Duhamel's Theorem, part 2

As an additional example of Duhamel’s theorem that we talked about in the last post, we’ll use the solution to a sudden change in temperature at the surface of a semi-infinite solid to examine the temperature in a thin layer in the wall of the cylinder of an internal combustion engine.
The response to a step change in temperature (to Ts) at the surface of a semi-infinite solid, initially at a uniform temperature, Ti, can be expressed like this:

where "erf" is the error function, "x" is the distance from the surface, alpha is the thermal diffusivity, and "t" is the time.since the change.
In dimensionless temperatures, the boundary has changed in a unit-step, and we can use that solution with Duhamel’s theorem to find the response to a more complicated boundary. This figure provides a notional boundary that might approximate (very roughly) the surface temperature of the cylinder in an internal combustion engine. The boundary (surface) temperature has a ramp-like rise followed by a steady temperature, and a ramp-like decline.

Using Duhamel’s theorem with the original solution, we can calculate the temperature as a function of depth and time. This figure shows the calculated temperature at three different depths, along with the specified boundary temperature. 

The blue dotted line shows the imposed surface temperature, the red line is the temperature at 0.1 mm below the surface, the orange line at 1 mm deep, and the green line at 4 mm deep. For the case shown, the variation in surface temperature has barely penetrated to a depth of 4 mm by the time the cycle would normally begin again. While the details of this example are only approximate, the important point is that with Duhamel's theorem, we can use a very simple known solution to calculate the heat transfer with a quite complicated boundary condition.