THE HITCHHIKERS GUIDE TO POPULATION BALANCES, BREAK-UP AND COALESCENCE. Lecture series by: Lars...

THE HITCHHIKERS GUIDE TO POPULATION BALANCES,

BREAK-UP AND COALESCENCE.

Lecture series by:

Lars Hagesaether

October 2002

NTNU

Overview - START LECTURE 2

population balance to be solved in CFD-program:population balance to be solved in CFD-program:

iCCBBiii DBDBnunt

][)()(

function of break-up function of break-up of ‘large’ particlesof ‘large’ particles

1

1

),()(i

kBB kiiD

break-up for class break-up for class ii into class into class kk

BBB P

break-up probabilitybreak-up probabilitycollision frequencycollision frequency

collision model for collision model for 2 fluid particles2 fluid particles

improved improved break-up break-up modelmodel

Overview

COMBINED MODELCOMBINED MODEL

CFD METHODSCFD METHODS

RESULTSRESULTS

BREAKUP BREAKUP MODELMODEL

INPUT DATAINPUT DATA

COALESCENCE COALESCENCE MODELMODEL

PB SIZE PB SIZE DISCRETIZATIONDISCRETIZATION

OTHER OTHER MODELSMODELS

•Note: .ppt file with lecture will be made available.

RR

dRDBndRdt

d)(

Population Balances

Particle number continuity equation:Particle number continuity equation:

GenerationNetOutputInputonAccumulati

Terms set to zeroTerms set to zero

For a sub-region, R, to move convectively with the particle phase-space velocity (i.e. Lagrangian viewpoint)

RRRR

dRndt

d

t

n

dt

dndR

t

nndR

dt

d xx

Population Balances

ei vvvx

dt

d

DBnnt

n

ei vv

x is the set of internal and external coordinates (x, y, z) comprising the phase space R.

Since R can be any region, the integration parts can be removed thus giving the differential form of the number continuity equation in particle space.

Reference for equations above is Randolph & Larsson (1988)

Population Balances

It is also possible to use the Bolzmann transport equation as a starting point.

•Time discretization by use of fractional time step Time discretization by use of fractional time step method. The convective terms are calculated by method. The convective terms are calculated by use of an explicit second order method (a TVD use of an explicit second order method (a TVD scheme was used)scheme was used)

iCCBBiii DBDBnunt

][)()(

By including the density and not including internal coordinates the transport equation for each class is:

Hagesæther (2002) and Hagesæther, Jakobsen & Svendsen (2000).

Berge & Jakobsen (1998) and Hagesæther (2002)

Birth and Death terms

Coalescence:Coalescence:

collision phasecollision phase film rupturefilm rupture

(coalescence)(coalescence)Death term

Death term

Birth term

collision phasecollision phase break-up break-up (uses energy)(uses energy)

Break-up:Break-up:

Death term

Birth term

Birth term


Column with coalescence and break-up:Column with coalescence and break-up:dispersed size dispersed size distributiondistribution

figure from Chen, Reese & Fan (1994)


Coalescence example:Coalescence example:

Death term

Death term

VolumeVolume

Number of Number of particlesparticles

Birth term

How to generate a finite How to generate a finite (small) number of classes (small) number of classes when coalescence and when coalescence and breakup can be between breakup can be between particles of any sizes?particles of any sizes?

Dispersed Size DistributionDispersed Size Distribution


Two methods for finite number of classes:Two methods for finite number of classes:•IntervalInterval size discretization size discretizationHounslow, Ryall & Marshall (1988) and Litster, Smit & Hounslow (1995)Hounslow, Ryall & Marshall (1988) and Litster, Smit & Hounslow (1995)

•Finite pointFinite point size discretization size discretizationBatterham, Hall & Barton (1981) and Hagesæther (2002)Batterham, Hall & Barton (1981) and Hagesæther (2002)

There are other methods beside There are other methods beside population balancespopulation balances that may be used that may be used to solve the to solve the transport equationtransport equation. These . These are not considered here.are not considered here.

Interval size discretization

Volume, (length, area or ...)Volume, (length, area or ...)


Dispersed Size DistributionDispersed Size Distribution

Class covers an intervalClass covers an intervalFirst classFirst class

Equal volume example:Equal volume example:


VolumeVolume


Dispersed Size DistributionDispersed Size DistributionDeath term

Death term

Birth term

Coalescence example:Coalescence example:

1073

21 43 65 7 8 9 1110 12 151413

Which class does Which class does this one belong to?this one belong to?

Second exampleSecond example(problem illustrator)(problem illustrator)

It should not matter as long as one is consistentIt should not matter as long as one is consistent


Two possible classes for a new particle:Two possible classes for a new particle:

n m

n + m - 1

n + m

Problem is how to differentiate between Problem is how to differentiate between the two possibilities.the two possibilities.

Question: Question: Does it matter?Does it matter?


Answer: Answer: YES!, because if the class placement YES!, because if the class placement process of the new particles is done incorrectly it process of the new particles is done incorrectly it willwill lead to a lead to a systematic decreasesystematic decrease or a or a systematic systematic increaseincrease in the total mass of the system. in the total mass of the system.(Easily seen if you assume that all particles are (Easily seen if you assume that all particles are initially ‘center particles’)initially ‘center particles’)


Example: Example: 4 particles of classes 2, 4 particles of classes 2, 2, 3 and 3, thus with initial 2, 3 and 3, thus with initial masses of 1.5, 1.5, 2.5 and 2.5. masses of 1.5, 1.5, 2.5 and 2.5. Assume class 2 and class 2 Assume class 2 and class 2 coalescence, class 3 and class 3 coalescence, class 3 and class 3 coalescence, then coalescence coalescence, then coalescence between the two new particles. between the two new particles. n+m-1 casen+m-1 case gives a gives a total mass total mass decrease of 1.5decrease of 1.5 and and n+m casen+m case gives a gives a total mass increase of 1.5total mass increase of 1.5..

2+2 3+32+2 3+3

3/4 5/63/4 5/6

7/107/10 7/107/10

6.5 9.56.5 9.5

88

ExampleExample

n+m-1n+m-1 n+mn+m

initial massinitial mass


Suggestion: Suggestion: 50% to class 50% to class n+m-1n+m-1 and 50% to class and 50% to class n+mn+m. This because a ‘center particle’ . This because a ‘center particle’ nn and a and a ‘center particle’ ‘center particle’ mm will give a particle on the will give a particle on the border between classes border between classes n+m-1n+m-1 and and n+mn+m..

Evaluation: Evaluation: This is based on the assumption of This is based on the assumption of (initial) flat profiles in the classes. Modifications (initial) flat profiles in the classes. Modifications are needed if class intervals varies in size.are needed if class intervals varies in size.

Conclusion: Conclusion: Suggested method must be tested Suggested method must be tested and analytical tools for such testing are needed.and analytical tools for such testing are needed.

Note: Note: Same problem exist for the break-up case.Same problem exist for the break-up case.


Question: Question: What physical properties do we want What physical properties do we want kept when using population balances, and why?kept when using population balances, and why?

There may be other properties too...There may be other properties too...

Some answers:Some answers:•Number of particles, break-up is O(n) and Number of particles, break-up is O(n) and coalescence is O(ncoalescence is O(n22).).•Length of particles, in crystallizing systems, Length of particles, in crystallizing systems, though ‘McCabe though ‘McCabe L law’ assumes growth is not a L law’ assumes growth is not a function of length of the particle.function of length of the particle.•Area of particles, when diffusion is important.Area of particles, when diffusion is important.•Mass of particles, in CFD simulations.Mass of particles, in CFD simulations.


How to intuitively check if properties are kept:How to intuitively check if properties are kept:

•Number of particles:Number of particles:

Assuming binary break-up and binary coalescence.Assuming binary break-up and binary coalescence.

Compare this to sum of particles in classes.Compare this to sum of particles in classes.

•Mass of particles:Mass of particles:

Compare this to sum of mass in classes.Compare this to sum of mass in classes.

ecoalescencofnumberbreakupofnumberNN 0

0mm

Note that Note that lengthlength and and areaarea are more complicated, I are more complicated, I do not know how to check these in a similar way...do not know how to check these in a similar way...


Scientific method - method of moments:Scientific method - method of moments:

0

)( dLLnLm jj

ii

jij NLm

given on integral form and discrete form.given on integral form and discrete form.number in each classnumber in each class

average valueaverage value

for each classfor each class

Hounslow, Ryall & Marshall (1988)Hounslow, Ryall & Marshall (1988)

Zero moment:Zero moment:

numbertotAreadLLnm .)(0

0


First moment:First moment:

0

1 )( dLLLnm

See also Edwards & Penney (1986)See also Edwards & Penney (1986)

0

1

0

0

)(

)(

m

m

dLLn

dLLLn

L

weighted middleweighted middle

length of particleslength of particles

total length of particlestotal length of particles


In general:In general:

0mNT 1mkL LT 2mkA AT 3mkV VT

numbernumber volumevolumeareaarealengthlength

We now have a method for tracking either the We now have a method for tracking either the total quantitiestotal quantities or or their averagetheir average in the population in the population balance system.balance system.

Total volume (or mass) should be constant.Total volume (or mass) should be constant.

How will the other quantities change How will the other quantities change with break-up and coalescence?with break-up and coalescence?


I am not going to try to find such formulas. See I am not going to try to find such formulas. See Hounslow, Ryall & Marshall (1988).Hounslow, Ryall & Marshall (1988).

With only aggregation (coalescence) they get:With only aggregation (coalescence) they get:

DBdt

dn well mixed batch systemwell mixed batch system

with constant volume.with constant volume.

jjj DB

dt

dm

0

BdLLB jj

0

DdLLD jj

moment equationmoment equation


It is thus found that:It is thus found that:

200

0

2

1m

dt

dm 03 dt

dm

collision frequencycollision frequency

Thus it is found that the number of particles Thus it is found that the number of particles decrease with coalescence and that there is decrease with coalescence and that there is no change in the total volume.no change in the total volume.

Discretization models should give the Discretization models should give the same result with the same assumptionssame result with the same assumptions


Why Hounslow, Ryall & Marshall (1988)?Why Hounslow, Ryall & Marshall (1988)?•‘‘Easy’ to understand this article.Easy’ to understand this article.•‘‘Standard’ reference for population balances.Standard’ reference for population balances.•Gives formulas for size discretization (coalescence).Gives formulas for size discretization (coalescence).

Discretization used: orDiscretization used: orii VV 21 ii LL 3/11 2

VolumeVolumeNum

ber

of p

arti

cles

Num

ber

of p

arti

cles

21 43


Volume, (length, area or ...)Volume, (length, area or ...)

Num

ber

of p

arti

cles

Num

ber

of p

arti

cles

Num

ber

of p

arti

cles

Num

ber

of p

arti

cles

VolumeVolume

Double volumeDouble volume intervals vs. intervals vs. equal sizedequal sized intervals: intervals:

•Generally dispersed Generally dispersed particles are of several particles are of several orders of magnitudeorders of magnitude(for example 1 mm to 10 cm)(for example 1 mm to 10 cm)

•Sometimes too few Sometimes too few classes with this classes with this methodmethod

•Mostly too many Mostly too many classes with this classes with this methodmethod

start of first interval must be set to > 0start of first interval must be set to > 0

Interval size discretizationN

umbe

r of

par

ticl

esN

umbe

r of

par

ticl

es

VolumeVolume

i-1i-1 ii i+1i+1i-2i-2

i2 12 i

Definition of sizes in the system:Definition of sizes in the system:

Size of class Size of class ii:: iiiiV 222 1

Density in class Density in class ii:: iii Nn 2/'


Mechanism for aggregation in interval Mechanism for aggregation in interval ii::•1:1: i-1i-1 and and 1 to i-11 to i-1 BIRTHBIRTH•2:2: i-1i-1 and and i-1i-1 BIRTHBIRTH•3:3: ii and and 1 to i-11 to i-1 DEATHDEATH•4:4: ii and and i to infinityi to infinity DEATHDEATH

coalescence betweencoalescence between


Details for mechanism 2, birth to class Details for mechanism 2, birth to class ii::•2:2: i-1i-1 and and i-1i-1 BIRTHBIRTH


Num

ber

of p

arti

cles

Num

ber

of p

arti

cles

VolumeVolume

i-1i-1 ii case one withcase one with

maximum valuesmaximum values

case two withcase two with

minimum valuesminimum values

Both cases give a new fluid particle in class Both cases give a new fluid particle in class ii. . Thus, coalescence between Thus, coalescence between any two particlesany two particles of of class class i-1i-1 gives a new fluid particle in class gives a new fluid particle in class ii..


1

21

2

2

11

1 2

1

22

1

1

iii

ii NdaN

NR

i

i

coalescence frequencycoalescence frequency3m

3/# m - particle density in class - particle density in class i-1i-1

2

source term forsource term for

mechanism number 2mechanism number 2

to avoid counting each to avoid counting each coalescence twicecoalescence twice

The result above is also easily seen without the The result above is also easily seen without the integration leading to it. The next mechanism integration leading to it. The next mechanism is a bit more difficult.is a bit more difficult.


Details for mechanism 1, birth to class Details for mechanism 1, birth to class ii::•1:1: i-1i-1 and and 1 to i-11 to i-1 BIRTHBIRTH


Num

ber

of p

arti

cles

Num

ber

of p

arti

cles

VolumeVolume

i-2i-2 i-1i-1

Only a fraction of the coalescence between particle Only a fraction of the coalescence between particle jj and particles in class and particles in class i-1i-1 result in a particle in class result in a particle in class ii..

ii

jj class particle, class particle, jj<<i-1i-1

minimum size minimum size needed of particleneeded of particle

in interval in interval i-1i-1 to to get the coalescedget the coalesced

particle in class particle in class ii..

Interval size discretizationN

umbe

r of

par

ticl

esN

umbe

r of

par

ticl

es

VolumeVolume

i-2i-2 i-1i-1 ii

ai 2a

Number of particles Number of particles available for coalescence:available for coalescence: 1

111 222

)2(2

ii

ii

ii

i

aNaN

i212 i

Above equation is based on Above equation is based on an assumption, what is it?an assumption, what is it?

Even (or flat) distribution within each intervalEven (or flat) distribution within each interval


Next step is to integrate over the class the Next step is to integrate over the class the particle of size particle of size aa belongs to belongs to

Summing over all possible Summing over all possible jj classes gives classes gives

jiij

j

j

ii

ji NNdaNaN

R

j

j

1

2

2

11

, 2322

1

coalescencecoalescence

frequencyfrequency

3m

3/# m - particle density in class - particle density in class jj

1

2

1123

i

jji

iji NNR 1


Details for mechanism 4, death of class Details for mechanism 4, death of class ii::•4:4: ii and and i to infinityi to infinity DEATHDEATH


Num

ber

of p

arti

cles

Num

ber

of p

arti

cles

VolumeVolume

i-1i-1 ii

All possible coalescence cases result in the All possible coalescence cases result in the removal of a particle in class removal of a particle in class ii..

i+1i+1case with case with minimum minimum valuesvalues


jij

jiji NNda

NNR

i

j

12

2

, 24

Summing over all possible Summing over all possible jj classes gives classes gives

ij

jii NNR

Integrate over Integrate over jj class gives class gives

4

When j=i, why is there no factor 0.5 included in When j=i, why is there no factor 0.5 included in order to avoid counting each coalescence twice?order to avoid counting each coalescence twice?

Trick question! It is included;) Also included is a factor 2 since two fluid particles are removed when i=jTrick question! It is included;) Also included is a factor 2 since two fluid particles are removed when i=j


Details for mechanism 3, death of class Details for mechanism 3, death of class ii::•3:3: ii and and 1 to i-11 to i-1 DEATHDEATH


Num

ber

of p

arti

cles

Num

ber

of p

arti

cles

VolumeVolume

i-1i-1 ii i+1i+1

jj particle, particle, jj<<i-1i-1

minimum size minimum size needed of particle needed of particle in interval in interval i i so that so that the new particle the new particle will be in classwill be in class i+1. i+1.

Only a fraction of the coalescence between particle Only a fraction of the coalescence between particle jj and particles in class and particles in class ii result in the net removal of a result in the net removal of a particle from class particle from class ii..


Same as for mechanism 1, just writing up the Same as for mechanism 1, just writing up the final resultfinal result

1

1

123i

jj

iji NR 3

Net rate of death for class Net rate of death for class ii is thus given as: is thus given as:

iiiiaggi RkRRkRR 3 421

NOTE: factor NOTE: factor kk added to first and third terms added to first and third terms

Why is there a factor included?Why is there a factor included?


Why factor is added:Why factor is added:

20

00 2

1m

dt

dN

dt

dmNm

i

ii

same result for any value of factor same result for any value of factor kk

03 dt

dmONLY when ONLY when kk = 2/3 = 2/3


Summary for Hounslow et al. (1988) article:Summary for Hounslow et al. (1988) article:•Geometric interval size discretization givenGeometric interval size discretization given•Factor 2 between each classFactor 2 between each class•Aggregation (coalescence) formula givenAggregation (coalescence) formula given•Nucleation and growth also formulatedNucleation and growth also formulated•Number balance and mass balance satisfiedNumber balance and mass balance satisfied•Assumes flat distribution in each classAssumes flat distribution in each class•Generally good results with model usedGenerally good results with model used

•Break-up not includedBreak-up not included


Further reading material:Further reading material:•Litster, Smit & Hounslow (1995)Litster, Smit & Hounslow (1995) give a refined give a refined geometric model for aggregation and growth geometric model for aggregation and growth wherewhere

•Hill & Ng (1995)Hill & Ng (1995) give a discretization give a discretization procedure for the breakage equation, allowing procedure for the breakage equation, allowing any geometric ratio.any geometric ratio.

•Kostoglou & Karabelas (1994)Kostoglou & Karabelas (1994) and and Vanni Vanni (2000)(2000) test several size discretization schemes on test several size discretization schemes on several test cases.several test cases.

qii VV /1

1 2/ whole positive integerwhole positive integer

Finite point size discretizationFinite point size discretization

Some literature for Some literature for finite pointfinite point size discretization: size discretization:•Batterham, Hall & Barton (1981),Batterham, Hall & Barton (1981), first to use first to use finite point size discretization. They made a finite point size discretization. They made a mistake in their balance though, see Hounslow, mistake in their balance though, see Hounslow, Ryall & Marshall (1988)Ryall & Marshall (1988)•Kumar & Ramkrishna (1996).Kumar & Ramkrishna (1996). Article series Article series starting with this one.starting with this one.•Ramkrishna (2000).Ramkrishna (2000). Book about population Book about population balances in general. No more details here than in balances in general. No more details here than in the article series.the article series.

Finite point size discretizationFinite point size discretization

Will show both versions, starting with the first one Will show both versions, starting with the first one since that one is simplest (easiest).since that one is simplest (easiest).

Own methods for Own methods for finite pointfinite point size discretization: size discretization:

•Geometric factor 2 increaseGeometric factor 2 increase

•Randomly increasing class sizesRandomly increasing class sizes

2)(/)1( imim

)()1( imim

Geometric factor 2 increaseGeometric factor 2 increase

Discretization used:Discretization used: )(2)1( imim

massmassNum

ber

of p

arti

cles

Num

ber

of p

arti

cles

21 43

Only fluid particles with these exact sizes are allowedOnly fluid particles with these exact sizes are allowed

What should What should be done with a be done with a fluid particle fluid particle in this area?in this area?


Particle between two classes:Particle between two classes:

)(im )1( imjm

divide particle into classes divide particle into classes ii and and i+1i+1

How to divide the particle into How to divide the particle into the two bounding allowed sizes?the two bounding allowed sizes?


Mass balance:Mass balance:

)1()1()()( iminiminmn jj

Number balance:Number balance:

)1()( ininn j

number density of particlenumber density of particle

mass of particlemass of particle

Combined:Combined:

)1()(

)()(

im

n

innim

n

inm

j

j

jj

the only unknown variablethe only unknown variablejx

With number balance and mass balance used there is With number balance and mass balance used there is only one possible split between the classes for each caseonly one possible split between the classes for each case


)()( kmmim j

Break-up into two daughter fragments with the smallest fragment of a population class size:

largest daughter largest daughter particle particle

Model requires that smallest daughter fluid particle is of a population class size, thus k<i.

Model requires break-up into two daughter fragments.


Break-up:Break-up:

)()()1()1()()( ,, kmimximxkmmim kikij

largest daughter particle is split into two classeslargest daughter particle is split into two classes

ikx ikki ,21

,

parent classparent class daughter classdaughter class

‘‘x’ is given by the mass balance x’ is given by the mass balance and the number balanceand the number balance

Why is fragment above split into classes Why is fragment above split into classes i-1i-1 and and ii??Largest fragment must be at least half the mass of the parent particle. Half the mass of the parent particle is the mass of the class Largest fragment must be at least half the mass of the parent particle. Half the mass of the parent particle is the mass of the class below. Thus the largest fragment must be in the interval between classes below. Thus the largest fragment must be in the interval between classes i-1i-1 and and i i..


Details forDetails for ikx ikki ,21

,

)()1()1()()( ,, imximxkmimm kikij

UsingUsing

)1(2)( 1mim i )1(2)( 1mkm k )1(2)1( 2 mim i

)1(2)1()1(2)1(2)1(2 1,

2,

11 mxmxmm iki

iki

ki GivingGiving

121, 2)22( kiikix )2(1

, 2 ikkix ik

kix 1, 2

particle balanceparticle balance class splittingclass splitting


Coalescence:Coalescence:

)1()1()()()( ,, imximxjmim jiji

largest parent particlelargest parent particle

jix ijji ,21,

parent classparent class parent classparent class

The largest parent particle is defined with index i

found same way as for break-up

Coalescence of two particles:


massmassNum

ber

of p

arti

cles

Num

ber

of p

arti

cles

21 43

)2,4(What are the break-up rates?What are the break-up rates?

total break-up rate, total break-up rate, sm3/1

parent particleparent particle

)2,4(

smallest daughtersmallest daughter

second daughtersecond daughterparticleparticle

volume balance gives:volume balance gives:


Finding break-up source terms by use of a test case:Finding break-up source terms by use of a test case:

If 4 classes, the possible break-ups are:If 4 classes, the possible break-ups are:)1,2(),2,3(),1,3(),3,4(),2,4(),1,4(

parent classparent class smallest daughter classsmallest daughter classtotal break-up rate, total break-up rate, sm3/1

Example:Example: )3,4(

)()()1()1()()( ,, kmimximxkmmim kikij

)3()3,4()4()1)(3,4()3()3,4()4()3,4( 3,43,4 mmxmxm

parent classparent class

amount breaking up of class 4 into class 3amount breaking up of class 4 into class 3total amount into class 3 from splitting the largest particletotal amount into class 3 from splitting the largest particle


All possible break-up cases listed:All possible break-up cases listed: 4)1)(3,4(3)3,4(3)3,4(4)3,4( 3,43,4 xx

4)1)(1,4(3)1,4(1)1,4(4)1,4( 1,41,4 xx

4)1)(2,4(3)2,4(2)2,4(4)2,4( 2,42,4 xx

3)1)(2,3(2)2,3(2)2,3(3)2,3( 2,32,3 xx

3)1)(1,3(2)1,3(1)1,3(3)1,3( 1,31,3 xx

2)1)(1,2(1)1,2(1)1,2(2)1,2( 1,21,2 xx

class numberclass number

death termsdeath terms

Note that death terms for class 3 are:Note that death terms for class 3 are: )2,3()1,3( and


Source term discretizations:Source term discretizations:

1

1

..2),,()(i

kBB NikiiD

death from break-up for class death from break-up for class ii

total number of classestotal number of classes

break-up rate for class break-up rate for class ii into smallest class into smallest class kk, 1/(m, 1/(m33s)s)

why no break-up why no break-up of smallest class?of smallest class?It is not possible to satisfy both number balance and mass It is not possible to satisfy both number balance and mass balance with break-up of class 1.balance with break-up of class 1.

Example with Example with ii=3:=3:

13

1

),3()2,3()1,3()3(k

BBBB kD


4)1)(3,4(3)3,4(3)3,4(4)3,4( 3,43,4 xx

4)1)(1,4(3)1,4(1)1,4(4)1,4( 1,41,4 xx

4)1)(2,4(3)2,4(2)2,4(4)2,4( 2,42,4 xx

3)1)(2,3(2)2,3(2)2,3(3)2,3( 2,32,3 xx

3)1)(1,3(2)1,3(1)1,3(3)1,3( 1,31,3 xx

2)1)(1,2(1)1,2(1)1,2(2)1,2( 1,21,2 xx

12

1,1,2

2

,1,12

,12

),2()1(),12()2,()2(ik

BkNik

Bk

N

NikBB kxkxkB

Example with Example with ii=2 for birth terms:=2 for birth terms:


Nikix

kixikiB

i

ikBki

i

NikBki

N

NiikBB

..1,),()1(

),1(),()(

1

1,1,

,1,1

,1


N

iBB imiDiB

1

0)]())()([(

Total mass balance:Total mass balance:

Why are these limits included?Why are these limits included?First one because the smallest particle in a break-up can not belong to the largest class. Second one is the lower boundary First one because the smallest particle in a break-up can not belong to the largest class. Second one is the lower boundary fragment of the largest fluid particle, it can never belong to the highest class. The upper boundary fragment can similarly fragment of the largest fluid particle, it can never belong to the highest class. The upper boundary fragment can similarly never belong to the lowest (first) class.never belong to the lowest (first) class.

Why is sum Why is sum of mass zero?of mass zero?

Mass balance kept in each break-up case, must thus be Mass balance kept in each break-up case, must thus be kept in sum of break-up cases.kept in sum of break-up cases.


1..1,),(),()(1

1

NiiijiiDN

jCCC


death from coalescence for class death from coalescence for class ii no coalescence no coalescence of largest classof largest class

The coalescence terms can be developed in theThe coalescence terms can be developed in thesame way as the break-up terms. Only showingsame way as the break-up terms. Only showingthe results here.the results here.

Why this term?Why this term?When both fluid particles are of same class two are lost, this When both fluid particles are of same class two are lost, this term accounts for the second oneterm accounts for the second one



birth from coalescence for class birth from coalescence for class ii no coalescence particleno coalescence particle

is possible in class 1is possible in class 1

NijixjixiB C

i

jji

i

NijCjiC ..2),,1()1(),()(

1

1,1

1

,1,

The The moment for the number balancemoment for the number balance will give the will give thesame result as for interval classes. This is expectedsame result as for interval classes. This is expectedsince the number balance is fulfilled in each casesince the number balance is fulfilled in each caseand must thus be similarly fulfilled in all cases.and must thus be similarly fulfilled in all cases.


Summary for geometric factor 2 increaseSummary for geometric factor 2 increase•Easy to implement both break-up and coalescenceEasy to implement both break-up and coalescence•Number balanceNumber balance and and mass balancemass balance fulfilled fulfilled•Possible to change to Possible to change to length-length- and/or and/or area balancesarea balances•Possible to use all balancesPossible to use all balances•Easy to include a growth termEasy to include a growth term

Suggest two ways to include growth?Suggest two ways to include growth?1 - redistribute each particle after each time step to new classes by using number balance and mass balance. 1 - redistribute each particle after each time step to new classes by using number balance and mass balance.

2 - let the size classes grow (must then recalculate ‘x’ used for coalescence and ‘x’ used for break-up, and similarly other variables 2 - let the size classes grow (must then recalculate ‘x’ used for coalescence and ‘x’ used for break-up, and similarly other variables that change with class size)that change with class size)

Randomly increasing class sizesRandomly increasing class sizes

Now we move onward to the more difficult topic of Now we move onward to the more difficult topic of randomly increasing class sizesrandomly increasing class sizes..

MAYBE A SMALL BREAK BEFORE THE MAYBE A SMALL BREAK BEFORE THE DIFFICULT PART? ;)DIFFICULT PART? ;)


massmassNum

ber

of p

arti

cles

Num

ber

of p

arti

cles

21 43

Only fluid particles with these exact sizes are allowedOnly fluid particles with these exact sizes are allowed

5 6 7

These class sizes can These class sizes can be of any mass sizebe of any mass size

The main constraints are:The main constraints are:• Each class must have higher mass than the class Each class must have higher mass than the class below.below.•No break-up of the lowest classNo break-up of the lowest class•No coalescence of the highest classNo coalescence of the highest class


)()( kmmim j

Break-up into two daughter fragments with smallest fragment of a population class size:

must be largest must be largest daughter particle daughter particle

The constraints are:

1,1 ik

)1()( mkmm j


)0

1(),(

disallowedisbreakupif

allowedisbreakupifkizB

Variable for when break-up is allowed:

when the second constraint is brokenwhen the second constraint is broken

If 4 classes, the theoretical possible break-ups are:If 4 classes, the theoretical possible break-ups are:)1,2(),2,3(),1,3(),3,4(),2,4(),1,4(

parent classparent class smallest daughter classsmallest daughter classparent classparent class

Each of these must be checked to see if they are valid.Each of these must be checked to see if they are valid.(well, if you start from (i,i-1), once you find the first valid one the rest will be valid too)(well, if you start from (i,i-1), once you find the first valid one the rest will be valid too)

Why have the constraint that Why have the constraint that mm(k) must be smallest?(k) must be smallest?To avoid the possibility of counting the same break-up case twiceTo avoid the possibility of counting the same break-up case twice


)),(()1()1),(( ,, kiymxkiymxm BkiBkij

Split of largest breakup daughter particle into classes yB(i,k)-1 and yB(i,k)

With geometric factor 2 classes we knew that the largest daughter fragment would be in interval [i-1,i].

Finding the bounding classes:


)),(()1),((

)),(()()(, kiymkiym

kiymkmimx

BB

Bki

Breakup fraction found from number and mass balances:

)()( kmimm j

)),(()),(1()1),((),( kiymkixkiymkixm BBBBj

Giving:


)),(0

),(1()),(,(

inmyif

inmyifnmyi

Kronecker delta (used for both breakup and coalescence):

used to place the break-up fragments in the used to place the break-up fragments in the right classes, same for the coalesced particleright classes, same for the coalesced particle


Break-up test case:Break-up test case:

If 4 classes, the possible break-ups are:If 4 classes, the possible break-ups are:)1,2(),2,3(),1,3(),3,4(),2,4(),1,4(

parent classparent class smallest daughter classsmallest daughter class

sm3/1

Example:Example: )3,4(

)3())3,4(()1(

)1)3,4(()3()4(

3,4

3,4

mymx

ymxmmm

B

Bj

parent classparent class


)3,4()1)(3,4(1)3,4()3,4(3)3,4(4)3,4( 3,43,4 BB yxyx

)1,4()1)(1,4(1)1,4()1,4(1)1,4(4)1,4( 1,41,4 BB yxyx

)2,4()1)(2,4(1)2,4()2,4(2)2,4(4)2,4( 2,42,4 BB yxyx

)2,3()1)(2,3(1)2,3()2,3(2)2,3(3)2,3( 2,32,3 BB yxyx

)1,3()1)(1,3(1)1,3()1,3(1)1,3(3)1,3( 1,31,3 BB yxyx

)1,2()1)(1,2(1)1,2()1,2(1)1,2(2)1,2( 1,21,2 BB yxyx

class numberclass number

death termsdeath terms

),( kizB

All possible break-up cases listed:All possible break-up cases listed:

Note that all terms must be multiplied with:Note that all terms must be multiplied with:

NikizkiiDi

kBBB ..2,),(),()(

1

1



death from break-up for class death from break-up for class ii

Example with Example with ii=3 (break-up death):=3 (break-up death):

),3(),3()2,3()2,3()1,3()1,3()3(13

1

kzkzzD Bk

BBBBBB


)3,4()1)(3,4(1)3,4()3,4(3)3,4(4)3,4( 3,43,4 BB yxyx

)1,4()1)(1,4(1)1,4()1,4(1)1,4(4)1,4( 1,41,4 BB yxyx

)2,4()1)(2,4(1)2,4()2,4(2)2,4(4)2,4( 2,42,4 BB yxyx

)2,3()1)(2,3(1)2,3()2,3(2)2,3(3)2,3( 2,32,3 BB yxyx

)1,3()1)(1,3(1)1,3()1,3(1)1,3(3)1,3( 1,31,3 BB yxyx

)1,2()1)(1,2(1)1,2()1,2(1)1,2(2)1,2( 1,21,2 BB yxyx

N

q

q

ikkqBBkq

N

q

q

NikkqBBkq

N

NikBBB

Niykqzkqx

ykqzkqxkzkB

1

1

1,1,,

1

1 ,1,1,1

,12

..1,),2(),(),()1(

))1,2(),1(),1()2,()2,()2(

Example with Example with ii=2 for birth terms (=2 for birth terms (zzBB(i,k)(i,k) not included): not included):


N

q

q

ikBBBB

N

q

q

NikBBBB

N

NiikBBB

Nikqyikqzkqkqx

kqyikqzkqkqx

ikzikiB

1

1

1,1

1

1 ,1

,1

..1,)),(,(),(),()),(1(

))1),1(,(),1(),1(),1(

),(),()(

Break-up birth (source term):

Total mass balance has been tested for these source terms (birth and death) and found to be correct for a number of random cases.


kmjmim )()(

Coalescence, i >= j:

The constraints are:

)(Nmmk

ji mm

can not have particles outside the can not have particles outside the population size discretization rangepopulation size discretization range

easier to code the model when one easier to code the model when one particle is always the largest oneparticle is always the largest one


)1),(()1()),(( ,, jiymxjiymxm CjiCjik

Split of coalesced particle into classes yC(i,j) and yC(i,j)+1


)1),(()),((

)1),(()()(,

jiymjiym

jiymjmimx

CC

Cji

Coalescence fraction found from number and mass balances:

)0

1(),(

disallowedisecoalescencif

allowedisecoalescencifjizC

Variable for when coalescence is allowed:


i

p

p

ijCCCjp

i

p

p

NijCCCjpC

Nijpyijpzjpx

jpyijpzjpxiB

2

1

1,1,1

1 ,1,

..1,)1),1(,(),1(),1()1(

)),(,(),(),()(

Coalescence birth (source term):

1..1,),(),(),(),()(1

1

NiijzijjizjiiDN

ijCC

i

jCCC

Coalescence death (source term):

coalescence rate (from coalescence model) coalescence rate (from coalescence model)

Total mass balance correct


Summary for randomly increasing class sizes:Summary for randomly increasing class sizes:

•Same as summary for geometric factor 2 increase Same as summary for geometric factor 2 increase •Implemented code about 40% slower than for Implemented code about 40% slower than for geometric factor 2 increase (flow calculations not geometric factor 2 increase (flow calculations not included)included)•Zero moment not tested, should be the same since Zero moment not tested, should be the same since number balance is usednumber balance is used

Population balancesPopulation balances

A framework is now established. What remains is to implement models for break-up and coalescence.

Need to find: ),( kiB ),( jiCcoalescence modelcoalescence modelbreak-up modelbreak-up model

Overview



RESULTSRESULTS






Break-up model

GOAL: FINDGOAL: FIND ),( kiB

Types of break-up:Types of break-up:•turbulent (deformation)turbulent (deformation)•viscous (shear)viscous (shear)•elongation (in accelerating flow)elongation (in accelerating flow)

this one is consideredthis one is considered

Turbulent break-up is assumed to be binary, Turbulent break-up is assumed to be binary, which means two daughter particleswhich means two daughter particles

This part is based on: Luo (1993) and This part is based on: Luo (1993) and Hagesæther (2002)Hagesæther (2002)

Break-up model

Turbulent breakage:Turbulent breakage:•Collisions between turbulent eddies and particlesCollisions between turbulent eddies and particles•Each collision Each collision maymay result in break-up result in break-up•Eddies have different sizesEddies have different sizes•Eddies have different energy levelsEddies have different energy levels

BBB P

Break-up rate written as:Break-up rate written as:

break-up probabilitybreak-up probability

collision frequency - based on gas collision theorycollision frequency - based on gas collision theory

Will focus on the break-up probabilityWill focus on the break-up probability

Break-up model

Number of eddies (inertial subrange):Number of eddies (inertial subrange):

))(1()(26

23 dkkEd

un GLL

turbulent energyturbulent energy

Spectral representationSpectral representationLagrangian representationLagrangian representation

3/53/2)( kkE mass of eddiesmass of eddies /2k

‘‘Differential’ Differential’ Eddy density:Eddy density:

42 /)1( Gcn eddy sizeeddy size

constantconstant gas fractiongas fraction

Break-up model

42 /)1( Gcn

Several possibilities exist for Several possibilities exist for the eddy classes, we use:the eddy classes, we use:

Must integrate nMust integrate n in order to in order to

find the number of particles find the number of particles in each interval (or class)in each interval (or class)

i

imtot DaDaDaaDDD 112 ...

total number of eddiestotal number of eddiessize of first classsize of first class size of second classsize of second class

Good values needed for Good values needed for aa and and m.m.

Break-up model

Eddy class discretization:Eddy class discretization:

)1/()1(... 12 aaDDaDaaDDD mmtot

series formula, see Barnett & Cronin (1986)series formula, see Barnett & Cronin (1986)

Could choose Could choose DD as either length or number of as either length or number of particles. The latter was chosen because:particles. The latter was chosen because:•Fewer particles wanted in each successive Fewer particles wanted in each successive class since larger particles are assumed to class since larger particles are assumed to generate more break-upgenerate more break-up•Do not know a good discretization with length Do not know a good discretization with length as as DD

max

min

dnDtot

Break-up model

So far we have a So far we have a fluid particle size discretizationfluid particle size discretization (population balance) and (population balance) and an eddy size discretizationan eddy size discretization..

Note that the eddy sizes have one general size Note that the eddy sizes have one general size discretization. It may be better to find a method that discretization. It may be better to find a method that fits each specific fluid particle case.fits each specific fluid particle case.

Such an example will be given for the Such an example will be given for the eddy energy eddy energy discretizationdiscretization..

Break-up model

Turbulent kinetic energy distribution in eddies:Turbulent kinetic energy distribution in eddies:

)(/)(),exp()( eepe

probability distributionprobability distribution mean energy mean energy of eddy of of eddy of size size (known)(known)

energy of eddy of size energy of eddy of size

C

all energy levels higher than the all energy levels higher than the critical level will cause break-upcritical level will cause break-up

Need to integrate in order to Need to integrate in order to find total amount of break-upfind total amount of break-up

Break-up model

Assume equal sized classes from 1 to 50 and Assume equal sized classes from 1 to 50 and assume that assume that CC=6.75=6.75, this gives:, this gives:

75.6

50

6

1.2)exp(/)exp( dd

this is the important part!this is the important part!

Even with 1000 Even with 1000 equal sized equal sized classes results classes results are off with up are off with up to about 5%to about 5%

Want equal sizes classes with respect Want equal sizes classes with respect to the number of eddies in each classto the number of eddies in each class

Why the discretization above?Why the discretization above?Each of the eddies result in a break-up. Since they each have the same influence it make sense to have the same Each of the eddies result in a break-up. Since they each have the same influence it make sense to have the same amount in each classamount in each class

Break-up model

Dividing interval Dividing interval CC to to CC+b+b into into nn classes the classes the

accuracy is thenaccuracy is then

C

C

CC

bb

eddd

)exp(/)exp()exp(

The The totaltotal accuracy wanted/needed defines accuracy wanted/needed defines bb..

The accuracy above has been called The accuracy above has been called ‘total’. What else needs to be considered?‘total’. What else needs to be considered?How many classes to divide the total integrated part into. This further affects the accuracy.How many classes to divide the total integrated part into. This further affects the accuracy.

One could also say that One could also say that bb above defines the maximum possible accuracy and that the number of classes, when less than infinite, above defines the maximum possible accuracy and that the number of classes, when less than infinite, will reduce it.will reduce it.

Break-up model

Total range can be written as:Total range can be written as:

nCbd CC

bC

C

)exp()exp()exp(

A single class can be written as:A single class can be written as:

Cbd kkk

bkk

k

)exp()exp()exp(

Combining above gives (latter for first class):Combining above gives (latter for first class):

)exp(

111ln1 b

nnb

Break-up model

By substitution a general formula for b is found:By substitution a general formula for b is found:

1

1

exp)1(

1

)1(

11ln

k

mmk bb

knknb

Note that this discretization is not dependent on Note that this discretization is not dependent on CC, thus the , thus the bb-values need only be calculated once-values need only be calculated once

Break-up model

We now have the following:We now have the following:•Fluid particle size discretizationFluid particle size discretization•Eddy size discretizationEddy size discretization•Eddy energy discretizationEddy energy discretization

The only thing left now is a break-up model...The only thing left now is a break-up model...

Let us take a voyage to such a modelLet us take a voyage to such a model

Break-up model

Old break-up criterion (surface energy criterion):Old break-up criterion (surface energy criterion):

eddyeddy fluid particlefluid particle break-up possibilitiesbreak-up possibilities

If the increase in surface energy (due to break-up) If the increase in surface energy (due to break-up) is less than the turbulent kinetic energy of the is less than the turbulent kinetic energy of the eddy, then break-up occurseddy, then break-up occurs

Break-up model

Increase in surface energy:Increase in surface energy:

)(),( 222ikjkii ddddde

diameter of diameter of parent particleparent particle

diameter of smallest diameter of smallest daughter particledaughter particle

symmetric figure if symmetric figure if volume fraction is volume fraction is used as axisused as axis

equal sized daughter equal sized daughter particles (highest energy)particles (highest energy)

surface tension and surface tension and surface areasurface area

Break-up model

Surface energy criterion:Surface energy criterion: Breakupddee kii ),()(

possible break-up sizespossible break-up sizes

eddy energy leveleddy energy level

equal sized daughter equal sized daughter particles (highest energy)particles (highest energy)

All particles will break upAll particles will break up

second daughter particle second daughter particle in this area is not shownin this area is not shown

Break-up model

Results - models by Luo (1993):Results - models by Luo (1993):

3 extra classes 3 extra classes included at included at lower end, slow lower end, slow decrease in decrease in amount of amount of bubblesbubbles

Break-up model

New break-up criterion (energy density criterion):New break-up criterion (energy density criterion):

The energy density of an eddy must be higher The energy density of an eddy must be higher or equal to the energy density of the daughter or equal to the energy density of the daughter particles resulting from the break-up.particles resulting from the break-up.

Breakupdww ksd )()(

eddy energy densityeddy energy density particle energy densityparticle energy density

Break-up model

volumevolumeareaarea

smallest daughter particlesmallest daughter particle

Energy density of fluid particle:Energy density of fluid particle:

]/[/6)2/()3/4(

)2/(4)( 3

3

2

mJdd

ddw k

k

kks

volumevolumeeddy sizeeddy size

Eddy energy density:Eddy energy density:

]/[)2/()3/4(

)()( 3

3mJ

ewd

volume specific volume specific surface energy?surface energy?

Break-up model

Minimum daughter size:Minimum daughter size:

)(/)()( 3, eddww minkksd

Quick recap:Quick recap:•Surface energy criterion Surface energy criterion may givemay give an upper an upper boundary to the daughter particle size.boundary to the daughter particle size.•Energy density criterion Energy density criterion givesgives a minimum a minimum daughter particle size.daughter particle size.

In order to find the break-up probability In order to find the break-up probability the lowest possible eddy energy level that the lowest possible eddy energy level that result in break-up must be found.result in break-up must be found.

Break-up model

Finding the critical energy density (CED):Finding the critical energy density (CED):

cminkCED de ,,3 /)( highest possible value is for highest possible value is for

equal sized daughter particlesequal sized daughter particles

Break-up for all Break-up for all energy levels higher energy levels higher than CED. Only the than CED. Only the energy density energy density criterion limits the criterion limits the break-up probability.break-up probability.

minimum energy level for the energy density criterionminimum energy level for the energy density criterion

Break-up model

Energy case with no break-up:Energy case with no break-up:

Surface energy Surface energy criterion fulfilledcriterion fulfilled

Energy density Energy density criterion fulfilledcriterion fulfilled

CEDe )(

CED here below max value CED here below max value for surface energy criterionfor surface energy criterion

)(e

No range where No range where both criteria are both criteria are fulfilled at the fulfilled at the same timesame time

Value higher Value higher than CEDthan CED

Break-up model

Finding the critical break-up point (CBP):Finding the critical break-up point (CBP):

First point where both First point where both criteria are fulfilled (CBP)criteria are fulfilled (CBP)



)(CBP

)(CED

Break-up model

)(CBP



Further increase in eddy energy level:Further increase in eddy energy level:

Only a range Only a range of specific of specific daughter sizes daughter sizes are allowed.are allowed.

Break-up model

Total amount of break-up, two cases:Total amount of break-up, two cases:

•e(e())CEDCED>e>eii(d(dii,d,dCEDCED)) (first case shown) (first case shown)

•e(e())CEDCED<e<eii(d(dii,d,dCEDCED))

eeii(d(dii,d,dCEDCED))

CEDCED de /)( 3

(surface energy)(surface energy)

Break-up model

•e (e () ) CEDCED>e>eii(d(dii,d,dCEDCED))

eeii(d(dii,d,dCEDCED))e (e () ) CEDCED

CEDCBP ee )()(

eeii(d(dii,d,dCEDCED))

e (e ()) CEDCED

),()( CBPiiCBP ddee

•e (e () ) CEDCED<e<eii(d(dii,d,dCEDCED))

e (e () ) CBPCBP

ddCBPCBP is the only unknown is the only unknown

Break-up model

)(/)(),exp()( eepe Total break-up probability:Total break-up probability:

)(/)( ee CBPC

C

dPB

)exp(

This is the probability of break-up for a collision This is the probability of break-up for a collision between a between a fluid particle of a specific sizefluid particle of a specific size and an and an eddy of a specific sizeeddy of a specific size..

What about the daughter size distribution?What about the daughter size distribution?

[0,1][0,1]

Break-up model

Surface energy criterion:Surface energy criterion:•Idea is that the more excess energy is available Idea is that the more excess energy is available the more probable the break-up is.the more probable the break-up is.

maxkd

kkii

kiikis

ddddee

ddeeddP

,

0

)()),()((

),()(),(

normalizing normalizing the probability the probability functionfunction(not really needed)(not really needed)

Normalizing means:Normalizing means:

max,

0

1)(),(kd

kkis ddddP

Break-up model

Surface energy plot and daughter probability plot:Surface energy plot and daughter probability plot:

an excess of an excess of energy gives a energy gives a higher probabilityhigher probability

Break-up model

maxk

mink

d

d

kksd

ksdkd

dddww

dwwdP

,

,

)())()((

)()()(

normalizing normalizing the probability the probability functionsfunctions(not really needed)(not really needed)

Energy density criterion:Energy density criterion:•Based on the same idea as surface energyBased on the same idea as surface energy

Break-up model

kdkiskiB dPddPddP ,,

Total break-up probability:Total break-up probability:

probability for surface probability for surface energy criterionenergy criterion

probability for energy probability for energy density criteriondensity criterion

C

dPB

)exp(

Note that the upper probability distribution Note that the upper probability distribution must be normalized so that it matches with must be normalized so that it matches with the lower one. the lower one. This is why the earlier This is why the earlier normalizing of Pnormalizing of Pss and P and Pdd was not needed was not needed..

Break-up model

Daughter size distribution:Daughter size distribution:

l

ljkljiBkjiB ededPddP ),(),,,(),,(

j

jiBkjiBkiB dddPdd ),(),,(),(

parent diameterparent diameter

daughter diameterdaughter diameter

eddy diametereddy diametereddy energy leveleddy energy level

fraction of fraction of eddies at eddies at specified size specified size with given with given energy levelenergy level

collision frequencycollision frequency

How the probability is implemented:How the probability is implemented:

Break-up model

Model assumes averages used.Model assumes averages used.

If too few cases, then use a If too few cases, then use a Monte Carlo methodMonte Carlo method..(Pål Skjetne and John Morud at SINTEF Chemistry have used a M. C. method)(Pål Skjetne and John Morud at SINTEF Chemistry have used a M. C. method)

For a short sensitivity analysis of the current model For a short sensitivity analysis of the current model see pages 126-127 in Hagesæther (2002).see pages 126-127 in Hagesæther (2002).

Break-up model

Results - system data:Results - system data:•Water/airWater/air•14 bubble classes, 0.375 mm to 7.5 mm (radius)14 bubble classes, 0.375 mm to 7.5 mm (radius)•80 eddy size classes, 0.75 mm to 300 mm80 eddy size classes, 0.75 mm to 300 mm•20 eddy energy classes20 eddy energy classes

32 /25.0 sm 12.0G mN /0726.0

eddy dissipationeddy dissipation void fractionvoid fraction surface tensionsurface tension

Break-up model

Results:Results: diameter class 13diameter class 13

log version of log version of the same plotthe same plot

20% increase in 20% increase in

Break-up model

Results - smaller bubble:Results - smaller bubble: diameter class 9diameter class 9


20% increase in 20% increase in has here a larger has here a larger effect than for effect than for diameter class 14diameter class 14

Results - smaller eddy:Results - smaller eddy:

Break-up model

diameter class 9diameter class 9


20% increase in 20% increase in has here a larger has here a larger effect than for effect than for eddy class 20eddy class 20

Break-up model

Results - importance of eddy diameter:Results - importance of eddy diameter:

eddies larger eddies larger than bubble are than bubble are important for important for the total amount the total amount of break-upof break-up

Break-up model

Results - importance of break-up criterions:Results - importance of break-up criterions:

surface energy surface energy criterion is here criterion is here important for important for the amount of the amount of break-upbreak-up

Break-up model

Hesketh, Etchells & Russell (1991) observed two Hesketh, Etchells & Russell (1991) observed two types of breakage:types of breakage:•Particles that undergo large scale deformations Particles that undergo large scale deformations resulting in a wide range of daughter sizesresulting in a wide range of daughter sizes•Tearing mechanism giving a local deformation, Tearing mechanism giving a local deformation, producing a very small and a large fragmentproducing a very small and a large fragment

surface energy criterionsurface energy criterion

Break-up model

Possible model refinements:Possible model refinements:•Activation energyActivation energy•Surface energy criterionSurface energy criterion•Inertial subrange of turbulenceInertial subrange of turbulence•Fluid particle at rest stateFluid particle at rest state•Number of daughter fragmentsNumber of daughter fragments•Collision frequencyCollision frequency

Break-up model

Activation energy:Activation energy:

Analogy to chemical reactions, a surplus of surface Analogy to chemical reactions, a surplus of surface energy may be needed for breakup to occur.energy may be needed for breakup to occur.

The intermediate step may have a larger The intermediate step may have a larger surface area than the final break-upsurface area than the final break-up

Breakupddee kii ),()( *AA**

AAii AAjj AAkk

Break-up model

Surface energy criterion:Surface energy criterion:

Break-up can not use more energy than what is Break-up can not use more energy than what is available in the eddy. When eddy is much larger available in the eddy. When eddy is much larger than particle this may not be realistic. than particle this may not be realistic. An An alternative when eddy is largestalternative when eddy is largest::

The relative size between particle and eddy is The relative size between particle and eddy is then taken into account.then taken into account.

Breakupdddee ikii 3/),()(

Break-up model

Inertial subrange of turbulence:Inertial subrange of turbulence:

An upper range should be included since the An upper range should be included since the turbulent intensity drops off toward it.turbulent intensity drops off toward it.

Fluid particle at rest state:Fluid particle at rest state:Risso & Fabre (1998) found that energy may Risso & Fabre (1998) found that energy may accumulate through successive collisions and finally accumulate through successive collisions and finally result in break-up. Model now assumes that prior result in break-up. Model now assumes that prior collisions does not have any effect.collisions does not have any effect.

Maybe use a rest state above zero would be a Maybe use a rest state above zero would be a solution (analogy to temperature)solution (analogy to temperature)

BreakupdddeEe ikii 30 /),()()(

energy level at restenergy level at rest

Overview



RESULTSRESULTS






Coalescence model

Total coalescence process:Total coalescence process:

collision phasecollision phase film drainagefilm drainage film rupturefilm rupture

(coalescence)(coalescence)

),(),(),( jiCjiCjiC ddddPdd Coalescence source term:Coalescence source term:

coalescence probabilitycoalescence probability collision frequencycollision frequency

This part is mostly from Luo (1993).This part is mostly from Luo (1993).

Coalescence model

Inertial subrange assumed:Inertial subrange assumed:

de

size of large energy size of large energy containing eddiescontaining eddies

(size of equipment)(size of equipment)

size of eddies where viscous size of eddies where viscous dissipation takes place dissipation takes place 4/13 / Ld

3/12/1 tu

Mean turbulent velocity of eddies of size Mean turbulent velocity of eddies of size ::

same equation used to find bubble velocities, same equation used to find bubble velocities, replace replace with with d.d.

Finding the collision frequency:Finding the collision frequency:

Coalescence model

Finding the collision frequency:Finding the collision frequency:

22jiij uuu

ijjijijiC unndddd 2

4),(

The collision frequency can then be written as The collision frequency can then be written as (compare to kinetic gas theory):(compare to kinetic gas theory):

Coalescence model

Finding the coalescence probability:Finding the coalescence probability:

I

CC t

tP exp

coalescence timecoalescence time

interaction timeinteraction time

Problems with above equation Problems with above equation (page 41 in Luo (1993))(page 41 in Luo (1993))::•It is empiricalIt is empirical

•Finding an expression for tFinding an expression for tCC

•Finding an expression for tFinding an expression for tII

Note that since the equation is empirical the correct Note that since the equation is empirical the correct values for tvalues for tCC and t and tII may not be the best values. may not be the best values.

Coalescence model

Finding the coalescence probability:Finding the coalescence probability:

2

2

/15.0

ji

iijLC

dd

dut

see Chesters (1991) for this equation see Chesters (1991) for this equation

3

32 /1/13

//1 iL

jiji

LGjiI

d

ddddddt

From theory by Luo (1993)From theory by Luo (1993)

Coalescence model

),(),( ijCjiC ddPddP Note that:Note that:

/2

ijiLij udWe

2/122

jiij uuu

Combined:Combined:

The above model has been implemented together The above model has been implemented together with the break-up modelwith the break-up model

})1()/(

)]1)(1(75.0[exp{),( 2/1

32/1

5.032

1 ijijLG

ijijjiC WeCddP

constantconstant added massadded mass Weber numberWeber numberdiameter size ratiodiameter size ratio

Coalescence model

A possible expansion of the coalescence rate:A possible expansion of the coalescence rate:

),(),(),( jiCjiCjiC ddddPdd

),(),(),( ,, jibCjitCjiC dddddd

turbulent collisionsturbulent collisionsbuoyancy collisionsbuoyancy collisions

(page 157 in Hagesaether (2002))(page 157 in Hagesaether (2002))

Coalescence model

I

CC t

tP exp

As noted the coalescence model is based on As noted the coalescence model is based on the following empirical equation:the following empirical equation:

Next part is a detailed collision model that Next part is a detailed collision model that hopefully will lead to a coalescence model.hopefully will lead to a coalescence model.

It is based on Luo (1993).It is based on Luo (1993).

Overview




COLLISION COLLISION MODELMODEL

Collision model for 2 fluid particles

Specifications for collision model:Specifications for collision model:•Particle oscillations (new)Particle oscillations (new)•Ellipsoid particles (new)Ellipsoid particles (new)•Exact volume balance (new)Exact volume balance (new)•Mass center correction (new)Mass center correction (new)•Particles of any sizeParticles of any size•Head on collisionsHead on collisions•Force balance for each particleForce balance for each particle•Form drag included (new)Form drag included (new)•Film drainage (new)Film drainage (new)

drainagedrainage

collision forcescollision forces

geometric geometric centercenter

mass mass centercenter

improvement neededimprovement needed


Head on collision:Head on collision:

What are the details of this What are the details of this region? Are they important?region? Are they important?


Flat interface assumption:Flat interface assumption:

Problem 1:Problem 1: Dimple in the film? Dimple in the film?

See Yiantsios & Davis (1990) for See Yiantsios & Davis (1990) for an example. Note that a dimple an example. Note that a dimple forms with relatively slow forms with relatively slow drainage. Turbulent collisions drainage. Turbulent collisions are fast (‘no’ dimple).are fast (‘no’ dimple).

Why is this not Why is this not really possible?really possible?

Drainage of film between particles require higher pressure in Drainage of film between particles require higher pressure in the middle. Since the film is flat the pressure must be the the middle. Since the film is flat the pressure must be the same along the collision interface. These two observations can same along the collision interface. These two observations can not be combined.not be combined.


Problem 2:Problem 2: Different sized particles Different sized particles

The interface area does not look The interface area does not look correct. What are the options?correct. What are the options?

one collision radius is larger one collision radius is larger than the other collision radius.than the other collision radius.


Test 1:Test 1: Same collision radius Same collision radius

The force on the smaller particle is The force on the smaller particle is much higher than on the large particle.much higher than on the large particle.

same collision interface radiussame collision interface radius

Test 2:Test 2: Same collision force Same collision force

One collision radius is much larger One collision radius is much larger than the otherthan the other


Solution:Solution: Curved interface Curved interface

Both collision force and collision radius Both collision force and collision radius can be the same on both fluid particles.can be the same on both fluid particles.

same collision interface radiussame collision interface radius


Basic assumption:Basic assumption: Constant volume during collision Constant volume during collision

Only one particle shown and equal sized collision Only one particle shown and equal sized collision assumed.assumed.

Luo (1993) assumes that the cut off volume is Luo (1993) assumes that the cut off volume is negligible compared to the rest of the particle. negligible compared to the rest of the particle. From Scheele & Leng (1971) I found that it may From Scheele & Leng (1971) I found that it may be about 15% of the total volume.be about 15% of the total volume.


Assume rotational ellipsoid:Assume rotational ellipsoid:

2000 3

4baV

bb00

aa00

2

32

33

2

a

hhabV

bb

aarr 2/1

2

2

)1(b

rah

And of course:And of course: 0VV

hh


Since volume is the same, Since volume is the same, aa and/or and/or bb must change: must change:

bb

aa bb

aa

bb

aa

Only Only bb increased increased due to due to cutoff partcutoff part

Only Only aa increased increased due to due to cutoff partcutoff part

None of the options above seem reasonableNone of the options above seem reasonable


Shape change due to cutoff:Shape change due to cutoff:0

0

b

a

b

a

bb00

aa00

aa

bb

Both length axis Both length axis have increasedhave increased

Even better would be:Even better would be:

Cutoff mass predominantly Cutoff mass predominantly collect in this area. Do not have collect in this area. Do not have equations for such a process.equations for such a process.


Mass center of particle Mass center of particle (used in force balance)(used in force balance)::

geometric centergeometric center and and mass centermass center of particle of particle

geometric centergeometric center of particleof particle

mass centermass center of particle of particle


How to find the difference between the geometric How to find the difference between the geometric center and the mass center:center and the mass center:

Shift of mass center is found by use of the Shift of mass center is found by use of the moment of the volume. The difference is:moment of the volume. The difference is:

22

2

4

200

2

4

1

2

1

44

3ah

a

h

ba

b

expansion of expansion of aa and and bb included included

I used MAPLE for the different integrations. I used MAPLE for the different integrations. MAPLE is a very nice tool (MAPLE is a very nice tool (alternative is MATHEMATICAalternative is MATHEMATICA).).


Oscillation of particle:Oscillation of particle:

tetaa )2/180/sin(1(0

amplitudeamplitude frequencyfrequency phase angle at contactphase angle at contact

damping factordamping factor

bb is found by using the volume balance is found by using the volume balance


faha

h

ba

bh

aha

h

ba

bhz

2

222

4

200

2

1

222

4

200

2

4

1

2

1

44

3

4

1

2

1

44

3

Distance between mass centers of particles:Distance between mass centers of particles:

Distance as function of velocity:Distance as function of velocity:

21 uudt

dz

film thicknessfilm thickness


Force balance for each particle:Force balance for each particle:

CformDDragb FFFF

dz

dum ,

added mass includedadded mass included steady form dragsteady form drag

DbcDrag CubF 22

2

1

brFC

2

restoring surface restoring surface forceforce

lubrication lubrication form dragform drag

parameter for extra parameter for extra pressure in the film (pressure in the film (=2)=2)

collision interface radiuscollision interface radius

particle radiusparticle radius


Lubrication form drag (part of force balance):Lubrication form drag (part of force balance):

rdrTFr

fzzzformD

02/, 2

total normal stress tensortotal normal stress tensor

z

uPT z

zz

2

pressure in the filmpressure in the film

viscous normal stresses viscous normal stresses (previously neglected)(previously neglected)

film interfacefilm interface


If only dissipation (asymptotic consideration):If only dissipation (asymptotic consideration):

2/

2, 2

fz

zc

ViscousformD z

urF

value needed for this parametervalue needed for this parameter

If only pressure loss (asymptotic consideration):If only pressure loss (asymptotic consideration):

3

4

, 2

/3

f

tfrF cPressure

formD

Expressions combined used as lubrication form dragExpressions combined used as lubrication form drag


Film drainage model (Bernoulli):Film drainage model (Bernoulli):

c

Frictionrr

c

rr

c A

FuPuP

22

2,

20,0

at centerat center at radius at radius rr

230 / frunF rcFriction

The above film drainage model is a simple one. After The above film drainage model is a simple one. After some figures are shown for the current model some some figures are shown for the current model some film drainage problems will be presented.film drainage problems will be presented.


No coalescence:No coalescence: Experimental dataExperimental data

from Scheele &from Scheele &

Leng (1971)Leng (1971)

film thicknessfilm thickness

distance between distance between mass centersmass centers

collision collision interface radiusinterface radius

oscillation of oscillation of particleparticle


Coalescence:Coalescence:dotted lines represent 10% increase in dotted lines represent 10% increase in collision velocitycollision velocity

thinner films for thinner films for cases that result cases that result in coalescencein coalescence


viscous termviscous term

pressure termpressure term

No coalescence case:No coalescence case:


Collision model conclusions:Collision model conclusions:•Good comparison with experimental collision radiusGood comparison with experimental collision radius•Good comparison with experimental contact timeGood comparison with experimental contact time•Approach process can be modeled independent of Approach process can be modeled independent of film drainagefilm drainage•No good coalescence criterion foundNo good coalescence criterion found

Overview




COLLISION COLLISION MODELMODEL

FILM DRAINAGE FILM DRAINAGE MODELMODEL

Film drainage for 2 fluid particles

Film problems - Lubrication theoryFilm problems - Lubrication theory

First assumptions made:First assumptions made:•Newtonian fluidNewtonian fluid and and are constants are constants•AxisymmetryAxisymmetry•Gravity is negligibleGravity is negligible

More assumptions:More assumptions:

0,0

z

vbutv z

z 0

t

vr 0

r

vv r

r

pseudo steady statepseudo steady state creeping flow, Recreeping flow, Repp<0.1<0.1

rh

Bird, Stewart & Lightfoot (1960)

Navier-Stokes and continuity equation:Navier-Stokes and continuity equation:

]))(1

([)(2

2

z

vrv

rrrr

P

z

vv

r

vv

t

v rrc

rz

rr

rc

])([)(2

2

z

v

r

vr

rz

P

z

vv

r

vv

t

v zzc

zz

zr

zc

0)(1

z

vrv

rrz

r

Assumptions are now used to remove terms!Assumptions are now used to remove terms!


Assumptions included:Assumptions included:

]))(1

([)(2

2

z

vrv

rrrr

P

z

vv

r

vv

t

v rrc

rz

rr

rc

])([)(2

2

z

v

r

vr

rz

P

z

vv

r

vv

t

v zzc

zz

zr

zc

0)(1

z

vrv

rrz

r

<<<<

Details:Details:

z

v

r

v

r

v

zz

v

rrv

rrrrzzz

r

)()())(1

(


Reduced equations (commonly used):Reduced equations (commonly used):

2

2

z

v

r

P rc

0

z

P0)(

1

z

vrv

rrz

r

Boundary conditions:Boundary conditions:

0rv at the surface (at the surface (z z = 0.5= 0.5hh))

thvz /5.0 at the surfaceat the surface

zr vr

hv

t

h

at the surfaceat the surface

(kinematic boundary condition)(kinematic boundary condition)

0/ zvr at at zz = 0 due to symmetry = 0 due to symmetry

0zv at at zz = 0 due to symmetry = 0 due to symmetry

flat surface is flat surface is not assumed!not assumed!


Integrate N.S. radial directionIntegrate N.S. radial direction

2

2

z

v

r

P rc

21

2

2

1CzCz

r

Pv

cr

0rv at the surface (at the surface (z z = 0.5= 0.5hh))

0/ zvr at at zz = 0 due to symmetry = 0 due to symmetry

Using the following boundary conditions:Using the following boundary conditions:

))2

((2

1 22 hz

r

Pv

cr

parabolic velocity profile caused by the pressure gradientparabolic velocity profile caused by the pressure gradient


Using velocity profile and integrating the continuity Using velocity profile and integrating the continuity equation:equation:

0)(1

z

vrv

rrz

r

2/

0

2/

0

222/

0

)))2

((2

(1

)(1 h h

z

c

h

r dzz

vdz

hz

r

Pr

rrdzrv

rr

Leibnitz theorem:Leibnitz theorem:

difficult to integratedifficult to integrate

)),(),((),()(

)(

11

22

)(

)(

2

1

2

1

ta

ta

ta

ta dt

dataf

dt

datafdx

t

fdxtxf

dt

d

ff

00

h/2h/2


Using Leibnitz:Using Leibnitz:

2/

0

22222

2/

0

22

0))

2(0())

2()

2((

2)))

2((

2(

1

)))2

((2

(1

h

cc

h

c

r

h

r

hhh

r

Prdz

hz

r

Pr

rr

dzh

zr

Pr

rr

zerozerothis can be integrated!this can be integrated!

)(24

1))

2(

3

1(

2

1 3

2/

0

23

r

Prh

rrz

hz

r

Pr

rr c

h

c


Right hand side of integrated continuity equation:Right hand side of integrated continuity equation:

02

12/

0

t

hdz

z

vhz

Combining equations gives:Combining equations gives:

)(12

1 3

r

Prh

rrt

h

c

this is the standard lubrication equationthis is the standard lubrication equation

Need the pressure gradient Need the pressure gradient (this is a problem)(this is a problem)


An alternative:An alternative:

0Uvr at the surface (at the surface (z z = 0.5= 0.5hh))

Using the following boundary conditions:Using the following boundary conditions:

)(12

1)(

1 30 r

Prh

rrhrU

rrt

h

c

plug flow partplug flow part pressure driven partpressure driven part

Gives:Gives:Need the pressure gradient Need the pressure gradient (this is a problem)(this is a problem)


An alternative, Charles & Mason (1960):An alternative, Charles & Mason (1960):

r

c

z

drr

F

dt

dhv

03

3

6

with curved interfacewith curved interface

Assuming a flat interface:Assuming a flat interface:

346

hr

F

dt

dh

c

an expression for an expression for the force is needed. the force is needed. (this is a problem)(this is a problem)


An alternative, modification of lubrication theory:An alternative, modification of lubrication theory:

Boundary conditions: Boundary conditions: (immobile films)(immobile films)

0,00 zr vvz

reference frame fixed reference frame fixed to lower fluid particleto lower fluid particle

dtdhvvthz zr /,0)(

00 rvr

Postulation:Postulation:),(),( 21 trftzfP

See Hagesæther (2002) for details and references.


From Navier-Stokes, radial direction:From Navier-Stokes, radial direction:

)(2

1 2 hzzr

Pv

cr

a little different from before a little different from before due to boundary conditionsdue to boundary conditions

Integrating continuity equation gives:Integrating continuity equation gives:

0)(1

z

vrv

rrz

r z

vrrv

rz

r

)( Cz

vr

rv z

r

2

1 2

00 rvr

Boundary condition:Boundary condition:

0Cz

vrv z

r

2


Using both expressions for radial velocity:Using both expressions for radial velocity:

)(2

1 2 hzzr

Pv

cr

z

vrv z

r

2

Gives:Gives:

232 )

3

1

2(

1Czz

h

r

P

rv

cz

zero from boundary zero from boundary conditioncondition

dtdhvvthz zr /,0)( Using boundary condition:Using boundary condition:

Gives:Gives:

3

6

h

r

dt

dh

r

P c pressure as a pressure as a

function of radiusfunction of radius


Axial velocity with pressure profile included:Axial velocity with pressure profile included:

)23(1 32

3zhz

dt

dh

hvz

Radial velocity with pressure profile included:Radial velocity with pressure profile included:

)(3 2

3hzz

dt

dh

h

rvr


Pressure equation by use of boundary condition:Pressure equation by use of boundary condition:

hf PPrr

outer edge of filmouter edge of film

hydrostatic pressurehydrostatic pressure

)(3

)( 223 fc

h rrdt

dh

hPrP

not consistent with a flat interfacenot consistent with a flat interface


Force fluid exerts on fluid particle:Force fluid exerts on fluid particle:

rdrSPPrdrTFff r

zzzd

r

zzz

00

00

)(22

total normal stress tensortotal normal stress tensor normal stressnormal stresspressure in fluid particlepressure in fluid particle

dt

dh

h

rPPrF cf

dhf 3

42

2

3)(2

SSzzzz is always zero for immobile interfaces is always zero for immobile interfaces

A similar expression can be found for mobile interfacesA similar expression can be found for mobile interfaces


Summary of what is needed:Summary of what is needed:•Transition to lubrication theoryTransition to lubrication theory•Transition between mobile and immobile filmsTransition between mobile and immobile films•Verification of assumptionsVerification of assumptions•Inclusion into collision modelInclusion into collision model•Coalescence criterion needed Coalescence criterion needed (probability function?)(probability function?)

•Curved interfaces rather than flat onesCurved interfaces rather than flat ones•Drainage for non head on collisionsDrainage for non head on collisions


References

Barnett, S. & Cronin, T. M. (1986). Mathematical formulae for engineering and science students, 4th ed., Longman Scientific & Technical, Bradford University Press, UK.

Batterham, R. J., Hall, J. S. & Barton, G. (1981). Pelletizing kinetics and simulation of full-scale balling circuits. Proc. 3rd Int. Symp. on Agglomeration, Nurnberg, W. Germany, A136.

Berge, E. & Jakobsen, H. A. (1998). A regional scale multi-layer model for the calculation of long-term transport and deposition of air pollution in Europe. Tellus, 50, 205-223.

Bird, R. B., Stewart, W. E. & Lightfoot, E. N. (1960). Transport Phenomena. John Wiley & Sons, New York, USA, 83-85.

Charles, G. E. & Mason, S. G. (1960). The coalescence of liquid drops with a flat liquid/liquid interface. Journal of Colloid Science, 15, 236-267.

Chen, R. C., Reese, J. & Fan, L.-S. (1994). Flow structure in a three-dimensional bubble column and three-phase fluidized bed. AIChE Journal, 40, 1093-1104.

References

Chesters, A. K. (1991). The modelling of coalescence processes in fluid-liquid dispersions. Trans. Inst. Chem. Eng., 69, 259-270.

Edwards, C. H. & Penney, D. E. (1986). Calculus and analytic geometry, 2nd ed., Prentice-Hall International Inc, Englewood Cliffs, USA, pages 338-345.

Hagesæther, L. (2002). Coalescence and break-up of drops and bubbles. Dr. ing. Thesis, Department of chemical engineering, Trondheim, Norway.

Hagesæther, L., Jakobsen, H. A. & Svendsen, H. F. (2000). A coalescence and breakup module for implementation in CFD codes, Computers- Aided Chemical Engineering, 8, 367-372.

Hesketh, R. P., Etchells, A. W. & Russel, T. W. F. (1991). Experimental observations of bubble breakage in turbulent flow. Ind. Eng. Chem. Res., 30, 835-.

References

Hill, P. J. & Ng, K. M. (1995). New discretization procedure for the breakage equation, AIChE Journal, 41, 1204-1216.

Hounslow, M. J., Ryall, R. L. & Marshall, V. R. (1988). A discretized population balance for nucleation, growth, and aggregation, AIChE Journal, 34, 1821-1832.

Hulburt, H. M. & Katz, S. (1964). Some problems in particle technology. Chemical Engineering Science, 19, 555-574.

Kostoglou, M. & Karabelas, A. J. (1994). Evaluation of zero order methods for simulating particle coagulation, Journal of Colloid and Interface Science, 163, 420-431.

Kumar, S. & Ramkrishna, D. (1996). On the solution of population balance equations by discretization - I. A fixed pivot technique. Chemical Engineering Science, 51, 1311-1332.

References

Litster, J. D., Smit, D. J. & Hounslow, M. J. (1995). Adjustable discretized population balance for growth and aggregation, AIChE Journal, 41, 591-603.

Luo, H. (1993). Coalescence, breakup and liquid circulation in bubble column reactors. Dr. ing. Thesis, Department of chemical engineering, Trondheim, Norway.

Ramkrishna, D. (2000). Population balances. Academic Press, San Diego, USA.

Randolph, A. D. & Larson, M. A. (1988). Theory of particulate processes. 2nd ed., Academic Press Inc., San Diego, USA.

Risso, F. & Fabre, J. (1998). Oscillations and breakup of a bubble immersed in a turbulent field. J. Fluid Mech., 372, 323-.

References

Scheele, G. F. & Leng, D. E. (1971). An experimental study of factors which promote coalescence of two colliding drops suspended in water - I. Chemical Engineering Science, 26, 1867-1879.

Vanni, M. (2000). Approximate population balance equations for aggregation-breakage processes, Journal of Colloid and Interface Science, 221, 143-160.

Yiantsios, S. G. & Davis, R. H. (1990). On the buoyancy-driven motion of a drop towards a rigid surface or a deformable interface. J. Fluid Mech., 217, 547-573.

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THE HITCHHIKERS GUIDE TO POPULATION BALANCES, BREAK-UP AND COALESCENCE. Lecture series by: Lars...

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