Question 4 of 8: Degree-day comparison of gas, electric and heat-pump heating

Question 4: Degree-day comparison of gas, electric and heat-pump heating (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 small commercial building in Regina, Saskatchewan, with the design loads and fuel prices tabulated, to be heated by one of three options.

Find. The annual heating cost of each option by the degree-day method, and a discussion of the environmental consequences, including the effect of coal-fired generation.

Approach. Use the modified degree-day equation to convert the design heat loss into annual delivered energy, then divide by each conversion efficiency and price the resulting fuel or electricity. Finally compare the options on primary energy rather than on delivered energy.

1. Assemble the degree-day inputs. Regina has approximately 10,806 °F-days below the 65 °F base (about 5660 °C-days below 18 °C in Environment Canada terms). The empirical correction factor $C_D$ allows for the fact that the building is not occupied at design conditions all winter and that internal gains offset part of the loss; for a design outdoor temperature as low as −29 °F a value of 0.60 is appropriate. The design temperature difference is $$\Delta t_d = 70 - (-29) = 99\ ^\circ\text{F}$$and the design heat loss to be made good is the total, $q_d = 350{,}000 + 45{,}000 = 395{,}000$ Btu/h (the latent term is real energy: the humidifier must evaporate water).
2. Annual energy delivered to the building. $$E = \frac{q_d \times 24 \times DD}{\Delta t_d}\,C_D = \frac{395{,}000 \times 24 \times 10{,}806}{99}\,(0.60) = \boxed{6.21 \times 10^{8}\ \text{Btu/yr}}$$That is equivalent to 1570 full-load hours per year, a sensible figure for a prairie climate.
3. Natural-gas furnace. Dividing by the 80% seasonal efficiency and by the heating value, $$V_{gas} = \frac{E}{\eta_{gas}\,HV} = \frac{6.21\times10^{8}}{0.80 \times 1000} = 7.76\times10^{5}\ \text{std ft}^{3} = 0.776\ \text{million ft}^{3}$$$$\text{Cost} = 0.776 \times 3311 = \boxed{\$2{,}570\ \text{per year}}$$
4. Electric resistance heating. All the delivered energy must be bought as electricity: $$E_{el} = \frac{6.21\times10^{8}}{3412} = 1.82\times10^{5}\ \text{kWh}, \qquad \text{Cost} = 1.82\times10^{5}\,(0.10) = \boxed{\$18{,}200\ \text{per year}}$$
5. Heat pump. The contractor's COP of 3.92 is a refrigeration-cycle figure; the electricity actually drawn from the meter is larger by the compressor-motor losses, so the effective seasonal COP is $$\mathrm{COP}_{eff} = 3.92 \times 0.82 = 3.21$$$$E_{hp} = \frac{6.21\times10^{8}}{3.21 \times 3412} = 5.66\times10^{4}\ \text{kWh}, \qquad \text{Cost} = \boxed{\$5{,}660\ \text{per year}}$$The heat pump costs about three times as much to run as the gas furnace but only about a third as much as resistance heating.
6. Primary-energy and emissions comparison. Delivered-energy costs hide where the energy really comes from. Per unit of heat delivered to the building the primary fuel consumed is $$\text{gas furnace: } \frac{1}{0.80} = 1.25, \qquad \text{resistance: } \frac{1}{0.38} = 2.63, \qquad \text{heat pump: } \frac{1}{3.21 \times 0.38} = 0.82$$so resistance heating on a coal grid burns more than twice the primary fuel of the gas furnace, while the heat pump burns less than either.

The ranking on operating cost is unambiguous: natural gas is cheapest, the heat pump costs about 2.2 times as much, and electric resistance heating is roughly seven times the cost of gas. On capital cost the order reverses — resistance elements are the cheapest equipment to install and the heat pump the most expensive — but the question directs that installation cost be neglected.

Environmentally, direct combustion of natural gas emits about 50 kg of carbon dioxide per gigajoule of fuel, together with nitrogen oxides from the burner and a risk of carbon monoxide and unburnt methane if the appliance is poorly maintained; venting and combustion-air requirements under the National Building Code and CSA B149.1 apply. Electric resistance heating emits nothing at the building, which makes it look clean, but the emissions are simply moved to the generating station. On a coal-fired system of 38% overall efficiency, every joule delivered to the building has required 2.6 joules of coal, and coal emits roughly twice the carbon dioxide of natural gas per unit of fuel energy — so resistance heating is by far the worst of the three options on greenhouse gas, and it also carries the sulphur dioxide, particulate and mercury burden of coal combustion. This is a real Saskatchewan consideration: SaskPower's grid has historically been coal-dominated, unlike the largely hydro and nuclear grids of Manitoba, Quebec or Ontario, where the same calculation would favour electricity far more strongly.

The heat pump reverses that verdict because it moves roughly three units of heat for each unit of electricity, bringing its primary-energy ratio below that of the gas furnace even on a coal grid. Its practical weakness in Regina is capacity: an air-source machine loses output exactly when the load peaks, and at a −29 °F design temperature it will need substantial supplementary heat, which erodes the seasonal COP well below the quoted 3.92. A ground-source installation, or a dual-fuel arrangement with a gas furnace below the balance point, would be the defensible recommendation for this climate.