ISSN: 2180-1053 Vol. 4 No. 1 January-June 2012

Counterflow Combustion of Micro Organic Particles

Counterflow Combustion of miCro organiC
PartiCles

mehdi bidabadi1, ali esmaeilnejad2, sirousfarshadi

Production technology department -Industrial Education
College - Beni-Suef University,

Industrial engineering department –jazan university.

Email: waleedshewakh@hotmail.com

ABSTRACT

The structure ofcounterflowpremixed flames in an axisymmetric
configuration, containing uniformly distributed volatile fuel particles,
with nonunity Lewis number of the fuel are examined. It is presumed that
the gaseous fuel, produced from vaporization of the fuel particles, oxidizes
in the gas phase and the fuel particles do not participate in the reaction. The
analysis is carried out in the asymptotic limit for large values of Zeldovich
number.A one-step reaction

Counterflow Combustion of Micro Organic Particles

Mehdi Bidabadi1, Ali Esmaeilnejad2, Sirousfarshadi

Faculty of Mechanical Engineering, Iran University Of Science and Technology, Tehran, Iran

Corresponding email: sirous.farshadi@gmail.com

ABSTRACT

The structure ofcounterflowpremixed flames in an axisymmetric configuration, containing
uniformly distributed volatile fuel particles, with nonunity Lewis number of the fuel are examined. It
is presumed that the gaseous fuel, produced from vaporization of the fuel particles, oxidizes in the
gas phase and the fuel particles do not participate in the reaction. The analysis is carried out in the
asymptotic limit for large values of Zeldovich number.A one-step reaction

is assumed. The flame position is determined andthe effect of Lewis number change on
the gaseous fuel mass fraction distribution is investigated.

KEYWORDS: counterflow combustion; nonunity Lewis number; organic particles; particle
combustion

1.0 INTRODUCTION

Many studies of dust clouds combustion have been published over the last few years; see, for
example, introduction of the article on flame propagation through micro-organic dust particles
(Bidabadi et al., 2010). But from the fact that in many practical applications the flow field is
appreciably strained, to yield realistic flame prediction under such conditions, counterflow
configuration is suitable for studying these cases.Over the last few decades, the counterflow
configuration has been extensively adopted in theoretical, experimental and numerical studies as a
means to investigate various physical effects on real flames on real flames, such as stretch,
preferential diffusion, radiation and chemical kinetics (Daou, 2011), (Thatcher et al.) and (Wang et
al., 2007). But these studies are done for gaseous fuels. Eckhoff clarified the differences and
similarities between dust and gases (Eckhof, 2006). It has been concluded that there are two basic
differences between dusts and gases which are of substantially greater significance in design of
safety standards than these similarities. Firstly, the physics of generation and up-keeping of dust
clouds and premixed gas/vapor clouds are substantially different. Secondly, contrary to premixed
gas flame propagation, the propagation of flames in dust/air mixtures is not limited to the
flammable dust concentration range of dynamic clouds. Thus here we modeled counterflow
combustion of dust clouds and the effect of Lewis number on gaseous fuel mass fraction
distribution is investigated.

A model is developed for describing a one dimensional, axisymmetric, premixed flame in a
counterflow configuration. Uniformly distributed volatile fuel particles in air are considered as the
entering materials. The initial number density of the particles, (number of particles per unit
volume) and the initial radius are presumed to be known.

vproduct [P] is assumed. The
flame position is determined andthe effect of Lewis number change on the
gaseous fuel mass fraction distribution is investigated.

KEYWORDS: Lapping, grinding, surface roughness, micro, nano scale.

1.0 introDuCtion

ISSN: 2180-1053 Vol. 4 No. 1 January-June 2012

Journal of Mechanical Engineering and Technology

dusts and gases which are of substantially greater significance in
design of safety standards than these similarities. Firstly, the physics
of generation and up-keeping of dust clouds and premixed gas/vapor
clouds are substantially different. Secondly, contrary to premixed gas
flame propagation, the propagation of flames in dust/air mixtures is not
limited to the flammable dust concentration range of dynamic clouds.
Thus here we modeled counterflow combustion of dust clouds and the
effect of Lewis number on gaseous fuel mass fraction distribution is
investigated.

A model is developed for describing a one dimensional, axisymmetric,
premixed flame in a counterflow configuration. Uniformly distributed
volatile fuel particles in air are considered as the entering materials.
The initial number density of the particles, nu (number of particles per
unit volume) and the initial radius ru are presumed to be known.

In the analysis it is presumed that the fuel particles vaporize to form
a known gaseous compound which is then oxidized. In other words
particles do not participate in the reaction. The kinetics of vaporization
are presumed to be of the form:

In the analysis it is presumed that the fuel particles vaporize to form a known gaseous compound
which is then oxidized. In other words particles do not participate in the reaction. The kinetics of
vaporization are presumed to be of the form:

Where the units of are mass of gaseous fuel vaporized per unit volume per second. The quantity
is constant which is presumed to be known, and denotes the gas temperature.For simplicity it is

assumed that gas and particle have same temperature.

The combustion process is represented by one –step irreversible reaction of the form

Where the symbols , and denote the fuel, oxygen and product, respectively, and the
quantities , , and denote their respective stoichiometric coefficients. The Zeldovich
number is presumed to be large and is defined as

In this paper subscript denotes conditions in the flame and the subscript denotes conditions at
the inlet. The quantities and R represent respectively the activation energy and the universal gas
number.

Figure 1 The counterflow configuration and twin planar premixed flames

In Figure 1 we have illustrated the considered configuration including the planar twin flames.
Reactants enter from direction and exhaust gases exit in the direction. The flame structure
consists of a broad preheat-vaporization zone, a thin reaction zone and a post flame zone. In the
formulation,subscripts , and are respectively used to show these zones. To determine the
structure of these zones, a number of approximations are introduced in the conservation equations
governing their structure.In the preheat-vaporization zone the rate of reaction between the fuel
andoxidizer is presumed to be small and the structure of this layer is determined from a balance
between the convective, diffusive, and vaporization terms in the conservation equation. In the thin
reaction zone the convective and vaporization terms are presumed to be small in comparison with

(1)

Where the units of wv are mass of gaseous fuel vaporized per unit
volume per second. The quantity Ais constant which is presumed to be
known, and T denotes the gas temperature. For simplicity it is assumed
that gas and particle have same temperature.

The combustion process is represented by one–step irreversible reaction
of the form

In the analysis it is presumed that the fuel particles vaporize to form a known gaseous compound
which is then oxidized. In other words particles do not participate in the reaction. The kinetics of
vaporization are presumed to be of the form:

Where the units of are mass of gaseous fuel vaporized per unit volume per second. The quantity
is constant which is presumed to be known, and denotes the gas temperature.For simplicity it is

assumed that gas and particle have same temperature.

The combustion process is represented by one –step irreversible reaction of the form

Figure 1 The counterflow configuration and twin planar premixed flames

(2)

Where the symbols F, O2 and P denote the fuel, oxygen and product,
respectively, and the quantities vF, vO2, and vproduct denote their respective
stoichiometric coefficients. The Zeldovich number is presumed to be
large and is defined as

In the analysis it is presumed that the fuel particles vaporize to form a known gaseous compound
which is then oxidized. In other words particles do not participate in the reaction. The kinetics of
vaporization are presumed to be of the form:

Where the units of are mass of gaseous fuel vaporized per unit volume per second. The quantity
is constant which is presumed to be known, and denotes the gas temperature.For simplicity it is

assumed that gas and particle have same temperature.

The combustion process is represented by one –step irreversible reaction of the form

Figure 1 The counterflow configuration and twin planar premixed flames

(3)

In this paper subscript f denotes conditions in the flame and the subscript

ISSN: 2180-1053 Vol. 4 No. 1 January-June 2012

Counterflow Combustion of Micro Organic Particles

u denotes conditions at the inlet. The quantities E and R represent
respectively the activation energy and the universal gas number.

In the analysis it is presumed that the fuel particles vaporize to form a known gaseous compound
which is then oxidized. In other words particles do not participate in the reaction. The kinetics of
vaporization are presumed to be of the form:

Where the units of are mass of gaseous fuel vaporized per unit volume per second. The quantity
is constant which is presumed to be known, and denotes the gas temperature.For simplicity it is

assumed that gas and particle have same temperature.

The combustion process is represented by one –step irreversible reaction of the form

Figure 1 The counterflow configuration and twin planar premixed flames

Figure 1 The counterflow configuration and twin planar premixed
flames

In Figure 1 we have illustrated the considered configuration including
the planar twin flames. Reactants enter from ±x direction and exhaust
gases exit in the ±y direction. The flame structure consists of a broad
preheat-vaporization zone, a thin reaction zone and a post flame zone.
In the formulation,subscripts1, 2 and 3are respectively used to show
these zones. To determine the structure of these zones, a number of
approximations are introduced in the conservation equations governing
their structure.In the preheat-vaporization zone the rate of reaction
between the fuel andoxidizer is presumed to be small and the structure
of this layer is determined from a balance between the convective,
diffusive, and vaporization terms in the conservation equation. In the
thin reaction zone the convective and vaporization terms are presumed
to be small in comparison with the diffusive and reactive terms.It is
assumed that all of the particles vaporizes just before the flame, thus in
the post flame zone the vaporization term is not considered.

As it has been mentioned in (Seshadri et.al,. 1992) for large values of
nu, ϕu>0.7, where ϕu is equivalence ratio based on fuel available in
the particles in the ambient reactant stream, the standoff distance of
the envelope flame surrounding each particle is much larger than
the characteristic separation distance between the particles. Thus, the
analysis developed here is only valid for ϕu>0.7.

The velocity field has components (-ax,ay,0)in the Cartesian directions,
where a is the (dimensional) strain rate. For small values of strain rate
we can consider the problem as one dimensional. All external forces are

ISSN: 2180-1053 Vol. 4 No. 1 January-June 2012

Journal of Mechanical Engineering and Technology

100

assumed to be negligible. Also diffusion caused by pressure gradient
is neglected.

It is also assumed that the ratio

the diffusive and reactive terms.It is assumed that all of the particles vaporizes just before the flame,
thus in the post flame zone the vaporization term is not considered.

As it has been mentioned in (Seshadri et al,. 1992) for large values of , , where is
equivalence ratio based on fuel available in the particles in the ambient reactant stream, the standoff
distance of the envelope flame surrounding each particle is much larger than the characteristic
separation distance between the particles. Thus, the analysis developed here is only valid for

The velocity field has components in the Cartesian directions, where is the
(dimensional) strain rate. For small values of strain rate we can consider the problem as one
dimensional. All external forces are assumed to be negligible. Also diffusion caused by pressure
gradient is neglected.

It is also assumed that the ratio is constant, where is the density of the mixture of gas and the
fuel particles.

The one dimensional governing equations are

In Eqs. 4-7, denotes density, and represent diffusion coefficients for heat and fuel
respectively, is the reaction rate and it's unit is mass of gaseous fuel consumed per unit volume
per second, stands for the heat release per unit mass of the fuel burned, is the heat associated
with vaporizing unit mass of the fuel, is the mass fraction of the fuel, is the combined heat
capacity of the gas and of the particles and denotes mass fraction of the particles. Further
approximation introduced in Eqs. 4-7 are that the mean molecular weight do not vary and that the
thermal conductivity of the mixture, , is constant and the diffusion coefficient, , is proportional
to . Also particles diffusion is neglected and the density of a fuel particle, , is presumed to be
constant.

Given the symmetry of the configuration about the plane , we only solve the problem for
with the boundary conditions

is constant, where ρis the density of
the mixture of gas and the fuel particles.
The one dimensional governing equations are

the diffusive and reactive terms.It is assumed that all of the particles vaporizes just before the flame,
thus in the post flame zone the vaporization term is not considered.

It is also assumed that the ratio is constant, where is the density of the mixture of gas and the
fuel particles.

The one dimensional governing equations are

Given the symmetry of the configuration about the plane , we only solve the problem for
with the boundary conditions

(4)

the diffusive and reactive terms.It is assumed that all of the particles vaporizes just before the flame,
thus in the post flame zone the vaporization term is not considered.

It is also assumed that the ratio is constant, where is the density of the mixture of gas and the
fuel particles.

The one dimensional governing equations are

Given the symmetry of the configuration about the plane , we only solve the problem for
with the boundary conditions

(5)

the diffusive and reactive terms.It is assumed that all of the particles vaporizes just before the flame,
thus in the post flame zone the vaporization term is not considered.

It is also assumed that the ratio is constant, where is the density of the mixture of gas and the
fuel particles.

The one dimensional governing equations are

Given the symmetry of the configuration about the plane , we only solve the problem for
with the boundary conditions

(6)

the diffusive and reactive terms.It is assumed that all of the particles vaporizes just before the flame,
thus in the post flame zone the vaporization term is not considered.

It is also assumed that the ratio is constant, where is the density of the mixture of gas and the
fuel particles.

The one dimensional governing equations are

Given the symmetry of the configuration about the plane , we only solve the problem for
with the boundary conditions

(7)

In Eqs. 4-7, ρ denotes density, DT and DF represent diffusion coefficients
for heat and fuel respectively, ωF is the reaction rate and it's unit is mass
of gaseous fuel consumed per unit volume per second, Q stands for the
heat release per unit mass of the fuel burned, Qv is the heat associated
with vaporizing unit mass of the fuel,YF is the mass fraction of the
fuel, C is the combined heat capacity of the gas and of the particles
and Ys denotes mass fraction of the particles. Further approximation
introduced in Eqs. 4-7 are that the mean molecular weight do not vary
and that the thermal conductivity of the mixture, λ, is constant and the
diffusion coefficient, D, is proportional to T. Also particles diffusion
is neglected and the density of a fuel particle, ρs, is presumed to be
constant.

Given the symmetry of the configuration about the plane x=0, we only
solve the problem for x>0 with the boundary conditions

the diffusive and reactive terms.It is assumed that all of the particles vaporizes just before the flame,
thus in the post flame zone the vaporization term is not considered.

It is also assumed that the ratio is constant, where is the density of the mixture of gas and the
fuel particles.

The one dimensional governing equations are

Given the symmetry of the configuration about the plane , we only solve the problem for
with the boundary conditions

Where YFu denotes the mass fraction of fuel available in the particles.

Nondimensionalization of governing equations

we define the following rescaled variables

ISSN: 2180-1053 Vol. 4 No. 1 January-June 2012

Counterflow Combustion of Micro Organic Particles

101

Where denotes the mass fraction of fuel available in the particles.

Nondimensionalization of governing equations

we define the following rescaled variables

where is the maximum temperature attained in the reaction zone, calculated neglecting the heat
of vaporization of the particles. The quantity is chosen such that

In Eq. 8, is mixing layer thickness and is defined by .

With introducing Eqs. 8 and 9 into governing equations, we obtain their non-dimensional form

Where the radius has been rewritten in terms of using the relation

Further assumption used in Eqs. 10-12 is that the rate of vaporization is expected to be dominant
near the reaction zone, for , the vaporization rate shown in Eq. was presumed to be

proportional to . The quantities and used above are given by

(8)

where Tf is the maximum temperature attained in the reaction zone,
calculated neglecting the heat of vaporization of the particles. The
quantity YFC is chosen such that

Where denotes the mass fraction of fuel available in the particles.

Nondimensionalization of governing equations

we define the following rescaled variables

where is the maximum temperature attained in the reaction zone, calculated neglecting the heat
of vaporization of the particles. The quantity is chosen such that

In Eq. 8, is mixing layer thickness and is defined by .

With introducing Eqs. 8 and 9 into governing equations, we obtain their non-dimensional form

Where the radius has been rewritten in terms of using the relation

Further assumption used in Eqs. 10-12 is that the rate of vaporization is expected to be dominant
near the reaction zone, for , the vaporization rate shown in Eq. was presumed to be

proportional to . The quantities and used above are given by

(9)

In Eq. 8, L is mixing layer thickness and is defined by

Where denotes the mass fraction of fuel available in the particles.

Nondimensionalization of governing equations

we define the following rescaled variables

where is the maximum temperature attained in the reaction zone, calculated neglecting the heat
of vaporization of the particles. The quantity is chosen such that

In Eq. 8, is mixing layer thickness and is defined by .

With introducing Eqs. 8 and 9 into governing equations, we obtain their non-dimensional form

Where the radius has been rewritten in terms of using the relation

Further assumption used in Eqs. 10-12 is that the rate of vaporization is expected to be dominant
near the reaction zone, for , the vaporization rate shown in Eq. was presumed to be

proportional to . The quantities and used above are given by

With introducing Eqs. 8 and 9 into governing equations, we obtain
their non-dimensional form

Where denotes the mass fraction of fuel available in the particles.

Nondimensionalization of governing equations

we define the following rescaled variables

where is the maximum temperature attained in the reaction zone, calculated neglecting the heat
of vaporization of the particles. The quantity is chosen such that

In Eq. 8, is mixing layer thickness and is defined by .

With introducing Eqs. 8 and 9 into governing equations, we obtain their non-dimensional form

Where the radius has been rewritten in terms of using the relation

Further assumption used in Eqs. 10-12 is that the rate of vaporization is expected to be dominant
near the reaction zone, for , the vaporization rate shown in Eq. was presumed to be

proportional to . The quantities and used above are given by

(10)

Where denotes the mass fraction of fuel available in the particles.

Nondimensionalization of governing equations

we define the following rescaled variables

where is the maximum temperature attained in the reaction zone, calculated neglecting the heat
of vaporization of the particles. The quantity is chosen such that

In Eq. 8, is mixing layer thickness and is defined by .

With introducing Eqs. 8 and 9 into governing equations, we obtain their non-dimensional form

Where the radius has been rewritten in terms of using the relation

Further assumption used in Eqs. 10-12 is that the rate of vaporization is expected to be dominant
near the reaction zone, for , the vaporization rate shown in Eq. was presumed to be

proportional to . The quantities and used above are given by

(11)

Where denotes the mass fraction of fuel available in the particles.

Nondimensionalization of governing equations

we define the following rescaled variables

where is the maximum temperature attained in the reaction zone, calculated neglecting the heat
of vaporization of the particles. The quantity is chosen such that

In Eq. 8, is mixing layer thickness and is defined by .

With introducing Eqs. 8 and 9 into governing equations, we obtain their non-dimensional form

Where the radius has been rewritten in terms of using the relation

Further assumption used in Eqs. 10-12 is that the rate of vaporization is expected to be dominant
near the reaction zone, for , the vaporization rate shown in Eq. was presumed to be

proportional to . The quantities and used above are given by

(12)

Where the radius r has been rewritten in terms of ys using the relation

Where denotes the mass fraction of fuel available in the particles.

Nondimensionalization of governing equations

we define the following rescaled variables

where is the maximum temperature attained in the reaction zone, calculated neglecting the heat
of vaporization of the particles. The quantity is chosen such that

In Eq. 8, is mixing layer thickness and is defined by .

With introducing Eqs. 8 and 9 into governing equations, we obtain their non-dimensional form

Where the radius has been rewritten in terms of using the relation

Further assumption used in Eqs. 10-12 is that the rate of vaporization is expected to be dominant
near the reaction zone, for , the vaporization rate shown in Eq. was presumed to be

proportional to . The quantities and used above are given by

(13)

Further assumption used in Eqs. 10-12 is that the rate of vaporization
is expected to be dominant near the reaction zone, for

Where denotes the mass fraction of fuel available in the particles.

Nondimensionalization of governing equations

we define the following rescaled variables

where is the maximum temperature attained in the reaction zone, calculated neglecting the heat
of vaporization of the particles. The quantity is chosen such that

In Eq. 8, is mixing layer thickness and is defined by .

With introducing Eqs. 8 and 9 into governing equations, we obtain their non-dimensional form

Where the radius has been rewritten in terms of using the relation

Further assumption used in Eqs. 10-12 is that the rate of vaporization is expected to be dominant
near the reaction zone, for , the vaporization rate shown in Eq. was presumed to be

proportional to . The quantities and used above are given by

, the
vaporization rate shown in Eq. 1 was presumed to be proportional to

Where denotes the mass fraction of fuel available in the particles.

Nondimensionalization of governing equations

we define the following rescaled variables

where is the maximum temperature attained in the reaction zone, calculated neglecting the heat
of vaporization of the particles. The quantity is chosen such that

In Eq. 8, is mixing layer thickness and is defined by .

With introducing Eqs. 8 and 9 into governing equations, we obtain their non-dimensional form

Where the radius has been rewritten in terms of using the relation

Further assumption used in Eqs. 10-12 is that the rate of vaporization is expected to be dominant
near the reaction zone, for , the vaporization rate shown in Eq. was presumed to be

proportional to . The quantities and used above are given by

. The quantities γ and ω used above are given by

Where denotes the mass fraction of fuel available in the particles.

Nondimensionalization of governing equations

we define the following rescaled variables

where is the maximum temperature attained in the reaction zone, calculated neglecting the heat
of vaporization of the particles. The quantity is chosen such that

In Eq. 8, is mixing layer thickness and is defined by .

With introducing Eqs. 8 and 9 into governing equations, we obtain their non-dimensional form

Where the radius has been rewritten in terms of using the relation

Further assumption used in Eqs. 10-12 is that the rate of vaporization is expected to be dominant
near the reaction zone, for , the vaporization rate shown in Eq. was presumed to be

proportional to . The quantities and used above are given by

(14)

Where denotes the mass fraction of fuel available in the particles.

Nondimensionalization of governing equations

we define the following rescaled variables

where is the maximum temperature attained in the reaction zone, calculated neglecting the heat
of vaporization of the particles. The quantity is chosen such that

In Eq. 8, is mixing layer thickness and is defined by .

With introducing Eqs. 8 and 9 into governing equations, we obtain their non-dimensional form

Where the radius has been rewritten in terms of using the relation

Further assumption used in Eqs. 10-12 is that the rate of vaporization is expected to be dominant
near the reaction zone, for , the vaporization rate shown in Eq. was presumed to be

proportional to . The quantities and used above are given by

(15)

ISSN: 2180-1053 Vol. 4 No. 1 January-June 2012

Journal of Mechanical Engineering and Technology

102

Where denotes the mass fraction of fuel available in the particles.

Nondimensionalization of governing equations

we define the following rescaled variables

where is the maximum temperature attained in the reaction zone, calculated neglecting the heat
of vaporization of the particles. The quantity is chosen such that

In Eq. 8, is mixing layer thickness and is defined by .

With introducing Eqs. 8 and 9 into governing equations, we obtain their non-dimensional form

Where the radius has been rewritten in terms of using the relation

Further assumption used in Eqs. 10-12 is that the rate of vaporization is expected to be dominant
near the reaction zone, for , the vaporization rate shown in Eq. was presumed to be

proportional to . The quantities and used above are given by

(16)

Where the quantity q is the ratio of the heat required to vaporize the
fuel particles to the overall heat release in the flame and is presumed
to be small in this analysis. If θ0 represents the nondimensionalized
temperature for q=0, then non-dimensional equations transform into

Where the quantity is the ratio of the heat required to vaporize the fuel particles to the overall heat
release in the flame and is presumed to be small in this analysis. If represents the
nondimensionalized temperature for , then non-dimensional equations transform into

With the boundary conditions

Where .

2.0 ASYMPTOTIC ANALYSIS

2.1. Preheat-vaporization zone

In the asymptotic limit , we can neglect chemical reaction between the gaseous fuel and
oxidizer in the preheat-vaporization zone. Thus the energy equation in this zone reduces to

Since, by definition is the flame temperature in the reaction zone, at . Where
is the flame position. Using boundary condition we have

Introducing Eq. 21 into the Eq. 19 using the boundary condition yields

Introducing Eqs. 21 and 22 into Eq. 18 and integratingit once, results in

(17)

With the boundary conditions

Where .

2.0 ASYMPTOTIC ANALYSIS

2.1. Preheat-vaporization zone

In the asymptotic limit , we can neglect chemical reaction between the gaseous fuel and
oxidizer in the preheat-vaporization zone. Thus the energy equation in this zone reduces to

Since, by definition is the flame temperature in the reaction zone, at . Where
is the flame position. Using boundary condition we have

Introducing Eq. 21 into the Eq. 19 using the boundary condition yields

Introducing Eqs. 21 and 22 into Eq. 18 and integratingit once, results in

(18)

With the boundary conditions

Where .

2.0 ASYMPTOTIC ANALYSIS

2.1. Preheat-vaporization zone

In the asymptotic limit , we can neglect chemical reaction between the gaseous fuel and
oxidizer in the preheat-vaporization zone. Thus the energy equation in this zone reduces to

Since, by definition is the flame temperature in the reaction zone, at . Where
is the flame position. Using boundary condition we have

Introducing Eq. 21 into the Eq. 19 using the boundary condition yields

Introducing Eqs. 21 and 22 into Eq. 18 and integratingit once, results in

With the boundary conditions

Where .

2.0 ASYMPTOTIC ANALYSIS

2.1. Preheat-vaporization zone

In the asymptotic limit , we can neglect chemical reaction between the gaseous fuel and
oxidizer in the preheat-vaporization zone. Thus the energy equation in this zone reduces to

Since, by definition is the flame temperature in the reaction zone, at . Where
is the flame position. Using boundary condition we have

Introducing Eq. 21 into the Eq. 19 using the boundary condition yields

Introducing Eqs. 21 and 22 into Eq. 18 and integratingit once, results in

Where

With the boundary conditions

Where .

2.0 ASYMPTOTIC ANALYSIS

2.1. Preheat-vaporization zone

In the asymptotic limit , we can neglect chemical reaction between the gaseous fuel and
oxidizer in the preheat-vaporization zone. Thus the energy equation in this zone reduces to

Since, by definition is the flame temperature in the reaction zone, at . Where
is the flame position. Using boundary condition we have

Introducing Eq. 21 into the Eq. 19 using the boundary condition yields

Introducing Eqs. 21 and 22 into Eq. 18 and integratingit once, results in

(19)

2.0 asYmPtotiC analYsis

2.1 Preheat-vaporization zone

In the asymptotic limit

With the boundary conditions

Where .

2.0 ASYMPTOTIC ANALYSIS

2.1. Preheat-vaporization zone

In the asymptotic limit , we can neglect chemical reaction between the gaseous fuel and
oxidizer in the preheat-vaporization zone. Thus the energy equation in this zone reduces to

Since, by definition is the flame temperature in the reaction zone, at . Where
is the flame position. Using boundary condition we have

Introducing Eq. 21 into the Eq. 19 using the boundary condition yields

Introducing Eqs. 21 and 22 into Eq. 18 and integratingit once, results in

, we can neglect chemical reaction
between the gaseous fuel and oxidizer in the preheat-vaporization
zone. Thus the energy equation in this zone reduces to

With the boundary conditions

Where .

2.0 ASYMPTOTIC ANALYSIS

2.1. Preheat-vaporization zone

In the asymptotic limit , we can neglect chemical reaction between the gaseous fuel and
oxidizer in the preheat-vaporization zone. Thus the energy equation in this zone reduces to

Since, by definition is the flame temperature in the reaction zone, at . Where
is the flame position. Using boundary condition we have

Introducing Eq. 21 into the Eq. 19 using the boundary condition yields

Introducing Eqs. 21 and 22 into Eq. 18 and integratingit once, results in

(20)

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Counterflow Combustion of Micro Organic Particles

103

Since, by definition Tf is the flame temperature in the reaction zone,

With the boundary conditions

Where .

2.0 ASYMPTOTIC ANALYSIS

2.1. Preheat-vaporization zone

In the asymptotic limit , we can neglect chemical reaction between the gaseous fuel and
oxidizer in the preheat-vaporization zone. Thus the energy equation in this zone reduces to

Since, by definition is the flame temperature in the reaction zone, at . Where
is the flame position. Using boundary condition we have

Introducing Eq. 21 into the Eq. 19 using the boundary condition yields

Introducing Eqs. 21 and 22 into Eq. 18 and integratingit once, results in

is the flame position. Using boundary condition we have

With the boundary conditions

Where .

2.0 ASYMPTOTIC ANALYSIS

2.1. Preheat-vaporization zone

In the asymptotic limit , we can neglect chemical reaction between the gaseous fuel and
oxidizer in the preheat-vaporization zone. Thus the energy equation in this zone reduces to

Since, by definition is the flame temperature in the reaction zone, at . Where
is the flame position. Using boundary condition we have

Introducing Eq. 21 into the Eq. 19 using the boundary condition yields

Introducing Eqs. 21 and 22 into Eq. 18 and integratingit once, results in

(21)

Introducing Eq. 21 into the Eq. 19 using the boundary condition yields

With the boundary conditions

Where .

2.0 ASYMPTOTIC ANALYSIS

2.1. Preheat-vaporization zone

In the asymptotic limit , we can neglect chemical reaction between the gaseous fuel and
oxidizer in the preheat-vaporization zone. Thus the energy equation in this zone reduces to

Since, by definition is the flame temperature in the reaction zone, at . Where
is the flame position. Using boundary condition we have

Introducing Eq. 21 into the Eq. 19 using the boundary condition yields

Introducing Eqs. 21 and 22 into Eq. 18 and integratingit once, results in
(22)

Introducing Eqs. 21 and 22 into Eq. 18 and integratingit once, results in

Where boundary conditions are used in writing Eq. 23. Also as the value of is sufficiently high, it
is assumed that , where is gaseous fuel mass fraction in the reaction zone.

In the post flame zoneas we have assumed that all of the particles vaporizes just before the flame,
the reaction and vaporization terms are not considered. Thus

Where the subscript denotes conditions at the interface between post flame zone and the reaction
zone.

The jump conditions across the reaction zone which are used in finding flame position are

Where the subscript denotes conditions at the interface between preheat-vaporization zone and
the reaction zone.

3.0 RESULTS AND DISCUSSION

For purpose of illustration it will be assumed that the gaseous fuel that evolves from the fuel
particles is methane. The values of , , , and are considered to be ,

, , and respectively, based on (Seshadri et al,. 1992).

Also , and are chosen from (Seshadri et al,.
1992)based on and assumption. Where is equivalence ratio based on fuel
available in the ambient reactant stream and is radius of fuel particle in the ambient reactant
stream.

The value of is calculated with trial and error method. A value is guessed. Then equation 22 is
solved and the answer is substituted in 18 neglecting reaction term. This equation is solved
numerically to find the first term of equation 23.At last Eq. 24 is used to find the next guess.This is
done until convergence. The value that is obtained is , which results in

(23)

Where boundary conditions are used in writing Eq. 23. Also as the
value of

Where boundary conditions are used in writing Eq. 23. Also as the value of is sufficiently high, it
is assumed that , where is gaseous fuel mass fraction in the reaction zone.

In the post flame zoneas we have assumed that all of the particles vaporizes just before the flame,
the reaction and vaporization terms are not considered. Thus

Where the subscript denotes conditions at the interface between post flame zone and the reaction
zone.

The jump conditions across the reaction zone which are used in finding flame position are

Where the subscript denotes conditions at the interface between preheat-vaporization zone and
the reaction zone.

3.0 RESULTS AND DISCUSSION

For purpose of illustration it will be assumed that the gaseous fuel that evolves from the fuel
particles is methane. The values of , , , and are considered to be ,

, , and respectively, based on (Seshadri et al,. 1992).

is sufficiently high, it is assumed that

Where boundary conditions are used in writing Eq. 23. Also as the value of is sufficiently high, it
is assumed that , where is gaseous fuel mass fraction in the reaction zone.

In the post flame zoneas we have assumed that all of the particles vaporizes just before the flame,
the reaction and vaporization terms are not considered. Thus

Where the subscript denotes conditions at the interface between post flame zone and the reaction
zone.

The jump conditions across the reaction zone which are used in finding flame position are

Where the subscript denotes conditions at the interface between preheat-vaporization zone and
the reaction zone.

3.0 RESULTS AND DISCUSSION

For purpose of illustration it will be assumed that the gaseous fuel that evolves from the fuel
particles is methane. The values of , , , and are considered to be ,

, , and respectively, based on (Seshadri et al,. 1992).

where

Where boundary conditions are used in writing Eq. 23. Also as the value of is sufficiently high, it
is assumed that , where is gaseous fuel mass fraction in the reaction zone.

In the post flame zoneas we have assumed that all of the particles vaporizes just before the flame,
the reaction and vaporization terms are not considered. Thus

Where the subscript denotes conditions at the interface between post flame zone and the reaction
zone.

The jump conditions across the reaction zone which are used in finding flame position are

Where the subscript denotes conditions at the interface between preheat-vaporization zone and
the reaction zone.

3.0 RESULTS AND DISCUSSION

For purpose of illustration it will be assumed that the gaseous fuel that evolves from the fuel
particles is methane. The values of , , , and are considered to be ,

, , and respectively, based on (Seshadri et al,. 1992).

is gaseous fuel mass fraction in the reaction zone.

In the post flame zoneas we have assumed that all of the particles
vaporizes just before the flame, the reaction and vaporization terms are
not considered. Thus

Where boundary conditions are used in writing Eq. 23. Also as the value of is sufficiently high, it
is assumed that , where is gaseous fuel mass fraction in the reaction zone.

In the post flame zoneas we have assumed that all of the particles vaporizes just before the flame,
the reaction and vaporization terms are not considered. Thus

Where the subscript denotes conditions at the interface between post flame zone and the reaction
zone.

The jump conditions across the reaction zone which are used in finding flame position are

Where the subscript denotes conditions at the interface between preheat-vaporization zone and
the reaction zone.

3.0 RESULTS AND DISCUSSION

For purpose of illustration it will be assumed that the gaseous fuel that evolves from the fuel
particles is methane. The values of , , , and are considered to be ,

, , and respectively, based on (Seshadri et al,. 1992).

Where the subscript, - denotes conditions at the interface between post
flame zone and the reaction zone.

The jump conditions across the reaction zone which are used in finding
flame position are

Where boundary conditions are used in writing Eq. 23. Also as the value of is sufficiently high, it
is assumed that , where is gaseous fuel mass fraction in the reaction zone.

In the post flame zoneas we have assumed that all of the particles vaporizes just before the flame,
the reaction and vaporization terms are not considered. Thus

Where the subscript denotes conditions at the interface between post flame zone and the reaction
zone.

The jump conditions across the reaction zone which are used in finding flame position are

Where the subscript denotes conditions at the interface between preheat-vaporization zone and
the reaction zone.

3.0 RESULTS AND DISCUSSION

For purpose of illustration it will be assumed that the gaseous fuel that evolves from the fuel
particles is methane. The values of , , , and are considered to be ,

, , and respectively, based on (Seshadri et al,. 1992).

(24)

Where the subscript + denotes conditions at the interface between
preheat-vaporization zone and the reaction zone.

ISSN: 2180-1053 Vol. 4 No. 1 January-June 2012

Journal of Mechanical Engineering and Technology

104

3.0 results anD DisCussion

For purpose of illustration it will be assumed that the gaseous fuel that
evolves from the fuel particles is methane. The values of A, nu, ρu, ρs
and Tu are considered to be

Where boundary conditions are used in writing Eq. 23. Also as the value of is sufficiently high, it
is assumed that , where is gaseous fuel mass fraction in the reaction zone.

In the post flame zoneas we have assumed that all of the particles vaporizes just before the flame,
the reaction and vaporization terms are not considered. Thus

Where the subscript denotes conditions at the interface between post flame zone and the reaction
zone.

The jump conditions across the reaction zone which are used in finding flame position are

Where the subscript denotes conditions at the interface between preheat-vaporization zone and
the reaction zone.

3.0 RESULTS AND DISCUSSION

For purpose of illustration it will be assumed that the gaseous fuel that evolves from the fuel
particles is methane. The values of , , , and are considered to be ,

, , and respectively, based on (Seshadri et al,. 1992).

Where boundary conditions are used in writing Eq. 23. Also as the value of is sufficiently high, it
is assumed that , where is gaseous fuel mass fraction in the reaction zone.

In the post flame zoneas we have assumed that all of the particles vaporizes just before the flame,
the reaction and vaporization terms are not considered. Thus

Where the subscript denotes conditions at the interface between post flame zone and the reaction
zone.

The jump conditions across the reaction zone which are used in finding flame position are

Where the subscript denotes conditions at the interface between preheat-vaporization zone and
the reaction zone.

3.0 RESULTS AND DISCUSSION

For purpose of illustration it will be assumed that the gaseous fuel that evolves from the fuel
particles is methane. The values of , , , and are considered to be ,

, , and respectively, based on (Seshadri et al,. 1992).

and 300K respectively, based on (Seshadri et.al,. 1992).

Also

Where boundary conditions are used in writing Eq. 23. Also as the value of is sufficiently high, it
is assumed that , where is gaseous fuel mass fraction in the reaction zone.

In the post flame zoneas we have assumed that all of the particles vaporizes just before the flame,
the reaction and vaporization terms are not considered. Thus

Where the subscript denotes conditions at the interface between post flame zone and the reaction
zone.

The jump conditions across the reaction zone which are used in finding flame position are

Where the subscript denotes conditions at the interface between preheat-vaporization zone and
the reaction zone.

3.0 RESULTS AND DISCUSSION

For purpose of illustration it will be assumed that the gaseous fuel that evolves from the fuel
particles is methane. The values of , , , and are considered to be ,

, , and respectively, based on (Seshadri et al,. 1992).

are chosen from
(Seshadri et.al,. 1992) based on ϕu=1 and ru=20μm assumption. Where
ϕu is equivalence ratio based on fuel available in the ambient reactant
stream and ru is radius of fuel particle in the ambient reactant stream.

The value of

Where boundary conditions are used in writing Eq. 23. Also as the value of is sufficiently high, it
is assumed that , where is gaseous fuel mass fraction in the reaction zone.

In the post flame zoneas we have assumed that all of the particles vaporizes just before the flame,
the reaction and vaporization terms are not considered. Thus

Where the subscript denotes conditions at the interface between post flame zone and the reaction
zone.

The jump conditions across the reaction zone which are used in finding flame position are

Where the subscript denotes conditions at the interface between preheat-vaporization zone and
the reaction zone.

3.0 RESULTS AND DISCUSSION

For purpose of illustration it will be assumed that the gaseous fuel that evolves from the fuel
particles is methane. The values of , , , and are considered to be ,

, , and respectively, based on (Seshadri et al,. 1992).

is calculated with trial and error method. A value is
guessed. Then equation 22 is solved and the answer is substituted in
18 neglecting reaction term. This equation is solved numerically to find
the first term of equation 23.At last Eq. 24 is used to find the next guess.
This is done until convergence. The value that is obtained is

Where boundary conditions are used in writing Eq. 23. Also as the value of is sufficiently high, it
is assumed that , where is gaseous fuel mass fraction in the reaction zone.

In the post flame zoneas we have assumed that all of the particles vaporizes just before the flame,
the reaction and vaporization terms are not considered. Thus

Where the subscript denotes conditions at the interface between post flame zone and the reaction
zone.

The jump conditions across the reaction zone which are used in finding flame position are

Where the subscript denotes conditions at the interface between preheat-vaporization zone and
the reaction zone.

3.0 RESULTS AND DISCUSSION

For purpose of illustration it will be assumed that the gaseous fuel that evolves from the fuel
particles is methane. The values of , , , and are considered to be ,

, , and respectively, based on (Seshadri et al,. 1992).

=1.71,
which results in

Where boundary conditions are used in writing Eq. 23. Also as the value of is sufficiently high, it
is assumed that , where is gaseous fuel mass fraction in the reaction zone.

In the post flame zoneas we have assumed that all of the particles vaporizes just before the flame,
the reaction and vaporization terms are not considered. Thus

Where the subscript denotes conditions at the interface between post flame zone and the reaction
zone.

The jump conditions across the reaction zone which are used in finding flame position are

Where the subscript denotes conditions at the interface between preheat-vaporization zone and
the reaction zone.

3.0 RESULTS AND DISCUSSION

For purpose of illustration it will be assumed that the gaseous fuel that evolves from the fuel
particles is methane. The values of , , , and are considered to be ,

, , and respectively, based on (Seshadri et al,. 1992).

(25)

Figure 2 effect of Lewis number on gaseous mass fraction distribution in preheat-vaporization zone

Eqs. 25 and 26 are used to find gaseous fuel mass fraction distribution in the preheat-vaporization
zone, which the result is plotted in Figure 2. This Figure shows that Lewis number has a dual effect
on gaseous fuel mass fraction distribution. First, since heat diffusion increases with increasing
values of Lewis number, gaseous fuel mass fraction can be expected to become larger near the
reaction zone, as Lewis number increases, because pyrolysis phenomena is increased. On the other
side, larger Lewis number means lower mass diffusion, which decreases gaseous fuel mass fraction
far from the reaction zone.

4.0 CONCLUSIONS

In this work the structure of a one dimensional, axisymmetric, premixed flame in a counterflow
configuration containing uniformly distributed volatile fuel particles is examined. Effect of Lewis
number on gaseous mass fraction distribution in preheat-vaporization zone is investigated, which
shows increasing in Lewis number has dual effect on gaseous mass fraction distribution based on
increasing in heat diffusion and decreasing in mass diffusion. This dual effect results a higher value
of gaseous mass fraction near the reaction zone and a lower value far from the reaction zone, in a
higher Lewis number value.

5.0 REFERENCES

M. Bidabadi, A.Haghiri, A. Rahbari. 2010, The effect of Lewis and Damk hler numbers on the
flame propagation through micro-organic dust particles, Int. J. Thermal Sci 49, pp. 534-542.

J. Daou. 2011, strained premixed flames: Effect of heat loss, preferential diffusion and reversibility
of the reaction, Combust. Theory Model. 15:4, pp. 437-454.

R. W. Thatcher, E.AlSarairah, Steady and unsteady flame propagation in a premixed counterflow,
Combust. Theory Model, 11:4, pp. 569-583.

H. Y. Wang, W. H. Chen, and C. K. Law. 2007, Extinction of counterflow diffusion flames with
radiative heat loss and nonunity Lewis numbers, Combust. Flame 148, pp. 100-116.

R. K. Eckhof.2006 , Differences and similarities of gas and dust explosions: a critical evaluation of
the European 'ATEX' directives in relation to dusts. J. Loss Prev. Process Ind. 19, pp. 553-
560.

(26)

Figure 2 effect of Lewis number on gaseous mass fraction distribution in preheat-vaporization zone

4.0 CONCLUSIONS

5.0 REFERENCES

M. Bidabadi, A.Haghiri, A. Rahbari. 2010, The effect of Lewis and Damk hler numbers on the
flame propagation through micro-organic dust particles, Int. J. Thermal Sci 49, pp. 534-542.

J. Daou. 2011, strained premixed flames: Effect of heat loss, preferential diffusion and reversibility
of the reaction, Combust. Theory Model. 15:4, pp. 437-454.

R. W. Thatcher, E.AlSarairah, Steady and unsteady flame propagation in a premixed counterflow,
Combust. Theory Model, 11:4, pp. 569-583.

H. Y. Wang, W. H. Chen, and C. K. Law. 2007, Extinction of counterflow diffusion flames with
radiative heat loss and nonunity Lewis numbers, Combust. Flame 148, pp. 100-116.

R. K. Eckhof.2006 , Differences and similarities of gas and dust explosions: a critical evaluation of
the European 'ATEX' directives in relation to dusts. J. Loss Prev. Process Ind. 19, pp. 553-
560.

Figure 2 effect of Lewis number on gaseous mass fraction distribution
in preheat-vaporization zone

Eqs. 25 and 26 are used to find gaseous fuel mass fraction distribution
in the preheat-vaporization zone, which the result is plotted in Figure
2. This Figure shows that Lewis number has a dual effect on gaseous
fuel mass fraction distribution. First, since heat diffusion increases with
increasing values of Lewis number, gaseous fuel mass fraction can be
expected to become larger near the reaction zone, as Lewis number
increases, because pyrolysis phenomena is increased. On the other side,

ISSN: 2180-1053 Vol. 4 No. 1 January-June 2012

Counterflow Combustion of Micro Organic Particles

105

larger Lewis number means lower mass diffusion, which decreases
gaseous fuel mass fraction far from the reaction zone.

4.0 ConClusions

In this work the structure of a one dimensional, axisymmetric, premixed
flame in a counterflow configuration containing uniformly distributed
volatile fuel particles is examined. Effect of Lewis number on gaseous
mass fraction distribution in preheat-vaporization zone is investigated,
which shows increasing in Lewis number has dual effect on gaseous
mass fraction distribution based on increasing in heat diffusion and
decreasing in mass diffusion. This dual effect results a higher value
of gaseous mass fraction near the reaction zone and a lower value far
from the reaction zone, in a higher Lewis number value.

5.0 referenCes

M. Bidabadi, A.Haghiri, A. Rahbari. 2010, The effect of Lewis and Damkohler
numbers on the flame propagation through micro-organic dust
particles, Int. J. Thermal Sci 49, pp. 534-542.

J. Daou. 2011, strained premixed flames: Effect of heat loss, preferential
diffusion and reversibility of the reaction, Combust. Theory Model.
15:4, pp. 437-454.

R. W. Thatcher, E.AlSarairah, Steady and unsteady flame propagation in a
premixed counterflow, Combust. Theory Model, 11:4, pp. 569-583.

H. Y. Wang, W. H. Chen, and C. K. Law. 2007, Extinction of counterflow
diffusion flames with radiative heat loss and nonunity Lewis numbers,
Combust. Flame 148, pp. 100-116.

R. K. Eckhof.2006 , Differences and similarities of gas and dust explosions:
a critical evaluation of the European 'ATEX' directives in relation to
dusts. J. Loss Prev. Process Ind. 19, pp. 553-560.

K. Seshadri, A. L. Berlad, and V. Tangirala. 1992, The structure of premixed
particle-cloud flames, Combust. Flame 89, pp. 333-342.