Optimization-Based Methods for Catalysis: Developing Improved Approaches to Microkinetic Modeling and Their Application to Methanol Synthesis

In the face of rapidly declining fossil fuel reserves and the prospect of global climate change, it is imperative that we develop renewable fuels and energy technologies.  Methanol is an important candidate as it is an industrially important chemical with significant promise as a transportation fuel. Furthermore, it has the potential to be produced from the CO₂, which would make it a carbon neutral fuel.  However, current industrial catalysts and processes are not optimized to function effectively in the absence of CO and our understanding of the reaction mechanism is incomplete. 

To improve our understanding of the reaction mechanism for methanol synthesis and other reactions we use density functional theory (DFT) and microkinetic modeling. DFT uses the principles of quantum mechanics to predict the properties of materials at the atomic and molecular levels.  Specifically, an approximation of the Schrödinger equation is solved and can be used to predict the binding sites and binding energies of atoms and molecules on catalytically active surfaces and the energy barriers to various potential elementary reaction steps on the surface.  It has provided us with unprecedented insight into the reactions on the surface of catalytic materials.  To use this information most effectively, we need to translate our new understanding of the elementary steps into predictions of macroscopic outcomes such as the rate of methanol production.  Microkinetic models provide a bridge between the elementary steps occurring at the molecular scale and production rates at the macroscopic scale.  DFT provides an excellent starting point for parameter values for modeling.  However, production rates can be very sensitive to the binding and activation energies (BE and Ea) of the surface species and elementary steps, respectively and the DFT values may contain inaccuracies due to selection of the wrong surface to model, or surface reconstruction under the reaction conditions.  Therefore, parameter estimation is a necessary component of successful microkinetic modeling. 

However, microkinetic models are computationally expensive to evaluate and highly non-linear, rendering optimization and parameter estimation difficult.  In the past, we had formulated microkinetic models as a system of ordinary differential equations (ODEs).  This is simple to set up and can be run in readily available software such as Matlab.  However, the ODE version of the model is computationally intensive to solve and problematic for parameter estimation as the gradient is not readily available.  Moreover, the optimization does not tend to result in large changes in the parameter values, so very good initial guesses are required.  This requires iterative adjustment of the parameter values by the user and considerable physical insight to obtain good fits to the experimental data.  Overall, parameter estimation with the ODE model is expensive in both computational and human time.

We have reformulated the microkinetic model as a system of non-linear equations (NLP).  This technique requires careful formulation and setting of appropriate limits on all variables to produce physically relevant surface coverages and gas phase partial pressures.  However, done correctly, it produces a microkinetic model that is much less computationally intensive to solve and for which an explicit formula for the gradient is available, which allows much more effective use of gradient-based optimization techniques.  The resulting NLP model requires 3 orders of magnitude less computational time for optimization.  Furthermore, the problem converges from a larger neighborhood of the local minimum.  Together these features allow the search for a good fit to the data to be substantially automated, removing the need for someone to iteratively adjust the initial parameter values by hand and dramatically reducing the human time required to obtain a comparably good fit to the data.