1. Introduction
Urban buildings are densely populated, with various types of buildings often containing large amounts of combustible furniture and decorative materials, resulting in high fire loads. Once ignited, such buildings can quickly spread fire, posing significant threats to life and property. Therefore, implementing effective fire safety measures and establishing relevant building design regulations is necessary. Understanding the mechanisms of fire spread and predicting fire development are of paramount importance for fire safety design.
Building fires often occur indoor environments. Indoor fire generates and accumulates a large amount of high- temperature smoke in the upper part of a room of fire origin. The surface temperatures of indoor walls increase significantly, and the wall surfaces emit radiation. Due to radiation feedback, fire growth is accelerated. Therefore, many researchers have recognized the significant differences between combustion in the open air and indoors. For example, Thomas et al.[1] found that the number and location of fire sources affect indoor combustion. Poulsen et al.[2] conducted pool fire experiments under free burn and room burn conditions. The experimental results showed that the thermal feedback from the enclosure enhanced the burning rate of the pool. Heidari et al.[3] also suggested that flame extension under ceiling has a significant impact on large-space fires and should be considered in structural fire resistance design. Therefore, when establishing indoor fire prediction models, it is necessary to consider the effects of thermal feedback by external radiation, i.e., radiation back to fuel surface from extended flame, smoke layer, heated walls, etc.
To quantify the effect of external radiation, calculation of the radiative heat flux is essential. To calculate the radiative heat flux, parameters such as flame shape, flame length, and emissive power are need to be determined. Many researchers have investigated this aspect. Pierce et al.[4] developed a model for predicting the radiative heat flux of indoor pool fire. Rew et al.[5] used a simple model to describe the shape of pool fire flames in windy conditions, providing calculation methods for flame length, flame drag, and flame surface emissive power to predict the radiative heat flux. He et al.[6] approximated the shape of the ceiling jet fire by a cone and a cylinder to simulate the hydrogen-blended natural gas pipeline leakage fire. Radiative heat flux to the ceiling and ground was predicted. As to the flame length, Lattimer et al.[7] conducted extensive experimental research, verifying the relationship between heat release rate (hereafter, HRR) and flame length.
About the mechanism of fire spread, Zhao et al.[8] proposed a comprehensive spread model for spill fires and conducted experimental verification. They suggested that the spread process can be divided into four combustion stages. During the shrink burning phase, the burning radius begins to decrease because the fuel consumption rate is greater than the discharge rate. However, the spread process of solid combustibles is different from that of spill fires. After ignition at a certain position of a solid combustible, the burning radius will continuously increase. When the burning front reaches the edge of the solid combustible, the burning radius remains constant. Since the flame spread mechanism of solid combustibles is completely different from that of liquids and gases, developing a flame spread model for solid combustibles is necessary. For use in building fire safety design, simple model is desirable that can be linked with zone type smoke transport models.
When the preheating surface of a combustible receives heat by external radiation, combustion accelerates. This is mainly because of the increase in flame spread rate and the burning rate. About the flame spread rate, Quintiere analyzed experimental results of lateral flame spread rate under external radiation and generalized the theory of fire spread[9,10]. He proposed that the inverse of square root of the flame spread speed is a linear function of the external radiation heat flux. Many researchers have studied flame spread rate based on this theory, such as by Zhou et al.[11] who validated the effect of external radiation on flame spread rate through experiments. They found that the greater the external radiation, the faster the flame spread.
About the burning rate, the mass burning rate depends on the fuel’s thermophysical properties and heat input vaporizing the fuel. Based on this, Wang et al.[12] developed a model of LNG pool fire controlled by material thermophysical properties. However, they did not consider the influence of external radiation on the increase in burning rate.
In summary, most researchers currently focus on liquid[1,2,4-6,8,12] and gaseous[3,7] fuels. The flame spread mechanism of solid combustibles is significantly different from that of liquid and gas. Since combustibles in building fires are often solid, this study focuses on solid combustibles. Additionally, as mentioned earlier, external radiation in indoor fire accelerates the combustion of solid combustibles. While some researchers[9,11] have quantified the effects of external radiation on flame spread rate and burning rate, these studies used fixed values for external radiation heat flux. For realistic indoor fires, the HRR of the fire and the external radiation heat flux change with time as fire develops and decays.
In this study, a simple fire growth model for solid combustibles that considers external radiation is proposed. The model is composed of two parts; flame spread model and two-layer zone model. Flame spread model was developed considering the effect of internal and external radiation from flame to fuel surface. Then the model was coupled with two-layer zone model to calculate the smoke layer temperature and wall surface temperatures. Using the calculated temperatures, heat fluxes from a smoke layer and enclosure walls were calculated. The calculated heat fluxes, together with the heat flux from extended flame, were applied to the flame spread model in order to consider thermal feedback to fuel surface. As a result, the increase of HRR in indoor burning compared with open burning were predicted.
2. Calculation Methods
2.1 Schematics of calculations
The calculation method is composed of two models. One is the fire growth model, which describes the changes in the flame shape, flame size, radiative heat flux from flame, HRR and so on. The other is the two-layer zone model, which describes the changes in the smoke layer temperature and radiation exchange between layers and enclosure surfaces.
Figure 1 is the composition of this model. Firstly, the physical property values and initial values are input to the model. Then the geometry of flame, flame spread rate and HRR are calculated by the fire growth model. Using the HRR value, smoke layer properties and radiation exchange between upper smoke layer, lower air layer and enclosure surfaces are calculated. Radiative heat flux calculated by the two-layer zone model is imposed on the fire growth model to calculate the fire spread and burning rate. As a result, the HRR is updated. Calculation continues until burnout when Q ≤ 0.
2.2 Fire growth model
Akao et al.[13] proposed a fire growth model by considering external radiation, but the accuracy of the model was not sufficient. This study proposed an improved fire growth model. The combustible is a square polyurethane foam slab of width W [m], thickness h [m]. It was assumed that the slab was ignited at the center of top surface.
2.2.1 Flame shape
Figure 2 shows the flame shape under a ceiling for predicting the radiative heat flux. The flame shape is modeled by a cylinder and a disk. In order to simulate a real flame as accurately as possible, the flame is divided into two regions: continuous flame region (shaded region) and intermittent flame region (blank region). In this model, the total flame length is approximated by that in open air, and the radius of the disk were used to approximate the flame radius. In Figure 2, Df represents the diameter of the fire source (m), while ri and rc denote the disk radius (m) of the intermittent and continuous regions.
The flame length in the open air is calculated using the equations proposed by Zukoski[14]. Eq. (1) is the equation for the mean flame length in the open air, and Eq. (2) is the equation for the continuous flame length in the open air.
where
Here, Q D *
2.2.2 Radiative heat flux
Figure 3 shows the radiative heat flux of internal radiation and external radiation in a compartment. qf is the radiative heat flux from the cylinder (kW/m2), which belongs to internal radiation. qw is the radiative heat flux from the hot ceiling where the flame touches (kW/m2). qd is the radiative heat flux from the disk of extended flame under ceiling (kW/m2). qf,res is the radiative heat flux from the smoke layer and enclosure walls and the indirect radiation heat flux from the flame (kW/m2). The heat fluxes qw, qd and qf,res belong to external radiation.
The emissivity of cylinder is calculated by the following equation following the model of Shintani et al.[16]:
where Acc,free is the surface area of the continuous flame region of the cylinder (m2), Aci,free is the surface area of the intermittent flame region of the disk (m2), Ecc (= 80.5) and Eci (= 51.2) are the radiant emittance of the cylinder continuous flame region and the cylinder intermittent flame region (m2), X is the radiative fraction from fire (-).
According to the radiative heat flux balance, the emissivity of disk can be expressed as:
where Acc and Aci are the surface areas of the cylinders of the continuous flame region and intermittent flame region (m2), Adc and Adi are the surface areas of the disks of the continuous flame region and intermittent flame region (m2), Edc (= 80.5) and Edi (= 47.7) are the radiant emittance of the disk continuous flame region and the disk intermittent flame region (kW/m2).
As shown in Figure 3(a), internal radiation to burning surface is equal to the sum of the heat flux from the continuous flame region and intermittent flame region:
where Fcc and Fci are the shape factors of the cylinder continuous flame region and the cylinder intermittent flame region relative to a target on the burning surface. τc is the transmittance through the cylinder flame.
As shown in Figure 3(b), qd is equal to the sum of the heat flux from the continuous flame region and intermittent flame region:
where Fdc and Fdi are the shape factors of the disk of continuous flame region and the disk of intermittent flame region relative to a target on the burning surface.
Similarly, qw is equal to the sum of the heat flux from the ceiling where continuous flame region and intermittent flame region touches:
where σ T w , c 4 σ T w , d 4
Since qf,res involves the smoke layer, the two-layer zone model is used for calculation. qf,res can be calculated by
where the heat flux incident on the lower layer’s wall qi,wa is the incoming radiation to lower layer wall calculated by the radiation exchange in two-layer zone model.
2.2.3 Heat release rate per unit area
The burning surface is subjected to internal radiation and external radiation. The burning rate increases by the radiation feedback. Figure 4 shows a schematic diagram of the heat flux to the burning surface. The burning rate per unit area (kg/s.m2) in the open air can be expressed by:
where, qf,free is the heat flux from the cylinder to the burning surface (kW/m2), L is the latent heat of gasification of fuel (kJ/kg).
Considering external radiation, the burning rate in a room can be expressed by:
where, qinc is the sum of external radiative heat fluxes (= qd + qw + qf,res).
The heat release rate per unit area q0 is equal to the burning rate multiplied by the heat of combustion per unit mass ∆H, which leads to the equation for HRR per unit area q0:
where, ∆H is the heat of combustion per unit mass (kJ/kg). For the same combustion, ∆ H L
Considering that the external radiation needs to pass through the cylinder to reach the burning surface, the corrected heat release rate per unit area can be expressed by:
where τc is transmissivity of the cylinder, ϵc is emissivity of the cylinder. The ϵc is the average transmittance of the flame, but the transmittance at the base of the flame is lower. Therefore, an adjusting coefficient f is introduced when calculating the τc. The value of f is set to 0.8 in this model.
2.2.4 Fire spread rate
The preheating surface is subjected to internal radiation and external radiation. As a result, the fire spread rate increases. Figure 5 shows a schematic diagram of the heat flux to the preheating surface. In the case of open air and the combustible is homogeneous, the fire spread rate[17] is given by:
where, δf is flame heat transfer length (m), tig is the time for the flame to move δf (s), Tigis the ignition temperature (K), T0 is the base temperature (K), kpc is the thermal inertia of the material (kW2.s/m4.K2), and qfl,free is the heat flux from the flame to the preheating surface in an open environment (kW/m2).
The fire spread rate when receiving the external radiation is calculated by
where qfl is the heat flux from the cylinder to the preheating surface (kW/m2), qinc,l is the external radiation heat flux to the preheating surface (kW/m2).
According to the burning experiments in an open air[18], the fire spread rate in the open air is fairly constant.
The internal and external radiation heat fluxes were calculated depending on flame geometry. The heat flux from the flame cylinder to the preheating surface were calculated by
where qf,free and qf are heat flux from flame cylinder to the flame base in open air and in compartment, respectively. As to the external radiation, radiation from extended flame qd, from heated ceiling qw and from smoke layer qf,res were considered. According to the flame shape model, part of qd and qw will pass through the cylinder, while the other part will not. This proportion changes by the flame shape, so it is complex to calculate. Therefore, this study assumes that 70% of qd and qw can pass through the cylinder, while 30% will not. Moreover, the lower part of the flame is a thick flame, so it is difficult for external radiation from smoke layer to pass through it and to reach the preheating surface. It was assumed that only half of the external radiation heat flux can pass through the thick flame. In summary, the external radiation to preheating surface is given by
In addition, this model also considered the downward vertical fire spread rate. When the burning surface reaches the edge of the top surface of the combustible, it starts to spread downward in vertical direction. According to experiments, the downward fire spread rate is approximately half of the horizontal flame spread rate. Finally, the flame spread rate can be expressed by Eq. (25):
where Df is the diameter of burning area (m), Wu is the equivalent edge length of top surface of the combustible material (m), Atop is the area of top surface of the combustible (m2).
2.2.5 Burning area and HRR
The burning area was assumed to be circular as shown in Figure 6. Using the flame spread rate as determined by Eq. (25), the radius of burning area rf (m) is
during initial stage. As the burning continues, burnout of material occurs in the initially ignited area. The time to burnout tb (s) is calculated by
Then the burnout radius is
The burning area is calculated by
Knowing the burning area Af and the HRR per unit area q0, it is simple to calculate the HRR from the fire source.
2.3 Improved two-layer zone model
When dealing with issues related to the early stages of a fire, the two-layer zone model[19] is often used, which assumes the formation of a hot smoke layer at the upper part of the room and a cold air layer at the lower part of the room. The original two-layer zone model assumed that all the radiative heat from the flame is absorbed by the upper layer gas. However, in reality, only part of the radiative heat from the flame is absorbed by the upper layer. Another part of the radiative heat incidents on the lower layer and enclosure surfaces. This proportion changes by the thickness of the upper layer and the position of the flame. Based on this problem, this study improved the radiation exchange calculation in the two-layer zone model.
Due to the large opening of the model room in this paper, the radiative heat from the flame does not entirely enter the upper layer and the lower layer, rather a part of it escapes to the outside through the opening. This ratio varies with the shape, size, and position of the opening. Based on the size of the model room used in this study, the ratio of the opening area to the interior surface area of the room is approximately 0.2 (rounded to one decimal place). Therefore, it is assumed that 20% of the radiative heat escapes through the opening. The radiative heat from the flame to the model room Qfrad,burner (kW) can be expressed as:
where x is the radiative fraction to HRR (-).
The proportion of radiative heat from the flame incident on the upper layer is designated by fs,burner, while the proportion of radiative heat from the flame incident on the lower layer is fa,burner. The radiative heat from the flame is assumed to be uniformly emitted over a hemisphere from the centroid of flame. The radiative heat from flame incident on the upper layer can be approximated by the portion passing through the surface of a spherical cap cut by a plane defined by the angle θb (rad.) determined by the edge of the upper layer and the center point of the flame. Therefore, the ratio can be calculated by:
where, Ad is the floor area of the room (m2), Zs is the height of the upper layer (m).
The radiative heat absorbed by the upper layer from the flame can be expressed as:
The radiative heat absorbed by the lower layer from the flame can be expressed as:
where, ϵs and ϵa are the emissivity of upper layer and lower layer which are equal to the absorptivity. The unit for Qfrad,s and Qfrad,a is (kW).
3. Model Validation
3.1 Experiment
Akao et al. carried out the experimental study on burning behavior of polyurethane foam in an open air and in a compartment[18]. The model is applied to their experiments and the results are compared with experimental data. The burning properties are derived from the results of open-air experiments. Then the model is applied to the conditions of experiments in a compartment. The degree of increase of HRR and fire spread rate is compared. The schematics of the experiments are described in the followings.
3.1.1 Burning experiments in an open air
The experimental setup is shown in Figure 7. Three load cells with a capacity of 50 kN were set under the specimen to measure the mass loss rate. The oxygen consumption method was used to measure the HRR. In addition, two cameras were set up for photography from two directions to record the burning behavior and the flame spread rate over horizontal surface.
The specimen size was 500 mm x 500 mm. Four experiments were carried out changing the thickness of the specimen. The thickness of the specimen is shown in Table 1. The density was 15.85 kg/m3. The heat release rates of the four experiments are shown in Figure 8. By dividing the total heat release by the specimen mass, the heat of combustion was calculated as ∆H = 29410 kJ/kg. The burning radii, rf (m), in the four experiments are shown in Figure 9. The fire spread rate, i.e., the rate of expansion of burning radius, is fairly unform and correlated as vf,free = 0.0035 m/s.
To derive the ratio of heat of combustion to heat of decomposition, ∆H/L, the HRR data and burning radius data were utilized. HRR per unit area q0 (kW/m2) is obtained by dividing HRR (kW) by burning area π r f 2
The radiation heat flux incoming to burning surface qf is calculated by Eq. (9) considering open-air flame configuration. Due to the heat balance at the burning surface, the ratio of heat of combustion to heat of decomposition can be approximated by the ratio of hear release per unit area to incoming radiation from flame,
3.1.2 Burning experiments in a compartment
The schematic drawings of the model room experiments are shown in Figure 11. The walls of the model room were made of ceramic fiber board. The specimen was set in the center of the model room. HRR was measured by oxygen consumption method. The mass loss rate was measured by a set of load cells under the specimen. The burning radii were measured in four directions using camera images installed close to the model room. Two water-cooled heat flux gauges (hfg) were installed on the floor.
3.2 Calculation conditions
Akao et al. carried out 12 experiments in all. Among them, 4 experiments were selected for comparison, differing in ceiling height and thickness of specimen. The configuration of compartment and specimen is the same as in the experiments. The main physical properties used in the calculations are shown in Table 2.
3.3 Results and discussion
3.3.1 Comparison of calculated and experimental results for experiment b1
The results of experiment b1 (ceiling height H = 0.8 m, specimen thickness h = 0.05 m) are compared with calculation results.
(1) HRR
Figure 12 shows the comparison of measured and calculated HRR. For comparison, the measured and calculate results of experiment o4 (open-air) are also plotted. In general, the calculated results are close to the experimental results. The increase in HRR in compartment from that in open-air is reproduced by the calculation. The maximum HRR in the compartment is higher than in the open air, but the overall progress of combustion agreed fairly with the experimental results. According to the calculated results in the compartment, the HRR rapidly increases at around 55 s, which is due to the sudden increase in external radiation caused by the flame extension under ceiling.
(2) Burning radius
Figure 13 shows the comparison of measured and calculated burning radius. The burning radius was defined as the distance from the center point to the flame spread front, measured approximately every 10 seconds by video image, until the flame spread front reached the edge of the top face of the specimen. So, the experimental values of the burning radius are limited up to 0.25 m. The calculated burning radius closely matches the experimental results. The flame extended below the ceiling during 55 and 85 s. The burnout began at 70 s, and totally burnt out at 160 s.
(3) Layer temperature
Figure 14 shows the comparison of measured and calculated layer temperatures. Regarding the experimental values of the upper layer temperature, it is difficult to accurately define the average temperature of the upper layer because the height of the smoke layer changes over time. So, the measured temperature of the ceiling jet was considered as the experimental value of the upper layer temperature for comparison.
The maximum upper layer temperature is higher than the experimental value. According to the experimental values, it can be seen that the temperature rises slowly during the first 40 s of burning, and the temperature at 40 s is about 50 ℃. In contrast, the calculated values increase quickly from the beginning but the time to reach the maximum is close to the experimental values. The calculated results for the lower layer temperature are in good agreement with the experimental results. About the calculated values, the sudden increase in the upper layer temperature at the end of burning is due to the abrupt drop in the mass flow rate of the fire plume, while the high temperature walls continue to release convective heat to the upper layer.
(4) Upper layer wall temperature
Figure 15 shows the comparison of measured and calculated upper layer wall temperature. Regarding the experimental values, the average ceiling surface temperatures at 250 mm and 550 mm horizontally from directly above the fire source were considered as the experimental values for the upper layer wall temperature. Since the ceiling surface temperature varies significantly with the distance from the fire source, the general tendency is compared. Overall, the calculated wall temperatures are higher than the experimental values, but the trend of the calculated values is consistent with the experimental values.
(5) Upper layer height (smoke layer height)
Figure 16 shows the comparison of measured and calculated upper layer height. In this study, the experimental values for the upper layer height were determined using the N% method (N = 10). Since a clear smoke layer is not formed in the early and late stages of burning, the experimental values for these periods have been deleted. In general, the experimental values are mostly around 0.5 m. The calculated values are higher than experimental values, and the upper layer height recovers to the opening height of 0.8 m after burnout.
(6) Heat flux to the lower layer wall qi,wa
Figure 17 shows the comparison of measured and calculated qi,wa. In the experiment, heat flux gauges were placed on the floor in the front and back to measure the heat flux incident on the lower layer wall qi,wa. The maximum qi,wa is higher than the experimental result typically during the period of flame extension.
3.3.2 Comparison with HRR of other experiments
Figure 18 shows the comparison of measured and calculated HRR in the open-air and in the compartment. When the specimen thickness is the same, HRR is increased as the ceiling is low. When the ceiling height is the same, the greater the specimen thickness, the greater the maximum HRR. For burning in the compartment, the prediction accuracy is better for higher ceiling heights, while HRR is slightly overestimated for the cases of lower ceiling heights as shown in Figure 14(d). This may be because the external radiation is calculated excessively when the ceiling is low.
4. Conclusions
In this study, the fire growth model and the two-layer zone model were improved. These two models were combined to predict the fire growth behavior of flexible polyurethane in a compartment. The improvements of this model are mainly as follows:
(1) The flame shape was improved in order to calculate the internal radiation and external radiation more accurately. All possible flame shapes are approximated by a cylinder and a disk.
(2) The calculation of heat release rate per unit area and fire spread rate has been improved considering external radiation.
(3) The two-layer zone model has been improved. The previous two-layer zone model assumed that all the radiative heat from the flame is absorbed by the upper layer gas. This study assumes that 20% of the radiative heat from the flame escapes to the outside through the openings, and the remaining 80% enters partly into the upper layer and partly into the lower layer.
By comparing the calculated and experimental values, it is found that the feedback effect is properly implemented and that the prediction accuracy of this model is good when the ceiling height is higher. When the ceiling height is lower, HRR is slightly overestimated.
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