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Elon _always_ lies and exaggerates during these things. He seemingly can't stop himself.
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I assume it's both...

it has its own pump, but if you don't want to wait 6 hours to recharge, you can fill it with a nitrogen cylinder.

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The Top Gear video mentions that kind of recharge time using an external system (though they do claim that they are trying to get it down to 30min).

I've seen some claims that the gas is stored at a pressure of 10000 psi/690 bar in a 200 l tank. If that's true, then that is a gas volume of 138000 l. A commercial, stationary scuba compressor compresses around 850 l/min [0] to 350 bar. A compressor like that would take around 160 min just to compress 138000 l to 350 bar (so significantly less than 690 bar), is a _large_ appliance, and _loud_. Compressing gas quickly also produces heat. An on-board compressor seems quite unrealistic to me unless I'm missing something.

[0] https://www.bauer-kompressoren.de/en_en/pe-ve1

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He's only 78% correct.
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promise everything now, work out everything else later.
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Perhaps they use liquid nitrogen and then just allow it to boil off inside the tanks? This would give them a safe and easy way to quickly recharge the tanks for testing.
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Air is basically nitrogen. It can probably use both.
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Based on a cursory search, propellant-grade nitrogen requires a high purity (99.99%) [0]. I am no aerospace engineer, but I'd be interested to know what the actual requirements are for a system like that to work reliably. If 99.99% nitrogen is actually required, then I'd assume that the corresponding compressor would neither be compact nor particularly maintenance-free.

[0] MIL-PRF-27401

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It's pretty simple, actually, Elon is a pathological liar.
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