Modelling of the transient diffusion of current and heat in railguns via an equivalent network

The coupled problem of the three-dimensional diffusion of current and heat in railguns with solid armature is formulated by means of a circuital approach. Rails and projectile are divided in a high number of small-size parallelepipeds, so that temperature and current density may be imagined uniform in every parallelepiped. Integrating the Ohm's law over each parallelepiped an equivalent R-L electrical network is obtained, whose resistive and inductive parameters are evaluated by means of explicit analytical expressions, possible in the case of discretization in parallelepipeds. The use of current filamentary parallelepipeds in the zone of the rails where only current directed in the sense of motion may be hyphotized, avoid to remesh in time the grid. The voltage fall in the boundary layers between rail and moving armature are taken into account by means of suitable driven generators. The integration of the Fourier's law over each parallelepiped leads to an R-C equivalent thermal network, whose external nodes are suitably connected to take into account the heat exchange by radiation and convection. The position of the projectile is deduced by the application of the virtual displacement principle, where the discretization in parallelepipeds allows to express the Lorentz's force again by means of explicit analytical relations. The non linear differential equations system so obtained fully simulates the behaviour of a solid armature railgun, and is solved in time, updating at each time-step the coefficients variable with current density, temperature, and projectile displacement. The proposed model has demonstrated good capability to simulate experimental railguns shots, as long as the frictional heat is negligible in comparison to the Joule heat and did not occur melting and arcing.