric

t

L

fric

xGc

Ddz

dp

ρ

22 ))1((2 −

Φ=− (5.3)

The symbol x represents the gas mass fraction, defined as:

1

)1(1

−

−+=

a

wxαρ

ρα (5.4)

2

LΦ (increasing with α ) and tD (constant) are positive numbers and fricc is defined as:

15

25.0

)1(−

−=

w

tfric

DxGc

µ (5.5)

In which µw (constant) is also a positive number. The pressure drop due to friction thus increases with

increasing gas mass fraction and thus void fraction.

The acceleration term in the pressure drop is defined as:

−

−+=−

)1(

)1( 222

αραρ waacc

xx

dz

dG

dz

dp (5.6)

Here G, the total mass flux, is a positive number and just like the pressure drop due to friction, the

pressure drop due to acceleration increases with increasing void fraction. From these two defined

pressure drops can be concluded that a two-phase pressure drop increases with the void fraction. The

pressure drop over the globe valve will increase with increasing void fraction.

5.2.2 Choking

Choking is a physical phenomenon that can appear in accelerated compressible fluids. Since the two-

phase flow, considered here, is compressible; choking is a phenomenon that should be taken in to

account. Here not an in depth analysis will be given. Solely the factors that can cause probable choking

will be discussed in a very simplified manner. Evidentially these factors are discussed in view of the

problem here: the valve.

Figure 5.1 contraction-pipe

Consider a gas that is applied to the contraction-pipe in figure 5.1. The mass flow can be increased by

increasing p1. As long as the Mach number at position 0 is smaller than 1, the mass flow will increase

with increasing p1. When M>1, the ratio p1/p0 remains constant with increasing p1. The flow is

choked and the mass flow is blocked. The difference between p0 and p2 (ambient pressure) will be

bridged with a shock-wave. For two-phase flows, another dimensionless number is used to see whether

choking occurs. This number is defined as [2]:

00DM tp α= (5.7)

where suffix 0 refers(in this analysis) to the position 0 in figure 5.1, suffix tp to ‘two-phase’, α is the

void fraction and D represents the momentum flux density, which is defined as [2]:

0

2

000

p

uD mmρ

= (5.8)

here ρ, u and p are respectively the density, velocity and pressure at position 0 in figure 1. The suffix m

denotes that these are averaged values for the whole two-phase flow. The two-phase flow is here

considered as homogenous. The dominating factor that thus can cause a two-phase flow to choke is the

velocity. This is the same as for gasses. If now the valve is taken in consideration (see figure 2.1). The

most critical position where choking can occur; is at the holes in the cage. The flow area in the holes is

the smallest and the velocity the highest. If choking appears, probably the downstream pressure

measurement gets influenced, because of the shock-wave. It is thus important to figure out; whether

there is a chance that choking has appeared in the experiments. Later with use of the numerical analysis

16

a first approximation will be made for the two-phase Mach-number. With this number the possible

appearing of choking can be determined. Note that this is only a first rough approximation. The model

for choking is strongly simplified, the valve has a complicated configuration and the numerical analysis

has only been done for single-phase flow.

5.3 Experiment Strategy

5.3.1 Operation points

Because of limits to the set-up, it is not possible to determine the local void fraction. In these

measurements the homogeneous void fraction is used. This is defined as the fraction of the air flow rate

to the total flow rate:

aw

a

QQ

QGVF

+= (5.9)

In the sequel of the report, void fraction refers to GVF and thus isn’t the actual local void fraction as

defined in formula 5.1. If different it will be denoted. The two-phase flow is applied to the set-up by

adding pressurized air in the water flow (12, figure 2.2). Void fraction is kept constant during a single

experiment. But experiments are done with different void fractions. With this method the same

experiments can be done for different void fractions. The only variable is the void fraction and thus the

effect of the void fraction on the measurements can be clearly studied. Since the void fraction and its

effect on the flow are of utmost importance in studying two-phase flows; this experimental strategy is

the most convenient. To maintain these constant void fractions during the experiments, where the water

flow rate is increased, the air flow rate (Qa) also has to be increased. The experiments are done for void

fractions of: 10%, 20% and 30%. The water flow rates used are the same as in the single-phase

experiments: from 100 l/min to 250 l/min with steps of 25 l/min. Only for GVF = 30% measurements

are done for water flow rates of 75 l/min to 175 l/min. It turned out that for air flow rates higher than 75

l/min the flow starts to oscillate and this is harmful for the set-up. Therefore only measurements on

GVF = 30% can be done for water flow rates up to 175 l/min. A maximum water flow rate of 175 l/min

namely requires a maximum air flow rate of 75 l/min. To have sufficient measurement points, here is

chosen to start measuring from a water flow rate of 75 l/min at GVF = 30%. The air flow rates (in

italic) for these different void fractions and water flow rates are presented below in table 5.1. The

values are in l/min.

Table 5.1 GVF, water- and air flow rates

GVF\ wQ 75

l/min

100

l/min

125

l/min

150

l/min

175

l/min

200

l/min

225

l/min

250

l/min

10% 11.11 13.89 16.67 19.44 22.24 25 27.78

20% 25 31.25 37.5 43.75 50 56.25 62.5

30% 32.14 42.86 53.57 64.28 75

These combinations of Qa and Qw are applied to Valve closings of 0mm, 10mm, 15mm, 20mm and

22.5mm. The empty spaces in table 5.1 are operation points which are not used.

Practicalities for two-phase flow experiments are same as for the single-phase experiments.

17

5.3.2 Tylan calibration The air is added to the water flow by an electronically controlled pump: Tylan. A set point for the

Tylan can be entered in Labview. This value has to be corrected, multiplied with a multiplication factor

(correction factor), to get the wished air flow rate into the set-up. In order to determine this correction

factor, the Tylan is calibrated with a rotameter. It has turned out that the correction factor is dependent

on the flow rate. The correction factor as a function of the air flow rate is given in figure 5.2.

10 20 30 40 50 60 70 80

2.5

3

3.5

4

Qa [l/min]

Corr

ection f

acto

r [-

]Tylan Calibration

Figure 5.2 Tylan calibration curve

The rotameter can only measure air flow rates up to a maximum of approximately 28 l/min. The graph

in figure 5.2 for Qa > 28 l/min is an extrapolation of the gradient measured around the value of 28

l/min. After figure 5.2 was determined, it is used in all two-phase flow experiments.

18

6. Two-phase Experiment Results

6.1 Pressure drop over the valve

The pressure drop over the valve is measured with two different methods, DpValve and DpASK92,

identical to the single-phase experiments (see chapter 4.2).

6.2 DpValve

Every single graph in figure 6.1 represents the measurements done for all void fractions and water flow

rates for one specific closing of the valve. Take note that all graphs have a different scale on the

vertical axis. Like with a single-phase flow the pressure drop over the valve increases with increasing

water flow rate and decreasing valve opening. The friction to the flow namely increases for increasing

flow rate and decreasing flow area. As a result the pressure drop over the valve increases. From figure

6.1 can be concluded that the effect of the void fraction on the pressure drop is not consistent. If the

valve is fully opened (0mm), a lower void fraction results in a higher pressure drop. For all other

closings of the valve (10mm to 22.5mm), a higher void fraction results in a higher pressure drop.

0 100 2000

50

100

Qw [l/min]

DpV

alv

e [

mbar]

DpValve 0mm

10%

20%

30%

0 100 2000

100

200

30010mm

Qw [l/min]

DpV

alv

e [

mbar]

0 100 2000

200

400

60015mm

Qw [l/min]

DpV

alv

e [

mbar]

0 100 2000

500

1000

1500

20mm

Qw [l/min]

DpV

alv

e [

mbar]

0 100 2000

1000

2000

3000

22.5mm

Qw [l/min]

DpV

alv

e [

mbar]

Figure 6.1 DpValve for different GVF and Valve Closings, Two-phase flow

The measurements for valve closings from 10mm to 22.5mm are in accordance to the theory of two-

phase flows (chapter 5.2.1). In a two-phase flow, the pressure drop namely increases with increasing

void fraction. Measurements on the fully opened valve are striking with theory.

Measurements for DpValve on the old set-up are only done for a fully opened valve. The results are

presented in Appendix 12.2.1. When the new measurements are compared to the old measurements, it

can be seen that DpValve for both is approximately equal. Though the values are approximately the

same, in the old measurements a higher void fraction results in a higher value for DpValve, like

expected from theory. This is in contrary to the new measurements on the fully opened valve but in

accordance to the new measurements when the valve is partially closed. With the results of the

numerical model (chapter 8) will be paid attention to the case of the fully opened valve. The flow field

can give insight in this remarkable behavior.

19

6.3 DpASK92 The results of the measurements on DpASK92 are presented in figure 6.2. Measurements up to

200mbar are possible with the DpASK92 sensor.

0 100 2000

50

100

Qw [l/min]

DpA

SK

92 [

mbar]

DpASK92 0mm

10%

20%

30%

0 100 2000

50

100

150

10mm

Qw [l/min]

DpA

SK

92 [

mbar]

0 100 2000

50

100

150

20015mm

Qw [l/min]

DpA

SK

92 [

mbar]

0 100 2000

50

100

150

200

20mm

Qw [l/min]

DpA

SK

92 [

mbar]

0 100 2000

50

100

150

200

22.5mm

Qw [l/min]

DpA

SK

92 [

mbar]

Figure 6.2 DpASK92 for different GVF and Valve Closings

The pressure drop over the valve is again plotted as a function of the water flow rate. It shows a similar

behavior as the measurements on DpValve. The pressure drop increases with increasing water flow rate

and increasing closing of the valve. Only for a fully opened valve again the pressure drop clearly

increases with decreasing void fraction. Measurements on DpASK92 for the old set-up are not

available. The measurements on DpASK92 are in principle the same as those for DpValve until

200mbar. The pressure taps used to determine the different pressure drops are the same. When

DpValve and DpASK92 in figures 6.1 and 6.2 are compared for all valve closings, indeed the pressure

drops have approximately the same values.

6.4 Pitot pressure drop

The measurements on the pitot pressure drop for two-phase flow are plotted in figure 6.3. In contrary to

the other pressure drops, DpPitot is plotted as a function of the total flow rate = water flow rate + air

flow rate. This is done because the measurements on the old set-up are also presented like this. A good

comparison is only possible if DpPitot in both cases is plotted as a function of the same variable. The

negative values for the pitot pressure drop for low flow rates are due to numerical reasons.

In figure 6.3 can be seen that a decreasing void fraction results in an increasing pitot pressure drop.

This can be understood from the following analysis. The pitot pressure drop can (simplified) be

expressed as:

2

2

1mixmixpitot vdp ρ= (6.1)

Here the suffix mix denotes that the variables, density and velocity, are average values for the total

mixture of water and air. Though in reality the mixture is compressible, in the analysis the density will

considered to be constant. The density and velocity of the total mixture can be expressed as:

wamix ρααρρ )1( −+= (6.2)

20

A

Q

A

QQv totwa

mix =+

= (6.3)

In which α is the void fraction, A the flow area and Qtot is the flow rate of the total mixture. Relations

6.2 and 6.3 substituted in 6.1 give as a result for the pitot pressure drop:

2

2))1((

2

1totwapitot Q

Adp ρααρ −+= (6.4)

Since water density ρw is about thousand times larger than air density ρa, dppitot will decrease with

increasing void fraction. This is in accordance to the graphs given in figure 6.3.

0 100 200 300

0

10

20

30

40

Qtot [l/min]

DpP

itot

[mbar]

DpPitot 0mm

10%

20%

30%

0 100 200 300

0

10

20

30

4010mm

Qtot [l/min]

DpP

itot

[mbar]

0 100 200 300

0

10

20

30

4015mm

Qtot [l/min]

DpP

itot

[mbar]

0 100 200 300

0

10

20

30

4020mm

Qtot [l/min]

DpP

itot

[mbar]

0 100 200 300

0

10

20

30

4022.5mm

Qtot [l/min]

DpP

itot

[mbar]

Figure 6.3 DpPitot for different GVF and Valve Closings

As stated in chapter 4.2 the pitot pressure drop for single-phase flow is hardly influenced by the closing

of the valve. This is also valid for two-phase flows (figure 6.3).

The big differences between the graphs in figure 6.3 for a valve closing of 22.5mm are caused by a lack

of measurement points. The pressure in the set-up namely became too high, for safety reasons the flow

rate was not increased. Safety restrictions to the pressure in the set-up are stated in chapter 3.1.

The measurements for the pitot pressure drop on the old set-up are presented in appendix 12.2.2.

Measurement results for single and two-phase flow show equal tendencies. Also the effect of the void

fraction is the same. The pitot pressure drop in both figures decreases with increasing void fraction.

The numerical values, in contrary, are totally different. The new measurements have resulted in much

higher values than the old measurements. New measured pitot pressure drops are approximately 5

times higher than the old ones. This difference between new and old measurements has already been

encoutered on the pitot pressure drop measurements for single-phase flow in chapter 4.2.

21

6.5 Fluid velocity in the globe valve An expression for the mean velocity of the fluid mixture can be derived from relation 6.1. It will be

adopted that the mixture density and velocity are homogeneous. The fluid velocity can be written as:

mix

pitot

mix

dpv

ρ

2= (6.5)

The graphs in figure 6.4 are determined with formula 6.5. Here is chosen to plot the velocity as a

function of Qw instead of Qtot. If the velocity is plotted as a function of Qtot, it’s namely not dependent

on the void fraction. Since the effect of void fraction is the main subject in this study; the choice for Qw

is convenient.

0 100 2000

0.5

1

1.5

2

2.5

3

3.5

Qw [l/min]

Velo

city [

m/s

]

GVF = 10%

0mm

10mm

15mm

20mm

22.5mm

0 100 2000

0.5

1

1.5

2

2.5

3

3.5GVF = 20%

Qw [l/min]

Velo

city [

m/s

]

0 100 2000

0.5

1

1.5

2

2.5

3

3.5GVF = 30%

Qw [l/min]

Velo

city [

m/s

]

Figure 6.4 Mean mixture fluid velocity as a function of water flow rate, different GVF and valve closings

In figure 6.4 can be seen that there exists a linear relation between the flow rate Qw and the velocity.

Moreover the gradient of the graph increases with increasing void fraction. This behavior can be

understood from the following simple analysis. The homogeneous void fraction can be written as:

aw

a

QQ

QGVF

+= (6.6)

This can be written differently as:

wa QGVF

GVFQ

−=

1 (6.7)

The total flow rate can be denoted, with use of 6.7, in terms of void fraction and water flow rate as:

GVF

QQ w

tot−

=1

(6.8)

22

Relation 6.3 and relation 6.8 together give:

)1( GVFA

Qv w

mix−

= (6.9)

Relation 6.9 shows exactly the behavior of the graphs in figure 5.6. The velocity increases linearly with

the water flow rate and the gradient )1(

1

GVFA − increases linearly with increasing void fraction.

23

7. Two-phase Multiplicator

7.1 Theory

Theoretical relations are available for predicting the pressure drop in single-phase flow through tubes

and appendices. With use of a correlation factor, the single-phase pressure drop can be correlated to the

two-phase pressure drop [3]:

liquidonly

Lodz

dp

dz

dp=Φ2

(7.1)

In which dz

dp is the pressure drop per unit length for a two-phase flow. The pressure gradient

liquidonlydz

dpis the pressure drop per unit length for a flow consisting solely of liquid. This can be

determined with classical relations for single-phase pressure drop. The correlation factor 2

LoΦ can be

expressed in terms of liquid properties, the geometry of the problem and liquid friction factors [3]. The

two-phase pressure drop can thus be determined with only liquid flow characteristics. This factor will

be determined in chapter 7.2 from the experimental results. If this factor is known for the valve, the

pressure drop in two-phase flow can easily be determined with only knowledge of the single phase

flow for different flow conditions.

7.2 Experiments

The correlation factor is determined by dividing the two-phase data for DpValve by the single-phase

data for DpValve for all valve closings:

svalve

tvalve

Lop

p

,

,2

exp,∆

∆=Φ (7.2)

In which ∆pvalve,t and ∆pvalve,s represent the pressure drop over the valve for two-phase and single-phase

flow respectively. Suffix exp indicates experimental. In relation 7.2 DpValve is not per unit length, like

in formula 7.1. This however is not a problem. DpValve is, both for single and two-phase flow,

measured over the same fixed distance. The derivative can thus be cancelled out. The way in which the

correlation factors are determined from all the data, taken from the experiments, is explained in the

sequel.

Measurements are done for different water flow rates, void fractions and valve closings. In order to

determine correlation factors which represent all these data, the correlation factors have to be averaged.

For a briefer notation the two-phase flow pressure drop over the valve will now be represented by ∆pt

and the single-phase flow pressure drop by ∆ps.

∆pt and ∆ps are row vectors. Every element of ∆pt and ∆ps is the pressure drop over the valve for a

specific water flow rate. First the mean is taken of

jis

t

p

p

,

∆

∆ . This mean is calculated for all

combinations of i and j which stand respectively for the different GVF (i = 1,2,3 = 10%, 20%, 30%)

and valve closings (j = 1,2,3,4 = 10mm, 15mm, 20mm, 22.5mm).

Next the root of this mean is taken in order to calculate the correlation factor:

24

5.0

,

,exp,,

∆

∆=Φ

jis

tjiLo

p

p (7.3)

In total there are now thus 3 GVF * 4 closings = 12 mean values for the correlation factor.

The measurements for a fully opened valve are not taken into account in these calculations because

those measurements are striking with theory. For all closings one mean is taken of

5.0

,

∆

∆

jis

t

p

pfor the three different GVF:

5.04

1 ,

1∑

=

=

∆

∆N

j jis

t

p

p

N for i = 1,2,3

These three remaining values represent the correlation factors for the three different GVF. The results

of these calculations are represented in figure 7.1 by the dots. Though there are only three measured

points (the point at GVF = 0 has to be 1 by definition of formula 7.1) a clear linear relation can be seen

between GVF and the correlation factor.

The line, in figure 7.1, is a 1st order polyfit of this data. The relation between the correlation factor and

GVF in the polyfit is:

GVF0.28241)(exp, ⋅+=Φ GVFLo (7.4)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.5

1

1.5

2

2.5

GVF [-]

ΦLo,e

xp [

-]

ΦLo,exp (GVF) for Mokveld Globe Valve

polyfit of mean data

mean measured data

Figure 7.1 Correction factor as a function of the void fraction.

The correction factors derived from the old data are given in appendix 12.2.3. For GVF up to 0.4 the

graph in figure 7.1 shows an identical behavior to the graph in appendix 12.2.3. For higher GVF

values, the gradient of the graph in appendix 12.2.3 increases. The correction factor for the new data

remains constant, but this is because the 1st order polyfit is only based on the points until GVF = 0.3. In

fact the correction factor for the new data for GVF>0.3 is thus not known and can’t be compared to the

old data. The line (for GVF>0.3) in figure 7.1 is only a first estimate.

25

8. Numerical analysis

The experiments have given insight in the behavior of the valve. The mean velocity in the valve and the

pressure drop are measured for different flow conditions. These data however don’t give information

about the flow field through the valve. Knowledge about the flow field can be helpful in explaining

certain experimental results, also for two-phase flow. To visualize the flow field through the valve, a

numerical model of the valve is made. At first only the incompressible, single-phase, flow will be

modeled. The same data (pressure drop, velocity) as measured during the experiments are calculated

with the model. The numerical model can thus be verified by the experimental results.

A straight tube is placed between the diverging section and the inlet of the valve (2, figure 2.2). The

goal of this tube is to achieve a fully developed flow after the diverging section before entering the

valve. The numerical model can also be used to visualize the flow field inhere. With the model can be

determined whether the tube is functioning like it is supposed to. Moreover the numerical single-phase

results together with the correlation factors, determined in chapter 7.2, enable to approximate the two-

phase pressure drop over the valve.

The finite element computer code that is used to calculate the flow field is XENIOS, written in

FORTRAN. This code, developed at University of Rome “La Sapienza” is particularly used for solving

flow problems. It’s developed for solving 2D/3D and laminar/turbulent flows [6]. The equations that

form the basic of the code are presented in the next chapter.

8.1 XENIOS

The content of this chapter is derived from Andrea Santoriello [5].

The flow problem considered in the numerical analysis is solved with use of a k-ε model in its low

Reynolds extension. Transport of turbulent kinetic energy k is of great importance in turbulent flows.

With a transport equation for k, the turbulent velocity scales can be determined. However the turbulent

length scale l* also has to be determined, locally. The most simple and widespread solution, in order to

determine l*, is to design a transport equation for the viscous dissipation rate ε. This gave rise to the

two equations of the k-ε model. The principle of this model is to take in account the equations for the

turbulent length and velocity scale: l*

and u*

with which the eddy viscosity νt can be calculated. The

expressions are:

ε

2/3* k

l = (8.1)

ku =* (8.2)

εν µµ

2**** ),(

kculcult == (8.3)

cµ is a model constant. The equations that complete the k-ε model are the transport equations for the

turbulent kinetic energy k and the viscous rate of turbulent kinetic energy dissipation, ε. Expression for

transport of turbulent kinetic energy:

jkv

uSku j

k

tjiijtjj ,2 ,,,

++−=

σνεν (8.4)

Where σk in an effective Prandtl number for diffusion, ju a velocity component and Sij the mean strain

rate tensor.

26

Transport equation for the viscous rate of turbulent kinetic energy dissipation:

jv

CuSCu jt

jiijtjj ,1

21

,2,1,

++−= ε

σνε

τν

τε

ε

εε (8.5)

Where Cε1 and Cε2 are constants of which the meaning goes beyond the discussion here, σε is an

effective Prandtl number for dissipation diffusion and τ is the turbulent time-scale. There are different

versions of turbulence models. A model that gives good solutions also near the walls for low, transient

and high Reynolds numbers is the k-ε model in its low Reynolds extension. This model makes use of a

modified dissipation rate ε~ which has homogenous boundary conditions on solid walls. This modified

dissipation rate is defined as:

( ) Dxk i −=∂∂−= ενεε2

/2~ (8.6)

The transport equations for the turbulent kinetic energy and the modified dissipation rate now read:

Djkv

uSku j

k

tjiijtjj −

++−= ,~2 ,,,

σνεν (8.7)

Ejv

fCuSfCu jt

jiijtjj +

++−= ,~1

21~

,22,11, εσ

νετ

ντ

εε

εεεε (8.8)

where fε1 = 1, fε2 and E are defined below:

[ ])Reexp(3.01 2

2 tf −−=ε (8.9)

22 )/(2 kjit xxuE ∂∂∂= νν (8.10)

To solve the complete set of equations, the boundary conditions have to be defined. This will be done

in the sequel of the report. The equations are solved in steady-state so no initial conditions are required.

All equations together with the boundary conditions have to be implemented in the Finite Element

Method which can be solved by XENIOS. For details on the numerical model, the reader is referred to

[1].

8.2 Grid generation

The configuration of the valve and the mesh are made with use of the program ‘Gridgen’. Since the

valve is rotational symmetric, the valve is modeled in only 2 dimensions. The total configuration

contains: a small inlet section, the diverging section, the straight tube for development of the flow, the

valve and an outlet section. In total the mesh consists of +-55.000 nodes. This number has turned out to

result in a sufficient accuracy. The grid is not everywhere equally spaced. Closer to the boundaries, the

grid becomes finer. At the boundaries the highest velocity gradients exist and to be able to calculate

these with a good accuracy, a local fine grid is required.

Circumferentially the cage (in the closing mechanism) has rows of five respectively six holes next to

each other (figure 2.1). Here is chosen to make the cross-section at a row with six holes. The

experiments are done with six different closings of the valve. In reality, with the closings used in the

experiments, the holes sometimes are partially closed. In the model, partially closing of the holes gives

big difficulties for the creation of the mesh. Therefore the different closings are modeled with only

fully closed and fully opened holes.

27

8.3 Boundary conditions

8.3.1 Velocity

Inlet profile

The Inlet velocity profile has to be defined to be able to calculate the flow field in the configuration.

The inlet of the configuration is the left side from the inlet section left from the diverging section (1,

figure 2.2). Fluid velocity in radial direction is in the inlet equal to zero. The velocity profile for the

axial velocity which will be adopted here is :

m

R

rvrv

/1

max 1)(

−= (8.11)

Here r is the radial coordinate, R the inner radius of the tube (R = 25.4mm at the inlet) and vmax the

maximum velocity at r = 0. The parameter m sets the thickness of the boundary layer. The flow is

considered to be turbulent and fully developed at the inlet. The velocity profile that fits best to these

specifications can be approximated with m = 8. The maximum velocity vmax is dependent on the flow

rate. The flow rate can be expressed in terms of velocity by:

θθππ

rdrdR

rvrdrdrvQ

mRR

w ∫ ∫∫ ∫

−==

2

0

/1

0

max

2

0 0

1)( (8.12)

Here θ is the tangential coordinate. The maximum velocity can be left outside the integral because it is

independent of both the radial and tangential coordinate. Relation 8.12 results in a linear relationship

between the maximum velocity and the flow rate:

∫ ∫

−

=π

θ2

0

/1

0

max

1 rdrdR

r

Qv

mR

w (8.13)

The integral in relation 8.13 is numerically approximated with use of a MATLAB routine. The relation

between the flow rate and the maximum velocity that follows from this integration is presented in

figure 8.1.

The velocity profile that has to be entered in XENIOS has to be normalized by the mean velocity. The

mean velocity can easily be determined by:

A

Qv w

mean = (8.14)

In which A is the flow area of the inlet tube. An expression for the velocity profile normalized by this

mean velocity is given in formula 8.13:

m

meanmean R

r

v

v

v

rvrv

/1

max 1)(

)(

−== (8.15)

Substitution of relations 8.13 and 8.14 in relation 8.15 give that the normalized velocity profile is

independent of the flow rate. The normalized velocity is plotted in figure 8.2. In figure 8.2 can be seen

that the boundary layer is very thin, like should be for a developed turbulent flow. The maximum

velocity is indeed at r = 0 and the corresponding value in figure 8.2 is greater than 1, thus vmax > vmean.

28

100 150 200 250

1

1.2

1.4

1.6

1.8

2

2.2

Maximum velocity inlet velocity profile

Qw [l/min]

v ma

x [

m/s

]

-25 -20 -15 -10 -5 0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

r [mm]

v/v

mea

n

Normalized velocity Inlet tube

Figure 8.1 Maximum velocity at the inlet tube as function

of flow rate.

Figure 8.2 Normalized velocity profile at inlet tube as

function of radius.

Walls

At the walls the no-slip condition is applied, both the vertical and horizontal velocities are equal to

zero.

8.3.2 Turbulent kinetic energy, viscous dissipation rate

Inlet profile

Both the kinetic energy and the viscous dissipation rate are normalized in the numerical model. The

kinetic energy inlet profile can be calculated with use of relation 8.2. At the inlet the normalized u* is

equal to the normalized velocity of formula 8.15. The inlet profile for the normalized kinetic energy

now reads:

2

)()(

=

mean

iv

rvrk (8.16)

where the suffix i refers to inlet. For determination of the dissipation rate at the inlet, next to the

turbulent velocity scale, the turbulent length scale is used. Dissipation rate can be calculated with:

*

2/3

l

k=ε (8.17)

The turbulent length scale l* at the inlet, has in the numerical simulation a length of 0.1*Lref. Lref is the

diameter of the inlet tube. The expression for the dissipation rate at the inlet now reads:

ref

mean

iL

v

rv

r1.0

)(

)(

3

=ε (8.18)

29

Walls

For the kinetic energy and viscous dissipation rate at the walls, the no-slip condition is applied. Kinetic

energy and viscous dissipation rate are zero at the walls.

8.4 Pressure and Velocity normalization

The pressure that can be read in the results of the numerical simulations calculated by XENIOS can be

expressed as:

CppXENIOS += (8.19)

where C is a constant value, different for each simulation. The data which can be read from the

numerical analysis is normalized. p is a normalized pressure which is defined as:

2

meanv

pp

ρ= (8.20)

where ρ is the density of water and vmean the actual mean velocity at the inlet of the configuration. The

horizontal bar on top indicates the normalization. Normalized velocity can be expressed as:

meanv

vv = (8.21)

where v is the actual velocity. This normalized velocity is also the velocity which is read in the results

of XENIOS: XENIOSvv = . Since two different flow rates are applied, two different mean velocities

exist. For Qw = 150 l/min, vmean = 1.2335 m/s and for Qw = 250 l/min, vmean = 2.0558 m/s. In the sequel

of the report, the normalized values will be used. It will be indicated if the flow rate is 150 l/min or 250

l/min which implies how the experimental values can be converted to actual values.

8.3 Flow rate and Valve closings

Because of the high number of grid points, the computation time for the numerical model is large.

Therefore with only two water flow rates (Qw = 150 l/min and Qw = 250 l/min) numerical experiments

are done. These flow rates will be applied to meshes which represent two closings of the valve: 0mm

and 20mm.

Next to the mesh, inlet velocity profile and boundary conditions the Reynolds number at the inlet has to

be defined in XENIOS. The Reynolds number here is based on the mean velocity and the diameter of

the inlet tube (D = 2”). The temperature of the water during the experiments was on average 25°C. For

these conditions the Reynolds numbers at the inlet are: Re = 5.9448e3 for Qw = 150 l/min and Re =

7.0214e4 for Qw = 250 l/min.

8.4 Results

The results of the numerical analysis give a view on the flow field in the valve. Special attention will

be paid to two interesting zones: the tube after the diverging section and the section after the cage. The

numerical model calculates: pressure, velocity, kinetic energy and dissipation at every grid point. In

order to determine whether the numerical model has succeeded in achieving a satisfactorily

convergence, some model results are compared to the results of the experiments. The valve pressure

drop and the pitot pressure drop are taken for this comparison.

30

8.4.1 Valve Pressure drop and Pitot pressure drop

Valve pressure drop is presented in table 8.1; numerical results at the left side and experimental results

at the right side. Pitot pressure drop is presented in table 8.2; numerical results at the left side and

experimental results at the right side.

Table 8.1 Valve pressure drop determined by numerical model (left), experiments (right)

Flow rate/Valve closing 0mm 20mm 0mm 20mm

150 l/min 63 mbar 105 mbar 150

l/min

50 mbar 140 mbar

250 l/min 216 mbar 293 mbar 250

l/min

460 mbar 1200 mbar

Table 8.2 Pitot pressure drop determined by numerical model (left), experiments (right)

Flow rate/Valve closing 0mm 20mm 0mm 20mm

150 l/min 24 mbar 14 mbar 150

l/min

5 mbar 16 mbar

250 l/min 80 mbar 40 mbar 250

l/min

10 mbar 22 mbar

From Table 8.1 can be concluded that the model succeeded in achieving a good convergence for Qw =

150 l/min. However for Qw = 250 l/min, the differences between the model and the experiments are too

large. The higher flow rate results in a higher Reynolds number and more iteration steps are needed for

the model to reach convergence. Numerical calculations for the pitot pressure drop give only a good

result for Qw = 150 l/min with a valve closing of 20mm. Reason for this is not encountered in the scope

of this study. Though the numerical model only partially achieved convergence, it will be used for

further analysis. Analysis focuses mainly on flow field, pressure and velocity contours and not on

numerical values.

8.4.2 Flow field

A streamline plot of the inlet section, diverging section and straight tube can be seen in picture 8.4.

Figure 8.3 streamline plot, Qw =150 m/s, fully opened, inlet, diverging section, straight tube

In figure 8.3 can be seen that the flow, for these flow conditions, is perfectly aligned with the

boundary. The diverging section doesn’t cause the boundary layer to get loose from the wall. The

straight tube between the diverging section and the valve is thus not necessary for Qw = 150 l/min. The

flow in this section of the configuration for a partially closed valve shows similar behavior. The flow

field upstream of the cage is not influenced by the closing of the valve. For Qw = 250 l/min the same

steam line profile in this section of the configuration is obtained. From these results can be concluded

that during all experiments the boundary layer doesn’t get loose from the walls of the diverging tube.

The stream line patterns for the different numerical experiments only differ around the cage, the

discussion focuses now on that specific part. Stream line plots for the four different experiments are

shown in figure 8.4. The most interesting feature is the big vortex that appears at the boundary right

from the most right hole of the cage. The vortex appears because the flow has to flow around the sharp

edge. The edge causes the boundary layer to get loose from the wall. Both for Qw = 150 l/min and Qw

31

= 250 l/min partially closing of the valve causes an enlargement of the dimensions of the vortex. The

closing of the valve causes a substantially increase of the fluid velocity at the exit of the holes in the

cage. As a consequence the tangential velocity increases and thus the size of the vortex increases.

When Qw = 150 l/min is compared to Qw = 250 l/min hardly any differences appear in the streamline

patterns. Increasing the flow rate evidently causes also an increase in the fluid velocity at the exit of the

holes. This increase however seems to be too less to cause visible enlargement of the vortex.

Figure 8.4 Stream line pattern, upper row: Qw = 150 l/min, lower row: Qw = 250 l/min, left column: opened valve,

right column: valve closing = 20mm

Pressure measurement

The pressure in the valve is circumferentially measured at four equidistant points in the wall of the

valve. Exit pressure measurement is located at X = 7.5. In the situation that the valve is closed for

20mm, the exit pressure tap measures inside the vortex. This however doesn’t obstruct the pressure

measurement. The pressure is radial approximately constant, see figure 8.4 – 8.6. Though the vortex,

the correct pressure is measured at the exit.

In chapter 6.2 the remarkable behavior of the pressure drop over the valve is encountered for zero

closing. The stream line pattern in figure 8.4 for the zero closing however doesn’t show deviate

behavior. The nature of the valve with zero closing: valve pressure drop increases with decreasing void

fraction can’t be explained with single-phase numerical simulation. Two-phase flow numerical

simulations are necessary for a more extensive study on this topic.

8.4.3 Pressure contour in valve

In this chapter an analysis is given of the pressure contours in the configuration for the different

experiments. Take notice that the values of the pressure are normalized. The read values have to be

multiplied by 2

meanvρ to obtain the actual pressure. The multiplication factor is evidentially different

for the two flow rates: Qw = 150 l/min: 15 mbar and for Qw = 250 l/min: 42 mbar.

In figures 8.5 and 8.6, the pressure contours are presented for both the fully opened and the partially

closed valve at Qw = 150 l/min. Stream upwards of the cage, the pressure is much higher at the partially

closed valve than is the case in the fully opened valve. Moreover the pressure gradient in the flow

stream upwards of the cage is much smaller in case of the partially closed valve. The pressure stream

downwards of the cage is for both open and partially closed approximately the same. These differences

are caused by the difference in pressure drop over the holes in the cage. In case of the partially closed

valve, the fluid is forced to fewer holes than in the case of the fully opened valve. As a consequence the

fluid has to be accelerated to a higher velocity and thus the pressure drop over the holes is much higher.

The higher pressure drop over the holes in the case of the partially closed valve requires a higher

pressure at the inlet of the holes. When all holes are opened the decrease in pressure over the valve is

32

more gradually. As a result a clearly visible pressure gradient in figure 8.5 (open valve) can be seen

and not in figure 8.6 (partially closed valve).

In figure 8.7 the pressure contour in the valve for Qw = 250 l/min is plotted. In absolute sense the

pressure in the valve at Qw = 250 l/min is higher than in the valve at Qw = 150 l/min. To maintain a

higher flow rate, a higher inlet pressure is required. Moreover the same difference between Qw = 150

l/min and Qw = 250 l/min can be seen as is the difference between the fully opened valve and the

partially closed valve. The pressure stream upwards of the cage is relatively higher to the pressure

stream downwards of the cage at Qw = 250 l/min than at Qw = 150 l/min. The velocity through the

holes at Qw = 250 l/min is higher than at Qw = 150 l/min which results in the higher pressure drop over

the holes. The results of the numerical experiment with Qw = 250 l/min and a partially closed valve are

not shown. The differences to the fully opened valve with Qw = 150 l/min are a sum up of the effects

which are explained with figure 8.6 and 8.7 in comparison to figure 8.5.

Figure 8.5 Pressure Contour, Qw = 150 l/min, valve fully opened

Figure 8.6 Pressure Contour, Qw = 150 l/min, valve 20mm closed

Figure 8.7 Pressure Contour, Qw = 250 l/min, valve fully opened

33

8.4.4 Velocity contour in valve

In figures 8.8 and 8.9 the velocity contours in the valve (with a flow rate of Qw = 150 l/min) are shown

for the fully opened valve and the valve with a closing of 20mm. The fluid velocity deceleration (-

12<X<0) of the partially closed valve is much higher than is the case in the opened valve. This is

caused by the higher pressure stream upwards of the cage for the partially closed valve in comparison

to the opened valve (chapter 8.4.3). The velocity in these holes in the case of the partially closed valve

is evidently higher. Stream downwards of the cage, the velocity profile (perpendicular to the flow

direction) in the case of the fully opened valve is more flat than in the partially closed valve. Like

stated in chapter 8.4.2, the closing of the valve causes an increase of the size of the vortex. Which

causes, downstream of the cage, a decrease of the flow area. As a consequence the velocity profile is

sharper when the valve is partially closed. At last the velocity contours of the fully opened valve with

Qw = 150 l/min and Qw = 250 l/min are compared. The velocity contour of Qw = 250 l/min is

presented in figure 8.10. There are hardly differences in the velocity contours. The most important

difference is the higher value for the velocity everywhere in figure 8.10 compared to figure 8.9.

Figure 8.8 Velocity Contour, Qw = 150 l/min, valve fully opened

Figure 8.9 Velocity Contour, Qw = 150 l/min, valve 20mm closed

Figure 8.10 Velocity Contour, Qw = 250 l/min, valve fully opened

Choking

With use of the results of the numerical analysis, probable appearance of choking can be determined.

This will be done with use of the characteristic value Mtp. Like noticed in chapter 5.2.2, this should be

done very carefully. First of all the fluid in the numerical analysis is not compressible. Moreover the

geometry is far more complicated than in figure 5.1, on which the choking criterion is based. The

situation with the largest chance on choking; is the case with high flow rate and small valve opening.

To examine Mtp, experimental data is needed because C (chapter 8.4) has to be determined. C is needed

to determine the normalized velocity and pressure. For Qw = 250 l/min and valve closing 20mm, this

data is not available. Therefore the case with Qw = 150 l/min and a valve closing of 20mm will be

analyzed.

34

The void fraction determines the compressibility of the fluid. The numerical calculations are done for

an incompressible fluid. A low value for α approaches best the incompressible fluid. Here, α = 0.1 is

taken. The constant C is determined with use of the experimental data. Sequentially the velocity and

pressure are determined with C. Normalized density is equal to 1. D is determined at several points

around and in the holes in the cage. In the cage the highest fluid velocities appear; as a consequence

here is the biggest chance on choking. It turned out that the highest value for Mtp (=0.062) appears at

the exit of the most right hole. This value is much smaller than 1 and thus no choking would appear

with these conditions. Since the circumstances are extremely idealized this conclusion may not be

taken. Nevertheless probably choking has the greatest chance to appear at the exit of the most right

hole. If later numerical calculations with compressible fluid are done, extra attention has to be paid to

this particular region.

35

9. Conclusions

Both for the single and the two-phase flows, the new data is compared to the old data. The pressure in

the set-up and the pressure drop over the valve turned out to be almost equal for the new and old data.

Only in two-phase flow for a fully opened valve an error is encountered. The pressure drop over the

valve for the new measurements showed opposite behavior to the old measurements. In the new

measurements the pressure drop decreases with increasing void fraction. This is both striking with the

old measurements and with theory. This striking behavior, for the fully opened valve, is both found for

the measurements DpValve and DpASK92. Both measurement methods use the same pressure taps, but

are differently connected to the computer. The behavior can’t thus be caused by electronics or the

computer. When the valve is, partially, closed (for all applied closings) the measurements are according

to theory. The difference is thus most likely caused by the set-up or valve itself and not by the sensors.

The pitot pressure drop and the there from derived fluid velocity show both in the single as in the two-

phase flow accordance to theory. The old and new measurements also show the same tendencies.

However the numerical values for the pitot pressure drop don’t show accordance. The difference

between the values, gained from the old and the new experiments, increases with increasing flow rate.

This difference can only be caused by the sensors, because the minor changes in the set-up can’t cause

these big differences in the pitot pressure drop. It can be concluded that in general the new

measurements are a reliable base for the development of a model. Beside of some exceptions the new

results showed good accordance to its references (old measurements and theory). The new

experimental results can be used for further calculations and as a reference for the numerical model.

For the considered void fraction range, the liquid only multiplication factor derived from both the old

and new measurements have turned out to be equal. This could also be expected when only the

measurements on the pressure drop over the valve for single and two-phase flow would have been

considered. The coefficient is only dependent on these pressure drops. For these pressure drops, the

same values are found in the old and new measurements and thus the coefficients have to be equal.

With this multiplication factor and single-phase flow data or model a first approximation for two-phase

flow pressure drop can be made.

The first attempt for a model of the valve for two-phase flow is made by means of the numerical model

in XENIOS. The numerical model has succeeded in approximating the pressure drop, in single-phase

flow for low flow rate, over the valve. For predicting the pressure drop, the currently obtained

convergence is sufficient. The numerical calculated pitot pressure drop in contrary differs too much

from the experimental obtained results.

The numerical model does not contain effects of compressibility and the multiplication factor is only

known for a small range of void fractions. The numerical model for single-phase flow however,

together with the multiplication factor can make an approximation of the pressure drop for the two-

phase flow. This is a good starting point for the final development of a model for a two-phase flow

through the valve.

36

10. Recommendations

The difference in measured pitot pressure between new and old data grows with increasing flow rate.

The pitot tube seems to generate an error dependent of the flow rate. This problem has to be

investigated in order to take in account for probable new measurements. If the error is constant (as a

function of the flow rate) this can be subtracted from new measurements. If not, the pitot pressure tube

has to be replaced.

The Tylan Calibration is limited by the limits of the rotameter. The rotameter can only measure until a

maximum flow rate of Qw= 28 l/min. For the measurements, air flow rates up to 75 l/min are needed.

The air flow rates from 28 l/min till 75 l/min are thus estimated. A rotameter with a larger maximum

air flow rate would result in more precise air flow rates in the set-up.

Though the void fraction is a local parameter, the homogenous void fraction is used in the report. It is

advised to use a local void fraction measurement for new experiments. This possibly gives more

precise relations between the void fraction and the different measured pressure drops and velocities.

When the set-up, in two-phase flow, is applied to higher air- and water flow rates the separation tank

starts to flow over. A larger separation tank would prevent the set-up from this problem. Also the water

tanks can better be enlarged. The tanks frequently overflow, because of there small capacity.

It turned out that the numerical model not has reached convergence for all experiments. At first the

number of iteration steps has to be increased for the experiments where convergence has not been

reached.

Only a numerical model for incompressible fluid has been developed. Together with the multiplication

factor (chapter 7) this gave a first attempt for the model of the valve. A numerical model with

compressible fluid however will give more insight in the actual behavior of the two-phase flow in the

valve. The remarkable behavior that the pressure drop over the valve increases with decreasing void

fraction (chapter 6.2) has to be studied more extensively. A numerical model of the valve for

compressible fluid can possibly give more insight in this behavior.

Possible choking can also be studied more carefully with a compressible fluid model. It can be

determined if the pressure measurements are affected by this possible choking. Moreover the vortex in

two-phase flow is an interesting feature to study. The vortex in single-phase flow doesn’t affect the

pressure taps, but for two-phase flow this can be different.

37

11. Bibliography

[1] Borello D, Corsini A and Rispoli F, A finite element overlapping scheme for turbomachinery flows

on parallel platforms, Computers and Fluids, Elsevier, 2003, 32/7, 1017-1047.

[2] Davis M.R, Wang D, Dual pressure drop metering of gas-liquid mixture flows, Int. J. Multiphase

Flow Vol. 20, No. 5, pp. 865-884, 1994

[3] Geld, C.v.d., Meerfasenstromingen met warmte-effecten, 2 November 2000.

[4] MOKVELD www.mokveld.com

[5] Santoriello A. Multiscale Finite Element Methods for turbulence modeling in turbomachinery CFD,

November 2005

[6] XENIOS code hand-book, DMA – University of Rome “La Sapienza”, November 2003

38

12. Appendix

12.1. Single-phase old measurements

12.1.1. Pa

0 50 100 150 200 2500

0.5

1

1.5

2

2.5Pa single phase old data

Pa [bar]

Qw

[l/m

in]

open

10mm

15mm

20mm

12.1.2. DpValve

0

200

400

600

800

1000

1200

0 50 100 150 200 250 300 350

Series1 0 1.5 2

2.5 2.25 2.75

39

12.1.3. DpPitot

0

2

4

6

8

10

12

14

16

18

0,0 50,0 100,0 150,0 200,0 250,0 300,0 350,0

Flow rate (l/min)

Pre

ssu

re d

iffe

ren

ce (

mb

ar)

Model 25.6-1 25.6-3 9.7 11.7 13.7

40

12.2 Two-phase old measurements

12.2.1 DpValve

0

20

40

60

80

100

120

140

160

180

200

0,00 50,00 100,00 150,00 200,00 250,00 300,00 350,00

Flow rate (l/min)

Pre

ssu

re d

iffe

ren

ce

(m

bar)

Liquid only GVF 10% GVF 20% GVF 30% GVF 40%

12.2.2 Pitot Pressure

0

1

2

3

4

5

6

7

8

0,0 50,0 100,0 150,0 200,0 250,0 300,0 350,0

Flow rate (l/min)

Pre

ssu

re d

iffe

ren

ce (

mb

ar)

Liquid only GVF 10% GVF 25% GVF 40% GVF 55%

12.2.3 Two-phase multiplicator

0

0,5

1

1,5

2

2,5

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7

GVF (-)

ΦΦ ΦΦtp

(-)

unfiltered data Chisholm model GVF 0% 100 l/min 167 l/min