Nov 15, 2016 - David N. Williams1, Zhenping Liu1,2, Christina Tang3,. Alberto Passalacqua2 and ..... Flory exponent good solvent. J. C. Cheng et al, Journal of ...
Modeling and Simulation of Flash NanoPrecipitation in a Multi-Inlet Vortex Reactor David N. Williams1, Zhenping Liu1,2, Christina Tang3, Alberto Passalacqua2 and Rodney O. Fox1 AIChE 2016 Annual Meeting November 15, 2016 2:10pm-2:35pm (1) Chemical and Biological Engineering, Iowa State University, Ames, IA (2) Mechanical Engineering, Iowa State University, Ames, IA (3) Chemical and Life Science Engineering, Virginia Commonwealth University, Richmond, VA
Flash NanoPrecipitation (FNP) •
Mixing sensitive process •
•
•
Occurs on order of μs
Two unique, miscible streams •
Solvent: includes copolymer and organic
•
Anti-solvent
Easily tuned particle size
Y. Shi et al, Journal of Micromechanics and Microengineering, 2013
B.K. Johnson and R.K. Prud’homme, Australian Journal of Chemistry, 2003
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FNP methods Confined Impinging Jets Reactor (CIJR)
http://ascomp.ch/uploads/page/nanoparticles-final.png
Multi-Inlet Vortex Reactor (MIVR)
J. C. Cheng and R.O. Fox, Industrial & Engineering Chemistry Research, 2010.
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FNP and scale-up
Z. Liu et al, Journal of Fluids Engineering, 2015
Macroscale Y. Shi et al, Journal of Micromechanics and Microengineering, 2013
Microscale •
Both scales have been simulated and studied experimentally for fluid study
•
CFD simulations aid in scale-up 4 of 43
Motivation and goals Simulate FNP at microscale and validate models such that simulations can be scaled up to macroscale Outline: 1.
2.
Continuous phase Disperse phase • Population balance equation • Quadrature-based moment methods • Experimental comparison
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FNP modeling Continuous phase
https://www.tonex.com/wp-content/uploads/cfd-640x372.jpg
Conservation equations, scalar mixing, etc
Disperse phase
http://www.me.iastate.edu/files/2011/06/flow2.jpg
Kinetic equations, population balances
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Continuous phase -> geometry and mesh
• •
MIVR has complex, swirling flow in mixing chamber Turbulence model uses hybrid RANS/LES method
Z. Liu, Ph.D Final Exam, Dept. of Mechanical Engineering, Iowa State University, Ames, IA, 2016
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Mean velocity in MIVR
Z. Liu, Ph.D Final Exam, Dept. of Mechanical Engineering, Iowa State University, Ames, IA, 2016
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Turbulence intensity
Z. Liu, Ph.D Final Exam, Dept. of Mechanical Engineering, Iowa State University, Ames, IA, 2016
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Disperse phase -> PBE 𝑝
𝑞
𝜕𝑛 𝑝, 𝑞 1 = 𝛽 𝑝 − 𝑖, 𝑞 − 𝑗; 𝑖, 𝑗 𝑛 𝑝 − 𝑖, 𝑞 − 𝑗 𝑛(𝑖, 𝑗) 𝜕𝑡 2 𝑖=0 𝑗=0
∞
∞
−𝑛(𝑝, 𝑞) 𝛽 𝑝, 𝑞; 𝑖, 𝑗 𝑛(𝑖, 𝑗) 𝑖=0 𝑗=0
• • •
Smoluchowski Coagulation Equation tracking two discrete variables No analytical solutions for complex 𝛽 −kernels Computationally expensive • p = 20, q = 100 -> 2,000 equations
https://en.wikipedia.org/w iki/Smoluchowski_coagul ation_equation
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Aggregation mechanisms 2) Unimer Insertion
1) Free Coupling
3) Large Aggregate fusion
J. C. Cheng et al, Journal of Colloid and Interface Science , 2010.
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Aggregation kernel equations •
Free Coupling 𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 𝛽 𝑓𝑟𝑒𝑒 (𝑝, 𝑞; 𝑖, 𝑗) = 4𝜋Θ(𝐷𝑝,𝑞 + 𝐷𝑖,𝑗 )(𝑅𝑝,𝑞 + 𝑅𝑖,𝑗 )
•
Unimer Insertion 𝛽 𝑖𝑛𝑠 (𝑝, 𝑞; 𝑖, 𝑗) = 4𝜋Θ 𝐷𝑝,𝑞 +
•
𝐷𝑖,𝑗 𝐴𝑖𝑛𝑠 𝑝,𝑞;𝑖,𝑗
∗ 𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 𝐷𝑝,𝑞;𝑖,𝑗 𝑅𝑝,𝑞 + 𝑅𝑖,𝑗 𝑅𝑝,𝑞 + 𝑅𝑖,𝑗 + 𝑅𝑖𝑐𝑜𝑟 𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 ∗ 𝐷𝑝,𝑞;𝑖,𝑗 𝑅𝑝,𝑞 + 𝑅𝑖,𝑗 + 𝐷𝑝,𝑞 + 𝐷𝑖,𝑗 𝑅𝑖𝑐𝑜𝑟
Large Aggregate Fusion 𝑓𝑢𝑠
𝛽 𝑓𝑢𝑠 (𝑝, 𝑞; 𝑖, 𝑗) = 4𝜋Θ 𝐷𝑝,𝑞 +
𝑓𝑢𝑠 𝐷𝑖,𝑗 𝐴𝑝,𝑞;𝑖,𝑗
𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 𝐷𝑝,𝑞;𝑖,𝑗 𝑅𝑝,𝑞 + 𝑅𝑖,𝑗 𝑅𝑝,𝑞 + 𝑅𝑝𝑐𝑜𝑟 + 𝑅𝑖,𝑗 + 𝑅𝑖𝑐𝑜𝑟 𝑓𝑢𝑠
𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 𝐷𝑝,𝑞;𝑖,𝑗 𝑅𝑝,𝑞 + 𝑅𝑖,𝑗 + 𝐷𝑝,𝑞 + 𝐷𝑖,𝑗 (𝑅𝑝𝑐𝑜𝑟 + 𝑅𝑖𝑐𝑜𝑟 )
J. C. Cheng et al, Journal of Colloid and Interface Science , 2010.
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PBE set-up and initial conditions 𝑝
𝑞
𝜕𝑛(𝑝, 𝑞) 1 = 𝛽 𝑝 − 𝑖, 𝑞 − 𝑗; 𝑖, 𝑗 𝑛 𝑝 − 𝑖, 𝑞 − 𝑗 𝑛(𝑖, 𝑗) 𝜕𝑡 2 𝑖=0 𝑗=0
𝑃
𝑄
−𝑛(𝑝, 𝑞) 𝛽 𝑝, 𝑞; 𝑖, 𝑗 𝑛(𝑖, 𝑗) 𝑖=0 𝑗=0
•
Initial concentrations •
𝑐𝑝0 = 5.91 𝑚𝑜𝑙/𝑚3
𝑐𝑞0 = 44.50 𝑚𝑜𝑙/𝑚3 t=0 • 𝑛(1,0) = 0.1172 • 𝑛(0,1) = 0.8828 •
•
•
Scaled to initial number concentration and characteristic aggregation time • 𝑛0 = (𝑐𝑝0 +𝑐𝑞0 )6.022 × 1023 #/𝑚3 • 𝑡 ∗ = 𝜂𝑠 /(𝑘𝑏 𝑇𝑛0 ) = 1.6822 × 10−8 𝑠 • 𝜏 = 𝑡/𝑡 ∗
J. C. Cheng et al, Journal of Colloid and Interface Science , 2010.
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Normalized number density function
𝑝
𝑞
𝑝 𝑞 14 of 43
Normalized number density function
𝑝
𝑞
𝑝 𝑞 15 of 43
Normalized number density function
𝑝
𝑞
𝑝 𝑞 16 of 43
Normalized number density function
𝑝
𝑞
𝑝 𝑞 17 of 43
Normalized number density function
𝑝
𝑞
𝑝 𝑞 18 of 43
Normalized number density function
𝑝
𝑞
𝑝 𝑞 19 of 43
Normalized number density function
𝑝
𝑞
𝑝 𝑞 20 of 43
Quadrature Based Moment Methods (QBMM) 𝑁
𝑀𝑘1 ,𝑘2 =
න න 𝑝 𝑘1 𝑞 𝑘2 𝑛
𝑝, 𝑞 𝑑𝑞 𝑑𝑝 ≈
𝑘1 𝑘2 𝑤𝛼 𝑝𝛼 𝑞𝛼
𝛼=1
• • •
Uses moments to describe distribution Assumes number density function (NDF) form Reduces number of equations to solve • N = 2 less than 12 equations • Down from 2,000 equations!
D.L. Marchisio and R.O. Fox, Computational Models for Polydisperse Particulate and Multiphase Systems, 2013.
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CQMOM set-up and initial conditions 𝑁𝑝 𝑁𝑞
𝑁𝑝
𝑁𝑞
𝑘
𝑘 𝑘 𝑘 𝑘2 2 𝜕𝑀𝑘1,𝑘2 1 𝑝𝑖 + 𝑝𝑚 𝑘1 𝑞𝑖𝑗 + 𝑞𝑚𝑛 − 𝑝𝑖 1 𝑞𝑖𝑗2 − 𝑝𝑚1 𝑞𝑚𝑛 = 𝜕𝑡 2 𝛽 𝑝𝑖 , 𝑞𝑖𝑗 ; 𝑝𝑚 , 𝑞𝑚𝑛 𝑤1;𝑖 𝑤2;𝑖𝑗 𝑤1;𝑚 𝑤2;𝑚𝑛 𝑖=1 𝑗=1 𝑚=1 𝑛=1
•
Initial concentrations • 𝑐𝑝0 = 5.91 𝑚𝑜𝑙/𝑚3 𝑐𝑞0 = 44.50 𝑚𝑜𝑙/𝑚3 t=0 • 𝑀0,0 = 1
•
•
•
•
𝑀𝑘1,0 = 0.1172
•
𝑀0,𝑘2 = 0.8828
Scaled to initial number concentration and characteristic aggregation time • 𝑛0 = (𝑐𝑝0 +𝑐𝑞0 )6.022 × 1023 #/𝑚3 𝑡 ∗ = 1.6822 × 10−8 𝑠 Nodes specified • 𝑁𝑝 = 2,3 •
•
•
𝑁𝑞 = 1,2
J. C. Cheng et al, Journal of Colloid and Interface Science , 2010.
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Moments 𝑁𝑝 = 2, 𝑁𝑞 = 1
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Moments 𝑁𝑝 = 3, 𝑁𝑞 = 1
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Moments 𝑁𝑝 = 2, 𝑁𝑞 = 2
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Average aggregation number 𝑁𝑝 = 2 𝑁𝑞 = 1
𝑀1,0 = 𝑀0,0
𝑀0,1 = 𝑀0,0 26 of 43
Average aggregation number 𝑁𝑝 = 3 𝑁𝑞 = 1
𝑀1,0 = 𝑀0,0
𝑀0,1 = 𝑀0,0 27 of 43
Average aggregation number 𝑁𝑝 = 2 𝑁𝑞 = 2
𝑀1,0 = 𝑀0,0
𝑀0,1 = 𝑀0,0 28 of 43
Computational expense Method
Simulation Time
CQMOM: Np = 2, Nq = 1
13 seconds
CQMOM: Np = 3, Nq = 1
32 seconds
CQMOM: Np = 2, Nq = 2
283 seconds
PBE
1.95 days
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Experimental comparison •
Carried out in CIJ mixer •
Residence time: 10 ms
•
Replaces organic particles with polymer
•
Uses Dynamic Light Scattering (DLS) to measure data
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FNP micelles
Micelles now contain polymeric core material 31 of 43
Nanoparticle cluster size equations 𝑐𝑜𝑟𝑒 𝑅𝑝,𝑞 =
1 𝜐𝐶3
𝑞𝑁𝐶
𝜈𝐴
+
1 𝜐𝐴3
𝑝𝑁𝐴
𝜈𝐴
1−𝜈𝐵 1 𝜈 𝑅𝑝𝑐𝑜𝑟𝑜𝑛𝑎 = 𝑁𝐵 𝐵 𝑝 2 𝜐𝐵3 𝑐𝑜𝑟𝑒 + 𝑅 𝑐𝑜𝑟𝑜𝑛𝑎 𝐿𝑝,𝑞 = 2 ∗ 𝑅𝑝𝑞 𝑝
• • •
𝑁𝐴 = number of hydrophobic monomers 𝑁𝐵 = number of hydrophilic monomers 𝑁𝐶 = number of core monomers
• • •
𝜐𝐴 = hydrophobic site volume 𝜐𝐵 = hydrophilic site volume 𝜐𝐶 = core material site volume
• •
𝜈𝐴 = Flory exponent bad solvent 𝜈𝐵 = Flory exponent good solvent
J. C. Cheng et al, Journal of Colloid and Interface Science , 2010.
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Polymers of interest block copolymer
core
Polystyrene (PS)
Polyethylene glycol (PEG)
Polyvinyl Phenol (PVPh)
Block A- hydrophobic
Block B- hydrophilic
C- hydrophobic core
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Copolymer and core sizes •
Block A (MW 1600): • •
•
𝑁𝐴 = 15 𝜐𝐴 = 3.16 × 10−27 𝑚3
Block B (MW 5000):
Site volumes calculated using Kuhn lengths (b) •
𝜐=
4𝜋 𝑏 3 3 2
𝑁𝐵 = 114 𝜐𝐵 = 2.04 × 10−28 𝑚3
•
𝑏𝐴 = 18 Å
Polymer C (MW 34000):
•
𝑏𝐵 = 7 Å
• •
•
•
• •
𝑁𝐶 = 284 𝜐𝐶 ≈ 𝜐𝐴
M. Rubinstein and R.H Colby, Polymer Physics, Oxford University Press, p. 53, 2003.
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Experimental conditions MW PS-b-PEG = 6600 MW PVPh = 34000
Mass conc. PS-b-PEG (mg/mL)
Mass conc. PVPh (mg/mL)
Mole fraction PS- Mole fraction b-PEG PVPh
10
10
0.8374
0.1626
20
10
0.9115
0.0885
30
10
0.9392
0.0608
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Comparison set-up and initial conditions 𝑁𝑝 𝑁𝑞
𝑁𝑝
𝑁𝑞
𝑘
𝑘 𝑘 𝑘 𝑘2 2 𝜕𝑀𝑘1,𝑘2 1 𝑝𝑖 + 𝑝𝑚 𝑘1 𝑞𝑖𝑗 + 𝑞𝑚𝑛 − 𝑝𝑖 1 𝑞𝑖𝑗2 − 𝑝𝑚1 𝑞𝑚𝑛 = 𝜕𝑡 2 𝛽 𝑝𝑖 , 𝑞𝑖𝑗 ; 𝑝𝑚 , 𝑞𝑚𝑛 𝑤1;𝑖 𝑤2;𝑖𝑗 𝑤1;𝑚 𝑤2;𝑚𝑛 𝑖=1 𝑗=1 𝑚=1 𝑛=1
•
•
Initial concentrations • 𝑐𝑝0 -> unimer • 𝑐𝑞0 -> core t=0 • 𝑀0,0 = 1 •
𝑀𝑘1,0 = 𝑐𝑝0 /(𝑐𝑝0 + 𝑐𝑞0 )
•
𝑀0,𝑘2 = 𝑐𝑞0 /(𝑐𝑝0 + 𝑐𝑞0 )
•
Scaled to initial number concentration and characteristic aggregation time • 𝑛0 = (𝑐𝑝0 +𝑐𝑞0 )6.022 × 1023 #/𝑚3 𝑡 ∗ = 1.6822 × 10−8 𝑠 Nodes specified • 𝑁𝑝 = 2 𝑘1 = 0, … , 3 •
•
•
𝑁𝑞 = 1 (𝑘2 = 0, 1)
J. C. Cheng et al, Journal of Colloid and Interface Science , 2010.
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NP sizes at 𝜏 = 1 × 106
Predicted NP sizes decrease slightly with copolymer concentration
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Average aggregation number
10 mg/mL PS-b-PEG
30 mg/mL PS-b-PEG 10 mg/mL PS-b-PEG
30 mg/mL PS-b-PEG
Average number of core per aggregate is less than or equal to one
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Size comparison summary PS-PEG Conc. (mg/mL)
• •
Exp. Size (nm) Sim. Size (nm)
10
311
52.61
20
302
49.52
30
311
48.11
Size disparity possibly due to high molecular weight of core Further studies with core sizes could help identify disparity 39 of 43
Summary •
Continuous phase turbulence model has been validated
•
QBMM helped to reduce computational cost and was validated with previous data
•
Simulation and experimental data were compared, although with discrepancies
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Future steps •
Test other calculations for core size
•
Test mixing performance of micro-MIVR using Bourne Reactions
•
Couple disperse phase equations with continuous phase equations in MIVR and CIJR
41 of 43
Acknowledgements •
Support from the the National Science Foundation of the United States, under the SI2 – SSE award NSF – ACI 1440443 is gratefully acknowledged
42 of 43
Thank you for your time!
43 of 43
Appendix QBMM • Aggregation Kernel Parameters •
• • •
•
Free Coupling Unimer Insertion Large Aggregate Fusion
RK2SSP Algorithm and Details
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Types of QBMM •
Direct QMOM 𝑁=2
•
Tensor Product QMOM
•
Brute Force QMOM
•
Conditional QMOM
𝜉2
𝜉1
𝑁
𝑛 𝜉1 , 𝜉2 ≈ 𝑤𝛼 𝛿 𝜉1 − 𝜉1;𝛼 𝛿(𝜉2 − 𝜉2;𝛼 ) 𝛼=1
D.L. Marchisio and R.O. Fox, Computational Models for Polydisperse Particulate and Multiphase Systems, 2013.
45 of 43
Types of QBMM •
Direct QMOM
•
Tensor Product QMOM
•
Brute Force QMOM
•
Conditional QMOM
𝜉2
𝑁𝜉1 = 2 𝑁𝜉2 = 2
𝜉1 𝑁𝜉1 𝑁𝜉2
𝑛 𝜉1 , 𝜉2 ≈ 𝑛 𝜉2 𝜉1 𝑛 𝜉1 ≈ 𝑤1;𝛼 𝑤2;𝛼𝛽 𝛿 𝜉1 − 𝜉1;𝛼 𝛿(𝜉2 − 𝜉2;𝛼𝛽 ) 𝛼=1 𝛽=1
D.L. Marchisio and R.O. Fox, Computational Models for Polydisperse Particulate and Multiphase Systems, 2013.
46 of 43
Conditional QMOM 𝑀𝑘1 ,𝑘2 = න න 𝑝𝑘1 𝑞 𝑘2 𝑛 𝑝, 𝑞 𝑑𝑞 𝑑𝑝 𝑀𝑘1 ,𝑘2 = න න 𝑝𝑘1 𝑞 𝑘2 𝑛 𝑝 𝑛(𝑞|𝑝)𝑑𝑞 𝑑𝑝 𝑁𝑝
𝑀𝑘1 ,𝑘2 = 𝑀𝑘1 ,𝑘2 =
𝑘1 𝑤1;𝑖 𝑝𝑖 𝑖=1 𝑁𝑝 𝑘1 𝑤1;𝑖 𝑝𝑖 𝑖=1
න 𝑛 𝑞 𝑝𝑖 𝑞 𝑘2 𝑑𝑞 < 𝑞 𝑘2 >𝑖
Conditional Moments
D.L. Marchisio and R.O. Fox, Computational Models for Polydisperse Particulate and Multiphase Systems, 2013.
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Vandermonde system 𝑀0,𝑘2 < 𝑞 𝑘2 >1 ⋮ ⋮ 𝑅𝑉 = 𝑀𝑁𝑝 −1,𝑘2 < 𝑞 𝑘2 >𝑁𝑝 𝑤1;1 𝑅=
⋱
,V= 𝑤1;𝑁𝑝
• •
𝑝10 ⋮
𝑁𝑝 −1 𝑝1
… 𝑝𝑁0 𝑝 ⋱ ⋮ 𝑁𝑝 −1 … 𝑝𝑁𝑝
Invert to solve for < 𝑞 𝑘2 >𝑖 Note that < 𝑞 0 >𝑖 = 1
D.L. Marchisio and R.O. Fox, Computational Models for Polydisperse Particulate and Multiphase Systems, 2013.
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PBE to moment transport equation 𝑝
𝑞
∞
𝜕𝑛 𝑝, 𝑞 1 = 𝛽 𝑝 − 𝑖, 𝑞 − 𝑗; 𝑖, 𝑗 𝑛 𝑝 − 𝑖, 𝑞 − 𝑗 𝑛(𝑖, 𝑗) − 𝑛(𝑝, 𝑞) 𝛽 𝑖, 𝑗 𝑛(𝑖, 𝑗) 𝜕𝑡 2 𝑖=0 𝑗=0
𝑖=0
𝑁𝑝 𝑁𝑞
𝑛 𝑝 ≈ 𝑤1;𝑖 𝑤2;𝑖𝑗 𝛿 𝑝 − 𝑝𝑖 𝛿(𝑞 − 𝑞𝑖𝑗 ) 𝑖=1 𝑗=1 𝑁 𝑘
𝑘
𝑀𝑘1,𝑘2 ≈ 𝑤1;𝑖 𝑤2;𝑖𝑗 𝑝𝑖 1 𝑞𝑖𝑗2 𝑖=1
𝑁𝑝 𝑁𝑞
𝑁𝑝
𝑁𝑞
𝜕𝑀𝑘1 ,𝑘2 1 = 𝜕𝑡 2
𝑖=1 𝑗=1 𝑚=1 𝑛=1
𝑝𝑖 + 𝑝𝑚
𝑘1
𝑞𝑖𝑗 + 𝑞𝑚𝑛
𝑘2
𝑘
𝑘
𝑘
𝑘
2 − 𝑝𝑖 1 𝑞𝑖𝑗2 − 𝑝𝑚1 𝑞𝑚𝑛 ∗
𝛽 𝑝𝑖 , 𝑞𝑖𝑗 ; 𝑝𝑚 , 𝑞𝑚𝑛 𝑤1;𝑖 𝑤2;𝑖𝑗 𝑤1;𝑚 𝑤2;𝑚𝑛
J. C. Cheng and R.O. Fox, Industrial & Engineering Chemistry Research , 2010.
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Free Coupling 𝑐𝑜𝑙𝑙 + 𝑅 𝑐𝑜𝑙𝑙 𝛽 𝑓𝑟𝑒𝑒 𝑝, 𝑞; 𝑖, 𝑗 = 4𝜋Θ∗ 𝐷𝑝,𝑞 + 𝐷𝑖,𝑗 𝑅𝑝,𝑞 𝑖,𝑗 𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 𝛽 𝑓𝑟𝑒𝑒 1,0; 0, 𝑗 = 4𝜋Θ∗ 𝐷𝑝,𝑞 + 𝐷𝑖,𝑗 𝑅𝑝,𝑞 + 𝑅𝑖,𝑗 𝑐𝑜𝑙𝑙 + 𝑅 𝑐𝑜𝑙𝑙 𝛽 𝑓𝑟𝑒𝑒 𝑝, 0; 𝑖, 0 = 4𝜋Θ𝑝 𝐷𝑝,𝑞 + 𝐷𝑖,𝑗 𝑅𝑝,𝑞 𝑖,𝑗
𝑐𝑜𝑙𝑙 + 𝑅 𝑐𝑜𝑙𝑙 𝛽 𝑓𝑟𝑒𝑒 0, 𝑞; 0, 𝑗 = 4𝜋Θ𝑜 𝐷𝑝,𝑞 + 𝐷𝑖,𝑗 𝑅𝑝,𝑞 𝑖,𝑗
𝐷𝑝,𝑞
𝑐𝑜𝑙𝑙 + 𝑅 𝑐𝑜𝑟 𝑅𝑝,𝑞 = 𝑅𝑝,𝑞 𝑝
𝑘𝐵 𝑇 = 6𝜋𝜂𝑠 𝑅𝑝,𝑞
𝑐𝑜𝑙𝑙 𝑅𝑝,𝑞 = 𝑞𝑁𝐶 𝜐𝑐 + 𝑝𝑁𝐴 𝜐𝐴
𝑅𝑝𝑐𝑜𝑟 =
1−𝜈𝐵 1 𝜈 𝑁𝐵 𝐵 𝑝 2 𝜐𝐵3
𝜈𝐴
Θ𝑝 = Θ(𝜉 − 𝜉𝑝 ) Θ𝑜 = Θ 𝜉 − 𝜉𝑜 Θ∗ = Θ𝑝 Θ𝑜
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Unimer Insertion 𝛽𝑝𝑖𝑛𝑠 (𝑞; 𝑖, 𝑗) = 4𝜋Θ 𝐷𝑝,𝑞 + 𝐷𝑖,𝑗 𝐴𝑖𝑛𝑠 𝑝,𝑞;𝑖,𝑗 𝑐𝑜𝑟 𝑐𝑝,𝑞 =
𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 ∗ 𝐷𝑝,𝑞;𝑖,𝑗 𝑅𝑝,𝑞 + 𝑅𝑖,𝑗 + 𝐷𝑝,𝑞 + 𝐷𝑖,𝑗 𝑅𝑖𝑐𝑜𝑟
3𝑝𝑁𝐵 𝑐𝑜𝑙𝑙 4𝜋 𝑅𝑝,𝑞 + 𝑅𝑝𝑐𝑜𝑟
∗ 𝐷𝑝,𝑞;𝑖,𝑗 = 𝐷𝑝,𝑞
𝑐∗ =
𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 ∗ 𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 𝐷𝑝,𝑞;𝑖,𝑗 𝑅𝑝,𝑞 + 𝑅𝑖,𝑗 𝑅𝑝,𝑞 + 𝑅𝑖,𝑗 + 𝑅𝑖𝑐𝑜𝑟
∗
𝑐 𝑐𝑜𝑟 𝑐𝑖,𝑗
3
𝑐𝑜𝑙𝑙 − 𝑅𝑝,𝑞
3
3 2
1 3𝜈 −1
𝜐𝐵 𝑁𝐵 𝐵 𝐴𝑖𝑛𝑠 𝑝,𝑞;𝑖,𝑗 = exp −𝛼 𝑞, 𝑗 𝑝 𝑖
J. C. Cheng et al. “A competitive aggregation model for Flash NanoPrecipitation.” Journal of Colloid and Interface Science ,vol. 351, pp. 330-342, Nov 15 2010.
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Large Aggregate Fusion 𝛽 𝑓𝑢𝑠 (𝑝, 𝑞; 𝑖, 𝑗) 𝑓𝑢𝑠
𝑓𝑢𝑠
= 4𝜋Θ 𝐷𝑝,𝑞 + 𝐷𝑖,𝑗 𝐴𝑝,𝑞;𝑖,𝑗 𝑓𝑢𝑠
𝐷𝑝,𝑞;𝑖,𝑗
𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 𝐷𝑝,𝑞;𝑖,𝑗 𝑅𝑝,𝑞 + 𝑅𝑖,𝑗 𝑅𝑝,𝑞 + 𝑅𝑝𝑐𝑜𝑟 + 𝑅𝑖,𝑗 + 𝑅𝑖𝑐𝑜𝑟 𝑓𝑢𝑠
𝑐𝑜𝑙𝑙 𝑐𝑜𝑙𝑙 𝐷𝑝,𝑞;𝑖,𝑗 𝑅𝑝,𝑞 + 𝑅𝑖,𝑗 + 𝐷𝑝,𝑞 + 𝐷𝑖,𝑗 (𝑅𝑝𝑐𝑜𝑟 + 𝑅𝑖𝑐𝑜𝑟 )
𝑅𝑝𝑐𝑜𝑟 + 𝑅𝑖𝑐𝑜𝑟 𝑘𝐵 𝑇 = 𝜂𝑠 𝑁𝑝,𝑞;𝑖,𝑗 𝜒𝑝,𝑞;𝑖,𝑗 𝐿2𝑝,𝑞;𝑖,𝑗
𝑅𝑝𝑐𝑜𝑟 + 𝑅𝑖𝑐𝑜𝑟 𝐿𝑝,𝑞;𝑖,𝑗 = 𝜒𝑝,𝑞;𝑖,𝑗 𝑁𝐵 𝑁𝑝,𝑞;𝑖,𝑗 = 5 𝜒𝑝,𝑞;𝑖,𝑗 3 1 𝜐𝐵3
2
2
1 𝜐𝐵3
𝜒𝑝,𝑞;𝑖,𝑗 = 𝑐𝑜𝑟 𝑐𝑝,𝑞 𝜐𝐵 𝑓𝑢𝑠
3 𝑐𝑜𝑟 + 𝑐𝑖,𝑗 𝜐𝐵 4
𝐴𝑝,𝑞;𝑖,𝑗 = exp −𝛼 𝑞, 𝑗 max
𝑝, 𝑖 min 𝑝, 𝑖
J. C. Cheng et al. “A competitive aggregation model for Flash NanoPrecipitation.” Journal of Colloid and Interface Science ,vol. 351, pp. 330-342, Nov 15 2010.
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RK2SSP 𝑑𝑦 𝑡 = 𝑓 𝑡, 𝑦 𝑡 𝑑𝑡
𝑠
𝑦 (𝑛+1) = 𝑦 (𝑛) + Δ𝑡 𝑏𝑖 𝑓(𝜏𝑖 , 𝜂𝑖 ) 𝑖=1
𝜏𝑖 = 𝑡𝑛 + 𝑐𝑖 Δ𝑡
𝑠−1
𝜂𝑖 = 𝑦 (𝑛) + Δ𝑡 𝑎𝑖𝑗 𝑓(𝜏𝑗 , 𝜂𝑗 ) 𝑖=1
𝑠′
𝑦 (𝑛+1) = 𝑦 (𝑛) + Δ𝑡 𝑏𝑖 𝑓(𝜏𝑖 , 𝜂𝑖 ) 𝑖=1 𝑠
𝑦ො (𝑛+1) = 𝑦
𝑛
+ Δ𝑡 𝑏𝑖′ 𝑓(𝜏𝑖 , 𝜂𝑖 ) 𝑖=1
Tan Trung Nguyen, Frederique Laurent, Rodney Fox, Marc Massot. Solution of population balance equations in applications with fine particles: mathematical modeling and numerical schemes. 2015.
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RK2SSP 𝑦𝑖𝑛 − 𝑦ො𝑖𝑛 =
Δ𝑡 𝑓 𝜏1 , 𝜂1 + 𝑓 𝜏2 , 𝜂2 − 2𝑓 𝜏3 , 𝜂3 3
𝑠𝑐𝑖 = 𝐴𝑡𝑜𝑙𝑖 + max 𝑦𝑖𝑛−1 , 𝑦𝑖𝑛 𝑅𝑡𝑜𝑙𝑖 𝑑
𝑒𝑟𝑟 =
1 𝑦𝑖 − 𝑦ො𝑖 𝑛 𝑠𝑐𝑖 𝑖=1
Δ𝑡𝑛𝑒𝑤 = Δ𝑡𝑜𝑙𝑑 min 1.1, max 0.001,
1.1 1
𝑒𝑟𝑟 3
Tan Trung Nguyen, Frederique Laurent, Rodney Fox, Marc Massot. Solution of population balance equations in applications with fine particles: mathematical modeling and numerical schemes. 2015.
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