Suspensions Processing Guide - Basic Principles & Test Methods

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Process Engineering Guide:
GBHE SPG PEG 302

Suspensions Processing Guide

Basic Principles & Test Methods
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Process Engineering Guide:

Basic Principles &
Test Methods

CONTENTS

0

GENERAL CONSIDERATIONS

1

DEFINITION OF THE COMPRESSIVE STRENGTH OF THE SOLID
PHASE – THE COMPRESSIVE YIELD STRESS

2

MEASUREMENT OF THE COMPRESSIVE YIELD STRESS

3

AN ALTERNATIVE APPROACH: MEASUREMENT OF THE NETWORK
MODULUS

4

SUMMARY OF METHODS OF ASSESSING THE COMPRESSIVE
STRENGTH OF SEDIMENTS AND CAKES

5

THE USE OF YIELD STRESS / MODULUS MEASUREMENTS –
OPERATIONS WITH LONG CONTACT TIMES

6

OPERATIONS WITH SHORT CONTACT TIMES

7

PREDICTION OF THE AMOUNT OF THICKENING FOR

8

PNEUMATIC DEWATERING


9

DETAILED THEORY OFBATCH CENTRIFUGATION / THICKENING

10

REFERENCES


0

GENERAL CONSIDERATIONS

In all of the operations and methods dealt with in this SPG, Suspensions
Processing Guide, a degree of solid-liquid separation is affected by expelling
liquid from a suspension of particles by mechanical means. The driving force is a
pressure gradient developed in the fluid, often called the capillary pressure
gradient. In general this may either be developed directly by the application of
positive or negative pressure to a cake, as in pressure or vacuum filtration, or
may result from the application of a body force to the particles (that is, a
gravitational, centrifugal or electrical force) which cause them to migrate relative
to the fluid. The forces opposing or retarding the concentration or consolidation of
the solid phase comprise the viscous drag associated with the outflow of liquid
from the particulate mass, and the direct resistance of the matrix of particles
forming the cake or sediment to densification or compression. The latter, the
resistance of the solid phase to compression, is very often, what limits the
ultimate degree of separation that can be achieved by mechanical means and
thus the level of solids content that can be obtained (prior, that is, to any blowing
or drying that may take place in order to displace trapped capillary moisture).
This is particularly the case when the particles are very small (< 5 µm) and/or
highly anisometric since cakes and sediments formed from fine particles and
anisometric particles can show considerable resistance to compression at
remarkably low solids contents. For this reason the compressive strength of the
solid matrix is of prime importance in fine particle separation and a good deal can
be said about the likely separation behavior of fine particulate materials from a
knowledge of this alone, as will be seen throughout this chapter.
The ideal way to approach the design, selection and scale-up of any solid/liquid
separation process would arguably be from a detailed kinetic model of the
process into which could be fed details of the appropriate physical properties of
the material of interest. Given such a model it would be possible in principle to
make predictions of the outcome of an operation and to develop detailed scaling
rules. Unfortunately a rigorous, quantitative approach of this type does not yet
exist and so empirical methods tend to be used, as do small and medium scale
trials. One of the main difficulties which has Inhibited progress in this area has
been the lack of a well-defined and accurate way of characterizing and
describing the properties of the solid matrix, and in particular its compressive
resistance. A method of doing this has, however, recently been developed within
GBHE [1, 2]. This new approach ascribes a compressive yield stress to the solid
phase and describes its behavior in terms of this. The yield stress approach has
several points in its favor:


(I)

it appears to describe the behavior of real cakes and sediments.

(ii)

It allows suspensions to be characterized experimentally in several
equivalent ways.

(iii)

It points to some useful scaling rules and In some cases provides
fundamental support for existing empirical rules.

(iv)

It is the starting point and key to the development of a manageable but
realistic kinetic theory of separation.

Progress on the latter front Is comparatively recent and the quantitative theory
has to date only been applied to a few very simple cases (e.g. batch
centrifugation). The yield stress approach has, however, been used to good
effect at a semi-empirical level and the use of yield stress measurements,
together with plausible semi-empirical scaling rules, underpins much of this
chapter in the first edition of the manual. This section is concerned with test
methods and principles. More detailed examples of how problems are
approached are given In the subsequent sections on specific operations.
1 DEFINITION OF THE COMPRESSIVE STRENGTH OF THE SOLID
PHASE – THE COMPRESSIVE YIELD STRESS
A simple method of describing the compressive resistance of the solid phase has
been developed and refined by several of the authors. The method was initially
developed for flocculated, coagulated or otherwise “structured” cakes but this is
not a restriction. The method Is based on the observation that if a structured
suspension is either filtered in a pressure-filter or allowed to sediment in a batch
centrifuge, the application of a particular pressure or acceleration leads (given
enough time) to a definite and limited contraction in cake volume (cf Figure 1a)
rather than a continued, inexorable consolidation (cf Figure lb). In other words,
equilibrium tends to be established. The establishment of a definite and limited
filtration or sedimentation equilibrium can be accounted for by supposing that the
cake or sediment has a yield stress in respect of compression that is a function of
and increases with solids concentration. Thus:
(i)

Consolidation only occurs if the compressive stress p acting on an
element of cake or sediment of solids volume fraction Ø satisfies p >
Py(Ø), where Py(Ø) is the yield stress in compression;


(ii)

Consolidation only proceeds until a new concentration Øz is reached
where now Py(Ø) = p and equilibrium Is established.

The simple idea of a concentration-dependent critical stress accounts for the
observed behavior of cakes and sediments of all types in the authors’
experience, and a good deal about the processing behavior of a suspended
material can be deduced from a knowledge of Py(Ø) or some equivalent
parameter as will be seen.

2

MEASUREMENT OF THE COMPRESSIVE YIELD STRESS

The yield stress Py and its dependence on solids content can be measured by
one of two methods, by filtration in a small-scale pressure filtration cell or by
centrifugation in a laboratory preparative centrifuge.
The pressure-filtration method is more direct since Py(Ø) is then simply the curve
of applied pressure against equilibrium solids content obtained by measuring the
latter at a series of applied pressures. The method does however suffer from a
number of disadvantages, including:
(i)

Filtration can stop as a result of blockage of the medium rather than full
consolidation of the cake.

(ii)

Leaks are always a possibility.

(iii)

It can be difficult to obtained a uniform cake in a small scale filter.

(iv)

Only one sample can be handled at a time

(v)

It is difficult to cover much more than about one decade of pressure
without undue complication.

The authors therefore prefer a centrifuge method which employs a standard
laboratory preparative centrifuge fitted with a swing-out rotor. The method has
the disadvantage that some analysis is required to obtain Py(Ø) from the raw
experimental data. Additionally It is sometimes problematical for certain biological
suspensions (see 3.81, where the suspended particles are close to neutrally
buoyant. In general, however, there are several advantages to the centrifuge
determination of Py(Ø), notably:


(i)

There is no possibility of leakage or blockage of a filter medium.

(ii)

Centrifuges capable of generating ~ 10 to 105 g are widely available; this
allows a wide range of methods and equipment to be simulated since a
wide range of stress can be applied.

(iii)

Several samples can be run at once, making the method suitable for the
evaluation of potential flocculants and the like.

The centrifuge method involves spinning samples at a range of speeds and
noting the equilibrium sediment height H at each speed. The time taken to reach
equilibrium can vary considerably (obviously it tends to decrease rapidly with
increasing centrifuge speed) but provided the acceleration is well in excess of 1 g
it is probably sufficient to spin for ~ 15 minutes at each speed for most
applications.
The condition for equilibrium at any point x in the sediment (cf Figure 2) is (1):

Equations (1) and (2) needs to be inverted in order to obtain Py(Ø) from the
experimental curve of H against ῳ. This can only be done exactly by an iterative
numerical method E21, there are, however, analytical approximations which are
sufficiently accurate to obviate the need for a precise analysis. A very good
approximation is [2]:

This method requires the slope of either a curve of H versus ῳ, or H versus log ῳ
to be taken so as to obtain S. This is often most easily done from a curve of H
versus loge ῳ since the slope (= 2S) is often approximately constant, or at least,
only slowly varying with log o. It has been shown [2] that this method can
generally be expected to be good to within 10%. This may sound a substantial
error, however the concentration-dependence of Py(Ø) is normally so strong that
a 10% error in Py(Ø) is equivalent to an insignificant variation in solids content.
An even simpler approximation [1] which is less accurate but avoids the need to
take slopes is:

This is not as accurate as the first method but is probably adequate for most
purposes.


3

AN ALTERNATIVE APPROACH: MEASUREMENT OF THE NETWORK
MODULUS

A quantity related to the compression yield stress Py(Ø) is the "network modulus"
defined by [1]

K(Ø) contains no more information than Py(Ø) since it is simply the rate of
change of Py(Ø) with respect to log (concentration). The network modulus is
however useful since it leads to an alternative method of estimating the
compressive strength of suspended materials in the laboratory.
It has been observed [1,3, 4] that the network modulus appears to be very similar
in magnitude, and identical in concentration-dependence, to the instantaneous
shear modulus of concentrated suspensions, G(Ø). This relationship is useful
because Gȸ(Ø) is easily measured. Shear modulus measurements thus give an
alternative way of estimating the compressive strength of cakes and sediments
via the implied relationship

Gȸ(Ø) can be readily measured using a commercial instrument known as the
Pulse Shearometer (Rank Brothers, Bottlsham, Cambridge). This device allows
the velocity of a small-amplitude, shear-wave propagating through the sample to
be determined. The shear modulus (or a good approximation to It) and the wavevelocity u are related by

where p is the overall density of the suspension. The main advantage of the
modulus method is that the measurements can be performed fairly rapidly, the
method Is thus useful for biological materials and other materials which are likely
to degrade or decay. Alternatively time dependent effects can be studied
systematically.


4

SUMMARY OF METHODS OF ASSESSING THE COMPRESSIVE
STRENGTH OF SEDIMENTS AND CAKES

The compressive yield stress can be estimated by one of three methods:
(i)

Pressure filtration, in which case a curve of yield stress against solids
content is given directly by the curve of applied pressure versus
equilibrium solids content.

(ii)

Centrifugation in a closed-tube laboratory centrifuge, in which case Py(Ø)
is obtained from a curve of equilibrium sediment height versus centrifugal
acceleration by means of an approximate analysis (e.g. equations (3)-(6)
or (7)-(9)).

(iii)

Measurement of the shear modulus as a function of solids content, in
which case Py can be estimated using equation (11)

Data for various materials illustrating the equivalence of the various methods is
shown in Figures 3 and 4.

5

OPERATIONS WITH LONG CONTACT TIMES

It is evident that the yield stress, concentration curve Py(Ø) Is equivalent to a
curve of applied pressure versus equilibrium solids content. Hence the solids
content that should be obtained in a mechanical separation operation (prior to
any air blowing that may take place) can be predicted, provided that the
compressive stresses operating on the material can be estimated. This is true
provided that:
(i)

There is time enough for equilibrium to be established.

(ii)

There are no spurious limits to performance such as blockage of the filter
medium.

Solids-content predictions are thus straightforward for operations with long
contact times provided that there are no complicating factors. A good example of
an operation with a long contact time is batch filtration. Even where the contact
time is too short to allow equilibration, it is clear that the solids content predicted
b y assuming that equilibrium is established is a useful benchmark since it
represents an absolute limit imposed by the material itself.

This limit can only be improved upon by increasing the stresses exerted on the
material, If the actual separation obtained is much lower than that suggested by
the prediction the implication is either:
(i)

that the separation is kinetically rather than compressibility (or
“structurally” )-limited; or

(ii)

that the separation is failing for some ancillary reason.

Predictions of the theoretical ultimate solids content (TUSC) thus have several
uses.
The following is a simple way of obtaining the TUSC for an operation; more
complicated and refined analyses are possible but probably not necessary: The
mean compressive stress p’ experienced by the material Is estimated. The
approximate TUSC is then read from a curve of Py(Ø) versus Ø by equating Py
with Ṗ and reading off the corresponding value of Ø. For example in a pressure
or vacuum filter the mean compressive stress Is simply the pressure-drop acting
across the cake.
Example
The method was applied to two grades of "Diakon" latex, Diakon XC32 and
Diakon XC37, Isolated by vacuum filtration. The pressure drop is thus one bar
and the predicted solids content taken to be the TUSC read off the curves of
Py(Ø) versus Ø obtained for each latex at Py = 1 bar. The predicted and actual
solids contents are compared below.

The agreement is very satisfactory. Note that this implies that equilibrium is either
established or very closely approached.


6


With many solid/liquid separation operations, particularly continuous operations
such as continuous centrifugation, the time of residence of the material in the
stress-field is often not long enough for consolidation to proceed to equilibrium.
The outcome of such operations might be said to be "kinetically-limited" rather
than "structurally-limited" and the rate of consolidation now needs to be known if
predictions about performance are to be made,
The yield stress approach has been developed into a kinetic model by White and
Buscall [2, 5]. The key assumption in the model is that the rate of consolidation of
a particulate mass of solids content Ø subjected to a normal stress p > Py(Ø) is
proportional to the excess stress p - Py(Ø) and not the total stress. The
consequences of this assumption and the scaling rules it leads to will now be
explored taking centrifugation by way of example.
The compressive stress developed in a uniform layer of thickness H and solids
concentration Ø subjected to a uniform acceleration g varies linearly across the
layer from zero on one side to a maximum value of Pmax = Δp g Ø H at the other
(cf Figure 5a). Thus for a material with a yield stress Py(Ø) there is a
corresponding level of acceleration which has to be exceeded before any
consolidation can occur (more generally a critical level of (gH) since H may also
be a variable). For a given layer thickness the critical level of acceleration is
given by:

in this simple case. If the acceleration exceeds g, by a small amount then
consolidation will occur in a thin layer at the outer edge where the stress exceeds
the yield stress. Increasing the acceleration further then has two effects, the zone
in which the stress exceeds Py(Ø) becomes thicker so that an increased fraction
of the layer actually consolidates, and the rate of consolidation at any point inside
this zone increases as a result of an Increase In the excess stress p - Py(Ø), The
effect of acceleration on the overall rate of consolidation, that is the rate of
change of mean concentration in the layer with time, thus looks like the
schematic curve shown In Figure 5b. Notice that for large enough accelerations
(g >> gc) the rate of increase of overall rate with g becomes nearly linear as the
thickness of the consolidating zone reaches a limiting plateau value. Again in
more general terms the compressive stress can be varied by varying both g and
H; the effect of these is, however, scaled onto a single curve by plotting mean
rate/acceleration against gH.

(Note that this provides a connection between centrifugation and gravity
thickening.) From Figure 5b it Is clear that this curve has the shape shown In
Figure 6. It is useful to divide this curve into two regimes, a regime where the
mean rate of consolidation depends upon g or gH in a complex way (regime I in
the diagram) and a regime (II) where the mean rate of consolidation is to all
intents and purposes linear in g. In the terminology used at the start of this
subsection, regime I is then the structurally-limited regime (hereafter denoted SL) and regime II is the kinetically-limited regime (K-L). The point of doing this is
that simple scaling rules can be constructed when the rate is more or less
proportional to acceleration, as It Is approximately in the kinetically-limited
regime:- doubling the acceleration should have the same effect on the degree of
thickening in a centrifuge operating in the kinetically-limited regime as doubling
the time the material is exposed to the centrifugal field. Thus defining the amount
of thickening as

It is clearly important to be able to identify within which regime a centrifuge is
running. Strictly speaking, there are three, not two, possibilities:


It is clearly useful to be able to tighten up on what is meant by ‘small” and “large”
and the kinetic yield stress model provides some guidance to this. Calculations
made for batch centrifugation (see Section 3) show that when the curve of meanrate/g versus gH is normalized by plotting:

The resulting curve does not depend upon the details of the suspended material.
The actual curve obtained is plotted in Figure 7. Also shown are some
preliminary data for Attapulgite clay suspensions; these support the theoretical
curve rather well.
It is clear from the curve that the rate only becomes linear in g asymptotically and
so a somewhat arbitrary decision has to be made as to where the K-L regime in
effect starts. The quantity plotted on the right-hand side is the actual rate
compared with the rate that would be obtained were the material to have no yield
stress. A value of, say, 0.8 might seem a reasonable definition of the effective
crossover between S-L and K-L behavior. From the curve this occurs at g/gc = z
~ 5 and so the latter is a possible criterion for the crossover. Note also that this
value is in line with intuition, one might naively expect the rate to reach 80% of its
maximum possible value at g/gc = z ~5.
With many types of centrifuge it is possible to vary speed and throughput, and
thus acceleration and residence time, independently over a limited range. There
is thus often a choice as to which regime the machine is operated in. Given this it
is clear that operation too far inside the kinetic regime, i.e. g/gc ,(Øoutput ) >> ~ 5
may mean that volumetric throughput is being obtained at the expense of
separation, whereas If the operation is In the structural regime In respect of
output, i.e. g/gc ,(Øoutput ) < 5, the opposite may be true. It can thus be argued
that continuous centrifuges are most efficiently run just outside of the structural
regime.

7

PREDICTION OF THE AMOUNT OF THICKENING FOR OPERATIONS WITH
SHORT CONTACT TIMES

From what has been said so far it is clear that precise predictions of the output
solids content can be made when it is safe to assume that equilibrium will be
reached. The solids content obtained Is then the TUSC calculated from Py (Ø).
Predictions can also be made for a centrifuge (or other continuous processor)
running in the kinetically limited regime using the scaling rule:

In this case a small-scale trial or batch centrifuge test is required in order to
establish the constant (see Section 3.4 on centrifuges for details of how this is
done). The general case is more difficult and a precise way of making predictions
has yet to be developed. (Note that the kinetic yield stress model provides a
promising starting point. 1 reasonably good “guesses” can however be made.
This is so because :
(i)

The solids content obtained cannot be greater than the TUSC.

(ii)

Py(Ø) and thus gc (Ø) normally increases rather rapidly with 8.

Given point (ii) it follows that if the predicted solids content is taken to be either
the TUSC, or that obtained using the scaling rule for kinetically-limited
performance, whichever the less, then a reasonably good estimate will inevitably
be obtained. Further, given that the steeper the curve of Py(Ø), the better this
estimate should be, with a little experience it should be possible to improve or
assess the quality of the guess in the light of the shape of the Py(Ø) curve.
Similar considerations apply to gravity-thickening (considered in more detail in 3).
It is now useful to speak of a critical layer thickness or height rather than a critical
acceleration. Clearly this is defined as

where now go is the acceleration due to normal gravity. The interest now is
making predictions from small-scale tests. This is straightforward in the
kinetically-limited regime, that is when both the small-scale and large-scale are
run under conditions where H/H crit (output) >> 1;


this however is not usually the case. More generally, the approximate rule - the
ultimate solids content will either be close to the TUSC or that predicted
assuming kinetic limitation, whichever the less - can be used. The kineticallylimited prediction is now made as follows:
Given that on the large scale H = HL and the residence time is t R, this prediction
is made by spinning a sample of depth Hs in a batch centrifuge (e.g. the Triton
stroboscopic centrifuge ) at a speed corresponding to an acceleration of g = go
HL/Hs for a time of t = tR/g and determining the mean concentration in the
sediment. The TUSC can be determined In the same experiment since this Is
simply the equilibrium sediment concentration obtained as t  ȸ.
The basis for this method of obtaining the kinetically-limited degree of thickening
should be fairly clear from Figure 7. Increasing the acceleration from g0 to g = go
HL/Hs mimics the normal stress obtained on a large scale and thus scales up in
one sense. At the same time, increasing g increases the pressure gradients
driving liquid through the pores and so the timescale has to be cut back
accordingly.
It is anticipated that gravity thickening will be one of the first operations to be
analyzed in detail using the kinetic yield stress model. The object of this will
however not be to provide complicated numerical routines for use in scaling and
prediction, but rather to define more precisely the validity of intuitively sensible
predictive methods such as those given above. The results of such calculations
will be Issued as and when.


8

PNEUMATIC DEWATERING

In certain types of filter and centrifuge there is the possibility that moisture
contained in the fully consolidated cake will be partially replaced by air. In certain
instances this is encouraged by prolonged blowing. In order for any displacement
to occur the air pressure drop in a filter or the difference between the normal
stress acting on the liquid in centrifuge and atmospheric pressure has to exceed
the capillary pressure ΔP. The latter is given by [6]

If the contact angle is not known then it is best assumed to be 0' so as to obtain
an upper bound to ΔP. ΔP is of order 0.1 bar for 1 µm size particles when the
liquid is water (Ɣ = 0.07 Nm-1).
Predictions of dewatering behavior tend to be difficult since slight inhomgeneities
in the cake can cause preferential breakthrough which in the filtration case can
lead to a loss of pressure drop. Further discussion of these features is provided
in Sections 3.


9

DETAILED THEORY OFBATCH CENTRIFUGATION / THICKENING

Numerical results (Figure 7) obtained by applying the yield stress theory to batch
centrifugation were used in Section 3 in order to identify the crossover between
kinetically-limited and structurally limited behavior. The theory is outlined here.
More detail can be found In references 2 and 5.
The geometry Is shown in Figure 2. The sediment is imagined to have an Initial
depth Ho and Initial uniform solids content lo. A uniform centrifugal field g is
acting down the bed in the x-direction.
A force balance on a volume element of the bed gives as the total force acting on
unit volume

where U and V are the particle and liquid velocities in the bed; R(Ø) is the
viscous drag coefficient for flow through the bed; p is the particle stress or normal
stress in the matrix; “, Is a unit vector pointing down the bed. The first term is
thus the viscous drag associated with the movement of liquid, the second the
gradient of the elastic stress in the network and the third the centrifugal driving
force.
A mass balance on the bed gives

which because in this case the bed Is closed at the bottom and there Is no net
flux, reduces to,

Note that equations (22) and (25) are just continuity equations, the only
assumption to have been made Is that the viscous drag is proportional to the
relative flow rate.

This would not be true, for example, if the liquid were highly non-Newtonian.
Equations (22), (24) and (25) are closed using the yield stress model which has
the local rate of consolidation related to the normal stress p by

Here λ(Ø) is a dynamic coefficient which represents the drag involved in
squeezing water out of the consolidating network. An approximate form for this
function is considered shortly.
Equations (22) and (26) combine to form a complicated second-order partial
differential equation which has yet to be solved in any general way. Initial rates
are however easier to get as the equation reduces to a pair of coupled first-order
differential equations for t = 0. These are easily solved and the initial rate of
boundary sedimentation is given by the coupled pair of algebraic equations:

and xe Is the point In the bed where p = Py(Ø), that is, below xe consolidation
occurs, above xe the bed simply falls. Equations (27) and (28) are solved by
eliminating xe. In order to do this It would appear at first sight that λ(Ø) and R(Ø)
and thus A need to be known rather well but this is not the case. Analysis of
these equations shows that the situation Is Independent of A for AH0 > 1. Order
of magnitude estimates of λ and R show that AHO is in fact very large for any
case of practical Interest, the length A-' being of the order of the particle size.
Given this the solution to equations (27), (28) can be written as:


and it is the function F that Is shown In Figure 7. R(Ø) can be obtained from the
Kozeny-Carman equation or some other empirical equation for the permeability
of a bed or swarm of particles. For example, an equation favored by one of the
authors gives

where ŋ s = liquid viscosity; á = mean particle-size; λ st = Stokes' drag factor < = 6
π for spherical particles); Ṽρ = mean volume per particle and β = 5. This works
well for deflocculated particles; It Is not yet clear, however, how sensitive R(Ø) is
to the precise structure of the solid phase and so In general R(Ø) from equation
(30) Is an order of magnitude estimate,


An estimate for λ(Ø) is


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