Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Reference texts: Straube & Burnett, Building Science for Building Enclosures; ASHRAE Handbook — Fundamentals (Ch. 25 Thermal and Water Vapor Transmission Data, Ch. 26 Heat, Air, and Moisture Control in Building Assemblies); National Building Code of Canada (NBCC), Part 5 (Environmental Separation); ASTM C1472, Standard Guide for Calculating Movement and Other Effects When Establishing Sealant Joint Width; CMHC Best Practice Guides for Building Envelopes (brick veneer, shelf angles, movement joints). This is a closed-book paper; the exam instructs that only the first five questions as they appear in the answer book are marked, but all six questions are answered in full below as a complete study resource.
Check: the Appendix Table 1 lists Tyvek's 4.37 under the Vapour permeability column (ng/m·s·Pa), the same column as brick/plywood/insulation/gypsum. Confirmed against the printed paper appendix page. Tyvek's permeance is therefore $M = \mu/l = 4.37/0.0002 = 21{,}850\ \text{ng/(s}\cdot\text{m}^2\text{}\cdot\text{Pa)}$, consistent with a real, vapour-open weather-resistive barrier.
Find. (1) The effective RSI by the parallel-path method; (2) whether interstitial condensation occurs, where, and at what rate; (3) the minimum vapour resistance an interior vapour retarder must supply to eliminate it.
Fig. 1 — Wall assembly, exterior to interior, showing the insulation (75% of area) and 2x6 stud (25% of area) parallel paths used in Eq. (4).
Approach. (1) Sum series resistances along the insulation path and the framing path separately, then area-weight the two conductances by Eq. (4); (2) build the wall's temperature profile from the R-values and its vapour-pressure profile from the layer vapour resistances $Z=1/M=l/\mu$ (Eq. 1), and compare actual vapour pressure to the saturation pressure at each interface to locate and size any condensation; (3) add an interior-side vapour resistance and solve for the value that just brings the actual pressure back to saturation at the critical interface.
Series resistance of each parallel path. The common layers (exterior air film, brick, air space, plywood sheathing, gypsum, interior air film) total $R_{common}=0.03+0.13+0.22+0.11+0.08+0.12=0.69\ \text{RSI}$. The stud thermal resistance is $R_{stud}=l/k=0.140/0.11=1.273\ \text{RSI}$. Adding the cavity fill to each:
$$R_{insulation\ path}=0.69+3.67=4.36\ \text{RSI}, \qquad R_{framing\ path}=0.69+1.273=1.963\ \text{RSI}$$
Area-weighted (parallel-path) U-value, Eq. (4). $U_{ins}=1/4.36=0.2294$, $U_{frame}=1/1.963=0.5095\ \text{W/(m}^2\text{}\cdot\text{K)}$. With a 25% framing factor,
$$U_{avg}=(1-0.25)(0.2294)+0.25(0.5095)=0.1720+0.1274=0.2994\ \text{W/(m}^2\text{}\cdot\text{K)}$$
Substituting, $\boxed{R_{eff}=1/U_{avg}=3.34\ \text{RSI (m}^2\text{}\cdot\text{K/W)}}$ — about 23% below the simple series value (4.36 RSI) that ignores the thermal bridge through the studs.
Vapour pressures at the boundary conditions. From the Appendix saturation table, $p_{sat}(22°C)\approx2645\ \text{Pa}$ and $p_{sat}(-10°C)\approx260\ \text{Pa}$ (over ice). Actual vapour pressures: $p_i=0.40\times2645=1058\ \text{Pa}$, $p_o=0.80\times260=208\ \text{Pa}$.
Temperature and vapour-pressure profile through the insulation path (75% of the wall, governs the moisture check). Using $T(x)=T_i-(R_{cum}/R_{total})(T_i-T_o)$ with $R_{total}=4.36$: the insulation/sheathing interface sits at cumulative $R=0.12+0.08+3.67=3.87$, giving $T=22-(3.87/4.36)(32)=-6.4°C$. The corresponding saturation pressure is $p_{sat}(-6.4°C)\approx356\ \text{Pa}$ (Magnus approximation, matched to the Appendix table to <1%).
Vapour resistances $Z=l/\mu$ (or $1/M$ where permeance is given directly): $Z_{gyp}=0.0125/21=5.95\times10^{-4}$, $Z_{ins}=0.140/172=8.14\times10^{-4}$, $Z_{ply}=0.0125/0.8=0.01563$, $Z_{tyvek}=0.0002/4.37=4.58\times10^{-5}$, $Z_{air\,space}=1/6960=1.44\times10^{-4}$, $Z_{brick}=0.100/5.12=0.01953$ (all Pa·s·m²/ng); $Z_{total}=0.0368$. The cumulative resistance up to the insulation/sheathing interface is only $Z_{gyp}+Z_{ins}=1.41\times10^{-3}$ (3.8% of the total), so the actual vapour pressure has barely dropped there: $p=1058-(1.41\times10^{-3}/0.0368)(1058-208)=1022\ \text{Pa}$.
Condensation check and rate. Since $p_{actual}(1022\ \text{Pa}) \gg p_{sat}(356\ \text{Pa})$ at the insulation/sheathing interface, condensation is predicted there — the coldest point the vapour reaches before crossing the wall's most vapour-resistant layers (plywood + brick, which together hold 96% of the total vapour resistance). Clamping the interface to saturation and splitting the flow each side of it (Eq. 1 form):
$$g_{in}=\frac{p_i-p_{sat}}{Z_{in}}=\frac{1058-356}{1.41\times10^{-3}}\approx4.96\times10^{5}\ \text{ng/(s}\cdot\text{m}^2\text{)}, \qquad g_{out}=\frac{p_{sat}-p_o}{Z_{out}}=\frac{356-208}{0.0353}\approx4190\ \text{ng/(s}\cdot\text{m}^2\text{)}$$
The net accumulation is $\boxed{\dot{m}_{cond}\approx 4.9\times10^{5}\ \text{ng/(s}\cdot\text{m}^2\text{)} \approx 42.5\ \text{g/(m}^2\text{}\cdot\text{day)}}$ — a severe rate, driven by the very low vapour resistance ahead of the insulation (no interior vapour barrier) against the much higher resistance of the sheathing/brick beyond it.
Minimum vapour retarder resistance (Part 3). Add a resistance $Z_{vr}$ on the interior face (inside the gypsum, ahead of the insulation). For the profile to just reach saturation — not exceed it — at the same critical interface, set $p_{actual}(Z_{crit}+Z_{vr})=p_{sat}=356\ \text{Pa}$ with the new total $Z_{total}+Z_{vr}$, and solve for $Z_{vr}$:
$$Z_{vr}=\frac{(p_i-p_o)Z_{crit}-(p_i-p_{sat})Z_{total}}{(p_i-p_{sat})-(p_i-p_o)}$$
Substituting the values above gives $\boxed{Z_{vr(min)}\approx0.165\ \text{Pa}\cdot\text{s}\cdot\text{m}^2\text{/ng}}$, i.e. a maximum permeance of $M_{max}=1/Z_{vr}\approx6.05\ \text{ng/(s}\cdot\text{m}^2\text{}\cdot\text{Pa)}$. Standard 6-mil polyethylene sheeting (permeance ≈ 2–3 ng/(s·m²·Pa)) comfortably satisfies this and is the conventional fix.
