Transcript Univ MD Pop Bal Lect..

Population Balance Modeling:
Solution Techniques &
Applications
Dr. R. Bertrum Diemer, Jr.
Principal Division Consultant
DuPont Engineering Research & Technology
Lecture Outline
Introduction
 Applications to Particles

 General
Balance Equation
 Aerosol Powder Manufacturing
 Design Problem

Solution Techniques
R. B. Diemer, Jr., 2003
Introduction
Definitions & Dimensions

The population balance extends the idea of mass and
energy balances to countable objects distributed in
some property.

It still holds!
In - Out + Net Generation = Accumulation

External & internal dimensions

external dimensions = dimensions of the environment:

internal dimensions = dimensions of the population:
3-D space (x,y,z or r,z,q or r,q,f) and time
diameter, volume, surface area, concentration, age, MW, number
of branches, etc.
R. B. Diemer, Jr., 2003
A Unifying Principle!

Objects of distributed size found everywhere...

particles:
– granulation, flocculation, crystallization, mechanical alloying, aerosol
reactors, combustion (soot), crushing, grinding, fluid beds

droplets:
– liquid-liquid extraction, emulsification

bubbles:
– fluid beds, bubble columns, reactors

polymers:
– polymerizers, extruders

cells:
– fermentation, biotreatment

Population balances describe how distributions evolve
R. B. Diemer, Jr., 2003
Multivariateness


Multivariate refers to # of internal dimensions
Univariate examples:


Bivariate examples:





particle size… polymer MW...cell age
particle volume and surface area (agglomerated particles)
polymer MW and # of branch points (branched polymers)
polymer MW and monomer concentration (copolymers)
cell age and metabolite concentration (biomanufacturing)
Trivariate example:

drop size and solute concentration and drop age for internal
concentration gradients (liquid-liquid extraction)
R. B. Diemer, Jr., 2003
General Differential Form, 1-D Population
  u p (V , x, t ) n(V , x, t ) 
   D p (V , x, t )n(V , x, t ) 


G (V , x, t ) n(V , x, t )
V
+ S (V ,x,t )
convection
diffusion
“In - Out” in
external
coordinates
growth (“In - Out” in internal coordinate)
sources & sinks (Net Generation)

n(V , x, t )
t
accumulation
Note: object’s velocity may differ from fluid’s velocity owing to
either slip or action of external forces
R. B. Diemer, Jr., 2003
Steady-state, Axisymmetric,
Incompressible Flow


Eliminates 2 physical dimensions, time dimension
Axial Dispersion Model:
With slip...

Without slip...
uz


 
n(V , z ) 
u pz (V , z )  n(V , z )  
D
(
V
,
z
)

p
z
z 
z 


G (V , z ) n(V , z ) +S (V , z )
V

n(V , z )  
n(V , z ) 


D
(
V
,
z
)


G (V , z ) n(V , z ) +S (V , z )
p


z
z 
z 
V
Plug Flow Model, No Slip:
uz
n(V , z )


G(V , z ) n(V , z ) +S (V , z)
z
V
R. B. Diemer, Jr., 2003
Ideally Mixed Stirred Tank

Eliminates 3 physical dimensions
n(V , t )

Batch:

G(V , t ) n(V , t ) +S (V , t )
t
V

Continuous:


Unsteady state…

Steady state (eliminates time dimension as well)...

   n(V , t )  Fo (t )no (V , t )  F (t )n(V , t ) 
t
 
 (t )  
G (V , t ) n(V , t )  +S (V , t ) 
 V

no (V )  n(V )


d
G(V ) n(V ) +S (V )  0;
dV
R. B. Diemer, Jr., 2003


F
Example: MSMPR Crystallizer



MSMPR = mixed suspension, mixed product removal
Same as continuous stirred tank
Steady state model… no particles in feed, size
independent growth rate, no sources or sinks (no
primary nucleation, coagulation, breakage)...
n o (V )  n(V )


n o (V )  0;
d
G (V ) n(V )  +S (V )  0
dV
G (V )  G;
S (V )  0
dn
n(V )

 n(V )  n(0)e V / G
dV
G
R. B. Diemer, Jr., 2003
Applications to Particles
General Balance Equation
Particle Formation, Growth &
Transformation
Precursor
Molecules
. .. . ..
.
... . . . ..
Nucleation
Nuclei
Growth
Agglomerates
Breakage
Coagulation
Singlets
Coalescence
R. B. Diemer, Jr., 2003
Partially Coalesced
Agglomerates
Sources and Sinks
Also known as Birth and Death terms
 Types of terms:

 Nucleation
(birth only)
 Breakage (birth and death terms)
 Coagulation (birth and death terms)
R. B. Diemer, Jr., 2003
Full 1-D Population Balance
(a partial integrodifferential equation)
n
nucleation term
    u p n      D p n  
t
growth term

N  (V  vo ) 
G n 
coagulation terms
V

1 V

b (v, V  v) n(v) n(V  v ) dv  n(V )
b (v,V )n(v )dv
0
2 0



V

G( ) b(V ;  ) n( ) d   G(V ) n(V )
breakage terms
R. B. Diemer, Jr., 2003
N = nucleation rate
G = accretion rate
b  coagulation rate
G  breakage rate
b = daughter distribution
vo = nuclei size
Applications to Particles
Aerosol Powder Manufacture
Gas-to-Particle Conversion
Aerosol Synthesis Chemistry Examples



M  OR  y  y H 2O  MO y / 2  y ROH
M= Si, Ti, Al, Sn… R=CH3, C2H5...
M  OR  y  MO y / 2   y 2  ROR
M= Si, Ti, Al, Sn… R=CH3, C2H5...
Halide Ammonation:

MX y   y 2  H2O  MO y / 2  y HX
M= Si, Ti, Al, Sn… X=Cl, Br...
Alkoxide Pyrolysis:


MX y   y 4  O2  MO y / 2   y 2  X2
M=Si, Ti, Al, Sn… X=Cl, Br...
Alkoxide Hydrolysis:


L=H, CO…
Halide Hydrolysis:


A=Si, C, Fe…
Halide Oxidation:


AL y  A  yL/  y 2  L2
Pyrolysis:
M=B, Al … X=Cl, Br...
MX y   y 3  NH3  MN y / 3  y HX
R. B. Diemer, Jr., 2003
General Aerosol Process Schematic
Feed #1
Preparation
Feed #2
Preparation
.
.
.




Vaporization
Pumping/Compression
Addition of additives
Preheating
Feed #N
Preparation
R. B. Diemer, Jr., 2003
General Aerosol Process Schematic
Feed #1
Preparation
Feed #2
Preparation
.
.
.
Feed #N
Preparation
Aerosol
Reactor






Mixing
Reaction Residence Time
Particle Formation/Growth Control
Agglomeration Control
Cooling/Heating
Wall Scale Removal
R. B. Diemer, Jr., 2003
General Aerosol Process Schematic
Feed #1
Preparation
Feed #2
Preparation
.
.
.
Aerosol
Reactor
Base Powder
Recovery

Gas-Solid Separation
Feed #N
Preparation
R. B. Diemer, Jr., 2003
General Aerosol Process Schematic
Vent or Recycle Gas
Feed #1
Preparation
Feed #2
Preparation
.
.
.
Treatment
Reagents
Offgas
Treatment
Waste

Aerosol
Reactor
Base Powder
Recovery
Feed #N
Preparation
R. B. Diemer, Jr., 2003

Absorption
Adsorption
General Aerosol Process Schematic
Vent or Recycle Gas
Feed #1
Preparation
Feed #2
Preparation
Aerosol
Reactor
.
.
.
Feed #N
Preparation
Offgas
Treatment
Treatment
Reagents



Waste
Base Powder
Recovery
Degassing
Desorption
Conveying
Powder
Refining


R. B. Diemer, Jr., 2003
Coarse
and/or Fine
Recycle
Size Modification
Solid Separations
General Aerosol Process Schematic
Vent or Recycle Gas
Feed #1
Preparation
Feed #2
Preparation
.
.
.
Offgas
Treatment
Treatment
Reagents
Waste

Aerosol
Reactor
Base Powder
Recovery


Powder
Refining
Feed #N
Preparation
Coarse

and/or Fine
Recycle 

Formulating
Reagents
Product
Formulation
R. B. Diemer, Jr., 2003


Coating
Additives
Tabletting
Briquetting
Granulation
Slurrying
Filtration
Drying
General Aerosol Process Schematic
Feed #1
Preparation
Feed #2
Preparation
.
.
.
Vent or Recycle Gas 
Offgas
Treatment
Treatment
Reagents
Waste


Aerosol
Reactor
Base Powder
Recovery
Powder
Refining
Feed #N
Preparation

Coarse
and/or Fine
Recycle
Bags
Super
Sacks
Jugs
Bulk
containers
 trucks
 tank
cars
Product
Formulating
Reagents
Product
Formulation
R. B. Diemer, Jr., 2003
Packaging
Product
General Aerosol Process Schematic
Vent or Recycle Gas
Feed #1
Preparation
Feed #2
Preparation
.
.
.
Offgas
Treatment
Treatment
Reagents
Aerosol
Reactor
Waste
Base Powder
Recovery
Powder
Refining
Feed #N
Preparation
Formulating
Reagents
Product
Formulation
R. B. Diemer, Jr., 2003
Coarse
and/or Fine
Recycle
Packaging
Product
TiO2
Processes
R. B. Diemer, Jr., 2003
Thermal Carbon Black Process
Carbon Generated by Pyrolysis of CH4
Johnson, P. H., and Eberline, C. R., “Carbon Black, Furnace Black”, Encyclopedia of Chemical Processing and Design,
J. J. McKetta, ed., Vol. 6, Marcel Dekker, 1978, pp. 187-257.
R. B. Diemer, Jr., 2003
Furnace
Carbon
Black
Process
Carbon generated by
Fuel-rich
Oil Combustion
Johnson, P. H., and Eberline, C. R.,
“Carbon Black, Furnace Black”,
Encyclopedia of Chemical Processing
and Design, J. J. McKetta, ed., Vol. 6,
Marcel Dekker, 1978, pp. 187-257.
R. B. Diemer, Jr., 2003
Applications to Particles
Design Problem
Design Problem Focus
Vent or Recycle Gas
Feed #1
Preparation
Feed #2
Preparation
.
.
.
Offgas
Treatment
Treatment
Reagents
Aerosol
Reactor
Waste
Base Powder
Recovery
Powder
Refining
Feed #N
Preparation
Formulating
Reagents
Product
Formulation
R. B. Diemer, Jr., 2003
Coarse
and/or Fine
Recycle
Packaging
Product
Gas to Recovery
Steps
The Design Problem
Baghouse
Flame Reactor
Feeds
8 psig
min
.25 mm particles
10 psig
Pipeline Agglomerator
What pipe diameter
and length?
Cyclone
What
cut size?
25% of
particle mass
max
75% of particle
mass min
R. B. Diemer, Jr., 2003
Design Problem Physics

Simultaneous Coagulation & Breakage


initial size = .25 micron
Coagulation via sum of:
2kT
b c (V ,  ) 
3m

continuum Brownian kernel:

Saffman-Turner turbulent kernel:
bt (V ,  )  .31


   1/ 3  V 1/ 3 
2       
  V 
   

V  3V 2/ 3 1/ 3  3V 1/ 3 2/ 3   


Power-law breakage, binary equisized daughters:
3/ 2
Fractal particles:
1/ D f
 6V 
d p  d0  3 
  d0 
 
G()  110 s /cm   1/ 3
 


b(V ; )  2  V  
2

8 2
; D f  1.8
R. B. Diemer, Jr., 2003
Design Problem Aims

Capture particles with a cyclone followed by baghouse

Need 75% mass collection in cyclone to minimize bag wear from
back pulsing

Agglomerate in pipeline… Initial pressure = 10 psig,

Maximum allowable DP = 2 psia

Need to design:


cyclone - “cut size” related to design

agglomerator - pipe diameter and length needed to get desired
collection efficiency
Optimize?… minimize the area of metal in pipe and cyclone to
minimize cost?
R. B. Diemer, Jr., 2003
Problem Setup
Steady-state, incompressible,
axisymmetric flow
 Plug flow, no slip
 Neglect diffusion
 Population Balance Model:


n
uz

G( ) b(V ; ) n() d   G(V ) n(V )
V
z


1 V

b (v, V  v) n(v) n(V  v) dv  n(V )
b (v,V )n(v)dv
0
2 0


R. B. Diemer, Jr., 2003
Moments
Moments of n(V):

j

V
n(V )dV continuous form

 0
Mj  
 niVi j discrete form with Vi  iV0
 i 1
Key Moments:
M0 
M1 

n
i 1
i
 particle number concentration

 nV
i 1
i
i
 particle volume fraction
(proportional to particle mass concentration)
R. B. Diemer, Jr., 2003
Solution Techniques
Partial List of Techniques
Discrete Methods
Will discuss
 Sectional Methods
 Similarity Solutions
 LaPlace Transforms
 Orthogonal Polynomial Methods
 Spectral Methods
 Moment Methods
 Monte Carlo Methods

R. B. Diemer, Jr., 2003
Will not
discuss
Discrete Methods
Size is integer multiple of fundamental size
 Write balance equations for every size
 Gives distribution directly
 Huge number of equations to solve
 Have to decide what the largest size is
 Example for coagulation and breakage:
 d 03
Vi  iV0 ; V0 
6


dni 1 i 1
uz
  b j ,i  j n j ni  j  ni  b i , j n j   G j b(i; j )n j  Gi ni
dz
2 j 1
j 1
j i 1

R. B. Diemer, Jr., 2003
Discrete Example Problem Setup
  i 1/ 3  j 1/ 3 

b c ,i , j
 2        ; b t ,i , j  .31 V0  i  3i 2 / 3 j1/ 3  3i1/ 3 j 2 / 3  j 

 i  
  j 
3/ 2
j  2i

 2,

 
8 2
1/ 3 1/ 3
110 s /cm   V0 i , i  1

Gi  
;
b
(
i
;
j
)

j  2i  1, 2i  1

1,
 

0, j  2i  1, j  2i  1
0, i  1


2kT

3m

dni 1 i 1
uz
   b c , j ,i  j  b t , j ,i  j  n j ni  j  ni   b c ,i , j  bt ,i , j  n j
dz
2 j 1
j 1
G 2i 1n2i 1  2G 2i n2i  G 2i 1n2i 1  Gi ni
Need slightly more than 2106 cells to cover
entire mass distribution range!
R. B. Diemer, Jr., 2003
Sectional Method


Best rendering due to Litster, Smit and Hounslow
Collect particles in bins or size classes, with
upper/lower size=21/q, “q” optimized for physics
i
i-3 i-2 i-1
i+1 i+2
vi
21/q vi 22/q vi
23/q vi
Balances are written for each size class reducing the
number of equations, but too few bins loses resolution
And… now the equations get more complicated to get
the balances right
Still have problem of growing too large for top class
Directly computes distribution
2-3/q vi




2-2/q vi
2-1/q vi
R. B. Diemer, Jr., 2003
Sectional Interaction Types

Type 1:


Type 2:


some particles land in the ith interval and some in a larger interval
Type 4:


all particles land in the ith interval
Type 3:


some particles land in the ith interval and some in a smaller interval
some particles are removed from the ith interval and some from
other intervals
Type 5:

particles are removed only from ith interval
R. B. Diemer, Jr., 2003
Sectionalization Example: q=1
Collision of Particle j with Particle k
Particle j, 16
V/vo
14
12
4
In which section goes the
daughter of a collision between
Particle j in Section i and
Particle k in Section n?
10
8
6
3
2
1
4
2
0
0
Section
number i:
2i-1vo<V<2ivo
2
1 2
4
6
3
8 10 12 14 16 18 20 22 24 26 28 30 32
4
R. B. Diemer, Jr., 2003
5
Particle k,
V/vo
Sectionalization Example: q=1
Particle j, 16
V/vo
14
4
12
j+k = constant
10
8
3
6
2
1
4
Any collision between these lines
produces a particle in Section 5
2
0
0
Section
number i:
2i-1vo<V<2ivo
2
1 2
4
6
3
8 10 12 14 16 18 20 22 24 26 28 30 32
4
R. B. Diemer, Jr., 2003
5
Particle k,
V/vo
Sectionalization Example: q=1
i,i collisions: map completely into i+1
Particle j, 16
V/vo
14
4
12
10
8
3
2
1
6
4
2
0
0
Section
number i:
i-1
2 vo<V<2ivo
2
1 2
4
6
3
8 10 12 14 16 18 20 22 24 26 28 30 32
4
R. B. Diemer, Jr., 2003
5
Particle k,
V/vo
Sectionalization Example: q=1
i,i+1 collisions: 3/4 map into i+2, 1/4 stay in i+1
Particle j, 16
V/vo
14
4
12
10
8
3
2
1
6
4
2
0
0
Section
number i:
i-1
2 vo<V<2ivo
2
1 2
4
6
3
8 10 12 14 16 18 20 22 24 26 28 30 32
4
R. B. Diemer, Jr., 2003
5
Particle k,
V/vo
Sectionalization Example: q=1
i,i+2 collisions: 3/8 map into i+3, 5/8 stay in i+2
Particle j, 16
V/vo
14
4
12
10
8
3
2
1
6
4
2
0
0
Section
number i:
i-1
2 vo<V<2ivo
2
1 2
4
6
3
8 10 12 14 16 18 20 22 24 26 28 30 32
4
R. B. Diemer, Jr., 2003
5
Particle k,
V/vo
Sectionalization Example: q=1
i,i+3 collisions: 3/16 map into i+4, 13/16 stay in i+3
Particle j, 16
V/vo
14
4
12
10
8
3
2
1
6
4
2
0
0
Section
number i:
i-1
2 vo<V<2ivo
2
1 2
4
6
3
8 10 12 14 16 18 20 22 24 26 28 30 32
4
R. B. Diemer, Jr., 2003
5
Particle k,
V/vo
Sectionalization Example: q=1
i,i+4 collisions: 3/32 map into i+5, 29/32 stay in i+4
Particle j, 16
V/vo
14
4
12
10
8
3
2
1
6
4
2
0
0
Section
number i:
i-1
2 vo<V<2ivo
2
1 2
4
6
3
8 10 12 14 16 18 20 22 24 26 28 30 32
4
R. B. Diemer, Jr., 2003
5
Particle k,
V/vo
Sectionalization Example: q=1
i,n collisions: 3/2n-i+1 map into n+1, n>i>0
i,i collisions: all map into i+1
Particle j,
V/vo 16
i,icollisions: all map into i+1
i,i+1 collisions: 3/4 map into i+2
14
i,i+2 collisions:
3/8 map into i+3
12
10
8
i,i+3 collisions:
3/16 map into i+4
6
4
i,i+4 collisions:
3/32 map into i+5
2
0
0
interval
number i:
voi-1<V<voi
2
4
1 2
6
3
8 10 12 14 16 18 20 22 24 26 28 30 32
Particle k,
4
R. B. Diemer, Jr., 2003
5
V/vo
Sectional Coagulation Model, q=1

Model Equation:
i 2

 i 1

dNi
1
2
uz
 Ni 1  bi 1, jq i 1, j N j  bi 1,i 1 Ni 1  Ni  bi , jq i , j N j   bi , j N j 
dz
2
j 1
j i
 j 1


Tentatively:
3
2
q i , j  C 2 j i

Can show (via 0th and 1st moments) that:




number balance gives correct general form for arbitrary q i,j
mass balance only closes for C=2/3 when Vi/Vj=2i-j
final expression:
q i , j  2 j i
kernels evaluated via:
V  2V0 3V0
Vi  2i 1V1 with V1  0

(recovers 3/2 factor)
2
2
R. B. Diemer, Jr., 2003
General 21/q Sectional Coagulation Model
i  S ( q ) 1
i S ( q )
dN i
1
2
 N i 1  b i 1, jq i 1, j N j  b i  q ,i  q N i  q  N i   b i , jq i , j N j 
dt
2
j 1
 j 1

bi, j N j 

j i  S ( q ) 1




b i  k , j q i 1, j   k  Ni  k N j 

k  2 j i  S ( q  k  2)  k 1

 2 new terms
q
i  S ( q  k 1)  k 1
1/ q


b
q

2

N
N



i  k 1, j
i, j
k 1
i  k 1
j

k  2 j i  S ( q  k  2)  k  2
q
i  S ( q  k 1)  k
q
S (q)   m ; q i , j
m 1
2( j i ) / q
2(1 k ) / q  1
 1/ q ;  k  1/ q
2 1
2 1
R. B. Diemer, Jr., 2003
Sectional Example Problem Setup (for q=1)
bi, j 
2kT
  3V 
 2  2(i  j ) / 3  2( j i ) / 3   .31  0   2i 1  3  2(2i  j ) / 31  2( i  2 j ) / 31   2 j 1 

3m
  2 


8 2
1

10
s
/cm


Gi  



3/ 2
1/ 3
  3V0 
 

  2 
0, i  1
2(i 1) / 3 , i  1
2,
; b(i; j )  
0,
j  i 1
j  i 1
i 2 b

 i 1 bi , j

dNi
1
i 1, j
2
uz
 Ni 1  i  j 1 N j  bi 1,i 1 Ni 1  Ni  i  j N j   bi , j N j 
dz
2
j 1 2
j i
 j 1 2

2Gi 1 Ni 1  Gi Ni
Need about 22 sections to cover entire
mass distribution range, suggest using 25-30
R. B. Diemer, Jr., 2003
Sectional Example Problem Setup (for q=1)
Calculation of Mass Collection Efficiency
1/ D f
 Vi 
 d 03
1/ D f
i 1  3V0 
Vi  2 
;
V


d

d
n

d

0
i
0 p
0

6
V
 2 
 0
i 
 di d pc 
2
1   di d pc 
2

 3  2i  2 
1  3 2
2/ Df

 d0 d pc 
i 2 2 / D f
d
0
 d 0  3  2i  2 
1/ D f
I
2
d pc 
2
;  w  100% 
 V N
i 1
I
example
grade efficiency
curve
i i
V N
i 1
i
i
i
I
M 1   Vi N i  M 0oV1
i 1
I
Ni
Vi
i 1
ni  o ;
 2   w  100%  i 2i 1 ni
M0
V1
i 1
1, i  1
Suggests to do calculation using ni with n  
0, i  1
o
i
I
Check mass closure via:
1. If not true, model is coded incorrectly!
 2R.nB. Diemer,
Jr., 2003
i 1
i 1
i
Sectional Example Problem Setup (for q=1)
Nondimensionalization
1/ 3
bi, j
 3V 
 3V 
 b co  c ,i , j  b to  0   t ,i , j ; Gi  G o  0 
 2 
 2 
2(i 1) / 3
2kT
 
b co 
;  c ,i , j  2  2(i  j ) / 3  2( j i ) / 3 ; G o  1 108 s 2 /cm  
3m
 
b to  .31
3/ 2

;  t ,i , j  2i 1  3  2(2i  j ) / 31  2( i  2 j ) / 31   2 j 1

1/ 3
t
2 b co
 b b co M 0o  2 
 b co M 0o z
 
; t  
; b  


o
c
uz
 c 3b t V0
c
G o  3V0 
 i , j   c ,i , j 
 t ,i , j
t
; ni 
Ni
2M1 4M 1
o
;
M


0
o
M0
3V0  d 03
( i 1) / 3
i 2 
i 1 



dni
1
2
i 1, j
i
,
j
 ni 1  i  j 1 n j   i 1,i 1ni21  ni  i  j n j    i , j n j  
24 / 3 ni 1  ni 

d
2
b
j 1 2
j i
 j 1 2

R. B. Diemer, Jr., 2003
Solution Technique Choices



If analytical method works use it! (rare)

similarity solution

Laplace transform
If it is crucial to get distribution detail right, and it is a 1-D
problem, and it is a stand-alone model (typical of research)

discrete

sectional

Monte Carlo

Galerkin (orthogonal polynomial… commercial code: PREDICI)
If an approximate distribution will do, or if the moments are
sufficient, or if the distribution is multivariate, or if the model will
be embedded in a larger model (typical of process simulation)

moments
R. B. Diemer, Jr., 2003
Concluding Remarks



Population balance applications are everywhere
The mathematics is difficult (unlike mass & energy
balances)
There are many solution techniques… choice
depends on object of model
R. B. Diemer, Jr., 2003
Backup Slides
Moments

Moments of n(V):

j

V
n(V )dV
 0
continuous form
Mj  
 niVi j discrete form with Vi  iV0
 i 1
M 0  particle number concentration
M 1  particle volume fraction (proportional to particle mass concentration)

Moments of b(V;):
  V j b(V ;  ) dV

 0
continuous form
b j    1
 b(i;  )Vi j ; V   discrete form

 i 1
b0  p daughters/breaking event  2 in binary breakage
b1   , the parent size
R. B. Diemer, Jr., 2003
Particle Number Balance
  n 
dM 0
d 
uz
dV  u z
n(V )dV  u z



0  z 
dz 0
dz
 



0 V
G( ) b(V ;  ) n( ) d dV 
0
G(V ) n(V ) dV
1  V

b (v, V  v) n(v) n(V  v) dvdV
2 0 0
 


  n(V )  b (v, V )n(v)dvdV
0
0
R. B. Diemer, Jr., 2003
Interchange of Limits
V




0 V
0 0
f ( , V )d dV 
f ( , V )dVd
V=
 goes from V to 
then V from 0 to 
V goes from 0 to 
then  from 0 to 

R. B. Diemer, Jr., 2003
Particle Number Balance (cont.)
Interchange limits of integration in
both coagulation and breakage terms





dM 0
b
(
V
;

)
dV


uz

G ( ) n (  ) 0
d 
G(V ) n(V )dV


0
0
dz
  b0  p 



 
1  

b (v,V  v) n(v) n(V  v) dVdv 
b (v,V )n(V )n(v)dVdv
0 0
2 0 v
 
 
 ( p  1)

0
G(V ) n(V )dV
 
1  

b (v,V  v) n(v) n(V  v) dVdv 
b (v,V )n(V )n(v)dVdv
0
v
0
0
2
 
 
R. B. Diemer, Jr., 2003
Particle Number Balance (cont.)
Change of variable in coagulation integral:
 = V v
dV = d at constant v

dM 0
uz
 ( p  1)
G(V ) n(V )dV
0
dz

 
1  

b (v,  ) n(v) n( ) d dv 
b (v,V )n(V )n(v)dVdv
0 0
2 0 0
 
 
General Number Balance for p Daughters

1  

( p  1) 0 G(V ) n(V )dV  2 0 0 b (v,V )n(V )n(v)dvdV
continuous
dM 0 
uz


 
1
dz

( p  1) Gi ni   b i , j ni n j
2 i 1 j 1

i 1
discrete


 
R. B. Diemer, Jr., 2003
Particle Volume (Mass) Balance
dM 1
d 
 n 
V u z
dV  u z
Vn(V )dV  u z


0
dz 0
dz
 z 





0 V V
G( ) b(V ;  ) n( )d dV 

0
V G(V ) n(V )dV
V
1 

V
b (v, V  v) n(v) n(V  v) dvdV
0
2 0



0

V n(V )

0
b (v,V )n(v)dvdV
R. B. Diemer, Jr., 2003
Particle Volume (Mass) Balance
(cont.)
Interchange limits of integration in
both coagulation and breakage terms
 


Vb
(
V
;

)
dV
dM 1



uz

G (  ) n ( ) 0
d 
V G(V ) n(V )dV


0
0
dz

b




1



 
1  

V b (v, V  v) n(v) n(V  v) dVdv 
V b (v,V )n(V )n(v)dVdv
0 0
2 0 v
 
 
 
dM 1 1  
uz

V b (v, V  v) n(v) n(V  v) dVdv 
V b (v,V )n(V )n(v)dVdv
0 0
dz
2 0 v
 
 
R. B. Diemer, Jr., 2003
Particle Volume (Mass) Balance
(cont.)
Change of variable in coagulation integral:
 = V v
dV = d at constant v
 
dM 1 1  
uz

(v   ) b (v,  ) n(v) n( ) d dv 
V b (v, V )n(V )n(v)dVdv
0 0
dz
2 0 0
 
 
 
 
   b (v,  ) n(v) n( ) d dv    V b (v, V )n(V )n(v)dVdv
0 0
0 0
b ( v , ) symmetric
General Mass Balance for p Daughters
uz
dM 1
 0 mass is conserved
dz
R. B. Diemer, Jr., 2003