Question

A \(1.0 \mathrm{~m} \times 1.5 \mathrm{~m}\) double-pane window consists of two 4-mm-thick layers of glass $(k=0.78 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\( that are separated by a \)5-\mathrm{mm}\( air gap \)\left(k_{\text {air }}=0.025 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)$. The heat flow through the air gap is assumed to be by conduction. The inside and outside air temperatures are \(20^{\circ} \mathrm{C}\) and \(-20^{\circ} \mathrm{C}\), respectively, and the inside and outside heat transfer coefficients are 40 and \(20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Determine \((a)\) the daily rate of heat loss through the window in steady operation and \((b)\) the temperature difference across the largest thermal resistance.

Short answer

Based on the given data, calculate the overall thermal resistance, the daily heat loss rate, the largest thermal resistance, and the temperature difference across the largest thermal resistance.

Step-by-step solution

Determine the overall thermal resistance

To find the total thermal resistance (\(R_{total}\)), we will consider the conduction resistance through the glass layers and the air gap, as well as the convection resistance at the inside and outside surfaces of the window. The conduction resistance of a uniform layer can be calculated as: \(R_{cond} = \frac{L}{kA}\), where \(L\) is the layer thickness, \(k\) is the thermal conductivity, and \(A\) is the area. The convection resistance can be calculated as: \(R_{conv} = \frac{1}{hA}\), where \(h\) is the heat transfer coefficient. Now, let's calculate the individual resistance of all the layers: Conduction resistance of glass layer 1 (\(R_{g1}\)): \(R_{g1} = \frac{L_{g1}}{k_g A} = \frac{0.004 m}{0.78 \frac{W}{m\cdot K} \cdot 1.0 m \cdot 1.5 m}\) Conduction resistance of glass layer 2 (\(R_{g2}\)): \(R_{g2} = \frac{L_{g2}}{k_g A} = \frac{0.004 m}{0.78 \frac{W}{m\cdot K} \cdot 1.0 m \cdot 1.5 m}\) Conduction resistance of the air gap (\(R_{air}\)): \(R_{air} = \frac{L_{air}}{k_{air} A} = \frac{0.005 m}{0.025 \frac{W}{m\cdot K} \cdot 1.0 m \cdot 1.5 m}\) Convection resistance of the inside surface (\(R_{in}\)): \(R_{in} = \frac{1}{h_{in} A} = \frac{1}{40 \frac{W}{m^2 \cdot K} \cdot 1.0 m \cdot 1.5 m}\) Convection resistance of the outside surface (\(R_{out}\)): \(R_{out} = \frac{1}{h_{out} A} = \frac{1}{20 \frac{W}{m^2 \cdot K} \cdot 1.0 m \cdot 1.5 m}\) Now we can calculate the total thermal resistance: \(R_{total} = R_{g1} + R_{g2} + R_{air} + R_{in} + R_{out}\).

Calculate the daily heat loss rate

Knowing the total thermal resistance, we can find the heat loss rate through the window, which is the rate of heat transfer from the inside to the outside, by using the formula: \(q = \frac{\Delta T}{R_{total}}\), where \(\Delta T\) is the temperature difference between the inside and outside air (20°C - (-20°C) = 40°C). Now, we can find the daily heat loss rate by multiplying the heat loss rate by the seconds in a day: \(q_{daily} = q \cdot 86400 s\)

Find the largest thermal resistance

Now, examine which of the resistance components calculated in Step 1 is the largest one, since it would have the greatest impact on the overall thermal resistance of the window. In this exercise, the largest resistance is \(R_{air}\).

Calculate the temperature difference across the largest thermal resistance

The temperature difference across the largest thermal resistance can be found using the following formula: \(\Delta T_{largest} = q \cdot R_{largest}\) Now, you can calculate the temperature difference across the air gap (largest resistance) using the previously calculated heat loss rate and the value of \(R_{air}\).