Mathematical Model of the Dynamics of Closure of Electrical Contacts in Vacuum

A mathematical model has been developed that describes the dynamics of thermal processes at closure of electrical contacts in vacuum. It describes all consecutive stages of the contact closure including contact touching, melting, evaporation of a contact micro-asperity and phase transformation in the contact zone of the main contact body. This model is based on a Stefan-type problem for a micro-asperity, which is considered as a body with a variable cross-section including three zones: solid, liquid and gaseous. The gas zone, due to its small thickness, is considered as a thermal resistance between the arc and the liquid zone with a linear temperature distribution. According to this model a touching micro-asperity of a contact explodes instantaneously due to the arc ignition with the power applied to the touching point, which can be described by the delta-function. The time of total vaporization of a micro-asperity calculated in this model can be considered as an upper estimate because touching of another micro-asperity may occur before this total vaporization. The other problems also consist in solving the Stefan problem, but in the spherical model of the contact zone introduced by R. Holm. The novelty of the formulation of these problems is that they have additional nonlinearity, taking into account the temperature dependence of the thermophysical and electrical parameters of the contact material. To solve these problems, a method based on the similarity theory has been developed, which allows us to reduce the Stefan problem under consideration to a nonlinear system of ordinary differential equations which can be solved using the Runge-Kutta method. An example of calculating the dynamics of evaporation of a micro-asperity is given, which makes it possible to estimate the error of the solution when taking into account the temperature dependence of the parameters of the problem.