Brian CritchfieldUchenna Paul

Prof. Calvin BartholomewProf. 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.

2

2+

2

m n

20.5 0.52

act H COC

H1 CO

( / )exp

1 ....

E RTA P Pr

K P K P

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 DesignCan 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, yi = f(xi,) where xi 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 Jacobianthe Jacobian of the rate function rFT is:

where fN,p is the set of partial derivatives of rFT with respect to the pth parameter at the Nth set of experimental conditions; the N+1 set is the new experimental conditions. For example, f1,1 = ∂rFT /∂A where A = the preexponential factor

1,1 1,2 1,

2,1 2,2 2,

1,1 1,2 1,

p

p

N N N p

f f f

f f fF

f f f

Sequential Design using DOD

Select conditional D-Optimal design

Obtain preliminary rate functions from literature

Define D-Optimal criteria

Determine D-Optimal experimental design conditional upon experiments completed to date

Response-surface

linear?

Perform experiment based on selected

design

Experimental data

Non-linear least squares fit of parameters

Initial points (Estimates of parameters)

Fitted parameters, std. dev., confidence intervals

Estimate refinement (optimum value far from data input)?

Conclude Optimal Settings

Yes (linear)

No

Yes

No (nonlinear)

Determine range of independent

variables

Select fractional factorial set of runs

Response surface (i.e., value of the determinant D as a function of PH2 and PCO indices) for D-optimal design of rate expression for C2+

hydrocarbons

Sequential Design Summary

1. Obtain initial estimates of parameters

2. Determine process condition that maximize D, i.e., minimize | FTF|-1/2

3. Run experiment at calculated optimal conditions

4. Nonlinear regression to estimate parameters

5. Repeat until |FTF|-1/2 reaches asymptotic value

Thus, statistical methods provide a map of the experiments, while

optimization serves as a compass.

Overall Research Approach

MicrokineticModel

Adsorption/Desorption TPD/TPH

Heats, Coverages

DFTElectronic structure of stablespecies, intermediates and

transition states

Detailed KineticsActivity, Selectivity, Stability

XPS, XRD, MössbauerAlloy formation, oxidation

states, surface composition

IRSurface species

MicroscopySurface morphologyand 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/Al2O3 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.

CO + S CO–S (1)

H2 + 2S 2 H–S (2)

CO–S + S C–S + O–S (3)

C–S + H–S CH–S + S RDS (4)

CH–S + H–S CH2–S + S (5)

O–S + H - S OH–S + S (6)

2 OH–S H2O + H–S + S RDS (7)

(Application of DOD to FTS on Fe)

2. Derive LH rate expression from reasonable mechanism.

3. Choose independent variables: temperature, PCO, and PH2

and model parameters: A, Eact, Aads, and Hads.

2

2+

2

2/3 5/ 6CO

22/3 1/ 3CO

HC

H1

k P Pr

K P P

(Application of DOD to FTS on Fe)4. Conduct scoping runs to obtain preliminary values of model

parameters.

2.5

3.5

4.5

5.5

6.5

7.5

8.5

0 50 100 150 200 250

Run Length, h

-rC

O, m

ol/g

-min

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

Model, mol/g-min

Ex

pe

rie

mn

tal

Ra

te,

mo

l/g

-min

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., PCO, and PH2 at a specified T) for the next set of experiments.

PCO

PH2

Steep gradient and maximum for D (snow-capped peak) is observed around PCO = 0.75 and PH2 = 10. Our next experiment should be in that region.

Run 5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

4 6 8 10 12

Run

|FTF

|-1/2

, x 1

0-4

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

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

4 6 8 10 12

Run

k, a

tm1.

5-m

ol/g

-min

x 10

5

0.150.17

0.190.210.23

0.250.270.290.31

0.330.35

K, /a

tm

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

k

K

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

2 2 3 3 4 4 5 5

Model, mol/g-min

Exp

erie

mn

tal

Rat

e, m

ol/

g-m

in

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

13 runs

8 runs

7 runs

5 runs

0

1

2

3

4

5

6

-0.2 0 0.2 0.4 0.6 0.8 1

K, /atm 0.222

k, a

tm0.

566-m

ol/g

-min

x 10

5

13 runs

8 runs

7 runs

5 runs

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 Atm1.5-mol/g-min

Eact

kJ/mol

Aads

atmx 103

Hads

kJ/mol

Parameter Estimate 1205 77.0 4.61 -18.4

Lower 95% Confidence Level

1137 76.8 4.36 -18.6

Upper 95% Confidence Level

1272 77.3 4.89 -18.2

Variation in parameters less than 5-10%

Conclusions A stable, well-dispersed 15% FePt/Al2O3-La2O3 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 C2+ production rate data well over a wide range of T and partial pressures of CO and H2.

Acknowledgments

Collaboration with Professors James Dumesic and Manos Mavrikakis of U. Wisconsin

Funding from DOE/NETL

Brian Critchfield and Uchenna Paul

Professor James A. DumesicACS Somorjai Award Recipient

Friend, colleague, and collaborator for 34 years

Pioneer and leader in catalysis research

Bright, whimsical, youthful, creative, and modest

Congratulations!