Question 6 of 8: U-factor of a 2 × 4 stud wall, and mould on an uninsulated solarium wall

Question 6: U-factor of a 2 × 4 stud wall, and mould on an uninsulated solarium wall (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A light wood-frame wall of conventional Canadian construction, with the fraction of the elevation occupied by each thermal path stated; and, for part (b), an enclosed balcony with an uninsulated concrete wall in Toronto.

Find. The area-weighted U-factor of the wall; and for the solarium, a remediation strategy with an explanation of how moisture moves in an enclosed space.

Approach. Use the parallel-path (isothermal-planes is not appropriate for wood framing, whose conductivity is close enough to the insulation that lateral heat flow is small) method: total the series resistances along each path, invert each to a U-factor, and weight them by the fraction of the elevation each path occupies.

1. Part (a), step 1 — resistance of the insulated cavity path. Adding the series resistances of the layers that make up the cavity path, $$R_{cav}=0.17+0.81+4.00+13.00+0.45+0.68=19.11\ \text{h}\cdot\text{ft}^{2}\cdot^{\circ}\text{F}/\text{Btu}$$ $$U_{cav}=\frac{1}{19.11}=0.05233\ \text{Btu}/\text{h}\cdot\text{ft}^{2}\cdot^{\circ}\text{F}$$
2. Resistance of the framing path. The stud, plate, sill and header are all 3.5 in of softwood, which at the ASHRAE value of 1.25 per inch is $R = 1.25\times3.5=4.375$, replacing the batt: $$R_{fr}=0.17+0.81+4.00+4.375+0.45+0.68=10.485 \;\Rightarrow\; U_{fr}=0.09537$$ The framing member conducts almost twice as fast as the insulated cavity.
3. Weight the paths by area. The question splits the elevation into three fractions; headers and studs share the same construction, so they share the same U-factor: $$U_{avg}=\sum a_{i}U_{i}=0.75(0.05233)+0.21(0.09537)+0.04(0.09537)$$ $$U_{avg}=0.03925+0.02003+0.00381=\boxed{0.0631\ \text{Btu}/\text{h}\cdot\text{ft}^{2}\cdot^{\circ}\text{F}}$$ which in SI is $0.0631\times5.678=\boxed{0.358\ \text{W}/\text{m}^{2}\cdot\text{K}}$.
4. Read what the answer means. The effective resistance of the wall is $1/0.0631=15.85$, against the 19.11 the insulated cavity alone would give. The framing has therefore consumed 17 % of the nominal insulation value, and the U-factor is 21 % higher than a cavity-only calculation would suggest. This is why Canadian energy codes now require either continuous exterior insulation or an effective-R calculation: the nominal batt rating overstates the wall. Here the R-4 foam sheathing is already doing that job — without it the framing penalty would be proportionally larger still.
5. Part (b) — diagnose the solarium before prescribing. The mould is condensation-driven, and the mechanism is worth naming precisely. An enclosed balcony has an uninsulated concrete wall and slab, both continuous with the exterior structure, so in a Toronto January the interior surface of that wall sits within a few degrees of the outdoor temperature. Meanwhile the apartment supplies the space with warm, moist indoor air. Whenever the surface temperature falls below the dew point of that air, water condenses; mould germinates on a surface held above about 80 % surface relative humidity for a few days, which is reached well before visible condensation. Two aggravations are specific to this case: the concrete slab and wall form a continuous thermal bridge to the exterior, so no amount of heating the air will lift the surface temperature much; and enclosing the balcony removed the ventilation that previously carried the moisture away.
6. Remediate in the right order. The professional sequence is (i) raise the surface temperature, (ii) lower the indoor vapour pressure, (iii) only then treat the mould. Concretely: insulate the wall and, as far as the structure allows, the slab edge and soffit, on the interior face, with a continuous layer of extruded polystyrene or spray polyurethane foam bonded to the concrete — the adhered foam is both the insulation and the vapour control layer, and because it is continuous there is no interior air space in which humid air can reach a cold surface. Exterior insulation would be thermally better but is rarely available on a condominium facade. Avoid the common error of fibrous batt against concrete behind a polyethylene sheet: any air leakage past the poly puts humid air directly onto the cold concrete and makes matters worse. Second, control the source: provide the solarium with conditioned supply air and a return path so that it is part of the apartment's ventilation system rather than a dead-end pocket, or fit a small continuous exhaust; verify the suite ventilation meets the ASHRAE 62.1 / National Building Code requirement and that the bathroom and kitchen fans actually discharge outdoors. Third, keep the winter indoor relative humidity in the 30 to 40 % band recommended for cold-climate buildings, which puts the dew point near $6$ to $9^{\circ}\text{C}$ and gives a real margin against the improved surface temperature. Only then remove the mould — clean non-porous surfaces, replace porous ones, following the CCOHS and Health Canada guidance — because remediation without fixing the surface temperature simply regrows.
7. Comment on moisture flow in an enclosed environment. Moisture moves through a building assembly by four mechanisms, and they differ by orders of magnitude. Bulk water — rain penetration, plumbing leaks — moves the most and is controlled by drainage and flashing. Air leakage is the dominant vapour transport in a Canadian winter: warm indoor air at higher pressure carries vapour through gaps in the enclosure, and a 1 mm crack can deposit far more water in a cavity than diffusion does through the entire adjacent wall area. Vapour diffusion follows the vapour pressure gradient through the materials themselves and is comparatively slow; it is controlled by placing a vapour-retarding layer on the warm side of the insulation, which in Canada means the interior. Capillary transport moves liquid water through porous materials such as concrete and masonry, which is why a slab edge can wick water even where no vapour path exists. Inside an enclosed space the vapour pressure is set by the balance of generation (occupants, cooking, showers, plants, drying laundry — several kilograms a day in a dwelling) against removal by ventilation and by condensation on cold surfaces. Reducing ventilation, as enclosing a balcony does, raises the vapour pressure until condensation on the coldest surface takes up the balance — the building finds its own equilibrium, and the coldest surface pays for it. The design rule that follows is the one applied above: keep every interior surface above the dew point of the air that will touch it, and control air leakage before worrying about diffusion.