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 run 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 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 derivation from Froude scaling, the units trap that catches the unwary, the fuel-coefficient variant, worked examples at real ship scale, and the boundaries where the estimate stops being trustworthy. 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 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 early studies still compute 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.

A cleaner way to state the same result uses the standard resistance form $R_T \propto \rho S V^n$ for small speed changes. With $S \propto \Delta^{2/3}$ and density fixed, $R_T \propto \Delta^{2/3} V^n$, and power $P \propto R_T V \propto \Delta^{2/3} V^{n+1}$. Setting the resistance exponent $n = 2$, which standard texts adopt for slow and medium-speed merchant hulls, recovers $P \propto \Delta^{2/3} V^3$ exactly. The choice $n = 2$ is the single assumption that fixes the cube, and it is the assumption that fails first as a ship speeds up, a point the exponent section returns to.

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, as the full resistance treatment in ship resistance and powering makes explicit.

Corresponding speeds and geometric similarity

Two hulls of the same form are at corresponding speeds when their Froude numbers match, so that $V \propto \sqrt{L}$, and since $L \propto \Delta^{1/3}$ this gives $V \propto \Delta^{1/6}$. That is the exact condition under which residual resistance per tonne is equal between them and $C$ stays constant. A designer comparing a 46,000-tonne handysize with a geometrically similar 180,000-tonne Capesize should not read one constant across both at the same absolute speed: the corresponding speed of the larger ship is higher by the sixth root of the displacement ratio, about 26 percent, and only at that matched Froude number does the shared constant hold. This corresponding-speed rule is the reason the method works between a ship and its true geometric sister and fails between ships of different length run at the same knots.

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. 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. This is also why some references that describe the coefficient as dimensionless are wrong: a dimensionless group would survive a change of units unchanged, and this one does not.

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 relationship between those stations runs through the shafting and stern tube that carries the power from engine to propeller. The second trap 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.

Typical values by ship type

The Admiralty constant is not universal. It depends on hull form, propulsion-train efficiency, hull-surface condition, displacement loading, and the speed regime. Published guidance gives the broad merchant band as about 350 to 600, higher being more efficient; Wartsila’s encyclopedia narrows this to 400 to 600. The figures below show the tendency by ship type for shaft power in kW, displacement in tonnes, and speed in knots, at design draft and service speed in calm water. They give an order of magnitude only. The constant should be determined individually from a basis ship, not read off a type table, because a type average hides the real propulsive efficiency and hull form.

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.

The fuel coefficient variant

The same algebra applies to fuel as to power. Daily fuel consumption at sea is delivered power times the specific fuel oil consumption times the hours run, so for a fixed engine and fuel it tracks power directly. Replacing power with tonnes of fuel per day in the relation gives the fuel coefficient, $C_{\text{fuel}} = \Delta^{2/3} V^3 / F$, where $F$ is the daily fuel burn. Rearranged, $F = \Delta^{2/3} V^3 / C_{\text{fuel}}$ estimates bunker burn straight from displacement and speed, which is why voyage estimators reach for it before any hydrodynamic model.

The fuel coefficient folds two things into the constant that the power coefficient keeps separate: the propulsive efficiency of the hull and the specific fuel oil consumption of the engine at its operating load. That makes it convenient for a quick voyage estimate and treacherous for a like-for-like hull comparison, because a change in $C_{\text{fuel}}$ can come from a fouled hull, a worn engine, a different fuel calorific value, or a shift in engine load, and the single number cannot separate them. An analyst who wants to isolate hull condition works in the power coefficient and applies the SFOC curve separately, so that a change in fuel burn splits cleanly into a hydrodynamic part and a machinery part.

Consider a handysize bulk carrier delivering 8,400 kW at sea with an SFOC of 170 g/kWh. Daily fuel is $8{,}400 \times 170 \times 24 / 10^6 = 34.3$ tonnes per day. If the same ship at the same speed later burns 37 tonnes per day, the fuel coefficient has fallen by about 8 percent, and the question is whether the hull fouled, the engine drifted off its clean SFOC, or the fuel changed. Only by holding SFOC and fuel constant does the fuel-coefficient trend become a clean hull-condition signal, which is why class and charter-party performance clauses specify the reference SFOC and fuel grade alongside the speed and consumption warranty.

Worked example: recovering the constant from a trial

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 inside the tendency band for a handysize, 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 displacement, the smallest of the three sensitivities at two-thirds power, is the term a surveyor worries about least.

Worked example: scaling power to a new speed

Hold the same hull at 46,500 tonnes and ask what it takes to make 15.0 knots instead of 14.2. Holding $C$ at 451, $P = \Delta^{2/3} V^3 / C = 1{,}292 \times 15.0^3 / 451 = 1{,}292 \times 3{,}375 / 451 = 9{,}670$ kW. The power climbs from 8,200 to 9,670 kW, about 18 percent, for a 5.6 percent speed increase, which is close to the cube-law expectation of $1.056^3 - 1 = 17.8$ percent. This is the arithmetic behind every slow-steaming and speed-up decision: the cube makes speed expensive, and a small knot gained near service speed costs a large slice of power. Whether the real cost is the clean cube or something steeper depends on how close the ship runs to its wave-making hump, which the exponent section takes up.

Worked example: scaling to a new displacement

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 451, $P = 48{,}000^{2/3} \times 14.5^3 / 451$. 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, 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 and lands close enough to bracket the engine selection and lay out the engine room before any model test.

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 estimate is later refined by the resistance chain described in ship resistance and powering and the hull form design study that fixes the lines.

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 regulatory minimum-power check runs.

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.

The word “corrected” carries weight here. A raw trial reading taken in real weather does not give a usable constant, because wind, waves, current, shallow water, and any deviation of displacement or trim from the contract condition all move the measured power. ITTC 7.5-04-01-01.1 and ISO 15016:2015 set the method that strips those effects out: runs are made on reciprocal headings to cancel current and tidal set, the wind and wave added resistance is subtracted, and the result is reduced to a still-water, deep-water, contract-displacement basis. Only the corrected power and speed belong in the coefficient. A constant computed from an uncorrected trial folds the day’s weather into the number and cannot be compared against a later docking, which defeats the whole purpose of tracking $C$ over time.

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 propeller side of that loss, roughness and blade wear, sits in the propeller theory and marine propeller articles.

EEXI and EEDI feasibility checks

The Energy Efficiency Existing Ship Index (EEXI) and the Energy Efficiency Design Index (EEDI) 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. The reference speed that feeds the EEDI computation is commonly derived from the displacement-speed-power relationship the coefficient expresses. 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 formal attained-index computation under MEPC.364(79). The interaction between installed power, the engine power limitation, 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 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 resistance components deep dive .

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 it is usually higher, not lower. Total resistance follows roughly $R_T \propto V^n$, where $n$ rises from about 2 at low Froude number toward 4 or more near the primary wave-making hump, so power scales as $P \propto V^{n+1}$. The classical Admiralty relation sets $n = 2$ and gets the clean cube, but T.H. Telfer argued in 1963 that no merchant ship is designed to run on a resistance varying as the square of the speed: he took $n = 3$ as the more realistic figure, which makes power vary as the fourth power of speed. Empirical guidance agrees that ships running above about 20 knots should use $V^4$ rather than $V^3$, because the cube underestimates the power demand once wave-making grows. A common container-ship convention takes the speed exponent as 3.5, 4.0, and 4.5 for feeder, medium, and jumbo sizes.

So the practical power exponent for displacement ships at service speed sits roughly between 3 and 4, which means a 10 percent speed increase demands 33 to 46 percent more power rather than the clean 33 percent the cube law alone predicts. A statistical study of in-service data found the resistance exponent $n$ climbing to between 3 and 6 above a threshold Froude number that depends on ship type and size. The Admiralty coefficient assumes the cube because that is the right average across the moderate-speed band, but a projection that spans a wide speed range, or reaches toward a hump, should use a fitted exponent instead of the fixed cube.

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 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 displacement exponent with hull-form coefficients so that the modified coefficient tracks the steeper power rise 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: 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 and 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 link from either method to the propeller and drivetrain runs through the marine propeller , propeller theory , and marine diesel engine articles.

The power chain the constant compresses

To see what the single number hides, follow the power from the water to the engine. Total resistance $R_T$ acting on the hull at speed $V$ sets the effective power $P_E = R_T V$, the power that would tow the bare hull at that speed with no propeller. The propeller cannot deliver that power for free: the delivered power at the propeller, $P_D$, exceeds $P_E$ by the quasi-propulsive coefficient $\eta_D$, so $P_D = P_E / \eta_D$. That coefficient is itself the product of three parts, the hull efficiency $\eta_H$ (which folds in the wake fraction and the thrust-deduction fraction), the propeller open-water efficiency $\eta_O$, and the relative rotative efficiency $\eta_R$. For a merchant hull $\eta_D$ typically runs 0.65 to 0.75, so the propeller and hull interaction alone accounts for a quarter to a third of the installed power.

From the propeller the power runs back up the line shaft to the engine. Shaft and bearing losses, and gearbox losses where a gearbox exists, sit between delivered power and the brake power $P_B$ at the engine flange, so $P_B = P_D / \eta_S$, with the transmission efficiency $\eta_S$ near 0.98 to 0.99 on a direct-coupled two-stroke and lower through a reduction gearbox. The Admiralty coefficient sits on top of this entire chain and swallows all of it into $C$. A constant fitted to shaft power carries the hull, the propeller, and the shaft losses of the reference vessel folded into one figure, which is why it reproduces a sister so well and a different hull form so poorly.

This is the practical reason the power basis has to travel with the constant. A $C$ recovered against $P_E$ describes the bare hull and would need the quasi-propulsive coefficient applied before it sized an engine; a $C$ recovered against $P_B$ already contains every loss down to the flange and sizes the engine directly. Mixing the two, taking a bare-hull constant and reading engine power off it, overstates the ship’s efficiency by the whole $\eta_D \eta_S$ product, roughly 30 percent. The ship resistance and powering article works the same chain quantitatively, and the marine propeller article covers where the open-water efficiency comes from.

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 fine, fast craft such as naval combatants. Shallow water and heavy weather break it too, because both add resistance the calm-water constant never saw. 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.

Related concepts

• Ship resistance and powering and resistance components deep dive
• Block coefficient and naval architecture coefficients
• Hull form design
• Marine propeller and propeller theory deep dive
• Marine propulsion shafting and stern tube and marine diesel engine
• Specific fuel oil consumption and SFOC curves
• Slow steaming and slow steaming and CII
• EEXI, EPL, and ShaPoLi and what is EEDI