Determination of the crystal growth, birth of crystal nuclei, dynamics and stability of the crystallizer

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Title:
Determination of the crystal growth, birth of crystal nuclei, dynamics and stability of the crystallizer
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xiv, 223 leaves : ill. ; 28 cm.
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Okonkwo, Charles Umejei Onwuegbunem
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Crystallization   ( lcsh )
Crystal growth   ( lcsh )
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Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1982.
Bibliography:
Includes bibliographical references (leaves 218-221).
Statement of Responsibility:
by Charles Umejei Onwuegbunem Okonkwo.
General Note:
Typescript.
General Note:
Vita.

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Full Text









DETERMINATION OF THE CRYSTAL GROWTH,
BIRTH OF CRYSTAL NUCLEI, DYNAMICS
AND STABILITY OF THE CRYSTALLIZER







By

CHARLES UMEJEI ONWUEGBUNEM OKONKWO


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY




UNIVERSITY OF FLORIDA


1982



































Copyright 1982

by



Charles Umejei Onwuegbunem Okonkwo

























This project
is
dedicated
to
my parents
for their continual love,
encouragement, and support
throughout the years

















ACKNOWLEDGMENTS


Glory be to God for seeing me through the many

years of my educational experience.

I wish to thank the chairman of my doctoral super-

visory committee, Dr. Hong Lee, for his guidance in this

research, and Dr. Robert Coldwell for his assistance

with the computer subroutine (simple).

Thank you to Dr. Robert Gould and Dr. Ranganatha

Narayanan for their encouragement.

Special thanks go to Dr. Charles Burnap and

Dr. Ulrich Kurzweg for the many fruitful discussions

I have had with them during the course of my studies

at this university.

Thank you to Linda McClintic and Elaine Everett

for their encouragement. Also sincere thanks go to

Vita Zamorano for her diligence and cooperation in

typing the manuscript.

In conclusion, a special word of thanks is extended

to my family for their moral support, prayers, love and

patience. Thank you Mom, Dad, Thomas, Elizabeth, Paul,

Rosaline, Francis and Fidelis for your understanding,

encouragement and assistance throughout my educational

experience.

















TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS ................................... iv

KEY TO SYMBOLS ............ ... .................... .. vii

ABSTRACT ............................................. xiii

CHAPTER

I INTRODUCTION ..... .......................... 1
Dynamics and Stability ..................... 6
Crystallizer Control ....................... 10

II LITERATURE REVIEW .......................... 12
Nucleation and Growth ...................... 12
Dynamics and Stability ..................... 19
Control ..................................... 30

III THEORY AND METHODOLOGY ..................... 33
Growth and Birth Rates in Steady State
Crystallizer ................. .... .......... 34
Growth and Birth Rates in Unsteady State
Crystallizer ...... ........................ 40
Dynamics and Stability of the Complex
Crystallizer .............................. 48
Complex Crystallizer Balance Equations ..... 49
Solute and Solid Balance ................... 53
Dynamics and Stability ..................... 56
Nucleation Models .......................... 73

IV RESULTS .. ................................. 81
Steady State Determination of Growth and
Birth Functions ........................... 81
Unsteady State Determination of Growth and
Birth Functions ............................ 93
Stability Criteria for the Complex
Crystallizer ...... ....................... 107
Discussion ............................... .. 141

V SUMMARY, IMPLICATIONS, CONCLUSIONS AND
RECOMMENDATIONS FOR FUTURE RESEARCH ....... 153
Summary ...................................... 153
Implications ................................. 154
Conclusions .... ........................... 156
Recommendations for Further Research ....... 157















Page


APPENDICES

A DERIVATION OF THE CRYSTALLIZER POPULATION
BALANCE ................ ................. 159

B ERROR ANALYSIS FOR THE DETERMINATION OF
GROWTH AND BIRTH FUNCTIONS ................ 166

C THE DETERMINATION OF SEPARABLE GROWTH AND
BIRTH FUNCTIONS .......................... 176

D COMPUTER PROGRAM FOR THE ILLUSTRATION OF THE
SUBROUTINE (SIMPLE) AND FOR THE MONITORING
OF THE STABILITY OF THE COMPLEX
CRYSTALLIZER .............................. 181

BIBLIOGRAPHY ................. ..... ..... .. ........ 218

BIOGRAPHICAL SKETCH ................................ 222

















KEY TO SYMBOLS


a Withdrawal rate of intermediate size product

a. Dimensional constants in balance equations
1 (3-53) and (3-54)

a. Constants in the fit for n(,t)

a Coefficients in characteristic equation
n
Al Constant in W(z), example data

A2 Constant in r (z), example data

A3 Constant in R(z), example data

A Constants in series for N(z)
n
AT Total crystal area

b Size dependent birth function

b Size dependent part of birth function in
equation (3-45)

b Net birth function in equation (A-20),
Appendix A

b* Dimensionless derivative of concentration
dependent part of birth function

B1 Constant in W(z), example data

B Birth function of crystals in equation (3-33)

B Birth rate at zero size

B Cummulative birth function measured in steady
state crystallizer and defined in equation
(3-17a)

c Concentration

c Constant in ASL model, equation (2-10)

c Saturation concentration











c

c






,C2,C
Cl
C.
1





CS1,CS2,CS3

C*

D

EM

EM2

EMH

ENINT


fn
FAi

Fi


F(z)

g


g(x)

g*


h.
1
h(z)

i


Steady state concentration

Reference concentration

Constants in the fit for number size distribu-
tion, steady state case

Constants in r3(z), example data

Absolute error in n, Appendix B

Concentration of the i-th stream

Metastable concentration

Inlet concentration in equation (3-37)

Constants in rl(z), example data

Dimensionless inlet concentration

Death rate of entities, Appendix A

Constant in W(z), example data

Constant in R(z), example data

Constant in no(z), example data

Integral of size distribution for seed
crystals

Appropriate set of orthonormal functions

Fitted values for steady state size distribution

Experimental data for steady state number size
distribution

Integral of h(z)

Concentration dependent part of growth rate in
equation (3-44)

Vector function in equation (2-22)

Dimension derivative of the concentration
dependent part of growth rate

Classification function

Quantity occurring in equation (3-65a)

Ratio of nucleation to growth rate
(nucleation-growth sensitivity parameter)

viii











I Quantities occurring in equation (3-90)

j Exponent to which suspension density is raised

J(z) Quantity occurring in equation (3-69a)

k Volumetric shape factor

K2,K3 Constants in B(c), equation (1-3)

K Constant in equation (3-95)
c
K Constant in B equation (1-1)
n
KN Constant in equation (2-16)

L Critical size of crystal

LD Dominant size

L* Maximum size of crystal fines

Lf Dimensionless maximum size of crystal fines

LK Discrete size

L Quantities occurring in equation (3-90)

L Maximum largest size measured
max
Lm Associate Laguerre polynomials
n
m Modified orthonomal set of functions, example
n data

L Lower limit of integration, equation (2-4)

L Maximum size for intermediate size crystals
p
L ,L1 Lowest size measured

M Matrix

m Amplitude of dimensionless time dependent
concentration

mk k-th moment of number size distribution

MT Suspension density

n Number size distribution

n. Inlet number size distribution for seed
1 crystals in inlet feed












n. Dimensionless number size distribution for
in
seeds

n Size distribution for the m-th stage
m
no Number of zero size crystals per unit volume

N Groups of quantities defined in equation (3-17c)

N(z) Size dependent part of dimensionless number
size distribution

P Nucleation-growth sensitivity parameter
occurring in equation (3-95)

P Quantities occurring in equation (3-90)

Q Volumetric flow rate

Q* Quantity defined in equation (3-75)

r Fraction change in concentration due to
dissolving and recycling fines

R Recycle ratio of dissolved fines

R(z) Dimensionless size dependent part of birth rate

S Sum of square error

t Time

t Time of particle growth after birth
g
t Discrete time
m
t ,tl Lowest time measured

T Reference time

u Dimensionless concentration dependent part of
birth function

v Dimensionless concentration dependent part of
growth function

v1 Growth rate, Appendix A

ve External velocity, Appendix A

v. Internal velocity, Appendix A
V Volume of crystallized's contents
V Volume of crystallizer's contents


1











V Crystallizer's volume, Appendix A

VL Volume of clear liquor volume

VS Slurry volume

W Dimensionless size dependent part of growth
rate

x Spatial coordinate, Appendix A

X Quantity defined in equation (3-17b)

y Spatial coordinate, Appendix A

y Dimensionless concentration

y Time independent dimensionless concentration
in perturbation analysis

z Spatial coordinate, Appendix A

Z Dimensionless size

Z* Withdrawal rate of oversize products



Greek Symbols

a Fractional error in n, Appendix B

a. Constant in expression in steady state birth
1 function, equation (3-8)

a. Constants relating residence times of crystals
1 to their corresponding flow rate in different
size ranges

a Quantities in equation (3-79)
mn
B Product recycle rate

n Quantities in equation (3-79)
mn
X Square root of chi-square error

Xn Quantities in equation (3-79)

6 Quantities in equation (3-79)

a Quantities in equation (3-90)
m











n Dimensionless number-size distribution

e Perturbation error

E Error in cumulative birth rate, Appendix B

Ec Ratio of clear liquor volume to slurry volume

Y Constant in equation (2-10)

y1,Y2 Constants in fit for n(), steady state case

yn Quantities in equation (3-79)

F1 Group of terms defined in (3-47)
I2 Group of terms defined in (3-49)

F3 Group of terms defined in (3-50)

X Eigen value in stability analysis

An Quantities in equation (3-90)

1m Microns

V5 Dimensionless residence time

R Quantity in equation (3-74)

( Size dependent part of growth rate

(Some entities, Appendix A

*mn Quantities in equation (3-90)

p Density of crystal

a Concentration dependent part of birth rate

T Residence time

T4 Average residence time of crystals in
crystallizers

Ti Residence time of seed crystals

T. Residence times of crystals in different size
1 ranges in equation (3-34)

9 Solid fines recycle rate

8 Dimension time


xii












Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



DETERMINATION OF THE CRYSTAL GROWTH,
BIRTH OF CRYSTAL NUCLEI, DYNAMICS
AND STABILITY OF THE CRYSTALLIZER

By

Charles Umejei Onwuegbunem Okonkwo

August 1982

Chairman: Hong H. Lee
Major Department: Chemical Engineering

The task of quantifying growth and nucleation rates

in a crystallizer grows increasingly difficult as one

demands more accuracy in the calculation of these qualities,

growth and nucleation rates. The associated stability

problem of the crystallizer becomes very complex as more

efficient and useful criteria are sought. The search for

very good criteria necessitates that the research work

with more complete balance equations.

Novel methods for quantifying growth and nucleation

rates in steady and unsteady state crystallizers are

developed. There were no assumptions made in the develop-

ment of these methods which utilize some minimization

procedure. The only approximation was in the minimization

procedure. When applied to some literature data,the

methods gave good results. In addition, the critical size,

Lc, at which the growth rate is zero could be determined

by these methods.


xiii











Very comprehensive and meaningful stability criteria

are developed for the complex crystallizer via perturba-

tion techniques. The complex crystallizer was a well-

mixed suspension crystallizer equipped with dissolving

and recycling of undissolved crystals and product clas-

sification. The criteria were derived from the complete

crystallizer's balance equations. The only approximation

was that the concentration dependent parts of the birth

and growth functions were linear in concentration. In

addition, the fourth order characteristic equation, from

which the criteria are derived, is used in monitoring the

effects of various crystallizer's parameters on the system's

stability. These effects were simulated via the Root-Locus

method. When applied to some example data,the charac-

teristic equation gave results that are in agreement with

experimental observations. The computer simulation and

the criteria agree in all cases. New ways of stably

operating a crystallizer under conditions in which the

crystallizer may be inherently unstable were obtained.


xiv

















CHAPTER I
INTRODUCTION



The preparation and handling of solids areinfluenced

not only by gross chemical composition, but also by particle

size distribution, morphology and surface chemistry. These

parameters are in turn often fixed by processes that occur

as the solid forms and is separated from its mother liquor.

Crystallization products include food materials, pharma-

ceuticals, fertilizers and commodity chemicals. A few uses

of crystallization include the following: (1) in process

development, (2) in separation techniques, (3) in poly-

merization processes, and (4) in controlling product quality

and shape. Thus if one wants to know about behavior of

solids--reaction rates, caking, blending, toxicology,

storage, dusting, bioavailability, etcetera--an adequate

knowledge of the crystallization process will prove worthy.

Sometimes a problem that is not initially viewed as a

crystallization problem becomes one in the end.

Unlike most chemical processes, particle conservation

balance is required in addition to mass, momentum, and energy

balances to determine process yields and energy requirements.

The population balance is an added complication which makes

solution of the balance equations extremely difficult, if

not impossible.











Primary and secondary nucleation are the two forms of

crystallization. Primary crystallization is further sub-

divided into homogeneous and heterogeneous nucleation.

Classical homogeneous nucleation is that which occurs only

because of supersaturation as the driving force. In other

words, it is conceived as the coming together of sufficient

solute molecules to form a critical mass capable of sur-

viving as a solid phase. An example of this is found in

simple cooling and temperature reduction of a solution or

in the evaporation of a portion of the solvent of a solu-

tion. Heterogeneous primary nucleation is similar to homo-

geneous nucleation except for the presence of a foreign

solid which acts as an active site for nucleation. This

foreign substance is of a different chemical composition.

Again heterogeneous nucleation has supersaturation as its

driving force. In general primary nucleation occurs at

high supersaturation levels and nucleation and growth

kinetics have a high order dependence on supersaturation.

On the other hand, secondary nucleation is one which can

occur at very low supersaturation provided there is some

form energy and a seed crystal of the same chemical com-

position as material being crystallized. Secondary nuclei

can form in saturated and undersaturated solution. For

the nuclei formed to survive the solution must be slightly

supersaturated. Secondary nucleation is not a strong func-

tion of nucleation and growth kinetics. The driving force

is the presence of the seed crystal and energy. This











energy might be one of several forms such as crystal im-

peller collisions, ultrasonic waves or fluid shear on the

crystal surface.

Crystallizers which operate at high and low super-

saturation levels irrespective of the throughput or process

conditions are classified as classI and class II systems

respectively. Class II systems are usually high yield and

fast growing while the contrary is true for class I sys-

tems. Primary and secondary nucleation are phenomena with

high probability of occurrence in class I and class II

systems respectively.

Though primary crystallization has been for a long time

a topic of interdisciplinary interest among various scien-

tists and engineers and has been extensively researched,

a better understanding of the phenomenon of secondary

nucleation awaits still much work to be at par with its

counterpart. It was no more than thirty years ago when

chemical engineers began work in secondary nucleation when

they realized that it was the dominant form of nucleation

in industrial crystallizers which usually operate at low

supersaturation. Since then, several authors (Miller and

Saeman, 1951; Murray and Larson, 1965; Timn and Larson,

1968; Clontz and McCabe, 1971; Randolph and Cise, 1972;

Strickland-Constable, 1972; Ottens and de Jong, 1973;

Sung et al., 1973; Bauer et al., 1974; Garside and Jancic,

1976; Lee, 1978; Garside and Jancic, 1979; Randolph and

Puri, 1981) have addressed one aspect or the other of











secondary nucleation. Some of the aspects of secondary

nucleation that have been of interest to chemical engineers

are the following: (a) factors such as the crystallizer

environment affecting the crystallizer (these may include

effects of impurities, crystallizer hydrodynamics, impeller

speed, surface regeneration time, hardness of contacting

surface--the effects of these factors are usually treated

via empirical correlations); (b) mechanisms involved in the

crystallization process such as that responsible for

nucleation and growth or the effect of impurities;

(c) dynamics, stability and control of crystallizer; and

(d) quantification of nucleation (birth of secondary nuclei)

and crystal growth. Of the aforementioned areas the last

is most critical in understanding the phenomenon of

secondary nucleation. Knowledge about accurate quantifica-

tion of nucleation and growth is important in understanding

other areas as well.

The mixed-suspension-mixed-product-removal (MSMPR)

pioneered by Randolph and Larson (1971) is a simple continuous

crystallizer with inlet and outlet flow and in many ways is

analogous to the continuous stirred tank reactor (CSTR).

The main difference is the simultaneous occurrence of nuclea-

tion and growth in the MSMPR crystallizer. The name MSMPR

crystallizer as used in the literature is a steady state

crystallizer of the above description but obeying certain

constraints that are rarely satisfied in practice.











Though the MSMPR crystallizer is valid for the large

size region of constant growth rate (that is the McCabe AL

law), researchers have used the MSMPR theory for the small

size region. Because of the complexity introduced by the

simultaneous occurrence of the two fundamental quantities

--birth and growth rates--authors working with the steady

state crystallizer have simply ignored the birth function

in the population balance equation even when there is

nuclei generation by contact nucleation. Contact nuclea-

tion is a form of secondary nucleation where energy is

provided by contacting the seed crystal. Khambaty and

Larson (1978) failed to account for nuclei generation in

their steady state crystallizer experiment involving contact

nucleation in addition to assuming that nuclei are born at

zero size. Garside and Jancic (1979), like their predeces-

sors, found it necessary to invoke the "zero size nuclei"

assumption in their attempt to determine birth and growth

rates in a steady state crystallizer. Authors have had

more problems in determining the two fundamental quantities

in an unsteady state crystallizer. To alleviate the problem

Randolph and Cise (1972), while working with an unsteady

state crystallizer,assumed zero birth rate in the population

balance in order to calculate growth rate and to calculate

birth rate they assumed zero growth. Similarly, Garside and

Jancic (1976) obtained birth and growth rates in an unsteady

state crystallizer via a differential technique, a method

which not only gives gross errors, but also is very











impractical considering the type of experimental data that

one works with. While the above list is by no means

exhaustive, it is representative of the general trend

adopted by most researchers. The above mentioned problems

are addressed in the first part of my work.



Dynamics and Stability


MSMPR crystallizers inherently produce a wide crystal

size distribution with a large coefficient of variation

(C.V.) and a dominant size LD = 3GT, where G and T are

constant growth rate and residence time respectively.

This wide distribution is industrially undesirable.

Design specifications for commercial products might require

a narrow or large distribution or somewhere between the two.

Sometimes one might desire some nucleation when there is

little or none, while at other times there may be un-

desirable excessive nucleation. These and other problems

dictate the need for controlling CSD. One way to change

the CSD is to incorporate some selective removal whereby

crystals of a certain size range are preferentially removed.

Preferential removal, dissolution and recycle of fines

generally lead to an increase in characteristic dimension

(such as dominant or average crystal size) and spread of

the distribution,while preferential removal of coarse

products leads to a narrower distribution with a concomitant

reduction in dominant size. Incorporation of clear liquor

advance has the effect of increasing crystal residence time











which in turn has little or no effect in increasing

dominant size. While it should be remembered that the

purpose of designing selective crystal removal into a

crystallizer is to force the CSD to a larger dominant size

or narrow the distribution, actual incorporation of selec-

tive removal often leads to sustained oscillations. This

action results in no improvement in product quality. It

would be desirable to be able to operate the crystallizer

without the incursion of instability even in regions where

the system is inherently unstable by choosing appropriate

control strategy. Ability to do this requires very

meaningful stability criteria.

Several researchers have investigated various aspects

of stability and control of one form of crystallizer or

the other. Some of these are complex in nature. Unlike the

MSMPR crystallizer which is analogous to the CSTR reactor

as used in reaction engineering, complex crystallizers have

no such counterpart. Complex crystallizers do not easily

lend themselves to mathematical description and solution.

The type of crystallization considered varied from slow

growth rate kinetics (class I systems), where residual

supersaturation is appreciable, to fast growth systems

(class II systems) with negligible residual supersaturation.

Miller and Saeman (1947) while working with an industrial

ammonium nitrate producing crystallizer observed cyclic

fluctuations in particle size distribution. Similar observa-

tions were made by Finn and Wilson (1954) during a fermentation











process and by Thomas and Mallison (1961) in the field of

continuous polymerization process. Yu and Douglas (1975)

showed in their paper that a class I MSMPR crystallizer

could be operated to give oscillations in the size distribu-

tion. These oscillations were the results of interactions

within the system and these interactions were in turn the

direct result of disturbances.

Randolph (1962) quantified stability of class II MSMPR
dL B
in the following manner: dL < 21. The logarithms of
n
nuclei birth rate and grow rate are Z B and Z G respectively.

By assuming the empirical power-law nucleation growth

kinetics as expressed in equation (1-1),


B = K Gi (1-1)
n


he obtained the following constraint:


dL Bo
n = i < 21. (1-2)
dL G


In equation (1-1) B is the nuclei birth rate defined via

the assumption of zero size nuclei; G is the size independent

growth rate and "i" is the kinetic order of birth to growth.

Oscillations generated by kinetic values as high as 21 are

referred to as high-order cycling.

Sherwin, Shinnar, and Katz (1967) derived stability

criteria for an isothermal class I MSMPR crystallizer using

Volmer's nucleation model in equation (1-2) and assuming

nuclei birth rate is size independent.











B(c) = K2exp(-K3/(Znc/s) 2) (1-3)


In equation (1-3), B(c) represents the concentration

dependence of birth function, K2 and K3 are constants, and

c and cs are the concentration and saturation concentration,

respectively. They claimed that instability was due to the

nonlinear nature of the dependence of nucleation on super-

saturation and that size dependency of the growth rate has

a stabilizing effect. Attempts to treat size dependence of

growth rate up to second order resulted in nonlinear moment

equations which were not closed. Similar attempts not to

linearize B(c) resulted in a closed set of nonlinear moment

equations which were solved numerically. The authors

concluded for the case of size independent growth rate and

B(c) as defined by equation (1-3) that the tendency towards

cycling increased with increasing particle size. Considering

the assumptions in their model these conclusions are

questionable.

Hulburt and Stefango (1969) modeled a double draw-off

crystallizer by assuming size independent growth and birth

rates and Volmer's nucleation model. They numerically

solved the coupled nonlinear population and mass balance

equations. The assumption of zero nuclei size was made in

deriving these equations. They concluded that one of the

draw-off streams, the clear liquid overflow,seemed to

enhance stability. They also concluded that both increased

seed addition and increased seed size in feed stream enhanced

stability for high growth rate conditions.











Randolph, Beer and Keener (1973) showed the birth-

growth stability constraint of a complex R-Z crystallizer

to be a function of "i," fines exponential decay factor,

X, product classification size, L and normalized

product classification rate, z. Their model assumes that

the birth and growth rates were size independent and that

nuclei were of zero size. Beckman (1976) experimentally

demonstrated sustained limit-cycle behavior in an R-Z type

crystallizer equipped with clear liquor advance and product

classification. He concluded that cycling behavior was

chiefly induced by product classification.



Crystallizer Control


Timn and Gupta (1970) investigated the stability control

of class II MSMPR crystallizer using fines seeding and

fines destruction plus recycle as manipulated variables.

They concluded that controlling the zero moment tended to

stabilize the system while controlling the second moment

tended to destabilize a normally stable system. Lei,

Shinnar, and Katz (1971a) investigated the stability of a

class I crystallizer with a point fines trap via spectral

method. The control variable was fines crystal area. They

were able to stabilize the system by manipulating the

throughput rate while by manipulating fines destruction and

recycle rate the system's stability was not enhanced. The

idea of a point fines trap (zero mass of fines in trap) has











no practical value since industrial crystallizers operate

with appreciable amounts of crystal mass in fines trap.

In order to avoid the excruciating mathematical com-

plexity that would result, the aforementioned researchers

found it necessary to make one assumption or the other

even in situations when the conditions of their assumptions

are never realized in practice. In consideration of the

above problems, the second part of this work is devoted to

obtaining more general stability criteria from which meaning-

ful control strategy can be obtained. The crystallizer in

this study will be discussed in detail in a later chapter.

While it is more general, it is in many ways physically

similar to the R-Z crystallizer modeled by Randolph, Beer,

and Keener (1973), and will henceforth be referred to as

"the Complex Crystallizer."
















CHAPTER II
LITERATURE REVIEW



The concept of the population balance is mainly

responsible for the progress and development witnessed in

crystallization thus far. The study of crystallization

has shifted from what was originally considered an art to

what is becoming, more and more, an engineering science.

The development of crystallization science started with

nucleation-growth kinetics and steady state system

analysis and progressed to dynamic system analysis. The

various aspects of crystallization have not been equally

embraced by the many works published on this subject.

More attention is now being directed to the very important

but least addressed areas of crystallization.



Nucleation and Growth


Saeman (1956) investigated the simplest continuous

crystallizer--the mixed-suspension-mixed-product-removal

(MSMPR) crystallizer. For this he essentially derived

equations for the steady state crystal size distribution

(CSD) in a mixed suspension. By first generating nuclei

he no longer had to consider the birth function as part of

the particle balance. Then he obtained the balance equation
(2-1) with respect to time of growth after particle birth.











dn + 0 (2-1)
dtg T


The number of particles per unit volume per unit size and

the residence time of these particles are n and T, respectively.

The variable t represents time (time of particle growth
g
after birth). The solution to equation (2-1) is given by


n(t ) = n(t =0) exp(-t /T). (2-2)


By assuming that growth was not a function of particle size,

this allowed him to relate time of growth, tg,to growth rate,

G, and crystal size, L, in the following manner:


L = Gt (2-3)


Substituting (2-3) in (2-2) and assuming that initial dis-

tribution of particles are nuclei of size L he obtained.


n(L) = n(Lo) exp (- -) (2-4)


In particular he chose Lo = 0 and obtained the standard

MSMPR crystallizer equation,


n(L) = n(0)exp ( ) = n exp(L) .(2-5)


Saeman then compared his theoretical distribution with that

obtained from an Oslo-Krystal type crystallizer producing

ammonium nitrate crystals under conditions of hindered

settling and found good agreement. Similar work was done

by Robinson and Roberts (1957) who obtained the theoretical











crystal size distribution resulting from a cascade of MSMPR

crystallizers with nucleation in the first stage only, but

same residence time in all stages. The size distribution

for the m-th stage could be represented by

m-1
n1(0) (G-) L
n (L) = (m-l exp(- L) ; m = 1,2,...
m (m-1) G-r
(2-6)


Randolph and Larson (1962) later extended the work of

Robinson and Roberts (1957) to include nucleation in all

stages (and equal residence time for each unit). By so

doing, they were able to extend application to more indus-

trial crystallizers. The size distribution of the m-th

stage for this type of crystallizer is


k-1
Sn(0) (L/GT)k
-L m+l-k
n (L) = () r ml-(2-7)
m GT (k-1)-2-
k=l


Abegg and Balakrishnan (1971) obtained the distribution in

equations (2-6) and (2-7) in their attempt to model the

mixing in different crystallizers. They found good agree-

ment between the theoretical distributions, equations (2-6)

and (2-7), and data taken from Draft Tube Battle (DTB) and

Forced Circulation (FC) crystallizers respectively.

Hulbert and Katz (1964) derived with great generality the

distribution of some entities over any associated property.

This entity and associated property can be, for instance,

particulate entity and particle size respectively. The











associated property can also be age or any other quantity

that varies with time. Their model was then applied to

particle nucleation and agglomeration, referred to as net

birth (birth and death) and growth in crystallization

terminology. Randolph (1964) in a "Note to the Editor"

derived essentially the same population balance and noted

that it would be of major importance in advancing the

theoretical understanding of continuous crystallization.

Murray and Larson (1965) designed and constructed a

continuous mixed suspension salting out crystallizer to

test both the MSMPR model as well as the unsteady state

model given by equation (2-8),


3n -r3n n (2-8)
at a C T *


They obtained nucleation and growth rates for the MSMPR model

and some of the first transient data in the literature. They

experienced great difficulty in the description of the very

small crystals which dominate the size distribution behavior

in terms of numbers. As a result determinations of nuclea-

tion and growth rates for this small size range were in

error. Timm and Larson (1968) obtained nucleation and growth

kinetics for three materials using steady state and transient

data. They found that unsteady state experiments have some

advantages over steady state experiments in determining the

kinetic order of nucleation rate to growth rate. McCabe and

Stevens (1951) found the growth rate of copper sulfate in a











suspension maintained by means of an agitator to be

dependent on crystal size if the relative velocities of

the crystals to solution velocity differ and if these

relative velocities are in the effective low range. They

found that larger crystals grew faster than small ones

because larger crystalshave higher relative velocity.

Youngquist and Randolph (1972) while working with a class

II system involving ammonium sulfate obtained with great

difficulty the nucleation rate after making several assump-

tions. They remarked that a rigorous quantitative defini-

tion of and use of secondary nucleation kinetics must await

quantitative separation of size dependent growth and birth

rates G(C), b(C) in the small size range. They said, "the

increased detail and accuracy of secondary nucleation

measurements made in this study have indicated the near

impossibility of quantitatively characterizing secondary

nucleation using the MSMPR technique as well as casting

doubt on the general applicability of gross secondary nuclea-

tion kinetis so obtained" (p. 429). It will become obvious

from this study, outlined in a later chapter, that one can

go much further with the quantitative determination of birth

and growth functions than they had thought. Randolph and

Cise (1972), while working with an unsteady state class II

system involving potassium sulfate, thought that it was

patently impossible to uniquely specify both size dependent

functions G(C) and b(C) using only the single size dependent

measurement n(C), without further simplifying assumptions











or other independent measurements. Faced with this dilemma

they assumed zero birth rate in the population balance in

order to calculate the growth rate and to calculate birth

rate they assumed zero growth rate, thus suppressing popula-

tion changes due to convective number flux. Similarly,

Garside and Jancic (1976) while working with potash alum in

an unsteady state crystallizer used a differential technique

to obtain birth and growth rates. This method is not only

in gross error, but also impractical as regards typical

experimental data.

Garside and Jancic (1979) found it necessary to invoke

the "zero size" assumption in their attempt to determine

birth and growth rates in a steady state crystallizer. They

rearranged the steady state population balance as shown in

equation (2-9):


dl n(S) 1 d G( )
b(C) = n(C)G() d + + d (2-9)
d G()T dC


They said that b() can be calculated if only G(C) is known.

Then they used surface integration rate and mass transfer

correlations to estimate the terms in equation (2-9) that

involve G(C). Their semi-log plot of particle number versus

size showed some curvature. They agreed that for small sizes,

particularly below 15pm, overall growth was strongly size

dependent. Khambaty and Larson (1978) in their experiment

with magnesium sulfate heptahydrate crystals obtained a

curved line on a semi-log plot of particle number versus size











and attributed the curvature to size dependency of the

small crystals. They ignored the birth function in the

population balance despite the fact that there was crystal

generation by contact. They also invoked the'zero size

assumption. Rousseau and Parks (1981) similarly studied

size dependency of the growth rate of magnesium sulphate

heptahydrate crystal and obtained some curvature in the

semi-log plot of particle number versus size. They

similarly concluded that growth rate was strongly dependent

on crystal size for the 44 to 10OOm size measured in the

class II MSMPR crystallizer. The size dependent growth

rate was obtained via the two parameter reduced version of

the Abegg-Stevens-Larson (ASL) model given in equations (2-10)

and (2-11):


G = go(l+yL)c ; c

y = 1/(GT) (2-11)


The parameter, G, is the growth rate of nuclei and the

parameters y and c are simply constants.

The purpose of studying nucleation and growth in crystal-

lizers is to be able to control crystal size distribution in

these crystallizers, which are often plagued by oscillatory

crystal size distribution dynamics. The associated dynamic

behavior of crystallizers presents serious industrial

problems.











Dynamics and Stability


Many investigators have dealt with the dynamic behavior

of crystallizers. Some have addressed problems associated

with class I crystallizers while others have directed

attention to class II systems. In addition, some of the

works have approached crystallizer instability from the

nucleation-growth-kinetics viewpoint while others have taken

the viewpoint that the mode of operation is the primary

cause of crystallizer instability.

Cycling of crystal size distribution has been reported

in class I crystallizers. Bennett (1962) observed cycling

with an industrial crystallizer with a 15-hour period and

a large amplitude swing of particle weight on a 12 SSM screen.

Song and Douglas (1975) designed a laboratory scale sodium

shloride crystallizer in which they produced oscillatory out-

puts using constant inputs. The experimental values of the

oscillatory output agreed slightly with their theoretical

predictions based on steady state measurements of growth

rate and nucleation rate kinetic parameters of an isothermal

MSMPR class I system. Only one cycle was observed in the

experiment. Their theoretical work for this system is

described in an earlier paper.

Sherwin, Shinnar and Katz (1967) carried out a theo-

retical investigation of the effect of feed seeding and

fines dissolving and recycling on the stability of a class

I system. These authors have expressed their stability

criterion in terms of ratio b/g which they defined as











dB
dc1-
b/g i (2-12)
dG



In equation (2-12) "bar" represents steady state values; G

is a linear function of supersaturation,


G = kl(c-cs) (2-13)


In equation (2-13) k1 is a constant. The quantity B was

approximated with Volmer's nucleation model defined earlier

in equation (1-2), and in which the quantity (c/c 1) is

assumed very small. By expanding B up to first order in

c/cs and utilizing the above assumption, equation (2-12) can

be written as


2K2
b/g 2- = i (2-14)
( 1)
cs


For a clear feed, it was found that the system becomes un-

stable when b/g > 21. Seeded feed was found to improve

stability strongly while volume fraction (void fraction) of

solution had an insignificant effect. Another observation

made was that the system stability increases significantly

if nuclei had a finite size and that the region of stable

operation enlarges with increasing nuclei size. The kinetic

order of nucleation rate to growth rate played a significant

role in determining the system stability.











Sherwin et al. (1969) were the first to uncover the

extreme importance of product classification in causing

crystallizer instability, in spite of the fact that their

model was too idealized to be of practical value for

engineering simulation. It was assumed in the model that

as soon as a particle reached product size it was instantly

removed thus producing mono-sized product crystals. It was

found that extreme classification enhanced the sensitivity

of the crystallizer to disturbances and increased the

tendency toward cyclic fluctuations. Hulbert and Stenfango

(1969) investigated a double draw-off crystallizer by

assuming size independent growth and birth rates, zero size

nuclei and Volmer's nucleation model. These assumptions

resulted in a coupled nonlinear population and mass balance

equations which were numerically solved. They found that

one of the draw-off streams, the clear liquor overflow,

tended to increase particle size and crystal solid content

due to the longer particle retention time. As Bennett and

Van Buren (1969) had already pointed out, increasing

particle retention time does not always result in larger

product crystals. In addition, the clear liquor overflow

seemed to enhance the system stability. Another conclusion

reached by Hulburt and Stefango was that increased feed

seeding and increased seed size in feed stream enlarged the

stability region for high growth conditions.

Nyvlt and Mullin (1970) observed damped oscillatory

behavior while experimenting with a 200-liter draft-tube











crystallizer fitted with an elutriating leg and containing

sodium thiosulfate as the specimen under investigation.

The data obtained were too scanty to conclusively demonstrate

sustained limit cycle behavior.

Lei et al. (1971a) carried out a theoretical study of

the stability and dynamic behavior of a continuous crystal-

lizer equipped with a fines trap. The fines had a fixed size.

Nucleation was assumed to occur at a fixed size and growth

rate was assumed independent of size. Their study showed

that cycling in crystallizers can be reduced or eliminated

by adjusting the operation of a fines trap, especially when

the size of the fines is increased slightly. They also noted

that fines dissolution and recycle do not always stabilize

a crystallizer system, and can, in fact, destabilize the sys-

tem depending on the conditions of operation.

Using perturbation methods Yu and Douglas (1975) carried

out a theoretical investigation of the stability of a class

I MSMPR crystallizer, and from which they concluded that,

in some cases, oscillatory behavior can produce yields that

exceed the predicted steady state value. The study enter-

tained many assumptions in the derivation of the balance

equations and process parameters used do not correspond to

typical industrial conditions.

Attention has been directed to class II crystallizers

by some investigators. Oscillatory behavior has been observed

in some class II systems. Miller and Saeman (1947) observed

oscillatory behavior of the crystal size distribution while











working with and industrial ammonium nitrate producing

crystallizer (an Oslo type crystallizer). Kerr-McGee (in

Beckman, 1976) observed cyclic fluctuations in an industrial

potassium chloride crystallizer as shown in Figure 1. The

severity of the fluctuations caused continuous rippling of

the weight distribution of product crystals. Cycling behaviors

have been observed in a continuous polymerization process by

Finn and Wilson (1954) and in a fermentation process by

Thomas and Mallison (1961).

Randolph and Larson (1965) investigated the problem of

stability in a class II MSMPR. The dynamic population

balance equation (2-15) in conjunction with boundary condi-

tion equation (2-16) and initial condition (2-17) were trans-

formed into moment equations, which were then solved on an

analog computer.

an Gan n
3n + = 0 (2-15)



n(=0) = KNGi (2-16)


n() = n(0) exp(-C/T) (2-17)

The coefficient, KN, of Gi is a constant and "T" is the

crystallizer residence time. All other symbols in equation

(2-15) are as defined in previous sections. The bar re-

presents steady state values. Step changes in production

rate and in nuclei dissolving rate were introduced into the

system, and the resulting disturbances in the zeroth,



























0









CCO








cd
r*H



0c
40

4-) *r





OQ













CO
4l-










i-1 w
a^s
+-

(3 F-
C 0


SCoO 0 0 0 0 0 0
N3 NO T a3N
N33HOS NO .3NIV.136 LN363ad LH913M











first, and second moments were plotted as functions of

time. The plots showed cyclic fluctuations.

Randolph and Larson (1971) modeled a class II MSMPR

crystallizer with equation (2-15) which was first trans-

formed into a closed set of moment equations in time. The

characteristic equation of this closed set of ordinary

differential equations was obtained via Laplace trans-

formation technique. Routh-Horwitz method was used to

establish stability criterion which guaranteed stability

of a crystallizer obeying equation (2-15), whenever the

slope of the log-log plot of nucleation rate versus growth

rate was less than 21. Randolph and Larson (1969) included

product classification and fines dissolution and recycle in

their crystallization. They concluded that the net effect

of fines dissolution and recycle is to force growth rate

to a higher level, producing the same production on larger

average size crystals having less total area. They noted

that if the higher supersaturation produced from the fines

dissolution and recycle resulted in a greater than propor-

tionate increase in nucleation rate (as would be the case

if nucleaction rate were a stronger function of supersatura-

tion than growth rate), then size improvement would be some-

what less than expected. They also noted that this internal

feedback (that is, an increase in supersaturation causing

an increase in nucleation rate) might affect the system in

such a way that disturbances to the system might not damp

out, thereby causing sustained oscillation of the crystal











size distribution. They further remarked that imperfect

classification of larger product crystals is almost always

accompanied by a reduction in average size, unless nuclea-

tion is independently controlled. Using the concept of a

classification function, Randolph et al. (1973) modeled

an R-Z crystallizer and showed the stability of such a

crystallizer to be a function of "i," fines dissolution and

recycle, product classification size, L and normalized

product classification rate, z. Beckman (1976) experimentally

demonstrated sustained limit cycle behavior in an R-Z type

crystallizer equipped with clear liquor advance and product

classification. He then solved the population balance equa-

tion (with a perturbed steady state solution as initial condi-

tion) via Laplace transform technique to obtain the dynamic

behavior of the crystallizer. He then compared his semianalytic

solution with a regular-falsi method of numerical solution.

Saeman (1956) in his study of the simplest continuous

MSMPR crystallizer noted that regardless of the classification

device used, a crystallizer cannot put out large crystals

unless conditions which are conducive to growth of large

crystals are maintained in the suspension. They also re-

marked that positive and direct means for size control lie

largely in the provision of effective means of segregation

and elimination of excess fines. He recommended that fines

should be eliminated at an average age that is less than or

equal to one-tenth that of the product crystals, and that

whatever the classification device used for removing fines,











it must not inadvertently overload with large intermediate

size crystals. Saeman further showed that classification

could yield a narrow distribution as well as one with

relatively larger dominant size.

Cohen and Keener (1975) used a multiple time scale

perturbation technique on a third order system of nonlinear

ordinary differential equations to predict the bifurcation

of time periodic solution of a class II MSMPR crystallizer.

Their model is essentially represented by equations (2-18),

(2-19) and (2-20).


3n + 1 an + n = 0 >0, t>O (2-18)
S20 n(C,t)d 2(



n(0,t) = [ t > 0 (2-19)
[c0 n(C,t)dC]1


n(S,0) = f(C) = e-5 (2-20)


The function, f(C), is the initial steady state distribution

of the CSD. By defining the K-th moment as in equation (2-21),

they transformed their equations into a set of nonlinear

ordinary differential equations of the moments.



m(t) = k- n(,t)dC (2-21)
k 0


In order to study their moment equations they investigated

a general system of nonlinear ordinary differential equa-

tions represented by equation (2-22):










dX = PX + 2AX + g(X) (2-22)



Here X is a three-component vector, P and A are constant matri-

ces, and g(X) is a smooth nonlinear vector function con-

taining no linear terms near the equilibrium point. In

other words, derivatives with respect to each component of

X varnish at the equilibrium point. In addition, the func-

tion g itself varnishes at the equilibrium point. If we re-

present XT = (x,y,z), then this condition is represented by

equation (2-23).


gi(0,0,0) = gix(0,0,0) = giy(0,0,0) = giz(0,0,0) = 0

i = 1,2,3 (2-23)


The parameter, E, is very small (0<<<1). The subscripts

x,y,z represent first order partial with respect to x, y,

and z respectively. By casting the nonlinear ordinary dif-

ferential equations of the moment into the form of equation

(2-22), they were able to show oscillation in the crystal

size distribution,n, and the growth rate, G, defined by

equation (2-24).



G(t) = 1/i( 2n( ,t)dl (2-24)



Their analysis predicted both amplitude and period of oscil-

lation. The authors noted that including the mass balance

would present a formidable task. In addition, they eliminated

the birth function from the population balance, equation











(2-18), and by representing it as a size independent func-

tion and assuming a power law type of nucleation-growth

kinetics, they included it as a boundary condition. Because

of the many assumptions made their whole analysis is

questionable.

Nyvlt and Mullin (1970) carried out a numerical study

of a class II MSMPR crystallizer equipped with an elutriator

via a Monte Carlo simulation technique. They used their

study to explain experimental data obtained from a pilot

plant. They concluded that crystallizers. with or without

product classification can exhibit periodic changes in pro-

duction rate, product crystal size, supersaturation, magma

density and other related parameters, and in some cases,

the steady state may not be reached. The cyclic period,

which is comparatively long in most cases, depends on the

supersaturation rate and on the fraction of product crystals

removed during classification. They noted that the stability

of the system increases with increasing growth rate, in-

creasing magma density, decreasing nucleation order, i,

decreasing minimum product size, and decreasing quantity of

crystals withdrawn per unit time. They also remarked that

seeding would have the same effect as decreasing the effec-

tive nucleation rate and should lead to a stabilization of

the system.

Good stability criteria is a prerequisite for an effective

control scheme for dynamic crystallizers.











Control


Publications in control of crystallizer dynamics are very

few. A good predictive model in terms of good stability

criteria is a prerequisite for effective control. Simple

proportional control ofsome crystallizer variable has

been used by most of the investigators.

Bollinger and Lamb (1962) and Luyben and Lamb (1963)

have outlined the basic design of feed forward control.

Han (1967) investigated a feed forward control of class I

MSMPR crystallizers by using feed concentration as the

disturbance, supersaturation as the controlled variable and

flow rate as the manipulated variable. By controlling the

supersaturation, he intended to control production rate.

Han's control scheme worked better when the system was

operated in the stable instead of the unstable region.of

birth-growth kinetics.

Lei, Shinnar and Katz (1971b) investigated a feedback

control of a class I MSMPR crystallizer equipped with a

fines trap. Because of its relatively easy accessibility

by light transmission measurement, the total surface area

of fines in the fines trap was used as the controlled

variable, with flow rate through the crystallizer as the

manipulated variable. This scheme showed good control

even when the crystallizer was operated in the unstable

region. However, attempts to use fines recirculation rate

(that is, the amount of fines destroyed) as the manipulated

variable did not readily stabilize an unstable operation.











The above results were derived from a linearized stability

analysis of a crystallizer equipped with point fines trap

and in which a simple proportional control was incorporated.

Timmand Gupta (1970) investigated a feedforward/feed-

back control of a class II MSMPR by using flow rate out of

the crystallizer as the disturbance, seeding and fines

destruction and recirculation rate as manipulated variables

and cumulative number of crystals as the controlled variable.

The control scheme monitoring the cumulative number of

crystals was demonstrated to be superior to the one monitoring

the total area of crystals within the crystallizer. The

control scheme with total area of suspended crystal magma

as the controlled variable was worse than no control. The

simple proportional control was capable of stabilizing the

system within an inherently unstable region. The feedforward/

feedback scheme showed remarkable improvement over conven-

tional feedback control.

Beckman (1976), using proportional control of nuclei

density with fines destruction and recirculation rate as the

manipulated variable, studied the control of a class I MSMPR

crystallizer equipped with fines destruction, clear liquor

advance and product classification. He theoretically pre-

dicted the proportional control constant necessary for

stability. The system was amenable to control. From a com-

puter simulation utilizing experimentally determined nuclea-

tion and process parameters, he obtained a value for the pro-

portional control constant which agreed with that predicted

theoretically by his model. Beckman assumed nuclei were of








32


zero size and growth rate was size independent. By

neglecting the birth function in the population balance,

he included it as a size independent function in the

boundary condition.

















.CHAPTER III
THEORY AND METHODOLOGY


As pointed out in earlier chapters, previous investiga-

tors encountered tremendous difficulty in their endeavor to

simultaneously quantify the two fundamental quantities,

growth and nucleation rates for all sizes. Apparently it

was realized by these researchers that the most important

tool towards understanding crystallization phenomenon in

steady state crystallizers is the ability to quantify growth

and nucleation rates. The MSMPR technique pioneered by

Randolph and Larson (1962, 1971) was not capable of explain-

ing experimental steady state data for the small size range

where most of the nucleation occurs. It was designed for

the large size region where growth rate is independent of

size and for the size region where particle birth is

negligible. Youngquist and Randolph (1972) in their work

with a class II system remarked that a rigorous quantitative

definition of and use of secondary nucleation kinetics must

await quantitative separation of size dependent growth and

birth rates G(C), b(i) in the small size range. They said

that their work indicated the near impossibility of quanti-

tatively characterizing secondary nucleation using the MSMPR

technique as well as casting doubt on the general applica-

bility of gross secondary nucleation so obtained. It is now











obvious that empirical correlations involving crystallizer

process parameters and environment must await the rigorous

quantification of growth and nucleation rates.

It is the unraveling of the simultaneous quantification

of growth and nucleation rates to which the next section is

devoted.



Growth and Birth Rates in Steady State Crystallizer


Consider the following crystal population balance equa-

tion (3-1), the derivation of which is included in detail

in the appendix:

1 3 3(Gn)
V -t (Vn) + b + (nQi n Q ) (3-1)


where

t = time

n = number of crystals per unit volume per unit size

in crystallizer, #/MZ-Im)

= crystal characteristic size, um

G = crystal growth rate, um/min

b = net birth rate of crystal nuclei, #/(m-jim-min)

n. = number of crystal seeds in inlet (feed) stream
1
per unit volume per unit size, #/mi-pm)

no = number of crystals in outlet stream per unit

volume per unit size, #/(mi.-m)

Qi = inlet volumetric flow rate, (mi/min)

Qo = outlet volumetric flow rate, (mi/min).











For a constant volume, well mixed, steady state

crystallizer with no seeds in inlet feed,the above equation

reduces to equation (3-2).:


a 1
(Gn) = b(S) n( ) (3-2)


V
where T = is the average crystal residence time. Most
Qo
researchers usually assume the McCabe AL Law to hold and

take growth rate, G, to be indendent of size and in addition

assume zero birth rate even in situations where secondary

nuclei are being generated by contacting the seed crystal.

When these assumptions are made, equation (3-3) is obtained;


Gnn n
-G5 + T = (3-3)



The solution to equation (3-3) is given by equation (3-4):

n(0) = n(Lo)exp(-(C-L )/(GT)) (3-4)


Further they assume nuclei are born at size zero (that is

L =0) so that equation (3-4) becomes



n(C) = noexp(C/(GT)) (3-5)


where n is an abbreviation for n(0).

A curved population balance on a semilog plot of n

versus size is of the form shown in Figure 2. Since equa-

tion (3-3) is not capable of explaining experimental data

of the form represented in Figure 2, it suggests that one













t



Z

;-4
0






Figure 2.


Lf size +

Plot of Population Density Versus Size.

+


Figure 3. Plot of N Versus X.


c=L- size -


Figure 5. Growth Rate Versus
Size.


G(L1)

Figure 4. Plot of Total
Chi Square Error Versus
G(L1) Values.

















size -

Figure 6. Birth Rate
Versus Size.











should work with the more complete equation (3-2) for the

steady state crystallizer.

Before proceeding, it is necessary to define a critical

size, L ,as that at which the growth rate is zero (that is

G(Lc)=0). This definition is motivated by the fact that

the growth rate, G, is a monotone increasing function of

size in the interval, I = (0,c). The shape of it is well

documented. Integrating equation (3-2) with respect to

size and applying the condition in equation (3-6) yields

equation (3-7).


G(Lc) = 0 (3-6)



G(C)n() = b(A)dA n(A)dA (3-7)
L L
c c


The shapes or functional forms of the functions b(C) and

n() are very well established from experimental as well

as theoretical considerations. The shapes of these func-

tions are exponential. For this reason, these functions

are represented by equations (3-8) and (3-9).

-ac2(5-L )
b(C) = ale ) (3-8)


n(i) = clexp(-yl,) + c2exp(-y2C) (3-9)


The parameters al, a2' cl, c2' Y1, and Y2 are constants.

Usually the parameters cl, c2, y1, and y2 can be determined

from a nonlinear least squares fit of number size











distribution data. Substitution of equations (3-8) and

(3-9) into equation (3-7) gives



G(C)n() = c exp(-a2(A-L ))dA L [clexp(-yIA)
C c

+ c2exp(-y2A)]dA (3-10)


We make the following definitions:


Fi = G(i)n(Ci) i = 1,2,...,n (3-11)




FA = lexp(-a2(A-L ))dA- [clexp(-ylA)
Ai fL LT L
c C

+ c2exp(-y2A)]dA ,

i = 1,2,...,n (3-12)


Experimental data are always reported for the size interval,

I = [Lw,Lmax], where L, and Lmax denote the respective

lowest and maximum experimentally measured sizes. The size

denoted as L in Figure 2 is that beyond which growth rate

is constant as indicated by the straight line portion of the

plot in the interval I = [L,Lmax]. It is for this same

size interval that the MSMPR technique usually applies.

Once n(C) is obtained by fitting experimental data as

represented by equation (3-9), it is used to interpolate for

more values so that n(.i) is available in much finer par-

tition; in other words, the interval, [L ,L ma], is sub-

divided into much finer partitions than the original











experimental data. The only unknown values al, a2, and L

can be obtained by minimizing the sum, S, defined by equa-

tion (3-13) over al, a2, and L .


N 2
S= (Fi--FAi) i (3-13)
i=l


Another version constitutes confining the minimization to

the interval, 12 = [L*,Lmax], for which Fi is known for
2 f maJ 1
each "i." The minimization procedure is carried out via

an appropriate nonlinear least square method. The determina-

tion of al, a2, and Le immediately yields the function,

b() and G(i) can be calculated from equation (3-10). The

integration in equation (3-10) are to be carried out

analytically. This concludes the analysis for the quantifica-

tion of the two fundamental quantities, G() and b(). It

should be noted that no assumption was made in the popula-

tion balance equation. The only approximation made is in

the minimization procedures. This approximation is in

general negligible since minimization procedures usually
-6
have built-in tolerances, some smaller than 106, that must

be satisfied before convergence is attained. I proceed to

the determination of G(C) and b(C) for the unsteady state

crystallizer.











Growth and Birth Rates in Unsteady State Crystallizer


The difficulty in quantifying the two fundamental

quantities in an unsteady state crystallizer is compounded

by the transient nature, which is an added dimension to

the steady state case. Randolph and Cise (1972) attested

to this difficulty in their experiment with an unsteady

state Class II crystallizer involving potassium sulphate.

They remarked that it was patently impossible to uniquely

specify both size dependent functions G(C) and b(C) using

only the single size dependent measurement n(C), without

further simplifying assumptions or other independent

measurements. Lee (1978) was the first to conceptualize

a method of attack that set the direction for determining

the two fundamental quantities in his study of the Single

Seeded Batch Crystallizer (SSBCR). The following study

will proceed along the same lines of thought.

Equation (3-1) shall form the basis of our analysis

for a constant volume, well mixed crystallizer with seeds

in inlet feed; equation (3-1) reduces to equation (3-14)

n.
n(C,t) + -[G(C)n(C,t)] = b(c) + -
Tt Ti T

(3-14)

where T. is the residence time of seed crystals in the feed.

Despite the fact that it is easier to start integration

of equation (3-14) at t=O and at =0 because of initial

and boundary conditions expressed by equations (3-15) and











(3-16), practical considerations dictate that we use the

lowest experimentally measured time, tw and size, Lw .


n(0,0) = 0


n(O,t) = 0 = Uim n(C,t)


G(O) = 0


(3-15)


(3-16)


First integrating with respect to time yields


t
n(C,tm) n(C,t ) + -- G() m n(C,T)dt = (tm-tw)b(C)
w

t t
1 1 rm
n(lT)dT + -1 n(C,T)dT.
T t i t
w w


A second integration with respect to size results in the

following expression:

Lk Lk tm
k n(,tm)d f -I n(C,tw)d; + G(Lk) n(Lk,T)dT
L L t
w w w


t
m
-G(L ) tm
w

Lk t
1 Lw w
w w


L
n(Lw,T)dT = (tm-t )
Lw


S Lk
n(,T)dTdi + 1-
i Lw


b( )dC


ni(C,T)dTdg.


After rearrangement and denoting tw and Lw by tl and L1

respectively, the following equation is obtained:







42

N(Lk'tm) = -G(Lk)X(Lktm) + B(Lk)


Lk
B =
L1


b( )dg


t
S1 ft m
= -- n(L T)dT

Lk Lk
L n(C,tm)d I n(C,tl)dC t
L1 1
= t-t t -tl--- G(L


Lk t
T(t -t l L1 t1 n(t ,T)dTdC

L t

T (t-i L n(,T)dTdc.
Ti~m-I)1 tlI1


(3-17b)




n(Ll, )dT


(3-17c)


Cognizant of the discrete nature of experimental data,

we let m = 2,3,...,M and k = 2,3,W. For each value of k,

N and X are M-dimensional vectors. A plot of equation (3-17)

in the form N(Lk,tm) versus X(Lk,tm) for each value of k

results in a straight line with a slope equal to -G(Lk) and

intercept, B(Lk), provided N(Lk,tm) and X(Lk,tm) can be

completely determined. Thus G and B can be generated at

corresponding sizes, Lk 's by repeating the above plot for

various values of k. Finally, G and B can each be plotted

against size.


where


(3-17)


(3-17a)











If we had started the integration at zero size (that is

L1=0), G(L1) would be zero and N would be completely

determined without much ado. Unfortunately N is yet un-

determined because G(L1)f 0. However, by following the

procedure enumerated below, N and hence G and B can be

determined.

Step 1. First fit particle size data via least squares

regression to obtain a functional form for n(,t). One

does not need to perform a fitting for ni.(,t) since this is

usually given.

Step 2. Calculate the various integrals that make up N

and X analytically, if possible.

Step 3. Because N cannot be calculated until G(L1)isknown,

guess a value for G(L1) and calculate N and X. For each

value of k>2, plot N(Lk,tm) versus X(Lk,tm). For example,

data for k=2 would be as shown below.


Data for k=2


N(L2,t2) X(L2,t2)

N(L2,t3) X(L2,t3)

N(L2,t4) X(L2,t4)

N(L2,t5) X(L2,t5)


N(L2,tM) X(L2,tM)


A plot of N versus X data is illustrated in Figure 2 for

the case k=2. The plot in Figure 3 is typical since every











value of k yields a straight line with negative slope and

positive intercept. An alternative approach is performing

a least squares linear regression on the data to determine

the slope and the intercept. For a given G(L1) record the

sum of square error or the chi-square error for each k and

the total chi-square error for all k's. Total chi-square
S2 2
error = I Xk where Xk is the chi-square error for a
k=2
particular value of k. For perfect data the coefficient

of determination of the fit for each k is one; in other

words, the chi-square error is zero.

Step 4. Repeat step 3 for different guesses of G(L1).

Step 5. Plot total chi-square error against G(L1)

values and record the G(L1) corresponding to the lowest

chi-square error. The value of G(L1) so obtained is more

accurate than other values of G(L1). A plot of total chi-

square error versus G(L1) values might look like that in

Figure 4.

Step 6. Using the value of G(L1) corresponding to the

minimum error, plot B(C) and G(C) each against C, where

takes discrete values, Lk, k = 2,3,...,W. Obtain G(Lc) by

extrapolation. Typical plots for B(C) and G(C) might look

like those in FiguIrs 5 and 6.

The first guess in step 3 can be generated by minimizing

equation (3-18), a rearranged version of equation (3-17),

over unknowns, G(L1), G(L2) and B(L2) when k=2.


N(L2,t ) + G(L2)X(L2,tm) B(L2) = 0, m = 2,3,...,M
(3-18)











For a crystallizer with no seeds in inlet feed equation

(3-17) is still valid with N no longer containing the term

in n.(C,T). Similarly,for a batch crystallizer N no longer

contains the term in n. as well as the double integral of
1
n(C,T). Thus for the batch crystallizer N is represented

by equation (3-19):
Lk Lk
n(,t m)d n(c,t )dc t
1 1
= G(L) n(L1,T)dm
t -t- t -t 1 l ^
(3-19)


Every other term in equation (3-17) remains the same.

This concludes our analysis for the determination of

growth and birth functions in an unsteady state crystallizer.

It should be noted that no assumptions were made in the above

analysis. The only approximations made were in the minimiza-

tion procedure used for data treatment.

The methods treated above for the determination of growth

and birth functions in steady and unsteady state crystallizers

also allow for the determination of these functions to be made

in terms of process conditions, such as supersaturation,

crystallizer impeller speed, etcetera. The growth and birth

functions are simply determined for each fixed value of a

process condition, and correlations could be obtained, if

desired. Good empirical correlations can only come after

an accurate method of quantification of growth and birth

rates has been established.









46













u-


rC



i1Y "? 1


3
c;
N
-rcu w
c) 1 L.il,,-L --------?
CU
C'
r


0


.Lt-


I
i

j I




I
I

j

j

I
I
I


i I /


I


~I
~9 (D
c Q ~I

L--C
- -- _. _i
--r _
--








47




























MII+
+ N













Z -4
-4-,



















__-_ *S-,
1I -4













"0
L 0 0 .
O 0
0 0 T-








4 -H 4-
S 0 0
II J -d



HH .,.q


4--) 4-) 0

C -I
i ^ (A*
=~~~~~ ~ J^FLrrqT .^^








48


A stability criteria would not be very practical unless

a method for the accurate quantification of growth and birth

rates was available. This is so because the very equations

from which stability criteria are derived contain the two

fundamental quantities, growth and birth rates.



Dynamics and Stability of the Complex Crystallizer


The problem associated with obtaining a good stability

criteria is much more complex mathematically than that

associated with the quantification of growth and birth

rates. Because of the associated mathematical intricacy

of crystallizers' stability problems, previous researchers

have oftentimes found it necessary to make oversimplifying

assumptions, thereby making the criteria so obtained of

less practical value.

The crystallizer under study is very similar to the R-Z

crystallizer studied by Randolph, Beer and Keener (1973),

and will be referred to as "the complex crystallizer." The

complex crystallizer, the schematic diagram of which is

shown in Figure 7, is that which is equipped with fines

destruction and recycle, seeding, clear liquor advance, and

product classification. It is obvious from Figure 7 that,

in addition to the population balance equation, other

material balance equations are required to completely

establish the balance equations of the complex crystallizer.

Because the conditions are isothermal and there is no











significant energy changes in the system, the balance

equations do not include an energy balance equation.



Complex Crystallizer Balance Equations


The various system boundaries, A, B, D, E, in Figure

7 are those around which material balances are made. The

flows P. and Qi (i = 0,1,2,...,6) represent solid and liquid

flow rates respectively. The classification functions,

h.(C),are size dependent removal functions representing the

ratio of crystals removed in a certain size range during

classification to that expected with ordinary mixed with-

drawal. The classification functions, hi.(),can also be

viewed as removal probability functions. As an example,

a classification function h1(5) is defined by equation (3-20).


R-l (O,Lf)
h1() = (3-20)
0 other sizes


Equation (3-20) indicates that crystals in the size range

(O,Lf) have a probability of removal that is (R-l) times

that in a mixed withdrawal case while the probability of

removal of crystals larger than size Lf is zero. The

functions h.(C) are represented below:

R-l [O,Lf]
hl() = (3-21)
0 otherwise


for all sizes


(3-22)


h2(0) = 0 ,











h3() = h1(?) (3-23)


1 [O,Lf]

h4() = a (Lf,L (3-24)

z (Lp O)


1-6 [O,Lf]

h5(c) = a (Lf'Lp] (3-25)

z(l-8) (L p,)


where 0 < e < 1

0 < 8 < 1




6 [0,Lf]

h6(o) = 0 (Lf'Lp] (3-26)

z6 (Lp' 0)


In the above equations Lf and L represent maximum sizes

for fines and intermediate size product. Sizes larger

than L represent the oversize product. The advantage of

the classification functions is that differences in crystal

distribution are completely accounted for, thereby eliminating

the need to separately indicate the different distributions,

ni's. It would be desirable that the volume on which the

birth function and crystal size distribution (CSD) is

based remains constant, even during transient operation.

To achieve this it is necessary to define the void fraction










within the crystallizer, Ec, in order to reexpress all

volume based quantities on a slurry volume basis. The

slurry volume remains constant during transient operations.

The void fraction, ec, is defined as the ratio of clear

liquor volume, VL, to slurry volume, VS.


c = VL/VS (3-27)


It suffices to specify four of the seven volumetric flow

rates in Figure 7. From Figure 7 the following relation-

ships among the various flow rates can be established:


Qo = Q3 + Q5 (3-28)

Q1 = Q3 + Q2 (3-29)

Qo = Q1 Q2 + 5 (3-30)

Q4 = 5 + Q6 (3-31)

Q = Ql Q2 + 4 Q6 (3-32)


The flow rates Q Q1' Q2' and Q5 are feed, fines removal,

dissolved fines recycle and product volumetric flow rates,

respectively. The clear liquor volumetric flow rate is

represented by Q3.

Applying the general population balance derived in the

appendix to Figure 7, the following particle number balance

equation results:


-T (Vsn) + (GVsn) = VBEc + -Qhl( 4h4

+ Q6h(C)ln(C) + Q no()










Since Vs is constant the above equation reduces to



-n + a (Gn) = BE: + 1_ + Q6 h (,t)
-+- (G) = B~c --V h1-h h h6(c) n( ,t)
s s s

Qo
+ no(W (3-33)
s

Let

Ti = V /Qi (3-34)

Ti = ai 5. (3-35)


By combining equations (3-33), (3-34) and (3-35), the

following equation results:


an + r -1 1 1
S+ (Gn) = BE + hl(C) h() + h6(') n(r)
t c 1T5 a4 t5 b+6 T5 j

+ n (3-36)
aOT5


The terms of equation (3-36) represent, in order, the

accumulation of crystals at size C, the net flux of crystals

away from size C due to growth, the input of particles due

to nucleation, the input or withdrawal of particles as a

result of classification, and the input of particles due to


solids in feed.











Solute and Solid Balance


As mentioned earlier the solute and solid balance is

required as it complements the crystal population balance

equation, both of which, together, form the governing equa-

tion for the dynamics of the complex crystallizer. The

solute and solid balance is derived by setting all inputs

in both dissolved and solid form equal to all outputs plus

accumulation.

A solute balance around boundary E of Figure 7 results

in the following expression:


dt (Vs cC) = QoC + QC [C+Q5] (k/T)Vs (,t)d
0
(3-37)


where T4 is taken to be the average residence time of

crystals in crystallizers. The quantities, Ci, i = 0,2,

denote the supersaturation of the i-th stream. The volu-

metric shape factor, crystal density and the supersaturation

of the well mixed suspension inside the crystallizer are

denoted by k p, and C, respectively.

Combining equations (3-35) and (3-37) yields the following

equation after rearrangement:


dE -E C C kp
c c dc +o + C2 r-1 rvp) v 3)
S d+ + + n(Ct)3 d
dt C dt ToC C2 T 1 5J J o0
(3-38)


A solute plus solid balance around boundary E of Figure 7

yields












dCd
V c dt = Vs(c-p) dt + QoCo + Q2C2 + (kvVs/T o)
Vs Ca- dt000


I1


n (c)3 dC


- Q1C + Q5C + (kvVsp/T5) h5(C)n(C,t) 3f


+ (k V P/T1) hl(C)n(C,t)C3d


where crystals in the feed have sizes in the range 1 to C2'

Upon rearrangement the above equation yields


dE
dc -(c-p) c
dt Ec dt


C
+ 0
T c


C
2
+ +
2EC


(k P/To) ) 2

Ec ~


n o()C ds
O


h5(C)n(C,t) C3 d
0


S(kv/T5)
E


(k P/T ) f 31
+ (kv hl(s)n(s,t)c3d
c 0 1

de
Substituting the expression for dT

in the following equation:


(3-39)


in -(c-p)
in c
c


de
results
dt


dE
c
dt


S-(c-p)
E
c


c dC +
C dt T-r
o


C2
+C2


1 1
-+ +-
T T5
1 5


[ O n(Ct) 3de



Upon rearrangement, the following equation results:


-c-p) c (c-p) dC
cc dt C dt


k p(c-p)
EC T4C


(c-p)Co
cCTo0


(c-p)C2 1(c-p) 1
Ec 2 c (1


n(c,t) 3dg
0


C +
1 c


C
T5:c


-(c-p)
E
C


1










Substituting the above equation into equation (3-39) yields


dC (c) dC (-p)Co (c-p)C2 (c-p) 1 ,1
dt c dt E CT E cCT2 E+ T

k p(c-p) n(, 3 C C
SC n( T,t)S o T +
Ec 4 0o


+ C2 n (W)3dC -c + C
E T c 5 c
1


(k P/T5)
c


3 (k vp/ l)
h (C)n(C,t)C3dI +
0 c


h0 l()n( Ct)3d


which after a slight algebraic manipulation becomes


p dC o C2 pL f + 1 1
(cdt ECo + 2
dt E CT E CT E Tl 5
co c2c 15


+ (kvP/o) I2
c 51


k p(c-p) 3
+ ETC 4 n(,t)3 d
c 4 0


3 kvP/T5
no0 de E c
c


h0 h5()n( r t). 3d3
f0c


k= VEP/ hl()n(C,t)3d .C
c 0

dC
Solving the above equation for d- results in equation (3-40).


dC o
-- +
dt E T
c 0


C2 C f kv(c-p)
C2 C 1 + 1 cT4
ECT2 c 1 5 c4


kvC n
n (O)C3d h5(C)n(C,t)C 3d
0 c T5 0


Shl()n(c,t) 3d
0


(3-40)


n(c,t)c3dC


k C
+ v
E T
C 0

k C
v
Ec T










Substituting equation (3-35) into (3-40) yields equation

(3-41):


dC Co C2 C 1 11 k (-) 3
dt aT + + n( X,t) dT
dt Ecao 5 c2 T5 Ec 15 5 c4T 5 0

k C 2 k C
+C n(O) i EC5- 0 h5(C)n(c,t)C3dC
Ec o o 0 c T 5 -0

kvC m 3
S hl( )n( ,t) dC (3-41)
Ec T10 1


Combining equations (3-33), (3-34) and (3-35) results in

equation (3-42):


n + (Gn) = B + hl() h4(c) + 1 h6( n(C,t)
1t U ( 5 BX a4 5 a6 5

+ n ( ) (3-42)
a oT5 o


This completes the derivation of the balance equations

(3-41) and (3-42) of the complex crystallizer. It must be

noted that there were no assumptions in the above derivation.


Dynamics and Stability


Most of the stability criteria for crystallizers have

been derived by making oversimplifying assumptions in the

balance equations because of the mathematical complexity.

Previous authors have found it very difficult if not im-

possible to deal with the size dependence of the birth and












growth functions. For this reason, they often assume that

birth occurs at one size and describe this as the "lumped

birth function." Moreover, this size (at which birth

occurs) is usually taken as the zero size. The criteria

derived with these assumptions, therefore, have little or

no practical value. In order to carry out a thorough

analysis one must use more complete equations such as (3-41)

and (3-42) as the basic equations to which stability methods

are applied. We shall apply our stability method to equa-

tions (3-41a) and (3-42), which are recapitulated below for

easy reference. Equation (3-41a) is a slightly modified

version of equation (3-41).



dC k_ (c-)
dC + C(l+r) 1 kv(p n(,t)
dt Ecoa T5 C25 Ec al5 5 45

kC 2 kC 3
+ k v C2n (C)C3dC -c5 0C h5()n(C,t)C3dC
Ecao 5 (1 oC T5


k C 3
-CT I hl(S)n(s,t) 3d( (3-41a)


1 1 1n(t)
+ (Gn) = BE + hl(C) h4(C) + h6(, n(,t)
t B c al 4) + a6 T5 6T6 5

+ no(() (3-42)


Equation (3-41a) was obtained by combining equations (3-41)

and (3-43).











C2 = (l+r)C 0 < r < 1 (3-43)


Because the mass of dissolved fines is usually very small,

C2 is often approximated by C, in which case, r equals zero.

However, the form of equation (3-43) above is chosen to

allow for conditions under which mass of dissolved fines is

not negligible.

The growth and birth functions not only depend on size

but also depend on the crystallizer environment as represented

by equations (3-44) and (3-45).


G = g(c)() (3-44)


B = a(c)b(C) (3-45)


where g(c) and -(c) denote the environmental dependence of

the growth and birth rates respectively, while c(C) and

b(C) similarly represent the size dependence. The crystal-

lizer environment has been taken to be the effective con-

centration that surrounds the various crystals. Incorporating

equations (3-44) and (3-45) into equation (3-42) yields the

following expression:


an + )1
3n + g(c) (n) = Ec(c)b(C)+rl()n + no () (3-46)
3t 3 o o 5


where

-1 ( 1 1
1 h() h4() + h6(5) (3-47)
a1 4 6











With the appropriate definitions of F2 and r3(C), equation

(3-41a) is rewritten as equation (3-48):



C C CT k-C r kp r
dC o 2 k v v 3d
+ + r (5)nC d n4 d
c o 5 c 5 c 5 0 c a4 T5 '0


k C 2
+ v
:caoT5 5 1


n (C)3d d


l+r 1
2 CL2 1


1 1
3 4 h5(4 I h5a l()1


(3-50)


Before applying our stability method equations (3-46) and

(3-48) are made dimensionless by defining the following

dimensionless variables:


c
c


v(y) = g(c) ;



R(z) = b(
b


C
C
c


u(y) -


t I n no()
6 = T ; n = n ; ni (z) -
n n

:(c) z = W(z) =
a


T5
S- T
5 T


where the "bars" denote some reference variables. The variable,

T, represents an appropriate reference time. Upon substitution

of the above dimensionless variables, equations (3-46) and

(3-49) can be transformed into the following equations:


where


(3-48)


(3-49)








60



d T v(y) (Wn) = b u(y)R(z) + l()
de dz c 5c


1 nin
+ 0 5
0o 5



d c + r2y + [k n().4] y 3(z)nz3dz
de cao5 a 5 [k v n E c V5 0 33


pn(C) 41 1 3
c ca4 5 0


y z2
+ [k n(c4] E:cN j
ca o5 z 1


terms in square brackets

dimensionless quantities

Let


n. (z)z dz



in equations (3-51) and

defined below.


SbaT
n


-4
a3 = k n( )
-4
k vn()4
a4 = -



Using the above definitions, equations (3-51) and (3-52) are

recasted into equations (3-53) and (3-54) respectively.

n
S+ alv(Y) (Wn) = a2u(y)R(z) + Fl(z) 5 + (3-53)
de 1 dz


(3-51)


The

are


(3-52)



(3-52)








61


dr c* 2y ay 3 a4 3
dy c* + 2 + a3Y r3(z)nz dz 4V5 nz3dz
de cao 5 c 5 c 5 0 45 0


S a 3y 2
Ecov5 z1


nin(z)z dz
in


Without loss of generality the a.'s, i = 1,2,3,4, are set

equal to one.


dn 8(wn) rl(z)n 1
+ v(y ) = u(y)R(z) + + -in.
d 9z V5 O+ o I5


(3-55)


c + y + y r3(z)nz3dz nz3dz
de ECao 5 c 5 Ec 5 J0 3c 4 c 5 J0


2
+ 1 co5 2
+ c ao
Eco5 z
1a V 1


n. (z)z3dz
in


The boundary and initial conditions for equations (3-55) and

(3-56) are given below:


n(z,O) = no(Z)


y(O) = yo


lim n(z,e) = 0
Z-om

n(o,e) = 0


(3-57)


(3-58)


(3-59)


(3-60)


As a reminder, we indicate below the change from the original

variables to the dimensionless variables:


(3-54)


(3-56)









n -+ # of particles

g v concentration dependent part of growth rate

W size dependent part of growth rate

a u concentration dependent part of birth rate

b R size dependent part of birth rate

C z particle size

c y supersaturation

c c* inlet concentration

t e 6 time

T5 -* 5 residence time


Equations (3-55) through (3-60) constitute the final forms

of the crystallizer governing equations to which our stability

method will be applied.

We proceed by taking a Taylor's series expansion of

some of the quantities in equations (3-55) and (3-56):


y(6) = y0 + EyI(6)

v(y) = vo + evo1(6)

u(y) = uo + cuoy1(6)

n(z,6) = n (z) + Enl(Z,6)


where n (z) is a time independent solution and E is a small

parameter. The quantities g' and a' are defined below;
O O0


v = dv(y) (3-61)
o dy y=Y



= du(y) (3-62)
o dy Y=
y=y
y O










Upon substitution of the above Taylor expansions into
equations (3-55) and (3-56), the following equations result:


-ae (no+Ey ) + (vo+EYlV -' (W(no+el)) = E R()(Uo+el)

F1(z) 1
+ (n +El) + n. (3-63)
o a 5 in



d c* 2 1
de (Y o+EY) =c + (Y +EYl) + (y +Ey)
S 0 ca V5 :cV5 1 Ec V5


Sr3(z)(n +E l)z3dz 1 ( +E z3dz
0 o Ec a45 0

+ o1 3z2
+ 1 (y +y ) nin(z)z dz (3-64)
ca V5 in


Equating coefficients of equal powers of E on both left
and right hand sides of equations (3-63) and (3-64) results

in the following equations:


d rl(z) 1
v d (Wn ) = ER(z)u + nr + -- ni (3-65)
o dz o c o v5 o av in
5 o 5


0 + y + 5 y 0 3(z)n z dz zdz
Coa 5 c 5 c 5 o 0 4 5 00
z
+ y 1 nin(z)z dz (3-66)
Eca V5 z in



( W d 1(
- I+ V 2- (W 1) + ylvo d- (WTIo uCoylR(z) + "1
S1 1 o dZ (3-67)
(3-67)












dyl F2Y1 1
d6 Vc5 EV5


3 1 45
0 3(z)(Ylo0+yn 1)z 3dz = E
0 3 E c L 4V


0 Tlz dz
^0


+ 1 z 2
+ caov
1


nin(z)z3dz
in


Rewriting equations (3-65) and (3-66) one obtains


Ec R(Z)Uo
c 0o 1
V + n.
voW a05VoW in


where


h(z) dW
h(z) = dz
TV d(z-


Yo c 51
YO aco 5


T 3
nl z dz
OQ o


C*1



1
+ 1
Eco5


z 2

1


2 + 1
n + c5
nc 5 E cV5


nin(z)z 3dz .


F3(z)n z3dz


(3-66a)


It should be observed that y is simply a constant. The

solution to equation (3-65a) is given by


no(z) = e


no(z ) + e-F()


Sz
z
0


+ 1
+ avW n. (x) dx
oV5 o n


where


dno
dz + h(z)no
dz 0


(3-68)


(3-65a)


(3-69)


I~cuoR(x)
e-F(x) EcuovRx)
[ ~V0o


Vrl W
V5 vow










z
F(z) = h(x)dx


By choosing zo equal to zero, equation (3-67) reduces to

equation (3-69):


o (z) = e-F(z) J(z)


(3-69a)


where


z -F(x) coR(x) 1
J(z) = e +j c (x) dx.
0 v o co 5 W in


Equations (3-66a) and (3-69a) constitute the pair of time

independent solutions, (no,Yo).

Considering equations (3-67) and (3-68) we make the

following substitutions to obtain equations (3-70) and (3-71),


nl(z,6) = N(z)e


yl(e) = moe


d -XT d
d (N(z)e ) + v dz

= u'm
coo



d -X _2
d (me = 2- m e
de o c 5 o


-XO -AG d
(WN(z)e ) + moe v' (Wdo)

0-X r0 (z) N -,
e -XR(z) + N(z)e- x
05


(3-70)


- _6 1 -Xe 3
-Xe + 1 T3(z)[m n +y N(z)]e z dz
c v5 f0 3 o o 0


1 -- 3 + m
c XN(z)e z dz e oe
coV5 oe
sca4 5 0 co 5

22
*z nin(z)z3dz.
z1


-Xe




(3-71)








66


Equation (3-71) reduces to equation (3-72):


3 1
N(z)z dz -
0 5c 5


1
Ec 5


f r 3() 3
S3(z)n z3dz
0 o


0 3(z)y N(z)z3dz
0 o


z2
a+ 1 I 2
E c ov5 fz 1
1


3


n. (z)z dz

(3-72)


Substitution of equation (3-72) into (3-70)results in the

following equation:


- A


S(W())
+ N(z) + v -(WN(z)) =
v5 o dz


1
Ec4 5


N(z)z3dz


1 1 (z)
5I 0rP3(z)y N(z)z dz z)
c' 5 0 0(A+Q*)


where

d
Q(z) = E u'R(z) v (W o)
c o o dz o


1- 1
3(z)noz3dz + 1
0 c o 5


z2


1


nin (z)z dz

(3-75)


Upon rearranging equation (3-73) one obtains equation


21 1I N1 d d
- 2 + + Q + Q* N(z) + Av d (WN(z))+ Q*v -d (WN(z))
V5 1 5


= -1 N(z)z3dz 1
cL4 5 0 c5


0P3(z)y N(z)z3dz 2(z).
(3-76)
(3-76)


1
Ec4V5


m =
o


2
cV 5


(3-73)


2
Q* =-
c v5


1
+ 1
EcV5


(3-74)










Since any piecewise continuous function with finite left

and right hand limits and with a square integrable first

derivative may be expanded in terms of a complete set of

functions, and since N(z) satisfies these restrictions, we

choose to expand N(z) as in equation (3-77):


N(z) = I Anf (z) (3-77)
n=0

where f (z) is a member of a complete and orthonomal set

of functions. Combining equations (3-76) and (3-77) one

obtains equation (3-78):


+ Q + + Q* = Af(z) + Xv d'z W Af (Z)
X v 5 v 5 j n0 o dz n-0n n
5o 5 n n=0


o dz n1 5 O n n
n=0 I 45 0 n=0n


J 03(z)yo (A nf(z))z3dz 2(z) (3-78)
c 5 0 0

Multiplying equation (3-78) by f (z), integrating with

respect to z from zero to infinity, and applying orthogonality

principle, one obtains equation (3-79) after a slight algebraic

manipulation.












( X2 AQ*)A (A+Q*) Aa + (A+Q*)v A B
m n=O mn mn


1 0 YO CO 0
c45 6m ~I A y + 6 A X 0
c 4 5 .n=0 c 5 n=O n


where


a = 1;

5mn =




m = lo





Y6 = o
xn =



nX, =


Fl(z)f (z)f (z)dz



(W'(z)f (z) + f'(z)W(z))fm(z)dz



f m(z)dz
m



f (z)z3dz




S3(z)fn(z)z3dz


It should be noted that stability is guaranteed

when the real part of X is positive and the system is said

to be asymptotically stable even if the imaginary part

of A is nonzero. A nonzero imaginary part of X gives rise

to an oscillatory system which may or may not be damped.

The expression representedby equation (3-79) is an

infinite dimensional system for the A 's and is represent-

able in matrix form by


(3-79)











MA = 0 ,


(3-80)


where M represents the matrix of coefficients, and I

denotes a vector of A 's. Obviously for nontriavial A 's
n n
the determinant of M must be zero, that is,


det M = 0 .


(3-81)


For a finite dimensional matrix, M equation (3-81) becomes

the characteristic equation which is a polynomial equation

in X, and of the form,


a + a n- + ...+a =0.
n n-1 o


(3-82)


Suppose Ao O, then the resulting characteristic equation

is a quadratic in A and is denoted below by the a 's.
n


a2 = 1


(3-83)


(3-84)


a = Q* + oo /V oo
1 00o 5 000o


a (Ca
o V5 00
5


(Yo-a4yoxo)
- v oo ) + 6 45
5 o oo' o sc4 5


(3-85)


For A Al 0, the resulting fourth order polynomial

equation is similarly denoted below.


a = 1


a3 = 2Q* + i0o


(3-86)


(3-87)


+ 11


a2 = Q* + 2Q*(+oo+Pll) + Aoo


- %01 0l (3-88)


+ A11 + 1ooPll










a = oo 11 + Q* oo+A11) + 2Q*( ool-110

+ 11o00 + 00oAll OA10 I10A01 (3-89)


a0 = Q*2 (oo110- 10 + Q*(QpljAVo+oo I- A10o-10oA1)

+ AoAll 01A10 (3-90)


where

amn
mn v 0o mn


A =6P
mn mn


Pn =(n-a4YoXn)/( ca 5)


6 = u'I v'L
m om om


Im = Ec fmR(z)dz
0

S= d
Lm fm(z) --[no(z)W(z)]dz.


For stability purposes, the quadratic equation usually

suffices. However, one can also utilize the quartic equa-

tion, which will show similar behavior as does the qua-

dratic, since higher and lower order characteristic equations

show similar trends with regard to stability.











By judiciously choosing f (z) to closely resemble

N(z) and by satisfying the boundary conditions of the

system, the similarity behavior mentioned above is made

even better. Applying Routh-Hurwitz criterion to the

polynomial equation (3-82) reveals that, in general, a

necessary condition for stability is for the a' s to have

alternating signs. An additional equation relating the

various a 's is usually required to obtain a sufficiency
n
condition. However, for the quadratic case, the alter-

nating signs are both a necessary and sufficient condi-

tion. Illustrating this for the quadratic case, one

obtains a2>0, al<0, a >0. Since equation (3-82) is in

such a form that a =1 always, a is therefore always

positive. The stability criteria for the quadratic case

are


a2 = 1>0 (3-83a)


a1 = [Q* + o/o5 Voo < 0 (3-84a)

0 5(Y0-a4Y0X0)

a = (a -v5Vo ) + 60 o4 J5 > 0.(3-85a)
0 v5 OO 5 ooo0 Ea S 5


Similarly for the quartic case the stability criteria are

given by


a4 > 0 (3-86a)


(3-87a)


a3 < 0











a2 > 0 (3-88a)


a1 < 0 (3-89a)


a > 0 (3-90a)


-al(-a3a2+al) aa > 0 (3-99)


It should be noted, however, that embedded in the a 's
n
are all the parameters of the complex crystallizer. As a

result the effects of the parameters on the stability of

the crystallizer can be monitored efficiently. It is pos-

sible that for some of the parameters, the effects are

reflective of the degree of positivity or negativity of

the expressions in the stability criteria. In other words,

if an increasing value of such a parameter shows an in-

creasing stabilizing trend on the system, this effect will

be reflected by a corresponding increase in the positivity

or negativity of the expressions. A better way to monitor

the stabilizing or destabilizing effect of a parameter is

to apply the Root-Locus method directly to the charac-

teristic polynomial equation (3-82).

The usual data available in the literature are not

suitable for a stability analysis such as that described

above. In addition to the time dependent number-

size distribution, one needs to obtain the growth and

birth functions in the separable forms shown in equation

(3-44) and (3-45).











G = g(c)(() (3-44)


B = a(c)b(). (3-45)


How one might obtain these separable functions is discussed

in the appendix. It should be noted that the above method

of analysis does not presume the existence of nucleation

models. Instead it requires that the analyst experimentally

determine b(c) and g(c). It accommodates the total size

dependent functions (() and b(C) and the complex crystal-

lizer operating parameters. The only assumption is that,

like its predecessors, it approximates g(c) and b(c) up to

first order in concentration, c. Therefore, in comparison

with others, this method is more thorough and is based on

very practical considerations. This ends the discussion

of our stability analysis. In Chapter IV we devise

example data that closely approximate reality to illustrate

how to utilize this method. Before proceeding to Chapter

IV some discussion about nucleation models is appropriate

inasmuch as these models form an integral part of other

stability methods.



Nucleation Models


The three main nucleation models that have been used

extensively are the (i) Volmers model, (ii) Mier's model,

and (iii) power-law model.











(i) The Volmer's model, based on thermodynamic

considerations, has an Arrehenius type of concentration

dependence as evidencedby equation (1-3) which is

recapitulated below for easy reference:


B(c) = K2exp(-K3/(nc/Cs)2) (1-3)


The concentration dependent part of the birth function,

B(c) represented by equation (1-3), decreases to zero

exponentially as the concentration decreases. In general,

the birth function depends both on particle size as well

as the environment which is represented by concentration.

Oftentimes researchers would simply ignore the size

dependence and represent the complete birth function by

B(c). Volmer's model was based on clear solution and

should, therefore, apply only to homogeneous nucleation.

To this effect Sherwin et al. (1967) mentioned that the

use of such a model in the design of continuous crystal-

lizers might seem questionable. However, they used the

same model in their study entitled "Dynamic Behavior of

the Well-Mixed Isothermal Crystallizer" because they felt

that in a bed of large amounts of crystals the dependence

of nucleation on total crystal surface is small compared

to the nonlinear dependence on saturation. To support

the use of the model they cited the work of Rumford and

Bain (1960) who claimed the dependence of nucleation on

supersaturation in a bed of large crystals of sodium











chloride was similar to that of homogeneous nucleation,

Rumford and Bain found the lower metastable concentration,

cm, for sodium chloride system to be 1.5 gm/liter. The

metastable concentration, cm, is that higher than the

saturation concentration, cs, and below which all secondary

nucleation ceases but above which nucleation increases

rapidly. Another version of equation (1-3) often used

consists of a one-term Taylor series expansion of Zn(c/c )

and is represented by equation (3-92):

B(c) = K2exp(-K3/(c/cs-l)) (3-92)

The above equation assumes (c/cs-l) is small.

(ii) Mier's metastable model is represented by equa-

tion (3-93):

B(c) = Kn(C-Cm) c>cm (3-93)

0 c
Mier's model shows the same discontinuity as Volmer's model.

Beckman(1976) claimed that both models are qualitatively

the same for small deviations about cm, that both are valid

for the lower metastable region and that Mier's model is

essentially a one-term Taylor expansion of Volmer's model.

It would be interesting to see some justification for this

claim.

(iii) The power-law model was obtained when Randolph

and Larson (1962) approximated cm by cs in equation (3-93),











thereby obtaining equation (3-94), by virtue of which B

can be expressed as a function of G defined in equation

(2-13)


B = K (c-c )P (3-94)


G = K1(c-Cs) (2-13)


Combining equations (3-94) and (2-13), one obtains equa-

tion (3-95):


B(G) = KNGP = k (c-cs) (3-95)


In support of the power law model, they argued that cm is

very close to cs for most inorganic systems. On the

contrary, however, Rumford and Bain (1960) argued that the

approximation, that (cm-cs) is very small compared to

(c-cs) might be valid only for precipitation of insoluble

compounds but hardly for crystallizers which, in most

cases, operate close to the metastable range. Equation

(3- 95 ) thus leads to a significant simplification of the

mathematics that would be involved in any analysis

involving B and G.

Another version of the power law model is represented

by equation (3-96) in which MT denotes the suspension

density,


B(c) = Kc(c-c )P Mi = K'GPM (3-96)
cc T (3-96)










The incorporation of MT accounts for the fact that

nucleation is by secondary mechanism. Unlike Mier's

model and Volmer's model, the power-law model is nicely

behaved and does not possess any discontinuity.

Researchers often use these models in defining steady

state dimensionless parameters, b* and g* used in

stability analysis. The quantities b* and g* are defined

essentially to be the steady state first derivatives with

respect to concentration of the birth and growth rates

with appropriate dimensionalizing quantities as follows:



(c -c) c c
= o dB(c)
b* = dc (3-97)


G d c

g* = -c) dG(c) (3-98)
Sdc



Hence b*/g* is defined by equation (2-12) :


dc c_
b*/g dc) (2-12)
dc c


The quantity, co, is inlet concentration as defined earlier.

However, any other appropriate concentration parameter

could be used in place of c The quantities, b*/g*, for

the various models are represented by the expressions

below.












Volmer's model (second version):


2k
b*/g* = i (2-14)
(C 2



Mier's model:


c-c )
b*/g* = P ( C-- c>cm (3-99)
(c-c )
m


Power-law model (both versions):


din B(c)I-
b*/g* = P = dn G( (3-100)
1 c


Randolph and Larson (1971) indicated that a value of 21

for the nucleation to growth sensitivity parameter, i,

would represent such an extreme kinetic order causing

discontinuity in nucleation rate, that the corresponding

higher concentration could be described as an upper meta-

stable threshold of homogeneous nucleation. Such a value

has never been observed in most kinetic studies. Of the

three nucleation models above Beckman (1976) claimed that

the power-law model is the most versatile inasmuch as it

is the only model good for both class I and class II

systems. However, attemptsbySong and Douglas (1975) to

explain their cyclic data with this model failed, thus

contradicting Beckman's claim. They were able to explain











the same data with Volmer's model by using long retention

times thereby approaching very low supersaturations.

Similarly Lei et al. (1971a claimed that Mier's model is

capable of approximating the behavior of crystallizers

with fine traps with either low or high b*/g* values.

A look at the above models reveals some striking

differences in the expressions for b*/g* as we let c

approach cs'


Volmer's model (second version): Rim b*/g* + (3-101)

s

Mier's model: Rim b*/g* = 0 (3-102)

ccs


Power-law model (both versions): Rim b*/g* = P. (3-103)
C-+C
s


The resulting b*/g* values show inconsistencies and hence

the models are not equivalent for all crystallizer condi-

tions. Researchers usually show the region of stability

on the graphs of the steady state parameters, g* and b*/g*.

Yu and Douglas (1975) showed plots with g* and b*/g* taking

values in the intervals (0,-) and (11,i ) respectively.

In practice, of course, one operates at one fixed point

(g*, b*/g*) which one hopes is in the stable region.

Because the parameters, g* and b*/g*, in theory, can only

be controlled inferentially and cannot easily be set to some

desired values, especially via the models discussed above,








80


their usefulness as stability parameters is very limited.

As a result of the inconsistencies arising in b*/g* for

the various models, it is not surprising that one model

can explain cyclic data while another cannot. It would

be useful to establish stability criteria which depend

on more readily accessible parameters. In addition,

there are some algebraic errors in the derivations of the

stability equations in the studies by Sherwin et al.

(1967) and Yu and Douglas (1975) which might invalidate

their results.

















.CHAPTER IV
RESULTS



Steady State Determination of Growth and Birth Functions


One of the purposes of this study is to devise a

method for accurately quantifying growth and birth rates

in a steady state crystallizer. This method developed

in the first section of Chapter III will be illustrated

with data taken from the literature. Kambaty and Larson

(1978) showed plots of particle number size distribution

for magnesium sulphate heptahydrate (MgSO4 7H20) crystals

as depicted in figures 9 and 10 Some of the

plots were for nuclei generation from the faces or the

edges of the crystals, while other plots were for dif-

ferent impurity concentration, equivalent to different

solute concentration. In all these plots nuclei genera-

tion was by contact. Repetitive contacting was carried

out for an interval of approximately ten residence times

after reaching steady state and sampling was done at one

residence time intervals (one residence time is 9 minutes).

The size distribution of the various samples was measured

with a model TA II Coulter Counter. The concentration of

each experimental run was fixed. Since the purpose of

this study is simply to use these data in exemplifying the











































+

2
E
3






a,
S
E
Z

e
o

(U
^2


Size -*


Figure 9.


Experimental and Fitted Curves for
Particle Number Versus Size for Both
Edge and Face of Magnesium Sulfate.









83











100.--------------~------








10


t





B

1.o- oA, Original Dat
A, Fitted Curve
S 1 B, Original Data
Z B, Fitted Curve
a *C, Fitted Curve
o *C, Original Data
"D, Fitted Curve
d OD, Original Data
o. I I I I
10 20 30 40 50 60 70
Size -


Figure 10. Experimental and Fitted Curves for
Crystal Size Distribution for
Magnesium Sulfate at Four Different
Concentrations. A,B,C,D Have
Concentration in Descending Order
of Magnitude.












Table 1


Data for Plot
C1=5407.625
y1=45.79739


E in Figure 10
C2=0.6569098
Y2=0.03146495


Original
Data


Fitted
Date


0.25000E
0.850000E
0.4700001
0.350000E
0.280000E
0.250000E
0.230000E
0.1900000
0.150000E
0.115000E
0.750000E
0.500000E

Data for Plot F
C1=890.0442
yl=20.33548

0.735000E
0.300000E
0.185000E
0.169500E
0.129000E
0.105000E
0.940000E
0.664000E
0.600000E
0.422000E
0.277000E
0.177000E


0.241687E
0.911892E
0.410214E
0.347251E
0.300283E
0.260334E
0.225640E
0.190965E
0.145233E
0.111151E
0.761961E
0.506159E


in Figure 10
C2=0.5652139
Y2=0.03509961

02 0.697932E
02 0.329321E
02 0.174400E
02 0.149199E
02 0.127664E
02 0.108451E
01 0.923421E
01 0.766462E
01 0.564760E
01 0.419077E
01 0.275024E
01 0.174263E


Size


5.0
7.0
10.0
11.5
14.0
18.0
22.5
27.8
36.5
45.0
57.0
70.0


5.0
7.0
10.0
11.5
14.0
18.0
22.5
27.8
36.5
45.0
57.0
70.0


Data Date











Table 2


Data for Plot
Ci=26044.52
Y1=34.03978


Original
Data


Size
7.2
9.2
10.09
14.2
18.3
23.3
28.8
36.2
44.4
56.8
71.7


A in Figure 11
C2=0.8731303
Y2=0.03112719


Fitted
Data
0.756792E 02
0.340184E 02
0.287520E 02
0.219863E 02
0.192605E 02
0.164820E 02
0.138886E 02
0.110312E 02
0.854617E 01
0.580956E 01
0.365359E 01


Sin Figure 11
C2=0.7708762
Y2=0.03808283


0.750000E
0.37000E
0.265000E
0.214300E
0.191700E
0.166700E
0.150000E
0.118500E
0.800000E
0.567000E
0.370000E

Data for Plot B
C1=5378.973
Y1=19.04890

0.35000E
0.200000E
0.137500E
0.118800E
0.106000E
0.800000E
0.625000E
0.413000E
0.350000E
0.229000E
0.129000E


0.353847E 02
0.178923E 02
0.152241E 02
0.111869E 02
0.949264E 01
0.784354E 01
0.636124E 01
0.479902E 01
0.351181E 01
0.219001E 01
0.124169E 01


7.2
9.2
10.09
14.2
18.3
23.3
28.8
36.2
44.4
56.8
71.7











Table 3


Data for Plot
C1=1621.810
Y1=15.34124


C in Figure 11
C2=0.7644494
Y2=0.05621562


Original
Size Data


Fitted
Data


0.166700E
0.118800E
0.850000E
0.700000E
0.567000E
0.413000E
0.300000E
0.200000E
0.129000E
0.330000E
0.270000E


0.168360E 02
0.105774E 02
0.942475E 01
0.693656E 01
0.548517E 01
0.414013E 01
0.303902E 01
0.200479E 01
0.129437E 01
0.329706E 00
0.272500E 00


Data for Plot D in Figure 11


C1=6608237.
Y1=13.46922


0.138000E
0.800000E
0.680000E
0.583000E
0.430000E
0.336000E
0.236000E
0.174000E
0.930000E
0.430000E
0.170000E


C2=1.955293
Y2=0.06041371


0.138004E
0.782786E
0.733949E
0.571179E
0.445860E
0.329619E
0.236432E
0.151200E
0.901307E
0.435574E
0.177062E


7.2
9.2
10.09
14.2
18.3
23.3
28.8
36.2
44.4
56.8
71.7


7.2
9.2
10.09
14.2
18.3
23.3
28.8
36.2
44.4
56.8
71.7