22-Mec-B2 Environmental Control in Buildings · May 2013

Question 4 of 8: Annual heating and cooling energy and cost for an Ottawa office (20 marks)

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. Professional Engineers of Ontario / EGBC annual examination, 07-Mec-B2 (now 22-Mec-B2) Environmental Control in Buildings, May 2013 sitting. Three hours, open book. Eight problems of 20 points each; the candidate is instructed to solve five and to nominate which five are to be graded. Psychrometric charts and a pressure–enthalpy diagram for ammonia (R-717) are appended to the paper, and candidates are expected to bring an environmental-control text and steam tables. Instruction 1 invites the candidate to state any interpretation assumptions with the answer — that latitude is used explicitly below where the printed data are redundant.

All eight problems are worked here. Every psychrometric state has been recomputed from the ASHRAE formulation for saturation vapour pressure rather than scaled off a chart, so the numbers are tighter than a graphical solution would be; chart-quality agreement (about ±0.2 K and ±0.0002 kg/kg) is all that an examiner expects.

Reference texts for this subject.

W. P. Jones, Air Conditioning Engineering, 5th ed., Butterworth-Heinemann — the standard reference for this examination code; Ch. 2–3 (psychrometry), Ch. 6 (cooling loads), Ch. 10 (cooling towers), Ch. 15 (duct design).
McQuiston, Parker & Spitler, Heating, Ventilating and Air Conditioning: Analysis and Design, 6th ed., Wiley — Ch. 3 (moist air), Ch. 8 (energy estimating and degree-day methods), Ch. 12–13 (fluid flow and duct design).
ASHRAE Handbook – Fundamentals (2021) — Ch. 1 (psychrometrics), Ch. 21 (duct design), Ch. 25–27 (heat, air and moisture transfer in the envelope).
Moran, Shapiro, Boettner & Bailey, Fundamentals of Engineering Thermodynamics, 9th ed., Wiley — Ch. 10 (vapour-compression and multistage refrigeration).
ANSI/ASHRAE Standard 55, Thermal Environmental Conditions for Human Occupancy, and ANSI/ASHRAE Standard 62.1, Ventilation for Acceptable Indoor Air Quality.
Canadian frame: National Building Code of Canada 2020, National Energy Code of Canada for Buildings 2020, and Environment and Climate Change Canada Canadian Climate Normals for degree-day data.

Psychrometric relations used throughout. At barometric pressure $p$, with saturation vapour pressure $p_{ws}(t)$ from the ASHRAE correlation,

$$W = 0.6220\,\frac{\phi\,p_{ws}(t)}{p - \phi\,p_{ws}(t)}, \qquad h = 1.006\,t + W\,(2501 + 1.86\,t)$$

in SI (kJ per kg of dry air), and in the inch-pound system $h = 0.240\,t + W\,(1061 + 0.444\,t)$ Btu per lb of dry air. The thermodynamic wet-bulb temperature is obtained from the adiabatic-saturation equation, which is what a chart's constant-wet-bulb lines represent.

Question 4: Annual heating and cooling energy and cost for an Ottawa 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. A small Ottawa office, its design loads and design temperature differences, its average winter internal gain, and three fuel prices.

Given data
Quantity	Value
Design heating load	250,000 Btu/h at 75/−5 °F
Design cooling load	95,000 Btu/h at 78/91 °F
Average winter internal heat gain	8 kW = 27,300 Btu/h
Natural gas	$0.075 per m³
Fuel oil	$1.22 per litre
Electricity	$0.07 per kWh

Find. the annual heating and cooling energy and its cost for (a) electric baseboard heating plus a high-efficiency air conditioner and (b) a gas furnace plus the same air conditioner.

Approach. Use the modified degree-day method: extract the building loss coefficient from the design heating load, shift the balance-point temperature down by the internal gain, take the annual heating load as $UA \times \text{HDD} \times 24$, divide by each plant's seasonal efficiency, and estimate the cooling energy from equivalent full-load hours because the cooling load is only partly driven by outdoor temperature.

Building loss coefficient. At the winter design condition the whole 250,000 Btu/h is transmission plus infiltration driven by an 80 F° difference, so $$UA = \frac{\dot{Q}_{design}}{t_i - t_o} = \frac{250{,}000}{75 - (-5)} = 3125\ \text{Btu/h per F}^\circ$$
Balance-point temperature. The 8 kW of internal gain offsets part of the loss, so heating is only needed below $$t_{bal} = t_i - \frac{\dot{Q}_{gain}}{UA} = 75 - \frac{27{,}297}{3125} = 66.3^\circ\text{F}$$ This is the physical justification for the conventional 65 °F degree-day base: the computed balance point is within 1.3 F° of it, so published base-65 heating degree-days may be used directly without a further correction (doing the integration at base 66.3 °F would raise the estimate by roughly 4%).
Annual heating load. Ottawa's Canadian Climate Normals give about 4600 °C-days below 18 °C, i.e. HDD $\approx 8300$ F°-days below 65 °F. Then $$\dot{Q}_{heat,annual} = UA \times \text{HDD} \times 24 = 3125 \times 8300 \times 24 = \boxed{6.23\times10^{8}\ \text{Btu}}$$ that is 623 MMBtu, or 182,400 kWh of delivered heat per year.
(a) Electric baseboard heating. Resistance heating is 100% efficient at the point of use, so the purchased electricity equals the delivered heat: $$E = \frac{6.225\times10^{8}}{3412} = 182{,}400\ \text{kWh}\;\Rightarrow\; \text{cost} = 182{,}400 \times 0.07 = \boxed{\$12{,}770\ \text{per year}}$$
(b) Gas furnace. Taking a conventional atmospheric furnace at AFUE 0.80, the fuel energy is $6.225\times10^{8}/0.80 = 7.78\times10^{8}$ Btu. Ontario natural gas has a higher heating value near 37.5 MJ/m³, i.e. 35,540 Btu/m³, so $$V = \frac{7.781\times10^{8}}{35{,}540} = 21{,}900\ \text{m}^{3} \;\Rightarrow\; \text{cost} = 21{,}892 \times 0.075 = \boxed{\$1{,}640\ \text{per year}}$$
Annual cooling energy (common to both cases). Cooling is driven as much by solar and internal gain as by outdoor temperature, so degree-days are a poor proxy; the equivalent-full-load-hour method is used instead. For an office in eastern Ontario, EFLH $\approx 600$ h: $$\dot{Q}_{cool,annual} = 95{,}000 \times 600 = 5.70\times10^{7}\ \text{Btu} = 16{,}700\ \text{kWh of cooling}$$ A high-efficiency unitary machine at SEER 16 has a seasonal COP of $16/3.412 = 4.69$, so $$E_{cool} = \frac{16{,}704}{4.69} = 3560\ \text{kWh}\;\Rightarrow\; \text{cost} = \boxed{\$249\ \text{per year}}$$
Totals and comparison. Adding the cooling cost to each heating option gives $13,020 per year for case (a) and $1,890 per year for case (b). At the prices quoted, gas heat costs about $0.0020 per MJ against $0.0194 per MJ for resistance electricity, so the gas furnace is roughly seven times cheaper to run overall even at only 80% seasonal efficiency. The oil price given in the question is not needed for either case; for reference, the same heat from an 80%-efficient oil furnace would need $7.781\times10^{8}/36{,}200 = 21{,}500$ litres, costing $26,200 per year — the most expensive of the three, which is why oil has all but disappeared from the Ottawa commercial market.

Final results
Quantity	(a) Baseboard + AC	(b) Gas furnace + AC
Annual heating load delivered	623 MMBtu (182,400 kWh)	623 MMBtu (182,400 kWh)
Purchased heating energy	182,400 kWh electricity	21,900 m³ natural gas (778 MMBtu)
Heating cost	$12,770	$1,640
Annual cooling energy	57.0 MMBtu (3560 kWh electricity)	57.0 MMBtu (3560 kWh electricity)
Cooling cost	$249	$249
Total annual energy cost	$13,020	$1,890

Check: the question says “estimate”, and three climate and equipment values are not printed on the paper. The values assumed here, and stated with the answer as instruction 1 requires, are: Ottawa HDD = 8300 F°-days base 65 °F (4600 °C-days base 18 °C, Environment and Climate Change Canada 1981–2010 normals); cooling equivalent full-load hours = 600 h; furnace AFUE = 0.80; air-conditioner SEER = 16 (seasonal COP 4.69); natural-gas HHV = 37.5 MJ/m³ and fuel-oil HHV = 38.2 MJ/L. The ranking of the two options is insensitive to all of them — gas wins by a factor near seven on the prices given — but the absolute costs scale directly with HDD and inversely with efficiency.

Topic. Degree-day and equivalent-full-load-hour energy estimating.

Key relations. $UA = \dot{Q}_{design}/\Delta t_{design}$; $t_{bal} = t_i - \dot{Q}_{gain}/UA$; $Q_{annual} = UA \cdot \text{HDD} \cdot 24 / \eta_{seasonal}$; $Q_{cool} = \dot{Q}_{design,c} \cdot \text{EFLH}$; seasonal $\text{COP} = \text{SEER}/3.412$; 1 kWh = 3412 Btu.

Why this works. The degree-day method is a linearisation: if the building load really is proportional to $(t_{bal} - t_o)$, then integrating that load over the year is exactly $UA$ times the integral of the temperature deficit, and that integral is what a degree-day total is. The internal-gain credit enters only through the balance point, which is why an office with large plug and lighting loads legitimately uses a lower degree-day base than a warehouse. The method fails for cooling because the summer load contains large solar and internal components that do not scale with $(t_o - t_i)$, and because the machine's efficiency itself varies with outdoor temperature — hence the separate EFLH or bin-method treatment.

Common pitfall. Applying the design-day efficiency to the whole season. AFUE and SEER are already seasonal ratings that include cycling and part-load losses; multiplying them by a further part-load factor double-counts. The mirror-image error is using steady-state thermal efficiency (about 0.92 for the same atmospheric furnace) in place of AFUE, which understates annual gas use by about 15%.

Source. McQuiston, Parker & Spitler, Heating, Ventilating and Air Conditioning: Analysis and Design, 6th ed., Wiley, Ch. 8 §8.2–8.5 (degree-day, bin and full-load-hour methods); ASHRAE Handbook – Fundamentals (2021), Ch. 19 (energy estimating and modelling methods); Environment and Climate Change Canada, Canadian Climate Normals 1981–2010, Ottawa Macdonald-Cartier International Airport.

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