Sequential Model Identification of Reaction Systems - The Missing Path between the Incremental and Simultaneous Approaches

Modeling chemical reaction systems is an important but complex task. The identified kinetic model must be able to explain all the underlying rate processes such as chemical reactions and heat and mass transfers. Traditionally, the modeling task is carried out using a simultaneous approach, which, for model prediction, requires having model candidates for all rate processes. The simultaneous approach leads to optimal parameter estimates, but it can be computationally expensive due to its combinatorial nature. The incremental approach, via either rates or extents, was introduced as an alternative to the simultaneous approach. It is characterized by the fact that each rate process can be modeled individually, that is, independently of the other rate processes. Hence, the incremental approach is computationally more attractive, however at the price of not guaranteeing bias-free estimates. This paper proposes a novel sequential approach that combines the advantages of the incremental and simultaneous approaches.

Sequential model identification of reaction systems-The missing path between the incremental and simultaneous approaches

Julien Léo Billeter, Dominique Bonvin, Sriniketh Srinivasan

Modeling chemical reaction systems is an important but complex task. The identified kinetic model must be able to explain all the underlying rate processes such as chemical reactions and heat and mass transfers. Traditionally, the modeling task is carried out using a simultaneous approach, which, for model prediction, requires having model candidates for all rate processes. The simultaneous approach leads to statistically optimal parameter estimates in the maximum-likelihood sense, but it can be computationally expensive due to its combinatorial nature. The incremental approach, via either rates or extents, was introduced as an alternative to the simultaneous approach. It is characterized by the fact that each rate process can be modeled individually, that is, independently of the other rate processes. Hence, the incremental approach is computationally more attractive, however, at the price of not guaranteeing statistically optimal parameter estimates. This article proposes a novel sequential approach that combines the advantages of the incremental and simultaneous approaches. (c) 2019 American Institute of Chemical Engineers AIChE J, 65: 1211-1221, 2019

2019

Invariant Relationships for Heterogeneous Chemical Reaction Systems in Open Reactors

Julien Léo Billeter, Dominique Bonvin, Sriniketh Srinivasan

Many chemical and biochemical industries utilize chemical reaction processes to get desired products from raw materials. Common examples include reaction systems used to manufacture pharmaceutical drugs, vaccines and other chemicals. A chemical reaction system is a complex combination of various rate processes. Apart from the chemical reactions, these systems may include (i) mass transfers between phases, and (ii) heat transfer due to heating and cooling. Also, the reactor can be operated in a continuous mode, which adds additional mass transport due to the inlet and outlet streams. The measured state variables, namely concentrations, temperature and mass, are functions of the underlying reactions, mass transfer between phases, and mass transport due to the inlet and outlet streams. Since these variables are highly coupled and contain effects of all the phenomena, the analysis of reactor performance based on these variables is highly complex. Proper understanding of the reaction system is necessary for process design, control and optimization. The analysis of the chemical reaction systems can be simplified, if a transformation can be made from the measured state variables into alternate variables (named variants) that each describe the dynamic behavior of the reactions, mass and heat transfers, inlets and outlets. The transformed states also include variables that are invariant with respect to time and remain constant during the course of the reaction. A number of applications of the reaction variants and invariants have been studied in the literature [1-7]. Srinivasan et al. [1] have discussed possible applications of reaction and flow variants/invariants for control-related tasks such as model reduction, state accessibility, state reconstruction, and feedback linearizability. Reaction invariants have been used to study the state controllability and observability of continuous stirred-tank reactors [2]. Reaction invariants have also been used to automate the formulation of mole balance equations for the non-reacting part of complex processes (mixing and splitting operations), thereby determining the degrees of freedom for process synthesis [3]. Furthermore, Waller and Makila [4] demonstrated the use of reaction invariants to control pH, assuming that equilibrium reactions are very fast. Gruner et al. [5] showed that, through the use of reaction invariants, the dynamic of reaction-separation processes with fast (equilibrium) reactions resembles the dynamic of corresponding non-reactive systems in a reduced set of transformed variables. Aggarwal et al. [6] considered multi-phase reactors operating at thermodynamic equilibrium and were able to use the concept of reaction invariants, which they called invariant inventories, to reduce the order of dynamic models and to design control strategies accordingly. Recently, Scott and Barton [7] have used reaction invariant relationships to compute bounds for kinetic models of a chemical reaction system operated under batch conditions. While the applications of reaction variants and invariants are clear, the challenge still remains in finding these relationships in the presence of different physical phenomena. Asbjornsen and co-workers [8] introduced a methodology for computing reaction variants and invariants and used it for reactor modeling and control. However, for open reactors, the reaction variants proposed are not fully representative of reactions since they are also affected by the inlet and outlet flows. Friedly [9] proposed to compute the variants of equivalent batch reactions, associating the remainder to transport processes, and used them to describe the dynamics of flow through porous media accompanied by chemical reactions. For open homogeneous reaction systems, Srinivasan et al. [1] developed a nonlinear transformation of the numbers of moles to reaction variants, flow variants, and reaction and flow invariants, thereby separating the effects of reactions and flows. Later, the same authors refined that transformation to make it linear [10] (at the price of losing the one-to-one property) and therefore more easily applicable. They also showed that, for a reactor with an outlet flow, the concept of vessel extent is most useful, as it represents the amount of material associated with a given process (reaction, exchange) that is still in the vessel. Bhatt et al. [11] extended that concept to heterogeneous G-L reaction systems for the case of no reaction and no accumulation in the film, the result being decoupled vessel extents of reaction, mass transfer, inlet and outlet, as well as true invariants, i.e. identically equal to zero. The use of this linear transformation for the task of kinetic identification has been clearly demonstrated [12] and this transformation has been extended to typical measured signals (calorimetry and spectroscopy) [13, 14]. Srinivasan et al. [15] have recently introduced a new transformation that gives the same variants and invariants as the one proposed in [10]. This new transformation, which is conceptually and computationally much simpler, also gives explicit relationships for reaction invariants in terms of the measured state variables. The invariant relationships from the transformation can now directly be used for process monitoring, control and optimization. This transformation also allows computing the reaction invariants in the presence of additional measurements (e.g. flow rates) and can be extended to heterogeneous systems. This contribution will first extend the computation of reaction invariants of homogeneous reaction systems when additional measurements (flow rates) are available. The procedure for writing the invariant relationships will then be adapted to F–F reaction systems with (i) reactions in the bulk of both phases, (ii) unsteady–state mass transfer between phases, and (iii) mass transport due to inlet and outlet streams in both phases. The procedure for writing the reaction invariant relationships will be demonstrated with suitable examples. [1] Srinivasan et al., AIChE J. 44(8), 1858-1867, 1998 [2] Fjeld et al., Chem. Eng. Sci. 29, 1917-1926, 1974 [3] Gadewar et al., Ind. Eng. Chem. Res. 41, 3771-3783, 2002 [4] Waller et al., Ind. Eng. Chem. Process Des. Dev. 20, 1-11, 1981 [5] Gruner et al., AIChE J. 52, 1010-1026, 2006 [6] Aggarwal et al., J. Process Contr. 21, 1390-1406, 2011 [7] Scott et al., Comp. Chem. Eng. 34(5), 717-731, 2010 [8] Asbjornsen et al., Chem. Eng. Sci. 27, 709-717, 1972 [9] Friedly, AIChE J. 37(5), 687-693, 1991 [10] Amrhein et al., AIChE J. 56, 2873, 2010 [11] Bhatt et al., Ind. Eng. Chem. Res. 49, 7704-7717, 2010 [12] Bhatt et al., Chem. Eng. Sci. 83, 24-38, 2012 [13] Srinivasan et al., Chem. Eng. J. 207-208, 785-793, 2012 [14] Billeter et al., Anal. Chim. Acta 767, 21-34, 2013 [15] Srinivasan et al., 1st IFAC Workshop on Thermodynamic Foundations of Mathematical Systems Theory (TFMST), Lyon (France), 2013

2014

Extent-based incremental identification of reaction systems – Minimal number of measurements for full state reconstruction

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)

2012

Invariant Relationships for Heterogeneous Chemical Reaction Systems in Open Reactors

Julien Léo Billeter, Dominique Bonvin, Sriniketh Srinivasan

2014

Extent-based incremental identification of reaction systems – Minimal number of measurements for full state reconstruction

Julien Léo Billeter, Dominique Bonvin, Sriniketh Srinivasan

2012