Extent-based incremental identification of reaction kinetics from spectroscopic data

Identification of kinetic models is an important step for the monitoring, control and optimization of chemical and pharmaceutical processes. Furthermore, the availability of a first-principles model often leads to substantial reduction in process development cost. Recent developments in Process Analytical Technology can help build kinetic models from large amounts of multivariate spectroscopic data [1]. Kinetic modeling from spectroscopic data typically relies on Beer’s law, A = C E, to decompose the observed data matrix A into the product of ns concentration profiles, C (nt x ns), and ns pure component spectra, E (ns x nw). In simultaneous identification approaches, the rate expressions are integrated simultaneously to predict Ĉ, E is estimated as E = Ĉ+A, while the rate parameters are determined by fitting the predicted absorbance Ĉ Ĉ+A to the measured absorbance A. As an alternative to simultaneous identification, incremental identification focuses on each reaction separately, with the main advantage that each rate can be fitted individually, thereby resulting in less correlation between the estimated rate parameters. The concentrations are estimated from absorbance measurements as C = A E+ or from a calibration C = f(A). In the rate-based (differential) approach, the reaction rates are first estimated by differentiation of concentrations, and the rate parameters are obtained by individually fitting each candidate rate expression to the corresponding estimated rate [2]. The difficulty with this approach lies in the differentiation of noisy and sparse concentration data. In order to avoid the differentiation step, an extent-based (integral) approach has been proposed [3], in which the extents of reaction Xr (nt x nr), i.e. the numbers of moles consumed or produced by each reaction, are computed from A. For this, the linear transformation S0 has been developed, which computes Xr = V C S0, where V is a diagonal matrix representing the volume at the nt time instants. When some of the species do not absorb or react, Xr can be calculated using a flow-based method [3]. The rate parameters are determined by fitting each predicted reaction extent to the corresponding extent computed from measurements. More recently, this extent-based approach has been extended to heterogeneous reaction systems [4]. This contribution presents the extent-based incremental identification of reaction kinetics based on spectroscopic data. The approach is illustrated through simulated examples. [1] Workman et al, Anal. Chem., 83, 4557 (2011). [2] Brendel et al, Chem. Eng. Sci., 61, 5404 (2006). [3] Amrhein et al, AIChE Journal, 56, 2873 (2010). [4] Bhatt et al, Ind. Eng. Chem. Res., 49, 7704 (2010).