Question

Consider a house in Atlanta, Georgia, that is maintained at \(22^{\circ} \mathrm{C}\) and has a total of \(20 \mathrm{~m}^{2}\) of window area. The windows are double-door type with wood frames and metal spacers and have a \(U\)-factor of \(2.5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) (see Prob. 1-125 for the definition of \(U\)-factor). The winter average temperature of Atlanta is \(11.3^{\circ} \mathrm{C}\). Determine the average rate of heat loss through the windows in winter.

Short answer

Answer: The average rate of heat loss through the windows during winter in the given house is 535 W.

Step-by-step solution

Understand the U-factor and the formula for heat loss

The U-factor (also called thermal transmittance) represents the rate at which heat is transferred through a building component (in this case, windows). The U-factor is expressed in W/m²·K. The formula for heat loss through the windows can be given as: Heat Loss = U-factor × Window Area × Temperature Difference Now that we know the U-factor, we can plug in the given values for window area and temperature difference and calculate the average rate of heat loss through the windows in winter.

Calculate the temperature difference

The temperature difference between the inside and outside of the house is crucial for calculating heat loss. Given the inside temperature of the house C_{in} = 22°C and the winter average outdoor temperature, C_{out} = 11.3°C. We can calculate the temperature difference (∆T) as: ∆T = C_{in} - C_{out} ∆T = 22°C - 11.3°C ∆T = 10.7°C

Calculate the average rate of heat loss through the windows in winter

Now that we have the temperature difference (∆T), the U-factor (given), and the window area (given), we can plug in the values into the Heat Loss formula: Heat Loss = U-factor × Window Area × Temperature Difference Heat Loss = (2.5 W/m²·K) × (20 m²) × (10.7 °C) Heat Loss = 535 W The average rate of heat loss through the windows in winter is 535 W.

Key concepts

U-factor

The U-factor is a key concept in understanding heat transfer through building materials. It measures how well a component, like a window, conducts heat from a warmer area to a colder one. Simply put, the U-factor indicates the efficiency of a material's insulating ability. The lower the U-factor, the better the material is at insulating. It is expressed in units of watts per square meter per Kelvin (

• U-factor, also known as thermal transmittance, represents how well a window prevents heat from escaping.
• A window with a high U-factor loses heat more quickly than a window with a low U-factor.
• In the example, the window's U-factor is 2.5 W/m²·K, meaning it allows 2.5 watts per square meter of heat to pass through with every degree of temperature difference between inside and outside.

Thermal transmittance

Thermal transmittance is another term for U-factor. It quantifies the rate of heat transfer through a structure. When assessing a building’s energy efficiency, understanding its thermal transmittance is crucial. It allows us to determine how much heat is being lost through materials, which can directly affect energy costs.

• Thermal transmittance measures the effectiveness of a material's capacity to conduct heat.
• A lower thermal transmittance value denotes better insulation and less energy loss.
• In practical terms, high thermal transmittance means more heat loss during the winter, leading to higher heating costs.

Heat loss calculation

Calculating heat loss involves determining how quickly heat energy passes through materials from the inside to the outside. The formula used is straightforward: Heat Loss = U-factor × Window Area × Temperature Difference. This equation allows us to quantify heat loss and is fundamental in energy assessments.

• To compute heat loss through windows, identify the U-factor, window area, and temperature difference.
• Using the given example, the equation becomes: Heat Loss = 2.5 W/m²·K × 20 m² × 10.7 °C.
• The calculated heat loss for the example is 535 watts, showing the rate at which heat exits through the windows during winter.

Temperature difference

The temperature difference ( ∆T) is a critical factor in the heat loss equation. It represents the difference between the warm indoor temperature and the cooler outdoor temperature. A greater difference means more heat loss because the drive for heat to flow from warmer to cooler areas increases.

• To find the temperature difference, subtract the outdoor temperature from the indoor temperature.
• In the example, the indoor temperature is 22°C, and the outdoor temperature is 11.3°C, resulting in a temperature difference of 10.7°C.
• The larger this difference, the more pronounced the heat loss, emphasizing the need for efficient insulation.