Type a line (displacement 45000 t 14 kn 9000 kW new speed 12 kn) or fill in the fields: the line and the fields stay in step.
Enter a known point, a speed and a power, with the displacement and the daily fuel if you have them, and a new speed, a new power or a new displacement, and the calculator gives the Admiralty coefficient, the power at the new condition and the change in per cent, or the speed for the new power, and with the daily fuel the fuel per day and per nautical mile. From an entered coefficient C it gives the power for a speed or the speed for a power. It shows the working with your numbers, draws the cube curve through the known point, and gives a line you can copy or add to a request.
The Admiralty coefficient and the cube law hold for small changes about a known condition of the same ship; the cube law is an approximation, and the real exponent differs with the hull, the speed range, fouling and the weather. The calculator gives no recommended speeds and no savings claims, and does not work out the resistance or the propeller: the arithmetic from your figures only.
C = D^(2/3) x V^3 / P, with the displacement D in t, the speed V in kn and the power P in kW. At 45 000 t, 14 kn and 9 000 kW: 45 000^(2/3) = 1 265.1, 14^3 = 2 744, C = 1 265.1 x 2 744 / 9 000 = 385.7. Shaft or brake power as entered, the same at both points.
At the same displacement P2 = P1 x (V2 / V1)^3: at 12 kn instead of 14 kn, 9 000 x 0.6297 = 5 668 kW, 37.0 % less. A new displacement adds the factor (D2 / D1)^(2/3); from 30 000 t in ballast to 52 000 t laden that is 1.443. The other way round: V2 = V1 x (P2 / P1)^(1/3).
The fuel per day scales as the power: 32 t/day at 14 kn gives 22.78 t/day at 12.5 kn. The fuel per mile is the fuel per day over 24 x V: 95.2 kg at 14 kn and 75.9 kg at 12.5 kn, so it goes as the square of the speed, (12.5 / 14)^2 = 0.797.
| Quantity | Relation | Note |
|---|---|---|
| Admiralty coefficient | C = D^(2/3) x V^3 / P | D displacement in t, V speed in kn, P power in kW, shaft or brake: the same basis at both points |
| New speed | P2 = P1 x (V2 / V1)^3 | the cube law, the same displacement |
| New displacement | P2 = P1 x (D2 / D1)^(2/3) x (V2 / V1)^3 | the same C at both points |
| Speed for a power | V2 = V1 x (P2 / P1)^(1/3) | from C: V = (C x P / D^(2/3))^(1/3) |
| Power from C | P = D^(2/3) x V^3 / C | on the basis C was worked out on |
| Fuel per day | F2 = F1 x P2 / P1 | t/day; scales as the power |
| Fuel per mile | F / (24 x V) | t per nautical mile; as V^2 at the same displacement |
| Units | 1 hp = 0.7456999 kW; 1 PS = 0.7354988 kW | 1 kn = 1852 m/h; 1 m/s = 3600 / 1852 kn |
The displacement in t, the speed in kn, the power in kW; horsepower is taken in kW and a speed in m/s in knots. The coefficient C depends on the units: other units give another figure. The relations hold for small changes about a known condition of the same ship.
The Admiralty coefficient C = D^(2/3) x V^3 / P, with the displacement in tonnes, the speed in knots and the power in kW, is defined in a marine technology encyclopedia and in a naval architecture reference tool; the displacement to the power 2/3 is part of that definition. That the power, and with it the daily fuel, goes roughly as the cube of the speed, and the fuel for a given distance as the square of the speed, is given in an independent 2012 study of ship speed reduction and in intergovernmental guidance on ship energy efficiency; a marine engineering article gives the daily fuel as proportional to the displacement to the power 2/3 and the cube of the speed. That the cube law holds near the design speed and less at other speeds is shown by a 2020 study of tanker noon reports, and the 2012 study adds that it cannot be taken for granted for ships of a fuller hull form; that fouling and the weather change the power for a speed is given in the marine encyclopedia (the sea margin) and in the same marine engineering article. The unit factors are from a national metrology institute guide to SI units and an encyclopedia entry on horsepower.
C = D^(2/3) x V^3 / P, with the displacement D in t, the speed V in kn and the power P in kW. At 45 000 t, 14 kn and 9 000 kW: 45 000^(2/3) = 1 265.1 and 14^3 = 2 744, so C = 1 265.1 x 2 744 / 9 000 = 385.7. The power can be the shaft power or the brake power: the calculator uses the one entered and says which, and the same basis has to be kept at both points, or the comparison mixes two things.
With the same displacement the cube law gives P2 = P1 x (V2 / V1)^3. From 9 000 kW at 14 kn, at 12 kn: (12 / 14)^3 = 0.6297, so P2 = 9 000 x 0.6297 = 5 668 kW, 37.0 % less. The same follows from C: with C the same at both speeds, the power goes with the cube of the speed. The figure is the arithmetic of the cube law, not a measured power.
Through the displacement to the power 2/3: P2 = P1 x (D2 / D1)^(2/3) x (V2 / V1)^3, which keeps C the same. From 8 500 kW at 15 kn in ballast at 30 000 t to laden at 52 000 t at the same speed: (52 000 / 30 000)^(2/3) = 1.443, so P2 = 12 265 kW. A change that large is shown with a caution: the coefficient is meant for small changes about a known condition.
The fuel per day scales as the power, so at the same displacement as the cube of the speed; the fuel per nautical mile is the fuel per day over 24 x V, so it scales as the square of the speed. From 32 t/day at 14 kn and 9 000 kW, at 12.5 kn: (12.5 / 14)^3 = 0.7118, so 22.78 t/day; per mile 32 / (24 x 14) = 95.2 kg at 14 kn and 22.78 / (24 x 12.5) = 75.9 kg at 12.5 kn, a ratio of (12.5 / 14)^2 = 0.797.
For small changes about a known condition of the same ship. The cube law is an approximation: the real exponent differs with the hull form and the speed range, and studies of noon reports find it holds best near the design speed. Fouling of the hull and propeller and the weather change the power for a speed as well. The calculator shows a caution when the new speed or displacement is more than 20 % from the known one, and gives no recommended speeds and no savings claims.
The displacement in t; the speed in kn or m/s; the power in kW, MW, hp (bhp, shp) or metric hp (PS), with 1 hp = 0.7456999 kW and 1 PS = 0.7354988 kW; the daily fuel in t/day; and the coefficient as C 480. A second speed, power or displacement, or one marked new, is the new condition. With C, one displacement and either a speed or a power. A shaft speed in r/min and the deadweight are not used.
The calculator gives the arithmetic of speed, power and fuel. When parts for the main engine, the shafting or the propeller are needed, send the line from the calculator with the engine details and a photograph of the nameplate, and Vessel Core checks the details before an offer goes back.
