Extinction point

The tangent indicated at point B also represents a critical reaction condition, but of a somewhat different type. In this case the reactor temperature corresponding to point B represents the minimum temperature at which autoignition will occur. In this sense it can be regarded as a minimum ignition temperature. Like the critical extinction point, this temperature should not be regarded as an absolute value but as a function of various operating parameters. 

An important outcome of these simulations is the location of HB points (largely ignored in previous work), which is important for the development of extinction theory. In particular, the turning point E lies on a locally unstable stationary solution branch and does not coincide with the actual extinction, as previously thought. The actual extinction point is the termination point of oscillations. Thus, local stability analysis is essential to properly analyze flame stability and develop extinction theory. 

As the pressure increases further, a second HB point (HB2) appears at the extinction point E and shifts toward the other HB HBi) point. An example is shown for 4 atm in Fig. 26.1c. Ignition Ii is no longer oscillatory, because the stationary partially ignited branch becomes locally stable in the vicinity of /i. Time-dependent simulations indicate that the two HB points are supercritical, i.e., self-sustained oscillations die and emerge at these points with zero amplitude. In this case, the first extinction Ei defines again the actual extinction of the system. 

It is expected that as the strain rate increases, the overall coupling between the surface and the gas-phase increases, since the flame is pushed toward the surface. Figure 26.6a shows the wall heat flux that can be extracted from the system, and the fuel mole fraction near the surface vs. the inverse of the strain rate for 28% inlet H2 in air, at two surface temperatures. The end points of the curves in Fig. 26.6, at high-strain rates, are the extinction points. The conductive heat flux exhibits a maximum as the strain rate increases from low values, which is at first counterintuitive. In addition, with increasing strain rate the fuel mole fraction increases monotonically, while the mole fractions of NOj, decrease, as seen in Fig. 26.66. The species mole fractions show sharper changes with strain rate near extinction, as the mole fractions of radicals decrease sharply near extinction. 

When P0 = i, the two roots of eqn (6.22) are exactly equal. The ignition and extinction points are coincident at ires = 64/27 multistability is lost. For larger inflow concentrations of B the stationary-state extent of reaction increases smoothly with the residence time and the distinction between the flow and thermodynamic branch is lost. 

If the rate constant k2 = 0.1, then the minimum gradient which L can have is 4k2 = 0.4. This is steeper than both the tangents for this system and might correspond to the flow line Ll in Fig. 6.16(a). There are no ignition or extinction points in the stationary-state locus (Fig. 16.6(b)) which has a unique solution for all residence times. 

If the CSTR is fed with both A and B, so p0 > 0, then a fifth pattern of response can also be found over a narrow range of experimental conditions. This is shown in Fig. 6.19(e) and has both a breaking wave and an isola. In total such a bifurcation diagram shows three extinction points and only one ignition. 

The flow diagram technique works very well for this adiabatic case. This is because, as noted above, the reaction curve R does not vary with the residence time, whereas the gradient of the flow line L does. The condition for an ignition or an extinction point is that R and L should become tangential, as shown in Fig. 7.3. We should simultaneously satisfy... 

For all p0 less than g, the ignition and extinction points in the x-rres locus are determined by the solutions of... 

Relaxation times near ignition and extinction points... 

When the residence time is varied so that we approach an ignition or extinction point in the stationary-state locus, then the flow and reaction curves L and R become tangential. The condition for tangency is R = L and 8R/da = SL/da. Thus the difference between the slopes of R and L decreases to zero. From eqn (8.17) we see that the tangency condition also causes the value of the eigenvalue A to tend to zero. An alternative interpretation, in... 

To determine the response of the cubic autocatalytic system to perturbations in the vicinity of a turning point in the locus, we must return to eqn (8.6). The first two terms (not involving A a) again cancel exactly, because of the stationary-state condition. If we are also at an ignition or extinction point, the tangency condition in any of its forms discussed above ensures that the coefficient of the A a term is also zero. Thus the first non-zero term is that involving (Aa)2 ... 

There is another type of time dependence possible in this system. If the inflow concentration of the autocatalyst is adjusted so that b0 - a0, then the ignition and extinction points merge at trcs = (k1ao) 1, with ass = Iu0 Under these special conditions, the coefficient of the term in (Aa)2 in the rate equation, and hence in the denominator of eqn (8.21), becomes zero as well as those of the lower powers in A a. Thus the inverse time dependence disappears, and the only non-zero term governing the decay of perturbation is that in (Aa)3 ... 

Thus we have an explicit formula in this case for the Hopf bifurcation points as a function of the decay rate constant for k2 = 20, Tres = 39.25 for k2 = Tres = 163.2. Figure 8.5 shows how the bifurcation point moves to longer residence times as k2 decreases, along with the locations of the extinction points t s from eqn (8.27). 

Fig. 8.5. The locus of Hopf bifurcation points t s-k2 described by eqn (8.42) for cubic autocatalysis with decay but no catalyst inflow. Also shown are the loci of the extinction points TreS, marking the ends of the isola, given by eqn (8.27). The three curves meet at the common...

The condition tr(J) = det(J) = 0 corresponds to a Hopf bifurcation point moving exactly onto the saddle-node turning point (ignition or extinction point) on the stationary-state locus. Above the curve A the system may have two Hopf bifurcations, or it may have none as we will see in the next subsection. Below A there are two points at which tr (J) = 0, but only one of... 

Figure 17.9 shows computed species profiles for two strain rates, a = 600 s-1 is far from extinction and a = 1260 s-1 is nearly at the extinction point [214], These profiles... 

Figure 17.12 shows some aspects of flame behavior that are revealed through sensitivity analysis (sensitivity analysis is discussed Section 15.5.4). For example, the maximum temperature is relatively insensitive to reaction rates, except very near the extinction point. At the extinction point, all sensitivities become unbounded because at the turning point the Jacobian of the system is singular. Near extinction, the hydrogen-atom concentration is... 

It is often important to predict and understand the flame extinction phenomenon in stagnation or opposed flows. As discussed briefly in Sect. 17.5 and illustrated in Fig. 17.11, the extinction point represents a bifurcation where the steady-state solutions are singular. Thus direct solution of the discrete steady problem by Newton s method necessarily cannot work because the Jacobian is singular and cannot be inverted or factored into its LU products. Moreover, in some neighborhood around the singular point, the numerical problem becomes sufficiently ill-conditioned as to make it singular for practical purposes. 

Computation at the extinction point is facilitated by arc-length continuation methods, which were developed by Keller [221,222], with early applications to flame stability by Heinemann, et al. [170]. The methods were further developed for combustion applications by Giovangigli and Smooke [145] and Vlachos [415,416], Recently Nishioka et al. [298] have developed an alternative continuation method that is motivated by and has much in common with arc-length continuation but provides increased flexibility for flame applications. It may also be somewhat more straightforward to implement in software. 

C) the mass fraction of methanol was set to the desired amount by switching a three-way tap. At the same time, the outlet temperature of the preheater was decreased gradually. At the moment the flame was extinguished, the characteristic temperatures of the last operating point before extinction were recorded. This procedure was repeated once or twice for every fuel mass fraction in order to estimate the accuracy of the results. Different extinction temperatures for one fuel mass fraction were mostly in the range of 10°C. A series of extinction points (WFuei.i ranging from 4% to 25%) resulted in a so-called extinction line. 

Eigenberger and Nieken demonstrated ignition-extinction hysteresis within the RFR. They illustrated that the maximum bed temperature extinguishes to a low, unignited cyclic steady state for high values of trev One must reduce the value of trev below this extinction point to reignite the reactor to the upper cyclic steady state. 

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