
Amp Hours = Amps x Hours
Hours = Amp Hours / Amps
Amps = Amp Hours / Hours.
OK 3.7 amps @ 120 VAC with and Inverter efficiency of 88% will draw 21 amps from a 24 volt battery. If that battery AMP Hour Capacity were to be 690 AH then use the formula above in RED for hours.
690 Amp Hours / 21 Amps = 32.8 Hours aka C/32.8.
There is a issue with your question. You asked until Depletion. You would never ever want to discharge a battery completely, you never want to discharge more than 50%. So instead of 32.8 hours becomes 16.4 hours. Even that is problematic because with just one cloudy day means you have to shut down and wait for a full sunny day to return to recharge the battery. So proper practice dictates your battery capacity should have 5 full days. In practice gives you about 3 cloudy days of use until you have to recharge. Thus is why you would want a generator or engine alternator to save your butt.
Based on that if you pull 500 watts 24 hours a day is 12,000 watts per day. That means the battery requirement is 60 Kwh or 60,000 watt hours. At 24 volts is a 60,000 wh / 24 volts = 2500 AH battery. That monster weighs 3600 pounds and will cost you your roughly $150/Kwh or 60 x $150 = $9000 every 3 to 5 years.
Now before you start ranting again understand your question is invalid, or at least it should be. If the AC is sized properly it does not run 24 hours a day. It cycles on/off. In fact a DC powered AC uses a VFD drive which means it does not use max power when it is running. It can vary the power to generate just enough cooling to do the job. That is why it is important for you to use a Watt hour meter for a week to see what your real Kwh usage really is.
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