The case study in this chapter involves finding the puddle length in a continuous casting operation. This involves calculating how fast molten steel solidifies. We introduce the Boltzmann similarity transformation as a way of solving the 1-D heat equation with a moving boundary (the boundary between molten and solidified steel, which changes with time). This technique reduces the PDE into an ordinary differential equation (ODE) and a parameter describing the moving boundary position is obtained as the solution of a transcendental equation.

Introduction to the case study problem

Of great interest in many industrial applications are those problems which involve a change of phase — from solid to liquid or liquid to solid, for example. These are also interesting mathematically because few exact solutions are known. Hence the ones for which analytic solutions are known give considerable insight into the physical processes involved. We develop a mathematical model to examine the feasibility of casting steel sheets by pouring molten metal onto a cooled rotating drum. This model involves the concept of a moving solidification boundary. The problem was brought to the 1985 Australian Mathematics in Industry Workshop by the Research Laboratories of BHP in Melbourne. It was reported in Barton (1985).

Background

A conventional method of producing steel sheets involves rolling steel billets down to the required thickness. This can be costly in both resources and time. Also it is difficult to produce very long sheets of steel by this method. One alternative is to pour molten steel onto a rotating drum which is cooled by water flowing through it (see Figure 2.1.1).

In this chapter we set the scene by introducing the case studies of the following chapters. We also introduce the main physical concepts for diffusion and heat conduction, and show how to formulate the main partial differential equations that describe these physical processes. Finally, dimensionless variables are introduced and it is shown how to scale differential equations and boundary conditions to make them dimensionless.

Heat and diffusion — A bird's eye view

Here we give a basic physical description of mass transport and heat transport by diffusion. This provides the physical ideas needed to formulate an appropriate differential equation, which is done in the next chapter.

Diffusion

Diffusion is a physical phenomenon involving the mixing of two different substances. Some examples include salt in water, carbon in steel and pollution in the atmosphere.

A fundamental quantity is the concentration of one substance in another. This may be defined in several diffierent ways. For example, the concentration can be measured as the ratio of the mass of one constituent to the total volume of the mixture (kilograms per litre). Another measure of concentration is the volume of one constituent to the total volume of the mixture.

Due to the random motion of constituent particles, concentrations tend to even out. Some molecules in a region of higher concentration move into a region of lower concentration. (See Figure 1.1.1).

Heat and temperature

An important thing to remember about modelling heat transport is that heat and temperature are not the same thing. Heat is a form of energy and may be measured in joules (the SI unit of energy).

At a pragmatic level, there are often no alternatives to mathematical models to test new industrial designs. Physical prototypes may be too expensive or too time consuming. In both the design phase and the operations phase, the direct measurement of important operational variables may be impossible or uneconomic. The regions of interest may be inaccessible because of mechanical barriers, high temperatures or hazardous chemical environments.

Mathematical modelling is an efficient and relatively inexpensive device for testing the effect of changing operating conditions in an industrial process. It is far easier and less costly to change a small number of parameters in a mathematical model than to shut down an industrial plant and modify its large-scale equipment. In most circumstances this should not be done as a trial-and-error experiment.

In this era of rapidly changing technology, the efficacy of mathematical modelling in industry should be appreciated more than ever before. Therefore, it is frustrating to note a recent trend towards reducing the amount of core mathematics subjects in engineering degree programmes. It is our expectation that this book could at least be used for an optional course for the more mathematically-oriented students of engineering. Such a course is even more important in the education of those with an interest in the logical design of new technology when the majority of graduates have insufficient mathematical training for this purpose. This course would meet one of the needs of a modern industrial mathematics course, namely that of relevance to real industrial problems.

In this final chapter we review the case studies examined in this book. We also explore (very briefly) some other areas of industry in which mathematics has been extensively used and mention some other mathematical techniques which commonly find application in industry.

Introduction

In the previous chapters we have explored several case studies from industry. All of these case studies have involved problems which use some variant of the diffusion equation. The case studies were deliberately chosen to use similar mathematics and physical backgrounds to make it easier for the reader and to allow the reader to see the links between the various case studies through mathematics.

In Chapter 2 we considered the problem of continuous casting. This problem introduced the mathematics of moving boundary problems and of similarity solutions using the Boltzmann similarity solution. The problem illustrated how a simplified model involving one dimensional heat flow yielded an exact solution in terms of error functions. Using this exact solution we were able to estimate the size of the puddle of molten steel, and showed that it was of the order of the size of the rotating drum, which meant that the process was not feasible.

Next, in Chapter 3, the case study was from the area of water filtration involving a process known as reverse osmosis. This was a diffusion problem with a non-constant advection coefficient. Here we continued with the idea of a similarity solution, and developed a technique (the method of stretching transformations) as a means for constructing similarity transformations which reduce the dimensionality of a partial differential equation.