During the week a series of talks was given by OCIAM members. On Tuesday afternoon Kevin Parrott gave a lecture entitled ``Particles, packages and parallelism''. John Ockendon gave a lecture on Wednesday, ``Mathematics-in-industry into the millennium'', and on Thursday Robert Leese talked about ``Applications of discrete methods''. On Monday evening the Tryfan Rogers Memorial Reception, held in the Reading Room of Somerville College, was very well attended.
The various problems tackled during the week are described below.
BT In crude terms, the problem is to determine the efficiency of a mobile phone antenna when it is placed in 1sq km of typical urban environment. More precisely, information was sought about the number N of rays that need to be considered at any particular point if the energy in the uncounted rays is at most 10\% of the total. Given a reflection coefficient of 1/2 on each ray bounce, and neglecting diffraction altogether, a study of some prototypical configurations suggested that N^(3N/2) needed to be less than some prescribed constant. It was also pointed out that in any street, say in a housing estate, there was probably only one dominant ray and that the identification of this ray could give much valuable information. Also several other representations of the solution of Helmholtz equation were proposed, ranging from ``hybrid'' schemes (which account for low frequencies in terms of discrete nodes and high frequencies in terms of rays) and ``diffusion'' approximations akin to those used in radiative transfer in optically thick media. Thus the problem suggested several new areas of applied mathematics as well as opening everyones eyes to the wonders of video conferencing.
The problems brought to the Study Group by Courtaulds Coatings
concerned the electrostatic deposition of powder paints onto an earthed metal
workpiece. The main questions considered were:
What factors affect deposition efficiency?
How is deposition efficiency maximised?
In general two types of ``gun'' are used in the painting process; the corona gun which charges particles by ionising the air in the vicinity of the gun nozzle and also the tribo gun which charges particles directly using friction inside the gun. As the corona gun involves many more complicated physical processes, only the tribo gun, with a single species of particle, was investigated at the meeting.
Non-dimensionalisation of the governing equations showed that electric and aerodynamic forces were in balance. The particular case of a narrow `jet' of particles impinging on an earthed workpiece was then analysed, showing that geometry is by far the most important factor affecting efficiency.
Pacific Northwest National Laboratories
Hydrogen gas is being produced by radioactive decay in large (10^3 m^3) nuclear waste storage tanks at the rate of 4 m^3/day. The waste/sludge has a yield stress of 10^3 Pa. The study group was asked to consider the mode in which the gas is released, whether as dangerous large eruptions or as acceptable small bubbles: there is danger if the concentration of gas in the roof of the tank exceeds 4\%.
The study group decided that bubbles larger than 10cm would overcome the yield stress and move. To examine how the bubbles might grow, an inflating crack model was constructed. Ideas of percolation through a network of cracks were abandoned when the originator of Percolation Theory told us that nothing was known (by which he meant proven), and that it would take 6 months to produce a computer estimate (probably more accurately than justified by the model). Looking at the problem as a very slow flow through a porous media, one rapidly concludes that the pores are nearly all sealed.
The main conclusions of the study group were that the gas should be released steadily, unless there is a seal or crust which traps the mobile 10 cm bubbles. Less happily, quite a large mass of gas is expected to be stored at the large hydrostatic (sludge-static?) pressure at the bottom of the tanks, so that it would be most unwise to stir the tanks.
The frequency of the national electricity grid is affected by fluctuations in supply and demand, and so continually ``judders'', in an essentially unpredictable fashion, around 50 Hz. At present such perturbations do not affect Nuclear Electric as their plant is run at more or less constant load, but they would like to be able to offer the national grid a mode of operation in which they ``followed'' the grid frequency: i.e., as the frequency rose above or fell below 50 Hz, the plant's output would be adjusted so as to tend to restore the frequency to 50 Hz. Such a mode of operation, however, would cause a certain amount of damage to plant components (e.g. the reactor fuel cladding) owing to the consequent continual changes in temperature and pressure. The problem was to devise a method which, given grid frequency data, could estimate this damage in reasonable computational time.
Two main approaches were considered: statistical prediction and analytical modelling via a low-order differential system. With the former approach it was difficult to reach concrete conclusions in the small amount of time available, but using ``typical'' data supplied by Nuclear Electric it was\/ possible to verify that a phase space predictor would in theory provide a feasible nonlinear statistical model. The analytical approach reached some very promising conclusions: a linear third-order system (apparently valid for all but the most rapid of grid changes) was obtained, and comparisons between the theoretical gain predicted by this model and the gain calculated from real data showed a remarkable degree of agreement. A good model of the behaviour of the grid was also derived based on Brownian motion.
The aim was to see how priorities employed by local authorities in allocating housing affected the numbers of households on registers for council housing. The study was motivated by prospective legislation of the Government which could mean councils giving less priority to homeless families in temporary accommodation. A system of differential equations was used to model the flow rates between different classes of population (e.g. the homeless, families content with private-sector accommodation, families in council property but on the register for more suitable homes). The rates between any categories were taken to be proportional to the population from which the flow originated, except that rates of rehousing were assumed to be jointly proportional to the number of spare council houses. Using the figures supplied (for a ``typical borough'') it was found that most category sizes were insensitive to the constant of proportionality controlling the rehousing of the homeless. The exception was the number of people in temporary accomodation which increased substantially as this constant was reduced towards those for the other rehousing rates (as the system was made ``fairer'').
DuPont wanted to understand the mechanisms for the formation and
evolution of defects in wet screen printed layers. Their
primary objective was to know
how best to alter the properties of the paste (rather than
the geometry of the screen printing process itself) in order
to eliminate the defects. With these goals in mind
the work done during the Study Group was as follows:
a simple model for the closure of craters,
a model for the partial closure of large, intentionally-created gaps in the paste layer,
a possible mechanism for the formation of pinholes which crucially involved the ``locking together'' of the relatively-large solid particles present in the paste
a more detailed consideration screen printing process, particularly attempting to understand the flow of the paste through the mesh.
Greycon are interested in pattern reduction in the one-dimensional stock cutting problem. They are concerned with cutting ``jumbo'' reels of paper into narrower ``customer'' reels, the quantities and widths of which are specified by the clients. They are satisfied with their current waste-minimisation program but given a minimum-waste solution they want to know if they can satisfy customer demand with the same minimum waste but with less settings of the paper slitting knives, or better still, the minimum number of settings of the knives.
We managed to show that there is no easy way to find the minimum number of knife settings for a given minimum-waste solution as the problem is NP-hard. However, we managed to construct an algorithm which was capable of detecting some possible reductions in the number of knife settings which the existing Greycon algorithm would not be able to detect. The algorithm was coded up and tested and managed to improve on Greycon's current algorithm by reducing the number of knife settings in some of the test problems that Greycon provided.
The first problem brought by Perkins was to calculate the temperature of diesel fuel as it is injected into the cylinder of a diesel engine. Fuel at a temperature of about 300 K arrives via a high-pressure line into a storage area in the injector; it has a residence time there of about 40 ms, before being injected into the cylinder in 1.5 ms. The quantity of fuel in one injection is about 85 cubic mm. The temperature inside the cylinder varies during the combustion cycle, but the temperature of the fuel is very hard to measure. Thermocouple readings in the wall of the injector give a temperature of about 550 K.
The heat transfer through the injector into the fuel was investigated. The major effects were conduction into the annular fuel storage region during the quiescent period and convection in the nozzle region during the injection period. Adiabatic cooling might be expected to cool the fuel slightly. An estimate of 10 K for the rise in temperature of the fuel was arrived at. More accurate temperature measurements in the body of the injector would help to make the figure more precise.
The second problem brought by Perkins Technology was to determine the feasibility of using a cyclone separator to remove small soot particles from exhaust emissions. By calculating the drift velocity of the particles relative to the air caused by the centrifugal force, we concluded that particles must have a minimum diameter of around one micron to be spun out by this kind of device. This is 1 to 2 orders of magnitude larger than the particles found in the exhaust.
We also investigated the mechanism for particle growth in the exhaust system. The dominant mechanism was found to be particle aggregation from Brownian motion. This predicts that the particles should grow to around 0.1 micron in diameter at the end of the exhaust system, in good agreement with experimental measurements of particle size.
Full reports will be sent out to all ``model'' participants, further copies are available on request. Finally, the organisers would like to thank all the participants and industrialists who came and made the week such an unqualified success. We are also grateful to the EPSRC and the LMS for their financial support and The Smith Institute for their assistance in organising the meeting.
May 10th Some problems from Schlumberger - P. Hammond (Schlumberger).
May 17th Surface tension driven flows - Dr T. Myers (OCIAM).
May 31st Paint Drying - Nick Hazel (BP Chemicals).
June 7th Foams - Prof. A.M. Kraynik (Sandia National Laboratories).
June 14th Filtering - Dr Agranovski (Brisbane).
Southampton University has an Industrial and Applied Maths seminar series advertised on the WWW at http://www.maths.soton.ac.uk/applied/ iam\_seminars.html.
ICCAM 96 The International Congress on Computational and Applied Mathematics will be held at the University of Leuven, from July 21-26, 1996. For information contact Prof. M. Goovaerts at email@example.com. 6651768.
Hermis 3 The Hellenic-European Conference on Mathematics and Informatics will be held in Athens from 26-28 September 1996. It incorporates a minisymposia entitled ``Industrial Mathematics in Europe- The ECMI Experience''. Contact: Prof. E.A. Lipitakis, fax + 1 8262204 or + 1 8676265, e-mail elip@iosun.ARIADNE-t.gr.
MISG 1997 The Australian Maths in Industry Study Group will be held at the University of Melbourne from 27th to 31st January 1997. Contact Dr Kerry Landman, firstname.lastname@example.org.
A complex ray approach to the acoustics of fluid-loaded structures Bridget Smith (1996), supervised by Richard Tew.
Mathematical analysis of novel magnetic recording heads. H.A.Shute (1996), supervised by Dr D.T. Wilton.
The next newsletter will be circulated by the AIMS Co-ordinator, Department of Mathematics, Strathclyde University. January 96 newsletter
October 95 newsletter
May 95 newsletter
January 95 newsletter