Model Reduction, Structure-property Relations and Optimization Techniques for the Production of Nanoscale Particles

  • Michael GröschelEmail author
  • Günter Leugering
  • Wolfgang Peukert
Part of the International Series of Numerical Mathematics book series (ISNM, volume 160)


The production of nanoscaled particulate products with exactly pre-defined characteristics is of enormous economic relevance. Although there are different particle formation routes they may all be described by one class of equations. Therefore, simulating such processes comprises the solution of nonlinear, hyperbolic integro-partial differential equations. In our project we aim to study this class of equations in order to develop efficient tools for the identification of optimal process conditions to achieve desired product properties. This objective is approached by a joint effort of the mathematics and the engineering faculty. Two model-processes are chosen for this study, namely a precipitation process and an innovative aerosol process allowing for a precise control of residence time and temperature. Since the overall problem is far too complex to be solved directly a hierarchical sequence of simplified problems has been derived which are solved consecutively. In particular, the simulation results are finally subject to comparison with experiments.


Population balance equations optimal control model reduction parameter identification 


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  1. 1.
    C. Artelt, H.-J. Schmid, W. Peukert, Modelling titania formation at typical industrial process conditions: effect of structure and material properties on relevant growth mechanisms. Chem. Eng. Sci., 61 (2006), 18–32.CrossRefGoogle Scholar
  2. 2.
    P.D. Christofides, Nonlinear and robust control of PDE systems: Methods and applications to transport-reaction processes, Birkhäuser-Verlag 2001, 250p.Google Scholar
  3. 3.
    M. Escobedo, P. Laurençot, S. Mischler, On a Kinetic Equation for Coalescing Particles, Communications in Mathematical Physics, 246 (2), 2004, 237–267.CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    T. Fischer, D. Logashenko, M. Kirkilionis and G. Wittum, Fast Numerical Integration for Simulation of Structured Population Equations, Mathematical Models and Methods in Applied Sciences, 16 (12), 2006, 1987–2012.CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    M. Fliess, J. Levine, P. Martin, P. Rouchon, Sur les systèmes non linéaires differentiellement plats, C.R. Acad. Sci. Paris, 1992, I/315, 619–624.MathSciNetGoogle Scholar
  6. 6.
    M. Fliess, J. Levine, P. Martin, P. Rouchon, Flatness and defect of nonlinear systems: Introductory theory and examples, 1995, International Journal of Control, 61 (6), 1327–1361.CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    J. Gradl, H.-C. Schwarzer, F. Schwertfirm, M. Manhart, W. Peukert, Precipitation of nanoparticles in a T-mixer: Coupling the particle population dynamics with hydrodynamics through direct numerical simulation, Chemical Engineering and Processing, 45 (10), 2006, 908–916.CrossRefGoogle Scholar
  8. 8.
    J. Gradl, W. Peukert, Simultaneous 3D observation of different kinetic subprocesses for precipitation in a T-mixer, Chemical Engineering Science (2009), 64, 709–720.CrossRefGoogle Scholar
  9. 9.
    W. Hackbusch, On the Efficient Evaluation of Coalescence Integrals in Population Balance Models, Computing 78, 2 (Oct. 2006), 145–159.CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    W. Hackbusch, Fast and exact projected convolution for non-equidistant grids, Computing 80, 2 (Jun. 2007), 137–168.CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    W. Hackbusch, Approximation of coalescence integrals in population balance models with local mass conservation, Numer. Math. 106, 4 (May, 2007), 627–657.CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    D.K. Henze, J.H. Seinfeld, W. Liao, A. Sandu, and G.R. Carmichael (2004), Inverse modeling of aerosol dynamics: Condensational growth, J. Geophys. Res., 109, D14201.CrossRefGoogle Scholar
  13. 13.
    M.J. Hounslow, R.L. Ryall, and V.R. Marshall, A discretized population balance for nucleation, growth and aggregation. AIChE Journal, 34 (1988), 1821–1832.CrossRefGoogle Scholar
  14. 14.
    J. Israelachvili, “Intermolecular and Surface Forces”, 2nd edition, Academic Press, London, Great Britain.Google Scholar
  15. 15.
    T. Johannessen, S.E. Pratsinis, and H. Livbjerg, Computational Fluid-particle dynamics for flame synthesis of alumina particles. Chem. Eng. Sci. 55 (2000), 177–191.CrossRefGoogle Scholar
  16. 16.
    A. Kalani, P.D. Christofides, Nonlinear control of spatially inhomogeneous aerosol processes, CES 54 (1999), 2669–2678.CrossRefGoogle Scholar
  17. 17.
    A. Kalani, P.D. Christofides, Simulation, estimation and control of size distribution in aerosol processes with simultaneous reaction, nucleation, condensation and coagulation, Com. and Chem. Eng. 26 (2002), 1153–1169.CrossRefGoogle Scholar
  18. 18.
    J. Koch, W. Hackbusch, K. Sundmacher, H-matrix methods for linear and quasilinear integral operators appearing in population balances, Computers and Chemical Engineering, 31 (7), July 2007, 745–759.CrossRefGoogle Scholar
  19. 19.
    J. Koch, W. Hackbusch, K. Sundmacher, H-matrix methods for quadratic integral operators appearing in population balances, Computers and Chemical Engineering, 32 (8), Aug. 2008, 1789–1809.CrossRefGoogle Scholar
  20. 20.
    J. Kumar, M. Peglow, G. Warnecke, S. Heinrich, E. Tsotsas, and L. Moerl, Numerical solutions of a two-dimensional population balance equation for aggregation, Proceedings of the 5th World Congress on Particle Technology, 2006.Google Scholar
  21. 21.
    J. Kumar, G. and Warnecke, Convergence analysis of sectional methods for solving breakage population balance equations-II: the cell average technique, Numer. Math. 110, 4 (Sep. 2008), 539–559.CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Ph. Laurencot, S. Mischler, The continuous coagulation-fragmentation equations with diffusion, Arch. Rat. Mech. Anal. 162, 2002, 45–99.CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    D. Logashenko, T. Fischer, S. Motz, E. D. Gilles, and G. Wittum, Simulation of crystal growth and attrition in a stirred tank, Comput. Vis. Sci. 9, 3 (Oct. 2006), 175–183.CrossRefMathSciNetGoogle Scholar
  24. 24.
    Y. Maday, J. Salomon, and G. Turinici. Monotonic time-discretized schemes in quantum control. Numerische Mathematik, 2006.Google Scholar
  25. 25.
    Ph. Martin, R. Murray, and P. Rouchon, Flat systems, equivalence and trajectory generation, technical report, 2003.Google Scholar
  26. 26.
    A. Mersmann, K. Bartosch, B. Braun, A. Eble, C. Heyer, “Möglichkeiten einer vorhersagenden Abschätzung der Kristallisationskinetik”, 2000, Chemie Ingenieur Technik 71(1-2), 17–30.CrossRefGoogle Scholar
  27. 27.
    H. Mühlenweg, A. Gutsch, A. Schild, and S.E. Pratsinis, Process simulation of gasto- particle-synthesis via population balances: Investigation of three models, Chem. Eng. Sci., 57 (2002), 2305–2322.CrossRefGoogle Scholar
  28. 28.
    Y. Qiu, S. Yang, ZnO Nanotetrapods: Controlled vapour-phase synthesis and application for humidity sensing, Adv. Functional Materials 2007, 17, 1345–1352.CrossRefGoogle Scholar
  29. 29.
    J.M. Roquejoffre, P. Villedieu, A kinetic model for droplet coalescence in dense sprays, Math. Models Meth. Appl. Sci., 11, 2001, 867–882.CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    J. Salomon, Contrôle en chimie quantique: conception et analyse de schémas d’optimisation, thesis, 2005.Google Scholar
  31. 31.
    A. Sandu, W. Liao, G.R. Carmichael, D.K. Henze, J.H. Seinfeld, Inverse modeling of aerosol dynamics using adjoints – theoretical and numerical considerations, Aerosol Science and Technology, 39 (8), 2005,Number 8, 677–694.CrossRefGoogle Scholar
  32. 32.
    H.-C. Schwarzer, W. Peukert, “Combined Experimental/Numerical Study on the Precipitation of Nanoparticles”, 2004, AIChE Journal 50 (12), 3234–3247.CrossRefGoogle Scholar
  33. 33.
    H. Schwarzer, W. Peukert, Tailoring particle size through nanoparticle precipitation, Chem. Eng. Comm. 191 (2004), 580–606.CrossRefGoogle Scholar
  34. 34.
    H.-C. Schwarzer, W. Peukert, Combined experimental/numerical study on the precipitation of nanoparticles, AIChE Journal 50 (2004), 3234–3247.CrossRefGoogle Scholar
  35. 35.
    H.-C. Schwarzer, F. Schwertfirm, M. Manhart, H.-J. Schmid, W. Peukert, “Predictive simulation of nanoparticle precipitation based on the population balance equation”, 2006, Chemical Engineering Science 61 (1), 167–181.CrossRefGoogle Scholar
  36. 36.
    D. Segets, J. Gradl, R. Klupp Taylor, V. Vassilev, W. Peukert, Analysis of Optical Absorbance Spectra for the Determination of ZnO Nanoparticle Size Distribution, Solubility, and Surface Energy, ACS nano (2009), 3(7), 1703–1710.CrossRefGoogle Scholar
  37. 37.
    D. Segets, L.M. Tomalino, J. Gradl, W. Peukert, Real-Time Monitoring of the Nucleation and Growth of ZnO Nanoparticles Using an Optical Hyper-Rayleigh Scattering Method, J. Phys. Chem. C 2009, 113, 11995–12001.CrossRefGoogle Scholar
  38. 38.
    E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, Berlin (2009).CrossRefzbMATHGoogle Scholar
  39. 39.
    V. Vassilev, M. Gröschel, H.-J. Schmid, W. Peukert, and G. Leugering, Interfacial energy estimation in a precipitation reaction using the flatness based control of the moment trajectories, Chemical Engineering Science (65), 2010, 2183–2189.CrossRefGoogle Scholar
  40. 40.
    R. Viswanatha, S. Sapra, B. Satpati, P.V. Satyam, B. Dev and D.D. Sarma, Understanding the quantum size effects in ZnO nanocrystals, J. Mater. Chem., 14, 2004, 661–668.CrossRefGoogle Scholar
  41. 41.
    U. Vollmer, J. and Raisch, Control of batch cooling crystallization processes on orbital flatness, Int. J. Control 76/16 (2003), 1635–1643.CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    M. Wulkow, A. Gerstlauer, U. and Nieken, Modeling and simulation of crystallization processes using parsival, Chem. Eng. Sci. 56 (2001), 2575–2588.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • Michael Gröschel
    • 1
    Email author
  • Günter Leugering
    • 1
  • Wolfgang Peukert
    • 2
  1. 1.Lehrstuhl für Angewandte Mathematik IIFriedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany
  2. 2.Lehrstuhl für Feststoff- und GrenzflächenverfahrenstechnikFriedrich-Alexander-Universität Erlangen-NürnbergErlangenGermany

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