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RATE 0 MEIVTATI

Nonflscculated Suspensions of Uniform Spheres

As an initial step in developing a more complete understanding of sedimentation i nconcentrated suspensions of fine powders, a study is presented of sedimentation in a

simple system, under conditions of laminar flow. The effect of concentration on the

rate of fall of uniform well-dispersed spheres is investigated both theoretically and

experimentally. Tests with suspensions of tapioca particles in oil provide empirical

solutions of functions of concentration left undetermined by the theoretical analy-

sis. Tests w ith fairly u niform microscopic glass spheres support the c onclusions drawn

from the tests wit h t he larger tapioca particles.

HAROLD H . S T E I N O U R , Por tland Cement Association, Chicago, 111.

HI S article is the first of a series on sedimentation phenom-

ena. Th e work was planned primarily to develop a betterunderstanding of the settling of fresh portland cement

pastes, an occurrence commonly called “bleeding”. An exten-

sive investigation of this property was made by Powers (16, 1 7 ) .

The present studies were undertaken to resolve some of the

questions raised by his analysis.

A cement paste is a concentrated, flocculated, aqueous sus-

pension of solid particles, of a wide range of sizes, slowly reactive

to water. To develop the theory of its sedimentation beyond

the stage to which it had been advanced, experiments were

made with simpler systems. Only the sedimentation of well-

dispersed uniform spheres is covered in this article, in which

th e effect of concentrat ion on the ra te of settlement is investigated

under conditions of laminar flow.

T

CONDITIONS IN SUSPENSIONS OF UNIFORM SPHERES

At Reynolds numbers, 2rV,p,/q , up to 0.6 11) a solid sphere

in an infinite expanse of fluid falls a t a uniform velocity given

by the Stokes law (16):

Within the given range of Reynolds numbers, the flow around a

sphere is laminar, or streamline, and inertial effects a re negligible.

In a suspension in which there are many spheres instead ofone, t he ra te of sedimentation is less tha n the velocity given by

the Stokes law. However, if the conditions are such tha t iso-

lated spheres will fall in accordance with Stokes’ law, and if the

spheres are of uniform size and densi ty and are well distributed

throughout the fluid, the rate can be represented by the Stokes

velocity multiplied by a term which is a function of concentration

only. This is shown by th e following study which also partiallyevaluates the new term; the restrictions that have been stated

here regarding the natur e of the suspension are assumed through-

out the development.

The spheres would necessarily all settle a t a common constant

rat e if they were in a stable uniform arrangement and if wall

and bottom effects were negligible. I n an actual mixture the

distribution of spheres cannot be strictly uniform, but under the

best conditions a fixed arrangement and constant velocity are

Hence, the fluid space can be as-

sumed to maintain a constant shape within which a steady

laminar flow patt ern is established. Relative to the spheres the

flow velocities increase from zero a t the sphere surfaces to maxima

in the intervening regions.

. rather closely maintained.

In order to make a general analysis, identical arrangemen ts of

the spheres in different suspensions will be assumed. At a givenconcentration of spheres by volume th e problem then becomes

one of comparing laminar flows in composite flow spaces having

the same shapes. When the sizes of the flow spaces are also

the same, as they are when the sphere sizes are the same, the

average velocities depend only on the velocity gradients at cor-

responding points, because equal gradients at such points in

flow spaces of th e same sizes and shapes obviously mean iden-

tical flows. Accordingly, a suitably defined velocity gradien t and

a characteristic length or dimension of the flow space are sufficient

to fix the average relative velocity of spheres and fluid when only

a particular concentration is concerned. Indeed , the velocity

must be proportional to the product of th e first powers of t he

gradient and the length, for only this combination of the vari-

ables has the dimensions of velocity. Hence, i t may be con-

cluded tha t a t a given concentration the average velocity is pro-

portional to the average velocity gradient or rate of shear at thesphere surface, and to the average spacing between spheres,

At constant concentration this spacing is proportional to sphere

radius T .

When the volume concentration is changed, the flow space

necessarily changes in shape. The spacing between spheres will

Figure 1. Fine-Pearl Tapioca Particles before TreatmenL(about 3 X )



July, 1944 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y

also change unless a compensating change is made in the size of

the spheres. These changes in the flow space affect the velocity,

but since the velocity at any one concentration is always pro-

portional to th e rat e of shear defined as above, and to the sphere

radius, a change in concentration simply alters the constant of

thi s proportionality. Accordingly, the average relative velocity

is given by

where (e) = size and shape factor which is a function only oft he proportion of fluid, e, and reduces to 1 a t infinite dilution.

0 2 4 6 8 10 12

Time, Hundreds of Seconds

Figure 2. Examples of Sedimentation Curves

Obtained for Tapioca in Oil

Since the concentration of solid by volume is ( I e), + , e )

represents a, function of concentration; another effect of con-

centration i s implicit in (dv/dn).., as will be shown.

VISCOUS RESISTANCE

The rate of shear at t he surface of a sphere, (dv/dn),,,may be

evaluated in terms of viscous resistance. This resistance results

from viscous forces both normal and tangential to t he surface of

the sphere. The resultant of the tangential forces is obtained

from the fundamental law by which the coefficient of viscosity is

commonly defined. As applied to the sphere this is:

2= h q 2 )

4rr2 av(3)

where 4702 = surface ar ea of sphere, sq. cm.h = dimensionless factor, constant for any given con-

centration, which corrects for the fact that tan-gential forces do not all act in line of motion

Because of the constancy of the flow pattern , the resultant of

th e tangential components of t he viscous force mainta ins a fixed

ratio to the resul tant of th e normal components, at a ny given

concentration. Thus, a t infinite dilution the resultant tangent ial

force is always twice the normal (16). However, as the concen-

tration is changed, both this ratio and h may change because of

th e change in shape of the flow space. Hence, a complete ex-

pression for the total viscous resistance, or fluid friction, de-

veloped by the motion of the sphere is

619

(4)

where kz = dimensionless pro ortionali ty constant which ex-presses ratio o? R / 4 d to q(dv/dn)., at infi-nite dilution

hape factor which is a function of E only, andbecomes equal to 1 at infinite dilution

Since +g(e) is purely a shape factor, Equation 4 shows that

changes in size of flow space caused by changes in E can affect

th e surface rate of shear, (dvldn),,, directly, only through apossible effect on viscous resistance R. When R is fully evaluated,

the only effect of size th at will remain undetermined will be that

embodied in +I ( €) of Equation 2.Eliminating (duldn)v between Equations 2 and 4,

The ratio & ( E ) / + ~ ( E ) may be replaced by a single function,

+ ( E ) , which, like its components, becomes equal to unity at in-

finite dilution. Also, th e combinat ion of constants 4kz/kl may

be replaced by a single term which can be evaluated from the

Stokes law, to which Equation 5 must reduce at infinite dilution.

Since the Stokes law in terms of t he viscous resistance is

R = 6?r7rV8 (6)

4k2/kl must equal 6, and Equation 5 becomes

' EFFECT OF CONCENTRATION ON BUOYANCY

Fluid friction R equals the motive force, which is the weight

of the sphere minus its buoyancy. The buoyancy depends on

the gradient of hydrostatic pressure and is therefore affected by

the presence of the other spheres. Th at is, since the spheres all

move without acceleration, their entire weight is supported by

the fluid, and this means th at the hydrostat ic pressure developed

by a layer of the mixture is determined b y th e density of the

mixture rath er tha n by the density of the liquid alone. Hence,

buoyancy is also determined by the density of the mixture. This

effect on the buoyancy is recognized in hydrometer pract iceII),and the principle has also been applied in some adaptations of the

0.20Elx=aLI

0.16

*kGi 0.125g

0.08

0.04

0 00

0.0 0.2 0.4 0.6 0.8 1 .o

Figure 3. [Q 1 - € ) ] ' / a US. for Sedimentation ofTapioca

E:

in Oil



620 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y Vol, 36, No. 7

4 43 d p m ) g Q = 3 71.7 P z ) g V

This equality reduces to

which, upon substitution from Equation 12, becomes e =

p , z ) / p , ,). Comparison with Equation 9 shows thatpz: = pm and confirms the previous formulation of the buoyancyas 4/3irrsgp,.

Substituting in Equation 12 the value of V given by Equa tion

11:

Figure 4. Log Q / t 2 z s. E for Sedimentation ofTapioca in Oil

Stokes law to suspensions (11, 18) though not always with con-

fidence ( 1 1 ) . That the usage is correct is further shown by the

development of Equation 13. Because of the augmented

buoyancy the equality between viscous resistance and motive

force for a sphere in a suspension is

8 )

(9)

4R = - M P . 3 Pmh

but,

P I Pm = Pa [ l ) P i P f l = e Pf )e

hence

R = ? 7 r r 3 p S m ) g e (10)

Substituting this value of R in Equation 7 and solving for V,

COMPARISON OF MEASURED VELOCITY AND RELATIVEVELQCITY OF SPHERES Ah-D FLUID

V was defined as the average relative velocity between spheres

and fluid whereas the measured velocity is th at of t he particles

relative to a fked horizontal plane, a velocity which will here be

represented by Q. The relation between Q and V may be de-

rived by equating th e volumes of solid and fluid tha t move in

opposite directions past a unit of horizontal cross section in

unit time.

Q = eV 12)

Another expression for the relation between Q and V can be

derived by equating the loss in potential energy attending the

fall of a sphere and the work done against viscous resistance.

That is,

Tha t is, 1 ) Q = E(V Q), or

In terms of the Stokes velocity, V,,

Q = Vae2+(e) (151

APPLICATION OF HYDRAULIC RADIUS

The function + ( E ) represents effects of both size and shape of

flow space. No complete theoretical solution of this function is

known but theoretical analyses aimed toward the solution o f

this problem were given by Cunningham 7) and 8moluchowski(20). Recent abstracts of papers not readily obtainable on

account of the war show that Burgers (b) has also contributed

to this subject. A theoretical study of th e effect of the spacing

upon axial flow between arrays of parallel cylinders was made

by Emersleben (8). A complete theoretical s o b t i o n for spheres

is not att empted here, but th e effect of size and part of t he

effect of shapeareevaluated by use of th e hydraulic radius; only

a residual undeterniined shape factor is left, which remains

nearly constant for concentrated suspensions.

0.16

01 2

5 0.08y1

:0.04

0.00

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

Figure 5. Shape Factor, 20- -1 . 8E I - -e ) u s . E

The hydraulic radius of a uniform length of conduit may be

defined as the flow volume per un it of wet ted surface. It hm

the dimension of length, is especially suitable, an d has long been

used, as a general radius term, for conduits of noncircular cross-

section. As applied t o a suspension,

hydraulic radius = (161(1 e u

For uniform spheres u = 3 / r and

(171re

hydraulic radius =

3(1 )

Previously, an T was placed in Equation 2 to represent the

felative spacing between spheres at constant concentration.



July, 1944 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 621

the hydraulic radius is now used to represent the spacing a t

any concentration, the remaining variable factor of Equation 17must be made explicit in Equation 15 by removal from b e).

Th at is,

wher;?@(e) represents those effects of shape tha t are not evaluated

by using the hydraulic radius. When +(e) is treated in this way,

Equations 14 and 15 become, respectively,

Since the choice of the hydraulic radius as t he spacing factor

is an arbitrary procedure, Equation 15 will still be considered,

along with Equation 20, in some of the further developments.

SHAPE FACTOR AND w i TERM

As the dilution is increased, Q must approach V as a limit.

Since @/ 1 E) approaches infinity, e € ) must approach zero;

bu t at high concentrations O(e) may remain practically constant,

and published work indicates that i t probably does. Kozeny

1 2 , IS)and Fair and Hatch (9) independently derived equivalent

forms of equat ion for the velocity of viscous flow through granu-

lar beds. Fair and Hatc h assumed the validity of the hydraulic

radius without any additional shape factor. Kozeny did not

write in terms of the hydraulic radius but his trea tmen t was

equivalent. These authors found reasonable experimental agree-

ment with the equation. Carman 3-6) applied it in tests on

many different kinds and shapes of particles and found excellent

experimental agreement over a range of porosities from 0.26

to 0.90. The same form of equation was also found applicable

t o flow through wads of textile fibers 83).

Powers (I?‘), starting with the Poiseuille law, developed an

equation for the rate of sedimentation or “bleeding” of concen-trated flocculated suspensions of cement and other powders, by

using the hydraulic radius and by determining the pressure

gradient from the hydrostatic pressure caused by the fall of the

particles. This was equivalent to adapting the Kozeny or the

Fair and Hatch equation to a sedimentation process. Fair and

Hatch had made a similar analysis in adapting their equation t othe flow through an “expanded” filter bed, in which the sand

grains become suspended in upward-flowing wash water. As

applied to uniform spheres, those adap tations were equivalent t o

Equation 20 except that constants were used instead of e(€),

and Powers introduced an additional, experimentally derived

term, as will be shown. Equation 20 might, therefore, have been

developed here by a slight modification of t he analysis based on

the Poiseuille law. The approach used was adopted instead, in

order to analyze conditions at the individual particles.

Although Powers’ theoretical analysis gave ~ 3 / ( l - ) as the

function of E in the rate equation, in order to represent his data

he had to subtract a constant, which he called w rom each E

th at appeared as an independent factor. This modification was

similar to one that Kozeny (I,%’) and Carman ( 4 ) had also found

necessary in a few cases, in permeabil ity tests on clays. Powers’

final equation in terms, of t he symbols used in thi s artic le was

in which the magnitude of toi was dependent upon the powder

under test but averaged about 0.28 for portland cements. The

numerical constant is the same as Carman (6, 6) found for flow

through porous media.

Equation 21 can be expressed in terms of V., as

For uniform spheres for which u = 3/r,

(22)

The first experiments of the present investigation were made

primarily to determine whether the w erm would be needed for

well dispersed (nonff occulated) suspensions of relatively large

particles, but they also provided opportunity for a study of .+ E)and e € ) . These te sts were made on suspensions of nearly

uniform spherical tapioca particles settling under conditions

characterized by low Reynolds numbers.

SEDIMENTATION O F TAPIOCA IN OIL

Fine-pearl tapioca was dried and soaked in SAE No. 50 lubri-cating oil under vacuum. Sedimentation tests were made in thesame oil at a series of concentrations. The oil-soaked tapiocagrains had a density of4.38 grams per CG. in the surface-dry condi-tion obtained by rolling them on absorbent paper. They werepractically uniform spheres about 0.174 em. in diameter. Formore than 80 of the grains, variations in diameter did not ex-ceed 10%. Figure 1 shows a representative sample beforetreatment. The oil had a density of 0.89 gram per cc. and aviscosity of 7.13 poises at 25’ C., the controlled laboratory tem-perature.

Figure 6. Photomicrograph of Glass Spheres ( X 500)Isolated black spots n the partialea indioate gas bubbles.

Most of the te sts were made in a 1000-ml. graduate d cylinderof t he glass-stoppered type, abou t 62 mm. in diameter. Thecylinder, filled to the shoulder with a test mixture, was evacuatedand closed. I t was supported manually; first one end and thenthe othe r was slowly elevated, and at the same time the cylinderwas rotated about the longitudinal axis. This was continueduntil the mixture appeared uniform; the cylinder was openedto the air after it was finally righted. This method of mixingproved very effective. The better tests showed a practicallylinear relation between time and amount of settlement through-out the sedimentation, exce t for a slight tapering off a t the finish.Figure 2 illustrates some o r he curves obtained.

An approximate determination of the Stokes velocity wasmade by dro ping single particles centrally into a 62-mm. diame-ter cylinder &led with the test oil. The average velocity of 152particles waa 0.1120 cm. per second. By applying the Francisformula (IO) or wall effect, the velocity at infinite dilution wascalculated to be 0.1194cm. per second. The corresponding Reyn-olds number is 0.0026.

The correction of the velocity consisted in mu1 tiplying the ex-perimental value by the factor (1 T / T ’ ) - ~ . ~ , here r and T ‘ arethe radii of sphere and tube, respectively. N o correction forwall effect was made in any of the other sedimentation tests.It was considered tha t the effect should become rapidly less as th econcentration was increased.



Vol. 36, No. 722 I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY

Th e results of sedimentation tests are given in Table I. As

th e final column showg, all sediments were practically constant in

porosity. Information on e € ) and wi is provided by Figure 3,

where [Q(1 d ]V: is plotted against e. It is evident that the

points up to E = 0.785 are adequately represented by a straight

line through the origin.

and from Equation 18,

2 6

The approxi

mate conformity of the factor to 0.123 at the lower values of

illustrates the close agreement between Equations 23 and 24 in

th at region. Figure 5 indicates, however, tha t th e shape facto

de(€) __ 1 O- L8 2 ( 1 - c )E

The equation of the line is

ea

Figure 5 is a plot of this Ahape factor against e .

(23)= 0.123V81

if t he corrected experimental value of 0.1194 cm. per second is

assigned to V,. Several conclusions may be drawn from this

equation. By comparison with Equa tion 20 it indicates that,

over a considerable range of high concentrat ions, the shape factor

e(€) remains practically constant at approximately 0.123. By

comparison with Equation 22 it shows that the wi factor which

Powers found to be necessary in evaluating the settling rates

of flocculated suspensions of part icles of microscopic size is not

needed for systems of the present type. Finally, it shows tha t

the proportionality constant, 0.123, is somewhat different from

the constant 0.10, derived from Powers' equation. Although the

data are not so precise but t ha t they could be represented fairly

well by a line having a slope based on the factor 0.10, the line

would pass to the left of t he origin. This is shown by the

dashed line of Figure 3 which has been drawn to this slope. Ifshifted horizontally until it passed through the data, it would

indicate a negative value of wi, hereas the values found by

Powers were all positive.

TABLE. SEDIMENTATIONESTS N TAPIOCAN OILVelocity, Av. Settlement Porosity

No. of &id, c X 10' Q X lo5 Initial Ht. SedimentConcn. Pro ortion &, Cm./Sec., Velocity, per Unit of of

139

127

2116a6

20

5

0.5020.508

0.5330.5390.5590 .5720.5840.592

0 .614

3 0.641

19 0.665

18 0.691

a Test in 92-mm.cylinder was used.

428 427 427346: 315,3 36,

307,336494,490474,483531,538,522642,657,666733,746,720797, 762, 753,

813 749

851, $64, 90 3,X48

1092, 1079,1094

1242, 1203,1180,1152

1562, 1600,1595,1597. . . . . .

2100,20902220,23003330,33005860,6050

6790.69506980,7340

. 120

. . . . . .diameter cylinder; in

427328

492479530655733775

867

1088

1194

1588

. . . .2095226033155955612068707160. . . .

all other

. . . . .0 IO99

0.1500.186

0.2250.245

0.279

0.331

0.374

0.420

. . . . .

. . . . .

0.4240.4900.5080.5990.7920.8050.8480.8610.904

tests a 62-n m.

. . . . .0.454

. . . . .0.4580 .458

0.4630 .460

0 .465

. . . . .

0.463

0.465

0.467

0.4640.4650.4560.4640.4710.4560.4870.4670.468

diameter

EMPIRICAL EXPRESSIONS FOR FUNCTIONS OF 6

In Figure 3 some of t he points a t high values of e fall far below

the solid line corresponding to Equation 23. Indeed, since Qcannot become infinite, [ (1 ) ] % must become zero at E = 1.An empirical equation which represents all the data, even the

limiting velocity at e = 1, is

Q V8e210-1.82(1-e) (24)

This equation is based on Figure 4 and was used to place the

dotted line in Figure 3. Comparison with Equation 15 shows

that it provides the following simple empirical expression for

d e ) :

2 5 )+(e) = 1 0 - 1 . 8 2 ( 1 - ~ ) , or e - 4 . 1 8 1 - 6 )

remains approximately constant up to E = 0.70 only. The ex

perimental data in Figure 3 show constancy up to e = 0.78

but tha t limit is indicated by only one experimental point. Othe

experimental data, to be presented in a later article, support the

inference from Figure 5. It appears that for nonflocculated

suspensions of spheres the s hape factor does not stay constant

up to such high values of -d as for beds of part icles, for which a

limiting value of 0.90 was quoted earlier.

SEDIMENTATIOh OF hlICROSCOPIC GLASS SPHERES IN WATER

The tests with tapioca appear to establish the effect of con

centration on the sedimentation of uniform spheres which are

so large that interfacial phenomena, such as manifest themselves

in colloidal systems, are negligible. Ordinary portland cements

have average particle diameters of about 10 to 12 microns, ascomputed from specific surfaces determined by the A.S.T.Mturbidimeter method ( 1 ) . Only a small fraction of the total

weight consists of particles as small as 0.5 micron in diameter.

For example, Lea and Nurse ( 1 4 ) reported the following per-

centages, by weight, of particles smaller tha n 0.6 micron in diam-

eter in various portland cements: 1.4, 0.5, 0.4, 0.7, 1.0, 0.8

0.6, 0.4, 1.3. Although such powders thus lie almost wholly

outside the conventional colloidal range, they are fine enough to

flocculate, to show significant adsorption, and to produce electro-

kinetic phenomena 62).

The impor tant effect of flocculation on sedimentation will be

discussed in later articles. The possibility th at other surface

effects might be capable of modifying the ra te of settlement was

investigated by making sedimentation tests with fairly uniform

glass spheres about 13.5 microns in diameter, designated as glass

spheres No. A . The test conditions were not, in general, so

satisfactory as in the work with tapioca, because the glass spheres

were much less uniform th an the tapioca and th e quantity was so

limited tha t th e tests had to be made on a small scale. However,

in the middle range of concentrations the dat a a re believed t o be

reliable.

By sedimentation in water, the spheres were separated from a

preparation made by essentially the method of Sklarew 19),as

modified by Sollner ( 1 ) . Figure 6 illustrates the quality of the

sized preparation. Some of the spheres carried enclosed gas.

This caused the average density to be 2.32 instead of 2.5 grams

per cc., th e density of the solid glass. Because of thi s gas not all

equal-settling par ticles were strictly the same size. However,

a sedimentation analysis in wate r was made by a special turbidim-

eter technique I ) , using hexametaphosphate as dispersant.

Diameters were calculated as though the particles all had theaverage density. The results are shown in Table 11. They

indica te th at for abou t three fourths of the sample the Traria-

tion from the average particle diameter did not exceed 20 .

The variation w s much greater than for the tapioca particles,

but it was believed that segregation would, in general, be pre-

vented by particle hinderance in th e suspensions.

The specific surface of the sample was obtained by th e air

permeability method of Lea and Nurse ( 1 4 ) which does not re-

quire that the particles all have a common density. The value

obtained was 4420 sq. em. per cc., corresponding to an average

particle diameter of 13.57 microns. These results are in reason-

able agreement with the da ta of Tab le 11, which indicate 4520

sq. cm. per cc. and 13.27 microns. By using the diameter of



July, 1944 I N D U S T R I A L A N D E N G I N E E R I N G C H 8 E M I S T R Y 623

TABLE1. SIZE NALYSISOF MICROSCOPICLASSSPHEIRESDiameter, Microns y Diameter, MioronsRange Variation Weight Range Variation e&

>21.3 I . . 6.7 13.9-12.8 1.1 19.721.3-19.2 2.1 1.3 12.8-11.7 1.1 25.819.2-18.1 1.1 1.9 11.7-10.7 1.0 18.518.1-17.1 1.0 1.2 10.7- 9.6 1.1 5.217.1-16.0 1.1 2.4 9.0- 8.5 1.0 0.616.0-14.9 1.1 4.8 8.5- 0 ... 0.314.9-13.9 1.0 11.6

13.57 microns and the limiting Stokes velocity calculated from itand from the average density, the Reynolds number for infinite

dilution was calculated to be 0.0025 at 27.5 C.

Sedimentation tests were made in a straight-walled glass Vial,

20 mm. in internal diameter. To ensure complete dispersion, a

0.1% aqueous solution of sodium hexametaphosphate was used

as the fluid medium. Th e preparations were mixed by slow manua l

manipulation of th e vial somewhat like th at adopted with the

cylinder of tapioca and oil. Readings of the height of each sus-

pension were taken at regular time intervals by a micrometer

microscope. Enough readings of final heights of sediments were

taken to establish that the porosities were essentially constant,

as was to be expected of nonflocculated material. Th e value of

E in these sediments was about 0.38.fluid contents (at e = 0.85 and 0.80) the upper boundary of he

suspension did not remain sharp during the settlement. A t the

other concentrations the boundary condition was satisfactory and

there appeared to be no segregation. The temperatures of the

suspensions were about 27.5 C.

Plot s of subsidence of th e upper boundary against time were

fair approximations to straight lines at all fluid contents from

E = 0.80 to 0.65, inclusive. In Figure 7 the curve for E = 0.70

is typica l of the be tter resUIts. The other curve, for E = 0.55,

illustrates an initial irregularity obtained in all tests at E = 0.60,0.55, and 0.50. At these fluid contents a steady rate was estab-

lished only after 10 to 15 minutes, and then i t was distinctly low

as compared with expectations based on othkr data. To avoid

attach ing undue significance to results obtained from curves

exhibiting such irregular phenomena, the rates for these three

lowest dilutions, although included in the tabulation of data,

were not plotted. The rate dat a are all presented in Table 111.

In the tests at the highest-

TABLE11. SEDIMENTATIONESTS N MICROSCOPICLASSSPHERES4 IE\' 0.1yo S O L~ TIO NF SODIUM EXAMETAPHOSPHATE

Fluid Velocity Fluid VelocityVolume e Q, Cm./Sec., X 106 Volume e Q,Cm./Sec., X 106

0.50 3924, 377 0.70 2170, 1800.55 6070, 325 0.75 3100, 0800.60 994a, 000 0.80 4170,4060,41200.65 1440, 1445, 1540 0.85 5400, 000, 530

a Test gave abnormal curve of height os. time (Figure 7).

COMPARISON OF DATA FOR GLASS SPHERES AND TAPIOCA

The average Stokes velocity for the glass spheres was found tobe 0.0156 cm. per second when calculated from the average sphere

radius determined by the air permeability test, t he average sphere

density, and the properties of water a t the test temperature of

27.5' C. This value of V ,when used in Equation 23 gives results

as follows:

e Q X 106

0.650.700.75

151022003250

These calculated values of Q are in good agreement with the

experimental values shown in Table 111, except that 3250 is a

little high, as it apparently should be since Equation 23 has been

indicated to hold strictly only up to e = 0.70.

Figure 8, like Figure 4, is a plot of log &/e2 against E , and is

therefore a test of Equation 24. The straight line has 8 slope of1.82, as required by the equation. It appears to be a fa ir re p

resentation of the data . Th e points% 6 = 0.85 fall distinctly

low, but the trend indicated by the line is sustained by the

point at e = 1.0which is based on the calculated Stokes velocity

The low points at e = 0.85 are less reliable tha n t he others be-cause th e suspensions developed diffuse upper boundaries, as d-ready mentioned.

16

14

12

9 108

EE* 8

8

v

3 6

8s2 43

2

0

0 5 10 15 20 25

Time, Minutes

Figure 7. Examples of Sedimentation Curves(Incomplete) Obtained wi th Aqueous Suspen-sions of Microscopic Glass Spheres

It appears from this study t ha t t he equations found applicable

to the sedimentation of the uniform spherical tapioca particles

are also applicable to these smaller glass spheres, which are

about the average size of portland cement grains. Evident ly

no new surface effects large enough to modify the rate of sedimen-

tation were developed by the reduction in size of sphere. In

drawing these conclusions, the tests at the highest concentra-

tions which developed the peculiarity shown by the curve for

e =: 0.55 in Figure 7 were disregarded. They do not appear to be

representative of th e phenomena to which the theory is to be

applied-namely, the sedimentat ion of cement pastes. Such

pastes, even highly concentrated ones, sta rt settling a t a uniform

rate.

SUMMARY

Theoretical considerations show that in nonflocculated sus-pensions of spheres of uniform size, settling under conditions suchtha t the Stokes law would apply a t infinite dilution, the rate ofsedimentation is given by the following equations:

Experiments with sus ension of approximately uniformtapioca particles provide tEe following empirical solutions of thefunctions of e:



6 2 4 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y Vol. 36, No. 7

where e € ) is approximately constant a t 0.123 for values of E lessthan 0.7.

Experiments with suspensions of microscopic glass spheresaveraging only 0.00136 em. in diameter support the conclusionsdrawn from the tests with the larger tapioca particles (except for

some work a t high concentrations of glass in which uniform ratesdid not establish themselves until after the sedimentations werewell advanced).

-1.8

-1.9

-2.0

-2.1

-2.2

-2.3

Figure 8. Logla Q/e’ u s . e for Sedimentationof Microscopic Glass Spheres in 0 . l ~ ~ S o d i u m

Hexametaphosphate Solution

At high concentrations, where e € ) is approximately constantat 0.123, Equation 20 conforms to a modification of t he Kozeny( l a , I S ) and Fair and Hatch (9) equations, obtained by evaluat-ing the pressure gradient in t erms of the hydrostatic pressureproduced by the fall of the particles; this was done by the latterauthors in treating the problem of flow through “expanded”filter beds, and by Powers 17) in deriving an equation for thesedimentat ion of flocculated pastes of por tland cement and otheipowders. It is noteworthy, however, that on the basis of Car-man’s work on flow through porous beds 5 ,6 ) ~ )n Equation 20would be 0.10 instead of 0.123.

A significant difference is shown in the effect of concentrationon the sedimentation of nonflocculated uniform spheres and onth at of the flocculaLed pastes investigated by Powers. To repre-sent his experimental data, Powers found it necessary to sub trac ta constant, w rom each E factor ; he thus obtained (e 0,)3 /

1 ) for the function of E instead of / ( l ) . The presentwork with a much simpler system was under taken as a first stepi n developing the significance of the zerm.

ACKNOWLEDGMENT

These studies on sedimentation Rere undertaken at th e sug-

gestion and with the cooperation of T. C. Powers, who showed

helpful interest throughout. The writer was assisted in various

phases of the experimental program by Lynn A . Brauer, Richard

G . Brusch, Herbert W . Schultz, and Ed\ in M. \Viler. Gerald

Picket t reviewed the applications tha t are made of fluid mechan-

ics in the theoretical analysis, and L. S. Brown made the

photomicrographs. To all of them the w r i h takes this op-

portunity to express his thanks and to aoknowledge his in-

debtednes.

NOMEh’C L.4TURE

(g)= average rate of shear in fluid at eurface of sphere,

reciprocal see. (the surface average of derivative ofvelocity at surface with respeot to the normal tothe surface)

g = acceleration of grav ity , cm./sec.2h = dimensionleks factor, constant for any given concen-

tration

Q = ra te of sett lement of top surface of suspension, cm./sec.

r = radius of a sphere, cm.r’ = radius of a sedimentation tube, em.R = viscous resistance developed by motion of sphere,

gram cm./sec.a, (dynes)R T = resultant of the viscous forces acting tangential to the

surface of a sphere, gram cm./sec.e, (dynes)p/ = density of a fluid, grams/cc.

pn = density of a mixture, grams/cc.p a = density of a solid, grams/cc.

p z = density used in computing buoyancy, grams/cc.V = average relative velocity between spheres and fluid,cm./sec.

V , = Ti for an isolated sphere, as given by the Stokes law,cm./sec.

20 = dimensionless constant used by Powers 1 7 ) .e = that part of total volume of suspension that is oc-

cupied by liquid, analogous to porosity in beds ofparticles (at infinite dilution E = 1)

+(e), +I €) = functions of e only, determined by size and shape of

e € ) , + * ( E ) = functions of e only, detymined solely by shape of

q = coefficient of viscosity of a fluid, grams/cm. sec.,

u = specific surface of a solid, sq. cm./oc.

5v

kl,’kl = dimensionless constants

flow space

flow space

(poises)

LITERATURE CITED

(1) -4m. SOC.or Testing Materials, Method for Finenessof Portland

(2) Burgers, J. M., Che m. A bs t r ac ts . , 37, 3652-3 (1943).

(3) Carman, P. C. , A.S.T.M., Symposium on New Met.hods forParticle Size Determination in Subsieve Range, 1941, 24-35;

J. SOC.C h e m . I n d . , 58, 1-7T (1939); T yans . Inst. Ch em .Engrs. (London), 16, 168-88 (19 38).

Cement, Designation C115-42.

(4) Carman, P. C., J . A g r . S c i . , 29, 262-73 (1939).

5 ) Carman, P. C., J . SOC. h e m . I n d . , 57,225-34T (1938).

(6) Carman, P. C., T r a n s . I n s t . Chem. E n g r s . (London),15, 150-61

(7) Cunningham, E., Proc . Roy. SOC.London),A83, 357-65 ( 1910 ).

(8) Emersleben, O ., P h y s i k . Z . , 26 , 601-10 (1925).(9) Fair, G. M., and Hatch, L. P., J. Am. W ater W o r k s A s s o c . , 25,

(1937).

1551-65 (1933).

(10) Francis, A. W., P h y s i c s , 4, 403-6 (1933).(11) Gaudin, A. M. “Principles of Mineral Dressing”, 1st ed., Chap.

(12) Kozeny, Josef, Kul tur te c hn ik e r , 35, 478-86 (193 2).

(13) Kozeny, Josef, Sitzber . A k a d . Wiss. Wien, 136, IIa, 27t-306

(14) Lea, F. M. and Nurse, R. W., J . SOC.Chem. I n d . , 5 8 , 277-83T

(15) Page, Leigh, “Introduction t o Theoretical Physics”, 2nd print-

(16) Powers, T. C., Proc . Am. Concrete Ins t . , 35,465-88 (1939).(17) Powers, T. C., Research Lab., Portland Cement Assoc., Bull. 2

(18) Robinson, C. S., IND. NQ. HEM.,8, 869-71 (1 926).

(19) Sklarew, Samuel, IND. NQ.CHEM.,~ S A L . ED.,6, 152-3 (1934).(20) Smoluchowski, M. S. 5th Intern. Congr. Mathematicians, 2,

8, New York, McGraw-HillBook Co., 1939.

(1927).

(1939).

ing, pp. 239-46, New York, D. Van Nostrand Co., 1928.

(1939).

192-201 (1912).

(21) Soher. K. , ND. ENG. EIEM., ANAL.ED.,11,48-9 (1939).

(22) Steinour, H. H., unpub. work for Riverside Cement Co.(23) Sullivan, R. R., and Hertel, K. L., in “Advances in Colloid

Science”, 1‘01. I, pp. 37-80, New York, Interscience Publish-ers, 1942.

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