Brian Critchfield Uchenna Paul Prof. Calvin Bartholomew Prof. Dennis Tolley Design of Kinetic Experiments for Fischer-Tropsch Synthesis on Supported Fe Catalysts Chemical Engineering and Statistics Brigham Young University Provo, Utah

Introduction Langmuir-Hinshelwood models derived from mechanisms are generally found to fit rate data well for a number of catalytic reactions, e.g., for Fischer-Tropsch synthesis: This model is nonlinear and, as a result, there is typically a high correlation between kinetic parameters.

Challenges in Collecting/Fitting Rate Data Collecting enough data to regress the model parameters can be time consuming. Without an appropriate experimental plan, parameter estimates may be poor; parameters may be highly correlated. Due to the nonlinear nature of the model, the best experimental design is not apparent.

Sequential/D-optimal Experimental Design Can Be Very Helpful

a form of response surface design – using optimization methods (Other forms include: A-Optimal, E-Optimal, G- Optimal, and V-Optimal). a proven tool for obtaining the most precise estimates of model parameters in the least number of experiments. enables selection of conditions that minimize the overall variances of the estimated parameters by spreading out design variables over available variable space. reduces the volume of the confidence region for estimated parameters. substantially reduces correlation among parameters. D-Optimal Design (DOD)

A rate function is specified, y i = f(x i,  ) where x i are the set of design parameter inputs and  is the set of kinetic coefficients. Calculus and matrix algebra are used to maximize the following determinant: where F is the Jacobian matrix and F T is the transpose of the Jacobian matrix, where ….. How Does DOD Work?

The Jacobian the Jacobian of the rate function r FT is: where f N,p is the set of partial derivatives of r FT with respect to the p th parameter at the N th set of experimental conditions; the N+1 set is the new experimental conditions. For example, f 1,1 = ∂r FT /∂A where A = the preexponential factor

Sequential Design using DOD

Response surface (i.e., value of the determinant D as a function of P H2 and P CO indices) for D-optimal design of rate expression for C 2+ hydrocarbons

Sequential Design Summary 1.Obtain initial estimates of parameters 2.Determine process condition that maximize D, i.e., minimize | F T F| -1/2 3.Run experiment at calculated optimal conditions 4.Nonlinear regression to estimate parameters 5.Repeat until |F T F| -1/2 reaches asymptotic value

Thus, statistical methods provide a map of the experiments, while optimization serves as a compass.

Overall Research Approach Microkinetic Model Adsorption/Desorption TPD/TPH Heats, Coverages DFT Electronic structure of stable species, intermediates and transition states Detailed Kinetics Activity, Selectivity, Stability XPS, XRD, Mössbauer Alloy formation, oxidation states, surface composition IR Surface species Microscopy Surface morphology and composition

Collaboration with Manos Mavrikakis and Jim Dumesic Objective: develop data for validation of microkinetic and LH models More than a dozen previous kinetic studies Most did not meet basic criteria of Ribeiro et al. (1997) for absence of heat/mass transfer effects, deactivation, etc. None used optimal statistical design of experiments. Data were fitted to power law and Eley-Rideal expressions mostly covering narrow ranges of operating conditions. Few reported TORs, thus preventing valid comparisons. Thus, much of previous work is unreliable or unusable FTS Reaction Kinetics on Fe

Derive LH and ER rate forms from a logical mechanism. Use D-optimal/sequential design to optimize experimental conditions, minimize errors in rate parameters, and minimize number of experiments Collect intrinsic rate data on a stable Fe-Pt/Al 2 O 3 catalyst in a Berty CSTR reactor over a wide range of commercially relevant conditions. Pt-promoter and La-stabilized alumina support facilitate Fe reduction and hydrothermal stability. Catalyst washcoated on monolith ensures high effectiveness, enabling operation over wide range of temperature. Use nonlinear regression methods to fit rate data to best mechanisms. Our Approach to Kinetic Study

Application of DOD to FTS on Fe 1. Select a reasonable mechanism.

(Application of DOD to FTS on Fe) 2. Derive LH rate expression from reasonable mechanism. 3.Choose independent variables: temperature, P CO, and P H 2 and model parameters: A, E act, A ads, and  H ads.

(Application of DOD to FTS on Fe) 4. Conduct scoping runs to obtain preliminary values of model parameters. Data are correlated well by the model. Catalyst is quite stable over > 150 h Run 11

(Application of DOD to FTS on Fe) 5.Set up Jacobian matrix with scoping runs and maximize determinant to obtain response surface for experimental parameters (i.e., P CO, and P H 2 at a specified T) for the next set of experiments. P CO P H2 Steep gradient and maximum for D (snow-capped peak) is observed around P CO = 0.75 and P H2 = 10. Our next experiment should be in that region. Run 5

Values of |FTF|-1/2 with respect to the number of runs at 239°C.

Values of k and K with respect to the number of runs at 239°C. k K

Experimental rates versus model predicted rates for sequentially designed experiments at 239°C.

Joint 95% likelihood confidence regions for k, and K at 239°C at different stages of the sequential design procedure

Results of 37 runs at 3 temperatures A Atm 1.5 -mol/g-min E act kJ/mol A ads atm x 10 3  H ads kJ/mol Parameter Estimate Lower 95% Confidence Level Upper 95% Confidence Level Variation in parameters less than 5-10%

Conclusions A stable, well-dispersed 15% FePt/Al 2 O 3 -La 2 O 3 wash- coated monolith catalyst in combination with a CSTR facilitates obtaining intrinsic FTS rates under commer- cially-relevant conditions. An LH rate expression based on C+H and OH + OH as RDSs provides the best fit to the data. A sequential design procedure using DOD resulted in precise parameter estimates in a minimal number of (10- 15) experiments at each of two temperatures. Three data sets at three temperatures (37 total runs) could be combined to obtain a rate law fitting C 2+ production rate data well over a wide range of T and partial pressures of CO and H 2.

Acknowledgments Collaboration with Professors James Dumesic and Manos Mavrikakis of U. Wisconsin Funding from DOE/NETL

Brian Critchfield and Uchenna Paul

Professor James A. Dumesic ACS Somorjai Award Recipient Friend, colleague, and collaborator for 34 years Pioneer and leader in catalysis research Bright, whimsical, youthful, creative, and modest Congratulations!