MATHEMATICAL MODEL OF THE TEMPERATURE FIELD OF A HOLLOW ROLL OF ROLLING MILL WITH DIFFERENT CONDITIONS HEAT EXCHANGE ON THE SURFACE

Abstract

The paper considers a mathematical model of radiation-convective heat transfer, which occurs during heat treatment between the roll of a roller mill and the heated metal. We consider the temperature field of a hollow cylindrical roll rotating around its axis at a constant angular velocity and heating up from a metal that has a constant temperature in the contact zone. Outside the contact zone with the metal, the roll gives off heat to its environment. A physical model of the heat transfer process is constructed, in which a thermally thin hollow cylinder is considered, the temperature field of which weakly depends on the radius of the cylinder. Outside of interaction with the metal, the roll gives off heat to the surrounding medium. The mathematical model is considered in the form of a boundary value problem for a homogeneous heat equation with nonlinear boundary conditions in a cylindrical coordinate system. The heat source that heats up the roll body is the moving belt, which transfers heat to the outside of the roll. At the initial moment of time on the surface and at the ends of the rolls have a constant initial temperature. On the surface in the contact zone, the temperature of the roll is equal to the temperature of the metal being processed, and on the other part of the roll surface, heat exchange with the environment takes place according to the Stefan-Boltzmann law. At a significant number of revolutions, the surface temperature function becomes periodic with the period of rotation of the roll around its axis. A simplified mathematical model of the temperature field of the section perpendicular to the axis of rotation of the hollow roll under the conditions of the above problem is considered. With such a simplification, the derivative with respect to the axial coordinate in the heat conduction equation vanishes. A method and an algorithm for solving the problem are proposed. The algorithm includes consideration of the radius-averaged temperature of the rolling mill roll. To find the temperature distribution, the solution to the boundary value problem is reduced to solving an equivalent nonlinear integral equation of the Hammerstein type with a kernel in the form of the Green's function. The Green's function is constructed in the form of a trigonometric series with coefficients - Bessel functions of the first kind of the n-th order, which is a solution to an own spectral problem with a parameter. As a simplification, a thermally thin hollow cylinder, whose temperature field weakly depends on the radius, is considered, and a transition is made to the consideration of the averaged temperature over the radius. A thermodynamic state is considered, which is established some time after the start of the process, as a result of which the Green's function turns out to be periodic in the angular coordinate and in time.