Introduction

A bird scare gun is a relatively simple device which produces impulsive noise by means of a gas explosion. It is used for scaring birds away from areas where their presence is unwanted, like orchards, airfields, or oilfields. It is meant to operate automatically for long periods of time, with little or no human intervention. The construction is simple and robust, with as few as possible moving parts, so that a lifecycle in the order of 100,000–200,000 explosions is attainable.

The mechanism is simple. Acarefully controlled mixture of air and propane or butane gas (stoichiometric mixture, or a little bit richer than that) is periodically (every 5 or 10 minutes) blown into a semi-open pot, which is the combustion chamber. This pot is connected via a small diaphragm or iris (a small hole in the wall of the combustion pot) to an exhaust pipe. After ignition, the gas burns quickly (but without detonation, i.e. with a subsonically moving flame front) so that pressure and temperature increase quickly. This high pressure drives the gas out of the pot via a hot jet, which issues from the diaphragm into the pipe. Acting like a piston, this jet creates a pressure wave in the cold exhaust pipe. Part of the wave reflects at the exit, and part radiates, nearly spherically, away into the open air.

An interesting detail in the design gives, without any further analysis, insight in the gasdynamic behavior. In order to vary the noise that is produced, the length of the exhaust pipe is made variable. The pipe consists of two shorter pipes, one of which slides inside the other, like a telescope.

Preface

This case study concerns the flow of gas in a so-called scroll compressor. In this device a number of chambers of gas at different temperatures and pressures are separated by narrow channels through which leakage can occur. Using compressible lubrication theory, an estimate for the leakage rate is found in terms of the material properties of the gas and the geometry of the compressor. Thus a simple functional is obtained which allows the efficiency of different compressor designs to be compared. Next we derive a set of ordinary differential equations for the temperature and pressure in each chamber; the coupling between them arises from the leakage. The numerical solution of these equations allows a realistic simulation of a working compressor, and suggests some interesting possibilities for future designs.

This problem arose at the 32nd European Study Group with Industry held in September 1998 at the Technical University of Denmark – the first to be held outside the United Kingdom. It was presented by Stig Helmer Jørgensen from DANFOSS, which is Denmark's largest industrial group and specialises in controls for refrigeration and heating. The Danish Study Group was a great success and is expected to be repeated annually henceforth. The feedback from DANFOSS has also been encouraging, and hopefully this represents the start of a long-term collaboration.

Introduction

The scroll compressor is an ingenious machine used for compressing air or refrigerant, which was invented in 1905 by Creux [1]. Unfortunately, technology was insufficiently advanced at the time for workable models to be manufactured, and it was not until the 1970s that commercial interest in the idea was revived – see e.g. [3].

This is a book about real industrial problems from real industries. The industries and the problems come from all parts of the globe and cover a wide variety of areas, so do the contributors.

We had two specific reasons for producing this book.

First, we believe that applied mathematics is a multifaceted discipline. One of the facets that has emerged and assumed an increasingly important role in the last 30 years has been the need of industrial scientists to use and extend results found by academics. At the same time, academics have become increasingly aware of new phenomena that are of interest to industry. These cross-fertilisations can lead to new and interesting mathematics. This book illustrates the process of industrial/academic interaction through case studies.

Second, we believe that many classical applied mathematics texts treat applications merely as platforms upon which to build mathematical structures. One unfortunate result of this can be that the original requirements which inspired development of the mathematics become hidden. Recently, in an effort to interest students in applied mathematics, and to expose them to the totality of the applied mathematics process, the mathematical modelling process itself (which precedes analysis and numerical considerations) is being taught. This book provides a set of real examples which could be useful for a high-level modelling course.

Preface

The following case study considers how tension varies in space and time in a continuous sheet of paper (commonly called a web) running at high speed through a newspaper press. The model will consider two different parts of the problem in isolation. First, those portions of the paper that are travelling over various rollers are treated, and second, the spans of paper between the rollers are considered. Modelling the first problem requires basic mechanics and considers the forces within the paper and the friction acting against the roller. Practical limiting cases result in a simple ordinary differential equation for the tension as a function of position. When we consider the spans of paper, the model is again mechanical in nature but requires modelling of the stretch within the paper. For practical situations, the resulting model is again an ordinary differential equation which is nowtime-dependent. The two models are then combined to show how rollers interact and how the tension changes along the paper's length. The mathematics used is relatively simple; the emphasis is on generating a model which is as simple as possible, but which mimics the physical situation. The problem is well suited to mathematical modelling courses because of the wide variety of different aspects of a printing machine that can be considered, and also because the mathematical models that emerge from the modelling process are similar to classical mechanics problems.

The models presented here will require some knowledge of basic mechanics. Hooke's law for a linear elastic material, conservation of mass, and conservation of linear and angular momentum give us the evolution equations of the systems.

Preface

The following case study develops several models to help understand how to uniformly and accurately cook whole grains for the manufacture of breakfast cereals. During the cooking process, heat and moisture must enter the grain. These two transport processes are modelled by diffusion equations. Linear diffusion is appropriate for the heat transport. A nonlinear diffusion model is examined for the more complex process of water uptake in the grain. The times to heat and wet the grain can be estimated using both numerical and analytic approaches. The mean action time proves to be a very powerful way of estimating the speed of the wetting front. The degree of overcooking in the present manufacturing process can also be estimated. Finally, some recommendations for improving the cooking process are suggested.

The work outlined here was part of the analysis carried out at the 1996 Mathematics-in-Industry Study Group at the University of Melbourne, Australia. Some extensions of the work have been carried out by various groups ([3], [5]). Only the basic analysis is presented here; extensions are suggested as projects. The collaboration with the company is ongoing.

Introduction

Starch crops, such as cereals, pulses, and tubers, account for over half the total food produced and consumed in the world. Humans find it easiest to digest starch after it has been cooked. The application of both heat and moisture to starch causes it to gelatinise – this allows the starch to be able to be digested by enzymes.

The cooking of intact whole grains (e.g. wheat, corn, rice) is industrially important in breakfast cereal manufacture.

Preface

A physical and mathematical model is developed for the study of the vapor-diffusion process of protein crystal growth. This is a process widely used to grow high-quality crystals of proteins for the rapidly expanding fields of structural biology and drug design. In this process, an aqueous solution of protein is dynamically concentrated via a passive evaporation from a water drop to a reservoir. The kinetics of the process greatly influence the quality of the crystalline phase, albeit in an unknown manner and with the underlying physical aspects little understood. The model is solved analytically using the method of multiple timescales, identifying and exploiting the disparity in the timescales associated with the various transport mechanisms. Full nonlinear transient numerical simulations are also performed and compared with the analytical results and the data obtained from a benchmark experiment. The roles of the controlling parameters in the process are identified, and the requirements for experimental repeatability are explored, especially with regard to the temperature boundary conditions. Finally, it is proposed to use the verified analytical solution for process optimization to reduce the number of independent process variables under consideration.

Introduction

Structural biology is an emerging-late 20th century science, combining biology, biochemistry, computational science, applied mathematics, and other disciplines, to explicitly decipher the operational characteristics of biomolecules. Hereafter the term protein will be used to describe large conformationally flexible biomolecules that are made by the genetic sequence. Proteins are the building blocks of life, and are responsible for the thousands of individual biochemical actions in living things. Proteins are transcribed from the genetic material using amino acids that are combined via chemical bonds (peptide bonds) to form large molecules.

The Continuum Model

Most of the modelling introduced in the following chapters uses the continuum approach. In this introductory chapter, we list the commonly used equations: those describing diffusion, convection, radiation, and fluid and solid mechanics. We do not attempt to give an even partially rigorous derivation of any of these equations; our purpose is to provide a ready resource, and to indicate source books of wider scope. Above all, this section should be seen as answering the question “why did they start from those equations?”.

Let us approach the continuum model by considering the example of diffusion. Diffusion is a molecular process. Consider the diffusion of heat: the diffusion happens because a hot region of a material has molecules of higher energy than those in cooler parts. Energy equalisation therefore takes place by molecular interaction – and the heat is said to “diffuse”. To enable us to view this at the continuum level, local averages (for example, over many molecules) are taken: the molecular picture is smeared. The concept of heat as a variable having a value only at molecular sites is replaced by a framework in which heat is regarded as a variable that has continuous values. The laws governing changes in the continuous functions to describe heat transfer are treated “phenomenologically”. Models are created at both levels; their usefulness, success, and relevance depends on the application.

Conservation Laws

Physical phenomena expressed in the continuum paradigm are usually phrased in terms of conservation laws. Let us introduce this in a generic fashion: we consider a substance, A, distributed continuously.

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.

Preface

The Institute for Industrial Mathematics, which will become the first Fraunhofer Institute for applied mathematics in Germany, has focused its work on practical problems coming from industry. The following case study deals with a very old problem in the production of glass. It is part of a project of the ITWM with Schott Glas, the most important producer of specialized glasses in Germany. The famous physicist Josef von Fraunhofer faced the problem when trying to produce large lenses for astronomical binoculars – almost 200 years ago. During the cooling of the glass, thermal tensions tended to produce fractures, which might be avoided by a proper control of the cooling process. This can be achieved by regulation of the surrounding temperature. However, in order to control the process, it is necessary to understand and to predict the thermal tensions, i.e. the stresses set up by nonhomogeneous thermal contraction. This is complicated, because heat is transported through radiation. Glass is a semitransparent medium – each point in the glass is a new source of radiation. This property is modelled in the radiation equation – but this equation is not easy to solve. In contrast to other case studies in this volume, there is in general nothing flat or thin. The thermal tensions depend heavily on the three-dimensional object as a whole, and the object very often has no symmetries. Therefore, numerical methods are unavoidable, but still pose difficult problems: a straightforward solution of the three-dimensional radiative equation, say by ray tracing, discrete ordinates, or even P1-approximation does not work in real (i.e. real industrial) problems.

Preface

The following case study concerns the resonant oscillations of an elastic plate in a fluid. At first sight one might expect that any analysis would get difficult quite quickly, forcing numerical solutions. However, simple beam theory suffices for the motion of the plate since it is thin, and potential theory for the motion of the fluid suffices since little viscous shear is generated. Moreover, amplitudes are very small, linear theory works, and a very simple solution generates the form of the answer that the client was seeking. The numerical results for this solution do not fit data at all well, but re-examination of the device indicates that a change in the boundary condition is required. This modification then provides results in remarkable agreement with data. The latter instructs on the importance of correctly modelling boundary conditions.

What follows is part of the analysis generated in projects completed by Claremont Mathematics Clinic teams for ITT-Barton, a company manufacturing the densitometer. The basic analysis was completed in 1982–3; various extensions were analyzed in 1983–4, see [8], [4]. A formal presentation was prepared, [2]. Only the basic analysis is presented here; extensions are suggested as projects for the interested student in section 3.7.

Introduction

As its name implies the densitometer is a device which measures density. (The reader is challenged to design a procedure which measures density to high accuracy – here the tolerance is 0.1% error – to operate safely in a hostile environment for long periods.)

Introduction

The engineering uses of rubber have expanded well beyond traditional products such as tires and seals. Today elastomeric, or rubber-like, components can be found in a diverse set of constructs including engine mounts, building foundations, belts, and fenders (see [13], [22]). Increasingly, the applications of rubber are becoming more sophisticated, as exemplified by the use of rubber bearings in bridges which allow for thermal expansions of the deck without placing excessive loads on the bridge supports (see [15]).

In current engineering applications, elastomer composites are typically filled with inactive particles such as carbon black or silica. If active fillers were used, such as piezoelectric, magnetic, or conductive particles, the resulting controllable elastomer could be used in products such as active or smart vibration suppression devices (e.g. see [8], [17], [18]). As these new materials are developed, the role of design will increase in both complexity and importance. In particular, the capability to predict the dynamic mechanical response of the components will become increasingly valuable.

The many desirable characteristics of rubber as a design component, which include the ability to undergo large elastic deformations and provide significant damping with near incompressibility, are also contributing factors to the complications arising in the process of formulating models. Damping is highly complex, and the strain history, rate of loading, environmental temperature, and amount and type of filler affect the mechanical response in a nontrivial manner. Additionally, many elastomers exhibit strong hysteresis characteristics similar to those found in shape-memory alloys and piezoceramic actuators. Hysteretic effects in an elastomer include a time-history-dependent stress–strain constitutive law.

Introduction

Most of us know that a catalytic converter is a device that is located in our car somewhere between the engine and the tail pipe, and that its basic function is to convert harmful components of the engine exhaust, such as carbon monoxide, to less harmful ones, such as carbon dioxide. It is an important device in pollution mitigation especially in urban areas where there is a large concentration of cars. For most of us the story ends there. However, for a manufacturer of catalytic converters, such as Allied Signal, there is an obvious need to gain a deeper understanding of the behavior of catalytic converters. At the Fifth Workshop on Mathematical Problems in Industry held at RPI in May, 1989, a representative of the research group at Allied Signal presented various mathematical problems related to the dynamic behavior of catalytic converters. The industrial representative had a set of differential equations modelling the behavior of the gas concentrations and temperatures in the device but was finding it difficult to obtain numerical solutions of the equations, due, in part, to the stiffness of the chemical reaction terms which can create rapid changes in the solution. The main goal, then, was to study the set of the equations and, if possible, find reduced models and use analytical methods to predict the solution behavior and give insight into appropriate numerical techniques. Atypical automotive catalytic converter consists of an inert ceramic monolith with many narrow channels running the length of the converter, through which the exhaust gas from the engine flows (see figure 7.1).

In single-pellet studies, it is assumed that no concentration or temperature gradients are present in the fluid phase surrounding the pellet. A reactor with external or internal recycle is one of the experimental realizations of the single-pellet concept. However, in a fixed-bed reactor, the fluid-phase composition and temperature vary with position. For this reason, the optimization problem becomes more complex. Thus, it is not surprising that relatively few reactor studies have appeared in the literature as compared with those for single pellets. In this chapter, we first discuss theoretical studies of single and multiple reactions under isothermal and nonisothermal conditions, and then present experimental work which supports the theoretical developments.

A Single Reaction

Isothermal Conditions

One of the earliest works in this area considered CO oxidation in excess oxygen over platinum catalyst, in monolith reactors for automobile converters (Becker and Wei, 1977a). Recall that this reaction exhibits a maximum in the rate as a function of CO concentration. The CO conversion to CO2 as a function of the Thiele modulus (or equivalently, the catalyst temperature) for a fixed inlet CO concentration is shown in Figure 3.1 for four monoliths with different catalyst distributions (cf. Figure 2.1). The conversion of CO below 650°F was significantly higher in the inner catalyst, while above 650°F the middle catalyst performed better and attained 100% conversion. The uniform and outer catalysts also showed 100% conversion at progressively higher temperatures.

The optimization of the isothermal fixed-bed reactor, where a bimolecular Langmuir–Hinshelwood reaction occurs, was performed analytically by Morbidelli et al. (1986a,b).