Using PCE for Uncertainty in Kinetics Models Cheng Wang

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

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.

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

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

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

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