1. Population Balance Techniques in Chemical Engineeringby
Richard Gilbert
&
Nihat M. Gürmen
September 29, 1999
Department of Chemical Engineering
University of South Florida
Tampa, USA

2. R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
Part I -- Overview
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

3. 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.
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

4. R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
(Dis)advantage of PBT
Advantage
Analysis of complex dispersed phase system
Disadvantage
Difficult integro-partial differential equations
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

5. R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
Application Areas
colloidal systems
crystallization
fluidization
microbial growths
demographic analysis
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

6. Origins of population balances: Demographic Analysis
Time = t
Age = q
Tampa
n(q,t)
Ni(q,t)
No(q,t)
Emigration
Immigration
Birth Rate
Death Rate
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

7. A Mixed Suspension, Mixed Product Removal (MSMPR) Crystallizer
Qi, Ci, ni
Particle
Size
Distribution
(PSD)
Qo, Co, n
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

8. Information diagram showing feedback interaction
Mass
Balance
Growth
Kinetics
Nucleation
Crystal
Area
Population
Growth Rate
Rate
PSD
Feed
Supersaturation
(from Theory of Particulate Processes, Randolp and Larson, p. 3, 1988)
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

9. Part II -- Mathematical Background
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

10. R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
Two common density distributions by particle number
Population Density, n(L)
Size, L
Exponential Distribution
Size, L
Population Density, n(L)
Normal Distribution
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

11. Exponential density distribution by particle number
Cumulative Population, N(L)
Population Density, n(L) N1 n1
Size, L
Size, L L1 L1
N1 is the number of particles less than size L1
n1 is the number of particles per size L1
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

12. Normal density distribution by particle number
Ntotal = Total number
of particles
Size, L
Population Density, n(L)
Ntotal
Lmax
Cumulative Population, N(L)
Lmax
Size, L
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

13. Normalization of a distribution
Size, L
Normalized Population Density, f(L)
Lmax 1
One way to normalize n(L)
normalized
Area under
the curve
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

14. Average properties of a distribution
The two important parameters of a particle size distribution are
* How large are the particles?
mean size,
* How much variation do they have with respect to the mean size?
coefficient of variation, c.v.
where 2 (variance) is
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

15. Moments of a distribution
j-th moment, mj, of a distribution f(L) about L1
Mean, = the first moment about zero
Variance, 2 = the second moment about the mean
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

16. Further average properties: Skewness and Kurtosis
j-th moment, j, of a distribution f(L) about mean
Skewness, 1 = measure of the symmetry
about the mean (zero for
symmetric distributions)
Kurtosis, 2 = measure of the shape of
tails of a distribution
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

17. Part III -- Formulation of Population Balance Technique
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

18. 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
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

19. R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
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
Space S
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

20. The particle number continuity equation
a subregion R1 from the Lagrangian viewpoint R1 S
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

21. Convenient variable and operator definitions
where
is the set of internal and external coordinates spanning the phase space R1
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

22. Applying the product rule of differentiation to the LHS
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

23. R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
Substituting all the terms derived earlier
As the region R1 is arbitrary
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

24. R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
In terms of m+3 coordinates
Micro-distributed Population Balance Equation
Averaging the equation in the external coordinates
Macro-distributed Population Balance Equation
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

25. R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
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
a death function consistent with the above birth function
would be
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

26. R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
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
2) Shear flow
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

27. Simplifications for a Solvable System
dynamic system => t
spatially distributed => x, y, z
single internal variable, size => L
Growth rate G is at most linearly dependent
with particle size =>
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

28. Moment Transformation
Defining the jth moment of the number density function as
Averaging PB in in the L dimension
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

29. R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
Microdistributed form of moment transformation
j = 0,1,2,... ³ k
Macrodistributed form of moment transformation
j = 0,1,2,... ³ k
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

30. 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
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

31. R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
Part IV -- Examples
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

32. Example 1 : Demographic Analysis
neglect spatial variations of population
one internal coordinate, age q
Ni(q,t)
No(q,t)
Immigration
Emigration
Tampa
n(q,t)
Set up the general population balance equation?
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

33. 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.
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

34. R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
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 , 1988
Hulburt H. M. and T. Akiyama, Liouville equations for agglomeration and dispersion processes, I&EC Fundamentals, 8:2, p , 1969
Ramkrishna D., The prospects of population balances, Chemical Engineering Education, p ,43, 1978
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999

35. R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
THE END
R. Gilbert & N. Gürmen, v.1.0, Tampa 1999