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  • Cambridge Centre forComputational Chemical

    Engineering

    University of CambridgeDepartment of Chemical Engineering

    Preprint ISSN 1473 4273

    Partially Stirred Reactor Model: Analytical

    Solutions and Numerical Convergence Study of

    PDF/Monte Carlo Method

    Amit Bhave, Markus Kraft1

    submitted: 16th July 2002

    1 Department of Chemical EngineeringUniversity of CambridgePembroke StreetCambridge CB2 3RAUKE-Mail: [email protected]: markus [email protected]

    Preprint No. 7

    c4e

    Key words and phrases. Partially stirred reactor model; analytical solutions; PDF/MonteCarlo method; numerical convergence.

  • Edited byCambridge Centre for Computational Chemical EngineeringDepartment of Chemical EngineeringUniversity of CambridgeCambridge CB2 3RAUnited Kingdom.

    Fax: + 44 (0)1223 334796E-Mail: [email protected] Wide Web: http://www.cheng.cam.ac.uk/c4e/

  • Abstract

    Partially stirred reactor (PaSR), a stochastic reactor model based on asimplified joint composition PDF transport equation, is investigated numer-ically. Analytical solutions for the first four moments of the mass densityfunction (MDF), for four different Cauchy problems obtained from the MDFtransport equation are presented. The Monte Carlo particle method withfirst order time splitting algorithm is implemented to obtain the first fourmoments of the MDF numerically. The dynamics of the stochastic particlesystem is determined by inflow-outflow, reaction and mixing events. Threedifferent inflow-outflow algorithms are investigated : algorithm based on theinflow-outflow event modelled as a poisson process; inflow-outflow algorithmmentioned in the literature and a novel algorithm derived on the basis of an-alytical solutions. It is demonstrated that the inflow-outflow algorithm usedin the literature can be explained by considering a deterministic waiting timeparameter of a corresponding stochastic process, and also forms a specific caseof the new algorithm. The number of particles in the ensemble, N , the non-dimensional time step, t(ratio of the global time step to the characteristictime of an event), and the number of independent simulation trials, L are thethree sources of the numerical error. Based on the split analytical solutions,the convergence with respect to t is studied. Numerical experiments arecarried out to study the convergence with respect to N and L. For a linearreaction, and the IEM model implemented, the investigation reveals that thesystematic error converges as N1 and t. The statistical error scales asL1/2 and N1/2. Finally the significance of the numerical parameters andthe inflow-outflow algorithms is studied, by applying the PaSR model to apractical case of premixed kerosene and air combustion.

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  • Contents

    1 Introduction 3

    2 PaSR model and analytical solutions 4

    2.1 Inflow-Outflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

    2.2 Inflow-Outflow + Reaction . . . . . . . . . . . . . . . . . . . . . . . . 6

    2.3 Inflow-Outflow + Mixing . . . . . . . . . . . . . . . . . . . . . . . . . 8

    2.4 Inflow-Outflow + Mixing + Reaction . . . . . . . . . . . . . . . . . . 9

    3 Numerical solution 11

    3.1 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    3.2 Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

    3.3 Inflow-Outflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    3.3.1 Algorithm 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    3.3.2 Algorithm 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

    3.3.3 Algorithm 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    3.3.4 Implementation of algorithm 3 in the time splitting algorithm 16

    3.3.5 Comparison of algorithm 2 with algorithm 3 . . . . . . . . . . 18

    4 Convergence studies 19

    4.1 Analysis of error with respect to time splitting . . . . . . . . . . . . . 20

    4.2 Analysis of error with respect to the number of particles . . . . . . . 22

    4.2.1 Inflow-Outflow . . . . . . . . . . . . . . . . . . . . . . . . . . 23

    4.2.2 Inflow-Outflow + Mixing + Reaction . . . . . . . . . . . . . . 24

    5 Combustion of kerosene 28

    5.1 Implementation of PaSR model . . . . . . . . . . . . . . . . . . . . . 28

    5.2 Results of the modelling study . . . . . . . . . . . . . . . . . . . . . . 28

    6 Conclusion 31

    7 Acknowledgements 31

    References 32

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  • 1 Introduction

    In many practical combustion devices, e.g., gas turbines, the characteristic timescales for mixing are of the same order of magnitude as the time scales for chem-ical kinetics. When modelling such practical combustion devices it is importantto account for both effects. However in order to include detailed reaction chem-istry, simplifying assumptions regarding the fluid flow description are necessary toavoid excessive computational and storage expenses. The partially stirred reactor(PaSR) is one such model based on the probability density function (PDF) trans-port equation of the physical quantities, assuming statistical spatial homogeneity.The model accounts for mixing and is computationally efficient for large coupledchemical reaction mechanisms involving many chemical species.

    Several applications of the PaSR model are mentioned in the literature. An early ex-ample of the simulation of a stirred tank reactor incorporating mixing, through-flowand reaction, is given in [13]. Over the last decade, with the substantial improvementin data storage and computational power, PaSR has received increased attention incombustion research. PaSR model was employed to investigate premixed methanecombustion and study the effect of mixing frequency on composition and tempera-ture [6]. PaSR was applied to evaluate reduced chemistry mechanisms for hydrogencombustion, and compared with detailed chemistry mechanisms [5]. Cannon et al.[4] used three reduced mechanisms to investigate their performance in the predic-tion of CO and NO in premixed turbulent combustion of methane and air simulatedin a PaSR. The PaSR model was used to study the coupling effect of chemistryand mixing for detailed and reduced chemical kinetics, and a projection schemewas introduced for mixing models to be used in conjunction with the intrinsic lowdimensional manifold (ILDM) method [2]. Thus, PaSR offers an ideal test bedfor evaluating reduced chemical mechanisms as well as mixing schemes, for use inPDF-methods-based models.

    The PaSR model can be derived from the one point joint scalar PDF transportequation, assuming statistical homogeneity [5]. The analytical solutions for somesimple cases of the PDF transport equation describing the PaSR model have beenpresented in [11]. The PDF equation has been solved numerically using a MonteCarlo particle method with time splitting techniques. This method involves approx-imating the PDF by an ensemble of stochastic particles, and has been successfullyexploited for solving high dimensional PDF equations [8, 16, 17]. However, withthe implementation of Monte Carlo particle techniques, a need for sufficiently smalltime step has been emphasised in previous works [6, 5, 4]. In order to remove thedependence of simulation on the time step, Chen [5] introduced an improved PaSRalgorithm based on theoretical age distribution of particles. However, the effect ofthis modified algorithm on the resulting PDF transport equation is not clear.

    To understand the influence of the numerical parameters in a numerical algorithm,a systematic investigation of the convergence of the algorithm is essential. Recently,in [18] a semi-analytic, steady state solution of the pdf for a PaSR model was

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  • presented and compared with a numerical solution obtained using a standard MonteCarlo particle algorithm as mentioned in [6, 5]. Overall, in the context of PDFmethods, Xu and Pope [19] noted that less attention has been paid to the accuracyof numerical algorithms. Particularly, the literature does not report convergencestudy of the Monte Carlo particle algorithms implemented for PaSR.

    The aims of the present work are as follows:

    To extend and develop the analytical solutions presented in [11], to a moregeneral case of the MDF transport equation and obtain the first four moments(zeroth to third) of the mass density function (MDF) describing the PaSRmodel.

    To present a novel inflow-outflow algorithm, derived on the basis of the analyt-ical solution, and to compare its performance with that of the inflow-outflowalgorithm mentioned in the literature, by implementing both in a Monte Carloparticle scheme with time splitting.

    Based on analytical solutions, understand the significance of the global timestep in the time splitting scheme and study the convergence of the MonteCarlo particle method with respect to the various numerical parameters.

    The paper is organised in the following way. The PaSR model is discussed andthe analytical solutions for the MDF and its first four moments are presented inSection 2. In Section 3, we describe a Monte Carlo particle scheme based onparticle approximation and a time splitting technique. Additionally the inflow-outflow algorithm based on a stochastic jump approach is described. A new inflow-outflow algorithm is derived based on the analytical solutions. These two inflow-outflow algorithms are compared with the inflow-outflow algorithm mentioned inthe literature. Section 4 is devoted to numerical experiments, dealing with theorder of convergence of the particle algorithm with respect to particle ensemble sizeN , number of independent simulation trials L and the non dimensional time stept (ratio of time step to the characteristic time of an event). In Section 5, apractical case of premix