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Re: [TCML] Re: Spark models, revisited
On 10/18/12 3:05 AM, Udo Lenz wrote:
Jim Lux wrote:
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).
The current will follow the path of least resistance. So as long as
there is
a steep rise in conductivity due to temperature the arc will be as thin as
possible since a thin arc will mean more conductivity. Or put in a
different
way: The energy spent in the arc is better put in to temperature than into
width if you want to increase conductivity.
But the curve flattens out. Once you get to "highly ionized"
(6000-7000K) going hotter doesn't increase the conductivity very much,
the ions may be more mobile, but they're also less dense. Basically by
the time you get that hot, you've stripped enough electrons off to make
it a "good" conductor. Going hotter doesn't increase it all that much.
You have to be careful when looking at some of the equations because
they might be normalized for "constant density" or something, and, as
you've noted there's lots of stuff going on.
So, what happens is that the core heats up to the magic temperature
where things are "highly ionized", then increased current just increases
the size of the spark channel, without increasing the temperature much.
The steep rise in conductivity will flatten out at some temperature,
though. From that point on the width would increase.
Exactly.
Goncz wrote about this in connection with Xenon flash lamps.
This kind of equilibrium would be in accordance with my assumption of
1/R ~ P^2 since an increase of power would mainly go into
the width of the arc. Conductivity would rise with r^2 and
power consumption proportional to r due to the surface area.
I estimate the power dissipation of my 20kW 70cm arc to be
about 600W/cm near the breakout point. At 6000K a black
body radiator puts out around 100MW/m^2 (!). That would
make my arc really thin, about 0.2 mm diameter. The arc certainly
looks fatter. That makes me wonder, whether the Stefan-Boltzmann
law really applies.
A lightning stroke is on the order of 20kA and has a diameter of about
0.5cm. 10-100kJ/meter dissipation. Yes, your sparks ARE thin, they just
look bigger because of the high contrast. Also, recall that air at
2000K will be noticeably glowing and bright, so the "visibly glowing"
size is bigger than the "conductive core" size.
Udo
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