B URNING V ELOCITY
2.3. AUTOIGNITIVE BURNING 1. I GNITION D ELAY T IME
2.3.4. A UTOIGNITIVE F RONT P ROPAGATING V ELOCITIES
AND
D
ETONATIONHere we consider how a reaction front can originate and spread through a gradient of reactivity within a single hot spot. If there is a spatial gradient in, say T, of a given
mixture, there also will be an associated spatial gradient inti. If ti increases with the radius,r, from the center of the hot spot, then autoignition will occur at increasing times at increasing distances. This will create a localized reaction front propagating at a velocity relative to the unburned gas ofua, equal to]r=]ti, or
ua¼ ]r ]T
]T ]ti
: (2:12)
For illustrative purposes, it is assumed ti can be expressed locally, at constant pressure, in Arrhenius form by
ti¼Aexp (E=RT), (2:13)
whereAis a numerical constant andEis a global activation energy. For a gasoline, the greater the fuel sensitivity, the greater the value of E becomes (Bradley and Head, 2006). From Equations (2.12) and (2.13):
ua¼(T2=(tiE=R))(]T=]r) 1: (2:14) This type of autoignitive reaction front propagation, in which the propagation velocity is inversely proportional to the product oftiand the temperature gradient, is very different from that in a laminar or turbulent flame. In those regimes where flame propagation also is possible, the two modes of burning can co-exist. This has been demonstrated for lean H2–air mixtures in the direct numerical simulations of Chenet al. (2006).
Values ofuaat 4 MPa withf¼0:5, given by Equation (2.14) and corresponding to the data for the two mixtures in Figure 2.13, are given in Figure 2.14 at the different temperatures. A hot spot temperature gradient of 1 K/mm is assumed. As the temperatures are increased above 1000 K, for both mixtures,uabecomes appreciable and eventually supersonic. Clearly, from Equation (2.14), an increase in ]T=]r decreases ua. As ]T=]r!0, ua! 1and, in the limit, athermal explosionoccurs throughout the mixture.
A special condition arises when the autoignition front moves into the unburned mixture at approximately the acoustic velocity,a (Zeldovich, 1980; Makhviladze and Rogatykh, 1991). The critical temperature gradient for this condition, from Equation (2.12), is
]T
]r c¼ 1
a(]ti=]T): (2:15)
0 500 1000 1500
0.8 0.9 1 1.1 1.2
1000/T
Propagation velocity (m/s)
0.5 H
2- 0.5CO-air
i-octane-air
Figure 2.14 Variations of propagation velocity, ua, with 1000/T at 4 MPa for the two mixtures of Figure 2.13, for]T=]r 1 K=mm.
This gradient, which is a function of the reactants, couples the leading edge of the pressure wave generated by the heat release with the autoignition front. The fronts are mutually reinforced and united to create a damaging pressure spike propagating at high velocity in adeveloping detonation. It is convenient to normalize temperature gradients by this critical value, to define a parameterx
x¼ ]T ]r
]T ]r
1 c
: (2:16)
Values of x are only known for the initial boundary condition. In practice, heat conduction, species diffusion, and some reaction modify the initial boundary before autoignition occurs. From Equations (2.12), (2.15), and (2.16),
x¼a=ua: (2:17)
The probability of a detonation developing at the outer radius of a hot spot,ro, also depends upon the chemical energy that can be unloaded into the developing acoustic wave during the timero=a, it takes to propagate down the temperature gradient. The chemical time for energy release defines an excitation time,te, which can be obtained from chemical kinetic modeling (Lutz et al., 1988). Here, it is taken to be the time interval from 5% to the maximum chemical power. The ratio of the two times yields (Guet al., 2003)
e¼(ro=a)=te: (2:18) This is an approximate indicator of the energy fed into the acoustic wave. It is similar to an inverse Karlovitz stretch factor (or Damköhler number), in that it is an aerody- namic time divided by a chemical time.
Shown in Figure 2.15, from Bradleyet al. (2002), is the temporal development of a detonation, x¼1, at a hot spot of radius of 3 mm for 0:5H20:5CO-air, f¼1:0, initial T ¼1200 K, and p ¼ 5.066 MPa. The initial hot spot maximum temperature elevation is 7.28 K. The radial profiles of temperature and pressure, some of which are numbered, are given for different time intervals. The times for the numbered profiles are given in the figure caption. The pressure and reaction fronts are soon coupled, with a steep increase in the temperature, immediately after the sudden increase in pressure.
The detonation wave becomes fully developed and travels with a speed of 1600 m/s (see Figure 2.15C). This is close to the Chapman–Jouguet velocity. Shortly afterwards, autoignition occurs throughout the remaining mixture in a thermal explosion, indicated by (i), atti¼39:16ms. At this time, the detonation wave has a radius of 4.6 mm. In practice, mostdeveloping detonationsare confined within the hot spot. However, with the more energetic mixtures at the highest temperatures, such as in the present case, they can propagate outside it. This creates the potential for damaging consequences, particu- larly when a significant amount of the remaining charge is consumed in the detonation front.
Shown in Figure 2.16 is a plot ofxagainste. This defines a peninsula, within which localized detonations can develop inside a hot spot. The points on the figure were generated computationally from chemical kinetic models of the development of auto- ignitive propagation for mixtures of 0:5H20:5CO-air, at equivalence ratios between 0.4 and 1.0 and for hot spots of different radii. These data were processed to identify the upper and lower limits for thedevelopment of detonations,xu andxl, and these define
3000 2200
1200
2000
1000
0 25
15
5
0.0 1.0 2.0 3.0 4.0 5.0
0.0 1.0 2.0 3.0 4.0 5.0
0.0 1.0 2.0 3.0 4.0 5.0
1 2 3 4 5 6 7 8
(i)
r (mm) A
B
C T (K) P (MPa)Wave speed (m/s)
Figure 2.15 History of a hot spot:ro 3 mm, x 1, stoichiometric 0:5H2 0:5CO-air, To 1200 K, Po 5:066 MPa, andti 39:16ms. Time sequence (ms) 1 35.81, 2 36.16, 3 36.64, 4 37.43, 5 37.72, 6 38.32, 7 38.86, 8 39.13. Plots show (A) temperature, (B) pressure, and (C) combustion wave speed (Bradleyet al., 2002).
0 5 10 15 20 25 30 35 40 45
0 10 15 20 25
x
B
Developing detonation
P
e xu
xl 5
Figure 2.16 Developing detonation peninsula on plot ofxagainste(Bradleyet al., 2002). Supersonic and subsonic autoignitive fronts propagate in regions P and B, respectively.
the limits of the peninsula shown in the figure. It was found that at sufficiently high values ofe, greater than about 80, the detonations could propagate outside the hot spot with the Chapman–Jouguet detonation velocity.
A variety of propagation modes of the reaction front are possible and these are discussed in more detail by Bradley et al. (2002) and Gu et al. (2003). Thermal explosionsoccur whenx¼0. At slightly higher values ofx, but less thanxl(regime P on the figure),supersonic autoignitive frontsoccur. Abovexu, in regime B, the fronts propagate subsonically. As fdecreases,teincreases,edecreases and, as can be seen from Figure 2.16, the detonation regime narrows appreciably. Conversely, as f increases from such low lean values, the detonation peninsula broadens appreciably.
This corresponds to increases infabove about 0.6.