Question

A single-pane window is a poor insulator. On a cold day, the temperature of the inside surface of the window is often much less than the room air temperature. Likewise, the outside surface of the window is likely to be much warmer than the outdoor air. The actual surface temperatures are strongly dependent on convection effects. For instance, suppose the air temperatures are \(21.5^{\circ} \mathrm{C}\) inside and \(-3.0^{\circ} \mathrm{C}\) outside, the inner surface of the window is at \(8.5^{\circ} \mathrm{C},\) and the outer surface is at \(4.1^{\circ} \mathrm{C} .\) At what rate will heat flow through the window? Take the thickness of the window to be \(0.32 \mathrm{~cm}\), the height to be \(1.2 \mathrm{~m},\) and the width to be \(1.4 \mathrm{~m}\).

Short answer

Answer: The rate of heat flow through the window is approximately 1840.8 W.

Step-by-step solution

Convert the dimensions to SI units

Convert the thickness of the window from cm to meters by dividing it by 100: $$d = 0.32 \frac{cm}{100} = 0.0032 \,m$$ Now, the dimensions are in SI units.

Calculate the surface area of the window

Multiply the height and width of the window to find its surface area: $$A = (1.2 \, m)(1.4 \, m) = 1.68 \, m^2$$

Write the temperature differences

We are given that: 1. The inside air temperature is \(21.5^{\circ} \mathrm{C}\) 2. The outside air temperature is \(-3.0^{\circ} \mathrm{C}\) 3. The inner surface of the window is at \(8.5^{\circ} \mathrm{C}\) 4. The outer surface of the window is at \(4.1^{\circ} \mathrm{C}\) Since we only need the difference in temperature between the inner and outer surfaces of the window, we have: $$(T_1 - T_2) = (8.5 - 4.1)^{\circ}C = 4.4^{\circ}C$$ Convert the temperature difference to Kelvin: $$\Delta T = 4.4 K$$

Calculate the rate of heat flow through the window

Using the given thermal conductivity for glass, \(k = 0.8 \frac{W}{m \cdot K}\), the surface area of the window \(A = 1.68 m^2\), the thickness of the window \(d = 0.0032 m\), and the temperature difference \(\Delta T = 4.4 K\), calculate the heat flow rate \(Q\) using the formula: $$Q = \frac{kA(T_1 - T_2)}{d}$$ Plug in the values: $$Q = \frac{(0.8 \, W/(m \cdot K))(1.68 \, m^2)(4.4 \, K)}{(0.0032 \, m)}$$ Solve for \(Q\): $$Q \approx 1840.8 \, W$$ So, the rate of heat flow through the window is approximately \(1840.8 \, W\).