Question 8 of 8: Internal heat gains at 4 p.m., and seasonal gas use by the degree-day method

Question 8: Internal heat gains at 4 p.m., and seasonal gas use by the degree-day method (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. An office floor with stated occupancy, lighting and equipment densities and schedules; and a Toronto building with a stated design heating load, furnace efficiency and fuel heating value.

Find. The sensible and latent heat gain of the space at 16:00; and the annual quantity of natural gas, in cubic feet, required to heat the Toronto building.

1. Part (a), step 1 — check the schedules at 16:00. This is why the question names a time. At 4 p.m. the occupants are still present (they leave at 17:00), the lights are still on (until 18:00), and the office equipment is in use. Every source is therefore at full value; had the question asked at 17:30 the people would have gone and only the lighting and equipment would remain.
2. People. For moderately active office work, ASHRAE gives an adjusted total of 475 Btu/h per person, split 250 sensible and 225 latent — the 200 Btu/h latent figure is used here, being the value tabulated for the seated, lightly-active office case: $$q_{s,people}=25\times250=6\,250\ \text{Btu/h},\qquad q_{l,people}=25\times200=5\,000\ \text{Btu/h}$$ People are the only latent source in the space.
3. Lighting. The fixtures are recessed and unvented, which is the point of that adjective: none of the lamp heat is carried away in a return-air plenum, so the whole electrical input appears in the space and the use factor and special-allowance factor are both 1.0: $$q_{lights}=2.5\ \frac{\text{W}}{\text{ft}^{2}}\times4\,000\ \text{ft}^{2} =10\,000\ \text{W}=10\,000\times3.412=34\,121\ \text{Btu/h}$$ all sensible. (If the 2.5 W/ft² were lamp wattage rather than input wattage, a ballast allowance of about 1.2 would raise this to 41 000 Btu/h; the figure is read here as installed input power.)
4. Equipment. Similarly, $$q_{equip}=1.5\times4\,000=6\,000\ \text{W}=20\,473\ \text{Btu/h}$$ also entirely sensible. Office equipment produces no moisture.
5. Part (a) result — total the gains. $$q_{s}=6\,250+34\,121+20\,473=\boxed{60\,844\ \text{Btu/h}\;(17.8\ \text{kW})}$$ $$q_{l}=\boxed{5\,000\ \text{Btu/h}\;(1.47\ \text{kW})}$$ The total internal gain is 65 844 Btu/h, or 5.49 tons of refrigeration, at a space sensible heat factor of 0.924 — and 4.8 W/ft², a figure worth remembering as the signature of a lit, occupied, equipment-heavy office floor.
6. Note what has and has not been calculated. These are instantaneous heat gains, which is what the question asks for. The cooling load at 16:00 would be smaller, because the radiant fraction of the lighting and people gains — roughly half — is absorbed by the floor, walls and furniture and released over the following hours. Applying cooling-load factors would reduce the lighting contribution at 16:00 to perhaps 0.85 of the gain while adding a tail after 18:00 when the lights are switched off. Envelope gains through the glazing and walls, and the ventilation load, would be added separately.

1. Part (b), step 1 — the degree-day method and its correction factor. The classical degree-day method assumes the heat requirement is proportional to the difference between the indoor base temperature and the outdoor temperature, so that the annual heat is the design load scaled by the ratio of annual degree-days to the design temperature difference: $$Q_{year}=\frac{24\,q_{design}\,\mathrm{HDD}}{\Delta t_{design}}\times C_{D}$$ The empirical factor $C_{D}$ corrects for the fact that a real building's efficiency falls at part load and that internal gains and solar gain are already embedded in the base-65 convention; the modified degree-day method gives $C_{D}$ between about 0.60 and 0.80, and 0.70 is taken here.
2. Equivalent full-load hours. With $\Delta t_{design}=70-(-12)=82\ ^{\circ}\text{F}$ and 6 827 $^{\circ}\text{F}\cdot$day for Toronto, $$\mathrm{EFLH}=\frac{24\times6\,827}{82}=1\,998\ \text{h}$$ which is squarely in the 2 000–2 500 h band expected for southern Ontario, and confirms that the degree-day figure and the design temperature difference are consistent. Applying the correction, the effective operating time is $0.70\times1\,998=1\,399$ h.
3. Annual heat delivered. $$Q_{year}=450\,000\times1\,399=6.294\times10^{8}\ \text{Btu}=\boxed{629\ \text{MBtu per year}}$$
4. Part (b) result — fuel. Dividing by the furnace efficiency and the heating value, $$V_{gas}=\frac{Q_{year}}{\eta_{f}\,\mathrm{HHV}} =\frac{6.294\times10^{8}}{0.80\times1\,000} =\boxed{787\,000\ \text{ft}^{3}\ \text{per year}}$$ which is 787 Mcf, or about 22 300 $\text{m}^{3}$, or 830 GJ of gas — roughly the consumption of twenty detached houses, consistent with a building whose design load is 450 000 Btu/h.
5. State the limits of the estimate. The method is a seasonal estimate, not a design calculation, and three cautions apply. It assumes the balance point is the 65 $^{\circ}\text{F}$ base implied by the published degree-days; a building with heavy internal gains has a lower balance point and will use less, while one held above 70 $^{\circ}\text{F}$ has a higher one and will use more, and in either case variable-base degree-days should be used instead. It takes the furnace efficiency as constant, whereas a modern condensing appliance gains efficiency at part load and an atmospheric one loses it. And it is silent on ventilation and infiltration scheduling, which on a commercial building can be a third of the seasonal heat. For a Canadian building being modelled for code compliance, CSA F280 or an hourly simulation would replace it — but as a first estimate, and as a way of checking a simulation's plausibility, the degree-day result is exactly what is wanted.