Earlier this year, in another Water Heater thread (about powering w/ 120 VAC) I mentioned that I had recently taken the time to document exactly how often a water heater in my house came "ON" with no water drawn - and how long it took for the thermostat to be satisfied during that recovery. The results surprised me.
I hooked up a cheap current sensing switch to a PLC and wrote a simple program to log the time(s) when that switch changed state.
With no water drawn from the tank, it takes roughly 11 hours before the thermostat kicks on. And it runs for about 8 minutes to get back up to temp. Pretty well insulated it would seem. Not just simple fiberglass blankets like 40 years ago. Now it's all foamed in and really well insulated from air flow. At least electric tanks anyway.
This water heater was a regular "nothing special" State 50 gallon water heater. Typical 4500 W elements (top and bottom). Not a Heat Pump model.
So - roughly - each recovery phase from simple tank heat loss uses ~ 0.6 kWh and happens 2x per day. At an electric rate of $0.15 per kWh that's 20 cents per day - so maybe like $6 per month.
And, as mentioned, part of that heat loss is mitigated in the winter if the "loss" is to indoor air you were trying to heat anyway.
If anybody is thinking that the On Demand Water Heater is going to pay for itself quickly by saving the heat loss from a modern water heater, I think those numbers say otherwise. Or, at least it won't be a quick payback