The overall \(U\)-factor of a fixed wood-framed window with double glazing is given by the manufacturer to be \(U=\) \(2.76 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) under the conditions of still air inside and winds of \(12 \mathrm{~km} / \mathrm{h}\) outside. What will the \(U\)-factor be when the wind velocity outside is doubled? Answer: \(2.88 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\)

Given that the wind velocity outside is doubled, we must first consider the influence of wind velocity on the \(U\)-factor. According to the convective heat transfer equation, the convective heat transfer coefficient, \(h\), is proportional to the wind velocity raised to a certain power, \(n\). Therefore, we can express the relationship between the convective heat transfer coefficient and wind velocity as: \(h=kV^{n}\), where \(k\) is a constant, \(V\) is the wind velocity, and \(n\) is the power. In the original situation, with a wind velocity of \(V_{1}\), the \(U\)-factor was \(U_{1}\). After the wind velocity is doubled, the new wind velocity will be \(V_{2} = 2V_{1}\), and we want to find the new \(U\)-factor, \(U_{2}\). Since we are assuming that all other factors except wind velocity influencing the \(U\)-factor remain constant, the relationship between the \(U\)-factors can be described as: $$\frac{U_{2}}{U_{1}} = \frac{h_{2}}{h_{1}} = \frac{kV_{2}^{n}}{kV_{1}^{n}}$$ Substitute \(V_{2} = 2V_{1}\): $$\frac{U_{2}}{U_{1}} = \frac{k(2V_{1})^{n}}{kV_{1}^{n}} = 2^{n}$$ Now, we can use the given information to solve for the new \(U\)-factor, \(U_{2}\): $$U_{2} = 2^{n}U_{1}$$ #Step 3: Solve for the Unknown \(n\) Value#

Using the equation derived in Step 2 and the value of \(n\) found in Step 3, calculate the new \(U\)-factor with the doubled wind velocity: $$U_{2} = 2^{0.8}U_{1}$$ Substitute the given value of \(U_{1}=2.76 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\): $$U_{2} = 2^{0.8}(2.76 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}) \approx 2.88 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$$ The \(U\)-factor of the fixed wood-framed window with double glazing when the wind velocity outside is doubled is approximately \(2.88 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Please note that this is an approximate solution based on the assumption that the convective heat transfer coefficient has a relationship with wind velocity raised to a power of \(0.8\). This value may vary for different structures and materials, and a more accurate answer would require further information or calculations beyond the scope of this exercise.