Admiralty Coefficient

Background and history

The relation that the Royal Navy’s Admiralty Department codified in the late nineteenth century, P = Δ^2/3 × V³ / C, predates dynamometer-instrumented sea trials and predates the experimental discovery of the friction-residual resistance split. It captured the practical observation that, for two ships of similar form operated in the same speed range, the ratio P × C / (Δ^2/3 × V³ ) is approximately one. The Admiralty’s interest was operational: given a planned displacement and a required speed, naval architects needed a power figure to specify boilers, engines, and bunker capacity at the early design stage when the hull lines were not yet drawn.

William Froude’s experimental work in the 1870s, conducted at the towing tank in Torquay, gave the relation a theoretical foundation. Froude established that ship resistance separates into a frictional component (a function of wetted surface area and the Reynolds number, scaling roughly with V^1.83 to V² ) and a residual or wave-making component (a function of the Froude number Fr = V / √(g·L), strongly nonlinear with speed). For geometrically similar ships operated at the same Froude number, the residual resistance per tonne of displacement is the same, and wetted surface scales as Δ^2/3 . The cubic speed term in the Admiralty relation captures the combined behaviour reasonably well in the moderate-speed regime where wave-making is significant but not yet dominated by hump speeds.

Through the twentieth century, with the development of systematic series (Taylor 1910, Series 60 from 1953, BSRA 1960s) and regression formulae (Holtrop-Mennen 1982, the present industry standard), the Admiralty Coefficient lost its role as the primary powering tool. It survived, however, as the rapid first-pass estimate that every early-stage design study still computes before committing model-tank or CFD resources. Modern usage acknowledges its limitations explicitly: it is a speed-similarity rule, valid only when the new vessel and the reference vessel operate at comparable Froude numbers and have similar block coefficients.

Formula and units

The Admiralty Coefficient relation in its standard form is:

P = Δ^(2/3) × V^3 / C

with the following conventions when C values are quoted from classical tables:

• P is delivered or shaft horsepower, traditionally in metric horsepower (mhp) and now usually in kilowatts. When converting between unit systems, both sides of the relation must use consistent units, since C is dimensional and absorbs the conversion factor.
• Δ is full-load displacement in tonnes (1 tonne = 1000 kg).
• V is speed through water in knots.
• C is the Admiralty constant, dimensionally complex (knots³ ·tonnes^2/3 /kW with kW power; different magnitude with mhp power) and tabulated by hull form.

Rearranging gives the constant directly from a known reference point:

C = Δ^(2/3) × V^3 / P

So for a ship of 50,000 tonnes displacement, 14 knots service speed, and 8,400 kW delivered power, C = 50,000^2/3 × 14³ / 8,400 = 1,357 × 2,744 / 8,400 ≈ 443. This figure can then be applied to a sister-ship at modestly different displacement or speed, recognising that the value drifts with deviations from design conditions.

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 varies, but the following ranges are commonly cited for delivered power at design draft and service speed in calm water:

The split between fuller-form vessels (low C) and finer-form vessels (high C) reflects the wave-making penalty paid by full hulls at typical service speeds. A bulk carrier with block coefficient C_b ≈ 0.85 sits firmly in the high-resistance regime relative to its displacement, while a container ship with C_b ≈ 0.65 cuts through the same speed range with lower residual resistance per tonne.

Use cases

Early-stage powering estimates

Before committing to a hull-form design study, naval architects use the Admiralty Coefficient to bracket the required engine MCR for a target displacement and speed. Selecting C from a peer-fleet table appropriate to the intended ship type, the relation P = Δ^2/3 × V³ / C gives a delivered-power figure that is generally within plus or minus 10 percent of the eventual model-tank result for conventional designs. This is enough precision to drive engine-room layout and bunker volume decisions in the concept design phase. The Admiralty power calculator implements this directly.

Sea-trial analysis

Following delivery, sea trials measure delivered power at contracted speeds. The Admiralty Coefficient is the natural metric for comparing the trial result against the design specification and against sister-ship trials. A C value below the contracted figure indicates the ship requires more power than predicted; above, less. Trends in C across a series of dockings (after hull cleaning, after propeller polishing) quantify the fouling penalty the operator pays between dry-dockings.

EEXI and EEDI back-of-envelope checks

The Energy Efficiency Existing Ship Index (EEXI) and the Energy Efficiency Design Index (EEDI) require demonstrating that a vessel’s attained efficiency is below a regulatory ceiling. Both indices depend on the engine MCR or limited MCR as input. When considering an EPL retrofit (engine power limitation), the Admiralty Coefficient projects what speed the ship will achieve at the limited MCR: V_limited ≈ V_original × (MCR_limited / MCR_original )^1/3 . This gives a fast feasibility filter before commissioning a full propulsion study. The EEXI attained calculator provides the regulatory computation; the Admiralty estimate informs whether a chosen MCR limit will produce an acceptable service speed.

Slow-steaming and operational savings

For an operator considering slow steaming, the Admiralty Coefficient projects the delivered-power saving from a given speed reduction. The cube-of-speed dependence means a 10 percent speed reduction yields roughly a 27 percent power reduction (1.0³ versus 0.9³ = 0.729), and a 20 percent reduction yields about 49 percent (0.8³ = 0.512). Real savings are smaller because C drifts as the ship moves out of its design speed regime, but the cube law sets the upper bound of the achievable saving and the SFOC sensitivity calculator translates the power change into fuel consumption.

Limitations

The Admiralty Coefficient is a similarity rule, not a physical model. Its accuracy degrades when:

1. The new ship operates at a Froude number different from the reference ship. The cubic speed term implicitly assumes residual resistance scales as V³ , which is a moderate-speed approximation. Near hump speeds (Froude number around 0.4 to 0.5) the actual exponent rises sharply.
2. The block coefficient differs significantly from the reference. A reference C drawn from a fine-form hull will overestimate the speed achievable for a full-form sister, and vice versa.
3. The ship is operating off design draft. Both wetted surface and the Froude number shift with displacement, and C drifts with both.
4. Wave-making is dominant. Twin-hull vessels, very fine-form fast craft, and ships near transition speeds violate the assumption of moderate residual resistance.
5. Hull condition deviates from the reference. Heavy fouling, propeller damage, or incorrect trim can shift C by 5 to 15 percent within a single docking interval.

For modern early-stage design, Holtrop-Mennen regression (Holtrop and Mennen 1982, with subsequent refinements) provides resistance estimates that respect Froude-number trends and block-coefficient sensitivity directly, and is the basis of the ship resistance and powering workflow at the detailed design stage. The Admiralty Coefficient remains the right tool when speed and simplicity matter more than precision.

Related Calculators

• Admiralty power calculator - primary implementation of the relation
• SHP to kW converter - historical horsepower units to kW
• HP to kW converter and kW to HP converter
• EEXI attained calculator
• SFOC sensitivity calculator
• Calculator catalogue

See also

• Ship resistance and powering
• Block coefficient
• EEXI, EPL, and ShaPoLi
• Slow steaming
• Marine propeller theory
• Bow thruster and stern thruster

Additional calculators:

• Admiralty Coefficient Power Calculator
• Service Allowance - Added Resistance
• EEDI Attained Calculator (MEPC.328(76))

References

• William Froude, “On the Comparison of Sailing Ships” (1874) and his Admiralty Reports on towing-tank experiments at Torquay (1872 to 1879).
• D. W. Taylor, The Speed and Power of Ships (1910; later editions through 1943) - the standard early reference, contains tabulated C values for the Taylor Standard Series. Full text available via HathiTrust and Internet Archive (search “Taylor Speed and Power of Ships”).
• J. Holtrop and G. G. J. Mennen, “An Approximate Power Prediction Method”, International Shipbuilding Progress, vol. 29, July 1982 - the regression-based alternative that displaced Admiralty as the early-design standard. Republished in TU Delft and MARIN technical archives.
• BSRA Series, Methodical Series Experiments (1960s) - British Ship Research Association systematic-series tank tests, source of much modern C tabulation for merchant hulls.
• E. V. Lewis (ed.), Principles of Naval Architecture, 2nd revision, vol. II Resistance, Propulsion and Vibration, published by SNAME (1988) - chapters 5 and 6 cover the Admiralty Coefficient in its modern context alongside Holtrop-Mennen and the systematic series.
• IMO Resolution MEPC.328(76), “Amendments to MARPOL Annex VI (EEDI Phase 3 and EEXI)” (2021) - the regulatory framework in which Admiralty estimates inform retrofit feasibility studies. Available from the IMO documents portal under Marine Environment Protection Committee resolutions.