PS02
相似則を用いた高分子の燃焼特性解明
Study on combustion properties of polymers utilizing asimilarity rule小川泰知 1, 松木大輝 2, 中村祐二 2,
Taichi OGAWA1, Daiki MATSUGI2, and Yuji NAKAMURA2
豊橋技術科学大学大学院工学研究科機械工学専攻，Department of Mechanical Engineering, Toyohashi University ofTechnology.
2 豊橋技術科学大学大学院工学研究科機械工学系，Department of Mechanical Engineering, Toyohashi University ofTechnology.
1

1. Introduction
In recent years, space development has been shifting to a commercialized business rather than national
project. Therefore, it is inevitable to reduce the cost under warranted safety regulation. Polymers have been
attracting attention because of their nature which fits for space use, such as inexpensive, lightweight, durable,
and have high thermal insulation properties, all of which are necessary elements for space use. Nevertheless,polymers are flammable, and there is a potential hazard of fire accidents, especially in the space environment;
namely, highly-enclosed, microgravity atmosphere. It is severely necessary to understand the “exact” hazard
including the burning characteristics.
Experiments in the space environment (hereinafter referred to as “microgravity experiments”) require
special “costly” equipment such as space station, parabolic flight, drop tower etc. If the cost is reasonable, wemust accept the limit of the microgravity time instead because they are trade-off. For example, theInternational Space Station (ISS) can offer several weeks to several months of microgravity time yet enormous
cost is needed1). A, aircraft offers 10 to 25 seconds with ten million JPY per one campaign2) and a drop tower
offers several seconds with hundreds thousand JPY per drop3). Not only the microgravity time, preparation
time and allowable size of the experimental apparatus are also involved. To end, it is natural to understandthat microgravity experiment is not easy task even less-costly method (cf. drop-tower experiment). Suppose ifalternative methodology is available, investigating the burning character at microgravity becomes handy andmore chance is opened to everybody with less cost.
In the previous study, a method to create a simulated microgravity environment by reducing Grashof number
using the similarity rule was proposed and validated4-7). This methodology allows us to achieve 1-D sphericalflame to be adoptive of ideal 𝑑2 -law8). With this, we can repeat the test in the laboratory with less limitations
as compared to the microgravity experiments. So far, validation was only made by PMMA sample whose
pyrolysis gas species, not only single monomer, suggesting that PMMA is quite special kind of polymericmaterial. In this study, we investigate the difference kinds of polymers, such as polypropylene (hereafter, PP)
and polyethylene (hereafter, PE) under lower pressure environment to achieve 1-D burning status.

1 of 92.

Design Rule2.1. Similarity Rule
Referred from the previous studies, it is effective to propose Grashof number (Gr [-]) similarity to mimic the
dynamics during the combustion of small specimen. Let us briefly recall the concept.
G𝑟 =𝜌 2 𝐺𝛽∆𝑇𝐿3~ 𝐿3 𝐺 1 𝑃2
𝜇2

(1)

where 𝜌 is density [𝑘𝑔⁄𝑚3 ], 𝐺 is gravitational acceleration [𝑚⁄𝑠 2 ], 𝛽 is coefficient of volume expansion[-], ∆𝑇 is temperature difference between flame and surroundings[𝐾], 𝐿 is representative length [𝑚], 𝜇 is
viscosity coefficient [𝑃𝑎 ∙ 𝑠] and 𝑃 is pressure [𝑃𝑎] . lltimately, the dependence of the representative
quantities on the Grashof number can be expressed by 𝐿, 𝐺, and 𝑃, as shown in Eq. (1), and microgravity
condition is regarded as “small Grashof number (namely, Gr << 1)”. Obviously, reduction of 𝐺 can fulfill suchrequirement, however, this is not the only way to achieve small Gr. That is, even maintain the normal gravity(𝐺 = 1), adopting small length scale (𝐿) and low pressure (𝑃) can reduce Gr drastically, suggesting to mimic
the dynamics in microgravity combustion. However, it is important emphasize that there is low-pressure limit
since too-much reduction of pressure can induce the natural extinction due to the lack of oxidizer in thechamber suggesting that it is achieved low-Damkohler number (Da [-]) limit. Hance, determination of the lowlimit and the range of low pressure to achieve the stable burning condition (beyond the extinction limit; Da >1) is important. We will investigate this for the case of PP and PE.
2.2. Target and Experiment
In order to observe the combustion of polymers, the experiment is performed on a spherical 1-D flame
formed over spherical specimen (polymers). The reason for this is to use the 𝑑2 -law to provide the burning
rate constant as eigenvalue of the system. Figure 1 shows (a) the combustion chamber diagram, (b) the
combustion zone, and (c) the sample diagram, respectively adopting in the present work. For detail, please
see Ref 7.

Figure 1. Experimental apparatus and sample, (a) Combustion chamber, (b) Inside the combustion chamber, (c) Magnified
picture of burning sample

The experiment was conducted in a combustion chamber with external dimensions of 335 mm x 335 mm x
335 mm. The chamber is equipped with a window through which the combustion process inside the chamber
can be observed. LEDs installed in the chamber are used as backlight during combustion to image the shapeof burning specimen clearly. The camera is equipped with a digital camera (Nicon D90: full-open for aperture,
24 fps for shutter speed). The sample is suspended by a vertically-oriented SiC fiber (Hi-NicalonTM
2 of 9manufactured by Nippon Carbon Co., Ltd., 14 µm in diameter) as shown in the figure. Ignition is induced bya nichrome wire, and the applied current is adjusted according to the conditions. The internal pressure iscontrolled by a vacuum and set initially as summarized in Table 1. Composition of the ambient gas is 21%
oxygen balanced with 79% of nitrogen.
Table 1. Internal pressure conditions for each test number (#1-#6). Unit: kPa

Sample

#1

#2

#3

#4

#5

#6

PP
PE

7.0
7.0

10.0
10.0

12.5
15.0

15.0
20.0

50.0
50.0

100.0
100.0

The ambient gas temperature was set to room temperature. Once the gas is loaded in the chamber, we shallkeep as is for a while to ensure that little gas flowed in the chamber before ignition. Then igniter is adopted tothe specimen to induce the spontaneous ignition. Once flaming is visually confirmed, the nichrome wire isthen detached from the specimen in order to avoid any interaction with flames. To ensure reproducibility, thetest was conducted at least three times.
The samples are PP and PE spheres with a diameter of 3.5 mm to 4.0 mm is completed by melting on the
ball.
3.

Results and Discussion

3.1. Burning Behavior at Each Pressure
Figure 2 is a cropped image of PP and PE flames at various pressures, whereas Fig. 3 shows a time series of
the combustion process during 1-D burning status at quasi-steady state. An ambient pressure is sufficiently
high (say, 100 kPa), both flames over PP and PE show typical “teardrop” shape, extending upwardly due tobuoyancy flow. A strong luminous zone was observed, indicating the formation of soot there. The flame length
in the vertical direction becomes shorter as the pressure is reduced. Accordingly, the flame shape becomes
nearly “spherical” when the pressure is reduced at 12.5 kPa for PP and at 10 kPa for PE.

Figure 2. Appearance of low-pressured flame at quasi-steady state

3 of 9Figure 3. Time-variation of burning polymers at sufficiently low pressure to exhibits 1-D flame

Formation of nearly spherical flame occurs regardless of the sample material, suggesting that this is dynamic
feature in the gas-phase over the specimen. Once the pressure is reduced to 7.0 kPa, no steady flame isobserved due to the oxygen gas is insufficient to have continuous quasi-steady-state combustion.
3.2. Observed Burning Rate Constants
According to droplet combustion theory8), the well-known 𝑑2 -law (Equation (2)) should be observed whena one-dimensional combustion is established.
𝑑(𝑡)2 = 𝑑02 − 𝐾𝑡(2)where 𝑑(𝑡) is the diameter of the burning specimen [𝑚𝑚], 𝐾 is the burning rate constant [𝑚𝑚2⁄𝑠], and𝑡 is the elapsed time [𝑠]. The subscript 0 indicates the initial state. Theoretically, the burning rate constant isrecognized as a physical constant under a given set of conditions and is claimed to be directly related to thetransfer coefficient, which indicates burning characteristics under a given environment. In this report, theburning rate constant is calculated and burning characteristics are evaluated.

Figure 4. Typical 𝑑2 - t relation study, (a) Changes of diameter squared in time, (b) Burning rate
constants for PP and PE.
4 of 9Figure 4 (a) shows the time evolution of the square of the sample diameter (𝑑2 ), from ignition to theextinction. The moment when the flame covers the sample is defined as the moment of ignition, 𝑡 = 0𝑠. In theinitial stage of combustion (named “Stage 1” in the figure), the square of the sample diameter (𝑑2 ) fluctuates
randomly with increase in time. Later (“Stage 2” in the figure), 𝑑2 decreases in the almost linearly, suggestingthat the combustion followed by 𝑑2 -law. In Stage 1, increase of 𝑑2 shall be due to the volume expansion of
the heated specimen. Because internal bubble is formed the specimen9-10), volume expansion may bepronounced further even though the specimen is covered by the flame (𝑡 > 0 𝑠) to be consumed (to deduce𝑑2 ). After the volume expansion, a constant decrease begins in Stage 2, suggesting that the bubble formationand expansion may stay constant. In this manner, we can observe quasi-steady state feature to have constantburning rate till extinction is experienced. The calculated burning rate constant defined in Stage 2 issummarized in Fig. 4 (b) for both specimens. It can be seen that the burning rate constant increases withdecreasing pressure, then eventually becomes constant when pressure is less than about 10 kPa. As depicted
in Fig. 2, it is interesting to note that the constant “maximum” burning rate constant is achieved when nearly
spherical (1-D) flame is formed. It is also noted that the burning rate constant of PP is larger than that of PE,
which will be discussed in the following section.
3.3. Relationship Between Burning Rate Constants of Different Polymers
From now on, let us examines why the burning rate constant for both polymers are not identical. The
pressure used for comparison is the highest pressure at which spherical flame can form (PP: 12.5 kPa, PE: 10kPa). The difference in the burning rate constants can be attributed to the difference in the heat transferred tothe sample surface, which can be responsible attributed to the difference in flame temperature and the distance
between the flame surface and the sample surface. Huggett 11) et al. found that the heating value per unit ofoxygen consumed during combustion is approximately constant over a wide range of organic materials
normally involved in combustion, and that the flame temperature does not vary significantly with the type ofsample. Thus, it is fair to assume that flame temperature of PP and PE is considered to be identical. Thus, it isconsidered that the difference in the burning rate for both specimens shall be regarded by the difference of theflame location to vary the heat flux to the molten specimen. To validate this assumption, we have attempted
to measure the distance from the sample surface and the flame location in Fig 5.

Figure 5. (a) Diagram of distance between flame surface and sample surface (b) Comparison of distance
between flame surface and sample surface

5 of 9Figure 5 (a) shows the flame model, 𝑇𝑓 is the flame temperature, 𝑇𝑠 is the temperature at the sample
surface, 𝑟𝑠 is the sample radius, 𝑣 is the outward advection velocity, and 𝛿 is the distance between the flameand the sample. Note that the location of the flame is middle of the luminous zone to define as “flame surface”
in this study. The difference in 𝛿 (distance between flame surface and specimen surface) shall directly affectthe burning rate constant to modify the conductive heat flux.
Let us estimate the heat flux at the sample surface which is responsible to the variation of the burning rate
constant. lnder quasi-state assumption, the heat conduction equation over the sample surface shall be written
in Eq. (3).

𝜌𝑐𝑝 𝑣

𝑑𝑇
𝑑2 𝑇
=𝜆 2
𝑑𝑟
𝑑𝑟

(3)

The convection and heat conduction terms are used from the steady-state heat conduction equation, where𝜌 is the density and 𝑐𝑝 is the specific heat. Solving the differential equation yields Eq. (4).

𝑇(𝑟) = 𝐴 + 𝐵𝑒

𝜌𝑐𝑝𝑣
(
𝑟)
𝜆

(4)

Where 𝐴 and 𝐵 are arbitrary constants. The boundary conditions are shown in Eq. (5).

𝑇 = 𝑇𝑠 → 𝑟 = 𝑟𝑠{
𝑇 = 𝑇𝑓 → 𝑟 = 𝑟𝑠 + 𝛿(5)lsing the boundary conditions in Eqs. (4) and (5), the calculation yields Eq. (6).
𝑟 𝛿
( − )

𝑟 𝛿
( 𝑠− )

𝑇(𝑟) = 𝑇𝑠 + (𝑇𝑓 − 𝑇𝑠 ) {𝑒 𝛼 𝛼 − 𝑒 𝛼 𝛼 }Where 𝛼 =

𝜆
𝜌𝑐𝑝 𝑣

(6)

as treated constant in this study. Finally, transforming the equation to find the heat flux, we

obtain Eq. (7).

𝑞𝑠 = 𝜌𝑐𝑝 𝑣(𝑇𝑓 − 𝑇𝑠 )𝑒𝑥𝑝 (𝑟𝑠 − 𝛿
)𝛼(7)From Eq. (7), it clearly revealed that the heat flux decreases with increasing 𝛿 and increases with decreasing
𝛿 . These results suggest that the heat flux transmitted to the sample surface is determined from theexperimental results of 𝛿 and the value of 𝛿 in the heat conduction equation.
Lastly let us obtain 𝛿 in case the burning specimen is varied based on the flame surface model developedin diffusion flame theory. The flame surface model implies that the flame is formed where the fuel and oxygenin the air mix well to reach stoichiometric condition, and the position of the flame surface can be defined by
the mixing fraction 𝑍 at the stoichiometric condition, 𝑍𝑠𝑡 , as shown in Fig. 6.

6 of 9Figure 6. Flame surface model over the burning specimen

Denoting the fuel and oxidizer before combustion as 𝑌𝐹,1 , 𝑌𝑜2,2 , respectively, the conditions for the part ofthe mixture of fuel and oxidizer are given by Eq. (8).

𝑌𝐹,𝑢 = 𝑌𝐹,1 𝑍{
𝑌𝑂2,𝑢 = 𝑌𝑂2,2 (1 − 𝑍)

(8)

Reactions are assumed as complete combustion to describe as following manner:

𝜈𝐹 ′[𝐶𝑚 𝐻𝑛 ] + 𝜈𝑂2 ′[𝑂2 ] → 𝜈𝐶𝑂2 "[𝐶𝑂2 ] + 𝜈𝐻2𝑂 "[𝐻2 𝑂](9)Where, 𝑣 represents the stoichiometric coefficient. The stoichiometric coefficient, expressed in terms of thechange between fuel and oxidizer, molar mass 𝑊𝑖 [𝑔⁄𝑚𝑜𝑙 ], is given by Eq. (10).

𝑑𝑌𝑂2 𝜈𝑂2 ′𝑊𝑂2=
=𝜈𝑑𝑌𝐹
𝜈𝐹 ′𝑊𝐹(10)

From Eq. (10), using the stoichiometric coefficients to show the scalar volume after and before combustion,
we obtain Eq. (11).

𝜈𝑌𝐹 − 𝑌𝑂2 = 𝜈𝑌𝐹,𝑢 − 𝑌𝑂2,𝑢(11)

In Eq. (11), the right side is before combustion whereas the left side is after combustion. From these equations,
𝑍 becomes Eq. (12) when considering the mixing state.

𝑍=𝜈𝑌𝐹 − 𝑌𝑂2 + 𝑌𝑂2,2
𝜈𝑌𝐹,1 + 𝑌𝑂2,2(12)

From Eq. (12), the stoichiometric mixing fraction 𝑍𝑠𝑡 is obtained as in Eq. (13) because 𝜈𝑌𝐹 − 𝑌𝑂2 = 0 is theflame surface, and 𝑍𝑠𝑡 is obtained as in Eq. (13) because various decomposition gases are generated in thepresent sample.

7 of 9∑𝑛𝑘=0 𝑌𝐹(𝑘+1),0 −1
𝑍𝑠𝑡 = [1 + 𝜈
]
𝑌𝑂2,0

(13)

𝑍𝑠𝑡 was calculated using these equations, and the decomposition gas species for PP and PE were taken fromreference12), since both PP and PE were pyrolyzed at around 400°C and the decomposition gases were
measured. The respective decomposition gas species are shown in Table 2. The oxygen concentration in theatmosphere was assumed to be 20%.
Table 2. Decomposition gases and calculated information for PP and PE

PPdecompositiongaschemicalformulaPE

decomposition
ratio [%]

molar
mass
[g/mol]decomposition

chemicaldecomposition

gas

formularatio [%]

molar
mass
[g/mol]butene

𝐶4 𝐻8

16.8

56.11

butene

𝐶4 𝐻8

24.6

56.11

butane𝐶4 𝐻10

12.8

58.12

butane𝐶4 𝐻10

19.1

58.12

pentene𝐶5 𝐻1019.7

70.135ethane

𝐶2 𝐻6

12.4

30.07

pentane𝐶5 𝐻12

11.2

72.15

propane

𝐶3 𝐻8

14.8

44.097

hexane𝐶6 𝐻1211.0

84.162The chemical reaction equations are shown in Eq. (14) for PP and Eq. (15) for PE, assuming complete
combustion.

𝐶4 𝐻8 + 𝐶4 𝐻10 + 𝐶5 𝐻12 + 𝐶6 𝐻12 + 37𝑂2 → 24𝐶𝑂2 + 26𝐻2 𝑂(14)

𝐶4 𝐻8 + 𝐶4 𝐻10 + 𝐶2 𝐻6 + 𝐶3 𝐻8 + 21𝑂2 → 13𝐶𝑂2 + 16𝐻2 𝑂(15)lsing Eqs. (10), (13), Table.2, and Eqs. (14) and (15) in the chemical reaction equation, 𝑍𝑠𝑡 for PP is
calculated as shown in Eq. (16) and 𝑍𝑠𝑡 for PE as shown in Eq. (17).

𝑍𝑠𝑡 ≅ 0.61

(16)

𝑍𝑠𝑡 ≅ 0.49

(17)

From Eqs. (16) and (17), it can be seen that the flame is formed on the sample side for PP compared to PE
because 𝑍 = 1 on the sample side and 𝑍 = 0 on the atmosphere side.
4.

Concluding Remarks
Similarity rule is adopted to general polymers, like polypropylene (PP) and polypropylene (PE) to check

whether they can achieve 1-D spherical flame mode. Results clearly showed that, even using PP and PE as
samples, one-dimensional flames were successfully observed under sufficiently low-pressure atmosphere,suggesting that we can reproduce the burning status in microgravity in the laboratory. The burning rate
constants increases as the pressure decreases and show maximum when the spherical flame is achieved. The
burning rate constants were found to be faster in PP than that of PE. This difference shall be attributed by thedifference in the decomposition gas species.
8 of 9

Acknowledgement
This work is supported by Adaptable and Seamless Technology transfer Program through Target-driven
R&D (A-STEP) from Japan Science and Technology Agency (JST) Grant Number JPMJTR22RA and JSPS
Kakenhi (#20H02397).
References
1)

R.A Schlagheck, B.L Trach, “Microgravity research results and experiences from the NASA/MIR space station
program”, Acta Astronautica, Vol. 53, (2003.12), pp. 983-996.
2) Vladimir Pletser, “Short duration microgravity experiments in physical and life sciences during parabolic fights: thefirst 30 ESA campaigns”, Acta Astronautica, Vol. 55, (2004), pp. 829-854.
3) K. Kogure, “8. Microgravity Experimental Facilities and Usage”, J. Plasma Fusion Res, Vol. 83, (2007), pp. 161-163.
4) Nakamura, Y., Yoshimura, N., Matsumura, T., Ito, H., and, Fujita, O, "Flame Spread over Polymer-Insulated Wire in
Reduced Pressure Environments: Similarity to Microgravity Phenomena", Progress in Scale Modeling, SpringerVerlag, New York NY, (2008), pp.17-28.
5) Nakamura, Y., Wakatsuki, K., and Hosogai, A., "Scale Modeling of Space Fire", Journal of the Japanese Society for
Experimental Mechanics (2013 Special Issue), Vol.13, (2013.7), pp.s69-s74.
6) Migita, T., Yamahata, T., Strempfl, P., Matsuoka, T., and Nakamura, Y., "Methodology to Achieve Pseudo 1-D
Combustion System of Polymeric Materials using Low-pressured Technique", Fire Technology (Special Issue;
Spacecraft Fire Safety), Vol.56 No.1, (2020.1), pp.229-245.
7) R. Matsumoto, E. Nagatsu, K. Nozaki, T. Migita, T. Yamahata and Y. Nakamura: Conference of the Japan Society ofMicrogravity Application, 25 (2021).
8) M.Mikami, “Vaporization and Combustion of Fuel Droplets”, Nagare 20, (2001), pp.106-115.
9) J. C. Yang, J.C., A. Hamins, and M. K. Donnelly, “Combustion of a Polymer (PMMA) Sphere in Microgravity,” NISTIR
6331, (1999).
10) J. C. Yang, J.C., A. Hamins, and M. K. Donnelly, “Reduced Gravity Combustion of Thermoplastic Sphere,” Combust.
Flame, Vol. 120: 61-74 (2000).
11) Huggett, C. : Fire and Materials, 4, pp.61-65 (1980).
12) K. Mima, E. Kunioka and H. Koshi: The Society of Safety Engineers of Japan “Safety Engineering”, 63229 (1967) (inJapanese)© 2024 by the authors. Submitted for possible open access publication under theterms and conditions of the Creative Commons Attribution (CC BY) license(http://creativecommons.org/li censes/by/4.0/).

9 of 9