Nonflocculated Suspensions o f Uniform
Spheres^---As an in itial step in developing a m ore co m p lete u n d erstan d in g o f sed im e n ta tio n in concentrated suspensions o f fine poivdcrs, a stu d y is presented o f sed im e n ta tio n in a sim p le sy ste m , u n der con d ition s o f la m in a r flow. T h e effect o f con centration on the rate o f fa ll o f u n ifo rm w ell-dispersed spheres is investigated b o th theoretically and experim en tally. T ests w ith suspensions o f tapioca particles in oil provide em pirical solu tion s o f fu n ctio n s o f con centration le ft u n d e te rm in ed by the theoretical an a ly sis. T ests w ith fairly u n ifo rm m icroscopic glass spheres su pp ort the con clu sions drawn fro m the tests w ith th e larger tapioca particles.
H A R O L D H . ST E IN O U R ,
Portland Cement Association, Chicago, III.T
HIS article is the first of a series on sedimentation phenomena. The work was planned primarily to develop a better understanding 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, 17).
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 o f sizes, slowly reactive to water. T o develop the theory of its sedimentation beyond the stage to which it- had been advanced, experiments were made with simpler systems. Only the sedimentation o f well- dispersed uniform spheres is covered in this article, in which the effect of concentration on the rate of settlement is investigated under conditions o f laminar flow.
C O N D IT IO N S IN S U S P E N S IO N S O F U N IF O R M S P H E R E S At Reynolds numbers, 2rV sp,/v, up to 0.6 (11) a solid sphere in an infinite expanse of fluid falls at a uniform velocity given by the Stokes law (IS) :
2g (p , — p/)r- ,,,,
Within the given range of Reynolds numbers, the flow around a sphere is laminar, or streamline, and inertial effects are negligible.
In a suspension in which there are many spheres instead of one, the rate o f sedimentation is less than the velocity given by the Stokes law. However, if the conditions are such that iso
lated spheres will fall in accordance with Stokes’ law, and if the spheres are o f uniform size and density and are well distributed throughout the fluid, the rate can be represented by the Stokes velocity multiplied by a term which is a function o f concentration only. This is shown by the following study which also partially evaluates the new term; the restrictions that have been stated here regarding the nature of the suspension arc assumed through
out the development.
T he spheres would necessarily all settle at a common constant rate if they were in a stable uniform arrangement and if wall and bottom effects were negligible. In an actual mixture the distribution o f spheres cannot be strictly uniform, but under the best conditions a fixed arrangement and constant velocity are rather closely maintained. Hence, the fluid space can be as
sumed to maintain a constant shape within which a steady laminar flow pattern is established. Relative to the spheres the flow velocities increase from zero at the sphere surfaces to maxima in the intervening regions.
In order to make a general analysis, identical arrangements o f the spheres in different suspensions will be assumed. At a given concentration of spheres by volume the problem then becomes one of comparing laminar flows in composite flow spaces having the same shapes. When the sizes o f 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 the same sizes and shapes obviously mean iden
tical flows. Accordingly, a suitably defined velocity gradient and a characteristic length or dimension of the flow space are sufficient to fix the average relative velocity o f spheres and fluid when only a particular concentration is concerned. Indeed, the velocity must be proportional to the product of the first powers of the gradient and the length, for only this combination o f thp vari
ables has the dimensions o f velocity. Hence, it may be con
cluded that at a given concentration the average velocity is pro
portional to the average velocity gradient or rate of shear at the sphere surface, and to the average spacing between spheres.
At constant concentration this spacing is proportional to sphere radius r.
When the volume concentration is changed, the flow space necessarily changes in shape. The spacing between spheres will
Figure 1. Finc-P earl Tapioca Particles before T r e a tm e n t (ab out 3 X )
618
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 619 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 the rate of shear defined as above, and to the sphere radius, a change in concentration simply alters the constant of this proportionality. Accordingly, the average relative velocity is given by
y (2)
where 0 i(e) = size and shape factor which is a function only of the proportion o f fluid, «, and reduces to 1 at infinite dilution.
most ré g u lai
4 6 8 10
Tim e, H u n dred s o f Second s
where ki = dimensionless proportionality constant which ex
presses ratio of R fiirr1 to rt(dv/dn)a» at infi
nite dilution
02(<) = shape factor which is a function of t only, and becomes equal to 1 at infinite dilution
Since 4>i(c) is purely a shape factor, Equation 4 shows that changes in size of flow space caused b y changes in < can affect the surface rate o f shear, (dv/dn)ap, directly, only through a possible effect on viscous resistance R. When R is fully evaluated, the only effect of size that will remain undetermined will be that embodied in 0i(e) of Equation 2.
Eliminating {dv/dn)av between Equations 2 and 4, 4JiiTri)rV
&t0i(f)
The ratio 0i(«)/<j>i(c) may be replaced by a single function, 0(e), which, like its components, becomes equal to unity at in
finite dilution. Also, the combination of constants 4fc2/Ti 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 o f the viscous resistance is R = 6 m,rV.
iki/k\ must equal 6, and Equation 5 becomes QirrjrV
E F F E C T O F C O N C E N T R A T IO N ON I1U O Y A N C Y
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. That is, since the spheres all move without acceleration, their entire weight is supported by the fluid, and this means that the hydrostatic pressure developed by a layer of the mixture is determined by the density of the mixture rather than by the density of the liquid alone. Hence, buoyancy is also determined by the density o f the mixture. This effect on the buoyancy is recognized in hydrometer practice {11), and the principle has also been applied in some adaptations of the
0 .2 0 .4 0 .6 0 .8 1.0
e
Figure 3. [Q (l — e)J'/» vs. e. for S e d im en ta tio n o f T apioca in Oil
Figure 2. E xam ples o f S e d im en ta tio n C urves O btain ed for T apioca in Oil
Since the concentration of solid by volume is (1 — «), 0 t(e) represents a function of concentration; another effect o f con
centration is implicit in (dv/dn)ai-, as will be shown.
V IS C O U S R E S IS T A N C E
The rate of shear at the surface o f a sphere, (dv/dn)a,, may be evaluated in terms of viscous resistance. This resistance results from viscous forces both normal and tangential to the surface of the sphere. The resultant of the tangential forces is obtained from the fundamental law by which the coefficient o f viscosity is commonly defined. As applied to the sphere this is:
4
- ‘ O . . »where 4irr2 = surface area 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 the tangential components of the viscous force maintains a fixed ratio to the resultant of the normal components, at any given concentration. Thus, at infinite dilution the resultant tangential force is always twice the normal {15). However, as the concen
tration is changed, both this ratio and h may change because of the change in shape o f the flow space. Hence, a complete ex
pression for the total viscous resistance, or fluid friction, de
veloped by the m otion of the sphere is
S o lid Line C o rre sp o n d s to E q u atio n 23
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
I
x r 3(p , — p ,)g Q =I
x r 3(p , — p , ) g VFigure 4. Log Q/t1 vs. t for S e d im en tation o f T apioca in Oil
Stokes law to suspensions ( 11, IS) though not always with con
fidence {11). 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
but
hence
R = ^ x r 3(p , — pm)g ( 8 )
P v i = P , — [(1 — e) p , + ip / ] = (p . — P f ) ( ( 9 )
R = g x r 3(p , —
Pf)gt
( 1 0 )Substituting this value of R in Equation 7 and solving for V, 2 g {p ,--p / )r 1e^{e)
This equality reduces to
Q _ Pm — P x
I Pm — P I (13)
which, upon substitution from Equation 12, becomes e = (Pm — Pi)/Pm ~ Pi) ■ Comparison with Equation 9 shows that px = pm, and confirms the previous formulation of the buoyancy as 4/3trr3g p „ .
Substituting in Equation 12 the value of V given by Equation 11:
n 2 g (p , — p /)r ! < V ( i ) „
<2 - ^ (14)
In terms of the Stokes velocity, V „
Q = V .* * U ) (15)
A P P L IC A T IO N O F H Y D R A U L IC R A D IU S
The function <j>(e) represents effects of both size and shape o f . flow space. N o complete theoretical solution of this function is known but theoretical analyses aimed toward the solution of this problem were given by Cunningham (7) and Smoluchowski (20). Recent abstracts of papers not readily obtainable on account of the war show that Burgers (2) has algo contributed to this subject. A theoretical study of the effect of the spacing upon axial flow between arrays of parallel cylinders was made by Emersleben (8). A complete theoretical solution for spheres is not attempted here, but the effect of size and part o f the effect o f shape are evaluated b y use of the hydraulic radius; only a residual undetermined shape factor is left, which remains nearly constant for concentrated suspensions.
O u<
V =
9i; (11)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Figure 3. Shape F actor, --- P0- t . s2( i - < ) rs> e
C O M P A R IS O N O F M E A S U R E D V E L O C IT Y A N D R E L A T IV E V E L O C IT Y O F S P H E R E S A N D F L U ID
V was defirifed as the average relative velocity between spheres and fluid whereas the measured velocity is that of the particles relative to a fixed horizontal plane, a velocity which will here be represented by Q. The relation between Q and V may be de
rived by equating the volumes o f solid and fluid that move in opposite directions past a unit of horizontal cross section in unit time. That is, (1 — e) Q = e(F — Q), or
Q (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,
The hydraulic radius of a uniform length o f conduit may be defined as the flow volume per unit of wetted surface. It has the dimension of length, is especially suitable, and has long been used, as a general radius term, for conduits o f noncircular cross- section. As applied to a suspension,
hydraulic radius =
For uniform spheres a = 3 /r and hydraulic radius =
(1 — e)c
3(1 - e)
( 1 6 )
(17)
Previously, an r was placed in Equation 2 to represent the relative spacing between spheres at constant concentration. If
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 at
any concentration, the remaining variable factor of Equation 17 must be made explicit in Equation 15 by removal from <£(<).
That is,
1 - (18)
where 0 ( e )represents those effects of shape that are not evaluated by using the hydraulic radius. When 4>(e) is treated in this way, Equations 14 and 15 become, respectively,
Q = 2g(p. — P/)r* e3 along with Equation 20, in some o f the further developments.
S H A P E F A C T O R A N D m T E R M
As the dilution is increased, Q must approach V, as a limit.
Since «5/ ( l ~ e) approaches infinity, 0(e) must approach zero;
but at high concentrations 0(e) may remain practically constant, and published work indicates that it probably does. Kozeny (12,13) and Fair and Hatch (9) independently derived equivalent forms of equation for the velocity of viscous flow through granu
lar beds. Fair and Hatch assumed the validity o f the hydraulic radius without any additional shape factor. Kozeny did not write in terms of the hydraulic radius but his treatment 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 o f equation was also found applicable to flow through wads of textile fibers (93).
Powers (17), starting with the Poiseuille law, developed an applied to uniform spheres, those adaptations were equivalent to Equation 20 except that constants were used instead of 9(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 the 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 — e) as the function o f e in the rate equation, in order to represent his data he had to subtract a constant, which he called u\, from each e that appeared as an independent factor. This modification was similar to one that Kozeny (12) and Carman (4) had also found necessary in a few cases, in permeability tests on clays. Powers’
final equation in terms o f the symbols used in this article was
Q _ g (p. — P/) (e — Wj)* well dispersed (nonflocculated) suspensions of relatively large particles, but they also provided opportunity for a study of <t>(e) and 6(e). These tests were made on suspensions of nearly uniform spherical tapioca particles settling under conditions characterized by low Reynolds numbers.
S E D IM E N T A T IO N O F T A P IO C A IN O I L
Fine-pearl tapioca was dried and soaked in SAE N o. 50 lubri
cating oil under vacuum. Sedimentation tests were made in the same oil at a series of concentrations. The oil-soaked tapioca grains had a density of 1.38 grams per cc. in the surface-dry condi
tion obtained by rolling them on absorbent paper. They were cylinder, filled to the shoulder with a test mixture, was evacuated and closed. It was supported manually; first one end and then linear relation between time and amount of settlement through
out the sedimentation, except for a slight tapering off at the finish.
Figure 2 illustrates some of the curves obtained.
An approximate determination of the Stokes velocity was made by dropping single particles centrally into a 62-mm. diame
ter cylinder filled with the test oil. The average velocity of 152 particles was 0.1120 cm. per second. B y applying the Francis formula (10) for wall effect, the velocity at infinite dilution was calculated to be 0.1194 cm. per second. The corresponding Reyn
olds number is 0.0026.
The correction of the velocity consisted in multiplying the ex
perimental value b y the factor (1 — r / r ') -2 -25, where r and r ' are the radii of sphere and tube, respectively. N o correction for wall effect was made in any of the other sedimentation tests.
It was considered that the effect should become rapidly less as the concentration was increased.
622 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 The results of sedimentation tests are given in Table I. As
the final column shows, all sediments were practically constant in porosity. Information on 6 (e) and Wi is provided by Figure 3, where [Q (l — <)]'/» is plotted against e. It is evident that the points up to < = 0.785 are adequately represented by a straight line through the origin. The equation of the line is
Q = 0.123V, (23)
if the corrected experimental value o f 0.1194 cm. per second is assigned to V,. Several conclusions may be draw'n from this equation. B y comparison with Equation 20 it indicates that, over a considerable range o f high concentrations, the shape factor 8 ( e) remains practically constant at approximately 0.123. By comparison with Equation 22 it shows that the u\ factor which Powers found to be necessary in evaluating the settling rates o f flocculated suspensions of particles of microscopic size is not needed for systems of the present type. Finally, it shows that 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 that they could be represented fairly In Figure 3 some o f the points at high values of e fall far below the solid line corresponding to Equation 23. Indeed, since Q that it provides the following simple empirical expression for 4>(e): but that limit is indicated by only one experimental point. Other experimental data, to be presented in a later article, support the so largo 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, as computed from specific surfaces determined by the A.S.T.M . turbidimeter 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 (14) reported the following per
centages, by weight, of particles smaller than 0.6 micron in diam
eter in various portland cements: 1.4, 0.5, 0.4, 0.7, 1.0, 0.S, 0.6, 0.4, 1.3. Although such pdwders 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 (22).
The important effect of flocculation on sedimentation will be discussed in later articles. The possibility that other surface effects might be capable of modifying the rate of settlement was investigated by making sedimentation tests with fairly uniform glass spheres about 13.5 microns in diameter, designated as glass spheres N o. A. The test conditions were not, in general, so satisfactory as in the work with tapioca, because the glass spheres were much less uniform than the tapioca and the quantity was so limited that the tests had to be made on a small scale. However, in the middle range of concentrations the data are believed to be reliable. equal-settling particles were strictly the same size. However, a sedimentation analysis in water was made by a special turbidim
eter technique (1), using hexametaphosphate as dispersant.
Diameters were calculated as though the particles all had the particle diameter of 13.57 microns. These results are in reason
able agreement with the data of Table II, which indicate 4520 sq. cm. per cc. and 13.27 microns. B y 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 E M I S T R Y 623
cause the suspensions developed diffuse upper boundaries, as al
ready mentioned.
13.57 microns and the limiting Stokes velocity calculated from it and 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. T o ensure complete dispersion, a 0.1 % aqueous solution of sodium hexametaphosphate was used as the fluid medium. The preparations were mixed by slow manual manipulation of the vial somowhat like that 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. The value of
< in these sediments was about 0.38. In the tests at the highest fluid contents (at e = 0.85 and 0.80) the upper boundary of the suspension did not remain sharp during the settlement. At the other concentrations the boundary condition was satisfactory and there appeared to be no segregation. The temperatures o f the attaching 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 data are all presented in Table III.
< in these sediments was about 0.38. In the tests at the highest fluid contents (at e = 0.85 and 0.80) the upper boundary of the suspension did not remain sharp during the settlement. At the other concentrations the boundary condition was satisfactory and there appeared to be no segregation. The temperatures o f the attaching 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 data are all presented in Table III.