The Effect of Separation Distance Between Informal Dwellings on Fire Spread Rates Based on Experimental Data and Analytical Equations

All dwellings had a floor area identical to that of the ISO 9705 room (3.6 m × 2.4 m), similar to ISDs tested by previous researchers [16], and had a height of 2.3 m, similar to [3, 6, 17]. The dwelling of fire origin (denoted as ISD1) was designed to be as simple as possible, in order to reproduce a consistent fire across all experiments. Note that ISD1 was kept the same for all experiments. Hence, ISD1 only had a single door opening with internal dimensions of 2.05 m (height) × 0.85 m (width), as depicted in Fig. 3. ISD1 was clad with galvanized IBR (Inverted Box Rib) sheeting (0.8 mm thick) and had a steel frame. The sheeting was fixed to the steel frame with self-drilling hex head screws. The flute height of the IBR sheeting is 36 mm and all leakages (i.e., openings created by the corrugation of the steel sheets) were closed with ceramic blanket and checked prior to each experiment. The steel frame consisted of 38 mm × 38 mm × 2 mm square hollow sections that were welded together on site.

The target dwellings (denoted as ISD2) were clad with galvanized corrugated steel sheeting (0.5 mm thick) and were fixed to a timber frame (where the cross-sectional area of the timber pieces used were 50 mm × 50 mm) using self-drilling hex head screws. The timber lengths were fixed together using self-tapping hex head screws. The flute height of the corrugated sheeting is 18 mm and all openings were closed with ceramic blanket. In both experiments, ISD2 had two openings, which included a door in the long wall with internal dimensions of 2.05 m (height) × 0.85 m (width) and a window in the short wall with internal dimensions of 0.6 m (height) × 0.85 m (width), placed 1.25 m from the ground, as depicted in Fig. 3. As depicted in Fig. 4, the separation distance between ISD1 and ISD2 for Experiment 1 (Exp 1) and Experiment 2 (Exp 2) were 1 m and 1.75 m, respectively. In addition to the change in separation distance, the window opening in the short wall was closed with a 3 mm single glazed glass, for Exp 2 (which does not affect the initial fire spread between dwelling that is being investigated in this paper). In both experiments, an isolated standing wall (ISD right wall denoted as RW) was erected at a distance from ISD2, as depicted in Fig. 4. The RW was also constructed from corrugated steel sheets, fixed to a steel frame. For Exp 1 and Exp 2 the distance between ISD2 and RW were 2 m and 1.5 m, respectively.

The fuel load in real ISDs varies substantially and can be anything from 410 MJ/m² to 2000 MJ/m² [18]. Even though the fuel load range significantly, it is clear from previous work that these dwellings are ventilation controlled [3, 6‐8, 16] and that these dwellings collapse before the fuel burns out [17], implying that fuel load alone would not have a substantial effect on the fire dynamics and heat fluxes produced once the fully developed stage is reached.

In this work, two pine timber cribs (each was approximately 130 kg) were placed 0.2 m from the long wall (see Fig. 4) with a separation distance of 0.4 m between the cribs and the short side walls, as depicted in Fig. 4. Each crib consisted of 10 layers of 10 sticks with a dimension of 0.04 m × 0.06 m × 1.2 m and an approximate density of 450 kg/m³. The moisture content of the timber was 4%.

For free burning conditions, these cribs would have had a maximum surface-controlled mass loss rate (MLR) of 0.39 kg/s combined according to [19]. The heat of combustion (HoC) of timber used was 17 MJ/kg as measured by a bomb calorimeter, thus giving a maximum surface-controlled heat release rate (HRR) of 6.7 MW. It is well-known that the ventilation-controlled MLR is given by the ventilation factor multiplied by a factor of 0.12 [19, 20]. Unlike pool fires than can burn in an enclosure in a highly fuel-rich manner, timber cribs have a burning limit of approximately 37% fuel rich [17]. The ventilation-controlled limit is thus presumed to account for the combined effects of heating and vitiating the crib intake air. In this case, the ventilation-controlled HRR of ISD1 and ISD2 are 5 MW and 5.99 MW, respectively. Hence, in both cases the dwellings were expected to be ventilation-controlled.

In addition to the timber cribs, ISD2 was lined with cardboard (for both experiments) to mimic reality [3]. In order to keep the fire as reproduceable as possible, ISD1 was not lined with cardboard. It was however regarded as important to line ISD2 with cardboard as the lining material is typically associated as the primary fuel of ignition [3, 6, 16], and/or significantly affect the growth phase of the fire. Figure 5 visually depicts cardboard and cribs in ISD2.

ISD1 was ignited by igniting 8 small bags filled with a 400 mm × 400 mm piece of kerosene-soaked hessian, with one bag placed in each of the bottom 4 corners of each crib.

In both experiments a variety of measurements were captured. In both ISD1 and ISD2, the MLR data was captured using scales with an accuracy of 10 g (the MLR of each timber crib was measure individually in these experiments). The gas temperatures that were recorded during these experiments were measured using K-Type thermocouples (1.5 mm tip diameter). Incident radiation was measured using Thin Skin Calorimeters (TSCs) that were calibrated against a water-cooled heat flux gauge, stationed at various distances from the door of ISD2 during the full-scale experiments. Gas flow velocities were captured at the door and window for ISD2 using bi-directional flow probes. The temperature data was captured at 0.2 Hz and the voltage data (flow probe and scale measurements) was captured at 1 Hz. The equipment layout, the equipment key and the equipment detailing of the right wall are depicted in Fig. 6. The abbreviation in brackets next to each instrumentation icon indicates the type of instrumentation, where Thin Skin Calorimeter is abbreviated as TSC, Thermocouple is abbreviated as TC and bi-directional Flow Probe is abbreviated as FP. The number next to the instrumentation abbreviation (Thin Skin Calorimeter → TSC, Thermocouple → TC, and Flow Probe → FP) indicates the number of instruments in a particular equipment tree location. The instrumentation spacings of the equipment trees are depicted in Fig. 7. Note that the flow probes were orientated perpendicular to all openings (i.e. from left to right of Fig. 4 for the door and top to bottom of Fig. 4 for the window) and parallel (i.e. from the bottom to top in Fig. 4) to the allay.

The sections above describe the experimental setup of Exp 1 (Fig. 4), the equipment configuration (Fig. 6), the wind conditions during the experiment (Fig. 2a) and the fuel content of both dwellings (Fig. 5). ISD1 was ignited at the 8 corners of the two cribs. After ignition, a fully developed fire stage was reached in approximately 4.3 min. For approximately 1 min before the fully developed fire stage was reached, flames were seen emerging out of the door causing flame impingement onto ISD2 within seconds after emerging (Fig. 8a). Approximately 30–60 s later (after the fully developed stage was reached), an exposed timber frame piece ignited externally on the left wall of ISD2 (Fig. 8a). This rapidly led (approximately 30 s) to the cardboard lining of ISD2 igniting (Fig. 8b). It should be noted that the ceramic blanket used to seal the leakages is not completely impermeable, hence some smoke did accumulate at the ceiling level, in ISD2, prior to ignition (Fig. 8b). At this stage flames spread across the cardboard (Fig. 8c), which generated enough heat to ignite the cribs and flashover ensued approximately 2.4 min after the cardboard ignited. ISD2 reached a fully developed fire stage approximately 2.8 min after the cardboard lining ignited (Fig. 8d).

Figure 9 depicts the HRR for both ISD1 and ISD2 along with the calculated ventilation-controlled HRR, and Fig. 10 depicts the ceiling temperatures for ISD1. The label ISD1_Roof1 refers to the ceiling thermocouple closest to the front wall of ISD1, and ISD1_Roof2 refers to thermocouple furthest from the front wall of ISD1. Figure 6 depicts both the location of the scales and the location of the ceiling thermocouples.

The calculated ventilation-controlled MLR is based on the ventilation factor, \(A_{0} \sqrt {H_{0} }\), where \(A_{0}\) is the total area of the openings and \(H_{0}\) is the area-weighted equivalent opening height), multiplied by a factor of 0.12 to account for the combined effects of heating and vitiating the crib intake air, as discussed in above. In this case, the calculated ventilation-controlled HRR (HRR = MLR × HoC) is lower than the maximum HRR measured (indirectly by the scales) during the experiments. This could be a result of some amount of leakage that was present during the experiment and that the ceramic blanket did not act as a perfectly impermeable material. The maximum recorded HRR in ISD1 and ISD2 were 6.5 MW (assuming a HoC of 17 MJ/kg, a ventilation factor of 3.2 is required to reach a HRR of 6.5 MW) and 8.4 MW (assuming a HoC of 17 MJ/kg, a ventilation factor of 4.1 is required to reach a HRR of 8.4 MW), respectively. The average ceiling temperature of ISD1 during the fully developed stage (between 4 min and 16 min) was 901°C and the maximum recorded ceiling temperature was 963°C. Real ISDs are often highly permeable (possibly far in excess of that observed here), due to imperfect materials and construction means available, meaning that ventilation in reality is very difficult to define. Furthermore, leakages may be stopped up with combustible draught-stoppers (e.g., newspaper pressed into holes), which burn through rapidly, leading to quick increases in ventilation.

Figure 11a–d depict the back left, back right, front left and front right thermocouple tree readings, respectively. The annotation that this paper uses for front wall and back wall is shown in Fig. 4. The positions of the thermocouple trees are depicted in Fig. 6. Each tree had 6 thermocouples labelled from 1 (top TC) to 6 (bottom TC), with the following spacing: 2200 mm, 2150 mm, 2050 mm, 1900 mm, 1700 mm and 1550 mm from floor level. The TC trees were constructed by fixing the TCs to a steel cable and suspending the cables from the roof perimeter beams of ISD2, by fixing the cables to the timber with screws. After thorough investigation of the data and video footages, it is clear that the sudden dips in temperatures across the height of the TC trees are as a result of the steel cables falling to ground level. Unfortunately, the charring of the timber resulted in the screws pulling out. Hence, temperatures after the trees falling are not depicted in Fig. 11a–d. Ignition shown on all graphs below references to the ignition of the cardboard lining and not ignition of the exposed timber frame.

The flow fields at the window and door were also captured with bi-directional flow probes and are depicted in Fig. 12a and b, respectively.

Considering the door velocities between 8 and 11 min (i.e., from the start of the fully developed stage up to 11 min where an increase in wind speeds causes the neutral plane to shift, which is describe in detail in the section below) the neutral plane is around 1180 mm (i.e., just above FP4). The neutral plane can be calculated and compared to the measured value. In “Appendix 1”, an analytical equation is derived to calculate the neutral plane for these ISDs. By substituting \(W_{w} = 0.85\) m, \(\rho_{g} = 353/T_{g}\), where the measured gas temperature is approximately 1223 K, \(H_{WT} = 1.85\) m, \(H_{WB} = 1.25\) m, \(W_{D} = 0.85\) m, \(\rho_{a} = 353/T_{a}\), where the measured ambient temperature is approximately 297 K, and \(H_{D} = 2\) m, into Equation (11) yields a neutral plane height of \(H_{N} = 0.94\) m. This is 320 mm lower than the measured neutral plane height of approximately 1180 mm. It should be noted that the neutral plane height calculated above (also see “Appendix 1”) did not account for the effect of wind. In this case, the wind direction (refer to Fig. 2a), which is in the -x direction as depicted in Fig. 30, increases the outside pressure on the face that it is blowing against (as depicted in Fig. 30) resulting in an increased neutral plane height, and causes suction on the other faces resulting in an decrease in neutral plane height on those faces.

Thoroughly investigating Fig. 12a and b a strange phenomenon can be seen, where approximately between 11 min and 13 min, there is a sudden increase in the outwards flow through the door and a sudden increase in inwards flow through the window. This phenomenon can also be explained in terms of the effect of wind on the pressure profile. By overlaying the wind speed with the window velocities (Fig. 13), one can clearly see that the spike in wind speed (blowing from south to north) increased the pressure at the window (as depicted in Fig. 30) which resulted in the ambient pressure rising at the window, thus retarding the flow of smoke and flames from the window opening and essentially holding the hot gases within the dwelling. However, a build-up of pressure is created within the dwelling, and this is sufficient to drive the flow of smoke and flames out of the door opening.

Figure 14a depicts the incident radiation measured at the left wall of ISD2 at roof height. Figure 14b depicts the incident radiation measured at the RW at a height of 2 m, 2 m away from the window at a height of 1.8 m and 4 m away from the window at a height of 1.8 m. For the locations of the TSCs refer to Fig. 6.

The heat fluxes measured at 2 meters away from the door (Fig. 14b) is similar to the heat fluxes measured by previous researchers [3, 16]. The average incident heat flux onto the left wall of ISD2 (Fig. 14a), during the fully developed stage of ISD1 until ignition of ISD2 was approximately 33 kW/m². It should be noted that the cable of the TSC placed on the left wall of ISD2, was exposed to flames once the cardboard of ISD2 ignited (at approximately 5.5 min), thus the results after the ignition of ISD2 should be interpreted accordingly. The RW (2 m from ground level), 2 m from the window of ISD2 (1.8 m from ground level) and 4 meters from the window of ISD2 (1.8 m from ground level) reached a maximum heat flux of 62 kW/m², 28 kW/m² and 17 kW/m², respectively.

The peak in heat flux onto the RW at approximately 8 min is as a result of the cardboard lining burning similar to what was observe in previous experimental work [3, 6]. At approximately 13–15 min, the roof sheets start to open (as seen from video footage taken from above) allowing the hot gases to escape and the flame sizes from the openings to reduce, and this is evident in the heat flux curve recorded by the RW TSC.

The sections above described the experimental setup of Exp 2 (Fig. 4), the equipment configuration (Fig. 6), the wind conditions during the experiment (Fig. 2b) and the fuel content of both dwellings (Fig. 5). ISD1 was ignited at the all 8 corners of the two cribs. After ignition a fully developed fire stage was reached in approximately 3.5 min. At this stage, flames were seen emerging out of the door causing some pulsed flame impingement onto ISD2. Within approximately 7 min (after ignition of ISD1), an exposed timber frame piece ignited on the left wall of ISD2 (Fig. 15a), which led to the cardboard lining of ISD2 igniting (Fig. 15b) at approximately 8.75 min. The flames spread across the cardboard (Fig. 15c) generating enough heat to ignite the cribs and flashover ensued within approximately 2 min after the cardboard ignited. ISD2 reached a fully developed fire state approximately 2.4 min after the cardboard lining ignited (Fig. 15d).

Figure 16 depicts the HRR for both ISD1 and ISD2 along with the calculated ventilation controlled HRR, and Fig. 17 depicts the ceiling temperatures for ISD1. The label ISD1_Roof1 refers to the ceiling thermocouple closest to the front wall of ISD1, and ISD1_Roof2 refers to thermocouple furthest from the front wall of ISD1. Figure 6 depicts both the location of the scales and the location of the ceiling thermocouples.

In this case, the calculated ventilation-controlled HRR is lower than the maximum HRR measured during the experiments. This most likely implies that some amount of leakage was still present during the experiment and that the ceramic blanket did not act as a perfectly impermeable material, as discussed for Exp 1. The maximum recorded HRR in ISD1 and ISD2 were 7.9 MW (for an extremely short-lived peak) and 7.9 MW (before collapse), respectively. Although, these peaks are extremely short and it is clear that the average MLR of ISD1 is lower than that of ISD2. The average ceiling temperature of ISD1 during the fully developed stage (between 4 min and 16 min) was 856°C and the maximum recorded ceiling temperature was 966°C.

Figure 18a–d depict the front left, front right, back left and back right thermocouple tree readings, respectively. The positions of the thermocouple trees are depicted in Fig. 6 and the spacings are the same as for Exp 1. As mentioned earlier, the sudden dips in temperatures across the height of the TC trees are as a result of the steel cables falling to ground level. Unfortunately, the charring of the timber resulting in the screws pulling out. Hence, temperatures after the trees falling are not depicted. Ignition shown on all graphs below references to the ignition of the cardboard lining and not the exposed timber frames.

The flow fields at the window and door were also captured with bidirectional flow probes and are depicted in Fig. 19a and b, respectively. It should be noted that the delay in gas flow through the window is as a result of the installed glass in the opening. Thus, looking at Fig. 19a, there is no flow through the window when the fully developed stage was reached. However, within approximately 1 min the glass window broke and flow started as one would expect, as depicted in Fig. 19a.

Considering the door velocities between 11 min and 15 min the neutral plane is around 900–1000 mm, with some fluctuations which is most likely as a result of the wind (Fig. 2). This is very similar to the calculated neutral plane of 940 mm, as for Exp 1 above (i.e., using Eq. 11).

Figure 20a depicts the incident radiation measured at the left wall of ISD2 at flute height. Figure 20b depicts the incident radiation measured at the RW at a height of 2 m, 2 m away from the window at a height of 1.8 m and 4 m away from the window at a height of 1.8 m. For the locations of the TSCs refer to Fig. 6.

The heat fluxes measured at 1.5 meters away from the door (Fig. 20b) is similar to the heat fluxes measured by previous researchers [3, 16]. The average incident heat flux onto the left wall of ISD2 (Fig. 20b), during the fully developed stage of ISD1 until ignition of ISD2 was approximately 27 kW/m². The RW, 2 m from the window of ISD2 and 4 m from the window of ISD reached a maximum heat flux of 52 kW/m², 23 kW/m² and 13 kW/m² (before 16 min), respectively.

Table 1 provides a summary of important parameters pertaining to both experiments, specifically those parameters and readings effecting and affected by separation distance, such as time-to-ignition of ISD2, time to flashover after ignition of ISD2, and the average heat fluxes emitted, by ISD2, during the fully developed stage. All incident heat fluxes listed are average heat fluxes during the fully-developed fire stage (until structural collapse), except for the average incident heat flux onto ISD2, which is the average between the fully developed stage of ISD1 and ignition of ISD2.

It is clear that an increase in separation distances leads to a decrease in the average heat flux emitted onto ISD2 (i.e. 33 kW/m² at 1 m to 27 kW/m² at 1.75 m), which results in a slower time-to-ignition. It should also be noted that ISD1 of Exp 2 reached a fully developed fire stage slightly faster than that of Exp1, implying that the time-to-ignition of ISD2 for Exp 2 is actually slightly more slower than that of Exp 1. The increase in separation distance from 1 m to 1.75 m had no noticeable effect on the time from ignition of ISD2 to flashover in ISD2. One would expect that the cardboard would preheat faster when closer to the fire source, hence leading to more rapid spread across the surface once ignited. However, from the results, this does not seem to be the case. The heat fluxes emitted from the window of ISD2 for Exp 2 are slightly less than that of Exp 1. It can relatively easily be seen that the contribution of ISD1 to the heat flux readings measured in front of the window decreases as ISD2 moves further away from ISD1.

The radiating surface of the hot gases will take the shape of the door opening of ISD1. Hence, the integration should be over the surface of the door. Using the coordinate system depicted in Fig. 33 (see “Appendix 2”), Equation (16) in “Appendix 2” now looks as follows:

By substituting \(y_{2} = 1.45 \,{\text{m}}\), \(z_{2} = 2.3 \,{\text{m}}\) and \(s = 1 \,{\text{m}}\) into Eq. (1) and solving the integrals in Matlab yields \(\emptyset = 0.13\). Since the height and width of a door is relatively standard, only the separation distance is varied in Fig. 21. This provides a brief parametric study useful for understanding the influence of the separation distance on the radiation received by ISD2 (at \(y_{2} = 1.45 \,{\text{m}}\) and \(z_{2} = 2.3 \,{\text{m}}\)) from the hot compartment gases.

Figure 21 shows that as the separation distance decreases, the configuration factor decreases approximately exponentially, implying that the incident heat flux would also decrease exponentially (the minimum distance of 0.5 m is an approximation of the minimum external dimensions out of a door).

Using the average temperature of the hot layer as measured in both Exp 1 and Exp 2 during the fully developed fire stage of ISD1 as 879°C, assuming the emissivity of the compartment gases is equal to 1 [17, 21], and the configuration factor of 0.13, the incident radiation due to the compartment gases is calculated as 13.1 kW/m² (using only the part of Equation (12) that applies to the compartment gases).

The configuration factor of the flames can be calculated by making some assertive assumptions. Firstly, it is assumed that the flames ejecting from the door only starts at the height of the neutral plane [0.79 m for ISD1, as calculated using Eq. (11)], secondly it is assumed that the width of the flame is approximately equal to the width of the door, and lastly, it is assumed that the height of the flames are equal to the average recorder flame height (2.5 m from video footage), i.e., during the fully developed stage. Equation (16) now looks as follows:

By substituting \(y_{2} = 1.45\,{\text{m}}\), \(z_{2} = 2.3\,{\text{m}}\) and \(s = 1 \,{\text{m}}\) into Eq. (2) and solving the integrals in Matlab yields \(\emptyset = 0.23\). It should be noted here that in this case it was assumed that s is from ISD1 to ISD2, but in reality, s should be taken from the center or edge of the flame. This however significantly complicates the equation and would lead to even more assumptions. That is, since the center of the flame varies with height and since the flame also fluctuates in height and length in reality. Hence, this would require more assumptions and would be less functional should this method be employed in a Monte Carlo-type simulation in the future. This assumption is slightly unconservative. Assuming the height and width of the flame remains relatively constant, and only varying the separation distance, Fig. 22 is obtained.

From Fig. 22, it is interesting to see that the configuration almost reduces to zero at approximately 4 m away from ISD1, implying that the external flame has virtually no effect on the incident radiation if the target is 4 m away from the burning dwelling. It should be noted that this is only the case for ignition at a height of z₂ = H. For other target heights, the configuration factor will change.

Using the average temperature of the door flames (measured at ISD2 next to each FP), as measured in both Exp 1 and Exp 2 during the fully developed fire state of 892°C (unfortunately the flame temperatures at the door of ISD1 were not measured, but it was measured at the door of ISD2, hence as a best alternative, it is assumed that the door flame temperatures for ISD1 and ISD2 are similar). Thus, the average temperature as measured by the FP thermocouples at the door of ISD2 for bot Exp 1 and Exp 2 are used here, assuming the emissivity of the flames is equal to 1 [17, 21] (which is a conservative assumption), and the configuration factor of 0.23, the incident radiation due to the door flames is calculated as 23.6 kW/m² (using only the part of Eq. (12) that applies to the door flames).

One can assume that the configuration factor for the IBR sheeting of the right wall of ISD1 (Fig. 31) is approximately the same as the configuration of IBR sheeting of right wall of ISD1, if the right wall of ISD1 had no door and was fully cladded with IBR sheeting minus the configuration factor of the door alone. Additionally, for simplicity, it has been assumed that the IBR acts as a flat emitter (i.e., the IBR does not have a corrugated surface but acts as a flat plate). Substituting \(y_{2} = 1.45 \,{\text{m}}\), \(z_{2} = 2.3 \,{\text{m}}\), \(H_{r} = 2.3 \,{\text{m}}\), \(L_{r} = 3.6 \,{\text{m}}\) and \(s = 1 \,{\text{m}}\) into Eq. (16) and solving the integrals in Matlab yields \(\emptyset = 0.41\) (i.e., the configuration factor of the right wall with no door). Figure 23 shows the configuration for a variety of separation distances, dwelling heights and dwelling lengths. This provides a brief parametric study useful for understanding the influence of the emitter’s geometry.

Figure 23 clearly shows the effect of the spacing between dwellings. The configuration factor can be halved by just increasing the separation distance from 1 m to 2.7 m. Figure 23 also immediately highlights the danger of double storey informal settlement dwellings. Increasing the dwelling height from 2.5 m (single storey) to 5 m (double storey) leads to a 64.8% increase in configuration factor.

The configuration factor of the IBR sheeting of the right wall of ISD1 can be approximated as:

Using the average temperature the IBR sheeting, as measured in both Exp 1 and Exp 2 during the fully developed fire stage of ISD1 as 585°C (where the measured temperatures of these experiments are relatively constant across the height of the dwelling), assuming the emissivity of the galvanized sheeting is equal to 0.32 (i.e., 0.22–0.28 for galvanized steel at 0–200°C [22] and 0.42 for galvanized steel at 1400°C [22]), thus assuming 0.2567 + (0.42 − 0.28)/(1400 − 200) × 585 = 0.32 for galvanized steel at 585°C), and the configuration factor of 0.28, the incident radiation due to the steel sheets is calculated as 3 kW/m² (using only the part of Eq. (13) that applies to the IBR sheets).

Based on the equations derived in “Appendix 2” and those used in the sections above, Figure 24 shows the contribution of each of ISD1’s radiating surfaces (components). It is clear that the flames from the door has a substantial effect on the total incident radiation and that the radiation from the sheets is almost negligibly small compared to the other components. Since the radiation emitted by the flame has such a dominant impact on calculations, the factors which affect its behavior most are of importance. During the experiments the wind did affect the flame orientation, although with the pulsating flame effect it is difficult to exactly quantify the effects of the wind. If the flame tilts towards the target ISD it will result in higher radiation, and a reduced time-to-ignition. This requires further research and the inclusion of methodologies to account for the influence of wind.

Over the past few decades a significant amount of research [23‐25] has focused on the development of correlations between time-to-ignition and incident radiation data for combustible materials, where most correlations are developed from Cone Calorimeter data. These correlations are generally based on a theoretical model of heat conduction into an inert solid as a result of the incident radiation and the losses due to radiation and convection to varying degrees of complexity. Comparing all the time-to-ignition methods is outside the scope of this work, however, a number of researchers have summarized these different correlations and it is available in the literature should the reader require further information [26‐28]. Based on work done by Baker et al. [29], it was decided to use the Flux-Time Product (FTP) method to determine the time-to-ignition in this work. For piloted ignition of combustible materials, the FTP method is a simplified way to predict time-to-ignition of the material under a time-varying incident radiation (\(\dot{q}_{inc}^{\prime \prime } )\). The theory behind the FTP method is that the FTP value of a material exposed to an external heat flux (that is greater that the critical heat flux (CHF) of the material) accumulates until it exceeds a threshold value at which point the material ignites. The method was originally derived by Smith and Satjia [24], which other researchers further developed upon. The method was then generalized by Shields et al. [21] such that:where \(t_{ig}\) is the time-to-ignition (s), \(\dot{q}_{inc}^{\prime \prime }\) is the incident heat flux (kW/m²) [as calculated using Eq. (12)], \(\dot{q}_{cr}^{\prime \prime }\) is the critical heat flux (kW/m²), and FTP index (n) is obtained by iteratively varying n to get the best linear fit for \(1/t_{ig}^{n}\) versus \(\dot{q}_{inc}^{\prime \prime }\). In this case, the only materials of importance are the timber used for the frames of the ISDs and the cardboard used for internal lining. Piloted ignition measurements from the cone calorimeter experiments for both the cardboard and timber used in the large-scale experiments can be seen in Fig. 25. By varying n, the trendline with the highest correlation factor (R²) can be obtained. In this case, the n value that gave the best correlation for the cardboard and timber are n = 1.1 and n = 1.6, respectively. The gradients of the trendlines are equal to the respective FTP^1/n values and the y asymptotes can be assumed to be the CHF of the materials.

As mentioned earlier, when a material is exposed to an external heat flux, the FTP value accumulates until it exceeds a threshold value at which point the material ignites. The mathematical formulation that describes this is a variant on Eq. (4) such that for every time step when \(\dot{q}_{inc}^{\prime \prime }\) exceeds \(\dot{q}_{cr}^{\prime \prime }\), the FTP value is calculated in a cumulative fashion:where \(\Delta t_{i}\) is the ith time increment.

Using the actual incident heat flux recorded at a height of 2.3 m the door of ISD1 for Exp 1 and Exp 2, the time-to-ignition can be predicted using Eq. (5). Figure 26a and b depicts the cumulative FTP value as a function of time for Exp 1 and Exp 2, respectively. Note that since the timber frame is the material exposed to the radiation and the material that ignited first, in this section the FTP and n values used are the values of the timber. Additionally, the time-to-ignition in this case will be the time-to-ignition of the timber and not the cardboard as used in the sections above.

From Fig. 26a and b (dwellings spaced at 1 m and 1.75 m, respectively) it can be seen that the FTP method slightly underpredicts (faster) the time-to-ignition by 6.3% and 6%, respectively. In both instances the calculated time-to-ignition deviates less than 7% of the actual time-to-ignition, implying that the correlation between the FTP method and reality is good. For further distances, the cooling effects of the wind might cause larger deviations, but since these dwellings are typically spaced at closed proximities this method is suitable. It should also be noted that if the spacing between dwellings become large, the assumption of piloted ignition might not hold, and thus adjustments should be made to Eq. (5). Baker et al. [30] discusses the alterations needed to account for auto ignition should the reader require further information, it is however not in the scope of this paper.

Given the temperatures and emissivity of the compartment gasses, flames and the IBR sheeting, the time-to-ignition can be calculated at various separation distances. The average temperatures of the hot layer, the IBR sheeting and the door flames, as measured for both Exp 1 and Exp 2 during the fully developed fire stage of ISD1 are 879°C (unfortunately the flame temperatures at the door of ISD1 were not measured, hence as a best alternative, it is assumed that the door flame temperatures for ISD1 and ISD2 are similar. Thus, the temperature as measured by the FP thermocouples at the door of ISD2 are used here), 585°C (where the measured temperatures of these experiments are relatively constant across the height of the dwelling) and 892°C, respectively. Assuming the emissivity of the flames and compartment gases are equal to 1 [31, 32], and that the emissivity of the galvanized sheeting is equal to 0.35 (i.e., 0.22–0.28 for galvanized steel at 0–200°C [22] and 0.42 for galvanized steel at 1400°C [22], thus assuming 0.2567 + (0.42 − 0.28)/(1400 − 200) × 585 = 0.32 for galvanized steel at 585°C, as discussed above). Substituting the knowns (i.e., calculating the configuration factor as per above, FTP = 6390 and n = 1.6) into Eqs. (12), (16) and (4) the time-to-ignition can be calculated for a variety of separation distances.

From Fig. 27, it is clear that the separation distance plays a significant role in the time-to-ignition. At a separation distance of 2.5 m, the calculations show that it would take an adjacent dwelling (built with timber and lined with cardboard) approximately 35 min to ignite. It can thus be said that there is a vertical asymptote at 2.9 m separation distance in this case. Hence, giving the input parameters above (i.e., for these specific dwellings), fire will not spread if there is a separation distance greater or equal to 2.9 m. At 1 m, Fig. 27 shows a heat flux of 39 kW/m² which is slightly greater than the measured heat flux at 1 m for Exp 1 of 33 kW/m². At 1.75 m, Fig. 27 shows a heat flux of 22 kW/m² which is slightly smaller than the measured heat flux at 1.75 m for Exp 2 of 27 kW/m². Thus, showing that the calculated heat fluxes are slightly more susceptible to the separation distance compared to the experiments. This could however be as a result of factors such as the assumed integration limits [Eq. (16)] and might be different from fire scenario to fire scenario, but with all the variables and unknowns, these calculations show promise for future use. With the high variability inherent with ISDs a robust, but simple, method may be preferable to a more advanced model that requires additional input parameters for those wishing to apply this in practice.

For interest, the effect of the ignitable material height and the CHF of the ignition source on the critical separation distance are depicted in Fig. 28a and b, respectively. It is clear that the CHF can significantly increase or decrease the critical separation distances needed for fire spread not to occur. Increasing the CHF from 10 kW/m² (as in this case) to 15 kW/m², reduces the critical separation distance needed from approximately 2.8 m to 2.1 m. From Fig. 28a it appears that the most critical positions are in the middle of the target dwelling and as the point of ignition moves up or down the height of the dwelling, the received radiation becomes less. However, combustibles closer to floor level would not necessarily have flame impingement, thus requiring a higher incident heat flux for spontaneous ignition.