september 29, 1999 department of chemical engineering university of south florida tampa, usa
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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. What is the Population Balance Technique (PBT)?. - PowerPoint PPT PresentationTRANSCRIPT
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September 29, 1999
Department of Chemical EngineeringUniversity of South Florida
Tampa, USA
Population Balance Techniquesin Chemical Engineering
by
Richard Gilbert&
Nihat M. Gürmen
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R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
Part I -- Overview
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R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
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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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
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R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
Application Areas
• colloidal systems
• crystallization
• fluidization
• microbial growths
• demographic analysis
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R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
Origins of population balances: Demographic Analysis
• Time = t• Age =
Tampan(,t)
Ni(,t) No(,t)
EmigrationImmigration
Birth RateDeath Rate
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R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
A Mixed Suspension, Mixed Product Removal (MSMPR) Crystallizer
Qi, Ci, ni
Qo, Co, n
Particle SizeDistribution(PSD)
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R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
Information diagram showing feedback interaction
MassBalance
GrowthKinetics
NucleationKinetics
CrystalArea
PopulationBalance
Growth Rate
NucleationRate
PSD
Feed
Growth Rate
Supersaturation
(from Theory of Particulate Processes, Randolp and Larson, p. 3, 1988)
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R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
Part II -- Mathematical Background
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R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
Two common density distributions by particle number
Size, LPo
pula
tion
Den
sity
, n(L
)
Normal Distribution
Popu
latio
n D
ensi
ty, n
(L)
Size, L
Exponential Distribution
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R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
Exponential density distribution by particle number
Size, L
Cum
ulat
ive
Popu
latio
n, N
(L)
Size, LPo
pula
tion
Den
sity
, n(L
)
dLL
0zddL
N1N1
N1 is the number of particles less than size L1
n1 is the number of particles per size L1
L1 L1
n1
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R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
Normal density distribution by particle number
Size, L
Cum
ulat
ive
Popu
latio
n, N
(L)
0
LzddL
Ntotal = Total number of particles
Lmax Size, LPo
pula
tion
Den
sity
, n(L
)
Ntotal
Lmax
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R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
Normalization of a distribution
f L dL f L dLL
L
( ) ( )min
max
zz 100
normalized
f L n L
n L dL( ) ( )
( )z
0
One way to normalize n(L)
Size, L
Nor
mal
ized
Pop
ulat
ion
Den
sity
, f(L
)
Lmax
1
0
Area under the curve
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R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
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.
L L f L dLz ( )0
2 2
0
zL L f L dLc h ( )
c vL
. . 2
where 2 (variance) is
L
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R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
Moments of a distribution
j-th moment, mj, of a distribution f(L) about L1
m L L f L dLj
j
z 10
b g ( )
Mean, = the first moment about zeroL
Variance, 2 = the second moment about the mean
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R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
Further average properties: Skewness and Kurtosis
j-th moment, j, of a distribution f(L) about mean
j
j
L L f L dL
zc h ( )
Skewness, 1 = measure of the symmetry
about the mean (zero for symmetric distributions)
1
33
2
4
22 3
FHGIKJKurtosis, 2 = measure of the shape of
tails of a distribution
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Part III -- Formulation of Population Balance Technique
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R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
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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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) coordinatesm internal coordinates (size, age, etc.)
Total number of particles is given by
N t n S t dSS
( ) ( , )Space S
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R. Gilbert & N. Gürmen, v.1.0, Tampa 1999
The particle number continuity equation
Accumulation Input Output Generation
R1
S
ddt
ndR B D dRR R1 1
a subregion R1 from the Lagrangian viewpoint
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x y z x jj
m
1
ddt
n R t dR B D dRR R
,b g b g1 1
z z
wherex is the set of internal and external coordinates spanning the phase space R1
d xdt
v v ve i
Convenient variable and operator definitions
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Applying the product rule of differentiation to the LHS
FHG
IKJznt dR n d x
dtR R11
FHG
IKJ
LNMM
OQPPz z
ntdR n d x
dtdR
R R1 1
FHG
IKJ
LNMM
OQPPz
nt
d xdt
n dRR1
ddt
n R t dRR
,b g1
z
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Substituting all the terms derived earlier
nt
v n v n D B dRe iR
1
0
As the region R1 is arbitrary
nt
vn D B 0
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nt x
v ny
v nz
v nx
v n D Bx y zjj
m
j
10
In terms of m+3 coordinates
Averaging the equation in the external coordinates
nt
v n nd V
dtB D Q n
Vik k
k
bg b gln
Micro-distributed Population Balance Equation
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
B v C u v u n u n v u duv
12 0
,
D v n v C v v n v dv
,0
the rate at which a bubble of volume u coalesces with a bubbleof volume v-u to make a new bubble of volume v is
a death function consistent with the above birth function would be
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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 K x y x ya, 1 3 1 3 1 3 1 3
2) Shear flow
C x y K x ya, 1 3 1 3 3
Coagulation kernel, C(x,y)
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Simplifications for a Solvable System
• dynamic system => t• spatially distributed => x, y, z• single internal variable, size => L
nt x
v ny
v nz
v nL
v n D Bx y z L 0
nt
v nL
Gn D Be 0
Growth rate G is at most linearly dependent with particle size => G G aLo 1
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Moment Transformation
Defining the jth moment of the number density function as
m x t n x t L dLj e ej, ,
0
Averaging PB in in the L dimension
L nt
v nL
Gn D B dLje
LNM OQP z bg0
0
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mt
v m B jG m am B Dje j
jj j 0 0
0 1( )
j = 0,1,2,...
Microdistributed form of moment transformation
Macrodistributed form of moment transformation
mt
m d Vdt
B jG m amQ m
VB Dj
jj
j jk j k
k
(log ) ( ) ,0 00 1
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
Tampan(,t)
Ni(,t) No(,t)
• neglect spatial variations of population• one internal coordinate, age
EmigrationImmigration
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-stateVolume of the tank : VOutflow equals the inflowFeed stream is free of particles
Growth rate of particles is independent of sizeThere are no particles formedby agglomeration or coalescnce
Derive the model equations for the system.
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ReferencesBOOK
• 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