Web-Published Supplement
This supplement presents a detailed derivation of the dimensionless model used in this
work. The method of moments is originally due to Hulbert and Katz.1 Additional
background information is available in reference texts.2-6
The units used in this derivation assume that the crystal size distribution function
and its moments are defined on a volume of suspension basis, i.e. 0 is the number of
crystals per unit volume of suspension (#/m3), kv 3 is the volume of crystals per unit
volume of suspension (m3/m3), and B is the rate of nucleation per unit volume of
suspension (#/m3s). We further assume that the total volume of the suspension does not
change significantly over the course of the batch. However, all of the analysis presented in
this paper will work regardless of the basis that is used (e.g. volume of clear liquid or
solvent mass) as long as that basis is used consistently.
1. Basic Equations
In the absence of agglomeration and breakage, a general statement of the
population balance for a well-mixed batch crystallization system is:
f  L, t   G  L, t  f  L, t  

0
t
L
(1)
where f  L, t  is the crystal size distribution (CSD) function, B is the nucleation rate
(#/m3s) and G is the crystal growth rate (m/s). Equation 1 is subject to an initial condition
based on the properties of the seeds at the beginning of the batch ( f  L,0 is given) and to
a left boundary condition:
f  0, t  
B  f  L, t  , t 
G  0, t 
(2)
The driving force for both nucleation and growth is the supersaturation:
S  C  Csat
(3)
In this work concentration is defined as the mass of dissolved solute per unit volume of
suspension. Common forms for the expressions for crystal growth rate and secondary
nucleation are:
G  kg S g
(4)
B  kb G   3
(5)
-1-
Where k g , g , kb and  are empirically determined kinetic parameters and  3 is the third
moment of the CSD. The moments of the CSD are determined according to:

i   Li f  L  dL i  0,1,2,...
(6)
0
And it can be shown (if the crystal growth rate is independent of size) that:
d 0
B
dt
(7)
d i
 iG i 1 i  1, 2,...
dt
(8)
A mass balance on the solute shows that the time rate of change of the concentration, C, is
given by:
dC
 3G c kv 2
dt
(9)
Where  c and kv are the crystal density and crystal volumetric shape factor, respectively.
The initial values of the moments are given by the seed properties and the initial
concentration is the starting batch concentration.
Generally speaking, it is desired to reduce the concentration from some initial
value C0 to some final value C f in some fixed batch time t f . This is accomplished by
reducing the saturation concentration Csat in some predetermined manner as a function of
time.
It is desirable to keep track of the nucleated crystals (subscript n) separately from
the seed-grown crystals (subscript s). In this case for the seed-grown crystals we can say:
d s ,0
0
dt
(10)
d s ,i
 iGs ,i 1 i  1,2,...
dt
(11)
And for the nucleus-grown crystals we can write:
d n ,0
B
dt
(12)
d n,i
 iGn,i 1 i  1,2,...
dt
(13)
Note that:
-2-
i   s ,i   n ,i
(14)
In order to generalize the analysis, we wish to non-dimensionalize Equations 4, 5
and 9 to 13. Table 1 summarizes all of the reference variables and dimensionless variables
used in this work.
Table 1: Definitions of reference variables and dimensionless variables
reference variables
dimensionless variables
1
C
G   kbt 4f  3
t  t t f
3   C0  C f   c kv 
B  B B
x0  x0
G  G G
L  L  Gt f

B  kb 3  kbt 4f  3
i  BGit if1 i  0,1, 2
C  C  C f
ms  ms
 C
0
Cf

0
Cf
 Gt 
f

f   Gf B
w  w  Gt f
 i   i  i


We start by defining all concentrations relative to the concentration at the end of the batch,
i.e. define
C†  C  C f
(15)
And
C 
C  Cf
(16)
C0  C f
Therefore C decreases from 1 to 0 over the course of the batch.
We also define:
t  t t f
3 
 C0  C f
C0†

 c kv
 c kv
(17)

(18)
and:
3 
3
3
(19)
We further define:
1
4  3
b f
G  k t
-3-

(20)
The reader can verify that G has units of velocity or linear crystal growth rate
(length/time). Then:
G 
G
G
(21)
Finally, we let:

B  kb 3  kbt 4f  3
B 
B

B
(22)
B
(23)

4  3
b f
kb 3  k t

We further define:
0  Bt f
(24)
1  BGt 2f
(25)
2  BG2t 3f
(26)
3  BG3t 4f
(27)
Note that Equation 27 is consistent with equation 18 after substitution of Equations 20 and
22:

4  3
b f
BG t  kb 3  k t
3 4
f
3
4  3 4
b f
f
 k t 
t  3
(28)
With these definitions, the differential equations for the time evolution of the moments of
the crystal size distribution become:
d 0

 B   G   3

dt
(29)
d 1
 G0
dt 
(30)
d  2
 2G 1
dt 
(31)
d 3
 3G2
dt 
(32)
The same treatment can also be applied to  s ,i and n ,i separately.
-4-
2. Monodisperse seeds
Next, consider the case where the crystallizer is seeded with monodisperse seeds
with initial mass per unit volume of suspension ms and length x0 . The initial seed crystal
size distribution is given by:
f0  L  
ms
  L  x0 
c kv x03
(33)
Where ms c kv x03 is the number of seed crystals per unit volume of suspension and   x 
is the Dirac delta function with the usual properties and the units of 1/length.
Define:
ms 
ms
C0†
(34)
x0 
x0
Gt f
(35)
L 
L
Gt f
(36)
Application of the scaling principle of the Dirac delta function (   x   1     x  )
gives:
f0  L  
ms
  L  x0 
Gt f c kv x03
(37)
ms
  L  x0 
x03
(38)
Then:
f 0  L  
Where:
f
Gf
B
(39)
The number of seeds (i.e. the zeroth moment of the initial seed crystal size distribution) is
given by:
 s ,0  0  
ms
 c kv x03
(40)
ms
x03
(41)
Whence:
s,0  0  
-5-
s,1  0  
ms
x02
(42)
s,2  0  
ms
x0
(43)
s,3  0  ms
(44)
At the end of the batch, the combined third moment of the CSD is:
3  t f  
3  t f  
ms   C0  C f

 c kv
3  t f 
3

ms  C0†
 ms  1
C0†
(45)
(46)
Thus, over the course of the batch, C decreases from 1 to 0, while  3 increases from ms
to ms  1 .
3. Polydisperse seeds
In any real system, it is not possible to achieve perfectly monodisperse seeds, and
some degree of polydispersity is expected. Our experience indicates that the degree of
polydispersity (i.e. the width of the distribution) has a less significant effect on the
operation of seeded batch crystallization processes than the average seed crystal size and
the total mass of seeds. Therefore in this work we have not considered the width of the
seed distribution as a major design variable. However in order to produce realistic plots
for the crystal size distribution, it is desirable to consider the case where the distribution
f(L)
has some finite width.
w
x0
L
Figure 1: Parabolic seed crystal size distribution function.
-6-
We consider the case where the initial seed crystal size distribution function f  L 
is parabolic with a distribution at the base equal to w  2 x0 , as shown in Figure 1. We
also desire that the total mass of the seeds will be equal to ms . Thus the function f  L  is
given by:
L  x0  w 2
0

w 
w 
 


f 0  L     a  L   x0     L   x0    x0  w 2  L  x0  w 2
2 
2 


 
0
L  x0  w 2
(47)
Where a is such that:

3
 L f0  L 
0
x0  w 2

L3 f 0  L  
x0  w 2
ms
 kv
(48)
Solving for a gives:
m 1
1

a  s  x03 w3 
x0 w5 
 c kv  6
40

1
(49)
And therefore:
f0  L   
1
ms  1 3 3 1
w  
w 
 


x0 w5   L   x0    L   x0   
 x0 w 
 c kv  6
40
2  
2 
 


(50)
for x0  w 2  L  x0  w 2 . Defining:
w 
w
Gt f
(51)
Gf  L, t 
B
(52)
And further defining:
f   L, t  
And substituting the definitions of the dimensionless variables into Equation 50 gives:
1
1
w   
w  
1
 


f 0  L  ms  x03w3 
x0 w5   L   x0     L   x0   
40
2 
2 
6
 


(53)
The dimensionless moments of the initial seed crystal size distribution are given by:
s,0  0  
20ms
x0  20 x02  3w2 
(54)
20ms
 20 x02  3w2 
(55)
s,1  0  
-7-
s,2  0  
ms  20 x02  w2 
x0  20 x02  3w2 
s,3  0  ms
(56)
(57)
4. Mullin-Nyvlt Trajectory
In 1971 Mullin and Nyvlt7 published a concentration trajectory for seeded batch
crystallization that will give a constant crystal growth rate in the absence of nucleation.
We briefly repeat their analysis to show the form of the trajectory within our
dimensionless framework.
If there is no nucleation, then all of the material that is crystallized out of solution
must contribute to seed growth. The seed size as a function of time will be given by:
x  t   x0  Gt
(58)
where by assumption the crystal growth rate G is constant. The third moment of the crystal
size distribution as a function of time is:
3  t   n0  x0  Gt 
3
(59)
where n0 is the number of seed crystals per unit volume of suspension. At the beginning
of the batch,
ms  c kv 3  0  c kv n0 x03
(60)
Whence:
n0 
3  t  
ms
 c kv x03
ms
3
x  Gt 
3  0
c kv x0
(61)
(62)
The concentration as a function of time is:
  Gt 3 
C  t   C0  c kv 3  t   ms  C0  ms  1    1


x0 


(63)
At the end of the batch:
  Gt f 3 
C f  C0  ms  1 
  1

x0 


Whence:
-8-
(64)
  Gt f 3 
C0  C f  ms  1 
  1

x0 


(65)
Subtracting C f from both sides of Equation 63, dividing both sides by  C0  C f  and
applying equation 65 gives:
  Gt 3 
ms   1    1 
3


x0 
1  at   1

C  t   C f C0  C f

  1


3
C0  C f
C0  C f
  Gt f 3 
1  a   1
ms   1 
  1

x0 






(66)
Where a  Gt f x0 and t   t t f .
1  at  1
C   t   1 
1  a   1
3
3
(67)
Rearrangement of equation 65 gives:
1
 C  Cf
3
 0
 1  1
x0  ms

Gt f
(68)
So:
1
 1
3
a
 1  1
 ms

(69)
Nomenclature
a
Parameter in Mullin-Nyvlt trajectory
B
nucleation rate (#/(m3 s))
C
concentration (mass of dissolved solute per unit volume of suspension) (kg/m3)
C0
initial batch concentration
Cf
final batch concentration
f
crystal size distribution function (#/m3 m)
G
crystal growth rate (m/s)
g
growth parameter (dimensionless)
kb
nucleation parameter (1/(m3 s)(m/s)-
kg
growth parameter (m/s)(kg/m3)-g
kv
volumetric shape factor (dimensionless)
-9-
L
crystal size (length) (m)
ms
mass of seeds per unit volume of suspension (kg/m3)
n0
number of seeds per unit volume of suspension (#/m3)
S
supersaturation (kg/m3)
t
time (s)
tf
final batch time (batch duration) (s)
x0
initial seed size (length) (m)
w
width of parabolic seed distribution (m)

nucleation parameter (dimensionless)
μi
ith moment of the crystal size distribution (mi/m3)
μs,i
ith moment of seed-grown crystal size distribution (mi/m3)
μn,i
ith moment of nucleated crystal size distribution (mi/m3)
ρ
crystal density (kg/m3)
References
1.
Hulburt HM, Katz S. Some problems in particle technology: A statistical
mechanical formulation. Chemical Engineering Science, 1964; 19: 555-574.
2.
Mersmann, A. Ed. Crystallization Technology Handbook. New York: Marcel
Dekker, 2001.
3.
Nyvlt, J. Design of Crystallizers. Boca Raton, FL: CRC Press, 1992.
4.
Ramkrishna D. Population Balances. San Diego: Academic Press, 2000.
5.
Randolph, AD. Larson MA. Theory of Particulate Processes. New York: Academic
Press, 1988.
6.
Tavare, NS. Industrial Crystallization: Process Simulation Analysis and Design.
New York: Plenum Press, 1995.
7.
Mullin JW, Nývlt J. Programmed cooling of batch crystallizers. Chemical
Engineering Science. 1971; 26: 369-377.
-10-