As Allen says (and you know) it is complicated.
There are lots of variables at play. The best I can do for you is give you a guess and the logic behind my guess.
A “Calorie” is a measure of energy contained in some food source, like sugar. One could think of Calories in a manner analogous to gallons of gasoline and a car engine. How long a car will go on a tank of gasoline depends upon how big the tank is and how much work the car engine is doing (HP). Going up some of those hills in PA is going to burn through the gasoline a lot faster than cruising through the flat corn fields of Iowa.
So how far does 1 pound of sugar get a colony of bees? First we need to know the Calories in 1 lbs of sugar. The web claims there is about 1500 Calories in a pound. Energy calculations are typically done in Watts so we have to convert those 1500 Calories into units that will end up giving wattage (power). We need to covert Calories to Joules at this point. 1 food Calorie = 4180 joules of energy. So 1 lbs of sugar is going to provide the bees fuel tanks with 6.27 million Joules of energy. Seems like a big number!
Now we just need to know how fast the bees are burning their fuel. The rate at which bees burn those Calories (Joules) is Watts; a unit of power. 1 watt of power (heat) = 1 joule of energy consumed / second.
This is where the guess work comes in:
How much energy does the cluster really generate to keep the center warm enough through the winter? I haven’t measured it, but the numbers various people throw around is 20 to 40 watts for a full sized colony when it’s cold outside. For a 5 frame nuc, my guess is they’ll max out at about 10 watts. On average, I would probably guess somewhere around 7 watts.
If we take a guess and say your 5 frame nuc of is generating 7 watts of power/heat to keep the inside of the cluster warm, then we can calculate how long 1 pound of sugar will last.
We know 1 watt = 1 j/s, so 7 watts = 7 joules/sec. There are 60 seconds per min, 60 min per hour, and 24 hours per day = 86400 seconds per day. 7 joules/sec x 86400 seconds/day = 0.604 million joules/day. Soooo….. a nuc cluster that is making 7 watts of heat to stay warm is going to consume 0.604 million joules of energy every day.
Earlier we found that a pound of sugar has 6.27 million joules of energy in it. If the bees are burning through 0.604 million joules a day, then 1 pound of sugar will last about 10 days.
So 12 lbs of sugar should keep your 5 frame nuc running for 120 days if their engines are idling at 7 watts on average.
How’s that for a guess?