Question

The inner and outer glasses of a 4-ft \(\times 4\)-ft double-pane window are at \(60^{\circ} \mathrm{F}\) and \(48^{\circ} \mathrm{F}\), respectively. If the \(0.25\)-in space between the two glasses is filled with still air, determine the rate of heat transfer through the window. Answer. $131 \mathrm{Btu} / \mathrm{h}$

Short answer

Answer: The rate of heat transfer through the double-pane window is 131 Btu/h.

Step-by-step solution

Identify the relevant parameters

We are given the following information: - Dimensions of window: 4 ft x 4 ft - Inner glass temperature: \(T_1 = 60^\circ\text{F}\) - Outer glass temperature: \(T_2 = 48^\circ\text{F}\) - Air space thickness: \(d = 0.25\text{ in}\) To determine the rate of heat transfer, we also need the thermal conductivity of air, which is approximately \(k_{air} = 0.015 \text{Btu}/(\text{h}\cdot\text{ft}\cdot\text{F})\).

Calculate the thermal resistance due to the air space

As the heat flows perpendicularly through the window, we can consider a 1-dimensional heat transfer. The thermal resistance of the air space can be calculated using the equation for conduction resistance, which is: $$R_{air} = \frac{d}{k A}$$ First, we need to convert \(d\) from inches to feet: \(d = 0.25 \text{ in} \cdot \frac{1\text{ ft}}{12\text{ in}} = \frac{1}{48} \text{ ft}\). Next, we calculate the area of the window, \(A = 4 \text{ft}\times 4 \text{ft} = 16 \text{ft}^2\). Now, we can find the thermal resistance: $$R_{air} = \frac{\frac{1}{48} \text{ft}}{0.015 \frac{\text{Btu}}{\text{h}\cdot\text{ft}\cdot\text{F}}\cdot 16 \text{ft}^2} = 0.18056\, \text{h}\cdot\text{F}/\text{Btu}$$

Calculate the rate of heat transfer

To find the rate of heat transfer \(q\), we can use the equation for heat transfer through a composite wall: $$q=\frac{\Delta T}{R_{total}}$$ Since we only have one layer of resistance (the air space), the total resistance is \(R_{total} = R_{air}\). The temperature difference across the window is \(\Delta T = T_1 - T_2 = 60^\circ\text{F} - 48^\circ\text{F} = 12^\circ\text{F}\). Substituting values, we get: $$q = \frac{12 \text{F}}{0.18056\,\text{h}\cdot\text{F}/\text{Btu}} = 131.00\, \text{Btu/h}$$ The rate of heat transfer through the window is \(131 \text{Btu/h}\).