Advertisement

Microgravity Science and Technology

, Volume 30, Issue 4, pp 419–433 | Cite as

Flame Spread and Group-Combustion Excitation in Randomly Distributed Droplet Clouds with Low-Volatility Fuel near the Excitation Limit: a Percolation Approach Based on Flame-Spread Characteristics in Microgravity

  • Masato Mikami
  • Herman Saputro
  • Takehiko Seo
  • Hiroshi Oyagi
Original Article

Abstract

Stable operation of liquid-fueled combustors requires the group combustion of fuel spray. Our study employs a percolation approach to describe unsteady group-combustion excitation based on findings obtained from microgravity experiments on the flame spread of fuel droplets. We focus on droplet clouds distributed randomly in three-dimensional square lattices with a low-volatility fuel, such as n-decane in room-temperature air, where the pre-vaporization effect is negligible. We also focus on the flame spread in dilute droplet clouds near the group-combustion-excitation 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 flame-spread limit distance, the flame-spread 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 flame-spread rate of randomly distributed droplet clouds is almost the same as the flame-spread rate of a linear droplet array except over the flame-spread limit.

Keywords

Flame spread Group combustion Droplet cloud Percolation Microgravity 

Introduction

Spray combustion is widely used in jet engines, oil-fired furnaces, diesel engines, etc. Stable operation of spray combustors requires so-called 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 well-known work on group combustion is the group combustion theory by Chiu and co-workers. Chiu et al. (1971) analytically studied the steady-state combustion of spherical mono-disperse 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< 102, and sheath combustion for 102<G. The group combustion theory, however, is based on the steady-state analysis and therefore cannot describe unsteady processes leading to the group-combustion excitation and excitation conditions. Our study employs a percolation approach to describe the unsteady group-combustion excitation based on findings obtained from microgravity experiments on the flame-spread of fuel droplets.

Figure 1 shows a group flame established in an ethanol spray jet in room temperature air. Since the spray jet is supplied continuously from upstream, the flame spread occurs near the flame base against the spray jet and thus stabilizes the group flame. As a fundamental research on flame spread in sprays, many researches on the flame spread of fuel droplet arrays have been conducted through theoretical analyses (Umemura 2002a, b; Mikami 2006), numerical simulation (Kikuchi et al. 2002, 2005), and experiments in microgravity (Okajima et al. 1981; Kato et al. 1998; Mikami et al. 2005b, 2006; Oyagi et al. 2009; Nomura et al. 2011), focusing on the droplet, which is an important element of the spray. Figure 2 shows typical flame-spread behavior of a linear droplet array in microgravity (Mikami et al. 2005b). Umemura (2002a) theoretically classified the flame spread along linear droplet arrays into the five modes depicted in Fig. 3 (Mikami et al. 2006):
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 auto-ignites 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.

Fig. 1

Spray combustion

Fig. 2

Flame-spread behavior of linear n-decane droplet array in microgravity (Mikami et al. 2005b)

Fig. 3

Flame-spread modes of linear droplet array (Mikami et al. 2006)

Kikuchi et al. (2005) numerically demonstrated that the flame-spread mode changes depending on the droplet spacing and ambient temperature. Mikami et al. (2006) performed flame-spread experiments of n-decane droplet arrays at temperatures ranging from 300 to 750 K in microgravity. They studied the dependence of the flame-spread rate on the droplet spacing and ambient temperature and obtained a mode map of flame spread and burning of droplet arrays in the plane of dimensionless droplet spacing S/d0 and ambient temperature, Ta, as shown in Fig. 4, where S is the inter-droplet distance and d0 is the initial droplet diameter. According to Fig. 4, as S/d0 increases at a relatively low Ta, the flame-spread mode changes in the order of Mode 1, Mode 2, Mode 3, and vaporization mode. The flame-spread-limit droplet spacing, (S/d0)limit, is defined as the boundary between Mode 3 and the vaporization mode. The premixed-flame propagation mode appears for relatively small S/d0 at Ta = 750 K.
Fig. 4

A map of flame spread and burning modes of n-decane droplet array in microgravity (Mikami et al. 2006)

Some researchers have tried to utilize findings obtained from fundamental flame-spread researches in order to improve understanding of the flame-spread characteristics in fuel sprays. Nunome et al. (2003) performed experiments of flame spread in quiescent n-decane sprays in microgravity and examined the dependence of the flame-spread rate on the mean droplet diameter considering the flame-spread characteristics of a droplet array. Mikami et al. (2009) elucidated the flame-stabilization mechanism in the counterflow of n-decane/air premixed-spray jet and air considering the dependence of the flame-spread 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 group-combustion excitation in randomly distributed droplet clouds. They developed a percolation model that describes the group-combustion excitation based on Mode 1 flame spread (Fig. 3) and site percolation. The droplets are randomly distributed at lattice points of a three-dimensional 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 flame-spread-limit droplet spacing, (S/d0)limit. The flame cannot spread to droplets outside (S/d0)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 n-decane/air premixed-spray jet and air in detail using a high-speed 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 flame-spread characteristics in microgravity (Oyagi et al. 2009) to study the flame spread and group-combustion excitation in droplet clouds in which the droplets are distributed randomly in three-dimensional square lattices. We focus on droplet clouds with a low-volatility fuel, such as n-decane in room-temperature air, where the pre-vaporization effect is negligible. We also focus on the flame spread in dilute droplet clouds near the group-combustion-excitation limit, where the occurrence probability of group combustion sharply varies with the number density of droplets.

Percolation Model Considering Flame-Spread-Limit Distance

According to the percolation theory, the local connection rule controls macroscopic connection characteristics of randomly distributed particle systems. In the case of basic site percolation, the local connection rule is that particles at adjacent lattice points can connect with each other and make a cluster, as shown in Fig. 5. If the occupation fraction, which is the ratio of the number of particles to the total lattice point number, is relatively small, the cluster size is small as shown in Fig. 5a. As the occupation fraction increases, the cluster size tends to increase and then a large-scale cluster reaching all sides of the lattice appears as shown in Fig. 5b. The occurrence probability of large-scale cluster increases rapidly around a specific occupation fraction as the occupation fraction is increased. This is for the case of a finite size of lattice. As the lattice size is increased up to infinity, the increase function of the occurrence probability of large-scale cluster finally becomes a step function increasing from 0 to 1 at the critical occupation fraction. The critical occupation fraction is 0.312 for a simple three-dimensional square lattice (Stauffer and Aharony 1994).
Fig. 5

Particle connection and appearance of large-scale cluster

Umemura and Takamori (2005) developed a percolation model of flame spread in randomly distributed fuel-droplet clouds with non- or low-volatility fuel. They extended the site percolation by setting the lattice-point 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/d0. On the other hand, there is the flame-spread limit, (S/d0)limit, below which Mode 3 flame spread appears for relatively large S/d0 (Kato et al. 1998, Mikami et al. 2006). Mode 3 flame spread was also observed near the leading flame region of the n-decane premixed-spray flame in counterflow (Mikami et al. 2009). Even if a very low-volatility 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 flame-spread limit, instead of Mode 1 flame spread. Therefore, it is necessary to develop a percolation model based on Mode 3 flame spread considering the flame-spread limit, especially near the critical occupation fraction.

Oyagi et al. (2009) introduced a simple percolation model considering the flame-spread-limit distance. As shown in Fig. 6, a burning droplet has the flame-spread-limit distance (S/d0)limit from the droplet center. The flame cannot spread to the droplets existing outside (S/d0)limit but spreads to the droplets existing inside (S/d0)limit in Mode 3. They used the flame-spread-limit droplet spacing of 14 obtained from flame-spread experiments of n-decane droplet array at room temperature in microgravity (Mikami et al. 2006) as (S/d0)limit. This model can be applied to flame spread over randomly distributed droplet clouds in three-dimensional, two-dimensional, and one-dimensional lattices. Oyagi et al. (2009), however, show the flame spread over randomly distributed droplets in a one-dimensional lattice, which is the simplest randomly distributed droplet system, in order to demonstrate the effect of local droplet interaction on the flame-spread probability. This study applies the percolation model by Oyagi et al. (2009) to the flame spread over randomly distributed droplet clouds in a three-dimensional lattice to study the group-combustion-excitation condition. Furthermore, we study unsteady flame-spread characteristics based on Mode 3 flame spread considering the normalized flame-spread time tf/\(d_{0}^{2}\) obtained from microgravity flame-spread experiments (Mikami et al. 2006).
Fig. 6

Percolation model considering flame-spread time, tf/\(d_{0}^{2}\), and flame-spread-limit distance, (S/d0)limit

Figure 7 shows the definition of the droplet cloud and lattice and flame-spread calculation procedure. Although we employ simple three-dimensional square lattices, a two-dimensional lattice is shown in Fig. 7 for easy understanding of the definition and procedure. M number of fuel droplets with the initial droplet diameter, d0, are randomly arranged at lattice points of an NxNxN square lattice with the lattice-point interval, L. The dimensionless lattice-point interval and lattice size are expressed as L/d0 and NL/d0, respectively. All the droplets on the bottom face are ignited first. If an unburned droplet exists within the flame-spread-limit, (S/d0)limit, of a burning droplet, the unburned droplet is ignited after the flame-spread time, tf/\(d_{0}^{2}\). This procedure is repeated until no more droplets are ignited. We define the group-combustion excitation as the state in which the effect of local droplet combustion, i.e., the burning region, extends beyond the scale of the droplet-cloud size. In the case of group-combustion occurrence, at least one droplet on each face of the lattice except for the bottom face is ignited through the flame spread. Therefore, the definition of group combustion is different from that of Chiu et al. (1971). The calculation is conducted for 1000 different droplet patterns for each occupation fraction or mean droplet spacing, (S/d0)m, to obtain the occurrence probability of group combustion, OPGC. The more the droplet patterns, the smaller the statistical error of OPGC. In the case of OPGC= 0.5, the statistical error is 3.2% for 1000 droplet patterns. The definition of occupation fractions and mean droplet spacing is explained later.
Fig. 7

Definition of droplet cloud and flame-spread calculation procedure

We use (S/d0)limit = 14, which is the flame-spread limit obtained from microgravity experiments on the flame spread of n-decane droplet arrays in atmospheric pressure and room-temperature air (Mikami et al. 2006). We calculate the normalized flame-spread time as tf/\(d_{0}^{2}=S/d_{0}\)/(Vfd0), where we use the experimental values of the flame-spread rate, Vfd0, as a function of the inter-droplet distance, S/d0, obtained in microgravity by Mikami et al. (2006). Here, the dimensionless time and flame-spread rate are respectively expressed as atf/\(d_{0}^{2}\) and Vfd0/a, where a is the thermal diffusivity of air (Mikami et al. 2006).

The occupation fraction in the percolation theory is usually defined as the fraction of number M of particles in the total number N3 of lattice points,
$$ P_{\mathrm{n}}=M/N^{3} $$
(1)
In this percolation model of group-combustion excitation, in which the flame can spread to the droplets inside the flame-spread limit, however, Pn is not appropriate because the critical occupation fraction changes with the lattice-point interval, L/d0. We introduce the occupation fraction Pv of the droplet volume in the total volume of the lattice,
$$ P_{v} =\frac{\pi {d_{0}^{3}}M}{6(NL)^{3}} $$
(2)
As shown in “Occurrence Probability of Group Combustion”, Pv in the critical condition is nearly independent of the lattice characteristics if the lattice-point interval, L/d0, is small and lattice size, NL/d0, is sufficiently large. Pv is the multiplication of the total droplet volume and the number density of droplets in spray. The number density of droplets is often used as one of the spray characteristics and expressed as M/(NL)3. Therefore, the volume occupation fraction, Pv is a dimensionless value considering both the occupation fraction Pn, used in the percolation theory and number density. By considering the liquid-to-gas density ratio, ρL/ρG, and stoichiometric fuel-air ratio, (F/A)st,Pv is related with the overall equivalence ratio φS of premixed-spray as,
$$ \phi_{S} =\left( \frac{P_{v}} {1-P_{v}} \right)\left( \frac{\rho_{L}} {\rho_{G}} \right)\left( \frac{F}{A}\right)_{st}^{-1} $$
(3)
Both Pv and ϕS are functions of the mean droplet spacing, (S/d0)m, which is explained later. In the case of three-dimensionally distributed droplet clouds, the volume occupation fraction Pv is expressed using the mean droplet spacing (S/d0)m as follows:
$$ P_{v} =\frac{\pi} {6}(S/d_{0} )_{m}^{-3} $$
(4)
We introduce the mean droplet spacing, (S/d0)m, of randomly distributed droplet clouds. We assume a new droplet cloud that has the same cloud volume, (NL/d0)3 as that of the randomly distributed droplet cloud and in which the same M number of droplets are arranged at the lattice points of the three-dimensional square lattice without any vacancy with an interval of (S/d0)m. The mean droplet spacing is also defined for droplet clouds randomly distributed on a two-dimensional square lattice The mean droplet spacing (S/d0)m is expressed as follows:
$$\begin{array}{@{}rcl@{}} &&\text{For droplet cloud on a three-dimensional square lattice:} \\&&\qquad (S/d_{0} )_{m} =P_{n}^{-1/3}L/d_{0} \end{array} $$
(5)
$$\begin{array}{@{}rcl@{}} &&\text{For droplet cloud on a two-dimensional square lattice:} \\&&\qquad (S/d_{0} )_{m} =P_{n}^{-1/2}L/d_{0} \end{array} $$
(6)
where the occupation fraction is Pn = M/N3 for droplet clouds on three-dimensional square lattices (Eq. 1) and Pn = M/N2 for droplet clouds on two-dimensional square lattices. Therefore, the mean droplet spacing, (S/d0)m has the advantage that the droplet-cloud characteristic can be expressed using the same dimensionless length (S/d0)m for different dimensions of lattice on which droplets are arranged randomly. The critical mean droplet spacing, (S/d0)mcr, which is the mean droplet spacing in the critical condition, can be compared between the two-dimensionally distributed droplet cloud and the three-dimensionally distributed one. It can also be compared with the flame-spread-limit droplet spacing, (S/d0)limit, of droplet arrays with even droplet spacing. As shown in “Occurrence Probability of Group Combustion”, (S/d0)mcr is nearly independent of lattice characteristics if the lattice-point interval, L/d0, is small and the lattice size, NL/d0, is sufficiently large.

Since the present percolation model employs the flame-spread limit and flame-spread 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 droplet-interaction effect. As shown in Fig. 4 (Mikami et al. 2006), Mode 3 flame spread appears for S/d0> 6 in the flame spread of n-decane droplet arrays at room temperature and atmospheric pressure. For S/d0< 11, however, flames that form around each droplet and separate at ignition merge into a group flame during burning. For 11<S/d0<(S/d0)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/d0)m> 11 in this research. Since the critical mean droplet spacing (S/d0)mcr is larger than (S/d0)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 group-combustion-excitation limit.

Results and Discussion

Occurrence Probability of Group Combustion

Figure 8 shows the dependence of occurrence probability of group combustion, OPGC, on the occupation fraction, Pn, where the lattice-point interval, L/d0, is set to be the same as the flame-spread limit, (S/d0)limit, in order to compare the results with the site percolation. In this case, the flame can spread to unburned droplets only when droplets exist at the nearest lattice points. This is the same as the site percolation. This graph approaches a step function as the lattice size is increased to infinity. As shown in Fig. 8, as the lattice size, NL/d0, is increased, the slope of the graph increases. The graph for NL/d0 = 420 is close to a step function. The critical occupation fraction for a three-dimensional simple square lattice is 0.312 (Stauffer and Aharony 1994). As shown in Fig. 8, OPGC sharply increases around the occupation fraction of this value.
Fig. 8

Dependence of occurrence probability of group combustion on occupation fraction of droplet number in total number of lattice points for different lattice sizes NL/d0 with L/d0 = (S/d0)limit

Figure 9a shows the dependence of OPGC on the occupation fraction of droplet number, Pn, for different lattice-point interval, L/d0. Here, we define the occupation fraction for OPGC= 0.5 as the critical occupation fraction, Pncr. While the flame-spread limit, (S/d0)limit, is constant, Pncr decreases as L/d0 is decreased. Figure 9b shows the dependence of OPGC on the occupation fraction of droplet volume, Pv, for different lattice-point interval, L/d0. As L/d0 is decreased from 14 to 7, the critical occupation fraction of droplet volume, Pvcr, increases greatly. As L/d0 is decreased from 7 to 2, however, Pvcr increases slightly and seems to approach a specific value.
Fig. 9

Dependence of occurrence probability of group combustion on occupation fraction, OPGC, of droplet for different L/d0 (NL/d0 is around 400) aOPGC vs. occupation fraction of droplet number in total number of lattice points bOPGC vs. occupation fraction of droplet volume in total volume of lattice

Figure 10 plots OPGC for the mean droplet spacing, (S/d0)m. As shown in Eq. 4, (S/d0)m is a function of the occupation fraction of droplet volume, Pv, alone. As L/d0 is decreased, the critical mean droplet spacing, (S/d0)mcr, conceivably approaches a specific value, too. This trend is clearly seen in Fig. 11.
Fig. 10

Dependence of occurrence probability of group combustion on mean droplet spacing, (S/d0)m, for different L/d0 (NL/d0 is around 400)

Fig. 11

Dependences of critical mean droplet spacing, (S/d0)mcr, on lattice-point interval, L/d0 (NL/d0 = is around 400)

Figure 12 shows the dependence of the critical mean droplet spacing, (S/d0)mcr, on the lattice size, NL/d0. As NL/d0 is increased, (S/d0)mcr approaches a specific value. (S/d0)mcr seems constant roughly for NL/d0> 150, which is about ten times as large as the flame-spread limit, (S/d0)limit, in this calculation and is a statistically significant cluster size, in which we can describe the flame-spread characteristics over an infinitely large cluster using finite-size clusters.
Fig. 12

Dependences of critical mean droplet spacing, (S/d0)mcr, on lattice size, NL/d0

Umemura and Takamori (2005) discussed the statistically significant lattice size and obtained the requirement condition, VfNL/D >> 1, by comparing the flame-spread time and oxygen diffusion time, where D is the diffusion coefficient. They concluded that NO(10) is sufficient to predict the transition behavior satisfactorily and employed N = 15 while L was set as the maximum flame radius. Since VfNL/D>> 1 can be converted as NL/d0>>(Vfd0/a)− 1 for the unity Lewis number, where the Lewis number is Le = a/D. According to Mikami et al. (2006), the dimensionless flame-spread rate, Vfd0/a, is around 0.1 near the flame-spread limit, (S/d0)limit. Therefore, NL/d0O(100) is required. As shown in Fig. 12, (S/d0)mcr seems nearly constant for NL/d0> 150, which also satisfies the requirement by Umemura and Takamori (2005). In the case of NL/d0 = 150,N is about 11 for L/d0 = 14, which is the same as (S/d0)limit, but 75 for L/d0 = 2. This study employs NL/d0 = 400 as a fully converged condition. Figures 911 also employ NL/d0 = 400. Hereafter, the results for NL/d0 = 400 and L/d0 = 2 are shown.

Figure 13 displays typical distributions of burned and unburned droplets for three different mean droplet spacings, (S/d0)m. Each burned droplet is illustrated with the flame. In the droplet cloud with (S/d0)m = 12.0, which is a relatively dense cloud, the burning area reaches all faces of the lattice and therefore the group combustion is excited as shown in Fig. 13a. On the other hand, in the droplet cloud with (S/d0)m = 18.0, which is a relatively dilute cloud, the combustion terminates on the way to the faces of the lattice and only a partial combustion occurs as shown in Fig. 13c. (S/d0)m = 15.8 is the critical condition of the group-combustion excitation. Although the burning area reaches all faces of the lattice, many unburned droplets are also seen in Fig. 13b.
Fig. 13

Typical distributions of burned and unburned droplets for different mean droplet spacings (NL/d0 = 400, L/d0 = 2)

Figure 14 plots the number density of unburned droplets that remained after completion of burning in the droplet cloud against the mean droplet spacing, (S/d0)m. The graph suggests that the number density of unburned droplets attains maximum around the critical mean droplet spacing, (S/d0)mcr = 15.8. The initial number density of droplets monotonically increases with the decrease in (S/d0)m. OPGC is nearly zero for (S/d0)m>(S/d0)mcr and increases sharply around (S/d0)mcr with the decrease in (S/d0)m. These dependencies contribute to the maximum number density of unburned droplets around (S/d0)mcr.
Fig. 14

Number density of unburned droplets (NL/d0 = 400, L/d0 = 2)

As shown in Fig. 12, the minimum value of statistically significant length scale of droplet clouds is about ten times greater than the flame-spread-limit distance, e.g., about 7 mm for 50 μ m n-decane droplets at atmospheric pressure. Since the flame-spread-limit distance, (S/d0)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/d0)mcr, is a good index for the stable combustion and UHC.

As mentioned in “Percolation Model Considering Flame-Spread-Limit Distance”, the effect of droplet interaction would be relatively small for (S/d0)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 1-D droplet array. The critical condition of the randomly distributed 3-D droplet cloud could also be affected by the local droplet interaction. This effect will be considered in a future study.

Flame-spread Behavior

This section reports flame-spread behavior. As explained in “Percolation Model Considering Flame-Spread-Limit Distance”, we neglect the effect of droplet interaction on the local flame-spread rate and limit because we analyze the flame spread with relatively large mean droplet spacing, (S/d0)m, where the effect of droplet interaction is relatively small. Thus, the local flame-spread rate, Vfd0, is assumed to be a function of dimensionless droplet spacing, S/d0, 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.

Figures 1516 and 17 respectively show the sequential figures of flame spread over droplet clouds with three different mean droplet spacings, (S/d0)m, the ignition time of each droplet, and the z-direction position of each ignited droplet vs. time. z-direction is perpendicular to the bottom face of the lattice. As shown in Fig. 15a, the flame spreads directly from the bottom face to the top face in the case of (S/d0)m = 12. This trend is also clearly seen in Figs. 16a and 17a. The flame reaches the top face, z/d0 = 400, at \(t/d_{0}^{2}=\) 45 s/mm2. On the other hand, in the case of (S/d0)m = 18, the flame spread terminates in the lower part of the droplet cluster. In the case of (S/d0)m = 15.8, which is the critical condition, the flame spreads through complicated paths and finally reaches all faces of the lattice. As shown in Figs. 16b and 17b, the flame spread starts from the bottom face, reaches close to the top face, then goes back and reaches close to the bottom face around \(t/d_{0}^{2}=\) 160 s/mm2. This result clearly shows that the flame in the critical condition of droplet cluster takes a long time to spread around.
Fig. 15

Flame-spread behavior for different mean droplet spacings (NL/d0 = 400, L/d0 = 2) a Dense droplet cloud, (S/d0)m = 12 b Critical condition, (S/d0)m = 15.8 c Dilute droplet cloud, (S/d0)m = 18

Fig. 16

Ignition time of each droplet during flame spread for different mean droplet spacings (NL/d0 = 400, L/d0 = 2)

Fig. 17

z-direction position of each ignited droplet during flame spread for different mean droplet spacings (NL/d0 = 400, L/d0 = 2) a Dense droplet cloud, (S/d0)m = 12 b Critical condition, (S/d0)m = 15.8 c Dilute droplet cloud, (S/d0)m = 18

Figure 18 plots the final ignition time in the flame spread of randomly distributed droplet clouds against the mean droplet spacing, (S/d0)m. The final ignition time has a sharp peak and attains maximum around the critical condition, (S/d0)mcr = 15.8. According to the percolation theory, the characteristic time scale diverges to infinity at the critical condition (Stauffer and Aharony 1994). Figure 18 clearly demonstrates that a characteristic time scale of flame spread over droplet clouds has a very large value in the critical condition, as suggested by the percolation theory.
Fig. 18

Dependence of final ignition time on mean droplet spacing (S/d0)m (NL/d0 = 400, L/d0 = 2). The red circle shows the averaged value

In the critical condition, some droplets have a very large ignition time over 150 s/mm2. In such a case with a relatively long waiting time for ignition, the pre-vaporization might be significant. Here, we estimate the droplet diameter change through 150 s/mm2 pre-vaporization. n-Decane is a low-volatility 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 pre-vaporization of the n-decane droplet does not produce a flammable mixture around the droplet in this condition. The vaporization-rate constant at room temperature in microgravity is estimated to be less than 0.0006 mm2/s based on our preliminary experiment. Considering the estimated pre-vaporization with this vaporization-rate constant in microgravity, the droplet diameter decrease is estimated to be very small, less than 5% even with the 150 s/mm2 pre-vaporization. When the spray burns at high temperatures as in diesel engines and jet engines, however, the pre-vaporization effect could be significant. Mikami et. al. (2006) conducted flame-spread experiments using linear n-decane droplet arrays in microgravity and reported that the flame-spread rate and limit increase with the ambient temperature. If the auto-ignition 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 pre-vaporization. This effect will be considered in a future study.

Figure 19 plots the overall flame-spread rate against the mean droplet spacing, (S/d0)m. We calculate the overall flame-spread rate as the lattice size divided by the time when the flame reaches the top face for the first time. If the flame does not reach the top face during the burning lifetime of any droplets, the overall flame-spread rate is zero. As shown in Fig. 19, the overall flame-spread rate decreases with the increase in (S/d0)m and scatters largely near the critical mean droplet spacing, (S/d0)mcr = 15.8.
Fig. 19

Dependence of overall flame-spread rates on mean droplet spacing (S/d0)m (NL/d0 = 400, L/d0 = 2). The flame-spread rate of linear droplet array in microgravity (Mikami et al. 2006) is also plotted

Figure 19 also plots the flame-spread rate of linear n-decane droplet arrays in microgravity (Mikami et al. 2006). The overall flame-spread rate of randomly distributed droplet clouds is almost the same as the flame-spread rate of a linear droplet array except over the flame-spread limit. As explained in “Occurrence Probability of Group Combustion”, the critical mean droplet spacing, (S/d0)mcr = 15.8, is greater than the flame-spread limit of a linear droplet array, (S/d0)limit = 14. Mikami et al. (2009) discussed the stabilized position of the leading flame of the n-decane spray/air mixture in counter flow and concluded that it can be predicted qualitatively by using the flame-spread rate of a linear droplet array obtained in microgravity as the overall flame-spread rate. As confirmed by Fig. 19, the overall flame-spread rate is not affected by the randomness of droplet distribution except over the flame-spread limit. The quantitative prediction of the overall flame-spread rate of spray in counter flow requires consideration of the effects of gas-liquid relative velocity and droplet-size distribution.

This study does not consider the effects of droplet interaction for relatively large mean droplet spacing, (S/d0)m. If (S/d0)m becomes smaller, the cooling effect of unburned droplets due to increased droplet interaction would become larger and make the overall flame-spread rate smaller than the flame-spread rate of a linear droplet array. If (S/d0)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 flame-spread limit and group-combustion-excitation limit, whereas this study focuses on the lean limit of the flame spread and group-combustion excitation in droplet clouds. Mikami et al. (2005a) observed the double flame structure of premixed n-decane spray/air jet and discussed the characteristics of the internal flame based on the flame-spread 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 group-combustion excitation in droplet clouds. To model such a rich limit of flame spread and group-combustion 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 group-combustion excitation based on findings obtained from microgravity experiments on the flame-spread of fuel droplets. This percolation model is based on Mode 3 flame spread. We focused on droplet clouds distributed randomly in three-dimensional square lattices with a low-volatility fuel, such as n-decane in room-temperature air, where the pre-vaporization effect is negligible. We also focused on the flame spread in dilute droplet clouds near the group-combustion-excitation 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 lattice-point interval is small and the lattice size is sufficiently large. If the lattice size is at smallest about ten times as large as the flame-spread limit distance, the flame-spread 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 flame-spread rate of randomly distributed droplet clouds is almost the same as the flame-spread rate of a linear droplet array except over the flame-spread 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 gas-liquid relative velocity, droplet-size distribution, droplet interaction and pre-vaporization. This study focuses on the lean limit of the flame spread and group-combustion 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 Grant-in-Aid 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.

References

  1. Chiu, H. H., Liu, T. M.: Group combustion of liquid droplets. Combust. Sci. Technol. 17, 127–142 (1971)CrossRefGoogle Scholar
  2. Chiu, H. H., Kim, H. Y., Croke, E. J.: Internal group combustion of liquid droplets. Proc. Combust. Inst. 19, 971–980 (1983)CrossRefGoogle Scholar
  3. Correa, S. M., Sichel, M.: The group combustion of a spherical cloud of monodisperse fuel droplets. Proc. Combust. Inst. 19, 981–991 (1982)CrossRefGoogle Scholar
  4. Imaoka, R. T., Sirignano, W. A.: Vaporization and combustion in three-dimensional droplet arrays. Proc. Combust. Inst. 30, 1981–1990 (2005a)CrossRefGoogle Scholar
  5. Imaoka, R. T., Sirignano, W. A.: Transient vaporization and burning in dense droplet arrays. Int. J. Heat Mass Transfer 48, 4354–4366 (2005b)CrossRefzbMATHGoogle Scholar
  6. Kato, S., Kobayashi, H., Mizuno, H., Niioka, T.: Experiments on flame spread of a fuel droplet array in a high-pressure ambience. JSME Int. J. B 41, 322–330 (1998)CrossRefGoogle Scholar
  7. Kikuchi, M., Arai, T., Yoda, S., Tsukamoto, T., Umemura, A., Uchida, M., Niioka, T.: Numerical study on flame propagation of a fuel droplet array in high temperature environment under microgravity. Proc. Combust. Inst. 29, 2611–2619 (2002)CrossRefGoogle Scholar
  8. Kikuchi, M., Wakashima, Y., Yoda, S., Mikami, M.: Numerical study on flame spread of an n-decane droplet array in different temperature environment under microgravity. Proc. Combust. Inst. 30, 2001–2009 (2005)CrossRefGoogle Scholar
  9. Labowsky, M., Rosener, D. E.: Group combustion of droplets in fuel clouds. I. Quasi-steady predictions. Adv. Chem. 166, 63–79 (1978)CrossRefGoogle Scholar
  10. Mikami, M., Yamamoto, K., Kojima, N.: Combustion of partially premixed spray jets. Proc. Combust. Inst. 30, 2021–2028 (2005a)CrossRefGoogle Scholar
  11. Mikami, M., Oyagi, H., Kojima, N., Kikuchi, M., Wakashima, Y., Yoda, S.: Microgravity experiments on flame spread along fuel-droplet arrays using a new droplet-generation technique. Combust. Flame 141, 241–252 (2005b)CrossRefGoogle Scholar
  12. Mikami, M., Oyagi, H., Kojima, N., Wakashima, Y., Kikuchi, M., Yoda, S.: Microgravity experiments on flame spread along fuel-droplet arrays at high temperatures. Combust. Flame 146, 391–406 (2006)CrossRefGoogle Scholar
  13. Mikami, M., Nakamoto, K., Kojima, N., Moriue, O.: Effects of overall equivalence ratio on flame structure of rich-premixed n-decane spray jet. J. Combust. Soc. Jpn. 50, 248–254 (2008). (in Japanese)Google Scholar
  14. Mikami, M., Mizuta, Y., Tsuchida, Y., Kojima, N.: Flame structure and stabilization of lean-premixed sprays in a counterflow with low-volatility fuel. Proc. Combust. Inst. 32, 2223–2230 (2009)CrossRefGoogle Scholar
  15. Nomura, H., Iwasaki, H., Suganuma, Y., Mikami, M., Kikuchi, M.: Microgravity experiments of flame spreading along a fuel droplet array in fuel vapor–air mixture. Proc. Combust. Inst. 33, 2013–2020 (2011)CrossRefGoogle Scholar
  16. Nunome, Y., Kato, Y., Maruta, K., Kobayashi, H., Niioka, T.: Flame propagation of n-decane spray in microgravity. Proc. Combust. Inst. 29, 2621–2626 (2003)CrossRefGoogle Scholar
  17. Okajima, S., Kimoto, T., Abe, K., Yamaguchi, S.: Experimental study on flame propagation of fuel droplet array under a zero gravity condition. JSME Trans. B 47, 2058–2065 (1981). (in Japanese)CrossRefGoogle Scholar
  18. Oyagi, H., Shigeno, H., Mikami, M., Kojima, N.: Flame-spread probability and local interactive effects in randomly arranged fuel-droplet arrays in microgravity. Combust. Flame 156, 763–770 (2009)CrossRefGoogle Scholar
  19. Ryan, W., Annamalai, K., Caton, J.: Relation between group combustion and drop array studies. Combust. Flame 80, 313–321 (1990)CrossRefGoogle Scholar
  20. Sano, N., Motomatsu, N., Saputro, H., Seo, T., Mikami, M.: Flame-spread characteristics of n-decane droplet arrays at different ambient pressures in microgravity. Int. J. Microgravity Sci. Appl. 330108, 33 (2016)Google Scholar
  21. Stauffer, D., Aharony, A.: Introduction to Percolation Theory, Revised 2nd edn., CRC Press (1994)Google Scholar
  22. Umemura, A.: Interactive droplet vaporization and combustion: Approach from asymptotics. Prog. Energy Combust. Sci. 20, 325–372 (1994)CrossRefGoogle Scholar
  23. Umemura, A.: Flame propagation along a linear array of liquid fuel droplets under micro-gravity condition (1st report, Inter-droplet flame propagation mode map). JSME Trans. B 68, 2422–2428 (2002a). (in Japanese)CrossRefGoogle Scholar
  24. Umemura, A.: Flame propagation along a linear array of liquid fuel droplets under micro-gravity condition (2nd report, Flame propagation speed characteristics). JSME Trans. B 68, 2429–2436 (2002b). (in Japanese)CrossRefGoogle Scholar
  25. Umemura, A., Takamori, S.: Percolation theory for flame propagation in non- or less-volatile fuel spray: A conceptual analysis to group combustion excitation mechanism. Combust. Flame 141, 336–349 (2005)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018
corrected publication March/2018

Authors and Affiliations

  • Masato Mikami
    • 1
  • Herman Saputro
    • 2
  • Takehiko Seo
    • 1
  • Hiroshi Oyagi
    • 1
  1. 1.Department of Mechanical EngineeringYamaguchi UniversityUbeJapan
  2. 2.Mechanical Engineering Education DepartmentSebelas Maret UniversityPabelanIndonesia

Personalised recommendations