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Article
Mathematical modeling of a Non-Premixed
organic Dust flame in a Counterflow configuration
Mehdi Bidabadi, Milad Ramezanpour, Alireza Khoeini Poorfar, Eliseu Monteiro, and Abel Rouboa
Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.6b01478 • Publication Date (Web): 13 Oct 2016
Downloaded from http://pubs.acs.org on October 17, 2016
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Energy & Fuels
Mathematical modeling of a Non-Premixed organic Dust flame in a
Counterflow configuration
Mehdi Bidabadia, Milad Ramezanpoura, Alireza Khoeini Poorfara1, Eliseu Monteirob, Abel Rouboac,d,
a
School of Mechanical Engineering, Department of Energy Conversion, Combustion Research Laboratory, Iran University
of Science and Technology (IUST), Tehran, Iran
b
C3i-Interdisciplinary Center for Research and Innovation, Polytechnic Institute of Portalegre, Lugar da Abadessa, Apartado
148, 7301-901 Portalegre, Portugal
c
d
Mechanical Engineering and Applied Mechanics Department, University of Pennsylvania, Philadelphia, PA, United States
CIENER/INEGI/Engineering Department, School of Science and Technology of University of UTAD, Quinta dos Prados,
Vila Real, Portugal
ABSTRACT:
ABSTRACT: In the present article an analytical approach is employed to study the counterflow non-premixed dust flame structure in air.
Lycopodium is assumed as the organic fuel in this paper. First, it is presumed that the fuel particles vaporize in a thin zone to form a gaseous
fuel to react with the oxidizer. The reaction rate is presumed to be of the Arrhenius type in first order respect to oxidizer and fuel. Mass conservation equations of dust particles, oxidizer, gaseous fuel and the energy conservation equation for non-unity Lewis numbers of oxidizer and
fuel are presented as the governing equations. Boundary conditions are applied for each zone for the purpose of solving the governing equations analytically. The flame temperature in terms of oxidizer and fuel’s Lewis numbers is calculated. Furthermore, the variation of flame
position based on the fuel and oxidizer’s Lewis numbers is evaluated. Also, mass fraction and temperature profiles of oxidizer and fuel are
presented. In addition, the variation of the ratio of critical extinction values of strain rates as a function of non-unity Lewis numbers of oxidizer
and fuel to the unity Lewis numbers of them is investigated.
Keywords:
Keywords Dust cloud combustion; Counterflow combustion; Non-Premixed dust Flame; Analytical solution; Extinction
1
Corresponding Author: Alirezapoorfar@iust.ac.ir
Tel: +98 21 77 226 116, Fax: 00982177240488
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1.
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the temperature distribution and flame propagation of
micron-sized lycopodium dust cloud with the consideration that the burning rate of a particle is controlled by the
oxygen diffusion’s process, was investigated by Rahbari et
al.[9].
In another experimental study, Yoshiie et al. investigated
the NOx emission in an oxyfuel coal combustion atmosphere, experimentally[10]. Furthermore, rudimentary
reaction kinetics for NOx formation studied numerically
under oxyfuel coal combustion.
Counterflow dust flame is of fundamental research interest since at the centerline of the nozzles gradients only exist
in the axial direction. Although these configurations are
somewhat more complex to establish experimentally, they
have definite advantageous. For instance, studying the effects of equivalence ratio, fuel type, strain rate and etc on
extinction, and the influence of positive stretch on the
structure of the flame, are the benefits of counterflow configuration [11,12]. In premixed counteflow dust flame,
fuel-oxidizer mixture exit from the nozzles and there is a
symmetry between two nozzles but in a non- premixed
counterflow dust flame, fuel and oxidizer flowing out of the
exhaust nozzles separately to mix and react in a position
between the two nozzles.
In modelling of counterflow combustion, Linan studied
the non-premixed gaseous fuel flames structure in a steady
state manner by studying the mixing and chemical reaction
of two opposed jets of oxidizer and fuel[13]. In another
work of gaseous fuel combustion in counterflow structure,
Seshadri et al. analyzed the stagnant and counterflow structure of non-premixed flames in the limit of high amount of
the non-dimensional activation energy, defining the reaction rate and the results were reported for low values of
the stoichiometric mass ratio of fuel to oxygen[14].
In resumption studies on flame propagation modelling in a
counerflow structure, a spray fuel is implemented by
Wichman and Yang [15]. They considered two streams
flowing out of two nozzles towards each other from opposite ways in a counterflow configuration in a spray model.
It was assumed that each of the two opposed streams carries a distribution of liquid droplets which after vaporization, oxidizer and fuel convect and diffuse in the direction
of the stagnation plane. In another work, a new analysis of
spray non-premixed flame in a counterflow configuration
was presented by Dvorjetski and Greenberg[16].
For mathematical and analytical modelling of gaseous fuel
combustion in a counterflow structure, some efforts are
accomplished by Daou and his coworkers. Daou et al. described the effects of strain and heat losses on premixed
flame-edges in a two-dimensional counterflow configuration[17]. A thermo-diffusive model was presented with a
single Arrhenius reaction and a mathematical description
of the flame-edge in the weak-strain limit was proposed. In
another study, Daou et al. investigated the effect of volumetric heat-loss on the propagation of counterflow triple
flames[18] .
NOx formation is of interest in some novel works. Fu et al.
investigated the emissions of NOx from pre-vaporized nHeptane and 1-Heptane in a counterflow configuration
Introduction
The burning rate and burning intensity of combustible
solid materials in oxidizer, increase with the rise of the subdivision degree of the material. Flame propagates by way of
combustion of dust particles and dust cloud explosions.
They are recognized as a serious industrial hazard and
threat to property and humans over the past 200 years[1].
A perfect knowledge of the origin and development of dust
explosion and its hazards is necessary in industries using
combustible dusts[2]. With the development of powder
technology, hazard assessment and the administration of
precautionary approaches for dust explosions, investigations over prevention and mitigation of dust explosions
have become of importance from safety issues point of
view[3].
Due to the importance of studies about the explosibility of
dusts, significant efforts were implemented but the basic
mechanisms involved in the flame spread in dust clouds
have not been investigated adequately. Experimental
difficulties in the generation of uniform dust suspensions,
the fact that particle size dispersions and particle size can
affect in a remarkable manner the combustion mechanisms
and experimental design challenges and interpretation of
experimental results, are the main obstacles in studying the
basic mechanisms of flame spread in dust suspensions[3,4].
These difficulties are expressed by Proust when measuring
the laminar burning velocities and the maximum flame
temperatures for combustible dust-air mixtures (starch
dust-air mixtures, lycopodium-air mixtures and sulphur
flour-air mixtures)[5]. The tube method and a tomography method are used and the results are compared with
those obtained with other devices such as resistors, pyrometers and with theoretical values. He conclude that the observed discrepancies seem mainly to come from the relatively poor efficiency of the burning processes inside the
flame front than to heat losses by radiation.
Due to the available difficulties in experimental evaluating
of dust flames, mathematical modelling are developed.
In a study by Rockwell and Rangwala [6] a model of combustible dust-air premixed flames is proposed. Burning
velocity, mixture temperature and mass fraction of coal
particles were calculated at high concentration of dust
cloud. In another work by Rockwell and Rangwala Hybrid
Flame Analyzer (HFA) as a new apparatus was developed
to investigate the turbulent premixed dust/air flames[7].
TGA method and tube reactor tests are implemented in a
work by Ren et al., to study the combustion features of coal
gangue under various oxidizers at atmospheric pressure
[8].
Due to aforementioned difficulties in experimental modelling of dust flame, mathematically approaches are implemented as substitute. There are some efforts in mathematically modelling of lycopodium dust combustion which in
one of them, a two-dimensional analytical model to define
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Energy & Fuels
with partially premixed flames, numerically[19].
Effect of / composition on extinction strain rates of
premixed and non-premixed syngas flames in a counterflow configuration is investigated experimentally and numerically in atmospheric pressure by Sahu et al. [20].
In another study by Wang et al., a numerical method is
implemented to perceive the flame structure in a , /
/ / counterflow diffusion flame with different
mole fractions of and [21]. It was found that the
flame temperature is decreased by thermal and chemical
effects of but the effect on flame temperature is
low.
A mathematical approach is utilized to investigate the
counterflow, non-premixed organic dust flame in air in this
article. First, it is presumed that the fuel particles vaporize
to yield a gaseous fuel to react with the gaseous oxidizer.
The governing conservation equations are proposed and
solved analytically. The flame temperature with the variation of different Lewis numbers of fuel and oxidizer is
proposed. Furthermore, the variation of flame position in
terms of oxidizer and fuel Lewis number is evaluated and
temperature and mass fraction profiles of fuel and oxidizer
are studied. At last, the variation of ratio of critical extinction values of strain rates in terms of non-unity values of
oxidizer and fuel Lewis numbers to unity value of them are
calculated.
Figure 1.
1 Schematic of counterflow combustion of nonpremixed dust flame.
In dust cloud combustion, particle vaporization rate states
as the produced mass of gaseous fuel per unit volume and
time is an essential parameter in controlling the combustion process. The vaporization rate of a particle is obtained
through the below relationship[22]:
(1 )
where is the constant characteristic time of vaporization, is the dust particles temperature, is the vaporization temperature of dust particles and is the Heaviside
function.
In addition to the vaporization rate of particles, another parameter which is the heat conduction to mass diffusivity ratio controls the combustion phenomenon. This ratio in
non-dimensional form is called Lewis number (Le) and defines as follows:
2. Governing equations
In the present study, the counterflow structure is formed
by the fuel stream flow (organic particles) from the direction -∞ and the oxidizer stream flow (air) from the direction +∞ toward the stagnation plane between these two
steady and axisymmetric streams of oxidizer and fuel
(Fig.1).The assumptions of equal velocity for both streams
and constant value for specific heat capacity, density and
other transport coefficients are considered to reduce the
complexity of the analytical approach [14]. It is supposed
that dust particles convert into vapor first to form a gaseous
fuel with specified chemical structure. Therefore, the heterogeneous combustion is disregarded in this study. Mixing and chemical reaction of oxidizer and gaseous fuel occur simultaneously near the stagnation plane. Flame position which is dependent on the initial conditions, could be
on either sides of the stagnation plane. By changing the initial conditions, the flame position moves near the stagnation plane. Fig. 1 shows a schematic of non-premixed
flame of organic dust cloud in a counterflow configuration.
As mentioned, dust particles vaporize in a thin zone called
vaporization front to form a gaseous fuel which Fig. 1
shows this zone. In Fig. 1, the flame position is in the left
side of the stagnation plane, which can be assumed in the
right side of the plane depending on the initial conditions.
(2 )
where , , and are the thermal conductivity of fuel or
oxidizer, the density of the mixture, the specific heat capacity of mixture and the characteristic mass diffusivity, respectively. The index can be replaced by subscripts, F or
O, which are the symbols for fuel and oxidizer.
Furthermore, the combustion procedure is modeled as a
one-step overall reaction:
! " # ! → %&'()* +!
(3 )
where , and + are symbols for oxidizer, fuel and product, respectively. , # and %&' are values which indicate their stoichiometric coefficients.
The velocity field is considered in the following form:
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,, -., -
where ?, ? and 9 are the heat of reaction, the latent
heat of vaporization and heat conduction coefficient
9 (4 )
/
/ " /.
/. 7
89
-.
(6 )
;
;
2222
# #
;
# ;#
(7 )
(9)
/
/
?
?
9
" /.
/.
(10)
J K B9M9N OPQRN
PU
PQRN K L
W
V
X
YZ[
PQ
PQRN
K& PS
TPQRN
(13)
which RN is the mass fraction of fuel at the distance ∞
,where coming out of the fuel nozzle. It should be noted
that for J as the non-dimensional form of temperature,
J 0 demonstrates the unburned temperature of fuel or
oxidizer when coming out of nozzles at ^ and
J J_ denotes the flame temperature at _ .
Using the parameters defined in equation (13) for temperature, mass fractions and length, the conservation equations of gaseous fuel, solid particles fuel, energy and oxidizer and substituting them into equations (5), (9), (10) and
(12), the equations can be rewritten in their new forms.
The definitions of the rate of chemical reaction in equation
(5), the vaporization rate of a particle in equation (1) and
the Lewis number in equation (2) can be applied in the old
form of the conservation equations to make the new
form[14,15] .
Where is the mass fraction of organic dust particles. In
that is the vaporization rate of particle defined in equation (1).
The energy conservation equation defined as follows:
-.
(12)
For the purpose of non-dimensionalizing the governing
equations, some dimensionless parameters are presented
in this study. J, K , K& , K and L are the dimensionless
forms of the temperature, mass fraction of gaseous fuel,
mass fraction of oxidizer, mass fraction of solid fuel and
length, respectively which define as follows:
(8 )
/
/.
/#
/ #
#
I
/.
/.
2.1 Dimensionless form of the governing equations
where # and are mass fraction of oxidizer and gaseous
fuel, respectively. m= and m> represent molecular weight
of oxidizer and fuel, respectively. ; is the molecular weight
of the mixture.
It is assumed that the diffusion of solid particles is neglected in this study and there is not any heterogeneous combustion, so solid particles do not take part in the combustion process. Based on the above assumptions, the solid
particles fuel conservation equation is considered as follows:
-.
(11)
where # , # and I are the diffusion coefficient of the oxidizer component, mass fraction of oxidizer and stoichiometric mass ratio of oxidizer to fuel, respectively.
where 0 is the factor of frequency characterizing rate of
222
gaseous fuel oxidation. 1 and # are defined as follows:
1 4DE F G
3
where , E and G are density of dust particle, particle radius and the number of dust particles per unit volume. For
uniformly distribution of dust particles in the combustion
chamber, the density of mixture is defined in form of
" DE F G .
F
The oxidizer conservation equation is given as below:
(5 )
:
, respectively. The parameter shown in the
"
where is the gaseous fuel mass fraction acquired from
the vaporization of lycopodium dust particles. is the
gaseous fuel diffusion coefficient. is the chemicalkinetic rate which is considered to be of first order with regards to the oxidizer and the fuel[13,23]:
0 # 1 222
# exp 6
@
AB
above equation is the combination of heat capacity of dust
particles and heat capacity of the gas mixture and can
be calculated from the following relation [24,25] :
where v and u display the velocity in Y and X directions, respectively. The parameter - is the gradient of velocity in
the stagnation plane and shows strain rate of streams.
Gaseous fuel conservation equation is given by:
-.
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Energy & Fuels
where L and L_ show the position of vaporization front
and flame front, respectively. Incd , c and cF , the reaction and vaporization terms are neglected compared to diffusion and convection terms. In fact, vaporization occurs at
L and reaction occurs at L_ , which both of these positions
are thin contrary to cd , c and cF , thereby the vaporization term is only present in the vaporization front zone
and the reaction term is present only in the reaction front
zone. The variable L specifies a position thatT ,
where vaporization of solid particles occurs at that position. According to the above description and different regions that are considered in this study, boundary and
matching conditions are defined as follows:
The non-dimensional form of gaseous fuel conservation
equation is given by:
1 / K
/K
K
"L
"
/L
/L -
(14)
) K K& exp The non-dimensional form of solid particles fuel conservation equation is defined as:
L
/K
K
/L -
(15)
The non-dimensional form of oxidizer conservation equation is defined as follows:
/K#
1 / K#
L
"
/L # /L (16)
) K K& exp K 0,K# 0,K
1,J
0@L → ∞
K 0,K# i,K
0,J
0@L
→ "∞
The non-dimensional form of energy conservation equation is shown as below:
/ J
/J
a
"L
K /L
/L - (17)
B K K& exp 'nQ
'o
p m p aL , m
'q
'o
'nS
'o
p
(21)
where i in equation (20) is the initial mass fraction of
oxidizer which defines as follows[14]:
7
i (18)
#,^
IM^
(22)
The square brackets denote the jump condition in the value enclosed by them. The jump condition of interface on a
variable for example X is expressed as [X]; in particular ,
[X]=0 means that X is continuous at the considered domain boundary[22].
As mentioned, vaporization of dust particles occurs in a
thin zone which is called vaporization front. In this thin
zone, reaction and convection terms are neglected compared to vaporization and diffusion terms in equations
(14), (16) and (17):
where I# and ; are the number of oxygen’s moles reacting with one mole of fuel under stoichiometric condition
and the molecular weight of fuel, respectively.
2.2 Boundary conditions
To solve the equations it is necessary to define different
zones of counterflow model with their boundary conditions in mathematical form. According to assumption of
the problem, three zones could be assumed in this study,
which define as follow:
Preheat zone
cd : ∞ < L
≤ L
Post-vaporization zone
c :L ≤ L ≤ L_
Oxidizer zone
m
(20)
0, K ! K# ! J! 0@L L
where is the activation temperature as .The pa8
rameter b is the overall activation energy defining the rate
of the one-step irreversible chemical reaction. Also, B as
the Damkohler number is defined as follows[14] :
B 0I# M^ /; -
j
klQ
(19)
/ J
a
K 0
/L
- 1 / K
1
"
K 0
/L
- 1 / K#
0
# /L cF :L_ ≤ L < ∞
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(23)
(24)
(25)
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Integrating equations (23), (24) and (25) from LM to Lr
and assuming that - 1, the following equation is
obtained:
a /K
/J
/K#
s
t s t aL , s
t
/L
/L
/L
0
counterflow structure zones. These equations are defined
for Preheat, Post-vaporization and oxidizer zones, so the
vaporization and reaction terms are neglected compared to
diffusion and convection.
1 / K
) K K& exp /L
1 /K
1 /K#
/J
ts
t s t
/L
# /L
/L
1 / K
/K
"L
0
/L
/L
(33)
1 / K#
/K#
"L
0
# /L
/L
(34)
According to the determination of Preheat, Postvaporization and Oxidizer zones and the boundary conditions related to each of these zones, equation (32) to (34)
as governing equations of energy and mass fractions conservation should be analyzed in each of these three zones.
(27)
2.3. Governing equations solution
(28)
2.3.1 Temperature equation in different zones
•
Zone∞ ≤ u ≤ uv
In the preheat zone, equation (32) as the non-dimensional
form of energy conservation equation should be solved to
achieve non-dimensional form of temperature profile. The
solution of differential equation (32) is as follows:
(29)
J wEx 6
Integrating equation (29) from L_M to L_r , the following
equation is obtained:
s
(32)
By adding both sides of the equation (27) to the relevant
sides of equation (28), the following expression is
achieved:
/J
1 / K
"
0
/L
/L /J
/J
"L
0
/L
/L
(26)
The approach employed to yield equation (26) and to derive the equation (21) is a method to achieving the jump
condition at L L which is expressed in equation (21).
It is assumed that there is a thin zone for flame which is
called flame front. This zone is located at L which varies
with changes in initial conditions. With regard to the flame
structure, there will be a jump condition in this position
like as vaporization point. To achieve this expression, vaporization and convection terms are neglected compared
to diffusion and reaction terms in equations (14), (16) and
(17):
/J
B K K& exp /L
Page 6 of 15
√
L: " 0
(35)
The above equation has two unknown parameters A and B.
To determine these two parameters, boundary conditions
at the exhaust of fuel nozzle and the vaporization zone
should be defined as
L → ∞ ∶ J 0
z
|
(36)
L L ∶ J J
By applying the above boundary conditions to equation
(35), A and B are determined and the non-dimensional
form of temperature equation in Preheat zone is acquired
completely as given below:
(30)
It should be noted that the same method can be applied on
equation (16) to achieve oxidizer part of equation (30).
Due to continuity of temperature and mass fraction of fuel
and oxidizer at L_ and the obtained expression in equation
(30), the boundary conditions at L_ are defined as:
1 /K
1 /K#
s
ts
t
/L
# /L
/J
s t, K !
(31)
/L
K
!
J!
# 0@L L_
J
√
drl%_~ o} €

q}
drl%_~
Equations (32), (33) and (34) should be solved to obtain
mass fraction of fuel and oxidizer and temperature profiles,
According to specified boundary conditions in all of the
q}
√
o €
 }
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Ex 6
√
L: "
(37)
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2.3.2 Oxidizer mass fraction equation of different zones
•
Zoneuv ≤ u ≤ u
As in Preheat zone, to obtaining temperature profile in
Post- vaporization zone, equation (32) as the nondimensional form of energy conservation equation should
be solved to achieve non-dimensional form of temperature
profile. The solution of the differential equation (32) is defined as:
J wEx 6
√
L: " 0
•
Zoneƒ ∞ < u ≤ uv , uv < u ≤ u
In the Preheat and Post-vaporization zones, there is not
any oxidizer. Hence, the amount of mass fraction of oxidizer in these two zones is zero.
The oxidizer mass fraction equation in Preheat and vaporization zones is expressed in equation (44):
K# 0
(38)
A and B are unknown parameters of equation (37). To determine these two parameters, boundary conditions at the
flame and vaporization positions should be defined as
L L ∶ J J z L L ∶ J J |
(39)
_
_
By applying the above boundary conditions to equation
(38), A and B are determined and the non-dimensional
form of temperature equation in Post- vaporization zone is
acquired completely as follows:
J
q‚ Mq}
√
√
l%_~ o‚ €Ml%_~ o} €


q‚ Mq}
√
√
l%_~ o‚ €Ml%_~ o} €


Ex 6
Ex 6
√
√
L: L_ : " J_
•
√
L: " 0
K# werfW
klS
(40)
q‚
√
l%_~ o‚ €Md

Ex 6
L: 1!
L " 0
(45)
By applying the above boundary conditions to equation
(45), A and B are determined and the non-dimensional oxidizer mass fraction equation in oxidizer zone is obtained
completely as follows:
(41)
√
The above equation has two unknown parameters A and B.
To determine these two parameters, boundary conditions
at the exhaust of oxidizer nozzle and the flame zone should
be defined as
L L_ ∶ K# 0
|
(46)
z
L → ∞ ∶ K# i
K# A and B are unknown parameters of equation (41). To determine these two parameters, boundary conditions at the
flame zone and the exhaust of oxidizer nozzle should be defined in the form of:
L → ∞ ∶ J 0
z L L ∶ J J |
(42)
_
_
By applying the above boundary conditions to equation
(41), A and B are determined and the non-dimensional
form of temperature equation in post- vaporization zone is
acquired completely as follows:
J Zoneu ≤ u ≤ ∞
In the oxidizer zone, equation (34) as the non-dimensional
form of oxidizer mass fraction conservation equation
should be solved to achieve non-dimensional form of oxidizer mass fraction. The solution of differential equation
(34) is as follows:
•
Zoneu ≤ u < ∞
The solution of differential equation (32) is as follows:
J wEx 6
(44)
†
dM‡ˆ‰W
†‡ˆ‰W
Š‹S
o  ‚
dM‡ˆ‰W
Š‹S
o  ‚
erf ŒW
klS
L (47)
Š‹S
o  ‚
Equations (44) and (47) are the solutions of oxidizer mass
fraction conservation equation to achieving nondimensional form of oxidizer mass fraction in Preheat,
Post- vaporization and oxidizer zones, respectively. With
extracting these equations, oxidizer mass fraction profile
would be obtained completely.
(43)
2.3.3 Gaseous fuel mass fraction equation in different
zones
Equations (37), (40) and (43) are the solutions of energy
conservation equation to achieving non-dimensional form
of temperature in Preheat, Post-vaporization and oxidizer
zones, respectively. With extracting these equations, temperature profile would be obtained completely.
•
Zone∞ ≤ u ≤ uv
In the Preheat zone, equation (33) should be solved to
achieve the gaseous fuel mass fraction in non-dimensional
form. The solution of differential equation (33) is as fol-
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K w erf ŒW
klQ
L " 0
K 0
(48)
Equations (50), (53) and (54) are the solutions of gaseous
fuel mass fraction conservation equation to achieving mass
fraction of gaseous fuel in a non-dimensional form in Preheat, Post- vaporization and oxidizer zones, respectively.
With extracting these equations, gaseous fuel mass fraction
profile would be obtained completely.
It is found from the solution of energy and mass fractions
conservation equations that there are four unknown parameters J_ , L_ , L and K in these extracted equations. It
is essential to determining these unknown parameters to
complete the solution process. A systems of four equations
is required to determinate these four unknowns parameters. These four equations are four boundary conditions
that have been defined in vaporization and flame positions
which are proposed in equations (26) and (31). Combination of extracted equations of temperature and mass fractions with equations (26) and (31) as four boundary conditions, determine the four unknown values J_ , L_ , L and
K .
The above equation has two unknown parameters A and B.
To determine these two parameters, boundary conditions
at the exhaust of solid fuel nozzle and the vaporization
zone is defined as:
L → ∞ ∶ K 0
z L L ∶ K K |
(49)
_
By applying the above boundary conditions to equation
(48), A and B are determined and the non-dimensional
fuel mass fraction equation in Preheat zone is obtained
completely as follows:
K
K
1 " erf ŒW L 
2
erf 
L‘ " 1!
2
(50)
•
Zoneuv ≤ u ≤ u
Similar to the preheat zone, equation (33) should be
solved to achieve gaseous fuel mass fraction equation in
non-dimensional form. The solution of differential equation (33) is as follows:
K w erf ŒW
klQ
L " 0
Combination of equation (31) as the boundary condition
at L_ with the obtained equations for temperature and
mass fractions in Post- vaporization and oxidizer zones,
yield to equations (49) and (50) as two essential equations
to specify the unknown parameters:
(51)
nQ}
erf ŒW
nQ} ‡ˆ‰ŒW
‡ˆ‰ŒW
klQ
L Š‹Q
o 
 ‚
1
”
K
erf ŒW L  erf ŒW L_ 
2
2
t
previous section. To determine these two parameters,
boundary conditions at the flame and vaporization positions should be defined as
L L ∶ K K ’ L L ∶ K K “
(52)
_
_
By applying the above boundary conditions to equation
(53), A and B are determined and the gaseous fuel mass
fraction equation in non-dimensional form in Post- vaporization zone is obtained completely as follows:
K Š‹Q
Š‹
o M‡ˆ‰ŒW Q o‚ 
 }

Jump condition at u
2.3.4
A and B are unknown parameters of equation (51) as in
‡ˆ‰ŒW
(54)
1
”#
(53)
Ex Œ
•
Zoneu ≤ u < ∞
In oxidizer zone, there is not any gaseous fuel. Therefore,
the amount of gaseous fuel mass fraction in these two
zones is zero. The mass fraction of gaseous fuel equation in
oxidizer zone is expressed in equation (54):
i
J_ J
1
L• s L_ 1
2
√2
√2
Ex Œ L_  Ex Œ L 
2
2
J_
√2
Ex Œ L_  1
2
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(55)
√2
L 1
2 _
1 erf ŒW # L_ 
2
# t
Š‹Q
Š‹
o M‡ˆ‰ŒW Q o‚ 
 }

J_ J
√2
√2
Ex Œ L_  Ex Œ L 
2
2
J_
L•
(56)
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Energy & Fuels
2.3.5
Jump condition at uv
Combination of equation (26) as the boundary condition
at L with the obtained equations for temperature and
mass fractions in Preheat and Post-vaporization zones,
yield to equation (57) and (58) as two of the essential
equations to specify the unknown parameters:
˜
2 —
K

—
D
—erf ŒW L  erf ŒW L_ 
2
2
–
›
1
š
š L• s 2 L t
1 " erf ŒW L š
2
™
L
K
˜
J_ J
2—
 —
D
—Ex Œ√2 L_  Ex Œ√2 L 
2
2
–
›
1 š
š L• s 2 L t aL
√2
1 " Ex Œ L š
2
™
J
unity and is the first term of an expansion δ ž "
1/œ.Which δ as the reduced Damkohler number is
defined as follows:
 (57)
(58)
#_
≡
# ¤L_ , # ¥
0.276 2.15
" ¦_
¦_
¦_ ”# 1 0.276 " 2.15
¦#_
¦#_ (63)
¦#_ 1 " 0.33333L_ ”# (64)
¦#_
1
(65)
As in equation (18) Damkohler number (D) has an inverse
relationship with strain rate (-).
Equation (62) shows the ratio of critical strain rate in nonunity Lewis numbers of oxidizer and fuel (-) to critical
strain rate in # _ 1 (-ž :
(59)
Analysis of Inner structure of flame and critical extinction
value which is called reduced Damkohler number, was performed completely previously [13, 14]. The details are not
mentioned and the results of previous work [14] to analyze
internal form of flame, is used here. The critical extinction
value of Damkohler number is defined as follows:
ž7 ≈ 2œ_ 2œ_ " 1.04œ_F " 0.44œ_ It is obvious from equation (62) that the value of ž7 is
dependent on flame position, L_ , flame temperature _ and
œ_ which is also dependent on L_ .
The variable #_ in equation (62) is defined as:
¦_ 1 " 0.33333L_ 2.3.6 Flame zone analysis
In analysis of flame zone of counterflow diffusion structure,
it is considered that the Zeldovich number is large enough
to acquire the thin reaction zone. The Zeldovich number is
defined as:
b?M^
c_
(61)
By substituting equation (60) into equation (61), ž7 as
the critical Damkohler number is obtained as follows:
i #_
œ F exp ~ L_ " 1€
_
ž7 ≈
1
4D# _ œ_
2œ_ " 1.04œ_
(62)
" 0.44œ_F Equations (55) to (58) should be solved simultaneously to
determine four unknown parameters. With substitution
these identified parameters in mass fractions and temperature equations, the temperature and mass fractions profiles
would be achieved completely. Simultaneously solving of
these nonlinear systems of equations is possible with use of
numerical methods[27].
œ 8DexpL_ # _ œ_
exp F
i #_ œ
_
_ž
# _ _ ª «
ž
L• ¬ ž Œ1 
-ž )%*
/
_
7
#_
_
_
" L_ L_ž ­
(66)
Where « and /7 are defined as follows:
« (60)
whereœ_ Ex¡L_ /√2. ž is a quantity of the order
d
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®‚
®‚¯
(67)
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1 2œ_ " 1.04œ_ " 0.44œ_F 1 2œ_ ž " 1.04¤œ_ ž ¥ " 0.44œ_ ž F Page 10 of 15
" 2 " 3.76 → " 2 " 7.52
(68)
(69)
For combustible mixture of air and gaseous fuel which is
formed from vaporization of dust particles, the equivalence
ratio of fuel is calculated as follows[6]:
3. Results
Present model calculates the temperature, mass fractions
of oxidizer and fuel and the critical strain rate for non-unity
oxidizer and fuel’s Lewis numbers in a counterflow configuration of organic dust cloud. To quantify the calculations
of this study, final equations of temperature and mass fractions in different zones are applied on lycopodium particles
as the organic dust cloud. Some of the constants which are
applied in equations are showed in Table 1 based on the
published data of previous studies [5,6,24].
µ( 17.18RN
1 RN
(70)
where RN in terms of number of particles G and their radius E is calculated as follows:
M^
Table1: Properties of the dust fuel studied.
4 F
DE G 3
(71)
Another definition for the concentration of fuel is mass
particle concentration which is as given below:
Property
Value
°±
;F
1000
[24
[24]
1.357
[6]
0.24
[6]
1.55 ´ 10
[6]
300
assumed
¡-²
±. ³
¡-²
±. ³
?
¡-²
±
^ ³
;-¶¶•-E¦¡²¡·G¡G¦E-¦·G
4
±E
DE F G F 3
;
Ref.
(72)
As shown in Figure 2, the flame temperature is calculated
based on the variation of fuel’s Lewis number in different
values of mass particle concentrations at # =1. Lewis
number is the heat conduction to mass diffusivity ratio.
Therefore, by increasing the fuel’s Lewis number, it will reduce the supply fuel mass fraction which will decrease the
flame temperature. As seen in Figure 2, the magnitude of
flame temperature goes down from 1943 K to 1473 K
when the value of fuel’s Lewis number rises from 0.1 to 1.4
¸%
at the value of 100 º for mass particle concentration. Also,
¹
it could be seen that the rise in mass particle concentration
¸%
¸%
from 67 º to 100 º increases the quantity of flame tem¹
¹
perature. The value of J as a known parameter is assumed
from the experimental published data of lycopodium flame
temperature for particles with 31 micrometer diameter[5].
In combustion of micron-sized organic particles with high
concentration of volatile materials, like Lycopodium, the
processes of drying, pyrolysis, combustion of volatile
materials in the gas phase, and combustion of heterogeneous char, occur in a sequence, along the combustion
chamber [26]. In the present study, the aforementioned
processes between the preheating and beginning of pyrolysis of lycopodium particles are neglected and it is assumed
that lycopodium dust clouds vaporize first to yield a gaseous fuel which is presumed to be methane based on previous works[22 , 24].
The methane combustion equation is as follows [6]:
Figure 2.
2 The variation of flame temperature as a
function of fuel Lewis number for different values of
mass particle concentration.
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The variation of flame temperature and flame position as a
function of oxidizer Lewis number in different values of
mass particle concentrations at 1.4 are illustrated in
Figures 3 and 4, respectively. Aforementioned, Lewis
number is the ratio of heat conduction to mass diffusivity,
so increasing in oxidizer Lewis number will decreases the
supply mass fraction of oxidizer which will decreases the
flame temperature and moves the flame position toward
¸%
the oxidizer nozzle. For 83 º as a constant value of mass
¹
particle concentration, the magnitude of flame temperature goes down from 1690 K to 1458 K and the flame position moves from -0.23 to -0.1 when the quantity of oxidizer
Lewis number rises from 0.6 to 1. By increasing in mass
particle concentration, the flame temperature increases
and flame position moves toward the oxidizer nozzle.
The increase of the fuel Lewis number causes the rise of
thermal diffusivity compared to mass diffusivity which, as a
result, the amount of available fuel would be decreased and
the vaporization process would be done faster due to the
available heat. Therefore, the flame location would be at
the position closer to the fuel nozzle’s location. As can be
concluded from Fig.5, for some points, the flame location
would be at the positive side or at the right side of the stagnation plane.
Due to the lack of theoretical and experimental data on
counterflow non-premixed organic dust flames, Fig. 4
also includes the variation of the flame position for nonpremixed methane-air flame in counterflow configuration obtained by Wang et al. [28] for 1for comparison. The comparison of both results shows a similar
trend, which allows to validate the developed model
qualitatively.
Figure 5.
5 The variation of flame position as a function of
fuel Lewis number for different values of mass particle
concentration.
The variation of vaporization front position as a function of
oxidizer Lewis number is illustrated in Figure 6. As for Figure
4, with decreasing of oxidizer Lewis number, the supply mass
fraction of oxidizer will decrease and the vaporization front
position moves towards the oxidizer nozzle.
In Fig. 4, the values of flame position in a non-premixed
counterflow system studied by Wang et al [28]are shown
on the right and above vertical and horizontal axes.
Figure 3.
3 The variation of flame temperature as a
function of oxidizer Lewis number for different values of
mass particle concentration.
Figure 6.
6 The variation of vaporization front position as a
function of oxidizer Lewis number for different values of mass
particle concentration.
The variation of flame temperature based on the variation of
equivalence ratio for 10, 20 and 50 micrometer as the values of
particle radius is investigated in Figure 7. The value of J as a
known parameter is assumed from the analytical published
data of lycopodium flame temperature[24]. It could be seen
that the increasing in equivalence ratio from 1 to 1.6 increases
the quantity of flame temperature. By increasing in particle
radius, the ratio of contact surface of particle with oxidizer to
particle volume decreases and this reduction yield to increasing
in required energy for vaporization of particle, which will decrease the flame temperature.
Figure 4. The variation of flame position as a
function of oxidizer Lewis number for different
values of mass particle concentration.
The variation of flame position as a function of fuel Lewis
number for different values of mass particle concentration
is illustrated in Fig. 5. As can be seen, with the increase of
the fuel Lewis number, it results in the flame location
moves toward the fuel nozzle.
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Figure 7. The variation of flame temperature as a
function of equivalence ratio for different values of
particle radius.
Page 12 of 15
Figure 9. Mass fraction profiles of oxidizer and fuel for
different values of mass particle concentrations.
Temperature profiles of fuel and oxidizer for three values
of oxidizer Lewis number as a function of distance from the
nozzle to flame position are illustrated in Figure 8. As
shown in this figure, the temperatures of fuel and oxidizer
which are located on the left and right sides of maximum
temperature, respectively, Increase gradually to reach the
flame temperature. Also increasing in oxidizer Lewis number from 0.6 to 1 will decreases the flame temperature and
moves the flame position toward the oxidizer nozzle.
For the purpose of observing the effect of non-unity Lewis
numbers on critical strain rate, the ratio of critical strains
rates for non-unity Lewis numbers of oxidizer and fuel to
critical strain rate for # _ 1 is illustrated in Figures 10 and 11 for different values of activation temperature .
The variation of
¯
based on the variation of fuel Lewis
number for three quantities of which are 30000, 35000
and 40000 is shown in Figure 10.
This figure implies that at # 1 decreases when the
¯
fuel Lewis number grows. For instance, the magnitude of
goes down from 28 to 0.6 when the quantity of fuel Lew¯
is number rises from 0.2 to 1.4 at 40000. Also, The
increase of associates with an increase in . With the
¯
reduction in magnitude of fuel Lewis number, the magnitude of L_ as the flame position rises which increases significantly the exponentially term of equation (66). This increase is superior to the reduction in magnitude of the
Figure 8. Temperature profiles of fuel and oxidizer for
different values of oxidizer Lewis number.
9
term9‚¯ª
‚
The profiles of gaseous fuel mass fraction which is formed
by vaporization of solid particles and oxidizer mass fraction
as a function of distance from the nozzles to flame position
for different values of mass particle concentrations are depicted in Figure 9. In the right side of profile which is relevant to oxidizer mass fraction, it could be seen that with rising in mass particle concentration, the maximum value of
oxidizer mass fraction decreases, because an increase in
mass particle concentration will decreases α as the initial
mass fraction of oxidizer. The initial mass fraction of oxidizer from the nozzle to flame position, decreases gradually
which is also valid for fuel. By increasing in value of mass
particle concentration, gaseous fuel mass fraction in vaporization point decreases and moves toward the oxidizer
nozzle.
kl‚ » 
'¼
¯
, resulting in an increase of the ratio .
In Fig.8 the obtained results are compared with the
methane-air non premixed counterflow combustion of
Seshadri et al. [14] due to the lack of any experimental/numerical data on non-premixed counterflow dust
flames. The comparison shows once again that the same
trend is obtained, which reinforces the qualitatively validation of the developed model. In Fig. 8, the values of flame
position in a non-premixed counterflow system studied by
Seshadri [14] are illustrated on the right vertical axes.
Figure 10.
10. Critical strains rates ratio for non-unity Lewis
numbers of fuel to critical strain rate for # _ 1.
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Energy & Fuels
The variation of
¯
Nomenclature
based on the variation of oxidizer Lew-
is numbers for three values of which are 10000, 18000
and 25000 is shown in Figure 11. This figure implies that
at 1 decreases when the oxidizer Lewis number
¯
grows. For instance, the magnitude of
¯
30 to 0.28 when the quantity of oxidizer Lewis number rises from 0.4 to 1.4 at 25000. Also, The increase of associates with an increase in . By increasing in magni-
resulting in a decrease of the ratio
¯
.
9‚
kl‚ » 
'¼
C
Mixture specific heat capacity
¯
tude of oxidizer Lewis number, the magnitude of L_ as the
flame position goes down which decreases significantly the
exponentially term of equation (66). This decrease pro9
Strain rate
0
goes down from
motes the reduction in magnitude of the term ‚¯ª
-
Heat capacity of dust particles
Characteristic mass diffusivity of fuel or oxidizer
Damkohler number
F
Gaseous fuel
Molecular weight of mixture
;#
Molecular weight of oxidizer
G
creases when the oxidizer or fuel Lewis number grows.
Number of dust particles per unit volume
Heat of reaction
E
Solid particle radius
S
Latent heat of vaporization
Solid fuel
Activation temperature
_
Flame temperature
Vaporization temperature of dust particles
K
Dimensionless form of the mass fraction of gaseous fuel
K
Dimensionless form of the mass fraction of solid fuel
K&
RN
#
¯
Molecular weight of fuel
Q
?
In this study, counterflow diffusion flame of organic dust
particles in air is studied analytically. Counterflow structure in
this study is that organic particles as fuel stream flow from the
direction -∞ and the oxidizer stream flow (air) as the oxidizer
stream flows from the direction +∞ to the stagnation plane,
which is constructed between two steady, axisymmetric
streams of oxidizer and fuel. It has been assumed that the fuel
particles vaporize first in a thin zone. After the vaporization, a
gaseous fuel forms to react with the oxidizer. The reaction rate
is considered to be of the Arrhenius type in first order respect
to oxidizer and fuel. The conservation equations of dust particles, gaseous fuel, oxidizer and energy for the non-unity Lewis
numbers of oxidizer and fuel are presented. Boundary conditions are applied to each zone for the purpose of solving analytically the governing equations. It has been shown that increasing in Lewis numbers of oxidizer and fuel will decreases the
flame temperature and increasing in oxidizer Lewis number
will moves the flame position toward the oxidizer nozzle. Also
it has been shown that by increasing in particle radius the flame
temperature decrease. At last it has been shown that
de-
Critical Damkohler number
m
;_
4. Conclusion
Heat conduction coefficient
B
ž7
Figure 11. The ratio of critical strains rates for non-unity
Lewis numbers of oxidizer to critical strain rate for
# _ 1.
Heat capacity of the gas
9
,
Frequency factor characterizing rate of gaseous fuel
oxidation
Z
Dimensionless form of the mass fraction of oxidizer
Mass fraction of fuel at the distance ∞
Mass fraction of the solid fuel
Mass fraction of the oxidizer
Mass fraction of the gaseous fuel
Secondary coordinate axis
13
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(5) C. Proust, “Flame propagation and combustion in some dustair mixtures,” J. Loss Prev. Process Ind., vol. 19, no. 1, pp. 89–100,
2006.
Greek Symbols
1
2
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ωÀ
Vaporization rate of a particle
τÀÂÃ
Constant characteristic time of vaporization
ω>
λ
ρ
ρÆ
ρÃ
The density of the reactant mixture
Density of gas
Dimensionless form of the temperature
θÀ
ϑ=
Le
(8) J. Ren, C. Xie, X. Guo, Z. Qin, J.-Y. Lin, and Z. Li, “Combustion
characteristics of coal gangue under an atmosphere of coal mine
methane,” Energy & Fuels, vol. 28, no. 6, pp. 3688–3695, 2014.
Dust particle density
θ
θ>
(7) S. R. Rockwell and A. S. Rangwala, “Influence of coal dust on
premixed turbulent methane–air flames,” Combust. Flame, vol. 160,
no. 3, pp. 635–640, 2013.
Thermal conductivity of fuel or oxidizer
Reduced Damkohler number
ϑ
(6) S. R. Rockwell and A. S. Rangwala, “Modeling of dust air
flames,” Fire Saf. J., vol. 59, pp. 22–29, 2013.
Rate of the chemical-kinetic
δ
(9) A. Rahbari, A. Shakibi, and M. Bidabadi, “A two-dimensional
analytical model of laminar flame in lycopodium dust particles,”
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