Abstract
This paper provides a review on the flow of free-flowing particles inside silos. We have previously reviewed in detail the experimental studies in this field. In the present work, the focus is placed on the theoretical approaches allowing numerical simulation and modeling of these systems. Modeling of granular flow in silos is very significant due to the advantages of modeling compared to experiments. The simulation methods are divided into four main groups: analytical methods, finite element method, discrete element method, and hybrid models. In each section, the most significant researches are reviewed. The drawbacks and advantages of each method are discussed, and the effects of different parameters are reviewed. Finally, the perspective of future work and the main challenges in this area are discussed.
Nomenclature
- A (m2)
cross-sectional area of the silo
- a (−)
Drucker-Prager model constant
- Bi (N/kg)
body force per unit mass in i direction
- dp (mm)
particle mean size
- E (Pa)
Young’s modulus
- F (N)
force
- Fyield (N/m2)
yield surface
- G (Pa)
shear modulus
- g (m/s2)
gravitational acceleration
- I (kg·m2)
moment of inertia
- J1 (N/m2)
first stress invariant
- J2 (N2/m4)
second stress invariant
- k (N/m2)
Drucker-Prager model constant
- kl (−)
lateral pressure ratio
- L (m)
distance between outlets of silo
- Mij (N·m)
torque
- mi (kg)
mass of particle i
- nij (−)
normal unit vector
- P (Pa)
pressure
- Q (m3/s)
volumetric flow rate
- Qu (m2/s)
flow rate per unit of thickness
- R (m)
radius
- T (s)
time
- tij (−)
tangential unit vector
- Vα, Vβ (m/s)
orthogonal projections of velocity vector on α and β
- Vi (m/s)
velocity in i direction
- x, y, z (−)
coordination axis
- Greek symbols
- α (−)
hopper angle
- δi (m)
displacement
- δij (mm)
overlap of particles
- εij (−)
strain
- Γ (kg/m3)
bulk density
- μ (−)
coefficient of sliding friction
- μr (−)
rolling friction of particles
- ν (−)
Poisson’s ratio
- η (kg/s·m0.25)
damping coefficient
- θ (−)
angular position
- dλ (−)
proportionality factor
- ρ (kg/m3)
density
- σ (N/m2)
stress
- φ (−)
angle of internal friction
- ψ (−)
angle of inclination
- ω (rad/s)
angular velocity
- Subscripts and superscripts
- Eff
effective
- N
normal
- p
plastic
- r
rolling
- T
tangential
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