Investigation and validation of void and pressure drop correlations in BWR fuel assemblies

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MASTER OF SCIENCE THESIS

Investigation and validation of void and

pressure drop correlations in BWR fuel

assemblies

MANUEL AULIANO

Supervisors

Prof. Henryk Anglart

Dr. Jean-Marie Le Corre

Prof. Bruno Panella

Division of Nuclear Reactor Technology

Royal Institute of Technology

Stockholm, Sweden, June 2014

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Preface

This thesis was submitted to Kungliga Tekniska Högskolan in fulfilment of the requirements for obtaining the double master degree from Kungliga Tekniska högskolan and Politecnico di Torino, under the supervision of Prof. Henryk Anglart, Dr. Jean-Marie Le Corre and Prof. Bruno Panella.

The present work was performed within the BWR Methods and Technology department (BTE) in Westinghouse Electric Sweden AB in Västerås in the period January - May 2014.

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Abstract

This thesis presents the review and the assessment of void and pressure drop correlations against experimental data (internal and external to Westinghouse) that cover a wide range of operating conditions typical of those in a nuclear reactor.

It confirms the drift-flux models as the most recommended choice for predicting void and it provides the opportunity to extend the applicability of void correlations to the high void region. The recommended void correlations are finally selected. A survey of correlations regarding one-phase and two-phase pressure drops has been conducted: an optimized correlation for the friction factor has been proposed and the grid pressure loss coefficients have thus been adjusted; a review of two-phase frictional pressure drop correlations has been performed and an optimized correlation for the friction two-phase multiplier has been proposed. Finally the two-phase pressure drops over the entire assembly have been evaluated comparing the grid two-phase multipliers derived from the homogeneous and the separated flow model.

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Acknowledgements

Professionally, I would like to thank my supervisors Prof. Henryk Anglart , Prof. Bruno Panella, and especially Dr. Jean-Marie Le Corre: all the work could not be done without his help and his patience. In addition, I should also thank the whole personal of the BTE department, particularly Mr. Juan Casal for having accepted me within Westinghouse

Electric Sweden AB and for his warm welcome: it was a pleasure for me to work with

all of you.

Personally, I would like to thank my Italian best friend Amir Al Ghatta for the last two years in Stockholm: it has been an amazing and unforgettable experience. Then I thank my colleagues in Västerås, Didier Bourgin, Raphal Barawnoski, Håkan Carlsson, Ante Hultgren and Karolina Olofsson for the enjoyable Swedish lunch and fika we took during this period in Westinghouse. I would like to express my gratitude to my uncle Daniele Auliano for his valuable advices during my academic path.

Of course it would not be possible to make this without the complete moral and economic support of my family under all my studies: ―Grazie di tutto, vi amo: senza di

voi non avrei conseguito questi risultati che dedico a voi‖.

Manuel Auliano Västerås, 6th June 2014

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Nomenclature

Latin notations

Parameter Description Definition Selected unit

A Area m2

Ablock Blocked area m2

Afuel Fuel pin area m2

a Coefficient in eq (5.3) and (5.4) -

b Coefficient in eq (5.3) and (5.4) -

C Coefficient introduced in eq. (2.17) -

CB Buoyancy force coefficient in eq. (2.15) -

Cc Coefficient in eq (2.52) -

CD Drag force coefficient in eq. (2.15) -

CDB Coefficient in eq (2.26) Eq. (2.27) -

CHN Coefficient in eq. (2.34) Eq (2.35) -

CS Surface force coefficient in eq. (2.15) -

C0 Drift-flux distribution parameter -

Cv Grid drag coefficient in eq. (2.41) -

C' Coefficient introduced in eq. (2.17)

cp Heat capacity at constant pressure J / kg / K

DH Equivalent heated diameter m

DW Equivalent wetted diameter

m

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err error -

f Darcy-Weisbach friction factor -

flo Darcy-Weisbach friction factor based on total flow assumed liquid

-

fgo Darcy-Weisbach friction factor based on total flow assumed vapor

-

G Mass flux kg / m2 / s

g Gravitational acceleration constant 9.81 m / s2

h Specific enthalpy J / kg

hDB Dittus-Bolter heat transfer coefficient Eq. (2.26) W / m2 / K hHN Hancol-Nicox heat transfer coefficient Eq. (2.34) W / m2 / K hevap Evaporation heat transfer coefficient Eq. (2.33) W / m2 / K

hlg Vaporization latent heat J / kg

hThom Thom heat transfer coefficient Eq. (2.29) W / m2 / K

j Superficial velocity m / s k Thermal conductivity W / m / K LC Characteristic length m P Perimeter m Pr Prandtl number cp μ / k - p Pressure Pa pc Critical pressure Pa q'' Heat flux W / m2 Re Reynolds number G DW / μ - rB Bubble radius m S Slip ratio ug / ul -

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s Ratio between the upstream flow area and downstream flow area

Aup / Adown -

T Temperature °C

TB Liquid temperature at bubble tip °C

TB+ Dimensionless liquid temperature at bubble tip

-

Tl (z) Bulk liquid temperature at the axial position °C

ΔTsub Subcooling temperature °C

u Phase velocity m / s

ugj Drift-flux velocity m / s

u* Friction velocity Eq. (2.18) m /s

W Mass flow kg / s

xa Actual flow quality Gg / G -

xe Thermodynamic equilibrium quality ( h – hls) / hfg

-

YB Distance between the wall and the tip bubble m

YB+ Dimensionless distance between the wall and the tip bubble

-

z Axial elevation m

zd Bubble detachment elevation m

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Greek notations

Parameter Description Definition Selected unit

α Void fraction Ag / A -

αB Void fraction on the point of the detachment Eq. (2.11) -

Γcond Condensation mass rate Eq. (2.31) kg / s

Γevap Evaporation mass rate Eq. (2.30) kg / s

ε Local pressure loss coefficient -

ε block Fraction of the unblocked flow area available for flow

Ablock/A -

εsurf Surface roughness -

ρ Density kg / m3

ρm Static mixture density ρls (1-α) +

ρgs α kg / m3 σ Surface tension N / m τ Shear stress N / m2 μ Dynamic viscosity Pa s Φ2

g Two-phase multiplier based on pressure gradient for gas alone flow

-

Φ2

go Two-phase multiplier based on pressure gradient for total flow assumed gas

-

Φ2

l Two-phase multiplier based on pressure gradient for liquid alone flow

-

Φ2

lo Two-phase multiplier based on pressure gradient for total flow assumed liquid

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Subscripts

Parameter Description

A Acceleration

A0 Acceleration

(single-phase only)

ArCh Area change

Cont Contraction down Downstream Exp Expansion e Thermodynamic equilibrium F Friction F0 Friction (single-phase only) G Gravity

g Local gas state

gs Saturated steam H Heated homo Homogeneous model Irr Irreversible K Local obstruction

K0 Local loss (single phase only)

l Local liquid state

ls Saturated liquid

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PhCh Phase change pred predicted Rev Reversible sat Saturation condition sp Single-phase tp Two-phase up Upstream W Wetted w wall

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Abbreviations and acronyms

Parameter Description

BHWR Boiling Heavy Water Reactor

BWR Boiling Water Reactor

DPRESS Pressure drop

EPRI European Power Research Institute

ext external

GTP Grid two-phase

homo Homogeneous flow model

LWHCR Light Water High Conversion Reactor

M-C Maier - Coddington

M-N Martinelli-Nelson

MS-N Muller-Stenhagen and Heck

meas measured

PSI Paul Sherrer Institute

PWR Pressurized Water Reactor

pred predicted

RBMK Reaktor Bolshoy Moshchnosti Kanalnyy ("High Power Channel-type Reactor")

RDIPE Research and Development Institute of Power Engineering

SCB Subcooled Boiling

SQP Sequential Quadratic Programming

separ Separated flow model

spdp Single-phase pressure drop

TH Thermal-hydraulic

tpdp Two-phase pressure drop

VF Void fraction

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List of figures

Figure 2.1: Void in subcooled boiling region [5] ... 8

Figure 2.2: Forces acting on a bubble [5] ... 10

Figure 3.1: Matlab TH code structure ... 18

Figure 3.2: Input ... 19

Figure 3.3: TH calculations ... 19

Figure 4.1: FRIGG loop [6] ... 21

Figure 4.2: The void measuring table [6] ... 22

Figure 4.3: Location of pressure taps [13] ... 23

Figure 5.1: Void experimental data ... 27

Figure 5.2: Statistical analysis - sf24va and sf24vb ... 29

Figure 5.3: Pred. Vs. Meas. - sf24a and sf24vb ... 29

Figure 5.4: Pred. vs. Meas. - sf24va and sf24vb ... 30

Figure 5.5: Pred. vs. Meas. - sf24va and sf24vb ... 30

Figure 5.6: Statistical analysis - ft36 of36 of64a of64b ... 31

Figure 5.7: Statistical analysis - psbt ... 32

Figure 5.8: Statistical analysis - rdipe ... 32

Figure 5.9: Subcooled boiling models - sf24va and sf24vb ... 34

Figure 5.10: Characteristic length in the subcooled boiling region ... 35

Figure 5.11: High void region ... 36

Figure 5.14: Different slip ratio ... 38

Figure 5.15: Different slip ratio ... 39

Figure 5.16: Bestion correlation - C0 ... 40

Figure 5.17: Pred. vs Meas - FRIGG data ... 42

Figure 5.18: Pred. vs Meas. - FRIGG data ... 42

Figure 5.19: Pred. vs Meas. - psbt ... 43

Figure 5.20: Pred. vs. Meas. - psbt ... 43

Figure 5.21: Pred. vs. Meas. - Steady-state experiments (PSI) ... 44

Figure 5.22: Pred. vs. Meas. - Steady-state experiments (PSI) ... 44

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Figure 5.24: Pred. vs. Meas. - Transient experiments (PSI) ... 45

Figure 5.25: aa69 void correlation - Error vs Mass flux ... 46

Figure 5.26: Recommended void correlations ... 47

Figure 5.27: Recommended void correlations ... 47

Figure 5.28: Statistical analysis - bfbt_spdp ... 51

Figure 5.29: Statistical analysis - sf24h ... 51

Figure 5.30: Pred. vs. Meas. - sf24h ... 52

Figure 5.32: Grid pressure loss coefficients - sf24x ... 55

Figure 5.33: Grid pressure loss coefficients - sf24s ... 55

Figure 5.34: Statistical analysis - sf24ec ... 57

Figure 5.35: Statistical analysis - sf24h ... 58

Figure 5.39: Grid two-phase multipliers - Pred. vs. Meas. ... 60

Figure A.1: Sub-bundle section [6] ... 66

Figure A.2: Sub-bundle section [6] ... 67

Figure A.3: Location of pressure taps at single-phase pressure drop measurements [6] 67 Figure A.4: Location of pressure taps at two-phase pressure drop measurements [6] ... 68

Figure A.5: Location of pressure taps at single-phase pressure drop measurements [6] 68 Figure A.6: Location of pressure taps at two-phase pressure drop measurements [6] ... 69

Figure A.7: Location of pressure taps - bfbt [13] ... 70

Figure A.8: Cross section of the experimental channel - rdipe1 [6]... 71

Figure A.9: Cross section of the experimental channel - rdipe2 rdipe3 [6] ... 71

Figure E.1: Statistical analysis - bfbt ... 80

Figure E.2: Statistical analysis - bwr8x8 neptun pwr5x5 tptf ... 80

Figure E.3: Recommended void correlations. - bfbt ... 81

Figure E.4: Recommended void correlations - bfbt ... 81

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Figure E.6: Recommended void correlations - bfbt ... 82

Figure E.7: Recommended void correlations - rdipe ... 83

Figure E.8: Recommended void correlations - rdipe ... 83

Figure E.9: Statistical analysis - sf24i ... 85

Figure E.10: Statistical analysis - sf24ec ... 85

Figure E.11: Statistical analysis - sf24et ... 86

Figure E.12: Grid pressure loss coefficients - sf24vc ... 87

Figure E.13: Grid pressure loss coefficients - bfbt ... 87

Figure E.14: Statistical analysis - sf24et ... 88

Figure E.15: Statistical analysis - sf24i ... 88

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List of tables

Table 2.1: Void correlations ... 6

Table 5.1: Experimental void databases ( FRIGG loop ) ... 25

Table 5.2: Experimental void databases ... 26

Table 5.3: Experimental void databases [2] ... 26

Table 5.4: Experimental pressure drop database ... 49

Table A.1: Geometry and Power distribution - bfbt [13] ... 69

Table A.2: Geometry and Power distribution - psbt [14] ... 70

Table E.1: Recommended void correlations - Mean Error [-] ... 84

Table E.2: Recommended void correlations - Standard deviation [-] ... 84

Table E.3: Homo vs Separ GTP multiplier - Mean error [bar]... 90

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1. Overview

1.1 Introduction

Presence of the steam in the nuclear reactors affects significantly the value of the coolant density, thus its moderation power decreases and in turn influences the local neutron flux and thus the local power. Due to the feedback between the local power and the local void fraction it is important to predict accurately its local value in order to predict the correct response of nuclear reactors [1] by using models predicting the energy transfer and the transport of the vapor phase along the system [2]. The void prediction is a required input for computing many key flow parameters, it is important in the modeling of the two-phase flow pattern transitions, heat transfer and pressure drops and it plays a crucial role in many thermal-hydraulic simulations.

1.2 Objectives and methodology

The main objective of this work is to review and optimize Westinghouse methods to compute void fraction and pressure drop in BWR fuel assemblies. In preparation of this project a large number of void fraction and pressure drop databases have been collected. In addition to the available internal FRIGG databases, other databases in rod bundle from the open literature have been compiled. Primarily the void benchmark analysis has been conducted in order to select the recommended void predictive correlations overall the whole range: it is important to predict accurately the void in order to predict accurately the pressure drop. Then the attention has been moved to the single-phase pressure drop: an optimized friction factor is proposed. Once the best friction factor is known, the grid pressure loss coefficients have been adjusted by minimizing the statistical objective functions such as mean error and standard deviation. A comparison between several two-phase friction multipliers has been conducted and an optimized correlation has been proposed. Finally the total pressure drops overall the fuel assembly have been computed including the grid two-phase pressure drop: the homogeneous and separated multipliers have been compared.

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1.3 Outline

The thesis is divided in three parts.

This chapter represents the introductory part (Part 0).

Part I presents the theory (predictive models and correlations) and the structure of the

Matlab code used in the work to perform the different thermal-hydraulics simulations.

Part II contains the description of experimental measurements and the numerical

investigations made as a part of this work.

Part III presents final remarks, conclusions and further works.

In the following, an overview of the topics discussed in each of the chapters is presented.

Part I: Theory and numerical framework

 Chapter 2: The main predictive correlations regarding void fraction and pressure drop are introduced and a general description of the subcooled boiling models is presented.

 Chapter 3: The structure of the Matlab code used to perform steady-state thermal-hydraulics calculations is schematically described: a user manual has been written during the project.

Part II: Experimental and numerical investigations

 Chapter 4: The test facility owned by Westinghouse Electric Sweden AB and the measurement techniques for void fraction and pressure drops are described.

 Chapter 5: The benchmark analysis regarding void and pressure drop is presented: the experimental data are compared with the results of the main correlations available from Westinghouse and open literature. The recommended void correlations are selected. The optimization is carried out for the friction factor, the grid pressure loss coefficients and the two-phase friction multiplier. Homogeneous and separated grid two-phase multipliers are compared.

Part III: Conclusions and further work

 Chapter 6: Final remarks and observations are presented, the main conclusions are drawn and further research directions are suggested.

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Part I

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2. Theory and models

2.1 Treatment of two-phase flow

Currently the two-phase flows are widely modeled by using the homogeneous and separated flow approaches. Furthermore empirical and semi-empirical approaches that model the hydrodynamics features of the flow have been developed. In the present work the simplest approach has been adopted for the TH steady-state calculations: the conservation equations are based on the homogeneous equilibrium model (that is three-equation model) and that the non-homogeneity and non-equilibrium is accounted for using additional constitutive relations.. The experience has taught that this simple approach supported by constitutive models approaches reasonably the separated flow model.

2.1.1 The homogeneous equilibrium model

The homogeneous equilibrium model is classified as one-fluid model of two-phase flows: the mixture is considered as a single phase with averaged properties of the liquid and vapor phase. The term homogeneous allows considering the flow as a homogeneous mixture with no relative motion between vapor and liquid (slip ratio equal to 1); the term equilibrium refers to the thermodynamic equilibrium between the two phases. Under the assumption of thermodynamic equilibrium the actual non equilibrium quality is equal to the thermodynamic equilibrium quality. The single-phase basic equations (mass, momentum and energy) are used for the mixture.[3]

The homogeneous approach computes the two-phase pressure drop by using the same formula for the single-phase flow but with the averaged properties defined by homogeneous models: the two-phase density is defined as

( ) (2.1)

for the mixture dynamic viscosity different models can be used as

McAdams ( ) (2.2)

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Cicchitti ( ) (2.3)

Dukler (

( )

) (2.4) . [4]

2.1.2 Separate flow model

The two phases vapor and liquid are considered separated into two streams each with a mean velocity. They have a constant but not necessarily equal velocity (slip ratio not equal to one) and are in thermodynamic equilibrium quality.[5]

The separated flow model introduces the phase multipliers to compute the two-phase pressure drops.

The most common approach (used in the present work) is to first compute the single-phase liquid pressure drop assuming that the two-single-phase mixture is entirely in the liquid phase and then to multiply it by the two-phase pressure drop multiplier ϕlo2 as

. /

.

/ (2.5)

A second approach less used is the ϕl, ϕg based method that computes the two-phase pressure gradient as

. /

.

/ (2.6)

where the single-phase pressure gradient is computed assuming the liquid phase to flow alone.

For both the approaches ϕ2lo and ϕ2l can be replaced respectively by ϕ2go and ϕ2g by considering the gas phase instead of the liquid phase. [4]

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2.2 Void fraction models

The table 2.1 shows the used void prediction correlations that have can be classified in three groups having in common the reference homogeneous model that often over-predicts the void fraction. The first group is represented by the slip ratio models based on empirical relationships to compute the slip between the two phases. The second group is given by the drift.-flux models (the most used and recommended) that compute the distribution parameter and the drift-flux velocity by using empirical relations. The third group is represented by the so-called miscellaneous correlations that are empirical relations not included into any of the other groups. A fourth group not used in the benchmark analysis is given by the so-called Kαhomo models that correct empirically the void fraction predicted by the homogeneous model by a factor K. [7]

Table 2.1: Void correlations

Homogeneous: 𝛼 ( ) (2.7) [1] Slip: ( ) (2.8) [1] Smith [5] EPRI, SCP [6] Drift-Flux: 𝛼 (2.9) [1] Zuber-Findlay (Z-F), Bestion, Chexal et al., Toshiba, Inoue,

Maier-Coddington (M-C) [2] aa69, aa78 [6]

Miscellaneous: experimental

vann96, vann97 [6]

All the void correlations are defined in the appendix B.

The drift-flux model

The drift-flux formulation developed by Zuber and Findlay for the void fraction is given by

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𝛼

(2.10)

where j and jg are respectively the mixture and vapor superficial velocity, C0 is the drift-flux distribution parameter that is a covariance coefficient for cross-section distributions of void fraction and total superficial velocity and ugj is the drift-flux velocity defined as cross-section averaged difference between gas velocity and total superficial velocity [1]: this model is able to take into account both the vapor production and the effect of the relative velocity between the two phases included respectively in jg and ugj. [2]

2.3 Subcooled boiling model

Although the bulk boiling is prevalent in the thermal-hydraulics performance of BWR reactors, accurate models are required to predict the void in the subcooled region. Considering a tube heated with axial uniform flux and introducing a subcooled liquid at the inlet, the void fraction will vary with the axial position as the curve ABCDE in figure 2.1.[5]

The subcooled boiling process can be divided into two regions, namely the wall voidage (region AB) and the detached voidage (region BCD): in the former with high degrees of subcooling (partial subcooled boiling) the vapor generated travels in a narrow bubble layer attached to the wall whilst growing and collapsing until the void departure point zd is achieved; in the latter with lower degree of subcooling (fully developed subcooled boiling) bubbles detach from the heated surface and an appreciable void fraction begins to appear into the subcooled core; the region DE represents the saturated nucleate boiling.

The mechanistic and profile-fit approaches have been analyzed in order to predict the forced convection subcooled void fraction: the former postulates a phenomenological description of the boiling heat transfer process and so computes the subcooled flow quality and void fraction, the latter postulates a convenient mathematical fit to the data for the flow quality or the enthalpy profile between the void departure point zd and the

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point at which thermodynamic equilibrium is reached ze. For steady-state calculations a profile-fit method is recommended since it is accurate though it is easier to use than a

Figure 2.1: Void in subcooled boiling region [5]

mechanistic method. The main drawback is that it is based on a fit to uniform axial heat flux data, so the predictions of subcooled void fraction in case of non-uniform axial heat flux have to be confirmed. For transient calculations the mechanistic model is recommended. Some of the more used references for each method are

1. Mechanistic models: Griffith et al., Bowring, Rouhani and Axelsson, Larsen and Tong, Hancox and Nicoll, EPRI.

2. Profile-Fit Models: Zuber et. Al., Staub, Levy, Saha and Zuber. [8]

2.3.1 Levy's model The highly subcooled region

The Levy´s model is based on the assumption that at point B (transition between the first and second subcooled regions) the bubbles are spherical with radius r and the

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distance between them is equal to SB. Around the heated perimeter there is a number of bubbles equal to PH/SB and the volume of vapor in a section of channel SB in length is (PH/SB)(4/3 π rB3), so the void fraction at the point B is given by

𝛼 . / . / . / . / (2.11)

Assuming the bubbles to be wrapped in a squared array and to interfere with each other, if rB/SB ≈ 0.25, then the void fraction on the point of detachment is given by

𝛼 (2.12)

where YB is the distance from the wall to the tip of the vapour bubble (shown in figure 2.2) given by the equation

0 1 (2.13)

where τw is the wall shear stress computed as

(2.14) . [5]

Departure of vapor bubbles from the heated surface

In order to compute the subcooling at the incipient bubble departure, primarily the size of the bubble has to be computed. Figure 2.2 depicts the forces acting on the bubble at the moment of the departure: the surface tension and the inertia forces (negligible) hold the bubble to the surface, while the buoyancy and frictional drag forces attempt to remove it. From the force balance

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the bubble radius in the incipient departure rB is computed as

. / . / (2.16)

and then the distance to the tip of the bubble YB, assumed to be proportional to rB, is computed as

0 1 0 . ( ) /1 (2.17)

where the constants C and C' were computed from the experimental data.

Figure 2.2: Forces acting on a bubble [5]

The dimensionless distance YB+ is expressed in terms of the parameter √ (2.18) as

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0 1 , - ₀ _.( ) _/1 _(2.19)

Necessary condition for the growing or equilibrium of the bubble is that the liquid temperature TB at the distance YB exceeds the saturation temperature: the Levy´s model simplifies the analysis and assumes that TB equals the saturation temperature.

By using the Martinelli's universal temperature profile the dimensionless temperature TB+ at the position YB+ is computed as:

, ( 2 3)- (2.20) , ( ) 23-

Under the assumption of thesingle-phase temperature profile in the liquid

( )

(2.21) and expressing the dimensionless temperature as

( ) (2.22)

where Tw is the wall temperature, q'' is the wall heat flux and hDB is the heat transfer coefficient computed by the Dittus-Bolter correlation, the subcooling at the bubble departure point can be expressed as:

( ) ( ) 0

1 (2.23) [5] If the fluid bulk temperature distribution is computed by applying the heat balance, the nonlinear equation can be solved to compute the bubble detachment elevation:

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( ) ( ) (2.24)

2.3.2 EPRI model

The mechanistic subcooled boiling model developed by the EPRI is based on the following physics: the evaporation process leads to the formation of bubbles at the surface of the cladding, then the condensation occurs due to the transport of the bubble in the subcooled water. The transition to the subcooled boiling occurs when the evaporation rate exceeds the condensation rate: heating comes only from fission, contributions from direct gamma and neutron heating are neglected. The saturated boiling occurs when the bulk temperature of the fluid reaches its saturation value. As long as the cladding wall superheat is negative, the single-phase subcooled heat transfer is described by the Newton's cooling law

( ) (2.25)

where the Dittus-Bolter's heat transfer coefficient is computed as

(2.26)

{

(2.27)

with εblock defined as fraction of the unblocked flow area available for flow.

When the cladding wall superheat is not negative anymore, the heat transfer is ruled by

( ) ( ) (2.28)

where the Thom heat transfer coefficient is given by

( ) (2.29)

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( )

(2.30)

( ) (2.31)

( ) ( ) (2.32)

where the evaporation and the Hancox-Nicol coefficient are computed respectively as

( ) (2.33) (2.34) with { (2.35)

When , the subcooled boiling begins. Net bulk boiling begins when : all the heat removed by the coolant is used to produce steam. [6] [9] [10]

2.4. Pressure drops

The total axial pressure loss for the two-phase mixture in a channel along the flow direction can be split into four contributions (gravity, friction, local flow obstructions and acceleration) as:

. / . / . / . / . / (2.36)

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The ϕ2lo based approach is used to compute the two-phase friction and grid pressure drop.

2.4.1 Gravity pressure drop

Assuming the gravity as the only external volume force, the gravitational contribution is expressed as:

. / (2.37)

2.4.2 Friction pressure drop

The frictional two-phase contribution is computed according to the separated flow model as

. / . /

(2.38)

where the pressure loss for the single-phase liquid is given by

. /

(2.39)

All the friction factor and two-phase friction multiplier correlations are defined respectively in the appendix A3 and A4.

2.4.3 Local pressure drop

The local pressure loss is due to a local geometric obstruction within the fluid flow region around a grid or an orifice. In single-phase it is computed as

. /

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where ε is the pressure loss coefficient for the local perturbation.

For rod bundles with part-length rods, to convert the spacer loss coefficients to different local area in each zone, the following "rule of thumb" is used:

. / (2.41)

where Cv is the grid drag coefficient and εblock is the relative blockage defined as the ratio between the area blocked in the axial direction Ablock that is the same for all the axial zones and the local flow area A is in each zone. [11]

The two-phase local pressure loss is calculated by using a two-phase spacer multiplier

. / . /

(2.42)

The two-phase local multipliers used are derived from the homogeneous and separated flow models and are respectively defined as:

[ ( )] (2.43) [1]

( _{( )}) (2.44) [12]

2.4.4 Acceleration pressure drop

Considering the flow incompressible, the flow changes velocity in a channel due to phase change and/or area change. The acceleration pressure contribution is due to the flow acceleration that affects the amount of net momentum in and out of the considered fluid volume.

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. /

( ) (2.45)

where ϕ2A0,PhCh is the phase change acceleration two-phase multiplier defined as

( )

( ) (2.46)

Under the conditions of constant mass flow rate and flow cross-section area, it becomes

. /

( ) (2.47) [1]

The acceleration contribution due to the area change (sharp expansion or sudden contraction) is computed as:

( ) (2.48)

where ϕ2A0,ArCh is the area change acceleration two-phase multiplier.

The pressure loss due to the area change has the reversible and the irreversible contributions.

For the reversible contribution, the term ε,ArCh is computed as

(2.49)

where the parameter s is the ration between the upstream flow area and the downstream flow area and the two-phase multiplier is computed as

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For a sharp expansion the reversible contribution is negative since s is lower than 1, therefore it represents a pressure gain.

The irreversible pressure drop due to sudden expansion is computed as

( ) {[( ) ( ) ] ( )[ ( ) ( ) ] [ ] } (2.51) [12]

For the irreversible pressure drop due to sudden contraction the local pressure loss coefficient is computed as

. / (2.52)

where the parameter Cc is function of the parameter s, and the homogeneous two-phase local multiplier is used. [12]

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3. Matlab steady-state TH code

The steady-state thermal-hydraulic code developed in Matlab presents a simple structure shown in figure 3.1.

Figure 3.1: Matlab TH code structure

The inputs shown in figure 3.2 are defined by the database, the channel and the geometry. The database contains information about the operating conditions (outlet pressure, mass flux, subcooling inlet temperature and total power) and the measurements (void fraction or pressure drops). The channel provides information about the geometry and the local axial power distribution. The model defines the axial grid along the channel, the models and correlations used in the calculations (void correlation, subcooled boiling model, friction factor, friction two-phase multiplier, grid two-phase multiplier) and the pressure option for the TH fluid property calculations, that is pressure can be kept constant along the channel and equal to the outlet pressure or the pressure distribution can be computed iteratively.

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Figure 3.2: Input

Once the input have been defined, for each experimental run the code reads the boundary conditions, performs the steady-state thermal-hydraulics calculations (shown in figure 3.3) and save the results (post-processing). As output it provides a comparison between the predictions and the measurements needed for the benchmark analysis.

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Part II

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4. Experimental measurements

4.1 Internal test facility

The internal databases have been collected from experiments performed at the FRIGG loop (only for internal databases) at the previously ABB Atom laboratories, now

Westinghouse, in Västerås (Sweden). The loop (shown schematically in figure 4.1) is

designed for a pressure of 10 MPa and a temperature of 311 °C and it covers all requirements for BWR fuel heat transfer and pressure drop testing at two-phase conditions including thermal-hydraulics stability: it consists of a main circulation loop including the test section, a cooling circuit and a purification system. The test section consists of a pressure vessel, a Zircaloy flow channel and a sub-bundle with heater rods representing the fuel design with full and part-length fuel rods. Pressure sensors are connected to the flow channel at different elevation taps and thermocouples are accommodated in the heater rods. The steam drum is used to separate the steam produced in the test section from the saturated water: the steam is transported to the condenser and the saturated water back to the main circulating pump. The cooling circuit is composed of the condenser, heat exchanger and a circulating pump.[6]

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4.2 Void measurements

Void measurements have been carried out at two-phase flow and different operating conditions defined by mass flux, the system pressure, the inlet subcooling temperature, the bundle power and the local power distribution. Databases from void measurement have been used to validate the void correlations needed in the core design methods. The technique used to detect the void distribution is the transmission tomography equipment shown in the figure 4.2: the intensity of the radiation beam emitted from a Cesium-137 source is attenuated as it passes through some material, particularly it decreases exponentially according to the Beer´s law and the coefficient of attenuation of a gamma ray through a bubbly flow may be determined by measuring the intensity before and after its passage through the channel flow. The measured coefficient of attenuation is directly proportional to the mean density of the mixture that is function of the void fraction. [6]

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4.3 Pressure drop measurements

In order to license a new fuel type pressure drop measurements are carried out at both single- and two-phase flow and different operating conditions defined by mass flux, the system pressure, the inlet subcooling temperature, the bundle power and differential pressures [6]. The main goal is to obtain pressure loss coefficients representative of the fuel assembly main components.

In the experiments the bundle pressure drop has been monitored at several locations as depicted in figure 4.3.

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5. Results and discussion

5.1 Void

Due to its relevance in characterizing two-phase flows, several void predictive correlations have been proposed and assessed by comparing the predictions against experimental data. Despite their use limited to co-current flow, the drift-flux models are considered the most recommended considering their simplicity and predictive accuracy. A review of a wide range of void correlations based on the Zuber-Findlay drift-flux model has been conducted by Paul Coddington and Rafael Macian, evaluating them against experimental PWR and BWR steady-state and transient (boil-off experiments) data obtained from facilities in France, Japan, Switzerland, the UK and the USA: the large size of the experimental database allowed a detailed statistical analysis that compared the different correlations and has pointed out that the iterative correlations do not increase so significantly the accuracy of the prediction. The present work assesses the predictive capability of the available void correlations (internal to Westinghouse and from open literature) against a much larger experimental database (including FRIGG data, RBMK data and channel geometry), extends their applicability to the high void region and investigates on the subcooled boiling region comparing two different models as Levy and EPRI.

5.1.1 Experimental void data

A wide range of experimental void fraction data internal (table 5.1) and external (table 5.2 and table 3.3) to Westinghouse at various pressure and mass flux has been collected and provides the opportunity to assess the predictive capability and the overall applicability of the void correlations (internal and external to Westinghouse). The data covers pressure from 0.1 MPa to 16.9 MPa and mass fluxes from 2.8 kg/m2/s to 4138.9 kg/m2/s and provides information on void fractions in sub-channels and rod bundles including BWR, PWR and RBMK normal operating conditions and small and large break transient conditions for both PWRs and BWRs. The experimental data can be split in 3 different groups according to the type of the experiment performed and are labeled as steady-state, boil-up and boil-down experiments. The majority of the

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experiments were performed under steady-state conditions with the inlet subcooling, mass flux and power at constant values: this first group includes all the databases from table 5.1, table 5.2 and some from table 5.3 (bwr8x8, neptun, pwr5x5, tptf )[15]. The boil-up experiments, where the inlet flow has been varied to keep constant the collapsed liquid level, include the pericles, thetis and lstf databases[15]. The last group of the boil-down experiments where the liquid inventory in the test facility is gradually boiled away includes the achilles and thetis data[15]. Figure 5.1 provides an indication of the wide range of pressure and mass fluxes covered by the experimental data.

Table 5.1: Experimental void databases ( FRIGG loop )

sf24va sf24vb of36 ft36 of64a of64b

Reference [6] [6] [6] [6] [6] [6] Type BWR BWR BWR BHWR BWR BWR Length [m] 3.74 2.37 –3.74 3.65 4.365 3.65 3.65 Rods (heated) 24(24) 24 (24) 36(36) 36(36) 64(64) 64(64) D r [mm] 9.62 9.62 - 10.32 12.27 13.8 11.78 – 12. 25 11.78 – 12. 25 D_w [mm] 10.22 9.88 13.5 26.9 14.07 14.07 Axial Power

distribution Uniform Uniform Uniform Uniform Top Peak Top Peak

ΔT_sub [K] 5.4 – 21.8 8.3 – 33.8 8.2 – 62.3 3.0 – 16.5 9.1 – 39.0 7.7 – 38.7 p [MPa] 5.5 – 7.1 5.4 - 8.0 3.0 – 9.0 7.0 4.8 – 6.8 6.7 – 6.9 G [kg/ m2/s] 374 - 1653 390 - 1730 548 - 2919 495 - 1967 494 - 2479 513 - 2006 q'' [kW/m2] 129 - 752 105 - 972 187 - 958 220 - 664 222 - 570 383 - 554

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Table 5.2: Experimental void databases

bfbt [11 - 21 - 31] bfbt [1071 - 2081 - 3091] bfbt [4101] psbt rdipe [1] rdipe [2,3] Reference _[13] _[13] _[13] _[14] _[6] _[6] Type _BWR _BWR _BWR _PWR _RBMK _RBMK Length [m] _3.71 _{3.71 /1.75/} 3.71 3.71 1.56 2.5 7.00 Rods (heated) ₆₄ (62/60/55) 64 (62) 64 (60) Subchannel [1-0.75-0.5-0.25] 1 7 D r [mm] 12.3 12.3 12.3 4.8 14 13.5 D w [mm] 13.0 13.0 12.4 5.1 – 7.8 8.8 7.7 Axial power

distribution _Uniform Cosine / _{Cosine /}

Inlet peak

Uniform Uniform Uniform Uniform

ΔT sub [K] 4.24 – 25.79 4.46 – 28.01 4.69 – 26.51 6.8 – 102.2 10.07 – 277.37 8.26-192.96 p [MPa] _{0.98 – 8.69} _{0.96 – 8.68} _{0.97 –} 8.71 5.0 – 16.91 2.99 – 10.03 3.01-14.38 G [kg/m2/s] 288 - 1987 284 – 1978 296 - 2046 500 - 4139 989 - 2038 491 - 2349 q'' [kW/m 2 ] 24 - 824 25 - 827 26 - 853 429 – 4301 486 - 1043 125 - 402

Table 5.3: Experimental void databases [2]

ACHILLES THETIS PERICLES NEPTUN PWR

5x5 BWR _8X8 LSTF TPTF

Reference [2] [2] [2] [2] - [2] [2] [2]

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Length [m] 3.7 3.6 3.7 1.7 3.66 3.7 3.7 3.7 Rods (heated) 69 (69) 49 (49) 357 (357) 37 (37) 25(25) 64 (62) 1104 (1008) 32 (24) D_r [mm] 9.5 12.2 9.5 10.7 9.5 12.3 9.5 9.5 D_w [mm] 13 13 11 4 15.6 13 13 10 Axial Power distribu-tion Chopped cosine Chopped cosine Chopped cosine Chopped cosine Uniform Uniform / Chopped cosine Chopped cosine Uniform ΔT_sub [K] 18 / 24 25-157 20/60 0.5-3 20.36 – 90.48 9-12 0 5-35 p [MPa] 0.1/0.2 0.2 – 4.0 0.3/ 0.6 0.4 7.4 – 16.6 1.0 – 8.6 1.0-7.3- 15.0 3.0/6.9/ 11.8 G [kg/m2/s] 0.08 2.5-3.1 21-48 42/91 2222 - 3056 284-1988 2.2-84 11-189 q'' [kW/m2] 11 11/12 11-40 5/10 1465 - 2014 225-3377 5-45 9-170

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5.1.2 Whole range

The experimental data have been used to assess the predictive capability of the void correlations used widely in thermal-hydraulic analysis codes. In order to determine the quality of the predictions for each experimental run the absolute error has been computed as the difference between the measured and predicted value

𝛼 𝛼 (5.1)

The comparison of the void prediction correlations is based on the mean absolute error and the standard deviation. The simulations have first been run by using the "reference" model (Levy subcooled boiling model with the equivalent wetted diameter and 50 axial nodes along the channel grid).

The statistical analysis has been applied firstly to the databases sf24va and sf24vb: the results for sf24va are slightly better than the ones for sf24vb. Figure 5.2 shows the void mean error and the void standard deviation over the whole void range: regarding the mean error the homogeneous model over-predicts the measurements as expected [1]; the original Zuber-Findlay model does not give good results when it is applied overall the void range; unlike the vann97, vann96 gives good results as expected since it is fitted to these experimental data: there are some deviations for some data around void fraction equal to 0.68. The iterative void correlations, such as EPRI and Chexal, increase the complexity of the solution without giving a dramatic increase in the quality of the prediction. Bestion (one of the simplest tested) and aa78 have a low mean error.

Regarding the void standard deviation, the unreliable behavior of the homogeneous model and vann97 is confirmed; the aa69, Bestion, aa78, scp, Toshiba,

Maier-Coddington and Inoue give good results and the iterative correlations do not provide a

significant improvement in the quality of the prediction.

Figures 5.3, 5.4 and 5.5 show the values predicted by the void correlations versus the measured values for the databases sf24va and sf24vb. It shows the under-prediction of the void in the subcooled region and the deviation for some correlations towards higher void fractions.

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Figure 5.2: Statistical analysis - sf24va and sf24vb

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Figure 5.4: Pred. vs. Meas. - sf24va and sf24vb

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Other FRIGG (see section 4.1) data regarding BWR rod bundles have been analyzed statistically in figure 5.6 and confirm the previous analysis.

Then the statistical analysis has been applied to the databases bfbt described in the appendix A (see table A.1): figure E.1 confirms the good statistic behavior from

Bestion, aa78, aa69, scp and the iterative correlations.

Figure 5.7 and figure 5.8 show the statistical analysis performed respectively for the databases psbt and rdipe (see table A.2 and figure A.8, A.9): it has been confirmed the unreliable behavior from correlations purely empirical like the vann96 that even presents not physical results by over-predicting the measurements more than the homogeneous model in both the cases. For PWR operating conditions the aa78 correlation has given not physical results by exceeding the predictions of the homogeneous model.

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Figure 5.7: Statistical analysis - psbt

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Figure E.2 (see appendix E) depicts the statistical analysis applied to the experimental data from PSI (only steady-state experiments are considered as bwr8x8, neptun, pwr5x5,

tptf): the good behavior from Bestion, Maier-Coddington, Inoue, scp, EPRI and Chexal

is confirmed, vann96 and aa78 exceed the homogeneous predictions in the case of high pressure operating conditions (pwr5x5).

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5.1.3 Subcooled boiling region

After assessing the predictive capability of the void correlations over the whole range of steady-state data, the attention has been focused on the subcooled boiling region in order to investigate the under-prediction pointed out in the section 5.1.2.

An open issue regarding the subcooled boiling models is the correct characteristic length to be used in the Nusselt number.

Figure 5.9 shows the deviation between predicted and measured values in the low void region by using equivalent wetted and heated diameter for both the Levy and EPRI subcooled boiling models. For the databases sf24va and sf24vb it seems that the characteristic length to be used in the Levy model is the heated equivalent diameter, for the EPRI model it is not clear which is the correct characteristic length to be used. A parametric study regarding the characteristic length has been performed: the database

psbt (see table A.2) has been considered owing to the significant variation of the

geometry.

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Figure 5.10: Characteristic length in the subcooled boiling region

The TH simulations have been run by using the EPRI subcooled boiling model and the

aa69 void correlation and varying the characteristic length in the Nusselt number.

Figure 5.10 depicts on the left the variation of the standard deviation and the mean error with the characteristic length: there is an optimum value for which the objective functions have a minimum value. On the right the optimum value found for each database has been plotted versus the heated equivalent diameter, both referred to the wetted equivalent diameter. This parametric study is intended to prove that the heated equivalent diameter works well in some geometry as shown for sf24va and sf24vb, but it is not always the correct characteristic length to be used.

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5.1.4 High void region

One of the most important goals is to extend the predictive applicability of the void correlations to the high void region.

Due to the lack of experimental data simulated experiments have been performed: the maximum total power in the void database sf24va and sf24vb has been increased in order to reach the saturated conditions, the power of the remainder experimental run has been proportionally increased and the void fraction has been "measured" at the outlet of the rod bundle so that the virtual void databank covers the high void region.

Figure 5.11, 5.12 and 5.13 show the void predicted by the correlations versus the void predicted by the homogeneous model taken as reference. It is interesting to look at how they approach the saturated vapor conditions. Most of them reach the unity, Bestion under-predicts slightly the void, vann96, Maier-Coddington, Toshiba and Inoue do not reach the unity and are quite far from the ideal behavior. Scatter is evident for Bestion,

aa78, scp, Inoue, Toshiba, Maier-Coddington correlations.

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Figure 5.12: High void region

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Figure 5.14 and figure 5.15 show the void fraction versus the non-equilibrium quality: the predicted curve and the curves with different slip ratios are plotted. By increasing the slip ratio the gas phase velocity becomes dominant over the liquid phase: the gas phase flows in the central part of the channel creating the gas core and the liquid phase flows as a thin film on the wall forming an annular ring of the liquid: as the slip ratio increases, the liquid film flows more slowly and it is more unlikely that liquid may be entrained in the gas core as small droplets. It is interesting to look at where the predicted void is located compared with the references: for most of the correlations the prediction is located in the expected slip ratio range (2 ; 3) in agreement with [16], the aa69 is located beyond S = 4 and the Bestion seems to have an irregular behavior for high void fractions.

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Figure 5.15: Different slip ratio

It is worth to investigate more on the Bestion correlation developed for use in the thermal hydraulic code CATHARE. Due to the absence of a value for the distribution parameter in the reference available to the authors it was set to 1. [2] A parametric study is shown in figure 5.16: other values of C0 lead to a decrease of the overall prediction quality.

The original drift flux correlation proposed and assessed by many CATHARE calculations is

√ (5.2) [2]

In agreement with [17], the Wallis annular flow correlation is used for very high void fractions: a new simple void correlation could be developed and extended to the whole range, thus avoiding specifying the transition between different correlations (one for the low void region, one for the high void region).

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5.1.5 Recommended void correlations

The statistical analysis performed overall the wide range of experimental data, supported by the extension to the high void region and the assessment against some boil-off experiments, allows selecting the recommended void correlations to be used in the TH codes.

The empirical correlations like vann96 and vann97 are not considered due to their unreliable behavior: particularly vann96 gives not reasonable predictions when it is applied to the rdipe and psbt as shown in figure 5.7 and 5.8 since it exceeds the homogeneous model. The aa78 correlation shows one of the best performances overall the benchmark analysis: it works not well for PWR since it gives a not physical behavior when it is applied to psbt as shown in figure 5.7.

On the one hand the void correlations aa69, Smith, Toshiba, scp, Maier-Coddington and

Inoue show a good behavior overall the statistical analysis, sometimes close to the

performance of the iterative correlations and even better (see figure 5.8); on the other hand figure 5.3, 5.4, 5.5, 5.17 and 5.18 point out deviation towards higher void fraction [0.8 ÷ 0.9].

According to the analysis performed in the section 5.1.4, Maier-Coddington, Inoue and

Toshiba cannot be used in the high void region.

The present section depicts the measured versus the predicted void for the void correlations: the simulations have been run by using the Levy subcooled boiling model with the equivalent heated diameter. Statistical results regarding the internal databases are included in the table E.1 and E.2.

Figure 5.19 and 5.20 confirm the under-prediction in the subcooled region and the investigation carried out in the section 5.1.3.

Figure 5.21 and 5.22 point out that the aa69, aa78, scp, Toshiba and Smith does not give reasonable predictions for several data from neptun and tptf database. For the

pwr5x5 experiments performed at high pressures, high mass fluxes and significant

subcooling (see table 5.3), all the correlations over-predict significantly the void in the subcooled region, that confirms the investigation regarding the characteristic length (see section 5.1.3).

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Figure 5.17: Pred. vs Meas - FRIGG data

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Figure 5.19: Pred. vs Meas. - psbt

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Figure 5.21: Pred. vs. Meas. - Steady-state experiments (PSI)

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Figure 5.23: Pred. vs. Meas. - Transient experiments (PSI)

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Transient data from boil-off experiments have finally been used to assess the predicitve capability of the correlations: figure 5.23 and 5.24 point out that aa69, aa78, scp and

Smith over-predict excessively the measurements, this can be explained due the fact that

the TH calculations performed assume a steady-state configuration. It has been verified that the bias increases as the inlet mass flux decreases as shown in figure 5.25 for the aa69 void correlation (as well for scp and Smith).

Despite the transient conditions, the Bestion, epri, Chexal, Maier-Coddington, Inoue and Toshiba give good results.

The correlations that have shown their robustness overall the benchmark analysis are those of Bestion, EPRI and Chexal: figure 5.26 and figure 5.27 show the statistical analysis for these correlations. It is quite clear that the iterative correlations do not improve too much the quality of the prediction increasing furthermore the complexity of the solution. Although not applicable with low mass fluxes and transient conditions,

aa69, aa78, scp and Smith are confirmed as reliable void correlations for BWRs fuel

assemblies; Maier-Coddington, Toshiba and Inoue have shown a good behavior overall the benchmark analysis even in the case of the boil-off experiments, but are not applicable to the high void region that plays a crucial role in BWRs operating conditions.

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Figure 5.26: Recommended void correlations

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5.2 Pressure drop

Due to the higher energy efficiency compared with the single-phase flow, the two-phase flow is applied to several fields, but the penalty to pay is represented by the higher pressure drops. A review of two-phase frictional pressure drop correlations has been conducted by the Institute of Air Conditioning and Refrigeration that evaluated them against experimental data covering operating conditions of industrial refrigerants [4]. The scope of the present work is to assess the predictive capability of the correlations (internal and external to Westinghouse) against experimental data covering BWR operating conditions and to adjust the grid pressure loss coefficients once the best friction factor has been found.

5.2.1 Experimental pressure data

A wide range of experimental pressure drop single- and two-phase data (internal and external) at various pressure and mass flux has been collected and provides the opportunity to assess the predictive capability and the overall applicability of the pressure drop correlations (internal and external to Westinghouse). The data covers pressure from 0.2 MPa to 8.6 MPa and mass fluxes from 291 kg/m2/s to 2560 kg/m2/s and provides information on pressure drops in BWR fuel bundles.

The single-phase databases sf24ec, sf24et, sf24h, sf24i and bfbt are those that allow removing the grid pressure loss so that only friction can be considered.

The single-phase databases sf24s, sf24x, sf24vc and bfbt have been used for evaluating the grid pressure loss coefficients.

Table 5.4 provides information about the operating conditions and geometry of the experimental databases.

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Table 5.4: Experimental pressure drop database

sf24ec sf24et sf24h sf24i sf24s sf24vc sf24x bfbt

Reference [6] [13] Type BWR Length [m] 3.77 3.77 3.77 3.77 3.74 3.75 3.76 3.71 Rods (heated) 24 (24) 24 (24/23/24) 64 (60) D r [mm] 9.66 9.66 9.66 9.66 9.84 9.85 9.85 12.30 D W [mm] 9.05 9.50 9.53 9.53 9.93 9.29 - 11.25 9.26 - 11-25 12.80 Axial Power distribution Cosine Top peak

Cosine Cosine Bottom / Cosine / Top Cosine Bottom / Cosine / Top Bottom peak ΔT_sub [K] 7.5 - 195 9.3 - 127 8.1-109.6 8.0 -120.6 50.7 - 198.2 6.6 - 142.6 80.3 - 157.6 1.7 - 85.8 p [MPa] 2.2 - 8.4 2.5 - 5.0 2.0 - 6.9 2.3 - 7.1 2.1 - 6.0 3.9 - 4.0 3.9 - 6.9 0.2 - 8.6 G [kg/m2/s] 530-2560 549-2516 533-2510 543-2511 924 - 2522 955 - 2483 915 - 2488 291 - 2061 q'' [kW/m2] 53 - 739 57 - 750 49 - 748 53 - 754 - 97 - 790

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5.2.2 Friction factor

The attention has been focused on the single-phase databases sf24h, sf24ec, sf24et, sf24i and bfbt that allow removing the local grid contribution from the measured pressure drops so that it is possible to compare the predicted frictional pressure drops against the experimental ones.

A multi-objective non-linear constrained optimization has been performed in order to minimize both the mean error and the standard deviation by using the bfbt single-phase database: the location of the pressure channel is depicted in figure 4.3, particularly the pressure taps T3-T1 and T4-T2 have been used to remove the grid pressure loss.

It is proposed to optimize a friction factor correlation under the explicit form

(5.3)

by using the optimization toolbox available in Matlab: the goal attainment method SQP (implemented function fgoalattain) has been used and it has given results very close to the ones provided by the least square method despite the different optimization criterion.

The coefficients a and b provided by the optimization are respectively equal to 0.3029 and 0.2521.

Figure 5.28 depicts the statistical analysis performed for the databases bfbt by computing the mean error and the standard deviation when different friction factor correlations are used, included the proposed correlation called optimum. The optimization has tried to minimize both the objective functions, achieving a trade-off: the Moody correlation remains the best considering the mean error, but the optimum correlation has the lowest standard deviation equal to 4.46e-4 bar.

Figure 5.29depicts the statistical analysis performed for the databases sf24h, figure 5.30 and figure 5.31 show the measured against the predicted values. Additive plots for the other considered databases are attached to the appendix E and highlight the best behavior of the optimum correlation.

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Figure 5.28: Statistical analysis - bfbt_spdp

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Figure 5.30: Pred. vs. Meas. - sf24h

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The statical analysis highlights that it is not convenient to use the Colebrook and

Nikuradse correlations that involve iterative calculations increasing the complexity of

the solution since there are many explicit correlations that can give better results as the

Investigation and validation of void and pressure drop correlations in BWR fuel assemblies

MASTER OF SCIENCE THESIS