tional,2005,Administration,1994). Following the work at the Bureau of Mines, other authors have performed MIE tests with methods similar to those used by Lewis and von Elbe (Calcote et al., 1952, Metzler, 1952a,b, Moorhouse et al., 1974) and found comparable results. Other authors have proposed improvements on the technique for determining MIE using capacitive spark discharge, most recently Ono et al. (Ono et al., 2005, 2007) and Randeberg et al. (Randeberg et al., 2006).
1.3.1 Analytical Models
Since the 1950s, several authors have attempted to develop analytic models to pre- dict the minimum ignition energy. Lewis and von Elbe (Lewis and von Elbe, 1961) proposed an empirical relationship for the required ignition energy,
Eign ≈ κuq
(cP/m)sL = κu
sL(Tb−Tu) (1.8)
where κu is the thermal conductivity of the unburned gas, q is the heat of reac- tion at constant pressure, cP is the specific heat, m is the mass, and Tu and Tb are the temperatures of the unburned and burned gas, respectively. This relation was also discussed in Strehlow (1979) and derived from unsteady conservation equations in Rosen (1959). The basis for the model was the idea that combustion waves have excess enthalpy that maintains the balance between the heat flow into the preheat zone by conduction and the heat release in the reaction zone. This excess enthalpy is required for the flame to grow spherically until it reaches a planar state. It was postulated that the excess enthalpy is usually provided by the burned gas, but when the diameter of the flame ball is less than the minimum diameter required for prop- agation, there is not enough excess enthalpy being generated by chemical reactions.
In this case, the temperature in the core would drop, reactions would stop, and the flame would be extinguished. Therefore, Lewis and von Elbe concluded that for the flame to grow to the minimum size, the required excess enthalpy must be provided by an ignition source. Hence, the minimum ignition energy would be equal to the excess enthalpy of the minimum diameter flame.
A second analytical model for the ignition energy is discussed in combustion text- books by Williams (1985), Glassman (1996), and Turns (2000). In this model the flame is considered to be a spherical volume of gas ignited by a point spark, and a critical radius is defined under which the spherical wave cannot propagate. To de- termine the critical radius, rcrit, it is assumed that there is a balance between the heat generated by chemical reactions inside the gas volume and the heat lost to the surrounding cold gas through conduction:
−dm000f uel dt ∆hc
4 3πrcrit3
≈ −κ dT dr rcrit
4πr2crit
(1.9)
where m000f uel is the fuel per unit volume, ∆hc is the heat of combustion, and κ is the thermal conductivity. The following approximations are made:
dT dr rcrit
≈ −(Tb −Tu)
rcrit (1.10)
∆hc ≈mcP(Tb−Tu) (1.11)
sL ≈
−2mα ρu
dm000f uel dt
12
(1.12)
pu =pb =p=ρuRuTu (1.13)
=ρbRbTb
(1.14) where
α= κ
ρcP (1.15)
is the thermal diffusivity, ρu andρb are the densities of the unburned and burned gas, respectively, m is the mass, and
Rb = Re
M Wb (1.16)
where Re is the universal gas constant and M Wb is the average molecular weight of the burned gas. Using these approximations, the critical flame radius is found to be
rcrit ≈√ 6α
sL ≈√
6δf lame
2 (1.17)
where δf lame is the flame thickness. It is then assumed that the required ignition energy is the energy needed to heat the critical gas volume to the adiabatic flame temperature, i.e.,
Eign =mcritcP(Tb−Tu) (1.18)
= 4
3πrcrit3
ρbcP(Tb −Tu) .
(1.19) Substituting Equation1.17 gives
Eign = 61.6 (p) cP
Rb
Tb−Tu Tb
α sL
3
. (1.20)
These analytical models greatly simplify the spark ignition process and do not include important aspects such as mass diffusion, geometry of the electrodes and spark gap, and turbulence in the surrounding gas. Therefore, determining ignition energy remains primarily an experimental issue. The oversimplification of analytical models for the minimum ignition energy is demonstrated by the comparison of calculations with experimental results in Section 3.4.3.
1.3.2 Ignition as a Statistical Phenomenon
The view of the ignition where the MIE is considered to be a single threshold value is the traditional viewpoint in combustion science and extensive tabulations of this kind of MIE data are available (Babrauskas, 2003, Magison, 1990). However, particularly in the aviation safety industry, a different approach to ignition characterization is being used that is more consistent with experimental observations of engineering
test data (Administration, 1994). In standardized testing guidelines published by the FAA and SAE International (Administration, 1994, International, 2005) ignition is not treated as a threshold phenomenon, but rather as a statistical event. The outcome of a series of ignition tests is used to define the probability of ignition as a function of stored energy, peak current, or some other characteristic of the ignition source. It is reasonable and useful to recognize that engineering test results have inherent variability, and hence using statistical methods to analyze these variable results provides a good basis for assessing the ignition hazard of flammable mixtures.
Simple statistical methods have been applied to Jet A ignition tests performed by Lee and Shepherd at the California Institute of Technology using a standard capacitive spark discharge system as the ignition source (Lee and Shepherd, 1999).
A set of 25 ignition tests were performed while varying only the spark energy, and the data points were then used to derive a mean value and standard deviation for the MIE, rather than a single threshold value. This data set is used in Section 3.3.2 as an example to illustrate statistical analysis resulting in a probability distribution for ignition versus spark energy and confidence intervals. Statistical analysis of ignition data has also been applied to ignition of automotive and aviation liquid fuels as a means of assessing the risk of accidental ignition by hot surfaces (Colwell and Reza, 2005). Taking on the viewpoint of ignition tests as being statistical in nature raises a key question: is the statistical nature of the data due to an intrinsic characteristic of the ignition process, or is it due only to variability in the test methods? To answer this question, the experimental variability must be minimized and quantified, and the ignition source must be well-controlled.
In ignition testing, there are many uncontrolled sources of variability in the ex- periment itself separate from the ignition energy. These uncertainties can lead to inaccurate test results and the appearance of variability in the results that has no correlation with the ignition energy. One major cause of variability in the test results is uncertainty in the mixture composition. Not only do changes in mixtures lead to changes in combustion characteristics (flame speeds, peak pressures, etc.), as shown in the previous MIE studies (Babrauskas, 2003, Magison, 1990), even small changes
in mixture composition can lead to large differences in MIE values. Therefore, it is important to precisely control and accurately measure composition during ignition experiments. Another cause of variability is the degree of turbulence near the ig- nition source, as the process of flame initiation and propagation can be affected by pre-existing turbulence. Finally, a third important source of variability in the test data is the method used to detect ignition. If the detection method is unreliable or unsuitable for the combustion characteristics of the mixture being tested, a given ignition energy may be perceived as not igniting a mixture when in fact combustion did occur. In this work test methods are proposed that minimize these uncertainties to isolate the statistical nature of the ignition process itself (Kwon et al.,2007). The sources of uncertainty are not limited to the three discussed here, but these three sources are major contributors to variability in the data that is unrelated to the igni- tion source. It is therefore necessary to quantify and minimize the uncertainties from these three sources before the variability of ignition with respect to ignition source energy can be examined.
1.3.3 Probability and Historical Spark Ignition Measurements
The large volume of historical minimum ignition energy data for capacitive spark dis- charge ignition has been extensively used in the chemical and aviation industry to set standards and evaluate safety with flammable gas mixtures. However, there is scant information on the experimental procedures, raw data or uncertainty consideration, or any other information that would enable the assignment of a statistical meaning to the minimum ignition energies that were reported. However, some researchers have claimed that the historical data can be interpreted as corresponding to a certain level of ignition probability. For example, in a paper by Moorhouse et al. (1974) the authors claim that the MIE results ofLewis and von Elbe (1961),Metzler (1952a,b), and Calcote et al. (1952) all correspond to an ignition probability of 0.01, i.e., 1 ig- nition in 100 tests. However, in all three studies the authors make no mention of
ignition probability and a specific probability of 0.01 is never discussed. In addition, the authors do not provide information about the number of tests performed nor the number of ignitions versus non-ignitions. Therefore, it is impossible to prescribe probabilities to historical minimum ignition energy data, as statistical analysis was never addressed in the literature, and there is not enough information on the number of tests performed and the experimental procedures. Also, obtaining a probability of ignition of only 0.01 with a reasonable level of confidence requires a large number of tests with very few ignitions, which does not appear to be consistent with the descriptions of the testing performed in the discussed literature (Calcote et al.,1952, Metzler,1952a,b,Lewis and von Elbe,1961). This issue of probability and historical MIE data is discussed in detail in AppendixA.