Free boundary problems and optimal control of axisymmetric polymer crystallization processes

The non-isothermal crystallization of a hollow cylindrical polymer sample with radial symmetry is studied. Three radial cooling strategies are considered: cooling from inside (outward cooling), cooling from outside (inward cooling), and cooling from both sides (double cooling). When the initial and boundary conditions are axisymmetric, the crystallization problem can be reduced to a one-dimensional formulation where a free boundary problem framework can be used. The solution is approximated by appropriate one-phase Stefan problems for which the analytical solution is provided. These results are compared to direct numerical simulations of the crystallization process, finding an excellent agreement in the approximation of the time-evolution of the crystallization front, the temperature distribution and the crystallization time. In a second part, the corresponding optimal control problems are formulated for a cost functional assessing the use of low temperatures and the duration of the crystallization process. Analytical expressions of the approximated optimal controls are derived for each cooling strategy. In particular, the double cooling case presents special difficulties that we are able to overcome by extending the technique that we previously developed for the case of homogeneous rectangular samples.