Ship Fuel Burn by the Cubic Speed Law

This is a scaling rule, and the cube in it is an assumption, not a measurement. The Fourth IMO Greenhouse Gas Study (2020) estimates a ship's main engine power with an Admiralty-type formula in which power varies with the speed ratio raised to a power $n$ that is assumed to be 3, and with the draught ratio to the 0.66 power. With a fixed specific fuel consumption the fuel burned per day follows the power, which gives $F = F_d\,(V/V_d)^{3}$. This page keeps only the speed term, so the draught is taken as unchanged.

$F_d$ is the fuel burned per day at the reference speed $V_d$, and $F$ is the burn at the new speed $V$; at the reference speed itself the burn is the reference burn. Because of the cube, a small change in speed moves the burn a lot. A ship that slows from 14 knots to 12, which is 14 percent, burns 31.5 tonnes a day instead of 50, a saving of 37 percent a day. The passage takes 17 percent longer, so the saving per nautical mile is about 27 percent, which is the fairer comparison because the slower ship is at sea longer.

Worked through, 50 t/day at 14 knots, slowed to 12 knots, gives $50 \times (12/14)^{3} = 31.4869$ t a day. Solved for the speed, 31.4868805 t/day against a reference of 50 t/day at 14 knots returns 12 knots; solved for the reference burn, it returns 50 t/day. The carbon dioxide follows from the fuel. Table 21 of the same study gives 3.114 g of CO2 per g of heavy fuel oil (and of low-sulphur heavy fuel), 3.206 for marine diesel, 2.750 for LNG and 1.375 for methanol, so the 31.5 tonnes a day of heavy fuel oil is 98.1 tonnes of CO2 a day, or 4.1 tonnes an hour. The fuel-combustion-co2 page does that step for any fuel.

Treat the result as a first estimate. Real engines burn a little more fuel per unit of work at low load, and a ship loses speed in weather, so a measured speed-and-consumption curve from the ship is better. The exponent is the study's assumed value, applied to the speed through the water. The ship-admiralty-power page gives the power the same law produces, before it is turned into fuel through the engine's specific consumption.