It's a function of charger capacity.  The number of batteries you need is a function of their charge rate.   If you use 2100 mA-hr per flight and fly at 9-minute intervals (each flight starts 9 minutes after the last flight), you need to be able to charge batteries at 2100 mA-hr * (1 A / 1000 mA) * (60 min / 1 hour) / 9 minutes = 14 A.  If you have batteries that will charge at 14 amps, you need 2.  I went last season with three batteries, charging them at 7 amps when flying continuously. 

You are lucky to be able to run the cord out the back door-- a situation I hope to have someday.  Then you can get a cheap server power supply that will put out plenty of power.  The charger is the costly part.  I have a Thunder Power TP 820CD that does the trick. 

Howard, I'm not sure how you're achieving what you're achieving, because your understanding of how to charge a LiPo battery is a bit off.

The control rule for charging a LiPo (and for a lead-acid battery, for that matter) is to push as hard as possible while holding the charge current and voltage within limits.  When you start charging the battery will accept more current, which is where your 20A limit comes in, but as the battery starts to charge up its voltage rises, at which point the charger must cut the current to the battery to keep the voltage within safe limits (usually 4.2V per cell, or 4.3V if you and the battery manufacturer are courageous or foolhardy).

I certainly see this when charging my RC packs -- the full current part of the charge cycle only lasts for ten or fifteen minutes, then the balance of the charge happens at the battery's own pace.  Keeping the charge to 4.2V/cell is critical, and is the reason for balancing chargers.

Charging at too high a voltage is believed to be one of the problems behind the fires in the Boeing 787, but if you chat with your old work buddies you may know more than I do at this point.

OK, I'm back.  I used my fancy FMA software and computer interface to see what happens.  I set the charger for 7A max. It put in 1963 mA-hr.  Current ramped up for a minute, then held 7A for 14 minutes, then approximately exponentially petered out.  Total charge time was 22:51.  How about assuming that charge time = 9 minutes + .83 * capacity in mA-hr * (1 A / 1000 mA) * (60 min / 1 hour) ?  So the number of batteries you need to fly continuously at 9-minute intervals = 2 + the minimum number it takes to put in .83 * capacity at the max charge rate in 9 minutes.  For the Hyperion example, assuming just one more battery, putting in .83 * 3200 mA-hr at 20A would take .83 * 3200 mA-hr  * (1 A / 1000 mA) * (60 min / 1 hour) / 20A = 8.0 minutes.  So 3 batteries should suffice.  This is kinda crude.  Does it look OK, Tim?

I also see from my log that I don't fly at 9-minute intervals when flying continuously.  It's more like 15-20 min.  I'd better pick up the pace.  

I don't know for sure but if LiPo batteries are like other batteries, then the speed at which they can accept charge depends on their construction.  So, "it depends".

It's probably safe to assume that all honestly-rated 20C batteries are pretty similar, but maybe not.

On the number of 12V batteries you need:

(3200 mA-hr)(0.001A/mA)(3600 sec/hr)(14.8V) = 170kJ.

(170kJ) / ((20 min)(60 sec/min)) = 142W

So if you have a little battery and a 150W solar panel, then you can fly all day (for Howard that's 150W on a cloudy day, I know).  Unless you pick up the pace -- then you need a 300W solar panel, and if you can fly more often than once every 10 minutes you're good.

Motorman, Howard and I are having fun, but this is your thread.  Slap us if we're too far afield.

on a slightly serious note,, one part of the equation to keep in mind,, IF your system puts a more intense load on the battery,, then you will potentially need to allow time to cool the battery before charging as well,,

  Motor's like to cool off once in awhile also.