Question 6 of 8: Degree-day estimate for a Winnipeg building — gas quantity, relative cost and primary energy

Question 6: Degree-day estimate for a Winnipeg building — gas quantity, relative cost and primary energy (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.

Find. The annual natural-gas volume, the equivalent electrical energy at 100 % efficiency, the ratio of annual heating costs, the primary-energy comparison between a gas furnace and an electric furnace, and a comment on the environmental consequences.

Approach. Convert the design load and the selected climate into equivalent full-load hours by the modified degree-day method, multiply out the annual delivered heat, then divide by each conversion chain in turn — furnace efficiency for gas, unity for electric resistance, and generating-plus-transmission efficiency for the primary-energy comparison.

1. Select and justify the design conditions. Environment and Climate Change Canada's 1981–2010 normals give Winnipeg Richardson International Airport HDD₁₈ = 5 670 °C·day, and the ASHRAE 99.6 % heating dry bulb for the same station is about −33 °C. With a 21 °C indoor design temperature the design temperature difference is $$\Delta T_{design} = 21 - (-33) = 54\ \text{K}$$ Winnipeg is one of the coldest major cities in Canada, and both figures matter: the design temperature difference sets how hard the plant works at its worst hour, and the degree-days set how long it works over the year.
2. Convert the design load and climate into equivalent full-load hours. The modified degree-day method writes the annual heating requirement as the design load multiplied by the hours the plant would have to run at full output to deliver it: $$\text{EFLH} = \frac{24\,\text{HDD}\,C_D}{\Delta T_{design}} = \frac{24 \times 5\,670 \times 0.70}{54} = \boxed{1\,764\ \text{h}}$$ The correction factor C_D ≈ 0.70 accounts for internal and solar gains, off-cycle losses and the fact that the base-18 °C convention is only an approximation to the building's true balance point. Between 2 000 and 2 500 full-load hours is typical for southern Ontario; 1 764 h for the colder but sunnier and drier Prairies is a sensible figure.
3. Annual delivered heat. $$Q_{annual} = q_{design} \times \text{EFLH} = 100 \times 1\,764 = \boxed{176\,400\ \text{kWh}} \;=\; 635\ \text{GJ}$$
4. The natural-gas quantity. Dividing by the furnace efficiency and the heating value, $$V_{gas} = \frac{Q_{annual}}{\eta_f \times HV} = \frac{635\,040\ \text{MJ}}{0.85 \times 37\ \text{MJ/m}^3} = \frac{635\,040}{31.45} = \boxed{20\,190\ \text{m}^3\!/\text{yr}}$$ which is about 747 GJ of gas purchased to deliver 635 GJ of heat.
5. The electrical alternative at 100 % efficiency. A resistance furnace converts every kilowatt-hour at the meter into a kilowatt-hour of heat in the building, so $$E_{elec} = \frac{Q_{annual}}{\eta_e} = \boxed{176\,400\ \text{kWh/yr}}$$
6. Compare the annual costs. $$\text{Cost}_{elec} = 176\,400 \times 0.10 = \boxed{17\,640\ \text{\$/yr}}$$ $$\text{Cost}_{gas} = 20\,190 \times 0.1175 = \boxed{2\,373\ \text{\$/yr}}$$ so electric resistance heating costs 7.4 times as much. Put on a common basis, gas delivers heat at 1.35 ¢/kWh against electricity's 10.0 ¢/kWh — a gap so large that no plausible variation in the degree-day estimate closes it, because the ratio depends only on the two prices, the heating value and the furnace efficiency, not on the annual quantity at all.
7. Compare on primary energy, which is what the 33 % is for. The exam supplies the generating efficiency precisely so that the two options can be referred to the same boundary — fuel at the mine or the wellhead, rather than energy at the building's meter: $$PE_{gas} = \frac{Q_{annual}}{\eta_f} = \frac{635\,040}{0.85} = 747\,100\ \text{MJ}$$ $$PE_{elec} = \frac{Q_{annual}}{\eta_{plant}\,\eta_{td}} = \frac{635\,040}{0.33 \times 0.93} = 2\,069\,000\ \text{MJ}$$ including a 7 % allowance for transmission and distribution losses. The electric furnace therefore consumes $$\frac{PE_{elec}}{PE_{gas}} = \boxed{2.8\ \text{times}}$$ the primary energy of the gas furnace — 2.6 times if transmission losses are ignored. The same arithmetic gives the threshold at which an electrically driven machine would break even with the gas furnace on primary energy: COP = η_f/(η_plantη_td) = 2.77. A resistance furnace at COP 1.00 fails that test by a wide margin; a heat pump at COP 3.5 passes it comfortably, which is the real conclusion buried in this part of the question.
8. Comment on the environmental consequences — and on why the 33 % plant is the wrong plant for Winnipeg. Burning 20 190 m³ of natural gas at 1.888 kg CO₂/m³ emits 38 tonnes of CO₂ a year. If the electricity really came from the 33 %-efficient thermal plant the question posits — a coal-fired station at roughly 1.0 kg CO₂e/kWh — the electric furnace would emit about 180 tonnes, nearly five times as much, so on that grid the primary-energy answer and the carbon answer point the same way. But the building is in Manitoba, whose grid is over 97 % hydroelectric at an emission factor near 1.2 g CO₂e/kWh, and the same 176 400 kWh would emit about 0.2 tonnes — roughly 180 times less than the gas furnace. The lesson is that primary-energy accounting and carbon accounting are different questions with different answers, and that an electric heating technology is exactly as clean as the generation behind it. Three consequences follow for a real Winnipeg design: gas remains far cheaper to run, so an operating-cost argument favours it; carbon pricing and Canada's Clean Electricity Regulations steadily erode that advantage; and the design that satisfies both criteria is neither of the two the question offers, but a ground-source heat pump on the Manitoba grid, which would cut the primary energy below the gas furnace's and the emissions to near zero.