Table 4.3: Thermal diffusivities of the hexane-air mixtures at the unburned temper- ature (Tu = 300 K) and at the average temperature (Tb+Tu)/2
Mixture αu Tave αave
(m2/s) (K) (m2/s) φ = 1 Hexane-Air 2.057×10−5 1289 2.200×10−4 φ= 1.72 Hexane-Air 1.930×10−5 1062 1.450×10−4
Table 4.4: Comparison of the ignition energy per length for the hexane-air mixtures calculated using the analytical cylindrical kernel model (Equation4.5) and the results from the statistical analysis of the experiments. The laminar burning velocities sL are by Davis and Law(1998).
Mixture sL ξ0 Eign/L (E/d)P=0.1 (E/d)P=0.9
(cm/s) (J·m/s2) (µJ/mm) (µJ/mm) (µJ/mm)
φ = 1 Hexane-Air 38.2 0.1547 1071 255 1057
φ = 1.72 Hexane-Air 11.3 0.0555 4346 149 535
Paschen’s law states that the breakdown voltage of the gap also scales approximately linearly with the spark gap size, therefore,
V ∼C2d (4.10)
where C2 is a second constant. Substituting Equation 4.10 into Equation 4.9 gives:
E ∼ 1
2CV (C2d)∼C1d (4.11)
and combining the constants results in:
CV =Q∼ constant. (4.12)
Therefore, it is hypothesized in von Pidoll et al. (2004) that the charge required for ignition does not vary when the voltage, V, or gap distance, d, is changed.
To investigate this hypothesis, the ignition test results for the 6% H2 test mixture and the two hexane-air test mixtures were sorted by the spark length. The minimum spark charge and energy that caused ignition for each spark length was identified, and these minimum ignition values are plotted versus the spark length in Figure4.7. The values shown in the plot are not necessarily the absolute minimum ignition charge or energy for that spark gap, only the minimum values from the tests performed in this work. However, this data can still provide insight into the dependence of the charge or energy required for ignition on the spark length. As the gap size increases the minimum charge required for ignition does increase by approximately 15%, 71%, and 39% for theφ= 1.72 hexane-air mixture,φ = 1 hexane-air mixture, and 6% hydrogen mixture, respectively. However, the percent increase in the required energy is 2.3 to 4 times larger than the percent increase in the charge. The minimum energy increases by approximately 51%, 160%, and 153% for the φ = 1.72 hexane-air mixture, φ = 1 hexane-air mixture, and 6% hydrogen mixture, respectively. These results, though only approximate given the limited number of tests, suggest that while the minimum charge for ignition may not remain exactly constant, it is less dependent on the voltage
and gap size than the spark energy, and therefore may be a more appropriate measure of the incendivity.
Probability distributions for ignition versus the spark charge were calculated for the three test mixtures, and are shown next to the probability distributions versus spark energy density in Figure 4.8(a) and (b) (6% H2 mixture), Figure 4.10(a) and (b) (φ = 1.0 hexane-air mixture), and Figure 4.9 (φ = 1.71 hexane-air mixture). To directly compare the broadness of the two distributions, and therefore the variability of the test results with respect to energy density versus charge, the energy density and charge must be normalized. We normalize the energy density and charge by dividing by the 50th percentiles (50% probability of ignition). This normalization results in the probability versus (E/d)/(E/d)P=0.50and Q/QP=0.50where (E/d) andQare the energy density and charge, respectively, and (E/d)P=0.50 and QP=0.50 are the energy density and charge corresponding to 50% ignition probability. The two probability distributions are then both centered at (E/d)/(E/d)P=0.50 = Q/QP=0.50 = 1.0 and can be shown on the same plot for comparison, as in Figures4.8(c),4.10(c), and4.9(c).
For all three test mixtures, the probability distribution versus charge is significantly more narrow than the distribution versus energy density, demonstrating that ignition is less variable with respect to the spark charge. For a more quantitative comparison of the two distributions, we can once again compare the broadness of curves using the relative width:
Relative Width = (E/D)P=0.90−(E/D)P=0.10
(E/D)P=0.50 (4.13)
= QP=0.90−QP=0.10
QP=0.50 . (4.14)
Using Equation4.14, the relative widths of the distributions for ignition versus energy density are 0.94, 1.13, and 1.22 for the 6% H2 mixture and rich (φ = 1.72) and sto- ichiometric hexane mixtures, respectively. The relative widths for the distributions versus charge, however, are 0.50, 0.82, and 0.77. Therefore the relative widths of the spark charge distributions are 27 to 47% smaller than the widths of the spark energy density distributions.
10 100 1000
0 2 4 6 8 10
Spark Gap (mm)
Minimum Spark Charge (nC)
Hexane-Air = 1.0
Hexane-Air = 1.71
6% H2
(a)
0.1 1 10
0 2 4 6 8 10
Spark Gap (mm)
Minimum Spark Energy (mJ)
Hexane-Air = 1.0
Hexane-Air = 1.71
6% H2
(b)
Figure 4.7: Approximate minimum spark charge (a) and spark energy (b) required for ignition versus spark gap length for the 6% H2 test mixture and the two hexane-air test mixtures
Figure4.11shows a comparison of the spark energy and spark charge distributions for the ignition tests using short, fixed-length sparks. In these cases, the probability distributions versus charge shows no improvement over the distributions versus spark energy due to the fact that in these tests both the voltage and the spark gap were held approximately constant. A comparison can also be made between the probability distributions for ignition versus spark charge for the short, fixed-length spark ignition tests and the variable-length spark ignition tests. The two distributions are shown in Figure 4.12, and the agreement between the results of the two sets of tests shows improvement over the comparisons using spark energy and even energy density. For example, the 50thpercentile spark charge obtained from the short, fixed spark ignition tests (94 nC) is only 15% larger than the value from the variable-length spark tests (80 nC). Also, the results from the two tests give a 99% probability of ignition at approximately the same spark charge (120 nC). All these results support the idea by von Pidoll et al. that the charge may be a better characterization of the incendivity of the sparks for tests with varying voltage and gap distance. The variability of the test results was reduced significantly when the probability was analyzed in terms of the spark charge versus the energy density. Also, the charge may be a more convenient quantity for comparing the incendivity of different electrostatic discharges because the charge transfer is often easier to measure directly than energy. However, there was still a considerable degree of variability of ignition with respect to charge, the possible sources of which are discussed in Section 4.5.