Computers and Chemical Engineering 58 (2013) 381– 389
Contents lists available at ScienceDirect
Computers and Chemical Engineering
jo u r n al homep age: www.elsev ier .com/ locate /compchemeng
ote
ime optimal cyclic crystallization
aim Bajcinca ∗
ax-Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany
r t i c l e i n f o
rticle history:eceived 13 December 2012eceived in revised form 25 March 2013ccepted 9 May 2013
a b s t r a c t
A time optimal cyclic control scheme for shape manipulation of multivariate crystal populations involvingsequences of subsequent growth and dissolution phases is proposed in this note. Such strategies employthe unequal growth and dissolution rates for attaining morphologies that do not result directly from apure growth or dissolution phase only. We prove that minimum time trajectories can be constructed by
Crystal shape is a critical aspect in sophisticated particle prod-cts in numerous industries (Patience & Rawlings, 2001). For
nstance, the surface structure and binding energies, and thus reac-ivity, varies with crystallographic orientation (Yang et al., 2008).rom the engineering point of view, manipulation of the particleorphology is therefore important. Industrial processes practiceidely utilization of chemical additives (Weissbuch, Popovitzbiro,
ahav, & Leiserowitz, 1995; Yang, Kubota, Sha, Louhi-Kultanen, &ang, 2006) for blocking or promoting of certain crystal faces.
owever, the related chemical design space may be limited, and,oreover, additional necessary purification steps may drive thehole design process to an un-economical one (Lovette, Muratore,
Doherty, 2012). Alternative or/and additional approaches con-eive optimal model-based cooling or heating policies for the sameurpose, as we propose in this note. In particular, we focus heren crystallization scenarios with desired particle morphologieshat can not be attained by a single growth or dissolution phasend hence necessitate a number of sequential growth and dissolu-ion phases. Such shaping strategies have been recently proposednd experimentally verified, e.g., see (Lovette et al., 2012; Snyder,tudener, & Doherty, 2007).
This work represents an extension of theoretical expositions in
ajcinca, Oliveira, Borchert, Raisch, and Sundmacher (2010) andajcinca, Perl, and Sundacher (2011) – providing algorithms foronstructing the minimum-time trajectories in the property space
of a single particle – to crystal population systems. We arguethat minimum-time switching trajectories consist of a number ofsubsequent growth and dissolution sections assuming constantsupersaturation levels. By means of a coordinate transformationin the property space of the underlying particles, we show that theproposed algorithms are directly extendable to an important classof systems with size-dependent growth rate kinetics. Optimal tra-jectories involving a single switching, i.e. consisting of a growth anddissolution phase only, are shown to be unique, provided that theswitching sequence is specified. Otherwise, infinitely many timeoptimal trajectories sharing the same cost exist, independently ofthe number of cycles. Furthermore, we provide an extension of thealgorithm aiming at the minimization of the number of switch-ing cycles. The proposed solutions are illustrated by a numericalexample.
2. Process dynamics
We consider a batch crystallizer made up of crystal particlesimmersed in a dispersed phase system, constituted by a continuoussolvent liquid medium (typically, water) and dispersed crystallinesolute entities (molecules or ions), whereby a permanent solutemass transfer between the liquid and the solid crystalline phaseoccurs. The fundamental “force”, driving such a mass transfer, ariseseffectively from the supersaturation �, which is a measure of thedifference between the solution concentration C and the saturationconcentration Csat, defined by
Subscriptsn ≥ 1 ∈ N number of switching phases� ∈ {1, 2, . . ., n} counter of switching phasesm ∈ N particle dimensionj ∈ {1, . . ., m} counter of particle dimensions (e.g., Vs) referring to crystal seedsf (e.g., tf) referring to final conditions
Coordinatest ∈ R+ timeLj ∈ R+ jth coordinate of property space�j displacement in the direction Lj�j ∈ R+ jth transformed coordinate of property space�j displacement in the direction �j
Process variablesT temperatureC concentration of the soluteCsat solubility concentration� supersaturation of the soluteG�,j growth rate in j in the phase �G0�,j size-independent growth rate��,j size-dependent growth rate factor
w
C
waiefglisf
s
wsoewajp
�
wtLiw�
Rwsn
Vc net particle volumeVc,s net seed particle volume
here
= mc
mw + mc, Csat = a0 + a1T + a2T2, (2)
ith mw standing for the solvent and mc for the solute mass; a0,1, and a2 are empirical parameters (Myerson, 2002). The solutions said to be supersaturated if � > 0, and undersaturated if � < 0; inquilibrium, � = 0. In a supersaturated solution the mass transferrom the liquid to the solid phase dominates, implying a particlerowth phase. In an undersaturated solution, the opposite disso-ution phase with mass transfer from the solid to the liquid phase,s dominant. The ODE model which describes the growth and dis-olution process of any particle in a population is adopted in theorm:
witching phase � :d�j
dt=: G�,j = k�,j|�|g�,j , (3)
here we use � = 1, . . ., n to index the evolution phase (with sub-equent growth and dissolution modes, i.e. � may refer to a growthr a dissolution mode), and j = 1, . . ., m the coordinates of the prop-rty space. It is important to observe that we consider trajectoriesith n + 1 switching sections (see Fig. 2), that is, in our notation
nd study, unless otherwise stated, we exclude single-phase tra-ectories. Moreover, �1, . . ., � m represent the displacements in theroperty space of an m-dimensional particle, i.e.,
(t) = L(t) − L0, (4)
ith � = [� 1, . . ., � m]T and L = [L1, . . ., Lm]T, the latter representinghe absolute coordinates (i.e., the morphology) at a moment t, and0 the initial morphology of the particle at hand; k�,j, g�,j are empir-cal parameters. We prefer to use the relative �-coordinates in this
ork, as at each moment a unique displacement coordinate vector can be associated with the whole particle population.
emark 1. (Conventions and notation) Without loss of generality,e will assume throughout this note that our switching scenario
tarts and closes with a growth mode, i.e., n is a fixed given evenumber n ≥ 2. The other three situations are handled analogously,
Engineering 58 (2013) 381– 389
and we will consider them when solving the optimization prob-lems in Section 4. Note that for a � corresponding to a growth phasewe shall occasionally use the index g, e.g., kgj
∧=k�,j and gj∧=g�,j . Sim-
ilarly, if � refers to a dissolution mode, we use the subscript d, asin kdj
∧=k�,j and dj∧=g�,j . Obviously, given (3), we have kgj
> 0 andkdj
< 0. Also, for the sake of notation and exposition simplicity ofthe key ideas, we shall consider throughout the note the bivariateparticulate systems with m = 2 and � = [� 1, � 2]. �
The mass transfer between the solid and liquid phase is gov-erned by the mass-balance law:
�C = −c
VtM · �Vc, (5)
where �Vc = Vc − Vc,0 refers to the net volume change of crystal par-ticles, and �C = C − C0 to the related change in the concentration;C0 and Vc,0 are initial values of the concentration and populationvolume, c stands for the crystal mass density, M for the solventmolar mass, and Vt for the solvent volume. In this article we shallassume that the nucleated mass is negligible, i.e., the volume ofthe solid phase is mainly contrbuted by the growing (or dissolv-ing) seed particles: Vc ≈ Vc,s. This assumption shall, however, notcontest our results essentially, as we deal with switching trajecto-ries with subsequent growth and dissolution phases. The amountof the nucleation mass is expected to be relatively low, as the borncrystal nuclei during a “cold” (i.e. growth) phase will dissolve dur-ing the next “hot” (i.e. dissolution) phase. We emphasize that suchan assumption is in agreement with our experimental observations(see Section 5). It is important to emphasize that the net populationvolume is then expressible as a function of the displacement vector�, i.e., Vc = Vc(�). In fact, it was shown in Bajcinca (2012) that theunderyling function is a multivariate polynomial of the form:
Vc,s(�) =v1,v2∑
j1=0,j2=0
˛j1,j2 �j11 �j2
2 , (6)
for some given ˛j1,j2 , v1 and v2. We use shortly this expression toformulate the constraints in the evolution of the population sys-tem in the property space, but in view of Eqs. (5) and (6), we canimmediately observe a physically motivated feasible region in the�-space, confined by the maximal and minimal net volume of thepopulation:
0 < Vc(�) < Vc,0 + VtMC0
c. (7)
Finally, the model setting, as suggested by (1)–(5), is furthemoresimplifying in that the kinetic parameters k�,j and g�,j are assumedto be constant in (3).
Remark 2. (Size-dependency) The ideas developed in this noteapply also to multiplying size-dependent growth rate models ofthe form:
G�,j(t, Lj) = ��,j(Lj)G0�,j(t), (8a)
where ��,j(Lj) > 0 and G0�,j stand for the size-dependent and size-independent factors of the particle located at L, respectively. Indeed,as ��,j(Lj) is typically a priori given, for a specified � we can introducethe transformation Lj = Lj(�j) by means of
dLj
d�j= ��,j(Lj), (8b)
leading to ODEs of the form (3) in the transformed property space:
d�j
dt=: G0�,j, (8c)
N. Bajcinca / Computers and Chemical Engineering 58 (2013) 381– 389 383
wauott
V
Cbfroissirdmc
Rtopicsl
3
pprttswtwwsto
positive or negative], yielding:
Fig. 1. Qualitative reachability analysis in the property space.
ith �j = �j − �j,0, representing the common displacement vari-ble to all particles in the population. As a consequence of annderlying polynomial expansion of Lj(�j) in the space of a familyf orthogonal polynomials (e.g., Taylor, Legendre, etc.), the summa-ion expression for the net population volume in (6), now convertso an infinite series in terms of �j:
c,s =∑j1,j2
˛j1,j2 �j11 �j2
2 . (8d)
learly, for computational purposes, (8d) has to be approximatedy a finite sum of the form (6). Hence, system evolution convertsormally to an equivalent scenario with size-independent growthate kinetics. Therefore, without loss of generality within the classf models described by (8a) (e.g., ASL-model, Myerson (2002)),n the remainder of this note we shall focus to the scenario withize-independent growth rate kinetics only. As an additional con-equence, we can introduce the single representative particle. Fornstance, for a Gaussian distribution, it is natural to define as aepresentative particle the one “living” at the center-point of theistribution. Every particle in the population will share the sameotion model as the representative one up to its specific initial
onditions. �
emark 3. (Input variable) Finally, note that for a given concen-ration C, Eqs. (1) and (2) can be solved for T in terms of �. Thus, �r T can be interchangeably used as the control variables. We shallrefer � as the input though (i.e., u
∧=�), whilst the temperature Ts considered as a computed output. This is a solely mathematicalonstruction, as temperature represents the physical input to theystem. Soon enough it will become apparent that this choice willead to a convex program. �
. Problem formulation
Particle shape manipulation can be formulated as a trajectorylaning problem in the property space R
m of the representativearticle. More specifically, we consider the minimum time trajecto-ies, starting at �(t0) = q0 (for simplicity, we let t0 = 0 and q0 = 0) anderminating at a prespecified desired morpohology �(tf ) = qf ∈ R
m
hat does not result directly from a pure growth or a dissolutioncenario. Therefore, we need to consider switching trajectoriesith several subsequent growth and dissolution phases. Indeed,
he growth rate vector G�,j, as defined in (3), is constrained to lieithin an “interval” parameterized by the supersaturation level �,
hich is confined to lie between a minimal and maximal value;
ee Fig. 1. The minimal and maximal value of the supersatura-ion are determined by the temperature constraints Tmin ≤ T ≤ Tmax
r are perhaps directly specified by �min ≤ � ≤ �max. Notice the
Fig. 2. A switching trajectory.
corresponding qualitative differences in the reachability region:supersaturation constraints produce a “conical” region [not a strictcone in the mathematical sense], whereas for the temperature con-straints the latter warps (dashed indicated for g2 > g1 with m = 2).The morphologies outside such reachability regions are then attain-able only by means of subsequent growth and dissolution sections.
In this work we consider minimization of the batch duration tf,while we constrain the trajectory of growing/dissolving particleswithin a prespecified region ⊂ R
m. We assume to be compactand closed, for instance, defined by constraining the net particlevolume between a minimal and maximal value Vmin and Vmax, whilerespecting (7):
:= {� ∈ Rm|Vmin ≤ Vc,s(�) ≤ Vmax}. (9)
Such �-space constraints could be of interest, as perhaps one wantsto keep the evolution of crystal particles within a domain of highmodeling accuracy and confedence. Note that with regard to theassumption Vc ≈ Vc,s and Eqs. (5) and (6), the domain (9) can bespecified by a minimal and maximal concentration level, see Fig. 1.A similar practice has been followed in Bajcinca et al. (2010) forshaping the single particles. In the case with populations, Eqs. (6)and (8d) come into play. Both provide a region ⊂ R
m, such that�(t) ∈ ˝, i.e., Vmin ≤ Vc,s(� (t)) ≤ Vmax, for all t ∈ [0, tf]. The controlproblem of interest in this note then reads:
minimize�(t)∈˝
tf
subject to�min ≤ |�| ≤ �max.(10)
We digress our discussion now shortly to minimum-time single-phase trajectories, in order to give the latter a more insightfulformulation. The decision variables will naturally arise in Section4.
3.1. Optimal single-phase trajectories
We construct first the minimum-time trajectory starting fromq0 = [q0,1, q0,2]T towards a target morphology qf = � (tf) = [qf,1, qf,2]T
within its feasible growth constraint cone; see Fig. 1. According tothe Pontryagin’s minimum principle (e.g., see Kirk (2004)), we seekfor the optimal supersaturation profile u = �* which minimizes theHamiltonian H defined by
H(u, p) = 1 + pT G�(u), (11)
where p = [p1, p2]T is the costate vector, and G� : = [G�,1, G�,2]T. Thecostates evolve as p1 = −∂H/∂�1 = 0 and p2 = −∂H/∂�2 = 0, i.e.,they are constant: p1 = p∗
1 and p2 = p∗2. Hence, H is a function of
� only, and, the optimal u∗ ∧=�∗ is obtained by solving ∂H/∂u = 0[non-smoothness in u in (3) need not to be considered as u is either
�∗ =(
−p∗1k�,1g�,1
p∗2k�,2g�,2
)(1/(g�,2−g�,1))
= const. (12)
3 mical
A
tme
q
Hdgs
tprg
a
�
wett
t
w
�
w
t
w
3
gjstF
n
I
84 N. Bajcinca / Computers and Che
s a consequence, we have H ≡ 0 (H = 0 must hold ture at t = tf),
he optimal control variable is constant (u∗ ∧=�∗ = const), and theotion of the representative particle in the property space is gov-
rned by a linear equation:
f = G�(�∗) · t∗f + q0. (13)
ence, we conclude that a feasible straight section (which weenote by) 〈q0, qf〉 in R
m represents the minimum-time single-phaserowth (dissolution) trajectory, corresponding to a constant super-aturation (undersaturation) level �* = const.
Next, let us compute the underying supersaturation level andhe process duration tf in terms of q0 and qf. Consider first a growthhase under a constant supersaturation �∗+ = const, driving the rep-esentative crystal from q0 to qf [the subscript + should indicate arowth phase!]. Then, we can write:
qf,2 − q0,2
qf,1 − q0,1= (qf,2 − q0,2)/(tf )
(qf,1 − q0,1)/(tf )= �2
�1
= kg2
kg1
· (�∗+)g2−g1
nd, hence
∗+ = K ·
(qf,2 − q0,2
qf,1 − q0,1
)(1/(g2−g1))
> 0, (14a)
here K := ((kg1 /kg2 ))(1/(g2−g1)). The latter equation gives thexplicit expression for the optimal supersaturation level in terms ofhe end points of the straight section. The corresponding minimalime t∗
As a consequence of the above elaboration, we conclude thativen a number n ≥ 1 of switching points, any minimum-time tra-ectory is composed of straight segments corresponding to constantupersaturation sections �∗
� = const, as indicated in Fig. 2, wherehe dashed areas at each corner depict the reachability “cones” fromig. 1.
Due to the subsequent phases of growth and dissolution, weow may claim:
q0 ≤ q1,
q2� ≤ q2�−1, � = 1, . . .,n
2− 1,
q2� ≤ q2�+1, � = 1, . . .,n
2− 1,
qn ≤ qn−1,
qn ≤ qf .
(16)
ntroducing the shorthand notation:
Qn := [q1,1, q1,2, . . ., qn,1, qn,2]T ,
Engineering 58 (2013) 381– 389
that is, with Qn including the coordinates of the all n ≥ 1 switchingpoints, we define, further, the set
Kn := {Qn ∈ Rmn; the components of Qn satisfy (16)},
which constrains the domain of the decision variable Qn in Rmn in
accordance with the fixed assumption of the sequence of switchings:growth-dissolution-. . .-growth. It is important to observe that Kn,being a cartesian product of n convex sets in R
m, represents itself aconvex set.
Remark 4. (Property space constraints) Note that the set inFig. 2 is meant to keep the evolution of the crystal population withina prespecified region in the property space, in agreement with theconstraints induced by (9). For the moment being, we have ignoredit by setting = R
m, which is obviously heading us towards a rathermathematical result. But it turns out, that such, possibly, infeasiblesolutions prove essentially useful in constructing the solutions thatobey the constraints set by ˝, which we illustrate when minimizingthe number of the switching cycles in Section 4. �
4. Main statements
4.1. Minimum time solutions
By following the approach in Section 3.1, it is now easy tocompute the traverse time of the trajectories consisting of a fixednumber n ≥ 1 of switching points, starting at q0 and ending at theend morphology qf = : qn+1, along q1, . . ., q�, . . ., qn with coordinatesq� = (q�,1, q�,2). This is revealed by Eqs. (14b) and (15b):
tf (Qn) =n+1∑�=1
t�(q�, q�−1) (17a)
where
t� = L� ·∣∣q�,1 − q�−1,1
∣∣(g�,2/(g�,2−g�,1)) ·∣∣q�,2 − q�−1,2
∣∣(−g�,1/(g�,2−g�,1))
(17b)
and
L�∧={
K, growth mode,
M, dissolution mode.
Note that by the expression tf = tf(Qn) in (17a), we emphasize thefact that tf is a pure function of the switching points only.
Now we are ready to formulate our first result.
Lemma 1. The function tf(Qn) is
(i) stricly convex for n = 1, and(ii) convex for n > 1
in the domain Kn.
Proof. See Appendix A. �
Eqs. (14a) and (15a) give explicit expressions for the optimallevels of supersaturation and undersaturation in terms of the endpoints of the optimal straight section in the property space. We usenow these expressions to translate the supersaturations contraints:
�min ≤ |�∗� | ≤ �max, � = 1, . . ., n + 1,
into state constraints. For a growth mode, we have
�min ≤ K
(q�,2 − q�−1,2
q�,1 − q�−1,1
)(1/(g2−g1))
,
mical
o
a
H
h
w
Cm
o
w
T
(
P
To
C
i
t
a
�
�
bi
b
N. Bajcinca / Computers and Che
r
�−g1minkg1
(q�,1 − q�−1,1) − �−g2minkg2
(q�,2 − q�−1,2) ≤ 0,
nd, analogously, for the maximal supersaturation level:
−�−g1max
kg1
(qi,1 − qi−1,1) + �−g2max
kg2
(qi,2 − qi−1,2) ≤ 0.
ence, we obtain the linear state constraints:
�(q(n)) := (hmin� (Qn), hmax
� (Qn)) ≤ 0, (18)
here
hmin� (Qn) := |�min|−g1
kg1
(q�,1 − q�−1,1)
−|�min|−g2
kg2
(q�,2 − q�−1,2),(19a)
hmax� (Qn) := −|�max|−g1
kg1
(q�,1 − q�−1,1)
+|�max|−g2
kg2
(q�,2 − q�−1,2).(19b)
learly, analogous expressions can be derived for the dissolutionodes, as well.Now, we can restate the optimization problem (10) as a convex
ptimization program in Rmn:
minimize tf (Qn)
subject to h�(qn) ≤ 0, � = 1, . . ., n + 1
Qn ∈ Kn,
(20)
hich leads us to the main result:
heorem 1. The optimization problem (20) has
(i) a unique solution for n = 1,ii) infinitely many solutions for n > 1.
roof. See Appendix B. �
he following statement provides the relationship between theptimal solutions with a single or multiple switching points.
orollary 1. Assume
q∗1 := argmin
Q1∈K1
tf (Q1)
s the solution to (20) for n = 1. Let
∗f := tf (q∗
1)
nd
∗+ := K
(q∗
1,2 − q0,2
q∗1,1 − q0,1
)(1/(g2−g1))
(21a)
∗− := −M
(q∗
1,2 − qf,2
q∗1,1 − qf,1
)(1/(d2−d1))
(21b)
e the required minimal time and optimal supersaturation levels dur-ng the growth and dissolution modes, respectively. Let further
Q ∗n := argmintf (Qn)
Qn∈Kn
e an optimal solution to (20) for a fixed n > 1. Then
�∗1 = �∗
3 = . . . = �∗n+1 = �∗+, �∗
2 = �∗4 = . . . = �∗
n = �∗−,
Engineering 58 (2013) 381– 389 385
and, moreover
tf (Q ∗n ) = t∗
f.
Proof. See Appendix C. �
In words, this result is stating that a minimum-time trajectorywith multiple switchings manifests two levels of supersaturationwhich are uniquely fixed by the optimal single-switching trajec-tory. As already indicated in Remark 4, while the single-switchingsolution itself may be physically infeasible, the optimal growingand dissolving supersaturation levels, �∗+ and �∗−, are important forthe construction of any minimum-time solution that obeys arbi-trary constraints in the property space.
4.2. Minimum number of switches
A natural and important problem in practice, consists in mini-mizing the number of switching sections subject to some specifiedconstraints in the property space. Due to the jumps in the tem-perature profile (see below Fig. 4(b)), minimization of switchingcycles entails a lower energy consumption. With regard to Fig. 2,we consider the following problem statement:
minimizen≥1
n
subject toQ ∗n ∈ ˝n,
(22)
where ˝n = × . . . × ⊂ Rmn and is a prespecified set as
defined by, e.g., (9). Notice that a solution to this problem will notprovide the minimum number of cycles, it will rather provide aminimum-time trajectory with a minimum number of switchingcycles (see the comments in the example in Section 5).
With reference to the algorithm presented in Appendix C, theproblem (22) consists essentially in breaking the minimum-timesingle-switching trajectory into a minimal number of sectionsassuming the same positive and negative slopes, such that the over-all trajectory resides within the domain ˝.
For the presentation of an algorithm which provides the solu-tion to this problem, let us introduce the rays R+(q) and R−(q)with the initial point q ∈ R
2 and the slopes kg2 /kg1 (�∗+)g2/g1 andkd2
/kd1|�∗−|d2/d1 , respectively. Thereby, let R+(q) point into the
growing coordinates and R−(q) into the decreasing ones. Moreover,let L+(q) and L−(q) be the corresponding line extensions of R+(q) andR−(q), respectively, and denote by ∂ the boundary of ˝.
The following intuitive algorithm, where for the sake of simplic-ity we assume q0 ∈ ˝, computes the number n+ [in response to theinitialization of the algorithm at the step S1 by a growth phase via“sgn =+”] by sequential intersections of a ray with a line and withthe boundary ∂ until the intersection point lies within ˝:
S0: Compute the optimal supersaturations �∗+ and �∗− from (21).S1: Set n = 1 and sgn =+.S2: Compute qn = Rsgn(qn−1)
⋂L−sgn(qf).
1 If qn ∈ ˝, set n+ = n and Stop.2 Else compute qn = Rsgn(qn−1)
⋂∂˝, set n = n + 1, sgn = − sgn
and Go to the step S2.
In a similar manner we can compute n− corresponding to the ini-tialization “sgn =−” at the step S1. The minimal number of cycles isthen given by nmin = min {n+, n−}. Dropping the assumptions q0 ∈ ˝can be easily carried out by a slight modification of the above algo-rithm.
5. Numerical example
We illustrate the proposed alogrithm in a case study withthe popular KDP crystal particles. The shape of KDP crystals is a
386 N. Bajcinca / Computers and Chemical Engineering 58 (2013) 381– 389
etragonal prism in combination with tetragonal bipyramid, andhe angle between the prism sides and pyramid faces is 45◦, see
ullin and Amatavivadhana (1967). The two internal coordinates1 and L2 represent the width and length of the KDP crystal,espectively, see Fig. 3. The model parameters, including the initialonditions, are listed in Table 1, which is adopted from Borchertt al. (submitted for publication). The initial seed distribution f0(L1,2) is taken to be
or L1 ∈ [190, 250], L2 ∈ [380, 440] (in [�m] units), and null oth-rwise. Note that f0 is centered at (220, 4200) [�m]. The desiredorphology of the representative particle (i.e. the center of the seed
ensity function) at the end of the batch is (1000, 711) [�m]; refer
o Fig. 4(a). The set ˝, implying the property space constraints (seeig. 2), is defined by means of (9) with Vmin = 6.087 × 104 [mm3] andmax = 3.854 × 105 [mm3], which are indicated by the dashed lines
2 4 6 8 10 12
2
4
6
8
10
L1 × 10−1, mm
L2,
mm
C = C+
C = C−
initialdistribution desired
distribution
(a)
0 10 20 30 40 50 60 70 80 90 100
25
28
31
34
tem
pera
ture
,◦ C
time, min(b)
ig. 4. Solution to problem (22): (a) minimum-time trajectory with minimal num-er of switching sections and (b) corresponding optimal temperature profile.
Initial conditionsInitial mass fraction C0 0.3
C = C+ and C = C−, respectively, in Fig. 4(a). Finally, the supersatu-ration constraints are: �min = 3 ×10−2 and �max = 13 × 10−2 duringthe growth phases, and �max = −10−2 and �min = −20 × 10−2 duringthe dissolution phases.
Following the algorithm from the previous section for minimiz-ing the number of switching cycles, in the first step we compute theoptimal single-switching trajectory, which is depicted by dashedlines in Fig. 4(a). The numerical outcomes read:
q∗1 = (2.1, 2.3) [mm], �∗
+ = 0.130, �∗− = 0.048.
Note that the optimal supersaturation level during the growingphase refers to the maximal value, while this is not the case duringthe dissolution, this indicating that it does not represent a bang-bang solution.
In the second step, we break the single-switching trajectory intoa minimal number of switching sections. The outcome, consisting ofsix (nmin = 8) sections, is diplayed in Fig. 4(a). In Fig. 4(b) we depictthe corresponding temperature profile. Note that the process ofshape manipulation is completed in about 96 min. It is importantto emphasize that for the same constraints in the property space,the shaping process could be completed with a smaller numberof switching sections, however this would evoke a longer batchduration.
Finally, in Fig. 5 we provide a schematic interpretation of theunderyling bimodal crystallization algorithm in the � − C space.Note that, in addition to the two optimal supersaturation levels�∗+ and �∗−, the algorithm is basically described by additional threeparameters referring to the concentration levels C+, C− and a con-centration value for entering the very last switching phase. In otherwords [under the modeling assumptions in Section 2], the prac-tical implementation of the proposed algorithm can be managed
by means of a concentration sensing only. Thereby, the switch-ing events (corresponding to the switching points) are generatedupon reaching the underlying parametric concentration levels, as
σ∗+
σ∗−
C
σ
dissolution
growth
C+
C− start
finish
Fig. 5. Cyclic bimodal algorithm in the � − C space.
mical
iueeaiB(ptAo|ap
6
tamgosTcqctadwsdtt
A
a
A
w
N. Bajcinca / Computers and Che
ndicated in Fig. 5. Yet, under real-world conditions, unmodelledncertainties (such as the nucleation mass, false kinetic param-ters, finite bandwidth in temperature feedback control loops,tc.) will most probably necessitate utilization of feedback controlnd real-time particle shape monitoring mechanisms by utiliz-ng sophisticated image processing techniques, e.g. as proposed inorchert and Sundmacher (2012), Eggers, Kempkes, and Mazzotti2008). Feedback control is, additionally, motivated by the fact thathysically it is impossible to decrease/increase instantaneouslyhe temperature as suggested by simulation results in Fig. 4.lternatively, a constraint in the absolute value of the derivativef temperature (or, supersaturation) could have been imposed:
dT/dt| < |dT/dt|max in the optimal control setting. Such deliberationsre beyond the scope of this article, whose message is solely theroof of a concept.
. Conclusions
Crystal shape manipulation for multivariate crystal popula-ions is formulated as a trajectory planning problem. We provide
convex optimization formulation which is used to constructinimum-time trajectories with subsequent multiple switching
rowth and dissolution phases. The solutions to the underylingptimal control problems are globally optimal and are efficientyolvable e.g. by numerical interior point methods (Bertsekas, 1999).he resulting strategy is a simple bimodal constant supersaturationontrol policy involving a concentration sensing only. As a conse-uence, it is easily implementable by common facilities in practicalrystallization engineering. It is fully described by few parame-ers, including two optimal supersaturation levels for the growthnd dissolution phases which are uniquely defined by the problemata. The algorithm is developed for a bivariate particle populationith size-independent growth rates. However, its basic lines can be
traightforwardly extended to multivariate populations of a higherimensions (e.g., see the discussion in Bajcinca et al. (2010)) and forhe size-dependent kinetics after introducing a suitable coordinateransformation in the property space.
cknowledgement
Thanks to Holger Eisenschmidt for the careful reading of therticle and the useful hints.
ppendix A. Proof of Lemma 1
(i) n = 1: to prove the (strictly) convexity of
t∗f
= K · (q1,2 − q0,2)(−g1/(g2−g1)) · (q1,1 − q0,1)(g2/(g2−g1))
implying direct paths as solutions, which are ruled out per theassumption n = 1.
(ii) n > 1: In view of Eq. (17), we need only to prove that the func-tions t�, � = 1, . . ., n + 1, from (17b) are convex. Therefore, we showthat the eigenvalues of the hessian of these functions are greater orequal to 0 in two different cases: (a) t� is a function of two variables,corresponding to the first and the last switching section, i.e. � = 1and � = n + 1, and (b) t� is a function of four variables correspondingto the intermediate phases.
he characteristic polynomial of the Hessian reads:
1(�1) = det(hess(t1(q1)) − �1E)
= det
(a − �1 −b−b c − �1
)= �2
1 − (a + c)�1 + (ac − b2) = 0.
s ac = b2, we have �1,1 = a + c and �1,2 = 0. Given that a ≥ 0 and c ≥ 0,nd the eigenvalues of the Hessian matrix ∂t1/∂Qn:
�1,i ={
tr(hess(t1)) i = 10 otherwise,
e conclude that t1 is convex in Kn. The calculations lines for tn+1ith (qn,1, qn,2) follow analogously and are obviated here.
(b) One has to discriminate again two scenarios for the dissolu-ion and for the growth phase. We explore here a dissolution modenly, as analogous calculation steps hold for the growth mode, too.e then have:
t� = M · (q�−1,2 − q�,2)(−d1/(d2−d1))(q�−1,1 − q�,1)(d2/(d2−d1)),
ith q�−1,2 − q�,2 > 0 and q�−1,1 − q�,1 > 0. Define first the seconderivates:
⎜⎝a − �� −b −a b−b c − �� b −c−a b a − �� −bb −c −b c − ��
⎟⎠ ,
Engineering 58 (2013) 381– 389
whose eigenvalues are
��,i =
⎧⎪⎨⎪⎩
0 i = 1, 2,
a + c −√
c2 − 2ac + a2 + 4b2 i = 3,
a + c +√
c2 − 2ac + a2 + 4b2 i = 4.
Since ac = b2, we obtain c2 − 2ac + a2 + 4b2 = c2 + 2ac + a2, yielding��,3 = 0 and ��,4 = 2(a + c) = tr(hess(t�)) ≥ 0. Finally, given the eigen-values of the Hessian matrix ∂t�/∂Qn:
��,i ={
tr(hess(t�)) i = 4� − 20 otherwise,
implying that t� is convex for all � = 2, . . ., n − 1.
Appendix B. Proof of Theorem 1
Consider the Lagrange dual function of (20):
(�) := minQn∈Kn
L(Qn, �)
where
L(Qn, �) := tf (qn) + �1hmin1 (qn) + �2hmax
1 (qn)+�3hmin
2 (qn) + �4hmax2 (qn).
Given the definition of the constraint functions, it is easy to seethat there exists at least one point Qn with h1(Qn) < 0 and h2(Qn) < 0.Since Kn is convex, the Slater condition holds true, and, hence, wehave the strong duality for our problem, implying:
minQn∈Kn,h1(Qn)≤0, h2(Qn)≤0
tf (Qn) = max�
minQn∈Kn
L(Qn, �)
and there exists at least one Lagrange multiplier �* such that (seeBertsekas (1999, pp. 511–512))
minQn∈Kn, h1(Qn)≤0, h2(Qn)≤0
tf (Qn) = minQn∈Kn
L(Qn, �∗)
see Bertsekas (1999, p. 480 Definition 5.1.1, p. 484 Proposition 5.1.1and p. 487 Proposition 5.1.4 (a)). In other words, we end up withthe problem:
minimizeQn∈Kn
L(Qn, �∗).
But, this completes the proof, as we know that
hessL(Qn, �∗) = hess(tf (Qn)).
Appendix C. Proof of Corollary 1
Fix an even number n > 1, and let
Qn := [q1,1, q1,2, . . ., qn,1, qn,2]T
mical
bp
Hrqtqpa
t
w
Ime
t
Tt
t
T
control by manipulating supersaturation in batch cooling crystallization. Crystal
N. Bajcinca / Computers and Che
e a feasible solution to (20). Introduce further new switchingoints:
r1 = q1,
r2 = r1 + q3 − q2,
...
rn/2 = rn/2−1 + qn+1 − qn,
rn/2+1 = rn/2 + q2 − q1,
rn/2+2 = rn/2+1 + q4 − q3,
...
rn = rn−1 + qn − qn−1.
(C.1)
ereof, it is easy to check that rn+1 = qf. The trajectory section 〈q0,1, . . ., rn/2〉 corresponds to a single growth mode, while 〈rn/2, . . ., rn,f〉 to a single dissolution one. Thus, it represents a single-switchingrajectory. Observe that during the growth phase we have: r2 − r1 =
˜3 − q2, . . ., rn/2+1 − rn/2 = qn+1 − qn, and during the dissolutionhase: rn/2+1 − rn/2 = q2 − q1, . . ., rn − rn−1 = qn − qn−1. Hence, as
consequence of (17), we have
f (Qn) = tf (Rn), (C.2)
here
Rn := [r1,1, r1,2, . . ., rn,1, rn,2]T .
n other words, (C.1) associates a single-switching trajectory to eachultiple switching trajetory sharing the identical end points and
lapse times.Let us next construct a trajectory with n switching points such
hat
�∗i
:={
�∗+ if ith mode is a growth mode,�∗− if ith mode is a dissolution mode.
satisfies �(0) = q0 and �(tf) = qf. Clearly, per construction, each sucha point Qn ∈ Kn is equivalent to the optimal single-switching tra-jectory in that
tf (Qn) = t∗f.
Denote the set of all such points by Qn ⊂ Kn. Then, Lemma 1 claimsthat if Qn is an optimal solution to (20), then Qn ∈ Qn. Indeed, letQn ∈ Kn be a minimizer of (20). Then, tf (Qn) ≤ tf (Q1). But, (C.2)reveals that tf(Rn) ≤ tf(Q1) is true only if the trajectory 〈r1, . . .,rn〉 is identical to the single-switching minimizer, this implyingQn ∈ Qn.
References
Bajcinca, N., Oliveira, V., Borchert, C., Raisch, J., & Sundmacher, K. (2010). Optimalcontrol solutions for crystal shape manipulation. In Proceedings ESCAPE 20 (pp.751–756).
Bajcinca, N., Perl, R., & Sundacher, K. (2011). Convex optimization for shape manip-ulation of multidimensional crystal particles. In Proceedings ESCAPE 21 (pp.855–859).
Bajcinca, N. (2012). Analytic solutions to optimal control problems in crys-tal growth processes. Journal of Process Control, http://dx.doi.org/10.1016/j.jprocont.2012.08.010
Bertsekas, D. (1999). Nonlinear programming. Belmont Massachusetts Athena Scien-tific.
Borchert, C., & Sundmacher, K. (2012). Morphology evolution of crystal popu-lations: Modeling and observation analysis. Chemical Engineering Science, 70,87–98.
Borchert, C., Temmel, E., Eisenschmidt, H., Lorenz, H., Seidel-Morgenstern, A., Sund-macher, K. Image-based in-situ identification of face specific growth rates fromcrystal populations. Crystal Growth & Design, submitted for publication.
Eggers, J., Kempkes, M., & Mazzotti, M. (2008). Measurement of size and shape distri-butions of particles through image analysis. Chemical Engineering Science, 63(22),5513–5521.
Kirk, D. E. (2004). Optimal control theory: An introduction. Mineola New York DoverPublications.
Lovette, M. A., Muratore, M., & Doherty, M. F. (2012). Crystal shape modificationthrough cycles of dissolution and growth: Attainable regions and experimentalvalidation. AIChE Journal, 58(5), 1465–1474.
Mullin, J. W., & Amatavivadhana, A. (1967). Growth kinetics of ammonium- andpotassium-dihydrogen phosphate crystals. Journal of Applied Chemistry, 17,151–156.
Myerson, A. S. (2002). Handbook of industrial crystallization. Boston, USA:Butterworth-Heinemann.
Patience, D. B., & Rawlings, J. B. (2001). Particle-shape monitoring and control incrystallization processes. AIChE Journal, 47(9), 2125–2130.
Snyder, R. C., Studener, S., & Doherty, M. F. (2007). Manipulation of crystal shape bycycles of growth and dissolution. AIChE Journal, 53(6), 1510–1517.
Weissbuch, I., Popovitzbiro, R., Lahav, M., & Leiserowitz, L. (1995). Understand-ing and control of nucleation growth habit dissolution and structure of2-dimensional and 3-dimensional crystals using tailor-made auxiliaries. ActaCrystallographica Section B-Structural Science, 51(Part 2), 115–148.
Yang, G., Kubota, N., Sha, Z., Louhi-Kultanen, M., & Wang, J. (2006). Crystal shape
Growth & Design, 6(12), 2799–2803.Yang, H. G., Sun, C. H., Qiao, S. Z., Zou, J., Liu, G., Smith, S. C., et al. (2008). Anatase
TiO2 single crystals with a large percentage of reactive facets. Nature, 453(7195),638–U4.
