6.6 Two-Species One-Step Model for Hydrogen-Air Systems
6.6.2 Second-Order Reaction
increasing by about 50 K from 1 bar to 8 bar, and the one-step model matched the detailed chemistry to within 1% over the entire range of pressure. The difference in the flame speed response to initial pressure calculated using the one-step model and detailed chemistry is more evident as the initial pressure increases. The one-step model flame speed clearly exhibits the dependence on pressure sL∼p−1/2, deviating from the detailed chemistry flame speed by up to 50%. The detailed chemistry flame speed does notably decrease with increasing pressure, a trend that is due to the increasing rate of the 3-body reaction H + O2 + M →HO2 + M slowing the energy release rate. This pressure dependence is a separate effect from the reaction order effect in Equation 6.60 and therefore cannot be reproduced with a simple one-step chemistry model.
0.0 5.0 10.0 15.0 20.0
250 350 450 550 650 750 850
Initial Temperature (K) sL (m/s)
Full Chemistry One-Step Model
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0 1 2 3 4 5 6 7 8 9
Initial Pressure (bar) sL (m/s)
Full Chemistry One-Step Model
(a) (b)
2000 2200 2400 2600 2800 3000
250 350 450 550 650 750 850
Initial Temperature (K)
Flame Temperature (K)
Full Chemistry One-Step Model
2000 2200 2400 2600 2800 3000
0 1 2 3 4 5 6 7 8 9
Initial Pressure (bar)
Flame Temperature (K)
Full Chemistry One-Step Model
(c) (d)
Figure 6.10: Flame speed response to large changes in initial (a) temperature and (b) pressure and flame temperature response to initial (c) temperature and (d) pressure calculated using first-order one-step models and detailed chemistry (Li et al., 2004) for 30% hydrogen-air
order one-step model compared with the detailed chemistry behavior for a 30%
hydrogen-air mixture. Small changes in initial temperature and pressure were consid- ered first by varying the temperature from 290 K to 350 K by 10K (with the pressure fixed at 1 bar) and the pressure from 0.8 bar to 1.3 bar by 0.05 bar (with the temper- ature fixed at 300 K). The resulting flame speed and flame temperature response is shown in Figure 6.12. Once again, for small changes in initial temperature both the flame speed and flame temperature response produced by the one-step model are in good agreement with the detailed chemistry, with the flame speed and temperature matching the detailed chemistry values to within 6% and 2%, respectively. Also, it
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0 20 40 60 80
H2 Concentration (%)
Laminar Burning Velocity (m/s)
Full Chemistry One-Step Model
(a)
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0 20 40 60 80
H2 Concentration (%)
Flame Temperature (K)
Full Chemistry One-Step Model
(b)
Figure 6.11: (a) Laminar flame speeds and (b) flame temperatures calculated us- ing second-order one-step models and the Li et al. mechanism (Li et al., 2004) for hydrogen-air compositions
was possible once again to match the one-step flame temperature response to small pressure change to within 1%. Most notable was the change in the flame speed re- sponse to small pressure changes using a second-order one-step model. The agreement with the detailed chemistry behavior was greatly improved and the one-step model values match the detailed chemistry results to within 2.5%.
1.0 1.5 2.0 2.5 3.0 3.5
280 290 300 310 320 330 340 350 360
Initial Temperature (K) sL (m/s)
Full Chemistry One-Step Model
1.0 1.5 2.0 2.5 3.0 3.5
0.8 0.9 1 1.1 1.2 1.3 1.4
Initial Pressure (bar) sL (m/s)
Full Chemistry One-Step Model
(a) (b)
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280 290 300 310 320 330 340 350 360
Initial Temperature (K)
Flame Temperature (K)
Full Chemistry One-Step Model
2000 2100 2200 2300 2400 2500 2600
0.8 0.9 1 1.1 1.2 1.3 1.4
Initial Pressure (bar)
Flame Temperature (K)
Full Chemistry One-Step Model
(c) (d)
Figure 6.12: Flame speed response to small changes in initial (a) temperature and (b) pressure and flame temperature response to initial (c) temperature and (d) pressure calculated using second-order one-step models and detailed chemistry (Li et al.,2004) for 30% hydrogen-air
As in Section 6.6.1 the flame speed and flame temperature response to large changes in initial temperature and pressure were also calculated, and the results are shown in Figure6.13. Like with the first-order one-step model, the one-step flame speed was about 5 to 15% larger than the detailed chemistry flame speed as the initial temperature was increased from 350 K to 550 K. For initial temperatures above 550 K the difference between the one-step and detailed chemistry flame speeds increases up to 33% until the one-step flame speed increases dramatically to 27 m/s and 76 m/s at initial temperatures of 750 K and 800 K, respectively. This phenomenon of rapidly increasing flame speed is a common issue with modeling high speed combustion waves
(seeSingh et al.(2003)). As the flame speed increases the effects of diffusion decrease and the pre-heat region of diffusion-convection balance disappears, resulting in a pre- dominant balance between convection and reaction. In this situation, the solution is no longer a typical flame, but rather the “fast flame” or “convected explosion”
solution. The flame temperature from the one-step model is 2 to 10% larger than the detailed chemistry flame temperature due to the same reasons as described in Section6.6.1 for the first-order one-step model.
0.00 5.00 10.00 15.00 20.00
250 350 450 550 650 750 850
Initial Temperature (K) sL (m/s)
Full Chemistry One-Step Model
1.0 1.5 2.0 2.5 3.0 3.5
0 1 2 3 4 5 6 7 8 9
Initial Pressure (bar) sL (m/s)
Full Chemistry One-Step Model
(a) (b)
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250 350 450 550 650 750 850
Initial Temperature (K)
Flame Temperature (K)
Full Chemistry One-Step Model
2000 2100 2200 2300 2400 2500 2600
0 1 2 3 4 5 6 7 8 9
Initial Pressure (bar)
Flame Temperature (K)
Full Chemistry One-Step Model
(c) (d)
Figure 6.13: Flame speed response to large changes in initial (a) temperature and (b) pressure and flame temperature response to initial (c) temperature and (d) pressure calculated using second-order one-step models and detailed chemistry (Li et al.,2004) for 30% hydrogen-air
Once again, the flame temperature only increased by about 50 K for initial pressure increasing from 1 bar to 8 bar, and the one-step model matched the detailed chemistry
to within 1% over the entire range of pressure. While increasing the effective reaction order in the one-step model from n = 1 ton = 2 gave better agreement between the flame speeds calculated using the one-step model and detailed chemistry, there was still a large (up to 40%) difference for high initial pressures due to the decrease in the detailed chemistry flame speed. As described before, this flame speed decrease is due to the increasing rate of the 3-body reaction H + O2 + M → HO2 + M as the initial pressure increases. Figure 6.14 shows the flame speed versus initial pressure calculated using detailed chemistry, the first-order one-step model, and the second-order one-step model for the 30% hydrogen case. Also shown is the flame speed calculated using the effective reaction order n = 1.8 (found previously using the constant pressure explosion method, Equation6.18) and the pressure dependence
sL∼p(n−22 ) = p−0.1 (6.63)
where the flame speed withn= 1.8 was scaled to match the detailed chemistry flame speed at 1 bar. The flame speed versus pressure calculated using detailed chemistry lies between the first-order one-step model with pressure dependence sL ∼p−1/2 and the second-order one-step model with no pressure dependence (sL∼p0). The scaled flame speed values generated using n= 1.8 match the detailed chemistry results very closely (within 16% over the entire range of pressure), however, fractional reaction orders cannot be used in the one-step model. Therefore, the flame speed dependence on pressure cannot be properly modeled using one-step chemistry for large changes in initial pressure. The pressure dependence could be improved by using multi-step models, but a second-order one-step model is much easier to implement and the flame speed response to small changes in initial pressure (up to 2 bar) is reasonable, within 5% of the detailed chemistry results.
0.00 0.50 1.00 1.50 2.00 2.50 3.00
0 1 2 3 4 5 6 7 8 9
Initial Pressure (bar) sL (m/s)
Full Chemistry One-Step Model with n=2 One-Step Model with n=1 One-Step Model with n=1.8
Figure 6.14: Flame speed response to changes in initial pressure calculated using detailed chemistry (Li et al., 2004), a first-order one-step model, and a second-order one-step model for 30% hydrogen-air. Also shown (red dashed line) is the predicted one-step model flame speed with effective reaction order n = 1.8 obtained from the constant pressure explosion model (Equation 6.18).