Finite Size Scaling of Flame Spread along Randomly Distributed Combustible Cubes

Flame spread in urban fire depends on quantity of combustible materials and their placement. The purpose of this study is to predict the flame-spread route by the rate of combustible area. In this experiment, we used combustible cube made from filter paper as a test sample, and combustible cubes and pores were randomly distributed in experimental area. Additionally, we examined non-uniform flame spread along combustible cubes in various porosities and experimental areas. Experimental results show that the flame-spread probability falls with increasing porosity, and despite uniform porosity, the flame-spread probability differs with the experimental area. When expanding the experimental area, less porosity is required to obtain zero flame-spread probability. To investigate based on finite size scaling of percolation theory for the change in porosity, we concluded that different thresholds exist for areas of all sizes. Because of the flame-spread progress stops in the thresholds, these porosities may be useful to predict fire hazard.


Introduction
After a huge earthquake, multiple fires proliferate simultaneously in urban areas, while secondary damage is exacerbated by the spread of fire.In fact, several fires broke out after the Great Kanto Earthquake, Great Hanshin Earthquake, Great East Japan Earthquake and others.Urban regions include combustible areas such as wooden structures and plants, as well as the non-combustible, e.g.concrete buildings, roads, parks, and open areas, which mean that flames spread non-uniformly.Moreover, it is difficult to predict the flame-spread route because it may also be affected by the wind direction and terrain and the density of combustible materials.Figure 1 shows a photograph of an example-urban area in Japan and a binary image, where the urban area is di-vided into combustible and non-combustible areas.As shown in Figure 1, combustible and non-combustible areas of various sizes are randomly distributed in an urban area.When a fire starts somewhere, the phenomenon is divided into two cases.One involves the flame spreading and combustible materials burning out, and the other involves the flame self-extinguishing on the way.The threshold for burning out or self-extinguishing may be determined by the quantity and placement of combustible materials.Establishing safety by predicting the flame-spread route is important to reduce secondary damage caused by urban fires.
Many studies have been conducted focusing on flame spread along a uniform solid fuel load as a fundamental component of fire research [1]- [3], but few have examined non-uniform flame spread in mixed combustible and non-combustible materials [4].Some studies have focused on the application of percolation theory to non-uniform flame spread [5]- [7], while a few numerical simulations have also been conducted using a square lattice model [8].However, discussing from an experimental perspective is rare.Accordingly, in our previous studies [9]- [11], we conducted experiments concerning flame spread along a thin combustible solid with randomly distributed pores of different shapes and sizes.We also discussed the relationship between porosity and flamespread probability based on percolation theory which treats connection of random media and refers to the "Introduction to Percolation Theory" written by Stauffer D. et al. about percolation theory [12] [13].The latter is a probability theory dealing with properties of the connection of random media.A square lattice is shown in Figure 2(a), and the dots are written in some squares in Figure 2(b).This dot is called a site, and the group of neighboring squares occupied by these dots is circled as shown in Figure 2(c).Many of the sites become connected into a single large cluster, which spans the entire area vertically or horizontally due to an increase in site occupancy.This is called critical probability P c and known as the percolation threshold.The value of P c differs depending on the lattice shape and is about 59.3% in a square lattice of infinite size.
In this paper, we studied the flame-spread probability along combustible cubes with randomly distributed pores of various porosities.In addition, performing flame-spread experiments by changing the experimental area, we studied the effect of the size of the experimental area on flame spread by investigating the relationship between porosity and flame-spread probability.Furthermore, we introduce the porosity of flame-spread probability set to zero, and discuss the hazards of urban fire.

Experimental Apparatus and Procedure
A schematic of the experimental apparatus and combustible cube are shown in Figure 3.In this study, we used combustible cubes made from filter paper (ADVANTEC, No. 131, 250 µm thick) as a test sample.The experimental area was divided into a square grid, combustible cubes and pores were randomly arranged in each grid and the sample which arranged corner of the experimental area was ignited by a pilot flame.Experiments were conducted in atmosphere and a video camera (SONY, HDR-11, 30 fps) was set up above the experimental area to record the top view image of the flame spread.The experiment apparatus was also covered in vinyl sheeting to eliminate ambient disturbance of the flame spread.The test sample was a hollow cube of sides 20 mm, while the experimental area was square lattice of equivalent size, like the sample width.The number of samples to be placed to obtain arbitrary porosity was specified and their position was determined, using a random function to produce an experimental area of non-uniformly distributed pores and samples.

Experimental Condition
In this study, we selected ten different experimental areas, while the number of grids  In this study, porosity was varied from 40% to 70% to investigate the relationship between flame-spread probability and changes in porosity.

Jump of Flame
The previous study [9]- [11] confirmed the phenomenon whereby the flame spread of a thin combustible solid jumped over a non-combustible area.In this study using combustible cubes, flame spread also featured jumping over non-combustible areas.A flame-jumping test was conducted using two combustible cubes with a noncombustible area as a preliminary test shown in Figure 5 and ignited one sample.It was defined as jumping success if the flame of ignited sample was spread to the unburned sample, and the flame-jumping probability is defined as Equation ( 3):

Number of jumping successes Flame-jumping probability
Number of flame-jumpingtests The relationship between flame-jumping probability and non-combustible distance was plotted in Figure 5.When the non-combustible distance is less than 5 mm, the flame can always propagate, but when the non-combustible distance exceeds 8 mm, the flame cannot propagate.Flame-jumping is related to the pre-heat length, L h , ahead of the flame leading edge, which means a flame can traverse a non-combustible area in d < L h , and not d > L h as shown in Figure 6.We introduced the scale ratio S ≡ d/L h (d: non-combustible distance, L h : pre-heat length) to classify whether the behavior was flame jumping or not, namely whether the flame could jump the noncombustible area for S < 1, and not for S > 1.From these results, in this study with the square lattice of 20 mm width, no flame spread to traverse a pore.

Assumption of S < 1
The condition to satisfy S < 1, namely requiring d < L h , was not met in this study.To study the effect of scale ratio on flame-spread probability, we assumed that the flame spread to the position shown in Figure 7, providing the condition of S < 1. Figure 7 is shown as the flame spread area of S > 1 and the flame-jumping area of S < 1.Where S < 1, in addition to the eight samples in contact with the combusting sample, we assumed the flame would spread to the four samples shown in orange color even if there were a pore between the samples.

Flame-Spread Trajectory
The cluster distribution for three different porosities and the flame-spread trajectory are shown in Figure 8.The image processing techniques of use for the flame-spread trajectory include normal speed photography (30 frames per second).In this figure, the black line is the flame-spread trajectory, which is depicted at intervals of ten seconds after ignition, while clusters mean area so finter connected pores, which prevent the flame-spread progressing.Pores form large clusters with increasing porosity until the flame is eventually extinguished when the cluster forms around the ignition point.For 40% porosity, the combustible area accounted for a large proportion, with little pore connection.For 50% porosity, the percentage combustible and non-combustible areas are half and half, with a large pore connection formed compared to the 40% porosity.Finally, in the case of 60% porosity, with a large pore proportion, the flame is swiftly extinguished.Figure 9 shows an example sample placement and the flame-spread trajectory at the time.Here, the L-shaped slit represents the cluster surrounding the flame source, where self-extinguishing occurred.In Figure 9, the red dashed line represents the L-shaped slit, while the samples shown in orange color are estimated to reach the flame spread.Accordingly, we conceived a method of calculating the extent of flame spread for the presence or absence of the L-shaped slit.In fact, there was no difference in the flame-spread areas to be calculated from the sample placement and observed from the flamespread test respectively, therefore the following results were obtained from this method.The flame-spread probability declines with increasing porosity in any number of grids.However, as the number of grids increases, the flame-spread probability becomes zero shows small porosity.To study the effect of experimental area in terms of its impact on flame-spread probability, we investigated porosity when the flame-spread probability become zero at any number of grids.Figure 10(b) shows the relationship of the number of grids and porosity when the flame-spread probability becomes zero.It can be seen that porosity when the flame-spread probability becomes zero declines with an increasing number of grids, and asymptotically approaches approximately 60%.It can be seen that in Figure 11(a), also S < 1, the flame-spread probability declines with increasing porosity as in the case of S > 1.However, the expanding flame-spread range means more porosity is needed to self-extinguish.In addition, the porosity when the flame-spread probability becomes zero declines with increasing number of grids, as in the case of S > 1 shown in Figure 11(b).

Scaling
We studied based on the percolation theory for the relationship shown in

Figure 1 .Figure 2 .
Figure 1.A photo of an urban area and a binary image of the urban area which is represented by combustible and noncombustible regions.

Figure 3 .
Figure 3. Schematic of the experimental apparatus.

Figure 4 .
Figure 4. Example of a sample placement and definition of the flame spread.

Figure 5 .
Figure 5. Relationship between flame jumping probability and noncombustible distance.

Figure 6 .
Figure 6.Schematic of flame jumping over a non-combustible distance.

Figure 7 .
Figure 7. Flame spread and flame jumping areas.

Figure 10 (
Figure 10(a) plots the flame-spread probability as a function of different porosities in the number of grids, 20 × 20, 40 × 40, 60 × 60, 80 × 80 and 100 × 100 when S > 1.The flame-spread probability declines with increasing porosity in any number of grids.However, as the number of grids increases, the flame-spread probability becomes zero shows small porosity.To study the effect of experimental area in terms of its impact on flame-spread probability, we investigated porosity when the flame-spread probability become zero at any number of grids.Figure10(b)shows the relationship of the number of grids and porosity when the flame-spread probability becomes zero.It can be seen that porosity when the flame-spread probability becomes zero declines with an increasing number of grids, and asymptotically approaches approximately 60%.It can be seen that in Figure11(a), also S < 1, the flame-spread probability declines with increasing porosity as in the case of S > 1.However, the expanding flame-spread range means more porosity is needed to self-extinguish.In addition, the porosity when the flame-spread probability becomes zero declines with increasing number of grids, as in the case of S > 1 shown in Figure11(b).

Figure 9 .Figure 10 .
Figure 9.Comparison of flame-spread test and flame-spread areas to be calculated from sample placement.