What is the propeller slip at this speed and rpm?

Formula, assumptions, and limits

A 4.5 m pitch propeller turning at 105 rev/min would advance 472.5 m a minute through a solid nut, which is 15.31 knots. At 13.5 knots through the water, the ship makes good 88.19% of that, so the apparent slip is 11.81%.

Formula

• $s_A$: apparent slip ratio, shown as a percentage.
• $s_R$: real slip ratio, shown as a percentage.
• $P$: propeller pitch, in meters, at 0.7 of the radius.
• $n$: rate of rotation in rev/s; the calculator takes $N$ in rev/min, with $n = N/60$.
• $V$: ship speed, in knots; 1 knot is 1,852 m an hour exactly.
• $V_A$: speed of advance, the speed of the water arriving at the propeller.
• $w$: Taylor wake fraction, $w = (V - V_A)/V$, dimensionless.

The engine speed is $V_E = 60\,P\,N/1852$ knots and the engine distance per day is $1440\,P\,N/1852$ nautical miles.

Derivation

A screw of pitch $P$ advances one pitch length per turn in a solid nut, so at $n$ turns a second it would move forward at $P\,n$. In water it moves forward less, because the water yields and accelerates aft under the thrust. Principles of Naval Architecture (SNAME, 1988, Vol. II, Ch. VI section 2.5) calls the shortfall the slip and its ratio to $P\,n$ the real slip ratio. The ITTC Dictionary of Hydromechanics (2024) gives the same definition, $s_R = (Pn - V_A)/(Pn)$, with $V_A$ the speed of advance.

$V_A$ is hard to measure on a ship in service, because the hull drags a layer of water forward and the propeller works in it. Taylor expressed that wake as a fraction of the ship’s speed, $V_A = V(1 - w)$. The apparent slip ratio replaces $V_A$ with the ship’s speed $V$, which the log or the receiver gives directly. Substituting one into the other:

So for any positive wake fraction, real slip exceeds apparent slip while the ship is moving; at bollard pull both are 100%. MAN Energy Solutions’ Basic principles of ship propulsion (5510-0004-05ppr, April 2023, p. 22) gives the same two equations as EQ 21 and EQ 22, and notes that the crew can work out apparent slip as an indication of propeller load, while real slip needs a speed of advance the crew does not know.

Assumptions

• Pitch is one figure for the whole blade. A propeller whose pitch varies with radius is represented by its pitch at 0.7R or its effective pitch, and the choice changes the result.
• Shaft speed and ship speed are averages over the same period.
• The wake fraction is constant at the loading condition entered.
• The ship speed basis is the one selected: through the water or over the ground. The calculator does not remove current from a speed over the ground.

Worked example

Take a pitch of 4.5 m, 105 rev/min and 13.5 knots through the water.

1. Engine distance per minute: 4.5 × 105 = 472.5 m.
2. Engine speed: 472.5 × 60 / 1852 = 15.3078 knots.
3. Apparent slip: 1 − 13.5 / 15.3078 = 0.1181, or 11.81%.
4. Over a day: engine distance 367.39 nm against 13.5 × 24 = 324.0 nm made good.
5. With a wake fraction of 0.30: $V_A$ = 13.5 × 0.70 = 9.45 knots, and real slip = 1 − 9.45 / 15.3078 = 38.27%.

The same answer comes through the advance coefficient. With a 6.0 m diameter, $J = V_A/(nD)$ = 4.8615 m/s / (1.75 rev/s × 6.0 m) = 0.4630, and $s_R = 1 - J/(P/D)$ = 1 − 0.4630 / 0.75 = 38.27%.

Edge cases and limits

Apparent slip below zero is a valid result, not an error. It appears whenever the speed entered exceeds $P\,n$: 16.0 knots on the same propeller at 105 rev/min gives −4.52%. The calculator flags a result below −10% or above 40% as unusual, a screening band of this site’s own and not an ITTC limit, since a pitch typed in feet and read as meters, or rev/s typed for rev/min, moves the figure by hundreds of percent.

A ship speed of zero gives an apparent slip of exactly 100%: the bollard pull case, where MAN’s paper notes that all the thrust goes into accelerating water aft. The calculator labels it as bollard pull rather than as a unit error.

A pitch of zero or a shaft speed of zero has no slip, because the engine distance is zero. A controllable pitch propeller at zero pitch is in this state.

Taylor’s wake values are from merchant ships of normal form, and Principles of Naval Architecture limits them to Froude numbers below 0.3. A fast twin-screw hull can have a wake fraction near zero or below it, in which case real and apparent slip are close or swap order.

Regulatory basis

No IMO convention sets a limit on propeller slip. The definitions come from the ITTC Dictionary of Hydromechanics, Version 2024, under “Slip ratio, apparent” and “Slip ratio, real”, unchanged from the 2017 edition. ITTC Recommended Procedure 7.5-04-01-01.1, Revision 08 (2024), which governs speed and power trials, defines the ship’s speed through the water as the speed over the ground less the current, $V_S = V_G - V_C$, and gives propeller pitch at 0.7R. ISO 19030-2:2016, the default method for monitoring hull and propeller performance, requires speed through the water from a log accurate to ±1% or ±0.1 knots, whichever is greater, and expresses performance as a change in speed at a given delivered power rather than as slip.

Common errors

• Pitch unit confused. A 4.5 m pitch is 14.76 ft, and a 4.5 ft pitch is 1.37 m.
• Shaft speed in rev/s where rev/min is asked for, or the reverse.
• Speed over the ground taken as speed through the water, so the current is counted as propeller performance.
• Reading a rise in slip as a propeller fault. MAN’s paper ties slip to propeller load: added resistance from fouling or weather raises the slip at a given rpm before anything is wrong with the propeller.
• Calling the logbook figure real slip. Real slip needs the wake fraction.
• Averaging rpm over the steaming time but speed over the whole day, including time stopped.

Log speed against GPS speed

The speed basis decides the answer more often than the arithmetic does. Take the worked example’s propeller, 4.5 m pitch at 105 rev/min, and the same 13.5 knots through the water on three days:

The propeller did the same work on all three days. Only the log figure shows that. A noon report that switches between log and GNSS speed moves the slip by several points with no change on board, which is why ITTC trials correct the speed over the ground to speed through the water before any analysis.

Which pitch to enter

The ITTC Dictionary of Hydromechanics separates five pitch figures, and the slip changes with the one chosen:

Principles of Naval Architecture (Ch. VI section 3.2) shows why it matters: on an open-water test of a propeller with a face pitch ratio of 1.00, thrust did not vanish at zero slip but nearer an advance ratio of 1.10, because the cambered blade sections lift at zero angle. Slip worked on face pitch reads low on such a propeller and can reach zero or below while it is still pushing the ship.

Taylor wake fractions

Real slip needs a wake fraction. Model tests for the hull give the best figure; without them, Principles of Naval Architecture, Vol. II, Table 4 reproduces Taylor’s values from the trials of more than 150 ships:

MAN Energy Solutions gives typical ranges by stern arrangement in Table 2.02 of Basic principles of ship propulsion (2023, p. 24), with the thrust deduction fraction alongside:

The two sources agree for single-screw merchant hulls: every Taylor value from a block coefficient of 0.50 to 0.85 sits inside MAN’s 0.20 to 0.45 range. Taylor’s twin-screw values run negative for fine hulls, where the propellers sit outboard of most of the wake. A full single-screw tanker or bulk carrier at a block coefficient of 0.85 works in water moving at 60% of its speed, so its real slip is far above its apparent slip.

Where to find each input

Pitch is on the propeller drawing, usually as a table of pitch by radius with the 0.7R value marked, and on the maker’s certificate. A controllable pitch propeller shows its running pitch on the bridge and engine room indicators; ITTC 7.5-04-01-01.1 has those indicators adjusted to measured ahead, design and astern pitch before trials. Shaft speed comes from the revolution counter, read as the counter difference over the period rather than an instant rpm. Speed through the water comes from the Doppler or electromagnetic log, and speed over the ground from the GNSS receiver or the distance between noon positions.