A chain of efficiencies from the towrope to the engine. Most questions come down to knowing which link is which, and to remembering that the propeller sees the speed of advance, not the ship's speed.
A propeller in open water is a well-understood machine. Put it behind a hull and two things change, in opposite directions, and both have names worth knowing precisely.
Wake. The hull drags a layer of water along with it, so the water arriving at the propeller disc is moving slower than the ship. The propeller therefore sees a reduced speed, the speed of advance.
Wake comes from three sources: the boundary layer (viscous wake), the orbital motion of the ship's own wave system (wave wake), and the streamline flow closing in around the stern (potential wake). Taylor's rough approximation for a single-screw ship is w ≈ 0.5CB − 0.05.
Thrust deduction. A working propeller accelerates water past the stern, which lowers the pressure there and effectively increases the hull's resistance. The thrust the propeller must produce therefore exceeds the resistance measured when towing.
It is important to see that this is a hull effect, not a propeller loss. The propeller has not become less efficient; the hull has become harder to push.
Wake helps — the propeller works in slower water, so it needs less power. Thrust deduction hurts — more thrust is needed than the bare resistance. Their combination is hull efficiency.
Start at the towrope and work back to the engine. Each link divides, so the power grows at every step.
A design instinct worth carrying: for a given power, a larger, slower-turning propeller is more efficient, because it accelerates a larger mass of water by a smaller amount. Draught and hull clearance are what stop designers from making it larger still.
A propeller's performance in uniform flow is captured by three non-dimensional numbers, and the whole of propeller selection rests on them.
Plot KT, 10KQ and ηO against J and you have the open-water chart. KT and KQ fall as J rises; efficiency rises to a peak and then collapses as thrust runs out.
Open-water efficiency is thrust power out divided by shaft power in, and writing that ratio in coefficient form gives:
Two unit conventions cause most of the lost marks here, and both are worth writing at the top of the page before you start: n is in revolutions per second, and Va, not V, appears in J. Note also the different powers of D — fourth for thrust, fifth for torque.
Propeller geometry sits alongside: pitch ratio P/D sets how coarse the blade is, blade area ratio AD/AO sets how much area carries the thrust, and skew staggers the blade so that it enters the wake peak gradually rather than all at once — which is a vibration measure, not an efficiency one.
If a propeller were a screw turning in a solid nut, one revolution would advance the ship by exactly the pitch. It does not, because water yields — and it must yield, or no thrust would be produced at all.
Apparent slip is what a bridge log reports. It is contaminated by current: a following current raises the observed speed and can even drive apparent slip negative, which looks impossible until you remember the ship is being carried along by the water.
Real slip is the physically meaningful one and is always positive. Trended over weeks at similar draught and weather, a rising real slip is genuine evidence of hull or propeller fouling.
Water boils when the local pressure drops to its vapour pressure — at ambient temperature, if the pressure falls far enough. On the back of a propeller blade, where the flow accelerates and pressure drops, that is exactly what can happen.
The vapour cavities collapse violently when they reach a region of higher pressure. Three consequences follow: thrust breakdown as the blade loses its low-pressure side, erosion as repeated implosions pit the metal, and noise and vibration.
Here h is the depth of the blade section — which is why cavitation is worst at the top of the disc, where immersion and hence pressure are least, and why a lightly ballasted ship in heavy weather is the classic condition for it.
The defence is blade area. Spreading the same thrust over more area lowers the mean thrust loading and raises the pressure on the back of the blade. Burrill's diagram plots thrust loading against cavitation number with limit lines for acceptable cavitation, and increasing the developed area ratio is how a designer moves back below the line.
Note the cost: more blade area means more frictional drag, so efficiency falls slightly. Blade area ratio is always a compromise between cavitation margin and efficiency.
The first walks the whole efficiency chain. The second is the open-water chart question that appears in some form almost every year. The third is the slip question that tests whether you understand what you are measuring.
A ship requires an effective power of 4000 kW at 16 knots. The wake fraction is 0.25, the thrust deduction fraction 0.18, the relative rotative efficiency 1.02 and the open-water efficiency 0.62. Shaft transmission efficiency is 0.98. Find the speed of advance, the hull efficiency, the delivered power and the brake power required.
P_E = 4000 kW, V = 16 kn w = 0.25, t = 0.18, η_R = 1.02, η_O = 0.62, η_S = 0.98
Find the speed of advance, the hull efficiency, the delivered power and the brake power required
Speed of advance first.
The hull drags water along with it, so the propeller works in a stream moving slower than the ship.
Hull efficiency.
It compares the useful work done on the hull with the work done by the propeller on the water it actually sees.
Note that it exceeds 1.
That is not an error and it is a favourite conceptual question: the propeller recovers some of the energy already lost into the wake, so the hull-and-propeller combination does better than the propeller alone would in open water.
Quasi-propulsive coefficient.
The product of the three efficiencies between towrope and propeller shaft.
Delivered power at the propeller.
Brake power at the engine.
| Stage | Divided by | Power (kW) |
|---|---|---|
| P_E effective | — | 4000 |
| P_D delivered | η_D = 0.691 | 5789 |
| P_B brake | η_S = 0.98 | 5907 |
After transmission losses in shafting and bearings.
AnswerV_a = 6.17 m/s, η_H = 1.093, P_D = 5789 kW, P_B ≈ 5910 kW
The trap: answering with the wrong power. P_E → P_D → P_S → P_B, each larger than the last. Read the question twice and note which one it wants before you start.
A propeller of 5.5 m diameter turns at 110 rpm behind a hull whose wake fraction is 0.28 at a ship speed of 15 knots. From the open-water chart at the resulting advance coefficient, K_T = 0.185 and K_Q = 0.0290. Find the thrust, the torque, the delivered power and the open-water efficiency.
D = 5.5 m, N = 110 rpm, V = 15 kn, w = 0.28 K_T = 0.185, K_Q = 0.0290, ρ = 1025 kg/m³
Find the thrust, the torque, the delivered power and the open-water efficiency
Get the units right before anything else.
The coefficients are defined with n in revolutions per second, not per minute, and with speed of advance, not ship speed.
Advance coefficient.
The number that located K_T and K_Q on the chart in the first place.
Thrust from K_T.
Torque from K_Q.
Note the fifth power of diameter — a common slip is to use the fourth.
Delivered power is torque times angular velocity.
Open-water efficiency — thrust power out over shaft power in.
The formula is just that ratio written in coefficient form.
Cross-check it directly, which is always worth thirty seconds.
AnswerT = 583 kN, Q = 503 kN·m, P_D = 5790 kW, η_O = 0.559
The trap: using ship speed in J. The wake fraction exists precisely because the propeller does not see the ship's speed, and substituting V for V_a shifts J by 28 % — which lands you on a completely different part of the chart.
A ship's propeller has a pitch of 4.8 m and turns at 95 rpm. The ship's log shows 13.5 knots. The wake fraction is 0.30. Find the apparent slip and the real slip. The next day, in the same conditions, the apparent slip is reported as −2 %. What does that tell you?
P = 4.8 m, N = 95 rpm, V = 13.5 kn, w = 0.30
Find the apparent slip and the real slip
What does that tell you?
Theoretical speed.
How fast the ship would advance if the propeller were a screw in a solid nut, losing nothing.
Convert the observed speed and find apparent slip.
Which uses ship speed.
Real slip uses speed of advance instead.
The water the propeller is actually working in.
Real slip is always the larger, and always positive.
The propeller must slip relative to the water to generate thrust at all. Apparent slip is a navigational quantity contaminated by whatever the water itself is doing.
Now the negative reading.
Apparent slip goes negative when the ship's observed speed over the ground exceeds the theoretical propeller speed — which cannot happen through the water, but happens easily with a following current.
This is why hull fouling is trended on real slip.
At similar draught and weather, over weeks. Apparent slip on any single day says more about the current than about the hull.
AnswerApparent slip = 8.6 %, real slip = 36.0 %; a negative apparent slip means a following current
The trap: treating a change in apparent slip as evidence about the hull. Only real slip, trended at comparable draught and weather, means anything about fouling.
V_a = V(1 − w)w = (V − V_a)/V; Taylor: w ≈ 0.5C_B − 0.05R = T(1 − t)t = (T − R)/T — a hull effect, not a propeller lossη_H = (1 − t)/(1 − w)Often greater than 1J = V_a/(n·D)n in rev/s, V_a not VK_T = T/(ρn²D⁴) K_Q = Q/(ρn²D⁵)Note the different powers of Dη_O = J·K_T/(2π·K_Q)Open-water efficiencyη_D = η_O·η_H·η_RQuasi-propulsive coefficient, typically 0.65–0.75P_D = 2π·n·Q = P_E/η_DDelivered power at the propellerApparent slip = (P·n − V)/(P·n)Real slip uses V_a; only real slip means anythingσ = (p₀ + ρgh − p_v)/(½ρV_R²)Cavitation number; cure with blade area