# Technical Report 7, c4e-Preprint Series, Cambridge

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

ref: Technical Report 7, c4e-Preprint Series, Cambridge

Associated Theme: Numerics

Partially stirred reactor (PaSR), a stochastic reactor model based on a simplified joint composition PDF transport equation, is investigated numerically. Analytical solutions for the first four moments of the mass density function (MDF), for four different Cauchy problems obtained from the MDF transport equation 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, reaction and mixing events. Three different inflow-outflow algorithms are investigated : algorithm based on the inflow-outflow event modelled as a poisson process; 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 non-dimensional time step, Dt* (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. Based on the split analytical solutions, the convergence with respect to Dt* is studied. Numerical experiments are carried out to study the convergence with respect to N and L. For a linear reaction, and the IEM model implemented, the investigation reveals that the systematic error converges as N-1 and Dt*. The statistical error scales as L-1/2 and N-1/2. Finally the significance of the numerical parameters and the inflow-outflow algorithms is studied, by applying the PaSR model to a practical case of premixed kerosene and air combustion.

Material from this preprint has been published in: SIAM Journal on Scientific Computing 25 (5), 1798-1823, (2004)

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