New approaches to wear optimization for a laying pipe

This project report reviews several extensions to prior research involving the optimal design of a steel laying pipe. A team of graduate and undergraduate researchers explored three different aspects of the pipe-laying problem, each of which attempt to minimize or evenly distribute the wear the pipe experiences during operation. These three areas are (1) functional minimization to determine a boundary value problem which the minimal-wear pipe should solve, (2) gradient-descent algorithm to find the minimal-wear pipe from a finite-dimensional search space, and (3) analysis of the force and torque balance equations to determine how internal moments in the pipe affect the wear the pipe suffers. Steel mills are used to form steel. One common rolling mill starts with a billet of hot steel and produces a thin steel rod [2]. In this particular mill, a billet of steel with a cross sectional area of approximately 180 square millimeters is formed into a rod with a cross sectional area of just a few square millimeters. During this process, the steel that is being formed accelerates and at one point is traveling at a speed exceeding 100 m/s. Near the end of the forming process the steel rod must slow down to walking speeds (∼ 5m/s). To accomplish this, the steel rod enters a pipe that is attached to a device called a laying head (aptly named because its job is to lay the steel onto the conveyor belt). The laying head is spinning at a high frequency (∼ 10 Hz). When the steel enters the laying head its linear momentum is converted to angular momentum. This change of momentum slows the rod back to an acceptable speed, and it continues its journey on the conveyor belt. As the steel rod moves through the laying pipe it collides with the sides of the pipe. Over time, these repeated collisions will wear out the pipe and eventually cause it to break. To avoid the dangerous situation of a broken laying head pipe (i.e. a hot steel rod escaping the laying head and moving at approximately 100 m/s through the factory), a steel mill will have to shut down production for an entire day to replace the current laying head pipe before it breaks. This operation costs the mill money and wastes precious production time (occurring almost weekly in some cases). We would like to solve the problem of finding the optimal laying head shape to minimize the wear on the pipe. We ask the question: what laying head pipe shape will make the pipe last the longest? The next three sections of this paper describe the work performed by the research group in the areas of (1) calculus of variations techniques to find a boundary value problem whose solution minimizes the wear in the pipe, (2) numerical algorithms to find a minimal wear in the pipe, and (3) exploration of the effect internal moment of the pipe has on the wear. Each section describes the techniques used, an analysis of the results obtained, and directions for further exploration. 1 Analytical Minimization Given some force ~ F (s), we would like to find the curve which minimizes the wear functional, the square of the wear function integrated over the length of the pipe. Here ~ F (s) is a function of the arc length s.