Nomenclature
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
22
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
Subscripts
- 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