Modeling of Crystallization Processes Final

7/28/2019 Modeling of Crystallization Processes Final

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BTP

REPORT

K.DHARMA TEJA07CH3011

[MULTI-DIMENSIONAL POPULATION BALANCE

MODELING OF CRYSTALLIZATION]


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Acknowledgement

I would like to thank my supervisor Prof. Debasis Sarkar for his constant guidance,

unwavering support and an objective criticism of my work that helped in shaping

up my subsequent project report.


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1. INTRODUCTION:

Crystallization is the (natural or artificial) process of formation of solid crystals precipitating from

a solution, melt or more rarely deposited directly from a gas. Crystallization is the first and probably the

main unit operation dealing with solids in chemical plants. Crystallization is also a chemical solid-liquid

separation technique, in which mass transfer of a solute from the liquid solution to a pure solid

crystalline phase occurs.

Crystallization is widely used to produce various solid-form specialty chemicals and

pharmaceuticals. . Finished solid products of chemical industries often come in the form of crystals i.e., in

an acceptable and granular form. Examples are numerous-sugar, sodium chloride, sodium hydrogen

phosphate, citric acid, ammonium sulphate, sodium chromate and a host of organic and inorganic

compounds.

In the industrial practice, the control of the chemical purity is no more sufficient and the control of

both crystal habit and crystal size distribution (CSD) can be critically important. Most past studies were

performed using products in which simple and isotropic shape (spherical, cubic, octahedral, etc.) could be

described by a single characteristic dimension. Obviously, such approaches become questionable in thecase of the crystallization of organics products presenting anisotropic morphologies that sometimes vary

during the crystallization The real challenge in industrial-scale crystallization is to obtain crystals with

specific shape, size distribution and polymorphic form to thus enable efficient downstream processing in

filtration, milling and grinding, as well as to deliver drugs with specific physicochemical characteristics

including dissolution rate, solubility and stability, as well as bioavailability features such as formulation

effects and pharmacokinetics. For CSD, population balance (PB) has been widely studied to simulate its

evolution with time subject to changing operating conditions. In traditional PB, the size of a crystal isoften defined as the volume equivalent diameter of a sphere. This simplified definition of crystal size has

ignored the fact that crystallization of organic products often produces crystals with anisotropic

morphologies reflecting in turn their anisotropic intermolecular packing structure in the solid state

Mindful of the fact that the fundamental process of crystal growth is surface controlled, the use of mono

size defined as the volume equivalent diameter in PB calculations therefore offers only limited value. Atwo-dimensional (2-D) PB model to simulate the time variations of two characteristic size dimensions

the length and width of needle-like crystals is developed.

The main difficulties of the analysis of the crystallization lie in the population balance equation

which governs the size variations of crystals involved in the crystallization. It is a partial differential

equation that has to be solved in connection with the other classical balances like mass balances for

example. Since the crystallization kinetics are dependent on the characteristics of the crystals and on theconcentrations of the solutes, the system becomes very complex, with a lot of feedback relationships

between the different equations. The modeling of the crystallization process is done and the method of

classes is proposed and using this method the further simulation and optimization shall be carried out.


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2. MODELLING MULTIDIMENSIONAL CRYSTALLIZATION PROCESSES

Let us now consider a well-mixed crystallizer with respect to liquid and suspension. This

simplification does not limit the generality of the method since associations of such ideal crystallizers are

useful tools for the description of actual crystallizers just as ideal chemical reactors in series or in paralle

are classical models for real industrial reactors.

2.1. Bi-dimensional population balance equations

Crystals exhibit a rod-like habit which can rather satisfactorily be represented as bi-dimensional

parallelepiped with dimensions L and W. An internal elongation shape factor can also be defined as the

ratio of the length L to the width W. To take into account either two sizes or one of these two sizes and

the elongation shape factor, a bi-dimensional population balance has to be established and solved. The

volume of the suspension is VT, and may vary during the crystallization. The inlet flow has a rate F in, and

the outlet flow F. In the following, the population of crystals is described by the number population

density function (L; W; t). It is assumed that the CSD does not depend on spatial coordinates in the

well-mixed crystallizer. During the time-interval dt, the balance of crystals in the bi-dimensional size

domain delimited by L; L + dL and W; W+ dW can be written as follows:

(1)

Here primary and secondary nucleation, and growth mechanisms should be considered while inthe following, breakage and agglomeration were found to be of second order of importance. Henceforth,

these phenomena were not taken into account in the PBE. Eq. (1) was then simplified to yield the

followingexpression:

(2)

With

(3)

Eq. (3) expresses possible primary and secondary nucleation rates, and , as the generation of newcrystals of critical size L*, W* upon formation, so that, in the following, the smallest class considered in the

population balance is delimited by 0, L*, W*.


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2.2. Kinetic nucleation and growth equations

Given a sufficient driving force, that is supersaturation in the case of solutions or supercooling in

the case of a melt, a liquid-to-solid phase transformation commences with the initial formation of

clusters, ordered collections of the crystallizing species. These clusters, or nuclei, are precursors to the

crystals eventually formed. In order to grow into a macroscopically detectable crystal, these nuclei have

to reach a certain, critical size. The critical nucleus size is governed by the excess free energy G of the

nucleus. If no crystal or any other particles (e.g. solid impurities) is present in solution, homogeneousnucleation takes place when a relatively high level of super saturation is obtained The presence of foreign

surfaces such as solid impurities or items of equipment in the crystallizer allows heterogeneous

nucleation to occur with lower supersaturation values.

In theevent that particles crystallizing from the solution are absent,the formation of particles on

such surfaces is referredto as primary nucleation. In real life this situation is only rarely achieved. Under

normal circumstances there are always impurities in the solution or melt, be the other chemical species

such as by-products from synthesis, or particulate impurities such as dust or particles resulting from

abrasion from the equipment. Mechanical disturbances result from agitation of the solution or vibrationsfrom ancillary equipment. Surface roughness of the equipment also falls into this classification. The more

realistic situation is therefore the case of primary heterogeneous nucleation which occurs at a much

lower supersaturation and where impurities or rough vessel walls function as nuclei.

The most frequently observed nucleation mechanism is called secondary nucleation. Secondary

nucleation requires the presence of crystals of the material to be crystallized and occurs at much lower

supersaturation than primary nucleation. As a rule, secondary nuclei are formed by the removal of

structured assemblies from the surface of the crystals. There are different mechanisms which lead to

those secondary nuclei, these are:

initial breeding: nuclei result from simply placing crystals into a supersaturated solution or

supercooled meltvia the washing off of dust particles from the surface of the crystals

collision breeding: nuclei result from fragments of existing crystals which are broken off due to

mechanical impact on crystal faces due to crystal-crystal, crystal-wall, or crystal-stirrer (-pump)

collisions

fluid shear: nuclei result from clusters or outgrowths being forced from the solid-liquid boundary

layer due to shear forces resulting from liquid motion. A prerequisite for this behavior is that the

growing crystal already has a size larger than the critical nucleus.

Collision breeding is the most frequently observed and dominant secondary nucleationmechanism, at least in the majority of industrial, mass production, crystallization processes.

There are two aspects to secondary nucleation.

The positive aspect results from the fact that without secondary nucleation as a permanent

source of new crystal nuclei, a continuous crystallizer with continuous crystal withdrawal

would rapidly experience a lack of growing crystals.

The negative aspect is the fact that many more secondary nuclei are produced in an

uncontrolled process than are required. As a result a very fine crystalline product is


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produced, unless measures are taken to reduce the power input into the crystallizer. More

important than the magnitude of the power input itself is the means by which the power is

brought into the equipment. Power is required to homogenize the suspension

(temperature and concentration, dispersion and circulation of solids) and to transport the

suspension. The key power input sources are pumps and impellers, and these are where

most secondary nuclei are produced. Secondary nucleation rates can be controlled via

diameter and tip speed of the impeller blade. Lower tip speeds and larger diameters result

in lower secondary nucleation rates.

Nucleation is also a problem concerning the start-up of a crystallizer. Primary nucleation is

difficult to control and unreliable, as it will not always occur at precisely the same supersaturation:

primary heterogeneous nucleation depends on the number and the nature of the impurities and is

therefore, within certain limits, a random event. As a consequence reproducibility of product quality

cannot be guaranteed and the performance of the crystallizer will vary. Assuming that nuclei start to

grow at high supersaturation, liquid inclusions or dendritic growth (vide infra) are likely, as well as

massive formation of small particles that tend to agglomerate. All of these phenomena lead to poor

product purity and quality. In addition, this may lead to strong tendency for caking in storage. Ifnucleation commences at a too low a supersaturation, the ensuing crystal growth may be slow and will

pose a problem with respect to production time or crystallizer size, respectively.

In order to produce high quality crystals in a reproducibly manner, secondary nucleation by

means of seeding is the preferred method of inducing the crystallization process. Here it is important

always to introduce the seed crystals at the same supersaturation. Keeping the driving force (the

supersaturation) constant subsequent to seeding normally results in high quality crystals, providing the

supersaturation selected coincides with the optimum growth rate for the system under consideration. In

order to maintain constant growth rates requires good control of the supersaturation and has to take into

account the constantly increasing crystal surface area, which is ultimately responsible for the reduction

in supersaturation.

If crystals are already dispersed in the crystallizing medium, secondary nucleation can occur at

supersaturation levels that are significantlylower than those at which primary nucleation takes place.In

this case, many various elementary processes are likely to be involved, including micro-breakage and

abrasion, contactswith existing particles and/or crystallizer surfacesall of which might result in the

production of secondarynuclei.For the system of concern, primary nucleation is assumedto initiate the

crystallization process according to the followingkinetic expression:

(4)

where A, B are coefficients that can be determined experimentally and have complex physical meaning

and is the degree of supersaturation C/C*.

In addition to an obvious dependence on the level of supersaturation, secondary nucleation is

generally considered to be promoted by the suspension density:


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(5)

where the kinetic parameter kN is generally assumed to be related to the stirring power and to exhibit a

temperature-dependency according to Arrheniuss law. Exponent n which does not depend on

temperature, lies between 0.5 and 2.5. Exponent k is generally assumed to be of the order of 1, and to

slightly depend on temperature. The growth of crystals in the supersaturated solution is a complex

process where both surface integration and diffusive mechanisms should be considered. In almost al

published studies, the relationship between the growth rates and the relative supersaturation is assumed

to be described through simple power laws.

Such approximation is far from being realistic:

It is well-known that several solid integration mechanisms can be observed during the crysta

growth which cannot easily be reduced to basic power laws.

As usual for mass transfer phenomena, diffusive limitations are likely to happen if the rate of solid

integration is very high. The overall growth rate can therefore be more or less limited by diffusion

during the crystallization process. In such conditions, the same growth model will neither be valid

during the whole crystallization process, nor for the various crystal surfaces considered in the

PBEs.

According to two different solid faces can be considered in the modeling, which have their own

growth rates GL and GW.

(6

The well-known film model can be used to relate the growth as a two-step mechanism where the

molecules of solute are transported by diffusion and or convection, and then incorporated onto the solidsurface through several possible mechanisms.

1. Diffusive/convective mass transfer through a film around the surface of the growing crystal is

assumed to be driven by the difference between the bulk concentration C and the concentration at

the crystal-solution interface, denoted below Ccs(t) and Ccs(t), respectively, for both faces:

( ) ( )(7)

where the mass transfer coefficient, kd, is assumed to be the same for both growth rates GL(t)

GW(t) as it mainly depends upon the hydrodynamic conditions around the crystal surface. kd canbe assessed using various published experimental correlations such as the dimensionless

relationship developed by Levins andGlastonbury (1972):

[]

(8)


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where Sh is the Sherwoodnumber(Sh=kdLeq/Diffuivity),Sc is the Schmidt number (Sc=/diffusivity)

Re is a solution/particle Reynolds number (Re = and Leq is the equivalentdiameter of the

sphere having the same crystal area

2. The incorporation of solid is represented with the assumption that the solution is saturated at the

crystal surface. The driving force of such incorporation is now given by the difference [Ccs(t)

C*(T(t))]; [Ccs(t) C*(T(t))], respectively.

As mentioned above, various growth mechanisms are available in the literature according to different

structures of crystalsolution interface. However, for practical purposes, empirical formulations such as

the following are more tractable:

( ) ( ) (9)

where are the flux density of the integration of solidon both crystal faces, kint; kint arethe corresponding kineticconstant and jintand jintare the order of integration, generallyfound between 1

and 2, depending on the mechanismin question.

The linear growth rates are directly related to the flux density of integration by

(10)

As for chemically reacting systems, it can be seen from the previous equations that the rate of

particle growth is a function of both diffusion and solid incorporation rates. In the context of mono-

dimensional modeling, an effectiveness factor was introduced by Garside (1971) to measure the rea

mass flux density _real, with respect to the maximum massflux density _max that would be obtained in

the absence of diffusive limitations. For the bi-dimensional case, we definetwo effectiveness factors,

and ' respectively :

intint *

max int

'( )

real s

jj

k C C

and

int'*

max int

' '

' ' ( ' )

real s

jk C C

(11)

and ' decrease if the crystal growth is strongly diffusion-controlled, while and ' = 1 if the growth

rate only depends on the kinetics of solids incorporation. One can easily show that the growth rates are

finally given by the following expressions which assimilate the real particles to bi-dimensiona

parallelepipeds:


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int

int

*

int

'*

int

( ) 2 .

'( ) 2 '. '

js

s

js

s

MG t k C t C T t

MG t k C t C T t

(12)

where the two effectiveness factors are solutions of the following equations

int int

intint

1 1/*int

' 11/ '*int

1 0

'' ' ' 1 0

j j

d

jj

d

k C t C T t k

kC t C T t

k

(13)

If the kinetic parameters are known, it follows that the effectiveness factors are given by Eq. (13) for

which numerical solutions may be obtained regardless of the value of the exponents jint and jint

Moreover, if the exponents are equal to 1 or 2, analytical solutions are available. It is worth noting that

the ratio G/Gmay depend:

on the level of supersaturation (if jint_= jint),

both sizes L and W(through kd, and ' ),

any parameter efficient in modifying selectively the individualkinetic constants kint and kint (e.g

the concentrationof impurities in the solution).

2.3. Mass balance equations

As an example, Eq. (14) describes the overall mass balance during an isothermal semi-continuous

crystallization operation fed with clear solution:

T Sinin t

V t C t V t C t

f C t dt

(14)

where the total volume of solution can easily be computed if one neglects the variation of volume related

to the generation of solid in the well-mixed crystallizer:

0

0t

inV t V f t dt

(15)

For a given time the solid concentration CS can be computed from the bi-dimensional CSD, as follows:

2

, ,

S

SS L W

C t LW L W t dLdW M

(16)

and the total volume of suspension is given by

1 ST SS

MV t V t C t

(17)

Finally, the solute concentration C(t) is computed through the integration of the balance equations (15)

to (17).


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3. THE NUMERICAL SOLUTION METHOD (METHOD OF CLASSES)

Only two size characteristic parameters, the length (L) and the width (W), are needed here in the M-PB

model formulation. Other assumptions made are that the agglomeration and breakage were negligible

only crystal growth is modeled, and both primary and secondary nucleation was not considered. The M-

PB modeling was performed for a time window with the starting time being x seconds since the start of

cooling, corresponding to a point where the mean values for the length and width of crystals were L and

W , respectively Under these assumptions, the 2-D PB can be written as follows:

1, , , , , , , , 0L W

T

L W t V t G L t L W t G W t L W tV t t L W

(18)

Fig.1 Simplified as a rectangular parallelepiped of length (L) and width (W).

The first term on the left-hand side is the population accumulation term. The second and third terms are

the population changes in both length and width directions, respectively.

Direct analytical solutions of PB equations can be obtained for a few simple cases, and numerical

techniques are regarded as the only available and realistic choice for most PB equations. The numerical

techniques studied in the literature typically involve three different approaches: method of moments

discretization techniques and finite element methods [1, 2735]. In this study, the method of classes,

which is one of the discretization techniques that integrate the PB models over small intervals of the

particle size domain, was used. This approach transforms the PB equations into a set of ordinary

differential equations (ODEs) via discretizing the particle size domains and each domain will form one

ODE. The formulations of these ODEs can be obtained through a population density, , or a population

number, N, function. The multi-dimensional ODEs, together with the mass balance equation of the

solution, are solved numerically by an iteration technique at each time step with the required growth rate

data derived from real-time in-process imaging experiments.


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Fig 2.Two-dimensional mesh distribution (n1 n2 classes) and net flows of crystals in class

Clj1,j2.

For the two parameters (L,W) in (1), the 2-D size domain (Fig. 2) can be discretized into n1, n2

classes, respectively, starting from the smallest class, L0,W0, and finishing at Ln1,Wn2. Taking into

consideration of the crystal length, L, n1 classes, CLj1 (j1 = 1, n1), can be formed with the extent of the

class CLj1 is _CLj1 = Lj1Lj11 and the corresponding characteristic length of the class CLj1 is Lj1 =

(Lj11 + Lj1)/2. Similar definitions can be obtained for crystal width, i.e. CWj2 (j2 = 1, n2),_CWj2,Wj2

Therefore, a system ofn1 n2 .2-D classes is formed with the class Clj1,j2 being delimited by (Lj1,Lj11)

(Wj2,Wj21). By integrating (1) over class Clj1,j2 ofL and W, the PB equation forms a set ofn1n2 ODEs

as [1]:

1, 2 1, 2 1, 2

10j j T j j j j

T

dN t V t FL t FW t

V t dt

(19)

where Nj1,j2(t) is the number of crystals in the class Clj1,j2:

1 1

1 1

1, 2

1 1

, ,

j j

j j

L W

j j

L W

N t L W t dLdW

(20)The net flow, FLj1,j2, of crystals in class Clj1,j2 in the length direction can be approximated using

first-order Taylor series expansion as follows:

1 1 11, 2 1 1, 2 1 1 1, 2 1 1 1 1, 2 1 1 1, 2, ,j jj j L j j j j j j L j j j j j jFL t G L t a N t b N t G L t a N t b N t

where


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1 1

1

1 1 1 1

j

j

j j j

CLa

CL CL CL

1 1

1

1 1 1 1 1

j

j

j j j

CLb

CL CL CL

(21)

A similar formulation can be obtained for the net crystal flow, FWj1,j2, in class Clj1,j2. The

corresponding boundary conditions for crystal flow fluxes in both length and width directions can be

written as:

1, 2 1,1 0j jFL t FW t

1, 2 1, 2 0o o

n j j nFL t FW t

(22)

where the superscripts I and O denote the crystals flows in letting into and out letting from the Clj1,j2 class

in each direction.

The first term in (ii) can be rewritten as:

1, 2 1, 2 1, 21, 2

11 j j j j T j j T j j T T

T T

N t N t V t N t V td dN t V t V t

V t dt V t dt t

(23)

For a well-mixed batch crystallizer, the solid concentration, CS(t), can be calculated by:

2

. . , ,S

S

S L WC t L W L W t dLdW M

(24)

whereas, with negligible effect of crystallization and temperature variation on the total volume, the

volume of suspension, VT(t), can be calculated by V (0)[1 Ms/sCs(t)], and the solute concentration, C(t)

can be estimated by:

0 / 1 /S S S S C t C C t M C t

(25)

Discretization sizing techniques appear to be robust and seem rather easy to extend in a second

dimension. This is why such method was selected in the present work.However two main drawbacks of the method should be outlined.

Firstly, the feasibility of the computation is likely to strongly depend on the complexity and

location of individual birth and death processes. Eg. for the crystallization of hydroquinone, only

birth through nucleation at low sizes has to be considered.

Secondly, the number of discretized equations increases as the number of classes to the square,

leading to high computational times. Implementing an adaptive bi-dimensional algorithm may

probably improve the situation which should be done further for the optimization.


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4. SIMULATION OF POPULATION SIZE DISTRIBUTION

In order to implement the numerical solution method, an initial population distribution of-form LGA

crystals was assumed, and the crystal growth rates in length and width directions were assumed. The PB

equation was solved in the time window between 915 and 2415 s. The crystal size ranges in length and

width directions were 0 250 and 0 100 m, respectively. The 2-D domain of length and width wasdiscretized into 2000 500 equal cells with each cell size being 0 .125 0.2 m, which generated 1

million discretized PB equations in ODE form. The ODEs were solved simultaneously using the ode[45]

solver of MATLAB. The 2-D PB distributions were obtained via solving the ODEs with a Dell laptop(core2duo CPU,@2.00 2.00 GHz processors and 3.0 Gb Ram) running Microsoft Windows seven operating

system.

Saturated solution (at 79C saturation temperature) was prepared with 31.5 g of LGA in 500 ml of

fresh distilled water and cooled down from 90 to 35 0C at constant agitation of 300 r.p.m. with a constantcooling rate of 0.1C/min. This proof of concept study will be restricted to a time window between 915 s

since the start of cooling and 2415 s, which correspond to reactor solution temperatures of 69.2 and

66.70C. In the current study, based on the obtained growth distributions of crystal length and width, the

growth rates of -form LGA in both length and width directions were estimated as a function of

supersaturation, and are shown in Fig. 5. The obtained growth rate equations as functions of

supersaturation are:

7 23.44 10 (( 0.49) / 0.02) / [1 (( 0.49) / 0.02)]L exp expG

2 6( 0.51 2.15 2.22 ) (10 )WG

(26)

where*

1tC

C

is the relative supersaturation

Figure 3. Crystal growth rates of-form LGA in length and width directions. Length growth rate

was calculated using (16); width growth rate was calculated using (17).

To perform M-PB modeling of-form LGA in the length and width directions over a time window

from 915 to 2415 s, one needs the initial condition of the population size distributions at 915 s, as well as

the growth rate equations derived above. From Fig. 6, the population size distributions for all crystals in

the whole reactor at 915 s can be deduced, as described below. The two Gaussian like functions as shown

in Fig. 6 could be extended to all the crystals in the reactor. The difficulty here is to know the total

number of crystals in the reactor. Fortunately, there is a constraint that the solid-phase volume


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concentration is known. Given the solid-phase volume concentration, and the formats of the size

distributions for the length and width in Fig. 6, there should be a unique solution for the population size

distributions for all crystals in the whole reactor.

Figure 4. Population size distributions of-form LGA crystals in length and width directions with Gaussian

like curve fitting (solid lines) at 915th s.

The 2-D population at 915 s for the whole reactor will have the following form:

2 2

0.5 / /m L m W L L W W

Ae

(26)

A can be calculated by using the equations 24, 25.

2 2

0

0.5 / /

1m L m W

t

L L W Ws

t

s L W

C CA

MC e dLdW

(27


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Using (27), the 2-D Gaussian population size distributions for the whole reactor at 915 s,1515 s is shown

in Fig. 5(a),5(b).

(a)

(b)

Figure 5. (a) Gaussian distribution of the initial 2-D population of-form LGA crystals at 915 s built up based

on the corresponding two 1-D population distributions (see Fig. 4) in length and width directions. (b) Population

distributions at 1515 s obtained after simulation.

Based on the population size distributions of the length and width at 915 s that was estimated, and

the growth rate equations, M-PB modeling was performed for the time window from 915 s to 2415 s

Figure 6 shows the simulated evolution with time for the length and width of-form LGA for the studied

time window. Also plotted in Fig. 6 are the length and width averages calculated from image analysis. It

shows that the crystals grow faster in the length direction and the predictions are in good agreement

with those of the experimental values. The small discrepancies may be caused by the coarse mesh size

and/or time step used.


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Figure 6. Length and width average sizes of-form LGA crystals.

The predicted population size distributions of-form LGA in the length and width directions at one

typical crystallization time of 1515 s were illustrated in Fig. 7(b). The population size distributions from

measurements in the length and width directions were plotted at the corresponding positions with themean values of width and length at the corresponding time, respectively. The population size

distributions from both simulation and measurement show good agreement. Figure 9 shows the

predicted population size distributions in the length and width directions at 1515, 1915, 2315 s. The

population distributions from measurements were very close to the Gaussian-type distribution at 1515 s

At 1915 and 2315 s, the population distributions in the width direction were still very close to a

Gaussian-type distribution.

a) Length and Width at time 1515 sec


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b) Length and Width at time 1915 sec

c) Length and Width at time 2315 sec


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Figure 7. Population size distributions of simulated results at times of (a) 1515, (b) 1915 and (c) 2315 s

5. CONCLUSIONS

The method of classes was presented and developed to simulate the time variations of

multidimensional crystal size distributions. Using the method of classes, the accurate simulation of thevery beginning of the generation of solid (i.e. at very short size and time scales) is bypassed as nuclei are

simply considered as a number of crystals belonging to the smaller size class of the size grid. The main

drawback of the technique is the requirement for small spacing in each size direction, leading to a very

large number of classes, which may result in high computational time.

PB modeling provides a useful tool for rapid simulation study of the evolution behavior of crystal

sizes in a reactor under varied operating conditions; however, the traditional simplified treatment o

crystal size as defined as the volume equivalent diameter misses important information associated with

crystal shape, especially for high aspect ratio crystals. M-PB modeling by incorporating the crystal shape

into PB offers a promising solution to address this issue, but a key element of success is the estimation of

the growth kinetics of individual face directions.

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Modeling of Crystallization Processes Final