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Control of Batch Crystallisation Processes

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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.

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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.

Batch Crystalliser.

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.

Temporal evolution of CSD in a batch crystalliser under linear cooling .
Temporal evolution of CSD in a batch crystalliser for optimal cooling.

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.


  1. 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.
  2. 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.

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