The collision branching process

We consider a branching model, which we call the collision branching process (CBP), that accounts for the effect of collisions, or interactions, between particles or individuals. We establish that there is a unique CBP, and derive necessary and sufficient conditions for it to be nonexplosive. We review results on extinction probabilities, and obtain explicit expressions for the probability of explosion and the expected hitting times. The upwardly skip-free case is studied in some detail.

Maximising heat transfer efficiency in the cold crucible induction melting process

Induction heating is an efficient method used to melt electrically conductive materials, particularly if melting takes place in a ceramic crucible. This form of melting is particularly good for alloys, as electromagnetic forces set up by the induction coil lead to vigorous stirring of the melt ensuring homogeneity and uniformity in temperature. However, for certain reactive alloys, or where high purity is required, ceramic crucibles cannot be used, but a water-cooled segmented copper crucible is employed instead. Water cooling prevents meltdown or distortion of the metal wall, but much of the energy goes into the coolant. To reduce this loss, the electromagnetic force generated by the coil is used to push the melt away from the walls and so minimise contact with water-cooled surfaces. Even then, heat is lost through the crucible base where contact is inevitable. In a collaborative programme between Greenwich and Birmingham Universities, computer modelling has been used in conjunction with experiments to improve the superheat attainable in the melt for a,number of alloys, especially for y-TiAl intermetallics to cast aeroengine turbine blades. The model solves the discretised form of the turbulent Navier-Stokes, thermal energy conservation and Maxwell equations using a Spectral Collocation technique. The time-varying melt envelope is followed explicitly during the computation using an adaptive mesh. This paper briefly describes the mathematical model used to represent the interaction between the magnetic field, fluid flow, heat transfer and change of phase in the crucible and identifies the proportions of energy used in the melt, lost in the crucible base and in the crucible walls. The role of turbulence is highlighted as important in controlling heat losses and turbulence damping is introduced as a means of improving superheat. Model validation is against experimental results and shows good agreement with measured temperatures and energy losses in the cooling fluid throughout the melting cycle.

A study of modelling approaches for rail vehicle collision behaviour

A rigid wall model has been used widely in the numerical simulation of rail vehicle impacts. Finite element impact modelling of rail vehicles is generally based on a half-width and full-length or half-length structure, depending on the symmetry. The structure and components of rail vehicles are normally designed to cope with proof loading to ensure adequate ride performance. In this paper, the authors present a study of a rail vehicle with driving cab focused on improving the modelling approach and exploring the intrinsic structural weaknesses to enhance its crashworthiness. The underpinning research used finite element analysis and compared the behaviour of the rail vehicle in different impact scenarios. It was found that the simulation of a rigid wall impact can mask structural weaknesses; that even a completely symmetrical impact may lead to an asymmetrical result; that downward bending is an intrinsic weakness of conventional rail vehicles and that a rigid part of the vehicle structure, such as the body bolster, may cause uncoordinated deformation and shear fracture between the vehicle sections. These findings have significance for impact simulation, the full-scale testing of rail vehicles and rail vehicle design in general.