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