Question 7 of 8: Annual heating cost by the degree-day method — Ottawa

Problem 7: Annual heating cost by the degree-day method — Ottawa (20 points)

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 small commercial building in Ottawa, three heating options, and a fuel and electricity tariff.

Find. The annual heating cost of each of the three options, and a comment on their environmental impact given a coal-fired grid.

Approach. Convert degree-days into equivalent full-load hours, multiply by the design load to get the annual heat delivered, then divide by each plant's efficiency (or multiply by its COP) to get purchased energy, and finally compare on cost, primary energy and carbon.

1. Equivalent full-load hours from the degree-day method. The modified degree-day form is $$Q_{\text{year}}=\frac{24\,q_d\,\text{HDD}\,C_D}{\Delta t_{\text{design}}}$$ where $C_D$ corrects the base-65 estimate for internal gain, distribution and part-load behaviour; $C_D = 0.70$ is the standard value for a commercial building. With Ottawa at about 8,100 °F·day (Environment and Climate Change Canada normals, 4,500 °C·day base 18 °C) and $\Delta t = 70-(-17)=87\ {}^{\circ}\text{F}$, $$\text{EFLH}=\frac{24(8100)}{87}=2{,}234\ \text{h (uncorrected)},\qquad \text{EFLH}_{\text{corrected}}=0.70(2234)=1{,}564\ \text{h}$$ The uncorrected 2,234 h sits squarely in the 2,000–2,500 h band typical of southern-Ontario heating, which is the cheapest available check that the degree-day figure and the design temperature are consistent with each other.
2. Annual heat delivered. $$Q_{\text{year}}=350{,}000(1564)=\boxed{5.47\times10^{8}\ \text{Btu/yr}=547\ \text{MMBtu/yr}}$$ The 55,000 Btu/h latent load is a humidification duty; it does not scale with outdoor temperature and is served by a humidifier rather than the heating plant, so it is excluded from the degree-day sum and noted separately (16 kW at design).
3. Natural-gas warm-air furnace. At 80 % seasonal efficiency, $$Q_{\text{fuel}}=\frac{5.474\times10^8}{0.80}=6.84\times10^{8}\ \text{Btu} \quad\Rightarrow\quad V=\frac{6.843\times10^{8}}{1000}=684{,}300\ \text{std ft}^3 = 684\ \text{Mcf}$$ $$\text{cost}_{\text{gas}}=684.3(5.50)=\boxed{\$3{,}760\ \text{per year}}$$
4. Electric resistance. At 100 % efficiency the delivered heat is the purchased energy: $$E=\frac{5.474\times10^8}{3412}=160{,}400\ \text{kWh} \quad\Rightarrow\quad \text{cost}=160{,}400(0.11)=\boxed{\$17{,}650\ \text{per year}}$$
5. Heat pump. The contractor's COP of 4.5 is at the shaft; the 82 % motor efficiency puts the system COP at $4.5(0.82)=3.69$: $$E_{hp}=\frac{160{,}400}{3.69}=43{,}500\ \text{kWh} \quad\Rightarrow\quad \text{cost}=43{,}500(0.11)=\boxed{\$4{,}780\ \text{per year}}$$
6. Comment on the cost ranking. Gas at $3,760 beats the heat pump at $4,780, which in turn beats resistance at $17,650 by nearly four to one. On operating cost alone the furnace wins, but only just — a 27 % margin that a modest gas price rise or a condensing furnace's higher capital would erase. The decisive objection to the contractor's proposal is not cost but climate: at the −17 °F (−27 °C) Ottawa design temperature an air-source heat pump is below its low-ambient cut-out, its capacity has collapsed and resistance back-up carries the coldest hours at COP 1. A rated COP of 4.5 belongs to a mild-weather rating point, not to a January night in Ottawa; either the machine must be ground-source, which holds its source near 5 °C all winter, or the seasonal average COP must be de-rated and back-up electricity added to the bill.
7. Environmental impact on primary energy. The exam supplies the 35 % generating efficiency precisely so the comparison is made at the fuel, not at the meter. The primary energy ratio of each option is $$\text{PER}_{\text{hp}}=\text{COP}_{\text{sys}}\times\eta_{pp}=3.69(0.35)=1.29,\qquad \text{PER}_{\text{furnace}}=0.80,\qquad \text{PER}_{\text{resistance}}=0.35$$ so even on a coal grid the heat pump delivers 61 % more heat per unit of fuel burned than the furnace, and resistance heating delivers less than half of what the furnace does. The break-even system COP against the gas furnace is $\eta_f/\eta_{pp}=0.80/0.35=2.29$, i.e. a shaft COP of 2.79 — well below the rated 4.5, so the heat pump keeps the primary-energy advantage even with substantial seasonal degradation.
8. Environmental impact on carbon. Primary energy and carbon point in opposite directions here, and that is the interesting part of the answer. Coal emits about 94.6 kg CO₂ per GJ of fuel, so at 35 % plant efficiency the grid carries $0.341/0.35 = 0.97$ kg CO₂ per kWh delivered; natural gas emits about 50.3 kg CO₂/GJ at the burner: $$\text{furnace}:\ 36\ \text{t CO}_2/\text{yr},\qquad \text{heat pump}:\ 42\ \text{t},\qquad \text{resistance}:\ 156\ \text{t}$$ On a coal grid the heat pump emits 17 % more carbon than the gas furnace despite using less primary energy, because coal's carbon intensity per unit of energy is nearly double that of natural gas. Resistance heating is indefensible on any measure.
9. The Canadian reading, and the recommendation. The coal grid in the question is a hypothetical: Ontario closed its last coal-fired station in 2014 and its grid now runs at roughly 30 g CO₂ per kWh on nuclear and hydro. On the real Ottawa grid the same heat pump emits about 1.3 t CO₂ a year against the furnace's 36 t — a factor of 28 the other way — and the whole carbon argument reverses. That is the point worth making: a heat pump is exactly as clean as the generation behind it, so the technology choice cannot be separated from the grid it is plugged into, and a national rule of thumb is worthless. For this building the recommendation is a ground-source heat pump if the site allows the loop field (best primary energy, near-zero emissions on the Ontario grid, and a source temperature that survives −27 °C), a condensing gas furnace if it does not, and electric resistance only as back-up.