Chemical Reactor Modeling

Hugo A. Jakobsen

Chemical ReactorModelingMultiphase Reactive Flows

Prof.Dr. Hugo A. JakobsenNorwegian Univ. of Science & TechnologyDept. of Chem. EngineeringN-7491 TrondheimNorwayhugo.jakobsen@chemeng.ntnu.no

ISBN: 978-3-540-25197-2 e-ISBN: 978-3-540-68622-4

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To Sara

Preface

This book is based on lectures regularly taught in the fourth and fifth yearsgraduate courses in transport phenomena and chemical reactor modeling,and in a post graduate course in modern reactor modeling at the NorwegianUniversity of Science and Technology, Department of Chemical Engineering,Trondheim, Norway. The objective of the book is to present the fundamentalsof the single-fluid and multi-fluid models for the analysis of single- and mul-tiphase reactive flows in chemical reactors with a chemical reactor engineer-ing rather than mathematical bias. Organized into 12 chapters, it combinestheoretical aspects and practical applications and covers some of the recentresearch in several areas of chemical reactor engineering. This book containsa survey of the modern literature in the field of chemical reactor modeling.

I hope this book can serve as a guide for my future Ph.D. students, aswell as other interested scientists, to get a thorough introduction to this fieldof research without spending too much of their invaluable time searching forand reading a large number of books and papers.

Comments on the contents of the book:In chap 1 a survey of the elements of transport phenomena for single phase

multicomponent mixtures is given. This theory serves as basis for the devel-opment of most chemical engineering models as well as the multiphase flowconcepts to be presented in the following chapters. The first part of the chapterconsiders laminar single phase flows for multicomponent mixtures. In the sec-ond part of the chapter the governing equations are applied to turbulentflows.

Chapter 2 contains a summary of the basic concepts of kinetic theory ofdilute and dense gases. This theory serves as basis for the development of thecontinuum scale conservation equations by averaging the governing equationsdetermining the discrete molecular scale phenomena. This method is an al-ternative to, or rather both a verification and an extension of, the continuumapproach described in chap 1. These kinetic theory concepts also determinethe basis for a group of models used describing granular flows, further out-lined in chap 4. A pedagogical advice basically for the students intending

VIII Preface

to obtain their very first overview of the content of reactor modeling on thegraduate level may concentrate on the continuum formulations first and, ifstrictly needed, go back to the chapters that are dealing with kinetic theory(i.e., chaps 2 and 4) after they feel confident with the continuum modelingconcepts.

Chapter 3 contains a survey of a large number of books and journal pa-pers dealing with the basic theory of multi-fluid flow modeling. Emphasisis placed on applying the multi-fluid model framework to describe reactiveflows. This is perhaps the main contribution in this book, as there exist notextbook on reactive multiphase flow modeling intended for reactor engineers.In the more advanced textbooks the basic multicomponent multiphase the-ory is introduced in a rather mathematical context, thus there is a need for aless demanding presentation easily accessible for chemical reaction engineeringstudents.

Chapter 4 contains a summary of the basic theory of granular flow. Theseconcepts have been adopted describing particulate flows in fluidized bed re-actors. The theory was primarily used for dense bed reactors, but modifiedclosures of this type have been employed for more dilute flows as well. Com-pared to the continuum theory presented in the third chapter, the granulartheory is considered more complex. The main purpose of introducing this the-ory, in the context of reactor modeling, is to improve the description of theparticle (e.g., catalyst) transport and distribution in the reactor system.

In chap 5 an outline of the basic theory of the required closure laws andconstitutive equations is provided. The first section presents the closures re-lated to averaged of products (i.e., the analogous to turbulence type of clo-sures). The following sections describe models for the interfacial transportphenomena occurring in multiphase reactive systems. An overview of the im-portant models for the different forces acting on a single particle, bubble ordroplet is given. Model modifications due to swarm or cluster effects are dis-cussed. The standard theories for interfacial heat and mass transfer are exam-ined. In the last section the literature controversy originating from the factthat with the present level of knowledge, there is no general mathematicaltheory available to determine whether the 3D multi-fluid model is well posedas an initial-boundary value problem, is examined.

In chap 6 the derivation of the classical reactor models is examined startingout from the microscopic heat and species mass balances. In chemical reac-tor engineering the idealized models like the plug flow reactor (PFR) - andcontinuous stirred tank reactor (CSTR) models are well known from basiccourses in chemical reaction engineering. For non-ideal flows the dispersionmodels (DMs) are frequently used. These standard models are deduced fromthe microscopic heat and species mass balances employing a cross-sectionalarea averaging procedure. Similar, but not identical, models can be obtainedby simplifying the governing microscopic transport equations.

In chap 7 a brief summary of the agitation and fluid mixing technology isgiven. The main emphasis is placed on examining the modern strategies used

Preface IX

to model the momentum transfer from the impeller to the fluid. The meth-ods are sketched and the basic equations are listed. A few model simulationexamples are presented.

In chap 8 the basic bubble column constructions and the principles ofoperation of these reactors are described. The classical models for two- andthree phase simple bubble column reactors are defined based on heat andspecies mass balances. The state of the art on fluid dynamic modeling ofbubble column reactors is then summarized including a few simulations ofreactive flows.

In chap 9 an outline of the basic theory of the population balance equationis provided. Three different modeling frameworks are defined, the macroscopicformulation, the microscopic continuum - and kinetic theory formulations. Themacroscopic model is formulated directly on the macroscopic scales, enablinga suitable framework for practical engineering calculations. In this frameworka simple and inaccurate numerical discretization scheme has become an inte-grated part of most closure laws. Since the numerical discretization schemescannot be split from the physical closure laws in a trivial manner, the morepopular closures for bubble coalescence and breakage rates are discussed in thischapter as well. The more rigorous microscopic formulations are presented andfuture reactor analysis should preferably be based on these concepts, enablingmore accurate closure laws to be formulated and more optimized solutionmethods to be used. The status on population balance modeling of bubblecoalescence and breakage phenomena is summarized.

Chapter 10 contains a literature survey of the basic fluidized bed reactordesigns, principles of operation and modeling. The classical two- and threephase fluidized bed models for bubbling beds are defined based on heat andspecies mass balances. The fluid dynamic models are based on kinetic theoryof granular flow. A reactive flow simulation of a particular sorption enhancedsteam reforming process is assessed.

In chap 11 an overview of the basic designs, principles of operation, andmodeling of fixed packed bed reactors is presented. The basic theory is ap-plied to describe the performance of particular chemical processes operatedin fixed packed bed reactors. That is, porous media reactive flow model simu-lations of particular packed bed sorption enhanced steam reforming processesare assessed.

In chap 12 a group of finite volume solution algorithms for solving themulti-fluid model equations is described. The basic single phase finite volumemethod solution strategies, spatial discretization schemes, and ODE solutionmethods in time are examined. The selected multiphase algorithms are ex-tended versions of the single-phase SIMPLE-like algorithms. However, alter-native algorithms can be found in the literature. Some of these methods arebriefly outlined in this chapter. Moreover, several numerical methods for solv-ing the population balance equation for dispersed flows are outlined. Finally,several solution methods for the resulting algebraic discretization equationsare mentioned.

X Preface

The book may be used as a reference book of the multi-fluid theory, or forteaching purposes at different educational levels. For example, at the graduatelevel, an introductory graduate course in single phase transport phenomenacan be based on chap 1 (and parts of chap 2). Suitable numerical solutionmethods for the governing single phase equations can be found in chapter 12.An introduction to reactor modeling can be based on chaps 6-11. The materialin chapters 2,3,4,5 and the multiphase parts of chap 12 may be better suitedat the post graduate level. Taking these three courses in sequence, I hope thePhD students get the necessary knowledge to give future contributions in thisfield of science.

I have received a great deal of help from numerous persons, over the nearlytwenty years association with this subject, in formulating and revising myviews on both reactor modeling and chemical reactor engineering. I wouldlike to acknowledge the inspiring discussions I have had with the colleaguesat NTNU during my work on this book. I am particularly incepted to thepresent and former members of the staff at the Chemical Engineering De-partment at NTNU. In addition, I wish to thank the PhD students that havetaken my graduate subjects and thus read the lecture notes carefully andsupplied me with constructive criticisms (among other comments) and sug-gestions for further improvements on the text. It is fair to mention that mystudents, especially Dr ing Carlos A Dorao, MSc Havard Lindborg, MSc HansKristian Rusten and MSc Cecilie Gotaas Johnsen, have contributed to thisbook in many ways. This includes technical contributions either in a director indirect way, and reading parts of the draft manuscript. I must also thankAssociate Professor Maria Fernandino for her valuable suggestions and com-ments regarding chapter 2. Finally, my thoughts are due to my wife, Jana, whostrongly believes quality is better than quantity. Her reviews and criticism ofthe contents surely improved the book.

Trondheim, November 2007 Hugo A. Jakobsen

Nomenclature

Latin LettersA Hamaker constant (J)A chemical component in general reactionA empirical model parameter (−)A macroscopic surface area defining the control volume (m2)A model parameter (−)A shorthand notation for advective terma coefficient in the FVM discretization equationa non-linear function in PDE classification theorya stoichiometric coefficients in general reaction (−)A(t, r) generalized variable dependent on time and spaceA0 catchment area (m2)A0 valve opening area (m2)a0 parameter in prescribed velocity profile in laminar boundary layer

theory (−)A1 surface of phase 1 in two phase system (m2)a1 parameter in prescribed velocity profile in laminar boundary layer

theory (−)A2 surface of phase 2 in two phase system (m2)a2 parameter in prescribed velocity profile in laminar boundary layer

theory (−)a3 parameter in prescribed velocity profile in laminar boundary layer

theory (−)aC(x, r;x′, r′,Y, t) coalescence frequency or the fraction of particle pairs of

states (x, r) and (x′, r′) that coalesce per unit time (1/s)Ah heat exchange surface of reactor (m2)AI interface area (m2)aI interfacial area density denoting the interface area per unit volume

(m2/m3)ai coefficient in generic algebraic equation in TDMA outlineai constants in MWR approximation of the solution

XII Nomenclature

AP projected area, average projected area of particle area distribution ona plane normal to the flow (m2)

Ar chemical affinity of reaction r (J/mol)AS particle surface, average surface calculated from a particle surface

distribution (m2)aij coefficients in algebraic system matrix AB baffle width (m)B chemical component in general reactionB coefficient consisting of inverted Maxwell-Stefan diffusivities (s/m2)B displacement factor (−)B model parameter (−)B model parameter in logarithmic velocity profile (−)b constant of integration in laminar boundary layer theory (−)b constant term in the FVM discretization equationb impact parameter (m)b non-linear function in PDE classification theoryb stoichiometric coefficients in general reaction (−)B(x, r,Y, t) net birth term in population balance equationB0 permeability (m2)bB(x, r,Y, t) particle breakup frequency (s−1)bi coefficient in generic algebraic equation in TDMA outlinebk total breakage rate of bubbles of group k in multi-group modelBB,i birth rate due to breakup in bubble class i ( 1

s m3 )BCi

birth rate due to coalescence in bubble class i ( 1s m3 )

Bdd(x) Kolmogorov second order velocity structure function (m2/s2)C chemical component in general reactionC clearance of turbulent impeller from the tank bottom (m)C constant in velocity structure function formula, C = 27

55Γ ( 13 )Ck ≈ 2.0

(−)C laminar impeller wall clearance (m)C model parameter (−)C shorthand notation notation for convective termC universal constant in the Kolmogorov two-third-law (−)c mole concentration of species (mol/m3)c non-linear function in PDE classification theoryc parameter in relation for the modulus of elasticity of the particulate

phase (−)c speed of electromagnetic radiation propagation in a medium (m/s)c stoichiometric coefficients in general reaction (−)c∗ mole concentration scale in turbulent boundary layer theory (−)c+ dimensionless mole concentration in turbulent boundary layer theory

(−)CM

L Magnus lift force coefficient (−)CS

L Saffman lift force coefficient (−)CT

L slanted wake transversal lift force coefficient (−)

Nomenclature XIII

C0 model coefficient (−)Cμ k-ε turbulence model parameter (−)Cω empirical model constant (−)Cb empirical parameter in two-phase k-ε turbulence model (−)Cc molar concentration of species c in mixture (kmol/m3)CD k-ε turbulence model parameter (−)CD drag coefficient (−)CE empirical constant in LES model (−)Cf (fVij

) surface area increase coefficient (−)Cf friction factor (−)ci coefficient in generic algebraic equation in TDMA outlineCK kinetic-energy velocity correction factor (−)Ck constant in the Kolmogorov five-third-law (−)CL lift force (net) coefficient (−)CM momentum velocity correction factor (−)CP specific heat at constant pressure (J/kgK)Cp laminar impeller off bottom clearance (m)CS Smagorinsky constant (−)CS speed of sound (m/s)CV specific heat at constant volume (J/kgK)CV virtual - or added mass force coefficient (−)c0 speed of light in the medium. In a vacuum c0 = 2.998 × 108 (m/s)Cε3 model parameter (−)Cε1 k-ε turbulence model parameter (−)Cε2 k-ε turbulence model parameter (−)Ckl rate of coalescence of bubbles of groups g and k in multi-group modelCW1 empirical wall lift force coefficient (−)CW2 empirical wall lift force coefficient (−)Cwb bubble wall friction force coefficient (−)Ca Capillary number, Ca = We/ReP (−)CFL Courant number, used in the Courant-Friedrichs-Lewy necessary sta-

bility condition for hyperbolic equationsD auxiliary factor in multiphase fractional step method implementationD bubble deformation factor (−)D chemical component in general reactionD diameter (m)D impeller diameter (m)D mass diffusion coefficient or diffusivity, binary or multicomponent

systems (m2/s)D model parameter (−)D shorthand notation notation for diffusive termd non-linear function in PDE classification theoryd particle diameter (m)d stoichiometric coefficients in general reaction (−)d′ diameter of daughter particle (m)

XIV Nomenclature

d′′ diameter of the smallest daughter particle (m)D′

s effective mass based diffusion coefficient in the explicit expression forthe Maxwell-Stefan flux (m2/s)

D′sm effective diffusion coefficient of species s in Wilke mass flux (m2/s)

D(x, r,Y, t) net death term in population balance equationd(i) diameter of particle in class, group or phase i (m)dS entropy of mixture (J/K)Dt

gp gas-particle turbulent dispersion coefficient (m2/s)dA surface average diameter (m)da major axis of an ellipsoidal bubble (m)db minor axis of an ellipsoidal bubble (m)dc critical bubble diameter (m)dD drag diameter (m)de equivalent bubble diameter (m)Dh hydraulic diameter (m)dH maximum horizontal dimension of a deformable particle (m)di coefficient in generic algebraic equation in TDMA outlinedi diameter of particle in interval i (m)dr reactor diameter (m)Ds effective diffusion coefficient for species s in explicit Maxwell-Stefan

flux expression (m2/s)dS Sauter mean diameter (m)De

s effective diffusivity of species s in explicit expression for the dusty gasmodel flux (m2/s)

DTs multicomponent thermal diffusion coefficients (kg/ms)

dV maximum vertical dimension of a deformable particle (m)dV volume average - or equivalent particle diameter (m)d12 distance between the centers of two hard spheres at collision (m)deff effective bubble diameter (m)dmax maximum stable fluid particle(m)DB,i death rate due to breakup in bubble class i ( 1

s m3 )DCi

death rate due to coalescence in bubble class i ( 1s m3 )

de,0 initial equivalent bubble diameter just above the distributor (m)dmax fixed maximum particle size (m)dmin fixed minimum particle size (m)Dsm Wilke effective diffusion coefficient for species s (m2/s)Dsr Fick’s law binary diffusivity for the species s and r mixture (m2/s)Da Damkohler number (−)da infinitesimal area on the sphere of influence denoting the face of the

collision cylinder (in kinetic theory) (m2)da infinitesimal surface element (m2)DaI Damkohler number, DaI = lr/u (−)dAp differential area used to define the radiation intensity, dAp = da cos θ

(m2)

Nomenclature XV

DEN auxiliary parameter (denominator) in kinetic expression for steamreforming

dl element of arc length (m)dl hight of collision cylinder (in kinetic theory) (m)dm/dt mass change rate (kg/s)ds arc length (m)dt infinitesimal increment in time (s)dv infinitesimal volume element (m3)dx infinitesimal increment in x-coordinate direction (m)dy infinitesimal increment in y-coordinate direction (m)dz infinitesimal increment in z-coordinate direction (m)E empirical wall law model parameter (−)E energy of each photon (J)E generalized total energy of a given system in classical mechanicsE heat flux emitted by a real surface (W/m2)E total emissive power of thermal radiation (W/m2)e non-linear function in PDE classification theorye thermal internal energy per unit mass of mixture (J/kg)e(λ) kinetic energy of eddy with size λ (J)e(di, λ) turbulent kinetic energy of an individual eddy of size λ breaking a

bubble of size di (J)E(k, t) three dimensional energy spectrum per unit mass (m3/s2)E(t) contact time distribution function in penetration theory (s−1)E(t) normalized element age distribution function in surface renewal theory

(s−1)EΓ

k interfacial energy transfer due to phase change (J/m3s)EE

k interfacial heat transfer (J/m3s)EW

k interfacial work by viscous and pressure forces (J/m3s)Eλ(λ) spectral emissive power of thermal radiation (W/m2 μm)Eeddies(λ) energy of discrete eddies of size λ ( J

m3[m] )Espectra(λ) energy of eddy wave function for eddies of size between λ and

λ + dλ ( Jm3[m] )

Etotal total energy associated with the center of mass of a thermodynamicsystem (J)

Etotal total energy content within an arbitrary volume V in the system (J)Ea activation energy of sorbent (J/kmol)Eb total thermal radiative power emitted by a blackbody (W/m2)Ec(k) scalar spectrum in wave number spaceEi internal energy associated with the center of mass of a thermodynamic

system (J)Ek kinetic energy associated with the center of mass of a thermodynamic

system (J)Ep potential energy associated with the center of mass of a thermody-

namic system (J)

XVI Nomenclature

Ep scalar potential energy function or potential (J)Ep(q, t) generalized potential energy of a given system in Lagrangian mechanicsEs(d) minimum energy required to deform a bubble of size d (J)es(di, dj) increase of bubble surface energy required breaking the parent bub-

ble di into a daughter bubble dj and a second corresponding daughterbubble (J)

ET total kinetic energy of the macroscopic fluid motion (J)Eλ,b spectral thermal radiative power emitted by a blackbody (W/m2 μm)ETotal total energy of two colliding particles (J)Ek,fluid kinetic energy (KE) of fluid surounding a particle in virtual mass

force analysis (J)Eo Eotvos number (−)f(t,x) longitudinal autocorrelation function (−)F dimensionless drag coefficient (−)F mass flux component in FVM discretizationF model parameter (−)F net flux of property ψ in elementary kinetic theoryf continuous number density probability in least squares method outlinef dimensionless constant in turbulent viscosity model (−)f friction factor (−)f non-linear function in PDE classification theoryf surface force component (N)f wave frequency associated with Taylor hypothesis (radians/s)f(...) distribution function in Hamiltonian mechanicsf(...) single distribution function in kinetic theoryf(x) quadratic function in CG definitionf(ζ, η) function defining a curve on a given surface, expressed in the curvi-

linear coordinatesf(m, r, t) particle distribution function with particle mass as inner coordinate

( 1m3[kg] )

f(t, r) general scalar, vector or tensor valued functionF (x, y) explicit function defining a surface in 3D space, expressed in Cartesian

coordinatesf(x, y, z) implicit function defining a surface in 3D space, expressed in Carte-

sian coordinatesf (1)(r, c, r1, c1, t) single distribution function in kinetic theory, f ≡ f (1)

f (1)(x, r, t) single number distribution function denoting the number of par-ticles per unit volume of the particle phase space at time t (general)

f (1)(d, r, t) average single particle number density function using particle di-ameter as inner coordinate ( 1

[m]m3 )f (1)(d, t) volume average particle number density probability with d as inner

coordinate ( 1m3[m] )

f (2)(r, c, r1, c1, t) pair distribution function in kinetic theoryf1 distribution function for molecule 1 in kinetic theory

Nomenclature XVII

f2 distribution function for molecule 2 in kinetic theoryfλ number density of eddies of size between λ and λ + dλ ( 1

m3s[m] )fb area (volume) fraction of bed gas taken by bubble phase gas (m2)fe area (volume) fraction of bed gas taken by emulsion phase gas (m2)f0

i fugacity of species i in the mixture at the standard state (Pa)fk auxiliary factor in PEA implementationFs molar flow rate of species s (mol/s)fw bubble wake fraction (−)fD Darcy friction factor (−)fF Fanning friction factor (−)Fin,ψ total inflow of property ψ into the calculation domain, used in conver-

gence criterionfsr proportionality or friction coefficient (kg/m3s)fVij

breakage volume fraction, fVij= d3

j/d3i (−)

ff fouling factor (Km2/W )Fr Froude Number, Fr = v2/gL (−)g(t,x) transverse autocorrelation function (−)G auxiliary factor in multiphase fractional step method implementationG non-dimansional shear rate (−)G particle growth rate (m/s)G specific Gibbs free energy expressed in terms of mole (J/kg)G total irradiation of thermal radiation (W/m2)g acceleration of gravity (m/s2)g magnitude of the relative velocity vector (m/s)g non-linear function in PDE classification theoryg source term in least squares method outlineG0 modulus of elasticity of the particulate phase (kg/ms2)Gλ(λ) spectral irradiation of thermal radiation (W/m2 μm)gI interface Gibbs free energy per unit mass (J/kg)gk Gibbs free energy per unit mass (J/kg)gαβ metric tensorGr Grashof number, Gr = l3Δρg/ρν2 (−)H H-property function in the Boltzmann H-theoremH liquid hight in standard turbulent stirred tank (m)H specific enthalpy, mixture enthalpy per unit mass expressed in terms

of temperature, pressure and the mass fractions of the species in themixture (J/kg)

H stagnation enthalpy (J/kg)H wall distance in square duct (m)h Plank’s constant = 6.6262 × 10−34 (Js)h film thickness (m)h grid spacing (m)h grid spacing in multigrid method outlineh head in head form of energy balance (m)

XVIII Nomenclature

h specific enthalpy, mixture enthalpy per unit mass expressed in termsof temperature, pressure and the mass fractions of the species in themixture, a fluid dynamic quantity (J/kg)

H(p,q, t) Hamiltonian function in Hamiltonian mechanicsh(ReP ) dimensionless function in particle drag expression (−)hΓ

k interfacial heat transfer due to phase change (J/m3s)h* specific enthalpy of ideal gas mixture (J/kg)hcond, conv

k combined convective heat transfer coefficient accounting for con-ductive and convective heat transfer (W/m2K)

hcondk convective heat transfer coefficient accounting for conductive heat

transfer (W/m2K)hexcess specific mixture excess enthalpy (J/kg)hexcess specific residual enthalpy of mixture (J/kg)hideal mixture specific mixture enthalpy expressed in terms of enthalpies of pure

real fluids (J/kg)hrad heat transfer coefficient accounting for radiantion transfer (W/m2K)h0 initial film thickness (m)hα Lame coefficients, metric coefficients, or scale factorshC effective swept volume rate (m3/s)hc specific enthalpy associated with chemical species/component c (J/kg)hf final film thickness (m)HI mean surface curvature (m−1)hI interface enthalpy per unit mass (J/kg)hk enthalpy per unit mass (J/kg)hv volumetric heat transfer coefficient (Jm−3K−1)hcd interfacial heat transfer coefficientHGL suspension height (m)hgp gas to particle heat transfer coefficient (Jm−2K−1)I integralI turbulence intensity (−)Ie total intensity of emitted thermal radiation (W/m2)Ii particle size interval (m)Ii total intensity of incident thermal radiation (W/m2)IR relative turbulence intensity (−)Iλ,b spectral radiation intensity of blackbody emission (W/m2 srμm)Iλ,e(λ, θ, φ) spectral intensity of emitted thermal radiation (W/m2 srμm)Iλ,i(λ, θ, φ) spectral intensity of incident thermal radiation (W/m2 srμm)J(ff) approximate collision term in the Boltzmann equation as given by the

Enskog expansionJ Jacobian determinantJ total radiosity of thermal radiation (W/m2)JΓ

k,s interfacial species mass transfer due to phase change (kg/m3s)Jj

k,s interfacial species mass transfer due to ordinary diffusion (kg/m3s)Jλ(λ) spectral radiosity of thermal radiation (W/m2 μm)

Nomenclature XIX

K equilibrium constantK generalized proportionality coefficient in the interfacial transfer flux

relationK model parameter (−)K number of FEM elements that constitute a part of the domainK parameter in bubble size model for bubbling beds (−)K permeability constant in the Davidson-Harrison model characteristic

of the particles and the fluidizing fluid (m3s/kg)k Boltzmann constant (J/K)k constant in capture kinetics (1/s)k mean turbulent kinetic energy per unit mass (m2/s2)k parameter in cyclone pressure drop relation (−)k reaction rate constantk thermal conductivity (W/mK)k wave number (m−1)K ′′ model parameter (−)K ′′′ model parameter (−)K(t− τ) kernel function in history force expression (−)Kcoll

p collisional diffusion coefficient (m2/s)Kt

p particle turbulent (or kinetic) diffusion coefficient (m2/s)k0 wave number for the integral scale of turbulence (m−1)K1 empirical parameter in drag coefficient parameterization (−)K2 empirical parameter in drag coefficient parameterization (−)K3 empirical parameter in drag coefficient parameterization (−)kdilute conductivity parameter for the dilute limit in granular theory (kg/ms)kC cutoff wave number in LES (m−1)kd viscous Cutoff wave number in LES (m−1)KG overall gas-side mass transfer coefficient (m/s)Kg empirical parameter in breakage kernel closure (−)KL overall liquid-side mass transfer coefficient (m/s)kp mass transfer coefficient associated with a pressure driving force

(mmole/Js)kp turbulent kinetic energy analogue of the particulate phase (m2/s2)Kr overall rate coefficient (kg/m3)kbc bubble-cloud mass interchange coefficient (m3/m3s)kc,s interfacial mass transfer coefficient (m/s)Keq chemical reaction equilibrium constantkgp gas-particle fluctuation covariance (m2/s2)kgp gas-particle fluctuation kinetic energy (covariance) (m2/s2)kp,kin kinetic thermal conductivities of the particle phase (W/mK)kp,m molecular thermal conductivities of the particle phase (W/mK)kSGS,t sub-grid scale kinetic energy per unit mass in LES (m2/s)Kn Knudsen number (−)L height (m)L length of reactor (m)

XX Nomenclature

L length scale of turbulence, or integral length scale of turbulence (m)L length scale, characteristic length (m)l mixing length (m)l turbulence length scale (m)l upper horizontal boundary in laminar boundary momentum balance

analysis (m)L(q,q, t) Lagrangian function in Lagrangian mechanicsLc integral scale of scalar segregation (m)LD height of solids in downcomer (m)le eddy size in the inertial subrange of the turbulence energy spectrum

(m)lI line formed by the intersection of AI(t) with A(t) (m)lj interfacial transport length for interface j (m)lS Smagorinsky lengthscale (m)lW intersection of CV wall interface with the cross sectional plane (m)LTotal total angular momentum of two colliding particles (kgm2/s)lk,s film thickness on the phase k side of the interface (m)Lmf hight of the fixed bed at minimum fluidization conditions (m)lW,k(t, z) closed curve of phase k in the cross section plane (m)Le Lewis number, Le = α/D (−)M mass (kg)M total mass of mixture in CV (kg)M total mass of particulate system (kg)m empirical model parameter (−)m1 parameter in Davidson-Harrison two-phase model (−)m2 parameter in Davidson-Harrison two-phase model (−)Mc mass of species c (kg)mc mass of fluid enclosed in the volume of the particle in the pressure

gradient force (kg)mc total mass associated with the center of mass of a binary particle

sustem (kg)mg mass of particles in group g (kg)MI net interface property termmi mass of one particle of type i (kg)mp mass of a single particle (kg)mV virtual mass of a particle in virtual mass force analysis (kg)Min,mass total inflow of mass into the calculation domain, used in convergence

criterionMwc

molecular weight of species c (kg/kmol)Ma Mach Number, Ma = v/CS (−)Mo Morton number (−)MTKE mean turbulent kinetic energy per unit mass, coincides with the k

quantity (m2/s2)N impeller stirring rate (RPM)N number of experiments, or realizations

Nomenclature XXI

N number of particles contained in a given system in classical mechanicsN total number of species in a mixturen empirical model parameter (−)n number density (number of particles/m3)n number of basis functions used in MWR solution approximation

functionn number of equations in matrix system, or number of unknowns in

algebraic equation systemn number of moles (mol)n1 parameter in capture kinetics (−)n2 parameter in capture kinetics (−)Na Avogadro’s numberNi number density of particles in the size interval i (1/m3)ni number density of particles in size class i ( 1

m3 )NP particle number density (Number/m3)Np Newton number, Np = P/ρN3D5 (−)NQ pumping number (−)Ns moles of species s (mol)Nu Nusselt number, Nu = hl/k (−)O origo, an arbitrary reference point in spaceP impeller power consumption (W )P wave period associated with Taylor hypothesis (s)p function in MWR examplep laminar impeller pitch (m)p pressure (Pa)p′k,I temporal deviation between the instantaneous pressure and the inter-

facial mean pressure variable (Pa)p(μ) probability density (−)p(ξ, r,vξ, c, t) extended distribution function in kinetic theoryP (r, c, t) normalized distribution function, or probability density functionp(x, r, c, t) advanced particle distribution function ( 1

[x](r)m )pB

C coalescence probability due to buoyancy processes (−)pT

C coalescence probability due to turbulence processes (−)p(1)i (m, r, c, ωc, T, t) advance particle distribution function with multiple inner

coordinates ( 1[kg,K](m/s)(r)m )

pLSC coalescence probability due to laminar shear (−)p0 constant in FEM examplepkin kinetic pressure in granular theory (N/m2)Pb term in k-ε turbulence model which represents the energy production

rate per unit volume due to bubble motion (J/m3s)pB(di, λj) breakage probability function, determining the efficiency of the

eddy-bubble collisions (−)pB(di : dj , λ) breakage probability function (−)pC coalescence probability (−)

XXII Nomenclature

pe(di, λ) normalized distribution function used to describe the turbulent ki-netic energy distribution of eddies of size λ (−)

Pi auxiliary coefficient in TDMA outlinePk term in k-ε turbulence model which represents the energy production

rate per unit volume due to fluid shear (J/m3s)PS pressure scale, or surface pressure (Pa)PC,kl probability of coalescence of bubbles of groups g and k in multi-group

modelpp,crit critical state frictional pressure for particle phase (Pa)pp,fric frictional pressure for particle phase (Pa)Pe Peclet number, Pe = vz,avl/D = RePr (−)Pr Prandtl Number, Pr = μCP /k (−)Q heat (J)Q impeller pumping capacity (m3/s)q mass of CO2 adsorbed divided by mass adsorbent (−)q number of components/species in the mixture (−)q number of independent reactionsqrad total radiation heat transfer flux (W/m2)qλ(λ) spectral heat transfer flux (W/m2μm)Qb volumetric gas flow rate in bubble phase (m3/s)Qi auxiliary coefficient in TDMA outlineQij Eulerian correlation function which represents a normalized velocity

correlation tensor (−)R impeller radius (−)R radius of pipe, tube or sphere (m)R radius of riser (m)R residual errorR universal gas constant (J/molK)r characteristic chemical reaction rate (s−1)r model parameter (−)r radial coordinate in the Cylindrical and spherical coordinate systems

(m)r smoothness monitor in TVD schemesr0 range of interaction defining a particle collision (m)Rb radius of bubble (m)rb bubble radius (m)Rc radius of cloud (m)Rc rate of generation of species c by chemical reaction per unit volume

(kg/m3s)rc radius of core (m)Rd radius of the liquid disk between two coalescing bubbles (m)RL(s) auto-covariance (m2/s2)Rn radius of curvature at the nose of the gas bubble (m)Rr reactor radius (m)

Nomenclature XXIII

rr rate of generation for reaction r defined independent of species, pro-portional to the extent of reaction (mol/m3s)

Rs dimensionless energy source due to mean force acting on particles ingranular flow (−)

rs mass rate of production of species s due to homogeneous chemicalreaction (kg/m3s)

rAB linear correlation coefficient (−)rad sorption rate (kmol/kgs)rij equivalent bubble radius (m)Rij(t,x) two-point correlation tensor (m2/s2)Rnk,s

volumetric species mass transfer rate (kg/m3s)RQk

volumetric heat transfer rate (W/m3)RQI,λ

volumetric heat transfer rate due to condensation/vaporization(W/m3)

Re Reynolds Number, Re = ρvL/μ (−)Reκ shear Reynolds number (−)ReΩ rotation Reynolds number (−)S action integral in classical or Lagrangian mechanicsS imaginary surface in phase space enclosing Ω, used in classical

mechanicsS specific entropy of mixture (J/kg K)s average surface renewal rate (s−1)s model parameter (−)s specific entropy of mixture (J/kg K)s time variable used calculating the autocorrelation, s = t′ − t. Time

difference between the two time instants at which the values of thetime dependent velocity fluctuations are measured (s)

Sψ generalized source termsA(ψ, g) scattering cross section expressed in terms of k (m2)SI perimeter (m)sI interface entropy per unit mass (J/kgK)Sk momentum source term of phase k (kg/m2 s2)sk entropy per unit mass (J/kgK)SW perimeter of the wall (m)SC,ψ constant part of the linearized source term in FVM discretizationSP,ψ constant part of the linearized source term in FVM discretizationSc Schmidt Number, Sc = μ/ρD (−)Sh Sherwood number, Sh = kcl/D (−)St Stanton number for heat, Sth = h/(CP ρvz,av) = Nu

RePr (−)St Stanton number for mass, Stm = kc/vz,av (−)T diameter of standard turbulent stirred tank (m)T temperature (K)T time period over which time averaging is performed (s)t time coordinate (s)T (q,q) generalized kinetic energy in Lagrangian mechanics

XXIV Nomenclature

T ∗ temperature scale in turbulent boundary layer theory (−)T+ dimensionless temperature in turbulent boundary layer theory (−)TShaft shaft torque (W )tb breakage time (s)tb bubble breakage time (s)tmax time between the first contact and the time when the film area between

two colliding bubbles reaches its maximum value (s)te contact time in surface renewal and penetration theories (s)U overall heat transfer coefficient (J/m2sK)ub average rise velocity of bubbles in a freely bubbling bed (m/s)ub average rise velocity of bubbles in bubbling bed (m/s)ue interstitial gas velocity in the emulsion phase (m/s)ubl,rise rise velocity of a single bubble in a liquid (m/s)ubr,0 ideal rise velocity of a single bubble in fluidized bed (m/s)Umf superficial gas velocity at minimum fluidization conditions (m3m−2s−1)V arbitrary macroscopic control volume fixed in space (m3)V combined abstract volume V = Vx + Vr

V speed of fluid flow; V =| v | (m/s)V tank volume (m3)V volume (m3)V volume over which volume averaging is performed (m3)v Kolmogorov micro velocity scale (m/s)v flow of gas through a bubble in bubbling bedv represents all admissible functions in the space X(Ω) of admissible

functions in least squares method outlinev speed of flow (m/s)v+ dimensionless velocity (−)vS superficial velocity in tubular reactor (m/s)V1 volume region of phase 1 in two phase system (m3)V2 volume region of phase 2 in two phase system (m3)v∞ terminal velocity (m/s)v* friction velocity (m/s)vbreakage characteristic velocity of the bubble breakage process (m/s)vrms root-mean-square of the fluctuating velocity components, rms-velocity

(m/s)Vb bubble volume (m3)Vb volume of bubble phase (m3)Vc volume of cloud phase (m3)Vi volume of a particle in class, group or phase i (m3)Vp volume of a single particle, or average of particle volume distribution

(m3)Vr volume in physical spaceVr volume of gas in bubbling bed (m3)vt impeller tip speed (m/s)Vw bubble wake volume (m3)

Nomenclature XXV

Vx abstract volume in internal property spacevequa equatorial speed parameter used in experimental data analysis (m/s)vslip fluid-particle velocity slip (m/s)Vb,i volume of bubble or particle in class i (m3)Vbs volume of solids in bubble phase (m3)Vcs volume of solids in cloud phase (m3)Ves volume of solids in emulsion phase (m3)vz,max maximum velocity at the center of a pipe (m/s)W impeller blade width (m)W weighting function in MWR discretizationWs solid feeding rate (kg/s)We Weber number, We = ρv2L/σ (−)x coordinate in Cartesian coordinate system (m)x fractional conversion of capture reaction (−)X(Ω) space of admissible functions, used in least squares method outlinexi pivotal points in Ii

Xk phase indicator functionxs mole fraction of species s in gas or liquid mixture (−)Xgkl intergroup transfer matrix in multi-group method distributing the

mass from groups k and l to group g in the coalescence processXgk matrix in multi-group method distributing the mass from the number

of group k bubbles broken to the number of group g bubbles formedin the breakage process

y coordinate in Cartesian coordinate system (m)y distance from wall (m)y+ distance from a wall measured in viscous lengths, or Reynolds number

(−)y0 distance between the wall and the particle (m)Z reactor height (m)z coordinate in Cartesian coordinate system (m)z position above the distributor (m)Zs−r collision frequency for one molecule of species type s colliding with

target molecules of type r (s−1)Zsr collision density, the number of collisions between pairs of molecules

s and r (m−3s−1)〈ωk〉ΓAI

interfacial mass flux weighted species mass fraction (−)〈hk〉ΓAI

interfacial mass flux weighted heat transfer (J/kg)〈Hs〉AI

surface average modified Henry’s law constant for species s in themixture (−)

〈kc〉L length average mass transfer coefficient (m/s)〈Ni〉te average mass transfer rate in surface renewal and penetration theories

(mol/m2s)〈vrel

n,k〉AInormal interface velocity due to phase change (m/s)

Gs partial mass Gibbs free energy for species s (J/kg)

XXVI Nomenclature

Hc partial specific enthalpy of species c in the mixture (J/kg)hc partial specific enthalpy of species c in the mixture, a fluid dynamic

quantity (J/kg)m total mass flow rate (kg/s)mk interface mass transfer rate (kg/m2s)ms mass flow rate of species s (kg/s)Q rate of heat added to the control volume V (J/s)Qcond, conv

k combined convective heat transfer rate due to conduction and con-vection (W )

Qcondk convective heat transfer rate due to conduction (W )

Qrad radiation heat transfer rate (W )Qλ spectral radiant heat transfer rate (W/μm)W rate of work done on the control volume V (J/s)dmdt particle growth rate related to mass change by condensation, evapo-

ration and dissolution (kg/s)�G Gibbs free energy expressed in terms of mole (J)Dsr symmetric Fickian multicomponent diffusivity for the s and r pair of

gases [17] [18] (m2/s)Csk multicomponent inverse diffusivities (s/m2)F Helmholtz energy (J)fi fugacity of species i in the mixture (Pa)G Gibbs free energy (J)G Gibbs free energy expressed in terms of mass (J)H total mixture enthalpy (or enthalpy) expressed in terms of tempera-

ture, pressure and the masses of the various species in the mixture(J); or enthalpy expressed in terms of temperature, pressure and themole numbers of the various species in the mixture (J)

h total mixture enthalpy (or enthalpy) expressed in terms of tempera-ture, pressure and the masses of the various species in the mixture,a fluid dynamic quantity (J); or enthalpy expressed in terms of tem-perature, pressure and the mole numbers of the various species in themixture, a fluid dynamic quantity (J)

T non-dimensional temperature (−)pk,I deviation between the local instantaneous pressure and the interfacial

area averaged pressure (Pa)ai activity of species i in the mixture (−)fpure

i fugacity of pure species i (Pa)De

sK effective Knudsen diffusivity of species s in porous medium (m2/s)De

sr effective bulk diffusivity of binary pair s−r in porous medium (m2/s)Dsk Maxwell-Stefan diffusivities (m2/s)˜Cij residual stress tensor in LES (Pa)˜Lij Leonard stress tensor in LES (Pa)˜Rij residual stress tensor in LES (Pa)

Nomenclature XXVII

˜Sij the large scale strain-rate tensor in LES (m2/s)A FEM characteristic matrixA system matrixD diagonal matrix containing interfacial coupling termsDsr generalized non-symmetric Fickian multicomponent diffusivity for the

s and r pair of gases [16] [39] [3] (m2/s)I source term in generalized Boltzmann type of equation representing

the effects of particle coalescence, breakage and collisionsJ (ψ(c)) collision term in the Boltzmann equationDsr generalized non-symmetric multi-component Fickian mass diffusion

coefficients (m2/s)A A : X×X, a symmetric, continuous bilinear form, used in least squares

method outlineA chemical reaction formula matrix (−)F F : X, a continuous linear form, used in least squares method outlineJ (f ; g) norm equivalent functional in least squares method outlineL lower triangular matrixM preconditioner in Krylov subspace methods outlineP processR space domainU upper triangular matrixe(di, λ) mean turbulent kinetic energy of an eddy of size λ breaking a bubble

of size di (J)Gs partial molar Gibbs free energy for species s (J/mol)p mean pressure, defined as the mean value of the normal stresses across

any three orthogonal planes (Pa)vλ average velocity of eddies of size λ (m/s)vdrops mean turbulent droplet velocity (m/s)vc continuous phase circulation velocity (m/s)vr,di

average rise velocity of particle of size class i (m/s)vrel,t,ij length of relative velocity between a pair of unlike bubbles (m/s)vt,d average speed of particles of size d due to turbulence (m/s)vt,i mean turbulent bubble velocity (m/s)dvc

dr average shear rate for the continuous phase (1/s)d diameter of complementary daughter particle (m)p′g fluctuation pressure component of the undisturbed flow (Pa)pg local instantaneous pressure of the undisturbed flow (Pa)CC symmetrical, traceless and non-divergent tensor in Enskog expansionA turbulence anisotropy tensor in extended k-ε model (−)B tensor function in Enskog expansione unit tensor with components eij

eI unit interface tensorP pressure tensor (Pa)pcoll collisional pressure tensor (Pa)

XXVIII Nomenclature

pkin kinetic pressure tensor (Pa)T total stress tensor (Pa); T = pe + σ˜Tg total stress tensor of the undisturbed flow (Pa)gαβ local metric tensorKαβ curvature tensor (m−1)G specific molar Gibbs free energy expressed in terms of mole (J/kmol)H specific molar enthalpy, mixture enthalpy per unit mole expressed in

terms of temperature, pressure and the mole fractions of the speciesin the mixture (J/mol)

h specific molar enthalpy, mixture enthalpy per unit mole expressed interms of temperature, pressure and the mole fractions of the speciesin the mixture, a fluid dynamic quantity (J/mol)

Fl interfacial coupling term in two-phase k-ε turbulence model (N/m3)〈nk,s〉AI

surface average combined species mass transfer flux (kg/m2s)〈v〉Ni

number average velocity representative for particle size class i (m/s)Mk interfacial momentum transfer to phase kP(p,q, t) a set of generalized momenta in Hamiltonian mechanicsQ(p,q, t) a set of generalized coordinates in Hamiltonian mechanicsq generalized velocities in Lagrangian MechanicsC vector function in Enskog expansionA vector function in Enskog expansiona acceleration of a single particle (m/s2)aξ generalized acceleration vector in property spaceB general vector or tensor valued functionb constant vector in algebraic equation systemb element-abundance vector (mole)C peculiar velocity (m/s)c velocity of a single particle, or velocity of a collection of mono-atomic

gas molecules (m/s)d generalized diffusional driving force (m−1)e error vector in multigrid method outlinee unit vector with components ei in the i (i = 1, 2,3) directionsF incident particle number flux or intensity (number/m2 s)F net force acting on a single particle (N)f FEM characteristic vectorf sum of forces acting on the mixture in the control volume (N)FW

L,d wall lift force acting on a collection of dispersed particles per unitmixture volume (N/m3)

FI surface force per unit area (N/m2)Fk generalized drag force per unit mixture volume (N/m3)FL lift force acting on a collection of particles per unit mixture volume

(N/m3)FP net force acting on a collection of particles per unit mixture volume

(N/m3)

Nomenclature XXIX

fS total hydrodynamic surface force exerted by a fluid on a particle (N)FV virtual mass force acting on a colliction of particles per unit mixture

volume (N/m3)fW net wall interaction force acting on a single particle (N)fBuo buoyancy force on a single particle (N)fB Basset history force acting on a single particle (N)fD steady drag force acting on a single particle (N)fE body forces (except gravity) acting on a single particle (N)fG body force due to gravity acting on a single particle (N)fhp hydrostatic pressure force acting on a single particle (N)FL,V combined lift-virtual mass force acting on a collection of particles per

unit mixture volume (N/m3)fL lift force acting on a single particle (N)fpg pressure gradient force acting on a single particle (N)fp force due to external pressure gradient acting on a single particle (N)FTD turbulent dispersion force per unit volume (N/m3)fV virtual- or added mass force acting on a single particle (N)G velocity of the center of mass expressed in the laboratory frame (m/s)g external force per unit mass, or gravity force per unit mass (m/s2)g relative particle velocity (m/s)gα tangent basis vectors in orthogonal coordinate systemsGc velocity of the center of mass expressed in the center of mass frame

(m/s)H angular momentum vector for a system (kgm2/s)h angular momentum vector for a particle (kgm2/s)Ik generalized interfacial transfer fluxJ JacobianJ local instantaneous diffusive flux of the generalized quantity ψJ∗

s instantaneous diffusive molar flux of species s in mixture (mol/m2s)jm volumetric flux of the mixture (m3/m2s)JS molecular entropy flux (J/m2sK)js instantaneous diffusive mass flux of species s in mixture (kg/m2s)J12 impulse of the force exerted by particle 1 on particle 2 (kgm/s)jgc multi-component mass diffusion flux due to external force (kg/m2s)joc ordinary multi-component mass diffusion flux (kg/m2s)jpc pressure gradient induced multi-component mass diffusion flux

(kg/m2s)jTc temperature gradient induced multi-component mass diffusion flux

(kg/m2s)JV d volumetric flux of the dispersed phase relative to the velocity of the

volume centre of the mixture (m3/m2s)k unit vector of the apse-lineMΓ

k interfacial momentum transfer due to phase change (N/m3)MT

k interfacial momentum transfer caused by stresses (N/m3)

XXX Nomenclature

mc instantaneous mass flux of molecules of species of type c with respectto stationary coordinate axes (kg/m2s)

MσI net surface tension force term (N/m2)

Mk,l interfacial momentum exchange between phases k and l (kg/m2 s2)Msr momentum transferred from species r to s per collision (kgm/

s, collision)n outwardly directed unit normal vectorn species-abundance vector (mole)NI unit vector normal to lI(t) that is both tangent to and outwardly

directed with respect to AI(t)Ns combined molar flux of species s in gas or liquid mixture (mol/m2s)ns combined mass flux of species s in gas or liquid mixture (kg/m2s)P momentum associated with a macroscopic CV (kgm/s)p generalized momenta in Hamiltonian mechanicspm search direction in m-th iteration in Krylov subspace methods outlinePsr diffusive force per unit volume exerted by species r on species s

(N/m3)q generalized coordinates in Lagrangian and Hamiltonian mechanicsq heat flux vector (W/m2)qrad total radiant energy flux arriving at a surface element (W/m2)qc ordinary conductive heat flux, defined by Fourier’s law (W/m2)qd energy flux resulting from inter-diffusion of the chemical species

(W/m2)qr radiative heat flux (W/m2)qx Dufour energy flux resulting from a temperature gradient induced by

mass diffusion of chemical species (W/m2)qI interface heat flux per unit length (J/m)qV volumetric radiant energy flux (W/m3)qcoll collisional granular heat flux (kg/s3)qkin kinetic granular heat flux (kg/s3)qAS

radiant surface energy flux (W/m2)r position vector, locating a point in space (m)r residual vectorrc definition of the center of mass point (m)rc position vector denoting the location of the center of mass (m)ri particle coordinates in Newtonian Mechanics (m)rp particle position vector, locating the center of mass (m)T total torque applied to fluid mass in control volume (Nm)t torque on a single sphere (Nm)t unit tangent vectort(k) tangent surface vectorTThrust propulsive force developed by a jet-propeller motor (N)u control volume surface velocity arising from the motion of the moving

grid (m/s)u generalized velocity in classical mechanics

Nomenclature XXXI

u velocity of the control surface with respect to the coordinate referenceframe (m/s)

v local instantaneous mixture velocity, mass average velocity (m/s)v′

d,s modified mass diffusion velocity for species s (m/s)v∗

s species velocity of molecules of species s generated by chemical reaction(m/s)

vdrift fluid particle drift velocity (m/s)vc instantaneous number mean velocity of molecules of species type c

with respect to stationary coordinate axes (m/s)vI interface velocity (m/s)vi mass average velocity of particle class i (m/s)vi surface average velocity of particle class i (m/s)vr external coordinate velocity (m/s)vx internal coordinate velocityvslip,d slip velocity for dispersed phase d (relative to the continuous phase c)

(m/s)vξ generalized velocity vector in property spacevc,d instantaneous diffusion velocity for species c, relative to the local mo-

tion of the mixture stream (m/s)vi,di

size average velocity of particle class i (m/s)vi,ni

number average velocity of particle class i (m/s)vi,Vi

volume average velocity of particle class i (m/s)vMk diffusion velocity of phase k, the velocity of phase k relative to the

velocity of the mass center of the mixture (m/s)vrel relative velocity between particles or phases (m/s)vrk relative velocity between dispersed phase k and continuous phase c

(m/s)vV k drift velocity of phase k, the velocity of phase k relative to the velocity

of the volume center of the mixture (m/s)w velocity of the fluid at the control surface with respect to the control

surface (m/s)x internal property vectorx position vector defining the separation of two points in space (m)x three coordinates in an arbitrary coordinate system fixed in spacex true solution of matrix system: Ax = bx vector of unknowns in algebraic equation systemxν intermediate solution of matrix system: Axν = b after ν iterationsximproved,ν corrected or improved solution in multigrid method outlinexm approximate solution in m-th iteration in Krylov subspace methods

outlineY vector representing the continuous phase variablesdc an infinitesimal element in a hypothetical velocity space containing

the velocity cdr an infinitesimal spatial space containing the point r

XXXII Nomenclature

Greek Lettersα(T ) total, hemispherical absorbtivity of a surface (−)αsr Onsager phenomenological coefficients (kgs/m3)α absorptivity, α = Gabs/G (−)α isothermal compressibility (Pa−1)α ratio of CO2-acceptor to catalyst massα thermal diffusivity of conducting medium (m2/s)α∗ parameter in relation for the modulus of elasticity of the particulate

phase (−)αk instantaneous volume fraction of phase k (−)αm auxiliary factor at iteration m in Krylov subspace methods outlineαP relaxation factor in FVM discretizationαλ,θ(λ, θ, φ, T ) spectral, directional absorbtivity of a surface (−)αλ(λ, T ) spectral, hemispherical absorbtivity of a surface (−)β bulk expansion coefficient (K−1)β constant in time-splitting methodβ empirical parameter (β ≈ 2.0) in the Kuboi mean square droplet ve-

locity relation (−)β interfacial friction coefficientβk time fraction (−)βm auxiliary factor at iteration m in Krylov subspace methods outlineχ Enskog free volume correction functionχ energy ratio, χ = e(di,λ)

e(di,λ) (−)Δhvap

lg,s latent heat of vaporization of pure species s in the multicomponentmixture (J/mol)

ΔE translational energy change during an inelastic collision (kgm2/s2)Δhvap

lg,mix latent heat of vaporization of the multicomponent mixture (J/kg)ΔL bubbling bed expansion (m)ΔpC pressure drop through cyclone (Pa)ΔpD pressure drop across downcomer (Pa)Δpf pressure drop due to friction (Pa)ΔpR pressure drop across riser (Pa)ΔpCD pressure drop through the solids flow control devices (Pa)δQ differential energy transfer to the thermodynamic system (J). The δ

symbol is used to indicate that the integration of δQ which is not astate function is dependent on the path.

Δt time increment (s)Δtcoal coalescence time interval (s)Δtcol collision time interval (s)δv turbulence velocity scale (m/s)δW differential work done on the thermodynamic system (J). The δ sym-

bol is used to indicate that the integration of δW which is not a statefunction is dependent on the path.

Δ filter width in LES (m)