Water droplet dynamics and evaporation in airtanker firefighting

Abstract

This study presents the first systematic investigation of the dynamics of individual water droplets in the context of airtanker firefighting. While previous work has focused on ground-deposition patterns measured in standardized field tests, the droplet-scale mechanisms governing evaporation and transport have remained largely unexplored. A tailored model of the coupled momentum, heat, and mass transfer of an isolated water droplet in ambient air is proposed and applied to examine the evolution of droplets under a wide range of atmospheric conditions. The results demonstrate that droplet size governs the effectiveness of water delivery, the release height emerges as the dominant operational parameter, and relative humidity is the key atmospheric property. Increasing the release height lengthens the flight time and increases evaporative losses, while low relative humidity accelerates evaporation, particularly for droplets smaller than one millimeter. Only droplets within a narrow range of initial radii, $150\,μ\mathrm{m} \lesssim r_{\mathrm{d},0} \lesssim 3\,\mathrm{mm}$, are able to reach the ground following an airtanker release, with smaller droplets fully evaporating during their fall and larger droplets being subject to secondary atomization. Although airtanker releases involve very large liquid volumes and complex spray dynamics, the present analysis deliberately isolates droplet-scale behavior and does not resolve collective spray effects, wake interactions, or turbulence. The findings therefore serve as a physically consistent baseline for droplet evaporation and transport, forming a foundation for spray-resolved modeling efforts aimed at improving airtanker delivery strategies.

Figures (9)

• Figure 1: (a) A McDonnell Douglas DC-10-30 airtanker releases fire retardant. U.S. Forest Service photo by Mike McMillan ("20130817-FS-UNK-0024" by U.S. Department of Agriculture, CC BY 2.0). (b) Droplet size distributions produced by the primary atomization process in airtanker firefighting defined by Eq. \ref{['eq:gamma']}, as proposed by Legendre2024. The solid line shows the probability density related to ligament breakup, the dashed and dotted lines show the probability density related to bag breakup.
• Figure 2: Evolution of the droplet surface area of static ($g=0$) or freely falling ($g<0$) water droplets with an initial radius of $r_\text{d,0} = 300 \, \upmu\text{m}$ in air under different ambient conditions. The solutions provided by the heat and mass transfer model presented in Section \ref{['sec:droplet-evaporation']} with the Nusselt number correction proposed in Eq. \ref{['eq:dNu']} are shown by the solid lines, the solutions obtained without the Nusselt number correction are shown by the dotted lines. The circular markers represent the reference solution obtained by solving Eq. \ref{['eq:r2']}.
• Figure 3: Evolution of the relative droplet surface area and relative droplet radius. The solutions provided by the heat and mass transfer model presented in Section \ref{['sec:droplet-evaporation']} with the Nusselt number correction proposed in Eq. \ref{['eq:dNu']} are shown by the solid lines, the solutions obtained without the Nusselt number correction are shown by the dotted lines. (a) A water droplet with initial radius $r_\text{d,0} = 600 \, \upmu\text{m}$ and initial temperature $\Theta_\text{d,0} = 15 \, ^\circ\text{C}$ in an air flow with velocity $|\mathbf{u}_\text{a}|\in \{0.8,2\} \, \text{m/s}$, temperature $\Theta_\infty = 30 \, ^\circ\text{C}$ and relative humidity $\phi_\infty = 0.3$. The circular markers show the experimental measurements of Fujita2010. (b) A water droplet with initial radii $r_\text{d,0} \in \{100,1000\} \, \upmu\text{m}$ and initial temperature $\Theta_\text{d,0} = 24 \, ^\circ\text{C}$ in an air flow with velocity $|\mathbf{u}_\text{a}|= 5 \, \text{m/s}$, temperature $\Theta_\infty=30\, ^\circ\text{C}$ and relative humidity $\phi_\infty = 0.5$. The circular markers show the reference results of Woo2011 and the time is scaled with the characteristic evaporation time $\tau_\text{ev}$ given in Eq. \ref{['eq:tauev']}. (c) A water droplet with initial radii $r_\text{d,0} \in \{150,500\} \, \upmu \text{m}$ and initial temperature $\Theta_\text{d,0} = 34 \, ^\circ\text{C}$ falling freely in air with temperature $\Theta_\infty = 20 \, ^\circ\text{C}$ and relative humidity $\phi_\infty = 0.5$. The time is scaled with the corresponding evaporation time $t_\text{CT}$ reported by Cavazzuti2023.
• Figure 4: Evolution of (a) the vertical velocity $w_\text{d}$ as a function of time and (b) the falling distance $\Delta z$ as a function of the radius $r_\text{d}$, of water droplets falling freely in a quiescent air atmosphere with temperature $\Theta_\infty = 25 \, ^\circ\text{C}$ and relative humidity $\phi_\infty = 0.5$. The droplets have an initial temperature of $\Theta_\text{d,0}=25 \, ^\circ\text{C}$ and initial radii of $r_\text{d,0} \in \{0.1, 0.15, 0.2, 0.5, 1, 3\} \, \text{mm}$, with thicker lines corresponding to larger radii. The typical release height of $30-100 \, \text{m}$ in airtanker firefighting is explicitly highlighted in (b).
• Figure 5: Phase maps of the time of flight of water droplets falling freely in an air atmosphere with a temperature of $\Theta_\infty = 25 \, ^\circ\text{C}$ for a fall from $z \in \{30,100\} \, \text{m}$, as a function of the initial droplet radius $r_\text{d,0}$ and the relative humidity $\phi_\infty$. The initial droplet temperature is $\Theta_\text{d,0} = 25 \, ^\circ\text{C}$.