PARTIALLY STIRRED REACTOR MODEL: ANALYTICAL ......NUMERICAL INVESTIGATION OF PaSR MODEL 1799 and reaction is given in [16]. Over the past decade, with a substantial improvement in

PARTIALLY STIRRED REACTOR MODEL: ANALYTICALSOLUTIONS AND NUMERICAL CONVERGENCE STUDY

OF A PDF/MONTE CARLO METHOD∗

AMIT BHAVE† AND MARKUS KRAFT†

SIAM J. SCI. COMPUT. c© 2004 Society for Industrial and Applied MathematicsVol. 25, No. 5, pp. 1798–1823

Abstract. We investigate the partially stirred reactor (PaSR), which is based on a simplifiedjoint composition probability density function (PDF) transport equation. Analytical solutions for thefirst 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 obtainthe first four moments of the MDF numerically. The dynamics of the stochastic particle system isdetermined by inflow-outflow, chemical reaction, and mixing events. Three different inflow-outflowalgorithms are investigated: an algorithm based on the inflow-outflow event modeled as a Poissonprocess, an inflow-outflow algorithm mentioned in the literature, and a novel algorithm derived onthe basis of analytical solutions. It is demonstrated that the inflow-outflow algorithm used in theliterature can be explained by considering a deterministic waiting time parameter of a correspondingstochastic process, and also forms a specific case of the new algorithm. The number of particles in theensemble, N , the nondimensional time step, ∆t∗ (ratio of the global time step to the characteristictime of an event), and the number of independent simulation trials, L, are the three sources ofthe numerical error. The split analytical solutions and the numerical experiments suggest that thesystematic error converges as ∆t∗ and N−1. The statistical error scales as L−1/2 and N−1/2. Thesignificance of the numerical parameters and the inflow-outflow algorithms is also studied by applyingthe PaSR model to a practical case of premixed kerosene and air combustion.

Key words. partially stirred reactor model, analytical solutions, probability density func-tion/Monte Carlo particle method, numerical convergence

AMS subject classifications. 65C20, 65C35, 65M25

DOI. 10.1137/S1064827502411328

1. Introduction. Methods based on probability density functions (PDFs) havebeen widely used to solve complex problems in such various fields as finance, physics,and engineering [7, 10, 15, 20]. These methods have great potential as they overcomecomplicated closure problems. This paper describes one such PDF-based partiallystirred reactor (PaSR) model particularly applicable to combustion and process engi-neering.

In many practical combustion devices, e.g., gas turbines and internal combustionengines, the characteristic time scales for mixing are of the same order of magnitudeas the time scales for chemical kinetics. When modeling such combustion devicesit is important to account for both effects. However, in order to include detailedreaction chemistry, simplifying assumptions regarding the fluid flow description arenecessary to avoid excessive computational and storage expenses. The PaSR modelbased on the PDF of the physical quantities assumes statistical homogeneity. Themodel accounts for mixing and is computationally efficient for large coupled chemicalreaction mechanisms involving many chemical species.

Several applications of the PaSR model are mentioned in the literature. An earlyexample of the simulation of a stirred tank reactor incorporating mixing, through-flow,

∗Received by the editors July 18, 2002; accepted for publication (in revised form) September 17,2003; published electronically May 20, 2004. This work was supported by the Cambridge Common-wealth Trust and the Overseas Research Studentship (ORS) committee.

http://www.siam.org/journals/sisc/25-5/41132.html†Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge

CB2 3RA, United Kingdom ([email protected], markus [email protected]).

1798

NUMERICAL INVESTIGATION OF PaSR MODEL 1799

and reaction is given in [16]. Over the past decade, with a substantial improvementin data storage and computational power, the PaSR has received increased attentionin combustion research. The model was employed to investigate premixed methanecombustion and study the effect of mixing frequency on composition and temperature[6]. It was also applied to evaluate reduced chemistry mechanisms for hydrogen com-bustion and compared with detailed chemistry mechanisms [5]. Premixed turbulentcombustion of methane and air was simulated in a PaSR and the performance of threereduced mechanisms in the prediction of NO and CO emissions was investigated [4].The model has also been used to study the coupling effect of chemistry and mixingfor detailed and reduced chemical kinetics, and a projection scheme was introducedfor mixing models to be used in conjunction with the intrinsic low-dimensional man-ifold (ILDM) method [2]. Thus, PaSR offers an ideal test bed for evaluating reducedchemical mechanisms as well as mixing schemes for use in PDF method–based models.

The PaSR model can be derived from the one-point joint scalar PDF transportequation, assuming statistical homogeneity [5]. For modeling the inhomogeneousvariable-density flows, the PaSR model is usually represented in terms of a massdensity function (MDF). The analytical solutions for some simple cases of such aPDF transport equation describing the PaSR have been presented in [13]. The PDFequation has been solved numerically using a Monte Carlo method with time split-ting techniques. This method involves approximating the PDF by an ensemble ofstochastic particles and has been successfully exploited for solving high-dimensionalPDF equations [9, 19, 20]. However, with the implementation of Monte Carlo particletechniques, a need for sufficiently small time steps has been emphasized in previousworks [4, 5, 6]. In order to remove the dependence of the simulation on the time step,an improved PaSR algorithm based on a theoretical age distribution of particles wasintroduced [5]. However, the effect of this modified algorithm on the resulting PDFtransport 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 [21] a semianalytic, steady state solution of the PDF for a PaSR model was pre-sented and compared with a numerical solution obtained using a standard MonteCarlo algorithm as mentioned in [5, 6]. The effect of time step on the numerical errorin a splitting algorithm has been investigated [23]. Overall, in the context of PDFmethods, it is noted that less attention has been paid to the accuracy of numericalalgorithms [22]. Particularly, the literature does not report a rigorous convergencestudy of the Monte Carlo algorithms implemented for the PaSR model.

The aims of the present work are as follows:• to extend and develop the analytical solutions presented in [13] to a more

general case of the MDF transport equation and obtain the first four moments(zeroth to third) of the MDF describing the PaSR model;

• 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 the analytical solutions, to understand the significance of the globaltime step in the time splitting scheme and study the convergence of the MonteCarlo method with respect to the various numerical parameters.

The paper is organized in the following way. The PaSR model is discussed and theanalytical solutions for the MDF and its first four moments are presented in section 2.In section 3, we describe a Monte Carlo particle scheme based on a particle approx-

1800 AMIT BHAVE AND MARKUS KRAFT

imation and a time splitting technique. Additionally the inflow-outflow algorithmbased on a stochastic jump approach is described. A new inflow-outflow algorithm isderived based on the analytical solutions. These two algorithms are compared withthe inflow-outflow algorithm mentioned in the literature. Section 4 is devoted to nu-merical experiments, dealing with the order of convergence of the particle algorithmwith respect to particle ensemble size, N , number of independent simulation trials, L,and the nondimensional time step, ∆t∗ (ratio of time step to the characteristic timeof an event). In section 5, a practical case of premixed kerosene and air combustionis investigated using the PaSR model. Conclusions are drawn in section 6.

2. PaSR model and analytical solutions. The PaSR model is described bythe following PDE, giving the evolution of the joint composition MDF:

∂F(ψ, t)∂t

+

R+1∑l=1

∂

∂ψl

[(Al(ψ) + Sl(ψ)

]F(ψ, t) = 1

τ(Fin(ψ) −F(ψ, t)) ,(2.1)

with the initial conditions

F(ψ, 0) = F0(ψ).(2.2)

The MDF is represented by F , and the variable ψ represents scalars such as massfraction of chemical species, temperature, density, etc. The right-hand side of (2.1)describes the inflow and outflow events in a PaSR, with the characteristic residencetime τ . Sl denotes the chemical source term for the scalars. In the present paper,we employ the interaction by exchange with the mean (IEM) mixing model. For theIEM model, the coefficients of (2.1) are

Al = −B(ψl − 〈φl〉), B =Cφ2τm

,(2.3)

where τm is the characteristic mixing time, Cφ is the mixing model constant, and 〈φl〉is the Favre average mean given by

〈φl〉 =∫∞−∞ ψlF(ψ, t)dψ∫∞−∞ F(ψ, t)dψ

.(2.4)

In order to carry out a systematic numerical investigation, we address four differ-ent Cauchy problems of ascending complexity in this section. These Cauchy problemsare obtained from (2.1) and describe the following events: inflow-outflow, inflow-outflow + reaction, inflow-outflow + mixing, and inflow-outflow + mixing + reaction.Only one scalar is considered while deriving the analytical solutions presented in thissection. Analysis involving multiscalar computation is presented in section 5.

The initial condition for the MDF is a double delta distribution:

F0(ψ) =[δψ(1)(ψ) + δψ(2)(ψ)

2

].(2.5)

In this investigation, we implement two different conditions for inflow.1. Delta distribution:

Fin(ψ) = δψ(in)(ψ).(2.6)

2. Uniform distribution:

Fin(ψ) ={

1 for 0 ≤ ψ ≤ 1,0 elsewhere.

(2.7)


The analytical solutions for the MDF, in case of inflow-outflow + mixing and inflow-outflow + mixing + reaction, as mentioned in [12], are valid only for a particular casewith inflow, Fin = 0. As an extension of the work carried out in [12], more generalanalytical solutions of the MDF and its first four moments for the different Cauchyproblems are presented next.

2.1. Inflow-outflow. The inflow-outflow term on the right-hand side of (2.1)makes the equation inhomogeneous. We first investigate the inflow-outflow eventswithout considering the mixing and the reaction. Thus (2.1) simplifies to

∂

∂tF(ψ, t) = 1

τ(Fin(ψ) −F(ψ, t)),(2.8)

with initial condition F(ψ, 0) = F0(ψ).The solution of the MDF is given by

F(ψ, t) = Fin(ψ)(1 − e(

−tτ ))

+ F0(ψ)e(−tτ ).(2.9)

The moments of the MDF are defined by

mj =

∫ ∞−∞

ψjF(ψ, t)dψ, j = 0, 1, 2, 3, . . . ,(2.10)

minj =

∫ ∞−∞

ψjFin(ψ)dψ, j = 0, 1, 2, 3, . . . .(2.11)

The analytical solution for the moments of the MDF is given by

mj(t) = minj + e

−t/τ (mj(0) −minj ), j = 0, 1, 2, 3, . . . .(2.12)

2.2. Inflow-outflow + reaction. The inflow-outflow and reaction events canbe described by

∂F(ψ, t)∂t

− ∂∂ψ

(kψ)F(ψ, t) = Fin(ψ)τ

− F(ψ, t)τ

.(2.13)

Employing the method of characteristics, two ODEs for ψ(t) and F(ψ, t) areobtained:

dψ

dt= −kψ,(2.14)

dF(ψ(t), t)dt

=Fin(ψ(t))

τ+

(k − 1

τ

)F(ψ(t), t).(2.15)

Solving (2.14), the inverse characteristic curve is

ψ∗ = ψekt.(2.16)

Solving the ODE for F(ψ(t), t),

F(ψ(t), t)e( 1τ −k)t =∫ [Fin(ψ(t))

τ

]e(

1τ −k)tdt + CI ,

where CI is a constant of integration. The solution for MDF is obtained by substi-tuting the conditions for the inflow, Fin.


Inflow as delta distribution. Substituting the condition for Fin, as givenin (2.6), we obtain

F(ψ(t), t)e( 1τ −k)t =∫ [

δ(ψ(t) − ψin)τ

]e(

1τ −k)tdt + ĆI(ψ

∗).(2.17)

From (2.16),

t = (−1/k) ln[ψ

ψ∗

].(2.18)

Thus

F(ψ, t)e( 1τ −k)t =(−1kτ

)∫δ(ψ − ψin)ψ∗(1−1/kτ)

ψ(−1/kτ)dψ + ĆI(ψ∗),

F(ψ, t)e( 1τ −k)t ={(−1

kτ

) (ψin)(−1/kτ)(ψ∗)1−1/kτ

+ ĆI(ψ∗), ψ ≥ ψin,

ĆI(ψ∗), ψ ≤ ψin,

which is equivalent to

F(ψ, t) =

⎧⎨⎩CI(ψ

∗)e(k−1τ )t, ψ ≥ ψin,(

1kτ

) (ψin)(−1/kτ)(ψ)1−1/kτ

+ CI(ψ∗)e(k−

1τ )t, ψ ≤ ψin.

Substituting the initial condition

t = 0, ψ = ψ∗, and F(ψ(0), 0) = F0(ψ∗),

we obtain

F0(ψ∗) ={CI(ψ

∗), ψ∗ ≥ ψin,CI(ψ

∗) + ψin(−1/kτ)

kτ ψ∗[(1/kτ)−1], ψ∗ ≤ ψin.

The value of the constant of integration, CI(ψ∗), is

CI(ψe(kt)) =

{F0(ψe(kt)), ψ ≥ ψine(−kt),F0(ψe(kt)) − ψin

(−1/kτ)

kτ

(ψ[(1/kτ)−1]e(

1τ −k)t

), ψ ≤ ψine(−kt).

(2.19)

Thus ψ will have three regimes: ψ ≥ ψin, ψine(−kt) ≤ ψ ≤ ψin, and ψ ≤ ψine(−kt).The corresponding MDF is

F(ψ, t) =

⎧⎪⎨⎪⎩F0(ψe(kt))e(k−

1τ )t, ψ ≥ ψin,

F0(ψe(kt))e(k−1τ )t + ψin

(−1/kτ)

kτ ψ[(1/kτ)−1], ψine

(−kt) ≤ ψ ≤ ψin,F0(ψe(kt))e(k−

1τ )t, ψ ≤ ψine(−kt).

(2.20)

Inflow as a uniform distribution. Substituting the inflow condition, we get

F(ψ(t), t)e( 1τ −k)t =∫ t

0

Fin(ψ(t))τ

e(1τ −k)tdt

=

∫ 1k lnψ

∗

0

Fin(ψ(t))τ

e(1τ −k)tdt +

∫ t1k lnψ

∗

Fin(ψ(t))τ

e(1τ −k)tdt.


Now, Fin is equal to 1.0 for the condition 0 < ψ < 1.0, which is equivalent to1k ln(ψ

∗) < t < ∞.

F(ψ(t), t) = F0(ψ∗)e(k−1τ )t +

1

(1 − kτ)

{1 − e(k− 1τ )t, ψ∗ < 1.0,1 − ψ∗( 1kτ −1)e(k− 1τ )t, ψ∗ > 1.0.

(2.21)

Analytical solutions for the moments of the MDF are obtained as follows: The pro-cedure to obtain the zeroth moment of the MDF is given below. The same procedureis followed to obtain first, second, and third moments.

m0 =

∫ ∞−∞

F(ψ, t)dψ,(2.22)

dm0dt

=

∫ ∞−∞

[Fin(ψ)

τ− F(ψ, t)

τ+ kψ

∂F(ψ, t)∂ψ

+ kF(ψ, t)]dψ,

m0(t) = min0 + (m0(0) −min0 )e(

−tτ ).

In general, the analytical solution for the moments of the MDF is given as

mj(t) =minj

(1 + jkτ)+

[mj(0) −

minj(1 + jkτ)

]e−(jk+

1τ )t, j = 0, 1, 2, 3, . . . .(2.23)

2.3. Inflow-outflow + mixing. The inflow-outflow and mixing events are de-scribed by

∂F(ψ, t)∂t

− ∂∂ψ

(B(ψ − 〈φ〉))F(ψ, t) = Fin(ψ)τ

− F(ψ, t)τ

,(2.24)

with the initial condition F(ψ, 0) = F0(ψ). In this case (inflow-outflow + mixing) andthe next (inflow-outflow + mixing + reaction), solving the characteristic curve doesnot yield an explicit relation between t and ψ. Thus the transformation of variablefrom dt to dψ is not possible, as it is in (2.18), and hence the analytical solution forthe MDF cannot be obtained by our method. However, the analytical solutions forthe first four moments of the MDF are obtained, as explained in the previous section(2.22). The analytical solutions for the moments of MDF are as follows:

m0(t) = min0 + (m0(0) −min0 )e−t/τ ,(2.25)

m1(t) = min1 + (m1(0) −min1 )e−t/τ ,(2.26)

m2(t) =min2

2Bτ + 1

(1 − e−(2B+ 1τ )

)+ e−(2B+

1τ )t

(m2(0) +

2B

m0(0)

×[

(m1(0) −min1 )2

(2B − 1τ )(e(2B−

1τ )t − 1) + m

in1 (m1(0) −min1 )

B(e2Bt − 1)

+

(min1)2

2B + 1τ(e2B+

1τ − 1)

]),

m3(t) =min3 (1 − e−(3B+

1τ ))

(3Bτ + 1)+ (m3(0) + 3BL)e

−(3B+ 1τ )t,

(2.27)


where

L = min1

[((m2(0) − 2m1(0)min1 + 2

(min1)2

)

B−

min2 + 2Bτ(min1)2

B(2Bτ + 1)

− (2τ(min1 −m1(0))

2)

(2Bτ − 1)

)[eBt − 1

]+

(2m1(0)min1 − 2min1

2)

3B

[e(3Bt) − 1

]

+2Bτ(min1 −m1(0))

2

(2Bτ − 1)(3B − 1τ )[e(3B−

1τ )t − 1

]+

(min2 + 2Bτ(min1)2

)

(2Bτ + 1)(3B + 1τ )

[e(3B+

1τ )t − 1

]]

+ (m1(0) −min1 )[(

(m2(0) − 2m1(0)min1 + 2(min1)2

)

(B − 1τ )−

min2 + 2Bτ(min1)2

(B − 1τ )(2Bτ + 1)

− (2Bτ(min1 −m1(0))

2)

(B − 1τ )(2Bτ − 1)

)[e(B−

1τ )t − 1

]+

(2m1(0)min1 − 2min1

2)

(3B − 1τ )[e(3B−

1τ )t − 1

]

+2Bτ(min1 −m1(0))

2

(2Bτ − 1)(3B − 2τ )[e(3B−

2τ )t − 1

]+

(min2 + 2Bτ(min1)2

)

(2Bτ + 1)(3B)

[e(3B)t − 1

]].

(2.28)

2.4. Inflow-outflow + mixing + reaction. Lastly we present the most com-plicated Cauchy problem. The inflow-outflow, mixing, and reaction events are de-scribed by

∂F(ψ, t)∂t

− ∂∂ψ

(B(ψ − 〈φ〉) + kψ

)F(ψ, t) = Fin(ψ)

τ− F(ψ, t)

τ,(2.29)

with the initial condition F(ψ, 0) = F0(ψ). The analytical solution for the momentsof MDF are as given below:

m0(t) = min0 + (m0(0) −min0 )e−t/τ ,(2.30)

m1(t) =min1 (1 − e−(k+

1τ )t)

(kτ + 1)+ m1(0)e

−(k+ 1τ )t,(2.31)

m2(t) =min2 (1 − e−(2B+2k+

1τ )t)

(2Bτ + 2kτ + 1)+ (m2(0) + 2BG)e

−(2B+2k+ 1τ )t,(2.32)

m3(t) =min3 (1 − e−(3B+3k+

1τ )t)

(3Bτ + 3kτ + 1)+ (m3(0) + 3BT )e

−(3B+3k+ 1τ )t,(2.33)

H=− (m1(0))2

(2B − 1τ )+

2m1(0)min1

τ(2B − 1τ )(2B + k)−

(min1)2

(2k2 + 4kτ +2τ2 )

(1 + kτ)2(2B + k)(2B − 1τ )(2B + 2k +

1τ )

,

(2.34)

G =

(m1(0) −

min1(1 + kτ)

)2e(2B−

1τ )t

(2B − 1τ )+

( (min1)2

(2B + 2k + 1τ )(1 + kτ)2

)e(2B+2k+

1τ )t

+

[2m1(0)m

in1

1 + kτ−

2(min1)2

(1 + kτ)2

]e(2B+k)t

(2B + k)+ H,

(2.35)


T =

(m1(0) −

min1(1 + kτ)

)[(m2(0) + 2BH − m

in2

(2Bτ+2kτ+1) )

(B − 1τ )[e(B−

1τ )t − 1]

+

(2B(min1)2

(1 + kτ)2 +

min2τ

) [e(3B+2k)t − 1

](2B + 2k + 1τ )(3B + 2k)

+2B

(2B − 1τ )

(m1(0) −

min1(1 + kτ)

) [e(3B−

2τ )t − 1

](3B − 2τ )

+4B

(2B + k)

(m1(0)m

in1

(1 + kτ)−(min1)2

(1 + kτ)2

) [e(3B+k−

1τ )t − 1

](3B + k − 1τ )

]

+

(min1

1 + kτ

)[(m2(0) + 2BH − m

in2

(2Bτ+2kτ+1) )

(B + k)

[e(B+k)t − 1]

+

(2B(min1)2

(1 + kτ)2 +

min2τ

) [e(3B+3k+

1τ )t − 1

](2B + 2k + 1τ )(3B + 3k +

1τ )

+2B

(2B − 1τ )

(m1(0) −

min1(1 + kτ)

) [e(3B+k−

1τ )t − 1

](3B + k − 1τ )

+4B

(2B + k)

(m1(0)m

in1

(1 + kτ)−(min1)2

(1 + kτ)2

) [e(3B+2k)t − 1

](3B + 2k)

].

(2.36)

The analytical solutions developed here are exploited for a detailed numerical inves-tigation of the Monte Carlo particle method. The following section deals with thedescription of a Monte Carlo algorithm implemented with a first order time splittingtechnique.

3. Numerical solution. A Monte Carlo particle method with a time splittingprocedure is adopted to obtain the first four moments of the MDF numerically. Theparticle algorithm involves an approximation of the initial density function by a no-tional particle ensemble.

For a unit mass, the initial MDF (F0) is approximated by

F0(ψ) ≈ FN0 =1

N

N∑n=1

δψ

(n)0

.(3.1)

These particles are then moved according to the evolution equation of the correspond-ing density function [18, 20]. In the present work, the system of stochastic particlesis introduced as

ψ(1)(t), ψ(2)(t), . . . , ψ(N)(t), t > 0.(3.2)

A time splitting algorithm is implemented in which the inflow-outflow, mixing, andreaction events are treated sequentially in a given time step ∆t. The Monte Carlomethod with first order time splitting algorithm is given as follows:

1. Initialize t = 0, tstop,∆t. Determine the state of the particle system (3.2) attime t = 0 according to the initial function F0.

2. Progress in time with a time step, t �→ t + ∆t. If t > tstop, then stop. Elsego to step 3.


3. Perform the inflow-outflow step.4. Move each particle according to the reaction step.5. Advance each particle according to the mixing step.6. Go to step 2.

The individual mixing, reaction, and inflow-outflow events and their implementationin the particle algorithm are discussed next.

3.1. Mixing. The IEM model (also known as the linear mean square estimation(LMSE) model) is a deterministic model that does not introduce any fluctuations.Simplicity and a low computational effort are the salient features of this model. Themodel works on the principle that the scalar value of each particle approaches themean scalar value of all the particles with a characteristic time τm. For a singlescalar, the IEM model is given by

dψ(t)

dt= −B (ψ(t) − 〈φ〉) .(3.3)

For the particle system (3.2), the Favre average 〈φ〉 in (3.3) is approximated by theempirical mean to give

dψ(i)(t)

dt= −B

⎡⎣ψ(i)(t) − 1

N

N∑j=1

ψ(j)(t)

⎤⎦ ,(3.4)

where i is the index for the particle number.For the IEM model,

d

dt

1

N

N∑i=1

ψ(i)(t) = 0.

Thus, given ψ at t = t0, the solution is given as

ψ(i)(t) = ψ(i)(t0)e−B(t−t0) +

[1 − e−B(t−t0)

] 1N

N∑j=1

ψ(j)(t0).(3.5)

At each time step, the particle ensemble is updated as per the mixing model.

3.2. Reaction. For the particle system given in (3.2), the reaction step is givenby

d

dtψ

(i)l (t) = Sl(ψ

(i)(t)).(3.6)

For a linear reaction A −→ B,

d

dtψ(i)(t) = −k(ψ(i)(t)).(3.7)

The rate equation can be solved analytically to obtain

ψ(i)(t) =(ψ(i)(t0)

)e−k(t−t0).(3.8)


3.3. Inflow-outflow. In [6], the number of inflow particles Nin within a timestep ∆t is given as

Nin ≈ṁ× ∆tmp

,(3.9)

where mp is the mass of a particle and ṁ is the mass flow rate. The total reactormass (M) and the residence time (τ) are incorporated in [5] as

Nin ≈∆t

(M/Nṁ)≈ ∆t×N

τ.(3.10)

The underlying assumption in the inflow-outflow models is that the particles which areselected according to uniform distribution flow out at the same rate. Three differentinflow-outflow algorithms are explained.

3.3.1. Algorithm 1. This section contains a stochastic algorithm for an inflow-outflow event modeled as a Markov jump process. In this paper, we adopt the notationof [8]. The particle system is as given in (3.2). The dynamics of the particle systemcan be described by a sequence of jump processes,

UN (t), t ≥ 0, N = 1, 2, 3, . . . ,

taking values in the space of discrete measures:

SN ={p =

1

N

N∑i=1

δψ(i) , ψi > 0, N = 1, 2, . . .

}.

The generator of a stochastic process relates time evolution of the PDF of a stochas-tic process to the time evolution of the process itself. The infinitesimal stochasticgenerator is given by

ANΦ(p) =N∑i=1

K(ψ(i))

∫R

[Φ(J(p, γ, i)) − Φ(p)]Fin(γ)dγ,(3.11)

where

Φ(p) =

∫ϕ(ψ)p(., dψ)

and ϕ is any test function. Thus the rate is

N∑i=1

K(ψ(i)) =

N∑i=1

1

τ=

N

τ(3.12)

and

J(p, γ, i) = p +δγN

−δψ(i)

N.(3.13)

Between each finite time step ∆t, the sequence of stochastic jump processes (Algo-rithm 1) is implemented in the time splitting algorithm as follows:


1. Given the state at time t0, UN (t0) = p ∈ SN .

2. Wait an exponentially distributed time step τ̂ with parameter

π(p) =N

τ,

where

Prob{τ̂ ≥ s} = e−π(p)s.

3. Update time t �→ t + τ̂ . If t > t0 + ∆t, then stop. Else go to step 4.4. Choose a particle index i according to the mass density

gi =1

N, i ∈ {1, 2, . . . , N}.

5. Generate an Fin deviate, γ, and replace the particle at position i.6. Go to step 2.

3.3.2. Algorithm 2. From section 3.3.1, the deterministic mean waiting timeis given as

Eτ̂ = π−1(p) =τ

N.(3.14)

With a specified time step ∆t, the number of particles to be replaced (Nin) in thistime step is given by

Nin =

[N∆t

τ

](3.15)

according to

∆t = NinEτ̂ = Ninτ

N,(3.16)

where the square brackets in (3.15) denote the integer part. Equation (3.15) is equiv-alent to (3.10), implemented to calculate the number of particles to be replaced (Nin)in a single time step (∆t). Thus Algorithm 2 can be explained by considering a deter-ministic waiting time parameter of a corresponding stochastic process.

3.3.3. Algorithm 3. In this section, Algorithm 3 is derived on the basis of theanalytical solution of the MDF describing the inflow-outflow event. The analyticalsolution of the split Cauchy problem, as stated in (2.24), is given by

F(ψ, t) = (1 −A)Fin(ψ) + AF0(ψ),(3.17)

where A = e−(t−t0)

τ . Representing the analytical solution as a measure, we obtain∫F(ψ, t)ϕ(ψ)dψ =

∫(1 −A)Fin(ψ)ϕ(ψ)dψ +

∫AF0(ψ)ϕ(ψ)dψ.

An approximation to this measure valued solution can be obtained by approximatingthe initial and the inflow distributions independently. Approximating the solutionwith an equiweighted particle ensemble for Fin and F0 distributions, we use Nin


particles to form the inlet distribution and N0 particles to form the initial distribution.According to (3.1),

Fin(ψ) ≈1

Nin

Nin∑n=1

δψ

(n)in

,

F0(ψ) ≈1

N0

N0∑n=1

δψ

(n)0

.

Substituting back,

F(ψ, t) ≈ (1 −A) 1Nin

Nin∑n=1

δψ

(n)in

+A

N0

N0∑n=1

δψ

(n)0

.

Thus, the approximation includes N = Nin +N0 particles. For the single approxima-tions of F0 and Fin, the number of particles is obtained as

N0(t) = [N ×A(t)],(3.18)Nin(t) = [N × (1 −A(t))],(3.19)

F(ψ, t) ≈ 1N

N0∑n=1

δψ

(n)0

+1

N

Nin∑n=1

δψ

(n)in

.

Therefore,

∫F(ψ, t)ϕ(ψ)dψ ≈ 1

N

N0∑n=1

ϕ(ψ(n)0 ) +

1

N

Nin∑n=1

ϕ(ψ(n)in ).

The time marching procedure is as follows: The particle representation of theanalytical solution is used to construct a numerical scheme. A representative timestep t −→ t + ∆t is considered for the demonstration: At time t the total numberof particles is N . With the time step ∆t, the number of particles at time t decayaccording to the relation,

N0(t + ∆t) = [A(t + ∆t) ×N ].(3.20)

Therefore, the number of inflow particles Nin is given by

Nin(t + ∆t) = N −N0(t + ∆t).(3.21)

In every time step, Nin = N [1−e−∆tτ ] particles are replaced from the particle ensemble

of size N according to uniform distribution.At this point we discuss three different ways in which Algorithm 3 can be imple-

mented in the Monte Carlo particle algorithm with time splitting.

3.3.4. Implementation of Algorithm 3 in the time splitting algorithm.Given a time step ∆t, Algorithm 3 gives the exact number of inflow particles Nin.Three different ways of implementing the inflow-outflow step are described below:

1. In a time step ∆t, Nin distinct particles are chosen from the ensemble ac-cording to uniform distribution and replaced by the inflow particles.


2. Consider a representative time step t −→ t+∆t. At time t, the ensemble con-sists of Nin,Fin distributed particles and N0,F0 distributed particles. Witha time step ∆t, the particle ensemble N representing FN is down-sampledto size N − Nin by choosing equivalent number of F0 and Fin distributedparticles from the ensemble at time t according to uniform distribution. Ninparticles are added, thus keeping the size of the ensemble limited to N .

3. n particles are chosen (allowing nondistinct particles to be chosen) in a timestep ∆t according to uniform distribution and replaced with the inflow par-ticles. The value of n is given according to the proposition below.Proposition.

n =−∆tτ

ln[1 − 1N

] .(3.22)Proof. The size of the particle ensemble is N . n indices are chosen accordingto uniform distribution U [1, N ]. Let Xi, where i = 1, 2, . . . , n, be the ran-dom variable that denotes the particle index. If Yn,N is the random variablewhich represents the number of distinct Xi values, then we can choose theexpectation of Y to be equal to the required number of inflow particles, Nin,as given by the algorithm 3. The expectation is obtained as follows:The probability of j distinct indices occurring in n is

p(n, j) = P (Yn,N = j) = p(n− 1, j) ×j

N+ p(n− 1, j − 1) × N − (j − 1)

N.

(3.23)

Now, the expectation is given by

En(j) = E(Yn,N = j) =n∑

j=1

j × p(n, j)

=

n∑j=1

jp(n− 1, j) jN

+n∑

j=1

jp(n− 1, j − 1)(N − (j − 1))N

=

n∑j=1

jp(n− 1, j) jN

+n∑

i=2

ip(n− 1, i− 1)(N − (i− 1))N

=n−1∑j=1

jp(n− 1, j) jN

+n−1∑j=1

(j + 1)p(n− 1, j) (N − j)N

=n−1∑j=1

[j2

N+

(j + 1)(N − j)N

]p(n− 1, j)

=n−1∑j=1

p(n− 1, j) +n−1∑j=1

(N − 1N

)jp(n− 1, j)

= 1 +

(N − 1N

) n−1∑j=1

jp(n− 1, j).


Therefore,

En(j) = 1 +(N − 1)

NEn−1(j).(3.24)

The solution is of the form

En = N − cxn.

Thus,

N − cxn = 1 + (N − 1)N

(N − cxn−1),

x = N−1N or 0. Using n = 0, E0 = 0. Therefore c = N . Thus,

En = N

[1 −(N − 1N

)n].(3.25)

Substituting the expectation as Nin,

Nin = N

[1 −(N − 1N

)n].(3.26)

Thus (3.22) follows. As N −→ ∞, neglecting the higher order terms, theequation becomes

n = N∆t

τ.(3.27)

The expression for n resembles the number of inflow particles as calculatedby Algorithm 2 in (3.15).

The three methods mentioned above are compared in terms of the CPU timeconsumed by each. At a value of ∆tτ < 1, very few particles are exchanged in a timestep. Hence, the CPU times consumed by methods 1 and 3 are same, whereas thatconsumed by method 2 is greater by more than a factor of two. In the condition∆tτ > 1, methods 2 and 3 are faster than method 1 by a factor of three. In this paper,

method 1 is implemented.

3.3.5. Comparison of Algorithm 2 with Algorithm 3. In the case of Al-gorithm 3, the number of particles exchanged (Nin) in a single time step ∆t is givenby

Nin = N −[Ne(−

∆tτ )].(3.28)

Expanding the exponential term using Taylor series expansion,

Nin = N −{N −N

(∆t

τ

)+ O

[(∆t

τ

)2]}

= N

(∆t

τ

)+ O

[(∆t

τ

)2].(3.29)


The first term on the right-hand side of (3.29) represents the number of particlesexchanged in the case of Algorithm 2. For the ratio (∆tτ ) 1, the higher order termsin the above equation become less significant; thus, Algorithm 2 is a special case ofAlgorithm 3. Algorithm 2 fails for the condition ∆t > τ , as the number of particlesto be replaced in a time step is greater than the number of particles in the ensemble.The following section includes the numerical experiments carried out to study theconvergence of the Monte Carlo particle method.

4. Convergence studies. For the Monte Carlo particle algorithm employed,there are three numerical parameters—the number of particles, N , the number ofindependent simulation trials, L, and the splitting time step, ∆t—that are the mainsources of numerical error. In this section we investigate the convergence of theparticle method with respect to these numerical parameters. First, we introducesome definitions and notation used in the numerical procedures.

The estimation of an interval, which will include the parameter being estimated,with known degree of uncertainty is advantageous as compared to point estimation[1, 14]. The confidence intervals for the expectation of a random variable are deter-mined, and the statistical and systematic errors are evaluated as follows:

Consider a functional, F , which represents, say, the scalar mass fraction,

F =

∫ϕ(ψ)F(ψ, t)dψ,

where ϕ is a test function. The functional is approximated (as N −→ ∞) by therandom variable,

ξN (t) =

∫ϕ(ψ)

1

N

N∑i=1

δψ(i)(dψ) =1

N

N∑i=1

ϕ(ψ(i)).

The expectation and the random fluctuations of the estimator are estimated by gen-erating L independent particle ensembles. The corresponding values of the randomvariable are denoted as ξN,1(t), . . . , ξN,L(t). The empirical mean value is given by

ηN,L1 (t) =1

L

L∑n=1

ξN,n(t).(4.1)

This mean is used to approximate the macroscopic quantity (mass fraction). Ac-cording to the weak law of large numbers, the empirical mean will asymptotically(L −→ ∞) converge to F . The variance of the random variable is estimated by theempirical variance

ηN,L2 (t) =1

L

L∑n=1

[ξN,n(t)]2 − [ηN,L1 (t)]

2.(4.2)

As a consequence of the central limit theorem, the standardized random variable has,asymptotically, a standard normal distribution. Then the confidence interval for theexpectation of the random variable is given by

Ip =

⎡⎣ηN,L1 (t) − ap

√ηN,L2 (t)

L, ηN,L1 (t) + ap

√ηN,L2 (t)

L

⎤⎦ .


The approximation as given in (4.1) leads to an error,

�(N,L)(t) = |η1(N,L)(t) − F (t)|,

with two components: statistical error and systematic error. Throughout the paper,confidence level p = 0.999 and ap = 3.29 (from standard normal density tables [1]) isused. With the evolution in time t such that ti = i∆t, the measure for the statisticalerror is

cstats = maxi

⎧⎨⎩ap

√ηN,L2 (ti)

L

⎫⎬⎭ .(4.3)

The statistical error scales as L−1/2, and its convergence with respect to the numberof particles, N , is investigated in section 4.2.2. The systematic error is approximatedas

ctot = maxi

{|η1(N,L)(ti) − F (ti)|

},(4.4)

where F (ti) is obtained from the analytical solutions. In the following sections, weinvestigate separately the influence of the size of the time splitting step and the numberof particles on the error.

4.1. Analysis of error with respect to time splitting. To study the effectof time splitting on the error, we split the analytical solutions for the moments ofthe MDF for the three events (inflow-outflow, reaction, and mixing). Thus the erroris dependent only on ∆t. We compare the moments obtained from the split analyt-ical solution with those from the nonsplit analytical solutions. For example, with arepresentative time step ∆t = t− t0, the second moment is given by

m2(t) = min2 + (m2(t0) −min2 )e−(

∆tτ ) (inflow-outflow),(4.5)

m2(t) = m2(t0)e−2(Cφ2 )(

∆tτm

) + m1(t0)2

(1 − e−2(

Cφ2 )(

∆tτm

)

)(mixing),(4.6)

m2(t) = m2(t0)e−2(∆tτr ) (reaction).(4.7)

The values for the various numerical parameters that are used in the simulations areas follows: The mixing model constant Cφ = 2 [20]. The initial condition for the MDFis given according to (2.5), where ψ(1) = 0.2 and ψ(2) = 0.8. The inflow condition, asgiven in (2.6), is set at ψin = 0.9. The strongest interactions between fluid mechanicsand chemical kinetics are expected to occur when the characteristic times for the threeevents are of the same order of magnitude. To relate the characteristic times, twodistinct Damköhler numbers can be defined, Dares =

ττr

and Damix =τmτr

. We choosethe values of the characteristic times such that the two Damköhler numbers approachunity. The residence time is τ = 3.0 s, whereas the characteristic time for mixing isτm = 2.0 s, and the rate of the reaction is k = 1 s

−1. For a linear reaction A −→ B,the reaction characteristic time τr =

1k . We define dimensionless time steps as

∆t∗event =time step

characteristic time of an event.


0.0010.01

0.11

∆t∗io 0.001

0.01

0.1

1

∆t∗r =∆tτr

0.0001

0.001

0.01

0.1

1

total error

Fig. 4.1. Total error for moment 2 with respect to the dimensionless time steps for inflow-outflow and reaction events (inflow as delta distribution).

10-5

0.0001

0.001

0.01

0.1

10-5 0.0001 0.001 0.01 0.1 1

To

tal e

rro

r

∆t*

Slope = 1

Fig. 4.2. Order of convergence with respect to ∆t∗ for moment 2 (inflow as delta distribution).

Considering the case for inflow-outflow + reaction, Figure 4.1 depicts the error withrespect to the dimensionless time steps for inflow-outflow and reaction events. Withdecrease in the dimensionless splitting time step, the error decreases.

To study the order of convergence we consider the case inflow-outflow + mixing+ reaction. ∆t∗ is set according to the relation

∆t∗ =∆t

min(τ, τr, τm).

Figure 4.2 shows the error convergence with respect to the dimensionless time step(∆t∗). The error scales as ∆t∗. Having investigated the influence of the nondimen-sional time step on the error, the effect of N is studied by employing the Monte Carlomethod with time splitting algorithm.


0

0.1

0.2

0.3

0.4

0.5

0 5 10 15

m1(numerical)

m2(numerical)

m3(numerical)

m1(analytic)

m2(analytic)

m3(analytic)

Mas

s fr

acti

on

: A

(m

1, m

2, m

3)

t (s)

[a]

0

0.1

0.2

0.3

0.4

0.5

0 5 10 15

m1(numerical)

m2(numerical)

m3(numerical)

m1(analytic)

m2(analytic)

m3(analytic)

Mas

s fr

acti

on

:A

(m

1, m

2, m

3)t (s)

[b]

Fig. 4.3. Evolution of moments for inflow as [a] delta and [b] uniform distributions (N = 104,∆t = 0.001, k = 1s−1, τ = 3.0s, τm = 2.0s).

4.2. Analysis of error with respect to the number of particles. We imple-ment the particle algorithm with time splitting. The statistical and systematic errorsare calculated, and the order of convergence with respect to number of particles, N ,is investigated in this section. First we illustrate the comparison of the evolution ofmoments obtained numerically and analytically for the Cauchy problem describinginflow-outflow + mixing + reaction events. Figure 4.3 depicts the evolution of thefirst, second, and third moments for the two different inflow conditions. For the valuesof N and ∆t chosen, an excellent agreement is observed for the numerically and an-alytically obtained moments with both inflow cases. To study the convergence withrespect to N we consider two Cauchy problems: one describing the inflow-outflowevent and the other inflow-outflow + mixing + reaction events.

4.2.1. Inflow-outflow. L independent runs (trials) of the particle ensemble arecarried out to construct the confidence bands with width cstats. The product L×N =200,000 is kept constant in order to maintain the confidence bands at roughly thesame width for different calculations. Thus the statistical error remains constant.The measured quantities (empirical moments) depend upon the dimensionless timestep, ∆t∗, the number of particles, N , and the number of trials, L. In this particularcase, since it is a single event with a fixed characteristic time, τ , there is no errordue to the dimensionless splitting time step ∆t∗. With ∆t∗ > 1, Algorithm 3 isimplemented, as Algorithm 2 can be implemented only up to ∆t∗ = 1. We study theorder of convergence with respect to N with ∆t∗ = 4/3.

Inflow as uniform distribution. The order of convergence can be studied inthe regime where the systematic error is greater than the statistical error. Figure 4.4shows the order of convergence for moment 1 with Algorithms 1 and 3 in case of inflowas a uniform distribution. The error bars denote the statistical error. With increasein the value of N , the systematic error reduces and the statistical error bounds of thenumerical solution encompass the analytical solution. Figure 4.4 suggests convergenceof order N−1 for both the algorithms.

Inflow as delta distribution. With the inflow as delta distribution, Figure 4.5gives the systematic and statistical errors with respect to N . As observed for inflowas a uniform distribution, the results suggest an order N−1 in this case as well.


0.0001

0.001

0.01

0.1

10 100 1000 104

To

tal e

rro

r, c

tot

N

slope = -1

[a]

0.0001

0.001

0.01

0.1

10 100 1000 104

To

tal e

rro

r, c

tot

N

slope = -1

[b]

Fig. 4.4. Order of convergence for moment 1 with [a] Algorithm 1 and [b] Algorithm 3 forinflow-outflow event (inflow as uniform distribution).

0.0001

0.001

0.01

10 100 1000

To

tal e

rro

r, c

tot

N

slope = -1

[a]

0.0001

0.001

0.01

10 100 1000

To

tal e

rro

r, c

tot

N

slope = -1

[b]

Fig. 4.5. Order of convergence for moment 1 with [a] Algorithm 1 and [b] Algorithm 3 forinflow-outflow event (inflow as delta distribution).

4.2.2. Inflow-outflow + mixing + reaction. Here we consider inflow-outflow,mixing, and reaction events with two subcases: one where all three characteristic timesare of the same order of magnitude, and the second where the characteristic time forinflow-outflow is much lower than the characteristic times for mixing and reaction.

For the first subcase, the characteristic times for these events are set as givenin section 4.2.1. The product N × L = 750,000 is kept fixed. From the error studyin section 4.1, the error obtained at ∆t∗ = 0.001 is comparable to the statisticalfluctuations obtained in the particle method, and hence ∆t∗ is fixed at 0.001. Asdescribed earlier, at such low values of ∆t∗, Algorithms 2 and 3 are identical. Weinvestigate the error with respect to the number of particles, N , for Algorithms 1and 2.

Figure 4.6 depicts the order of convergence for the systematic error for first andthird moments using Algorithm 1. The systematic error is observed to scale as N−1.


0.0001

0.001

0.01

10 100 1000

To

tal e

rro

r, c

tot

N

slope = -1

[a]

0.0001

0.001

10 100 1000

To

tal e

rro

r, c

tot

N

slope = -1

[b]

Fig. 4.6. Order of convergence for [a] moment 1 and [b] moment 3 inflow-outflow + mixing +reaction events (inflow as uniform distribution) with Algorithm 1.

0.0001

0.001

0.01

0.1

100 1000 104 105

To

tal e

rro

r, c

tot

N

[b]

0.0001

0.001

0.01

10 100 1000

To

tal e

rro

r, c

tot

N

slope = -1

[a]

Fig. 4.7. Order of convergence for moment 2 inflow-outflow + mixing + reaction events (inflowas uniform distribution) with [a] Algorithm 1 and [b] Algorithm 2.

For the same case, the order of convergence for moment 2 using Algorithms 1 and 2is compared in Figure 4.7. For the case of Algorithm 2, the number of inflow particlesto be replaced, Nin, is calculated according to (3.15). As Algorithm 2 considers amean waiting time parameter of the stochastic process, it requires a certain minimumnumber of particles, N , in order to let at least one particle be replaced and thusallow the inflow-outflow event to occur. Therefore at the value of ∆t∗ implemented,it needs approximately N = 3000, so that the inflow-outflow event occurs. However,for Algorithm 1, with L trials, the inflow-outflow event can occur at a lower valueof N , as depicted in Figure 4.7. Similar results are obtained for second and thirdmoments for both conditions of inflow. Thus, Algorithm 1, based on a stochastic jumpprocess, can be implemented at low values of N at the expense of a greater number ofrepetitions.


0.0001

0.001

1000 104 105

m1

m2

m3

Sta

tist

ical

err

or,

cst

ats

N

slope = -1/2

Fig. 4.8. Order of convergence for statistical error, cstats, with Algorithm 2 (inflow as deltadistribution).

0.0001

0.001

0.01

10 100 1000

To

tal e

rro

r, c

tot

N

slope = -1

[a]

0.0001

0.001

0.01

10 100 1000

To

tal e

rro

r, c

tot

N

slope = -1

[b]

Fig. 4.9. Order of convergence for moment 1 with [a] Algorithm 1 and [b] Algorithm 3 (inflowas delta distribution).

At this point, the statistical error is also investigated. The number of simulationruns L = 60 is kept constant for this test. Figure 4.8 shows the order of convergence ofthe statistical error for first, second, and third moments for inflow as delta distribution.The statistical error clearly scales as N−

12 in agreement with an observation given

in [22].For the second subcase, the characteristic time for mixing is set to τm = 5000, and

that for the reaction is τr = 5000. The residence time is 3.0, and the dimensionlesstime step ∆t∗ = ∆tτ =

43 . With ∆t

∗ > 1, Algorithms 1 and 3 are compared. As shownin Figure 4.9, the order of convergence obtained with both algorithms is the same,and thus the systematic error scales as N−1.


Similar results are obtained for the second and third moments for both inflowconditions. With ∆t∗ > 1, even a small number of particles (say 100–200) are sufficientto let the inflow-outflow event occur, contrary to the earlier subcase, where ∆t∗ =0.001 (Figure 4.7). The CPU time is proportional to the nondimensional time splittingstep. Adopting Algorithm 3 enables the use of larger time steps as compared toAlgorithm 2, thus saving computational expense in terms of CPU time. This has beenvalidated for the practical combustion case study given in the next section.

5. Combustion of kerosene. In this section, the PaSR model is applied tostudy a practical combustion case: premixed kerosene-air combustion.

Case study. A premixed kerosene-air stream flows into a reactor at a high flowrate (small residence time, τ = τ1). With a small residence time, the kerosene-airmixture fails to ignite. After the steady state is reached, the flow rate of the inflowstream is reduced, resulting in a large enough residence time (τ = τ2) and ignition ofthe kerosene-air mixture.

5.1. Implementation of the PaSR model. To investigate the premixed com-bustion case systematically, we split it into two parts, the first part with residencetime (τ = τ1) and the second one where ignition occurs, with residence time (τ = τ2).

In the first part the time step can be set greater than τ1, as shown in section 4.2.2.Hence the inflow-outflow event is modeled by implementing Algorithm 3. In addition,to compare the computational effort of Algorithm 3, the inflow-outflow event is alsomodeled using Algorithm 2. In the second part the ratio of ∆tτ2 should be chosen to be

much less than unity, as described in section 4.2.2. At a low value of ∆tτ2 , Algorithms 2and 3 are identical.

The combustion chemistry is modeled by a reaction mechanism consisting of 57chemical species and 273 chemical reactions [3]. The initial conditions for the concen-trations are Xn−C12H26 = 6.624×10−3, XO2 = 2.086×10−1, and XN2 = 7.848×10−1,with the concentration of the remaining 54 species being zero. The inflow stream hasthe same composition as mentioned above. The chemical kinetics are computed withthe Chemkin-III package [11] and the stiff system of ODEs is solved using the Senkinpackage [17]. The air/fuel ratio is set to 25.4, and the pressure is kept constant at1.5133 × 106Pa. The other numerical parameters are given in Table 5.1.

Table 5.1Numerical parameters for part one and part two.

Quantity Part one Part two

τ 1 × 10−2µs 4 × 10−3sTinflow 2000K 1000KT (0) 1200K 2000K

τm 1 × 10−4s 1 × 10−4s

5.2. Results of the modeling study. In part one, no ignition occurs. Thetemperature evolution is shown in Figure 5.1. Due to the small residence time (τ1),the temperature in the reactor reaches a steady state at 2000 K (temperature of theinflow stream) in 0.05 µs.

The ratio ∆tτ for Algorithm 2 and Algorithm 3 is set to 0.2 and 2.0, respectively.The CPU time consumed by Algorithm 2 is an order of magnitude greater than thatconsumed by Algorithm 3, thus proving to be computationally more expensive. In thispaper, the small residence time, τ1, is considered for theoretical interest. The residence


1200

1400

1600

1800

2000

0 5 10-8 1 10-7 1.5 10-7 2 10-7

PaSR - (algorithm 2)CPU time = 56500 sPaSR - (algorithm 3)CPU time = 5650 s

Tem

per

atu

re (

K)

t (µs)

Fig. 5.1. Comparison of the temperature evolution obtained with Algorithms 2 and 3.

2000

2200

2400

2600

2800

0 0.005 0.01 0.015

Tem

per

atu

re (

K)

t (s)

0

0.02

0.04

0.06

0.08

0.1

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.005 0.01 0.015 0.02

NO2

CONO N

O2

mo

le f

ract

ion

x 1

04

CO

, NO

mo

le fraction

t (s)

Fig. 5.2. Temperature profile and CO and NOx emissions for part two.

time τ2 is of practical relevance. Algorithm 3 offers a computational advantage onaccount of the high ∆tτ ratio in practical cases where the residence times are similarto τ2 but the temperatures are not high enough to ignite the mixture.

Figure 5.2 depicts the temperature profile and the NOx and CO emissions fromthe combustion in part two. Ignition is indicated by the rapid rise in temperatureand a significant peak due to CO formation. The temperature then settles down to asteady state at 2200 K, as shown in Figure 5.2.

In the literature, various researchers have used fixed time steps ∆t of the order10−6s and the number of particles N in the range of 100–700 for various PaSR studies[4, 6, 21]. Although these values have been fixed depending on the steady stateconvergence, the effect of the chosen parameters on transient dynamics is equallyimportant.


2160

2180

2200

2220

2240

2260

2280

0.01 0.012 0.014 0.016 0.018 0.02

∆t = 1 x 10-5 s

∆t = 5 x 10-6 s

N = 1000

Tem

per

atu

re (

K)

t (s)

[a]

2150

2200

2250

2300

0.0112 0.0128 0.0144 0.016 0.0176

N = 500N = 1000

Tem

per

atu

re (

K)

t (s)

∆t = 1 x 10-5 s

[b]

Fig. 5.3. Steady state: varying [a] ∆t and [b] N .

2000

2200

2400

2600

2800

0 2 4 6 8 10

∆t = 1.15 x 10-6 s

∆t = 5.75 x 10-7 s

Tem

per

atu

re (

K)

t (s) x 105

N = 14000

[a]

2000

2200

2400

2600

2800

0 2 4 6 8 10

N = 7000N = 14000

Tem

per

atu

re (

K)

t (s) x 105

∆t = 5.75 x 10-7 s

[b]

Fig. 5.4. Transient state: varying [a] ∆t and [b] N .

Hence convergence with respect to N and ∆t was studied for two time regimes:steady state (as shown in Figure 5.3) and transient state (Figure 5.4). The convergencestudy for the steady state regime indicated that the parameters N = 1000 and ∆t =5 × 10−6s yield a 0.1% error calculated for mean temperature. The ignition delay asgiven by the PaSR simulation depends upon the size of the time step. In the transientstate, ∆t = 1.15×10−6s and N = 7000 were required for a reasonable accuracy (error1%). Thus, while selecting N and ∆t, a balance between accuracy and computationalexpense has to be obtained on the basis of transient and steady state convergence.

6. Conclusion. In this paper we presented a detailed numerical investigation ofthe PaSR model. The need for such a numerical study was motivated by the dearthof analytical solutions for the PDF-based methods, and by the problems mentionedin the literature regarding the size of the time step in a Monte Carlo particle methodwith time splitting algorithms applied to PaSR.


The analytical solutions were developed for the first four moments of the MDF ob-tained from the PaSR model. An inflow-outflow algorithm was derived (Algorithm 3)on the basis of the analytical solutions. It was demonstrated that the algorithm men-tioned in the literature (Algorithm 2) is a special case (for the condition ∆tτ < 1)of the more general Algorithm 3. An inflow-outflow algorithm (Algorithm 1) basedon a stochastic jump approach was also presented. Algorithm 2 was explained byconsidering a deterministic parameter of the waiting time in Algorithm 1.

The analytical solutions for the moments were split according to first order split-ting and the splitting error scales as ∆t∗(ratio of time step to the characteristic timeof an event). On implementing the Monte Carlo particle method with time splitting,the convergence studies suggest that the systematic error is inversely proportional tothe number of particles, N . The statistical error was observed to converge at the rateN−

12 .The three inflow-outflow algorithms were implemented in the Monte Carlo particle

method. As compared to Algorithms 2 and 3, Algorithm 1 requires fewer particles N ,but at the expense of L simulation trials. From the model reaction A −→ B and thecombustion of kerosene studies, it can be concluded that the dimensionless time step∆t∗ must be less than unity when the characteristic times for inflow-outflow, reaction,and mixing are of comparable order of magnitude. However, with the residence timevery small compared to the characteristic times for reaction and mixing, Algorithm 3can be employed with the condition ∆tτ > 1 and is an order of magnitude faster thanAlgorithm 2, thus saving the computational expense.

From the view point of a compromise between accuracy and computational ex-pense while applying PDF/Monte Carlo methods for simulating a practical combus-tion case, convergence at steady as well as transition states should be studied forsetting N and ∆t.

Acknowledgments. The authors thank Mike Goodson, Uri Zarfaty (Universityof Cambridge), and Wolfgang Wagner (Weierstrass Institute for Applied Analysis andStochastics, Germany) for their mathematical expertise and help.

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PARTIALLY STIRRED REACTOR MODEL: ANALYTICAL ......NUMERICAL INVESTIGATION OF PaSR MODEL 1799 and reaction is given in [16]. Over the past decade, with a substantial improvement in