2.4.1 Nozzle Arc Theory Based on LTE in Laminar Flow
In the first half of nineteen seventies, algorithms for the solution of arc conservation equations were not sufficiently robust and the computation cost with the then available computer power in terms of storage and speed was very high. Ad hoc assumptions were, therefore, introduced by theBBC group to simplify the governing equations for the analysis of the nitrogen arc shown in Figure 2.1. These assumptions include artificially dividing the arc into two zones radially with top hat temperature distribution in the radial direction for each zone as used in [2.2] or dividing the arc axially with assumed effective lengths, one section being laminar and the other turbulent and assuming three radial zones with a fixed radial temperature profile for each zone for the current zero period as done in [2.1]. Thus, the introduction of turbulence [2.1, 2.2] cannot be fully justified.
The investigation by Zhang et al. [2.19] based on LTE and laminar flow and the solution of the full arc conservation equations with a proper account for radiation transport especially the radiation absorption shows that the calculated electrical field (Figure 2.7), temperature (Figure 2.8), arc radius, velocity and pressure are in excellent agreement with those measured for a nitrogen arc at 2 kA DC. The RRRV of the BBC nitrogen arc predicted by the arc model with laminar flow [2.22] is in agreement with that measured [2.1] within the error expected for the shot to shot variation (Figure 2.9). It appears that, with the limited experimental results,
especially the reproducible RRRV test data, the LTE arc model with laminar flow can describe nitrogen arc behaviour adequately in the high current phases as well as in the current zero period.
Figure 2.7. Computed [2.19] and measured [2.2] electrical field intensity as a function of axial distance. Curves: A, Computed using measured axial pressure distribution. Thermodynamic and transport properties are dependent on pressure; B, Computed using measured pressure distribution; C, Pressure is iteratively computed assuming isothermal external flow; D, Pressure is iteratively computed assuming adiabatic external flow, E, Pressure is computed from nozzle area. For curves B, C, D and E the thermodynamic (except density) and transport properties at 10atm are used. The open circles are the experimental results after [2.2].
Figure 2.8. The axis temperature as a function of z. The differences between A, B, C, D and E are very small, hence only a single curve is shown. The open circles are the experimental results after [2.2] and the solid curve is computational results from [2.19].
Figure 2.9. Relationship between RRRV and di/dt at P0 = 23atm. The arc consists of
two sections: a self-similar arc section of 1 cm long [2.1] and the nozzle section of 4 cm. Key to the curves: (a) Results of [2.22] with p(z) corresponding to a direct current of 1 kA, (b) p(z) calculated from nozzle geometry [2.22], (c) air [2.1], (d) nitrogen [2.23], (e) Lowke and Lee [2.24]; (x) experimental points [2.1].
When the LTE laminar flow arc model is used to predict the radial temperature profile of an SF6 nozzle arc, there is a large discrepancy between the predicted and
those measured by the Aachen Group (Figure 2.10) [2.7]. As a consequence, the predicted RRRV for SF6 nozzle arcs in laminar flow is two orders of magnitudes
lower than that measured (Figure 2.11) [2.7].
Figure 2.10. Comparison of computed temperature profiles [2.7] (solid lines) and measured results after [2.4, 2.5] (triangles) at Z = 3 cm, P0 = 9 atm, di/dt = 16 Aμs-1,
Figure 2.11. (a) Logarithmic relationship between RRRV and di/dt at stagnation pressure 37.5atm. (b) Logarithmic relationship between RRRV and stagnation pressure at a di/dt=27 Aμs-1. Full lines: experiments (after [2.11]) and broken lines: predictions[2.7].
2.4.2The Effects of Non-LTE on the Switching Arc Behaviour
From the previous discussions, it is apparent that an LTE arc model in laminar flow cannot give a satisfactory account of the experimentally measured RRRV for an SF6
arc. It is natural to query if the assumption of LTE is correct. It is known that the rapid variation of discharge conditions during current zero period can make an arc depart from LTE [2.8, 2.9, 2.10]. Those processes which are unlikely to attain LTE are summarized below:
(a) The characteristic time of temperature variation during current zero period is about 1 μs. There are many chemical reactions in SF6 which are not fast enough
to reach chemical equilibrium
(b) High electrical field within the arc during current zero period especially after current zero results in higher electron temperature than that of heavy particles, due to inefficient energy transfer between electrons and heavy particles [2.8, 2.10].
(c) Electrical conductivity depends on electrical field because of the departure of electron velocity distribution function from Maxwellian in the direction of electrical field. This becomes appreciable after current zero when recovery voltage is imposed on the breaker.
(d) When current decays towards current zero, the axis temperature can be below 9000 K, and the electron number density is too low to maintain LTE.
Gleizes and his colleagues [2.8, 2.9, 2.10] investigated the combined effects of the aforementioned factors on arc conductance decay during current zero and on RRRV. It has been found that, in the region where electron temperature departs from that of heavy particles, electron temperature is maintained at a higher value due to inefficient exchange of energy with heavy particles for temperature below 8000 K [2.10]. Electrical field usually increases electrical conductivity due to increased ionisation rate and higher electron number density compared with the case where the influence of electric field is not considered [2.8]. These combined effects result in a slower conductance decay (Figure 2.12) and orders of magnitude lower values of RRRV in comparison with those measured and even with those predicted by LTE theory assuming laminar flow [2.10].
Figure 2.12. Evolution of the conductance during the decay, turbulent case. Reproduced from [2.10].