Population Balance Techniques
in Chemical Engineering
by
Richard Gilbert
&
Nihat M. Gürmen
September 29, 1999
Department of Chemical Engineering
University of South Florida
Tampa, USA
Part I -- Overview
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What is the Population Balance
Technique (PBT)?
PBT is a mathematical framework for an accounting
procedure for particles of certain types you are interested in.
The technique is very useful where identity of individual
particles is modified or destroyed by coalescence or
breakage.
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(Dis)advantage of PBT
Advantage
• Analysis of complex dispersed phase system
Disadvantage
• Difficult integro-partial differential equations
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Application Areas
• colloidal systems
• crystallization
• fluidization
• microbial growths
• demographic analysis
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Origins of population balances:
Demographic Analysis
• Time
• Age
Ni(q,t)
=
=
t
q
Tampa
Immigration
n(q,t)
Birth Rate
Death Rate
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No(q,t)
Emigration
A Mixed Suspension, Mixed Product Removal
(MSMPR) Crystallizer
Qi, Ci, ni
Particle
Size
Distribution
(PSD)
Qo, Co, n
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Information diagram showing feedback interaction
Growth Rate
Growth
Kinetics
Feed
Growth Rate
Mass Supersaturation
Balance
Nucleation Nucleation Population PSD
Rate
Kinetics
Balance
Crystal
Area
(from Theory of Particulate Processes, Randolp and Larson, p. 3, 1988)
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Part II -- Mathematical Background
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Population Density, n(L)
Population Density, n(L)
Two common density distributions by particle number
Size, L
Exponential Distribution
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Size, L
Normal Distribution
d
dL
N1
z
L
dL
0
L1
Size, L
Population Density, n(L)
Cumulative Population, N(L)
Exponential density distribution by particle number
N1
n1
L1
N1 is the number of particles less than size L1
n1 is the number of particles per size L1
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Size, L
Ntotal = Total number
of particles
d
dL
z
L
0
Size, L
Lmax
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Population Density, n(L)
Cumulative Population, N(L)
Normal density distribution by particle number
Ntotal
Size, L Lmax
Normalization of a distribution
Normalized Population Density, f(L)
1
0
One way to normalize n(L)
f ( L) 
n( L )
z

n( L)dL
z

0
f ( L)dL 
0
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f ( L)dL  1
Lmin  0
normalized
Area under
the curve
Size, L
z
Lmax
Lmax
Average properties of a distribution
The two important parameters of a
particle size distribution are
z

* How large are the particles?
mean size,
L  L f ( L) dL
L
0
* How much variation do they have with
respect to the mean size?
coefficient of variation, c.v.
where 2 (variance) is
 
2
zc

L L
0
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c. v.

2
L
hf ( L) dL
2
Moments of a distribution
j-th moment, mj, of a distribution f(L) about L1
zb

mj 
L  L1
j
gf ( L) dL
0
Mean,
L
= the first moment about zero
Variance, 2 = the second moment about the mean
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Further average properties: Skewness and Kurtosis
j-th moment, j, of a distribution f(L) about mean
j 
zc

L L

j
hf ( L) dL
3
1  3

Skewness, 1 = measure of the symmetry
about the mean (zero for
symmetric distributions)
Kurtosis, 2
= measure of the shape of
tails of a distribution
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2
F
 I
 G J 3
H K
4
2
2
Part III -- Formulation of Population Balance
Technique
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Basic Assumptions of PBT
(Population Balance Technique)
• Particles are numerous enough to approximate
a continuum
• Each particle has identical trajectory in
particle phase space S spanned by the chosen
independent variables
• Systems can be micro- or macrodistributed
Check these
Assumptions
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Basic Definitions
Number density function n(S,t) is defined in an
(m+3)-dimensional space S consisted of
3 external (spatial) coordinates
m internal coordinates (size, age, etc.)
Total number of particles is given by
N (t )   n( S , t )dS
S
Space S
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The particle number continuity equation
Accumulation  Input  Output  Generation
R1
a subregion R1
from the
Lagrangian
viewpoint
S
d
ndR    B  D dR

dt R1
R1
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Convenient variable and operator definitions
zb g zb
g
d
n R, t dR  B  D dR
dt R1
R1
x
is the set of internal and
external coordinates spanning
the phase space R1
where
dx
 v  ve  vi
dt
m

 


  
x y z j 1 x j
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Applying the product rule of differentiation to the LHS
zb g
d
n R, t dR
dt R1
F I
G
J
H K
z
n
dx

dR  n
t
dt
R
1
z
R1
L
F
G
zM
M
NH
IO
P
J
KP
Q
n
dx

dR    n
dR
t
dt
R
R
1
1
L
F
n
d x IO
M
P
z
   G nJ
dR
t
dt K
M
P
H
N
Q
R1
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Substituting all the terms derived earlier
 n

R  t    ven    vi n  D  BdR  0
1
As the region R1 is arbitrary
n
    vn  D  B  0
t
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In terms of m+3 coordinates
m
n 



 v x n 
v y n  v z n  
v jn  D  B  0
t x
y
z
j 1 x j
 
 
Micro-distributed Population Balance Equation
Averaging the equation in the external coordinates
b
g
bg
d lnV
n
Qk nk
  vi n  n
 B D
t
dt
V
k
Macro-distributed Population Balance Equation
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B - D terms represent the rate of coalescence
conventionally collision integrals are used for B and D
the rate at which a bubble of volume u coalesces with a bubble
of volume v-u to make a new bubble of volume v is
v
1
B v    Cu, v  un u n v  u du
20
a death function consistent with the above birth function
would be

D v   n v   Cv , v n v  dv 
0
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Coagulation kernel, C(x,y)
C(x,y) : the rate at which bubbles of volumes x and y
collide and coalesce.
in the modeling of aerosols two of the functions used for
C(x,y) are where Ka is the coalescence rate constant
1) Brownian motion
C x, y  Ka  x  y
13
13
 x
2) Shear flow
C x, y   Ka  x  y
13
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
13 3
1 3
y
1 3

Simplifications for a Solvable System
• dynamic system => t
• spatially distributed => x, y, z
• single internal variable, size => L
n 



 v x n 
v y n  v z n 
v L n  D  B  0

t x
y
z
L
 
n

 Gn  D  B  0
   ve n 
t
L
Growth rate G is at most linearly dependent
with particle size => G  G 1  aL
o
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Moment Transformation
Defining the jth moment of the
number density function as

m j  xe , t    n xe , t  Lj dL
0
Averaging PB in in the L dimension
n

L
O
  v n  b
Gng
 D  BP
dL  0
zL M
L
Nt
Q

j
e
0
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Microdistributed form of moment transformation
m j
   ve m j  0 j B 0  jG0 (m j 1  am j )  B  D
t
j = 0,1,2,... 
Macrodistributed form of moment transformation
m j
Qk m j ,k
d (logV )
j 0
 mj
 0 B  jG0 (m j 1  am j )  
BD
t
dt
V
k
j = 0,1,2,... 
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If Assumptions Do Not Allow Moment
Transformations
You have to use other methods of solving PDEs like
• method of lines
• finite element methods
difficult if both of your
variables go to infinity
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Part IV -- Examples
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Example 1 : Demographic Analysis
• neglect spatial variations of population
• one internal coordinate, age q
Ni(q,t)
Immigration
No(q,t)
Tampa
Emigration
n(q,t)
Set up the general population balance equation?
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Example 2: Steady-state MSMPR Crystallizer
Qi, Ci, ni
Qo, Co, n
The system is at steady-state
Volume of the tank : V
Outflow equals the inflow
Feed stream is free of particles
Growth rate of particles is
independent of size
There are no particles formed
by agglomeration or coalescnce
Derive the model equations for the system.
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References
BOOK
• Randolph A. D. and M. A. Larson, Theory
of Particulate Processes, 2nd edition, 1988,
Academic Press
PAPERS
• Hounslow M. J., R. L. Ryall, and V. R. Marhsall, A
discretized population balance for nucleation, growth, and
aggregation, AIChE Journal, 34:11, p. 1821-1832, 1988
• Hulburt H. M. and T. Akiyama, Liouville equations for
agglomeration and dispersion processes, I&EC
Fundamentals, 8:2, p. 319-324, 1969
• Ramkrishna D., The prospects of population balances,
Chemical Engineering Education, p. 14-17,43, 1978
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THE END
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