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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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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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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