Question

Determine the winter \(R\)-value and the \(U\)-factor of a masonry cavity wall that consists of \(100-\mathrm{mm}\) common bricks, a \(90-\mathrm{mm}\) airspace, \(100-\mathrm{mm}\) concrete blocks made of lightweight aggregate, 20 -mm airspace, and \(13-\mathrm{mm}\) gypsum wallboard separated from the concrete block by \(20-\mathrm{mm}\)-thick (1-in \(\times 3\)-in nominal) vertical furring whose center-to-center distance is \(400 \mathrm{~mm}\). Neither side of the two airspaces is coated with any reflective films. When determining the \(R\)-value of the airspaces, the temperature difference across them can be taken to be \(16.7^{\circ} \mathrm{C}\) with a mean air temperature of $10^{\circ} \mathrm{C}$. The airspace constitutes 84 percent of the heat transmission area, while the vertical furring and similar structures constitute 16 percent.

Short answer

Answer: The winter R-value for the masonry cavity wall is approximately 5.16 m²K/W, and the U-factor is approximately 0.19 W/m²K.

Step-by-step solution

Determine the R-values of individual layers

To calculate R-values for individual layers, we need to know their thermal conductivity (k) and thickness (d) values. The following values are provided in the problem description: 1. Common bricks: 100mm thickness, k = 0.69 W/mK 2. Airspace (90mm): n/a 3. Concrete blocks (lightweight aggregate): 100mm thickness, k = 0.19 W/mK 4. Airspace (20mm): n/a 5. Gypsum wallboard: 13mm thickness, k = 0.17 W/mK 6. Furring: 20mm thickness, k = 0.8 W/mK To calculate their respective R-values, use the formula R = d/k for each material.

Determine the R-value of the airspaces

For the airspaces, we need to consider the temperature difference and mean air temperature provided in the problem description. Use the values given: ΔT = 16.7°C and mean air temperature Tm = 10°C. For the 90mm airspace, R-value = 1.64 + 0.079ΔT = 1.64 + 0.079(16.7) ≈ 2.96 m²K/W For the 20mm airspace, R-value = 1 + 0.077ΔT = 1 + 0.077(16.7) ≈ 2.29 m²K/W

Calculate the R-value for the assembly with vertical furring

Since we have both the furring and the airspaces in parallel, we need to calculate the R-value for these two components in parallel. We are given that the airspace constitutes 84% of the heat transmission area, and the furring constitutes 16%. R-value for airspace and furring in parallel = 0.84(R-value of airspace) + 0.16(R-value of furring) = 0.84(2.29) + 0.16(20/0.8) ≈ 2.52 m²K/W

Calculate the overall R-value for the entire cavity wall

Now, we can calculate the overall R-value for the entire cavity wall by adding the R-values of each layer: R_total = R_bricks + R_90mm_airspace + R_concrete_blocks + R_20mm_airspace_furring + R_gypsum_wallboard R_total = (100/0.69) + 2.96 + (100/0.19) + 2.52 + (13/0.17) ≈ 5.16 m²K/W

Calculate the U-factor for the cavity wall

The U-factor is the inverse of the total R-value. So, to calculate the U-factor, simply take the reciprocal of the total R-value: U-factor = 1/R_total ≈ 1/5.16 ≈ 0.19 W/m²K The winter R-value for the masonry cavity wall is approximately 5.16 m²K/W, and the U-factor is approximately 0.19 W/m²K.