Model-Based Design of Experiments for Temporal Analysis of Products (TAP): A Simulated Case Study in Oxidative Propane Dehydrogenation

Temporal analysis of products (TAP) reactors enable experiments that probe numerous kinetic processes within a single set of experimental data through variations in pulse intensity, delay, or temperature. Selecting additional TAP experiments often involves an arbitrary selection of reaction conditions or the use of chemical intuition. To make experiment selection in TAP more robust, we explore the efficacy of model-based design of experiments (MBDoE) for precision in TAP reactor kinetic modeling. We successfully applied this approach to a case study of synthetic oxidative propane dehydrogenation (OPDH) that involves pulses of propane and oxygen. We found that experiments identified as optimal through the MBDoE for precision generally reduce parameter uncertainties to a higher degree than alternative experiments. The performance of MBDoE for model divergence was also explored for OPDH, with the relevant active sites (catalyst structure) being unknown. An experiment that maximized the divergence between the three proposed mechanisms was identified and provided evidence that improved the mechanism discrimination. However, reoptimization of kinetic parameters eliminated the ability to discriminate between models. The findings yield insight into the prospects and limitations of MBDoE for TAP and transient kinetic experiments.


Model parameter sensitivities for MBDoE for precision
To narrow the parameters explored during the MBDoE for precision analysis, we looked at the sensitivity of all parameters to determine which are identifiable.Not all parameters will be experimentally observable due to their extreme values (i.e. if the energy is too high or too low, varying the energy in the model will not alter reaction rates).Using the initial parameter guesses and the seven-parameter fit, we show all the parameter sensitives in Table S1.The parameters excluded from the analysis have sensitives at and below 1e-3, whereas the included parameters had higher values.For this reason, we performed the analysis with these seven parameters.
Table S1: The sensitives of the parameters found with the initial parameter guesses and the final fit of the seven parameter optimization (the local minimum) .

Correlation between predicted D and actual D criteria
We compared different methods for distilling the Fisher information matrix and calculated covariance matrices as the A (trace), D (determinant), and E (eigenvalue) criteria.Although each is used in the literature, we observed the strongest correlation for the D-optimal criteria and therefore use it for MBDoE.

Initial conditions used during the precision refinement process
The MBDoE approach for precision used an initial, arbitrary experiment (1 st experiment in Table S2), followed by three additional experiments.The 2 nd experiment is the first experiment predicted using MBDoE, while the 3 rd experiment is the second experiment predicted using MBDoE.The Alt. experiment involves the adjusted approach for designing the experiments involving only the most uncertain parameter.

MBDoE experiment selection correlation
Although we only considered mechanism two as the black box experiment in the primary text, we also explored the use of mechanism 1 and 3 as the black box experiment (shown in the top and bottom plots of Figure S6).These experiments showed clear divergence between each other, so we focused on mechanism 2, which has non-trivial divergence for some experiments (e.g.there exist experiments for which it is not possible to clearly distinguish mechanism 1 and 2 based on BIC).

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Figure S2 :
Figure S2:The predicted A, D, and E criteria for designing experiments were compared to their actual A, D, and E criteria for each of the possible experiments.The D criteria, which stands for the determinant, is the only criteria that had a strong correlation and, for that reason, is the only criteria thoroughly explored in this paper.

Figure S3 :Figure S4 :Figure S5 :
Figure S3: The predicted improvement to kinetic understanding using D-optimality for additional parameters fitted following the first experiment.

Figure S6 :
Figure S6: Divergence plots for the three oxidative propane mechanisms provided in the primary text.The top, middle, and bottom plots represent divergence when the experimental data is generated using mechanism 1, 2, and 3, respectively.

Table S2 :
The experimental conditions used to constrain the parameters.Experiment 1 was selected arbitrarily, while experiments 1 and 2 were selected through MBDoE for precision.