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


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.


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


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)


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


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


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


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


Coagulation

BreakageAgglomerates

Singlets

Nucleation

Growth

Nuclei

CoalescencePartially CoalescedAgglomerates

PrecursorMolecules

.... . ..

. . .. ...

.

Particle Formation, Growth & Transformation


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)


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


Aerosol Powder Manufacture

Gas-to-Particle Conversion


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…


Vaporization Pumping/Compression Addition of additives Preheating

General Aerosol Process Schematic

Feed #1Preparation

Feed #2Preparation

.

.

.

Feed #NPreparation


Mixing Reaction Residence Time Particle Formation/Growth Control Agglomeration Control Cooling/Heating Wall Scale Removal


AerosolReactor

Feed #1Preparation

Feed #2Preparation

.

.

.

Feed #NPreparation


Gas-Solid Separation


AerosolReactor

Feed #1Preparation

Feed #2Preparation

.

.

.

Feed #NPreparation

Base PowderRecovery


Absorption Adsorption


AerosolReactor

Feed #1Preparation

Feed #2Preparation

.

.

.

Feed #NPreparation

Base PowderRecovery

OffgasTreatment

TreatmentReagents Waste

Vent or Recycle Gas


Size Modification Solid Separations

Degassing Desorption Conveying


AerosolReactor

Feed #1Preparation

Feed #2Preparation

.

.

.

Feed #NPreparation

Base PowderRecovery

OffgasTreatment


Vent or Recycle Gas

PowderRefining

Coarseand/or FineRecycle


Coating Additives Tabletting Briquetting Granulation Slurrying Filtration Drying


AerosolReactor

Feed #1Preparation

Feed #2Preparation

.

.

.

Feed #NPreparation

Base PowderRecovery

OffgasTreatment


Vent or Recycle Gas

PowderRefining


ProductFormulation

FormulatingReagents


Product

Bags Super

Sacks Jugs Bulk

containers trucks tank cars


AerosolReactor

Feed #1Preparation

Feed #2Preparation

.

.

.

Feed #NPreparation

Base PowderRecovery

OffgasTreatment


Vent or Recycle Gas

PowderRefining


ProductFormulation

FormulatingReagents Packaging Product



AerosolReactor

Base PowderRecovery

OffgasTreatment

ProductFormulation

Packaging

PowderRefining

Feed #1Preparation

Feed #2Preparation

.

.

.

Feed #NPreparation

FormulatingReagents


Product


Vent or Recycle Gas


TiO2 Processes


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


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


Design Problem


Design Problem Focus

AerosolReactor

Base PowderRecovery

OffgasTreatment

ProductFormulation

Packaging

PowderRefining

Feed #1Preparation

Feed #2Preparation

.

.

.

Feed #NPreparation

FormulatingReagents


Product


Vent or Recycle Gas


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?


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


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?


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


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


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


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


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!


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


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


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


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 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


Sectionalization Example: q=1i,i collisions: map completely into i+1


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


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 j, V/vo

Sectionnumber i:

2i-1vo<V<2ivo

4

3

2

1




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


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



Particle j, V/vo

Sectionnumber i:

2i-1vo<V<2ivo

4

3

2

1


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 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




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


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


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


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


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


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


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


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


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


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


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


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


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


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


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

Population Balance Modeling: Solution Techniques & Applications Dr. R. Bertrum Diemer, Jr. Principal...

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Transcript of Population Balance Modeling: Solution Techniques & Applications Dr. R. Bertrum Diemer, Jr. Principal...