Population Balance Modeling: Solution Techniques & Applications Dr. R. Bertrum Diemer, Jr. Principal...
-
date post
21-Dec-2015 -
Category
Documents
-
view
217 -
download
0
Transcript of Population Balance Modeling: Solution Techniques & Applications Dr. R. Bertrum Diemer, Jr. Principal...
Population Balance Modeling:Solution Techniques &
Applications
Dr. R. Bertrum Diemer, Jr.Principal Division Consultant
DuPont Engineering Research & Technology
R. B. Diemer, Jr., 2003
Lecture Outline
Introduction Applications to Particles
General Balance Equation Aerosol Powder Manufacturing Design Problem
Solution Techniques
Introduction
R. B. Diemer, Jr., 2003
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:
3-D space (x,y,z or r,z,or r,,) and time internal dimensions = dimensions of the population:
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:
particle size… polymer MW...cell age
Bivariate examples: 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
( , , ) ( , , )
( , , ) ( , , )
( , , ) ( , , )
+ ( , , )
( , , )
p
p
V t n V t
D V t n V t
G V t n V tV
S V t
n V t
t
u x x
x x
x x
x
x
convection
diffusion
“In Out” inexternalcoordinates
growth (“In Out” in internal coordinate)
accumulation
sources & sinks (Net Generation)
Note: object’s velocity may differ from fluid’s velocity owing toeither slip or action of external forces
R. B. Diemer, Jr., 2003
Eliminates 2 physical dimensions, time dimension Axial Dispersion Model:
With slip...
Without slip...
Plug Flow Model, No Slip:
Steady-state, Axisymmetric, Incompressible Flow
( , ) ( , )( , ) ( , ) ( , ) + ( , )z p
n V z n V zu D V z G V z n V z S V z
z z z V
( , )( , ) ( , ) ( , )
( , ) ( , ) + ( , )
pz p
n V zu V z n V z D V z
z z z
G V z n V z S V zV
( , )( , ) ( , ) + ( , )z
n V zu G V z n V z S V z
z V
R. B. Diemer, Jr., 2003
Ideally Mixed Stirred Tank Eliminates 3 physical dimensions Batch:
Continuous: Unsteady state…
Steady state (eliminates time dimension as well)...
( , )( , ) ( , ) + ( , )
n V tG V t n V t S V t
t V
( , ) ( ) ( , ) ( ) ( , )
( ) ( , ) ( , ) + ( , )
oon V t F t n V t F t n V t
t
t G V t n V t S V tV
( ) ( )( ) ( ) + ( ) 0;
on V n V dG V n V S V
dV F
R. B. Diemer, Jr., 2003
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)...
/
( ) ( )( ) ( ) + ( ) 0
( ) 0; ( ) ; ( ) 0
( )( ) (0)
o
o
V G
n V n V dG V n V S V
dV
n V G V G S V
dn n Vn V n e
dV G
Applications to Particles
General Balance Equation
R. B. Diemer, Jr., 2003
Coagulation
BreakageAgglomerates
Singlets
Nucleation
Growth
Nuclei
CoalescencePartially CoalescedAgglomerates
PrecursorMolecules
.... . ..
. . .. ...
.
Particle Formation, Growth & Transformation
R. B. Diemer, Jr., 2003
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 rateG = accretion ratecoagulation ratebreakage rateb= daughter distributionvo = nuclei size
0 0
( )
1( , ) ( ) ( ) ( ) ( , ) ( )
2
( ) ( ; ) ( ) ( ) ( )
p p
o
V
V
nn D n
t
N V v G nV
v V v n v n V v dv n V v V n v dv
b V n d V n V
u nucleation term
growth term
coagulation terms
breakage terms
Applications to Particles
Aerosol Powder Manufacture
Gas-to-Particle Conversion
R. B. Diemer, Jr., 2003
Aerosol Synthesis Chemistry Examples
2 / 2M OR H O MO ROHyyy y Alkoxide Hydrolysis:
M= Si, Ti, Al, Sn… R=CH3, C2H5...
2 2 / 2MX H O MO HXyy y y Halide Hydrolysis:
M= Si, Ti, Al, Sn… X=Cl, Br...
4 22 / 2 2MX O MO Xy yy y Halide Oxidation:
M=Si, Ti, Al, Sn… X=Cl, Br...
2/ 2M OR MO RORyyy
Alkoxide Pyrolysis: M= Si, Ti, Al, Sn… R=CH3, C2H5...
3 3 /3MX NH MN HXyy y y
Halide Ammonation: M=B, Al … X=Cl, Br...
2 2AL A L/ Lyy y Pyrolysis:
A=Si, C, Fe… L=H, CO…
R. B. Diemer, Jr., 2003
Vaporization Pumping/Compression Addition of additives Preheating
General Aerosol Process Schematic
Feed #1Preparation
Feed #2Preparation
.
.
.
Feed #NPreparation
R. B. Diemer, Jr., 2003
Mixing Reaction Residence Time Particle Formation/Growth Control Agglomeration Control Cooling/Heating Wall Scale Removal
General Aerosol Process Schematic
AerosolReactor
Feed #1Preparation
Feed #2Preparation
.
.
.
Feed #NPreparation
R. B. Diemer, Jr., 2003
Gas-Solid Separation
General Aerosol Process Schematic
AerosolReactor
Feed #1Preparation
Feed #2Preparation
.
.
.
Feed #NPreparation
Base PowderRecovery
R. B. Diemer, Jr., 2003
Absorption Adsorption
General Aerosol Process Schematic
AerosolReactor
Feed #1Preparation
Feed #2Preparation
.
.
.
Feed #NPreparation
Base PowderRecovery
OffgasTreatment
TreatmentReagents Waste
Vent or Recycle Gas
R. B. Diemer, Jr., 2003
Size Modification Solid Separations
Degassing Desorption Conveying
General Aerosol Process Schematic
AerosolReactor
Feed #1Preparation
Feed #2Preparation
.
.
.
Feed #NPreparation
Base PowderRecovery
OffgasTreatment
TreatmentReagents Waste
Vent or Recycle Gas
PowderRefining
Coarseand/or FineRecycle
R. B. Diemer, Jr., 2003
Coating Additives Tabletting Briquetting Granulation Slurrying Filtration Drying
General Aerosol Process Schematic
AerosolReactor
Feed #1Preparation
Feed #2Preparation
.
.
.
Feed #NPreparation
Base PowderRecovery
OffgasTreatment
TreatmentReagents Waste
Vent or Recycle Gas
PowderRefining
Coarseand/or FineRecycle
ProductFormulation
FormulatingReagents
R. B. Diemer, Jr., 2003
Product
Bags Super
Sacks Jugs Bulk
containers trucks tank cars
General Aerosol Process Schematic
AerosolReactor
Feed #1Preparation
Feed #2Preparation
.
.
.
Feed #NPreparation
Base PowderRecovery
OffgasTreatment
TreatmentReagents Waste
Vent or Recycle Gas
PowderRefining
Coarseand/or FineRecycle
ProductFormulation
FormulatingReagents Packaging Product
R. B. Diemer, Jr., 2003
General Aerosol Process Schematic
AerosolReactor
Base PowderRecovery
OffgasTreatment
ProductFormulation
Packaging
PowderRefining
Feed #1Preparation
Feed #2Preparation
.
.
.
Feed #NPreparation
FormulatingReagents
TreatmentReagents Waste
Product
Coarseand/or FineRecycle
Vent or Recycle Gas
R. B. Diemer, Jr., 2003
TiO2 Processes
R. B. Diemer, Jr., 2003
Thermal Carbon Black Process
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.
Carbon Generated by Pyrolysis of CH4
R. B. Diemer, Jr., 2003
Furnace Carbon Black Process
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.
Carbon generated by Fuel-rich
Oil Combustion
Applications to Particles
Design Problem
R. B. Diemer, Jr., 2003
Design Problem Focus
AerosolReactor
Base PowderRecovery
OffgasTreatment
ProductFormulation
Packaging
PowderRefining
Feed #1Preparation
Feed #2Preparation
.
.
.
Feed #NPreparation
FormulatingReagents
TreatmentReagents Waste
Product
Coarseand/or FineRecycle
Vent or Recycle Gas
R. B. Diemer, Jr., 2003
The Design Problem
Feeds
Flame Reactor
.25 m particles
Cyclone25% of
particle massmax
10 psig
Gas to RecoverySteps
Baghouse
Pipeline Agglomerator
8 psigmin
75% of particle mass min
What pipe diameterand length? What
cut size?
R. B. Diemer, Jr., 2003
Simultaneous Coagulation & Breakage initial size = .25 micron
Coagulation via sum of: continuum Brownian kernel:
Saffman-Turner turbulent kernel:
Power-law breakage, binary equisized daughters: Fractal particles:
Design Problem Physics
1/31/32
( , ) 23c
kT VV
V
2/3 1/3 1/3 2 /3( , ) .31 3 3t V V V V
3/ 28 2 1/3( ) 1 10 s /cm
( ; ) 22
b V V
1/
0 30
6; 1.8
fD
p f
Vd d D
d
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 P = 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:
0 0
( ) ( ; ) ( ) ( ) ( )
1( , ) ( ) ( ) ( ) ( , ) ( )
2
z VV
nu b V n d V n V
z
v V v n v n V v dv n V v V n v dv
R. B. Diemer, Jr., 2003
Moments
0
01
( ) continuous form
discrete form with
j
jj
i i ii
V n V dVM
nV V iV
Moments of n(V):
Key Moments:
01
11
particle number concentration
particle volume fraction
(proportional to particle mass concentration)
ii
i ii
M n
M nV
Solution Techniques
R. B. Diemer, Jr., 2003
Discrete Methods Sectional Methods Similarity Solutions LaPlace Transforms Orthogonal Polynomial Methods Spectral Methods Moment Methods Monte Carlo Methods
Partial List of Techniques
Will discuss
Will notdiscuss
R. B. Diemer, Jr., 2003
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:
30
0 0
1
, ,1 1 1
;6
1( ; )
2
i
ii
z j i j j i j i i j j j j i ij j j i
dV iV V
dnu n n n n b i j n n
dz
R. B. Diemer, Jr., 2003
Discrete Example Problem Setup
1/3 1/3
2 /3 1/3 1/3 2 /3, , , , 0
3/ 28 2 1/3 1/3
0
22 ; .31 3 3
3
2, 21 10 s /cm , 1
; ( ; ) 1, 2 1, 2 1
0, 2 10, 1
c i j t i j
i
kT i jV i i j i j j
j i
j iV i i
b i j j i i
j ii
, 2 1j i
1
, , , , , , , ,1 1
2 1 2 1 2 2 2 1 2 1
1
2
2
ii
z c j i j t j i j j i j i c i j t i j jj j
i i i i i i i i
dnu n n n n
dz
n n n n
Need slightly more than 2106 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
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
vi2-3/q vi 2-2/q vi 2-1/q vi 21/q vi 22/q vi 23/q vi
i i+1 i+2i1i2i3
R. B. Diemer, Jr., 2003
Sectional Interaction Types
Type 1: some particles land in the ith interval and some in a smaller interval
Type 2: all particles land in the ith interval
Type 3: some particles land in the ith interval and some in a larger interval
Type 4: 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
0
2
4
6
8
10
12
14
16
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
Sectionalization Example: q=1Collision of Particle j with Particle k
Particle k, V/vo1 2 3 4 5
Particle j, V/vo
Sectionnumber i:
2i-1vo<V<2ivo
In which section goes the daughter of a collision between
Particle j in Section i and Particle k in Section n?
4
3
2
1
R. B. Diemer, Jr., 2003
0
2
4
6
8
10
12
14
16
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
Sectionalization Example: q=1
Particle k, V/vo1 2 3 4 5
Particle j, V/vo
Sectionnumber i:
2i-1vo<V<2ivo
4
3
2
1
j+k = constant
Any collision between these linesproduces a particle in Section 5
R. B. Diemer, Jr., 2003
Sectionalization Example: q=1i,i collisions: map completely into i+1
Particle k, V/vo1 2 3 4 5
Particle j, V/vo
Sectionnumber i:
2i-1vo<V<2ivo
4
3
2
10
2
4
6
8
10
12
14
16
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
R. B. Diemer, Jr., 2003
0
2
4
6
8
10
12
14
16
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
Sectionalization Example: q=1i,i+1 collisions: 3/4 map into i+2, 1/4 stay in i+1
Particle k, V/vo1 2 3 4 5
Particle j, V/vo
Sectionnumber i:
2i-1vo<V<2ivo
4
3
2
1
R. B. Diemer, Jr., 2003
Sectionalization Example: q=1i,i+2 collisions: 3/8 map into i+3, 5/8 stay in i+2
Particle k, V/vo1 2 3 4 5
Particle j, V/vo
Sectionnumber i:
2i-1vo<V<2ivo
4
3
2
10
2
4
6
8
10
12
14
16
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
R. B. Diemer, Jr., 2003
Sectionalization Example: q=1i,i+3 collisions: 3/16 map into i+4, 13/16 stay in i+3
Particle k, V/vo1 2 3 4 5
Particle j, V/vo
Sectionnumber i:
2i-1vo<V<2ivo
4
3
2
10
2
4
6
8
10
12
14
16
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
R. B. Diemer, Jr., 2003
0
2
4
6
8
10
12
14
16
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
Sectionalization Example: q=1i,i+4 collisions: 3/32 map into i+5, 29/32 stay in i+4
Particle k, V/vo1 2 3 4 5
Particle j, V/vo
Sectionnumber i:
2i-1vo<V<2ivo
4
3
2
1
R. B. Diemer, Jr., 2003
0
2
4
6
8
10
12
14
16
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
Sectionalization Example: q=1i,n collisions: 3/2n-i+1 map into n+1, n>i>0
i,i collisions: all map into i+1
Particle k, V/vo1 2 3 4 5
Particle j, V/vo
intervalnumber i:vo
i-1<V<voi
i,i+1 collisions: 3/4 map into i+2i,icollisions: all map into i+1
i,i+2 collisions: 3/8 map into i+3
i,i+3 collisions: 3/16 map into i+4
i,i+4 collisions: 3/32 map into i+5
R. B. Diemer, Jr., 2003
Sectional Coagulation Model, q=1 Model Equation:
Tentatively:
Can show (via 0th and 1st moments) that: number balance gives correct general form for arbitrary i,j
mass balance only closes for C=2/3 when Vi/Vj=2i-j
final expression: kernels evaluated via:
2 12
1 1, 1, 1, 1 1 , , ,1 1
1
2
i ii
z i i j i j j i i i i i j i j j i j jj j j i
dNu N N N N N N
dz
,
32
2j i
i j C
, 2 j ii j
1 0 0 01 1
2 32 with (recovers 3/2 factor)
2 2i
i
V V VV V V
R. B. Diemer, Jr., 2003
General 21/q Sectional Coagulation Model
( ) 1 ( )2
1 1, 1, , , , ,1 1 ( ) 1
( 1)
, 1,2 ( 2) 1
1/1, , 1 1
( 2)
1
2
2
i S q i S qi
i i j i j j i q i q i q i i j i j j i j jj j j i S q
q i S q k k
i k j i j k i k jk j i S q k k
qi k j i j k i k j
j i S q k k
dNN N N N N N
dt
N N
N N
( 1) 1
2 2
( ) / (1 ) /
, 1/ 1/1
2 2 1( ) ; ;
2 1 2 1
q i S q k k
k
j i q k qq
i j kq qm
S q m
2 new terms
R. B. Diemer, Jr., 2003
Sectional Example Problem Setup (for q=1)
( ) /3 ( ) /3 1 (2 ) /3 1 ( 2 ) /3 1 10,
1/33/ 28 2 ( 1) /30
322 2 2 .31 2 3 2 2 2
3 2
32, 11 10 s /cm 2 , 1
; ( ; )20, 1
0, 1
i j j i i i j i j ji j
i
i
VkT
Vj ii
b i jj i
i
Need about 22 sections to cover entire mass distribution range, suggest using 25-30
2 11, ,2
1 1, 1 1 ,11 1
1 1
1
2 2 2
2
i ii j i ji
z i j i i i i j i j ji j i jj j j i
i i i i
dNu N N N N N N
dz
N N
R. B. Diemer, Jr., 2003
Sectional Example Problem Setup (for q=1)Calculation of Mass Collection Efficiency
1/31/1/1 20 0
0 0 0 00
2/ 22 20 1
2 2/ 220
example 1grade efficiency curve
11
32 ; 3 2
2 6
3 2; 100%
1 1 3 2
f
ff
f
f
DDDi ii
i i p
IDi i i i
pci pc ii wD Ii
i pc pc i ii
I
i ii
V d VV V d d n d d
V
V Nd dd d
d d d d V N
M V N
0 1
1 1
10 1
1
1
; 2 100% 2
1, 1Suggests to do calculation using with
0, 1
Check mass closure via: 2 1. If not true, model is coded incorrectly!
o
Ii ii i
i w i ioi
oi i
Ii
ii
M V
N Vn n
M V
in n
i
n
R. B. Diemer, Jr., 2003
Sectional Example Problem Setup (for q=1)Nondimensionalization
1/3( 1) /30 0
, , , , ,
3/ 2( ) /3 ( ) /3 8 2
, ,
1 (2 ) /3 1 ( 2 ) /3 1 1, ,
0
3 3; 2
2 2
2; 2 2 2 ; 1 10 s /cm
3
.31 ; 2 3 2 2 2
2;
o o o ii j c c i j t t i j i
o i j j i oc c i j
o i i j i j jt t i j
o oc t
tc z c
V V
kT
M z
u
1/3
0
0 0
, , 1 1, , , 0 3
0 0 0
2;
3 3
2 4; ;
3
o o oc b c
bo ot c
t i j oii j c i j i o
t
M
V V
N M Mn M
M V d
( 1) /32 1
1, ,2 4 /31 1, 1 1 , 11
1 1
1 22
2 2 2
ii ii j i ji
i j i i i i j i j j i ii j i jj j j i b
dnn n n n n n n n
d
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
Backup Slides
R. B. Diemer, Jr., 2003
Moments
0
01
0
1
( ) continuous form
discrete form with
particle number concentration
particle volume fraction (proportional to particle mass concentration)
j
jj
i i ii
V n V dVM
nV V iV
M
M
Moments of n(V):
Moments of b(V;):
0
1
1
0
1
( ; ) continuous form
( ; ) ; discrete form
daughters/breaking event 2 in binary breakage
, the parent size
j
jj
ii
V b V dV
bb i V V
b p
b
R. B. Diemer, Jr., 2003
Particle Number Balance
0
0 0
0 0
0 0
0 0
( )
( ) ( ; ) ( ) ( ) ( )
1( , ) ( ) ( )
2
( ) ( , ) ( )
z z z
VV
dMn du dV u n V dV u
z dz dz
b V n d dV V n V dV
v V v n v n V v dvdV
n V v V n v dvdV
R. B. Diemer, Jr., 2003
Interchange of Limits
V
V=0
0 0
( , )
( , )
Vf V d dV
f V dVd
V goes from 0 to then from 0 to
goes from V to then V from 0 to
R. B. Diemer, Jr., 2003
Particle Number Balance (cont.)
0
0
00 0
0 0 0
0
0 0 0
( ; )( ) ( )
1( , ) ( ) ( ) ( , ) ( ) ( )
2
( 1) ( ) ( )
1( , ) ( ) ( ) ( , ) ( ) ( )
2
( ) ( )z
v
v
dM b V dVu d V n V dV
dzb p
v V v n v n V v dVdv v V n V n v dVdv
p V n V dV
v V v n v n V v dVdv v V n V n v dVdv
n
Interchange limits of integration in both coagulation and breakage terms
R. B. Diemer, Jr., 2003
Particle Number Balance (cont.)Change of variable in coagulation integral:= Vv dV = dat constant v
0
0
0 0 0 0
( 1) ( ) ( )
1( , ) ( ) ( ) ( , ) ( ) ( )
2
z
dMu p V n V dV
dz
v n v n d dv v V n V n v dVdv
0
,1 1 1
0 0 0
1( 1) ( ) ( ) ( , ) ( ) ( )
2
1( 1)
2
continuous
z
i i i j i ji i j
discrete
p V n V dV v V n V n v dvdV
dMu
dz p n n n
General Number Balance for p Daughters
R. B. Diemer, Jr., 2003
Particle Volume (Mass) Balance
1
0 0
0 0
0 0
0 0
( )
( ) ( ; ) ( ) ( ) ( )
1( , ) ( ) ( )
2
( ) ( , ) ( )
z z z
VV
dMn dV u dV u Vn V dV u
z dz dz
b V n d dV V V n V dV
v V v n v n V v dvdV
n V v V n v dvdV
V
V
V
R. B. Diemer, Jr., 2003
Particle Volume (Mass) Balance (cont.)
1
1
1
00 0
0 0 0
0 0 0
( ) ( )
1( , ) ( ) ( ) ( , ) ( ) ( )
2
1( , ) ( ) ( ) ( , ) ( ) ( )
2
( ; )( ) ( )z
z
v
v
dMu d V V n V dV
dz b
V v V v n v n V v dVdv V v V n V n v dVdv
dMu V v V v n v n V v dVdv V v V n V n v dVdv
dz
Vb V dVn
Interchange limits of integration in both coagulation and breakage terms
R. B. Diemer, Jr., 2003
Particle Volume (Mass) Balance (cont.)
Change of variable in coagulation integral:= Vv dV = dat constant v
1
( , ) symmetric
0 0 0 0
0 0 0 0
1( ) ( , ) ( ) ( ) ( , ) ( ) ( )
2
( , ) ( ) ( ) ( , ) ( ) ( )
z
v
dMu v v n v n d dv V v V n V n v dVdv
dz
v n v n d dv V v V n V n v dVdv
1 0 mass is conservedz
dMu
dz
General Mass Balance for p Daughters