Investigation of Flame Propagation in a Model with Competing Exothermic Reactions H.S. Sidhu1, I.N. Towers1, V.V.Gubernov2, A.V. Kolobov2, A.A. Polezhaev2 1
Applied and Industrial Mathematics Research Group School of Physical Environmental and Mathematical Sciences University of New South Wales Canberra, 2600, ACT, Australia
[email protected],
[email protected] 2 I.E. Tamm Theory Department P.N. Lebedev Physical Institute of Russian Academy of Sciences 53 Leninskii Prospect, Moscow 119991, Russian Federation
Abstract— We consider a diffusional-thermal model with twostep competitive reactions for premixed combustion wave propagation in one spatial dimension. Numerical investigation of the flame speed of the system for different fuel types shows the existence of regions where two stable travelling combustion wave solutions co-exist - solutions pertaining to the fast and slow branches. A hysteresis type phenomena exists, and we show that by altering the initial conditions, one can move from one solution branch to the other. Regions of complex pulsating combustion waves were also uncovered.
Keywords-two-step reactions; combustion waves; wave speed; multiplicity; hysteresis; pulsating waves I.
INTRODUCTION
Applications of combustion processes in today’s industries are the result of numerous studies and experiments conducted by past researchers (Merzhanov and Rumanov, 1999). Flames propagating through a reactive media, for example in the production of exotic materials using Self-propagating Hightemperature Synthesis (see Makino, 2001), are of interest to industry from a commercial point of view. However, from a safety perspective, understanding the properties of propagating flames in a dust-filled atmosphere within a coalmine shaft is of paramount importance. A combustion wave is essentially a propagating reaction front delineating the change from the initial chemical reactants to the reaction products. Whilst most observed combustion processes consist of multiple (perhaps tens or hundreds) chemical reactions occurring consecutively or simultaneously, it is often useful to ‘lump’ reactions into different ‘categories’ describing the dominant reaction kinetics. The simplest of these is the onestep irreversible reaction models which have contributed greatly to our understanding of combustion phenomena. In these models it is assumed that the reaction is well described by a single step of fuel ( F ) and oxidant ( O2 ) combining to produce products ( P ) and heat. The generic kinetic schemes of models with one-step chemistry are: k (T )
F → P + heat
k (T )
or
F + O2 → 2 P + heat,
where the temperature dependent rate constant
k (T ) is
given
by Arrhenius kinetics k (T ) = exp(−Te / T ) with Te representing the activation temperature, and T corresponding to the
temperature of the reaction. These models have proven their usefulness since they are relatively simple and allow analytical investigation using asymptotic methods in the limit of infinitely large activation temperature (Zeldovich et al., 1985). However, these authors have also noted that in the overwhelming majority of cases, chemical reactions in flames proceed according to a complex mechanism that involves a variety of different steps. Moreover, for many reactions, models with simple one-step kinetics may lead to erroneous conclusions as noted by Westbrook and Dryer (1981). Hence there is a need to develop and study multi-step reaction schemes. Around twenty years ago, many researchers such as Seshadri et al. (1994) and Sanchez et al. (1996, 2000) developed and investigated reduced systems for specific reactions. These authors showed that the remarkable feature of these types of models is that they allow analytical investigation to be successfully undertaken, while still being able to produce excellent quantitative results (Sanchez et al., 1996; 2000) such as predicting flame characteristics accurately for some specific reactions. In reduced systems where thermal effects are prominent, detailed kinetics can be reduced to simple and useful models by describing the different reactions paths - ‘sequential’ where both reactions occur consecutively (Please et al., 2003); ‘parallel’ where both reactions occur at the same time whilst consuming different reactants (Ball et al., 1999); and ‘competitive’ where both reactions are simultaneously occurring and feeding on the same reactant (Gubernov et al., 2012 and Sharples et al., 2012). In all of the above quoted studies, one reaction is exothermic while the other is endothermic. In this paper we focus on the investigation of premixed combustion waves in a model with two-stage competing exothermic reactions. Such a scheme has been studied experimentally for the combustion of Me-C-H2, where Me is either Ti or Zr (Martirosyan et al., 1983a; 1983b). Such a scheme was also studied analytically by several authors (such as Clavin et al., 1987) using activation energy asymptotic (AEA) methods. However, our earlier studies (Gubernov et al., 2010, 2012; Sidhu et al., 2009a, 2009b) have shown that, while the AEA approach can provide insights into the generic
behavior of a system, properties of the flame for finite activation energies are usually not reflected by such an approach. The aim of the present paper is thus to further investigate the behaviour of the competitive exothermic systems and to explore the properties, stability and behaviour of competitive systems. In particular we focus on one of the most important parameters of a combustible mixture – the flame speed. As stated by Chen et al. (2009), the “determination of laminar flame speed is extremely important for the development and validation of kinetic mechanisms...”. As in Towers et al. (2013), this study will neglect physical heat loss processes such as radiative cooling or the conduction of heat away from the system, that is we consider adiabatic conditions. II.
MODEL FORMULATION
We consider a reaction scheme with two-step kinetics for combustion wave propagation in a pre-mixed one-dimensional reactive medium under adiabatic conditions. It is assumed that the reactant undergoes two competitive exothermic reactions where both reactions are competing for the same initial reactant (fuel) and that the reaction products are chemically inert and do not change the physical properties such as the heat capacities, density or diffusivity of the system. An example of such a configuration would be a long insulated cylinder containing a fuel undergoing decomposition, with an appropriate a priori averaging over the transverse spatial dimension of the reaction front. With the aim of analysing the properties of a combustion wave front in a competitive reaction-diffusion scheme in one spatial dimension, we propose a generalised description of the scheme’s reaction pathways, assuming Arrhenius kinetics: Reactant → Product 1 + heat (Q1),
- E1 / RT k1(T ) = Ae 1
Reactant → Product 2 + heat (Q2 ),
k2 (T ) = A2e
- E2 / RT
(exothermic) (exothermic) (1)
Here the exothermic reactions drive the combustion wave front and are characterised by the activation energies E1 (J mol1 ) and E2 (J mol-1). The corresponding pre-exponential rate constants are given by A1 (s-1) and A2 (s-1), while the heat release are denoted by Q1 (K kg-1) and Q2 (K kg-1). The temperature, reaction rates of the exothermic reactions are represented by T (K), k1 (s-1) and k2 (s-1) respectively. The governing equations for the above scheme can be obtained by applying heat and mass balance to the reaction and diffusion of reactant and heat. Similar equations can be found in Weber et al. (1997), Gubernov et al. (2010, 2012):
(J s-1 m-1 K-1) is the thermal conductivity, D (m2 s-1) is the coefficient of mass diffusion, R is the universal gas constant (equal to 8.314 J mol-1 K-1) and cp (J kg-1 K-1) is the specific heat capacity at constant pressure of the reactant. To simplify the system, we follow Sharples et al. (2012) and introduce the dimensionless temperature and space and time coordinates:
u=
RT , x% = E2
ρ Q2 Α 2 R kE2
Q AR , t% = 2 2 . c p E2
The dimensionless system of equations can be shown to reduce to:
∂u ∂ 2u = 2 + ve −1/ u + qrve − f / u , % ∂t ∂x% ∂v 1 ∂ 2 v = − vβe −1/ u − β rve − f /u , 2 % % ∂t Le ∂x
(3)
where the new parameters q, f, r, Le and β are defined by:
q=
c p E2 Q1 E A k , f = 1 , r = 1 , Le = , β= . Q2 E2 A2 RQ2 ρcp D
We refer to q as the ratio of the reaction enthalpies, f as the ratio of activation energies, r as the ratio of pre-exponential rate constants, Le as the Lewis number and β is often described as the exothermicity parameter, and in the above nondimensionalization this parameter is associated with the reaction producing Product 2. The two parameters which will play a big role in our study is the Lewis number Le (the ratio of thermal to mass diffusivities of the fuel), and the exothermicity parameter β. Increasing β has the same effect as making the mixture less exothermic. For very large β, the reaction is essentially a one-step reaction where by the second reaction in (1) is “switched off”. Intermediate values of β causes interplay between the two competitive reactions in scheme (1). The boundary conditions for the system (1) are:
x% → L : x% = 0 :
u = ua ,
v = 1,
∂u ∂v = 0, = 0, ∂x% ∂x%
(4)
corresponding to a reaction front propagating in the positive
∂T ∂ 2T ρcp = k 2 + ρ ( Q1 A1e − E1 / RT + Q2 A2 e − E2 / RT ) v, ∂t ∂x ∂v ∂ 2v ρ = ρ D 2 − ρ ( A1e − E1 / RT + A2e − E2 / RT ) v, ∂t ∂x
x% direction (that is, flame fronts moving from left to right). On
(2)
where the reactant mass fraction and temperature are denoted by v and T respectively and the time and space coordinates are t and x respectively. Ρ (kg m-3) is the density, k
the right boundary ( x% →L) , we have the cold (u = ua) and unburnt state (v = 1), where ua refers to the ambient temperature. Here L is taken to be large to ensure that no transient behaviour exists, in other words steady state behaviour is observed. On the left boundary ( x% = 0 ) corresponding to the burnt product mixture however, both the concentration and temperature of the product mixture cannot be
explicitly stated. We know that there are no reactions occurring on the left boundary where the wave front has already passed, thus setting the derivatives of u and v to zero here. III.
RESULTS
NUMERICAL SCHEME
The governing partial differential equations (PDEs) for the competing exothermic reaction scheme (3) subject to the boundary conditions (4) were solved numerically using FlexPDETM (http://www.pdesolutions.com), a commercial finite-element package for obtaining numerical solutions to partial differential equations in one, two and three dimensions for both steady state and time dependent solutions. Due to its adaptive nature, errors in both space and time are minimized as mesh points are added in regions of large gradients particularly at the regions close to the travelling wave front. All solutions to the governing PDEs in this paper were obtained using this package, with most of the error tolerance limits set to 0.001 (noting that for cases where complex oscillations exist, this tolerance level was dropped to 0.0001). The solutions obtained by FlexPDETM were also validated by using an independent approach to solving PDEs. In our case, we employed the Method of Lines (MoL) approach which is a technique for obtaining numerical solutions to PDEs, and has become wellused in multiple studies (see Schiesser, 1991). In the MoL approach, the spatial partial derivatives in the governing PDEs (1) are discretised by finite-difference approximations, transforming the PDEs into ordinary differential equations (ODEs), which are continuous in the time variable. The resulting system of ODEs may then be solved as an initial value problem. The results were found to be within 0.1% of the solutions obtained via FlexPDETM. Due to convenience and speed of calculations, we have used FlexPDETM for most of the results presented in this paper (with occasional validation using the MoL). For the numerical integration of the governing partial differential equations, the initial condition for the reactant mass fraction was taken to be v = 1, while the initial temperature profile was taken in the form of a Gaussian pulse
u = A exp(−0.01x% 2 ),
IV.
(5)
simulating a hot wire ignition (or an ignition spark) at the starting end of the domain, x% = 0 . The amplitude of the Gaussian pulse is given by A, and we will show later that this will be an important parameter in regions where multiplicity exists. The integration domain was set to 0 ≤ x% ≤ L where L is typically taken to be very large, namely L = 10000 in most of our simulations. Also the integration time was set to be very large (of the order of 106) to ensure the decay of transient behaviour. As in our previous study (Towers et al., 2013), we will assume q=5, r=25 and f=3 throughout this paper. In order to circumvent the cold boundary problem, which has been a matter of discussion by many authors (see for example, Weber et al., 1997), we shall also take ua=0 in our simulations.
Figure 1. The speeds of the flame fronts with varying β for Le=1 (green squares joined by red dashed lines) and Le=2 (yellow squares joined by the blue curve). The lower curves are associated with the slow branch solutions while the upper curves correspond to the fast branch solutions.
Fig.1 illustrates the dependence of the flame speed upon the exothermicity parameter β for two values of Lewis numbers (Le=1 and Le=2). For both of these cases, there are two disjointed branches: the lower branch associated with combustion fronts with slow speeds (often referred to as the slow branch), and the upper branch (sometimes called the fast branch). These solution branches appear disjointed, however there exist curves joining these branches corresponding to unstable travelling waves solutions. Since we are solving the governing PDEs, we do not obtain these unstable branches. We also note that from Fig.1, there are regions of bistability regions where both the fast and slow stable solutions co-exist. It must be said that the existence of bistability regions for flame fronts are relatively rare and to the best of our knowledge, we have not seen such phenomenon in the earlier studies for sequential, parallel, or competitive schemes. For Le=1 the region of bistability is narrower (3.28≤β≤3.374) than that for Le=2 (3.14≤β≤3.42). Fig. 2 shows combustion wave profiles for the temperature and reactant mass fraction for the case Le=2 and β=3.3. The general shape of these profiles are typical of combustion waves - ahead of the front where there is no reaction v=1, and u=ua, and behind the reaction front, the reactant mass fraction decreases to v=0, and the temperature increases to the burnt temperature, and remains high since we consider adiabatic conditions. However, the speeds of the two flame fronts are significantly different - Fig 2(a) shows the profile from the fast branch with speed c=1.03, whereas Fig2 (b) displays the profile for the slow branch with speed c=0.152. Furthermore, by comparing the temperature flame wave front profiles for the fast and slow branch solutions, we can see that the burnt temperature is lower for the slow branch case (Fig 2b) when compared to the fast branch solutions (Fig 2a).
(a)
that not all initial conditions will evolve into flame fronts, and that some will decay to the ambient conditions. There has to be sufficient energy to "kick start" the reaction in order for the flame front to be formed and consequently propagate. The distinction between those initial conditions that do evolve into combustion waves and those that do not is sometimes referred to as the "watershed initial conditions" (Watt et al. 1999). However, in regions of bistability, there is another watershed, whereby solutions move from one stable solution branch to the other. Table 1 displays that varying A, the amplitude of the initial Gaussian profile in (5), we can move from the slow branch to the fast branch. TABLE I. SUMMARY OF RESULTS WHEN THE AMPLITUDE OF THE INITIAL CONDITION GIVEN BY EQUATION (5) ARE VARIED FOR THE CASE WHEN LE=2 AND β =3.3. THE VALUE FOR THE INITIAL MASS FRACTION IS FIXED AT V=1. Amplitude of Gaussian (initial spark) 0
