With the higher rolling speeds used in modern cold-rolling mills, proper roll cooling has become a critical factor in avoiding problems of excessive roll spalling and poor thermal crowning. Poor thermal crowning of rolls can severely affect the shape and profile of sheet and strip products. To determine the influence of cooling practices on roll temperature, a mathematical model was developed that determines the two-dimensional (radial and circumferential) steady-state temperature distribution in a rotating roll subject to constant surface heat input over one portion of the circumference and convective cooling over another portion of the circumference. The model is analytical in nature, as opposed to a direct numerical simulation, which enables extensive parametric studies to be performed conveniently. The solution technique can be used to solve numerous problems involving any combination of surface boundary conditions that have, at most, a linear dependence with respect to the surface temperature. With the use of the principle of superposition, the present solution can be utilized to solve problems where various regions of the surface have constant heat fluxes. Results of the present analysis indicate that for normal cold-rolling situations during steady operation, the penetration of the effects of the surface heating and cooling that occur during every roll revolution is usually less than 4 percent of the radius. Furthermore, the bulk of the roll is at a uniform temperature that can be calculated quite accurately by neglecting all internal temperature gradients. The location of the cooling regions relative to the heat-input regions has little effect on the bulk roll temperature in this situation. This approximation would be useful for computing bulk roll temperature, which could be utilized in future models for determining thermal crowns, but would not be suited for determining accurate temperatures at the roll surface.
Determination of surface location between two sub-domains using the temperature distribution on the external surface
