To reduce the threat of predators, the sand grouse, a bird of Kenya, will lay its eggs in locations well removed from sources of groundwater. To bring water to its chicks, the grouse will then fly to the nearest source and, by submerging the lower part of its body, will entrain water within its plumage. The grouse will then return to its nest, and the chicks will imbibe water from the plumage. Of course, if the time of flight is too long, evaporative losses could cause a significant reduction in the water content of the plumage, and the chicks could succumb to dehydration.
To gain a better understanding of convective transfer during flight, wind tunnel studies were performed using molded models of the grouse. By heating the portion of the model that corresponds to the water-encapsulating plumage, an average convection heat transfer coefficient was determined. Results for different air speeds and model sizes were then used to develop an empirical correlation of the form
$$
\overline{N u}_{L}=0.034 \operatorname{Re}_{L}^{4 / 5} \operatorname{Pr}^{1 / 3}
$$
The effective surface area of the water-encapsulating portion of the plumage is designated as $A_{x}$, and the characteristic length is defined as $L=\left(A_{s}\right)^{1 / 2}$.
Consider conditions for which a grouse has entrained $0.05 \mathrm{~kg}$ of water within plumage of $A_{x}=0.04 \mathrm{~m}^{2}$ and is retuming to its nest at a constant speed of $V=30 \mathrm{~m} / \mathrm{s}$. The ambient air is stagnant and at a temperature and relative humidity of $T_{x}=37^{\circ} \mathrm{C}$ and $\phi_{x}=25 \%$, respectively. If, throughout the flight, the surface $A_{s}$ is covered with a liquid water film at $T_{s}=32^{\circ} \mathrm{C}$, what is the maximum allowable distance of the nest from the water source, if the bird must return with at least $50 \%$ of its initial water supply?