ok randy here's the answer

whatever you saved by not
turning it on earlier, has to be paid again on the way back up.

absolutely correct, you've got part of the answer


the lower you let it drop before turning back
on, the more thermal inertia to overcome


no, the energy lost during the cool down is the only energy that needs to be
replaced. Your thermal inertia concept is flawed if you think there is a extra
penalty for letting the temp fall further.

If the tank cools 20 degs & it takes 10000 btu's to bring it back up to temp;
then it will only take twice that energy 20000 btu's if the tank is allowed to
fall 40 degs.

The small (if not trivial) energy savings is had because a cooler tank (one
that is closer to room temp) looses less enery than a hotter tank. If the tank
lost all its heat (fell to room temp) it would stop loosing heat to the
environment.

Electrical water heaters typically are well insulated & loose heat rather
slowly. The only energy saved during the "off period" would be the energy
consumed in the cycling (if any) that would have occured to "maintain" temp
during the day.


By your logic I wouldn't save any energy if I turned the thing off for a week
because "I had to make up all that thermal inertia"?

A cooler house (or water heater) looses less heat to the surroundings.

That's why I turn the heater WAY down when I'm away from the mountain cabin