The Population Balance as a Tool for Understanding Particle Rate Processest

This article discusses the use of population balance techniques in the understanding of particulate processes. Population Balance Equations (PEEs) describe how properties of a group of particles change with time and position. These "properties" are frequently some measure of particle size, so PEEs are most often used to describe how particle size distributions (PSDs) change during the processing of particulate materials. After a brief review of how PEEs are formulated and solved, three case studies, from the author's research, are considered. Crystallisation: Aggregation During Precipitation From Solution. This case study reports a decade of work on the aggregation of calcium oxalate monohydrate crystals in supersaturated solutions. Our knowledge of the dependence of growth and aggregation rates on solution composition is reported and physical models explaining the dependencies are discussed. The main conclusion for this study is, apart from a description of COM, that substantial microscopic lessons may be drawn from observations made at a mesoscopic (i.e. many particle) scale. Fluidisation: Spray Coating Of Grass Seeds The objective of this work was a description of how seed particles increase in size as new material is sprayed on. This study is unusual in that the PBE was solved analytically and gave rise to a model with no adjustable parameters. At first sight the model is capable of very high fidelity predictions; it is not until each particle is considered to have two properties, and not just one, in size, that failings in the model become apparent. Granulation: Extracting Kinetics This study reports a substantial experimental and numerical investigation of the rates at which size enlargement occurs in high-shear granulation equip-

ment.It is shown that breakage is extremely important and that its rate can be quantified.It is also shown that it is not possible to uncouple the-effects of particle size, age and other properties without careful experimental design.
The main conclusions of this work are: • It is possible to pose and solve PEEs for a variety of problems • It is possible to extract physical information about particle rate processes by means of PEEs • There are indications that this type of work must move on from representing particles by a single property, i.e. their size, and recognise that most particles have a variety of properties that affect their behaviour.

One particle
The starting perspective of this paper is that the first property of a particle is its size.We will view "size" perhaps as a linear dimension, l, or with less ambiguity, as a particle volume, v.There are cases where a single size co-ordinate can completely and un-ambiguously describe a particle.It is more common, however, for descriptions of particle shape, composition and even age to be required for a complete description.Each of these descriptions is, after the work of Hulburt and Katz 1 l, termed an internal co-ordinate.The properties of a particle may then be described as a list, or vector, of internal co-ordinate, Xi.
The other properties of a particle that might determine the rates with which it changes are its co-ordinates in physical space (x,y,z) -its external co-ordinates,

Groups of particles
A description of the rates of particle processes is of necessity a description of many particles at once.In so doing we most naturally seek a statistical description of a very large number of particles; so large in fact that we may assume that the distribution of particle properties is continuous and so we may use continuous probability density functions to describe how these properties are distributed.
In this work we use number density functions, n, and mass density functions, w.These relate the differential number or mass of material in a differential region of phase space as dN = n(x, t)dx and d W = w(x, t)dx For clarity, consider the simplest case where the only phase-space co-ordinate is size: the number of particles with sizes in the range (1, l + dl) is dN = n(l)dl.
The usual expectation when using probability density functions is that they will be normalised in such a way that integrating them over the domain of their arguments leads to a value of one.It is quite common, however, when modelling particulate systems, to allow some other value that contains information on the amount of material present.For example, in the study of crystallisation, it is usual to use a basis of 1 m 3 of suspension, so that dN in equation (1) is the number of crystals per unit volume of suspension in the indicated size range.

The Population Balance Equation (PBE)
The PBE is described by many authors -most notably Hulburt and Katzll, Randolph and Larson2l, and Ramkrishna 3 l -as a statement of continuity, or of conservation, for particulate systems.Just as conservation statements lie at the heart of descriptions of chemical rate processes, so they do in the description of particle rate processes.
At every point in phase space, the local rate of destruction of particles r(x, t) must exactly balance the local rate of accumulation and divergence in the flow field.The PBE is then an(x, t) -a-t-+ Y'•(u(x,t)n(x,t)) + r(x,t) = 0 (2) In this equation -termed the micro-continuity equation by Randolph and Larsonu is the vector of velocities in phase space.It is this equation that must be used in systems where there is dependence on spatial position, i.e. on the external co-ordinate position.If, however, the system under consideration is sufficiently well mixed for there to be no spatial dependence, equation (2) can be averaged over the extent of the external co-ordinates to yield the so called macro-continuity equation: where the Qk are flow rates of streams in to or out of the well-mixed region, which has a size V.If the basis used for n is 1 m 3 of suspension, the Q will be volumetric flow rates and V the volume.

The Rate Processes
Of the many processes that might change the behaviour or characteristics of a particle we concentrate here on those that will change the particle size distribution.In so doing, we are already making the implicit assumption that the particles are described by a single internal co-ordinate i.e. size.
Once this assumption is made, we need only consider four distinct mechanisms: nucleation, growth, aggregation and breakage.

Nucleation
Nucleation is the process by which new particles are formed from matter that is recognised as non-particulate.In fields such as crystallisation, precipitation or aerosol science, this usually entails the formation of a new particle from some clustering together of ions or molecules.In some other fields, such as granulation, it is allowed that the very fine primary powder is not, in fact particulate, and grains of this powder must clump together, i.e. nucleate, before they can be reckoned particulate.
Quantitative descriptions of the rate of nucleation date from the time of Gibbs 4 l; in general these take account of the energy barriers that exist to generating new particles and then compare this barrier with the energy available.
The rate at which this process takes place is given by the nucleation rate B 0 (m-3s-1 ) i.e. the rate of appearance of particles per unit of, typically, bulk volume.This rate must then be combined with information on the size distribution of nuclei, fN.It is quite common to assume that the nuclei will be mono-disperse in size so thatfN= 8(1-lo) where 8 is the Dirac distribution and lo is the size of the nuclei.In general however, the appropriate rate term for inclusion in the PBEis (4) Growth is the process by which non-particulate matter becomes incorporated within a particle.Since the distinction between particulate and non-particulate matter is that the former is available in very small units only, growth must of necessity be a differential process.The growth rate of a particle can then be described by a rate of change of size; as such the growth rate is simply a velocity along the particle-size axis.So if linear dimension is used to measure size, the growth rate is Gi(Xe, l, f) = .!!:!:._ (5) dt

and if volume
Gv(Xe, V, t) = dv dt (6) Values for these growth rates are usually deduced from rates of mass transfer or rates of surface reactions and are well known for very many types of systems.In those systems where the rate of reaction (per unit surface area of particle) is independent of size, the linear rate of growth defined in equation ( 5) is independent of particle size -which makes for considerable analytical convenience when seeking solutions to the PBE.

Aggregation
Aggregation is the other size-enlargement mechanism; new large particles are formed by combinations of smaller particles.By contrast with nucleation and growth, aggregation is a poorly understood phenomenon.It is usual to assume that aggregation involves only two-body interactions and so may be described by a second order rate constant, or kernel, as it is often known.The rate of aggregation events, per unit volume, for particles of types 1 and 2 is given by (7) where N1 and Nz are the number concentrations of particles of each type.By restricting the internal co-ordinates to particle size, using volume as the measure of size, this definition may be integrated over appropriate size ranges to give the net rate of aggregation Little is known about the prediction of aggregation rate constants.In some applications it is often assumed that the rate of aggregation is independent of particle size, while in others quite complicated KONA No. 16 (1998) empirical dependencies are fitted to experimental results.Two well-known functional forms are those of Von Smoluchowski 5 l for ortho-and for peri-kinetic aggregation.

Breakage
Breakage is that set of processes that reduce the size of a particle while producing more, smaller particles.The description of the rate of breakage requires definition of two functions: the selection rate constant S(x, t) with units of s-1 and a breakage function, b (Xe, Xi, new, Xi, old, t). 5 is SOmetimes termed the specific rate of breakage and is, in fact, the first order rate constant for the selection of particles.The breakage function, b, gives the distribution of "daughter" particles formed when a "mother" particle is broken.The subscript "old" denotes the co-ordinates of the mother particle and "new" those of the daughter particles being born.It is worth noting the convention that the first set of internal co-ordinates is those of the daughter particles and the second that of the mother.
Once more, it is very unusual to use more than size as the measure of the internal co-ordinate, in which case the rate of breakage may be reckoned as For conservation of mass (or more strictly, volume, as equation ( 9) uses volume as the internal co-ordinate) it is necessary that It is possible to define forms of b corresponding to certain kinds of limiting behaviour.For example attrition, the process by which a single small daughter particle of volume vo is broken off a mother particle has Uniform fragmentation where the mother particle is broken entirely into daughter particles of size vo has v There is little theoretical basis for the choice of either Sorb.

Using the PBE
There are very, very few useful analytical solutions to the PBE.Indeed, the author knows of no solution when all four mechanisms are active, even when the most simplistic assumptions are made for the various rate constants.It is frequently necessary, then, to use numerical methods; these fall into two broad groups: finite difference techniques6.?,SJ (so-called discretised population balances, DPBs) and finite element meth-ods9•10l.It is not the intention here to explore these numerical methods, but simply to report that they exist and that it is now possible to solve PBEs (with one internal co-ordinate and at most one external coordinate) relatively routinely.
By solve, here, it is meant simulate a size distribution; i.e. given initial and/ or feed information and a complete set of kinetic parameters, PSDs can be calculated as functions of time or spatial position.The so-called inverse problem presents a far greater challenge.In their full form, inverse problems require that the kinetic parameters be determined as functions of size and time from experimental measurements.Muralidar and Ramkrishna 11 l show that even with quite elaborate treatment, data of an exceptional, probably unattainable, quality are required.
It is, however, possible to address a much reduced form of the inverse problem.Bramley et al 12 l have recently shown that if the dependence on size of the kinetic parameters is known, it is possible to extract the dependence on time, spatial position or some other driving force.

Case Study 1. Crystallisation: Aggregation During Precipitation From Solution
The author's group has for approximately 10 years studied the aggregation of calcium oxalate monohydrate crystals while they grow in supersaturated saline solutions.The processes active in this case are growth and aggregation; so, for the well mixed batch systems studied, the PBE is The purpose of this study is to develop knowledge of how G and ~ depend on particle size and on solution conditions, such as supersaturation (which accounts for the dependence on time).

Experimental Detail
The experimental techniques employed is an adaptation of that of Ryall 1 3l.A saline metastable (i.e.supersaturated, but non-nucleating) is prepared by mixing NaCI, Na2C204 and CaC]z solutions with buffer at 37oC.At t = 0, a thermostated agitated flask of the metastable solution is inoculated with a small 182 volume of seed crystals.There after small volumes are extracted at regular intervals for particle size analysis by Coulter Counter.

Typical Results
Two key results are shown in Figures 1 and 2. In Figure 1, the size distribution of crystals at the start and at the end of a typical one hour experiment are shown.It is clear from the location of the curves that the crystals have increased in size; from the area under each curve it is also clear that the number of crystals has decreased substantially.This immediately suggests that crystal aggregation is occurring.In Figure 2 -which shows results corresponding to an extended, 24 hour experiment -it is very clear that aggregates are present.
--' d X 2.510 11 --, ---------------------, 2.0 10 11 L510 11 LO 10 11   5.0 10 10 0.0 10° 2 /; . ' • 0 minutes • 60 minutes 3.3 Data Extraction, Modelling & Simulation For a batch system in which growth and aggregation are active, the PBE is given by equation (10).We expect that growth will be size independent, but have little knowledge of how the aggregation rate will depend on size.The method we adopted 14 l uses an iterative technique in which we first assumed a size dependence of the aggregation rate, then extracted the values of the growth rate G1, and the size independent portion of the kernel, ~o.using the method of Bramley and Hounslow 12 l.We then simulated the size distribution using the method of Hounslow et at 6 >.We considered initially a great many kernels (some of which can be found in Hartel and Randolph 1 5l).Of these, some kernels could be eliminated immediately as they show gelling behaviour (as described by Smit et a]l 6 l).Of those that remained, the most frequently used kernels are those shown in Table 1.
We then characterised the ability of each kernel to describe our results by considering the sum of square error in predicting the PSDs.Complete results are tabulated in Bramley et a]l 4 l, but as Figure 3 ill us-Table 1 The three kernels used in Figure 3 Kernel, ~(11, /2) Physical rationale None; but Mumtaz et a/ 17 !show that when the effect of particle size on Size Independent, ~o collision rate and on collision efficiency is considered, only weak size dependence should be expected.:..-,,

L().lm)
Fig. 3 Predicted and experimental PSDs for three assumed forms ofthe aggregation kernel trates, the shear kernel is incapable of describing the results while the Brownian and size-independent kernels do quite well.In what follows, and in our further work, we adopt the size-independent kernel as giving the best fit and, as indicated in Table 1, as best corresponding to our recent a priori predictions 17 l.

The effect of process conditions
Having established that we can describe our experiments in terms of a size-independent growth rate, G, and a size independent aggregation rate constant, ~o, we then explored how these kinetic parameters depend on operating conditions.In Bramley and Hounslow 14 l, and Bramley et al 18 J we report the effect of changing supersaturation S, and initial ion ratio, ao.

Growth Rates
In Figure 4 the square root of growth rate is plotted against supersaturation.There is an excellent correlation between the variables from which it is apparent that growth rate is directly proportional to the square of supersaturation and is independent of the ion ratio.This result is entirely expected: being equivalent to the results of Nancollas et al 19 J.We conclude from this that the rate of growth is determined by the rate of reaction, or incorporation, of ions on the surface of the crystals.

Aggregation Rates
Figure 5, is the aggregation analogue of Figure 4. We see now that while the aggregation rate constant also increases with increasing supersaturation, there is very clear dependence of the rate on the ion ratio.' '(' Relative Supersaturation, S //+ ,// 12.0 J0-14 Relative Supersaturation, S Fig. 5 The effect of relative supersaturation and initial ion ratio on the aggregation rate constant On the basis of these data it can be seen that the further the ratio is from 1:1, the lower the rate of aggregation.Furthermore, it is apparent that this effect is symmetrical, so that a system with a ratio of 1:5 behaves the same as one with 5:1.These observations have been confirmed by Hounslow et a]1 8 l who considered ratios from 1:10 through to 10:1.These results are, to us, reminiscent of sequential chemical reactions of the first and second order.This observation led us to propose a pore diffusion reaction mode]l 8 l.In this model we propose that in order for two particles to aggregate they must first meet and then form a sufficiently strong bridge of new material, or cement as we have it, before they experience disruptive forces (induced by fluid motion).The process of laying down this cement involves diffusion (a first order process) to a cementing site (down some pore) and then reacting (a second order process) to form the new, solid, cement.
If we assert that the transport down the pore and the reaction at the cementing site are in pseudo steady state, we may write where rc is the rate of reaction in mol m-2 s-1 , the symbols T refer to total concentration of dissolved ions, D is the diffusivity, X is the pore length, k is the reaction rate constant and sc the relative supersaturation at the cementing site.From this equation we can deduce that the rate of deposition at the cementing site and the rate at the crystal surface are related by  -,-----,---,--~--.-----,--1O.OW' 2.0 10-4 4.0 10-4 6.0 10-4 8.0 1Q-4 Driving force at cementing site r• (S-1)' Fig. 6 The dependence of aggregation rate constant on the calculated driving force at the cementing site where k* is a Thiele modulus.
Figure 6 shows the results of Figure 5 plotted now against the calculated value for the driving force at the cementing site.The results have now collapsed onto a single curve showing no dependence on the ion ratio.

The effect of agitation
The results shown above were obtained in simple shaken flasks; those reported in Hounslow et a]1 8 l, were collected using a one litre baffled stirred tank vessel.For that system we find that the results are described by a Thiele modulus of k* = 0.3, rather than the diffusion limited value of k* = =.Our most recent work, Barrick 20 l, shows that for the stirred vessel, the value of k* is very strongly dependent on the stirrer speed.

Observations on crystallisation
This case study represents, by far, our most mature work.We now very routinely use the PBE -equation (10), in this case -to extract the growth rate G1 and the aggregation rate constant ~o as functions of time.Since we are also capable of determining the solution composition and therefore supersaturation, as functions of time, we can very easily explore the kinetics of both processes.The whole thrust of our work is the investigation of how these kinetics change under various conditions.This approach of using the PBE to reduce extensive experimental data to just a few fundamental kinetic parameters is the perfect illustration of what is meant by the title of this paper: The Population Balance as a Tool for Understanding Particle Rate Processes.

Case Study 2. Fluidisation: Spray Coating of Grass Seeds
Of the three case studies presented in this paper, the first represents approximately a decade's work and the last approximately three year's.By contrast the analysis presented here stems from about two week's work.
The problem to be considered is one of coating grass seeds with a layer of fertiliser in a bottomsprayed spouted fluidised bed -as shown in Figure 7. Details of the experimental arrangements can be found in Litster et aF 1 J.The aim of the work is to predict first how the distribution of particle sizes varies with time and then to consider the variability of the coating thickness.
Fluidising Air Fig. 7 Schematic of the seed coating device We adopt a simple model: we take the bed to consist of a plug flow "coating zone" representing the spout where we expect to find droplets of binder, and a well mixed "bed"; as shown in Figure 8.

Plug flow model for the coating zone
The coating-zone model will, in due course, be required to relate the size distribution of particles coming into the zone (from the bed) to the distribution leaving the zone and passing back to the bed.In order to do so an expression for the population density is required as a function of height, x, in the zone, and of particle size, in this case the particle volume, v.In principle this would require a description of the aggregation of binder droplets and particles.Rather than proceed along these lines, we treat the binder as non-particulate developing an effective rate of growth of the larger particles.To do so we now develop a binary population balance; we consider two populations, population 1 being the spray droplets, and 2 the seeds.We then allow that the numbers of particles per unit volume are N1 and Nz.
We first calculate the rate of collisions per seed: where the velocities U and V are as shown in Figure 9, U has been taken to be much greater than V, and az is the projected area of a seed.The population balance for droplets may then be reckoned as where where £ is the void fraction in the coating zone.In this last result we see that the concentration of spray droplets decays exponentially with a decay length proportional to the diameter of the seeds and inversely proportional to the volume fraction of seeds.We deduce from this that collisions between seeds and droplets occur only over a few, perhaps 10, particle The coating zone diameters of length.This observation requires that attention be paid to the region at the very bottom of the bed where we expect the particles to have relatively low velocity.The population balance for the coating region may now be deduced by application of equation (2), with two co-ordinates (x, v) and r= 0. a a (dxn(x,v))+j!_(dvn(x,v))=o(11)

X dt av dt
In what follows, it is more convenient to work with a basis for the population density of 1 second rather than 1m 3 , as naturally arises in equation (11).Defining n (x, v) as the number density passing a point at a height, x, per second, equation (11) becomes an(x, v) + j!_ ( dv n(x, v)jv) = 0 (12) ax av dt To calculate the dependence of particle velocity, V, on position, the equation of motion, allowing for size and slip dependence of drag, was solved numerically to give The velocity the size axis may be calculated as Equation ( 12) becomes ( 14)

an(x,v) kv 0 • 87 e-A,x an(x,v)
where k is a combined constant of proportionality from equations ( 13) and ( 14).This equation can be solved with considerably greater ease if the exponent of v, 0.87 is replace with a value of 1, in which case the PBE is an kve-A,x an kve-A,x which has the solution n(x, v) = iz ( 0, ve-kert(.JA;X))e-kerf(.JA;X) (15) If we allow that the length of the coating zone is much greater than the decay length of the droplet concentration, the error function in this last equation tends to 1, so .() izb(v/K) nz X = K (16) where K = e-k.
Consequently, for each pass through the zone a particle's volume increases by a size-independent factor: the coating parameter, K.We can now relate the number densities entering and leaving the coating zone by

2 Two-compartment population balance model
We now write a PBE for the well mixed fluidised bed, and then combine it with that for the plug-flow coating zone.We take equation (3), allow that there is no change in particle size in the bed, and use a basis of unit time to obtain anb + i!±_ dVs = izz-izb at Vs dt Vs/ Qs (18) where Vs is the volume of solid in the bed and Qs is the volumetric flow rate of solid from the bed into the coating zone.Combining this last result with equation ( 17) yields Vs dt K Vs Defining the moments of the size distribution as mb,j = f~vjnb(v)dv allows (19) to be transformed to Before proceeding, some assumption concerning the flow of material from the bulk to the coating zone is required.The obvious choices are constant number flow of particles or constant mass flow.Since it is likely that the rate of entrainment is dependent on momentum transport from the gas to the particles, it seems most appropriate to assume constant mass flow.In which case, the first moment leaving the bulk will be constant.From (20), with dmbd dt = 0, it follows that We may now solve (20) Preliminary calculations, and common sense, suggest that K « 1.In which case (23) becomes cr(t) = ( Vs~~)) cr (0) (24) Quit unusually, it is also possible to solve the PBE itself, (19).If the initial condition is nb(v, Vs(O)) = izt(v), some time later the size distribution in the bulk is v; .:.::..!S... izb(V Vs) = ( -5 -)K-l.
Although deduced directly from the population balance, the form of this solution and some of the terms it contains may be given physical significance.Each time a particle passes through the coating zone its size increases by a factor, K.It follows, that in determining the number of particles currently of size v, one may add up the numbers of particles that have been through the coating zone no times, once twice etc. (as counted by J) which initially had sizes v, v/K, vi K 2 etc.The maximum number of passes through the coating zone CPmax) is the largest integer satisfying Vmin::::: v/ KPmax.The remaining terms account for the probability of a particle having passed through the zone j times and normalisation of the density function to maintain a constant third moment.

Comparison with Experiment
The striking thing about the predictions of the previous section is that according to equations (22) and (24) the mean size and standard deviation are directly proportional to the volume of solids added and further, the model has no adjustable parameters.
KONA No.16 (1998) Figure 10 shows results from 14 different experi-ments22), from which we may conclude that the model gives a good description of the mean and standard deviation.
Further evidence of the quality of the model may be obtained by comparing an observed and predicted size distribution; as shown in Figure 11.Here equation (25) has been used with K = 1, so that once again, the prediction is free of adjustable parameters_ Both the position of the curve shown in Figure 11 and its shape depend only on the amount of solid added.
For most problems in Particle Technology, predictions of the quality shown in Figure 10 and Figure 11 would be considered very strong verification of the model proposed_ Hawever, in this case it is also possible to test the model further by considering the thickness of coating applied to each seed_ It is not shown here, but by means of a two-dimensional population balance, it is possible to show that taking a value of K = 1 requires the coating thickness (expressed as a volume of coating material) to depend only on the initial seed size for each particle.A plot of coating volume versus seed size (such as Figure 12) should show no dispersion.Quite clearly, the model prediction is not verified in this case_ What is seen is that the amount of coating a seed of a given size receives depends quite strongly on the seed's initial size (as predicted) but also on some random variable.This is likely to be the number of passes through the coating zone and indicates, perhaps, a more detailed description of how particles flow from the bulk of the bed into the coating zone is v (mm 3 ) Fig. 11 Predicted and observed particle size distribution for spray coating.The prediction is made using equation (25), and is free of adjustable parameters.
required.Prediction of coating thickness is of some importance in this type of equipment as it is often desirable that uniform coatings be achieved.Two points of importance emerge here: a complete model requires better knowledge of the flow pattern and the particle's state needs to be described by two internal co-ordinates (seed size and coating size).

Observations on spray coating
The development of the PBE in this case study took place while Dr ].D. Litster was on sabbatical leave from the University of Queensland, Australia, at the University of Cambridge.We sought first to explain how size increase proceeds -we conclude that it is by means of a constant coating factor, K-and then to predict the overall size distribution.Both of these objectives were quickly met, but we came to the conclusion that explanation of the discrepancies shown in Figure 12 would require considerable additional experimental work.

Experimental
We have studied the granulation of finely divided calcium carbonate powders (mean size 15 J.!m) with the addition of liquid polyethylene glycol binder (PEG 1600) at 50°C in a Fukae 30 litre high-shear mixer.For further details see Knight et a/ 2 5) and Pearson et at 2 6l.Samples of granules are removed from the mixer at regular intervals, cooled to solidify the binder and then sieved on a fourth-root-of-two series of sieves.
We also conduct tracer studies, in which additional granules -marked with blue dye -are added to the mixer, typically eight minutes after commencement of operation.The distribution of dye with time and size can then be followed by means of sieving and spectrophotometry to determine dye concentrations.

Key Results
The two most important results are that the PSDs observed are clearly bi-modal and that dye added in large tracer particles rapidly becomes distributed to much smaller sizes.This last point is most readily illustrated by considering the fraction of dye that remains in particles larger than the smallest tracer particle added.An example of these data is shown in Figure 13.It may be seen that the fraction decays exponentially with a time constant of approximately 1000 s.

Modelling
The standard approach to PBE modelling of granulating systems is to assume a functional form of the aggregation kernel -i.e.how it depends on granule size, and perhaps time -and then to choose parame- ters for the kernel (perhaps exponents of the size dependence) so that the model fits the data well in some least-squares sense.In so doing the differential equations of the PBE are solved -integrated, as is often said -using varying values of the parameters until the fit is optimal.
The approach adopted here is a different one.We take the differential approach that we developed for our crystallisation work (Bramley et al 12 )) which requires a priori knowledge of the size dependence of the rate constants and extracts the time dependence.At this early stage of our work we simply propose, from first principles, the forms of the size dependencies and asses their performance.The two mechanisms to be considered are size enlargement by aggregation and size reduction by breakage.

Aggregation
Visual observation of the bed of granules in a highshear mixer reveals that while the granules adopt significant rotational velocities they also display noticeable deviations from the local average velocity.This may be likened to turbulent flow of a fluid or to molecular motion in a gas.
Consider the rate of collisions between particles of size l1 and lz, each with time varying velocities V1 and Vz.We take Smoluchowski's approach and write Rate of collisions= Collsion cross-section x average relative speed oc (II +lz) 2 1Vl-Vzl We borrow from the study of turbulence and write that V(t) = V + V' (t), where we assume that the average velocity of a granule does not depend on its size (being the average tangential velocity of the bed of solid) but that the random component does.Making this assumption allows In the kinetic theory of gases it is assumed that kinetic energy is equally distributed between differing molecular species, so that the square of random velocity is inversely proportional to the molecular mass of the species.That assumption couples naturally with the assumption that all collisions are elastic and that all energy is manifested as translational kinetic energy.In the current situation such an assumption seems rather implausible.Instead we assume that each granule is subject to the same random fluctuating impulses (in the mechanical sense) so that the random component of translational momentum will be constant, and that the random component of velocity will be inversely proportional to a granules mass, or the cube of its size V' oc 1/[3 Combining these last results, and allowing that the aggregation rate constant and the rate of collisions differ by an efficiency that will at least be a function of operating conditions, and thus time, we write ~(II,lz,t) = ~o(t)(l1+lz)2 j 1 ~ + 1 ~ (26)

Breakage
To define the breakage rate we need to specify a breakage selectiort function S(l, t) and a breakage distribution b(ll,{z).Our tracer data indicates that the breakage rate is not strongly size or time dependent and from the slope of the line in Figure 13, S = 0.001 s-1 .We attribute the presence of the small-size mode to the breakage process and make the assumption that breakage proceeds by fragmentation.We then take b to be a log-normal distribution with a massweighted geometric mean size of 125 11m (the location of the samll-size mode) and geometric standard deviation of 1.15 (the width of that mode).

Simulations
Figure 14 below shows how well the PBE is able to predict the observed size distributions.
The input to the simulation is the dependence of the aggregation rate constant, ~o. on time.The values for ~o.extracted from the measured PSDs, and the curve used in the simulations are shown in Figure 15.
These data, then, suggest that the aggregation rate constant increases with time; which must almost  Fig. 14 Measured and predicted mass density functions certainly mean that some property of the granules is changing with time.The current focus of our research is the identification of what that property might be.We conclude, once again, that the behaviour of each particle is governed by more than just its size -that a physicaily sound model must incorporate additional internal co-ordinates.

Observations on granulation
Our experiments and population balance analysis leave us confident that breakage is important in this system.We have proposed that the rate of breakage is independent of time and size and that the fragments formed are nearly mono-disperse.We have developed a coilision model to predict the effect of particle size on aggregation rate.

Conclusions
It is possible now to solve population balance equations routinely when particles can be described by one internal co-ordinate (i.e.size) and at most one external co-ordinate.In the three case studies reported, size was variously characterised as equivalent volume diameter, particle volume and sieve size.In the second case study it was ailowed that the size distribution varied with position in the coating zone of a spray coating apparatus.
It is also possible to extract rate constants, by means of PEEs, provided the size dependence of the rates is known.This approach was used in the first and third case studies.It is perhaps this activity that can reveal most about the processes under study.For in this way ail of the behaviour of a system -or at least, such of it as is captured by the model -is reduced to a simple list of rate constants.In the first case study it was shown that simultaneous growth and aggregation can be described by standard supersaturation dependence of the growth rate, but that a rather more elaborate description was needed for aggregation.Without taking this rate-based approach, it would have been quite impossible to interpret our experimental data.The second case study, however, revealed that the PSDs produced are entirely independent of any rate constant; the spray coating system behaves as the analogue of the equilibrium chemical reactor, the product being determined in this case solely by the amount of material sprayed on.In the final case study a second internal co-ordinate (dye concentration) was added to the experimental data (but not, at this stage, the model) in order to identify the active mechanisms and determine their rates.
It is surely noteworthy that ail of the modelling reported here has one common flaw.In every case it has been shown that description of particles by one internal co-ordinate -size -is, eventuaily, inadequate.In the first case study, it eventuaily becomes apparent that some internal structure of the aggregates (presented here as a Thiele modulus) determines the rate of aggregation.In this case it seems possible to assume that ail particles have the same value of the modulus, so no difficulty is encountered.In the second case study, inspection of a second internal co-ordinate (coating size) revealed that the model was not capable of predicting the observed dispersion in coating thickness -despite the exceilent predictions of the overail size distribution.In the last case study it was shown that the aggregation rate constant is strongly dependent on time -which must arise out of some changing property of the granules other than their size.
The need to represent a particle by more than just its size is a reflection that particles are heterogeneous and that the heterogeneity is in general not uniform over different particle sizes.A different kind of heterogeneity, this time external to the particles,imperfect mixing -was also suspected of being important in the eventual shortcomings of the model in the second case study.It is these areas -heterogeneity within and without particles -that are the focus of our current research in population balance techniques.

Fig. 1
Fig. 1 Number density functions of the seed and of the product after one hour Brownian motion,Corresponds to perikinetic motion as ~ox (11 +i2ll( i+ L) described by Von Smoluchowski Shear, ~ox (/1 +12) 3 Corresponds to orthokinetic motion as described

Fig. 12
Fig.12 The coating size as a function of initial seed size.The model incorrectly predicts no dispersion of coating size for any given seed size.

0. 4 Fig. 13
Fig.13 The fraction of dye found ia granules of size greater than or equal to the size of tracer granules added

Fig. 15
Fig.15 The dependence of the aggregation rate constant on time