Sequential Crystallization Parameter Estimation Method for

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A sequential crystallization parameter estimation method for determination of nucleation, growth, breakage and agglomeration kinetics Atul H Bari, and Aniruddha Bhalchandra Pandit Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b03995 • Publication Date (Web): 04 Jan 2018 Downloaded from http://pubs.acs.org on January 4, 2018

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Industrial & Engineering Chemistry Research

1 2

A sequential crystallization parameter estimation method for determination of nucleation, growth, breakage and agglomeration kinetics.

3 4 5 6 7 8

Atul H. Bari and Aniruddha B. Pandit*

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Department of Chemical Engineering, Institute of Chemical Technology, Mumbai-40019, India

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* Corresponding Author, Email: [email protected] Phone: +91 22 33611111 Fax: +91 22 33611020

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1 ACS Paragon Plus Environment

Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Abstract:

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In this paper a method to estimate crystallization kinetics considering simultaneous

3

nucleation, growth, breakage and agglomeration is discussed. The crystallization process is

4

modelled mathematically using population balance equation (PBE). Fixed pivot technique of

5

discretization is used for solving PBE and gPROMS parameter estimation tool is used to

6

estimate various kinetic parameters. Instead of estimating all the kinetic parameters in one

7

step, the present method focuses on estimating kinetic parameters in three successive steps. In

8

first step, primary nucleation and growth parameters can be estimated using simple induction

9

time experiments. In second step, secondary nucleation parameters can be found out by

10

gPROMS parameter estimation tool by optimizing supersaturation profile. In third and final

11

step, the breakage and agglomeration parameters can be found out by gPROMS parameter

12

estimation tool by optimizing crystal size distribution. The developed method was applied to

13

estimate the crystallization kinetics of potassium sulphate-water system.

14

Keywords: Crystallization, gPROMS, Nucleation, Growth, Breakage, Agglomeration.

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1

1. Introduction:

2

Crystallization is one of the most widely used unit operations in chemical industries.

3

It finds extensive applications in separation of pharmaceuticals, photo materials and fine

4

chemicals [1]. Crystal purity and crystal size distribution (CSD) are the properties particularly

5

targeted during designing and operation of crystallization process. CSD from crystallizer

6

have critical influence on the downstream processes like filtration, drying, packaging and

7

transport. Hence, it is important to have control over the CSD during the crystallization

8

process. However, the CSD is a result of several phenomena viz. nucleation, growth,

9

breakage and agglomeration, occurring simultaneously during the crystallization. Because of

10

which the overall process becomes very much complex and makes it difficult to monitor CSD

11

during crystallization.

12

Evolution of CSD can be described mathematically by using population balance

13

equations (PBE). Complete description of crystallization process can be accomplished by

14

coupling mass and energy balance equations with PBE. However PBE’s have analytical

15

solutions in rare cases, hence needs to be solved using numerical methods. Various numerical

16

methods have been proposed in literature for solving PBE’s which were effectively used for

17

some theoretical as well as practical processes. These methods include method of

18

discretization, method of moments, finite volume method and Monte-Carlo simulation

19

method

20

parameters and some initial condition. However for the kinetic analysis our aim is to estimate

21

the kinetic parameters from the available CSD data.

[2]

.

Generally, aim of solving PBE is to obtain the CSD with known kinetic

22

In the past decades several kinetic studies have been reported focusing on

23

crystallization parameter estimation. The parameters associated with the different phenomena

24

can be estimated either simultaneously or sequentially. In simultaneous estimation, either all

25

of the phenomena occurring during crystallization (or some of the selective phenomena)

26

considered alongside. Then the parameters are estimated by fitting all the available

27

experimental data in a single step [3-6]. This approach is convenient if the kinetic parameters to

28

be estimated are fewer. However as the number of parameters to be estimated increases,

29

optimization becomes difficult with increasing the computation efforts. Also, this approach

30

might just give the numerical values of the kinetic parameters with no physical significance.

31

Nevertheless, sequential parameter estimation methods were also applied to crystallization

32

process. A thoughtful combination of experimental strategy, analytical methods and 3 ACS Paragon Plus Environment


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mathematical operations allows determination of the rates of individual processes, even if

2

those phenomena occur simultaneously. Sequential parameter estimation approach has been

3

successfully applied to estimate nucleation and growth kinetics[7-9], nucleation, growth, and

4

agglomeration kinetics[10-13], growth and breakage kinetics

[14]

, growth and agglomeration

[15]

,or nucleation, growth, agglomeration and breakage kinetics[16-17]. Any reliable

5

kinetics

6

sequential parameter estimation method for crystallization comprise of three main aspects: 1)

7

Experimental protocol i.e deciding process operation domain to neglect/consider particular

8

phenomena 2) Numerical method to solve PBE and 3) Optimization algorithm. Based on the

9

different combination of above three aspects, different parameter estimation approaches can

10

be formulated. It is necessary to explore them so as to have a simple and robust approach for

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the parameter estimation of crystallization process. The objective of the current study is to

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develop such an approach for the estimation of kinetic parameters of crystallization process.

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This paper is organized as follows: The mathematical modelling of batch

14

crystallization (PBE coupled with mass balance) is discussed in next section. In subsequent

15

sections, the discussion of method of solution of PBE is given which is followed by the

16

discussion of gPROMS parameter estimation. Subsequently the new sequential parameter

17

estimation approach is discussed, followed by its application to K2SO4 crystallization.

18 19

2. Mathematical Modelling of crystallization:

20

2.1 Batch crystallization PBE:

21

The PBE of a batch crystallization process involving simultaneous nucleation, growth,

22

breakage and agglomeration, is given as [18].

23

24

() () () () + = + + 1

25

Where G is the size independent growth rate and Bnucleation is the crystal nucleation rate.

26

The form of PBE which we have considered is one dimensional. For nucleation and growth,

27

the one dimensional form of PBE considers the size of crystal in length. While on the other

28

hand, the mass conservative form of PBE of agglomeration and breakage considers size in

29

volume. To have dimensional consistency of size, PBE of agglomeration and breakage have

30

been modified by assuming the shape of crystal to be cubical. Thus, if length of each side is

31

considered as ‘L’, the volume of crystal can be replaced by L3 in the conventional PBE of 4 ACS Paragon Plus Environment

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1

breakage and agglomeration. The each term involved in equation 1 is discussed in detail in

2

following section.

3 4

Nucleation: The first event to occur during crystallization is nucleation and it indicates the

5

formation of tiny solid particles (nuclei) from the solution. Supersaturation is the driving

6

force for the nucleation. There are two types of nucleation 1) Primary and 2) Secondary.

7

Primary nucleation: It is the 'classical' form of nucleation which occurs mainly at high level

8

of supersaturation. This mode of nucleation is subdivided into homogeneous and

9

heterogeneous nucleation.

10

Homogeneous nucleation: The process of homogeneous nucleation occurs in pure solution

11

without aid of foreign particles or external surfaces.

12

Heterogeneous nucleation: Heterogeneous nucleation is induced by foreign particles or

13

external surfaces present in the supersaturated solutions.

14

Primary nucleation rate is given as[1] :

15

= ∆ 2

16

18

Where; is primary nucleation constant,

19

Secondary nucleation: Secondary nucleation describes the nucleation that takes place with the

20

help of preexisting solute crystals. It can occur at a low level of supersaturation, which is also

21

the prevailing process in the most industrial crystallizations. Secondary nucleation often

17

22 23

24 25 26

supersaturation.

!

is primary nucleation order and ∆ is

depends on the suspension density or magma density " .Secondary nucleation is given by [1] # = $ ∆ $ " 3

Where, $ is the secondary nucleation constant and The magma density is given by:

&

is the secondary nucleation order.

28

" = ' ( )0 3 (, )+ 4

29

Where ' the density of solute is, ( is volume shape factor of crystals and L is size of crystal.

The effective nucleation rate is given by the summation of primary nucleation and secondary

30

nucleation rate. Thus total nucleation rate is given as [1]:

27

∞

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1

= + # 5

2

Growth: During crystallization, due to the addition of solute on existing crystals, size of

3

crystals increases. This increase in size is referred as crystal growth. The crystal growth is a

4

function of supersaturation and is a mass transfer controlled process. Growth rate can be a

5

function of crystal size or it can be a size independent function. Here for the ease of solving

6

PBE, we have used size independent form of growth rate and is expressed as [1]:

7

8

= ∆ 6

9

Breakage: During crystallization, crystal breakage may occur for variety of reasons viz.

10

intense turbulence, collision of crystals with each other (intra-particle collision), collision of

11

crystal with wall of container or collision of crystals with impeller etc. This may lead to

12

formation of two or more small crystals. Mathematically, breakage process can be explained

13

by PBE using two terms: 1) Birth term 2) Death term. For any size ‘L3’, birth term denotes

14

the creation of crystal of size ‘L3’ because of breakage of crystals of all possible sizes greater

15

than ‘L3’. Whereas death term for size ‘L3’ represents the disappearance of crystals after the

16

breakage of crystal of size ‘L3’. These two terms can be expressed as [2].

17 18 19 20 21 22

23 24

1(23 )

1

= )2 4( 5 , 6 5 ) × 9:6 5 ; × :6 ;+(6 5 ) − 9: 5 ; × ( ) 7 7 3

Here, : 5 , 6 5 ; is the breakage distribution function and 9: 5 ; is the specific breakage rate which is given by power law model:

9(5 ) =

5 =

( ) 8

The breakage distribution function, 4: 5 , 6 5 ; gives the number of daughter crystals of size

26

5 produced when a single crystal of size 6 5 breaks. Various breakage distribution

27

distribution function which assumes the formation of two particles after breakage of single

28

particle is widely used. It also suggests that the probability of formation of particle of all

25

29

functions have been suggested in the literature. Among those, uniform binary breakage

possible smaller size is same. Thus, if the particle of size 6 5 breaks down, the probability of


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1 2 3 4

formation of particle of size 5 is same for all 5 < 6 5 . The uniform binary breakage distribution function is given as[2]:

4: 5 , 6 5 ; = 2 A 9 & @

5 6

Agglomeration: During crystallization, crystals might collide with each other due to

7

turbulence. Simultaneously, growth of crystals is also taking place. If the micro-turbulence is

8

insufficient to detach the agglomerated crystal in a short period and these crystals grow at the

9

point of contact, the agglomerate will behave as a single crystal. Thus the agglomerated

10

crystals won’t separate as the solute crystallized out at the point of contact, will provide

11

sufficient strength to the agglomerate. In PBE, agglomeration adds one birth and one death

12

14

crystals of size 6 5 and B6 5 agglomerate. While death terms for size 5 represents the rate

15

other sizes. These two terms are given as[2]:

13

term. Birth term for size 5 represents the rate of formation of crystal of size 5 when two of loss of number of crystals of size 5 when crystal of size 5 agglomerate with crystals of

16

23

( ) 1 = C D(B6 5 , 6 5 ) × (B6 ) × (6 ) × +(6 5 ) 2 E

7

− ( ) C D( 5 , 6 5 ) × :6 ; × +(6 5 ) 10 E

18

Here D( , 6 ) is the agglomeration kernel for agglomeration of two crystals of size 5

19

have considered size independent agglomeration kernel. Thus we have:

17

20

5

5

and 6 5. Agglomeration kernel can be size dependent or size independent. In this study, we

21

D( 5 , 6 5 ) = DE 11

22

2.2 Mass balance equations: As mentioned earlier, a batch crystallization model can be

23

completed by coupling mass balance equation with PBE. The mass balance term appearing in

24

PBE’s of crystallization is supersaturation. Nucleation and growth rate are dependent on the

25

supersaturation and it is given by;

26



1 2 3 4 5

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∆ = − ∗ 12

Where ∗ is the equilibrium concentration corresponding to the operating temperature (T)

and C is the actual concentration inside crystallizer. The equilibrium supersaturation ∗ can be expressed as a function of temperature. For K2SO4 water system, ∗ is given as [3];

6

∗ = 6.29 × 10B& + 2.46 × 10B5 I − 7.14 × 10BK I & 13

7

On the other hand, experimental concentration of solute can be obtained from mass of solute

8

that is crystallized out. Its rate of change can be expressed by the following equation:

9

10 11 12

13 14 15

+ +" + = 0 14 + +

Here " is the magma density and its rate of change is given by:

+" = 3' ( L5 + ' ( : + # ;E 5 15 +

Where ' is the density of solute, ( is volume shape factor of crystals and L5 is the third moment of CSD. Typical ith moment of CSD is given by; 7

L = C (, )+ 16 E

16 17

3. Method of solution of PBE:

18

3.1 Discretization technique

19

As discussed earlier, PBE are partial integro differential equations having analytical

20

solution in rare cases. Moreover, PBE involving breakage and agglomeration is very complex

21

which often requires numerical method to solve. Since it is necessary to solve equations

22

involving parameters with utmost accuracy for any parameter estimation, it is necessary to

23

identify the effective method for solving PBE involving breakage and agglomeration. Method

24

of discretization is regarded as the simple and effective method for solving PBE of breakage

25

and agglomeration.


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1

In the method of discretization, the PBE in the form of partial differential equation are

2

converted into ordinary differential equation by discretizing size domain in different size

3

intervals. The number density function, which is a continues function is replaced by M , the

4 5 6 7

number of particles in any interval.

M = )2 3N (). + 17 2

3

8

The first obvious choice of discretization is linear discretization. However, the typical

9

crystal size range in crystallization would result is very high number of intervals (considering

10

the minimum size equal to that of nuclei), and would increase computational cost. Hence

11

numerous discretization techniques for the PBE were developed and applied to various

12

particulate systems [19-22]. Some methods, such as the fixed pivot method (FP)[23], moving

13

pivot technique[24] and the cell average (CA) technique[25], are specially developed for

14

processes with agglomeration and breakage. CA technique is a modification of FP method

15

with improved accuracy. The FP technique has emerged over the years as a widely used one

16

for its easy implementing ability, flexibility, and effectiveness to accommodate both for

17

breakup and aggregation of particles [26]. However, CA method (modified FP method) has

18

advantage of higher accuracy for agglomeration problem. Hence we have used the CA

19

technique to solve the PBE of agglomeration and breakage. The details of FP and CA

20

technique can be found out in Kumar et al., 2008.

21 22

Discrete form of PBE:

23

We have used the method discretization for solving PBE. The discrete form of equation 1 is

24

given as:

25 26

+M +M +M +M +M =O P +O P +O P +O P 18 + + + QR + +

Where S T3V TU

, S T3V TU

QR

, S T3 V TU

and S T3V TU

are the change in number of

27

crystal in ‘ith’interval due to nucleation, growth, breakage and agglomeration respectively.

28

These are given by discretizing equations 4, 5, 6 and 9 respectively. As per FP technique,

29

different terms of equation 17 are given as:

30



O

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+M [\ [ = 1 P = Z 19 0 [\ [ ≠ 1 +

1

+M! M! O P = 20 + QR & − ! 2

O

+M MB! M P = − 21 + QR − B! ^! −

3

4

The expression for S T3V

5

information.

6

The change in number of crystals in ‘ith’ interval i.e

7

calculated in two step and is given as:

8

9

TU

10

O

12

+ S T3 V TU

as per CA technique is given in supporting TU3 T

which is given by equation 17, is

+M +M +M =O P +O P 22 + + U_ + à O

11

+M +M +M P =O P +O P 23 + U_ + + QR

+M +M +M P =O P +O P 24 + à + +

Where S and S

TU3 T

TU3 T

V

V

à

U_

is change in the crystal number in ‘ith’ interval due to nucleation and growth

is change in crystal number in ‘ith’ interval due to breakage and agglomeration.

13

The two step approach was used since the dimension of sizes in PBE of nucleation and

14

growth is different than that of breakage and agglomeration. To solve these equations, we

15

used two discretization domain; one based on length and other based on volume. The only

16

thing we need to ensure is the superimposition of grids of length domain with those of

17

volume domain. This means, the ‘ith’ interval formed after discretizing length domain will be

18

same as the ‘ith’ interval formed after discretizing volume domain. This can be done by 10 ACS Paragon Plus Environment

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1

discretizing the volume coordinate by a grid ratio, equal to the third power of grid ratio used

2

for discretizing the length co-ordinate. This means if we discretized the length co-ordinate by

3

ratio of ‘x’, we need to discretize the volume co-ordinate by ratio of ‘x3’.

4 5 6

4. gPROMS parameter estimation: In recent years, rigorous parameter estimation methods, utilizing non-linear [27-28]

7

optimization techniques, have been employed in crystallization kinetics

. These

8

optimization techniques are now employed in commercial softwares which have special

9

dedicated solvers to estimate parameter. These software’s are quite user friendly and

10

bypasses the need of writing the optimization algorithm. One such software having its own

11

tool for parameter estimation is gPROMS, developed by Process System Enterprise (PSE),

12

UK. The parameter estimation tool in gPROMS can be effectively used for the simultaneous

13

estimation of multiple parameters.

14

gPROMS is an equation-oriented modelling software that provides a general purpose

15

modelling environment for process and equipment design, optimization of operating

16

processes, as well as parameter estimation. The parameter estimation tool of gPROMS has

17

been successfully used for the estimation of nucleation and growth parameters

18

addition, it has been applied for the estimation of kinetic parameters for breakage process [30].

19

Based on the experimental data inputs, gPROMS parameter estimation tool tries to obtain the

[29]

. In

21

values for the b’s, the parameters to be estimated. A solver named MXLKHD is available for

22

likelihood optimization. The objective function used for capturing the maximum likelihood

23

goal is given by [31]

20

parameter estimation and is explicitly designed for solving the problems of maximum

24 x

w3 u3,@

&

:r^6 − r6 ; " 1 & ∅ = ln(2j ) + k[l m n n n op:q6 ;+ ty 25 & 2 2 q6 v! 6v! v!

25

27

∅ is the objective function,

28

M is the total number of measurements taken during all the experiments,

26

29 30 31

b is set of parameters to be estimated,

z is the number of experiments performed,

{ is the number of variables measured in the ith experiment,

|,6 is the number of measurements of the jth variable in the ith experiment, 11 ACS Paragon Plus Environment


1 2

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& q6 is the variance of the kth measurement of variable j in experiment i,

r^ is the kth measured value of variable j in experiment i,

4

r is the predicted value of variable j in experiment i.

5

global optimum. It calculates the derivative of the objective function with respect to the

6

parameters to be estimated and then uses this information to determine its search direction.

7

gPROMS can be used to estimate the values of model parameters from the results of one or

8

more experiments. Both steady state and dynamic experiments can be used for this purpose.

9

The data can be taken from experiments with different initial conditions and/or operating

10

sequences; the sets of variables measured by each experiment, as well as the times at which

11

these measurements are taken.

3

MXLKHD solver uses a sequential quadratic programming method to obtain the

12

The PBE can be solved using FP technique or CA technique of discretization and

13

gPROMS parameters estimation tool can be employed for parameter estimation. The efficacy

14

of this approach is checked for some analytical cases. Although these cases may or may not

15

have any physical significance, they are capable of representing (mathematically) the type of

16

equations governed by respective processes. Thus, we would get an enough idea about any

17

solution technique and computational tool employed for parameterizing the PBE’s indicating

18

the physics of these processes. Six different cases having analytical solutions are considered

19

and are given in supporting information (Table S1-S6). From these cases, it can be concluded

20

that gPROMS parameter estimation tool can be successfully employed for the estimation of

21

parameters involved in complex PBE.

22

5. Sequential parameter estimation:

23

The various parameters associated with crystallization which needs to be estimated

24

are given in table 1. Although we have considered the simplest type of agglomeration and

25

breakage kernels and temperature independent nucleation and growth rates, the number of

26

parameters need to be estimated are 9. As mentioned earlier, simultaneous estimation will

27

increase the computational efforts and errors associated with optimization will also be higher.

28

Also, because of the complexity of crystallization and interference of one phenomena with

29

another, it is impossible to estimate kinetic parameters separately for each phenomena. Hence

30

a step by step parameter estimation procedure needs to be developed.

31 32

Table 1. Various parameters associated with different processes during crystallization. Process

Parameters


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Primary nucleation

Secondary nucleation

!

$ &

Growth

}

Breakage

Agglomeration 1

k

DE

2

The first event that occurs during any crystallization operation is nucleation. As

3

discussed earlier, two types of nucleation occur: Primary and secondary. However for

4

unseeded crystallization, initially only primary nucleation will occur and secondary

5

nucleation will take place only after a substantial amount of solute is crystallized out. Also

6

the breakage and agglomeration which largely depends on the suspension density are

7

negligible in initial stage. Thus the data of crystallization (unseeded) during initial phase can

8

be used for the estimation of primary nucleation. This approach has been already used for the

9

estimation of primary nucleation[32-34].

10

Metastable zone width (MSZW) and induction time experiments has been used in past

11

for estimation of primary nucleation kinetics. MSZW is determined by conventional

12

polythermal method i.e. by increasing supersaturation, usually at constant cooling rate. One

13

side of the MSZW is regarded as the temperature corresponding to equilibrium concentration

14

(typically obtained from the solubility curve), while the other side is considered as the

15

temperature at which the crystals are detected. This temperature can be referred as the MSZ

16

limit. The difference between equilibrium concentration temperature and MSZ limit is

17

referred as MSZW. On the other hand, induction period is determined by the isothermal

18

method by maintaining constant supersaturation. It is defined as the time required for the

19

detection of first nucleation event after the creation of supersaturation.

20

Recently Kubota [34] proposed a new method for the estimation of nucleation kinetics

21

from MSZW and induction time experiments. Kubota suggests that the nucleation is not an

22

instantaneous phenomenon. Nucleation gets started as soon as the supersaturation is achieved

23

and its get detected only when numbers of primary nuclei reaches some fixed value. Growth



Page 14 of 32

1

of crystals takes place as soon as nuclei get crystallized out. However, secondary nucleation,

2

breakage and agglomeration are neglected during induction time or within MSZW. This

3

method relies only on the number of crystals for the estimation of nucleation parameters and

4

not on concentration. Hence, for the method proposed by Kubota, consideration or exclusion

5

of growth of crystals doesn’t affect the estimated parameters. However, if we consider the

6

growth of crystals during induction time or within MSZW and get the experimental data of

7

crystal number, CSD and concentration; we can easily estimate growth kinetics in addition to

8

that of primary nucleation kinetics. In doing so, we can neglect secondary nucleation,

9

breakage and agglomeration since the crystal suspension density is very low during induction

10

time or within MSZW. Thus primary nucleation and growth kinetics can be found out from

11

MSZW and/or induction experiments.

12

During crystallization, the mass transfer processes occurring are nucleation and

13

growth. Agglomeration and breakage are mass conserving process and hence their presence

14

or absence will not affect the mass that have crystallized out. Thus concentration profile

15

during crystallization can be said to be unaffected by breakage and agglomeration and only

16

phenomena affecting concentration profile are nucleation and growth. Hence the

17

concentration profile or supersaturation profile can be optimized to estimate the kinetic

18

parameters of nucleation and growth. Once the primary nucleation and growth parameters are

19

estimated, the only parameters related to mass transfer phenomena remained unknown are the

20

parameters of secondary nucleation. Thus supersaturation profile can be used to estimate

21

secondary nucleation kinetics. For this gPROMS parameter estimation tool can be employed.

22

Evolution of CSD is a result of all the four phenomena occurring simultaneously.

23

After estimating parameters associated with nucleation and growth, only parameters left to

24

estimate are those of breakage and agglomeration. Thus CSD profile can be optimized to

25

obtain the parameters associated with breakage and agglomeration. Once again, gPROMS

26

parameter estimation can be employed for this. Thus, overall we can estimate all the

27

parameters associated with different process occurring during crystallization in three steps.

28

The overall method can be summarized as follows:

29 30

1. In first step, induction time and/or MSZW experiments can be carried out and primary nucleation and growth kinetics can be estimated.

31

2. In second step, gPROMS parameter estimation tool can be used to optimize

32

concentration profile by estimating secondary nucleation kinetics. While doing

33

this, the primary nucleation and growth parameters estimated in step 1 will be


Page 15 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1

used in PBE. In this step, we will neglect the breakage and agglomeration term in

2

PBE.

3

3. In third step, parameters associated with breakage and agglomeration can be

4

estimated. Here we will consider breakage and agglomeration term in PBE,

5

incorporating estimated nucleation and growth kinetics. This time gPROMS

6

parameter estimation tool will be used to optimize CSD by estimating breakage

7

and agglomeration kinetics. Overall strategy for estimation of crystallization

8

kinetics is shown in figure 1.

9


Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Page 16 of 32

Figure 1. Three step parameter estimation strategy for estimation of crystallization kinetics. 16

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1

6. Application of three step parameter estimation approach for K2SO4 crystallization:

2

6.1 Materials and experimental setup:

3

K2SO4 (99.9%) was purchased from S.D. Fine Chem., Ltd., Mumbai, India. A 250mL

4

jacketed glass vessel was used as a crystallizer. A temperature controlled cooling system with

5

accuracy of ±0.10C was used for controlling the temperature of the solution/suspension

6

present in the crystallizer. Silicon oil was circulated through the jacket of the glass vessel.

7

Universal PMDC RQG-126D motor (1/8 hp) with a four pitched blade turbine of size 30mm

8

was used for stirring. For intermediate removal of slurry, a sample withdrawal system was

9

provided. Schematic of the experimental setup is shown in figure 2. Concentration of K2SO4

10

was determined by using conductivity meter. CSD was obtained by image analysis of

11

crystals. Images of K2SO4 crystals were captured using Leica microscope provided with a

12

camera. The crystal images were processed and analyzed using ImageJ software to obtain

13

CSD.

14

15 16

Figure 2. Schematic of experimental setup.

17 18 17 ACS Paragon Plus Environment


Page 18 of 32

1

6.2 Method:

2

Induction time experiment:

3

The induction time experiments (constant supersaturation experiments) were carried out for

4

equilibrium concentration corresponding to 60 0C. In these experiments, the required amount

5

of water and K2SO4 corresponding to saturation solubility was taken into the crystallizer.

6

Then the stirring was employed and the temperature was raised slightly (2-3 0C) above the

7

saturation temperature. This was done so that whole solute gets dissolve. After a clear

8

solution was obtained, its temperature was brought down to the desired supersaturation

9

temperature. This cooling was done as fast as possible (within 2 min) to make sure that the

10

time required to achieve the desired supersaturation is negligible as compared to the

11

induction period. Once the temperature is achieved corresponding to the desired

12

supersaturation, it was kept constant within ± 0.10C to determine the induction time.

13

The onset of crystallization was detected visually and then the sample was taken out

14

immediately. The sample which was in the form of slurry was vacuum filtered instantly.

15

Filtration was carried out rapidly to arrest the growth of crystals, during filtration. Filtrate

16

was analyzed by conductivity meter to obtain the solute concentration. Mass of crystals that

17

have crystallized out was calculated from the concentration of solute. Mean crystal size was

18

obtained from the solid residue of the filtration which was analyzed for CSD using Leica

19

microscope. In order to eliminate the experimental errors induced due to visual detection and

20

other experimental protocols, each experiment was carried out in triplicate.

21

Constant cooling rate experiment:

22

Cooling crystallization experiments with constant cooling rate were carried out for different

23

cooling rates and with initial temperature of 60 0C. The initial K2SO4 concentration was kept

24

equivalent to its solubility at 600 C. Similar procedure as that of induction time experiments

25

was followed to ensure that all K2SO4 is dissolved. Once all the solute gets dissolved at 630C,

26

solution was cooled to 60 0C and was kept constant at this temperature for 2-3 min. Then the

27

solution was cooled at a constant rate till the crystallizer temperature reaches 200C.

28

Intermediate sampling was done by taking out a small amount of suspension from the

29

crystallizer. Similar to the induction period experiment, analysis were carried out for CSD

30

and concentration. Here also, each of the experiments was performed in triplicate.

31

6.3 Results and Discussion: 18 ACS Paragon Plus Environment

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1

6.3.1 Estimation of Primary nucleation and growth:

2

In the current approach, instead of using the polythermal method (MSZW

3

experiment), we will use the isothermal method (Induction period). Here, we assume that the

4

relaxation time, defined as the time required in bringing solution from saturation temperature

5

to a constant supersaturation temperature is negligible as compared to the induction time.

6

Since the temperature over the induction period experiment is kept constant and the mass of

7

the crystals precipitating out during the induction time is negligible, we can say that

8

supersaturation has remained nearly constant. Thus, both the nucleation and growth rates are

9

assumed constant (equations 4 and 5). Since we have assumed no secondary nucleation,

10

breakage and agglomeration, size distribution would be parallel to the length axis at number

12

equal to ∆ . The distribution would be from nuclei size (L0) to maximum size (Lmax). The

13

given as;

11

14

maximum size will be of those crystals which got crystallized out at t=0, and hence it will be

15

= ~ = E + × T 26

16

Since crystals are uniformly distributed between nuclei size (L0) to maximum size (Lmax),

17

mean size is given as

18

= = 19 20

= ~ + E 27 2

From equation 25 and 26 we will get

21

= ∆ = 22 23

Assuming shape of crystal to be cubical V=L3, total mass of crystals is given as

" = '9 C 24

2= − 2E 28 T

2

2

. 5 + 29

25

Where ρ is density of crystals in kg/m3, SV is volumetric shape factor. Putting the values of

26

nucleation rate and integrating, we get the total crystallized mass of the crystals as

27 19 ACS Paragon Plus Environment


" = 1 2

Page 20 of 32

' 9 (T + E ) − E 30 4

hence

3

= ∆ = 4

4" 31 (T + E ) − E ' 9

5

We can find out experimentally mean size and the total mass of crystals at the point of

6

detection. Thus, from these two values we can obtain Growth rate and the nucleation rate for

7

different ∆C using equation 27 and 30 respectively. Subsequently, different rate constants can

8

be obtained. Table 2 Experimental Induction time at various supersaturation.

9 Temp

Solubility

∆C

Indction

Mass

Mean

Growth

Nucleation

0

(kg/kg

(kg/kg

time

crystal

length

rate

rate

solvent)

solvent)

(sec)

(kg/kg

(m)

(m/sec)

(#/sec)

C

solvent)

53

0.173

1.16E-02

6537±43 8

6.25E2.86E03±4.21E 04±1.32E 8.71E-08 -4 -5

5.15E+00

3.78E1.96E03±2.76E 04±8.54E 1.07E-07 -4 -6

1.72E+01

52

0.172

1.33E-02

3652±31 2

51

0.170

1.50E-02

1918±12 1

7.91E1.14E04±4.45E 04±5.63E 1.17E-07 -5 -6

3.55E+01

50

0.168

1.67E-02

894±154

1.60E6.49E04±7.63E 05±2.54E 1.43E-07 -6 -6

8.30E+01

613±59

4.75E4.62E05±2.16E 05±9.73E 1.47E-07 -6 -7

1.01E+02

49

0.166

1.85E-02

10 11

The induction period, mass of crystals crystallized out at the induction period and mean size

12

of crystals at the end of the induction period for different supersaturation experiments is

13

given in table 2. From the experimentally obtained mean size of crystals and mass of crystals, 20 ACS Paragon Plus Environment

Page 21 of 32

1

by using equation 27 and 30, growth rate and nucleation rate were found out for each

2

supersaturation (Table 2). To obtain the growth kinetic parameters, a graph of logarithm of

3

growth rate was plotted against logarithm of supersaturation, figure 3. The slope of which

4

gives the growth order and intercept as logarithm of the growth rate constant. Similarly, a

5

graph of logarithm of nucleation rate was plotted against logarithm of supersaturation, figure

6

4. The slope of which gives the primary nucleation order and intercept as logarithm of

7

nucleation constant. The various values of kinetic constants thus obtained are given in table 3.

ln(∆C) -4.5

-4.4

-4.3

-4.2

-4.1

-4

-3.9

ln(Growth Rate)

-15.6

-15.8

-16 ln(G) = 1.15 ln(∆C) - 11.08 R² = 0.97

-16.2

-16.4

8

Figure 3. Estimation of growth rate parameters.

9 10

6 5 ln(Nucleation rate)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ln(B) = 6.50 ln(∆C) + 30.81 R² = 0.97

4 3 2 1 0 -4.5

-4.4

-4.3

-4.2 ln(∆C)

-4.1

-4

-3.9

11 12 13

Figure 4. Estimation of primary nucleation parameters. 6.3.2 Estimation of secondary nucleation kinetics: 21 ACS Paragon Plus Environment


1

After estimating primary nucleation and growth kinetics, parameters affecting

2

supersaturation profile remained unknown are those of secondary nucleation. To check

3

whether secondary nucleation kinetics can be estimated from the supersaturation profile, PBE

4

was solved for supersaturation with estimated primary nucleation and growth parameters. In

5

these simulations secondary nucleation, breakage and agglomeration were neglected. The

6

simulated supersaturation profile was compared with that of experimental and is shown in

7

figure 5.

Experimental Simulated with no secondary nucleation Supersaturation (kg solute/kg solvent)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 32

0.05 0.04 0.03 0.02 0.01 0 0

200

400

600 Time (sec)

800

1000

1200

8 9 10

Figure 5 (a)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Supersaturation (kg solute/kg solvent)

Page 23 of 32

0.025

Experimental

0.02

Simulated with no secondary nucleation

0.015

0.01

0.005

0 0 1

2000

4000 6000 Time (sec)

8000

10000

2

Figure 5 (b)

3 4 5

Figure 5. Simulated supersaturation profile without secondary nucleation and

6

Experimental supersaturation profile. (a) cooling of 20C/min (b) cooling of 0.250C/min

7 8

The simulated supersaturation profile for higher cooling rate matches well with that of

9

experimental (figure 5a). This means secondary nucleation does not have a significant effect

10

on supersaturation profile for higher cooling rate. However, simulated supersaturation profile

11

for lower cooling rate is not matching well with that of experimental (figure 5b). In fact,

12

experimental supersaturation is lower than that of simulated, suggesting significant effect of

13

secondary nucleation in later phase of crystallization. So, for the estimation of secondary

14

nucleation rate, the supersaturation profile for lower cooling rate was used and was optimized

15

using gPROMS parameter estimation. The estimated kinetic parameters of secondary

16

nucleation are given in table 3. Also the simulated profile of supersaturation using primary

17

nucleation, secondary nucleation and growth is compared with that of experimental in figure

18

6. Thus it can be concluded that the supersaturation profile for higher cooling rate can be

19

used for estimation of secondary nucleation kinetics if primary nucleation and growth

20

kinetics are already known.

21



0.025 Supersaturation (kg solute/kg solvent)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 32

Experimental 0.02 Simulated with no secondary nucleation 0.015 Simulated with secondary nucleation 0.01

0.005

0 0

1

2000

4000 6000 Time (sec)

8000

10000 \

2 3

Figure 6. Simulated supersaturation profile with secondary nucleation and

4

Experimental supersaturation profile for cooling of 0.250C/min

5

6.3.3 Estimation of agglomeration and breakage kinetics:

6

After estimation of nucleation and growth parameters, the parameters left unknown

7

were those of agglomeration and breakage. Since the evolution of CSD is result of all the four

8

processes occurring simultaneously, CSD profile can be used to estimate the remaining

9

parameters. It was observed that moment profile was sufficient for gPROMS parameter

10

estimation to estimate kinetic parameters (Refer to supporting information). Hence instead of

11

CSD, we have used mean length profile for estimation of breakage and agglomeration

12

parameters. Before that, simulations were carried out for the mean length profile with known

13

nucleation and growth parameters, neglecting breakage and agglomeration. Figure 7 shows

14

the comparison of experimental mean length with that of simulated with no breakage and

15

agglomeration. It can be clearly seen from figure 7 that simulated mean length profile

16

deviates significantly from that of experimental.

17


Page 25 of 32

Simulated with Agglomeration and breakage

7.5E-04

Simulated with no Agglomeration and breakage

mean length (m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


6.0E-04

Experimental

4.5E-04 3.0E-04 1.5E-04 0.0E+00 0

2000

4000 6000 Time (sec)

8000

10000

1 2

Figure 7. Simulated mean length profile and Experimental mean length profile for

3

cooling of 0.250C/min

4

During this parameter estimation step, nucleation and growth parameters, which were

5

estimated earlier, were used as known parameters. The breakage and agglomeration

6

parameters were estimated using gPROMS parameter estimation and are given in table 3. All

7

the estimated parameters were then used for the simulation of time evolution of mean length

8

profile. Figure 7 shows the comparison of simulated mean length profile using breakage and

9

agglomeration with that of experimental. It can be clearly seen that the estimated mean length

10

profile matches quite well with that of experimental.

11 12

6.3.3 Kinetic parameter validation:

13

In order to validate the kinetic parameters, simulations were carried out for the cooling

14

crystallization of K2SO4 at 0.750C, the experimental data of which was not used for parameter

15

estimation. Figure 8 (a) shows the comparison of simulated supersaturation with experimental

16

supersaturation. Figure 8 (b) shows the comparison of simulated mean length profile with that

17

of experimental. It can be seen from both the supersaturation and mean length profile that the

18

kinetic parameters describe the crystallization process correctly.



0.035

Experimental

Supersaturation

0.03

Simulated

0.025 0.02 0.015 0.01 0.005 0 0

400

800

1200 1600 2000 2400 2800 3200 Time (sec)

1

8(a)

2

Simulated with Agglomeration and breakage

4.0E-04

Simulated with no Agglomeration and breakage

mean length (m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 32

Experimental

3.0E-04 2.0E-04 1.0E-04 0.0E+00 0

700

1400 2100 Time (sec)

2800

3500

3 4

8(b)

5

Figure 8. Model validation for cooling of 0.750C/min. 8(a) Supersaturation profile. 8(b)

6

Mean length profile.

7

7. Conclusions:

8

We have proposed a sequential parameter estimation method to estimate the crystallization

9

kinetic parameters for simultaneous nucleation, growth, breakage and agglomeration.

10

Effectiveness of the developed method was checked for the crystallization of K2SO4 from its

11

saturated solution in water. Although the equations considered for different phenomena were 26 ACS Paragon Plus Environment

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1

simple, some more complex equations can also be considered. This includes size dependent

2

growth rate, temperature dependent nucleation and growth rate, size dependent agglomeration

3

kernel etc. This will increase the number of parameters to be estimated; however, as we have

4

estimated almost two parameters for each phenomenon, consideration of any complex

5

equation won’t be much difficult. Nevertheless, incorporation of these types of equations are

6

employed only for deducing the mechanistic behavior of particular phenomena and not for

7

approximating the course of crystallization process and hence are not used frequently. Thus

8

the method developed can be effectively used for the estimation of crystallization kinetics

9

over a wide domain.

10 11

Table 3. Kinetic parameters for different phenomena during cooling crystallization of

12

K2SO4: Estimated in three sequential steps.

13

Process

Parameters

Primary nucleation

2.40 × 10!5 (

$

1.14 × 10!5 (# . #( .=A )

!

Secondary nucleation

&

Growth

}

Breakage

k

DE

Agglomeration

Value (unit)

6.5

#

# . #(

2.25

)

#

1.54 × 10B (k/) 1.15

8.55 (#/) 0.493

5.71 × 10B (#/)

14 15 16

Supporting information: Application of gPROMS parameter estimation tool for PBE’s

17

having analytical solutions, parameter estimation for pure growth, pure breakage, pure

18

agglomeration, simultaneous growth-nucleation, simultaneous nucleation-agglomeration and

19

simultaneous breakage-agglomeration.



Page 28 of 32

1

Acknowledgement: Authors would like to acknowledge University Grants Commission

2

(UGC), New Delhi, India for financial support.

3 4

Nomenclature:

!

&

4(, ’)

∗

∆

}

DE

$

Breakage rate constant (no/sec) Primary nucleation order Secondary nucleation order Breakage distribution function Nucleation rate (no/sec) Birth rate due to agglomeration (no/sec) Birth rate due to breakage (no/sec) Primary nucleation rate (no/sec)

Secondary nucleation rate (no/sec) Solute concentration (kg solute/kg of solvent) Equilibrium solute concentration ((kg solute/kg of solvent) Super-saturation (kg solute/kg of solvent) Death rate due to agglomeration (no/sec)

Death rate due to breakage (no/sec) Growth rate order Crystal growth rate (m/sec) Constant agglomeration kernel (1/(no×sec)) Growth rate constant (m/sec)

Primary nucleation rate constant (no/sec) Secondary nucleation rate constant (no/sec)


Page 29 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Length of crystal (m)

E

Size of nuclei (m)

= k

Mean size (m) Breakage rate exponent

"

Suspension density (kg solute/kg of solvent)

M

Number of crystal in ‘ith’ interval

( , )

Number of crystals of size at time .

9(

Volume shape factor

9(5 )

Breakage rate of particle of size ‘L’ (no/sec) Time (sec)

T

Induction time (sec)

Volume of crystal (m3)

I

Temperature (0C)

Greek letters:

L

‘ith’ moment

' 1 2 3 4 5 6 7 8 9 10 11 12 13

Density of crystals kg/m3

References: 1. 2. 3. 4.

5.

Mullin, J. W. (2001). Crystallization. Butterworth-Heinemann. D. Ramkrishna, Population Balances :Theory and Applications to Particulate Systems in Engineering, 1st edition, Academic Press, 2000, p 117. Hu, Q., Rohani, S., and Jutan, A. (2005). Modelling and optimization of seeded batch crystallizers. Computers & chemical engineering, 29(4), 911-918. Pohar, Andrej, and Blaž Likozar. (2014) "Dissolution, nucleation, crystal growth, crystal aggregation, and particle breakage of amlodipine salts: Modeling crystallization kinetics and thermodynamic equilibrium, scale-up, and optimization." Industrial & Engineering Chemistry Research 53 (26), 10762-10774. Tadayon, Abdolsamad, Sohrab Rohani, and Michael K. Bennett. (2001). "Estimation of nucleation and growth kinetics of ammonium sulfate from transients of a cooling batch seeded crystallizer." Industrial & engineering chemistry research 41 (24) 6181-6193. 29 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

6.

7.

8.

9.

10. 11.

12.

13.

14. 15. 16.

17.

18. 19. 20.

21. 22. 23.

Page 30 of 32

Himawan, C., and G. J. Witkamp. (2006). "Crystallization kinetics of MgSO4· 12H2O from different scales of batch cooling scraped crystallizers." Crystal Research and Technology 41 (9), 865-873. Peng, Y., Zhu, Z., Braatz, R. D., and Myerson, A. S. (2015). "Gypsum crystallization during phosphoric acid production: modeling and experiments using the mixed-solventelectrolyte thermodynamic model." Industrial & Engineering Chemistry Research 54 (32) 7914-7924. Luo, Yang-Hui, Guo-Gan Wu, and Bai-Wang Sun. (2013). "Antisolvent crystallization of biapenem: estimation of growth and nucleation kinetics." Journal of Chemical & Engineering Data 58 (3), 588-597. Kougoulos, E., A. G. Jones, and M. W. Wood-Kaczmar. (2005). "Estimation of crystallization kinetics for an organic fine chemical using a modified continuous cooling mixed suspension mixed product removal (MSMPR) crystallizer." Journal of crystal growth 273 (3), 520-528. Lindenberg, C., Schöll, J., Vicum, L., Mazzotti, M., and Brozio, J., (2007). "L-glutamic acid precipitation: agglomeration effects." Crystal Growth and Design 8 (1), 224-237. Wang, Y., Ma, S., and Lv, X., (2012). “Determination of reactive crystallization kinetics of 5-(difluromethoxy)-2-mercapto-1H-benzimidazole in a batch crystallizer” Crystal Research and Technology, 47(8), 848-862.. Pérez‐Calvo., José‐Francisco., Somnath S. Kadam., and Herman JM Kramer., (2016) "Determination of kinetics in batch cooling crystallization processes—A sequential parameter estimation approach." AIChE Journal 62 (11), 3992-4012. El Moussaouiti, M., Boistelle, R., Bouhaouss, A., & Klein, J. P. (1997) "Crystallization of calcium sulphate hemihydrate in concentrated phosphoric acid solutions." Chemical Engineering Journal 68 (2-3), 123-130. Bao, Y., Zhang, J., Yin, Q., & Wang, J. (2006). "Determination of growth and breakage kinetics of l-threonine crystals." Journal of crystal growth 289 (1), 317-323. Andreassen, Jens‐P., and M. J. Hounslow. (2004). "Growth and aggregation of vaterite in seeded‐batch experiments." AIChE journal 50 (11), 2772-2782. Zauner, Rudolf, and Alan G. Jones. (2000). "Determination of nucleation, growth, agglomeration and disruption kinetics from experimental precipitation data: the calcium oxalate system." Chemical Engineering Science 55 (19), 4219-4232. Wohlgemuth, Kerstin, and Gerhard Schembecker., (2013). "Modeling induced nucleation processes during batch cooling crystallization: A sequential parameter determination procedure." Computers and Chemical Engineering 52, 216-229. Randolph, A. D., and M. A. Larson. (1971). "Theory of particulate processes: analysis and techniques of continuous crystallization." Gelbard, F., Tambour, Y., and Seinfeld, J. H. (1980). “Sectional representations for simulating aerosol dynamics”. Journal of Colloid and Interface Science, 76(2), 541-556 Batterham, R. J., Hall, J. S., and Barton, G. (1981). “Pelletizing kinetics and simulation of full scale balling circuits”. In Proceedings of the 3rd International Symposium on Agglomeration (Vol. 136). Nurnberg W. Germany. Hounslow, M. J., Ryall, R. L., and Marshall, V. R. (1988). “A discretized population balance for nucleation, growth, and aggregation”. AIChE Journal, 34(11), 1821-1832. Hounslow, M. J. (1990). “A discretized population balance for continuous systems at steady state” AIChE journal, 36(1), 106-116. Kumar, S., and Ramkrishna, D. (1996a). “On the solution of population balance equations by discretization—I. A fixed pivot technique”. Chemical Engineering Science, 51(8), 1311-1332.


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24. Kumar, S., and Ramkrishna, D. (1996b). “On the solution of population balance equations by discretization—II. A moving pivot technique”. Chemical Engineering Science, 51(8), 1333-1342. 25. Kumar, J., Peglow, M., Warnecke, G., Heinrich, S., and Mörl, L. (2006). “Improved accuracy and convergence of discretized population balance for aggregation: The cell average technique”. Chemical Engineering Science, 61(10), 3327-3342. 26. Vanni, M. (2000). “Approximate population balance equations for aggregation–breakage processes”. Journal of Colloid and Interface Science, 221(2), 143-160. 27. Rawlings, J. B., Miller, S. M., and Witkowski, W. R. (1993). “Model identification and control of solution crystallization processes: a review”. Industrial and Engineering Chemistry Research, 32(7), 1275-1296. 28. Mignon, D., Manth, T., and Offermann, H. (1996). “Kinetic modelling of batch precipitation reactions”. Chemical engineering science, 51(11), 2565-2570. 29. Shaikh, L., Pandit, A., and Ranade, V. (2013). “Crystallisation of ferrous sulphate heptahydrate: Experiments and modelling”. The Canadian Journal of Chemical Engineering, 91(1), 47-53 30. Bari, Atul H., and Aniruddha B. Pandit. (2014). “Ultrasound-facilitated particle breakage: Estimation of kinetic parameters using population balance modelling”. The Canadian Journal of Chemical Engineering 92 (12), 2046-2052. 31. P.S.E. Ltd., gPROMS Advanced User Guide-Release 2.3, Process Systems Enterprise Ltd., London (2001) 32. Nývlt, J., Rychlý, R., Gottfried, J., and Wurzelová, J. (1970). “Metastable zone-width of some aqueous solutions”. Journal of crystal growth, 6(2), 151-162. 33. Nývlt, J. (1968). “Kinetics of nucleation in solutions”. Journal of Crystal Growth, 3, 377-383. 34. Kubota, N. (2008). “A new interpretation of metastable zone widths measured for unseeded solutions”. Journal of Crystal Growth, 310(3), 629-634.


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