Question

How do the relative magnitudes of \(U\)-factors of windows with aluminum, wood, and vinyl frames compare? Assume the windows are identical except for the frames.

Short answer

Question: Arrange the frame materials - aluminum, wood, and vinyl - in ascending order of their U-factor values based on their insulating properties. Answer: Aluminum, Vinyl, Wood

Step-by-step solution

Understanding the U-factor

The \(U\)-factor, also known as the thermal transmittance, is a measure of how effective a building material is as an insulator. The lower the \(U\)-factor, the better the insulating properties of the material. Windows with a low \(U\)-factor will transfer less heat, providing better energy efficiency.

Thermal conductivity values for aluminum, wood, and vinyl

The thermal conductivity (\(k\)) of a material is a measure of its ability to conduct heat. The lower the thermal conductivity, the lower the rate of heat transfer through the material. The thermal conductivity values for the three frame materials are as follows: - Aluminum: \(k_{Al} \approx 237\, W/(m\cdot K)\) - Wood: \(k_{wood} \approx 0.14\, W/(m\cdot K)\) - Vinyl: \(k_{vinyl} \approx 0.19\, W/(m\cdot K)\)

Comparing the U-factors of windows with different frame materials

Now that we know the thermal conductivity values for aluminum, wood, and vinyl, we can compare the \(U\)-factors of windows with these frame materials. Since a lower thermal conductivity value corresponds to a lower rate of heat transfer (and better insulating properties), we can expect that the \(U\)-factor of windows should follow the same trend. This means that the window with better insulating frame material will have a lower \(U\)-factor. By comparing the thermal conductivity values, we can see that wood has the lowest thermal conductivity value, followed by vinyl, and then aluminum has the highest thermal conductivity value. Therefore, we can expect that the windows with wood frames will have the lowest \(U\)-factor, followed by vinyl frames, and then aluminum frames with the highest \(U\)-factor. In conclusion, for windows of identical design except for their frame materials, the relative magnitudes of their \(U\)-factors can be compared as follows: - Wood frames have the lowest \(U\)-factor - Vinyl frames have the second lowest \(U\)-factor - Aluminum frames have the highest \(U\)-factor