Insulation

For many practical applications, we want to reduce the rate of conductive heat transfer from a system to its surroundings, or vice versa. Going back to the conductive heat transfer equation:
Q•=kAΔTΔx\overset{•}{Q}=kA\frac{ΔT}{Δx}
There are several ways we can manipulate the variables in this equation to reduce Q•\overset{•}{Q}. We can choose a material with a lower conductivity, we can reduce the cross-sectional area that the material conducts through, we can decrease the temperature gradient between the system and its surroundings, or we can increase the thickness of the conducting substance.

Reducing ΔΔT isn't usually feasible, as we are trying to keep our system at a constant temperature, and while changing the temperature of the surroundings can be possible, it is usually beyond our control.

Reducing surface area (A) can reduce the conductive rate of heat transfer. The total energy of a system is mass-dependent, and by extension-volume dependent (volume and mass are related through density). By minimizing the ratio between surface area and volume, we can reduce the surface area (A), which leads to a lower Q•\overset{•}{Q} for the same volume. For example, a sphere has a surface-area volume ratio of 4.83598 per unit volume, while a cube has a ratio of 6 per unit volume. Because the sphere has a lower surface area than a cube per volume, the sphere would have a lower rate of conductive heat transfer to its surroundings than the cube if all other variables and conditions were the same. The surface area isn't always feasible to change, however. We could design a spherical cooler that would have a lower rate of heat transfer than a conventional cooler design, but we would lose the ability to efficiently store a cooler with other objects.

It isn't often practical to change the physical design of an insulator in order to reduce the rate of heat transfer, but if we want to increase the rate of conductive heat transfer, increasing the surface area is a great idea. For example, many radiators will be made of layers of high-conductivity, spaced sheets that maximize the surface area to volume ratio, and maximize convective heat transfer (the design of radiators like these will be talked about more in the section on convection).

Heat sink and fan photo By fir0002flagstaffotos [at] gmail.comCanon 20D + Canon 70-200mm f/2.8 L - Own work, GFDL 1.2, https://commons.wikimedia.org/w/index.php?curid=1403949
Image 4: Heat sink and fan system for increasing the rate of heat transfer.
By fir0002flagstaffotos [at] gmail.comCanon 20D + Canon 70-200mm f/2.8 L - Own work, GFDL 1.2, https://commons.wikimedia.org/w/index.php?curid=1403949

For many insulating jobs, changing the conductivity (k-value) or changing the thickness Δx are often the easiest to change to reduce or increase the rate of heat transfer. Using a low k-value and thick material will reduce the rate of heat transfer, while a high k-value and thin material will increase the rate of heat transfer between a system and it's surroundings. These are the properties that are usually easiest to control, and how they are manipulated will be discussed further in the next section.


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