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Publication# Maximum-likelihood estimation of kinetic parameters via the extent-based incremental approach

Julien Léo Billeter, Dominique Bonvin, Diogo Filipe Mateus Rodrigues

*PERGAMON-ELSEVIER SCIENCE LTD, *2019

Journal paper

Journal paper

Abstract

For lumped homogeneous reaction systems, this paper presents a kinetic model identification scheme that provides maximum-likelihood parameter estimates and guarantees convergence to global optimality. The use of the extent-based incremental approach allows one to (i) identify each reaction individually, and (ii) reduce the number of parameters to be identified via optimization to the ones that appear non-linearly in the investigated rate law. The approach results in maximum-likelihood parameter estimation if the experimental extents are uncorrelated and the rate estimates used to compute the modeled extents are unbiased. Furthermore, the identification problem can be rearranged via Taylor series expansion as a polynomial optimization problem. This optimization problem is then reformulated as a convex optimization problem that can be solved efficiently to global optimality. Different aspects of the approach are demonstrated via simulated examples. (C) 2018 Elsevier Ltd. All rights reserved.

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Chemical reaction

A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the posit

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Julien 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)

2012Julien 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, Diogo Filipe Mateus Rodrigues

The identification of reaction kinetics represents the main challenge in building models for reaction systems. The identification task can be performed via either simultaneous model identification (SMI) or incremental model identification (IMI), the latter using either the differential (rate-based) or the integral (extent-based) method of parameter estimation. This contribution presents an extension of extent-based IMI that guarantees convergence to globally optimal parameters. In SMI, a rate law must be postulated for each reaction, and the model concentrations are obtained by integration of the balance equations. The procedure must be repeated for all combinations of rate candidates. This approach is computationally costly when there are several candidates for each reaction, and convergence problems may arise due to the large number of parameters. In IMI, the identification task is decomposed into several sub-problems, one for each reaction [1]. Since IMI deals with one reaction at a time, only the rate candidates for that reaction need to be compared. In addition, convergence is facilitated by the fact that only the parameters of a single reaction rate are estimated. In rate-based IMI, the parameters are estimated by fitting the simulated rates to the experimental rates obtained by differentiation of measured concentrations. In extent-based IMI, the simulated rates are integrated to yield extents, and the parameters are estimated by fitting the simulated extents to experimental extents obtained by transformation of measured concentrations [2]. The simulated rates are functions of concentrations. Hence, since each reaction is simulated individually, the simulated rates must be computed from measured concentrations. Most parameter estimation methods converge to local optimality, which may result in an incorrect model. It turns out that extent-based IMI is particularly suited to global optimization since each estimation sub-problem (i) involves only a small set of parameters, and (ii) can be rearranged as an algebraic problem, where the objective function is polynomial in the parameters with coefficients computed only once prior to optimization using a Taylor expansion. These features facilitate the task of finding a global optimum for each reaction. Instead of the classical branch-and-bound approach, this technique relies on reformulating the estimation problem as a convex optimization problem, taking advantage of the equivalence of nonnegative polynomials and conical combination of sum-of-squares polynomials on a compact set to solve the problem as a semidefinite program [3]. A simulated example of an identification problem with several local optima shows that extent-based IMI can be used to converge quickly to globally optimal parameters. References: [1] Bhatt et al., Chem. Eng. Sci., 2012, 83, p. 24 [2] Rodrigues et al., Comput. Chem. Eng., 2015, 73, p. 23 [3] Lasserre, SIAM J. Optim., 2001, 11(3), p. 796

2017