The rate of heat loss through a unit surface area of a window per unit temperature difference between the indoors and the outdoors is called the \(U\)-factor. The value of the \(U\)-factor ranges from about \(1.25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (or \(0.22 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F}\) ) for low-e coated, argon-filled, quadruple-pane windows to \(6.25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (or \(1.1 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F}\) ) for a single-pane window with aluminum frames. Determine the range for the rate of heat loss through a \(1.2-\mathrm{m} \times 1.8-\mathrm{m}\) window of a house that is maintained at \(20^{\circ} \mathrm{C}\) when the outdoor air temperature is \(-8^{\circ} \mathrm{C}\).

Step by step solution

01

Identify the given data

We are given the following information: - U-factor of a quadruple-pane window = 1.25 W/m²·K - U-factor of a single-pane window = 6.25 W/m²·K - Window dimensions: 1.2 m × 1.8 m - Indoor temperature (T_indoor) = 20°C - Outdoor temperature (T_outdoor) = -8°C

02

Calculate the surface area of the window

To find the total surface area of the window, multiply its length and width: Surface_area = Length × Width Surface_area = 1.2 m × 1.8 m Surface_area = 2.16 m²

03

Calculate the temperature difference

We need to find the temperature difference between the indoor and outdoor temperatures: Temperature_difference = T_indoor - T_outdoor Temperature_difference = 20°C - (-8°C) Temperature_difference = 28°C

04

Calculate the range of heat loss for the single-pane and quadruple-pane windows

To find the range of heat loss, we'll use the U-factor formula: Heat_loss = U-factor × Surface_area × Temperature_difference For the single-pane window: Heat_loss_single_pane = 6.25 W/m²·K × 2.16 m² × 28°C Heat_loss_single_pane = 386.4 W For the quadruple-pane window: Heat_loss_quadruple_pane = 1.25 W/m²·K × 2.16 m² × 28°C Heat_loss_quadruple_pane = 77.28 W

05

Present the range of heat loss

The range of heat loss through the 1.2 m × 1.8 m window is between 77.28 W (quadruple-pane window) and 386.4 W (single-pane window).

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