Turbulent combustion of carbonair aerosols.
NIKITIN V.F.(1), PARKHOMENKO A.S. (1), YANUSHKEVICH V.N. (1), LEGROS J.C. (2),
(1) Faculty of Mech. and Math., Moscow M.V.Lomonosov State University, Moscow 119992,
Russia
(2) Microgravity Research Centre, Free University of Brussels, CP 165/62 Brussels 1050, Belgium
Abstract:  A mathematical model for ignition and turbulent combustion of polydispersed dustair
mixtures in cylindrical vessel is developed. It uses both deterministic methods of continuum mechanics of
multiphase flows to determine the mean values of the gaseous phase parameters, and stochastic methods to
depict the evolution of polydispersed particles in it and fluctuations of paremeters. The equations of
motions for particles consider the influence of random turbulence pulsations in gas flow. Thermal
destruction of dust particles, vent of volatiles, chemical reactions in the gas phase, carbon skeleton
heterogeneous oxidation by , and chemical reaction with are the processes essential for
describing dust and particulate phases. The mathematical model developed makes it possible to investigate
the peculiarities of polydispersed organic dusts ignition and combustion and the influence of flow
nonuniformities on the ignition limits.
2
O
2
CO
2
H O
KeyWords:  Combustion, turbulence, heat flux, dust, chemical reactions, diffusion, mathematical simulation
1 Introduction
Investigation of the ignition and combustion of
polydispersed mixtures is important for the
description of dust explosion accidents, developing
preventive measures in industry, and working out
methods for testing ignitability and explosibility of
dusts.[14] To describe the processes taking place in
motor chambers and burners of different types and to
develop measures for improving their perfomance
characteristics the models for ignition and
combustion of polydispersed mixtures are created.
The processes of polydispersed mixtures
combustion are rather complex due to the effects of
turbulent interaction of gaseous and dispersed
phases, heat and mass transfer, thermal destruction
and volatiles extraction, gaseous and heterogeneous
chemistry [511]. All of the processes are greatly
influenced by turbulence [1217]; and a
polydispersed phase which is distributed
stochastically makes the experimental investigations
very difficult because of nonreproducibility of the
initial conditions at microscales and the lack of
possibility to vary a single model parameter keeping
the rest of them constant in the experiments. The
ignition limits sensitivity to the properties of mixture
– composition, size and shape of particles,
turbulence, dust concentration and distribution, gas
phase composition  can hardly be investigated
experimentally as in this case only groups of
parameters vary. Besides, the stabilization of the
mixture initial properties in different experiments is
hardly possible. Hence, numerical investigations
provide a unique possibility to study the influence
of dust parameters on the ignition process.
The present research aims at obtaining
numerical solutions for turbulent combustion in the
heterogeneous mixtures of gas with polidispersed
particles, and studying theoretically the influence of
governing parameters on the mixture
characteristics. It uses both deterministic methods
of continuum mechanics of multiphase flows (to
determine the mean values of the gaseous phase
parameters) and stochastic methods (to depict the
evolution of polydispersed particles in it and
fluctuations of parameters). Thermal destruction of
dust particles, vent of volatiles, chemical reactions
in the gas phase, and carbon heterogeneous
oxidation by and and chemical reaction
with are the processes essential for the
models of phase transitions and chemical reactions
[18]. Thus accounting for these effects was
incorporated into the current numerical model.
2
O
2
CO
2
H O
2 Mathematical model
The characteristic peculiarities of the
mathematical model are as follows:
• The Euler approach is used to describe the gas
phase, and the Lagrange method enables to
model that of dust.
• The turbulent flows take place into both gas
and particulate phases. To simulate the gas
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ISSN: 17905095
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ISBN: 9789606766978
phase flow we use the modified kepsilon model
and investigate combustion and heat and mass
transfer in the phase [1920].
• Combustion is developing both on the particle
surface and in the gas phase. Three reactions are
assumed to take place on a particle surface.
Futhermore, there is a volatiles (L) extraction
from the carbon dust.
( )
,
p
L →
L
,22
2
COOC →+
2
2C CO CO+ →
2 2
C H O CO H
+
→ +
Here L denotes generalized volatile component.
It is supposed that the volatiles being extracted from
particles consist of the following components:
L O CO CO HO N CH H
O CO CO HO N CH H
= + + + + + +ν ν ν ν ν ν ν
2 2 2 2 4
2
2
0
2
0 0
2
0
2
0
2
0
4
0
We use the modified kepsilon model to describe the
gas phase behavior. The generalization of this model
concentrates on the influence of other phases (mass,
momentum, energy fluxes from the particles phase)
as well as the combustion and heat and mass transfer
in the gas phase. The system of equations for the gas
phase was obtained by Favre averaging the system
of multicomponent multiphase media with
weight
α
ρ
. [8, 20].
To simulate the polydispersed particulate phase
a stochastic approach is perfomed, for that a group
of representative particles is distinguished. The
effect of the gas mean stream and pulsations of
parameters in the gas phase on the motion of
particles is considered. Gas flow properties (the
mean kinetic energy and the rate of pulsations
decay) make it possible to model the particles
stochastic motion determining the pulsations change
frequency under assumptions of the Poisson flow of
events.
We model a great amount of real particles by
assembling model particles (their number is of the
order of thousands). Each model particle is
characterized by a vector of values, representing its
location, velocity, mass and other properties. The
following vector is determined for each model
particle:
{
}
,,,,,,,
s
s
i
N m m r u w Tω
,
i N
.
1,...,
p
=
When a particle is burnt out, its mass
m
0
i
=
, and
this particle is excluded from calculations. The
processes of adhesion and fragmentation are not
considered.
The parameter is used to model dust phase
turbulent pulsations. It represents the pulsation
velocity vector added to the gas velocity vector
i
w
v
,
thus a stochastic force resulting from the
interaction with turbulized gas is calculated
jointly with a resistance force. The laws of
particle motion are as follows:
( )
( )
i
f
i si ti fi si ti
i
i
du
m m m m m m g f
dt
dr
u
dt
ri
+
+ = + + +
=
where the force affecting the particle consists of
gravity, Archimedean forces, resistance force and a
stochastic force resulting from the interaction with
turbulized gas. We neglect several forces such as
associate mass forces, Magnus force, Basse force.
Fig. 1. Scheme of particle burning.
It is supposed that the volatiles are extracted
from dust (Fig. 1), and three brutto reactions are
assumed to take place on a particle surface:
( )
,
p
L L→
,22
2
COOC →
+
2
2C CO CO
+
→
2 2
C H O CO H
+
→ +
.
To calculate the volatiles extraction rate
and carbon decay rates from
i
th
model particle, the solutions obtained for quasi
steady heterogeneous combustion of spherical
particles are applied. To generalize the solutions
certain modifications were perfomed. First, in order
to take into account the interacion between
particles, boudary conditions are constructed at the
characteristic radius , where
L
m
1 2
, ,
c c c
m m m
3
r R=
R
is determined
by the formulas:
3 3
0
4 4
 particle volume,
3 3
1
 particles volume content in the nthcell
n
n i i
i
n
r R
N
ω π π β
β ω
∈Ω
= =
=
Ω
∑
Introducing new variables and modifying the
diffusion coefficient we obtain the solution for
the complicated problem with the finite boundary
conditions. Second, one more correction is applied
to the diffusion coefficient to take under
D
6th IASME/WSEAS International Conference on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT (HTE'08)
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ISSN: 17905095
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ISBN: 9789606766978
consideration unsteady condition and diffusion
modifications
1 3
0,5
1
D N
D
u
α
⋅
→
−
,
2 3 1 3
2 0.16Re PrNu = +
The solution obtained for
makes it possible to calculate particles mass rate as
follows:
1 2
,,,
L c c c
m m m m
5
1 2 5
, , 0
fi
si ti
L c c c
dm
dm dm
m m m m
dt dt dt
= − = − − − =
The internal energy of the th model particle is
determined by the formula:
i
(
)
(
)
(
)
0 0
0
si
E
s
i s si s fi L si L ti T si T
m c T h m c T h m c T h= + + + + +
where
,,
s
L T
c c c
denote the heat capacity of carbon
skeleton, condensed volatile component and tar
respectively,
0 0 0
,,
s
L T
h h h
denote the enthalpy of the
gasphase chemical reactions.
To model the internal energy balance the
following equation is used:
( )
si
f
i si ti i si
de
m m m q Q e
dt
+ + = + +
ei
si
,
where is a heat flux between the gas and the
particle:
i
q
( )
i i i
q d Nu T T
π λ= −
(
λ
denotes a thermal conductivity calculated while
the particle temperature equals
s
i
T
; is the
temperature of a surrounding gas).
T
The specific internal energy of particle
s
i
e
is equal
to the ratio of full energy
s
i
E
to the particle mass:
0
fi L fi
si si s ti T
s
i
i i i i i
m c m
E m c m c
e
m m m m m
⎛ ⎞
= = + + +
⎜ ⎟
⎝ ⎠
si L
T h
.
The term Q
si
represents heat extraction due to
chemical reactions on the
i
th model particle surface.
The term determines the energy influx to the
th model particle from the energy source.
ei
e
i
Further the fluxes from model particles are
considered, and their recalculation into gasphase
equations is performed.
The boundary conditions for the gas phase are
constructed in accordance with the following
considerations: the walls of the cylindrical domain
are noncatalytic, the gas velocity is zero at the walls,
and the averaged gas motion has cylindrical
symmetry. This leads to Neumann’s conditions for
temperature and mass fractions at the vessel walls:
0, 0, 0
k
x r
Y T
v v
n
∂ ∂
= = =
∂
n
=
∂
,
where n is the normal vector to the wall. The
boundary conditions for turbulent parameters and
k
ε
are constructed according to the LamBremhorst
model:
0
=
k
,
0
n
ε
∂
=
∂
.
Boundary conditions at the symmetry axis
( 0
)r
=
are constructed to satisfy those of the flow
characteristics continuous differentiability:
0, 0, 0, 0, 0, 0
k x
r
Y v T k
v
r r r r r
ε
∂
∂ ∂ ∂ ∂
=
= = = = =
∂ ∂ ∂ ∂ ∂
.
The following boundary conditions for
particulatephase motion are considered: particles
are bouncing against the solid walls of the
cylindrical vessel without kinetic energy loss. It
means that a particle velocity normal component
changes its sign after the particle collides with the
wall, and the tangential component keeps its value
and direction.
The initial conditions for the gas phase include:
zero average velocity, given gas temperature
distribution, the species mass fraction, and the
turbulence parameters (taken uniform in our
investigations).
As to initial conditions for the particulate
phase, each model particle is distributed within the
cylindrical computational domain using a given
density of the particulate phase. The exact model
particle localization is generated by random
counters. The particle temperature, shape, and
condensed volatiles quantity are constant, thus we
determine mass and other model particle properties.
Ignition process was modelled as an energy
release in a relatively small volume inside the
vessel with the power as
a given function of time.
Calculations are
performed in terms of
cylindrical geometry
with the uniform grid
61x41 and 10000 model
particles. Each time step
contains the model
particles motion calculations, determining fluxes
from particles to the gas phase and recalculating
them to the grid. Particle motion was computed
using an iterative implicit algorithm for each
particle independently. Then two iterations were
undertaken to calculate gas dynamics parameters
concerning fluxes from the particulate phase. At
each iteration the space splitting in x and r
coordinates is performed as well as the splitting in
the following physical processes: chemistry, source
terms and turbulent energy production (local part of
the equations), convection (hyperbolic part),
diffusion (parabolic part). The applied operator
splitting techniques represent the general operator
(
)
L
t
∆
transferring the parameters vector
n
P
to
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ISSN: 17905095
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ISBN: 9789606766978
the next time step: .
1n
P
+
( )
1n n
P L t P
+
= ∆
(
)
L t
∆
is then represented in a special form by means of
two operators, each split into three parts: parabolic,
hyperbolic and local. The local part is solved
implicitly using an iterative algorithm independently
for each grid node. The hyperbolic part was solved
using explicit fluxcorrected transport (FCT)
techniques; the parabolic part was solved implicitly
using 3diagonal matrix solvers for linear equations.
The scheme gas dynamic part was validated by
comparing with standard solutions. The physical
model for gasparticles flows and phenomenological
laws for phases interactions were validated by
comparing the numerical modeling results of
multiphase hydrocarbonair mixtures combustion
with the shocktube experiment results. The dust
combustion model was compared with experiments
on organic dusts combustion [811].
3 Results
The numerical simulations were performed for
carbon dustair mixtures combustion in a closed
cylindrical 1.25 m
3
vessel. The mixture was ignited
in the vessel center by energy release in a ball
shaped volume. The flame boundaries were detected
along five central rays (in horizontal, vertical
directions and 45º to the horizon) as the points of
volatiles oxidizing intensity maximal gradients on
each ray.
According to the numerical computations the
combustion process can be divided into the
following stages: the initial flame ball formation at
1020 ms just after switching off an ignitor; the
developed flame stage at 2050 ms; the final flame
propagation stage at 5080 ms and further processes
evolution. Time limits strongly depend on the
mixture composition, initial turbulization, initial
dustphase density, oxygen concentration, etc.
Figures 2, 3, 4, 5 illustrate combustion parameters
dynamics in time: temperature
T
, dustphase mass
( )
P
M
t
, mean pressure in the vessel
(
)
P t
and
flame front position
( )
f
R
t
respectively.
( )
P t
and
( )
P
M
t
graphs correlate greatly:
pressure increases with dustphase mass diminution
due to its converting into the gas phase. The
pressure maximum is observed at 7080 ms after
flame front passing. It is explained by the fact that
almost all dust phase is heated at the moment, and
pyrolysis amounts to its maximum. As time
exceeds 80 ms, the pressure growth stops abruptly,
and dustphase mass approximates to the value of
tar. The effect is caused by the completion of
volatiles pyrolysis and carbon skeleton oxidation.
Fig. 2. Temperature evolution in time.
Fig. 3. Dustphase mass evolution in time.
Fig. 4. Pressure evolution in time.
Fig. 5. Flame front evolution in time.
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Considering the front position graph (figure 5),
blue and red lines (being in close agreement)
correspond with the flame front position according
to the gradient maximum of oxidizer and
temperature. Discrepancy is only essential at the
ignition stage as a result of energy source action.
By 65 ms fore front approaches to the vessel wall.
Up to 57 ms it is moving behind the flame central
axis, and front width is nearly constant. Then the
front return to the vessel center is registered, and
after 65 ms its motion is unstable, because as the
flame fore front comes to the walls, the flame
width and intensity increase rapidly, while
intensity peak decreases. As a result the
position=1/10 of intensity maximum moves back.
It is derived from the graphs that at the
beginning the velocity of flame propagation
equals 9,6 m/s, and as the flame approaches the
walls it equals 5,8 m/s. The mean velocity value
is 7,9 m/s.
The set of numerical simulations was
performed for different initial density values:
ρ
= {0,1; 0,4;} kg/m
3
. Figures 67 illustrate the
evolution of the mean pressure and total
( )
P t
dustphase mass
( )
f
M
t
with the time for the
pointed densities correspondingly.
Fig. 6. Pressure and dust phase mass evolution in
time for
0.1
ρ
=
.
Fig. 7. Pressure and dust phase mass evolution in
time for
0.4
ρ
=
.
All experiments testify a strong correlation
between the pressure and the particles mass;
pressure increases first of all due to gasephase
mass augmentation (influencing gasphase
temperature). For the case of not very high initial
density (0.10.2 kg/m
3
) the pressure increases up
to 12 bar, for higher density – up to 911 bar.
When
0.4
ρ
=
kg/m
3
, the pressure abrupt
decrease is observed after its maximum value.
This is a result of irreversible reactions
2
2C CO CO
+
→
2 2
C H O H CO+ → +
,
becoming more intensive due to fuel surplus
and the lack of oxidizer. Experiments provide
the same effect mainly because of heat fluxes
through walls.
4 Conclusions
• A computational model describing the turbulent
combustion dynamics in heterogeneous
mixtures of gas with polydispersed particles is
modified. The present model takes into account
complicated kinetics thus specifying solutions.
The model enables one to determine
peculiarities of turbulent combustion of
polydispersed mixtures.
• The integral combustion parameters dynamics
was analyzed for the case of airdust mixture.
• The initial dustphase density effect on pressure
and dustphase mass were invstigated. It is
shown that after pressure equals its maximum
6th IASME/WSEAS International Conference on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT (HTE'08)
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ISBN: 9789606766978
value, it decreases rapidly (for the numerical
simulations featuring dustphase density
exceeding 0.2 kg/m
3
). The effect is explained by
irreversible endothermic reactions occuring due
to fuel abundance and lack of oxygen. Hence
mean temperature in the vessel goes down
producing the pressure decrease.
• The comparison of flame propagation character
is performed for the set of num
erical simulations
with different initial dustphase density. The
mean flame propagation velocity is calculated;
its increase resulting from density augmentation
is discovered.
Acknowledgements
We gratefully acknowledge the support of the
Russian Foundation for Basic Research (RFBR 06
0800029)
and NATO Science for Peace Program
CBP.NR.NRCLG 983264.
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6th IASME/WSEAS International Conference on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT (HTE'08)
Rhodes, Greece, August 2022, 2008
ISSN: 17905095
54
ISBN: 9789606766978
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