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Publication# Incremental Identification of Kinetic Models by Online Spectroscopy and Multivariate Calibration

Abstract

Extent-based kinetic identification is a modeling technique that uses number of moles / concentrations measurements to compute extents and identify reaction kinetics (partial orders of reactions and reaction rate constant) by the integral method of parameter estimation. This project uses infrared spectroscopic data with calibration models to predict moles / concentrations via a PLS1 regression method. The calibration set (design of experiment) is constructed from a seven-level design for multivariate calibration in molar fraction with reacting material. The extent-based kinetic identification using number of moles predicted from spectroscopic data is tested through experiments in a homogeneous liquid-phase reaction system. The experiments were conducted in an RC1 calorimeter from Mettler Toledo with two inlet streams, without outlet stream, and a ReactIR 10 FT-IR probe. The reaction studied was the transesterification of methanol and hexanol with acetic anhydride producing methyl, hexyl acetate and acetic acid. No solvents or catalyst were used, only pure species at a reaction temperature of 50°C. Results in terms of calibration design and model, computation of extents and extent-based kinetic identification are discussed.

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Julien Léo Billeter, Konrad Hungerbühler

Process Analytical Chemistry/Technology has tremendously evolved in the last decades due to the development of multivariate online sensors that are able to monitor the properties of industrial processes in real time [1, 2]. Online monitoring of product quality and the detection of process upsets are important for the pharmaceutical and fine chemical industry in order to maintain product specifications and meet their commitments regarding safety, health and environment. Many methods exist to extract useful information from the vast amount of data produced by online sensors. Chemometric methods, such as Principal Component Regression (PCR) and Partial Least Squares (PLS) or Black Box modelling (e.g. Neural Networks) are commonly used during the monitoring of batch processes [3, 4]. However, for these data-driven methods, calibration conditions need to be maintained during the actual process and the calibration generally behaves poorly when extrapolated to different operating conditions. On the other hand, kinetic modelling techniques [5], based on first principal models, describing the kinetics of main and side products, do not encounter such drawbacks and can be adapted for the monitoring of highly fluctuating processes, e.g. under semi-batch conditions. During batch and semi-batch processes, deviations from standard operating conditions can have various origins. Most frequent sources of deviations are due to slightly imprecise initial conditions (e.g. initial concentrations) or impurities in the initial reactants causing unexpected side reactions [6]. In this contribution, we propose a method for the online monitoring of semi-batch processes based on a kinetic modelling approach in order to optimise operating conditions and reduce “batch to batch” deviations. To our knowledge, this option has not yet been considered in literature. The proposed method requires the kinetic model and the associated rate constants to be known, i.e. determined in an early phase of R&D. In the following, the different steps of the algorithm, currently implemented into Matlab, are outlined. The algorithm assumes a first small amount of reagent to be dosed into the reaction mixture inside the reactor. Corrected initial concentrations are then determined by fitting the kinetic model to measurements, such as UV-vis, IR or heat power, using the Newton-Gauss-Levenberg/Marquardt (NGL/M) optimiser. If the optimiser fails the operator has the option to dose more reagent. Possible failure can be due to an early process upset, or to the fact that too little reagent was dosed in order to follow the kinetics reliably. The corrected initial concentrations are then fed back into the kinetic model and the algorithm optimises the flow rate for the dosed reagent or the operating temperature in order to maximise under constraints user-defined properties of the process, such as yield, selectivity or conversion. For this constrained optimisation, nonlinear programming (NLP) is employed (Matlab’s fmincon function). As soon as optimum operating conditions are obtained by the algorithm, the reactor will automatically run at these improved settings. As an option, flow rate and temperature can continuously be re-optimised to adapt to possible fluctuations in operating conditions. During the whole procedure, the algorithm also tests for possible process upset. If such an incident is detected the operator is asked to take appropriate action, for example a reactor shut down. The algorithm will be demonstrated using simulated data from mid-IR and UV-vis spectroscopy as well as from calorimetry. [1] P. Gemperline, G. Puxty, M. Maeder, D. Walker, F. Tarczynski, M. Bosserman, Analytical Chemistry 76 (2004) 2575-2582. [2] J. Workman, M. Koch, D. Veltkamp, Analytical Chemistry 77 (2005) 3789-3806. [3] M. Spear, Chemical Processing 70 (2007) 20-26. [4] T.J. Thurston, R.G. Brereton, D.J. Foord, R.E.A. Escott, Journal of Chemometrics 17 (2003) 313-322. [5] M. Maeder, Y.M. Neuhold, Practical Data Analysis in Chemistry, Elsevier, Amsterdam NL, 2007. [6] E.N.M. van Sprang, H.J. Ramaker, H.F.M. Boelens, J.A. Westerhuis, D. Whiteman, D. Baines, I. Weaver, Analyst 128 (2003) 98-102.

2008Julien Léo Billeter, Konrad Hungerbühler

In recent years, chemometric methods for the analysis of multivariate kinetic data have considerably progressed. Kinetic hard-modelling is one of these methods that is based on the rate law and used to determine the kinetic parameters (e.g. rate constants) of chemical reactions by non-linear optimisation. Applied to spectroscopy, kinetic hard-modelling relies on Beer’s law to decompose the time and wavelength resolved data into the concentration profiles and the molar spectra of the pure components. In direct implicit kinetic hardmodelling, the concentration profiles are obtained by numerical integration of the rate laws and pure spectra are linearly estimated at each iteration using the pseudo-inverse of the concentration matrix [1]. Direct implicit kinetic hard-modelling of spectroscopic data allows the validation of the kinetic mechanism by comparing the estimated component spectra with independently measured ones. A severe drawback, however, is that this implicit method fails when concentrations profiles are linearly dependent, as the pseudo-inverse and thus the component spectra cannot be computed. Different Strategies have been proposed as a remedy to this rank deficiency problem, such as (1) defining some absorbing species as uncoloured, (2) providing some component spectra for the analysis, (3) dosing one or more species or (4) analysing simultaneously several experiments recorded under different initial concentrations (3-way analysis). In absence of a systematic method, the appropriate species to be included in these four Strategies are selected by experience or trial and error. Spectral validation of the kinetic mechanism can also be difficult when Strategy (1) is employed, as the fitted component spectra are complex linear combinations of the true pure spectra. We have recently proposed a systematic method for the experimental and data analytical design of kinetic data measured by spectroscopy that allows identifying the species to be incorporated in Strategies (1) – (4) and calculating the linear combinations of the true pure spectra when Strategy (1) is used, an important step for spectral validation [2]. This systematic method is based on a time-invariant matrix that avoids the numerical integration of the time-variant concentration profiles and allows the experimental design of chemical reactions, even if the associated rate constants are not yet known, i.e. before optimisation. This time-invariant matrix uses the entire set of kinetic reactions and only requires a reduction to independent reactions if linear combinations of the true pure spectra are desired (Strategy 1). For this, a method has also been developed. In this presentation, the systematic method of species selection is presented using simulated data and, from this, appropriate experimental designs (Strategies) are suggested. The method is also presented using experimental results obtained from the reaction of benzophenone with phenylhydrazine in THF (under catalysis of acetic acid), for which the postulated kinetic mechanism has been successfully validated via the comparison between fitted and measured component spectra in mid-IR and UV-vis [3]. [1] M. Maeder, A.D. Zuberbühler, Anal. Chem., 62 (1990), 2220-2224. [2] J. Billeter, Y.M. Neuhold, K. Hungerbühler, Chemom. Intell. Lab. Syst., 95 (2009), 170-187. [3] J. Billeter, Y.M. Neuhold, K. Hungerbühler, Chemom. Intell. Lab. Syst., 98 (2009), 213-226.

2009Julien Léo Billeter, Dominique Bonvin, Sriniketh Srinivasan

The identification of kinetic models is an important step for the monitoring, control and optimization of industrial processes. This is particularly the case for highly competitive business sectors such as chemical and pharmaceutical industries, where the current trend of changing markets and strong competition leads to a reduction in the process development costs [1]. Moreover, the PAT initiative of the FDA advocates a better understanding and control of manufacturing processes by the use of modern instrumental technologies and innovative software solutions [2]. Reaction systems can be represented by first-principles kinetic models that describe the time evolution of states – numbers of moles, temperature, volume, pressure – by means of conservation and constitutive equations of differential and algebraic nature. These models are designed to include all kinetic phenomena, whether physical or chemical, involved in the reaction systems. Generally, such kinetic phenomena include the dynamic effects of reactions (stoichiometry and reaction kinetics), transfer of species between phases (mass-transfer rates), and operating conditions (initial conditions as well as inlet and outlet flows). The identification of reaction and mass-transfer rates as well as the estimation of their corresponding rate parameters represents the main challenge in building first-principles models. The task of identification is commonly performed in one step via ‘simultaneous identification’, in which a dynamic model comprising all rate effects is postulated, and the corresponding model parameters are estimated by comparing the measured and modeled concentrations [3]. This procedure is repeated for all combinations of model candidates, and the combination with the best fit is usually selected. The main advantage of this identification method lies in its capability to model complex dynamic effects in a concomitant way and thus to generate enough constraints in the optimization problem so that indirect measurements such as spectroscopic and calorimetric data can be modeled without the use of a calibration step [4, 5]. However, the simultaneous approach can be computationally costly when several candidates are available for each dynamic effect. Furthermore, this method often leads to high parameter correlation with the consequence that any structural mismatch in the modeling of one part of the model can result in errors in all estimated parameters and, in addition, convergence problems can arise from a poor choice of initial guesses [6, 7]. As an alternative, the incremental approach decomposes the identification task into a set of sub-problems of lower complexity [8]. The approach consists in transforming the measured concentrations into decoupled rates or extents, which can then be modeled individually. When needed, prior to the modeling step, the missing or unmeasured states can be reconstructed using the computed rates or extents. In the ‘rated-based incremental identification’ [9], rates are first obtained by differentiation of concentration measurements. Then, postulated rate expressions and rate parameters are estimated one at a time by comparing the measured and modeled rates. However, because of the bias introduced in the differentiation step, the rate parameters estimated by this method are not statistically optimal. That is why, another approach, termed ‘extent-based incremental identification’ [10], that is based on the integral method of parameters estimation has been introduced. In this approach, extents are first computed from measured concentrations, and then postulated rate expressions are integrated individually for each extent and the corresponding rate parameters estimated by comparing the measured and modeled extents. The extent-based identification can also be adapted to analyze calorimetric and spectroscopic data using a calibration step [11, 12]. The transformation to rates or extents reduces the dimensionality of the dynamic model since all redundant states (invariants) can be discarded. More importantly, the remaining states (variants) isolate the effects of the reactions, mass transfers and operating conditions, which can then be analyzed individually [13]. This allows substantially reducing the computational effort, the convergence problems and the correlation between the estimated rate parameters. Recently, the extent-based incremental identification has been extended to fluid-fluid reaction systems undergoing unsteady-state mass transfer and reactions at the interface of the two immiscible phases. This situation is commonly encountered in reaction systems that are limited by diffusion, such as CO2 post-combustion capture and nitration reactions. Such reaction systems can be modeled using the film theory, where the two bulks are separated by a spatially distributed film, located in either of the two phases, in which diffusing species can accumulate and react. In both bulks, the mass balance equations describing the dynamics of chemical species are expressed as ordinary differential equations (ODE) and serve as boundary conditions for the film. The dynamic accumulation in the film is described by Fick’s second law combined with a reaction term, thus leading to partial differential equations (PDE), which can be solved by appropriate spatial discretization and rearrangement in ODEs. The extent-based model identification of fluid-fluid reaction systems with unsteady-state mass transfer and reactions requires a large number of measurements for reconstructing all the states and modeling the dynamics of the film [14]. The difficulty lies in the fact that, with the current state of sensor technology, such measurements can only come from the two homogeneous bulks, which provide information from a well-mixed reactor region and consequently are resolved only in time and not in space. Nevertheless, extents of reaction and extents of mass transfer can be extracted from these bulk measurements. These extents of reaction represent the effect of slow reactions that take place in the bulks of the two phases and can be modeled as before. On the other hand, the extents of mass transfer represent now the combined effect of mass transfer by diffusion through the film and of fast reactions taking place at the interface or in the film. Hence, both the diffusion coefficients and the rate constants of the fast reactions can be estimated by comparing the measured extents of mass transfer and the extents obtained by solving the corresponding PDEs. In the absence of coupling terms in the PDEs due to interactive diffusion and/or reactions, the diffusion coefficients of each species transferring through the film can be estimated incrementally. However, in the case of interactive diffusion and/or reactions, the interdependence of species via the coupling terms of the PDEs calls for a simultaneous identification of the diffusion coefficients and rate constants within the film. This contribution extends the extent-based incremental identification to the analysis of reaction systems with dynamic accumulation and reactions in the film. In particular, the question of whether to use incremental or simultaneous estimation of the diffusion coefficients and rate constants within a diffusion layer will be addressed. [1] Workman et al., Anal. Chem. 83, 4557-4578, 2011 [2] Billeter et al., 100th AIChE Annual Meeting, Philadelphia, 2008 [3] Hsieh et al., Anal. Chem., http://dx.doi.org/10.1021/ac302766m, 2013 [4] Billeter et al., Chemom. Intell. Lab. Syst. 95(2), 170-187, 2009 [5] Zogg et al., Thermochimica Acta 419, 1-17, 2004 [6] Billeter et al., Chemom. Intell. Lab. Syst. 93(2), 120-131, 2008 [7] Billeter et al., Chemom. Intell. Lab. Syst. 98(2), 213-226, 2009 [8] W. Marquardt, Chem. Eng. Res. Des., 83(A6), 561–573, 2005 [9] Brendel et al., Chem. Eng. Sci. 61, 5404-5420, 2006 [10] Bhatt et al., Ind. Eng. Chem. Res. 50, 12960-12974, 2011 [11] Srinivasan et al., Chem. Eng. J. 207-208, 785-793, 2012 [12] Billeter et al., Anal. Chim. Acta 767, 21-34, 2013 [13] Srinivasan et al., IFAC Workshop TFMST, Lyon, 2013 [14] Billeter et al., 104th AIChE Annual Meeting, Pittsburgh, 2012

2013