Multiple fire interactions: A further investigation by burning rate data of square fire arrays

Available online at www.sciencedirect.com Proceedings of the Combustion Institute xxx (2012) xxx xxx Proceedings of the Combustion Institute www.elsevier.com/locate/proci Multiple fire interactions: A further investigation by burning rate data of square fire arrays Naian Liu a,, Qiong Liu a,b, Jesse S. Lozano c, Linhe Zhang a, Zhihua Deng a, Bin Yao a, Jiping Zhu a, Kohyu Satoh a a State Key Laboratory of Fire Science, University of Science and Technology of China, Hefei, Anhui 230026, PR China b School of Resources and Safety Engineering, Central South University, Changsha, Hunan 410083, PR China c Mechanical Engineering Department, University of California, Riverside, USA Abstract This paper presents the first effort to explore the spatial distributions of the burning rates in group fires consisting of a large number of fire points, by analyzing burn-out time data from experimental square fire arrays ranging from 3 3 to 15 15. A new concept termed fire layer is introduced and defined to characterize the spatial locations of fire points by which the complex spatial variations of burning rates, under different conditions, are analyzed and physically interpreted. Analysis shows that the fire layer burning rates vary from outer to inner in definite nonlinear modes. This indicates that the two fire interaction effects, heat feedback enhancement and air supply restriction, involve distinct spatial fluctuations in fire arrays. The spatial fluctuations of the two interaction effects are significantly affected by the two major parameters, fire spacing and fire array size. Definite spatial regions and parameter ranges for the spatial fluctuations and high competitions of the two interaction effects are clearly distinguished. It is demonstrated that the average burning rates of all fire layers involve consistent variations versus fire spacing or fire array size, especially with high comparability to the entire fire array. It is found that by varying fire spacing, the average burning rates for all fire layers vary linearly versus the fire area ratio, within the same ranges as the entire fire array, while there exist different fluctuation modes of fire layer burning rates with respect to fire array size. Furthermore, analysis shows that the burning rates of all fire layers will be significantly affected by fire merging when it occurs. Finally, a new approach is presented to simulate fire propagation among discrete fuel sources, by which the positive effect of the surrounding new fire points on the burning rates of the original ones is definitely indicated. Ó 2012 Published by Elsevier Inc. on behalf of The Combustion Institute. Keywords: Group fire; Pool fire; Burning rate; Spatial distribution; Fire merging 1. Introduction Group fires normally consist of a number of pool fires which burn simultaneously on separate Corresponding author. Fax: +86 551 3601669. E-mail address: liunai@ustc.edu.cn (N. Liu). fuel beds, and generally involve intense burning and even remarkable dynamical behaviors such as fire merging and fire whirl, due to fire interactions by thermal radiation and convection. The burning rate of pool fires has attracted extensive investigations. Blinov and Khydyakov [1] pioneered measurements for various pool diameters and fuel types. Their experimental data was later 1540-7489/$ - see front matter Ó 2012 Published by Elsevier Inc. on behalf of The Combustion Institute. http://dx.doi.org/10.1016/

2 N. Liu et al. / Proceedings of the Combustion Institute xxx (2012) xxx xxx interpreted by Hottel [2], who established a systematic scheme of the variation of burning rate versus pool size. Since then numerous measurements for different fuels and pool sizes have been reported (see detailed review by Hamins et al. [3,4]). In physics, the heat feedback from the flame to the fuel surface and the gas evaporation from the pool constitute a positive feedback loop that controls the mass burning rate of any single fire. However for multiple fires, because the fire interactions may exert notable impact on both the heat feedback to fuel surface and the air flow in the burning zone, the laws and models for a single fire cannot be directly extrapolated to multiple fires. Despite the importance of fire interactions, there is limited literature focusing on the physical interpretation and quantitative characterization of multiple burning fires. Some authors proposed correlations of merged flame height versus different arrangement of multiple fires by physical modeling [5,6] or by empirical analyses [7 10]. Also, several criteria establishing the critical conditions required for flame merging were developed through experiments [11]. Nevertheless, note that all the developed physical or empirical models only apply to a limited number of multiple fire sources. Additionally, in order to achieve quasi-steady fuel burning rates most experiments were conducted using gaseous or liquid fuels stabilized on burner. A limited number of reports are available which discuss the simulation of practical group fires consisting of numerous fire points that burn naturally with mass flux not manually controlled. Our study which has been persistently pursued for several years was motivated by the limitations of previous investigations, and intended to systematically examine the burning rates of fire arrays consisting of a large number of fire points (up to 225 pools in this work). Experiments of square fire arrays with different values of fire spacing D and fire array size n (corresponding to a n n fire array) were conducted, and in our previous work [12,13] the theoretical analyses were conducted in two ways. The first is in a local sense for which the interaction effects for local fire points were quantitatively characterized and compared, and definite critical conditions for the onset of fire merging were clarified. Secondly, we examined the global average burning rate of the entire fire array versus fire spacing and fire array size, and the complex variation of burning rates was physically interpreted by the mechanisms of fire interactions. In this paper, we proceed to present the first effort to explore the spatial distributions of the burning rates in group fires, by analyzing the burn-out time data from experimental square fire arrays. A new approach for data analysis is proposed by which the complex spatial distributions of the burning rates for different parameters of fire spacing and fire array size are extensively analyzed. Such spatial distributions of burning rates are compared with the global average burning rate of the entire fire array, then physically interpreted by the variations of fire interactions under different conditions. 2. Experimental A total of 40 experiments (Table 1) of square fire arrays (3 3 to 7 7, 9 9, and 15 15) were performed in a large test hall with ceiling exhaust vents. The doors and windows remained closed during each experiment. All fires originated from equidistant identical circular steel fuel pans, 5 cm in diameter and 2 cm in height. All the pans were completely filled with n-heptane (98%). The difference between ignition times was maintained below 10 s. The minimum fire spacing D was 5 cm, while the maximum was 50 cm. Note that fire spacing D was defined as the centerline distance between adjacent fuel pans, thus the minimum fire spacing was equal to the diameter of the fuel pan (d = 5 cm). All tests were recorded using several video cameras with time synchronism that captured video from various view angles. The videos were used to fully observe all burning behaviors and to accurately extract the burn-out time data for all the fire points. During each test a single free-burning reference fire, with the same test fuel condition, was located far from the array. All the reference fires had an average burn-out time of 600 s. 3. Theoretical consideration In multiple burning fires, there exist two major fire interaction mechanisms which may affect the burning rates significantly (Fig. 1). One is heat feedback enhancement, which means that the fuel surface of any fire not only receives heat feedback from its own flame, but also experiences heat transfer from the surrounding fires, mainly by radiation. This will in response induce a burning rate increase. The accelerated burning fire will then in turn release more heat to the fuels of the surrounding flames. Another interaction mechanism is air supply restriction, which specifies that at smaller fire spacing, the air supply for inner fires may be suppressed, decreasing combustion efficiency and thus reducing the heat feedback to the fuel surface, which will in turn decrease the burning rate. The two mechanisms obviously depend on fire spacing and fire array size. Within certain ranges of these parameters, the two mechanisms may have comparable or competitive effects on burning rate. Regarding the fire array as a whole, we define the dimensionless global average burning rate for any n n fire array as

N. Liu et al. / Proceedings of the Combustion Institute xxx (2012) xxx xxx 3 Table 1 Average burning rates of fire layers and the entire fire arrays. BR *,3 3 BR *,4 4 BR *,5 5 D * rs(%) Di =0 Di = 1 Global rs(%) Di =0 Di = 1 Global rs(%) Di =0 Di =1 Di = 2 Global 1 78.54 2.09 2.23 2.10 78.54 2.03 1.96 2.02 78.54 2.64 2.73 2.73 2.67 2 28.27 2.79 3.08 2.82 25.65 3.02 4.35 3.27 24.24 3.31 4.11 4.11 3.56 3 14.43 2.21 2.61 2.25 12.57 2.48 2.90 2.57 11.62 2.80 3.59 4.00 3.05 4 8.73 1.63 2.28 1.69 7.44 2.17 2.98 2.33 6.79 2.11 2.92 3.18 2.35 6 4.18 1.46 1.67 1.48 3.48 1.58 1.96 1.66 3.14 1.76 2.25 2.45 1.92 8 2.45 1.17 1.62 1.35 2.01 1.47 1.89 1.56 1.80 1.53 1.80 1.65 1.61 10 1.60 1.29 1.43 1.30 1.31 1.30 1.65 1.37 1.17 1.62 1.74 1.80 1.62 BR *,6 6 BR *,7 7 D * r s (%) D i =0 D i =1 D i = 2 Global r s (%) D i =0 D i =1 D i =2 D i = 3 Global 1 78.54 2.50 2.51 2.48 2.50 78.54 3.08 3.23 3.23 3.23 3.15 2 23.37 3.78 4.37 4.44 4.03 22.77 3.97 4.65 5.26 5.41 4.38 3 11.05 3.03 4.32 4.69 3.52 10.66 3.10 4.21 4.44 4.44 3.61 4 6.41 2.32 3.66 4.20 2.80 6.16 2.39 4.05 4.51 4.51 3.06 6 2.94 1.83 2.29 2.55 2.03 2.81 1.62 2.20 2.63 2.96 1.92 8 1.68 1.46 1.64 1.79 1.55 1.60 1.53 1.81 2.09 2.24 1.70 10 1.09 1.49 1.66 1.77 1.57 1.03 1.42 1.55 1.70 1.94 1.51 BR *,9 9 BR *,15 15 D * rs(%) Di =5 Di = 6 Global D * rs(%) Di =0 Di =1 Di =2 Di =3 Di =4 Di =5 Di =6 Di = 7 Global 6 2.65 2.10 2.38 1.71 4 5.44 2.22 3.98 4.62 4.62 5.46 6.00 6.00 6.00 3.50 10 0.97 1.43 1.58 1.32 6 2.45 1.57 2.04 2.37 2.65 2.90 3.42 4.19 5.12 2.18 8 1.38 1.39 1.61 1.78 2.04 2.05 2.10 2.17 2.41 1.71

4 N. Liu et al. / Proceedings of the Combustion Institute xxx (2012) xxx xxx Fig. 1. Typical burning behavior of multiple fires. BR BR ¼ NM= P N m¼1bot ðmþ BR r M=BOT r ¼ N=P N m¼1bot ðmþ ð1þ 1=BOT r where BR ¼ NM= P N m¼1bot ðmþ is the global average burning rate of the fire array, and BR r ¼ M=BOT r is the time-averaged burning rate of the single reference fire. Here BOT r denotes the burn-out time of the reference fire (600 s). M is the initial constant fuel mass for each pool, BOT ðmþ is the burn-out time of the fire m, and N ¼ n 2 is the number of fire points.our previous work [13] clearly indicated that BR * could be well correlated with a dimensionless parameter r s, called fire area ratio, which denotes the ratio of the total fuel surface area to the whole fire array area, formulated as r s ¼ n2 ðp=4þd 2 ½d þðn 1ÞDŠ 2 p=4 ¼ ð2þ ½1=n þðn 1Þ=nD Š 2 where D D=d is the dimensionless fire spacing. By definition, r s decreases with D * or n monotonously, and it physically characterizes the fuel load, namely the total amount of fuel in a defined two-dimensional space. It was indicated that when the fire spacing is varied within 4 6 D 6 10, BR * linearly depends on r s for any specific fire array size. This linear correlation was physically interpreted by the variations of fire interactions.for the purpose of investigating the spatial distributions of burning rates in any fire array, a new concept termed fire layer is introduced here and defined to denote any group of fire points with equal distance to the boundary of the fire array (Fig. 2). The number of fire layers is thus (1 + n)/2 for odd n and n/2 for even n. The spatial location of any fire layer can be characterized by a dimensionless location index D i, defined by Fig. 2. The fire layers of 5 5 fire array. D i ¼ S i =D ð3þ where S i is the distance from the ith fire layer to the boundary fire layer. The average burning rate of the ith fire layer can then be defined as BR i BR i ¼ N P N i i= m BOT ðmþ ð4þ BR r 1=BOT r where N i is the number of fire points in the fire layer. This quantity allows us to use the fire layer burning rate data from all the experimental fire arrays (Table 1) to examine the spatial distributions of burning rates versus fire spacing and fire array size. Note that for convenience of discussion, we also use BR * to denote the fire layer burning rate without ambiguity. 4. Results and discussion 4.1. Nonlinear spatial distributions of fire layer burning rates in fire arrays Figure 3 presents the fire layer burning rates versus the spatial location index D i for 7 7 and 15 15 fire arrays, with fire spacing D * = 6 and 8. The two arrays are representative because they involve maximum numbers of fire layers in this work. As can be seen, the average burning rates of fire layers vary nonlinearly with D i. The air supply for the outmost fire layer (D i = 0) is naturally considered to be unrestricted. The increase of fire layer burning rates versus D i is obviously due to the enhancement of heat feedback. However, it is observed that for certain ranges of increasing D i (nearly within 0 3) the fire layer burning rates increase but with deceleration for all the curves. This indicates that the air supply restriction becomes more and more significant, and that the two fire interaction effects are competitive in

N. Liu et al. / Proceedings of the Combustion Institute xxx (2012) xxx xxx 5 Fig. 3. Fire layer burning rates versus D i for 7 7 and 15 15 fire arrays (D * = 6, 8). Fig. 4. Fire layer burning rates versus D i for 7 7 fire arrays (D * = 1 10). intensity. Especially for the 15 15 fire array with D * = 8, the fire layer burning rates remain nearly constant on D i = 3 5, which suggests high competition between the two interaction effects. When D i continues to increase, the heat feedback enhancement again becomes highly dominant, as evidenced by the accelerated increase of fire layer burning rates for the 15 15 fire arrays. The above analysis clearly reveals that the two fire interaction effects involve spatial fluctuations in any fire array. From outer to inner, the air supply becomes more and more restricted and within certain ranges the two fire interaction effects are highly competitive, leading to a comparable impact on burning rates. At the fire points adjacent to the fire array center, the heat feedback enhancement plays a dominant role in the variation of burning rates. 4.2. Fire layer burning rates versus fire spacing Fire spacing is one key parameter controlling the behavior of multiple burning fires. In physics, when fire spacing increases, the decayed flame heat feedback from the surrounding fires will naturally result in a burning rate decrease. At the same time, the burning may be enhanced due to the increased amount of space available for the air supply from ambient surroundings. However, the enhancement of air supply can also affect the air pressure gradient in the burning zone, resulting in the flames tending to stand vertically with weak flame stretching and leaning, which will further decrease the flame heat feedback. Therefore, due to the competition between the two fire interaction effects, the burning rate may involve complex variation versus fire spacing. Figure 4 shows the fire layer burning rates versus D i with different fire spacings (D * = 1 10) for 7 7 fire arrays. Other fire arrays have a similar variation mode. For small fire spacings like D * =1, BR * remains nearly constant versus D i, which suggests a high competition of the two interaction effects (with comparable intensities) in all fire layers. When D * increases, the fire layer burning rates monotonically increase versus D i, which indicates that although the air supply gradually becomes restricted from outer to inner, the enhancement of the heat feedback is more significant, and the range of high competition between the two fire interaction effects shifts to higher values of D i. For large fire spacings, the fire layer burning rates would again become to be constant versus D i, because the two fire interaction effects would decay gradually and finally disappear. It can thus be summarized that fire spacing significantly affects the fluctuations of the two interaction effects within the fire array. This causes different spatial ranges of high competitions between the two interaction effects for different fire spacings, leading to different spatial distributions of fire layer burning rates. Although the fire layer burning rates involve complex spatial distributions, the average burning rates of all fire layers show consistent variations versus fire spacing. Figure 5 shows the average burning rates versus fire spacing D *, for different Fig. 5. Average burning rates versus D * for different fire layers and the entire fire arrays. (a) 6 6; (b) 7 7.

6 N. Liu et al. / Proceedings of the Combustion Institute xxx (2012) xxx xxx Fig. 6. Average burning rates versus r s for different fire layers and the entire fire arrays, by varying fire spacing (4 6 D 6 10). (a) 6 6; (b) 7 7; (c) 15 15. fire layers and the entire fire arrays of 6 6 and 7 7, which indicates that the burning rates of fire layers vary with fire spacing similarly as the entire fire array. For D * = 1, as indicated above, the air supply is significantly restricted with strong competition with the heat feedback enhancement effect, leading to relatively lower values of BR *. BR * reaches maximum values at D * = 2 3, and then decreases monotonically versus D *, which clearly indicates that the heat feedback decays more rapidly than the enhancement of air supply. It is observed that the BR * values vary little for D * of 8 10, suggesting that the two interaction effects are comparable within that range. It is important to note that the average burning rate of the outmost fire layer is most close to that of the entire fire array. This implies that the burning intensity of the entire fire array can be estimated by monitoring the outmost fire layer. This experimental finding seems to be applicable for all the fire array sizes in this work. In fact, using the data in Table 1, it can be suggested that for 15 15 fire arrays the global burning rate is also most close to the burning rate of the boundary fire layer. Figure 6 shows the fire layer burning rates versus fire area ratio r s, by varying fire spacing within 4 6 D 6 10, for 6 6, 7 7 and 15 15 fire arrays. It is found that the average burning rates for all fire layers vary linearly versus r s, same as the entire fire array. This reveals that the entire fire array has high comparability to different fire layers in variations of burning rates, implying that the burning behavior of any fire layer can be used to reflect the variation trend of the entire fire array. Note that the linear variations versus r s appear within the same range for the entire fire array and different fire layers. 4.3. Burning rates of the entire fire array and fire layers versus fire array size Fire array size n is another key parameter for the behavior of multiple burning fires. When n increases for any specified fire spacing, the heat feedback will naturally be enhanced since any fire will receive heat from more surrounding flames. On the other hand, the air supply may be further restricted, since more fires consume the same available oxygen. Therefore the two fire interaction mechanisms may also have competitive effects on the burning rate of fire array. Figure 7 shows the variations of the average burning rates versus fire array size, for the first and second fire layers to center. Obviously the two layers show similar variation modes versus fire array size. Especially, with different values of D *, there exist different fluctuation modes of BR * with respect to n, however, this can be interpreted in physics. Take the first fire layer from center as an example. For D * = 6 and n = 3 7, the heat feedback enhancement is more significant than the air supply restriction, leading to an increase of BR *. However, the decay of the increasing rate implies that the air supply becomes more and more restricted at this range. For n =7 9, the two interaction effects show high competition, even causing a decrease of BR *. For higher values of n, BR * continues to increase, indicating that the heat feedback enhancement becomes dominant. For D * = 10 and n > 5, both fire layers

N. Liu et al. / Proceedings of the Combustion Institute xxx (2012) xxx xxx 7 Fig. 7. Fire layer burning rates versus n. (a) The first fire layer to center; (b) The second fire layer to center. show decreasing BR * versus n, which clearly reflects the dominant role of air supply restriction in the burning rate variations. We found that the variation trends versus fire array size n can be more clearly identified by the data of burning rates versus the fire area ratio r s. Recall that r s varies monotonically with n. Figure 8(a) shows the global average burning rate of the entire fire array versus r s by varying fire array size. The fire array sizes are marked on the data points. When n = 3, generally the air supply in the fire array would experience little restriction. Then for n = 3 5, the heat feedback is enhanced due to the increasing number of fires, leading to increased burning rates. Especially for n = 4 5, the burning rates would further increase due to the effect of fire merging. As indicated in [12], fire merging would occur for D * < 8. The effect of fire merging is clearly identified by the persistent increase of burning rates versus n on the curves for D * = 6, 8 and 10 within n = 3 5, and remarkably by the rapid increase for D * = 10 within n = 4 5. In another aspect, note that with increasing n, the air supply restriction also becomes increasingly significant, this in turn suppresses the increase of burning rates. This fact is evidenced by the nearly constant burning rate within n = 4 5 for D * = 4, and also by the decreasing burning rates within certain ranges for D * =6,8 and 10. For larger n values with D * = 4 and 6 (when fire merging would occur), the fire merging has a great effect on the burning of the fire array, leading to a persistent increase of the burning rates. For D * = 10, which would not induce fire merging, no increase of burning rate is observed for larger n (or lower r s ) values. This further demonstrates that the persistent increase of burning rates for higher values of n, in the other cases of D *, are mainly due to the role of fire merging. Figure 8(b) shows BR * versus r s for the first and second fire layers from center. In contrast to Figure 8(a), Figure 8(b) indicates that the fire layers and the entire fire array involve nearly consistent variation processes. Three ranges of increasing n, with different dominant mechanisms of fire interactions, can be definitely distinguished. For lower values of n, the burning rates of all the fire layers and the entire fire array first undergo an increase versus n due to heat feedback enhancement. Within certain ranges of further increasing n, the air supply would be more and more restricted and would gradually become highly competitive with the heat feedback enhancement effect, leading to constant or even decreased Fig. 8. Average burning rates versus r s for different fire layers and the entire fire arrays, by varying fire array size for specific fire spacings. The fire array sizes n are marked on the data points.

8 N. Liu et al. / Proceedings of the Combustion Institute xxx (2012) xxx xxx Table 2 Average burning rates of the original 3 3 and 5 5 fire arrays. D * =2 D * =3 D * =4 D * =6 D * =8 D * =10 r s (%) BR * r s (%) BR * r s (%) BR * r s (%) BR * r s (%) BR * r s (%) BR * 3 3 original array 3 3 28.27 2.82 14.43 2.25 8.73 1.69 4.18 1.49 2.45 1.35 1.60 1.30 5 5 24.24 4.11 11.62 3.63 6.79 2.95 3.14 2.27 1.80 1.78 1.17 1.75 7 7 22.77 5.28 10.66 4.44 6.16 4.51 2.81 2.66 1.60 2.11 1.03 1.73 9 9 NA NA NA NA NA NA 2.65 2.36 NA NA 0.97 1.56 15 15 NA NA NA NA 5.44 6.00 2.45 4.28 1.38 2.20 NA NA 5 5 original array 5 5 24.24 3.56 11.62 3.05 6.79 2.35 3.14 1.92 1.80 1.61 1.17 1.66 7 7 22.77 4.89 10.66 4.30 6.16 4.20 2.81 2.35 1.60 1.91 1.03 1.61 9 9 NA NA NA NA NA NA 2.65 2.19 NA NA 0.97 1.47 15 15 NA NA NA NA 5.44 6.00 2.45 3.69 1.38 2.131 NA NA Fig. 9. Global average burning rates of the original fire arrays versus r s for fire array expansion in four directions. (a) 3 3; (b) 5 5. The fire array sizes n are marked on the data points. burning rates for the entire fire array and different fire layers. In this range, the air supply restriction obviously plays a dominant role in the variation of burning rates. For large values of n, if fire merging occurs, then the burning would be significantly enhanced by fire merging. It is important to note that the ranges of the distinguished stages for different fire layers are nearly the same as those of the entire fire array. 4.4. Simulation of fire propagation among discrete fuel sources In practical urban or wildland group fires, fire propagation can be regarded as a series of fire ignitions among discrete fuel sources, mainly by radiation and firebrands [14]. For the fire propagation by radiation ignition, one question is that when one group of fire points act as the original sources to induce new ignitions, how about the reverse effect of new fire points acting on the original sources. This is obviously related to the spatial distribution of burning rates in multiple fires. The data in this work can be used for initial simulation analysis. It is assumed here that a group fire consists of multiple fire points which constitute one initial square fire array, which in series induces ignitions of its surrounding fuel sources in four directions leading to continual size expansion of the fire array. The ignitions are assumed to occur rapidly with negligible ignition times. By these assumptions, in this work, data with different fire array sizes can be used by regarding one smaller fire array as a benchmark to investigate the effect of new discrete fire points on the original fire array. We take 3 3 and 5 5 fire arrays as the original arrays and examine their global average

N. Liu et al. / Proceedings of the Combustion Institute xxx (2012) xxx xxx 9 burning rates when they are undergoing expansion to generate new fire points. The data for the two original fire arrays is listed in Table 2. In physics, the original array is always located in the center of burning zone, and thus receives thermal radiation from its surrounding new fire points leading to a burning rate increase. At the same time, fresh air entrainment to the center of the fire array is hindered by the external fires which act as a barrier preventing air from reaching the innermost fires, thus burning of the original array is suppressed. As shown in Fig. 9, BR * increases with decreasing fire area ratio r s. Since r s decreases with fire array size n monotonously, the result indicates that although the two fire interaction effects cause complex spatial fluctuations of burning rates, the burning rates of the original fire arrays show persistent increases by the expansions of the fire arrays. In other words the new ignited fire points have positive effects on the burning intensities of the original fire points. 5. Conclusions As we continue investigating the multiple fires problem, this paper presents the first effort to explore the spatial distributions of the burning rates in group fires consisting of a large number of fire points, by analyzing burn-out time data from experimental square fire arrays of 3 3 15 15. A new concept termed fire layer was defined to characterize the spatial locations of fire points by which the complex spatial variations of the fire layer burning rates for different values of fire spacing and fire array size were extensively analyzed. The roles of the fire interaction effects in the spatial variations of burning rates were physically interpreted. The major conclusions are summarized as follows. (1) The average burning rates of fire layers show definite nonlinear spatial distributions from outer to inner regions of any fire array. This indicates that the two fire interaction effects, heat feedback enhancement and air supply restriction, involve distinct spatial fluctuations in the fire array. The spatial regions with high competitions between the two effects were clearly distinguished. (2) The spatial fluctuations of the two interaction effects in fire arrays are significantly affected by the two major parameters of fire spacing and fire array size. Definite parameter ranges for the spatial fluctuations and the high competitions of the two interaction effects were distinguished. (3) The average burning rates of all fire layers involve consistent variations versus fire spacing or fire array size, and the fire layers involve high comparability to the entire fire array for variations of burning rates. This was evidenced by the consistent parameter ranges for the spatial fluctuations between the two fire interaction effects. This implies that the burning behavior of any fire layer can be used to reflect the variation trend of the entire fire array. Especially by varying fire spacing, the average burning rates for all fire layers vary linearly versus the fire area ratio, within the same ranges as the entire fire array. By varying fire array size, there exist different fluctuation modes of average burning rates for different fire layers. Furthermore, the burning rates of all fire layers will be significantly affected by fire merging when it occurs. (4) A new approach is presented for initial simulation analysis for fire propagation among discrete fuel sources by addressing the reverse effect of the new fire points on the original points. The results suggest positive effects of the new ignited fire points on the burning intensities of the original fire points. Acknowledgements This work was sponsored by the National Natural Science Foundation of China under Grant 51036007, 51120165001 and 51076148, and National Key Technology R&D Program under Grant 2011BAK07B01-02. Naian Liu was supported by the Fundamental Research Funds for the Central Universities (No. WK2320000014). Qiong Liu was supported by Open Fund of SKLFS (HZ2010-KF13). References [1] I.V. Blinov, G.N. Khudiakov, Diffusive Burning of Liquids (English translation by US Army Engineering and Research Laboratories), T-1490a-c, ASTIA, AD296762, 1961. [2] H.C. Hottel, Fire Res. Abstracts Rev. 1 (1959) 41 44. [3] A. Hamins, S.J. Fischer, T. Kashiwagi, M.E. Klassen, J.P. Gore, Combust. Sci. Technol. 97 (1-3) (1994) 37 62. [4] A. Hamins, J. Yang, T. Kashiwagi, A Global Model for Predicting the Burning Rates of Liquid Pool Fires, US Department of Commerce, September 1999. [5] A.A. Putnam, C.F. Speich, Proc. Combust. Inst. 9 (1963) 867 877. [6] P.H. Thomas, R. Baldwin, J.M. Heselden, Proc. 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