The attachment length in orificed hollow cathodes

A first-principles approach to obtain the attachment length within a hollow cathode with a constrictive orifice, and its scaling with internal cathode pressure, is developed. This parameter, defined herein as the plasma density decay length scale upstream of (away from) the cathode orifice, is critical because it controls the utilization of the hollow cathode insert and influences cathode life. A two-dimensional framework is developed from the ambipolar diffusion equation for the insert-region plasma. A closed-form solution for the plasma density is obtained using standard partial differential equation techniques by applying an approximate boundary condition at the cathode orifice plane. This approach also yields the attachment length and electron temperature without reliance on measured plasma property data or complex computational models. The predicted plasma density profile is validated against measurements from the NSTAR discharge cathode, and calculated electron temperatures and attachment lengths agree with published values. Nondimensionalization of the governing equations reveals that the solution depends almost exclusively on the neutral pressure-diameter product in the insert plasma region. Evaluation of analytical results over a wide range of input parameters yields scaling relations for the variation of the attachment length and electron temperature with the pressure-diameter product. For the range of orifice-to-insert diameter ratio studied, the influence of orifice size is shown to be small except through its effect on insert pressure, and the attachment length is shown to be proportional to the insert inner radius, suggesting high-pressure cathodes should be constructed with larger-diameter inserts.

Induction Heating of the Filling Conveyor Molds

In the smelting and foundry production of aluminum ingots, filling conveyors are widely used. Aluminum ingots of a certain shape and weight are obtained by crystallizing liquid aluminum (melt) in the molds of the filling conveyor. As the mills move along the conveyor, the melt gradually hardens in them. In high-performance conveyors, the mills move through the water to increase the cooling rate of the melt. Therefore, after the mill is freed from the hardened ingot, water enters it. In order to avoid temperature shock and possible release of liquid metal, the molds must be dried and heated before pouring. At present, gas burners are used in aluminum plants for this purpose [1]. The purpose of this work is to study the possibility of induction heating of the filling conveyor molds. The calculation is carried out using Fourier series in complex form and approximate boundary conditions on the surface of ferromagnetic molds. The approximate boundary conditions avoid the need to calculate the electromagnetic field in a nonlinear ferromagnetic medium. In the heated object, the energy of the induced alternating electric field is irreversibly converted into thermal energy. This dissipation of thermal energy, which leads to the heating of the object, is determined by the presence of conduction currents (eddy currents). Induction heating is widely used in metallurgy for melting, heating and mixing of electrically conductive bodies. The method is based on the absorption of electromagnetic energy by bodies of an alternating magnetic field created by an inductor. The heated product is located in the immediate vicinity of the inductor. There are many publications on analytical and numerical, analysis of physical processes in the inductor-heated billet system. In this paper, an analytical calculation of electromagnetic processes in the system of inductor – ferromagnetic molds of the filling conveyor is carried out. The analytical solution is obtained by using the approximate boundary condition of L. R. Neumann on the surface of nonlinear ferromagnetic molds

Free‐boundary conditions of arbitrary polygonal topography in a two‐dimensional explicit elastic finite‐difference scheme

We present a simple method for simulating 2-D elastic waves in a model with free‐surface topography of polygonal shape, i.e., a continuous but irregular surface composed of line segments. Our method requires special treatment for each of the six specific cases involving line segments of various slopes as well as transition points between the sloping segments. For brevity, only nonnegatively sloping segments are specifically included. On an inclined free surface, vanishing stress conditions are implemented using a rotated coordinate system parallel to the inclined boundary. At transition points on the topography between line segments, we use a first‐order approximate boundary condition in a locally rotated coordinate system aligned with the bisector of the corner. As in the classical one‐sided explicit approximation scheme widely used for the flat free‐surface case, these extrapolation formulas are accurate to first order in spatial increment. Numerical tests indicate that the present scheme is stable over a range of Poisson’s ratios of practical interest (v > 0.3) for fairly complicated geometric shapes consisting of ridges and valleys with both steep and gentle slopes. Stability for complicated shapes enables us to study realistic problems for which the topography plays a significant role in shaping the wave field and for which analytical solutions are not generally available.

Free Vibrations of a Circular Plate

The Free and damped vibrations of a circular plate with given initial conditions, which is either free or elastically restrained along its outer edge, are discussed using the method developed by Reid and some numerical results are also given.The vibration of a circular plate has been studied considerably and most of the authors seem to be interested in determining the frequencies of vibration and the nodal patterns. A few investigators have studied the problem of determining the subsequent motion of a thin plate started with some given initial displacement or velocity. Sneddon has attempted this problem, taking into account an approximate boundary condition. Recently Reid has discussed this problem without taking recourse to any approximate boundary condition. He has developed a method of analysis to determine the displacement of a freely vibrating thin circular plate with the given initial conditions, using the norm of the eigen functions.