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

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.

[1]  Markus Kraft,et al.  Some analytic solutions for stochastic reactor models based on the joint composition PDF , 1999 .

[2]  Stephen B. Pope,et al.  An investigation of the accuracy of manifold methods and splitting schemes in the computational implementation of combustion chemistry , 1998 .

[3]  J. Hull Options, Futures, and Other Derivatives , 1989 .

[4]  Sanjay M. Correa,et al.  Parallel simulations of partially stirred methane combustion , 1993 .

[5]  Stephen B. Pope,et al.  A Monte Carlo Method for the PDF Equations of Turbulent Reactive Flow , 1981 .

[6]  Jinchao Xu,et al.  Assessment of Numerical Accuracy of PDF/Monte Carlo Methods for Turbulent Reacting Flows , 1999 .

[7]  R. J. Kee,et al.  Chemkin-II : A Fortran Chemical Kinetics Package for the Analysis of Gas Phase Chemical Kinetics , 1991 .

[8]  Jing Chen,et al.  Stochastic modeling of partially stirred reactors , 1997 .

[9]  J. Bendat,et al.  Random Data: Analysis and Measurement Procedures , 1971 .

[10]  Wolfgang Wagner,et al.  An Efficient Stochastic Algorithm for Studying Coagulation Dynamics and Gelation Phenomena , 2000, SIAM J. Sci. Comput..

[11]  Mikhael Gorokhovski,et al.  An Application of the Probability Density Function Model to Diesel Engine Combustion , 1999 .

[12]  Robert J. Kee,et al.  CHEMKIN-III: A FORTRAN chemical kinetics package for the analysis of gas-phase chemical and plasma kinetics , 1996 .

[13]  Octave Levenspiel,et al.  A Monte Carlo treatment for reacting and coalescing dispersed phase systems , 1965 .

[14]  Ulrich Maas,et al.  Coupling of detailed and ILDM-reduced chemistry with turbulent mixing , 2000 .

[15]  Luís Fernando Figueira da Silva,et al.  Partially stirred reactor: study of the sensitivity of the Monte-Carlo simulation to the number of stochastic particles with the use of a semi-analytic, steady-state, solution to the pdf equation , 2002 .

[16]  W. Wagner,et al.  Numerical study of a stochastic particle method for homogeneous gas-phase reactions , 2003 .

[17]  L. D. Smoot,et al.  Stochastic Modeling of CO and NO in Premixed Methane Combustion , 1998 .

[18]  U. Maas,et al.  Monte Carlo PDF modelling of a turbulent natural-gas diffusion flame , 1997 .

[19]  S. Pope PDF methods for turbulent reactive flows , 1985 .

[20]  Karl Sabelfeld,et al.  Direct and Adjoint Monte Carlo Algorithms for the Footprint Problem , 1999, Monte Carlo Methods Appl..