Control of Continuously Operated Crystallisers
This research work investigates control synthesis for continuously operated crystallisers on the basis of so called population balance models. It follows a "late lumping" philosophy, implying that the infinite dimensional character of the model is maintained during the actual synthesis step.
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. This makes model-based controller synthesis a challenging task, both for continuously operated and batch crystallisers.
In industrial continuous crystallisation plants, dissolution of small crystals (fines dissolution) is frequently used to improve the product CSD. This effectively shifts the CSD towards larger crystal sizes and often makes the distribution narrower. However, these benefits are being paid for by a deterioration of the dynamic process behaviour: fines dissolution often leads to sustained oscillations of CSD and solute concentration around the designated operating point. Clearly, this interferes with the aim of providing constant product quality and therefore motivates the design of suitable feedback controllers to stabilise the crystalliser at high fines dissolution rates.
A detailed population balance model for an evaporative crystalliser was developed by 'A. Mitrovic and S. Motz [1]. In a first step, based on physical considerations, this model is simplified to allow analytical determination of the steady state. This step does not include lumping of the model and hence preserves the infinite-dimensional nature of the system. We can now linearise around the steady state to obtain a transfer function relating the control input (the fines dissolution rate) to the third moment of the CSD, which is assumed to be the measured output. Reflecting the infinite-dimensional character, the transfer function is irrational with infinitely many poles and zeros. Clearly, as we use a linearised version of an already simplified model, we need to emphasise robustness during the control design, i.e., we seek a controller that will tolerate a large degree of model uncertainty. H∞-theory provides a framework for the synthesis of robust controllers, which has recently been extended to a class of infinite-dimensional systems. Applying this method to the problem at hand yields an irrational controller transfer function [2] [3], which, for implementation purposes, needs to be approximated by a rational, i.e., finite-dimensional, transfer function corresponding to a finite set of ordinary differential equations. Simulations of the resulting controller in closed loop with the detailed population balance model from [1] are shown in the figure below. They demonstrate the effectiveness of our approach.
Temporal evolution of CSD without control.
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Temporal evolution of CSD under control.
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Nevertheless, one might ask what we gain by pursuing a late lumping philosophy, i.e., by lumping the controller instead of the process model. The gain is twofold. First, late lumping preserves the infinite-dimensional nature and, as a consequence, physical intuition during the entire design process. This is especially important in H∞-synthesis, where a proper choice of weighting factors in the cost function is extremely difficult without that sort of intuition. Second, with late lumping, it is possible to quantify the degree of suboptimality implied by the lumping-related approximation.
Publications
- ↑ 1.0 1.1
- Stefan Motz, Aleksandar Mitrovic, Ernst-Dieter Gilles, Ulrich Vollmer, Jörg Raisch. Modeling, simulation, and H∞-control of an oscillating continuous crystallizer with fines dissolution. Chemical Engineering Science, 58 (15):3473–3488, 2003.
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BibtexAuthor : Stefan Motz, Aleksandar Mitrovic, Ernst-Dieter Gilles, Ulrich Vollmer, Jörg Raisch| Abstract | PDF
Title : Modeling, simulation, and H∞-control of an oscillating continuous crystallizer with fines dissolution
In : Chemical Engineering Science,
58 (15):3473–3488, 2003.
Date : 2003
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↑
- Ulrich Vollmer, Jörg Raisch. H∞-Control of a continuous crystallizer. Control Engineering Practice, 9 pages 837-845, 2001.
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BibtexAuthor : Ulrich Vollmer, Jörg Raisch| Abstract | PDF
Title : H∞-Control of a continuous crystallizer
In : Control Engineering Practice,
9 pages 837-845, 2001.
Date : 2001
-
↑
- Ulrich Vollmer, Jörg Raisch. Population balance modelling and H∞-controller design for a crystallization process. Chemical Engineering Science, 57 (20):4401–4414, 2002.
-
BibtexAuthor : Ulrich Vollmer, Jörg Raisch| Abstract | PDF
Title : Population balance modelling and H∞-controller design for a crystallization process
In : Chemical Engineering Science,
57 (20):4401–4414, 2002.
Date : 2002