... be certain to insulate all hot water lines

How much energy would we save by adding R2 insulation to 100' of 1/2" pipe,
with 6 10 gallon hourly 110 F water uses per day at 3 gpm (an SRCC standard
test condition)?

The pipe is only used for about 3 minutes per hour...

Hint: not much, since there isn't much water in 1' of pipe,

.... 62.33Pi(0.5/24)^2 = 0.085 pounds.

so it cools fast between uses, even with lots of insulation.

L' of ri' radius pipe with an r' insulation radius and a k Btu/h-F-ft
insulation conductivity and an airfilm conductance h Btu/h-F-ft^2 would have
thermal resistance R = (1/(hr)+ln(r/ri)/k)/(2PiL), so 1' of bare 1/2" pipe
with ri = r = 0.3125/12 = 0.026' and an h = 1.5 slow-moving airfilm would
have R = 1/(1.5x0.026x2Pi) = 4.1 F-h/Btu, so RC = 4.1x0.085 = 0.35 hours,
ie 21 minutes. After 57 minutes in 70 F house air the 8.5 pounds of water
would cool from 110 to 70+(110-70)e^(-57/21) = 72.7 F, losing (110-72.7)8.5
= 317 Btu of heat.

With 1/2" of R4 per inch (R48 per foot, ie k = 0.021) foam insulation,
r = 0.6875/12 = 0.0573' and R = (1/(1.5x0.0573)+ln(0.0573/0.026)/0.021)/2Pi
= 7.8, so RC = 0.67 hours, ie 40 minutes, and the water cools from 110 to
70+(110-70)e^(-57/40) = 79.6 F, losing (110-79.6)8.5 = 258 Btu.

The net savings is 6(317-258) = 352 Btu per day or 129K Btu (38kWh) per year,
worth $3.80 at 10 cents/kWh.

OTOH, some pipe insulation near the heater where the pipes are warm most of
the time could pay for itself quickly.

Nick