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Solid-liquidmixinganalysisinstirredvessels
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BaharakSajjadi
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Rev Chem Eng 2015; aop
*Corresponding author: Abdul Aziz Abdul Raman: Faculty of
Engineering, Department of Chemical Engineering, University of
Malaya, 50603 Kuala Lumpur, Malaysia,
e-mail: azizraman@um.edu.my
Raja Shazrin Shah Raja Ehsan Shah and Baharak Sajjadi: Faculty
of Engineering, Department of Chemical Engineering, University of
Malaya, 50603 Kuala Lumpur, Malaysia
http://orcid.org/0000-0001-5147-7667 (Raja Shazrin Shah Raja
Ehsan Shah)
Shaliza Ibrahim: Faculty of Engineering, Department of Civil
Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia
Raja Shazrin Shah Raja Ehsan Shah , Baharak Sajjadi , Abdul Aziz Abdul Raman *
and Shaliza Ibrahim
Solid-liquid mixing analysis in stirred vessels
Abstract : This review evaluates computational fluid
dynamic applications to analyze solid suspension quality
in stirred vessels. Most researchers typically employ either
Eulerian-Eulerian or Eulerian-Lagrangian approach to
investigate multiphase flow in stirred vessels. With suf-
ficient computational resources, the E-L approach simu-
lates flow structures with higher spatial resolution for
dispersed multiphase flows. Common turbulence models
such as the two-equation eddy-viscosity models ( k - ε ),
Reynolds stress model, direct numerical simulation,
and large eddy simulation are described and compared
for their respective limitations and advantages. Litera-
ture confirms that k- ε is the most widely used turbulence
model, but it suffers from some inherent shortcomings
due to assumption of isotropy of turbulence and homog-
enous mixing. Subsequently, the importance of different
forces concerning solid particle flotation is concluded.
Studies on dilute systems take into account only drag and
turbulence forces while other forces have always been
ignored. The simulations of off-bottom solid suspension,
solid drawdown, solid cloud height, solid concentration
distribution, and particle collision are considered for
studies involving solid suspension. Different models and
methods applied to investigate the abovementioned phe-
nomena are also discussed in this review.
Keywords: computational fluid dynamics (CFD); drag
force; solid suspension; stirred vessel.
DOI 10.1515/revce-2014-0028
Received July 2 , 2014 ; accepted December 16 , 2014
Abbreviations
ALE Arbitrary Lagrangian-Eulerian
CARPT Computer automated radioactive particle
tracking
CFD Computational fluid dynamics
DNS Direct numerical simulation
E-E Eulerian-Eulerian
E-G Eulerian-Granular
E-L Eulerian-Lagrangian
k - ε Two-equation Eddy-viscosity
LES Large eddy simulation
MRF Multiple reference frames
PDF Probability density function
RANS Reynolds averaged Navier-Stokes
RNG Renormalization group k - ε
RSM Reynolds stress model
SG Sliding grid
1 Introduction Turbulently agitated vessels where a solid-liquid sus-
pension is produced account for approximately 80%
of all industries and are common in leaching process,
crystallization process, catalytic reactions, bio-slurry
processes, mineral processing, precipitation, coagula-
tions, dissolution, water treatment, and a variety of other
applications ( Š pidla et al. 2005 ). Optimum performance,
accurate control, and reliable design of these equip-
ment are highly dependent on thorough understanding
of several phenomena such as turbulence, local dis-
persed concentration distribution, fluid flow dynamics,
system composition, and different equipment configura-
tions ( Leng and Calabrese 2004 ). At present, computa-
tional fluid dynamics (CFD) has emerged as an effective
and powerful mean in both applied and fundamental
research to predict local fluid dynamics, chemical reac-
tions, and related phenomena such as heat and mass
transfer in tanks ( Van den Akker 2006 ). However, hydro-
dynamic simulation in stirred vessels is very complex
due to the interaction between the rotating impeller with
multiphase dispersion and highly unsteady field of flow.
2 R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels
Therefore, progress on multiphase flow simulations and
selection of appropriate models in stirred vessels are far
from being fully understood.
Liquid-particle and particle-particle interactions
are dependent on dispersed phase volume fraction and
should be considered in CFD simulations. The interaction
between phases in multiphase stirred vessels is modeled
according to the coupling strength between them.
For dilute particle concentrations ( α p < 10 -6 ), particle
flow can be neglected but fluid flow should be considered
(one-way coupling). For intermediate particle concentra-
tions (10 -6 < α p < 10 -3 ) and above, the effects of continuous
phase on particles may be considered regardless of par-
ticle interaction (two-way coupling). For high concentra-
tions ( α p > 10 -3 ), the interaction between the particles and
the particle action on fluid must be accounted for (four-
way coupling) ( Elghobashi 1994 ). Hence, by increasing the
particle volume fraction, the interaction between the par-
ticles and fluid increases, as shown in Table 1 . Generally,
in systems containing two solid phases along with a liquid
as continuous medium, the solid-solid interaction should
be considered in the computational model. The methods
of Arastoopour et al. (1982) and Gidaspow et al. (1986) are
the most famous approaches for investigating these rela-
tionships, where the former is applied to the dilute phase
and the latter to the dense bed. It is also worth mentioning
that Gidaspow ’ s method ( Gidaspow et al. 1986 ) was modi-
fied by Bell et al. (1997) for simulating a two-dimensional
(2D) gas-solid fluidized bed containing two different par-
ticle sizes.
Tamburini et al. (2009) investigated the transient
behavior of an off-bottom solid-liquid suspension from
start-up to steady-state conditions inside a mechanically
baffled stirred tank equipped with a standard Rushton
turbine using the classical Eulerian-Eulerian (E-E) mul-
tifluid model approach. They tested different computa-
tional approaches in order to compute the particle-fluid
and particle-particle interactions since these terms affect
both solid suspension and distribution significantly. An
excess particle concentration treatment was therefore
adopted to compute particle-particle interaction. Particle
fluid interactions were assumed to be limited to drag force
exchange. These authors observed satisfactory agreement
between the modeled and experimental data, suggest-
ing that their modeling approaches were able to capture
the main factors affecting particle distribution in stirred
dense systems.
The main focus of this review is on collision-free dis-
persed phase by considering two-way coupling effects.
This work is the third part of a review series where the
previous two parts focused on gas-liquid ( Sajjadi et al.
2012 ) and liquid-liquid ( Sajjadi et al. 2013 ) systems. This
review provides an insight into different approaches in
multiphase flow simulation and investigates two terms
of momentum balance in solid-liquid systems in stirred
vessels. The first term was momentum and stress transfer
from continuous to dispersed phase, which was investi-
gated through different turbulence models. The second
term was momentum transferred between dispersed and
continuous phase, which depends on the force balance
on each particle, and the dispersed phase volume frac-
tion. Based on the force balance, the effects of drag and
nondrag forces on particles were investigated. The cou-
pling and interactions were, on the other hand, investi-
gated according to the dispersed phase volume fraction.
Results from these models for solid-liquid stirred vessel
could be used for analyzing suspension quality, draw-
down of floating solids, off-bottom complete solid sus-
pension, cloud height, and solid distribution. There are
different review papers on experimental methods and
geometrical parameters in stirred vessels ( Gogate et al.
2000 , Van den Akker 2006 , Ochieng et al. 2009 , Meyer
and Deglon 2011 ).
2 General approaches
2.1 Black-box
The “ black-box ” viewpoint was the first approach used to
study solid flow dynamics in stirred vessels. However, it
was not fully predictive and it required experimental infor-
mation at all conditions. Algebraic slip mixture model
was the other approach that assumed that both solid and
liquid phases existed at all points of vessel in the form of
interpenetrating continua and move at different velocities
( Guha et al. 2008 ).
2.2 Eulerian-Eulerian
The E-E and Eulerian-Lagrangian (E-L) approaches are
frequently found in literature. In the former, both of the
continuous and dispersed phases are considered as con-
tinuum, which can interpenetrate each other. In this
approach, several averaging methods (such as time,
volume, or ensemble averaging) are used to formulate
governing equations, so the particles are modeled with
a certain mass to impose the momentum exchange at the
point locations of the particles ( Ding et al. 2010 ). Hence,
R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels 3
Tabl
e 1
Co
nti
nu
ou
s a
nd
dis
pe
rse
d p
ha
se
in
tera
ctio
ns
.
Pa
rtic
le m
otio
n Co
uplin
g Pa
rtic
le vo
lum
e fra
ctio
n In
tera
ctio
n
Incr
ea
sin
g p
art
icle
volu
me
fra
ctio
n
On
e-w
ay
cou
pli
ng
Dis
pe
rse
d f
low
Sp
ars
e f
low
α p < 1
0 -6
1.
Co
nti
nu
ou
s f
luid
aff
ect
s p
art
icle
s
Two
-wa
y co
up
lin
gD
isp
ers
ed
flo
w
Dil
ute
flo
w
10
-6 <
α p < 1
0 -3
1.
Co
nti
nu
ou
s f
luid
aff
ect
s p
art
icle
s
2.
Pa
rtic
le m
oti
on
aff
ect
s c
on
tin
uo
us
-flu
id m
oti
on
Thre
e-w
ay
cou
pli
ng
Dis
pe
rse
d f
low
10
-3 <
α p < 1
0 -2
1.
Co
nti
nu
ou
s f
luid
aff
ect
s p
art
icle
s
2.
Pa
rtic
le m
oti
on
aff
ect
s c
on
tin
uo
us
-flu
id m
oti
on
3.
Pa
rtic
le d
istu
rba
nce
of
the
flu
id l
oca
lly
aff
ect
s a
no
the
r p
art
icle
’ s m
oti
on
Fou
r-w
ay
cou
pli
ng
Dis
pe
rse
d f
low
10
-2 <
α p < 1
0 -1
1.
Co
nti
nu
ou
s f
luid
aff
ect
s p
art
icle
s
2.
Pa
rtic
le m
oti
on
aff
ect
s c
on
tin
uo
us
-flu
id m
oti
on
3.
Pa
rtic
le d
istu
rba
nce
of
the
flu
id l
oca
lly
aff
ect
s a
no
the
r p
art
icle
’ s m
oti
on
4.
Pa
rtic
le c
oll
isio
n a
ffe
cts
mo
tio
n o
f in
div
idu
al
pa
rtic
le
Fo
ur-
wa
y co
up
lin
gD
en
se
flo
w
Co
llis
ion
-do
min
ate
d f
low
α p > 1
0 -1
1.
Co
nti
nu
ou
s f
luid
aff
ect
s p
art
icle
s
2.
Pa
rtic
le m
oti
on
aff
ect
s c
on
tin
uo
us
-flu
id m
oti
on
3.
Pa
rtic
le d
istu
rba
nce
of
the
flu
id l
oca
lly
aff
ect
s a
no
the
r p
art
icle
’ s m
oti
on
4.
Pa
rtic
le c
oll
isio
n a
ffe
cts
mo
tio
n o
f in
div
idu
al
pa
rtic
les
5.
Pa
rtic
les
mo
ve a
s a
gro
up
wit
h h
igh
fre
qu
en
cy o
f co
llis
ion
s
Fo
ur-
wa
y co
up
lin
gD
en
se
flo
w
Co
nta
ct-d
om
ina
ted
flo
w
1.
Co
nti
nu
ou
s f
luid
aff
ect
s p
art
icle
s
2.
Pa
rtic
le m
oti
on
aff
ect
s c
on
tin
uo
us
-flu
id m
oti
on
3.
Pa
rtic
le d
istu
rba
nce
of
the
flu
id l
oca
lly
aff
ect
s a
no
the
r p
art
icle
’ s m
oti
on
4.
Pa
rtic
le c
oll
isio
n a
ffe
cts
mo
tio
n o
f in
div
idu
al
5.
Pa
rtic
les
ha
ve a
hig
h f
req
ue
ncy
of
con
tact
4 R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels
one calculates the average local velocity, volume fraction,
and other parameters, not the properties of each indi-
vidual particle ( van Wachem and Almstedt 2003 ). In the
Eulerian approach, most researchers have used monosize
particle. Ochieng and Lewis (2006a) used a polydisperse
approach to investigate particle size distribution and
solid concentration for high-density (nickel) particles and
clearly showed that this assumption could not accurately
predict experimental results for high-density particles
( Ochieng and Lewis 2006a ).
2.3 Eulerian-Lagrangian
In the E-L approach, the continuous phase is treated in
a Eulerian framework whereas the motion of dispersed
phase is simulated by solving the force balance of each
particle.
Briefly, since the E-L approach tracks the particles
individually, it produces more reflective results and can
handle complex phenomena involving particle size,
composition, distribution agglomeration, and disag-
gregation ( Farzpourmachiani et al. 2011 ). However, the
E-L approach faces three problems, which are i) false
fluctuations of the continuous phase velocity that occur
when dispersion passes through the interface of the com-
putational grid and causes sudden changes in the local
volume fraction, ii) the amplitude of false velocity fluc-
tuation increases when a small grid is applied, and iii) as
the volume fraction of the dispersed phase increases, the
interaction between the two phases increases too, which
cannot be accounted for by this method due to consider-
ably high demand of computational cost. As a solution
for the second problem, Kitagawa et al. (2001) presented
a calculation method to convert dispersion to continuous
phase interaction, which was accompanied by spherical
dispersion migration. They used Gaussian and sine wave
filtering function to convert discrete dispersion volume
fractions to a spatially differentiable distribution, so that
3D continuous phase flow structures induced by rising
spherical bubbles and/or settling solid particles were dem-
onstrated ( Kitagawa et al. 2001 ). For the third problem, it
potentially worsens as the calculation requires the time
history of the stationary mean of two-phase flow for a
long period of time, which increases the computational
demand ( Ochieng and Onyango 2008 , 2010 ). In view of
these problems, 3D flow simulation of this approach is
more applicable in dilute systems ( Derksen 2003 , Yeoh
et al. 2005 ). In the E-E approach, the interaction between
the phases is accounted for through source terms and flow
field is solved for both phases.
2.4 Arbitrary Lagrangian-Eulerian
It should be noted that in Eulerian description of solids,
the expression x x u= + holds and the mathemati-
cal models could be derived using either the Eulerian
description ( x , t ) or the Lagrangian description ( , ).x t
The two descriptions are similar in terms of displace-
ment, u . However, there would be a restriction in fluid
at the fixed location, ,x material transport, and the
current position of some material particle, but without
knowledge of displacement u . Many recent published
works switched to arbitrary Lagrangian-Eulerian (ALE)
as a starting point for designing various computational
strategies for computing liquid-solid interaction evo-
lution. There is no basic dependence on particles, and
ALE treats the computational mesh as a reference frame
that may be moving with arbitrary velocity, .ν In the
choice of ,ν Eulerian and Lagrangian descriptions are
derived from a single mathematical model, which is sub-
sequently applied for numerical computations. Since
1981, ALE approaches have been used in a wide range
of solid-liquid-based aspects, i.e., fluid-solid interaction,
free surface problems, deforming-spatial-domain, stabi-
lized space-time, etc. In most recent works using ALE,
Sankaran et al. (2009) developed a coupled and unified
solution method for fluid-structure interactions. Another
work by Farhat et al. (2010) considered various time
integration schemes for fluid-solid interaction prob-
lems using this approach. A dual-primal finite element
tearing and interconnecting method was considered
by Li et al. (2012) in order to solve a class of fluid-solid
interaction problems in the frequency domain using
the ALE approach. Wang et al. (2011) improved various
algorithms for load computation in embedded boundary
methods and interface treatment for fluid-structure and
fluid interaction problems through the ALE approach.
Gr é tarsson et al. (2011) investigated numerically stable
fluid-structure interactions between compressible flow
and solid structures. Amsallem and Farhat (2012) consid-
ered the stability of linearized reduced-order models with
application to fluid-structure interaction. Subsequently,
Wang et al. (2012) considered computational algorithms
for tracking dynamic fluid-structure interfaces in embed-
ded boundary methods.
In mentioned approaches, there are two basic equa-
tions that should be solved: continuity and momentum
balance equations, as given in Eqs. (1) and (2), respectively.
( ) ( U )k k k k kt x
α ρ α ρ Γ∂ ∂+ =∂ ∂
(1)
and
R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels 5
( ) ( ) - ,k k k k k k k k k k k Ik
Pu u u g T F
t x x xα ρ α ρ α α ρ α
∂ ∂ ∂ ∂+ = + + ⋅ +∂ ∂ ∂ ∂
(2)
where T k is the stress tensor from the liquid and F
Ik is the
momentum transferred from bubbles to liquid.
The Eulerian-Granular (E-G) model is the other
approach based on the Eulerian model wherein the solids
are considered as a pseudo-fluid. Therefore, the pseudo-
fluid physical properties, such as solid stress, pressure,
and viscosity, are derived from the kinetic theory of gran-
ular flow. Continuity and momentum equations are also
solved in this approach. It also includes an additional
transport equation for granular temperature. The solid
phase momentum equation includes an additional solid
pressure term that can be written in the form of
1
( ),n
fs fs fsk
R m u=
+∑ �
(3)
where the subscript f s accounts for the exchange between
the operational phases. The coupling between the phases
is obtained through interphase exchange coefficients and
pressure term, which are determined from the kinetic
theory of granular flow ( Ding and Gidaspow 1990 ). This
approach is more accurate for processes containing high
local particle concentrations.
3 Stress tensor from continuous phase
Turbulence treated through the terms of stress tensor ( T k )
in the momentum equation balance is shown in Eq. (4):
α μ μ δ α ρ⎛ ⎞∂∂ ⎛ ⎞ ∂ ∂= + + ′ ′⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠
,
2( ) - - ( - ).
3
ji k
k k t k ij k i jk
j i k j
uu uT u u
x x x x (4)
The term ρ ′ ′i j
u u introduces Reynolds stresses:
ρ ρ δ μ⎛ ⎞∂∂
= + +′ ′ ⎜ ⎟∂ ∂⎝ ⎠
2( ) .
3
ji
i j ij t
j i
UUu u k
x x
(5)
Two main categories of turbulence models are avail-
able through coupling of this term with the others in the
momentum balance equation. The first category under
the Lagrangian approach includes direct numerical simu-
lation (DNS), large eddy simulation (LES), and stochas-
tic modeling. The second category includes Reynolds
averaged Navier-Stokes (RANS) and probability density
function (PDF) modeling, which are collectively called the
Eulerian approach.
Since the Lagrangian framework suffers from limi-
tations of high number of dispersed phase, a Eulerian
framework is more useful. RANS and PDF are two impor-
tant models in this regard. The related equations for
these models for dispersed phase statistical properties
are in the form of “ fluid ” equations based on the Eulerian
framework.
RANS equations are divided into two categories:
Reynolds stress model (RSM) and eddy-viscosity model.
The two-equation eddy-viscosity models ( k- ε ) are based
on turbulent kinetic energy ( k ) and energy dissipation
rate ( ε ). Standard k- ε , renormalization group k- ε (RNG),
and realizable k- ε are three different appearances of the
k- ε family ’ s model; k- ω is another member of this group,
where ε has been replaced by turbulence frequency ( ω ).
In the situation of low dispersed phase, the RANS
method represents a good tradeoff between the accuracy
and computational costs ( Petitti et al. 2010 ). However, it
suffers from some shortcomings due to assumption of iso-
tropic turbulence and homogeneous and underprediction
of energy dissipation rate. Osman and Varley (1999) found
that the predicted mixing time was two times higher than
the experimental results due to underestimation of mean
velocity near the Rushton turbine. Jaworski et al. (2000)
showed an overpredicted value of mixing time using
standard and RNG k- ε due to high underprediction of mass
exchange between the four distinct axial-radial circulation
loops at the boundary between the circulation loops of
upper and lower impeller, whereas k- ε yielded the smallest
values of effective diffusivity in the impeller region.
However, k- ε can accurately predict the length and
intensity of the trailing vortices generated off impeller
blades in stirred tanks but requires a large number of
modeled revolutions to obtain good statistical average
( Singh et al. 2011 ). Guha et al. (2008) carried out a system-
atic experimental investigation of solid hydrodynamics
in dense solid-liquid suspensions by the computer auto-
mated radioactive particle tracking (CARPT) technique
and CFD simulation. They validated the CFD results of
averaged solid velocities, turbulent kinetic energy, and
solid sojourn time distribution using the results from
CARPT technique. They also investigated the ability of
LES along with the Euler-Euler approach in modeling
solid-liquid stirred vessels. They reported that LES and
the Euler-Euler model could not capture the difference
between the top and bottom flow patterns. In spite of the
observed mismatch, the simulation results were in rea-
sonably good agreement with the experimental data for
mean and standard deviation of solid sojourn times in
6 R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels
the stirred vessel. PDF modeling is the most recent tech-
nique. In this method, the transition between the Lagran-
gian and Eulerian frames is modeled initially by using the
Liouville theorem ( Monaghan 2005 ). The closure prob-
lems are then solved to achieve closed PDF equation and
the moments of the PDF equation should be taken. These
equations are related to baro-diffusion, turbulent thermal
diffusion, and turbophoresis. The phase space variables
make up various PDF models. The applied closure scheme
and the researcher ’ s ability to predict selected statistical
properties of the fluid are the determining factors on accu-
racy of PDF equation. Generally, the result of PDF method
contains more information than that of RANS models.
Some researchers have also used the LES method,
which is able to overcome k- ε limitations for investigations
of unsteady behavior in turbulent flow. In LES, large-scale
eddies are resolved, and the small scales, which are iso-
tropic in nature, are modeled using subgrid-scale models.
The major role of subgrid-scale models is to provide proper
dissipation for the energy transferred from the large-scale
to the small-scale eddies ( Joshi et al. 2011 ). LES is not fine-
tuned for quick process design validation and it requires
rather long computational time. There has not been any
validation for slurries with high solid concentrations
using LES, although it is still easier to recognize fluctua-
tions in comparison to RANS ( Hartmann et al. 2004 ).
DNS is another option that resolves all time scales for
which the basic equations are constructed without turbu-
lence and interface models. This is done by direct inte-
gration of continuity, momentum, and energy equations
in high-order numerical schemes on a sufficiently fine
grid. Besides, the assumptions involved in Navier-Stokes
equation are still arguable especially in two-phase flows.
Thus, solving of flow around the particles is not possi-
ble in this model. Therefore, physical models should be
considered to describe the interactions between the two
phases. This argument becomes much stronger when
the two-way coupling between the phases is considered.
Generally, DNSs of two-phase turbulent flows are catego-
rized in three different groups: isotropic homogeneous,
anisotropic homogeneous, and inhomogeneous flows. In
isotropic homogeneous flows, the average properties of
random motion do not change under reflection or rotation
of the coordinate system. Besides, they are independent
of position in the fluid. Therefore, only one component is
presented for Reynolds stress tensor. In anisotropic homo-
geneous flow, the turbulence of all components should be
considered, whereas in inhomogeneous flow, these values
can differ from two different locations in the flow. This
method also provides details on fluctuations and particle
flow properties.
In contrast to DNS, LES provides more satisfying
results at high Reynolds numbers. LES can provide details
of flow field that cannot be obtained by RANS, but it cannot
offer details on small-scale fluctuations ( Derksen and Van
den Akker 1999 ). In other words, LES provides a solution
for turbulence with larger scales while smaller scales are
presented in the form of statistical average. Therefore, the
interaction between different scales of turbulence and
particles remains an open question. Since dissipation rate
cannot be solved during LES, it is considered posteriori
and is strongly dependent on subgrid-scale model used
in the simulations ( Delafosse et al. 2008 ). Generally, lit-
erature confirms that RANS-based models (especially k- ε )
are the most widely used turbulence model in spite of its
shortcomings because RANS-based models assume the
isotropy of turbulence and homogenous mixing, which
are suitable particularly for very high Reynolds number in
unbaffled stirred reactors. In addition, it is worth mention-
ing that LES and DNS approaches are also more accurate
for turbulent fluid mixing but more computing demand-
ing ( Hartmann et al. 2004 , Singh et al. 2011 ). The related
equations considering the mentioned turbulence models
are in Table 2 .
In conclusion, the k - ε turbulence model is the most
widely used for solid-liquid systems. Dispersed, per phase,
homogeneous, and asymmetric are the main extensions of
the standard k - ε turbulence model for two-phase turbu-
lent fluid fields.
In the per phase approach (also named phase-spe-
cific), the turbulence equations, along with the additional
terms referring to the modeling of interphase transport of
k and ε , are considered for each phase.
In the homogeneous formulation, only one k and
one ε equations are considered and the values are shared
between the phases while the physical properties of the
mixture are adopted. No interphase turbulence transfer
terms are employed for the transport equations for k and ε .
The dispersed approach is more suitable for dilute
suspensions. In this expression, fluctuating quantities
of the particles are solved as functions of the mean prop-
erties of the carrier phase and the ratio of eddy-particle
interaction time and the particle relaxation time. Carrier
phase turbulence is modeled through the standard k - ε
model involving extra terms that account for the effect of
the particles on the carrier phase ( Feng et al. 2012 ).
The asymmetric k- ε is more applicable for dense solid-
liquid suspensions where N ≤ Nj. In this formulation, many
particles may be unsuspended under partial suspension
conditions, and therefore, no turbulent viscosity was
accounted for the particles and only the turbulence of the
continuous phase was calculated ( Tamburini et al. 2011 ).
R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels 7
Table 2 Summary of turbulence models.
Model Parameters
RANS
Two eddy viscosity models
Standard k - ε
1, 2,-k
S S
GC C
k kε ε
ρ ερ
⎛ ⎞⎜ ⎟⎝ ⎠
C μ , S
= 0.09, C 1, S
= 1.44, C 2, S
= 1.92, σ k , S
= 1, σ ε , S
= 1.314
RNG k - ε
1, 2, RNG- -kG
C Ck k kε ε ε
ρ α ε ερ
⎛ ⎞⎜ ⎟⎝ ⎠RNG
3 0
3
1- /,
1C
μ
η ηα η
βη=
+ C μ ,RNG
= 0.0845, C 1,RNG
= 1.42, C 2, S
= 1.68, σ k,S
= σ ε , S
= 0.719, η 0 = 4.8,
β = 0.012,
,kEηε
=
E 2 = 2 E ij E
ij ,
0.5ji
ijj i
uuE
x x⎛ ⎞∂∂
= +⎜ ⎟∂ ∂⎝ ⎠
Realizable k - ε model 2
1,Re 2,Re-C E C
kε
ρ ενε
⎛ ⎞⎜ ⎟⎝ ⎠+
μ
ηη
⎡ ⎤= ⎢ ⎥+⎣ ⎦
,Remax 0.43; ,
5C C
1,Re = 0.09, C
2, Re = 1.9, σ
k ,Re = 1, σ
ε ,Re = 1.2, η
ε= ,
kE E 2 = 2 E ij E
ij ,
⎛ ⎞∂∂
= +⎜ ⎟∂ ∂⎝ ⎠0.5
jiij
j i
uuE
x x
k - ω ( )
-j
kj j j
u kk k kP kt x x x
β ω ν σω
∗ ∗∂ ⎡ ⎤⎛ ⎞∂ ∂ ∂+ = + +⎜ ⎟⎢ ⎥⎝ ⎠∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦
2( )
-j
kj
d
j j j j
uP
t x kk k
x x x x
ωω ωα βω
σω ων σ
ω ω
∂∂ + =∂ ∂
⎡ ⎤⎛ ⎞∂ ∂ ∂ ∂+ +⎜ ⎟⎢ ⎥⎝ ⎠∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦
lim
2max ,
ij ijt
S Sk Cν ω ωω β∗
⎛ ⎞⎜ ⎟= = ⎜ ⎟⎝ ⎠
��
α = 0.52, β = β 0 f
β , β 0 = 0.0708, β * = 0.09, σ = 0.5
3
1 850.6, , ,
1 100 ( )
ij jk kiSf ω
β ωω
χσ χ
χ β ω∗
∗
Ω Ω+= = =
+
0 0, 0.125 0d dj j j j
k kx x x x
ω ωσ σ
∂ ∂ ∂ ∂= ≤ = >∂ ∂ ∂ ∂
for for
RSM
2
-
- -
ij ij j ik ik jk
k k k
ijk ij ij ijk
u uu
t x x x
Cx
τ ττ τ
ε ν τ
⎛ ⎞∂ ∂ ∂⟨ ⟩ ∂⟨ ⟩+⟨ ⟩ = +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠
∂ +Π + ∇∂
j iij
i j
u uPx xρ
⎡ ⎤∂ ′ ∂ ′′Π = +⎢ ⎥∂ ∂⎢ ⎥⎣ ⎦
2ji
ijk k
uux x
ε ν∂ ′∂ ′
=∂ ∂
1( )ijk i j k i jk j ikC u u u P u P uδ δ
ρ=⟨ ⟩+ ⟨ ⟩ +⟨ ⟩′ ′ ′ ′ ′ ′ ′
σ k = 0.82, C μ = 0.09
LES
SGS 2
2
1( ) - -
Re
iji ii j
j i j j
u upu ut x x x x
τ∂∂ ∂∂ ∂+ = +∂ ∂ ∂ ∂ ∂
SGS -ij i j i ju u u uτ =
SGS SGS SGS1- -23ij kk ij T ijSτ τ δ ν=
2 2| | 2T S S ij ijL S L S Sν = =SGS
1
2
jiij
j i
uuS
x x⎛ ⎞∂∂
= +⎜ ⎟∂ ∂⎝ ⎠
1/3min( , )S S iL kd C V=
Comparing the mentioned models, the homogeneous
k - ε turbulence model provides satisfactory representation
of the particle distribution throughout the system in cases
including dense suspensions in stirred vessels ( Montante
et al. 2001a , Micale et al. 2004 , Montante and Magelli
2005 , Khopkar et al. 2006 , Kasat et al. 2008 , Tamburini
et al. 2009 ).
Further details about the turbulence models are dis-
cussed in Sajjadi et al. (2012) .
4 Dispersed and continuous phase interaction
Another important term in momentum balance equations
is the momentum transfer between the particles (dis-
persed phase) and liquid (continuous phase), as shown
by Eq. (6). The value is dependent on the dispersed phase
volume fraction ( α ) and the momentum transferred ( F IK
):
8 R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels
cell
-(1/V ) ( ) ,k Ik i L
n k
F tE F tα Δ Δ= = ∑∑
(6)
where V is the volume of the cell and Δ t is the time step.
The last term is the summation of drag and nondrag
forces. Additionally, the particle velocities can be calcu-
lated using Eq. (7) by solving the force balance on each
particle:
= + + + + .b
b G P D L VM
t
dum F F F F F
d
(7)
The m b and u
b on the left-hand side of Eq. (7) represent
particle mass and velocity, respectively. The right-hand
section is the summation of all forces acting on the parti-
cle that are divided into two major groups including drag
and nondrag forces ( Buffo et al. 2011 ).
4.1 Nondrag forces
Nondrag forces include turbulent dispersion ( F TD
), lift
force, ( F L ) and virtual mass ( F
VM ). Turbulent dispersion
force denotes the effect of turbulent fluctuations on
effective momentum transfer. In the simulation of solid
suspension in stirred vessels, turbulent dispersion force
is significant when the particle size is smaller than the
turbulence eddies ( Wadnerkar et al. 2012 ). Turbulent
dispersion force appears in the momentum equation for
Favre averaging approach, but it appears as a function of
Schmidt number in the continuity equation for time-aver-
aging approach ( Ochieng and Onyango 2010 ).
The lift force represents traverse force caused by rota-
tional strain and is significant when there is a large veloc-
ity gradient or strong vorticity in the continuous phase
( Taghavi et al. 2011 ). In stirred vessels, velocity gradients
in vicinity to the impeller are more than those in the bulk
region, so the magnitude of lift force is much smaller at
the region far away from the impeller.
Virtual mass force denotes an inertial force that is
caused by relative acceleration of phases due to move-
ment of particles and is significant only when there is
strong acceleration. In other words, when the dispersed
phase density is much smaller than the continuous
phase density ( Khopkar et al. 2006 ), the mentioned
forces are significant in vicinity to the impeller blades
and wall lubrication force tends to push dispersed
phase away from the wall. Detailed descriptions on
nondrag forces and turbulent dispersion force were
given by Lahey et al. (1993) and Lopez de Bertodano
(1998) .
4.2 Drag force
Drag force represents interphase momentum transfer
due to disturbance. Drag and turbulence forces over-
come other forces in stirred vessels due to the mechani-
cal energy of liquid bulk and its turbulence. These forces
cause solid particles to draw down, and other forces may
be considered depending on the fluid flow properties as
well as particle and fluid physical properties ( Khazam and
Kresta 2008 ). The relative magnitude of drag and gravi-
tational forces is given by Archimedes number ( Ar ). The
drag force is more important for smaller particles, which
have lower values of Ar , where terminal velocity has less
influence on the flow field ( Ochieng and Lewis 2006b ).
Drag models affect CFD simulations for solid-liquid stirred
reactors. In these systems, the interphase drag coefficient
( C D ) is a complex function of a drag coefficient in a stag-
nant liquid ( C Do
), prevailing turbulence level, and solid
holdup present in the reactor. If the surrounding liquid is
turbulent, the prevailing turbulence is expected to influ-
ence effective drag coefficient on particles. By assuming
standard drag coefficient (quiescent flow), the effect of
turbulent eddies on motion of dispersed phase is often
ignored. This results in large errors in simulation of dis-
persed phase concentration profiles for a turbulent flow. It
is worth noting that the magnitude of the effect increases
with both mean turbulent energy dissipation rate and par-
ticle size. Flow field around solid particles is also affected
by microscale turbulence instead of inertial-scale turbu-
lence, so interphase drag coefficient is affected by micro-
scales ( Tamburini et al. 2011 ).
Drag coefficient is also influenced by the relative inten-
sity of turbulence in a free stream turbulent flow and is
defined as the ratio of the root-mean-square of fluid veloc-
ity to mean particle-liquid relative velocity. Brucato et al.
(1998) and Mersmann et al. (1998) explained that if the
ratio of particle diameter to Kolmogoroff eddy size ( d p / λ )
was more than 5, the influence of free stream turbulence
on drag force had to be considered; but if the ratio was < 5,
the interaction between the energy dissipating eddies and
particles was negligible. A constant drag coefficient may
be used for such particle size because the drag coefficient
is not affected by turbulence (Ochieng and Onyango 2008).
Generally, three methods are available to compute
drag forces. The first and simplest approach, namely,
fixed- C D , is based on standard drag curve that computes
the value for a single particle settling at its terminal
velocity in a silent fluid. This approach is more suitable
for dilute systems and low-Reynolds-number flows. The
second is slip- C D , which considers C
D variables in each cell
R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels 9
Table 3 Drag models.
Model Note References
0.687
0
24max (1 0.15Re ),0.44 ,Re
Re
p rD P P
d UC
ν
⎛ ⎞= + =⎜ ⎟⎝ ⎠
– Applicable on spherical and single particle immersed in
unidirectional flow
– Applicable on particles moving in a stagnant liquid
– Provided satisfactory results at low impeller speed
– Applicable on spherical particles in an infinite fluid phase
Schiller and
Naumann 1935
3
-4
01 8.76 10
pD D
dC C
λ
⎡ ⎤⎛ ⎞⎢ ⎥= + × ⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦
– Accounts for free stream turbulence through the
Kolmogoroff length scale
– Gives reasonable prediction in dilute solid-liquid systems
– Gives the best prediction in the impeller region, in which
mixing is due to turbulent eddies or turbulent dispersion
– Reliable in modeling particle drag under partial-to-
complete suspension conditions
Montante et al.
2001b
α α⎛ ⎞
= + =′⎜ ⎟′⎝ ⎠-1.65 0.68724
max (1 0.15 ),0.44 , Re ReReD L P P L p
p
C – Applicable for dilute systems
– Applicable for immersed single particle
– Applicable for unidirectional flow
Wen and Yu 1966
α μ α ρ
α= +
2
2
s
7 | |150
4(1- )
s L s L rD
pp
UC
dd
– Applicable for immersed single particle
– Applicable for unidirectional flow for single particle
immersed in a unidirectional flow
– Provided satisfactory results for suspended solids only
Ergun 1952
0.7524(1 0.1Re )
ReD ss
C = + – Provided satisfactory results for cell slip velocity calculation
– Influenced by the gravitational and surface tension forces
on the interface between the phases
Ishii and Zuber 1979
-2.65
| - |3
4s l l s l
sl D ls
K Cd
α α ρ ν να=
� �
α α
α= + <0.68724
(1 0.15( Re ) ) 0.2ReD l P s
l s
C
α α μ α ρ ν να
α= + >
� �(1- ) | - |
150 1.75 0.2s l l s l s lsl s
l s s
Kd d
– Combination of the Wen and Yu model and the Ergun Model
– Valid when the internal forces are negligible, which means
that the viscous forces dominate the flow behavior
– Applicable for both high and low solid loading
Gidaspow 1994
22
, ,
3 3
3(1 ) ( )2 8 | - |
2 ( )
ss fr ss s s s s s s o ss
ss s ls s s s
e C d d gK v v
d d
π πα ρ α ρ
π ρ ρ
′ ′ ′ ′ ′ ′
′′ ′
⎛ ⎞+ + +⎜ ⎟⎝ ⎠
=+
� � – Applicable for solid-solid interaction in liquid-solid-solid
systems of the interphase momentum transfer between two
dispersed solid phases
Syamlal 1987
-2
00.6 0.4tanh 16 -1D D
p
C Cdλ⎡ ⎤⎛ ⎞
= +⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦
– Does not account the influence of particle density
– Reliable in modeling particle drag under partial-to-
complete suspension conditions
Penilli et al. 2001
3
-5
01 8.76 10
pD D
dC C
λ
⎡ ⎤⎛ ⎞⎢ ⎥= + × ⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦
– Accounts for free stream turbulence through the
Kolmogoroff length scale
– Gives reasonable prediction in dilute solid-liquid systems
– Gives the best prediction in the impeller region, in which
mixing is due to turbulent eddies or turbulent dispersion
– Developed Brucato Model for solid-liquid system in stirred
tank
Khopkar et al. 2006
0.5
-0.6 0.32tanh 16 -1S P l
t p l
UU d
ρ ρλρ
⎛ ⎞⎛ ⎞⎜ ⎟= + ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
– Gives reasonable prediction in dilute solid-liquid systems Fajner et al. 2008
in relation to slip velocity. The third approach is turb- C D
( Brucato et al. 1998 ), which considers the influence of free-
stream turbulence upon drag interphase force. Table 3
shows a number of drag models available to estimate drag
force in solid-liquid systems. In many CFD simulations,
these models are used in RANS equations.
10 R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels
Tamburini et al. (2009) investigated dense solid-liquid
off-bottom suspension inside a stirred tank and focused
on different computational approaches to compute drag
interphase force. They found that a simple modeling of
interphase drag was practical and sufficient to correctly
predict the suspension height of a solid-liquid suspen-
sion. They used three different equations to account for
the effects of solid concentration on interphase drag force.
The reported turb- C D simulation results were in better
agreement with the experimental result in comparison
to the fixed- C D simulations ( Tamburini et al. 2009 ). In
the same work, they also reported that Brucato ’ s model
( Brucato et al. 1998 ) predicted a much higher drag coef-
ficient for d p / λ ratios ( Tamburini et al. 2011 ). The equa-
tions cater for low solid fractions in the upper clear liquid
layer, intermediate volume fractions inside the suspen-
sion, and high solid concentrations near the tank bottom,
respectively.
4.2.1 Drag models
A variety of drag models available in the literature are
summarized in Table 3 . Dependency of drag on volume
fraction and density are always taken into considera-
tion ( Lane et al. 2005 , Fajner et al. 2008 ), except for low-
Reynolds-number and dilute systems, in which particle
drag is given by Stokes law. Turbulent fluctuations increase
and significantly affect the predictions at high Reynolds
number. The drag and turbulence thus become important
for stirred tank systems. Besides, the contribution of tur-
bulent dispersion force is significant only when the parti-
cle size is smaller than the turbulent eddies. Therefore, in
stirred vessels where the largest energy containing eddy (in
mm) is 10 times more than the particle size, the role of the
turbulent dispersion is entirely prominent ( Sardeshpande
and Ranade 2012 ). There should then be a model that takes
turbulence into account, since the impact of turbulence on
the drag increases with increasing Reynolds number. In
view of this, Brucato ’ s model ( Brucato et al. 1998 ) is more
suitable in stirred vessel. This model is based on the ratio
of particle diameter and Kolmogorov length scales, which
is dependent entirely on turbulence.
Therefore, change in turbulence changes the drag.
This model has been successfully employed by some
authors ( Montante et al. 2001a , Montante and Magelli
2005 ). However, this model does not capture complete
suspension of solid particles in stirred vessels and when
there are more than 10% of solid particles. In view of
this, the predicted value may deviate from the real value,
with overprediction of the height of solid suspension.
A mismatch in the radial and tangential components
of velocity at impeller plane was also reported by Pan-
neerselvam and Savithri (2008) . Some other authors
( Ljungqvist and Rasmuson 2001 , Montante et al. 2001a )
have also reported that this model gives the best results
in systems with low particle hold-up. Considering these
problems, Khopkar et al. (2006) modified Brucato ’ s model
( Brucato et al. 1998 ) by decreasing the value of the pro-
portionality constant by about 10 times ( K = 8.76 × 10 -5 ) for
systems containing particle size < 655 μ m and solid hold-
up < 16%. Therefore, the model clearly indicates that the
effective drag coefficient depends on the ratio of the par-
ticle diameter to the Kolmogorov length scale. Khopkar
et al. (2006) reported that suspension height was well pre-
dicted by the modified correlation, substantiated by the
experimental data. The other authors have also indicated
this result ( Sardeshpande et al. 2010 ). Hence, the propor-
tionality constant appearing in Brucato ’ s model ( Brucato
et al. 1998 ) needs to be reduced for higher solid loading
and larger particle Reynolds numbers ( Khopkar et al.
2006 ). Additionally, this model can be combined with the
k - ε model; thus, it is more appropriate for stirred tanks.
Gidaspow ’ s (1994) model is another widely used
model for solid-liquid systems. The model is a combina-
tion of Wen-Yu ’ s ( Wen and Yu 1966 ) model for low solid
loading rate ( ϕ s < 0.2) and Ergun ’ s (1952) model for high
solid loading rate ( ϕ s > 0.2). Gidaspow ’ s (1994) model is
more appropriate for systems with relatively higher solid
loading rate compared to other models. Ochieng and
Lewis (2006b) compared Gidaspow ’ s (1994) model and
Brucato ’ s model ( Brucato et al. 1998 ) in both dilute and
dense systems (1 – 20%w/w with density of 9800 kg/m 3 )
and reported that both models produced similar results.
Ochieng and Onyango (2008) compared different
drag models that account for the effect of free stream
turbulence, including those based on Reynolds number
only. They also studied those models that account for
solid volume fraction for predicting solids suspension in
stirred vessels and found that drag models based on solids
volume fraction produced comparatively better results.
Ljungqvist and Rasmuson (2001) used two phase flow
field in an axially stirred vessel containing nickel particles
of 75 μ m diameter to investigate slip velocity distribution
and compared the performance of four different drag
models: Ishii-Zuber model ( Ishii and Zuber 1979 ), Ihme ’ s
model ( AEAT 2003 ), Brucato ’ s model ( Brucato et al. 1998 ),
and Schiller-Naumann ’ s model ( Schiller and Naumann
1935 ). In their study, λ / d p ratio was high enough so that C
D
did not increase relatively to C Do
; thus, they did not observe
any difference in prediction by those models. Besides,
they reported that radial and tangential components were
R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels 11
severely underestimated due to overestimation of axial
slip. They further estimated slip velocity from the set-
tling velocity of particles in a quiescent liquid (Andersson
theory; Andersson 1986 ). However, slip velocity derived
from this method deviated from the measured values. This
was because drag in viscous Reynolds number region was
increased by free stream turbulence. Besides, enhanced
drag also resulted in lower slip velocities and overestima-
tion of the axial slip ( Ljungqvist and Rasmuson 2001 ). In
other words, the magnitude of drag increases with turbu-
lence and influences slip velocity between the primary
and secondary phases. Hence, underprediction of turbu-
lent kinetic energy eventually results in underprediction
of slip velocities. Derksen (2003) reported dominance
of high slip velocity in the impeller zone by comparing
slip velocities in terms of linear and rotational Reynolds
number. The author also pointed out that although all
drag models were able to capture high slip velocities in
this region, only modified Brucato ’ s drag model ( Brucato
et al. 1998 ) yielded reasonable predictions. Wadnerkar
et al. (2012) also discussed the significant differences in
the magnitude of slip velocity between Gidaspow ’ s (1994)
model and Brucato ’ s drag model ( Brucato et al. 1998 ) in
the impeller zone and reported underprediction of slip
velocities using both the models. Recently, Sardeshpande
et al. (2010) has simulated dimensionless axial slip veloc-
ity ( V slip
/ U tip
) for 1% and 7% (v/v) solid loadings in a stirred
vessel. They indicated that drag correlations proposed by
Brucato et al. (1998) and Khopkar et al. (2006) overpre-
dicted experimental axial slip velocity data in the vicin-
ity of impeller region. Therefore, it was more appropriate
to measure local slip velocity in solid-liquid stirred tanks
and use these data to evaluate different interphase drag
correlations.
5 Particle collision Particle collision defines the agglomeration and/or
breakup of particles in systems. This may involve a con-
tinuous phase and one or more dispersed phases. Particle
collision plays a vital role in multiphase turbulent flows
especially turbulent solid-liquid flows as it controls/influ-
enced by the process.
Formulation of the collision process starts very
simply, chosen for their mathematical convenience until
highly involved models are obtained that can analyze
complex fluid-particle interactions. The practical investi-
gation of particle collision is difficult and problem arises
as the complexity of the carrier fluid flow field rises, i.e.,
in turbulent flows. Table 4 lists the main particle collision
models, which are discussed in the following.
The boundaries of the particle collision in turbulent
flow regimes were fairly well determined in the formu-
lations of Saffman and Turner (1956) and Abrahamson
(1975) . The former considered particles that were remark-
ably smaller than the smallest scale of turbulence and rep-
resented velocities that were perfectly correlated with the
surrounding continuous phase, while the latter applied
to particles with significantly high inertias, representing
velocities that are entirely decorrelated from that of the
continuous phase.
The collision models have been considerably improved
and validated over the entire range of particle inertias by
Williams and Crane (1983) and Kruis and Kusters (1997) .
The expression of Williams and Crane investigated the
fluctuating relative motion of two liquid drops or solid
particles with intermediate sizes in a turbulent gaseous
system that exhibits velocities that are neither well cor-
related nor completely independent. Hence, this model is
not applicable in solid-liquid systems. Furthermore, it is
erroneously based on a cylindrical as opposed to a spheri-
cal formulation.
Yuu (1984) expressed a formulation for the fluctuating
relative velocity of two inertial particles in viscous systems
that are subrange of turbulence. This expression involved
the added mass effect experienced by particles in turbu-
lent liquid systems. However, it is not applicable for very
small particles. Neither the model of Yuu (1984) nor that of
Williams and Crane (1983) is valid for liquid systems with
particle sizes in the inertial subrange of turbulence.
Kruis and Kusters (1997) derived a universal formu-
lation for the collision kernel, which included the added
mass effect. This makes the expression valid for both the
viscous as well as the inertial subrange of turbulence.
Meanwhile, Hu and Mei (1997) stated that the colli-
sions between identical particle expressions have been
ignored in all previous formulations. Considering the
collision between moderately inertial particles, Hu and
Mei (1997) derived an expression; however, the effect of
gravity was not included in their formulation. The Hu and
Mei (1997) model was extended by Wang et al. (1998) con-
sidering a finite density-ratio correction as well as gravity
effects.
The preferential concentration of particles, in regions
of low vorticity and high strain rate, exhibited relaxation
times close to the Kolmogorov time scale. Increased colli-
sion frequency range from one to two orders of magnitude
is observed. Considering the effect of preferential concen-
tration, attempts at formulating a collision kernel was
initiated primarily by Sundaram and Collins (1997) , who
12 R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels
Tabl
e 4
Co
llis
ion
mo
de
ls.
Mod
el
Com
men
ts
Refe
renc
es
1/2
38
()
15i
jr
rπ
εβ
ν⎛⎞
=+
⎜⎟
⎝⎠
– V
ali
d f
or
Ga
us
sia
n,
iso
tro
pic
tu
rbu
len
ce f
low
re
gim
es
– I
nva
lid
as
lo
ng
as
th
e p
art
icle
s r
em
ain
in
th
e S
tok
es
flo
w r
eg
ime
– U
nd
erp
red
icti
on
fo
r s
ma
ll p
art
icle
siz
es
Sa
ffm
an
an
d T
urn
er
19
56
1/2
22
2
2
2
22
2
1-(
-)
8(
)
11-
(-
)(
)3
9xi
jij
ij
ij
ij
p
Du Dtr
r
gr
r
ρτ
τρ
βπ
ρε
ττ
ρν
⎡⎤
⎛⎞
⎛⎞
⎢⎥
+⎜
⎟⎜
⎟⎢
⎥⎝
⎠⎝
⎠⎢
⎥=
+⎢
⎥⎛
⎞⎢
⎥+
+⎜
⎟⎢
⎥⎝
⎠⎣
⎦
– V
ali
d f
or
flo
ws
wit
h s
ign
ific
an
t e
ffe
ct o
f p
art
icle
in
ert
ia o
r g
ravi
ty
– T
he
un
de
rlyi
ng
as
su
mp
tio
n i
s u
sin
g a
cyl
ind
rica
l a
s o
pp
os
ed
to
a s
ph
eri
cal
form
ula
tio
n
for
the
co
llis
ion
ke
rne
l, w
hic
h i
s i
nva
lid
fo
r tu
rbu
len
t fl
ow
s
– O
vere
sti
ma
tio
n b
y a
bo
ut
20
% f
or
a s
imp
le u
nif
orm
sh
ea
r fl
ow
an
d b
y 2
5%
in
is
otr
op
ic
turb
ule
nce
Sa
ffm
an
an
d T
urn
er
19
56
1/2
2
22
22
2
22
22
22
(-
)8
ex
p- 2
()
()
(-
)(
-)
-2
()
tjti
ij
ij
ij
tjti
ij
tjti
tjti
ij
ww
rr
ww
ww
erf
ww
πν
νν
νβ
νν
πν
ν
⎡⎤
⎛⎞
⎢⎥
⟨⟩+
⟨⟩
+⎜
⎟⟨
⟩+⟨
⟩⎢
⎥⎝
⎠⎢
⎥=
+⎛
⎞⎢
⎥+⟨
⟩+⟨
⟩⎜
⎟⎢
⎥⎜
⎟⎢
⎥⟨
⟩+⟨
⟩⎝
⎠⎣
⎦
– V
ali
d f
or
pa
rtic
les
wit
h c
om
ple
tely
un
corr
ela
ted
ve
loci
tie
s
– V
ali
d f
or
mo
no
dis
pe
rse
he
avy
pa
rtic
les
in
an
is
otr
op
ic t
urb
ule
nt
flo
w i
n t
he
ab
se
nce
of
glo
ba
l s
he
ar
an
d g
ravi
ty
– O
verp
red
icts
th
e c
oll
isio
n k
ern
el
Ab
rah
am
so
n 1
97
5
22
8(
)3
3i
jx
rr
wπ
β=
+⟨
⟩ –
On
ly a
cce
lera
tive
or
ine
rtia
l e
ffe
cts
in
clu
de
d i
n t
his
fo
rmu
lati
on
, w
ith
no
pro
vis
ion
be
ing
ma
de
fo
r tu
rbu
len
t s
he
ar
eff
ect
s
– 1
D f
orm
ula
tio
n w
ith
is
otr
op
y a
ss
um
pti
on
– I
nva
lid
fo
r li
qu
id s
yste
ms
wit
h p
art
icle
siz
es
in
th
e i
ne
rtia
l s
ub
ran
ge
of
turb
ule
nce
– N
ot
incl
ud
ed
th
e a
dd
ed
ma
ss
ca
us
ed
by
pa
rtic
les
mo
vin
g
– P
art
icle
dra
g b
as
ed
on
Sto
ke
s f
low
an
d t
hu
s p
lace
s a
lim
it o
n t
he
ma
xim
um
pa
rtic
le s
ize
– I
nva
lid
fo
r p
art
icle
siz
es
in
th
e i
ne
rtia
l s
ub
ran
ge
of
turb
ule
nce
– U
nd
erp
red
icts
th
e n
orm
ali
zed
co
llis
ion
ke
rne
l
Wil
lia
ms
an
d C
ran
e 1
98
3
1/2
2
22
2
22
2
22
()
()
152
()
4(
)
xi
ji
j
ij
ij
ij fDu
rr
Dtr
rr
rDu Dt
ετ
τπ
νπ
βπ
ττ π
λ
⎛⎞
⎛⎞
⎜⎟
++
++
⎜⎟
⎜⎟
⎝⎠
⎜⎟
=+
⎜⎟
+⎛
⎞⎜
⎟⎜
⎟⎝
⎠⎜
⎟⎝
⎠
– I
nva
lid
fo
r li
qu
id s
yste
ms
wit
h p
art
icle
siz
es
in
th
e i
ne
rtia
l s
ub
ran
ge
of
turb
ule
nce
.
– I
mp
rove
d v
ers
ion
of
the
fo
rmu
lati
on
of
Sa
ffm
an
an
d T
urn
er
(19
56
)
– V
ali
d f
or
the
co
llis
ion
be
twe
en
mo
de
rate
ly i
ne
rtia
l p
art
icle
s
– E
ffe
ct o
f g
ravi
ty n
ot
incl
ud
ed
– A
dd
ed
ma
ss
eff
ect
s n
ot
incl
ud
ed
– I
nva
lid
fo
r s
yste
ms
in
clu
din
g n
on
un
ifo
rm p
art
icle
co
nce
ntr
ati
on
du
e t
o f
low
-pa
rtic
le
mic
ros
tru
ctu
re i
nte
ract
ion
Hu
an
d M
ei
19
97
22
28
()
3i
jr
rw
wπ
β=
+⟨
⟩+⟨
⟩a
cce
lsh
ea
r
– I
nva
lid
fo
r ve
ry s
ma
ll p
art
icle
s a
s t
he
ex
po
ne
nti
al
as
op
po
se
d t
o t
he
pa
rab
oli
c fo
rm o
f th
e
Lan
gra
ng
ian
co
rre
lati
on
– D
oe
s n
ot
incl
ud
e t
he
in
flu
en
ce o
f n
on
un
ifo
rm p
art
icle
co
nce
ntr
ati
on
du
e t
o f
low
-pa
rtic
le
mic
ros
tru
ctu
re i
nte
ract
ion
– A
n i
mp
rove
d v
ers
ion
of
Sa
ffm
an
an
d T
urn
er
(19
56
) fo
rmu
lati
on
– I
nva
lid
fo
r in
ert
ial
an
d v
isco
us
su
bra
ng
e o
f tu
rbu
len
ce
Yu
u 1
98
4
R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels 13
Mod
el
Com
men
ts
Refe
renc
es
22
28
()
3i
jr
rw
wπ
β=
+⟨
⟩+⟨
⟩a
cce
lsh
ea
r
– I
ncl
ud
ed
th
e a
dd
ed
ma
ss
eff
ect
– V
ali
d f
or
ine
rtia
l a
nd
vis
cou
s s
ub
ran
ge
of
turb
ule
nce
– O
n t
he
ba
sis
of
Wil
lia
ms
an
d C
ran
e (
19
83
) fo
r la
rge
r p
art
icle
s a
nd
Yu
u (
19
84
) fo
r s
ma
ll
pa
rtic
les
– H
ow
eve
r, i
s n
ot
un
ive
rsa
l s
ince
th
ere
is
no
sin
gle
ex
pre
ss
ion
th
at
ho
lds
fo
r b
oth
la
rge
an
d s
ma
ll p
art
icle
s
– U
nd
erp
red
icts
th
e n
orm
ali
zed
co
llis
ion
ke
rne
l
Kru
is a
nd
Ku
ste
rs 1
99
7
1/2
22
22
22
2
2
2
2
22
1(
)1-
()
15
()
22
()
21-
()
1-8
xi
ji
jij
ij
xi
ji
jij
D
ij
ij
Dur
rDt r
rDu
rr
Dt g
ερ
ττ
πν
ρ
ρβ
πτ
τρ
λ
πρ
ττ
ρ
⎛⎞
⎛⎞
⎛⎞
⎜⎟
++
+⎜
⎟⎜
⎟⎜
⎟⎝
⎠⎝
⎠⎜
⎟⎜
⎟⎛
⎞+
⎛⎞
⎜⎟
=+
++
⎜⎟
⎜⎟
⎜⎟
⎝⎠
⎝⎠
⎜⎟
⎜⎟
⎛⎞
⎜⎟
+⎜
⎟⎜
⎟⎝
⎠⎝
⎠
– I
mp
rove
d v
ers
ion
of
Hu
an
d M
ei
(19
97
) to
in
clu
de
gra
vity
an
d f
init
e d
en
sit
y-ra
tio
corr
ect
ion
– I
nva
lid
fo
r s
yste
ms
in
clu
din
g n
on
un
ifo
rm p
art
icle
co
nce
ntr
ati
on
du
e t
o f
low
-pa
rtic
le
mic
ros
tru
ctu
re i
nte
ract
ion
– V
ali
d f
or
dil
ute
co
nce
ntr
ati
on
sys
tem
s
Wa
ng
et
al.
19
98
21
22
1-e
xp
-1
1-1-
ex
p-
pp
dθ
βπ
νθ
θθ
⎡⎤
⎛⎞
=⎢
⎥⎜
⎟⎛
⎞⎝
⎠⎛
⎞⎣
⎦⎜
⎟⎜
⎟⎝
⎠⎝
⎠
– I
mp
rove
d v
ers
ion
of
Wa
ng
et
al.
(2
00
0)
to i
ncl
ud
e b
idis
pe
rse
as
op
po
se
d t
o
mo
no
dis
pe
rse
pa
rtic
les
– I
ncl
ud
es
bid
isp
ers
e a
s o
pp
os
ed
to
mo
no
dis
pe
rse
pa
rtic
les
– V
ali
d f
or
pa
rtic
les
wit
h t
wo
dif
fere
nt
siz
es
Zh
ou
et
al.
20
01
2
1/2
22
,,
2(
)|
|
2|
|(
)
r
rr
r
dg
dw
ww
w
βπ π
=⟨
⟩
⎛⎞
⟨⟩=
⟨⟩+
⟨⟩
⎜⎟
⎝⎠
acc
ed
she
ar
– E
ffe
ct o
f p
art
icle
lo
ad
ing
an
d p
art
icle
dia
me
ter
no
t co
ns
ide
red
– I
mp
rove
d v
ers
ion
of
Su
nd
ara
m a
nd
Co
llin
s (
19
97
) to
in
clu
de
th
e s
ph
eri
cal
form
ula
tio
n o
f
the
co
llis
ion
ke
rne
l
Wa
ng
et
al.
20
00
2
1/2
2(
)|
|
2|
|
r
rpl
l
dg
dw
wS
βπ π
=⟨
⟩
⎛⎞
⟨⟩=
⎜⎟
⎝⎠
– R
ela
tive
me
an
-sq
ua
re v
elo
city
, th
e r
ad
ial
dis
trib
uti
on
fu
nct
ion
, a
nd
oth
er
two
-pa
rtic
le
sta
tis
tica
l ch
ara
cte
ris
tics
ca
n b
e o
bta
ine
d b
y th
is f
orm
ula
tio
n
Za
ich
ik e
t a
l. 2
00
6
2
1/2
2(
)(
)|
|
2|
|
ij
iji
jr
rpl
l
rr
gr
rw
wS
βπ π
=+
+⟨
⟩
⎛⎞
⟨⟩=
⎜⎟
⎝⎠
– I
mp
rove
d v
ers
ion
of
Za
ich
ik e
t a
l. (
20
06
) to
bid
isp
ers
e i
ne
rtia
l p
art
icle
s
– G
oo
d a
gre
em
en
t is
ob
tain
ed
wit
h t
he
DN
S d
ata
– R
es
po
nd
s w
ell
wit
h c
ha
ng
es
in
Re
yno
lds
nu
mb
er
Za
ich
ik e
t a
l. 2
00
6
(Tab
le 4
C
on
tin
ue
d)
14 R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels
used radial distribution function at contact to account for
the relative velocity PDF and the concentration effect for
the decorrelation of adjacent particle motion.
The collision kernel by Sundaram and Collins (1997)
was based on a cylindrical formulation that was later con-
sidered erroneous, and therefore, Wang et al. (2000) cor-
rected it using a spherical approach.
The improved model allowed Wang et al. (2000) to
isolate the effect of turbulent transport from the prefer-
ential or accumulation concentration effect as quantified
by the radial distribution function. Afterward, Zhou et al.
(2001) extended Wang ’ s model to include bidisperse as
opposed to monodisperse particles.
However, the mentioned models are all based on col-
lision data obtained through DNS. Different limitations
arise from the numerical simulation on the magnitude of
the Reynolds number. Therefore, the effect thereof on the
collision process in turbulent flow remains elusive.
Moving away from DNS, several researchers
( Alipchenkov and Zaichik 2003 , Zaichik and Alipchenkov
2003 , Zaichik et al. 2006 ) succeeded to model accumula-
tion of inertial particles and binary dispersion in an iso-
tropic turbulent flow field through a kinetic equation for
the PDF of the relative velocity of a pair of particles. In
conclusion, numerical modeling of the collision process
has the advantage of higher accuracy and ability to control
the primary variables.
6 Suspension quality The aim of the equations and relations mentioned above
is to investigate the quality of solid suspension, which can
be quantified through some important criteria including
drawdown of floating solids, just off-bottom suspension
( N js ), suspension height, complete suspension, and homo-
geneous suspension ( Ochieng and Lewis 2006b ).
6.1 Drawdown of floating solids
Generally, particle flotation depends on three different
phenomena: i) surface tension effects and poor wettabil-
ity, causing the surface tension force to be greater than the
gravitation force; ii) powders with low bulk density, which
tend to agglomerate and trap air; and iii) low true density,
whereas flotation is caused by buoyancy force ( Waghmare
et al. 2011 ).
Drawdown of particles in stirred vessels happens when
the downward forces (turbulence and drag) overcome the
upward forces (buoyancy and the surface tension) when
Buoyancyforce (FB) Surface force
(F Surf)Surface force(F Surf)
Gravitational force (Fg)
Turbulent force
Drag forceDrag force
Figure 1 Force balance around floating particle on liquid surface.
the impeller is turned on, which is shown in Figure 1 . Gen-
erally, CFD simulations clarify three mechanisms of solid
drawdown: mean drag, turbulent fluctuations, and single
vortex formation. In the first mechanism, large waves and
swirls break up stagnant zones and ingest solids. Besides,
some macro-instabilities are produced over the entire free
surface of liquid initially. They also lead to breakage of
solid clumps and assist distribution of particles in strong
liquid circulation. In this mechanism, the particle slip
velocity should be lower than the axial component of the
near surface of liquid. In the second mechanism, energetic
surface eddies along with the splashy and wavy surface
pull the solids down. In this mechanism, very large eddies
tend to be oriented against the mean circulation loops and
eddies smaller than the particles are not able to draw down
particles due to energy deficiency. Generally, eddies that
are 3 – 10 times larger than the particle are more applicable
in this mechanism. The last mechanism is applicable for
nonbaffled stirred vessels. In these systems, large stable
vortex is produced at the center of liquid bulk and breaks
up stagnant regions at the surface. Therefore, the centrifu-
gal forces pull the solid particles to the vortex surface. The
best solid distribution in this mechanism has been found
in systems with one single baffle. However, particles still
tend to concentrate around the vortex.
Khazam and Kresta (2008) investigated the interac-
tions between impeller and tank geometry by flow visu-
alization as well as CFD simulations. They investigated
one axial and two mixed flows over a solid concentration
range of 2 – 10 vol%. They found that a stable central vortex
acted as a centrifuge to concentrate a layer of solids close
to the liquid surface. Multiple reference frames (MRFs)
were used in their study, which were suitable for both
the down-pumping pitched blade turbine and the up-
pumping pitched blade turbine. This selection was due to
the strong momentum exchange between the whole bulk
and impeller zone ( Sajjadi et al. 2012 ). In the same study,
turbulence and flow were predicted more accurately via
R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels 15
LES model within a long simulation time (1 month). The
k - ε model decreased the simulation time to about 1 h, but
there was an underprediction due to overcalculation of
diffusive term.
In CFD simulation of fully baffled systems, the fluctu-
ation of free surface is negligible. However, the medium-
scale transient eddies and the behavior of surface
turbulence in the system containing one singular baffle
should be considered. In the study of Khazam and Kresta
(2008) , CFD simulation showed similar behavior for large
changes in turbulence and mean flow in single phase with
and without existence of solid particles. This observa-
tion was also independent of particle concentration. This
assumption (or observation) was confirmed by Montante
et al. (2001a,b) .
Recently, Waghmare et al. (2011) simulated solid
drawdown phenomenon in nonstandard tanks with differ-
ent geometries, which included tilted shaft, nonstandard
impellers, as well as multiple impellers, and developed
a scale-up correlation using CFD. Their correlation was
based on fundamental hydrodynamic parameter, velocity,
and operating parameters. The goal was to calculate resi-
dence time of particles on the liquid surface. They used
two important assumptions: i) the surface of the liquid
was a thin layer of liquid at the top of the tank without
adversely affecting the quality of the grid and ii) the dis-
tortion of free liquid surface was negligible; hence, free
surface was treated as symmetry boundary. The move-
ment of inert solid particles was simulated using discrete
phase modeling. This model assumes only one-way cou-
pling between liquid and solid and performs Lagrangian
tracking of particles. The drawdown rate also depends on
solid properties such as particle density, diameter, parti-
cle-particle interactions (stickiness), and particle-liquid
interaction (wettability). The results of the developed CFD
were accurate, but the correlation failed to predict draw-
down rate for cases with strong vortex formation. The
authors also found the second mechanism to be dominant
for cases with low submergence. However, the correlation
based on mean drag showed satisfactory agreement with
the experimental data.
6.2 Off-bottom solid suspension and complete suspension
There are two other main operational requirements for
investigating solid suspension in stirred vessels, which
are critical impeller speed for N js and critical impeller
speed for just complete suspension ( N s ) ( Armenante and
Nagamine 1998 ). N js causes particles to just suspend from
the bottom of the vessel completely and not stay stationary
at the bottom for more than 1 – 2 s. It is also dependent on
the ratio of the impeller diameter to vessel diameter and
impeller clearance. Optimum stirrer velocity is required for
obtaining complete off-bottom suspension and uniform
dispersion in mixing operation. The impeller speed must
not be too high in order to allow for an acceptable energy
consumption, and it is very important for solid slurrying,
efficient mass transfer, industrial mixing process optimi-
zation, and design ( Abu-Farah et al. 2010 ). CFD simula-
tion can produce accurate prediction of N js . Table 5 shows
the prediction models for off-bottom suspension.
Kee and Tan (2002) introduced a new CFD approach
for prediction of minimum impeller speed by observation
of transient profiles of the solid phase volume fraction for
the layer of cells adjacent to the bottom of the vessel. They
also characterized the effect of C and D on N js .
Ochieng and Lewis (2006b) provided quantitative and
qualitative insight into solid suspension by investigating
the results of CFD and laser Doppler velocimetry. They
used three different criteria to investigate N js based on CFD
simulation results. They also indicated that transient sim-
ulation produced more accurate results than steady simu-
lation did. Besides, CFD produced better results of cloud
height and off-bottom suspension for low solid loading
( < 6%) and small particles ( d p < 150 μ m).
Murthy et al. (2007) studied the influence of geomet-
rical parameters on critical impeller speed in both solid-
liquid and gas-solid-liquid systems. The work covered
solid loading from 0.34 to 15 wt%. They further investi-
gated the effect of particle size and solid loading. They
confirmed that for a fixed set of operational conditions
and constant configuration of impeller and tank, the uni-
formity of solids decreased with an increase in diameter of
particle. They could predict the effect of solid loading on
liquid velocity successfully using CFD simulation.
Lea (2009) focused on a similar problem in a system
containing dense solid-liquid interaction. They devel-
oped a design heuristic that can be useful in industrial
processes.
The study of Micheletti et al. (2003) is noteworthy in
order to further clarify the importance of this parameter.
Micheletti et al. (2003) found that the liquid phase mixing
process showed a complex interaction with suspension
quality and fluid mixing in stirred slurry reactors with high
solid loading. Meanwhile, the trend of the mixing time
depends on the impeller rotational speed and suspension
quality. So, mixing time for incomplete suspension regime
increased until a maximum value was achieved and grad-
ually decreased with higher impeller speed until the N js
condition ( Micheletti et al. 2003 ). Kasat et al. (2008) also
16 R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels
Tabl
e 5
Pre
dic
tio
n m
od
els
of
off
-bo
tto
m s
us
pe
ns
ion
.
As
sum
ptio
ns a
nd re
mar
ks
Refe
renc
es
0.4
20
.14
0.1
70
.12
5
0.5
80
.28
1.8
9
()
Pjs
Lgd
TN
XPo
Dρ
νρ
⎛⎞
Δ=
⎜⎟
⎝⎠
Ba
sic
as
su
mp
tio
n:
– B
as
ed
on
th
e t
ran
sfe
rre
d k
ine
tic
en
erg
y o
f tu
rbu
len
t e
dd
ies
to
th
e p
art
icle
s a
nd
th
e p
ote
nti
al
en
erg
y g
ain
ed
by
the
pa
rtic
le
– C
an
no
t ju
sti
fy t
he
hig
he
r e
ffe
ctiv
en
es
s o
f th
e i
mp
ell
er
tha
t cr
ea
tes
ma
ss
cir
cula
tio
ns
th
an
th
at
of
cre
ate
s a
lo
t o
f tu
rbu
len
ce
– C
an
no
t co
ns
ide
r th
e e
ffe
ct o
f s
oli
d c
on
cen
tra
tio
n a
nd
vis
cos
ity
Ba
ldi
et
al.
19
78
0.5
0.5
-1
jsp
gN
dD
Lρρ
⎛⎞
Δ= ⎜
⎟⎝
⎠ –
Ba
se
d o
n t
he
ba
lan
ce b
etw
ee
n t
he
do
wn
wa
rd a
nd
up
wa
rd f
orc
es
act
ing
on
pa
rtic
les
– C
an
no
t ju
sti
fy t
he
as
su
mp
tio
n o
f h
om
og
en
ou
s d
istr
ibu
tio
n o
f p
art
icle
s n
or
tha
n n
o s
lip
co
nd
itio
n b
etw
ee
n s
oli
d a
nd
liq
uid
– A
pp
lica
ble
ju
st
for
very
dil
ute
d s
oli
d c
on
cen
tra
tio
ns
Na
raya
na
n e
t a
l. 1
96
9
0.4
20
.16
0.2
5-0
.99
-0.5
8js
pl
Nd
Dρ
μρ
⎛⎞
Δ= ⎜
⎟⎝
⎠ –
Ba
se
d o
n t
he
Ba
ldi
en
erg
y b
ala
nce
wit
h p
ayi
ng
att
en
tio
n t
o e
dd
y s
ize
s
– C
an
no
t ju
sti
fy t
he
hig
he
r e
ffe
ctiv
en
es
s o
f th
e i
mp
ell
er
tha
t cr
ea
tes
ma
ss
cir
cula
tio
ns
th
an
on
e t
ha
t cr
ea
tes
a l
ot
of
turb
ule
nce
– C
an
no
t co
ns
ide
r th
e e
ffe
ct o
f s
oli
d c
on
cen
tra
tio
n a
nd
vis
cos
ity
Rie
ge
r a
nd
Dit
l 1
99
4
1/2
-1/3
1/6
1/3
-5/3
jsp
L
gN
APo
dC
TDν
ρρ
⎛⎞
Δ=
⎜⎟
⎝⎠
– B
as
ed
on
th
e f
orc
es
act
ing
on
pa
rtic
les
an
d l
iqu
id v
elo
city
ne
ar
the
bo
tto
m o
f th
e v
es
se
l in
cip
ien
t m
oti
on
of
pa
rtic
les
– C
an
no
t ju
sti
fy t
he
arr
an
ge
me
nt
pa
ram
ete
rs o
f s
oli
d p
art
icle
s
Sh
am
lou
an
d Z
olf
ag
ha
ria
n
19
87
, M
ak
19
92
Fin
e p
art
icle
( Ar
≤ 4
0)
0.5
6
-0.1
1-1
0.1
1D
pjs
L
dg
NT
ρν
ρ
⎛⎞
Δ∝
⎜⎟
⎝⎠
Co
ars
e p
art
icle
s (
Ar >
40
)
0.1
4
0.5
0.3
6-1
()
()
pjs
LdN
gC
HD
νρ
ρ
⎛⎞
∝Δ
⎜⎟
⎝⎠
– B
as
ed
on
th
e A
rch
ime
de
s n
um
be
r vi
a t
wo
me
cha
nis
ms
– F
irs
t, c
om
ple
te s
us
pe
ns
ion
of
fin
e p
art
icle
s a
t h
igh
sh
ea
r s
tre
ss
es
of
wa
ll b
ou
nd
ary
la
yer
– S
eco
nd
, b
as
ed
on
th
e a
na
lys
is o
f th
e p
um
p c
ha
ract
eri
sti
cs o
f a
sti
rre
d v
es
se
l
– R
eq
uir
es
acc
ura
te p
red
icti
on
of
sh
ea
r ra
te a
t th
e b
ou
nd
ary
la
yer
of
the
ag
ita
ted
ta
nk
Mo
leru
s a
nd
La
tze
l 1
98
7
4/9
-1/9
2-4
/3
jsjs
LN
ND
Tgρ
νρ
∗⎛
⎞=
⎜⎟
Δ⎝
⎠ –
Ba
se
d o
n t
he
dif
fere
nce
be
twe
en
th
e l
iqu
id v
elo
city
an
d t
erm
ina
l s
ett
lin
g v
elo
city
of
pa
rtic
le
– V
elo
city
ch
ara
cte
riza
tio
n c
alc
ula
ted
by
me
as
uri
ng
act
ua
l va
lue
s o
f s
he
ar
rate
– P
red
icti
on
of
N JS b
as
ed
on
th
e r
ati
o b
etw
ee
n s
ett
lin
g v
elo
city
an
d N
JS
Wic
hte
rle
19
88
0.5
0.5
-1
jsp
L
gN
dD
ρ
ρ
⎛⎞
Δ∝
⎜⎟
⎝⎠
– P
ow
er
inp
ut
dis
sip
ate
d b
y tw
o p
he
no
me
na
: co
ns
um
pti
on
of
po
we
r to
avo
id s
ett
lin
g a
nd
ge
ne
rati
ng
dis
cha
rge
flo
w f
or
su
sp
en
sio
n
– I
na
ccu
rate
pre
dic
tio
n o
f fl
uct
ua
tin
g v
elo
city
at
the
bo
tto
m o
f th
e v
es
se
l th
at
led
to
un
de
r p
red
icti
on
of
N JS
Me
rsm
an
n e
t a
l. 1
99
8
R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels 17
reported similar observations while using CFD to inves-
tigate complex interactions between suspension quality
and fluid mixing. They reported the maxima in mixing
time, which was nearly 10 times of the minimum value at
N = 5, rps = 13 N CS
. The delayed mixing in the top clear liquid
was due to very low liquid velocities in this section and
was responsible for the increase in mixing time.
Recently, Fletcher and Brown (2013) used two differ-
ent CFD approaches to study solid suspension, which were
E-E and algebraic slip modeling. Both of these approaches
provided computationally efficient results. The authors
further investigated the importance of turbulent disper-
sion forces and the role of turbulence model (homogene-
ous or phasic). Both of these produced similar results, but
there was less solid dispersion in the homogeneous case.
Prediction methods of N js through CFD can be clas-
sified into five groups: tangent-intersection method,
particle axial velocity method, transient profile method,
variation of particle concentration, and Zwietering (1958)
correlation.
6.2.1 Tangent-intersection method
This method was proposed by Mak (1992) and could be
applied both experimentally and computationally:
υΔρρ
⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
0.45 0.13 0.2 0.1
0.85,P
js
L
X dgN S
D
(8)
where S is constant for a given system geometry, Δ ρ = ρ s - ρ
l .
In this method, a horizontal plane is assumed with a small
displacement from the tank bottom. The distance from the
tank bottom for measurement of particle concentration
is arbitrary. Then, mean particle concentration is plotted
versus impeller speed at this height. N js can be identified
from the tangents to this curve drawn at the points having
minimum and maximum slopes. Hosseini et al. (2010)
employed this method to investigate N js and reported sat-
isfactory agreement between the CFD and experimental
data. However, Tamburini et al. (2012) reported an under-
estimation using this method.
6.2.2 Particle axial velocity method
Wang et al. (2004) employed a method based on axial
velocity of solid phase in the computational cells near
the tank bottom at different impeller speeds. Based on
this method, all particles are considered to be suspended
when the axial velocity of the solid phase in some cells in
specific places becomes positive. These cells are located
at areas where it is difficult to suspend the solid particles.
Wang et al. (2004) made the position the locus of accu-
mulation for the last particles to be suspended, since the
main flow was from the periphery towards the center at
the bottom of the tank.
Ochieng and Lewis (2006b) proposed a CFD simula-
tion method to determine N js in a flat-bottomed tank using
high solid loading. In their method, the simulation was
initiated with settled particles at the vessel bottom and
they found that drag curves and turbulence models were
the most important factors for estimation accuracy.
6.2.3 Transient volume fraction profile method
Kee and Tan (2002) adopted a method based on solid
transient volume fraction profile in a 2D transient simula-
tion. They proposed two criteria to identify N js at different
impeller speeds. First, all transient volume fraction profiles
versus time exhibited steady-state behavior; second, all
steady-state values of volume fraction were approximately
50% of the initial packed volume fraction. These criteria
were used because all solid particles are entrained and cir-
culated in the continuous phase at complete suspension
state, so the volume fraction profiles of cells at the tank
bottom would eventually attain steady-state behavior. Tam-
burini et al. (2011) reported that transient profile provided
a strong overestimation of N js for all investigated systems.
6.2.4 Variation of particle concentration
A method was proposed by Wang et al. (2004) to monitor
the variation in particle concentration versus impeller
speed. In this method, complete off-bottom suspension
was considered achieved when solid volume fraction at
the place where the last particle was suspended showed
some values lower than the maximum value. However, the
definition of the exact volume fraction corresponding to
complete suspension is subjective.
6.2.5 Applying Zwietering correlation in CFD simulation
Zwietering correlation in CFD simulation is the other
method. The basic correlation was developed by Zwietering
(1958) , and most of the recent works on the critical impel-
ler speed have been based on this correlation:
-0.85
jsN SD A,= (9)
18 R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels
where N js depends on solid-liquid mixture properties [by
geometry-dependent coefficients A , Eq. (7)] and vessel
geometry (by coefficient S ):
ρ ρν
ρ
⎛ ⎞= ⎜ ⎟⎝ ⎠
0.45
0.1 0.2 0.13( - ).s l
p s
l
gA d w
(10)
In this equation, ν is liquid kinematic viscosity, d p is parti-
cle diameter, ρ is density, and w is defined as
ρ α
ρ α ρ α=
+,
( 1- )s s
s
l s s s
w
(11)
where α is the solid volume fraction.
Ochieng and Lewis (2006b) estimated just-suspended
speed and cloud height using the Zwietering (1958) model
and Bittorf and Kresta (2003) correlation, respectively.
They employed a Eulerian-based multiphase approach
along with MRF approach to provide an initial flow field
for fully predictive sliding grid (SG) to investigate off-
bottom suspension of nickel solids and cloud height in
a stirred reactor. However, Zwietering (1958) correlation
is rarely used in E-E simulation because of its complex-
ity. Derksen (2003) employed this model using the EL
approach along with LES. In this simulation, 6.7 million
particles were tracked in a liquid flow field. It was pro-
posed that a realistic particle distribution throughout the
tank could be obtained by considering particle-particle
collisions. The collision algorithm was improved to reach
this aim because it inhibited particles to be at mutual
distance smaller than the particle diameter. The CFD
simulation underpredicted N js especially for larger parti-
cles ( d p > 150 μ m), and this disparity was attributed to the
drag force, which decreased with an increase in surface.
For larger particles ( d p > 300 μ m), the deviation was 6%.
The simulation was also in excellent agreement with the
experimental results for solid loading up to 10%, but this
model failed to show the cloud height for higher loading
rate. This was attributed to empirical constants that were
used in the formula (such as k- ε , drag curves, nondrag
forces, and solid pressure). The author explained that
CFD simulation yielded the best result for smaller parti-
cles along with low loading rate.
Suspension curves show the amount of suspended
solids and power consumption at different impeller
speeds, including the partial ( N < N js ) or complete ( N > N
js )
suspension conditions. Tamburini et al. (2011) devel-
oped an original CFD modeling approach with the aim of
numerically predicting suspension curves with two differ-
ent types of experimental data (Pressure Gauge Technique
and snapshot data) for validation. They implemented
a posteriori algorithm, namely, excess solid volume
correction with an iterative approach for particle bed lying
at the bottom to describe all partial suspension regimes.
The best results came from modeling employing asym-
metric k - ε for turbulence and standard drag model with
two-way coupling without any additional term arising
from turbulence closure in continuity equations. Zwieter-
ing (1958) also offered another method that could be used
for this purpose.
6.3 Cloud height
Another criterion that provides a qualitative indication
of suspension quality is height of homogeneous zone.
However, it cannot consider local mixing quality and pro-
vides only satisfactory information on a global parameter.
The height of homogeneous zone is also called cloud
height. Mixing is vigorous below this interface but limited
above this interface (in clear volume), which results in
increased mixing time and large amount of unreacted
fluid in this area. Several authors have simulated solid
distribution in mixing systems to investigate this param-
eter. However, most of the published works focused on 1D
sedimentation of particles, resulting in a function based
on Peclet number. The solid concentration was < 6 wt%,
whereas no well-defined interface was formed. At concen-
tration above 10 wt%, clear interface was formed at the
place where the upward and downward velocity balance
each other.
Stratified flow, hindered settling, negatively buoyant
jets, and solid iso-surface are four available phenomena
to investigate the formation of this interface. The differ-
ence between unconfined and confined flows is estab-
lished by an examination of literature on stratified flows.
Hindered settling happens in confined 1D flows with
high solid concentration, where the presence of parti-
cles significantly reduces the area available for flow and
increases particle-particle collisions. These collisions
may increase drag force on the particles. Since flow in
stirred tank is 3D, hindered settling models cannot truly
predict solid flow.
In stirred vessels, a negatively buoyant jet penetrates
upward. As the upward jet expands, velocity decreases.
Meanwhile, the heavier fluid in the upward jet stops and
returns to the bottom of the vessel. Negatively buoyant jets
flow to a maximum height during the initial period and
then drop to a lower steady-state height due to the down-
ward drag on the free jet, refer to Figure 2. The penetra-
tion height depends on the momentum flux and buoyancy
flux. Generally, three criteria are required for forming a
clear interface: first, sufficient density difference, which
R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels 19
is provided at high solid concentration; second, momen-
tum, which causes break up of a stratified layer; and third,
additional momentum of wall jet, which results in higher
penetration of jet. Therefore, jet penetration increases by
increasing the momentum of wall jet. This leads to full
suspension at higher impeller speed.
Larson and Jonsson (1994) indicated that increasing
jet nozzle speed in a two-layer stratified fluid increased
the penetration distance of jet into the second layer. This
is similar to the stirred tank where an increase in impel-
ler speed raises the interface height, which may result in
fully suspended situation. Any additional momentum in
the wall jet increases the penetration height. This behav-
ior proposes a mechanism for formation of cloud height.
In other words, the location of clear interface can be
estimated by this technique if the particle distribution is
driven by upward jet. Bittorf and Kresta (2001) showed
that the flow at the wall of a stirred vessel containing axial
impellers could be characterized using four 3D wall jets in
front of each baffle.
In another work, Bittorf and Kresta (2003) modeled
this technique by CFD simulation. They defined an inter-
face at the location where the upward velocity of fluid at
the wall was exactly balanced by the downward veloc-
ity of particles. Their model was based on the relation
between the maximum velocity in wall jet and solid cloud
height, which is the flow created when fluid is blown
tangentially along a wall. The authors assumed that
once particles were lifted and prevented from settling,
a force depending on the wall jet must move them away
from the bottom. Their model assumed that the cloud
height was determined only by mean flow conditions,
UR
Um
ym
y
ym
y
UR
Um
ym
y
Ucore
x
y
Figure 2 Negatively buoyant jets.
not turbulence or macro-instabilities. The mean flow can
also be described by 3D wall jet model introduced previ-
ously. However, wall jet model in stirred tanks is sensi-
tive to geometry. This criterion was clarified by Bittorf and
Kresta (2003) . Other authors ( Hicks et al. 1997 , Bujalski
et al. 1999 ) have reported data for solid distribution and
cloud height in a form suitable for model development
based on wall jets. Hicks et al. (1997) examined cloud
height with different solid types with constant concentra-
tion. Bujalski et al. (1999) also varied the solid concen-
trations (20 – 40 wt%). Satisfactory agreement between
the CFD and experimental result was observed in both
studies. This technique has also been successfully applied
by Ochieng and Lewis (2006b) . An excellent agreement
between the modeled and the experimental data for solid
loading up to 10% was reported in their study. However,
a poor prediction was obtained for higher solid loading.
The authors attributed this deviation to some factors.
First, the k- ε model is more applicable in simple systems
compared to high-solid-containing systems. Second is
the density difference between the models developed by
Bittorf and Kresta (2003) and Ochieng and Lewis (2006b) .
Third is the higher interaction between solid particles in
the system and effect of drag and nondrag forces, which
depends on the particle interactions. The solid interface
method was introduced by Kasat et al. (2008) . The cloud
height in this method was defined as the maximum verti-
cal coordinate of a solid iso-surface. The location of this
iso-surface was defined as the average of solid phase
volume fraction. Gr é tarsson et al. (2012) also applied this
method in E-G approach and found reasonable predic-
tions. Micale et al. (2004) used the opposite direction of
the last model to find the solid cloud height. They used
the location of particle-free liquid layer to find the solid
suspension height. However, they obtained poor predic-
tion results. Recently, Wadnerkar et al. (2012) have also
used this method to investigate cloud height and reported
satisfactory agreement. In CFD simulation, most of the
other works in this area have focused on comparing the
experimental data with the simulation results using dif-
ferent frame grids (MRF or SG), approaches, turbulence
models, or drag models ( Montante et al. 2001b , Kee and
Tan 2002 , Derksen 2003 ).
6.4 Solid distribution
Uniformity of a multiphase system is another important
parameter especially in mass transfer. This criterion can
be determined by investigating solids concentration dis-
tribution in the entire vessel.
20 R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels
6.4.1 Settling velocity model
In this context, polydisperse approach is one of the emerg-
ing multiphase simulation concepts to visualize particles
of a specific size range in systems with high solid holdup
( Sharma and Shaikh 2003 , Ochieng and Lewis 2006b ). The
first attempt by Magelli et al. (1990) evaluated dispersed
phase behavior using terminal settling velocity and mass
balance. The continuous phase was obtained by 1D axial
dispersion model. A more common procedure was to
assume a single-phase flow field for the continuous phase
as a basis for the dispersed phase calculations. It was
clear that there was no coupling back from the dispersed
to the continuous phase in this procedure. Smith (1997)
used this approach to calculate Lagrangian particle tracks.
Brucato et al. (1996, 1997) used single-phase results for the
continuous phase as initial data for their Settling Velocity
Model along with E-E approaches in two separate investi-
gations containing single and multiple impeller systems.
They focused on the calculation of a solid concentration
profile based on the mass balance of each control volume.
A similar method was employed by Myers et al. (1995) and
Bakker et al. (1994) . They introduced a term of turbulent
dispersion calculation for dispersed phase into a continu-
ity equation and assumed a constant slip equal to terminal
settling velocity for obtaining the dispersed phase veloci-
ties. Decker and Sommerfeld (1996) and Barrue et al. (1997)
introduced two-way coupled 3D E-L calculations account-
ing for particle turbulent dispersion through instantane-
ous fluid velocity that acts on a particle at a certain time.
Hamill et al. (1995) introduced a full two-way coupled cal-
culation along with multifluid E-E approach on a propeller
stirred crystallizer. It was challenging to conduct measure-
ment and modeling using such systems, especially with
high solid loading, even with a variety of simplifications.
Recently, Klenov and Noskov (2011) simulated this
model to study solid distribution in liquid-solid stirred
vessel equipped with five impellers mounted on a single
shaft, which created four separated domains in the reactor
with 3D complex swirling streams.
They observed parameters such as specific density of
solid phase and place of solid injection and found these to
be very influential on the distribution of solid suspension
in the slurry reactor. The mechanical energy of a revolving
impeller transforms to energy of turbulent rotation flow
and then the turbulence energy dissipates through small-
scale curls. They also observed minimal nonuniformity
of axial distribution for high-density solid particles. They
used correlation for settling velocity in stirred tank and
still liquid to estimate the order of magnitude of solid set-
tling velocities ( Pinelli et al. 2009 ).
6.4.2 Standard deviation model
Bohnet and Niesmak (1980) presented the most widely
used criteria that quantified suspension quality in stirred
vessels by means of standard deviation. This method was
based on a standard deviation, which was suitable to be
used with the E-E approach, although standard deviation
reduced with increment of homogenization degree:
2
2
1 2,
1-1 ,
i
ni
n avgn
ασ
α=
⎛ ⎞= ⎜ ⎟
⎝ ⎠∑
(12)
where n is the number of sampling locations used for allo-
cating the solid volume fraction. On the basis of standard
deviation, three ranges of suspension quality are identi-
fied. For nonuniform (incomplete) suspensions, the value
of standard deviation is found to be more than 0.8. The
value of standard deviation was between 0.2 and 0.8
(0.2 < σ < 0.8) for N js condition and σ < 0.2 for homogeneous
condition.
Khopkar et al. (2006) used this method along with
the modified Brucato model ( Brucato et al. 1998 ) to simu-
late turbulent solid-liquid flow in a stirred vessel. They
observed reasonable agreement between the modeled
data and the experimental data for N js , critical impel-
ler speed, and cloud height. The same methodology was
successfully employed by Oshinowo and Bakker (2002) .
However, Montante et al. (2003) reported that σ varied
with operational conditions, fluid properties, and tank
configurations, although it lacked physical significance.
Recently, Wang et al. (2010) used this method along
with the Syamlal-O ’ Brien-symmetric drag force model
( Syamlal 1987 ) of the interphase momentum transfer
between two dispersed solid phases and predicted solid
concentration in different sections of stirred vessel.
6.4.3 Size-dependent model
Modeling of size-dependent classification function in CFD
was done by Sha et al. (2001) through a Eulerian simula-
tion approach to model six phases of solid particles using
the following model:
τ
=
⎡ ⎤× ⎢ ⎥
⎣ ⎦∫
( ) [ , ( , ), ]
1exp - .
( ) [ , ( , ), ]
o
p
o
L
oo o
nn
f L h Z f N D L
dL
G f L h Z f N D L
(13)
However, there was very little comparison between
the simulation results with the experimental data even
R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels 21
though this aspect is crucial for validating the adopted
models and procedures ( Sha et al. 2001 ). In another
work by Sha and Palosaari (2002) , they used the model
to estimate the population-density distribution and
simulation of local particle size distribution in a crystal-
lizer with imperfect mixed suspension. They calculated
the classification by population density and suspension
density based on the volume fraction in each cell with the
CFD program successfully. Ochieng and Lewis (2006b)
employed the fully predictive Eulerian simulation to
investigate the effect of solid loading and particle size on
solid distribution in a stirred vessel containing high-den-
sity particles (nickel). They reported that high-density
particles tend to settle at the bottom of the tank so particle
density affects particle size distribution in both the radial
and axial directions, and assumption of mono-disperse-
particle may result in incorrect prediction. However, they
validated the modeling approach only for monodispersed
solid fractions and dilute conditions.
Š pidla et al. (2005) measured axial and radial par-
ticle concentration gradient for a solid-liquid suspen-
sion (with average particle concentration of 5 vol%) in a
just-suspended state in a baffled stirred vessel equipped
with pitched blade turbine. They performed E-E mul-
tiphase model along with the mixture of k - ε turbulence
model ( Š pidla et al. 2005 ). They used three different drag
models: Schiller-Naumann ( Schiller and Naumann 1935 ),
Pinelli ( Pinelli et al. 2009 ), and Brucato ( Brucato et al.
1998 ), and suggested higher drag coefficient for obtaining
better results.
Wang et al. (2006) successfully simulated the distribu-
tion of dispersed oil phase under high agitation rate using
the E-E multifluid method to simulate a liquid-liquid-solid
system in a stirred tank. However, the method/simulation
failed under low impeller speed. In another work ( Wang
et al. 2010 ), they applied CFD model to investigate a liq-
uid-solid-solid two-phase flow field to predict distribu-
tions of two dispersed solid phases in the flow field. This
was done based on the E-E multifluid approach, where
they studied crystallization of monosodium aluminate
hydrate from a super saturated aluminate solution con-
taining red mud as leaching liquor of bauxite. They used
two methods for their liquid-solid-solid dispersion system
(glycerite, red mud, and sand). The first method was based
on first-order upwind discretization with 245 – 258 K grid
sizes scheme, and the second was based on the Quadratic
Upstream Interpolation for Convective Kinematics discre-
tization scheme with 690 – 740 K grid sizes. Within these
approaches, they employed Gidaspow ’ s (1994) drag force
and Syamlal-O ’ Brien-Symmetric ’ s drag force models for
the interphase momentum exchange for liquid-solid and
solid-solid, respectively. The liquid and two solid phases
were all considered as different continua, interpenetrat-
ing and interacting with each other in the whole computa-
tional domain. The obtained simulation result was in good
agreement with experimental results. They found that the
simulated and distributions in the flow fields by the two
methods were basically the same ( Wang et al. 2010 ).
Montante and Magelli (2007) applied CFD simulation
to calculate solid concentration distribution and meas-
ured flow field of all intervening phases and the vertical
distribution of solid concentration by an optical tech-
nique. However, they considered only two solid fractions
with equal size. They used three different sets of RANS
equations: one for continuous phase and the other two
for the two dispersed solid fractions. They also modeled
momentum transfer between the liquid phases and each
solid phase by drag forces with influence of turbulence
and obtained realistic predictions of solid distribution
along the vessel ’ s vertical coordinates.
Tamburini et al. (2009) summarized the most impor-
tant published work on solid-liquid stirred vessel until
2009 in a table in their review on turbulence models. In
this work, a number of relevant information after 2009
was added into the table ( Table 6 ).
7 Conclusions This review shows that CFD models can obtain useful
results for transport phenomena and flow field in stirred
vessels containing solid particles. There is a general trend
toward development of CFD simulation methods to predict
solid concentration distribution, particle trajectory, parti-
cle size distributions, cloud height, and minimum stirrer
speed, which have traditionally been determined using
experimental methods only. However, the turbulence
model still remains constrained in obtaining accurate
CFD simulation data especially in systems with high solid
loading.
1. Generally, the review shows that the E-E approach
is more popular than E-L, in spite of the momentum
exchange between the operational phases due to drag
and lift. Virtual mass through the interface can be mod-
eled more accurately by E-L in comparison with E-E.
This is caused by the huge demand of computational
cost of E-L especially at a high dispersed phase vol-
ume fraction, where the interactions between the two
phases are increased as well ( Ochieng and Onyango
2010 ). In addition, the combination of k - ε model with
the modified Bruccato model has proposed satisfying
22 R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels
Tabl
e 6
Ma
jor
res
ea
rch
es
on
so
lid
-liq
uid
sti
rre
r ve
ss
els
.
Auth
ors y
ear/
year
/sys
tem
im
pelle
r typ
e, s
peed
, T,
D/T
, C/T
, N, W
/T, Z
/T, o
pera
ting
mat
eria
l, Re
Grid
size
, typ
e,
impe
ller m
otio
n m
odel
Fl
ow va
riabl
e Re
mar
ks
Tam
bu
rin
i e
t a
l. 2
01
1 /2
01
1
Six
-bla
de
d R
us
hto
n t
urb
ine
Wa
ter
an
d g
las
s b
all
ott
ini,
10
0 –
60
0
α = 1
1.9
– 5
.95
% v
/v
ρ s =
24
80
kg
/m 3
d p = 2
12
an
d 2
50
μ m
0.1
9,
T/2
, T/
3,
4,
T/1
0,
T
43
0,0
80
(h
exa
he
dra
) fo
r h
alf
of
the
ta
nk
SG
, M
RF
CFD
co
de
: C
FX4
.4 (
US
RIP
T)
Liq
uid
fre
e s
tre
am
tu
rbu
len
ce,
rad
iall
y
ave
rag
ed
axi
al
pro
file
of
so
lid
vo
lum
etr
ic
fra
ctio
ns
Pu
rpo
se
: p
red
icti
on
of
su
sp
en
sio
n c
urv
es
Turb
ule
nce
mo
de
l: s
tan
da
rd k -
ε
Dra
g:
sta
nd
ard
, G
ida
sp
ow
’ s d
en
se
pa
rtic
le,
Tam
bu
rin
i
No
tes
: E
ule
ria
n/E
ule
ria
n
Wa
gh
ma
re e
t a
l. 2
01
1 /2
01
1
Va
rio
us
im
pe
lle
rs (
a,
b,
c)/5
0 –
80
0
Fum
ed
Dif
fere
nt
no
ns
tan
da
rd v
es
se
ls
Sil
ica
ρ s =
2.2
g/c
c
ρ b =
0.0
52
g/c
c (b
ulk
)
d p = 0
.1 m
m
30
0,0
00
– 8
00
,00
0
He
xah
ed
ral
MR
F,
CFD
Co
de
: Fl
ue
nt
6.3
.26
Liq
uid
su
rfa
ce v
elo
city
, d
raw
do
wn
ra
teP
urp
os
e:
sca
lin
g u
p l
iqu
id-
lig
htw
eig
ht
so
lid
s m
ixin
g
Turb
ule
nce
mo
de
l: R
NG
k - ε
Dra
g:
–
No
tes
: E
ule
ria
n/L
ag
ran
gia
n
Kle
no
v a
nd
No
sk
ov
20
11
/20
11
Five
-sto
ry i
mp
ell
ers
/10
0
2,
0.6
5T,
0.0
6T,
6,
0.0
5T,
T
α = 0
.4 –
0.7
% v
/v
ρ l =
81
0 k
g/m
3
ρ s =
2,
5,
8 *
10
3 k
g/m
3
d p = 0
.1 m
m
2.6
8 ×
10
6
Tetr
ah
ed
ral
MR
F, S
M
CFD
co
de
: Fl
ue
nt
(6.1
– 6
.3)
Dis
trib
uti
on
of
vort
icit
y, v
elo
city
,
dis
trib
uti
on
of
so
lid
ph
as
e
Pu
rpo
se
: in
ves
tig
ati
on
of
so
lid
dis
trib
uti
on
Turb
ule
nce
mo
de
l: s
tan
da
rd k -
ε
Dra
g:
Bru
cato
No
tes
: E
ule
ria
n/E
ule
ria
n
Tam
bu
rin
i e
t a
l. 2
00
9 /2
00
9
A s
tan
da
rd R
us
hto
n t
urb
ine
/38
0 –
12
00
0.1
9 m
, T/
2,
0.1
7T,
4,
0.1
T, 1
.5T
Wa
ter,
sil
ica
α = 2
1.5
%w
/w
ρ s =
25
80
kg
/m 3
d p = 2
31
μ m
Re
p =
57
,00
0
24
× 6
9 ×
32
42
3,9
36
ce
lls
SM
CFD
co
de
: A
ns
ys-C
FX4
.4
Loca
l a
xia
l p
rofi
le o
f p
art
icle
co
nce
ntr
ati
on
,
rad
iall
y a
vera
ge
d a
xia
l p
rofi
les
Pu
rpo
se
: in
ves
tig
ati
on
of
tra
ns
ien
t b
eh
avi
or
of
an
off
-bo
tto
m s
oli
d-l
iqu
id s
us
pe
ns
ion
du
rin
g s
tart
-up
Turb
ule
nce
mo
de
l: s
tan
da
rd k -
ε
Dra
g:
Gid
as
po
w,
Erg
un
No
tes
: E
ule
ria
n/E
ule
ria
n
Sa
rde
sh
pa
nd
e e
t a
l. 2
01
0 ,
Wa
dn
erk
ar
et
al.
20
12
/20
10
Six
-bla
de
d P
itch
ed
Bla
de
/15
0 –
60
2 r
pm
0.7
m,
T/1
0,
T/3
, 4
, 0
.1T,
T
D = 5
0 –
25
0 μ
m
ro =
10
00
kg
/m 3
ro =
25
00
kg
/m 3
1 –
7%
50
16
42
He
xah
ed
ral
cell
s
CFD
co
de
: Fl
ue
nt
6.3
.26
Mix
ing
tim
e,
Clo
ud
he
igh
t
Pu
rpo
se
: in
ves
tig
ati
on
of
so
lid
su
sp
en
sio
n
Turb
ule
nce
mo
de
l: s
tan
da
rd k -
ε
Dra
g:
Kh
op
ka
r M
od
el;
Bru
cato
mo
de
l
No
tes
: E
-E a
pp
roa
ch
R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels 23
Auth
ors y
ear/
year
/sys
tem
im
pelle
r typ
e, s
peed
, T,
D/T
, C/T
, N, W
/T, Z
/T, o
pera
ting
mat
eria
l, Re
Grid
size
, typ
e,
impe
ller m
otio
n m
odel
Fl
ow va
riabl
e Re
mar
ks
Wa
dn
erk
ar
et
al.
20
12
/20
12
A s
ix-b
lad
ed
Ru
sh
ton
tu
rbin
e/1
00
T, D
/T,
C/T
, N
, W
/T,
Z/T
,
0.2
m,
T/3
, T/
3,
4,
0.1
T, T
Wa
ter,
sm
all
gla
ss
d p = 0
.3 m
m
ρ s =
25
50
kg
α = 1
%,
Re
= 7
5,0
71
.85
, P
o =
4.9
50
α = 7
%,
Re
= 8
1,9
46
.97
, P
o =
4.6
33
22
42
80
(MR
F)
CFD
co
de
: A
NS
YS
12
.1 F
LUE
NT
Ve
loci
ty c
om
po
ne
nts
Sli
p v
elo
city
Turb
ule
nt
kin
eti
c e
ne
rgy
(TK
E)
Clo
ud
he
igh
t a
nd
ho
mo
ge
ne
ity
Pu
rpo
se
: a
na
lyzi
ng
dra
g m
od
els
Turb
ule
nce
mo
de
l: s
tan
da
rd k -
ε
Dra
g:
Gid
as
po
w,
Bru
cato
, m
od
ifie
d B
ruca
to
dra
g m
od
els
Wa
ng
et
al.
20
10
/20
10
Six
-bla
de
d R
us
hto
n,
20
0 –
30
0
0.1
4 m
, T/
2,
0.3
57
T, 0
, T
Alu
min
ate
cry
sta
l
α = 1
% V
ρ ac =
23
40
kg
/m 3
d ac =
10
2 μ
m
Re
d M
ud
ρ rm =
22
42
.7 k
g/m
3
d ac =
10
.7 μ
m
24
5,0
00
– 2
58
,00
0 H
exa
he
dro
n
an
d t
etr
ah
ed
ral
MR
F
CFD
co
de
: Fl
ue
nt
6.3
Su
sp
en
sio
n q
ua
lity
Mix
ing
tim
e
Po
we
r n
um
be
r
Flo
w p
att
ern
Pu
rpo
se
: in
ves
tig
ati
on
so
lid
su
sp
en
sio
n
qu
ali
tie
s o
f b
oth
liq
uid
-so
lid
an
d l
iqu
id-
so
lid
-so
lid
sys
tem
s
Turb
ule
nce
mo
de
l: R
NG
k - ε
Dra
g:
Sya
mla
l-O
’ Bri
en
-sym
me
tric
dra
g f
orc
e
mo
de
l
No
tes
: E
-E
Qi
et
al.
20
12
/20
12
Sm
ith
tu
rbin
e
0.4
76
, 0
.4T,
4,
0.1
T, T
,
liq
uid
M =
1 ×
10
-3 ,
2 ×
10
-2
d = 1
× 1
0 -4
, 3
× 1
0 -4
ro =
25
00
– 4
00
0 k
g/m
3
2.5
– 1
1%
12
6,9
28
me
sh
es
CFD
co
de
: A
NS
YS
CFX
10
.0
Po
we
r n
um
be
r
Flo
w p
att
ern
Pu
rpo
se
:
Turb
ule
nce
mo
de
l: s
tan
da
rd k
- ε
Dra
g:
Sch
ille
r N
au
ma
nn
dra
g
No
tes
: E
-E
Tam
bu
rin
i e
t a
l. 2
01
2 /2
01
2
Six
-bla
de
d R
us
hto
n t
urb
ine
, 2
00
– 8
00
0.1
9 m
, T/
2,
T/3
,4,
T/1
0,
H =
T
Gla
ss
ba
llo
ttin
i p
art
icle
s
ro =
25
00
kg
/m 3
α = 1
6.9
– 3
3.8
wt%
ro =
24
70
kg
/m 3
MR
F a
nd
SG
CFD
co
de
: C
FX4
.4
No
rma
lize
d s
oli
ds
vo
lum
e f
ract
ion
, p
ow
er
nu
mb
er,
clo
ud
he
igh
t
Pu
rpo
se
: in
ves
tig
ati
on
of
the
N js
Turb
ule
nce
mo
de
l: s
tan
da
rd k -
ε
Dra
g:
Bru
cato
, Pin
ell
i
No
tes
: E
-E
(Tab
le 6
C
on
tin
ue
d)
24 R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels
Auth
ors y
ear/
year
/sys
tem
im
pelle
r typ
e, s
peed
, T,
D/T
, C/T
, N, W
/T, Z
/T, o
pera
ting
mat
eria
l, Re
Grid
size
, typ
e,
impe
ller m
otio
n m
odel
Fl
ow va
riabl
e Re
mar
ks
Mo
nta
nte
et
al.
20
12
/20
12
α = 0
.2 v
ol%
23
.2 c
m,
T/3
, T/
10
, H
= T
ro =
24
70
kg
/m 3
d = 1
15
– 7
74
μ m
N = 1
4.2
1/s
Re
= 8
.8 ×
10
4
12
6,9
28
me
sh
es
MR
F
CFD
co
de
: A
NS
YS
CFX
10
.0
Turb
ule
nce
mo
du
lati
on
an
d a
xia
l s
oli
d-
liq
uid
Sli
p
Ve
loci
tie
s
Pu
rpo
se
: th
e e
ffe
ct o
f p
art
icle
pro
pe
rtie
s
(de
ns
ity
an
d d
iam
ete
r) a
nd
eff
ect
ive
slu
rry
vis
cos
ity
on
pa
rtic
le s
us
pe
ns
ion
be
ha
vio
r
Turb
ule
nce
mo
de
l: s
tan
da
rd k -
ε
Dra
g:
The
Sch
ille
r a
nd
Na
um
an
n
No
tes
: E
-E
Tam
bu
rin
i e
t a
l. 2
01
4
α = 1
6.9
– 3
3.8
%w
/w
19
cm
, T/
2,
T/1
0,
H =
T
ro =
25
00
, 2
47
0 k
g/m
3
d = 2
12
– 6
00
μ m
N = 2
31
– 6
55
rp
m
MR
F
CFD
co
de
: A
NS
YS
CFX
4.4
Axi
al
pro
file
s o
f p
art
icle
co
nce
ntr
ati
on
sP
urp
os
e:
infl
ue
nce
of
dra
g a
nd
tu
rbu
len
ce
mo
de
lin
g o
n p
art
icle
su
sp
en
sio
n b
eh
avi
or
Turb
ule
nce
mo
de
l: s
tan
da
rd k -
ε
Dra
g:
Bru
cato
Mo
dif
ica
tio
n o
f th
e i
nte
rph
as
e d
rag
fo
rce
us
ing
th
e D
PE
an
d t
he
Pw
C a
pp
roa
che
s
No
tes
: E
-E
Tam
bu
rin
i e
t a
l. 2
01
3
α = 1
6.9
– 3
3.8
%w
/w
29
cm
, T/
3,
T/1
0,
H =
T
ro =
24
70
kg
/m 3
d = 6
00
– 7
10
μ m
N = 4
00
– 1
10
0 r
pm
MR
F
CFD
co
de
: A
NS
YS
CFX
4.4
So
lid
vo
lum
etr
ic f
ract
ion
s (
rad
ial
an
d a
xia
l
pro
file
s),
su
sp
en
sio
n c
urv
e d
ata
Pu
rpo
se
: p
red
icti
on
of
so
lid
pa
rtic
le
dis
trib
uti
on
Turb
ule
nce
mo
de
l: a
sym
me
tric
k - ε ,
ho
mo
ge
ne
ou
s k -
ε
Dra
g:
Bru
cato
, P
ine
lli
No
tes
: E
-E
(Tab
le 6
C
on
tin
ue
d)
R.S.S. Raja Ehsan Shah et al.: Solid-liquid mixing in stirred vessels 25
results in most cases. In the Eulerian approach, most
researchers have assumed monodispersed size of par-
ticles. Ochieng and Lewis (2006b) showed that this
assumption could not predict practical applications
accurately especially for high-density particles.
2. Assuming standard drag coefficient (quiescent flow)
results in large errors in the simulation of dispersed
phase concentration profiles of a turbulent flow. Drag
coefficients in turbulent fluids depend on both flow
field and particle characteristics.
3. In solid suspension studies, CFD makes it possible
to obtain detailed information on the flow field and
simulation of off-bottom solid suspension, solid cloud
height, and solid concentration distribution. Satisfac-
tory agreement between the experimental findings
and the numerical predictions has been observed.
4. In particle collision, the most recent models regard-
ing the direct numerical solutions were in good agree-
ment with the experimental results.
5. There are much more work that can be done at the
particle scale for further understanding of the rela-
tionship between the flow field in the reactor and the
physiology.
6. Majority of the CFD studies for solid-liquid stirred
tanks have focused on prediction of particle suspen-
sion height in a stirred vessel or have been devoted
to improved prediction of solid concentration profiles.
There were not enough publications for investigation
of mass and heat transfer, particle-particle and par-
ticle-impeller collisions, vortex structure, power con-
sumption, circulation and macro-mixing times, and
the distribution of turbulence quantities using CFD.
7. CFD simulation is a powerful and often indispensible
tool for modeling mixing systems such as solid-liquid
flows in stirred vessel, similarly true for any hydrody-
namic systems. However, one of the most important
aspects of this type of simulations that the compu-
tational fluid dynamicists must understand are the
limitations of the methods and results need to be vali-
dated before such predictions can be relied on.
Nomenclature Ar Archimedes number 3 2( / [( - ) / ])
p p L LAr gd ν ρ ρ ρ=
C Impeller bottom clearance, m
C D Drag coefficient
d p Particle size, m
D Impeller diameter, m
H Liquid depth, m
N Number of baffles
T Tank diameter, m
u Velocity vector
X Solid weight fraction
Greek symbols α Volume fraction of disperse phase
β Smagorinsky eddy viscosity ( β = 2/9)
ν Kinematic viscosity
ρ Density
μ ( t ) Turbulent viscosity, kg/m s
μ Dynamic viscosity
λ Kolmogoroff length scale λ = ( ν 3 / ε ) 1/4
τ Stress tensor
ϕ Phase hold up
r Particles of size
τ η Scales of fluid motion
ε Dissipation rate of turbulent kinetic
U x Velocity components of a turbulent flow field
u x Fluctuating velocity
E t Total local energy dissipation
⟨ u x ⟩ Mean velocity
2
xu⟨ ⟩ Mean-square velocity fluctuations
w ′ or 2
xw⟨ ⟩ Particle root-mean-square velocity
η Kolmogorov microscale
Subscripts and superscripts L Liquid
S Solid
P Particle
Acknowledgments: The authors are grateful for the Uni-
versity of Malaya High Impact Research Grant (HIR-MOHE-
D000038-16001) from the Ministry of Education Malaysia
and University of Malaya Bright Spark Unit, which finan-
cially supported this work.
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Bionotes Raja Shazrin Shah Raja Ehsan ShahFaculty of Engineering, Department of
Chemical Engineering, University of Malaya,
50603 Kuala Lumpur, Malaysia
Raja Shazrin Shah Raja Ehsan Shah graduated with a Chemical Engi-
neering degree at the University of Malaya, Malaysia, in 2004 and
obtained his Master ’ s degree from the same university in 2010. He
worked in applied research for resource recovery for waste streams
coming from natural rubber and palm oil industries in Malaysia,
with particular emphasis on membrane technologies. In 2012, he
joined the University of Malaya as a doctoral candidate working on
hydrodynamic studies on multiphase systems in stirred reactors.
Baharak SajjadiFaculty of Engineering, Department of
Chemical Engineering, University of Malaya,
50603 Kuala Lumpur, Malaysia
Baharak Sajjadi graduated with a Chemical Engineering degree at
Arak University, Iran, in 2008 and obtained her Master ’ s degree
from the same university in 2010. She worked on hydrodynamics
and mass transfer investigations in airlift bioreactors with compu-
tational fluid dynamics (CFD). In 2011, she joined the University of
Malaya, Malaysia, as a doctoral candidate working on biochemical
processing with particular interest in biofuel production. Other
interests include development, validation, and optimization in
unconventional geometries.
Abdul Aziz Abdul RamanFaculty of Engineering, Department of
Chemical Engineering, University of Malaya,
50603 Kuala Lumpur, Malaysia
azizraman@um.edu.my
Abdul Aziz completed his PhD in the area of three-phase mixing.
Currently, he is a professor and holds the position of Deputy Dean
at the Faculty of Engineering, University of Malaya, Malaysia. His
research interests are in mixing in stirred vessels and cleaner pro-
duction technologies. He is also active in consultancy projects and
has supervised numerous PhD candidates. He is also member of
professional and learned societies such as the Institution of Chemi-
cal Engineers (IChemE, UK), Institution of Engineers Malaysia (IEM),
and American Chemical Society (ACS).
Shaliza IbrahimFaculty of Engineering, Department of Civil
Engineering, University of Malaya, 50603
Kuala Lumpur, Malaysia
Shaliza Ibrahim is attached to the Department of Civil Engineering
at the University of Malaya (UM), under the Environmental Engineer-
ing Programme. A chemical engineer by training, her interest in
mixing research stemmed from her PhD study back in 1988 – 1992
at the University of Birmingham, in the area of multiphase mixing
in stirred vessels. She has been instrumental in keeping mixing
research going at UM, with particular interest in solid-liquid mixing,
while more recent projects are applied to biological and adsorption
processes in wastewater treatment.
Rev Chem Eng 2015 | Volume xx | Issue x
Graphical abstract
Raja Shazrin Shah Raja Ehsan
Shah, Baharak Sajjadi, Abdul Aziz
Abdul Raman and Shaliza Ibrahim
Solid-liquid mixing analysis in stirred vessels
DOI 10.1515/revce-2014-0028
Rev Chem Eng 2015; xx(x): xxx–xxx
Review: This review evaluates
CFD applications to analyze
solid suspension quality in
stirred vessels.
Keywords: computational fluid
dynamics (CFD); drag force;
solid suspension; stirred vessel.
Buoyancyforce (FB) Surface force
(F Surf)Surface force(F Surf)
Gravitational force (Fg)
Turbulent force
Drag forceDrag force
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