OR4-3
二液滴の力学系モデルにおける無次元数が冷炎振動に及ぼす影響
Influence of Dimensionless Numbers
in a Dynamics Model of a Droplet Pairon the Cool Flame Oscillation
家村和輝 1, 齋藤麟太郎 1, 髙畑優星 1, 齊藤允教 2, 田辺光昭 2Kazuki IEMURA1, Rintaro SAITO1, Yusei TAKAHATA1, Masanori SAITO2, and Mitsuaki TANABE2
1 日本大学大学院理工学研究科航空宇宙工学専攻, Department of Aerospace Engineering, Graduate School of Scienceand Technology, Nihon University,
2 日本大学理工学部航空宇宙工学科, Department of Aerospace Engineering, College of Science and Technology, NihonUniversity,1.

Introduction
Combustion engines such as jet engines and liquid rocket engines use spray combustion. The operational

range of these engines involves a two-stage ignition process, occurring both cool flame and hot flame. The
smallest element of the spray is a droplet. Tanabe et al.1-3) conducted droplet combustion studies using a drop
tower and numerical simulations, clarifying the occurrence of cool flame during the spontaneous ignition
process of droplets. One interesting behavior of cool flames is their oscillation4-5). Griffiths et al.5) studied a 0dimensional dynamics system of low-temperature reactions with diffusion in detail. Tanabe et al.6-7) simulatedcool flame oscillation with the simplest dynamical system that covers the characteristics of cool flame basedon reactions. In his study, the amount of the chain agent and temperature influenced the dissipation term. This
enables the simulation of the cool flame in the droplet system. It was reported that cool flame oscillation in thedroplet system depends on the rate of reaction and dissipation. Saito et al.8) used Tanabe's dynamical systemmodel to simulate differences in the relative dissipation of substance and heat. They reported that as the valueof the heat dissipation term is increased, the chain branching reaction is reactivated with decreasing
temperature, resulting in a limit cycle. These studies considered diffusion to the ambient but did not consider
the interaction phenomena that occur around the droplet pair system or diffusion to the other droplet.
In practical spray combustion, numerous droplets exist. Moriue et al.9) clarified two interaction effects
during the spontaneous ignition process of droplet pairs, before and after the occurrence of cool flame. In the“FLEX” project in the United States, they discovered the cool flame occurrence during the extinction processof the hot flame in the droplets10). Mikami et al.11) conducted the first combustion experiment at theInternational Space Station's Kibo module titled “Group Combustion” as a bridge to fill the gap between
droplets and sprays. Recent research has shown the occurrence of cool flame even in the flame spread
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process12). In the spontaneous ignition process, “PHOENIX-2”6) is an underway collaborative project between
Japan and Germany. This project aims to clarify the cool flame dynamics including cool flame oscillation13),cool flame spread, and so on near the spontaneous ignition limit.
In this paper, we describe cool flame oscillations that occurred around droplet pair using the simplest
dynamics model. The model is based on Tanabe's system. We introduce dimensionless numbers resembling
Damköhler and Lewis numbers and discuss their influence on cool flame oscillations.
2.

Modeling of cool flame oscillation occurred around droplet pair
Equations (1) to (4) define the rates of concentration and temperature rise with time at droplets 1 and 2.

Subscripts of T and C denote droplets 1 and 2. Subscripts of 0 indicate initial conditions. Equations (5) to (9)express Arrhenius-type reaction rate constants. The “k*” is the reaction rate constant when the production andconsumption terms are equal where the temperature is defined as “T*” at that moment. The “Da” and “Le”
represent dimensionless numbers that imitate the Da and Le. The Da and Le are defined in equations (10) and(11). The “hs” corresponds to the coefficient indicating the diffusion of the substance. Also, the “hh”
corresponds to the coefficient indicating the diffusion of the heat. The “Ca” and “Ta” denote concentration andtemperature in the ambient conditions. The Ca was determined based on the cool flame oscillation that occurs
around the droplet in 2D simulation. The Ta was determined based on the cool flame temperature at 1 (750 K).
The “a” is defined as the interaction coefficient and larger values influence the other droplet. The “Te”
corresponds to the activation temperature. The “Te1” was determined from the slope of an Arrhenius plot
consisting of cool flame delay time and temperature. The “q” is the energy value. The q was determined fromthe dT/dC which consists of the gas phase maximum temperature and the maximum molar concentration of
OH in the cool flame oscillation that occurs around the droplet in 2D simulation.

∗∗

−−

=−

−

=−

=−

=exp −,(5)=exp −,(6)=exp −,(7)=exp −,(8)=exp −,

(9)

∗∗

∗

∗

1−−−1−−−∗=1−−−1−−−

∗∗

∗

−,(1)

−,(2)

−,(3)

−,

(4)

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==

∗,

(10)

ℎℎ,
ℎ(11)

ℎTable 1 shows the values used in the dynamics model.
Table 1. The values in the system equations (1) through (9).

3.

A1

1Ta1，Ta2

0.8

A2148

Ca1，Ca2

2.0×10Da

1~10Te1

Le

0.1~5

Te2

T01

1.15

T02

1.14

13C01

018C020-9a0.3, 0.7

q107

ResultFigure. 1 is spanned by the Da and Le. Red represents no oscillation, green represents damping, and blue

represents the limit cycle. The left hand of the figure shows the interaction coefficient “a” at 0.3, and the otherhand shows it at 0.7. The combination of the Da and the Le indicates whether the oscillation phenomenon
occurs or not. When the Da is greater than 3 and the Le is greater than 0.3, there are regions of oscillation inboth cases. Limit cycle oscillations also occur as the Da and Le increases. In this system, an equilibrium point
reaches as increasing the Le, and the oscillation phenomenon does not occur. Increasing the a expands theoscillation region. An increase in the a from 0.3 to 0.7 also leads to an expansion of the limit cycle region.

Figure 1. Mapping of the occurrence of oscillation phenomena by dimensionless numbers.

4.

Summary
The combination of the Damköhler number Da and the Lewis number Le determines whether cool flame

oscillation phenomena will occur. Considering the diffusion into the other droplet, an interaction coefficient ais introduced. When the a is 0.3 and 0.7, there is an oscillating region if the Da is greater than 3 and the Le isgreater than 0.3. As the Le increases, an equilibrium point is reached, and the oscillation does not occur.
Increasing the a expands the damping and the limit cycle region.

3 of 4References
This study is supported by “Small-Scale Project” by ISAS-JAXA, JSPS KAKENHI Grant Number #JP19K04843, and#JP21K14347, and by Nihon Univ. President Grant Initiative.
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