Getting Crystallization Parameters from

Experimental Data

by

Terry A. Ring

 Department of Chemical Engineering

University of Utah

50 S. Central Campus Drive

Salt Lake City, UT 84112

Population Balances in Engineering

• Crystallization, Precipitation• Coagulation, Agglomeration and

Flocculation• Particle Breakage, Particle Attrition• Dissolution/Leaching• Cell Growth

– Fermentation– Tissue Engineering

Modeling Industrial Crystallization

Sponsor

U.S. Department of Energy

Office of Industrial Technology

Industrial Participants

Academic Participants

• Prof. T.A. Ring,

• Prof. R. Fox,

• Prof. M. Hounslow, Sheffield U.

• Prof. A. Myerson,

• Prof. Ramkrishna,

www.che.utah.edu/~ring/CrystallizationWeb/CrystallizationModelingandValidation.doc

Constant Stirred Tank Crystallizer (MSMPR)

• On-line Analysis– Feed Flow Rate Controlled– Product Flow Rate Controlled– Liquid Level Controlled – Reactor

• pH - • Ionic Strength - • Temperature - • Stirrer RPM - • Stirrer Torque - • Heat Balance -

– Particle Size• Movie/Stills of Particles

• Off-line Analysis• Beckman-Coulter LS230• AA + ICP of feed and output• Yield

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0:00:00 1:12:00 2:24:00 3:36:00 4:48:00 6:00:00

Time

E(t

)

CaCl2-NaCl-KCl• 45% CaCl2

– NaCl+KCl

• Baffled Stirred Tank– Cooling– Impurity Removal

Residence TimeDistribution

outlet

CaCl2 Results - Sample #10-01

NaCl

0

1000

2000

3000

4000

5000

6000

7000

8000

0 200 400 600 800 1000 1200

Wave Length

Inte

nsi

ty

100100Total

0.811.11Ca

50.9261.25Cl

48.2637.65Na

At %Wt %Element

Na

Cl

ClCa

Particle Size Analysis with Beckman Coulter LS230

0

1

2

3

4

5

6

7

8

9

0.01 0.1 1 10 100 1000 10000

Particle Diameter (micro-meter)

Vo

lum

e (

%)

PSA 01

PSA 02-01

PSA 03-01

Sample #11-02

30 min. 90 min.

6 hr. 6 hr.07/25/03 Reaction Temp: 30.1oC, Flow Rate: 45 ml/min, Mixing Power: 287.7 rpm, Reactor Volume: 1400 ml

Particle Size Distribution with Time

x y z( ) x y z( )

07/25/03Reaction Temp: 30.1oC, Flow Rate: 45 ml/min, Mixing Power: 287.7 rpm, Reactor Volume: 1400 ml

Particulate Mechanisms

• Nucleation– Heterogeneous– Secondary

• (crystal impact with impeller creates nuclei)

• Particle Growth– Diffusion Limit– Other mechanisms

• Particle Breakage/Attrition• Particle Aggregation

Traditional Population Balance – differential number

.0)(),(

RateNucleation),()],(),([

0),(

'),'()',(),('),'(),'()'(2

1

),()('),'()'()'()'(

)],(),([),(

,

0 0

tatxftxf

conditioninitialand

xsizeatt

txxf

x

txftxG

xastxf

conditionsboundarywith

nAggregatiodxtxfxxatxfdxtxftxxfxxa

Breakagetxfxbdxtxfxxpxbx

DeathBirthx

txftxG

t

txf

oNN

ooN

xx

N

N

x

NNNN

Nx N

NN

o

Nucleation

• primary homogeneous– Classical Nucleation

• J(#/m3sec)=Jmax exp(-G(r*)/RT)

• primary heterogeneous– Classical Nucleation + new surface energy

• J(#/m3sec)=Jmax exp(-Ghet(r*)/RT)

• secondary• J (#/m3sec) = f(rpm, S, Solids Mass Fraction)

Crystal Growth Rate Mechanisms

Particle Collision Rate due to TurbulenceR e g i m e

s i z e a i + a j C o l l i s i o n V e l o c i t y Δ u ( a i , a j )

C o l l i s i o n F r e q u e n c y F a c t o r ( a i , a j , r , t )

B r o w n i a n D i f f u s i o n

0 > a i + a j >

)aa

)Tk

)(a )(a3

2(

ji

Bji

T u r b u l e n t s u b r a n g e

V i s c o u s > a i + a j > 6

)(257.02/1

ji aa

1 . 2 9 ( < 1 / 2 > / 1 / 2 ) ( a i + a j )3

E q u i v a l e n t t o

( 4 / 3 ) ( a i + a j ) 3

T r a n s i t i o n 6 a i + a j < 2 5

3/24/1

12/5

)(471.0 ji aa

2 . 3 6 < 5 / 1 2 > - 1 / 4 ( a i + a j )

8 / 3

I n e r t i a l 2 5 a i + a j < L / 2

3/13/1 )(37.1 ji aa 6 . 8 7 < 1 / 3 > ( a i + a j )7 / 3

M a c r o L / 2 > a i + a j ~ L 3/12 L 7 . 0 9 < 1 / 3 > L 1 / 3 ( a i + a j )2

Kusters, K.A. "The Influence of Turbulence on Aggregation of Small Particles in Agitated Vessels," Technical University Eindhoven, The Netherlands, 1991.

Breakage Kinetics

• Breakage Rate = Collision Frequency * Target Efficiency * Breakage Probability

Collisions Collision Efficiency

Particle-wall

Particle-impeller

Particle-baffle

Particle-Particle due to either diffusion or turbulence

Crystallizers are Complicated

• Size Increases by– Aggregation– Growth

• Size Decreases by– Breakage– Dissolution

• Number of Particles Increases by– Nucleation– Breakage– Less Aggregation

HOW TO DECOUPLE?

Do not try to solve all at once!

Decomposition of the Problem

• Using standard operating conditions for tank,– Study aggregation separately– Study breakage separately

• Focus on aggregation and breakage under batch conditions after crystallization has been run to steady state

Traditional Population Balance – differential number

.0)(),(

RateNucleation),()],(),([

0),(

'),'()',(),('),'(),'()'(2

1

),()('),'()'()'()'(

)],(),([),(

,

0 0

tatxftxf

conditioninitialand

xsizeatt

txxf

x

txftxG

xastxf

conditionsboundarywith

nAggregatiodxtxfxxatxfdxtxftxxfxxa

Breakagetxfxbdxtxfxxpxbx

DeathBirthx

txftxG

t

txf

oNN

ooN

xx

N

N

x

NNNN

Nx N

NN

o

Stieltjes Formulation1 – Cumulative Mass

0)(),(

1),(,00),(

,sizeat RateNucleation Mass),(

),(),(),(

)(

),'()'()'(

ˆˆ),'('

),'(

'

),(),'(

),(

ˆˆ''

),(

'

),'(

''

),(

,

01

0

0

tatxFtxF

conditioninitialand

xattxFxattxF

xt

txF

conditionsboundarywith

ionAgglomerattsFtuFu

usat

BreakagetxFxxPxb

DBtxFx

txg

x

txFtxg

t

txF

DBdxx

txF

x

txg

xx

t

txF

omm

mm

oxx

m

xx

x

s sxu

x xm

x

mxMm

Mm

xmMm

o

Ramkrishna, D., “Population Balances,” Academic Press, 2000, p.56 and private communications.

Sinc Approximation

N

Nj

M

Mkkjm

txm txFk

h

tj

h

xtxF ),()

)((sinc)

)((sinc),(

z

zz

)sin(

)(sinc

N

ch

jhjhx

xx

x

xxj

1

1 )exp()(

)ln()(

M

ch

kh

khTkht

tTtt

t

t

ttk

2

1

)exp(1

)exp()(

)]/(ln[)(

0 5 100.5

0

0.5

1

sinc x( )

x

Final Formulation

ionAgglomeratdttsFtuFu

usat

BreakagedttxFxxPxb

growthdttxFx

txgdt

x

txFtxg

txFtxF

t

xx

x

s sxu

t

x xm

t x

mxM

tm

M

omm

0 01

0

0 00

,

')',()',(),(

)'(

')','()'()'(

')','('

)','('

'

)','()','(

)0,(),(

ionAgglomeratPXtdiagAt

BreakageFAxbPAxAt

GrowthFAxx

gAxAtFAxgAt

FF

I

mDmII

mDM

IImDMI

omm

~)]([

][

][][][][

1

,

Sinc Results – Batch

• Power Law Breakage• S(x)=x1.2

– Kt=0.1

0.01 0.1 1 10 1000

0.5

1Breakage Only

Diameter (micons)

Cum

ulat

ive

Mas

s F

ract

ion

Sinc Result -Batch

• Aggregation Only– Sum Kernal– Kt=0.5– μ = 0.1

0.01 0.1 1 10 1000

0.5

1Agglomeration Only

Diameter (microns)

Cum

ulat

ive

Mas

s F

ract

ion

Agglomeration and Breakage

Sinc Result- Batch

• Aggregation & Breakage– Compensating Effects

0.01 0.1 1 10 1000

0.5

1Agglomeration & Breakage

Diameter (microns)

Cum

ulat

ive

Mas

s F

ract

i

Need to Know

• Aggregation Rate Constant

• Breakage Rate Constant

• Breakage Selectivity

• Breakage Daughter Distribution

• Where are we going to get these?

Focused Experimentation

• Using standard operating conditions for tank,– Study aggregation separately– Study breakage separately

• Forced Fitting of the Data

Diatomite Breakage• Batch Experiment for Breakage

– 10% wt. Commercial Diatomite

• 1.4 L baffled batch stirred tank– 300 rpm

– Sampling every hr.

• Progeny Function

• Rate Function

)1exp(1

)'/exp(1)'(

xx

xxPmA j k

a Zxj Zxk Zxk

0.1 1 10 100 1 103

1 104

1 105

0

500

1000

Diameter (microns)

Bre

akag

e R

ate

Con

stan

t

1 10 100 1 1030

0.5

1Attrition Data

Diameter (micons)

Cum

ulat

ive

Mas

s Fr

actio

n

Increasing Time

DATA□ Feed◊ 1 hr.x 2 hr.o 3 hr.+ 4 hr.

Diatomite Attritionikmaxt hr 240min

1 10 100 1 1030

0.5

1Attrition Sinc Simulation

Diameter (micons)

Cu

mula

tiv

e M

ass

Fra

ctio

n

T hr 240min

1 10 100 1 1030

0.5

1Attrition Data

Diameter (micons)

Cu

mula

tiv

e M

ass

Fra

ctio

n

Increasing Time

Increasing Time

DATA

Simulation

□ Feed◊ 1 hr.x 2 hr.o 3 hr.+ 4 hr.

□ Feed◊ 1 hr.x 2 hr.o 3 hr.+ 4 hr.

0.1 1 100

0.5

1Aggregation Data

Diameter (micons)

Cumu

lative

Mas

s Frac

tionPolystyrene Latex

Aggregation • Batch Experiment for

Aggregation– 1% wt. Commercial

Polystyrene Latex (1 micron) in 10-3 M NaCl

• 1.4 L baffled batch stirred tank– 100 rpm– Sampling every 10 min.

• Dynamic Balance – Breakage (diatomite expt.)– Aggregation

• a(x,y) = x + y

□ Feed◊ 20 min.x 40 min.o 60 min.+ 80 min.

Time

Dynamic Balance for Polystyrene Latex

0.1 1 10 1000

0.2

0.4

0.6

0.8

1Agglomerat ion & Breakage Sinc Simulat ion

Diameter (microns)

Cum

ulat

ive

Mas

s F

ract

i

0.1 1 10 1000

0.2

0.4

0.6

0.8

1Aggregation/Breakage Data

Diameter (micons)

Cum

ulat

ive

Mas

s F

ract

ion

Simulation

Expt. Data□ Feed◊ 20 min.x 40 min.o 60 min.+ 80 min.

x Feedo 20 min.

□ 40 min.◊ 60 min.o 80 min.

Polystyrene Latex Aggregation

Data = PointsSinc Simulation = Lines

0.1 1 100

0.2

0.4

0.6

0.8

1Polystyrene Agglomeration

Diameter (microns)

Cum

ulat

ive

Mas

s F

ract

ion

Focus on Aggregation and Breakage after Crystallization

• Stop Feed – Batch Tank– Constant RPM, Constant Temperature

• Supersaturation is Constant– No nucleation

– No particle growth

– Measure Changes in Particle Size Distribution

NaCl Aggregation

• 1.4 L Batch, Baffled Stirred Tank– 287 rpm

• Charge from Crystallization– 2% NaCl particles in

47% CaCl2 solution

– Aggregation/Attrition over 3 hrs.

Aggregation/Breakage Data

x y z( )

07/25/03Reaction Temp: 30.1oC, Flow Rate: 45 ml/min, Mixing Power: 287.7 rpm, Reactor Volume: 1400 ml

30 min. 60 min.

3 hr.2 hr.

07/25/03 Reaction Temp: 30.1oC, Flow Rate: 45 ml/min, Mixing Power: 287.7 rpm, Reactor Volume: 1400 ml

Aggregation/Breakage

Traditional Population Balance – differential number

.0)(),(

RateNucleation),()],(),([

0),(

'),'()',(),('),'(),'()'(2

1

),()('),'()'()'()'(

)],(),([),(

,

0 0

tatxftxf

conditioninitialand

xsizeatt

txxf

x

txftxg

xastxf

conditionsboundarywith

nAggregatiodxtxfxxatxfdxtxftxxfxxa

Breakagetxfxbdxtxfxxpxbx

DeathBirthx

txftxg

t

txf

oNN

ooN

xx

N

N

x

NNNN

Nx N

NN

o

Particle Collision Rate due to TurbulenceR e g i m e

s i z e a i + a j C o l l i s i o n V e l o c i t y Δ u ( a i , a j )

C o l l i s i o n F r e q u e n c y F a c t o r ( a i , a j , r , t )

B r o w n i a n D i f f u s i o n

0 > a i + a j >

)aa

)Tk

)(a )(a3

2(

ji

Bji

T u r b u l e n t s u b r a n g e

V i s c o u s > a i + a j > 6

)(257.02/1

ji aa

1 . 2 9 ( < 1 / 2 > / 1 / 2 ) ( a i + a j )3

E q u i v a l e n t t o

( 4 / 3 ) ( a i + a j ) 3

T r a n s i t i o n 6 a i + a j < 2 5

3/24/1

12/5

)(471.0 ji aa

2 . 3 6 < 5 / 1 2 > - 1 / 4 ( a i + a j )

8 / 3

I n e r t i a l 2 5 a i + a j < L / 2

3/13/1 )(37.1 ji aa 6 . 8 7 < 1 / 3 > ( a i + a j )7 / 3

M a c r o L / 2 > a i + a j ~ L 3/12 L 7 . 0 9 < 1 / 3 > L 1 / 3 ( a i + a j )2

Kusters, K.A. "The Influence of Turbulence on Aggregation of Small Particles in Agitated Vessels," Technical University Eindhoven, The Netherlands, 1991.

Energy Dissipation Rate [w/kg] (ε) Profile

Energy Dissipation Rate on Individual Particle Track

• High ε when passes through impeller zone

• Effectively– ε =0 elsewhere

0 1000 20001 10

41 10

30.01

0.1

1

10

100

1 103

Time

Ene

rgy

Dis

sipa

tion

Rat

e (w

att/

kg)

Energy Dissipation Rate Statistics

• εmax = 429 w/kg• Multimode

Distribution

• Probability α ε-3/2

– Decay of Isotropic Turbulence

• V= 1.4 L, Q=40 ml/min., 80 rpm

1 104

1 103

0.01 0.1 1 10 100 1 103

1 105

1 104

1 103

0.01

0.1

1PDF of Cfd Epsilon Data

Energy Dissipation Rate (watts/kg)

Pro

babi

lity

lower upper

0.1 1 10 100 1 1031 10

7

1 106

1 105

1 104

1 103

0.01

0.1PDF Histogram of Cfd Epsilon Data

Energy Dissipation Rate (watts/kg)

Pro

babi

lity

upper

Aggregation Rate

1 10 100 1 103

1 104

1 105

1 106

1 104

1 103

0.01

0.1

1

10

100

1 103

1 104

1 105

1 106

1 107

1 108

radius(micron)

Agg

lom

erat

ion

Rat

e C

onst

. (H

z)

07/25/03Reaction Temp: 30.1oC, Flow Rate: 45 ml/min, Mixing Power: 287.7 rpm, Reactor Volume: 1400 ml

Breakage Kinetics

• Breakage Rate = Collision Frequency * Target Efficiency * Breakage Probability

Collisions Collision Efficiency

Particle-wall

Particle-impeller

Particle-baffle

Particle-Particle due to either diffusion or turbulence

Use Fluid Dynamics with Sinc Methods

• Particle Breakage– Particle Collisions with

• Particles (same equations as aggregation)– depend upon turbulent energy dissipation rate

• Baffles/walls – impaction efficiency

• Impeller – collision frequency

– Sufficient Energy of Collision to Break Particle

1.2

arg 32.0

St

Stett N

N

352

32

32

min, 64r

pKH

GW

Gahn, C. and Mersmann, A., Chem. Eng. Sci. 54,1274-82(1999).

25.2max

25.2min

25.325.2)(

xx

xxp

Particle Breakage Efficiency

• Baffle

• Wall

• Impeller

• Other Particles

1 10 100 1 103

1 104

0

0.5

1

Particle Diameter (micron)

Targ

et C

olli

sion

Eff

icie

ncy

07/25/03Reaction Temp: 30.1oC, Flow Rate: 45 ml/min, Mixing Power: 287.7 rpm, Reactor Volume: 1400 ml

Breakage Rate

1 10 100 1 103

1 104

1 105

1 106

10

100

1 103

1 104

radius(micron)

Bre

akag

e R

ate

Con

st.

(Hz)

07/25/03Reaction Temp: 30.1oC, Flow Rate: 45 ml/min, Mixing Power: 287.7 rpm, Reactor Volume: 1400 ml

Model Results – Aggregation/Breakage

A_Factor=400, a colloid stability parameter

Conclusions

• Getting Crystallization Parameters from Experimental Data– Requires

• Decomposition of the multiplicity of phenomenon

• Knowledge of fundamental particle mechanisms

• Simple calculation tool for stirred tank