A Semi-Discretized Heat Transfer Model for Optimal Cooling of Steel Profiles

Several generalized state-space models arising from a semi-discretization of a controlled heat transfer process for optimal cooling of steel profiles are presented. The model orders differ due to different levels of refinement applied to the computational mesh. 1 The model equations We consider the problem of optimal cooling of steel profiles. This problem arises in a rolling mill when different steps in the production process require different temperatures of the raw material. To achieve a high production rate, economical interests suggest to reduce the temperature as fast as possible to the required level before entering the next production phase. At the same time, the cooling process, which is realized by spraying cooling fluids on the surface, has to be controlled so that material properties, such as durability or porosity, achieve given quality standards. Large gradients in the temperature distributions of the steel profile may lead to unwanted deformations, brittle-ness, loss of rigidity, and other undesirable material properties. It is therefore the engineers goal to have a preferably even temperature distribution. For a picture of a such cooling plant see Figure 1. The scientific challenge here is to give the engineers a tool to pre-calculate different control laws yielding different temperature distributions in order to decide which cooling strategy to choose. We can only briefly introduce the model here; for details we refer to [Saa03] or [BS04]. We assume an infinitely long steel profile so that we may restrict ourselves to a 2D model. Exploiting the symmetry of the workpiece, the computational domain Ω ⊂ R 2 is chosen as the half of a cross section of the rail profile. The heat distribution is modeled by the instationary linear heat equation on Ω: