∑
ω
=+
t
MM
d
d
()
i
i
ij ijt, r,
157
(2)
158 where m
i
, I
i
. v
i
, and ω
i
represents the mass, the moment of
159 inertia, the translational velocity, and angular velocity of particle
160 i, respectively. F
c,ij
, F
lr,ik
, F
pf,i
, and F
g,i
indicate the contact force,
161 the noncontact (long-range) forces, the interaction forces
162 between particle−fluids, and a gravity force. The tangential and
163 rolling friction moments are represent by M
t,ij
and M
r,ij
,
164 respectively. The equations of the solid−fluid interaction forces
165 rely upon what interaction forces are considered. This is
166 explained in Section 2.3.1.
167 The key of the DEM model is the contact model between
168 particles. The contact forces F
c,ij
include the normal (F
cn,ij
) and
169 tangential (F
ct,ij
)
4,5
component, which is described with the
170 following equations
=+
FF
ij ij i
c, cn, ct,
171
(3)
δγδ δγδ=− −
−−kk
ij ij ij
nij
nij ij ij
ij
ti
c, n, n,
,
,t,n,
t,
,
172
(4)
173 where k
n,ij
, k
t,ij
, γ
n,ij
, γ
t,ij
, δ
n,ij
, and δ
t,ij
indicate the normal stiffness
174 coefficients, the tangential stiffness coefficients, the normal
175 damping coefficients, the tangential damping coefficients, and
176 the normal and tangential particle overlaps, respectively. δ
n,ij
177 and δ
t,ij
represent their corresponding derivative terms in time.
178 In this work, the normal forces are calculated by the Tsuji
179 model
23−25
and the tangential forces are calculated by the
180 Mindlin model.
26,27
These models are described by t he
181 equations in Table S1 (in the Supporting Information),
182 where Y represent the material ’ s Young’s modulus, and v and
183 e
r
represent the Poisson ratio and coefficient of restitution.
184 2.2. VOF Model. In order to track the phase interface
185 between two phases, the volume fraction of each fluid phase is
186 added. In this study, there are only the primary phase and the
187 secondary phase, and thus their volume fractions are denoted as
188 α
1
and α
2
, respectively. Hence, there are three possible
189 situations for the cell: α
i
= 0, indicating that the ith fluid is
190 empty in the mesh cell; α
i
= 1, indicating that the ith fluid is full
191 in the mesh cell; 0 < α
i
< 1, meaning that the two fluids exist in
192 the mesh cell. The dynamic behavior of the phase interface is
193 the solution of the equation of continuity with the second fluid
194 phase volume fraction
28
α
α
∂
∂
+·∇ =
t
u()
2
2
195
(5)
196 The volume fraction of primary fluid phase will be calculated by
197 the equation below
α+=1
12
198
(6)
199 The explicit and implicit time discretization can both be used
200 to solve the equation of volume fraction. For transient VOF
201 calculations, the explicit scheme should be used. In this work,
202 the geometric reconstruction approach is used, which is the
203 most accurate scheme in ANSYS FLUENT.
28
204 The fluid density and viscosity are linearized by the present
205 composition phases in each mesh cell
αρ α ρ=+
1
1
2
2
206
(7)
αμ αμ=+
1
1
2
2
207
(8)
208 The well-known volume averaged governing equations
29
are
209 used to describe the fluids phase motion. The continuity
210 equation is
ε
ε
∂
∂
+∇· =
t
u()
f
f
211
(9)
212Because of highly swirling flows in unbaffled stirred tank or
213single-baffle stirred tank in which the important flow character-
214istics are influenced by the Reynolds stress anisotropy, the
215Reynolds stress model (RSM)
30−32
was adopted in this study.
216And the momentum equation is
τ
ε
ε
ερρ
∂
∂
+∇·
=−∇+∇·
−
′′
++ +
⎛
⎝
⎜
⎞
⎠
⎟
t
puu
u
uu
fgf
()
()
(() )
ij
f
f
f
fs
f
p
217
(10)
218where ε
f
represents the void fraction. The fluid density and
219velocity are represented by ρ
f
and u, respectively. f
pf
represents
220the reacting force of the particle−fluid interaction term (1).
221The viscous stress tensor τ is defined as
μδ
=∇+∇−∇·
⎜⎟
⎛
⎝
⎞
⎠
uu u()()
2
3
()
T
k
222
(11)
223where μ indicates the dynamic viscosity, δ
k
represents the
224
identity tensor and ρ
′
u
ij
represents the Reynolds stresses.
225The CSF model are used to calculated the surface tension f
s
226of free surfaces
33−35
σκ
=··∇f
s
227
(12)
228where σ represents the surface force coefficient and κ represents
229the free-surface curvature, which is calculated by
=
|| ||
·∇ | | − ∇·
⎜⎟
⎡
⎣
⎢
⎛
⎝
⎞
⎠
⎤
⎦
⎥
n
n
n
nn
1
()
230
(13)
231where n = ∇·α
2
is the normal vector.
232f
pf
is a term of the momentum exchange between particle
233phase and fluid phase, and the expression of the momentum
234exchange term is
∑
=
Δ
−− − −
τρ∇∇·∇·
′′
V
fFFFFF
1
i
n
ii i uui
pf pf, p, , ( ), s,
ij
p
235(14)
236where
=+ + + ++ + +
+
τρ∇∇·∇·
′′
FF F F FF F F
F
ii i i uuii i i i
i
pf, d, p, , ( ), s, vm, B, Saff,
Mag,
ij
237
(15)
238and where ΔV is the corresponding mesh cell volume and n
p
239indicates the particles number. F
pf,i
represents the total of all
240interaction forces between particle−fluids: drag (F
d,i
), pressure
241gradient (F
∇p,i
), viscous stress force (F
∇·τ,i
), Reynolds stress
242
force (
ρ∇·
′′
uu
(),
ij
), capillary force (F
s,i
), virtual mass (F
vm,i
),
243Basset force (F
B,i
), Saffman lift (F
Saff,i
) and Magnus lift (F
Mag,i
).
2442.3. Particle −Fluids Interactions. 2.3.1. Interaction
245Forces. In DEM−VOF model, the expression of each
246interaction force contained in the term of fluid-particle
247interaction needs to be given. The drag force, pressure gradient
248force, viscous stress force, Reynolds stress force, capillary force,
249and lift forces, like Saffman lift force, Magnus lift force and
250fluid-induced torque, are all considered in this work. Table S2
251(in the Supporting Information) lists the expression of these
252interaction forces.
2532.3.2. Calculation of Mesh Porosities. The mesh porosity is
254important which could be used to calculate the drag force.
Industrial & Engineering Chemistry Research Article
DOI: 10.1021/acs.iecr.7b04833
Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
C