Re: [TCML] Spark models, revisited

[Jim Lux <jimlux@xxxxxxxxxxxxx>]

On 10/17/12 7:45 AM, Udo Lenz wrote:
> I've tried to make the spark model proposed by Jim Lux


Very cool..


> There are mainly two effects, which generate conductivity
> in arcs, thermal ionisation and ionisation by electric fields.
> Thermal ionisation has a very strong dependence on
> the temperature since the air molecules must have enough energy
> to ionise on impact. Conductivity begins at around 6000K and rises
> up steeply. See e.g the diagram on plasma conductivity in
>
> http://www.zeuz.ch/doc/itet/sem5/ensy/227-0122-00-autography_systemtechnology.pdf
>
>
> on page 21. I'm no expert on plasma physics and I don't have
> a clear view of where a balance between conductivity, temperature
> and power dissipation lies. Maybe some of you can help there.

Most of the literature on sparks indicates that because the curve of 
conductivity vs temperature has a real steep step in it (around 
6000-7000K) that you can model the spark as basically a uniform 
isothermal conductor.  There's a sort of dynamic balance between power 
dissipation and spark diameter: more current dissipates more heat, which 
makes a larger channel, which then loses more heat because the radiating 
surface area goes as the diameter.  The heat lost from the channel is by 
radiation, and you pick a convenient temperature (e.g. 7000K).



> Ionisation by electric fields requires fields of about 30kV/cm
> at room temperature and pressure, which is much more than the
> the voltage drop along RF arcs.
> But, as Paschens law states, the breakdown voltage is proportional
> to the density of the gas and that depends on the temperature.
> At 6000K, which is 20 times room temperature, the gas density
> is 20 times lower, reducing the field requirement to about 1.5kV/cm,
> which is much closer to the observed values. A rough estimate of
> this effect is:
>
> V ~ density of gas ~ 1/T (Paschens law and gas law)
>
> P ~ T^4 (Power needed due to loss by thermal radiation)

THrow in the radius factor.. Plost/length = k*r*T^4
> V^2 / R = P (Power dissipated by the arc)
>
> Putting these together results in:
>
> 1/R ~ P^(3/2)
>
> This amounts to a n of 3/2. I have some doubts, though,
> whether the T^4 law really holds for gases, which are no black
> radiators as the Stefan-Boltzmann law demands.

I think they are (or close enough)



Possibly
> the temperature dependence is slower, maybe T^3 or T^2. That
> would make n larger. Anyway, at voltage drops in the arc
> below 1.5kV/cm electrical ionisation cannot be anymore the
> main source of conductivity.
> I believe, that there is a mix of both thermal and electrical
> ionisation in a typical RF arc.

Yes.. in fact, I would think that it is almost ALL thermal ionization.

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