Just to stoke up the discussion about heat loss from an electric (or
gas) hot water tank/heater, versus those instant-on type, herewith
some current numbers.

Left the house at around 2.00 PM Thursday Sept 4th. turning off both
the water supply and the circuit breaker for the electric hot water
tank. Returned today, Sept 9th. at around 2.00 PM and noting that
water was still quite warm, after an absence from a completely vacant
house except for neighbour checking, of 5 days, measured the water
temperature.

It was 80 deg F.

The foam insulated 40 US gallon tank is located in the basement which
has an air temperature at this time of year of about 60 deg F. The
water is normally heated to about 150 deg F.

So in 5 days the 40 gallons lost heat to the 60 deg. basement dropping
in temp by 150 - 80 = 70 degrees.

Leaving aside for the moment that the 'rate' of heat loss was probably
directly proportional to the difference in temperatures; i.e. highest
at the start when the tank was at full temperature, what are members
opinions on the following.

40 gallons x 8.33 = 333 pounds of water.
333 x 70 = 23,324 BTUs of heat lost during the 5 days.
23,324 / 3.41 = 6,840 watts of electricity. Previously used to heat
water.
6,840 = 6.84 kilowatts of electrcity
Which at 0.1 $ per k.watt hr. is/was a heat cost of 68.4 cents.
68.4 / 5 = 13.6 cents, per day.
So at a rough approximation 14 cents per day (from a tank that was
cooling down gradually over the 5 days).
When the same tank is being maintained at 'normal' full temperature
the heat loss; could we assume no more than twice that? Perhaps of
the order of say 25 cents per day? **

Maybe someone with better mathematical skills could to do a more
informed calculation? Using dy/dt etc. Also btw anything wrong with
the numbers used?????

** Also noting that during the winter at least, the lost heat adds
very slightly to warming this all electric house which consumes a year
round average of about $8 per day (all costs including sales taxes
included). Again, just hypothesising, if heating is say 60% of the $8
(say $4.80?), the rest being hot water, lighting computers, TV, radio,
work shop etc. (some of which also aid heating) then the 'lost' heat
from the hot water tank is of the order of, say 0.25 / 4.8 = about 5%
of the house heating cost! i.e. Not very significant?