Combustion, Explosion, and Shock Waves, Vol. 48, No. 1, pp. 10–16, 2012. c M.M. Kabilov. Original Russian Text 

Diffusion-Thermal Stability of Gas Filtration Combustion Waves in an Inert Porous Medium M. M. Kabilova

UDC 536.46

Translated from Fizika Goreniya i Vzryva, Vol. 48, No. 1, pp. 14–20, January–February, 2012. Original article submitted February 16, 2011.

Abstract: A mathematical model for the filtration gas combustion taking into account thermal conductivity, diffusion, and intense interfacial heat transfer is presented. The temperature dependence of the chemical reaction rate is approximated by a δ-function and the thermal-expansion coefficient of gases behind the combustion front is taken into account. Unsteady combustion regimes are analyzed using the method of small perturbations. The boundaries of the longitudinal and spatial stability for steady regimes of the filtration combustion wave are obtained. The dependence of the Lewis number on the thermal-expansion coefficient of the gas mixture along the boundary of stability is derived, along with other relations. Keywords: filtration combustion, stability of combustion wave. DOI: 10.1134/S0010508212010029

The problem of stability of filtration combustion of gases in inert porous media is of practical importance. As shown by analysis, the problem has not been solved to a sufficient degree [1]. The fact of the existence of steady filtration combustion of gases has been studied theoretically and experimentally proved [1–5]. Numerical studies [6] of the steady combustion wave structure have shown that in the absence of gas filtration to the zone of the exothermic chemical reaction, the combustion wave has a one-temperature and a two-temperature structure, depending on the pressure of the initial gas mixture. In this work, gas combustion was studied by the method of small perturbations [7–9] assuming very intense interfacial heat transfer [10], which leads to degeneration of the two-temperature wave structure to the single-temperature one. The mathematical model used in this work differs from other models in the form of the energy conservation equation for the two-phase medium. The temperature dependence of the chemical reaction rate is approximated by a δ-function [7, 11] taking into account the thermal-expansion coefficient of gases behind the combustion front. The transfer coeffia

Institute of Mathematics of the Republic of Tajikistan, Dushanbe, 734063 Tajikistan; [email protected].

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cients and thermophysical characteristics of the phases are assumed to be independent of ambient temperature and concentrations of the deficient component. Heat losses to the ambient medium are absent, and the densities of the gas mixture on both sides of the surface of the combustion front are constant but not identical. It should be noted that the differential equations of the diffusion- thermal stability of the flame front by the method of small perturbations [12] and numerical methods [13, 14] have been extensively studied (see the references in [15–17]). In [18, 19], the boundaries of stability of the steady combustion wave structure of the reacting gases and condensed material were determined and attempts were made to determine the stability margin with respect to one-dimensional and two-dimensional perturbations, taking into account the thickness of the combustion zone. The effect of longitudinal and spatial perturbations on the stability of a steady wave of filtration combustion of gases and the stability conditions obtained in different ways were examined in [20–22]. Following these studies, we consider the dimensionless equations of heat and material transfer for the filtration combustion of gases in a coordinate system moving along the x axis at a rate of unsteady combustion φ . The following equations are identical to the equations

c 2012 by Pleiades Publishing, Ltd. 0010-5082/12/4801-0010 

Diffusion-Thermal Stability of Gas Filtration Combustion Waves in an Inert Porous Medium of heat and mass transfer for reacting gases (ϕ = 0 and φ = 1) and condensed material (v1 = 0, φ = 1, and Le = 0) in other combustion models:   ∂T v1  ∂T = ΔT + +φ + QJ, ∂t 1+ϕ ∂x (1) ∂η ∂η = LeΔη + (v1 + φ ) − J. ∂t ∂x Here J = AN η exp[N (1 − 1/T )] is the chemical reaction rate, Le is the Lewis number defined, according to [12], as the ratio of the diffusion coefficient of the deficient component to the coefficient of effective thermal diffusivity of the medium. Here and below, the following dimensionless variables and parameters are used: t=

u2n tr , æeff z=

v1 =

v1r , un

x=

un (−xr − φr ), æeff

un zr , æeff

T =

φ =

φr , un

Tr , Te Q=

η=

Le =

k0 æeff , u2n N eN

D , æeff

σ=

N= T0 , Te

E , RTe æeff =

u n yr , æeff

ηr , η0

Qr η0 , cp Te (1 + ϕ)

Qr η0 , Te = T0 + cp [1 + ϕ/(1 + u0 )] A=

y=

v10r u0 = , un ϕ=

ρ20 c2 , ρ10 cp

α1 λ1 + α2 λeff . ρ10 cp + ρ2 c2

Here T and η are the dimensionless ambient temperature and the relative mass concentration of the deficient component, v1r is the unsteady gas filtration velocity, un the steady propagation velocity of the filtration combustion wave, Te the equilibrium temperature of the phases, Qr the heat of the reaction, E the activation energy, R the universal gas constant, k0 the pre-exponential factor, æeff the coefficient of effective thermal protection, D the diffusion coefficient of the deficient component, ρ20 , ρ10 , c2 , and cp are the normalized density and specific heat of the porous medium and gas mixture, respectively, T0 is the ambient temperature, η0 the concentration of the deficient component, v1r0 the rate of filtration of the gas before the reaction, λ1 the thermal conductivity of the gas mixture, λeff the effective thermal conductivity of the porous medium, and α1 and α2 are the volumetric concentrations of the gas and solid phases; the subscript r denotes dimensional quantities. The first equation of (1) is obtained by combination and transformation of the equations of heat input in the phases with equal temperatures of the

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porous medium and gas. The conditions at infinity for system (1) are given by x = +∞:

T =σ=

T0 , Te

η = 1, (2)

dη dT = 0, = 0. x = −∞: dx dx The velocities are approximated as follows [7]:   N (T (0, y, z, t) − 1) δ(x), J ≈ AN exp 2 AN = 1 + u0 ,   N (T (0, y, z, t) − 1) , v1 = u0 exp 2   N φ = exp (T (0, y, z, t) − 1) . 2 In this case, the matching conditions derived from (1) by integrating over x in the neighborhood of x = 0 are as follows: [T ] = 0,       ∂T u0 N + 1+ exp (T (0, y, z, t) − 1) = 0, ∂x 1+ϕ 2 [η] = 0, (3)     ∂η N Le + (1 + u0 ) exp (T (0, y, z, t) − 1) = 0, ∂x 2 [f ] = f (x + 0, y, z, t) − f (x − 0, y, z, t) = 0. After switching to the variable θ = (T − σ)/(1 − σ) in system (1)–(3) and linearization of the obtained equations and the matching conditions (3) around the steady-state solution  exp(−uϕ x), x > 0, θs = 1, x < 0, (4)  1 − exp(−wl x), x > 0, ηs = 0, x < 0, where uϕ = 1 + u0 /(1 + ϕ), wl = (1 + u0 )/Le, ∂θ ∂θ ∂θs = Δθ + u + uαθ (0, y, z, t) , ∂t ∂x ∂x θ  = θ − θs , 1 ∂η  ∂η  ∂ηs = Δη  + w + wαθ (0, y, z, t) , Le ∂t ∂x ∂x η  = η − ηs ,

(5)

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Kabilov

[θ ] = 0,





G3 ω 3 + (3G3 k 2 + G2 )ω 2 + (3G3 k 4 + 2G2 k 2 + G1 )ω



∂θ + uϕ αθ (0, y, z, t) = 0, ∂x

θ (±∞) = 0,   ∂η − wl αθ (0, y, z, t) = 0, [η  ] = 0, ∂x N (1 − σ) . η  (±∞) = 0, α = 2 Here



u=

uϕ ,

x > 0,

βuϕ , x < 0,

 w=

wl ,

x > 0,

βwl , x < 0,

+ G3 k 6 + G2 k 4 + G1 k 2 + G0 = 0,

(7)

where Gj = 4u8−2j gj , ϕ

j = 0, 1, 2, 3,

g0 = α2 β − α2 , g1 = αβ 2 + (−α3 + 5α2 − α)β − 5α2 + 2α3 , g2 = α(3 − α)β 2 + α(−2α2 + 7α − 1)β − α4 + 6α3 − 5α2 − 2α, g3 = β 2 + (4α − 2)β + 4α2 − 4α + 1.

β = ρ10 /ρ1e is the thermal-expansion coefficient of the gas mixture, where ρ10 and ρ1e are the densities of the gas phase before and after the combustion front. Assuming small perturbations in the form

Since instability of the steady-state solutions occurs when the sign of the real part of ω changes from negative to positive, applying the theory of polynomial algebra to Eq. (7), we obtain the relation

θ (x, y, z, t) = ς(x) exp(ωt + iky y + ikz z),

− 8g32 k 6 − 8u2ϕ g3 g2 k 4 − 2u4ϕ (g3 g1 + g22 )k 2

η  (x, y, z, t) = ξ(x) exp(ωt + iky y + ikz z) and taking into account the conditions at infinity (5), we obtain ⎧ u2 α ⎪ ⎪ ⎨ Aς exp(μ1 x) − ω + k 2 ς(0) exp(−ux), ς(x) = x  0, ⎪ ⎪ ⎩ x < 0, ς(0) exp(μ2 x), ⎧ w2 α ⎪ ⎪ Aξ exp(μ3 x) − ς(0) exp(−ux), ⎪ ⎨ ω/Le + k 2 ξ(x) = x  0, ⎪ ⎪ ⎪ ⎩ x < 0, ξ(0) exp(μ4 x), where

   1 2 2 −u ± u + 4(ω + k ) , μ1,2 = 2    1 −w ± w2 + 4(ω/Le + k 2 ) , μ3,4 = 2 k 2 = ky2 + kz2 .

The matching conditions in (5) give a system of linear homogeneous equations for ς(0), ξ(0), Aς , and Aξ . Requiring a nontrivial solution of this system, we obtain the following dispersion relation for the perturbation frequency ω:     u3ϕ α u2 α (μ4 − μ3 ) μ1 1+ +uϕ α − μ2 + = 0. (6) ω+k 2 ω+k 2 Since μ4 −μ3 = 0, after transformation of the expression in square brackets, we write the following cubic equation for ω:

+ u6ϕ (g3 g0 − g1 g2 ) = 0.

(8)

In the case where the left side of Eq. (8) is positive, the complex roots of Eq. (7) have positive real parts, and, hence, the perturbations that arise do not decay with time and the combustion wave becomes unstable. The region of parameters for which relation (8) is smaller than zero corresponds to the region of stability of the steady combustion wave. From relation (8) (k 2 = 0), we obtain the following condition for the boundary of longitudinal stability of the combustion wave: g3 g0 − g1 g2 = 0; in the case β = 1, after Fourier and Laplace transforms [19], this condition takes the form ω = ω(α, u). In addition, the condition of longitudinal stability of the combustion wave g3 g0 − g1 g2 < 0 for √ β = 1 and u = 1 implies the condition α < α0 = 2 + 5, studied in [7]. Relation (8) is a cubic equation for k 2 , where k 2 is a positive quantity. Then, the conditions for the existence of positive real roots of Eq. (8) are conditions of the boundary of stability of the steady regime with respect to spatial perturbations: g2 < 0, or g2 > 0, g0 g3 − g1 g2 > 0. We first consider the region g2 > 0, g0 g3 − g1 g2 > 0. For g2 = 0, we have β = β∗ (α), k 2 = k∗2 (α, uϕ ), where    1 2 3 2 2α −7α+1− 8α − 39α +38α + 25 , β∗ = 2(3 − α)   3  2

3 g0 g1 g0 2 2  + + (9) k∗ = uϕ 16g3 12g3 16g3 

3  2 

3 g0 g1 g0  , − + + 16g3 12g3 16g3

Diffusion-Thermal Stability of Gas Filtration Combustion Waves in an Inert Porous Medium and Eq. (7) has the following roots:  ω1 = −3k∗2 , ω2,3 = ±i 3k∗4 + u4ϕ g1 /g3 .

13

(10)

Relations (9) were obtained from an analysis of the spatial stability of the steady combustion wave [22] and are analogs of the longitudinal stability of the combustion wave. Performing a continuous transition from the spatial perturbation frequencies (10) to longitudinal frequencies, we arrive at the expressions  g1 ω1 = 0, ω2,3 = ±iu2ϕ . (11) g3 It is seen that the perturbation frequency depends not only on the injection velocity u0 , but also on the dimensionless activation energy α and the thermal expansion coefficient β. The formula for ω2,3 (11) was obtained from an analysis of the longitudinal stability of steady regimes of the combustion wave. Comparing the spatial (10) and longitudinal (11) perturbation frequencies for the same values of the parameters uϕ , α, and β, we see that the oscillation period of the combustion front is longer in the longitudinal case than in the spatial one. (This is in the range of parameters corresponding to curves 1 and 2 in Fig. 1. In all subsequent figures, uϕ = 1.) Figure 1 shows curves of α = α(β) on the boundaries of stability. The dependence β = β∗ (α) (curve 2 in Fig. 1) is satisfactorily approximated by the relation α = 4.55 − 1.25

β−1 β

Fig. 1. Dimensionless activation energy (α) versus the thermal-expansion coefficient of a gas mixture (β) on the boundaries of stability: (1) Re ω = 0 (longitudinal perturbation); (2) the wave number cal2 culated  4 by formula (9) for ω1 = −3k∗ and ω2,3 = 4 ±i 3k∗ + uϕ g1 /g3 ; (3) numerical study of the pulsating regime; (4) wave numbers are calculated by formula (13); 3g3 g1 − g22 = 0; (5) 2g23 − 9g3 g2 g1 + 27g0 g32 = 0; oscillation boundary on which the wave number is not uniquely defined.

(12)

(curve 3 in Fig. 1). Relation (12) was obtained in a numerical study of pulsating combustion regimes of condensed material [13] and is given in our notation. In accordance with (12), above the curve β = β∗ (α) is an unstable region of parameter values, and below this curve is a stable region. Curve β = β∗∗ (α) (curve 1 in Fig. 1; k 2 = 0) which is the boundary of one-dimensional stability of the steady wave, is also approximated by relation (12). Below curves 1–3, stable ranges of the parameters are observed only in a one-dimensional consideration of the problem, and in a spatial consideration, there is √ a stability margin. As α decreases from α0 = 2 + 5 to three, the values of β∗ increase from one to infinity, and for β → ∞, we have the following orders of magnitude: g0 ≈ α2 β∗ , g1 ≈ αβ∗2 , and g1 ≈ β∗2 , where k∗2 ≈ 0. In addition, for α = α0 and β∗ = 1, we also obtain k∗2 = 0 and, hence, in the interval (3, α0 ) there is a maximum of the function k∗2 = k∗2 (α, uϕ ): k∗2 max = 4u2ϕ /25, αmax = 3.116, and β∗ = 17.215. In this case, the region of stability of the steady wave respect to spatial perturbations in the plane (β, k 2 ) is bounded by the

Fig. 2. Wave number of the thermal-expansion coefficient of a gas mixture on the boundaries of stability (the designations of the curves correspond to those in Fig. 1).

straight line k 2 = 0 (the β axis, Fig. 2) and the curve of k∗2 = k∗2 (β∗ , uϕ ) (curve 2 in Fig. 2), which increases for uϕ > 1 (cocurrent flow) and decreases in the case uϕ < 1 (countercurrent flow). Note that a study of the spatial stability reveals a boundary (curve 4 in Fig. 1) on which the perturbation, which takes negative values, is proportional to the parameter uϕ , except for a pair of complex conjugate imaginary solutions. This means that the resulting perturbations decay in time purely

14

Kabilov

exponentially (more uϕ , the faster). Dangerous spatial perturbations (k 2 = k∗2 max = 0.16) correspond to a value α = 3.116, which is smaller than that in [10, 18], where α = 3.91 (k 2 = 1)) and α = 4 (k 2 = 1), respectively, i.e., the combustion front is sensitive to perturbations whose wave vectors lie within 0 < k 2 < k∗2 max , which is also contained in the interval obtained in a study of the stability of the steady front of the exothermic condensed-phase reaction [18]. We now consider the region g2 < 0. From the theory of the algebra of polynomials, it is known that the cubic equation (8) has one or three positive real roots. Requiring the existence of three roots of Eq. (8) for which Eq. (7) has three corresponding roots, we obtain: in the case 3g1 g3 − g22 = 0 (curve 4 in Fig. 1),    3  g2 g2 g1 g0 g2 2 2 1 3 , k1,2,3 = uϕ 2 − + − 2 3g3 3g32 g3 3g3 3u2ϕ ω1 = − 2

 3



g2 2 3g3

3 −

g2 g1 g0 + , 3g32 g3

Fig. 3. Perturbation frequency (ω) on the boundaries of stability (notation the same as in Fig. 1).

(13)

 4 ω2,3 = ±i 3k1,2,3 + u4ϕ g1 /g3 and in the case 3g1 g3 −g22 < 0, 2g23 −9g3 g2 g1 +27g0g32 = 0 (curve 5 in Fig. 1), g2 2 k12 = − u , 3g3 ϕ    1 g2 2 2 2 k2,3 = − u 1± g2 − 3g3 g1 , 3g3 ϕ 2g2 ω1 = 0,

 4 + u4 g /g . ω2,3 = ±i 3k2,3 ϕ 1 3

Curve 4 in Fig. 1 lies in the region of stability since ω1 < 0 although the other two complex roots are complex imaginary. The oscillatory boundary  is reached on 4 + u4 g /g . curve 5, where ω1 = 0 and ω2,3 = ±i 3k2,3 ϕ 1 3 On this boundary, the wave number is ambiguous. We note that in the consideration of spatial perturbations, the boundary of longitudinal stability (curves 1–3 in Fig. 1) is moved to the position of curve 5. Figure 2 shows curves of k 2 = k 2 (β) (curves 2, 4, and 5) that correspond to the curves in Fig. 1; furthermore, along curves 5 in the plane (β, k 2 ), the combustion front oscillates and curve 4 (Fig. 2) is between the upper two curves 5, along which the oscillations of the combustion front decay. Consequently, the region of the plane (β, k 2 ) between curves 5 belongs to the region of stability. The region between curve 5 (the lower curve in Fig. 2) and the line k 2 = 0, along which the combustion flame only oscillates, is also a region of stability because

Fig. 4. Lewis number versus thermal-expansion coefficient of the gas mixture along curve 4 in Fig. 1.

this region contains curve 2, along which oscillations of the combustion front decay. Curves of the perturbation frequency ω versus thermal-expansion coefficient β are shown in Fig. 3; curves 1 and 2 correspond to the imaginary values of ω, and curves 4 and 5 to real values (less than zero and equal to zero, respectively). Since μ4 > 0, we have ω (14) − 2 < Le. k The dependence Le(β) calculated by formula (14) sing values of ω and k 2 from (13), i.e., along curve 4 in Fig. 1 is shown in Fig. 4. It has the form 3 Le >   3 − 1 . (15) g2 2g2 g2 g1 g0 3 2 1− − + 3g3 3g3 3g32 g3 The region defined by inequality (15) is above the curve in Fig. 4. In curve 5 in Fig. 1, the right side of inequality (15) is equal to zero, i.e., in approaching the oscillatory boundary, the curve in Fig. 4 descends to the β axis. Note that in this approximation, the number Le does not depend on the parameter u0 —the dimensionless velocity of forced filtration of the gas mixture.

Diffusion-Thermal Stability of Gas Filtration Combustion Waves in an Inert Porous Medium Equating the exponents in (4), i.e., assuming symmetry of the temperature and concentration distributions, we obtain 1 + u0 Le = . 1 + u0 /(1 + ϕ)

(16)

This is the case of coincidence of the ranges of temperature and concentration variations of the deficient component of the gas mixture. Note that in the absence of a solid phase (ϕ = 0), the Lewis number is unity. From (16), it is easy see that countercurrent flow (−1 < u0 < 0) corresponds to the Le < 1, and cocurrent flow (u0 > 0) to Le > 1. Consequently, according to classical combustion theory (Le < 1 corresponds to stable propagation and Le > 1 to unstable propagation), the front is unstable in the case of cocurrent flow. Note that the parameter β was first introduced in a study [20] of the stability of a gas flame in inert porous media. Our analysis of the diffusion-thermal stability of steady wave regimes of filtration combustion of gases has showsn the following: • In the case of countercurrent flow, the propagation of the flame front is more stable than that in the case of cocurrent flow, and this confirms the results obtained previously [1]. Countercurrent flow corresponds to a relatively broad zone of chemical reaction compared to that in the case of concurrent flow; • Cocurrent flow corresponds to a Lewis number greater than unity, and countercurrent flow to a Lewis number less than unity; • The obtained boundaries of oscillatory instability are satisfactorily approximated by the condition of stability of pulsating combustion regimes of condensed media; • On the boundary of oscillatory stability, the perturbation frequency is higher in the spatial case of the problem than in the one-dimensional case; • Spatial perturbations are less sensitive to the direction and value of the forced filtration velocity in the gas mixture than longitudinal perturbations; • An advantage of the study is that to determine stability of the combustion front, it is sufficient to know the values of two parameters: the activation energy and the equilibrium temperature, from which the parameters α and β are determined. Thus, the study of the diffusion-thermal stability of the filtration combustion front of gases with intense interfacial heat transfer and no heat losses to the ambient medium fills the gaps available in the theory of stability of the combustion front.

15

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