Incremental Identification of Reaction and Mass–Transfer Kinetics Using the Concept of Extents

This paper proposes a variation of the incremental approach to identify reaction and masstransfer kinetics (rate expressions and the corresponding rate parameters) from concentration measurements for both homogeneous and gas-liquid reaction systems. This incremental approach proceeds in two steps: (i) computation of the extents of reaction and mass transfer from concentration measurements without explicit knowledge of the reaction and mass-transfer rate expressions, and (ii) estimation of the rate parameters for each rate expression individually from the computed extents using the integral method. The novelty consists in using extents that are computed from measured concentrations. For the computation of the individual extents, two cases are considered: if the concentrations of all the liquid-phase species can be measured, a linear transformation is used; otherwise, if the concentrations of only a subset of the liquid-phase species are available, an approach that uses flowrate and possibly gas-phase concentration measurements is proposed. The incremental identification approach is illustrated ∗To whom correspondence should be addressed 1 in simulation via two reaction systems, namely the homogeneous acetoacetylation of pyrrole and the gas-liquid chlorination of butanoic acid. Introduction Dynamic models are used to analyze, monitor, control and optimize reaction systems. These models are often based on first principles and describe the evolution of the states (number of moles, temperature and volume) by means of conservation equations of differential nature and constitutive equations of algebraic nature. The models include information regarding the underlying reactions (stoichiometry and kinetics), the transfer of species between phases (mass-transfer kinetics), and the operation of the reactor (initial conditions, inlet and outlet flows, operational constraints). The identification of reaction and mass-transfer kinetics (rate expressions and the corresponding rate parameters) represents the main challenge in building first-principles models for reaction systems. The rate expressions, which are typically chosen from a set of candidates, need to be confronted to measured data. The identification task can be performed globally in one step via a simultaneous approach, or successively over several steps via an incremental approach, as discussed next. Simultaneous identification proceeds as follows. From a library of reaction pathways and rate expressions, one chooses a rate expression candidate for the reaction system and estimates the rate parameters by comparing model predictions and measured data. The approach is termed ‘simultaneous identification’ since all reaction and mass transfer kinetics are identified simultaneously. The procedure needs to be repeated for all rate expression candidates. The candidate with the best fit is usually selected. Issues like parameter and structural identifiability1,2 and experimental planning3,4 are important to guarantee parameter estimates with little correlation and narrow confidence intervals. The main advantage of simultaneous identification is that it can deal with complex reaction and mass-transfer kinetics and lead to optimal parameters in the maximum-likelihood sense.5 However, simultaneous identification can be computationally costly when several candidates are available for each rate expression. Furthermore, since the global model is fitted so as to reduce the 2 prediction error, structural mismatch in one part of the model will typically result in errors in all the rate parameters. Finally, it is often difficult to choose suitable initial parameter values, which may lead to convergence problems.6 On the other hand, incremental identification decomposes the identification task into a set of subproblems.7–10 First, the reaction stoichiometry is identified from measured concentrations. For this, each reaction can be determined individually without explicit knowledge of reaction kinetics using target factor analysis (TFA).11,12 Incremental TFA has been proposed to remove the variability associated with a reaction, once it has been accepted, before continuing target testing for the other reactions.13 The next step computes the rate profiles of reaction and mass transfer from measured data and the known stoichiometry. Finally, the rate parameters are estimated from the computed rate profiles. For each subproblem, the number of model candidates can be kept low. This approach is also termed ‘individual identification’ since each reaction and mass transfer can be dealt with individually. Regarding the identification of reaction and mass-transfer kinetics from measured data, two methods can be distinguished depending on the way data are handled, namely the differential and the integral methods.14,15 These two methods are detailed next. In the differential method, reaction rate profiles are computed through differentiation of concentration data. Furthermore, individual rate profiles can be computed upon knowledge of the stoichiometry. Then, for a given reaction, a rate expression is proposed and its parameters are estimated by fitting the simulated rate profile to the computed values. Note that the differentiation of measured concentrations is a difficult task due to noise and the sparsity of measurements.16 In the integral method, the rate expressions are integrated analytically or numerically to predict concentrations, and the unknown rate parameters are estimated by fitting these predictions to measured concentrations. The integral method is computationally intensive because of the need to integrate the rate expressions for each set of parameter values proposed by the optimization algorithm. However, in the absence of structural uncertainty and for Gaussian mea3 surement noise, the integral method leads to optimal estimates in the maximum-likelihood sense.6,17 The simultaneous and incremental approaches, which use the integral and differential methods, respectively, are illustrated in Figure 1 for homogeneous reaction systems: Path "1" indicates the simultaneous identification approach that uses the integral method, whereby the rate expressions for all reactions are integrated to predict concentrations that are fitted to measured values via a least-squares problem. Path "2" represents the incremental identification approach that uses the differential method, whereby the rate profile of the ith reaction is computed by differentiation of concentration measurements and use of information regarding the stoichiometry, the inlet composition, the volume, and the inlet and outlet flowrates. The ith rate model, which is chosen from a library of rate expressions, is fitted to the computed rate profile via a least-squares problem. Unfortunately, numerical differentiation introduces a bias in the computed rate profiles, thus leading to suboptimal parameters.6,7 Hence, as part of a final adjustment step, simultaneous identification using the model structure identified by incremental approach is often performed to obtain unbiased parameter estimates. For the sake of completeness, two special cases of generalized simultaneous and incremental approaches available in the literature are briefly mentioned next. A framework for automatic modeling of chemical/biochemical reaction systems (TAM-C/B) based on concentrations and calorimetric data has been proposed.18 TAM uses an automatic iterative procedure that imitates the human expert in modeling reaction systems. From measured data, TAM first generates a qualitative description of the dynamic behavior of the reaction system using a fuzzy interval identification method.18 Then, based on the resulting qualitative description, prior knowledge regarding the reaction stoichiometry and a rule-based library, TAM postulates possible rate expressions and fits the global model to the concentrations and calorimetric data. 4