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Solid-liquidmixinganalysisinstirredvessels

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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

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on

of

so

lid

pa

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le

dis

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uti

on

Turb

ule

nce

mo

de

l: a

sym

me

tric

k - ε ,

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ou

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, P

ine

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: E

-E

(Tab

le 6

  C

on

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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