Looking Down: Particle Technology at the Macroscale

Transcription

1 Looking Down: Particle Technology at the Macroscale James B. Rawlings Department of Chemical and Biological Engineering University of Wisconsin, Madison, WI UK Particle Technology Forum Particle Technology Across the Length Scales 24 July

2 Outline Particle technology at the macroscale A brief history A systems engineering perspective Modeling of populations Virus infections Crystallization A stochastic population balance alternative Experimental applications GlaxoSmithKline pharmaceutical Control of particle size and shape Looking to the future: multiscale modeling 2

3 A brief history... Crystal Size Distribution Dynamics, Stability and Control (A Review Paper), Randolph, Technologies do not exist for the on-line instrumental control of CSD. The literature is rife with theoretical studies of CSD stabilization and/or control, but typically the suggested control variable cannot presently be measured and in no case were any of the control algorithms experimentally investigated. The current industrial practice of CSD in industrial crystallizers can be summed up in three words: Hold Everything Constant. 3

4 A brief history... Crystal Size Distribution Dynamics, Stability and Control (A Review Paper), Randolph, Technologies do not exist for the on-line instrumental control of CSD. The literature is rife with theoretical studies of CSD stabilization and/or control, but typically the suggested control variable cannot presently be measured and in no case were any of the control algorithms experimentally investigated. The current industrial practice of CSD in industrial crystallizers can be summed up in three words: Hold Everything Constant. 3

5 A brief history... Crystal Size Distribution Dynamics, Stability and Control (A Review Paper), Randolph, Technologies do not exist for the on-line instrumental control of CSD. The literature is rife with theoretical studies of CSD stabilization and/or control, but typically the suggested control variable cannot presently be measured and in no case were any of the control algorithms experimentally investigated. The current industrial practice of CSD in industrial crystallizers can be summed up in three words: Hold Everything Constant. 3

6 A brief history... Crystal Size Distribution Dynamics, Stability and Control (A Review Paper), Randolph, Technologies do not exist for the on-line instrumental control of CSD. The literature is rife with theoretical studies of CSD stabilization and/or control, but typically the suggested control variable cannot presently be measured and in no case were any of the control algorithms experimentally investigated. The current industrial practice of CSD in industrial crystallizers can be summed up in three words: Hold Everything Constant. 3

7 A brief history years later Model Identification and Control of Solution Crystallization Processes: A Review, Rawlings, Miller, Witkowski, Considerable research effort has been devoted to CSD control, and we are now seeing the advances in measurement and computing technologies necessary for successful industrial implementation. It is reasonable to expect closed-loop CSD control to become part of accepted industrial practice in the near future. 4

8 A brief history years later Model Identification and Control of Solution Crystallization Processes: A Review, Rawlings, Miller, Witkowski, Considerable research effort has been devoted to CSD control, and we are now seeing the advances in measurement and computing technologies necessary for successful industrial implementation. It is reasonable to expect closed-loop CSD control to become part of accepted industrial practice in the near future. 4

9 A brief history years later Model Identification and Control of Solution Crystallization Processes: A Review, Rawlings, Miller, Witkowski, Considerable research effort has been devoted to CSD control, and we are now seeing the advances in measurement and computing technologies necessary for successful industrial implementation. It is reasonable to expect closed-loop CSD control to become part of accepted industrial practice in the near future. 4

10 Measurement technology review Technique Description Reference Video Microscopy On-line images captured and digitized Patience and Rawlings (2001) and Image Analysis Limited to dilute slurries Plummer and Kausch (1995) Shape information present Bharati and MacGregor (1998) Lasentec particle and In-situ probe coupled with strobing Braatz and Hasebe (2001) vision measurement (PVM) Numerous images are captured lasentec.com Dense slurries possible Randomly oriented particles Forward laser light scattering Energy pattern of scattered light Heffels et al. (1994,1995) Ill-conditioned inversion of scattering Rawlings et al. (1993) pattern to CSD Transmittance In-situ probe van de Hulst (1981) Second moment of CSD Matthews and Rawlings (1998) Limited to single scattering Patience et al. (2001) Backward laser light scattering In-situ probe Monnier et al.(1997) Lasentec focussed beam reflected Chord length of CSD lasentec.com measurement (FBRM) Dense slurries possible Electro-zone Sensing Electronically counts particles Allen (1997) (Coulter counter) Requires a conducting solution Rovang and Randolph (1980) Third moment of CSD 5

11 A brief history... projecting 10 years in the future Tailoring Size, Shape and Composition of the Solid Phase, some future researcher, A large investment in process technologies required to achieve reliable manufacturing processes for new products in high value-added industries, such as pharmaceuticals, took place in the 1990s. This investment resulted in a revolution in methods for measuring, monitoring, modeling and controlling the creation of the solid phase from solution. All aspects of this formation are now tailored to meet product quality specifications including the size and shape of particles, as well as the multicomponent composition in the solid mixture. It is not clear why the earlier researchers were such pessimists... 6

12 A brief history... projecting 10 years in the future Tailoring Size, Shape and Composition of the Solid Phase, some future researcher, A large investment in process technologies required to achieve reliable manufacturing processes for new products in high value-added industries, such as pharmaceuticals, took place in the 1990s. This investment resulted in a revolution in methods for measuring, monitoring, modeling and controlling the creation of the solid phase from solution. All aspects of this formation are now tailored to meet product quality specifications including the size and shape of particles, as well as the multicomponent composition in the solid mixture. It is not clear why the earlier researchers were such pessimists... 6

13 A brief history... projecting 10 years in the future Tailoring Size, Shape and Composition of the Solid Phase, some future researcher, A large investment in process technologies required to achieve reliable manufacturing processes for new products in high value-added industries, such as pharmaceuticals, took place in the 1990s. This investment resulted in a revolution in methods for measuring, monitoring, modeling and controlling the creation of the solid phase from solution. All aspects of this formation are now tailored to meet product quality specifications including the size and shape of particles, as well as the multicomponent composition in the solid mixture. It is not clear why the earlier researchers were such pessimists... 6

14 A brief history... projecting 10 years in the future Tailoring Size, Shape and Composition of the Solid Phase, some future researcher, A large investment in process technologies required to achieve reliable manufacturing processes for new products in high value-added industries, such as pharmaceuticals, took place in the 1990s. This investment resulted in a revolution in methods for measuring, monitoring, modeling and controlling the creation of the solid phase from solution. All aspects of this formation are now tailored to meet product quality specifications including the size and shape of particles, as well as the multicomponent composition in the solid mixture. It is not clear why the earlier researchers were such pessimists... 6

15 A systems engineering perspective First we need a model... mathematical abstractions of real processes predict behavior over some operating range of interest Systems theory can... provide general tools for extracting information from complex models and experimental data examples steady-state analysis parameter estimation optimization on-line estimation and control 7

16 Outline Particle technology at the macroscale a brief history A systems engineering perspective Modeling of populations Virus infections Crystallization A stochastic population balance alternative Experimental applications GlaxoSmithKline pharmaceutical Control of particle size and shape Looking to the future: multiscale modeling 8

17 Modeling of Populations Dynamics of populations are determined by applying the equation of continuity. Conservation Equation d f (z, t)dτ = dt V [ ] x z = = y (B D)dτ V [ external characteristics internal characteristics Assumes that V is large enough to contain a statistically significant portion of the population. ] Microscopic Equation of Continuity f (z, t) t ( + f (z, t) z t ) = B D 9

18 Modeling of Populations Crystallization: segregate crystals by particle length (l) f (l, t) t + l ( f (l, t) l t ) = B D Emulsion polymerization: segregate polymers by particle size (r ) f (r, t) t + ( f (r, t) r ) t Viral infections: segregate infected cells by age (τ) f (τ, t) t + ( f (τ, t) τ ) t = B D = B D See D. Ramkrishna, Population Balances [18] for a recent monograph 10

19 Application to Viral Infections: Intracellular Level nucleotides + gen k 1 V1 tem nucleotides + amino acids nucleotides str k 2 V 2, tem k 3 tem k 4 str gen degraded gen + str k 5 secreted virus K 1 V 1 + I 1 V I 1 K 2 V 2 + I 2 V I 2 k 11 I 1 k 12 I 2 secreted secreted 11

20 Application to Viral Infections: Physiological Level virus + uninfected cell k 6 infected cell virus k 7 infected cell uninfected cell k 8 k 9 degraded death death precursors k 10 uninfected cell Generation Healthy Cells Infected Cells death k 13 I 1 degraded / secreted I 1 + unc k 14 I 1 (adsorbed) + unc Free Virus k 15 I 2 degraded / secreted I 2 + unc k 16 I 2 (adsorbed) + unc 12

21 Initial Infection of a Healthy Host Extracellular Components ( 10 7 #/host) uninfected Time (Days) virus infected Extracellular Components ( 10 7 #/host) uninfected Time (Days) virus infected Population Balance Model Optimal Fit of Purely Extracellular Model Population balance models can incorporate multiple levels of description. 13

22 Evaluation of Drug Therapies Extracellular Virus (#/host) increasing I Intracellular I 2 (#/cell) Extracellular Virus (#/host) increasing I Intracellular I 2 (#/cell) Nominal Steady State Steady State After Mutation With population balance models, we can quantify effects of intracellular perturbations on extracellular components. Analysis of population balance models may yield counterintuitive results. 14

23 Application to Crystallization Population Balance: f (L, t) t f (L, t) = G L in which f (L, t) is the number of crystals of size L at time t, and G is the crystal growth rate. Mass and Energy Balances: dĉ dt = 3ρ ck v hg f L 2 dl 0 dt ρv C p dt = 3 H cρ c k v V G f L 2 dl UA(T T j (t)) 0 B = k b (Ĉ Ĉsat (T ) Ĉ sat (T ) Nucleation and Growth in the Bulk: ) b (Ĉ ) g = k b S b Ĉsat (T ) G = k g = k g S g Ĉ sat (T ) 15

24 Outline Particle technology at the macroscale a brief history A systems engineering perspective Modeling of populations Virus infections Crystallization A stochastic population balance alternative Experimental applications GlaxoSmithKline pharmaceutical Control of particle size and shape Looking to the future: multiscale modeling 16

25 A Stochastic Population Balance Alternative Reaction Mechanism for Nucleation and Growth 2M kn N 1 N n + M k g N n+1 H n rxn H g rxn N n is the number of crystals of size (n + 1) per volume. Mass Balance M tot = M sat + M Energy Balance dt dt = UA ρc p V (T j T ) (between reaction events) Solubility Relationship log 10 M sat = a log 10 T + b T + c 17

26 Simulating reactions stochastically Stochastic kinetic models treat reactions as molecular events Consider the well-mixed reaction: A o B o C o = A + B k 1 k 1 C molecules 0 Randomly select [6, 7, 25]: 1. When the next reaction occurs τ = log(p 1) r tot 2. Which reaction occurs p 2 Scale probabilities by reaction rates r 1 = k 1 AB r 2 = k 1 C r tot = r 1 + r 2 0 r 1 r tot r 2 r tot 1 18

27 Stochastic Simulation of A + B C One simulation Deterministic Stochastic Average of many simulations Extent Probability Time 0.02 Time Extent 19

28 Implications for Crystal Populations Distributions Scalars N 1. AVERAGE N 1. TABULATE N j. N j. N n Stochastic Trajectory N n Stochastic Average Distribution of Stochastic Averages 20

29 Connection to Deterministic Kinetics (Ω = System Size) for A + B C Ω Ω 10 1 Deterministic Stochastic 1 Deterministic Stochastic Extent Extent Time Time Ω 100 Extent Deterministic Stochastic As Ω increases: Stochastic Deterministic Computing burden increases! Time 21

30 Implications for Crystal Populations Distributions Scalars N 1. Ω N 1. 0 N j. N j. N n Stochastic N n Deterministic dn j ( ) dt = km N j 1 N j η(l, t) t Deterministic = k M η(l, t) l 22

31 Isothermal, Size-Independent Nucleation and Growth Stochastic Solution Average of 100 Simulations Deterministic Solution Via Orthogonal Collocation Discrete particle sizes Integer-valued particle accounting Continuous particle sizes Real-valued particle accounting 23

32 Nonisothermal, Size-Independent Nucleation and Growth Stochastic Solution Average of 500 Simulations Deterministic Solution Via Orthogonal Collocation 24

33 Isothermal, Size-Independent Nucleation, Growth, and Agglomeration Stochastic Solution Add one reaction: N p + N q N p+q Average of 500 Simulations k a Deterministic Solution Via Adaptive Mesh Methods? Large time investment! 25

34 Stochastic Models for Populations: Conclusions Stochastic models provide a general, flexible solution technique for examining many possible nucleation and growth mechanisms Stochastic models are equivalent to deterministic models in the large volume, small monomer unit limit 26

35 Outline Particle technology at the macroscale a brief history A systems engineering perspective Modeling of populations Virus infections Crystallization A stochastic population balance alternative Experimental applications GlaxoSmithKline pharmaceutical Control of particle size and shape Looking to the future: multiscale modeling 27

36 Pharmaceutical crystal measurements 3 min 11 min Images show no apparent secondary nucleation, only growth of seeds Experiment min 140 min 265 min 1 mm Standard deviation [µm] Time [min] Manual digitizing of images shows the pharmaceutical system has growth-dependent dispersion 28

37 Modeling apparent growth-rate dispersion f (L, t) t = t 0 t > t 0 L Population Balance The population balance equation describes no changes in the seed CSD width. f (L, t) t f (L, t) = G L 29

38 Modeling apparent growth-rate dispersion f (L, t) t = t 0 t > t 0 L Population Balance: Growth-dependent dispersion Particles grow in discrete sizes, in which is much larger than a molecule. f (L, t) t = G f (L, t) +G 2 f (L, t) L 2 L 2 30

39 Measurements C : solution concentration I/I 0 : slurry transmittance L : CSD mean length σ : CSD standard deviation Parameters k g : growth rate constant g : growth rate order d w : depth-to-width ratio L max : maximum seed size : size of crystal growth unit 31

40 Model predictions - GSK pharmaceutical Concentration Transmittance Concentration [g pharmaceutical/g solvent] Experiment 16 Experiment 16 fit Transmittance [%] Experiment 16 Experiment 16 fit Time [min] Time [min] 32

41 Model predictions - GSK pharmaceutical CSD mean length CSD standard deviation 200 Experiment 16 Experiment 16 fit Experiment 16 Experiment 16 fit 3 Mean length [µm] Standard deviation [µm] Time [min] Time [min] 33

42 Parameter estimates ln(k g ) g ln(d w ) L max [µm] [µm] Estimate Interval ± 0.2 ± 0.08 ± 0.1 ± 0.5 ±3.3 3 min 265 min 34

43 Models for dispersion Class Mechanism Author Size-dependent Growth rate increases Kashchiev (2000) [11] Growth with more units Abegg et al. (1968) [1] G(L, t) = k g S g (1 + γ 1 L) γ 2 Random Growth Same mean growth rate Kashchiev (2000) [11] Diffusivity but each crystal s growth Randolph & White (1977) [19] f t = G f L + D 2 f L 2 randomly fluctuates White & Wright (1971) [27] Growth-dependent Crystals grow in McCoy (2001) [13] Dispersion discrete sizes Randolph & White (1977) [19] f t f = G L + G 2 f 2 L 2 find D = 50G Intrinsic Growth Each crystal exhibits Jones & Larson (1999) [10] an individual growth rate Rojkowski (1993) [23] f (L, g, t)dg Berglund & Larson (1984) [3] f (L, t) = 0 35

44 Intrinsic growth mechanism Population Balance Equation: l 1, g 1 : f (L, g, t) t = [ ] ggf (L, g, t) L l 1, g 2 : l 1, g 3 : f (L, g, t)dldg is the number of crystals of size L with intrinsic growth rate g at time t G is the crystal growth rate Assumes that the intrinsic growth rate g does not change with time. 36

45 Can the intrinsic growth rate model fit the data? Mean length [µm] CSD mean length Time (min) Standard deviation [µm] CSD standard deviation Time (min)

46 Finding the optimal cooling profile For this system, the crystals with the greatest bioavailability are the ones with the greatest solubility in administered form For this system, the crystals with the greatest stability are the ones with the lowest solubility The desired CSD for this pharmaceutical is the one with a µm range

47 Finding the optimal cooling profile For this system, the crystals with the greatest bioavailability are the ones with the greatest solubility in administered form For this system, the crystals with the greatest stability are the ones with the lowest solubility The desired CSD for this pharmaceutical is the one with a µm range f (L, t) desirable CSD undesirable CSD requires milling and sieving L

48 Finding the optimal cooling profile For this system, the crystals with the greatest bioavailability are the ones with the greatest solubility in administered form For this system, the crystals with the greatest stability are the ones with the lowest solubility The desired CSD for this pharmaceutical is the one with a µm range f (L, t) f (L, t) t=0 desirable CSD t=1 undesirable CSD requires milling and sieving t=2 L L = 110 L 38

49 Finding the optimal cooling profile For this system, the crystals with the greatest bioavailability are the ones with the greatest solubility in administered form For this system, the crystals with the greatest stability are the ones with the lowest solubility The desired CSD for this pharmaceutical is the one with a µm range f (L, t) f (L, t) t=0 desirable CSD t=1 t=1 t=2 undesirable CSD requires milling and sieving t=3 L L = 110 L 39

50 Cooling profiles to produce L=110µm Temperature Supersaturation no dt/dt constraints dt/dt 10 C/h dt/dt 10 C/h, S no dt/dt constraints dt/dt 10 C/h dt/dt 10 C/h, S 0.1 Temperature [ C] Supersaturation Time [min] Time [min] 40

51 Resulting L and CSD coefficient of variation from cooling profiles to produce L=110µm CSD mean length CSD standard deviation no dt/dt constraints dt/dt 10 C/h dt/dt 10 C/h, S no dt/dt constraints dt/dt 10 C/h dt/dt 10 C/h, S 0.1 Crystal length [µm] Coefficient of variation Time [min] Time [min] 41

52 Implemented cooling profiles Concentration CSD mean length Concentration [g pharmaceutical/g solvent] Experiment 18 model prediction Experiment 18 Experiment 20 model prediction Experiment Mean length [µm] Experiment 18 model prediction Experiment 18 Experiment 20 model prediction Experiment Time [min] Time [min] 42

53 Pharmaceutical conclusions New use of video images in model identification and parameter estimation for crystallization kinetics.

54 Pharmaceutical conclusions New use of video images in model identification and parameter estimation for crystallization kinetics. Successfully identified kinetics for a pharmaceutical crystallization, with narrow parameter confidence intervals.

55 Pharmaceutical conclusions New use of video images in model identification and parameter estimation for crystallization kinetics. Successfully identified kinetics for a pharmaceutical crystallization, with narrow parameter confidence intervals. For this seeded pharmaceutical crystallization without secondary nucleation, the minimization of CSD properties is infeasible. Instead we minimize the batch operating time.

56 Pharmaceutical conclusions New use of video images in model identification and parameter estimation for crystallization kinetics. Successfully identified kinetics for a pharmaceutical crystallization, with narrow parameter confidence intervals. For this seeded pharmaceutical crystallization without secondary nucleation, the minimization of CSD properties is infeasible. Instead we minimize the batch operating time. Calculated optimal cooling profiles to manufacture crystals into a desired size range by minimizing t f. 43

57 Particle shape High value-added products in the chemical industry are becoming increasingly complicated in structure. Pharmaceutical compounds are very complex: multiple crystal habits and multiple crystal structures. 44

58 Particle shape industrial characteristics and challenges Particle shape is affected by unmeasured disturbance variables that cannot be modeled nor controlled. Online sensing is available in the form of video images. The images are replete with bad data. Particles are fused or broken; particle boundaries overlap. It is difficult to obtain representative samples. It is difficult to sample enough images to remove the effects of noise through averaging. Standard image analysis software provides simple shape measures such as particle boxed area and aspect ratio. These simple measures are inadequate signals for feedback control. We can manipulate a process variable that also influences particle shape. We seek a feedback policy with which we maintain the desired habit in the face of the unmeasured disturbances. 45

59 Macroscopic sodium chlorate (NaClO 3 ) habit The 110, 111 and faces grow fast under relatively high supersaturation and grow out leaving only the 100 faces visible under microscopy. 46

60 Macroscopic sodium chlorate (NaClO 3 ) habit In the presence of sodium dithionate (Na 2 S 2 O 6 ) the faces are blocked by the impurity and the 100, 110 and 111 faces grow relatively faster, leaving only the faces visible, seen as a tetrahedron under microscopy. See Sherwood s and coworkers papers for further discussion [22, 21] 47

61 Experimental apparatus Impurity-Free NaClO 3 solution Contaminated NaClO 3 solution Impurity Na 2 S 2 O 6 Photo-microscopy TT Image Analysis System Hot Stream TT Cold Stream Controller Cascaded MPC-PI temperature controller. Automated image analysis system provides on-line information about particle size and shape by binarizing images of samples of the crystal slurry. 48

62 Sodium chlorate shape manipulation Figure i 15 min Figure ii 49 min no habit modifier 225ppm Na 2 S 2 O 6

63 Sodium chlorate shape manipulation Figure i 15 min Figure ii 49 min no habit modifier 225ppm Na 2 S 2 O 6 200µm 200µm Figure iii 70 min Figure iv 135 min 225ppm Na 2 S 2 O 6 no habit modifier

64 Sodium chlorate shape manipulation Figure i 15 min Figure ii 49 min no habit modifier 225ppm Na 2 S 2 O 6 200µm 200µm Figure iii 70 min Figure iv 135 min 225ppm Na 2 S 2 O 6 no habit modifier 200µm 200µm 49

65 Image analysis - raw data Boxed area shows trends in the correct direction Aspect ratio Boxed area Time [min] Time [min] 50

66 Perturbations to squares and equilateral triangles square triangle boxed area aspect ratio roundness object area bounding box area Lmax L min per 2 4π area 4 π 9 π 3 Saw tooth waves of increasing amplitude are added to the boundaries of a square and triangle and sized to asses the reliability of the sensor µm 51

67 Classification of shape The size of the elliptical regions is chosen to classify the shapes. Classification regions Classification regions and data Boxed area squares 1-7, 9-10 triangles squares 8 triangles triangle region square region analytical w/l trajectory Aspect ratio

68 Classification of shape The size of the elliptical regions is chosen to classify the shapes. Classification regions Classification regions and data Boxed area squares 1-7, 9-10 triangles squares 8 triangles triangle region square region analytical w/l trajectory 8 8 Boxed area data triangle region square region Aspect ratio Aspect ratio 52

69 Processed image data Sensor detects clearly shape change using square and triangle regions alone Boxed area Time [min]

70 Processed image data Sensor detects clearly shape change using square and triangle regions alone Boxed area Boxed area Time [min] Time [min] 53

71 Feedback control of particle shape to steady state Number of cubes [%], Impurity concentration [ppm] Percentage of cubes Impurity concentration Setpoint zone Time [min] By maintaining the habit modifier at a level near the critical impurity concentration, simple PI control algorithms are able to keep a desired habit in the presence of a disturbance. The controller finds the critical dosage ( ppm) required to achieve 40% cubes in a slurry without any knowledge of a model. This value agrees with literature values of ppm (Sherwood 1993). 54

72 If you want to understand data... Conclusions

73 Conclusions If you want to understand data... you have to model them

74 Conclusions If you want to understand data... you have to model them Good models enable good engineering: design, monitor, forecast, control

75 Conclusions If you want to understand data... you have to model them Good models enable good engineering: design, monitor, forecast, control Crystal engineering poses challenging modeling problems, but progress is steady and breakthroughs are possible off-the-shelf data/model combinations are often incompatible these applications require state-of-the-art modeling tools for numerical solution, parameter estimation, optimization and control

76 Conclusions If you want to understand data... you have to model them Good models enable good engineering: design, monitor, forecast, control Crystal engineering poses challenging modeling problems, but progress is steady and breakthroughs are possible off-the-shelf data/model combinations are often incompatible these applications require state-of-the-art modeling tools for numerical solution, parameter estimation, optimization and control We are capable of modeling, monitoring and controlling many systems with current technology 55

77 Looking to the future: multiscale modeling

78 Looking to the future: multiscale modeling More and better sensors in-situ video imaging for on-line particle analysis appears feasible commercial instrument vendors are very active complex product specifications require better sensors

79 Looking to the future: multiscale modeling More and better sensors in-situ video imaging for on-line particle analysis appears feasible commercial instrument vendors are very active complex product specifications require better sensors More and better models couple ab initio molecular, chemical models to process models move from simple scaling heuristics to mixing models based on computational fluid dynamics real-time compression of large sets of video data into useful information stochastic simulation of complex population behavior 56

80 References [1] C. F. Abegg, J. D. Stevens, and M. A. Larson. Crystal size distributions in continuous crystallizers when growth rate is size dependent. AIChE J., 14(1): , January [2] T. Allen. Particle Size Measurement. Chapman and Hall, London, fifth edition, [3] K. A. Berglund and M. A. Larson. Modeling of growth rate dispersion of citric acid monohydrate in continuous crystallizers. AIChE J., 30: , [4] M. H. Bharati and J. F. MacGregor. Multivariate image analysis for real-time process monitoring and control. Ind. Eng. Chem. Res., 37: , [5] R. D. Braatz and S. Hasebe. Particle size and shape control in crystallization processes. In Chemical Process Control CPC 6, Tucson, Arizona, January [6] D. T. Gillespie. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys., 22: , [7] D. T. Gillespie. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem., 81: , [8] C. M. G. Heffels, D. Heitzmann, E. D. Hirleman, and B. Scarlett. The use of azimuthal intensity variations in diffraction patterns for particle shape characterization. Part. Part. Syst. Charact., 11: , [9] C. M. G. Heffels, D. Heitzmann, E. D. Hirleman, and B. Scarlett. Forward light scatter- 57

81 ing for arbitrary sharp-edged convex crystals in Fraunhoffer and anomalous diffraction approximations. Applied Opt., 34(26): , [10] C. M. Jones and M. A. Larson. Characterizing growth-rate dispersion of NaNO 3 secondary nuclei. AIChE J., 45(10): , [11] D. Kashchiev. Nucleation. Butterworth-Heinemann, Oxford, England, 1st edition, [12] H. B. Matthews and J. B. Rawlings. Batch crystallization of a photochemical: Modeling, control and filtration. AIChE J., 44: , [13] B. J. McCoy. A new population balance model for crystal size distributions: reversible, size-dependent growth and dissolution. J. Colloid Interface Sci., 240(1): , [14] O. Monnier, G. Fevotte, C. Hoff, and J. P. Klein. Model identification of batch cooling crystallizations through calorimetry and image analysis. Chem. Eng. Sci., 52(7): , [15] D. B. Patience, H. A. Mohameed, and J. B. Rawlings. Crystallization of para-xylene in scraped-surface crystallizers. AIChE J., 47(11): , [16] D. B. Patience and J. B. Rawlings. Particle-shape monitoring and control in crystallization processes. AIChE J., 47(9): , [17] C. Plummer and H. Kausch. Real-time image analysis and numerical simulation of isothermal spherulite nucleation and growth in polyoxymethylene. Colloid Polym. Sci., 273: , [18] D. Ramkrishna. Population Balances. Academic Press, San Deigo,

82 [19] A. D. Randolph and E. T. White. Modeling size dispersion in the prediction of crystal-size distribution. Chem. Eng. Sci., 32: , [20] J. B. Rawlings, S. M. Miller, and W. R. Witkowski. Model identification and control of solution crystallization processes: A review. Ind. Eng. Chem. Res., 32(7): , July [21] R. Ristic, B. Y. Shekunov, and J. N. Sherwood. Growth of the tetrahedral faces of sodium chlorate crystals in the presence of dithionate impurity. J. Cryst. Growth, 139(3-4): , [22] R. Ristic, J. N. Sherwood, and K. Wojciechowski. Morphology and growth kinetics of large sodium chlorate crystals grown in the presence and absence of sodium dithionate impurity. J. Phys. Chem., 97(41): , [23] Z. H. Rojkowski. Crystal growth rate models and similarity of population balances for size-dependent growth rate and for constant growth rate dispersion. Chem. Eng. Sci., 48(8): , [24] R. D. Rovang and A. D. Randolph. On-line particle size analysis in the fines loop of a KCl crystallizer. AIChE Symp. Ser., 76(193): , [25] B. H. Shah, D. Ramkrishna, and J. D. Borwanker. Simulation of particulate systems using the concept of the interval of quiescence. AIChE J., 23(6): , [26] H. van de Hulst. Light Scattering by Small Particles. Dover Publications, New York,

83 [27] E. T. White and P. G. Wright. Magnitude of size dispersion effects in crystallization. Chem. Eng. Progr. Symp. Ser., 67(110):81 87,