In an effort to prevent the formation of ice on the surface of a wing, electrical heaters are embedded inside the wing. With a characteristic length of \(2.5 \mathrm{~m}\), the wing has a friction coefficient of \(0.001\). If the wing is moving at a speed of \(200 \mathrm{~m} / \mathrm{s}\) through air at 1 atm and \(-20^{\circ} \mathrm{C}\), determine the heat flux necessary to keep the wing surface above \(0^{\circ} \mathrm{C}\). Evaluate fluid properties at \(-10^{\circ} \mathrm{C}\).

Firstly, we need to calculate the Reynolds number for the given flow conditions to analyze the flow regime and find the heat transfer coefficient. The Reynolds number is calculated as: Re = \(\frac{\rho V L}{\mu}\) where: - Re is the Reynolds number, - \(\rho\) is the density of the fluid, - \(V\) is the flow velocity, - \(L\) is the characteristic length, - \(\mu\) is the dynamic viscosity of the fluid. Since we have to evaluate fluid properties at -10°C, we can refer to air property tables to obtain the required values. For air at -10°C, the properties are: - Density, \(\rho = 1.34 \mathrm{~kg/m^3}\) - Dynamic viscosity, \(\mu = 1.63 \times 10^{-5} \mathrm{~Pa \cdot s}\) Now we can calculate the Reynolds number for the given flow: Re = \(\frac{(1.34)(200)(2.5)}{1.63 \times 10^{-5}} = 2.06 \times 10^6\)

Now that we have the Reynolds number, we will use the given friction coefficient (\(C_f\)) to calculate the heat transfer coefficient (h) using the following relation: \(h = C_f \cdot \frac{1}{2} \cdot \rho V^2 \cdot \frac{Pr}{\left(\frac{c_p \mu}{k} \right)}\) where: - Pr is the Prandtl number, - \(c_p\) is the specific heat at constant pressure, - k is the thermal conductivity of fluid. Again, referring to the property tables for air at -10°C, we have: - Pr = 0.72, - \(c_p = 1008 \mathrm{~J/kg \cdot K}\), - k = 0.0235 \(\mathrm{W/m \cdot K}\). Now we can calculate the heat transfer coefficient: \(h = 0.001 \cdot \frac{1}{2} \cdot (1.34)(200)^2 \cdot \frac{0.72}{\left(\frac{1008 \cdot 1.63 \times 10^{-5}}{0.0235} \right)} = 341.6 \mathrm{~W/m^2 \cdot K}\)

Finally, we can calculate the heat flux per unit area needed to keep the wing surface above 0°C. We know the heat transfer coefficient (h) and the temperature difference between the surface and the fluid: \(\Delta T = T_{surface} - T_{fluid} = 0^{\circ}C - (-20^{\circ}C) = 20^{\circ}C = 20 \mathrm{~K}\) Now, using the convective heat transfer equation: \(q_{unit} = h \cdot \Delta T = (341.6)(20) = 6832 \mathrm{~W/m^2}\) So, the heat flux necessary to keep the wing surface above 0°C is 6832 W/m².