A Systems Approach to the Modeling and Control of Molecular, Microparticle, and Biological Distributions

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Processes with distributions are pervasive:
- Molecular: molecular weight distribution in polymerization
- Microparticle: particle size distribution in suspension polymerization
- Biological: rupture frequency distributions in single- molecule pulling experiments

This thesis presents a systematic approach to the modeling and control of these processes

Systematic approach applied to diverse processes
-Molecular distributions
-Microparticle distributions
-Biological distributions

Common approach
- Experiments/equipment
- Parameter estimation
- Sensitivity and uncertainty analysis
- Model selection
- Optimal control

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  • Provide a picture of the MWD
  • The second item needs to be clearer Fourth item should list computer
  • Is this section really simulation? Isn’t it really model validation? title should say what the purpose of the slide is, e.g., example comparison of model predictions with experimental data
  • Figures are too small. It is not clear what the purpose of this slide is. Why not just use only the second set of plots?
  • Isn’t this really the distributional results, not the worst-case results (see title)?
  • A Systems Approach to the Modeling and Control of Molecular, Microparticle, and Biological Distributions

    1. 1. A Systems Approach to the Modeling and Control of Molecular, Microparticle, and Biological Distributions Eric J. Hukkanen Dept. of Chemical & Biomolecular Engineering University of Illinois at Urbana-Champaign
    2. 2. Introduction <ul><li>Processes with distributions are pervasive </li></ul><ul><ul><li>Molecular: molecular weight distribution in polymerization </li></ul></ul><ul><ul><li>Microparticle: particle size distribution in suspension polymerization </li></ul></ul><ul><ul><li>Biological: rupture frequency distributions in single- molecule pulling experiments </li></ul></ul><ul><li>This thesis presents a systematic approach to the modeling and control of these processes </li></ul>
    3. 3. Components of the Systematic Approach NO YES design constraints & performance criteria Robust Optimization optimized design Is model accurate? Hypothesis Mechanism Selection Experimental Data Collection Multiple Models Parameter Estimation Experimental Design experimental constraints Sensitivity & Uncertainty Analysis
    4. 4. Uncertainty Analysis <ul><li>Parameter uncertainties lead to variations in product quality </li></ul><ul><ul><li>Full molecular weight distribution </li></ul></ul><ul><ul><li>Particle size distribution </li></ul></ul><ul><li>Quantified these variations in 2 ways </li></ul><ul><ul><li>Minimum and maximum deviations </li></ul></ul><ul><ul><li>Distributions </li></ul></ul><ul><li>See thesis for detailed mathematical expressions </li></ul>
    5. 5. Molecular Distribution Outline NO YES design constraints & performance criteria Robust Optimization optimized design Hypothesis Mechanism Selection Experimental Data Collection Multiple Models Parameter Estimation Experimental Design experimental constraints Sensitivity & Uncertainty Analysis Is model accurate?
    6. 6. Introduction – Free Radical Polymerization <ul><li>Common process for plastics </li></ul><ul><ul><li>Acrylonitrile-butadiene-styrene (ABS) </li></ul></ul><ul><ul><li>Poly(vinyl chloride) (PVC) </li></ul></ul><ul><ul><li>Poly(methyl methacrylate) (PMMA) </li></ul></ul><ul><ul><ul><li>Shatter-resistant glass, optical lenses, dentistry </li></ul></ul></ul><ul><li>End-use properties depend on the molecular weight distribution (MWD) </li></ul>
    7. 7. Introduction – Free Radical Polymerization <ul><li>General reaction mechanism </li></ul>Initiator decomposition Initiation Propagation Monomer transfer Termination by combination Termination by disproportionation
    8. 8. Experiments/Equipment – Sensors ATR-FTIR Spectroscopy
    9. 9. Experiments/Equipment – Sensors <ul><li>IR spectra recorded during polymerization </li></ul><ul><li>Differential spectra used for calibration </li></ul><ul><li>Monomer concentration determined </li></ul>
    10. 10. Modeling of MWD <ul><li>Method of moments (most widely used) </li></ul><ul><ul><li>Average molecular weight properties </li></ul></ul><ul><ul><li>Simplified system of ODEs (computationally efficient) </li></ul></ul><ul><li>Monte Carlo simulations </li></ul><ul><ul><li>Probability of reaction events considered </li></ul></ul><ul><ul><li>Construct full MWD during simulation </li></ul></ul><ul><ul><li>Computationally expensive and time-consuming </li></ul></ul><ul><li>Empirical distributions </li></ul><ul><ul><li>Normal and log-normal distributions </li></ul></ul><ul><ul><li>Easy to implement </li></ul></ul><ul><ul><li>Does not consider physics of problem (bimodality) </li></ul></ul>
    11. 11. Modeling of MWD <ul><li>Direct simulation – parallel CVODE </li></ul><ul><ul><li>ODE solver on parallel computers (MPI) (Byrne and Hindmarsh, 1998) </li></ul></ul><ul><ul><li>System size determined by gel permeation chromatography </li></ul></ul><ul><ul><li>System size = 200,000 equations </li></ul></ul><ul><ul><li>Solution time = ~2-4 minutes on 41 1.0 GHz Pentium III processors </li></ul></ul><ul><ul><li>First direct simulation of MWD </li></ul></ul>
    12. 12. Modeling of MWD <ul><li>Parallel CVODE offers several advantages over other methods </li></ul><ul><ul><li>Does not require approximating material balance eqns. </li></ul></ul><ul><ul><li>No prior information of distribution is required </li></ul></ul><ul><ul><li>Able to model/predict multi-modal distributions </li></ul></ul><ul><li>Semi-batch polymerization </li></ul>Polymer chain transfer
    13. 13. Parameter Estimation <ul><li>Maximum likelihood estimates from conversion and average molecular weights </li></ul><ul><li>Kinetic parameters determined: </li></ul><ul><li>Poorly conditioned covariance matrix: </li></ul>
    14. 14. Model Validation
    15. 15. Uncertainty Analysis – Worst-case Output <ul><li>Dynamic simulation: worst-case parameters </li></ul>
    16. 16. Worst-case Analysis of the Molecular Weight Distribution <ul><li>These plots show the p.d.f. for the MWD </li></ul>
    17. 17. Summary <ul><li>Direct simulation of full MWD enables for interesting investigations (e.g., bimodality, kinetic studies, distributional control) </li></ul><ul><li>Maximum likelihood parameter estimates determined from monomer conversion and average molecular weights </li></ul><ul><li>Quantified effect of “nasty” parameters on the MWD </li></ul><ul><li>Presented distribution of the MWD </li></ul>
    18. 18. Microparticle Distribution Outline NO YES design constraints & performance criteria Robust Optimization optimized design Hypothesis Mechanism Selection Experimental Data Collection Multiple Models Parameter Estimation experimental constraints Sensitivity & Uncertainty Analysis Is model accurate? Experimental Design
    19. 19. Why Study Suspension Polymerization? <ul><li>Commercial resins </li></ul><ul><ul><li>Poly(vinyl chloride) </li></ul></ul><ul><ul><li>Acrylonitrile-Butadiene-Styrene (ABS) </li></ul></ul><ul><ul><li>Poly(methyl methacrylate) </li></ul></ul><ul><ul><li>Ion-exchange resins, chromatographic material </li></ul></ul><ul><li>Advantages </li></ul><ul><ul><li>Easy temperature control </li></ul></ul><ul><ul><li>Low impurity levels </li></ul></ul><ul><ul><li>Low separation costs </li></ul></ul><ul><ul><li>Final product in particle form </li></ul></ul>
    20. 20. Why Study Suspension Polymerization? <ul><li>Suspension polymerization reactors often do not produce sufficiently narrow particle size distribution (PSD) </li></ul><ul><li>Empirical knowledge and expensive experimental efforts are still used to design reactors </li></ul><ul><li> Global competition is making this less acceptable </li></ul><ul><li>How to accurately predict & control the final particle size distribution? </li></ul>
    21. 21. What is Suspension Polymerization? Monomer Reaction Aqueous Phase: Water, Surfactant Monomer Droplets: Monomer, Initiator Heat of Reaction Time Final Particles
    22. 22. Suspension Polymerization Droplets Particles In Situ Video Microscopy Optical Microscopy
    23. 23. Sensor Technology – Suspension Polymerization <ul><li>In situ video microscopy and advanced image analysis </li></ul><ul><li>Laser backscattering and inverse modeling </li></ul><ul><li>Presented at Ph.D. Oral Prelim </li></ul>
    24. 24. Simulation of Polymerization Within Droplets
    25. 25. Simulation of Droplet Size Distribution using Population Balance Equation (PBE) Accounts for droplet breakage and coalescence and polymerization kinetics (i.e., increase in viscosity and volume contraction/expansion)
    26. 26. Simulation of Droplet Size Distribution <ul><li>Numerical problems, such as oscillations and numerical diffusion, occur in standard finite difference approaches for solving the PBE </li></ul><ul><li>High resolution finite-volume methods are computationally efficient while giving second-order accuracy and no undesirable oscillations </li></ul><ul><li>First time high resolution methods applied to suspension polymerization </li></ul><ul><li>Simulated on parallel computers using MPI </li></ul>
    27. 27. Parameter Estimation for Droplet Size Distribution <ul><li>Measurements of droplet size distribution </li></ul><ul><ul><li>In situ video microscopy & advanced image analysis </li></ul></ul><ul><ul><li>Laser backscattering </li></ul></ul><ul><li>5 unknown parameters being estimated </li></ul><ul><ul><li>Breakage frequency and efficiencies </li></ul></ul><ul><ul><li>Coalescence frequency and efficiency </li></ul></ul><ul><li>Parameter sensitivity analysis determines most sensitive parameters </li></ul><ul><ul><li>Moments (1 st – 6 th ) </li></ul></ul><ul><ul><li>Breakage kernel most sensitive </li></ul></ul>
    28. 28. Comparison of Experiments and Model Predictions
    29. 29. Optimal Control Trajectories and Uncertainty Analysis <ul><li>Objective function considered to minimize/maximize Sauter mean diameter </li></ul><ul><li>Determined nominal parameter vector and covariance matrix, V θ </li></ul>
    30. 30. Uncertainty Analysis – Moments Output
    31. 31. Summary <ul><li>Rigorous approach presented for modeling and control of suspension polymerization </li></ul><ul><li>Sensitivity analysis indicates that breakage is dominant phenomenon </li></ul><ul><li>Model in agreement with observed data </li></ul><ul><li>Uncertainty analysis estimate upper/lower bounds on states </li></ul>
    32. 32. Biological Distribution Outline NO YES design constraints & performance criteria Robust Optimization optimized design Hypothesis Mechanism Selection Experimental Data Collection Multiple Models Parameter Estimation experimental constraints Sensitivity & Uncertainty Analysis Is model accurate? Experimental Design
    33. 33. Introduction <ul><li>Atomic force microscopy can be used to identify kinetic parameters associated with bond dissociation </li></ul><ul><li>Kinetic parameters typically determined from most probable force measurements </li></ul><ul><ul><li>Force at which most dissociations occur </li></ul></ul><ul><ul><li>Binning effects ignored </li></ul></ul><ul><ul><li>Ignores shape of distribution </li></ul></ul><ul><ul><li>Does not consider effect of multiple bonds formed </li></ul></ul><ul><li>This thesis provides a systematic investigation of the analysis of data from single-molecule pulling experiments </li></ul>
    34. 34. Introduction – Model Selection <ul><li>Microscopic model (Hummer and Szabo, 2003) </li></ul><ul><li>Phenomenological model (Evans, 2001) </li></ul>
    35. 35. Experiments/Equipment
    36. 36. Experiments/Equipment
    37. 37. Maximum Likelihood Parameter Estimates for Single-bond Microscopic Model Model not accurate (total residual = 0.4194)
    38. 38. Proposed Double-bond Microscopic Model <ul><li>derived from single-bond model & sensitivity analysis </li></ul>
    39. 39. Maximum Likelihood Parameter Estimates for Double-bond Microscopic Model Model much more accurate (total residual = 0.0687)
    40. 40. Parameter Estimation – Model Identification <ul><li>F-test justifies double-bond model (at 99.5%) </li></ul><ul><li>F-test statistically eliminates more complicated models (x 1 ,x 2 ,k 1o ,k 2o ) </li></ul><ul><li>Double-bond model has one additional parameter </li></ul>
    41. 41. Model Validation <ul><li>Motivation for the systematic approach </li></ul>
    42. 42. Summary <ul><li>AFM data not described by the single-bond models </li></ul><ul><li>Proposed a double-bond microscopic model </li></ul><ul><ul><li>Sensitivity analysis – conf. intervals and model development </li></ul></ul><ul><ul><li>Maximum likelihood parameters determined </li></ul></ul><ul><li>F-test statistically justifies proposed model </li></ul><ul><li>Double-bond model fits experimental data an order-of-magnitude better than single-bond model </li></ul><ul><li>These conclusions are consistent with NCAM having two binding sites, which was also seen in SFA data (Johnson et al., 2004, PNAS ) </li></ul>
    43. 43. Overall Summary <ul><li>Systematic approach applied to diverse processes </li></ul><ul><ul><li>Molecular distributions </li></ul></ul><ul><ul><li>Microparticle distributions </li></ul></ul><ul><ul><li>Biological distributions </li></ul></ul><ul><li>Common approach </li></ul><ul><ul><li>Experiments/equipment </li></ul></ul><ul><ul><li>Parameter estimation </li></ul></ul><ul><ul><li>Sensitivity and uncertainty analysis </li></ul></ul><ul><ul><li>Model selection </li></ul></ul><ul><ul><li>Optimal control </li></ul></ul>
    44. 44. Acknowledgments <ul><li>Richard D. Braatz </li></ul><ul><li>Richard C. Alkire </li></ul><ul><li>Deborah E. Leckband </li></ul><ul><li>Timothy O. Drews </li></ul><ul><li>Braatz Group </li></ul><ul><li>Mitsuko Fujiwara </li></ul><ul><li>Zoltan Nagy </li></ul><ul><li>Julie A. Wieland </li></ul><ul><li>Chii-wey Chen </li></ul><ul><li>Alkire Group </li></ul>

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