Advances in Materials Science and Engineering

Volume 2017, Article ID 2435079, 8 pages

https://doi.org/10.1155/2017/2435079

## Modeling the Influence of the Penetration Channel’s Shape on Plasma Parameters When Handling Highly Concentrated Energy Sources

^{1}Department of Welding, Metrology, and Materials Technology, Perm National Research Polytechnic University, Komsomolskiy Prospekt, No. 29, Perm 614099, Russia^{2}Department of Automation and Telematics, Perm National Research Polytechnic University, Komsomolskiy Prospekt, No. 29, Perm 614099, Russia^{3}Department of Mechanical Engineering, Rapid Manufacturing Laboratory, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

Correspondence should be addressed to Dmitriy N. Trushnikov; ur.xednay@rtimidrt

Received 2 July 2017; Accepted 24 August 2017; Published 12 October 2017

Academic Editor: Michael Aizenshtein

Copyright © 2017 Dmitriy N. Trushnikov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In our work to formulate a scientific justification for process control methods when processing materials using concentrated energy sources, we develop a model that can calculate plasma parameters and the magnitude of the secondary waveform of a current from a non-self-sustained discharge in plasma as a function of the geometry of the penetration channel, thermal fields, and the beam’s position within the penetration channel. We present the method and a numeric implementation whose first stage involves the use of a two-dimensional model to calculate the statistical probability of the secondary electrons’ passage through the penetration channel as a function of the interaction zone’s depth. Then, the discovered relationship is used to numerically calculate how the secondary current changes as a distributed beam moves along a three-dimensional penetration channel. We demonstrate that during oscillating electron beam welding the waveform has the greatest magnitude during interaction with the upper areas of the penetration channel and diminishes with increasing penetration channel depth in a way that depends on the penetration channel’s shape. When the surface of the penetration channel is approximated with a Gaussian function, the waveform decreases nearly exponentially.

#### 1. Introduction

An important problem in the development of flawless electron beam and laser welding technologies is ensuring that high-quality welds can be consistently reproduced. Work to automate the process using secondary waveform parameters has been pursued for a long time with varied success. Certain successes have been achieved using methods involving registration of the secondary X-ray radiation emitted during welding with highly concentrated energy sources. However, these methods have limitations due to the difficulty placing additional sensors within the vacuum chamber, which creates complications when used in industrial conditions. Moreover, a number of studies indicate that electron beam welding is accompanied by high-frequency processes [1–4] that often carry the most information about the thermal characteristics of the electron beam’s interaction with the metal in the penetration channel. Frequencies up to 20 kHz are inherent to these processes, which makes it possible to study them using X-ray radiation sensors. Methods based on registration of the secondary plasma current originating above the welding zone [5] are promising. Several methods have been developed to study and control electron beam welding [1–5] processes. Similar works in the area of laser welding have been widely conducted recently. However, we must note that the problem of controlling penetration during electron beam welding using the parameters of the secondary plasma current cannot yet be considered fully solved. There are no models that describe the relationship between the parameters of the secondary waveforms in the plasma. To date, there are no universal, reliable, and sufficiently responsive process control systems for making a welded joint with electron beam welding. The accumulated knowledge is frequently tentative for now.

References [1–3] describe the results of experimental studies on generating secondary current waveforms in plasma during EBW. Several hypotheses have been put forward to describe the observed phenomena. However, the proposed theories are insufficient to fully interpret the obtained results [4, 5]. We need a detailed model of the processes occurring in the plasma. There are well-known works that estimate plasma parameters using simplified methods [6–10]. The authors of these works propose one-dimensional models that can be used to estimate the concentration and energy of electrons in the plasma above the EBW zone, but they do not make it possible to calculate the parameters of the non-self-sustained discharge. References [11–13] describe a two-dimensional model of the formation of the non-self-sustaining discharge in plasma during EBW. However, for practical purposes, [14] is of interest in determining how the waveform’s magnitude changes during a change in the position of the beam in the penetration channel and as a function of the channel’s geometric dimensions. Direct modeling requires a three-dimensional formulation, which entails unjustifiably large computational costs. This article proposes a method and numeric implementation based on a preliminary solution to the problem in a two-dimensional axisymmetric formulation with subsequent use of the results of this solution for a full description. The results obtained from this solution for measuring secondary electric current in plasma during EBW may be extended to laser welding.

#### 2. Description of the Model

At the heart of the model are transfer equations for the concentration of electrons and the average electron energy in the plasma over the EBW zone [14–17]: where is the electron stream density, is the volumetric energy density of the electrons, , are the electrons’ diffusion coefficient and mobility coefficient for the energy,* R*_{e} is the intensity of the electron source, is the electrons’ energy source (describes the energy loss due to inelastic collisions), and is the electric field vector.

To describe the mass transfer of heavy plasma particles (ions, neutral unexcited and excited atoms), we use the mass transfer equation for a multicomponent mixture [18–20]:where expresses the density of the mass stream of the th component; is the mass concentration of the th component; is the intensity of the source of the th component; is the average velocity vector; is the mixture density.

The electric field is determined from the Poisson equation

The plasma becomes collisionless above the penetration channel. When this happens, the diffusion coefficients in (1) go to zero and the equations degenerate into continuity equations for charge and energy.

A full system of equations and description is given in [14].

Calculations are performed in cylindrical coordinates in an axisymmetric formulation (Figure 1).