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Here's another option for a heat pump. Over the last couple of years, almost every major heat pump brand has come out with a variable speed compressor in their high end units. They are pricey, but very efficient with ratings up to 20 SEER and an almost nonexistent start surge. Available in sizes of 2 to 5 tons.
2600 sq. feet is a lot of space to heat and cool, even when well insulated. If might be a good idea to have an A/C service company come in at some point in construction and do a proper evaluation to size the unit. That, along with other loads will contribute a lot to properly sizing the battery.
This is one brand, like I said, there are many, including Carrier and Trane.
I had a 1500 watt 24 volt Cotek inverter and it called for 4 AWG cable from inverter to battery and a 100 amp fuse, but I believe you will find the cable and fuse requirements in the inverter manual.
Also, keep in mind, these batteries need to be fully charged every three days or so. If you just continuously recharge only the the amount of power that you use then battery life will be shortened. Since the charger is 16 amps, you will get about 100 amps hours back into the batteries in 8 hours including losses due to internal resistance. Probably shouldn't take the batteries down to a point were they can't be fully charged by the end of the day.
If four in series equal 48 volts and the charge current is 22 amps then, I would think, for two in series it would be 11 amps. A 5% charge rate is common with gel batteries.0
The neutral wire is a ground wire, however it is the only designated ground that should normally be carrying current to ground. As such it is grounded in the main panel only. No other sub panel or circuit should have a neutral to ground bond.
So a 120 volt circuit is the electromotive force alternating between a 120v feed and ground. A 240 volt circuit alternates between 2 120 volt feeds. So, it's not like it goes back to the electric power company. Imagine we are holding an 8 foot rod between us. I push it then you push it. This goes back and forth. This is like a 120v circuit. If we hold 2 rods and do the same except the right hand pushes and the left hand pulls, this like a 240v circuit. So the actual power that you are using is in distance that your hands travel in the analogy.