chemical engineering science...1 is seeded in tank a, whereas enantiomer e 2 is seeded in tank b....

Chemical Engineering Science 88 (2013) 48–68

Contents lists available at SciVerse ScienceDirect

Chemical Engineering Science

0009-25

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/ces

Solutions to inversion problems in preferential crystallization ofenantiomers—Part II: Batch crystallization in two coupled vessels

Steffen Hofmann a,n, Jörg Raisch a,b

a Technische Universität Berlin, Fachgebiet Regelungssysteme, Sekretariat EN 11, Einsteinufer 17, 10587 Berlin, Germanyb Max Planck Institute for Dynamics of Complex Technical Systems, Systems and Control Theory Group, Sandtorstr. 1, 39106 Magdeburg, Germany

H I G H L I G H T S

c Focus on control of a setup of two coupled vessels for preferential crystallization.c Liquid exchange between the tanks lowers supersaturations of the counter enantiomers.c Dynamic inversion for realization of two final-time crystal size distributions.c Efficient adaptation and extension of recently published results.c Numerical example allows a comparison with our results for a single crystallizer.

a r t i c l e i n f o

Article history:

Received 27 June 2012

Received in revised form

25 September 2012

Accepted 4 October 2012Available online 20 November 2012

Keywords:

Enantiomers

Preferential crystallization

Population balance

Moment model

Dynamic inversion

Orbital flatness

09/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.ces.2012.10.030

esponding author. Tel.: þ49 30 314 79281; fail address: [email protected] (S.

a b s t r a c t

In this series of two papers, we investigate inversion techniques for models describing the crystal-

lization of conglomerate forming enantiomers, with application for preferential crystallization. In Part I,

population balance and moment model equations for crystallization in a single vessel were analyzed

and inverted. Here, in Part II, a configuration consisting of two crystallizers coupled by the exchange of

crystal-free liquid is considered. A main problem addressed in this series concerns the use of previous

results on orbital flatness of a model for single substance crystallization. An analysis presented in Part I

for the single crystallizer moment model, which is a single-input system, is extended in this paper to

show that also the moment model for the coupled crystallizer configuration, which is a multi-input

system, is not orbitally flat. Despite the absence of orbital flatness, it is possible to obtain inversion

results for the simultaneous realization of crystal size distributions (CSDs) in both vessels. The

techniques we use build on a time transformation introduced in previous work. We also investigate

the effect of idealizations, which are motivated by the design principles of the coupled crystallizer

configuration. In combination with idealizing assumptions that were introduced in Part I, solutions to

inversion problems are greatly simplified. In particular, the resulting idealized systems are orbitally flat.

We extend a numerical example presented in Part I to the coupled crystallizer configuration. In this

example we compare the exact and simplified solution techniques and demonstrate how the idealizing

assumptions can be justified a posteriori.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In Part I of this series of two papers (Hofmann and Raisch,2012), we have investigated solutions to inversion problems forpreferential crystallization in a single vessel. In this context, weprovided a brief introduction to preferential crystallization ofconglomerate forming enantiomers out of solution, and to twotechnical realizations of the process. As Part II deals with thecoupled process configuration, we first provide a more detailed

ll rights reserved.

ax: þ49 30 314 21137.Hofmann).

discussion of this new concept. It has been described in Matsuoka(1997) and Elsner et al. (2005) was analyzed by Elsner et al.(2007) and shown to be a feasible approach by experimentalstudies presented in Elsner et al. (2009). Further analysis andexperimental studies have recently been presented in Elsner et al.(2011).

The principle of this concept is depicted in Fig. 1. Twopreferential crystallization processes are carried out simulta-neously in two vessels. In the following, we will assume thatenantiomer E1 is seeded in tank A, whereas enantiomer E2 isseeded in tank B. This implies that E1 is the preferred (desired)enantiomer in tank A, and E2 in tank B. As discussed in Part I, aslong as no spontaneous nucleation of the respective (unwanted)

www.elsevier.com/locate/ceswww.elsevier.com/locate/cesdx.doi.org/10.1016/j.ces.2012.10.030dx.doi.org/10.1016/j.ces.2012.10.030dx.doi.org/10.1016/j.ces.2012.10.030mailto:[email protected]/10.1016/j.ces.2012.10.030

Fig. 1. Preferential crystallization in two coupled vessels.

S. Hofmann, J. Raisch / Chemical Engineering Science 88 (2013) 48–68 49

counter enantiomers, i.e., E2 in tank A and E1 in tank B, occurs, thesolid phases will stay pure. Without liquid exchange, the solu-tions in tanks A and B would get depleted with respect to E1 andE2, and the difference between supersaturations of the preferredand counter enantiomers would increase. This is obstructive ifone wants to prevent crystallizations of the preferred enantio-mers from gradually slowing down: keeping supersaturations ofthe preferred enantiomers constant can cause supersaturations ofthe counter enantiomers to significantly exceed their initiallevels, resulting in excessive primary nucleation and impurity ofthe product. This situation can be mitigated by exchangingcrystal-free liquid between the two tanks: the process in tank Acan benefit from liquid phase consumption of E2 in tank B, andvice versa. In other words, the solutions in tanks A and B now getdepleted with respect to both the preferred and counter enantio-mers. In fact, in the hypothetical case that the liquid exchange isinfinitely fast, and crystallizations in tanks A and B are perfectlysynchronous, initially racemic solutions will always stay racemic.This idealized case will be revisited later in this paper. However,experimental studies have also shown that moderate asymme-tries do not generally have a negative impact on purity (Eickeet al., 2010b).

Recall that Part I and II deal with the control of processes forpreferential crystallization which are limited to the conglomerateforming type of enantiomers. This is particularly true here.As explained in Part I, enantiopure crystals of the other type,i.e., racemic compounds, can only coexist in equilibrium with asolution having a certain minimum enantiomeric excess. Bringingthe solution close to a racemic mixture by exchanging crystal-freemother liquids between two vessels would be counter-productivein this case (Elsner et al., 2009).

For the technical realization of the coupled crystallizer setup, itis essential to prevent the (growing) seeds and small nuclei of thepreferred enantiomers in each tank from traveling into the othertank, as this would result in high impurity of the product in theother tank. One way to achieve this is to put simple, sufficientlyfine filters into the exchange paths (Elsner et al., 2005), asindicated in Fig. 1. Experimental results (Elsner et al., 2009)clearly support the effectiveness of the coupled crystallizer setupin this relatively simple form.

Another possibility is the introduction of fines dissolutionunits into the liquid exchange paths (Eicke et al., 2009), whichpermits the use of coarser filters and can provide additionalbenefit for purity and productivity. However, including finesdissolution makes modeling and simulation of the process arelatively cumbersome task. It can be treated in the context ofnumerical techniques based on discretization of the populationbalance equation, see, e.g., Qamar et al. (2008, 2009). Especiallyfor controller design, it is desirable to approximate the respectivepopulation balance equations via reasonably sized ordinary dif-ferential equation (ODE) systems, see, e.g., Chiu and Christofides(1999, 2000), El-Farra et al. (2001), Grosch et al. (2007), Qamaret al. (2010), and references therein. Note that we do not considerfines dissolution in this paper.

In Part I, we have given a literature overview related to modelingand control problems in single substance crystallization as well aspreferential crystallization in a single tank. This overview is com-plemented here by a short discussion of results related to feedfor-ward control of batch crystallization processes. Frequently,temperature is treated as the control input, and the control profileis determined as the solution to an optimal control problem wherethe cost function expresses certain properties of the final-timecrystal size distribution (CSD) (e.g., Miller and Rawlings, 1994;Lang et al., 1999; Mohameed et al., 2003; Corriou and Rohani,2008). Optimal control solutions may also be useful in the context offeedback control schemes, like model predictive control (e.g., Shiet al., 2005, 2006). In other circumstances, e.g., in the case of highlyspecific demands on filterability or mechanical properties of thecrystalline material, it can make more sense to fully specify thefinal-time CSD. Computing an appropriate feedforward controlprofile in this case has also been treated as an optimal controlproblem by some authors (e.g., Worlitschek and Mazzotti, 2004;Nagy, 2009; Aamir et al., 2009). While this approach aims at onlyapproximate achievement (e.g., in terms of the mean squared error)of CSDs, it has the advantage that process requirements (like themaximum allowable process duration) can be explicitly considered.In Nagy (2009), repeated computation of robust feedforward opti-mal control profiles is part of a feedback control scheme involvingnonlinear model predictive control and supersaturation control.

A different approach aims at the exact realization of specifiedfinal-time CSDs. In Vollmer and Raisch (2003, 2006) this inversionproblem was solved by introducing a time transformation thatturns population balance equation (PBE) models describing singlesubstance crystallization into simple transport equations, andwhich transforms the corresponding moment models into differ-entially flat systems. The reader is referred to Part I for adefinition of these concepts. These ideas were also adopted byHounslow and Reynolds (2006) where, additionally, the forward(simulation and reconstruction of final-time CSDs) problem wastreated based on the method of moments and the method ofcharacteristics. In Zhang and Xu (2011), related ideas were usedfor realization of specified final-time CSDs in the context of adiscretized population balance equation, and conditions for theirrealizability were given; additional feedback control, whichmakes use of measurements of concentration and the zerothmoment of the CSD, was demonstrated to compensate foruncertainties in the nucleation parameters.

In the introduction of Part I, we have also addressed existingapproaches for control of preferential crystallization. The majorityof them focus on feedforward control (i.e., optimization of controlprofiles and other operating conditions). To the best of theauthors’ knowledge, control of the specific coupled processdiscussed here, with the exception of Hofmann et al. (2010), hasnot been investigated in other papers. In Hofmann et al. (2010), alow-level feedback controller was designed with the aim of,loosely speaking, synchronizing crystallization in the two vessels.

Following a tradition of work which focuses on the solution offeedforward control problems in batch crystallization, this paper

S. Hofmann, J. Raisch / Chemical Engineering Science 88 (2013) 48–6850

and its companion (Hofmann and Raisch, 2012) strive to extendthese ideas to preferential crystallization of enantiomers. Here, inparticular, the approach of exactly realizing prescribed final-timecrystal size distributions (CSDs) for single substance crystalliza-tion processes (Vollmer and Raisch, 2006) shall be extended topreferential crystallization of enantiomers in two coupled vessels.This problem is illustrated in Fig. 2: in each tank, the final-timeCSD of one of the enantiomer species is specified, and onesearches for the temperature profiles in tanks A and B whichhave to be applied in order to achieve the specification. Detailsfrom Fig. 2 will be explained in Sections 2 and 3.1. As discussed inPart I, crystallization (or slurry) temperatures can be treated asthe manipulated variables. Note that it would also be possible toinfluence the crystallization process by other means, like anti-solvent addition, continuous seeding, etc. In particular, for thecoupled crystallizer configuration, controlling the liquid exchangerates would be another option. This paper, however, focuses ontemperature control only. Note that the problem is different fromthe one in Part I from a control theoretic point of view: we nowdeal with a multi-input system, i.e., there are more degrees offreedom for control than in Part I. Nevertheless, the analysis andinversion techniques developed in the following are closelyrelated to the results in Part I.

The paper is organized as follows: in Section 2, populationbalance and moment models are given for the process of crystal-lization of two conglomerate forming enantiomers in two coupledcrystallizer vessels. In Section 3.1, we show how parts of theresults by Vollmer and Raisch (2003, 2006) can be used in theprocess of system inversion that is necessary to realize prescribedfinal-time CSDs. In Section 3.2, we show that, normally, momentmodels for crystallization of enantiomers in the coupled crystal-lizer setup are not orbitally flat, i.e., they cannot be turned flat byan appropriate time scaling. This statement is based on anargument from Part I which, as we are now dealing with amulti-input system, needs to be extended appropriately. Despitethe absence of orbital flatness, we solve the problem of inverting amoment model of the coupled setup in Section 3.3, and cantherefore simultaneously realize two prescribed final-time CSDsin the two vessels. The solution process is quite cumbersome,however, and involves the integration of ODEs.

Therefore, in Sections 4.1–4.3, we introduce simplified solu-tions for the case when crystallizations in the two vessels run in aperfectly symmetric way, and the liquid exchange is assumed tobe infinitely fast. We show that this results in an equivalent singlecrystallizer model, which is not orbitally flat. Thereafter, inSections 4.4.1 and 4.4.2 we present simplified solutions for thecases when, additionally, nucleation of the counter enantiomersor the preferred enantiomers, respectively, is neglected. In bothcases, the resulting simplified systems are orbitally flat.

The numerical example in Section 5 closely follows the onestudied in Part I. For maximum comparability, all parametersand initial conditions are identical to the ones used there. We

Fig. 2. Illustration of the inversion problem and the crystal populations involvedin preferential crystallization in two coupled vessels. In this example, the

two highlighted final-time CSDs shall be realized by controlling the temperatures

in tanks A and B. The meaning of tE1 A , tE2A , tE1 B and tE2 B will be explained inSection 3.1.

compare the idealized and non-idealized solution methods.We also investigate the results when temperature trajectoriesobtained using the idealized models are used as inputs to thenon-idealized models. In Section 6, we draw some overall con-clusions for Part I and II, and give a perspective on possiblefuture work.

2. Model

Like in Part I, we consider only the mechanisms of growth andnucleation. Other effects, such as conglomeration and breakage,are neglected. It is assumed that the dimensions of the particlescan be described by only one internal coordinate, L. Growth ratesare assumed to be independent of particle size, and newlynucleated particles are assumed to initially have size zero. Simple,ideal filters are supposed to enable the exchange of crystal freeliquid, i.e., to prevent any crystals from entering the liquidexchange paths. Also, no fines dissolution is performed in theliquid exchange path or elsewhere. Modeling crystallizationsystems with fines dissolution (see, e.g., Qamar et al., 2008,2009 can be considerably more complex. In particular, it mayno longer be possible to use the method of moments.

Population balance equations (PBEs) are frequently used todescribe the dynamics of particulate processes (e.g., Randolph andLarson, 1988; Ramkrishna, 2000). There are now four crystalpopulations involved, which are distinguished by the enantiomerspecies, and by the tank they reside in. The species are denoted byE1 and E2, and the tanks by A and B. Consequently, thesepopulations will have indices E1A, E2A, E1B and E2B. The completesystem model involves four PBEs, i.e.,

@f E ðL,tÞ@t

¼� @ðGEðL,tÞf EðL,tÞÞ@L

, ð1aÞ

with boundary and initial conditions

f Eð0,tÞ ¼BE ðtÞ

GEð0,tÞ, ð1bÞ

f EðL,0Þ ¼ f E,seedðLÞ, ð1cÞ

where EAfE1A,E2A,E1B,E2Bg, t represents time and the scalarL crystal length, f E is a number density function, the so-calledcrystal size distribution (CSD), and GE ðL,tÞ, BEðtÞ, respectively, aregrowth and nucleation rates of the respective enantiomer speciesin the respective tank.

Because of the size-independency of the growth rates, we canwrite GEðtÞ instead of GEðL,tÞ. Also, dependency of the growth andnucleation rates on time is implicit rather than explicit. This willoften be denoted by writing BEð�Þ and GEð�Þ.

The right side of Fig. 2 illustrates the involved crystal popula-tions at final time, i.e., after crystallizations in both tanks havebeen stopped at times ts, respectively tf. This stopping will beexplained in more detail in Section 3.3. The picture looksqualitatively the same at any time t40. Note that the CSDsbelonging to E1 in tank A, respectively E2 in tank B, have each beensplit into two parts. One part represents grown seed crystals, e.g.,f E1A,s, and the other one, e.g., f E1A,n, represents grown nucleatedcrystals. In the following, these parts will be referred to as theseed CSDs, and the nucleated CSDs, respectively. In this scenario,E1 and E2 are the desired enantiomers, respectively, in tanks Aand B.

Like in Part I (following, e.g., Hulburt and Katz, 1964; Myerson,2002) it is possible to derive a closed moment model,which describes the evolution of a number of moments of allinvolved crystal populations. In total, 16 ordinary differentialequations (ODEs) are needed to describe the evolution ofall relevant moments in both tanks, i.e., four ODEs for each


enantiomer species in each tank, EAfE1A,E2A,E1B,E2Bg_m iE ¼ iGE ð�Þmði�1ÞE , i¼ 1 . . .3, ð2aÞ

_m0E ¼ BEð�Þ, ð2bÞ

where the moments are defined as

miEðtÞ ¼Z 1

0Lif E ðL,tÞ dL, i¼ 0 . . .3: ð3Þ

The associated initial conditions are

miEð0Þ ¼ miE,seed ¼Z 1

0Lif E,seedðLÞ dL, i¼ 0 . . .3: ð4Þ

Note that E2 and E1, respectively, are the counter enantiomers intanks A and B. Therefore, miE2A,seed ¼ 0 and miE1B,seed ¼ 0.

Up to now, the dynamics pertaining to the solid phases havebeen described by PBEs and, alternatively, by a moment model.These equations must be augmented by the dynamics of theliquid phases in order to form a complete model. In particular, theliquid exchange between the crystallizers must be taken intoaccount. We can define FAB and FBA, respectively, as the volumeflow rates of (crystal free) liquid being pumped through theexchange path from tank A to tank B, and vice versa. As discussedin the introduction, we do not consider the volume flow rates ascontrol inputs in this paper. Instead, we require that they beregulated using some simple inner control loop. For example, theycould be regulated to be equal and constant. In that case,however, due to changing densities of the solutions in tanks Aand B, the masses of the solvent (frequently water) in the tankswould change over time. In order to concentrate on the mainideas, the following exposition uses the simplistic idea that thevolume flow rates FAB and FBA are regulated such that the massflow rates of solvent are the same and constant, i.e.,

_mW ,ABðtÞ ¼ _mW ,BAðtÞ ¼: _mW ,exð ¼ const:Þ: ð5Þ

This simplifies the model in that the same constant mW can beused for the total mass of solvent in each tank, and no extra stateshave to be added for the solvent masses. The analysis andinversion results in the later sections could easily be extendedto more realistic cases, such as for constant (and equal) volumeflow rates, or when a controller adjusts the flow rates in order tomaintain equal volumes of slurry in tanks A and B. The dynamicsfor the evolution of the liquid phases in tanks A and B thenbecomes

_ml,E1A ¼�3GE1Að�Þrskvm2E1Aþ _mW ,exml,E1B�ml,E1A

mW, ð6aÞ

_ml,E2A ¼�3GE2Að�Þrskvm2E2Aþ _mW ,exml,E2B�ml,E2A

mW, ð6bÞ

_ml,E1B ¼�3GE1Bð�Þrskvm2E1Bþ _mW ,exml,E1A�ml,E1B

mW, ð6cÞ

_ml,E2B ¼�3GE2Bð�Þrskvm2E2Bþ _mW ,exml,E2A�ml,E2B

mW, ð6dÞ

where rs is the density of the crystals, and kv is a volume shapefactor. The initial conditions of the dissolved masses are

ml,Eð0Þ ¼ml0,E , EAfE1A,E2A,E1B,E2Bg: ð7Þ

In contrast to crystallization in a single vessel, see Part I, thisdynamics cannot be entirely converted into algebraic massbalance equations. Four additional state variables are introducedin (6). Note that two of these could be eliminated by consideringmass balances covering both tanks. Here, we decided not to dothis to preserve the symmetry of the model.

Because of (5), the masses of solvent in tanks A and B areconstant. Hence, the liquid mass fractions wl,E1A, wl,E2A, wl,E1B and

wl,E2B are uniquely determined by the dissolved masses ml,E1A,ml,E2A, ml,E1B and ml,E2B. E.g., in tank A, we have

wl,E1AðtÞ ¼ml,E1AðtÞ

ml,E1AðtÞþml,E2AðtÞþmW, ð8aÞ

wl,E2AðtÞ ¼ml,E2AðtÞ

ml,E1AðtÞþml,E2AðtÞþmW, ð8bÞ

and likewise in tank B.As discussed in Part I, it is not trivial to obtain accurate

expressions for the growth and nucleation rate functions. Thismay involve first principles modeling as well as empirical terms.To account for current and future model developments, theinvestigations in the main part of this paper will be based ongeneric functional dependencies of the growth and nucleationrates. For tank A, we have

GE1Að�Þ ¼ GE1AðTA,wl,E1A,wl,E2AÞ, ð9aÞ

GE2Að�Þ ¼ GE2AðTA,wl,E1A,wl,E2AÞ, ð9bÞ

BE1Að�Þ ¼ BE1AðTA,wl,E1A,wl,E2A, m0E1A,m1E1A,m2E1A,m3E1A,

m0E2A,m1E2A,m2E2A,m3E2AÞ, ð9cÞ

BE2Að�Þ ¼ BE2AðTA,wl,E1A,wl,E2A, m0E1A,m1E1A,m2E1A,m3E1A,

m0E2A,m1E2A,m2E2A,m3E2AÞ, ð9dÞ

and analogous expressions hold for tank B. The slurry tempera-tures in tanks A and B (also called the crystallization tempera-tures) are denoted by TA and TB, respectively. Note that, since themasses of solvent in the tanks are assumed constant, it issufficient to consider the mass fractions wl,E1A, wl,E2A in (9). Notealso that the rates in one tank do not depend on variablesreferring to the other tank.

As in Part I, we need some basic assumptions on the growthand nucleation rates to establish existence and uniqueness ofsolutions to the inversion problem. Essentially, we now have twocopies of the set of dynamic equations for a single crystallizer, andthe only link between these is through the state variables for theliquid phase, i.e., ml,E1A, ml,E2A, ml,E1B and ml,E2B. The assumptionsfrom Part I can therefore be adapted in a straightforward manner.(a) The physically meaningful domain D of the overall state andinput space must be defined in such a way that all growth andnucleation rates are strictly positive. (b) Within D, the growth andnucleation rate functions must be continuously differentiablew.r.t. all the state variables. (c) Within D, the growth andnucleation rate functions pertaining to the crystal populationsin tank A must be continuously differentiable twice w.r.t. TA, andthe growth and nucleation rate functions pertaining to the crystalpopulations in tank B must be continuously differentiable twicew.r.t. TB. Then, because the rate functions in tank A do not dependon TB, and vice versa, this implies that all the growth andnucleation rate functions are C2 functions of the two-dimensional input ½TA,TB�T . (d) For the crystal populations whosefinal-time CSDs are specified we require that within D, thederivatives of the nucleation to growth rate ratios, e.g.,BE1A=GE1Að�Þ, w.r.t. the appropriate temperature, e.g., TA, shouldbe non-zero.

Note that, in contrast to Part I, we now have two controlinputs, i.e., the crystallization temperatures in tanks A and B. Themodel is therefore a MIMO system. The availability of two controlinputs enables the simultaneous realization of specified final-timeCSDs for two of the involved crystal populations. This will bedemonstrated in the following sections.


3. Analysis of the coupled crystallizer configuration

Like in Part I, the main goal for inversion is the realization ofprescribed, final crystal size distributions (CSDs). Compared to thesingle crystallizer, there are more degrees of freedom for control,i.e., two crystallization temperatures, TA and TB. This should makeit possible to simultaneously realize CSD profiles for two of thefour involved crystal populations.

3.1. Time scaling

In Part I, it was argued that a time scaling, according to thegrowth rate of one of the two enantiomer species (in a singletank), can be applied to the population balance equation pertain-ing to that species. The same can be done here, for the PBEpertaining to the evolution of any of the involved CSDs, i.e., forEAfE1A,E2A,E1B,E2Bg we can scale time with GE ð�Þ, and denote theresulting scaled time tE ,

dtE ¼ GEð�Þ dt: ð10Þ

Then, the population balance equation for the respective enantio-mer in the respective tank becomes

@f EðL,tEÞ@tE

¼� @f EðL,tEÞ@L

, ð11aÞ

with boundary and initial conditions

f E ð0,tEÞ ¼BE ðtEÞGEðtEÞ

, ð11bÞ

f E ðL,0Þ ¼ f E,seedðLÞ: ð11cÞ

Note that if E corresponds to an undesired (counter) enantiomerin the respective tank, then f E,seedðLÞ � 0, as no seeds are added forthat species.

The characteristic curves that describe the solutions of the PBEbecome straight lines, compare Vollmer and Raisch (2003),Hofmann and Raisch (2012), and see Fig. 3. Thus, the requiredfinal-time shape of the nucleated part of the CSD f EðL,tE,f Þ,LA ½0,Lmax ¼ tE,f � can trivially be converted to a temporal evolu-tion of the boundary condition, i.e.,

BEGEðtE Þ ¼ f E ð0,tEÞ ¼ f EðtE,f�tE ,tE,f Þ, tEA ½0,tE,f �: ð12Þ

The part of the final-time CSD f E ðL,tE,f Þ with L4tE,f is merely ashifted version of the CSD of seed crystals, and can therefore notbe freely prescribed,

f E ðL,tE,f Þ ¼ f E,seedðL�tE,f Þ, L4tE,f : ð13Þ

This shows that tE,f corresponds to the gain in length of crystalsbelonging to the seed distribution of the respective enantiomer in

Fig. 3. Characteristic curves of the population balance equation for one enantio-mer in one tank; (a) without time-scaling: f E ðL,tÞ ¼ const:; (b) with time scaling(10): f E ðL,tE Þ ¼ const:

the respective tank, and it is also the maximum length of anycrystals belonging to the nucleated CSD.

The ratio BE=GE directly (i.e., statically) depends on the state,i.e., some or all moments and the state variables describing theliquid phase in the respective tank, as well as on the respectivetemperature input TA or TB. According to the model structuregiven in Section 2, it does not directly depend on the temperaturein the other tank. However, both temperatures mutually influencethe evolution of the state in both tanks. Therefore, both tempera-ture profiles in scaled, respectively original time cannot becomputed independently. In Section 3.3, we show how thetemperature profiles can be obtained by solving an overall systemof differential equations.

Like in Part I, one can apply the time transformation (10) alsoto the moment model. The part of the moment model pertainingto the evolution of f E then becomes

d

dtEmiE ¼ imði�1ÞE , i¼ 1 . . .3, ð14aÞ

d

dtEm0E ¼

BEGEð�Þ, ð14bÞ

with the associated initial conditions

miE ð0Þ ¼ miE,seed: ð15Þ

Note that the differential equations for the moments of the otherinvolved crystal populations will of course not be in the simpleform (14).

In analogy to Part I, if BE=GE is considered as the input tosubsystem (14), then m3EðtEÞ is a flat output. Given a sufficientlysmooth profile m3EðtEÞ, tEA ½0,tE,f � for that output, the trajectoriesmiE ðtEÞ, i¼ 0 . . .3, as well as the input trajectory, ðBE=GE ÞðtE Þ, arealgebraically determined by the flat output and its derivativesw.r.t. tE . Given a specified final CSD f E,des, (12) can be used tocompute ðBE=GEÞðtEÞ in a first step, and the trajectory of theoutput m3E ðtEÞ is obtained by integrating ðBE=GEÞðtEÞ four timesw.r.t. tE .

3.2. Lack of orbital flatness

In Part I (Hofmann and Raisch, 2012), it was argued thatmoment models describing preferential crystallization in onevessel, where the temperature of the solution is the control input,typically do not possess the orbital flatness property. In AppendixA the derivations presented in Part I are extended to cover thedynamics of preferential crystallization in two coupled vessels.Note that we are now dealing with a multi-input system.

It is clear that the model presented in Section 2 conforms tothe structure required in Appendix A and (for typical growth andnucleation kinetics) can be shown to lack orbital flatness. Alsonote that the system structure (A.1) is more general than the oneassumed in Section 2. In particular, it is able to cover morerealistic liquid exchange dynamics. In the context of coupledpreferential crystallization, (A.1a) would correspond to the evolu-tion of the moments of both enantiomer species in tank A, and(A.1b) to the evolution of the moments in tank B. The derivativesin these subsystems are only directly influenced by one input, i.e.,u1 ¼

4TA or u2 ¼

4TB, respectively. On the other hand, (A.1c) would

represent the dynamics of the liquid phase. In a more realisticmodel (e.g., for constant volume exchange rates between thetanks), the temperatures can impact the mass exchange ratesvia the solution densities. In this way, TA can directly influence thederivatives of liquid phase state variables for tank B, and viceversa. Furthermore, the masses of solvent (e.g., water) in the tanksmay change during the process, making it necessary to includethem in the system state. In the special case of the model


assumed in Section 2, (A.1c) consists of the four equations (6), twoof them depending, in the face of (9), only on u1, the other twoonly on u2.

3.3. Dynamic inversion

Following the aforementioned scenario, it will be assumed thatCSD-profiles f E1A,des and f E2B,des are specified for enantiomer E1 intank A, and enantiomer E2 in tank B, respectively. Other cases, e.g.,when profiles for E1, both in tanks A and B, are specified, can betreated in a similar way. Only the numerical results will changeand, of course, it always has to be checked whether a solutionexists (i.e., the specified profiles are realizable) for a particularcase, with given combination of specified CSDs, growth andnucleation rate functions, parameters and initial conditions.

In Section 3.1, we have described how a specified finalcondition for a PBE (i.e., a specified final CSD for one enantiomerspecies in one tank) can be converted to the correspondingboundary condition for this PBE. For our chosen scenario, wecould perform this step twice, for enantiomer E1 in tank A andenantiomer E2 in tank B. Note that in general this would involvetwo different time scalings. More specifically, one of the resultingboundary conditions is a profile in tE1A, i.e., BE1A=GE1AðtE1AÞ, whilethe other one is a profile in tE2B, i.e., BE2B=GE2BðtE2BÞ.

A constructive way to deal with this problem is to expand thestate vector with variables which keep track of tE1A and tE2B, andthen to solve the resulting set of differential equations in originaltime, similar to the integration of the zero dynamics in Part I.Therefore, we define the overall state as

x¼4 ½m3E1A,m2E1A,m1E1A,m0E1A,m3E2A,m2E2A,m1E2A,m0E2A,m3E1B,m2E1B,m1E1B,m0E1B,m3E2B,m2E2B,m1E2B,m0E2B,

ml,E1A,ml,E2A,ml,E1B,ml,E2B,tE1A,tE2B,�T , ð16Þ

and the control input as

u¼4 ½TA,TB�T : ð17Þ

We arrive at the following complete set of differential equa-tions, which involve the moment model, liquid phase dynamicsand growth and nucleation kinetics, (2) to (9)?, and dynamics forthe evolution of the scaled times

_xi ¼ ð4�iÞGE1Aðu1,xÞxiþ1, i¼ 1 . . .3, ð18aÞ

_x4 ¼ BE1Aðu1,xÞ, ð18bÞ

_xi ¼ ð8�iÞGE2Aðu1,xÞxiþ1, i¼ 5 . . .7, ð18cÞ

_x8 ¼ BE2Aðu1,xÞ, ð18dÞ

_xi ¼ ð12�iÞGE1Bðu2,xÞxiþ1, i¼ 9 . . .11, ð18eÞ

_x12 ¼ BE1Bðu2,xÞ, ð18fÞ

_xi ¼ ð16�iÞGE2Bðu2,xÞxiþ1, i¼ 13 . . .15, ð18gÞ

_x16 ¼ BE2Bðu2,xÞ, ð18hÞ

_x17 ¼�3rskvGE1Aðu1,xÞx2þ _mW ,exx19�x17

mW, ð18iÞ

_x18 ¼�3rskvGE2Aðu1,xÞx6þ _mW ,exx20�x18

mW, ð18jÞ

_x19 ¼�3rskvGE1Bðu2,xÞx10þ _mW ,exx17�x19

mW, ð18kÞ

_x20 ¼�3rskvGE2Bðu2,xÞx14þ _mW ,exx18�x20

mW, ð18lÞ

_x21 ¼ GE1Aðu1,xÞ, ð18mÞ

_x22 ¼ GE2Bðu2,xÞ, ð18nÞ

coupled with two algebraic equations, which follow from (12):

BE1AGE1Aðu1,xÞ ¼ f E1AðtE1A,f�x21,tE1A,f Þ ¼ f E1A,desðtE1A,f�x21Þ, ð18oÞ

BE2BGE2Bðu2,xÞ ¼ f E2BðtE2B,f�x22,tE2B,f Þ ¼ f E2B,desðtE2B,f�x22Þ: ð18pÞ

Note that (18) is valid only as long as

x21rtE1A,f4x22rtE2B,f : ð19Þ

This issue will be addressed later on in this section.According to the basic assumptions in Section 2, (18o) and

(18p) can be solved for u1 and u2, respectively. That is, within thephysically meaningful domain D, the conditions of the implicitfunction theorem are satisfied. Slightly abusing notation, thesesolutions can be written as

u1 ¼BE1AGE1A

� ��1ðx,f E1A,desðtE1A,f�x21ÞÞ, ð20aÞ

u2 ¼BE2BGE2B

� ��1ðx,f E2B,desðtE2B,f�x22ÞÞ: ð20bÞ

The specified final-time CSDs must be continuously differen-tiable functions of L. Then the right hand sides of (18o)and (18p) are continuously differentiable functions of x21 and x22,respectively.

Note that as soon as tE1AðtÞ ¼ tE1A,f or tE2BðtÞ ¼ tE2B,f , the targetCSD in the respective tank is achieved. Then, crystallization in thistank needs to be stopped immediately, as further growth would‘‘destroy’’ the accomplished CSD. In a practical setup, this wouldbe achieved by turning off the liquid exchange, draining therespective tank and letting the controlled crystallization continuein the other one. This implies that at this instant in time themodel structure changes instantaneously to a system of ODEsdescribing the isolated crystallization in the remaining tank.

After substituting (20), and using the basic assumptions on themodel given in Section 2, continuous differentiability of the righthand sides of the differential Eqs. (18) with respect to all the statevariables can be established using the implicit function theoremand other, basic rules of differentiation. In the same way asdescribed in Part I, the existence of a unique solution to thedifferential Eqs. (18) can then be checked, up to the time, called ts,when either x21 ¼ tE1A or x22 ¼ tE2B reaches its final value tE1A,f ,respectively tE2B,f . These values are directly determined by thespecified final-time CSDs. If a unique solution exists, the state attime ts is also uniquely determined. At time ts, the describedswitch in the model structure occurs, and it has to be checkedwhether the ‘‘new’’ set of ODEs has a unique solution from time tsto the time tf, when the remaining one of the state variableskeeping track of scaled time (i.e., tE1A or tE2B) reaches its finalvalue. The initial conditions for this second part of the solutionare given by the relevant part of the state of the first part of thesolution at time ts.

Fig. 4 summarizes the main steps in the complete inversionscheme for the case when desired final CSDs are given forenantiomer E1 in tank A, and for enantiomer E2 in tank B.As mentioned before, solutions analogous to the presented oneare possible, if CSD-profiles for a different pair of enantiomerpopulations are specified. However, we do not consider here thecase when profiles for two enantiomers in one tank are specified.

To reconstruct the final-time CSDs of the remaining popula-tions, we can proceed as follows: the respective scaled times, e.g.,tE2A and tE1B, are included into the ODE system (18). Then, from

Fig. 4. Complete inversion scheme for simultaneous realization of two desired final CSDs, f E1A,des and f E2 B,des, for enantiomers E1 in tank A and E2 in tank B.


the corresponding nucleation and growth rate profiles in time t,we can compute the respective nucleation to growth rate ratios inthe appropriate scaled time, e.g., BE2A=GE2AðtE2AÞ and BE1B=GE1BðtE1BÞ. The corresponding CSDs are then easily determined by(12) and, where applicable (13). This argument shows that thetwo non-specified final-time CSDs are completely determined bythe specified ones. Following these ideas, it is not hard to adaptthe results to the pure forward (i.e., simulation) problem: giventhe temperature profiles, determine the final-time CSDs of allinvolved crystal populations.

Note also that once a solution is obtained, the temporalprofiles of all state variables are known. This allows one to checkwhether these profiles are acceptable with respect to otherprocess and product requirements. In particular, when the speci-fication consists in the CSDs of the preferred enantiomers, itshould be checked whether the final impurity w.r.t. the counterenantiomers is sufficiently low. A possible way to deal with thesituation that either no solution exists for a given CSD specifica-tion or that the unique solution violates additional requirementsis to adjust the desired CSDs in some appropriate manner andthen to repeat the solution procedure.

Recall that in Part I (Hofmann and Raisch, 2012) the dynamicinversion of the moment model was performed in transformedtime. However, the procedure used here (making use of theadditional state variables, e.g., tE1A and tE2B) could also be easilyadopted for the single crystallizer case from Part I. It furtherpermits the following intuitive description of the entire inversionscheme. Assume (like above) that the specification consists in (thenucleated part of) the crystal populations for E1 in tank A and E2 intank B. For the size-independent growth considered here, thegrowth rates of crystals belonging to these distributions are GE1Aand GE2B, respectively. A crystal belonging to the distribution f E1Ainitially having length (characteristic dimension by which the sizeof a particle is described) LE1Að0Þwill, until time t, grow to a lengthof LE1A ¼ LE1Að0Þþ

R t0 GE1Aðt

0Þ dt0 ¼ LE1Að0ÞþtE1A. The values of tE1Aand tE2B are thus equal to the length difference by which anycrystal initially belonging to the distributions f E1A and f E2B,respectively, has grown since the beginning of the batch. On theother hand, crystals nucleated at the beginning of the batch willhave length tE1A,f , respectively tE2B,f , when the process is finished.As these crystals are the largest ones belonging to the nucleatedCSD parts, tE1A,f and tE2B,f are known a priori from the specifica-tion. Furthermore, knowing the current values of tE1AðtÞ and

tE2BðtÞ, the final lengths of crystals nucleating at any time instant tcan be predicted, i.e., these crystals will grow to a final length oftE1A,f�tE1AðtÞ, respectively tE2B,f�tE2BðtÞ. Considering these lengths inrelation to the specified CSDs gives the required nucleation intensitiesat current time. More precisely, as the CSDs represent numberdensities rather than absolute numbers, the nucleation to growthrate ratios, BE1A=GE1AðtÞ and BE2B=GE2BðtÞ, are determined. Consideringalso the properties of the solid and liquid phases at current time, thetemperatures (in each tank in the coupled mode) must then beadjusted appropriately to realize BE1A=GE1AðtÞ and BE2B=GE2BðtÞ.

The solution procedure can be thought of as a (feedback)controller manipulating the temperatures in the tanks. It isreasonable to assume that the relevant liquid and solid phaseproperties (in the numerical example in Section 5, these will bethe liquid mass fractions and the second and third moments ofthe CSDs) can be accurately measured in this hypothetical setting.On the other hand, the values of tE1A and tE2B can also be linked to(at least hypothetically measurable) physical facts of the system:one could, e.g., track a set of particular, initially selected crystalsand monitor their growth throughout the process. For numericalcomputation a moment model can be used to simulate theprocess and obtain the ‘‘measurements’’, and the growth of twoindividual crystals of enantiomer species E1 in tank A and E2 intank B, respectively, can be described by the two additional ODEs(18m) and (18n).

Of course, developing these ideas into a practically implemen-table controller concept requires more thought. For example, allmeasurements are noisy, and direct measurement of CSD proper-ties is often considered a difficult task. Observers are frequentlyused to obtain this information (e.g., Chiu and Christofides, 1999;Shi et al., 2005). Observers (e.g., similar to the one suggested inHofmann et al., 2010) could also be a practical means of keepingtrack of the variables tE1A and tE2B. Furthermore, a straightforwardimplementation of such a controller would be highly model-based: an integral part of it would be the inversion (i.e., solutionfor the temperature) of the ratios of kinetic expressions (i.e.,nucleation and growth rates). Therefore, even if all the variablesjust mentioned can be measured or observed with high precision,accurate achievement of CSDs will strongly depend on thecorrectness of these kinetic expressions. The transfer of relatedfeedback approaches from the context of single substance crystal-lization (e.g., following Nagy, 2009; Zhang and Xu, 2011) mayprovide more robustness against model uncertainties.


4. Analysis of the coupled crystallizer configuration usingidealizing assumptions

In Part I, we discussed the implications on the inversionproblem for a single crystallizer, when nucleation of one of thetwo enantiomer species is neglected. Inversion was then per-formed for that enantiomer species, for which nucleation is notneglected, meaning that its entire nucleated CSD is the specifica-tion for the inversion problem. Here, we introduce two moreidealizations, which are specific to the coupled crystallizer con-figuration, and are motivated from its design principles: liquidexchange between two crystallizers was introduced, so that thecrystallizations can take mutual benefit from a balanced decreaseof the concentrations of both enantiomer species. The newidealizations, and their combination with the ones described inPart I, will be discussed in the following. Fig. 5 illustrates theireffect on crystal size distributions encountered during the

Fig. 5. Effect of possible idealizations; (a) original case without idealizations;(b) perfect symmetry; (c) additionally ideal liquid exchange; (d) additionally

neglecting crystallization of the counter enantiomers or (e) neglecting nucleation

of the preferred enantiomers.

process. Also, the reason behind the depicted order of theirapplication will become clear.

When applying the idealizations, one of course has to checkwhether they are justified for the specific problem, possibly aposteriori, i.e., after temperature profiles TAðtÞ and TBðtÞ have beenobtained. This will be demonstrated for a numerical example inSection 5.

4.1. Perfect symmetry

The benefit from liquid exchange is most balanced if the twoprocesses are run in a totally symmetric way. By this we mean, onthe one hand, that the CSDs of the seed crystals of E1 in tank A,respectively E2 in tank B, are exactly equal, the initial liquid massfractions are equal, i.e., wl,E1Að0Þ ¼wl,E2Bð0Þ and wl,E2Að0Þ ¼wl,E1Bð0Þ,and the temperature profiles are identical, i.e., TAðtÞ ¼ TBðtÞ ¼: TðtÞ.However, more conditions must be met to make the two pro-cesses not only start, but also evolve symmetrically. In particular,all functions and parameters pertaining to tank A should besymmetric to the respective functions and parameters pertainingto tank B, i.e., the roles of E1 and E2 have to be interchanged. This,of course, is an idealization, since it was assumed in Section 2 thatparameters (more generally, growth and nucleation rate func-tions) can differ between the two enantiomers in the mostaccurate models. We will refer to this idealization as perfectsymmetry. Of course, it involves identical geometries of the twotanks. With respect to the process design principles, it makessense to also require that all initial liquid masses (respectivelymass fractions) be equal. However (for this phase of modelidealization), they will not stay equal, i.e., for t40,

ml,E1AðtÞ ¼ml,E2BðtÞaml,E2AðtÞ ¼ml,E1BðtÞ: ð21Þ

Analogous equations hold for any respective pair of variables intanks A and B, e.g., m3E1AðtÞ ¼ m3E2BðtÞ.

When we write the differential equations pertaining to theliquid (dissolved) masses in tank A, and substitute the aboveassumption, we get (compare (5) and (6) in Section 2)

_ml,E1A ¼�3rskvGE1Að�Þm2E1Aþ _mW ,exml,E2A�ml,E1A

mW, ð22aÞ

_ml,E2A ¼�3rskvGE2Að�Þm2E2Aþ _mW ,exml,E1A�ml,E2A

mW: ð22bÞ

Now it is clear that, with perfect symmetry, a closed system ofdifferential equations can be set up for one tank, and thereforehalf of the state variables can be saved, compared to (18). Perfectsymmetry also implies that the CSDs produced in tanks A and B,with the roles of E1 and E2 interchanged, are equal. Hence, theycannot be specified independently. Also, it is clear that, unlike inSection 3.3, crystallizations in both tanks have to be stopped atthe same time.

The resulting set of equations can be interpreted as anequivalent single crystallizer system, augmented by the particularliquid phase dynamics (22). Remember that, for a true singlecrystallizer (see Part I), it was possible to replace the dynamics ofthe liquid phase by algebraic mass balance equations. This is notthe case here.

In Part I, it was shown that a moment model for crystallizationof enantiomers in a single vessel, in general, is not orbitally flat.It is clear that these results apply here.

4.2. Ideal liquid exchange

The second idealization specific to coupled operationis concerned with the rate of liquid exchange. In the limit casewhere there is no liquid exchange between the crystallizers,


i.e., _mW ,ex-0, we arrive at two completely separated processes,which can only be treated ‘‘one by one’’, using the approaches inPart I (Hofmann and Raisch, 2012). No further insight can begained. On the other hand, the benefit from the coupled setupappears most clearly in the opposite limit case, when_mW ,AB ¼ _mW ,BA ¼ _mW ,ex-1. This ideal liquid exchange implies

an ideal mixing of the liquid phases. The two tanks then,effectively, share the same solution. The liquid phase dynamicscan thus be replaced by the following mass balance equations:

ml,E1AðtÞ ¼ml,E1BðtÞ ¼ml0,E1A�0:5rskvðm3E1AðtÞ�m3E1Að0Þþm3E1BðtÞ�m3E1Bð0ÞÞ, ð23aÞ

ml,E2AðtÞ ¼ml,E2BðtÞ ¼ml0,E2A�0:5rskvðm3E2AðtÞ�m3E2Að0Þþm3E2BðtÞ�m3E2Bð0ÞÞ: ð23bÞ

In the following we will focus on the combination of ideal liquidexchange and perfect symmetry.

4.3. Perfect symmetry and ideal liquid exchange

In this case, the mass balance equations become even simpler

ml,E1AðtÞ ¼ml,E1BðtÞ ¼ml,E2AðtÞ ¼ml,E2BðtÞ¼ml0,E1A�0:5rskvðm3E1AðtÞ�m3E1Að0Þþm3E2AðtÞ�m3E2Að0ÞÞ: ð24Þ

We can define ml:¼ ml,E1A ¼ml,E2A ¼ml,E1B ¼ml,E2B, respectivelywl:¼ wl,E1A ¼wl,E2A ¼wl,E1B ¼wl,E2B. Furthermore, miE1AðtÞ ¼ miE2BðtÞ ¼:mipðtÞ, i¼ 0 . . .3, and miE2AðtÞ ¼ miE1BðtÞ ¼: micðtÞ, i¼ 0 . . .3. Theindices p and c stand for the preferred and counter enantiomers,respectively. Consequently,

mlðtÞ ¼mlð0Þ�0:5rskvðm3pðtÞ�m3pð0Þþm3cðtÞ�m3cð0ÞÞ: ð25Þ

This equation shows that, as in the case with perfect symmetryonly, an equivalent model for a single crystallizer can be identified.This time, however, the liquid phase can be described by algebraicmass balance equations, i.e., there is no need to include extra statevariables for the liquid phase.

With the perfect symmetry and ideal liquid exchange assumptionsand (9), we get GE1AðT ,wlÞ ¼ GE2BðT,wlÞ and GE2AðT ,wlÞ ¼ GE1BðT,wlÞ.Note that the growth rates depend only on the temperature andproperties of the liquid phase, each of which are equal in both tanks.It is reasonable to further restrict the following discussion to modelswhere (naturally) the growth rates in tank A depend on thetemperature and liquid phase properties in tank A in exactly thesame way as the growth rates in tank B depend on the temperatureand liquid phase properties in tank B. Then, in fact, all four growthrates in the system are equal, GðtÞ:¼ GE1AðtÞ ¼ GE2AðtÞ ¼ GE1BðtÞ ¼GE2BðtÞ. On the other hand, nucleation rates of the two enantiomerspecies in one tank are, in general, not equal. We therefore defineBpðtÞ:¼ BE1AðtÞ ¼ BE2BðtÞ and BcðtÞ:¼ BE2AðtÞ ¼ BE1BðtÞ.

Note that the algebraic mass balance equation (25) can beincorporated into the expressions for nucleation and growth ratessuch that, by defining the state and input as

x¼4 ½m3p,m2p,m1p,m0p, m3c ,m2c ,m1c ,m0c�T , ð26Þ

and

u¼4 T ¼ TA ¼ TB, ð27Þ

the entire model can be written as

_xi ¼ ð4�iÞGðx,uÞxðiþ1Þ, i¼ 1 . . .3, ð28aÞ

_x4 ¼ Bpðx,uÞ, ð28bÞ

_xi ¼ ð8�iÞGðx,uÞxðiþ1Þ, i¼ 5 . . .7, ð28cÞ

_x8 ¼ Bcðx,uÞ: ð28dÞ

Note that, in contrast to the single tank scenario discussed in PartI, both species exhibit the same growth rate in (28).

4.3.1. Lack of orbital flatness

Unfortunately, even with only one common growth rateappearing in (28), this system is still not orbitally flat. This isargued in the following, by referring to typical expressions for thegrowth and nucleation rates.

In Part I, it was shown for simple nucleation rate expressionsthat the model for crystallization of enantiomers in a single vesselis not orbitally flat. Only secondary nucleation was considered,with the same nucleation exponent for both enantiomer species.Following the same lines of argument as in Part I is only possiblewhen considering a more detailed model structure, for exampleone with primary and secondary nucleation,

Bpð�,TÞ ¼ Bprimð�,TÞþBsec,0ð�,TÞm3p, ð29aÞ

Bcð�,TÞ ¼ Bprimð�,TÞþBsec,0ð�,TÞm3c: ð29bÞ

Here, the primary nucleation rate, Bprim, as well as Bsec,0, whichcharacterizes secondary nucleation, can be equal for both enan-tiomer species (supersaturations for the preferred and counterenantiomers are equal due to the idealizing assumptions). Differ-ences in the total nucleation rates are (naturally) caused bydifferent amounts of crystalline material of the preferred andcounter enantiomers, i.e., m3pam3c .

To apply the orbital flatness test from Part I, recall the mainingredients of this test: given a system _x ¼ f ðx,uÞ of n ODEs withscalar input u, if for each of the n right hand sides, say f a, we canfind two other right hand sides, say f b and f c , such that thefollowing inequalities hold in at least one (initially selected) pointðxp,upÞ, orbital flatness can be excluded:

f af0b�f

0af ba0, ð30aÞ

and

f a f0b f00c�f a f

00b f0c�f

0a f b f

00cþ f

0a f00b f cþ f

00a f b f

0c�f

00a f0b f c a0, ð30bÞ

where ð�Þ0 :¼ d=duð�Þ and ð�Þ00:¼ d2=du2ð�Þ.After substituting (29) into (28), with u¼4 T, it is clear how to

choose f b and f c , depending on f a: when f a corresponds to one ofthe right hand sides of (28a) or (28c), then f b and f c are chosen asthe right hand sides of (28b) and (28d); when f a corresponds toeither (28b) or (28d), then either f b or f c must be chosen as theother nucleation rate function. Then, and choosing a point in thestate space where mipa0, i¼ 0 . . .3, and mic a0, i¼ 0 . . .3, andm3pam3c , the conditions (30) will translate into, e.g.,

BprimB0sec,0�B

0primBsec,0a0, ð31aÞ

and

B0primB00sec,0G�B

00primB

0sec,0G�BprimB

00sec,0G

0 þB00primBsec,0G0

þBprimB0sec,0G00�B0primBsec,0G

00a0: ð31bÞ

Now the results from Part I can be applied. We choose specific(but very basic) examples of growth and nucleation kinetics,which are adapted from Angelov et al. (2008)

Gð�Þ ¼ kgðS�1Þg , ð32aÞ

Bprimð�Þ ¼ kb,prime�bprim=ln2 S, ð32bÞ

Bsec,0ð�Þ ¼ kb,secðS�1Þbsec , ð32cÞ

where the supersaturation is given by

S¼ wlweqðTÞ

, ð32dÞ


and weq(T) is a static, invertible function of temperature T. For thekinetics (32), the conditions (31) can be verified numerically forspecific choices of bprim, bsec and g. The exact values of growth andnucleation coefficients are not important as long as kg 40,kb,prim40 and kb,sec40. For example, the conditions can beverified for g¼1, bprim ¼ 0:06 and bsec ¼ 4, values taken fromAngelov et al. (2008).

In conclusion, we have demonstrated for typical growth andnucleation kinetics that the idealized model (28), which isarguably even simpler than the single tank model from Part I,lacks the orbital flatness property.

4.3.2. Time scaling

In preparation for the following sections, we apply the timescaling:

dt :¼ Gð�Þ dt, ð33Þ

to system (28). This gives

d

dt xi ¼ ð4�iÞxðiþ1Þ, i¼ 1 . . .3, ð34aÞ

d

dt x4 ¼BpGðx,uÞ, ð34bÞ

d

dtxi ¼ ð8�iÞxðiþ1Þ, i¼ 5 . . .7, ð34cÞ

d

dtx8 ¼

BcGðx,uÞ: ð34dÞ

Because (28) has been shown to be not orbitally flat, it is clearthat a flat output does not exist for (34).

4.4. Combination with the idealizations from Part I

In the following, we investigate the additional effect of theidealizations introduced in Part I, on the model described in Section4.3, i.e., the one resulting from the combination of the perfectsymmetry and ideal liquid exchange assumptions. Recall thatthe idealizations introduced in Part I are neglecting nucleation ofthe unseeded (counter) enantiomer, or neglecting nucleation of theseeded (preferred) enantiomer. The former seems especially justifiedhere, since nucleation of the counter enantiomer(s) is expected to besuppressed to a greater extent than in the single crystallizer config-uration. We will show that the latter case is interesting in two ways:first, the system resulting from this particular combination ofidealizations is indeed an orbitally flat system. Second, as will becomeclear below, by specifying a single final-time CSD, a direct tradeoff canbe made between the resulting crystallized amounts of the preferredand counter enantiomers.

4.4.1. Neglecting nucleation of the counter enantiomers

If nucleation (and therefore crystallization entirely) isneglected for the unseeded (counter) enantiomers, representedby the state variables x5 to x8 in (34), then a system consistingonly of the first four equations in (34) remains, describing theevolution of the moments of the preferred enantiomers. To obtainthis system, one has to set x5 to x8 to zero to give

d

dt xi ¼ ð4�iÞxðiþ1Þ, i¼ 1 . . .3, ð35aÞ

d

dtx4 ¼

BpGðx,uÞ: ð35bÞ

Under the basic assumptions in Section 2, the system (35) canthen be verified to be flat, with y :¼ x1 ¼

4 m3p as a flat output. Thisstep is analogous to checking the flatness of the single crystallizersystem with neglecting nucleation of the unseeded enantiomer,

only that now the mass balance equation reads differently(before, the dissolved mass of the unseeded enantiomer wasconstant; now, it is always equal to the dissolved mass of theseeded enantiomer). Using flatness-based inversion of this idea-lized model, a prescribed final nucleated CSD of the preferredenantiomers can then be realized. This especially makes sense ifthe mass of the grown seed crystals is required to be lowcompared to the nucleated mass of the preferred enantiomers.Of course, it should be checked a posteriori (by simulating thenon-idealized model) that nucleation of the counter enantiomersis sufficiently low.

4.4.2. Neglecting nucleation of the preferred enantiomers

The dual case, when nucleation of the seeded (preferred)enantiomers is neglected, is also interesting from an applicationpoint of view. This is especially true if the specification requiresthe mass of grown seed crystals to be high compared to thenucleated mass of the preferred enantiomers.

For this case, in the single crystallizer scenario, it was shown inPart I how the moments of the preferred enantiomer can beexpressed as polynomial functions of the respective scaled time.Here, as all growth rates are equal, and we have applied a timetransformation, resulting in (34), this implies

m0pðtÞ ¼ m0pð0Þ ¼ const:; ð36aÞ

m1pðtÞ ¼ m0pð0Þtþm1pð0Þ, ð36bÞ

m2pðtÞ ¼ m0pð0Þt2þ2m1pð0Þtþm2pð0Þ, ð36cÞ

m3pðtÞ ¼ m0pð0Þt3þ3m1pð0Þt2þ3m2pð0Þtþm3pð0Þ: ð36dÞ

When x1 to x4 in (34) are substituted by these polynomialfunctions, one arrives at a system with state dimension four andexplicit (t)-time dependency

d

dtxi ¼ ð8�iÞxðiþ1Þ, i¼ 5 . . .7, ð37aÞ

d

dtx8 ¼

BcGðx,t,uÞ: ð37bÞ

System (37) looks very similar to (35), and it can be verified thaty :¼ x5 ¼

4 m3c is a flat output. All state variables algebraically dependon y and its first three derivatives w.r.t. t. The input algebraicallydepends on the fourth derivative of y via a (t)-time dependent law.Such notions of flatness have already been used in examples given inthe literature (see, e.g., Antonov et al., 2008; Sira-Ramı́rez, 2004). Notethat this explicit (t)-time dependency poses no extra difficulty for ourinversion scheme, since the values of t are inherently known when afinal-time CSD is specified, compare (12).

5. Numerical example

The following case study builds on and extends the one from PartI. The equations and parameters are based on ongoing work regardingthe modeling of the crystallization of L-/D-threonine in the PCF groupat the Max Planck Institute in Magdeburg (Elsner et al., 2011; Eickeet al., 2010a). In the following, L-threonine corresponds to E1, andD-threonine to E2. The kinetic expressions for growth and nucleationrates have been summarized in Part I. As stated in Section 2, themodel consists of PBEs and additional mass balance differentialequations. Essentially the same model has also been used byHofmann et al. (2011). Here, however, we assume size-independentgrowth rates, so that conversion of the PBE model into a momentmodel is justified. As explained in Section 2, for simplicity, a constantsolvent mass stream is assumed instead of a constant volumeexchange rate. Only one additional parameter, the solvent mass

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 109

Fig. 6. Scenario one, final CSDs, tank A, enantiomer E1.

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5x 108

Fig. 7. Scenario one, final CSDs, tank A, enantiomer E2.


exchange rate _mW ,ex, must be specified in the context of modeling thecoupled crystallizer configuration.

The values of parameters and initial conditions can be found ina table in Part I. Recall that the two tanks must share the same setof parameters. In this case study, additionally, we assume sym-metrical (i.e., equal) parameters for E1 and E2. Further, we usesymmetrical initial conditions, i.e., the initial conditions given inthe table in Part I for enantiomer E1 now pertain to enantiomer E1in tank A as well as to enantiomer E2 in tank B. In this case study,differing specifications for the final-time CSDs in tanks A and B areconsidered the only source of asymmetry.

In the following, in order to cover most of the ideas presentedin Sections 3 and 4, we have chosen three different scenarios. Ineach of these, the final-time CSDs of two of the involved nucleatedcrystal populations are specified.

5.1. Scenario one: identical final-time CSDs specified for the

preferred enantiomers

First, we aim at realizing identical prescribed (desired) final-time CSDs for the preferred enantiomers, i.e., the seeded enantio-mers, E1 in tank A and E2 in tank B,

f E1A,desðLÞ ¼ f E2B,desðLÞ ¼

m3f ,n,des � exp �20L

Lmax

� �

R Lmax0 L

3 exp �20LLmax

� �dL

, LA ½0,Lmax�,

f E1A,seedðL�LmaxÞ, L4Lmax:

8>>>>><>>>>>:

ð38Þ

Note that f E1A,seedðLÞ � f E2B,seedðLÞ. The free parameters m3f ,n,des andLmax determine the resulting third moment of the nucleated partof the specified final CSD, and the maximum length of crystalsbelonging to that part, respectively. The same values used inscenario one of the numerical example in Part I are chosen

m3f ,n,des ¼ 10:0 ½cm3�; Lmax ¼ 0:5 ½mm�: ð39Þ

Recall that the respective upper CSD parts are merely shiftedversions of the CSDs of the seed crystals.

We compare the exact solution, Section 3.3, to the one basedon the orbital flatness property of the system resulting fromneglecting the nucleation of the counter enantiomers, and theassumptions of perfect symmetry and ideal mixing of the liquidphases in the two crystallizers, Section 4.4.1. Note that perfectsymmetry holds for the chosen values of parameters and initialconditions, and because the desired final-time CSDs of thepreferred enantiomers are equal.

It will be seen that a high enough liquid exchange rate is mostcritical for justification of the idealized, flatness-based inversiontechnique. To demonstrate this, the analysis is repeated with threedifferent solvent mass exchange rates, i.e., the rate is either_mW ,ex ¼ 34:1 ½g=min�, _mW ,ex ¼ 341 ½g=min� or _mW ,ex ¼ 1706 ½g=min�.

Comparison between exact and simplified solutions is alwaysconducted in the following manner: the input (temperature)profiles computed by the exact and simplified solution techni-ques, respectively, are compared. Furthermore, the temperatureprofile corresponding to the simplified solution is applied as aninput to the original model, and the resulting deviations of thefinal-time CSD from the specification are shown. In addition toplots of CSDs of final crystal populations, plots of mass densityfunctions (MDFs) are shown, which may be more appropriate tojudge errors. To obtain these CSDs, respectively MDFs, theforward (i.e., simulation) problem is solved.

The numerical results of scenario one are shown in Figs. 6–14.We only show plots pertaining to tank A. Due to symmetry, theplots for tank B look identical, after switching the roles of E1 and

E2. In the temperature plots, multiple lines are shown for theexact scheme (labeled ‘‘exct. inv.’’), each one for a different liquidexchange rate. Recall that the idealized scheme (labeled ‘‘flatinv.’’) is not concerned with the actual exchange rate, but ratherassumes an infinitely fast one.

In all other plots, the final-time CSDs, final-time MDFs ortemporal profiles resulting from application of the temperatureprofiles computed using the exact as well as simplified inversiontechniques to the original, non-idealized model are shown.In these plots, the value of the liquid exchange rate has an impactfor both the exact and simplified inversion schemes. The onlyexceptions are the final-time CSDs, respectively MDFs, of thepreferred enantiomers. The exact inversion scheme adjusts to theactual liquid exchange rate and, thus, always leads to the desiredCSDs, respectively MDFs. With the simplified inversion schemethe target is never precisely achieved. However, the faster theactual liquid exchange is, the closer the result gets to thespecification. In the plots of the third moments, we distinguishparts from nucleation (index n) and growing seeds (index s). Forexample, m3E1A ¼ m3E1A,nþm3E1A,s. This splitting is, of course, only

0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14

Fig. 8. Scenario one, final MDFs, tank A, enantiomer E1.

0 0.1 0.20

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.3 0.4 0.5

Fig. 9. Scenario one, final MDFs, tank A, enantiomer E2.

0 10 20 30 40 500

5

10

15

20

Fig. 10. Scenario one, third moments, tank A, enantiomer E1.

0 10 20 30 40 500

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Fig. 11. Scenario one, third moments, tank A, enantiomer E2.

0 10 20 30 40 501.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

Fig. 12. Scenario one, supersaturation profile, tank A, enantiomer E1.

0 10 20 30 40 501.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

Fig. 13. Scenario one, supersaturation profile, tank A, enantiomer E2.


0 10 20 30 40 5026

27

28

29

30

31

32

Fig. 14. Scenario one, temperature profile, tank A.

0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14x 107

Fig. 15. Scenario two, final CSDs, tank A, enantiomer E1.

0 0.1 0.2 0.3 0.4 0.50

1

2

5

6

7

8

9x 104

3

4

Fig. 16. Scenario two, final CSDs, tank A, enantiomer E2.


relevant for the preferred enantiomer species, for which seedcrystals are initially added.

When compared to the results of scenario one in Part I, thesuperiority of the coupled setup clearly turns out. In particular, forthe same specified CSD of the preferred enantiomer, the resultingnucleation of the counter enantiomer is greatly reduced in eachtank, both for the exact and simplified inversion schemes. Asexpected, higher liquid exchange rates provide additional benefit.

From the above observation it is clear that neglecting nuclea-tion of the counter enantiomers is very well justified. Thesimulations also indicate that the most critical of the idealizingassumptions is the infinitely fast liquid exchange. Hence, it needsto be checked whether the deviations caused by this assumptionare acceptable for the specific application.

Computing the solution to the exact inversion problem (for thelowest liquid exchange rate) took approximately 85 s. Computing thesolution to the idealized problem took approximately 1.22 s. Notethat reducing computation time may be possible in either casethrough more optimized implementations. The numerical resultswere computed using MATLABs on an AMD AthlonTM 64X2 DualCore Processor 3800þ running at 1000 MHz, with 2.94 GB RAM.

5.2. Scenario two: identical final-time CSDs specified for the counter

enantiomers

Now we consider the case when identical final CSD profiles arespecified for the counter enantiomers, i.e., the unseeded enantio-mers, E2 in tank A and E1 in tank B. We compare the exactsolution, Section 3.3, to the approximate one based on theassumptions of perfect symmetry, ideal liquid exchange andneglecting nucleation of the preferred enantiomers. As describedin Section 4.4.2, the moment model resulting from these idealiz-ing assumptions is orbitally flat, making its inversion a trivialtask. Again, we demonstrate the impact of the liquid exchangerate on the quality of the idealized solution.

As before, exponential final-time CSDs are specified,

f E2A,desðLÞ ¼ f E1B,desðLÞ ¼

m3f ,n,des � exp �20L

Lmax

� �

R Lmax0 L

3 exp �20LLmax

� �dL

, LA ½0,Lmax�,

0, L4Lmax,

8>>>>><>>>>>:

ð40Þ

with

m3f ,n,des ¼ 0:0001 ½cm3�; Lmax ¼ 0:5 ½mm�: ð41Þ

Note that Lmax corresponds to the maximum length of any crystalbelonging to the specified CSD. The numerical results are shownin Figs. 15–23. Again, the plots for tank B look identical, afterswitching the roles of E1 and E2.

The first thing to note is that the final-time third momentcorresponding to the CSD of the grown seed crystals is almost thesame as in scenario one, especially for the faster liquid exchangerates. This was expected for the following reasons: (a) this part ofthe final-time CSD of the preferred enantiomer is solely deter-mined by the CSD of the seed crystals and tE1A,f (respectivelytE2B,f ), i.e., the length by which the seed crystals have grown;(b) all growth rates in the system are almost equal, i.e.,tE1A,f ¼ tE2B,f � tE1B,f ¼ tE2A,f . The higher the liquid exchange rate,the closer they get. If the specifications are met,tE2A,f ¼ tE1B,f ¼ Lmax. The above implies that also the growth ofthe seed crystals of the preferred enantiomers is approximately

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

Fig. 17. Scenario two, final MDFs, tank A, enantiomer E1.

0 0.1 0.2 0.3 0.4 0.50

1

2

x 10−4

Fig. 18. Scenario two, final MDFs, tank A, enantiomer E2.

0 10 20 30 40 50 60 700

5

10

15

20

Fig. 19. Scenario two, third moments, tank A, enantiomer E1.

0 10 20 30 40 50 60 700

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8 x 10−4

Fig. 20. Scenario two, third moments, tank A, enantiomer E2.

0 10 20 30 40 50 60 701.05

1.1

1.15

1.2

1.25

1.3

Fig. 21. Scenario two, supersaturation profile, tank A, enantiomer E1.

0 10 20 30 40 50 60 701.05

1.1

1.15

1.2

1.25

1.3

Fig. 22. Scenario two, supersaturation profile, tank A, enantiomer E2.


0 10 20 30 40 50 60 7027

28

29

30

31

32

33

Fig. 23. Scenario two, temperature profile, tank A.

3.5

4

4.5x 109


determined by the parameter Lmax, and the accuracy improveswith higher liquid exchange rate; (c) both the seed CSD and Lmaxare the same as in scenario one.

The above argument shows that, by specifying the final-timeCSDs of the counter enantiomers, one can directly influence theresulting final-time third moments of both the counter enantio-mers and the grown seeds of the preferred enantiomers. In thisway, the level of impurity of the preferred enantiomers w.r.t. thecounter enantiomers can be controlled.

Note that we have specified a final-time third moment of thecounter enantiomers which is only one tenth of that in thecorresponding single tank scenario (scenario two) in Part I. Onthe other hand, the resulting final third moments of the grownseeds of the preferred enantiomers are approximately the same,and the process duration tf is slightly shorter. This backs theobservation from scenario one on the superiority of the coupledcrystallizer configuration, when compared to crystallization in asingle tank.

Again, for low liquid exchange rates, relatively large errorsresult from the idealized inversion solution based on orbitalflatness. Note also that increasing the liquid exchange rate from341 ½g=min� to 1706 ½g=min� does not imply noticeable improve-ment. Rather than that, the error in the final-time CSD seems toincrease, see Figs. 16 and 18. This indicates that neglectingnucleation of the preferred enantiomers is the paramount sourceof errors when the liquid exchange rate is high.

Computing the solution to the exact inversion problem (for thelowest liquid exchange rate) took approximately 78 s. Computingthe solution to the idealized problem took approximately 1.25 s.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.5

1

1.5

2

2.5

3

Fig. 24. Scenario three, final CSDs, tank A.

5.3. Scenario three: non-identical final-time CSDs specified for the

preferred enantiomers

Finally, we demonstrate the solution for the asymmetric case,i.e., we require different specifications for tanks A and B,

f E1A,desðLÞ ¼

m3E1A,f ,n,des � exp �20L

Lmax

� �

R Lmax0 L

3 exp �20LLmax

� �dL

, LA ½0,Lmax�,

f E1A,seedðL�LmaxÞ, L4Lmax:

8>>>>><>>>>>:

ð42aÞ

f E2B,desðLÞ ¼

m3E2B,f ,n,des � exp �20L

Lmax

� �

R Lmax0 L

3 exp �20LLmax

� �dL

, LA ½0,Lmax�,

f E2B,seedðL�LmaxÞ, L4Lmax:

8>>>>><>>>>>:

ð42bÞ

Here, we choose the same Lmax for the specifications in both tanks.Again, note that f E1A,seedðLÞ � f E2B,seedðLÞ. The asymmetric part ofthe specification consists in requiring the double amount ofproduct in tank B than in tank A, i.e., we choose

m3E1A,f ,n,des ¼ 10:0 ½cm3�,

m3E2B,f ,n,des ¼ 20:0 ½cm3�,

Lmax ¼ 0:5 ½mm�: ð43Þ

Asymmetric specifications may naturally come along with differ-ent purposes of the two simultaneously produced pure enantio-mer species. For example, one of the two species may be intendedfor use as a drug, involving tight CSD specifications due tobioavailability requirements, while the other species may bemerely a by-product. On the other hand, the following resultsmay also apply qualitatively when asymmetry is caused by otherreasons which, for simplicity, are not considered in this numericalexample. For instance, seed crystals of the two species may bemanufactured in a different way (possibly by different manufac-turers) and thus have different CSDs.

Note that in this scenario, the evolution of the state variablesin both tanks is necessarily asymmetric. Hence, the idealizedsolutions described in Section 4 are not applicable. Therefore, thenumerical results shown in Figs. 24–32 only reflect the exactinversion technique described in Section 3.3. Computing thesolution took approximately 100 s.

In the plots of supersaturation and temperature profiles, thelines corresponding to tank B finish earlier than those of tank A,indicating that crystallization in tank A is continuing whilecrystallization in tank B has already been stopped. This corre-sponds to the period where the third moments in tank B stayconstant in Fig. 29.

Somewhat surprisingly, the crystallization has to be stoppedfirst in tank B, where a higher amount of preferred enantiomercrystalline material is specified. This is explained by the fact thatthe supersaturation levels in tank B resulting from the exactinversion procedure are higher than in tank A. This is also the

0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12x 109

Fig. 25. Scenario three, final CSDs, tank B.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

5

10

15

Fig. 26. Scenario three, final MDFs, tank A.

0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

25

30

Fig. 27. Scenario three, final MDFs, tank B.

0 10 20 30 40 500

2

4

6

8

10

12

14

16

18

Fig. 28. Scenario three, third moments, tank A.

0 10 20 30 40 500

5

10

15

20

Fig. 29. Scenario three, third moments, tank B.

0 10 20 30 40 501

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

Fig. 30. Scenario three, supersaturation profiles, tank A.


0 10 20 30 40 501

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

Fig. 31. Scenario three, supersaturation profiles, tank B.

0 10 20 30 40 5025

26

27

28

29

30

31

32

Fig. 32. Scenario three, temperature profiles.


reason why the final amount of counter enantiomer in tank B ishigher than in tank A.

Also note the kink which is visible in the supersaturation plotfor E2 in tank A. This non-differentiability is caused by the instantswitching-off of the liquid exchange between the tanks, which is aconsequence of stopping crystallization in tank B.

6. Conclusion

In this paper, we have considered population balance andmoment models describing preferential crystallization in twocoupled vessels, when crystallization temperatures in the twotanks are the manipulated variables. In a technical application, ineach tank one species, the preferred enantiomer, is seeded,while nucleation of the other species, the counter enantiomer, isundesired.

In Part I, we have summarized the concepts of flatness andorbital flatness, which are properties of nonlinear systems thatgreatly facilitate analysis and control. Based on a well knownnecessary condition for flatness, we have also derived a test toexclude orbital flatness of typical moment models for preferentialcrystallization. Here, in Part II, we have extended this test to also

show that typical models for the coupled crystallizer setup, whichare multiple-input systems, are not orbitally flat.

We then have presented a technique for inversion of thesystem, i.e., for computation of the temperature profiles whichsimultaneously realize one specified final crystal size distribution(CSD) in each tank. Time transformations according to the growthrates of the enantiomer species targeted by the specified CSDs areessential ingredients of the inversion techniques presentedin Part I and II, which essentially proceed in two steps: exploita-tion of straight characteristic curves of time-scaled populationbalance equations (PBEs) is followed by the solution of a setof ordinary differential equations (ODEs) corresponding to amoment model and, in the case of the coupled crystallizers, liquidphase dynamics. While in Part I both steps can be carried outwithin a common transformed time, the situation gets slightlymore complicated for the simultaneous realization of two finalCSDs in Part II. Here, we suggest the introduction of extra statevariables, which ‘‘keep track’’ of two different scaled times,and the addition of respective differential equations to the setof ODEs. Besides determination of the temperature profiles,our results also involve reconstruction of the final-time CSDsof the remaining crystal populations, which are not part of thespecification.

In Part I and II, this type of inversion problem has beenaddressed for the first time in the context of preferential crystal-lization. Two different crystallizer configurations were consid-ered. Although (as shown by formal arguments in both Part I andII) moment models for preferential crystallization are not orbi-tally flat, our approach still follows the same lines (combinationof the method of characteristics and nonlinear inversion of amoment model based on geometric methods) as previous, orbitalflatness-based work for the single substance crystallization case.This is beneficial in that the resulting problem can be stated as aninitial value problem for a system of a small number of ordinarydifferential equations. The numerical part of the solution can thusbe carried out efficiently using standard software. In a numericalexample, we have shown that the computation times for theidealization-based schemes are drastically smaller than for theexact schemes. The latter, however, are still reasonable (e.g.,when seen in relation to the time scale of the actual process).

While prescribing the final-time CSDs of the preferred enan-tiomer (in a single tank), respectively the preferred enantiomers(in coupled mode) can have various applications regarding down-stream processing (e.g., filtration) or bioavailability (e.g., dissolu-tion of drugs in the human body), the dual case, i.e., prescribingfinal-time CSDs of the counter enantiomer(s) can also be inter-esting from an application point of view. In this case, one cancontrol the final amount of the counter enantiomer(s), which canbe important when considering purity requirements. However,purity also depends on the amount of crystallization of thepreferred enantiomer(s). In the coupled case, a qualitative rela-tion between the specified and the resulting non-specified final-time CSD in each tank can be exploited a priori: under symmetricconditions and for sufficiently high liquid exchange rates, thegrowth rates of both enantiomer species in one tank are almostequal. Consequently, by specifying the final-time CSD of thecounter enantiomers, one also approximately determines thegrowth of the seeds of the preferred enantiomers. Althoughnucleation of the preferred enantiomers is only known aftercomputing the solution to the inversion problem, a certainminimum purity of the preferred enantiomers can thus be guar-anteed (because the additional nucleation of the preferred enan-tiomers can only increase purity).

Idealizing assumptions on the models have been investigatedin both Part I and II, some of them turning the models intoorbitally flat systems, which can be efficiently inverted. It has


been demonstrated in numerical examples that neglectingnucleation of counter enantiomers is suitable for obtainingtemperature profiles which realize desired final-time CSDs ofpreferred enantiomers. In particular, if it can be justified aposteriori that counter enantiomer crystallization is sufficientlylow for typical purity constraints to be met (e.g., an impurity ofless than 1%), then one can at the same time expect highlyaccurate achievement of the preferred enantiomer CSD specifica-tion. Note that due to the benefits arising from liquid exchange,nucleation of the counter enantiomers is suppressed to a higherextent in the coupled crystallizer setup treated in this paper.

Note that, whereas in the single crystallizer case an orbitallyflat moment model could not be obtained by neglecting nuclea-tion of the preferred enantiomer, the situation is different in thecoupled case. There, an orbitally flat moment model can beobtained by combining the idealizations of perfect symmetryand ideal liquid exchange with either neglecting nucleation of thecounter or the preferred enantiomers. The latter case is encoura-ging in the face of the above comments on the impact of thespecification on product purity: trading off the final-time massesof the preferred enantiomers (seed growth) and the counterenantiomers poses a meaningful optimization problem whichmay be tackled via efficient, orbital flatness-based techniquessuggested in Vollmer and Raisch (2003) or Hofmann and Raisch(2010).

In future work, the developed feedforward control schemesshould be complemented by appropriate feedback control. Ourtreatment naturally suggests the use of methods from geometriccontrol theory, such as exact, respectively input–output linear-ization. In Vollmer and Raisch (2006), this was already demon-strated in the context of single substance crystallization. We willalso investigate to which extent the idealization-based inversionschemes, or optimal control solutions based on this, can be usefulin the context of model predictive control, a field extensivelystudied in the context of single substance crystallization (e.g., Shiet al., 2005, 2006; Nagy, 2009). Another possible starting point forthe addition of feedback has been indicated near the end ofSection 3.3.

Another line of future research concerns more complex PBEmodels for crystallization (and preferential crystallization), forwhich closed moment models cannot be obtained. This can, e.g.,be due to size-dependent growth rates or the addition of finesremoval units (especially in a coupled preferential crystallizationscenario, see Section 1). For the extension of the inversionmethods presented in Part I and II to these cases, solutiontechniques which involve approximation of PBE models by loworder ODE models are very promising.

Acknowledgments

We gratefully acknowledge financial support by the GermanResearch Foundation for this project (DFG RA 516/7-1). We alsoacknowledge the productive cooperation with our project part-ners from the PCF group at the Max Planck Institute for Dynamicsof Complex Technical Systems: Matthias Eicke, Martin PeterElsner and Andreas Seidel-Morgenstern.

Appendix A. Test for lack of orbital flatness

Here, we derive a test to show that moment models describingpreferential crystallization in two coupled vessels typically do notpossess the orbital flatness property. In contrast to Part I, we arenow dealing with a system with two inputs. A flat output, if itexisted in a suitable, transformed time, would therefore have to

be two-dimensional as well (Fliess et al., 1999). Nevertheless, wecan efficiently extend the test derived in Part I by considering amodel which has the following structure:

_x1 ¼ f 1ðx,u1Þ,^

_xn1 ¼ f n1 ðx,u1Þ, ðA:1aÞ

_xn1þ1 ¼ f n1þ1ðx,u2Þ,^

_xn1þn2 ¼ f n1þn2 ðx,u2Þ, ðA:1bÞ

_xn1þn2þ1 ¼ f n1þn2þ1ðx,u1,u2Þ,^

_xn1þn2þn3 ¼ f n1þn2þn3 ðx,u1,u2Þ: ðA:1cÞ

The derivatives in (A.1a) and (A.1b) are only directly influ-enced by one input, i.e., u1 or u2, respectively. In the following, wewill refer to (A.1a) or (A.1b) using the term subsystem. By this wemean that the remaining state variables, whose derivatives do notappear in the respective sets of equations, are treated as con-stants, and that these systems are treated as having only oneinput, u1 or u2, respectively. In the following argument we willshow that if, after choosing an overall point ðxp,upÞ (see below),the test for single-input systems derived in Appendix of Part I(Hofmann and Raisch, 2012) establishes the absence of orbitalflatness for each of these subsystems, then the overall system(A.1) cannot be orbitally flat. The conditions of the test given inPart I are also briefly summarized in Section 4.3.1.

In Part I, we give a short review of flatness and orbital flatness.This includes the ruled manifold criterion (Sluis, 1993; Rouchon,1994; Fliess et al., 1995; Martin et al., 199

chemical engineering science...1 is seeded in tank a, whereas enantiomer e 2 is seeded in tank b....

Documents