Question

Consider a \(1.5-\mathrm{m}\)-high and 2 -m-wide triple pane window. The thickness of each glass layer $(k=0.80 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\( is \)0.5 \mathrm{~cm}$, and the thickness of each airspace \((k=0.025 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) is \(1.2 \mathrm{~cm}\). If the inner and outer surface temperatures of the window are $10^{\circ} \mathrm{C}\( and \)0^{\circ} \mathrm{C}$, respectively, the rate of heat loss through the window is (a) \(3.4 \mathrm{~W}\) (b) \(10.2 \mathrm{~W}\) (c) \(30.7 \mathrm{~W}\) (d) \(61.7 \mathrm{~W}\) (e) \(86.8 \mathrm{~W}\)

Short answer

Based on the calculation, the rate of heat loss through the triple pane window is approximately 0.2926 W. When comparing this result to the given options, it is closest to (a) 3.4 W. Thus, the correct answer is (a) 3.4 W.

Step-by-step solution

Calculate the thermal resistance of each layer

We can start by calculating the thermal resistance of each layer. The thermal resistance of a layer can be calculated using the formula: \(R = \frac{d}{kA}\) where \(R\) is the thermal resistance, \(d\) is the thickness of the layer, \(k\) is the thermal conductivity, and \(A\) is the area of the layer. We will calculate the thermal resistance of each glass layer and airspace layer separately.

Calculate the total thermal resistance

Next, we need to find the total thermal resistance of the triple pane window. Since the layers are arranged in a series, we can add the thermal resistances of each layer to find the total thermal resistance: \(R_{total} = R_{glass_1} + R_{glass_2} + R_{glass_3} + R_{airspace_1} + R_{airspace_2}\)

Calculate the rate of heat loss

Now, we can calculate the rate of heat loss through the window using the formula: \(Q = \frac{\Delta T}{R_{total}}\) where \(Q\) is the rate of heat loss and \(\Delta T\) is the temperature difference between the inner and outer surfaces of the window. In this case, the temperature difference is \(10^\circ \mathrm{C} - 0^\circ \mathrm{C} = 10^\circ \mathrm{C}\).

Compare the result with the given options

After calculating the rate of heat loss, we can compare our result with the given options to find the correct answer.

Answer

We will calculate the thermal resistance of the glass layer and airspace layer: - Glass layer: \(R_{glass} = \frac{0.005\text{ m}}{0.80\frac{\text{W}}{\text{m}\cdot\text{K}} \cdot 3\text{ m}^2}= 20.83\text{ m}^2 \frac{\text{K}}{\text{W}}\) - Airspace layer: \(R_{airspace} = \frac{0.012\text{ m}}{0.025\frac{\text{W}}{\text{m}\cdot\text{K}} \cdot 3\text{ m}^2} = 20\text{ m}^2 \frac{\text{K}}{\text{W}}\)

Answer

Now, we will calculate the total thermal resistance of the triple pane window: \(R_{total} = 3R_{glass} + 2R_{airspace} = 3(20.83\text{ m}^2 \frac{\text{K}}{\text{W}}) + 2(20\text{ m}^2 \frac{\text{K}}{\text{W}}) = 102.49\text{ m}^2 \frac{\text{K}}{\text{W}}\)

Answer

We can now calculate the rate of heat loss through the window: \(Q = \frac{10\text{ K}}{102.49\text{ m}^2 \frac{\text{K}}{\text{W}}} = 0.09752\text{ W/m}^2 \cdot 3\text{ m}^2 = \bold{0.2926 \text{ W}}\)

Answer

Our calculated rate of heat loss (0.2926 W) is closest to option (a) 3.4 W. Therefore, the correct answer is: (a) \(3.4 \mathrm{~W}\)