Partially Stirred Reactor Model: Analytical Solutions and Numerical Convergence Study of a PDF/Monte Carlo Method

Abstract

We investigate the partially stirred reactor (PaSR), which is based on a simplified joint composition probability density function (PDF) transport equation. Analytical solutions for the first four moments of the mass density function (MDF) obtained from the PaSR model are presented. The Monte Carlo particle method with first order time splitting algorithm is implemented to obtain the first four moments of the MDF numerically. The dynamics of the stochastic particle system is determined by inflow-outflow, chemical reaction, and mixing events. Three different inflow-outflow algorithms are investigated: an algorithm based on the inflow-outflow event modeled as a Poisson process, an inflow-outflow algorithm mentioned in the literature, and a novel algorithm derived on the basis of analytical solutions. It is demonstrated that the inflow-outflow algorithm used in the literature can be explained by considering a deterministic waiting time parameter of a corresponding stochastic process, and also forms a specific case of the new algorithm. The number of particles in the ensemble, N, the nondimensional time step, $\Delta t^{*}$ (ratio of the global time step to the characteristic time of an event), and the number of independent simulation trials, L, are the three sources of the numerical error. The split analytical solutions and the numerical experiments suggest that the systematic error converges as $\Delta t^{*}$ and N-1 . The statistical error scales as L-1/2 and N-1/2 . The significance of the numerical parameters and the inflow-outflow algorithms is also studied by applying the PaSR model to a practical case of premixed kerosene and air combustion.