COMPUTATIONAL FLUID DYNAMICS ANALYSIS OF AEROSOL DEPOSITION IN PEBBLE BEDS

Transcription

1 COMPUTATIONAL FLUID DYNAMICS ANALYSIS OF AEROSOL DEPOSITION IN PEBBLE BEDS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Margaret Msongi Mkhosi, M.S. ***** The Ohio State University 2007 Dissertation Committee: Dr. Richard Denning, Advisor Dr. Shoichiro Nakamura Dr. Audeen Fentiman Dr. Xiangdong Sun Approved by Advisor Nuclear Engineering Graduate Program

2 ABSTRACT The Pebble Bed Modular Reactor is a high temperature gas cooled reactor which uses helium gas as a coolant. The reactor uses spherical graphite pebbles as fuel. The fuel design is inherently resistant to the release of the radioactive material up to high temperatures; therefore, the plant can withstand a broad spectrum of accidents with limited release of radionuclides to the environment. Despite safety features of the concepts, these reactors still contain large inventories of radioactive materials. The transport of most of the radioactive materials in an accident occurs in the form of aerosol particles. In this dissertation, the limits of applicability of existing computational fluid dynamics code FLUENT to the prediction of aerosol transport have been explored. The code was run using the Reynolds Averaged Navier-Stokes turbulence models to determine the effects of different turbulence models on the prediction of aerosol particle deposition. Analyses were performed for up to three unit cells in the orthorhombic configuration. ii

3 For low flow conditions representing natural circulation driven flow, the laminar flow model was used and the results were compared with existing experimental data for packed beds. The results compares well with experimental data in the low flow regime.for conditions corresponding to normal operating of the reactor, analyses were performed using the standard k-ε turbulence model. From the inertial deposition results, a correlation that can be used to estimate the deposition of aerosol particles within pebble beds given inlet flow conditions has been developed. These results were converted into a dimensionless form as a function of a modified Stokes number. Based on results obtained in the laminar regime and for individual pebbles, the correlation developed for the inertial impaction component of deposition is believed to be credible. The form of the correlation developed also allows these results to be applied to pebble beds of different porosities. The effect of turbulence on the deposition of aerosols was analyzed using the discrete random walk model. The results obtained with k-ε turbulence model show high deposition of aerosols at low particle diameters. To validate the results in this regime, detailed experimental work is needed. iii

4 DEDICATION Dedicated to my Family and Friends iv

5 ACKNOWLEDGMENTS I would like to thank my husband, Keaobaka, for his wonderful support and encouragement throughout my studies and during the preparation of this dissertation. I also acknowledge the patience he and my children, Reratile and Ratanang exercised, especially during those times when I had to work endless hours and miss being with the family when they needed me. Special appreciations to my parents and siblings who believed in me and made me believe in myself. I would also like to thank all my friends for the support and help they offered when I really needed it. I wish to thank my supervisor and academic advisor, Dr Richard Denning, for his guidance throughout my studies here at The Ohio State University, as well as his advice and intellectual support during the preparation, and review of this dissertation. I would like to thank Prof. S. Nakamura for his guidance in the consideration of computational fluid dynamics methods, and for carefully reviewing contents of the dissertation. v

6 I extend my thanks to Dr Audeen Fentiman who has proved to be a wonderful mentor and advisor throughout my studies at Ohio State. I also thank her for taking time off her busy schedule in order to be part of my dissertation committee. I also thank Dr Xiangdong Sun for being a member of my candidacy and dissertation committee. I wish to thank Nuclear Engineering faculty members at Ohio State University for their encouragement and the support they provided. I would like to thank Dr Alayne Parson and Patrice Alan for their support and encouragement, as well as for being wonderful coworkers. and assistance. It is a great pleasure to express thanks to Vijayalakshmi Krishnan for her support I thank all the sponsors and staff members of the Tertiary Education Linkage Program (USAID/UNCF) for providing initial funds for my studies. This research was carried out using the Ohio Supercomputer Center computing resources at The Ohio State University. I would like to thank the staff of OSC for the support they offered. vi

7 VITA November 19, 1970 Born - Boschpoort, South Africa 1995 B Sc (Honors) Physics, University of North-West, South Africa 2000 M Sc Physics, University of North-West, South Africa 2003 M.S. The Ohio State University Science Teacher Physics Lecturer, University of North-West Graduate Research Associate, The Ohio State University PUBLICATIONS 1. M. M. Mkhosi, R. Denning and S. Nakamura, The Application of Advanced Computational Methods to the Modeling of PBMR Source Term, ICONE12, Arlington, VA, April 25-29, R. Denning, M. Mkhosi, S. Nakamura, R. Christensen and V. Kogan, Aerosol Transport in PBMR Accidents, ICAPP 04, Pittsbhurg, PA, June 13-17, FIELDS OF STUDY Major Field: Nuclear Engineering vii

8 TABLE OF CONTENTS ABSTRACT... ii DEDICATION... iv ACKNOWLEDGMENTS... v VITA... vii LIST OF FIGURES...xiii LIST OF ABBREVIATIONS AND SYMBOLS... xv 1. OVERVIEW Statement of the Problem PBMR Design Characteristics Radionuclide Release and Transport Objectives Technical Approach Scope of Dissertation AEROSOL TRANSPORT BEHAVIOR AND DEPOSITION MECHANISM IN PEBBLE BEDS Aerosol Transport Modeling Aerosol Trajectory Analysis Saffman Lift Force Dispersion of Aerosols in Turbulent Flows Treatment of Aerosol Characteristics Aerodynamic Shape Factor Aerodynamic Diameter Aerosol Growth Mechanisms Vapor Deposition Aerosol Agglomeration Aerosol Transport and Deposition Mechanisms Brownian Diffusion viii

9 2.4.2 Thermophoresis Gravitational Settling Inertial Impaction Particle Bounce Turbulence-Enhanced Inertial Deposition Re-entrainment Electrical Charge Summary FLUID FLOW BEHAVIOR Fluid Flow Modeling: Navier-Stokes Equations Nature of Fluid Flows Laminar Flows Turbulent Flows Turbulent Kinetic Energy Large and Small Scales of Turbulence The Turbulence Intensity Scale Turbulence Modeling Approaches Direct Numerical Simulation Reynolds Averaged Navier-Stokes Equations Standard k-ε Model Standard k-ω Model Reynolds Stress Model (RSM) Large Eddy Simulation Near-Wall Modeling for Turbulent Flows Standard Wall functions Approach Enhanced Wall Treatments TURBULENT FLUID FLOW AND AEROSOL DEPOSITION IN A PIPE Problem Description Modeling Procedure The Geometrical Modeling for CFD Fluid Flow Modeling Aerosol Deposition Model Results Pressure Drop in a Pipe Aerosol Deposition Assessment FLUID FLOW AND AEROSOL DEPOSITION ON A SINGLE AND LINEAR ARRAYS OF SPHERES Problem Description FLUENT Modeling Procedure: Single Sphere ix

10 5.2.1 The Geometrical Model Fluid Flow Modeling Aerosol Deposition Model Results: Single Sphere Fluid Flow Model Velocity Flow Field Turbulence Intensity Reynolds Stresses Pressure Field Aerosol Deposition Efficiency on a Single Sphere Turbulent Deposition of Aerosols Effect of Lift Force on Particle Deposition Aerosol Deposition on a Linear Array of Spheres CFD Model: Linear Spheres Fluid Flow Modeling Aerosol Deposition Model Results: Fluid Flow Results: Aerosol Deposition Assessment Discussion INERTIAL DEPOSITION OF AEROSOLS IN GRANULAR BEDS Overview of Flow in Granular Beds Aerosol Collection Efficiency Fluid Flow Models in Granular Beds Fluid Flow Models in a Dense Cubic Bed Pressure Drop in Packed Beds of Spheres Dimensionless Aerosol Trajectory Equation Description of Experimental System Computational Model Geometric Model The Velocity Field Results Comparison with Experiments Assessment of the Results INERTIAL DEPOSITION OF AEROSOLS IN PEBBLE BEDS Pebble Bed Characteristics Orthorhombic Array Cubic Array Body Centered Cubic Computational Model Model Geometries Fluid Flow Modeling Aerosol Deposition Model Results x

11 7.3.1 Low Reynolds Number Regime High Reynolds Number Regime SUMMARY AND CONCLUSIONS Comparison of CFD Analysis with Existing Experiments Aerosol Deposition in Smooth Pipes Aerosol Deposition for Single and Linear Arrays of Spheres Aerosol Deposition in Pebble Beds Aerosol Deposition in Laminar Flow Regime Aerosol Deposition in Turbulent Flow Regime Lessons Learned from the CFD Analysis Future Work APPENDIX A JOURNAL FILE FOR THE PIPE APPENDIX B JOURNAL FILE FOR SINGLE SPHERE APPENDIX C JOURNAL FILE FOR LINEAR SPHERES APPENDIX D JOURNAL FILE FOR PEBBLE BEDS (D S = CM) APPENDIX E JOURNAL FILE FOR PEBBLE BEDS (D S = 6 CM) APPENDIX F JOURNAL FILE FOR PEBBLE BEDS (D S = 6 CM, VOID FRACTION = 0.35). 197 xi

12 LIST OF TABLES Tables Page Table 4.1: Pressure Drop Per Unit Length: Re = 10, Table 4.2: Pressure Drop per Unit Length: Re = 50, Table 5.1: Models Analyzed for a Single Sphere Table 5.2: Drag Coefficient on a Sphere Table 6.1: Mesh Size for a 4-Unit System Table 7.1: Unit Cell Flow Parameters Table 7.2: Helium Gas Properties Table 7.3 : Reynolds Numbers for Laminar Regime Table 7.4: Nodalization for the Different Unit Cells in Laminar Flow Table 7.5: Cases Analyzed for the High Reynolds Number Flow Table 7.6: Pressure Drop Results across a 3-Unit Cell at Different Velocities and Low Density Table 7.7: Nodalization for the Different Unit Cells xii

13 LIST OF FIGURES Figure Page Figure 3.1: The Subdivision of Near Wall Region [5] Figure 4.1 : Experimental Apparatus of Liu and Agarwal Figure 4.2: Geometrical Model Figure 4.3: Fluid Flow Vectors Figure 4.4: Velocity Profile for Pipe at Re = 10, Figure 4.5: Velocity Profile for Pipe at Re = 50, Figure 4.6: Experimental Data of Particle deposition from Turbulent Flows [38] Figure 4.7: Comparison between CFD and Experimental Results at Re = 10, Figure 4.8: Comparison between CFD and Experimental Results at Re = 50, Figure 5.1: Experimental Setup for Deposition Efficiency at Low Stokes Numbers Figure 5.2: Experimental set-up for Deposition Efficiency at High Stokes Numbers Figure 5.3: Axisymmetric Grid and Boundary Conditions Figure 5.4: Flow Past a Sphere at (a) Re = and (b) Re = (Flow visualizations by H. Werlé, courtesy of J. Delery of ONERA) Figure 5.5: Typical Velocity Vectors behind the Sphere using RSM: Re = 5, Figure 5.6: Velocity Vectors for (a) k-ε (b) k-ω turbulence models Figure 5.7: Contours of Turbulence Intensity for the RSM Figure 5.8: Contours of Turbulence Intensity for (a) k-ε and (b) k-ω turbulence models Figure 5.9: Contours of Reynolds stress for RSM (a) UU stress (b) UV stress Figure 5.10: Contours of Static Pressure using RSM Figure 5.11: Contours of Static Pressure using (a) k-ε and (b) k-ω Turbulence Models.. 85 Figure 5.12: Pressure Coefficient as a function of Position for a smooth sphere Figure 5.13: Single Sphere Collection Efficiency for various RANS Turbulence Models Figure 5.14: Collection Efficiency on a Single Sphere: FLUENT vs. Experimental Data Figure 5.15: Collection Efficiency on a Single Sphere: Stochastic Tracking Figure 5.16: Collection Efficiency on a Single Sphere Figure 5.17: Collection Efficiency on a Single Sphere: RSM with Saffman Lift Force. 93 Figure 5.18: Linear Array of Spheres Figure 5.19: Mesh for Linear Arrangement with L/D s = Figure 5.20: Wake Region between Spheres (a) L/D s = 2 and (b) L/D s = Figure 5.21: Collection Efficiency on Linear Array: N Stk = 1.2 and L/D s = Figure 5.22: Collection Efficiency on Linear Array: N Stk = 1.2 and L/D s = xiii

14 Figure 5.23: Collection Efficiency on Linear Array: L/D s = 2 and Figure 6.1 : Schematic Representation of a dense Cubic Array of Spheres Figure 6.2 : Schematic of Experimental Setup of Gal et al Figure 6.3 : Three-dimensional view of the Dense Cubic Unitcell Figure 6.4 : Efficiency as a Function of Reynolds Number Figure 6.5 : Single Sphere Efficiency as a Function of Modified Stokes Number Figure 7.1 : Orthorhombic Packing of Pebbles Figure 7.2 : Simple Cubic Packing of Pebbles Figure 7.3 : Body Centered Cubic Array of Pebbles Figure 7.4 : Orthorhombic Unit Cell Figure 7.5 : View of the Computational Domain: 1unit cell Figure 7.6 : Contours of Velocity Magnitude in Laminar Flow for Case Figure 7.7 : Contours of Pressure for Laminar Flow for Case Figure 7.8 : Aerosol Particle Source Points at the Inlet Figure 7.9 : Effect of Nodalization on Aerosol Deposition for a 2-UnitCell for Case Figure 7.10 : Size-Dependent Deposition as a Function of Unit-cell Size Figure 7.11 : Asymptotic Deposition Fraction vs. Particle Diameter for Case Figure 7.12 : Deposition as a Function of Stokes Number for Various Densities Figure 7.13 : Deposition as a Function of Stokes Number for Various Velocities Figure 7.14 : Asymptotic Deposition as a Function of Stokes Number for Different Porosities Figure 7.15 : Deposition as a Function of Modified Number for a 3Unitcell Figure 7.16 : Comparison of CFD Results with Experimental Data of Gal et al Figure 7.17 : Contours of Velocity for a 3-UnitCell for High Reynolds Number Case Figure 7.18 : Contours of Turbulent Kinetic Energy Figure 7.19 : Contours of Pressure for Case 3 for a 3UnitCell Figure 7.20 : The Pressure Drop vs. Velocity for a 3-unit cell for Cases 2, 3 and Figure 7.21 : Effect of Nodalization on Aerosol Deposition for a 2-UnitCell for Case Figure 7.22 : Size-Dependent Deposition as a Function of Unit-cell Size Figure 7.23 : Asymptotic Deposition Fraction vs. Particle Diameter for Case Figure 7.24 : Deposition as Function of Stokes Number for Various Densities Figure 7.25 : Deposition as Function of Stokes Number for Various Velocities Figure 7.26 : Aerosol Collection Efficiency as a Function of N St, for a 3-Unit Cell Figure 7.27 : Aerosol Deposition for Stochastic and Non-stochastic Tracking Figure 7.28 : Eddy Enhanced Deposition in Unit Cell based on Liu and Agarwal xiv

15 LIST OF ABBREVIATIONS AND SYMBOLS ABBREVIATIONS AVR CFD DNS DPM DRW FHSH HTR HTGR LES PBMR RANS RNG RSM Arbeitsgemeinschaft Versuchsreactor Computational fluid dynamics Direct numerical simulation Discrete phase model Discrete random walk Fuel Handling & Storage System High temperature reactor High temperature gas-cooled reactor Large eddy simulation Pebble Bed Modular Reactor Reynolds Averaged Navier-Stokes Renormalization Group Reynolds stress model xv

16 NOMENCLATURE Latin Characters A [---] Hammaker constant A p [m 2 ] Projected area of the spherical particle A s [m 2 ] Surface area of control volume C [#/m 3 ] Concentration of aerosol particles in volume C1 ε, C2ε, Cμ [---] Constants in k-ε C c [---] Cunningham correction factor C D [---] Drag coefficient C μ [---] Empirical constant d a [m] Aerodynamic diameter d e [m] Equivalent volume diameter d p [μm] Diameter of an aerosol particle D [cm 2 /sec] Diffusion coefficient D id.. [m] Inner pipe diameter D s [m] Diameter of a sphere e [---] coefficient of restitution E n [---] Unit cell efficiency E( κ ) [J] [J] Energy spectrum function * E [---] Single sphere efficiency f [---] Friction factor f p [---] Friction factor for the bed F D [N] Drag Force F E [N] Electrostatic force F n [N] Form drag F τ [N] Viscous drag F [N/kg] Force per unit mass acting on an aerosol particle x g [g/s 2 ] Acceleration due to gravity G [---] Filter function in LES h [---] Adhesion fraction H [m] Height of a unit cell J [#/m 2 s] Number of particles deposited per area per time J ijk [---] Transport of Reynolds stresses by diffusion J x [# particles/m 2 ] Particle flux k [m 2 /s 2 ] Turbulent kinetic energy xvi

17 k B [m 2 kg/s 2 K] Boltzmann s constant k c [Nm/C 2 ] Coulomb force constant K [---] Thermophoretic coefficient th K C [---] Collision kernel between aerosol particles K [---] Coagulation coefficient 0 K th [---] Thermophoretic coefficient k r [---] Local velocity gradient L 0.51m [m] Length of the deposition section that is 0.51 m long n [---] Total number of aerosol particles n in [---] Average particle concentration at the inlet n [---] Number of unit cells l n out [---] Average particle concentration at the outlet ˆn [---] Normal unit vector N [---] Number of aerosol particles deposited on a surface deposited N total [---] Total number of upstream particle concentration N St [---] Stokes number N St [---] Modified Stokes number P [Pa] Pressure P 0.51m [----] Fraction penetrating the deposition section P [---] Rate of production of Reynolds stresses ij P L [---] Fraction penetrating a pipe section Q [m 3 /s] Volumetric flow rate r [m] Position vector r [m] Particle radius p R [#/s} Aerosol particle removal rate Re [---] Reynolds number Re p [---] Reynolds number for a particle Svmt (,,) [---] Extrinsic source of aerosol particles ˆt [---] Tangential unit vector T [K] Temperature T [s] Eddy lifetime e T [---] Aerosol particle transmission fraction f T [---] Turbulence intensity u [m/s] Average fluid velocity in the x-direction u [m/s] Fluid velocity xvii

18 u () t [m/s] Fluid fluctuating velocity in the x-direction u i [m/s] Fluid velocity in the i-th direction u [m/s] Aerosol particle velocity p u τ [m/s] Fluid friction velocity u + [---] Dimensionless velocity V ave [m/s] Bulk average velocity U [m/s] Free stream mean-flow velocity e U z [m/s] Average fluid velocity in the z-direction V 0 [m/s] Superficial velocity V d [m/s] Deposition velocity V [m/s] Terminal settling velocity s V + d [---] Dimensionless particle deposition velocity v [m/s] Instantaneous fluid velocity in the y-direction v () t [m/s] Fluid fluctuating velocity in the y-direction v [m/s] Average fluid velocity in the y-direction x i [m] Coordinate in the ith direction V th [m/s] Thermophoretic velocity w [m/s] Instantaneous fluid velocity in the z-direction w () t [m/s] Fluid fluctuating velocity in the z-direction w [m/s] Average fluid velocity in the z-direction x [m] Separation distance between the charges c y + [---] Dimensionless distance from the wall Greek Symbols α [---] Porosity *, α α [---] Constants in k-ω turbulence model ω * ω β [---] Constant in k-ω turbulence model δ ij [---] Kronecker delta ε [m 2 /s 3 ] Dissipation rate ε [---] Rate of dissipation of the Reynolds stresses ij Φ ij [---] Reynolds stress transport by turbulence interactions ω [---] Specific dissipation rate χ [---] Dynamic shape factor μ [Pa s] Fluid viscosity μ t [Pa s] Turbulent viscosity Δ P [Pa] Pressure drop xviii

19 T [K/m] Temperature gradient ρ 0 [g/cm 3 ] Unit density ρ [kg/m 3 ] Density of fluid f ρ [kg/m 3 ] Density of an aerosol particle p σ, σ [---] Constants in k-ε k ε ν [m 2 /s] Kinematic viscosity τ [Pa] Viscous stresses τ t [s] Kolmogorov scale of time τ [Pa] Viscous stress tensor ij τ w [Pa] Wall shear stress τ [s] Particle relaxation time p τ + p [---] Dimensionless particle relaxation time η [---] Total bed collection efficiency η [m] Kolgomorov scale of length l υ [m/s] Kolgomorov scale of velocity ω [] Specific dissipation rate xix

20 CHAPTER 1 1. OVERVIEW The Pebble Bed Modular Reactor (PBMR) is an advanced high-temperature gascooled, and graphite moderated reactor. The fuel elements are spherical graphite pebbles of about 6 cm in diameter. The pebbles contain microspheres of uranium dioxide. Helium gas, which is both chemically and radiologically inert, flows over the pebbles and through the gaps between the pebbles and acts as a coolant. The graphite fuel has high thermal conductivity and high heat capacity. Because of that, the plant can withstand a broad spectrum of accidents without the need for operation of active safety systems and with very limited release of radionuclides to the environment [1]. The PBMR concept is being developed for either high-efficiency production of electricity or process heat applications, such as the production of hydrogen. The PBMR design relies on the excellent heat transfer properties of graphite and a fuel design that is inherently resistant to the release of radioactive material up to high temperatures. The PBMR designers believe that sufficient retention of radioactive material would occur under all credible accident conditions such that a leak-tight reactor containment structure is not required [1]. Despite the safety features of the concepts, 1

21 these reactors still contain large inventories of radioactive materials. The safety of the reactor must be proved if a new generation of reactors is to be accepted by a safetyconscious public. The computer codes, which are used to predict the progression of severe accidents and the release of radioactive materials to the environment (e.g., MELCOR [2], MAAP [3] and SCDAP/RELAP [4] for the current generation of light water reactors, required years of developmental work and experimentation. Although the models in these codes have evolved in complexity over time, they rely on major simplified model assumptions. The fluid flow models are based on the use of control volumes and the radionuclide deposition models are based on deposition velocities. Thus, the physical processes can only be crudely estimated. However, over the past twenty years major improvements have been made in computational abilities. The severe accident behavior models that will be used to examine the safety characteristics of future generations of reactors will be based on the first principles. Although there will still be a need for experimental validation of these models, because the technical bases of the models will be sounder, the extent of validation required should be reduced. The objective of this dissertation is to examine the effects of different computational models on the prediction of the deposition of aerosol particles in a pebble bed, in case of a severe accident. The analyses were performed using the computational fluid dynamics (CFD) computer code FLUENT [5] to describe the details of fluid flow through the pebble bed. FLUENT has the capability of predicting three dimensional flow velocities and the forces that arise between the fluid and the entrained aerosol particles. 2

22 CFD codes are now being used in a wide variety of multi-dimensional fluid flow problems. However, the potential for misuse or over-confidence in the results of CFD analysis is very great, as is the case with many computer models. This dissertation explores the limits of applicability of existing CFD models in FLUENT to the prediction of aerosol transport and deposition. Two flow conditions for analysis were selected for : normal operating conditions of the PBMR [1] and low flow conditions in the later stages of a loss of coolant accident. The CFD models are run using the Reynolds Averaged Navier-Stokes (RANS) turbulence models to determine the impact of the turbulence model on the predicted deposition of aerosols. Once the flow fields were computed, the trajectories of the aerosol particles are obtained by superimposing particles on the flow fields. Deviation of the trajectory of the particle is determined by solving momentum equation for the particle. Particles that are predicted to contact the surface are assumed to be captured. 1.1 Statement of the Problem In the PBMR design, pebbles are continually circulated through the core by extracting pebbles from the bottom of the core and adding pebbles at the top. As the surfaces of the pebbles rub against each other, very large quantities of graphite dust are produced during normal operation. This dust is transported around the reactor coolant system and is deposited on surfaces. Although the PBMR design has inherent safety features that are expected to reduce the likelihood and extent of release of radioactive materials in an accident, assessment of the safety of PBMR relies on the methods for estimating the magnitude of radionuclide source terms for a range of possible accident 3

23 scenarios. Although radionuclides are released from fuel in the form of vapors, the transport and deposition of radionuclides typically are dominated by aerosol processes. Although a large number of processes affect the transport and deposition of aerosols in pebble beds, the dominant deposition process is likely to be inertial deposition in the tortuous flow channels in the pebble bed [6]. The physical process, Stokes law, which determines the force on a particle associated with an accelerating fluid, is well known. Typically, inertial deposition is very effective for large diameter aerosols but very ineffective for small diameter aerosols. However, the eddies associated with turbulent flow can dramatically enhance the inertial deposition of small aerosols [7] PBMR Design Characteristics The PBMR design is based on the German high temperature reactor (HTR) technology demonstrated in the Arbeitsgemeinschaft Versuchsreaktor (AVR), which operated for 21 years and reached a utilization factor of 21%, and the thorium high temperature reactor (THTR)[1]. The PBMR fuel consists of pebbles of about 6 cm in diameter [1]. Each pebble has two zones: about 5 cm layer of the fuel zone and an external graphite shell of about 0.5 cm. A large number of triso micro-spheres in a graphite matrix make up the fuel particle. The graphite and silicon carbide barriers surround the micro-spheres which contain uranium dioxide fuel. The uranium used in the fuel is enriched to about 8%. These protective barriers prevent the release of fission products from the fuel region during the reactor operation. 4

24 The PBMR core consists of about 440,000 spheres, one-fourth of which are unfueled graphite making up the central reflector column. An annular core of about 330,000 fuel spheres surrounds the central reflector. Helium gas, which is both chemically and radiologically inert, flows through the gaps between the pebbles as a coolant. The pebbles in the PBMR are not in a constant fixed position since the reactor is refueled while in operation. The pebbles are continuously cycled during operation as the fuel, new or reusable is added from the top while the used fuel is constantly extracted from the bottom of the reactor core [1]. During handling and recirculation of the pebbles, damage occurs to the graphite layer coating the pebbles. Also, during motion of the pebbles, friction occurs between pebbles as well as between the pebbles and the metallic surfaces. Based on the AVR experiments, this abrasion and collision of pebbles can result in the production of graphitic dust or aerosol particles. The size of these particles is between µm [8]. Most of the dust particles would be expected to be produced in the Fuel Handling & Storage System (FHSS) and the pebble bed core. There is a possibility that some of this dust can be transferred into the reactor core together with the fuel spheres. Friction amongst the spheres as well as the mechanical wear of the bottom reflector could also increase the rate of production of dust to the main circuit [1]. However, the PBMR will include a built-in on-line system that will limit the buildup of dust with operating time Radionuclide Release and Transport During normal operation of the reactor, a small quantity of fission products is released in the core. These fission products can be release from micro-spheres with 5

25 barriers that are imperfectly manufactured or from those that fail. The fission products can also be from uranium contamination on the surface of the fuel pellets or they can be formed from the decay of short-lived noble gases. Some of these radionuclides are in vapor form when released from overheated fuel. However, as they cool in the reactor core flow area the radionuclides can condense to form aerosols. The condensed aerosols can then deposit on the surface of preexisting aerosols as well as on the graphite dust. Condensation on the reactor coolant system surfaces is also a possibility. Some of the radioactive materials can even chemically react with the reactor coolant system surfaces. At the time of hypothetical pipe break, rapid depressurization of the system may cause the release of radioactive materials airborne at the time of the accident as well as in the re-suspension and release of the aerosols from the reactor coolant system. The quantity of radioactively contaminated dust deposited on surfaces within the reactor coolant system is very large and a significant fraction of this dust is in a respirable size range. As the core heats slowly over an extended time period after the pipe break, an additional release of radionuclides occurs but at a reduced flow rate,, potentially involving natural circulation driven flow patterns within the core. The delayed release of radionuclides would occur at high temperature and low radionuclide concentration. Within the core region, some of these radionuclides would initially transport as vapors during this period of release. At cooler regions within the core or reactor coolant system these vapors would interact with pre-existing airborne dust or structural surfaces. The 6

26 release of radionuclides to the containment or confinement region external to the reactor coolant system will be in the form of noble gases, radionuclide vapors, and aerosols. 1.2 Objectives The PBMR is developed by a team led by South Africa [9]. The developers of the concept claim sufficient retention of radioactive material under all conditions such that containment for the reactor is not necessary and that an exclusion boundary can be established at a distance as small as 400 meters. Substantial degradation of the barriers to the release of radionuclides from the fuel is not anticipated beyond design basis accidents because the graphite core has high heat capacity and thermal conductivity. This characteristic feature will allow passive removal of heat from the core in an unmitigated loss of coolant accident with very small estimated release of radioactive materials (source terms). To enable the General Design Requirement for a containment system to be withdrawn, a validated source term methodology must be developed and a strong safety case must be made. There are a number of aerosol deposition mechanisms that can contribute to the retention of aerosols in pebble beds. The objective of this dissertation is to investigate the ability of CFD methods to predict the inertial deposition of aerosols in pebble beds under both laminar and turbulent conditions. For turbulent flow, the enhancement of inertial impaction which results from interaction with eddies will be examined. The results of this dissertation are expected to support the development of advanced source term models that rely on more fundamental characterization of fluid 7

27 flow and aerosol transport mechanisms. The results of these analyses will also be used to develop correlations that can be used in less rigorous codes that rely on control volume and deposition velocity approximations. The correlations developed will also have broad applicability for the prediction of aerosol deposition in other pebble bed applications. 1.3 Technical Approach In recent years, high-speed computers and the availability of computational fluid dynamics software provide a means to examine flow problems such as those in a pebble bed reactor in a level of three-dimensional detail that was not previously practical. This dissertation relies on the use of the FLUENT code [5], not only to predict the flow behavior within the pebble bed but also to track the trajectories of aerosols. In this analysis, flow fields are estimated using the FLUENT, which has the capability of predicting three dimensional flow velocities. Conditions are selected that correspond to the normal operation of the PBMR and the initial transient period of an accident, as well as for low flow conditions corresponding to the laminar flow in the latter stages of a loss of coolant accident. The CFD models are run using the RANS approximations to determine the time-averaged flow field. Once the flow fields are established, the trajectories of the aerosol particles are obtained by superimposing them on the flow fields. Deviation of the trajectory of the particle will be determined by applying Stokes law. Particles that are predicted to contact the surface will be assumed to be captured. 8

28 1.4 Scope of Dissertation This dissertation composes of eight chapters. Chapter 1 is a brief overview of the dissertation. In this chapter, the problem to be analyzed is described. The statement of the problem, objectives of the dissertation as well as the technical approach to the problem is outlined. Chapter 2 gives a brief overview of a number of aerosol deposition mechanisms that can contribute to retention of aerosols in pebble beds. The chapter also discusses two important phases of radionuclide release from the reactor coolant system during a pipebreak accident. An introduction to the nature of fluid flows is discussed in Chapter 3. Basic fluid flow governing equations as well as approaches to turbulence modeling. Turbulent flow models that are used in this investigation are discussed in details. The chapter also includes a description of the discrete phase model of FLUENT that tracks the trajectories of aerosols. Chapter 4 analyzes the deposition of aerosol particles in turbulent pipe flows using the CFD code. The results obtained are then compared with experimental data of Liu and Agarwal [10]. Inertial deposition of aerosol particles in single and linear array of spheres is analyzed in Chapter 5 and the results are compared with the experimental data of Waldenmaier [11]. The turbulence-enhanced deposition using the stochastic tracking is also analyzed. 9

29 In Chapter 6, an analysis of aerosol filtration efficiency in granular beds is analyzed. The flow is modeled as laminar and the results are compared with experimental data obtained by Gal et al [12]. Chapter 7 analyses inertial deposition of aerosols in a pebble bed modular reactor. The analysis is done using the standard k-ε turbulence model in high Reynolds number regime to obtain the fluid flow field as well as to calculate aerosol trajectories. In this regime, the stochastic tracking of aerosols is used to examine the effect of eddies on aerosol deposition in pebble beds. In the final chapter, Chapter 8, some conclusions and remarks are drawn from all the analyses performed. Recommendations as well as future work directions are also suggested. 10

30 CHAPTER 2 2. AEROSOL TRANSPORT BEHAVIOR AND DEPOSITION MECHANISM IN PEBBLE BEDS As aerosol particles transport through a reactor core system, the particles undergo a number of processes such as growth, deposition, and resuspension mechanisms that change the characteristics of the aerosol particles. A general dynamic equation describes the balance of mechanisms that result in the formation of aerosols, transfer of aerosols between size groups, the removal of aerosols from the flow, and the resuspension of aerosol particles from surfaces. 2.1 Aerosol Transport Modeling Historically, the transport of aerosol particles in severe accidents has been analyzed by simplified solutions to the fluid flow and aerosol transport within control volumes, such as the treatment of aerosol transport in the CONTAIN code[13] for large volumes of a reactor containment. In these analyses, the conditions within a volume are assumed to be uniform and well-mixed. The fluid flow equations are formulated to conserve mass, energy and momentum for the control volume. Typically, it is assumed that the concentration of aerosols is sufficiently small that their effect on the flow is negligible. 11

31 There is no direct characterization of the local conditions, such as the concentration of aerosols or velocity of the fluid in the vicinity of the wall. Aerosol deposition on surfaces is treated using the concept of deposition velocity, which is the proportionality constant relating aerosol concentration in the volume to the removal rate, V RA C s d = Equation 2-1 where V d = deposition velocity (m/s) R = Removal rate (#/s) A s = surface area of control volume (m 2 ) C = number concentration of aerosols in volume (#/m 3 ) However, because deposition on surfaces is highly dependent on local conditions, the values used for deposition velocities must be obtained either from experiments in which the conditions closely simulate the conditions of the analysis or from a more detailed analysis that does account for flow regimes and aerosol concentrations across the control volume. Frequently the size distribution of aerosols has been treated as having a log normal distribution. However, in the more mechanistic analyses, a time-dependent balance equation, Boltzmann Equation, is solved for the concentration of aerosols as a function of their volume and mass. 12

32 n( v, m, t) = R( v, m, t) n( v, m, t) t du dw dq n( v, m, t) 0 du 0 ds n( u, q, t) n( w, s, t) K dqk C ( u, q v, m) + S( v, m, t) C ( u, q w, s) δ ( v u w) δ ( m q s) where Equation 2-2 nvmt (,,) = number of aerosols with volume v and mass m as a function of time t R(, vmt,) = removal rate by deposition of aerosols K (,, ) C u q w s = collision kernel between aerosols with volume u and mass q with aerosols of volume w and mass s Svmt (,,) = extrinsic source of aerosols The first term on the right hand side of the equation represents the removal or aerosols from the control volume by deposition. The second term represents the production of new aerosols by the coagulation of aerosols with initial mass q and volume u with those of initial volume w and mass s to produce aerosols with mass m and volume v. The delta functions select the combinations of q and s that add to m and combinations of u and w that add to v. The collision kernel K must either be obtained empirically or by a more detailed theory. These collisions also represent a loss term for the mass and volume being considered. This is represented by the third term on the right hand side. The final term is an extrinsic source of aerosols. 13

33 Clearly, the control volume approach to the analysis of fluid flow and aerosol transport has severe limitations. The more rigorous approach would be to use computational fluid dynamics to treat aerosol transport as a multi-phase flow problem in which aerosols are treated as a discrete phase. There are two problems with this approach. The first is a practical problem of computational size and running time. The second problem is the determination of the constitutive relationships that describe the detailed interactions among aerosols that have historically been treated as separate effects and in some cases with diffusion theory Aerosol Trajectory Analysis In FLUENT, the trajectory of an aerosol particle is predicted by integrating the force balance on an aerosol. This force balance, written in a Lagrangian frame of reference, balances the particle s inertia with the forces acting on it [5]. For the motion of the particle in the x-direction, the force balance can be written as, du dt where C Re 18μ = ( u up) + Fx Equation ρ d p D p p 2 p C D = drag coefficient μ = fluid viscosity d p = aerosol particle diameter ρ p = aerosol particle density 14

34 u = fluid velocity u p = particle velocity F x = force per unit mass acting on the aerosol particle The drag force per unit mass is represented by the first term on the right hand side of Equation 2.3, CD Re p 18μ FD = ( u up) Equation ρ d where p 2 p Re p = the relative Reynolds number For a spherical particle, the relative Reynolds number is given by Re p ρ pdp u up = Equation 2-5 μ At various Reynolds numbers, the drag force is based on the drag coefficient as C D defined C D = 1 ρ 2 F D ( ) 2 u u A f p p Equation 2-6 where 2 π d p Ap = = projected area of the spherical particle. 4 15

35 2.1.2 Saffman Lift Force The velocity gradient or particle rotation can cause lift forces on particles. When a particle is moving in the shear layer, it can experience lift force due to non-uniform relative velocity over the particle and the resulting non-uniform pressure distribution. This lift force due to shear is also referred to as Saffman lift force, and it is included as an option in FLUENT. Saffman [14], found the magnitude of this force to be 2 r FS = 1.62μd k p ( u up) Equation 2-7 ν where k r = local velocity gradient The lift force becomes negative when the particle leads the fluid motion and the particle moves down the velocity gradient towards the wall. In the case where the particle lags behinds the fluid, the Saffman lift force becomes positive. In this case, the particle moves away from the wall and up the velocity gradient Dispersion of Aerosols in Turbulent Flows The discrete phase model (DPM) in FLUENT has the capability of predicting the effect of turbulence on the dispersion of aerosols [5]. The effect of the instantaneous turbulent velocity fluctuations on the aerosol trajectories is determined by the discrete random walk (DRW) model. In this approach, the trajectories of individual particles are determined by the instantaneous fluid velocityu, v and w given respectively by u = u + u (), t v= v + v () t and w= w+ w () t Equation 2-8 where 16

36 u, v and w are fluid velocity averages and u () t, v ( t) and w ( t) are the fluctuating fluid velocity components [5]. The kinetic energy of turbulence is known at each point on the flow, therefore, assuming isotropy, the values of the root mean square fluctuating velocity components are given as k u = v = w = Equation '2 '2 '2 2 In order to account for the random effects of turbulence on the dispersion of aerosols, the trajectories computed must be for a sufficiently large number of representative particles [5]. 2.2 Treatment of Aerosol Characteristics Aerosols exist in different sizes, shapes and densities and these influence the way in which the particles are deposited on the surfaces. The transport behavior of nonspherical particles is much more difficult to predict than the behavior of spherical particles. Thus, most models of aerosol behavior treat the particles as spherical but apply correction factors for their non-spherical behavior Aerodynamic Shape Factor Aerosol particles exist in different shapes and sizes. The motion and deposition probabilities of aerosols depend on the shape of the particle. The drag force and settling velocity of the aerosols are affected by their shape. In order to account for the effect of shape on the particle motion, a correction known as dynamic shape factor is applied to Stokes law. This correction is defined by Hinds [15] as the ratio of the resistance force 17

37 of the nonspherical particle to the resistance force of a sphere having the same volume and velocity as the nonspherical particle. Using this definition, the dynamic shape factor is then given by FD χ = Equation πμVd s e where d e = equivalent volume diameter or diameter of a spherical aerosol with the volume equal to that occupied by an irregular particle V s = terminal settling velocity Aerodynamic Diameter The mass of an aerosol also has a significant effect on its behavior. Aerosol physicists typically describe an aerosol by its aerodynamic diameter, which is the diameter for a spherical particle with unit density, ρ 0, of 1 gm/cm 3 ( or 1000 kg/m 3 ) and the same inertial properties in the gas as the particle of interest, rather than its physical diameter [15]. For spherical particles greater than 0.5 μm, this aerodynamic diameter can be approximated as d a ρ p = dp Equation 2-11 ρ0 where ρ 0 = unit density (1 g/cm 3) 18

38 Particles with the same aerodynamic diameter follow the same trajectory. For the purpose of this study, the aerosol particles will be treated as spherical and with unit density. 2.3 Aerosol Growth Mechanisms There are a number of aerosol deposition mechanisms that can contribute to retention of aerosols in pebble beds. These mechanisms include gravitational settling, agglomeration, diffusion, thermophoresis and inertial impaction, with diffusion, gravitational settling and inertial impaction as the major deposition mechanisms Vapor Deposition One of the means by which aerosols form and grow is by the condensation of vapors. Typically in a severe accident, radioactive and inert structural materials are vaporized within the over-heated core. They then migrate to a region of lower temperature at which they can condense to form new aerosols or condense on the surface of pre-existing aerosols. The condensation of water vapor on aerosols can also be a very important growth mechanism for aerosols in containment [15] Aerosol Agglomeration Relative motions between aerosol particles cause them to collide with each other. The collision between particles in inelastic and the attractive forces between aerosols particles can be large. As a result, the collisions between aerosol particles can cause them to stick to each other and form larger particles or agglomerates [15]. This interparticle phenomenon is referred to as agglomeration for solid particles, and the resulting particle 19

39 clusters are known as agglomerates [15, 16]. The process of agglomeration results in the number density of the particles in a system decreasing, whereas average particle sizes increase. Relative motion between particles caused by Brownian motion results in coagulation referred to as thermal coagulation and this occurs for particles with diameter greater than 0.1μm. The type of agglomeration that occurs as result of motion between particles caused by gravitational, electrical or aerodynamic effects is termed kinematic agglomeration. The rate of agglomeration depends on the range of sizes present in a system, and occurs faster between particles of different sizes than those of the same size [15]. 2.4 Aerosol Transport and Deposition Mechanisms Brownian Diffusion Brownian motion is irregular motions of particles suspended in a fluid. This random motion is a result of random forces exerted on the particles by surrounding molecules. The random motion results in a process known as diffusion, which is the net transport of aerosols from a region containing high concentration of particles to that of a lower concentration [15, 16]. The net flux of particles transported through a unit area per unit time is proportional to the concentration gradient dn dx and is given by Fick s law: J x dn = D Equation 2-12 dx where 20

40 n = aerosol particle concentration (#/cm 3 ) J x = flux (# particles/cm 2 sec) D = diffusion coefficient (cm 2 /sec) Brownian motion is characterized by the diffusion constant D. The larger the value of D, the more vigorous is the Brownian motion and the more rapid the mass transfer in a concentration gradient [15, 16]. The diffusion coefficient is a function of particle size with smaller particles diffusing more rapidly. For spherical particle of diameter d, the diffusion coefficient is given by the Stokes-Einstein relation, p D ktc πμd B c = Equation 2-13 p where k B = Boltzmann s constant T = temperature in K C C = Cunningham correction factor Brownian diffusion depends mainly on the diffusive diameter of the aerosol rather than on its aerodynamic diameter. As a result, Brownian diffusion increases with decreasing particle size. Brownian diffusion is the primary transport mechanism for small 21

41 particles of diameter smaller than 0.1μm, and it is important when the transport distance is small. The diffusion coefficient depends on the particle size, temperature and concentration [15] Thermophoresis Deposition of aerosols can be induced by the presence of external forces the magnitude of which depends largely on the properties of the gas and the aerosols as well as the temperature gradient. The forces due to the difference in temperatures are referred to as thermophoretic forces and the phenomenon is called thermophoresis [15].Once a temperature gradient is established, aerosols tend to move towards the direction of the decreasing temperature where deposition may occur. The thermophoretic velocity is given as: u p Vth = Kth T where T K th = thermophoretic coefficient T = temperature gradient Equation 2-14 The thermophoretic coefficient depends, among other parameters, on the aerosol particle radius, gas properties and thermal conductivity.the negative sign indicates the migration of particles towards the direction of decreasing temperature. From the above equation, the thermophoretic velocity is inversely proportional to the temperature of the aerosols and directly proportional to the temperature gradient [15-17]. 22

42 Thermophoresis is more effective for small particles than for large particles because small particles are more affected by the momentum exchange that occurs when a gas molecule hits a particle. It is dominant for particles with sizes ranging between 0.01 and 0.3 μm. This equation holds for particles with diameters less than the mean free path of the gas, and it does not depend on the size of the particle Gravitational Settling Aerosols in motion can fall under the influence of gravity. The velocity of a particle falling through a fluid due to gravity increases until there is a balance between the gravity and the frictional forces. When this balance is reached, the velocity of the aerosol particle remains constant and this velocity is termed terminal velocity or gravitational settling velocity [15-17]. This gravitational settling velocity can be obtained from equating the force of gravity to the Stokes force for a spherical particle, such that V s where 2 dpρ pgcc = Equation μ g = acceleration due to gravity This equation shows that the gravitational settling velocity is directly proportional to the square of particle diameter and its mass, causing larger particles to settle faster than smaller particles. Hence deposition through gravitational settling becomes more efficient for larger particle rather than smaller ones. 23

43 2.4.4 Inertial Impaction Inertial impaction is a mechanism by which particles with large inertia get deposited on surfaces. Because of their inertia, the trajectories of these aerosols may deviate from the fluid streamline since they are unable to follow the motion of an accelerating gas. The larger aerosols will remain in their trajectories rather than following the trajectory of the fluid as it flows around a solid object, and will be deposited on the surface. Deposition can occur if the center of the aerosol particle comes within one particle radius of the surface. The effect is most significant for larger particles [15-17]. The parameter that governs inertial deposition is the Stokes number, N st, which is the ratio of the stopping distance to the characteristic dimension of the obstacle (D), and is given by: N St where 2 dpρ purelcc = Equation μD u rel = relative velocity of the aerosol and fluid The Stokes number gives the ratio of the inertial force to the drag force. The amount of deposition resulting from inertial impaction correlates with the Stokes number such that deposition increases with the Stokes number [15, 16] Particle Bounce When an aerosol particle contacts a surface, adhesive forces tend to cause the particles to adhere to the surface. The main adhesive force is van der Waals force, which 24

44 arises from the formation of random charge concentrations in the particle and corresponding complementary concentrations of oppositely, charged regions in the contacted surface, electrostatic force, and surface tension, if the aerosol is liquid [15, 18]. However, depending on the kinetic energy and coefficient of restitution of the particle, it is possible that the particle will elastically recover from the impact with sufficient energy to escape the energy well associated with adhesion energy. During the deformation process, part of kinetic energy is dissipated and part is converted into kinetic energy of rebound [18]. If the incident velocity of the particle is very high and the energy of rebound is higher than the adhesion energy, the particle will bounce off the surface. For the particle to bounce off a surface after collision, the kinetic energy required is given by 2 dp A(1 e ) Ek = Equation xe where A = Hammaker constant, and it depends on the materials used e = coefficient of restitution or ratio of rebound velocity to incident velocity x = separation distance Adhesion probability as a function of impact energy was measured by Ellenberger et al [19] for fly ash particles impacting stainless steel fibers. These data can be fit to the expression h = E Equation 2-18 where 25

45 h = adhesion fraction E = kinetic energy (J) Turbulence-Enhanced Inertial Deposition Turbulent deposition of aerosols occurs when a turbulent gas carrying particles larger than 1μm flows parallel to the surface, and aerosols get deposited onto surfaces as a result of the fluctuating velocity components normal to the collecting surface. The aerosols which cannot follow the eddying motion will be projected onto the wall [16] Re-entrainment Re-entrainment or blow-off refers to the resuspension of particles as a result of a jet of gas. The process of resuspension itself can be defined as the detachment of an aerosol from the wall and its subsequent transport away from the surface. This process is important for build-up and removal of dust in conduits. Before resuspension, a particle may roll or slide before it becomes airborne, in which case and very little force is required to remove the particles [15]. Even though it is easy to determine the jet velocity as well as to determine whether an aerosol particle has been removed from the fluid stream, it is a challenge to determine the fluid velocity and the resultant drag force on a particles that adhere to surfaces [15, 17]. Due to stochastic nature of the reentrainment process, for a given air velocity, only a fraction of particles of a given size can be estimated to be removed from a surface. The probability of reentrainment increases with increasing particle size and increasing fluid velocity. Aerosols embedded in the laminar sub-layer at the surface of the collector cannot be easily removed from the surface. However, eddies can penetrate through the 26

46 boundary layer and detach these particles. Reentrainment of a given size of aerosols will depend on how long is the surface exposed to the turbulent fluid jet [15] Electrical Charge A fraction of aerosol particles larger than 0.1µm carry an electrostatic charge, which has some influence on particle deposition [15, 18]. If a particle is charged, it can be attracted to the surface of the collector. This force can be written as F c kq x 2 c = Equation 2-19 Q where k c = Coulomb force constant x c = separation distance between the charges 2.5 Summary There are two important phases of radionuclide release from the reactor coolant system in a pipe-break accident. At the time of depressurization, radioactivelycontaminated aerosols in suspension at the time of the accident can be released from the reactor coolant system in addition to previously deposited aerosols that are resuspended by the blow-down forces. The rapid depressurization of the system at the time of a hypothetical pipe break results not only in the release of radioactive materials airborne at the time of the accident but also in resuspension and release of the aerosols from the reactor coolant system. The quantity of radioactively contaminated dust deposited on 27

47 surfaces within the reactor coolant system is very large and a significant fraction of this dust is in a respirable size range. The second phase of release occurs much later in the accident when the system is depressurized, is in natural circulation, and the core has heated up due to decay heat. Even at temperatures below that at which the integrity of microspheres would be lost, fission products can be released from micro-spheres with imperfectly manufactured or failed barriers, from uranium contamination on the surface of the pellets or formed from the decay of short-lived noble gases. When released from overheated fuel, these radioactive materials are in vapor form, but as they cool in the core flow area they can condense to form aerosols, deposit on the surface of preexisting aerosols including the graphite dust, condense on reactor coolant system surfaces or chemically react with reactor coolant system surfaces. The delayed release of radioactive nuclides would occur at high temperature and low radionuclide concentration. Radionuclides will probably transport as vapors during this period of release within the core region. A number of aerosol deposition mechanisms can contribute to the retention of aerosols in the reactor coolant system. These mechanisms include Brownian diffusion which may be important for small particle diameter, while larger aerosols may be deposited through gravitational settling or by inertial effects. Migration and deposition of aerosols to adjacent surfaces may also be affected by external forces such as thermophoresis resulting from the presence of thermal gradients or the effects of electrostatic charge. Because of the tortuous flow channels through the pebble bed, inertial deposition of aerosols is expected to be a very effective means of trapping 28

48 aerosols including dust within the bed. Typically, inertial deposition is very effective for large diameter aerosols but very ineffective for small diameter aerosols. However, eddies associated with turbulent flow can dramatically enhance the inertial deposition of small aerosols. 29

49 CHAPTER 3 3. FLUID FLOW BEHAVIOR Fluid flows can be classified in different ways depending on the various physical characteristics of the flow fields. A fluid flow can be characterized as steady if all its properties remain constant as time changes at any point in space. Flows which are dependent on both time and space are referred to as unsteady. A flow can be classified as compressible if its density changes as the flow progresses. If the density variations are negligible, then the flow is treated as incompressible [20-22]. 3.1 Fluid Flow Modeling: Navier-Stokes Equations Fluid flows are governed by conservation laws system known as Navier-Stokes equations. These governing equations are conservation laws of energy, momentum and continuity. The continuity equation describes how the macroscopic property of mass per unit volume, density of a fluid, changes as position and time change. For an incompressible fluid with constant density, the continuity equation can be written as i x ρu i ( ) = 0 Equation

50 where ρ = the density of the fluid x i = ith direction coordinate u i = velocity in the x i coordinate direction. The conservation of momentum for laminar flows is described by the equation u u P τ ρ ρu t x x x i i ij + j = + j i j Equation 3-2 where P = static pressure and τ ij = viscous stress tensor The viscous stress tensor for Newtonian fluids is given by τ ij u μ x i = + j u i j x Equation 3-3 where μ is the dynamic viscosity. In conservation form, the convective term can be written as u u u = u u u = u u j ( ) ( ) i j j i i j i xj xj xj Equation

51 and this leads to the Navier-Stokes equation in conservation form [23-25] written as u P + = + t x x x ( uu j i) ( ij) i ρ ρ τ j i j Equation Nature of Fluid Flows The nature of the flow regime depends largely on the ratio of the inertial forces to the viscous forces in the fluid. This ratio is a dimensionless parameter called Reynolds number [20], given by Re ρud = Equation 3-6 μ The parameter μ is the fluid viscosity; ρ is its density, U is the average velocity of the flow and D is the characteristic length of the geometry of interest. Inertial forces are proportional to the fluid s density and its velocity Laminar Flows At low Reynolds number, viscous forces dominate the inertial forces and keep the fluid in line. These flows, characterized by highly ordered motion, are referred to as laminar flows. The flow consists of smooth streamlines caused by the fact that the fluid is flowing in layers. Under these conditions, the Navier-Stokes equation can be solved numerically [20, 22]. 32

52 3.2.2 Turbulent Flows When Reynolds number increases, the laminar flow becomes unstable and this gives rise to turbulent flow. Turbulent flows are irregular, unsteady and highly threedimensional. To model these flows, appropriate modeling procedures are required in order to describe the effect of turbulent fluctuations of velocity on the basic fluid flow governing equations. For steady state flows, a time-averaging procedure, also known as Reynolds averaging is used [5, 26]. Turbulent flows are also characterized by high diffusivity. The transfer of mass including aerosols in multi-component flow, energy and momentum is increased greatly by turbulence diffusivity; with larger eddies being responsible for the enhanced diffusivity. The rate of transport and mixing by turbulence is several orders of magnitude greater than the transfer rates due to molecular diffusion. The wall friction in internal flows through channels is also increased by diffusivity [27, 28]. 3.3 Turbulent Kinetic Energy The energy spectrum E( κ ), gives an indication of the distribution of turbulent kinetic energy among the eddies of different sizes [27]. In the wave number range (κ, κ+dκ), the energy can be expressed as k = E( κ) dκ Equation where E( κ ) = energy spectrum function 33

53 3.3.1 Large and Small Scales of Turbulence One characteristic of turbulent flows is the different eddy sizes. The larger eddies are comparable in size to the flow geometry, whereas the smallest eddies are smaller than the largest eddies by many orders of magnitude. However, these smallest scales often exceed the molecular scales by three or more orders of magnitude; hence turbulent flows can be treated as a continuum. With this assumption, the fluid properties can be considered to be continuous functions of position and time. As the Reynolds number increases, the ratio of the smallest to largest scales decreases. During the energy cascade process, large scales transfer their turbulence kinetic energy to smaller scales by inviscid processes until energy is dissipated by viscous action at the smallest scales. The largest scale eddies of characteristic length scale l 0, comparable to flow lengthscale L, have characteristic velocity u u( l ) on the order of the root mean square turbulence intensity given by 0 0. This velocity is 2 u k 3 1/2 Equation 3-8 These large eddies have a very high Reynolds number comparable to that of the flow such that direct effect of viscosity can be ignored. The energy cascade ends when the Reynolds number for the small eddies becomes so small such that eddy motion is stable and the viscosity is effective in dissipating the kinetic energy. At this low Reynolds number, the smallest scales depend only upon the kinematic viscosity v (in m 2 /s) and the rate ε (in m 2 /s 3 ), at which larger scales supply energy to the smaller scales [24, 27, 29] is given by, 34

54 ε = dk dt Equation 3-9 The following scales can be formed given ε and v; 3 v η ε v τ t ε υ ( vε) 1/4 1/2 1/4 Equation 3-10 where η, τ t and υ are Kolmogorov scales of length, time and velocity respectively characterizing the smallest dissipative eddies. These Kolmogorov scales estimate the length, time and velocity scales for small eddies in the turbulent flow [24, 27] The Turbulence Intensity Scale Turbulence intensity measures how intense the turbulent fluctuations are. In turbulent flows, the velocity field fluctuates in the x, y and z directions. These velocities are given respectively byu, v and w [5]. From the Reynolds stresses, the turbulent kinetic energy is defined as '2 '2 '2 ( ) 1 1 k = u + v + w = u u Equation i j Assuming isotropic fluctuations, the turbulence intensity percentage is given as 2 k T = 100 Equation U where U is the free stream mean-flow velocity [5]. 35

55 3.4 Turbulence Modeling Approaches Direct Numerical Simulation Direct numerical simulation (DNS) refers to the direct solution of solving Navier- Stokes equation even for turbulent flows. The velocity field has to be resolved on lengthscales down to the Kolmogorov scale. The DNS model requires the resolution of all lengthscales and timescales, thus making it computationally expensive. The model is practical for simple flows with low-to-moderate Reynolds numbers since computational costs increases as Re 3 [5, 29, 30] Reynolds Averaged Navier-Stokes Equations Although turbulent flows could be simulated in principle by DNS, it is not practical because of the requirements for very fine mesh in space and time [5, 26, 27]. For most fluid flow measurements performed, however, mean values are of more importance than time histories. When modeling turbulent flows, it is therefore necessary to decompose instantaneous variables into mean values and fluctuating values [23] In Reynolds averaged model, the velocity is expressed as the sum of a timeaveraged component u i, which is independent of time[5, 26, 27], and a fluctuating component u i such that: ui = ui + u i Equation

56 Substituting this velocity into the instantaneous continuity and momentum equations and taking the time average yield the Reynolds averaged equations in conservation form as u x i i = 0 Equation 3-14 and P ( uu j i uu j i) ( ij) u + + = + t x x x i ρ ρ τ j i i Equation 3-15 The term ρ uu j i is the time-averaged rate of momentum transfer due to turbulence. It represents the correlations between fluctuating velocities but it is unknown. The Reynolds stress tensor is symmetric and is given byτ ij = uu j i. There are a total of six Reynolds stress terms, making the total number of unknowns ten, which is larger than the number of available equations namely; the mean continuity equation and the three components of Navier-Stokes equation. The Reynolds averaged Navier-Stokes equation requires closure relations to determine the Reynolds stresses. To achieve this, an assumption known as Boussinesq hypothesis is used. In this hypothesis the Reynolds stresses are related to the mean velocity gradients with the turbulent viscosity μ t being the coefficient [5, 24, 26, 27] and the Reynolds stresses are given by 37

57 μ t u u i j 2 uu i = j + + kδij ρ xj x i 3 Equation 3-16 according to Boussinesq hypothesis where δ ij = Kronecker delta The eddy viscosity depends strongly on the state of the turbulence. The turbulent kinetic energy k is defined as half the trace of the Reynolds stress tensor such that k 1 = uu i j Equation The eddy viscosity is considered to be proportional to the velocity characterizing the fluctuating motion and the characteristic length such that 2 k μt = ρc μ Equation 3-18 ε where C μ = empirical constant The above equation is in analogy to the molecular viscosity. The analogy presumes that the turbulent eddies behave like molecules that collide and exchange momentum. This turbulence model provides eddy viscosity and the turbulent kinetic energy [5, 24, 27]. 38

58 3.4.3 Standard k-ε Model The k ε model belongs to the class of two-equation models. This model solves one transport equation for turbulent kinetic energy k and another for dissipation rate. The solution allows the independent determination of the turbulent velocity and the length scales. The transport equations for both k and ε are written respectively as Dk μ k t 2 ρ = μ + + μts ρε Dt x j σ k x j Equation 3-19 and Dε μ ε ε = + + Dt x j σ ε x j k 2 ( C1 ε ts C2ε ) t ρ μ μ ρ ε Equation 3-20 The coefficients 1 2 σ k, σ, C, C and C μ are constants determined empirically ε ε ε [5]. The k ε model is an eddy-viscosity model. The turbulent viscosity is obtained by assuming that it is proportional to the product of the turbulent velocity and length scales which can be obtained from the turbulent kinetic energy and its dissipation rate. The turbulent viscosity can be computed from Equation The k-ε model has been used in the wide range of applications. The model is robust, economic and provides reasonable accuracy for simple flows, but can be inaccurate for complex flows [5]. 39

59 3.4.4 Standard k-ω Model The standard k-ω model on the other hand, has the capability of addressing some of the deficiencies of the k-ε model. The k-ω model solves the standard k equation but uses ω as a length-determining factor. The quantity ω is sometimes often referred to as the specific dissipation, and by definition ε ω or k written respectively as * ε = βωk.the standard k-ω transport equation for k and ω can be Dk U μ k ρ i * t = ij f * k Dt τ x ρβ ω + β j x μ + σ j k x j and Equation 3-21 Dω ω U μ k i * 2 t ρ = αω τij ρβ f * kω + β μ + Dt k x j x j σ k x j Equation 3-22 The turbulent viscosity for the standard k-ω can then be written as k μ = Equation 3-23 t * αωρ ω * and α ω depends on the turbulent Reynolds number, Re T [5, 27] Reynolds Stress Model (RSM) The Reynolds stress model is more elaborate than the k-ε and the k-ω turbulence model. Rather than using the eddy-viscosity model, the RSM closes the Reynolds- 40

60 averaged Navier-Stokes equations by solving transport equations for Reynolds stresses along with an equation for dissipation rate [5]. In three dimensions, this leads to seven additional transport equations that need to be solved. The Reynolds Stress transport equation has the form Dτ Dt ij ε J ijk = Pij +Φij ij + Equation 3-24 xk where τ ij = Reynolds stresses P ij = rate of production of Reynolds stresses Φ ij = transport of Reynolds stresses due to turbulent pressure-strain interactions ε ij = rate of dissipation of the Reynolds stresses J ijk = transport of Reynolds stresses by diffusion Unlike the k-ε and k-ω models, the RSM accounts for the effects of curvature, rotation, swirling flows, acceleration or retardation and therefore it has a higher potential to give accurate predictions for complex flows. The closure assumptions employed to model various terms in the exact transport equations for Reynolds stresses, however, can limit the fidelity of the Reynolds stress model. Also, the accuracy of the model can be 41

61 compromised by the modeling of the pressure-strain and the dissipation rate. In general the RSM requires more CPU resources than the simpler models [5, 27] Large Eddy Simulation In turbulent flows, the large scales have more energy than the smaller ones, and they are therefore most effective in the transport of conserved properties. The large eddies depend strongly on the flow configuration, boundary conditions and the flow parameters. Small eddies have much lower energy and these provide little transport of the conserved properties. In LES, the larger three-dimensional time-dependent turbulent scales are resolved directly whereas the small-scale turbulence is modeled [5, 29, 30]. In order to get the velocity field that contains only the large scale components of the total field, the scales that are smaller than the mesh size are filtered. The filtering is carried out such that the resulting filtered velocity U ( x, t) can be resolved on a relatively coarse grid, with the required grid spacing h proportional to the specified filter with Δ, such that large eddies are those of size larger than Δ, and smaller eddies of size smaller than Δ need to be modeled. Filtering The filtered velocity is defined by Equation 3-25 uxt (, ) = Grxux (, ) ( rtdr, ) where G(r,x) = filter function. 42

62 defined by For simplest case, the filter function is independent of x and the residual field is u ( x, t) = u( x, t) u( x, t) Equation 3-26 The velocity field has the decomposition given by uxt (, ) = uxt (, ) + u ( xt, ) Equation 3-27 This equation is analogous to Reynolds decomposition except that U ( x, t) is a random field and the filtered residual u ( x, t) 0 [5, 24, 29, 30]. The filtering of the Navier-Stokes equations for incompressible flow yields equations that are of the same form as the RANS equations. 3.5 Near-Wall Modeling for Turbulent Flows The turbulence models discussed above, namely, k-ε, RSM and LES, are primarily valid for turbulent flows in regions away from the wall. The k-ω model, however, was designed to be applied throughout the boundary layer. The near wall region of the turbulence boundary layer near the wall can be subdivided into three sub-regions: Viscous sub-layer: this is the layer closest to the wall where viscous effects dominate and the flow is laminar since the turbulent eddying motion must stop due to zero velocity at the wall; 43

63 Outer layer: this is the fully-developed turbulent boundary layer where the effects of inertia are dominant; Buffer layer: in this region between the viscous sub-layer and the turbulent layer where the effects of molecular viscosity and turbulence are equally dominant The presence of solid boundaries affects turbulent flows. Closer to the solid boundaries viscous effects are dominant and the mean velocity depend on the distance y + from the wall, the fluid density, its viscosity and the wall shear stress [5, 24, 27]. This mean velocity varies logarithmically with distance from the solid boundaries, and the behavior is referred to as the law of the wall, and is written as u 1 = ln( Ε y ) Equation 3-28 κ + + where Ε and κ are constants and y ρuy τ μ + = Equation 3-29 and y + and u + are dimensionless quantities. In FLUENT, for the values of y + < , the flow is modeled as laminar. For the buffer layer with y + range 30 < y + < 60, the flow is modeled as turbulent and the viscous forces are included in the calculations. In this region, shear stresses are assumed to be constant and equal to the wall shear stresses. Figure 3.1 shows the sub-divisions of the near wall regions [5, 31]. 44

64 In the viscous sub-layer, the fluid closer to the wall is dominated by viscous shear. The viscous forces in this layer play a dominant role in the transfer of heat and momentum. The viscous sub-layer is valid for y and the solutions for both laminar and turbulent flow equations are taken to be laminar [5, 31]. Figure 3.1: The Subdivision of Near Wall Region [5] Standard Wall functions Approach The viscosity-affected inner region (viscous sublayer and buffer layer) is not resolved. Instead, semi-empirical formulas called "wall functions'' are used to bridge the viscosity-affected region between the wall and the fully-turbulent region. The use of wall 45

65 functions obviates the need to modify the turbulence models to account for the presence of the wall [5] Enhanced Wall Treatments The turbulence models are modified to enable the viscosity-affected region to be resolved with a mesh all the way to the wall, including the viscous sublayer. For RANS modeling it is recommended that, when using enhanced wall treatment, the y+ value should be about 1 [5]. 46

66 CHAPTER 4 4. TURBULENT FLUID FLOW AND AEROSOL DEPOSITION IN A PIPE Before applying CFD to the problem of fluid flow and aerosol deposition in the complex geometry of a pebble bed, the simpler problem of turbulent flow and turbulenceenhanced deposition in a pipe is examined. Turbulent flow in a smooth pipe has been examined in great detail both theoretically and experimentally, particularly with regard to the flow velocity distribution across the pipe and the pressure drop per unit length of pipe at distances far from entrance effects [32-35]. Also, an experimental data base exists for turbulence enhanced deposition of aerosol in pipes. The specific conditions modeled are those of the experimental setup of Liu and Agarwal [10] for experiments to examine turbulence-enhanced deposition in a pipe. 4.1 Problem Description In their experiments, Liu and Agarwal measured the rate of deposition to the wall of olive oil droplets from turbulent air flow inside a smooth pipe. The experiment was performed with the nominal pipe for Reynolds number of 10,000 and 50,000 and particle diameters of 1.4 to 21 μ m. The experimental apparatus used by Liu and Agarwal is shown in Figure 4.1. Uniform, spherical droplets of olive oil were generated using a 47

67 vibrating-orifice monodisperse aerosol generator at its rated output of 1.5 x 10-3 m 3 /s of air. The aerosols were then transported vertically upward through a nominal m diameter copper pipe to a plenum chamber at the top. From the plenum chamber, the particles were transported down a nominal 0.032m diameter copper pipe before entering a 1.02 m long and m inner diameter glass deposition pipe. The glass deposition pipe was connected to the copper pipe through a m copper reducer. A filter holder containing 90 mm diameter glass filter was placed downstream followed by a volumetric flow transducer, a regulating valve, and a suction blower. Clean air was added at point (B) when the air flow required was higher than 1.5 x 10-3 m 3 /s. Olive oil particles used in the experiment contained less than 10 percent by weight of uranine, which is used as a fluorescent tracer for quantitative analysis of deposited particles. Once they reached the surface of the glass pipe, the aerosols would stick or get trapped. Therefore, the rate of deposition of deposition of the aerosols onto the wall of the pipe was measured. 48

68 Plenum Chamber Secondary Dilution Air Transition Pipe Flexible Hose Neutralizer Aerosol Solution Compressed Air A Monodisperse Aerosol Copper Reducer Deposition Pipe Filter Flow Indicator To Vacuum Pump Flow Transducer Figure 4.1 : Experimental Apparatus of Liu and Agarwal 49

69 4.2 Modeling Procedure The Geometrical Modeling for CFD A pipe of length 1.02 m and m inner diameter was modeled. To reduce computational time, this configuration was modeled with a two-dimensional, r-z coordinates (axisymmetric). The deposition pipe was divided into eight sections, each of length m. A structured dense grid of 16 x 400 cells was generated. Figure 4.2 shows an enlarged portion of the mesh with details of the mesh near the wall. The different boundaries were specified on the domain and the mesh was then exported into the FLUENT solver. Figure 4.2: Geometrical Model 50

70 4.2.2 Fluid Flow Modeling The problem involves air at standard properties flowing through a 1.02 m long pipe of circular cross-section with a radius of m. The fluid flow field is modeled as two dimensional, incompressible, steady and both the laminar viscosity and density are assumed to be constant. The model assumes rotational symmetry so that an axisymmetric analysis can be performed. To satisfy the flow at the inlet, the velocity inlet boundary condition was specified. For the Reynolds numbers of 10,000 and 50,000 considered in the computations, two separate simulations with a constant velocity 11.5 m/s and 58 m/s were specified at the inlet. Values for the turbulent kinetic energy and dissipation are also specified at the inflow. The fluid flow outlet is located at the end of the pipe so as to allow the development of the flow in the downstream direction. At the outflow, no velocity boundary conditions are explicitly imposed, resulting in zero normal stress at the outflow. The turbulent kinetic energy and dissipation are not specified at the outflow resulting in zero flux boundary conditions for both these quantities. The walls of the pipe are modeled as smooth, rigid and impermeable. The no-slip boundary condition is assumed at the walls; thus, each of the velocity components assumes a zero value at the wall. The fluid flow was analyzed using RANS turbulence models in order to obtain the fluid flow field. The standard k-ε and its variation, RNG; as well as the RSM were used. 51

71 As the turbulence model is not valid in the viscosity affected regions close to the wall, use is made of a near-wall modeling approach in the close proximity of the solid walls. The mesh was refined all through the viscous layer such that the enhanced wall treatment applies. In this case, the y + values were below 1 to satisfy this near wall modeling requirement Aerosol Deposition Model The effect of turbulent fluctuating velocity on the trajectories of the aerosol particles was simulated in the Lagrangian particle trajectory based on DRW where the fluctuating velocities are drawn randomly from a Gaussian random distribution of the turbulent kinetic energy. Samples of 10,000 aerosol particles of diameters range from 1.4 μm to 21 μm and with the density of 920 kg/m 3 were injected at uniformly spaced positions at the inlet to the flow. The boundary conditions were set such that the aerosols were assumed to be trapped once they strike the wall of the pipe. For each of the turbulence models, the number of aerosols transported to each section of the pipe was determined. Deposition measurements were carried out in the middle region of the pipe, comprising of section 2 to Results The turbulent flow data for pipe flows are abundantly available in literature. For positions far from the entrance of the pipe, expressions have been developed to predict the radial distribution of the time-averaged velocity and pressure drop per unit length that is in good agreement with experiment. 52

72 The Velocity Field Figure 4.3 shows a picture of how the velocity profile develops downstream the inlet, with length and color of the arrows representing the velocity as the boundary layer grows; the flow near the wall is retarded by viscous friction. The arrows in the near wall region close to the inlet indicate that the slowing of the flow in the near-wall region results in an injection of fluid into the region away from the wall to satisfy mass conservation. Thus, the velocity outside the boundary layer increases. Figure 4.3: Fluid Flow Vectors 53

73 Reichart [36] has developed an expression in dimensionless terms that predicts the radial profile of the velocity in each flow region. Equation 4.1 provides this expression [37] in dimensional terms for comparison with the results of FLUENT analyses. ( + r a) ( r a) 2 τ τwρ / w Uz = ln( a r) ρ μ 1+ 2 / Equation 4-1 Figure 4.4 and Figure 4.5 show the results of FLUENT analyses performed for the experimental setup of [10] at Reynolds s numbers of 10,000 and 50,000 respectively in comparison with Equation 4.1. The results are seen to be in good agreement. The results indicate that the axial velocity is a maximum at the centerline and zero at the wall to satisfy the no-slip boundary condition for viscous flow. vz/vz_max Reichart RSM ske RNG r/r0 Figure 4.4: Velocity Profile for Pipe at Re = 10,000 54

74 vz/vz_max Reichart RSM ske RNG r/r0 Figure 4.5: Velocity Profile for Pipe at Re = 50, Pressure Drop in a Pipe For fully-developed flow, the flow lines represent lines of constant time-averaged velocity and the friction resistance on each flow element becomes 4l Δ P = w d τ Equation 4-2 and the friction pressure drop per unit length is Δ P 4 = w l d τ Equation 4-3 where τ w = shear stress at the wall. 55

75 The Darcy friction factor f used in predicting friction pressure drop in a pipe is well represented by the transcendental expression 1 f ( f ) = 2.0log Re 0.8 Equation 4-4 ( U ) 2 z dp f = dz 2d ρ Equation 4-5 Table 4.1 and Table 4.2 compare the values for friction pressure drop per unit length obtained from the FLUENT analyses for pipe flow with the value from Equations 4.4, for Re = 10,000 and 50,000 respectively. The FLUENT results indicate that the turbulent flow pattern is established by the second section of the pipe, which is between 10 and 20 pipe diameters from the entrance. The pressure drop is higher in this region than after the boundary layer has been established. After the turbulent flow pattern has been established, the FLUENT results are in reasonable agreement with the empirical estimate. For the flow in the high Reynolds number, Table 4.2 shows that the flow pattern is established by the fifth section of the pipe. For both Reynolds numbers, the k-ε results are actually in better agreement than the Reynold s Stress Model. 56

76 Pipe Sections and Length Pressure Drop for Different Models (in Pascal) Section Number from inlet to outlet) Pipe length (l) in meters Empirical RSM Standard k-ε RNG k-ε Total Table 4.1: Pressure Drop Per Unit Length: Re = 10,000 57

77 Pipe Section and Length Pressure Drop for Different Models (in pascal) Section Number from inlet to outlet) Pipe length (l) in meters Empirical RSM Standard k-ε RNG k-ε Total Table 4.2: Pressure Drop per Unit Length: Re = 50,000 58

78 4.3.2 Aerosol Deposition Aerosol deposition from turbulent flow has been studied by a number of investigators. Experiments have been performed on the deposition of aerosol particles from turbulent air streams in pipes. From these experiments, it has been observed that particle size, degree of air turbulence, as well as roughness of the deposition surface influence particle deposition [38]. Figure 4.6 shows a summary of some of the experimentally measured aerosol particle deposition onto walls of small circular pipes. The measured depositions are usually presented as plots of dimensionless deposition velocity V + d against dimensionless relaxation timeτ + p. The deposition velocity is the effective velocity at which particles migrate toward the surface of the pipe. The deposition velocity is a function of particle size. The dimensionless deposition velocity V + d is given by V V + d d = Equation 4-6 u τ where V d = particle transport velocity to the pipe wall u τ = fluid friction velocity The fluid friction velocity is given by τ ρ w u τ = Equation 4-7 f 59

79 The dimensionless particle relaxation time τ + p is given by 2 τ pu τ τ + p = Equation 4-8 ν Figure 4.6: Experimental Data of Particle deposition from Turbulent Flows [38]. Figure 4.6 is divided into three regimes: diffusion, diffusion-impaction, and the inertia moderated regime. The results indicate that small particles in the diffusion regime have small inertia and therefore follow the turbulent eddies. These particles rely mainly on Brownian diffusion to deposit onto the walls. In the diffusion-impaction regime, V + d increases substantially for small particles. The particles may deposit from the inertia imparted by eddies without relying on Brownian diffusion in order to reach the wall. In the inertia moderated region, transport to the wall results from momentum transferred by eddies but the extent to which momentum can be transferred from an eddy to the particle is limited by the inertia of the particle. 60

80 The rate at which particles deposit onto a surface can be expressed as a function of deposition velocity V d. This deposition velocity is defined as V d J = Equation 4-9 n 0 where J = number of particles deposited per area per time n 0 = number of particles in the undisturbed flow per unit volume. For a circular pipe, penetration P L of a particle through a section of length L is related to the deposition velocity by P L n out 4LV d = = exp nin Di. d. Vave Equation 4-10 where n out = average particle concentration at the outlet n in = average particle concentration at the inlet L = length of the pipe section D id.. = inner diameter of the pipe 61

81 V ave = bulk average air velocity in the pipe equation as For this analysis, the deposition velocity can be calculated from the above V d D V 1 id.. ave = ln 4Ls PL D Q 1 id. = 2 ln 4 L0.51 m π Did../4 P0.51 m Q 1 = ln π Did.. L0.51m P0.51 m Equation 4-11 where : Q = the volumetric flow rate L 0.51m = length of the deposition section that is 0.51 m long P 0.51m = fraction penetrating the deposition section Figure 4.7 shows the comparison of particle deposition velocity from the fully developed turbulent pipe flow calculated using the CFD code FLUENT, with experimental data of Liu and Agarwal, at Reynolds numbers of 10,000. The k-ε models are able to predict behavior in the high relaxation time range but do not predict the decrease in deposition that occurs at lower relaxation times. From the graph, the RNG k- ε model is shown to substantially over-predict the deposition of aerosols with dimensionless relaxation times τ + p below

82 The results with the RSM are in approximate agreement with the experimental data over a wide range of particle relaxation time, 1 < τ + p < 100. In this range, the effects of inertia are significant on the deposition of particle onto the walls. When particles reach the boundary layer, they continue to move towards the wall of the pipe due to the inertia imparted by the turbulent eddies. However, in the diffusion dominated regime, the results of the RSM analysis show a substantial increase in deposition velocity with decreasing relaxation time, which does not appear to have a physical basis. In the FLUENT analyses, Brownian diffusion, which is the controlling mechanism for small relaxation time, was not modeled. 1 Dimensionless Deposition Velocity Liu & Agarwal RSM RNGke Dimensionless Relaxation Time Figure 4.7: Comparison between CFD and Experimental Results at Re = 10,000 63

83 1 Dimensionless Deposition Velocity Liu & Agarw al RSM Dimensionless Relaxation Time Figure 4.8: Comparison between CFD and Experimental Results at Re = 50, Assessment The deposition of aerosol particles from turbulent pipe flow onto the walls of the pipe has been investigated using FLUENT. The objective of the analyses was to determine a turbulence flow model that can best describe the inertial deposition of aerosols in simple pipe flows. From the fluid flow analysis, the results of the analyses performed in FLUENT indicate that the axial velocity is a maximum at the centerline and zero at the wall to satisfy the no-slip boundary condition for viscous flow, which is consistent with theory. The results also indicate that the turbulent flow pattern is established by the second section of the pipe, for Re = 10,000 and the fifth section of the pipe for Re = 50,

84 The aerosol deposition from the fully developed turbulent pipe flow was also calculated within FLUENT, and the results were compared with experimental data of Liu and Agarwal, at Reynolds numbers of 10,000 and 50,000. The results indicate that the k-ε models are able to predict behavior in the high relaxation time range; however, this turbulence model did not predict the decrease in deposition that occurs at lower relaxation times. El-Batsh [39] studied deposition of aerosols in turbulent pipe flow using an earlier version of FLUENT and obtained better agreement with the k-ε model than the results obtained in this analysis. At that time FLUENT did not have a model for the Saffman lift force and El-Batsh included the effect with a user-defined function. The difference between the results of El-Batsh and our experience has not been resolved. The RSM results, on the other hand, show approximate agreement with the experimental data over a wide range of particle relaxation time, 1 < τ + p < 100. However, in the region of dimensionless relaxation time less than unity, the results of the RSM analyses show an anomalous increase in deposition velocity with decreasing relaxation time. 65

85 CHAPTER 5 5. FLUID FLOW AND AEROSOL DEPOSITION ON A SINGLE AND LINEAR ARRAYS OF SPHERES Inertial deposition of aerosols on spheres has been studied experimentally. Jacober and Matteson [40] measured the inertial deposition of monodisperse 10 μm aerosols on arrays on one to six spheres of 6 mm diameter arranged linearly to the axis of the flow. The deposition was measured as a function of spacing between spheres. Matteson and Prince[41] performed a similar experiment but included the effect of electrostatic attraction. In both experiments, only one particle diameter and one sphere diameter were considered, which limits the scope of the conclusions drawn from experimental results. Hahnër et al. [42] performed experimental analysis on deposition of aerosols on single and linear array of spheres. Waldenmaier [11] performed a similar experiment but also studied deposition of aerosols on a staggered array of spheres. In their experiments, they studied inertial deposition of aerosols for the low Stokes numbers between 0.03 and 0.5, and mid-range of Stokes numbers between 0.4 and 5. 66

86 The experiments were performed in closely spaced linear arrays of equal spheres ranging from diameter m to 0.009m. Aerosol deposition on single spheres was also measured and the results compared to theoretically predicted efficiencies based on potential flows. This chapter focuses on CFD analysis of aerosol deposition on single and linear spheres and the results are compared with those of Waldenmaier. 5.1 Problem Description The main focus of Waldenmaier s study was on inertial deposition of aerosols on linear array of spheres at low Stokes and high Stokes numbers. Deposition experiments for single and multiple spheres were carried out in a circular channel of 0.1 m diameter at gas velocities ranging between 5 and 28 ms -1, and steel spheres ranging in diameter from to m. The spheres were mounted on thin wires of 150 μm thickness stretched across the channel. The wires were fixed in frames in order to facilitate the change and handling of the test arrays. Figure 5.1 shows an experimental setup for low Stokes numbers. In this setup, deposition measurements were carried out in a horizontal flow channel using monodisperse test aerosols in a diameter range from 1.5 to 4 μm. 67

87 P Velocity Measurement Prandtl Tube P Standard Orifice P P OPC ADC Computer P T Isokinetic Sampling Particle Size Analysis Figure 5.1: Experimental Setup for Deposition Efficiency at Low Stokes Numbers A modified Sinclair-LaMer condensation aerosol generator was used to produce monodisperse test aerosols. The carrier gas stream was first saturated with di-ii-ethylhexyl-sebacate (DES) vapor by means of a bubbler. The formation of monodisperse aerosol was then initiated by controlled cooling of the saturated carrier gas stream. Heterogeneous nucleation was obtained onto condensation nuclei produced from a dilute solution of sodium chloride by means of a Collision generator. The resulting particle size 68

88 was varied by adjusting the temperature of the bubbler, regulating the flow rate of the carrier gas, and by setting the nuclei concentration to a suitable level. Figure 5.2 shows the experimental setup for high Stokes numbers between 0.4 and 5 with aerosol particles in the size range of 5 to 15 μm. In this setup, a vertical flow channel was used. A spinning top generator was used to produce monodisperse test aerosols. The liquid to be atomized was delivered to the center of a flat surfaced rotor through a hypodermic needle and the rotor was driven by compressed air. A film of liquid spread outwards over the rotor to the edge where the surface tension caused droplets to form. The particle size was controlled by varying the speed of the rotor, adjusting the liquid flow rate and by diluting DES with a volatile solvent. 69

89 Particle size analysis Computer ADC OPC Standard orifice Blower P P Deposition section Velocity and concentration measurement Electrostatic neutralization Kr-85 Spinning top aerosol Compressed air Figure 5.2: Experimental set-up for Deposition Efficiency at High Stokes Numbers 70

90 In this dissertation, analyses were performed with various turbulence models available in the FLUENT code to evaluate the ability of these models to predict the results of the experimental studies described above, before applying to complex geometries, particularly that of PBMR. For single spheres, the flow Reynolds number 5 below the drag crisis ( 4 10 ) is unsteady laminar flow. Therefore, it could be well simulated by using DNS. In this analysis, FLUENT with turbulence model is used because DNS is not practical for complicated geometries and in FLUENT, using turbulent models is the only way to simulate unsteady flows. The analyses were performed using the Lagrangian approach in which the trajectories of aerosols are calculated according to the force balance imposed by the flowing fluid. The analyses were performed for single spheres in the Stokes number range between 0.03 and 10, as well as for a linear arrangement of eight spheres at air velocities ranging from 5 to 28 ms FLUENT Modeling Procedure: Single Sphere The Geometrical Model Single spheres of diameters 3.2 mm, 6.5 mm and 9 mm respectively were modeled using the preprocessor Gambit. For each sphere diameter, D s, a 2-D, axisymmetric grid was created. Each sphere was placed in a channel of diameter 5D s and of length 4Ds.The sphere was centered such that both the inlet and outlet of the channel were 2D s from the center of the sphere. Figure 5.3 shows the axisymmetric sphere mesh and boundary conditions. 71

91 Figure 5.3: Axisymmetric Grid and Boundary Conditions The velocity inlet boundary condition was specified at the inlet. This inlet boundary condition was used with incompressible flow with the magnitude and direction of the inlet velocity specified. At the centerline of the geometry, an axis boundary condition was used. No-slip boundary conditions were specified on the wall on the sphere. The outflow boundary condition was used to model the flow exit Fluid Flow Modeling The standard k-ε and its variant RNG k-ε, the standard k-ω (better for modeling separation and reattachment) and the RSM turbulence models were used for the analysis. The Reynolds number of the flow based on the sphere diameter is 5,

92 Table 5.1 is a summary of fluid flow analysis as applied to a single sphere. Sphere Diameter, D s Axial Velocity, U 0 (Re = 5,000) Turbulence Models Examined m 23 m.s -1 k-ε, k-ω, RSM m 11 m.s -1 k-ε, k-ω, RSM m 8 m.s -1 k-ε, k-ω, RSM Table 5.1: Models Analyzed for a Single Sphere Aerosol Deposition Model The effect of turbulent fluctuating velocity on the trajectories of the aerosols was simulated in the discrete random walk. Deposition was analyzed for 100 source points uniformly distributed at the inlet. An analysis was performed for aerosol diameters ranging in size from 1 μm to 15 μm and with unit density. For the cases with stochastic tracking, 100 samples were taken at each source point. The source region extended from the axis to twice the radius of the sphere. The boundary conditions were set such that the aerosols were assumed to be trapped once they strike the wall of the sphere. For each of the turbulence models, the number of aerosols trapped on the walls of the sphere was determined. The aerosol deposition fraction was determined for each sphere diameter and the results were plotted against Stokes Number. 73

93 5.3 Results: Single Sphere Fluid Flow Model The net drag force experienced by a sphere in a fluid is regarded as a sum of two components: the pressure or form drag F n S Pnds ˆ = Equation 5-1 and the viscous drag F τ S τtds ˆ = Equation 5-2 where: ˆn = the normal unit vectors on the sphere surface ˆt = the tangential unit vectors on the sphere surface P = pressure on the sphere surface τ = viscous stresses on the sphere surface For bluff bodies that create large wakes, the pressure force dominates; whereas the friction force dominates for streamlined bodies that do not create a large wake. For a spherical object in a fluid, flow separation occurs and a wake is formed downstream of the body. The net pressure on the front of the sphere is higher compared to that downstream, resulting in a pressure drag on the sphere. Figure 5.4 is the snapshot of the flow illustrated by dye injected into the boundary layer [29]. 74

94 a b Figure 5.4: Flow Past a Sphere at (a) Re = and (b) Re = (Flow visualizations by H. Werlé, courtesy of J. Delery of ONERA) 75

95 At low values of the Reynolds Number in the turbulent regime, a laminar boundary layer is established at the front surface of the sphere. As the Reynolds Number increases, the laminar boundary transition to a turbulent boundary. When the boundary layer on flow becomes turbulent, the pressure drag on the sphere is lowered. The net drag on a sphere is produced by both the pressure and shear stress effects. These two effects are considered together in most cases and an overall drag coefficient is used. For a sphere, this drag coefficient, C D is given by Equation 2-6. For a sphere, it has been found that the drag coefficient depends on Reynolds number, C D = C (Re) [20]. D For Re < 1, the drag coefficient is given by 24 C D = Equation 5-3 Re In the region, 1 < Re < 1000, the drag coefficient is given by C D = ( Re ) Equation 5-4 Re For Reynolds number in the range 1000 < Re < 5 x 10 5, the drag coefficient is roughly a constant C D 0.44 [20]. Using this value and rearranging Equation 5.3, the drag force on the sphere is found to be F D = N. Table 5.2 shows the results for the drag coefficient and drag force on a sphere for the different RANS model, as calculated using FLUENT. The CFD results are consistent with experimental results with the SST k-ω and RSM giving better results. 76

96 Model Drag coefficient Drag Force (N) k-ε RNG k-ε k-ω SST k-ω RSM Empirical Table 5.2: Drag Coefficient on a Sphere 77

97 5.3.2 Velocity Flow Field Figure 5.5 shows typical velocity vectors behind the single sphere as obtained with the RSM model at a Reynolds Number of 5,000. The figure shows a recirculation zone behind the sphere with length of x 1.3 D s from the center of the sphere. Figure 5.5: Typical Velocity Vectors behind the Sphere using RSM: Re = 5,000. The figure shows a maximum velocity of 28.2 ms -1 at the top of the sphere. The velocity vectors for the k-ε and k-ω models are shown in Figure 5.6. The k-ε turbulence model shows a smaller separation bubble and a slightly higher maximum velocity of 31.2 ms -1 with a recirculation zone at x = 0.8D s. The velocity vectors for the k-ω turbulence model show a recirculation zone at x = 1.2D s, shown in Fugre 5.6 (b). 78

98 (a) (b) Figure 5.6: Velocity Vectors for (a) k-ε (b) k-ω turbulence models. 79

99 5.3.3 Turbulence Intensity The contours of turbulence intensity for the RSM are shown in Figure 5.7. The maximum turbulence intensity is found behind the sphere at about x = 0.9D s. Figure 5.7: Contours of Turbulence Intensity for the RSM. The contours of turbulence intensities of the k-ε and k-ω turbulence models are shown in Figure 5.8. The figure shows maximum turbulence intensity for both models occurring at about x = 0.5D s. Although the fluid flow results for the different models are qualitatively similar, the characteristics of fluid flow that affect aerosol deposition have quantitative differences. 80

100 (a) (b) Figure 5.8: Contours of Turbulence Intensity for (a) k-ε and (b) k-ω turbulence models. 81

101 5.3.4 Reynolds Stresses Figure 5.9 shows contours of UU Reynolds stress and UV Reynolds stresses respectively. The values of the UU stresses are low in front of the sphere and high near the wake region. In the case of of UV stress, the values are positive before transition and then become negative in the wake region of the sphere. 82

102 (a) (b) Figure 5.9: Contours of Reynolds stress for RSM (a) UU stress (b) UV stress 83

103 5.3.5 Pressure Field Figure 5.10 shows the contours of static pressure around the sphere using the RSM. The figure shows a maximum pressure of 345 Pa at the front of the sphere and a minimum in the wake region. Figure 5.10: Contours of Static Pressure using RSM The contours of static pressure as obtained by using the k-ε and the k-ω models are shown in Figure The k-ε model shows a pressure slightly higher of 385 Pa than that given by the both the RSM and the k-ω. The results of the k-ω are consistent with RSM, showing a pressure of about 345 Pa. 84

104 (a) (b) Figure 5.11: Contours of Static Pressure using (a) k-ε and (b) k-ω Turbulence Models. 85

105 A plot of the pressure coefficient along the surface of the sphere obtained using the RSM is shown in Figure The plot shows high coefficient on the sphere front and a minimum value just before the top of the sphere. Figure 5.12: Pressure Coefficient as a function of Position for a smooth sphere. 86

106 5.3.6 Aerosol Deposition Efficiency on a Single Sphere Deposition efficiencies of aerosols on single spheres of size m, m and m spheres were measured by Waldenmaier for aerosols in the range of 1 to 15 μm and Stokes Numbers between 0.01 and 10. The collection efficiency, η, is calculated by the following expression: where: Ndeposited η = Equation 5-5 N ( A / A ) total p c N deposited = number of aerosol particles deposited on the sphere N total = upstream aerosol particle concentration 2 π D A p = projected area of the sphere, s 4 A c = area of the channel 87

107 Figure 5.13 shows deposition efficiency for a m diameter sphere for the time-averaged flow using different RANS models for Re= 5,000. The particle trajectories were simulated for low aerosol concentration assuming that no particle interactions occurred. The figure shows agreement among all of the models across the range of Stokes number. Collection efficiency ske rng rsm skw sstkw Stokes number Figure 5.13: Single Sphere Collection Efficiency for various RANS Turbulence Models 88

108 Figure 5.14 shows comparison of the inertial deposition model with the experimental data of Waldenmaier for the m, m and m diameter spheres. The FLUENT results indicate that deposition is insensitive to turbulence model. Single Sphere Deposition FLUENT 3.2mm Collection efficiency mm 9mm Stokes number Figure 5.14: Collection Efficiency on a Single Sphere: FLUENT vs. Experimental Data 89

109 5.3.7 Turbulent Deposition of Aerosols For the RANS models, the effect of turbulent fluctuating velocity on the particle trajectory was investigated using the discrete random walk (DRW) model. The different RANS models investigated are: standard k-ε and its variant RNG k-ε; standard k-ω and its variant SST k-ω, as well as the RSM. Samples of 10,000 particles were released from the inlet plane. The diameter of the particles ranged from 1 to 15 micrometers with density of 1000 kg/m 3. The particles were injected with velocity equal to the fluid inlet velocity. The trajectories were calculated for each particle. Figure 5.15 shows the simulation results for DRW model for the different turbulence models in comparison to the results presented without the effect of eddies ske rng Collection Efficiency skw rsm sstkw nonstoch Stokes Number Figure 5.15: Collection Efficiency on a Single Sphere: Stochastic Tracking. 90

110 In the Stokes number range 0.01 < N St < 0.7, the standard k-ω turbulence model shows particle deposition of about 10 to 30%. For N St > 0.4, the SSTk-ω model shows enhanced deposition relative to the standard k-ω and other RANS models. These results indicate that the k-ω models may not be applicable to the analysis of aerosol deposition on pebbles. The turbulence-enhanced deposition of particles was analyzed using RSM at the k default value of characteristic eddy lifetime of Te = The results show an ε increased deposition of about 2 to 7% in the range 0.01 < reasonable agreement with experiment N St < 0.2. These results are in The manual for the FLUENT code recommends a coefficient of 0.3 to determine the eddy lifetime for the Reynolds stress model. A comparison of results for the two coefficients is shown in Figure The RSM with a coefficient of 0.3 shows an increase in deposition for decreasing Stokes Number for with experimental data. N St < 0.1 which is inconsistent 91

111 Collection Efficiency rsm_t=0.15 rsm_t= Stokes Number Figure 5.16: Collection Efficiency on a Single Sphere. 92

112 5.3.8 Effect of Lift Force on Particle Deposition Figure 5.17 shows the turbulent dispersion FLUENT results with inclusion of the Saffman lift force. These results show a considerable increase in deposition by lift. The results with the lift force included are inconsistent with experimental results. Collection Efficiency rsm rsm_lift Stokes Number Figure 5.17: Collection Efficiency on a Single Sphere: RSM with Saffman Lift Force 93

113 5.4 Aerosol Deposition on a Linear Array of Spheres The aerosol deposition efficiency from a single sphere cannot be applied directly to an array of spheres. The particle concentration and velocity approaching the trailing spheres in an array are influenced by the wakes of the leading spheres. The objective of this part of the study is to take a small step from deposition on a single sphere to deposition for a linear array of spheres before examining deposition in a bed of spheres CFD Model: Linear Spheres Waldenmaier measured deposition on linear arrays of spheres. Eight spheres are equally spaced linearly in the direction of the flow. The arrangement is characterized by the ratio of the distance L between two successive spheres to the diameter of the sphere, D s. The first sphere facing the undisturbed flow is labeled sphere number one. The linear arrangement is shown in Figure D s V f L Figure 5.18: Linear Array of Spheres 94

114 Analyses were performed for spheres of diameter D s = m and collector spacing L/D of 2 and 6. The spheres were arranged in a channel of diameter 5D s and the leading sphere placed 4Ds from the inlet. Figure 5.19 shows the mesh generated for the collector spacing L/D s = 2, as well as the boundary conditions used. Figure 5.19: Mesh for Linear Arrangement with L/D s = 2. 95

115 5.4.2 Fluid Flow Modeling The FLUENT code has been used to model steady and incompressible fluid flow for the linear array of spheres. The problem involves air flowing over spheres of diameter m and the Reynolds number of 5,000 based on the sphere diameter and undisturbed fluid velocity. The analysis was performed in radial geometry using RSM turbulence model for collector spacing of 2 and 6 diameters and the fluid flow velocity of 23 m/s Aerosol Deposition Model Stochastic tracking of aerosols in a linear array of spheres was analyzed using the RSM. The analyses were performed at a constant Stokes number of 1.2 for the collector spacing of 2 and 6. Source points at the inlet were distributed inversely proportional to the radius to reflect a uniform source distribution. The problem was divided into two inlet source regions from the origin to twice the sphere radius and from twice the sphere radius to four times the sphere radius. In each region 100 source points were used with 100 samples per source point. The deposition results were weighted appropriately and superimposed Results: Fluid Flow As illustrated for the single sphere, for Reynolds numbers of 5,000, the wake covers a region of about 1 to 1.5 sphere diameters. This implies that for a collector spacing of 2, the second sphere is in contact with the near wake region. Figure 5.20 (a) 96

116 and (b) shows the velocity vectors between the leading sphere and the second sphere for L/D s = 2 and L/D s = 6, respectively. As the collector spacing increases, the second sphere moves out of the wake region. The velocity vectors for the wake region between the first and second spheres for the collector spacing of 6 are shown in Figure

117 (a) (b) Figure 5.20: Wake Region between Spheres (a) L/D s = 2 and (b) L/D s = 6. 98

118 5.4.5 Results: Aerosol Deposition Figure 5.21 compares FLUENT results with the experimental data of Waldenmaier for a spacing of 2. The figure shows maximum deposition of aerosols on the leading sphere which is the same as the deposition on the single spheres at the same Stokes number (Figure 5.16). Minimum deposition of aerosols is found on the second sphere. In this case, the leading sphere acts as an obstacle and shield preventing more particles to be onto the second sphere. Analyses performed without the stochastic tracking model indicate no deposition on any of the trailing spheres. Results with the DRW model are, however, consistent with experiment. In this case, the effect of stochastic tracking is to increase the diffusion of aerosols into the flow path impinging onto the downstream spheres rather than eddy-enhanced deposition Efficiency Experiment FLUENT Sphere Number Figure 5.21: Collection Efficiency on Linear Array: N Stk = 1.2 and L/D s = 2. 99

119 Figure 5.22 shows the collection efficiency of aerosol in a linear array for the collector spacing of 6. In this arrangement, the shielding effect on the second sphere by the leading sphere is also apparent. The FLUENT model shows a higher deposition on the second sphere than that of the experiment. The model, however, shows reasonable agreement with experimental data considering the simplicity of the stochastic tracking approximation Efficiency exp FLUENT Sphere Number Figure 5.22: Collection Efficiency on Linear Array: N Stk = 1.2 and L/D s =

120 Comparison of collection efficiencies for the spacing of 2 and 6 as obtained from the FLUENT analysis is shown Figure The results show higher deposition rates at larger collector spacing. For the L/D s = 2, the second sphere is in contact with the near wake region whereas for L/D s = 6, it lies in the far wake region. The results show that for closely spaced spheres, significant shadowing occurs of subsequent spheres in the linear array. For larger spacing, eddy diffusion allows particles introduced at larger values of the radius at the source boundary to migrate toward the origin and reduce the shadowing effect of preceding spheres Efficiency L/D = 2 L/D = Sphere Number Figure 5.23: Collection Efficiency on Linear Array: L/D s = 2 and

121 5.4.6 Assessment Inertial deposition of aerosol on a linear array of spheres for the N Stk = 1.2 has been investigated using computational fluid dynamics code FLUENT at two different collector spacings of 2 and 6. The results show that collection efficiency increases at the downstream spheres with increasing collector spacing. 5.5 Discussion Inertial deposition of aerosols on single and linear array of spheres has been investigated. The fluid flow field was determined using the RANS turbulence model within the CFD code FLUENT. Inertial deposition on single spheres was determined using the discrete phase model in FLUENT for the Stokes number range between 0.02 and 5. The single sphere results show good agreement with experimental data at Stokes numbers greater than 0.1. At lower Stokes numbers, where deposition results from eddy effects, the uncertainty in the experimental data is large. Nevertheless, the FLUENT results appear to be in qualitative agreement with the experimental results. Analyses on linear arrays of spheres show that the collection efficiency increases with increasing collector spacing. From the results, it can also be seen that the collection efficiency by the second sphere depends on the flow conditions, particle transport into the projection area downstream of the leading sphere and the spacing between the spheres. The agreement between FLUENT and experiment is quite reasonable considering the simplicity of the stochastic tracking model. 102

122 CHAPTER 6 6. INERTIAL DEPOSITION OF AEROSOLS IN GRANULAR BEDS Granular bed filters can be used to remove dust particles from high temperature as well as corrosive gases where other filters cannot be used. High filtration efficiency for aerosols in submicron range can be obtained in granular beds. However, the prediction of such efficiency is very difficult. The experimental and theoretical work that has been performed for these filters is applicable to the analysis of aerosol deposition in pebble beds in the laminar regime. Gal, Tardos and Pfeffer [12] performed experimental studies on the inertial deposition of aerosols in a granular bed of spheres. Their results are directly applicable to the study of deposition in a pebble bed reactor. In their study, they developed a filtration model predicting both single sphere and total bed efficiency for a dense orthorhombic packing. FLUENT analyses were performed to reproduce the experimental results. Analyses were then performed for the pebble bed reactor conditions and compared with the Gal et al. correlation. 103

123 6.1 Overview of Flow in Granular Beds A randomly packed bed of granules can be considered to be made up of a number of unit bed elements connected together. The mechanisms and nature of transport and deposition of aerosols in granular media need to be well understood in order to accurately predict the collection efficiency of such a bed. The flow processes need to be well prescribed and the assumed expressions should be valid in order for the predictions to be accurate. where In a bed of n l identical unit cells, the total efficiency is given by η = 1 (1 ) nl Equation 6-1 E n E n = efficiency of a unit cell In a square duct of cross-sectional area L 2, the characteristic length L of the unit cell for a granular bed of spherical particles of diameter D s and porosity α is given by 1/3 L π = 6(1 α) D s Equation 6-2 This characteristic length is a function of porosity and grain diameter. The number of unit cells is related to L by n l H = Equation 6-3 L where 104

124 H = the total bed height Aerosol Collection Efficiency The collection efficiency of aerosols is governed by Stokes number N st. In order to determine the characteristic efficiency curve, the Navier-Stokes equations are solved for the fluid flow field and the trajectories of the particles are determined for each streamline for each particle diameter [12, 18]. The fraction of the trajectories that intercept the collector gives the efficiency fraction is given by En for that particle size, and this deposition E n N deposited = Equation 6-4 N total The transmission fraction T f or fraction of the aerosol particles that pass through the collector is given by T f Ntotal Ndeposited = = 1 En Equation 6-5 N total Fluid Flow Models in Granular Beds A complete description of the flow in a granular medium can be achieved if the collector size, its geometry as well as flow field around the collector is specified. Appropriate porous media models can be used in order to attain the complete model description of the granular media and the flow fields. The porous media models can be grouped into two kinds: internal flows and external flows [12, 43]. 105

125 Internal Flow Models In internal flows, the aerosol carrying fluid flows through the spaces between the grains in the medium. The capillary model, where the medium is modeled as cylindrical capillaries of uniform sizes, is the simplest of the internal flow models. The walls of the capillaries act as collectors for the aerosols. The unit bed element is assumed to have N capillaries per unit bed area and for particles larger than 0.5 micron; inertial deposition is the main mechanism for aerosol filtration [12, 18]. A constricted tube model has been developed by Payatakes, Tien and Turian [44]. In this model, a unit bed is characterized by the maximum diameter of the tube D max, the minimum diameter D c of the constricted tube as well as the height H of the tube. In this model, the filtration occurs when particles deposit on the surfaces of the constricted tube. The increase in the minimum to maximum diameter ratio leads to increase in filtration efficiency. The constricted tube model, however, predicts higher deposition when compared to other internal flow models External Flow Models In external flow models, the fluid is assumed to be flowing around the collectors. In this kind of flow, the granules are approximated as spheres, and the granular medium is considered as a bed of homogeneous spherical collectors of uniform size. The model, assumed that each sphere is isolated and located at the center of a cell surrounded by the flowing fluid. The unit cells are all the same such that they experience the same flow in the voids. The flow fields in this model are given in terms of stream function. Some of 106

126 these models assume that the flow is axisymmetric and the flow to be either creeping or potential [12, 18] Fluid Flow Models in a Dense Cubic Bed Since simplified geometries are assumed, the flow models discussed above are just approximations. The aerosol trajectories obtained are far from representative of a real bed. The models usually predict pressure drops that agree well with experimental data but the velocity profiles tend to differ significantly with the actual velocity profile. The reason for this deviation is that the sphere-in-cell models assume that the spheres do not touch each other [12]. The tortuosity of the streamlines is increased by the contact points between the grains and this affects the flow of the particles through the bed. The contact points can then increase the amount of aerosols trapped by the collectors. The aerosol trajectories and the filtration efficiency of the granular bed depend strongly on the actual three-dimensional flow profile of the fluid flowing through the bed. Gal et al developed a filtration model for a dense orthorhombic packing using the flow field developed by Snyder and Stewart [45]. In their model, Snyder and Steward used a dense orthorhombic packing of spheres as shown in Figure

127 TOP VIEW x = -1 x = 0 x = +1 y = +1 y = 0 y = -1 x = -1 x = 0 x = +1 z = + 2 SIDE VIEW z = 0 z = 2 Figure 6.1 : Schematic Representation of a dense Cubic Array of Spheres 108

128 The flow is assumed to be along the z coordinate. The basic unit for characterizing the regular packed bed is a sphere with its center located at the origin of the coordinates (0, 0, 0) plus portions of the eight adjacent spheres, with porosity α = Pressure Drop in Packed Beds of Spheres A packed bed of spheres can be thought of as a large number of solid spheres in contact with each other. The spaces between the spheres form a tortuous passage of the fluid in the bed. A fluid flowing through such a packed bed does so under the influence of a pressure drop Δ P over a bed of unit length L. The pressure drop required for the liquid to flow through a granular bed can be calculated using the friction factor correlation attributed to Ergun [46]. For a packed bed containing spherical granules of diameter D s per unit volume, the Ergun equation can be written as 150 f p = Equation 6-6 Re p where f p is the friction factor for the bed is given by f p 3 ΔP D s α = 2 L ρv0 1 α Equation 6-7 and Re p is the Reynolds defined as Re p ρdv s 0 = (1 α) μ Equation

129 Rearranging Equation 6.7 and substituting for the friction factor and the Reynolds number, the pressure drop across a granular bed can be written as 2 ( ) V ( ) Δ P 150μV 1 α 1.75ρ 1 α = + Equation 6-9 L D D sα sα where V 0 Q = : Superficial velocity A Dimensionless Aerosol Trajectory Equation The motion of an aerosol in granular bed depends on the boundary conditions, its geometry and the flow field around it, external forces acting on it as well as the obstacles in its path. The trajectory of a particle moving around an obstacle can be determined once these flow parameters are known. In granular beds, diffussional and gravitational effects are significant only for very small particle diameters less than about 0.5 μm at low velocities. For an aerosol flowing in a fluid where only inertial effects are significant, the trajectory equation can be written as πμ 2 dr 6 rp dr = u 2 dt mpcc dt Equation 6-10 where r = position vector 110

130 u = fluid velocity r p = particle radius m p = mass of the particle The terms 2 dr dt 2 and dr dt are the acceleration and velocity of the particle respectively. Once the solution of Equation 6.10 is known, the collection efficiency due to inertial effects can be predicted. form By non-dimensionalizing Equation 6.10, the trajectory of the aerosol takes the 2 d R 1 dr = U 2 dt NStF dt where R = r / a, T = tuf/ a, U = u/ UF and Equation 6-11 N St are dimensionless parameters. At any Reynolds number different from zero, F is the ratio of the dimensionless superficial velocity V 0 at Re = 0 to the superficial velocity at that Reynolds number [12]. This factor is given as V = = Δ Δ Equation 6-12 ( / ) ( / ) 0 F U P U P Re= 0 Re 0 V where Δ P is the pressure drop across the bed and F can be obtained as a function of Reynolds number, F(Re). For any granular arrangement, F can be obtained using Equation 6.8. At low Reynolds number, Gal et al developed the expression 111

131 Re F = + 150(1 α ) where Equation 6-13 α = porosity Gal defined a modified Stokes number, N St, as 1.75 Re N St = NSt (1 α Equation 6-14 By performing trajectory analysis with a numerical approximation to the flow field, Gal et al [12]obtained a correlation for single cell efficiency as a function of for low values of Reynolds number N Stk E * 2N = St N St Equation 6-15 where * E is the fraction of aerosols deposited in a unit cell. 6.2 Description of Experimental System Gal et al conducted experiments in which a densely packed bed was used to filter solid latex particles from both air and a helium stream. Figure 6.2 shows the schematic representation of the experimental set up. 112

132 Air Square duct Particle Analyzer Multichannel Monitor Clay Dense cubic bed of stainless spheres Screen Particle Analyzer Square duct Particle Neutralizer Particle Charger Humidifier Rotameter Rotameter Particle Generator Compressed air Figure 6.2 : Schematic of Experimental Setup of Gal et al. 113

133 In this experiment, a dense cubic array of uniform metallic spheres was arranged in a rectangular duct. In the array, each layer is a square array of spheres in contact. Consecutive layers are offset by one pebble radius in each direction filling the gaps in the preceding layer. The granules were of diameter D g = cm and the cross-sectional area of the duct was 5.23 x 5.23 cm 2 to allow for exactly 11 x 11 spheres in all odd layers. There were 10 x 10 spheres in even layers and the excess voidage near the wall was blocked by clay to avoid fluid flow along the walls. This gives the actual crosssectional area of the bed to be 4.76 x 4.76 cm 2. The experiments were conducted with a bed of thickness equivalent to layers n = 3, 4 and 9 layers of spheres, i.e. the number of unit cells were m = 1, 4, and 7 respectively. To avoid buildup of electrostatic charge on the spheres, the bed was grounded. In order to reduce bouncing of aerosols, the surfaces of the granules were coated with a DOP layer. Monodisperse latex particles were generated from a solution of pure methanol and the concentration of particles was measured simultaneously at the inlet and at the outlet of the bed. In order to allow for fully developed flow in a duct and good mixing of aerosols in the gas, the total length of the square duct was 58 cm below and 30 cm above the bed. The experiments were carried out with the gas flowing upward and with gas flowing downward with no apparent difference in efficiency, indicating that capture by gravitational settling was negligible. The gas velocity, controlled using rotameters, ranged between 40 and 100 cm/s for air, corresponding with Re = For helium, the gas velocity ranged between 90 and 270 cm/s (Re = ). 114

134 6.3 Computational Model A unit cell approach, in which packed beds of spheres is represented by a periodic unit cell consisting of only a few collectors is considered Geometric Model The model is a dense cubic bed containing nine layers of spheres, i.e. four layers of unit cells. A three dimensional view of such a model is shown in Figure 6.3. The spheres used are of diameter D g = m. In order to reduce computational effort, only one-eighth of the geometry was modeled with symmetry boundary conditions [47]. Figure 6.3 : Three-dimensional view of the Dense Cubic Unitcell. 115

135 6.3.2 The Velocity Field The laminar fluid flow model in FLUENT was used to determine the fluid flow field. The flow was modeled using helium gas at Reynolds numbers range between 35 and 110. Once the flow fields were obtained, aerosol trajectories were calculated by superimposing them on the flow field. Approximately 200 aerosols were distributed uniformly at the inlet to the flow channel, with the velocity equal to that of the fluid, and unit density. The aerosol deposition efficiency was calculated for an aerosol diameter of 2.35 μm. 6.4 Results The fluid flow was modeled for a 4-unit cell system with three nodalization schemes. The mesh sizes are shown in Table 6.1. Mesh Size for 4unitcell coarse Fine Finest 767,920 2,585,974 10,251,275 Table 6.1: Mesh Size for a 4-Unit System 116

136 6.4.1 Comparison with Experiments The CFD results were compared with experimental data of Tardos et al and the results are shown in Figure 6.4. The results of CFD show good agreement with experimental data. The results show an increase in aerosol deposition as Reynolds number increases Collection Efficiency Experiment CFD Reynolds Number Figure 6.4 : Efficiency as a Function of Reynolds Number 117

137 Gal et al. developed a correlation E = f( N ) predicting single sphere efficiency. Using this correlation the experimental results for helium and air for a nine layer bed were compared with CFD single sphere efficiency results. Deposition as a function of modified Stokes number is shown in Figure 6.5. The CFD results agree well with experiment. St 1 Single Sphere Efficiency helium data air data CFD Modified Stokes Number Figure 6.5 : Single Sphere Efficiency as a Function of Modified Stokes Number. 118

138 6.5 Assessment of the Results The ability of the CFD code FLUENT to predict aerosol deposition in a granular bed has been analyzed using a nine layer dense cubic packing of spheres of diameter m. The fluid flow field was obtained using laminar flow model in FLUENT. The discrete phase model was used to determine collection efficiency for aerosols of diameter d p = 2.35 μm. The comparison of experimental results and CFD results with single sphere efficiency was also performed. The inertial deposition results agree well with experimental data of Gal et al. 119

139 CHAPTER 7 7. INERTIAL DEPOSITION OF AEROSOLS IN PEBBLE BEDS The Pebble Bed Modular Reactor designers claim that sufficient retention of radioactive material would occur under all possible accident scenarios such that a leaktight reactor containment structure is not necessary [1]. Despite the safety features of the concepts, these reactors still contain large inventories of radioactive materials which can attach themselves to aerosols. For this new reactor design to be introduced to a concerned public a strong safety case will have to be made and a validated source term methodology must be developed [48]. As flow circulates through a pebble bed reactor in normal operation or during an accident, the process of inertial deposition is expected to dominate the trapping of aerosols within the pebble bed. Numerous researchers have recently used computational methods to study fluid flow in packed beds of spheres. The analyses have been performed for ordered and random [49] arrays of spheres where the entire bed is simulated (Logtenberg et al [50], Nijemeisland and Dixon [51], Calis et al [52]; Nijemeisland and Dixon [53]. Other researchers [54] used CFD to simulate flow in unit cells of spheres. Gunjal et al [55] used CFD code FLUENT to characterize fluid flow in repeating unit cells for various geometries and the results compared well with experimental data of Suekane et al [56] 120

140 The analysis performed by Gunjal shows the ability of this code to predict flow fields very well. Since the CFD code is capable of predicting 3-D fluid flow in packed beds of spheres, the objective of the analyses to described in this section of the dissertation is to evaluate the ability of computational fluid dynamics codes, like FLUENT to predict the magnitude of aerosol deposition within the pebble bed. The results of these analyses are used to develop a correlation for the deposition of aerosols within an asymptotic unit cell removed from the boundary conditions at the entrance of the array. Conditions are selected for the study that corresponds to the normal operation of the PBMR, during accidents, and for low flow conditions corresponding to natural circulation in the latter stages of a loss of coolant accident. The FLUENT models are run using the RANS turbulence approximation to determine the time-averaged flow field. In forced flow, the Reynolds number through the pebble bed is high and unsteady. For conditions under which the flow is driven by natural circulation (i.e. the pumps are not operating) the flow velocity will be much lower. A low flow condition representing a natural circulation driven flow is analyzed using the laminar model in FLUENT. 121

141 7.1 Pebble Bed Characteristics The packing of pebbles in a PBMR occurs at random, and the patterns within the core vary. Because of the tortuous flow path through the pebbles, the potential for both impaction and turbulence enhanced deposition within a pebble bed array is very high Orthorhombic Array The orthorhombic array is the tightest arrangement of identical spheres. Each layer is a square array of spheres in contact. Consecutive layers are offset by one pebble radius in each direction filling the gaps in the preceding layer, as illustrated in Figure 7.1. The porosity of an orthorhombic array is Although typical random arrays have higher porosity, approximately 0.35, the orthorhombic array is convenient for studying regular arrays of pebble beds. Other regular arrays which have porosities closer to that of a random bed have unimpeded channels that penetrate the entire bed, potentially allowing a streaming of particles for extended distances. Random arrays do not have these streaming pathways. The orthorhombic configuration is the focus of this investigation. 122

142 Figure 7.1 : Orthorhombic Packing of Pebbles Cubic Array In a cubic array, the second layer of spheres is packed directly on top of the first layer such that there is central void space, as illustrated in Figure 7.2. The porosity in the cubic packing is This is the loosest regular packing of identical spheres. 123

143 Figure 7.2 : Simple Cubic Packing of Pebbles Body Centered Cubic In this array a sphere is inserted at the center of a cubic array. Within each plane, the spheres are not in contact but the center sphere touches the eight spheres in the neighboring planes. The porosity for this array is 0.32, which is intermediate between the orthorhombic and simple cubic array. The body centered array of pebbles is shown in Figure

144 Figure 7.3 : Body Centered Cubic Array of Pebbles. 125

145 7.2 Computational Model Model Geometries The geometry modeled is a unit cell of total area 3.6 x 10-3 m 2 and the flow area of x 10-3 m 2 at the inlet to the unit cell. The unit cell flow parameters are based on the following core parameters for the pebble bed modular reactor as shown in Table 7.1. Parameter Core Diameter Core Height Dimension 3.7 m 11 m Core Total Area m 2 Pebble Diameter 0.06 m Table 7.1: Unit Cell Flow Parameters 126

146 Standard Orthorhombic Array In order to overcome problems in obtaining a convergent solution, the pebbles were separated a distance of about m [47]. Three-dimensional modeling is required for this geometry and a unit cell representing such geometry is assumed, as shown in Figure 7.4 below. Figure 7.4 : Orthorhombic Unit Cell 127

147 To reduce computational efforts, only one-eighth of the geometry was modeled. Figure 7.5 shows an isometric view of the model for a 1 unit cell. The model to be used was meshed in Gambit using the Tetrahedral/Hybrid scheme, for 1-, 2-, and 3-unitcells. Figure 7.5 : View of the Computational Domain: 1unit cell Fluid Flow Modeling FLUENT is used to estimate the flow fields in the pebble bed and then to follow the trajectories of aerosols based on the forces imposed by the flow field. Boundary conditions were specified as follows: velocity inlet, no slip at the wall, symmetry at the unit cell side boundaries, and outflow boundary conditions. The interior of the domain 128

148 was specified as the fluid zone and the mesh was then exported to the FLUENT solver. The flow is modeled as incompressible and steady i.e. the applied boundary conditions do not change with time. The three-dimensional steady state solution is carried out using the segregated solver formulation. This approach solves the flow governing equations segregated from each other. Materials Table 7.2. The operating fluid is helium gas. The fluid flow properties are tabulated in Parameter Dimension Temperature Operating pressure 1173 K 8.5 MPa Density 3.49 kg/m 3 Viscosity Mass flow rate 4.7 x 10-5 Pa.s 189 kg/s Table 7.2: Helium Gas Properties 129

149 7.2.3 Aerosol Deposition Model Once the flow fields are established, the trajectories of the aerosol particles are obtained by superimposing them on the flow fields. Deviation of the aerosol s trajectory is determined by applying Stokes law according to the discrete phase model (DPM). Particles that are predicted to contact the surface are assumed to be captured. Approximately 200 aerosol particle source points are uniformly distribute along the inlet. The velocity of each source particle is initialized the same as the fluid at the inlet. The aerosols are defined as particles of unit density, 1 g/cm 3, and as inert particles. The boundary conditions for the DPM are specified such that the aerosol particle is to be trapped once it strikes the wall zones, and escapes once it comes into contact with the outlet. The initial velocity vector is assumed to be parallel to the z-axis. For turbulent flows, the stochastic behavior of eddies is simulated. In FLUENT, stochastic tracking accounts for the random effects of turbulence on the particle dispersion using the discrete walk model (DRW) for a large number of particles. In the DRW, the fluctuating velocity components are discrete piecewise constant functions of time, with the random values kept constant over an interval of time given by the characteristic lifetime of eddies. The turbulent velocity fluctuations are included in the particle force balance equation. 130

150 7.3 Results The objective of the analysis is to determine the fractional deposition that occurs in a unit cell for flow conditions that are characteristic of normal operating and accident conditions. Particular emphasis is placed on flow conditions that would be expected to be turbulent. Because the PBMR concept is still under development, reference values have been established by the design teams but these values can be expected to change with time. The reference full power flow conditions were based on conceptual design values: 189 kg/s for a core diameter of 3.7 m. These conditions were used to obtain the nominal core averaged approach velocity of 5.0 m/s. In order to cover a range of possible operating and accident transient flow conditions, this thesis covers a range of core velocities in the high Reynolds number regime from 0.31 m/s to 5.0 m/s. An intermediate value of 1.2 m/s is used as the reference case from which sensitivity studies have been performed. In order to examine deposition behavior in the low Reynolds number regime, characteristic of natural convection conditions, a core averaged approach velocity of m/s was used Low Reynolds Number Regime In order to examine deposition behavior in the low Reynolds number regime, characteristic of natural convection conditions, an inlet velocity of 0.1 m/s was used. In order to cover a range of possible accident flow conditions, a range of core averaged velocities in the low Reynolds number regime from m/s to m/s. The velocity at the inlet to the flow channel is a factor of 4.66 higher than the core average approach 131

151 velocity. These velocities as well as the corresponding Reynolds numbers based on the core average approach velocity and sphere diameter are shown in Table 7.3. Inlet Velocity (m/s) Core Average Velocity (m/s) Reynolds Number Case Case Case Table 7.3 : Reynolds Numbers for Laminar Regime Fluid Flow Results The fluid flow fields are estimated using the CFD code FLUENT, which has the capability of predicting three dimensional flow velocities. The laminar flow model was used for the analysis. The low flow corresponding to laminar flow was modeled at a coreaveraged inlet velocity of m/s. The contours of velocity magnitude for the flow are shown in Figure 7.6. The contours of pressure are shown in Figure

152 Figure 7.6 : Contours of Velocity Magnitude in Laminar Flow for Case 1. Figure 7.7 : Contours of Pressure for Laminar Flow for Case

153 Aerosol Deposition Analysis domain. Figure 7.8 shows the distribution of the source points to the inlet of the flow Figure 7.8 : Aerosol Particle Source Points at the Inlet For each of these source points, the velocity of the particle was set equal to the fluid velocity entering the flow channel. The velocity vector is assumed to be parallel to the z-axis. Because this initial flow pattern is inconsistent with flow behavior in the asymptotic unit cell, it was necessary to perform cases for increasing numbers of unit cells (1unit cell, 2 unit cells and 3 unit cells), until the results converged. 134

154 Grid Independence Studies The fluid flow was modeled for a 1-unit cell, 2-unit cell, and 3-unit cell system to examine the approach toward asymptotic flow behavior. Three different nodalization schemes were examined, as indicated in Table 7.4. Unit-cell Mesh Sizes Coarse Fine Finest 1-355, ,290 2,706, ,645 1,458,700 4,677, ,803 2,805,782 6,553,955 Table 7.4: Nodalization for the Different Unit Cells in Laminar Flow. The effect of nodalization on the deposition fraction of the aerosols is shown in Figure 7.9 for a 3-unit cell. At higher values of particle diameter near the region of complete deposition, the results are similar for the different degrees of nodalization. However, at small particle diametersthe truncation error for the coarse nodalization is substantial. At very small diameters (or Stokes number), the amount of attenuation is expected to be negligibly small. The substantial attenuation predicted in the coarse mesh analysis is an artifact of the analysis as indicated by the decreasing amount of attenuation 135

155 predicted, as shown in Figure 7.9 as the mesh is refined. The same effect was observed for 1- and 2-unit cells Deposition Fraction coarse fine finest Particle Diameter in Micrometers Figure 7.9 : Effect of Nodalization on Aerosol Deposition for a 2-UnitCell for Case

156 Size Dependent Deposition The results for deposition of aerosols as a function of unit cell size for the 1-, 2- and 3- unit cells are shown in Figure Deposition Fraction UnitCell 2-UnitCell 3-UnitCell Particle Diameter in Micrometers Figure 7.10 : Size-Dependent Deposition as a Function of Unit-cell Size. Note the unexpected behavior of the one unit cell having greater deposition than the 2- and 3-unit cells under some conditions. Because the boundary conditions for fluid velocity and aerosol velocity imposed at the inlet to the system are different from the asymptotic conditions achieved deep within the pebble bed, the deposition fraction in the first few cells will be different from the asymptotic value. However, for a large number of unit cells n, the asymptotic transmission fraction will be approximately equal to 1/n power of the total transmission factor for the n cells. 137

157 Figure 7.11 shows the deposition fraction for the 1, 2 and 3 unit cell using the square root and cubic root of transmission fractions for the 2 and 3 unit cells respectively. This plot shows that results converge to an asymptotic behavior. The single cell results are different because the boundary conditions for the first unit cell are not characteristic of the asymptotic conditions Deposition Fraction UnitCell 2-UnitCell UnitCell Particle Diameter in Micrometers Figure 7.11 : Asymptotic Deposition Fraction vs. Particle Diameter for Case Deposition as a function of Gas density The density of a fluid affects the magnitude of the forces that act within it, as well as the rate at which particles settle through it. In this analysis, the density of the helium gas was reduced to half and one-fourth of its value. The aerosol analyses were performed 138

158 at these densities and the base velocity and the results are shown in Figure There is clearly a density effect on the deposition that is not accommodated by the Stokes number Deposition Fraction rho= kg/m^3 rho=1.745 kg/m^3 rho=3.49 kg/m^ Stokes Number Figure 7.12 : Deposition as a Function of Stokes Number for Various Densities Deposition as a Function of Velocity Inertial deposition of aerosols was analyzed at a range of velocities in the low Reynolds number regime and the results are shown in Figure There is clearly a velocity effect that is not accommodated by the Stokes number. 139

159 1.00 Deposition Fraction v=0.025 m/s v = 0.05 m/s v = 0.1 m/s Stokes Number Figure 7.13 : Deposition as a Function of Stokes Number for Various Velocities 140

160 Effect of Porosity In order to assess the effect of porosity on the deposition of aerosol particles, a geometry with a porosity of 0.32 was modeled. Figure 7.14 shows the comparison of deposition fraction for a 3 unit cell at different void fractions but for the same velocity at the inlet plane. The results indicate maximum efficiency of about 70% for the larger porosity, at Stokes number of about 2. At the lower porosity, 100% efficiency occurs at a Stokes number of 0.6. There are two features of interest in this figure. For the spread geometry, as indicated earlier, streaming paths exist through the regular array, such that multiple cells become ineffective in further reducing the amount of transmitted aerosols. The second feature is the Stokes number at which deposition begins. As shown in Equation 7.1, a single correlation for deposition in a unit cell can be developed for which the ratio of inlet velocity to approach velocity is calculated as v v inlet approach π = 1 ( 4 3 π ) 2(1 α) Equation 7-1 For α = 0.26, the velocity ratio is and for α = 0.32, the velocity ratio is Thus, if the inlet velocity of the case for the spread geometry had been doubled, the onset of deposition would have occurred at ½ the Stokes number and would have coincided with the onset of deposition for the compact geometry, as expected. 141

161 Deposition Fraction Porosity = 0.26 Porosity = Stokes Number Figure 7.14 : Asymptotic Deposition as a Function of Stokes Number for Different Porosities Comparison with Experiments The attenuation for a 3-unit cell in the pebble bed reactor configuration was plotted against N St, as shown in Figure In this case the value of F is obtained using the formula from Gal et al, which was developed for flows in this region of low Reynolds number. 142

162 1.00 Deposition Fraction CFD: 3Unitcell Modified Stokes Number Figure 7.15 : Deposition as a Function of Modified Number for a 3Unitcell A comparison of the above results with experimental results of Gal et al is shown in Figure The CFD results show good agreement with the experimental results for the laminar flow conditions. 143

163 1.00 Deposition Fraction CFD Experiment: helium Experiment: Air Modified Stokes Number Figure 7.16 : Comparison of CFD Results with Experimental Data of Gal et al. 144

164 7.3.2 High Reynolds Number Regime In this regime, four inlet velocities were chosen to span the range of possible design velocities and transient conditions within the high Reynolds number regime. The fluid flow analysis was performed using the standard k-ε turbulence model. Table 7.5 shows these velocities as well as the Reynolds number based on the approach velocity and the diameter of the sphere. Inlet Velocity (m/s) Core Approach Velocity (m/s) Reynolds Number Case ,381 Case ,762 Case ,525 Case ,004 Case ,028 Table 7.5: Cases Analyzed for the High Reynolds Number Flow. 145

165 Fluid Flow Analysis The velocity contours for a 3-unit cell are shown in Figure The figure shows acceleration of the flow as the fluid passes through the gaps in between the pebbles. From these velocity vectors, it can be seen that the flow appears to have reached an asymptotic behavior by the end of the first unit cell. Figure 7.17 : Contours of Velocity for a 3-UnitCell for High Reynolds Number Case

166 Figure 7.18 shows the filled contours of kinetic energy. The figure indicates a high amount of turbulence in front of the center sphere. Figure 7.18 : Contours of Turbulent Kinetic Energy The Pressure Drop Each of the pebbles offers resistance to the flow of helium through the pebble bed. The pressure drop through a 3-unit cell configuration is shown in Figure The plot shows higher pressure at the inlet and the pressure decreasing as the flow progresses through the channel. 147

167 Figure 7.19 : Contours of Pressure for Case 3 for a 3UnitCell. The values for the total pressure drop between the inlet and the outlet for a 3-unit cell are tabulated in Table 7.6 and illustrated in Figure 7.20 below. The plot of the pressure drop against velocity show that the dependence of pressure on velocity varies as V