Optimal Control of the Cooling of Steel Profiles

Ralf Lezius Chemnitz, Germany

We consider the cooling of steel profiles in cooling sections. This process has a great influence on the quality of the material, so that an optimal control is desirable. In a cooling section the profile passes a number of cooling segments, where water is sprayed on it’s surface by a large number of spray nozzles. The cooling segments are followed by zones of air cooling, where an equalizing process takes place. The aim of our work is to find an optimal control of the water flow, so that the temperature of the profile is minimal at the end of the cooling process, and some restrictions on the temperature’s gradient hold.

The heat transfer within the profile is described by the (two-dimensional) nonlinear heat equation

ρ(ϑ(t, x)) c(ϑ(t, x)) \frac{∂ϑ(t, x)}{∂t} = div(λ(ϑ(t, x)) grad ϑ(t, x)).

For the heat exchange at the boundary we apply a boundary condition of third type

λ(ϑ) \frac{∂ϑ}{∂n} = u α(ϑ) [ϑ_{fl} − ϑ].

The control u is a measure for the activity of each nozzle, ϑ_{fl} denotes the temperature of the cooling fluid, and α is the (temperature dependent) heat exchange coefficient.

We select some points P_{i} and Q_{j} and regard the objective functional

∑_{j=1}^{N_{p}} α_{j} ϑ(T, P_{j}) = min!

with box constraints for the control and the state constraints

\frac{ϑ(t, P_{i}) − ϑ(t, Q_{j})}{dist(P_{i}, Q_{j})} ≤ θ_{ij}, i = 1, ..., N_{p}, j = 1, ..., N_{q}.

This optimization problem is solved by means of a method of feasible directions. In each step of this method a large-dimension linear optimization problem is to solve. This is done by the Simplex method. Both the nonlinear and the linearized heat equations are solved by a finite element multigrid method.

Several numerical results on this problem are presented. In contrast to former presentations, now all nonlinearities are included in the model.