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 THE USE OF YIELD STRESS / MODULUS MEASUREMENTS –

OPERATIONS WITH SHORT CONTACT TIMES

7 PREDICTION OF THE AMOUNT OF THICKENING FOR

OPERATIONS WITH SHORT CONTACT TIMES 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 wave-velocity 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 THE USE OF YIELD STRESS / MODULUS MEASUREMENTS –

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

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 S-L) 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 mean-rate/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 kinetically-limited 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

















