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Publication# Concept of Variants and Invariants for Reaction Systems, with Application to Estimation, Control and Optimization

Abstract

The concept of reaction variants and invariants for lumped reaction systems has been known for several decades. Its applications encompass model identification, data reconciliation, state estimation and control using kinetic models. In this thesis, the concept of variants and invariants is extended to distributed reaction systems and used to develop new applications to estimation, control and optimization.

The thesis starts by reviewing the material and heat balances and the concept of variants and invariants for several lumped reaction systems. Different definitions of variants and invariants, in particular the vessel extents, are presented for the case of homogeneous reaction systems, and transformations to variants and invariants are obtained. The extension to systems with heat balance and mass transfer is also reviewed.

The concept of extents is generalized to distributed reaction systems, which include many processes involving reactions and described by partial differential equations. The concept of extents and the transformation to extents are detailed for various configurations of tubular reactors and reactive separation columns, as well as for a more generic framework that is independent of the configuration.

New developments of the extent-based incremental approach for model identification are presented. The approach, which compares experimental and modeled extents, results in maximum-likelihood parameter estimation if the experimental extents are uncorrelated and the modeled extents are unbiased. Furthermore, the identification problem can be reformulated as a convex optimization problem that is solved efficiently to global optimality.

The estimation of unknown rates without the knowledge or the identification of the rate models is described. This method exploits the fact that the variants computed from the available measurements allow isolating the different rates. Upon using a Savitzky-Golay filter for differentiation of variants, one can show that the resulting rate estimator is optimal and obtain the error and variance of the rate estimates.

The use of variants and invariants for reactor control is also considered. Firstly, offset-free control via feedback linearization is implemented using kinetic models. Then, it is shown how rate estimation can be used for control via feedback linearization without kinetic models. By designing an outer-loop feedback controller, the expected values of the controlled variables converge exponentially to their setpoints.

This thesis presents an approach to speed up steady-state optimization, which takes advantage of rate estimation without rate models to speed up the estimation of steady state for imperfectly known dynamic systems with fast and slow states. Since one can use feedback control to speed up convergence of the fast part, rate estimation allows estimating the steady state of the slow part during transient operation.

The application to dynamic optimization is also shown. Adjoint-free optimal control laws are computed for all the types of arcs in the solution. In the case of reactors, the concept of extents allows the symbolic computation of optimal control laws in a systematic way. A parsimonious input parameterization is presented, which approximates the optimal inputs well with few parameters. For each arc sequence, the optimal parameter values are computed via numerical optimization.

The theoretical results are illustrated by simulated examples of reaction systems.

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Julien Léo Billeter, Dominique Bonvin, Diogo Filipe Mateus Rodrigues

Abstract of the conference paper The kinetic identification of chemical reaction systems often represents a time-consuming and complex task. This contribution presents an approach that uses rate estimation and feedback linearization to implement effective control without a kinetic model. The reaction rates are estimated by numerical differentiation of reaction variants. The approach is illustrated in simulation through the temperature control of a continuous stirred-tank reactor. Extended abstract Model identification and controller design are often seen as closely related tasks, since the control law is calculated using the plant model. Previous control approaches based on extensive variables or inventories are examples of this strong dependence on the model [1, 2]. Since the identification of chemical reaction systems can be a time-consuming and complex task, one would ideally like to avoid it as much as possible. The concept of variant and invariant states allows isolating the different rates in chemical reaction systems, thereby facilitating analysis, monitoring and control [3-5]. Using this concept, one can estimate dynamic effects without the need of identifying the corresponding kinetic models. This contribution presents a feedback linearization approach that is based on the estimation of unknown rates, such as the rates of reaction and mass transfer, thus allowing efficient control without the use of kinetic models. Rate estimation uses the numerical differentiation of appropriately transformed extensive variables called rate variants that are invariant with respect to the manipulated variables. A rate variant contains all the information about the corresponding rate and, as such, is decoupled from the other unknown rates. Since it is possible to estimate the unknown rates this way, the controller does not require kinetic information. However, because of the differentiation step, the controller is most effective with frequent and precise measurements of several output variables. Feedback linearization sets a rate of variation for the controlled variables, thereby guaranteeing quick convergence of these variables to their set points. For open chemical reactors, the parameters of the feedback linearization controller are determined by readily available information, such as the reaction stoichiometry, the heats of reaction, the inlet composition or the inlet and outlet flow rates. This novel control strategy is illustrated in simulation for the control of both concentration and temperature in a continuous stirred-tank reactor. [1] Georgakis, Chem. Eng. Sci., 1986, 41, 1471 [2] Farschman et al., AIChE J., 1998, 44, 1841 [3] Asbjørnsen and Fjeld, Chem. Eng. Sci., 1970, 25, 1627 [4] Bhatt et al., Ind. Eng. Chem. Res., 2011, 50, 12960 [5] Srinivasan et al., IFAC Workshop on Thermodynamic Foundations of Mathematical Systems Theory, Lyon, 2013.

Julien 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

2013Julien Léo Billeter, Dominique Bonvin, Sriniketh Srinivasan

The identification of kinetic models is an essential step for the monitoring, control and optimization of industrial processes. This is particularly true for the chemical and pharmaceutical industries, where the current trend of strong competition calls for a reduction in process development costs [1]. This trend goes in line with the recent initiative in favor of Process Analytical Technology (PAT) launched by the US Food and Drug Administration, which advocates a better understanding and control of manufacturing processes with the goal of ensuring final product quality. Reaction systems can be represented by first-principles models that describe the evolution of the states (typically concentrations, volume and temperature) by means of conservation equations of differential nature and constitutive equations of algebraic nature. These models include information regarding the reactions (stoichiometry and reaction kinetics), the transfer of species between phases (mass-transfer rates), and the operation of the reactor (initial conditions, inlet and outlet flows, operational constraints). The identification of reaction and mass-transfer rates represents the main challenge in building these first-principles models. Note that first-principles models can include redundant states because the modeling step considers balance equations for more quantities than are necessary to represent the true variability of the process. For example, when modeling a closed homogeneous reaction system with R independent reactions, one typically writes a mole balance equation for each of the S species, whereas there are only R < S independent equations, that is S - R equations are redundant. The situation is a bit more complicated in open and/or heterogeneous reaction systems.The identification of reaction systems can be performed in one step via a simultaneous approach, in which a kinetic model that comprises all reactions and mass transfers is postulated and the corresponding rate parameters are estimated by comparing predicted and measured concentrations [2]. The procedure is repeated for all combinations of model candidates and the combination with the best fit is typically selected. This approach is termed 'simultaneous identification' since all reactions and mass transfers are identified simultaneously. The advantages of this approach lie in the capability to handle complex reaction rates and in the fact that it leads to optimal parameters in the maximum-likelihood sense. However, the simultaneous approach can be computationally costly when several candidates are available for each reaction, and convergence problems can arise for poor initial guesses. Furthermore, structural mismatch in one part of the model may result in errors in all estimated parameters.As an alternative to simultaneous identification, the incremental approach decomposes the identification task into a set of sub-problems of lower complexity [3]. With the differential method [2], reaction rates are first estimated by differentiation of transient concentrations measurements. Then, each estimated rate profile is used to discriminate between several model candidates, and the candidate with the best fit is selected. This approach is termed 'rate-based incremental identification' since each reaction rate and each mass-transfer rate is dealt with individually. However, because of the bias introduced in the differentiation step, the estimated rate parameters are not statistically optimal. With the integral method [4-5], extents are first computed from measured concentrations. Subsequently, postulated rate expressions are integrated individually for each reaction, and rate parameters can be estimated by comparing predicted and computed extents. Since each extent of reaction and mass transfer can be investigated individually, this approach is termed 'extent-based incremental identification' [6, 7].The context of this work is the extent-based incremental identification of rate laws for fluid-fluid (F-F) reaction systems on the basis of process measurements. Process measurements are available for some of the species only, as it is difficult to measure the concentrations of all species due to limitations in the current state of sensor technology. Hence, it is necessary to reconstruct the unmeasured concentrations that appear in the rate laws from the available measurements. If a process model were available, this reconstruction could be done via state estimation using observers or Kalman filters. In the absence of such a reaction model, the idea is to perform instantaneous reconstruction by having as many measured quantities as there are non-redundant states. Hence the key question: How many measurements are needed to be able to reconstruct the full state? R measurements suffice in the case of a homogeneous batch reactor, whereas R + 2 pm + pl + pg + 2 measurements are needed in the case of an open gas-liquid reaction system without reaction/accumulation in the film [8], where pm is the number of mass transfers, pl the number of liquid inlets and pg the number of gas inlets, and there is one outlet in each phase.After a review of the extent-based incremental identification, this contribution will extend the results on the minimal number of measured species required for reconstructing all states to the cases of F-F reaction systems with reactions taking place in one or two bulks, without and with accumulation/reactions in the film. For the case where the number of measured species is insufficient to compute all the states, this presentation will address the possibility of using additional measurements, such as calorimetry and gas consumption, to augment the number of measured quantities [9]. These theoretical results will be illustrated through simulated examples of F-F reaction systems.[1] J. Workman et al, Anal. Chem. 83, 4557 (2011) [2] A. Bardow et al, Chem. Eng. Sci. 59, 2673 (2004) [3] M. Brendel et al, Chem. Eng. Sci. 61, 5404 (2006) [4] M. Amrhein et al, AIChE Journal 56, 2873 (2010) [5] N. Bhatt et al, Ind. Eng. Chem. Res. 49, 7704 (2010) [6] N. Bhatt et al, Ind. Eng. Chem. Res. 50, 12960 (2011) [7] N. Bhatt et al, Chem. Eng. Sci. 83, 24 (2012) [8] N. Bhatt et al, ACC, Montreal (Canada), 3496 (2012) [9] S. Srinivasan et al, Chem. Eng. J. 208, 785 (2012)

2012