Hull efficiency is the ratio of the effective power (the power required to tow the hull) to the thrust power (the power developed by the propeller in producing thrust). It accounts for the wake and the thrust deduction:

Hull efficiency = (1 - t)/(1 - w)

where t is the thrust deduction factor and w is the wake fraction. It reflects how well the hull and propeller interact: the wake reduces the speed of advance of the propeller (reducing the power needed), while the thrust deduction increases the thrust required.

Propeller efficiency (open-water efficiency) is the ratio of the thrust power to the delivered power:

Propeller efficiency = T x Va / (2 pi n Q)

where T is the thrust, Va the speed of advance, n the revolutions and Q the torque. It represents the efficiency of the propeller itself in converting the delivered power into thrust power, and depends on the propeller design (pitch, diameter, blade area) and the loading.

A propeller of 4.8 m pitch turns at 110 rev/min. The apparent slip is -S% and the real slip is +1.5S%. The wake speed is 25% of the ship speed. Calculate the ship speed, the apparent slip and the real slip.

Pitch speed = pitch x rev/s = 4.8 x 110/60 = 8.8 m/s.

Let the ship speed be V (m/s). The speed of advance Va = V x (1 - 0.25) = 0.75 V (wake = 25% of ship speed).

Apparent slip = (pitch speed - V)/pitch speed = -S/100.

Real slip = (pitch speed - Va)/pitch speed = 1.5 S/100.

From the apparent slip: (8.8 - V)/8.8 = -S/100, so V = 8.8(1 + S/100).

From the real slip: (8.8 - 0.75 V)/8.8 = 1.5 S/100, so 8.8 - 0.75 V = 8.8 x 1.5 S/100 = 0.132 S.

Substitute V = 8.8(1 + S/100): 8.8 - 0.75 x 8.8(1 + S/100) = 0.132 S.

8.8 - 6.6(1 + S/100) = 0.132 S.

8.8 - 6.6 - 0.066 S = 0.132 S.

2.2 = 0.198 S, so S = 11.11.

Apparent slip = -11.11%; real slip = 1.5 x 11.11 = 16.67%.

Ship speed V = 8.8(1 + 0.1111) = 8.8 x 1.1111 = 9.78 m/s = 9.78 x 1.944 = 19.0 knots.

Answer: ship speed about 19.0 knots; apparent slip -11.1%; real slip +16.7%.