Question

Determine the \(U\)-factor for the center-of-glass section of a double-pane window with a \(13-\mathrm{mm}\) airspace for winter design conditions. The glazings are made of clear glass having an emissivity of \(0.84\). Take the average airspace temperature at design conditions to be $10^{\circ} \mathrm{C}$ and the temperature difference across the airspace to be \(15^{\circ} \mathrm{C}\).

Short answer

Answer: The U-factor of the center-of-glass section of the double-pane window for winter design conditions is approximately 1.68 W/m²K.

Step-by-step solution

Understand the different layers in the window

In this case, the window comprises three layers: (1) the first pane of glass, (2) the airspace, and (3) the second pane of glass. The glass panes have the same properties (emissivity), while the airspace has its own properties (such as width and temperature).

Calculate resistance for each glass pane

To determine the \(U\)-factor, we first need to calculate the resistance for each glass pane. For each pane, we will use the equation: Resistance_glass = 1 / (emissivity_glass * h_out) where emissivity_glass is the emissivity of the glass, h_out is the convection coefficient at the outside surface of the glass and can be taken as 25 W/(m^2 K) for vertical surfaces. Resistance_glass = 1 / (0.84 * 25) = 0.0476 m^2 K/W

Calculate resistance for the airspace

To determine the resistance of the airspace, we can use the equation: Resistance_air = width_air / (k_air * A) where width_air is the width of the airspace (13 mm or 0.013 m), k_air is the conductivity of air (approximately 0.026 W/mK), and A is the area of the airspace. Resistance_air = 0.013 / (0.026 * A) = 0.5 / A (assuming that A = 1 m^2 for simplification)

Calculate total resistance and U-factor

To find the \(U\)-factor, we must first determine the total resistance of the window. Since the resistance of each layer is in series, we can simply add their resistances together: Total_resistance = Resistance_glass1 + Resistance_air + Resistance_glass2 Since the glass panes have the same properties, their resistance is the same. Total_resistance = 2 * Resistance_glass + Resistance_air Total_resistance = 2 * 0.0476 + 0.5 = 0.5952 m^2 K/W Now we can calculate the \(U\)-factor, which is the inverse of the total resistance: U-factor = 1 / Total_resistance U-factor = 1 / 0.5952 = 1.6797 W/m^2 K The \(U\)-factor of the center-of-glass section of the double-pane window for winter design conditions is approximately 1.68 W/m^2 K.