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Is there an equation that can be used to determine the air temperature around a given spark discharge? Let's say I produce a 50K volt spark between two wires using a standard HV generator. I'm sure the air in close proximity to the discharge must also rise in temperature, but how to get a practical idea of its temperature and more importantly, the temp gradients around this air? An equation would produce more useful data. Thanks!

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    $\begingroup$ Near the spark, radiation temperature is going to dominate material heat transport as the source of local temperature. You can approximate radiation temperature by assuming air is transparent, drawing a symmetric surface around your arc, and treating the symmetric surface as a blackbody with a radiant flux of $IV$. Note that arcs are non-ohmic resistor so you can't just use the breakdown voltage, you'll need to actually measure, or find an experimentally curve-fit formula to estimate from. $\endgroup$
    – g s
    Commented Sep 11, 2021 at 20:38

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Determining temperature is never easy because of the heat equation which is a differential equation. Temperature is determined by rate of energy input vs rate of dissipation into the surrounding environment, and the rate is controlled by the difference between the two, but the difference changes if they have not reached equilibrium.

But in your case, you might be in luck because if the spark is large and fast enough, the flow out could be neglected since the rate of energy in is significant compared to the rate of energy out. Then you can use adiabetic heat transfer where all the energy transferred in, stays in.

So you would then just need voltage across and current in the spark gap to calculate input energy. Then use that to calculate how much a volume air would heat up after accounting for the energy required to ionize (similar to phase change energy when something evaporates) which won't contribute to temperature rise. Air volume seems tricky to determine though.

No gradients though. That is the first principles heat equation and simulation territory with differential equations.

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