Experimental investigation of fire behaviours and heat transfer in single cypress tree crown fires

Hanwen Guo; Yunji Gao; Ziqun Ye; Zhengyuan Yang; Yuchun Zhang; Zijian Lei; Ao Sun

doi:10.1071/WF24030

RESEARCH ARTICLE (Open Access)

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Experimental investigation of fire behaviours and heat transfer in single cypress tree crown fires

Hanwen Guo ^A ^B , Yunji Gao ^A ^* , Ziqun Ye ^A , Zhengyuan Yang ^A , Yuchun Zhang ^A , Zijian Lei ^A and Ao Sun ^A

+ Author Affiliations

- Author Affiliations

^A Department of Fire Protection Engineering, Southwest Jiaotong University, Chengdu, 610031, China.

^B College of Forestry, Northeast Forestry University, Harbin, 150040, China.

^* Correspondence to: gyj119@swjtu.edu.cn

International Journal of Wildland Fire 34, WF24030 https://doi.org/10.1071/WF24030

Submitted: 12 February 2024 Accepted: 21 January 2025 Published: 25 February 2025

© 2025 The Author(s) (or their employer(s)). Published by CSIRO Publishing on behalf of IAWF. This is an open access article distributed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND)

Abstract

Background

Crown fires are among the most perilous and challenging types of forest fires to combat, posing a significant hazard to forest ecosystems and the wildland–urban interface.

Methods

Crown fire experiments were carried out using cypress trees with five different crown heights from 0.4 to 1.2 m to investigate the fire behaviours and heat transfer mechanisms.

Key results and conclusions

The characteristic values of flame height, mass loss rate and heat flux increased with crown height, whereas flame width increased with crown width. Empirical correlations of mass loss rates were established. Flame temperature varied with flame height and relationships between these were developed based on four dimensionless temperature zones. Flame emissivity was related to crown height and increased exponentially with the heat release rate. This study proposes a modified model to predict the radiation heat flux from crown fires, taking the variations of flame temperature and flame emissivity into account.

Implications

The findings of this work enhance understanding of fire behaviours and controlling mechanisms of crown fire under different crown heights, and also improve the accuracy of prediction of wildland fire development and spread in the wildland–urban interface.

Keywords: combustion, crown fire, fire behaviour, fire plume temperature, flame height, heat transfer, mass loss rate, radiation heat flux.

Introduction

Wildland fires not only burn and desiccate trees, directly reducing forest area, but also severely damage forest structure and environment. This disruption leads to an imbalance in forest ecosystems, a decrease in forest biomass, a decline in productivity, a decrease in beneficial birds, and even human and animal casualties (Motazeh et al. 2013; Guo et al. 2024). Wildland fire can generally be categorised into underground, surface and crown fire (Wagner 1977). Surface fire has the highest occurrence probability and frequency (Rothermel 1991; Castle 2015), whereas crown fires occur less frequently, but account for ~90% of total forest area burned owing to their rapid spread and difficulty to control by direct action (Albini and Stocks 1986; Mell et al. 2009; Keyser and Smith 2010; Ruiz-González and Álvarez-González 2011; Thompson et al. 2020). Furthermore, most of the commercial timber value lost annually to wildland fires comes from crown fires. Therefore, it is essential to investigate burning and spread behaviours, which can provide valuable data for management measures to prevent and reduce the risk of crown fires.

Several studies have investigated the spread rate and behaviours of crown fire. A fire intensity model was first proposed by Ryan (1981) to quantify the relationship between fire spread rate and flame height. Albini and Stocks (1986) developed a prediction model for the spread rate of crown fires using immature jack pine. Wagner carried out two sets of crown fire experiments with ponderosa pine trees and established modified models for predicting the spread rate of surface fires and the degree of crown depletion (Stocks 1987, 1989; Wagner 1993). Cruz et al. (2005) studied the spread rate of crown fires over flat to gently undulating terrain, identifying active and passive crown fires.

Recently, Hoffman et al. (2015) conducted a coarse assessment of predicted crown fire spread rate values using the Wildland Fire Dynamics Simulator or FIRETEC. A series of brush-burning experiments were performed by Rossa et al. (2016) to investigate the effect of moisture content on the rate of spread of crown fire. Evans et al. (2004) established a relationship between average flame height and crown height ratio for landscape trees. Zhou and Simeoni (2022) proposed an analytical model to predict the crown scorch height based on experimental data. However, almost all the above studies have focused on flame behaviours such as crown fire spread rate and flame height. The control mechanism of heat transfer from crown fires has rarely been mentioned.

As is well known, thermal radiation plays a dominant role in the development and spread of crown fires. Mell et al. (2009) experimentally and numerically studied the impact of tree height and fuel moisture content on mass loss rate and radiant heat flux in crown fires of Douglas fir trees. For the mechanism of radiative heat transfer from crown fires, Cohen and Butler (1996) and Cohen (2000a) developed a building ignition assessment model based on a rectangular flame plane and conducted a series of experiments to investigate the ignition caused by crown fires of wooden walls at various distances (10, 20, 30 m). Further, Evans et al. (2004) developed a straight-cylinder model for crown fires under windless conditions, which was applied in numerical simulations of single-tree burning to study fire spread. Modarres et al. (2024b) experimentally and numerically investigated the fire behaviours of Eucalyptus globulus trees, including flame height, rate of mass loss and heat flux, and found the flammability of eucalyptus varied with growth stage. In addition, the determination of radiation and convection for firefronts was studied by Modarres et al. (2024a) to evaluate fire risk for unprotected individuals, firefighters and buildings. All of the above models simplified the crown flame surface to a single structure and a uniform flame surface temperature was used to calculate radiation heat flux.

Moreover, previous studies on pool fires by McCaffrey (1979), Wan et al. (2018) and Ji et al. (2021) observed that the fire plume could be divided into three regions: the continuous flame region, intermittent flame region and plume region. It was stated that the temperature was approximately constant in the continuous flame region and decreased with height in the plume region. Further, recent work (Guo et al. 2024) also found that the fire plume temperatures at different heights of the crown fire varied, with the centre temperature being the highest. In summary, variable flame temperature should be considered when calculating the heat transfer mechanism of crown fires.

In view of the above problems, this paper is devoted to studying combustion characterisation and mechanisms of heat flux to the surroundings of crown fires using single cypress trees with different crown heights (0.4–1.2 m). A series of well-controlled crown fire experiments were conducted in a specially designed experiment set-up under no-wind conditions, and various crown fire spread behaviours and heat flux control mechanisms were analysed and compared in detail. A modified heat transfer model is proposed to predict the radiation heat flux from a single cypress crown fire to the surroundings, taking the variations of emissivity and flame temperature into account. However, the applicability of this finding needs to be further verified, taking into account more factors such as wind speed, water content and scale effects. By investigating the combustion characteristics and heat transfer of crown fires, the accuracy of prediction of wildland fire development and the timeliness of fire warnings in wildland–urban interface areas can be improved.

Materials and methods

Heat transfer model

The initiation of a crown fire to surrounding trees or buildings is a complex process, where the unburned fuel is heated and ignited by the burning tree. Heat received by unburned fuel from a crown fire mainly consists of convection and radiation heat fluxes, expressed as (Guo et al. 2023):

(1)

{\dot{q}}_{t}^{″} ~ {\dot{q}}_{r}^{″} + {\dot{q}}_{c}^{″}

where ${\dot{q}}_{t}^{″}$ is the total heat flux, ${\dot{q}}_{r}^{″}$ is the radiation heat flux and ${\dot{q}}_{c}^{″}$ is the convection heat flux.

For crown fire spread, the radiation exchange between individual elements like trees and buildings can be assessed using an approach used by the Society of Fire Protection Engineers (SFPE) (Luke and McArthur 1978; Pyne et al. 1996) to evaluate thermal radiation on an external target from a pool fire. The radiation assessment approach for pool fire mainly consisted of two types of semi-empirical models, which were the point source mode (Modak 1977) and the solid flame model (Mudan 1987). The point source model assumed that flame radiation came from a single point, and radiation heat flux received by a nearby target was expressed by following equation:

(2)

{\dot{q}}_{r}^{″} = τ X_{r} \dot{Q} \cos η / 4 π R_{0}^{2}

where τ is atmospheric transmissivity, Q̇ is heat release rate, X_r is the proportion of heat emitted in the form of thermal radiation, which is chosen to be the mid value of 0.35 in the present work, similarly to previous work on crown fires (Evans et al. 2004; Wan et al. 2018), R₀ is the distance between the target and the virtual origin, η is the angle between the connecting line and the normal direction of the target. The point source model has commonly been used to evaluate the radiation heat flux when the measured distance is greater than 2.5 times the flame diameter (Ji et al. 2021). However, this model was relatively ineffective for predicting radiation heat flux at closer distances.

Unlike the point source model, the solid flame model assumed that the thermal radiation was emitted uniformly over the entire flame. According to the solid flame model, thermal radiation heat flux can be expressed as:

(3)

{\dot{q}}_{r}^{″} = E F τ

where E is the average emissive power at the flame surface; F is the geometric view factor, τ is the atmospheric transmissivity. The average emissive power at the flame surface can be written as E = εσT⁴, where ε σ, and T are the flame emissivity, Stephan–Boltzmann constant and flame temperature, respectively. Flame emissivity is one of the important parameters that determines flame radiation to external targets. Therefore, the thermal radiation heat flux of crown fire can be expressed as:

(4)

{\dot{q}}_{r}^{″} = F τ ε σ T^{4}

In previous work, ε and T in Eqn 4 were usually assumed to be constant in the whole flame for simplicity in the calculation of radiant heat flux values, so that E was also regarded as constant, as used in Mudan’s model (1984) and Shokri and Beyler’s model (1989). In fact, these parameters are not fixed in crown fires under different conditions. Therefore, modelling the heat transfer of crown fire taking the variation of parameters into account is the main motivation for this research.

Experimental set-up

The crown fire experiments in this work were conducted on a laboratory-scale experimental set-up, schematically shown in Fig. 1. The experimental trees with various crown heights were secured on the fixed stand to prevent them from tilting during combustion. A specially designed circular ignitor, filled with alcohol, was used to ignite the base of the tree crown, similarly to previous work (Evans et al. 2004). The ignitor flame was promptly extinguished and removed after ignition of the tree. The set-up, consisting of a tree and an ignitor, was placed on a fireproof board supported by a load cell. An electronic balance (MS12002TS/02, with a maximum range of 12,200 g, accuracy of 0.01 g and response time of 1.0 s) was used to measure mass loss during the crown fire over time. The electronic balance was levelled and calibrated by the weighing method before each experiment to measure accurately. Twenty-one thermocouples (K-type) with a diameter of 1.0 mm were arranged at the central line of the crown to obtain the flame plume temperature of the crown fires, with a range and frequency of 0–1100°C and 1.0 Hz, respectively. Twelve thermocouples (t₁–t₁₂) were spaced at 10 cm intervals from the base to the top of the crown, while the remaining nine thermocouples (t₁₃–t₂₁) were arranged from the top of the crown at intervals of 20 cm. To ensure the measured data’s accuracy, thermocouples were calibrated prior to each experiment with errors less than 3% (Guo et al. 2023). One Sony FDR-AX60 camera (25 frames/s) was positioned 3.0 m away from the front of the tree, and the other was 1.5 m away from the side of the tree, and they were used to record the flame geometry and the process of the crown fire from two directions. Before each experiment, calibration of the cameras was conducted for white balance and exposure. An architectural wood panel, simulating nearby combustible materials like buildings in the wildland–urban interface, was set up near the tree to evaluate heat transfer from the crown fires. The horizontal distance between the wooden panel and the experimental tree was adjustable to simulate different spacing. Two pairs of heat flux sensors (Gardon and water-cooled), including total heat flux and radiation heat flux sensors, were fixed on the wood panel, and arranged at the height of the middle and the top of the crown to observe the heat flux from the crown fire. The ranges for the total heat flux and radiation heat flux sensors were 0–100 and 0–50 kW/m². The heat flux sensors were regularly calibrated before experiments through a standardised calibration process, similarly to previous work (Xie et al. 2020). Moreover, two pairs of heat flux sensors were tested before the experiment to ensure their measurement error was less than 5%. For burning experiments with different crown heights, the vertical positions of heat flux sensors should be adjusted before each experiment. A set of steel rulers was installed on the board to determine flame geometry.

Fig. 1.

Twelve thermocouples (t₁–t₁₂) were spaced at 10 cm intervals from the base to the top of the crown, while the remaining nine thermocouples (t₁₃–t₂₁) were arranged from the top of the crown at intervals of 20 cm. Schematic diagram of experimental set-up: ➀ thermocouple; ➁ experimental tree; ➂ ignitor; ➃ support; ➄ wood panel; ➅ heat flux sensors; ➆ camera; ➇ load cell; ➈ steel ruler.

In previous work, cypress trees were selected as the tree species for crown fire experiments. Based on field surveys, cypress is a typical tree species in southwest Sichuan, where wildfires are frequent in China. In addition, cypress trees are highly flammable and characterised by large amounts of volatile oils and waxes, which can be easily ignited by high temperatures or hot sparks from fires in nearby areas. The combustion heat of cypress trees is 19.8 MJ/kg measured by an oxygen bomb calorimeter. The oils and waxes produce a lot of toxic smoke in the fire, which seriously affect fire rescues. Moreover, the common cypress has been reported to be the most flammable of several Mediterranean species tested by Liodakis et al. (2002). Comparing different Mediterranean species, Petriccione et al. (2006) classified the flammability of cypress litter as moderate–high. In summary, compared with other tree species, higher forest fire hazard is associated with cypress trees owing to their nearly cylindrical shape, lower ignition point and higher combustion heat. Therefore, cypress trees were selected as the tree species for crown fire experiments in the present paper, as they were also chosen in previous work (Guo et al. 2024; Wang et al. 2024). The experimental trees were heated and dried in an oven at 60°C for 24 h until the mass of the trees no longer changed.

Five crown heights were selected in this work to investigate the effects of crown height on crown fire behaviours, namely 0.4 ± 0.05, 0.6 ± 0.08, 0.8 ± 0.0, 1.0 ± 0.10 and 1.2 ± 0.15 m. The mass of the trees was 78 ± 10, 131 ± 15, 257 ± 30, 506 ± 40 and 765 ± 50 g, respectively. Fourteen distances between the wooden panel and the experimental tree of 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.5 and 1.8 m were chosen to study the heat transfer from crown fires to surrounding buildings near the wildland–urban interface. The experimental conditions, including crown height, crown width, mass of tree and distance between the tree and wooden panel, are tabled in Appendix 1. All the experiments were performed in a large test hall in still air, in which the ambient temperature and relative humidity were 11.5 ± 2℃ and 71 ± 3%, respectively.

Results and discussion

Flame geometry

Fig. 2 shows the evolution of the flame shape of crown fire for different crown heights (H_c). Note that the flame height of the crown fire initially increases over time, reaches a maximum and then falls gradually toward extinction. Moreover, the larger the crown height, the higher the flame height.

Fig. 2.

Evolution of flame shape with time for typical experimental conditions: (a) H_c = 60 cm; (b) H_c = 100 cm.

Flame height and flame width are essential indicators of crown fires, and greatly influence the heat transfer process. Flame height is defined as the distance from the base to the top of the flame, while flame width is the maximum distance of flame lateral extension. In this work, flame height and flame width were obtained by converting the RGB image of the experimental video to a grayscale image frame by frame in a coded manner (Yan et al. 2012). In previous work, this method has been successfully applied to obtain flame geometry of both surface fire (Li et al. 2021) and crown fire (Wang et al. 2024). Subsequently, the pixel values for flame height and width were then converted to actual measurements using a proportional relationship established using the steel ruler on the board, providing the geometric parameters of the flame at each moment. Finally, the flame length and width were determined by the method of flame presence probability, which was first proposed by Zukoski et al. (1985). The flame presence probability at a given point in the fire plume was obtained from the ratio of the number of pixels with flame presence to the total pixel number during the steady stage (Wei et al. 2023) (Fig. 3).

Fig. 3.

Schematic diagram of flame geometry acquisition method.

Fig. 4 plots the evolutions of flame height and flame width over time for different crown height and width conditions. It was found that the flame height and flame width first increased with time, reached a peak at some time, and then decreased until the crown fire was extinguished. As crown height increased, the flame height grew accordingly. Additionally, the flame height and flame width fluctuated around the maximum value for a short period of time, reflecting the fully developed stage of combustion in the crown fire. Using the method described above, the characteristic values of flame height and length were obtained by reading the contours with a presence probability of 0.5 at the steady stage.

Fig. 4.

Evolutions of flame height and width with time for different crown heights: (a) flame height; (b) flame width.

Fig. 5a, b represents the variations of characteristic values of flame height and flame width with crown height and crown width, respectively. The results of experiments by Evans et al. (2004) and Wang et al. (2024) are also shown in Fig. 5a. From Fig. 5a, a linear relationship can be observed between flame height and crown height, which can be expressed as follows:

(5)

H_{f} = 2.27 H_{c}

where H_f is the flame height, H_c is the crown height. In the present work, the flame height was 2.27 times the crown height, which is greater than the results in previous work (Evans et al. 2004; Wang et al. 2024). At present, the relationship between flame width and crown width has attracted little attention, but it plays an important role in the heat transfer process. According to the experimental data in this work, the relationship between the flame width and crown width can be stated as:

(6)

W_{f} = 1.51 W_{c}

where W_f is the flame width, W_c is the crown width. The ratio of flame width and crown width was 1.51, less than the ratio of flame height and crown height. As the analysis mentioned above shows, the predicted values of flame height and width can be obtained according to the size of the tree.

Fig. 5.

Variations of flame height and width with crown height and width: (a) flame height; (b) flame width.

In previous work, the empirical relationship between heat release rate and flame height has been developed as follows:

(7)

H_{f} = 0.027 {\dot{Q}}^{2 / 3}

(8)

H_{f} = 0.0347 {\dot{Q}}^{0.608}

(9)

H_{f} = 0.0235 {\dot{Q}}^{2 / 5} - 1.02 W_{f}

Eqns 7 and 8 were developed by Kim (2009) and Modarres et al. (2024b) to evaluate the flame height for Pinus densiflora and eucalyptus trees. Eqn 9 was established by Heskested (1983) for calculating the height of a pool fire.

According to previous work, the heat release rate of the flame can be expressed as:

(10)

\dot{Q} = φ Δ h \dot{m}

where φ is the combustion efficiency factor, Δh is the heat value of the fuel, ṁ is the mass loss rate, which is analysed in the Mass loss rate section below.

The relationship between heat release rate and flame height in this work can be expressed as:

(11)

H_{f} = 0.196 {\dot{Q}}^{0.481} - 0.94 W_{f}

The form of the relationship between heat release rate and flame height in this work is similar to Heskested’s, which can be attributed to the similarity of the flame shape. The comparison between experimental and calculated values of flame height from Eqns 7–9 is shown in Table 1. The comparison of the numerical and absolute errors reveals that Eqns 9 and 11 better evaluate flame heights of cypress trees.

Table 1.Comparison between experimental and calculated values from Eqns 7–9 and Eqn 11.

Conditions	H_f (m) [absolute error (m)]
Conditions	H_c = 0.4 m	H_c = 0.6 m	H_c = 0.8 m	H_c = 1.0 m	H_c = 1.2 m
Eqn 7	0.25 [−0.8]	0.45 [−1.03]	0.55 [−1.07]	0.73 [−1.08]	1.24 [−1.52]
Eqn 8	0.26 [−0.79]	0.44 [−1.04]	0.53 [−1.09]	0.68 [−1.13]	1.11 [−1.65]
Eqn 9	0.9 [−0.15]	1.3 [−0.18]	1.45 [−0.17]	1.65 [−0.16]	2.42 [−0.34]
Eqn 11	0.94 [−0.11]	1.42 [−0.06]	1.54 [−0.08]	1.72 [−0.09]	2.81 [0.05]

Crown fire plume temperature

Fig. 6 illustrates the variation in plume temperature of crown fires over time under typical experimental conditions. As can be seen, the fire plume temperature gradually increased to a peak and then decreased to ambient temperature. Three temperature positions were selected to compare the crown fire plume temperature in the crown portion: the base, middle and top of the crown height. Note that the plume temperatures in the middle of the crowns were greater than those at the top and the base of the crowns, which is consistent with our previous work (Guo et al. 2024). However, the correlation of flame temperature variation with flame height was not modelled in previous studies (Guo et al. 2024; Wang et al. 2024). In addition, the fire plume temperature of crown fires was assumed to be constant for heat transfer analysis in previous work (Evans et al. 2004). Nevertheless, the experimental data indicated that the fire plume temperature of the crown fires varied with vertical position (Table 2).

Fig. 6.

Evolution of fire plume temperature with time for typical experimental conditions: (a) Case 22; (b) Case 40.

Table 2.Fitting results for a, b, c, d.

Conditions	a	b	c	d
H_c = 0.4 m	–	–	−0.71	0.79
H_c = 0.6 m	0.41	1.22	−0.84	0.84
H_c = 0.8 m	0.65	1.32	−0.93	0.89
H_c = 1.0 m	0.57	1.43	−1.11	0.94
H_c = 1.2 m	0.92	2.24	−1.21	1.07

In previous work on pool fire by McCaffrey (1979), the axisymmetric fire plume above the fire surface was divided into three zones, namely the continuous flame, intermittent zone and the plume zone, to evaluate the central line temperature rise versus height. For further investigation, a similar dimensionless analysis of height was carried out as follows:

(12)

k = \frac{H_{t}}{H_{f}}

where k is the dimensionless ratio of the thermocouple height to the tree height; H_t is the thermocouple height. A dimensionless analysis of fire plume temperature was also conducted to investigate the temperature at different positions as follows:

(13)

t = \frac{Δ T}{Δ T_{\max}}

where t is the dimensionless ratio of the fire plume temperature at different positions to a maximum value of temperature; ΔT is the fire plume temperature at different positions; ΔT_max is the maximum value of the fire plume temperature for each experiment. The evolution of dimensionless flame temperature with dimensionless height for different crown height conditions is represented in Fig. 7. It can be seen that the dimensionless temperature ΔT/ΔT_max increases sharply from the crown base height to the middle of the crown (k ≤ 0.5). Remarkably, the maximum value of ΔT/ΔT_max is obtained at k = 0.5, indicating the maximum temperature appears in the middle of the crown. Previous studies of pool fires had typically divided the axisymmetric flame plume above the fuel surface into three zones: the continuous flame region, the intermittent flame region and the plume region. Among these, the flame temperature in the continuous flame region was essentially constant, while the flame temperature in the intermittent and plume regions decreased with increasing height (McCaffrey 1979; Ji et al. 2021). Unlike pool fires, a significant temperature rise was observed from the base to the middle of the crown. Additionally, little change in dimensionless temperature was found from the middle to the top of the crown (0.5 < k ≤ 1), demonstrating that the temperature in this zone was approximately constant. This phenomenon was more pronounced in the cases of greater crown height owing to a longer flame-stabilised combustion phase. From the top of the crown to the flame tip (1 < k ≤ 2.27), the values of temperature decreased rapidly, but with further increase in k (2.27 < k), the values of t decreased slowly. Therefore, the axisymmetric fire plume for a crown fire can be divided into four zones based on the variation of dimensionless temperature: Zone I: increase stage (k ≤ 0.5); Zone II: stable stage (0.5 < k ≤ 1); Zone III: rapid decline stage (1 < k ≤ 2.27) and Zone IV: slow decline stage (2.27 < k).

Fig. 7.

Evolution of fire plume temperature with k for different crown heights: (a) H_c = 0.8 m; (b) H_c = 1.2 m.

In previous work on pool fires, the temperature rise of individual regions with normalised height was fitted with power functions (Ji et al. 2021). In the present work, similar fitting methods were used to evaluate the dimensionless temperature rise ΔT/ΔT_max at different zones with dimensionless height H_t/H_f.

(14)

t = \frac{Δ T}{Δ T_{\max}} = {\begin{cases} b \times {(H_{t} / H_{f})}^{a}, & 0 < H_{t} / H_{f} \leq 0.5 \\ 1, & 0.5 < H_{t} / H_{f} \leq 1 \\ d \times {(H_{t} / H_{f})}^{c}, & 1 < H_{t} / H_{f} \leq 2.27 \end{cases}

where a, b, c and d are constants. The average values of a, b, c and d for different experiments with the same kind of crown height were selected as the characteristic values for each crown height, with R² greater than 0.9. The fitting results are shown in Table 2. It can be seen that greater values of a and b were observed under conditions of higher crown height, indicating a faster temperature increase in Zone I for taller crowns. Similarly, the higher the crown, the faster the temperature dropped at Zone III. Therefore, the different fire plume temperatures of crown fires at different heights should be taken into account especially when considering heat transfer processes. Owing to the limited number of thermocouples (only two) positioned from the bottom to the middle of the 40 cm crown, fitting parameters a and b were not available for this height.

Mass loss rate

Fig. 8 shows the evolution of fuel mass over time measured by the load cell under typical experimental conditions. It was noted that the fuel mass experienced a rapid decline after ignition, which corresponded to the rapid combustion from the crown fire. Subsequently, a slow decline in fuel mass was observed due to the transition from rapid combustion to the smouldering stage. Once the crown portion of the tree was fully burned, leaving only the trunk, the fuel mass stabilised and no longer changed. In addition, a more significant mass loss and longer duration of rapid combustion were found for conditions with greater crown heights owing to the availability of more fuel.

Fig. 8.

Evolution of fuel mass over time for typical conditions.

The mass loss rate was determined by curve fitting in the rapid decline stage of the fuel mass. Fig. 9 plots the mass loss rate as a function of crown height for different experimental conditions. It can be seen that the mass loss rate grows exponentially with crown height, which can be expressed as:

(15)

\dot{m} = 0.92 e^{0.45 H_{c}}

Comparing with previous work by Modarres et al. (2024b), the mass loss rate values were relatively larger than in the present work owing to the effects of crown size and tree species, and the mass loss rate was 39.9, 56.8 and 70.1 g/s for eucalyptus trees under different conditions.

Fig. 9.

Variation of mass loss rate with crown height for different experimental conditions.

Heat flux

Fig. 10 plots the evolution of three kinds of heat fluxes, namely total, radiation and convection heat fluxes over time at the middle and top of the crown under typical experimental conditions. As can be seen from Fig. 10, the three kinds of heat fluxes at different positions increased sharply at first up to a maximum value and then gradually decreased to similar values to the initial state. Compared with the radiation heat flux, the convection heat flux was much smaller and accounted for a lower percentage of heat transfer, which indicates that radiation is a dominant mode for fire spread of crown fire. Previous studies indicated that radiation heat transmission and spot fires were the primary modes of crown fire spread (Maranghides 1993; Cohen 2000b), which aligns with experimental observations. In the present work, radiation heat flux was considered the primary mechanism of heat transfer from the crown fires to the surroundings. Therefore, radiation heat transfer will be analysed and compared in detail in the following section. Moreover, greater values of total, radiation and convection heat fluxes were observed at the middle than that at the top of the crown, in agreement with our previous work (Guo et al. 2024). Further, larger crown heights and shorter distances between the crown and the heat flux sensors resulted in higher total and radiation heat flux values.

Fig. 10.

Evolution of different heat fluxes with time for typical experimental conditions: (a) Case 23; (b) Case 37; (c) Case 38.

Heat transfer model and control mechanism of crown fire

In previous work, models of heat transfer for crown fire have been proposed (Cohen 2004; Evans et al. 2004) in which the crown flame surface was simplified to a single structure and the flame surface temperature was assumed to be uniform. However, based on the experimental observations mentioned above, the temperature varied with height during crown fire propagation. In addition, it was noted that soot production increased as the heat release rate increased, resulting in an increase of emissivity (Markstein 1976; Wan et al. 2018). Therefore, the heat transfer model was explored based on the assumption that the flame surface temperature varied with crown height and the emissivity was related to the heat release rate.

Based on Eqn 10 and above assumption, the total energy that radiates to the surroundings for the entire flame height is given by:

(16)

{\dot{Q}}_{r} = \dot{Q} X_{r} = X_{r} φ Δ h \dot{m}

where Q̇_r represents the total energy radiated.

Fig. 11 shows a schematic diagram of the heat transfer mechanism from a crown fire to the surroundings, modelled as a vertical cylinder with a total height of H_f and a diameter of W_f. Based on this assumption, the energy radiated to the surroundings by an arbitrary circular cylinder element with the thickness of dz and height of z is given by:

(17)

d {\dot{Q}}_{r} = d A_{f} E = P d z ε σ T^{4}

where A_f is the area of the flame, P is the perimeter of the circular cylinder base; in this work, P = πW_f.

Fig. 11.

Schematic diagram of the crown flame radiation model.

The total energy radiated to the surroundings for the entire flame height can be expressed as:

(18)

{\dot{Q}}_{r} = \int_{0}^{H_{f}} P ε σ T^{4} d z = P ε σ \int_{0}^{2.27} T^{4} d (\frac{z}{H_{f}})

Based on analysis of the results above, the fire plume temperature T was a piecewise function of z/H_f where the temperatures were divided into upward stage, stable stage, rapid decline stage and slow decline stage. Eqn 13 can be rewritten by integrating z in the flame length zone from 0 to H_f, shown as:

(19)

{\dot{Q}}_{r} = \dot{Q} X_{r} = P ε σ {\int_{0}^{0.5} {[b {(\frac{z}{H_{f}})}^{a} Δ T_{\max}]}^{4} d (\frac{z}{H_{f}}) + \int_{0.5}^{1.0} Δ {T_{\max}}^{4} d (\frac{z}{H_{f}}) + \int_{1.0}^{2.27} {[d {(\frac{z}{H_{f}})}^{c} Δ T_{\max}]}^{4} d (\frac{z}{H_{f}})}

The model development in this work is based on indoor single-tree crown fire experiments under no-wind conditions, with some differences from the real situation. For example, it was pointed out that wind exerts an important influence on the spread and propagation of forest fires (Morandini et al. 2001; Guo et al. 2023). Other factors, such as bulk density and humidity, could affect crown fire combustion, but the effects of variations in these factors were not thoroughly investigated in the present work. Therefore, the impact of fuel density and atmospheric variations on heat transfer should be fully considered when the model is applied in real wildland–urban interface (WUI) fires.

According to the above analysis, flame emissivity can be calculated by expanding the quartic polynomial, and the calculated flame emissivity against heat release rate is plotted in Fig. 12. From Fig. 12, flame emissivity increased exponentially with heat release rate, which is similar to the results of the pool fire from Wan et al. (2018). It was noticed that the fitting parameters were different from previous studies owing to the difference between crown fires and pool fires.

(20)

ε = 1 - \exp (- 0.15 {\dot{Q}}^{0.37})

Based on the schematic relationship between the flame surface and the unburnt infinitesimal element shown in Fig. 11, the geometric view factor F in Eqn 4 can be expressed as (Weber 1989; Morandini et al. 2005; Rossi et al. 2010; Guo et al. 2023):

(21)

F = F_{d A_{1} - A_{2}} = \int_{A_{2}}^{} \frac{\cos θ_{1} \cos θ_{2}}{π S^{2}} d A_{2}

where S is the distance between the flame front and the cell target, dA₁ is an unburnt infinitesimal element, A₂ is the flame surface, θ₁ is the angle between the connecting line and the flame surface (A₂) normal orientation $(\vec{n_{1}})$ and θ₂ is the angle between the connecting line and the target surface (A₁) normal orientation $(\vec{n_{2}})$ .

Fig. 12.

Relationship between flame emissivity and heat release rate.

Based on the assumption of a cylindrical flame in Fig. 11, the total height and radius of the bottom surface of the crown fire cylinder are H and W_f/2, respectively. For a vertical target on the ground, the view factor F_dA1 − A2 between a cylinder and the target element determined according to Evans et al. (2004), Ji et al. (2021) can be expressed as:

(22)

F_{d A_{1} - A_{2}} = \frac{1}{π s} \tan^{- 1} (\frac{l}{\sqrt{s^{2} - 1}}) - \frac{1}{π S} \tan^{- 1} (\sqrt{\frac{s - 1}{s + 1}}) + \frac{A l}{π s \sqrt{A^{2} - 1}} \tan^{- 1} \sqrt{\frac{(A + 1) (s - 1)}{(A - 1) (s + 1)}}

s = \frac{l}{r}

h = \frac{H}{r}

A = \frac{s^{2} + h^{2} + 1}{2 s}

where s is the dimensionless horizontal distance of the target element from the centreline of the fire, l is the distance between tree and wooden panel (m), h is the dimensionless cylindrical flame height, r is the equivalent diameter of the fire source.

In previous work (Evans et al. 2004), the crown fire surface temperature was always assumed to be constant to evaluate the radiation heat flux. However, in the experimental observations of crown fire plume temperature in the present work, temperature varied with height during crown fire propagation. Consequently, different fire plume temperatures based on crown fire height should be taken into account when calculating radiation heat flux. In this work, the flame surface of crown fires was assumed to be a cylinder, divided into multiple cylinders along the vertical direction with different temperatures as shown in Fig. 11. This approach accounts for the non-uniform temperature distribution within the flame volume, representing the flame as multiple small cylindrical segments, each with distinct surface emissive powers. Consequently, the radiation heat flux received by a vertical target from a crown fire is the cumulative sum of the radiation from all these small cylindrical segments. The view factor of the jth cylinder to the vertical and targets can be expressed as:

(23)

F_{d A_{1} - A_{2}} = F (| z (j) - h | + Δ z, d (I), r) - F (| z (j) - h |, d (I), r)

The radiation heat flux received by the target from the crown fire surface can be represented as:

(24)

{\dot{Q}}_{t}^{″} = \sum_{j = 1}^{n} τ ε σ F_{d A_{1} - A_{2}} T_{j}^{4}

where n is the total layer number, T_j is the flame surface temperature of the jth layer.

Fig. 13 shows the comparison results of predicted radiation heat flux calculated by the new model for different layers (n = 10, 20, 30). It can be seen that different layers have a certain impact on the accuracy of prediction models. As the number of layers increases, the predicted results are more consistent with the experimental data. For more accurate results, 30 layers were chosen in this work to evaluate the radiation heat flux from crown fire.

Fig. 13.

Comparison of the radiation heat flux profile with the multi-layer cylindrical flame model with different layer numbers and experimental data for typical conditions: (a) the middle of the crown; (b) the top of the crown.

The comparisons of the predicted radiation heat flux calculated by the solid flame model and the modified model in this work are shown in Fig. 14. It can be seen that the solid flame model overestimates the radiation heat flux under most experimental conditions. For a detailed comparison of the model’s predictions with the solid model, the uncertainty parameters for radiation heat flux, standard error (σ_e) and standard deviation (σ_d), are shown in Table 3. It was noted that the values of σ_e and σ_d increased with crown height in both the middle and top of the crown. The values of σ_e and σ_d for the model proposed in this paper are smaller than that for the solid flame model, which indicates that the modified model proposed in this work better predicted effects. This is the result of taking the variations in emissivity and flame temperature into account.

Fig. 14.

Comparisons of predictive radiation heat flux using the modified model and the classical solid flame model for typical conditions: (a) the middle of the crown; (b) the top of the crown.

Table 3.The uncertainty parameters for different models.

Positions	Models	σ_e (kw/m²) [σ_d (kw/m²)]
Positions	Models	H_c = 0.4 m	H_c = 0.6 m	H_c = 0.8 m	H_c = 1.0 m	H_c = 1.2 m
Middle	Solid	2.58 [0.86]	3.11 [1.04]	3.79 [1.26]	4.26 [1.42]	5.19 [1.73]
Middle	Modified	0.31 [0.1]	0.43 [0.14]	0.52 [0.17]	1.67 [0.56]	2.71 [0.9]
Top	Solid	1.96 [0.65]	2.36 [0.79]	2.77 [0.93]	3.34 [1.11]	3.84 [1.28]
Top	Modified	0.28 [0.09]	0.42 [0.14]	0.63 [0.21]	1.42 [0.44]	1.96 [0.65]

In addition, Fig. 15 plots the comparison between experimental and predicted radiation heat flux using the solid flame model and the new modified model. It can be seen from Fig. 15a that the predicted values of heat flux obtained with the solid flame model are overall greater than the experimental values. However, the predicted values of radiation heat flux from the modified model were in fairly good agreement with the experimental data, which indicates that the model proposed above for radiation heat flux, considering the variations in emissivity and flame temperature at different positions, can effectively predict the radiation heat flux for a crown fire of a single tree.

Fig. 15.

Comparisons between predicted and experimental radiation heat flux values: (a) solid flame model; (b) modified model.

According to results of radiation heat flux in previous work (Barry 2002), the threshold values of 4.7, 7.0 and 12.5–15.0 kW/m² represent the maximum value for causing injuries, for firefighters with bunker gear and for wood starting to ignite, respectively. The modified model proposed in the present work can be used to assess the radiation heat flux from a single cypress crown fire to surrounding combustibles and firefighters at different distances. Likewise, a reference value for the safe distance between firefighters and combustibles during firefighting operations can be predicted. Meanwhile, it is suggested the results of the study be integrated into predictive models or fire management systems to further enhance the practical value of the findings.

The modified model involves many parameters, including view factor, emissivity and flame temperature. When applied at the field scale for wildland fires, the model can be further simplified. For instance, emissivity can be considered as a constant value based on tree height, fire plume temperature can be substituted by the characteristic temperature, flame geometry can be calculated by empirical formulas based on a tree’s dimensions. In addition, the modified model predicted radiation heat transfer using flame geometry and temperature data, making it applicable to other fire categories.

In summary, accurate prediction of a crown fire’s radiation heat transfer by the modified model can improve not only wildland fire warning capability and optimise firefighting strategies, but also promote the coordinated development of forest ecology and urban areas and enhance emergency response capability.

Conclusions

In this work, a set of crown fire burning experiments using real trees was conducted on a customised experimental set-up. Crown fire behaviours, including flame geometry, flame temperature, mass loss rate and heat flux under different crown heights, were compared and analysed. In addition, the control mechanisms of heat flux for crown fire were explored, and a modified model for radiation heat flux considering variations of emissivity and flame temperature is proposed. The major conclusions are summarised as follows:

The characteristic values of flame height, mass loss rate and heat flux increased with increasing crown height, and flame width increased with crown width. Empirical correlations of flame height, flame width and mass loss rate were established.
The flame temperature of crown fires varied with flame height, and relationships for flame temperature and flame height were developed for different zones. Moreover, the flame emissivity was related to crown height and increased exponentially with heat release rate.
A modified model is proposed, taking the variations of flame temperature and flame emissivity into account, to predict radiation heat flux from crown fires. The predictions of the new modified model of radiation heat flux agree well with experimental data.

The findings in this study can be used to predict forest flammability and assess the radiation heat flux from a single cypress crown fire to surrounding combustibles and to firefighters at different distances. Integration of the findings into predictive models or fire management systems provides valuable insights for further effective WUI fire risk assessment and mitigation strategy development. Further studies are in progress to investigate the effects of moisture content variation and different tree species on the behaviour and heat transfer of crown fires. It is also planned to study the spread of single crown fires to multiple crown fires and the combustion characteristics of multiple crown fires in order to validate the application of the model at a real scale.

Nomenclature

a	constant (–)
b	constant (–)
c	constant (–)
d	constant (–)
A_f	area of the flame (m²)
E	average emissive power at flame surface (kW/m²)
F	geometric view factor (–)
H	total height of the crown fire cylinder (m)
H_c	crown height (m)
H_f	flame height (m)
H_t	thermocouple height (m)
h	dimensionless cylindrical flame height (–)
k	dimensionless ratio of thermocouple height to tree height (–)
L	height and diameter of the cylinder (m)
l	distance between tree and wooden panel (m)
ṁ	mass loss rate (g/s)
n	total layer number (–)
P	perimeter of circular cylinder base (m)
${\dot{q}}_{c}^{″}$	convection heat flux (kW/m²)
${\dot{q}}_{r}^{″}$	radiation heat flux (kW/m²)
${\dot{q}}_{t}^{″}$	total heat flux (kW/m²)
Q̇	heat release rate (kW)
Q̇_r	energy radiated by total energy (kW)
r	equivalent diameter of the fire source (m)
R₀	length of the connecting line between the target and virtual origin (m)
S	distance between the flame front and the cell target (m)
s	dimensionless horizontal distance of the target element from the centreline of the fire (–)
T	flame temperature (K)
T_j	flame surface temperature of the jth layer (K)
t	dimensionless ratio of fire plume temperature to a maximum value of temperature (-)
W_c	crown width (m)
W_f	flame width (m)
X_r	radiative fraction (–)
z	constant (–)
Δh	heat value of the fuel (J/kg)
ΔT	fire plume temperature (K)
ΔT_max	maximum value of fire plume temperature (K)
ε	flame emissivity (–)
η	angle between the connecting line and the normal direction of the target (°)
θ₁	angle between the connecting line and flame surface normal orientation (°)
θ₂	angle between the connecting line and target surface normal orientation (°)
σ	Stephan–Boltzmann constant (W/(m² K⁴))
σ_e	standard error of heat flux (kW/m²)
σ_d	standard deviation of heat flux (kW/m²)
τ	atmospheric transmissivity (–)
φ	combustion efficiency factor (–)

Data availability

Data are available on request from the authors.

Conflicts of interest

The authors declare that they have no conflicts of interest.

Declaration of funding

This work was supported by National Key R&D Program of China (No. 2022YFC3005704), Sichuan Science and Technology Program (No. 2024YFFK0111), Sichuan Fire Research Institute of MEM Science and Technology Projects (No. 20248815Z).

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Appendix 1

Table A1.Experimental conditions.

Case no.	Crown height (m)	Crown width (m)	Distance between tree and wooden panel (m)	Mass of tree (g)
1	0.36	0.22	0.20	85
2	0.41	0.19	0.30	79
3	0.37	0.18	0.40	83
4	0.34	0.11	0.50	74
5	0.37	0.21	0.60	69
6	0.40	0.30	0.70	76
7	0.46	0.16	0.80	88
8	0.45	0.15	0.90	84
9	0.39	0.22	1.00	68
10	0.59	0.20	0.30	116
11	0.57	0.14	0.40	126
12	0.60	0.19	0.50	150
13	0.59	0.19	0.60	129
14	0.56	0.30	0.70	124
15	0.56	0.13	0.80	138
16	0.57	0.22	0.90	129
17	0.64	0.25	1.00	121
18	0.60	0.26	1.20	146
19	0.72	0.17	0.20	287
20	0.68	0.20	0.30	278
21	0.73	0.18	0.40	249
22	0.73	0.23	0.50	246
23	0.69	0.24	0.60	239
24	0.62	0.22	0.70	248
25	0.70	0.19	0.80	253
26	0.70	0.18	0.90	247
27	0.72	0.22	1.00	265
28	0.92	0.18	0.20	522
29	0.82	0.19	0.30	496
30	0.85	0.22	0.40	488
31	0.84	0.21	0.50	546
32	0.82	0.17	0.60	478
33	0.89	0.19	0.80	515
34	0.96	0.29	1.00	526
35	0.97	0.25	1.20	474
36	1.22	0.38	0.50	728
37	1.16	0.45	0.60	714
38	1.18	0.38	0.80	726
39	1.28	0.39	1.00	775
40	1.27	0.38	1.10	812
41	1.12	0.42	1.30	789
42	1.25	0.26	1.50	775
43	1.27	0.36	1.70	796
44	1.27	0.42	1.80	803