Thermal Conductivity

In keeping with the theme of material properties from the last post, today we’ll talk about thermal conductivity which is a material property that relates to the conduction of heat.

Fundamentals of Thermal Resistance

The Thermal Resistance Analogy
Thermal resistance is a convenient way of analyzing some heat transfer problems using an electrical analogy in order to make complicated systems easier to visualize and analyze.  It is based on an analogy with Ohm’s law which is:

In Ohm’s law for electricity, “V” is the voltage which drives a current of magnitude “I”.  The amount of current that flows for a given voltage is proportional to the resistance (R_elec).  For an electrical conductor, the resistance depends on the material properties (copper tends to have a lower resistance than wood, for example) and the physical configuration (thick short wires have less resistance than long thin wires).

Tire Heating on Landing, Part 2

In the last post, we talked about the moments of sliding/rolling motion when a tire first contacts a surface with a mismatched velocity.  Using the assumptions and simplifications outlined in that post, today we will make estimates of heat generation at the tire/runway interface, and temperature profiles inside the tire material based on a hypothetical airplane landing scenario.

Tire Heating on Landing

When an airplane first touches the runway, there is an interesting moment of drama because the ground (relative to the airplane’s wheel) is moving past quite rapidly, while the wheel just at the instant of touchdown, is not moving at all.  Of course, there is a great deal of friction between the non-rotating wheel and the moving ground, so the wheel is accelerated rapidly in rotation until it “catches up” to the ground speed and rolls without sliding.  However, the wheel has some rotational inertia and so requires a little bit of time to get up to speed.  During that time, the tire is sliding across the ground and generating a lot of heat. Today we’ll talk about a rough approximation for the heat generated and transferred to the tire during an airplane landing.

Isothermal and Isoflux Boundary Conditions

In analytical conduction problems, two commonly used boundary conditions are isothermal, meaning that the temperature at the boundary is fixed, and isoflux, which means that the heat flux at the boundary is fixed. The two are mutually exclusive (you can’t specify both the heat flux and the temperature on a single boundary) and lead to very different thermal behaviors inside the body. In this post, we’ll look at the effects of these two boundary conditions.

How big is the heat flow?

Sometimes it is useful to have a ballpark idea of the size of a heat flow before even starting a more detailed heat transfer analysis.  It is also kind of fun to have a rough idea of the magnitude of different heat flows. To those ends, we present this figure: