Experimental implementation of a Quality-by-Control (QbC) framework using a mechanistic PBM-based nonlinear model predictive control involving chord length distribution measurement for the batch cooling crystallization of l-ascorbic acid
Introduction
l-ascorbic acid (AA), often called as vitamin C, was extracted from paprika. It is a very potent anti-oxidant effect which leads to increased metabolic flux in the Krebs cycle which strengthens the immunological system (Berg et al., 2010). Since extraction of l-ascorbic acid from natural sources is not cost-effective, it is synthetized at industrial scale through a biotechnological route from glucose with an estimated worldwide production rate of 110 metric kilotons. AA is used as an ingredient in the pharmaceutical industry (50%), food industry (25%), beverage sector (15%) and for animal feed uses (Pappenberger and Hohmann, 2014). The overall yield for the glucose transformation to l-ascorbic acid is between 50 and 60% thus it is important to reduce the separation and purification losses (Wierzbowska et al., 2011). Since l-ascorbic acid is mainly marketed in solid crystalline form, the particle formation step is very important since the CSD influences numerous macroscopic material properties, such as dissolution rate and mechanical properties (Myerson, 2002). In the production of l-ascorbic acid, crystallization takes the critical role of simultaneously being the purification as well as the particle formation stage. The separation and purification of l-ascorbic acid has been investigated intensively (Beuzelin-Ollivier et al., 2012) and this paper focuses on controlling the CSD during crystallization.
The state-of-the-art control strategies of cooling crystallizers can be divided into two major groups: model free and model based control approaches (Nagy et al., 2013). Direct nucleation control (DNC) is a model free control method, which controls the crystal size through the relative solution crystal number density via repeated and controlled dissolution-growth (heating-cooling) stages (Abu Bakar et al., 2009). Supersaturation control (SSC) is another model free control strategy, which is based on the principle that the crystallization process should be conducted in the metastable zone to suppress nucleation (Nagy and Braatz, 2012). These two well established control strategies have found numerous applications, both as solo control strategies (Barrett et al., 2010, Kacker et al., 2016) as well as combined control approach (Griffin et al., 2015)
Model based control techniques use real time simulation based on a process model to predict the effects of inputs and disturbances on the systemic output. In essence this translates to real time optimization of the control signal, which in the case of cooling crystallization translates into the prediction of the future temperature profile (Moldoványi et al., 2005, Qin and Badgwell, 2000). The basic requirement of a model based controller is the existence of an adequate process model in terms of complexity, accuracy and solution time (Agachi et al., 2007). The population balance (PB) framework is the most widely accepted modeling approach in the mathematical description of crystallization processes (Hulburt and Katz, 1964). The PB equations, from mathematical point of view, are hyperbolic partial differential equations (PDEs), which becomes an integro-PDE, if agglomeration and/or breakage is also considered. Due to the inherent complexity of these equations, the solution might become complicated, inaccurate and time consuming. Numerous mathematical and numerical techniques have been addressed to the solution of PB equations (PBEs). Moment based methods (Randolph and Larson, 1973) and their quadrature extensions (McGraw, 1997) are computationally efficient, accurate and widely applicable techniques (Grosch et al., 2007) to calculate the moments of distribution. A major problem with these methods is that the estimation of CSD based on its moments is not exact thereby limiting their applicability. In contrast, the combined method of moments and method of characteristics (MOM-MOC) is able to accurately solve the PBE with nucleation and growth with reduced computational burden (Aamir et al., 2009). However, when nucleation is also modeled using the PBEs, more adaptations becomes necessary using MOM-MOC which also makes it practically infeasible as a solution procedure (Mesbah et al., 2009). High resolution finite volume method (HR-FVM) is a solution technique of hyperbolic PDEs (LeVeque, 2002), providing a generic framework for the numerical simulation of PBEs even in multiple dimensions (Gunawan et al., 2004). FVM is a discretization based technique: finer discretization entails higher accuracy but this also translates into higher computational costs. Significant reduction in the time to solve PBEs based on FVM can be achieved by efficient computer implementation (Majumder et al., 2010, Szilágyi and Nagy, 2016).
The vast majority of literature reports controlling crystallization using underlying optimization based strategies employ moment based PBE solution (Acevedo et al., 2015, Bötschi et al., 2018, Paz Suárez et al., 2011, Su et al., 2017, Zhang and Rohani, 2004). However, a full PBE based approach would enable the direct manipulation of CSD, instead of only its moments. Moreover, to use the valuable distributional information of the CLD, the calculation of the CSD would be required, but in the field of control applications the literature is meager (Ma and Wang, 2012, Mesbah et al., 2012). A major limitation of real time full PBM based control is the difficulty of the state estimation, since the actual CSD is required as the initial condition of the process optimization and simulations (in addition to the increased computational time). Since the CSD cannot be accurately measured with on-line tools, it has to be estimated through state estimation using the measurable data (Porru and Özkan, 2017, Szilagyi et al., 2018). The state estimator is also required for robust control (Nagy and Braatz, 2003, Rawlings and Mayne, 2012) and it represent a crucial part of the predictive control system (Mesbah et al., 2011b). To the best of the authors knowledge, the experimental implementation of an NMPC system that uses the CLD as direct feedback information was not reported in the literature before.
Even though the crystallization modeling and control is an intensively investigated topic and the fact l-ascorbic acid is an industrially important product produced in large quantities, only a few papers deal with its crystallization. Eggers et al. investigated the shape and size variations during l-ascorbic acid crystallization using state-of-the-art process analytical technology (PAT) tools assuming two dimensional crystal shape (Eggers et al., 2009). Wierzbowska et al. investigated the solubility, nucleation and growth kinetics have been investigated for ethanol-water systems (Shalmashi and Eliassi, 2008, Wierzbowska et al., 2008a, Wierzbowska et al., 2008b, Wierzbowska et al., 2008a, Wierzbowska et al., 2008b). In a theoretical investigation it was shown that the shape of l-ascorbic acid exhibits complex habit changes during the crystallization (Uesaka and Kobayashi, 2002). However, a control oriented study on l-ascorbic acid crystallization has not been published yet.
Quality-by-Control (QbC) has been proposed recently as a novel paradigm to design processes, whereby feedback control is used to obtain suitable operating procedures by direct adaptation of the operating conditions during the process using a feedback control based approach; hence, it provides a much faster methodology to find the optimal operating procedure, requiring less material as traditional Quality-by-Design (QbD) approaches based on standard design-of-experiments techniques (Simone et al., 2015, Yang et al., 2015).
The aim of the current work is to provide and experimental proof-of-concept and comparison between two state-of-the-art QbC frameworks: (i) a model-free QbC, based on direct nucleation control (DNC), which is one of the most advanced model-free control strategies and (ii) a model-based QbC using a full PBM based nonlinear model predictive control (NMPC), which is one of the most complex model based control approaches. These approaches are implemented and investigated for the cooling batch crystallization of l-ascorbic acid from aqueous solutions, used as the model system with large industrial significance, however the methodology presented is generic and applicable for most batch cooling crystallization processes. For the model system, a kinetic investigation has also been carried out to estimate the nucleation and growth kinetics based on concentration and CLD measurements, which is then used in the NMPC. It was shown that the system is characterized by a large time delay to time constant ratio with the particular crystallization kinetics, which causes the DNC to oscillate. However, the NMPC approach produced good quality crystals with significantly better CSD than the corresponding linear cooling, and with considerably shorter batch time than the DNC. This work provides an exemplary case study for the comprehensive experimental implementation and investigation of a high-resolution finite volume PBM based NMPC with the novel features of using real-time kinetic parameter adaptation through a growing horizon estimation and applying direct CLD measurement and FBRM sensor model with a forward CSD to CLD conversion. The proposed framework also provides a simultaneous model identification and kinetic parameter refinement as well as optimal model based temperature design approach through a single batch experiment, offering a platform for rapid and automated model identification and model based optimal design for batch cooling crystallization systems.
Section snippets
Direct nucleation control of l-ascorbic acid crystallization
The main idea of DNC is to keep the relative particle number between some pre-defined limits while the system is gradually cooled to the desired final temperature. For a predefined difference between the initial and final temperatures, the produced crystal number is inversely proportional with the mean crystal size. The actual crystal number density in solution, however cannot be directly measured but a relative particle number can be tracked, traditionally with Focused Beam Reflectance
Model development
A key requirement of the model based controller is the existence of an adequate process model, which can be solved computationally much faster than the time scale of the actual process evolution, thus enabling real time optimization. The trade-off among model complexity, dimensionality and solution accuracy are all dictated by the objectives of the NMPC system. Thus, the control-relevant model used in the NMPC is not necessarily the most comprehensive model, which should be used for fundamental
NMPC system for CLD control of l-ascorbic acid crystallization
The idea behind model based control approaches is the use of process model in real time to optimize input control signal(s) with respect to the objective function subject to imposed constraints. Measured output data is processed by the state estimator, which estimates the unmeasurable system states, which is required as initial condition for the NMPC optimizations, and it also mitigates the effects of disturbances. The procedure is repeated in every discrete time step.
Fig. 7 presents the
Conclusions
Two Quality-by-Control (QbC) approaches for the batch cooling crystallization of l-ascorbic acid were analyzed in this work. First, a model-free QbC based on direct nucleation control (DNC) was applied as a quick crystallization design approach via feedback control. The DNC however provided oscillatory control performance due to the particular crystallization kinetics of the systems, which introduces an apparent time delay in the nucleation rate during the cooling stages. Despite the
Acknowledgements
The financial support of the International Fine Particle Research Institution is acknowledged gratefully. Funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement No. [280106-CrySys] is also acknowledged.
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