Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 8.5.3 - A very simple relationship for the voidage function in binary-solid suspensions
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Title: 8.5.3 - A very simple relationship for the voidage function in binary-solid suspensions Particle-Laden Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Di Felice, R.
Rotondi, M.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: drag force
voidage function
suspensions
binary-solid mixtures
 Notes
Abstract: The drag force on a particle in a multiparticle suspension is a function of the particle-fluid relative velocity and of the particle volume concentration. Its determination heavily relies on experimental observations, as theoretical support is still limited to viscous flow regime and dilute solid concentrations (Batchelor, 1982). When uniform particle suspensions are considered, there is a certain abundance of experimental data available which has permitted the proposition of simple and reliable relationships for the determination of the drag force: these relationships are normally expressed through the use of the so-called “voidage function”, i.e. a function by which the drag force on an isolated particle has to be multiplied in order to obtain the drag force on a particle in a multiparticle suspension. The extension of the approach mentioned above to suspensions made up of particles differing in size and density has been attempted here and new simple relationships are presented, limited to the case of binary-solid systems. We assumed here that particles of the same type have identical influence on the drag force as in uniform particle type systems, whereas for the estimation of the effect on the drag force on a particle exerted by the other particle type we have utilized a previous published analogy. The simple relationships obtained for the estimation of the drag force on a particle in a binary-solid suspension have been tested, with satisfactory success, against experimental data available in literature.
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7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


A very simple relationship for the voidage function in binary-solid suspensions


Renzo Di Felice and Marco Rotondi

University degli Studi di Genova, Dipartimento di Ingegneria Chimica e di Processo "GB. Bonino"
via Opera Pia, 15, Genova, 16145, Italy
renzo.difelice unige.it; marco.rotondi@unige.it


Keywords: drag force, voidage function, suspensions, binary-solid mixtures




Abstract

The drag force on a particle in a multiparticle suspension is a function of the particle-fluid relative velocity and of the particle
volume concentration. Its determination heavily relies on experimental observations, as theoretical support is still limited to
viscous flow regime and dilute solid concentrations (Batchelor, 1982). When uniform particle suspensions are considered, there
is a certain abundance of experimental data available which has permitted the proposition of simple and reliable relationships for
the determination of the drag force: these relationships are normally expressed through the use of the so-called "voidage
function", i.e. a function by which the drag force on an isolated particle has to be multiplied in order to obtain the drag force on a
particle in a multiparticle suspension. The extension of the approach mentioned above to suspensions made up of particles
differing in size and density has been attempted here and new simple relationships are presented, limited to the case of
binary-solid systems. We assumed here that particles of the same type have identical influence on the drag force as in uniform
particle type systems, whereas for the estimation of the effect on the drag force on a particle exerted by the other particle type we
have utilized a previous published analogy. The simple relationships obtained for the estimation of the drag force on a particle in
a binary-solid suspension have been tested, with satisfactory success, against experimental data available in literature.


Introduction

The last few years have witnessed an exponential increase in
the use of numerical computation in fluid dynamic problems.
A large number of commercial or in-house codes have been
developed and are widely available, both for single phase or
multiphase systems. It goes without saying that the
reliability of the code numerical output depends on the
correctness of the equations utilized to describe the
phenomenon on hand, which therefore requires a
satisfactory physical understanding of the phenomenon
itself. The aspect is particular relevant when multiphase
flow is considered, given that the interaction between the
different phases must be included in order to close the
constitutive equations.
Attention will be limited here to the most important
interphase interaction existing in a solid-fluid system, the
fluid dynamic drag, which arises when a relative velocity is
imposed between the two different phases. It is common use
to express the drag force as the product of two terms: the
first is some sort of "reference" drag force, obtained for
cases where it is easily estimated, such as the case of a
single solid particle in an infinite extent of fluid, whereas
the second is a "voidage function" which take care of the
effect brought about by the presence of the other particles.
In this paper first a short review of the state the of art for
monocomponent solid systems is presented, given that
enough information have been collected to confidently
quantify the drag force for that situation, then a similar
approach is extended to systems where the solid phase is


represented by a binary mixture of particles of different
diameter, for which no universally accepted quantification
of the fluid dynamic drag force exists as yet.

Nomenclature


drag coefficient
particle diameter (m)
drag force (N)
acceleration due to gravity (ms-2)
Reynolds number, -
velocity (ms-1)


Greek letters
e voidage, -
0 volume concentration, -
vt viscosity (Pa s)
p density (kgm-3)

Subsripts
F fluid
L large particle
0 reference
p particle
S small particle


The voidage function for monocomponent
solid-fluid systems









Before considering binary-solid fluid systems, a short
summary is presented here on the voidage function for
monocomponent solid-fluid systems, as this will represent
then the basis for the extension to more complicate
situations.
The system considered consists in layer of solid particles, of
the same diameter, homogenously dispersed in a fluid phase.
The particles will possess a uniform and constant velocity up,
the fluid will possess a velocity uf and 0 is the particle
volume fraction (the voidage a is the volume fraction
occupied by the fluid, with + e = 1)
We are interested here in quantifying the fluid dynamic drag
force exerted by the fluid on the particle. Although it would
have been much more preferable to have theoretical
expressions for the determination of the drag in a
multiparticle system, these approaches are limited to viscous
flow regime and dilute solid concentrations (Batchelor, 1982,
Batchelor & Wen, 1982) so that we are left with no
alternative but to resort to experimentally derived
relationship, which make use of published data on
sedimenting, fixed and fluidized suspensions.
For a given system, this drag force, FD, will depend on the
particle-fluid relative velocity and on the particle volume
concentration:

FD = f( p -Uf ) (1)

It is common practice to consider these two effects
separately, by writing


FD = FDog()


FDO is the drag force for a single particle, when the effect of
the other particles has been removed. The reference fluid
velocity, relative to the particle, is given by


uo = u -f = (up -U X1 -)


so that the knowledge of the single particle drag coefficient


CDO =f(Re) = pu (4)


allows the numerical determination of this reference drag
force


1 2 P2
FD = CDO P1uO p


(in the present work the numerical expression for the single
particle drag coefficient suggested by Dellavalle (1948) is
used throughout)


CDO 0.63 +4-- (6)
Reo

Results from investigations on experimentally measured
drag force on multiparticle suspensions have unexpectedly


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

demonstrated that, in order take into account the effect of
the neighboring solid on the drag, acceptable estimations
are produced by the use of a very simple voidage function
given by (Wen & Yu, 1966)


g() = (1- )-3 8


regardless of the flow regime and for the whole range of
solid fractions, from packed bed to isolated particle. Further
refinements of the above relationship, such as the one
suggested by Di Felice (1994), improve its predictive
capability when tested against experimental evidence
confirming however at the same time its main features.


The voidage function for binary solid-fluid systems

The binary-solid system of interest is schematically depicted
in Figure 1: it consists in a layer of bidispersed particles, the
larger being characterized by a volume fraction 0L, the
smaller by a volume fraction Os, the remaining volume
being occupied by the fluid,


Pr Lc


o*O O *



0 0

0 *.
O o *
0 .


Figure 1: Schematic representation of a binary-solid
suspension

In the most general case, the two solid phases will possess
different velocity, uL and Us, whereas the actual fluid
velocity is uf. As before, the drag force on a large particle is
expected to be a function of the particle-fluid relative
velocity and, in this case, of both particle volume
concentrations


FD,L =fUL -f, L OS)


and for the smaller particles

FDS = fUS u f, LS)


Again we can assume the effect of relative velocity and
particle concentration to be separated by writing


FD,L = FDO,L L (LL OS)

FDS =FDOSSg(bLOS)









Obviously the expressions above are of little use when
determining the drag force, if the voidage functions g are
not known. It is not surprising that for the case of
binary-solid mixtures also very little help can be drawn
from theoretical analysis for the determination of the
voidage. Moreover, unlike the case of monocomponent
suspensions, experimental evidence is also scarce and
certainly not sufficient to attempt a purely empirical
determination of the voidage function.
The approach proposed here for the estimation of the drag
force in a binary-solid suspension is very simple and is
based on two main aspects: the utilization of previously
proposed semi-theoretical approaches which have allowed a
partial description of the main fluid dynamic behaviour of
these systems, namely the pseudo-fluid approach for the
estimation of the effect of the presence of the smaller
particles on the larger and the particle-in-a-tube analogy for
the estimation of the effect of the larger particles on the
smaller, and, whenever possible, an extrapolation of well
accepted results regarding the drag on a monocomponent
particle layer.

A simple voidage function for the larger in binary
solid-fluid systems


*) _


00 00
0 00 00


0



0


0
00


O0 00
0


0
0
O


Figure 2: Schematic representation of a binary-solid
suspension where the smaller solid and the fluid form a
pseudo-homogeneous suspension


If we concentrate our attention on the larger particles only,
the smaller particles and the fluid can be thought as a
pseudo-homogeneous mixture (Figure 2), possessing some
convenient physical characteristics. A general relation for
the drag force experienced by a large sphere can be written
as


FD,L = FD,Lg(Ob)


where F D,L is the drag force for the isolated large sphere
and the voidage function g takes into account the effect of
the neighboring large spheres. By assuming that the effect
of the same particle type is identical as for the
monocomponent suspension, then we have


g(L) = (1-L)-


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

In order to have the possibility to estimate FD.L using the
above relationships, the drag force on the isolated large
sphere in a suspension of fluid and smaller spheres needs
however to be quantified. The system is the one sketched in
figure 3, where the smaller particles occupy a fraction of the
total volume given by


* "-
1- L


and has been extensively studied, in viscous flow regime,
both theoretically (Batchelor, 1982, Batchelor & Wen, 1982)
and experimentally (Poletto and Joseph, 1995, Di Felice and
Pagliai, 2007).

4,
1- L


*
**



S* O
.* *

** *


. .
* *


Figure 3: Schematic representation of a single large particle
in a suspension of smaller solid and fluid.


The results clearly indicate that the drag force on the large
particle is increased by a factor due the presence of the
smaller particles function of the smaller particle volume
concentration. It is common procedure to associate this
factor to a suspension "effective" viscosity for which
Gibilaro et al (2" i") gave a semi-theoretical quantification
valid for any particle volume concentration


l =(f (- )28
/ it'0 ) _


Outside the viscous flow regime experimental evidence is
much more scarce. Poletto and Joseph (1995) produced data
on a single sphere velocity falling in a fluidised suspension
at high Reynolds number: they carried out a thorough
analysis of their experimental observation and concluded
that, in analogy with the work of Barea and Mizrahi (1972),
the correction factor applicable for the drag force coincided
with that found for the viscous flow regime. Therefore,
using these conclusions, we can write a general expression
for the drag force experienced on a single large sphere in a
suspension of smaller spheres and fluid


F *, *iY2.8 (16)
FDL = FDOL (1)
with





7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010


1 2 ;af2
FDO,L CDO.L 2 PO.L 4


"UOL =(HL If)(1-L)


CDOL = f(ReL)J dL7L (19)-
The use of the above
equations together with Equation (12) and (13) allow for the
determination of the drag force on a large particle in a
suspension of large and smaller particles at any flow
conditions.

A simple voidage function for the smaller in binary
solid-fluid systems

The previous approach is not valid here any longer as, from
the smaller particles point of view, the larger particles can
not be assimilated with the fluid. The analogy attempted for
this case, as already done in a similar context (Di Felice,
1996) assumes the smaller particle to "feel" the presence of
the larger particles as equivalent of a "container wall", as
depicted in Figure 4. The "equivalent" wall diameter is
going to be dependent on the large particle volume fraction
(Di Felice, 1995). Di Felice & Parodi (1996) however
experimentally demonstrated that for particles sedimenting
in a column with a solid volume concentration larger than
0.05, the effect of the wall on the sedimentation velocity, i.e.
on the interaction drag between solid and fluid, was
negligible. In other word a smaller particle will feel the
presence of the other smaller particles only, and the effect of
the larger will limited to the reduction the overall volume
available.

4.
1- )L


















Therefore by drawing the analogy with the monocomponent
.
* .' .
9 *















case, the drag force experienced by the smaller solid species
is suggested as
FD FO 3.8
FD,S = FDO,S 1- 3.) ,m


with


1 2 ;d 2
DO,S DO.S 2po.s 4
2 4

UOS =(s- Uf f d)


CDOS f(Res) -. fdsos


(21)


(22)


(23)


Again, for the time being, there relationships are assumed to
be valid for any flow regime.


Validation of the proposed voidage functions with
published experimental observations

The relationships presented in the previous paragraph for
the drag force on the particles in a binary-solid fluid
suspension were the results of working hypotheses and
adaptation of previous findings relative to monocomponent
systems rather than exact theoretical approaches. As a
consequence some confidence on the proposed relationships
can only be gained through a positive comparison with
experimental. The main problem here is the scarcity of
published experimental works from which the drag on a
solid particle can be estimated: it is important to stress that
we are looking here for situation where larger and smaller
solid are present at the same time in the test section and they
must be also homogeneously dispersed. Both solid type
velocity and fluid velocity, together with phase volume
concentrations and, obviously, the experimental drag force
have to be known in order to carry out the comparison.
A thorough analysis of the relevant literature has pointed to
the fact that very little usable information is readily
available for the problem in hand. Usable data are from
systems concerning batch sedimentation of binary-solid
mixtures, liquid fluidization of binary-solid exhibiting the
inversion phenomenon, vertical solid transport by liquid of
solids and a closed loop circulating system. In all the four
cases the solid particles are fully suspended by the fluid so
that the net force on each particle must be zero. Drag force
can therefore be easily estimated as difference between
weight and buoyancy


FD,L = -d{PL [qLPL + SPS +(- L -- s)P]}g
6


FDS = S LPL + SPS + ( SL SS)}
6

Data from batch sedimentation of binary-solid mixtures
in liquids

There is a relative abundance of data published in literature
reporting settling velocities of solid during batch
sedimentation experiments. It is well known that when an
initial homogenous solid mixture is left sedimenting in a
vertical column four distinct zones form. Starting from the
top of the column we can observe clear liquid, a zone made
up of only smaller particle type, a zone made up of both









large and small particle type and at the bottom, a static final
sediment, with consequently three different interfaces
moving at a constant velocity. The region of interest for the
present work is in the third zone, where both particle types
are suspended. The volumetric concentration of the solids
there it is taken as the same of the initial concentrations, and
the large solid velocity is assumed coincident with the
interface velocity whereas the fluid velocity (and the smaller
solid velocity) are calculated from an overall volumetric
balance. Here the data produced by Mirza and Richardson
(1979) have been utilized as source data. They investigated
three different systems of binary glass spheres settling in a
viscous liquid so that Reynolds number was always very
low. Smaller spheres have always a diameter of 115 ptm,
whereas larger spheres had a diameter of 231, 327 and 462
ptm. Large solid concentration was varied in the range
0.18-0.38, and small solid concentration in the range
0.07-0.16. Figure 5 reports the comparison between
experimental and calculated drag force as experienced by a
large particle from that experimental work. Unfortunately
the same set of data could not be used to test model
prediction relative to the smaller particles, as their velocity
in the mixed suspended zone was not measured and
reported.


1 UUU -


0C
Z
500.
0
E

11 250-


0 250 500 750
FDL exp, N*109


1000


Figure 5: Experimental and calculated drag force on a large
particle for a batch sedimentation system of binary-solid.
Experimental data from Mirza and Richardson (1978).



Data from fluidization of binary-solid mixtures exhibiting
inversion

It is well known that batch fluidization of binary-solid
mixtures by liquid yield a bed which is in the vast majority
of cases made up of two distinct layers (Di Felice, 1995),
with the larger/more dense solid occupying the bottom layer
and the smaller/less dense solid occupying the top layer. A
notable exception of this simple rule is observed when the
binary-solid mixture is made up of larger/less dense
particles together with smaller/more dense ones. For
specific cases of diameter and solid ratio, coupled with
specific values of the fluidizing velocity, the bed will be
completely mixed, with both particle types present, and this


o 2(

a3
E
E 1(
U-


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

is termed as the inversion point (Di Felice, 1995). This
situation can be easily utilized for the present purpose, given
that several data have been published in the literature for
systems showing mixed bottom layer at the inversion point.
The average velocity of both solid phases will be zero and
the liquid velocity is calculated from the overall liquid flow
rate. Particle volume concentrations are also easily inferred
from the knowledge of the mixed layer bed height and the
amount of material charged into the column.
Table 1 summarizes the main characteristics of the systems
used for comparison in the present work (for every
investigation ambient water was used as fluidizing medium).
All the systems reported in Table 1 are in the intermediate
flow regime.
Experimental particle volume concentrations were as high as
0.3, both for the large and the smaller solid type.
Figure 6 depicts experimental drag force for the larger
particle type compared with the model prediction, whereas
Figure 7 show the correspondent comparison for the smaller
particle type

Table 1 Inverting system characteristics utilized in this
work.

Reference PL Ps dL d
Reference (kg/) (kg/m') (pm) (pm)
Moritomi et
M tomet 1500 2450 385-775 163-214
al. (1982)
Di Felce 3800 8800 700 135
(1988)
Di Felice
Fece 1270 2500 2000 195
(1988)
Jean & Fan
a1509 2510 778 193
(1986)


FD,L exp, N*109


Figure 6: Experimental and calculated drag force on a large
particle for binary-solid systems fluidised by liquid at the
inversion point.


dL (pm)
a 452
0 327
A 231





/


























S 50 7
FDs exp, N*109


Figure 7: Experimental and calculated drag force on a small
particle for binary-solid systems fluidised by liquid at the
inversion point.

Data from vertical transport of binary-solid mixtures

Lockett and Al-Habbooby (1973) presented an interesting
experimental work which is perfectly suited to test the
problem in hand. They studied the vertical transport
characteristics of binary-solid mixtures in the intermediate
and turbulent flow regimes: the experimental setup
consisted of a vertical column where ambient water was fed
at the bottom and two solid types, both glass ballotini, were
continuously fed at the top. The direct measurement of the
phases flow rates and volume concentrations (large particle
concentrations varied in the range 0.01 0.44 whereas
smaller particle concentrations varied in the range 0.03 -
0.36) in the test section allows a comparison of
experimental and calculated drag force for both the larger
and smaller particle types. This comparison is reported in
Figures 8 and 9 respectively.


IUUI
dL (pm) ds(pm)
E 1960 682
0 1960 965
S 75- A 1145 682


50
a

E
O 25
LL /


0
0 25 50 75 100
FD,L exp, N*106

Figure 8: Experimental and calculated drag force on a large
particle for vertical transport of binary-solid systems.


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010



10
dL (pm) ds(pm)
E 1960 682
0 1960 965
A 1145 682
0 /




E /
LL
O



0 5 10
FD,L exp, N*106

Figure 9: Experimental and calculated drag force on a small
particle for vertical transport of binary-solid systems.

Data from a liquid circulating bed of binary-solid
particles

The last set of data, by Di Felice et al. (1988), utilizes
experimental observation on a liquid circulating bed, where
the circulating stream was made up of water and small
particles, which suspended larger particles in the vertical
test sections. The larger solid has a zero velocity relative to
the column wall, and the smaller solid velocity was assumed
to be practically coincident with the liquid velocity. The
larger solid was in every case 5 mm plastic spheres, whereas
the smaller solid was either plastic (2 or 4 mm in diameter)
or 420 ptm glass balottini. The maximum large particle
volume fraction investigated was 0.10 for the larger solids
and 0.21 for the smaller solids, Figure 10 reports the
comparison between experimental and calculated drag force
as experienced by a large particle from that experimental
work. The same set of data could not be used to test model
prediction relative to the smaller particles, as their velocity
in the mixed suspended zone was not directly measured.



200


150-
O
'-
z
S100
-D

E
LLU 50-


100 150
FD.L exp, N*106


Figure 10: Experimental and calculated drag force on a
large particle for a circulating liquid fluidized system of
binary-solid mixtures. Experimental data from Di Felice et
al. (1988).






7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

performance of this work seems to be beyond discussion.


Discussion


In the previous section simple relationships for the
estimation of the solid-fluid drag force in binary-solid
suspensions have been presented and tested against data
available in literature. The theoretical support for the
proposed approach is rather thin, coming mainly from
analogies between a fluidised multiparticle suspension and
the flow in pipes and from some extension of previous work
on monocomponent suspension, so that the only confidence
on the proposed relationships originates from a comparison
with experimental evidence. This exercise has been carried
out and presented in the last section of the paper and the
results can be considered encouraging. However the scarcity
of experimental data does not allow, at the present moment,
to fully test the simple expression for the drag force and
work is being carried out at the moment in our laboratory to
fill this gap.
There are certain aspects that are in obvious need of some
experimental support. The first one concerns the rather
crude assumption of ignoring the effect of the particle
diameter ratio on the determination of the drag force: this
simplification may sound extremely dangerous although it
should not be forgotten that such simplification has been
successfully used on the pseudo-fluid approach when the
effect of the presence of the smaller particles on the larger
has been quantified (Di Felice et al., 1991). The second
aspect concerns the assumption of a voidage function which
is independent of the flow regime. Again there is no support
for such assumption, the only comforting evidence coming
from a similar result for the drag force on a monocomponent
suspension.
Given the difficulty of finding or producing experimental
data in literature, same authors have recently proposed
correlations for the drag force using lattice-Boltzmann
simulation data. The lattice-Boltzmann model is basically
a finite-difference scheme to solve the Boltzmann equation.
First, a randomly distributing spherical particle are inserted
in cubic domains, with, for example, the Montecarlo method,
and then, the lattice-Boltzmann method is used to solve the
hydrodynamic equations, with stick boundary rules applied
on the solid phase surface. The advantage is that the
material and/or the flow can be perfectly controlled.
The data produced by such simulation have been used
instead of proper experimental data to propose new
correlations: an example has been presented for low
Reynolds number binary-solid suspensions by Van der Hoef
et. al. (2005), and then extended to intermediate flow by
Beestra et al. (" )1). A similar work has been carried out by
Yin & Sundaresan (2009a and 2009b), who, starting from
the first result of Van der Hoef et al.(2005), developed a
model for mono and polydisperse systems for low Reynolds
number gas solid suspension.
Figure 11 and Figure 12 depicts calculated drag force on a
large and a small particle respectively, for one specific
suspension in viscous flow regime. It seems that the
relationships proposed by Beestra et al. 2. I) always
indicate drag forces, both for the larger and the smaller
particles, definitely larger that those calculated from the
present work. Also in Figure 11 experimental data extracted
from Mirza and Richardson (1979) are reported: although
this is not by any means a comprehensive comparison
between the two approaches, the superior predictive


R ichardson (1979)
LL
(c 15-



E.
0 5- -----

0-
00 01 02 03 04 05
L' -

Figure 11: The drag force on a large particle in a
binary-solid suspension in viscous flow regime. Os = 0.05.


Beestra et al. (2007)
this work
20-

LL





o
(f 15-








00 01 02 03 04
10- /,





00 01 02 03 04
S' -

Figure 12: The drag force on a small particle in a
binary-solid suspension in viscous flow regime. OL = 0.20.


Conclusions

In this work the extension of the established approach for the
estimation of the drag force on a suspended particle in a
multiparticle system to suspensions made up of particles
differing in size (and density) has been attempted, limited to
the case of binary-solid systems. We have assumed that
particles of the same type have identical influence on the drag
force as in uniform particle type systems, whereas for the
estimation of the effect on the drag force on a particle exerted
by the other particle type we have utilized a previous
published analogy. The simple relationships obtained for the
estimation of the drag force on a particle in a binary-solid
suspension have been tested, with satisfactory success,
against experimental data available in literature. Further
validation is still required, but this further work is hampered
by the lack of suitable experimental evidence.









References

Barnea, F. & Mizrahi, J. A generalized approach to the fluid
dynamics of particulate systems. Part I. General correlation
for fluidization and sedimentation in solid multiparticle
systems. Chemical Engineering Journal, 5, 171-189
(1973).

Batchelor, G. K.; Sedimentation in a dilute polydisperse
system of interacting spheres. Part 1. General theory;
Journal of Fluid Mechanics 119, 379-408 (1982)

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