8 Building Physics
We are now ready to address some of the fundamental technical concepts in building science, which are needed for us to reason about (plan) the data that our autonomous technologies will be gathering (i.e., what they sense). As we saw in the previous chapter, energy use in buildings is dominated by heating and cooling services. Thus, we will begin by reviewing basic concepts of thermodynamics leading up to the establishment of simple dynamic models of the temperature inside buildings and the energy required to maintain it.
8.1 Building Thermodynamics
In a very general sense, buildings loose (and gain) heat through interactions with the outdoor environment, and through internal processes that compensate for that. During a whole season (say, a full year), the energy lost through the building envelope is (again, at a high level) proportional to the leakiness of the building envelope, and to the temperature demand (i.e., the average difference between the indoor and outdoor temperatures). So, if we were to write an equation for the total (say, heating) losses \(Q\) that need to be compensated to keep the building at a given internal temperature (\(T_i\)), we can write the following high-level formula:
\[Q = \text{leakiness} \times \text{temperature-demand}\]
This leakiness is a function of the properties of the building’s envelope. A bit more specifically, it is mostly affected by the way the envelope handles heat conduction and air infiltration and is usually expressed in units of energy per time per degree of temperature difference. The temperature demand is a function of the accumulated difference between the exterior temperature \(T_o\) and the average temperature we wish to keep the indoor space \(T_i\) over the season in question. This is usually described and tabulated as Heating Degree-Days (HDD) or Cooling Degree-Days (CDD), where Degree-Days refers to the accumulation (e.g., integral) of the difference in temperature ($T_o - T_i) over the season (expressed as number of days).
Read Appendix E of MacKay (2008) for more details on this topic. See if you can re-calculate the numbers in Figure E.12 that refer to “my house, before” and “my house, after”.
8.1.1 Heat transfer
Aside from air exchanges (e.g., infiltration), the leakiness defined above is caused by not just conduction but also convection and radiation. Together (conduction, convection and radiation) they form the basis for all modes of heat transfer. It is worth studying these in a little bit more detail.
8.1.1.1 Conduction
Joseph Fourier (1768 - 1830) came up with the law of conduction heat transfer, which relates the rate at which heat transfer occurs through a material, to the temperature difference it is subjected to, and the distance through which conduction is occurring. It is actually very similar to Ohm’s law relating current (the rate at which electric charge flows through a conductor, to the voltage difference and the resistance). In particular, Fourier’s law shows that:
\[\dot{Q} = -k A \frac{dT}{dx}\]
Where \(k\) is the thermal conductivity of the material, \(A\) is the area through which heat is flowing, and \(\frac{dT}{dx}\) is the temperature gradient at a specific point \(x\) in the material.
Through a plane wall:
\[\dot{Q} = k A \frac{T_1 - T_2}{\Delta x}\]
(Assumming many things, including \(T_1 > T_2\), no heat sources within the material, constant thermal conductivity, etc).
Which can also be stated as:
\[\dot{q} = \frac{T_1 - T_2}{\Delta x / (kA)}\]
Which is similar to how one would define Ohm’s Law (i.e., \(I = V/R\)), and shows that the term \(\Delta x / (kA)\) can be thought of as the resistance \(R\) of the wall.
Define fundamental concepts needed before proceeding futher:
- Temperature
- Heat Capacity: \(C = cm\), \(c = \frac{Q}{m\Delta T}\)
- Heat transfer
- Conduction
- Radiation
- Convection
Work on an example calculation of heat losses through a whole building envelope (problem set and spreadsheet on Canvas).