Tribological Journal BULTRIB Vol. 6, 2016 Papers from the International Conference BULTRIB '16 27-29 October 2016, Sofia, Bulgaria Society of Bulgarian Tribologists FIT – Technical University of Sofia
MODELING OF THE PROCESSES IN THE CATHODE REGION OF THE ELECTRIC ARC DURING TIG WELDING eng. Alexander GECHEV Technical university – Sofia, Faculty of industrial technology, Bulgaria Abstract: This work presents a mathematical model of the cathode region of the electric arc during TIG welding. The processes in the cathode region and the arc column are considered. The electric arc is sustained by electron emission from a tungsten cathode. The shielding gas is argon represented by a mechanism consisting of 3 species (electrons, ions and atoms). The solution is timedependent, calculated in a Cartesian coordinate system. In the present work solutions for the electrical potential, space charge and densities of electron, ion current and the particles constituting the model are presented. Key Words: mathematical modeling, arc welding, TIG, electrical processes, thermal processes.
1. INTRODUCTION The electric arc is known for a long time and over the years finds application in various manufacturing processes such as welding, cutting and melting of metals. The behavior of the electric arc directly affects the quality of the production so it’s very important to study the processes characterizing the arc and gain in depth knowledge about its behavior as a function of the operating conditions. The current work focuses on modeling the processes defining the electric arc column and cathode region during TIG welding. The electrons are emitted from the cathode surface through thermionic emission from a small, high temperature zone known as the cathode spot. There the electrons acquire enough energy to start the process of ionization and the concentration of electrons increases. The ions (as a product of ionization) are drawn to the negatively charged cathode surface and a positively charged layer is created called a space charge layer. The modeling of electric arc welding is a complex task because of the various physical processes taking part in different areas of the arc. Therefore the electric arc is usually divided into subregions (fig. 1), in which the various physical processes are dominating. The literature addressing the modeling of the electric arc can be divided into two different groups. The first group focuses on describing the processes in the cathode region by a shielding fluid consisting of electrons, ions and atoms, and the cathode region is divided into two layers (space charge layer and an ionization layer). Lowke [1], Hsu and Pfender [2], Zhou and Heberlein [3, 4] and Benilov and Marotta [5] contributed in the development of modeling the cathode region of the electric arc. Lowke’s theory [1] treats the ionization layer as a one temperature region and does not take into account the space charge. The calculation of the electron density distribution in the cathode vicinity allows the electric conductivity to be calculated. As a result the current flow between the cathode and the plasma is ensured in a self consistent way. The main drawback of this model is the absence of a space charge layer which Fig. 1. Schematic representation of the leads to untrue values for the current density. subregions in the arc
Zhou and Heberlein [3, 4] don’t consider the ionization layer. However, they calculate the plasma composition at the space charge layer/ionization layer interface due to a two-temperature composition model. Two other important features of this model are to include the flow of electrons moving back to the cathode and to consider a heavy particle temperature at the space charge layer/ionization layer interface equal to the cathode surface temperature. The main drawback of this model is that it requires setting several parameters like cathode voltage drop, cathode spot radius and electron temperature. Benilov and Marotta [5] describe the processes in the space charge and ionization layer. They implement the heat fluxes of the particle flows thus calculating the current density. The electric field at the cathode surface is used to calculate the Schottky correction in the Richardson emission formula to account for the enhancement of the thermionic emission. A drawback of this model is that it does not consider secondary emission. The second group of models describes the shielding fluid as a continuum with defined properties, thus the temperature of the species composing the fluid is the same. This type of models is often used because it’s simple and doesn’t require a two-temperature model of the arc, but it’s not valid for the cathode region. Therefore the equations defining the processes in the cathode region are added as boundary conditions. 2. MODEL In this part of the current work a model of the electric arc column and cathode region during TIG welding is presented. Electrons are emitted from the cathode surface through thermionic emission. The current density due to thermionic emission is calculated by the Richardson-Dushman equation [6].
𝑗ТЕ = 𝐴𝑇 2 𝑒𝑥𝑝 −
ФС КВ Т
(1)
where KB is the Boltzman constant; Т – temperature of the cathode surface; 2 2 A – quantum coefficient, for W it’s 0.7 [A/(mm .K )]; ФС – effective work function. After a collision with an ion the cathode surface may emit a secondary electron with a fixed probability. The current density due to secondary emission of electrons is calculated by the equation:
𝐽𝑠𝑒𝑚 = 𝛾𝑖 𝐽𝑖
(2)
where ϒi is a reflection coefficient; JTE – electron current density. The emitted electrons are accelerated by the strong electric field near the cathode. There they acquire enough energy to start ionization. Electrons that possess lower energy are attracted back to the cathode. The result is a rapid increase in the concentration of electrons near the cathode in an area which is called a cathode region. These processes are summarized by the boundary condition for the flow of electrons:
−𝑛 ∙ Г𝑒 =
1 𝜈 𝑛 − 2 𝑒,𝑡ℎ 𝑒
and electron energy [7] 5 −𝑛 ∙ Г𝜀 = 𝜈𝑒,𝑡ℎ 𝑛𝜀 − 6
𝛾𝑖 Г𝑝 ∙ 𝑛
(3)
𝜀𝑝 𝛾𝑖 Г𝑝 ∙ 𝑛
(4)
𝑝
𝑝
where εp is the mean energy of the electrons; νе, th – thermal velocity; ne – electron density; Гр – ion flux; nε – electron energy density;
The electron flux vector is calculated by:
Ге = − 𝜇𝑒 ∙ 𝐸 𝑛𝑒 − 𝐷𝑒 ∙ ∇𝑛𝑒
(5)
where µе is the electron mobility; Е – electric field; De – electron diffusivity; and the electron energy density:
Г𝜀 = − 𝜇𝜀 ∙ 𝐸 𝑛𝜀 − ∇ 𝐷𝜀 𝑛𝜀 where µε is the electron energy mobility; Dε – electron energy diffusivity; The reactions at the cathode surface are shown in table 1, and the reactions that describe the processes in the arc column are shown in table 2.
№ 1 2 3 4 5 6 7
(6)
№
Table 1. List of boundary reactions Reaction Coefficient
1 2
𝐴𝑟𝑠 → 𝐴𝑟 𝐴𝑟 + → 𝐴𝑟
Table 2. List of the reactions in the plasma Reaction Type Elastic reaction 𝑒 + 𝐴𝑟 → 𝑒 + 𝐴𝑟 Excitation 𝑒 + 𝐴𝑟 → 𝑒 + 𝐴𝑟𝑠 Superelastic reaction 𝑒 + 𝐴𝑟𝑠 → 𝑒 + 𝐴𝑟 + Ionization 𝑒 + 𝐴𝑟 → 2𝑒 + 𝐴𝑟 + Ionization 𝑒 + 𝐴𝑟𝑠 → 2𝑒 + 𝐴𝑟 + Ionization 𝐴𝑟𝑠 + 𝐴𝑟𝑠 → 𝑒 + 𝐴𝑟 + 𝐴𝑟 Returning to normal energy state 𝐴𝑟𝑠 + 𝐴𝑟 → 𝐴𝑟 + 𝐴𝑟
1 1
∆ε(еV) 0 11.5 -11.5 15.8 4.24
The electron density and the mean electron energy are calculated by the equations [7]:
𝜕 𝑛 + ∇ ∙ −𝑛𝑒 𝜇𝑒 ∙ 𝐸 − 𝐷𝑒 ∙ ∇𝑛𝑒 = 𝑅𝑒 𝜕𝑡 𝑒 𝜕 𝑛 + ∇ ∙ −𝑛𝜀 𝜇𝑒 ∙ 𝐸 − 𝐷𝜀 ∙ ∇𝑛𝜀 + 𝐸 ∙ Г𝑒 = 𝑅𝜀 𝜕𝑡 𝜀
(7)
(8)
where Re is an electron source; Rε – energy loss; The electron source and the loss of energy due to inelastic collisions are defined by the equations: 𝑀
𝑅𝑒 =
𝑥𝑗 𝛼𝑗 𝑁𝑛 Ге
(9)
𝑗 =1 𝑃
𝑅𝜀 =
𝑥𝑗 𝛼𝑗 𝑁𝑛 Ге ∆𝜀𝑗 𝑗 =1
where xj is the mole fraction of the target species; αj – the Townsend coefficient for the reaction; Nn – the electron number density; ∆εj – loss of energy due to the reaction;
(10)
The electron diffusivity, energy mobility and energy diffusivity are computed from the electron mobility using:
𝐷𝑒 = 𝜇𝑒 𝑇𝑒 5 𝜇𝜀 = 𝜇 3 𝑒 𝐷𝜀 = 𝜇𝜀 𝑇𝑒
(11) (12) (13)
The electrostatic field is computed using the following equation:
−∇ ∙ 𝜀0 𝜀𝑟 ∇𝑉 = 𝜌
(14)
where ε0 is the electric constant; εr – relative permittivity; The space charge density ρ is automatically computed based on the plasma chemistry specified in the model using the formula [7]: 𝑁
𝜌=𝑞
𝑍𝑘 𝑛𝑘 − 𝑛𝑒
(15)
𝑘=1
where q electron charge; Zk – degree of ionization; nk – ion number density; 3. RESULTS In this part of the article results for the characteristics of the electric arc as a function of the current density being used during TIG welding are shown. Figure 2 illustrates the mean electron energy as a function of the electron current density and on figure 3 the ionization reaction rates can be seen.
Fig. 2. Mean electron energy as a function of the current density
Fig. 3. Ionization reaction rates as a function of the mean electron energy Usually authors define the thickness of the cathode layer as the sum of the space charge layer and the ionization layer, but in this model the processes of ionization occurs mostly in the arc column when using larger current densities. Therefore the thickness of the cathode region is defined as the thickness of the space charge layer. As shown in fig. 4, the thickness of the cathode region depends 2 2 on the current density, which is varied from 2.6 [A/mm ] to 15.6 [A/mm ].
Fig. 4. Cathode region thickness Figure 5 illustrates the electric potential drop in the cathode region. With the electron current density rising, the electric potential is decreasing because the larger number of electrons increases the electrical conductivity.
Fig. 5. Electric potential in the cathode region as a function of the electron current density 4. CONCLUSION In the current work a model of the cathode region and arc column during TIG welding is presented. The model is able to provide information about the space charge density, electric potential and the ionization processes as a function of the electron current density. The results obtained from the model are in agreement with literature data in this area. After analyzing the results the following conclusions can be made: Modeling the arc column is necessary when researching the cathode layer, because the space charge density is sustained mainly by the ionization occurring in the arc column. At low current densities the mean electron energy of the emitted electrons is enough to initiate direct ionization (a reaction between an electron and an argon atom). At higher current densities the mean electron energy decreases so the ionization is realized in two stages (a reaction between an electron and an excited atom). 2 The thickness of the space charge layer depends on the current density. At 2,6 [A/mm ] the -5 -5 2 space charge is 4,64.10 [m] thick, and 6,85.10 [m] thick at 15,6 [A/mm ]. ACKNOWLEDGMENTS The current work is created by the support of contract № 152ПД0025-05 financed by the research sector of Technical University – Sofia. REFERENCES 1. Lowke, J J, P Kovitya, and H P Schmidt. "Theory of free-burning arc columns including the influence of the cathode." Journal of Physics D: Applied Physics (Journal of Physics D:Applied Physics), 1992. 2. Hsu, K C, K Etemadi, and E Phender. "Analysis of the cathode region of a free-burning, high intensity argon arc." J. Appl. Phys 54 (Journal of Applied Physics), 1983: 1293-1301. 3. Zhou, X, and J Heberlein. "Analysis of the arc-cathode interaction of freeburning arcs." Plasma Sources Science and Technology, 1999. 4. Zhou, X, J Heberlein, and E Pfender. "Theoretical study of factors influencing arc erosion of cathode." Components, Packaging, and Manufacturing Technology, Part A, IEEE Transactions on, 1994: 107–112. 5. Benilov, M S, and A Marotta. "A model of the cathode region of atmospheric pressure arcs." Journal of Physics D: Applied Physics, 1995: 1869. 6. Тонгов, М. "Заваряване, част първа - процеси." 2009. 7. Hagelaar, G J, and L C Pitchford. "Solving the Boltzmann equation to obtain electron transport coefficients and rate coefficients for fluid models." Plasma Sources Science and Technology, 2005.
CORRESPONDENCE Alexander Gechev Technical University - Sofia, Bulgaria e-mail:
[email protected]
