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(1)ISSN 1403-1701 K¨arnkrafts¨akerhet 9 TRITA EKS.Avhandling ISRN 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 Finnish Power Companies (MSWI Project). Royal Institute of Technology Stockholm, Sweden.

(2) ii. ¨ KUNGL TEKNISKA HOGSKOLAN Royal Institute of Technology Ecole Royale Polytechnique Kgl. Technische Hochschule. Postadress: S-100 44 STOCKHOLM Drottning Kristinas v¨ag. 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: askar@ne.kth.se. Royal Institute of Technology 100 44 STOCKHOLM January 2002.

(3) Abstract Natural convection is a subject of interest in fundamental fluid dynamics and heat transfer, as well as in a great number of engineering applications. In particular, knowledge of thermal convection driven by internal heat sources is of vital importance to reactor safety research. Severe accidents in light water reactors have been a subject of intense study and the focal point of reactor safety in the last two decades. Recently, the problem of reactor pressure vessel integrity has received particular attention. This issue is closely related to the coolability and stabilization of the damaged molten core of the reactor. The main mechanisms for coolability are radiative and convective heat transfer. The present work aims to investigate natural convection heat transfer and mixing phenomena in a two-fluid density-stratified pool, which may exist in the reactor vessel lower 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 problem of in-vessel melt retention is presented. The major findings of the present work are outlined. In the second chapter, mathematical models for single- and double-layer thermal convection are introduced. An extensive literature review is carried out. State-of-art of research in natural convection of heat-generating fluids is presented. In the third chapter, the results of an analytical investigation are presented. General correlations are suggested to estimate the heat transfer coefficient as a function of appropriate dimensionless parameters. The predicted results are compared against published experimental data. The fourth part presents methods and results of a numerical study. Computational fluid dynamics (CFD) analysis is performed to study the effect of fluid stratification on heat transport in two-layer pools. The Volume-of-Fluid interface tracking method is employed. It is found that interface remains undisturbed by convective motion. Dif-. iii.

(4) iv. Abstract. ferent cases are considered. The predicted results are used to validate the suggested analytical correlation. As a next step, the liquid-liquid interface is assumed to be horizontal and non-deformable (fixed interface model). This model is implemented into CFX-4.1 code by means of user routines. The results of extensive validation are presented. The model is applied to a complex case of semicircular geometry. It is found that a heat flux peak occurs just below the interface and that its value can be significantly higher than that in the much-studied uniform pool benchmark. The fraction of heat transferred upwards is found to be much less in a double-layer system than in a uniform pool. The predicted average Nusselt numbers and temperature distributions are compared against SIMECO experimental data. The effect of physical properties is studied. A numerical analysis is performed for a two-layer salt-stratified system, destabilized and mixed either by lateral heating in a square cavity, or by internal heating in a semicircular vessel. The predicted mixing time and heat transfer data for the rectangular geometry are compared to those reported in the literature. The motion of the initially planar interface between the two stably stratified layers of fluid is computed once convection begins. The development of interfacial instabilities leading to a rapid mixing of the layers are predicted. The shape of the interface during mixing becomes highly nonlinear and is characterized by multiple vortices. Physical mechanisms responsible for mixing of stratified layers in an internally-heated system are elucidated. Both fixed interface and double-diffusion models are applied to investigate natural convection phenomena in semicircular pools. Calculations have been performed for the cases of ”thin”, 4:22, and ”thick”, 8:18, upper layers with heat sources in the lower layer or both layers. The comparison of the results for the immiscible and miscible  ) between the systems demonstrates that there is very little difference (less than average Nusselt numbers. The instantaneous local Nusselt number distribution at the side wall is practically identical for both cases. The side wall heat flux gradients over the interface are less sharp in the miscible system due to the diffused nature of the interface. This is the first computational attempt to compare different stratification conditions with immiscible fluids and a double-diffusive system in semicircular cavities with 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 process. It is found that the presence of a mushy layer of low permeability has a small effect on average heat transfer characteristics at steady state. In addition, a semiempirical correlation for an estimate of crust thickness in a molten pool is presented and validated against published experimental data. The turbulent characteristics inherent to the convection at Rayleigh numbers of in-.

(5) Abstract. v. terest were obtained by direct numerical simulation (DNS). The results are compared against experimental data. The underlying physics of confined buoyancy-driven turbulent convection is elucidated. The shortcomings of existing turbulence models are discussed. In the fifth part, the results of an experimental study are reported. Several experiments with water/water and paraffin oil/water separated by a fixed interface are conducted. The results compared favorably to previously reported tests with water/salt water and paraffin oil/water without ”housing”, and confirmed the conclusions drawn from the CFD study. experiments employing an eutectic salt mixture of A series  

(6) of15high  andtemperature the Cerrobend alloy are performed in the SIMECO facility. Top boundary conditions are found to have a significant impact on heat transfer. In the case of an adiabatic top boundary, the ”focusing effect” can be of a factor of 3-4 in comparison with a uniform pool. The ”focusing effect” vanishes when the top surface of the metal layer is cooled. The presence of a crust between the upper layer and the lower pool uncouples heat transfer, so that the temperature field in the salt pool depends little on the thermal processes in the upper layer. In addition, tests employing Glycerol and Cerrobend alloy were conducted. In these tests, the upper and the lower pools are coupled through the interface, since there is no crust. Heat transferred upwards is less in the case of metal stratification. Despite the high conductivity of the metal layer, its presence imposes an additional thermal resistance, and thus increases sidewall thermal loads just below the interface. Numerical analysis is performed by a code developed in-house under the name of MVITA (Melt Vessel Interactions Thermal Analysis). The summary and technical accomplishments are presented in the sixth part of the thesis. Keywords: computational fluid dynamics, heat transfer, natural convection, light water reactor, severe accident, numerical modeling, multiphase flow, doublediffusion, mixing, turbulence, solidification.

(7) vi. Abstract.

(8) Preface Research results, presented in the present thesis, were obtained by the author during the period between 1996 and 2001. The work was performed at the Nuclear Power Safety Division of Royal Institute of Technology (KTH). The results of these research activities have been summarized and described in the following publications:. 1. Gubaidullin, A. A., ”Implementation and Testing of a Multifluid Model for Multiphase 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 Melt Spreading in One-Dimensional Channel”, RIT/NPS Research Report for European 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 LWR  Containment 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 and DNS Methods: Exploring Capabilities and Limitations”, Proceedings of the Annual 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 and  Drop Behavior under Shock Wave”, Proceedings of the 3 International Conference 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 Convection Heat Transfer and Flows in Internally Heated Stratified Liquid”, Proceedvii.

(9) Preface. viii. ings of the 33 15-17, 1999. . National Heat Transfer Conference, Albuquerque, NM, August. 7. Gubaidullin, A. A. and Sehgal, B. R., ”Natural Convection in a Double-Layer Pool with Internal Heat Generation”, Proceedings of the 8 International Conference on Nuclear Engineering (ICONE-8), Baltimore, MD, April 1-6, 2000 8. Gubaidullin, A. A. and Sehgal, B. R., ”Numerical Analysis of Mixing in a  Double-Diffusive System”, Proceedings of the 34 National Heat Transfer Conference, 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 on the Surface of a Thermally Convecting Liquid-Metal Pool”, to appear in Nuclear Technology, vol. 138, April, 2002.

(10) 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 fluid system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. Literature review. Convection in a single fluid . . . . . . . . . . . . . .. 9. 2.3. ix.

(11) CONTENTS. x. 2.3.1. 2.3.2. 2.4. Heat transfer correlations for Rayleigh-B´enard convection in a single layer . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. Heat transfer correlations for convection in a volumetrically heated pool . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Literature review. Convection in a two-fluid system . . . . . . . . . . . 18. 3 Analytical Study. 23. 3.1. Correlation for convection in two stratified layers with internal heat generation (top side cooled) . . . . . . . . . . . . . . . . . . . . . . . . 23. 3.2. Correlation for convection in two stratified layers with the internal heat generation (top and bottom sides cooled) . . . . . . . . . . . . . . . . . 29. 3.3. An estimate of the crust thickness on the surface of a thermally convecting liquid-metal pool . . . . . . . . . . . . . . . . . . . . . . . . . 32. 4 CFD Study 4.1. 4.2. 34. 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-B´enard convection . . . . . . . . . . . . . 35. 4.1.3. Two-fluid convection with internal heat generation . . . . . . . 36. 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 sources in the lower layer in rectangular cavities . . . . . . . . . . . . . 41.

(12) CONTENTS. 4.2.5 4.3. 4.4. 4.5. xi. Simulation of two-layer convection in semicircular pools . . . . 41. 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. 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-B´enard convection with solidification 52. 4.4.4. Heat transfer in an internally heated pool with solidification . . 53. 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-B´enard 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.

(13) CONTENTS. xii. 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 Internally 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-B´enard 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 Heat Generation 123 B.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127.

(14) 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.

(15) CONTENTS. xiv. D Paper 4: Numerical Analysis of Natural Convection in a Double-Layer Immiscible 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 ThermallyConvecting 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.3. F.2.1. Description of SIPHRA-3D code . . . . . . . . . . . . . . . . . 201. F.2.2. Validation of the Shock Calculations . . . . . . . . . . . . . . . 202. RESULTS OF COMPUTATIONS . . . . . . . . . . . . . . . . . . . . 203.

(16) CONTENTS. F.4. xv. F.3.1. Liquid drop deformation . . . . . . . . . . . . . . . . . . . . . 203. F.3.2. Gas bubble deformation . . . . . . . . . . . . . . . . . . . . . 205. CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . 207.

(17) xvi. CONTENTS.

(18) Nomenclature  C c. 1.

(19)  

(20)   !#"%$ !#"67. @ J. A 2CB&2;D K. Y. M J  J

(21) PO Q. Z. L  L GR S L S 7 S 7-.   . Biot number,  constant specific heat, J/kg K solute diffusivity, m /s deformation tensor,

(22)  .     , s           , s  fluctuating deformation tensor,

(23)   !#"&$' )(,* + Fourier number, !#".-/ )(,0 + Fourier number, ,2 43&5 2  gravity vector, ( ), m/s - > =  6 7 9  ; 8 % : < Grashof number, ?+ height, m 2GFH2;I directions unit vectors in E characteristic length scale, m turbulent kinetic energy,  , m /s L latent heat of solidification, J/kg ratio of i-layer height to j-layer height J  N0 Lewis number,

(24) mass flow rate, kg/s normal vector    Nusselt number, L Nusselt number at the upper surface Nusselt number at the curved surface ST turbulent kinetic energy production, S 7  ? Prandtl number, 0 S 7 -/ ?GX X turbulent Prandtl number, 0 dynamic pressure, N/m electric resistance, [. xvii. VU .W  L L.

(25) NOMENCLATURE. xviii. Z  Z  W  . .  . . Z  Z  Z  Z >$ Z  W W   W

(26) "!  #.  %2 $ 2%&  L L  %2 $ 2%&  (L' 2 ) 2*  ) ' $ L E 2GFH2;I  E F +. Greek ,. -. , 0 1. 1 2 7. 3. 5    5  8  Richardson number,    + -  Z.    ; :.8  5 flux Richardson number,  

(27)       

(28) V 

(29) ,  mean rate of strain, W . heat transfer rate, W . volumetric heat generation rate, W/m ratio of heat transferred upwards to heat transferred downwards heat flux, W/m Z 98;:?   internal Rayleigh number, 0 Rayleigh number based on the reference- >values = of the lower layer Z  98;:%< external Rayleigh number, ?0 critical Rayleigh number Rayleigh number (internal or external) solute concentration  N? Schmidt number, W  $ < Stefan number, W

(30) temperature, K fluctuating temperature, K melting temperature, K time, s velocity vector, m/s velocity vector components, m/s  velocity vector component, , m/s fluctuating components of velocity vector, m/s average components of velocity vector, m/s voltage, V  mean velocity vector component, , m/s turbulent shear stress, m /s Cartesian coordinates, m  coordinate, , m dimensionless distance from the wall. . . thermal diffusivity, m /s turbulent diffusivity for heat transfer, m /s  coefficient of thermal expansion, /.  - , 1/K coefficient of fractional expansion due to solute dynamic boundary layer thickness, m thermal boundary layer thickness, m difference between two values of some parameter 546

(31)  

(32)   ,m /s dissipation of turbulent kinetic energy, 3 fraction of heat generated within the layer that is transferred;. . .

(33) xix.   . .  .   .  6. 6. U. .  . downward; dimensionless distance dimensionless temperature heat conductivity, W/m K ratio of i-layer to j-layer conductivity  -  * <-  stability number,   :%< * <  + stability number,  :   ratio of i-layer to j-layer dynamic viscosity dynamic viscosity, Pa s     vector operator, = i  + j  5 + k  kinematic viscosity, m /s turbulent diffusivity for momentum transfer, m /s permeability, m

(34)  +  , a dimensionless number  density, kg/m angle, degree; solid fraction porosity. . Subscripts 4. $.

(35) EY   " P Y  " 7%7  7 .    . P E " 7 P  " &.  . top layer lower layer average experimental coolant computational, computed correlation critical interface liquidus liquid maximum reference value root-mean-square solidus wall averaged over surface. Superscripts. !. fluctuating quantity dimensionless quantity. Overbar. . List of acronyms. space average; time mean.

(36) NOMENCLATURE. xx. Z. *. W J W J * Z T !  . ). S. Z S Z. * ). Z. . W 6W W W     W TS#J   ). ).  !. Asea Brown Boveri Boiling Water Reactor Direct Numerical Simulation Large-Eddy Simulation Light Water Reactor Molten Fuel - Coolant Interaction Melt Vessel Interactions Thermal Analysis Pressurized Water Reactor Reactor Pressure Vessel Reynolds-Averaged Navier Stokes Sub-Grid Scale SImulation of MElt COolability Semi-Implicit Method for Pressure Linked Equations Three Mile Island Total Variation Diminishing Volume-Of-Fluid.

(37) Acknowledgements First of all, I would like to thank my advisor Professor Bal Raj Sehgal for providing excellent support and guidance during my graduate studies. He always endorsed my initiatives and gave me substantial freedom to try new ideas and methods on the long and 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 my studies. 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 when it 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, Tuan and Zbigniew. NPS division provided me with a stimulating environment. I would like to thank Dr. A. Theerthan (Ananda) and Dr. P. Yakubenko (Petya) for their comments and discussions on some parts of the thesis. Special thanks to Kajsa Bergman whose administrative skills no one can beat. I am grateful to Dr. M. Vynnycky who did most of the editing of the manuscript. Finally, I would like to thank my parents and my fianc´ee Nadia for their love and support during the last five years. The work performed in this thesis was financially supported, in the framework of the following projects:. 1. Melt-Vessel Interactions MVI Project funded by the European Union (Fourth Framework 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.

(38) xxii. NOMENCLATURE. by the Swedish Nuclear Power Inspectorate (SKI) and the US Nuclear Regulatory Commission (US NRC)..

(39) Chapter 1. Introduction Natural convection heat transfer has been the subject of extensive experimental and numerical investigation over the years. Research interest in buoyancy-driven convection has been motivated by its relevance in many applications including geophysical, chemical, and nuclear. In particular, nuclear reactor safety has impelled research in thermal convection in volumetrically heated layers in the past three decades. The research in this area of heat transfer is of paramount importance in the evaluation of core damage accidents. A severe accident in a nuclear power plant can be defined as an event that involves the melt-down of the reactor core. Its most serious consequence is the release of radioactive 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 element of human error that was the major reason behind two severe accidents (TMI-2 and Chernobyl) that occurred in the last two decades. Beginning with the TMI-2, severe accidents became the focus of activity in reactor safety research. The main issues are the survivability of the containment and the vessel for various accident scenarios. Severe accident 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, creep and failure of reactor pressure vessel) and ex-vessel (i.e. melt-fuel-coolant interactions, spreading of melt, debris coolability, hydrogen accumulation) stages. These complex stages involve extreme conditions, such as high pressures and high temperatures, and occur at large scales. Consequently, direct experimental studies at these scales are not feasible. Thus, mechanistic models, separate-effect experiments and numerical simulations are required. The results provide insights and estimates, and lead to the deterministic assessments of consequences in particular phases of a severe accident, as well 1.

(40) Chapter 1. Introduction. 2. as methods of accident mitigation and prevention. This thesis is devoted to the study of aspects relevant to in-vessel melt pool convection heat transfer (Papers 1-5). The work relevant to ex-vessel molten fuel coolant interaction (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 pressurized water reactor (PWR) or drywell of a boiling water reactor (BWR) is one of the severe accident management strategies for in-vessel melt retention (IVMR). It involves the decay heat removal through the vessel wall by the external surface. The effectiveness of 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 been adopted as an accident management strategy for AP600 reactor [111]. The strategy of reactor 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 invessel 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.  2 7  '. In the last phase of the core degradation, an oxidic melt pool of mainly. , and unoxidized Zircaloy and stainless steel will form in the lower head of the RPV !  7 [111]. A molten metal layer (composed mainly of

(41) and ) will rest on the top of the crust of the oxidic pool. A thin oxidic crust layer of frozen core material is formed on the vessel’s inside wall. The configuration is illustrated schematically in Fig.1.1. In this bounding configuration, thermal loads to the RPV walls are determined by natural convection heat transfer driven by internal heat sources. Decay heat from fission.

(42) 3. Molten Metal Layer. Water. Oxidic Pool Boiling Crust. Figure 1.1: Schematic of the IVMR.. products is assumed to be generated uniformly within the oxidic pool and generally no heat generation is considered in the upper metallic layer. For example, in a hypothetical severe accident scenario for an AP600-like reactor, the following values can 

(43) be expected: volumetric heat generation 1 MW/m , volume of the oxidic pool Z 

(44) )

(45) 10 m , radius 2 m, temperatures in the oxidic pool 2700 C, temper

(46) atures in the metal layer 2000 C=, maximum depth ratio of the metal layer to J

(47)    , properties of the oxidic pool, depending on melt composithe oxidic pool  S 7

(48)    , properties of the metallic layer tion, as characterized by the Prandtl number, S 7    , the intensity of convective motion, as characterized by the Rayleigh number, Z

(49)      [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 the accident scenario and the type of reactor.. . . . . The presence of the highly conductive metallic layer raises some additional concerns for the vessel integrity, since large heat fluxes could be focused on vessel wall next to of the metallic layer (”focusing effect”) and a melt-through at that location can result because of critical heat flux (CHF) conditions on the outside of the RPV. Moreover, 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 boundaries of volumetrically heated stratified pools, for different, configurations is essential for predicting or preventing failure of the RPV in a hypothetical severe accident scenario in a LWR. In addition, the success of the external vessel cooling, for in-vessel melt retention (IVMR), depends upon the evaluation of thermal loadings imposed by the convecting melt pool for different scenarios..

(50) 4. Chapter 1. Introduction. 1.2 Present Work The present work aims to investigate natural convection heat transfer and mixing phenomena in a two-fluid density-stratified pool. The thesis consists of six parts. In the first chapter a brief introduction to the problem of in-vessel melt retention is given. The major findings of the present work are outlined. In the second chapter, mathematical models for single- and double-layer thermal convection are introduced. An extensive literature review is carried out. In the third chapter, the results of an analytical investigation are presented. General correlations are suggested to estimate the heat transfer coefficient as a function of appropriate dimensionless parameters. The predicted results are compared against published experimental data. The fourth part presents methods and results of a numerical study. Computational fluid dynamics (CFD) analysis is performed to study the effect of fluid stratification on heat transport in two-layer pools. 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 process. The turbulent characteristics inherent to convection 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 experiments with water/water and paraffin oil/water separated by a rigid interface are conducted. A series of high temperature experiments employing an eutectic salt mixture and the Cerrobend alloy are performed in the SIMECO facility of for different boundary conditions and geometry. Numerical analysis is performed by a code developed in-house under the name of MVITA (Melt Vessel Interactions Thermal Analysis)..  

(51) . The summary and technical accomplishments are presented in the last part of the thesis..

(52) Chapter 2. Literature Review 2.1 Mathematical formulation of natural convection for a singlelayer fluid. In this section, governing equations describing natural convection, that is the motion of a fluid in a gravitational field by temperature-induced density gradients, are introduced. We consider an incompressible Newtonian fluid of constant thermophysical properties, except for the density variation in the body force (the Oberbeck-Boussinesq J approximation). The fluid is bounded by two horizontal surfaces at distance . Boundaries 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   .. U  U  U . The density variation U is small compared with the constant   -  VU - .  Y  Y with Y satisfying the hydrostatic equation Y  U 1    "       . (3) Y. (2) U U density . So. The variables are scaled as. . !. . . J 2 #!  #. J . 6. 2 ! .  6. . J 2 5. !.  . . . . Y !  Y J    U 6. ..

(53) Chapter 2. Literature Review. 6. Then the Navier-Stokes equations along with the energy conservation equation can be written in the following dimensionless form (without asterisks):. #  ,2   #   #    Y  67 B   # 2 #   S 7     . # . 1. (2.1) (2.2) (2.3). The two independent dimensionless parameters -  appear are the Prandtl number, S 7  ? , and the Grashof number, 67  8;: >=  ? - + +that   . The Prandtl number provides 0 a measure of the relative effectiveness of momentum and energy transport by diffusion in the dynamic and thermal boundary layers [53]. For laminar boundary layers 1 1.  . S 7R2

(54). (2.4). . . . The Grashof number provides ratio of the buoyancy force to the viscous where Z T 67 S 7 , is used force acting on the fluid [53]. Often, the Rayleigh number, Z Z > $  instead. When the fluid layer becomes unstable, and a convective motion gradually develops.. . . !#" . ,. J.   , that is a The transients are characterized by the Fourier number # ! "

(55) dimensionless time. The time corresponding to is usually called the relaxation time for thermal conduction..  J. The boundary conditions give a dimensionless quantity   , the Nusselt L number, which characterizes the heat transfer between the solid bodies and the fluid. In 1915, W. Nusselt applied dimensionless analysis to natural convection heat transfer. He considered the Boussinesq assumption which holds for small temperature differences  and derived the following equation for the heat flow, :.  J . . 2.   67 2 S.     . . 7.  2J U 3 . - 2. . (2.5). The expression  is widely accepted not only in the limit of small L temperature difference, but is applied also to conditions at larger temperature differences, when average values for the fluid properties are introduced [32]. For pure  . When a fluid layer with conduction in fluid bounded by isothermal walls, L uniformly distributed volumetric heat sources is insulated at the bottom and cooled 1. Dissipation of energy by viscosity is neglected..

(56) 7. . 4. isothermally at the top, one has . When convection becomes turbulent, the  L Nusselt number is approximately proportional to the  -power of the Rayleigh number. The thermal boundary layer thickness can be estimated via the Nusselt number: in 1  J 4 J fluid layers of thickness , heated from below,  layers heated 1 J L , and inbyfluid  . These estimates can be easily derived from within, dividing the fluid L layer into the thermal boundary conductive sublayer where the local Nusselt number is unity and the isothermal core, where the temperature takes its maximum value.. . External heat transfer from the surface to the ambient is characterized by the Biot   J   . Adiabatic boundaries correspond to   , and number defined by    . 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     number. In  G , appears. This group is referred to as the Dammk¨ Z o hler 4 67 S 7 this case, two Rayleigh numbers are usually employed: the external and Z  Z . the internal .. . . .  . .  )Z  2 Z  2 S . 7. The heat transfer is given by  . Under steady-state quiescent L conditions, the temperature distribution is non-linear, and given by.  F    . F . 4  . F . 2. (2.6). where the constants and can be evaluated from the thermal boundary conditions at F   and F  J . In internally heated systems when  (e.g. a uniformly cooled  semicircular pool) or when only is present (e.g. a horizontal layer of fluid cooled at    -  -   )Z 2 S 7  . one surface and insulated at the other),  . L. .  . 2.2 Mathematical formulation of natural convection for a doublelayer fluid system Consider two immiscible fluids bounded by horizontal, isothermal or adiabatic solid surfaces of infinite extent. Uniform heat sources can be considered in the lower layer. J and The upper fluid is designated as 1 and the lower fluid as 2. The fluid depths are  J , respectively, (see Fig.2.1). The interfacial forces due to surface tension variations produced by temperature gradients (Marangoni effects) are assumed to be negligible. The interface is assumed to be horizontal and non-deformable (sufficiently large surface tension). We are interested in cases when thermal convection is established in individual layers. In a double-layer system, the governing parameters are.

(57) Chapter 2. Literature Review. 8.  . T1. fluid 1. L1. g. T int. fluid 2  . L2. T2. Figure 2.1: Diagram of the two-layer system. #. 2  2 Y  U 2 2 U 2 6 2 , 2 - 3 2 6 2 , 2 - 3 2    2 J 2 J , t.       . The independent variables are scaled as. 6 2    2Y J  U 6 .     6 7 Z  4 (or ) is defined on the reference values of fluid and on the temHere,    . So the investigation of the two-layer problem becomes. perature difference  significantly more complicated.. E J 2. J. #. The conservation equations for fluid. are. #  ,2   #   #  VU    Y  -  67 B  , 6   # 2 #   S 7    . . # . and for fluid. 4.  , 2. . # . #   #   #    Y  67 B  #   S 7  .  . (2.8) (2.9). (2.10). # 2. (2.7). . (2.11) (2.12). Continuity of temperature, velocity, heat flux and shear stresses must be satisfied at 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.

(58) 9. equation will be .  #.  S 7   S 7 . . . (2.13).  ? 8  >=  @ 0. It is convenient to introduce a dimensionless measure of the gravity as  %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-B´enard convection in a single layer Fluid motion between two rigid planes kept at different constant temperatures is commonly referred to as Rayleigh-B´enard convection, in honor of H. B´enard, who in 1900 observed 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 of stress-free boundaries. In fact, the thermal convection studied by B´enard was driven by the variation of surface tension with temperature, and instabilities of this kind are usually referred to as Marangoni effects. The stability of the Rayleigh-B´enard system was first studied by H. Jeffreys in 1928 who predicted the condition for the onset of Z >$4    At sufficiently high Z numbers a transition to turbulence convection at S 7 , (e.g. Z

(59)   for S 7

(60)  ). occurs, depending on. . . . Table 2.1: Correlations for Rayleigh-B´enard convection. References G  D(1959) [40] O  S(1961) [107] C  G (1973) [22] Castaing A1BC+D7 (1989) [14] Cioni A1BC+D7 (1997) [16] Niemela A1BC+D7 (2000) [77]. .  

(61)    "!#  %$-5(   476  $$-   828  .1-:%   "2

(62)    "4#  .1(5   8!4  $-5 8"!  . :%   86  .$@:%    ;.

(63)   %$'&)(+*-,-.

(64)  =<>*

(65) =<E * 

(66)  =<F %$,

(67)  =<E *.  . ,0/-.123&) (/-.14 . *'/-.1  &)5 ,0/-.1  5 ,0/-.1  &9.1 2 .1 2 &9.1; $? *@0/-.1 2 &9. %,'/.1 4. .1 6 &9.1 7!. There exists a whole body of literature on Rayleigh-B´enard convection (e.g. see [17]). Here we summarize of most well-known correlations, given in the form   Z R S 7 ! , in Table some 2.1.. L. .

(68) Chapter 2. Literature Review. 10. . Most of the known correlations for turbulent convection give values of between     yield  and  . Arguments based on the dimensional reasoning  , and this  relation can be easily derived (e.g. see Bejan [5]). The  -law means that the actual J heat transfer rate  is independent of the layer thickness . 4. . . Based on theoretical analysis of the boundary layer or/and experimental observa tions, Heslot

(69)  [49] identified different types of turbulence, ”soft” and ”hard” turZ   . It was found that bulence, the distinction between the two coming at Z relation in log-log coordinates retains   value only in ”soft” the slope of the 4  L turbulence, and that it decreases to about  in the ”hard” turbulence regime. The  basis of this proposal was an experiment in He at 7 K. Moreover, Chavanne

(70)  [17] Z reported experiments using cryogenic helium gas at low temperatures for 4     rang  at ing up to  at S 7   Z  and found an increase in up to about  very high Rayleigh numbers ( G ). This change was interpreted as the transition to an ”asymptotic regime”. It was noted that the features of the observed new regime matched those of the ultimate regime predicted by Kraichan [63] at moderate S 7 (S 7    ). However, the questions of the existence of a strict power law in the Z relation and the asymptotic state remain open. The effects of the Prandtl L number, 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

(71)   [77] investigated thermal transport over eleven orders of magnitude of the Rayleigh Z     ), using cryogenic helium gas as a working number ( Their    fluid.   data were correlated by a single power law with scaling exponent , and no evidence of any transition was found.. . .  . . . .  . . . . . . . Numerical studies: Among innumerable computational studies of Rayleigh-B´enard convection, it is necessary to quote the work of Deardorff Willis [24], Gr¨otzbach [42], and Kerr Herring [58].. . . Gr¨otzbach [42] performed a direct numerical simulation 2 (DNS) at moderate Rayleigh Z       , the grid of  4  nodes was employed. The numbers. For the highest analysis of the spatial mesh resolution requirements for DNS of the Rayleigh-B´enard convection was presented in [43].. . . . Kerr and Herring [58] investigated by means of DNS the dependence of the Nusselt   Z   and     S 7   number on the Rayleigh and Prandtl number for 4 4? 4  Z   to resolve  in a box. The mesh of  nodes was used for

(72) Z  necessary turbulence scales. The resulting Nusselt number scaled as at S 7     and S 7   , and

(73) Z   at lower S 7 . L. . L. . 2. . . . . . by 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..

(74) 11. Different subgrid-scale (SGS) models in large eddy simulations (LES) for buoyancydriven flows are discussed in [83]. It should be pointed out that computation of turbulent buoyancy-driven flows is very complicated due to the existence of the large range of scales. The presence of a rigid boundary affects the condition of heat flux and stress complicating the matter further. The available Reynolds-averaged Navier-Stokes (RANS) turbulence models (e.g. the K 3 models) are not always satisfactory [47]. On the other two-equation eddy-viscosity hand, DNS or LES cannot yet handle high Rayleigh number turbulent convection due to limitations in computer capacity. However, since developments in computer hardware and numerical algorithms have been outspacing advances in turbulence modeling, the future for DNS and LES looks bright [84].. . 2.3.2 Heat transfer correlations for convection in a volumetrically heated pool Theoretical studies: The earliest theoretical investigation of the onset of convective motion in horizontal fluid layers bounded by isothermal surfaces with a uniformly dis tributed heat source was reported by Sparrow, and Jonsson [101]. The >= - Goldstein ; 8 : + Z , ' $   values of the critical Rayleigh number ?  were calculated as a function  J  4   0  of the lower boundary is of the parameter    . If the),temperature Z >$ decreases greater than that of the upper boundary ( then from its limiting   , the critical Rayleigh   value of as  increases. For the case of  is found to be equal to number Z >$ , defined in terms of the volumetric heat generation, 37 325.. . . . . Roberts [87] solved the marginal stability problem for a horizontal layer of fluid cooled 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 turbulent thermal convection in a horizontally infinite layer of fluid confined between a rigid isothermal upper plate and a rigid adiabatic lower plate, driven by volumetric heat sources. From dimensional considerations, the Nusselt number was found to be proportional to  of the Rayleigh number. No account was made for the effect of Prandtl number. A correlation for the mean Nusselt number was obtained for steady heat trans  Z   G : fer for. . L. .  4  . Z  . . (2.14). A set of differential equations for average temperature distributions, production of therZ    were derived & &  -  and turbulent mal variance transport at various. .

(75) Chapter 2. Literature Review. 12. and solved numerically.. . Bergholz, Chen Cheung [7] examined the basic differences and similarities between the upward heat transfer in bottom-heated and internally-heated fluid layers. It was assumed that far-field effects are negligible on average upward heat transfer, and the interaction between lower and upper boundaries is very small. The modified Nus! selt number defined in terms of near-field parameters: 2 !  !  boundary   :%? <  -   was  L at the upper the implicit length scale and the new temperature scale . The 2 !   4 temperature scale is defined as for bottom-heated layers and   The   length 2 !  "!   J  for internally-heated layers. scale for bottom-heated fluid was defined as half of the layer depth. For internally-heated layers with isothermal  J boundaries at the same temperature, the length scale was defined as   +   + , whereas for internally heated layers with an adiabatic lower boundary, the length scale was defined as total layer depth. Experimental data for Rayleigh-B´enard convection and for an internally-heated fluid was correlated in terms of the modified Nusselt number and the Rayleigh number. The modified Nusselt number was found to have an extremely weak dependence upon the Rayleigh number. The correlated heat transfer results for the two types of heating were similar. However, experimental data for internally heated water layers having insulated lower boundary and a cooled upper boundary showed considerably more scatter than that for Rayleigh-B´enard convection or an internally-heated layer with both boundaries at the same temperature. The authors concluded that the surface heat-transfer coefficient for turbulent convection in horizontal layers depends primarily upon the near-field parameters, regardless of the method of heating. In the turbulent convective regime, the heat transfer characteristics of Rayleigh-B´enard convection and convection with internal heat generation can be derived one from the other ! by defining the appropriate boundary layer Nusselt number .. . . L. Cheung [19] investigated turbulent thermal convection in a horizontally infinite layer of fluid confined between a rigid isothermal upper plate and a rigid adiabatic lower plate, driven by volumetric heat sources. The dependency of the upper surS 7 and Z was obtained. It was indicated that the turface Nusselt number upon bulent structure (the r.m.s. vertical velocity, the variation of mean turbulent temperature, the production of turbulent energy in the core region) in internally heated fluid layers is quantitatively different from that of Rayleigh-B´enard convection. It was Z (about  G ),

(76) Z  S 7   , and found that at sufficiently high  for 7

(77) Z  S 7  S   L Rayleigh numbers, L

(78) Z      )S 7 for %)S 7  Z . At  lower L is found to vary as    .. . . . L. . . . . .

(79). . Recently, Arpaci [3] proposed a dimensionless number +  for buoyant  turbulent flows driven by the internal heat generation. The theory developed, generalizes the model previously proposed by Cheung. The new model yields the Nusselt.

(80) 13. number correlated as. . L. . .   . .  . . Z        S 7 . . .  . . 2. (2.15) . . . Z . . (2.16). Good agreement with experimental data was reported for G . Experimental studies (rectangular cavity). The frst experimental studies of natural convection with volumetric heat sources, motivated by issues related to the post-accident heat removal in liquid-metal-cooled fast breeder reactors, were reported in early 1970s. . . qualitative experiments on convection in a horizontal layer of water S( 7 The

(81)  earliest   ) cooled from above, and internally heated nearly uniformly by electrolytic. . . currents, were carried out by Tritton and Zarraga [106]. Cellular convection patterns Z up to   Z ,$ . Two striking differences from B´enard convection were visualized for Z , in the flow patterns were observed. First, the cell structure was, for moderate hexagonal with motion downwards at the center of each cell; second, the horizontal Z . It was observed that the scale of the convection pattern grew larger with increase of Z Z , $ flow remains non-turbulent to values  high compared with the corresponding ratio for the B´enard configuration.. .  . . The same problem was experimentally investigated by Fiedler and Wille [36], Kulacki and Nagle [67] and Kulacki and Emara [64]. The heat transfer correlations for volumetrically heated rectangular cavities with the top boundary cooled are summarized in Tables 2.2. Table 2.2: Correlations for internally-heated horizontal fluid layers with an adiabatic J  lower wall and an isothermal upper boundary, with being the height and being the length of the fluid layer. References F  W(1970) [36] K  N(1975) [67] K  E(1977) [64]. .  ::%   884  $,-(   8 ;  55(   88!.

(82)  & *  $'&)  $? (0&) .  : /-.1 2 &9.1; 5 /-.1 2 & ,0/.1;  5 (0/.1  &  :  50/.1 78. .  $-& 1.65  %,?& 0.25  %$,?& 0.5. The first heat transfer measurements for turbulent thermal convection in a fluid bounded by two rigid isothermal planes of equal temperature were reported by Kulacki and Goldstein [65]. In their study, the volumetric energy source was provided by Joule 4    heating. The fluid height was varied from  to   cm, and the length and depth of the channel were both 25.4 cm. Horizontally-averaged temperature profiles were determined optically. Heat transfer correlations for the upper and lower boundaries were obtained from 67 observations. From the empirical correlation, the critical Rayleigh.

(83) Chapter 2. Literature Review. 14. Z >$'. . number Z to occur at. .  . .  . . .  4 . was deduced. The transition to turbulence was observed. .. Experimental studies in a similar configuration were reported by Mayinger [74], Jahn and Reineke [54], Kulacki and Emara [64]. Table 2.3 summarizes the heat transfer correlations obtained from these investigations. Table 2.3: Correlations for internally heated horizontal fluid layers with isothermal lower and upper boundaries. K J S K. References G(1972) [65] R(1974) [54] R(1978) [103] E(1980) [33]. .   5%$-+  86  *@(   8  5-:+,-   8  5(%,-   8. .

(84) . :%5%   ; # $? . :%    . 5(    ; 2 . .15.1    ;. : /-.1-# .1 6 .1 7 .1 2.  & $0/.1! &9.1; &9.1  # &9.1 4.

(85)  <E < * > < * > < * >. Configurations with different temperatures at the upper and lower surfaces were experimentally investigated by Suo-Antilla and Catton [102] and analytically by Baker,  Faw, and Kulacki [4]. Details of the correlation of Baker

(86)  are given in Paper 1.. . The review of other work relevant to natural convection in plane layers and cavities can be found in Ref.[69]. Similarity between convective heat transfer in fluid layers heated from below and heated from within. It is interesting that the power exponent heat transfer correlations for rectangular internally heated pools can be derived from the correlations for Rayleigh-B´enard convection. In the following, let us consider heat transfer in an J internally-heated fluid layer of depth cooled from above and insulated from below 2 with maximum temperature difference and Rayleigh-B´enard convection in a layer 4%J 4 2 with the driving temperature difference . The internal Rayleigh numof depth Z Z  and the can be rewritten as the product of the external Rayleigh ber number +  Z  Z   < - . Then, considering a heat Nusselt (or Dammk¨ohler) number :

(87) Z R  L  for internally heated layer, and assuming that a cortransfer correlation

(88) Z R L relation for Rayleigh-B´enard convection can be applied, we obtain the  L   following relation between the exponents and :. . . . . . . . . .  .  . (2.17). . Taking  , we will get a value for the exponent  , which is close to values obtained from the experiments (Table 2.2) and equal to the theoretical value derived by Cheung. A similar approach is presented in Refs. [10] and [9] reported recently.. . . . .

(89) 15. R φ. L. R. L. a). b) W. L. c) Figure 2.2: a) Spherical pool b) Semicircular pool c) Two-Dimensional Slice Geometry of a Torospherical Bottom of a VVER-440 Reactor. Circular and spherical cavities. Mayinger et al. [74] were the first to report experimental and computational results for thermal convection in volumetrically-heated semicircular pools. In the last decade, a number of research programs have appeared world-wide: USA (UCLA facility, [1], ACOPO at UCSB, [108]), Russia (OECD RASPLAV project, [2]), France (BALI, [11]), Finland (COPO, [70]), and Sweden (FOREVER, [94], SIMECO, [93]). The main objective of these tests is to obtain the heat transfer characteristics applicable to prototypical configurations and simulants. The heat transfer correlations for different geometries (Figs. 2.2) are summarized in Tables 2.4-2.5 below. Table 2.4: Correlations for internally-heated semicircular and spherical pools. no. 1 2 3 4 5 6 7 8 9. Reference Mayinger et al. (1975) [74] Jahn  Reineke (1974) [54] Gabor et al. (1980) [38] Asfia et al. (1996) [1] mini-ACOPO (1995) [111] ACOPO (1995/97) [108] ACOPO (1997) [108] COPO (1994) [70] BALI (1998) [92]. . . .

(90) .  5   8.  ,@:%   74 @   82     8  ,,-   72 @ "   ,@:%   8 @   82.  5-:+,-   8 . %,-   74.  5   88  ,@:%   74 @   86  ..1   82 @   8.

(91)

(92)  . ,0/-.1  .

(93). . .1 ! &9.1  $0/-.1   &.

(94) . $0/.1 .

(95). $'/.1   &9.1  # .1 78 &.  $-   "

(96)    "4#  5-:+,-   8.  .1 ! &. *'/.1 72. .1 7 &9.1 76  <. .1 76. . .1 72. .1 7 &9.1 7!. 4. . . 4.

(97). .  .   .

(98) Chapter 2. Literature Review. 16. Table 2.5: Correlations for internally-heated semicircular and spherical pools (contd). no. 1 2 3 4. Pool configuration semicirc. semicirc. hemispher. hemispher.. Method of heating. Scale. Joule heat. microwave. 1/8. W   @  Water/Freon-113. 5 6 7 8. hemispher. hemispher. hemispher. torospher.. trans. cool-down trans. cool-down trans. cool-down Joule heating. 1/8 1/2 1/2 1/2. Water/Freon-113 Water Water Water. 9. 2D slice semicirc.. Joule heat.. 1/1. Water. 1/2 . . Working fluid Water Water. Boundary conditions cooled rigid surf. free surf. free surf. insul. rigid surf. cooled rigid surf. cooled rigid surf. cooled rigid surf. cooled rigid surf. cooled rigid surface. Numerical studies. The earliest two-dimensional (2D) simulations of thermal convection in rectangular enclosures with internal heat generation were performed in the early 1970s by Thirlby [104] Peckover and Hutchinson [82] and Jahn and Reineke  [54]. Mayinger

(99)   [74] predicted natural convection heat transfer in a hemispherical cavity.. . Emara and Kulacki [33] performed simulations for Prandtl numbers ranging from     on a grid of . The boundary con0.05 to 20 and Rayleigh numbers from  to ditions for the cooled upper surface were either those of zero slip (a rigid boundary) or zero shear (free boundary). No turbulence model was employed. A zero-shear upper boundary resulted in larger average Nusselt numbers. The computed L and horizontally averaged temperature profiles agreed fairly well with the experimental data S 7 was found. It was reaof Kulacki and Emara [64]. Weak dependency of on L soned that the average heat transfer coefficient is relatively insensitive to the details of the turbulent flow field, and 2D simulations give reasonably accurate solutions at the moderate Rayleigh numbers considered.. K . 3 model was applied by Steinberner and A low-Reynolds-number two-equation Reineke [103] to numerical treatment of turbulent convection in a cavity with vertical cooled walls, and the upper cooled wall by Farouk [35].. . Z . . . S 7 .  . The spatial grids with Gr¨otzbach [44] applied DNS for up to and  4  up to 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..

(100) 17. Counter-gradient heat fluxes were observed in the core of the flow. The predicted statistical turbulence data showed that first order models, like the eddy conductivity and K 3 models, cannot be the proper tools to model turbulent heat flux.. . In a recent work by Gr¨otzbach and W¨orner [45], DNS and LES applications in nuclear engineering were reviewed. The role of DNS as a tool to provide data for developing statistical turbulence models, and for models and codes validation was disS 7   , in the fully turbulent cussed. The results of DNS for the model fluid water, Z   were presented. The number of mesh cells were   ? . It was regime at also stressed that the eddy conductivity concept is not adequate for the thermal energy equation, and development and applications of LES or second-order models should be pursued. It was argued, however, that ”the progress of LES towards engineering applications would strongly depend on the availability of adequate SGS models and boundary conditions”, as well as that ”real retardation comes mainly from the numerics”.. . . . Dinh and Nourgaliev [28] carried out an extensive numerical study for different geZ (up to   ). The available turbulence models were critometries and wide range of ically reviewed. The finite-difference two-dimensional code (named NARAL) based on the SIMPLE solution procedure and the commercial CFX-4.1 software package were employed. First, the Navier-Stokes equations were solved in 2D without any turbulence model. The resulted averaged Nusselt numbers compared well with Jahn and Reineke’s Z up to   G . Good [54] experimental correlation for a semicircular cavity with Z   G . agreement with experimental data was also reported for square cavities at Secondly, the performance of a number of turbulence models was examined in application 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 and Z    , the uppwards heat flux was consistently Reineke’s [103] correlations. For   underpredicted (by about ). The analysis showed the deficiency of existing firstorder turbulence models in application to unstably stratified buoyancy-driven flows. Developing the ideas of Gibson and Launder [41], corrections for the near boundary S 7 were proposed. The expresturbulent viscosity  and turbulent Prandtl number 7 S  )   Z  2GF +   )   Z.   G 2. F + sions for  and   were obtained. Here, Z  is the flux Richardson number which is a ratio of the rate of turbulent energy removal by buoyant forces to the rate of turbulent energy creation by mean shear. The above mentioned corrections for turbulent transport coefficients along with wall damping functions were K 3 model. Reasonable agreement with COPO, and Steinberner and added to the Z    . Reineke’s experimental data was achieved with these modifications for. . . . . . . . Nourgaliev and Dinh [78] conducted three-dimensional (3D) computations for an internally-heated fluid contained in a channel with an isothermal upper boundary and an adiabatic lower boundary. The Navier-Stokes equations were solved directly (without.

(101) Chapter 2. Literature Review. 18. . 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 and Z up to  Emara [64] for the averaged heat transfer coefficient was achieved for      Ra    , and S 7     and data was obtained for S 7   . Turbulent . The average temperature distribution in the core region, as well as turbulent heat fluxes, compared well with those predicted by Cheung’s analytical model [18]. S 7 were similar, a strong dependency of Whereas thermal fields obtained for different 7 S turbulence parameters on was reported. The analysis of the Reynolds stresses and turbulent heat fluxes revealed significant anisotropy of turbulent transport properties. It was concluded that the isotropic eddy diffusivity approach cannot be used to describe turbulent convection in unstably stratified flows. The 3D calculations were compared   with 2D predictions on a grid of . It was shown that a 2D approach provides Z reasonably accurate solutions, in terms of average heat transfer characteristics, for  Z >$ up to .. . . . S 7. Nourgaliev, Dinh and Sehgal [79] presented a CFD analysis of the effect of on thermal convection. The computations were performed for 2D square, semicircular and elliptical geometries, and for 3D semicircular and hemispherical enclosures with isothermal boundaries. The Prandtl number was varied from 0.2 to 7, and the Rayleigh    numbers up to  were considered. The grid sizes were for square, 3200 nodes for elliptical and 3988 for semicircular 2D domains. For 3D calculations, a mesh of 44   (semicircular) and 70464 (hemispherical) nodes were employed. The effect 7 S of on the average value of upper cooled boundaries was found to be small. S 7 , a significant increaseL inat the At low local was predicted at the bottom boundary of L semicircular, elliptical and hemispherical cavities, and near the lower corners of square cavities. This drastic change in local was attributed to more vigorous, in the case L of 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 numerical studies 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 much studied single-layer problem. Most of interest in a multiple-layer convection has been inspired by problems of the convection of the Earth’s mantle and the encapsulated crystal growth..

(102) 19. Early work on the linear stability problem for the two-layer B´enard problem was reported by Zeren and Reynolds [119] and Ruehle [89]. An extensive listing of other studies can be found in Joseph and Renardy [56]. Most of the studies are focused on theoretical predictions of the critical states (when convection rolls form) and are not pertinent to the present work. Some numerical and experimental work have been reported on viscous and thermal coupling between the layers at the onset of convection [99].. .  [100] performed natural convection experiments in a rectangular Sparrow

(103)  enclosure (aspect ratio of unity) containing either a single fluid or two immiscible fluids 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 the   experiments were distilled water and research grade (99 pure) hexadecane paraffin, with respective Prandtl numbers of 5 and 39.2. Single-fluid experiments were also carried out, and it was found that the reduced two-fluid heat transfer data agreed with the 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 the nondimensional turbulent heat flux. The model was validated for water/water and silicon oil/water systems heated from below and separated from each other by a thin aluminum 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 defined 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 the individual layers can be correlated as.  >= -   -   ; 8 : Z  . where 0 ? L.  . . L.   Z  ! S 7 R 2. (2.19). , the following expression is obtained:. !    + ! , ,    !    !   +  ,  ! !   !  +    ! 2 , / + / +       . where for each layer. . ,.   J     !  S 7 R . . (2.20). ! . (2.21).

(104) Chapter 2. Literature Review. 20. . The term inside the braces in Eq.2.20 is called the ”modified interfacial Rayleigh numZ R  !  . The constants were determined from the experiments for turbulent ber”, 9       2 P       , natural convection in a single fluid layer and taken equal to T    and . The correlation of the nondimensional heat flux through a densitystratified interface becomes then. L. .    . %Z  R    ! .  . .  . (2.22). It should be noted that the value of the coefficient in Haberstroh and Reinders’ study   is 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 the convection in two immiscible fluids heated from below. The numerical method was based on the vorticity-stream function formulation. The effect of fluid property ratios 67  3 - 2 J  6 on the onset of instabilities was investigated for Grashof numbers      between and .. . . Recently, Prakash and Koster [85] performed an experimental and numerical study of two-dimensional convection in a system of two immiscible liquids heated from below. The objective of the study was to understand the coupling physics at a liquid-liquid interface at onset of convection. The effect of the ratio of the fluid depths on different coupling modes between the layers was considered. It was found that when buoyancy forces 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. Mechanical (viscous) coupling, identified by counterrotating rolls, dominates when buoyancy forces are very different in both layers. . Z   The   numerical work reported in literature has been performed for very low (.

(105). ) 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 internal heat sources. Fieg [34] was, perhaps, the first to investigate the natural convection characteristics of two stratified immiscible liquid layers with the lower one internally heated. The temperature was maintained equal at the top and bottom boundaries. Heptane and water were used as lighter and heavier liquids, respectively. The important conclusion was that the two layers behaved as if separated by a rigid highly conductive wall. The correlation for the B´enard problem in the upper layer and the one for the layer bounded by isothermal walls at different temperatures and volumetric heating were applied and the calculated values agreed with the experimental data to within   10 accuracy. Temperature profile measured by Fieg seemed to support this assumption. However, experimental data obtained by Fieg for a double-layer configuration is.

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