Resolving the inverse problem in pulse response analysis of TAP reactors

Abstract

Pulse experiments in the temporal analysis of products (TAP) reactor are one of the most important methods for studying transient kinetics of gas-solid catalytic reactions. The Y-procedure (Yablonsky et al., Chem. Eng. Sci. 62, 6754, 2007) is a model-free analysis framework for inferring the relationship between the reaction-rate $R$ and the reactant concentration $C$ from measurements of the outlet flux of gas. While elegant in conception, its application is hindered by the amplification of measurement noise that results from having to backtrack diffusive transport from the outlet to the reaction zone. Here, we explicitly recognize the inverse problem inherent in the Y-procedure and treat it using well-developed tools from the field of inverse problems. While previous implementations of the Y-procedure used Fourier-based filtering, we do not pre-process the measurements with an ad hoc noise-filter. Instead, we use a basis of localized square pulses to formulate a discrete inverse problem, whose regularized solution is obtained via the truncated singular value decomposition (TSVD) method. This method requires one to select a cutoff mode number; while we show how the choice of this regularization parameter can be guided by a Picard plot, we also develop an objective selection strategy for state defining experiments, for which $R(C)$ is a single-valued function. We apply our proposed inverse-problem approach to synthetic data corresponding to linear and nonlinear reactions and compare the results with the Fourier-filtration method. The former produces better reconstructions of the $R$ vs $C$ relationship, especially for nonlinear reactions. Our work facilitates the automation of pulse response analyses and enables the application of other discrete inverse-problem techniques, such as Tikhonov regularization or machine-learning methods.