1

Crystallization Process Modeling

Marco Mazzotti,1 Thomas Vetter,2 David R. Ochsenbein1

1.1

Introduction

The properties of crystalline products are not only defined by their compositionand crystal structure, but also by their size and shape. The influence of thesefeatures is particularly important for downstream processing operations, suchas filtration, drying, milling, granulation, blending, etc. For instance, one cansurmise that thin needle-like crystals are more prone to break in an agitateddryer than compact crystals of the same material, or that the time required toseparate mother liquor from crystals by cake filtration depends on how tightlycrystals in the filter cake are packed, which in turn depends on the crystal sizeand shape.

Crystals owe their characteristic sizes and shapes to an interplay betweencrystal structure, thermodynamics and kinetics – in short: to some inherentproperties and to the process they were manufactured in. In fact, at the levelof individual crystals, it is a particle’s history, the environments and events ithas encountered, that determines those features. Given the variety of possibletrajectories within a process, it is no surprise that crystals also exhibit adiversity of sizes and shapes, typically described by a particle size and shapedistribution (PSSD).

Simple approaches to modeling crystallization processes, e.g., a yield calculationbased solely on thermodynamics, are not able to successfully describe, much lesspredict properties that are connected to the crystal size and shape distribution.

1Institute of Process Engineering, ETH Zürich, Switzerland2School of Chemical Engineering and Analytical Science, University of Manchester,

United Kingdom

1

CHAPTER 1. CRYSTALLIZATION PROCESS MODELING

Yet, as illustrated in the drying and filtration example above, this capabilitywould be highly desirable. The remainder of this chapter is therefore focused onintroducing modeling concepts for crystallization processes that allow keepingtrack of the properties of the liquid phase and of the solid phase, specifically thePSSD. After briefly deriving the underlying concepts in the following paragraphs,selected case studies showcasing applications of this modeling methodology arepresented. Among the examples covered are polymorph transformations, crystalgrowth and agglomeration rate estimation, as well as examples of model-basedprocess optimization.

1.1.1

Population Balance Equations

For the purpose of an accessible introduction to crystallization process modeling,we first assume that crystals can be described by a single characteristic lengthL. This implies the existence of a one-dimensional particle size distribution(PSD), denoted here as f(L). Formally, f is a number density function, sothat f(L)dL corresponds to the number of crystals per volume of suspensionwith characteristic lengths between L and L + dL. The evolution of thisdistribution over time can be described using population balance models [1–3].The corresponding equations account for changes in the number of particleswithin a given control element, i.e., they describe how many particles are in thecontrol element, how many are entering it, and how many are leaving it. Toillustrate this concept, we derive the population balance equation (PBE) for anidealized tubular crystallizer with constant cross-section A (cf. Figure 1.1(a)).We will then show how this model can be modified to apply also to continuousstirred tanks as well as batch crystallizers.

In the tubular crystallizer considered here, we assume that its content is per-fectly mixed in radial direction, but that no mixing in axial direction occurs.Collectively, these assumptions lead to a “plug flow” behavior, however wehighlight that these simplifying approximations are by no means a necessarycriterion for the modeling framework, i.e., non-idealities could easily be ac-counted for if necessary. Regardless, here, the PSD is hence not only a functionof the time t, and of the internal coordinate (characteristic crystal size) L, butalso of the external coordinate (the position along the crystallizer axis) x, thatis, f(t, x, L). We consider a control element stretching from x to x+ ∆x andstretching from L to L+ ∆L (cf. Figure 1.1). In the external coordinate thecontrol element can be visualized as a slice of the plug flow crystallizer (drawnas the green disk in Figure 1.1(a)), while the internal coordinate is not visible

2

1.1. Introduction

a) b)

flow in, Q

x x+Δx

Aflow out, Q

x

external

coordinate x L

L+ΔL

x+Δx

Particles

entering due to

growth

Particles in

control element

A Δx ΔL f

Particles

leaving due to

growth

Particles

leaving due to

flow

Particles

entering due to

flow

flow in, Q

x x+Δx

Aflow out, Q

x

external

coordinate x L

L+ΔL

x+Δx

Particles

entering due to

growth

Particles in

control element

A Δx ΔL f

Particles

leaving due to

growth

Particles

leaving due to

flow

Particles

entering due to

flow

Figure 1.1: a) Conceptual drawing of a plug flow crystallizer with controlelement highlighted as a slice of the reactor in green; b) drawing of the controlelement and fluxes into and out of it.

in this representation. Acknowledging this, the control element is redrawn inFigure 1.1(b) to visualize both internal and external coordinates. A populationbalance equation is obtained by accounting for all fluxes, drawn as arrows inFigure 1.1(b), and possible source or sink terms. In abstract terms, we canwrite for the control element:

Accumulation = In−Out + Birth−Death (1.1)

Fluxes in x-direction represent the transport of fluid and particles along theaxial coordinate of the crystallizer, while fluxes in L-direction account for crystalgrowth or dissolution. Nucleation, breakage or agglomeration are importantexamples of mechanisms that can be described using birth and death terms.For the case of a plug flow crystallizer with volumetric flow Q, Eq. (1.1) can

3

CHAPTER 1. CRYSTALLIZATION PROCESS MODELING

be written as:

A∆x∆L([f ]t+∆t − [f ]t

)= Q∆L

([f ]x − [f ]x+∆x

)∆t

+A∆x([Gf ]L − [Gf ]L+∆L

)∆t

+A∆L∆x(B −D)∆t

(1.2)

where we have deliberately omitted the arguments of f as well as those of thegrowth (or dissolution) rate G for brevity’s sake. The subscripts in Eq. (1.2)indicate at which point the terms in square brackets have been evaluated. Bydividing Eq. (1.2) by ∆L, ∆t, ∆x and A and by letting ∆L→ 0, ∆t→ 0 and∆z → 0, we obtain

∂f

∂t+ Q

A

∂f

∂x+ ∂ (Gf)

∂L= B −D (1.3a)

However, the above considerations do not yet describe changes in concentrationin the liquid phase, a crucial property due to its influence on the driving forcefor nucleation and growth. The necessary material balance for the solute yields:

∂c

∂t+ Q

A

∂c

∂x= −dmc

dt (1.3b)

where mc is the crystal mass per volume of suspension, typically given bymc = kvρcµ3, where kv is a shape factor and ρc is the crystal density. Here, µ3is the third moment of the particle size distribution, in general defined as

µi =∫ ∞

0Lif dL. (1.4)

Eq. (1.3) forms the general description of a plug flow crystallizer with knowntemperature profile, both in transient phases as well as during steady state. Ina similar manner, the equations describing a (well-mixed) continuously stirred

4

1.1. Introduction

tank reactor (CSTR) with volume V can be derived, yielding

∂ (fV )∂t

+ V∂ (Gf)∂L

= +V (B −D) +Qinfin −Qf (1.5a)

d (cV )dt + d (mcV )

dt = Qincin −Qc, (1.5b)

which in the case of clear input stream (fin = 0) and at steady state re-duces to the well-known mixed suspension mixed product removal (MSMPR)formulation:

∂ (Gf)∂L

+ Q

Vf = B −D (1.6a)

Q

V(c− cin) = −Q

Vmc. (1.6b)

Notably, Eq. (1.6b) is not a differential, but only an algebraic equation. Finally,we consider the case of a batch crystallizer, which we find to be described by:

∂f

∂t+ ∂ (Gf)

∂L= B −D (1.7a)

dcdt = −dmc

dt (1.7b)

In order to solve any one of Eqs. (1.3) and (1.5) to (1.7), additional informationis needed. First, a set of initial and boundary conditions is required; for thecase of the batch crystallizer and assuming zero-sized nuclei these can, forexample, be written as:

f(t = 0, L) = f0(L), f(t, L = 0) = J

G(1.8a)

c(t = 0) = c0 (1.8b)

5

CHAPTER 1. CRYSTALLIZATION PROCESS MODELING

where J is the rate of nucleation and f0 and c0 are a seed distribution and theinitial solute concentration, respectively. Second, we need some knowledge re-garding the constitutive equations that describe the kinetics of the system, thatis, we need expressions for J , G, B and D. This requires some understandingof the underlying phenomena which is often not trivial to obtain, but has beenaccomplished—at least to some degree—for many of the major crystallizationmechanisms [4], e.g., nucleation [5, 6], growth [7–9], agglomeration [10–13] andbreakage [14, 15]. For the sake of simplicity, here, we assume that the necessaryexpressions are available.

1.1.2

Notes regarding Population Balance Models

Energy balances and Fluid DynamicsThe models presented in Section 1.1.1 represent useful descriptions in thecase of comparably slow crystallization processes, whose temperature can beadequately controlled by some low-level feedback controller. For fast processesor those that are strongly exo- or endothermic, an additional heat balance whichis coupled to the other equations is necessary for a complete model. Nevertheless,it should be noted that the assumption of perfect temperature control is oftena reasonable approximation, particularly for organic compounds grown at lowsupersaturations, as is often the case in pharmaceutical production.

In a similar vein, the assumption of well-mixedness, be it partial (e.g., in theradial direction in Eq. (1.3)) or complete (cf. Eqs. (1.5a), (1.6a) and (1.7a))may be violated for systems where uniform mixing is difficult (e.g., largetanks) or where crystallization occurs with very short characteristic times(e.g., precipitation). In these cases, mixing aspects need to be taken intoconsideration explicitly, resulting again in more complex descriptions of theprocess [11, 16, 17].

Solution of population balance equationsAn application of the above models requires an accurate solution of the set of(integro-)partial differential equations derived above. Unfortunately, analyticalsolutions are only available for the simplest cases and in general numericaltools are necessary to compute model outputs. Fortunately, there exists avast literature on the fast and efficient numerical solution of PBEs [18–21],together with various strategies to reduce the complexity of the resulting modelequations making simplifying assumptions. Regarding the latter, particularlythe various method of moments that have been developed deserve mention[1, 22].

6

1.2. System Characterization and Optimization

ApplicationsThe population balance models outlined in Section 1.1 represent a flexibleframework to describe particulate processes and can be useful for a variety oftasks. For instance, it is possible to characterize systems whose behavior hasnot yet been identified by fitting parameters in population balance models toexperimental data; we present examples for this application in Sections 1.2.1to 1.2.3. Once these kinetics are known, processes can be optimized usingcomputational studies (cf. Section 1.2.4) and become candidates for model-based control strategies, such as model predictive control. In addition, suchsystems can also be realistically investigated on a process design level, allowingthe analysis of different flowsheets in terms of, e.g., reachable sets [23]. Finally,extensions to systems with multiple internal states (e.g., multiple characteristicsizes) are possible and represent an important new research direction, ashighlighted in Section 1.3.

1.2

System Characterization and Optimization

Identifying the kinetic parameters that form part of the constitutive equations inthe population balance model is vital in order to obtain a truly predictive modelof a process. To some extent, this can be done through independent experiments,whose goal it is to extract information about the rate of individual mechanisms.Important examples are induction time (nucleation rate) measurements aswell as growth rate studies, which may be conducted using setups that aresubstantially different from a standard crystallizer.

However, due to the complexity of the process, it may sometimes be moremeaningful to estimate kinetics from experiments that are closer to how thecrystallization process would be carried out in production, thereby taking intoaccount nonidealities that occur due to imperfect mixing, particle-particleinteractions, etc. A generally applicable pathway for system identification isto run simulations and then compare the model output to experimental data(cf. route A in Figure 1.2). By defining an objective function—for examplethe sum of squared residuals between model predictions and experimentalmeasurements—and embedding the process model in a higher-order optimiza-tion routine, the difference between experimental data and model outcomecan be minimized through iterative adaptation of the kinetic parameters. Thenecessary experimental data is typically acquired by using process analyticaltechnology (PAT) tools, which permit on-line monitoring of the continuousphase (e.g., the solution concentration through infrared spectroscopy, Raman

7

CHAPTER 1. CRYSTALLIZATION PROCESS MODELING

spectroscopy) and/or the particles (e.g., the particle size distribution or someproperties of the PSD through Raman spectroscopy, focused beam reflectancemeasurements (FBRM), imaging probes, in-situ laser diffraction). While suchdata alone already helps to understand a process in greater detail, using them inthe PBE modeling framework allows drawing more in-depth and more generalconclusions regarding the process behavior.

Analogously to parameter estimation, one can optimize the outcome of aprocesses whose kinetics are already known, the main difference lying in thefact that the comparison is then made by comparing the model output withsome target outcome (B route in Figure 1.2) rather than the experiments, andthat the decision variables are related to the operating policy instead of thekinetic parameters.

Critically, it must be noted that—due to the nature of the problem—thereis generally no certainty supported by theory that any optimization (A orB in Figure 1.2) converges toward a local, much less a global optimum inreasonable time. Nevertheless, we will show in the following that, even withoutthis guarantee, valuable results can be obtained with this approach.

1.2.1

Crystal Growth

The characterization of systems under conditions for which crystal growth playsa dominant role is probably the simplest, yet also most important application forthe approach described in Section 1.2. One such example is given by Vetter et al.[7], in which the population balance model shown in Eq. (1.7) with r.h.s. zerowas used to fit the growth rate of ibuprofen to measured concentration profilesin seeded batch desupersaturation experiments. By applying the estimationprocedure to experiments at different concentration of a polymeric additive,the latter’s influence on the growth rate could be determined.

Likewise, Codan et al. [24] used a similar system of equations and experimentalprocedure to determine the growth kinetics of S-mandelic acid in the presenceof its counter enantiomer within the two phase region in water, indicating theapplicability of the population balance framework also for chiral systems. Thegrowth inhibiting effect of R-mandelic acid at various concentration levels wasquantified and its dependence on supersaturation demonstrated.

As for continuous crystallizers, similar strategies can be used to obtain infor-mation regarding the process kinetics. An important simplification occurs forMSMPRs: in the absence of agglomeration and breakage, the identification of

8

1.2. System Characterization and Optimization

Figure 1.2: Schematic overview of parameter estimation (A) and model-basedprocess optimization (B). Compound data (thermodynamic/crystallographicproperties, etc.), kinetic parameters and information regarding the operatingparameters are generally necessary for the simulation of a crystallization process.

9

CHAPTER 1. CRYSTALLIZATION PROCESS MODELING

both nucleation and growth rates simultaneously for a single operating condi-tion can be done rapidly by comparing the obtained particle size distributionwith the one computed from Eq. (1.6). In fact, under the above assumptionsand assuming the growth rate is size-independent, that equation possesses asimple analytical solution, given by:

fss(L) = Jss

Gssexp(−LQV Gss

)(1.9)

which indicates that, ideally, nucleation and growth rates at the steady statecan be computed from the y-intercept and slope of the line drawn by a plotof ln(fss) vs. L. An important discussion regarding the usefulness of suchexperiments is provided by Garside and Shah [25].

1.2.2

Polymorph Transformation

The modeling of solvent-mediated polymorph transformation can be achievedby extending the standard formulation in Section 1.1, which considers onlyone crystal species, to the case where there exist multiple solid state forms. Inparticular, population balance equations for all relevant species in the systemneed to be written and solved in parallel, taking into account the fact that theequations for the different solid forms are coupled through the concentrationof the solute, which is one and the same for all of them. For the case of awell-mixed batch reactor with m different polymorphs and no source or sinkterms, Eq. (1.7a) becomes

∂fi(t, L)∂t

+ ∂ [Gi(L, S)fi(t, L)]∂L

= 0 i = 1, . . . ,m (1.10a)

10

1.2. System Characterization and Optimization

Note that growth, dissolution and nucleation kinetics differ for different speciesand are typically expressed as functions of the corresponding supersaturationSi = c/c∗i . The associated material balance, too, is rewritten to account forchanges in the solution concentration due to the various polymorphs

dcdt = −

m∑1

dmc,i

dt (1.10b)

The system of partial and ordinary differential equations formed by Eqs. (1.10a)and (1.10b) is valid regardless of the underlying thermodynamics and can besolved in a similar way as in the case of a single species. A number of authorshave reported system characterization results using this model and its variants(e.g., for continuous systems) [26, 27].

In particular, Cornel et al. [28] have investigated the solvent-mediated poly-morph transformation from α to the monotropically stable β l-glutamic acid,that is, a system for which m = 2. In particular, they performed seeded experi-ments and solved the above equations to fit the secondary nucleation kineticsof the β form; nucleation rate of α l-glutamic acid as well as the growth anddissolution rates had been determined or estimated independently [29, 30].The results obtained from the seeded experiments were further used to predictthe behavior of the system in the unseeded case with acceptable success asillustrated in Figure 1.3. The model finally demonstrated its ability to forecasttotal transformation times for experiments starting from clear solutions atdifferent supersaturation levels.

It is important to highlight the role of the two spectroscopic techniques usedin that work: in situ attenuated total reflection Fourier transform infrared(ATR-FTIR) and Raman. The two PAT tools allowed insight into the mainmechanisms even before the rigorous kinetics determination. Namely, the factthat the dissolution of the metastable α form is not the rate-determining stepwas established through qualitative analysis of the data alone [31].

1.2.3

Agglomeration

The characterization of systems that exhibit effects besides nucleation andgrowth has been performed in literature as well. Focusing on the case ofagglomeration, it is convenient to first rewrite Eq. (1.7a) in the volume-based

11

CHAPTER 1. CRYSTALLIZATION PROCESS MODELING

Figure 1.3: Evolution of solid composition and liquid concentration over timefor an unseeded polymorph transformation experiment. Markers indicate exper-imental data (composition: Raman; solute concentration: ATR-FTIR), whilesolid lines show model fits. Reprinted with permission from Cornel et al. [28].Copyright 2009 American Chemical Society.

12

1.2. System Characterization and Optimization

form, where the characteristic length L is replaced by a characteristic volumev = kvL

3. In the case of a well-mixed batch reactor this yields

∂f(t, v)∂t

+ ∂ [Γ(v, S)f(t, v)]∂L

= B(t, v, S)−D(t, v, S) (1.11)

where the length-based growth rate G was further replaced by its volume-basedequivalent, Γ = 3k1/3

v v2/3G. Clearly, the mass balance as well as the initialand boundary conditions previously presented in Eqs. (1.7b) and (1.8) can beeasily rewritten to reflect this change in internal coordinate. If it is assumedthat agglomeration is an irreversible process, that is, agglomerated particlesare cemented together via a stable bridge that is strong enough to withstandall forces acting on it, the birth term can be written as (see, e.g, Ramkrishna[3] for a detailed derivation)

B(t, v, S) = 12

∫ v

0β(v − v′, v′, S)f(t, v − v′)f(t, v′)dv′ (1.12)

while the death term is given by

D(t, v, S) = f(t, v)∫ ∞

0β(v, v′, S)f(t, v′)dv′. (1.13)

The newly-included kinetics are governed by the agglomeration kernel β, itselfoften assumed to be the product of two factors: a collision frequency βc and anagglomeration efficiency (or sticking probability) Ψ. While different derivationsand expressions exist for the two factors [10, 11, 32–36], there is a generalconsensus that the agglomeration kernel depends on fluid viscosity, energydissipation rate as well as supersaturation; theoretical derivations furtherpredict a dependence of the agglomeration rate on particle size, althoughseveral authors have chosen to neglect this effect, being still able describeexperiments satisfactorily [12, 37, 38].

The above model as well as variations thereof have been used to describe orcharacterize multiple agglomerating systems in literature [39–42]. Lindenberg

13

CHAPTER 1. CRYSTALLIZATION PROCESS MODELING

et al. [12] tested different models for βc and Ψ together with Eqs. (1.11) to (1.13)to describe the agglomeration behavior of α l-glutamic acid in water undervarying process conditions. The fitted model showed excellent agreement withthe experimental results with respect to its prediction of the supersaturationprofiles and the particle size distribution (cf. Figure 1.4). It is hence suitablefor further use during process design development. The same work, in whichadditional computational fluid dynamics (CFD) simulations were performedto investigate shear rate variations in the stirred batch vessel (cf. Figure 1.5),further serves to demonstrate a number of key issues that play a role in themodeling of agglomeration.

First, process characterization for these systems becomes inherently morecumbersome due to the increased complexity of the process. Second, theunderstanding of the fluid dynamics of suspensions in stirred vessels and itsrole in agglomeration is still limited, particularly for the higher suspensiondensities and larger vessel sizes that are of industrial interest. Third, there isa difficulty to experimentally distinguish agglomeration from other processesacting on the PSD, such as growth. The reason for this is that typically only atotal particle size distribution is measured, that is, a distribution that includesboth agglomerates and primary particles. The problem is compounded bythe fact sample preparation, such as sonication, can affect the population bybreaking up otherwise stable particles.

In response to the former two issues, a trend toward reduced models, whoselower computational cost allows for faster computation of process outcomesable to be integrated in CFD software, is evident [22]. With regard to thethird point, image analysis approaches have shown potential in distinguishingagglomerates from primary particles, a step that might help to obtain thehigher resolution data sets necessary for experimental validation [43–47].

14

1.2. System Characterization and Optimization

a) b)

c) d)

Figure 1.4: Comparison of experimental (markers) and fitted modeling (solidlines) results. a) & b): effect of supersaturation; c) & d): effect of seed particlesize. Reprinted with permission from Lindenberg et al. [12]. Copyright 2008American Chemical Society.

15

CHAPTER 1. CRYSTALLIZATION PROCESS MODELING

a) b)

Figure 1.5: a) Contour plot of velocity magnitude in m/s for a stirring rate of200 rpm; b) distribution of energy dissipation in the stirred reactor for the samestirring rate. Reprinted with permission from Lindenberg et al. [12]. Copyright2008 American Chemical Society.

1.2.4

Optimization

Here we shall give a brief introduction to model-based optimization in thefield of crystallization, although it should be noted that several model-freeapproaches to optimize or control processes exist as well (e.g., [48–50]). A morecomprehensive review of the current state of the art in both fields can be foundelsewhere [51–53].

Schematically, model-based optimization of a (crystallization) process withrespect to some generic objective, Φ, can be viewed as the following genericproblem

minimize/maximize Φsubject to

Model equationsConstraints

(1.14)

16

1.2. System Characterization and Optimization

where the objective function Φ generally depends on the time (for batch systemsoften the end-time only), the state of the system (e.g., density function and itsmoments or nucleation/growth rates) and the inputs (temperature, antisolventaddition rate, etc.); as indicated also in Figure 1.2. Further, the model equations(e.g., Eqs. (1.7) and (1.8)) define the dynamics of the system, while constraintsare set to make sure the solution remains within a physically and potentiallyeconomically sensible domain (feasible cooling rates, limited supersaturation,etc.).

The mathematical problem stated in Eq. (1.14) is extremely difficult due to itsnonlinearity and nonconvexity, which allows for the existence of many localextrema. Consequently, there is no single pathway that guarantees successfulconvergence to an optimum, nor is there a simple way of demonstrating ingeneral global optimality of an already found solution; different authors havehence used different numerical methods to optimize Eq. (1.14) with no singlestrategy showing clear superiority. Regardless, the optimization of crystal-lization processes can yield insights that are well worth the additional effort.Sheikholeslamzadeh and Rohani [54] conducted an investigation of the optimalcontrol policy for the polymorphic transformation of l-glutamic acid basedon kinetics identified using the procedure outlined in Section 1.2.2. In anotherstudy, Lindenberg et al. [55] performed multi-objective optimization to improveprocess time and fine fraction for the combined cooling and antisolvent crystal-lization of aspirin in an ethanol-water mixture. By allowing temperature andanti-solvent fraction to change simultaneously, the reachable set of possibleoutcomes is expected to be significantly larger [56]. Specifically, the followingobjective function was used:

Φ =

[tp∫ tp

0 Jdt

](1.15)

with the first element of Φ referring to the total process time and the second rowreferring to the number of new crystals formed during the process. The solubility,nucleation and growth kinetics were identified beforehand as function of T , Sand antisolvent concentration. In the case of multi-objective optimization, theobjective function Φ is not scalar but rather a vector-valued function, whoseelements are to be optimized simultaneously. This leads to a set of so-calledPareto optimal solutions, for which none of the objectives can be improvedwithout degradation of another; an example of such a Pareto front for the

17

CHAPTER 1. CRYSTALLIZATION PROCESS MODELING

Figure 1.6: Pareto-optimal set of the two-objective optimization problem treatedin Lindenberg et al. [55] (solid line). The dashed line is obtained with constantanti-solvent addition and cooling rate (see inset for operating policies). Theregion below the Pareto set is not feasible and the set is limited at low processingtimes due to constraints on the maximum cooling rate. The circles representdifferent process alternatives shown in detail in Figure 1.7: anti-solvent additionfirst, then cooling (case 1) or the reverse (case 2). The filled circle is a Pareto-optimal point that was experimentally implemented (case 3). Reprinted withpermission from Lindenberg et al. [55]. Copyright 2009 American ChemicalSociety.

above work is shown in Figure 1.6. Focusing on one point within the Paretoset, the authors could demonstrate the superiority of the resulting cooling andantisolvent profile as compared to two alternative strategies (“cooling first”and “antisolvent first”; cf. Figure 1.7), highlighting that the overall parameterestimation and process optimization strategy was successful.

1.3

Multidimensional population balance modeling

While one-dimensional population balance models have served researchers wellin the description of many systems, it is widely recognized that in many cases,the properties of the solid are not well-characterized by a single descriptor.Particularly, this is the case for systems where dynamic impurity incorporation

18

1.3. Multidimensional population balance modeling

a) b)

Figure 1.7: Different combined cooling and anti-solvent processes as inves-tigated in Lindenberg et al. [55]; (1) anti-solvent first, (2) cooling first, (3)optimized process. (a) Process trajectories in the phase diagram, (b) particlesize distribution obtained at the end of each process. The figures highlightthat the non-optimized processes result in an uncontrolled supersaturationprofile, which leads to undesirable particle size distributions, while the opti-mized process results in a unimodal particle size distribution of medium sizedparticles. Reprinted with permission from Lindenberg et al. [55]. Copyright2009 American Chemical Society.

and evolving crystal shapes are observed. Indeed, in order to capture crystalshape or impurity content in particles, one requires additional internal states inthe model. The resulting, generalized form of the population balance equationwith n internal states, e.g., for the case of a well-mixed batch reactor, is givenby

∂

∂tf(t,x) +

n∑i=1

∂ [Gi(x, S)f(t,x)]∂xi

= B(t,x, S)−D(t,x, S). (1.16)

The main difference between the above equation and Eq. (1.7) is the substitutionof L with the more general n× 1 state vector x and the corresponding use of

19

CHAPTER 1. CRYSTALLIZATION PROCESS MODELING

the summation operator. As in the 1D case, the mechanisms that affect thepopulation can be modeled through G, B and D, although the former hasbecome an n×1 vector, too. In the important case where particle shape alone isof interest, the internal state vector corresponds to the vector of characteristicsizes, i.e., x = L, and Gi = dLi/dt, typically the normal growth rate of facet i.

While the larger number of internal states in Eq. (1.16) grants the ability tosimulate processes in greater detail, the availability of techniques to accuratelymeasure properties such as particle shape for a statistically significant sampleof particles—preferably in real-time—is limited. Furthermore, even though mul-tidimensional population balance models have been used in multiple instancesfor parameter estimation and system characterization [57–59], multidimen-sional population balance modeling, and in particular appropriate and fastsolution techniques, are an ongoing topic of research [60–65]. It is mostly forthese two reasons that the use of multidimensional modeling has been verylimited outside of academia, despite the fact that it represents an importantresearch direction for the future of the application of population balances tocrystallization processes.

1.4

Conclusion

Having provided the theoretical background as well as a broad outline ofpotential applications in this chapter, a brief discussion of the benefits anddrawbacks of crystallization models together with an outlook is in order.Population balance modeling can be a powerful and versatile tool that allows adeep and quantitative understanding of crystallization processes. However, thisinsight typically comes at the cost of a time- and labor-intensive examinationof individual systems; with guidelines for transferability of the lessons learntbeing rarely investigated in detail. Similarly, issues of model distinguishabilityand an incomplete understanding of statistics on the side of experimenterslead to parameterized models whose predictive capabilities outside of an oftennarrow operating region may be unsatisfactory. All these problems are beingcompounded by the fact that real processes often deal with complex mixtureswith significant batch-to-batch variations. Consequently, population balancemodels have not yet met with widespread acceptance in industry.

Nevertheless, we believe that a turning point has been reached for severalreasons. First, the presence of a mathematical framework intrinsically providesa structure that facilitates systematic analysis and understanding of the pro-cess. This stands in contrast with a more qualitative understanding, which

20

1.4. Conclusion

lacks organization and is more prone to misinterpreting or completely missingimportant interactions. Undoubtedly, this approach brings with it a largerdependency on the know-how and experience of the person evaluating theresults and varying interpretations between different experts are hence to beexpected. Second, with growing availability of fast computational methods andsoftware tools, accurate solutions to the complex set of equations presentedhere become more and more accessible to non-modelers. This frees up time thatcan be used for devising shorter and more statistically robust experimentalplans. Third, once a satisfactory description of a process has been established,process models possess crucial advantages over experiments: simulations canbe obtained at virtually no cost and process models can be easily integratedinto higher-order hierarchies. This means that novel ideas or process designscan be tested quickly and without great financial investment, thus allowing toexplore a much larger set of alternatives than can be done in a laboratory orpilot scale.

Crystallization processes are determined by the behavior of myriads of solidparticles interacting with at least one fluid phase and with each other. Whilestudies on single particles, such as studies on growth mechanisms, are necessaryand important to gain fundamental insight, a single crystal does not makea crystallizer. To believe that this is sufficient for an understanding of theentire process is to fool oneself, as the variability in the history of particulatesmust not be neglected. Population balance modeling represents a scientific andmathematically sound pathway of dealing with these properties while providingsufficient flexibility to deal with problems of varying complexity. Thus, itremains an indispensable instrument for those seeking to further improve ordevelop crystallization technology.

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