(284e) 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.