(369e) A Stochastic Approach to Catalyst Optimization: Robust Design of Experiments and Catalysts

The use of detailed quantitative multiscale microkinetic models for the prediction of the behavior of heterogeneous catalysts, and the model-based optimal design of the catalysts themselves, have received much attention in the recent past. This is especially true of catalytic systems for energy production, such as hydrogen generation. However, there are serious challenges in the optimization of these systems, which must be overcome before these methods can replace the intuitive or trial-and-error approaches currently used.

A crucial step in building accurate quantitative models of catalyst behavior is the optimal design of experiments for parameter estimation. This involves identifying optimal input sequences (temperature, flow, pressure, inlet composition, catalyst dispersion, etc.) to conduct experiments such that the kinetic parameters can be identified with small confidence intervals. Since these are highly nonlinear systems with multiple local minima, an additional question to be answered is the optimal design of experiments for discrimination between competing models.

Once quantitative models have been built for specific catalysts, the optimal catalyst for the particular chemistry being considered can be found by solving the inverse problem, that of finding the optimal parameter values (in this case, binding energies of species to the catalyst) such that quantities such as conversion and selectivity are maximized.

In both of these optimization problems, an unappreciated fact is the effect of the uncertainty in the parameter values may lead to the deterministically identified solution not being the true optimal solution. In the design of experiments, uncertainty in the parameter values leads to uncertainty in the sensitivity calculations used in D optimal design. In the optimal design of catalysts, kinetic parameters have a degree of uncertainty due to their variation between different catalyst materials, and this, too, leads to a stochastic optimization problem. Another consideration is that the models used for experiment design and catalyst optimization need to satisfy kinetic and thermodynamic constraints to ensure that the choices of the kinetic parameters do not violate conditions on the overall entropy and enthalpy of the process.

Given that both of these problems involve optimization under uncertainty, we employ stochastic optimization techniques to solve them. Stochastic optimization approaches optimize the expectation of the objective function of the decisions and the uncertainties, and thus come up with a decision that will perform best on average. These techniques take advantage of the fact that probability distributions governing the uncertain parameters are known or can be assumed reasonably. The solution from stochastic optimization provides more realistic predictions of performance than that of the certainty equivalence (CE) approach, where uncertain parameters are replaced by their means. The design of experiments and selection of the optimal catalyst involve complex models solved using numerical integration, given the initial conditions and parameters in general. Moreover, reaction systems generally exhibit nonlinearity due to the exponential term in kinetic expressions. These make it impossible to convert the stochastic problem into a deterministic counterpart using the CE approach.

In this work, computationally efficient stochastic approaches to design of experiments and optimal selection of a catalyst are proposed and demonstrated. In the proposed scheme, the expectation of the objective value is evaluated using a sample average approximation (SAA) method, due to its simplicity. However, the exponential increase in the number of samples with the number of uncertain parameters defies using all possible combinations of the uncertain parameters.

In this work, linearity analysis, together with partial least squares, is proposed to further reduce the number of random variables (uncertain parameters) given the fact that a CE solution is optimal when the objective function has only a linear dependence on the uncertain variables. In other words, the CE approach is used for uncertain parameters whose effect on the output is found to be approximately linear, and the parameters showing nonlinear effects are treated with the SAA approach. In the optimization step, a gradient-free sample-based method, particle swarm optimization (PSO), is employed to circumvent issues involving local optima and computational time associated with gradient-based approaches.

The system we use to demonstrate our approach is the catalytic decomposition of ammonia to produce hydrogen. The proposed stochastic approach is applied to the robust design of experiments (involving seven experimentally manipulated variables: pressure, temperature, residence time, catalyst surface area per unit reactor volume, and inlet gas composition of reactants and products) and the optimal design of catalysts (involving the specification of optimal binding energies of hydrogen and nitrogen on the catalyst surface).

Our approach to the robust design of experiments results in the identification of regions of optimal experiments, and subsequent parameter estimation (with tight confidence intervals) for a Ru/γ-Al2O3 catalyst. Further, our computational results for the optimal catalyst design identify the binding energies for the optimal catalyst, and show that the average conversion corresponding to the stochastic and deterministic optimal solutions are 0.23 and 0.19, respectively, which means that the value of stochastic solution (VSS) is 0.0369. The average performance based on the stochastic solution is improved by about 20% compared with that of the deterministic solution, and the optimal binding energies are different, indicating that the deterministic solution does not identify the true optimal catalyst. The proposed approach holds promise for designing the optimal catalyst for complex reaction systems with significant uncertainty and nonlinearity.