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A Avogadro’s number

c Velocity space, m/s

cs Instantaneous particle velocity in i direction, m/s

Cs Peculiar particle velocity, m/s

dip = (dt + dp)/2 Average diameter of particles i and p, m

ds and, dp Diameter of particle, m

E Internal energy, J

e Restitution coefficient

esw Particle-wall restitution coefficient frequency distribution of

velocities of particles

FDip Momentum source (drag between solid phases), N

f Maxwellian velocity distribution function

f Single velocity distribution function

fpp Pair velocity distribution function

g Gravity acceleration, m/s2

g0 Radial distribution function

gip Radial distribution function at contact between particles i and p

gss Radial distribution function at contact between particles of the

same phase

H Gap between two plates, m

I Identity tensor

kB The Boltzmann constant (1.3805 x 10~23 J/K)

k=k- Unit vector connecting the centers of the two particles

= ip


Kf Turbulent kinetic energy of fluid, m /s

ks Granular conductivity, kg/m s

M Molecular weight, kg/kmol

ms Mass of particle of phase s, kg

m0 = mi + mp Total mass of two colliding particles, kg

N Number of moles

n Compact factor

ns Number density of phase s

Nip Energy dissipation

Pc Collisional pressure tensor, N/m2

Pk Kinetic pressure tensor, N/m2

pc Critical state pressure, N/m2

pf Frictional pressure, N/m2

Q Momentum flux, N/m2

qc Collisional flux, J/s m2

qk Kinetic flux, J/s-m2

Rj Rate of heterogeneous reaction between phases, mol/m2 s

Rep Particle Reynolds number

(S) Average strain rate

r Position

S Deviatoric stress tensor

T Average of random kinetic energy

v Hydrodynamic velocity, m/s

v Center of mass mean velocity, m/s

vs Mean velocity of phase s, m/s

V0 Velocity of plates, m/s

vs,para Particle slip velocity parallel to wall, m/s

vts Solid particle terminal velocity, m/s

yi Species mass fraction


  • 1 Large
  • 2 Small

cr Critical

fr Frictional

g Gas phase

col Collisional

kin Kinetic

max Maximum

T Total

s Solid phase

w Wall

Greek Symbols

Xip Collisional flux, J/s m2

es Solid volume fraction of phase s

eT Total solid volume fraction

(ps Perturbation to Maxwellian distribution function

Yip Collisional source of granular temperature, J/m2 s

p Viscosity, kg/m. s

pss Granular viscosity of phase s, kg/m. s

iip Mixture granular viscosity, kg/m. s

ps Solid density of phase s, kg/m3

ds Fluctuating granular energy m2/s2

dm Mixture fluctuating granular energy, m2/s2

tn Mixture normal stress, N/m2

ts Solid-phase shear stress, N/m2

Tt Mixture shear stress, N/m2

ys Property of particle

в Granular temperature (random kinetic energy per unit mass), m2/s2

ф Specularity coefficient Ф Angle of internal friction

gs Granular energy exchange between phases

p Bulk density, kg/m3

es Volume fraction of solids

ps Solid density, kg/m3

112D Second invariant of the deviatoric stress tensor

Y Shear rate, 1/s

в Interface momentum exchange coefficient

Y Collisional energy dissipation, J/m3 s

? Solid-phase bulk viscosity, kg/m-s

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