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Chemical Engineering and Processing 46 (2007) 1096–1106

Optimization in seeded cooling crystallization: A parameterestimation and dynamic optimization study

S. Mostafa Nowee a, Ali Abbas b,1, Jose A. Romagnoli c,∗a School of Chemical and Biomolecular Engineering, University of Sydney, Australia

b School of Chemical and Biomedical Engineering, Nanyang Technological University, Singaporec Department of Chemical Engineering, Louisiana State University, USA

Available online 3 March 2007

bstract

In this solution crystallization study, a population balance model that predicts the crystal size distribution (CSD) is used for targeting, byptimization, the product mean size and the coefficient of variation of the final CSD. The model is robust over a wide range of conditions andredicts the effects of heating during a batch since the kinetics of dissolution have been identified and incorporated into the model. The dynamic

emperature profile and the initial seed size distribution are optimized while the initial seed induction time is conferred through knowledge ofaturation conditions. This is quite a highly developed method that provides a systematic approach towards optimal operation. Results fromptimizations under different objective functions are presented and validated experimentally.

2007 Elsevier B.V. All rights reserved.

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eywords: Crystallization; Cooling; Seeding; CSD; Nucleation; Growth; Param

. Introduction

Crystallization has been referred to as an art rather than a sci-nce. The main objective of crystallization systems engineerings to develop advanced understanding of crystallization phe-omena enabling the efficient bulk processing of high qualityrystalline products. Crystallization, in many industries, is theost common way of production of high value chemicals with

igh purity and desired size and shape. Crystal size is typicallyhe most important factor to be controlled because of its closeelation to the characteristics of the crystalline product, suchs flowability and bulk density, and also different applicationsequire different particle size. Crude methods are still being usedn industrial operations to operate/control their crystallizationrocesses and those industries that implement any control rarelyo this at optimal conditions. Nevertheless, there have been aumber of successful research investigations as evidenced in

he literature on the control of the crystallization process with

ost of these studies aiming to control the product crystal sizer size distribution.

∗ Corresponding author. Tel.: +1 225 578 1377; fax: +1 225 578 1476.E-mail address: jose@lsu.edu (J.A. Romagnoli).

1 Present address: Division of Engineering Science and Technology Universityf New South Wales Asia, Singapore.

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255-2701/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.cep.2007.02.020

estimation; Dynamic optimization

Crystallization processes are complex in nature and arenown to be highly non-linear and thus pose difficulties in theirodeling and control. They also carry liquid as well as solid

particulate) phases adding to the process’s complexity. Theirathematical description is based on population balance equa-

ions, coupled with integral DAEs. In addition, the kinetic historyf the process largely influences the end product quality. Anynanticipated disturbances during process operation could leado irreversible changes to product particle characteristics. It ishus evident and based on research observations that the use ofhe model is crucial for optimizing the crystallization operationsllowing for end-product crystal size reproducibility.

Researchers have recently recognized the significant role arystallization model can play in optimizing operations. Manip-lated variables that would be candidates for optimizationecision variables are traditionally the working temperature andeeding. Considerable effort has been made on particle size con-rol in batch cooling crystallization [1]. Most of these worksonsider supersaturation as the key variable to be optimized.his strategy which is based on the determination of the opti-al temperature profile was first discussed by [2]. It was though

ased on a key understanding namely that constant nucleationate is required in order to keep the supersaturation low andithin the metastable zone. Other works dealing with an opti-al cooling profile generation intended to keep supersaturation

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ow in the beginning stages of the process inhibiting nucleationut promoting crystal growth.

Although seeding was believed to be an important factoror controlling crystal size (and allowing for the production ofono-dispersed CSDs), the amount of seed to be added was

argely based on empirical knowledge. Bohlin and Rasmuson [3]nalyze, via simulations, the effect of seed load on product sizeharacteristics. Their seeding loads are chosen arbitrarily pro-iding empirical trends about the effect of seeding. They showhat the product weight mean size and coefficient of variation

ay increase or decrease at increasing seed amounts attributinghis variation to variations in kinetics. Only in recent years somexperimental research appeared considering seeding as a factoro be optimized [4–9]. In these investigations the seed chart wassed to determine optimal seeding policies. These works demon-trated the impact seeding has on the crystallization showingven under un-controlled profile of cooling (e.g. natural cool-ng), the rate of nucleation could be kept low. It could also behown via simple mass balance calculations and the assumptionhat only seeds consume the generated supersaturation duringhe batch, the predicted final crystal mean size correlates wellith the experimental ones. Other finds included the ability toroduce an end-product with narrow monomodal CSD and fur-her due to secondary nucleation, the final product could have aimodal distribution.

A model-based dynamic optimization of seeded coolingrystallization was discussed by [10]. Seeding parameters inddition to supersaturation were optimized and the objectiveunction used was maximization of final mean size with mini-um coefficient of variance. They concluded that the effect of

eeding variables on the final product characteristics are muchore pronounced than that of supersaturation. Other workers

11,12] incorporated factorial experimental design using a sur-ace response model in determining initial seed concentrationnd seed introduction time as well as the temperature profile.heir target was a specific crystal yield, size, shape and purity.ther optimization studies based on factorial design [13] related

he input parameters of seed load and seed type (dry or slurry)mongst others directly to the output parameter ‘filtration effi-iency’. Their experimental model showed that in order to reducehe filtration time, the seed amount and addition time should bencreased while dry seeds should be used. Another investigation14] that dealt with model-based optimization of seeded coolingrystallizations determined the optimal seeding characteristics,ean size and seed mass, by maximization of product mean

ize and minimization of CSD coefficient of variance as objec-ive functions, respectively. Recent research trends, as evident inhe literature, show an increase in the use of the dynamic crys-allization model for achieving various optimal objectives. Theimultaneous use of the two variables, temperature and seeding,s evident in recent works [10,15]. A more comprehensive study16] solves the population balance equation (PBE) for a growth-ependent dispersive crystal system and uses temperature profile

s well as two seeding variables as decision variables. They [16]lso validate their optimization results experimentally.

In this study, we extend on a previously developed dynamicrystallization model [17] to include a mathematical description

wewu

nd Processing 46 (2007) 1096–1106 1097

f (1) both super- and under-saturation crystallization regimesnd (2) seeding. In (1) dissolution and disappearance kineticates are introduced to the model. We describe and report theesults of a model-based parameter estimation study undertakeno determine the kinetic parameters and thus identify the crys-allization model. Finally an offline model-based optimizationnalysis was conducted to determine the optimal cooling profilen conjunction with the optimal initial seed size distribution. Theatter being a unique determination that is novel to this work andxtends on model-based optimization work by previous work-rs. We further present results of experimental validation of suchual decision variable offline optimization.

. The crystallization model

Since the crystallization process is a particulate one, the popu-ation balance equation (PBE) as proposed by Hulburt and Katz18] is used accounting for the evolution of crystal particlescross temporal and size domains. For a batch crystallizationystem such as the ammonium sulfate one we are dealing withn this study, crystal growth is assumed to be non-dispersed andndependent of crystal size and agglomeration and attrition areonsidered negligible, the population balance simplifies to

∂n(L, t)

∂t= −G∂n(L, t)

∂L+ B (1)

here n(L, t) is the number density of crystals, t the time, Lhe characteristic crystal size, G the growth rate of the crystalshile B is the nucleation rate. The first term in the equation

epresents the accumulation of crystal numbers, while the seconderm represents the net flux of particles through the size rangey crystal growth (G). The third term (B) represents birth orroduction (by nucleation) of crystals in the first size range,rom which it follows that

= B0δ(L) (2)

here B0 is the nucleation rate and δ(L) is the Dirac delta func-ion. In the case where crystal dissolution by heating exists suchs when the process is operating under temperature control giv-ng rise to temperature increases, the PBE will change and termsenoting dissolution (D) and disappearance (Da) of particles willeed to be incorporated (Eq. (3)). The structure of PBE remainshe same as in Eq. (1), but the dissolution and disappearanceerms are introduced to replace the growth and nucleation ones,espectively, and with opposite signs.

∂n(L, t)

∂t= D

∂n(L, t)

∂L−Da (3)

The switch from supersaturation state to undersaturation stateesults in elimination of growth and nucleation processes, andn the occurrence of dissolution and disappearance processes.he reverse occurs when the process switches from under- toupersaturation (e.g. switching from heating to cooling). In other

ords, where there are high supersaturation levels, one would

xpect to observe nucleation and growth of crystals and Eq. (1)ould hold true. Where the process is going in the direction ofnder-saturation, i.e. heating, one would observe dissolution and

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isappearance of crystals rendering Eq. (3) to be true. Clearly,his is only true for those solute systems that have a solubilityhat increases with temperature.

In crystallization processes some phenomena that may occurre nucleation and growth. These phenomena govern the processnd ultimately dictate the product characteristics and particularlyhe shape of the product PSD. In this study, two additional kineticates are considered namely dissolution and disappearance. Ashe objective of this research is to identify optimal operationalolicies, the kinetic model structure is kept simple and confinedo a power law structure.

= k�C� (4)

here R is the vector of kinetic rates [B G D Da] representing theates of nucleation, growth, dissolution and disappearance, k theector of kinetic rate parameters [kb kg kd ka],�C the supersatu-ation which is equal to the difference between the concentrationf the solute C and the saturated concentration C*. Finally � rep-esent the vector of rate orders [b g d a] corresponding to eachf the kinetic rates, respectively.

To complete the model, the mass and energy balances, afterawlings et al. [19], for the solute and crystallizer temperature,re incorporated. The model was solved by finite differenceiscretization method. For the supersaturated and undersatu-ated states, the backward and forward finite techniques weresed, respectively, providing stability. The initial condition forhe system is the size distribution of seeds. The solution wasmplemented and simulated using the process modeling packagePROMS (Process Systems Enterprise Ltd., UK) which exe-uted the solution within a timeframe of a few seconds using aomputer running a Pentium 4 processor.

. Crystallization kinetic model identification

The model kinetic system of Eq. (4) contains parameters thateed to be identified via a parameter estimation step, the lat-er being an important prerequisite step for simulation, dynamicptimization and model-based control. Essentially the parame-er estimation problem is an optimization one where the models fitted to the experimental data. Once parameters are estimated,he identified model may be used for predictive simulation anal-sis where the various process aspects may be elucidated, and forptimization and control scheme development. In the followingections, parameter estimation in the context of crystallizations reviewed followed by a presentation of optimization-basedarameter estimation with specific address to the optimizationormulation towards identification of the parameters of the crys-allization kinetic sub-model presented in Section 2.

Rigorous parameter estimation methods have been employedn crystallization model identification but of all techniques, pos-ng the estimation as an optimization problem has been regardeds most relevant [19,20]. Initial works in crystallization param-

ter estimation relied on population density data fitting [21]. Aommon method presented throughout literature is the use of thenalytical solution of the PBE along with experimental data fromixed suspension mixed product removal (MSMPR) crystalliz-

otge

nd Processing 46 (2007) 1096–1106

rs. This mode of crystallization as the name suggests operatesn a state of ideal mixing, where the solid phase is ideally dis-ersed in the liquid phase. Removal of ideally mixed productrom such crystallizers occurs iso-kinetically for both phases.his is so as to avoid the segregation of the suspension into theifferent phases. If segregation does occur, it renders the use ofhe PBE impossible. The analytical solution of the PBE is giveny

= n0 exp

(− L

Gτ

)(5)

here τ is the residence time. Experiments are performedt different residence times. The size distribution analysis, ifhen plotted semi-logarthmically, reveals a linear relationshipetween number density, n, and crystal size, L, then the meaninear crystal growth rate can be determined from the slope of thetraight line. In addition the y-intercept of the slope represents0—the number density of the zero-sized crystals.

Once the growth rate is identified, it is then possible to deter-ine the nucleation rate after assuming that the nuclei form in

he nano-size range (i.e. as L → 0). The nucleation rate is theniven by the following boundary condition

0 = n0G (6)

The use of MSMPR crystallizers has been a popular methodor kinetic parameter estimation simply because of the ease ofomputing the parameters in question. The disadvantages of thisechnique are that the use of the population balance assumeshe experimental results are drawn from a suspension at steadytate conditions. This implies that the crystallization experi-ents need to be brought to steady state before samples are

aken. Further, estimates of parameters made from steady stateata would not be suitable for the description of the varyingynamics occurring during the batch time. Another disadvantagerises when the semi-logarithmic plot of f versus L exhibits non-inearities. An upward curvature may appear in some systemshich do not obey McCabe’s �L law.The MSMPR technique is now becoming outdated and this is

videnced by its gradual absence from the crystallization litera-ure. This decline has been catalyzed by the increasing computerrocessing power and the development of other techniques par-icularly optimization based ones. These are discussed next.

.1. Optimization based parameter estimation

Rigorous non-linear optimization techniques are now avail-ble paving the way for more efficient estimation and with thedvances in power of modern computers, optimization methodsre becoming more prevalent in the literature. The mathemat-cal background behind these techniques is intricate and therere several text available detailing the mathematical derivations22,23]. In this work, a short explanation of the formulation

f the parameter estimation problem is given in the context ofhe application to the crystallization system under study. TheEST function in gPROMS is also discussed as it is used in thestimation of the parameter sets.

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The crystallization model described in Section 2 consists of aoupled set of non-linear integro-differential equations and cane represented by Y (t) = [L(t), C(t)] being the differential out-ut variables, U(t) = [Tj(t), N0] being the time-varying inputsariables, and k and � being the vectors of parameters to bedentified which in this case are [kb kg kd ka] and [b g d a],espectively. We can represent the model as follows:

(Y (t), U(t), θ) = 0 (7)

Eq. (7) is accompanied by a set of initial conditions for theifferential equation of the model and these can be collated intohe vector Y(0) = y, where y is a set of values in this case repre-enting (N(0),C(0)). Measured variables from experiments aressential in the parameter estimation exercise and are denotedy the vector Y (t) = [L(t), C(t)]. Two general criteria may besed in the estimation, namely the least squares or maximumikelihood. The least squares case is effectively a special case ofhe maximum likelihood method as will be explained. Consid-ring the problem specified, the maximum likelihood criterionhat describes the highest probability of the model predicting theeal data is given by the following objective function:

(k, θ) = M

2ln (2π)

+1

2mink,θ

⎧⎨⎩

α∑i=1

βi∑j=1

γij∑k=1

[ln(σ2

ijk) + (Yijk − Yijk)2

σ2ijk

]⎫⎬⎭(8)

here M is the total number of measurements taken during allxperiments, α, βi and γ ij are, respectively, the number of exper-ments, the number of variables measured in the ith experimentnd the number of measurements of the jth variable in the ithxperiment, σ2

ijk is the variance of the kth measurement of vari-

ble j in experiment i. σ2ijkcan be described by any of a number

f models available [24] when the error structure of the data isnown. These are either homoscedastic where the variance errors assumed to be constant, or heteroscedastic where the variancerror is a function of the measured and predicted values and

ay be proportional to∧Y (t)λ/2 and Y(t)λ/2, where λ is a param-

ter determined in the optimization and ranges between 0 and.

In the homoscedastic case, the variance error may be deter-ined by analysing similar sets of experimental data. The value

f the variance can be specified in the estimation as

2ijk = ω2

ijk (9)

here ω is the constant standard deviation of the measurementrror. This specification reduces the maximum likelihood prob-em into the least squares criterion⎧⎨ βi∑ γij∑ [

2 (Yijk − Yijk)2 ]⎫⎬

(k, θ) = mink,θ ⎩

j=1 k=1

ln(ωijk) +ω2ijk

⎭ (10)

For some cases, more information is available about the data,nd the error structure may be known to depend on the magnitude

tose

nd Processing 46 (2007) 1096–1106 1099

f the measured or predicted values. For instance, it may benown that as Y (t) increases in magnitude the variance of Y (t)lso increases. For these cases, the heteroscedastic model ofonstant relative variance

2 = ω2ijkY

2ijk (11)

s used and the maximum likelihood criterion becomes aeighted least squares problem represented by

(k, θ) = mink,θ

⎧⎨⎩

βi∑j=1

γij∑k=1

[ln(ω2

ijkYijk) + (Yijk − Yijk)2

ω2ijkYijk

]⎫⎬⎭ (12)

By this definition, the error weights,ωijk, are chosen to reflecthe relative level of confidence in the measurements or predictedalues. This leads to the minimization of the objective functionith more weight given to the data that has less error.For pure heteroscedastic variance models, the structure of the

rror takes the more complex form

2 = ω2ijk

(Y2ijk

)λ(13)

If λ has the value 0, Eq. (13) becomes the same as Eq. (9)nd so the variance model reduces to the constant variance one.t the other extreme, where λ has the value 1, Eq. (13) becomes

qual to Eq. (11) and thus the variance model reduces to theonstant relative type.

The measured variables L(t) and C(t) are assumed to be bothescribed by constant relative variance models on the measuredalues. The choice of this model is justified as the measurementrrors in both variables are constant and do not depend on theagnitude of the measurement. It is possible using the gEST

arameter estimation facility in gPROMS to specify the param-ter estimation problem according to any one type of varianceodels mentioned above. gEST also allows for the determina-

ion of ω along with k and � as part of the optimization. Theoncentration of the solute and the mean size data are describedy constant relative variance models.

After its validation against experimental data, the model maye used in the subsequent step of dynamic optimization that willrovide optimum cooling and seeding for a desired objectiveriterion such as maximizing the mean crystal size. Dynamicptimization of the crystallization model is discussed next.

. Dynamic optimization

The optimization of the model was performed using the gOPTunction in gPROMS. Two objective functions were posed; ones to maximize the mean crystal size, L, in the shortest time whileeeping the distribution variance (Var) at a minimal value and thether is to minimize the distribution variance while constraining

¯ to a specific end-point value again in minimum time. The opti-ization variables are the temperature of the crystallization and

eed loading profile. Ten control intervals were implemented in

his dynamic optimization. Temperature being the time-variantptimization variable was specified as piecewise-linear while theeed distribution was specified as a time-invariant variable forach class of crystal size. Finding the optimal seed distribution

1 ring and Processing 46 (2007) 1096–1106

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Table 1The values of optimization constraints parameters

Variable Value Units

N0,max 9 × 108 no./m3

Tmin 20 ◦CTmax 55 ◦CRmin −0.006 ◦C s−1

Rmax 0.006 ◦C s−1

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100 S.M. Nowee et al. / Chemical Enginee

as possible by applying anecdotal knowledge, bounding theeed distribution to a size range known to provide adequate seed-ng conditions. For instance the seed size must be above a criticalize determined from simulation investigations [25] otherwisehe seeds will disappear before realizing their effect, in addi-ion the number of seed crystals should be chosen adequatelyo prevent secondary nucleation and directing the supersatura-ion to the growth of seeds [7], while the seed mass being inpractical and economical range (e.g. to be less than 10% ofnal crystal mass [10]). A number of constraints were imple-ented and consisted of the path rate of cooling constraint and

nd-point constraints reflecting the end-product characteristics.he formulation of these objective optimization problems haseen discussed elsewhere [19,26] and for our specific case isummarized as follows:

ptimize ΦT (t),N0

ubject to model and the following constraints:

f1(0) : N0 ≥ 0f2(0) : N0 −N0,max ≤ 0f3(t) : Tmin − T (t) ≤ 0f4(t) : T (t) − Tmax ≤ 0f5(t) : Rmin − dT (t)

dt ≤ 0

f6(t) : dT (t)dt − Rmax ≤ 0

f7(tf) : Tf,min − T (tf) ≤ 0f8(tf) : T (tf) − Tf,max ≤ 0f9(tf) : Cs,min − Cs(tf) ≤ 0

f10(tf) : ψf,min − ψ(tf) ≤ 0

hereΦ is the desired characteristic of the product; either meanize (L) or distribution variance (Var), whileψ is the other char-

po

Fig. 1. Schematic illustrating set-up

f,min 20 C

f,max 25 ◦C

s,min 30 kg m−3

cteristic which an endpoint constraint was applied to it to satisfyhe desired product characteristics. N0 is the vector for numberf seed crystals in each size domain class, which is the initialondition for crystal number distribution, so as a result this opti-ization variable was optimized as a time invariant variable.onstraints f1 and f2 are the ranges for seed crystal number dis-

ribution. Constraints f3–f6 are path constraints to make sure theooling rate and temperature of the crystallizer stay in the prac-ical operating condition. Finally constraint f7–f11 are endpointonstraints. Temperature and crystal mass concentration end-oint constraints were introduced to keep the final temperaturen a range so that the process yield meets the minimum require-

ent. The initial temperature and solution concentration of therystallizer are fixed for all studied cases. Parameter values ofhe constraints are presented in Table 1.

. Experimental work

Experiments for the purpose of identifying the model kineticarameters, validation of the model and for the purpose ofptimization validation were conducted in a glass jacketed

of the experimental apparatus.

S.M. Nowee et al. / Chemical Engineering a

Fig. 2. Temperature profile of a batch cooling run—A: initial heating to dissolveacp

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and thus provides closed loop control.

The implementation for validation of the optimal operationalpolicies within the experimental rig described earlier and for this

ll solids; B: soaking of the solution to confirm complete dissolution; C: slowooling to bring the temperature to saturation; D: crystallisation along a coolingrofile.

rystallizer unit with a volume of 1 l. Cooling/heating was imple-ented by means of a circulator (Julabo, USA) passing heat

xchange water through the jacket. This circulator is connectedo the Honeywell industrial distributed control system (DCS).he experimental setup is illustrated in Fig. 1. A standard PIDontroller was used to track temperature setpoints. Fig. 2 illus-rates the temperature recipe segments with heating, soakingnd cooling phases. Such cooling profiles are in their entiretyre-programmed as ramp and soak piecewise-linear segmentsithin the Honeywell Control Builder environment (see Section.1). This strategy matches well the dynamic optimization out-ut given by gOPT which is also piecewise-linear. Such outputan be readily migrated and implemented into the Honeywellnvironment.

The feed to the crystallizer was prepared from nano-pureater and Laboratory reagent grade ammonium sulfate. The

mmonium sulfate solution was prepared so as to have a concen-ration corresponding to the saturation temperature of 50 ◦C andvolume of 0.85 l. We use a conductivity probe (Orion conduc-

ivity probe model 018020A, USA) and temperature to track C(t)the solution concentration while CSD was determined offline

sing laser diffraction (Malvern Mastersizer-S, UK) providings the mean size data – L(t).

The experiments designed for parameter estimation consistedf two phases of linear temperature change; one cooling andne heating. During the cooling phase supersaturation developsffecting the crystallization. This is then followed by a heatinghase affecting the dissolution of the crystals formed during theooling phase. Solution concentration and CSD data under theseooling/heating regimes are recorded providing information onhe kinetic rates and allows the quantification of the parameterets. For experiments leading to offline optimization validation,he temperature profiles and seed determinations from optimiza-ion results were implemented.

.1. Implementation issues

This work is geared towards the closed-loop optimal con-rol of crystal size where we, after determining the optimalolicies (temperature and seeding), can use these profiles aseal-time moving set-points that direct the regulatory control. F

nd Processing 46 (2007) 1096–1106 1101

his involves the definition and implementation of a multi-ayer control architecture within the DCS environment [27].his advanced strategy is discussed briefly here to show the

eader how the optimization work presented in this paper can be,n general, exploited in full real-time optimizing crystallizationontrol. In a higher order layer, off-line optimization (as shown inection 4), uses the crystallizer model and process optimization

o calculate optimal CSD properties which become the set-pointor an intermediate layer. Model predictive control, in the inter-ediate layer, determines the real-time optimal temperature

et-point trajectory of the plant given the production require-ents (CSD) and operational constraints, and keeps the process

perating near optimum efficiency by constantly adjusting theet-points and responding to plant disturbances while keepingonstraints within given bounds. This layer utilizes the processodel to guide its actions which consequently become the set-

oints to the lower layer. In the lower layer, regulatory controlakes action, where the process variables, namely the crystallizeremperatures are kept at their set-points. In essence the role ofhe regulatory layer is to follow the directives (setpoints) of thepper control layers. The flexibility in the design of this controlrchitecture allows the lower level regulatory control set pointo be inputted from either one of the following three sources:

1) Manual set-point: this is used when the temperature of thecrystallizer is required to be set at a constant value. This ispure regulatory control.

2) Ramp-soak set-point: this is a set-point varying with timeand it changes according to an operator built-in and pre-defined temperature profile. This constitutes open-loopcontrol.

3) External set-point: this is used when an external program isused to provide the set-point. A model predictive controller,for example, could be used as the external set point source

ig. 3. Experimental temperature profiles for parameter estimation experiments.

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102 S.M. Nowee et al. / Chemical Enginee

pecific application, involves the use of the Option 2 that is theamp-soak set point.

.2. Technical issues

Another issue is the characterization of the crystallizationrocess which can not be achieved without the measurement ofolution concentration. This measurement provides informationbout the solute mass transfer and in turn the rate of crystalass evolution. The concentration data is also essential for the

arameter estimation study. A number of techniques have beeneported in the literature for the measurement of solution con-entration. These include refractive index measurement [28–30],ensitometry [31–34], Fourier Transform Infrared (FTIR) Spec-roscopy [35–37]. FTIR measurement is an advanced techniquend has become a standard method of in situ solution concentra-ion measurement. In this work the concentration of the solution

s determined by simultaneously measuring the solution conduc-ivity and temperature being the concentration measurementsvailable to the authors. This is an alternative cheaper measure-ent technique which has been exploited by a number of workers

ig. 4. (a) Model predictions vs. experiment data for solution concentration. (b)ean crystal size.

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nd Processing 46 (2007) 1096–1106

28,38,39] to measure the solution concentration. This methods suitable for high-concentration crystallization operations [40]here most other sensors would fail. It constitutes a soft-sensor

nferring concentration from the primary measurements, con-uctivity and temperature. One must note that since this is aechnique based on a concentration prediction model, it inher-ntly introduces errors into the parameter estimation step. Suchrrors are subject to experimental variance considerations.

. Results and discussion

The data gathered for the parameter estimation was fromxperiments undertaken in the 1 l laboratory scale batch crys-allizer described in Section 5 under various cooling regimes.he problem was split in two sections, cooling and heating,xperimental temperature profiles of which are shown in Fig. 3.

ated using the experimental data from the cooling phasehile the dissolution and disappearance kinetics were estimated

rom the heating phase. The estimated parameter values are

ig. 5. (a) Optimal temperature profiles for the two optimization objectives. (b)orresponding optimal mean size profiles.

S.M. Nowee et al. / Chemical Engineering and Processing 46 (2007) 1096–1106 1103

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ig. 6. (a) Optimal seed distributions for the two optimization objectives. (b)orresponding optimal end-product size distributions.

og(k) = [3.25 −6.55 −10.04 −6.83] and � = [10.8 6.0 2.0 1.0].he experimental data Y (t) = [L(t), C(t)] and the model pre-ictions Y (t) = [L(t), C(t)] are shown in the overlay plots inig. 4 showing good agreement between the model predictionsnd the experimental measurements. The solution of the non-inear dynamic optimization problem represented the optimalombination of tf, T(t) and initial n(L, t) that gives the maximumr minimum value of the objective function while satisfying theiven constraints. The significance of the dynamic optimizationroblem is that a path to optimality is drawn. In this case theptimal path is described by a temperature trajectory and a cor-esponding mean size trajectory (Fig. 5a and b). The optimalemperature profiles are very similar and comprise of three dis-inct regions. In region one, we observe an initial decrease in

emperature that can be explained by the drive to cross overhe saturation temperature to generate supersaturation. Region

sees a phase of slow cooling in which the supersaturation iseing stabilized and growth of crystals is occurring under con-

sstF

ig. 7. (a) Optimal temperature profile for objective function: maximum meanize. (b) Corresponding optimal mean size profile vs. experimental.

rolled nucleation towards the maximum size. In the third andnal region, a sudden decrease in temperature is observed and

s the result of the optimization attempting to minimize tf ando forcing the temperature down sharply. The correspondingean size profiles simulated under the optimal temperature pro-les are shown in Fig. 5b. Again similarities between the twoxist though the results of the maximization of the mean sizere slightly better. No size reduction of the seeds is observedecause the induction time of the seeds is fixed at a point verylose to saturation conditions rendering the rate of dissolution toe slow so that the reduction in seed size is not appreciable. Therystals continue to grow and stabilize at about the same size foroth optimizations leading to suggest that the two optimizationroblems provide similar end-size solutions and thus are similarn the way they are posed. As for the time of the batch, no signif-cant difference is observed between the two optimizations. Theptimal seed distributions for the two optimization problems are

hown in Fig. 6a. The two optimizations observed very similareeding optima. The end product size distribution was foundo vary for the two optimization problems which are shown inig. 6b showing different distribution heights and shapes. These

1104 S.M. Nowee et al. / Chemical Engineering and Processing 46 (2007) 1096–1106

Fa

otgp(rnobtmtmaiHhitv

Fc

7

cepopttf

A

abBBC�

C

CCdDDgGkkkkd dissolution rate coefficient (m/s)

ig. 8. (a) Optimal temperature profile for objective function: minimum vari-nce. (b) Corresponding optimal mean size profile vs. experimental.

utputs from the optimizations where implemented experimen-ally using the bench scale crystallizer and the results show aood agreement. The temperature profiles consist of four linearrofiles reasonably close to the profile given by optimizationsFig. 7a and b). As the optimal seed size distributions were nar-ow and only for two size classes, to prepare the seeds it wasot practical to select the exact number of crystals suggested byptimization at each size class, so the equivalent amount of seedsetween the nearest ASTM sieves were weighed and introducedo the crystallizer in the early stages of the process. Mean size

easurements were close to what it was expected from simula-ion studies, although there is a very minor fluctuation from the

ean size growing profiles presented from simulations (Fig. 8and b), which could be mainly due to the oscillatory behaviorn the controlled measured temperature inside the crystallizer.owever, as could be seen in Fig. 9 the final size distributionas a very narrow shape compared with a process without seed-

ng and with linear cooling profile. Note that the distributions inhis figure are from Mastersizer and unlike Fig. 6, are based onolume percentage of particles at each size.

kLL

ig. 9. Comparison between final product distribution of optimal and linearooling profiles.

. Conclusions

In this work we (1) presented a model of the cooling seededrystallization with dissolution kinetics, (2) identified the param-ters of the kinetic sub-model, (3) developed dynamic optimalrofiles for cooling and seed size distribution under two differentbjective functions and (4) validated the model-based optimalolicies experimentally. We were able to define and experimen-ally achieve a particle mean size objective. This approach suitshe problem of crystal size control very well and paves the wayor real-time optimal control.

ppendix A. Nomenclature

disappearance rate ordernucleation rate ordernucleation rate (no./m3 s)

0 nucleation rate of zero-sized crystals (no./m3 s)solution concentration (g/100 g soln.)

C supersaturation (g/100 g soln.)ˆ measured concentration (g/100 g soln.)

s mass concentration of crystals (g/100 g soln.)* saturated solution concentration (g/100 g soln.)

dissolution rate orderdissolution rate (m/s)

a disappearance rate (no./m3 s)growth rate indexgrowth rate (m/s)vector of kinetic rate parameters

a disappearance rate coefficient (no./m3 s)b nucleation rate coefficient (no./m3 s)

g growth rate coefficient (m/s)crystal size (m)

ˆ measured mean size (m)

ring a

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σ

σ

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S.M. Nowee et al. / Chemical Enginee

¯ mean size (m)total number of measurements

(L, t) particle number density (no./m3 m)0 number density of zero-sized crystals (no./m3 m)

vector of number of particles in each size class (no./m3)0 vector of number of seed in each size class (no./m3)

vector of kinetic ratescooling rate (◦C/s)time (s)

f batch time (s)temperature (◦C)time varying input variables

ar crystal size distribution variance (m)differential output variables

ˆ vector of measured variables

reek lettersnumber of experiments

i number of variables measured in the ith experimentij number of measurements of the jth variable in the ith

experimentDirac delta functionvector of kinetic parametersheteroscedastic variance model parameterstandard deviation of a measured variable

2ijk variance of the kth measurement of jth variable in the

ith experimentresidence time (s)objective functionoptimization variablestandard deviation of the measurement error

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