37 Numerical Analysis – Case studies

V.D. Pathak

1. Introduction:

In earlier modules, various numerical methods were introduced for solving some mathematical problems. Some simple applications of these methods were also discussed. However, in real world problems, these numerical methods seldom work in isolation. In fact, any real world application starts with mathematical modeling to convert the problem under consideration into a tractable Mathematical form. Such mathematical problems rarely have analytical solutions and hence we look for a numerical solution. Prior to computing numerical solution one has to do the numerical modeling of the mathematical problem, so as to make it amenable for applications of numerical methods. Often it is required to apply combinations of more than one numerical methods or work out some innovative improvisation of the existing methods, in order to get a meaningful solution. We will illustrate these techniques by presenting two case studies in this Module.

These case studies are based on the problems which were partially tackled in the Modelling week organized by ITM Universe, Vadodara in collaboration with the Department of Applied Mathematics, M. S. University of Baroda, Vadodara and Industrial Mathematics Group at IITB during 10 – 14 March at ITM Universe, Vadodara under the National Program on Differential Equations: Theory, Computation & Applications. (NPDE-TCA 2016). The first problem was posed by one of the industries in Vadodara. And the second problem was posed by Mr. Amit Patel from Mechanical Engineering department of M. S. University of Baroda.

2. Case Study 1: Determining relationship between the quality parameters of the Bundy tube and wastage:

Bundy tube, is a double wall brazed steel tube which is constructed from copper coated steel strip which is rolled twice around laterally, then brazed in a furnace to produce a tube of double wall structure, with a clear, scale free coppered bore, a plated external surface and a consistently uniform wall thickness.

Manufacturing of the tube:

Raw steel given by the supplier is coated with copper on both the sides in the Electroplating division of the plant. Making of tube is divided into 3 stages as follows:

Stage-1: Making of un-brazed Tubes:

In stage-1, the un-brazed tube is made by the 7-mill process. These 7-mill processes are as follows: