91Ó°ÊÓ

First, we will set up an energy balance equation for the wing on the ground: \[ q_{\text{in}} - q_{\text{out}} = q_{\text{net}} \] Where \(q_{\text{in}}\) is the solar irradiation, \(q_{\text{out}}\) is the radiation exchange between the wing and the sky, and \(q_{\text{net}}\) is the net heat transfer. We consider the wing and the sky as blackbodies, so we can apply the Stefan-Boltzmann law: \[ q_{\text{in}} = G \qquad q_{\text{out}} = \epsilon \sigma (T^{4}-T_{\text{sky}}^{4}) \] Where \(\epsilon\) is the emissivity, \(\sigma\) is the Stefan-Boltzmann constant, \(T\) is the temperature of the wing, and \(T_{\text{sky}}\) is the effective sky temperature. Plugging the values given in the problem statement, we have: \[ 800 - \epsilon \sigma (T^{4}-270^{4}) = 0 \] Next, we will solve for the temperature \(T\).

In this part of the problem, we'll consider both radiation and convection heat transfer between the wing and the surrounding air. We'll use the energy balance equation and the convective heat transfer equation: \[ q_{\text{in}} - q_{\text{rad}} - q_{\text{conv}} = 0 \] The radiation term is the same as in part (a), and the convection term is given by: \[ q_{\text{conv}} = hA(T-T_{\infty}) \] Here, \(h\) is the convective heat transfer coefficient, and \(T_{\infty}\) is the air temperature. We need to calculate the convective heat transfer coefficient \(h\), for which we need to determine the flow conditions (laminar or turbulent) by calculating the Reynolds number: \[ Re = \frac{\rho vL_c}{\mu} \] Plugging the values given in the problem statement, we have: \[ Re = \frac{0.470(200)(4)}{1.50 \times 10^{-5}} \] Determine the Reynolds number and decide if the flow is laminar or turbulent. Then, calculate the Nusselt number using the suitable correlation and find the convective heat transfer coefficient \(h\) using: \[ Nu = \frac{hL_c}{k} \] Finally, plug the values of \(q_{\text{in}}\), \(q_{\text{rad}}\), and \(q_{\text{conv}}\) into the energy balance equation and solve for the temperature \(T\).