SUPPLEMENTAL MATERIAL
To
Effects of reaction reversibility on ignition and flame propagation
Cong Li 1, Yunchao Wu 1, and Zheng Chen 1, 2
1 SKLTCS, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China
2 Department of Aeronautics and Astronautics, College of Engineering, Peking University, Beijing 100871, China
This supplementary document is provided to show the details on mathematical model and asymptotic analysis.
S1. Mathematical model
We consider one-dimensional, adiabatic, premixed, spherical flame initiation and propagation. In order to include reaction reversibility, a single reversible reaction in the form of is considered. The overall reaction rate is [1]
(1)
in which is the density, the fuel mass fraction, the product mass fraction, the temperature, and the universal gas constant. The pre-factor of the Arrhenius law and activation energy of the forward reaction are and , respectively; and those of the backward reaction are and .
For the sake of simplicity, we employ the diffusive-thermal model [2], according to which the density of the mixture , the heat capacity , the heat conductivity , and molecular diffusivities of fuel and product, and , are all constant. In a one-dimensional spherical coordinate, the governing equations for temperature, mass fraction of fuel, and mass fraction of product are
(2a)
(2b)
(2c)
where and are time and radial coordinate, respectively. The parameter, , is the reaction heat-release per unit mass of fuel. Since thermal expansion is neglected in the diffusive-thermal model, the flow is static and there is no convection term in these governing equations. In practical ignition and flame propagation process, the flow is not static due to thermal expansion. Nevertheless, numerical study [3] demonstrated that qualitative prediction on ignition and flame propagation can be achieved by the diffusive-thermal model neglecting thermal expansion.
We introduce the following non-dimensional variables
, , , , , (3)
where and denote the temperature and fuel mass fraction in the fresh mixture. The characteristic speed , characteristic length , and characteristic temperature are, respectively, the laminar flame speed, flame thickness, and flame temperature of an adiabatic planar flame with one-step irreversible reaction (i.e. and ).
We study the propagating spherical flame in the coordinate (τ, ξ) attached to the moving flame front r=R(t) [4, 5]. In the new coordinate (τ=t, ξ=r−R(t)), the flame can be considered as in a quasi-steady state (∂/∂τ=0). This quasi-steady assumption has been used in previous studies [4, 6, 7] and validated by transient numerical simulation [4, 8]. Consequently, the non-dimensional governing equations are simplified to
(4a)
(4b)
(4c)
where U=dR(t)/dt is the non-dimensional flame propagation speed (normalized by , the adiabatic planar flame speed for the irreversible case). The Lewis numbers of fuel and product are respectively defined as and . The non-dimensional reaction rate becomes
(5)
in which is the Zel’dovich number and is the temperature ratio. Both Zad and σ are defined under the irreversible condition since is defined for the irreversible case. The ratios of the pre-exponential factors and activation energies of the backward and forward reactions are, respectively:
, (6)
Similar to the work of Daou [1], the variable Γ is the main parameter in this study and it is referred to as the reversibility parameter. For the irreversible condition, we have Γ=0 (i.e. ). The difference in the activation energies, , is equal to the enthalpy of the reaction in the case of an elementary reaction [1] and thereby we have Θ≈[1+(1−σ)2/Zad].
In this study, the impact of external energy deposition on spherical flame kernel development is investigated. The ignition power, Q, is provided as a heat flux at the center [4, 5, 7, 9] (the limitation on this assumption is discussed in Ref. [4]). The boundary conditions are
(7a)
(7b)
The above model extends previous analytical description [4, 7] of spherical flame initiation and propagation beyond the common framework of a one-step irreversible Arrhenius reaction. The previous analytical description [4, 7] is the limiting case of zero reversibility parameter (Γ=0) in the present study.
S2. Asymptotic analysis
The spherical flame kernel generated by energy deposition is highly stretched. Its propagation speed depends strongly on the Markstein length which characterizes the variation in local flame speed due to the influence of external stretching [10]. Therefore, understanding the stretched flame propagation speed and Markstein length is helpful for examining the critical ignition condition. Here we first consider the freely propagating spherical flame without ignition energy deposition at the center (i.e. Q=0).
Fig. 1 The schematic flame structure of a freely propagating spherical flame.
The freely propagating spherical flame with a reversible reaction is analyzed using the large-activation-energy asymptotic method [2, 10, 11]. As shown in Fig. 1, the flame structure consists of the upstream preheat zone (ξ>0) and downstream equilibrium zone (−R≤ξ<0), which are connected by the thin reaction zone located around ξ=0. At large activation energy, the ratio between the thickness of the inner reaction zone and that of the outer preheat zone is a small parameter, ε, which is the inverse of the Zel’dovich number (i.e. ε=1/Zad) [10]. Following the procedure for asymptotic analysis of planar and spherical flames with a one-step irreversible reaction [10, 12], the asymptotic solution is obtained in ascending powers of this small parameter and then asymptotically matched.
In the preheat zone (ξ>0), the reaction term in Eq. (4) can be neglected due to the low temperature and high activation energy. With the boundary conditions in Eq. (7b), the asymptotic solutions in the outer preheat zone are obtained as
(8a)
(8b)
(8c)
where Ci (i=1~6) are integration constants to be determined through matching with the inner solution in the reaction zone.
In the downstream equilibrium zone (−R≤ξ<0), the solutions are
, , (9)
where Tb is the flame temperature determined later (see Eqs. 16-18). In the equilibrium zone, the net reaction rate is zero (i.e. ω=0). According to Eq. (5), we have the following relationship between the mass fraction of fuel, YF,b, and that of product, YP,b, in the equilibrium zone
(10)
Due to the large activation energy, the inner reaction zone is much thinner than the preheat zone and equilibrium zone. In order to adequately resolve the reaction zone, the spatial coordinate is stretched by introducing the coordinate transform: X=ξ/ε [10-12]. In the inner reaction zone, the asymptotic solutions are assumed to be
(11a)
(11b)
(11c)
where θ0, φ0, and ψ0 are the leading-order solutions; while θ1(X), φ1(X), and ψ1(X) are perturbation functions. In order to determine the solutions in the inner reaction zone, matching conditions between the inner and outer solutions should be used. The matchingconditions for the inner and outer solutions at the downstream boundary (i.e. X=ξ/ε→−∞) require the continuity of the temperature and mass fractions as well as their first-order derivatives [10, 11], which yields:
(12a)
(12b)
(12c)
Similarly, using the matchingconditions for the inner and outer solutions at the upstream boundary of the reaction zone (i.e. X=ξ/ε → +∞), we have
, , (13a)
, , (13b)
, , (13c)
Substituting Eq. (11) and the coordinate transform X=ξ/ε into Eq. (4), we find that the convective term can be neglected since it is one-order smaller than the diffusion term. Consequently, the inner reaction zone is governed by the following reaction-diffusion equations
(14)
After integrating Eq. (14) with the boundary conditions at X→−∞ in Eq. (12), we have
, (15)
Substituting Eq. (13c) into Eq. (15) yields the following relationships among Tb, YF,b, and YP,b
(16)
The expressions for mass fractions of fuel and product, YF,b and YP,b, in the equilibrium zone are obtained from Eq. (16) in terms of Tb and they are substituted into the equilibrium condition in Eq. (10), which yields the following relationship among the flame propagation speed U, flame temperature Tb, and flame radius R
(17)
Equation (14) for the temperature perturbation, θ1, can be solved analytically in the inner reaction zone. Using the reaction rate in Eq. (5) and the boundary conditions at in Eqs. (12) and (13), we get the following expression by integrating Eq. (14)
(18)
in which the mass fractions of product in the equilibrium zone, YP,b, can be obtained from Eq. (16). Equation (18) is another relationship among the flame propagation speed U, flame temperature Tb, and flame radius R. Therefore, the flame propagation speed, U, and flame temperature, Tb, as functions of flame radius, R, are determined by Eqs. (17) and (18).
We now turn to the influence of reaction reversibility on spherical flame initiation. We consider the case with ignition power deposition at the center (i.e. Q0) and study the ignition kernel propagation with and without the reverse reaction. The asymptotic analysis of spherical flame initiation is similar to that of freely propagating spherical flame presented in the previous section and to that of spherical flame ignition analyzed by Deshaies and Joulin [13]. For non-zero ignition power at the center (Q>0), we can get the counterpart correlations to Eqs. (16-18) as follows
(19)
(20)
(21)
As expected, Eqs. (19-21) reduce to Eqs. (16-18) in the limit of Q=0. By solving Eqs. (19-21) numerically, the change of flame kernel propagation speed U with flame radius R at different ignition power Q, reversibility parameter Γ, and fuel Lewis number LeF can be obtained and thereby the critical ignition condition can be investigated.
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