# Sedimentation Velocity of Solids in Finite Size Vessels

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Sedimentation Velocity of Solids in Finite Size Vessels
By Renzo Di Felice* and Ralf Kehlenbeck
The Richardson-Zaki equation is by far the most popular empirical equation used to describe the velocity-voidage relationship
for sedimenting solid-liquid homogeneous suspensions,using only two empirical parameters.In this work some of Richardson
and Zaki suggestions for the two parameters are challenged on the basis of new and old experimental evidence.
1 Introduction
The timeneededfor asolidsuspensiontosettleinavessel of
finite size is an important parameter for the correct design of
the unit.It is well known that a single particle in a large vessel
settles under steady-state conditions at a velocity,the
unhindered settling velocity u
t
,is easily estimated by
balancing the weight of the particle with the buoyancy and
dragforces.Smaller vessel dimensions or thepresenceof other
particles will lead to a reduction of the settling velocity,with
this reduction being more pronounced as the particle
diameter,d,becomes comparable to the vessel diameter,D,
or as the suspension voidage,e,decreases
1)
.
In both cases the main reason for the reduction in the
settling velocity is similar.As the solid particles,and some
fluid attached to it,move downwards,some fluid must move
upwards,increasing in this way the drag force and conse-
quently reducing the equilibriumsettling velocity.
For the important practical cases of concentrated solid
suspensions (e smaller than 0.95) no exact theoretical
treatment is available,so that we must resort to experiments.
Richardson and Zaki [1] extensively studied sedimentation of
liquid-solid suspensions of spherical particles;they investi-
gatedthedependencyof thesettlingvelocity,u,onthevoidage
fraction.Their results were summarized with the relationship
which today is known worldwide as the Richardson-Zaki
equation (although the same equation had been used,in a
somewhat different form,earlier by Lewis and Bowermann in
1952 [2])
u  u
i
e
n
(1)
Eq.(1) simplysays that inalog-logplot velocityandvoidage
are linked by a linear relation;therefore only two parameters
are needed to represent the observed behavior regardless of
the system investigated.Furthermore,based on their own
experimental investigation and on their theoretical analysis,
expressions for the two parameter n and u
i
were given as
follow.
The parameter n,reported in Tab.1,was found to be a
function of the flow regime,expressed by the terminal
Reynolds number Re
t
,and of the particle to column diameter
ratio d/D.
Table1.Values of the parameter nas recommendedby RichardsonandZaki [1].
Re
t
<0.2 n=4.65+19.5d/D
0.2< Re
t
<1 n=(4.35+17.5d/D) Re
t
±0.03
1< Re
t
<200 n=(4.45+18d/D) Re
t
±0.1
200< Re
t
<500 n=4.45 Re
t
±0.1
Re
t
> 500 n=2.39
The parameter u
i
,which graphically represents the extra-
polation of the velocity to voidage equal to 1 and therefore is
easilyrelatedtothesingleparticle terminal settlingvelocityu
t
,
was found to be coincident with the single particle terminal
velocity,i.e.,u
i
=u
t
.The simplicity of Eq.(1) is probably its
most striking feature,where the complex influence of the fluid
and particle physical characteristics on the particle-fluid
interaction forces is magically condensed into only two
parameters.Efforts to reproduce the Richardson-Zaki equa-
tion from basic fluid dynamic considerations are today still
only partially successful [3].
There are some observations to be made regarding the
proposed expressions for n and u
i
.Let us first consider cases
for which the particle diameter is much smaller than the
container diameter so that any possible effect of d/D can be
safelyignored.RichardsonandZaki suggestedthat inthis case
n is a function only of the terminal Reynolds number,
decreasing from 4.65 to 2.39 as we move from viscous to
inertial flowregime.This is easily justifiable as the amount of
fluid dragged down by the solid decreases with Reynolds
number,reducing in this way the overall effect of the
suspension voidage on the settling velocity.
The analysis,when wall effects are taken into account,is
somewhat less straightforward.Richardson and Zaki sug-
gested that n increases with d/D.As a consequence,identical
sedimenting systems differing only as far as the factor d/D
would possess the characteristics represented in Fig.1;this
figure seems to suggest that the effect of the wall is more
pronounced as the suspension becomes concentrated rather
than when it is diluted which does not appear to be correct.
Chem.Eng.Technol.23 (2000) 12,Ó WILEY-VCH Verlag GmbH,D-69469 Weinheim,2000 0930-7516/00/1212-1123 \$ 17.50+.50/0
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±
[*] R.Di Felice andR.Kehlenbeck,Dipartimentodi IngegneriaChimica e di,
Processo ªG.B.Boninoº,Università degli Studi di Genova,via Opera Pia
15,16145 Genova,Italy;e-mail difelice@istic.unige.it
1) List of symbols at the end of the paper.
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Figure 1.Velocity function of voidage as predicted by Richardson and Zaki.
System:Re
t
=10.
Overall,there is roomfor doubt,doubt which have already
been expressed and investigated,only as far as sedimentation
in viscous flowconditions,in a previous paper [4].In this work
we extend the investigation to higher Reynolds number
systems.However,we will still use the limit of spherical
particles of nearly constant diameter.
2 Experimental
Particle sedimentation velocities were investigated as a
function of the suspension voidage.Experimental runs were
carried out in cylindrical columns 500 mmtall with an internal
diameters of 24,30 and40 mm,respectively.The fluids utilized
were sugar solutions of different concentrations (up to 58%in
weight) at 20 C,with the density ranging from998 to 1275 kg/
m
3
and the viscosity ranging from 0.001 to 0.042 kg/m/s.An
important practical problem concerns the evaluation of the
single particle settling velocity,with which the parameter u
i
is
subsequently compared (a significant range of u
t
values can
occur for adjacent sieve sizes).The use of solids with the
smallest size range is therefore a must,and these solids can
only be obtained in practice for either large sizes (1 mm or
more) or very small (of the order of 1 lm).The solid particles
utilized (reported in Tab.2) were all spherical,with a very
narrow size range (the plastic particle had a tolerance of 0.01
mm,which was verified by measuring a batch of 100 spheres)
and,for each particle type the single particle settling velocity
was measured experimentally:it was obtained in a vessel of
300 mmdiameter in order to minimize any possible effect of
the wall.
Details of the experimental procedure for the sedimenta-
tion runs are analogous to those reported more at length in a
previous work [4].
Table 2.Solid physical characteristics.
Solid material Diameter (mm) Density (kg/m
3
)
Acetate 2.95 1280
Acetate 4.93 1280
Glass 1.69 2500
Glass 3.00 2500
Zirconia 1.17 3800
Teflon 1.96 2100
Delrin 5.00 1400
3 Results
All the systems investigated exhibited an expansion
characteristic law which,when plotted on logarithmic
coordinates,yielded a straight line:this is to say that Eq.(1)
is indeed a very good representation of the experimental
observations.The values of n and u
i
were determined with the
help of a standard error minimization routine.
Of course,there is not much novelty in this finding;more
important for this work was to verify some of the specific
results suggested by Richardson and Zaki.Fig.2 shows 3 mm
acetate particles sedimenting in vessels of different diameters
in sugar solutions.No noticeable differences appear in the
experimental data whereas the Richardson and Zaki predic-
tions (reported as continuous lines) exhibit a quite different
slope.This result was quitegeneral:for everycaseinvestigated
where only the value of the parameter d/Dwas changed,the
slope of the expansion characteristic law was constant,with
only a little difference,generally smaller than 0.2,being
measured,which is attributable to experimental uncertainty.
Figure 2.Experimental settling velocity function of voidage (points) and
corresponding Richardson-Zaki predictions (lines).The solid diamonds
represent the experimental values of u
t
.System:3 mm acetate;upper points
41 %sugar solution,lower points 51%sugar solution.
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The experimental values of nfor all thesystems investigated
are reported in Fig.3,where it can be seen that,once the wall
effect has beenremoved,the RichardsonandZaki correlation
is quite satisfactory.
Figure 3.Experimental values of n as a function of Re
t
(points) andRichardson-
Zaki predictions (line) ignoring wall effects.
Let us turn our attention now to the second numerical
parameter in Eq.(1):the extrapolation of the velocity to a
voidageequal toone,u
i
,andmorespecificallytoits ratiotothe
single particle settling velocity,u
t
,expressed by k
k 
u
i
u
t
(2)
Fig.4 depicts experimental values of k function of d/Dfrom
this and published works [4±9] (unlike the present investiga-
tion,published works are performed in the viscous flow
regime);all values are actually smaller than 1 and no specific
trend is evident.It must be said here that we tried to correlate
k with other parameters (Reynolds number,Archimedes
number,particle diameter,solid density to fluid density ratio)
but in every case we were unsuccessful,always obtaining plots
similar to Fig.4.
The present experimental results can be summarized as
follows:
l
Eq.(1) is an excellent representation of the expansion
characteristic of concentrated sedimenting suspensions
l
The parameter n is a function of the Reynolds number but
not a function of the particle to wall diameter ratio.It may
be easier to use the Rowe relationship [10],where only one
equation covers the entire flow regime for n
4:7ÿn
nÿ2:35
 0:175Re
0:75
t
(3)
or the relationship proposed by Khan and Richardson [11]
which relates n to Archimedes number
4:8ÿn
nÿ2:4
 0:043Ar
0:57
(4)
so that we can estimate n directly without first calculating
Re
t
.Both Eqs.(3) and (4) yield values of n very close to those
calculated fromTab.1.
l
The extrapolation to voidage equal to 1 of the fluid velocity
is not a function of d/Deither,its value being about 0.8±0.9
times the single particle terminal settling velocity.
Figure 4.Experimental values of the k function of d/Dfor sedimenting systems
fromthis and frompublished works.
This last remark needs some discussion.The velocity of the
particle will certainlyapproachu
t
as thevoidageapproaches 1,
so that the validity of the Richardson-Zaki equation must be
somewhat restricted to an upper limiting voidage;if the whole
voidage spectrumis experimentally investigated then one can
actually find a behavior of the type reported in [4] for the
sedimentation of 4.96 acetate particles in a water-glycerol
mixture where the expansion characteristic is not represented
by a straight line anymore but there is a break occurring at a
voidageof about 0.95.Howgeneral this findingis,is still,inour
opinion,anopenquestion;interestingly a similar behavior has
been reported at the other end of the particle diameter
spectrum:the sedimentation velocity of very small solid
particles (1.5 lm plastic particles) reported in [12] (for that
specific case the value of d/Dis equal to 0.000075 so that the
wall effect can certainly not be invoked to explain the
difference between u
i
and u
t
).No equivalent analysis is
possible at the present moment with intermediate diameter
spheres (let say0.1mm) giventhat theyareavailableonlywith
a relatively large size distribution and whose experimental
determinationof the settling velocity indilute andunhindered
conditions poses difficult practical problems [13]
4 Conclusions
We have confirmedthat the Richardson-Zaki equationis an
excellent tool in describing the characteristics of sedimenting
concentrated suspensions.Some small adjustments of the
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empirical parameters areneeded,especiallyas far as theeffect
of the container dimension is concerned.Specifically,we have
found no effect on the slope of the log-log voidage-velocity
expansion characteristics,n,and on its extrapolation to a
voidage equal to one,u
i
.
Acknowledgments
The financial support from the University of Genova is
gratefully acknowledged.Ralf Kehlenbeck would also like to
thank the European Union for supporting his stay in Genova
through an Erasmus student mobility grant.
Symbols used
Ar [d
3
(r
p
±r)rg/l
2
] Archimedes number
d [m] particle diameterer
D [m] column diameter
g [m/s
2
] acceleration due to gravity
k [±] defined by Eq.(3)
n [±] parameter in Eq.(1)
Re
t
[dru
t
/l] Reynolds number
u [m/s] settling velocity
u
i
[m/s] parameter in Eq.(1)
u
t
[m/s] single particle terminal settling
velocity
Greek letters
e [±] voidage
r [kg/m
3
] fluid density
r
p
[kg/m
3
] solid density
m [kg m
±1
s
±1
] fluid viscosity
References
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