Question 8 of 8: Annual heating and cooling energy and cost for a Toronto office

Question 8: Annual heating and cooling energy and cost for a Toronto office (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 heating and cooling energy requirements, and the annual energy cost for (a) electric baseboard heating with a high-efficiency air conditioner and (b) a gas furnace with the same air conditioner.

Approach. Use the modified degree-day method. Convert the design loads into a building conductance, translate the published degree days into equivalent full-load hours with the empirical correction factor, multiply to get delivered energy, then divide by each plant's seasonal efficiency and price the resulting fuel and electricity.

1. Convert the design loads into rates per degree. The heating load is by definition the conductance times the design temperature difference: $$UA=\frac{q_{h,\text{design}}}{\Delta T_{h,\text{design}}}=\frac{280\,000}{75-(-5)}=3500\ \text{Btu}\,\text{h}^{-1}{}^{\circ}\text{F}^{-1}$$ For the cooling side $\Delta T = 90-78 = 12\,{}^{\circ}\text{F}$, so the apparent cooling conductance is $95\,000/12 = 7917$ Btu h$^{-1}$°F$^{-1}$ — more than twice the heating value, because the summer load also contains solar and internal gains that have nothing to do with the outdoor temperature. That discrepancy is a warning about the cooling estimate, addressed in step 6.
2. Get the degree days. Environment and Climate Change Canada normals for Toronto City — the appropriate station for a downtown site — give 3520 heating degree-days base $18\,{}^{\circ}\text{C}$ and 356 cooling degree-days base $18\,{}^{\circ}\text{C}$. Multiplying by 1.8 converts to the Fahrenheit base-65 convention: $$HDD_{65}=3520\times 1.8 = 6336\ {}^{\circ}\text{F}\,\text{day},\qquad CDD_{65}=356\times 1.8 = 641\ {}^{\circ}\text{F}\,\text{day}$$
3. Convert degree days to equivalent full-load hours. The modified degree-day method applies an empirical correction $C_D$, here taken as 0.70, which accounts for internal and solar gains and for the fact that the real balance point is not exactly the base temperature: $$EFLH=\frac{24\,DD\,C_D}{\Delta T_{\text{design}}}$$ $$EFLH_h=\frac{24\times 6336\times 0.70}{80}=1331\ \text{h},\qquad EFLH_c=\frac{24\times 641\times 0.70}{12}=897\ \text{h}$$
4. Annual energy delivered to the building. Multiplying each design load by its equivalent full-load hours: $$\boxed{\;Q_h = 280\,000\times 1331 = 3.73\times 10^{8}\ \text{Btu},\qquad Q_c = 95\,000\times 897 = 8.52\times 10^{7}\ \text{Btu}\;}$$ Normalised by floor area these are 77.6 and 17.8 kBtu ft$^{-2}$yr$^{-1}$ respectively — plausible for a small, older Toronto office, where a heating-dominated total in the region of 100 kBtu/ft$^2$ yr is typical.
5. Part (a) — electric baseboard plus a high-efficiency air conditioner. Resistance heating converts electricity to heat one for one, so $$E_{\text{base}}=\frac{3.726\times 10^{8}}{3412}=109\,186\ \text{kWh}\ \Rightarrow\ 109\,186\times 0.11=\$12\,010$$ Taking "high efficiency" as SEER 16 for the air conditioner, $$E_{\text{AC}}=\frac{8.523\times 10^{7}}{16\times 1000}=5327\ \text{kWh}\ \Rightarrow\ 5327\times 0.11=\$586$$ $$\boxed{\;\text{Annual cost (a)} = 12\,010+586 = \$12\,596\;}$$
6. Part (b) — gas furnace plus the same air conditioner. Canadian natural gas has a higher heating value of about 37.5 MJ/m$^3$, that is 35 543 Btu/m$^3$, and a modern condensing-capable furnace achieves a seasonal efficiency of 0.90: $$V_{\text{gas}}=\frac{3.726\times 10^{8}}{0.90\times 35\,543}=11\,647\ \text{m}^3\ \Rightarrow\ 11\,647\times 0.10=\$1165$$ The cooling cost is unchanged at CAD 586, so $$\boxed{\;\text{Annual cost (b)} = 1165+586 = \$1751\;}$$ Gas heating costs CAD 10 845 per year less than baseboard — a factor of 10.3 on the heating bill alone at the prices given. For completeness, since the question supplies an oil price: No. 2 fuel oil at 138 500 Btu/US gallon and 85% seasonal efficiency needs 3165 gallons, costing CAD 10 285, for a total of CAD 10 871 — better than baseboard but nearly nine times the cost of gas.
7. Interpret the comparison. The ranking is unambiguous and does not depend on the finer assumptions: gas is cheapest, oil next, resistance electricity by far the most expensive, because baseboard heating pays the electricity price for every unit of heat while the furnace pays the gas price for about 1.11 units of fuel. The engineering conclusion is that the choice between (a) and (b) is a fuel-price decision, not an equipment-efficiency decision, and that the only way to make electric heating competitive at these prices would be a heat pump, which would divide the 109 186 kWh by a seasonal coefficient of performance of roughly 2.5–3 in Toronto and bring the electric option to CAD 4000–CAD 4800 — still above gas at CAD 0.10/m$^3$, but much closer, and better on carbon given Ontario's largely non-emitting grid.