Bayesian Framework for Building Kinetic Models of Catalytic Systems Shuo-Huan Hsu, Stephen D. Stamatis, James M. Caruthers, W. Nicholas Delgass,* and Venkat Venkatasubramanian Center for Catalyst Design, School of Chemical Engineering, Purdue UniVersity, West Lafayette, Indiana 47907 Gary E. Blau, Mike Lasinski, and Seza Orcun E-enterprise Center, DiscoVery Park, Purdue UniVersity, West Lafayette, Indiana 47907 Recent advances in statistical procedures, coupled with the availability of high performance computational resources and the large mass of data generated from high throughput screening, have enabled a new paradigm for building mathematical models of the kinetic behavior of catalytic reactions. A Bayesian approach is used to formulate the model building problem, estimate model parameters by Monte Carlo based methods, discriminate rival models, and design new experiments to improve the discrimination and fidelity of the parameter estimates. The methodology is illustrated with a typical, model building problem involving three proposed Langmuir-Hinshelwood rate expressions. The Bayesian approach gives improved discrimination of the three models and higher quality model parameters for the best model selected as compared to the traditional methods that employ linearized statistical tools. This paper describes the methodology and its capabilities in sufficient detail to allow kinetic model builders to evaluate and implement its improved model discrimination and parameter estimation features. 1. Introduction A validated kinetic model is a critical and well-recognized tool for understanding the fundamental behavior of catalytic systems. Caruthers et al. 1 and others 2-6 have suggested that the fundamental understanding captured by the model can also be used to guide catalyst design, where a microkinetic 7 analysis is a key component of any fundamental model that intends to predict catalytic performance. While the building of such kinetic models has seen many advances, 8-40 the complexity of real reaction systems can still overwhelm the capabilities of even the most recent existing optimization-based model building procedures. The difficulty in discriminating rival models and in determining kinetic parameters even from designed experi- mental data is exacerbated by kinetic complexity as well as the fact that all experimental data contain error. Most approaches assume that the proposed model is true and that the data has a certain error structure, e.g., constant error or constant percentage error over the entire experimental range. Linear or nonlinear regression techniques are used to generate point estimates of the parameters. The uncertainty of the parameters is subse- quently computed under the assumption that the model is linear in the neighborhood of these point estimates. There are several limitations with this approach. First, the model may be wrong and thus parameter estimates are corrupted by model bias. Even when the model is adequate, linearization of nonlinear models can lead to spurious confidence and joint confidence regions. 41 An additional complication is the potential for generating multiple local optimal parameter sets depending on the starting “guesses” of the parameters supplied to the regression algorithm. The bottom line is that the existing point estimation methods frequently give the wrong parameters for the wrong model. In addition, they do not take advantage of the considerable a priori knowledge about the reaction system to design the experimental program. In contrast to traditional point estimate methods, Bayesian approaches make full use of the prior knowledge of the experimenter, do not require linearization of nonlinear models, and naturally develop probability distributions of the parameter estimates that are a more informative description of parameter values than point estimates. When combined with effective modeling of experimental error, Bayesian methods are ideally suited to kinetic modeling. The value of Bayesian approaches has been known for some time, 11,12,14,16,18,22,42,43 but they require significant computational resources. However, recent advances in computational power have made the implementation viable. 41 The purpose of this paper is to introduce the application of Bayesian methods for the development of microkinetic models. The emphasis is on identification of the most appropriate description of the reaction chemistry, i.e. sequence of elementary steps and on obtaining the best quantification for the associated kinetic parameters. While this level of detail may not be essential for some levels of reactor design, it is critical for catalyst design, where the objective of the research is to connect the various kinetic parameters in a microkinetic model with the molecular structure of the catalyst. In order to illustrate the principles and capabilities of the Bayesian approach and its differences from currently practiced point estimate approaches, we will analyze simulated sets of experimental data generated from known reaction equations. In an attempt to separate the capabilities of the method from the mathematical details of its execution, we will first present the modeling framework, introduce the models to be discriminated, and summarize the results. Then, when the advantages of the Bayesian approach are clear, we will describe the details of its utility. Its ability to treat nonlinearity without approximation, its inherent recognition of error, the effect it has on the probability distribution of kinetic parameters are the reasons it can differentiate models and improve confidence intervals for the parameters. The price for these advantages is a higher, but manageable, computational load. Our intent is to make the computational requirements clear, but to emphasize the benefits in analysis of catalytic kinetics that are the return on the investment in the mathematics. * To whom correspondence should be addressed. E-mail: delgass@ ecn.purdue.edu. Ind. Eng. Chem. Res. 2009, 48, 4768–4790 4768 10.1021/ie801651y CCC: $40.75  2009 American Chemical Society Published on Web 04/23/2009