Heating (and Cooling) Buildings

For this task you will need your home’s electricity and gas bills (if these are your primary energy inputs). Using the analysis methods outlined in these links, find:

• Electrical energy used each month, in GJ.
• Natural gas used each month, in GJ.
• Any other means of heating (oil, wood).
• Human occupation (about 100 W per resident adult)

Plot a graph of monthly energy use (electrical, gas, other, total) against month of the year. Calculate a mean power (averaged over the year) for each type of energy input, in kW. Values for the author’s house are shown in Fig.1., together with the external temperatures in Fig.2.

Fig. 1. Combined gas and electricity use (monthly averages) for ten years in the author’s house.

Fig.2. External temperatures (monthly averages) for the same ten years depicted in Fig.1.The annualized mean power use for the author’s house 2003-2012 is 3.5 kW. Now let us examine the obvious correlation with external temperature to see how well the insulation is performing.

Glazing and insulation materials like styrofoam are rated by their ability to insulate (“R-factor” in North America) or the inverse, to conduct heat (“U-factor” in Europe). How can you find the R-factor for an entire building. With a year of energy bills and external temperature data, the task is straightforward.

Understanding how well your building is insulated depends on the idea that heat transfer through a surface depends on the quality of the wall material, and is proportional to its thickness, the surface area exposed to the outside world, and the interior-exterior temperature difference. This much is correct for the thermal conduction[note] Thermal Conduction, https://en.wikipedia.org/wiki/Thermal_conduction [2019-10-08].[/note] and is approximately true for radiative heat transfer.

We can check this for our building by plotting the energy use – gas and electricity in this case – as a function of mean outside temperature (helpfully given on the gas bill). Fig.3. shows the raw data.

Fig.3. Ten years of gas and electricity use (monthly means) for the author’s house, plotted against mean outside temperature

It is plain from Fig.3. that there is a component that is strongly dependent on outside temperature, and is the result of deliberate heating of the building. There is also a background component that persists even into the dog days of summer that is associated with temperature-independent activities such as washing, lighting and electronics (including standby power[note] Standby Power, https://en.wikipedia.org/wiki/Standby_power [2019-10-08].[/note]). These two components are marked as lines in Fig.4.

Fig.4. Same data as Fig.3, with fit lines showing the slope of the heating component, and the baseline of the constant contribution predominantly from hot water, electronics and lighting.

The quality of insulation material is given by the resistance to heat flow called the R-value in North America (larger means better) and the conductance to heat flow U-value in Europe (smaller means better)[note] R-value, https://en.wikipedia.org/wiki/R-value_(insulation)#U-value [2019-10-08].[/note]. The U-value is given in SI units so we will do this first; it is heat flow per unit area per unit temperature difference, measured in W/m$^2$/K.

Plainly the slope of the diagonal line in Fig.4. is measured in W/K; it is 0.36 kW/K (negative as heat is flowing out of the house). The straightness of the line is a consequence of Newton’s Law of Cooling[note] Newton’s Law of Cooling, https://en.wikipedia.org/wiki/Newton’s_law_of_cooling [2019-10-09].[/note]. Now all that is needed is the area of the house wall to get the U-factor; that I estimate to be 450 m$^2$. Dividing one by the other gives U = 0.80 W/m$^2$/K.

The R-value is measured in ft$^2$$^\circ$F/(BTU/h) (“Customary Units”). Beware of online conversions, many of which will give you U-values in BTU/h/(ft$^2$$^\circ$F) without saying so. A correct conversion to R-value in customary units gives a value of 7.1. This result is nothing to brag about; the house is old and only partially upgraded to modern insulation standards.

One can also see the benefits of turning down the thermostat. The house loses 0.36 kW per degree (C or K) of internal-external temperature difference, to be compared to a total annualized mean of 3.5 kW. Lowering the internal temperature by one degree will thus save about 10% of the energy bill.

It is also important to note that the energy loss through the walls and windows of a house is proportional not only to the internal-external temperature difference but also to the time over which any given difference persists. Hence it makes no sense at all to keep a house at 20C while there are no occupants.

A commonly held view is that you should not turn off the heat in your apartment or house when you go out, because although you may save a bit of energy, it will take even more energy to heat the space back up to normal temperature. Fact or rubbish?

The first point to note is that whatever heat is generated inside the house will eventually be lost to the environment. The second point is that the rate of heat loss to the environment is proportional to the temperature difference, as is plain in Fig.4. Already you might guess that this means keeping the mean temperature of the building as low as possible, for example by turning the heat off when you go out, but let’s examine a real situation more carefully.

Below is a graph of temperatures inside and outside the author’s house for one week in October/November 2007, a typical autumnal week in Vancouver, not bitterly cold but none-too-warm either. The data were taken with a small, cheap temperature logger[note] Temperature logger, https://www.dataq.com/data-logger/temperature/ [2019-10-09].[/note] and collected on a spreadsheet. The graph runs from midnight Friday/Saturday for seven days. You can tell a lot from this graph: when we go to work (4 days a week M-Th for one of us), go to bed, get up etc. We have forced-air gas heating so it takes very little time to heat up the house, but the house cools down in a nice exponential with a thermal time constant[note]The temperature decays exponentially with a characteristic time: the “time constant”, see https://en.wikipedia.org/wiki/Exponential_decay [2019-10-09].[/note] of 1.9 days.

Fig.5. The temperatures inside and outside of our house in Vancouver for one week in October/November starting from Friday/Saturday midnight and running for seven days.

For this house the rate of heat loss is 360 W/C so we can calculate the difference in temperature (inside-out) for each instant in time. We can now figure out the energy loss in each time interval by multiplying the temperature difference by 360 W/C and by the time interval between data points, in seconds: this gives the heat loss in Joules. Summing the entire column gives the energy loss for the week, in this case 2.66 GJ.

To estimate the heat loss if we did not ever turn off the heat, imagine the the red data points to be all at 20.5 C, the mean “heat-on” temperature. Repeat the calculation above with this new inside temperature. The weekly sum will obviously be higher than what we calculated above; it is 3.08 GJ. Keeping the house at a constant 20.5 C would mean a 16% increase in our gas bill.

More importantly, the GHG to energy ratio of methane is 50 kg/GJ and 2.66 GJ/week is 139 GJ/year or 6.9 tonnes CO$_2$ per year. This is not particularly good, but if we didn’t turn the heat off when we go out or go to bed, it would be 8.0 tonnes, with no additional benefit to ourselves.

You can make a similar measurement yourself. It is easier if you have some means of recording temperature automatically, unless you like staying up all night in a cold house.

A note on windows

Glass is poor insulator, but “low-E” glass cuts down on radiation losses. See this experiment with a sample of low-E glass that illustrates how it works.

Roofs and pavements together comprise a majority of the surfaces found in urban areas. The most common type of roofing used in North America today is Asphalt Shingle. Asphalt is generally very dark and quite unreflective, and hence has a low albedo, ranging between 0.05 and 0.2[note]Albedo of Concrete, http://www.concretepromotion.com/pdf/PCA%20Albedo%20of%20concrete.pdf [2019-10-11].[/note]

What does Albedo mean?
Albedo refers to the fraction of incoming radiation that gets reflected off a surface. For our purposes, it is the fraction of incoming solar radiation (light and heat) that is reflected off roofs and pavements. It ranges on a scale from 0 to 1, with 1 being 100% reflective and zero being unreflective. Black surfaces absorb most of the visible light incident upon them, and thus have low albedo. White, or lighter surfaces have a higher albedo and reflect most of the incident visible light. One must not forget that about half of “sunlight” is in the near infrared (NIR), and it is possible for surfaces that reflect visible light to absorb NIR, or vice versa.

How does Albedo work to offset CO₂ emissions?
Carbon dioxide emissions into the Earth’s atmosphere contribute to the greenhouse effect – greenhouse gases allow solar radiation to pass through into the Earth but trap long wavelength infrared that is radiated back from the Earth’s surface. It thus has a positive radiative forcing– it allows more radiation to be absorbed by the Earth than is re-emitted.

Raising the surface albedo, on the other hand, has a negative radiative forcing, enabling more radiation to escape out of the Earth’s atmosphere. Recent calculations have shown that every extra tonne of CO₂ put in the atmosphere increases the radiative forcing by 910 W_, whereas raising the surface albedo by 0.01 has a radiative forcing of -1.27 Wm^-2 of surface area[note]Menon, S., Akbari, H., Mahanama, S., Sednev, I., & Ronnen, L. (2010). Radiative forcing and temperature response to changes in urban albedos and associated CO₂ offsets. Environ. Res. Lett. 5 (2010) 014005 [/note]. Raising the surface albedo thus has the ability of offset CO₂ emissions.

What effect does raising the albedo of roofs and pavements have?
If the roof of an average house has a surface area of approximately 100 m², raising its albedo by 0.25 would offset (0.25/0.01)(100 m²)(1.27 Wm^-2)/(910 W/tonne) = 3.5 tonnes of CO₂. Note: this is a ONE TIME benefit, and should be compared to the typical emissions of a single family dwelling in Vancouver BC of several tonnes per year (see above). Also note that it makes no sense to reduce passive solar heating of a building that you are burning fossil fuels to heat it. Vancouver BC had 2050 heating degree days[note] Heating Degree Day,  https://en.wikipedia.org/wiki/Heating_degree_day [2019-10-11].[/note] in the last 12 months, but only 313 cooling degree days[note] Degreedays.net, https://www.degreedays.net [2019-10-11].[/note], calculated from a base mean daily temperature of 15.5 C, for which no heating reckoned to be needed.

Cool roofs only make sense if you are spending more resources (and money) in cooling than on heating, but this is rapidly becoming the case for large swathes of the developed world.

Are electric heaters really 100% efficient?

When people hear that electric heaters are 100% efficient it is natural to assume they will be cheaper and less polluting than gas furnaces. However the story isn’t always that simple. To figure out which kind of heat is better, we need to understand what efficiency means, and also where electricity comes from.

Efficiency of Electric and Gas Heaters

Efficiency is broadly defined to be an output to input ratio:

$\textnormal{Efficiency}, \eta=\dfrac{\textnormal{useful output}}{\textnormal{total input}}\tag{1}$

In the case of a heater:

$\textnormal{Efficiency}, \eta=\dfrac{\textnormal{heat energy output}}{\textnormal{fuel energy input}}\tag{2}$

A gas furnace produces heat by burning a fuel (e.g. natural gas) and then directing that heat into your home. However, exhaust gases that carry some of the heat from burning the gas are vented to the outside. Because of this heat loss, gas furnaces are never 100% efficient. An old furnace may be as little as 60% efficient, but modern furnaces have an efficiency of 78% – 84%, while new condensing gas furnaces are 90% – 97% efficient [note]Heating with Gas, https://www.nrcan.gc.ca/sites/oee.nrcan.gc.ca/files/pdf/publications/infosource/pub/home/Heating_With_Gas.pdf [2019-10-11].[/note]

By contrast, an electric heater is just a big resistor that converts electrical energy into heat. Because it can convert ALL of the incoming electricity into heat, we would say that it is 100% efficient. However, one cannot make a direct comparison like this; we should consider the fact that this heater needs to be fuelled by electricity, and therefore we need to look at the efficiency and pollution associated with where that electricity comes from.

Sources of Electricity in BC

To consider the efficiency and pollution of our electrical system we need to know how that electricity is generated. In British Columbia, around 90% of the electricity that is produced within the province comes from hydroelectric dams[note]Canada Energy Regulator, https://www.cer-rec.gc.ca/nrg/ntgrtd/mrkt/nrgsstmprfls/bc-eng.html [2019-10-11].[/note], with 6% coming from thermal stations powered by natural gas.

The efficiency of generating electricity in a thermal station is typically about 50%, and then there are transmission and transformer losses depending on the distance from the station. Plainly this overall thermodynamic efficiency is going to be much less than even an old home furnace.