Principles And Construction 2c:
Tech: Insulation

Thermal Conductance And Thermal Resistance

Substances do more than just store heat. Heat also moves through them. 'Thermal conductance' and 'thermal conductivity' are two closely-related ways of describing how easily heat moves through a substance. This relates directly to how we rate insulation (the well-known "R-value").

Insulation, such as one might put on a house, is a substance through which heat flows slowly. Heat is energy, and (as mentioned before) quantities of energy are measured in joules.

Looking at a wall, one sees that it has an area. Imagine heat energy escaping. If everything else was the same, more energy would escape across a larger wall, one with more surface area. So one thing we would want to take into account is the area of the wall, in square metres.

Heat flows from warm areas to cold. If we take a unit temperature difference of 1 kelvin as an example, and keep everything else the same, the more energy escaping per unit of time, the more conductive the wall is for heat. So two other things we take into account is a difference in temperature (in kelvins), and a rate of energy flow (in joules per seciond, or watts).

So we end up with a quantity, thermal conductance, described in watts per square meter kelvin (W/m²·K). It's abbreviated by the letter C. This has been measured for many building materials and types of insulation. As the insulation becomes more efficient, the thermal conductance goes down.

Often, you will see a related number on packages of insulation. This is the thermal resistance, abbreviated R, and it's simply 1/C. The thermal resistance is measured in m²·K/W, and it goes up as the insulation becomes more efficient.

On packages of insulation in Canada, this is labeled as "RSI", for "metric R value".

People in Canada, influenced by the USA, often still speak of R-values in the Imperial and US Customary system. These use the unit of BTU/ft²·hr·°F and are numerically 5.678 times larger. When you hear of someone putting "R-40" insulation in their attic, that's a non-metric R-value.

Thermal Conductivity and Thermal Resistivity

There are a couple of related values as well. Thermal conductivity is expressd in W/m·K, and is abbreviated by the letter k. Its inverse, thermal resistivity, is 1/k and is expressd in m·K/W.

These two values relate more to thickness of a material, rather than area. They assume one square metre of a substance one metre thick.

So what does this mean for our 0.5 m x 3 m x 6 m wall?

Let's imagine placing a thickness of insulation on the outside of the wall. Rock wool insulation, like the Potters used, has been measured with a C-value (thermal conductivity) of 0.81 W/m²·K for a thickness of 50 mm. (That's a metric R-value of 1/0.81 = 1.23, or an Imperial/US R-value of 1/0.81 x 5.678 = 7.) The area of the side of the wall is 3 m x 6 m, or 18 m².

If we multiply these two values together, we get 18 m² x 0.81 W/m²·K = 14.58 W/K. This means that our insulated wall will lose energy at a rate of 14.58 watts for every kelvin of temperature difference across the insulation.

If the insulation is protecting against a temperature difference of 10 K, it will lose energy at a rate of 145.8 watts. And a difference of 50K, like at the Potters' in midwinter? 729 W.

Now, let's take the 10-kelvin temperature difference described before. The wall is losing heat at a rate of 145.8W. It has 185 400 000 J of energy stored in it, which must be removed to eliminate the temperature difference. Since one W is 1 J/s, how long will that take?

Assuming linear heat flow and no replenishment, that would be 185 400 000 J / (145.8 J/s) = 1 271 604 s = 14.7 days.

(In actuality, I suspect the heat flow over time would not be linear, but would follow a curve like the voltage from a discharging capacitor. And in real walls, heat goes sideways within the wall, between different building materials. And the sun shines on it during the day, heating things up. And there are convection effects that move air past a wall surface, moving even more energy through the wall than one would expect. And in a real house, there are air leaks and open windows and fireplace drafts that exchange air with the surroundings. And then there're radiation losses through the windows... But those are just details. For now.)

14.7 days. That's quite a long time. But that's a testimony to how much heat can be stored in these massive walls. Traditional stick-built houses have almost no heat storage inside their walls' insulation. They may pump up their R-values, but with almost no heat storage in the house, their furnaces must provide heat as quickly as it is lost during the winter, and their cooling systems must remove heat as quickly as it enters during the summer.

In the Potters' house, the thick insulation on the walls loses heat slowly, slowly enough that it can be replenished by entering sunlight. And the thick walls themselves store enough heat that the house will not cool off a lot during cloudy days.

Next: how climate governs the amount of heat leaving (and entering) the house--in which the mystery of the degree-day is explained.

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