A Population Balance Model for Agglomeration

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A Population Balance Model for Agglomeration

• M. Goodson, M. Kraft, S. Forrest, J. Bridgwater

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Powder Granulation

• Dry calcium carbonate powder particles
• Add aqueous polymer solution as binder
• Irregular aggregates formed
• Coagulation (sticking together)
• no well-established rate law
• Compaction (porosity reduction)
• first order porosity reduction

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Experimental Results Size Distributions
Size distributions are very similar after the
same number of blade revolutions
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Results Qualitative Observations
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Existing Population Balance Model 1

• Characterise particle with three independent
  properties
• solid volume, s,
• liquid volume, l, and
• air volume, a.
• Particle can then be characterised according to
  whether it is
• big (sla gt 500 ?m) or small
• wet ( ) or dry
• Conservation laws for two coagulating particles
  are simple

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Existing Population Balance Model 2

• Propose a property-independent collision rate
  with a property-dependent coagulation
  probability.
• Observations lead to a table of coagulation
  probabilities
• PhD
  Thesis PAL Wauters, TU Delft

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Existing Population Balance Model 3

• Compaction reduction of particle porosity, ?
• Treated by using empirically derived rate law
• In terms of the parameters of interest, porosity
  can be written as
• So the rate of compaction can be expressed as
• Which can lead to unphysical results

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A New Population Balance Model

• How do we characterise an aggregate?
• solid volume, s
• liquid volume, l
• pore volume, p
• total volume, v
• porosity, ?
• surface area, a
• How many of these are independent?
• How many of these are conserved?

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Two types of Coagulation
(1)
(2)

• Interactions can preserve surface area (1) or
  pore volume (2).
• Real interactions may fall between these two
  extremes.
• Need some way of predicting pore volume if area
  is known (and vice versa)

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Characterising an Aggregate
Define theoretical radius that includes a total
volume equal to s p. Use fractal
dimensions Assume Relate pore volume to
surface area Now solid volume (and liquid
volume) conserved on aggregation so if surface
area is conserved, pore volume can be found (and
vice versa)
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New Coagulation Probabilities
Include a consideration of whether particle is
either soft ( ) or hard
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Experimental Results Size Distributions
Size distributions are very similar after the
same number of blade revolutions
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Proposed Coagulation Rate

• Set collision rate and compaction rate
  proportional to blade rotation speed
• Expect similar size distributions after same
  number of rotations at different speeds
• Make critical porosity (above which a particle
  is considered to be soft) a function of blade
  rotation speed
• Expect to see differences in other particle
  properties (e.g. porosity) at different rotation
  speeds

?crit
?
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Multi-dimensional Population Balances

• pbe often size-dependent only
• can be readily solved using standard
  deterministic techniques
• This model is based on three independent
  particle properties
• Solving the population balance equation by
  standard numerical methods gets prohibitively
  computationally expensive

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Stochastic Simulation

• Convert rate equations into probabilities that a
  certain event will happen within a given waiting
  time
• This is a function of the whole population of
  particles at any given time
• Update particle population according to either
  coagulation or compaction jump
• Stochastic treatment of coagulation can be
  inefficient due to the need to consider n(n-1)/2
  possible coagulating pairs.
• Increment timestep for every collision rather
  than every coagulation
• Use table of probabilities to determine which
  collisions result in coagulations
• Complexity of problem significantly reduced

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Stochastic Particle System
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Simulation Results 1
After 120 revolutions, very similar size
distributions are observed at different critical
porosities (to model different blade rotation
speeds)
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Simulation Results 2
but differences can be observed in other
properties, such as the average particle porosity.
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Conclusions

• Some problems cannot be solved with a
  one-dimensional population balance
• Standard numerical methods become prohibitively
  computationally expensive when applied to
  multi-dimensional population balances
• Stochastic methods can be extended from one
  dimension to many without a significant loss of
  efficiency
• Stochastic simulation has been shown to
  qualitatively predict the behaviour of
  granulating powders
• Refinements to the model will then lead to
  quantitative prediction