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Separation and Purification Technology 53 (2007) 153–163

Multiscale modeling, simulation and validationof batch cooling crystallization

Ali Abbas ∗, Jose A. Romagnoli 1

Department of Chemical Engineering, University of Sydney, Sydney 2006, Australia

bstract

This study investigates issues relating to the modeling of the batch cooling crystallization process. A systems model is presented where athe microscale, a population balance model is used and solved numerically with the aim of predicting end-product properties under the effectf different cooling conditions affected by mesoscale fluid heat transfer and macroscale flow/temperature control. Model validation is carriedut experimentally. Technical issues arising from these implementations including the aspects of the numerical solution, generalization of therystal size domain and connectivity with an advanced distributed control system are discussed and results presented. The model is proposed for

pplications as a soft-sensor for the prediction of the crystal size and in the design of the model-based control scheme. This paper presents annvironment for advanced simulation and operation of crystallization processes. This environment supports the ultimate challenge of crystallizationhich is control towards efficient separations and purifications. 2006 Elsevier B.V. All rights reserved.

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eywords: Crystallization; Multiscale modeling; Population balance; Paramete

. Introduction

Crystallization is an important unit operation in the fieldf chemical engineering, which can obtain solid products ofigh purity at low costs. Crystallization is employed to obtainigh value products, which are essentially characterized by therystal size distribution (CSD). The crystallization process isseparation process as it produces solid particles from liquid

r from vapour phase. Many manufacturing industries rely onrystallization for the production of their materials. These indus-ries include the pharmaceutical, the agro-chemical, and generalhemical manufacturers.

Crystallization is a transport process and the driving force isupersaturation, which drives the solid phase out of the solution.upersaturation is achieved by either cooling or evaporation. In

his study the cooling method of crystallization is researched.he rate of cooling, and thus the cooling profile in a batch, hasirect influence on the nucleation and growth kinetics of crys-

∗ Corresponding author. Present address: School of Chemical and Biomedicalngineering, Nanyang Technological University, Singapore.el.: +65 6316 8811; fax: +65 6794 7553.

E-mail addresses: aabbas@ntu.edu.sg (A. Abbas), jose@lsu.eduJ.A. Romagnoli).

1 Present address: Department of Chemical Engineering, Louisiana State Uni-ersity, Baton Rouge, LA 70803, USA.

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383-5866/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.seppur.2006.06.027

ation; Hybrid simulation; Pilot plant

allization and is important in shaping the final product’s crystalize. How a crystallizer is designed and operated influences theuality of the end-product and particularly the crystal size. Its therefore important to research the relationship between therystallization product and the crystallization operating con-itions. Understanding the dynamic behaviour of crystallisersllows model-based control schemes to be implemented and thusptimize the operating conditions that aim at the production ofrystal products of desired size and quality. This is a key incen-ive for this study which presents the development of a completeybrid simulation for crystallising ammonium sulfate in a pilotlant.

. Multiscale modeling of industrial crystallizers

The ultimate purpose behind the modeling of crystallizationrocesses is to develop mathematical models that adequatelyescribe the dynamic behaviour of crystallisers and conse-uently, use these models in simulation, optimization andodel-based control of real industrial applications. In order to

evelop models that accurately describe the setting in which the

eal physical process lies, it becomes necessary to understandhe multiscale nature of such systems ranging from fundamentalsf crystallization phenomena at the microscale, to the dynamicehaviour of crystallization plants at the meso- and macroscale.

1 d Purification Technology 53 (2007) 153–163

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54 A. Abbas, J.A. Romagnoli / Separation an

ere we define the microscale to represent the size scale inhe viscinity of the particle physical dimension. The mesoscales here defined as the size scale at which fluid dynamics areonsidered in the crystalliser and its jacket and where theseynamics affect heat transfer. The macroscale is lastly defineds the scale at which activities relating to changes/manipulationsn lumped macroscale variables, such as the temperature of theacket inlet, occur. This interconnection between the differentcales is a key feature of this study. Applicably important heres the manipulation of the microscale properties by inducinghanges at the scales readily accessible to the engineer, namelyhe macroscale (this includes variables like flowrate). We decom-ose the system that spans across spatial scales into models forach scale and thus phenomena are modeled at their respectivecales.

The model developed in this study is primarily aimed at arystallization pilot plant currently in operation in our labora-ory and shown in Fig. 1. The model describes in detail therystallization unit consisting of a jacketed crystallizer having aolume of 75 L. The plant is supported by a specially designedtilities unit described in [1] providing heating and coolingequirements to the jacket. The plant is fully instrumentednd is interfaced to an industrial distributed control systemDCS).

The modeling of particulate systems like crystallization andheir interactions in the solid–liquid phases has been addressedy a number of modeling tools yet inadequacies prevail espe-ially when one is looking at developing a multiscale dynamicodel of a pilot plant. It is not intended in this study to analyse

he different modeling packages but it suffices to say that, whereossible, the integration of packages, such that each packagean offer its strengths, within a hybrid simulation environments a robust solution to complex multiscale process modeling.ere, we use the modeling package gPROMS (Process Sys-

ems Enterprise Ltd., UK) to represent, at the microscale, the

istributed paramter crystallization process. The flowsheetingackage HYSYS (Aspentech Ltd., USA) is used to represent,t the macroscale, the utilities unit of the pilot plant. A previ-us study [2] discussed the fundamentals of the architecture of

Fig. 1. Photo of the 75-L pilot crystallization plant.

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ig. 2. Multiscale communication between the components of the hybrid envi-onment.

his hybrid simulation environment and demonstrated success-ul integration of the mentioned packages which resulted in aomplementary hybrid multiscale modeling environment illus-rated in Fig. 2. The spreadsheeting package Excel (Microsoft,SA) is used as a bridge and also to solve a detailed jacketeat transfer mesoscale model. The communication protocols,CS and gFPI, used in this architecture while proving to be

dequate are ones that the authors had access to. Protocolsill continue to evolve improving and streamlining connectivityetween packages. We next look at the microscale modeling ofhe crystallization process in some detail. The mesoscale modelomponent is presented in Appendix A. For a description of theowsheeting macroscale model component, the reader is refered

o Ref. [2].

.1. Population balance modeling theory

Since the crystallization process is a particulate one, Hul-ert and Katz [3] proposed the population balance theory asn approach to modeling of the crystallization process. Thisheory proposes the use of a balance to account for the num-er of crystals in the crystallizer. The output of such a balances a distribution of the number of crystals across both timend size domains. In their approach, Hulbert and Katz char-cterize the crystallization system using internal and externaloordinates. The external coordinates are simply the ordi-ary rectangular coordinates specifying the location of thearticle, while the internal coordinates represent the particleize and other aspects of particle quality as may be relevant.his theory leads to the deterministic description of the fullynamic distribution of particle sizes of the process unit beingnalysed.

In their prominent book “Theory of Particulate Processes”,andolph and Larson [4] devote a whole chapter to the the-ry and the development of the population balance model. Foratch cooling crystallization with size independent growth, theesulting crystal population balance equation is shown below. Itescribes the evolution of crystals in both time (t) and size (L)istribution domains.

∂f (L, t)

∂t= −G

∂f (L, t)

∂L+ B − D (1)

he first term in Equation (1) represents the accumulation ofrystal numbers in the size range Li to Li+1, while the second

erm represents the net flux of particles through the size range byrystal growth (G). The third and fourth terms, B and D representirth or production (by nucleation) of crystals and death (byissolution) of crystals, respectively, in the size range. We will

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A. Abbas, J.A. Romagnoli / Separation

ssume that crystallization will occur in the cooling temperatureirection only, i.e. neglecting the dissolution term D.

.2. Kinetics of crystallization

In any crystallization event, a number of different phenomenaay be observed including, nucleation, growth, agglomera-

ion, breakage and dissolution. These phenomena constitute theinetic sub-models that effectively drive the crystal formationrocess resulting in a population of crystals with a certain sizeistribution. The type of phenomena occurring determines theype of kinetic expressions used in the population balance equa-ion.

Nucleation and growth rate kinetics are the two predominanthenomena and most crystallization models found in literatureevolve around these two kinetic rates. These rates are com-eting simultaneously [5,6], and are governed by complicatedunctions. Various nucleation and growth rate model structuresave been reported in literature [7,8]. The parameters of theseodels vary according to the type of material being crystallized.

n this study the ammonium sulfate–water system will be theystem used. A number of researchers [9–12] have studied thisystem. In this work, the authors opt to use the power law modelsEqs. (2) and (3)) for both kinetic rates, for their adequacy andimplicity.

= kb(C − C∗)b (2)

= kg(C − C∗)g (3)

here C is the solute concentration in the liquid phase and C* ishe saturation concentration. Kb, b, kg and g are unknown kineticarameters.

.3. Mass and energy balances

To complete the model, the mass balance can be written afterawlings et al. [13] for a batch system as:

dC

dt= −3ρcαVFG

∫ ∞

0fL2 dL (4)

here ρc is the density of the crystal, αV the volume shapeactor and F is a conversion constant. The crystallizer volume,, is heated and cooled by way of a syltherm oil flowing in the

acket. The energy balance for a batch system, Eq. (5), accountsor the heat being removed by the cooling medium in the jacketf the crystallizer and also for the heat of crystallization.

CpVdT

dt= −3�HcρcαVVG

∫ ∞

0fL2 dL − UA�Tlm (5)

The heat of crystallization, �Hc, for ammonium sulfate is1.46 kJ mole−1 [14]. Simulation studies reveal that this energyffect is negligible. We can, hence, ignore the first term on the

ight hand side of Eq. (5) and simplify the energy balance to

dT

dt= −UA

ρCpV�Tlm (6)

m

m

urification Technology 53 (2007) 153–163 155

q. (6) is used to determine the temperature of the crystallizerontents. The temperature of the jacket outlet, Tj, the overall heatransfer coefficient, U, and other parameters are determined fromeat transfer relations which are presented in Appendix A. Theodel equations for a batch cooled crystallization process are

resented in Appendix B.

. Modeling solution and related issues

The population balance equation is a partial differential equa-ion in both time and size. Researchers have reported difficultyn obtaining an analytical solution to the above equation sug-esting the use of numerical methods for a solution. It is alsorequirement that the population balance be solved efficiently

nd accurately so as to allow advanced optimization and controltrategies to be implemented. There are two primary solutionethods found in literature and these are the application of theethod of moments and the discretised population balance.

.1. The method of moments

The method of moments reported by Hulburt and Katz [3]nd later by Randolph and Larson [4] is commonly used andts application has been reported by many workers [11,15–17].his method reduces the partial differential equation that is theopulation balance into a set of coupled ordinary differentialquations (ODEs). The advantage becomes the ease of solu-ion, as ODE solvers are readily available. This method has

disadvantage in that a unique CSD cannot be reconstructedrom a finite set of moment equations as different CSDs maye formulated from the same moment equations. A second dis-dvantage is that population balances carrying size dependentrowth functions may not be solved.

The population density is a function of crystal size and we canse the density function, f(L), to express the number of crystals,i, in a size class Li to Li+1 per unit volume of slurry as

i =∫ Li+1

Li

f (L) dL (7)

rom Eq. (7), a series of relationships evolves representing theoments of the density function. We write the jth moment of

he density function as

i =∫ ∞

0Lj(L) dL (8)

here the 0th moment, m0, of the distribution, at j = 0, andxpressed as m0 = ∫ ∞

0 f (L) dL characterizes the total num-er of crystals. The first, second and third moments also providenformation about the distribution and represent the total length,he total area and the total volume of crystals, respectively. Thats, ∫ ∞

1 =0

Lf (L) dL (9)

2 =∫ ∞

0L2f (L) dL (10)

1 d Pur

m

3

uvtmmaingot

D

waIsmIsbom

D

3

pdcmfprtTHepeati

3

f

dy

T

iE

f

Ia

oTuf

φ

toaestrQdiscretisation in equal size intervals rather than in equal timeintervals generate better allocation of the computational results.

Si = Li−1 + Li

2(19)

56 A. Abbas, J.A. Romagnoli / Separation an

3 =∫ ∞

0L3f (L) dL (11)

.1.1. Different mean sizesIt is worthwhile to understand the various types of means

sed to describe particles. Three commonly used means are theolume weighted mean diameter (De Brouckere mean diame-er), D[4,3], the surface area weighted mean diameter (Sauter

ean diameter), D[3,2], and the number mean diameter (arith-etic mean), D[1,0]. It is important that the modeling takes into

ccount the type of measurement being used, where validations a subsequent step to modeling. Different particle sizing tech-iques measure different properties of the particle and usuallyenerate a derived mean based on either volume, surface, lengthr number. The various derived diameters can be described byhe following general equation [18]

[m, n] =[∑

dm−3i Vi∑

dn−3i Vi

]1/m−n

(12)

here Vi is the relative volume of particles with size di. mnd n are integer values that describe the mean being used.t is important to select the correct mean size to represent theample of particles for the particular application. In this study,easurements are made by the Malvern Mastersizer (Malvern

nstruments Ltd., UK) which reports a D[4,3] as the Master-izer works by laser diffraction and calculates a size distributionased around volume terms. Eq. (12) is related to the momentsf the distribution and D[4,3] can be got by dividing the fourthoment of the distribution by the third moment, that is

[4, 3] = m4

m3=

∫ ∞0 L4f (L) dL∫ ∞0 L3f (L) dL

(13)

.2. Discretisation of the population balance

The discretisation of the population balance equation is sim-ly a finite difference method of solution, where the partialifferential equation is sectioned along the size domain into finitelasses. This method overcomes the disadvantage found in theethod of moments as it permits the discretisation of the growth

unction along the size domain. Furthermore, this method allowshenomena, such as agglomeration and attrition to be incorpo-ated in the solution. Two principal researchers who pioneeredhis method are Hounslow et al. [19] and Marchal et al. [16].he disadvantage of this method is its inaccuracy as reported byounslow et al. [19]. This method was overlooked by Rawlings

t al. [13] who deems it inefficient. The advancement of com-uters in recent years may alleviate this disadvantage allowingfficient solution of the population balance equation. Mignon etl. [20] report that the use of discretisation formulae gives riseo unstable numerical solutions. Stability of numerical solutionss discussed in the next section.

.2.1. Discretisation issuesDiscretisation transforms the PBE into a family of linear dif-

erential equations by discretising along the size domain. The

ification Technology 53 (2007) 153–163

iscretisation along the independent variable (L) from the PBEields

∂f

∂t(Li, t) ∼= −G

1

�L{f (Li, t) − f (Li−1, t)} + B,

i = 1, 2, . . . , n − 1 (14)

The time derivative is now considered of the ordinary type.hus, a system of ordinary differential equations

dfi

dt= −G

1

�L(fi − fi−1) + B, i = 1, 2, . . . , n − 1 (15)

s produced, where Eq. (15) represents the discretised form ofq. (14).

Marchal et al. [16] performed the discretisation of the PBEor an adipic acid crystallization study and is as follows:

dNi

dt+ [

G(Li)f (Li) − G(Li−1)f (Li−1)] − Bi + Ni

1

V

dV

dt

= −Ni

∑k

Qk

V(16)

ncorporating McCabe’s assumption and neglecting the volumeccumulation and input–output terms leaves

dNi

dt+ G

[f (Li) − f (Li−1)

] = Bi (17)

Fig. 3 is a schematic illustrating the division of the continu-us density function into classes (intervals) along the size axis.he width of the ith size interval is the difference between thepper and lower sizes of that interval and is represented by theollowing equation

i = Li − Li−1 (18)

While the arithmetic mean size, or the characteristic size ofhe crystals of the ith interval, Si, is taken as the arithmetic meanf that size interval represented by Eq. (19). The scale of the sizexis showing the relationship between the different axis param-ters L, S and φ is represented by Fig. 4. The discretisation of theize domain meant that intervals of fixed size were employed,hus �L was held constant and �t was varied accordingly to theequirements. The choice of this methodology was promoted byiu and Rasmuson [21], as they concluded in their research that

Fig. 3. The discretised density function.

A. Abbas, J.A. Romagnoli / Separation and P

da

f

Ac

N

Rt

f

a

f

S

Iao

Tfieφ

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w

A

d

N

weisowp

N

w

A

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Fig. 4. The discretised size domain.

Now assuming that the value of the density function f at eachiscrete size class in the size domain is constant with value N̄i,nd the value of f at Li is the following arithmetic mean

(Li) = N̄i + N̄i+1

2(20)

lso the value of Ni(t) is the area of the rectangle at that sizelass and may be represented by

i(t) = N̄i(t)φi (21)

ewriting the above equation as N̄i(t) = Ni(t)/φi, and substi-uting into Eq. (21) yields

(Li) = Ni/φi + Ni+1/φi+1

2(22a)

nd

(Li−1) = Ni−1/φi−1 + Ni/φ1

2(22b)

ubstituting these into Eq. (17) gives

dNi

dt+ G

2φi+1Ni+1 − G

2φi−1Ni−1 = Bi (23)

ntroducing the Dirac delta function to the nucleation termllows the nucleation rate to represent the rate of appearancef crystals of zero size, and equation becomes

dNi

dt+ G

2φi+1Ni+1 − G

2φi−1Ni−1 = Bi�(Li − L0) (24)

his forces the nucleation rate to become zero at all but therst interval. The above equation is a set of ordinary differentialquations, dNi/dt = c(NL(t))i, where c is equivalent to −G/φ and, the interval width, is constant for all intervals. This may be

ritten as follows:

dN1

dt= − G

2φ2N2 + Bi�(Li − L0)

dNi

dt= − G

2φi+1Ni+1 + G

2φi−1Ni−1, i = 2, 3, . . . , n − 1

dNn

dt= G

2φn−1Nn−1

(25)

ta⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

urification Technology 53 (2007) 153–163 157

ssuming L ∈ [0,1] and φ = 1/m, where m is the number ofequired intervals, the system of equations can be expressed as

d �Ndt

= A �N + �B�(�L − L0),

{ �N = (N1, N2, · · ·, Nn)T

�B = (B1, 0, 0, · · ·, 0)T

(26)

here

= G

2φ

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 −1 0 0 · · · 0

1 0 −1 0 · · · 0

0 1 0 −1 · · · 0

0 0 1 0 · · · 0...

......

.... . .

...

0 · · · · · · · · · 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(27)

The above formulation is a discretisation along the sizeomain based on the finite central difference approximation

i(t) = Ni+1 − Ni−1

2φ+ E(φ2) (28)

here E(φ2) is the approximation error. With this formula theigenvalues of the resulting system are predominantly imag-nary, requiring a suitable integration technique of absolutetability near the complex plane’s imaginary axis. Fortunately,ne can overcome this drawback by applying the back-ard finite difference approximation, often called an upwindrocedure

i(t) = Ni − Ni−1

2φ+ E(φ2) (29)

ith matrix A becoming

= G

2φ

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−1 0 0 0 · · · 0

1 −1 0 0 · · · 0

0 1 −1 0 · · · 0

0 0 1 −1 · · · 0...

......

.... . .

...

0 · · · · · · · · · 1 −1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(30)

ith this backward finite approximation, matrix A now has onlyne eigenvalue, (−G/2φ), and is easily solved using conventionalunga–Kutta techniques. It is also worth noting that the finiteifference approximation for NL is only first order renderinghe solution inaccurate for large interval widths. To overcomehis and hence reduce E(φ2), small interval φ is chosen duringhe simulation. The system of ODE’s is revised and rewrittens

dN1

dt= − G

2φ1N1 + B1

dNi

dt= − G

2φi

Ni + G

2φi−1Ni−1, i = 2, 3, · · ·, n − 1

dNn

dt= − G

2φn

Nn + G

2φn−1Nn−1

(31)

158 A. Abbas, J.A. Romagnoli / Separation and Purification Technology 53 (2007) 153–163

f disc

dim

3

raoc

i

L

wwcrL

L

Amsmo

t

φ

Tf

d

S

wbocr

odmIottdao

idiTwnw

n

Fig. 5. The effect o

Fig. 5 shows the effect of the discretisation methods on theensity function. The instability of the central finite techniques clearly seen in contrast with the more stable backward finite

ethod.

.3. Generalising the size axis

This section summarizes the methodology and results fromesearch into generalising the X-axis to enable it to representny desired total range. The significance of this is that this setf equations can be used to set (and modify) the range of anyrystal size distribution.

The general equation describing the size (length) of a crystals:

i = abi, i = 0, 1, 2, . . . , n (32)

here a is the size of the nucleate, which is the lowest size athich the measuring device registers a reading, b the geometric

onstant to be determined and Li is the size (m) of the ith sizeange. Ln is the largest crystal size to be considered in the model.0 can be described by equation

0 = a (33)

s can be seen from above, a geometric series is used to deter-ine the distance between discrete crystal sizes. A geometric

eries is following researchers [22] who reported that the geo-etric series is the most suitable for determining the sequence

f size ranges.Eq. (18) characterising the size of each interval can be rewrit-

en as:

i = abi − abi−1 (34)

his is simply the length of the (i − 1)th size range subtractedrom the length of the ith size range. While the midpoint for each

cs

n

retisation methods.

iscrete crystal size range, Si, is characterized by equation:

i = abi + abi−1

2(35)

hich is equivalent to Eq. (19) and represents the averageetween the ith and the (i − 1)th size ranges. The upper boundaryf the total range can be varied by either, varying the geometriconstant, b, or varying in the number, n, of discrete crystal sizeanges. The effect of each is discussed in turn.

Varying the geometric constant increases the upper boundaryf the total size range but also changes the distance between theiscrete crystal size ranges. This is done by assessing a desiredaximum size Ln and keeping a and n constant in Eq. (32).

ncreasing the geometric constant may have the unwanted effectf decreasing the accuracy of the results. However, it would havehe desirable effect of decreasing the time necessary to carry outhe computation. Kilpio and Norden [6] report that changing theiscretisation size of the particle axis by a factor of 25 greatlyffects the simulation time, however it only had minor effectsn the accuracy of the resulting crystal size distribution.

Changing the number of size ranges has the effect of increas-ng the upper boundary of the total crystal size range. This can beone by setting the desired maximum crystal size, Ln, and vary-ng n while keeping the size of the nucleate, a and b, constant.he advantage of increasing the number of size ranges comparedith increasing the geometric constant is that increasing n willot make the simulation any less accurate. Computational timeill increase slightly.Both increasing the geometric constant and increasing the

umber of discrete size ranges give the same result. The resulting

rystal size distributions are identical, as long as the range is theame.

An example is now given to illustrate the method. Say theucleate size is 0.4563 �m, Ln = 880 �m and n = 101. The geo-

and Purification Technology 53 (2007) 153–163 159

m

b

3

ib

b

n

g

3

ic

b

Sectd

4

tttkrpacinIftp

scPltFooi

Fig. 6. Temperature of the crystallizer under three cooling regimes.

ldetlother two profiles. To illustrate this, the results shown in Fig. 8suggest that nuclei are produced early in the batch when cool-ing with Profile 1. The other two profiles approach the labile

A. Abbas, J.A. Romagnoli / Separation

etric constant, b, can be found by using Eq. (32).

=(

880

0.4563

)1/101

= 1.0777

.3.1.1. Case 1: calculating the number of size intervalsIf it is desired to increase the maximum size to 2000 �m by

ncreasing the number of size intervals, the new value for n cane found as follows:

n = 2000

0.4563

log(1.0777) = log

(2000

0.4563

)

iving n = 112.

.3.1.2. Case 2: calculating the geometric constantIf it is desired to increase the maximum size to 2000 �m by

ncreasing the geometric constant, b, the new constant can bealculated as follows:

=(

2000

0.4563

)1/101

= 1.08657

o, these new parameters can now be calculated speedily andntered into the simulation to vary the upper boundary of therystal size range in the particle size distribution. In effect,his method generalises the crystallization model to any systemesired.

. Simulation results

In cooling crystallization the main manipulated variable isemperature. Cooling regimes are key for attainment of crys-als with desired end-product sizes. It is well established thathe rate of cooling directly affects the nucleation and growthinetics and thus the eventual CSD. For instance, a fast coolingegime results in the build of supersatuation to levels that havereference for nucleation over growth, consequently leading tolarge population of crystals of small sizes. On the other hand, aontrolled cooling regime can employ the supersaturation driv-ng force to grow the nucleated crystals rather than produce newuclei, resulting in large-sized crystals of a small population.n essence as will be seen later, nucleation and growth competeor the supersaturation driving force. For illustrative purposes,he results of simulations under three cooling conditions areresented next.

Three temperature profiles are implemented in three differentimulation runs. The results show that the effect of the rate ofooling is evident throughout the figures. In Fig. 6, temperaturerofile 1 represents a natural cooling policy, Profile 2 is for a

inear cooling policy while Profile 3 represents a policy wherehe cooling occurs at a slower rate than the other two policies.

ig. 7 shows the supersaturation profiles corresponding to eachf the cooling profiles. Examining Profile 1, we see that the ratef desupersaturation is high in the beginning when the cool-ng rate is highest. The other two supersaturation profiles have

Fig. 7. Supersaturation profiles under the three cooling regimes.

ess variation and the degree of supersaturation decreases in theirection of Profile 3. A fast cooling rate is translated in high lev-ls of supersaturation as the concentration of the solution movesowards the labile region. Nucleation is expected to occur ear-ier under Profile 1 as the labile region is penetrated before the

Fig. 8. Mean crystal size profiles under the three cooling regimes.

160 A. Abbas, J.A. Romagnoli / Separation and Purification Technology 53 (2007) 153–163

ztcisncudtitpt

mmtTuipsAr

Fig. 11. Contour plot view of the CSD of Fig. 10.

Fig. 9. End-product CSDs under the three cooling regimes.

one slower and so exhibit delayed nucleation. The influence ofhe temperature profile is also seen when comparing the productrystal size distribution profiles generated. These are presentedn Fig. 9. Because nucleation occurs early under Profile 1, theupersaturation is consumed in the production of new crystaluclei eventually resulting in a large population of small sizedrystals. In contrast to the slower cooling where the supersat-ration is not immediately used up on nucleation but is betteristributed so that growth receives a reasonable share. This even-uates in a smaller population of larger sized crystals. From this,t can be understood that there is an optimum cooling profilehat exists for a given crystal size objective function. It is thusossible to control the size by controlling the cooling rate duringhe batch.

The linear cooling regime provides a crystal product of largerean size but because of the supersaturation in Profile 3 beingore constant throughout the batch, the results shows that crys-

al growth was more dominant than under Profiles 1 and 2.his is identified by the lower crystal number density producednder Profile 3. The three-dimensional crystal size distributions shown in Fig. 10 as a surface plot and in Fig. 11 as a contour

lot. Both plots complement each other and show the crystalize distribution across the two temperal and spatial domains.s mentioned earlier the model is capable of presenting the

esults in volume percent (vol%) to be comparable with size

Fig. 10. The CSD surface resulting from cooling with Profile 3.

idd

5

shafos(Ttuos

Fig. 12. The vol% CSD surface resulting from cooling with Profile 3.

nstrumentation for analysis purposes. As such The numberensity distribution shown in Fig. 10 is converted to a volumeistribution as shown in Fig. 12.

. Parameter estimation

The driving force of crystallization as described earlier is theolution supersaturation and it becomes of interest to quantifyow much of this driving force is committed towards nucleationnd how much is committed towards growth. This is importantor kinetic model identification—a prerequisite step for dynamicptimization and model-based control. The parameter set θ con-isting of kb, b, kg and g in the nucleation and growth modelsEqs. (2) and (3)) needs to be estimated from experimental data.he data gathered for this purpose were from experiments under-

aken in an un-seeded 1-L laboratory scale batch crystallizernder various cooling regimes given in Fig. 13. The durationf each experiment, the initial concentration of solute in theolution and the cooling regime are not the same in each exper-

A. Abbas, J.A. Romagnoli / Separation and Purification Technology 53 (2007) 153–163 161

Fig. 13. Experimental temperature profiles.

Table 1The optimal parameter estimates

Experiment ln(kb) b ln(kg) g

1 7.66 4.42 −18.72 3.374 4.59 5.40 −19.60 3.406 7.66 9.24 −15.43 3.927 3.53 5.92 −18.16 2.388

x

iactfaatwop

6

sti

Fig. 15. New experimental cooling profile used for validating parameter esti-mates.

F

ssoatmmt

7.66 9.41 −15.42 4.67

¯ 7.17 6.88 −16.28 3.55

ment while agitation rate is kept constant across experimentsnd infrequent measurements are made of temperature, solutiononductivity and crystal size. By determining θ, we maximisehe probability that the model will predict the values obtainedrom experiments. Rigorous non-linear optimization techniquesre now available paving the way for more efficient estimationnd with the advances in power of modern computers, optimiza-ion methods are becoming more prevalent in literature. In thisork, the gEST function in gPROMS is used which is basedn the maximum likelihood criterion that captures the highestrobability of the model predicting the real data.

. Results and validation

This estimation problem is easily specified in gPROMS andolved by the gEST facility. Optimal parameter estimates forhe five selected experiments are given in Table 1. The exper-mental data and the model predictions for Experiment 7 are

stwe

Fig. 14. Model fit for Experiment 7: (a) me

ig. 16. Mean size predictions versus experimental data for new experiment.

hown in Fig. 14. The current model is now ready for use in theubsequent step of dynamic optimization that will provide anptimum cooling profile for a desired objective criterion, suchs maximising the mean crystal size. An Initial experimentalest using an estimated optimal cooling profile (Fig. 15) for sizeaximisation is performed to validate the feasibility of on-lineodel application. The implementation of the model in real-

ime with the plant demonstrates its practicality as an on-lineensor for inferring the mean crystal size. It is useful to have

he model-generated crystal size data available on-line as thisould circumvent the problems associated with unreliable or

xpensive on-line particle size instrumentation. The results of

an size and (b) solute concentration.

1 d Pur

tyTc

tsmsh

7

aawamsErc

Ah

te

r

T

Q

wct

�

Ej

Q

T

wc

d

N

d(

N

N

bn

N

wttd

D

wj

A

F

V

ab

N

e

N

62 A. Abbas, J.A. Romagnoli / Separation an

he predictions are compared (Fig. 16) with off-line sieve anal-sis of crystal samples taken from the crystallizer infrequently.he model is shown to satisfactorily predict the course of therystallization.

A predictive control scheme for optimal operation, based onhe proposed model environment is anticipated and will be theubject of our current research. It is this control aim that initiallyotivated the development of the proposed model framework

ince the efficiency of any model-based control strategy relieseavily on the efficiency of the model predictions.

. Conclusions

This work demonstrated the key steps in development ofcrystallization systems model. Model solution environment

nd multiscale and other modeling issues were discussed. Itas shown that the process can adequately be modeled withinhybrid simulation environment that addresses the multiscaleodeling challenge. Successful decomposition and synchroni-

ation of spatial scales was successful in a hybrid simulation.xperimental validation was shown to be modestly successful in

epresenting the multiscale crystallization process. Such modelan be utilised within inferential model-based control schemes.

ppendix A. Mesoscale model of crystallizer–jacketydrodynamic heat transfer relations

The determination of heat transfer in jacketed agitated crys-allizer must begin with the film theory. The basic film theoryquation is

ate = driving force

resistance(A.1)

his can be rewritten as

= UA�Tlm (A.2)

here Q is the heat transfer rate, U the overall heat transferoefficient, A the heat transfer area and �Tlm is the log meanemperature difference calculated by Eq. (A.3).

Tlm = (T − T0) − (T − Ti)

ln[(T − T0)/(T − Ti)

] (A.3)

The heat transfer rate can be simultaneously determined fromq. (A.4) incorporating the jacket flow, Fj and the inlet and outlet

acket temperatures, Ti and Tj.

= FjCp(T0 − Ti) (A.4)

he overall heat transfer coefficient, U, is found from equation

1 1 x 1

U=

hi+

k+

hj(A.5)

here x is the thickness of the vessel wall and k is the thermalonductivity of the wall.

stt

ification Technology 53 (2007) 153–163

The internal (hi) and jacket (hj) heat transfer coefficients areetermined from the dimensionless Nusselt number, NNu,

Nu = hDT

k(A.6)

The values hi and hj are got by rearranging Eq. (A.6) andetermining the value of the Nusselt number by Eqs. (A.7) andA.8) for hi and hj, respectively.

Nu,i = 0.85(NRe)0.66(NPr)0.33

(Z

DT

)−0.56(D

DT

)0.13

×(

μ

μW

)0.14

(A.7)

Nu,j = 0.027(NRe)0.8(NPr)0.33

(μ

μW

)0.14 (1 + 3.5De

Dc

)(A.8)

Eq. (A.7) assumes that the crystallizer is baffled and agitatedy a flat-blade turbine in the turbulent region. The Reynoldsumber for Eq (A.7) is found from

Re = D2Nρ

μ(A.9)

here D is the impeller diameter and N is the rotational speed ofhe agitator. Eq (A.8) assumes that the jacket has no baffles withurbulent flow. This correlation uses an equivalent heat transferiameter, De, found from the following equation

e = D2jo − D2

ji

Dji(A.10)

here Djo and Dji are the outside and inside diameters of theacket, respectively. And the cross-sectional area, Acs, equals

cs = πDe

2(A.11)

rom (A.11), the velocity of flow in the jacket can be found from

= Fj

Acs(A.12)

llowing the determination of Reynolds number for jacket flowy equation

Re = DVρ

μ(A.13)

The Prandtl numbers for Eqs. (A.7) and (A.8) are found fromquation

Pr = Cpμ

k(A.14)

Two assumptions are made in this discussion. Firstly, thepecific heats of the fluids are constant and secondly the heatransfer coefficients, hi and hj, are constant throughout the heatransfer area.

and P

A

P

N

G

M

E

I

R

[

[

[

[

[

[

[

[

[

[

[

[21] Y. Qiu, A.C. Rasmuson, Crystal growth rate parameters from isother-mal desupersaturation experiments, Chem. Eng. Sci. 46 (1991) 1659–

A. Abbas, J.A. Romagnoli / Separation

ppendix B

Model summary for a batch cooled crystallization process

opulation balance equation∂f (L, t)

∂t= −G

∂f (L, t)

∂t+ B

ucleation kinetic rate B = kb(C − C*)b

rowth kinetic rate G = kg(C − C*)g

ass balancedC

dt= −3ρcαVFG

∫ ∞

0

fL2 dL

nergy balancedT

dt= −UA

ρCpV�Tlm

nitial conditions f(L,t) = 0, t = 0

C(t) = C0, t = 0

T(t) = T0, t = 0

eferences

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[

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