Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problems 1.56-1.57.
We can derive heuristically the Fourier heat conduction law as follows. Suppose we have a flat slab of area and thickness of some substance with the temperature on one side held at and on the other side at , with . At what rate does heat flow through the slab?
If we consider an analogous situation with a ball rolling downhill, the rate at which the ball moves (its velocity) depends on the gradient of the slope. In the case of heat flow, we might therefore expect that the rate of heat flow depends on the temperature gradient across the slab, that is on . If the temperatures are constant over the area , then we’d also expect the heat flow to be proportional to , so if is the amount of heat that crosses the slab in time interval , we get
where is a constant called the thermal conductivity, which depends on the material of which the slab is made. The minus sign means that heat flows down the temperature gradient, so that if , then heat flows from to . [In more complex situations, could also depend on the temperature and other things, but we’ll ignore that for now.] In principle, it is possible to derive from the atomic nature of the material, but for now we’ll assume that it’s just a property of a material that must be measured. The units of are therefore .
Example 1 for air. For a layer of still air with a surface area of , a thickness of 1 mm and , we get
Example 2 In the building trade, the thermal conductivity is often given in terms of the value, which is defined as
Since depends on the inverse of the thermal conductivity and directly on the thickness of the insulating material, a larger means a better insulator. For the 1 mm layer of still air in the previous example, we have
Given that for glass 0.8, the value of a 3.2 mm thick sheet of glass (a typical window) is
Thus if there is a 1 mm layer of still air next to a window, it actually provides more insulation than the window glass itself.
For some reason, Schroeder makes us use the convoluted so-called ‘English’ units for (even though we in the UK now use the metric system for most building quantities now). In the US, the units of are where a British thermal unit (Btu) is defined as the energy required to raise one pound of water by . To convert to SI units of , we need to convert 1 Btu to Joules. One Fahrenheit degree is of a Kelvin, there are 453.592 grams per pound, and 4.186 joules are required to raise 1 gram of water by 1 K. Therefore
One foot is 0.3048 m and 1 hour is 3600 seconds, so
The previous values in English units are therefore
The values of a compound layer of two different materials is the sum of the individual values. We can see this as follows. Suppose we have a compound layer composed of two materials: material 1 and material 2. The temperature on the exposed side of material 2 is and on the material 1 side is . The temperature at the point where the two materials join is . In the steady state, the rate of heat flow must be the same across the two layers (otherwise heat would build up somewhere) so from 1 (taking for convenience; it drops out of the calculation anyway)
Now the overall rate of heat flow across the compound layer must also be the same. If we define to be the effective value of the compound layer, then
We thus have 2 equations in 2 unknowns ( and ). From the first equation, we can solve for :
Substituting into the second equation we can solve for :
Example 3 Using this compound value, we can estimate the rate of heat loss from a single-pane window of thickness 3.2 mm, but with a 1 mm layer of still air on each side. The effective value of this system is
When the temperature difference is , the rate of heat loss is
This compares with the heat loss through the glass on its own of . Thus the air layer actually provides most of the insulation.