1. THE HITCHHIKERS GUIDE TO POPULATION BALANCES, BREAK-UP AND COALESCENCE. Lecture series by: Lars Hagesaether October 2002 NTNU

2. Overview - START LECTURE 2 population balance to be solved in CFD-program: function of break-up of ‘large’ particles break-up for class i into class k break-up probability collision frequency collision model for 2 fluid particles improved break-up model

3. Overview COMBINED MODEL CFD METHODS RESULTS BREAKUP MODEL INPUT DATA COALESCENCE MODEL PB SIZE DISCRETIZATION OTHER MODELS Note:.ppt file with lecture will be made available.

4. Population Balances Particle number continuity equation: Terms set to zero For a sub-region, R, to move convectively with the particle phase-space velocity (i.e. Lagrangian viewpoint)

5. Population Balances 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)

6. Population Balances It is also possible to use the Bolzmann transport equation as a starting point. Time discretization by use of fractional time step method. The convective terms are calculated by use of an explicit second order method (a TVD scheme was used)Time discretization by use of fractional time step method. The convective terms are calculated by use of an explicit second order method (a TVD scheme was used) 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)

7. Birth and Death terms Coalescence: collision phase film rupture (coalescence) Death term Birth term collision phase break-up (uses energy) Break-up: Death term Birth term

8. Birth and Death terms Column with coalescence and break-up: dispersed size distribution figure from Chen, Reese & Fan (1994)

9. Birth and Death terms Coalescence example: Death term Volume Number of particles Birth term How to generate a finite (small) number of classes when coalescence and breakup can be between particles of any sizes? Dispersed Size Distribution

10. Birth and Death terms Two methods for finite number of classes: Interval size discretizationInterval size discretization Hounslow, Ryall & Marshall (1988) and Litster, Smit & Hounslow (1995) Finite point size discretizationFinite point size discretization Batterham, Hall & Barton (1981) and Hagesæther (2002) There are other methods beside population balances that may be used to solve the transport equation. These are not considered here.

11. Interval size discretization Volume, (length, area or...) Number of particles Dispersed Size Distribution Class covers an interval First class Equal volume example:

12. Interval size discretization Volume Number of particles Dispersed Size Distribution Death term Birth term Coalescence example: 1073 214365789111012151413 Which class does this one belong to? Second example (problem illustrator) It should not matter as long as one is consistent

13. Interval size discretization Two possible classes for a new particle: nm n + m - 1 n + m Problem is how to differentiate between the two possibilities. Question: Does it matter?

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

15. Interval size discretization Example: 4 particles of classes 2, 2, 3 and 3, thus with initial masses of 1.5, 1.5, 2.5 and 2.5. Assume class 2 and class 2 coalescence, class 3 and class 3 coalescence, then coalescence between the two new particles. n+m-1 case gives a total mass decrease of 1.5 and n+m case gives a total mass increase of 1.5. 2+2 3+3 3/4 5/6 3/4 5/6 7/10 7/10 6.5 9.5 6.5 9.5 8 Example n+m-1n+m initial mass

16. Interval size discretization Suggestion: 50% to class n+m-1 and 50% to class n+m. This because a ‘center particle’ n and a ‘center particle’ m will give a particle on the border between classes n+m-1 and n+m. Evaluation: This is based on the assumption of (initial) flat profiles in the classes. Modifications are needed if class intervals varies in size. Conclusion: Suggested method must be tested and analytical tools for such testing are needed. Note: Same problem exist for the break-up case.

17. Interval size discretization Question: What physical properties do we want kept when using population balances, and why? There may be other properties too... Some answers: Number of particles, break-up is O(n) and coalescence is O(n 2 ).Number of particles, break-up is O(n) and coalescence is O(n 2 ). Length of particles, in crystallizing systems, though ‘McCabe  L law’ assumes growth is not a function of length of the particle.Length of particles, in crystallizing systems, though ‘McCabe  L law’ assumes growth is not a 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.

18. Interval size discretization How to intuitively check if properties are kept: Number of particles:Number of particles: Assuming binary break-up and binary coalescence. Compare this to sum of particles in classes. Mass of particles:Mass of particles: Compare this to sum of mass in classes. Note that length and area are more complicated, I do not know how to check these in a similar way...

19. Interval size discretization Scientific method - method of moments: given on integral form and discrete form. number in each class average value for each class Hounslow, Ryall & Marshall (1988) Zero moment:

20. Interval size discretization First moment: See also Edwards & Penney (1986) weighted middle length of particles total length of particles

21. Interval size discretization In general: numbervolumearealength We now have a method for tracking either the total quantities or their average in the population balance system. Total volume (or mass) should be constant. How will the other quantities change with break-up and coalescence?

22. Interval size discretization I am not going to try to find such formulas. See Hounslow, Ryall & Marshall (1988). With only aggregation (coalescence) they get: well mixed batch system with constant volume. moment equation

23. Interval size discretization It is thus found that: collision frequency Thus it is found that the number of particles decrease with coalescence and that there is no change in the total volume. Discretization models should give the same result with the same assumptions

24. Interval size discretization 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: or Volume Number of particles 2143

25. Interval size discretization Volume, (length, area or...) Number of particles Volume Double volume intervals vs. equal sized intervals: Generally dispersed particles are of several orders of magnitudeGenerally dispersed particles are of several orders of magnitude (for example 1 mm to 10 cm) Sometimes too few classes with this methodSometimes too few classes with this method Mostly too many classes with this methodMostly too many classes with this method start of first interval must be set to > 0

26. Interval size discretization Number of particles Volume i-1ii+1 i-2 Definition of sizes in the system: Size of class i: Density in class i:

27. Interval size discretization Mechanism for aggregation in interval i: 1:i-1 and 1 to i-1BIRTH1:i-1 and 1 to i-1BIRTH 2:i-1 and i-1BIRTH2:i-1 and i-1BIRTH 3:i and 1 to i-1DEATH3:i and 1 to i-1DEATH 4:i and i to infinityDEATH4:i and i to infinityDEATH coalescence between

28. Interval size discretization Details for mechanism 2, birth to class i: 2:i-1 and i-1BIRTH2:i-1 and i-1BIRTH coalescence between Number of particles Volume i-1i case one with maximum values case two with minimum values Both cases give a new fluid particle in class i. Thus, coalescence between any two particles of class i-1 gives a new fluid particle in class i.

29. Interval size discretization coalescence frequency - particle density in class i-1 source term for mechanism number 2 to avoid counting each coalescence twice The result above is also easily seen without the integration leading to it. The next mechanism is a bit more difficult.

30. Interval size discretization Details for mechanism 1, birth to class i: 1:i-1 and 1 to i-1BIRTH1:i-1 and 1 to i-1BIRTH coalescence between Number of particles Volume i-2i-1 Only a fraction of the coalescence between particle j and particles in class i-1 result in a particle in class i. i j class particle, j<i-1 j class particle, j<i-1 minimum size needed of particle in interval i-1 to get the coalesced particle in class i.

31. Interval size discretization Number of particles Volume i-2i-1i Number of particles available for coalescence: Above equation is based on an assumption, what is it? Even (or flat) distribution within each interval

32. Interval size discretization Next step is to integrate over the class the particle of size a belongs to Summing over all possible j classes gives coalescencefrequency - particle density in class j

33. Interval size discretization Details for mechanism 4, death of class i: 4:i and i to infinityDEATH4:i and i to infinityDEATH coalescence between Number of particles Volume i-1i All possible coalescence cases result in the removal of a particle in class i. i+1 case with minimum values

34. Interval size discretization Summing over all possible j classes gives Integrate over j class gives When j=i, why is there no factor 0.5 included in 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=j

35. Interval size discretization Details for mechanism 3, death of class i: 3:i and 1 to i-1DEATH3:i and 1 to i-1DEATH coalescence between Number of particles Volume i-1ii+1 j particle, j<i-1 j particle, j<i-1 minimum size needed of particle in interval i so that the new particle will be in class i+1. Only a fraction of the coalescence between particle j and particles in class i result in the net removal of a particle from class i.

36. Interval size discretization Same as for mechanism 1, just writing up the final result Net rate of death for class i is thus given as: NOTE: factor k added to first and third terms Why is there a factor included?

37. Interval size discretization Why factor is added: same result for any value of factor k ONLY when k = 2/3

38. Interval size discretization 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

39. Interval size discretization Further reading material: Litster, Smit & Hounslow (1995) give a refined geometric model for aggregation and growth whereLitster, Smit & Hounslow (1995) give a refined geometric model for aggregation and growth where Hill & Ng (1995) give a discretization procedure for the breakage equation, allowing any geometric ratio.Hill & Ng (1995) give a discretization procedure for the breakage equation, allowing any geometric ratio. Kostoglou & Karabelas (1994) and Vanni (2000) test several size discretization schemes on several test cases.Kostoglou & Karabelas (1994) and Vanni (2000) test several size discretization schemes on several test cases. whole positive integer

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

41. Finite point size discretization Will show both versions, starting with the first one since that one is simplest (easiest). Own methods for finite point size discretization: Geometric factor 2 increaseGeometric factor 2 increase Randomly increasing class sizesRandomly increasing class sizes

42. Geometric factor 2 increase Discretization used: mass Number of particles 2143 Only fluid particles with these exact sizes are allowed What should be done with a fluid particle in this area?

43. Geometric factor 2 increase Particle between two classes: divide particle into classes i and i+1 How to divide the particle into the two bounding allowed sizes?

44. Geometric factor 2 increase Mass balance: Number balance: number density of particle mass of particle Combined: the only unknown variable With number balance and mass balance used there is only one possible split between the classes for each case

45. Geometric factor 2 increase Break-up into two daughter fragments with the smallest fragment of a population class size: largest daughter 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.

46. Geometric factor 2 increase Break-up: largest daughter particle is split into two classes parent class daughter class ‘x’ is given by the mass balance and the number balance Why is fragment above split into classes i-1 and i? 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 i-1 and i.

47. Geometric factor 2 increase Details for Using Giving particle balance class splitting

48. Geometric factor 2 increase Coalescence: largest parent particle parent class The largest parent particle is defined with index i found same way as for break-up Coalescence of two particles:

49. Geometric factor 2 increase mass Number of particles 2143 What are the break-up rates? total break-up rate, parent particle smallest daughter second daughter particle volume balance gives:

50. Geometric factor 2 increase Finding break-up source terms by use of a test case: If 4 classes, the possible break-ups are: parent class smallest daughter class total break-up rate, Example: parent class amount breaking up of class 4 into class 3 total amount into class 3 from splitting the largest particle

51. Geometric factor 2 increase All possible break-up cases listed: class number death terms Note that death terms for class 3 are:

52. Geometric factor 2 increase Source term discretizations: death from break-up for class i total number of classes break-up rate for class i into smallest class k, 1/(m 3 s) why no break-up of smallest class? It is not possible to satisfy both number balance and mass balance with break-up of class 1. Example with i=3:

53. Geometric factor 2 increase Example with i=2 for birth terms:

54. Geometric factor 2 increase Source term discretizations: Total mass balance: 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 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. Why is sum of mass zero? Mass balance kept in each break-up case, must thus be kept in sum of break-up cases.

55. Geometric factor 2 increase Source term discretizations: death from coalescence for class i no coalescence of largest class The coalescence terms can be developed in the same way as the break-up terms. Only showing the results here. Why this term? When both fluid particles are of same class two are lost, this term accounts for the second one

56. Geometric factor 2 increase Source term discretizations: birth from coalescence for class i no coalescence particle is possible in class 1 The moment for the number balance will give the same result as for interval classes. This is expected since the number balance is fulfilled in each case and must thus be similarly fulfilled in all cases.

57. Geometric factor 2 increase Summary for geometric factor 2 increase Easy to implement both break-up and coalescenceEasy to implement both break-up and coalescence Number balance and mass balance fulfilledNumber balance and mass balance fulfilled Possible to change to length- and/or area balancesPossible to change to length- and/or area 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? 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 that change with class size)

58. Randomly increasing class sizes Now we move onward to the more difficult topic of randomly increasing class sizes. MAYBE A SMALL BREAK BEFORE THE DIFFICULT PART? ;)

59. Randomly increasing class sizes mass Number of particles 2143 Only fluid particles with these exact sizes are allowed 567 These class sizes can be of any mass size The main constraints are: Each class must have higher mass than the class below. Each class must have higher mass than the class 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

60. Randomly increasing class sizes Break-up into two daughter fragments with smallest fragment of a population class size: must be largest daughter particle The constraints are:

61. Randomly increasing class sizes Variable for when break-up is allowed: when the second constraint is broken If 4 classes, the theoretical possible break-ups are: parent class smallest daughter class parent class 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) Why have the constraint that m(k) must be smallest? To avoid the possibility of counting the same break-up case twice

62. Randomly increasing class sizes Split of largest breakup daughter particle into classes y B (i,k)-1 and y B (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:

63. Randomly increasing class sizes Breakup fraction found from number and mass balances: Giving:

64. Randomly increasing class sizes Kronecker delta (used for both breakup and coalescence): used to place the break-up fragments in the right classes, same for the coalesced particle

65. Randomly increasing class sizes Break-up test case: If 4 classes, the possible break-ups are: parent class smallest daughter class Example: parent class

66. Randomly increasing class sizes class number death terms All possible break-up cases listed: Note that all terms must be multiplied with:

67. Randomly increasing class sizes Source term discretizations: death from break-up for class i Example with i=3 (break-up death):

68. Randomly increasing class sizes Example with i=2 for birth terms (z B (i,k) not included):

69. Randomly increasing class sizes 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.

70. Randomly increasing class sizes Coalescence, i >= j: The constraints are: can not have particles outside the population size discretization range easier to code the model when one particle is always the largest one

71. Randomly increasing class sizes Split of coalesced particle into classes y C (i,j) and y C (i,j)+1

72. Randomly increasing class sizes Coalescence fraction found from number and mass balances: Variable for when coalescence is allowed:

73. Randomly increasing class sizes Coalescence birth (source term): Coalescence death (source term): coalescence rate (from coalescence model) Total mass balance correct

74. Randomly increasing class sizes Summary for randomly increasing class sizes: Same as summary for geometric factor 2 increaseSame as summary for geometric factor 2 increase Implemented code about 40% slower than for geometric factor 2 increase (flow calculations not included)Implemented code about 40% slower than for geometric factor 2 increase (flow calculations not included) Zero moment not tested, should be the same since number balance is usedZero moment not tested, should be the same since number balance is used

75. Population balances A framework is now established. What remains is to implement models for break-up and coalescence. Need to find: coalescence model break-up model

76. Overview COMBINED MODEL CFD METHODS RESULTS BREAKUP MODEL INPUT DATA COALESCENCE MODEL PB SIZE DISCRETIZATION OTHER MODELS

77. Break-up model GOAL: FIND Types of break-up: turbulent (deformation)turbulent (deformation) viscous (shear)viscous (shear) elongation (in accelerating flow)elongation (in accelerating flow) this one is considered Turbulent break-up is assumed to be binary, which means two daughter particles This part is based on: Luo (1993) and Hagesæther (2002)

78. Break-up model Turbulent breakage: Collisions between turbulent eddies and particlesCollisions between turbulent eddies and particles Each collision may result in break-upEach collision may result in break-up Eddies have different sizesEddies have different sizes Eddies have different energy levelsEddies have different energy levels Break-up rate written as: break-up probability collision frequency - based on gas collision theory Will focus on the break-up probability

79. Break-up model Number of eddies (inertial subrange): turbulent energy Spectral representation Lagrangian representation mass of eddies ‘Differential’ Eddy density: eddy size constant gas fraction

80. Break-up model Several possibilities exist for the eddy classes, we use: Must integrate n in order to find the number of particles in each interval (or class) total number of eddies size of first class size of second class Good values needed for a and m.

81. Break-up model Eddy class discretization: series formula, see Barnett & Cronin (1986) Could choose D as either length or number of particles. The latter was chosen because: Fewer particles wanted in each successive class since larger particles are assumed to generate more break-upFewer particles wanted in each successive class since larger particles are assumed to generate more break-up Do not know a good discretization with length as DDo not know a good discretization with length as D

82. Break-up model So far we have a fluid particle size discretization (population balance) and an eddy size discretization. Note that the eddy sizes have one general size discretization. It may be better to find a method that fits each specific fluid particle case. Such an example will be given for the eddy energy discretization.

83. Break-up model Turbulent kinetic energy distribution in eddies: probability distribution mean energy of eddy of size (known) energy of eddy of size energy of eddy of size all energy levels higher than the critical level will cause break-up Need to integrate in order to find total amount of break-up

84. Break-up model Assume equal sized classes from 1 to 50 and assume that  C =6.75, this gives: this is the important part! Even with 1000 equal sized  classes results are off with up to about 5% Want equal sizes classes with respect to the number of eddies in each class 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 amount in each class

85. Break-up model Dividing interval  C to  C +b into n classes the accuracy is then The total accuracy wanted/needed defines b. The accuracy above has been called ‘total’. What else needs to be considered? How many classes to divide the total integrated part into. This further affects the accuracy. One could also say that b above defines the maximum possible accuracy and that the number of classes, when less than infinite, will reduce it.

86. Break-up model Total range can be written as: A single class can be written as: Combining above gives (latter for first class):

87. Break-up model By substitution a general formula for b is found: Note that this discretization is not dependent on  C, thus the b-values need only be calculated once

88. Break-up model 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... Let us take a voyage to such a model

89. Break-up model Old break-up criterion (surface energy criterion): eddy fluid particle break-up possibilities If the increase in surface energy (due to break-up) is less than the turbulent kinetic energy of the eddy, then break-up occurs

90. Break-up model Increase in surface energy: diameter of parent particle diameter of smallest daughter particle symmetric figure if volume fraction is used as axis equal sized daughter particles (highest energy) surface tension and surface area

91. Break-up model Surface energy criterion: possible break-up sizes eddy energy level equal sized daughter particles (highest energy) All particles will break up second daughter particle in this area is not shown

92. Break-up model Results - models by Luo (1993): 3 extra classes included at lower end, slow decrease in amount of bubbles

93. Break-up model New break-up criterion (energy density criterion): The energy density of an eddy must be higher or equal to the energy density of the daughter particles resulting from the break-up. eddy energy density particle energy density

94. Break-up model volumearea smallest daughter particle Energy density of fluid particle: volume eddy size Eddy energy density: volume specific surface energy?

95. Break-up model Minimum daughter size: Quick recap: Surface energy criterion may give an upper boundary to the daughter particle size.Surface energy criterion may give an upper boundary to the daughter particle size. Energy density criterion gives a minimum daughter particle size.Energy density criterion gives a minimum daughter particle size. In order to find the break-up probability the lowest possible eddy energy level that result in break-up must be found.

96. Break-up model Finding the critical energy density (CED): highest possible value is for equal sized daughter particles Break-up for all energy levels higher than CED. Only the energy density criterion limits the break-up probability. minimum energy level for the energy density criterion

97. Break-up model Energy case with no break-up: Surface energy criterion fulfilled Energy density criterion fulfilled CED here below max value for surface energy criterion No range where both criteria are fulfilled at the same time Value higher than CED

98. Break-up model Finding the critical break-up point (CBP): First point where both criteria are fulfilled (CBP) Surface energy criterion fulfilled Energy density criterion fulfilled

99. Break-up model Surface energy criterion fulfilled Energy density criterion fulfilled Further increase in eddy energy level: Only a range of specific daughter sizes are allowed.

100. Break-up model Total amount of break-up, two cases: e( ) CED >e i (d i,d CED ) (first case shown)e( ) CED >e i (d i,d CED ) (first case shown) e( ) CED <e i (d i,d CED )e( ) CED <e i (d i,d CED ) e i (d i,d CED ) (surface energy)

101. Break-up model e ( ) CED >e i (d i,d CED )e ( ) CED >e i (d i,d CED ) e i (d i,d CED ) e ( ) CED e i (d i,d CED ) e ( ) CED e ( ) CED <e i (d i,d CED )e ( ) CED <e i (d i,d CED ) e ( ) CBP d CBP is the only unknown

102. Break-up model Total break-up probability: This is the probability of break-up for a collision between a fluid particle of a specific size and an eddy of a specific size. What about the daughter size distribution? [0,1]

103. Break-up model Surface energy criterion: Idea is that the more excess energy is available the more probable the break-up is.Idea is that the more excess energy is available the more probable the break-up is. normalizing the probability function (not really needed) Normalizing means:

104. Break-up model Surface energy plot and daughter probability plot: an excess of energy gives a higher probability

105. Break-up model normalizing the probability functions (not really needed) Energy density criterion: Based on the same idea as surface energyBased on the same idea as surface energy

106. Break-up model Total break-up probability: probability for surface energy criterion probability for energy density criterion Note that the upper probability distribution must be normalized so that it matches with the lower one. This is why the earlier normalizing of P s and P d was not needed.

107. Break-up model Daughter size distribution: parent diameter daughter diameter eddy diameter eddy energy level fraction of eddies at specified size with given energy level collision frequency How the probability is implemented:

108. Break-up model Model assumes averages used. If too few cases, then use a Monte Carlo method. (Pål Skjetne and John Morud at SINTEF Chemistry have used a M. C. method) For a short sensitivity analysis of the current model see pages 126-127 in Hagesæther (2002).

109. Break-up model 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 eddy dissipation void fraction surface tension

110. Break-up model Results: diameter class 13 log version of the same plot 20% increase in 

111. Break-up model Results - smaller bubble: diameter class 9 20% increase in  20% increase in  has here a larger effect than for diameter class 14

112. Results - smaller eddy: Break-up model diameter class 9 20% increase in  20% increase in  has here a larger effect than for eddy class 20

113. Break-up model Results - importance of eddy diameter: eddies larger than bubble are important for the total amount of break-up

114. Break-up model Results - importance of break-up criterions: surface energy criterion is here important for the amount of break-up

115. Break-up model Hesketh, Etchells & Russell (1991) observed two types of breakage: Particles that undergo large scale deformations resulting in a wide range of daughter sizesParticles that undergo large scale deformations resulting in a wide range of daughter sizes Tearing mechanism giving a local deformation, producing a very small and a large fragmentTearing mechanism giving a local deformation, producing a very small and a large fragment surface energy criterion

116. Break-up model 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

117. Break-up model Activation energy: Analogy to chemical reactions, a surplus of surface energy may be needed for breakup to occur. The intermediate step may have a larger surface area than the final break-up A*A*A*A* AiAiAiAi AjAjAjAj AkAkAkAk

118. Break-up model Surface energy criterion: Break-up can not use more energy than what is available in the eddy. When eddy is much larger than particle this may not be realistic. An alternative when eddy is largest: The relative size between particle and eddy is then taken into account.

119. Break-up model Inertial subrange of turbulence: An upper range should be included since the turbulent intensity drops off toward it. Fluid particle at rest state: Risso & Fabre (1998) found that energy may accumulate through successive collisions and finally result in break-up. Model now assumes that prior collisions does not have any effect. Maybe use a rest state above zero would be a solution (analogy to temperature) energy level at rest

120. Overview COMBINED MODEL CFD METHODS RESULTS BREAKUP MODEL INPUT DATA COALESCENCE MODEL PB SIZE DISCRETIZATION OTHER MODELS

121. Coalescence model Total coalescence process: collision phase film drainage film rupture (coalescence) Coalescence source term: coalescence probability collision frequency This part is mostly from Luo (1993).

122. Coalescence model Inertial subrange assumed: size of large energy containing eddies (size of equipment) size of eddies where viscous dissipation takes place Mean turbulent velocity of eddies of size : same equation used to find bubble velocities, replace with d. Finding the collision frequency:

123. Coalescence model Finding the collision frequency: The collision frequency can then be written as (compare to kinetic gas theory):

124. Coalescence model Finding the coalescence probability: coalescence time interaction time Problems with above equation (page 41 in Luo (1993)) : It is empiricalIt is empirical Finding an expression for t CFinding an expression for t C Finding an expression for t IFinding an expression for t I Note that since the equation is empirical the correct values for t C and t I may not be the best values.

125. Coalescence model Finding the coalescence probability: see Chesters (1991) for this equation From theory by Luo (1993)

126. Coalescence model Note that: Combined: The above model has been implemented together with the break-up model constant added mass Weber number diameter size ratio

127. Coalescence model A possible expansion of the coalescence rate: turbulent collisions buoyancy collisions (page 157 in Hagesaether (2002))

128. Coalescence model As noted the coalescence model is based on the following empirical equation: Next part is a detailed collision model that hopefully will lead to a coalescence model. It is based on Luo (1993).

129. Overview BREAKUP MODEL COALESCENCE MODEL PB SIZE DISCRETIZATION COLLISION MODEL

130. Collision model for 2 fluid particles 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) drainage collision forces geometric center mass center improvement needed

131. Collision model for 2 fluid particles Head on collision: What are the details of this region? Are they important?

132. Collision model for 2 fluid particles Flat interface assumption: Problem 1: Dimple in the film? See Yiantsios & Davis (1990) for an example. Note that a dimple forms with relatively slow drainage. Turbulent collisions are fast (‘no’ dimple). Why is this not really possible? Drainage of film between particles require higher pressure in the middle. Since the film is flat the pressure must be the same along the collision interface. These two observations can not be combined.

133. Collision model for 2 fluid particles Problem 2: Different sized particles The interface area does not look correct. What are the options? one collision radius is larger than the other collision radius.

134. Collision model for 2 fluid particles Test 1: Same collision radius The force on the smaller particle is much higher than on the large particle. same collision interface radius Test 2: Same collision force One collision radius is much larger than the other

135. Collision model for 2 fluid particles Solution: Curved interface Both collision force and collision radius can be the same on both fluid particles. same collision interface radius

136. Collision model for 2 fluid particles Basic assumption: Constant volume during collision Only one particle shown and equal sized collision assumed. Luo (1993) assumes that the cut off volume is negligible compared to the rest of the particle. From Scheele & Leng (1971) I found that it may be about 15% of the total volume.

137. Collision model for 2 fluid particles Assume rotational ellipsoid: b0b0b0b0 a0a0a0a0 b a r And of course: h

138. Collision model for 2 fluid particles Since volume is the same, a and/or b must change: b a b a b a Only b increased due to cutoff part Only a increased due to cutoff part None of the options above seem reasonable

139. Collision model for 2 fluid particles Shape change due to cutoff: b0b0b0b0 a0a0a0a0 a b Both length axis have increased Even better would be: Cutoff mass predominantly collect in this area. Do not have equations for such a process.

140. Collision model for 2 fluid particles Mass center of particle (used in force balance): geometric center and mass center of particle geometric center of particle mass center of particle

141. Collision model for 2 fluid particles How to find the difference between the geometric center and the mass center: Shift of mass center is found by use of the moment of the volume. The difference is: expansion of a and b included I used MAPLE for the different integrations. MAPLE is a very nice tool ( alternative is MATHEMATICA ).

142. Collision model for 2 fluid particles Oscillation of particle: amplitudefrequency phase angle at contact damping factor b is found by using the volume balance

143. Collision model for 2 fluid particles Distance between mass centers of particles: Distance as function of velocity: film thickness

144. Collision model for 2 fluid particles Force balance for each particle: added mass included steady form drag restoring surface force lubrication form drag parameter for extra pressure in the film (  =2) collision interface radius particle radius

145. Collision model for 2 fluid particles Lubrication form drag (part of force balance): total normal stress tensor pressure in the film viscous normal stresses (previously neglected) film interface

146. Collision model for 2 fluid particles If only dissipation (asymptotic consideration): value needed for this parameter If only pressure loss (asymptotic consideration): Expressions combined used as lubrication form drag

147. Collision model for 2 fluid particles Film drainage model (Bernoulli): at center at radius r The above film drainage model is a simple one. After some figures are shown for the current model some film drainage problems will be presented.

148. Collision model for 2 fluid particles No coalescence: Experimental data from Scheele & Leng (1971) film thickness distance between mass centers collision interface radius oscillation of particle

149. Collision model for 2 fluid particles Coalescence: dotted lines represent 10% increase in collision velocity thinner films for cases that result in coalescence

150. Collision model for 2 fluid particles viscous term pressure term No coalescence case:

151. Collision model for 2 fluid particles 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 film drainageApproach process can be modeled independent of film drainage No good coalescence criterion foundNo good coalescence criterion found

152. Overview BREAKUP MODEL COALESCENCE MODEL PB SIZE DISCRETIZATION COLLISION MODEL FILM DRAINAGE MODEL

153. Film drainage for 2 fluid particles Film problems - Lubrication theory First assumptions made: Newtonian fluidNewtonian fluid  and  are constants  and  are constants AxisymmetryAxisymmetry Gravity is negligibleGravity is negligible More assumptions: pseudo steady state creeping flow, Re p <0.1 Bird, Stewart & Lightfoot (1960)

154. Navier-Stokes and continuity equation: Assumptions are now used to remove terms! Film drainage for 2 fluid particles

155. Assumptions included: << Details: Film drainage for 2 fluid particles

156. Reduced equations (commonly used): Boundary conditions: at the surface (z = 0.5h) at the surface (kinematic boundary condition) at z = 0 due to symmetry flat surface is not assumed! Film drainage for 2 fluid particles

157. Integrate N.S. radial direction at the surface (z = 0.5h) at z = 0 due to symmetry Using the following boundary conditions: parabolic velocity profile caused by the pressure gradient Film drainage for 2 fluid particles

158. Using velocity profile and integrating the continuity equation: Leibnitz theorem: difficult to integrate f 0 h/2 Film drainage for 2 fluid particles

159. Using Leibnitz: zero this can be integrated! Film drainage for 2 fluid particles

160. Right hand side of integrated continuity equation: Combining equations gives: this is the standard lubrication equation Need the pressure gradient (this is a problem) Film drainage for 2 fluid particles

161. An alternative: at the surface (z = 0.5h) Using the following boundary conditions: plug flow part pressure driven part Gives: Need the pressure gradient (this is a problem) Film drainage for 2 fluid particles

162. An alternative, Charles & Mason (1960): with curved interface Assuming a flat interface: an expression for the force is needed. (this is a problem) Film drainage for 2 fluid particles

163. An alternative, modification of lubrication theory: Boundary conditions: (immobile films) reference frame fixed to lower fluid particle Postulation: See Hagesæther (2002) for details and references. Film drainage for 2 fluid particles

164. From Navier-Stokes, radial direction: a little different from before due to boundary conditions Integrating continuity equation gives: Boundary condition: Film drainage for 2 fluid particles

165. Using both expressions for radial velocity: Gives: zero from boundary condition Using boundary condition: Gives: pressure as a function of radius Film drainage for 2 fluid particles

166. Axial velocity with pressure profile included: Radial velocity with pressure profile included: Film drainage for 2 fluid particles

167. Pressure equation by use of boundary condition: outer edge of film hydrostatic pressure not consistent with a flat interface Film drainage for 2 fluid particles

168. Force fluid exerts on fluid particle: total normal stress tensor normal stress pressure in fluid particle S zz is always zero for immobile interfaces A similar expression can be found for mobile interfaces Film drainage for 2 fluid particles

169. 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 (probability function?)Coalescence criterion needed (probability function?) Curved interfaces rather than flat onesCurved interfaces rather than flat ones Drainage for non head on collisionsDrainage for non head on collisions Film drainage for 2 fluid particles

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174. 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.

175. The end SO LONG AND THANKS FOR ALL THE GREATFUN.