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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.

Received:January 7,2000 [RN17]

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

[1] Richardson,J.F.;Zaki,W.N.,Sedimentationandfluidisation:Part I.Trans.

Inst.Chem.Eng.32 (1954) pp.35ff.

[2] Lewis,E.W;Bowerman,E.W.,Fluidization of solid particles in liquids.

Chem.Eng.Prog..48 (1952) pp.603ff.

[3] Foscolo,P.U.;Gibilaro,L.G.;Waldram,S.P.,A unified model for

particulate expansion of fluidised beds and flow in fixed porous media.

Chem.Eng.Sci.38 (1983) pp.1251ff.

[4] Di Felice,R.;Parodi,E.,Wall effects on the sedimentation velocity of

suspensions in viscous flow.AIChE J.42 (1996) pp.927ff.

[5] Chong,Y.S;Ratkowsky,D.A.;Epstein,N.,Effect of particle shape on

hindered settling in creeping flow.Powder Technol.23 (1979) pp.55ff.

[6] Steinour,H.H.,Rate of sedimentation ± non flocculated suspensions of

uniformspheres.Ind.Eng.Chem.36 (1944) pp.618ff.

[7] Mertes,T.S.;Rhodes,H.B.,Liquid-particle behavior.Chem.Eng.Prog.51

(1955) pp.429ff.

[8] Whitmore,R.L.,The sedimentation of suspensions of spheres.British J.

Appl.Phys.6 (1955) pp.239ff.

[9] Oliver,D.R.,The sedimentation of suspensions of closely sized spherical

particles.Chem.Eng.Sci.15 (1961) 230.

[10] Rowe,P.N.,A convenient empirical equation for estimation of the

Richardson-Zaki exponent.Chem.Eng.Sci.43 (1987) pp.2795ff.

[11] Khan,A.R;Richardson,J.F.,Fluid-particle interactions and flow

characteristics of fluidized beds and settling suspensions of spherical

particles.Chem.Eng.Comm.78 (1989) pp.111ff.

[12] Buscall,R.;Goodwin,J.W.;Ottewill,R.H.;Tadros,Th.F.,The settling of

particles through Newtonian and non-Newtonian media.J.Colloid

Interface Sci.85 (1982) pp.78ff.

[13] Davis,R.H.,Velocities of sedimenting particles in suspensions.in:E.M.

Tory (editor) Sedimentation of Small Particles in a Viscous Fluid,(p.161)

1996.

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