a b s t r a c tModels of chemical reaction systems can be quite complex as they typically include information regarding the reactions, the inlet and outlet flows, the transfer of species between phases and the transfer of heat. This paper builds on the concept of reaction variants/invariants and proposes a linear transformation that allows viewing a complex nonlinear chemical reaction system via decoupled dynamic variables, each one associated with a particular phenomenon such as a single chemical reaction, a specific mass transfer or heat transfer. Three aspects are discussed, namely, (i) the decoupling of reactions and transport phenomena in open non-isothermal both homogeneous and heterogeneous reactors, (ii) the decoupling of spatially distributed reaction systems such as tubular reactors, and (iii) the potential use of the decoupling transformation for the analysis of complex reaction systems, in particular in the absence of a kinetic model.

A novel method is presented for the rigorous propagation of uncertainties in initial concentrations and in dosing rates into the errors in the rate constants fitted by multivariate kinetic hard-modelling of spectroscopic data using the Newton-Gauss-Levenberg/Marquardt optimisation algorithm. The method was successfully validated by Monte-Carlo sampling. The impact of the uncertainties in initial concentrations and in the dosing rate was quantified for simulated spectroscopic data based on a second and a formal third order rate law under batch and semi-batch conditions respectively. An important consequence of this study regarding optimum experimental design is the fact that the propagated error in a second order rate constant is minimal under exact stoichiometric conditions or when the reactant with the lowest associated uncertainty in its initial concentration is in a reasonable excess (pseudo first order conditions). As an experimental example, the reaction of benzophenone with phenylhydrazine in THF was investigated repeatedly (17 individual experiments) by UV-vis and mid-IR spectroscopy under the same semi-batch conditions, dosing the catalyst acetic acid. For all experiments and spectroscopic signals, reproducible formal third order rate constants were determined. Applying the proposed method of error propagation to any single experiment, it was possible to predict 80% (UV-vis) and 40% (mid-IR) of the observed standard deviation in the rate constants obtained from all experiments. The largest contribution to this predicted error in the rate constant could be assigned to the dosing rate. The proposed method of error propagation is flexible and can straightforwardly be extended to propagate other possible sources of error.

h i g h l i g h t s" Extent-based incremental identification models extents computed from concentrations. " Calorimetry is related to extents of reaction and mass transfer via enthalpies. " Extents are computed by augmenting rank-deficient concentrations with calorimetry. " Extents computed from full-rank conc. allow estimating enthalpies from calorimetry. " These concepts are illustrated via homogeneous and gas-liquid reaction systems. a r t i c l e i n f oArticle history: Available online 21 July 2012 Keywords:Reaction kinetics Mass-transfer rates Extents of reaction Extents of mass transfer Incremental identification Calorimetry a b s t r a c t Extent-based incremental identification uses the concept of extents and the integral method of parameter estimation to identify reaction kinetics from concentration measurements. The approach is rather general and can be applied to open both homogeneous and gas-liquid reaction systems. This study proposes to incorporate calorimetric measurements into the extent-based identification approach for two main purposes: (i) to be able to compute the extents in certain cases when only a subset of the concentrations is measured and (ii) to estimate the enthalpies when all concentrations are measured. The two approaches are illustrated via the simulation of a homogeneous and a gas-liquid reaction system, respectively.

Hard-modelling Concentration matrix Rank deficiency and augmentation A priori information and experimental designA novel method is presented for the systematic identification of the minimum requirements regarding mathematical pre-treatment, a priori information, or experimental design, in order to allow optimising rate constants and pure component spectra associated with a kinetic model via multivariate kinetic hardmodelling of spectroscopic data. Rank deficiencies in the kinetic concentration matrix represent a major problem for the calibration free method developed by Maeder and Zuberbühler, as its pseudo-inverse, required for the optimisation process, is not defined. In this contribution, the underlying linear dependencies in the concentration profiles are systematically elucidated and appropriate strategies are discussed in order to break them. Also, conditions are predicted for which full spectral resolution can be expected. The method is based on the kernel of a time invariant augmented matrix covering potential rank deficiency due to stoichiometry and rate laws, also relevant for the concentration matrix. Compared to employing the full concentration matrix, this augmented matrix does not require a numerical integration of the differential equations describing the kinetic model and thus can easily be set up. The kernel can be calculated numerically by Singular Value Decomposition (SVD) or determined in a symbolical way, the latter allowing the detection of particular stoichiometric conditions leading to spectral resolution of species. The capabilities of the method are demonstrated analysing three kinetic mechanisms of increasing complexity covering consecutive and parallel reactions.

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The identification of kinetic models for multiphase reaction systems is complex due to the simultaneous effect of chemical reactions and mass transfers. The extentbased incremental approach simplifies the modeling task by transforming the reaction system into variant states called vessel extents, one for each rate process. This transformation is carried out from the measured numbers of moles (or concentrations) and requires as many measured species as there are rate processes. Then, each vessel extent can be modeled individually, that is, independently of the other dynamic effects. This paper presents a modified version of the extent-based incremental approach that can be used to identify multiphase reaction systems in the presence of instantaneous equilibria. Different routes are possible depending on the number and type of measured species. The approach is illustrated via the simulated example of the oxidation of benzyl alcohol by hypochlorite in a batch reactor.

Slurries are often used in chemical and pharmaceutical manufacturing processes but present challenging online measurement and monitoring problems. In this paper, a novel multivariate kinetic modeling application is described that provides calibration-free estimates of timeresolved profiles of the solid and dissolved fractions of a substance in a model slurry system. The kinetic model of this system achieved data fusion of time-resolved spectroscopic measurements from two different kinds of fiber-optic probes. Attenuated total reflectance UV−vis (ATR UV−vis) and diffuse reflectance near-infrared (NIR) spectra were measured simultaneously in a small-scale semibatch reactor. A simplified comprehensive kinetic model was then fitted to the time-resolved spectroscopic data to determine the kinetics of crystallization and the kinetics of dissolution for online monitoring and quality control purposes. The parameters estimated in the model included dissolution and crystal growth rate constants, as well as the dissolution rate order. The model accurately estimated the degree of supersaturation as a function of time during conditions when crystallization took place and accurately estimated the degree of undersaturation during conditions when dissolution took place. S ignificant progress in the area of multivariate batch process monitoring, modeling, and control has been made over the last 2 decades; 1 however, strategies for monitoring and modeling of slurries have not been widely reported, despite the fact that slurries are often used in chemical and pharmaceutical manufacturing processes. Many of the early developments in process analysis can be attributed to groundbreaking work of Nomikos and MacGregor 2,3 and Wold and co-workers.4−6 These efforts were largely focused on the use of principal component analysis (PCA) and partial least-squares (PLS) to develop multivariate statistical process control (MSPC) models for characterization of process operating conditions and thereafter the definition of normal operating conditions for the production of batches fulfilling the desired specifications.1 These models were then used to monitor future batches, product quality, and yield, as well as detect faults and diagnose process deviations.An alternative to this approach called multivariate kinetic modeling has also seen significant development over the last 2 decades. In these approaches, first-principles physical models are fitted directly to multivariate spectroscopic measurements where, typically, the adjustable model parameters are rate constants.7−10 Recently, kinetic model fitting methods were extended to achieve fusion of calorimetric measurements of univariate nature with multivariate spectroscopic measurements, 11−14 extended to incorporate chemical equilibria, 15,16 used for estimation of additional parameters such as activation energies and reaction enthalpies, 17,18 and for fitting of extents of reaction in gas−liquid systems. 19 These modeling approaches offer some advantages and some drawbacks compared to...

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