data acquisition and post-processing of a high time ... › smash › get › diva2:823454 ›...

DEGREE PROJECT, IN , SECOND LEVELNUCLEAR ENERGY ENGINEERING

STOCKHOLM, SWEDEN 2015

Data acquisition and post-processingof a high time resolution local phasesignal at the Westinghouse FRIGGfacility

BOEL MORENIUS

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ENGINEERING SCIENCES

Master Thesis

Data acquisition and post-processing of a

high time resolution local phase signal at

the Westinghouse FRIGG facility

Boel Morenius

Reactor Technology, Department of Physics,School of Engineering Sciences

Royal Institute of Technology, SE-106 91 Stockholm, Sweden

Stockholm, Sweden 2015

Akademisk uppsats för avläggande av civilingenjörsexamen inom ämnesområdetmaskinteknik.

Scienti�c thesis for the degree of Master of Science in Engineering in the area ofMechanical Engineering.

TRITA-FYS 2015:37ISSN 0280-316XISRN KTH/FYS/--15:37--SE

Abstract

In this work, a procedure has been developed to compute local �ow informationfrom measurements in the steam/drop �ow of a thermal-hydraulic test loop, simu-lating the core of a Boiling Water Reactor (BWR). The determined �ow parametersare local void fraction, local velocity, drop occurrence frequency, drop sizes, inter-facial area density and Stokes number.

The measurements have been performed using optical probes for phase recog-nition and Pitot probes for dynamic pressure measurements, in addition to thetest loop's existing instrumentation which records power, mass �ow rate, pressure,temperatures, etc. The Pitot probe signal was imported into the loop's main acqui-sition system, while the optical probe instrumentation had a dedicated acquisitionsystem as provided by the probe manufacturer.

Through this project, a Matlab script has been generated, which inputs data�les from the two systems, matches and time-synchronizes the data and performsthe adequate calculations.

The computed results could be used to validate local void predictions fromthermal-hydraulic simulation codes. It can also be used as benchmark data forComputational Fluid Dynamics (CFD) codes, which need information such as dropsizes for input.

iii

Acknowledgments

I would like to thank my supervisors Jean-Marie LeCorre and Jan Dufek for allthe guidance and instructions, and for making it possible for me to undertake thischallenging and rewarding project. Many thanks also to the helpful and welcomingsta� at the lab and at the o�ce, for technical assistance in the lab, theoretical andpractical help when I have needed it, and enjoyable lunch-break conversations. Lastbut not least, I also want to thank my partner David Aceituno for his support andvaluable insight in C++ programming and the Evolutionary Monte Carlo algorithm.

v

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgments v

Contents vii

1 Introduction 1

2 Background 32.1 The Westinghouse FRIGG facility . . . . . . . . . . . . . . . . . . 32.2 Optical �ber probes and Pitot probes for void and velocity measure-

ments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.1 Optical probes functionality . . . . . . . . . . . . . . . . . . 52.2.2 Pitot probe functionality . . . . . . . . . . . . . . . . . . . 7

2.3 Separate acquisition systems . . . . . . . . . . . . . . . . . . . . . . 8

3 Theoretical considerations 93.1 Void fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Drop velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Drop size - CLD and DSD . . . . . . . . . . . . . . . . . . . . . . . 10

3.3.1 CLD to DSD transformation . . . . . . . . . . . . . . . . . 103.3.2 DSD characterization . . . . . . . . . . . . . . . . . . . . . 123.3.3 Interfacial area density . . . . . . . . . . . . . . . . . . . . . 153.3.4 Stokes number . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 Upper Limit Lognormal distribution . . . . . . . . . . . . . . . . . 173.5 Fast model �tting with Evolutionary Monte Carlo . . . . . . . . . 17

4 Measurements and post-processing 194.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Tested fuel geometries . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Data post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3.1 Time-synchronization . . . . . . . . . . . . . . . . . . . . . 214.3.2 Selective reading . . . . . . . . . . . . . . . . . . . . . . . . 214.3.3 Data sorting . . . . . . . . . . . . . . . . . . . . . . . . . . 21

vii

viii Contents

4.3.4 Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3.5 Presentation of results . . . . . . . . . . . . . . . . . . . . . 26

4.4 Measurement uncertainties . . . . . . . . . . . . . . . . . . . . . . 26

5 Results 27

6 Summary and conclusions 336.1 Validity of calculations . . . . . . . . . . . . . . . . . . . . . . . . . 336.2 Outlook and extensions . . . . . . . . . . . . . . . . . . . . . . . . 34

A Additional plots 35

Bibliography 43

Nomenclature

n Refractive index [−]θ Indicent or refracting angle [◦]α Void fraction [-]Ag Total area of gas phase in �ow [m2]Atot Total �ow area [m2]∆tg Gas residence time [s]∆tl Drop residence time [s]Tacq Total acquisition time [s]pt Total pressure [Pa]ps Static pressure [Pa]pd Dynamic pressure [Pa]u Overall �ow velocity [m s−1]ug Gas velocity [m s−1]ul Liquid velocity [m s−1]ρ Density [kg/m3]ρmix Mixture density [kg/m3]ρg Saturated steam density [kg/m3]ρl Saturated liquid density [kg/m3]L Chord length [m]D Drop diameter [m]gn(D) Number distribution of dropsgs(D) Surface distribution of dropsgv(D) Volume distribution of dropsNd Number of detected drops [−]fd Drop frequency [Hz]ai Interfacial area concentration [m−1]St Stokes numer [−]Dh Hydraulic diameter of �ow channel [m]τ Drop relaxation time [s]µg Dynamic viscosity of steam [Pa s]

ix

Chapter 1

Introduction

The FRIGG facility is a test loop where BWR operating conditions are createdfor evaluation of new fuel designs. The loop has recently been supplemented withsome new instrumentation; optical probes for phase recognition and Pitot probesfor dynamic pressure measurements. The purpose of the new instrumentation is tomeasure local void fraction, �ow velocity, probe-drop interference frequency and tocharacterize the drop population in terms of size.

The optical probe manufacturer provides a dedicated acquisition software. Inaddition to recording the probe signal, this software can also be used to obtainthe �ow information mentioned above. However, a separate post-processing scriptis needed, in order to adapt and automate the process speci�cally for the needs ofthe FRIGG tests (numerous test conditions, coupling with the Pitot measurements,two-phase �ow parameters speci�c to drop �ow, etc). The development of the scriptis descibed in this report, including background, theoretical aspects and calculationsteps. Finally, some results from conducted FRIGG tests are presented.

At the time of writing this report, two test campaigns have been conductedusing the new instrumentation. During the �rst, two optical probes were installedat di�erent positions in the bundle - though one of them proved dysfunctional.During the second test, only one probe was installed. The maximum number ofoptical probes that can be used at the same time is 16 (limited by the number ofrecording channels in the acquisition system). The developed post-processing scriptmust therefore be able to process data from numerous probes.

1

Chapter 2

Background

2.1 The Westinghouse FRIGG facility

The FRIGG facility is a full-scale thermal-hydraulic test loop for BWR fuel as-semblies. It was built in Västerås, Sweden, by ABB Atom and has been operatingsince the mid-1960s, with an upgrade in 1995 [1]. The loop enables evaluation offuel rod bundle pressure drop and dryout performance under operating conditions(steady-state and transient) and is particularly valuable in the evaluation of fuelwith complex geometrical features such as part-length rods and mixing vanes, whichcreate an inhomogeneous �ow that is di�cult to predict using core simulation soft-ware or CFD tools.

The fuel rods that are used in the FRIGG loop are mock-ups, electrically heatedby means of a spiral resistor within the rod wall, and equipped with up to eightthermocouples at di�erent axial elevations to warn of local overheating. In a typ-ical FRIGG experiment, annular two-phase �ow is created around the test bundleand the power is successively increased until �lm dryout is obtained. The mainobjective with this procedure is identi�cation of the critical heat �ux (CHF), i.e.the maximum power that the bundle can have, while the coolant system is stillable to remove the heat e�ciently. The CHF varies with certain conditions suchas pressure, coolant �ow and inlet temperature, and these can be varied in the testloop. Some FRIGG loop speci�cations are shown in Table 2.1 and a schematic ofthe loop is shown in Figure 2.1a and 2.1b (images based on [3]).

Other FRIGG objectives are to test the hydraulic stability of assemblies andto measure the pressure drop distribution as well as void distribution. Up untilthe late 1990s, the void measurements in FRIGG were performed using an X-ray tomography system. This technique was based on irradiating the test sectionwith gamma rays from a Cs-137 source, then interpreting the exiting beams using

3

4 Chapter 2. Background

Thermal rod power 15 MWPressure 10 MPaTemperature 311 ◦CFlow 25 kg/s

Table 2.1: Some FRIGG loop characteristics [2]

1

2

3 4

5

6

7

8

(a) Sketch of whole loop

1

2

3 4

5

6

7

8

(b) Detail of test section

Figure 2.1: Illustration of the FRIGG loop. 1: Steam separator. 2: Test section. 3: Maincirculation pump. 4: Heat exchanger. 5: Coolant loop circulation pump. 6: Condenser. 7:Pressure vessel. 8: Test bundle. (Images based on [3])

scintillator detectors and deducing the steam/water relation from the di�erencein gamma ray attenuation between liquid water and steam [4]. Performing thesemeasurements at various angles and axial positions, it was possible to generategraphical images of the steam/drop mixture inside. The equipment and its shieldingwas however heavy and inconvenient to work with, not least due to the radioactivesource it contained.

2.2. Optical �ber probes and Pitot probes for void and velocity measurements 5

2.2 Optical �ber probes and Pitot probes for void

and velocity measurements

The new means of performing void measurements in the FRIGG loop is with theuse of optical probes. They rely on the fact that the probe tip can recognize thephase of its surrounding medium. The probe functionality is explained in moredetail in Section 2.2.1. The probes are installed at the top of the bundle, beinginserted via a tapped hole in the top �ange. Graphite gaskets are used to ensurethe pressure sealing of the vessel.

The probes are custom-made by the French company RBI Instrumentation.The delicate tip is made out of sapphire, designed to withstand the challangingconditions of the loop and it is about 30 µm wide. This size is one order of magnitudesmaller than the expected drop sizes, which is important for avoiding disturbanceof the �ow conditions. A probe tip is shown in Figure 2.2a and a probe installationusing two probes is shown in Figure 2.2b.

(a) Probe tip (b) Probe installation

Figure 2.2: Optical probe photos

On symmetrical positions to the optical probes in the bundle, there are alsoPitot tubes installed, which provide the dynamic pressure. From this, and withknowledge of the void fraction, the �ow velocity can be computed, as described inSection 2.2.2.

2.2.1 Optical probes functionality

The utilized phenomenon in the optical probe is that its tip is sensitive to the re-fractive index of the surrounding medium; the probe can thus recognize if the tip


is enclosed by gaseous or liquid-phase water.

Inside the probe is an opto-electronic module that sends out a light beam. Asthe beam reaches the probe tip - at the interface between the tip material andsurrounding medium - it can either be emitted from the tip at a certain refractionangle or it can be re�ected back within the probe again. The process follows theSnell's law of refraction:

n1 sin θ1 = n2 sin θ2, (2.1)

where the angles θ of incidence and refraction are de�ned in relation to thenormal of the interface surface and n denotes the media's refraction indices. Aspreviously mentioned, the probe tip is for the case at hand made of sapphire andthe surrounding media are liquid water and steam. The refractive indices of thesematerials1 are presented in Table 2.2.

Medium n [-]Steam 1.01Liquid water 1.25Sapphire 1.77

Table 2.2: Refractive indices

In the case that there is no possible angle θ2 which satis�es Equation 2.1, i.ewhen the situation is:

n1n2

sin θ1 > 1,

then there will be a total internal re�ection. Considering the values in Table2.2, this means for the case at hand that the beam is re�ected back into the probeif it is surrounded by steam (Figure 2.3a), and refracted away if it is surrounded byliquid water (Figure 2.3b2), given that the incident angle θ1 is: 34.8◦ < θ1 < 44.9◦.

The light that has travelled back into the probe again is received by the opto-electronic module and a voltage signal is generated. An example signal is shownin Figure 2.4 (blue), where a high signal corresponds to steam and a low signalcorresponds to liquid. The raw voltage signal is converted into a binary signal (redin Figure 2.4) by applying a threshold voltage. When the raw signal is above the

1Example values taken from:For steam and liquid water: rounded average from table (280 deg C, 6.4 MPa): Table 8, RefractiveIndex of Water and Steam as Function of Wavelength, Temperature and Density, J. Phys. Chem.Ref. Data, Vol. 19, No. 3, 1990.For sapphire: From table: http://hyperphysics.phy-astr.gsu.edu/hbase/tables/indrf.html

2Please note that the incident angle is in reality smaller than sketched in Figure 2.3, c.f. thereal probe tip in Figure 2.2a.

2.2. Optical �ber probes and Pitot probes for void and velocity measurements 7

𝜃1 = 𝜃2

𝑛1 = 𝑛2

(a) Probe tip in gas

𝜃2

𝜃1

𝑛1

𝑛2

(b) Probe tip in liquid

Figure 2.3: Visualization of probe function.

threshold, the binary signal is 1 and when the raw signal is below the threshold,the binary signal is 0.

Figure 2.4: Example of a raw signal (blue) and its binary signal (red) generated by the raw datacrossing a threshold value (yellow dot). The ordinate is in voltage units and the abscissa is intime units.

It is the binary signal that is the output received from the optical probe equip-ment, together with the corresponding time entries for the changes in signal (tran-sitions from 1 to 0 and 0 to 1).

2.2.2 Pitot probe functionality

A Pitot probe is a device which can be used to measure the local velocity in a �uid�ow. It consists of a tube which is placed with its inlet facing the direction of the�ow. There is no outlet on the tube, from which the �ow could continue, so the�uid is brought to a rest (i.e. it stagnates) inside it. This makes it possible to


measure the stagnation pressure of the �ow at the tube's position.The stagnation pressure pt of a �ow (also called total pressure) is the sum3 of thestatic pressure ps and the dynamic pressure pd (also called impact pressure):

pt = ps + pd (2.2)

The Pitot probe is also provided with a means of measuring the static pressureps, for instance by the use of a coaxial tube placed around the Pitot tube withadditional ori�ces on its walls, orthogonal to the �ow direction. The output froma Pitot probe is the di�erence between stagnation and static pressure, i.e. thedynamic pressure.The dynamic pressure can be expressed in terms of the �uid density ρ and itsvelocity u:

pd =ρu2

2(2.3)

The �ow velocity can thus be obtained if the �uid density is known.

2.3 Separate acquisition systems

The main FRIGG acquisition system records quantities such as water �ow, totalpower, pressures and temperatures at inlet and outlet, and fuel rod temperaturefrom thermocouples at di�erent axial elevations. The Pitot probes are also inte-grated into the main FRIGG acquisition system. These quantities are measured ata frequency of 1 Hz for steady-state tests and 25 Hz for transient tests.

The new optical probes measurements are performed at a frequency of 20 MHzand has its own dedicated acquisition system driven by the software ISO, providedby the probe manufacturer. In order to enable time synchronization between thetwo systems, a digital signal is triggered by the start and stop of an ISO acquisitionand this digital signal is recorded by the main FRIGG acquisition system.

3This is true for incompressible �uids, such as water.

Chapter 3

Theoretical considerations

3.1 Void fraction

Void fraction α is de�ned as the amount of gas phase within a �uid. A commonphysical interpretation is the area fraction that is occupied by gas phase, Ag, withina total area section Atot:

α =Ag

Atot(3.1)

From a line measurement performed with an optical probe, the local time-averaged void fraction can be obtained as the amount of time

∑∆tg for which

the probe encounters steam, divided by the total acquisition time Tacq.

α =

∑∆tg

Tacq(3.2)

3.2 Drop velocity

The dynamic pressure pd in a �uid �ow, as obtained for instance from a Pitotprobe, can be expressed in terms of the �uid density ρ and velocity u, as seen inEquation 2.3. For the case of a two-phase �ow with gas and liquid, and under theassumption that both phases have the same velocity ug = ul = u, the density ρ inEquation 2.3 should be replaced by the density ρmix of the mixture. This densitycan be expressed in terms of the void fraction α and the densities ρg and ρl of gasand liquid respectively:

ρmix = αρg + (1− α)ρl (3.3)

This gives the �ow velocity:

9

10 Chapter 3. Theoretical considerations

u =

√2pdρ

=

√2pd

αρg + (1− α)ρl(3.4)

3.3 Drop size - CLD and DSD

One of the challenges in drop measurements with an optical probe is estimatingthe sizes of detected drops. Using the measured time duration ∆tl, for which theprobe traverses a certain drop, and the measured drop velocity u as obtained fromthe Pitot probe calculation, the length L of the traversed path can be found as:

L = u∆tl (3.5)

It is however important to note that the length obtained by Equation 3.5 is thelength of an arbitrary chord within the drop, and not necessarily its diameter D.An illustration is shown in Figure 3.1.

𝐷 𝐿

Figure 3.1: Drop geometry.

Making a histogram of all measured chord lengths from a recording, a chordlength distribution (CLD) can be directly obtained by normalizing the histogram.Theactual drop size distribution (DSD) will however not be the same as the sampledCLD. The transformation from CLD to DSD is explained in the following section.

3.3.1 CLD to DSD transformation

An algorithm described by Li and Wilkinson[5] can be used to compute the DSDfrom a CLD. The following summarizes their approach.

Having measured a set of chord lengths Lj , these are sorted into n bins to formthe CLD:

3.3. Drop size - CLD and DSD 11

c =

c(L1, L2)...

c(Ln, Ln+1)

(3.6)

where c(Lj , Lj+1) is the number of chords which are in the length interval[Lj , Lj+1].

The unknown DSD from which the measured chords were sampled, can beexpressed:

f =

f(L1, L2)...

f(Ln, Ln+1)

(3.7)

where each element f(Lj , Lj+1) contains the number of drops in each corre-sponding interval. Now introducing a transition matrix P which maps the CLD cto the corresponding DSD f , we can write:

c = Pf (3.8)

(My way: Probability of measuring a certain chord of a sphere can be reduced to a 2D problem;

probability of hitting a certain area of a circle.)

ℎ𝑗 = √𝐷𝑖2

4−𝐿𝑗2

4

Probability of measuring a chord with the minimum length 𝐿𝑗:

𝑃𝐷𝑖(𝐿𝑗) =ℎ𝑗

𝐷𝑖/2 =

{

√1 − (

𝐿𝑗

𝐷𝑖)2

, 𝐿𝑗 ≤ 𝐷𝑖

0 , 𝐷𝑖 < 𝐿𝑗

Probability of measuring a chord length within the interval [𝐿𝑗, 𝐿𝑗+1]:

𝑃𝐷𝑖(𝐿𝑗, 𝐿𝑗+1) =

{

√1 − (𝐿𝑗

𝐷𝑖)2

−√1 − (𝐿𝑗+1

𝐷𝑖)2

, 𝐿𝑗+1 ≤ 𝐷𝑖

√1 − (𝐿𝑗

𝐷𝑖)2

, 𝐿𝑗 ≤ 𝐷𝑖 < 𝐿𝑗+1

0 , 𝐷𝑖 < 𝐿𝑗

Mitt icke-fungerande sätt: 𝜋𝐷2

4 är hela arean.

𝐷𝑖

ℎ𝑗

ℎ𝑗+1

𝐿𝑗

ℎ𝑗

𝐷𝑖

ℎ𝑗

𝐿𝑗

𝐿𝑗

ℎ𝑗

𝐷𝑖

Figure 3.2: Drop geometry.

The transition matrix P is composed ofconditional probabilities. That is, the prob-ability of sampling a particular chord lengthgiven a certain drop diameter. Equation3.11, that describes the generation of the ma-trix P , is motivated by the following deriva-tions.

The closest distance hj from the drop cen-ter to the chord, as seen in Figure 3.2, can beexpressed:

hj =1

2

√D2

i − L2j (3.9)

The probability PDi(Lj) that a chord that is measured from a drop with diam-

eter Di will have at least the length Lj is:


PDi(Lj) =2hjDi

=

{ √1− (

Lj

Di)2, Lj ≤ Di

0, Di < Lj

(3.10)

The probabilities PDi(Lj , Lj+1) that make up the elements in the transitionmatrix P , i.e. the probability that a measured chord be within an interval [Lj , Lj+1]are then written in the form:

PDi(Lj , Lj+1) =

√1−

(Lj

Di

)2−√

1−(

Lj+1

Di

)2, Lj+1 ≤ Di√

1−(

Lj

Di

)2, Lj ≤ Di < Lj+1

0, Di < Lj

(3.11)

Having the experimentally obtained chord lengths c and the analytically gen-erated transition matrix P , there may not be an exact solution f for the drop sizedistribution which satis�es Equation 3.8. However, f can be approximated by thevector f̂ which best �ts the equation, using the least squares method:

f̂ = (PTP )−1PTc (3.12)

Problems may arise when trying to solve Equation 3.12 when the matrix P isill-conditioned, meaning that it is non-invertible. This problem can be overcomeusing Ridge regression, introducing a smoothing parameter λ:

f̂ = (PTP + λI)−1PTc (3.13)

where I is the identity matrix. Typically, λ is chosen to be a small positive value(λ < 1). Performing the calculation as in Equation 3.13 means a trade-o� betweenaccuracy and matrix conditioning. A big smoothing parameter (e.g. λ = 1) willguarantee a well-conditioned transition matrix P but it will also give highly biasedresults (the DSD will be too �at).

3.3.2 DSD characterization

Di�erent kinds of size distributions

When it comes to describing a particle size distribution it can be done on either anumber, surface or volume basis. That is, the distribution can show either the num-ber of particles within certain size intervals, or it can show how the particle area orvolume is distributed on these intervals. Most importantly, di�erent measurementtechniques will produce data already belonging to one of these distributions.

Some examples:


• A microscope is used to produce a 2D image of sand grains which are thencounted and sorted into size categories. What is obtained in this way is anumber distribution, since the number of grains are counted.

• The laser di�raction technique is used to evaluate the sizes of particles withina cloud of dust. This method relies on the fact that a laser beam hitting aparticle will be di�racted at a certain angle depending on the particle size.The obtained data corresponds to a volume distribution, since the three-dimensional shape of a particle determines the dispersion angle.

• An optical probe is used to detect drops in a steam/drop �ow. The probetip performs a line sampling through the volumetric �ow and will hence onlypierce some of the drops. The probability of detection is related to the cross-sectional area of the drops, which is why the measured drops will belong to asurface distribution.

In summary, since a measurement technique utilizes some characteristic of thesample to perform the measurement, that characteristic is re�ected in the recordeddata.

Conversion between di�erent distributions

Having data belonging to one type of distribution, i.e. number fn(D), surface fs(D)or volume distribution fv(D), it is possible to convert it into one of the other types.

Consider the number n(D), surface s(D) and volume v(D) of drops with thediameter D. We may write the above mentioned distributions:

fn(D) =n(D)∫n(D)dD

(3.14)

fs(D) =n(D)s(D)∫n(D)s(D)dD

=D2n(D)∫D2n(D)dD

(3.15)

fv(D) =n(D)v(D)∫n(D)v(D)dD

=D3n(D)∫D3n(D)dD

(3.16)

where the integrals in the denominators are the total respective number, surfaceand volume of the drops in the population, and the last equalities in Equations 3.15and 3.16 are under the assumption of spherical drops.

In order to �nd the relation between fn(D) and fs(D) we re-write Equation3.15 in terms of n(D) and insert in into Equation 3.14:


n(D) =fs(D)

D2

∫D2n(D)dD

⇒ fn(D) =fs(D)

D2∫ fs(D)

D2 dD(3.17)

Thus we have the relation:

fn(D) =Kns

D2fs(D) (3.18)

where Kns is a constant:

Kns =1∫ fs(D)

D2 dD(3.19)

Similarly, the relation between number and volume distribution can be ex-pressed:

fn(D) =Knv

D3fv(D) (3.20)

with the constant Knv:

Knv =1∫ fv(D)

D3 dD(3.21)

Finally, the surface and volume distributions are related as:

fv(D) = KsvDfs(D) (3.22)

with the constant Ksv:

Ksv =1∫

Dfs(D)dD(3.23)

Sauter mean and other diameter measures

Although many ensembles of particles consist of a variety of sizes - a size distribution- it is often desired to describe the typical size of the ensemble with a single value.Some conventional ways of describing a distribution are by providing its mode, me-dian or mean value. When it comes to characterizing the drop �ow within a reactor


core, a common drop diameter measure is the Sauter mean, because it describesan average diameter based on a surface distribution, which is of highest interest forheat transfer (see section 3.3.3).

The arithmetic mean of a surface distribution is often called the Sauter mean.It is de�ned as the diameter of a sphere which has the same surface to volume ratioas the particle distribution that it aims to describe. For a distribution of drops,the physical meaning of the Sauter diameter is that if all drops in the populationwere replaced with spheres having the Sauter mean as their diameters, the totalarea and total volume of the drop population would remain the same.

The Sauter mean diameter is often denoted D3,2 since it can be computed asthe ratio of the third and second moment of a number distribution fn(D):

D3,2 =

∫D3fn(D)dD∫D2fn(D)dD

(3.24)

If the surface distribution fs(D) is known, the Sauter mean can be directlycalculated as the arithmetic mean:

D3,2 =

∫Dfs(D)dD (3.25)

Similarly, the arirthmetic meanD4,3 of a volume distribution, called de Brouckeremean, is a measure of interest for mass/volume transfer events. Examples of suchevents for a dispersed annular �ow in a channel, are drop entrainment or evapora-tion from the liquid �lm on the channel wall, and drop deposition onto the wall. Thede Brouckere mean diameter can be obtained from the number fn(D) or volumefv(D) distributions as follows:

D4,3 =

∫D4fn(D)dD∫D3fn(D)dD

(3.26)

D4,3 =

∫Dfv(D)dD (3.27)

3.3.3 Interfacial area density

The interfacial area density ai of a particle population is a measure of the totalamount of surface area that the population has, per unit volume. It is a factor inthe equations governing mass, momentum and heat transfer, and a�ects for instancethe heat transfer at a steam/liquid surface. It is of importance for situations such


as the cooling of fuel rods in a reactor, since it is the surface of liquid in contactwith the steam that is able to transfer the heat away. For instance, in a post-dryoutsituation where the steam is super-heated, the interfacial area will control the rateof drop evaporation.

The interfacial area density can be obtained as the sum of all particle surfacesdivided by the volume in which they are enclosed. For the case of N sphericaldrops in a volume Vtot, where the size of each drop is represented by the sautermean diameter D3,2 to preserve the total area to volume ratio, we get:

ai =NπD2

3,2

Vtot(3.28)

The volume of the system can be expressed in terms of the void fraction α:

α =Vtot − Vdrops

Vtot(3.29)

where the total volume Vdrops occupied by the drops is:

Vdrops =NπD3

3,2

6(3.30)

The interfacial area density for a drop �ow1 can hence be expressed:

ai =6(1− α)

D3,2(3.31)

3.3.4 Stokes number

Another important parameter in a �uid-particle �ow is the dimensionless Stokesnumber, which describes how well the particle trajectories follow the �uid motion.It is de�ned as the product of the particles's relaxation time τ and the �uid �owvelocity u, divided by a characterizing dimension of the �ow channel. For the caseof a steam-drop �ow in a channel with hydraulic diameter Dh the Stokes numbercan be expressed:

St =τu

Dh=

ρlD2u

18µgDh(3.32)

where ρl is the liquid water density, D is a representative diameter of the dropsin the �ow and µg is the steam dynamic viscosity. For very low Stokes number(St� 1), the particles are prone to follow the �uid �ow very well, while for a highStokes number (St� 1), the particles have a more ballistic movement.

1For a bubbly �ow, the factor (1− α) in Equation 3.31 is replaced by α.

3.5. Fast model �tting with Evolutionary Monte Carlo 17

3.4 Upper Limit Lognormal distribution

The upper limit lognormal distribution (ULLN) is a three-parameter function devel-oped by Mugele and Evans[6], which has been found to adequatly describe particlesize distributions ([7], [8]). The three parameters are then the mean µ, standarddeviation σ and the maximum particle diameter xmax:

f(x) =1√2πσ

· xmax

x(xmax − x)exp

(−1

2

[log x

xmax−x − µσ

]2)(3.33)

The ULLN distribution has two main strengths in describing size frequencies.The �rst is its capability to adopt a variety of shapes ranging from symmetricalto right or left skewed. The other strength is that it has a limiting maximumvalue; whereas many other distributions used to describe size frequency have not(for instance the Rosin-Rammler[9] or gamma distributions).

3.5 Fast model �tting with Evolutionary Monte

Carlo

The technique of Evolutionary Monte Carlo (EMC) can be used to e�ciently samplelarge and complex parameter spaces to �nd a su�ciently good combination. Suchproblems often arise in optimization problems and in model �tting of experimentaldata to target distributions. The EMC approach as used here is described in thebook by Liang et al. [10].

In the EMC algorithm a population of individuals is simulated, each with its ownDNA encoded in binary. A DNA sequence corresponds to a proposed set of �ttingparameters and the idea is that a certain DNA sequence will yield the best solution,or �t, to a minimization problem. Initially, all individuals start with random DNAsequences that map to ill-�tting parameters. At each generation, operators areapplied to evolve the DNA of all individuals, such that good DNA is kept and poorDNA is mutated. The evolutionary operators mimic natural selection and followthe familiar scheme of Metropolis Monte Carlo algorithms.

The advantage of using the EMC algorithm is its performance when dealingwith many �tting parameters. Scanning through combinations sequentially is com-putationally expensive, while other conventional methods such as gradient descentdo not guarantee �nding the global minimum. The EMC algorithm provides suchguarantees because of its ergodic property and theorems that follow from being aMarkov Chain Monte Carlo simulation. Judging by the fact that the convergencetime is about the same with 2, 6 or even 8 dimensional parameter spaces, a coarse


estimation is that the EMC algorithm scales much better than linearly with numberof parameters.

Chapter 4

Measurements and

post-processing

Post-processing is performed using Matlab. The �ow information to be determinedis:

• Void fraction

• Probe-drop interference frequency

• Drop velocity

• Drop size

• Interfacial area density

• Stokes number

The way in which these are computed will be explained in this section.

4.1 Assumptions

The assumptions made throughout this analysis are:

• Unidirectional motionThe steam �ow is considered to have constant direction; vertically up throughthe �ow channel.

• Spherical dropsThe drops are assumed to be spherical. This is motivated by the high pressureand small drop sizes, which result in a high surface tension.

19

20 Chapter 4. Measurements and post-processing

• Negligible steam/drop velocity slipThe liquid velocity is considered the same as the gas velocity.

• Probe has no in�uence on dropsThe probe's presence in the steam is assumed not to disturb the �ow ve-locity nor the drop shapes. This is motivated by the fact that the probe issigni�cantly smaller than the expected drop sizes.

4.2 Tested fuel geometries

The data that has been treated in this project comes from two FRIGG scoping testcampaigns during which two di�erent fuel geometries have been tested; FRIGG2and FRIGG3B. Each of the campaigns involve three types of measurements: dry-outs, pressure drops and transients. The post-processing work and results explainedin this report are limited to the dry-out measurements.

Both the FRIGG2 and the FRIGG3B test bundles are 5x5 rod bundles, simu-lating a quarter of a real assembly. Three of the rods are part-length rods. Themain di�erence between the two test bundles is that the probe is surrounded byhot walls in FRIGG2 and mainly by cold walls in FRIGG3B.

For both test campaigns, the optical probe and Pitot probe were installed atend of heated length (EOHL) at the top of the bundle, in the open region above theend of part-length rods (EOPLR). The hydraulic diameter is for both geometriesapproximately 9 mm at the inlet and 11 mm at the outlet.

4.3 Data post-processing

As mentioned, the data to be treated comes from dry-out measurements. Duringsuch a measurement, a range of operating conditions are tested (by varying the �ow,pressure, axial power distribution and sub-cooling temperature of the water inlet).For each of the con�gurations, the thermal power is successively increased untildry-out takes place in the top of the fuel bundle. After a test has been completed,the quantites recorded by the main FRIGG acquisition system (such as pressure,temperatures, �ow etc), are processed using the Westinghouse software STAT95 intime intervals of 20 s, at the time where dry-out is reached. The STAT95 output isa text �le where the measured quantites are presented by their mean values duringthe 20 s time period, and it is recently modi�ed to also include the time stamps forstart and stop of the optical probe recording, read from the digital signal describedin Section 2.3.

The data output from the optical probe acquisition system is also a text �lecontaining the detected events and corresponding time stamps. The events arerepresented as a binary signal, where 0 indicates start of liquid phase detection and

4.3. Data post-processing 21

1 indicates start of gas phase detection. An example signal can be seen in Figure2.4.

Having these data �les from STAT95 and the probe system, the post-processingcan begin. A Matlab script has been written, which performs the steps descibedbelow, for each of the 20 s time-intervals in the STAT95 output.

4.3.1 Time-synchronization

As mentioned in Section 2.3, the main FRIGG data acquisition system is sepa-rate from the optical probe acquisition system, so a �rst action is to time-matchmeasured properties and events between the two systems.

4.3.2 Selective reading

The next step is to identify and read only the part of the optical probe data which isrelevant for the actual time interval. The probe data �le often contains recordingsfor a time span of about 500 s, so in order to speed up the data reading, it is ofinterest to read only the necessary part.

4.3.3 Data sorting

Drop residence times are identi�ed from the probe binary data, i.e. the time dura-tions for which the probe has traversed individual drops. The corresponding timevector is also identi�ed, i.e. at what time each new drop were encountered. If thereare more than one probe in the recording, this is recognized and the data belongingto each of the probes is put into an own structure array.

Finally, the data sorting function also identi�es if the evaluated �le contains abig amount of unreasonably short drop residence times, which can occur due tomeasurement noise. When the number of suspected faulty measurements exceedsa threshold value (set as a user-de�ned fraction of the total number of drops), the�le is discarded.

4.3.4 Calculations

The imported data is then ready for calculations.

1. Void fraction α is computed as explained in section 3.1, using Equation 3.2.

2. The void fraction, together with the dynamic pressure pd obtained by thesymmetric Pitot probe enables to compute the drop velocity u by Equation3.4, where saturated densities for steam and liquid water at the prevailingpressure are found using the steam table XSteam [11] for Matlab.


3. Probe-drop interference frequency fd, also called the collision frequency,indicates how often droplets interact with the probe. It is thus obtainedby dividing the number Nd of detected drops with the duration Tacq of theacquisition:

fd =Nd

Tacq(4.1)

4. The procedure for determining drop size distribution DSD is performedas described in Section 3.3 and brie�y in the following.

Drop residence times and �ow velocity are used to compute chord lengths byEquation 3.5. Figure 4.1 shows an example of a chord length histogram fromone recording.

Figure 4.1: Example of typically obtained chord lengths. This recording was made for 40 s duringwhich time around 92 000 drops were detected. Most of the chord lengths range from 0 to 0.2mm.

The chord length histogram is normalized, creating a chord length distribution(CLD). The CLD is then transformed into a drop size distribution (DSD) bythe algorithm described in Section 3.3.1. The result for the same examplerecording as in Figure 4.1 is shown in Figure 4.2, where the used smoothingparameter is λ = 0.5.

The DSD that is directly obtained in this way is a surface distribution, butit may also be converted into distributions on number or volume bases, asexplained in section 3.3.2. A demonstration of the DSD from Figure 4.2plotted as a number, surface and volume distribution is shown in Figure 4.3.


Figure 4.2: Resulting DSD from CLD using the Li and Wilkinson[5] method.

To characterize the drop population with a few diameter measures, the Sautermean diameter D3,2 is computed directly from the surface distribution (Equa-tion 3.25) and the de Brouckere mean D4,3 from the volume distribution(Equation 3.27). The minimum and maximum diameters Dmin and Dmax

are also identi�ed. They are de�ned as the diameters that 99 % of the distri-bution on a volume/mass basis is above (for Dmin), and below (for Dmax).

In Figure 4.3 these diameter measures are plotted on the respective distribu-tion from which they have been generated.

5. Interfacial area density is obtained as from Equation 3.31.

6. Stokes number is computed by Equation 3.32 for the four diameter valuesD3,2, D4,3, Dmin and Dmax.

Additional post-processing for FRIGG3B: model �tting

Analyzing the data from the two test campaigns, it can be observed that the chordhistogram for FRIGG3B (Figure 4.4) often has two peaks, while the other fuel ge-ometry FRIGG2 (Figure 4.1) has only one peak. Hence it seems like the di�erencein geometry gives rise to two di�erent populations of drops. In order to character-ize each of the populations, it is not possible to use the whole DSD as describedabove. Therefore, an analytical expression is sought, to describe the whole DSD asa superposition of two functions. As discussed in 3.4, the ULLN distribution is wellsuited to describe a single population. Hence the regression algorithm assumes amodel where the DSD is composed of two ULLN distributions.


Figure 4.3: Di�erent characterizing diameters plotted on the respective distribution from whichthey were computed.

Figure 4.4: Example histogram from FRIGG3B.

This distribution contains three parameters µ, σ and xmax, as seen in Equation3.33. However, the superposition of two ULLN functions means the need for seven


parameters; one µ and one σ for each peak, one xmax for the �rst distribution, onefraction parameter to preserve the relative area between the functions, and �nallyone shift parameter to indicate the onset of the second function.

The �tting is performed using the least-squared method. To �nd the sevenparameters, the Evolutionary Monte Carlo method (EMC) is employed1, since itwould be very computationally expensive to scan through all parameter combina-tions with su�cient resolution for each parameter. The EMC method is describedin section 3.5.

The EMC model �tting procedure is performed in C++. The Matlab script iswritten to automatically compile and execute the C++ source �le and then importthe resulting output �les back into Matlab. An example of the decomposition of aDSD into separate ULLN functions is shown in Figure 4.5.

Figure 4.5: Separation of DSD into a superposition of two ULLN distributions.

1The author of this report has not written the EMC algorithm, but merely altered an existingC++ code to �nd the parameters of ULLN distributions.


Figure 4.6: Cumulative average void fraction vs drop count.

4.3.5 Presentation of results

Finally, the computed �ow information is used to generate graphs showing howthey vary with �ow rate, super�cial steam velocity, vapour quality and enthalpyrise over the bundle; all of which are obtained from the main FRIGG acquisitionsystem.

4.4 Measurement uncertainties

The Pitot probe measures dynamic pressure pd with a standard deviation rang-ing from 7 % for low pressures (pd ≈ 2 kPa) to around 1 % for high pressures(pd ≈ 40 kPa). Since the velocity has an inversely quadratic dependence on dy-namic pressure (c.f. Equation 3.4), this leads to a velocity uncertaintly of 0.2 %,averaged over the whole dynamic pressure range.

The optical probe data is treated in time intervals of typically 20 s. Normalcollision frequencies are from around 2 kHz to 6 kHz (see Section 5), which leadsto typical drop numbers of between 40 000 and 120 000. Figure 4.6 shows thecumulative void fraction as function of drop counts, to visualize how many dropsare needed in a recording in order to reach good statistics. The precision accuracyseems to be less than 0.1 % void fraction after 2000 drops.

Chapter 5

Results

Some results from the post-processing of optical probe and Pitot probe measure-ments are presented in this section. Additional graphs of the same results versusother �ow conditions are presented in Appendix A. All results are at steady-statedryout power, with a certain con�guration of �ow conditions (�ow, water sub-cooling, power).

Figures 5.1 through 5.3 show FRIGG2 results and Figures 5.4 through 5.7 showFRIGG3B results. The same overall trends are visible in results from both testcampaigns. A general di�erence between the two is that the FRIGG3B resultshave a larger spread, probably due to di�culties with the acquisition system at thetime of recording, which lead to noisier data.

For both tests, the void fraction is relatively high; ranging between 0.86 and 0.99.Flow velocities are also high; from 5 m s−1 (FRIGG2) and 8 m s−1 (FRIGG3B) upto 35 m s−1 (both tests).

Figures 5.1, 5.4 and 5.6 show computed �ow results as function of super�cialsteam velocity. This velocity is derived from the recorded mass �ow rate, usingthe simpli�cation that the �ow consists of pure gas-phase (the approximation be-ing motivated by the high void fraction). A linear dependency is seen between thesuper�cial steam velocity and the local velocity as computed from Pitot probe dy-namic pressure and optical probe void fraction. Collision frequency and interfacialarea increase with increased super�cial steam velocity.

Figure 5.2 shows that the void fraction increases monotonously with steam ther-modynamic quality. It also shows the perhaps confusing occurrence of decreasinglocal velocity with an increase in vapour quality. The explanation for this peculiar-ity is the fact that all processed data is at critical power conditions; there is hence acertain �ow/power dependence. Therefore, the decrease in velocity may be caused

27

28 Chapter 5. Results

by other �ow characteristics; not by the increase in vapour quality.

As can be seen in Figures 5.3 and 5.6, most drop diameter measures seem ratherinsensible to variations in �ow conditions, and show relatively constant values forminimum, Sauter mean and De Brouckere mean diameters, while the maximumdiameters show a larger variation. Comparing the two tests, the diameters seemsomewhat larger for FRIGG3B. However, it should be noted that Figure 5.6 showsthe diameters as obtained from the whole drop size distribution (DSD); withouttaking into account that FRIGG3B often has two peaks in its DSD (c.f. Figure 4.4).Figure 5.7 shows diameter values computed from the separated �rst and secondpeaks of the EMC-�tted analytical DSD expression. Average diameter values arelisted in Table 5.1.

FRIGG2 FRIGG3B,whole DSD

FRIGG3B,1st peak

FRIGG3B,2nd peak

Minimum 0.11 0.13 0.15 0.74Sauter mean 0.65 0.76 0.57 0.97De Brouckere mean 1.1 1.4 1.1 1.7Maximum 3.6 5.8 3.5 4.5

Table 5.1: Average diameters [mm]

Stokes numbers are also presented in Figures 5.3 and 5.6, as computed from theminimum, Sauter mean and De Brouckere mean diameters displayed in the respec-tive overlying plots.

29

Figure 5.1: Computed �ow information as function of super�cial steam velocity, FRIGG2.

Figure 5.2: Computed �ow information as function of vapour quality, FRIGG2.


Figure 5.3: Drop diameters and Stokes numbers as function of vapour quality, FRIGG2.

Figure 5.4: Computed �ow information as function of super�cial steam velocity, FRIGG3B.

31

Figure 5.5: Computed �ow information as function of vapour quality, FRIGG3B.

Figure 5.6: Drop diameters and Stokes numbers as function of super�cial steam velocity,FRIGG3B.


Figure 5.7: Drop diameters for the �rst (1) and second (2) peaks as function of vapour quality,FRIGG3.

Chapter 6

Summary and conclusions

A Matlab script has been written, which performs the steps described in this report,to compute and present the �ow parameters of interest. The resulting data can beused for validation and benchmark of local void predictions by TH sub-channelcodes, and two-phase CFD models, which need the drop size data as input.

6.1 Validity of calculations

Some indications regarding the validity of the computations performed by the Mat-lab script can be drawn from the results (Section 5):

• Figures 5.1 and 5.4 show a linear dependency between computed velocity(from Pitot probe dynamic pressure and optical probe void fraction), andthe super�cial steam velocity (as derived from the recorded mass �ow rate).It is worth noting that the computed velocity is rougly twice as high as therecorded one; but it should also be noted that the latter is a cross-sectionalaveraged velocity, while the �rst is a local �ow velocity in an open region ofthe core. Less obstruction and less wall friction means that a higher velocityis expected in the open region. It stands to reason that the two velocitiesincrease linearly, which speaks in favour of the calculation's validity.

• Void fraction plotted as function of vapour quality shows a curve which ischaracteristic[12] to this relation, as seen in Figures 5.2 and 5.51. This alsospeaks in favour of the computed void fraction's credibility.

• The obtained drop size results are di�cult to validate, since no similar exper-imental studies at high pressures have been found in the literature. However,the Sauter mean diameter under BWR conditions was estimated by [13], using

1although less apparent in the latter, from FRIGG3B.

33

34 Chapter 6. Summary and conclusions

a correlation by [14], to be 0.6-0.7 mm, which is consistent with the obtainedSauter diameters in Table 5.1.

• However, Table 5.1 also shows an indication of calculation anomalies for theminimum and maximum diameters of the two separated peaks. It is notreasonable that the minimum diameter as computed from the �rst (separated)peak be larger than that obtained from the whole (double-peaked) diameterdistribution. Similarly, it is not reasonable that the maximum of the isolatedsecond peak be smaller than the maximum of the whole distribution. TheSauter and De Brouckere mean diameters for the �tted distributions seem tobe within reasonable values, so a conclusion might be that the �tted functionsrepresents the middle of each peak better than the beginning and end of it.

6.2 Outlook and extensions

The new optical and Pitot probe instrumentation are planned to be used in fu-ture FRIGG tests as well. The prospect is to use up to 8 optical probes, withsymmetrically positioned Pitot probes, and to enable for automatic, on-line dataprocessing. It is also anticipated to process data from fast transient measurements.An improvement that could readily be implemented to the post-processing proce-dure elaborated in this work, is to include the slip ratio to account for the di�erencein velocity between steam and drops. That change is however not expected to sig-ni�cantly alter the drop velocity.

Appendix A

Additional plots

35

36 Appendix A. Additional plots

Figure A.1: Computed �ow information as function of �ow rate, FRIGG2.

Figure A.2: Computed �ow information as function of enthalpy rise, FRIGG2.

37

Figure A.3: Drop diameters and Stokes numbers as function of �ow rate, FRIGG2.

Figure A.4: Drop diameters and Stokes numbers as function of super�cial steam velocity,FRIGG2.


Figure A.5: Drop diameters and Stokes numbers as function of enthalpy rise, FRIGG2.

Figure A.6: Computed �ow information as function of �ow rate, FRIGG3B.

39

Figure A.7: Drop diameters and Stokes numbers as function of �ow rate, FRIGG3B.

Figure A.8: Drop diameters and Stokes numbers as function of vapour quality, FRIGG3B.


Figure A.9: Computed �ow information as function of enthalpy rise, FRIGG3B.

Figure A.10: Drop diameters and Stokes numbers as function of enthalpy rise, FRIGG3B.

41

Figure A.11: Drop diameters for the �rst (1) and second (2) peaks as function of �ow rate,FRIGG3B.

Figure A.12: Drop diameters for the �rst (1) and second (2) peaks as function of super�cialsteam velocity, FRIGG3B.


Figure A.13: Drop diameters for the �rst (1) and second (2) peaks as function of enthalpy rise,FRIGG3B.

Bibliography

[1] Nylund, O., Upgrade of the FRIGG test loop for BWR fuel assemblies, ABBReview, 6, 1997.

[2] Andersson, S., Mingo, A., Majed, M.,Westinghouse BWR and PWR Fuel Ther-

mal Hydraulic Test Facility in Sweden, conference proceedings; Nuclear Energyfor New Europe 2008, Slovenia, September 8-11 2008.

[3] Figure 1, �Basic arrangement of the modernized FRIGG BWR test facility anddetail of the test section with size comparison�, Nylund, O., Upgrade of the

FRIGG test loop for BWR fuel assemblies, ABB Review, 6, 1997.

[4] Ahnesjö, M., Tomographic reconstruction of subchannel void measurements of

nuclear fuel geometries, diploma work; Uppsala University, 2015.

[5] Li, M., Wilkinson, D., Determination of non-spherical particle size distribution

from chord length measurements. Part 1: Theoretical analysis, article; ChemicalEngineering Science 60, 2005, 3251-3265.

[6] Mugele, R. A., Evans, H. D., Droplet Size Distribution in Sprays, Industrial andEngineering Chemistry, Vol.43, 1951.

[7] Bezdek, J. C., Solomon, K. H., Upper limit lognormal distribution for drop size

data, Journal of Irrigation and Drainage Engineering, Vol. 109, No. 1, 1983.

[8] Azzopardi, B. J., Drops in annular two-phase �ow, International Journal ofMultiphase Flow, 12/1997.

[9] Alderliesten, M., Mean Particle Diameters. Part VII. The Rosin-Rammler

Size Distribution: Physical and Mathematical Properties and Relationships to

Moment-Ratio De�ned Mean Particle Diameters, Particle and Particle SystemsCharacterization, Vol.30, 2013.

[10] Liang, F., Liu, C., Carroll, R., Advanced Markov Chain Monte Carlo Meth-

ods: Learning from Past Samples, chapter 5.5.1, Wiley Series in ComputationalStatistics, 2010.

43

44 Bibliography

[11] Steam table for Matlab, Copyright Magnus Holmgren, 1999, www.x-eng.com.

[12] Collado et al., Thermodynamics of Void Fraction in Saturated Flow Boiling,Journal of Heat Transfer, Vol.128, 2006.

[13] Xie, H., Koshizuka, S., Oka, Y.,Numerical Simulation of Liquid Drop Deposi-

tion in Annular-Mist Flow Regime of Boiling Water Reactor, Journal of NuclearScience and Technology, Vol.41, 2004.

[14] Ambrosini, W., Andreussi, P., Azzopardi, B. J., A physically based correlation

for drop size in annular �ow, International Journal of Multiphase Flow, Vol.17,1991.

TRITA FYS 2015:37

ISSN 0280-316X

ISRN KTH/FYS/--15:37--SE

www.kth.se

data acquisition and post-processing of a high time ... › smash › get › diva2:823454 ›...

Documents