A heat loss calculation answers one question: how bad is the worst hour. A fuel bill asks a different one: what does the whole season add up to. The bridge between them is the degree-day, and the idea behind it is simple — a building's heat loss is very nearly proportional to how far the outside is below the inside, so if you know the average size of that gap you can scale the design load straight through the season.

$E = \dfrac{\dot{Q}_d \Delta T_m \, t}{\Delta T_d \, \eta}$ — read aloud E equals Q-design delta-T-mean t, over delta-T-design eta. $E$ is the seasonal fuel energy in kilowatt-hours; $\dot{Q}_d$ is the design heat loss in kilowatts, the worst-hour number; $\Delta T_d$ is the design temperature difference that load was calculated at, in kelvins; $\Delta T_m$ is the average temperature deficit over the season — degree-days divided by days — in the same kelvins; $t$ is the season length in hours; and $\eta$ — the Greek letter eta — is the seasonal efficiency of the plant. The subscripts are the convention to fix: d is design, the worst case, and m is mean, the season's average. Their ratio is a bare fraction, and it is typically a third — which is the quiet message of the whole method.

The efficiency divides, and it divides for a reason worth saying out loud. $\dot{Q}_{out} = \dot{Q}_{in}\,\eta$ says a boiler delivers less than it burns; read backwards, $\dot{Q}_{in} = \dot{Q}_{out}/\eta$ says you must always buy more fuel than heat. Multiply where you should divide and the bill comes out smaller than the heat delivered, which would be a very interesting boiler indeed.

Then the last step is the easiest and the most persuasive: $C = E p$, energy times the utility rate. It is the line that turns a thermal argument into a business case, and it is usually the only line anyone reads.