OR4-2
炭素繊維強化プラスチックの下方燃え拡がり挙動のモデル化
Modeling the downward flame spread over carbon fiber
reinforced plastics
花本慶 1, 岡村康希 1, 小林芳成 2, 高橋周平 2
Kei HANAMOTO1, Kouki OKAMURA1, Yoshinari KOBAYASHI2 and Shuhei TAKAHASHI2
岐阜大学大学院自然科学技術研究科エネルギー工学専攻，Department of Energy Engineering, Graduate School of
Natural Science and Technology, Gifu University,
2 岐阜大学工学部機械工学科，Department of Mechanical Engineering, Faculty of Engineering, Gifu University
1

1. Introduction
Flame spread over solid materials has been extensively studied to investigate the effects of shape,
thermophysical properties, and ambient atmospheres on the flame spread characteristics, such as flame spreadrate, flammability limit, and so on. For example, Fernandez-Pello et al. investigated the effect of thickness ofmaterials on flame spread rate1). Takahashi et al. examined the effect of ambient atmospheres and found thatthe flame spread rate reached the maximum under oxygen and argon mixtures 2). These studies addressedpolymers ─e.g. polymethyl methacrylate (PMMA) ─which are thermally isotropic mono-materials. Suchmaterials have low thermal conductivity of less than 1 W/m/K, and therefore the solid-phase heat transfer isvery small. However, there are some objects where the solid-phase heat transfer should be considered, forexample, electrical wires consisting of polymer insulations and a metal core 3). Flame spread over the electricalwires has been studied, while that over flat high-thermal-conductivity materials has not been well understood.
In recent years, carbon fibers (CFs) attract a lot of attention and are used extensively because they havesuperior properties, e.g., high strength, high electric conductivity, and low thermal expansion. Carbon fiber
reinforced plastics (CFRPs) are an example of composite materials that consists of the CFs and applied to avariety of products. The CFs are found to have very high thermal conductivity, and therefore the CFRPs arealso a high-thermal-conductivity material. However, the literature on flame spread over the CFRP is very
limited, and therefore the flame spread behaviors are still not well understood. In recent years, Kobayashi etal addressed the flame spread over thermally thin CFRP sheets and found that the preheat zone of CFRP ismuch larger than that of PMMA due to the high thermal conductivity of the CFs 4,5). Matsukawa et al.
investigated flame spread on CFRP sheets with different CF orientations, focusing on the effect of CF
orientation on the flame spread characteristics6). It was found that the CF orientation angle controls theforward heat transport and that the effect of this angle has a significant influence on the flame spread. In
addition, the effect of the size of the no combustion zone in the side was clarified by Okamura et al7). In thesestudies, the simplified flame spread model proposed by Kobayashi et al.4,5) used experimental values for flame
length when determining the flame spread rate. This work derived a theoretical equation for the flame spread
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rate of the CFRP by employing the predicted flame length on PMMA under natural convection developed by
Bhattacharjee et al8). That equation was numerically solved to obtain the theoretical value of the flame spread
rate. Finally, by comparing the theoretical and experimental values of the flame spread rate, the validity of thepredicted equation of flame length and the flame spread model of CFRP was verified.
2. Experiment2.1. Carbon fiber reinforced plastic sheets with different carbon fiber orientations
The CFs are classified into two main types: polyacrylonitrile (PAN)-based CFs and petroleum pitch-based
CFs. Their chemical structures are different, thereby resulting in different physicochemical properties. The
pitch-based CFs have higher thermal conductivity than the PAN-based ones, and therefore the solid-phaseheat transfer is greater in the flame spread over pitch-based CFRPs. This study then selected the pitch-based
CFRPs as a test sample. A variety of differently oriented CFRP sheets, as illustrated in Fig. 1, were fabricatedby laminating two unidirectional CF sheets impregnated with epoxy resins, i.e., prepregs (Nippon Graphite
Fiber, NT91500-525S) in different direction and curing them in a high temperature furnace (Yamato Scientific,
FO810) at 403 K (130 ℃) for 1h. CF orientations of those CFRP sheets are symmetric for longitudinal directionof CFRP sheets. Specifications of the used prepreg are listed in Table 2.1. The prepared CFRP sheets can be
categorized as “thermally thin” because of the high thermal diffusivity of ~2.3 × 102 mm2/s.

θ

Figure 1. Schematic of carbon fiber reinforced plastic (CFRP) sheets with different carbon fiber (CF)orientation: (a) CFRP [0°] and (b) [40°].
Table 1. Specifications of unidirectional CF sheets impregnated with epoxy resin.

Manufacturer / Model

Nippon Graphite Fiber / NT91500-525S

Type of thermosetting resins

Mesophase pitch-based continuous
carbon fibers
Epoxy resins

Thickness

0.11 mm per sheet

Fiber areal weight

150 g/m2

Resin content

25 wt.%

Thermal conductivity（CF）
Thermal conductivity（epoxy resin）

500 W/m/K
0.3 W/m/K

Type of carbon fibers

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2.2. Apparatus for downward flame spread tests
The fabricated CFRP sheets were cut 120 mm long by 22 mm wide and inserted vertically into a stainlesssteel sample holder (Fig. 2). In addition, in this experiment, the heat transport through the CFRP to the sample
holder was limited by placing a heat insulator between the sample holder and the CFRP. Specifications of theused heat insulator are listed in Table 2. Note that this work studied the downward flame spread to facilitate
an analysis as much as possible. The size of combustion part was 120 mm long by 20 mm wide. A 0.1-mmthick nichrome wire was equipped with the sample holder to ignite the CFRP sheets. The nichrome wire wasenergized with 150 W (20 V × 7.5 A) and then turned off once a self-sustaining flame spread was recognized.
Locating the sample holder with the CFRP sheets in the glovebox allowed oxygen concentration in theatmosphere to vary. The flame spread tests were conducted in variable oxygen concentrations at a totalpressure of 0.1 MPa. Pressure and oxygen concentration were constantly monitored during the flame spread
tests via a manometer (SIBATA, DM-1) and an oxygen meter (JIKCO, JKO-25LD3), respectively. Note that the
glovebox was so large (0.8 × 0.8 × 1.2 m3) that a decrease in oxygen concentration due to combustion was
small enough to be negligible. This work defined “flame spread” if a flame could spread and reach the point
which was 4 cm above from the bottom edge of the CFRP sheets and “no flame spread” if a flame was extinct
before reaching the above point. Flame spread behaviors were recorded via a video camera (Sony, HDR-CX470,
FDR-AX45), and the in-plane temperature distribution during flame spread was visualized via an infrared
camera (Nippon Avionics, InfRec S25). The flame spread tests were repeated at least three times for eachcondition to quantitatively assess the experimental uncertainty.
Igniter (NiCr coil)

Heat
insulator

130 mm

120 mmFlame

CFRP
Sample15 mm 20 mm
22 mm

holder(SUS)

Figure 2. Schematic of the downward flame spread tests.

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Table 2. Specifications of heat insulator.

Manufacturer / Model
Blending ratio of ingredients

ISOLITE INSULATING PRODUCTS Co. Ltd. / BSF paper S
SiO2: 70-80%, MgO: 18-27%

Thickness

1 mm

Density

210 kg/m3

Thermal conductivity

0.10 W/m/K @ 673.15 K,
0.22 W/m/K @ 1073.15 K

0.15 W/m/K @ 873.15 K

3. Basic theory of flames
3.1 Derivation of theoretical values for flame length
The simplified flame spread model in buoyant flow, shown in Fig. 3, was proposed by Subrata Bhattacharjee
et al8). To obtain an expression for the flame length, we assumed that the fuel is completely pyrolyzed andconsumed within the extent of the flame length Lf. Therefore, the amount of fuel consumption can be obtained
from a solid phase mass balance as 𝑚𝑓̇ ′ = 𝜌𝑠 𝑉𝑓 𝜏. Assuming theoretical combustion, the oxygen consumption
rate per unit width (the problem is 2D) is given as:
𝑚̇ ′𝑂𝑥 = 𝑠𝜌𝑒 𝑉𝑓 𝜏.

(1)

The length of the flame is dependent by the transport by diffusion of just enough oxygen for the complete
combustion of the evaporated fuel. The oxygen concentration gradient in the perpendicular direction (ydirection in Fig. 3) can be scaled as:𝜕𝑥𝑂𝑥 𝑥𝑂𝑥,∞ − 0 𝑥𝑂𝑥,∞
∼
~
𝜕𝑦
𝛿𝑂𝑥
𝛿𝑓

(2)

where 𝛿𝑂𝑥 is the characteristic oxygen diffusion length in the y direction. We will use Hf to represent thecharacteristic diffusion length in the perpendicular direction, which can be estimated by equating theconvection time scale in the x-direction with the diffusion time scale in the y-direction:
𝐿𝑓 𝐻𝑓 2
𝛼𝑔 𝐿𝑓
𝛼𝑔 𝐿𝑓
∼
→ 𝐻𝑓 ~√
~√
.
𝑉𝑓
𝛼𝑔
𝑉𝑒𝑓𝑓
𝑉𝑔

(3)

Although the effective velocity seen by the flame Veff depends on the boundary layer development length xd,
we ignore this effect by using Veff ~ Vg in simplifying Eq. (3). The oxygen diffusing toward the flame, therefore,
can be estimated by using Eqs. (2) and (3) as:𝑥𝑂𝑥,∞

̇ ′ ~𝜌𝑔 𝐷𝑂𝑥 (
𝑚𝑂𝑥,𝑑
𝛿

𝑓

) 𝐿𝑓 ~𝜌𝑔 𝑥𝑂𝑥,∞ √𝛼𝑔 𝑉𝑔 𝐿𝑓 .

(4)

where we assumed the Lewis number to be unity. Equating the oxygen consumed from Eq. (1) with theestimate of oxygen diffusion of Eq. (4), we obtain:2𝑠𝜌𝑒 𝑉𝑓 𝜏
1
𝐿𝑓 =
(
) .
𝛼𝑔 𝑉𝑔 𝜌𝑔 𝑥𝑂𝑥,∞(5)

To obtain an expression for flame length for downward flame spread and flame height in a quiescent
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environment, the buoyancy-induced velocity can be determined from the scaling of the vertical inertial term:𝜌𝑔 𝑉𝑔,𝐵 2 /𝐿𝑓 ~ (𝜌∞ − 𝜌𝑔 ) 𝑔 , which produces 𝑉𝑔,𝐵 ~√∆𝑇𝑔𝐿𝑓 /𝑇∞ , where the flame length is assumed to be therelevant length scale over which the characteristic induced flow develops, and ∆𝑇~𝑇𝑓 − 𝑇∞ is thetemperature difference driving the buoyant flow. Substituting this expression for Vg in Eq. (5) and solving for
Lf,B, we obtain:
4

1
3

𝑇∞

𝑠𝜌𝑒 𝑉𝑓 𝜏 3
𝐿𝑓,𝐵 = ( 2) () .
𝛼𝑔 𝑔∆𝑇
𝜌𝑔 𝑥𝑂𝑥,∞(6)

Similarly, to obtain an expression for flame height Hf,B for downward flame spread in a quiescent environment,
substituting this Lf,B and Vg in Eq.(3) :1

1

𝑇∞ 𝛼𝑔 3 𝑠𝜌𝑒 𝑉𝑓 𝜏 3
𝐻𝑓,𝐵 = (
) (
) .
𝑔∆𝑇
𝜌𝑔 𝑥𝑂𝑥,∞(7)

Thus, theoretical equations for flame length and flame height for downward flame spread in a quiescentenvironment are obtained. However, if this equation is used as is, the theoretical and experimental values
differ significantly. Therefore, the experimental constants were given so that the theoretical values of flame
length and flame height matched the experimental values, respectively.
Vg

Lf

Flame

Lg
Veff

Hf

Lg

y

δf

τ

CFRP

Vf

x
Figure. 3. Schematic of the buoyant-flow flame spread in flame-fixed coordinates.

3.2 Flame height analysis method
In general, the flame height (Hf) is a distance between the bottom of the flame and the CFRP surface, as
shown in Fig. 3. However, as shown in Fig. 4, the thickness of the sample holder prevents us from observing
the underside of the flame in the experiment, and therefore we attempted to approximate the flame height as
the distance between the flame top and the CFRP surface (δf in Fig. 3). However, this attempt failed. As theoxygen concentration increases, the flame reached the sample, increasing the heat transfer to the CFRP.
However, as the oxygen concentration increased, the distance between the flame top and the specimen
increased, and δf increased (Fig. 3.4). The experimental results also show that Vf has no effect on δf we measured(Fig. 5). Moreover, even if we had applied our measured flame height to Eq. (7), Vf would have disappearedin the energy balance equation that ultimately determines Vf. Therefore, we will solve the energy balance
5 of 14equation using Hf, which is independent of Vf, devised by Kobayashi et al5). in the next chapter.

Figure 4. Example of flame height analysis.

Figure 5. Correlation between experimental flame height and flame spread rate.

3.3. Thermal conductivity of CFRP
The thermal conductivity of CFRP (s) varies with the magnitude of θ. The thermal conductivity of CFRP
here is the thermal conductivity in the longitudinal direction of the sample. As θ increases, more heat flows
laterally along the direction of the carbon fiber, so the thermal conductivity in the longitudinal direction (𝜆𝑠,𝜃 )must be shown with the magnitude of θ. 𝜆𝑠,𝜃 is geometrically formulated as follows:𝜃(7)𝜆𝑠,𝜃 = 𝜆0,𝜃 × 2 .

where 𝜆𝑠,𝜃 is 347 W/m/K at θ = 0deg. The thermal conductivities for each crossing angle are shown in Table
3.
Table 3. Thermal conductivity of CFRP sheets in the direction of flame spread.

𝜆𝑠,0

𝜆𝑠,10

𝜆𝑠,20

𝜆𝑠,30

(W/m/K)(W/m/K)(W/m/K)(W/m/K)

347

345.7

341.7

335.2

4. Results
4.1. Experimental results of flame length
𝐿𝑓 was measured by applying the in-house Python image-processing code. As shown in Fig. 6, 𝐿𝑓 became
longer as oxygen concentration are increased. 𝐿𝑓 would be proportional to mass flux of fuel vapors. Increasing
6 of 14

oxygen concentration elevates the flame temperature to facilitate the pyrolysis. Consequently, the mass fluxof fuel vapors is increased with oxygen concentration, resulting longer 𝐿𝑓 . On the other hand, the net
conductive heat transfer rates, which contribute to the flame spread, decrease with increasing 𝜃 because ofincreased heat losses sideways. In other words, lower heat transfer rates only produce lower mass flux of fuelvapors, so we expected 𝐿𝑓 becomes shorter as 𝜃 is increased, but the results show that θ has little effect on
𝐿𝑓 .

Figure 6. Flame length of CFRP sheets at CF orientation angles of 0 deg., 10 deg., 20 deg., and 30 deg.

4.2. Experimental results of flame spread rate and correlation between flame length and flame spread rate
Flame spread rate (𝑉𝑓 ) was measured by tracking the flame leading edge via an in-house Python imageprocessing code. Vf decreased as θ increased and increased as the oxygen concentration increased in Fig. 7. As
θ increases, the Vf is considered to decrease because the flame spread in the longitudinal direction of thesample is suppressed due to the decrease in pre-heat zone at CFs (Fig. 8). In this study, to understand how theCFs work in the flame spread, the in-plane temperature distribution during flame spread was visualized via
the IR camera. To quantify the solid-phase preheat zone, the length of the solid-phase preheat zone, i.e., thesolid-phase preheating length, was measured by processing the IR images via an in-house Python imageprocessing code. Note that this work defined the distance from the flame leading edge to the point where thenon-dimensional temperature 𝛩 = (𝑇 − 𝑇∞ )/(𝑇𝑣 − 𝑇∞ ) reached a value of equal to 20% of Tv as therepresentative preheat length. If the pyrolysis and ambient temperatures (𝑇𝑣 and 𝑇∞) are 670 K and 293 K,
respectively, then the preheat length is the distance to the point at the temperature of 352K. Vf rate is also
considered to increase as the oxygen concentration increases, because the higher oxygen concentration
increases the flame temperature and the amount of heat transferred through the carbon fiber. Next, thecorrelation between Lf and Vf is shown in Fig. 9. As can be seen from this figure, as Lf increases, so does Vf. This
7 of 14

result compares favorably with Eq. (6) above.

Figure 7. Flame spread rate of CFRP sheets at CF orientation angles of 0 deg., 10 deg., 20 deg., and 30 deg.

Figure 8. Preheat length of CFRP sheets at CF orientation angles of 0 deg., 10 deg., 20 deg., and 30 deg.

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Figure 9. The correlation between flame length and spread rate.

5. Discussion
5.1. Comparison of theoretical flame length with experimental values and determination of experimental
constants
Figure 10 compares the theoretical and experimental values of flame length calculated by Eq. (6). From thisfigure, the theoretical and experimental values are positively correlated. In response to this result,experimental constants were determined so that the theoretical values match the experimental values. In thepresent study, the intercept was set to 0 when Vf = 0, i.e., when there is no flame spread, the flame length isalso considered to be 0:
𝑇∞

1
3

4

𝑠𝜌𝑒 𝑉𝑓 𝜏 3
𝐿𝑓 = 0.0339 ( 2) () .
𝛼𝑔 𝑔∆𝑇
𝜌𝑔 𝑥𝑂𝑥,∞(9)

Using this corrected 𝐿𝑓 , the energy balance equation is solved for Vf in the next section.

Figure 10.

Comparison of theoretical flame length with experimental values
9 of 14

5.2. Energy balance equations in the burned and pyrolysis regions
Equations for energy balance in the burned and pyrolysis regions are formulated to determine the rate of
spread of combustion for each oxygen concentration at each crossing angle(θ). The following simplified flame
spread model was proposed by Kobayashi et al5)．The possible heat transfer rates involve.
𝑄1 , 𝑄2 : incident heat transfer rates from flame (convection and radiation), and𝑄3 , 𝑄4 : heat transfer rates through the material.
Note that this model does not consider the forward gas-phase heat transfer because it is low enough to be
negligible as compared to the solid-phase heat transfer (𝑄2 and 𝑄4 ).

yBurnedResiduezone

Pyrolysis
Lf

Airflow

Lpy

Vr = Vf +Vg ~ Vg
Lg
Q1

Q3

Hf

Q2

CFRP

Lg

τ

Q4

x

T
Tmax
Tv
T∞

−Lf

−Lpy

0

x
Ls

Figure 11. Schematic of simplified flame spread model involving the solid-phase heat transfer withsimulated surface temperature profile.

Formulating energy balance in each zone gives the following equations:
In burned zone:
𝑄1 = 𝑄2 .

(10)

In pyrolysis zone:
𝜌𝐶 𝜏𝑉𝑓 𝐿𝑣 = 𝑄2 +𝑄3 − 𝑄4 .

(11)

Eq. (10) does not involve conductive heat loss backward from the flame, i.e., downstream, because this model
assumes that the high temperature of the burned material is still maintained behind the flame. The flame
incident heat in the burned zone (𝑄1 ) is therefore transferred to the pyrolysis zone through the material.
Substituting Eq. (10) into Eq. (11) to eliminate the conductive heat transfers (𝑄2 ) yields an analytical solutionof flame spread rate as:

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(𝑄1 + 𝑄3 ) − (𝑄4 ),𝜌𝑠 𝜏𝐿𝑣(12)

𝑄4 = 𝜌𝐶 𝑐𝐶 (𝑇𝑣 − 𝑇∞ )𝜏𝑉𝑓 ,

(13)

𝑉𝑓 =

𝑄1 + 𝑄3,
𝜌𝐶 {𝑐𝐶 (𝑇𝑣 − 𝑇∞ ) + 𝐿𝑣 }𝜏

(14)

𝑄1 = 𝜆𝑔𝑇𝑓 − 𝑇𝑚𝑎𝑥(𝐿𝑓 − 𝐿𝑝𝑦 ),
𝐻𝑓

(15)

𝑄3 = 𝜆𝑔𝑇𝑓 − 𝑇𝑣
𝐿𝑝𝑦 ,
𝐻𝑓

(16)

𝑉𝑓 =

which suggests that the CFs just work as a heat conductor to transfer the heat flux forward, and that the flame
spread over CFRP is driven by the flame heat flux transferred by the CFs. We then have only to consider theincident heat transfer rates from the flame. Here, the solid-phase heat transfer rate from the burned zone tothe pyrolysis zone (𝑄2 ) is represented as:

𝑄2 = 𝜆𝐶

𝑇𝑚𝑎𝑥 − 𝑇𝑣
𝜏.
𝐿𝑝𝑦

(17)

From Eq. (10), (14), and (16), the maximum temperature (𝑇𝑚𝑎𝑥 ) is then:𝑻𝒎𝒂𝒙 =𝝀𝒈 (𝑳𝒇 − 𝑳𝒑𝒚 )𝑳𝒑𝒚 𝑻𝒇 + 𝝀𝑪 𝝉𝑯𝒇 𝑻𝒗
𝝀𝒈 (𝑳𝒇 − 𝑳𝒑𝒚 )𝑳𝒑𝒚 + 𝝀𝑪 𝝉𝑯𝒇

.

(18)

The flame length (𝐿𝑓 ) in Eq. (18) is obtained by multiplying Eq. (6) by an experimental constant: As for theflame height (𝐻𝑓 ), as mentioned above, attempts to define 𝐻𝑓 as the distance between the flame top and thesample surface (δf) have failed. Therefore, a closed-form equation for flame shape in counter-current flame
propagation was proposed by Bhattacharjee et al.9) and the following equations were adopted for 𝐻𝑓 andpyrolysis region (𝐿𝑝𝑦 ).

𝐻𝑓 =1 1 𝑇𝑓 − 𝑇𝑣
𝐿 ,5 𝑥𝑂𝑥,∞ 𝑇𝑣 − 𝑇∞ 𝑔

(19)

𝑇𝑓 − 𝑇𝑣
1
𝐿𝑣
𝐿 .
2 𝑐𝑒 (𝑇𝑣 − 𝑇∞) 𝑇𝑣 − 𝑇∞ 𝑔

(20)

𝐿𝑝𝑦 =

Substitute Eqs. (13), (15), (16), and (18) into Eq. (13) to obtain Vf at each crossing angle and oxygen concentration.
5.3. Validation of the revised flame spread model
To validate the developed flame spread model, the flame spread rate (𝑉𝑓 ) of CFRP sheets (CF and epoxy) is
calculated via that model with the physicochemical properties listed in Table 4 and compared with themeasured 𝑉𝑓 . Here, the air temperature was taken as the average of the pyrolysis temperature of CFRP, 670 K,
and the adiabatic flame temperature, and the physical properties in Table 5 at these temperatures were
substituted when determining 𝑉𝑓 . Figure 12 shows a comparison of the calculated and experimental values of
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𝑉𝑓 calculated by the Eq. (14) in Section 5.2. The calculated value is considerably smaller. As can be seen in thisresult, the calculated value of Vf does not change at all when the crossing angle (θ) of the carbon fibers ischanged. This indicates that the change in the crossing angle has little effect on the effective thermal
conductivity of the CFRP in the present model. Since there is still little data on the thermal conductivity of
CFRP with different crossing angles, as described in Section 3.3, the thermal conductivity in the longitudinal
direction was formulated geometrically from the blend angle in this study. The effective thermal conductivity
was formulated geometrically based on θ. It is necessary to collect research data on the thermal conductivityof composite materials in order to elucidate to what extent this effective thermal conductivity (𝜆𝑒𝑓𝑓 ) affects
flame length and spread rate, and to improve the model. Assuming Lg = αg/Vg,B for the flame height (Lf) in Eq.
(19), the flame height (Hf) approaches zero as Lf increases. This means that the flame gradually approaches the
sample, but, as it gets closer, the sample loses heat (wall quenching effect) and eventually extinguishes theflame. In other words, there is a distance at which the sample cannot be approached any closer (quenching
distance) in order to sustain the spread of flame. We will add a constraint condition. The quenching distanceis about 0.9 mm based on the laminar flame speed of methane, the main volatile component of epoxy 10). This
was added as an intercept to the equation for flame height, and the flame spread rate was recalculated.
𝐻𝑓 =1 1 𝑇𝑓 − 𝑇𝑣
𝐿 + 0.0009.
5 𝑥𝑂𝑥,∞ 𝑇𝑣 − 𝑇∞ 𝑔

(21)

Figure 13 compares the calculated and experimental Vf derived by correcting for Hf. When θ is 0 deg. or 10
deg., there is no significant difference between the calculated and experimental values, indicating that themodel has been successfully corrected. On the other hand, as θ increases, the error between the calculated andexperimental values increases. As mentioned above, the lateral heat loss due to θ is a significant factor in Vf,
so accurate measurement of 𝜆𝑒𝑓𝑓 is still necessary for modeling CFRP with large crossing angles.
Table 4. Physicochemical properties of CF (0 deg.) and epoxy
𝜏

𝜌𝐶

𝜌𝑒

𝑐𝐶

𝑐𝑒

𝜆𝐶

𝜆𝑒

𝑎𝑒

𝑇𝑣

𝐿𝑣

(kg/m )

(J/kg/K)(J/kg/K)(W/m/K)(W/m/K)2(mm)(kg/m )

(m /s)(K)(MJ/k)

CF

0.22

1780

_

870

_

335.2~347

_

_

_

_

epoxy

_

_

1255

_

1046

_

0.188

1.43×10-7

670

1.417

3

3

Table 5. Physicochemical properties of air
𝜌𝑔

𝑐𝑔

𝜆𝑔

𝑎𝑔

(kg/m )

(J/kg/K)(W/m/K)(m2/s)

0.203~0.216

1762~1881

0.0856~0.0888

2.25×10-4~2.326×10-4

3Air

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Figure 12. Comparison of theoretical and experimental flame spread rate.

Figure 13. Comparison of corrected theoretical and experimental flame spread rate.

6. Conclusion
In this study, a theoretical equation for flame spread rate was derived from the theoretical equation for flame
length, and the validity of the downward flame spread model was examined by comparing it withexperimental data. The theoretical flame length equation was obtained from the scale analysis and comparedwith the experimental data to develop a flame length equation with a good tendency. The flame spread rate
was derived from the flame length equation and the flame spread model. However, the calculated flame
spread rate was considerably smaller than the experimental value. The validity of the flame spread model was
confirmed when the crossing angle of CFs was small (0 deg.~10 deg.) by setting a constraint of quenching
distance on the flame height. On the other hand, when the crossing angle is large (20 deg.~), the flame spread
velocity is greatly affected by the decrease in thermal conductivity of the carbon fiber in the flame spread
direction and heat loss to the side. It is important to collect data on this effective thermal conductivity in orderto modify the model accurately. In this study, the flame length, which was previously represented byexperimental values in previous models, is represented by theoretical values. The findings of this study have
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a significant role in the development of theoretical downward flame spread models.
Nomenclature
𝑉𝑓flame spread rate𝜏material thickness

Veff

effective velocity seen by the flame

𝑇𝑓

flame temperature𝜌𝑔gas-phase density

𝑇𝑚𝑎𝑥

maximum temperature𝜌𝑠solid-phase density𝑇𝑣pyrolysis temperature𝑐𝑔

gas-phase specific heat𝑇∞ambient temperature𝑐𝐶

solid-phase (CF) specific heat𝐿𝑣

latent heat of vaporization𝑐𝑒solid-phase (epoxy) specific heat𝐿𝑓flame length

𝜆𝑔gas-phase thermal conductivity

𝐿𝑝𝑦

pyrolysis zone length

𝜆𝐶solid-phase thermal conductivity

𝐻𝑓flame height

𝜆𝑒solid-phase thermal conductivity

𝛿𝑓distance between top of flame and CFRP

𝛼𝑔gas-phase thermal diffusivity

𝐿𝑔thermal diffusion length𝛼𝐶solid-phase thermal diffusivitysstoichiometric ratio

𝛼𝑒solid-phase thermal diffusivity

𝑥𝑂𝑥,∞

mass fraction of oxygen

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