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Use of a Monte Carlo PDF method in a study of the influence of turbulent fluctuations on selectivity in a jet-stirred reactor

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Abstract

A model for single-phase turbulent reacting flow is presented and a solution algorithm is described. The model combines the standardk - ε model for the velocity field with a transport equation for the probability density function (PDF) of the thermochemical variables. In this equation terms describing spatial transport by velocity fluctuations and mixing on the smallest scales are modelled. The essential advantage of this approach is that the effect of nonlinear kinetics appears in closed form and that the influence of turbulent fluctuations on mean reaction rates is included. A stochastic algorithm for the solution of the PDF transport equation, essentially due to Pope, is described. Cylindrical symmetry is assumed. The PDF is represented by ensembles ofN representative values of the thermochemical variables in each cell of a nonuniform finite-difference grid and operations on these elements representing convection, diffusion, mixing and reaction are derived. A simplified model and solution algorithm which neglects the influence of turbulent fluctuations on mean reaction rates is also described. Both algorithms are applied to a selectivity problem in a real reactor studied earlier by Liu and Barkelew. Spatial profiles of mean species mole fractions and of relative selectivity to the target product are obtained. The profiles are clearly different in both models but at the end of the reactor the same selectivity is predicted.

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Abbreviations

cx :

index referring to convection inx-direction

cy :

index referring to convection iny-direction

dx :

index referring to diffusion inx-direction

dy :

index referring to diffusion iny-direction

f φ :

composition probability density function (PDF)

f Uφ :

velocity-composition joint PDF

\(\tilde f_\phi \) :

composition PDF for density-weighted averages

\(\tilde f_{U\phi } \) :

velocity-composition joint PDF for density-weighted averages

g :

body force per unit mass

h :

specific enthalpy

i :

index referring to direction in position space

i :

index referring to value of axial coordinate

j :

index referring to direction in position space

j :

index referring to value of radial coordinate

m :

number of species

m :

index referring to mixing

n cx(i,j):

defined by equation (4.6)

n cy(i,j):

defined by equation (4.8)

n Wdx (i,j):

defined by equation (4.10)

n Edx (i,j):

defined by equation (4.12)

n Sdy (i,j):

defined by equation (4.14)

n Ndy (i,j):

defined by equation (4.16)

n m(i,j):

defined by equation (4.18)

p :

pressure

p 0 :

reference pressure

t max :

upper bound of time step associated with any process

x i :

coordinate in position space

x i :

x-coordinate of cell center

\(\bar x_i \) :

x-coordinate of cell face

y j :

y-coordinate of cell center

\(\bar y_j \) :

y-coordinate of cell face

A :

distribution function in mixing model

C µ :

model constant

J α :

flux vector of scalar φα

N :

number of elements in Monte Carlo ensemble

N l :

number of grid cells in axial direction

N J :

number of grid cells in radial direction

Q :

any fluctuating quantity

〈Q〉:

expectation value ofQ

\(\tilde Q\) :

Favre average ofQ

Q :

Favre average ofQ

Q″:

Favre fluctuation ofQ

\(\tilde Q\) :

source of speciesα

T :

temperature

U :

radial component of density-weighted mean velocity

U:

velocity vector

V :

axial component of density-weighted mean velocity

V :

coordinate vector in velocity space

X A :

mole fraction of speciesA

Y A :

mass fraction of speciesA

α :

index referring to scalar variable

ε :

dissipation of turbulent energy

ɛφ :

dissipation of scalar

μ T :

turbulent viscosity

ρ :

density

σ :

number of scalar variables

σ T :

turbulent Schmidt number

τ :

turbulence time

τ ij :

stress tensor

φα :

scalar variable, also called composition variable

φ :

vector of scalar variables

φ ij :

scalar variables in grid celli, j

\(\underline \phi _{ij}^n \) :

elementn of ensemble of scalar variables in celli, j

ψ :

coordinate in composition space

ω :

mixing frequency

Γ T :

turbulent diffusivity

Δt :

time step

\(\underline \Phi _{ij} \) :

ensemble of scalars in grid celli, j

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Authors and Affiliations

  1. Koninklijke/Shell-Laboratorium, Amsterdam, (Shell Research B.V.), P.O. Box 3003, 1003 AA, Amsterdam, The Netherlands

    D. Roekaerts

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Roekaerts, D. Use of a Monte Carlo PDF method in a study of the influence of turbulent fluctuations on selectivity in a jet-stirred reactor. Appl. Sci. Res. 48, 271–300 (1991). https://doi.org/10.1007/BF02008201

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