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Accessible Unlicensed Requires Authentication Published by De Gruyter August 1, 2019

Modeling methods for gravity flow of granular solids in silos

Shahab Golshan, Reza Zarghami and Khashayar Saleh


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.


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)


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)


mi (kg)

mass of particle i

nij (−)

normal unit vector

P (Pa)


Q (m3/s)

volumetric flow rate

Qu (m2/s)

flow rate per unit of thickness

R (m)


T (s)


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)


δij (mm)

overlap of particles

εij (−)


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


proportionality factor

ρ (kg/m3)


σ (N/m2)


φ (−)

angle of internal friction

ψ (−)

angle of inclination

ω (rad/s)

angular velocity

Subscripts and superscripts











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Received: 2019-02-06
Accepted: 2019-07-01
Published Online: 2019-08-01
Published in Print: 2021-05-26

© 2019 Walter de Gruyter GmbH, Berlin/Boston