The effects of jet fuel composition on ignition probability have been studied in a flowfield that is relevant to turbine engine combustors, but also fundamental and conducive to modeling. In the experiments, a spark kernel is ejected from a wall and propagates transversely into a crossflow. The kernel first encounters an air-only stream before transiting into a second, flammable (premixed) stream. The two streams have matched velocities, as verified by hot-wire measurements. The liquid fuels span a range of physical and chemical kinetic properties. To focus on their chemical differences, the fuels are prevaporized in a carrier air flow before being injected into the experimental facility. Ignition probabilities at atmospheric pressure and elevated crossflow temperature were determined from optical measurements of a large number of spark events, and high-speed imaging was used to characterize the kernel evolution. Eight fuel blends were tested experimentally; all exhibited increasing ignition probability as equivalence ratio increased, at least up to the maximum value studied (∼0.8). Statistically significant differences between fuels were measured that have some correlation with fuel properties. To elucidate these trends, the forced ignition process was also studied with a reduced-order numerical model of an entraining kernel. The simulations suggest ignition is successful if sufficient heat release occurs before entrainment of colder crossflow fluid quenches the exothermic oxidation reactions. As the kernel is initialized in air, it remains extremely lean during the initial entrainment of the fuel–air mixture; thus, richer crossflows lead to quicker and higher exothermicity.

Motivation and Background

A robust, efficient, and effective energy economy requires multiple energy sourcing options. Combustion of liquid fuels, especially in the transportation sector, is common and will continue to play an important role for the foreseeable future [1], with increased diversity of liquid fuel sources in the coming years [2]. When integrating these fuels into our current and evolving infrastructure, it is important to maintain reliable combustion performance. For aero-engine operations, one important combustion metric is reliable ignition, especially for challenging requirements such as high altitude relight [3], standby protection [4], and smooth ignition on cold startup at ground conditions [5].

Some aspects of ignition have been addressed for various high-order hydrocarbon fuels. For example, previous work examined potential differences in auto-ignition between low- and high-temperature chemistry [6], as well as the pathways by which these fuels are converted to products [7]. Though auto-ignition can be used to characterize chemical differences in fuel conversion, there are important differences between auto-ignition and forced ignition, which is used in most practical devices. Forced ignition, where an external source of thermal energy or radical species is introduced [8], can be highly dependent on the coupling of ignition chemistry with fluid dynamics [9]. Also, the forced ignition is more strongly correlated to fuel–air ratio compared to auto-ignition delay times [10].

The forced ignition of premixed gases, mainly in quiescent conditions, has been studied in great detail for a variety of fuels [9,1113]. Work focused on jet fuels has identified the chemical effects and challenges associated with igniting complex fuels, including mixed phase issues [14,15]. A number of experimental studies examined flowing combustor configurations [1619]; more recent efforts have examined the importance of capturing the effects of mixing of less reactive fluid into the ignition kernel, prior to mixing with flammable fluid [20].

In this work, we examine the effects of fuel composition and chemistry on ignition performance in a test facility that captures important aspects of gas turbine combustor flowfields. In addition to ignition probabilities for different fuels over a range of fuel–air ratios, results from high-speed imaging and hot-wire velocity measurements are presented that are useful for developing and comparing low-order modeling or high-fidelity simulations of the experiments. We also examine the interplay between the chemical evolution of the ignition kernel and the competing fluid entrainment using a reduced-order model.

Approach

Experimental Facility.

The facility used in this work was developed previously to study the flow effects on the probability of ignition [20]. The rectangular (54.0 mm × 85.7 mm) experimental section incorporates a movable splitter plate that divides the flow into upper and lower regions as shown in Fig. 1. A common inflow air plenum feeds the two flows resulting in matched velocities downstream. For this work, the upper region (main flow) was fed a prevaporized fuel and carrier air gas mixture through fuel-injection bars. The test section features full quartz windows on the sides and small windows on the top and bottom for optical access. A commercial igniter with a 15 Hz discharge frequency and an energy discharge of roughly 1.25 J/pulse is located at the bottom wall. The spark discharge, which occurs within a cavity, creates a pressure rise that ejects the hot kernel into the crossflow. The fuel supply and vaporization system is illustrated in Fig. 2. Liquid fuel is placed in a cylinder and pressurized using compressed nitrogen regulated to 200 psi. The liquid flow rate was regulated by a needle valve and metered by an AW-Lake JVM-20KG positive displacement flow meter. Fuel was injected into a carrier air stream through a 0.41 mm orifice atomizer. Heating tape was placed along the carrier air lines to improve vaporization of the fuel. The fuel and carrier air flow was routed into a Bronkhorst heating unit that was controlled to 473 K. With the fuel flowing at the desired rate, the carrier air was adjusted to 50 slm to ensure repeatability. Complete vaporization of the fuel was verified by passing a HeNe laser beam through the test section, with no observable scattering indicating the absence of liquid droplets. Though the spatial uniformity of the fuel distribution across the main flow region was not verified here, previous work [20] using methane and the same fuel-injection system with a similar momentum ratio between the injected gas and main flow air produced a variation of <5%.

In the facility, the height of the splitter plate was set to 6.35 mm and the igniter was raised 3.18 mm in order to produce reasonable ignition probabilities for the conditions studied. The inflow air temperature was set to 477 K for all experiments to prevent liquid fuel condensation. The air velocity in the test section was set to 12.0 ± 0.2 m/s for all tests, unless otherwise noted.

Fuel Properties.

A set of eight liquid fuels were used in this study to observe the effects of their properties on the ignition process. The fuels are divided into two groupings: category “A,” which are conventional fuels, and category “C,” which are test fuels. These fuels have varying physical and chemical properties as listed in Table 1. Fuels A-1, A-2, and A-3 correspond to JP8, Jet-A, and JP5. Test fuel C-1 has the lowest C/H ratio, aromatic content, and cetane number of all the fuels; C-2 was designed to have a bimodal boiling point temperature range and has the highest cetane number; C-3 is a high viscosity fuel; C-4 has a low cetane number and low aromatic content; and C-5 was designed to have a nearly constant boiling point and has the highest aromatic content. For this work, we denote A-2 as the reference for comparison between the fuels.

Hot-Wire Velocity Profiles.

Hot-wire anemometry measurements were made to characterize the inflow to the test section and determine the velocity profile in the region above the igniter and through which the ejected kernel passes. A calibrated 5 μm diameter tungsten hot-wire probe (Dantec) was used to acquire data at a 100 kHz sampling rate, with a 30 kHz low-pass filter.

Profiles near and downstream of the splitter plate, along the midplane of the facility, were gathered at an operating velocity of 6 m/s, with no preheating and no fuel added. Normalized, the profile is expected to be similar at the actual operating condition. The resulting profiles, shown in Fig. 3, were taken at two streamwise locations. RMS values of the velocities are shown as horizontal error bars on the plots. The upstream profile (x = 0) shows the wake of the splitter plate, set to a height of (6.35 mm). The wake is nearly dissipated at the downstream location. The crossflow velocity fluctuations are <0.5 m/s, which is much less than the shear velocity between the kernel and the crossflow at early times in the kernel's evolution, as was observed through schlieren visualization. Thus, the turbulence generated by the kernel likely dominates the influence of freestream turbulence.

High-Speed Schlieren Imaging.

The hot gases of the ignition kernel were visualized using a schlieren optical configuration employed previously [21]. This configuration uses a 50 W halogen lamp as a continuous light source and two 20 cm parabolic mirrors for collimating and focusing. A Photron SA 1.1 high-speed camera was used to capture the kernel ejection process at 50 kHz framing rate. Images were postprocessed in matlab using threshold edge detection to track volume growth and trajectory as can be seen in Fig. 4 by the outline. Trajectory information confirmed the kernel does not impinge on the walls of the facility during operation at the desired flow rate, and that no interaction occurred between successive kernels with v = 12 m/s and the igniter operating at 15 Hz. Additionally, schlieren imaging indicated that the mixing layer is roughly 1 mm thick where the kernel crosses it.

Flame Detection.

A high-speed camera (Photron SA3) was used to capture broadband emission of flame kernel growth and propagation to determine ignition success. The 15 Hz igniter pulsing was captured by a nanosecond response photodiode which triggered the SA3 via an SRS DG535 pulse generator with a 2 ms delay. Thus, an image of the test section was captured 2 ms following each spark discharge with a 2 ms exposure. If the spark kernel successfully ignited the flammable mixture, a growing flame kernel was observed, as shown in Fig. 5. No signal was observed for failed attempts, though an image was saved. Edge tracking was used in postprocessing to rapidly analyze the presence of a flame and calculate ignition probability for each experimental run, as according to Eq. (1), where Nsucc is the number of successful ignition events, and Ntotal is the number of ignition attempts or spark discharges from the igniter

 
P(ign)=Nsucc/Ntotal
(1)

High-Speed Chemiluminescence Imaging.

Development of the flame kernel was observed through high-speed imaging of broadband chemiluminescence. This image signal correlates strongly with the presence of OH* chemiluminescence as detected with a 307 nm filtered photomultiplier tube in methane experiments [20]. The high-speed camera (SA1) was set to a looping record mode and post-triggered by a photodiode observing the flame luminescence at the end of the test section. This triggering method ensured that image sequences were only recorded for successful ignition events. The camera recorded single sequences of images at 15,000 fps. The collected image sequences were then analyzed in matlab to characterize the growth of the signal using edge tracking methods.

Reduced-Order Model.

A previously developed [10] reduced-order reactor model for an entraining ignition kernel, implemented in cantera [22], was adapted to accept a chemical mechanism for larger hydrocarbon fuels. The model (Fig. 6) employs a constant-pressure (volume-expanding) perfectly stirred reactor (PSR), which was chosen based on observations that mixing rates of the experimental kernel were very rapid and that flame chemistry begins in the regions of high vorticity [20]. Additionally, rapid fluid entrainment of the surrounding fluid into the kernel was observed and was included in the model through a mass entrainment term. The simulations here used a mass entrainment of 60 mg/s, determined from experimental measurements presented below.

The air plasma mechanism used in the first portion of the simulation [2326] uses thermodynamic coefficients from NASA CEA [27]. To characterize the state of the initial plasma, the internal energy of the volume of air contained within the igniter cavity (O(0.1cm3)) at the desired temperature and pressure (T = Ti, p=1atm,χN2=0.79,andχO2=0.21) was raised by deposition energy of the igniter (1.25 J) and allowed to expand to p = 1 atm and equilibrate. The assumption of a high discharge efficiency is supported by the short discharge duration [9]. The state of this gas was then used to initialize the PSR whose volume was set to the initial kernel volume (0.02 cm3) as previously estimated from schlieren results [10]. The simulation progresses as new ambient fluid is added at the inflow rate. Following the specified transit time (τtransit), the state of the reactor was used to continue the simulation with a fuel mechanism. Fuel A-2 was simulated using a chemical mechanism developed by Wang et al. [28], which does not contain ionized or nitrogen oxides. Therefore, transfer to the new mechanism required all ionized species to be recombined and NO to be converted to N2 and O2.

Results

Schlieren Kernel Imaging.

Schlieren images were collected in order to obtain approximate kernel trajectories for an unfueled 12 m/s mean flow with no preheat, and to estimate entrainment rates at early times. The edge tracking postprocess gathered the number of pixels representing the line-of-sight integrated area of the kernel. This projection was registered into units of cm2. It was assumed that the kernel had symmetry along its vertical axis and a revolved volume was calculated. The calculated volume growth of 44 ejected kernels is shown in Fig. 7. Using an average early time equilibrium density of the air kernel at 4000 K and p = 1 atm, the mass entrainment of the kernel was estimated to be m˙=60mg/s, as was performed in earlier work [9]. This value is double that of previous estimates [10] that included data from later times in the kernel's growth. This suggests, not surprisingly, that the kernel's mass entrainment varies over time. Future characterization of this variation would be helpful for improved modeling results.

Ignition Probabilities.

Each ignition trial consisted of ∼135 spark events; each event is an independent ignition attempt, as the kernel fluid from one event has convected far downstream before the subsequent igniter pulse. Tests were conducted at main flow equivalence ratios of ϕ=0.50.8. At all the conditions tested, the single event ignition probabilities were low, not exceeding 20%. Providing statistically reliable results required a larger number of ignition attempts then could be obtained from a single ignition trial, which was limited by the acquisition system. Furthermore, small experimental variations occurred between trials. Therefore, binning of data at similar operating conditions was necessary.

Bin widths were set to 0.04 in ϕ space in order to average probability values from each experimental run contained within the bin. The resulting ignition probabilities for each of the fuels over a range of binned ϕ values are shown in Fig. 8. The error bars depict the 95% confidence interval of the data variability in that particular bin. Each fuel also has a quadratic fit plotted using least-squares regression. These curves depict good agreement over this range of ϕ values.

The relative ignition probabilities of each fuel were compared for the ϕ=0.75 condition as seen in the bottom right axes of Fig. 8. These are plotted relative to the baseline fuel (A-2). The values were obtained using the quadratic fits of each set of binned probabilities, and nominal uncertainty bars come from the variability in the binned data clustered near ϕ=0.75. The relative rankings of the fuel performance depict significant differences in ignition performance between several of the fuels.

Probability Correlations.

Ignition performance variation as a result of fuel properties is a metric of interest, and therefore, a correlation of P(ign) to the fuels was desired. The fuel vaporizing feature of the experimental facility was designed to highlight the chemical effects over the physical properties of the fuels. The interrogated properties, found in Table 1, are C/H, % hydrogen (%H), % aromatics (%Aro), % isoparaffins (%isoP), and MW.

The ϕ -binned data for each fuel was used to build a polynomial model with a least-squares regression method. The model includes linear terms, interaction terms, and a squared term for ϕ. The number of terms was limited by the small number of fuels available, and therefore a limited set of fuel property inputs to include in the regression. Consequently, the model was built by including additional terms if they were influential on the predictive capability of the model, performed using jmp statistical software. The model that was found to predict the ignition probability well is of the form found in Eq. (2), and the coefficients for this polynomial model are found in Table 2. The actual-by-predicted plot for this polynomial model is shown in Fig. 9 and results in R2 = 95%

 
p̂(ign)=a0+a1ϕ2+a2ϕ+a3C/H+a4MW+a5%Aro+a6ϕ×%Aro+a7ϕ×C/H+a8ϕ×MW+a9C/H×MW+a10%Aro×MW+a11%Aro×C/H+a12%Aro×MW×C/H+a13MW×ϕ×C/H
(2)

Significance for each parameter in the polynomial model is represented by the t-ratio of that parameter, which tests the coefficient value against the null hypothesis. The tornado plot in Fig. 10 presents the parameters in decreasing significance. The significance level of α = 0.05 is represented by the vertical lines. Experimentally, varying ϕ strongly influences ignition probability and dominates over fuel composition, as would be expected. Furthermore, the strength in the quadratic ϕ term reinforces the goodness in fit of the curves in Fig. 8 as ignition probability is plotted against ϕ for each fuel.

Despite the dominant ϕ influence, fuel effects are still evident, for example, through the interaction terms with ϕ. These terms indicate that an improvement in ignition performance is enhanced by varying both the fuel property and ϕ. For example, an increase in ϕ and using a fuel with a high %Aro content improve P(ign) beyond what improvement would result from increasing only one of the input parameters. This development of the polynomial model shows that beyond ϕ, fuel properties that largely influence ignition probability are %Aro, C/H, and MW.

Flame Growth and Propagation.

Three fuels: A-2, C-1 and C-5, were chosen for further characterization as they represent the range of ignition performance shown in Fig. 8. Flame growth was measured with high-speed chemiluminescence images for the three fuels; for each, approximately 40 flame kernels were processed to develop statistical growth curves. The background spark profile was subtracted to remove the effect of the bright discharge. Additionally, the baseline signal following the discharge was subtracted from each fuel profile to remove artifacts of the variability in the discharge event. The resulting signals for each fuel with 95% uncertainty bars are presented in Fig. 11. The A-2 and C-5 flame kernels have quite similar growth histories, and their mean values are within the individual variabilities. These area growths are presumed to be related to a flame propagation speed, as this time scale is beyond the scale of the initial ignition process. The area of the C-1 flame appears to grow more slowly than the other fuels, which would suggest a difference in the flame propagation speed. However, the turbulent flame speeds are likely similar for these fuels. More likely, the growth of the C-1 flame is simply delayed; this is confirmed by the shifted C 1 data, which was advanced by 1.33 ms. This shifted C-1 curve is similar in profile to the other fuels, within the bounds of variability. This suggests that the C-1 fuel exhibits some chemical difference that requires more time to reach a flame propagation controlled state.

Reduced-Order Model Results.

While the ignition trends for the fuels show some correlation with bulk fuel properties, the correlations provide little insight into the processes and specific fuel differences that control ignition probability. Therefore, simulations were conducted using the entraining, constant-pressure PSR ignition model in order to better understand the kernel evolution. The primary inputs to this reduced-order model are (1) the initial kernel mass and energy density, (2) the mass entrainment rate, (3) the temperature of the crossflow, (4) the transit time across the nonflammable air layer, and (5) the fuel–air ratio of the main (flammable) flow region. The Approach section describes how the first two inputs were determined, and they are held constant in the simulations presented here.

As an example of the kernel evolution for conditions similar to those in the experiments, Fig. 12 presents the kernel's temperature history for a crossflow temperature of 420 K and a transit time of 90 μs. The initial kernel temperature exceeds 4000 K, but it drops rapidly as the kernel entrains ambient air. At the time the kernel begins to entrain the flammable mixture, its temperature has dropped to nearly 1700 K. While not as hot as the initial kernel, this temperature is still quite high compared to values typically considered necessary to ignite hydrocarbon–air mixtures.

Figure 12 includes the results for four main flow equivalence ratios (0.5–1.5). In all four cases, the kernel temperature continues to drop as the flammable mixture begins to be introduced to the kernel PSR. For the first 10–15 μs after fuel is introduced, the kernel temperature drops at nearly the same rate as if the kernel had continued to entrain pure air (light dashed line in the figure). After that time, however, the temperature decay for all the fueled kernels is less than for the pure air case. For the two leanest mixtures, the temperature continues to decrease with time. For the two richest cases, the kernel temperature increases. This bifurcation in the kernel temperature response as a function of equivalence ratio distinguishes a successful ignition from a kernel that fails to ignite. This behavior is investigated further by examining the change in the chemical composition of the kernel with time, which is presented in Fig. 13. Results are shown for the two equivalence ratios closest to the temperature bifurcation and for three species: ethylene (C2H4), and the formyl (HCO) and hydroxyl (OH) radicals. The rapid rise in the ethylene mole fractions indicates that the jet fuel is quickly decomposed, within a microsecond or so as it mixes into the high-temperature (1700 K) kernel. There is also significant partial (exothermic) oxidation of the decomposition products within 10 μs, as evidenced by the rise in the OH and HCO levels. The heat release from partial oxidation is partly responsible for the slower drop in the kernel temperature during this time. After 50–100 μs, there is a marked difference between the cases with the richer and leaner entrained mixture. For the richer cases, the ethylene is rapidly consumed and there is a parallel increase in the radical concentrations. This sudden heat release is the cause of the rise in the kernel temperature and a further sign of successful ignition. Conversely, the leaner mixture experiences a continued growth in ethylene and a drop in the radical levels. This indicates newly entrained fuel is still decomposing, but the oxidation reaction rates are dropping, thus leading to ignition failure. The richer mixture provides more heat release per mass of entrained fluid compared to the leaner mixture, preventing the kernel temperature from dropping as much and thereby allowing the exothermic reaction rates to outweigh the effect of diluting the kernel with cold entrained gas. It is also worth noting that during these times, the overall equivalence ratio of the kernel is fuel-lean. Even when rich mixtures are being entrained, they enter a kernel that is initially air; it takes much longer than the times shown before the kernel becomes overall fuel-rich.

Given the ability of the model to predict conditions that can lead to successful ignition, the model can also be used to study ignition limits. An example of this is shown in Fig. 14. Simulation results are shown for a range of crossflow temperatures and transit times. For each combination of temperature and transit time, simulations were performed for a range of main flow equivalence ratios (0.1–2.0). If any of these cases lead to successful ignition, it is indicated on the figure with a green circle. If none of the equivalence ratio cases ignited, the point is marked with a red square. The results provide a clear set of ignition limits; as the transit time is increased, and therefore the kernel temperature is lower at the end of the transit time, a higher crossflow temperature is required for successful ignition. The trend is roughly linear, with a 50–75 K increase needed for every 10 μs more of transit time. This suggests that forced ignition in a turbine engine combustor should be highly sensitive to shot-to-shot variations in the time required for a kernel produced in a fuel-free zone to begin entraining fuel.

These simulation results follow the trends observed in the experiments. First, both the experiments and the model suggest a significant sensitivity to increasing the main flow equivalence ratio (at least up to a value 0.8 in experiments and 2.0 in simulations), which makes it easier for the kernel to lead to successful ignition. Second, both indicate a strong sensitivity to kernel transit time. At the fixed crossflow temperature used in the experiments, tests were run for longer transit times by increasing the distance between the igniter and splitter plate. The probability of ignition dropped rapidly as this distance was increased; the probability was nearly zero when the distance was increased by just a few millimeters. While the PSR model oversimplifies the entrainment and mixing processes occurring between the kernel and the crossflow, it does appear to capture the controlling physics of forced ignition, or at least processes that determine if the kernel is able to transition into a growing flame.

Conclusions

Liquid fuel composition effects on ignition probability were studied in a flow relevant to gas turbine ignition. Specifically, prevaporized liquid fuels were studied in a stratified flow facility in order to isolate the influence of fuel chemistry when the forced ignition kernel is produced in a pure air flow and must convect to a flammable mixture. In this study, crossflow temperature and velocity were fixed, as was the distance between the igniter and the flammable flow region. Inflow velocity profiles were acquired as boundary conditions for future high-fidelity models. Additionally, schlieren imaging of the kernel ejected into the crossflow was acquired at high framing rates to estimate the kernel entrainment rate and to obtain the kernel trajectory for model comparisons.

Ignition probability data were collected for eight fuels at several ϕ values, comprising approximately 47,000 individual ignitor events. Across all fuels, successful ignition increased with the fuel–air ratio of the main flow, at least to up to the maximum equivalence ratios tested, with a nearly quadratic dependence. The results also showed statistically meaningful differences between the eight fuels, with the A-1 and C-5 fuels having ignition probabilities nearly three times higher than the C-1 fuel at an equivalence ratio of 0.75. The measured ignition probabilities, when examined against fuel properties, show strong correlations to aromatic content, C/H ratio, and molecular weight. Physical fuel properties (viscosity, boiling points, etc.) were also analyzed against ignition performance and showed little correlation.

Modeling results indicate that even when the flammable environment is very fuel-rich, e.g., ϕ=2, the kernel remains overall fuel-lean prior to ignition due to the fact that it was initialized in an air-only environment. The simulations also indicate rapid fuel decomposition (e.g., into ethylene) as it is entrained into the high-temperature kernel. This is followed after a short time by some heat release from exothermic oxidation reactions. Thus, the rich mixtures improve ignition probability because the additional heat release at early times offsets the rapid kernel cooling due to entrainment, which in turn tends to quench the oxidation rates. The fuels with enhanced ignition probabilities likely facilitate these early heat release processes.

At a given crossflow temperature and transit time, simulations at varying equivalence ratios also reveal the stark difference in kernel evolution between successful and failed ignition cases. The narrow boundary between success and failure within (Ti, τtransit) space reflects similar trends in experiments that allowed for ignition to be observed when the separation between the igniter and splitter plate was reduced by just 3 mm.

Additionally, the high-speed chemiluminescence imaging shows a similarity in the long-term growth rate of the ignited flame for all the fuels. This suggests they have similar turbulent flame speeds. For the fuel with the lowest ignition probability (C-1), the flame growth is delayed, suggesting it takes longer for flame propagation to become important.

Future work is planned that will focus on performing and comparing simulation results with different input fuel compositions to elucidate the subtle contrasts in chemical pathways that influence forced ignition performance. Additionally, expanding both experiments and simulations to include liquid (nonprevaporized) fuel physics will allow us to study fuel characteristics that likely influence successful ignition in practical devices.

Acknowledgment

This work was funded by the U.S. Federal Aviation Administration (FAA) Office of Environment and Energy as a part of ASCENT Project 13-C-AJFE-GIT-008 under FAA Award No. 27A.

Nomenclature

     
  • ai =

    probability of ignition regression model coefficients

  •  
  • C/H =

    average carbon-to-hydrogen ratio of fuel

  •  
  • m˙flam =

    mass entrainment rate of flammable gas in simulation

  •  
  • m˙nonflam =

    mass entrainment rate in simulation

  •  
  • MW =

    effective molecular weight of the fuel

  •  
  • Nsucc =

    number of successful ignition events

  •  
  • Ntotal =

    number of ignition attempts (spark discharges)

  •  
  • p =

    pressure

  •  
  • p̂(ign) =

    predicted probability of ignition

  •  
  • P(ign) =

    probability of ignition

  •  
  • t =

    time since the discharge event

  •  
  • Ti =

    inflow temperature

  •  
  • α =

    significance level for t-statistic

  •  
  • μ40C =

    dynamic viscosity of fuel taken at 40 °C

  •  
  • ρ15C =

    density of fuel taken at 15 °C

  •  
  • σ =

    surface tension

  •  
  • τtransit =

    transit time of kernel in nonflammable fluid

  •  
  • ϕ =

    main flow equivalence ratio

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