Flame Spread and GroupCombustion Excitation in Randomly Distributed Droplet Clouds with LowVolatility Fuel near the Excitation Limit: a Percolation Approach Based on FlameSpread Characteristics in Microgravity
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Abstract
Stable operation of liquidfueled combustors requires the group combustion of fuel spray. Our study employs a percolation approach to describe unsteady groupcombustion excitation based on findings obtained from microgravity experiments on the flame spread of fuel droplets. We focus on droplet clouds distributed randomly in threedimensional square lattices with a lowvolatility fuel, such as ndecane in roomtemperature air, where the prevaporization effect is negligible. We also focus on the flame spread in dilute droplet clouds near the groupcombustionexcitation limit, where the droplet interactive effect is assumed negligible. The results show that the occurrence probability of group combustion sharply decreases with the increase in mean droplet spacing around a specific value, which is termed the critical mean droplet spacing. If the lattice size is at smallest about ten times as large as the flamespread limit distance, the flamespread characteristics are similar to those over an infinitely large cluster. The number density of unburned droplets remaining after completion of burning attained maximum around the critical mean droplet spacing. Therefore, the critical mean droplet spacing is a good index for stable combustion and unburned hydrocarbon. In the critical condition, the flame spreads through complicated paths, and thus the characteristic time scale of flame spread over droplet clouds has a very large value. The overall flamespread rate of randomly distributed droplet clouds is almost the same as the flamespread rate of a linear droplet array except over the flamespread limit.
Keywords
Flame spread Group combustion Droplet cloud Percolation MicrogravityIntroduction
Spray combustion is widely used in jet engines, oilfired furnaces, diesel engines, etc. Stable operation of spray combustors requires socalled group combustion. Many researches on group combustion have been conducted (Chiu et al. 1971, 1983; Labowsky and Rosener 1978; Correa and Sichel 1982; Ryan et al. 1990; Umemura 1994; Imaoka and Sirignano 2005a, b). A wellknown work on group combustion is the group combustion theory by Chiu and coworkers. Chiu et al. (1971) analytically studied the steadystate combustion of spherical monodisperse droplet clouds and proposed the group combustion number, G, which classifies the combustion into several modes; single droplet combustion for G< 10^{−2}, internal group combustion for 10^{−2}<G< 10^{− 1}, external group combustion for 10^{− 1}<G< 10^{2}, and sheath combustion for 10^{2}<G. The group combustion theory, however, is based on the steadystate analysis and therefore cannot describe unsteady processes leading to the groupcombustion excitation and excitation conditions. Our study employs a percolation approach to describe the unsteady groupcombustion excitation based on findings obtained from microgravity experiments on the flamespread of fuel droplets.
 Mode 1:

the vaporization of the next droplet becomes active after the leading edge of an expanding group diffusion flame passes the next droplet and pushes the leading edge forward;
 Mode 2:

the leading edge of the diffusion flame reaches the flammable mixture layer formed around the next unburned droplet and the premixed flame propagates in the mixture layer to form the new diffusion flame around the next droplet;
 Mode 3:

the next unburned droplet autoignites through heat from the diffusion flame, whose leading edge does not reach the flammable mixture layer of the next droplet;
 Premixed flame propagation mode:

the flame propagates in a continuous flammable mixture layer formed around the droplet array;
 Vaporization mode:

flame spread does not occur.
Some researchers have tried to utilize findings obtained from fundamental flamespread researches in order to improve understanding of the flamespread characteristics in fuel sprays. Nunome et al. (2003) performed experiments of flame spread in quiescent ndecane sprays in microgravity and examined the dependence of the flamespread rate on the mean droplet diameter considering the flamespread characteristics of a droplet array. Mikami et al. (2009) elucidated the flamestabilization mechanism in the counterflow of ndecane/air premixedspray jet and air considering the dependence of the flamespread rate of a droplet array on the droplet spacing. In order to consider the random dispersity of droplets in spray, Umemura and Takamori (2005) applied the percolation theory, which can describe the connection characteristics of a randomly distributed particle system, to an analysis of the groupcombustion excitation in randomly distributed droplet clouds. They developed a percolation model that describes the groupcombustion excitation based on Mode 1 flame spread (Fig. 3) and site percolation. The droplets are randomly distributed at lattice points of a threedimensional square lattice. Adjacent droplets interactively burn to establish a group flame, and the expanded group flame swallows unburned droplets. A new group flame is established through interactive droplet burning including swallowed droplets and their adjacent droplets in the same cluster. As shown in Fig. 4, however, Mode 1 flame spread appears only for very narrow droplet spacing, such as in dense sprays. Oyagi et al. (2009) proposed a percolation model based on Mode 3 flame spread focusing on the flamespreadlimit droplet spacing, (S/d_{0})_{limit}. The flame cannot spread to droplets outside (S/d_{0})_{limit} but can spread to droplets within it. They applied this model to the flame spread along randomly distributed droplet arrays. Mikami et al. (2009) examined the flame structure in the counterflow of ndecane/air premixedspray jet and air in detail using a highspeed photography and revealed that Mode 3 flame spread to unburned droplets occurs near the leading flame region. The group flame is established through interaction between the single flames surrounding each droplet. These researches will bridge the gap between the combustion of a small number of droplets and spray combustion.
This study is an extension of Oyagi’s percolation model based on Mode 3 flamespread characteristics in microgravity (Oyagi et al. 2009) to study the flame spread and groupcombustion excitation in droplet clouds in which the droplets are distributed randomly in threedimensional square lattices. We focus on droplet clouds with a lowvolatility fuel, such as ndecane in roomtemperature air, where the prevaporization effect is negligible. We also focus on the flame spread in dilute droplet clouds near the groupcombustionexcitation limit, where the occurrence probability of group combustion sharply varies with the number density of droplets.
Percolation Model Considering FlameSpreadLimit Distance
Umemura and Takamori (2005) developed a percolation model of flame spread in randomly distributed fueldroplet clouds with non or lowvolatility fuel. They extended the site percolation by setting the latticepoint interval to be the same as the maximum flame radius and by assuming that the flame spreads in Mode 1. As shown in Fig. 4, Mode 1 flame spread appears in a limited condition with very narrow droplet spacing, S/d_{0}. On the other hand, there is the flamespread limit, (S/d_{0})_{limit}, below which Mode 3 flame spread appears for relatively large S/d_{0} (Kato et al. 1998, Mikami et al. 2006). Mode 3 flame spread was also observed near the leading flame region of the ndecane premixedspray flame in counterflow (Mikami et al. 2009). Even if a very lowvolatility fuel is used, sufficient heating of the unburned droplet by the flame will cause vaporization from the droplet surface or thermal decomposition inside the droplet and thus Mode 3 flame spread will occur near the flamespread limit, instead of Mode 1 flame spread. Therefore, it is necessary to develop a percolation model based on Mode 3 flame spread considering the flamespread limit, especially near the critical occupation fraction.
We use (S/d_{0})_{limit} = 14, which is the flamespread limit obtained from microgravity experiments on the flame spread of ndecane droplet arrays in atmospheric pressure and roomtemperature air (Mikami et al. 2006). We calculate the normalized flamespread time as t_{f}/\(d_{0}^{2}=S/d_{0}\)/(V_{f}d_{0}), where we use the experimental values of the flamespread rate, V_{f}d_{0}, as a function of the interdroplet distance, S/d_{0}, obtained in microgravity by Mikami et al. (2006). Here, the dimensionless time and flamespread rate are respectively expressed as at_{f}/\(d_{0}^{2}\) and V_{f}d_{0}/a, where a is the thermal diffusivity of air (Mikami et al. 2006).
Since the present percolation model employs the flamespread limit and flamespread time obtained from the linear droplet array with even droplet spacing in microgravity, the model is suitable for the flame spread in droplet clouds with negligible local dropletinteraction effect. As shown in Fig. 4 (Mikami et al. 2006), Mode 3 flame spread appears for S/d_{0}> 6 in the flame spread of ndecane droplet arrays at room temperature and atmospheric pressure. For S/d_{0}< 11, however, flames that form around each droplet and separate at ignition merge into a group flame during burning. For 11<S/d_{0}<(S/d_{0})_{limit} = 14, the flames around each droplet never merge, and thus the interactive effect between burning droplets is small. Therefore, the interactive effect is conceivably small on the mean characteristics of flame spread over randomly distributed droplet clouds with the mean droplet spacing (S/d_{0})_{m}> 11 in this research. Since the critical mean droplet spacing (S/d_{0})_{mcr} is larger than (S/d_{0})_{limit} = 14, as explained in “Occurrence Probability of Group Combustion”, most droplets are not affected by droplet interaction near the critical condition, i.e., the groupcombustionexcitation limit.
Results and Discussion
Occurrence Probability of Group Combustion
Umemura and Takamori (2005) discussed the statistically significant lattice size and obtained the requirement condition, V_{f}NL/D >> 1, by comparing the flamespread time and oxygen diffusion time, where D is the diffusion coefficient. They concluded that N ∼ O(10) is sufficient to predict the transition behavior satisfactorily and employed N = 15 while L was set as the maximum flame radius. Since V_{f}NL/D>> 1 can be converted as NL/d_{0}>>(V_{f}d_{0}/a)^{− 1} for the unity Lewis number, where the Lewis number is Le = a/D. According to Mikami et al. (2006), the dimensionless flamespread rate, V_{f}d_{0}/a, is around 0.1 near the flamespread limit, (S/d_{0})_{limit}. Therefore, NL/d_{0} ∼ O(100) is required. As shown in Fig. 12, (S/d_{0})_{mcr} seems nearly constant for NL/d_{0}> 150, which also satisfies the requirement by Umemura and Takamori (2005). In the case of NL/d_{0} = 150,N is about 11 for L/d_{0} = 14, which is the same as (S/d_{0})_{limit}, but 75 for L/d_{0} = 2. This study employs NL/d_{0} = 400 as a fully converged condition. Figures 9–11 also employ NL/d_{0} = 400. Hereafter, the results for NL/d_{0} = 400 and L/d_{0} = 2 are shown.
As shown in Fig. 12, the minimum value of statistically significant length scale of droplet clouds is about ten times greater than the flamespreadlimit distance, e.g., about 7 mm for 50 μ m ndecane droplets at atmospheric pressure. Since the flamespreadlimit distance, (S/d_{0})_{limit}, decreases with the increase in the ambient pressure (Sano et al. 2016), the minimum value of the statistically significant length scale also decreases. In practical sprays, the spray size is greater than the minimum value of the statistically significant length scale, and thus we can define the mean droplet spacing locally in the spray. The mean droplet spacing is not uniform over the entire spray; it is small in the core region and large in the peripheral region. The peripheral region whose mean droplet spacing is larger than the critical value will cause unburned droplets, resulting in unburned hydrocarbon (UHC) in the exhaust gas. Therefore, the critical mean droplet spacing, (S/d_{0})_{mcr}, is a good index for the stable combustion and UHC.
As mentioned in “Percolation Model Considering FlameSpreadLimit Distance”, the effect of droplet interaction would be relatively small for (S/d_{0})_{m}> 11, and thus, this study disregards the effect of droplet interaction. Oyagi et al. (2009), however, reported that the local droplet interaction affects the critical condition of a randomly distributed 1D droplet array. The critical condition of the randomly distributed 3D droplet cloud could also be affected by the local droplet interaction. This effect will be considered in a future study.
Flamespread Behavior
This section reports flamespread behavior. As explained in “Percolation Model Considering FlameSpreadLimit Distance”, we neglect the effect of droplet interaction on the local flamespread rate and limit because we analyze the flame spread with relatively large mean droplet spacing, (S/d_{0})_{m}, where the effect of droplet interaction is relatively small. Thus, the local flamespread rate, V_{f}d_{0}, is assumed to be a function of dimensionless droplet spacing, S/d_{0}, between the nearest droplet pair. The burning lifetime of each droplet is also assumed not to be affected by the droplet interaction and set as one second.
In the critical condition, some droplets have a very large ignition time over 150 s/mm^{2}. In such a case with a relatively long waiting time for ignition, the prevaporization might be significant. Here, we estimate the droplet diameter change through 150 s/mm^{2} prevaporization. nDecane is a lowvolatility fuel with a boiling point of 447 K at 101 kPa. The equivalence ratio in the gas phase at the droplet surface is about 0.1 at room temperature and atmospheric pressure, which is much lower than the lower flammability limit (Mikami et al. 2006, 2009). Therefore, the prevaporization of the ndecane droplet does not produce a flammable mixture around the droplet in this condition. The vaporizationrate constant at room temperature in microgravity is estimated to be less than 0.0006 mm^{2}/s based on our preliminary experiment. Considering the estimated prevaporization with this vaporizationrate constant in microgravity, the droplet diameter decrease is estimated to be very small, less than 5% even with the 150 s/mm^{2} prevaporization. When the spray burns at high temperatures as in diesel engines and jet engines, however, the prevaporization effect could be significant. Mikami et. al. (2006) conducted flamespread experiments using linear ndecane droplet arrays in microgravity and reported that the flamespread rate and limit increase with the ambient temperature. If the autoignition or forced ignition of spray occurs prior to the completion of droplet vaporization, the flammable mixture is not always spatially continuous, and therefore, the present percolation model can be extended to such a case with prevaporization. This effect will be considered in a future study.
Figure 19 also plots the flamespread rate of linear ndecane droplet arrays in microgravity (Mikami et al. 2006). The overall flamespread rate of randomly distributed droplet clouds is almost the same as the flamespread rate of a linear droplet array except over the flamespread limit. As explained in “Occurrence Probability of Group Combustion”, the critical mean droplet spacing, (S/d_{0})_{mcr} = 15.8, is greater than the flamespread limit of a linear droplet array, (S/d_{0})_{limit} = 14. Mikami et al. (2009) discussed the stabilized position of the leading flame of the ndecane spray/air mixture in counter flow and concluded that it can be predicted qualitatively by using the flamespread rate of a linear droplet array obtained in microgravity as the overall flamespread rate. As confirmed by Fig. 19, the overall flamespread rate is not affected by the randomness of droplet distribution except over the flamespread limit. The quantitative prediction of the overall flamespread rate of spray in counter flow requires consideration of the effects of gasliquid relative velocity and dropletsize distribution.
This study does not consider the effects of droplet interaction for relatively large mean droplet spacing, (S/d_{0})_{m}. If (S/d_{0})_{m} becomes smaller, the cooling effect of unburned droplets due to increased droplet interaction would become larger and make the overall flamespread rate smaller than the flamespread rate of a linear droplet array. If (S/d_{0})_{m} is too small, the vaporization of each droplet occurs in a limited space and thus the local equivalence ratio at the outer boundary of the limited space could exceed the rich flammability limit, resulting in another flamespread limit and groupcombustionexcitation limit, whereas this study focuses on the lean limit of the flame spread and groupcombustion excitation in droplet clouds. Mikami et al. (2005a) observed the double flame structure of premixed ndecane spray/air jet and discussed the characteristics of the internal flame based on the flamespread mechanism of a droplet array (Mikami et al. 2008). As the equivalence ratio of the premixed spray jet is increased, the internal flame disappears at a specific equivalence ratio. This finding suggests that there is a rich limit of the flame spread and groupcombustion excitation in droplet clouds. To model such a rich limit of flame spread and groupcombustion excitation in droplet clouds, the effect of droplet interaction should be considered. This effect will be considered in a future study.
Conclusions
Our study employs a percolation approach to describe the unsteady groupcombustion excitation based on findings obtained from microgravity experiments on the flamespread of fuel droplets. This percolation model is based on Mode 3 flame spread. We focused on droplet clouds distributed randomly in threedimensional square lattices with a lowvolatility fuel, such as ndecane in roomtemperature air, where the prevaporization effect is negligible. We also focused on the flame spread in dilute droplet clouds near the groupcombustionexcitation limit, where the droplet interactive effect is assumed negligible.
The results show that the occurrence probability of group combustion sharply decreases with the increase in mean droplet spacing around a specific value, which is termed the critical mean droplet spacing. The critical mean droplet spacing is nearly independent of lattice characteristics if the latticepoint interval is small and the lattice size is sufficiently large. If the lattice size is at smallest about ten times as large as the flamespread limit distance, the flamespread characteristics are similar to those over an infinitely large cluster. The number density of unburned droplets remaining after completion of burning attains maximum around the critical mean droplet spacing. Therefore, the critical mean droplet spacing is a good index for stable combustion and unburned hydrocarbon. In the critical condition, the flame spreads through complicated paths, and thus the characteristic time scale of flame spread over droplet clouds has a very large value. The overall flamespread rate of randomly distributed droplet clouds is almost the same as the flamespread rate of a linear droplet array except over the flamespread limit. It scatters around the critical condition.
This study will be extended to predict the flame spread characteristics and group combustion occurrence of fuel sprays by considering the effects of gasliquid relative velocity, dropletsize distribution, droplet interaction and prevaporization. This study focuses on the lean limit of the flame spread and groupcombustion excitation in droplet clouds. In order to model the rich limit, the effects of droplet interaction will be considered in future studies.
Notes
Acknowledgments
This research was partly subsidized by JSPS KAKENHI GrantinAid for Scientific Research (B) (24360350 and 15H04201). We would like to acknowledge the assistance by Mr. Hisashi Shigeno, Mr. Yuki Tsuchida and Mr. Daiki Azakami. This research was also conducted as a part of “Group Combustion” project by JAXA.
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