Question

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: $2.88 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$

Short answer

Answer: The new U-factor is 2.88 W/m²·K.

Step-by-step solution

Recall the definition of U-factor

The U-factor, also known as the thermal transmittance, is a measure of how effective a building material is as an insulator. A lower U-factor indicates better insulating properties, and a higher U-factor indicates poorer insulating properties. In this exercise, we are given the U-factor for a specific window under certain conditions and need to determine the new U-factor when the wind velocity is doubled.

Identify important relationships between U-factor and wind velocity

The U-factor can be affected by several factors, including temperature, convection, and wind velocity. As wind velocity increases, the rate of heat transfer between the interior and exterior of the window is generally increased, leading to a higher U-factor. In this exercise, we are focused on the relationship between the wind velocity and the U-factor.

Use provided formula to find the U-factor for doubled wind velocity

The U-factor for doubled wind velocity can be found using the Formula: \(U' = U \cdot \sqrt{\frac{v'}{v}}\) where \(U\) = initial U-factor, \(U'\) = new U-factor, \(v\) = initial wind velocity, \(v'\) = new wind velocity. In this case, we have: \(U = 2.76 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) \(v = 12 \mathrm{~km} / \mathrm{h}\) \(v' = 2 \cdot v = 2 \cdot 12 \mathrm{~km} / \mathrm{h} = 24 \mathrm{~km} / \mathrm{h}\)

Calculate the new U-factor

Substitute the given values into the formula and compute the new U-factor: \(U' = 2.76 \cdot \sqrt{\frac{24}{12}}\) \(U' = 2.76 \cdot \sqrt{2}\) \(U' = 2.88 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) The new U-factor when the wind velocity outside is doubled is \(2.88 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\).