Some analytic solutions for stochastic reactor models based on the joint composition PDF

Kraft, Markus; Fey, Harald

doi:10.1088/1364-7830/3/2/308

Cited by 5 publications

(4 citation statements)

References 19 publications

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“…and are integrated analytically to give (Kraft and Fey, 1999;Bhave and Kraft, 2004) The advantage of this solver is that it is very fast. It is general in the sense that it may be applied to any number of scalars, although it is restricted to N ¼ 2 particles.…”

Section: Eq (19) Is Rewritten Asmentioning

confidence: 99%

Numerical investigation of DQMoM-IEM as a turbulent reaction closure

Akroyd

Smith

McGlashan

et al. 2010

Chemical Engineering Science

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Section: Eq (19) Is Rewritten Asmentioning

confidence: 99%

Numerical investigation of DQMoM-IEM as a turbulent reaction closure

Akroyd

Smith

McGlashan

et al. 2010

Chemical Engineering Science

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“…Note that, with the notations (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) representation (2.12), (2.13) coincides with formulas (14)- (15) in [13]. Moments…”

Section: The Stochastic Processmentioning

confidence: 95%

An efficient stochastic chemistry approximation for the PDF transport equation

Kraft¹,

Wagner²

2002

Monte Carlo Methods and Applications

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In this paper we present an efficient algorithm for the numerical treatment of the PDF transport equation. Using the partially stirred plug flow model in conjunction with the IBM mixing model we construct a numerical scheme that is based on a time Splitting technique and a stochastic chemistry approximation. For this purpose a particle/sub-particle System is introduced. The dynamics of this particle System is determined by a mixing step and a chemistry step. The chemistry step is solved by a Jump process where forward and reverse reactions are combined. Various numerical experiments are carried out to study convergence with respect to particle number and sub-particle number. In case of a linear reaction, the comparison between analytical solution and numerical approximation of the third moment reveals that the systematic error is inversely proportional to the number of particles and sub-particles, respectively. The performance of the algorithm is evaluated by studying the combustion of a premixed stoichiometric mixture of n-heptane and air. The stochastic chemistry algorithm is compared with a deterministic approach using the ODE solver DASSL and it is found, for the examples studied, that the stochastic algorithm is more efficient than the deterministic approach.

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“…The PaSR model can be derived from the one point joint scalar PDF transport equation, assuming statistical homogeneity [5]. The analytical solutions for some simple cases of the PDF transport equation describing the PaSR model have been presented in [11]. The PDF equation has been solved numerically using a Monte Carlo particle method with time splitting techniques.…”

Section: Introductionmentioning

confidence: 99%

“…• To extend and develop the analytical solutions presented in [11], to a more general case of the MDF transport equation and obtain the first four moments (zeroth to third) of the mass density function (MDF) describing the PaSR model.…”

Section: Introductionmentioning

confidence: 99%

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

Bhave¹,

Kraft²

2004

SIAM J. Sci. Comput.

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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 nondimensional time step, ∆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. Based on the split analytical solutions, the convergence with respect to ∆t * 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 ∆t *. 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.

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Some analytic solutions for stochastic reactor models based on the joint composition PDF

Cited by 5 publications

References 19 publications

Numerical investigation of DQMoM-IEM as a turbulent reaction closure

Numerical investigation of DQMoM-IEM as a turbulent reaction closure

An efficient stochastic chemistry approximation for the PDF transport equation

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

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