Say, this is a great science project! We got thermal mass, thermal resistance, heat energy and all that good stuff.
I'm not sure if you actually need to get up at 3 am to run another hot water circulation. It all depends on the thermal resistance of the tank. Here's where a little more testing comes into play. We need some temperature measurements through the night (or some of it). Measure water tank and outside temperature over time. Run test with tank 1/2 full. This is where automated temperature data gathering comes in handy.
