Two heat conduction problems from crystal growth technology are considered. The growth process studied as the first problem comprises two phases (solid crystal and liquid melt) which are separated by a moving interface; affine temperature dependencies of the parameters are taken into account. Planning trajectories for finite time transitions between two regimes is considered. This is possible using recently developed “flatness based” methods leading to series representations. The series are directly parameterized by the flat output trajectories. A proof of series convergence is given. The interest of the second problem, called tempering process, resides in the fact that here a fixed temperature at a moving point is prescribed. An “indirect” solution is proposed where a flat output trajectory is designed such as to approximately achieve that goal.