The second stage in the model development is to gain an idea of the energy dissipated through the contact due to arcing and the subsequent heating and resulting damage through material vaporization. The procedure follows the work done on mass loss by (Swingler and McBride, 1996), by developing a model of the arc energy transfer and the heat flow through the contact. From this, vaporization of material may be estimated and subsequent mass loss.
5.2.1 Calculation of Arc Energy
The physics of arcing is extremely complex and not fully understood. Over the years numerous models have been proposed in literature to enable the effect of arcing in power systems to be simulated and understood. A great deal of the early work revolved around defining the volt-ampere characteristics of the arc experimentally and was dependent upon the test conditions, gap width and current magnitude.
A low current, well stabilized arc can be constrained as a cylindrical shape and would look similar to geometry in Figure 5.5 below, in higher current arcs, the convection of heat forces the arc to bow upwards, an attribute that lends itself to the name ‘arcing’. The arc is made up of three main regions; the anode, the plasma column and the cathode region. The anode and cathode regions are commonly known as the electrodes where the solid metal regions transpires into a gaseous plasma.
Figure 5.5. Showing the approximate geometry of a low current arc and associate fall in voltage.
Between these electrodes, a voltage drop or gradient, which is dependent upon the arc length, develops (Ammerman, 2010). Due to the complexity and nature of how the arc behaves, theoretical models become very difficult to develop that accurately describe the physics of the arcing process. This leads to most models resembling a ‘black box’ approach. Equations over the years have been developed by (Nottingham,
Anode Cathode
IArc
Anode Region Cathode Region
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1926), (Ayrton, 1902), (Steinmetz, 1906), (Nottingham, 1923), (Van and Warrington, 1931), (Miller and Hildenbrand, 1973), (Hall, 1978), (Myers, Vilicheck, Stokes and Oppenlander, 1991), that approximates the fall in voltage between the anode and cathode.
5.2.2 Energy Transferred by Arcing
The energy released during arcing is subject to the law of energy conservation, and hence the electrical energy input is equal to losses encountered in the form of heat, light, pressure, sound and electromagnetic radiation. This led to the development of arc-resistance models that take into account the above to form an estimate of the electrical energy delivered during arcing (Gammon and Matthews, 2003).
For steady state dc power systems, power is defined as
𝑃 = 𝑉𝑑𝑐𝐼𝑑𝑐 (123)
and the power for dc and single phase ac arcs can be described as 𝑃𝐴𝑟𝑐 = 𝑉𝐴𝑟𝑐𝐼𝐴𝑟𝑐 = 𝐼𝐴𝑟𝑐2 𝑅
𝐴𝑟𝑐 (124)
as energy is a function of time, the energy associated with arcing can be approximated by
𝐸𝐴𝑟𝑐 ≈ 𝐼𝐴𝑟𝑐2 𝑅
𝐴𝑟𝑐𝑡𝐴𝑟𝑐 (125)
Where time is measured in seconds.
An arc model therefore needs to incorporate the fall in voltage between the anode and cathode of the contacts, and from this, along with the power, an estimate of the energy transferred can be approximated.
(Swingler and McBride, 1996) provided a model to estimate the amount of energy brought by the electrical arc to the electrode surface under study for each time t of the breaking process under DC conditions. The input parameters such as current intensity, circuit voltage and opening velocity related to experimental conditions are used. From this, the power flux density through the arc is used to produce the output which will be used as the input to the thermal model used later on.
The anode region, the plasma region and the cathode region are modelled in terms of the arc energy transport regions.
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Figure 5.6. Illustrating the anode, plasma and cathode transport regions
Within each of these three regions, the power dissipation is computed from the current through the region and the voltage drop across it.
The power dissipation from the anode, through the arc region to the cathode surface is calculated by considering the two ways of energy transport processes that encompass all the mechanisms involved at a microscopic scale in the energy transport process: radial and channel transport processes. The Radial transport processes represent the processes which radiate energy equally in all directions such as thermal energy from random bombardment of particles, radiation from de-excitation of particles, etc. The channel transport processes account for mechanisms which transport energy (channel energy) toward the cathode or anode. The energy is channelled to neighbouring regions by, for instance, positive ion or electron bombardment as they are accelerated through the electric field.
This leads to the amount of energy being available from any arc region at a given time t being equal to the energy transported from any neighbouring regions by the transport processes plus the energy generated within that region itself. The power flux density out of the plasma region transporting energy towards the cathode is then given by (Swingler and McBride, 1996):
𝑞𝑘 =(𝑃𝑝𝑙𝑎𝑠𝑚𝑎𝜋𝑟2+𝑃𝑎)× (𝑘1
𝑟
2√𝑟2+𝑟𝑥2+ 𝑘2) (126)
Where:
qk is the power flux density out of any particular region transporting energy
towards the cathode
Power flux through radius of 𝑟𝑎 ra Pa Plasma region Cathode Anode Px rx
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K1 is the proportion of energy radially dissipated
K2 is the channel transport process and is assumed to be 50%
Pa is the power input at the anode region
Pplasma is the power dissipated in the plasma region
ra is the radius of the arc to the axis
rx is the distance between the point source and the plasma boundaries
Instead of K2, another control term may be inserted, K3, which allows the power flux
density out of the plasma region transporting energy towards the anode to be found.