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Determine the overall \(U\)-factor for a double-doortype wood-framed double-pane window with \(13-\mathrm{mm}\) air space and metal spacers, and compare your result with the value listed in Table 9-6. The overall dimensions of the window are \(2.00 \mathrm{~m} \times 2.40 \mathrm{~m}\), and the dimensions of each glazing are \(1.92 \mathrm{~m} \times 1.14 \mathrm{~m}\).

Short Answer

Expert verified
Question: Calculate the overall U-factor for a double-pane wooden window and compare it with the value given in Table 9-6. Answer: The overall U-factor for the double-pane wooden window is 6.78 W/m²·K. Compare this value to the one listed in Table 9-6 to check if it is within the expected range for this type of window.

Step by step solution

01

Calculate the glazing U-factor

We will use the following formula to calculate the glazing U-factor: \(U_{glazing} = \frac{1}{R_{out} + R_{gap} + R_{in}}\), where \(R_{out}\) and \(R_{in}\) are the resistance of the outer and inner surfaces of the glazing, and \(R_{gap}\) is the resistance of the air gap between the panes. We can find these values in Table 9-6 or similar reference material. For example, we can use \(R_{out} = 0.04 \mathrm{~m^2·K/W}\), \(R_{gap} = 0.18 \mathrm{~m^2·K/W}\), and \(R_{in} = 0.04 \mathrm{~m^2·K/W}\). Now, calculate the glazing U-factor: \(U_{glazing} = \frac{1}{0.04 + 0.18 + 0.04} = \frac{1}{0.26} = 3.85 \mathrm{~W/m^2·K}\)
02

Calculate the spacer U-factor

The spacer U-factor is given by: \(U_{spacer} = \frac{k_{spacer}}{t_{spacer}}\), where \(k_{spacer}\) is the thermal conductivity of the metal spacer and \(t_{spacer}\) is the thickness of the spacer. We can find these values in reference material or assume typical values, such as \(k_{spacer} = 20 \mathrm{~W/m·K}\) and \(t_{spacer} = 0.013 \mathrm{~m}\) (13 mm). Now, calculate the spacer U-factor: \(U_{spacer} = \frac{20}{0.013} = 1538.46 \mathrm{~W/m^2·K}\)
03

Calculate the overall window area and glazing area

The overall window area is given by: \(A_{window} = 2.00 \mathrm{~m} \times 2.40 \mathrm{~m} = 4.8 \mathrm{~m^2}\). The glazing area is given by: \(A_{glazing} = 1.92 \mathrm{~m} \times 1.14 \mathrm{~m} = 2.1888 \mathrm{~m^2}\).
04

Calculate the overall U-factor

Finally, we will use the following formula to calculate the overall U-factor: \(U_{overall} = \frac{A_{glazing}}{A_{window}} \cdot U_{glazing} + \frac{A_{spacer}}{A_{window}} \cdot U_{spacer}\), where \(A_{spacer} = A_{window} - A_{glazing}\). Now, calculate the overall U-factor: \(U_{overall} = \frac{2.1888}{4.8} \cdot 3.85 + \frac{(4.8-2.1888)}{4.8} \cdot 1538.46 = 1.7629 \cdot 3.85 + 0.5435 \cdot 1538.46 = 6.788 \mathrm{~W/m^2·K}\) The overall U-factor for the double-pane wooden window is 6.78 W/m²·K. Compare this value to the one listed in Table 9-6 to check if it is within the expected range for this type of window.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

U-factor calculation
The U-factor is a measure of how well a building element, such as a window or door, can conduct heat .
It's also known as the thermal transmittance coefficient. Low U-factor values indicate better insulating properties as they allow less heat to pass through the window.
In the context of windows, the U-factor is a crucial element in evaluating thermal efficiency because it tells us about the rate of heat transfer through the window.
To calculate the U-factor for a window, we consider components such as glazing, spacers, and the frame.
For glazing, the U-factor can be calculated using the formula:
\[ U_{glazing} = \frac{1}{R_{out} + R_{gap} + R_{in}} \]
where:\
  • \( R_{out} \) is the thermal resistance of the outer surface.
  • \( R_{gap} \) is the thermal resistance of the air gap between the panes.
  • \( R_{in} \) is the thermal resistance of the inner surface.
This formula helps us understand that the heat transfer through glazing is inversely related to the sum of these resistances.
Likewise, the spacer U-factor calculation considers the thermal conductivity and thickness of the metal spacers. Finally, to find the overall U-factor of a window, we must take into account the contributions of both glazing and spacers, proportionate to their respective areas within the total window area. This comprehensive consideration ensures an accurate representation of the window's ability to insulate against heat transfer.
Double-pane window
A double-pane window consists of two layers of glass separated by an air gap. This structure is designed to improve energy efficiency by enhancing the window's insulation capacity. The air gap acts as a barrier to heat loss.
One of the main benefits of using double-pane windows is their ability to reduce energy consumption within a building. By minimizing heat transfer, they help maintain a comfortable indoor temperature, thereby reducing the need for heating or cooling.
The effectiveness of a double-pane window in insulating against heat flow is linked to its physics design. A critical component in this design is the thickness of the air gap between the panes. A common gap thickness is around 13 mm, which provides a balance between insulation effectiveness and overall window thickness.
This air space, often filled with air or an inert gas like argon, further lessens thermal conductivity compared to single-pane windows.
Also, the materials used for the window frame and spacers play a significant role in overall performance, as these parts can form paths for heat to bypass the insulating air gap. By understanding how each component contributes to the U-factor, one can make informed decisions about window selection based on climate and energy efficiency goals.
Thermal resistance
Thermal resistance is a key concept in understanding how well a material or assembly, such as a window, resists the flow of heat. Higher resistance values mean a material is better at insulating.
In our window scenario, thermal resistance is considered for the outer and inner glazing surfaces, as well as the air space between the panes.Given as \(R\), thermal resistance values depend largely on the material properties and thickness.
  • For glazing, surface resistances \(R_{out}\) and \(R_{in}\) are influenced by surface treatments and coatings that reduce heat transfer.
  • The resistance of the gap \(R_{gap}\) is a pivotal aspect of the overall window U-factor, governed by the thermal properties of the air or gas fill.
Understanding these resistances offers valuable insights into the window’s insulating behavior. The concept of thermal resistance aids in calculating thermal transmittance (U-factor), emphasizing the importance of each layer's insulating properties.
It further illustrates how small changes in materials and construction can substantially impact energy efficiency.

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Most popular questions from this chapter

Consider laminar natural convection from a vertical hot-plate. Will the heat flux be higher at the top or at the bottom of the plate? Why?

A vertical double-pane window consists of two sheets of glass separated by a \(1.5-\mathrm{cm}\) air gap at atmospheric pressure. The glass surface temperatures across the air gap are measured to be \(278 \mathrm{~K}\) and \(288 \mathrm{~K}\). If it is estimated that the heat transfer by convection through the enclosure is \(1.5\) times that by pure conduction and that the rate of heat transfer by radiation through the enclosure is about the same magnitude as the convection, the effective emissivity of the two glass surfaces is (a) \(0.47\) (b) \(0.53\) (c) \(0.61\) (d) \(0.65\) (e) \(0.72\)

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