Mathematical study of threshold of thermal-diffusive instability in counter-flow non-premixed biomass-fueled flames considering effective parameters

Computers and Mathematics with Applications xxx (xxxx) xxx

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Mathematical study of threshold of thermal-diffusive instability in counter-flow non-premixed biomass-fueled flames considering effective parameters Sadegh Sadeghi, Mehdi Bidabadi

∗

School of Engineering, Mechanical Engineering Department, Iran University of Science and Technology, Narmak, Tehran, Iran

article

info

Article history: Available online xxxx Keywords: Thermal-diffusive instability Analytical study Mathematical software Counter-flow design Non-premixed flames Biomass combustion

a b s t r a c t Identifying the thermal-diffusive instability boundaries of flames is of great significance in the field of combustion. In this paper, a detailed mathematical analysis is performed to detect the threshold of thermal-diffusive instability in planar counter-flow non-premixed flames using an asymptotic concept. Uniformly-scattered micron-sized lycopodium particles and air are applied as organic fuel and oxidizer, respectively. In order to suggest a basic combustion structure, preheat, vaporization, flame and oxidizer zones are considered. In the first phase of this investigation, time-dependent forms of mass and energy conservation equations are derived considering appropriate boundary and jump conditions. In each of the considered zones, governing equations are solved by Matlab and Mathematica software using perturbation method. To predict the onset of instability, critical values of frequency of the wrinkled flame front, Zeldovich and Lewis numbers are calculated. Eventually, the effects of wave number on the critical Zeldovich and Lewis number are explained. For validation purposes, results of this analysis obtained for critical flow strain rate (corresponding to extinction of the counterflow non-premixed flame) are compared to those reported in prior studies. Based on the comparisons, proper compatibility is observed between the mathematical results of current study and the data reported in the literature. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Thermal-diffusive instabilities (TDIs) originating from the destabilization effect of species occur in both premixed and non-premixed combustion systems [1,2]. In these systems, TDIs play a pivotal role in soot formation, reduction of emissions and extinction of flames [2]. In non-premixed flames, it is possible to decline or practically remove the negative effects of TDIs [2]. Moreover, from a structural point of view, non-premixed flames are safer than premixed flames [3]. In the last couple of decades, lycopodium particles have extensively been applied as a reference biofuel in laboratory experiments for studying and testing the structure of flames [3]. Up to now, a sheer volume of experimental and mathematical efforts have been devoted to detecting the characteristics of bio-fueled premixed and non-premixed flames [4,5]. Bidabadi et al. [6] explained the impacts of several dimensionless parameters on the initiation of instability in premixed combustion of a moisty organic fuel. Bidabadi et al. [7] mathematically modeled the premixed combustion of randomly-distributed lycopodium particles by assuming large values of Zeldovich number. Rahbari et al. [8] analytically ∗ Corresponding author. E-mail address: [email protected] (M. Bidabadi). https://doi.org/10.1016/j.camwa.2019.06.014 0898-1221/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: S. Sadeghi and M. Bidabadi, Mathematical study of threshold of thermal-diffusive instability in counterflow non-premixed biomass-fueled flames considering effective parameters, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.06.014.

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Nomenclature a

Strain rate, ( 1S )

Ca

Gaseous phase specific heat,

Cp

Solid particle specific heat,

DC DF

Damkohler number 2 Mass diffusivity coefficient of gaseous fuel, ( ms )

DO D0E DT E erfi(x) H Le m

Mass diffusivity coefficient of oxidizer, ( ms ) Critical Damkohler number 2 Thermal diffusivity coefficient, ( ms ) Activation energy, (kJ) Error function Heaviside function Lewis number kg Mixture molecular mass, ( mol )

mF

Fuel molecular mass, ( mol )

mO np Q

Oxygen molecular mass, ( mol ) Number of particle per volume unit ( ) kJ Reaction heat per unit of fuel mass, kg

R

Universal constant of gases,

rp t Ta Tf Tv Uxf

Particle radius, (m) Time, (s) Activation temperature, (K) Flame temperature, (K) Particle start temperature of (vaporization, (K) ) Velocity field in x-direction, m S

WF x xf xv YF YO Ys Ze

Molecular weight of fuel, mol Position, (m) Flame position, (m) Vaporization front position, (m) Gaseous fuel mass fraction Oxidizer mass fraction Particle mass fraction Zeldovich number

(

(

kJ kg K

kJ kg K

)

)

2

kg

kg

(

(

kg

m3 Pa mol K

)

)

calculated the flame speed and temperature profile of lycopodium particles during a premixed combustion considering a two-dimensional structure. Bidabadi et al. [9] mathematically predicted the effects of radiation and heat losses on propagation of a premixed flame fed with lycopodium particles. Han et al. [10] experimentally scrutinized the characteristics of an upward-propagating premixed laminar flame fueled by lycopodium particles. Bidabadi et al. [11] analytically described the influences of Lewis and Damkohler numbers on the premixed flame expansion through lycopodium particles. Motahari Nejad et al. [12] numerically simulated the dynamic behavior of one-dimensional premixed flames in counter-flow configuration. Bidabadi et al. [13] mathematically investigated the volatization and combustion behavior of randomlydistributed lycopodium particles in premixed mode. Proust [14] employed an experimental method to measure the laminar burning velocity and maximum flame temperature in premixed combustion of lycopodium–air mixture. Rahbari et al. [15] perused the profiles of velocity and density across the premixed flame propagation of lycopodium particles in presence of thermophoretic, gravity and buoyancy forces. Seshadri et al. [16] characterized the structure of steady premixed flames fueled by lycopodium particles using an asymptotic approach. Bidabadi et al. [17] used a theoretical model to peruse the effect of heat loss on combustion structure of counter-flow premixed flames through two-phase lycopodium–air mixture. In the field of non-premixed combustion, Bidabadi et al. [18] used an analytical model to describe the structure of counter-flow non-premixed flames in mixture of lycopodium and air. In another investigation, Bidabadi et al. [19] introduced a basic analytical model to analyze the steady multi-zone non-premixed flames in counter-flow configuration taking into account drying and vaporization processes. Bidabadi et al. [20] presented a theoretical model to obtain Please cite this article as: S. Sadeghi and M. Bidabadi, Mathematical study of threshold of thermal-diffusive instability in counterflow non-premixed biomass-fueled flames considering effective parameters, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.06.014.

S. Sadeghi and M. Bidabadi / Computers and Mathematics with Applications xxx (xxxx) xxx

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Greek symbols

α θ λ υF υO υproduct ρ

Initial mass fraction of oxidizer Dimensionless temperature ( kJ ) Thermal conductivity of fuel or oxidizer, m s K Fuel stoichiometric coefficient oxidizer stoichiometric coefficient Product stoichiometric ( ) coefficient Mixture density,

kg m3

(

ρa

Gaseous phase density,

ρp τvap ω ωv

Density of solid particle,

ωF

Rate of chemical reaction,

kg m3

(

)

kg m3

)

Constant time characteristic of vaporization Wave number ( ) kg Particle vaporization rate, m s2

(

kg m s2

)

the influences of particle porosity and thermophoresis on the structure of counter-flow non-premixed flames. Bidabadi et al. [21] asymptotically studied the combustion behavior of non-adiabatic counter-flow non-premixed flames burning lycopodium particles considering drying and vaporization processes. Seshadri and Trevino [22] revealed the influence of reactants Lewis number on asymptotic design of stagnant non-premixed flames in counter-flow arrangement. Rasam et al. [3] proposed a mathematical model to investigate the effects of radiation and particle porosity on the combustion behavior of counter-flow non-premixed flames fed with uniformly-distributed lycopodium particles. According to the literature review, most of prior studies on combustion of lycopodium particles demonstrated the structure of premixed and non-premixed flames under steady conditions; however, instabilities could considerably influence the combustion behavior of these flames [1,2]. Furthermore, in previous analytical and numerical investigations, large quantities of Zeldovich number are assumed while variation of Zeldovich number can affect the thickness of the flame. Up to now, a detailed mathematical analysis on counter-flow non-premixed flames which evaluates the threshold of thermal-diffusive instabilities has not yet been reported. In this respect, fundamental concepts of thermal-diffusive instabilities, can be described by critical Zeldovich and Lewis numbers, are still ambiguous for non-premixed bio-fueled flames in counter-flow design. In this study, a basic mathematical analysis is conducted to reveal the threshold of thermal-diffusive instability in planar counter-flow non-premixed flames using an asymptotic concept. Uniformly-scattered micron-sized lycopodium particles and air are considered as organic fuel and oxidizer, respectively. Preheat, vaporization, flame and oxidizer zones are taken into account. In the first phase of this analysis, time-dependent forms of mass and energy conservation equations are derived considering accurate boundary and jump conditions. In each of the aforementioned zones, governing equations are solved by Matlab and Mathematica software applying perturbation method. To detect the threshold of thermal-diffusive instability, critical values of Zeldovich and Lewis numbers are measured. Finally, the influences of wave number on the critical Zeldovich and Lewis number are demonstrated. For validation purposes, results of this investigation obtained for critical strain rate (corresponding to extinction of the flame) are compared to those provided by Seshadri and Trevino [22]. 2. Theoretical investigation 2.1. Flame structure In this study, effect of thermal-diffusive instability on non-premixed combustion behavior of lycopodium particles in counter-flow design is analyzed. Schematic of the combustion system is sketched in Fig. 1. As suggested in Fig. 1, the flame structure is divided into preheat zone (−∞ < X < Xv ap ), vaporization zone (Xv ap < X < Xf ), reaction zone (Xf− < X < Xf+ ) and oxidizer zone (Xf < X < +∞). Lycopodium particles, accompanied by an inert gas, and oxidizer escape from two distinct nozzles located at −∞ and +∞, respectively. Initial temperature of lycopodium particles and oxidizer is equal to the ambient temperature. As momentum of biofuel and oxidizer flows is presumed to be equal, stagnation line is located in the middle of the nozzles. It should be pointed out that location of the stagnation line is employed as the reference coordinate. Lycopodium particles heat up in the preheat zone and temperature of the particles enhances until reaching the vaporization front. At the vaporization point, solid lycopodium particles are rapidly devolatized and are converted into a gaseous fuel with a certain chemical composition, namely methane [3]. The gaseous flow is mixed with the oxidizer Please cite this article as: S. Sadeghi and M. Bidabadi, Mathematical study of threshold of thermal-diffusive instability in counterflow non-premixed biomass-fueled flames considering effective parameters, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.06.014.

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Fig. 1. Schematic of the counter-flow non-premixed combustion system fueled by lycopodium particles.

stream and combust in a very thin zone, reaction zone. A great amount of heat is released from the reaction zone and by-products exit the system as shown in Fig. 1. It is notable that no solid remains are produced upon the combustion of lycopodium particles [19]. In this paper, Arrhenius one-stage reaction is used to model the counter-flow non-premixed combustion of lycopodium particles [3]:

vF [F ] + vO [O] → vproduct [P]

(1)

where [F ], [O] and [P ] demonstrate fuel, oxidizer and products, respectively. Also υF , υO and υP denote stoichiometric coefficients of fuel, oxidizer and products, respectively. In this investigation, it is presumed that a gaseous fuel with certain composition, namely methane, evolves from the vaporization process so pyrolysis is disregarded [16]. Complete reaction of methane is considered as below [3]: CH4 + 2 (O2 + 3.76N2 ) → CO2 + 2H2 O + (7.52)N2

(2)

Velocity field in x-direction is considered as below [3]: Uxf = −aX

(3)

In the above equation, a denotes the flow strain rate. In order to evaluate the instability, critical Lewis and Zeldovich numbers are obtained. Lewis number expresses the ratio of thermal diffusivity to mass diffusivity, which is written as below [3]: Le =

α

(4)

D

where α and D describe thermal and mass diffusivities, respectively. Also, α is formulated as below [3]:

α=

λ ρC

(5)

where λ, ρ and C describe thermal conductivity of fuel or oxidizer, mixture density and mixture specific heat capacity, respectively. Zeldovich number providing a quantitative measurement for activation energy of a chemical reaction is defined as follows [3]: Ze =

EQYF−∞ RCT2f

(6)

Please cite this article as: S. Sadeghi and M. Bidabadi, Mathematical study of threshold of thermal-diffusive instability in counterflow non-premixed biomass-fueled flames considering effective parameters, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.06.014.

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In the above equation, E, Q, C, R and Tf are overall activation energy, heat released from the reaction zone, specific heat capacity of mixture, universal gas constant and flame temperature, respectively. Specific heat capacity of lycopodium/oxidizing gas mixture is calculated by the following correlation [3]: C = Ca +

4ρp 3ρ

π rp 3 Cp np

(7)

where ρp , np , Cp , Ca , rp denote density of lycopodium particles, number density of particles per unit volume, specific heat capacity of lycopodium particles, specific heat capacity of oxidizing gas and radius of the particles, respectively. In order to obtain the density of the mixture, following correlation is used [3]: 4

ρ = ρa + π rp 3 np ρp

(8)

3

where ρa describes the density of oxidizing gas. 2.2. Modeling of the asymptotic processes 2.2.1. Mathematical modeling of the vaporization process According to Fig. 1, lycopodium particles are transformed into gaseous fuel over a vaporization process. As the devolatization process occurs rapidly, rate of vaporization is mathematically modeled by a Heaviside function [18]:

ωv =

Ys

τvap

H(T − Tv )

(9)

In the above equation, Ys , τvap and Tv describe mass fraction, constant characteristic time of vaporization and vaporization temperature of lycopodium particles, respectively. Also, the Heaviside function is formulated as below: H (T − Tv ) =

{

0

T < Tv

1

T ≥ Tv

(10)

2.2.2. Mathematical modeling of the flame zone In former studies, investigations were performed for large values of Zeldovich number so the flame zone was presumed to be very slender. In this paper, critical Zeldovich number and Damkohler number which affect the instability are measured. Damkohler number is defined as below [3]: DC = ρ BϑO YF −∞ /WF a

(11)

where B, ϑO , WF and YF −∞ denote frequency constant, number of oxidizer’s moles that combust with one mole of fuel in the reaction zone, fuel molecular weight and mass fraction of lycopodium particles at x = −∞ when the particles are completely transformed into methane, respectively. In order to assess the onset of extinction for the flame, critical Damkohler number can be formulated as below [3]:

δ0E ≈ 2e(Sf − 2S2f + 1.04S3f + 0.44S4f ) where Sf =

x 1 erfc( √f ) 2 2

δ = δ0 + O(

1 Ze

(12)

in which xf represents the position of flame front [3]. Also, δ0 is approximated as follows [3]:

)

(13)

δ , known as reduced Damkohler number, can be obtained by the following correlation [3]: ( ) 8π exp(x2f )DLeO Lef S2f Ta δ= exp − Tf α 2 F2Of Z3e

(14)

where LeO and Lef express the oxidizer and fuel Lewis numbers, respectively. Also, FOf is defined as below [3]:

⎛ tof

1−

⎝ Fof ≡ Fo (xf , Leo ) = √ t Leof 1 −

0.276 tf

+

2.15 t2f

⎞

0.276 tof

+

2.15 tof 2

⎠

(15)

√

In the above equation, tof = 1 + 0.33333xf Leo and tf = 1 + 0.33333xf . By applying the aforementioned definitions into Eq. (14), critical Damkohler number at extinction is rewritten as below [3]:

D0E ≈

α 2 F2Of Z3e exp

(

Ta Tf

) − x2f + 1 (

4π LeO Lef Sf

1 − 2Sf + 1.04S2f + 0.44S3f

)

(16)

Please cite this article as: S. Sadeghi and M. Bidabadi, Mathematical study of threshold of thermal-diffusive instability in counterflow non-premixed biomass-fueled flames considering effective parameters, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.06.014.

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Accordingly, ratio of critical flow strain rate (corresponding to flame extinction) to the reference flow strain rate (at which fuel and oxidizer Lewis numbers are unity) is presented below [3]:

(

a a0

) =

LeO Lef F2Of

crit

(

Tf

)6

η2

T0f

In the above equation, η =

dE Sf S0f

[ exp

Ta

(

T0f

and dE = η

1−

T0f Tf

) +

(x2f

( 0 )2

− xf

] )

(1−2Sf +1.04S2f +0.44Sf3 )

( )

(1−2Sf 0 +1.04 Sf 0

2

+0.44(Sf 0 ) ) 3

(17) . It can be implied from Eq. (17) that critical flow

strain rate depends mainly on flame front position and flame temperature. 2.3. Governing equations In this subsection, general forms of mass and energy conservation equations under unsteady conditions are derived. It should be noted that in the preheat zone, interactions between lycopodium particles are disregarded so diffusion term in mass conservation equation of solid lycopodium particles, which is written below, is removed:

∂ Ys ∂ Ys − Uxf = −ωv ∂t ∂X

(18)

Mass conservation equation of gaseous fuel under unsteady condition is presented below:

∂ YF ∂ 2 YF ∂ YF ωC − Uxf = DF 2 − + ωv (19) ∂t ∂X ∂X ρ where YF , DF and ωC denote mass fraction of gaseous fuel, mass diffusion of fuel and reaction rate, respectively. Rate of reaction for the gaseous methane is formulated as follows [3]:

ωC = Bρ 2 vF vO YF YO exp(−

E RT

)

(20)

In the above equation, YF and YO are formulated as follows [3]: m YF = YF vF mF YO = YO

m

(21)

(22)

vO mO

where YO and YF describe oxidizer and fuel mass fraction, respectively. Also, mF , mO and m are fuel, oxidizer and mixture molecular weight, respectively. Mass conservation equation of oxidizer under unsteady condition is derived as below: dYo d2 Yo ωF ∂ Yo − Uxf = Do 2 − ϑ ∂t dX dX ρ

(23)

where Do describes the mass diffusion of oxidizer. With regard to Eq. (2), ϑ is equal to 2. General form of energy conservation equation under unsteady condition is written as follows:

∂T dT d2 T Q Qv − Uxf = DT 2 + ωF − ωv ∂t dX dX ρC C

(24)

where DT and Qv represent the thermal diffusivity coefficient and latent heat of vaporization, respectively. In order to solve the time-dependent governing equations in the counter-flow non-premixed combustion system, perturbation method is employed. Dimensionless coordinate of time-space (ξ , η, τ ) is defined as below:

(ξ , η, τ ) =

{ (

) Uxf Uxf Uxf2 x − xdf (y, t) ,y ,t DT

DT

}

DT

(25)

where xdf (y, t ) is defined as follows: xdf (y, t) = −Uxf t + ε Φ (y, t)

(26)

In the above equation, Φ and ε represent the perturbation coefficient and amplitude of the flame front corrugations, 1 respectively, where ε = Ze . Φ can be rewritten as follows using variable change technique:

Φ (y, t) =

DT Uxf

Φ (η, τ )

(27)

where Φ (η, τ ) is considered to possess the following form:

Φ (η, τ ) = exp (ωτ + ikη) = exp (ωτ ) [cos (kη) + isin (kη)]

(28)

Please cite this article as: S. Sadeghi and M. Bidabadi, Mathematical study of threshold of thermal-diffusive instability in counterflow non-premixed biomass-fueled flames considering effective parameters, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.06.014.

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In the above equation, ω and k describe the frequency and wave number of perturbation, respectively. By applying the aforementioned parameters into the mass and conservation governing equations, the equations are rewritten as follows:

[ ] ∂ Φ (η, τ ) ∂ Ys ∂ Ys + 1−ε = −ˆ ω˙ v ∂τ ∂τ ∂ξ [ ] ∂ YF ∂ Φ (η, τ ) ∂ YF 1 + 1−ε = ∆YF + ˆ ω˙ v − ˆ ω˙ c ∂τ ∂τ ∂ξ LeF [ ] ∂ Φ (η, τ ) ∂ Yo 1 ∂ Yo + 1−ε = ∆Yo − ˆ ω˙ c ∂τ ∂τ ∂ξ LeO [ ] ∂T ∂ Φ (η, τ ) ∂ T + 1−ε = ∆T − ˆ ω˙ v + ˆ ω˙ c ∂τ ∂τ ∂ξ

(29) (30) (31) (32)

where ˆ ω˙ v and ˆ ω˙ c are the rates of vaporization and reaction in the unsteady perturbed system, respectively (see Eqs. (9) and (20)). Also, ∆ is expressed as below:

∆=

∂2 + ∂ξ 2

(

∂Φ ∂ ∂ −ε ∂η ∂η ∂ξ

)2 (33)

In order to solve the governing equations, perturbation method is used. The variables are now defined as the summation of steady-state solution and a harmonic perturbation term: Ys = Ys (ξ ) + ε Φ (η, τ ) Y˜ s (ξ ) + O ε 2

(34)

( ) YF = YF (ξ ) + ε Φ (η, τ ) Y˜ F (ξ ) + O ε 2 ( ) YO = YO (ξ ) + ε Φ (η, τ ) Y˜ O (ξ ) + O ε 2 ( ) T = T (ξ ) + ε Φ (η, τ ) T˜ (ξ ) + O ε 2

(35)

( )

(36) (37)

By substituting Eqs. (34)–(37) into Eqs. (29)–(32), following equations are obtained:

∂ Y˜ s ∂ Ys ωY˜ s + =ω − ω˜ v ∂ξ ∂ξ ( ) ) ( 2 ∂ Y˜ F ∂ YF 1 ∂ 2 Y˜ F k2 ˜YF ω + k + − = ω+ + ω˜ v − ω˜ c LeF ∂ξ LeF ∂ξ 2 LeF ∂ξ ( ) ( ) k2 ∂ Y˜ O k2 1 ∂ 2 Y˜ O ∂ YO Y˜ O ω + + ω + − = − ω˜ c LeO ∂ξ LeO ∂ξ 2 LeO ∂ξ ( ) ∂ T˜ ( ) ∂T ∂ T˜ 2 T˜ ω + k2 + − 2 = ω + k2 − ω˜ v + ω˜ c ∂ξ ∂ξ ∂ξ

(38) (39) (40) (41)

where ω ˜ v and ω˜ c are defined in the following:

ω˜ v =

Ys a τvap

ω˜ c = T˜

∂ˆ ω˙ c ∂T

Ys |ξ =ξvap

⏐

⏐ H(T − Tv ) + T˜ ⏐ + Y˜ F

ξ =ξv ap

a τvap

δ (T − Tv )

∂ˆ ω˙ c ∂ YF

(42) (43)

Location of thermal perturbation corresponding to the vaporization process in the counter-flow non-premixed combustion system is determined by the following formula:

ξ˜vap

⏐ ⏐ − T˜ ⏐ ξ =ξ ⏐ vap = ∂T ⏐ ∂ξ ⏐

(44)

ξ =ξv ap

Please cite this article as: S. Sadeghi and M. Bidabadi, Mathematical study of threshold of thermal-diffusive instability in counterflow non-premixed biomass-fueled flames considering effective parameters, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.06.014.

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Fig. 2. Variation of critical strain rate with oxidizer Lewis number for unity fuel Lewis number.

2.4. Boundary and jump conditions

In this subsection, appropriate boundary and jump conditions for solving the governing equations are presented. In the preheat zone (ξ → −∞), following boundary conditions are considered: Y˜ s = 1 Y˜ F = 0 Y˜ O = 0

(45)

T˜ = 0 At the vaporization front (ξ = ξv ap ), boundary conditions are defined as below: Y˜ s = 0 Y˜ F = Yv Y˜ O = 0

(46)

T˜ = Tv At the flame front (ξ = ξf ), following boundary conditions are used: Y˜ s = 0 Y˜ F = 0 Y˜ O = 0

(47)

T˜ = TF In the oxidizer zone (ξ → +∞), following boundary conditions are applied: Y˜ s = 0 Y˜ F = 0 Y˜ O = α

(48)

T˜ = 0 where α is expressed as follows:

α=

YO,∞

ϑ YF−∞

(49)

Please cite this article as: S. Sadeghi and M. Bidabadi, Mathematical study of threshold of thermal-diffusive instability in counterflow non-premixed biomass-fueled flames considering effective parameters, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.06.014.

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Fig. 3. Variation of critical strain rate with fuel Lewis number for unity oxidizer Lewis number.

Fig. 4. Variation of critical Zeldovich number with wave number for different fuel Lewis numbers and mass particle concentrations.

In the above equation, YF−∞ is formulated as below [3]: YF−∞ =

4 3

π rp 3 np ρp ρ

(50)

In the above correlation, MPC = 34 π rp 3 np ρp is known as mass particle concentration [19]. It is notable that in the present analysis, initial value of MPC is presumed to be less than 1 kg/m3 ; hence, flame sheet is initially placed at the left-hand side of the stagnation plane (see Fig. 1). YF−∞ is expressed as the mass fraction of gaseous fuel at x = −∞ when lycopodium particles are completely in gaseous phase (gaseous methane) [19]. In order to calculate ξf , jump condition at this position is required. In the reaction zone, convection and vaporization terms can be neglected. Therefore, by disregarding the aforementioned terms from Eqs. (39) and (41), and integrating the equations from ξf− to ξf+ , following jump condition is obtained at the flame front:

[(

∂ T˜ 1 ∂ Y˜ F ∂ T˜ 1 ∂ Y˜ F )+{ . }] − = [( ) + { . }] + ∂ξ LeF ∂ξ ξf ∂ξ LeF ∂ξ ξf

By considering ε =

( )

Y˜ F = Y˜ F

0

( +

(51)

expanding T˜ (ξ ), Y˜ F and Y˜ O , and using perturbation approach, following equations are achieved: )( ) ( ) 1 1 Y˜ F + O (52) 2 1 , Ze

Ze

1

Ze

Please cite this article as: S. Sadeghi and M. Bidabadi, Mathematical study of threshold of thermal-diffusive instability in counterflow non-premixed biomass-fueled flames considering effective parameters, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.06.014.

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S. Sadeghi and M. Bidabadi / Computers and Mathematics with Applications xxx (xxxx) xxx Table 1 Characteristics of lycopodium particles and input parameters used in this analysis. Parameter

Value

Unit

E

96.2

kJ/mole

[19]

ρa

1.16

kg/m3

[3]

ρp

1000

kg/m3

[3]

rp

12

µm

[3]

np

12 × 109

–

[3]

Cp

5.67

kJ/kg-K

[3]

Ca

1.004

kJ/kg-K

[3]

B

3.5 × 106

mol−1 s−1

[19]

λ

1.46 × 10−4

kJ/(m s K)

[3]

Q

64 895.40

kJ/kg(fuel)

[3]

Qvap

500.09

kJ/kg(water)

[19]

T∞

300

K

Assumed

YO,∞

0.13

–

[3]

) ( )( ) ( 1 1 + Y˜ O + O 0 1 Ze Ze2 ( ) ( ) 1 1 T˜ (ξ ) = T˜ O + T˜ 1 (ξ ) + O 2 (

Y˜ O = Y˜ O

)

Ze

Ze

Ref.

(53) (54)

Sivashinsky demonstrated the relationship between T˜ 0 (ξ ) and T˜ 1 (ξ ) as below [23]:

(

∂ T˜ 0 (ξ ) ∂ξ

) = 0−

1( 2

)

T˜ 1 (ξ )

0−

(55)

3. Results and discussion In this section, results obtained for critical flow strain rate (associated with flame extinction) and critical Lewis and Zeldovich numbers (associated with thermal-diffusive instability) are presented. Characteristics of lycopodium particles and input parameters considered in this study are presented in Table 1. In order to demonstrate the reliability and accuracy of the model suggested for counter-flow non-premixed flames with thermal-diffusive instability, results of this analysis for critical flow strain rate are compared to those provided by Seshadri and Trevino [22]. Fig. 2 demonstrates the effect of oxidizer Lewis number on the critical flow strain rate for different magnitudes of activation temperature. In this plot, fuel Lewis number is taken to be unity. With respect to Fig. 2, critical flow strain rate descends with an increment in oxidizer Lewis number. The variation can be justified by definition on critical flow strain rate presented in Eq. (17). Based on this equation, critical flow strain rate mainly depends on flame temperature and flame front position. With an increment in oxidizer Lewis number, the amount of oxidizer penetrates into the reaction zone. Accordingly, flame temperature decreases leading to a decrement in value of critical flow strain rate. It can also be concluded that raising the activation temperature would result in an increment in critical flow strain rate. As plotted in this figure, results of this study are compared to data provided by Seshadri and Trevino for the case in which fuel Lewis number is unity and activation temperature is equal to 10 000 K. As can be implied from Fig. 2, current results are appropriately compatible with the results given in Ref. [22] under the aforementioned conditions, especially for oxidizer Lewis numbers greater than 0.6. Changes in critical flow strain rate with fuel Lewis number for different magnitudes of activation temperature are plotted in Fig. 3. It should be expressed that in this figure, oxidizer Lewis number is considered to be unity. Regarding Fig. 3, magnitude of critical flow strain rate descends with an increment in fuel Lewis number. Similar to prior explanation for Fig. 2, with increasing the fuel Lewis number, penetration of fuel into the reaction zone would reduce. Thus, the flame temperature descends so the exponential term in Eq. (17) declines leading to reduction of critical flow strain rate. Similar to the conclusion presented for Fig. 2, increasing the activation temperature would result in increment of critical flow strain rate. Obtained results are also compared to those that can be found in Ref. [22] for the case in which oxidizer Lewis number is unity and activation temperature is 30 000 K. It can be implied from Fig. 2 that there is an acceptable agreement between the results for the considered case, particularly for fuel Lewis numbers larger than 0.4. In order to assess the boundaries of thermal-diffusive instability for the counter-flow non-premixed flame, relationship between critical Lewis and Zeldovich numbers can be extracted by solving the jump condition presented in Section 2.4 using solution of the governing equations and the considered boundary conditions. The effect of wave number on critical Zeldovich number for a wrinkled flame front considering different fuel Lewis numbers and mass particle concentrations Please cite this article as: S. Sadeghi and M. Bidabadi, Mathematical study of threshold of thermal-diffusive instability in counterflow non-premixed biomass-fueled flames considering effective parameters, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.06.014.

S. Sadeghi and M. Bidabadi / Computers and Mathematics with Applications xxx (xxxx) xxx

11

Fig. 5. Variation of critical frequency with wave number for different fuel Lewis numbers and mass particle concentrations.

Fig. 6. Variation of critical Lewis number with wave number for different Zeldovich numbers and mass particle concentrations.

is plotted in Fig. 4. Regarding Fig. 4, with an increment in the value of wave number, value of critical Zeldovich number ascends. In other words, possibility in occurrence of the thermal diffusive instability (threshold of the instability) would increase with increasing the wave number in the counter-flow non-premixed system. For the considered interval of wave number, critical Zeldovich number varies between 11.78 and 28.48. It can also be concluded from Fig. 4 that with a decrement in value of fuel Lewis number, critical Zeldovich number increases at the same value of wave number. Moreover, critical Zeldovich number ascends with enhancing the value of mass particle concentration. Variation of critical frequency of the wrinkled flame front, corresponding to threshold of the instability, with wave number is depicted in Fig. 5 for several values of fuel Lewis number and mass particle concentration. As can be observed in Fig. 5, with an increment in the value of wave number, value of critical frequency enhances. In other words, the greater the value of wave number, the higher the frequency of the wrinkled flame front. In addition, for a certain wave number, magnitude of critical flow strain rate rises with a decrement in fuel Lewis number and an increment in mass particle concentration. For the investigated range of wave number, values of critical frequency at threshold of the thermal-diffusive instability change between 1.88 and 5.97. Fig. 6 describes the effect of wave number on critical fuel Lewis number (which determines the ratio of thermal to mass diffusivities) for several values of Zeldovich number and mass particle concentrations. With regard to Fig. 6, with an increment in value of wave number, critical fuel Lewis number enhances. In other words, greater wave number increases the thermal-diffusive instability boundary represented by critical Lewis number. It can also be implied from this figure that at a certain value of wave number, threshold of the instability occurs in larger values of fuel Lewis number when Zeldovich number decreases and mass particle concentration increases. As can readily be observed in Fig. 6, for the considered range of wave number, critical Lewis number varies between 0.85 and 2.95. Please cite this article as: S. Sadeghi and M. Bidabadi, Mathematical study of threshold of thermal-diffusive instability in counterflow non-premixed biomass-fueled flames considering effective parameters, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.06.014.

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S. Sadeghi and M. Bidabadi / Computers and Mathematics with Applications xxx (xxxx) xxx

Fig. 7. Variation of critical Zeldovich number with Lewis number for different mass particle concentrations.

Fig. 8. Variation of critical frequency with Lewis number for different mass particle concentrations.

Variation of critical Zeldovich number with fuel Lewis number for mass particle concentrations of 1 and 0.87 is illustrated in Fig. 7. It can be observed from Fig. 7 that enhancing the fuel Lewis number would result in reduction of critical Zeldovich number. In fact, increment of fuel Lewis number causes that the thermal-diffusive instability occurs in lower values of Zeldovich number. It can also be extracted from Fig. 7 that increasing the mass particle concentrations at a fixed value of fuel Lewis number enhances the critical Zeldovich number. With respect to this figure, for the examined range of fuel Lewis number and mass particle concentration, critical Zeldovich number ranges from 42.78 to 5.1. Fig. 8 plots the influence of fuel Lewis number on critical frequency of the wrinkled flame in the counter-flow nonpremixed system for mass particle concentrations of 1 and 0.87. As can be seen in Fig. 8, with an increment in value of fuel Lewis number, critical frequency decreases. In other words, increment of fuel Lewis number due to rise in thermal diffusivity enhances the amount of heat flux from the reaction zone to the preheat zone leading to reduction of critical frequency. At the same value of fuel Lewis number, increase of mass particle concentration increases the critical frequency of the wrinkled counter-flow non-premixed flame. With respect to Fig. 8, for the considered intervals of fuel Lewis number and mass particle concentration, critical frequency varies between 12.02 and 2.55. 4. Conclusion This mathematical study presented a detailed mathematical analysis to obtain the onset of thermal-diffusive instability in planar counter-flow non-premixed flame fueled by an organic fuel. In this paper, an asymptotic technique was enforced. Preheat, vaporization, flame and oxidizer zones were analyzed. Time-dependent forms of mass and energy conservation equations were written and solved considering necessary boundary and jump conditions. To obtain boundaries of thermaldiffusive instability, critical frequency of the wrinkled counter-flow non-premixed flame, critical Zeldovich number, critical Please cite this article as: S. Sadeghi and M. Bidabadi, Mathematical study of threshold of thermal-diffusive instability in counterflow non-premixed biomass-fueled flames considering effective parameters, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.06.014.

S. Sadeghi and M. Bidabadi / Computers and Mathematics with Applications xxx (xxxx) xxx

13

fuel Lewis number and critical flow strain rate (represented by Damkohler number) were investigated taking into account wave number parameter. Main conclusions of this study are presented below:

• Critical flow strain rate responsible for extinction of the counter-flow non-premixed flame descends with enhancements in oxidizer and fuel Lewis numbers in presence of thermal-diffusive instabilities.

• Increase of wave number causes an enhancement in value of critical Zeldovich number leading to increase of • • • •

the possibility of the thermal diffusive instability initiation in the counter-flow non-premixed system. For the investigated range of wave number, critical Zeldovich number varies between 11.78 and 28.48. With a rise in value of wave number, critical frequency of the wrinkled flame front grows. When the wave number ranges between 0 and 1, the magnitude of critical frequency at threshold of the thermal-diffusive instability varies between 1.88 and 5.97. Increase of wave number enhances the critical fuel Lewis number which represents the ratio of thermal to mass diffusivities in the system. For the considered interval of wave number, critical Lewis number ranges between 0.85 and 2.95. Increment of fuel Lewis number causes that the thermal-diffusive instability occurs in lower values of Zeldovich number. With an increment in value of fuel Lewis number, critical frequency of the wrinkled flame front lessens.

In the future works, variations of density, heat capacity and particle diameter time in presence of convective and radiative heat losses can be examined. Moreover, a CFD approach can be used to simulate the results. Appendix

A.1. Solution of the governing equations In this subsection, solution of the afore-derived governing equations (Eqs. (38)–(41)) is presented. In order to solve the equations, Mathematica software is employed. Since changes in the reaction zone are highly intense, reaction zone is considered as the inner region in the perturbation method. In this respect, following zones are considered to evaluate the variations of mass fraction and temperature before and after the reaction zone: R1 : ξ | ξ f − ε < ξ < ξ f

}

(A.1)

R2 : ξ | ξ f < ξ < ξ f + ε

}

(A.2)

{

{

Solutions of Eqs. (38)–(41) for R1 and R2 zones are obtained as follows using Mathematica software: • Zone R1 : y˜ s (ξ ) = C1 exp (−ωξ ) y˜ F (ξ ) = (

(

) √ξv

erf

× exp

LeF 2

(√

1

ξ √f

− erf

2 LeF

((

+

(A.3)

( )√ k2 yF v ω + Le 2 F ( ) √ (

2LeF

√ −

LeF 2

LeF

1

(√

+

+

k2 LeF

2 ω+

2

k2

)

LeF

LeF

( Le ω+ F ⎜ −erf ⎜ ⎝ ω+

√ (

k2 LeF

+

)

+

LeF

LeF 2

2

2

LeF

2

) ξ

2

√ ( )2 ) k2 LeF 2 ω+ +

⎛√

(

2 2 ω+

2 LeF

√ ( [ ( ( ) ) LeF 1 LeF k2 ) erf + (2ξ + 1) − 2 ω +

k2 LeF

)

(

+ √

2

2

LeF 2

)2

+ LeF ξ +

LeF 2

LeF 2

⎞

⎛ (√ √ ) ( ) ⎟ LeF k2 LeF LeF ⎟ exp ⎝ 2 ω+ + + x ⎠ 2 LeF 2 2

)

2

2

LeF 2

√ (

2 ω+

k2 LeF

) +

LeF 2

+

LeF 2

)

2

Please cite this article as: S. Sadeghi and M. Bidabadi, Mathematical study of threshold of thermal-diffusive instability in counterflow non-premixed biomass-fueled flames considering effective parameters, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.06.014.

14

S. Sadeghi and M. Bidabadi / Computers and Mathematics with Applications xxx (xxxx) xxx

⎛√

( Le ω+ F ⎜ +erf ⎜ ⎝

+

1

√

LeF

2

√

2

√

2 ω+

k2

+

2 ω+

2 1

(√

LeF

2LeF

+C1 exp

LeF 2

((√ +C2 exp

) +

k2

−

LeF

LeF 2

√ (

2

LeF

+

3LeF 2

)

LeF

√ (

2 ω+

2 ω+

k2 LeF

2

LeF

+

−

)

k2

)

LeF

))

LeF 2

+

+

(( exp

2

LeF

+

( ) ⎛(√ √ 2 ) ) ( ω + Lek ⎟ LeF LeF LeF k2 F ⎟ exp ⎝ + + ξ+ 2 ω+ ⎠ 2 LeF 2 2 2

⎠

2

−3

2

⎞

⎞

√

LeF

k2

LeF 2

+

2

+

LeF

2 ω+

√

)2

LeF 2

)

√ (

2

((

(

LeF 2

LeF

( (√ ( +erf

+

2

(

1

)

k2 LeF

LeF 2

11 −

LeF 2

√ (

2 ω+

k2 LeF

) +

LeF 2

) ξ

)2 ⎞⎞⎞⎞⎤ ⎠⎠⎠⎠⎦

LeF 2 LeF 2

√

) ) ξ

) ) ξ

(A.4)

y˜ O (ξ ) = 0

(A.5)

⎛√ ( ⎞ ) ⎡⎛ )√ 2 ( ) ω + k2 θf − θv 4 ω + k2 + 1 − 2ξ − 1 8a+2 ⎠ ( )) ⎣⎝−erf ⎝ θ˜0 (ξ ) = ( ( ) √ ξ ξ 2 2 erf √f − erf √v 2 2 (

) (√ [ (√ ( ) ( ) 1 2 4 ω+k +1−1 4 ω + k2 + 1 × exp 8

−4ξ − 1

)]

⎛√ ( ⎞ ) [ (√ ) (√ )] 4 ω + k2 + 1 − 3 ( ( ) ) ⎠ exp 1 + erf ⎝ 4 ω + k2 + 1 − 1 4 ω + k2 + 1 − 4ξ − 1 √ 2 2

8

(A.6)

⎛√ ( ⎞ ) [ (√ ) (√ )] 4 ω + k2 + 1 + 2ξ + 1 ( ) ( ) 1 ⎠ − exp 4 ω + k2 + 1 + 1 4 ω + k2 + 1 + 4ξ + 1 erf ⎝ √ 8

2 2

⎛√ ( ⎞ ⎞⎤ ) [ (√ ) (√ )] 4 ω + k2 + 1 + 3 ( ) ( ) 1 ⎠ exp 4 ω + k2 + 1 + 1 4 ω + k2 + 1 + 4ξ + 1 ⎠⎦ +erf ⎝ √ 2 2

8

[ (√ ) ] [ (√ ) ] ( ) ( ) 1 1 4 ω + k2 + 1 − 1 ξ + C2 exp 4 ω + k2 + 1 − 1 ξ + C1 exp − 2

2

• Zone R2 : y˜ s (ξ ) = 0

(A.7)

y˜ F (ξ ) = 0

(A.8)

Please cite this article as: S. Sadeghi and M. Bidabadi, Mathematical study of threshold of thermal-diffusive instability in counterflow non-premixed biomass-fueled flames considering effective parameters, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.06.014.

S. Sadeghi and M. Bidabadi / Computers and Mathematics with Applications xxx (xxxx) xxx

( ) √ 2 ω + Lek α 2 F ( ) √ y˜ O (ξ ) = ( ( ξ 1 − erf √ f2 2 2 ω+

)

k2

LeO

( (

) erf

1

LeO

2

2

LeO 2

+

LeO

[

√ (2ξ + 1) −

2α +

LeO

15

)

2

⎛( √ ( (√ ) )2 √ √ ) LeO LeO 1 LeO k2 LeO LeO LeO ⎝ × exp − 2α + ξ+ 2 ω+ + − 2

2

⎛√

(

ω+

1

(√

+

+

k2 LeO

(

+ √

2

)2

LeO 2

2LeO

+ LeF ξ +

2

LeO

2

2

⎞

LeO 2

⎛ (√ √ ( ) ) ⎟ LeO LeO LeO k2 ⎟ exp ⎝ + + xξ 2 ω+ ⎠ 2 LeO 2 2

LeO 2

)

2

2

√

2

(

+

√

2

LeO

√

(

1

2 ω+

2

+

2

LeO 2

)2

+

)

LeO 2

2

⎞

( ) ⎛ (√ 2 ) √ ω + Lek ⎟ LeO LeO LeO O ⎟ exp ⎝ + ξ+ 2α + ⎠ 2 2 2 2

3LeO 2

LeO 2

+

LeO

( (√ (

(

)

k2

2 ω+

2

+ erf

)

LeO

+

LeO

k2 LeO

LeO ω +

)

k2

2 ω+

2

⎜ + erf ⎜ ⎝

1

√ (

LeO

⎛√

+

)

k2 LeF

⎜ LeO ω + − erf ⎜ ⎝ (

2

LeO

)

k2 LeO

+

2

+

LeO 2

⎞

LeO 2

⎠

2

√ −3

LeO

))

2

⎛( √ ( √ ( ) (√ )2 ⎞⎞⎞⎞⎤ √ ) ) LeO 1 LeO LeO LeO k2 LeO k2 ⎠⎠⎠⎠⎦ + ξ+ + − × exp ⎝ 11 − 2 ω+ 2 ω+ 2

(( + C1 exp

LeO 2

√ −

LeO

LeO 2

√ (

2 ω+

2

k2 LeO

) +

2LeO

LeO

2

) ) ξ

2

((√ + C2 exp

LeO 2

LeO

2

√ (

k2

2 ω+

LeO

2

) +

LeO 2

+

LeO 2

) ) ξ (A.9)

⎛√ ( ⎞ ) √ 2 ( ) ⎡⎛ ) θ ω + k2 4 ω + k2 + 1 − 2ξ − 1 f 8a+2 ⎠ ) ⎣⎝−erf ⎝ θ˜0 (ξ ) = ( ( ) √ ξ 2 2 erf √v − 1 2 (

× exp

[ (√ 1

8

⎛√ (

(

)

4 ω + k2 + 1 − 3

+ erf ⎝

) (√

4 ω + k2 + 1 − 1

)

√ 2 2

4 ω + k2 + 1 − 4ξ − 1

(

)

)]

⎞

[ (√ ) (√ )] ( ) ( ) ⎠ exp 1 4 ω + k2 + 1 − 1 4 ω + k2 + 1 − 4ξ − 1 8

Please cite this article as: S. Sadeghi and M. Bidabadi, Mathematical study of threshold of thermal-diffusive instability in counterflow non-premixed biomass-fueled flames considering effective parameters, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.06.014.

16

S. Sadeghi and M. Bidabadi / Computers and Mathematics with Applications xxx (xxxx) xxx

− exp

[ (√ 1 8

⎞ ⎛√ ( ) )( )] 4 ω + k2 + 1 + 2ξ + 1 √ ( ) ⎠ 4 ω + k2 + 1 + 1 4a + 1 + 4ξ + 1 erf ⎝ √ 2 2

⎞⎤ ⎞ ⎛√ ( ) ) (√ )] [ (√ 4 ω + k2 + 1 + 3 ( ) ( ) 1 ⎠ exp 4 ω + k2 + 1 + 1 4 ω + k2 + 1 + 4ξ + 1 ⎠⎦ +erf ⎝ √ 2 2

8

(A.10)

[ (√ ) ] ( ) 1 + C1 exp − 4 ω + k2 + 1 − 1 ξ 2

[ (√ ) ] ( ) 1 + C2 exp 4 ω + k2 + 1 − 1 ξ 2

In order to find C1 and C2 , the boundary conditions (Eqs. (45)–(48)) must be applied. References [1] B. Denet, P. Haldenwang, Numerical study of thermal-diffusive instability of premixed flames, Combust. Sci. Technol. 86 (1–6) (1992) 199–221. [2] E. Robert, P.A. Monkewitz, Thermal-diffusive instabilities in unstretched, planar diffusion flames, Combust. Flame 159 (3) (2012) 1228–1238. [3] H. Rasam, M. Nematollahi, S. Sadeghi, M. Bidabadi, An asymptotic assessment of non-premixed flames fed with porous biomass particles in counter-flow configuration considering the effects of thermal radiation and thermophoresis, Fuel 239 (2019) 747–763. [4] F. Al-Malki, Numerical simulation of the influence of partial premixing on the propagation of partially premixed flames, Comput. Math. Appl. 66 (3) (2013) 279–288. [5] F. Al-Malki, Influence of transverse temperature gradient on the propagation of triple flames in porous channels, Comput. Math. Appl. 76 (2) (2018) 406–418. [6] M. Bidabadi, F.F. Dizaji, H.B. Dizaji, M.S. Ghahsareh, Investigation of effective dimensionless. [7] M. Bidabadi, P. Aghajannezhad, M. Harati, E. Yaghoubi, G. Shahriari, Modeling random combustion of lycopodium particles and gas, Int. J. Spray Combust. Dyn. 8 (2) (2016) 100–111. [8] A. Rahbari, A. Shakibi, M. Bidabadi, A two-dimensional analytical model of laminar flame in lycopodium dust particles, Korean J. Chem. Eng. 32 (9) (2015) 1798–1803. [9] M. Bidabadi, A. Shakibi, A. Rahbari, The radiation and heat loss effects on the premixed flame propagation through lycopodium dust particles, J. Taiwan Inst. Chem. Eng. 42 (1) (2011) 180–185. [10] O.S. Han, M. Yashima, T. Matsuda, H. Matsui, A. Miyake, T. Ogawa, Behavior of flames propagating through lycopodium dust clouds in a vertical duct, J. Loss Prev. Process Ind. 13 (6) (2000) 449–457. [11] M. Bidabadi, A. Haghiri, A. Rahbari, The effect of Lewis and Damköhler numbers on the flame propagation through micro-organic dust particles, Int. J. Therm. Sci. 49 (3) (2010) 534–542. [12] M.M. Nezhad, M. Beheshti, S. Shahraki, H. Ghassemi, R. Shahsavan-Markadeh, On dynamic behavior of premixed counterflow flame propagation, Thermochim. Acta 632 (2016) 86–90. [13] M. Bidabadi, M. Harati, Q. Xiong, E. Yaghoubi, M.H. Doranehgard, P. Aghajannezhad, Volatization & combustion of biomass particles in random media: Mathematical modeling and analyze the effect of lewis number, Chem. Eng. Process.-Process. Intensification 126 (2018) 232–238. [14] C. Proust, Flame propagation and combustion in some dust-air mixtures, J. Loss Prev. Process Ind. 19 (1) (2006) 89–100. [15] A. Rahbari, K.F. Wong, M.A. Vakilabadi, A.K. Poorfar, A. Afzalabadi, Theoretical investigation of particle behavior on flame propagation in lycopodium dust cloud, J. Energy Resour. Technol. 139 (1) (2017) 012202. [16] K. Seshadri, A.L. Berlad, V. Tangirala, The structure of premixed particle-cloud flames, Combust. Flame 89 (3–4) (1992) 333–342. [17] M. Bidabadi, M.A. Vakilabadi, A.K. Poorfar, E. Monteiro, A. Rouboa, A. Rahbari, Mathematical modeling of premixed counterflow combustion of organic dust cloud, Renew. Energy 92 (2016) 376–384. [18] M. Bidabadi, M. Ramezanpour, A.K. Poorfar, E. Monteiro, A. Rouboa, Mathematical modeling of a Non-Premixed organic Dust flame in a Counterflow configuration, Energy & Fuels 30 (11) (2016) 9772–9782. [19] M. Bidabadi, P. Panahifar, S. Sadeghi, Analytical development of a model for counter-flow non-premixed flames with volatile biofuel particles considering drying and vaporization zones with finite thicknesses, Fuel 231 (2018) 172–186. [20] M. Bidabadi, P.G. Nejad, H. Rasam, S. Sadeghi, B. Shabani, Mathematical modeling of non-premixed laminar flow flames fed with biofuel i n counter-flow arrangement considering porosity and thermophoresis effects: An asymptotic approach, Energies 11 (11) (2018) 2945. [21] M. Bidabadi, S. Hosseinzadeh, M. Setareh, P. Panahifar, S. Sadeghi, Theoretical study of non-adiabatic counter-flow diffusion flames propagating through a volatile biomass fuel taking into account drying and vaporization processes, Fuel Process. Technol. 179 (2018) 184–196. [22] K. Seshadri, C. Trevino, The influence of the lewis numbers of the reactants on the asymptotic structure of counterflow and stagnant diffusion flames, Combust. Sci. Technol. 64 (4–6) (1989) 243–261. [23] G.I. Sivashinsky, Diffusional-thermal theory of cellular flames, Combust. Sci. Technol. 15 (3–4) (1977) 137–145.

Please cite this article as: S. Sadeghi and M. Bidabadi, Mathematical study of threshold of thermal-diffusive instability in counterflow non-premixed biomass-fueled flames considering effective parameters, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.06.014.