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Flame-spread probability and local interactive effects in randomly arranged fuel-droplet arrays in microgravity
Combustion and Flame 156 (2009) 763–770
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Combustion and Flame www.elsevier.com/locate/combustflame
Flame-spread probability and local interactive effects in randomly arranged fuel-droplet arrays in microgravity Hiroshi Oyagi, Hisashi Shigeno, Masato Mikami ∗ , Naoya Kojima Graduate School of Science and Engineering, Yamaguchi University, 2-16-1 Tokiwadai, Ube 755-8611, Japan
a r t i c l e
i n f o
Article history: Received 31 October 2007 Received in revised form 13 December 2008 Accepted 16 December 2008 Available online 14 February 2009 Keywords: Fuel-droplet arrays Flame spread Unevenly-spaced droplet array Flame-spread probability Droplet interaction Microgravity
a b s t r a c t We investigated the flame-spread characteristics of randomly arranged fuel-droplet arrays in microgravity. Flame-spread probability was calculated based on a percolation model with the flame-spread-limit distance of evenly-spaced n-decane droplet arrays in microgravity. Flame-spread probability depends on the occupation fraction of droplets in a lattice and rapidly increases with the occupation fraction. The local flame-spread-limit distance of unevenly-spaced n-decane droplet arrays was experimentally investigated in microgravity. The droplets were arranged in a straight line at uneven intervals. The local flame-spread-limit distance of the unevenly-spaced droplet arrays depended on the droplet arrangement and increased from the flame-spread-limit distance of the evenly-spaced droplet arrays due to interactive effects. The flame-spread probability considering the increase in local flame-spread-limit distance is larger than that without it. © 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction Flame spread in fuel spray occurs immediately after ignition in diesel engines and occurs near the flame base in gas turbine engines. Flame spread plays an important role in heat release in the initial stage in the former case and in the stability of flame in the latter case. In order to improve understanding of the flame-spread mechanism, the flame spread along evenly-spaced droplet arrays has been investigated experimentally [1–8], theoretically [9,10] and numerically [11,12]. In spray combustion, however, fuel droplets distribute at random in the combustion chamber. The findings of researches on flame spread along droplet arrays have not been well utilized for elucidation of spray combustion. The present paper investigates the flame-spread characteristics of randomly arranged droplets using a percolation model based on the findings of microgravity experiments on flame spread along droplet arrays. Percolation theory gives us probabilistic characteristics of connections of particles randomly arranged in a lattice [13]. In the case of a small number of particles, the size of particle connection is small, as shown in Fig. 1(a). But, in the case of a large number of particles, as shown in Fig. 1(b), the size of particle connection is large and a large cluster is formed. Some studies have applied the percolation theory to the elucidation of spray combustion [14–16]. When percolation theory is applied to spray combustion, a droplet is characterized as a particle. Kerstein
*
Corresponding author. Fax: +81 836 85 9101. E-mail address: [email protected] (M. Mikami).
et al. analyzed the percolate combustion zone in non-premixed sprays using percolation theory [14,15]. Umemura and Takamori [16] calculated the flame-spread probability of a randomly distributed droplet cloud based on Mode 1 flame spread, in which the vaporization of the next droplet becomes activated after the leading edge of an expanding group diffusion flame passes the next droplet and pushes the leading flame forward [7,9,12]. However, microgravity experiments on flame spread along droplet arrays show that Mode 1 flame spread occurs only for a small droplet spacing S /d0 [7], where S is the droplet spacing and d0 is the initial droplet diameter. For a relatively large S /d0 , Mode 3 flame spread occurs, in which the next droplet autoignites through heating by the diffusion flame, whose leading edge does not reach the flammable-mixture layer around the next droplet [7,9,12]. In view of the general site percolation, only when two particles exist on adjacent lattice points, these two particles are connected as shown in Fig. 1 [13]. So, the inter-particle distance is meaningless. In flame spread, however, the dimensionless droplet spacing S /d0 is important. Mikami et al. [7] have shown that there exists a flame-spread limit ( S /d0 )limit in terms of the dimensionless droplet separation distance, below which the leading flame can spread to the next droplet, but over which it cannot spread. Therefore, the flame-spread-limit distance is the threshold of the connection. The present paper introduces a percolation model considering the flame-spread-limit distance based on Mode 3 flame spread, as shown in Fig. 2. In spray combustion, if the droplet number density is small, the leading flame spreads with difficulty, as shown in Fig. 3(a). But, if the droplet number density is large enough, the leading flame spreads easily to the next un-
0010-2180/$ – see front matter © 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2008.12.013
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(a) Small number of particle
(a) Droplet spacing between burned droplet and unburned droplet is less than flame-spread-limit distance
(b) Droplet spacing between burned droplet and unburned droplet is larger than flame-spread-limit distance Fig. 2. Percolation model considering local flame-spread-limit distance. (b) Large number of particle Fig. 1. Two different site percolation images.
burned droplet, and the flame spreads to droplets on all the lattice sides, resulting in group flame formation, as shown in Fig. 3(b). If the local droplet spacing is small enough, droplet interaction occurs [17–19], which affects the local flame-spread characteristics. In order to elucidate it, we experimentally investigated the local flame-spread-limit distance of unevenly-spaced droplet arrays in microgravity. The flame-spread probability of randomly arranged droplet arrays was calculated considering the local flame-spreadlimit distance of unevenly-spaced droplet arrays. 2. Flame-spread probability based on the flame-spread limit of evenly-spaced droplet arrays This section introduces a percolation model considering the flame-spread-limit distance and demonstrates flame-spread characteristics of randomly arranged droplets for the simplest droplet system with the flame-spread-limit distance ( S /d0 )limit of evenlyspaced droplet arrays in microgravity. The flame-spread-limit distance along evenly-spaced droplet arrays was investigated in microgravity [3,7]. Mikami et al. [7] reported that the flame-spread-limit distance ( S /d0 )limit of evenlyspaced droplet arrays is about 14 in 300 K air with n-decane as a fuel. As shown in Fig. 4(a), if every droplet spacing is less than the flame-spread-limit distance, the flame can spread to the last droplet. On the other hand, if the droplet spacing is larger than the flame-spread-limit distance, the flame cannot spread to the next droplet, as shown in Fig. 4(b). We introduced a simple criterion for local flame spread; for S /d0 smaller than ( S /d0 )limit ,
the flame can spread to the next droplet, but for S /d0 larger than ( S /d0 )limit , the flame cannot spread to the next droplet, as shown in Fig. 2. In Fig. 4(b), the mean droplet spacing is the same as that in Fig. 4(a), so the occupation fraction of droplets in a lattice is the same. The occupation fraction is the ratio of the number of droplets to the total number of lattice points. In three-dimensional droplet clouds, the droplet number density and equivalence ratio are proportional to the occupation fraction. There are many droplet patterns for the same occupation fraction of droplets in a lattice. The flame-spread probability is the proportion of the pattern number showing flame spread to droplets on all sides of the lattice to the total pattern number. We show the flame-spread probability of randomly arranged droplet arrays as the droplet system with the lowest dimension showing flame spread. In this calculation, as shown in Fig. 5, M droplets are randomly arranged at N lattice points, so the occupation fraction of droplets is M / N. Two imaginary droplets are also placed next to both edges of the lattice; the left imaginary droplet is for ignition and the right one is for judging the flame spread to the end of the lattice. The flame can spread to the last droplet only when the local droplet spacing is less then the flame-spread limit ( S /d0 )limit . Fig. 6 shows the flame-spread probability for the number of lattice points N = 100 and dimensionless lattice-point interval L /d0 = 2. The flame-spread-limit distance of evenly-spaced droplet array, ( S /d0 )limit = 14 [7] was used and prevaporization of unburned droplets was ignored. Calculation was conducted for 3000 different droplet patterns at each occupation fraction. As shown in Fig. 6, the flame-spread probability rapidly increases with the occupation fraction. In the general site percolation theory, only when particles exist on adjacent lattice points, particles connect
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Fig. 5. Droplet array model.
(a) Small number of droplets
Fig. 6. Dependence of flame-spread probability on occupation fraction of droplets in a lattice (N = 100, L /d0 = 2, ( S /d0 )limit = 14).
(b) Large number of droplets Fig. 3. Two different percolation images considering the local flame-spread-limit distance.
(a) Every droplet spacing less than flame-spread-limit distance
array. In spray, however, droplet spacing is uneven. The flamespread-limit distance of unevenly-spaced droplet arrays is possibly different from that of evenly-spaced droplet arrays. Researches on the interactive effect of two droplets reported that the burning rate, burning time and flame radius depend on the droplet spacing [17–19]. When the droplet spacing is small, a group flame forms around the two droplets and the flame radius of the group flame is larger than that of the single droplet. When the droplet spacing is large, however, an individual flame forms around each droplet. The following sections describe our experimental investigation of the local flame-spread-limit distance in microgravity. In addition, the influence of the local flame-spread-limit distance on the flamespread probability is investigated. 3. Experimental apparatus and method
(b) One droplet spacing larger than flame-spread-limit distance Fig. 4. Typical flame-spread behavior for two different flame-spread patterns in microgravity for mean droplet spacing ( S /d0 )m = 8. The white circles are inserted to show the droplet position.
to each other. So, all the particles arranged in one dimension connect only when the occupation fraction of particles in a lattice is unity [13]. However, the present percolation model considering the flame-spread-limit distance can explain the flame-spread probability of randomly arranged droplets even if the droplets are arranged in one dimension. In this section, the flame-spread probability was calculated based on the flame-spread-limit distance of evenly-spaced droplet
Fig. 7 represents the droplet array generation apparatus. This apparatus was designed based on the multiple droplets generation technique proposed by Mikami et al. [6]. It was composed of a droplet-array holder, a fuel-supply system, a droplet array generator, a hot wire ignitor and a droplet remover. The droplets are suspended on the intersection of X-shape SiC fibers of 14 μm in diameter. n-Decane was supplied to the intersections of the fibers through fine glass tubes whose outer diameter at the tip is 40 μm. The multiple droplets generation technique is described in detail in Ref. [6]. There are two differences between this apparatus and the apparatus in Ref. [6]. First, each glass tube was connected through a Teflon tube to the corresponding syringe pushed by a steppingmotor-driven stage. Second, a droplet remover was used. Before droplet generation, the fuel was supplied to glass tube tips, and surplus fuel was removed by the droplet remover. In most cases, the initial droplet diameter d0 was set to 1.0 mm for different combinations of droplet spacing. In order to vary the dimensionless droplet separations S /d0 finely, the initial droplet diameter var-
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ied from 0.93 to 1.14 mm. If there exists one-pulse error in the displacement of the stepping-motor driven stage, the droplet diameter has about 1.1% error for a 1 mm droplet. The microgravity experiments were performed at atmospheric pressure and at room temperature in Micro-Gravity Laboratory Japan (MGLAB). The microgravity duration is 4.5 s and the microgravity level is 10−5 G. Flame-spread behavior was recorded by a digital video camera (SONY, DCR-HC90), whose framing rate is 30 fps. The experimental sequence is as follow: (1) Fuel was supplied to the glass tube tips. (2) Surplus fuel was removed by gas jet from the droplet remover. (3) The glass tube tips were brought to the intersections of SiC fibers. (4) The fuel was supplied to generate the droplets. (5) The drop capsule that carried the experimental apparatus was dropped.
Fig. 7. Droplet array generator.
Fig. 8. Droplet array for examining the effect of two-droplet interaction on the local flame-spread-limit distance. This pattern is called Pattern S BA /d0 –S AL /d0 .
(6) The glass tubes were drawn back in microgravity. (7) The droplet at one end of the array was ignited by a hot-wire ignitor. (8) Flame-spread behavior occurred and was recorded. 4. Local flame-spread limit of unevenly-spaced droplet arrays This section shows the experimental results of the local flamespread limit of unevenly-spaced droplet arrays in microgravity. 4.1. Local flame-spread limit influenced by two-droplet interaction Fig. 8 shows the definition of the droplet array for examining the effect of two-droplet interaction on the local flame-spreadlimit distance. Four droplets were placed linearly with uneven droplet spacing. The flame spread started by igniting Droplet I. We investigated whether the leading flame spreads from Droplet A to Droplet L, with the interaction between Droplet B and Droplet A. In this research, the droplet array pattern as shown in Fig. 8 was called Pattern S BA /d0 –S AL /d0 . For example, if the dimensionless droplet spacing between Droplet B and Droplet A S BA /d0 is 2.0 and that between Droplet A and Droplet L S AL /d0 is 16, the pattern name is 2.0–16. The droplet spacing between Droplet B and Droplet A S BA was varied from 2 mm to 14 mm, and that between Droplet A and L S AL was varied from 14 mm to 18 mm. Initial-droplet diameter d0 was varied from 0.93 mm to 1.14 mm. Hereby, the dimensionless droplet spacing between Droplet B and Droplet A S BA /d0 was varied from 1.9 to 14, and that between Droplet A and Droplet L S AL /d0 was varied from 14 to 18. Droplet I was used to reduce the disturbance from the hot-wire igniter. The droplet spacing between Droplet I and Droplet B S IB was 12 mm, so the dimensionless droplet spacing between Droplet I and B S IB /d0 was from 10.5 to 12.9. Mikami et al. [7] reported that individual flames occurred in two-droplet combustion in microgravity for S /d0 > 11. The influence of the hot-wire igniter on Droplet B, A, and L was small. S IB /d0 should not be too large due to the limitation of the microgravity duration as well as the existence of the flame-spread-limit distance ( S /d0 )limit = 14 [7]. Fig. 9 shows the flame-spread behavior for different droplet patterns. Elapsed time t /d20 is from the ignition of Droplet B. In Pattern 1.9–17 shown in Fig. 9(a), a large group flame formed around Droplet B and Droplet A. It did not spread to Droplet L. In Pattern 2.0–16 and Pattern 4.0–16 shown in Figs. 9(b) and 9(c), the flame spread to Droplet L. But, if S BA /d0 was increased to 6.0, the flame did not spread to Droplet L as shown in Fig. 9(d). Thus, the local flame-spread-limit distance ( S AL /d0 )limit was influenced by S BA /d0 . Fig. 10 shows the local flame-spread-limit dis-
(a) Pattern 1.9–17
(b) Pattern 2.0–16
(c) Pattern 4.0–16
(d) Pattern 6.0–16
Fig. 9. Typical flame-spread behavior for different droplet-array patterns in the case of two-droplet interaction. The white circles are inserted to show the droplet position.
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Fig. 12. Droplet array for examining the effect of three-droplet interaction on the local flame-spread-limit distance. This pattern is called Pattern S CB /d0 –S BA / d0 –S AL /d0 .
Fig. 10. Flame-spread-limit distance of fuel-droplet arrays with uneven droplet spacing in the case of two-droplet interaction. The flame-spread-limit distance with even droplet spacing [7] is also shown.
the flame spread to Droplet L requires heating of Droplet L by heat from the flame surrounding Droplet A. In the near field of burning Droplet A, nearly quasi-steady field forms where the convection term and the diffusion term are of the same order. In the far field, however, unsteady field forms where the diffusion term and the unsteady term are of the same order. The thermal layer affecting flame spread to Droplet L exists in the far field, and therefore, the temperature in the thermal layer is transient in nature and does not approach a quasi-steady limit asymptotically. The thermal layer initially develops with time, but decreases eventually. The radius of the thermal layer is therefore finite, resulting in the flame-spread limit ( S AL /d0 )limit . In unsteady droplet combustion, the mixture fraction Z in the far field is expressed as, Z=
Mf
ρ (4π at )3/2
exp −
r2
4at
,
according to analyses by Spalding [20] and Crespo and Liñan [21]. Here, the temperature T has the linear relationship with Z . The radius of the thermal layer is a function of time as
r C = 6at ln
ρ ZC
−1 1/2
2/ 3
(4π at )
Mf
,
where a is thermal diffusivity, ρ is the gas density, M f is the initial fuel-droplet mass, and Z C is the mixture fraction corresponding to the representative temperature in the thermal layer. rC attains a maximum value
rCm = Fig. 11. Maximum flame radiuses of Droplet A.
tance of unevenly-spaced droplet arrays of all the experimental results. The flame-spread-limit distance of evenly-spaced droplet arrays [7] is also shown. As shown in Fig. 10, when S BA /d0 is less than 6, the local flame-spread-limit distance ( S AL /d0 )limit increased with decreasing S BA /d0 . For S BA /d0 > 6, ( S AL /d0 )limit is close to the flame-spread-limit distance of evenly-spaced droplet arrays ( S /d0 )limit . Fig. 11 shows the maximum radiuses of a luminous and a blue flame of Droplet A, which were measured from the center of Droplet A in the direction to Droplet L. If S BA /d0 is less than 6, the maximum flame radiuses of Droplet A increased with decreasing S BA /d0 . This is similar dependence to the local flame-spread limit shown in Fig. 10. Since the flame diameter becomes larger due to the interactive effect between droplets [18,19], the flame-spread-limit distance might also become larger due to the interactive effect. We will discuss the flame-spread-limit distance in view of the droplet interaction in the following. The equivalence ratio of fuel vapor/air mixture at the n-decanedroplet surface is about 0.1 at 300 K, which is much less than the lean flammability limit [7]. Therefore, there is no flammable mixture around unburned n-decane droplets at room temperature, so
2π e
at time t Cm =
1/2
3
1 4π ea
1/3
Mf
ρ ZC Mf
ρ ZC
2/ 3 .
These equations show that both rCm and t Cm increase with M f . If there exists another droplet near one droplet inside the flame, the equivalent initial mass increases as compared to the single droplet combustion, and the maximum radius rCm of the thermal layer increases, resulting in the increase in the flame-spread-limit distance. 4.2. Local flame-spread limit influenced by three-droplet interaction This section investigates the influence of three-droplet interaction. Fig. 12 shows the definition of the droplet array configuration for examining the effect of three-droplet interaction on the local flame-spread-limit distance. Five droplets were placed linearly. The droplet array pattern as shown in Fig. 12 is called Pattern S CB /d0 –S BA /d0 –S AL /d0 . For example, if S CB /d0 is 1.9, S BA /d0 is 1.9 and S AL /d0 is 17, the pattern name is 1.9–1.9–17. S AL was 18 mm and the initial droplet diameter d0 was 1.06 mm under all conditions. So, S AL /d0 was 17, for which the flame did not spread
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(a) Pattern 1.9–1.9–17
(b) Pattern 3.8–1.9–17
(c) Pattern 5.7–1.9–17
(d) Pattern 7.5–1.9–17
(e) Pattern 9.4–1.9–17 Fig. 13. Flame-spread behavior of different droplet-array patterns in the case of three-droplet interaction. The white circles are inserted to show the droplet position.
(a) Pattern 1.9–3.8–17
(b) Pattern 3.8–3.8–17 Fig. 14. Flame-spread behavior of different droplet-array patterns in the case of three-droplet interaction. The white circles are inserted to show the droplet position.
to Droplet L with two-droplet interaction as shown in Figs. 9 and 10. For S BA = 2 mm, S CB was varied from 2 mm to 10 mm. For S BA = 4 mm, S CB was varied from 2 mm to 4 mm. In most cases, the droplet spacing between Droplet I and Droplet C S IC was 12 mm. In Pattern 9.4–1.9–17, however, S IC was 10 mm. The reason for this is to secure sufficient time to investigate whether the flame can spread to Droplet L in the limited microgravity duration of 4.5 s. Fig. 13 shows the flame-spread behavior for S BA /d0 = 1.9 and S AL /d0 = 17. S CB /d0 was varied to investigate the effect of Droplet C. Elapsed time t /d20 is the time from the ignition of Droplet C. In Pattern 1.9–1.9–17 and 9.4–1.9–17, the flame did not spread to Droplet L. In Pattern 3.8–1.9–17, 5.7–1.9–17 and 7.5– 1.9–17, however, the flame spread to Droplet L. S AL /d0 = 17 is larger than the flame-spread limit ( S AL /d0 )limit for S BA /d0 = 1.9 with two-droplet interaction between Droplet B and Droplet A as shown in Fig. 10. Fig. 14 shows the flame-spread behavior for S BA /d0 = 3.8 and S AL /d0 = 17. In these patterns, the flame did not spread to Droplet L. Fig. 15 shows the local flame-spread limit in S CB /d0 –S BA /d0 plane for S AL /d0 = 17. As shown in Fig. 15, there is a combination of S CB /d0 and S BA /d0 to increase the flame-spread
Fig. 15. Flame-spread modes of fuel-droplet arrays with uneven droplet spacing in the case of three-droplet interaction for S AL /d0 = 17.
limit ( S AL /d0 )limit . It is interesting that for Pattern 1.9–1.9–17, the flame did not spread to Droplet L although this pattern seems to have the strongest interaction. Fig. 16 shows the maximum blue-flame radius of Droplet A, R A , which was measured from the center of Droplet A to Droplet L. R A for the cases that the flame spread to Droplet L is somewhat larger than the others. So, R A for Pattern 1.9–1.9–17 is smaller than Pattern 3.8–1.9–17. This is possibly related to the flame-spread mode around Droplets C, B and A. The relation between the flame-spread-limit distance and the droplet pattern is examined in view of the flame-spread mode, which is introduced by Umemura [9]. First, in Pattern 1.9–1.9– 17, the flame-spread mode from Droplet C to Droplet B was Mode 1, in which vaporization of the next unburned droplet is activated after the diffusion flame passes it and then the active vaporization pushes the leading flame forward. In Patterns 3.8– 1.9–17 and 5.7–1.9–17, the flame-spread mode from Droplet C to
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Droplet B was Mode 2, in which the flammable mixture layer forms around the unburned droplet and the premixed flame propagates in the flammable mixture layer immediately after the leading flame reaches the flammable mixture. Since the flammable mixture layer forms before the flame spreads to the unburned droplet in Mode 2, the initial flame radius with the Mode 2 flame spread is larger than that with the Mode 1 flame spread. The flamespread limit ( S AL /d0 )limit possibly increases with the initial flame radius of Droplet A. Hereby, the local flame-spread-limit distance after the Mode 2 flame spread became larger for the same S BA /d0 . In Pattern 7.5–1.9–17, the flame-spread mode from Droplet C to Droplet B is Mode 3, in which the next unburned droplet autoignites through heating by the leading flame, which does not reach the flammable mixture layer around the next unburned droplet. Since the flammable mixture layer formed before the flame spreads to the unburned droplet in Mode 3, the initial flame radius with Mode 3 is larger than that with Mode 1, too. In Patten 9.4–1.9–17, the flame-spread mode from Droplet C to Droplet B was Mode 3. However, the flame did not spread to Droplet L. In this pattern, since the droplet spacing between Droplet C and Droplet B is too large, the interaction between Droplet C and B is small. Hereby, the local flame-spread-limit distance of unevenly-spaced droplet arrays is influenced by the flame-spread mode as well as the interaction effect. In Pattern 3.8–3.8–17, interaction between Droplet B and Droplet A is probably not large enough although the flame-spread mode from Droplet C to Droplet B is Mode 2. Nomura et al. [8] investigated the gas flow caused by droplet ignition during the flame spread along n-heptane droplet arrays in normal gravity. They reported that burned-gas flow was caused by the droplet ignition and its influence extended 12 times further than the initial droplet diameter. For higher-volatility fuel, the flammable mixture layer around the unburned droplet is thicker, so the burned gas flow by the premixed-flame propagation through the mixture layer is stronger. In such a case, convective heat transfer would play an important role in the flame-spread limit.
Fig. 16. Maximum flame radius of Droplet A.
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5. Flame-spread probability based on local flame-spread limit of unevenly-spaced droplet array The local flame-spread-limit distance increases by interaction effect and flame-spread mode. In this section, the flame-spread probability considering the local flame-spread-limit distance of unevenly-spaced droplet arrays is calculated. First, the appearance frequency of some droplet-array patterns is investigated. In the case of lattice number N = 100 and lattice-point interval L /d0 = 2.0, the appearance frequency of two- or three-droplet-array patterns just before S /d0 = 16 was calculated in 1,000,000 patterns. As shown in the previous section, the flame can spread to the next droplet even with S /d0 = 16 only if the droplet patterns meet the requirement. Up to the sixth most appearance frequency patterns are shown in Tables 1 and 2. As shown in Table 1, Pattern 2.0–16 has the highest frequency, 11%, in the case with two-droplet interaction. However, the highest frequency with the three-droplet interaction is 4.8% for Pattern 2.0–2.0–16. The local flame-spreadlimit distance would be the largest for Pattern 4.0–2.0–16, as shown in the previous section, but the appearance frequency of this pattern is only 2.7%. Hereby, the flame-spread probability was calculated considering the local flame-spread-limit distance only with two-droplet interaction. In the case of lattice number N = 100, lattice-point interval L /d0 = 2.0, the flame-spread probability was calculated as shown in Fig. 17. Calculation was conducted for 3000 different droplet Table 1 Appearance frequency of droplet-array patterns in the case of two-droplet interaction for S AL /d0 = 16 (N = 100, L /d0 = 2). Pattern S BA /d0 –S AL /d0
2.0–16
4.0–16
6.0–16
8.0–16
10–16
12–16
Proportion [%]
11
6.1
3.3
1.7
0.9
0.5
Fig. 17. Dependence of flame-spread probability on the occupation fraction of droplets (N = 100, L /d0 = 2).
Table 2 Appearance frequency of droplet-array patterns in the case of three-droplet interaction for S AL /d0 = 16 (N = 100, L /d0 = 2). Pattern S CB /d0 –S BA /d0 –S AL /d0
2.0–2.0-16
2.0–4.0–16
4.0–2.0–16
2.0–6.0–16
4.0–4.0–16
6.0–2.0–16
Proportion [%]
4.8
2.7
2.7
1.5
1.5
1.5
770
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patterns at each occupation fraction. As shown in Fig. 17, the flame-spread probability considering the local flame-spread-limit distance of unevenly-spaced droplet arrays is larger than that considering the flame-spread-limit distance of evenly-spaced droplet arrays. It is important to consider the local interaction effect to accurately calculate the flame-spread probability. The present paper demonstrates flame-spread characteristics of randomly arranged droplets using a percolation model considering the local flame-spread-limit distance. This model can be applied to each of one-, two-, and three-dimensional droplet clouds. As mentioned above, the local flame-spread-limit distance is affected by droplet interaction. If the effect of flame-spread direction after the droplet interaction on the flame-spread-limit distance were negligible, the local flame-spread-limit distance based on onedimensional unevenly-spaced droplet array could be used to the flame spread in two- or three-dimensional droplet clouds without modification. The extension of the model to higher dimension droplet clouds, and the effects of L /d0 and lattice size will be reported in the next article. 6. Conclusions We investigated the flame-spread probability in randomly distributed droplet arrays and the local flame-spread-limit distance of unevenly-spaced droplet arrays. A percolation model considering the flame-spread-limit distance was proposed to utilize the findings from droplet array combustion to spray combustion and to calculate the flame-spread probability. The local flame-spread-limit distance of unevenly-spaced droplet arrays was experimentally investigated in microgravity. The following conclusions were drawn: (1) The local flame-spread-limit distance of unevenly-spaced droplet arrays is larger than the flame-spread-limit distance of evenly-spaced droplet arrays. The local flame-spread-limit distance increases with decreasing the droplet spacing between two droplets just before the limit distance. (2) The local flame-spread-limit distance of unevenly-spaced droplet arrays is influenced by the droplet pattern of the three droplets just before the limit distance. Strongly interacting droplet burning initiated by Mode 2 flame spread extends the flame-spread-limit distance to the next droplet.
(3) The flame-spread probability of randomly arranged droplet arrays considering the local flame-spread-limit distance of unevenly-spaced droplet arrays is larger than that of evenlyspaced droplet arrays. Three-droplet interaction has a minor effect on the flame-spread probability. Acknowledgments We wish to thank Mr. M. Asano and Mr. K. Yagi for their valuable help in experiments. This study was carried out as a part of the “Ground-Based Research Announcement for Space Utilization” promoted by Japan Space Forum. References [1] R. Reichenbach, D. Squires, S.S. Penner, Proc. Combust. Inst. 8 (1962) 1068– 1073. [2] T.A. Brzustowski, A. Sobiesiak, S. Wojcicki, Proc. Combust. Inst. 8 (1962) 265– 273. [3] S. Kato, H. Mizuno, H. Kobayashi, T. Niioka, JSME Int. J. B 41 (1998) 322–330. [4] S. Okajima, T. Kimoto, K. Abe, S. Yamaguchi, JSME Int. J. B 47 (2002) 2058–2065. [5] H. Kobayashi, J. Park, T. Niioka, Proc. Combust. Inst. 29 (2002) 2603–2610. [6] M. Mikami, H. Oyagi, N. Kojima, M. Kikuchi, Y. Wakashima, S. Yoda, Combust. Flame 141 (2005) 241–252. [7] M. Mikami, H. Oyagi, N. Kojima, Y. Wakashima, M. Kikuchi, S. Yoda, Combust. Flame 146 (2006) 391–406. [8] H. Nomura, M. Takahashi, Y. Ujiie, H. Hara, Proc. Combust. Inst. 30 (2005) 1991– 1995. [9] A. Umemura, JSME Trans. B 68 (2002) 24242–24248 (in Japanese). [10] A. Umemura, JSME Trans. B 68 (2002) 2429–2436 (in Japanese). [11] M. Kikuchi, T. Arai, S. Yoda, T. Tsukamoto, A. Umemura, M. Uchida, T. Niioka, Proc. Combust. Inst. 29 (2002) 2611–2619. [12] M. Kikuchi, Y. Wakashima, S. Yoda, M. Mikami, Proc. Combust. Inst. 30 (2005) 2001–2009. [13] D. Stauffer, A. Aharony, Introduction to Percolation Theory, revised 2nd ed., CRC Press, Boca Raton, FL, 1994. [14] A.R. Kerstein, C.K. Law, Proc. Combust. Inst. 19 (1982) 961–969. [15] A.R. Kerstein, Combust. Sci. Technol. 37 (1984) 47–57. [16] A. Umemura, S. Takamori, Combust. Flame 141 (2005) 336–349. [17] K. Miyasaka, C.K. Law, Proc. Combust. Inst. 18 (1981) 283–292. [18] M. Mikami, H. Kato, J. Sato, M. Kono, Proc. Combust. Inst. 25 (1994) 431–438. [19] A. Umemura, Proc. Combust. Inst. 18 (1981) 1355–1363. [20] D.B. Spalding, ARS J. 29 (1959) 828–835. [21] A. Crespo, A. Liñan, Combust. Sci. Technol. 11 (1975) 9–18.
Contents lists available at ScienceDirect
Combustion and Flame www.elsevier.com/locate/combustflame
Flame-spread probability and local interactive effects in randomly arranged fuel-droplet arrays in microgravity Hiroshi Oyagi, Hisashi Shigeno, Masato Mikami ∗ , Naoya Kojima Graduate School of Science and Engineering, Yamaguchi University, 2-16-1 Tokiwadai, Ube 755-8611, Japan
a r t i c l e
i n f o
Article history: Received 31 October 2007 Received in revised form 13 December 2008 Accepted 16 December 2008 Available online 14 February 2009 Keywords: Fuel-droplet arrays Flame spread Unevenly-spaced droplet array Flame-spread probability Droplet interaction Microgravity
a b s t r a c t We investigated the flame-spread characteristics of randomly arranged fuel-droplet arrays in microgravity. Flame-spread probability was calculated based on a percolation model with the flame-spread-limit distance of evenly-spaced n-decane droplet arrays in microgravity. Flame-spread probability depends on the occupation fraction of droplets in a lattice and rapidly increases with the occupation fraction. The local flame-spread-limit distance of unevenly-spaced n-decane droplet arrays was experimentally investigated in microgravity. The droplets were arranged in a straight line at uneven intervals. The local flame-spread-limit distance of the unevenly-spaced droplet arrays depended on the droplet arrangement and increased from the flame-spread-limit distance of the evenly-spaced droplet arrays due to interactive effects. The flame-spread probability considering the increase in local flame-spread-limit distance is larger than that without it. © 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction Flame spread in fuel spray occurs immediately after ignition in diesel engines and occurs near the flame base in gas turbine engines. Flame spread plays an important role in heat release in the initial stage in the former case and in the stability of flame in the latter case. In order to improve understanding of the flame-spread mechanism, the flame spread along evenly-spaced droplet arrays has been investigated experimentally [1–8], theoretically [9,10] and numerically [11,12]. In spray combustion, however, fuel droplets distribute at random in the combustion chamber. The findings of researches on flame spread along droplet arrays have not been well utilized for elucidation of spray combustion. The present paper investigates the flame-spread characteristics of randomly arranged droplets using a percolation model based on the findings of microgravity experiments on flame spread along droplet arrays. Percolation theory gives us probabilistic characteristics of connections of particles randomly arranged in a lattice [13]. In the case of a small number of particles, the size of particle connection is small, as shown in Fig. 1(a). But, in the case of a large number of particles, as shown in Fig. 1(b), the size of particle connection is large and a large cluster is formed. Some studies have applied the percolation theory to the elucidation of spray combustion [14–16]. When percolation theory is applied to spray combustion, a droplet is characterized as a particle. Kerstein
*
Corresponding author. Fax: +81 836 85 9101. E-mail address: [email protected] (M. Mikami).
et al. analyzed the percolate combustion zone in non-premixed sprays using percolation theory [14,15]. Umemura and Takamori [16] calculated the flame-spread probability of a randomly distributed droplet cloud based on Mode 1 flame spread, in which the vaporization of the next droplet becomes activated after the leading edge of an expanding group diffusion flame passes the next droplet and pushes the leading flame forward [7,9,12]. However, microgravity experiments on flame spread along droplet arrays show that Mode 1 flame spread occurs only for a small droplet spacing S /d0 [7], where S is the droplet spacing and d0 is the initial droplet diameter. For a relatively large S /d0 , Mode 3 flame spread occurs, in which the next droplet autoignites through heating by the diffusion flame, whose leading edge does not reach the flammable-mixture layer around the next droplet [7,9,12]. In view of the general site percolation, only when two particles exist on adjacent lattice points, these two particles are connected as shown in Fig. 1 [13]. So, the inter-particle distance is meaningless. In flame spread, however, the dimensionless droplet spacing S /d0 is important. Mikami et al. [7] have shown that there exists a flame-spread limit ( S /d0 )limit in terms of the dimensionless droplet separation distance, below which the leading flame can spread to the next droplet, but over which it cannot spread. Therefore, the flame-spread-limit distance is the threshold of the connection. The present paper introduces a percolation model considering the flame-spread-limit distance based on Mode 3 flame spread, as shown in Fig. 2. In spray combustion, if the droplet number density is small, the leading flame spreads with difficulty, as shown in Fig. 3(a). But, if the droplet number density is large enough, the leading flame spreads easily to the next un-
0010-2180/$ – see front matter © 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2008.12.013
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(a) Small number of particle
(a) Droplet spacing between burned droplet and unburned droplet is less than flame-spread-limit distance
(b) Droplet spacing between burned droplet and unburned droplet is larger than flame-spread-limit distance Fig. 2. Percolation model considering local flame-spread-limit distance. (b) Large number of particle Fig. 1. Two different site percolation images.
burned droplet, and the flame spreads to droplets on all the lattice sides, resulting in group flame formation, as shown in Fig. 3(b). If the local droplet spacing is small enough, droplet interaction occurs [17–19], which affects the local flame-spread characteristics. In order to elucidate it, we experimentally investigated the local flame-spread-limit distance of unevenly-spaced droplet arrays in microgravity. The flame-spread probability of randomly arranged droplet arrays was calculated considering the local flame-spreadlimit distance of unevenly-spaced droplet arrays. 2. Flame-spread probability based on the flame-spread limit of evenly-spaced droplet arrays This section introduces a percolation model considering the flame-spread-limit distance and demonstrates flame-spread characteristics of randomly arranged droplets for the simplest droplet system with the flame-spread-limit distance ( S /d0 )limit of evenlyspaced droplet arrays in microgravity. The flame-spread-limit distance along evenly-spaced droplet arrays was investigated in microgravity [3,7]. Mikami et al. [7] reported that the flame-spread-limit distance ( S /d0 )limit of evenlyspaced droplet arrays is about 14 in 300 K air with n-decane as a fuel. As shown in Fig. 4(a), if every droplet spacing is less than the flame-spread-limit distance, the flame can spread to the last droplet. On the other hand, if the droplet spacing is larger than the flame-spread-limit distance, the flame cannot spread to the next droplet, as shown in Fig. 4(b). We introduced a simple criterion for local flame spread; for S /d0 smaller than ( S /d0 )limit ,
the flame can spread to the next droplet, but for S /d0 larger than ( S /d0 )limit , the flame cannot spread to the next droplet, as shown in Fig. 2. In Fig. 4(b), the mean droplet spacing is the same as that in Fig. 4(a), so the occupation fraction of droplets in a lattice is the same. The occupation fraction is the ratio of the number of droplets to the total number of lattice points. In three-dimensional droplet clouds, the droplet number density and equivalence ratio are proportional to the occupation fraction. There are many droplet patterns for the same occupation fraction of droplets in a lattice. The flame-spread probability is the proportion of the pattern number showing flame spread to droplets on all sides of the lattice to the total pattern number. We show the flame-spread probability of randomly arranged droplet arrays as the droplet system with the lowest dimension showing flame spread. In this calculation, as shown in Fig. 5, M droplets are randomly arranged at N lattice points, so the occupation fraction of droplets is M / N. Two imaginary droplets are also placed next to both edges of the lattice; the left imaginary droplet is for ignition and the right one is for judging the flame spread to the end of the lattice. The flame can spread to the last droplet only when the local droplet spacing is less then the flame-spread limit ( S /d0 )limit . Fig. 6 shows the flame-spread probability for the number of lattice points N = 100 and dimensionless lattice-point interval L /d0 = 2. The flame-spread-limit distance of evenly-spaced droplet array, ( S /d0 )limit = 14 [7] was used and prevaporization of unburned droplets was ignored. Calculation was conducted for 3000 different droplet patterns at each occupation fraction. As shown in Fig. 6, the flame-spread probability rapidly increases with the occupation fraction. In the general site percolation theory, only when particles exist on adjacent lattice points, particles connect
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Fig. 5. Droplet array model.
(a) Small number of droplets
Fig. 6. Dependence of flame-spread probability on occupation fraction of droplets in a lattice (N = 100, L /d0 = 2, ( S /d0 )limit = 14).
(b) Large number of droplets Fig. 3. Two different percolation images considering the local flame-spread-limit distance.
(a) Every droplet spacing less than flame-spread-limit distance
array. In spray, however, droplet spacing is uneven. The flamespread-limit distance of unevenly-spaced droplet arrays is possibly different from that of evenly-spaced droplet arrays. Researches on the interactive effect of two droplets reported that the burning rate, burning time and flame radius depend on the droplet spacing [17–19]. When the droplet spacing is small, a group flame forms around the two droplets and the flame radius of the group flame is larger than that of the single droplet. When the droplet spacing is large, however, an individual flame forms around each droplet. The following sections describe our experimental investigation of the local flame-spread-limit distance in microgravity. In addition, the influence of the local flame-spread-limit distance on the flamespread probability is investigated. 3. Experimental apparatus and method
(b) One droplet spacing larger than flame-spread-limit distance Fig. 4. Typical flame-spread behavior for two different flame-spread patterns in microgravity for mean droplet spacing ( S /d0 )m = 8. The white circles are inserted to show the droplet position.
to each other. So, all the particles arranged in one dimension connect only when the occupation fraction of particles in a lattice is unity [13]. However, the present percolation model considering the flame-spread-limit distance can explain the flame-spread probability of randomly arranged droplets even if the droplets are arranged in one dimension. In this section, the flame-spread probability was calculated based on the flame-spread-limit distance of evenly-spaced droplet
Fig. 7 represents the droplet array generation apparatus. This apparatus was designed based on the multiple droplets generation technique proposed by Mikami et al. [6]. It was composed of a droplet-array holder, a fuel-supply system, a droplet array generator, a hot wire ignitor and a droplet remover. The droplets are suspended on the intersection of X-shape SiC fibers of 14 μm in diameter. n-Decane was supplied to the intersections of the fibers through fine glass tubes whose outer diameter at the tip is 40 μm. The multiple droplets generation technique is described in detail in Ref. [6]. There are two differences between this apparatus and the apparatus in Ref. [6]. First, each glass tube was connected through a Teflon tube to the corresponding syringe pushed by a steppingmotor-driven stage. Second, a droplet remover was used. Before droplet generation, the fuel was supplied to glass tube tips, and surplus fuel was removed by the droplet remover. In most cases, the initial droplet diameter d0 was set to 1.0 mm for different combinations of droplet spacing. In order to vary the dimensionless droplet separations S /d0 finely, the initial droplet diameter var-
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ied from 0.93 to 1.14 mm. If there exists one-pulse error in the displacement of the stepping-motor driven stage, the droplet diameter has about 1.1% error for a 1 mm droplet. The microgravity experiments were performed at atmospheric pressure and at room temperature in Micro-Gravity Laboratory Japan (MGLAB). The microgravity duration is 4.5 s and the microgravity level is 10−5 G. Flame-spread behavior was recorded by a digital video camera (SONY, DCR-HC90), whose framing rate is 30 fps. The experimental sequence is as follow: (1) Fuel was supplied to the glass tube tips. (2) Surplus fuel was removed by gas jet from the droplet remover. (3) The glass tube tips were brought to the intersections of SiC fibers. (4) The fuel was supplied to generate the droplets. (5) The drop capsule that carried the experimental apparatus was dropped.
Fig. 7. Droplet array generator.
Fig. 8. Droplet array for examining the effect of two-droplet interaction on the local flame-spread-limit distance. This pattern is called Pattern S BA /d0 –S AL /d0 .
(6) The glass tubes were drawn back in microgravity. (7) The droplet at one end of the array was ignited by a hot-wire ignitor. (8) Flame-spread behavior occurred and was recorded. 4. Local flame-spread limit of unevenly-spaced droplet arrays This section shows the experimental results of the local flamespread limit of unevenly-spaced droplet arrays in microgravity. 4.1. Local flame-spread limit influenced by two-droplet interaction Fig. 8 shows the definition of the droplet array for examining the effect of two-droplet interaction on the local flame-spreadlimit distance. Four droplets were placed linearly with uneven droplet spacing. The flame spread started by igniting Droplet I. We investigated whether the leading flame spreads from Droplet A to Droplet L, with the interaction between Droplet B and Droplet A. In this research, the droplet array pattern as shown in Fig. 8 was called Pattern S BA /d0 –S AL /d0 . For example, if the dimensionless droplet spacing between Droplet B and Droplet A S BA /d0 is 2.0 and that between Droplet A and Droplet L S AL /d0 is 16, the pattern name is 2.0–16. The droplet spacing between Droplet B and Droplet A S BA was varied from 2 mm to 14 mm, and that between Droplet A and L S AL was varied from 14 mm to 18 mm. Initial-droplet diameter d0 was varied from 0.93 mm to 1.14 mm. Hereby, the dimensionless droplet spacing between Droplet B and Droplet A S BA /d0 was varied from 1.9 to 14, and that between Droplet A and Droplet L S AL /d0 was varied from 14 to 18. Droplet I was used to reduce the disturbance from the hot-wire igniter. The droplet spacing between Droplet I and Droplet B S IB was 12 mm, so the dimensionless droplet spacing between Droplet I and B S IB /d0 was from 10.5 to 12.9. Mikami et al. [7] reported that individual flames occurred in two-droplet combustion in microgravity for S /d0 > 11. The influence of the hot-wire igniter on Droplet B, A, and L was small. S IB /d0 should not be too large due to the limitation of the microgravity duration as well as the existence of the flame-spread-limit distance ( S /d0 )limit = 14 [7]. Fig. 9 shows the flame-spread behavior for different droplet patterns. Elapsed time t /d20 is from the ignition of Droplet B. In Pattern 1.9–17 shown in Fig. 9(a), a large group flame formed around Droplet B and Droplet A. It did not spread to Droplet L. In Pattern 2.0–16 and Pattern 4.0–16 shown in Figs. 9(b) and 9(c), the flame spread to Droplet L. But, if S BA /d0 was increased to 6.0, the flame did not spread to Droplet L as shown in Fig. 9(d). Thus, the local flame-spread-limit distance ( S AL /d0 )limit was influenced by S BA /d0 . Fig. 10 shows the local flame-spread-limit dis-
(a) Pattern 1.9–17
(b) Pattern 2.0–16
(c) Pattern 4.0–16
(d) Pattern 6.0–16
Fig. 9. Typical flame-spread behavior for different droplet-array patterns in the case of two-droplet interaction. The white circles are inserted to show the droplet position.
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Fig. 12. Droplet array for examining the effect of three-droplet interaction on the local flame-spread-limit distance. This pattern is called Pattern S CB /d0 –S BA / d0 –S AL /d0 .
Fig. 10. Flame-spread-limit distance of fuel-droplet arrays with uneven droplet spacing in the case of two-droplet interaction. The flame-spread-limit distance with even droplet spacing [7] is also shown.
the flame spread to Droplet L requires heating of Droplet L by heat from the flame surrounding Droplet A. In the near field of burning Droplet A, nearly quasi-steady field forms where the convection term and the diffusion term are of the same order. In the far field, however, unsteady field forms where the diffusion term and the unsteady term are of the same order. The thermal layer affecting flame spread to Droplet L exists in the far field, and therefore, the temperature in the thermal layer is transient in nature and does not approach a quasi-steady limit asymptotically. The thermal layer initially develops with time, but decreases eventually. The radius of the thermal layer is therefore finite, resulting in the flame-spread limit ( S AL /d0 )limit . In unsteady droplet combustion, the mixture fraction Z in the far field is expressed as, Z=
Mf
ρ (4π at )3/2
exp −
r2
4at
,
according to analyses by Spalding [20] and Crespo and Liñan [21]. Here, the temperature T has the linear relationship with Z . The radius of the thermal layer is a function of time as
r C = 6at ln
ρ ZC
−1 1/2
2/ 3
(4π at )
Mf
,
where a is thermal diffusivity, ρ is the gas density, M f is the initial fuel-droplet mass, and Z C is the mixture fraction corresponding to the representative temperature in the thermal layer. rC attains a maximum value
rCm = Fig. 11. Maximum flame radiuses of Droplet A.
tance of unevenly-spaced droplet arrays of all the experimental results. The flame-spread-limit distance of evenly-spaced droplet arrays [7] is also shown. As shown in Fig. 10, when S BA /d0 is less than 6, the local flame-spread-limit distance ( S AL /d0 )limit increased with decreasing S BA /d0 . For S BA /d0 > 6, ( S AL /d0 )limit is close to the flame-spread-limit distance of evenly-spaced droplet arrays ( S /d0 )limit . Fig. 11 shows the maximum radiuses of a luminous and a blue flame of Droplet A, which were measured from the center of Droplet A in the direction to Droplet L. If S BA /d0 is less than 6, the maximum flame radiuses of Droplet A increased with decreasing S BA /d0 . This is similar dependence to the local flame-spread limit shown in Fig. 10. Since the flame diameter becomes larger due to the interactive effect between droplets [18,19], the flame-spread-limit distance might also become larger due to the interactive effect. We will discuss the flame-spread-limit distance in view of the droplet interaction in the following. The equivalence ratio of fuel vapor/air mixture at the n-decanedroplet surface is about 0.1 at 300 K, which is much less than the lean flammability limit [7]. Therefore, there is no flammable mixture around unburned n-decane droplets at room temperature, so
2π e
at time t Cm =
1/2
3
1 4π ea
1/3
Mf
ρ ZC Mf
ρ ZC
2/ 3 .
These equations show that both rCm and t Cm increase with M f . If there exists another droplet near one droplet inside the flame, the equivalent initial mass increases as compared to the single droplet combustion, and the maximum radius rCm of the thermal layer increases, resulting in the increase in the flame-spread-limit distance. 4.2. Local flame-spread limit influenced by three-droplet interaction This section investigates the influence of three-droplet interaction. Fig. 12 shows the definition of the droplet array configuration for examining the effect of three-droplet interaction on the local flame-spread-limit distance. Five droplets were placed linearly. The droplet array pattern as shown in Fig. 12 is called Pattern S CB /d0 –S BA /d0 –S AL /d0 . For example, if S CB /d0 is 1.9, S BA /d0 is 1.9 and S AL /d0 is 17, the pattern name is 1.9–1.9–17. S AL was 18 mm and the initial droplet diameter d0 was 1.06 mm under all conditions. So, S AL /d0 was 17, for which the flame did not spread
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(a) Pattern 1.9–1.9–17
(b) Pattern 3.8–1.9–17
(c) Pattern 5.7–1.9–17
(d) Pattern 7.5–1.9–17
(e) Pattern 9.4–1.9–17 Fig. 13. Flame-spread behavior of different droplet-array patterns in the case of three-droplet interaction. The white circles are inserted to show the droplet position.
(a) Pattern 1.9–3.8–17
(b) Pattern 3.8–3.8–17 Fig. 14. Flame-spread behavior of different droplet-array patterns in the case of three-droplet interaction. The white circles are inserted to show the droplet position.
to Droplet L with two-droplet interaction as shown in Figs. 9 and 10. For S BA = 2 mm, S CB was varied from 2 mm to 10 mm. For S BA = 4 mm, S CB was varied from 2 mm to 4 mm. In most cases, the droplet spacing between Droplet I and Droplet C S IC was 12 mm. In Pattern 9.4–1.9–17, however, S IC was 10 mm. The reason for this is to secure sufficient time to investigate whether the flame can spread to Droplet L in the limited microgravity duration of 4.5 s. Fig. 13 shows the flame-spread behavior for S BA /d0 = 1.9 and S AL /d0 = 17. S CB /d0 was varied to investigate the effect of Droplet C. Elapsed time t /d20 is the time from the ignition of Droplet C. In Pattern 1.9–1.9–17 and 9.4–1.9–17, the flame did not spread to Droplet L. In Pattern 3.8–1.9–17, 5.7–1.9–17 and 7.5– 1.9–17, however, the flame spread to Droplet L. S AL /d0 = 17 is larger than the flame-spread limit ( S AL /d0 )limit for S BA /d0 = 1.9 with two-droplet interaction between Droplet B and Droplet A as shown in Fig. 10. Fig. 14 shows the flame-spread behavior for S BA /d0 = 3.8 and S AL /d0 = 17. In these patterns, the flame did not spread to Droplet L. Fig. 15 shows the local flame-spread limit in S CB /d0 –S BA /d0 plane for S AL /d0 = 17. As shown in Fig. 15, there is a combination of S CB /d0 and S BA /d0 to increase the flame-spread
Fig. 15. Flame-spread modes of fuel-droplet arrays with uneven droplet spacing in the case of three-droplet interaction for S AL /d0 = 17.
limit ( S AL /d0 )limit . It is interesting that for Pattern 1.9–1.9–17, the flame did not spread to Droplet L although this pattern seems to have the strongest interaction. Fig. 16 shows the maximum blue-flame radius of Droplet A, R A , which was measured from the center of Droplet A to Droplet L. R A for the cases that the flame spread to Droplet L is somewhat larger than the others. So, R A for Pattern 1.9–1.9–17 is smaller than Pattern 3.8–1.9–17. This is possibly related to the flame-spread mode around Droplets C, B and A. The relation between the flame-spread-limit distance and the droplet pattern is examined in view of the flame-spread mode, which is introduced by Umemura [9]. First, in Pattern 1.9–1.9– 17, the flame-spread mode from Droplet C to Droplet B was Mode 1, in which vaporization of the next unburned droplet is activated after the diffusion flame passes it and then the active vaporization pushes the leading flame forward. In Patterns 3.8– 1.9–17 and 5.7–1.9–17, the flame-spread mode from Droplet C to
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Droplet B was Mode 2, in which the flammable mixture layer forms around the unburned droplet and the premixed flame propagates in the flammable mixture layer immediately after the leading flame reaches the flammable mixture. Since the flammable mixture layer forms before the flame spreads to the unburned droplet in Mode 2, the initial flame radius with the Mode 2 flame spread is larger than that with the Mode 1 flame spread. The flamespread limit ( S AL /d0 )limit possibly increases with the initial flame radius of Droplet A. Hereby, the local flame-spread-limit distance after the Mode 2 flame spread became larger for the same S BA /d0 . In Pattern 7.5–1.9–17, the flame-spread mode from Droplet C to Droplet B is Mode 3, in which the next unburned droplet autoignites through heating by the leading flame, which does not reach the flammable mixture layer around the next unburned droplet. Since the flammable mixture layer formed before the flame spreads to the unburned droplet in Mode 3, the initial flame radius with Mode 3 is larger than that with Mode 1, too. In Patten 9.4–1.9–17, the flame-spread mode from Droplet C to Droplet B was Mode 3. However, the flame did not spread to Droplet L. In this pattern, since the droplet spacing between Droplet C and Droplet B is too large, the interaction between Droplet C and B is small. Hereby, the local flame-spread-limit distance of unevenly-spaced droplet arrays is influenced by the flame-spread mode as well as the interaction effect. In Pattern 3.8–3.8–17, interaction between Droplet B and Droplet A is probably not large enough although the flame-spread mode from Droplet C to Droplet B is Mode 2. Nomura et al. [8] investigated the gas flow caused by droplet ignition during the flame spread along n-heptane droplet arrays in normal gravity. They reported that burned-gas flow was caused by the droplet ignition and its influence extended 12 times further than the initial droplet diameter. For higher-volatility fuel, the flammable mixture layer around the unburned droplet is thicker, so the burned gas flow by the premixed-flame propagation through the mixture layer is stronger. In such a case, convective heat transfer would play an important role in the flame-spread limit.
Fig. 16. Maximum flame radius of Droplet A.
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5. Flame-spread probability based on local flame-spread limit of unevenly-spaced droplet array The local flame-spread-limit distance increases by interaction effect and flame-spread mode. In this section, the flame-spread probability considering the local flame-spread-limit distance of unevenly-spaced droplet arrays is calculated. First, the appearance frequency of some droplet-array patterns is investigated. In the case of lattice number N = 100 and lattice-point interval L /d0 = 2.0, the appearance frequency of two- or three-droplet-array patterns just before S /d0 = 16 was calculated in 1,000,000 patterns. As shown in the previous section, the flame can spread to the next droplet even with S /d0 = 16 only if the droplet patterns meet the requirement. Up to the sixth most appearance frequency patterns are shown in Tables 1 and 2. As shown in Table 1, Pattern 2.0–16 has the highest frequency, 11%, in the case with two-droplet interaction. However, the highest frequency with the three-droplet interaction is 4.8% for Pattern 2.0–2.0–16. The local flame-spreadlimit distance would be the largest for Pattern 4.0–2.0–16, as shown in the previous section, but the appearance frequency of this pattern is only 2.7%. Hereby, the flame-spread probability was calculated considering the local flame-spread-limit distance only with two-droplet interaction. In the case of lattice number N = 100, lattice-point interval L /d0 = 2.0, the flame-spread probability was calculated as shown in Fig. 17. Calculation was conducted for 3000 different droplet Table 1 Appearance frequency of droplet-array patterns in the case of two-droplet interaction for S AL /d0 = 16 (N = 100, L /d0 = 2). Pattern S BA /d0 –S AL /d0
2.0–16
4.0–16
6.0–16
8.0–16
10–16
12–16
Proportion [%]
11
6.1
3.3
1.7
0.9
0.5
Fig. 17. Dependence of flame-spread probability on the occupation fraction of droplets (N = 100, L /d0 = 2).
Table 2 Appearance frequency of droplet-array patterns in the case of three-droplet interaction for S AL /d0 = 16 (N = 100, L /d0 = 2). Pattern S CB /d0 –S BA /d0 –S AL /d0
2.0–2.0-16
2.0–4.0–16
4.0–2.0–16
2.0–6.0–16
4.0–4.0–16
6.0–2.0–16
Proportion [%]
4.8
2.7
2.7
1.5
1.5
1.5
770
H. Oyagi et al. / Combustion and Flame 156 (2009) 763–770
patterns at each occupation fraction. As shown in Fig. 17, the flame-spread probability considering the local flame-spread-limit distance of unevenly-spaced droplet arrays is larger than that considering the flame-spread-limit distance of evenly-spaced droplet arrays. It is important to consider the local interaction effect to accurately calculate the flame-spread probability. The present paper demonstrates flame-spread characteristics of randomly arranged droplets using a percolation model considering the local flame-spread-limit distance. This model can be applied to each of one-, two-, and three-dimensional droplet clouds. As mentioned above, the local flame-spread-limit distance is affected by droplet interaction. If the effect of flame-spread direction after the droplet interaction on the flame-spread-limit distance were negligible, the local flame-spread-limit distance based on onedimensional unevenly-spaced droplet array could be used to the flame spread in two- or three-dimensional droplet clouds without modification. The extension of the model to higher dimension droplet clouds, and the effects of L /d0 and lattice size will be reported in the next article. 6. Conclusions We investigated the flame-spread probability in randomly distributed droplet arrays and the local flame-spread-limit distance of unevenly-spaced droplet arrays. A percolation model considering the flame-spread-limit distance was proposed to utilize the findings from droplet array combustion to spray combustion and to calculate the flame-spread probability. The local flame-spread-limit distance of unevenly-spaced droplet arrays was experimentally investigated in microgravity. The following conclusions were drawn: (1) The local flame-spread-limit distance of unevenly-spaced droplet arrays is larger than the flame-spread-limit distance of evenly-spaced droplet arrays. The local flame-spread-limit distance increases with decreasing the droplet spacing between two droplets just before the limit distance. (2) The local flame-spread-limit distance of unevenly-spaced droplet arrays is influenced by the droplet pattern of the three droplets just before the limit distance. Strongly interacting droplet burning initiated by Mode 2 flame spread extends the flame-spread-limit distance to the next droplet.
(3) The flame-spread probability of randomly arranged droplet arrays considering the local flame-spread-limit distance of unevenly-spaced droplet arrays is larger than that of evenlyspaced droplet arrays. Three-droplet interaction has a minor effect on the flame-spread probability. Acknowledgments We wish to thank Mr. M. Asano and Mr. K. Yagi for their valuable help in experiments. This study was carried out as a part of the “Ground-Based Research Announcement for Space Utilization” promoted by Japan Space Forum. References [1] R. Reichenbach, D. Squires, S.S. Penner, Proc. Combust. Inst. 8 (1962) 1068– 1073. [2] T.A. Brzustowski, A. Sobiesiak, S. Wojcicki, Proc. Combust. Inst. 8 (1962) 265– 273. [3] S. Kato, H. Mizuno, H. Kobayashi, T. Niioka, JSME Int. J. B 41 (1998) 322–330. [4] S. Okajima, T. Kimoto, K. Abe, S. Yamaguchi, JSME Int. J. B 47 (2002) 2058–2065. [5] H. Kobayashi, J. Park, T. Niioka, Proc. Combust. Inst. 29 (2002) 2603–2610. [6] M. Mikami, H. Oyagi, N. Kojima, M. Kikuchi, Y. Wakashima, S. Yoda, Combust. Flame 141 (2005) 241–252. [7] M. Mikami, H. Oyagi, N. Kojima, Y. Wakashima, M. Kikuchi, S. Yoda, Combust. Flame 146 (2006) 391–406. [8] H. Nomura, M. Takahashi, Y. Ujiie, H. Hara, Proc. Combust. Inst. 30 (2005) 1991– 1995. [9] A. Umemura, JSME Trans. B 68 (2002) 24242–24248 (in Japanese). [10] A. Umemura, JSME Trans. B 68 (2002) 2429–2436 (in Japanese). [11] M. Kikuchi, T. Arai, S. Yoda, T. Tsukamoto, A. Umemura, M. Uchida, T. Niioka, Proc. Combust. Inst. 29 (2002) 2611–2619. [12] M. Kikuchi, Y. Wakashima, S. Yoda, M. Mikami, Proc. Combust. Inst. 30 (2005) 2001–2009. [13] D. Stauffer, A. Aharony, Introduction to Percolation Theory, revised 2nd ed., CRC Press, Boca Raton, FL, 1994. [14] A.R. Kerstein, C.K. Law, Proc. Combust. Inst. 19 (1982) 961–969. [15] A.R. Kerstein, Combust. Sci. Technol. 37 (1984) 47–57. [16] A. Umemura, S. Takamori, Combust. Flame 141 (2005) 336–349. [17] K. Miyasaka, C.K. Law, Proc. Combust. Inst. 18 (1981) 283–292. [18] M. Mikami, H. Kato, J. Sato, M. Kono, Proc. Combust. Inst. 25 (1994) 431–438. [19] A. Umemura, Proc. Combust. Inst. 18 (1981) 1355–1363. [20] D.B. Spalding, ARS J. 29 (1959) 828–835. [21] A. Crespo, A. Liñan, Combust. Sci. Technol. 11 (1975) 9–18.