Impact of Kinetic Uncertainties on Accurate Prediction of NO Concentrations in Premixed Alkane-Air Flames

ABSTRACT

Accurate thermochemical mechanisms that can predict the formation of nitrogen oxides (NO${ }_x$) are important design tools for low-emissions engines. The lack of accurate direct measurements of reaction rates and the associated measurement scatter have resulted in recommended rate parameters for individual chemical reactions that have large uncertainty intervals. In an effort to quantify the impact of these parametric uncertainties on emissions predictions, forward uncertainty propagation is performed with five spectral methods. Sparse grids are identified as the optimal technique to rapidly construct accurate surrogate models. Subsequent polynomial expansions with sparse grids, performed in one-dimensional atmospheric laminar flames for only the 30 uncertain reactions that greatly affect NO formation, produce uncertainty intervals two orders of magnitude larger than nominal predictions. Primary uncertainty sources were identified with reaction pathway analyses to evaluate the contribution of individual formation routes and the uncertainties in prompt NO were found to propagate mostly from the CH chemistry. These results highlight the necessity of a comprehensive approach, using experimental measurements with uncertainty quantification and inference techniques, to reduce uncertainty and develop predictive NO${ }_x$ models.

KEYWORDS:

• Uncertainty quantification
• NO modelling
• reaction pathway analysis
• predictive modelling

Nomenclature

$I$    Integration operator

$K$   Number of terms in the expansion

MC  Monte Carlo

$P_i$  Orthogonal polynomial basis of order i

PCE  Polynomial chaos expansion

PDF  Probability density function

$Q$  Quadrature operator

$R$  Response, quantity of interest

RPA  Reaction pathway analysis

SD San Diego mechanism

$S_{\mathrm{u}}$  Reference flame speed

$T$  Temperature

$X_i$  Mole fraction of species $i$

$f$  Reaction rate uncertainty factor

$[i]$  Concentration of species $i$

$k$  Specific reaction rate constant

$\mathrm{\ell }$  Level of accuracy of the quadrature rule

$n$  Number of variables in the expansion

$p$  Order of the polynomial expansion

$w^r$  Weight of the quadrature point $r$

$\mathbf{x}$ and $x_i$ Variable studied in the spectral expansion; vector and scalar elements

$\alpha$   Coefficients of the polynomial expansion

${\mathrm{\Delta }}_{\mathrm{\ell }}$   Nested quadrature operator at level $\mathrm{\ell }$

$\mu$   1${ }^{\mathrm{s}\mathrm{t}}$moment, average

${\rho }_x$   Joint probability density function

$\sigma$   2${ }^{\mathrm{n}\mathrm{d}}$ moment, standard deviation

$\tau$   Residence time

$\varphi$   Equivalence ratio

$\psi$  One-dimensional polynomial basis

$\mathrm{\Psi }$   Multivariate polynomial combinations

Subscripts

low   Lower uncertainty limit

high  Upper uncertainty limit

Superscripts

(1)   One-dimensional operator

(2)   Two-dimensional operator

(n)   N-dimensional operator

$r$   Node of the quadrature rule

Additional information

Funding

The authors wish to acknowledge the support of the Fonds de Recherche du Québec - Nature et Technologies (FRQNT), the Natural Sciences and Engineering Research Council of Canada (NSERC) and Siemens Canada Limited, a subsidiary of Siemens.