Theory for the Arc Discharge Multiphysics Interface
Theory
Arc discharges are widely utilized across various industrial applications due to their unique properties and capabilities. These applications include cutting and welding, where the intense heat of the electric arc melts and fuses materials with high precision. In spraying processes, electric arcs are used to vaporize and deposit materials onto surfaces, enabling the creation of specialized coatings. Arc discharges are also employed in waste destruction technologies, where the high temperatures break down hazardous or complex materials into simpler, safer forms (Ref. 1).
While electric arcs have many beneficial uses, they are also a significant concern in electrical power systems. In components like circuit breakers, unintended arc formation can lead to equipment damage, power outages, and even safety hazards. Electric arcs in these contexts are undesirable because they can cause rapid degradation of materials, overheating, and potential failure of the entire electrical system.
Accurately simulating and predicting the dynamics of electric arcs is essential. Advanced simulations enable engineers to model the complex interactions that drive arc formation, propagation, and extinction, which in turn helps optimize industrial processes and improve the safety of electrical systems. The Arc Discharge interface is specifically designed to make the modeling of these complex arc discharges more predictable and reliable.
Fully ionized electric arcs are assumed to be under partial to complete local thermodynamic equilibrium (LTE) conditions (Ref. 1). At a macroscopic level, these kind of plasma can be considered as conductive fluid mixtures, which leads to the magnetohydrodynamics (MHD) equations. The latter combine the Navier–Stokes, heat, and Maxwell’s equations to describe the motion of the conducting fluid in an electromagnetic field.
The Arc Discharge interfaces are based on a set of assumptions that lead to simplifications of the MHD equations:
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The plasma is fully ionized (two-component mixture, that is, electrons and ions).
•
The plasma is under local thermodynamic equilibrium (LTE) conditions (the electron temperature Te is approximately equal to the ion temperature, Ti).
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The plasma is considered to be a locally neutral Newtonian fluid mixture.
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The displacement current is negligible (quasistatic approximation, that is, low frequencies).
•
The plasma is optically thin.
Domain equations
Considering the assumptions stipulated above, the Arc discharges can be described by a set of equations defined in the physics interfaces constituting the different multiphysics interfaces. Note that all the physics features available in the individual physics interfaces are also available to the multiphysics interface in use.
Energy Conservation (Heat Transfer in Fluids Interface and Arc Discharge Heat Source Multiphysics Coupling Feature)
The energy conservation equation is used for the Arc Discharge interface.
(3-4)
The source Q (W/m3) defined in the Equilibrium Discharge Heat Source multiphysics coupling feature includes three source/sink components:
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Resistive heating (ohmic heating)
(3-5)
The definition of the electric field is:
(3-6)
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Volumetric net radiation loss Qrad defined by the total volumetric emission coefficient, which is a material property from Ref. 1.
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Enthalpy transport (energy carried by the electric current)
(3-7)
Note that the enthalpy transport term prevails, for example, in the boundary layers close to electrodes in a fully ionized electric discharge, Ref. 1.
Momentum Conservation (Laminar Flow Interface)
The momentum conservation equation is expressed as:
(3-8)
where μ is the dynamic viscosity of the fluid, I is the identity matrix and
the Lorentz force acting on the fluid is defined in the Magnetohydrodynamics coupling feature.
Ampère’s Law and Current Conservation (Magnetic and Electric Fields Interface)
Ampère’s Law and the current conservation equations are used for the Arc Discharges interface:
(3-9)
(3-10)
where the total current density is expressed as
(3-11)
transport properties
The above equations require specification of material properties, which are functions of temperature. The Equilibrium Discharge folder in the Material Browser contains properties for density, specific heat, viscosity, thermal conductivity and electric conductivity as a function of temperature up to 25,000 K. Available gases include air, argon, helium, hydrogen, nitrogen, and oxygen. The data is taken from the tables in the Appendix of Ref. 1.
Boundary equations
The Arc Discharge interface also includes cooling/heating of electrodes in contact with the electric arc (Ref. 2).
Ion Bombardment and Thermionic Emission at the Cathode
Positive ions from the plasma are accelerated toward the cathode and generates heat at the electrode surface. As the electrode heats up, more electrons are emitted by thermionic emission, which cause cooling of the cathode. The associated cathode heat flux is defined as:
(3-12)
Where k is the thermal conductivity (W/(m·K)), ϕs is the surface work function of the electrode (V), and Vion is the ionization potential of the plasma (V). The ion current density norm is defined by
(3-13)
where
is the normal current density at the interface and where
(3-14)
is the electron current density norm. The latter is defined by Richardson–Dushman current density if the total normal current density is larger than
(3-15)
Where AR is the Richardson’s constant (A/(m2·K2)), q is the electronic charge (C), kB is the Boltzmann’s constant (J/K), and ϕeff is the effective work function of the surface (V). Note that the ion current density norm Jion = 0 if the Richardson–Dushman current density is larger than the total normal current at the interface.
Resistive Heating at the Anode
Electron entering the anode generates heat. Following the approach presented in Ref. 2, you can assume that there is no ion current and hence no ion heating at the anode. Accordingly the anode heat flux is defined as:
(3-16)
Where
is the normal current density at the interface, k is the thermal conductivity, and ϕs is the surface work function of the anode. Note that all the physics features available in the individual interfaces are also available to the multiphysics interface in use. This include, for instance, radiation heat losses, and wall boundary conditions.