Ansys Chemkin includes a generalized method for performing uncertainty analysis on reacting-flow simulations. The purpose is to determine the uncertainty of the solution as derived from the known uncertainties of one or more input parameters. In addition, the Uncertainty Analysis Facility determines the quantitative variance contribution from different inputs. This chapter provides the theoretical background for the methods used by the Uncertainty Analysis Facility.

A generic system model for chemical or materials processing can be described mathematically as:

(20–1)

where is a vector of dependent variables and is a set of differential equations that involve , the first derivative , as well as N parameters Θi. The parameters may be specified operating conditions, such as inlet temperatures or pressures, or they may be chemistry parameters such as reaction-rate coefficients. The dependent variables for this type of system would usually be chemical state variables, including pressure, temperature, species composition, and velocity components. In a deterministic approach to modeling the system, single values of the N parameters would be chosen, and a single result for would be calculated. However, if Θi is not a known constant but is better represented by a probability distribution function, f(Θi), then the resultant vector would also be a distribution of unknown shape and mean. Traditional methods for dealing with these stochastic problems are based on Monte-Carlo methods, where statistics on the mean and variance of are collected based on solving using random samplings of the distributions of Θi values. Such methods are extremely time consuming and typically require very large numbers of model simulations (that is, evaluations of ) to produce meaningful statistics. Moreover, as the number of parameters N grows, the number of simulations required increases exponentially with N. For such complex systems, then, the cost of Monte Carlo-based analyses is often prohibitive. In this chapter, a more computationally efficient method using polynomial chaos expansion is developed to solve the stochastic problems with large numbers of parameters.