1. Population Balance Modeling: Solution Techniques & Applications Dr. R. Bertrum Diemer, Jr. Principal Division Consultant DuPont Engineering Research & Technology

2. Ó R. B. Diemer, Jr., 2003 Lecture Outline l Introduction l Applications to Particles n General Balance Equation n Aerosol Powder Manufacturing n Design Problem l Solution Techniques

3. Introduction

4. Ó R. B. Diemer, Jr., 2003 Definitions & Dimensions l The population balance extends the idea of mass and energy balances to countable objects distributed in some property. l It still holds! In - Out + Net Generation = Accumulation l External & internal dimensions n external dimensions = dimensions of the environment: 3-D space (x,y,z or r,z,  or r, ,  ) and time n internal dimensions = dimensions of the population: diameter, volume, surface area, concentration, age, MW, number of branches, etc.

5. Ó R. B. Diemer, Jr., 2003 A Unifying Principle! l Objects of distributed size found everywhere... n particles: –granulation, flocculation, crystallization, mechanical alloying, aerosol reactors, combustion (soot), crushing, grinding, fluid beds n droplets: –liquid-liquid extraction, emulsification n bubbles: –fluid beds, bubble columns, reactors n polymers: –polymerizers, extruders n cells: –fermentation, biotreatment l Population balances describe how distributions evolve

6. Ó R. B. Diemer, Jr., 2003 Multivariateness l Multivariate refers to # of internal dimensions l Univariate examples: n particle size… polymer MW...cell age l Bivariate examples: n particle volume and surface area (agglomerated particles) n polymer MW and # of branch points (branched polymers) n polymer MW and monomer concentration (copolymers) n cell age and metabolite concentration (biomanufacturing) l Trivariate example: n drop size and solute concentration and drop age for internal concentration gradients (liquid-liquid extraction)

7. Ó R. B. Diemer, Jr., 2003 General Differential Form, 1-D Population convection diffusion “In  Out” in external coordinates growth (“In  Out” in internal coordinate) accumulation sources & sinks (Net Generation) Note: object’s velocity may differ from fluid’s velocity owing to either slip or action of external forces

8. Ó R. B. Diemer, Jr., 2003 l Eliminates 2 physical dimensions, time dimension l Axial Dispersion Model: n With slip... n Without slip... l Plug Flow Model, No Slip: Steady-state, Axisymmetric, Incompressible Flow

9. Ó R. B. Diemer, Jr., 2003 Ideally Mixed Stirred Tank l Eliminates 3 physical dimensions l Batch: l Continuous: n Unsteady state… n Steady state (eliminates time dimension as well)...

10. Ó R. B. Diemer, Jr., 2003 Example: MSMPR Crystallizer l MSMPR = mixed suspension, mixed product removal l Same as continuous stirred tank l Steady state model… no particles in feed, size independent growth rate, no sources or sinks (no primary nucleation, coagulation, breakage)...

11. Applications to Particles General Balance Equation

12. Ó R. B. Diemer, Jr., 2003 Coagulation Breakage Agglomerates Singlets Nucleation Growth Nuclei Coalescence Partially Coalesced Agglomerates Precursor Molecules............... Particle Formation, Growth & Transformation

13. Ó R. B. Diemer, Jr., 2003 Sources and Sinks l Also known as Birth and Death terms l Types of terms: n Nucleation (birth only) n Breakage (birth and death terms) n Coagulation (birth and death terms)

14. Ó R. B. Diemer, Jr., 2003 Full 1-D Population Balance (a partial integrodifferential equation) N = nucleation rate G = accretion rate  coagulation rate  breakage rate b  = daughter distribution v o = nuclei size nucleation term growth term coagulation terms breakage terms

15. Applications to Particles Aerosol Powder Manufacture Gas-to-Particle Conversion

16. Ó R. B. Diemer, Jr., 2003 Aerosol Synthesis Chemistry Examples l Alkoxide Hydrolysis: n M= Si, Ti, Al, Sn… R=CH 3, C 2 H 5... l Halide Hydrolysis: n M= Si, Ti, Al, Sn… X=Cl, Br... l Halide Oxidation: n M=Si, Ti, Al, Sn… X=Cl, Br... l Alkoxide Pyrolysis: n M= Si, Ti, Al, Sn… R=CH 3, C 2 H 5... l Halide Ammonation: n M=B, Al … X=Cl, Br... l Pyrolysis: n A=Si, C, Fe… L=H, CO…

17. Ó R. B. Diemer, Jr., 2003 l Vaporization l Pumping/Compression l Addition of additives l Preheating General Aerosol Process Schematic Feed #1 Preparation Feed #2 Preparation...... Feed #N Preparation

18. Ó R. B. Diemer, Jr., 2003 l Mixing l Reaction Residence Time l Particle Formation/Growth Control l Agglomeration Control l Cooling/Heating l Wall Scale Removal General Aerosol Process Schematic Aerosol Reactor Feed #1 Preparation Feed #2 Preparation...... Feed #N Preparation

19. Ó R. B. Diemer, Jr., 2003 l Gas-Solid Separation General Aerosol Process Schematic Aerosol Reactor Feed #1 Preparation Feed #2 Preparation...... Feed #N Preparation Base Powder Recovery

20. Ó R. B. Diemer, Jr., 2003 l Absorption l Adsorption General Aerosol Process Schematic Aerosol Reactor Feed #1 Preparation Feed #2 Preparation...... Feed #N Preparation Base Powder Recovery Offgas Treatment Reagents Waste Vent or Recycle Gas

21. Ó R. B. Diemer, Jr., 2003 l Size Modification l Solid Separations l Degassing l Desorption l Conveying General Aerosol Process Schematic Aerosol Reactor Feed #1 Preparation Feed #2 Preparation...... Feed #N Preparation Base Powder Recovery Offgas Treatment Reagents Waste Vent or Recycle Gas Powder Refining Coarse and/or Fine Recycle

22. Ó R. B. Diemer, Jr., 2003 l Coating l Additives l Tabletting l Briquetting l Granulation l Slurrying l Filtration l Drying General Aerosol Process Schematic Aerosol Reactor Feed #1 Preparation Feed #2 Preparation...... Feed #N Preparation Base Powder Recovery Offgas Treatment Reagents Waste Vent or Recycle Gas Powder Refining Coarse and/or Fine Recycle Product Formulation Formulating Reagents

23. Ó R. B. Diemer, Jr., 2003 Product l Bags l Super Sacks l Jugs l Bulk containers n trucks n tank cars General Aerosol Process Schematic Aerosol Reactor Feed #1 Preparation Feed #2 Preparation...... Feed #N Preparation Base Powder Recovery Offgas Treatment Reagents Waste Vent or Recycle Gas Powder Refining Coarse and/or Fine Recycle Product Formulation Formulating Reagents Packaging Product

24. Ó R. B. Diemer, Jr., 2003 General Aerosol Process Schematic Aerosol Reactor Base Powder Recovery Offgas Treatment Product Formulation Packaging Powder Refining Feed #1 Preparation Feed #2 Preparation...... Feed #N Preparation Formulating Reagents Treatment Reagents Waste Product Coarse and/or Fine Recycle Vent or Recycle Gas

25. Ó R. B. Diemer, Jr., 2003 TiO2 Processes

26. Ó R. B. Diemer, Jr., 2003 Thermal Carbon Black Process Johnson, P. H., and Eberline, C. R., “Carbon Black, Furnace Black”, Encyclopedia of Chemical Processing and Design, J. J. McKetta, ed., Vol. 6, Marcel Dekker, 1978, pp. 187-257. Carbon Generated by Pyrolysis of CH 4

27. Ó R. B. Diemer, Jr., 2003 Furnace Carbon Black Process Johnson, P. H., and Eberline, C. R., “Carbon Black, Furnace Black”, Encyclopedia of Chemical Processing and Design, J. J. McKetta, ed., Vol. 6, Marcel Dekker, 1978, pp. 187-257. Carbon generated by Fuel-rich Oil Combustion

28. Applications to Particles Design Problem

29. Ó R. B. Diemer, Jr., 2003 Design Problem Focus Aerosol Reactor Base Powder Recovery Offgas Treatment Product Formulation Packaging Powder Refining Feed #1 Preparation Feed #2 Preparation...... Feed #N Preparation Formulating Reagents Treatment Reagents Waste Product Coarse and/or Fine Recycle Vent or Recycle Gas

30. Ó R. B. Diemer, Jr., 2003 The Design Problem Feeds Flame Reactor.25  m particles Cyclone 25% of particle mass max 10 psig Gas to Recovery Steps Baghouse Pipeline Agglomerator 8 psig min 75% of particle mass min What pipe diameter and length? What cut size?

31. Ó R. B. Diemer, Jr., 2003 l Simultaneous Coagulation & Breakage n initial size =.25 micron l Coagulation via sum of: n continuum Brownian kernel: n Saffman-Turner turbulent kernel: l Power-law breakage, binary equisized daughters: l Fractal particles: Design Problem Physics

32. Ó R. B. Diemer, Jr., 2003 Design Problem Aims l Capture particles with a cyclone followed by baghouse l Need 75% mass collection in cyclone to minimize bag wear from back pulsing l Agglomerate in pipeline… Initial pressure = 10 psig, Maximum allowable  P = 2 psia l Need to design: n cyclone - “cut size” related to design n agglomerator - pipe diameter and length needed to get desired collection efficiency l Optimize?… minimize the area of metal in pipe and cyclone to minimize cost?

33. Ó R. B. Diemer, Jr., 2003 Problem Setup l Steady-state, incompressible, axisymmetric flow l Plug flow, no slip l Neglect diffusion l Population Balance Model:

34. Ó R. B. Diemer, Jr., 2003 Moments Moments of n(V): Key Moments:

35. Solution Techniques

36. Ó R. B. Diemer, Jr., 2003 l Discrete Methods l Sectional Methods l Similarity Solutions l LaPlace Transforms l Orthogonal Polynomial Methods l Spectral Methods l Moment Methods l Monte Carlo Methods Partial List of Techniques Will discuss Will not discuss

37. Ó R. B. Diemer, Jr., 2003 Discrete Methods l Size is integer multiple of fundamental size l Write balance equations for every size l Gives distribution directly l Huge number of equations to solve l Have to decide what the largest size is l Example for coagulation and breakage:

38. Ó R. B. Diemer, Jr., 2003 Discrete Example Problem Setup Need slightly more than 2  10 6 cells to cover entire mass distribution range!

39. Ó R. B. Diemer, Jr., 2003 Sectional Method l Best rendering due to Litster, Smit and Hounslow l Collect particles in bins or size classes, with upper/lower size=2 1/q, “q” optimized for physics l Balances are written for each size class reducing the number of equations, but too few bins loses resolution l And… now the equations get more complicated to get the balances right l Still have problem of growing too large for top class l Directly computes distribution vivi 2 -3/q v i 2 -2/q v i 2 -1/q v i 2 1/q v i 2 2/q v i 2 3/q v i i i+1i+2 i1i1 i2i2i3i3

40. Ó R. B. Diemer, Jr., 2003 Sectional Interaction Types l Type 1: n some particles land in the i th interval and some in a smaller interval l Type 2: n all particles land in the i th interval l Type 3: n some particles land in the i th interval and some in a larger interval l Type 4: n some particles are removed from the i th interval and some from other intervals l Type 5: n particles are removed only from i th interval

41. Ó R. B. Diemer, Jr., 2003 Sectionalization Example: q=1 Collision of Particle j with Particle k Particle k, V/v o 12345 Particle j, V/v o Section number i: 2 i-1 v o <V<2 i v o In which section goes the daughter of a collision between Particle j in Section i and Particle k in Section n? 4 3 2 1

42. Ó R. B. Diemer, Jr., 2003 Sectionalization Example: q=1 Particle k, V/v o 12345 Particle j, V/v o Section number i: 2 i-1 v o <V<2 i v o 4 3 2 1 j+k = constant Any collision between these lines produces a particle in Section 5

43. Ó R. B. Diemer, Jr., 2003 Sectionalization Example: q=1 i,i collisions: map completely into i+1 Particle k, V/v o 12345 Particle j, V/v o Section number i: 2 i-1 v o <V<2 i v o 4 3 2 1

44. Ó R. B. Diemer, Jr., 2003 Sectionalization Example: q=1 i,i+1 collisions: 3/4 map into i+2, 1/4 stay in i+1 Particle k, V/v o 12345 Particle j, V/v o Section number i: 2 i-1 v o <V<2 i v o 4 3 2 1

45. Ó R. B. Diemer, Jr., 2003 Sectionalization Example: q=1 i,i+2 collisions: 3/8 map into i+3, 5/8 stay in i+2 Particle k, V/v o 12345 Particle j, V/v o Section number i: 2 i-1 v o <V<2 i v o 4 3 2 1

46. Ó R. B. Diemer, Jr., 2003 Sectionalization Example: q=1 i,i+3 collisions: 3/16 map into i+4, 13/16 stay in i+3 Particle k, V/v o 12345 Particle j, V/v o Section number i: 2 i-1 v o <V<2 i v o 4 3 2 1

47. Ó R. B. Diemer, Jr., 2003 Sectionalization Example: q=1 i,i+4 collisions: 3/32 map into i+5, 29/32 stay in i+4 Particle k, V/v o 12345 Particle j, V/v o Section number i: 2 i-1 v o <V<2 i v o 4 3 2 1

48. Ó R. B. Diemer, Jr., 2003 0 2 4 6 8 10 12 14 16 02468101214161820222426283032 Sectionalization Example: q=1 i,n collisions: 3/2 n-i+1 map into n+1, n>i>0 i,i collisions: all map into i+1 Particle k, V/v o 12345 Particle j, V/v o interval number i: v o i-1 <V<v o i i,i+1 collisions: 3/4 map into i+2i,icollisions: all map into i+1 i,i+2 collisions: 3/8 map into i+3 i,i+3 collisions: 3/16 map into i+4 i,i+4 collisions: 3/32 map into i+5

49. Ó R. B. Diemer, Jr., 2003 Sectional Coagulation Model, q=1 l Model Equation: l Tentatively: l Can show (via 0 th and 1 st moments) that: number balance gives correct general form for arbitrary  i,j mass balance only closes for C=2/3 when V i /V j =2 i-j n final expression: n kernels evaluated via:

50. Ó R. B. Diemer, Jr., 2003 General 2 1/q Sectional Coagulation Model 2 new terms

51. Ó R. B. Diemer, Jr., 2003 Sectional Example Problem Setup (for q=1) Need about 22 sections to cover entire mass distribution range, suggest using 25-30

52. Ó R. B. Diemer, Jr., 2003 Sectional Example Problem Setup (for q=1) Calculation of Mass Collection Efficiency

53. Ó R. B. Diemer, Jr., 2003 Sectional Example Problem Setup (for q=1) Nondimensionalization

54. Ó R. B. Diemer, Jr., 2003 Solution Technique Choices l If analytical method works use it! (rare) n similarity solution n Laplace transform l If it is crucial to get distribution detail right, and it is a 1-D problem, and it is a stand-alone model (typical of research) n discrete n sectional n Monte Carlo n Galerkin (orthogonal polynomial… commercial code: PREDICI) l If an approximate distribution will do, or if the moments are sufficient, or if the distribution is multivariate, or if the model will be embedded in a larger model (typical of process simulation) n moments

55. Ó R. B. Diemer, Jr., 2003 Concluding Remarks l Population balance applications are everywhere l The mathematics is difficult (unlike mass & energy balances) l There are many solution techniques… choice depends on object of model

56. Backup Slides

57. Ó R. B. Diemer, Jr., 2003 Moments Moments of n(V): Moments of b(V;  ):

58. Ó R. B. Diemer, Jr., 2003 Particle Number Balance

59. Ó R. B. Diemer, Jr., 2003 Interchange of Limits V  V=  V goes from 0 to  then  from 0 to   goes from V to  then V from 0 to 

60. Ó R. B. Diemer, Jr., 2003 Particle Number Balance (cont.) Interchange limits of integration in both coagulation and breakage terms

61. Ó R. B. Diemer, Jr., 2003 Particle Number Balance (cont.) Change of variable in coagulation integral:  = V  vdV = d  at constant v General Number Balance for p Daughters

62. Ó R. B. Diemer, Jr., 2003 Particle Volume (Mass) Balance

63. Ó R. B. Diemer, Jr., 2003 Particle Volume (Mass) Balance (cont.) Interchange limits of integration in both coagulation and breakage terms

64. Ó R. B. Diemer, Jr., 2003 Particle Volume (Mass) Balance (cont.) Change of variable in coagulation integral:  = V  vdV = d  at constant v General Mass Balance for p Daughters