computational models for polydisperse particulate and ...€¦ · cambridge series in chemical...

Computational Models for Polydisperse Particulate andMultiphase Systems

Providing a clear description of the theory of polydisperse multiphase flows,with emphasis on the mesoscale modeling approach and its relationshipwith microscale and macroscale models, this all-inclusive introduction isideal, whether you are working in industry or academia. Theory is linked topractice through discussions of key real-world cases (particle/droplet/bubblecoalescence, breakup, nucleation, advection and diffusion, and physical- andphase-space), providing valuable experience in simulating systems that canbe applied to your own applications. Practical cases of QMOM, DQMOM,CQMOM, EQMOM, and ECQMOM are also discussed and compared, asare realizable finite-volume methods. This provides the tools you need to usequadrature-based moment methods, choose from the many available options,and design high-order numerical methods that guarantee realizable momentsets. In addition to the numerous practical examples, Matlab scripts for sev-eral algorithms are also provided, so you can apply the methods described topractical problems straight away.

Daniele L. Marchisio is an Associate Professor at the Politecnico di Torino,Italy, where he received his Ph.D. in 2001. He has held visiting positions at theLaboratoire des Sciences du Genie Chimique, CNRS–ENSIC (Nancy, France),Iowa State University (USA), Eidgenossische Technische Hochschule Zurich(Switzerland), and University College London (UK), and has been an invitedprofessor at Aalborg University (Denmark) and the University of Valladolid(Spain). He acts as a referee for the key international journals in his field ofresearch. He has authored more than 60 scientific papers and 5 book chapters,and co-edited the volume Multiphase Reacting Flows (Springer, 2007).

Rodney O. Fox is the Anson Marston Distinguished Professor of Engineeringat Iowa State University (USA), an Associate Scientist at the US-DOE AmesLaboratory, and a Senior Research Fellow in the EM2C laboratory at the EcoleCentrale Paris (France). His numerous professional awards include an NSFPresidential Young Investigator Award in 1992 and Fellow of the AmericanPhysical Society in 2007. The impact of Fox’s work touches every technologi-cal area dealing with multiphase flow and chemical reactions. His monographComputational Models for Turbulent Reacting Flows (Cambridge UniversityPress, 2003) offers an authoritative treatment of the field.

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Cambridge Series in Chemical Engineering

Series Editor

Arvind Varma, Purdue University

Editorial Board

Christopher Bowman, University of ColoradoEdward Cussler, University of MinnesotaChaitan Khosla, Stanford UniversityAthanassios Z. Panagiotopoulos, Princeton UniversityGregory Stephanopolous, Massachusetts Institute of TechnologyJackie Ying, Institute of Bioengineering and Nanotechnology, Singapore

Books in Series

Baldea and Daoutidis, Dynamics and Nonlinear Control of Integrated Process SystemsChau, Process Control: A First Course with MATLAB

Cussler, Diffusion: Mass Transfer in Fluid Systems, Third EditionCussler and Moggridge, Chemical Product Design, Second EditionDenn, Chemical Engineering: An IntroductionDenn, Polymer Melt Processing: Foundations in Fluid Mechanics and Heat TransferDuncan and Reimer, Chemical Engineering Design and Analysis: An IntroductionFan and Zhu, Principles of Gas–Solid FlowsFox, Computational Models for Turbulent Reacting FlowsLeal, Advanced Transport Phenomena: Fluid Mechanics and Convective TransportLim and Shin, Fed-Batch Cultures: Fundamentals, Modeling, Optimization, and Control of

Semi-Batch BioreactorsMarchisio and Fox, Computational Models for Polydisperse Particulate and Multiphase SystemsMewis and Wagner, Colloidal Suspension RheologyMorbidelli, Gavriilidis, and Varma, Catalyst Design: Optimal Distribution of Catalyst in Pellets,

Reactors, and MembranesNoble and Terry, Principles of Chemical Separations with Environmental ApplicationsOrbey and Sandler, Modeling Vapor–Liquid Equilibria: Cubic Equations of State and Their

Mixing RulesPetyluk, Distillation Theory and its Applications to Optimal Design of Separation UnitsRao and Nott, An Introduction to Granular FlowRussell, Robinson, and Wagner, Mass and Heat Transfer: Analysis of Mass Contactors and Heat

ExchangersSchobert, Chemistry of Fossil Fuels and BiofuelsSlattery, Advanced Transport PhenomenaVarma, Morbidelli, and Wu, Parametric Sensitivity in Chemical Systems






Computational Models forPolydisperse Particulate andMultiphase Systems

DANIELE L. MARCHIS IOPolitecnico di Torino

RODNEY O. FOXIowa State University






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c© Daniele L. Marchisio and Rodney O. Fox 2013

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Library of Congress Cataloging in Publication DataMarchisio, Daniele L.Computational models for polydisperse particulate and multiphase systems / Daniele L.Marchisio, Politecnico di Torino, Rodney O. Fox, Iowa State University.

pages cm. – (Cambridge series in chemical engineering)ISBN 978-0-521-85848-91. Multiphase flow – Mathematical models. 2. Chemical reactions – Mathematical models.3. Transport theory. 4. Dispersion – Mathematical models. I. Fox, Rodney O., 1959–II. Title.TA357.5.M84M37 2013532′.56–dc23

2012044073

ISBN 978-0-521-85848-9 Hardback

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a Giampaoloa Roberte






Contents

Preface page xiii

Notation xvii

1 Introduction 11.1 Disperse multiphase flows 11.2 Two example systems 3

1.2.1 The population-balance equation for fine particles 31.2.2 The kinetic equation for gas–particle flow 8

1.3 The mesoscale modeling approach 141.3.1 Relation to microscale models 161.3.2 Number-density functions 181.3.3 The kinetic equation for the disperse phase 191.3.4 Closure at the mesoscale level 201.3.5 Relation to macroscale models 20

1.4 Closure methods for moment-transport equations 231.4.1 Hydrodynamic models 231.4.2 Moment methods 25

1.5 A road map to Chapters 2–8 27

2 Mesoscale description of polydisperse systems 302.1 Number-density functions (NDF) 30

2.1.1 Length-based NDF 322.1.2 Volume-based NDF 332.1.3 Mass-based NDF 332.1.4 Velocity-based NDF 34

2.2 The NDF transport equation 352.2.1 The population-balance equation (PBE) 352.2.2 The generalized population-balance equation (GPBE) 372.2.3 The closure problem 37

2.3 Moment-transport equations 382.3.1 Moment-transport equations for a PBE 382.3.2 Moment-transport equations for a GPBE 40

2.4 Flow regimes for the PBE 432.4.1 Laminar PBE 432.4.2 Turbulent PBE 44

2.5 The moment-closure problem 45

vii






viii Contents

3 Quadrature-based moment methods 473.1 Univariate distributions 47

3.1.1 Gaussian quadrature 493.1.2 The product–difference (PD) algorithm 513.1.3 The Wheeler algorithm 533.1.4 Consistency of a moment set 55

3.2 Multivariate distributions 623.2.1 Brute-force QMOM 633.2.2 Tensor-product QMOM 683.2.3 Conditional QMOM 74

3.3 The extended quadrature method of moments (EQMOM) 823.3.1 Relationship to orthogonal polynomials 833.3.2 Univariate EQMOM 843.3.3 Evaluation of integrals with the EQMOM 913.3.4 Multivariate EQMOM 93

3.4 The direct quadrature method of moments (DQMOM) 99

4 The generalized population-balance equation 1024.1 Particle-based definition of the NDF 102

4.1.1 Definition of the NDF for granular systems 1024.1.2 NDF estimation methods 1054.1.3 Definition of the NDF for fluid–particle systems 107

4.2 From the multi-particle–fluid joint PDF to the GPBE 1104.2.1 The transport equation for the multi-particle joint PDF 1114.2.2 The transport equation for the single-particle joint PDF 1124.2.3 The transport equation for the NDF 1124.2.4 The closure problem 113

4.3 Moment-transport equations 1144.3.1 A few words about phase-space integration 1144.3.2 Disperse-phase number transport 1164.3.3 Disperse-phase volume transport 1164.3.4 Fluid-phase volume transport 1174.3.5 Disperse-phase mass transport 1184.3.6 Fluid-phase mass transport 1214.3.7 Disperse-phase momentum transport 1234.3.8 Fluid-phase momentum transport 1244.3.9 Higher-order moment transport 127

4.4 Moment closures for the GPBE 130

5 Mesoscale models for physical and chemical processes 1365.1 An overview of mesoscale modeling 136

5.1.1 Mesoscale models in the GPBE 1375.1.2 Formulation of mesoscale models 1415.1.3 Relation to macroscale models 145

5.2 Phase-space advection: mass and heat transfer 1475.2.1 Mesoscale variables for particle size 1495.2.2 Size change for crystalline and amorphous particles 1525.2.3 Non-isothermal systems 1555.2.4 Mass transfer to gas bubbles 156






Contents ix

5.2.5 Heat/mass transfer to liquid droplets 1585.2.6 Momentum change due to mass transfer 160

5.3 Phase-space advection: momentum transfer 1615.3.1 Buoyancy and drag forces 1625.3.2 Virtual-mass and lift forces 1715.3.3 Boussinesq–Basset, Brownian, and thermophoretic forces 1735.3.4 Final expressions for the mesoscale acceleration models 175

5.4 Real-space advection 1775.4.1 The pseudo-homogeneous or dusty-gas model 1795.4.2 The equilibrium or algebraic Eulerian model 1805.4.3 The Eulerian two-fluid model 1815.4.4 Guidelines for real-space advection 182

5.5 Diffusion processes 1835.5.1 Phase-space diffusion 1845.5.2 Physical-space diffusion 1875.5.3 Mixed phase- and physical-space diffusion 188

5.6 Zeroth-order point processes 1895.6.1 Formation of the disperse phase 1895.6.2 Nucleation of crystals from solution 1915.6.3 Nucleation of vapor bubbles in a boiling liquid 191

5.7 First-order point processes 1925.7.1 Particle filtration and deposition 1935.7.2 Particle breakage 195

5.8 Second-order point processes 2025.8.1 Derivation of the source term 2035.8.2 Source terms for aggregation and coalescence 2055.8.3 Aggregation kernels for fine particles 2065.8.4 Coalescence kernels for droplets and bubbles 212

6 Hard-sphere collision models 2146.1 Monodisperse hard-sphere collisions 215

6.1.1 The Boltzmann collision model 2176.1.2 The collision term for arbitrary moments 2186.1.3 Collision angles and the transformation matrix 2216.1.4 Integrals over collision angles 2236.1.5 The collision term for integer moments 230

6.2 Polydisperse hard-sphere collisions 2366.2.1 Collision terms for arbitrary moments 2376.2.2 The third integral over collision angles 2426.2.3 Collision terms for integer moments 243

6.3 Kinetic models 2466.3.1 Monodisperse particles 2466.3.2 Polydisperse particles 248

6.4 Moment-transport equations 2506.4.1 Monodisperse particles 2516.4.2 Polydisperse particles 255

6.5 Application of quadrature to collision terms 2616.5.1 Flux terms 2616.5.2 Source terms 263






x Contents

7 Solution methods for homogeneous systems 2667.1 Overview of methods 2667.2 Class and sectional methods 269

7.2.1 Univariate PBE 2697.2.2 Bivariate and multivariate PBE 2797.2.3 Collisional KE 283

7.3 The method of moments 2897.3.1 Univariate PBE 2907.3.2 Bivariate and multivariate PBE 2967.3.3 Collisional KE 297

7.4 Quadrature-based moment methods 3007.4.1 Univariate PBE 3017.4.2 Bivariate and multivariate PBE 3077.4.3 Collisional KE 314

7.5 Monte Carlo methods 3157.6 Example homogeneous PBE 319

7.6.1 A few words on the spatially homogeneous PBE 3197.6.2 Comparison between the QMOM and the DQMOM 3237.6.3 Comparison between the CQMOM and Monte Carlo 324

8 Moment methods for inhomogeneous systems 3298.1 Overview of spatial modeling issues 329

8.1.1 Realizability 3308.1.2 Particle trajectory crossing 3328.1.3 Coupling between active and passive internal coordinates 3358.1.4 The QMOM versus the DQMOM 337

8.2 Kinetics-based finite-volume methods 3408.2.1 Application to PBE 3418.2.2 Application to KE 3458.2.3 Application to GPBE 347

8.3 Inhomogeneous PBE 3498.3.1 Moment-transport equations 3498.3.2 Standard finite-volume schemes for moments 3508.3.3 Realizable finite-volume schemes for moments 3538.3.4 Example results for an inhomogeneous PBE 358

8.4 Inhomogeneous KE 3628.4.1 The moment-transport equation 3638.4.2 Operator splitting for moment equations 3638.4.3 A realizable finite-volume scheme for bivariate

velocity moments 3648.4.4 Example results for an inhomogeneous KE 366

8.5 Inhomogeneous GPBE 3738.5.1 Classes of GPBE 3738.5.2 Spatial transport with known scalar-dependent velocity 3768.5.3 Example results with known scalar-dependent velocity 3778.5.4 Spatial transport with scalar-conditioned velocity 3818.5.5 Example results with scalar-conditioned velocity 3888.5.6 Spatial transport of the velocity-scalar NDF 396

8.6 Concluding remarks 401






Contents xi

Appendix A Moment-inversion algorithms 403A.1 Univariate quadrature 403

A.1.1 The PD algorithm 403A.1.2 The adaptive Wheeler algorithm 404

A.2 Moment-correction algorithms 405A.2.1 The correction algorithm of McGraw 405A.2.2 The correction algorithm of Wright 407

A.3 Multivariate quadrature 408A.3.1 Brute-force QMOM 408A.3.2 Tensor-product QMOM 410A.3.3 The CQMOM 412

A.4 The EQMOM 413A.4.1 Beta EQMOM 413A.4.2 Gamma EQMOM 416A.4.3 Gaussian EQMOM 418

Appendix B Kinetics-based finite-volume methods 421B.1 Spatial dependence of GPBE 421B.2 Realizable FVM 423B.3 Advection 427B.4 Free transport 429B.5 Mixed advection 434B.6 Diffusion 437

Appendix C Moment methods with hyperbolic equations 441C.1 A model kinetic equation 441C.2 Analytical solution for segregated initial conditions 442

C.2.1 Segregating solution 442C.2.2 Mixing solution 443

C.3 Moments and the quadrature approximation 444C.3.1 Moments of segregating solution 444C.3.2 Moments of mixing solution 446

C.4 Application of QBMM 447C.4.1 The moment-transport equation 447C.4.2 Transport equations for weights and abscissas 448

Appendix D The direct quadrature method of moments fully conservative 450D.1 Inhomogeneous PBE 450D.2 Standard DQMOM 450D.3 DQMOM-FC 453D.4 Time integration 455

References 459

Index 488






Preface

This book is intended for graduate students in different branches of science and engineering(i.e. chemical, mechanical, environmental, energetics, etc.) interested in the simulation ofpolydisperse multiphase flows, as well as for scientists and engineers already working inthis field. The book provides, in fact, a systematic and consistent discussion of the basictheory that governs polydisperse multiphase systems, which is suitable for a neophyte,and presents a particular class of computational methods for their actual simulation, whichmight interest the more experienced scholar.

As explained throughout the book, disperse multiphase systems are characterized bymultiple phases, with one phase continuous and the others dispersed (i.e. in the form ofdistinct particles, droplets, or bubbles). The term polydisperse is used in this context tospecify that the relevant properties characterizing the elements of the disperse phases, suchas mass, momentum, or energy, change from element to element, generating what are com-monly called distributions. Typical distributions, which are often used as characteristicsignatures of multiphase systems, are, for example, a crystal-size distribution (CSD), aparticle-size distribution (PSD), and a particle-velocity distribution.

The problem of describing the evolution (in space and time) of these distributions hasbeen treated in many ways by different scientific communities, focusing on aspects mostrelevant to their community. For example, in the field of crystallization and precipitation,the problem is described (often neglecting spatial inhomogeneities) in terms of crystal orparticle size, and the resulting governing equation is called a population-balance equation(PBE). In the field of evaporating (and non-evaporating) sprays the problem is formu-lated in terms of the particle surface area and the governing equation is referred to as theWilliams–Boltzmann equation. In this and other fields great emphasis has been placed onthe fact that the investigated systems are spatially inhomogeneous. Aerosols and ultra-fineparticles are often described in terms of particle mass, and the final governing equation iscalled the particle-dynamics equation. Particulate systems involved in granular flows haveinstead been investigated in terms of particle velocity only, and the governing equation isthe inelastic extension to multiphase systems of the well-known Boltzmann equation (BE)used to describe molecular velocity distributions in gas dynamics.

Although these apparently different theoretical frameworks are referred to by differentnames, the underlying theory (which has its foundation in classical statistical mechanics) isexactly the same. This has also generated a plethora of numerical methods for the solutionof the governing equations, often sharing many common elements, but generally with aspecific focus on only part of the problem. For example, in a PBE the distribution repre-senting the elements constituting the multiphase system is often discretized into classes orsections, generating the so-called discretized population-balance equation (DPBE). Amongthe many methods developed, one widely used among practitioners in computational fluiddynamics (CFD) is the multiple-size-group (MUSIG) method. This approach resembles,

xiii






xiv Preface

in its basic ideas, the discretization carried out for the BE in the so-called discrete-velocitymethod (DVM). Analogously, the method of moments (MOM) has been used for the solu-tion of both PBE and BE, but the resulting closure problem is overcome by followingdifferent strategies in the two cases. In the case of the BE the most popular moment clo-sure is the one proposed by Grad, which is based on the solution of a subset of 13 or26 moments, coupled with a presumed functional form for the velocity distribution. Incontrast, in the case of a PBE the closure strategy often involves interpolation among theknown moments (as in the method of moments with interpolative closure, MOMIC). Giventhe plethora of approaches, for the novice it is often impossible to see the connectionsbetween the methods employed by the different communities.

This book provides a consistent treatment of these issues that is based on a generaltheoretical framework. This, in turn, stems from the generalized population-balance equa-tion (GPBE), which includes as special cases all the other governing equations previouslymentioned (e.g. PBE and BE). After discussing how this equation originates, the differentcomputational models for its numerical solution are presented. The book is structured asfollows.

• Chapter 1 introduces key concepts, such as flow regimes and relevant dimensionlessnumbers, by using two examples: the PBE for fine particles and the KE for gas–particle flow. Subsequently the mesoscale modeling approach used throughout thebook is explained in detail, with particular focus on the relation to microscale andmacroscale models and the resulting closure problems.

• Chapter 2 provides a brief introduction to the mesoscale description of polydispersesystems. In this chapter the many possible number-density functions (NDF), formu-lated with different choices for the internal coordinates, are presented, followed by anintroduction to the PBE in their various forms. The chapter concludes with a shortdiscussion on the differences between the moment-transport equations associatedwith the PBE, and those arising due to ensemble averaging in turbulence theory.

• Chapter 3 provides an introduction to Gaussian quadrature and the moment-inversionalgorithms used in quadrature-based moment methods (QBMM). In this chapter,the product–difference (PD) and Wheeler algorithms employed for the classicalunivariate quadrature method of moments (QMOM) are discussed, together withthe brute-force, tensor-product, and conditional QMOM developed for multivari-ate problems. The chapter concludes with a discussion of the extended quadraturemethod of moments (EQMOM) and the direct quadrature method of moments(DQMOM).

• In Chapter 4 the GPBE is derived, highlighting the closures that must be introducedfor the passage from the microscale to the mesoscale model. This chapter also con-tains an overview of the mathematical steps needed to derive the transport equationsfor the moments of the NDF from the GPBE. The resulting moment-closure problemis also throughly discussed.

• Chapter 5 focuses on selected mesoscale models from the literature for key phys-ical and chemical processes. The chapter begins with a general discussion of themesoscale modeling philosophy and its mathematical framework. Since the numberof mesoscale models proposed in the literature is enormous, the goal of the chapter isto introduce examples of models for advection and diffusion in real and phase space






Preface xv

and zeroth-, first-, and second-order point processes, such as nucleation, breakage,and aggregation.

• Chapter 6 is devoted to the topic of hard-sphere collision models (and related simplerkinetic models) in the context of QBMM. In particular, the exact source terms forinteger moments due to collisions are derived in the case of inelastic binary collisionsbetween two particles with different diameters/masses, and the use of QBMM toovercome the closure problem is illustrated.

• Chapter 7 is devoted to solution methods of the spatially homogeneous GPBE,including class and sectional methods, MOM and QBMM, and Monte Carlo meth-ods. The chapter concludes with a few examples comparing solution methods forselected homogeneous PBE.

• Chapter 8 focuses on the use of moment methods for solving a spatially inhomo-geneous GPBE. Critical issues with spatially inhomogeneous systems are momentrealizability and corruption (due to numerical advection and diffusion operator) andthe presence of particle trajectory crossing (PTC). These are discussed after introduc-ing kinetics-based finite-volume methods, by presenting numerical schemes capableof preserving moment realizability and by demonstrating with practical examplesthat QBMM are ideally suited for capturing PTC. The chapter concludes with anumber of spatially one-dimensional numerical examples.

• To complete the book, four appendices are included. Appendix A contains theMatlab scripts for the most common moment-inversion algorithms presented inChapter 3. Appendix B discusses in more detail the kinetics-based finite-volumemethods introduced in Chapter 8. Finally, the key issues of PTC in phase space,which occurs in systems far from collisional equilibrium, and moment conservationwith some QBMM are discussed in Appendix C and Appendix D, respectively.

The authors are greatly indebted to the many people who contributed in different waysto the completion of this work. Central in this book is the pioneering research of Dr. RobertL. McGraw, who was the first to develop QMOM and the Jacobian matrix transformation(which is the basis for DQMOM) for the solution of the PBE, and brought to our atten-tion the importance of moment corruption and realizability when using moment methods.The authors are therefore especially grateful to Professor Daniel E. Rosner, who in 1999directed their attention to the newly published work of Dr. McGraw on QMOM. Theywould also like to thank Professor R. Dennis Vigil for recognizing the capability of QMOMfor solving aggregation and breakage problems, and Professor Prakash Vedula for provid-ing the mathematical framework used to compute the moment source terms for hard-spherecollisions reported in Chapter 6.

A central theme of the solution methods described in this book is the importance ofmaintaining the realizability of moment sets in the numerical approximation. On this point,the authors are especially indebted to Professor Marc Massot for enlightening them onthe subtleties of kinetics-based methods for hyperbolic systems and the general topic ofparticle trajectory crossings. Thanks to the excellent numerical analysis skills of ProfessorOlivier Desjardins and a key suggestion by Dr. Philippe Villedieu during the 2006 SummerProgram at the Center for Turbulence Research, Professor Massot’s remarks eventuallypointed us in the direction of the realizable finite-volume schemes described in Chapter 8.In this regard, we also want to acknowledge the key contributions of Professor Z. J. Wang






xvi Preface

in the area of high-order finite-volume schemes and Dr. Varun Vikas for the developmentand implementation of the realizable quasi-high-order schemes described in Appendix B.

The idea of publishing this book with Cambridge University Press is the result ofthe interest shown in the topic by Professor Massimo Morbidelli. The contribution ofmany other colleagues is also gratefully acknowledged, among them Antonello A. Barresi,Marco Vanni, Giancarlo Baldi, Miroslav Soos, Jan Sefcik, Christophe Chalons, FrederiqueLaurent, Hailiang Liu, Alberto Passalacqua, Venkat Raman, Julien Reveillon, and ShankarSubramaniam. All the graduate students and post-doctoral researchers supervised by theauthors in the last ten years who have contributed to the findings reported in this book aregratefully acknowledged and their specific contributions are meticulously cited.

The research work behind this book has been funded by many institutions and amongthem are worth mentioning the European Commission (DLM), the Italian Ministry of Edu-cation, University, and Research (DLM), the ISI Foundation (DLM, ROF), the US NationalScience Foundation (ROF), the US Department of Energy (ROF), the Ecole Centrale Paris(ROF), and the Center for Turbulence Research at Stanford University (ROF). The constantstimulus and financial support of numerous industrial collaborators (ENI, Italy; BASF,Germany; BP Chemicals, USA; Conoco Phillips, USA; and Univation Technologies, USA)are also deeply appreciated.






Notation

Upper-case Roman

A generic particle acceleration due to buoyancy, gravity, and dragA coefficient matrix in DQMOM and brute-force QMOM for

determining the quadrature approximationA coefficient matrix constituted by mixed moments for calculating

velocity parameters un in inhomogeneous systems

AD particle cross-section surface areaAeq area of equivalent sphereAH Hamaker parameterAp particle surface area

Af fluid acceleration due to body forcesAfp pure particle acceleration due to fluid–particle momentum exchangeAp pure particle acceleration due to body forcesAp,0 mean particle-acceleration termApf pure fluid acceleration due to fluid–particle momentum exchange

A(n) acceleration acting on nth particle due to body forces(for particles in vacuum)

A(n)f acceleration acting on the fluid in the neighborhood of the

nth particle due to pressure, body, and viscous forcesA(n)

fp acceleration acting on the nth particledue to fluid–particle forces

A(n)p acceleration acting on the nth particle due to body forces

(for particles suspended in a fluid)A(n)

pf acceleration acting on the fluid in the neighborhood of thenth particle due to fluid–particle forces

A∗fp global particle acceleration due to fluid–particle momentumexchange (including diffusion terms)

A∗i coefficient matrix constituted by moments m∗j,k;i for calculatingvelocity parameters un in inhomogeneous systems with FVM

Ai jkl collision frequencies between particle-velocity classes in DVM

〈Af〉1 multi-particle conditional expected fluid accelerationdue to body forces

〈Afp〉1 multi-particle conditional expected particle accelerationdue to fluid–particle forces

xvii






xviii Notation

〈Ap〉1 multi-particle conditional expected particle accelerationdue to field forces

〈Apf〉1 multi-particle conditional expected fluid accelerationdue to fluid–particle forces

〈A(n)fp 〉1 single-particle conditional expected continuous particle acceleration

due to fluid–particle forces〈A(n)

p 〉1 single-particle conditional expected continuous particle accelerationdue to field forces

�Af�f total acceleration of the fluid seen by the particlesdue to forces in the fluid phase

�Afp�N total acceleration acting on monodisperse particles(of constant size and mass) due to drag, lift, and pressure forces

�Afp�p total acceleration acting on particles due to drag, lift,and pressure forces

�Ap�N total acceleration acting on monodisperse particles(of constant size and mass) due to body forces

�Ap�p total acceleration acting on particles due to body forces�Apf�f total acceleration of fluid seen by the particles

due to momentum transfer between phasesArp Archimedes number for disperse phase

B(x, y) beta functionB(g, x) hard-sphere collision kernelBi j rate of change of the particle-number density in intervals I(i)

1 and I( j)2

due to a generic point process (in CM for bivariate systems)Bagg

i rate of change of the particle number in interval Ii due toaggregation (in CM for univariate systems)

Bbreaki rate of change of the particle number in interval Ii due to

breakage (in CM for univariate systems)

B coefficient matrix constituted by mixed moments for calculatingvelocity parameters un (in inhomogeneous systems)

Bfv mixed phase-space diffusion tensor for fluid velocityand particle velocity

Bfvf pure phase-space diffusion tensor for fluid velocityBfξ mixed phase-space diffusion tensor for fluid velocity

and particle internal coordinateBfξf mixed phase-space diffusion tensor for fluid velocity

and fluid internal coordinate

Bpv pure phase-space diffusion tensor for particle velocityBpvf mixed phase-space diffusion tensor for particle velocity

and fluid velocityBpξ mixed phase-space diffusion tensor for particle velocity

and particle internal coordinateBpξf mixed phase-space diffusion tensor for particle velocity

and fluid internal coordinateBo Bond number (equal to Eotvos number)

C constant appearing in the parabolic daughter distribution functionfor particle breakage

C(ψ) collisional source term for monodisperse system






Notation xix

C1 coefficient appearing in definition of the fluid effective(laminar plus turbulent) viscosity μf,eff

C1–C7 constants appearing in breakage kernel

CD particle-drag coefficientC∗D particle-drag coefficient including the Cunningham correction

factor for rarefied continuous phaseCi concentration of the potential-determining ionsCL lift-force coefficientCm momentum exchange coefficient appearing in

thermophoretic forceCs thermal slip coefficient appearing in thermophoretic forceCt thermal exchange coefficient appearing in

thermophoretic force

Cvm virtual-mass force coefficient identifying the fraction offluid volume moving with a particle

Cα coefficients appearing in the functional expansion of the NDFCαβ(ψ) collisional source term for polydisperse system

(particles of types α and β)C(m)αβ (ψ) terms appearing in the expansion of the collisional source term

for polydisperse systems (particles of types α and β)C∗γi jk approximate collision source term for velocity moments

of global order γ (monodisperse systems)C∗γi jk,α approximate collision source term for velocity moments

of global order γ of particle type α (polydisperse systems)

C generic collisional source termCαβ collisional source term for particles of types α and βCNp Np-particle collision operatorCfv mixed phase-space diffusion tensor for fluid internal

coordinate and particle velocityCfvf mixed phase-space diffusion tensor for fluid internal

coordinate and fluid velocityCfξ mixed phase-space diffusion tensor for fluid internal

coordinate and particle internal coordinate

Cfξf pure phase-space diffusion tensor for fluid internal coordinateCpv mixed phase-space diffusion tensor for particle internal

coordinate and particle velocityCpvf mixed phase-space diffusion tensor for particle internal

coordinate and fluid velocityCpξ pure phase-space diffusion tensor for particle internal coordinateCpξf mixed phase-space diffusion tensor for particle internal

coordinate and fluid internal coordinateCpzc

i concentration of potential determining ions at pointof zero charge

C(n)D drag coefficient for the nth particle

C(n)fξ rate of change of the internal coordinate vector for the fluid

surrounding the nth particle due to discontinuous eventsC(n)

pU rate of change of velocity for the nth particledue to collisions (particles suspended in fluid)






xx Notation

C(n)pξ rate of change of particle internal coordinate vector for nth particle

due to collisions (particles suspended in fluid)

C(n)U rate of change of velocity for the nth particle

due to discontinuous particle collisions (in vacuum)C(n)ξ rate of change of particle internal coordinate vector for nth particle

due to particle collisions (in vacuum)C(n)

1 single-particle collision operatorC(m)

l1l2l3collision source terms for integer moments of orders l1, l2, and l3

with respect to the three velocity components

D solute molecular diffusion coefficientDb diameter of a stable bridge between two aggregating particlesDf particle fractal dimensionDG average size of objects constituting a porous mediumD0 cut-off distance for calculating the Hamaker parameterD symmetric N ×N diffusion matrixD volume-average symmetric N ×N diffusion matrixD∗ dimensionless normalized diffusion matrixDaa aggregation Damkohler numberDab breakage Damkohler number

E bubble aspect ratioEαβ energy scaling factor in polydisperse systems (particle types α and β)Ep total particle granular energyEo Eotvos number

F inter-particle forceF(ζ) dimensionless normalized NDFFt cumulative probability distribution for the quiescence time in

MC methodsFγ

i,l1l2l3ith component of spatial flux for moment of global order γ

F+i,l1l2l3ith component of spatial flux for moment of global order γcorresponding to positive velocity

F−i,l1l2l3ith component of spatial flux for moment of global order γcorresponding to negative velocity

F(M) generic moment flux functionFfp drag and buoyancy fluid–particle forceFγ

l1l2l3spatial flux for moment of global order γ

F+l1l2l3spatial flux for moment of global order γcorresponding to positive velocity

F−l1l2l3spatial flux for moment of global order γcorresponding to negative velocity

Fl flow number for particle aggregationFrg Froude number for the continuous phase

G(nl, nr) numerical flux function for inhomogeneous systemsdiscretized with FVM

Gf fluid shear rate






Notation xxi

Gi gain rate of particles with velocity ξi due to collisions in DVMGi jk,αβ Gaussian moments with mean velocity Uαβ and

covariance matrix σαβGL continuous rate of change of particle size (growth rate)GL,k average particle growth rate for moment of order k

Gm(ψ) component m of the collisional-flux termGm,l1l2l3 collisional-flux term for integer moments of orders l1, l2, and l3

with respect to the three velocity componentsGp pure advection component for the rate of change of crystal sizeGo

f correction for fluid-dynamic interactions between particlesGk

n coefficients appearing in the calculation of the velocity parameters un

for inhomogeneous systemsG∗p global rate of change of crystal sizeG∗p1 mass-transfer rate from fluid to particle

G vectorial numerical flux function for inhomogeneous systemsdiscretized with FVM

G(ψ) collisional-flux term for monodisperse systemsGf pure mesoscale advection model for the fluid internal coordinateGl1l2l3 collisional-flux vector for velocity moment of order l1l2l3

Gp pure mesoscale advection model for particle internal coordinateGαβ(ψ) collisional-flux term for polydisperse systems constituted by

particle types α and β

G(n) continuous rate of change of the internal coordinate vectorfor the nth particle (particles in vacuum)

G(n)f continuous rate of change of the internal coordinate vector

for the fluid surrounding the nth particleG(n)

p continuous rate of change of the internal coordinate vectorfor the nth particle (particles suspended in fluid)

G(m)αβ (ψ) terms appearing in the expansion for the collisional-flux term

for polydisperse systems with particle types α and β

〈Gf〉1 multi-particle conditional expected continuous rate of changeof fluid internal coordinate vector

〈G(n)f 〉1 single-particle conditional expected continuous rate of change

of fluid internal coordinate vector〈Gp〉1 multi-particle conditional expected continuous rate of change

of particle internal coordinate vector〈G(n)

p 〉1 single-particle conditional expected continuous rate of changeof particle internal coordinate vector

�Gf� rate of change of fluid-phase mass density due tocontinuous processes

�Gp� rate of change of disperse-phase mass density due tocontinuous processes

�Gf�V rate of change of fluid-phase volume fraction due tocontinuous processes

�Gp�V rate of change of disperse-phase volume fraction dueto continuous processes

�Gf�f global fluid momentum rate of change due to mass transferfrom fluid phase






xxii Notation

�Gp�p global particle momentum rate of changedue to mass transfer from liquid phase

H(ξ) Heaviside step functionH functional appearing in the general definition of the pair

distribution function (function of the moments)Ho distance between two primary particles within an aggregateHm,l1l2l3 symmetric change of variable involving I(m)

l1l2l3

Ii finite ith internal coordinate interval used in CMIr relative turbulence intensityI[x1 ,x2](x) indicator function equal to unity if x ∈ [x1, x2] and zero otherwiseI(i)

j finite ith interval for the jth internal coordinatein CM when extended to multivariate problems

I(m)l1l2l3

factor appearing in collision term for integervelocity moments of orders l1, l2, and l3

I(p,q)l1l2l3

factor appearing in the collision term for polydisperse systemsfor integer velocity moments of orders l1, l2, and l3

J molar flux of solute molecules at particle surfaceJ(φf ) rate of particle formationJ(η, η) Jacobian of variable transformation relating phase-space

variables before and after collisionJf rate of particle formationJf volume-average rate of particle formationJk kth moment of the rate of particle formation in univariate GPBEJk kth moment of the rate of particle formation in multivariate GPBEJ∗ dimensionless and normalized rate of particle formation

KB history-force kernelK(m)

i jk integral over collision anglesK(mn)

i jk coefficients appearing in the third integral over collision anglesK exponent matrix used to build the quadrature approximationK moment vector used in the definition of the moment set MKm

i,α reconstructed K in the ith cell at time step m employed in FVMK±α,l/r K evaluated with v+α,l or v−α,r

Kn Knudsen number for continuous phase (relative to particle diameter)Kn∗ Knudsen number for continuous phase (relative to particle radius)Knp Knudsen number for disperse phase

L characteristic length of the system under investigationL particle lengthLi loss rate of particles with velocity ξi due to collisions in DVMLv latent heat of evaporationL10 number-average mean particle lengthL32 area-average mean particle length or Sauter diameterL43 volume-average mean particle lengthL transformation matrix between laboratory and collision framesLe Lewis number






Notation xxiii

M number of internal coordinates appearing in the NDFM number of sections or classes used in CM and DVM

Mf number of fluid internal coordinates appearing in the NDFMG ratio between particle and collector sizeMk number of intervals used for the kth internal

coordinate in multivariate CMMi mass of the particles in the interval Ii in CMMp number of particle internal coordinates appearing in the NDFMp particle massMw molecular weight (relative molecular mass) of chemical speciesMw1 molecular weight of the evaporating componentMw2 molecular weight of the stagnant component

Maggi rate of change of particle mass in interval Ii due to aggregation in CM

Mbreaki rate of change of particle mass in interval Ii due to breakage in CM

Mγi jk velocity distribution moment of global order γ = i + j + k

Mγ∗i jk velocity equilibrium moment of global order γ = i + j + k

Mγi jk,α velocity distribution moment of global order γ = i + j + k

for particles of type α in polydisperse systemsM+i,l1l2l3

positive integer moment of the velocity distributionin the ith direction

M−i,l1l2l3

negative integer moment of the velocity distributionin the ith direction

M vector defining the tracked moment setM+ positive half-moment set (integration for positive velocity)M− negative half-moment set (integration for negative velocity)Mm

i volume-averaged moment set in the ith cell at time step mdefined in FVM for a 1D grid

Mmi jk volume-averaged moment set at cell Ωi jk and time step m

defined in FVM for a structured 3D grid

(Mmi jk)p updated moment set at cell Ωi jk and time step m defined

in FVM for a structured 3D grid calculated with permutation pM(h)

i jk moment set at cell Ωi jk updated after advection in the h directionin FVM for a structured 3D grid

M(1)i first-stage moment set in the ith cell at time step m

defined in FVM for a 1D gridM+

i jk positive half-moment set at cell Ωi jk in FVM (3D)M−

i jk negative half-moment set at cell Ωi jk in FVM (3D)M∗

i second-stage moment set in the ith cell at time step mdefined in FVM for a 1D grid

Ma Mach number for continuous phaseMap Mach number for disperse phaseMo Morton number

N order of the quadrature approximationN± number of quadrature nodes used in the calculation

of the positive and negative fluxes






xxiv Notation

N(t, x) total particle-number concentration (or density)N(V |V ′) volume-based daughter distribution functionN(ξ′|ξ) daughter-particle conditional NDF

Nd global number of degrees of freedom of the multiphase systemNi number of particles belonging to the ith interval in CM and DVMNi j number of particles belonging to intervals I(i)

1 and I( j)2

for bivariate CMNi number density of particles with velocity equal to ξi in DVMNp number of particles considered in NDF definitionNp number of primary particles forming a fractal objectNt total particle-number densityN′ number of weights and abscissas for Laguerre-

polynomial recursion coefficients in EQMOMNu Nusselt number

P(ξ′|ξ) daughter-particle conditional PDFPα(ξ) polynomial of order α orthogonal to the NDFPα,β elements of the matrix used in the PD algorithmPi j components of the second-order pressure tensorPi j probability for the encounter of particles i and j in MC simulationsP matrix used in the PD algorithmP total particle stress tensor and second-order pressure tensorPα second-order pressure tensor in polydisperse systems

for particles of type αPe Peclet numberPr Prandtl number

Q+ positive moment flux (integration for positive velocity)Q− negative moment flux (integration for negative velocity)

R ideal-gas constantRg radius of gyration of the particleReg continuous (gas)-phase Reynolds numberRep disperse (particulate)-phase Reynolds numberRec

p critical particle Reynolds numberReM

p meta-critical particle Reynolds numberRe∗p modified disperse-phase Reynolds number

S supersaturationS comprehensive source term in the GPBE including drift, diffusion,

and point processesS c particle collisional cross-sectional areaS k,i reconstructed slope in the ith cell for the solution of the

moment transport equation discretized with FVM (1D)S nαβ,i reconstructed slope in the ith cell at time instant n

from EQMOM discretized with FVM (1D)S moment set source term

S generic source term due to discontinuous events for the GPBESfV source term due to discontinuous events for

the fluid-phase volume fractionSM source term due to discontinuous events for

the disperse-phase mass density






Notation xxv

SN source term due to discontinuous events for the totaldisperse-phase number density

SV source term due to discontinuous events for the disperse-phasevolume fraction

S1 generic source term due to discontinuous events for GPBESf viscous and pressure stress tensor for fluid phase

S k source term for the kth moment of the NDFS k source term for the moment of order k of the multivariate NDFS i jk source term of the moments of orders i, j, and k with respect to the

particle-velocity components for the NDFS (m)

l1l2l3factor appearing in the collision term for velocity moments of theNDF of orders l1, l2, and l3 with respect to the three components

S γl1l2l3

source terms for moment of global order γ = l1 + l2 + l3

�S�p rate of change for particle momentum due to discontinuous eventsS+ collision cross section

Sc Schmidt numberSh Sherwood numberSt Stokes numberStp particle Stokes number

Tf continuous (fluid) phase temperatureTp disperse (particulate) phase temperatureTref reference temperature for liquid boilingTs temperature on particle surfaceTsat saturation temperature for the continuous phase

U characteristic continuous phase velocityU(ξ) particle velocity conditioned on particle size ξU characteristic particle velocityUg continuous phase velocityUp mean particle velocityUr impact, or relative, velocity for fragmenting particlesU∞ particle terminal velocity

Uf mean fluid velocity fieldUfV volume-average fluid velocityUM mass-average particle velocityUmix mass-average mixture velocityUN number-average particle velocityUp mean particle velocityUp,2 second-order particle velocity-moment tensorUp,k conditional particle velocity for ξp = ξpk

UpM fluid-mass-average particle velocityUV volume-average particle velocityUαβ mean velocities for polydisperse Gaussian distributionsU†p characteristic disperse-phase velocity

U(n) velocity of the nth particle (in vacuum)U(n)

f fluid velocity in the neighborhood of the nth particle






xxvi Notation

U∗f mesoscale variable describing fluid velocityU(n)

p velocity of the nth particleU∗p mesoscale variable describing particle velocity

〈Uf〉 Reynolds-average fluid velocity field〈Up|ξ = ζ〉 particle conditional velocity for ξ = ζ〈Up|ξp〉 expected mean particle velocity for internal coordinates equal to ξp

�Uf�p particle-mass-average fluid velocity

V particle volumeVf fluid volume seen by the particleVL length-based volume density functionVp particle volumeVW sample volume used in the estimation of the NDF

V∗αβ;i particle velocity with EQMOM in the ith cell after the advection stepwhen using FVM (1D) with time splitting

V†αβ;i particle velocity with EQMOM in the ith cell after the advection stepwhen using FVM (1D) with time splitting

V dimensionless and normalized particle velocityV(n)

f fluid-velocity space for fluid surrounding the nth particleV(n)

p particle-velocity space for the nth particle

W(t) generic Wiener processWαβ product of wα and wαβ in EQMOMW∗

αβ product of w∗α and w∗αβ in EQMOMand calculated after the advection step with time splitting

W(t) generic vectorial Wiener processWe Weber number

X dimensionless and normalized spatial coordinateXT abscissa (or node) matrixX(n) center of mass of the nth particleX∗p mesoscale variable describing particle position

Yf1 molar fraction of evaporating component in the gas phaseYs1 molar fraction of evaporating component on droplet surfaceY1 gas-phase molar fraction of evaporating componentY2 gas-phase molar fraction of stagnant component

Yf fluid-phase species mass fractionsYp particle species mass fractionsYα = wαξα weighted node (or abscissa) of the M-dimensional

quadrature approximationYT weighted-abscissa (or weighted-node) matrix

Lower-case Roman

a aggregation kernela breakage kernel (Chapter 7)a0 constant-breakage kernel (Chapter 7)






Notation xxvii

ai j affinity parameter for i– j aggregation (with i = A, B and j = A, B)aα coefficients of recursive formula for orthogonal polynomialsaα source term for the evolution equation of weight wα in DQMOMa′α coefficients of recursive formula appearing in Wheeler algorithm〈a〉 volume-average aggregation kernelam

i jk slope vector employed in second-order spatial reconstructionsfor FVM (3D)

b frequency of first-order process (breakage kernel)b(ξ′|ξ) daughter distribution function (Chapter 7)bi,α source term for the evolution equation of weighted node α

for the ith internal coordinate in DQMOMbα coefficients of recursive formula for orthogonal polynomialsbα source term for the evolution equation of weighted node α

in univariate DQMOMb′α coefficients of recursive formula appearing in Wheeler algorithmb∗ dimensionless and normalized kernel for first-order process

b volume-average frequency of first-order process〈b〉 volume-average breakage kernel

bk

α moment transform of order k of the daughter distributionfunction for ξα in univariate problems

bkα moment transform of order k of the daughter distribution

function for ξα in multivariate problems

bfvf fluid velocity coefficient for fluid fluctuationsbk response vector for third-order differences to a unit incrementbpvf particle-velocity coefficient for fluid fluctuations

ceq equilibrium solute concentrationcpξp self-diffusion component of crystal size growth ratecαβ constant appearing in the definition of the pair distribution functionc± coefficients appearing in upwind reconstruction schemescp

f specific heat of fluid phasecp

p particle specific heatcα lattice velocities used in LBM

d degree of accuracy of the quadrature approximationd molecular diameter of a solute moleculedo diameter of primary particles in an aggregatedp particle size

d10% size corresponding to 10% of the smaller particlesd90% size corresponding to 10% of the larger particlesdα size of the particles of type α taking part in a collisiondβ size of the particles of type β taking part in a collisiondαβ arithmetic average of colliding particle size (polydisperse systems)

d∗p characteristic particle lengthdi j

k net flux of particles for the kth internal coordinatedue to phase-space diffusion in bivariate CM

d vector containing the moments source terms in DQMOMdn difference vector of order n of natural logarithm of NDF moments






xxviii Notation

e coefficient of restitution for particle–particle collisionse elementary chargeei particle specific energy for the ith velocity componentep particle specific energyeαβ restitution coefficient in polydisperse systems for

collisions between particles of types α and β

f (t, x, v) particle-velocity NDF for monodisperse systemf (ξ1, ..., ξM−1) marginal NDF used in CQMOMf (ξM |...) conditional NDF used in CQMOMf (δ) shape function for colliding particlesff one-point PDF for the fluidfGf volume distribution of the fluid shear ratefNp multi-particle joint PDFft probability density function for the quiescence time in MC methodsfeq equilibrium velocity NDF

fα weights used in reconstructing the NDF in LBMfα(t, x, v) particle velocity NDF for particle type αfβ(t, x, v) particle velocity NDF for particle type βfε volume distribution of the fluid turbulent dissipation rate

f (2) pair correlation velocity NDFf ∗(v) equilibrium distribution function in the BGK

kinetic model and in Grad’s moment closuref (n)1 single-particle joint PDF

f i jk net flux of particles for the kth internal coordinate

due to phase-space drift in bivariate CMf ∗α weights used in LBM corresponding to the equilibrium NDF

g gravity acceleration constantg(ε) Kuwabara function for particle deposition in porous mediag0(αp/α

∗p) particle radial distribution function

gαβ pair correlation function for particles of types α and βg0,αβ constant appearing in the pair correlation functiongn(ξ) velocity parameters used in conjunction with EQMOM〈g〉 mean velocity difference used to approximate |v1 − v2|

h size of the regular discretization used in CMh(ξ) function used to model the second-order tensor for mixed advectionhi(ξ) numerical NDF in the ith cell for FVM (1D)hL discontinuous event term for length-based formulationhL,k moment of order k of source term for discontinuous eventhW constant kernel function used as filter to estimate NDFh1 particle collisional acceleration termh+ rate of particle formation due to discontinuous eventsh− rate of particle disappearance due to discontinuous events

i index vector identifying a discrete particle velocity in DVM

k order of moment for univariate NDFkA particle surface shape factor






Notation xxix

kB Boltzmann constantkc corrective growth crowding factorkco coordination number for an aggregatekd particle mass-transfer coefficient

kf fluid-phase turbulent kinetic energykf fluid-phase thermal conductivitykg fractal scaling factor of order unitykh particle heat-transfer coefficientki(ξ) polynomial used to represent the NDF in the Ii interval in CMkp thermal conductivity of particlekV particle volumetric shape factor

k∗A equivalent particle surface shape factork∗V equivalent particle volumetric shape factork exponent vector for the order of moment in multivariate NDFkξ internal coordinate exponent vector

m particle mass (used in daughter NDF for breakage)m(k) moment of order k of univariate NDFm(k) moment of order k = (k1, . . . , kM) of multivariate NDFmc mass of newly formed particle (nucleus mass)mcp mass of liquid evaporating per unit volume and unit timemk moment of order k of NDFmi jk moment of order i, j, and k with respect to the three velocity

components of the particle velocity NDFmj,k;i mixed moment of orders j and k for the ith cell in FVM (1D)

mL,k kth moment of length-based NDFmM,k kth moment of mass-based NDFmU,k kth moment of velocity-based NDFmV,k kth moment of volume-based NDFmξ,k moment of order k = (k1, . . . , kM) of the multivariate NDF

mα mass of particle of type α taking part in collisionmβ mass of particle of type β taking part in collisionm∗k moment definition used in EQMOMmn

k,i reconstructed moment for the ith cell at time instant n in FVMm∗j,k;i mixed moment of order j and k for the ith cell in FVM

when using time splitting after convectionm†j,k;i mixed moment of orders j and k for the ith cell in FVM

when using time splitting after drag

n generic NDF appearing in GPBEneq equilibrium NDFnL length-based NDFnM mass-based NDFnU velocity-based NDFnV volume-based NDFnξ generic NDF〈nξ〉 Reynolds-average NDFn volume-average NDF






xxx Notation

n∗ reconstructed NDFn∗ equilibrium Maxwellian NDFn[13] NDF reconstructed in Grad’s 13-moment closurenm

i (ξ) NDF in the ith cell at time step m reconstructedin FVM (1D) from Mm

i

n±α weights of the quadrature approximation calculated from thepositive and negative moments of the velocity distribution

n(h)i jk NDF in Ωi jk updated after advection in the h direction for FVM

nmi jk NDF in cell Ωi jk at time t = mΔt for FVM

n+i jk contribution for updating the NDF due to advection frompositive velocity in cell Ωi jk for FVM

n−i jk contribution for updating the NDF due to advection fromnegative velocity in cell Ωi jk for FVM

pf pressure of the fluid phasepf1 partial pressure of evaporating component in gas phasepg pressure of the gas phasepref reference pressure for boiling liquidps1 partial pressure of evaporating component on droplet surfacepα orthogonal polynomials of order α used

in functional expansion of NDFpα granular pressure of particles of type α

q heat flux to surface of particleqi skewness of the NDF with respect to the ith velocity componentq total particle energy flux

s specific surface area of the porous mediums ratio of geometric grids employed in CMs particle surface area

tαβ abscissa computed from Laguerre-polynomial recursioncoefficients used in EQMOM

u generic known disperse-phase velocity (1D)u(ξ) known particle velocity conditioned on internal coordinate ξuk (with k = 0, 1, 2) flow-dependent velocity parametersuα velocity node α of the quadrature approximationu+i,l positive i-component of velocity evaluated at left face of

cell Ωi jk in FVM (1D)u−i,r negative i-component of velocity evaluated at right face of

cell Ωi jk in FVM (1D)

�upup�N disperse-phase stress tensor

v disperse-phase velocity (1D)vf fluid-phase velocity (1D)v+i positive i-component of velocity evaluated at left face

of cell Ωi jk in FVMv−i negative i-component of velocity evaluated at right face

cell of Ωi jk in FVM






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