A method of determining flame radiation fraction induced by interaction burning of tri-symmetric propane fires in open space based on weighted multi-point source model

Abstract

The interaction of multiple fires may lead to a higher flame height and more intense radiation flux than a single fire, which increases the possibility of flame spread and risks to the surroundings. Experiments were conducted using three burners with identical heat release rates (HRRs) and propane as the fuel at various spacings. The results show that flames change from non-merging to merging as the spacing decreases, which result in a complex evolution of flame height and merging point height. To facilitate the analysis, a novel merging criterion based on the dimensionless spacing S/z_c was proposed. For non-merging flames (S/z_c >0.368), the flame height is almost identical to a single fire; for merging flames (S/z_c ≤0.368), based on the relationship between thermal buoyancy B and thrust P (the pressure difference between the inside and outside of the flame), a quantitative analysis of the flame height, merging point height, and air entrainment was formed, and the calculated merging flame heights show a good agreement with the measured experimental values. Moreover, the multi-point source model was further improved, and radiation fraction of propane was calculated. The data obtained in this study would play an important role in calculating the external radiation of propane fire.

Graphical abstract

Keywords

flame interaction / air entrainment / flame height / multi-point source model / thermal radiation

1 Introduction

In recent years, many accidents caused by pool fires have been widely reported, which attracts widespread concern in society [1–4]. Compared with a single-pool fire, a multiple-pool fire (MPF) is more uncontrollable and hazardous, making firefighting difficult. The MPF is mainly affected by two mechanisms, air entrainment and thermal feedback. A simultaneous quantitative analysis of two mechanisms is very complicated [5]. For gaseous fuels, the radiation feedback from the flame to burner surface can be ignored [6].

When multiple fires are sufficiently close, air entrainment is limited. The pressure difference is generated between the inside and the outside of the flame, which causes flames to tilt inward or even merge [5,7–10]. The interaction among multiple fires significantly affects flame height, which is an important parameter to describe flame behavior [9,11–14]. Thomas and colleagues [8] derived the correlation of flame height according to equilibrium relationship between pressure and buoyancy. However, due to rough approximation, the effect of heat release rate (HRR) was neglected, and calculation results greatly deviated from experimental data. Sugawa and Takahashi [9] obtained the flame heights of three fires at different spacings but did not provide the correlation of flame height, spacing, and HRR.

When multiple fires interact, the change of flame shape alters radiation to the surroundings and makes it easily ignite nearby combustibles [15]. The study of external radiation is critical to the criteria for possibility of radiation ignition and fire spread and the determination of safety distance [4,16–19]. Two typical analytical models have been used to assess fire radiation hazards [20], point source models [17,21,22] and solid flame models [17,21]. Liu et al. [18] analyzed external radiation of a heptane fire array by using the point source model. When the target was closer to the center of point source, the calculated results had a large deviation from the radiation values measured. Weng et al. [14] calculated multi-fire external radiation based on the assumption that multiple flames did not tilt, but ignored merging behavior of real flames, which may lead to poor prediction accuracy of this model. Wan et al. [15] proposed a rectangular model to calculate the external radiation of two fires, the accuracy of which was verified by a comparison with experimental data. However, the flame transmittance cannot be solved for multiple fires, which limits the adaptation range of this model.

Since the flow field of surrounding environment is quite complicated when the fires burn, three fire sources were symmetrically placed to simplify the analysis. In this paper, the effects of air entrainment on flame shape were studied by combining experimental observations with theoretical analysis. Moreover, the calculated formula of flame height was established, based on which, a novel calculation method was proposed for the multi-point source weights and the variation rule of propane radiation fraction was obtained.

2 Experiments

As shown in Fig. 1, burners A, B, and C whose cross-sectional shape is square, were identical and symmetrically placed with a side length (D) of 15 cm, and their surfaces were 53 cm above the ground, the effect of the ground on air entrainment being ignored. Propane gas was supplied through mass flow meters with a range of 0–100 L/min and an accuracy of 0.01 L/min. Each fire source was provided with five HRRs of 10, 20, 30, 40, and 50 kW. For convenient description, the dimensionless power of each burner {\dot{Q}}_{\mathrm{d}}^{*}=\dot{Q}/\left({\rho }_{\infty }{c}_{\mathrm{p}}{T}_{\infty }{g}^{\mathrm{1}/\mathrm{2}}{D}^{\mathrm{2}/\mathrm{5}}\right) is utilized in the subsequent analysis. The calculated {\dot{Q}}_{\mathrm{d}}^{*} is 1.03, 2.06, 3.10, 4.13, and 5.16. In the experiments, the burner moved along the track, and spacing (S) changed from 2 to 60 cm. In addition, burner C burned separately for comparison. All experimental conditions are listed in Table 1.

A column of K-type thermocouples with 1 mm in diameter at a sampling interval of 1 s were placed at the symmetric center of three burners. The height above the ground was 60–205 cm. The thermocouples from the bottom to the top were numbered as 1–21. The interval between thermocouples 1–12 was 5 cm, and the interval between thermocouples 12–21 was 10 cm. Four water-cooled radiometers facing the fire source, whose height was flushed with burner surface, were used to measure the total radiation. As spacing changed, r₁–r₄ with a range of 0–10 kW moved with the burner, but the position relative to the burner remained unchanged. DV1, 2, and 3 recorded flame shape from three different angles with a shooting time of 4 min. The repeatability of experiments was verified by repeated tests under identical conditions. The image processing software written by Yan et al. [23] was used to analyze the flame image captured, thereby the shape data such as flame width, height and inclination were obtained. In the experiments, flames were very stable. Therefore, the average flame heights extracted from quasi-steady burning stages and radiation values measured by radiometer differ slightly.

3 Results and discussion

3.1 Flame interaction

Interaction occurs among fire sources when they are near one another. The flame merging process has three distinct regimes, a fully continuous merging regime, an intermittent merging regime, and a complete non-merging regime. Figure 2 demonstrates that when the spacing is sufficiently small (S/D = 0.2), three flames merge into a single one, which denotes a completely continuous merging regime. When the spacing gradually increases (S/D = 0.8), the flames sometimes merge and sometimes separate, which denotes an intermittent regime. When the spacing continues to increase (S/D = 2), multiple fires gradually separate and finally form three completely separated flame, which denotes a complete non-merging regime.

To quantify the flame-merging behavior, flame merging probability (P_m) is defined as the ratio of the number of merging flame frames to the number of total sampling frames, which was calculated by utilizing specific treatment methods [24]. Figure 3 plots the variation of flame merging probability P_m with spacing S. P_m increases as \dot{Q} increases and S decreases. Namely, P_m is dependent on \dot{Q} and S. Then, an attempt was made to establish the relationship among \dot{Q}, S, and P_m. A characteristic length z_c is defined as z_{\mathrm{c}}={\left({\displaystyle \frac{\dot{Q}}{{\rho }_{\infty }{c}_{\mathrm{p}}{T}_{\infty }{g}^{\mathrm{1}/\mathrm{2}}}}\right)}^{\mathrm{2}/\mathrm{5}} [25], and the following function can be expected to represent the merging probability P_{\mathrm{m}}=f\left({\displaystyle \frac{S}{z_{\mathrm{c}}}}\right). The data of different HRRs exhibit good convergence in Fig. 4. The segmentation function is divided into three states, a fully merging region (S/z_c<0.113), an intermittent merging region (0.113≤S/z_c<0.658), and a separated region (0.658≤S/z_c). P_m = 0.5 is used as the criterion for flame merging [6], i.e., when S/z_c≤0.368, the flames merge; whereas when S/z_c>0.368, the flames do not merge.

3.2 Flame height

3.2.1 Non-merging flame height

In this paper, the averaged flame height of three flames was regarded as non-merging flame height. The variation of the dimensionless flame height with the dimensionless spacing is provided in Fig. 5. As the spacing increases, flame height gradually decreases. When spacing increases to a certain extent, the flames do not merge, and flame height no longer changes with increasing spacing. When the HRR is constant, the variance of non-merging flame height is calculated by utilizing Eq. (1), whose results are shown in Table 2.

(1)s^{\mathrm{2}}={\displaystyle \frac{\mathrm{1}}{n-\mathrm{1}}}{\displaystyle {\sum }_{i=\mathrm{1}}^n}{(L_{\mathrm{f}i}-{\overline{L}}_{\mathrm{f}})}^{\mathrm{2}},

where n is the number of samples, L_{\mathrm{f}i} and {\overline{L}}_{\mathrm{f}} are the flame height at each spacing and the mean value of all non-merging flame heights. The error of image-processing method to obtain flame height is within 5% [26]. It can be considered that flame height remains unchanged when s²<0.05. The maximum s² of non-merging flame heights in Table 2 is 0.036, namely, non-merging flame heights are independent of the spacing.

The data of non-merging flame heights from Fig. 5 is rearranged to show the evolution with normalized HRRs in Fig. 6 containing a single fire and two non-merging fires [6]. As the spacing increases, the degree of limited entrainment is reduced. But air entrainment restriction does not completely disappear, and non-merging flame height increases (compared to single fire) to obtain more air. Due to the pressure difference, the flame tilts inward, and flame height reduces. Under the action of two mechanisms, the non-merging flame height is similar to the single flame height. Through exponential fitting, the relationship between the dimensionless non-merging flame height and the dimensionless HRR is established, as expressed in Eq. (2).

(2){\displaystyle \frac{L_{\mathrm{f}}}{D}}=\mathrm{3.46}{\dot{Q}}_{\mathrm{d}}^{*\mathrm{0.49}}.

3.2.2 Merging flame height

The curvature radius of the flame is much larger than the length of the flame when flames merge. Therefore, the axis of the flame can be considered as a straight line. The shape of the flame is an interaction result of two forces, upward buoyancy B and pressure P of vertical flame axis. The angle between flame axis and vertical direction is defined as q in Fig. 7. Solving in the direction of vertical axis, the equilibrium equation is obtained, as expressed in Eq. (3).

(3)B\sin \theta =P.

The buoyancy is the density difference between the flame and surrounding environment and acts on the entire flame. Assuming that flame shape is a rectangular parallelepiped [12], then

(4)B={\displaystyle {\int }_{\mathrm{0}}^{L_{\mathrm{f}}}}({\rho }_{\infty }-{\rho }_{\mathrm{f}})g{D}^{\mathrm{2}}\mathrm{d}z.

The ideal gas equation can be written as {\rho }_{\infty }{T}_{\infty }={\rho }_{\mathrm{f}}{T}_{\mathrm{f}}. Substituting it into Eq. (4), Eq. (5) can be obtained.

(5)B={\rho }_{\infty }g{D}^{\mathrm{2}}{\displaystyle {\int }_{\mathrm{0}}^{L_{\mathrm{f}}}}{\displaystyle \frac{\mathrm{\Delta }{T}_{\mathrm{f}}}{T_{\mathrm{f}}}}\mathrm{d}z.

It is difficult to measure the temperature of flame axis due to the variation of tilt angle with S. The change in temperature along the single-fire centerline with dimensionless height z/L_f is consistent with the change in temperature along the centerline of two fires with z/L_f [15]. Thus, this paper considers that the change in temperature along the flame axis with z/L_f is also consistent with the change in temperature along the single-fire centerline with z/L_f. Taking the average flame height of a single fire as the characteristic length, ΔT_f/T_f (temperature increase/temperature) along the single-fire centerline varies with the dimensionless height z/L_f as displayed in Fig. 8. The temperatures are corrected for the radiation error [27]. Figure 8 indicates that the continuous flame height/ average flame height is 0.5, which is consistent with that of single fire [28] and two fires [15]. Therefore, z/L_f = 0.5 is used as the boundary between continuous flame and intermittent flame. The continuous flame zone can be divided into the core zone in which ΔT_f/T_f increases as z/L_f increases due to the heat absorption of low-temperature fuel vapor near burner surface, and the constant zone. The exponential function is fitted to obtain the ΔT_f/T_f distribution model.

(6)

Since ΔT_f/T_f is a piecewise function of z/L_f in Eq. (6), the integral of ΔT_f/T_f over entire flame height can be expressed as

{\displaystyle {\int }_{\mathrm{0}}^{L_{\mathrm{f}}}}{\displaystyle \frac{\mathrm{\Delta }{T}_{\mathrm{f}}}{T_{\mathrm{f}}}}\mathrm{d}z=L_{\mathrm{f}}{\displaystyle {\int }_{\mathrm{0}}^{L_{\mathrm{f}}}}{\displaystyle \frac{\mathrm{\Delta }{T}_{\mathrm{f}}}{T_{\mathrm{f}}}}\mathrm{d}\left({\displaystyle \frac{z}{L_{\mathrm{f}}}}\right)

=L_{\mathrm{f}}[{\displaystyle {\int }_{\mathrm{0.049}}^{\mathrm{0.25}}}\mathrm{0.98}{\left({\displaystyle \frac{z}{L_{\mathrm{f}}}}\right)}^{\mathrm{0.18}}\mathrm{d}\left({\displaystyle \frac{z}{L_{\mathrm{f}}}}\right)

+{\displaystyle {\int }_{\mathrm{0.25}}^{\mathrm{0.50}}}\mathrm{0.764}\mathrm{d}\left({\displaystyle \frac{z}{L_{\mathrm{f}}}}\right)

+{\displaystyle {\int }_{\mathrm{0.50}}^{\mathrm{1}}}\mathrm{0.59}{\left({\displaystyle \frac{z}{L_{\mathrm{f}}}}\right)}^{-\mathrm{0.43}}\mathrm{d}\left({\displaystyle \frac{z}{L_{\mathrm{f}}}}\right)]

(7)=\mathrm{0.67}{L}_{\mathrm{f}}.

Substituting Eq. (7) into Eq. (5) provides

(8)B=\mathrm{0.67}{\rho }_{\infty }g{D}^{\mathrm{2}}{L}_{\mathrm{f}}.

Figure 9 illustrates the pressure distribution of each zone. A, B, and C are flame zones, the zone enclosed by dotted line is named as the additional zone, and outside of the additional zone is the environmental zone. The Bernoulli equation in Ref. [6] is also applicable to this study. Assuming that the pressures in regions A, B, and C are identical, and the pressure difference is approximately the flow momentum that passes through the boundary.

p_{\infty }-p_{\mathrm{f}}\approx {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\rho }_{\infty }{v}_{\mathrm{1}}^{\mathrm{2}},

p_{\infty }-p\approx {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\rho }_{\infty }{v}_{\mathrm{2}}^{\mathrm{2}},

(9)p-p_{\mathrm{f}}\approx {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho v_{\mathrm{3}}^{\mathrm{2}},

where p and ρ are the pressure and density of additional region, p_{\infty } and {\rho }_{\infty } are these of environmental region, p_f is the pressure of flame region, and v_{\mathrm{1}}, v_{\mathrm{2}}, and v_{\mathrm{3}} are the flow velocities from environmental zone to flame zone, from environment zone to additional zone, and from additional zone to flame zone.

When the flames are in contact, the air flows into additional zone through a section of an approximately triangular shape. The air flows into flame zone from additional zone through a section of an approximately rectangular shape. The distance from the lowest point of the merging portion to burner surface is defined as the flame merging point height L_m. When the flames merge (P_m = 0.5), i.e., S/z_c = 0.386, flame merging point height L_m has the meaning of existence. Figure 10 provides the solving formula of L_m.

(10){\displaystyle \frac{L_{\mathrm{m}}}{D}}={\displaystyle \frac{\mathrm{2.39}S}{z_{\mathrm{c}}\times \mathrm{0.386}}}+\mathrm{0.51.}

The additional area satisfies the mass conservation equation as expressed in Eq. (11).

(11){\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\rho }_{\infty }{v}_{\mathrm{2}}S{L}_{\mathrm{m}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho v_{\mathrm{3}}D{L}_{\mathrm{m}}.

Substituting Eq. (11) into Eq. (9) gives

(12)v_{\mathrm{2}}=v_{\mathrm{1}}/\sqrt{\mathrm{1}+\left({\displaystyle \frac{{\rho }_{\infty }}{\rho }}\right){\left({\displaystyle \frac{S}{D}}\right)}^{\mathrm{2}}}.

The ideal gas equation is {\rho }_{\infty }/\rho =T/T_{\infty }. The temperature of additional zone is the average temperature of measurement points, from number 1 to the corresponding number at flame-merging point height, along the centerline of three fires. Considering T=f(\dot{Q},S,D), T/T_{\infty } can be expressed as T/T_{\infty }=f({\dot{Q}}_{\mathrm{d}}^{*},S/D). Figure 11(a) depicts the relationship between T/T_{\infty } and {\dot{Q}}_{\mathrm{d}}^{*}. By exponential fitting, the average of three indices is 0.38 as the index of {\dot{Q}}_{\mathrm{d}}^{*}, and Fig. 11(b) shows that the relationship between T/\left(T_{\infty }{\dot{Q}}_{\mathrm{d}}^{*\mathrm{0.38}}\right) and S/D is

(13){\displaystyle \frac{T}{T_{\infty }{\dot{Q}}_{\mathrm{d}}^{*\mathrm{0.38}}}}=\mathrm{1.25}{\left({\displaystyle \frac{S}{D}}\right)}^{-\mathrm{0.23}}.

The characteristic entrainment velocity of the flame is \sqrt{g{L}_{\mathrm{f}}} [12]. Then, v_{\mathrm{1}}=\sqrt{g{L}_{\mathrm{f}}}, and substituting Eq. (13) into Eq. (12) yields

(14)v_{\mathrm{2}}=\sqrt{g{L}_{\mathrm{f}}}/\sqrt{\mathrm{1}+\mathrm{1.25}{\dot{Q}}_{\mathrm{d}}^{*\mathrm{0.38}}{(S/D)}^{\mathrm{1.77}}}.

The pressure difference between the inside and the outside of the flame is

(15)\mathrm{\Delta }{P}_{\mathrm{1}}=p_{\infty }-p={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\rho }_{\infty }{v}_{\mathrm{2}}^{\mathrm{2}}.

Substituting Eqs. (14) and (15) into the formula of pressure P (\mathrm{\Delta }{P}_{\mathrm{2}}=\mathrm{0}) yields

(16)P=\mathrm{\Delta }{P}_{\mathrm{1}}D{L}_{\mathrm{m}}+\mathrm{\Delta }{P}_{\mathrm{2}}D(L_{\mathrm{f}}-L_{\mathrm{m}})={\displaystyle \frac{{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\rho }_{\infty }gD{L}_{\mathrm{m}}{L}_{\mathrm{f}}}{\mathrm{1}+\mathrm{1.25}{\dot{Q}}_{\mathrm{d}}^{*\mathrm{0.38}}{(S/D)}^{\mathrm{1.77}}}}.

When the flames merge, the tilt angle q is very small. Then,

(17)\sin \theta \approx \tan \theta ={\displaystyle \frac{D/\mathrm{2}+S/\sqrt{\mathrm{3}}+\sqrt{\mathrm{3}}D/\mathrm{6}}{L_{\mathrm{f}}}}={\displaystyle \frac{S/\sqrt{\mathrm{3}}+\mathrm{0.118}}{L_{\mathrm{f}}}}.

Substituting Eqs. (8), (16), and (17) into Eq. (3) yields the merging flame height L_{\mathrm{f}}

(18)L_{\mathrm{f}}=\mathrm{1.34}\times (S/\sqrt{\mathrm{3}}+\mathrm{0.118})\times {\displaystyle \frac{\mathrm{1}+\mathrm{1.25}{\dot{Q}}_{\mathrm{d}}^{*\mathrm{0.38}}{(S/D)}^{\mathrm{1.77}}}{L_{\mathrm{m}}}}.

Figure 12 indicates that flame heights calculated are consistent with experimental values, suggesting that Eq. (18) can be reliable for predicting the merging flame height of three fires in open space.

3.3 Calculation of radiation fraction

The radiation fraction is a key parameter to calculate flame radiation to an external target. The radiation fraction is a characteristic of the fuel and is constant under certain conditions for each fuel [29]. Markstein et al. believed that radiation fraction of buoyant dominated propane flame was a constant which is independent of burner size and HRR [30]. However, these values were obtained when a single fire burned. When multiple fires simultaneously burn, the change of propane radiation fraction is worth further examination.

3.3.1 Flame tilt angle

The flame shape and view factor are related to flame tilt angle in calculating external radiation. Figure 13 displays that tilt angle of merging flame gradually increases as the spacing increases, and that tilt angle of non-merging flame gradually decreases. For merging cases, as the spacing increases, each fire would tilt more to form a merging flame. For non-merging cases, the interaction among the fires decreases with increasing spacing, resulting in a smaller tilt angel. When spacing is equal to 0.3 and 0.6 m, three fires are independent of one another, and flame inclination is no longer considered in calculating external radiation.

3.3.2 Radiation fraction

When the flames are close to one another, the solution to view factor becomes more complicated. There is occlusion among the flames, and flame transmittance is also unknown. Thereby, it is very difficult to solve radiation fraction using cylindrical model and cuboid model. In addition, the point source model is suitable for far field and the prediction of near field is not accurate. Therefore, the multi-point source model is considered to solve radiation fraction.

The multi-point source model was first proposed by Hankinson and Lowesmith [31] based on the single-jet-fire study. It considers that the radiation is emitted from multiple point sources distributing along vertical central axis of the flame, and incident radiation received by the target is the vector sum of radiation from each individual point source. The position of each point source is determined by flame height L_f and tilt angle. For three fires as plotted in Fig. 14, the vector sum of incident radiation received by the target is

(19)q_{\mathrm{total}}={\displaystyle {\sum }_{j=\mathrm{1}}^N}j\left({\displaystyle {\sum }_{i=\mathrm{1}}^n}\overrightarrow{q_i}\right)={\displaystyle {\sum }_{j=\mathrm{1}}^N}j\left({\displaystyle {\sum }_{i=\mathrm{1}}^n}{\displaystyle \frac{w_i{x}_{\mathrm{r}}\dot{Q}{\tau }_i}{\mathrm{4}\mathrm{\pi }\overrightarrow{s_i^{\mathrm{2}}}}}\cdot \cos {\varphi }_i\right),

where q_total is the total incident radiation received by external target; N, the total number of fire sources; n, the corresponding thermocouple number at the average flame height; q_i, the incident radiation of the ith point source received by the target; w_i, the weight of the ith point source; x_r, radiation fraction; τ_i, atmospheric transmittance between the ith point source and radiometer (τ_i =1); s_i, the distance from the ith point source to radiometer (the position of the ith point source is the position of the ith thermocouple); {\varphi }_i, the angle between the radiometer normal and s_i; and L_i, the height of the ith point source to burner surface. s_i is calculated by

m=L_{\mathrm{f}}/\tan \left({\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}-\theta \right)-L_i/\tan \left({\displaystyle \frac{\mathrm{\pi }}{\mathrm{2}}}-\theta \right),

s_{\mathrm{t}}=m^{\mathrm{2}}+L_{\mathrm{s}}^{\mathrm{2}}-\mathrm{2}m{L}_{\mathrm{s}}\cos {\displaystyle \frac{\mathrm{\pi }}{N}},

(20)s_i^{\mathrm{2}}=s_{\mathrm{t}}^{\mathrm{2}}+L_i^{\mathrm{2}}.

Assuming that the flame is a gray body, and that the flame emissivity is independent of the wave length [16], the total energy radiated by a unit area of the gray body with absolute temperature T in each direction of the space is

(21)E=\epsilon E_{\mathrm{b}}=\epsilon \sigma T^{\mathrm{4}},

where ε is the graybody emissivity; σ, Stephan-Boltzmann constant; and T, the absolute temperature of the gray body. Based on Eq. (21), a new multi-point source weight calculating method is proposed as

(22)w_i={\displaystyle \frac{T_i^{\mathrm{4}}}{{\displaystyle {\sum }_{i=\mathrm{1}}^n{T}_i^{\mathrm{4}}}}},

where {\displaystyle {\sum }_{i=\mathrm{1}}^n{w}_i=\mathrm{1}}. Assuming that L_i/L_{\mathrm{f-three}} of inclined flame is equal to z/L_{\mathrm{f-single}} of single fire at the same HRR, and that the temperature at height L_i of inclined flame is equal to the temperature at height z of single fire at the same HRR, the formula for x_r is expressed as

(23)x_{\mathrm{r}}={\displaystyle \frac{q_{\mathrm{total}}}{{\displaystyle {\sum }_{j=\mathrm{1}}^{N}j}({\displaystyle {\sum }_{i=\mathrm{1}}^n}{\displaystyle \frac{w_{i}Q{\tau }_i}{\mathrm{4}\mathrm{\pi }\overrightarrow{s_i^{\mathrm{2}}}}}\cdot \cos {\varphi }_i)}\mathrm{.}}

Radiation fractions x_r1, x_r2, x_r3 and x_r4 were obtained by substituting the radiation value q_total measured by r₁, r₂, r₃ and r₄ into Eq. (23). Finally, radiation fraction x_r is the mean of x_r1, x_r2, x_r3 and x_r4. Figure 15 presents the variation of radiation fraction with HRR. As HRR increases, radiation fraction first increases and subsequently decreases. Souil et al. [32] used the point source model to obtain radiation fraction variation of single propane fire and found that it first increased with increasing HRR and subsequently tended to be an asymptotic value. The maximum HRR is 39.7 kW [32] so that the evolution of radiation fraction at a higher HRR is not discussed. In this paper, radiation fraction begins to decrease when smoke begins to obscure the flame from view and blocks the radiation from luminous flame regions [33]. In addition, the spacing hardly affects radiation fraction, which further verifies that the multi-point source model is suitable for the near and far field. Finally, the radiation fraction range of propane obtained in this paper is 0.27–0.33 when multiple fires simultaneously burn.

4 Conclusions

This paper mainly studied the combustion characteristics of three symmetric fires in open space. Based on a series of propane fire experiments, the merging behavior and flame height were quantitatively analyzed. The radiation fraction of propane was solved by measured radiation values. The main conclusions are as follows:

The flame-merging probability based on the statistical concept was used to quantify the degree of flame merging. A new merging criterion based on the dimensionless spacing S/z_c was proposed. When S/z_c>0.368, the flames do not merge. When S/z_c≤0.368, the flames merge.

Based on the relationship between buoyancy B and thrust P, the correlation between flame height and air entrainment was established, and the calculation formula of flame height was obtained.

A new weight determination method was proposed based on the previous multi-point source model. The new weight formula was w_i=T_i^{\mathrm{4}}/{\displaystyle {\sum }_{i=\mathrm{1}}^n}{T}_i^{\mathrm{4}}, and the radiation fraction was calculated, whose range is 0.27–0.33.

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