Question

Consider a window glass consisting of two 4-mmthick glass sheets pressed tightly against each other. Compare the heat transfer rate through this window with that of one consisting of a single 8 -mm-thick glass sheet under identical conditions.

Short answer

Answer: Yes, the heat transfer rates of both window glass configurations are equal under identical conditions.

Step-by-step solution

Understand the concept of thermal conductivity

Thermal conductivity (k) is a property of materials that is a measure of their ability to conduct heat. A higher thermal conductivity means a material is better at conducting heat, whereas a lower thermal conductivity indicates lesser heat transfer capabilities. For this problem, both the window glass made out of two 4-mm sheets and the single 8-mm sheet are the same material, so their thermal conductivity values will be the same.

Use the heat transfer formula for solids

The formula to calculate the heat transfer rate (Q) through a solid is given by: Q = kA(T1 - T2) / d, where k is the thermal conductivity, A is the surface area, (T1 - T2) is the temperature difference between the two sides, and d is the thickness of the material. Since we need to compare the heat transfer rate of two different window glass configurations, we will first calculate the heat transfer rate for each case and then compare their values.

Calculate the heat transfer rate for the 4-mm window glass configuration

Let's denote the heat transfer rate for the 4-mm window glass configuration as Q1. According to the formula, we have: Q1 = kA(T1 - T2) / d1, where d1 = 4 mm for both sheets pressed together tightly. However, since the two sheets are placed together tightly, the total thickness becomes 8 mm. Thus, we can rewrite the formula as: Q1 = kA(T1 - T2) / 8 mm.

Calculate the heat transfer rate for the 8-mm window glass configuration

Similarly, for the 8-mm window glass configuration, we denote the heat transfer rate as Q2. The formula is: Q2 = kA(T1 - T2) / d2, where d2 = 8 mm for the single glass sheet. So, we have: Q2 = kA(T1 - T2) / 8 mm.

Compare the heat transfer rates of both configurations

Given that the conditions are identical for both window glass configurations, the values of k, A, and (T1 - T2) are the same for both cases. Therefore, the only differences in the formula are the thickness values - d1 and d2. Since both window glass configurations have the same thickness (8 mm) and have identical conditions, the ratios between heat transfer rates can be obtained by: Q1 / Q2 = (kA(T1 - T2) / 8 mm) / (kA(T1 - T2) / 8 mm) = 1 This means the heat transfer rates of both the 4-mm window glass configuration and the 8-mm window glass configuration are equal under identical conditions.