Firebrand burning under wind: an experimental study

Weidong Yan; Naian Liu; Hong Zhu; Haixiang Chen; Xiaodong Xie; Wei Gao; Zhihao Du

doi:10.1071/WF23151

RESEARCH ARTICLE (Open Access)

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Firebrand burning under wind: an experimental study

Weidong Yan ^A ^B , Naian Liu ^A ^B ^* , Hong Zhu

^A ^B ^* , Haixiang Chen

^A ^B , Xiaodong Xie ^A ^B , Wei Gao

^A ^B and Zhihao Du ^A ^B

+ Author Affiliations

- Author Affiliations

^A State Key Laboratory of Fire Science, University of Science and Technology of China, Hefei, Anhui 230026, China.

^B Ministry of Emergency Management Key Laboratory of Forest Fire Monitoring and Warning, University of Science and Technology of China, Hefei 230026, China.

^* Correspondence to: liunai@ustc.edu.cn, zhuh15@ustc.edu.cn

International Journal of Wildland Fire 33, WF23151 https://doi.org/10.1071/WF23151

Submitted: 19 September 2023 Accepted: 15 March 2024 Published: 12 April 2024

© 2024 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

Spot fires play a significant role in the rapid spread of wildland and wildland–urban interface fires.

Aims

This paper presents an experimental and modelling study on the flaming and smouldering burning of wood firebrands under forced convection.

Methods

The firebrand burning experiments were conducted with different wind speeds and firebrand sizes.

Key results

The burning rate of firebrands under forced convection is quantified by wood pyrolysis rate, char oxidation rate and a convective term. The firebrand projected area is correlated with firebrand diameter, char density, wind speed, and flaming or smouldering burning. A surface temperature model is derived in terms of condensed-phase energy conservation. We finally establish a simplified firebrand transport model based on the burning rate, projected area and surface temperature of firebrands.

Conclusion

The mass loss due to wood pyrolysis is much greater than that due to char oxidation in self-sustaining burning. The burning rate is proportional to U^1/2, where U is wind speed. The projected area for flaming firebrands decreases more rapidly than that for smouldering ones. The firebrand surface temperature is mainly determined by radiation.

Implications

Knowledge about firebrand burning characteristics is essential for predicting the flight distance and trajectory in firebrand transport.

Keywords: burning rate, firebrand density, firebrands, flaming and smouldering, forced convection, projected area, surface temperature, transport trajectory.

Introduction

Spot fire denotes the phenomenon that firebrands (e.g., burning barks and twigs) are lofted by the buoyancy of the flame plume, and may be carried by the wind. Firebrands can be transported for hundreds or even tens of thousands of metres by wind (Hall et al. 2015). Spot fire may form continuous showers (Fernandez-Pello 2017), and normally cause numerous new ignitions beyond the primary burning zone (Tarifa et al. 1965; Manzello et al. 2020). Spot fire plays a significant role in the rapid spread of wildland and wildland–urban interface (WUI) fires (Suzuki et al. 2015; Manzello 2020) and poses a severe risk of igniting large areas of wildland fuels and even WUI buildings (Sardoy et al. 2007; Song et al. 2017; Santoso et al. 2019). At least 50% of ignitions are attributable to firebrands in WUI areas (Mell et al. 2010; Fang et al. 2021).

During the past several decades, there has been much attention on firebrand generation, transport and ignition (for more details, refer to the comprehensive reviews by Manzello et al. (2020) and Wadhwani et al. (2022)). Prediction of the flight distance and trajectory of self-sustained burning firebrands highly depends on understanding firebrand burning. However, despite considerable research on firebrand transport, the firebrand burning mechanism under wind (forced convection) is still poorly understood. Most relevant studies fall into empirical modelling or experimental analysis. Tarifa et al. (1965) pioneered wind tunnel experiments on spherical and cylindrical firebrand transport and developed empirical models to discuss the influence of firebrand size, density, type and moisture on transport trajectories. Some studies investigated the trajectories of firebrands initially elevated from fire plumes by theoretical calculations (Lee and Hellman 1969; Woycheese et al. 1999) or numerical simulations (Anthenien et al. 2006; Oliveira et al. 2014). Albini 1983 developed an empirical burning rate model for transporting firebrands in wind-driven fires by line thermals. Himoto and Tanaka (2005) proposed a model for wind-driven firebrands without considering firebrand burning. Ellis (2013) measured the duration of flaming burning (flameout time), total burning time during flight (burnout time) and mass at burnout of 192 samples in a CSIRO blower-type vertical wind tunnel (Knight 2001). However, the burning mechanism of firebrands was not discussed. Almeida et al. (2011) also established an empirical mass model for flaming and smouldering firebrands with a vertical wind tunnel at different speeds.

Very limited research has touched on the firebrand burning mechanism under forced convection through experimental and theoretical analyses. Mukunda et al. (1985) conducted burning experiments on wooden spheres under natural convection and developed a wood burning model, showing mass loss and diameter reduction with increasing initial diameter of firebrand samples. They predicted the burning time, mass, diameter, surface temperature and core temperature of firebrands. Song et al. (2017) analysed the flight distance and mass change of firebrands experimentally. However, firebrand burning was not fully self-sustaining, and the experimental transport range was limited. Urban et al. (2019a) provided a method to measure the surface temperature of firebrands, but did not correlate the surface temperature with the firebrand burning rate.

Moreover, firebrands under forced convection may be in a mode of flaming or smouldering burning. However, burning modes have rarely been distinguished in evaluating firebrand burning rate and size. During firebrand burning, the pyrolysis front propagates inward and forms char gradually. When wood pyrolysis is complete, the inhibitory effect of pyrolysis gas on oxygen diffusion is weakened, allowing more oxygen to diffuse to the firebrand surface. As firebrand temperature increases, char oxidation occurs (Sardoy et al. 2006) on the surface rather than through the whole sample (Sardoy et al. 2007). Tse and Fernandez-Pello (1998) suggested that volume reduction may be attributed to char oxidation on the particle surface. Based on liquid fuel burning theory (Spalding 1953; Turns 2000), the mass transfer number B is an important parameter affecting the burning rate of firebrands (Pagni 1981; Turns 2000; Lattimer et al. 2022). Woycheese and Pagni (1999) determined the maximum propagation distance of combusting firebrands exposed to the fire plume and constant horizontal wind, without considering the mode of flaming or smouldering burning. Lattimer et al. (2022) used analytical equations to evaluate firebrand temperature, mass loss rate, char diameter and burning duration based on char oxidation. However, wood pyrolysis was neglected.

Considering the above problems, this work studies the flaming and smouldering burning of wood firebrands under forced convection by experimental and modelling approaches. Firebrand density, burning rate, projected area and surface temperature are analysed. Correlations for firebrand burning characteristics are developed and used in a firebrand transport model.

Experimental methods

Sample

Reflective of the firebrand shape and size observed in WUI fires and typically used in experimental studies (Xiong et al. 2020; Manzello et al. 2020), spherical wooden firebrands (Schima Superba) with diameters of 10, 15, 20 and 25 mm were used as the samples in the present work. The sphere has been regarded as a typical shape for firebrand research (Wadhwani et al. 2022; Dal-Ri dos Santos and Yaghoobian 2023) because natural firebrands sometimes undergo a gradual transition into near-spheroids through burning (Urban et al. 2019b). Moreover, the burning model and transport model for spherical wooden firebrands can be extended for cylinder firebrands by simple geometric transformation (Lattimer et al. 2022). Before the experiments, the samples were dried in a drying oven (Jinghong electric heating blast drying oven, XMTD-8222) at 100–120°C for 24 h (Mukunda et al. 1985) and were then stored in a sealed bag. After drying, the moisture contents of the samples were 0.73–1.56%. The densities of the oven-dry samples were 0.520–0.599 g cm⁻³. The sample masses were 0.267–4.705 g, with a relative standard deviation <5%.

Experiment

The experimental set-up is illustrated in Fig. 1. A jet fan (Suitai, China, model SDS4.0-2.2kw-2p), equipped with two layers of damping nets inside the outlet end, provided a stable airflow. The wind speed was adjusted through a frequency converter. A hot-wire anemometer (Kanomax, CTA/HWA7000) was placed 5 cm above the centreline of the firebrand to measure the wind speed. The wind speeds varied between 0 and 8 m s⁻¹ with a fluctuation of ±5%. A digital video camera (Sony FDR-AX60, 50 frames s⁻¹, 1920 × 1080 pixels) and an infrared thermal imaging camera (American FLIR T650sc, 30 frames s⁻¹, 640 × 480 pixels) recorded firebrand burning and surface temperature from the front view. Firebrands were ignited by a propane igniter, which was turned off immediately when the firebrands reached self-sustaining burning. Self-sustaining burning means that after the igniter is removed, the firebrand continues to burn until firebrand breakage occurs. Under the same experimental conditions, the ignition times are identical. Burned mass was measured with an electronic balance (Jingfei Instrument Technology, DHS-10A, resolution 0.005 g) after the sample was quenched with water and then dried. As a stationary firebrand burns fast on the windward side and slowly on the leeward side, the firebrand in each test of this work was rotated on a fixing nail at a speed of 5 s per circle for uniform burning. The rotating samples under forced convection are close to the conditions of natural firebrands under wind (Manzello et al. 2020). Burning tests of rotational and non-rotational firebrands under natural convection showed comparable temporal variations of firebrand masses. Therefore, the speed relative to air caused by rotating (calculated to be lower than 0.0126 m s⁻¹) is ignored in this work.

Fig. 1.

Experimental setup for firebrand burning under forced convection.

We quenched the firebrand for each set of experimental conditions at approximately 20 different moments and measured the instantaneous features such as sample mass and projected area. The quenched firebrands were dried to a moisture content of 2.21–3.76%. For each set of experimental conditions, a large number of repeated tests were performed before the experiment to obtain the burning duration of firebrands. Then, the entire burning duration was divided into ~20 instants for firebrand quenching. This step is consistent with Mukunda et al. (1985). The electronic balance was utilised for larger firebrand masses (>12.95 g (Almeida et al. 2011)) or lower wind speeds (<2 m s⁻¹ (Lattimer et al. 2022)) without quenching the firebrand. However, the sample masses (<4.70 g) and wind speeds (>2 m s⁻¹) in the present study, in accord with those of firebrands in natural fires (Hall et al. 2015; Storey et al. 2020b), caused unacceptable fluctuation in pre-experiments without quenching. The firebrand surface temperature was measured by an infrared thermal imaging camera, calibrated using a thermocouple contacting the surface of non-rotating firebrands. An emissivity of 0.9 (Sardoy et al. 2007; Lattimer et al. 2022) was adopted for the same temperature as the thermocouple. The firebrand image recorded by a digital video camera was converted to a greyscale image and then into a binary image by setting a cut-off threshold. The projected area of the firebrand was calculated with pixels and a reference scale, and then the firebrand diameter was obtained. Each experiment was repeated at least four times.

Model description

At low wind speeds, the firebrand is first flaming and then smouldering. At high wind speeds, the firebrand is always smouldering. The mass and volume of a firebrand during burning decrease with burning time. At low wind speeds, the firebrand density decreases monotonically with burning time during flaming, whereas it does not change significantly during smouldering.

Flaming and smouldering firebrands involve different char oxidation mechanisms, with the main products being CO and CO₂. The firebrands in this work are assumed to be homogeneous wood. A mechanism of firebrand burning (consisting of wood pyrolysis and char oxidation) is proposed, and the ratio of wood pyrolysis rate to char oxidation rate is determined. A burning rate model incorporating the wood pyrolysis rate, char oxidation rate and a convective term is developed. Based on mass conservation, the reduction rate of the firebrand projected area is evaluated from the char oxidation rate under forced convection. The firebrand surface temperature is derived in terms of condensed-phase energy conservation. The surface temperature of the firebrand is mainly determined by radiation, and the relationship between surface temperature and wind speed is then obtained. Finally, based on the above, and in combination with the principles of mechanics, a firebrand transport model is developed to calculate the transport trajectory and essential parameters affecting the burning of firebrands.

Results and discussion

Experimental observations

Fig. 2 shows the burning phenomena of a firebrand with a diameter of 20 mm under different wind speeds. The dotted white lines denote the specific position of the fixing nail. For each wind speed, the first image corresponds to the moment when the propane igniter was just turned off, and the last image of the smouldering stage corresponds to firebrand breakage. As observed, the firebrands first burned with a flame, and after the flame was extinguished, residue smouldering continued at wind speeds of 2 and 4 m s⁻¹. The flame was extinguished quickly and then residue smouldering continued at wind speeds of 6 and 8 m s⁻¹.

Fig. 2.

Firebrand burning under different wind speeds.

The firebrand burning results in a reduction in mass and size, as well as eventual breakage along the wood grain. The firebrand breaks off and forms multiple smaller firebrands that are blown away by the wind. The higher the wind speed, the shorter the firebrand breakage time. The breakage of the firebrand significantly promotes mass and heat transfer and reduces the firebrand mass. Although firebrand burning was suggested to be affected by firebrand breakage (Thunman et al. 2002; Kuo and Hwang 2003; Almeida et al. 2011) and ash formation (Hall et al. 2015), we do not consider these effects in this work because firebrand breakage occurred only in the last few seconds and the ash accumulated only at the wind speed of 2 m s⁻¹ during char oxidation. Additionally, the thin ash layer (Fig. 2) is considered to have a minor effect on surface energy transfer and gas diffusion (Wong et al. 2022).

Firebrand density and burning mechanism

The temporal variation of firebrand density under different wind speeds and particle diameters is shown in Fig. 3. In Fig. 3a, under wind speeds of 2 and 4 m s⁻¹, Regimes I–III correspond to the flaming stage, flame extinction and the smouldering stage, respectively. The dotted lines denote the time of flame extinction.

Fig. 3.

Firebrand density under (a) 2 and 4 m s⁻¹, (b) 6 and 8 m s⁻¹ wind speeds.

Mukunda et al. (1985) carried out a combustion experiment on spheroid wood particles with different diameters under natural convection. Experimental results showed that the diameter squared decreases by 20–25%, and the particle mass decreases by 75–80% during flaming. The mass loss of a firebrand is caused by pyrolysis and char oxidation, while the volume loss is mainly attributable to char oxidation (Tse and Fernandez-Pello 1998; Porteiro et al. 2007). The firebrand density decreases in flaming (Regime I) owing to wood pyrolysis and char oxidation. The firebrand density almost remains constant in smouldering (Regime III), in which char oxidation is the primary reaction. Char oxidation can exist with flaming here because the flame cannot fully cover the firebrand surface. Thus, the reduction in firebrand density is considered mainly attributable to wood pyrolysis. Pyrolysis and char oxidation occur simultaneously during flaming. The squared diameter loss is ~25%, density loss is ~70%, and mass loss is ~80%, of which pyrolysis accounts for ~69%. The experimental results are consistent with those of Mukunda et al. (1985). In Fig. 3b, smouldering firebrand density decreases under wind speeds of 6 and 8 m s⁻¹, which differs from the almost constant firebrand density in the smouldering stage under 2 and 4 m s⁻¹ (Regime III in Fig. 3a). This difference may be attributable to the fact that the smouldering firebrand under lower wind speeds is dominated by char oxidation, and under higher wind speeds, undergoes pyrolysis and char oxidation simultaneously.

The volatiles released from wood pyrolysis support flaming combustion. Wood char normally contains small amounts of oxygen, hydrogen and nitrogen, which can be neglected. Thus, char is assumed to be made of pure carbon (Porteiro et al. 2007). The flame then affects the char oxidation reaction. Porteiro et al. (2007) assumed that char oxidation occurring in flaming and smouldering firebrands results in the simultaneous generation of CO and CO₂. However, the oxygen diffuses to the firebrand surface with difficulty in flaming burning, whereas it directly contacts the char surface in smouldering burning. Thus, for a flaming firebrand, CO₂ and CO are generated on the flame and firebrand surfaces, respectively (Caram and Amundson 1977). For a smouldering firebrand, CO and CO₂ are produced on the firebrand surface, and the overall rate of char consumption is calculated using Eqn 4 as the ratio between CO and CO₂ Porteiro et al. (2007). The above considerations lead to the following firebrand burning mechanism:

(1)

Wood pyrolysis : Wood \to char + pyrolysate (g)

(2)

Flame surface : Pyrolysate + O_{2} \to {CO}_{2}

(3)

Firebrand surface : 1 kg C + ζ_{fla} kg O_{2} \to CO

(4)

Firebrand surface : \begin{array}{l} 1 kg C + ζ_{smo} kg O_{2} \to 2 (η - 1) CO + (2 - η) {CO}_{2} \\ η = 2 [1 + 4.3 \exp (- 3390 / T)] / [2 + 4.3 \exp (- 3390 / T)] \end{array}

where ζ is the stoichiometric coefficient, and η is the CO-to-CO₂ ratio. The subscripts fla and smo denote flaming and smouldering stages, respectively. The temperature T (K) is 950–1150 K, and the corresponding η is 1.057–1.101. The η = 1.078 for a temperature of 1050 K is chosen, and the resulting error is negligible. ζ_fla = 1.333, and ζ_smo = 2.474. For wind speeds of 2 and 4 m s⁻¹, the flaming stages follow Eqns 1–3, and the smouldering stages follow Eqn 4. For wind speeds of 6 and 8 m s⁻¹, the flaming stages are ignored, and the smouldering stages follow Eqns 1 and 4.

Burning rate

The burning rate (mass loss rate) of a firebrand is

(5)

\frac{d m_{fb}}{d t} = - (1 + φ_{pyr}) m_{char}^{'}

where m_fb is the firebrand mass (g), t the burning time (s), $m_{char}^{'}$ the char oxidation rate (g s⁻¹), φ_pyr the ratio of wood pyrolysis rate to char oxidation rate, and the subscript pyr denotes wood pyrolysis.

For the flaming stages at 2 and 4 m s⁻¹:

(6)

m_{pyr} = ρ_{fla, 0} Ω_{fla, 0} - ρ_{fla, fe} Ω_{fla, 0}

(7)

m_{oxi} = (ρ_{fla, 0} Ω_{fla, 0} - ρ_{fla, fe} Ω_{fla, fe}) - m_{pyr} = ρ_{fla, fe} Ω_{fla, 0} - ρ_{fla, fe} Ω_{fla, fe}

(8)

\bar{φ_{pyr}} \equiv \frac{m_{pry}}{m_{oxi}} = \frac{ρ_{fla, 0} Ω_{fla, 0} - ρ_{fla, fe} Ω_{fla, 0}}{ρ_{fla, fe} Ω_{fla, 0} - ρ_{fla, fe} Ω_{fla, fe}}

where $\bar{φ_{pyr}}$ is the mean ratio of wood pyrolysis rate to char oxidation rate in the burning stage, Ω the firebrand volume (m³), ρ the firebrand density (kg m⁻³). The subscript oxi denotes the char oxidation, 0 the initial value, and fe the flame extinction.

For the smouldering stages at 2 and 4 m s⁻¹:

(9)

\bar{φ_{pyr}} = 0

For the smouldering stages at 6 and 8 m s⁻¹:

(10)

m_{pyr} = ρ_{smo, 0} Ω_{smo, 0} - ρ_{bre} Ω_{smo, 0}

(11)

m_{oxi} = ρ_{smo, 0} Ω_{smo, 0} - ρ_{bre} Ω_{bre} - (ρ_{smo, 0} Ω_{smo, 0} - ρ_{bre} Ω_{smo, 0}) = ρ_{bre} Ω_{smo, 0} - ρ_{bre} Ω_{bre}

(12)

\bar{φ_{pyr}} = \frac{ρ_{smo, 0} Ω_{smo, 0} - ρ_{bre} Ω_{smo, 0}}{ρ_{bre} Ω_{smo, 0} - ρ_{bre} Ω_{bre}}

where the subscript bre denotes firebrand breakage. Table 1 presents firebrand density (ρ) and volume (Ω), where D is firebrand diameter (mm), and U is the wind speed (m s^-1). The $\bar{φ_{pyr}}$ value is 4.51 for the flaming stages at 2 and 4 m s⁻¹ and the smouldering stages at 6 and 8 m s⁻¹, indicating mass loss due to wood pyrolysis is much greater than that caused by char oxidation, consistent with Zhu and Liu (2020).

Table 1.Density (ρ) and volume (Ω) for firebrand burning.

D (mm)	U (m s⁻¹)	Ω × 10⁻⁷ (m³)				ρ ^A (kg m⁻³)
D (mm)	U (m s⁻¹)	Ω_fla,0	Ω_fla,fe	Ω_smo,0	Ω_bre	ρ_fla,0	ρ_fla,fe	ρ_smo,0	ρ_bre
15	2	14.3	7.0	8.7	7.0	457.4	142.0	142.0	142.0
15	4	13.6	7.1	8.8	5.8	457.4	142.0	142.0	142.0
20	2	34.0	19.8	22.8	6.0	457.4	142.0	142.0	142.0
20	4	37.2	17.3	20.1	7.2	457.4	142.0	142.0	142.0
20	6	–	–	36.9	17.7	–	–	457.4	142.0
20	8	–	–	38.5	17.1	–	–	457.4	142.0
25	6	–	–	73.3	37.5	–	–	457.4	142.0
25	8	–	–	73.1	38.9	–	–	457.4	142.0

^A ρ is calculated by dividing m_fb by Ω in each single test.

The sample in this work is assumed to be homogeneous wood with uniform mass loss. The gas diffusivity D_m under air is 1.6 × 10⁻⁵ m² s⁻¹ for CO₂ at 298 K and 2.08 × 10⁻⁵ m² s⁻¹ for CO at 293 K. Thermal diffusion α is 2.058 × 10⁻⁵ m² s⁻¹ for air at 300 K. Thus, the Lewis number Le = α/D_m ≈ 1 (Turns 2000; Lattimer et al. 2022). The power law relationship between the Nusselt (Nu) and Reynolds (Re) numbers has been fully studied for external flow (Incropera et al. 2007). They are correlated by a multiplicative correction of an additional convective term under forced convection for droplet combustion (Faeth 1977; Incropera et al. 2007):

(13)

Nu = 2 + 0.6 {Re}^{1 / 2} \Pr^{1 / 3}

where the Prandtl number Pr = 0.707 (Incropera et al. 2007; Lattimer et al. 2022). As the Pr value varies from 0.707 to 0.728 within 300–1200 K (Incropera et al. 2007; Lattimer et al. 2022), the temperature-induced error is within 3%. Re = D₀ρ_gU/μ (Lattimer et al. 2022), where D₀ is the initial firebrand diameter (mm), and the dynamic viscosity μ = 1.846 × 10⁻⁵ (N s m⁻²) (Incropera et al. 2007). The diameter-induced mean error in Re^1/2 is ~5.5%.

Based on the char oxidation rate ( $m_{char}^{'} = 2 π r ρ_{g} D_{m} Nu \ln (1 + B) L / D$ , L is firebrand length (mm)) (Lattimer et al. 2022), the burning rate of spherical wooden firebrands under forced convection is formulated as

(14)

m_{fb}^{'} = 2 π D ρ_{g} D_{m} \ln (1 + B) (1 + \bar{φ_{pyr}}) \times [1 + fb ({Re}^{1 / 2} \Pr^{1 / 3}) / 2]

where the gas diffusion constant D_m = 1.84 × 10⁻⁵ m² s⁻¹ (Lattimer et al. 2022), and the air density ρ_g = 1.1614 kg m⁻³ at 300 K (Incropera et al. 2007). The mass transfer number $B_{smo} = Y_{O_{2}, \infty} / ζ_{smo}$ and $B_{fla} = Y_{O_{2}, \infty} / ζ_{fla}$ , where Y is the mass fraction $(Y_{O_{2, \infty}} = 0.233 and Y_{O_{2}, s} \approx 0)$ . The firebrand diameter D (see Fig. 4) is obtained by linear fitting (Tarifa et al. 1965).

(15)

D^{2} = D_{0}^{2} - (β_{D} + δ_{D} U) t

Here, the parameters β_D and δ_D depend on wood type and moisture content.

(16)

\frac{d S}{d t} = - (β + δ U)

where S is the projected area of the sample (mm²), and the parameters β = β_Dπ/4 and δ = δ_Dπ/4. Thus, the burning rate depends on firebrand diameter, flaming or smouldering burning, pyrolysis rate and relative wind speed.

Fig. 4.

Firebrand diameter change under (a) 2 m s⁻¹, (b) 4 m s⁻¹, (c) 6 m s⁻¹, and (d) 8 m s⁻¹ wind speeds.

Then, only parameter fb is unknown in Eqn 14. The firebrand mass (Fig. 5) is introduced to solve for fb, and expressed as:

(17)

m_{fb} = m_{0} - \int_{0}^{t} m_{fb}^{'} d t

where m₀ = Ω_fla,0ρ_fla,0 or m₀ = Ω_smo,0ρ_smo,0 (Table 1). We substitute Eqn 14 (combined with Eqn 15) into Eqn 17, and integrate the expression. Then, the correction factor fb value (Fig. 6) is determined by non-linear fitting Eqn 17 in integral form with $\bar{φ_{pyr}} = 4.51$ and different experimental m_fb, m₀ and D₀ values. The trends of burned mass with quenching over burning time agree well with previous reports (Almeida et al. 2011; Lattimer et al. 2022) at wind speeds of 2 and 4 m s⁻¹. Spherical wooden firebrand burning, involving wood pyrolysis and char oxidation, significantly differs from droplet combustion in heat and mass transfer processes. Thus, the fb values for flaming and smouldering burning of wood firebrands obtained by the above regression analysis, rather than the empirical value of 0.6 concerning droplet combustion, are used for the following calculations.

Fig. 5.

Firebrand mass change under (a) 2 m s⁻¹, (b) 4 m s⁻¹, (c) 6 m s⁻¹, and (d) 8 m s⁻¹.

Fig. 6.

Ratio of wood pyrolysis rate to char oxidation rate $(\bar{φ_{pyr}})$ and flow parameters (fb).

Projected area

The variation of firebrand projected area is mainly attributable to char oxidation (Tse and Fernandez-Pello 1998; Porteiro et al. 2007). The char oxidation rate is:

(18)

\frac{d m_{char}}{d t} = - m_{char}^{'}

The char mass is formulated as:

(19)

m_{char} = ρ_{char} Ω = ρ_{char} π D^{3} / 6

Based on the char oxidation rate $(m_{char}^{'} = 2 π D ρ_{g} D_{m} \ln (1 + B) [1 + fb ({Re}^{1 / 2} \Pr^{1 / 3}) / 2])$ from Eqns 14, 18, and 19, the reduction rate of the projected firebrand is:

(20)

\frac{d S}{d t} = - \frac{2 π ρ_{g} D_{m}}{ρ_{char}} \ln (1 + B) \times [1 + \frac{fb ({Re}^{1 / 2} \Pr^{1 / 3})}{2}]

The theoretical dS/dt values for flaming and smouldering firebrands are indicated in Fig. 7, and agree with the experimental data. Based on Eqn 20,

(21)

\frac{d S}{d t} = - f ({ρ_{char}}^{- 1}, \ln (1 + B), [1 + \frac{fb (D_{0}^{1 / 2} {ρ_{g}}^{1 / 2} \Pr^{1 / 3})}{2 μ^{1 / 2}} U^{1 / 2}])

A higher dS/dt value corresponds to a higher char density, a lower ln(1 + B), a lower firebrand diameter, or a lower wind speed. The ln(1 + B) value in flaming is much higher than that in smouldering. Thus, dS/dt in flaming is slightly lower than in smouldering under 2 and 4 m s⁻¹ wind speeds in Fig. 7.

Fig. 7.

Theoretical and experimental dS/dt values at wind speeds of (a) 2 m s⁻¹, (b) 4 m s⁻¹, (c) 6 m s⁻¹, and (d) 8 m s⁻¹.

Surface temperature

Fig. 8 shows the experimental firebrand surface temperature (extracted as the average value in the first 10 s) under forced convection. The higher the wind speed, the higher the surface temperature, indicating that the heat generated by firebrand burning exceeds the heat loss when the wind speed is increased. As presented in Fig. 9, the condensed-phase energy conservation at the firebrand surface is:

(22)

α_{f}^{char} m_{fb}^{'} Δ h_{char} = - k_{sol} {4 π r}_{s}^{2} {\frac{d T}{d r} |}_{r_{s}} + h_{conv} {4 π r}_{s}^{2} (T_{s} - T_{film}) + ε_{s} {4 π r}_{s}^{2} σ (T_{s}^{4} - T_{\infty}^{4})

where T_s denotes the firebrand surface temperature, the mass fraction of char in firebrand $α_{fb}^{char} = 0.182$ $(\bar{φ_{pyr}} \approx 4.51)$ , the heat of char oxidation Δh_char is 21.0 MJ kg⁻¹ (Lattimer et al. 2022), and the thermal conductivity of the solid phase k_sol is 0.1 W m⁻¹ K⁻¹ (Sardoy et al. 2006). The firebrand radius r = D/2 and the subscript s denotes the surface. The temperature of the gas phase film T_film = (T_s + T_∞)/2 ≈ 700 K. The convective heat transfer coefficient h_conv = Nuk_film/D, where the thermal conductivity of air k_film is 5.24 × 10⁻² W m⁻¹ K⁻¹ at 700 K. The Stefan–Boltzmann constant is σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴. The temperature T_∞ = 293.15 K. The emissivity ε_s = 0.9 (Sardoy et al. 2007; Lattimer et al. 2022). It is reasonable to assume that the emissivity remains constant because the temperature values are measured in the first 10 s, during which the firebrand surface is always covered with char (Fig. 2). The temperature gradient at the firebrand surface (Turns 2000) is

(23)

{\frac{d T}{d r} |}_{r_{s}} = \frac{c_{p} m_{char, nc}^{'} (T_{\infty} - T_{s}) \exp (- c_{p} m_{char, nc}^{'} / 4 π k_{sol} r_{s})}{4 π k_{sol} r_{s}^{2} [1 - \exp (- c_{p} m_{char, nc}^{'} / 4 π k_{sol} r_{s})]}

where the subscript nc denotes natural convection, and the specific heat capacity at constant pressure c_p is 1075 J kg⁻¹ K⁻¹. The particle diameter varies insignificantly in 10 s and is assumed as a constant D₀. Then, based on Eqn 14, $m_{fb, nc}^{'}$ satisfies

(24)

\begin{array}{l} m_{fb}^{'} = f (D, In (1 + B), (1 + \bar{φ_{pyr}}), [1 + \frac{fb ({D_{0}}^{1 / 2} {ρ_{g}}^{1 / 2} \Pr^{1 / 3})}{2 μ^{1 / 2}} U^{1 / 2}]) \\ = m_{fb, nc}^{'} (1 + K \times U^{1 / 2}) \end{array}

where K is the function coefficient. The larger the firebrand diameter and wind speed, the greater the firebrand burning rate. As illustrated in Fig. 10, the firebrand burning rate significantly depends on the wind speed. The K and $m_{fb, nc}^{'}$ values are obtained by fitting the experimental firebrand burning rate and Eqn 24. The reliability of Eqn 24 is verified by good agreement with experimental data under natural convection. Then, the firebrand surface temperature is calculated with Eqn 22. It is found that the firebrand surface temperature is mainly determined by radiation, with a contribution of more than 90%. $T_{s}^{4} - T_{\infty}^{4} \approx T_{s}^{4}$ because $T_{s}^{4} ≫ T_{\infty}^{4}$ . Based on Eqns 22 and 24, the relation between firebrand surface temperature and wind speed is:

(25)

T_{s} = {(α_{f}^{char} m_{fb, nc}^{'} Δ h_{char} / ε_{s} {4 π r}_{s}^{2} σ)}^{1 / 4} {(1 + K \times U^{1 / 2})}^{1 / 4}

The experimental and calculated surface temperatures are in good agreement, as shown in Fig. 11. The surface temperature increases with wind speed, as the heat from the firebrand burning overcomes the heat loss.

Fig. 8.

Relationship between surface temperature and wind speed.

Fig. 9.

Energy fluxes on the firebrand surface.

Fig. 10.

Burning rate of firebrands under (a) forced convection and (b) natural convection (D = 20 mm).

Fig. 11.

Relationship between surface temperature and wind speed (D = 20 mm).

Transport trajectory

The firebrand trajectory is analysed with some assumptions: (1) firebrand uplift, rotation and vibration are neglected. (2) Firebrands do not affect each other. (3) Only the relative velocity of a firebrand and the surrounding air is considered during the firebrand transport. The firebrand transport model is presented in Fig. 12, where H is the height of the firebrand (m), L_x the horizontal transport distance (m), U_h the horizontal wind speed (m s⁻¹), V_fb the firebrand velocity (m s⁻¹), F the force (N), F_y,G = mg the force exerted by gravity on the firebrand, F_y,res the air resistance, V_x the horizontal speed of the firebrand, and V_y the vertical speed of the firebrand. The drag force of wind on the firebrands is equal to the air resistance in the horizontal direction (F_w = F_x,res). Based on Newton’s second law and discretisation technique,

(26)

\frac{d (m_{i} V_{fb i})}{d t} = ρ_{fb i} Ω_{fb i} g - \frac{1}{2} ρ_{0} C_{D} S_{i} ‖ V_{fb i} - U ‖ (V_{fb i} - U)

where C_D is the drag coefficient, subscript i denotes iteration and the relative velocity U_r,i≈||V_fbi − U||. Based on Eqns 14, 16, 20 and 25, ${m'}_{fb} = ξ (S_{i - 1}^{1 / 2} + K_{con} \times {U_{r, i - 1}}^{1 / 2} S_{i - 1}^{3 / 4})$ , where $ξ = {4 π}^{1 / 2} ρ_{g} D_{m} \ln (1 + B) (1 + \bar{φ_{pyr}})$ , $K_{con} = fb {(4 / π)}^{1 / 4} {(\Pr or Sc)}^{1 / 3} / 2 μ^{1 / 2}$ , $S_{i} = S_{i - 1} - (β_{con} + δ_{con} U_{r, i - 1}) Δ t$ , $T_{s i} = {(α_{f}^{char} {m'}_{fb i} Δ h_{char} / ε_{s} 4 π r_{s i}^{2} σ)}^{1 / 4}$ , $m_{i} = m_{i - 1} - {m'}_{fb i} Δ t$ , and Δt is the time step.

Fig. 12.

Firebrand transport model.

The height of firebrand and horizontal flight distance are:

(27)

H = H_{\max} - \sum_{i = 1}^{n} (V_{y, i} Δ t), L_{x} = \sum_{i = 1}^{n} (V_{x, i} Δ t)

The drag coefficient (Ganser 1993) is calculated by:

(28)

C_{D} = \frac{24}{Re K_{1}} [1 + 0.1118 {(Re K_{1} K_{2})}^{0.6567}] + \frac{0.4305 K_{2}}{1 + 3305 / (Re K_{1} K_{2})}

where K₁ is Stokes’ shape factor, and K₂ is Newton’s shape factor. Here, K₁ = K₂ = 1. The transport trajectory is calculated for firebrands of D = 15, 20 and 25 mm in self-sustaining burning. The initial values of the mass (Eqn 17, Fig. 5) and the projected area (Eqn 20, Fig. 7) are the mean values at t = 0 s under 6 and 8 m s⁻¹ wind speeds. Both β and δ are evaluated based on the slopes for 6 and 8 m s⁻¹ in Fig. 7. The ambient wind speed is set as 20 m s⁻¹. The theoretical maximum height of firebrands in self-sustaining burning (Albini 1979) is:

(29)

H_{\max} = 0.39 \times 10^{5} D

Fig. 13 illustrates the calculated results of the transport model. The burning rate, projected area, relative velocity between firebrands and the surrounding fluid, and firebrand surface temperature are presented in Fig. 13a. The calculated firebrand flight trajectory is indicated in Fig. 13b. The wind speed and transport distance correspond with those in wildfires (Storey et al. 2020a, 2020b). As shown, firebrands with diameters of 15 and 20 mm burned out before landing, whereas the firebrand with a diameter of 25 mm remained burning on landing. The calculated descent trajectory is similar to that in previous work (Lee and Hellman 1970). The maximum flight height is ~1 km for diameters of 20 and 25 mm, consistent with Tarifa et al. (1965). The maximum horizontal transport distance is ~1600 m, comparable with previous works (Tarifa et al. 1965; Lee and Hellman 1970). It should be noted that the residual mass is zero for calculations, whereas it is non-zero for experiments owing to firebrand breakage. Firebrands in self-sustaining burning may land with flames (Fig. 2) when the relative velocity is less than 6 m s⁻¹, which poses a significant fire hazard in igniting WUI combustibles (Caton et al. 2017; Santoso et al. 2019; Xiong et al. 2020; Wadhwani et al. 2022). The burning and transport of firebrands depend on the relative velocity and direction of the firebrands and the surrounding air flow. Combined with meteorological conditions, the developed models have promising potential for engineering applications.

Fig. 13.

Transport of firebrands: (a) variations of parameters (D = 20 mm); (b) transport trajectory (U_h = 20 m s⁻¹).

Conclusions

In this work, wooden firebrand burning was conducted under forced convection. The density, mass, projected area and surface temperature of the firebrands were investigated. A firebrand burning mechanism, including wood pyrolysis and char oxidation, is proposed based on the firebrand density and wind speed. The mass loss due to wood pyrolysis is much greater than that due to char oxidation in self-sustaining burning. Firebrand burning rate is correlated with firebrand diameter, wind speed, pyrolysis rate and flaming or smouldering. The burning rate is proportional to U^1/2. The reduction rate of the firebrand projected area is correlated with char density, firebrand diameter, wind speed, and flaming or smouldering of firebrands. The projected area for flaming firebrands decreases more rapidly than that for smouldering ones. The surface temperature model is established with firebrand burning rate and wind speed in terms of energy conservation. The firebrand surface temperature is mainly determined by radiation. A simplified firebrand transport model is established to predict the transport trajectory and essential parameters affecting the burning of firebrands.

Nomenclature

B	mass transfer number (–)
c_p	specific heat capacity at constant pressure (J kg⁻¹ K⁻¹)
C_D	drag coefficient (–)
D	firebrand diameter (mm)
D_m	gas diffusivity (m² s⁻¹)
fla	flaming
fb	correction factor (–)
F	force (N)
F_w	drag force of the wind (N)
F_y,G	force exerted by gravity on the firebrand (N)
F_y,res	air resistance (N)
Δh_char	heat of char oxidation (MJ kg⁻¹)
h_conv	convective heat transfer coefficient (W m⁻² K⁻¹)
H	height of the firebrand (m)
K_film	thermal conductivity of air (W m⁻¹ K⁻¹)
K_sol	thermal conductivity of solid phase (W m⁻¹ K⁻¹)
K	function coefficient (–)
K₁	Stokes’ shape factor (–)
K₂	Newton’s shape factor (–)
Le	Lewis number (–)
L_x	horizontal transport distance (m)
m_fb	firebrand mass (g)
${m'}_{char}$	char oxidation rate (g s⁻¹)
Nu	Nusselt number (–)
Pr	Prandtl number (–)
r	firebrand radius (mm)
Re	Reynolds number (–)
smo	smouldering
S	projected area (mm²)
t	burning time (s)
T_film	temperature of gas phase film (K)
T_s	firebrand surface temperature (K)
U	wind speed (m s⁻¹)
U_h	horizontal wind speed (m s⁻¹)
V_fb	firebrand velocity (m s⁻¹)
U_r,i	relative velocity (m s⁻¹)
V_x	horizontal speed of firebrand (m s⁻¹)
V_y	vertical speed of firebrand (m s⁻¹)
Y	mass fraction (–)

Greek symbols

α	thermal diffusion (m² s⁻¹)
$α_{fb}^{char}$	mass fraction of char in firebrand (–)
β	parameter (–)
δ	parameter (–)
ε_s	emissivity (–)
φ_pyr	ratio of wood pyrolysis rate to char oxidation rate (–)
$\bar{φ_{pyr}}$	mean ratio of wood pyrolysis rate to char oxidation rate (–)
η	CO-to-CO₂ ratio (–)
μ	dynamic viscosity (N s m⁻²)
ρ	firebrand density (kg m⁻³)
ρ_g	air density (kg m⁻³)
σ	Stefan–Boltzmann constant (W m⁻² K⁻⁴)
ζ	stoichiometric coefficient (–)
Ω	firebrand volume (m³)

Subscripts

bre	firebrand breakage
fe	flame extinction
fla	flaming
smo	smouldering
i	iteration
nc	natural convection
oxi	char oxidation
pyr	wood pyrolysis
s	surface
0	initial value
∞	ambient or infinity

Data availability

Data that support this study will be shared upon reasonable request to the corresponding author.

Conflicts of interest

Dr Naian Liu is an Associate Editor of the International Journal of Wildland Fire. All authors hereby declare that they do not have any conflict of interest, and have participated in (a) experimental design, and analysis of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version. This manuscript has not been submitted to, nor is under review at another journal or other publishing venue. The authors have no affiliation with any organisation with a direct or indirect financial interest in the subject matter discussed in the manuscript.

Declaration of funding

This research is sponsored by the National Key Research and Development Plan (No. 2022YFC3003000), the National Natural Science Foundation of China (Nos. 52321003 and 51936011), the China Postdoctoral Science Foundation (2021M703083) and the Foundation of State Key Laboratory of Laser Interaction with Matter (SKLLIM2001).

Acknowledgements

We thank the following people for help in the laboratory: Fengkai Liu and Hongming Zhang for general advice and help.

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