# Nomenclature

*A* Avogadro’s number

c Velocity space, m/s

*c _{s}* Instantaneous particle velocity in i direction, m/s

*C _{s}* Peculiar particle velocity, m/s

*d _{ip} = (d_{t} + d_{p})/2* Average diameter of particles i and p, m

*d _{s and}, d_{p}* Diameter of particle, m

*E* Internal energy, J

*e* Restitution coefficient

*e _{sw}* Particle-wall restitution coefficient frequency distribution of

velocities of particles

*F _{Dip}* Momentum source (drag between solid phases), N

f Maxwellian velocity distribution function

f Single velocity distribution function

*fp _{p}* Pair velocity distribution function

*g* Gravity acceleration, m/s^{2}

*g** _{0}* Radial distribution function

*g _{ip}* Radial distribution function at contact between particles i and p

*g _{ss}* Radial distribution function at contact between particles of the

same phase

*H* Gap between two plates, m

*I* Identity tensor

*k _{B}* 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

*k _{s}* Granular conductivity, kg/m s

*M* Molecular weight, kg/kmol

*m _{s}* Mass of particle of phase s, kg

m_{0} = *m _{i} + m_{p}* Total mass of two colliding particles, kg

*N* Number of moles

*n* Compact factor

*n _{s}* Number density of phase s

*N _{ip}* Energy dissipation

*P _{c}* Collisional pressure tensor, N/m

^{2}

*P _{k}* Kinetic pressure tensor, N/m

^{2}

*p _{c}* Critical state pressure, N/m

^{2}

*pf* Frictional pressure, N/m^{2}

*Q* Momentum flux, N/m^{2}

*q _{c}* Collisional flux, J/s m

^{2}

*q _{k}* Kinetic flux, J/s-m

^{2}

*Rj* Rate of heterogeneous reaction between phases, mol/m^{2} s

*Re _{p}* 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

*v _{s}* Mean velocity of phase s, m/s

*V _{0}* Velocity of plates, m/s

*v*_{s},_{para} Particle slip velocity parallel to wall, m/s

*v _{ts}* Solid particle terminal velocity, m/s

*y _{i}* 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

X_{ip} Collisional flux, J/s m^{2}

*e _{s}* Solid volume fraction of phase s

*e _{T}* Total solid volume fraction

*(p _{s}* Perturbation to Maxwellian distribution function

*Y _{ip}* Collisional source of granular temperature, J/m

^{2}s

*p* Viscosity, kg/m. s

*p _{ss}* Granular viscosity of phase s, kg/m. s

*i _{ip}* Mixture granular viscosity, kg/m. s

*p _{s}* Solid density of phase s, kg/m

^{3}

*d _{s}* Fluctuating granular energy m

^{2}/s

^{2}

*d _{m}* Mixture fluctuating granular energy, m

^{2}/s

^{2}

t_{n} Mixture normal stress, N/m^{2}

t_{s} Solid-phase shear stress, N/m^{2}

*T _{t}* Mixture shear stress, N/m

^{2}

*y _{s}* Property of particle

*в* Granular temperature (random kinetic energy per unit mass), m^{2}/s^{2}

*ф* Specularity coefficient Ф Angle of internal friction

gs

Granular energy exchange between phasesp Bulk density, kg/m^{3}

* e_{s}* Volume fraction of solids

* p_{s}* Solid density, kg/m

^{3}

* 11_{2D}* Second invariant of the deviatoric stress tensor

* Y* Shear rate, 1/s

* в* Interface momentum exchange coefficient

Y Collisional energy dissipation, J/m^{3} s

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