Critical to the success of the circuit breaker is that it is designed to cause the arc to move away from the contacts, into a widening wedge-shaped region. This lengthens the arc, and then moves it onto a series of separator plates called an arc divider or splitter.

The arc divider raises the voltage required to sustain the arcs across it, above the voltage that is provided across the breaker, so that the circuit is broken and the arcing dies away. This entire process occurs in milliseconds, and is usually associated with a sound like an explosion and a bright ash from the arc. Parts of the contacts and the arc divider may melt and/or vapourise.

The question to be addressed by the Study Group was to mathematically model the arc motion and extinction, with the overall aim of an improved understanding that would help the design of a better circuit breaker.

Further discussion indicated that two key mechanisms are believed to contribute to the movement of the arc away from the contacts, one being self-magnetism (where the magnetic field associated with the arc and surrounding circuitry acts to push it towards the arc

divider), and the other being air flow (where expansion of air combined with the design of the chamber enclosing the arc causes gas flow towards the arc divider).

Further discussion also indicated that a key aspect of circuit breaker design was that it is desirable to have as fast a quenching of the arc as possible, that is, the faster the circuit breaker can act to stop current flow, the better. The relative importance of magnetic and air pressure effects on quenching speed is of central interest to circuit design.

VR Technology is a leading supplier of technical dive computers. The company is interested in expanding upon an existing algorithm (the Variable Gradient Model - VGM), which is used to design ascent profiles/decompression schedules and thereby mitigate the risk of decompression sickness in divers.

The Study Group took the approach of trying to extend the existing Haldane model to account more explicitly for the formation of bubbles. By extending the model to include bubble dynamics it was expected that some physical understanding could be gained for the existing modifications to some of the parameters. The modelling that occurred consisted of first looking at the Haldane model and then considering a single small isolated bubble in each of the compartments and interpreting the predictions of the model in terms of decompression profiles.

An optimal stopping problem that minimises the total jerk was formulated and solved. This model was extended by including a linear relationship between the brake pressure and the acceleration of the car where the coefficients are estimated by linear regression. Finally, a Kalman filter estimates the state of the car using the tone wheel.

The study group concluded that while ‘prediction’ of price in any meaningful sense was not viable, a model for scenario analysis could be realised. The model did not incorporate all of the factors of interest, but did model important time lags in the response of market players’ future behaviour to current oil prices. Consideration of the optimisation of supply through new capacity in the telecoms industry led to a generalisation of the standard Cournot-Nash equilibrium. This indicates how an output-constrained competitive market might operate. It enables identification of different pricing regimes determined by the level of competition and the resource limitations of particular supplier firms. Two models were developed sufficiently to enable simulation of various conditions and events. The first modelled oil price as a mean reverting Brownian motion process. Strategies and scenarios were included in the model and realistic simulations were produced. The second approach used stability analysis of an appropriate time-delayed differential equation. This enabled the identification of unstable conditions and the realisation of price oscillations which depended on the demand scenarios.

This report describes some initial models, two of which are developed in more detail: one for the propagation of torsional waves along the drill string and their reflection from contact points with the well bore; and one for the dynamic coupling between the underreamer and the drill bit during drilling.

In these circumstances, Dstl wish to have mathematical models that give an understanding of the process, and can be used to choose the parameters to give adequate removal of the contaminant. Mathematical models of this have been developed and analysed, and show results in broad agreement with the effects seen in experiments.