3.1. Flame Shape and Structure
Figure 2 shows flame shapes of PMMA with different overlap lengths, i.e., 20 mm, 30 mm, 40 mm and 50 mm. It can be seen from this figure that during the process of flame spread over vertical discrete PMMA, the flame has experienced an incremental process from the ignited sample plates mainly relying on discontinuous flame spread in heat radiation to stable and continuous combustion. In the initial stage of flame spread, the flame adheres closely to the PMMA plate and there is no significant flame fluctuation, which is consistent with Jiang et al.’s results concerning upward flame spread over continuous PMMA plate [
17]. As the burning time extends, the shape of the flame changes from an initial rectangle to a jagged shape, a significant periodic flame pulsation can be seen and a great increase in the intensity of flame fluctuation can also be observed. Similar phenomena were observed in the study of Meng et al. [
18], who investigated upward flame spread characteristics of discrete thermal insulation material under the influence of porosity. When two or three PMMA plates are in combustion simultaneously, the phenomenon of flame fusion is observed. This phenomenon was also found in the study of Zhang et al. [
19], who investigated the flame characteritics of double pool fire.
As the overlap length increases, the PMMA combustion becomes more intense. At the same time, it can be seen that the flame height and flame area become larger correspondingly. The reason is that a larger overlap length of the PMMA plate leads to higher pyrolysis rate, and thus there is sufficient combustible gas released, causing the fully intense burning. Moreover, a decrease in the intensity of combustion at the same time period is observed when the overlap length is larger than 40 mm, compared with that under other experimental conditions. Firstly, this might be attributed to the fact that the overlap length is so large that it can be ignited by the heat radiation of inner flame of the front plate under the lower part of the spreading plate. Secondly, when the overlap length is larger than 40 mm, the end of the spreading plate will be directly ignited by the outer flame of the front plate, leading to its pyrolysis and flame spread.
When the PMMA sample burns to the end, entrainment phenomenon will appear in the top flame. Especially when the overlap length is larger, the phenomenon becomes more significant. Then part of the fire plume bends because of the air entrainment, which is also one of the factors affecting the flame height.
3.2. Flame Height
The pyrolysis height of vertical PMMA plate array with different overlap lengths are shown in
Figure 3. The flame height is mainly determined by the rising height of pyrolysis gas under the effect of buoyancy and the ignition point when it mixes with the air [
20,
21].
Figure 3 shows that the flame height was small, and the fluctuation of the flame was not obvious in the initial stage. It increased gradually, while sudden changes happened under some experimental conditions. As the PMMA flame continued to spread, the flame height fluctuated more and more significantly. Especially in the stage when the flame of one PMMA plate spreads to the adjacent plate, i.e., the discrete flame spread stage, the flame height changed obviously. The overall increasing of flame height was observed. This is mainly because in the discrete flame spread stage, the adjacent PMMA plate is ignited by heat radiation, and the flame is not in the stable combustion stage, intensifying the fluctuation of the flame height.
In addition, the videos recording flame spread over discrete PMMA were processed using a MATLAB program proposed by us to obtain the real-time flame height. Further, the average flame height of vertical PMMA plate array with different overlap lengths was calculated and presented in
Table 5 [
22]. Moreover, the standard deviation of flame height is calculated from
Figure 3, and the results are also shown in
Table 5. The standard deviation could reflect the dispersion degree of the data, and then reflect the fluctuation of the flame height.
It can be deduced from
Table 5 that with the increase of overlap length, the average flame height increases gradually first, and then decreases obviously after 40 mm. This is different from the conclusions of Zhao et al. [
23], who investigated the effects of sample thickness on the flammability of continuous PMMA plate. They found that the average flame height increased with an increase in sample thickness. Miller et al. [
14] studied the flame spread behavior over discrete PMMA plates, which were separated with several noncombustible plates. They defined a parameter named “fuel coverage” as the ratio of noncombustible plate length to the total length of discrete PMMA plates. They also found that the average flame height increased first and then decreased with the increase of the fuel coverage. It demonstrates that when the overlap length is large, the flame height is significantly affected by the overlap length. The difference in the average flame height between 20 mm and 30 mm overlap length is 10.79 mm. This means that when the overlap length is smaller than 30 mm, the influence of the overlap length on the flame height is not significant. However, the difference is 79.31 mm between 30 mm and 40 mm overlap length, and the difference is 39.02 mm between 40 mm and 50 mm overlap length. This indicates that when the overlap length is larger than 30 mm, the effects of the overlap length on the flame height is more significant. In addition, as the overlap length increases, the standard deviation of flame height approximately increases first and then decreases after 40 mm, indicating the similar changing trend of flame fluctuation. This demonstrates that the flame fluctuation is more significant for
D = 40–50 mm, compared to that when
D = 20–40 mm.
The non-linear change of the average flame height with the overlap length can be attributed to the coupling of stack effect and heat release rate. Firstly, a partially enclosed vertical channel is formed when two adjacent plates overlap with a certain spacing, and thus the stack effect occurs, which could promote the air flow in the channel. According to An et al.’s work [
24], the flow velocity of the induced airflow in the vertical channel is:
where
H is the height of the vertical channel,
is the hydraulic diameter of the vertical channel, and
. In this paper, the height of the channel is equal to the overlap length, i.e.,
H =
D. As the overlap length rises, the flow velocity of the induced airflow increases according to Equation (1), further leading to the increase of the average flame height. However, when the overlap length increases to a certain value, the heat release rate may become the dominant factor determining the average flame height. Zukoski et al. [
25] proposed the relation between the flame height and heat release rate, as shown in Equation (2).
where
is the total flame height, and
h is sample characteristic length. In this paper,
h = 100 mm.
is the dimensionless heat release rate,
denotes heat of combustion, and
is specific heat.
is ambient temperature, and
is radiation fraction. When the overlap length continues to increase after 40 mm, mutual limiting effects of two adjacent PMMA plates on air entrainment cause inadequate oxygen supply, which leads to inadequate combustion and a lower heat release rate. According to Equation (2), the lower heat release rate corresponds to the lower flame height. Therefore, the average flame height decreases with the increase of overlap length after 40 mm.
3.3. Flame Spread Rate
The pyrolysis height of vertical PMMA plate array with different overlap lengths are shown in
Figure 4. The mothed employed to obtain
Figure 4 is explained as follows. Firstly, an infrared video was obtained using an infrared camera. Secondly, the infrared pictures were selected 25 frames per second from the infrared video. Thirdly, the pyrolysis height of each infrared picture would be obtained through the CAD ratio method. Finally, the pyrolysis heights under different time were obtained and are depicted in
Figure 4.
Conducting linear curve fitting of
Figure 4, it can be obtained that the flame spread rates (i.e., the slope of the fitting line) of PMMA plate array under 20 mm, 30 mm, 40 mm and 50 mm overlap lengths are 1.32 mm/s, 1.34 mm/s, 1.67 mm/s, 1.55 mm/s, respectively. With the increase of overlap length, the discrete flame spread rate increases first and then decreases. The flame spread rate for 40 mm overlap length is much larger than that of 20 mm and 30 mm, while there is little change in flame spread rate as the overlap length rises from 20 mm to 30 mm. This phenomenon demonstrates that the flame spread rate will be influenced significantly with the overlap length when it increases to a certain value [
26]. This influence can be maximized as the overlap length is 40 mm, while there is a significant reduction in the impact between the overlap length from 20 mm to 30 mm.
When the overlap length is 40 mm, the flame spread rate reaches the maximum. From the flame shape in
Figure 2, it is found that when the overlap length reaches 40 mm, the flame of the front plate begins to touch the end of the spreading plate. The temperature of the outer flame is about 500 °C, while the pyrolysis temperature of PMMA plate in this experiment is 350 °C. Therefore, the PMMA plate end will be ignited by direct heating on the basics of the heat radiation, leading to a sharp rise in the flame spread rate.
Further, the reason for the changing trend of flame spread rate is explained. The discrete flame spread rate is closely corelated with the ignition time (
) of the adjacent PMMA plate. The ignition times under different experimental conditions obtained from infrared video are listed in
Table 6. As the overlap length increases, the ignition time first decreases and then increases. When
D = 40 mm, the ignition time is the shortest. This demonstrated that the discrete flame spread rate is negatively corelated with the ignition time.
The prediction model of ignition time of PMMA plate exposed to constant uniform heat flux was proposed by previous researcher [
27]:
where
is heat conduction coefficient of air.
and c denote the density and specific heat capacity of PMMA, respectively.
is the ignition temperature of PMMA,
is the ambient temperature, and
is the external heat flux.
From Equation (3), the ignition time is negatively corelated with the heat flux. Therefore, it is deduced that the flame spread rate of discrete PMMA plate is mainly determined by the heat flux transferred to its surface. The heat flux under different experimental conditions will be calculated in
Section 3.4.
Further, the model establishing for predicting flame spread rate of discrete PMMA array is presented as follows. Based on the phenomena observed in the experiment and the continuous fire spread model established by previous researchers, the physical model of vertical flame spread over discrete PMMA array is established, as shown in
Figure 5.
For both discrete flame spread and continuous flame spread, the length of the preheating zone is the flame height subtracting the pyrolysis height, i.e., Equation (4). The relation between the flame height and pyrolysis height is Equation (5) [
28].
where
is the preheating zone length,
is the flame height, and
is the pyrolysis front height, respectively. The values of
k and
n can be found in Hasemi ’s work [
28].
Based on the flame spread rate model of a continuous solid proposed by Quintiere [
29], this paper establishes the mathematical model for flame spread rate of discrete PMMA array, assuming that the flame heat flux (
) received by PMMA surface within the preheating zone is constant. Quintiere proposed the following equation to calculate the flame spread rate of continuous thermal-thick solid.
Substituting Equation (3) into Equation (6), Equation (7) is obtained.
Miller et al. investigated the correlation between discrete and continuous flame spread rate [
14]. In their work, several noncombustible plates were inserted into the PMMA plate, and they defined a parameter named “fuel coverage” as the ratio of noncombustible plate length to the total length of discrete PMMA plates. Further, the relation between discrete flame spread rate
and
was proposed by Miller et al., as shown in Equation (8).
where
is the fuel coverage. In this work, noncombustible plates were not inserted into the PMMA plate. On the contrary, the discrete PMMA plates overlapped with a certain length (
D). Therefore,
in this work is defined as follows.
where
h is the PMMA plate length. Substituting Equations (4), (5), (7), and (9) into Equation (8), Equation (10) is obtained.
Employing Equation (10), the average value of
x in
Table 5, and the data of
in
Table 6, the predicted flame spread rate of discrete PMMA array can be calculated. Both the calculated and experimental flame spread rate are shown in
Figure 6. From
Figure 6, it is found that the changing trend of predicted flame spread rate with the overlap length is consistent with the experimental one. The prediction error is smaller than %, within the acceptable error range of engineering.
3.4. Temperature Field
Figure 7 shows a front view of the temperature field of vertical PMMA plate array with different overlap lengths during combustion. The green part in
Figure 7 is the pyrolysis zone of PMMA plate, with a temperature of 350 °C. When the PMMA is ignited, the pyrolysis zone gradually expands and spreads to adjacent PMMA plate [
30]. At 120 s, for the overlap lengths of 40 mm and 50 mm, the phenomenon that the pyrolysis area spreads to the third PMMA plate was observed. Among them, the pyrolysis area accounted for about half of the third plate area for 50 mm overlap length, while the third plate in the 40 mm overlap length have completely become the pyrolysis area. For the overlap length of 40 mm, the width of the pyrolysis area of the first two plates was obviously larger than other experimental conditions, and the PMMA plate was sufficiently pyrolyzed. At the 120 s, for the overlap lengths of 20 mm and 30 mm, there is no pyrolysis area spreading to the third plate. It indicates that larger overlap length corresponds to higher flame spread rate, which leads to the increase of the flame height and flame area. At 100 s, only the pyrolysis area under the 40 mm overlap length spreads to the third plate while the pyrolysis area of other three experimental conditions remains in the second plate. it can explain that the discrete flame spread rate decreases as the overlap length rises when it is relatively large [
31].
After the PMMA plate is ignited, the pyrolysis area gradually increases, and the flame height and flame area also gradually rise. In addition, the flame of PMMA plate array with a larger overlap length can spread to adjacent plates without complete pyrolysis [
32]. For example, for 40 mm overlap length at 40 s, the green area of the first plate only accounts for half of the overall plate area, while in the second plate the green area has already spread to the top of the plate. On the contrary, for the experimental condition with a smaller overlap length, the discrete flame spread is very difficult to keep. For 20 mm and 30 mm overlap length at 120 s, the first two plates were nearly completely pyrolyzed, while the pyrolysis was not observed in the third plate.
As shown in
Figure 8, for the 40 mm overlap length, the difference in the time for temperature rising between thermocouple 2 and thermocouple 1 was smaller among the four experimental conditions. The temperature change indicates the flame spread rate, which shows the maximum value of the discrete flame spread rate is observed when the overlap length is 40 mm.
The temperature field can be further processed with the software of the infrared camera. The flame area was selected, and the average temperature of this area could be obtained using the IR software. The results are shown in
Figure 9, which demonstrates that the average temperature increases first and then decreases as the overlap length rises, reaching the maximum value when
D = 40 mm.
Further, the average temperature can be used to calculate the radiative heat flux from the flame to the preheating zone of PMMA. According to the Stefan–Boltzmann’s law, the radiant heat flow received by the preheating zone from the opposite flame can be expressed as:
where
is the view factor,
is the Boltzmann’s constant,
is the emissivity of PMMA.
= 5.67 ×
and
= 0.92 [
33]. In this work, the average value of the average temperature shown in
Figure 9 is deemed as
.
The view factor can be obtained using the following integration:
where
and
denote the angle between the radiation ray and the straight line perpendicular to the elements
and
,
R is the distance between the elements
and
. For two parallel plates with a certain spacing (
L) and overlap length (
D), Equation (12) could be transformed to the following equation:
where
= 8 mm and
h = 100 mm. Substituting Equation (13) into Equation (11), the radiative heat flux from the flame to the preheating zone of PMMA can be obtained and shown in
Table 7. As discussed in
Section 3.3, the flame spread rate of discrete PMMA plate is mainly determined by the heat flux transferred to its surface, including radiative and convective heat flux. Zhao et al. [
34] indicated that for upward flame spread over vertical PMMA plate, the convective heat transfer plays an important role when the spacing is small, while the radiant heat transfer is dominant when the spacing is larger. The plate spacing in this work is 8 mm, and thus the flame spread rate is mainly determined by radiant heat transfer. From
Table 7, the radiative heat flux increases first and then decrease as the overlap length rises, reaching the maximum value when
D = 40 mm. This could explain the changing trend of the flame spread rate in
Section 3.3.