Preface

In the following case study, the slow viscous flow of blocks of “carbon paste” is analysed. The paste blocks are essential components of an electric smelting process by which a variety of ferro-alloys and other substances are produced. The problem is first proposed in its most general form. A nondimensionalisation using typical parameter values of the process then shows that a much simpler set of equations may be used to analyse the flow. After examining the qualitative details of the fluid motion in various key regions of the flow, an asymptotic analysis of a long thin block of paste allows us to develop a good general understanding of the main principles of slow viscous flow in paste blocks. The theory highlights the key differences between various tests that are used to determine the viscosity of carbon paste. Finally, some fairly straightforward numerical analysis is used to show that the general problem is amenable to simple methods; the numerical results also show that in many cases the “long thin” analysis used earlier can produce remarkably accurate results.

The work outlined below is part of research that was produced during and after the 1988 European Study Group with Industry, which was held at the University of Heriot-Watt, Scotland. Some extensions to the work presented here are suggested as projects for the interested student in section 2.6. Close links have continued to be maintained between industrial mathematicians and the ELKEM ASA, the Norwegian company that originally proposed the problem. A number of other problems (see [1], [4], and [3]) concerning various aspects of the electric smelting industry have also been considered in detail and have led to some interesting mathematical problems.