Question 2 of 6

Question 2 (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.

Part (A) — required physical properties of a thermal insulator. A thermal insulation material is selected primarily for a low, STABLE thermal conductivity (k) across the temperature range and moisture conditions it will actually see in service, since several common insulations lose a large share of their rated R-value when even lightly wetted (fibrous battings especially) or when a foam's low-conductivity blowing gas diffuses out and is replaced by air over the years (long-term thermal resistance, LTTR, versus the initial rating). It must be dimensionally and thermally stable — no significant shrinkage, sagging out of a stud cavity, or embrittlement over decades — and it must have mechanical properties matched to its application: adequate compressive strength if it will carry load (under a slab, below grade, under a roof membrane) but no particular strength requirement in a stud cavity. Fire performance appropriate to the assembly and code (flame-spread/smoke-developed ratings, or a required thermal barrier such as gypsum board over foam plastic) is mandatory, as is compatibility with the air and vapour control layers it sits beside so the assembly does not trap moisture. Finally, a modern specification also weighs the environmental profile — low VOC off-gassing and, for foam insulations, a low-global-warming-potential blowing agent.

Part (B).

Given.

Find. (i) An expression/value for the total thermal resistance including surface films; (ii) the total heat loss through the 300 m² wall; (iii) the percentage increase in heat loss when the outside wind rises to 45 mph; (iv) which layer controls the heat flow.

Approach. Model the wall as four conduction resistances plus two convective surface films, all in series (1-D steady-state conduction); sum the resistances, then apply Q = A·ΔT/Rₜₖₜₕ for the base case and repeat with a revised outside film coefficient for the wind-speed change.

1. Surface (convective) resistances. ASHRAE Fundamentals gives standard winter-design values for a vertical wall: inside, still air, hᵢ = 8.29 W/(m²·K) ⇒ Rᵣᵢ = 1/8.29 = 0.1206 m²·K/W; outside, "typical" 24 km/h (15 mph) winter design wind, hᵢ = 34 W/(m²·K) ⇒ Rᵣᵢ = 1/34 = 0.0294 m²·K/W.
2. Layer conduction resistances, R = L/k: $$R_{ply}=\frac{0.020}{0.12}=0.1667,\quad R_{fg}=\frac{0.100}{0.04}=2.500,\quad R_{gyp}=\frac{0.010}{0.16}=0.0625\ \ (\text{m}^2\text{K/W})$$
3. (i) Total resistance, base case. Summing all six terms in series, $$\boxed{R_{total} = R_{si}+R_{ply}+R_{fg}+R_{gyp}+R_{so} = 0.1206+0.1667+2.500+0.0625+0.0294 = 2.879\ \text{m}^2\text{K/W}}$$
4. (ii) Total heat loss, base case. With ΔT = 20−(−15) = 35 K and A = 300 m², $$\boxed{Q = \frac{A\,\Delta T}{R_{total}} = \frac{300\times 35}{2.879} = 3647\ \text{W}\approx 3.65\ \text{kW}}$$
5. (iii) Revise the outside film coefficient for 45 mph wind. Using the standard forced-convection wind correlation h₀ = 5.7 + 3.8V (V in m/s) to scale the ASHRAE base value proportionally: base wind 24 km/h = 6.67 m/s gives a correlation value of 5.7+3.8(6.67) = 31.0; 45 mph = 20.12 m/s gives 5.7+3.8(20.12) = 82.2. Scaling the tabulated h₀=34 W/(m²K) by this ratio, $$h_{o,new} = 34\times\frac{82.2}{31.0} = 90.0\ \text{W/(m}^2\text{K)}\ \Rightarrow\ R_{so,new}=\frac{1}{90.0}=0.0111\ \text{m}^2\text{K/W}$$
6. Recompute R and Q at the higher wind speed. $$R_{total,new} = 0.1206+0.1667+2.500+0.0625+0.0111 = 2.861\ \text{m}^2\text{K/W}$$ $$Q_{new} = \frac{300\times 35}{2.861} = 3670\ \text{W}$$ $$\boxed{\%\ \text{increase} = \frac{3670-3647}{3647}\times 100 = 0.64\%}$$
7. (iv) Controlling resistance. The fibreglass batt alone (R = 2.500) is 86.8% of Rₜₖₜₕ, so it is by a wide margin the CONTROLLING resistance — consistent with the near-negligible 0.64% change in heat loss found in (iii), since even tripling the outside film coefficient barely dents a total dominated by the insulation layer.