Model Reduction of Particulate Processes

Introduction

Particle processes are of paramount importance in chemical and pharmaceutical industry. The majority of chemical products and even more than 90% of pharmaceutical agents are produced in form of particles. Therefore the main task of design and control of particle processes is to generate particle populations with desired property distributions. Property distributions change due to various physical phenomena like nucleation, i.e. formation of new particles, particle growth, breakage or agglomeration. All these effects depend strongly on the interaction between fluid flow and particle phase. A realistic mathematical process model must account for transport processes in the liquid phase and in the solid phase as well as for the coupling between both phases. The model has to describe fluid dynamic effects in up to three space coordinates or external coordinates and in addition the development of particle populations along one or several internal coordinates like e.g. the characteristic particle size. Such a model is computationally very demanding. For process design as well as advanced process control, models of low system order are required that can be solved in real time but nevertheless capture the nonlinear properties of the system.

Objectives and results of this project

In this project, proper orthogonal decomposition techniques have been applied to the model reduction of various crystallization and granulation processes. The aim is to provide a tool that performs the model reduction step automatically for a large class of reference systems.
Currently, a setup for continuous crystallization and separation of enantiomers is considered as an application example. This work is done in cooperation with the PCF group within a DFG funded project, which is a part of the SPP 1679 research network. A nonlinear reduced model has been developed for a system consisting of a fluidized bed crystallizer and a ultrasonic attenuator. The reduced model shows good agreement with the reference system. An a posteriori error estimator implemented in cooperation with the CSC group is found to provide a reliable error bound for the reduced model.
A tool for automatic model reduction has been implemented in the process modeling tool ProMoT. The tool is applicable to arbitrary model systems. Via an interface to Diana and Octave, numerical basis functions are generated from test simulations / snapshots with the reference model. Nonlinearities are treated by empirical interpolation. The resulting reduced model is available in symbolic form in ProMoT.

Periodic solutions in a fluidized bed crystallizer connected to a ultrasonic attenuator [1]; graphs show number size density of particles at different times; vertical coordinate = geometrical
coordinate in direction of fluid flow; horizontal coordinate = crystal size (property coordinate).
Comparison between reference model (order 18400) and reduced model (order 7). The blue line shows the true relative error, the red line shows the error bound provided by a a-posteriori error estimator [1].

Current Research

The project is continued under the guidance of former group member Prof. Michael Mangold at University of Applied Sciences in Bingen/Germany.

Cooperation Partners

  • PCF group
  • Dr. Lihong Feng (CSC group)
  • Dr. Gabor Janiga’s group (OVGU)

Publications

  1. Khlopov D. and Mangold. M. (2017): Automatic model reduction of differential algebraic systems by proper orthogonal decomposition. Comput. Chem. Eng., 97:104–113.
  2. Khlopov D. and Mangold M. (2016): Automatic model reduction of population balance models by proper orthogonal decomposition. In Computer Aided Chem. Eng., 38:163-168
  3. Khlopov D. and Mangold. M. (2016): Automatic model reduction of linear population balance models by proper orthogonal decomposition. In Proc. Vienna Conference on Mathematical Modeling. IFAC-Papers-Online, Volume 28, Issue 1, p. 11-16.
  4. Mangold, M., Feng, L., Khlopov, D., Palis, S., Benner, P., Binev, D., und Seidel-Morgenstern, A. (2015): Nonlinear model reduction of a continuous fluidized bed crystallizer. Journal of Computational and Applied Mathematics 289:253–266.
  5. Khlopov, D. und Mangold, M.: Automatic model reduction of linear population balance models by proper orthogonal decomposition. IFAC Papers Online 48(1):11-16, DOI 10.1016/j.ifacol.2015.05.019.
  6. M. Krasnyk, C. Borchert, and M. Mangold (2012). Model reduction techniques for the simulation of particle populations in fluid flow. Mathematical and Computer Modelling of Dynamical Systems 18:427-438.
  7. M. Krasnyk, M. Mangold, S. Ganesan, and L. Tobiska (2011). Reduction of a crystallizer model with internal and external coordinates by proper orthogonal decomposition. Chemical Engineering Science (in press)
  8. M. Krasnyk, M. Mangold, and A. Kienle (2010). Reduction procedure for parametrized fluid dynamics problems based on proper orthogonal decomposition and calibration. Chemical Engineering Science, 65:6238-6246
  9. M. Krasnyk and M. Mangold (2010). Reduction of a urea crystallizer model by proper orthogonal decomposition and best point interpolation. Industrial & Engineering Chemistry Research, 49:9887-9898
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