NATURAL CONVECTION HEAT TRANSFER IN TWO-FLUID …9069/FULLTEXT01.pdf · The effect of a mushy layer on convective heat transfer is investigated numerically. A ﬁxed-grid enthalpy-porosity

ISSN 1403-1701 Kärnkraftsäkerhet 9TRITA EKS.AvhandlingISRN KTH/NPS/MSWI-0201-SE

NATURAL CONVECTION HEAT TRANSFER

IN TWO-FLUID STRATIFIED POOLS WITH

INTERNAL HEAT SOURCES

Askar Amirovich Gubaidullin

Doctoral Thesis

January 21, 2002

SPONSORS: The European Union (Projects: MVI, ARVI, MFCI),SKI, US NRC, HSK, Swedish and FinnishPower Companies (MSWI Project)

Royal Institute of Technology

Stockholm, Sweden

ii

KUNGL TEKNISKA HÖGSKOLANRoyal Institute of TechnologyEcole Royale PolytechniqueKgl. Technische Hochschule

Postadress: S-100 44 STOCKHOLMDrottning Kristinas väg. 33A

Telephone: 08-790 9251 (National)+46 8-790 9251 (International)

Fax: 08-790 9197 (National)+46 8-790 9197 (International)

URL: http://www.egi.kth.se/nps/Email: [email protected]

Royal Institute of Technology100 44 STOCKHOLMJanuary 2002

Abstract

Natural convection is a subject of interest in fundamental fluid dynamics and heat trans-fer, as well as in a great number of engineering applications. In particular, knowledgeof thermal convection driven by internal heat sources is of vital importance to reactorsafety research. Severe accidents in light water reactors have been a subject of intensestudy and the focal point of reactor safety in the last two decades. Recently, the prob-lem of reactor pressure vessel integrity has received particular attention. This issue isclosely related to the coolability and stabilization of the damaged molten core of thereactor. The main mechanisms for coolability are radiative and convective heat transfer.

The present work aims to investigate natural convection heat transfer and mixingphenomena in a two-fluid density-stratified pool, which may exist in the reactor vessellower head due to the chemical interactions in a corium melt.

The thesis consists of six parts. In the first chapter a brief introduction to the prob-lem of in-vessel melt retention is presented. The major findings of the present work areoutlined.

In the second chapter, mathematical models for single- and double-layer thermalconvection are introduced. An extensive literature review is carried out. State-of-art ofresearch in natural convection of heat-generating fluids is presented.

In the third chapter, the results of an analytical investigation are presented. Gen-eral correlations are suggested to estimate the heat transfer coefficient as a functionof appropriate dimensionless parameters. The predicted results are compared againstpublished experimental data.

The fourth part presents methods and results of a numerical study. Computationalfluid dynamics (CFD) analysis is performed to study the effect of fluid stratification onheat transport in two-layer pools. The Volume-of-Fluid interface tracking method isemployed. It is found that interface remains undisturbed by convective motion. Dif-

iii

iv Abstract

ferent cases are considered. The predicted results are used to validate the suggestedanalytical correlation. As a next step, the liquid-liquid interface is assumed to be hor-izontal and non-deformable (fixed interface model). This model is implemented intoCFX-4.1 code by means of user routines. The results of extensive validation are pre-sented. The model is applied to a complex case of semicircular geometry. It is foundthat a heat flux peak occurs just below the interface and that its value can be signifi-cantly higher than that in the much-studied uniform pool benchmark. The fraction ofheat transferred upwards is found to be much less in a double-layer system than in auniform pool. The predicted average Nusselt numbers and temperature distributionsare compared against SIMECO experimental data. The effect of physical properties isstudied.

A numerical analysis is performed for a two-layer salt-stratified system, destabi-lized and mixed either by lateral heating in a square cavity, or by internal heating in asemicircular vessel. The predicted mixing time and heat transfer data for the rectangu-lar geometry are compared to those reported in the literature. The motion of the initiallyplanar interface between the two stably stratified layers of fluid is computed once con-vection begins. The development of interfacial instabilities leading to a rapid mixingof the layers are predicted. The shape of the interface during mixing becomes highlynonlinear and is characterized by multiple vortices. Physical mechanisms responsiblefor mixing of stratified layers in an internally-heated system are elucidated.

Both fixed interface and double-diffusion models are applied to investigate naturalconvection phenomena in semicircular pools. Calculations have been performed forthe cases of ”thin”, 4:22, and ”thick”, 8:18, upper layers with heat sources in the lowerlayer or both layers. The comparison of the results for the immiscible and misciblesystems demonstrates that there is very little difference (less than

��) between the

average Nusselt numbers. The instantaneous local Nusselt number distribution at theside wall is practically identical for both cases. The side wall heat flux gradients overthe interface are less sharp in the miscible system due to the diffused nature of theinterface. This is the first computational attempt to compare different stratificationconditions with immiscible fluids and a double-diffusive system in semicircular cavitieswith internal heat generation.

The effect of a mushy layer on convective heat transfer is investigated numerically.A fixed-grid enthalpy-porosity method is applied for analysis of the solidification pro-cess. It is found that the presence of a mushy layer of low permeability has a smalleffect on average heat transfer characteristics at steady state. In addition, a semiem-pirical correlation for an estimate of crust thickness in a molten pool is presented andvalidated against published experimental data.

The turbulent characteristics inherent to the convection at Rayleigh numbers of in-

Abstract v

terest were obtained by direct numerical simulation (DNS). The results are comparedagainst experimental data. The underlying physics of confined buoyancy-driven tur-bulent convection is elucidated. The shortcomings of existing turbulence models arediscussed.

In the fifth part, the results of an experimental study are reported. Several ex-periments with water/water and paraffin oil/water separated by a fixed interface areconducted. The results compared favorably to previously reported tests with water/saltwater and paraffin oil/water without ”housing”, and confirmed the conclusions drawnfrom the CFD study.

A series of 15 high temperature experiments employing an eutectic salt mixture of��and the Cerrobend alloy are performed in the SIMECO facility.

Top boundary conditions are found to have a significant impact on heat transfer. In thecase of an adiabatic top boundary, the ”focusing effect” can be of a factor of 3-4 incomparison with a uniform pool. The ”focusing effect” vanishes when the top surfaceof the metal layer is cooled. The presence of a crust between the upper layer and thelower pool uncouples heat transfer, so that the temperature field in the salt pool dependslittle on the thermal processes in the upper layer.

In addition, tests employing Glycerol and Cerrobend alloy were conducted. In thesetests, the upper and the lower pools are coupled through the interface, since there is nocrust. Heat transferred upwards is less in the case of metal stratification. Despite thehigh conductivity of the metal layer, its presence imposes an additional thermal resis-tance, and thus increases sidewall thermal loads just below the interface. Numericalanalysis is performed by a code developed in-house under the name of MVITA (MeltVessel Interactions Thermal Analysis).

The summary and technical accomplishments are presented in the sixth part of thethesis.

Keywords: computational fluid dynamics, heat transfer, natural convection,light water reactor, severe accident, numerical modeling, multiphase flow, double-diffusion, mixing, turbulence, solidification

vi Abstract

Preface

Research results, presented in the present thesis, were obtained by the author duringthe period between 1996 and 2001. The work was performed at the Nuclear PowerSafety Division of Royal Institute of Technology (KTH). The results of these researchactivities have been summarized and described in the following publications:

1. Gubaidullin, A. A., ”Implementation and Testing of a Multifluid Model for Mul-tiphase Flow Simulation”, RIT/NPS Research Report, Stockholm, May 1997

2. Sehgal, B. R., Dinh, T. N., Green, J. A., Konovalikhin, M. J., Paladino, D.,Leung, W. H., and Gubaidullin, A. A., ”Experimental Investigation of MeltSpreading in One-Dimensional Channel”, RIT/NPS Research Report for Euro-pean Union EU-CSC-1D1-97, 86p., November, 1997

3. Dinh, T. N., Konovalikhin, M. J., Paladino, D., Green, J. A., Gubaidullin, A. A.,and Sehgal, B. R., ”Experimental Simulation of Core Melt Spreading on a LWRContainment Floor in a Severe Accident”, Proceedings of the 6 �

�International

Conference on Nuclear Engineering (ICONE-6), San Diego, CA, USA, May,1998

4. Sehgal, B. R., Bui, V. A., Nourgaliev, R. R., Dinh, A. T., Gubaidullin, A. A.,and Dinh, T. N., ”Simulation of Intense Multiphase Interactions by CFD andDNS Methods: Exploring Capabilities and Limitations”, Proceedings of the An-nual Meeting of Institute for Multifluid Science and Technology, Santa Barbara,February 26-28, 1998

5. Gubaidullin, A. A., Bui, A. V., and Sehgal, B. R., ”Simulation of Bubble andDrop Behavior under Shock Wave”, Proceedings of the 3

��International Confer-

ence on Multiphase Flow (ICMF’98), Lyon, France, June 8-12, 1998

6. Gubaidullin, A. A., Dinh, T. N., and Sehgal, B. R., ”Analysis of Natural Con-vection Heat Transfer and Flows in Internally Heated Stratified Liquid”, Proceed-

vii

viii Preface

ings of the 33��

National Heat Transfer Conference, Albuquerque, NM, August15-17, 1999

7. Gubaidullin, A. A. and Sehgal, B. R., ”Natural Convection in a Double-LayerPool with Internal Heat Generation”, Proceedings of the 8 �

�International Con-

ference on Nuclear Engineering (ICONE-8), Baltimore, MD, April 1-6, 2000

8. Gubaidullin, A. A. and Sehgal, B. R., ”Numerical Analysis of Mixing in aDouble-Diffusive System”, Proceedings of the 34 �

�National Heat Transfer Con-

ference, Pittsburgh, PA, August 20-22, 2000

9. Gubaidullin, A. A. and Sehgal, B. R., ”Numerical Analysis of Natural Convec-tion in a Double-Layer Immiscible System”, Proceedings of the 9 �

�International

Conference on Nuclear Engineering (ICONE-9), Nice, France, April 8-12 , 2001

10. Gubaidullin, A. A. and Sehgal, B. R., ”An Estimate of the Crust Thickness onthe Surface of a Thermally Convecting Liquid-Metal Pool”, to appear in NuclearTechnology, vol. 138, April, 2002

Contents

Abstract iii

Preface vii

Contents ix

Nomenclature xvii

Acknowledgements xxi

1 Introduction 1

1.1 In-Vessel Core Melt Retention . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Literature Review 5

2.1 Mathematical formulation of natural convection for a single-layer fluid . 5

2.2 Mathematical formulation of natural convection for a double-layer fluidsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Literature review. Convection in a single fluid . . . . . . . . . . . . . . 9

ix

x CONTENTS

2.3.1 Heat transfer correlations for Rayleigh-Bénard convection in asingle layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.2 Heat transfer correlations for convection in a volumetricallyheated pool . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Literature review. Convection in a two-fluid system . . . . . . . . . . . 18

3 Analytical Study 23

3.1 Correlation for convection in two stratified layers with internal heatgeneration (top side cooled) . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Correlation for convection in two stratified layers with the internal heatgeneration (top and bottom sides cooled) . . . . . . . . . . . . . . . . . 29

3.3 An estimate of the crust thickness on the surface of a thermally con-vecting liquid-metal pool . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 CFD Study 34

4.1 Computations with Volume-of-Fluid Method . . . . . . . . . . . . . . 34

4.1.1 Description of the modeling approach and solution algorithm . . 34

4.1.2 Two-fluid Rayleigh-Bénard convection . . . . . . . . . . . . . 35

4.1.3 Two-fluid convection with internal heat generation . . . . . . . 36

4.2 Computations with a fixed interface model . . . . . . . . . . . . . . . . 38

4.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.2 Computational method . . . . . . . . . . . . . . . . . . . . . . 38

4.2.3 Simulation of laminar Couette flow . . . . . . . . . . . . . . . 40

4.2.4 Simulation of two-layer convection with internal heat sourcesin the lower layer in rectangular cavities . . . . . . . . . . . . . 41

CONTENTS xi

4.2.5 Simulation of two-layer convection in semicircular pools . . . . 41

4.3 Computations with a double-diffusion model . . . . . . . . . . . . . . 44

4.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . 44

4.3.3 Simulation of step-wise density formation . . . . . . . . . . . . 46

4.3.4 Mixing in a two-layer salt-stratified system . . . . . . . . . . . 47

4.3.5 Comparison with the case of a non-diffuse interface . . . . . . . 49

4.4 Effects of a mushy layer on convective heat transfer . . . . . . . . . . . 49

4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4.2 Model Description and Implementation . . . . . . . . . . . . . 50

4.4.3 Heat transfer for Rayleigh-Bénard convection with solidification 52

4.4.4 Heat transfer in an internally heated pool with solidification . . 53

4.5 Turbulent Characteristics of Heat-Generating Fluid Layers . . . . . . . 55

4.5.1 Two-dimensional computations . . . . . . . . . . . . . . . . . 56

4.5.2 DNS of a heat-generating fluid layer. . . . . . . . . . . . . . . 58

4.5.3 DNS of Rayleigh-Bénard convection. . . . . . . . . . . . . . . 62

5 Experimental Study 71

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3 Uniform Pool Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3.1 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . 75

xii CONTENTS

5.4 Low Temperature Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.5 High Temperature Salt-Cerrobend Tests . . . . . . . . . . . . . . . . . 78

5.5.1 Numerical Analysis of Salt-Cerrobend Tests . . . . . . . . . . . 81

5.6 High Temperature Glycerol-Cerrobend Tests . . . . . . . . . . . . . . . 82

5.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6 Conclusions 92

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

A Paper 1: Analysis of Natural Convection Heat Transfer and Flows in In-ternally Heated Stratified Liquid 105

A.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

A.2 NUMERICAL SOLUTION METHOD . . . . . . . . . . . . . . . . . 110

A.2.1 Numerical Method and Procedure . . . . . . . . . . . . . . . . 111

A.3 MODEL VALIDATION . . . . . . . . . . . . . . . . . . . . . . . . . 112

A.3.1 Rayleigh-Bénard Convection Benchmark . . . . . . . . . . . . 112

A.4 CONVECTION IN SUPERPOSED LAYERS . . . . . . . . . . . . . . 116

A.4.1 Heat Generation in Lower Layer . . . . . . . . . . . . . . . . . 116

A.4.2 Heat Generation Occurs in Both Layers . . . . . . . . . . . . . 118

A.5 SUMMARY AND CONCLUDING REMARKS . . . . . . . . . . . . 119

B Paper 2: Natural Convection in a Double-Layer Pool with Internal HeatGeneration 123

B.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

CONTENTS xiii

B.2 MODEL FORMULATION . . . . . . . . . . . . . . . . . . . . . . . . 128

B.2.1 Double-Diffusive Convection Model . . . . . . . . . . . . . . . 128

B.2.2 Model For Immiscible Layers . . . . . . . . . . . . . . . . . . 130

B.3 NUMERICAL PROCEDURE . . . . . . . . . . . . . . . . . . . . . . 131

B.3.1 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . 131

B.3.2 Computational Mesh . . . . . . . . . . . . . . . . . . . . . . . 131

B.4 RESULTS OF COMPUTATIONS . . . . . . . . . . . . . . . . . . . . 132

B.4.1 Uniform Pool . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

B.4.2 Stratified Pool with Heat Generation in Both Layers . . . . . . 134

B.4.3 Stratified Pool with Heat Generation in Bottom Layer . . . . . . 138

B.5 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

C Paper 3: Numerical Analysis of Mixing in a Double-Diffusive System 147

C.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

C.2 MODEL FORMULATION . . . . . . . . . . . . . . . . . . . . . . . . 152

C.3 NUMERICAL PROCEDURE . . . . . . . . . . . . . . . . . . . . . . 153

C.3.1 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . . . 153

C.4 RESULTS OF COMPUTATIONS . . . . . . . . . . . . . . . . . . . . 154

C.4.1 Mixing in a Rectangular Cavity . . . . . . . . . . . . . . . . . 154

C.4.2 Mixing in a Semicircular Pool with Internal Heat Generation . . 159

C.5 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

xiv CONTENTS

D Paper 4: Numerical Analysis of Natural Convection in a Double-LayerImmiscible System 167

D.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

D.2 CFD SIMULATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

D.2.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . 171

D.2.2 CFD Model Validation . . . . . . . . . . . . . . . . . . . . . . 172

D.2.3 Results of Computations for Semicircular Pools . . . . . . . . . 174

D.2.4 Effect of the upper layer physical properties . . . . . . . . . . . 178

D.3 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

E Paper 5: An Estimate of the Crust Thickness on the Surface of a Thermally-Convecting Liquid Metal Pool 187

E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

E.2 Scaling Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

E.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

E.4 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

F Paper 6: Simulation of Bubble and Drop Behavior under Shock Wave 197

F.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

F.2 NUMERICAL MODELING APPROACH . . . . . . . . . . . . . . . . 201

F.2.1 Description of SIPHRA-3D code . . . . . . . . . . . . . . . . . 201

F.2.2 Validation of the Shock Calculations . . . . . . . . . . . . . . . 202

F.3 RESULTS OF COMPUTATIONS . . . . . . . . . . . . . . . . . . . . 203

CONTENTS xv

F.3.1 Liquid drop deformation . . . . . . . . . . . . . . . . . . . . . 203

F.3.2 Gas bubble deformation . . . . . . . . . . . . . . . . . . . . . 205

F.4 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . 207

xvi CONTENTS

Nomenclature

��Biot number,

�� C constantc� specific heat, J/kg K

solute diffusivity, m � /s�� deformation tensor, �� , s �� fluctuating deformation tensor, �� , s ��!#"%$

Fourier number,!#"&$'� �)( *�,+!#"�-

Fourier number,!#".-/� �)( 0�,+1 gravity vector, ( �,2 �43&5 2 � ), m/s �6�7

Grashof number,6�7 �98;:%< - �>=? +@

height, mA 2CB&2;Dunit vectors in E 2GFH2;I directionsJcharacteristic length scale, mKturbulent kinetic energy, �� L � , m � /s �M latent heat of solidification, J/kgJ � ratio of i-layer height to j-layer heightJ � Lewis number, J � � 0NOP mass flow rate, kg/sQ normal vector

�L Nusselt number,

�L� ��

L � � Nusselt number at the upper surface�L �GR Nusselt number at the curved surfaceS

turbulent kinetic energy production,ST� �VU

L L �.W �S 7Prandtl number,

S 7 � ?0S 7 - turbulent Prandtl number, S 7 -/� ?GX0 XY dynamic pressure, N/m �Z

electric resistance, [

xvii

xviii NOMENCLATURE

Z��Richardson number, � 8 ��

5�� 5�� +Z��

flux Richardson number,Z�� ; - :.8 � -� � �� 5W � mean rate of strain, W � � �� V� �� ,��

heat transfer rate, W� � volumetric heat generation rate, W/m�

�� ratio of heat transferred upwards to heat transferred downwards� heat flux, W/m �Z � internal Rayleigh number, Z � �98;:�� ? 0��Z � � Rayleigh number based on the reference values of the lower layerZ ��

external Rayleigh number,Z �� 98;:%< - �>=? 0Z �>$ critical Rayleigh numberZ �

Rayleigh number (internal or external)W solute concentrationW�� Schmidt number, W�� ?NW�� Stefan number, W�� $�� < - temperature, K fluctuating temperature, K "!melting temperature, K� time, s# velocity vector, m/s�

L2%$ 2%& � velocity vector components, m/s

L velocity vector component,�, m/s�

L 2%$ 2%& � fluctuating components of velocity vector, m/s�(' 2�) 2�* � average components of velocity vector, m/s)

voltage, V' mean velocity vector component, � , m/sL $ turbulent shear stress, m � /s �� E 2GFH2;I � Cartesian coordinates, mE coordinate, � , mF+

dimensionless distance from the wallGreek,

thermal diffusivity, m � /s, -turbulent diffusivity for heat transfer, m � /s-coefficient of thermal expansion,

� ��/. �� - , 1/K0 coefficient of fractional expansion due to solute1 � dynamic boundary layer thickness, m1� thermal boundary layer thickness, m2

difference between two values of some parameter3 dissipation of turbulent kinetic energy, 3 �54�6 �� ,m � /s�

7 fraction of heat generated within the layer that is transferred;

xix

downward; dimensionless distance�dimensionless temperature

� heat conductivity, W/m K� � ratio of i-layer to j-layer conductivity� -

stability number,� - � *

xx NOMENCLATURE

� � �Asea Brown Boveri� * ZBoiling Water Reactor � W Direct Numerical SimulationJ�� W Large-Eddy SimulationJ * ZLight Water Reactor

�T!��Molten Fuel - Coolant Interaction

� ) � �Melt Vessel Interactions Thermal AnalysisS * ZPressurized Water ReactorZ S )Reactor Pressure VesselZ � � W Reynolds-Averaged Navier StokesW 6 W Sub-Grid ScaleW �� SImulation of MElt COolabilityW ��TS#J�� Semi-Implicit Method for Pressure Linked Equations ��Three Mile Island )

Total Variation Diminishing) � !Volume-Of-Fluid

Acknowledgements

First of all, I would like to thank my advisor Professor Bal Raj Sehgal for providingexcellent support and guidance during my graduate studies. He always endorsed myinitiatives and gave me substantial freedom to try new ideas and methods on the longand winding road of research. I would like to express my special gratitude to Dr. T. N.Dinh (Nam) for his help, stimulating discussions and constructive criticism during mystudies. I am particularly happy to have had the opportunity to work with Dr. V. A. Bui(Anh), Dr. R. R. Nourgaliev (Robert) and Dr. J. Green (Joe) who always gave help whenit was needed. I have learned quite a bit from them. I feel lucky to work with Andrei,Aram, Asis, Audrius, Domenico, Gilles, Jose, Gunnar, Ivan, Lin, Maxim, Oskar, Tuanand Zbigniew. NPS division provided me with a stimulating environment. I would liketo thank Dr. A. Theerthan (Ananda) and Dr. P. Yakubenko (Petya) for their commentsand discussions on some parts of the thesis. Special thanks to Kajsa Bergman whoseadministrative skills no one can beat. I am grateful to Dr. M. Vynnycky who did mostof the editing of the manuscript.

Finally, I would like to thank my parents and my fiancée Nadia for their love andsupport during the last five years.

The work performed in this thesis was financially supported, in the framework ofthe following projects:

1. Melt-Vessel Interactions MVI Project funded by the European Union (FourthFramework Programme);

2. Assessment of Reactor Vessel Integrity ARVI Project funded the European Union;

3. Molten Fuel-Coolant Interactions MFCI Project funded by the European Union(Fourth Framework Programme).

4. Analysis of Natural Convection in Volumetrically-Heated Melt Pools jointly funded

xxi

xxii NOMENCLATURE

by the Swedish Nuclear Power Inspectorate (SKI) and the US Nuclear Regula-tory Commission (US NRC).

Chapter 1

Introduction

Natural convection heat transfer has been the subject of extensive experimental and nu-merical investigation over the years. Research interest in buoyancy-driven convectionhas been motivated by its relevance in many applications including geophysical, chem-ical, and nuclear. In particular, nuclear reactor safety has impelled research in thermalconvection in volumetrically heated layers in the past three decades. The research inthis area of heat transfer is of paramount importance in the evaluation of core damageaccidents.

A severe accident in a nuclear power plant can be defined as an event that involvesthe melt-down of the reactor core. Its most serious consequence is the release of ra-dioactive fission products and subsequent contamination of the environment. The de-sign goals for core damage frequency set by nuclear facilities range from

�� to �� per reactor year. However, this probabilistic criterion does not take into account the ele-ment of human error that was the major reason behind two severe accidents (TMI-2 andChernobyl) that occurred in the last two decades. Beginning with the TMI-2, severe ac-cidents became the focus of activity in reactor safety research. The main issues are thesurvivability of the containment and the vessel for various accident scenarios. Severeaccident progression in light water reactors (LWR) can be classified into in-vessel (i.e.core heat-up/degradation, melt relocation, debris bed/core melt pool formation, creepand failure of reactor pressure vessel) and ex-vessel (i.e. melt-fuel-coolant interactions,spreading of melt, debris coolability, hydrogen accumulation) stages. These complexstages involve extreme conditions, such as high pressures and high temperatures, andoccur at large scales. Consequently, direct experimental studies at these scales are notfeasible. Thus, mechanistic models, separate-effect experiments and numerical simu-lations are required. The results provide insights and estimates, and lead to the deter-ministic assessments of consequences in particular phases of a severe accident, as well

1

2 Chapter 1. Introduction

as methods of accident mitigation and prevention.

This thesis is devoted to the study of aspects relevant to in-vessel melt pool con-vection heat transfer (Papers 1-5). The work relevant to ex-vessel molten fuel coolantinteraction (MFCI) is presented in Paper 6.

1.1 In-Vessel Core Melt Retention

The external passive cooling of the RPV by the flooding of the cavity of a pressurizedwater reactor (PWR) or drywell of a boiling water reactor (BWR) is one of the severeaccident management strategies for in-vessel melt retention (IVMR). It involves thedecay heat removal through the vessel wall by the external surface. The effectivenessof external reactor vessel cooling was first assessed for the Loviisa plant (Finland)(VVER-440 reactor) [113], and later for Westinghouse’s AP600 design. It has beenadopted as an accident management strategy for AP600 reactor [111]. The strategy ofreactor cavity flooding for large PWRs has also been considered [48].

The phenomenology of the IVMR involves several inter-related issues:

� melt pool natural convection heat transfer

� radiative heat transfer from the pool top surface and also boiling in case of in-vessel flooding

� mechanical behavior of the RPV (thermal stresses, creep)

� chemistry of melt components

� possible pool stratification

� metal layer focusing of heat flux

� ex-vessel boiling heat transfer

In the last phase of the core degradation, an oxidic melt pool of mainly'�

� 2��7 �

� ,and unoxidized Zircaloy and stainless steel will form in the lower head of the RPV[111]. A molten metal layer (composed mainly of

! � and � 7 ) will rest on the top ofthe crust of the oxidic pool. A thin oxidic crust layer of frozen core material is formedon the vessel’s inside wall. The configuration is illustrated schematically in Fig.1.1. Inthis bounding configuration, thermal loads to the RPV walls are determined by natu-ral convection heat transfer driven by internal heat sources. Decay heat from fission

3

Crust

Molten Metal Layer

Oxidic Pool

Boiling

Water

Figure 1.1: Schematic of the IVMR.

products is assumed to be generated uniformly within the oxidic pool and generallyno heat generation is considered in the upper metallic layer. For example, in a hypo-thetical severe accident scenario for an AP600-like reactor, the following values canbe expected: volumetric heat generation

� � � 1 MW/m�, volume of the oxidic pool) �

10 m�, radius

Z �2 m, temperatures in the oxidic pool

�2700 � C, temper-

atures in the metal layer �

2000 � C=, maximum depth ratio of the metal layer tothe oxidic pool

J� �� , properties of the oxidic pool, depending on melt composi-

tion, as characterized by the Prandtl number,S 7 � � � � , properties of the metallic layerS 7 � � � � , the intensity of convective motion, as characterized by the Rayleigh number,Z � � �� [111]. The time scale of core melt pool formation is estimated as

1/2 to 1 hour [91]. Indeed, these estimates could vary, depending very much on theaccident scenario and the type of reactor.

The presence of the highly conductive metallic layer raises some additional con-cerns for the vessel integrity, since large heat fluxes could be focused on vessel wallnext to of the metallic layer (”focusing effect”) and a melt-through at that location canresult because of critical heat flux (CHF) conditions on the outside of the RPV. More-over, the oxidic pool itself may be stratified into two layers of different components of� 7 � � and

'�� , due to the combined effects of buoyancy and chemical processes [2].

Thus, a knowledge of the natural convection heat transfer coefficient on the bound-aries of volumetrically heated stratified pools, for different, configurations is essentialfor predicting or preventing failure of the RPV in a hypothetical severe accident sce-nario in a LWR. In addition, the success of the external vessel cooling, for in-vesselmelt retention (IVMR), depends upon the evaluation of thermal loadings imposed bythe convecting melt pool for different scenarios.

4 Chapter 1. Introduction

1.2 Present Work

The present work aims to investigate natural convection heat transfer and mixing phe-nomena in a two-fluid density-stratified pool. The thesis consists of six parts. In thefirst chapter a brief introduction to the problem of in-vessel melt retention is given. Themajor findings of the present work are outlined.

In the second chapter, mathematical models for single- and double-layer thermalconvection are introduced. An extensive literature review is carried out.

In the third chapter, the results of an analytical investigation are presented. Gen-eral correlations are suggested to estimate the heat transfer coefficient as a functionof appropriate dimensionless parameters. The predicted results are compared againstpublished experimental data.

The fourth part presents methods and results of a numerical study. Computationalfluid dynamics (CFD) analysis is performed to study the effect of fluid stratification onheat transport in two-layer pools. The effect of a mushy layer on convective heat trans-fer is investigated numerically. A fixed-grid enthalpy-porosity method is applied foranalysis of the solidification process. The turbulent characteristics inherent to convec-tion at the Rayleigh numbers of interest were obtained by direct numerical simulation(DNS). The results are compared against experimental data.

In the fifth part, the results of experimental study are reported. Several experi-ments with water/water and paraffin oil/water separated by a rigid interface are con-ducted. A series of high temperature experiments employing an eutectic salt mixtureof��

and the Cerrobend alloy are performed in the SIMECO facilityfor different boundary conditions and geometry. Numerical analysis is performed by acode developed in-house under the name of MVITA (Melt Vessel Interactions ThermalAnalysis).

The summary and technical accomplishments are presented in the last part of thethesis.

Chapter 2

Literature Review

2.1 Mathematical formulation of natural convection for a single-layer fluid

In this section, governing equations describing natural convection, that is the motion ofa fluid in a gravitational field by temperature-induced density gradients, are introduced.

We consider an incompressible Newtonian fluid of constant thermophysical prop-erties, except for the density variation in the body force (the Oberbeck-Boussinesqapproximation). The fluid is bounded by two horizontal surfaces at distance

J. Bound-

aries are kept at constant temperatures � and � . The following assumptions are made:

(1) the variable temperature is written in the form � �� . Here, �� is

some constant mean temperature and is the temperature variation which is such that �� .

(2)U � U � � U . The density variation U is small compared with the constant

densityU �

. SoU � �� - � �VU � - .

(3) Y � Y � � Y with Y � satisfying the hydrostatic equation Y � � U � 1 �� " � � � � � � .The variables are scaled as

� ! � �� J 2 # ! � # J � 6 2 � ! � � 6 � J � 2 ! � � � �� Y

! � Y J �� U � 6 � � .5

6 Chapter 2. Literature Review

Then the Navier-Stokes equations along with the energy conservation equation 1

can be written in the following dimensionless form (without asterisks):

� # � �,2 (2.1)� #� � � � # � � # � � � Y � 6�7 B � � � # 2 (2.2)

� � � � # � �

�

S 7 � � � (2.3)The two independent dimensionless parameters that appear are the Prandtl number,S 7 � ?

0 , and the Grashof number,6�7 � 8;: �>= � - + �

-��

? + . The Prandtl number providesa measure of the relative effectiveness of momentum and energy transport by diffusionin the dynamic and thermal boundary layers [53]. For laminar boundary layers

1 �1�

� S 7 R 2(2.4)

where� � � . The Grashof number provides ratio of the buoyancy force to the viscous

force acting on the fluid [53]. Often, the Rayleigh number,Z �T� 6�7 S 7

, is usedinstead. When

Z � � Z �>$ the fluid layer becomes unstable, and a convective motiongradually develops.

The transients are characterized by the Fourier number!#" � � , � J � , that is a

dimensionless time. The time corresponding to!#" � �

is usually called the relaxationtime for thermal conduction.

The boundary conditions give a dimensionless quantity�L�� J � � , the Nusselt

number, which characterizes the heat transfer between the solid bodies and the fluid. In1915, W. Nusselt applied dimensionless analysis to natural convection heat transfer. Heconsidered the Boussinesq assumption which holds for small temperature differencesand derived the following equation for the heat flow,

�:

� � � J 2 ��

2 J� U � 3 - 2 � � � � (2.5)

The expression�L� � � 6�7 2 S 7 � is widely accepted not only in the limit of small

temperature difference, but is applied also to conditions at larger temperature differ-ences, when average values for the fluid properties are introduced [32]. For pureconduction in fluid bounded by isothermal walls,

�L� �

. When a fluid layer withuniformly distributed volumetric heat sources is insulated at the bottom and cooled

1Dissipation of energy by viscosity is neglected.

7

isothermally at the top, one has�L� 4

. When convection becomes turbulent, theNusselt number is approximately proportional to the

� � � -power of the Rayleigh num-ber. The thermal boundary layer thickness can be estimated via the Nusselt number: influid layers of thickness

J, heated from below,

1�� J � 4 � L , and in fluid layers heatedfrom within, 1 �

��J � � L . These estimates can be easily derived by dividing the fluidlayer into the thermal boundary conductive sublayer where the local Nusselt number isunity and the isothermal core, where the temperature takes its maximum value.

External heat transfer from the surface to the ambient is characterized by the Biotnumber defined by

�� J � � . Adiabatic boundaries correspond to �� , and

isothermal boundaries correspond to�� .

In the case of internal volumetric heat generation� � when both boundaries are

cooled at different temperatures � and � , one more dimensionless parameter,

� �� J � � � � � �

� � �G� , appears. This group is referred to as the Dammköhler number. Inthis case, two Rayleigh numbers are usually employed: the external

Z ��4� 6�7 S 7and

the internalZ � � Z ��

.

The heat transfer is given by�L� � �)Z �� 2 Z � 2 S 7 � . Under steady-state quiescent

conditions, the temperature distribution is non-linear, and given by � F � � �� 4 � � F � � � F � � 2 (2.6)where the constants

�and

�can be evaluated from the thermal boundary conditions atF � �

andF � J

. In internally heated systems when �

� � (e.g. a uniformly cooledsemicircular pool) or when only

� is present (e.g. a horizontal layer of fluid cooled atone surface and insulated at the other),

�L� � �� - ��

-�� )Z � 2 S 7 � .

2.2 Mathematical formulation of natural convection for a double-layer fluid system

Consider two immiscible fluids bounded by horizontal, isothermal or adiabatic solidsurfaces of infinite extent. Uniform heat sources can be considered in the lower layer.The upper fluid is designated as 1 and the lower fluid as 2. The fluid depths are

J� andJ

� , respectively, (see Fig.2.1). The interfacial forces due to surface tension variationsproduced by temperature gradients (Marangoni effects) are assumed to be negligible.The interface is assumed to be horizontal and non-deformable (sufficiently large sur-face tension). We are interested in cases when thermal convection is established inindividual layers.

In a double-layer system, the governing parameters are


gfluid 1

fluid 2

��T

L 1

T

1

T

int

L2

��

2

Figure 2.1: Diagram of the two-layer system.

# 2 � 2 Y � U � 2 2U� 26� 2,� 2-�3 2 6 � 2

,� 2-�3 2 � �

� � � 2J� 2J� , t.

The independent variables are scaled as

E�� J � 2 #J� �6� 2 �

� �� 2 Y

J �� U�6 �� .

Here,6�7 � (or

Z � � ) is defined on the reference values of fluid4

and on the tem-perature difference

� �� . So the investigation of the two-layer problem becomes

significantly more complicated.

The conservation equations for fluid�

are

� # � �,2 (2.7)� #� � � � # � � # � �VU ��

� Y � - � �6�7 �

B � 6 � �� # 2 (2.8)

� � � � # � �

,� �S 7�� (2.9)

and for fluid4

� # � �,2 (2.10)� #� � � � # � � # � � � Y � 6�7 �

B � � � # 2 (2.11)� � � � # � �

�

S 7�� (2.12)

Continuity of temperature, velocity, heat flux and shear stresses must be satisfiedat the fluid-to-fluid interface. Continuity of heat flux gives the dimensionless group� � �

� K� �K� .

In the case of heat generation� �� in the lower layer, the dimensionless energy

9

equation will be

� � � � # � �

�

S 7��

� �S 7�� (2.13)

It is convenient to introduce a dimensionless measure of the gravity as � � 8 �>=�? � 0 �( �� %2 4 ) and a dimensionless measure of the heat generation in the � -layer as @ �� +�� - .

2.3 Literature review. Convection in a single fluid

2.3.1 Heat transfer correlations for Rayleigh-Bénard convection in a sin-gle layer

Fluid motion between two rigid planes kept at different constant temperatures is com-monly referred to as Rayleigh-Bénard convection, in honor of H. Bénard, who in 1900observed the formation of hexagonal cells in a thin layer of fluid heated from below,and Lord Rayleigh, who in 1916 derived the criterion for instability in the case ofstress-free boundaries. In fact, the thermal convection studied by Bénard was drivenby the variation of surface tension with temperature, and instabilities of this kind areusually referred to as Marangoni effects. The stability of the Rayleigh-Bénard systemwas first studied by H. Jeffreys in 1928 who predicted the condition for the onset ofconvection at

Z �>$4� �� At sufficiently high Z � numbers a transition to turbulenceoccurs, depending on

S 7, (e.g.

Z � � �� for S 7 �� ).Table 2.1: Correlations for Rayleigh-Bénard convection.

References � �� G � D(1959) [40] �� "!�# �� %$'&)(+*-,-� .�� ,0/-.1��23&)�� (�/-.1��4O � S(1961) [107] �� %$-5�(�� 4��76 .�� *'/-.1� � &)5�� ,0/-.1� �

�� $�$-�� 8�2�8 5�� ,0/-.1� � &9.1� 2�� .1�-:%�� "2 �� "4�# .1� 2 &9.1��;

C � G (1973) [22] �� .1(�5�� 8�!�4 ��=* $?� *@�0/-.1� 2 &9.�� %,'/�.1� 4Castaing A1BC�+D7� (1989) [14] �� $-5�� 8"��! ��=


Most of the known correlations for turbulent convection give values of�

between4 � � and � � � . Arguments based on the dimensional reasoning yield � � � � � , and thisrelation can be easily derived (e.g. see Bejan [5]). The

� � � -law means that the actualheat transfer rate � is independent of the layer thickness J .

Based on theoretical analysis of the boundary layer or/and experimental observa-tions, Heslot � � � � � [49] identified different types of turbulence, ”soft” and ”hard” tur-bulence, the distinction between the two coming at

Z � �� . It was found thatthe slope of the

�L -Z �

relation in log-log coordinates retains� � � value only in ”soft”

turbulence, and that it decreases to about4 � � in the ”hard” turbulence regime. The

basis of this proposal was an experiment in He at 7 K. Moreover, Chavanne � � � � � [17]reported experiments using cryogenic helium gas at low temperatures for

Z �rang-

ing up to4 �� at S 7 � � � � and found an increase in � up to about � � � � � at

very high Rayleigh numbers (Z � � �� G� ). This change was interpreted as the tran-

sition to an ”asymptotic regime”. It was noted that the features of the observed newregime matched those of the ultimate regime predicted by Kraichan [63] at moderateS 7

(S 7 � � � � ). However, the questions of the existence of a strict power law in the�

L� Z �

relation and the asymptotic state remain open. The effects of the Prandtlnumber, i.e. the relative values of the thermal and velocity boundary layer thicknesses,and the aspect ratio of enclosures are not well understood either. Recently, Niemela � �� [77] investigated thermal transport over eleven orders of magnitude of the Rayleighnumber (

�� Z � � �� ), using cryogenic helium gas as a working fluid. Theirdata were correlated by a single power law with scaling exponent

� � � � � � , and noevidence of any transition was found.

Numerical studies: Among innumerable computational studies of Rayleigh-Bénardconvection, it is necessary to quote the work of Deardorff � Willis [24], Grötzbach[42], and Kerr � Herring [58].

Grötzbach [42] performed a direct numerical simulation 2 (DNS) at moderate Rayleighnumbers. For the highest

Z � � � � � �� , the grid of � 4 �� nodes was employed. Theanalysis of the spatial mesh resolution requirements for DNS of the Rayleigh-Bénardconvection was presented in [43].

Kerr and Herring [58] investigated by means of DNS the dependence of the Nusseltnumber on the Rayleigh and Prandtl number for

�� Z � � �� and � � �� S 7 � �

in a� �� 4 box. The mesh of 4?� � �� 4 � nodes was used for Z � � �� to resolve

necessary turbulence scales. The resulting Nusselt number scaled as�L� Z � � � � atS 7 � � � � and S 7 � � , and � L

� Z � � � � at lower S 7 .2by DNS it is understood a complete solution of three-dimensional, non-steady-state equations of the

conservation of mass, momentum and energy where all scales of motion are resolved.

11

Different subgrid-scale (SGS) models in large eddy simulations (LES) for buoyancy-driven flows are discussed in [83].

It should be pointed out that computation of turbulent buoyancy-driven flows is verycomplicated due to the existence of the large range of scales. The presence of a rigidboundary affects the condition of heat flux and stress complicating the matter further.The available Reynolds-averaged Navier-Stokes (RANS) turbulence models (e.g. thetwo-equation eddy-viscosity

K � 3 models) are not always satisfactory [47]. On the otherhand, DNS or LES cannot yet handle high Rayleigh number turbulent convection due tolimitations in computer capacity. However, since developments in computer hardwareand numerical algorithms have been outspacing advances in turbulence modeling, thefuture for DNS and LES looks bright [84].

2.3.2 Heat transfer correlations for convection in a volumetrically heatedpool

Theoretical studies: The earliest theoretical investigation of the onset of convectivemotion in horizontal fluid layers bounded by isothermal surfaces with a uniformly dis-tributed heat source

� � was reported by Sparrow, Goldstein and Jonsson [101]. Thevalues of the critical Rayleigh number

Z �,$'� 8;: �>=�� - + �-� �

0 ? were calculated as a functionof the parameter � � � � � J � � 4 � � �� . If the temperature of the lower boundary is

greater than that of the upper boundary ( � � � ), then

Z �>$decreases from its limiting

value of��

as� � increases. For the case of �

� � , the critical Rayleigh numberZ �>$, defined in terms of the volumetric heat generation, is found to be equal to 37 325.

Roberts [87] solved the marginal stability problem for a horizontal layer of fluidcooled from above, thermally insulated from below, and heated uniformly from within.The critical Rayleigh number of 2772 was obtained.

Cheung [18] developed a phenomenological model of eddy transport in turbulentthermal convection in a horizontally infinite layer of fluid confined between a rigidisothermal upper plate and a rigid adiabatic lower plate, driven by volumetric heatsources. From dimensional considerations, the Nusselt number was found to be pro-portional to

� � � of the Rayleigh number. No account was made for the effect of Prandtlnumber. A correlation for the mean Nusselt number was obtained for steady heat trans-fer for

�� Z � � �� G� :�L� � � 4 �� Z � � � � � (2.14)

A set of differential equations for average temperature distributions, production of ther-mal variance

& � -�� and turbulent transport & at various Z � � �� were derived


and solved numerically.

Bergholz, Chen � Cheung [7] examined the basic differences and similarities be-tween the upward heat transfer in bottom-heated and internally-heated fluid layers. Itwas assumed that far-field effects are negligible on average upward heat transfer, andthe interaction between lower and upper boundaries is very small. The modified Nus-selt number

�L!

at the upper boundary was defined in terms of near-field parameters:the implicit length scale

� ! � � ? �� :%< -��

�and the new temperature scale

2 !. The

temperature scale is defined as2 ! � � � � � � �

4for bottom-heated layers and2 ! � "! � � � � for internally-heated layers. The length scale J for bottom-heated

fluid was defined as half of the layer depth. For internally-heated layers with isothermalboundaries at the same temperature, the length scale was defined as

� �� + � + J , whereas

for internally heated layers with an adiabatic lower boundary, the length scale was de-fined as total layer depth. Experimental data for Rayleigh-Bénard convection and for aninternally-heated fluid was correlated in terms of the modified Nusselt number and theRayleigh number. The modified Nusselt number was found to have an extremely weakdependence upon the Rayleigh number. The correlated heat transfer results for the twotypes of heating were similar. However, experimental data for internally heated waterlayers having insulated lower boundary and a cooled upper boundary showed consid-erably more scatter than that for Rayleigh-Bénard convection or an internally-heatedlayer with both boundaries at the same temperature. The authors concluded that thesurface heat-transfer coefficient for turbulent convection in horizontal layers dependsprimarily upon the near-field parameters, regardless of the method of heating. In theturbulent convective regime, the heat transfer characteristics of Rayleigh-Bénard con-vection and convection with internal heat generation can be derived one from the otherby defining the appropriate boundary layer Nusselt number

�L!.

Cheung [19] investigated turbulent thermal convection in a horizontally infinitelayer of fluid confined between a rigid isothermal upper plate and a rigid adiabaticlower plate, driven by volumetric heat sources. The dependency of the upper sur-face Nusselt number upon

S 7and

Z �was obtained. It was indicated that the tur-

bulent structure (the r.m.s. vertical velocity, the variation of mean turbulent tempera-ture, the production of turbulent energy in the core region) in internally heated fluidlayers is quantitatively different from that of Rayleigh-Bénard convection. It wasfound that at sufficiently high

Z �(about

�� G� ), � L� Z � � � � for S 7 � � � , and�

L� Z � � � � S 7 � � � for S 7 � � � . At lower Rayleigh numbers, � L is found to vary as�

L� Z � � � � � � � �)S 7 � ��%�)S 7 � Z � �� .

Recently, Arpaci [3] proposed a dimensionless number

� ��

� +�� for buoyantturbulent flows driven by the internal heat generation. The theory developed, gener-alizes the model previously proposed by Cheung. The new model yields the Nusselt

13

number correlated as

�L� � � � � � � �

�

� � � � � � � �� 2

(2.15)

� � Z �� S 7 �� (2.16)

Good agreement with experimental data was reported for�� Z � � �� G� . Experi-

mental studies (rectangular cavity). The frst experimental studies of natural convec-tion with volumetric heat sources, motivated by issues related to the post-accident heatremoval in liquid-metal-cooled fast breeder reactors, were reported in early 1970s.

The earliest qualitative experiments on convection in a horizontal layer of water(S 7 � � � � ) cooled from above, and internally heated nearly uniformly by electrolytic

currents, were carried out by Tritton and Zarraga [106]. Cellular convection patternswere visualized for

Z �up to

� � Z �,$. Two striking differences from Bénard convection

in the flow patterns were observed. First, the cell structure was, for moderateZ �

,hexagonal with motion downwards at the center of each cell; second, the horizontalscale of the convection pattern grew larger with increase of

Z �. It was observed that the

flow remains non-turbulent to valuesZ � � Z �,$ high compared with the corresponding

ratio for the Bénard configuration.

The same problem was experimentally investigated by Fiedler and Wille [36], Ku-lacki and Nagle [67] and Kulacki and Emara [64]. The heat transfer correlations forvolumetrically heated rectangular cavities with the top boundary cooled are summa-rized in Tables 2.2.

Table 2.2: Correlations for internally-heated horizontal fluid layers with an adiabaticlower wall and an isothermal upper boundary, with

Jbeing the height and

�being the

length of the fluid layer.

References � �� F � W(1970) [36] �� :�:%�� 8�8�4 ��& * : /-.1� 2 &9.1��; �� $-��& 1.65K � N(1975) [67] �� $�,-(�� 8�� ; �� $'&)�� 5�/-.1� 2 & ,0/�.1��; �� %,?& 0.25K � E(1977) [64] �� 5�5�(�� 8�8�! $?� (0&)�� 5�� (0/�.1� � & :�� 50/�.1� �78 �� %$�,?& 0.5

The first heat transfer measurements for turbulent thermal convection in a fluidbounded by two rigid isothermal planes of equal temperature were reported by Kulackiand Goldstein [65]. In their study, the volumetric energy source was provided by Jouleheating. The fluid height was varied from

� � 4 � to � � � � cm, and the length and depth ofthe channel were both 25.4 cm. Horizontally-averaged temperature profiles were de-termined optically. Heat transfer correlations for the upper and lower boundaries wereobtained from 67 observations. From the empirical correlation, the critical Rayleigh


numberZ �>$'� �� 4 � was deduced. The transition to turbulence was observed

to occur atZ � � � �� .

Experimental studies in a similar configuration were reported by Mayinger [74],Jahn and Reineke [54], Kulacki and Emara [64]. Table 2.3 summarizes the heat transfercorrelations obtained from these investigations.

Table 2.3: Correlations for internally heated horizontal fluid layers with isothermallower and upper boundaries.

References ��

�� K � G(1972) [65] �� 5%$-��+� � 8��6 .�� :%5��%� � � ; # : /-.1�-# & $0/�.1��!

15

L R

W

L

φ

R

L

a)

c)

b)

Figure 2.2: a) Spherical pool b) Semicircular pool c) Two-Dimensional Slice Ge-ometry of a Torospherical Bottom of a VVER-440 Reactor.

Circular and spherical cavities. Mayinger et al. [74] were the first to report ex-perimental and computational results for thermal convection in volumetrically-heatedsemicircular pools. In the last decade, a number of research programs have appearedworld-wide: USA (UCLA facility, [1], ACOPO at UCSB, [108]), Russia (OECD RAS-PLAV project, [2]), France (BALI, [11]), Finland (COPO, [70]), and Sweden (FOR-EVER, [94], SIMECO, [93]). The main objective of these tests is to obtain the heattransfer characteristics applicable to prototypical configurations and simulants. Theheat transfer correlations for different geometries (Figs. 2.2) are summarized in Tables2.4-2.5 below.

Table 2.4: Correlations for internally-heated semicircular and spherical pools.

no. Reference �

��

�� 1 Mayinger et al. (1975) [74] �� 5�� 8�� ,@:%�� 74�� @�� 8�2 .1� ! & ,0/-.1� � � � �2 Jahn � Reineke (1974) [54] �� 8 .1� ! &9.1� �� 3 Gabor et al. (1980) [38] �� ,�,-�� 72�� @�� "� � $0/-.1� � � & $0/�.1� �� 4 Asfia et al. (1996) [1] �� ,@:%�� 8�� @�� 8�2 $'/�.1� � � &9.1� � # � �5 mini-ACOPO (1995) [111] �� 5-:+,-�� 8�� .1� �78 & *'/�.1� �72 4 � �6 ACOPO (1995/97) [108] .�� %,-�� 74 �� 5�� 8�8 .1� �7� &9.1� �76 � � � �7 ACOPO (1997) [108] �� $-�� "� �� "4�# �� < .1� �76 4 � �8 COPO (1994) [70] �� 5-:+,-�� 8�� ,@:%�� 74�� @�� 8�6 �� .1� �72 � �9 BALI (1998) [92] �� .�.1�� 8�2�� @�� 8 .1� �7� &9.1� �7!


Table 2.5: Correlations for internally-heated semicircular and spherical pools (contd).

no. Pool Method Scale Working Boundaryconfiguration of heating fluid conditions

1 semicirc. 1/2 Water cooled rigid surf.2 semicirc. Water3 hemispher. Joule heat.

� � W �� @

��

free surf.4 hemispher. microwave 1/8 Water/Freon-113 free surf.

insul. rigid surf.cooled rigid surf.

5 hemispher. trans. cool-down 1/8 Water/Freon-113 cooled rigid surf.6 hemispher. trans. cool-down 1/2 Water cooled rigid surf.7 hemispher. trans. cool-down 1/2 Water cooled rigid surf.8 torospher. Joule heating 1/2 Water cooled rigid surface

2D slice9 semicirc. Joule heat. 1/1 Water

Numerical studies. The earliest two-dimensional (2D) simulations of thermal con-vection in rectangular enclosures with internal heat generation were performed in theearly 1970s by Thirlby [104] Peckover and Hutchinson [82] and Jahn and Reineke[54]. Mayinger � � � � � [74] predicted natural convection heat transfer in a hemisphericalcavity.

Emara and Kulacki [33] performed simulations for Prandtl numbers ranging from0.05 to 20 and Rayleigh numbers from

�� to �� on a grid of � � � . The boundary con-ditions for the cooled upper surface were either those of zero slip (a rigid boundary) orzero shear (free boundary). No turbulence model was employed. A zero-shear upperboundary resulted in larger average Nusselt numbers. The computed

�L and hori-zontally averaged temperature profiles agreed fairly well with the experimental data

of Kulacki and Emara [64]. Weak dependency of�L on

S 7was found. It was rea-

soned that the average heat transfer coefficient is relatively insensitive to the details ofthe turbulent flow field, and 2D simulations give reasonably accurate solutions at themoderate Rayleigh numbers considered.

A low-Reynolds-number two-equationK � 3 model was applied by Steinberner and

Reineke [103] to numerical treatment of turbulent convection in a cavity with verticalcooled walls, and the upper cooled wall by Farouk [35].

Grötzbach [44] applied DNS forZ �

up to��

andS 7 � � . The spatial grids with

up to �� 4 nodes were used to resolve all relevant scales of turbulence in a heated

fluid layer. Most of the flow region (about� ��

) was found to be stably stratified.

17

Counter-gradient heat fluxes were observed in the core of the flow. The predicted sta-tistical turbulence data showed that first order models, like the eddy conductivity andK � 3 models, cannot be the proper tools to model turbulent heat flux.

In a recent work by Grötzbach and Wörner [45], DNS and LES applications innuclear engineering were reviewed. The role of DNS as a tool to provide data fordeveloping statistical turbulence models, and for models and codes validation was dis-cussed. The results of DNS for the model fluid water,

S 7 � �, in the fully turbulent

regime atZ � � ��

were presented. The number of mesh cells were� � � � � �?� . It was

also stressed that the eddy conductivity concept is not adequate for the thermal energyequation, and development and applications of LES or second-order models shouldbe pursued. It was argued, however, that ”the progress of LES towards engineeringapplications would strongly depend on the availability of adequate SGS models andboundary conditions”, as well as that ”real retardation comes mainly from the numer-ics”.

Dinh and Nourgaliev [28] carried out an extensive numerical study for different ge-ometries and wide range of

Z �(up to

�� ). The available turbulence models were crit-ically reviewed. The finite-difference two-dimensional code (named NARAL) based onthe SIMPLE solution procedure and the commercial CFX-4.1 software package wereemployed. First, the Navier-Stokes equations were solved in 2D without any turbulencemodel. The resulted averaged Nusselt numbers compared well with Jahn and Reineke’s[54] experimental correlation for a semicircular cavity with

Z �up to

� �� G� . Goodagreement with experimental data was also reported for square cavities at

Z � � �� G� .Secondly, the performance of a number of turbulence models was examined in applica-tion to a COPO geometry (Fig.2.2 c) and a square enclosure. The non-uniform mesh of� � � was utilized. The averaged Nusselt numbers were compared with Steinberner andReineke’s [103] correlations. For

Z � � �� , the uppwards heat flux was consistentlyunderpredicted (by about

�� ). The analysis showed the deficiency of existing first-

order turbulence models in application to unstably stratified buoyancy-driven flows.Developing the ideas of Gibson and Launder [41], corrections for the near boundaryturbulent viscosity � � and turbulent Prandtl number

S 7� were proposed. The expres-

sions forS 7

�� )Z�� 2GF+ � and � �

� � �)Z�� 2GF+ � were obtained. Here, Z�� is the fluxRichardson number which is a ratio of the rate of turbulent energy removal by buoyantforces to the rate of turbulent energy creation by mean shear. The above mentionedcorrections for turbulent transport coefficients along with wall damping functions wereadded to the

K � 3 model. Reasonable agreement with COPO, and Steinberner andReineke’s experimental data was achieved with these modifications for

Z � � �� .Nourgaliev and Dinh [78] conducted three-dimensional (3D) computations for an

internally-heated fluid contained in a channel with an isothermal upper boundary and anadiabatic lower boundary. The Navier-Stokes equations were solved directly (without


an additional turbulence model) by means of a high-order numerical scheme on a non-uniform mesh of

� � � � � � . Good agreement with experimental data of Kulacki andEmara [64] for the averaged heat transfer coefficient was achieved for

Z �up to

� �� . Turbulent data was obtained for � �� Ra � � �� , and S 7 � � � � andS 7 � �

. The average temperature distribution in the core region, as well as turbulentheat fluxes, compared well with those predicted by Cheung’s analytical model [18].Whereas thermal fields obtained for different

S 7were similar, a strong dependency of

turbulence parameters onS 7

was reported. The analysis of the Reynolds stresses andturbulent heat fluxes revealed significant anisotropy of turbulent transport properties. Itwas concluded that the isotropic eddy diffusivity approach cannot be used to describeturbulent convection in unstably stratified flows. The 3D calculations were comparedwith 2D predictions on a grid of

�� . It was shown that a 2D approach providesreasonably accurate solutions, in terms of average heat transfer characteristics, for

Z �up to

�� Z �>$

.

Nourgaliev, Dinh and Sehgal [79] presented a CFD analysis of the effect ofS 7

on thermal convection. The computations were performed for 2D square, semicircularand elliptical geometries, and for 3D semicircular and hemispherical enclosures withisothermal boundaries. The Prandtl number was varied from 0.2 to 7, and the Rayleighnumbers up to

�� were considered. The grid sizes were � � � for square, 3200 nodesfor elliptical and 3988 for semicircular 2D domains. For 3D calculations, a mesh of�� 4�4 (semicircular) and 70464 (hemispherical) nodes were employed. The effectofS 7

on the average value of�L at the upper cooled boundaries was found to be small.At low S 7 , a significant increase in local � L was predicted at the bottom boundary ofsemicircular, elliptical and hemispherical cavities, and near the lower corners of square

cavities. This drastic change in local�L was attributed to more vigorous, in the caseof low viscosity fluids, flows descending along cooled side boundaries which lead to

mixing in, otherwise, stagnant fluid.

In addition to the above-mentioned CFD studies, a number of recent numericalstudies on melt pool heat transfer at conditions close to prototypical were reported in[13], [8], [23] and [72].

2.4 Literature review. Convection in a two-fluid system

Natural convection in a stratified two-fluid system is a natural extension of the muchstudied single-layer problem. Most of interest in a multiple-layer convection has beeninspired by problems of the convection of the Earth’s mantle and the encapsulatedcrystal growth.

19

Early work on the linear stability problem for the two-layer Bénard problem wasreported by Zeren and Reynolds [119] and Ruehle [89]. An extensive listing of otherstudies can be found in Joseph and Renardy [56]. Most of the studies are focusedon theoretical predictions of the critical states (when convection rolls form) and arenot pertinent to the present work. Some numerical and experimental work have beenreported on viscous and thermal coupling between the layers at the onset of convection[99].

Sparrow � � � � � [100] performed natural convection experiments in a rectangularenclosure (aspect ratio of unity) containing either a single fluid or two immiscible flu-ids in a layered configuration. The vertical walls of the enclosure were, respectively,heated and cooled, while the horizontal walls were adiabatic. The liquids used in theexperiments were distilled water and research grade (99

�pure)

� �hexadecane paraf-

fin, with respective Prandtl numbers of 5 and 39.2. Single-fluid experiments were alsocarried out, and it was found that the reduced two-fluid heat transfer data agreed withthe single-layer heat transfer correlation to within a tolerance of no greater than 5%.

Haberstroh and Reinders [46] presented a simple physical model to predict thenondimensional turbulent heat flux. The model was validated for water/water and sili-con oil/water systems heated from below and separated from each other by a thin alu-minum sheet, with individual Rayleigh numbers ranging from 10

�to 10 � � and Prandtl

number ranging from 3.3 to 920.

The main idea of the model is to derive an expression for the Nusselt number de-fined as the ratio of the actual heat transfer to the heat transfer by conduction alone,i.e.

�L� � �)J � � � � �

J� � � � ��

� � �� (2.18)

From the steady state heat balance and the assumption that the Nusselt numbers of theindividual layers can be correlated as

�L � � Z � ! S 7 R 2 (2.19)

whereZ � � 8;: � �>=� � - � �

- � �� 0 � ? � , the following expression is obtained:

�L� � � � � �� +� +�� ! , � , � � � � � �� ,

� �! �/� � + !

��

,� �! �/� � + !

� � � ! + � � � !��!2

(2.20)

where for each layer, �� J � � �

!�� S 7 R �

! � (2.21)


The term inside the braces in Eq.2.20 is called the ”modified interfacial Rayleigh num-ber”,

Z � R� �!�� . The constants were determined from the experiments for turbulent

natural convection in a single fluid layer and taken equal to�9� � � � � � � 2 P � � � � � � ,

and�T� � � �� . The correlation of the nondimensional heat flux through a density-

stratified interface becomes then

�L� � � � � � �%Z � �� R

� �!�� (2.22)

It should be noted that the value of the coefficient�

in Haberstroh and Reinders’ studyis about

� ��higher than in the well-known Globe and Dropkin correlation for a single

fluid [40].

Simanovskii [99] was the first to report the results of a numerical study of theconvection in two immiscible fluids heated from below. The numerical method wasbased on the vorticity-stream function formulation. The effect of fluid property ratioson the onset of instabilities was investigated for Grashof numbers

6�7 � 3 -�2 J � � 6 ��

between��

and� �� .

Recently, Prakash and Koster [85] performed an experimental and numerical studyof two-dimensional convection in a system of two immiscible liquids heated from be-low. The objective of the study was to understand the coupling physics at a liquid-liquidinterface at onset of convection. The effect of the ratio of the fluid depths on differentcoupling modes between the layers was considered. It was found that when buoyancyforces in both layers are of similar strength (equal layer heights), thermal coupling,identified by co-rotating rolls on both sides of the interface, is preferred. Mechani-cal (viscous) coupling, identified by counterrotating rolls, dominates when buoyancyforces are very different in both layers.

The numerical work reported in literature has been performed for very low (Z � �

�� ) Rayleigh numbers. No numerical study has been reported for semicircular

enclosures or for moderately high Rayleigh numbers.

There are very few studies that report heat transfer in double-layer pools with inter-nal heat sources. Fieg [34] was, perhaps, the first to investigate the natural convectioncharacteristics of two stratified immiscible liquid layers with the lower one internallyheated. The temperature was maintained equal at the top and bottom boundaries. Hep-tane and water were used as lighter and heavier liquids, respectively. The importantconclusion was that the two layers behaved as if separated by a rigid highly conduc-tive wall. The correlation for the Bénard problem in the upper layer and the one forthe layer bounded by isothermal walls at different temperatures and volumetric heat-ing were applied and the calculated values agreed with the experimental data to within�

10�

accuracy. Temperature profile measured by Fieg seemed to support this assump-tion. However, experimental data obtained by Fieg for a double-layer configuration is

21

very scarce.

Schramm and Reineke [90] studied experimentally and numerically natural con-vection in a rectangular channel filled with two immiscible fluids of different physicalproperties. The MAC (”Marker and Cell”) numerical technique was applied to the two-fluid problem. In the experiment the upper layer was silicon oil and the lower layerwas water. Internal heating was present in the lower layer. Upper and lower bound-aries were cooled at constant temperature. Measured isotherms in the upper fluid werereported to be similar to Rayleigh-Bénard convection and in the lower layer to thoseobtained in an internally heated fluid bounded by rigid walls at the top and the bottom.No heat flux data was obtained in the experiment. In computations, similar isothermswere obtained. The numerical results were obtained for modified Archimedes numbers,� 7 � 3>J � � 6 �

,� , of �� to �� , upper sublayer fractions,

J� �J

, of 0.11 to 0.75, anda dimensionless number

�9� � � J � - � �K� , of 0.09 to 1.9, and � � �

� � � �� . Their

numerical results agree qualitatively with their experimental data obtained with the aidof holographic interferometry.

Kulacki and Nguen [68] studied hydrodynamic instability and thermal convectionin a horizontal layer of two immiscible fluids with internal heat generation in the lowerlayer. In their study, the systems of heptane-water or silicone oil-water was boundedin a square cavity from below by a rigid, insulated surface and from above, by anisothermal wall. The heat was generated internally in the lower layer. Experimentalmeasurements of transient and steady state convection up to Rayleigh numbers of

�� G�were presented. The Nusselt number based on the average heat transfer coefficientfor different layer thickness ratios were obtained from the experiments and correlated.The uncertainty in the measured

�L and

Z �is reported to be less than

� � 4 � . Thenumerical predictions in the vorticity-stream function formulation for a uniform com-putational domain of

4 � � � � were found to be in general agreement with the data. Theinterface was treated as a semi-solid conducting sheet with interfacial shear transmit-ted from the lower layer to the upper layer. In addition, simulations were done withthe interface assumed to be hydrodynamically rigid. It was concluded that for eitherinterface, the overall flow pattern was not much affected by the hydrodynamics at theinterface. However, predicted Nusselt number for the case of a hydrodynamically rigidinterface was about

��lower.

Recently, the SIMECO experimental program at NPS/RIT [93] was initiated tostudy the effects of density stratification on heat transfer characteristics. A numberof tests were performed in a semicircular slice geometry with paraffin oil as the toplayer, and water as the lower one (immiscible fluids) [59]. The experiments were con-ducted for

Z � � �� , various depth ratios

J� �� 2 � � � � 2 � � 4 �� with the lower

layer heated, andJ� �� -�>2 � � � � 2 � � � � � for the case when both layers are heated.

Temperature distributions, local and averaged heat fluxes, ratio of energy transferred


upwards to downwards,�� were presented. The fluid stratification appeared to signif-

icantly decrease the upwards heat transfer. The local heat flux peak below the interfacewas found to be greater than in corresponding single layer experiments. No depen-dency on

J� � was observed for cases of heat generation in the lower layer. No mixing

between the two fluids has been observed, and the interface did not deform.

Tests employing salt water for the lower layer, and fresh water for the upper layer(miscible fluids) were conducted as well [105]. The density difference between theupper and the lower layer was varied from

2 U � 4 �to

2 U � � ��. For

2 U � � � ,a blurred interface was observed, and for

2 U � � � the interface was defined by adistinct boundary. Mixing time was found to be dependent on

2 Uand heat generation

rate.�� took values less than in the corresponding uniform pool and did not change

significantly during quasi-steady state, when partial mixing occurs at the interface andside boundaries. As rapid total mixing occurred,

�� increased and the temperature of

the pool decreased to the values of the corresponding single-layer pool. Temperaturedistributions, local and averaged heat fluxes, ratio of energy transferred upwards todownwards (sidewards),

�� were measured and the mixing process was recorded on

video.

Chapter 3

Analytical Study

3.1 Correlation for convection in two stratified layers withinternal heat generation (top side cooled)

We consider a horizontal layer of two immiscible fluids with internal heat generationin the lower layer, as studied by Kulacki and Nguen [68]. The top boundary is kept atconstant temperature

� , and the bottom boundary is insulated. The depth of the lowerlayer is

J� and the depth of the top layer is

J� . The physical properties of the top and

the bottom fluids are denoted by indices�

and4

respectively.

The experimental results of Kulacki and Nguen [68] are summarized in Table 3.1.The thermophysical properties of heptane and silicone oil are given in Table 3.2. How-ever, these experimental correlations are valid only for the fluid systems and heightratios chosen in the tests, and are not universal.

Here, we suggest a conservative semiempirical correlation to estimate dimension-less heat fluxes in the system described above. The expression obtained will be vali-dated with the experimental data of Kulacki and Nguen and may be used for any im-miscible fluid system and height ratio.

We will treat the interface as a thin, highly thermally conductive, solid (non-slip)boundary. We assume that

(1) the correlation of the following form is valid in the lower heated layer:�L �

��

��)S 7

� �Z � R +

� . The constants�

��)S 7

� � and�� can be taken from known heat trans-

fer correlations for horizontal fluid layers with an adiabatic lower boundary and an

23

24 Chapter 3. Analytical Study

Table 3.1: The experimental Nusselt number correlations of Kulacki and Nguen [68]

top layerJ� �

�L range of

Z ��

silicon oil 0.035 0.186 Z � �� Z � � � �� G�

silicon oil 0.111 0.183 Z � �� Z � � � �� G�

silicon oil 0.433 0.115 Z � �� G� � ��

Z �� G�

heptane 0.04 0.126 Z � �� Z � � � �� G�

heptane 0.111 0.112 Z � ��

�� Z � � � �� G�heptane 0.433 0.135 Z � ��

��

��

Z �� G�

Table 3.2: Physical properties (taken at � � �

��

)U - � � � � S 7

water 995 3 �� 4200 0.61 8 �� 6silicon oil 900 1 ��

�1900 0.12 4 ��

�67

heptane 670 7 �� 2050 0.13 4 �� 6

isothermal upper wall (e.g. see [18])

(2) the heat transfer in the upper non-heat generating layer is described by an ex-pression of the type

�L �

� ��)S 7

� �Z � R �� . The constants

�� and

�

� can be takenfrom some known correlations (e.g. O’Toole � Silveston [107], Globe � Dropkin [40]or Chu � Goldstein [22]) for Rayleigh-Bénard convection. We estimate the Nusseltnumber, defined as

�L� � �)J � �

J� �K

� � "! � � � � �

2(3.1)

where the heat flux through the interface is � � � � J � .The Rayleigh numbers

Z �� and

Z �� are defined as

Z �� 3 - �

J �� R

�� ,

�6�

2(3.2)

Z �� 3 -�� J ��,

�6� � �

� (3.3)

Introducing the notation

� � 3 J �, 6 2 (3.4)

@ � � � J �� -�K�2

(3.5)

25

the expressions for the Nusselt numbers in both layers can be written as� � J �

J��

� R��

� ��

R �� -�� R

�� G� R � 2 (3.6)

� � J �� "! � � � R � �

� �� -�

R +�

@ R + 2

� R�� - ��

� @ J � ��

R � - � � �� 2 (3.7)

� "! � � � R � �� - ��

� �� R +�

@ �G� R + �Adding

� R�� to � "! � � � R � � , and taking into account Eq. 3.1, we obtain the

following expression for the Nusselt number:

�L� @ � � � J � � �� +

� +�� ++ � � � ( � � �� : � + 2 (3.8)where

� � J� �

: � +� � + � � � � �� . The Nusselt number depends on the following dimen-

sionless parameters: � � , � � ,@

, � � � ,-� � and

J� � where � �

� � ��>= � +0 � + ? � + . One caninterpret � as a dimensionless measure of the gravity in each layer [56], and @ as a

dimensionless measure of the heat generation.

It is assumed that the upper fluid is in a turbulent convection regime. The constantsare taken from O’Toole � Silveston [107] for the upper layer,

�

�� S 7 �� and

�� ,

and the Kulacki � Emara [64] correlation for the lower layer,�

�� ?� and � �

� � � 4�4 � , for � � �+� �� Z � � � � � � �� and 4 � � � � S 7 �

� � � � . Their data has an experimental uncertainty of � � � � for Z � and � � � � for � L .The Nusselt number is calculated as a function of

Z ��

@in order to com-

pare with the correlated experimental data of Kulacki and Nguen. The comparison forvarious values of

J� � is presented in Figs. 3.1 - 3.2. The Nusselt number values given

by the correlation

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