Most efficient method I believe:
Loadout-
-Chaos set obviously
Combuster
Electron Vortex
Blitz Needle
Elite Slash Module
Elite Trueshot module (
Swiftstrike Buckler
(Barbarous thorn shield + penta if you want a 5* shield, the ASI isnt crucial

That is a very effective loadout, I averaged 27 mi.s solo on my last acct with this. Lvl 10 combuster and blitz, 5+ EV

For each depth:
-D24
Skip. Crowns here are garbage (about 800 total on average)

-D25
I find doing each room takes about 1-2 minutes. I do all the right rooms, and the northwest rooms on the left. Kill all enemies in middle, skip the crap loot that needs water, proceed.

-D26
Just EV + combuster charge, take left path

-D27
EV + combuster charge for skellies, skip side slags before final room, get loot, boss

Boss usually takes 3 mins with max dmg and ctr blitz. Circle and blitz in a clockwise fashion, to have the best chance of avoiding the mace. Use water on the left side to water mask. Ignore slags and guards.

With this, I average 27 mins, 24 at the very best and 30 at worst. D25 is the time sink.

27 mins (round up to 30 for consistency), 8k crowns at least. Counting vana tokens as 1k each, plus materials, estimating 15k crowns per run. 15k over 30 mins is 500 crowns per minute (sell mats on AH), or 400 cpm if you keep the mats.

There is a less tedious method to earn 450-550 crowns per minute, but you wouldnt get the radiants

FSC isnt the easiest or even the fastest way to get crowns, especially if you aren't good at it.

If you have trouble beating FSC, I recommend joining an active guild that if full of experienced 5* players. Ask them to help you. If you have trouble making Crowns in general, just search some methods online.

From my observations, there are two large factors controlling how quickly you beat Firestorm Citadel: not wasting time and your gear.

For gear, its less about what you bring and more about that it works and is quality. For instance, if everyone else has combuster, don't spam shiver. Generally people will bring some combination of blitz needles, shivermist buster, brandishes and other elemental weaponry. The only one of these that actually matters is the blitz needle, as it can be used to do literally the entire mission. Everything else you bring is really just to make it more interesting and perhaps kill the slags a bit faster. Bringing shivermist buster to compliment it works very well, but isn't necessary. As long as you have some combination of those, it becomes about quality of weapons.

The second thing is not wasting time. By that, I mean "brb", "afk", and "one sec". If one person goes to the restroom, then the entire party will end up doing so, and one of them might make some food, and then the rest get back and he's still gone so why not go afk some more and that's how your party will take two hours to beat Firestorm Citadel, even with all of the gear etc. for it. Avoid these situations by taking care of your business prior to the run and you can go from hour and a half long runs to forty five minute runs or even half hour runs.

And as Skepticraven said, don't limit yourself to the citadel. Arenas and other low depth areas can pay out more per time spent. In fact, the only real advantage that grinding Firestorm Citadel has over grinding elsewhere is that there are far more people prepared for it that believe in it than say, a fiend arena. This means that you can easily get parties going and grind, whereas to grind the lower half of tier 3 in the arcade you'll need to get friends or lucky randoms that actually know what they're doing in on it... which isn't impossible, as many experienced knights prefer the arcade. Rather, it is just significantly harder than pressing "Join party" and giving the citadel a whirl.

Of course if you don't need radiant fire crystals, just grind wherever you please because no one can stop you.