Admiralty coefficient

What the Admiralty coefficient is

The Admiralty coefficient is a single number that ties together a ship’s displacement, its speed, and the power its machinery must deliver. Wartsila’s marine encyclopedia gives the working definition in one line: “a coefficient used in the preliminary estimations of the power required in a new design to attain the desired speed.” Read the relation backward, with power and speed measured on an existing ship, and it becomes a performance index: a higher $C$ means the hull moves more displacement at a given speed for each kilowatt installed.

The number is not a physical constant. It is a similarity parameter that holds approximately steady across ships of comparable form operated in the same speed range, and it drifts as soon as the hull form, the loading, or the speed regime departs from the reference. That property is exactly what makes it useful. A naval architect who knows $C$ for a delivered vessel can predict the power of a sister at modestly different displacement without rerunning the resistance chain, and a superintendent who tracks $C$ across dry-dockings can read off the fouling penalty the ship pays between cleanings.

This article covers where the relation comes from, the full derivation from Froude scaling, the units trap that catches the unwary, a worked example at real ship scale, and the boundaries where the estimate stops being trustworthy. The Admiralty power calculator and the Admiralty coefficient power calculator apply the formula in both directions, and the block coefficient and ship resistance and powering articles cover the hull-form and resistance theory that the coefficient compresses into one figure.

Background and history

The relation that the Royal Navy’s Admiralty codified in the nineteenth century, $P = \Delta^{2/3} V^3 / C$, predates dynamometer-instrumented sea trials and predates the experimental split of resistance into frictional and residual parts. It captured a practical observation: for two ships of similar form run in the same speed range, the group $P \cdot C / (\Delta^{2/3} V^3)$ stays near unity. The Admiralty’s interest was operational. Given a planned displacement and a required speed, naval constructors needed a power figure to size boilers, engines, and bunker capacity at the stage before the lines plan existed.

William Froude’s towing-tank work in the 1870s gave the rule its physical footing. Froude built the first controlled tank at Torquay and ran tests there from 1868, then worked at the Admiralty tank at Haslar from 1872. His 1874 paper to the Institution of Naval Architects, reporting the towing trials of HMS Greyhound, established that total resistance separates into a frictional part, governed by wetted surface and the Reynolds number, and a residual part, governed by the Froude number $Fn = V / \sqrt{gL}$ and strongly nonlinear in speed. For geometrically similar hulls at the same Froude number, residual resistance per tonne of displacement is the same, and wetted surface scales as $\Delta^{2/3}$. The cube-of-speed term in the Admiralty relation captures the combined behavior over the moderate-speed band where wave-making matters but has not yet run away into the humps.

Through the twentieth century the systematic series and regression methods displaced the Admiralty coefficient as the primary powering tool. The Taylor Standard Series, compiled by Admiral David W. Taylor of the US Navy from around 1910 and revised by Gertler in 1954, gave families of residual-resistance curves against speed-length ratio. The British Ship Research Association series of the 1950s and 1960s extended methodical testing to full merchant forms. The Holtrop-Mennen regression, published by Jan Holtrop and G.G.J. Mennen of MARIN in International Shipbuilding Progress in 1982 and re-analyzed by Holtrop in 1984, became and remains the standard analytical method for preliminary resistance. The Admiralty coefficient survived all of this as the rapid first-pass estimate that every early study still computes before committing tank or CFD resources. Modern texts treat it as a speed-similarity rule, valid only when the new vessel and the reference run at comparable Froude numbers and carry similar block coefficients. The full resistance treatment sits in the ship resistance and powering and resistance components deep dive articles.

Derivation from Froude similarity

The relation follows from splitting resistance the way Froude did and then asking what stays constant between geometrically similar hulls run at the same Froude number. Effective power is resistance times speed, $P_E = R_T V$, and total resistance divides into a frictional part and a residual, wave-making part, $R_T = R_F + R_R$. At equal Froude number $Fn = V / \sqrt{gL}$, the residual resistance coefficient $C_R = R_R / (\tfrac{1}{2} \rho S V^2)$ is the same for both hulls, so residual resistance scales with $\rho S V^2$. Wetted surface $S$ scales with the two-thirds power of displacement, since area goes as length squared and displacement as length cubed, which gives $S \propto \Delta^{2/3}$.

Collecting the residual term, $R_R \propto \Delta^{2/3} V^2$, and effective power $P_E = R_R V \propto \Delta^{2/3} V^3$. Dividing through by a quasi-propulsive coefficient that is itself roughly constant across similar forms turns effective power into delivered or shaft power without changing the proportionality, and the inverse of that whole proportionality constant is the Admiralty coefficient, $P = \Delta^{2/3} V^3 / C$. The cube of speed is therefore not an empirical fit but the product of the $V^2$ in residual resistance and the extra $V$ that converts resistance into power.

The frictional part is what the derivation glosses over. Frictional resistance scales with the Reynolds number, not the Froude number, so it does not obey the same clean similarity, and at full scale it is the larger share of the total for a slow, full ship. The Admiralty relation absorbs this by letting $C$ carry the frictional regime of the reference hull, which is why the constant holds only between ships of similar length and speed where the frictional fraction is comparable. Where the two ships differ enough in size that the Reynolds-number regime shifts, the frictional scaling breaks the similarity and the estimate drifts, in the direction the full resistance treatment in ship resistance and powering makes explicit.

Units and the constant’s dimensions

The Admiralty coefficient is dimensional, so its numerical value depends entirely on the units used for power, displacement, and speed, and a constant quoted without its unit basis is unusable. The modern merchant convention is power in kilowatts, displacement in tonnes, and speed in knots, which is the basis of every value in the table below and of both calculators on this site. An older British convention used imperial or metric horsepower for power and gave numerically different constants for the same ship, which is the most common reason a value copied from an old table does not reproduce a measured trial.

The dimensions follow from the relation. With $P$ in kW, $\Delta$ in t, and $V$ in kn, the constant carries units of $\text{t}^{2/3}\,\text{kn}^3 / \text{kW}$, a mixed group with no physical meaning of its own. That is the price of compressing a resistance calculation into one number: the constant is a bookkeeping device tied to a unit system, not a property of the water or the hull. Converting a constant from one unit system to another means substituting the conversion factor for each quantity separately, not applying a single multiplier, because power, displacement, and speed each enter at a different power.

Two further traps catch the unwary. The first is the power station in the drivetrain: a constant fitted to shaft power differs from one fitted to delivered power at the propeller or to brake power at the engine, because the shaft and gearbox losses sit between them, so the power basis has to be recorded alongside the value. The second is the displacement basis: the relation uses the full displacement in tonnes, not the deadweight or the lightship, and a constant computed against the wrong displacement is off by the two-thirds power of the ratio. The SHP to kW converter and the HP to kW converter keep the power term consistent when reconciling a constant against a trial reported in horsepower.

Typical values by ship type

The Admiralty constant is not universal. It depends on hull form, the propulsion-train efficiency, hull-surface condition, displacement loading, and the speed regime. Wartsila’s encyclopedia gives the broad band as 400 to 600 for merchant ships, higher being more economic. Published design guidance refines this by ship type; the figures below are commonly cited for shaft power in kW, displacement in tonnes, and speed in knots, at design draft and service speed in calm water.

The split between fuller-form vessels with low $C$ and finer-form vessels with high $C$ reflects the wave-making penalty that full hulls pay at typical service speeds. A bulk carrier with block coefficient $C_b \approx 0.85$ sits in the high-resistance regime relative to its displacement, while a container ship with $C_b \approx 0.65$ cuts through the same speed range with lower residual resistance per tonne. The container-ship numbers run high because the constant rewards moving displacement cheaply at the design Froude number; the same hull slow-steamed well below its design speed no longer earns that reward, which is the matching problem the slow steaming and CII article covers.

These ranges are guides for sanity-checking, not design values. A constant recovered from the actual sister ship’s trial, corrected under ISO 15016:2015, beats any table value, because it carries the real propulsive efficiency and the real hull form rather than a type average.

Worked example: a sister-ship power estimate

Take a delivered handysize bulk carrier whose corrected speed trial recorded 8,200 kW shaft power at 14.2 knots and a trial displacement of 46,500 tonnes. The constant is $C = \Delta^{2/3} V^3 / P = 46{,}500^{2/3} \times 14.2^3 / 8{,}200$. The displacement term is $46{,}500^{2/3} = 1{,}292$, the speed term is $14.2^3 = 2{,}863$, so $C = 1{,}292 \times 2{,}863 / 8{,}200 = 451$. That sits squarely in the 420 to 490 band the table gives for a handysize, which is the first check that the trial and the arithmetic are sound. Because speed enters cubed, the recovered constant is sensitive to the trial speed above all else: a 0.1-knot error at 14.2 knots shifts $C$ by about 2.1 percent, which is why the trial speed is averaged over reciprocal runs to cancel current and tidal set, and why the displacement, the smallest of the three sensitivities at two-thirds power, is the term a surveyor worries about least.

Now size a sister built to the same lines but loaded to 48,000 tonnes and required to make 14.5 knots. Holding $C$ at the measured 451, the power is $P = \Delta^{2/3} V^3 / C = 48{,}000^{2/3} \times 14.5^3 / 451$. The displacement term is $48{,}000^{2/3} = 1{,}321$, the speed term is $14.5^3 = 3{,}049$, so $P = 1{,}321 \times 3{,}049 / 451 = 8{,}930$ kW. The sister needs about 8,930 kW shaft power at the heavier displacement and higher speed, roughly 9 percent above the reference, of which the speed accounts for about 6 percent through the cube term and the displacement about 2 percent through the two-thirds term.

Adding a sea margin of 15 percent for weather and a fouled hull lifts the design power to about 10,270 kW, and a shaft-line efficiency near 0.99 puts the engine brake power near 10,370 kW, which points to a two-stroke frame rated around 10,500 kW MCR. The whole estimate runs in a minute on a pocket calculator and lands close enough to bracket the engine selection and lay out the engine room before any model test. The Admiralty power calculator recovers the constant from a trial in one direction and predicts the sister’s power in the other, and the speed-power cubic fit calculator refines the speed term once more than one trial point is available.

Where the relation is used

Early-stage powering estimates

Before committing to a hull-form study, naval architects use the Admiralty coefficient to bracket the engine maximum continuous rating for a target displacement and speed. Selecting $C$ from a peer-fleet table for the intended ship type, $P = \Delta^{2/3} V^3 / C$ gives a delivered-power figure that for conventional designs lands within roughly plus or minus 10 percent of the eventual model-tank result. That is enough to size the engine room and the bunker tanks at the concept stage, where the lines plan does not yet exist and a full Holtrop-Mennen run would be premature. The Admiralty power calculator does this directly, and the estimate is later refined by the resistance chain described in ship resistance and powering.

The estimate also drives the minimum-power check. A design that trims installed power to improve EEDI must still clear the IMO minimum propulsion power guidelines for safe maneuvering in adverse conditions; the Admiralty estimate gives a fast first read on whether the chosen rating is in the right neighborhood before the minimum propulsion power calculator runs the regulatory check.

Sea-trial analysis and sister-ship comparison

After delivery, the speed trial measures shaft power at the contracted speeds. The Admiralty coefficient is the natural metric for comparing the trial result against the design specification and against earlier sister-ship trials. A constant below the contracted figure means the ship needs more power than predicted; above it, less. Because the trial follows ITTC 7.5-04-01-01.1 and is scheduled within 30 days of undocking, the measured $C$ reflects a clean hull, which makes it the reference against which all later in-service values are read.

Tracking $C$ across a series of dockings quantifies the cost of hull and propeller deterioration. A constant that has fallen 5 to 10 percent since the last clean trial signals that the ship is burning that much more power for the same speed, which translates directly into added fuel through the specific fuel oil consumption relationship. The SFOC sensitivity calculator turns the power change into a fuel-rate change.

EEXI and EEDI feasibility checks

The Energy Efficiency Existing Ship Index (EEXI) and the Energy Efficiency Design Index require demonstrating that a vessel’s attained efficiency sits below a regulatory ceiling, and both depend on the engine MCR or the limited MCR as input. When an operator considers an engine power limitation to bring an existing ship into EEXI compliance, the Admiralty coefficient projects the speed the ship will reach at the limited rating. Because power scales as the cube of speed, the speed at a limited MCR follows:

A 20 percent power limit, taking $\text{MCR}_{\text{lim}} / \text{MCR}_{\text{orig}} = 0.80$, drops the service speed by a factor $0.80^{1/3} = 0.928$, about 7.2 percent. For a 14.5 kn ship that is a drop to roughly 13.5 kn. This gives a fast feasibility filter before a full propulsion study, and the EEXI attained calculator and the EEDI attained calculator then run the regulatory computation. The interaction between installed power, the EPL, and the shaft power limiter is set out in the EEXI, EPL, and ShaPoLi article.

Slow steaming and operational savings

For an operator weighing slow steaming, the Admiralty coefficient projects the power saving from a given speed cut. The cube-of-speed dependence means a 10 percent speed reduction yields about a 27 percent power reduction, since $0.9^3 = 0.729$, and a 20 percent cut yields about 49 percent, since $0.8^3 = 0.512$. Real savings are smaller because $C$ drifts as the ship leaves its design speed regime and the added resistance from weather does not scale the same way, but the cube law sets the achievable upper bound. The SFOC sensitivity calculator converts the projected power change into a fuel figure, and the commercial and regulatory drivers behind speed reduction are covered in slow steaming and CII, EU ETS for shipping, and FuelEU Maritime explained.

Displacement scaling in voyage planning

The displacement term alone gives a useful scaling for the power penalty of carrying excess weight. At constant speed, power scales with displacement to the two-thirds, so moving from $\Delta_1$ to $\Delta_2$ changes power by $(\Delta_2 / \Delta_1)^{2/3}$. A ship sailing 5 percent heavier on excess ballast pays about $1.05^{2/3} - 1 = 3.3$ percent more power for the same speed. Voyage planners use this to weigh the fuel cost of retained ballast against the operational reason for carrying it, and the same scaling appears in the displacement-correction step of the resistance treatment in ship resistance and powering.

Bunker and engine sizing at the concept stage

The Admiralty estimate feeds two early decisions that are expensive to revisit. The first is engine selection. A delivered-power figure within 10 percent of the eventual model result is enough to pick a two-stroke engine frame size from a maker’s program, because the standard ratings step in coarse increments and a 10 percent bracket usually lands inside a single frame. A handysize bulk carrier estimated at 8,400 kW shaft power, allowing a shaft and transmission efficiency near 0.99 and a sea margin of 15 percent on trial power, points to an engine rated near 9,700 kW MCR, which a designer can match to a real frame before the lines exist. The second decision is bunker volume. Daily fuel burn is the delivered power times SFOC times 24 hours, so the same 8,400 kW at an SFOC of 170 g/kWh burns about 34.3 tonnes of fuel per day at sea, and a 30-day voyage range then fixes a bunker capacity near 1,030 tonnes plus margins. Both numbers are first cuts, refined later by Holtrop-Mennen and the marine diesel engine load profile, but they are accurate enough to lay out the engine room and tank arrangement in the concept phase.

The speed-power exponent in service

The cube law is the headline, but in service the exponent that links power to speed is rarely exactly 3, and knowing the real number sharpens every projection made from the coefficient. Total resistance follows roughly $R_T \propto V^m$, where $m$ rises from about 2 at low Froude number to 4 or more near the primary wave-making hump, so brake power scales as $P \propto V^{m+1}$. For most displacement ships at service speed the power exponent sits between 2.5 and 3.5, which means a 10 percent speed increase demands 28 to 41 percent more power rather than the clean 33 percent the cube law predicts. The Admiralty coefficient assumes the exponent is 3 because that is the right average across the moderate-speed band, but a projection that spans a wide speed range should use a fitted exponent instead.

This is why a single trial point is weaker than a curve. A ship that records power at 12, 14, and 16 knots gives three points from which the real exponent can be fitted, and that fitted curve predicts intermediate and modestly extrapolated speeds far better than any single constant. The speed-power cubic fit calculator performs exactly this fit, and the result feeds the carbon-intensity correction that operators apply when they project a CII rating at a planned average speed. The Tu et al. (2018) study took this further for container ships, replacing the fixed exponent with hull-form coefficients so that the modified coefficient tracks the steeper power rise that fine, fast hulls show as they approach their humps, and reporting a closer match to measured power curves than the classical form.

For a bulk carrier at 15 knots on a 200-metre waterline the Froude number is near 0.30, which places it close to the primary hump where the exponent is already climbing. A container ship at 22 knots on a 330-metre hull sits near a Froude number of 0.22, where wave-making is appreciable but the exponent is more moderate. The same 10 percent speed cut therefore saves a different amount on each: more on the container ship sitting on the steeper part of the curve, less on the bulk carrier whose resistance is more friction-dominated. The cube law is the common starting point; the real exponent, read from the ship’s own data, is the refinement, and the resistance theory behind it is set out in ship resistance and powering.

Relationship to block coefficient and the resistance chain

The Admiralty constant is a compressed image of the full resistance and propulsion calculation, and its value moves with the same variables that move resistance. Block coefficient is the dominant driver. A fuller hull at a given displacement and set of principal dimensions has more wetted surface and more residual resistance per tonne at service speed, so it needs more power and shows a lower $C$. This is why the table above places VLCCs and Capesize bulk carriers at the bottom of the range and fine-form naval craft at the top.

The wake field works in the other direction and partly offsets the resistance penalty of fullness. A high-$C_b$ hull carries more of the surrounding water along with it, raising the wake fraction and the hull efficiency, so the propulsive chain returns a little of what the higher resistance took. The net of these two effects is what the single constant captures, which is why a constant lifted from a sister of the same form is far more reliable than one borrowed across forms. The naval architecture coefficients article sets the full coefficient family in context, and the hull form design article covers how designers trade fullness against resistance.

Against the Admiralty coefficient, the Holtrop-Mennen regression respects Froude-number trends and block-coefficient sensitivity directly, predicting total bare-hull resistance from the principal dimensions and the form coefficients rather than from a single fitted number. It is the basis of the detailed-design powering workflow, implemented in the Holtrop-Mennen resistance calculator, and it is the method the Admiralty estimate hands off to once the lines exist. The two are not rivals: the Admiralty coefficient brackets the answer in seconds, and Holtrop-Mennen refines it once there is a hull to feed the regression. The relationship of both to engine selection runs through the marine diesel engine and marine propeller articles.

Limitations

The Admiralty coefficient is a similarity rule, not a physical model, and its accuracy degrades in predictable ways. The new ship must operate at a Froude number close to the reference, because the cube-of-speed term assumes residual resistance scales as $V^3$, a moderate-speed approximation that fails near the humps where the exponent climbs above 3. The block coefficient must match the reference, because a constant from a fine hull overpredicts the speed a full hull will reach on the same power. The loading condition must match, because both wetted surface and the Froude number shift with displacement and the two-thirds term covers only part of that shift. Wave-making must not dominate, which rules out catamarans, SWATH forms, and very fine fast craft. And the hull condition must match the reference, because fouling, propeller roughness, or wrong trim can move the apparent constant 5 to 15 percent within a single docking interval.

These are not reasons to discard the method but reasons to use it for what it is. For a conventional displacement ship near its design speed, with a constant drawn from a true sister corrected under ISO 15016:2015, the estimate is reliable to roughly 10 percent, which is why it remains the standard concept-stage and sea-trial tool. When precision matters more than speed, the work moves to Holtrop-Mennen, the systematic series, model tests, or CFD, the methods set out in ship resistance and powering. When speed and a defensible first number matter more than precision, the Admiralty coefficient is still the right tool, and it has been for more than a century.

One last caution applies to the constant’s provenance. A value copied from a textbook table carries the average propulsive efficiency and the average hull form of whatever fleet built that table, and a value recovered from a CFD prediction carries the assumptions of the simulation rather than a measurement. The strongest constant is the one read off the actual sister ship’s corrected trial, because it folds the real quasi-propulsive coefficient, the real shaft losses, and the real hull into a single measured number. A designer who records the loading condition, the power unit, the speed basis, and the trial standard alongside every quoted $C$ keeps the method honest, and an operator who logs the same four items each docking turns the coefficient into a clean fouling-trend signal rather than a number that drifts for reasons no one can reconstruct.

See also

• Admiralty power calculator - delivered power and the constant from displacement, speed, and power
• Admiralty coefficient power calculator - the same relation in the technical calculators suite
• SHP to kW converter - shaft horsepower to kilowatts for unit-consistent constants
• HP to kW converter and kW to HP converter
• Speed-power cubic fit calculator - fit a power curve through several trial points
• Froude speed scaling calculator - place a speed and length in the right Froude regime
• Holtrop-Mennen resistance calculator - the detailed-design method the estimate hands off to
• EEXI attained calculator and EEDI attained calculator
• Minimum propulsion power calculator
• SFOC sensitivity calculator
• Ship resistance and powering
• Block coefficient and hull form coefficients
• Naval architecture coefficients
• EEXI, EPL, and ShaPoLi
• Slow steaming and slow steaming and CII
• Specific fuel oil consumption
• Marine diesel engine and marine propeller
• ShipCalculators.com calculator catalogue

References

• William Froude, “On experiments with HMS Greyhound,” Transactions of the Institution of Naval Architects, vol. 15 (1874): 36-73. The towing-tank work at Torquay and Haslar that gave the frictional-residual split its experimental footing.
• K.J. Rawson and E.C. Tupper, Basic Ship Theory, 5th ed. (Oxford: Butterworth-Heinemann, 2001), Vol. 2, resistance chapter, which presents the Admiralty coefficient in its modern form.
• E.V. Lewis (ed.), Principles of Naval Architecture, 2nd revision, Vol. II: Resistance, Propulsion and Vibration (Jersey City: SNAME, 1988), the standard treatment of the coefficient alongside Holtrop-Mennen and the systematic series.
• A.F. Molland, S.R. Turnock and D.A. Hudson, Ship Resistance and Propulsion: Practical Estimation of Ship Propulsive Power, 2nd ed. (Cambridge: Cambridge University Press, 2017).
• J. Holtrop and G.G.J. Mennen, “An approximate power prediction method,” International Shipbuilding Progress, vol. 29, no. 335 (1982): 166-170, and J. Holtrop, “A statistical re-analysis of resistance and propulsion data,” International Shipbuilding Progress, vol. 31, no. 363 (1984): 272-276.
• H. Tu, Y. Yang, L. Zhang, D. Xie, X. Lyu, L. Song, Y. Guan and J. Sun, “A modified admiralty coefficient for estimating power curves in EEDI calculations,” Ocean Engineering, vol. 150 (2018): 309-317.
• ISO 15016:2015, Ships and marine technology: Guidelines for the assessment of speed and power performance by analysis of speed trial data. International Organization for Standardization.
• ITTC Recommended Procedures and Guidelines 7.5-04-01-01.1, Preparation, Conduct and Analysis of Speed/Power Trials. International Towing Tank Conference.
• IMO Resolution MEPC.364(79), 2022 Guidelines on the Method of Calculation of the Attained Energy Efficiency Design Index (EEDI) for New Ships, adopted 16 December 2022.