ХИМИЧЕСКАЯ ФИЗИКА, 2010, том 29, № 7, с. 9-14
ГОРЕНИЕ, ВЗРЫВ ^^^^^^^^^^^^ И УДАРНЫЕ ВОЛНЫ
УДК 541.126
EXISTENCE OF PRESSURE DRIVEN WAVE © 2010 N. Krapivnik*, V. Gol'dshtein**
Ben-Gurion University of the Negev, Departments of Mathematics P.O.B.653, Beer-Sheva 84105, Israel Received 22.04.2009
The phenomenon of pressure driven flames in inert porous media filled with a flammable gaseous mixture is considered in the present work. We focus on the travelling wave solutions arising in the adiabatic model of this phenomenon that was developed and studied in [2, 3, 5, 6]. We start with the simplest possible model which describes the combustion process in inert porous media in the absence of inertial effects, hydraulic resistance, and thermo conductivity. For this model, the existence and uniqueness of the sub-sonic reaction wave were proved under reasonable assumptions. This simplest model is characterized by the linear dependence of friction force on the velocity of gaseous mixture.
Keywords: flame, porous media.
1. INTRODUCTION
Recently, there has been an interest in the combustion in inert porous media filled with combustible gaseous mixture. The distinguishing feature of such medium is the well-known fact that, under definite conditions, speed of pressure perturbations in it may be significantly lower than the sound velocity in the open space. In these conditions, local elevation of pressure may lead to the formation of self-sustaining combustion wave controlled by pressure diffusion ([1—4]). In the frame of reference attached to the flame front spreading in porous media, the set of equations describing its aero-thermo-chemical structure reads [5,6],
—(pe) + — (pue + Pu) = Qw,
dt dx
—(pC) + — (puC) = -w, dt dx
д (pu) + (pu2 + P) = -f,
dt dx
д (p)(pu)=о,
dt dx P = (cp - cv)pT.
(1.1) (1.2)
(1.3)
(1.4)
(1.5)
♦Electronic address: mordeev@bgu.ac.il ♦♦Electronic address: vladimir@bgu.ac.il
ence of porous media is accounted for by friction force (term f) added to the momentum (1.3) equation. The expression for friction force is taken in the form which corresponds to the Darcy—Forcheimer law:
f = KDpu + KFpu|u| (1.6)
and the reaction rate w is taken in the conventional form of the Arrhenius law for a one-step reaction of the first order:
w
ApC exp(-
M-)■
\ rt!
(1.7)
The first and the second equations in (1.1)—(1.5) represent conservation equations for energy and for deficient reactant. Equations (1.3), (1.4), (1.5) represent the equation of momentum, the continuity equation and the equation of state for ideal gas, respectively. Here e is the energy equal to cvT + u2/2. The pres-
The following notations were used: T — temperature, P — pressure, C — concentration of the deficient reactant, E — activation energy, c — specific heat capacity, u — gas velocity in the laboratory frame of reference, Q — combustion energy, w — reaction rate, KD, KF — the coefficients that describe permeability of the medium, A — pre-exponential (frequency) factor, R — universal gas constant. The subscripts mean: " p" — under constant pressure, "v " — under constant volume, "0 " — undisturbed state, "b " — burnt (behind the combustion wave front). We are interested in the proof of existence of traveling wave solutions for equations (1.1)—(1.7) in the form T(x - Dt), P(x - Dt), C (x - Dt ), p(x - Dt), u ( x - Dt ), where D is the a priori unknown front speed. After substituting these functions into original system (1.1)—(1.7) we obtain the following system of ODE's for the travelling wave solutions
d_
dx
(p(u - d
- D)\cT +1 u2
+ Pu) = Qw,
d(p (u - D)С) = -w, dx
d(p(u - D) + P) = -f,
dx
(1.8) (1.9) (1.10)
d(p(u - D)) = 0,
dx
(Cp - cv)T
(1.11)
(1.12)
C(x ^ -<x>) = 0, u(x ^ -<x>) = 0.
(1.14)
Ç = _xAexp\ —— ], p = ^ D I 2p/ E
C
n = —,
Co
IT - To
n = 1
(1.15)
(1.16)
de_ A n-e
- — A D-
D i + pe
+ i + pn
+ --— exp
î + pe \i + pe
(2.1)
dn . n-e
a— = A D-,
D î + pe
dn 1 + pn
—1 = -6 2n--— exp
d^ 1 + p0 \l + p0
(2.2)
(2.3)
The system (1.8)—(1.12) is subject to the boundary conditions: upstream boundary condition (fresh mixture far before the flame front) is given by
T (x ^+cx>) = To , C (x ^+cx>) = Co,
P(x ^ +o>) = P0, p(x ^ +o>) = p0
and downstream boundary condition (far behind the wave ) is
where the following dimensionless parameters s1, 6 2, ct are given by
f , \ f
_ C0Q
s1 _ —— exp CpTlP
s1 < 1,
1 ,
— I, s 2 _ exp
2P. 2
J_ 2ft
1
(2.4)
Y _ *
The system (1.8)—(1.14) was investigated for different purposes: see a recent paper [7] for references. The paper [7] considers and examines different asymptotical methods for the problem of pressure driven flames in inert porous media. But these approximate methods can be applied with confidence only if the solution existence is assured in advance. The goal of the present paper is to prove that the model (1.8)—(1.12) of the pressure driven flames in porous media admits travelling wave solutions.
Dimensionless system
To simplify the analysis further, let us introduce di-mensionless variables along the lines of the Semenov approach, which is well-accepted in thermal-explosion theory [8, 9] .These dimensionless variables are defined as follows:
S2 « 1 , G_ 1 -- ,
Y cv
and the initial conditions are defined by the equation
0(^-w) = 00 = 0, = n = 1,
n(% ^ -w) = n0 = 0.
Expression of the speed velocity has the following form
(2.5)
A D = Kd
D2
ATçCp
exp|è
(2.6)
The (2.1)—(2.3) system is adiabatic, therefore energy integral exists and is easily derived
n- 1 + — (0-an) = 0. (2.7)
Sl
Because of the existence of energy integral (2.7) the system (2.1)—(2.3) that is composed of three equations can be reduced to the system of two ordinary differential equations
de_ * n-e
- — A d--
d% D1 + pe
+ e1| 1 -^(e-an)
S1
_dn
1 + pn 1 + pe
n-i
exp
1 + pe.
P To p Po
where n is dimensionless pressure, 9 is dimensionless temperature, n is dimensionless concentration of deficient reactant and 2, is dimensionless self-similar variable. Here p is reciprocal dimensionless activation temperature and is supposed to be small compared to unity at high activation energies.
2. LINEAR FRICTION
Let us consider the simplest version of the model (1.8)—(1.12) that describes the given phenomena (pressure driven flame in porous media). Approximation is characterized by the friction force linearly dependent on gas velocity in absence of inertia. The system of governing equations (1.8)—(1.12) becomes:
' d% AD1 + pe
with boundary conditions
(2.8)
(2.9)
0o = 0(0) = s, 0 <s < 1, n0 =n(0) < 1 (2.10)
a
at the initial stage (fresh mixture). The choice of the boundary conditions in the (2.10) form, allows us to ignore the cold boundary problem. Characteristic values of the system parameters are
0.1 <ct< 0.3, 10-6 <s2 < e1 < 10-4. System (2.8)—(2.9) has a singular point
n» - ° °
62(1 -a) 62(1 -a) We note that this point is a saddle point. In order to prove the existence of the travelling wave solution we used the method that is based on the analysis of phase portrait. So, existence of the travelling wave solution is reduced to proving the existence of the path that connects two equilibrium states of the system. In combustion theory, we assume that the chemical reaction level is low for low temperatures
with respect to the maximum temperatures of combustion wave. Hence, the first equilibrium state is a stationary state (singular point) while the second state is introduced artificially. Therefore, we are interested in finding a trajectory n (2,), 0( 2) that leaves the point n (0) = s/g , 0(0) = 6 and reaches the singular point
n = ■ ° °
e 2(1 -a) s 2(1 -a)
In order to prove the existence of the pressure-driven wave, it is enough to check two following conditions for parametric family of paths 0(n, AD) that are solutions of the equation dQ
+ 8^
dn
1 -^(0-an)
. ei
= ct +
1 + ßn
exp
AD(n - 0) + ß0
(2.11)
point. So, we will divide the proof into two parts: we will investigate (2.8)—(2.9) system on the phase plane outside a singular point (0b, n b) vicinity and then, we will look at the system behavior in a vicinity of point ( 0b, n b). We will fix the ( 0b, n b) point vicinity. Let 5 be the radius of ball B(pb, 8) with the center in the singular point pb = (0b,nb) and 5 < 1.
In order to prove the continuity of solution 0(n,Ad) of the system (2.8)—(2.9) outside the singular point vicinity B(pb,8) according to AD parameter, we will use the following theorem
Theorem 1 (see [10]). Assume that function f(t,y,z) is continuous in an open set E = {(t,y,z)}, assume also that for every (t0, y0, z0) e E for Cauchy problem y ' = f( t, y, z )
The first condition is the existence of two A d1, A d2 values of the AD parameter (wave velocity) for which the corresponding paths 0(n, AD1), 0(n, A d2) are located on different sides of the singular point (stationary state) (n b, 0b). The second condition is the continuity of the family ofpaths according to AD parameter. When those two conditions are met, there will exist a path that goes to the singular point (n b, 0b).
In order to prove the uniqueness of the fast travelling wave in addition to continuity, we have to show that the equation solution (2.11) 0(n,Ad) is monotonous according to AD parameter, where 0(n, Ad) is the solution of (2.8)—(2.9) system on the phase plane. First, we will give asymptotic solutions of the system (2.8)—(2.9) that correspond to AD ^ to and AD ^ 0. If AD ^ 0, we get dn / d0 = 0. After the integration we get n (0) = const = e/a. If AD ^ to, equation (2.11) gets the following form d 0 / d n = a, therefore 0 = an. Hence, the singular point (0b, n b) is located between these lines.
The transition from (2.8)—(2.9) system to (2.11) equation in the singular point is incorrect, because the right-hand sides of equations (2.8), (2.9) vanish at that
with a constant z there is a single solution
y (t0) = y0
y (t) = n(t,t0,y0,z); < t < is a maximum interval of y(t) = n(t,t0,y0,z)
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