Turbulent combustion of carbon-air 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 dust-air

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

Key-Words: - 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.[1-4] 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 [5-11]. All of the processes are greatly

influenced by turbulence [12-17]; 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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phase flow we use the modified k-epsilon model

and investigate combustion and heat and mass

transfer in the phase [19-20].

• 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 k-epsilon 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 n-thcell

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

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ISBN: 978-960-6766-97-8

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

gas-phase 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 gas-phase

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 Lam-Bremhorst

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

particulate-phase 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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ISBN: 978-960-6766-97-8

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 flux-corrected transport (FCT)

techniques; the parabolic part was solved implicitly

using 3-diagonal matrix solvers for linear equations.

The scheme gas dynamic part was validated by

comparing with standard solutions. The physical

model for gas-particles flows and phenomenological

laws for phases interactions were validated by

comparing the numerical modeling results of

multiphase hydrocarbon-air mixtures combustion

with the shock-tube experiment results. The dust

combustion model was compared with experiments

on organic dusts combustion [8-11].

3 Results

The numerical simulations were performed for

carbon dust-air 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

10-20 ms just after switching off an ignitor; the

developed flame stage at 20-50 ms; the final flame

propagation stage at 50-80 ms and further processes

evolution. Time limits strongly depend on the

mixture composition, initial turbulization, initial

dust-phase density, oxygen concentration, etc.

Figures 2, 3, 4, 5 illustrate combustion parameters

dynamics in time: temperature

T

, dust-phase 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 dust-phase mass diminution

due to its converting into the gas phase. The

pressure maximum is observed at 70-80 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 dust-phase 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. Dust-phase 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 6-7 illustrate the

evolution of the mean pressure and total

( )

P t

dust-phase 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 gase-phase

mass augmentation (influencing gas-phase

temperature). For the case of not very high initial

density (0.1-0.2 kg/m

3

) the pressure increases up

to 12 bar, for higher density – up to 9-11 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 air-dust mixture.

• The initial dust-phase density effect on pressure

and dust-phase mass were invstigated. It is

shown that after pressure equals its maximum

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value, it decreases rapidly (for the numerical

simulations featuring dust-phase 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 dust-phase 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-

08-00029)

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 20-22, 2008

ISSN: 1790-5095

54

ISBN: 978-960-6766-97-8

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