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1/11

A

THEORY OF

SEDIMENTATION

BY

G.

J.

KYNCH

Department of Mathematical Physics, The University, Birmingham

Received 22nd May, 1951

;

in fim form, 6th September, 1951

The theory assumes that the speed of fall of particles in a dispersion is determined by

the local particle density only. The relationship between the two can be deduced from

observations on the fall o f the top of the dispersion. It is shown that discontinuous

changes in the particle density

can

occur under stated cond itions.

1. INnioDumoN.--The process of sedimentation of particles dispersed in

a fluid is one of great practical importance, but it has always proved extremely

difficult o examine it theoretically. The hydrodynamical problem of one particle

falling through a fluid has been solved (Stokes' law), and a formula has been

obtained by Einstein,l Smoluchowski and many others when the density of

particles is very small and their distance apart is much greater than their size.

This formula states that the speed of fall is

= u l -

rp)

1)

where

a =

2.5 for hard spheres,

u

is the Stokes' velocity, and

p

is the volume

concentration. The same problem has never been satisfactorily solved when

the density of particles is great. In fact no theory has yet been given which even

suggests

how

to interpret the experimental results when the concentrations are

relatively large.

In this paper it is hoped to remedy this particular omission by showing that

a considerable amount can be learned by the single main assumption that at any

point in a dispersion the velocity of fall of a particle depends only on the local

concentration of particles. The settling process is then determined entirely from

a continuity equation, without knowing the details of the forces on the particles.

We find that the theory then predicts the existence of an upper surface to the dis-

persion in the liquid and that the motion of this surface together with a knowledge

of the initial distribution of particles is sufficient to determine the variation of

the velocity of fall with density for that particular dispersion.

A complication which is dealt with fully, as far as fairly uniform initial dis-

tributions are concerned, is that due to the formation and existence of layers

where the density suddenly changes its value. Observations of dispersions

suggest that these do occur in dilute solutions, and it is satisfactory that the

theory not only predicts their occurrence but gives in addition the necessary

conditions to be satisfied. Using these results we are able to suggest various

quite different modes of settling which may occur. It is fortunate that we can

handle the discontinuities without knowing the precise mechanism by which

they are maintained. This mechanism is indeed a subject for further examina-

tion. This aspect of the process

is

also discussed in detail because the mathe-

matical technique of using the characteristics of

a

partial differential equation,

as the density lines are technically called, is not one which is generally known

to chemists.

The assumption that the local conditions determine the settling process is by no

means necessary. Changes in particle density are propagated through a dispersion

just as sound is propagated through air, and it is only if either the speed of

propagation is relatively slow or the damping

is

great that our assumption can

166

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G .

J .

K Y N C H

167

be justified. Until the details of the forces on the particles can

be

specified it is

impossible to state when our hypothesis is valid, even for a dispersion of identical

particles. It is probably true for dilute or concentrated ones but not for those

of intermediate concentrations. Nevertheless the theory is a first step in the

analysis of experimental data. The velocity against concentration curve deduced

from one experiment by this theory is a property of that particular dispersion.

Unless the dispersion can be accurately reproduced it may not be obtained again

in exactly the same form, but the character of each curve could

be

a guide to a

more detailed knowledge of the types of particle which occur in the dispersion

or to the physical and chemical processes which occur, when a comparison between

a number of curves can be made.

We leave further discussion of this and of other assumptions made in this

paper and the lines on which extensions to the theory might be made, until the

last section, when the method of treating the problem has been outlined on detail.

consider the settling of a dispersion of similar particles. It is assumed that the

velocity

v

of any particle

is

a function only of the local concentration p of particles

in its immediate neighbourhood. The concentration here means the number of

particles per unit volume of the dispersion. As the particles have the same size

and shape it is proportional to the volume fraction. It is convenient to introduce

the particle flux

which is the number of particles crossing a horizontal section per unit area per

unit of time. It is assumed everywhere that the concentration is the same across

any horizontal layer. The particle flux

S

therefore at any level determines, or

is determinzd by, the particle concentration.

As

p increases from zero to its

maximum value pm he velocity

v

of fall presumably decreases continuously from

a finite value

u

to

zero. The variation of S is more complicated, but a simple vari-

ation is assumed in the following sections for convenience of exposition.

Let x be the height of any level above the bottom of the column of dispersed

particles. If S varies with x the concentration must vary as well and, in a region

where the variation is continuous, the relation between the two is called the

continuity equation. Consider two layers at x and x + dx. In time dt the ac-

cumulation of particles between the two is the difference between the flow of

particles

S x

+ dx) in through the upper layer and the flow

S x )

out through the

lower layer, per unit area.

2. CONTINUITY EQUATION AND LINES

OF

CONSTANT CONCENTRATION.-we

s= pv, (2)

d

pdx)dt = S X

+

dx)dt - S(x)dt.

at

Dividing by dxdt, we derive the continuity equation

JP - 3s

3t ax'

- _ -

On account of the relation (2) this is written as

JP

V@)-

= 0,

at 3 X

3)

(4)

where V(p) = - dS/dp. 5 )

This equation is interpreted in the following way. On a graph where position x

is plotted against time t, curves are drawn through points with the same value of

the concentration. The co-ordinates

x , t )

and

x

+ dx, t

+

dt) of two adjacent

points on such a curve are related by the equation

P(X dx, t dt) = p ( ~ ,),

3P 3P

ax at

i.e. -dx

+

-dt =

0.


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168 THEORY OF SEDIMENTATION

Combining eqn.

(4)

and (6) the slope of such a curve is given by

As

p ,

and therefore V, is a constant along the curve, it must be a straight line.

Therefore, on an x against t diagram, the concentration is constant along straight

lines whose slope

V

depends only on the value of the concentration. One such

line passes through every point in the diagram below the top of the dispersion,

and in a region where the density is continuous the correct pattern of lines is such

that no two lines intersect. This simple result forms the basis of our analysis

of the settling process using this diagram. It can be expressed in another form,

which is discussed more fully in

4 .

This states that a particular value of the con-

centration is propagated upwards through the dispersion with a velocity

V

given

by eqn. 5).

mentation process in detail for a dispersion where the initial concentration in-

creases towards the bottom and V decreases with increasing

p

in the concentration

range covered during the settling. The reasons for these limitations appear later.

The x against

t

diagram, together with lines of constant concentration for

such a process. is shown in fig. l(a). These lines have been drawn according

to the following arguments. The initial values

of

the concentration determine

p

along the x-axis. Then a line K P of constant concentration has a slope dx/dt = V

determined by p at the point K where it intersects the x-axis. If the top of the

dispersion

is

at x

=

H, and the concentration increases from p

= pa

at x

=

H

to p =

pb

where

x

= 0 in a known manner, then all the lines crossing the x-axis

can be drawn. Since V decreases with increasing p they diverge as they leave the

x-axis. The line OB in fig. l(a) is the line of concentration

pb.

The equation of

any line KP, which crosses the x-axis at XO the value of

x

where the concentration

is

p

at

t=

0

is

( 8 4

if

pa

< p < pb. Since xg is a known function of p this equation gives the con-

centration at any point

x

in the dispersion at time

t,

provided that x, ) lies in

the region AOB.

We now calculate where these lines

of

constant concentration terminate, that

is to say, the position of the curve AB representing the fall of the top of the dis-

persion. At any point P, since the speed of fall of the surface is that of the par-

ticles in it, then along AB

Expressing

p

in terms of x and t by means of eqn. (8a) we obtain

a

differential

equation for

x

in terms of t which can be integrated to give the curve of fall.

However, the following method leads to the integral in a more direct manner.

The line KP represents the rise through the dispersion with velocity V of a

level, across which particles of concentration p fall with velocity ~ p ) ownwards.

In time t from the start the number of particles which have crossed this level

is

p ( V

+ v)t per unit area. The level reaches the surface at the point

P

when this

number equals the total number of particles It originally above the level

K .

Using

the initial distribution of particles this is

dx/dt = V(p).

7)

3. THE

EDIMENTATION OF A DISPERSION.-In this section we describe the sedi-

x = xo +

VG)t

(dxldt) =

- ( p ) .

9)

H

xo

n xo>= 1 pdxo.

1 0 4

1 la)

We thus derive the equation

where n can be expressed as a function of p . To determine the co-ordinates

of

P x, t) in the surface we now have two equations (8a) and (l la).*

* The fall of any other layer of particles not at the

top

can be found in the same way,

using instead

of

n the amount

of

material above the levelxo and below the

given

layer.

n(xo)

=

p

. V v)t ,


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G .

J .

KYNCH 169

We have now to consider the lines of constant concentration starting from the

t-axis, which cover the region in the diagram below the line OB. It is worth

noting at this point that these lines are determined in position only by the end

conditions at

=

0 at the bottom of the container, and not by the initial con-

ditions in the suspension. The initial conditions determine what happens above

OB.

Similarly the fall of the surface is determined by the initial conditions only

down to the point B.

A physically reasonable assumption about conditions along the t-axis is that

near 0 there is a continuous but extremely rapid increase of concentration from

pb

to the maximum possible concen-

tration p and that, subsequently,

the concentration remains at pm.

Since V decreases with increasing

p the lines crossing the t-axis near

0 form a spray of lines, as shown

in

fig.

la, between

OB

correspond-

ing to concentration

pb

and OC

corresponding to concentration p m .

The line OC and other parallel

lines starting from the t-axis at later

times all have slopes

Vm =

V(p,).

The equations of these lines be-

tween OB and OC are clearly

where

To find the curve of fall BC of

the surface we use the same argu-

ment as before. The number of

particles crossed by each level of

constant density is

now the total

number

N

of particles in the dis-

persion, where

x =

V(p)t,

86)

pb < p < pm.

H

N

=

pdxo, lob)

and

N

= p .

v

+

V )

t.

llb)

Combining 8b) and lob)we find

0

C

t

FIG. 1.-Fall

of

surface of dispersion,

showing

lines of density propagation (dV/dp> 0).

(a) when initial density increases from top to

bottom.

(b )

when initial density is uniform.

that for this part of the fall since

u

and

V

are functions of

p

alone, that

12)

where

f

represents some function depending on the law of fall.

Below

OC

the concentration is pmand hence along CD the whole suspension

has settled to its maximum concentration and is no longer moving. Its depth

is

now h where

N

=

t

x/t).

N

= pmh. (13)

These equations can now be used to discuss a problem of more immediate

practical importance, where it is required to deduce the properties of a dispersion

from observations of the settling. Thus assuming that the velocity of fall is a

function only of the concentration, we wish to find the relation between

S

and

p

given that the initial concentration increases in a known manner from po to p b

as

x

decreases from

H

to 0, and given the law of fall of the surface

ABCD

from

its initial height

= H

We do this by using our equations to calculate the values of p and u at points

on

the curve ABC. At any point

P

on this curve the value of v is given, according

to

eqn. 9), by the slope of the curve at P. Moreover, if P lies close to A we assume


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6/11

G . J . KYNCH

171

initially.

The experimental results are given in table

1 .

This should not

be

taken

to mean that

our

assumptions are necessarily valid for this suspension, as further

experiments are necessary to verify this. In this example, the initial constant

rate of fall only exists for a very short time, which would mean that the speed of

propagation of density changes,

V

=

dS/dp, is initially large.

4.

DISCONTINUTI~ES

F

FIRST

AND

SECOND

ORDERS.-Adiscontinuity of the first

kind in the particle concentration is

a

sudden finite change of concentration at

a

certain level. The differential equation of continuity (eqn.

(2))

no longer applies.

It is replaced by an equation stating that the flow of particles into one side of the

layer equals the flow out on the other side.

If

the suffix 1 denotes the layer above

the discontinuity and the suffix

2

the layer below, and

U

s the upwards velocity

of the discontinuity, this equation is

16)

This makes it clear that in general the discontinuity is not at rest but moves through

the dispersion with

a

velocity

17)

where S

=

pv. On an S against

p

diagram the speed U s the slope of the line

joining the points

pl , S l )

and p2,

5 2).

A

discontinuity of the second kind is a very small change in the particle con-

centration. If p2 -

1

= dp is small, the expression for U educes to

P l h + w =

p2@2

u .

Sl

-

s2

p2

- P I ,

u=-

U -

dS/dp=

V @ ) . 18)

The velocity Vintroduced in 2 now appears as the velocity of a discontinuity

between concentrations

p

and

p +

dp.

A

small change dp, if maintained, is pro-

pagated through

a

dispersion of concentration

p

with velocity

Y

ust as sound is

propagated through

air

with

a

definite velocity.

A

line of constant concentration

in the

x

against

t

diagram therefore describes the motion of a boundary between

dispersions of density p and p + dp so that its slope is necessarily equal to this

velocity V. The whole adjustment of concentration which occurs when a dis-

persion settles (fig.

l a),

(b))

can

be

described as a series of small discontinuities

propagated through the fluid.

The final settling of a dispersion into a layer of maximum concentration p m

is an interesting application of these results. The velocity

U

s now that rate

of increase of the thickness of the deposit. If there is a sudden change of con-

centration

U

s given by eqn, (17) with p2

= pm

and

S2 = 0

on the lower side : f

there is no sudden change then

U

=

V,.

5. STABILITYF DIsco rmurrm.-The possibility of discontinuities having

been demonstrated, it remains to explain why a dispersion of any concentration

does not always settle discontinuously into a layer of maximum concentration.

A

discussion of the formation and stability of these sudden changes shows that

this is indeed possible, but is not necessary.

For dispersions where the concentration increases downwards towards the

bottom the condition for the formation of a first-order discontinuity can be

expressed in the following equivalent ways

:

a) the lines of constant concentration in the

x

against t diagram, if continued

(6)

the propagation velocity

Y

ncreases with concentration

;

c ) the S against p curve is concave to the p-axis.

If these conditions are not satisfied a first-order discontinuity is not formed.*

This assertion can be proved in terms of second-order discontinuities.

If

V

increases with

p

small concentration changes from the denser regions below

away from the x-axis, would intersect ;

* The first of these three is the most general.


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7/11

172 T H E O R Y

OF

S E D I M E N T A T I O N

move faster upwards than those in the less dense regions above, and overtake

them.

This means that the concentration gradient increases until a first-order

discontinuity is formed.

If

V

decreases with p the reverse takes place, the con-

centration gradient decreases and any discontinuity is dispersed.

The construction

of

concentration-line diagrams for a few selected problems

is sufficient to show that these arguments can

be

made quantitative. Thus fig.

3a

shows that when

V

increases with

p ,

intersection

of

the lines can be prevented only

by stopping them at a discontinuity. Moreover the diagram shows how this is

gradually built up along the envelope of the concentration lines. This envelope,

Y

FIG.3.-Production and stability of discontinuities.

a)

V ncreases with p : initially p varies continuously.

b) V

increases

with p : p increases suddenly at

A.

c) instability when V decreases with increasing p.

d) physically impossible solution with same initial conditions as

(c).

(e) V

ncreases then decreases with increasing

p.

therefore, is the curve whose equation is needed to determine precisely the con-

centration variations.

If

the initial region of varying concentration is sufficiently small, there is

effectively an initial discontinuity on the x-axis which is propagated along the

line AE (fig.

3b

with a speed determined by eqn.

(17).

In both of these examples

the discontinuity is stable and is fed by the lines running into it.

In contrast

to

these two, both

fig.

3c and

3d

have been drawn to fit the initial

conditions that p =

p1

above A and p

=

p below A with a dispersion in which

V

decreases with increasing concentration. In

(c)

it has been assumed that the

sudden alteration at A is the limit of a very rapid change. No such assumption

has been made in Cd).


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G . J .

K Y N C H 173

Of the two, (c) is physically more reasonable and is the one which should

be

chosen; but both satisfy the initial conditions and are mathematically correct.

The dficulty is that the initial conditions are only sufficient to determine the

solution between the x-axis and the lines at A and B, but they are not sufficient

to determine the solution between the two. This difficulty and the stability

problem associated with it have never been solved mathematically, although

the correct procedure is clear physically.

Finally in fig. 3(e) a diagram has been drawn for a dispersion with the property

that Vat first increases to

a

maximum and then decreases, as the concentration

increases from

p1

to

p2.

The discussion of the previous paragraphs suggests that

the increase of Vnear p1 requires the formation of a discontinuity

AB,

whereas the

subsequent decrease near p2 requires a spray of lines

BAC.

A more careful exam-

ination shows that one discontinuity from p1 to a concentration p3 can be formed,

followed by a continuous increase of concentration to

pz.

Just below AB the

concentration is everywhere p3 so that this line is also the concentration line

a )

cc>

FIG.

4.-Modes of sedimentation distinguished

by S

agajnst p curves.

through A for this density. This condition, expressed in the following equation,

determines the value of p3 :

-

u = - - - -

1

-

3-

(&

P3

- 1

i.e. on an S against

p

diagram the chord joining the points ( p l , S1) and p3, S3

is a tangent to the curve at the latter point.

6. MODES

OF

SEDIMENTATION.-The examples and calculations of the previous

sections have shown the main settling processes and the construction and use

of lines of constant concentration. It is now possible to compare the modes of

sedimentation of

dilute and concentrated dispersions initially

of

a constant and

uniform concentrationp1. The modes depend entirely on the form of the S against

p curve.

The simplest S against p curve (fig. 4 4) is everywhere concave downwards,

i.e. it has a maximum and

no

point of inflexion. According to the discussion of

4, the concentration during the sedimentation process changes suddenly to p m

whatever the value of p l , as shown in the diagram by the lineP1

N.

A concentrated

layer

of

Concentration

p n ,

is built up on the bottom of the contain.=r.

In fig. 4(b) a point

of

inflexion has been introduced in the curve at C after the

maximum, and the curve is made to touch the axis at

N.

Provided

p1 < p c

a

tangent can

be

drawn to the curve from the point PI, touching the curve at Pz.

6 *


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174 T H E O R Y OF S E D I M E N T A T I O N

The situation is that discussed at the end of

4.

The bottom of the dispersion

settles discontinuously into a layer rising from the bottom with a speed given by

the slope of the chord PlP2. Immediately below this layer the concentration is

p 2

and it increases continuously to

p m

on the bottom. As

v =

0 at p

=

pnt this

final settling is relatively

slow.

However, if p1 > p c there is no discontinuity in

concentration and

a

continuous settling takes place.

In fig. 4 c) there is still one point of inflexion but the curves come to the

point N at an angle so that V m, the final layer rate, is not zero, as in b). The

tangent at N meets the curve at

T

where the density is PT. There are now three

possible modes :

(a)

p1

< pT: sudden increase of concentration to p m ;

(6) p e

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