### Inhalt des Dokuments

# Control of Batch Crystallisation Processes

### From Fachgebiet Regelungssysteme TU Berlin

## Contents |

### Abstract

For a batch crystallisation process described by a population balance model, we investigate the problems of trajectory planning and feedback control. The approach is based on a specific time transformation, which translates the (infinite-dimensional) population balance equation into a simple transport equation. Applying the same transformation to the corresponding (finite-dimensional) moment model yields a differentially flat dynamic system.

### People involved

- Ulrich Vollmer (now with Robert Bosch GmbH)
- Jörg Raisch

### Cooperation

- A. Seidel-Morgenstern and M.-P. Elsner, PCF Group, MPI Magdeburg

### Description

In the chemical and pharmaceutical industries, crystallisation is used for the production of solids from liquids. Supersaturation, which is generated either by cooling or by evaporation of solvent, represents the driving force for the two processes dominating crystallisation dynamics: nucleation, i.e. the production of new crystals, and crystal growth. Furthermore, phenomena such as attrition, breakage and agglomeration of crystals may occur.

Since nucleation, growth, etc. take place
simultaneously, crystals of different sizes are present in a
crystalliser. Product quality depends heavily on *crystal size distribution* (CSD), i.e., the distribution of crystals with respect to
crystal size. The evolution of the CSD over time is usually modelled by a
*population balance equation* (PBE). This is a partial
differential equation, sometimes with an additional integral part representing
breakage, attrition, and agglomeration phenomena.
It is coupled to one or more ordinary differential equations (ODEs)
resulting from a solute mole balance of the liquid phase and, if necessary, an
energy balance of the system. Hence, commonly accepted models for
crystallisation processes are relatively complex, nonlinear,
infinite-dimensional systems.

In batch mode, the crystalliser is initially filled with undersaturated solution. Supersaturation is generated by gradual cooling. The CSD obtained at the end of the batch is determined by the temperature-time profile applied to the process. This, essentially, defines an open-loop control, or trajectory planning, problem, namely, how to find a temperature signal producing a predefined CSD.

A solution to this problem has been developed based on a standard population
balance model from the literature. This model allows the
derivation of a closed set of ordinary differential equations for a finite
number of leading moments of the CSD. The solution makes use of the
flatness concept from nonlinear control
theory: a dynamic system is called *differentially flat* if there exists a
"flat output", which completely parametrises the system state and its input.
This can be interpreted as an invertibility property and is extremely useful
for the solution of open loop control problems. Although the system of moment
equations derived from the PBE form is not flat, it can be made so by applying
state dependent scaling of time . Such systems are called *orbitally flat*. Applying the same scaling of time to the PBE yields a simple transport equation. Exploiting these two properties -
orbital flatness of the moment equations and the simple structure of the time
scaled PBE - the open loop control problem can be solved in a
very elegant way. A procedure has been developed which enables the
analytic computation of the corresponding temperature profile for any desired
(and physically meaningful) final CSD ^{[1]}.
Based on these results, it is also possible to determine a control policy that
optimises the final CSD by solving a *static* optimisation
problem ^{[1]}.
Simulation results for a typical optimisation problem - maximising the ratio of
final seed crystal mass and nucleated crystal mass - are shown in the right
half of the following figure. The left half of the figure shows the temporal
evolution of the CSD obtained from a conventional linear cooling policy.

Finally, in ^{[2]}, we use the fact that flat systems are feedback linearisable to design a closed loop control scheme that tracks the
previously designed trajectory in the presence of modelling errors and
disturbances.

### Publications

- ↑
^{1.0}^{1.1}- Ulrich Vollmer, Jörg Raisch.
**Control of batch cooling crystallizers based on orbital flatness**.*International Journal of Control*, 76 (16):1635–1643, 2003. - Bibtex| Abstract | PDF
**Author :**Ulrich Vollmer, Jörg Raisch**Title :**Control of batch cooling crystallizers based on orbital flatness**In :***International Journal of Control*,**Date :**2003

- Ulrich Vollmer, Jörg Raisch.
- ↑
- Ulrich Vollmer, Jörg Raisch.
**Control of Batch Crystallization - a System Inversion Approach**.*Chemical Engineering & Processing (Special issue on particulate processes)*, 45 (10):874–885, 2006. - Bibtex| Abstract | PDF
**Author :**Ulrich Vollmer, Jörg Raisch**Title :**Control of Batch Crystallization - a System Inversion Approach**In :***Chemical Engineering & Processing (Special issue on particulate processes)*,**Date :**2006

- Ulrich Vollmer, Jörg Raisch.