Question

Consider three similar double-pane windows with air gap widths of 5,10 , and \(20 \mathrm{~mm}\). For which case will the heat transfer through the window be a minimum?

Short answer

Answer: The heat transfer through the double-pane window is at a minimum in Case 3, where the air gap width is 20 mm.

Step-by-step solution

Understanding the relation between heat transfer and air gap width

In a double-pane window, the main mode of heat transfer through the air gap is conduction. As the air gap width increases, the overall resistance to heat transfer increases, which means that the heat transfer will decrease. Consequently, we expect the case with the largest air gap width to have a minimum heat transfer.

Comparing different cases

We can compare the heat transfers for each case, based on the air gap widths: - Case 1: 5 mm - Case 2: 10 mm - Case 3: 20 mm As discussed in Step 1, the heat transfer decreases as the air gap width increases. Therefore, we can say that the heat transfer in Case 3 (with the 20 mm air gap) would be less than that in Case 2 (with the 10 mm air gap), and the heat transfer in Case 2 would be less than that in Case 1 (with the 5 mm air gap).

Determining the case with the minimum heat transfer

As we found in Step 2, the heat transfer through the double-pane window decreases as the air gap width increases. Consequently, the case with the minimum heat transfer will be the case with the largest air gap width. In our three cases, the largest air gap width is 20 mm, which is Case 3. Therefore, the heat transfer through the double-pane window will be a minimum for Case 3.