1. Using PCE for Uncertainty in Kinetics Models
Cheng Wang

2. Why We Need Uncertainty Analysis
Sensitivity analysis commonly used to treat uncertainty in kinetics models
Fully integrated with models through ODE solvers
Apply principal component analysis to gain more insight
Uncertainty analysis can provide more information
Variance analysis provides contributions from each uncertain input
Provide error bars on target parameters
PDF can provide confidence level for target parameters

3. Uncertainty Analysis Using PCE
Using Deterministic Equivalence Modeling Method developed by Tatang*
Use orthogonal polynomials for each distribution type
Use roots of polynomials to construct collocation points
Independent
Random Variable
Known Distribution
(e.g., N(m,s))
Functional
(e.g., Hermite Polynomials)
*Tatang, M.A., Ph.D. Thesis, Department of Chemical Engineering, MIT, 1995.

4. Uncertainty Analysis in CHEMKIN-PRO
Identify uncertain inputs
2. Define input PDFs
3. Select target outputs
4. Provide max order of PCE
Input NO CO
% Cat. Disp.
53%
39%
Equiv. Ratio
47%
61%
5. Run model cases
6. Analyze results

5. Using Collocation Method in Kinetics Models
Collocation method treats kinetics model as a black box
Easy to integrate with existing models
Uncertainty analysis serves as post-processor
Collocation method uses limited number of sampling points
Kinetics models can take a long time to run
Collocation points are reusable during uncertainty analysis
Use collocation points for future runs to determine errors in current PCE approximation

6. Limitations of Collocation Method
Collocation points impose restriction on model sampling
Model may not converge on collocation points
Using probability distribution to describe uncertain input/output is inadequate
Probability distribution extends to unphysical domain of input/output parameter
Lack of prior knowledge on the feasible domain of input parameters
Difficult to represent correlations and dependencies between input parameters

7. Wish List for Better PCE Methods
Kinetics models are complex and time-consuming
Treat the models as black-box
Use limited number of sampling points
Uncertainty in model input/output is complex
Describe the physical domain of input/output parameter
Represent correlations and dependencies between input parameters
Provide flexible and extensible representation of model input due to lack of prior knowledge