Reaction and flow variants/invariants for the analysis of chemical reaction data
This dissertation is concerned with the development of a methodology and appropriate tools for the investigation of chemical reaction systems using measured data. More specifically, the determination of reaction stoichiometry and kinetics from concentration or, preferably, spectral measurements is considered. The main contribution of this work is the derivation of a nonlinear transformation of the dynamic model that enables the separation of the evolution of the states into three parts: (i) the reaction-variant part (related to the reactions), (ii) the reaction-invariant and flow-variant part (related to the inlet and outlet streams), and (iii) the reaction- and flow-invariant part (related to the initial conditions). This transformation is very helpful in the analysis of concentration and spectral data. Dynamic model First-principles models of reaction systems are gaining importance in chemical and biotechnological production. They can considerably reduce process development costs and be used for simulation, model-based monitoring, control, and optimization, thus leading to improved product quality, productivity, and process safety. These models include information regarding both the chemical reactions and the operational mode of the reaction system. For the analysis of these models, it is important to distinguish between the states that depend on the reactions and those which do not. The concept of reaction invariants is extended to include the flow invariants of reaction systems with inlet and outlet streams. A nonlinear transformation of the first-principles dynamic model to normal form is proposed. Model reduction, state accessibility, and feedback linearizability are analyzed in the light of this transformation. Concentration data Concentration data collected from reaction systems are highly structured, a result of the underlying reactions and the presence of material exchange terms. It is shown that concentration data can be analyzed in the framework of the three-level decomposition provided by the transformation to normal form. The resulting factorization, termed the factorization of concentration data, enables (i) the separation of the reaction and flow variants/invariants and (ii) the segregation of the dynamics (extents of reaction, integral of flows) from the static information (stoichiometry, initial and inlet concentrations). Using the factorization of concentration data, it is possible to isolate the reaction variant part by subtracting the reaction-invariant part from measured concentrations. The reaction-variant part is often unknown, since it depends on the kinetic description (typically the main difficulty in modeling chemical reaction systems). In contrast, the reaction-invariant part is usually known or measured. It is shown that, in cases where the reaction variants can be computed from the concentrations of a few measured species, the concentrations of the remaining species can be reconstructed using the known reaction-invariant part. Target factor analysis has been used successfully with concentration data to determine, without knowledge of reaction kinetics, the number of reactions and the corresponding stoichiometries. It is shown that, when only the reaction-variant part of the data is considered, existing target factor-analytical techniques can be readily applied. However, if target factor analysis needs to be applied directly to measured concentrations, knowledge of reaction-invariant relationships is required to specify necessary and sufficient conditions for the acceptance of stoichiometric targets. Spectral data In current practice, concentration measurements during the course of a reaction are generally not available, neither on-line nor off-line. Owing to new measurement technologies, spectral measurements are now available in both the laboratory and production. Various spectral instruments enable non-destructive indirect concentration measurement of most of the species in-situ/on-line during the course of a reaction. Measurements are available at high sampling rates and delay-free at low costs. Furthermore, in most cases, the spectral data are linear, i.e., the mixture spectrum is a linear combination of the pure-component spectra weighted by the concentrations. It is shown that the three-level interpretation provided by the transformation to normal form is applicable to spectral data from reacting mixtures. Similarly to traditional wet-chemical analysis methods, a calibration model must also be estimated that provides concentration estimates from spectral measurements. All calibration methods require that a new spectrum lies in the space spanned by the calibration spectral data (space-inclusion condition). To verify this space-inclusion condition, it is proposed to build a calibration model for the reaction-variant part only. Once the reaction variants are predicted from a new spectrum, the (known) reaction invariants can be added to reconstruct the concentrations. Concentration measurements for some species of interest are often not available due to difficulties/costs in sampling, sample preparation, and development of analytical techniques. Thus, traditional calibration of spectral measurements for the purpose of concentration estimation is not possible. Instead, explicit or implicit knowledge about the kinetic structure will be used (prior knowledge about the reaction-variant part), thus enabling the formulation of factor-analytical methods as a calibration problem. For pedagogical reasons, the results are developed for isothermal, constant-density reaction systems with inlet and outlet streams. The results are then extended to various scenarios such as reaction systems with varying density and temperature. Furthermore, factorizations of concentration data are presented that include temperature or calorimetric measurements. Several special cases are considered, encompassing continuous stirred-tank reaction systems, semibatch and batch reaction systems, systems with reactions in quasi-equilibrium conditions, and non-reacting mixtures with closure.