The stability of a mixed suspension crystallizer with classified product withdrawal.

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Title:
The stability of a mixed suspension crystallizer with classified product withdrawal.
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Mixed suspension crystallizer with classified product withdrawal
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xi, 74 leaves. : ill. ; 28 cm.
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English
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Metha, Ashish Jayant, 1944-
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Subjects / Keywords:
Crystallization   ( lcsh )
Stability   ( lcsh )
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (M.S. in Engin.)--University of Florida.
Bibliography:
Bibliography: leaves 67-68.
Statement of Responsibility:
By Ashish Jayant Mehta.
General Note:
Manuscript copy.
General Note:
Vita.

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University of Florida
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Full Text














THE STABILITY OF A MIXED SUSPENSION
CRYSTALLIZER WITH CLASSIFIED
PRODUCT WITHDRAWAL










By

ASHISH JAYANT MEHTA


A THESIS PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE IN ENGINEERING










UNIVERSITY OF FLORIDA
1969


















ACKNOWLEDGMENTS


The author wishes to thank Dr. A.D. Randolph of the University

of Arizona, Tuscon, Arizona, for help in developing the basic equations

used for analyzing the problem presented in this work. He would also

like to thank Dr. Thomas M. Reed of the Department of Chemical Engi-

neering of the University of Florida for assisting him in completing

the present thesis.



















TABLE OF CONTENTS


Page


ii

v

vii

xi


ACKNOWLEDGMENTS

LIST OF FIGURES

NOTATIONS

ABSTRACT

CHAPTER

I INTRODUCTION TO CRYSTALLIZATION

Theoretical Approach

Dynamic Behavior

Problem of Stability

The Classified Product Withdrawal Crystallizer

II THE POPULATION BALANCE AND THE MIXED SUSPENSION
MIXED PRODUCT WITHDRAWAL CRYSTALLIZER

Conservation of Particulate Entities

Nucleation and Growth

Kinetic Growth Models

Kinetic Nucleation Models

Nucleation and Growth Rate Interaction:
The Power Law

The Mixed Suspension Mixed Product Withdrawal
Crystallizer

Steady State Size Distribution

Stability Analysis












Page

III THE MIXED SUSPENSION CRYSTALLIZER WITH 21
CLASSIFIED PRODUCT WITHDRAWAL

Description of the Model 21

Population Balance 23

Steady State Population Density 23

Dimensionless Equations 26

Moment Transformation 28

Steady State Moments 32

Constraint on the Growth Rate 35

Expressing the Quantities y and y as 36
Functions of the Moments

IV STABILITY ANALYSIS 39

Linearization of Moment Equations 39

Stability Analysis of the Matrix 42

V RESULTS OF THE ANALYSIS 44

Root-Loci 44

Effect of Suspension Void 53

Discussion 62

Summary 65

REFERENCES 67

APPENDIX I LEIBNITZ RULE 70

APPENDIX II CRYSTAL PRODUCTION RATE 71

APPENDIX III TRANSFORMATION MATRIX [A] 73

BIOGRAPHICAL SKETCH 74



















LIST OF FIGURES


Figure

1 Population Density Plot for a Mixed Suspension
Mixed Product Withdrawal Crystallizer

2 The Mixed Suspension Crystallizer with Classified
Product Withdrawal


3 Population Density Plot for 0 < a < 1


Population

Population

Root-Locus

Root-Locus

Root-Locus

Root-Locus

Root-Locus

Root-Locus

Root-Locus

Root-Locus

Root-Locus

Root-Locus

Root-Locus

Root-Locus

Root-Locus

Root-Locus

Root-Locus


Density Plot

Density Plot


for a = 0

for a > 1


Plot for a = 1.0


Plot

Plot

Plot

Plot

Plot

Plot

Plot

Plot

Plot

Plot

Plot

Plot

Plot

Plot


for

for

for

for

for

for

for

for

for

for

for

for

for

for


= 0.5, B = 1

= 0.1, B = 1

= 0.05, B = 1

= 0.01, B = 1

= 0.005, 8 = 1

= 0.001, B = 1

= 0.0, B = 1

= 1.2, = 1

= 0.005, B = 2

= 0.005, 8 = 3

= 0.005, B = 4

= 0.001, B = 2

= 0.001, B = 3

= 0.001, B = 4


Page


17












Figure


Page


21 Block Diagram Depicting the Interrelationship 63
Between the Variables in a Mixed Suspension
Crystallizer with Classified Product With-
d awal


















NOTATIONS


a Constant relating growth rate to size

Ag, ..., A4 Constant coefficients in the fifth order characteristic
equation

[A] Nonsymmetric transformation matrix

Co Solute concentration in feed, lb./cu.ft.

Cl Solute concentration in outflow, Ib./cu.ft.

C Solute concentration in crystallizer, Ib./cu.ft.
s
D Dissipation function

f nt undersize moment (normalized)
n
f Zero thundersize moment (normalized)

fZ First undersize moment (normalized)

f2 Second undersize moment (normalized)

f3 Third undersize moment (normalized)

gn n oversize moment (normalized)
go Zero othment (normalized)

g1 First oversize moment (normalized)

g Second oversize moment (normalized)

g3 Third oversize moment (normalized)
g3 Third oversize moment (normalized)

G Generation function

i Sensitivity parameter in the power law, p 1

kI Constant relating growth rate to supersaturation

k2, k3 Constants in Volmer's nucleation model


vii


























,K2,K3,K4,K5


Constant in Mier's metastable model

Crystal area shape factor, sq.ft./crystal/sq.ft.

Constant relating-the void fraction to the third moments

Constant relating nucleation rate to growth rate

Crystal volume shape factor cu.ft./crystal/cu.ft.

Steady state parameter relating the growth rate to the
second moments

Steady state parameters in the matrix [A]

Steady state parameter relating the perturbation in
dimensionless void fraction to perturbation in the
third moments

Characteristic crystal size, ft.

Mean crystal size, ft.

Crystal classification size, ft.

Total crystal mass, lb.

Nuclei population density, number/ft-cu.ft. of mother
liquor

Undersize population density, number/ft-cu.ft. of mother
liquor

Oversize population density, number/ft-cu.ft. of mother
liquor

Crystal population density, number/ft-cu.ft. of mother
liquor

Number of nuclei per unit volume, number/cu.ft. of mother
liquor

Sensitivity ratio of nucleation rate to growth rate in
the power law

Volumetric outflow rate in mixed product withdrawal
crystallizer, cu.ft./sec,

Volumetric feed rate in mixed suspension crystal-
lizer with classified product withdrawal, cu.ft./sec.

Volumetric underflow rate in mixed suspension crystal-
lizer with classified product withdrawal, cu.ft./sec.

viii





















s

s
m


S

t

u

v

v
xi

v


V

W1

W2

X1, ..., X
n

x, y, z

y
o
y

Y


Y2


a

B

Y


Volumetric overflow rate, cu.ft./sec.

Crystal growth rate, ft./sec.

Region of n-dimensional space and m-dimensional
property space

Supersaturation in solution, lb./cu.ft.

Critical supersaturation below which the solution is
metastable

Total crystal surface area, sq.ft.

Time, sec.

Region of space

Velocity vector in the spatial region, ft./sec.

Velocity component in spatial region along xi direction,
ft./sec.

Velocity component in property space along i direction,
ft./sec.

Crystallizer volume, cu.ft.

Undersize product weight distribution, lb.

Oversize product weight distribution, lb.

Coordinates of n-dimensional spatial region

Coordinates of three-dimensional space

Dimensionless crystal population density, n/n0
o

Dimensionless nuclei population density, n /n0
O

Dimensionless undersize crystal population density,
n1/no

Dimensionless oversize crystal population density,
n2/n0

Ratio of undersize to oversize outflow rates, QI/Q2

Dimensionless classification size, L /r r
c oo

Dimensionless crystal population density of the
classification size












r Average residence time of oversize particles,
V/Q2, sec.

r sAverage residence time of-solids, sec.

E Crystallizer void fraction

y Point population density of particulate entities

8 Dimensionless time coordinate, t/r

X Root of the characteristic equation
th
Pn n moment of distribution in a classified product
withdrawal crystallizer
th
y0 Zeroth moment of distribution in a classified product
withdrawal crystallizer

pI First moment of distribution in a classified product
withdrawal crystallizer

U2 Second moment of distribution in a classified product
withdrawal crystallizer

13 Third moment of distribution in a classified product
withdrawal crystallizer

v Dimensionless void fraction, E/E

p crystal density, Ib./cu.ft.

a Variance of weight distribution

Dimensionless growth rate, r/r

X Mean value

A1, *...' m Coordinates of the m-dimensional property space



Superscript

Perturbation

Subscript

o Steady state



















Abstract of Thesis Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Master of Science in Engineering

THE STABILITY OF A MIXED SUSPENSION CRYSTALLIZER
WITH CLASSIFIED PRODUCT WITHDRAWAL

By

Ashish Jayant Mehta

March, 1969

Chairman: Thomas M. Reed

Major Department: Chemical Engineering


The effect of size classification on the dynamic behavior of a

mixed suspension crystallizer with classified product withdrawal is

investigated by analyzing a simple model of such a crystallizing

unit.

It is observed that the stability of the crystallizer depends on

the relative kinetic rates of nucleation and growth. The results show

that classification lowers the sensitivity of the crystallizer to dis-

turbances and reduces the tendency toward cyclic fluctuations.
















CHAPTER I

INTRODUCTION TO CRYSTALLIZATION


Theoretical Approach

In continuous crystallization processes, a considerable importance

is attached to the technically complex problem of controlling the crystal

size distribution. The form of the size distribution depends on the

physico-chemical characteristics of the crystallizing material and on the

constraints imposed upon the crystallizing unit itself. Industrially and

otherwise, the study and control of crystallization processes and, hence,

of crystal size distribution have been carried out for several years [1].

However, even up to the present time, most of the process techniques are

still something of an art and depend to a considerable extent on the ex-

perience and intuitive skill of the operator.

Theoretical approach to industrial crystallization has been late in

coming, and papers written on the matter deal mainly with the steady state

behavior of operating systems. In 1956, Saeman [2] derived equations for

the steady state size distribution in a mixed suspension. He compared the

theoretical distribution with that obtained from an Oslo-Krystal type unit

producing ammonium nitrate crystals under conditions of hindered settling,

and found a fairly good correlation between the theoretical prediction and

experimental results. A similar work was done by Robinson and Roberts [3]

who obtained the theoretical crystal size distribution resulting from a

series of well-mixed units with nucleation in the first stage only. A












comparison of the size distribution obtained from a single unit with

that obtained from an industrial ammonium sulfate producing crystallizer

showed good quantitative agreement.

The special feature that distinguishes continuous crystallization

processes from other continuous reactors is the simultaneous occurrence

of nucleation and growth. This point was brought out clearly in the

work of Saeman and in that of Robinson and Roberts; and, indeed, it has

been observed that the control of nucleation rate is the key to con-

trolling the size distribution in a crystallizer. Saeman [2] has dis-

cussed the problem of controlling the nucleation rate by removal of

crystal nuclei in order to control the size distribution for the case

of a crystallizer with classified product withdrawal.

Dynamic Behavior

Papers dealing with the dynamic behavior of crystallization

processes are yet fewer in number. This is not surprising in view of

the fact that the processes of nucleation and growth in a crystal-

lizing system are of a complex nature and do not lend themselves easily

to mathematical description. In 1962, Randolph and Larson [4] derived

time dependent equations relating crystal population density to size in

a mixed suspension of arbitrary volume. These equations are based on

the concept of conservation of particle population, and it was pointed

out that a particulate system must conserve particles in addition to

satisfying laws describing the conservation of mass and energy. Ana-

lytically, the concept of population balance is of importance inasmuch

as it provides a way of characterizing the size distribution by taking

into account the effect of nucleation on the size distribution. The












steady state form of the population balance equation was solved for the

case of a series of crystallizers with nucleation in each unit. The

size distribution thus obtained was reduced to that obtained by Robinson

and Roberts [3] when nucleation was assumed to occur in the first stage

only. Further, for the case of a single unit, the size distribution was

found to be identical to that derived by Saeman earlier [2].

In another study, Randolph and Larson [5] solved the transient

population balance for the case of an ammonium sulfate producing mixed

suspension crystallizer. The transient response of the size distri-

bution to upsets in nuclei dissolving rate and in production rate was

studied by simulating the system equations on an analog computer. The

results were interesting inasmuch as it was observed that the size

distribution undergoes long term cyclic fluctuations due to pertur-

bations affecting the system. In an earlier study, Miller and Saeman [6]

reported similar observations in an industrial ammonium nitrate producing

crystallizer. In the field of continuous polymerization and in fermen-

tation processes, such observations have also been made [7, 8].

Problem of Stability

In view of the observed dynamic characteristics of crystallizing

systems involving long term cyclic fluctuations in particle size distri-

bution, the question of the stability of such systems becomes one of

interest inasmuch as it has a bearing on the problem of controlling the

size distribution. Randolph and Larson [5] have investigated the sta-

bility limits of a mixed suspension crystallizer with mixed product

withdrawal. These authors point out the importance of the relative

kinetic rates of nucleation and growth in defining the stability limits









4

of the system. The stability criterion was expressed in terms of the

ratio of nucleation rate to growth rate which, as the authors point

out, would be transgressed in the event of a discontinuous jump in

the nucleation rate. Such a situation would arise for instance when

the supersaturation becomes so high that sudden mass nucleation occurs

as the metastable solubility boundary is crossed.

The Classified Product Withdrawal Crystallizer

Many crystallizers are operated with some kind of a classification

device, either in the crystallizer bed itself or in the outlet of the

crystallizer. The types of classifiers in general use are either those

that segregate particles by gravity separation or those that use centri-

fugal action. An Oslo-Krystal type unit may be cited as an example of

the first type wherein only the larger particles that settle down at the

bottom of the crystallizer bed are withdrawn. Devices using the elutri-

ative principle belong to the same class. On the other hand, devices

such as hydroclones, which make use of the centrifugal principle, fall

into the second category. Theoretically, an ideal classifying device

would cause a perfect separation of oversize and undersize particles.

However, in most of the known classifying devices in which a certain

amount of mother liquor is withdrawn to maintain overall material balance,

a proportionate amount of undersize particles cannot be prevented from

being withdrawn with the mother liquor. Further, a factor that prevents

the product from being perfectly uniform is that product withdrawal takes

a finite amount of time, and, therefore, the oversize particles exhibit

a certain degree of size distribution. An ideal classifier would require

an infinite recycle rate for an instantaneous withdrawal of oversize

crystal product.







5
The steady state aspects of a crystallizer with classified product

withdrawal have been studied previously [2, 9, 10]. Miller and Saeman [6]

observed that classification tends to produce crystals of a poor quality.

They found that it was impossible to operate a-completely classified

ammonium nitrate producing-unit continuously, but that it was necessary

to withdraw the product periodically in a cyclic manner.

SSherwin et al. [11] have analyzed the dynamic behavior and the

stability limits of a crystallizer with ideal classifying action. These

authors found that,for the classified case, the stability criterion is

considerably relaxed in relation to the case of mixed product withdrawal,

or in other words, as was pointed out, size classification introduces

greater instability into the system. However, the validity of their re-

sults is questionable inasmuch as, for reasons given earlier, the concept

of an ideally classified crystallizer is not realistic when applied to

practice. The mathematical description of such a system, wherein there

is no withdrawal of undersize particles and the withdrawal rate of over-

size particles is infinite, is not suitable for a realistic evaluation of

the actual situation. The present work attempts to analyze the stability

problem of a classified product withdrawal crystallizer with a more real-

istic model wherein both the undersize and the oversize particles are

removed at a finite rate, and there is.-,preferential withdrawal of

oversize particles. A linearized stability analysis of the system e-

quations is made, and the analytical approach follows closely that for

a crystallizer with mixed product withdrawal as given by Randolph and

Larson [5].

The theoretical background, including the concept of population









6

conservation, is developed in Chapter II. The stability problem of the

mixed withdrawal crystallizer is also discussed, and,in Chapter III, the

classified withdrawal case is analyzed. In Chapter IV, the linearized

system equations are investigated for the stability criterion, and the

results are discussed subsequently in Chapter V.


















CHAPTER II

THE POPULATION BALANCE AND THE MIXED SUSPENSION
MIXED PRODUCT WITHDRAWAL CRYSTALLIZER


Conservation of Particulate Entities

In Chapter I, it was mentioned that the primary factor controlling

the particle size distribution in continuous crystallization processes

is the kinetics of nucleation and growth. Now, inasmuch as the product

crystal size distribution is also dependent on the residence time proba-

bilities of particles within the system, a theoretical development at-

tempting to describe the size distribution in various systems must take

into account the interaction between the residence time distribution of

particles and the kinetics of nucleation and growth.

Now, a particulate system must conserve numbers in addition to con-

serving mass, momentum, and energy if the size distribution is to be

fully defined; and, therefore, equations of the conservation of the last

three quantities are not sufficient for formulating the problem of

particle size distribution. In general, one can speak of the distri-

bution of particulate entities over any associated property. This

property can be, for instance, the size of a particle, or age, or any

quantity that varies as a function of time. A population balance of

particulate entities derived with sufficient generality can be used to

describe the property distribution in various constrained systems

[12, 13, 14]. Such a general derivation of a population balance due

to Randolph [15] is given in the sequel.

7











Consider the distribution of particulate entities that are continu-

ously distributed through a region U of n-dimensional space


Xn(xi, X2, ..., xn)

and through a region V of m-dimensional property space


'(V l 2, ..., m)


A point number density in the region R = U + V can, therefore, be de-

fined as


n = n(x1, x2, ..., Xn, 1i, P2, -2, ),


Next, the velocities of propagation of the entities along the ith

spatial axis and along the ith property axis may be expressed as

axi .i
V =- = t
x. at at


Further, consider the generation and dissipation functions operating

throughout the region R

Generation:

G = G(X1, X2, ..., x I, 2>2, ..**, n' t)

Dissipation:

D = D(x1, x2, ..., xn, 1i, P2, ., ** ', t)


which, of course, are functions of time. Now, a particulate balance

may be written as

{Accumulation} = {Input} {Output} (2-1)

If a Lagrangian approach is taken, that is, if it is assumed that the










region R translates with local velocities v the particulate balance

Equation (2-1) becomes


d ndR = (G D) dR (2-2)

R R

where

dR = dxI dx2 ... dxn dil dp2 ... dmi

or equivalently,



dt // /// dxm ... d*1 dxn "' dxl

R X1 x2 ... xn i 2 ...*** m (2-3)

Equation (2-3) may be differentiated by repeated application of the

Leibnitz rule (See Appendix I) to give

n m
(a + a (Vx n) + (a p + D G} dR = 0 (2-4)
at ax a 8
R i=l i=l

which further reduces to

n m
S)+ D G = 0 (2-5)
7t ax
S+ xi i
i=l i=l

inasmuch as R is an arbitrary region. Equation (2-5) is a population

balance for a particulate system but is of a form that is too general

for application to the size distribution problem in a crystallizer.

However, before Equation (2-5) is constrained to represent population












density distribution in a crystallizer, it is essential to discuss

the role of nucleation and growth in a crystallizing system.

Nucleations and Growth

The driving force in the process of crystallization is the super-

saturation of the solute material in suspension. Further, inasmuch as

the nucleation rate and the growth rate both are functions of super-

saturation, one would expect a functional relationship of the following

type, expressing the dependence of nucleation rate and growth rate on

supersaturation.

Growth rate:

dNo
dt = f(s) (2-6)

Nucleation rate:


r = g(s) (2-7)

where No is the total number of nuclei per unit suspension volume. The

functions f(s) and g(s) must, of course, be determined experimentally.

Kinetic Growth Models

Miller and Saeman [6] found a linear function of the form


r = kls (2-8)

describing the dependence of growth rate on supersaturation. The work

of Saeman [2] and of Van Hook [16] has supported this kind of relation-

ship.

Kinetic Nucleation Models

Volmer found an Arrhenius type kinetic law [17] describing nu-

cleation behavior. His model is based on thermodynamic considerations









and may be expressed as


k3
En(s + C )
dN k2 e .s (2-9)


where k2 and k3 are constants, and C is the solubility concentration.

Another nucleation model, which is a simplified representation of

Equation (2-9), is Mier's metastable model [17]


dN_
d- = k4(s s )P (2-10)

where k4 is a constant, and sm is the critical supersaturation below

which the solution is metastable. There is some experimental evidence

supporting this model [4].

At this point, it should be mentioned that the process of nu-

cleation in a mixed suspension crystallizer is a rather complex phe-

nomenon involving both homogeneous and heterogeneous nucleation, the

latter process being the formation of nuclei on a crystal surface.

Volmer's model was derived for homogeneous nucleation only, and,

therefore, its applicability to a mixed suspension crystallizer is

questionable. In a heterogeneous nucleation process, the nucleation

rate should depend both on concentration and total crystal surface. In

the presence of a large number of crystals, the dependence on surface

area is relatively less important than the dependence on supersaturation.

Rumford and Bain [18] have investigated the nucleation of sodium chloride

in a bed with large crystals. These authors found that the dependence

of nucleation rate on supersaturation is very similar to that of homo-

geneous nucleation. For such a system, Volmer's model would well












express the process of nucleation in a crystallizing system. The ad-

vantage of using Equation (2-9) is that it is smooth over the entire

concentration range.

Nucleation experiments on an ammonium sulfate system have indi-

cated that the metastable supersaturation level in such a system is

almost nonexistent [4]. In this event, Mier's nucleation model would

reduce to


dN = k4SP (2-11)


where the value of p has been given as 4 for the ammonium sulfate

system. Sherwin et al. [19] have offered criticism for the use of

Equation (2-11) in dynamic studies on the grounds that in a large

number- of systems metastable supersaturation cannot be neglected. It

will, however, be shown in the sequel that this problem does not arise

in determining the interactive relationship between nucleation and

growth since in a-well-mixed crystallizer both the nucleation rate and

growth rate depend on the same level of supersaturation.

Nucleation and Growth-Rate Interaction: The Power Law

The nucleation rate can be related to the nuclei density n (which

is actually the population density of particles of "zero" size) as

follows


dN= L --) = rn0 (2-12)
dt at L


where r = 3L/3t is the crystal growth rate expressed as the rate of

change of the characteristic size L of the particle; the characteristic

size being any one spatial dimension of a collection of geometrically












similar particles that can be used to characterize the size of the

particles.

Combining Equations (2-8) and-(-2-11) results in the power law


dN= kNr (2-13)
dt- -


where kN is another constant. In Equation (2-13), inasmuch as the

nucleation rate is expressed as a function of the growth rate, the

level of supersaturation on which the former quantity depends does

not enter into the expression.

The advantage of using the- nucleation-growth power model over

that of Volmer is clearly seen from the form of Equation (2-13)

which- is mathematical simplification of Equation (2-9). The ex-

ponent- p in Equation (2-9) is actually the sensitivity ratio of nu-

cleation rate to growth rate when both are expressed as functions

of-supersaturation. As was mentioned in Chapter I, this ratio is im-

portant in defining the stability limits of a mixed suspension crystal-

lizer.

Substituting Equation (2-12) into Equation (2-13) results in the

expression


n = kNr (2-14)

which is another form of the power law and where i = p 1. Equation

(2-14) expresses the nuclei density in terms of the growth rate and is

a convenient boundary condition for solving the population balance

equation as will be shown later, and it is known to hold over a wide

range-of concentrations for many inorganic systems [20].












The Mixed Suspension- Mixed Product Withdrawal Crystallizer

Prior to investigating a crystallizer with classified product re-

moval, the case of a crystallizer with mixed product withdrawal must

be reviewed since the problem of-the stability of the former is ana-

lyzed along lines similar to the latter.

The general population balance Equation (2-5) may now be reduced

further to suit the description of a mixed suspension mixed product

withdrawal crystallizer.

In a crystallizer of arbitrary volume V, the processes of particle

nucleation and growth can be characterized by three orthogonal spatial

coordinates x, y, and z, and a size coordinate L defining the particle

size. Hence, in this case, Equation (2-5) reduces to

an a__(rn)
an + V.(,n) + ( + D G = 0 (2-15)
at dL

where n is the population density in a mixed suspension crystallizer.

v is the vector velocity of the particles in the spatial region, and

r = 3L/8t is the growth rate.

If it is further assumed that the generation and dissipation

terms are absent, Equation (2-15) reduces to


n + V. ) + (rn) = (2-16)
at 3L

This equation can now be integrated over the volume V of the crystal-

lizer, giving


S+ -(rn)} V + V-.(n) dV = 0 (2-17)
at 3L V
V












Then, by the divergence theorem, the volume integral in Equation (2-17)

can be converted to a surface integral


I V- (n) dV = f ndS


(2-18)


where the velocity v is orthogonal to the surface S. The following

two contributions to this integral are important.

a. The term due to the change in the volume of the system,

which is


ndV (2-19)
dt

b. The integral over an arbitrary number of inflows and out-

flows, which yields a term involving the flow rates


n


(2-20)


- Qini

i=1


where by convention inflows are considered positive. Equation (2-17)

thus becomes


n
n + _(rn)
at L
i=l


d(anV) = 0
Qini + dt


The following constraints may now be applied without much loss of

generality.

1. The feed is clear, that is, there are no nuclei in the feed.

2. The crystallizer volume is constant.


(2-21)












Under these conditions, Equation (2-21) reduces to


Dn + () + qn = 0 (2-22)
at aL V

where Q is the volumetric withdrawal rate.

In order to uncouple the second term in Equation (2-22), it is

necessary to assume a certain functional dependence of the growth

rate on the size. Sherwin et al. [19] have assumed a growth rate

model of the form


r = 1 + aL (2-23)


where a is a constant. Canning and Randolph [21] have used the same

growth model to derive expressions for the size distribution in a

mixed suspension crystallizer under various constraints.

Steady State Size Distribution

Under conditions of a steady state and assuming that McCabe's AL

law [24] holds, that is, the growth rate is independent of size, Equation

(2-22) can be integrated to give

*QL
n = n e V (2-24)
o o

where the subscript o refers to the steady state condition. Equation

(2-24) suggests that in a mixed suspension crystallizer with mixed

product withdrawal the population density decays exponentially with

size. In Figure 1, the logarithm of the population density is plotted

as a function of size. The slope of the straight line is QL/V, and

the intercept on the y-axis is, of course, the nuclei density n

Several systems are known to exhibit this kind of behavior [6, 22].





























O
0







O-







O
FIGURE -
0






a.
a-
0
0






FIGURE I:


CRYSTAL SIZE


POPULATION DENSITY PLOT FOR A MIXED SUS-
PENSION MIXED PRODUCT WITHDRAWAL CRYSTALLIZER












SStability Analysis

The form of Equation (2-22) suggests that even under conditions

of steady state energy and material balance transients in the size

distribution are expected. Randolph and Larson [5] investigated the

problem of the stability of size distribution in a mixed suspension

mixed product withdrawal crystallizer. The linearized moment trans-

forms of the system equations, obtained from the population balance

Equation (2-22) in conjunction with the initial condition Equation (2-24)

and the boundary condition Equation (2-14), were solved on an analog

computer. Step changes in production rate and in nuclei dissolving rate

were introduced into the system, and the resulting disturbances in total

suspension population, crystal length, and area were plotted as functions

of time. The plots showed wave-like cyclic fluctuations in these quanti-

ties propagating with time, and the periods of the disturbances were

found to be three or four times the order of magnitude of residence times

of particles in industrial units. Transients have been observed in

commercial crystallizers [6]; and, in view of this observation, the

question of the stability of an operating crystallizer becomes one of

interest. The same authors analyzed the system equations of a mixed

product withdrawal crystallizer by application of the Routh-Hurwitz

stability criterion and found that the system exhibits instability when

p > 21, that is, when the nucleation rate varies with a power of growth

rate greater than twenty-one.

Sherwin et al. [19] have also investigated the same problem. Their

treatment of the situation is more general since they have considered

the effects of seeded feed and of nuclei dissolving on the system











stability. These authors have expressed the stability criterion in

terms of a ratio b/g which they have defined as


d dNo
ds dt o o (2-25)
b/g = dr dN
ds o dt 0

Substituting Equation (2-12) into Equation (2-25) gives




b/g = kNprop do ro p(2-26)
dr
()o kN ro

For a clear feed, it was found that the system becomes unstable when

b/g > 21, an observation which is in agreement with that of Randolph

and Larson. A seeded feed was found to improve the stability in certain

cases. An interesting observation in connection with the effect of

liquid voidage was that change in the liquid void fraction does not have

a significant bearing on the stability criterion. In a qualitative

sense, this observation appears to be correct. As it will be shown in

Chapter V, a fluctuation in the voidage causes a perturbation in the

rate of nucleation in the opposite sense, such that the two effects tend

to nullify each other, so that the dynamic behavior of the system is not

altered significantly. Another observation made was that the system sta-

bility increases significantly if the nuclei have a finite size and that

increasing the nuclei size enlarges the region of stable operation.

The important point brought out in these studies is that the rela-

tive kinetic order of nucleation rate to growth rate plays a significant

role in determining the stability limits in a mixed product withdrawal









20


crystallizer; and that,for unstable operation, the criterion of p > 21

establishes the stability limit of the system wherein the feed is clear,

and the nuclei have a size of "zero" dimension.


















CHAPTER III

THE MIXED SUSPENSION CRYSTALLIZER WITH
CLASSIFIED PRODUCT WITHDRAWAL


Description of the Model

The concept of population conservation, developed in Chapter II,

may be applied to a mixed suspension crystallizer with classified

product withdrawal. Any model describing such a system should conform

to simple mathematical formulation and, at the same time, must be

flexible enough to fit a realistic situation. Consider a mixed sus-

pension crystallizer with a classifying device as shown in Figure 2,

wherein particles below the classification size Le (undersize) are

withdrawn at a rate Q1, and particles above the classification size

(oversize) are removed at a rate Q2 such that


Qi = aQ2

where a is the ratio of the two flow rates, and Qi is also the volumetric

withdrawal rate of the clear liquor for maintaining material balance

within the system. The volumetric feed rate is Q0, and the crystal-

lizer has a constant volume V. The classifying device C with a recycle

is shown as an external component of the crystallizer; however, size

classification may occur internally as in an Oslo-Krystal type unit.

In either case, the basic mechanism of size classification remains un-

altered and, thus, lends itself to the same mathematical description.


































FIGURE 2: THE MIXED
CLASSIFIED


QI ,C
SUSPENSION CRYSTALLIZER WITH
PRODUCT WITHDRAWAL












It is-further assumed that the-crystallizer is operating-under conditions

of steady energy throughput; that is, the process is isothermal, and is

under a steady state mass balance with constant material densities.

Population Balance

Two population balances are needed to describe the particle popu-

lation distribution within such a system, one for the undersize and one

for the oversize. These are


9an S3n Qin1
+ r + = 0 L in [0, L ] (3-1)
at aL V c

and

an2 an2 Q2n2
-- + r + = +0 L in [L m] (3-2)
at aL V c

where nj and n2 are the population densities of the undersize and over-

size-particles, respectively, and where the growth rate is assumed to

be independent of size.

Further, let F = V/Q2 be the average residence time of the particles

based on the drawdown of oversize particles. Equations (3-1) and (3-2)

then become


3nI Sn1 an1
--+ r -- 0 L in [0, L ] (3-3)
at aL r c

and

an2 an2 n2
---+ r + -- + 0 L in [Lc, =] (3-4)
at aL r c

Steady State Population Density

At this point, it is of interest to analyze the steady state












behavior of Equations (3-3) and (3-4). At steady state, these equations

become


dnio anlo
dL r r
oo




dn2o n2o
dL r r
oo


L in [0, Lc]






L in [Lc, m]


(3-5)


(3-6)


which may be integrated using the following boundary conditions

o
B.C. 1 at L = 0, nlo = nlo


B.C. 2 at L = L C, nlo = n2o


Thus,


0
n10 = no


- aL
r r
0o
e


(3-7)


L (l-a)-L
rr
n20 = n20 e (3-8)


Several cases of these expressions are of interest.

Case 1 (a < 1)

In this case, as shown in Figure 3, the drawdown rate of oversize

particles is greater than that of undersize particles. In other words,

there is size classification,and the slope of Equation (3-8) is greater

than the slope of Equation (3-7). In Case a = 0, there would be no

withdrawal of undersize particles so that Equation (3-7) would be repre-

sented by a horizontal line. This idealized situation is shown in

Figure 4.















z
0




O
0-
0
0-
0
0-
0
-i


CRYSTAL


FIGURE 3: POPULATION


DENSITY


PLOT FOR O< < I.


SI -



0
0- -




Lc

CRYSTAL SIZE

FIGURE 4: POPULATION DENSITY PLOT FOR


SIZE


C= 0












Case 2 (a = 1)

This, of course, is the same as the case of mixed product with-

drawal inasmuch as the rates of removal of oversize and undersize

particles are equal. Figure 1 depicts this case.

Case 3 (a > 1)

This case would arise if the undersize particles were prefer-

entially removed from the system. As is shown in Figure 5, in such an

event,the slope of Equation (3-8) would be less than that of Equation

(3-7).

Case 4 (L = 0, L c+ o)

In both of these cases, there is really no size classification,

and, therefore, Equations (3-7) and (3-8) are reduced to those for a

mixed product withdrawal.

Dimensionless Equations

The following dimensionless groups may be defined in order to ex-

press Equations (3-3) and (3-4) in dimensionless forms

t n L r
S y = x and = -
r no rr r
0 00 0


Hence, Equations (3-3) and (3-4) become


3yl yil
-- + + ayl = 0 x in [0, B] (3-9)

and


ay2 aY2
-_- + X- + Y2 = 0 x in [8, ] (3-10)


where 8 = L /r r is the dimensionless classification size.
c oo





















no





z
w
o

z
o x
03



-JI
0


0-



Lc

CRYSTAL SIZE


FIGURE 5: POPULATION DENSITY PLOT FOR C( > I












Moment-Transformation

The nth moment of the population distribution is defined as the

integral


I (Population density) LndL, n = 0, 1, 2, ...


(3-11)


which for a classified product withdrawal crystallizer may be written

as the sum of two integrals


L 0

pn f niLndL + f


n2L dL, n = 0, 1, 2, ...


The physical significance of the moments is as follows


L
c
1'O = f

0

L
c

.1 =


njdL + f

L c


00

njLdL +

L
c


L

kA12 = kA f nL2dL + kA

0


(3-13)


n2dL = The total number
of particles in
the system


n2LdL = The total length
all the crystals
the system


(3-14)


n2L2dL = The total crystal
area in suspension

(3-15)


where kA is the area shape factor.

L
c
pk v3 = pk f nL3dL + pk f n2L3dL = The total mass
of crystals in
0 L suspension
c (3-16)


(3-12)











where k is the crystal volume shape factor, and p is the density.

Now the weight distributions of the product crystals may be written

as

Wi = pk nlL3 (3-17)


W2 = pkn2L (3-18)

so that the mean crystal size, with respect to the weight distribution,

is

L
c
SWILdL + WLdL

0 L
c
L = -- (3-19)
L
c
fWdL + / W2dL

0 L
c

which may be written in terms of the moments as


L = -- (3-20)
P3

Similarly, the variance of the weight distribution is

L
c
f (L-L)2 WIdL + / (L-L)2 W2dL

0 0
a = (3-21)
c
SWdL + WdL
0 L











L L 00 L CO
c C c
f W1L2dL + f W2L2dL 27 { WIdL + W2dL} + 2 {j WdL + W2dL}

0 L 0 L 0 L
c C C
L 00
C
f WidL + f W2dL

0 L
c

or in terms of the moments

P5 V4 2
0 = -- M(3-22)


Finally, the coefficient of variation may be written as

V5 42 2
{-- }
C.V. (3-23)
11I4
13

The moments in Equations (3-20), (3-22), and (3-23) can be deter-

mined by a sieve analysis of the product, and, conversely, a knowledge

of these moments is sufficient to characterize the performance of a

crystallizer. Further, as it will be shown, the partial differential

Equations (3-9) and (3-10) can be transformed into a set of ordinary

differential equations involving the moments. These moments, which

represent the total properties of crystal number, length, area, and mass

through Equations (3-13), (3-14), (3-15), and (3-16), respectively, de-

fine the stability of the system. Normalizing Equation (3-12) about the

steady state gives B 0

normalizedd ylxndx + y2xndx fn + g (3-24)












Next, multiplying Equation (3-9) by x" and integrating over the size

range x in [0, 8] gives


nayl ayl
{x -+x -- + axny} dx = 0
ae ax


(3-25)


0

which reduces to a set of ordinary differential equations involving

the undersize moments fn This set of equations is


dfo o
- + 4(y-yY)
de

df
d- + 0(n -
de


+ afo0 0



nf )n + af = 0
n-1 n


where


0
n o
yl -" Y
1x0 n
o


Yx=8


Similarly, Equation (3-10) may be moment transformed to yield a

set of ordinary differential equations involving the oversize moments gn.

Co


f yY2 aY2
{x -;0- + axn 7- + xny2} dx = 0
86 n Ya yax


(3-28)


which gives


dg0
d6 *Y + go = 0


(n = 0)


(n = 0)



(n # 0)


(3-26)



(3-27)


(3-29)












dgn + ng
dg a (n Y + ngn1)+ g = 0


(n # 0)


Steady State Moments

Under conditions of a steady state, Equation (3-24) becomes

8

(u ) yf x dx + fyx dx = f + g
(no)normalized = f xndx + f2xndx = fno + no
0 8


(3-30)


(3-31)


The steady state normalized population distributions yl and Y2 are the

dimensionless forms of Equations (3-7) and (3-8). Substituting these

into Equation (3-31) gives


(nonormalized
no normalized =


e-ax ndx+ e (1-a)-x xndx


0 8

Expanding the integrals on the right of Equation (3-32) gives


f e- [, + 2+n n(n-1)8 +n-2
no n+1 a a2 a3
a+


+ n n ]},
n n+l
a a


n = 0, 1, 2, ...


(3-33)


gno ea [Sn + n8nl + n(n-l) 8n-2 + ... + n!8 + n!],
no


n = 0, 1, 2, ...


(3-34)


so that the first few moments may be written as


foo = (-1 e-u )
f 0 o =


(3-32)










flo 1= 72 e".- (I.+E as)l


f2o l [1 ( + a0 + 2a")]

f3o = e-6 (1i + ac + 2i + 2 ( )] (3-35)

and


-aB
go80o = e


glo = e-a (B + 1)


g2o = e-a8 (82 + 28 + 2)


g3o = e-a (83 + 382 + 60 + 6) (3-36)

At this point, it is interesting to note the relation between the

solids residence time rs and the liquor residence time in a crystal-

lizer with classified product withdrawal.

The average solids residence time is defined as

r = crystal holdup volume
s crystal volumetric production rate


L =
c
k { nloL3dL +f n2oL3dL}

0 L
s "L
'c
S -

k r
v aIQ2 noL3dL + Q2 n2oL3dL} (3-37)

0 L

which upon normalization of the involved quantities becomes











fo + 83of y io + g3o
r=f30 + 930 I [. r (3-38)
af30 + 30 Q2 af30 + g83


For the case of mixed product withdrawal, a is unity, hence,


r r (3-39)

or in other words, the solids residence time and the liquor

residence time are equal, which is expected.

Alternatively, the production rate can be written in terms of

the growth rate and the total surface area as (See Appendix II for

the derivation)

Production rate = ( k) r kA 2oP (3-40)

so that the solids residence time may be expressed as

L
c
k{ nloL3dL + fn2oL3dL} p

0 L
r = c (3-41)
s L
c

kA nloL2dL + n2oL2dL} (o) p
0 L
c
Further, if the shape factors kA and k are defined in terms

of a unit cube then

k = 6k (3-42)
A v

so that Equation (3-41) is reduced to

1 f3o + g30o
r I2o +g2 (3-43)
s 3ry tf2o + 820


Now, for the case of a crystallizer with an ideal classifying action,












Equation (3-43) is further simplified, giving


1 f3o L
r = c (3-44)
s 3ro f2o 4r o

or

L 4F r (3-45)
C s

Thus, the classification size is four times the growth rate per solids

residence time, which was derived by Saeman [2].

Constraint on the Growth Rate

The growth rate can be expressed in terms of the moments inasmuch

as it is constrained by material balance. An overall material conser-

vation for the crystallizer may be expressed as


{mass in) {mass out} = {accumulation} (3-46)

Here, the accumulation term involves two factors, namely, the accumu-

lation of mass in supersaturation and the accumulation of mass on the

crystals. The former factor is small compared to the latter and,

therefore, may be neglected. Thus, the material balance becomes


QoCo (a + 1) Q1C1 = ( ) kA2VP (3-47)

and Equation (3-47) reduces to


QoC (a + 1) Q1C1 () kAn (r ro)3 (f2 + g2) Vp (3-48)
A QoC 00

or
Kg
r = 2 + 2 (3-49)
f2 + 82


where












Q0CO (a + 1) Q1CI
Ko (3-50)
() kAno (roro)3 Vp


Thus, the growth rate is inversely proportional to the total crystal

area.

The normalized growth rate becomes

f20 + 820
r f2 + g20 (3-51)
r 0 2 +982

Expressing the Quantities y and yo as Functions of the Moments

Now that the growth rate is a known function of the moments,

Equations (3-26), (3-27), ,3-29), and (3-30) can be used to solve the

stability problem. These equations form a closed set at n = 2 inasmuch as

the growth rate involves the second moments of distribution. However, the

quantities y and y or the values of the normalized population density at

the classification size and the nuclei density, respectively, are still

unknowns in these equations; and their behavior as functions of the moments

must be known.

Hong's approximation

The following approximation expressing the relationship between y

and the moments is due to Hong and Larson [23].

Consider the integral

8 8 8

x dx = xyi ydx Y fo (3-52)

0 0 0

wherein the right-hand side is obtained by the method of integration by








37


parts. Further, changing the limits of the function on the left gives


f Yl f
x dx = xdyl
o y0


(3-53)


so that


/ xdyI
o
y


= BY fo


(3-54)


The mean value theorem may

and yl is bounded. Thus,


now be used since x is continuous in [0, 8],


Y

Y fo f xdyl (Y yO) X
0
y

x may now be approximated as

fl
x = -

so that

S- fo Y fl
By fo = (y yO)


and, therefore,


2 0
fo y fi
Y afo f


(3-58)


which expresses y in terms of the zeroth and the first moment of distri-

bution. Alternatively, y may be expressed as a function of the oversize

moments inasmuch as


Y = y1(B) =" y2(B)


(3-55)


(3-56)


(3-57)


(3-59)











and, therefore, following the same approach

00

Jx T By go Yx (3-60)



and letting X = gl/g0 gives

2
go
( ag go ) (3-61)
(B80 gl)

One way to establish the validity of Equations (3-58) and (3-61) is to

obtain the steady state value of y by substituting the corresponding

values of the undersize and oversize moments in these equations. In

both cases,


yo e-a8 (3-62)

which is correct.

yv and the power law

In order to express the normalized nuclei density yO in terms of

the moments, the power law of Equation (2-14) is used since it is

assumed to relate correctly the nuclei density to the growth rate.

Normalizing Equation (2-14) gives

0 i
y =i (3-63)

and the value of 0 may be substituted from Equation (3-51) to give

f20 + g2o0i
S= 2 +2 2 ) (3-64)

The stability problem can now be analyzed since all the quanti-

ties are now known in terms of the moments of distribution.


















CHAPTER IV

STABILITY ANALYSIS


Linearization of Moment Equations

In Chapter III, the basic population balance equations, describing

the distribution of crystal entities within a mixed suspension crystal-

lizer with classified product withdrawal, were converted into a set of

linear differential equations in terms of the moments of distribution.

It was also mentioned that a set of these equations involving up to the

second moments close satisfactorily, and that this set is essentially

sufficient for a stability analysis. This set of equations is


df0
de- + (Y y0) + afo = 0 (4-1)


dfI
d- + (Byo foo) + afl = 0 (4-2).


df2
d- + (82y 2flo) + af2 = 0 (4-3)


dgo
d- yo + go = 0 (4-4)


dgl
do- 4(8o + goo) + gl = 0 (4-5)


dg2
d-- (82yo + 2glo) + g2 = 0 (4-6)

39











where

f02 y0fo
S= fo f (4-7)

and

Sf20 + 820 i
S f2 + g2 (4-8)

The set of Equations (4-1) through (4-8) are now linearized by

introducing linear perturbations in the system variables. A linearly

perturbed quantity may be expressed as the sum of the steady state

value and the perturbation. Thus,


fn (t) fno + n(t) (4-9)


gn(t) gno + g(t) (4-10)


y(t) = Yo + Y(t) (4-11)


y(t) = 1 + y(t) (4-12)


4(t) = 1 + $(t) (4-13)

where the quantities with ~ are the perturbations, and the steady state

values of the normalized (about the steady state) nuclei density and

growth rate, of course, are unity. The linear equations therefore be-

come

dfo
de + (1 yo) y + y auf (4-14)








41

dfl -
-d y + (2flo 8Yo) + f ofl (4-15)


df2
d- 82 + (2flo 82y) 0 + 2f1 af2 (4-16)



-- = y + y 80 (4-17)



dgi
d- -= 8 + (8o + goo) + g0 g1 (4-18)



dg2
d- 2 27 + (2yo + 2glo) 4 + 2g -g2 (4-19)


S= K1(f2 + g2) (4-20)


yo i (4-21)


y = K2fo + K3Y0 + K1fI (4-22)

where

K 1 (4-23)
f2o + g20

2foo (4-24)
K2 o (4-24)
K2 = foo flo

flo
K3 = (4-25)
8foo flo

and
y 1
K4 = (4-26)
Bfoo flo

It is interesting to note that all the quantities involved in Equations

(4-14) through (4-26) are dimensionless, and, therefore, the stability

of the system can be defined entirely in terms of dimensionless quantities.








42

Equations (4-14) through (4-19) are linearly dependent inasmuch as

the moments fo and fl are related to each other through Equation (4-22).

Thus, the five independent equations may be expressed in the vector form

as


fl f1

f2 f2
d = [A] (4-27)
de go go

gl gl

82 82


where the transformation matrix [A] is obtained by substituting the
-o
values of y and y from Equations (4-21) and (4-22) into Equations (4-15)

through (4-19).

The matrix elements are shown in Appendix III. The rank of the non-

symmetric matrix is five. In the matrix, the d6oeffic~int K5 is the

ratio flo/f0o.

Stability Analysis of the Matrix

The matrix [A] has a'fifth order characterist-ic equation of the form


A5 + AAX4 + A3X3 + A2A2 + AjX + A0 0 (4-28)

in which the coefficients of the terms are functions of the dimensionless

steady state parameters.

A stability test can be made by applying the Routh-Hurwitz criterion

to Equation (4-28). However, inasmuch as the process of evaluating the

coefficients of Equation (4-28) is somewhat tedious, an alternative method

consisting of evaluating the roots of the characteristic equation, or the

eigenvalues, was used in the present study. As will be seen in the sequel,









43

an investigation of the root-locus plots obtained from the eigenvalues

is sufficient in defining the stability limits of the system. The

digital analysis was made on an IBM 360/50 computer.


















CHAPTER V

RESULTS OF THE ANALYSIS


Root-Loci

The root-locus plots obtained for various chosen values of the para-

meters a and B are shown in Figures 6 through 20. Only the complex roots

are plotted since they are important in determining the stability limit.

In a stable system, all the roots lie on the left-hand side of the com-

plex plane, and a crossover to the right-hand side implies that the

system is exhibiting instability. In the present analysis, only the com-

plex roots crossover to the right-hand plane at the stability limit.

The following cases are of particular interest:

Mixed product withdrawal (a = 1)

Figure 6 shows that when the value of a is unity, the observed be-

havior is that of a mixed product withdrawal crystallizer; that is, the

system exhibits instability at p > 21. The value of the dimensionless

classification size B is, of course, arbitrary in this case.

Size classification (a < 1, 0 = 1)

Figures 7 through 13 show plots for a values ranging from 0.5 to

0.0. In all of these cases, the system is found to be more stable than

when the withdrawal is mixed. It is seen from these figures that all

the root-loci are nearly identical. From this observation, it may be

concluded that the system stability increases rapidly when a takes on a

value that is less than unity, that is, when the oversize withdrawal

rate is larger than the undersize withdrawal rate. However, little
44







45

IMAGINARY AXIS

0* 0 c

o -

















m






w 0






w
O

I I
0

II.

D 0
UI D












CL
4 0L
U-





-I I




4--
IdO


































0

I)



II

16



O
0


Co
0

I-
LL




0
o
OL

0
0






U:
































0







Sa
0
U-

0
-J
LL




0
0
0



0J
0L




























0


Q,

0
0




U.
0
-j
CL


U-


O

0


o
0
0







CD
0t

























0


g

0





I-O
.0



0



o

0
I-



0
0




0L
3
CD
ti




























0



0


0




I-
0
U-
0
0.
C,

O
0

I
O
I-


w



LL



























O



0
0




0
LL
I-


0
a.
0-
O



0
0o
0
O
..I


O
0:





























0










IL
I-






cu
0

0






0
U-
1-



0.



a0

=3












change in the stability criterion occurs once a is less than unity.

Another point of interest is the case of no-undersize withdrawal (a = 0).

Figure 13 shows that even for this idealized case, the system continues

to exhibit a high degree of stability.

Preferential removal of undersize particle (a > 1. 8 = 1)

This case would arise if the classifying action was reversed, that

is, if the underflow was made higher than the overflow. Figure 14 shows

that, inthis event, the stability criterion is relaxed, and the cross-

over occurs at P = 17 for a = 1.2.

Effect of varying the dimensionless classification size

The following cases were investigated:

For a 0.005; 6 = 2, 3, and 4

For a = 0.001; 8 = 2, 3, and 4

The root-loci for these cases are shown in Figures 15 through 20.

As expected from the previous results for values of a less than unity,

the stability criterion is found to be independent of the value of a,

but changes with increasing value of the dimensionless classification

size. A comparison of Figures 15 and 18 with Figures 11 and 12 shows

that for a given value of a the system stability increases greatly when

8 changes from a value of 1 to 2. For higher values of B, there is little

change in the stability criterion.

Effect of Suspension Void

It would be interesting to investigate the effect of fluctuations

in the suspension void on the stability of the system. The void fraction

e in the crystallizer may be expressed in terms of the moments as


e = 1 kM(f3 + g3)


(5-1)





























0

II










I-
0
-*
oC


C-)
O
0
h-


0
-J




0
0
IL

UF-
































II
0
oj



0






LL


0



O
0

0
-I

0

Q:



w
a:
0


































arj
0




in




Z-
o








U-
o













0

-J
0





cr
:3












LL
O




1-
O




LJ

L0.








37


IMAGINARY AXIS

0 0 0

O









N c;









O




to
a:4 0
_\_.___













CL
= 0
c.

___ Ix
z u







2 .J
< Oa










r 0





























CLj
of





it


0
LL
I-


0
-i



ao
CO
0








3
at
0









































z
w
h:
w
41
2C
a-




0-


oL
0
w
-J

$


0
r-r



0

0
II


0



0
-o
0-


0
-1
0



0
0



Ow
O
LJ







r.
D,






























0

II


0


It


Z5
O
O
U-




U)
-j



C)
0







-J
L-
0
0




CD
Oh)
w?
cr












where

k = k nrr (5-2)
M V 0 0

Further, the nucleation-growth rate law of Equation (2-14) must be re-

defined inasmuch as the nuclei density is based on the free volume.

Thus,


nO = e i (5-3)

Introducing the dimensionless void fraction

V -(5-4)
EO

Equations (5-1) and (5-3) may be made dimensionless to give

1 kM(f3 + g3)
1 kM(f3o + g30)

and

y = i (5-6)

The linearized forms of Equations (5-5) and (5-6) are


v = K6(f3 + 83) (5-7)

where
kM
Kk (5-8)
K6 1 M(f3 + g3o)5-8)


and

y = + i0 (5-9)

So that, substituting Equation (5-7) into Equation (5-9) gives


yo i K6(f3 + g3)


(5-10)












If Equation (5-10) were used instead of Equation (4-21) in

stability analysis, a matrix of rank seven would result inasmuch

as the third moments of distribution are now required to close the

set of moment equations.

Discussion

Figure 21 is a block diagram depicting the interrelationship be-

tween the various quantities involved in defining the size distribution.

The level of supersaturation s is determined by the feed rate Qo and

the total crystal surface area A. The rate of nucleation and growth,

in turn, depend on the supersaturation. Finally, as is seen on the

right, the size distribution is defined by nucleation and growth through

the population balances. However, since the total surface area is a

function of the size distribution, there is an internal feed back in

the system as shown. This feed back is the cause of the cyclic insta-

bility in the crystallizer [4].

The important point brought out in the present analysis of the

stability problem is that size classification tends to stabilize a

crystallizer. The residence time probabilities of the undersize and

of the oversize particles are important in defining the stability limit

of the system. If the residence time of the oversize particles is re-

duced relative to that of the undersize, the system shows increase

stability. On the other hand, if the smaller particles are drawn down

at a relatively lower rate, the stability criterion is relaxed.

Sherwin et al. [11] with their model of an ideal classifying device

having no drawdown of undersize particles and with an infinitely high

rate of withdrawal of oversize particles find that classification







63


OLJ






W
(0

a-

0

ZW



z




--z
SIz
0

-

u,
z
0 w
I X


CO :3


w


sZ
<0




a O
CO
0











actually introduces a high degree of instability into the system.

They give a value of 2.3 for the sensitivity ratio of nucleation

rate to growth rate as compared to a value greater than 21 found

in the present analysis as the criterion for stability. How-

ever, for reasons given in Chapter I, the validity of their work is

in question inasmuch as their model does not describe the function

of an actual classifying device in a realistic manner.

It has been shown [4] that in a single mixed suspension crystal-

lizer, the dominant crystal size, with respect to the weight distri-

bution, occurs at 0 = 3. From the present analysis, it appears that

the classified product withdrawal crystallizer operates with a high

degree of stability when size classification occurs near the region of

dominant crystal size.

The stability criterion defined in terms of the sensitivity ratio

p would be violated, as it was mentioned earlier, in the event of a

discontinuous jump in the nucleation rate. This would happen, as was

pointed out, at high supersaturations when a fluctuation in the feed

concentration causes the supersaturation to cross the metastable limit,

resulting in mass nucleation. Industrial crystallizers do operate near

this region of mass nucleation in order to achieve a maximum growth

rate consistent with low nucleation rate. Mass nucleation is therefore

likely to occur when there are upsets in the feed or in the temperature

of the system. Miller and Saeman [6] have made such observations in a

crystallizer with a classified bed producing ammonium nitrate crystals.

Referring again to the block diagram of Figure 21, it is apparent

that a positive perturbation in the suspension void in the system will












cause an increase in the level of supersaturation inasmuch as the

suspension area is reduced. This, in turn, will increase the nu-

cleation rate so that the surface area will tend to increase again.

Qualitatively, it therefore appears that the inclusion of the effect

of suspension void in the analysis will not have a significant bearing

on the stability criterion.

Summary

A realistic model of a mixed suspension crystallizer with classi-

fied product withdrawal was analyzed for stability limits. The moment

transforms of the population balance equations for the model were solved

for their characteristic roots on a digital computer for a range of

classification sizes and at different residence times of undersize and

oversize particles. The stability criterion was expressed in terms of

the sensitivity ratio of nucleation rate to growth rate, and it was

found that, in general, size classification tends to stabilize the

classified product crystallizer. The following points are relevant to

the current discussion:

1. The form of the nucleation-growth rate law used for the analy-

sis is of considerable importance in predicting the dynamic behavior

of the system inasmuch as the stability limit of the system is defined

in terms of the relative order of nucleation rate to growth rate. Un-

fortunately, as was pointed out earlier, experimental information on

this important interactive effect is scarce, and the power law used in

the present study is just an approximation.

2. The residence time was arbitrarily based on the drawdown rate

of oversize particles. It is likely that a more definitive way of

parameterizing the problem exists.









66


3. Hong's approximation for correlating the population density

at the classification size with the moments of distribution may not

correctly represent the actual dynamic behavior.

In view of the above considerations, it is clear that the present

work needs further elaboration. Also lacking, among other things, are

essential experimental results for gaining insight into the dynamic

behavior of the classified product crystallizer. More information in

this area is necessary for a greater clarification of the problem.


















REFERENCES


1. Perry, J.H., Chemical Engineers' Handbook, 4th Ed., p. 17-7,
McGraw-Hill, New York, 1963.

2. Saeman, W.C., A.I.Ch.E. Journal, 2, p. 107, 1956.

3. Robinson, J.N. and R.C. Roberts, Can. J. Chem Eng., 35, p. 105,
1957.

4. Randolph, A.D. and M.A. Larson, A.I.Ch.E. Journal, 8, p. 639,
1962.

5. Randolph, A.D. and M.A. Larson, Chem. Eng. Progr. Symp. Series,
61, p. 147, 1965.

6. Miller, P. and W.C. Saeman, Chem. Eng. Progr., 43, p. 667, 1947.

7. Finn, R.K. and R.E. Wilson, Agr. Food Chem., _, p. 66, 1954.

8. Thomas, W.M. and W.C. Mallison, Petrol. Refiner, p. 211, 1961.

9. Bransom, S.H., Brit. Chem. Eng., p. 838, 1960.

10. Han, H.D. and R. Shinnar, Paper presented at the 60th National
Meeting of the A.I.Ch.E., April, 1967.

11. Sherwin, M.B., R. Shinnar, and S. Katz, Paper presented at the
A.I.Ch.E. Symposium on Crystallization, Houston, February, 1967.

12. Cha, L.C. and L.T. Fan, Can. J. Chem. Eng., 41, p. 62, 1963.

13. Frederickson, A.G. and H.M. Tsuchiya, A.I.Ch.E. Journal, 9,
p. 459, 1963.

14. Hulburt, H.M. and S. Katz, Chem. Eng. Sci., 19, p. 555, 1964.

15. Randolph, A.D., Note to the Editor, Can. J. Chem. Eng., p. 280,
1964.

16. Van Hook, A., Crystallization: Theory and Practice, ACS Mono-
graph 152, p. 94, Reinhold, New York, 1961.

17. Shinnar, R., J. Fluid Mech., 10, p. 259, 1961.












18. Rumford, F. and J. Bain, Trans. Inst. Chem. Engrs., 38, p. 10,
1960.

19. Sherwin, M.B., R. Shinnar, and S. Katz, A.I.Ch.E. Journal, 13,
p. 1141, 1967.

20. Randolph, A.D., A.I.Ch.E. Journal, 11, p. 424, 1965.

21. Canning, T.F. and A.D. Randolph, A.I.Ch.E. Journal, 13, p. 5,
1967.

22. Murray, D.C. and M.A. Larson, A.I.Ch.E. Journal, 11, p. 728,
1965.

23. Hong, K.C. and M.A. Larson, Paper presented at the A.I.Ch.E.
Joint Automatic Control Conference, Minneapolis, May, 1963.

24. McCabe, W.L., Ind. Eng. Chem., 21, p. 30, 1929.
































APPENDICES















APPENDIX I


LEIBNITZ RULE


Leibnitz's rule for the differentiation of definite integrals

is given as


bt(t)


b(t)


f(x,t) dx =


a(t)


{(x.t) + d dx t l
t "xd[ f(x,t)]A dx
at dx -dt
(1A-1)


where the function f(x,t) is integrated over the variable limits


a(t) to b(t)..


a(t)
















APPENDIX II

CRYSTAL PRODUCTION RATE


The total mass in the crystallizer is

L L L

M WIdL + / W2dL = k n1L3dL + k p n2L3dL (3A-1)

0 L 0 L
c c

Now at specified drawdown rates and at a given classification size,

the total mass is constant.- Therefore,

L O
c

dt dt nL3dL + n2L3dL 0 (3A-2)
0 L
c
or

L
anL an2
SL3 dL + L3 -- dL = 0 (3A-3)

0 L

Substituting the values of the partial differential quantities within

.. the integral from Equations (3-3) and-(3-4) gives

L
an1 can1 p an2 n2
L3{- r --- } dL + L3{- r } dL = 0 -4)

0 Lc











These integrals upon evaluation-by-the-method-of-integration

by parts give

Lc

a f nlL3dL +f n2LdL

0 L
r = L (3A-5)
c
3r f njL2dL + n2L2dL

0 L
c

So that the production rate may- be expressed as

L

Production rate a nL3dL + n23dL rk (3A-6)

0 L


where the shape factors kA and k are expressed in terms of a unit

cube so that


kA = 6kv
A v


(3A-7)















r1-






o -o
o 0

0
SI + 0
I 0 0 N
o N -_ +
SboN + 1 4

+ ,a ca
N m crv m
m CY *


H -
M M I I







E-44




00 0 0 0 0 N


H +


I O
Iy g~ ,6 O <-0
0









N +
44 1 + 0
0 0 C




ca +-









0N % +
(M% '-.- +


E-4-
Scn cN

















c ar
e( -a -t t^
m (nl 1t b





















H Ln N1 Ni N
I I






txi I M MG 3r


















BIOGRAPHICAL SKETCH


Ashish Jayant Mehta was born July, 1944, at Bombay, India. He

was graduated from Fellowship High School, Bombay, in June, 1960.

In June, 1964, he received the degree of Bachelor of Science in

Chemistry from the University of Bombay. He then studied at the

University of California at Berkeley, and there received the degree

of Bachelor of Science with a major in Chemical Engineering in

December, 1960. In January, 1967, he enrolled in the Graduate School

of the University of Florida where he had a departmental assistant-

ship. From June, 1967, until the present time, he has been working

for the Department of Coastal and Oceanographic Engineering of the

University of Florida. During this period, he also completed the

final phase of the present work.

Ashish Jayant Mehta is a member of the student chapter of the

American Institute of Chemical Engineers.












This thesis was prepared under- the direction of the chairman of

the candidate's supervisory committee-and-has-been-approved by all

members of that committee. It was submitted to the Dean of the College

of Engineering and to the Graduate Council, and was approved as partial

fulfillment of the requirements for the degree of Master of Science in

Engineering.



March, 1969


Dean, Graduate School


Supervisory Committee:








-------





































UNIVERSITY OF FLORIDA
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3 1262 08554 0341