The problem statement, in general, was discussed earlier with two-dimensional heat-conduction modeling. There is no need, therefore, to repeat the analysis, except to point out that the problem at hand is much more involved. Computerwise, the problem requires a much greater amount of memory and CPU time consumption. In fact, the mere completion of the solution-domain discretization model, using one of the finite element categories, such as the curved-boundary isoparametric element, is perhaps half the work there.

It is the author's opinion that exploring a complex-geometry three-dimensional heat-conduction problem would serve students much more than just casting some bulky matrix equations. The chosen problem, in this chapter, has to do with, perhaps, one of the most complicated bodies there can be. The conduction “body” here is a “slice” in a radial inflow turbine rotor, one that includes one blade and the corresponding segment of the center body (Figure 8.1). The variational approach is, once again, used, and the simplest three-dimensional finite element, meaning the tetrahedron, is used to discretize the problem domain. Naturally, source (actually sink) terms will have to appear in the governing equations because they represent the heat loss due to the passing cooling air, wherever that applies in the blade unit. This in no way eliminates the heat convection through the blade unit. The only difference, in fact, is that heat convection will be the means by which the flowing hot gases interact with the rotor, whereas heat sinks model the effect of the cooling air. Note that in calculating the heat-convection coefficient h, the relative velocity between the hot gases and the rotating body will, and should be, used.