The Simulation of Polymer Aggregation/Breakage at High Solid
Fraction in Turbulent Flow by Population Balance
A dissertation submitted to
Murdoch University
Department of Engineering
In partial fulfilment of the
Requirements for the degree of
DOCTOR OF PHILOSOPHY
Alex Heath
2002
Supervisors
Dr Parisa Bahri
Dr Phillip Fawell
1
I declare that this thesis is my own account of my research and contains as its main
content work which has not previously been submitted for a degree at any tertiary
education institution.
………………………………….
Alex Heath
2
Acknowledgements
I would like to thank the following people for their contributions towards this Ph.D.
project:
Dr John Farrow - Identifying and setting up the project, overall direction
Dr Parisa Bahri - Primary supervisor, help with maths, coding,
engineering, and general support
Dr Phillip Fawell - Co-supervisor, chemistry, references, thesis preparation
Mrs Jean Swift - Laboratory & pilot plant trials, flocculation and settling
Mrs Evalyn Beaumont - Library
Dr John Rumball - General support, reading thesis
Dr Peter Koh - CFD, end user
3
Abstract A mathematical model has been developed to describe the size and settling rate of
aggregates formed by flocculation in thickeners/clarifiers used in the mineral processing
industry. The aggregation process was simulated with a population balance model,
which calculates the aggregate size distribution through time as a function of the
competing processes of aggregation and breakage. The population balance was written
specifically to describe aggregation by high molecular weight polymer flocculants that
have now largely replaced coagulant salts in mineral processing operations, and the
model includes terms to describe flocculant/slurry mixing and irreversible aggregate
breakage (Table 1). The model was written to form part of a full computational fluid
dynamics (CFD) model of a thickener/clarifier, allowing the eventual simulation and
optimisation of the full-scale unit.
In addition to the effects of fluid shear and residence time normally described by
population balance models, additional process variables have been considered, with the
model correctly accounting for changes in the flocculant dosage, primary particle size
and solid fraction. The model has also been extended to describe the hindered settling
rate under the same range of conditions, considerably increasing its usefulness by
forming a link between the aggregation kinetics in the feedwell and the subsequent
setting rate as the aggregates enter the settling zone of the thickener.
The model was fitted to experimental data from a turbulent pipe reactor. A variety of
pipe sizes, lengths, and flow rates were used to give a range of fluid shear rates and
mean residence times, with the aggregate size distribution measured by an on-line
sizing probe placed at the end of the pipe, immediately before a settling column.
Experimental data was collected under a sparse matrix of experimental conditions, with
the fluid shear rate (G), flocculant dosage (θf), primary particle size (dp) and solid
fraction (φ) varied independently away from a common baseline. Additional data was
collected from conditions in the gaps of the experimental matrix, and was used to check
the predictivity of the model.
The population balance model is based on the discretised balance by Hounslow et al.
(1988) and Spicer and Pratsinis (1996), in this case using 35 channels covering the size
4
range: 0.2-3500 μm. The aggregation rate is described by Saffman and Turner’s (1956)
turbulent collision kernel, used in conjunction with a capture efficiency term. The
capture efficiency is initially taken to be zero (no successful collisions) before
flocculant addition, but increases to unity (all collisions successful) as described by a
flocculant-suspension mixing term.
The breakage rate is described as a function of the aggregate size, energy dissipation
rate, suspension viscosity and effective flocculant surface coverage. The effective
flocculant coverage is taken to decrease through time, reflecting the loss of flocculant
activity caused by repeated aggregation/breakage. Aggregate porosity is also included,
using fractal geometry, and is used to calculate the aggregate capture radius and
effective suspension solid volume fraction. The solid volume fraction is used to
determine the suspension viscosity, accounting for experimental data showing the
pressure drop in the pipe is a function of the aggregate size.
The model equations were solved numerically as an initial value problem with a
commercial dynamic simulation package (gPROMS), which was also used to estimate
the unknown model parameters. The model was found to be robust and stable, and gave
good predictions of the additional experimental data sets.
The extension of the model to also describe the hindered settling rate allowed a dynamic
optimisation to give the maximum settling flux, and hence unit throughput. Various
optimisation strategies were investigated, in particular the use of recycle to find the
optimal feed solid fraction. The inclusion of the model within a full CFD simulation
will allow further optimisation strategies to be investigated, in particular changes to the
feedwell geometry to give good mixing but without subjecting fully formed aggregates
to regions of destructive high fluid shear.
5
Table 1: Model Summary Process Equation Comment
D/Lfk1e1M −−=
where: (Equation 5-6)
M = Mixing index [0,1], used to estimate α k1 = Fitted parameter # 1 (0.343) f = Pipe friction factor L = Pipe length (m) D = Pipe diameter (m)
Simplified macro-scale mixing equation for turbulent pipe flow by Etchells & Short (1988), Godfrey & Amirtharajah (1991). Assumes flocculant/particle collision and adsorption are rapid. Likely to be highly scale-up dependent, requires further work.
( )3jiij aa294.1 +ν∈
α=β where: (Equation 2-7)
βij = Collision kernel) (m3s-1) ∈ = Energy dissipation rate (m2s-3) μ = Fluid viscosity (N s m-2) ai = Radius of the ith particle (m) α = Capture efficiency [0,1] (taken as = M)
Saffman & Turner’s (1956) collision kernel. ∈ and ν determined experimentally (Equations 4-2, 4-3, 4-7, Table 4-1) and predicted in the model (Equations 5-8, 5-16). Particle radius incorporates aggregate porosity (Equation 5-23), with the aggregate size given by an on-line, in-pipe sizing probe (FBRM).
dK
Sf
i,aggk
2i
3
θ
μ∈=
where: (Equation 5-19)
Si = Breakage rate of the ith sized particle (s-1) θf = Flocculant surface coverage (kg m-2) k2 = Fitted parameter # 2 := 38.1 k3 = Fitted parameter # 3 := 0.677 M = Mixing index [0,1] mf = Mass of flocculant (kg) dagg,i = Diameter of the ith particle (m)
Breakage kernel, similar to others in the literature (Table 2-6) incorporates flocculant dosage term (Equations 5-18, 5-20), i.e. more flocculant produces stronger aggregates and reduced breakage. Partially irreversible breakage introduced via reduced effective flocculant coverage (Equation 5-20), containing parameter k4.
∑∑
∑∑∞
=
∞
=
−
=−−−
−
=−−
+−
Γ+−β−
β−β+β= −
ijjjj,iii
ijjj,ii
1i
1jjj,ii
21i1i,1i
2i
1j21
j1ij,1i1iji
NSNSNN
N2NNNN2dt
dN ij
where: (Equation 2-45)
Ni = Number of ith sized particles (m-3) Γij = Daughter fragment distribution function
Population balance for aggregation and breakage, after Hounslow et al. (1988), Spicer & Pratsinis (1996) (see Appendix). Solved numerically with gPROMS on SUN mainframe.
( ) 65.4D3
p
aggs
3D
p
aggls
2agg
h
f
f
d
d1
18
d
dgd
U⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛φ−
μ
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ρ−ρ
=
−
−
where: (Equation 6-10)
Uh = Settling velocity (m s-1) d = Diameter (p = primary particle, agg = aggregate) (m) g = Gravity (9.8 m s-2) ρ = Density (s = solid, l = liquid) (kg m-3) μ = Fluid viscosity (N s m-2)
Hindered settling velocity based on Richardson & Zaki’s (1955) extension of Stokes’ (1851) law (Equations 6-1, 6-2). Aggregate porosity introduced using fractal geometry (Equations 6-8, 6-9). Equation used to estimate fractal dimension (Df = 2.4) from experimental data (Equation 6-11, Figure 6-6).
Dilute flocculant stream
Particle Suspension
Pipe flow →
Flocculant/particle mixing & adsorption
Particle or aggregate collision & aggregation
+
Aggregate breakage
Initial hindered settling velocity
Simulation/computing
6
List of terms Aggregate Generic term describing an assemblage of smaller particles; may be
formed by the addition of a coagulant or flocculant. Capture efficiency (α) Ratio of the number of collisions resulting in stable aggregates to
the total number of particle collisions. Coagulant Soluble salt, usually a multivalent cation salt like alum
(Al2(SO4)3.16H2O), which reduces the surface charge resulting in aggregation (specifically: coagulation).
Coalescence Analogous to aggregation, except with liquid droplets. Sometimes
used to approximate aggregation due to the simplifications of the spherical shape and no porosity.
Compression Compaction or de-watering of sediment in thickener bed region.
Occurs when the height of the bed builds up to the point where the overburden weight exceeds the compressive yield stress, resulting in the restructuring of the network and the forcing of displaced fluid back up through the structure.
Fractal Dimension (Df) Exponent (typically 2-2.5) of power law relationship between
aggregate size and mass. Used to describe aggregate porosity by assuming the aggregate volume increases as the cube of the diameter.
Flocculant High molecular weight (∼ 15-20 × 106 g mol-1) polymer with
charged or polar functional groups to make it water-soluble and to attract it to the particle surface. Results in aggregation (specifically: flocculation) through a combination of surface charge neutralisation and polymer bridging.
Kernel Size dependent rate equation describing the kinetics of aggregation
or breakage. Solved within the population balance. Permeability Ability of fluid to pass through porous structure. Body may be
porous, but impermeable, i.e. if the pores do not join up. Porosity Void space (ε) in aggregate or sediment network due to the packing
efficiency of the primary particles. Population balance Dynamic model describing the change in the aggregate size
distribution through time. Usually solved in discrete form, where the continuous particle size distribution is sectioned, resulting in a closed set of ordinary differential equations that can be solved numerically as an initial value problem.
Sedimentation As per settling, but usually describing the network settling in the
hindered or compression regimes. Settling Generic term describing effect of gravity to drag particles down
through fluid of lower density.
7
Contents
1. INTRODUCTION 9 1.1 Aggregation of suspended particles 9 1.2 Solid/liquid separation 10 1.3 Modelling aggregation kinetics by population balance 11 1.4 Aggregate settling 12 1.5 Project aim 13 1.6 Thesis overview 13
2. LITERATURE REVIEW 15 2.1 Introduction 15 2.2 Aggregation 16
2.2.1 Particle collision mechanisms 16 2.2.2 Turbulent collision, particles smaller than micro-scale 18 2.2.3 Turbulent collision, particles larger than micro-scale 21 2.2.4 Summary - particle collision 23 2.2.5 Effect of short range inter-particle forces/bonding mechanisms 26 2.2.6 Coagulation with salt 28 2.2.7 Flocculation with polymer 28 2.2.8 Loss of flocculant activity by polymer degradation 31
2.3 Aggregate porosity 32 2.4 Fluid behaviour 38
2.4.1 Laminar flow, shear stress (τ), viscosity (μ), and fluid shear (γ), 38 2.4.2 Turbulent flows, G, Kolmogoroff micro-scale, eddy spectrum 38 2.4.3 Deviations from ideal Newtonian behaviour 40 2.4.4 Effect of suspended solid on viscosity 42 2.4.5 Multi-phase mixing 44
2.5 Aggregate breakage 46 2.5.1 Cause of aggregate breakage 46 2.5.2 Effect of aggregate size on breakage 49 2.5.3 Daughter particle distribution 51
2.6 Population balance models 54 2.6.1 Smoluchowski’s equation, aggregate breakage 54 2.6.2 Discrete models, numerical solution 55
2.7 Solid-liquid separation by gravity settling 59 2.7.1 Particulate settling (low solid fraction) 61 2.7.2 Hindered settling (intermediate solid fraction) 68 2.7.3 Compression (high solid fraction) 77
2.8 Conclusions to literature review 81
3. ON-LINE PARTICLE SIZING 83 3.1 Particle sizing techniques 83 3.2 Principle of FBRM 85 3.3 Estimating particle size from FBRM data 86
3.3.1 Theoretical methods 86 3.3.2 Empirical methods 88
3.4 Experimental 90 3.4.1 Equipment 90 3.4.2 Samples 91
3.5 Results and discussion 91 3.5.1 Effect of applying weightings to FBRM chord distributions 91 3.5.2 Comparison of average particle size by various techniques 93 3.5.3 Effect of solid volume fraction on indicated size 94 3.5.4 Comparison between C and F-electronics 97 3.5.5 Effect of particle material 98 3.5.6 Minor effects 101 3.5.7 Effect of solid fraction on particle counts 104
3.6 Conclusions 106
8
4. CALCITE FLOCCULATION IN TURBULENT PIPE FLOW 107 4.1 Experimental 109 4.2 Results and discussion 114
4.2.1 Effect of mean fluid shear rate 114 4.2.2 Effect of flocculant dosage 122 4.2.3 Effect of primary particle size 124 4.2.4 Effect of suspension solid fraction 125
4.3 Conclusions 127
5. POPULATION BALANCE MODEL 128 5.1 Model development 128
5.1.1 Aggregation 128 5.1.2 Aggregate breakage 133 5.1.3 Simplification for coding/simulation 137
5.2 Comparison with experimental data 140 5.2.1 Effect of flow regime (shear rate) 140 5.2.2 Effect of flocculant dosage 147 5.2.3 Effect of primary particle size 148 5.2.4 Effect of solid fraction 150 5.2.5 Additional runs to check model predictivity 153 5.2.6 Effect of changing fractal dimension 154
5.3 Conclusions 155
6. RELATING HINDERED SETTLING RATE WITH AGGREGATE SIZE 156 6.1 Experimental 158 6.2 Results and discussion 159
6.2.1 Effect of fluid shear 159 6.2.2 Effect of flocculant dosage 161 6.2.3 Effect of suspension solid fraction 162 6.2.4 Effect of primary particle size 164 6.2.5 Fractal geometry 164
6.3 Conclusions 169
7. DYNAMIC OPTIMISATION AND EXTRAPOLATION TO FULL-SCALE 170 7.1 Predicting settling behaviour –Dynamic optimisation 170 7.2 Extrapolation to Industrial scale 174 7.3 Conclusions 176
8. CONCLUSIONS AND FUTURE WORK 177 8.1 Conclusions 177 8.2 Suggested future research 181
9. NOTATION 182
10. REFERENCES 185
11. APPENDIX 209 11.1 Hounslow et al.’s (1988) population balance 209 11.2 Fractal geometry, calculation of dagg and ρagg from dm, dp & Df 216 11.3 Fractal geometry, calculation of effective mean aggregate density 217
9
1. Introduction
1.1 Aggregation of suspended particles Solid-liquid separation in mineral processing and wastewater treatment is frequently
performed by gravity sedimentation (Amirtharajah 1991, Williams 1992, Shamlou
1993, Rushton et al. 1996, Bustos et al. 1999, Svarovsky 2000). Fine particles settle
slowly in viscous flow, and are usually aggregated to increase their settling rate. The
particles typically carry a surface charge that causes electrostatic repulsion between the
particles, preventing aggregation. However, this repulsion can be overcome by the
addition of soluble ions (coagulants), which are attracted to the particle surface where
they neutralise and compress the electrical double layer. The particles are then able to
approach each other closely enough that van der Waals attractive forces can hold the
particles together as stable aggregates (Kohler 1993, Hughes 2000).
However, in mineral processing circuits coagulants have been largely superseded by
high molecular weight (∼ 20 × 106 g mol-1) polymer flocculants (Bagster 1993, Farinato
& Dubin 1999). Flocculants have polar or charged functional groups attached to a
hydrocarbon backbone, making the polymer water-soluble and attracting it to the
mineral surface. In addition to the effect of surface charge neutralisation, the length of
high molecular weight flocculants allows them to span the gap between fine particles,
and polymer bridging (Figure 1-1) is usually assumed to be the dominant bonding
mechanism (Healy 1961, Gregory 1989).
Figure 1-1: Stages (somewhat concurrent) of flocculation
Aggregate breakage
Particle collision & aggregation
+
→ +
→
→
Flocculant/particle mixing & adsorption
10
1.2 Solid/liquid separation For solid/liquid separation on an industrial scale, the concept of a simple settling-pond
is extended to continuous (under) flow by adding a conical base and a rotating rake
assembly (Figure 1-2) (Dahlstrom & Fitch 1985, Bustos et al. 1999). The rake aids
sediment bed compaction as it encourages the solid down towards the underflow pump
at the base (Pearse 1977, Svarovsky 2000). The unit is termed a thickener or clarifier,
depending whether the primary process objective is to increase or decrease the solid
fraction of the feed stream (Williams & Simons 1992, Perry & Green 1997). Several
units can be connected in series, forming a washer train or counter current decantation
(CCD) circuit (Dahlstrom & Emmet 1985).
A clarifier typically receives a feed with a low solid concentration (typically < 1 %
w/v), and may benefit from underflow solid recycle to increase fines capture during
flocculation. Conversely, a thickener is likely to have a relatively high feed solid
concentration (typically 2-50 % w/v) (Pearse 1977, Perry & Green 1997, Dahlstrom &
Fitch 1985) and may benefit from dilution by overflow recycle, either via the overflow
weir or within the feedwell, according to the propriety design (Amirtharajah 1991,
Svarovsky 2000).
Figure 1-2: Cross section of thickener/clarifier.
Dilute flocculant solution (< 1 % w/v) is mixed with the feed suspension, usually in the
feedwell or feed pipe (Dahlstrom & Fitch 1985, Perry & Green 1997). The primary
purpose of the feedwell is to dissipate the momentum of the feed stream, which might
otherwise disrupt the settled sediment. However, the agitation intensity generated in the
feedwell provides good mixing, making it a suitable flocculant addition point.
Flocculant
Overflow weir
Hindered settling region
Free settling (clarification)
Underflow
Feedwell Feed suspension
Sediment compaction Rotating rake
11
Adequate mixing is required, initially to disperse the flocculant evenly through the feed
stream, preventing localised overdosing (Farinato & Dubin 1999, p. 18). Further
mixing is then required to cause flocculant/particle collision and adsorption, followed
by particle/particle collision to form aggregates. However, if the agitation is too intense
the aggregates will break, irreversibly degrading the polymer flocculant by chain
scission or rearrangement (Healy 1961, Williams & Simons 1992, Bagster 1993,
Gregory 1993).
The complex flow behaviour in industrial vessels can now be described by
computational fluid dynamics (CFD) models (Lainé et al. 1999, Ducoste & Clark 1999,
Farrow et al. 2000), which have been developed since the introduction of powerful
computers and efficient numerical routines. However, although many fluid flow
characteristics (kinetic and dissipative energy, fluid shear, multi-phase mixing, recycle,
etc) are readily calculated, CFD cannot currently predict the effect of these parameters
on the kinetics of aggregation or breakage.
1.3 Modelling aggregation kinetics by population balance Aggregation kinetics have been extensively studied on a laboratory scale, and widely
modelled by population balance. The population balance approach was first described
by Smoluchowski (1916, 1917):
∑∑∞
=
−
=+=
β−β=1i
kiik
1k
kji,1ijiij2
1k NNNNdt
dN 1-1
However, current population balance models for flocculation typically also include a
aggregation capture efficiency (α) term, and terms to describing aggregate breakage,
giving (Argaman & Kaufman 1968, Randolph & Larson 1988, Lick & Lick 1988,
Spicer & Pratsinis 1996, Serra & Casamitjana 1998b, Kramer & Clark 1999):
∑∑∑∞
+=
∞
=
−
=+=
Γ+−αβ−αβ=1kl
lllkkk1i
kiik
1k
kji,1ijiij2
1k NSNSNNNNdt
dN 1-2
12
where: Ni = Number of ith sized particles (m-3) t = Time (s) α = Capture efficiency [0,1] βij = Rate of collision between i and j sized particles (aggregation kernel) (m3s-1) Sk = Breakage rate (kernel) of kth sized particles (s-1) Γlk = Breakage distribution function (number of k size particles produced from the breakage of a l sized particle)
Despite the body of work already published in this area, few current population balance
models simultaneously account for changes to the process variables encountered in
practice (fluid shear rate, flocculant dosage, primary particle size, solid fraction). There
is also a need to link the predicted aggregate size back to the primary process objective:
solid-liquid separation.
1.4 Aggregate settling The velocity of an individual spherical particle settling under gravity was described by
Stokes (1851), who assumed creeping flow (Re → 0) and solved the remainder of the
Navier-Stokes equation, giving (Happel & Brenner 1973, Seville et al. 1997):
( )μρ−ρ
=18
gdU ls2
1-3
where: U = Settling velocity (m s-1) d = Particle diameter (m) g = Gravity (9.8 m s-2) ρ = Density (s = solid, l = liquid) (kg m-3)
μ = Fluid viscosity (N s m-2) However, aggregates are a loose assemblage of primary particles, with an irregular
shape and porous structure. These reduce the settling velocity by increasing the
hydrodynamic drag force, and reducing the effective particle density (Clift et al. 1978).
The settling velocity is also reduced as a function of the suspension solid volume
fraction, initially due to inter-particle hindrance, but eventually also from mechanical
support propagating up through the compressing sediment (Richardson & Zaki 1955,
Tory 1996, Bustos et al. 1999).
13
1.5 Project aim The aim of this work was to develop a mathematical description of the kinetics of
aggregation/breakage, providing a suitable sub-model for a CFD model of a thickener
feedwell. This will ultimately allow CFD to explore various feedwell design options
(e.g. flocculant dosing point(s), feed entry points and/or velocity, baffle size/position,
recycle, etc) towards an improved design.
The model needed to describe the various stages of aggregation, from the initial
flocculant/suspension mixing and adsorption, the subsequent rate of aggregation, and
finally the (partially irreversible) aggregate breakage. In addition, the model had to
adequately account for the effects of the process variables found in operating plant, for
example: the fluid shear rate (G), the flocculant dosage (θ), the primary particle size
(dp) and solid fraction (φ).
The model was also extended to describe the hindered settling rate of the aggregates
formed under the same process conditions, considerably increasing the usefulness of the
model by allowing the calculation and optimisation of the unit throughput.
1.6 Thesis overview Aggregation and settling are mature areas of research. Chapter 2 reviews this material,
with separate sections for aggregation, aggregate structure, fluid behaviour, aggregate
breakage, population balance models, and finally gravity settling. While these topics
are individually well described, there is a surprising lack of synthesis between the
topics, in particular the effect of aggregation on the settling rate. Workers studying
aggregation have usually implicitly assumed that a larger aggregate will lead to a faster
settling rate, and although this is typically the case, it represents a limited view of
reasonably complex settling behaviour. Conversely, workers studying particle settling
generally continue to assume that the settling rate is a unique function of the solid
fraction, ignoring the effects of aggregation. This situation has resulted partially from
the previous lack of instrumentation available to give on-line, in-pulp aggregate size
measurement.
Chapter 3 considers the difficulty of in-pulp aggregate sizing, and describes the
Lasentec FBRM (Focussed Beam Reflectance Measurement) instrument. FBRM gives
14
a particle chord distribution, as opposed to a conventional particle diameter distribution,
and the major thrust of this section was to develop a method to convert chord size data
to diameter data suitable for the population balance (Chapter 5) and settling (Chapter 6)
models.
Chapter 4 describes the experimental methods used to collect aggregation kinetics data.
Aggregation was performed in a turbulent pipe reactor, with a range of pipe lengths,
diameters, and suspension flow rates used to give a range of fluid shear rates and
residence times. The FBRM aggregate sizing probe was placed at the end of the pipe
reactor, immediately before the isolatable column used to take hindered settling rate
measurements. The effect of flocculant dosage (θ), primary particle size (dp) and solid
fraction (φ) were also studied under conditions approximating those found in full-scale
plant.
Chapter 5 describes the population balance model used to simulate the aggregation
kinetics measured in the experimental campaign. The model describes flocculant
mixing, aggregation, and (partially irreversible) breakage as a function of the process
variables covered by the experimental runs. Wherever possible the model equations
were kept physically realistic, describing the dominant aspects of the process, but
without including so many terms and effects that the model became unwieldy and
difficult to incorporate into CFD.
Chapter 6 describes the relationship between the aggregate size and the hindered
settling rate as a function of the process variables considered above. A relationship was
developed from established descriptions of the hindered settling rates of non-aggregated
suspensions (Stokes 1851, Richardson and Zaki 1955), with additional terms describing
the effect of aggregate porosity.
Chapter 7 brings the various aspects together, and explores ways that the model may be
used to optimise the performance of a full-scale unit. Various possibilities were
considered, in particular the use of recycle to alter the feed solid concentration and
maximise the solid settling flux (or unit throughput).
15
2. Literature Review 2.1 Introduction This review covers a range of topics directly relating to the aggregation and settling of
particles, as occur in thickeners/clarifiers in mineral processing circuits. In addition
there is a considerable quantity of relevant material relating to similar processes, in
particular aggregation and settling behaviour in water treatment plant, or occurring
naturally in river estuary systems. These systems have been actively studied for over a
century, and intensively since the 1950s, resulting in a daunting body of literature.
Despite this, recent advances in computing and instrumentation make this an active area
of research.
This review begins with a description of different collision mechanisms that may cause
aggregation, and considers how the addition of flocculants or coagulants leads to the
formation of stable aggregates. The second section considers the porosity of the
aggregates and how it may be described using fractal geometry. Aggregates are usually
highly porous, which significantly impacts on several key areas. Porous aggregates
have an increased collision radius, increasing the aggregation rate compared to solid
particles of the same mass. However, highly porous aggregates may have a reduced
number of primary particle contact points, reducing aggregate strength, leading to an
increased breakage rate. In addition, porous aggregates occupy an increased volume,
increasing the suspension viscosity, while also decreasing the subsequent hindered
settling rate.
Fluid dynamics is of central importance to both aggregation and settling and is
discussed at various points throughout this thesis. However, a separate section is
included in the review, describing fundamental relationships like viscosity and shear
rate, and then considering some of the deviations from idealised behaviour that are
likely to impact on the aggregation and settling rates.
Aggregate breakage is reviewed next, with separate sections describing the causes of
breakage, the effect of aggregate size on the settling rate, and the number and size of the
daughter fragments produced. Once the aggregation and breakage rates have been
described, the overall process can be simulated to describe the change in aggregate size
distribution through time. This is usually done with a population balance, a
16
mathematical model that accounts for the number of particles in each size range while
also conserving mass. A population balance is a dynamic model, written as a closed set
of differential equations. These equations are usually intractable analytically, and are
normally solved numerically as an initial value problem.
The final section describes the gravity settling of the solid particles. This section is
divided into the usual categories depending on the solid fraction. At a low solid fraction
the particles are taken to settle independently of each other (free-settling), which is
normally taken to be the case in the clarification zone at the top of a thickener/clarifier.
At an intermediate solid fraction the particles are constrained to settle as a loosely
connected network, reducing the settling velocity. This is referred to as
hindered-settling, and is frequently taken to limit the thickener throughput in the
thickener settling-zone beneath the feedwell. At a high solid fraction, as found in the
thickener bed region, the solid particles are in intimate contact, resulting in a network
structure that can transmit force up through the sediment network, further reducing the
settling rate.
2.2 Aggregation 2.2.1 Particle collision mechanisms
Aggregation results from successful particle or aggregate collision. Particle collision
was first described by Smoluchowski (1916, 1917) as a combination of three possible
mechanisms: differential settling, Brownian motion and fluid shear (Figure 2-1). The
settling velocity of a particle in viscous fluids increases as a function of its diameter
(Equation 1-3), with larger particles settling more rapidly, sweeping smaller particles
from the suspension. Larger particles also have a larger collision radii, giving
(Smoluchowski 1916, 1917, as referenced in: Stumm 1992, Gregory 1989):
( )( ) ji3
jilsij aaaa9
g2−+ρ−ρ
μπ
=β 2-1
where: βij = Aggregation kernel (m3s-1) ai = Radius of the ith particle (m) g = Gravity (9.8 m s-2) ρ = Density (s = solid, l = liquid) (kg m-3) μ = Fluid viscosity (N s m-2)
17
Figure 2-1: Mechanisms of particle collision.
Smoluchowski also described particle collision due to Brownian motion (perikinetic
aggregation) (Stumm 1992, Gregory 1989):
ji
ij aa3KT2μ
=β 2-2
where: K = Boltzmann’s constant (1.38 ×10-23 J K-1) T = Absolute temperature (K) and for laminar fluid shear (orthokinetic aggregation) (Stumm 1992, Gregory 1989): ( )3ji3
4ij aa +γ=β 2-3
where:
γ = Laminar shear rate (s-1) The overall rate of particle collision will be a combination of the above three kernels
(Smoluchowski 1917, Swift & Friedlander 1964, Feke & Showalter 1983, Han and
Lawler 1992, Adachi et al. 1994), however, in practice one mechanism is likely to
dominate and to reduce mathematical complexity usually only one kernel is used.
The Peclet number (Pe) is the ratio of the viscous and thermal forces and is commonly
used to demonstrate that shear aggregation dominates Brownian aggregation in
experimental studies (Oles 1992):
Differential settling
Brownian motion (perikinetic)
Fluid shear (orthokinetic)
18
KT
Ga6Pe3πη
=
2-4
where: η = Dynamic viscosity (N s m-2) G = Average shear rate (s-1) a = Particle radius (m) K = Boltzmann’s constant (1.38 × 10-23 J K-1) T = Temperature (K)
A Peclet number greater than unity indicates that shear is dominant (Hansen & Ottino
1996), with most workers reporting much higher numbers: >120 (Serra & Casamitjana
1998), >160 (Oles 1992), 1300-21000 (Krutzer et al. 1995), >1 Wistrom and Farrell
(1998) (however they acknowledge some Brownian aggregation).
Differential settling only becomes significant in quiescent systems and when the
particles are widely differing in size (Lawler 1993), and it is generally accepted (Tambo
& Watanabe 1979, Wagberg & Lindstroem 1987), that fluid shear is the dominant
aggregation mechanism in industrial-scale turbulent flows (de Boer et al. 1989a, Brunk
et al. 1998). However, Smoluchowski’s description (Equation 2-3) of collision due to
fluid shear assumes laminar flow, and is not appropriate for turbulent systems.
2.2.2 Turbulent collision, particles smaller than micro-scale
Camp and Stein (1943) substituted a term, G, the spatially averaged shear rate
(see section 2.4.2) and proposed:
( )3ji3
4ij aaG +=β 2-5
where:
μ=
ΦG 2-6
and: Φ = Energy dissipation rate per unit volume (kg m-1s-3)
μ = Viscosity (kg m-1s-1, or N s m-2)
[units G = ( (kg m-1s-3)/(kg m-1s-1) )0.5 = (s-2)0.5 = s-1 ]
Camp and Stein’s kernel has been critiqued repeatedly. Glasgow and Kim (1986) point
out that the equation only strictly applies to isotropic turbulence (where the shear is the
19
same in all directions) and may not be applicable to systems like stirred tanks. The
shear rate in vessels like stirred tanks may vary considerably, both in direction and
magnitude (Koh et al. 1984). For example, the shear rate at the impeller tips is typically
5-10 times the amount in the rest of the tank (Perry and Green 1997, 18-6). Koh et al.
developed a multi-component model for the effective shear rate through a stirred tank
using the first moment of the shear rate distribution. They suggest that the type of shear
(laminar or turbulent) is not important, but rather it is that the extent of mixing is
critical.
Cleasby (1984) critiqued Camp and Stein’s work on the basis that the absolute viscosity
(μ) is not an appropriate parameter for use in turbulent systems and goes on to suggest
that the use of G be restricted to when the particle size is below the Kolmogoroff micro-
scale. Above the micro-scale, Cleasby suggests βij ∝ ∈2/3, where ∈ is the local energy
dissipation rate per unit mass, as suggested by Saffman and Turner (Equation 2-7).
Camp and Stein’s work has also received criticism from Clark (1985), who performed
tensorial analysis and concluded that Camp and Stein’s conceptualisation is
“fundamentally incorrect, since they essentially require that a three-dimensional flow in
general be represented by a single two-dimensional flow.” (Their italics).
Despite these criticisms, a similar but more rigorously derived formulation by Saffman
and Turner (1956)(Table 2-1):
( )3jiij aaG294.1 +=β 2-7 and:
ν∈
=G 2-8
where: ∈ = Local energy dissipation rate per unit mass (m2s-3) ν = Kinematic viscosity (m2s-1) [units G = (m2s-3 / m2s-1)0.5 = s-1] has reached general acceptance (Clark 1985, Cleasby 1984, Gregory 1989), provided
(Clark 1985):
20
1. The particles are small relative to the Kolmogoroff micro-scale. 2. The particles are neutrally buoyant and spherical. 3. The shear is high Reynolds number isotropic turbulence. 4. There are no large spatial differences in the local energy dissipation rate. 5. The coagulation is slow. 6. There are no hydrodynamic or colloidal interactions between particles.
Particles smaller than the micro-scale are contained within even the smallest eddies, and
particle collision is primarily caused by localised fluid shear (De Boer 1989a). For
particles in this range there is general agreement (Clark 1985, Cleasby 1984, Spicer
1997, Kusters 1991, Krutzer et al. 1995) that Saffman and Turner’s Kernel
(Equation 2-7) is appropriate, except if there is a significant hydrodynamic interaction
(Section 2.1.2). Saffman and Turner’s kernel appears to be the most widely accepted
(Tables 2-1 & 2-2).
The condition of neutral buoyancy is typically largely ignored, even by Saffman and
Turner, who described the motion of water droplets in clouds. Experimental studies are
typically performed with neutrally buoyant latex particles (Oles 1992, Flesch et al.
1999), although aggregates are also typically highly porous (see Sections 2.2 & 2.6.1),
reducing their effective density considerably.
Various other collision kernels have also been proposed, with different constants. For
example: Chin et al. (1998) calculated the constant as 2.3, apparently in agreement with
Pearson et al. (1984). Delichatsios and Probstein (1975) calculated from mean free
paths a constant of 0.4 (de Boer 1989b). Levich (1962) calculated a much larger
constant of 13.8 (Delichatsios & Probstein 1975) or 9.24 (de Boer 1989a) using a
turbulent diffusion model. Delichatsios and Probstein (1975) suggest that the
discrepancy is likely to be caused by: “the inability of a steady diffusion model to
correctly describe the relative motion of particles in turbulent flows due to the non-
Markovian nature of the turbulent dispersion process”.
de Boer (1989a) compared various aggregation kernels with experimental results from
stirred tank experiments where 0.74 μm polystyrene particles were coagulated with
sodium chloride. In all cases the theoretically predicted aggregation rates exceeded the
21
experimental result. In contrast, Spicer and Pratsinis (1996) used Saffman and Turner’s
aggregation kernel in conjunction with a breakage kernel to successfully model
coagulation of 0.87 μm polystyrene particles with alum (Figure 2-11) without requiring
a capture efficiency term.
Although Saffman and Turner’s kernel is only strictly correct for particles smaller than
the Kolmogoroff micro-scale, it has been used successfully for somewhat larger
particles. Delichatsios (1980) suggests that the particles may be as much as ten times
the micro-scale, compared to Kusters (1991) who suggests six times larger.
2.2.3 Turbulent collision, particles larger than micro-scale
Particles significantly larger than the Kolmogoroff micro-scale are too large to be
completely contained in the small scale eddies, and inertial effects become important.
For particles larger than the micro-scale Cleasby suggests:
32
ij ∝∈β 2-9 may be more appropriate than:
21
ij ⎟⎠⎞
⎜⎝⎛ν∈
∝β 2-10
and goes on to suggest that the energy used to mix water treatment flocculators could be
reduced as a cost saving. Cleasby’s arguments were both theoretical and based on the
correlation with experimental data from Argaman and Kaufman (1968, 1970). Cleasby
also suggested that turbulent eddies larger than the micro-scale were responsible for
flocculation in industrial systems. Conversely, Casson and Lawler (1990) performed
experiments with an oscillating grid and concluded that flocculation was caused by
eddies of a similar size to the particles, and that larger eddies had little effect, except to
prevent the settling of aggregates in the flocculator.
Cleasby’s suggestion of ∈2/3 is at odds with various other calculations that generally
suggest ∈1/3 in the inertial sub-range (Delichatsios & Probstein 1976, Kusters 1991).
Kusters also comments that no analytical expression for inertial effects has been
22
formulated for the inertial sub-range for solid/liquid systems, but quotes Kuboi et al.
(1972) that the collision frequency can be calculated from:
( )3
731
jiij aa87.6 +∈=β 2-11 where: ai = radius of the ith particle which is similar to the kernel proposed by Delichatsios and Probstein (1975):
37
31
31
31
jiij vv41.1 ⎟⎠⎞⎜
⎝⎛ +∈=β 2-12
where:
vi = Volume of ith particle Kruis and Kusters (1997) recently proposed the general case:
( ) 2shear
2.accel
2ji)shear.accel(ij wwaa
38 2
1
++⎟⎠⎞
⎜⎝⎛ π
=β + 2-13
where: waccel. = Relative particle velocity of particles due to acceleration (inertial effect) wshear = Relative particle velocity of particles due to fluid shear
In general it appears that there is far less agreement about the aggregation rate in the
inertial sub-range than in the viscous sub-range. A full description becomes even more
complex if the particles are significantly heavier than water and particle inertia also
affects the collision rate. The problem is exacerbated by a general lack of experimental
evidence. Delichatsios and Probstein (1975) did not get good experimental agreement
with their proposed equations and concluded that the error was due to aggregate
breakage.
Abrahamson (1975) considered the turbulent macro-scale and suggested:
( ) 2
j2i
2jiij VVaa5 ++=β 2-14
for large particles in industrial level turbulent flow, i.e. in the macro sub-range. Vi is
the particle’s mean square velocity:
23
2i
2
i V/5.11VV∈τ+
= 2-15
and: V = mean square velocity of the fluid (m s-1) τi = particle relaxation time (s)
The relaxation time is the time it takes a particle to accelerate to the same velocity as
the fluid after a change in the fluid velocity (Brunk et al. 1998).
2.2.4 Summary - particle collision
Particle collision can be caused by a combination of Brownian motion, differential
settling and fluid shear. However, it is generally accepted that in turbulent industrial
feed streams fluid shear is the dominant collision mechanism. While the collision rate
in laminar flow is relatively easy to describe (Equation 2-3), collision in turbulent flow
is far more complex and is influenced by the size of the particle in relation to the size of
the turbulent eddies. Despite this uncertainty, the aggregation kernel by Saffman and
Turner (1956) (Equation 2-7) is widely accepted in the literature (Table 2-2). However,
Saffman and Turner’s kernel typically overestimates the aggregation rate compared to
experimental data, and a collision capture efficiency term is normally introduced to
account for the discrepancy.
24
Table 2-1: Aggregation kernels
Reference Kernel Comment/References Smoluchowski 1917
( )3ji3
4ij rr +γ=β #1, Table 2-2
Laminar, widely referenced
Camp & Stein 1943
( )3ji34
ij rr21
+⎟⎟⎠
⎞⎜⎜⎝
⎛μΦ
=β #2, Table 2-2 Widely referenced
Saffman & Turner 1956
( )3jiij rr158 2
1
+⎟⎠⎞
⎜⎝⎛ν∈π
=β #3, Table 2-2
Widely referenced
Levich 1962
( )3jiij rr24.921
+⎟⎠⎞
⎜⎝⎛ν∈
=β Shamalou & Titchener-Hooker 1993
Kuboi et al. 1972
( )37
31
jiij aa87.6 +∈=β Inertial, Kusters 1991
Delichatsios & Probstein 1975
( )3jiij rr77.021
+⎟⎠⎞
⎜⎝⎛ν∈
=β
37
31
31
31
jiij vv41.1 ⎟⎟⎠
⎞⎜⎜⎝
⎛+∈=β
Viscal Shamalou & Titchener- Hooker 1993 Inertial
Abrahamson 1975
( ) 2j
2i
2jiij rr0.5 ω+ω+=β
ω = mean sq. velocity of particle
De Boer 1989 a, b
( )3jiij aa4.021
+⎟⎠⎞
⎜⎝⎛ν∈
=β
( )37
31
jiij aa15.2 +∈=β
Viscal Kuboi et al. 1972 is similar except 2.89 rather than 2.15. Inertial Delichatsios & Probstein 1975
Shamalou & Titchener-Hooker 1993
( )37
31
ji2ij rrA +∈=β
Inertial.
Kruis & Kusters 1997
( ) 2shear
2acc
2jiij rr
38
ω+ω+π
=β Viscal & inertial
Chin et al. 1998
3
jiij31
312
1
vv3.2 ⎟⎠⎞⎜
⎝⎛ +⎟
⎠⎞
⎜⎝⎛ν∈
α=β Pearson 1984
Flesch et al. 1999
3D3
3oij
fD1
j
fD1
i
21
f rrkR294.1 ⎟⎠⎞
⎜⎝⎛ +⎟
⎠⎞
⎜⎝⎛ν∈
=β−
Jiang & Logan 1991
Vanni 2000
( ) ( ) 2j
2i
2ji
3fp
ij rrrr27.1 4
1
−+⎟⎟⎠
⎞⎜⎜⎝
⎛ν∈
μ
ρ−ρ=β
Inertial, attributed to Saffman & Turner 1956
25
Table 2-2: References to simple shear kernels (Table 2-1, #1, #2 & #3) Reference Comment Abrahamson 1975 Effect of larger aggregates Adachi et al. 1994 Akers et al. 1987 Argaman & Kaufman 1968, 1970 Atteia 1998 α = 1 × 10-4 - 0.1 Brunk et al. 1998 φexp = 0.002-0.12, α = 0.05-0.6 Burban et al. 1989 α = 0.15-0.3 Casson & Lawler 1990 Cleasby 1984: ∈2/3 rather than G, φexp = 1.05 × 10-4 Chin et al. 1998 α = 0.05-0.65, φexp = 1 × 10-4 Cleasby 1984 Suggests ∈2/3 rather than G De 1989a, b α ∼ 0.25, range 0.14-0.51, φexp = 2 × 10-5 – 1 × 10-4 Delichatsios & Probstein 1975 φexp = 1 × 10-3 – 6 × 10-3 Ducoste and Clark 1998 α = 0.006 Flesch et al. 1999 φexp = 1.4 × 10-5 Gardner & Theis 1996 Table, α = 0.0003-1 Glasgow & Lueche 1980 Gonzales & Hill 1998 α = 0.015-0.021 Gregory 1989 α = f(λ)CA
0.18 = 0.1-0.5 Gruy & Saint Raymond 1997 α = 0.05-1.4 Han & Lawler 1992 α = f(hydrodynamic effect) = 1 × 10-5 – 1 Higashitani et al. 1983 Kramer & Clark 1999 α = 0.005-0.025 Kruis & Kusters 1997 Krutzer et al. 1995 φexp = 1 × 10-4 Koh et al. 1984, 1987 α = 0.5 × 10-4 – 1 × 10-3 Kusters et al. 1987 α = 0.05-1, φexp = 3.1 × 10-5 – 2.1 × 10-4 Lawler 1993 α = 1E-4 – 0.5, φexp = 2 × 10-4 – 6 × 10-4 Lick & Lick 1988 Lu et al. 1998 α = 0 – 0.5, φexp = 0.5 % ∅degaard 1979 Serra & Casamitjana 1998 φexp = 2.5 × 10-5 Shamlou & Titchener-Hooker 1993 Spicer & Pratsinis 1996 α = 1, φexp = 8.3 × 10-5 Swift & Friedlander 1964 φexp = 3.45 × 10-5 – 8.6 × 10-4 Tambo & Watnabe 1979 α = 0 – 0.02 Thomas et al. 1998 Tomi & Bagster 1978 Whittington & George 1992 φexp = 1.18 g L-1
Wiesner 1992 φexp = 2.1 × 10-6 Wigsten & Stratton 1984 α = 0.2 – 0.33 Wistrom & Farrell 1998 α = 3 × 10-7 – 1 × 10-5, φexp = 50-100 mg L-1
Zeichner & Schowalter1979 φexp = 5 × 10-6 – 2.5 × 10-5 Vanni 2000
26
2.2.5 Effect of short range inter-particle forces/bonding mechanisms
All the above kernels predict the rate of collision by assuming that there are no inter-
particle forces; the so-called rectilinear model (Lawler 1993). This may not be correct,
and various inter-particle forces may increase or diminish the effective collision radius
of the particles by forcing the particles to deviate from their previous trajectories.
These inter-particle forces are often used to calculate a capture efficiency term (α) for
insertion into one of the aggregation kernels above. However, inter-particle forces have
the greatest impact on solid spherical particles being coagulated with salt where the
particles must come very close together before adhesion. Inter-particle forces are less
important to bridging aggregation with polymers (Gregory 1989).
van der Waals and electrostatic forces
Prior to the addition of a flocculant, a suspension may already be aggregated to some
extent due to coagulation. Several forces can act between the particles, with the extent
of aggregation dependent on which forces are dominant. Deryagin and Landau (1941)
and Verwey and Overbeek (1948) independently combined the effects of van der Waals
and electrostatic forces between particles, leading to the so-called DLVO (Deryagin,
Landau, Verwey and Overbeek) theory. The electrostatic repulsion initially repels the
particles (first term of Equation 2-16), but on close approach van der Waals force
dominates (second term)(Figure 2-2).
d12aA
zeKTa32V 111d
2
1t −γ⎟⎠⎞
⎜⎝⎛∈π= κ− 2-16
where: Vt = Interaction energy a1 = Radius of particle
∈ = Permittivity of the medium z = Valency (symmetrical z-z electrolyte assumed) γ1 = Dimensionless function of surface charge κ = Deybe-Huckel parameter d = Separation distance
27
VA
VT
Energybarrier
Secondaryminimum
Primaryminimum
Inte
ract
ion
Ener
gyd
VE
Figure 2-2: Energy of interaction between two particles, VA is the van der Waals attraction, VE is the electrical repulsion, and VT the total interaction energy. (adapted from Gregory 1989).
Hydrodynamic force
Another force that may cause repulsion between aggregating particles is the so-called
hydrodynamic (or viscous) interaction (Lawler 1997, Gregory 1989). When the
particles come close together the final approach is resisted by a thin film of fluid
separating them. The hydrodynamic force has been used to calculate a capture
efficiency term (van de Ven & Mason 1977, Gregory 1989, Potanin & Uriev 1991,
Adachi et al. 1994):
18.0
AC)(f λ=α 2-17 where:
λ = Characteristic wavelength (m) CA = A/36πμGa3 A = Hamaker constant μ = Viscosity (N s m-2) G = Mean shear (s-1) a = Particle radius (m)
The hydrodynamic force is largely overcome by the van der Waals force of attraction at
very close approach (van de Ven & Mason 1977a, Gregory 1989, Spielman 1978,
Delichatsios 1980). Also, in practice the particles are irregularly shaped and rapidly
aggregate into porous aggregates that allow the fluid to flow through the structure,
reducing the hydrodynamic effect considerably (Adachi 1994, Wolynes & McCammon
1977, Adler 1981b, Torres et al. 1991a, Chellam and Wiesner 1993).
28
2.2.6 Coagulation with salt
The electrostatic repulsion can be dramatically reduced by the addition of an electrolyte
to the solution, a phenomena that has been used widely to aggregate particles in water
treatment plants. The electrostatic repulsion is reduced because the additional
electrolyte increases the ion concentration in the double layer, compressing it (Stumm
1992). In addition, specific ion adsorption may also occur, directly altering the surface
charge. The critical minimum concentration of salt required for aggregation can be
calculated approximately from (Gregory 1989):
62
4
f ZAKc γ
= 2-18
where: K = Constant related to the fluid γ = Dimensionless function of surface charge A = Hamaker's constant Z = Valency
The suspension is completely destabilised when enough salt is added that the
electrostatic repulsion becomes negligible and all collisions result in aggregation
(Gregory 1989). The fact that the critical concentration is inversely proportional to the
valency raised to the sixth power (it drops to Z2 with lower surface potential - Stumm
1992, p. 266) is exploited by using multivalent ions like Ca2+, Al3+ or Fe3+. The cation’s
charge is more important than the anion’s charge for negatively charged particles, i.e.
CaCl2 is a better coagulant than Na2SO4 (Gregory 1989).
2.2.7 Flocculation with polymer
Polymer flocculants give better performance than salt coagulants and have largely
replaced them in mineral processing applications (Gregory 1989, Adachi et al. 1994,
Kim & Glasgow 1987). Two models of flocculation by polymers have been proposed.
In 1952 Ruehrwein and Ward proposed the bridging model (Figure 2-3) where high
molecular weight polymers can bridge the gap between two particles and cause
aggregation (Chen 1998). This is supported by the evidence that the polymer can be
uncharged, or even carry the same charge as the particles, and still cause aggregation
(Chen 1998, Stumm 1992). However, cationic polymers can also be used so that charge
neutralisation also aids the aggregation. The particular mechanism may be pH
29
dependent (Lindquist & Stratton 1976, Ditter et al. 1982) with bridging dominating
above pH 9 due to the low cationic charge of the polymer at higher pHs (Chen 1998).
(a) (b) Figure 2-3: Bridging model of polymer flocculation (a), and re-stabilisation from excess polymer (b). (adapted from Gregory 1989)
The second model for polymer flocculation was proposed independently by Kasper
(1971) and Gregory (1973) and is referred to as the electrostatic patch model
(Figure 2-4). In this model a polymer of opposite charge to the particle is thought to
adsorb onto the particle, resulting in “patches” of opposite charge (Gregory 1989).
+--
--
--
-
--
--
-
---+++++++++++ - -
-- --
-
-+++
+++
+++++-
--
- ++
+-
++++++++
++++-- -
---
--
-
-
+++++++++++
- ----- -
---
- ---
-
--
-
- - --
-
-
-- -
+++++++++++
+++++++++
+++++++++
++--- -
-----
- - --
--
-- ---
---- -- - ----------- -
Figure 2-4: Electrostatic patch model (adapted from Gregory 1989)
Various relationships between surface coverage (θ) and capture efficiency (α) have
been proposed. For example La Mer and Healy (1963) proposed:
α = θ(1-θ) 2-19
implying that the aggregation rate should be highest at 50 % coverage. Unfortunately,
this proposal is not supported by experimental evidence (Hsu et al. 1995, Glasgow &
30
Lueche 1980). Hogg (1984) took into account the possibility of particle re-orientation
and proposed:
α = 1 - θ2n - (1-θ)2n 2-20
where n is the number of adsorption sites per particle. When n = 1, Equation 2-20
collapses to (Hsu et al. 1995):
α = 2θ(1-θ) 2-21 which Hsu et al. (1995) state is the correct formulation of 2-19 when the statistics of
collision between bare and occupied patches are properly taken into account. Various
even more complex relationships have been proposed, see for example Berlin and
Kislenko (1995) or Hsu et al. (1995).
There appears to be experimental evidence to support both the bridging and the
electrostatic patch models. Mabire et al. (1984) flocculated silica with a copolymer of
acrylamide and acrylate and found maximum flocculation performance at intermediate
flocculant dosages, concluding that the electrostatic patch model was appropriate. The
optimal performance was found at intermediate coverage due to steric stabilisation
(Figure 2-3b) where at high polymer dosages the particles become totally covered with
polymer (Gregory 1989). Adachi et al. (1994) found that polymer (PEO) flocculation
occurred faster than coagulation by salt and concluded that the polymer chains were
bridging the gap between particles and increasing the collision radius.
Lindquist and Stratton (1976), and Ditter et al. (1982) found that the bridging
mechanism dominated above pH 9 and 9.5 respectively, while the electrostatic
attraction dominated at lower pHs. Leu and Ghosh (1988) found that the optimal
flocculant dosage was predominantly related to the charge density of the polymer rather
than the length of the polymer, supporting the electrostatic patch model.
Ditter et al. (1982) suggest the mechanism can be determined by briefly increasing the
fluid shear and noting whether the aggregate size returns to its original value. If the
original aggregate size returns, the electrostatic patch mechanism can be assumed, if
not, bridging is assumed. Chen and Doi (1989) suggest a similar method involving
31
modelling. Both these suggestions are based on the assumption that polymer
breakage/re-conformation affect bridging but not electrostatic patch aggregation.
In summary, although the electrostatic effect of the polymer is likely to contribute to the
bonding mechanism, polymer bridging is usually considered to be dominant. This is
supported by experimental evidence of the irreversible effect of shear on the aggregate
size (see Section 2.2.8) and the improved performance generally given by a flocculant
of higher molecular weight.
2.2.8 Loss of flocculant activity by polymer degradation
Although aggregates formed by coagulation with salt are comparatively weak,
aggregate breakage is generally found to be reversible, allowing the daughter fragments
to re-aggregate. However, with polymer flocculants there is considerable evidence that
broken aggregates have a reduced ability to re-aggregate (Pelton 1981, Sikora &
Stratton 1981, Glasgow & Lueche 1980, Chen et al. 1990) except when the mechanism
is charge neutralisation (Chen et al. 1990, Ditter et al. 1982). The bridging performance
of polymers decreases due to scission or re-conformation of the polymer due to shear
(Gregory 1987a, Stratton 1983) and Ditter et al. (1982) suggested that the breakage
reversibility in shear may be used to determine the mechanism.
The strength of the carbon-carbon bond is only about 10-8-10-9 N, and Gregory (1997)
suggests that several bridges are required between particles to provide sufficient
strength to prevent breakage. Pelton (1981) has proposed a decreasing aggregate
strength depending on the history of the particle surface through time. Initially, after
polymer addition all the surface is considered active. However, after aggregation and
subsequent breakage the previously bonded surface is considered inactive (Figure 2-5).
As the aggregation/breakage process continues the adsorbed flocculant progressively
loses its activity, decreasing the aggregate strength.
32
Figure 2-5: Diagram showing polymer degradation after contact and subsequent breakage (adapted from Pelton 1981).
The loss of bonding strength after breakage has been observed experimentally by Yeung
et al. (1997), who measured the strengths of individual polymer flocs with a cantilever
balance arrangement and reported that the re-aggregated flocs had a strength of only
10 % that of the original flocs. The decrease in the aggregate size with continued shear
appears to have received little attention in the literature, particularly the modelling
literature that has typically considered coagulation rather than flocculation.
2.3 Aggregate porosity Aggregates are a loose assemblage of primary particles, enclosing considerable void
space within the aggregates structure (Figures 2-6 & 2-7). The porosity of the
aggregate has several important effects on the aggregation process. Firstly, it increases
the effective collision radius of the aggregate, thereby increasing the collision rate
(Flesch et al. 1999). Secondly, it decreases the settling rate of the aggregate, compared
to a more compact structure (Gregory 1989). Thirdly, it affects the strength of an
aggregate, since a more compact structure is likely to have more particle-particle
contacts (Gregory 1989, Dobias 1993, Bagster 1993). Finally, a porous aggregate may
also increase the thickener bed height for a given underflow solid (Healy et al. 1994,
Svarovsky 2000).
ATTACHED DETACHED
Large sphere Large sphere
Polymer flocculant Inactive
area
33
Figure 2-6: Metallic aluminium particles (~ 5 μm) flocculated with high molecular weight polyacrylate, image is 880 × 660 μm.
Aggregate porosity is typically found to increase with the aggregate size (Figure 2-7)
(see also section 2.7.1) and is usually described by fractal geometry. Various
approaches are possible, however the one most widely used (and directly useful) is
based on the aggregate’s mass and diameter. In this case the aggregate volume is taken
to increase with the cube of the diameter as with solid objects:
3
aggagg kdV = 2-22 where: Vagg = Volume of aggregate (m3) k = Constant, (π/6 for spheres) dagg = Diameter of the aggregate (m) but the aggregate mass increases at some lower, fractional, (hence fractal) power,
giving:
fDaggs dm ∝ 2-23
where: ms = Mass of solid in aggregate (kg) Df = Mass-length fractal dimension (1 [thin rods] ≤ Df ≤ 3 [spheres etc])
The aggregate density (ρagg) is (see also Section 2.7.1 & Appendix, Figure 2-18) (Mills
et al. 1991, Jiang & Logan 1991, Kusters et al. 1997, Serra & Casamitjana 1998):
34
3D
p
aggsagg
f
dd
−
⎟⎟⎠
⎞⎜⎜⎝
⎛ρ=ρ 2-24
where: dagg = Effective diameter of aggregate (m) dp = Primary particle diameter (m) Df = Fractal dimension
ρ = Density (agg = aggregate, p = primary particle) (kg m-3)
The decreased density with aggregate size is usually rationalised on the basis of
increasing voidage as large aggregates are formed from aggregate/aggregate collisions
later in the aggregation process (Figure 2-7).
Figure 2-7: Pictorial (2D) representation of self-similar fractal aggregation showing that the density drops with the ratio of the aggregate to primary particle size (adapted from Meakin 1988).
Typically, the fractal dimension is assumed to be D3 (3-dimensional, i.e. mass-length),
however other systems are used (e.g. area, D2, or perimeter based, Dpf) depending on
the method measurement or simulation. Clearly it is important to distinguish between
them, for example Serra and Casamitjana (1998a) calculated: D3 = 2.6, D2 = 1.98,
D1 = 1.06, Dpf = 1.12 for the same aggregate structure.
Computer simulation of fractal aggregates has been popular and several models have
been proposed. Vold (1963) used what has become referred to as a “diffusion limited”
type model (Meakin 1988) where a single primary particle has successive primary
particles randomly added to build up an aggregate using a Monte-Carlo type simulation.
35
This type of simulation typically gives a Df of 2.75 (Gregory 1989), higher than
typically found in practice (Table 2-3).
Meakin wrote a series of papers describing a diffusion limited cluster-cluster
aggregation that leads to a lower Df of around 1.75 (Gregory 1989), the same as found
experimentally by Weitz et al. (1985) because extra voidage is introduced when
aggregates combine, as opposed to single particle addition. The fractal dimension is
somewhat lower than typical, and the model has been extended to allow for
reorganisation during or after aggregation (Meakin 1985), leading to higher density
aggregates. This has been supported experimentally by Sonntag and Russel (1986),
who formed aggregates very slowly under Brownian conditions, and then increased the
mass-size fractal dimension from 2.2 to 2.5 by shearing in a Couette device.
Higher fractal dimensions are also produced if the aggregation is chemically rather than
diffusion limited, as attaching particles have a chance to try various positions
(Weitz et al. 1985), presumably preferring a position that leads to higher particle-
particle contact and hence higher density. Aubert and Cannell (1986) aggregated silica
slowly to get Df = 2.08, and rapidly to give 1.75. Similarly, Weitz et al. (1985) found
Df = 2.05 and 1.75 for reaction or diffusion limited aggregation, respectively.
The above models assume that aggregation is performed under Brownian conditions.
Since shear aggregation is the dominant mechanism for flocculation on the
industrial-scale, various ballistic models have also been developed where the colliding
particles are assumed to travel in straight lines (Gregory 1989). Tence et al. (1986)
achieved good agreement between the measured Df (1.88) of aggregated iron particles
and a ballistic cluster-cluster model that predicted Df = 1.91. Conversely, Serra and
Casamitjana (1998a) found that the fractal dimension was independent of the shear rate
or volume fraction, and Gregory (1989) notes that it is not affected by polydispersity of
particles.
Fractal dimensions have also been measured experimentally for a variety of different
systems (Table 2-3), generally giving results in the range Df = 1.8-2.5. However,
detailed comparisons are difficult due to the wide range of measurement techniques and
substrates used.
36
Table 2-3: Fractal dimensions reported in the literature Reference Fractal
dimension⊗ Type Method Material
Adachi and Ooi 1990
2.08-2.4 (area based) 2.00-2.27 (max diameter based)
D3 from total number of primary particles
Brownian DLA microphotography to measure area and max. diameter. Aggregate destruction by ultrasonication followed by particle counting
PSL particles 0.804 μm
Adachi & Kamico 1994
1.6-2.3 D2(?) from max. floc. φ
Video Latex, 0.93 & 2.02 μm
Allain et al. 1996 2.1-2.3 D3 from settling rates Aggregation by Brownian and differential settling. Settling rate (800 mm jar) vs. radius by microphotography
CaCO3.
Amal et al. 1990a 2.3 D3 from Malvern and rate change in PSD
Brownian DLA. Malvern and rate data to calculate Df
Haematite
Amal et al. 1990b 2.3 diffusion limited 2.8 reaction limited
As above As above As above
Danielson et al. 1991
1.58-1.72 D2 from number of primary particles and radius
Shear aggregation in chaotic and regular flows
Model
Haw et al. 1997 1.42 (average) D2 from weight (area) average and characteristic radius
2D CCDLA Model
Horne 1987 2.21 D3 from mass-length Brownian DLA. Light scattering
Skimmed milk/ calcium ions
Horwatt et al. 1992
2.5 - DLA 2.6 - RLA 2.0 - HCCA ∼ 3.0 - LTA ∼ 3.0 – Eden
D3 from mass and radius of gyration
Compares Df for variety of different models
Model
Huang 1994 1.83-1.97 D3 Couette flow, Malvern and settling data
Estuarine sediment
Jiang & Logan 1991
2.4-3.75 (not a typo) (shear) 1.61-2.31 (DS)
D3 from mass-diameter
Shear and DS using dimensional analysis of fractal scaling relationships
Model
Johnson et al. 1996
1.78-2.25 D3 from size - total particle No.
Gentle shear in stirred tank for different times to give different Df. Microscopy and ultrasonication/ particle counting
0.87 or 2.6 μm latex spheres/ NaCl
Jullien & Meakin 1989
1.89-2.13 D3 from 3D model
Ballistic CCA with restructuring
Model
Kolb et al. 1983
1.38 D2 from 2D model Brownian CCA Model
Kusters et al. 1997
2.5 ± 0.1 D3 from laser scattering
Stirred tank, Malvern particle sizer.
1 μm polystyrene spheres/ NaCl
37
Table 2-3: Continued Reference Fractal
dimension⊗ Type Method Material
Lin et al. 1990 1.86 (DLCA) 2.13 (RLCA)
D3 from laser scattering
Brownian RLA or DLA
Colloidal gold or silica
Lin et al. 1989 2.10 (gold) 2.12 (silica) 2.13 (polystyrene)
D3 from laser scattering
Brownian RLA Colloidal gold, polystyrene or silica
Meakin 1983 1.45 -1.5 D2 from 2D model 2D Diffusion limited cluster-cluster/particle
Model
Meakin & Jullien 1988
1.80-2.18 (DL cluster-cluster) 1.95-2.19 (ballistic cluster-cluster) 2.09-2.24 (RL cluster-cluster)
D3 from 3D model Various models and effect of restructuring
Model
Neinmark et al. 1996
1.75 D3 from TEM. Combustion generated soot
Soot particles
Oles 1992 2.1 (initially) 2.5 (extended shear)
D3 from light scattering
Shear flow in Couette, Malvern Mastersizer
2.17 mm polystyrene spheres/ NaCl
Serra & Casamitjana 1998a
2.24 D3 from laser scattering and microphotography
Aggregation in shear (Couette) flow (laminar)
2 μm latex/ NaCl
Sonntag and Russel 1986
2.48 D3 from light scattering
Sheared aggregates in Couette (laminar)
Polystyrene/KCl
Spicer & Pratsinis 1996b
1.19-1.29 Dpf from microscopy and image analysis
Stirred tank (turbulent). Perimeter based fractal dimension
0.87 μm polystyrene/ alum.
Spicer et al. 1998 2.1-2.5 D3 from light scattering
Stirred tank. Malvern Mastersizer 0.87 μm polystyrene/ alum.
Stoll & Buffle 1998
2.03 (RLA - no polymer) 2.03-2.52 (polymer in rod conformation) 1.97-2.08 (polymer in coiled conformation
D3 from radius of gyration and number of primary particles
CCA with various polymer conformations
Model
Torres et al. 1991a
1.75-1.85 D3 PCDLA & CCDLA Model
Weitz et al. 1985 1.7-1.8 (DLA) 2.0-2.1 (RLA)
D3 calculated from TEM
Brownian aggregation either DLA or RLA depending on coagulant dosage
Gold particles/ pyridine
Wiesner 1992 Table Literature review ⊗. DLA = diffusion limited aggregation, RLA = reaction limited aggregation, PCDLA = particle cluster DLA, CCDLA = cluster-cluster DLA, LTA = linear trajectory aggregates, EDEN = Model for biological systems after Eden 1961.
38
2.4 Fluid behaviour 2.4.1 Laminar flow, shear stress (τ), viscosity (μ), and fluid shear (γ),
The shear rate of a fluid is the difference in velocity between adjacent parts of the fluid
(Figure 2-8)(Perry & Green 1997) due to an applied shear stress.
x
y
Figure 2-8: Laminar shear
The shear rate (γ) is calculated from:
γ = dU/dy 2-25 where:
U = velocity (m s-1) i.e. dx/dt The shear stress (τ) is:
τ = F/A 2-26 where: F = Force (N or kg m s-2) A = Area (m2) and the viscosity (μ) is (Perry & Green 1997): μ = τ /γ 2-27 [units μ = kg m s-2 / m2 / m s-1 = kg m-1s-1, or N s m-2]
2.4.2 Turbulent flows, G, Kolmogoroff micro-scale, eddy spectrum
Turbulent flows occur when fluid inertia dominates viscous dissipation, leading to the
formation of turbulent eddies. Since turbulent shear is the dominant aggregation
mechanism on an industrial-scale, some way is required to describe the turbulence and
its effect on the flocculation process. Kolmogoroff (or Kolmogorov 1941) envisaged a
39
cascade of turbulent eddies that transfer energy from the large, energy-containing
eddies, down to the small energy dissipating eddies (Figure 2-9).
1/(Eddy size)
Ene
rgy
spec
trum
Large eddies,Eulerian macro-scale
Kolmogoroff micro-scale
Viscous dissipation
Inertial subrange
Figure 2-9: Eddy energy spectrum (adapted from Shamlou, Titchener-Hooker 1993, see also Klute & Amirtharajah 1991, Glasgow & Lueche 1980).
The Kolmogoroff eddy scale is divided into three regions. The large macro-scale eddies
are of a similar size to the stirrer (Kusters 1991, Spicer 1997) or vessel (Gregory 1989)
and contain the bulk of the energy (Parker et al. 1972). However, they dissipate little
energy directly; instead transferring it down to smaller eddies in the inertial sub-range
(Kusters 1991). These eddies in turn pass the energy down to the smallest eddies
(Kolmogoroff micro-scale of turbulence), where the bulk of the energy is released as
heat through viscous dissipation (Cleasby 1984). The smallest eddies also have the
highest shear rate (Brunk et al. 1998). The Kolmogoroff micro-scale marks the
boundary between the inertial and viscous sub-ranges (Parker et al. 1972).
The smallest eddies have a characteristic length scale referred to as the Kolmogoroff
micro-scale of turbulence, which Kolmogoroff calculated from dimensional reasoning
(Kusters 1991):
41
3
⎟⎟⎠
⎞⎜⎜⎝
⎛∈ν
=η 2-28
Kolmogoroff time and velocity scales are also sometimes quoted (Kusters 1991):
40
21
k ⎟⎠⎞
⎜⎝⎛∈ν
=τ 2-29
( )41
kv ∈ν= 2-30
where: η = Kolmogoroff micro-scale of length (m) τk = Kolmogoroff micro-scale of time (s) vk = Kolmogoroff micro-scale of velocity (m s-1)
ν = Kinematic viscosity (m2s-1)
∈ = Energy dissipation rate (m2s-3)
Similar relationships can be found by Delichatsios (1980), and Lu et al. (1998).
2.4.3 Deviations from ideal Newtonian behaviour
Fluids do not always display Newtonian behaviour, where the shear stress is
proportional to the shear rate. Various deviations are possible (Figure 2-10). This of
course means that it is important that the viscosity (τ/γ) is measured at an appropriate
shear rate, since the viscosity is only a constant for Newtonian fluids.
Figure 2-10: Rheograms of various fluid behaviour (adapted from Perry & Green 1997).
A variety of equations have been proposed to describe non-Newtonian behaviour
(Table 2-4), with the Herschel-Bulkey equation being widely used:
n
y Kγ+τ=τ 2-31
Where: τ = Shear stress (N m-2 or kg m-1s-2) τy = Yield stress (N m-2 or kg m-1s-2) K, n = Fitted parameters
Pseudo plastic (shear thinning)
Dilitant (shear thickening) Sh
ear
stre
ss
Newtonian
Bingham plastic
Shear rate
41
Table 2-4: Equations used to describe stress(τ)/shear(γ) relationships.
Reference: Relationship: Source/comments:
Newtonian
μγ=τ
1, 2, 3 Newtonian fluids
Bingham 1922
γμ+τ=τ ∞y
1, 3 Bingham plastics
nKγ=τ 1, 2, 3 Shear thinning
ηγ=τ−τ y 2, Bingham plastic
Herschel-Bulkey ny Kγ+τ=τ
2, 3, Bingham plastic
Casson ( )2cy K γ+τ=τ
3, Non-linear plastic
Prandtl 1928, Eyring 1936 ⎟
⎠⎞
⎜⎝⎛ γ=τ −
BsinhA 1
2, Pseudo plastics
Philippoff 1935
( )2s
o
/1 ττ+
μ−μ+μ
τ=γ
∞∞
2, Pseudoplastics
Powell and Eyring 1944 ( )
nPE
PEoa t
)1tln(γ+γ
μ−μ+μ=μ ∞∞ 3, Time dependent
Sisko 1958 cba γ+γ+τ
2, Hydrocarbon greases
Moore 1959
( )
( )λγ+−=θλ
γλ+μ=τ
baadd
co
2, Thixotropic
Ritter 1961, Ritter & Govier 1970 DR
,ss
,so,sD
s2
o,s
,ss Kloglogo,
Klog −θ⎟⎟⎠
⎞⎜⎜⎝
⎛
τ−τ
τ+τ−=⎟
⎟⎠
⎞⎜⎜⎝
⎛
τ−τ
τ−τ
∞
∞∞ 2, Thixotropic
Meter 1963 ( ) 1
m
oa
/1 −α∞
∞ττ+
μ−μ+μ=μ
2, 3 Pseudoplastics in laminar pipe flow
White & Metzner 1963 t
d2 j,irj,iaj,i ∂
τ∂θ+μ−=τ
2, Viscoelastic
Cross 1965
32
1o
aαγ+
μ−μ+μ=μ ∞
∞ 2, 3 Pseudoplastics
Cheng and Evans 1965 ( ) ( )λγ=
θλ
γγλ=τ 21 fdd ,f
2, Thixotropic
Ellis 1967 ( )ττφ−φ=γ −α 11o
2, Empirical
Sestak 1988 ( )λγ+τ+γ+τ=τ n11y
nyo KK
3, Thixotropic Liu & Masliyah 1996 μγ+γ=τ G 3, Viscoelastic Kranenburg 1999
)d3/(2
pym
f
25
−
⎟⎟⎠
⎞⎜⎜⎝
⎛
φφ
μγ=τ Fractal aggregates
1: Perry and Green 1997 2: Govier and Aziz 1972 3: Liu and Masliyah 1996
42
2.4.4 Effect of suspended solid on viscosity
The viscosity of a suspension increases as a function of the solid volume fraction. In
1908 Einstein proposed a relationship based on the first terms of a series expansion:
( )...K1 Eos +φ+μ=μ 2-32
where: μs = Viscosity of suspension (kg m-1s-1, N s m-2) μo = Fluid viscosity (kg m-1s-1, N s m-2) KE = Constant (2.5) φ = Solid volume fraction [0,1] Einstein originally proposed that KE = 1, but later, famously, revised it to = 2.5.
Equation 2-32 is only correct for very dilute solutions, and various alternative equations
(Table 2-5) have been formulated by adding additional terms.
For suspensions of high solid fraction the viscosity is typically given as a function of
the maximum possible solid fraction (φm) in the form (Govier & Aziz 1972, Liu &
Masliyah 1996):
k
mos 1
−
⎟⎟⎠
⎞⎜⎜⎝
⎛φφ
−μ=μ 2-33
The maximum solid volume fraction is typically taken as 0.6-0.7 (Fleer & Scheutjens
1993, Strenge 1993, Schramm 1996, Bustos et al. 1999). At this point the particles are
taken to be in close contact (gelation) and form a continuous network that resists shear,
causing the viscosity to rise exponentially.
The particle shape and surface charge may also affect the viscosity. Various workers
(Greenburg et al. 1965, Mishra et al. 1970, Hiemenz 1986) report that smaller particles
tend to increase the viscosity, due to a film of water bound to the particle that increases
the effective volume occupied by the particle. Smaller particles have a larger surface
area (per unit mass), leading to an increased viscosity.
Irregularly shaped particles also tend to increase the viscosity (Shaw 1993, Hiemenz
1986), because of the increase in surface area and reduced packing efficiency. Ionic
strength and pH may also alter the viscosity of a suspension by modifying the surface
charge and particle-particle interaction (Eirich 1960, Greenburg 1965, Horsley et al.
1984, Farrow et al. 1989, Johnson et al. 2000).
43
Table 2-5: Equations relating solid fraction to viscosity Reference: Equation: Solid fraction: Source/comments: Einstein 1908 ( )φ+μ=μ k1os k = 2.5 φ < 2-3 % Govier and Aziz 1972 Kunitz 1926
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛
φ−φ+
μ=μ 4os 15.01.0
φ < 10 %
Purchas 1981
Guth et al. 1936 ⎟⎟
⎠
⎞⎜⎜⎝
⎛
φ−φ−φ−φ+
μ=μ 2
2
os 6.9215.05.01
φ < 8 %
Vand 1949
Guth et al. 1936 ( )2os 1.145.21 φ+φ+μ=μ φ < 20 % Govier and Aziz 1972, Kruyt
1952 Vand 1947 ( ) ( ) ...KkrQ2KK
!21k1 2
122111os +⎟⎟⎠
⎞⎜⎜⎝
⎛φ⎥⎦⎤
⎢⎣⎡ −+++φ+μ=μ
low-intermediate Vand 1949
Vand 1948 ( )2
os 349.75.21 φ+φ+μ=μ Govier and Aziz 1972
Burgers & De Bruyn 1949
⎟⎟⎠
⎞⎜⎜⎝
⎛φ+
φ−μ=μ
5.111
os Kruyt 1952
Vand 1948 ( )32
os 2.1617.75.21 φ+φ+φ+μ=μ Kruyt 1952
Chong et al. 1971 ⎟⎟
⎠
⎞⎜⎜⎝
⎛φφ−
φφμ=μ
m
mos /1
/
Chiu and Don 1989
Gillespie 1983
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛
φ−φ+
μ=μ 2os 12/1
Thomas 1965
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛
φ+φ+φ+
μ=μ6.16exp00273.0
05.105.21 2
os 0 < φ < φm
Govier and Aziz 1972, Mishra et al. 1970
Frankel & Acrivos 1967
( )( ) ⎟
⎟⎠
⎞⎜⎜⎝
⎛
φφ−
φφμ=μ
31
31
m
mos
/1/125.1
0.8φm < φ < φm
Govier and Aziz 1972
Maron-Pierce-Kitano 1956, 1981
2
mos 1
−
⎟⎟⎠
⎞⎜⎜⎝
⎛φφ
−μ=μ
Liu and Masliyah 1996
Landel et al. 1963
5.2
mos 1
−
⎟⎟⎠
⎞⎜⎜⎝
⎛φφ
−μ=μ 0.4φm < φ < 0.95φm
Govier and Aziz 1972
Turian et al. 1997 ( )
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛
φφ−φφ
μ=μ β
α
m
mos /1
/
Mooney
φ−φ
=μs1
kln s Liu and Masliyah 1996, Kao
et al. 1975, Van Diemen et al. 1985
Krieger 1972
φ−
⎟⎟⎠
⎞⎜⎜⎝
⎛φφ
−μ=μ5.2
m
fos 1
Liu and Masliyah 1996, Van Diemen and Stein 1984
Taylor 1932
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
μ+μ
μ+μφ+μ=μ
p
52
pos 5.21
emulsions < 2-3 %
Govier and Aziz 1972
Shaw 1993 ( ) ( )φ+μ=μ k1ratio axial 2os non-spherical
Hiemenz 1986
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⎥⎦
⎤⎢⎣
⎡ρρ
+φ+μ=μ1
2
2
b,1os m
m15.21
bound water
Phan-Thien & Graham 1991 ( ) ⎟
⎟⎠
⎞⎜⎜⎝
⎛
φφ−φφ−
φ+μ=μ 2m
mEos /1
/5.01K1
Liu and Masliyah 1996
Kranenburg 1999
)D3/(2
m
ps
f
52
−
⎟⎟⎠
⎞⎜⎜⎝
⎛φ
φμ=μ
Fractal aggregates
Liu & Masliyah 1996 ⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛
φ−φ
φ+μ=μ
m
3y
osa
1
Liu and Masliyah 1996, Flocculated suspensions
44
2.4.5 Multi-phase mixing
Mixing can be divided into macro-scale and micro-scale mixing. Macro-scale mixing is
the large-scale mixing of the bulk-phases in the mixing vessel. It is dependent on the
vessel size and geometry, and also the flow behaviour in the vessel. Most macro-scale
mixing theory has been developed for stirred tanks where the pumping capacity of the
impeller (Perry & Green 1997) determines the macro-scale mixing.
Due to the effect of geometry, macro-scale mixing is difficult to scale-down from
industrial to laboratory scale. Laboratory experiments in small vessels have short
circulation times and good macro-scale mixing (Perry & Green 1997). Laboratory
vessels also have lower spatial variation in shear rates and have far lower Reynolds
numbers (typically 5-25 times lower) (Perry & Green 1997).
However, to get good mixing on a molecular or small particle scale (< 100 μm),
micro-scale turbulence is also required (Perry & Green 1997). This is primarily
determined by the energy input for a given volume, simplifying scale-up from
laboratory to full-scale. Since micro-scale mixing is determined primarily by the power
input, laboratory systems are typically scaled on the mean energy dissipation rate: the
familiar G. While this may adequately describe the collision rate of particles in a
homogeneous suspension (micro-scale mixing), the rate of macro-scale mixing of the
flocculant and feed suspensions is likely to be overestimated by laboratory-scale
studies.
Mixing can also be described as a blend time. This is somewhat arbitrary (Perry &
Green 1997), but can be measured experimentally, for example by adding an acid or
alkali to a transparent vessel already containing a pH indicator. This is essentially
limited to laboratory situations, but for plant studies an electrolyte can be added and
conductivity measurements taken. Hence the blend time would be the time taken to
reach a given stability of the conductivity probe, for example 5 % of the original change
when the electrolyte was added (Perry & Green 1997). Electrical impedance
tomography (Brown et al. 1985, Webster 1990, Salkeld 1991, Dickin et al. 1992,
Williams & Simons 1992) systems are now becoming available and are likely to
45
become useful for validating computational fluid dynamics predictions of macro-scale
mixing.
Uhl and Gray (1967) suggest the usual rule of thumb: “If the reaction half-time is five
times greater than the average loop circulation time, then perfect mixing is justified for
usual engineering precision”.
Nagata (1975) defines a mixing index as:
0
1Mσσ
−= 2-34
where: M = Mixing index σ = Concentration variation σo = Concentration variation at time = 0 and:
( )∫ −=L
0
22 dx)m(C)x(Cσ 2-35
where: C(x) = Concentration at point x C(m) = Mean concentration through vessel
Macro-scale mixing appears to have received little attention in the flocculation
literature, although there is general recognition that the polymer has to be well mixed
with the pulp (Gregory 1987, Bajpai & Bajpai 1995, Wigsten & Stratton 1984) to
prevent local overdosing and possibly steric re-stabilisation (Figure 2-3b). This usually
requires a reasonable level of fluid shear, however this tends to result in aggregate
breakage, and Elimelech et al. (1995) concluded that an intermediate shear level is the
best.
Elimelech et al. (1995) also note that aggregation is sensitive to the flocculant addition
point and suggest staged addition to improve flocculant distribution (as do Gregory &
Guibai 1991). Tapered shear rates are also suggested (Guibai & Gregory 1991,
Gregory & Guibia 1991, Ives & Bhole 1973). Flocculant solution used on an industrial-
scale is usually quite dilute, 0.1-1.0 % w/v, (Gregory 1987), which reduces its viscosity
and helps improve mixing (Stratton 1983).
46
2.5 Aggregate breakage 2.5.1 Cause of aggregate breakage
Aggregate breakage is generally assumed to be caused by hydrodynamic forces on the
particle due to the turbulent flow, although occasionally breakage is attributed to
particle collision (Ham & Christman 1969, Burban et al. 1989). Two mechanisms are
proposed for breakage by fluid shear; fragmentation or surface erosion. As with
aggregation, the mechanism is thought to be dependent on the aggregate’s size in
relation to the Kolmogoroff micro-scale. When the aggregate is larger than the micro-
scale, fragmentation is assumed, due to the fluctuating dynamic pressure (Gregory
1989, Glasgow & Lueche 1980). Below the micro-scale, viscous forces dominate and
cause either fracture or erosion of small particles from the aggregate surface (Gregory
1989, Glasgow & Lueche 1980).
Healy and La Mer (1964) suggested primary particle erosion based on the surface drag
using Stokes’ equation. Glasgow and Lueche (1980) point out that Stokes’ equation
was derived for creeping flows, not the turbulent flow that Healy and La Mer
considered. Around the same time, Thomas (1964) proposed bulgy pressure
deformation based on existing theories of emulsion droplet breakage. Later, Parker
et al. (1972) suggested that aggregation would continue up to a characteristic stable
aggregate size for a given system. Breakage was assumed to be via erosion, with the
maximum aggregate size (dmax) and rate equations being:
nmax GCd = 2-36
where: G = Mean shear (s-1) n = Exponent, (typically 0.5-2, Tambo & François 1991) C = Floc strength coefficient Or, alternatively as a differential:
mBGK
dtdN
= 2-37
where: KB = Floc breakage coefficient (equations given by Parker et al. 1972) m = Exponent (typically 1.5-2, see text) N = Number of particles
47
Kim and Glasgow (1987) argue against the erosion mechanism, suggesting that a range
of fragment sizes is more likely, and that eddies comparable to the Kolmogoroff micro-
scale contain too little energy to remove primary particles.
However, the general form of Equation 2-37 has proved popular in the literature (Akers
et al. 1987, Bemer & Zuiderweg 1980, Brakalov 1987, Chung et al. 1998, ∅degaard
1979, Serra & Casamitjana 1998b, Spicer & Pratsinis 1996a). Experimental values of
m vary from 1.6 (Spicer et al. 1996), 2.07 (Chung et al. 1998) and 1.89-5.62 depending
on solid fraction (Serra & Casamitjana 1998b).
This has a major impact on the aggregation process, because the aggregation rate is
generally regarded as being proportional to Gn where n ≤ 1 (i.e. Saffman & Turner
1956). That is, an increased shear rate increases the rates of both aggregation and
breakage, but breakage is dominant. This results in the typical observation
(Figure 2-11) that the aggregate particle size drops with increased fluid shear (Fair &
Gemmel 1964, Smith & Kitchener 1978, Spicer & Pratsinis 1996a, Serra & Casamitjana
1998b, Chung et al. 1998).
Data of Oles, 1992Population balancemodelα = 1y = 1.6±0.18
0 20 40 600
5
10
15
20
25
G=25 s-1
50 s-1
75 s-1
100 s-1
125 s-1
150 s-1
Dimensionless Time, Gφt
Dim
ensi
onle
ss M
ass M
ean
Floc
Dia
met
er, d
mm
/d1
A' = 0.0047±0.0002
Figure 2-11: Effect of shear on the aggregate size adapted from Spicer and Pratsinis 1996. Polystyrene particles coagulated with KCl.
48
Aggregate porosity also has a significant effect on the strength of aggregates. As the
aggregates become more porous (lower fractal dimension) there are fewer particle-
particle contact points and the aggregate becomes more fragile (Gregory 1989). For a
given number of primary particles the aggregate also becomes larger and more likely to
break. Tomi and Bagster (1975) used a structural link model to simulate the
hydrodynamic forces on various aggregate structures and suggested dmax ∝ s-0.5, where s
is the velocity gradient. They also report that both Fair and Gemmel (1964) and Ritchie
(1955) suggested dmax ∝ 1/s.
Horwatt et al. (1992) modelled the breakage of a variety of fractal structures (DLA,
RLA, HCCA, LTA, EDEN – see footer Table 2-3) in simple shear flow. Fracture was
modelled to give either a planar or irregular fracture. Not surprisingly, the planar model
predicted a higher aggregate strength, and was considered less realistic. In general,
aggregates became weaker as the fractal dimension was reduced (more porous), but the
arrangement of the primary particles also influenced the aggregate strength.
Blunt (1989) simulated the hydrodynamic forces on fractal aggregates using a steady-
state Navier-Stokes model. The model used a series of small-amplitude sinusoidal
buckles and suggested that the hydrodynamic forces act mostly on the outer protruding
tips of the aggregate, with little effect on the inner core. This generated very high local
forces at specific points of the aggregate making “breakage or deformation of the
structure very likely”. Blunt also comments that as the shear force acts mostly on the
surface of the aggregate, and since there is a broad distribution of forces on the
aggregate, the aggregate may continue to break down over a considerable length of
time.
Figure 2-12 shows the results by Lee and Brodkey (1987), who viewed wood pulp
aggregate breakage in Couette flow with a variable speed camera (400-8000 fps). They
describe two main mechanisms. Firstly, global deformation, breakage and
fragmentation; and secondly a surface erosion mechanism. In 1991 Glasgow and Liu
used a stainless steel mesh to break polymer aggregates and followed the process using
a high-speed camera. Their results essentially mirrored those of Lee and Brodkey
(1987), and they observed that in virtually all cases aggregate breakage was initiated by
49
vertical compression. This was often followed by surface erosion, however Glasgow
and Liu report that the eroded fragments were too large to be primary particles.
break
fragment
shed
stretch
disintegrate
string
doublet
thread
dispersed
fragments
aggregate
Figure 2-12: Breakage phenomena observed by Lee and Brodkey (1987).
In another photographic study, Glasgow and Lueche (1980) studied the breakage of pre-
formed clay-polymer aggregates by pumping them through a rectangular section glass
pipe. They modelled the decrease in number of the largest aggregate size range as
measured by a camera. Based on the fit between the modelled and experimental data
Glasgow and Lueche concluded that shear was responsible for breakage.
∅degaard (1979) used population balances to model aggregate breakage and concluded
that the shear breakage model based on that by Parker et al. (1972) (i.e. erosion) gave a
better fit to experimental data than a collision model based on Ham and Christman
(1969).
2.5.2 Effect of aggregate size on breakage
In Section 2.4.1 the concept of a maximum stable aggregate size (Parker et al. 1972)
was mentioned briefly (Equation 2-36). Parker et al. envisaged that aggregation would
continue until some point where the forces on the aggregate exceeded its strength, at
which point the aggregate would break. While the concept of a maximum stable
aggregate size is attractive in its mathematical simplicity, it has several limitations. It is
50
not supported by experimental evidence, where particle size distributions invariably
form some sort of smooth bell-shaped curve, and not a distinct upper size limit. It is
also inconvenient for use in a population balance model, since the particles will tend to
pile up at the largest stable size.
An alternative approach takes the breakage rate to be a continuous function of aggregate
size. This can be rationalised since there will be a wide range of fluid fluctuations and
particle orientations through time, leading to a spectrum of breakage events. Spicer and
Pratsinis (1996) and Chung et al. (1998) both used the breakage kernel (2-38) based on
earlier work by Kapur (1972), Broadway (1978) and Peng and Williams (1994):
a
iY
i VAGS = 2-38 where: Si = Breakage rate of the ith sized particle (s-1) A = Breakage rate constant (cm-3ay) G = Shear rate (s-1) Vi = Volume of ith particle a = Constant (⅓) y = Fitted parameter
The constant of ⅓ is in agreement with the model result from Pandya and Spielman
(1982).
Various attempts have been made to directly measure the strength of aggregates. Smith
and Kitchener (1978) used MgSO4 or polymer to attach glass spheres to a silica surface.
The surface was either inverted or rotated in a centrifuge and the number of spheres
remaining counted. By calculating the force on the spheres by gravity (or centrifugal
acceleration) they concluded that the force of adhesion was roughly proportional to the
size of the particle.
More recently Yeung et al. (1997) trapped individual aggregates between two capillary
tubes and measured the force (around 100 nN) required to break the aggregates using a
calibrated cantilever beam. They reported to previously (Yeung et al. 1996) finding no
correlation between aggregate size and strength in the size range 15-48 μm. They did,
however, note that the maximum aggregate strength occurred when the aggregates were
formed at intermediate shear rates. They proposed this was due to an initial
51
densification of the aggregate at low shear rates leading to stronger aggregates,
followed by weakening of the polymer links at higher shear rates as the polymer
became degraded. They suggest that the re-flocculated aggregates had a strength only
10 % that of the original aggregates.
2.5.3 Daughter particle distribution
Modelling aggregate breakage requires both a knowledge of the rate of breakage, and of
the size and number of daughter fragments (Ilievski 1996). Most workers suggest that
aggregates break into two or three similarly sized daughter aggregates (Fair & Gemmel
1964, Coulaloglou 1977, Glasgow & Lueche 1980, Pandya & Spielman 1982, Lu &
Spielman 1985, Hill 1996, Chung et al. 1998).
Breakage is frequently assumed to result in two equally sized fragments (Spicer &
Pratsinis 1996a, Flesch et al. 1999, Randolph & Larson 1988), simplifying the solution.
Alternatively, the fragments can be taken to be distributed according to some
distribution such as: normal (Valentas & Amundson 1966, Coulaloglou & Tavlarides
1977), log-normal (Valentas et al. 1966, Spicer & Pratsinis 1996a, Lu & Spielman
1985, Cheng & Redner 1988, Ducoste & Clark 1998), or binomial (Lick & Lick 1988,
Burban et al. 1989, Serra & Casamitjana 1998b).
The erosion of primary particles from the aggregate surface is occasionally suggested as
an alternative breakage mechanism. Experimental evidence using cameras has shown a
variety of breakage mechanisms (Lee & Brodkey 1987, Glasgow & Liu 1991)
(Figure 2-12) including breakage, fragmentation and surface erosion. Glasgow and Liu
(1991) reported seeing small particles eroded from the surface of a clay-polymer
aggregate, but note that the eroded particles were too large to be primary particles. This
is in qualitative agreement with Kim and Glasgow (1987), who suggest that turbulent
eddies of the size of the Kolmogoroff micro-scale have too little energy to erode
primary particles from the aggregate. Pandya and Spielman (1982) found an increase in
particles in a lower size range (although above the size of primary particles) and argued
that it was evidence of erosion.
52
Table 2-6: Breakage equations reported in the literature Reference Kernel Comment/References Thomas 1964
25
23
29
p
fi D
S −− ∈ν⎟⎟⎠
⎞⎜⎜⎝
⎛ σ∝ Inertial
212
1
p
fi D
S ⎟⎠⎞
⎜⎝⎛∈ν
⎟⎟⎠
⎞⎜⎜⎝
⎛ σ∝ Viscal
σ = surface tension term
Parker et al. 1972
mB
i Gkdt
dN=
Inertial, m = 4,Viscal, m = 2
Coulaloglou & Tavlarides 1977
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
ν∈ρ
σ−∈ν= −
95
52
31
92
d
21
kexpk
dtdn
Droplet splitting
Tomi & Bagster 1978
( ) )d(dC)d(p 3
2
1 ∈ρ= η < d < L 21
2C)d(p ⎟⎠⎞
⎜⎝⎛ν∈
μ= d < η
32
23 dC)d(p ⎟
⎠⎞
⎜⎝⎛ν∈
ρ= d ∼ η
η = Kolmogoroff micro-scale L = Eulerian macro-scale
∅degaard 1979
p
Bi GkS φ= P = 2 or 4, Attributed to Parker, Kaufman & Jenkins
Glasgow & Lueche 1980
i2ii
ii NdfS
ψτ
= Viscal
( )i
ii
NdS
34
65
21
ψ∈μρ
≈ Inertial
τ = shear stress ψ = strength parameter fi = eddy freq. scale
Pelton 1981
( ) 2i
i r32kNS
θ−θ−θ=
θ = surface thickness of flocculant covering
Pandya & Spielman 1982
mii VS ∝
M fitted to ≈ 0.33 from experimental data, i.e. breakage ∝ diameter
Lu & Spielman 1985
ii KGVS =
Akers et al. 1987
kGdtdv
−=
Erosion kernel attributed to Spielman 1970
Leu & Ghoshi 1988
c3DG
dtdN r1
nv2
nv
π=
+
Shamalou & Titchener-Hooker 1993
v
2
7i CAS ⎟⎠⎞
⎜⎝⎛ν∈
= Inertial
v7i CAS ⎟⎠⎞
⎜⎝⎛ν∈
= Viscal
Cv = aggregate volume
53
Table 2-6: Continued
Huang & Helums 1993
m
ii 1xkS ⎟⎟
⎠
⎞⎜⎜⎝
⎛φ−
= m = 0.1 φ = void fract.
Spicer & Pratsinis 1996
y
ii GAVS 31
= y = 1.6 ± 0.18 G = 25-150 s-1, (Oles 1992)
Berlin et al. 1997
( )
⎟⎠⎞
⎜⎝⎛ −= − N
oN1b e1NK
dtdN
Serra & Casamitjana 1998
31
ib
Bi VGkS =
From Spicer & Pratsinis (1996)
Lu et al. 1998
32
31
21
iSη
∈δ−=
δ = double layer η = K.M
Chung et al. 1998
ya
ii GAVS = From Chen (1990), Spicer & Pratsinis (1996)
Ducoste & Clark 1998
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
ρ=
21
21
21
dNDN
CNerfcNCS
p
3Qii
Stirred tank reactor
Kobayashi et al. 1999
21-
4max,f fNcGCd ⎟⎠⎞
⎜⎝⎛ μ= Viscal
83
32 -
5max,f fNcCd ⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛ ∈ρ= Inertial
Kramer & Clark 1999
( ) fD1
)k(ra2kNS 1b'
max'fsi μ=
Reich & Vold, Lu & Spielman, Thomas, Ray & Hogg
Flesch et al. 1999
y
i G'AS = Kusters (1991), Spicer & Pratsinis (1996)
Young et al. 2000
L'
Bi GkS = L = 2 (<KM) or 4 (>KM)
54
2.6 Population balance models 2.6.1 Smoluchowski’s equation, aggregate breakage
In earlier sections, kinetic equations (kernels) for aggregation and breakage were
described. In order to simulate the overall aggregation process, some way is needed to
keep track of the number and size of the aggregates through time. In 1917
Smoluchowski proposed the first population balance model, describing aggregation
only:
∑∑∞
=
−
=+=
β−β=1i
kiik
1k
kji,1ijiij2
1k NNNNdt
dN 2-39
where: Nk = Number of k sized particles βij = Rate of collision between i and j sized particles (aggregation kernel) t = Time (s) The first term on the right of Equation 2-39 describes the increase in number of k-sized
particles through the aggregation of i and jth particles. The constant of ½ is included to
avoid double-counting. The second term describes the loss of k-sized particles through
aggregation of a k-sized particle with another particle.
To model aggregate breakage another two terms are needed (Argaman & Kaufman
1968, Randolph & Larson 1988, Lick & Lick 1988, Spicer & Pratsinis 1996a, Serra &
Casamitjana 1998b, Kramer & Clark 1999):
∑∑∑∞
+=
∞
=
−
=+=
Γ+−αβ−αβ=1kl
lllkkk1i
kiik
1k
kji,1ijiij2
1k NSNSNNNNdt
dN 2-40
where: Sk = Breakage rate (kernel) of kth sized particles
Γlk = Breakage distribution function (number of k size particles produced from the breakage of a l sized particle) αij = Capture efficiency
In this case the third term on the right describes the loss of k-sized particles due to
breakage. The final term describes the gain in k sized particles through the breakage of
an i sized particle to give a k-sized daughter fragment.
55
A population balance is a system of differential equations, with one equation for each
size particle. However, a particle size distribution is effectively a continuous function
(many particles), and covers a wide range of sizes (typically several orders of
magnitude). Various analytical solutions have been formulated, but usually only for
breakage or aggregation separately (Patil and Andrews 1998), and only for the simplest
aggregation or breakage kernels e.g. size-independent (Hounslow et al. 1988). Patil and
Andrews (1998) describe an analytical solution for simultaneous aggregation/breakage,
but the solution requires the assumption that the total particle number does not change
with time.
2.6.2 Discrete models, numerical solution
Analytical solutions are generally unavailable for physically realistic kernels, but
simplified kernels (e.g. size-independent) are used to check the accuracy of numerical
solutions (Hounslow et al. 1988, Litster et al. 1995, Smit et al. 1994). For a numerical
solution, the continuous spread of particle sizes is sectioned (discretised), in much the
same way that particle sizing instruments give histograms where each bar covers a
range of sizes.
Discretisation allows the population balance to be written as a closed set of ordinary
differential equations, which can be readily solved numerically. Initially workers
writing population balances had to write their own numerical routines, typically
Runge-Kutta methods coded in Fortran (Ilievski 1996), but recently commercial
packages have become available e.g. SPEEDUP (Johnson & Cresswell 1996), NIMBUS
(Ilievski 1996), MATLAB (Biggs & Lant 2002), gPROMS (this work).
In 1964 Fair and Gemmel coded one of the earliest discretised population balances
based on Equation 2-38 by Smoluchowski (1917). The size range was discretised into
20 equally spaced bins by volume. The balance was solved using an iterative technique
in Fortran. Aggregate sizes were limited either by stopping aggregation that would
cause oversize particles, or by breakage of oversized particles.
Batterham et al. (1981) wrote a population balance for pelletization kinetics
(aggregation only) in balling circuits, which Hounslow et al. (1988) simplified to:
56
( )∑ ∑−
=
−
−−−−−−−−−
β−β++
β+β+β=
2i
1j j
2jijjij,i
ij
21i1i,1ii1ii,1i4
31i2i1i,2i8
3i
NNN21
NNNNNdt
dn
2-41
Hounslow et al. (1988) noted that the model correctly predicts particle volume (third
moment) but does not accurately predict total particle number (zeroth moment). Kumar
and Ramkrishna (1996) pointed out that Batterham’s model double counts equal sized
particle aggregation, although the errors are somewhat self-cancelling. Despite the
criticisms, Batterham et al. proposed an elegant method of discretisation, following a
simple numerical progression where volume (or mass) doubles for each size range:
2V
Vi
1i
=+
2-42
or: 3
i
1i
2L
L=
+
2-43
where: Vi = Volume of the ith channel Li = Length of the ith channel This method of discretisation has several advantages. The spacing is finer for smaller
particles, giving additional accuracy for the large number of small particles which
would be almost completely lost if constant width discretisation was used (Hounslow
et al. 1988). The discretisation also covers a wide range of particle sizes with
comparatively few channels, reducing the computation time. This discretisation was
also previously consistent with the output from a Coulter counter (Hounslow et al.
1988) and Reynard (R10) series sieves (Allen 1990, p. 192), simplifying comparisons
between modelled and experimental data. Batterham et al.’s discretisation method has
been used by other workers, notably Hounslow et al. (1988), and Spicer and Pratsinis
(1996a) (Equations 2-44 & 2-45).
A variety of population balance models have subsequently been described (Pandya and
Spielman 1982, Lu & Spielman 1985, Hill 1996, Randolph & Larson 1988, Hounslow
et al. 1988, Marchal et al. 1988, Lick & Lick 1988, Serra & Casamitjana 1998a, Litster
et al. 1995, Wynn 1996, Chung et al. 1998, Nickmanis & Hounslow 1998).
57
Hounslow et al. (1988) used the discretisation described by Batterham et al. (1981) and
derived (see Appendix):
∑∑∑∞
=
−
=−−−
−
=−−
+− β−β−β+β= −
ijjj,ii
1i
1jjj,ii
21i1i,1i
2i
1j21
j1ij,1i1iji NNN2NNNN2
dtdN ij 2-44
Hounslow et al.’s model describes aggregation only. The first two terms on the right
describe the aggregation of smaller particles into the ith size range, and the second two
terms describe the loss of particles in the ith size range as they aggregate with other
particles (see Appendix).
Hounslow et al.’s balance has been widely used and referenced (Atteia 1998, Cresswell
et al. 1994, Ilievski & White 1994, Ilievski & White 1995, Kumar and Ramkrishna
1997a, b, c, Patil & Andrews 1997, Simons 1996, Spicer & Pratsinis 1996a, Spicer et al.
1996, Spicer 1997, Flesch et al. 1999) primarily because it accurately predicts both
particle number (zeroth moment) and volume (third moment) (Hounslow 1988, Kumar
& Ramkrishna 1997) and is simple to solve, with a minimum of terms containing
summations rather than integrals.
However, Hounslow et al.’s population balance has a relatively coarse discretisation,
and gives slight errors in the first and second moments (length and area weightings).
Kostoglou and Karabelas (1994) compared population balances by Batterham 1981,
Hounslow et al. 1988, Marchal et al. 1988, and Gelbard et al. 1990 and found that
Hounslow et al.’s balance gave the best performance. Similar results are shown by
Litster et al. (1995), who showed that for particles coalesced from 50 primary particles
the errors in the first (length) and second (area) moments were in the order of 2 %, with
no error in the zeroth or third moments.
Hounslow et al.’s population balance was written primarily to describe crystallisation
systems, and the paper also describes crystal growth kinetics. For flocculation
applications growth terms are not needed, but breakage terms are. Spicer and Pratsinis
(1996a) added two terms to Hounslow et al.’s balance:
58
∑
∑∑∑∞
=
∞
=
−
=−−−
−
=−−
+−
Γ+−
β−β−β+β= −
ijjjj,iii
ijjj,ii
1i
1jjj,ii
21i1i,1i
2i
1j21
j1ij,1i1iji
NSNS
NNN2NNNN2dt
dN ij
2-45
These terms describe the loss of particles from the ith channel by breakage (second last
term) and increase in the number of ith particles due to the breakage of larger aggregates
(final term). In this case Si is the breakage kernel and Γi,j is the breakage distribution
(see Section 2.4.3). The simplicity of the discretisation used by Hounslow et al. makes
the coding of Equation 2-45 relatively straightforward.
Solving population balances numerically allows physically realistic kernels to be solved
relatively simply. However, solving discretised systems of differential equations
numerically typically leads to numerical errors. Using finer particle size discretisation
and decreasing the step size is likely to help, but other methods are required to check
the accuracy of the results. With aggregation, the mass of the solid is constant through
time (as opposed to a crystallisation with surface deposition), and the population
balance should conserve mass (third moment).
However, a model may correctly conserve mass but not correctly predict particle
number (e.g. Batterham et al. 1981). Particle sizing is frequently performed using
Coulter counters, hence the correct prediction of the total particle number (zeroth
moment) is frequently described in the population balancing literature (Kumar and
Ramkrishna 1996, Hounslow et al. 1988, Litster 1995). Typically the moments from
zero through to the third (particle number, length weighted, area weighted and volume
or mass weighted) are compared to analytical solutions, usually for a very simple
formulation of the aggregation kernel, e.g. size independent (Hounslow et al. 1988,
Litster et al. 1995).
Higher moments are also sometimes checked, usually the sixth. The physical
interpretation of such a high moment is obscure, but it is used to check the mathematical
stability and accuracy of the model. Weighting the system so heavily with size tends to
reveal the errors in the few larger particles in their wider channels. Smit et al. (1995)
59
used the sixth moment as an indication of mathematical gellation, where the population
balance rapidly “runs away” producing a very few super-sized aggregates. In practice,
the use of a breakage kernel effectively limits the aggregate size.
Population balance models are usually compared to experimental data, with the
experimental data used to estimate unknown parameters of the model. This is usually
performed with a sum-of-squares minimisation routine (Ilievski et al. 1993, Ilievski &
White 1994). The data is also used as evidence for various formulations of aggregation
and breakage kernels; the “inverse problem” (Muralidar & Ramkrishna 1986, Ilievski &
White 1994). However, caution is required in interpreting the fit between modelled and
experimental data, because even a physically unrealistic model may give a reasonable
agreement with enough fitted parameters, or if there is limited experimental data.
In conclusion, the architecture of population balances is well established, with several
successful models already being available, in particular the models by Hounslow et al.
1988, and Spicer and Pratsinis 1996a. Although other, slightly more accurate,
population balance models are also available, they are also considerably more complex
reducing the incentive to use them, especially as a sub-model within CFD simulations.
In addition, the level of inaccuracy in the numerical population balance models is
probably relatively minor in comparison to the inaccuracy in the experimental data
(Ilievski & White 1995). The availability of existing population balance models, and
commercial simulation packages capable of solving the differential equations, will
allow future research efforts to focus on the determination of appropriate aggregation
and breakage kernels for specific systems.
2.7 Solid-liquid separation by gravity settling The settling of solid particles in a viscous fluid can occur in several different regimes,
depending on the solid volume fraction and the aggregate size and density (Figure
2-13). At very low solid fractions, individual particles are well separated in the fluid
and settle freely with little, or no, interaction with each other. The settling velocity of
solid spherical particles in creeping flow was first described by Stokes (1851), but
Stokes’ equation (Equation 1-3) has since been extended to suit other particle
geometries and higher settling velocities.
60
Figure 2-13: Effect of solid fraction and degree of flocculation on settling behaviour (adapted from Perry and Green 1997).
At higher solid fractions (Figure 2-13) individual particles begin to interact with each
other (Bhatty et al. 1982, Perry & Green 1997), until a critical point when the particles
are constrained to settle together as a loosely connected mass (Govier & Aziz 1972,
Tong et al. 1998, Perry & Green 1997, p. 18-60). This mass subsidence is referred to as
hindered or zone settling, (or even more confusingly, sometimes free-settling, Fitch
1975A, Pearse 1977) and is characterised by a distinct mudline separating clear
supernatant liquor from the settling mass of solids (Tong et al. 1998, Pearse 1977,
p. 15). Although the settling velocity is slower than the average free-settling velocity of
the same particles (Wesely 1985, Dahlstrom & Fitch 1985, Pearse 1977),
hindered-settling has the advantage that it captures fine particles that would otherwise
report to the overflow (Wesely 1985). In this region, solid settling is still restrained
only by hydrodynamic forces, and the settling velocity is a function of solid fraction and
aggregation state (Pearse 1977).
Eventually, the mass of settling solid will contact the base of the vessel or thickener and
sediment compaction begins. The rate of solid settling is now restrained by both
hydrodynamic and mechanical forces. Some sediment compaction is usually necessary
to achieve the required underflow concentration. Compaction is often encouraged by
Solid
frac
tion
Highly aggregated
particles
High concentration
Low concentration
Degree of particle coherence
Hindered settling regime
Particulate settling regime
Compression regime
Totally discrete particles
61
allowing the bed to build up (typically to ∼ 1 m, Galvin & Waters 1987, Pearse 1977,
Perry & Green 1997) to increase the overburden weight of the bed. The action of
raking the solids towards the underflow pump at the base of the thickener also helps de-
water the sediment by disrupting the bed structure, and by forming channels that
increase the drainage rate of the displaced fluid (Perry & Green 1997, Farrow et al.
2000).
2.7.1 Particulate settling (low solid fraction)
Solid spherical particles
The terminal settling velocity of individual particles is determined by the various forces
acting on the particle, as shown by Figure 2-14 (Stokes 1851, Clift et al. 1978, Seville
et al. 1997, Happel & Brenner 1973).
Figure 2-14: Forces acting on a free-settling particle (adapted from Wesely 1985).
where: Fg = force of gravity (N, or kg m s-2) Fd = force of drag (N, or kg m s-2) Fb = force of buoyancy (N, or kg m s-2) The general equation for the settling velocity of solid spheres is:
dl
ls2
C3)(dg4
Uρ
ρ−ρ= 2-46
where: U = Settling velocity (m s-1) d = Particle diameter (m) g = Gravity (9.8 m s-2)
ρ = Density (s = solid, l = liquid) (kg m-3) Cd = Coefficient of drag
Fb
Fg
Fd
Particle
62
Stokes solved the Navier-Stokes equation and showed that for creeping flows past rigid
spheres (see Clift et al. 1978):
Re24Cd = 2-47
Where Re, Reynolds number for spheres is:
μρ
= lUdRe 2-48
where: μ = Fluid viscosity (kg m s-2) Combining Equations 2-45, 2-46 and 2-47 gives Stokes’ law:
μρ−ρ
=18
)(gdU ls
2
2-49
Stokes’ law rapidly breaks down (Concha & Almendra 1979) at higher Reynolds
numbers as fluid inertial effects become significant in comparison to the viscous
resistance. Alternative drag coefficients have been proposed for higher Reynolds
numbers, usually as a function of the Reynolds number (Figure 2-15). The settling
velocity is overestimated by Stokes’ law by 2 % at Re = 0.1, 4 % at Re = 0.2 and 20 %
at Re = 1 (Seville et al. 1997, Rushton et al. 1996). Figure 2-16 shows the effect of
particle diameter and settling velocity on the Reynolds number, indicating regions
(Re < 1) where Stokes’ law can be used to describe the settling velocity.
63
ψ = 0.125ψ = 0.220ψ = 0.600ψ = 0.806
ψ = 1.000
10000
100060040020010060402010
6421
0.60.40.20.1
0.10.010.001 1 10 100 1000 10000 105 106
Reynolds number Dpvρ=NRe
μ
Dra
g co
effic
ien t
2F
d g c
=C
dv2ρS
Figure 2-15: Effect of Reynolds number and sphericity (ψ) on the drag coefficient (adapted from Wesely 1985).
0
10
20
30
40
50
60
70
80
90
100
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Particle size (µm)
Sett
ling
rate
(m h
r-1)
Re = 10
Re = 3
Re = 1
Re = 0.1
Re = 0.3
Re = 0.01
Figure 2-16: Iso-plot of particle Reynolds number as a function of size and settling rate, calculated from Stokes’ law for the free-settling rate of spherical particles. Stokes’ law is only accurate for low Reynolds numbers (see text) and hence is only appropriate for particles bounded by the lower lines in the figure. (constructed from Equation 1-3, 2-49).
64
A variety of alternative drag coefficients have been published (Table 2-7), generally in
the (dimensionless) form of Equation 2-50 for particles settling at higher Reynolds
numbers when inertial effects become significant (Pearse 1977).
nd RekC = 2-50
where: Cd = Coefficient of drag (dimensionless) Re = Reynolds number for spheres (dimensionless) k = Constant n = Exponent Very high Reynolds number settling (1000 < Re < 350 000) is described by Newton’s
(1687) equation (Equation 2-51), where the drag coefficient is a constant of 0.445,
independent of the viscosity and settling velocity, but highly shape dependent
(Figure 2-15) (Clift et al. 1978, Perry & Green 1997):
( )lN
ls2N C3
dg4U
ρρ−ρ
= 2-51
where: UN = Settling velocity given by Newton’s equation (m s-1)
ρ = Density (s = solid, l = liquid) (kg m-3) g = Acceleration due to gravity (9.8 m s-2) CN = Drag coefficient given by: 5.31 - 4.88ψ ψ = Sphericity, ratio of surface area of equivalent sphere to actual surface area (see also: Amirtharajah et al. 1991)
Gregory (1997) comments that in most cases porous aggregates settle sufficiently
slowly for Stokes’ law to apply.
65
Table 2-7: Coefficient of drag (Cd) as a function of Reynolds number (Re)
Reference Cd Re Source Stokes 1851 24/Re Re → 0 1 Oseen 1927 (24/Re)(1+3Re/16) Re << 1 1 Goldstien 1929
(24/Re)(1 +3Re/16 –19Re2/1280 + 71Re3/2480 – 30179Re4/34406400 + 122519Re5/56742400)
Re << 1
1
Clift et al. 1978
3/16 + 24/Re
Re < 0.01
3
Clift et al. 1978
(24/Re)(1 + 0.1315Re(0.82-0.05logRe))
0.01 ≤ Re ≤ 20
1, 3
Pruppacher & Steinberger 1968, Beard & Pruppacher 1969
(24/Re)(1 + 0.102Re0.955)
0.02 ≤ Re ≤ 3
1
Kurten et al.1966
0.28 + 6/Re0.5 + 21/Re
0.1 < Re < 4000
3
Gilbert et al. 1955 0.48 + 28Re-0.85
0.2 < Re < 2000 3
Landmuir & Blodgett 1948 (24/Re)(1 + 0.197Re0.63 + 2.6x10-4Re1.38)
1 < Re < 100 3
Lin & Lee 1973 (24/Re)(1 + 0.2207Re0.5 + 0.0125Re)
1 ≤ Re ≤ 1000 1
Allen 1900 30Re-0.625
1 < Re < 1000 3
Allen 1900 10Re-0.5 2 < Re < 500
3
Schiller & Naumann 1933
(24/Re)(1 + 0.15Re0.687) 2 ≤ Re ≤ 800 1, 3
Pruppacher & Steinberger 1968, Beard & Pruppacher 1969
(24/Re)(1 + 0.115Re0.802) 3 ≤ Re ≤ 20 1
Proudman & Pearson 1957
(24/Re)(1 + 3Re/16 + 9Re2ln(Re/2)/160) Re ≤ 1.3 1
Kurten et al.1966
2 + 24/Re Re < 10 3
Clift et al. 1978 (24/Re)(1+ 0.1935Re0.6305) 20 ≤ Re ≤ 260
1, 3
Pruppacher & Steinberger 1968, Beard & Pruppacher 1969
(24/Re)(1 + 0.189Re0.632)
0 ≤ Re ≤ 4000
1
Kurten et al.1966
1 + 24/Re Re < 100 3
Lapple 1951
(24/Re)(1 + 0.125Re0.72) Re < 1000 3
Abraham 1970
0.2924(1 + 9.06Re-0.5)2 Re < 6000 3
Matsumo & Mori 1975
K/NRe0.82 2, Fractal
Tambo & Watnabe 1979, Moudgil & Vasudevan 1989
K/NRe
2, Fractal
Dirican 1981, Klimpel & Hogg 1986, Hogg et al. 1987
K(1+K’/NRe
0.5)2
2, Fractal
Newton 1687 0.445 (accurate to 13%) 1E+3 ≤ Re ≤ 3.5E+5 4 1: Nguyen-Van et al. 1994 2: Moudgil and Vasudevan 1989 3: Clift et al. 1978 4: Perry and Green 1997
66
Other shapes
Stokes’ equation was written for solid spherical particles, however aggregates formed in
thickeners have highly irregular shapes. Various workers (Pettyjohn and Christiansen
1948, Happel and Brenner 1973, Kousaka et al. 1981, Seville et al. 1997) have studied
the effect of shape on free-settling. Pettyjohn and Christiansen followed the settling
behaviour of a wide range of regular metal shapes in glucose/water solution and
calculated a wide range of correction factors (k) under various conditions. The
correction factors were typically in the range 0.7-1.0, i.e. the particles settled slightly
slower than predicted by Stokes’ equation (Heiss & Coull 1952). Happel and Brenner
(1973) and Seville et al. (1997) considered shapes of various aspect ratio, settling with
various orientations (Figure 2-17). Again, both showed that shapes with aspect ratios
typical of aggregates settle slightly slower than predicted by Stokes’ equation.
0.1 0.5 5 10
0.50
0.60
0.70
0.80
0.90
1.00
Constant circularity ofprojected area
Variable circularity ofprojected area
- Cylinders- Rectangular parallelograms
- Cylinders- Rectangular parallelograms
Spheroidedgewise
Spheroidflatwise
Height-diameter ratio, h/d
Settl
ing
f act
or, K
1.0
Figure 2-17: Effect of particle shape on free-settling velocity in creeping flows (Heiss and Coull 1952, adapted from Happel and Brenner 1973).
At high Reynolds numbers the particle shape has a larger impact on the particle drag
(Figure 2-15).
Aggregate Porosity
Aggregate porosity has two counteractive effects on the settling velocity of an
aggregate. Firstly, it may allow fluid to pass through the structure, reducing the drag.
Secondly, it dramatically reduces the aggregate density.
67
Veerapaneni and Wiesner (1996) compared various models that assume constant
internal permeability that suggest that the flow through the aggregate structure only
affects the settling velocity for fractal dimensions lower than approximately 2.0, less
than found in practice (Table 2-3). Gregory (1997) used Brinkman’s (1948)
permeability model to model the effect of fractal dimension under typical industrial
conditions and also showed that flow through the structure had little impact with a
fractal dimension of over 2.
The major effect of porosity on particle settling is from the reduction in the effective
density at lower fractal dimensions (Figure 2-18), given by (Gonzales & Hill 1998,
Atteia 1998, Ellis & Glasgow 1999, Manning & Dyer 1999):
( ) ( )3D
p
agglslagg
f
dd
−
⎟⎟⎠
⎞⎜⎜⎝
⎛ρ−ρ=ρ−ρ 2-52
where: d = Diameter (agg = aggregate, p = primary particle)(m) Df = Mass-length fractal dimension
ρ = Density (agg = aggregate, l = liquid, s = solid) particle (kg m-3)
0
500
1000
1500
2000
2500
3000
0 10 20 30 40 50 60Relative aggregate size (dagg/dp)
Agg
rega
te d
ensi
ty (k
g m
-3)
Figure 2-18: Effect of aggregate porosity on density. Assuming: ρs = 2710 kg m-3, ρl = 1000 kg m-3, Df = 2.4.
Numerous experimental studies have related the particle settling velocity to the particle
size under various conditions, flocculant/coagulant types, particle material etc. These
studies invariably show significant experimental scatter, but usually produce a straight
line relationship between log (settling velocity) vs. log (particle diameter) (Sternberg
68
et al. 1996, Matsumoto & Mori 1975, Kusuda et al. 1981, Allain et al. 1996, Adachi &
Kamiko 1994, Huang 1994, Adachi & Tanaka 1997, Dyer & Manning 1999), although
some workers have plotted settling velocity vs. size (Farrow & Warren 1989, 1993).
2.7.2 Hindered settling (intermediate solid fraction)
Effect of solid fraction on hindered settling
The transition from free-settling to hindered (or zone) settling is gradual, and generally
poorly defined (Bhatty et al. 1982, Perry & Green 1997). However, hindered settling is
characterised experimentally by a distinct settling line (mudline), leaving a clear
supernatant above (Fitch 1975a, 1987, Tong et al. 1998). This situation may be brought
about by two separate conditions (Pearse 1977), firstly, if the bulk settling velocity of
the suspension is slower than the free-settling velocity of the slowest (probably
smallest) particles. In this situation the small particles will catch up to the mudline,
leaving a clear supernatant. The onset of hindered-settling is a function of the particle
size distribution and the bulk-settling velocity at a given concentration. In the artificial
situation where all the particles are the same size and shape, a distinct settling line will
appear regardless of the solid fraction.
The second condition that will cause a clear supernatant is when differential settling
results in aggregation (Pearse 1977). In this situation large, fast-settling aggregates
collide with slower settling particles and sweep them from solution (Fitch 1975a). The
onset of a distinct mudline in this situation will be a function of the capture efficiency
(Bhatty et al. 1982), the collision radius, the fractal dimension and the solid
concentration.
The hindered-settling velocity decreases as a function of solid fraction (Table 2-8), due
to the increase in the hydrodynamic force (Govier & Aziz 1972, Perry & Green 1997),
which increases as the suspension behaves increasingly like a permeable network.
Settling is also slowed by an increase in the rise velocity of displaced fluid (Perry &
Green 1997). Flocculation has the effect of creating channels that ease the fluid flow
through the solid network, increasing the settling velocity (Govier & Aziz 1972).
69
Various empirical and theoretical relationships between settling velocity and solid
fraction have been proposed (Table 2-8), with the relationship by Richardson and Zaki
(1955) being the most widely used:
( )n
o 1UU φ−= 2-52 where: φ = Solid volume fraction (m3m-3)
n = Exponent, usually taken as 4.65 (Table 2-9) U = Suspension settling velocity (m s-1) Uo = Free settling velocity of same size particle (m s-1)
Richardson and Zaki’s (1955) equation has been fitted successfully to a wide range of
suspensions, including flocculated suspensions (Table 2-9), with the exponent typically
found to be 4.65 in Stoke’s regime (Re < 0.2).
70
Table 2-8: Effect of solid fraction on the hindered-settling velocity Reference Settling velocity (U, m s-1) Particle
size (μm) Solid fraction (φ)
Comments
Burgers 1941, 1942
φ+=
88.61UU o
0 < φ < 0.1 1 , 2
Steinour 1944a,b,c )(UU 2
s εφε= 13.5 & 1740
3, widely used
Uchida 1952
31
1.21
UU o
φ+=
0 < φ < 0.1 2
McNown & Lin 1952
31
6.11
UU o
φ+=
0 < φ < 0.1 2
Richardson & Zaki 1955, Maude and Whitmore 1958
( )no 1UU φ−= 100-1000
0 < φ < 0.35
3, widely used
Michaels & Bolger 1962 ( )nfloc k1UU φ−=
0.2-2 3
Thomas 1963
φ−= 9.5oeUU
0 < φ < 1 1, Empirical
Famularo & Happel 1965
31
1
UU o
γφ+= γ = 1.3 ± 0.24
0 < φ < 0.1 1, 2, 3
Smith 1965, 1966 )(3FU)(2FU)(1FU iiiiii i∑ φ+φ+φφ= φ
2900-6000
0.008 < φ < 0.250
3
Dollimore & McBride 1968 ( )noUU φ=
widely used
Batchelor 1972 )66.61(UU o φ−=φ
3 Govier & Aziz 1972
( )φφ−
=101U
U3
o
0 < φ < 1 1
Lockett & Al-Habbooby 1973 s,ls1o1 )1(UU φ−φ−= 600-2000 0.05 < φ < 0.5
Barnea & Mizrahi 1973 [ ])1(3/5xpε)1(
)1(UU31
o
φ−φφ+
φ−=φ
5-1740
3
Garside & Al-Dibouni 1977
( ) 2.0
R
R Re06.0U)(B
AU +∈=−∈
∈−
170-3100
0.05 < φ < 0.5
3
Ishii & Zuber 1979 Re),('FUU o φ=φ 360-6000 0.05 < φ < 0.5 3
Masliyah 1979 μ
∈ρ−ρ∈=
18)(F)(gxU ms
2i
70-400
0.1 < φ < 0.5
3
Concha & Alemendra 1979 )(f1)(fx0921.01
x52.20U 1
2
2dd
* 21
32
φ⎥⎦⎤
⎢⎣⎡ −⎟
⎠⎞⎜
⎝⎛ φ+=
3
Reed & Anderson 1980 )0.41(
)88.11(UU o
φ+φ−
=φ
3
Bhatty et al. 1982 [ ])11/(1oUU ∈−∈∈= 0.05 < φ < 0.6 3
Buscall et al. 1982 P'k
P1UU o ⎟
⎠⎞
⎜⎝⎛ φ−=φ
3
0.01 < φ < 0.5
3
Glendinning & Russel 1982 )55.61(UU o φ−=φ φ < 0.2 3
Patwrdhan & Chi Tien 1985 ⎟⎟
⎠
⎞⎜⎜⎝
⎛ρ−ρρ−ρ
=∈ −φ
fsi
msi)2n(i
oUU
i
0.15 < φ < 0.5 3
Patwrdhan & Chi Tien 1985 φφ μμ∈= /UU o 2000-5000 0.1 < φ < 0.5 3
Mandersloot et al. 1987
D/x8.1770.4so
*)K1(UU +φ+=
0.05 < φ < 0.8 3
Daves & Gecol 1994 ( ) ( )( )∑ φ−+φ−= jiiij
Ss SS11UU ij
U = settling velocity (m s-1), Us = settling velocity given by Stokes’ law (Equation 1-3, 2-46) (m s-1), Uo = settling velocity at infinite dilution (m s-1) (may = Us), φ = solid fraction (1-ε) (dimensionless), ε = liquid volume fraction (1-φ) (dimensionless), Sij = function of polydispersity. 1: Govier and Aziz (1972). 2: Famularo and Happel (1965). 3: Williams and Amarasinghe (1989).
71
Table 2-9: Exponents reported for Richardson & Zaki’s Equation Reference Exponent Comment Bhatty et al. 1982 Not stated
Davies & Gecol 1994 5.1 Attributed to Garside & Al-Dibouni 1977
Dollimore & Horridge 1971 Not stated
Farrow et al. 1985 Not stated
Fitch 1975B 4.65 References Michaels & Bolger 1962, can use for
aggregates provided use effective solid fraction
Font 1991 4.65
Hogg & Bunnaul 1992 Not stated Suggests maximum effective solid volume fraction approx 0.5-0.6
Kanungo & De 1970 4.7
Maude & Whitmore 1958 ∼ 5 Large table of exponents for different systems, 4.15-9.35
Michaels & Bolger 1962 4.65 Aggregates
Perry & Green 1997 4.65
2.33 Stokes’ regime Newton’s regime
Turian et al. 1997
4.8 5.4 5.8
Spheres Cube Brick shape
Di Felice et al. 1993
4.65 4.4Re-0.03
4.4Re-0.1 2.4
Re < 0.2 0.2 < Re < 1 Re = particle Reynolds number 1 < Re < 500 Re > 500
Rushton et al. 1996
4.65 4.35Re-0.03
4.44Re-0.1
4.45Re-0.1 2.39
Re < 0.2 0.2 < Re < 1 1 < Re < 200 200 < Re < 500 Re > 500
Estimating the continuous settling flux from batch data
Thickeners usually operate under approximately steady-state conditions, with
continuous feed, underflow, and overflow rates. However, experimental settling data is
normally collected from a batch-settling test in a measuring cylinder, or similar settling
column. Various methods are available for estimating the continuous settling flux (and
hence required thickener area) from batch data.
72
Most methods for sizing thickeners assume that some critical solid fraction will limit
the settling flux. The settling flux is defined (Pearse 1977, Rushton et al. 1996) as the
settling velocity multiplied by the solid fraction:
φ=ψ U 2-54 where: ψ = Settling flux (m s-1) U = Settling velocity (m s-1) φ = Solid volume fraction (m3m-3) Multiplying by the solid density (ρs) gives the mass settling flux (kg s-1m-2, or t m-2h-1).
Mishler (1912) pioneered the use of laboratory tests to predict settler performance by
assuming that the initial feed settling velocity was balanced by the overflow fluid rise
velocity. Coe and Clevenger (1916) suggested measuring the initial settling velocity of
various suspensions, ranging from the feed concentration to the desired underflow
concentration. The settling flux relative to the underflow withdrawal flux is then
calculated from:
u
11U
φ−
φ
=ψ 2-55
where: U = Settling velocity (m s-1) φ = Solid fraction (m3m-3) φu = Underflow solid fraction (m3m-3)
Coe and Clevenger assumed that the settling velocity was a unique function of the solid
fraction, which may not be true for flocculated suspensions if the degree of flocculation
is also dependent on the solid fraction (Williams & Simons 1992, Svarovsky 2000).
Equation 2-55 is then used to calculate the settling flux from feed to underflow dilution,
with the concentration giving the minimum flux rate used to calculate the required
thickener area from:
l
ul
U
11FA
⎟⎟⎠
⎞⎜⎜⎝
⎛φ
−φ
= 2-56
73
where: A = Thickener area (m2) F = Solid feed rate (m3s-1) Ul = Settling velocity of limiting concentration (m s-1) φl = Limiting solid fraction (m3m-3) φu = Underflow solid fraction (m3m-3)
The Coe and Clevenger method for sizing thickeners was the preferred method for
nearly 40 years (Pearse 1977) until Kynch (1952) produced his analysis of the batch
settling curve, allowing the calculation of the flux curve from a single settling test. In
this method a large (e.g. 1 litre) measuring cylinder is filled with a suspension of the
expected feed concentration and the fall of the mudline is measured through time.
Kynch proposed that layers of higher concentration would propagate upwards from the
base of the container (Figure 2-19) at constant rates until they intersected the mudline,
at which point the mudline settling velocity would drop to that given by the new
interface concentration. As with Coe and Clevenger, Kynch also assumed that the
settling velocity is a unique function the solid fraction. However, since Kynch’s
method starts with a suspension at feed concentration, it is less influenced by the effect
of solid fraction on the aggregation process.
Time (min.)
Sedi
men
t hei
ght (
m)
Constant settling region (concentration constant at φf)
Kynch isoconcentraion lines (concentration increasing at interface)
H1
Ho
40 % w/v45 % w/v50 % w/v55 % w/v60 % w/v
Figure 2-19: Kynch analysis of the batch settling curve, where layers of higher concentration propagate upward at constant rates until they intersect the interface where they cause a reduction in the settling velocity (see Figure 2-20). Line drawn from H1 is used in Equation 2-58.
74
The rate of upward propagation of the isoconcentration lines is given by (see Figures
2-19 & 2-20):
φψ
−=dd
dtdHk 2-57
where: HK = Height of Kynch isoconcentration line ψ = Settling flux (m h-1), see Figure 2-20
The batch experimental test will not give settling flux data covering the entire solid
fraction range shown in Figure 2-20. Initially the solid fraction just below the mudline
will be the initial feed concentration, provided the suspension is initially mixed to
homogeneity. This concentration will initially be maintained as the mudline falls.
Eventually the mudline will meet higher concentration Kynch layers propagating
upwards from the base of the vessel (Figure 2-19). At this point the solid fraction at the
mudline will increase rapidly and the settling velocity of the mudline will begin to
decrease.
0 10 20 30 40 50 60 70 80 90 100
Solid concentration (% w/v)
Sett
ling
flux
(m h
r-1)
φf
Kynch isoconcentration lines
40 % w/v45 % w/v50 % w/v55 % w/v60 % w/v
Figure 2-20: Kynch analysis of the settling flux curve, where layers of higher concentration propagate upward at rates given by lines drawn at a tangent to the flux curve. The lowest concentration Kynch line is found by drawing a line at a tangent to the flux curve such that it intersects the flux curve at the feed concentration (φf) (see Figure 2-19).
The solid fraction of the Kynch layers is likely to be considerably greater than the feed
concentration. The first Kynch layer intersected by the mudline is the Kynch layer with
75
the fastest rise velocity. This is given by drawing a tangent to flux curve such that it
also intersects the flux line at the feed concentration (e.g. φ = 40 % w/v in Figure 2-20).
Faster rising Kynch layers are not possible because insufficient feed flux is available to
maintain them, i.e., φ = 40 % w/v is the most direct pathway in this case.
Talmage and Fitch (1955) extended Kynch theory to thickener sizing by calculating the
concentration at each point on the settling curve by drawing a tangent to the batch
settling curve (Figure 2-19), with the intercept giving H1 (Pearse 1977):
1
oo
HHφ
=φ 2-58
where: φ = Solid fraction (φo at t = 0)
H = Height of suspension (Ho at t = 0)
Drawing the tangent to the settling curve accurately from experimental data is difficult
(Pearse 1977), often leading to significant errors in the final result. Kynch did not
consider sediment compression (see Section 2.6.3), so analysis of the settling curve is
limited to concentrations lower than the compression point (φg). Talmage and Fitch
(1955) suggested that the compression point gives the lowest settling velocity within the
hindered-settling region (Pearse 1977).
Yoshioka (1957) and Hasset (1965) extended the ideas of Kynch by using the flux curve
rather than the settling velocity curve (Figure 2-21). This simplifies the interpretation
compared to the Talmage and Fitch method by allowing the experimental and required
flux (also, and somewhat misleadingly, referred to as the operating line) curves to be
plotted together.
The operating line (see Figure 2-21) is drawn from the intercept of the solid feed rate
(y-axis) to the desired underflow concentration (x-axis), such that it is never above the
observed flux rate over the expected range of solid fraction (φf to φv). The operating
line is described by (Pearse 1977):
φ−ψ=φ ur UU 2-59
76
where: Ur = Settling velocity required to prevent solid reaching the overflow (m s-1)
Uu = Settling velocity due to underflow removal (m s-1)
Equation 2-59 uses the solid settling flux due to bulk fluid flow to the underflow to
calculate the required additional flux from hindered particle settling. If the feed flux or
required underflow concentration (φu) are raised further, a bed of concentration φcrit will
build up and overflow the thickener (Fitch 1975a). In this situation the underflow solid
fraction will also be affected, since not all of the feed solid reaches the underflow.
Figure 2-21: Yoshioka/Hasset combined flux curve, showing the effect of underflow concentration on the possible throughput.
The Yoshioka/Hasset construction has several advantages. By changing the desired
flux and underflow concentration the relationship between throughput and underflow
density can be rapidly investigated, allowing the selection of a suitable compromise.
This construction also identifies a lower conjugate concentration, limiting the possible
feed dilution. If the feed is diluted further the overflow will contain solid, because feed
dilution increases the rise velocity in the clarification zone, even if the thickener
throughput has not otherwise been exceeded. As discussed above, the feed should
ideally be close to the lower conjugate concentration (rather than the maximum of the
flux curve, for example) because a lower solid fraction typically improves the mixing
during flocculation (Pearse 1977).
77
Fitch (1975a) and Pearse (1977) argue that the lower conjugate concentration is
unstable, requiring thorough mixing of fresh feed with displaced liquor flowing
upwards towards the overflow. In practice it is more likely that the feed slurry will
plunge from the feedwell and spread out as a layer at its hydrostatic equilibrium.
Caution is required when using the combined flux curve (Figure 2-21) to interpret
thickener operation. The only hindered settling concentrations possible in steady-state
operation are the upper and lower conjugate concentrations. These are the only points
where the required, and available, flux rates are equal. Pearse (1977) describes steady
state thickener operation including the behaviour when the unit is under or overloaded.
When the thickener is under-loaded, the effective settling area will naturally contract to
its steady-state equilibrium position, as the hindered settling and compression zones
contract into the thickener’s conical base. Hindered settling has also been modelled by
dynamic simulation (Fitch 1990, Diehl 1997, Bustos et al. 1999).
2.7.3 Compression (high solid fraction)
The settling solid will eventually build up on the base of the vessel, forming a sediment
bed. In this region settling is resisted by both hydrodynamic and mechanical forces.
The critical concentration (φg – gelation) where compression begins is the concentration
where the suspension forms a continuous network that can transmit mechanical force
through the structure (Perry and Green 1997, Chandler and Hogg 1987, Healy et al.
1994).
Highly flocculated suspensions will enter compression at relatively low solid fractions
because highly porous aggregates have a large effective volume. In extreme cases,
flocculated suspensions may be in compression from the onset (Pearse 1977).
However, due to their high porosity and few particle/particle contact points, the
compressive yield strength may initially be very low, and settling may essentially still
follow hindered-settling behaviour as predicted by Kynch type theory (Pearse 1977). In
many cases the transition from hindered to compression settling is so gradual that it is
difficult to detect. In these cases Pearse (1977) and Fitch (1975b) suggest plotting
log(H - H∞) vs. time to help detect the transition (Figure 2-22).
78
It was realised very early in the development of thickener technology (Pearse 1977) that
increasing the height of the bed increased the underflow concentration, although the
mechanism remained unclear. Coe and Clevenger (1916) proposed that the rate of
compression was a function of time only, and that increasing the height of the bed
merely increased the residence time in the bed at a constant underflow rate. Roberts
(1949) suggested that the rate of removal of interstitial water was also proportional to
the amount present, and proposed:
( )KD
dtDDd
−=− ∞ 2-60
where: D = Units of water per unit solid D∞ = D at infinite time K = Constant
giving a plot as shown by Figure 2-22.
Figure 2-22: Roberts’ plot (adapted from Pearse 1977).
Roberts’ assumptions were not correct (Yoshioka et al. 1957, Fitch 1975b), however
most suspensions (Fitch 1975b, Pearse 1977) give linear relationships on a Roberts’ plot
and Roberts’ work forms the basis of some of the current thickener sizing techniques
(Dahlstrom & Fitch 1985).
Michaels and Bolger (1962) proposed that both the compressive yield strength (P) and
permeability (κ) are functions of the concentration:
Time
Hindered settling regime
Compression regime L
og (H
-H∞)
79
)(PP y φ= 2-61 )(φκ=κ 2-62 where: P = Bed overburden (N m-2) Py(φ) = Compressive yield strength (N m-2) κ = Permeability (m kg-1) κ(φ) = Dynamic compressibility
They proposed that when the weight of the bed exceeds the compressive yield stress of
the sediment, it will compress at a rate limited by the rate of upward fluid drainage for
the given sediment permeability. These assumptions form the basis of current
compression theory (Buscall & White 1987, Healy et al. 1994, Green et al. 1996):
( )φ≤=φ
yPP ; 0dtd 2-63A
( )[ ] ( )φ>φ−φκ=φ
yy PP ; PP)(dtd 2-63B
The permeability of the sediment is typically calculated from Darcy’s (Wakeman &
Holdich 1984, Perez et al. 1998, Diplas & Papanicolaou 1997, Concha et al. 1996,
p. 62) or Kozeny’s (Fitch 1975b) law of fluid flow through packed beds. Permeability
may also be a function of the dynamic pressure gradient (dP/dL)(Fitch 1975b,c, Kos
1980), which may cause the formation of drainage channels in the sediment structure.
Rakes with vertical pickets have also been used to improve sediment compression
(Dahlstrom & Fitch 1985).
Equation 2-63 is not solvable analytically (Perez et al. 1998), but various numerical
solutions have been described (Wakeman & Holdich 1984, Williams & Amarasinghe
1989, Barker & Grimson 1990, Font 1988, 1990, 1991, Concha et al. 1996, Diehl 1997,
Diplas & Pypanicolaou 1997, Burger & Concha 1998, Perez et al.1998, de Kretser et al.
2000).
Simulating bed behaviour is currently an active topic in settling research. This subject
has become far more important since the introduction of polymer flocculants that
dramatically improve free and hindered-settling velocities, but also bring forward the
80
onset of compression by forming voluminous porous aggregates (Kos 1980, Hogg &
Bunnaul 1992). Also, the modelling of compression has only been possible since
efficient computerised numerical methods have become available.
The investigation of the compressive yield stress (Py(φ)) is currently an active research
area. Two methods of measuring Py(φ) are popular, centrifugation or pressure filtration.
The use of centrifugation was suggested by Buscall and White (1987), who proposed a
method for determining Py(φ) by measuring the equilibrium sediment height at various
rotation speeds (Landman & White 1992). Green et al. (1996) propose an alternative,
and apparently more accurate, numerical analysis.
Alternatively, the sediment concentration profile can be measured, either by x-ray or
γ-ray gauge, or manually by sectioning the sediment (Berström et al. 1992). This
simplifies the calculation of Py(φ) from experimental data (Green et al. 1996).
Liu and Masliyah (1996) used data from Auzerais et al. and proposed:
φ−φ
φ=φ
max
no
yP
)(P 2-64
where: Po = 30 - 160 kPa φmax = 0.72 n = 3 - 6 Py(φ) can also be determined by pressure filtration at various pressures (Landman et al.
1995), a method that has the additional advantage of also giving sediment permeability
data (via the fluid flow rate).
An increasing solid fraction increases the yield strength rapidly (Fitch 1987). Polymer
flocculants are likely to affect both the sediment yield strength and permeability (Healy
et al. 1994, Figure 2-23).
81
120000
100000
60000
40000
20000
00.10 0.15 0.20 0.25 0.30 0.35 0.40
Volume Fraction
Com
pres
sive
Yie
ld S
tres
s (Pa
)
80000
ZrO2(0.2 µm diam.)
pH=6.0 7.1 2.9
polyacrylic acidMw~750000
polyacrylic acidMw~2000
no additive
Figure 2-23: Effect of flocculant type on Py(φ) (adapted from Healy et al. 1994).
Increasing the bed height to aid compression is the basis for the deep cone thickener,
which may have a bed height up to 5 m (Healy et al. 1994). However, these units may
require rakes with 3 to 4 times the torque, and underflow pumps capable of pumping
against 3 to 4 kPa m-1 of pipe (Perry & Green 1997).
While high dosages of polymeric flocculants may improve the initial sedimentation rate
by increasing the permeability, they also increase the yield stress, leading to lower final
underflow concentrations (Healy et al. 1994). In this case the action of rakes may be
beneficial, by disrupting the sediment structure. Rakes with vertical pickets have been
used (Dahlstrom & Fitch 1985) to create drainage channels in the sediment, increasing
its permeability.
2.8 Conclusions to literature review Aggregation and settling are both mature areas of research with most phenomena
already well understood. However, the incentive to further improve unit performance
remains, and research and development continues in several key areas, particularly
computer modelling. Although the basic architecture of the various models (e.g. CFD,
population balancing, settling simulations) is already well established, many current
models assume simplified behaviour and do not adequately describe real systems. This
is largely hampered by a lack of experimental data suitable for model construction, with
few of the models supported by credible experimental data, and with many modelling
papers containing no experimental data at all.
82
The population balance approach to aggregation kinetics modelling was first proposed
by Smoluchowski in 1917, however the majority of the work in this area has been done
in the last decade, since efficient numerical techniques have become available to solve
the equations. This has allowed the population balances to become sufficiently
complex to describe real systems. However, the full potential of population balancing
has yet to be realised and several aspects require more work.
Population balances have already been used successfully to simulate the
aggregation/breakage of aggregates formed by coagulation (e.g. Spicer & Pratsinis
1996a), but apart from the limited work by Pelton (1981) polymer flocculation has been
largely ignored. Flocculation and coagulation are similar in many respects, but the
breakage of flocculated aggregates is generally found to be irreversible, with the
aggregate size decreased permanently by extended shear. This may be a significant
practical issue in some cases, for example in feedwells which subject fully formed
aggregates to high levels of shear (perhaps due to less-than-optimal baffle placement).
Fluid shear has a major impact on the rates of both aggregation and breakage, and most
current population balance models (e.g. Spicer & Pratsinis 1996a) are written to
accommodate changes to the shear rate. However, other process variables (e.g.
flocculant dosage, primary particle size and solid fraction) also impact on the
aggregation process, and require consideration.
Population balance modelling has previously been treated as an isolated research area,
with little effort made to link the predicted aggregate size to important performance
parameters like the settling rate. There also exists the opportunity to begin to couple
various models (e.g. CFD-population balance, or population balance-settling
simulation) and create larger process models to describe the wider process. This would
then allow the unit performance to be optimised as a whole, for example feedwell
design could be altered to give optimal settling performance. In addition to
optimisation, sophisticated dynamic process models are likely to have other
applications, for example in the development of suitable process control strategies.
83
3. On-line particle sizing
3.1 Particle sizing techniques The measurement of particle or aggregate size distributions has already been well
reviewed elsewhere (Herbst & Sepulveda 1985, Allen 1990, Barth & Sun 1991,
Stanley-Wood & Lines 1992, Farrow & Warren 1993, Perry & Green 1997) and only
brief general comments are offered here.
In practice, particle size distributions tend to be broad, usually several orders of
magnitude, and although there may be only a few large particles, they may contain the
bulk of the mass. For these reasons a particle size distribution is usually plotted as the
particle volume weighted counts per channel (y-axis) against log (size) (x-axis). This
usually produces a distribution that may be approximated by a log-normal distribution,
although alternative distributions are available (Perry & Green 1997, Dahlstrom & Fitch
1985, Randolph & Larson 1988).
Although numerous techniques and instruments are available for determining particle
size, few can be applied in-situ/on-line, or over a wide range of solid concentrations
(Williams 1992). In-situ ultrasonic techniques have been developed (Coghill et al.
1997), but require lengthy measurement times, making them unsuitable for dynamic
systems. Laboratory instruments like the Malvern Mastersizer have also been modified
(Bale & Morris 1987, Hobbel et al. 1991, Bale 1996, Phillips & Walling 1995a) to suit
in-stream sizing, but are bulky and relatively expensive for field or plant use.
Alternatively, in-stream measurements can be made by focussed beam reflectance
measurement (FBRM). The FBRM probe has a wide range of applications, frequently
in the process environment (Table 3-1). The instrument gives a particle chord length
distribution, which is a function of the true particle diameter distribution. For many
process applications (e.g. control) this distinction is not important because the relevant
process variable (e.g. crystal size) can be correlated with some aspect of the chord data,
and accurate particle sizing is not specifically required.
84
Table 3-1: Summary of common FBRM process applications Application Reference Control
Heffels et al. 1998, Fakatselis 1994, Farrell and Tsai 1995, Peng and Williams 1993, Dijkstra et al. 1996, Reid and Stachowicz 1990
Crystallisation/ precipitation
Sparks and Dobbs 1993, Farrell and Tsai 1995, Dijkstra et al. 1996, Monnier et al. 1997, Johnson et al. 1997, Poilov et al. 1997, Bongartz et al. 1999, Barrett and Glennon 1999, Hildred et al. 2000
Estuarine solids Bale 1996, Phillips and Walling 1995a,b, 1996, 1998, 1999, Law et al. 1997
Flocculation Williams et al. 1992, Peng and Williams 1994, Spears and Stanley 1994, Sengupta et al. 1997, Caron-Charles and Gozlan 1996, Murphy et al. 1994,1995, Sharma et al. 1994, Fawell et al. 1997, Hokanson and Preikschat 1992, Honaker et al. 1994, Hecker et al. 1999, Alfano et al. 2000
Grinding Hokanson and Preikschat 1992, Reid and Stachowicz 1990, Thomas et al. 1998
Paper manufacture Hanseler and McKean 1988, Blanco et al. 1996, Anderson 1996, Alfano et al. 1999
Petroleum Cowie and Schmoll 1994, El-Hamouz and Stewart 1996, Simmons et al. 1999
Pharmaceutical Vo 1994, Daniels and Barta 1992
The instrument’s rapid on-line analysis, and conventional log-normal size distribution
make it attractive to workers modelling the kinetics of flocculation (Peng & Williams
1994, Sharma et al. 1994) or crystallisation (Farrell & Tsai 1995, Monnier et al. 1997,
Poilov et al. 1997) by population balance. For dynamic modelling, accurate sizing and
counting of particles is desired for model parameter estimation. Both theoretical
(see Section 3.3.1) and empirical (see Section 3.3.2) methods for estimating particle
diameter distributions from FBRM chord distribution are reported in the literature.
The aim of the work described in this section was to find a calibration method for
FBRM, allowing the aggregate chord data to be converted to aggregate diameter data
suitable for the population balance modelling described later (Section 5).
85
3.2 Principle of FBRM The Lasentec FBRM instrument directs a 789 nm laser through a lens rotating at
4410 rpm (Sparks and Dobbs 1993). This results in a rotating beam of light highly
focussed at a point near the instrument’s window. The position of the focal point is
adjustable, and sometimes (Monnier et al. 1996) adjusted depending on the particle
size. The diameter of the circular path of the laser beam is 8.5 mm and the beam is
focussed to a section of approximately 0.7 × 2 μm with a focal depth of 10 μm
(Hokanson & Preikschat 1992). However, the exact width of the beam appears to be
dependent on both the focal position, and the size of the particle and its distance from
the focal position (Reid & Stachowicz 1990, Sparks & Dobbs 1993).
Laser beam
Opticsrotating ata fixed highvelocity
Sapphirewindow
Scanning directionof focusedlaser beam
Reflec
ted ch
ord
Reflec
ted ch
ord
Reflec
ted ch
ord
Probe
Laser
Figure 3-1: Schematic of the FBRM probe.
When the moving laser beam crosses the path of a particle, some of the light is reflected
back to the instrument detector. Since the tangential velocity of the beam is known
(∼ 1.9 m s-1), the duration of the reflected light pulse is directly proportional to the width
of the particle intersected. The tangential beam velocity is assumed (Section 3.3.1) to
be much larger than the particle velocity relative to the instrument window. However,
since it is unlikely that the laser beam will pass directly across the centre of the particle
(i.e. diameter), a chord length is measured.
Several descriptions are given of previous versions of the FBRM instrument and data
processing regimes (Hokanson & Preikschat 1992, Sparks & Dobbs 1993, Reid &
Stachowicz 1990). Data processing of the detector response uses both the signal
strength and slope. Current instrument data processing is adjustable through the use of
either F (fine) or C (coarse) electronics modules depending on the particle size.
86
3.3 Estimating particle size from FBRM data 3.3.1 Theoretical methods
Several papers deal with the transformation of chord to diameter distributions from first
principles (Clark & Turton 1988, Jakeman & Anderssen 1975, Hobbel et al. 1991,
Williams et al. 1992, Liu & Clark 1995, Liu et al. 1998, Tadayyon & Rohani 1998,
Barrett & Glennon 1999, Simmons et al. 1999). Most workers assumed that the
particles are spherical to simplify the mathematics, and most analysis of non-spherical
particles is generally limited to two dimensional ellipses (Jakeman & Anderssen 1975,
Clark & Turton 1988, Liu & Clark 1995, Liu et al. 1998, Tadayyon & Rohani 1998).
Assuming the particles are spherical dramatically simplifies the mathematics, because
regardless of the particle’s orientation it always presents a circular profile (Figure 3-2).
This assumption may be valid for suspensions of spherical latex or glass beads, or
emulsions of two immiscible liquids. The spherical-particle assumption may also be
essentially valid for suspensions of particles with an aspect ratio (ratio major:minor
axis) approaching 1:1. As the aspect ratio increases and the particles or crystals tend
towards needles/plates, the chord distribution broadens (Williams et al. 1992, Barrett
and Glennon 1999).
Most descriptions assume that the focussed laser beam has no width, and that light is
reflected immediately after the beam reaches the edge of the particle, however Hobbel
et al. (1991) introduce a parameter (k) to account for the “arch of rejected reflection”.
Sparks and Dobbs (1993) considered the physics of the laser/lens interaction and
suggest that the beam may be broadened in some conditions. Spreading of the beam
away from the optimal focal point may also affect the chord measurement (Reid &
Stachowicz 1990), and beam spreading is likely to be a function of solids loading, fines,
solution and particle surface characteristics.
87
Figure 3-2: Method of determining the chord distribution expected from a spherical particle of diameter Dp.
Where: Dp = Diameter of pth particle (μm) Ci,u = Length of the upper boundary of the ith chord channel (μm) Ci,l = Length of the lower boundary of the ith chord channel (μm)
Wp,i = Width of pth particle giving chords in the ith channel (μm) Figure 3-2 shows that the likelihood of a particle being detected is proportional to its
diameter, introducing a bias. Therefore, for modelling purposes FBRM is assumed to
measure the first diameter moment of the chord distribution. Calculating a chord
distribution from a given spherical particle distribution is relatively straightforward,
however the reverse algorithm is more difficult, usually requiring a numerical solution
in the form (Tadayyon & Rohani 1998, Simmons et al. 1999):
( )∑=
−=φmax
1 i
2*ii
NNNmin
*p
3-1
where: Ni = Number of i sized chord counts observed (s-1)
Np* = Number of pth sized particles predicted (s-1)
Ni* = Number of i sized chords predicted (s-1):
( ) ( )∑=
−−−=max
1i
2u,i
2p
2l,i
2p
*p
*i
21
21
CDCDNN 3-2
where:
Ci,u Dp Ci,l
2 × 0.5Wp,i
Particle
Path taken by laser beam at Ci,l
88
Dp = Diameter of pth particle (μm) Np
* = Number of pth sized particles predicted (s-1) Ci,u = Length of the upper boundary of the ith chord channel (μm)
Ci,l = Length of the lower boundary of the ith chord channel (μm) The resulting calculated diameter distribution is a number distribution, and is converted
to the conventional particle volume distribution by applying a length-cube weighting in
the form of Equation 3-3 (see Section 3.5.1).
3.3.2 Empirical methods
FBRM results are also compared with alternative particle sizing techniques (Table 3-2),
for a variety of particle sizes and materials. Early Lasentec FBRM software included a
spherical-equivalent size algorithm (Sparks & Dobbs 1993, Peng & Williams 1993,
1994, Caron-Charles & Gozlan 1996, Law et al. 1997, Phillips & Walling 1998).
Comparisons between the resulting average size and alternative sizing methods showed
that the FBRM oversized small particles (<150 μm Law et al. 1993, <300 μm Phillips &
Walling 1998) and undersized larger ones (>500 μm Phillips & Walling 1998).
Current software computes various chord statistics: median, mean and mode with
various chord weightings. Daymo et al. (1999) found good agreement between the
cube-weighted mean size and sieve analysis of kaolin suspensions with a mean size of
around 55 μm. Alfano et al. (2000) show a good correlation between Malvern
Mastersizer volume-average size and un-weighted FBRM chord lengths for suspensions
of aggregated silica 20-200 μm. However, the un-weighted FBRM results
underestimated the size of the larger particles. El-Hamouz and Stewart (1996)
compared FBRM and Malvern square-weighted mean drop sizes of oil/water emulsions
and found the FBRM overestimated smaller (-180 μm) drops, but underestimated larger
(+180 μm) drops.
89
Table 3-2: FBRM comparisons with other sizing techniques Reference Particle material Alternative sizing
method (size) Size by FBRM
Hobbel et al. 1991
Glass beads
Helos, laser forward scattering (66 μm)
170 μm
Sparks & Dobbs 1993
Duke standards, carbon, sand, alumina, hydrate.
Duke certified standards, Microtrac (laser forward scattering) (0.8-410 μm)
Spherical equivalent, FBRM(μm)=52.353+1.0362Duke(μm) 10-162 μm
Murphy et al. 1994
Ground silica.
BET, SEM, Micrometrics sedigraph, (2.1 μm)
Oversized considerably, could not be calibrated in this range
Monnier et al. 1995
Glass beads, latex, micronized pharmaceutical product. various supporting fluids
Cilas (laser diffraction) Coulter Counter Microscopy (4.1-396 μm)
Oversizes < 100 μm Undersizes > 100 μm
Dijkstra et al. 1996
Sugar. Sieved (30-250 μm)
Sieve(μm)=2.044E-3(FBRM μm)2.2844
El-Hamouz & Stewart 1996
Oil/water emulsions Malvern Mastersizer (d32 = 60-350 μm)
Sauter mean (d32 = 130-210 μm), Undersized > 180 μm Oversize < 180 μm
Law et al. 1997 Pollen, glass and latex Duke standards, fractionally settled sediment, sieved sand. 6.5-766 μm
Coulter Counter Malvern Mastersizer Certified standards Optical Microscopy
FBRM Par-Tec spherical equivalent, Undersized > 500 μm Oversized < 150 μm
Phillips & Walling 1998
Sieved river sediment
Coulter laser diffraction (1-355 μm)
FBRM Par-Tec spherical equivalent, Coulter d50(μm)=-19.239+1.064FBRM(μm)
Tadayyon & Rohani 1998
Sieved ion-exchange resin.
Sieve (212-800 μm). FBRM undersized, better with cube-weighting
Daymo et al. 1999
Graphite, gibbsite, bentonite, mica, silica, kaolin, plastic beads
Sieve, Horiba sedigraph. (0.3-663 μm)
5 % w/v kaolin = 53 μm by cube-weighted FBRM, 57 μm by sedigraph. Other samples gave week correlation (order of magnitude)
Alfano et al. 2000
Mircrocrystalline cellulose.
Malvern Mastersizer (20-180 μm)
FBRM(μm)=10.09+0.586Malvern(μm)
90
3.4 Experimental 3.4.1 Equipment
Suspensions of various particle size and solid fraction were presented to a Lasentec
M500 FBRM probe as shown by Figure 3-3. The focal point was set to 0 μm (the
window) as calibrated by the manufacturer unless otherwise stated. This position was
also confirmed experimentally by placing a film of marking pen ink on the window and
adjusting for maximum counts. The impeller (Figure 3-3) was set to 650 rpm unless
otherwise stated. Data was collected using 90 log-channels unless otherwise stated.
Window cleaning was performed regularly (∼ every 5 minutes measurement time) to
give a total count rate < 50 s-1 in air.
Figure 3-3: Schematic of the laboratory equipment.
A MS10 Malvern Mastersizer and a Coulter Multisizer II were used as alternative sizing
techniques. Particles were dry sieved using conventional Endecotts screens (38, 45, 53,
63, 75, 106, 125, 150, 180, 212, 250, 300, 425, 500, 600, 710, 850, 1000, 1180 μm) in a
Rotap RX-29-10 mechanical shaker, and smaller size fractions obtained by cyclosizing
with a Warman M-4 cyclosizer. SEM (scanning electron microscope) micrographs are
from a GEOL JSM-5800LV.
FBRM probe Stirred suspension
Instrument
Focal position adjusting knob
Stirrer
Fiber optic cable
91
3.4.2 Samples
Most samples were either aluminium (Valimet, Stockton California) or calcite (Omya
Southern, Commercial Minerals) particles that were sized to narrow fractions by
screening or cyclosizing. The solids were weighed out on a 0.1 mg balance and made
up to concentrated stock suspensions in a volumetric flask. A series of more dilute
suspensions were prepared by serial dilution of the stock suspension from a stirred and
baffled tank to prevent stratification. The calcite samples suffered from fines and where
possible were washed and decanted. Suspensions of aluminium particles were reactive,
producing a gas (presumably hydrogen) and were kept on ice to reduce the reaction rate.
A single suspension of mono-sized latex spheres (Dynaspheres, Duke Scientific) with a
specified size of 19.9 μm was also measured.
3.5 Results and discussion 3.5.1 Effect of applying weightings to FBRM chord distributions
The raw chord lengths measured by the FBRM are sorted into a series of size intervals
(channels) covering the range 1 – 1000 μm (Figure 3-4). The channel widths (μm) are
discretised according to a numerical progression (following Ci,l = 1.08i-1, i = [1, 90]),
although the FBRM software has various options, including constant width channel
spacing.
0
200
400
600
800
1000
1200
1 10 100 1000Chord length (µm)
Cou
nts p
er c
hann
el (s
-1)
Sieved 38-45 µmSieved 45-53 µmSieved 53-63 µmSieved 63-75 µmSieved 75-106 µmSieved 106-125 µmSieved 125-150 µm
Figure 3-4: Effect of particle size on chord distribution. Sieved aluminium particles, 10 % w/v, F-electronics.
92
Figure 3-5 shows the effect of applying length weightings (moments) to the chord
distribution given by the FBRM, i.e.:
n
A,io,in,i CNN = 3-3 where: Ni,n = number of n-weighted counts in the ith channel (μmn s-1) Ni,o = raw un-weighted counts in the ith channel (s-1)
n = exponent, 0 = un-weighted, 1 = length-wtd, 2 = square-wtd, 3 = cube-wtd, etc Ci,A = geometric average length of ith channel (μm) = (Ci,u × Ci,l)1/2 Ci,u = length of the upper boundary of the ith channel (μm)
Ci,l = length of the lower boundary of the ith channel (μm)
0
1
2
3
4
5
6
7
8
1 10 100 1000Length (µm)
% c
ount
s per
cha
nnel
(FB
RM
)
0
5
10
15
20
25
30
% c
ount
s per
cha
nnel
(M
alve
rn)
FBRM (un-weighted)
Malvern
FBRM (square-weighted)
FBRM (length-weighted)FBRM (cube-weighted)
Figure 3-5: Effect of applying various length weightings (moments) to the FBRM chord distribution. Malvern Mastersizer volume weighed distribution also shown. Aluminium particles, sieved 45-53 μm.
This is analogous to applying a length-cube weighting to a particle number distribution
to give a conventional volume-weighted distribution. For particles between
approximately 50 to 400 μm, Figures 3-5 and 3-6 show that the square-weighted (n =
2) chord distribution gave the best correlation with Malvern Mastersizer results,
although the FBRM distributions remained broader.
A previous version of the FBRM software (Sparks & Dobbs 1993, Peng & Williams
1993, 1994, Caron-Charles & Gozlan 1996, Law et al. 1997, Phillips & Walling 1995A)
included a chord-to-spherical-equivalent algorithm that both narrowed the distribution
and increased the length weighting (Phillips & Walling 1995a).
93
3.5.2 Comparison of average particle size by various techniques
Particle size distributions can be weighted and averaged in a variety of ways (Randolph
& Larson 1988, Rushton et al. 1996, Seville et al. 1997, Perry & Green 1997). FBRM
software provides a range of options including: mean, median (d50) and mode averages
of length-weighted chord distributions.
Figure 3-6 compares various length-weighed median FBRM average sizes with Malvern
D50 and sieve sizing for a range of sieved or cyclosized fractions. Generally, smaller
particles (< 50 μm) were best estimated by a length-weighed FBRM average. Particles
in the range approximately 50 to 400 μm were best estimated by the square-weighted
average. Suspensions of very large particles (> 400 μm) benefited from cube-
weighting, which has the effect of further weighting the distribution towards larger
particles. The Malvern (as opposed to the sieve) results are plotted on the x-axis
because the smaller fractions were prepared by cyclosizing.
y = 1.1032x + 21.386R2 = 0.9881
y = 0.8936x + 12.453R2 = 0.9927
y = 0.8847x + 4.8684R2 = 0.9918
y = 0.7492x + 5.6964R2 = 0.9921
y = 0.5304x + 2.5324R2 = 0.9808
0
50
100
150
200
0 50 100 150 200 250
Malvern D50 (µm)
Mea
sure
d si
ze (µ
m)
FBRM chord cube-weighted
FBRM chord un-weighted
FBRM chord length-weighted
FBRM chord square-weighted
Sieve
Figure 3-6: Comparison of median average sizes given by various techniques and chord weightings, showing the square-weighted chord gives the best agreement with Malvern and sieve analysis. Sieved/cyclosized aluminium fractions. F-electronics, FBRM suspensions 5 % w/v.
The mean, and to a lesser extent also the median, averages were affected by the FBRM
instrument’s sensitivity to fine particles (Figures 3-11 & 3-13). However, the mode
(distribution maxima) were less affected (Figure 3-7). Figure 3-7 shows direct
comparison of average sizes by the techniques for a series of sieved or cyclosized
94
suspensions. Linear correlation equations (Figure 3-7) could be re-arranged to give a
2-parameter calibration, to give reasonable agreement over 20-500 μm.
y = 0.8108x + 8.0231R2 = 0.9983
y = 0.8752x - 10.783R2 = 0.9903
y = 0.8157x + 0.5694R2 = 0.9927
0
100
200
300
400
500
600
700
800
0 100 200 300 400 500 600 700 800
Malvern D50 (µm)
Mea
sure
d av
erag
e pa
rtic
le si
ze (µ
m)
FBRM mode of square-weight
Coulter
Sieve
Figure 3-7: Comparison of mode average sizes from the FBRM square-weighted distributions, with median values from Malvern and Coulter analysis. Sieved/cyclosized calcite particles, 5 % w/v. F-electronics.
Large particles can be obscured if the suspension contains large quantities of fines
(see Section 3.7). In this case even the square-weighted distribution may become
bimodal, and applying the cube-weighting and manually assigning the mode to the right
hand peak may extend the measurable range. In some situations, decanting the fines
may be feasible.
3.5.3 Effect of solid volume fraction on indicated size
Increasing the solid volume fraction has the effect of increasing the FBRM instrument’s
sensitivity to fine particles, probably due to fine particles crowding the measurement
zone near the window (see Section 3.5.7). However, this effect is largely overcome by
applying the square-weighting, (Figures 3-8 & 3-9). Applying a cube-weighting may
lead to further improvements, especially for suspensions of large particles.
95
0
50
100
150
200
250
300
350
400
1 10 100 1000
Chord length (µm)
0.0E+00
5.0E+05
1.0E+06
1.5E+06
2.0E+06
2.5E+06
3.0E+060.1 % w/v, un-weighted0.5 % w/v, un-weighted1 % w/v, un-weighted2 % w/v, un-weighted5 % w/v, un-weighted10 % w/v, un-weighted20 % w/v, un-weighted0.1 % w/v, square-weighted0.5 % w/v, square-weighted1 % w/v, square-weighted2 % w/v, square-weighted5 % w/v, square-weighted10 % w/v, square-weighted20 % w/v, square-weighted
Figure 3-8: Effect of solid fraction on the un-weighted and square-weighted FBRM distributions. Aluminium particles sieved 125-150 μm, F-electronics.
Previous FBRM software calculated a spherical-equivalent size. Various studies
showed that the spherical-equivalent size was essentially unaffected by solid fraction
(Sparks & Dobbs 1993, Law et al. 1997). Daymo (1999) used the chord cube-
weighting and also found little variation in the indicated size as the solid fraction was
varied.
0
100
200
300
400
500
600
0.1 1 10 100Weight fraction (% w/v)
Cal
cula
ted
size
from
FB
RM
(µm
) sieved 38-45 µmsieved 45-53 µmsieved 53-63 µmsieved 63-75 µmsieved 75-106 µmsieved 106-125 µmsieved 125-150 µmsieved 150-180 µmsieved 180-212 µmsieved 212-250 µmsieved 250-300 µmsieved 300-425 µmsieved 425-500 µm
Figure 3-9: Effect of solid fraction on the square-weighted mode of the chord length, calcite particles, F-electronics.
Figure 3-8 shows the influence of solid fraction on the un-weighted FBRM chord
distributions. The chord-to-diameter models described in Section 3.3.1 (Equation 3-2)
96
rely on the un-weighted distribution, and gave results that were dependent on the solid
fraction (Figure 3-10). This effect limits the usefulness of algorithms like Equation 3-
2, and further model parameters would be required to compensate for this effect. The
calculated distributions are also noticeably broader than the Malvern volume-weighted
distribution, despite the particles (Figure 3-16) approaching the sphericity assumed by
the model.
A high flow velocity past the probe window could be suspected to cause this effect, via
both positive and negative Doppler shifts depending on the direction of the particle
relative to the circular path of the laser beam. However, changing the impeller speed
had little effect on the FBRM distribution, despite the impeller passing 2 mm from the
instrument window at a velocity of approximately 2 m s-1 at 750 rpm. The cause of
FBRM distribution broadening remains unclear, but may be a combination of particle
shape/reflectivity, width of the viewing zone (affected by particle concentration), and
data processing (C or F electronics).
0
5
10
15
20
25
30
1 10 100 1000Particle size (µm)
% p
er c
hann
el (M
alve
rn)
0
1
2
3
4
5
6
7
8
9
10
% P
er c
hann
el (F
BR
M)
Malvern
FBRM calc. vol. dist. (0.1 % w/v)
FBRM calc. vol. dist. (0.5 % w/v)
FBRM calc. vol. dist. (1 % w/v)
FBRM calc. vol. dist. (2 % w/v)
FBRM calc. vol. dist. (5 % w/v)
FBRM calc. vol. dist. (10 % w/v)
FBRM calc. vol. dist. (20 % w/v)
Figure 3-10: Effect of solid concentration on transformed chord data, showing that the model (Equation 3-2) overestimated the particle size in dilute suspensions. Aluminium particles sieved 38-45 μm, F-electronics.
97
3.5.4 Comparison between C and F-electronics
As mentioned previously, the FBRM instrument is available with either C (coarse) or F
(fine) electronics modules. As the names suggest, F-electronics is preferred for
suspensions of fine particles, and C-electronics for coarse fractions or aggregates.
Modified signal processing using C-electronics allows aggregates to be measured as
single, large particles, rather than a series of fine particles.
Figure 3-11 shows the results from F and C-electronics using the same suspension of
calcite particles sieved between 38 and 45 μm. Fine particles abraded from the solid
surface during mixing are readily detected with F-electronics, with the C-electronics
recording more counts in the higher channels.
0
100
200
300
400
500
600
700
800
1 10 100 1000
Length (µm)
Un-
wei
ghte
d co
unts
per
ch
anne
l (s-1
)
0.0E+00
5.0E+05
1.0E+06
1.5E+06
2.0E+06
2.5E+06
Squa
re-w
eigh
ted
coun
ts p
er
chan
nel (
µm2 s-1
)
F-electronics, unweighted
C-electronics,square-weighted
F-electronics, square-weighted
C-electronics, unweighted
Figure 3-11: Calcite particles sieved 45-53 μm, 5 % w/v, F-electronics.
Figure 3-12 shows the square-weighted median sizes given by F and C-electronics for a
range of sieved or cyclosized calcite particles. As expected, smaller particles
(< ∼250 μm) were more accurately sized by F-electronics, with C-electronics being
slightly preferable for larger fractions.
98
y = 0.6837x + 36.903R2 = 0.994
y = 0.7161x + 15.658R2 = 0.9953
y = 0.8119x - 2.5398R2 = 0.9961
0
50
100
150
200
250
300
350
400
0 50 100 150 200 250 300 350 400 450
Malvern D50 (µm)
Mea
sure
d si
ze (µ
m)
Sieve (calcite)
Calcite, F-electronics
Calcite, C-electronics
Figure 3-12: Comparison between square-weighted median FBRM results given by C and F-electronics for various sieved or cyclosized calcite fractions. FBRM suspensions 5 % w/v.
3.5.5 Effect of particle material
Particle shape (Williams et al. 1992, Barrett & Glennon 1999) and reflectivity (Reid &
Stachowicz 1990, Sparks & Dobbs 1993) were expected to affect the reflection of the
laser beam to some extent. Figure 3-13 shows a comparison between aluminium and
calcite particles of the same size and solid fraction. Aluminium and calcite powders
have the advantage of similar densities (ρal. = 2 702, ρcal. = 2 710 kg m-3), giving them
similar volume fractions when made up by weight, and similar flow requirements to
keep the particles suspended.
The increased fine counts shown by the (un-weighted) calcite distribution (Figure 3-13)
are again probably caused by fine material abraded from the surface during suspension
mixing. The counts rate increased in this region with time, but was reduced by washing
and decantation. Aluminium particles were also experimentally inconvenient, as they
were reactive, producing a gas, (presumably hydrogen) and required careful
preparation, storage and disposal (see Section 3.4.2).
Generally however, little difference was observed between either the square-weighted
distributions (Figure 3-13), their median sizes (Figure 3-14), or the overall count rate
(Figure 3-13), despite the two samples having different morphologies, and presumably
99
also different reflectivities (see SEM images, Figures 3-15 & 3-16). A colour
difference between the aluminium and calcite particles was visible to the naked eye, but
not clear in the SEM images (Figures 3-15 & 3-16) where the calcite particles were gold
plated for imaging. Daymo et al. (1999) added food colouring to a suspension of 4 %
w/v bentonite and reported no change in the average size. However, Sparks and Dobbs
(1993) report that transparent particles (oil drops, latex particles) were generally poorly
sized by FBRM.
0
100
200
300
400
500
600
700
800
1 10 100 1000
Length (µm)
Un-
wei
ghte
d co
unts
per
ch
anne
l (s-1
)
0.0E+00
2.0E+05
4.0E+05
6.0E+05
8.0E+05
1.0E+06
1.2E+06
1.4E+06
1.6E+06
Squa
re-w
eigh
ted
coun
ts p
er
chan
nel (
µm2 s-1
)
Calcite particals, un-weighted
Calcite particals,square-weighted
Aluminium particals,square-weighted
Aluminium particals, un-weighted
Figure 3-13: Comparison between FBRM distributions of aluminium or calcite. Sieved 45–53 μm, 5 % w/v, F-electronics.
0
50
100
150
200
250
300
350
400
0 50 100 150 200 250 300 350 400 450
Malvern D50 (µm)
Mea
sure
d si
ze (µ
m)
Sieve (aluminium)Sieve (calcite)Aluminium, F-electronicsCalcite, F-electronics
Figure 3-14: Comparison between square-weighted median FBRM results given by various sieved or cyclosized calcite and aluminium fractions. FBRM suspensions 5 w/v, F-electronics.
100
Figure 3-15: SEM secondary electron image of the un-sieved calcite particles.
Figure 3-16: SEM secondary electron image of un-sieved aluminium particles
Mono-sized (19.9 μm) spherical latex spheres (Duke Dynaspheres) were also sized by
FBRM, and gave mean sizes of 26 μm (F-electronics, un-weighted), 51 μm
(F-electronics, square-weighted), 36 μm (C-electronics, un-weighted), 66 μm
(C-electronics, square-weighted). Similar observations have been reported previously
101
(Sparks & Dobbs 1993), where transparent particles (laser = 789 nm) are sized poorly
by FBRM, presumably due to complex reflection/refraction effects.
3.5.6 Minor effects
A number of additional minor effects may alter the instrument response. The position
of the focal point of the laser can be manually adjusted in relation to the instrument’s
window. This has the effect of moving the position of the measurement region further
into the pulp, or, as negative measurements indicate, behind the window/suspension
interface. Larger particles may have trouble entering the measurement region when it is
set to the window (= 0 μm, Figure 3-17), leading to an exaggerated sensitivity to fine
particles. Monnier (1996) and Law et al. (1997) suggest setting the focal position
further into the suspension for larger particles. In some cases setting the focal position
approximately 20 μm behind the window’s outer face will also improve the sizing of
large particles (Becker 1999).
0
50
100
150
200
250
300
350
400
1 10 100 1000Chord length (µm)
Cou
nts p
er c
hann
el (s
-1)
focal posn. -100 µm, un-weighted
focal posn. -50 µm, un-weighted
focal posn. -20 µm, un-weighted
focal posn. -10 µm, un-weighted
focal posn. 0 µm, un-weighted
focal posn. 10 µm, un-weighted
focal posn. 20 µm, un-weighted
focal posn. 50 µm, un-weighted
focal posn. 100 µm, un-weighted
Figure 3-17: Effect of laser focus position on chord distribution calcite, sieved 106-125 μm, 20 % w/v, 500 rpm impeller speed. Un-weighted. (See Figure 3-18 for square-weighted).
However, as was the case previously, applying a square-weighting (Figure 3-18)
normalises the response considerably. Adjusting the focal position to the window
( = 0 μm) increased the count rate and so improved the statistical accuracy. Except
102
Figures 3-17 and 3-18, all other data presented is with the focal position set to the
window.
0.0E+00
1.0E+06
2.0E+06
3.0E+06
4.0E+06
5.0E+06
1 10 100 1000Chord length (µm)
Cou
nts p
er c
hann
el (µ
m2 s-1
)focal posn. -100 µm, square-weighted
focal posn. -50 µm, square-weighted
focal posn. -20 µm, square-weighted
focal posn. -10 µm, square-weighted
focal posn. 0 µm, square-weighted
focal posn. 10 µm, square-weighted
focal posn. 20 µm, square-weighted
focal posn. 50 µm, square-weighted
focal posn. 100 µm, square-weighted
Figure 3-18: Effect of laser focus position on chord distribution of calcite, sieved 106-125 μm, 20 % w/v, 500 rpm impeller speed. square-weighted. (See Figure 3-17 for un-weighted).
The tangential velocity of the focussed laser beam is approximately 1.9 m s-1, and is
assumed (see Section 3.3.1) to be far greater than the particle velocity with respect to
the instrument window. The tip velocity of the supplied 50 mm diameter pitched blade
impeller reaches 2 m s-1 at approximately 750 rpm. Other flow geometries such as pipes
or process streams may also give flow velocities in this range. However, since the
viewing region is adjacent to the window, the flow velocities in the viscous sub-layer
are likely to be considerably lower than for the bulk of the fluid flow.
In practice, changing the impeller speed had little effect (Figure 3-19) on the observed
square-weighted FBRM distribution. Larger particles (> 500 μm) tended to stratify at
low stirring speeds (< 500 rpm), leading to a reduction in the average measured size.
Extended high-speed stirring of the calcite particles lead to increased fines via surface
abrasion. Except Figure 3-19, all measurements were made with the impeller set to
650 rpm.
The effect of fluid flow rate on the measured size has already been investigated, with
contradictory results reported. Law et al. (1997) showed no significant effect of stirring
103
speed on the indicated size in the range 500-1000 rpm. Monnier et al. (1996) found that
increasing the impeller speed from 500-1000 rpm dramatically increased both the
measured size and total counts of glass beads. Daymo et al. (1999) reported that
increasing the flow velocity of a kaolin suspension from 1.3 to 2.4 m s-1 in a 76 mm ID
pipe reduced the indicated size by 13 %. However, these changes may have been due to
particle setting at low flow rates, or particle breakage at high flow rates.
0.0E+00
1.0E+07
2.0E+07
3.0E+07
4.0E+07
5.0E+07
1 10 100 1000
Length (µm)
Squa
re-w
eigh
ted
coun
ts
per
chan
nel (
µm2 s-1
)
250 rpm375 rpm500 rpm750 rpm1000 rpm
Figure 3-19, extra figure showing effect rpm. Calcite particles sieved 106-125 mm, 20 % w/v. Duplicate results performed with rpm decreasing with time showed less fines at higher rpm.
A short (30 s) period of stirring was sufficient to disperse the solid. The measurements
were then taken rapidly before excessive fines were produced from surface abrasion.
The effect of dispersants on the suspension was also investigated, but gave no
significant improvement over the short period of stirring. In other systems the use of
dispersants (Monnier et al. 1995) can be advantageous.
Some particles are prone to adhere to the window (Coghill et al. 1997), especially fine
particles, or when flocculant is added. The particles build up on the window, increasing
the count rate and influencing the measured size. All results shown in this work were
obtained with regular (approx. every 5 min.) window cleaning to give a total
background count (in air) of less than 50 s-1. In many situations such regular cleaning
may not be feasible, although it is recommended that the probe is positioned so that the
104
fluid flow helps keep the window clear (Sparks & Dobbs 1993, Lasentec Users
Manual).
3.5.7 Effect of solid fraction on particle counts
Estimation of total particle number is desirable for mass balancing or estimation of
aggregate porosity. Unfortunately, total FBRM counts did not correlate well with solid
fraction, tapering off at high particle concentrations (Figure 3-20). Similar results are
reported by Barrett and Glennon (1999) and Daymo et al. (1999). This effect is
probably due to an increase in the instrument dead-time at higher solid fraction, because
the instrument only counts one particle at a time. At a low solid fraction only a small
portion of the scan time is used detecting particles, and the likelihood of particles
overlapping in the viewing zone is low. However, at a higher solid fraction, a
significant portion of the time is spent measuring particles, reducing the time available
to detect other particles in the measurement zone (e.g. behind the measured particle).
This dead-time will be slightly larger than the time spent traversing the particle, because
the reflected light must return to zero for a short period between particles. The signal
processing algorithm also requires that the reflected light pulse to rise and fall quickly
before and after each particle, and counts from overlapping particles (or particles
outside the viewing zone) are rejected, further increasing the dead-time.
0
5000
10000
15000
20000
25000
0 5 10 15 20 25 30
Solid fraction (% w/v)
Tot
al c
ount
s (s-1
) Sieved 38-45 µmSieved 45-53 µmSieved 53-63 µmSieved 63-75 µmSieved 75-106 µmSieved 106-125 µmSieved 125-150 µmSieved 150-180 µmSieved 180-212 µmSieved 212-250 µm
Figure 3-20: Relationships between total counts and solid fraction for various sieved calcite fractions. F-electronics.
105
Figure 3-21 shows the effect of correcting for instrument dead-time according to:
( )
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛+−×
×=
∑=
max
1 iA,ii
6
6
t*t
k2CN109.1
109.1NN 3-4
where: Nt
* = Corrected total particle counts (s-1) Nt = Total observed particle counts (s-1) 1.9×106 = Velocity of laser spot (μm s-1) Ni = Number of ith sized chord counts observed (s-1) Ci,A = Geometric average length of ith channel = (Ci,u × Ci,l)1/2 (μm) k = Additional dead time required before (and after) particle (μm) Equation 3-4 includes a parameter (k) to estimate the additional dead time required
before and after the laser intersects the particle. In this case k = 26.3 μm (calcite,
F-electronics) was found to produce the most linear relationships, and was found
numerically by maximising the sum of the correlation coefficients. Performing the
same optimisation on the other possible data sets gave values of k = 33.3 μm (calcite,
C-electronics), 10.9 μm (aluminium, F-electronics), 21.3 μm (aluminium,
C-electronics).
0
20000
40000
60000
80000
100000
120000
0 5 10 15 20 25 30
Solid fraction (% w/v)
Cor
rect
ed to
tal c
ount
s (s-1
)
Sieved 38-45 µmSieved 45-53 µmSieved 53-63 µmSieved 63-75 µmSieved 75-106 µmSieved 106-125 µmSieved 125-150 µmSieved 150-180 µmSieved 180-212 µmSieved 212-250 µm
Figure 3-21: Relationships between corrected total counts and solid fraction for various sieved calcite fractions. F-electronics. k = 26.3 μm.
106
3.6 Conclusions The FBRM instrument gives rapid, on-line, in-situ, particle size analysis of concentrated
pulps. While the instrument gives a particle chord distribution, as opposed to a
conventional diameter based distribution, mean or mode averages of the square-
weighted FBRM sizes are comparable to conventional sizing techniques over the range
approximately 50-400 μm.
This range may be extended by using the length-weighted average for smaller particles,
and by applying a cube-weighting to larger particles. However, using a lower
weighting increases the instrument’s susceptibility to changes in solid fraction, while
large particles may be difficult to size accurately due to obscuration by fines. This
effect also appears to result in a poor correlation between total particle counts and solid
fraction. However, FBRM square-weighed average sizes were found to be essentially
independent of suspension bulk flow velocity, focal position, or solid fraction in the
range 0.1-20 % w/v. Although the FBRM technique may have slightly less accuracy
and require more calibration compared to other sizing techniques, it is currently the
most suitable commercially available on-line and in-stream sizing probe. It is therefore
the best technique for aggregation studies where sub-sampling and dilution will disrupt
fragile aggregates (see Section 4).
107
4. Calcite flocculation in turbulent pipe flow
This section describes experimental work to measure aggregation kinetics in turbulent
flow, providing data suitable for the population balance model described in Section 5.
Experimental studies of aggregation kinetics are relatively difficult to perform, and may
suffer a range of scale-related problems (Keys & Hogg 1978, Shamlou 1993). Several
laboratory-scale reactor designs have been used successfully, with a simple stirred and
baffled tank being the most popular (Ives 2000). A stirred tank reactor has the
advantage of readily suspending solid particles at a comparatively low average shear
rate (Shamlou 1993), however the shear rate varies considerably through the tank, and
is typically 5-10 times higher around the impeller (Koh et al. 1984, Shamlou &
Titchener-Hooker 1993, Perry & Green 1997). The spatially averaged shear rate (G) is
typically calculated from the mean energy dissipation rate (Cleasby 1984, Bird et al.
1960, p. 157, Davies 1972, p. 7, Clark 1985, p. 749, Perry & Green 1997, p. 6-45, Chen
& Jaw 1998, p. 6-18):
ν∈
=G 4-1
where: ∈ = Energy dissipation rate per unit mass (J s-1kg-1, m2s-3) ν = Kinematic viscosity (m2s-1)
In a stirred tank the energy input can be estimated from the impeller power number
(Spicer & Pratsinis 1996, Perry & Green 1997, Ives 2000), or measured directly from
the motor torque and rpm (Lu et al. 1998, Svarovsky 2000, p. 141). However, a stirred
tank reactor is not suited to continuous flow for aggregation kinetics experiments, due
to the broadening of the residence time distribution (Nauman & Clark 1991).
Restricting the laboratory vessel to a batch process leads to difficulties if aggregate
sizing or settling rate measurements have to be taken ex-situ. The action of removing
the suspension of fragile aggregates (e.g. by syringe or peristaltic pump) can alter the
aggregate size (Williams 1992, Spicer et al. 1998). Further, if the required sub-sample
is large the volume remaining in the tank reactor may be significantly reduced,
increasing the power dissipation rate per unit mass.
108
At the relatively high fluid shear (~10-1000 s-1) and solid fraction (~1-10 % w/v)
(Dahlstrom & Fitch 1985, Williams & Simons 1992) found in mineral processing
thickener feedwells, aggregation proceeds rapidly after flocculant/feed mixing and
adsorption. The short reaction time serves as a further incentive to avoid time-
consuming sub-sampling.
Another popular laboratory vessel for aggregation studies is a Couette device, where the
suspension fills the gap between concentric rotating cylinders (Fair & Gemmell 1964,
Ritchie 1965, Akers et al. 1987, Oles 1992, Shamlou 1993). Either the inner or outer,
or both, cylinders may rotate. In this case the laminar fluid shear rate can be calculated
directly from the relative rotation speed, the cylinder radii, and the width of the gap
(Mühle 1993, Krutzer et al. 1995). However, the Couette device may suffer end effects
(Ives 2000), although an open, upright design effectively removes one end. Both batch
(Ives 2000) and continuous (Farrow & Swift 1996) Couette reactors are reported, and
tapered shear rate can be generated using tapered cylinders (Ives & Bhole 1975, 1977)
with continuous flow.
Aggregation studies have also been performed in pipe flow, either laminar (Gregory
1981, Eisenlauer & Horn 1984, Whittington & George 1992, Suharyono & Hogg 1996),
or turbulent (Delichatsios & Probstein 1975, Klute & Amirtharajah 1991, Wigsten &
Stratton 1994, Ives 2000). However, particles tend to settle out of laminar flow unless
the pipe is vertical, or the particles are neutrally buoyant. Turbulent pipe flow has the
advantage of relatively homogeneous and isotropic turbulence in the core (Delichatsios
& Probstein 1975, Koh et al. 1984), and the mean shear rate can be estimated from the
pressure drop along the pipe (Thomas 1964, Matsuo & Unno 1981, Gould 1985,
Krutzer et al. 1995, Ives 2000), for example by manometer. The average residence time
is readily calculated from the flow rate and the pipe length/diameter, by assuming plug
flow (Nauman & Clark 1991 p. 127) in the turbulent regime. However, small scale
turbulent pipe flow generates a relatively high mean fluid shear rate (Suharyono &
Hogg 1996, Ives 2000, p. 152), and requires a large feed volume, which cannot be re-
cycled because it is irreversibly altered by the flocculant.
109
As outlined before, the aim of this section is to provide aggregation kinetics data
suitable for a population balance model (see Section 5) of flocculation. In addition to
fluid shear, several other process variables may impact on the industrial flocculation
process, and the effects of flocculant dosage, solid volume fraction and primary particle
size are also been investigated.
4.1 Experimental Aggregation/breakage kinetics were studied in turbulent pipe flow (Figures 4-1, 4-2 &
4-3). A combination of pipe diameters (25.4 or 38.1 mm ID) and flow rates
(14-40 L min-1, 0.21-1.29 m s-1) were used to produce a range of fluid shear rates. A
series of flocculant injection nipples along the pipe produced a range of effective pipe
lengths, and hence mean residence times assuming plug flow.
The pipe reactors were assembled from lengths (1.83 m, BW Plastics) of transparent
acrylic pipe, with care taken to produce seamless joints and ensure a smooth pipe inner
surface. Conveniently, the pipes are produced in a range of sizes where the outside
diameter (OD) of each pipe is the same as the inside diameter (ID) of the next largest
size. This allowed a butt joint to be easily fabricated by gluing the ends together inside
a short length of larger pipe. Disassembly was by saw.
Similarly, the flocculant injection nipples were attached to the outside of the pipe so
that the inside surface of the pipe was only altered by only drilled (2.4 mm) holes. The
pipe was suspended approximately 1 m above the floor from a tensioned horizontal
suspension cable arrangement (Figure 4-1), ensuring that the pipe reactor was kept
straight and horizontal.
110
Figure 4-1: Schematic of the laboratory equipment.
Figure 4-2: View of manometer bank and Figure 4-3: View of pipe reactor experimental apparatus
Flocculant dosing points
Sizing probe
Pipe reactor 15 m x 25.4 mm ID 20 m x 38.1 mm ID
Feed suspension storage tank
To waste
Settling column
Flocculant pump
Flocculant storage tank
Manometer
Feed pump
Linked ball valves
111
A Lasentec FBRM (C-electronics) particle sizing probe was placed in the flow at the
end of the pipe reactor, followed by a vertical column used to take hindered settling rate
measurements. The FBRM instrument was described in Section 3, and was calibrated
with the same solid substrate (Calcite, Commercial Minerals).
The feed suspension was made up quantitatively in a 0.8 m3 stirred/baffled feed tank,
and confirmed gravimetrically by drying. A range of feed suspensions of different solid
fraction (3.33 – 16.7 % w/v, φExp = 0.012-0.062, ρcalcite = 2710 kg m-3) and/or mean
primary particle size (2.36 – 24.3 μm) were used to simulate changes to the feed
(Table 4-1). The various grades of calcite were sized independently by Laser
diffraction (Malven Mastersizer), which also provided particle surface area
measurements. Surface area measurements were also made by an alternative technique
(BET) for comparison.
The feed suspension was delivered to the pipe reactor by a variable ratio drive (metal
belt type) positive displacement pump (Mono-pump), and the flow rate confirmed by an
on-line (Admac AE) flow meter. The pressure drop along the pipe was measured by a
manometer bank (Figures 4-1 & 4-2). The manometers were equilibrated from above,
after flushing with water. This removed any solid from the manometer tubes, ensuring
a known density (taken as 1000 kg m-3) and accurate pressure drop measurement.
Flocculant (Nalco 9902, 30 % anionic acrylate/acrylamide copolymer with a nominal
molecular weight of 15 million g mol-1) stock solution (0.02 % w/v) was made up
reproducibly (stirrer intensity, volume and time) the day before use, minimising any
effect from flocculant aging (Owen et al. 2002). The flocculant was dosed
quantitatively on a mass (flocculant) for mass (solid) basis into the pipe reactor with a
peristaltic pump (Masterflex) and confirmed by flow meter (Magflow MAG 300).
Settling rate measurements were made on the aggregated suspension using an isolatable
(note valves) and detachable settling column, also made from acrylic tube (38.1 mm
ID). The column was marked at regular intervals (1 & 5 cm) and lit from behind by
fluorescent tube, allowing the measurement of the hindered settling rate by following
the fall of the mudline through time with a stopwatch.
112
The pipe work and fittings were designed to keep the top of the settling column as close
as possible to the particle sizing probe, reducing the additional residence time. This
was the primary reason for mounting the pipe reactor above floor level (note Figure 4-
1). The fittings around the bend and valve were also constructed to produce a smooth
inner surface in an effort to reduce additional turbulence and pressure drop. Despite
this, the additional effective residence time was still considerable (see Section 6).
The addition of flocculant irreversibly alters the suspension, precluding the possibility
of re-cycle. Consequently, after size and settling rate measurement the flocculated
suspension was fed to a 10 m3 waste storage sump.
A matrix of experimental runs was performed according to Figure 4-4. The process
variables of fluid shear, flocculant dosage, solid fraction and primary particle size were
altered independently away from a common baseline point: 25.4 mm ID pipe,
14.01 L min.-1. (mean velocity 0.461 m s-1), 10 % w/v (3.69 % v/v) solid, Omya-carb 5
(volume-weighted average diameter 6.59 μm by Malvern), and a flocculant dosage of
20 g t-1 (solid).
Figure 4-4: Experimental matrix
Flocculant dosage (g t-1)
5
80
3.33
10
40
2.36
3.47
15.0
24.3
Mean primary particle size (μm)
Flow velocity (m s-1) [pipe ID (mm)]
6.67
16.67
0.554 [38.1]
0.207 [38.1]
Solid volume fraction (% w/v)
Baseline: 0.461 m s-1 [25.4 mm], 10 % w/v, 6.59 μm, 20 g t-1
0.343 [38.1]
1.294 [25.4]
0.781 [25.4]
13.33
113
Aggregate size and settling rate measurements were made with flocculant injected at 13
positions (including a zero residence time - no flocculant) along the pipe, producing a
range of residence times. The injection points were spaced according to a numerical
progression, so that they were closer together at short residence times where the
aggregate size increases rapidly (e.g. Figure 4-5).
This range of pipe lengths was used for each of the experimental conditions described
by the sparse matrix described by Figure 4-4, giving a total of 13 × (6 + 4 + 4 + 4) data
points. A complete 13 × 5 × 4 × 4 × 4 experimental matrix was clearly not feasible.
However, data was also collected under additional conditions, in the gaps of the matrix,
by changing two process variables simultaneously. These additional data points were
used to test the predictivity of the population balance model described in Section 5.
Table 4-1: Matrix of pipe reactor experimental runs Run No. Pipe ID Flow rate Solid φ Mean dp Floc. dose Viscosity Susp. Density Kin. visc. Re ∈ G f Comment
(m) (m3s-1) (m3m-3) (m) (kg kg-1) (N s m-2) (kg m-3) (m2s-1) (m2s-3) (s-1)211100A 0.0254 2.34E-04 0.037 6.59E-06 2E-05 0.0024 1063 2.21E-06 5291 0.071 179.6 0.0093 Baseline # 1211100B 0.0254 2.34E-04 0.037 6.59E-06 4E-05 0.0165 1063 1.56E-05 752 0.116 86.4 0.0151 Settling211100C 0.0254 2.34E-04 0.037 6.59E-06 1E-05 0.0016 1063 1.52E-06 7681 0.065 206.6 0.0084221100A 0.0254 2.34E-04 0.037 6.59E-06 8E-05 0.0509 1063 4.79E-05 245 0.154 56.7 0.0200 Settling221100B 0.0254 2.34E-04 0.037 6.59E-06 5E-06 0.0014 1063 1.34E-06 8759 0.063 217.0 0.0082231100A 0.0254 2.34E-04 0.025 6.59E-06 2E-05 0.0018 1042 1.72E-06 6821 0.067 197.6 0.0087231100B 0.0254 2.34E-04 0.062 6.59E-06 2E-05 0.0030 1105 2.73E-06 4294 0.075 166.1 0.0098231100C 0.0254 2.34E-04 0.012 6.59E-06 2E-05 0.0015 1021 1.42E-06 8222 0.064 211.9 0.0083241100A 0.0254 2.34E-04 0.049 6.59E-06 2E-05 0.0027 1084 2.52E-06 4637 0.074 170.9 0.0096271100A 0.0254 2.34E-04 0.037 6.59E-06 2E-05 0.0026 1063 2.47E-06 4739 0.073 172.3 0.0095 Baseline # 2281100A 0.0254 3.96E-04 0.037 6.59E-06 2E-05 0.0015 1063 1.45E-06 13700 0.274 435.0 0.0073281100B 0.0254 6.56E-04 0.037 6.59E-06 2E-05 0.0012 1063 1.17E-06 28173 1.039 944.1 0.0061281100C 0.0381 6.32E-04 0.037 6.59E-06 2E-05 0.0019 1063 1.80E-06 11736 0.068 194.2 0.0076011200A 0.0381 3.91E-04 0.037 6.59E-06 2E-05 0.0330 1063 3.10E-05 421 0.037 34.5 0.0174 Settling011200B 0.0381 2.36E-04 0.037 6.59E-06 2E-05 0.1675 1063 1.58E-04 50 0.014 9.4 0.0297 Settling041200A 0.0254 2.34E-04 0.037 1.51E-05 2E-05 0.0058 1063 5.42E-06 2158 0.089 128.3 0.0116041200B 0.0254 2.34E-04 0.037 2.36E-06 2E-05 0.0019 1063 1.81E-06 6478 0.068 193.8 0.0088051200A 0.0254 2.34E-04 0.037 3.47E-06 2E-05 0.0016 1063 1.46E-06 7996 0.064 209.7 0.0084051200B 0.0254 2.34E-04 0.037 2.43E-05 2E-05 0.0101 1063 9.46E-06 1237 0.103 104.1 0.0133 Settling061200A 0.0254 2.34E-04 0.025 1.51E-05 2E-05 0.0028 1042 2.72E-06 4307 0.075 166.3 0.0098061200B 0.0254 3.96E-04 0.037 6.59E-06 4E-05 0.0012 1063 1.17E-06 16934 0.260 471.0 0.0069071200A 0.0381 3.91E-04 0.037 6.59E-06 1E-05 0.0029 1063 2.77E-06 4715 0.020 85.3 0.0095071200B 0.0381 3.91E-04 0.025 6.59E-06 2E-05 0.0136 1042 1.31E-05 1000 0.030 47.7 0.0140 Settllng
114
4.2 Results and discussion 4.2.1 Effect of mean fluid shear rate
Figure 4-5 shows the effect of changing the flow rate and pipe reactor inside diameter
(ID) on the volume-average aggregate size. The flow rate and pipe ID were altered to
produce a range of spatially-averaged shear rates (~100-1000 s-1, Table 4-1) calculated
from the pressure drop measured by the manometer bank. A small pipe and high flow
rate increases the flow resistance, the energy dissipation, and hence also increases the
mean fluid shear rate. This initially leads to an increased aggregation rate due to faster
flocculant/suspension mixing and particle collision, but ultimately leads to a reduced
aggregate size due to an increased aggregate breakage rate. The reduction in final
aggregate size at higher shear rates is typical, and other systems show a similar response
(Curtis & Hocking 1970, Oles 1992, Keys & Hogg 1978, Mühle 1993, p. 382, Spicer &
Pratsinis 1996, Serra & Casamitjana 1998, Chin et al. 1998, Flesch et al. 1999).
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
0 10 20 30 40 50 60 70
Time (s)
Mea
n ag
greg
ate
diam
eter
(m)
38.1 mm ID, 0.343 m sֿ¹, G = 52.6 sֿ¹, (settling obs. in pipe)
25.4 mm ID, 0.461 m sֿ¹, G = 174.1 sֿ¹, (Baseline)
38.1 mm ID, 0.554 m sֿ¹, G = 175.2 sֿ¹
25.4 mm ID, 0.781 m sֿ¹, G = 488.1 sֿ¹
25.4 mm ID, 1.294 m sֿ¹, G = 935.5 sֿ¹
Figure 4-5: Effect of pipe size and fluid flow rate on mean aggregate size.
The gentle reduction in the aggregate size at extended residence times is typical
behaviour when polymer flocculants are used (Keys & Hogg 1978, Williams et al.
1992, Leu & Ghosh 1988, Bagster 1993, Hogg 2000), and is usually taken as evidence
of flocculant degradation by scission or rearrangement (Sikora & Stratton 1981,
Gregory 1993, Stratton 1983, Ditter et al. 1982). Conversely aggregates formed by
coagulation with soluble salt typically reach a steady state aggregate size, because in
115
that case breakage is reversible (Oles 1992, Flesch et al. 1999, Spicer & Pratsinis
1996a, Serra & Casamitjana 1998).
A manometer bank (Figures 4-1 & 4-2) was used to measure the pressure drop along the
pipe reactor. This allowed the calculation of the energy dissipation rate per unit mass of
fluid (∈) from (Perry & Green 1997):
LPV
fρΔ
∈= 4-2
where: V = Mean flow velocity along pipe (m s-1)
△P = pressure drop along pipe (N m-3) given experimentally from: ghP ρ=Δ ρf = Density of the fluid (kg m-3) L = Pipe length (m)
The mean pipe flow velocity (V) is known experimentally via the volumetric flow rate
and pipe ID. The fluid density (ρf) is readily calculated from the solid fraction and
density:
( )φ−ρ+φρ=ρ 1wsf 4-3
where: ρ = Density (f = fluid, s = solid, w = water) (kg m-3) φ = Solid volume fraction [0,1]
In this case the solid volume fraction (φ) is known experimentally because the feed is
made up quantitatively in batches in the feed tank, and confirmed gravimetrically by
drying.
The mean dissipation rate is readily calculated from Equation 4-2, however in order to
calculate the mean fluid shear rate (Equation 4-1), the viscosity is also required. In
most studies (Williams & Simons 1992) the viscosity is taken to be unchanged from
pure water at the same temperature. This assumption is usually valid because most
studies (Oles 1992, Spicer et al. 1996, Serra & Casamitjana 1998, Flesch et al. 1999,
Manning & Dyer 1999) have simulated conditions in water treatment clarifiers or river
estuary system, which are characterised by a low solid fraction (Dahlstrom & Fitch
1985, Perry & Green 1997).
However, feed streams to thickener units used in mineral processing have a
considerably higher solid volume fraction, typically in the percent range (Pearse 1977,
Dahlstrom & Fitch 1985, Perry & Green 1997), leading to an increased fluid viscosity
116
(Williams et al. 1992, see Section 5). The volume of liquid enclosed within porous
aggregates has the effect of further increasing the effective solid volume fraction (Mills
et al. 1991, see Section 6). This effect is demonstrated experimentally (Figure 4-7) by
an increased flow resistance and larger pressure drop along the pipe. The use of a
positive displacement feed and flocculant pumps ensured constant flow rates, with the
additional required energy readily supplied by the pumps since the overall backpressure
along the pipe was comparatively minor at ∼ 1-10 × 103 N m-2 (0.01- 0.1 atm.).
Estimating the pressure drop through pipes is a common engineering problem, e.g.
when specifying pumps for pipelines. This has lead to the development of the concept
of a pipe Fanning friction factor (f), given by (Bird et al. 1960, Govier & Aziz 1972,
Wasp et al. 1977, Perry & Green 1997):
32 V2D
LV2PDf ∈
=ρΔ
= 4-4
where: f = Fanning friction factor D = Pipe diameter (m) ΔP = Pressure drop along pipe (N m-3)
ρ = Fluid density (kg m-3) ∈ = Energy dissipation rate (J s-1kg-1, m2s-3) V = Mean flow velocity (m s-1) L = Pipe length (m)
An alternative system is based on the Darcy friction factor (Daugherty et al. 1989),
differing only by a factor of 4.
Equation 4-4 is usually under-specified (ΔP or ∈), and efforts have been made
(Table 4-2) to find alternative descriptions of the friction factor, usually as a function of
the pipe roughness and pipe Reynolds number (Re), given by:
μρ
=DVRe 4-5
For turbulent flow (4000 < Re < 100 000) in smooth pipes the friction factor is described by the
Blasius (1913) equation (Table 4-2):
41
Re
0791.0f = 4-6
117
Table 4-2: Equations relating pipe friction factor to Reynolds number Reference Equation Comment Govier & Aziz 1972 LV2
PgDf 2ρΔ
≡ Definition of friction factor (similar to Darcy’s friction factor)
Blasius 1913
41
Re
079.0f = 4 000<Re<100 000, smooth pipes (1)
Colebrook 1939 (see below)
⎥⎦
⎤⎢⎣
⎡+
∈−=
fRe256.1
D7.3log4
f1
Re>4 000, rough pipes, (1)
Churchill 1977
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+
∈−=
9.0
Re7
D27.0log4
f1
Re>4 000, rough or smooth pipes, ∈ = surface roughness (mm), (1)
Knudsen & Katz 1958 2.0Re
046.0f = (2)
Nikuradse 1932
[ ] 4.0fRelog4f
1−=
3 000<Re<3 000 000, (2)
Drew, Koo & McAdams 1932 32.0Re
125.000140.0f +=
3 000<Re<3 000 000, (2)
Govier & Aziz 1972
[ ] 6.0fRelog06.4f
1−=
(2)
Nikuradse 1932
48.3k2
Dlog4f
1+⎥⎦⎤
⎢⎣⎡=
Rough pipes (2)
Von Karman 1930 36.3
k2Dlog06.4
f1
+⎥⎦⎤
⎢⎣⎡=
Rough pipes (2)
Colebrook 1939 (see above)
⎟⎟⎠
⎞⎜⎜⎝
⎛+−+⎥⎦
⎤⎢⎣⎡=
fRek2D35.91log448.3
k2Dlog4
f1
Partially rough pipe (2)
Buckingham 1921 ⎟
⎟⎠
⎞⎜⎜⎝
⎛
+−ρ= 3
31
34 xx1
1DV16f where:
PDL4
x y
w
y
Δ
τ=
τ
τ=
Non-Newtonian fluid (2)
Govier & Winning 1948
⎟⎟⎠
⎞⎜⎜⎝
⎛μ
τ
μρ
φ=V
gD,DVf cy
1 Non-Newtonian fluid (2)
Hedstrom 1952
⎟⎟⎠
⎞⎜⎜⎝
⎛
μ
ρτ
μρ
φ= 2cy
2
2gD
,DVf Non-Newtonian fluid (2)
Tarrance 1963 n
65.20.6kRlog
n07.4
f1
−+⎥⎦⎤
⎢⎣⎡=
Non-Newtonian fluid (2)
Clapp 1961 [ ] 3.2fRelog53.4
f1
−= Non-Newtonian fluid (2)
(1) Perry & Green 1997 (2) Govier & Aziz 1972
118
Equation 4-6 was used to calculate the pressure drop along the pipe reactor with water
only (no solid), allowing the estimation of the fluid viscosity (0.00105 N s m-3, compared
to 0.00102 N s m-3 at 20 oC, Daugherty 1989) from the experimental pressure drop.
y = x - 1.0461R2 = 1
0
100
200
300
400
500
600
700
800
900
0 100 200 300 400 500 600 700 800 900
Measured pressure drop (N m-3)
Est
. pre
ss. d
rop
Bla
sius
eq.
(N m
-3)
Figure 4-6: Comparison of measured and predicted (Equation 4-7) pressure drop of water only under various flow regimes (as per Figure 4-5).
Equation 4-6 is tentatively assumed to hold for the turbulent flow of the aggregated
suspensions, allowing the calculation of the fluid viscosity by combining Equations 4-4,
4-5 & 4-6:
4734
54
LV)0791.0*2(DPρ
Δ=μ 4-7
where: μ = Suspension viscosity (N s m-2) D = Pipe diameter (m)
ΔP = Pressure drop along pipe (N m-3) ρ = Fluid density (kg m-3) ∈ = Energy dissipation rate (J s-1kg-1, m2s-3) V = Mean flow velocity (m s-1) L = Pipe length (m)
Figure 4-7 shows the increase in the average suspension viscosity flocculated at
different shear rates, as calculated from the pressure drop using Equation 4-7 (see also
Table 4-1). The viscosity increases at lower shear rates, because the increased
aggregate size (Figure 4-5) leads to increased aggregate porosity (see Sections 5 & 6)
and hence effective solid fraction and suspension viscosity. The data is truncated to
119
lower shear rates because stratification/settling in the pipe reactor throttled the flow,
leading to an increased pressure drop.
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0 100 200 300 400 500 600 700 800 900 1000Mean shear rate (s-1)
Vis
cosi
ty (N
s m
-2)
Figure 4-7: Aggregated suspension fluid viscosity estimated (Equation 4-7) for the pressure drop along the pipe (see Figure 4-5 for aggregate size data).
Solid particles are suspended by turbulent eddies, however, if the turbulence is
insufficient for a given particle settling rate, the solid may stratify, or form a stationary
bed on the bottom of the pipe reactor (Figure 4-8, Table 4-3). This has the effect of
throttling the flow, increasing the flow velocity and turbulence in the remaining clear
portion of the pipe, preventing further settling. Eventually a steady state condition is
reached with a uniform stationary layer. The use of a transparent acrylic pipe allows the
observation of stratification/settling, especially if a fluorescent tube is mounted behind
the pipe to cast a shadow. Data collected when there was visible stratification/settling is
of limited value, because the settled solid reduced the effective pipe diameter and
decreased the average residence time.
Minimum suspension flow velocities in pipelines are usually described by the Durand
(1953) equation as a function of the mean flow velocity in the pipe, and the particle size
and density, although there is a variety of alternative relationships to choose from
(Spells 1955, Zandi & Govatos 1967, Kao & Wood 1974 , Turian & Yuan 1977, Wasp
et al. 1977, Oroskar & Turian 1980, Anon. 1989, Shah & Lord 1991, Table 4-3).
120
( )[ ] 5.0L2M 1sgD2FV −= 4-8
where: VM2 = Minimum velocity required to keep solid suspended (m s-1) (Figure 4-8) d = Particle diameter (m) D = Pipe diameter (m)
g = Gravity (9.8 m s-2) s = ratio of the solid to liquid density FL = function of solid volume fraction and particle size
Equation 4-8 is typically used for estimating the minimum transport velocity of
relatively coarse particles (>1000 μm) in full-scale pipelines, and has limited
applicability to the current work with aggregated suspensions of fine particles. In the
current case, stratification/settling was observed around the transition to laminar flow
(Re ∼ 2000-4000, see Table 4-1).
Figure 4-8: Critical flow velocities, see Table 4-3 below (adapted from Perry and Green 1997).
Log suspension velocity
Log
Pre
ssur
e gr
adie
nt VM3
VM1 VM4
MixturePure liquid
Sym
met
ric
Susp
ensi
on
Sym
met
ric
Susp
ensi
on
Stat
iona
ry B
ed
Mov
ing
Bed
VM2
121
Table 4-3. Equations describing critical minimum flow velocities of slurries Reference Equation Comments Durand 1953
( )[ ] 5.0
L2M 1sgD2FV −=
Widely referenced
Spells 1955 )1s(gd
DV075.0V 85
775.0M1M2
1M −⎟⎟⎠
⎞⎜⎜⎝
⎛μρ
=
Spells 1955 )1s(gd
DV025.0V 85
775.0M1M2
2M −⎟⎟⎠
⎞⎜⎜⎝
⎛μρ
=
Oroskar and Turian 1980
( )[ ] 30.009.0Re
378.050
3564.0v
1536.0v
5.0502M N)D/d()C1(C1sgd85.1V χ−−= − [ ]
f)1S(gd2fD
N21
50Re μ
−ρ=
Wasp et al. 1977 ( )( )
31
21
dDF1sgd2V L2M ⎟⎠⎞
⎜⎝⎛−=
Zandi and Govatos 1967
( )( )21
21
dD
CC401sgdVd
2M ⎟⎠⎞
⎜⎝⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−=
Shah and Lord 1991
( )[ ] zRe
w50
3564.0v
1536.0v
5.0502M N)D/d()C1(C1sgdYV −−−=
Turian and Yuan 1977
( )( )21
3225.0065.1w
5906.d2M CfC1sgD4608.0V −−−−=
Kao and Wood 1974 ( )( )
21
n1
21
dD
Dd
381sgdV 22M ⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
αβ−=
D = Pipe diameter (m) d = Particle diameter (m) ρ = Density (kg m-3) s = Ratio of the solid to liquid density Cv = Particle drag coefficient FL = Function of solid volume fraction and particle size The relatively high shear rate produced by small-scale turbulent pipe flow
(approx 100-1000 s-1 here) acts as an incentive to reduce the flow velocity and increase
the pipe size. However, reducing the flow reduces the Reynolds number, causing the
solid to settle out, while increasing the pipe size dramatically increases the required
feed and disposal volumes (~ 5 tonnes in this case). Alternatively, settling can be
suppressed by substituting a smaller, or less dense primary particle to reduce the settling
rate. In this case a range of relatively fine primary particles were used
(mean dp = 2.36 - 24.3 μm). Using a lower density solid substrate (latex perhaps) would
have compromised the hindered settling rate measurements (Section 6), and may also
have given a poor approximation of a mineral surface.
122
4.2.2 Effect of flocculant dosage
Figure 4-9 shows the increase in aggregate size with flocculant dosage. The increase in
size can be attributed to either an increased capture efficiency (α) and/or increased
aggregate strength leading to reduced breakage. Although the data show a faster initial
aggregation rate at higher dosages, the difference must be at least partially attributed to
the increased initial mixing at higher dosages. The dilute (0.02 %) flocculant stream is
injected perpendicularly into the pipe reactor via a small hole in the pipe wall
(see Section 4.1), causing a visible (acrylic pipe) increase in the turbulence around the
injection point. The additional initial turbulence increased with flocculant dosage,
because the dosage was altered by changing the flocculant stream flow rate (rather than
by increasing the flocculant stream concentration).
Flocculant/suspension macro-scale mixing is likely to be highly scale dependent, and its
importance is evidenced by the industrial practice of using a very dilute flocculant
stream (0.01-0.1 %, Perry & Green 1997, Dahlstrom & Fitch 1985) and various
proprietary feedwell designs (e.g. multi-point addition) to produce efficient mixing. It
is difficult to replicate industrial-scale mixing conditions on a laboratory-scale, however
the use of emerging techniques like computational fluid dynamics (Lainé et al. 1999,
Ducoste & Clark 1999, Farrow et al. 2000), or electrical impedance tomography
(Brown et al. 1985, Webster 1990, Salkeld 1991, Dickin et al. 1992, Williams &
Simons 1992) is likely to increase the understanding in this area.
Figure 4-9 shows an increased aggregate size with flocculant dosage. There is evidence
to suggest (Healy 1961, Fleer & Scheutjens 1993, Svarovsky 2000) that very high
flocculant dosages may lead to steric re-stabilisation as the particle becomes overloaded
with flocculant. However, it is not clear that such a situation would occur
commercially, where flocculant is the major operating cost for many units (Perry &
Green 1997, Hogg 2000), and excessive flocculant may reduce the unit’s compression
performance (Healy et al. 1994, Svarovsky 2000).
123
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
2.5E-04
3.0E-04
3.5E-04
0 5 10 15 20 25 30 35
Time (s)
Mea
n ag
greg
ate
diam
eter
(m)
Dosage = 80 g tֿ¹, (settling obs. in pipe)Dosage = 40 g tֿ¹, (settling obs. in pipe)Dosage = 20 g tֿ¹, (Baseline)Dosage = 10 g tֿ¹Dosage = 5 g tֿ¹
Figure 4-9: Effect of flocculant dosage on the aggregate size.
Figure 4-10 shows the increase in viscosity and corresponding reduction in fluid shear
as a function of the flocculant dosage. Again, this is attributed to the increased effective
solid volume fraction as the aggregate size is increased. The possibility of the
flocculant alone increasing the viscosity was discounted by performing an additional
experiment performed under the same conditions, except without solid (water and
flocculant only). Somewhat surprisingly, the pressure drop actually decreased
fractionally (not shown), a phenomenon widely reported (Paterson and Abernathy 1970,
Moussa and Tiu 1994, Rho et al. 1996, Perry & Green 1997) and exploited to reduce
the pumping costs in long pipelines.
0
50
100
150
200
250
0 5 10 15 20 25Flocculant dosage (g t-1)
Shea
r ra
te (s
-1)
0
0.001
0.002
0.003
0.004
0.005
Vis
cosi
ty (N
s m
-2)
Shear rateViscosity
Figure 4-10: Effect of flocculant dosage on the measured fluid viscosity and shear rate. Data is truncated due to settling in pipe reactor.
124
4.2.3 Effect of primary particle size
Figure 4-11 shows the increase in aggregate size with primary particle size. This may
be caused by a combination of effects. As the primary particle size is increased, the
total particle surface area per unit mass is decreased, leading to an increased flocculant
surface coverage per area (Hogg 1999a). The increase in primary particle size will also
increase the solid packing efficiency within the aggregate, decreasing the porosity,
hence increasing the aggregate strength (Dobias 1993, Bagster 1993). The decreased
porosity also leads to a decrease in the effective suspension solid fraction (Equation 6-
9), leading to a decreased fluid viscosity, dissipation rate and breakage.
Overall however, the increased strength of aggregates formed from larger primary
particles leads to an increased overall degree of aggregation, a higher suspension
viscosity (Figure 4-12) and dissipation rate. In this case the viscosity is the dominant
term of Equation 4-1, so the fluid shear is reduced with a larger primary particle.
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
2.5E-04
0 5 10 15 20 25 30 35
Time (s)
Mea
n A
ggre
gate
size
(m)
dp = 24.3 µm, (settling obs. in pipe)dp = 15.0 µmdp = 5.59 µm, (Baseline)dp = 3.47 µmdp = 2.36 µm
Figure 4-11: Effect of mean primary particle size on the measured fluid viscosity.
125
0
50
100
150
200
250
0 5 10 15 20 25 30
Mean primary particle diameter (µm)
Shea
r ra
te (s
-1)
0
0.002
0.004
0.006
0.008
0.01
0.012
Vis
cosi
ty (N
s m
-2)
Shear rateViscosity
Figure 4-12: Effect of primary particle size on the measured fluid viscosity and shear rate. The mean size is the volume weighted mean as determined by laser diffraction (Malvern Mastersizer).
4.2.4 Effect of suspension solid fraction
Figure 4-13 shows the reduction in aggregate size with increased solid fraction, a result
that has been observed elsewhere (Lick & Lick 1988, Williams et al. 1992), although
not well understood. Figure 4-14 shows the increase in suspension viscosity and
reduction in the shear rate as determined by manometer measurements. The reduction
in the aggregate size at high solid fraction is attributed to a reduced particle collision
rate at lower shear, and an increased breakage rate due to the increase in viscosity and
energy dissipation. At very low solid fraction, there is evidence (Thomas 1964, de Boer
et al. 1989, Kobayashi et al. 1999) to suggest that the aggregate size may be reduced,
perhaps due to a decreased collision rate (see Section 5).
126
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
0 5 10 15 20 25 30 35
Time (s)
Mea
n A
ggre
gate
size
(m)
3.33 % w/v6.67 % w/v10.0 % w/v (Baseline)13.33 % w/v16.67 % w/v
Figure 4-13: Effect of suspension solid fraction on mean aggregate size. The flocculant dosing rate was maintained at 20 g (flocculant)/t (solid) as per the baseline condition.
0
50
100
150
200
250
300
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Absolute solid fraction (v/v)
Shea
r ra
te (s
-1)
0
0.001
0.002
0.003
0.004
Vis
cosi
ty (N
s m
-2)
Shear rateViscosity
Figure 4-14: Effect of feed suspension solid fraction on the measured fluid viscosity.
127
4.3 Conclusions The aggregation/breakage kinetics of calcite flocculation have been studied
experimentally in turbulent pipe flow under a range of conditions, reflecting changes in
likely process variables: fluid shear, flocculant dosage, primary particle size, and solid
volume fraction. Reaction residence times were altered by varying the length of pipe
between the flocculant injection point and the aggregate sizing probe. The pressure
drop along the pipe was measured by manometer, allowing the calculation of the energy
dissipation, fluid viscosity, and the mean shear rate.
Efforts were taken to match the conditions in the pipe reactor to those expected in the
full-scale plant, although scale-up remains an issue. Flocculant/feed suspension mixing
is faster with small volumes, and it is likely to be considerably overestimated by
small-scale equipment. However, the major limitation is due to the tendency of solid to
settle out of small-scale low Reynolds number flows, restricting this study to a smaller
primary particle and higher mean shear rate than are expected in practice.
The initial aggregation rate was found to increase at higher shear rates, however the
final aggregate size was ultimately limited by aggregate breakage which is significantly
increased at a higher shear rate. The aggregate size is increased as a function of the
flocculant dosage and primary particle size, but reduced at higher solid fraction.
128
5. Population balance model
5.1 Model development 5.1.1 Aggregation
The experimental data presented in Section 4 was used to develop a population balance
model for aggregation and breakage of calcite flocculated by high molecular weight
polymer flocculant. The population balance is based on the successful population
balance described by Hounslow et al. (1988) and Spicer and Pratsinis (1996a) (see also
Equation 2-45):
∑
∑∑∑∞
=
∞
=
−
=−−−
−
=−−
+−
Γ+−
β−β−β+β= −
ijjjj,iii
ijjj,ii
1i
1jjj,ii
21i1i,1i
2i
1j21
j1ij,1i1iji
NSNS
NNN2NNNN2dt
dN ij
5-1
where: Ni = Number of ith sized particles (m-3) t = Time (s) βij = Rate of aggregation of i and j sized particles (aggregation kernel) (m3s-1) Sj = Breakage rate (kernel) of jth sized particles (s-1) Γij = Breakage distribution function (number of i size particles produced from the breakage of a j sized particle) The widely accepted (Tables 2-1 & 2-2) turbulent collision kernel by Saffman and
Turner (1956) was used to describe the aggregation rate:
( )3jiij aaG294.1 +α=β 5-2
where: α = Capture efficiency [0,1] (taken as = M)
G = Average turbulent shear rate (s-1) ai = Radius of the ith particle (m)
Before flocculant addition, the primary particles are taken to be well dispersed as a
stable suspension, although in many systems there the primary particles may naturally
coagulate to some extent due to a low zeta potential. However, no large stable
aggregates are formed at this stage, despite their collision in the turbulent flow. In this
situation the particles are repelled by electrostatic repulsion (Gregory 1989, Kohler
1993, Hughes 2000) and the effective capture efficiency (α) is taken to be zero.
Alternatively, every collision could be considered to be successful (α = 1), but with the
breakage rate equal to the collision rate. Both methods were tried during model
129
construction, but in this case the capture efficiency was taken to be zero before
flocculant addition, giving a simpler and more stable model, and a better fit with the
experimental data.
On the addition of flocculant, the aggregation rate exceeds the breakage rate, producing
the increase in the aggregate size shown in Figure 5-1. In this case the capture
efficiency must be greater than zero, however, if the capture efficiency is set to unity
immediately on flocculant addition, Equation 5-2 considerably overestimates the
aggregation rate compared to the experimental data. This is a common finding
(Table 2-2) and the capture efficiency is frequently taken to be in the range 0.05-0.5. In
this case, if the capture efficiency was set to a constant value after flocculant addition, it
had to be relatively small (~ 0.06) to give a reasonable fit to the data, suggesting that
particle collision (Equation 5-2) was not the rate limiting step. Alternatively, the
capture efficiency was taken to increase rapidly after flocculant addition
(Equations 5-3 - 5-6), reflecting the mixing/adsorption time required for the polymer to
reach the particle surface.
The mixing and adsorption behaviour of polymer flocculants is complex and poorly
described, although there is general agreement that adsorption is rapid under normal
industrial conditions (Gregory 1993, Bagster 1993, Mühle 1993, Hogg 1999a, 1999b),
and that the bulk mixing of the flocculant and feed streams is rate limiting. This effect
is likely to be highly scale-dependent, but its importance is evidenced by the complexity
of commercial feedwells, containing various design elements (Dahlstrom & Fitch 1985,
Perry & Green 1997, Svarovsky 2000) to produce efficient flocculant mixing.
Mixing is usually described in terms of a mixing index (Nagata 1975, Etchells & Short
1988, Godfrey & Amirtharajah 1991):
o
1Mσσ
−= 5-3
and assuming: α = M where:
σ = Concentration variation (σo at t = 0), given by:
130
( )∫ −=L
0
22 dx)m(C)x(Cσ 5-4
and: M = Mixing index [0,1]
C(x) = Concentration at point x (or m = mean). However, experimental data for Equation 5-4 is generally unavailable, and alternatively
Etchells and Short (1988) suggest for turbulent pipe flow:
D/Lfk
o
1e−=σσ
5-5
where: k1 = Constant, Etchells and Short (1988) suggest = 0.5 (Fitted parameter # 1 := 0.3431) f = Pipe friction factor (Equation 4-4) L = Pipe length (m) D = Pipe diameter (m)
Combining Equations 5-3 and 5-5 gives:
D/Lfk1e1M −−= 5-6 Equation 5-6 rapidly approaches unity (by ~ 100 pipe diameters; Figure 5-2)
independent of the flocculant dosage, although this is unlikely in practice and
inconsistent with Equations 2-17 and 2-19. However, in this case the population
balance was made dosage dependent via the breakage kernel (Section 5.1.2,
Equation 5-16) reflecting an increase in aggregate strength with flocculant dosage
(Smith & Kitchener 1978, Ray & Hogg 1987). Both approaches were tried during
model construction, but making the breakage kernel (rather than the capture efficiency)
dosage dependent gave a better fit with the experimental data.
Equation 5-6 is a reasonably crude approximation of the overall mixing/adsorption
process and further work is required to better describe these effects, perhaps by a
combination of CFD and electrical impedance tomography. When the flocculant
mixing/adsorption process is better understood it should be possible to further refine the
capture efficiency term by including the effects of flocculant surface coverage and/or
the hydrodynamic effect.
131
It would have been preferable to begin the experimental and model with the flocculant
already well mixed, however this is not possible with polymer flocculants. Studies of
aggregation using salt coagulants can be performed (Spicer & Pratsinis 1996a) by
adding coagulant to a stirred tank with a high impeller speed to mix the coagulant and
disperse the primary particles. The stirrer may then be turned down to give the
appropriate shear rate, and aggregation allowed to begin. However, this approach is not
possible with fragile polymer flocculants, and any initial rapid stirring results in
irreversible polymer degradation.
Aggregation kernels for turbulent flow (Tables 2-1 & 2-2) are usually written in terms
of the mean shear rate (G):
21
fG ⎟⎟⎠
⎞⎜⎜⎝
⎛μρ∈
= 5-7
and the dissipation rate (∈) is given by Equation 4-4, which rearranges to:
DfV2 3
∈= 5-8
and, for turbulent pipe flow in smooth pipes, the friction factor (f) is given by the Blasius equation:
41
Re
0791.0f = 5-9
where the pipe Reynolds number is:
μρ
= fDVRe 5-10
The slurry density (ρf) is readily calculated from experimental data given the solid
fraction (φ) and the solid density (ρs):
( )φ−ρ+φρ=ρ 1wsf 5-11 leaving the viscosity (μ) the only unknown in Equation 5-7. The increase in fluid
viscosity with solid fraction was described by Einstein (1908), using the first terms of a
series expansion in the form:
132
( )...K1 Eos +φ+μ=μ 5-12 where: μs = Viscosity of suspension (N s m-2) μo = Fluid viscosity (no solid) (N s m-2) KE = Constant = 2.5 φ = Solid fraction [0,1]
Einstein originally proposed that KE = 1, but later, famously, revised it to KE = 2.5.
Equation 5-12 is only correct for very dilute solutions, and various alternative equations
(Table 2-5) have been proposed for higher solid fractions by adding additional terms.
For suspensions of high solid fraction the viscosity is typically given as a function of
the maximum solid fraction in the form (Govier & Aziz 1972, Liu & Masliyah 1996):
k
mos 1
−
⎟⎟⎠
⎞⎜⎜⎝
⎛φφ
−μ=μ 5-13
where: φm = Maximum solid fraction. k ≈ 2
μs = Suspension viscosity (N s m-2) μo = Viscosity of water (1.02 × 10-3 N s m-2 at 20 oC)
The maximum solid volume fraction is typically taken as 0.6 - 0.7 (Fleer & Scheutjens
1993, Strenge 1993, Schramm 1996, Bustos et al. 1999). At this point the particles are
taken to be in close contact and form a continuous network that resists shear, causing
the viscosity to rise exponentially.
The effective solid fraction will be higher than the actual solid fraction because the
aggregates are porous, incorporating fluid into the structure and increasing the enclosed
volume, and hence the suspension viscosity (Mills et al. 1991, Potanin & Uriev 1991).
The effective volume fraction (φeff) is described by fractal geometry (Mills et al. 1991,
Jiang & Logan 1991, Oles 1992, Kusters et al. 1997, Flesch et al. 1999) (see
Appendix):
⎟⎟⎠
⎞⎜⎜⎝
⎛
ρρ
φ=φagg
sseff 5-14
where: φs = Actual suspension solid volume fraction [0,1]
133
ρs = Density of the solid (calcite = 2710 kg m-3) ρagg = Effective aggregate density (kg m-3), given by:
3D
p
aggsagg
f
dd
−
⎟⎟⎠
⎞⎜⎜⎝
⎛ρ=ρ 5-15
where: dagg = Diameter of aggregate (m) dp = Primary particle diameter (m) Df = Mass-diameter fractal dimension (see Appendix) i.e. substituting and using mean sizes gives:
kD3
p
agg
m
effos
f
d
d1μμ
−−
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
φφ
−= 5-16
In this case the maximum packing fraction (φm = 0.65) and exponent (k = 2) are
effectively model parameters estimated from the literature. During model construction
φm was initially treated as a fitted parameter [0.6,0.7], however the model was
surprisingly insensitive to φm over that range, and was subsequently fixed at an
intermediate value of 0.65, removing a degree of freedom. The fractal dimension (2.4)
was estimated from the hindered settling data (see Section 6).
5.1.2 Aggregate breakage
Aggregate breakage is included by the final two terms in the population balance
(Equation 5-1). In this case breakage is taken to be binary, producing two equal sized
daughter fragments, a common modelling approach (Randolph & Larson 1988, Chen
et al. 1990, Chatzi & Kiparissides 1995, Spicer & Pratsinis 1996, Flesch et al. 1999)
although more fragments covering a range of sizes (Valentas & Amundson 1966,
Valentas et al. 1966, Coulaloglou & Tavlarides 1977, Lu and Spielman 1985, Cheng
and Redner 1988, Lick and Lick 1988, Burban et al. 1989, Spicer & Pratsinis 1996a,
Ducoste & Clark 1998, Serra & Casamitjana 1998b) have also been used. In the case of
the population balance used here, the method of discretisation is relatively coarse
(aggregate mass doubles every channel, Equation 5-22), in an effort to reduce the
number of size fractions and equations. Because of this, most of the particles in a
distributed breakage function (Spicer & Pratsinis 1996a) will naturally appear in the
next smallest size range, unless the distribution is very broad or the number of
134
fragments is assumed to be ≫2. In the case of a broad distribution, problems may arise
unless the distribution is truncated so that the daughter fragments must be smaller than
the parent aggregate.
Although there is general agreement that Saffman and Turner’s aggregation kernel is
appropriate (provided that the particle is spherical, neutrally buoyant, and is smaller
than the Kolmogorov micro-scale; Tables 2-1 & 2-2), there is less agreement on the
form of the breakage kernel. A typical modelling approach is to use Saffman and
Turner’s aggregation kernel in conjunction with a capture efficiency term, and a
breakage kernel that gives an acceptable fit with the experimental data.
The breakage kernel is usually taken to be a function of the aggregate size and the mean
energy dissipation or shear rate. Most of the theoretical work on aggregate breakage
(Parker et al. 1972, Shamlou & Titchener-Hooker 1993) has assumed that the
aggregates will grow up to some limiting size, when the aggregate will break. Although
this approach is attractive in its mathematical simplicity, aggregate distributions
typically cover a range of sizes (often described by some distribution like log-normal)
and not a distinct upper size limit.
For population balance models the breakage rate is usually (Glasgow & Lueche 1980,
Leu & Ghosh 1988, Spicer & Pratsinis 1996a, Serra & Casamitjana 1998b) taken to be
some continuous function of the aggregate size, reflecting the distribution of aggregate
strengths, orientations and the instantaneous fluid fluctuations in turbulent flow. Most
workers (Table 2-6) have found that the breakage rate is directly proportional to the
aggregate size, although higher exponents are also suggested (Glasgow & Lueche 1980,
Shamlou & Titchener-Hooker 1993). Pandya and Spielman (1982) found
experimentally that aggregate breakage was directly proportional to the aggregate size.
The breakage rate is usually also taken to be a function of the energy dissipation or
shear rate (Table 2-6). A higher shear rate typically provides rapid initial mixing and
fast initial aggregation (Figure 5-1), but eventually results in a smaller aggregate size
due to increased breakage. This is a typical experimental result (Curtis & Hocking
1970, Keys & Hogg 1978, Oles 1992, Chin et al. 1998) and is usually incorporated into
the population balance by assuming:
135
x
i GS ∝ 5-17 where x is a higher exponent than in the aggregation kernel. That is, an increased shear
rate increases both the aggregation and breakage rates, but the breakage term is
dominant. Aggregation is usually taken as being directly proportional to G (e.g.
Equation 5-2) and x is typically reported to be 1.5-2 (Spicer & Pratsinis 1996a, Serra &
Casamitjana 1998b, Chung et al. 1998).
Equation 5-17 suggests (note Equation 5-7) that the breakage rate is decreased as a
function of viscosity, although some workers have suggested the more intuitive
response that the breakage will increase with the viscosity (Glasgow & Lueche 1980,
Kramer & Clark 1999). In most cases the form of Equation 5-17 has been successful,
because most studies have focused on coagulation in water treatment plants or river
estuaries which are characterised by a low solid fraction where the viscosity is
essentially unchanged from pure water (Williams & Simons 1992, Stumm & Morgan
1996, Spicer and Pratsinis 1996a, Serra & Casamitjana 1998b, Flesch et al. 1999,
Manning & Dyer 1999).
However, feed streams to thickener units used in mineral processing have a
considerably higher solid volume fraction, typically in the percent range (Pearse 1977,
Dahlstrom & Fitch 1985, Perry & Green 1997). In this case the breakage rate was
initially considered to be ∝ ∈yμz where μ, the viscosity, is given by Equation 5-10.
During model construction, parameter estimation repeatedly gave y = 0.7 ± 0.1, and
z = 1 ± 0.05. The value of ~ 0.7 is equivalent to x ~ 1.4 in Equation 5-17 (note
Equation 5-7) similar to other studies, and z was set to unity, removing a degree of
freedom. The final estimate of y was 0.677, perhaps suggesting Si ∝ ∈2/3.
Flocculant dosage dependency was also introduced into the breakage kernel to account
for the formation of larger aggregates at higher dosages (Figure 5-11). The dosage was
expressed as a surface coverage (θf) on a mass/area basis calculated from the
experimental addition rate and measured particle surface area (see Section 4):
136
( )Θ−=θ 1MAm
s
ff 5-18
and the breakage kernel as:
paggf
i,aggk
2i dd
dkS
3
>θ
μ∈= 5-19A
pagg dd 0 ≤= 5-19B
where: θf = Effective flocculant surface coverage (kg m-2) k2 = Fitted parameter # 2 := 38.1 k3 = Fitted parameter # 3 := 0.677 M = Mixing index given by Equation 5-6 mf = Mass of flocculant (kg) As = Surface area of solid (m2) Θ = Flocculant degradation [0,1](Equation 5-20)
Studies of coagulation typically show an initial increase in the aggregate size after
coagulant addition, followed by the attainment of a stable steady-state size (Oles 1992,
Spicer & Pratsinis 1996a, Serra & Casamitjana 1998b, Flesch et al. 1999). However,
polymer flocculants do not give a steady-state size except if flocculation occurs
primarily through surface charge neutralisation. Aggregation by polymer bridging
typically results in a decreased aggregate size on extended shearing due to flocculant
degradation through chain scission or re-arrangement (Keys & Hogg 1978, Sikora &
Stratton 1981, Leu & Ghosh 1988, Williams et al. 1992, Bagster 1993, Hogg 2000).
Polymer flocculant degradation could be incorporated into the model with a decreasing
capture efficiency (Heath et al. 1999), or by an increased breakage rate via a weaker
aggregate. The latter approach was taken here, keeping the terms together in the
breakage kernel:
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛φφ
Θ−μφ∈=Θ 3
1
3
s
effs
k4 1Mk
dtd 5-20
with initial condition: Θ = 0 at t = 0. and: k4 = Fitted parameter # 4 := 1224 The rate of polymer degradation is taken to be a function of the breakage rate, although
additional solid fraction dependence was required to achieve an acceptable fit with the
data. The dimensionless term (φeff/φs)1/3 is introduced to prevent the polymer becoming
137
completely degraded. The term can be replaced by ≈ 2 (allowing 50 % degradation)
giving only a slight increase in the residual. However, the term used has the additional
advantage that it collapses to 1 (complete degradation) under unfavourable conditions
(e.g. extreme shear), allowing the aggregate size distribution to eventually return to the
primary particle size.
The physical interpretation of this effect is not clear, and could be taken as evidence of
flocculation by surface charge neutralisation. However in this case an anionic polymer
was used, and the calcite surface is also expected to be negatively charged (Geffroy
et al. 2000). Alternatively it could be taken as evidence that the flocculant binding
together the core of aggregates is not significantly degraded, because the aggregates are
not broken down to that extent. The polymer degradation would then be expected to be
a function of the aggregate surface area, volume, packing efficiency etc. The term used
is entirely empirical, although it is in a similar form to that used by Strenge (1993) to
describe the mean distance between particles in the aggregate.
5.1.3 Simplification for coding/simulation
Equations 5-6 and 5-16 effectively switch off the population balance when no flocculant
has been added. This is necessary to ensure that the primary particles do not aggregate
(or break), and the mean size is maintained when no flocculant is present. However,
although gPROMS (commercial simulation package described later) will accept step
functions (e.g. Equation 5-19) in the ordinary simulation mode, it will not perform
parameter estimations on codes with case functions. Hence Equation 5-19 is replaced
by the continuous approximation:
( )⎟⎠⎞
⎜⎝⎛ −
θ
μ∈= −
5pagg
3d/d
f
i,aggk
2i e1
dKS 5-21
where: dp = Volume weighted mean primary particle diameter (m)
In this case dp was fixed at 30 × 10-6 m, which is larger than any of the experimental
primary mean sizes, and below any of the mean aggregate sizes in the breakage
dominant region.
138
If equations describing aggregate porosity (e.g. Equations 5-11, 5-12 & 5-20) are
inserted directly into the population balance (Equation 5-1), the model’s complexity
increases significantly and the convergence time becomes far longer. In order to
simplify the population balance, the fractal geometry describing aggregate porosity and
size can be calculated externally, bringing them before the integral.
The Hounslow/Spicer population balance conserves mass based on the assumption:
2V
Vi
1i
=+
5-22
where: Vi = Volume of the particles in the ith size range (m3) and with the initial size range set to some suitable value below the smallest primary
particle.
Equation 5-22 is only suitable for non-porous particles, e.g. for droplet coalescence in
solvent extraction or clouds. However, Equation 5-22 does provide a basis for mass
conservation via the assumption that Vi can be substituted by Vm,i, the mass equivalent
volume giving (see Appendix):
fD
3
p
i,mpi,agg d
ddd ⎟
⎟⎠
⎞⎜⎜⎝
⎛= 5-23
where: dagg,i = Diameter of the ith sized aggregate (m) dp = Diameter of the primary particle (m) dm,i = Mass effective diameter of the ith sized aggregate (m) In this case 35 size channels were used, covering a size range 0.2-3500 μm.
The population balance model described above contains 4 fitted parameters (Equations
5-6, 5-19 & 5-20) that are estimated from the experimental size data using a
conventional sum-of-squares minimisation:
( )2t,,d,,G
Exp,aggPB,aggk,k,k,k
∑p4321
d-dminθφ
=ψ 5-24
where: kn = Fitted parameters G = Spatially averaged shear rate (s-1) φ = Solid fraction (m3m-3)
θ = Flocculant surface coverage (kg m-2) dp = Primary particle size (m)
139
dagg,PB = Modelled (PB = population balance) mean aggregate size (m) dagg,Exp = Experimental (FBRM) mean aggregate size (m) t = Time (s)
Simulations and parameter estimation were performed using gPROMS, a commercial
UNIX based dynamic simulation package, running on a SUN Enterprise 3000
mainframe. Individual simulations took in the order of 10 seconds to converge, while
parameter estimation using the 17 available data sets took several hours depending on
the level of accuracy required.
The fitted parameters are used to fit the model to the experimental data, with the
parameters adjusting various aspects of the model behaviour. The first parameter (k1)
alters the initial flocculant/suspension mixing rate in the pipe, changing the capture
efficiency and hence the initial rate of aggregation. The fitted value of 0.343 compares
favourably with the literature value of 0.5 suggested by Etchells and Short (1988).
However in its current form the mixing equation (Equation 5-6) only describes mixing
in turbulent pipe flow, and requires modification for other systems. Ultimately,
flocculant/suspension mixing and adsorption in thickener feedwells is likely to be
described by CFD, perhaps with validation by electrical impedance tomography.
The second model parameter (k2) effectively scales the breakage rate, and is likely to be
a complex function of the various factors affecting the aggregate strength. For example:
the flocculant/particle surface bonding chemistry, the flocculant molecular weight, the
presence of coagulating salts (if any), or the primary particle packing structure within
the aggregate.
The third fitted parameter (k3) essentially alters the effect of the energy dissipation rate
on the final aggregate size, producing a smaller aggregate at higher shear rates as
described above. The final parameter (k4) sets the rate of polymer degradation at
extended residence times. Due to the complex interrelated nature of the population
balance equations the parameters are somewhat interactive, with any deficiencies in the
model effectively accommodated elsewhere in the model. However, the model was
found to be robust and stable, converging to the same parameter values from a range of
initial estimates.
140
5.2 Comparison with experimental data 5.2.1 Effect of flow regime (shear rate)
Figure 5-1 compares the modelled and experimental mean aggregate size data for
calcite under various flow conditions (see Section 4). A combination of pipe sizes (25.4
or 38.1 mm ID) and flow rates gave different fluid shear rates. As has been discussed
previously (Section 4), a higher mean fluid shear results in a rapid initial aggregation
rate, but ultimately results in a smaller mean aggregate size due to increased breakage.
The population balance model (lines) follows the same trends due to the interplay
between various factors. At a higher shear rate the flocculant mixing (Equation 5-6,
Figure 5-2) and particle collision (Equation 2-12) rates are higher, resulting in a higher
initial aggregation rate. However, a higher energy dissipation rate increases the
breakage (Equation 5-19) and polymer degradation (Equation 5-20) rates, resulting in a
smaller average aggregate size.
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
0 10 20 30 40 50 60 70 80
Time (s)
Mea
n ag
greg
ate
diam
eter
(m)
Model, 38.1 mm ID, 0.343 m sֿ¹Experimental, 38.1 mm ID, 0.343 m sֿ¹, G = 52.6 sֿ¹, (settling obs. in pipe)Model, 38.1 mm ID, 0.554 m sֿ¹Experimental, 38.1 mm ID, 0.554 m sֿ¹, G = 175.2 sֿ¹Model, 25.4 mm ID, 0.461 m sֿ¹ (Baseline)Experimental, 25.4 mm ID, 0.461 m sֿ¹, G = 174.1 sֿ¹ (Baseline)Model, 25.4 mm ID, 0.781 m sֿ¹Experimental, 25.4 mm ID, 0.781 m sֿ¹, G = 488.1 sֿ¹Model, 25.4 mm ID, 1.294 m sֿ¹Experimental, 25.4 mm ID, 1.294 m sֿ¹, G = 935.5 sֿ¹
Figure 5-1: Modelled (lines) and experimental (dots) change in mean aggregate size under different pipe flow conditions. At the lowest shear rate (open circles) solid settling was observed in the pipe, and this data should be interpreted with caution. Other conditions as per baseline.
Figure 5-2 shows the modelled flocculant/suspension mixing index (M, Equation 5-6)
as a function of time and the flow regime, allowing an estimation of the capture
efficiency (α), which is also taken to approach 1 when the flocculant becomes well
mixed. The actual capture efficiency may be lower than 1, although the real value is
unknown and any discrepancy is effectively compensated by the breakage kernel.
141
0
0.2
0.4
0.6
0.8
1
1.2
0 10 20 30 40 50 60 70
Time (s)
Mix
ing
Inde
x ( α
=M)
Model, 25.4 mm ID, 1.294 m sֿ¹Model, 25.4 mm ID, 0.781 m sֿ¹Model, 25.4 mm ID, 0.461 m sֿ¹ (Baseline)Model, 38.1 mm ID, 0.554 m sֿ¹Model, 38.1 mm ID, 0.343 m sֿ¹
Figure 5-2: Variation in modelled mixing index with time under different flow conditions.
Figures 5-1 and 5-2 suggest that the initial aggregation rate is limited by
flocculant/suspension mixing, rather than the particle collision rate (Equation 2-12).
This is supported by experimental evidence where the mixing condition was changed,
either by flocculant dilution or by changing the flocculant injection velocity by
changing the injection nipple size. Figure 5-3 shows the increase in the aggregate size
with flocculant dilution with a fixed length of short pipe (compare Figure 5-1, 38.1 mm
ID, 0.55 m s-1). Using a short length of pipe and gentle shear rate allowed the particle
size measurement to be taken where the aggregate size was still increasing rapidly and
was likely to be influenced by the mixing condition. The overall dosage was
maintained at 20 g t-1, but a dilute flocculant stream produced better mixing due to a
higher flocculant stream flow, and lower viscosity.
Increasing the flocculant injection velocity by changing the nipple diameter (not shown)
gave a similar result, with a smaller nipple increasing the flocculant stream momentum,
the turbulence at the injection point, producing an increased aggregate size. In all cases
the flocculant stream was minor (0.25-2 %) compared to the main slurry flow in the
pipe reactor.
142
0.0E+00
2.0E-05
4.0E-05
6.0E-05
8.0E-05
1.0E-04
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09Flocculant concentration (% w/v)
Figure 5-3: Effect of flocculant stream dilution on the measured aggregate size for a fixed (short) length of pipe, giving a mean residence time of 3 s, see Figure 5-1. 38.1 mm ID pipe, 0.34 m s-1 mean flow velocity.
Figure 5-4 shows the variation in the modelled fluid viscosity under the conditions
described by Figure 5-1. The fluid viscosity is taken to be a function of the effective
solid fraction (Equation 5-16), including aggregate porosity using fractal geometry.
Since aggregates become increasingly porous as they increase in size, the total enclosed
volume of the aggregates also increases, leading to a higher effective solid fraction
(Equation 5-14) and hence fluid viscosity (Williams et al. 1992, Barnes & Holbrook
1993). This effectively prevents gelation, where the aggregates might grow to the point
where they fill the container, because as the effective solid fraction approaches the
maximum (φm), the viscosity tends towards infinity (Equation 5-13). This leads to an
increased pressure drop (as measured by manometer), and an increased energy
dissipation rate (the flow velocity is fixed via a positive displacement pump), and hence
an increased breakage rate (Equation 5-19).
Since the fluid viscosity is dependent on the aggregate size, various other fluid
parameters also change and are calculated within the population balance. Figures 5-5,
5-7 & 5-8 show changes to the mean shear rate, pipe Reynolds' number and the friction
factor respectively. The use of manometers allows experimental estimates of the same
fluid properties, with the predicted and experimental average shear rates compared in
Figure 5-6. These fluid parameters are described using standard pipe flow equations
(Equations 5-7 – 5-10) incorporating the effect of solid fraction on the suspension
143
viscosity (Equations 5-13 – 5-16), achieving an acceptable fit to the data (Figure 5-6)
without the need for any additional fitted parameters.
0
0.001
0.002
0.003
0.004
0 10 20 30 40 50 60 70
Time (s)
Vis
cosi
ty (N
s m
-2)
Model, 38.1 mm ID, 0.343 m sֿ¹
Model, 38.1 mm ID, 0.554 m sֿ¹
Model, 25.4 mm ID, 0.461 m sֿ¹ (Baseline)
Model, 25.4 mm ID, 0.781 m sֿ¹
Model, 25.4 mm ID, 1.294 m sֿ¹
Figure 5-4: Variation in the modelled fluid viscosity under different flow regimes. Other conditions as per baseline.
0
200
400
600
800
1000
1200
0 10 20 30 40 50 60 70
Time (s)
Mea
n tu
rbul
ent s
hear
rat
e (s
-1) Model, 25.4 mm ID, 1.294 m sֿ¹
Model, 25.4 mm ID, 0.781 m sֿ¹
Model, 25.4 mm ID, 0.461 m sֿ¹ (Baseline)
Model, 38.1 mm ID, 0.554 m sֿ¹
Model, 38.1 mm ID, 0.343 m sֿ¹
Figure 5-5: Variation in modelled spatially averaged shear rate under various flow regimes.
144
10
100
1000
10 100 1000
Measured mean shear rate (s-1)
Mod
elle
d m
ean
shea
r ra
te (s
-1)
dp = 2.36 µm
dp = 3.47 µm
dp = 15.1 µm
Dosage = 5 g tֿ¹
Dosage = 10 g tֿ¹
38.1 mm ID, 0.554 m sֿ¹
Baseline
25.4 mm ID, 0.781 m sֿ¹
25.4 mm ID, 1.294 m sֿ¹
3.33 % w/v
6.67 % w/v
13.33 % w/v
16.67 % w/v
10 g tֿ¹, 38.1 mm ID, 0.343 m sֿ¹
10 µm, 7 % w/v
40 g tֿ¹, 25.4 mm ID, 0.781 m sֿ¹
Figure 5-6: Comparison between modelled (Equation 5-7 – 5-16) and experimental (Equation 4-1 – 4-7) spatially averaged shear rates.
0
5000
10000
15000
20000
25000
30000
35000
0 10 20 30 40 50 60 70
Time (s)
Pipe
Rey
nold
s Num
ber
(Re) Model, 25.4 mm ID, 1.294 m sֿ¹
Model, 38.1 mm ID, 0.554 m sֿ¹
Model, 38.1 mm ID, 0.554 m sֿ¹
Model, 25.4 mm ID, 0.461 m sֿ¹ (Baseline)
Model, 38.1 mm ID, 0.343 m sֿ¹
Figure 5-7: Variation in the modelled pipe Reynolds’ number under different flow regimes.
145
0
0.002
0.004
0.006
0.008
0.01
0.012
0 10 20 30 40 50 60 70
Time (s)
Pipe
fric
tion
fact
or (f
)
Model, 38.1 mm ID, 0.343 m sֿ¹
Model, 25.4 mm ID, 0.461 m sֿ¹ (Baseline)
Model, 38.1 mm ID, 0.554 m sֿ¹
Model, 25.4 mm ID, 0.781 m sֿ¹
Model, 25.4 mm ID, 1.294 m sֿ¹
Figure 5-8: Variation in the modelled pipe Fanning friction factor under different flow regimes.
Figure 5-9 shows the population balance correctly conserved mass through time
according to:
∑ ρ=φ
i s
iit
mN 5-25
Where: ρs = Density of the solid (2710 kg m-3) mi = Mass of ith aggregate (kg) Ni = Number of ith aggregates (m-3)
9.99
10
10.01
0 10 20 30 40 50 60 70
Time (s)
Solid
vol
ume
frac
tion
(% w
/v)
Model, 38.1 mm ID, 0.343 m sֿ¹
Model, 38.1 mm ID, 0.554 m sֿ¹
Model, 25.4 mm ID, 0.461 m sֿ¹ (Baseline)
Model, 25.4 mm ID, 0.781 m sֿ¹
Model, 25.4 mm ID, 1.294 m sֿ¹
Figure 5-9: Variation in the modelled absolute mass fraction under different flow regimes. Conservation of mass.
146
Aggregation with polymer flocculant typically results in partially irreversible breakage,
manifesting as a gentle reduction in the aggregate size after an initial peak as the
polymer becomes degraded (Bagster 1993). Figure 5-10 shows the extent of flocculant
degradation predicted by the model under various flow conditions. Due to the
incorporation of the (φeff/φs)1/3 term in Equation 5-20, the degradation tends towards a
value of less than 1 (where 1 would represent complete degradation). This improves the
fit with the experimental data, and is rationalised on the basis that the repeated
aggregation/breakage process will not affect flocculant binding the core of aggregates,
because the aggregates are not broken down to that extent. It could also be interpreted
as evidence supporting the electrostatic patch model of flocculation (Kasper 1971,
Gregory 1973), since polymer chain scission is unlikely to affect the flocculant’s
charge. However, if the flow conditions are very aggressive, aggregation will only
proceed to a limited extent and φeff → φs, therefore the flocculant could be completely
degraded. Such a situation is clearly undesirable industrially, but may occur around
feed entry points or baffles with high local shear rates.
0
0.1
0.2
0.3
0.4
0.5
0.6
0 10 20 30 40 50 60 70
Time (s)
Poly
mer
deg
rada
tion
inde
x Θ
Model, 25.4 mm ID, 1.294 m sֿ¹
Model, 25.4 mm ID, 0.781 m sֿ¹
Model, 25.4 mm ID, 0.461 m sֿ¹ (Baseline)
Model, 38.1 mm ID, 0.554 m sֿ¹
Model, 38.1 mm ID, 0.343 m sֿ¹
Figure 5-10: Variation in the modelled polymer degradation index under different flow regimes.
147
5.2.2 Effect of flocculant dosage
Figure 5-11 shows the predicted and experimental mean aggregate sizes as a function of
time and flocculant dosage. The dosage does not affect the aggregation term in the
population balance (except for a slight change in the initial mixing due to a slight
change in the flocculant stream momentum) and the capture efficiency is taken to
approach 1 as the flocculant/suspension mixing becomes complete, regardless of the
actual flocculant dosage. However, flocculant dosage is incorporated into the breakage
kernel of the population balance (Equation 5-19), where a higher dosage leads to a
stronger aggregate, a decreased breakage rate, and hence a larger aggregates. The
flocculant dosage is known experimentally on a mass basis, but is incorporated into the
population balance as a surface coverage, i.e. kg(floc)/m2(solid) (Equation 5-18). The
surface area of the primary particles was measured by laser diffraction (Malvern
Mastersizer) and BET, and hence the effective surface coverage also depends on the
primary particle size (Figure 5-12).
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
0 5 10 15 20 25 30 35 40
Time (s)
Mea
n ag
greg
ate
diam
eter
(m)
Model, dosage = 20 g tֿ¹, (Baseline)Experimental, dosage = 20 g tֿ¹, (Baseline)Model, dosage = 10 g tֿ¹Experimental, dosage = 10 g tֿ¹Model, dosage = 5 g tֿ¹Experimental, dosage = 5 g tֿ¹
Figure 5-11: Modelled (lines) and experimental (dots) change in mean aggregate size with change in flocculant dosage. Other conditions as per baseline.
Higher dosages were also tried (see Section 4), but resulted in stratification/settling in
the pipe reactor, leading to uncertainty in the solid fraction and residence time. Both
the modelled and experimental results gave larger aggregates at higher dosages,
although the data match was poor (not shown). At extremely high flocculant dosages
there is data to suggest (Healy 1961, Fleer & Scheutjens 1993, Svarovsky 2000) that the
aggregate size can be reduced by steric re-stabilisation as the particle surface becomes
148
completely covered, although it is unlikely that such a high dosage would be reached in
industrial applications (Hogg 2000).
5.2.3 Effect of primary particle size
The primary particle size distribution is also likely to be a process variable, depending
on the feed (milling, crystallisation, ore etc). This effect was incorporated into the
population balance model by assuming that the aggregate breakage rate is a function
(Equation 5-18) of the effective flocculant surface coverage. That is, a larger primary
particle, with a smaller effective surface area (per unit mass) has a higher coverage, a
higher strength, and hence results in larger aggregates (Figure 5-12).
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
2.5E-04
0 5 10 15 20 25 30 35 40
Time (s)
Mea
n ag
greg
ate
diam
eter
(m)
Model, dp = 15.1 µmExperimental, dp = 15.1 µmModel, dp = 5.59 µm, (Baseline)Experimental, dp = 5.59 µm, (Baseline)Model, dp = 3.47 µmExperimental, dp = 3.47 µmModel, dp = 2.36 µmExperimental, dp = 2.36 µm
Figure 5-12: Modelled (lines) and experimental (dots) change in mean aggregate size using differently sized primary particles. Other conditions as per baseline.
Figure 5-12 shows that using the effective flocculant dosage gives the correct model
response with respect to primary particle size, although the modelled size
underestimates the experimental data for small primary particles. An alternative
particle surface area measurement (BET) was also tried, giving similar results, but no
improvement in the fit. Although the discrepancy is clearly noticeable to the eye, the
residual error is comparatively minor and did not justify the addition of a further fitted
parameter.
Alternatively, the aggregate strength could be considered to be a function of the packing
efficiency within the aggregate (Gregory 1989, Mühle 1993), which would change with
149
primary particle size according to Equation 5-23. Attempts were made along these lines
during model construction, but ultimately did not fit the experimental data as well as
relating the aggregate strength to the flocculant surface coverage. The latter also gave a
simpler model.
The incorporation of fractal geometry into the population balance to account for
aggregate porosity complicates the modelling of different primary particle sizes. Fractal
geometry describes the increase in aggregate porosity with aggregate size. However,
since the primary particles are distributed (known via Malvern Mastersizer) and the
population balance does not distinguish between aggregates and primary particles, the
smaller primary particles will be undersized by the population balance. Conversely, the
size of large primary particles will be overestimated, as they cannot be distinguished (by
the population balance) from small aggregates of the same mass.
This issue was corrected by changing the number of primary particles in each channel
of the population balance so that the correct number of collision and breakage events
occurred. The distribution was still based on the Malvern volume averaged data, and
the total number of primary particles normalised to give the correct solid fraction
(10 % w/v for most runs). The particle porosity was taken to be zero at the mean
primary particle size (taken as 6.59 × 10-6 m), however in situations when the mean
primary particle size was different (i.e. Figure 5-12), the normalisation step resulted in
the total solid fraction being different from the actual 10 % w/v. Altering the calculated
total mass fraction to give the correct aggregation/breakage rate was seen as preferable
to the reverse. Since the effective viscosity was based on the effective solid volume
fraction (which is given correctly), rather than the absolute solid fraction, the fluid
parameters (shear rate, Reynolds number, friction factor etc) are also unaffected by the
above correction.
The above issue could also be tackled more rigorously using a two-dimensional
population balance, where the distribution of primary particles within each aggregate
was accounted for explicitly. However, this would considerably complicate the model,
requiring the use of 1225 (352) model equations rather than the 35 used here, almost
certainly making the model too unwieldy for use as a sub-model within a CFD code.
150
5.2.4 Effect of solid fraction
The solid fraction is also a process variable and can be altered in some processes by
overflow or underflow recycle (Perry & Green 1997). Figure 5-13 shows a comparison
between modelled and experimental mean aggregate sizes as a function of time and
solid fraction. The reduction in aggregate size with solid fraction was somewhat
unexpected, given that aggregation (Equation 5-2) is usually taken as second order with
respect to particle number, whereas breakage is taken as first order (Equation 5-1 &
5-19). However, by making the breakage rate a function of viscosity (Figure 5-14) and
the energy dissipation rate (which also increases with the viscosity;
Equations 5-13 - 5-16) the breakage rate was increased at high solid fraction, producing
the desired reduction in the aggregate size (Figure 5-13). In addition, an increased solid
fraction gives a reduced shear rate (Figure 5-15) reducing the aggregation rate. A
reduction in aggregate size with solid fraction has been observed elsewhere (Lick &
Lick 1988, Williams et al. 1992).
At very low solid fraction (< 3 % w/v), the viscosity approaches that of pure water, and
the model predicts a reduction in aggregate size as the solid fraction is further decreased
due to the difference in the order of the aggregation and breakage kernels. The increase
in aggregate size with increased solid fraction has been observed previously in studies
at very low solid fraction (Thomas 1964, De Boer et al. 1989, Kobayashi et al. 1999).
Lower solid fractions were not investigated here due to the difficulty of measuring the
hindered settling velocity (see Section 6) at low solid fraction where the mudline is no
longer distinct, and the settling rate is rapid.
151
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
0 5 10 15 20 25 30 35 40
Time (s)
Mea
n ag
greg
ate
diam
eter
(m)
Model, 3.33 % w/v Experimental, 3.33 % w/vModel, 6.67 % w/v Experimental, 6.67 % w/vModel, 10.0 % w/v, (Baseline) Experimental, 10.0 % w/v, (Baseline)Model, 13.33 % w/v Experimental, 13.33 % w/vModel, 16.67 % w/v Experimental, 16.67 % w/v
Figure 5-13: Modelled (lines) and experimental (dots) change in mean aggregate size using different solid fractions. Other conditions as per baseline.
0
0.001
0.002
0.003
0.004
0.005
0 5 10 15 20 25 30 35 40
Time (s)
Vis
cosi
ty (N
s m
-2)
Model, 16.67 % w/vModel, 13.33 % w/vModel, 10.0 % w/v, (Baseline)Model, 6.67 % w/vModel, 3.33 % w/v
Figure 5-14: Variation in modelled fluid viscosity through time with various solid fractions.
152
0
50
100
150
200
250
300
0 5 10 15 20 25 30 35 40
Time (s)
Mea
n tu
rbul
ent s
hear
rat
e (s
-1)
Model, 3.33 % w/vModel, 6.67 % w/vModel, 10.0 % w/v, (Baseline)Model, 13.33 % w/vModel, 16.67 % w/v
Figure 5-15: Variation in modelled spatially averaged shear rate through time with various solid fractions.
Figure 5-16 again shows the population balance conserved mass correctly. Although
conservation of mass does not guarantee that the population balance is working
correctly, it is simple to check and picks up gross coding and numerical errors that
frequently result in the model gaining or losing mass.
0
5
10
15
20
0 5 10 15 20 25 30 35 40
Time (s)
Solid
vol
ume
frac
tion
(% w
/v) Model, 16.67 % w/v
Model, 13.33 % w/vModel, 10.0 % w/v, (Baseline)Model, 6.67 % w/vModel, 3.33 % w/v
Figure 5-16: Variation in the modelled absolute mass fraction with different solid fractions. Mass is conserved.
153
5.2.5 Additional runs to check model predictivity
Figures 5-1, 5-11, 5-12 and 5-13 show results from a sparse matrix of experimental pipe
reactor runs (see Section 4, Figure 4-4) performed using the baseline as the central
point, and varying the process variables (fluid shear, flocculant dosage, solid fraction,
primary particle size) independently away from that the centre. However, additional
experimental runs were performed in the gaps of the matrix, by changing two process
variables simultaneously. These were not included in the parameter estimation for the
population balance, but were used to assess the predictivity of the model (Figure 5-17).
The predictions appear reasonable, except for the latter stages of the 15.1 μm,
6.67 % w/v run. This appears to be due to the effect of the primary particle size, which
shows some deviation from the experimental values (see Figure 5-12).
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
2.5E-04
3.0E-04
0 10 20 30 40 50 60 70Time (s)
Mea
n ag
greg
ate
diam
eter
(m)
Model, dp = 15.1 µm, 6.67 % w/vExperimental, dp = 15.1 µm, 6.67 % w/vModel, dosage = 40 g tֿ¹, 25.4 mm ID, 0.781 m sֿ¹Experimental, dosage = 40 g tֿ¹, 25.4 mm ID, 0.781 m sֿ¹Model, dosage = 10 g tֿ¹, 38.1 mm ID, 0.343 m sֿ¹Experimental, dosage = 10 g tֿ¹, 38.1 mm ID, 0.343 m sֿ¹
Figure 5-17: Modelled (lines) and experimental (dots) change in mean aggregate size with change under various additional conditions. Other conditions as per baseline.
154
5.2.6 Effect of changing fractal dimension
In all the above figures a fractal dimension of 2.4 was assumed, as per the estimation
from the experimental settling data (Section 6). This allowed a dramatic simplification
of the population balance, by reading the effective aggregate sizes into the population
balance as an array. However, an alternative model was also coded, where fractal
geometry was incorporated directly into the population balance’s differential equations.
The convergence time was increased by an order of magnitude, and simply changing the
array values to suit the fractal dimension would probably have been a simpler modelling
path in retrospect.
Figure 5-18 shows the effect of changing the fractal dimension on the predicted mean
aggregate size. A higher fractal dimension (lower porosity) results in an initially
reduced aggregation rate due to a reduced aggregate capture radius. However, the
lower porosity gives a lower effective solid fraction, reducing the viscosity and the
breakage rate, hence the increased aggregate size later in the aggregation process.
Unfortunately, there are no experimental data available to confirm the trend in Figure 5-
18, and it is not clear how the fractal dimension could be changed in isolation.
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
0 5 10 15 20 25 30 35 40
Time (s)
Mea
n A
ggre
gate
size
(m)
Df = 2.6Df = 2.5Df = 2.4Df = 2.3Df = 2.2
Figure 5-18: Modelled variation in the mean aggregate size with fractal dimension (i.e. aggregate porosity). Other conditions are as per baseline.
155
5.3 Conclusions A population balance model has been developed to describe the kinetics of
aggregation/breakage of calcite particles flocculated with high molecular weight
flocculant in turbulent flow. In addition to the effect of fluid shear normally
incorporated into population balance models of aggregation, the model accounts for the
important full-scale process variables, namely: flocculant dosage, primary particle size
and solid fraction, representing a significant advance on the population balance models
described previously in the literature.
The population balance is based on the successful population balance proposed by
Hounslow et al. (1988), later modified to incorporate aggregate breakage by Spicer and
Pratsinis (1996a). Saffman and Turner’s (1956) collision kernel is used in conjunction
with a capture efficiency (α) term based on the degree of flocculant/suspension mixing.
This accounts for the stability of the suspension before flocculant addition, and prevents
the overestimation of the aggregation rate during the initial mixing period.
The breakage kernel is a function of the mean energy dissipation rate, suspension
viscosity and aggregate diameter. Flocculant dosage dependence is also incorporated
into the breakage term, with a higher dosage increasing the aggregate strength, leading
to a reduced breakage rate and ultimately a larger aggregate size. The partial non-
reversibility of polymer aggregate breakage is included with a decay term that accounts
for the flocculant degradation due to repeated aggregation/breakage.
The effects of aggregate porosity are incorporated by the use of fractal geometry,
increasing both the effective capture radii and the effective solid fraction. The increase
in the suspension viscosity with solid volume fraction is also incorporated, and results
in changes in the energy dissipation rate, fluid shear, pipe friction factor and Reynolds
number. The inclusion of these additional terms allow the population balance to
describe aggregation and breakage kinetics in suspensions of high solid fraction as
found in mineral processing thickeners, as opposed to the low solid fractions considered
by most workers studying estuarine and wastewater systems.
156
6. Relating hindered settling rate with aggregate size
Previous sections have covered the effect of various process variables on the resulting
aggregate size. However, the primary objective of aggregation is to increase the settling
rate and hence efficiency of solid-liquid separation, with the actual aggregate size being
of secondary importance. This section investigates the relationship between the
aggregate size and the resulting hindered settling rate, ultimately allowing the settling
rate to be predicted from the population balance model.
The gravity sedimentation velocity of individual solid spheres settling in creeping flow
(Re → 0) was first described theoretically by Stokes (1851) (Happel & Brenner 1973,
Seville et al. 1997):
( )μ
ρ−ρ=
18gdU ls
2
6-1
where: U = Settling velocity (m s-1) d = Particle diameter (m) g = Gravity (9.8 m s-2) ρ = Density (s = solid, l = liquid) (kg m-3)
μ = Fluid viscosity (N s m-2)
In practice, mineral particles/aggregates are rarely spherical, and large aggregates may
be highly porous. Non-spherical particles generally settle slightly slower (~ 0.7-1.0)
than spheres (Pettyjohn & Christiansen 1948, Heiss & Coull 1952, Happel & Brenner
1973, Kousaka et al. 1981, Seville et al. 1997), although needle shaped particles settle
slightly faster if the major axis remains vertical. Porous aggregates have larger
hydrodynamic profiles than equivalent mass solid particles, considerably reducing their
settling velocity (Gregory 1997). However, highly porous aggregates may allow some
fluid flow through the structure, counteracting this effect somewhat (Veerapaneni &
Wiesner 1996).
Large, dense particles may settle too rapidly to be accurately described by Stokes’ law,
and by a particle Reynolds number above about 0.1 inertial effects become significant
(Rushton et al. 1996, Seville et al. 1997) and the drag coefficient increases faster than
Stokes’ law describes, reducing the settling velocity (Table 2-7). Both empirical and
theoretical relationships have been proposed for higher Reynolds number settling
157
(Moudgil & Vasudevan 1989, Clift et al. 1978, Nguyen-Van et al. 1994, Perry & Green
1997), although in most cases aggregates settle sufficiently slowly to be described by
Stokes’ equation (Gregory 1997)
As the suspension solid fraction is increased there is a gradual, but poorly defined,
transition to hindered-settling. The hindered settling regime is characterised by a
distinct solid/liquid interface (mudline) that settles to leave a clear supernatant above
(Bhatty et al. 1982, Fitch 1975a, 1987, Tong et al. 1998). The settling velocity of the
mudline decreases as a function of the solid fraction, due to the decreasing permeability
of the settling layer and an increase in the upward velocity of the displaced fluid
(Govier & Aziz 1972, Pearse 1977, Perry & Green 1997, Bustos et al. 1999), as
described by Richardson and Zaki (1955) (Tables 2-8 & 2-9):
( )no 1UU φ−= 6-2 where:
U = Settling velocity (m s-1) Uo = Settling velocity at infinite dilution (m s-1) n = Exponent, usually taken as 4.65 (Table 2-9). φ = Solid volume fraction [0,1]
Aggregation increases the hindered-settling velocity by increasing the particle size,
forming channels that increase the sediment permeability. Settling behaviour in the
hindered settling regime is usually described with Kynch (1952) theory, but it does not
account for the effect of flocculation (Williams & Simons 1992, Svarovsky 2000)
because the settling velocity is usually taken as a unique function of the solid fraction.
As the solid loading is increased further, the particles will eventually form a continuous
network and settling will also be restrained by mechanical support from below
(Michaels & Bolger 1962, Pearse 1977, Chandler & Hogg 1987, Healy et al. 1994,
Perry & Green 1997). The mechanical strength of the network is a function of the
solid/packing fraction and the strength of inter-particle bonding. The sediment will
compress when the weight of sediment overburden exceeds the compressive yield stress
(Buscall & White 1987, Healy et al. 1994, Green et al. 1996, Bustos et al. 1999). The
excess overburden weight not supported mechanically is restrained hydrodynamically,
and is the force required to squeeze the fluid back up through the collapsing sediment
158
(Fitch 1975b, Wakeman & Holdich 1984, Concha et al. 1996, Diplas & Papanicolaou
1997, Perez et al. 1998). In practice channel formation may occur spontaneously, or
can be encouraged by rakes with vertical pickets (Pearce 1977, Dahlstrom & Fitch
1985, Williams 1992, Svarovsky 2000).
The aim of this section is to develop a validated mathematical relationship between the
aggregate size and the initial hindered settling rate under a range of conditions likely in
a mineral processing thickener. The influence of fluid shear, flocculant dosage, feed
solid volume fraction, primary particle size and residence time are addressed.
6.1 Experimental Aggregation of calcite (Omya-carb, Commercial Minerals) with polymeric flocculant
(Nalco 9902) was performed in a linear horizontal pipe reactor, as described in
Section 4. Size measurements were taken using the Lasentec FBRM probe described in
Section 3, and hindered settling rates taken in a graduated 0.5 m settling column
(Figures 4-1 & 6-1).
The settling column was constructed from acrylic tube (38.1 mm ID), and the flow was
isolatable (note valves) and removable for extended settling measurements. The valve
arrangement allowed a portion of the flow to be captured in the settling column, without
disrupting the flow through the pipe reactor and past the sizing probe, allowing
continuous operation. The column was marked at regular intervals (1 & 5 cm) and lit
from behind by fluorescent tube, allowing the measurement of the hindered settling rate
by following the fall of the mudline through time using a stopwatch.
The pipe work and fittings were designed to keep the top of the settling column as close
as possible to the particle sizing probe, reducing the additional residence time. This
was the primary reason for mounting the pipe reactor above floor level (note
Figure 6-1). The fittings around the bend and valve were also constructed to produce a
smooth inner surface in an effort to reduce turbulence. Despite this, the time offset was
still considerable (see Section 6.2.1).
159
Figure 6-1: Schematic valve arrangement to capture portion of flow in settling column.
6.2 Results and discussion 6.2.1 Effect of fluid shear
Figure 6-2 shows the development of the mean aggregate size and corresponding
hindered settling velocity as a function of the spatially averaged fluid shear rate and
residence time. As described previously (Sections 4 & 5), a high turbulent shear rate
initially increases the aggregation rate by increased mixing and particle collision.
However, aggregate breakage is dramatically increased at high shear rates, and larger
aggregates are ultimately formed at a lower shear rate. The hindered settling velocity
follows a similar trend, but there is a time offset due to the additional pipe-work and
fittings between the sizing probe and top of the settling column (Figure 6-1).
Linked ball valves
FBRM sizing probe
FBRM sizing probe
Linked ball valves
To waste
Pipe reactor
To waste
Pipe reactor
Settling column Settling
column
160
-1.0E-04
-5.0E-05
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
0 10 20 30 40 50 60 70
Time (s)
Mea
n ag
greg
ate
diam
eter
(m)
0
5
10
15
20
Hin
dere
d se
ttlin
g ve
loci
ty (m
hr-1
)
38.1 mm ID, 0.343 m sֿ¹, (settling obs. in pipe)
38.1 mm ID, 0.554 m sֿ¹
25.4 mm ID, 0.461 m sֿ¹, (Baseline)
25.4 mm ID, 0.781 m sֿ¹
25.4 mm ID, 1.294 m sֿ¹
Figure 6-2: Effect of mean pipe shear rate on the development of the mean aggregate size and hindered settling velocity. Other conditions as per baseline.
The additional time offset was calculated by the velocity-head method (Perry & Green
1997), using the Bernoulli equation for incompressible fluids:
2
VK)zz(g2V
2VP 2
12
211
222 +−+
α−
α=
ρΔ 6-3
where: α = Velocity profile factor g = Gravity (9.8 m s-2) z = Elevation (m), unchanged in this case with horizontal pipe
ΔP = Pressure drop (N m-2) V = Mean fluid velocity (m s-1)
ρ = Fluid density (kg m-3) K = Velocity head, tabulated values for fittings, or for pipes:
DfL4K = 6-4
where: f = Fanning friction factor (= 0.079/Re0.25 for turbulent flow, smooth pipes) L = Pipe length (m), 0.27 m in this case D = Pipe diameter (m), 0.040 m in this case
Figure 6-1 shows that in this case several fittings are present between the particle sizing
probe and the settling column. The valve on the settling tube was assumed to be fully
open with K = 0.17 as suggested by Perry and Green (1997). The protruding sizing
probe was also assumed to have K = 0.17, and the tee given as K = 1.0 (ibid.). In
addition the expansion loss when using the smaller diameter pipe reactor (25.4 mm ID)
was given by the Borda-Carnot equation (ibid.):
161
2
2
12
1
AA1
2VP
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
ρΔ 6-5
where: A = Cross sectional area of the pipe (m2) The contribution from the various fittings and pipe were combined, and converted to an
equivalent time depending on the flow velocity and size of the main pipe. The
additional equivalent time between the FBRM sizing probe and the top of the settling
column varied from 0.44 s (25.4 mm pipe, 39.33 L min.-1) to 7.31 s (38.1 mm pipe,
14.16 L min.-1). In Figure 6-3 below (25.4 mm pipe, 14.01 L min.-1) the offset was
1.86 s. The additional time offset is unfortunate since the data is lost from low
residence times where the aggregate size increases rapidly. However, it does serve to
reinforce the observation that flocculation is rapid in well mixed suspensions of high
solid fraction.
6.2.2 Effect of flocculant dosage
Figure 6-3 shows the increase in the aggregate size and hindered settling velocity with
flocculant dosage. The increased size is attributed to a higher flocculant surface
coverage and increased particle bridging (Bagster 1993), leading to stronger aggregates
and a reduced breakage rate. However, the flocculant is gradually degraded through
time due to the repeated aggregation/breakage. This results in a reduced flocculant
activity and the gentle reduction in the aggregate size at extended residence times. If
the particles were coagulated, or if charge neutralisation (patch model) was the
dominant flocculation mechanism, a stable steady state aggregate size would be
produced (Bagster 1993) where the aggregation and breakage rates were balanced.
Higher flocculant dosages (40 & 80 g t-1) were also tried (not shown), increasing the
aggregate size and settling rate further, however the increased settling rate caused the
solid to settle prematurely in the pipe reactor (see Section 4).
Although the flow rate could have been increased, allowing the suspension of larger
aggregates formed at a higher flocculant dosage, this would also result in a higher shear
rate and increased aggregate breakage (Figure 6-2). Since flocculant is the major
operating cost in many units (Perry & Green 1997), and excessive flocculant addition
162
can increase the sediment yield stress (Healy et al. 1994), reducing the fluid shear and
hence the flocculant requirement may lead to process improvements.
-1.5E-04
-1.0E-04
-5.0E-05
0.0E+00
5.0E-05
1.0E-04
1.5E-04
0 5 10 15 20 25 30 35 40
Time (s)
Mea
n ag
greg
ate
diam
eter
(m)
0
2
4
6
8
10
12
14
16
Hin
dere
d se
ttlin
g ve
loci
ty (m
hr-1
)
Dosage = 20 g tֿ¹, (Baseline)Dosage = 10 g tֿ¹Dosage = 5 g tֿ¹
Figure 6-3: Effect of flocculant dosage on the development of the mean aggregate size and hindered settling velocity. Other conditions as per baseline.
6.2.3 Effect of suspension solid fraction
Figure 6-4 shows the effect of the solid fraction on the mean aggregate size and
corresponding hindered settling velocity. The decreased size at high solid fraction is
attributed (Sections 4 & 5) to an increase in fluid viscosity and energy dissipation rate,
leading to an increased breakage rate and smaller aggregates. The hindered settling
velocity is further reduced by greater hindrance at an increased solid fraction, as
described by Equation 6-2, although the effective thickener throughput is limited by the
solid settling flux given by:
sUφρ=ψ 6-6 where:
ψ = Settling flux (kg m-2s-1) U = Settling velocity (m s-1) φ = Solid volume fraction [0,1] ρs = Solid density (kg m-3)
163
-1.5E-04
-1.0E-04
-5.0E-05
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
0 5 10 15 20 25 30 35 40
Time (s)
Mea
n ag
greg
ate
diam
eter
(m)
0
10
20
30
40
50
60
70
Hin
dere
d se
ttlin
g ve
loci
ty (m
hr-1
)
3.33 % w/v6.67 % w/v10.0 % w/v, (Baseline)13.33 % w/v16.67 % w/v
Figure 6-4: Effect of suspension solid fraction on the development of the mean aggregate size and hindered settling velocity. Other conditions as per baseline.
164
6.2.4 Effect of primary particle size
Figure 6-5 shows the effect of primary particle size on the aggregate size and
subsequent hindered settling velocity. As described previously (Sections 4 & 5), a
larger primary particle has a lower surface area (per unit mass) and hence a higher
flocculant surface coverage. This leads to a stronger aggregate and an increased size
due to a reduction in the breakage rate. In addition, a larger primary particle also leads
to a lower aggregate porosity (Equation 6-8), further increasing the settling velocity.
-1.0E-04
-5.0E-05
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
2.5E-04
0 5 10 15 20 25 30 35 40
Time (s)
Mea
n ag
greg
ate
diam
eter
(m)
0
10
20
30
40
50
60
70
80
90
100
Hin
dere
d se
ttlin
g ve
loci
ty (m
hr-1
)
dp = 24.3 µm, (settling obs. in pipe)dp = 15.1 µmdp = 5.59 µm, (Baseline)dp = 3.47 µmdp = 2.36 µm
Figure 6-5: Effect of primary particle size (dp) on the development of the mean aggregate size and hindered settling velocity. Other conditions as per baseline.
6.2.5 Fractal geometry
Aggregate porosity is usually described by fractal geometry (Meakin 1988), where
although the total enclosed volume of the aggregate increases as a cube of its diameter,
its mass increases at some lower fractional (hence: fractal) power:
fD
p
agg
dd
m ⎟⎟⎠
⎞⎜⎜⎝
⎛∝ 6-7
where: m = Aggregate mass (kg) d = Diameter (agg = aggregate, p = primary particle) (m) Df = Mass-length fractal dimension The increase in porosity with aggregate size is usually rationalised on the basis of the
increasing voidage produced later in the aggregation process where large aggregates
result from aggregate-aggregate collision (Figure 2-7).
165
Aggregate porosity has two major impacts on the hindered settling velocity. Firstly, it
dramatically reduces the aggregate density, reducing the driving force for the particle to
settle under gravity (Equation 6-1), giving (Gonzales & Hill 1998, Atteia 1998, Ellis &
Glasgow 1999, Manning & Dyer 1999):
( ) ( )3D
p
agglslagg
f
dd
−
⎟⎟⎠
⎞⎜⎜⎝
⎛ρ−ρ=ρ−ρ 6-8
where: ρ = Density(s = solid, l = liquid, agg = aggregate) (kg m-3)
Secondly, the effective suspension solid volume fraction is increased, increasing the
inter-particle hindrance as described by Richardson and Zaki (Equation 6-2) (Potanin &
Uriev 1991, Flesch et al. 1999):
fD3
p
aggseff d
d−
⎟⎟⎠
⎞⎜⎜⎝
⎛φ=φ 6-9
where: φ = Solid volume fraction (s = solid, eff = effective) [0,1] Combining equations 6-1, 2, 8 and 9, and substituting mean sizes gives:
( ) 65.4D3
p
aggs
3D
p
aggls
2agg
h
f
f
d
d1
18
d
dgd
U⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛φ−
μ
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ρ−ρ
=
−
−
6-10
The settling velocity may be further reduced by the effect of particle shape as the
particles deviate from the sphericity assumed by the above equations. However, the
effect of shape is relatively minor at low particle Reynolds number (∼ 0.7-1.1 times the
rate, Pettyjohn & Christiansen 1948, Heiss & Coull 1952, Happel & Brenner 1973,
Kousaka et al. 1981, Seville et al. 1997, Figure 2-17), and is partially offset by the
permeability of porous aggregates, allowing some fluid flow through the structure,
decreasing the drag slightly (Veerapaneni & Wiesner 1996, Gregory 1997). Hence, the
effects of particle shape, permeability, and Reynolds’ number have been ignored as
unnecessary complications at this stage, and Equation 6-10 incorporates the dominant
effects (suspension solid fraction and aggregate density) into Stokes’ Equation to
predict the hindered settling rate.
166
In this case the fluid viscosity is taken to be a constant 0.00102 N s m-2 (water), and is
unaffected by the aggregation process. During hindered settling the particles are
essentially joined together as a continuous, permeable network, with the displaced water
passing through channels in the sediment. Previously (Sections 4 & 5) the overall
suspension viscosity was described as a function of the solid fraction and degree of
aggregation, however in that situation the flow is fundamentally different, with a high
degree of relative particle movement in the turbulent pipe flow.
Equation 6-10 was used to estimate the fractal dimension (Df) from the experimental
data via :
2
t,d,,,G Exp,h
106.Eq,hExp,h
Dp
f UUU
min ∑θφ
−
⎟⎟⎠
⎞⎜⎜⎝
⎛ −= 6-11
where: Uh,Exp = Experimental hindered settling rate (m s-1)
Uh,Eq.6-10 = Predicted hindered settling rate from Equation 6-10 (m s-1) In this case the fractal dimension (Df) was found to produce the best fit with a value of
2.42 (Figure 6-6), which is a typical value for aggregates formed by flocculation at high
shear rates (Table 2-3).
0.1
1
10
100
0.1 1 10 100
Experimental hindered settling velocity (m hr-1)
Pred
icte
d hi
nder
ed se
ttlin
g ve
loci
ty (m
hr-1
)
dp = 2.36 µm
dp = 3.47 µm
Baseline # 1
dp = 15.08 µm
dp = 24.3 µm
Dose = 5 g tֿ¹
Dose = 10 g tֿ¹
3.33 % w/v
6.67 % w/v
13.33 % w/v
16.67 % w/v
38.1 mm ID, 0.343 m sֿ¹
38.1 mm ID, 0.554 m sֿ¹
Baseline # 2
25.4 mm ID, 0.781 m sֿ¹
25.4 mm ID, 1.294 m sֿ¹
38.1 mm ID, 0.343 m sֿ¹, 10 g tֿ¹
dp = 15.0 µm, 6.67 % w/v
25.4 mm ID, 0.781 m sֿ¹, 40 g tֿ¹
Figure 6-6: Comparison between measured and predicted (Equation 6-10) hindered settling rates.
Aggregate settling data typically displays considerable scatter, however a sensitivity
analysis was performed to determine if the fractal dimension (Df) was a function of any
167
of the experimental variables (shear rate G, flocculant dosage θf, primary particle size dp
and solid fraction φ). Both linear and non-linear sensitivities were tried as per
Equations 6-12 and 6-13, the linear according to:
⎟⎟⎠
⎞⎜⎜⎝
⎛φ
φ−φ+⎟
⎟⎠
⎞⎜⎜⎝
⎛ −+⎟⎟
⎠
⎞⎜⎜⎝
⎛θ
θ−θ+⎟⎟
⎠
⎞⎜⎜⎝
⎛ −+=
BL
BLExp
BL,p
BL,pExp,p
BL
BLExp
BL
BLExp*ff D
ddd
CBG
GGADD 6-12
with: G = Mean shear rate (s-1)
θ = Flocculant dosage (kg m-2) dp = Primary particle size (m)
φ = Solid fraction [0,1] Exp = Experimental condition BL = Experimental condition @ baseline Df
* = 2.43 A = 0.0244 B = 0.0842 C = -0.0060
D = -0.0786
and the non-linear:
z
BL
Expy
BL,p
Exp,px
BL
Expw
BL
Exp*ff d
dGG
DD ⎟⎟⎠
⎞⎜⎜⎝
⎛
φ
φ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
θ
θ⎟⎟⎠
⎞⎜⎜⎝
⎛= 6-13
with: Df
* = 2.44 W = 0.01949 X = 0.0337 Y = 0.00314 Z = -0.0337 Figure 6-7 shows the fit using the linear method (Equation 6-12), with the residual
reduced from 19.4 (Figure 6-6) to 11.9. The non-linear method (not shown, Equation
6-13) reduced the residual fractionally further, to 10.5. The improvement of Figure 6-7
over Figure 6-6 is minor, considering the number of degrees of freedom increased from
1 to 5, indicating that Equation 6-10 holds reasonably well within the experimental
scatter.
168
0.1
1
10
100
0.1 1 10 100
Experimental hindered settling velocity (m hr-1)
Pred
iced
hin
dere
d se
ttlin
g ve
loci
ty (m
hr-1
)
dp = 2.36 µm
dp = 3.47 µm
Baseline # 1
dp = 15.08 µm
dp = 24.3 µm
Dose = 5 g tֿ¹
Dose = 10 g tֿ¹
3.33 % w/v
6.67 % w/v
13.33 % w/v
16.67 % w/v
38.1 mm ID, 0.343 m sֿ¹
38.1 mm ID, 0.554 m sֿ¹
Baseline # 2
25.4 mm ID, 0.781 m sֿ¹
25.4 mm ID, 1.294 m sֿ¹
38.1 mm ID, 0.343 m sֿ¹, 10 g tֿ¹
dp = 15.0 µm, 6.67 % w/v
25.4 mm ID, 0.781 m sֿ¹, 40 g tֿ¹
Figure 6-7: Comparison between measured and predicted (Equation 6-12) hindered settling rates.
The sensitivity analysis indicates some increase in the fractal dimension (less porous)
with shear and flocculant dosage, but a decrease with solid fraction. However, caution
may be required interpreting these observations, since none of the effects are very
strong (i.e. Df ∼ 2.4 ± 0.1) and do not change the overall observation that the initial
hindered settling rate is increased with flocculant dosage (Figure 6-3) and primary
particle size (Figure 6-5), but decreased with solid fraction (Figure 6-4) and shear rate
(Figure 6-2).
Figures 6-2, 3, 4 and 5 show a slight reduction in aggregate size on extended shearing,
typical of aggregates formed by polymer flocculants (Keys & Hogg 1978, Pelton 1981,
Williams et al. 1992, Bagster 1993). This effect could be interpreted as either aggregate
compaction, or partially irreversible breakage (or both) (Mills et al. 1991, Oles 1992,
Ellis & Glasgow 1999). However, aggregate breakage appears to be the dominant
mechanism for this system, because in all cases the hindered settling rates also drop
with aggregate size on extended shearing. If compaction was the dominant size
reduction mechanism, the hindered settling velocity would be expected to rise due to the
increased density (Equation 6-8) and reduced effective solid fraction (Equation 6-9).
169
6.3 Conclusions The initial hindered settling velocities of flocculated calcite suspensions have been
studied under various process conditions (fluid shear, flocculant dosage, primary
particle size, solid fraction) and related to the on-line aggregate size measurements. As
would be expected a larger aggregate size leads to a higher hindered settling velocity,
however the effects of inter-particle hindrance must also be considered. Richardson and
Zaki’s (1955) relationship describing the hindered settling velocity of un-flocculated
suspensions has been adapted to flocculated suspensions by incorporating fractal
geometry to describe aggregate porosity. This has enabled the estimation of the fractal
dimension (2.4) from the experimental data.
170
7. Dynamic optimisation and extrapolation to full-scale 7.1 Predicting settling behaviour –Dynamic optimisation Computer modelling may lead to process improvements through an increased
understanding of the system, however, in this case the primary objective is to allow the
prediction of optimal process conditions. These are typically determined by a model
optimisation; in this case performed within the software (gPROMS). However, care is
required in choosing the correct manipulated and objective variables. For example,
attempting to optimise the mean aggregate size as a function of flocculant dosage (note
Figure 6-3) would be meaningless, since the optimisation would rapidly converge to the
maximum allowable dosage.
An alternative optimisation might use the data from Figure 6-2 to predict the flow
conditions (i.e. shear rate, perhaps tapered) to give the largest aggregate in the shortest
possible time. However, aggregation at high solid fractions is fundamentally different
from, say, a slow chemical reaction, and sufficient time is probably already available for
complete aggregation. In this situation the optimisation would probably be better
performed to yield the largest possible aggregate, and the process changed (e.g. via a
change in the flocculant dosing point) to suit.
Another optimisation strategy would be to alter the feed solid fraction (e.g. by overflow
or underflow recycle). Figure 7-1 shows the effect of solid fraction on the predicted
average aggregate size through time. Figure 7-2 shows the variation in the maximum
mean aggregate size with solid fraction. A dynamic optimisation gave the peak at
4.32 % w/v. However, the primary objective of a thickener is to separate solid from
liquid, and the actual aggregate size is essentially irrelevant (accepting that larger
aggregates typically lead to a higher settling rate) and the objective function should be
written in terms of the solid settling rate, rather than the aggregate size. Figure 7-3
shows that the maximum hindered settling velocity is predicted to occur at a lower
3.32 % w/v for this system. The maximum hindered settling velocity occurs at a lower
solid concentration because the hindered settling rate is also a function of the solid
fraction (Equation 6-2).
171
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
0 5 10 15 20 25 30 35 40
Time (s)
Mea
n A
ggre
gate
size
(m)
1.00 % w/v1.93 % w/v3.32 % w/v4.32 % w/v5.18 % w/v6.67 % w/v10 % w/v, (Baseline)
Figure 7-1: Modelled mean aggregate size as a function of time and solid fraction.
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
0 10 20 30 40 50 60
Solid concentration (% w/v)
Peak
pre
dict
ed m
ean
aggr
egat
e si
ze (m
)
Peak @ 4.32 %
Figure 7-2: Modelled maximum aggregate size as a function of the solid fraction. The maximum aggregate size occurs somewhere between 5-10 seconds after flocculant addition, see Figure 7-1.
172
0.0E+00
5.0E-04
1.0E-03
1.5E-03
2.0E-03
2.5E-03
3.0E-03
0 10 20 30 40 50 60
Solid concentration (% w/v)
Peak
pre
dict
ed h
inde
red
sett
ling
velo
city
(m s-1
)
Peak @ 3.32 %
Figure 7-3: Modelled maximum hindered settling velocity as a function of the solid fraction.
While Figure 7-3 indicates that the highest initial hindered settling velocity occurs at
3.32 % w/v for this system, the total solid throughput is of primary interest for a
thickener (with boundary values for acceptable overflow clarity and underflow solid),
hence the solid mass settling flux rate (kg m-2s-1 or t m-2hr-1) is calculated as the product
of the settling rate and the solid mass fraction (Equation 6-6) (Figures 7-4 & 7-5) with
the optimal value of 5.18 % w/v, occurring after 9.79 s (Figure 7-4).
If the optimisation is static rather dynamic, the results change fractionally. For
example, if the residence time in the pipe reactor is limited to 3 s, then the predicted
optimal solid fraction becomes 6.4 % w/v, because a higher solid fraction leads to a
faster initial aggregation rate via an increased collision rate. Alternatively, if the mean
residence time is taken to be 30 s, the optimal feed solid decreases fractionally to 4.8 %
w/v, due to a decrease in the polymer degradation at lower solid fraction.
173
0
0.02
0.04
0.06
0.08
0.1
0.12
0 5 10 15 20 25 30 35 40
Time (s)
Solid
flux
rat
e (k
g m
-2s-1
)
4.80 % w/v5.18 % w/v6.40 % w/v
Figure 7-4: Effect of residence time on maximum initial settling flux.
0.0E+00
2.0E-02
4.0E-02
6.0E-02
8.0E-02
1.0E-01
1.2E-01
0 10 20 30 40 50 60
Solid concentration (% w/v)
Peak
pre
dict
ed fl
ux r
ate
(kg
s-1m
-2) Peak @ 5.18 %
Throughput limiting solid fraction
Figure 7-5: Modelled maximum flux rate as a function of the solid fraction. The maximum flux rate occurs between 5-10 seconds, see Figure 7-4.
The optimisation described above maximises the initial settling flux, since only the
initial hindered settling velocity was collected during the experimental sessions.
However, a thickener is unlikely to be limited by the initial settling flux, and Figure 7-5
includes the required settling flux (operating line) to give a dynamic form of the
Yoshioka/Hasset combined flux curve (see Figure 2-21). The line is drawn tangentially
to the flux curve such that it also passes through the desired underflow solid
concentration (taken as 50 % w/v here). The y-axis intercept gives the upper limiting
feed flux for the thickener (i.e. about 6 × 10-2 kg m-2s-1 in this case). The limiting solid
174
concentration is about 28 % in this case, a Kynch settling layer that would occur lower
in hindered settling region.
Although Figure 7-5 has the conventional form, the flux curve was produced by running
the population balance simulation with a range of initial solid fractions (as per the Coe
and Clevenger experimental method). The combined flux curve is usually constructed
from the analysis of a single batch-settling test (Section 2.7.2, Kynch 1952, Talmage &
Fitch 1955), avoiding the effect of solid fraction on the aggregation process.
The combined flux curve makes various assumptions based on Kynch theory
(Section 2.7.2), significantly that the thickener is limited by the hindered settling rate.
If, in fact, the compaction/bed region is rate limiting, then some alternative model is
required (Svarovsky 2000).
7.2 Extrapolation to Industrial scale Fluid dynamics are notoriously difficult to scale accurately, and this work is no
exception. Industrial flows are characterised by large geometries and hence high
Reynolds number. The size of the macro-scale turbulent eddies gives them a high
tangential velocity, enabling the suspension of large, rapidly settling particles, but at a
comparatively low fluid shear rate. Wastewater treatment plants typically aim for a
mean shear rate of ∼ 60 s-1 (Amirtharajah et al. 1991).
However, it is difficult to replicate these conditions on a small scale. The reduced
geometry leads to a lower Reynolds number and the tendency for the solid to settle out
in the experimental apparatus. The Reynolds number can be increased by increasing the
flow velocity, but that would also increase the shear rate, causing irreversible aggregate
breakage (Figure 6-2). Alternatively, solid settling can be suppressed by using less
dense, and/or smaller primary particles.
Figure 7-6 shows the development of the mean aggregate size predicted by the
population balance, with appropriate conditions changed to better reflect the industrial
scale. Primary particle size distributions vary considerably depending on the process
(Pearse 1977, Perry & Green 1997), varying from μm to mm scale, with the product of
175
grinding circuits typically reasonably coarse (∼ 10-1000 μm) compared to crystallisers
(∼ 0.1-100 μm). The primary particle distribution was altered to give a mean primary
particle size of 80 × 10-6 m (80 μm), and the flow velocity reduced to give a (assumed
still turbulent) shear rate of about 40-50 s-1.
The increased primary particle size also lead to an increased effective dosage via a
reduction in the solid surface area, but the absolute dosage was kept at 20 g t-1.
Figure 7-6 shows that under these conditions the final aggregate size is in the range
1000 × 10-6 m (1000 μm); approximately correct for full-scale.
0.0E+00
2.0E-04
4.0E-04
6.0E-04
8.0E-04
1.0E-03
1.2E-03
0 10 20 30 40 50 60 70 80 90 100
Time (s)
Figure 7-6: Modelled change in mean aggregate size under simulated plant conditions – see text.
It should be noted that the conditions assumed in the population balance used to
generate Figure 7-6 are not actually achievable experimentally in the pipe reactor. The
coarse solid would rapidly settle out of the flow, which would have become laminar by
such a low flow rate.
Figure 7-7 shows the resulting settling rate and flux rates, which also appear to be
plausible for the full-scale plant, accepting that the flux rate is the initial settling flux,
not the limiting flux (see discussion around Figure 7-5). It is also interesting to note
that the model predicts very little polymer degradation at a low shear rate, with the
settling rate effectively constant after the first 10 s.
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0
10
20
30
40
50
60
70
80
90
100
0 10 20 30 40 50 60 70 80 90 100
Time (s)
Hin
dere
d se
ttlin
g ra
te (m
hr-1
)
0
2
4
6
8
10
12
14
Solid
flux
rat
e (t
hr-1
m-2
)
Hindered settling rateSolid flux rate
Figure 7-7: Modelled hindered settling rate and solid flux rates under simulated plant conditions – see text & Figure 7-6.
7.3 Conclusions This section has briefly investigated how the population balance and associated settling
model can be used to optimise full-scale thickener operation, in particular by altering
the solid fraction to give optimal aggregation to produce the maximum settling flux.
The optimisations were also dynamic, as the values used represented the maximum
values achieved during the aggregation process (around 5-10 seconds in this case).
Ultimately, the model will be used as a sub-model within the full CFD model of a
thickener, allowing the scope of the optimisation to be expanded considerably. In that
case the feedwell design (feed entry velocity, flocculant dosing point(s), baffle size and
position etc) will be able to be optimised to give the best shear profile and residence
time. The optimal feed solid fraction can also be determined, giving the highest settling
flux and hence unit throughput.
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8. Conclusions and future work
8.1 Conclusions The aggregation/breakage kinetics of calcite flocculation in turbulent pipe flow have
been modelled by population balance. The model draws heavily on successful work
previously described in the literature, but extends it by combining various aspects into a
single model that describes both the aggregate size and settling rate as a function of the
key process variables. The model is robust and stable, and although it adequately
describes complex aggregation and settling behaviour, it remains simple enough to form
a sub-model within a larger CFD simulation of a thickener/clarifier.
The population balance model is based on the descriptions by Hounslow et al. (1988)
and Spicer and Pratsinis (1996a), with the collision rate described by Saffman and
Turner’s (1956) turbulent shear kernel. However, Saffman and Turners kernel
(Equation 2-7) considerably overestimates the initial aggregation rate immediately after
flocculant addition, and a capture efficiency (α) term is introduced to account for the
discrepancy. If the capture efficiency was treated simply as a fitted parameter, it had to
be relatively small (α ∼ 0.06) to give a reasonable fit to the experimental data,
suggesting that particle/particle collision is not initially rate limiting.
Experimental evidence (Figure 5-3) shows that changing the initial
flocculant/suspension mixing (e.g. by flocculant stream dilution and/or velocity)
significantly changed the initial aggregation rate, suggesting that the initial capture
efficiency was a function of the macro-scale flocculant/suspension mixing. In this case,
the capture efficiency (Equation 5-6) is taken to be zero before flocculant addition
(stable suspension), and rapidly (Figure 5-2) approach unity as the
flocculant/suspension become well mixed. Equation 5-6 is a relatively crude
approximation of the overall flocculant mixing, collision and adsorption process. In this
case, no account is taken for polymer/particle collision or adsorption, as they are taken
to be rapid (Gregory 1993, Bagster 1993, Mühle 1993, Hogg 1999) compared to the
bulk macro-scale flocculant/suspension mixing. Macro-scale mixing is likely to be
highly scale-dependent, and much slower on the full scale. Industrial-scale mixing is
difficult to replicate on a laboratory scale, although emerging techniques like
computational fluid dynamics (CFD) (Lainé et al. 1999, Ducoste & Clark 1999, Farrow
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et al. 2000), or electrical impedance tomography (Brown et al. 1985, Webster 1990,
Salkeld 1991, Dickin et al. 1992, Williams & Simons 1992) are likely to increase the
understanding of this area in the future.
Equation 5-6 suggests that the capture efficiency approaches unity as the flocculant
becomes well mixed. This is unlikely to be the case in practice, and the capture
efficiency is likely to be a function of the hydrodynamic interaction (Equation 2-17) and
the flocculant dosage (Equation 2-19). However, in this case the model was made
dosage dependent via the breakage kernel (Equation 5-19), where a higher surface
coverage gave a stronger aggregate, less breakage, and hence a larger final aggregate
size. The effective active flocculant surface coverage was taken to decrease through
time due to the repeated aggregation/breakage process. The flocculant is degraded due
to polymer chain scission or rearrangement, and is shown experimentally (Figures 4-5,
4-9, 4-11 & 4-13) as a gentle reduction in the aggregate size at extended residence
times. Flocculant degradation is clearly undesirable industrially, but may occur around
feed or flocculant entry points, baffles etc. Although sufficient fluid shear is required to
provide effective mixing, excessive shear results in irreversible aggregate breakage.
In addition to fluid shear (G) and flocculant dosage (θf), the effects of the suspension
solid volume fraction (φ) and primary particle size (dp) were also considered. These
may also be process variables depending on the system (feed, milling, crystallisation
etc), and the solid fraction can be altered in some units via overflow or underflow
recycle. The final aggregate size was found to increase with primary particle size, and
although a full description is likely to be a complex function of the primary particle
packing and bonding within the aggregate structure, in this case a reasonable
relationship was found based on the effective flocculant coverage (Equation 5-18).
That is, a larger primary particle has a lower surface area (per unit mass) and therefore a
higher flocculant surface coverage, increased strength, and hence a larger final
aggregate size.
The effect of solid fraction has received little attention in the literature due to the
previous difficulty of on-line aggregate sizing in concentrated suspensions. Also, most
workers have considered coagulation in river estuary systems or water treatment
clarifiers, which are characterised by a low solid fraction. At low solid fraction (< 1 %),
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there is evidence (Thomas 1964, de Boer et al. 1989, Kobayashi et al. 1999) to suggest
that the aggregate size increases with solid fraction, probably reflecting an increased
particle collision rate. This is consistent with the aggregation and breakage kernels,
because aggregation (Equation 2-7) is taken to be second order with respect to particle
number, whereas breakage (Equation 5-19) is first order. That is, doubling the number
of primary particles doubles the breakage rate, but quadruples the aggregation rate,
leading to a higher degree of aggregation.
However, at the high solid fractions (3.33-16.7 % w/v) studied here, the final aggregate
size is reduced with solid fraction. This is attributed to the increased suspension
viscosity at high solid fraction (Equation 5-16), leading to a higher energy dissipation
rate (flow rate fixed by positive displacement pump) and increased breakage. Breakage
kernels (Table 2-6) are typically written in terms of G (as per aggregation kernels),
suggesting that the breakage rate would decrease at higher viscosity (Equation 5-7).
However, most workers have previously considered systems with a solid fraction that is
so low that the viscosity is effectively unchanged from water. In this case, during
model construction the breakage kernel was initially taken as ∝ ∈yμz where y and z
were fitted parameters. Parameter z repeatedly converged to 1 ± 0.05 and was
subsequently set = 1, removing a degree of freedom from the model. Parameter y was
typically in the range 0.7 ± 0.1 (final estimate = 0.677, perhaps Si ∝ ∈2/3).
The four model parameters were estimated from experimental data collected in a pipe
reactor. A range of pipe diameters, lengths and suspension flow rates gave a range of
mean shear rates (G) and residence times. The shear rate and suspension viscosity were
calculated from the pressure drop along the pipe reactor, as measured by a manometer
bank. The manometer showed an increase in fluid viscosity (larger ΔP) as a function of
the aggregate size, and also served to confirm the lower limiting flow rate when the
solid settled out in the pipe reactor. The relatively high mean shear rate (∼ 100-1000 s-1
here) in small-scale turbulent pipe flow is an incentive to reduce the flow rate, but is
ultimately limited by the transition to laminar flow, with poor mixing and solid
stratification/settling in the pipe.
A sparse matrix (Figure 4-4, Table 4-1) of experimental runs was performed by
changing the process variables (G, φ, θf and dp) independently away from a common
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baseline. Some of the data were of limited value due to settling, but the remaining data
were used for model parameter estimation. Several additional runs were performed
(Figure 5-17) in the gaps of the experimental matrix by changing two process variables
simultaneously. These data were not used for parameter estimation, but were used to
confirm the predictivity of the model.
In addition to the on-line, in-pipe particle sizing (FBRM), hindered settling
measurements were taken using a vertical settling column (Figure 6-1). This allowed a
relationship to be found between the mean aggregate size and the hindered settling
velocity (Equation 6-10), ultimately allowing the population balance to predict the
initial settling flux. The relationship is based on Richardson and Zaki’s (1955)
extension of Stokes’ (1851) law for particulate settling in viscous flow. A variety of
other factors also influence the hindered settling velocity (aggregate shape and
permeability, fluid inertial effects at higher particle Reynolds number), but the
dominant effect is due to the increased volume (and reduced density) of porous
aggregates. Aggregate porosity was described with fractal geometry, allowing the
fractal dimension to be estimated (Df = 2.4) from the experimental data (Equations 6-10
& 6-11).
Although this work has provided some insight into the size of aggregates ultimately
produced by the competing effects of aggregation and breakage, the initial macro-scale
mixing of the flocculant/suspension is described in a simplified form appropriate for
only turbulent pipe flow and requires further investigation for other geometries like
feedwells. The practical importance of macro-scale mixing is evidenced by various
industrial design practices to encourage efficient flocculant mixing (e.g. flocculant
stream dilution, multi-point addition, baffles, in-line mixers etc).
There is also a need to consider the down-stream effects of changes to the flocculation
process. Although the initial hindered settling velocity has been investigated here, the
subsequent compaction of the sediment will be important in many applications. For
example, increasing the flocculant dosage to increase the throughput may not be viable
if it produces a voluminous sediment with a high yield stress that limits the final
underflow solid concentration. Issues may also arise further down-stream, and the
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filterability and pumping requirement of the underflow may also be compromised (or
improved) by changes to the flocculation conditions.
8.2 Suggested future research 1. Code the population balance as a sub-model into a CFD model describing a
mineral processing thickener/clarifier. This was the overall aim of the work
presented here, and should allow process improvements (throughput, clarity,
perhaps also underflow solid) through improved flocculation. The crude
flocculant mixing/adsorption described will require replacing by a 3-dimensional
description provided by the CFD model.
2. Expand the aggregation kinetics experimental using the pipe reactor to other feed
materials, flocculant types, liquor temperature/chemistry etc. Fit the population
balance model to these new data sets, and build up a matrix of parameter values of
different systems. Ultimately the parameters could be replaced by functions
accounting for the differences between the various systems.
3. Further investigate the relationship between the aggregation state and the
subsequent sedimentation, in particular yield stress and sediment dewaterability,
this would greatly increase the ability to use the population balance model to
improve the unit performance. Similarly, the link between aggregation and
overflow clarity should be investigated, allowing the optimisation of units limited
by overflow clarity.
4. Various other techniques could be used to further characterise the
aggregation/breakage process, in particular the use of CFD to investigate
fluid/particle interactions on a small scale, e.g. the hydrodynamic force that acts
between colliding particles, or the action of fluid shear to break aggregates.
Micromechanical devises might be used to further understand the forces of
attraction/repulsion between particles, and the influence of flocculants on these
forces.
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9. Notation A = Area (m2) A = Hamaker constant Ab = Thickener area from batch experiment (m2) As = Surface area of solid (m2) ai = Radius of the ith particle (m) BL = Experimental condition at baseline C = Floc strength coefficient C(x) = Concentration at point x (or m = mean) CA = A/36πμGa3 Cd = Coefficient of drag (dimensionless) Ci,A = Geometric average length of ith channel (μm) = (Ci,u × Ci,l)1/2 Ci,l = Length of the lower boundary of the ith chord channel (μm) Ci,u = Length of the upper boundary of the ith chord channel (μm) Cm = Mean concentration through vessel CN = Drag coefficient in Newton’s equation D = Units of water per unit solid D∞ = D at infinite time D = Pipe diameter (m) Df = Mass-length fractal dimension (1 [thin rods] ≤ Df ≤ 3 [spheres etc]) d = Separation distance d = Particle diameter (m) dp = Diameter of the primary particle (m) dagg = Diameter of aggregate (m) dagg,i = Diameter of the ith sized aggregate (m) dagg,Exp = Experimental (FBRM) mean aggregate size (m) dagg,PB = Modelled (population balance) mean aggregate size (m) dm = Mass equivalent diameter (m) dm,i = Mass effective diameter of the ith sized aggregate (m) Exp = Experimental condition F = Force (N or kg m s-2) F = Solid feed rate (m3s-1) Fb = Force of buoyancy (N or kg m s-2) Fd = Force on particle due to fluid drag (N, or kg m s-2) Fg = force of gravity (N or kg m s-2) f = Fanning friction factor (dimensionless) G = Mean turbulent shear rate (s-1) g = Acceleration due to gravity (9.8 m s-2) H = Expected full scale compression zone depth (m) H = Mudline height (m) H = Height of suspension (Ho at t = 0) HK = Height of Kynch isoconcentration line. K = Boltzmann’s constant (1.38 × 10-23 J K-1) K = Velocity head KB = Floc breakage coefficient KE = Einstein’s constant (2.5) K = Constant k = Constant k1 = Fitted parameter # 1 = 0.3431 k2 = Fitted parameter # 2 = 38.1 k3 = Fitted parameter # 3 = 0.677 k4 = Fitted parameter # 4 = 1224 L = Length (m) L = Characteristic aggregate length (m) L = Pipe length (m) Li = Length of the ith channel
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M = Mixing index [0,1] m = Exponent m = Mass (kg) mf = Mass of flocculant (kg) mi = Mass of ith aggregate (kg) ml,i = Mass of liquid in the pores of aggregate, size i (kg) mp = Mass of primary particle (kg) ms = Mass of solid in aggregate (kg) N = Number of particles Ni = Number of ith sized particles (m-3) Ni = Number of i sized chord counts observed (s-1) Ni,n = Number of n-weighted chord counts in the ith channel (μmn s-1) Ni,o = Raw un-weighted counts in the ith channel (s-1) Np
* = Number of pth sized particles predicted (s-1) Nt = Total observed particle counts (s-1) Nt
* = Corrected total particle counts (s-1) n = Exponent P = Pressure (N m-2) P = Fitted parameter(s) Py(φ) = Compressive yield strength of the sediment (N m-2) Re = Reynolds number (dimensionless) Si = Breakage rate of the ith sized particle (s-1) Sv = specific surface area of solid per unit volume of particles (m2m-3 = m-1) T = Absolute temperature (K) t = Time (s) tc = Time required to reach required sediment concentration (days) tm = Metric Tonnes (1000 kg) U = Velocity (m s-1) Uf = Feed settling velocity (m s-1) Uh = Hindered settling velocity (m s-1) Uh,Eq.6-10 = Predicted hindered settling rate from Equation 6-10 (m s-1) Uh,Exp = Experimental hindered settling rate (m s-1) Ul = Settling velocity of limiting concentration (m s-1) UN = Settling velocity given by Newton’s equation (m s-1) Uo = Free settling rate of the particle (m s-1) Ur = Settling velocity required to prevent solid reaching the overflow (m s-1) Us = Settling velocity given by Stokes’ equation (m s-1) Uu = Settling velocity due to underflow removal (m s-1) u = Velocity component in the x direction (m s-1) V = Mean flow velocity (m s-1) VB = Velocity of the bulk pipe flow (m s-1) VF = Velocity of the flocculant stream (m s-1) Vi = Volume of the particles in the ith size range (m3) Vo = Initial volume of the suspension (m3) Vsed = Equilibrium volume of the sediment (m3) Vt = Interaction energy VT = Total suspension volume (m3) vx = Instantaneous velocity of component x (m s-1) v = Velocity component in the y direction (m s-1) v = Volume of particle (m3) vi = Volume of ith particle
xv = Time averaged velocity (m s-1) 'xv = X-component velocity fluctuation (m s-1)
Wp,i = Width of pth particle giving chords in the ith channel (μm) w = Velocity component in the z direction (m s-1) waccel. = Relative particle velocity of particles due to acceleration (inertial effect) wshear = Relative particle velocity of particles due to fluid shear. z = Height of compression zone (m)
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z = Valency z = Elevation (m) z∞ = Height of compression zone at infinite time (m) zc = Height of compression zone at critical time (m) α = Capture efficiency [0,1] α = Velocity profile factor β = Ratio of apparent to actual volume fraction βij = Rate of collision between i and j sized particles (aggregation kernel) (m3s-1) ∈ = Energy dissipation rate per unit mass (J s-1kg-1, m2s-3) Φ = Power input per unit volume (kg m-1s-3) θ = Flocculant surface coverage (kg m-2) σ = Concentration variation σo = Concentration variation at time = 0 η = Dynamic viscosity (N s m-2) η = Kolmogoroff micro-scale of length (m) κ = Deybe-Huckel parameter κ = Permeability (m2) κ(φ) = Dynamic compressibility π = Pi (∼3.14) ε = Porosity, i.e. liquid fraction (1-φ) (m3m-3) Ω = Relative drag of an aggregate compared to solid sphere of same diameter τi = Particle relaxation time (s) τk = Kolmogoroff micro-scale of time (s) Γjk = Breakage distribution function ΔP = Pressure drop along pipe (N m-3) μs = Suspension viscosity (N s m-2) μs = Viscosity of suspension (kg m-1s-1, N s m-2) μo = Viscosity of water (1.02 × 10-3 N s m-2) φ = Solid volume fraction (1-ε) (m3m-3) φo = Initial solid fraction (m3m-3) φu = Underflow solid fraction (m3m-3) φm = Maximum solid fraction, (taken as 0.65) φeff = Effective solid fraction φl = Limiting solid fraction (m3m-3) φs = Actual suspension solid volume fraction [0,1] φ∞ = Solid fraction after infinite time (m3m-3) φcrit = Critical solid fraction that limits thickener throughput (m3m-3) φf = Feed solid fraction (m3m-3) φg = Solid fraction where gelation (compression) begins (m3m-3) φ(ε) = Function of ε φ(z) = Volume fraction at position z ρ = Density (kg m-3) ρs = Density of the solid (calcite = 2710 kg m-3) ρeff = Effective aggregate density (kg m-3) ρl = Density of the liquid (water = 1000 kg m-3) ρsl = Density of slurry (kg m-3) Θ = Flocculant degradation index [0,1] λ = Characteristic wavelength (m) θf = Effective flocculant surface coverage (kg m-2) γ1 = Dimensionless function of surface charge γ = Laminar shear rate (s-1) ν = Kinematic viscosity (m2s-1) ψ = Sphericity, ratio of surface area of equivalent sphere to actual surface area ψ = Settling flux (kg m-2s-1) ψm = mass settling flux (kg h-1m-2)
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11. Appendix
11.1 Hounslow et al.’s (1988) population balance Hounslow et al. discretised the size interval according as per Batterham et al. (1981):
vv
i
i
+
=1
2 11-1
or
LL
i
i
+
=1
3 2 11-2
where: v = Particle volume (m3) L = Particle length (m) I.e. 2i ≤ vi ≤ 2i+1 The number density within the channel is taken to be constant across the channel, and is given by: n’ = Ni/(2i+1-2i) = Ni/2i The average particle volume within the channel is then given by:
v diii
i
i
i
i
i
i i
i i
=
=
= −
=−
+
+
+
+
∫ 2
22
22
22
2 22
1
1
1
1
ln
ln ln
ln
=
22
i
ln 11-4
Which is slightly less than the geometric mean (2i/0.6666). *N.B. note from text, if the average channel volume was assumed to be the geometric mean, then a normal distribution of three daughter fragments would have a mean at the boundary of the i-2 and i-1 channels, giving a 2:1 breakage distribution. But since 11-4 < geometric mean the distribution becomes greater than 2:1, with the final distribution being a function of the standard deviation of the distribution.
210
Mechanism 1 Collision involving a small particle with a large particle to give an aggregate in the ith channel. The larger particle has to be in the i-1th channel (2*2i-k < 2i, when k >1), and probably at the upper end of that channel (see Figure 11-1):
Figure 11-1: Mechanism 1
The number of available particles in the i-1th channel is:
aNii−
−1
12 11-5
i.e. the shaded area in Figure 11-1. The rate of aggregation is then given by:
R aN Ni i ji
i a( )
,1
11
12= −
−−β 11-6
I.e., consider the interval a to da, the population density will be:
Naa2
11-7
I.e. the total number of particles will be:
N daaa2
11-8
I.e. the total number of interactions that will cause a new ith sized particle is:
2a
2i 2j+1 2j 2i-1 2i+1
2i-2a
+
211
dR aN N dai a i j
ii
aa,
( ),
11
112 2
= −−
−β
Hounslow substitutes Njj2for Na
a2 since j ≤ a ≤ j+1, and 11-7:
dR aN N da
R N Nada
R N N a
R N N
R N N
i j i ji
ij
j
i j i jii
jj
i j i jii
jj
i j i jii
jj
j j
i j i jii
j
j
j
j
j
i
i
j
j
,( )
,
,( )
,
,( )
,
,( )
,
,( )
,
( ) ( )
1
2
2
12
21
1
11
11
2
2
11
11
2
2
2
11
11
1 2 2
11
11
1 1
1
1
2 2
2 2
2 2 2
2 22
222
2
+ +
+
+
∫ ∫
∫
=
=
=
= −⎛⎝⎜
⎞⎠⎟
=
−−
−
−−−
−−−
−−−
+
−−−
β
β
β
β
β ( )
( )
( )
22 2 2 2
2 22 2
2 22 2
21
2 22 2 2 1
2 22 2
2 2 1 2 1
11
11
2 1 2 1
11
11
2 12 1
2 1
11
11
2 1 2 1 2 1
11
11
2 1 2
jj j
i j i jii
jj
j j
i j i jii
jj
jj
j
i j i jii
jj
j j j
i j i jii
jj
j
R N N
R N N
R N N
R N N
+ − −
−−−
+ −
−−−
−+
−
−−−
− + − +
−−−
−
−
= −
= −⎛⎝⎜
⎞⎠⎟
= −
= −
,( )
,
,( )
,
,( )
,
,( )
,
β
β
β
β ( )1
32 2
2
3 2
11
11
2 1
11 1
R N N
R N N
i j i jii
jj
j
i j i j i jj i
,( )
,
,( )
,
=
=
−−−
−
− −−
β
β
but, want to sum all particles from j = 1 to i-2, i.e.:
R N Ni i j i jj i
j
i( )
,1
1 11
2
3 2= − −−
=
−
∑ β 11-9
212
Mechanism 2 This is similar to mechanism 1, but aggregation is between two particles in i-1th size. All aggregation events between two i-1th particles must give an ith sized particle since 2(2i-1) = 2i (i.e. bottom of the ith channel), and 2(2i) = 2i+1 (i.e. the top of the ith channel) - see Figure 11-2.
Figure 11-2: Mechanism 2
( )
( )
dR N N da
dR N da
R N a
R N
R N
R N
i i i iii
i i iii
i i iii
i i iii
i i
i i iii
i
i i iii
i
i
i
i
( ),
( ),
( ),
( ),
( ),
( ),
2 12 1 1 1
11
2 12 1 1
12
12
2
2 12 1 1
12
1 2
2
2 12 1 1
12
11
2 12 1 1
12
112
2 12 1 1
12
2
2
2
22 2
22 1
2
1
1
=
=
=
= −
= −
=
− − −−−
− −−−
− −−−
− −−−
−
− −−−
− −−
−
−
∫∫
β
β
β
β
β
β −−
112i
R Ni i i i
( ),
2 12 1 1 1
2= − − −β 11-10 Once again the half stops the double counting as per 11-1
2i 2i-1 2i+1
213
Mechanism 3 Loss of particles in the ith size channel when they collide with particles in i-1th size range or smaller. I.e. forming particles in the i+1th channel.
Figure 11-3: Mechanism 3
dR aN Nda
dRNN
a da
RNN
R NN
i j i ji
ijj
i j i ji ji j
j
j
i j i ji ji j
j
i j i j i jj i
,( )
,
,( )
,
,( )
,
,( )
,
.
3
31
3 2 1
3 1
2 2
2
23 2
3 2
=
=
=
=
∫ ∫+
+
+−
− −
β
β
β
β
as per mechanism 1 11-11
Summing for all j:
∑−
=
−−β=1
1
13 23i
j
ijjij,i
)(j,i NNR 11-12
2a
2i+1 2j+1 2j 2i 2i+2
+
214
Mechanism 4 Death of an ith sized particle by collision with a particle in ith or larger size range. All successful collisions will cause death of ith sized particle.
Figure 11-4: Mechanism 4
R NNi j i j i j,( )
,4 = β
I.e. summing over all j:
∑∞
=
β=ij
jij,i)(
i NNR 4 11-12
In this case the constant of ½ for j = i isn’t needed because each successful collision removes 2 particles. It would be expected then that the overall rate of change in the ith channel would be the sum of 11-9, 11-10, 11-11 and 11-12:
)(i
)(i
)(i
)(i
i RRRRdt
dN 4321 −−+= 11-13
However as the figures above show there is a tendency for the particles to be taken from the top of each channel, and through aggregation with small particles be moved to the bottom of the next channel in mechanisms 1 and 3 (Kumar and Ramkrishna 1997c). Since the model assumes that the distribution is even across the channel (11-3), some adjustment is required to compensate. To do this, Hounslow et al. multiplied mechanisms 1 and 3 by 2/3 giving:
)(i
)(i
)(i
)(i
i RRRRdt
dN 433221
32 −−+= 11-14
2i 2j+1 2j 2i+1
+
215
Alternatively (Mechanism 1):
111
111
1
111
2
22
22
1
1
+−−−
−−−
−
−−
β=
⎟⎟⎠
⎞⎜⎜⎝
⎛β=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛β=
ijijj,i
j
iijj,i
j
i
ijj,i)(
ij
NN
lnln
NN
N.VolN.Vol
NNR
but, want to sum all particles from j = 1 to i-2, ie:
1
2
11
11 2 −
−
=−
+−∑ β= ij
i
jj,i
ij)(ij NNR 11-15
216
11.2 Fractal geometry, calculation of dagg and ρagg from dm, dp & Df Given (Abel et al. 1994, Cohen & Wiesner 1990, Manning & Dyer 1999, Flesch et al.1999, Gregory 1997): fD
agg5s dkm = -A1 where: ms = Mass of solid in aggregate (kg) dagg = Effective diameter of the aggregate (m) Df = Mass – Length fractal dimension and for solid objects: 3
ms6s dkm ρ= -A2 where: k6 = Shape dependent constant (e.g. spheres = π/6) ρs = Solid density (calcite = 2710 kg m-3) dm = Mass equivalent diameter (m) Taking the common point at dp the primary particle size, combining Equations A1 & A2 gives:
3p
sDp5p d
6dkm f
πρ== -A3
where: mp = Mass of primary particle (kg) i.e. (Jiang & Logan 1991):
fD3p
s5 d
6k −πρ
= -A4
i.e., combining A1 & A4
ff Dagg
D3p
ss dd
6m −πρ
= -A5
and, equating with A3:
3m
sDagg
D3p
s d6
dd6
ffπρ
=πρ − -A7
i.e.:
3
p
mDpD3
p
3mD
agg dd
dd
dd f
f
f
⎟⎟⎠
⎞⎜⎜⎝
⎛==
− -A8
i.e.:
fD/3
p
mpagg d
ddd ⎟
⎟⎠
⎞⎜⎜⎝
⎛= -A9
Now, considering ρagg, where:
217
agg
sagg V
m=ρ -A10
substituting A5 and with the effective volume simply given by (again, assuming spherical): 3
aggagg d6
V π= -A11
i.e.:
3agg
Dagg
D3p
s
agg
sagg
d6
dd6
Vm
ff
π
πρ
==ρ
−
-A12
i.e. (Mills et al. 1991, Jiang & Logan 1991, Kusters et al. 1997, Serra & Casamitjana 1998):
3D
p
aggsagg
f
dd
−
⎟⎟⎠
⎞⎜⎜⎝
⎛ρ=ρ -A13
11.3 Fractal geometry, calculation of effective mean aggregate density For the settling rate equation (Equation 6-10) the density of the aggregate relative to the fluid is required. Starting with equation A10:
lagg
lslagg V
mmρ−
+=ρ−ρ -A14
where: ml = Mass of liquid in the pores of aggregate, size i (kg) ρl = Density of the liquid (water = 1000 kg m-3) The mass of water in the pores is:
( ) ⎟⎠⎞
⎜⎝⎛ π
−π
ρ=−ρ= − ff Di,agg
D3p
3i,aggli,mi,aggli,l dd
6d
6VVm -A15
Combining Equations A5, A12, A14 and A15:
3
i,agg
Di,agg
D3p
3i,aggl
D3p
li,agg
d6
dd6
d6
d6
fff
π
⎟⎠⎞
⎜⎝⎛ π
−π
ρ+ρπ
=ρ−ρ
−−
-A16
i.e.:
218
( ) ( )3D
p
agglslagg
f
dd
−
⎟⎟⎠
⎞⎜⎜⎝
⎛ρ−ρ=ρ−ρ -A17
and, substituting volume weighted mean sizes (Jiang & Logan 1991, Manning & Dyer 1999, Ellis & Glasgow 1999, Huang 1994):
( ) ( )3D
p
agglslagg
f
d
d−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ρ−ρ=ρ−ρ -A18
