Thermal camber model for hot and cold rolling

The MDS profile and flatness control system is based on many physical part models. The thermal camber model is the only one of these part models which, apart from the spatial dimensions, inherently depends on the time dimension as opposed to part models like roll bending or even roll wear. This means that work roll temperature and shape keep changing even ii boundary conditions like water distribution or strip contact stay constant. The exact mathematical solution of the resulting dynamic equations is a very time consuming task usually not suitable for on-line purposes. Here a new mathematical approach is presented, which solves the 2-dimensional Fourier heat conduction equation with 3-dimensional boundary conditions at a speed suitable for on-line applications. Furthermore an approximate solution of the Hooke stress strain relations is derived, which translates the temperature distribution of the roll into an expansion distribution. This thermal camber model has been implemented in several hot and cold rolling mills. Data collected there show good agreement with the model.