Marine Resistance and Propulsion Exercises

These exercises practise marine resistance and propulsion calculations as naval engineering evidence. They cover Froude number, Reynolds number, effective power, delivered power, service margin, propeller advance ratio, thrust and torque coefficients, cavitation screening, fuel consumption, and sea-trial interpretation.

The goal is not only to calculate a power value. The goal is to decide whether hull, propulsor, machinery, operating profile, loading condition, and validation evidence support the same performance claim.

Assume simplified screening models unless an exercise states otherwise. Real vessel design needs hull-specific resistance prediction, propulsion data, wake measurements, sea-margin policy, machinery limits, cavitation analysis, classification rules, and trial correction procedures.

How to Use These Exercises

For each exercise, define:

1. vessel speed, displacement, draft, trim, and water density;
2. hull condition and propeller condition;
3. calm-water, service, or trial context;
4. efficiency assumption and its boundary;
5. evidence needed before accepting the result.

The common mistake is comparing numbers from different boundaries: effective power, delivered power, brake power, electrical power, and fuel power are not interchangeable.

Exercise 1: Froude Number

A vessel has waterline length:

and speed:

Compute Froude number:

Solution

Compute denominator:

Then:

Engineering Comment

Froude number indicates the importance of wave-making effects. A value near 0.4 can be demanding for many displacement hulls, so resistance may rise strongly with speed.

Exercise 2: Reynolds Number

Use seawater density:

dynamic viscosity:

speed:

and length:

Compute Reynolds number:

Solution

Substitute:

Engineering Comment

The full-scale Reynolds number is very high and the boundary layer is turbulent over most of the hull. Model tests cannot match Froude and Reynolds scaling simultaneously, so viscous corrections matter.

Exercise 3: Effective Power

At a trial condition, estimated total resistance is:

and vessel speed is:

Compute effective power:

Solution

Convert:

Compute:

Therefore:

Engineering Comment

Effective power is the power needed to tow the hull at that speed. It is not the engine brake power, shaft power, or fuel power. Propulsive and mechanical losses must be included separately.

Exercise 4: Delivered Power from Propulsive Efficiency

Using effective power:

and overall propulsive efficiency:

estimate delivered power:

Solution

Compute:

Engineering Comment

The delivered-power estimate depends strongly on the efficiency boundary. It should be clear whether gearbox, shaft, propeller, hull interaction, and appendage effects are included.

Exercise 5: Service Power Margin

A clean calm-water delivered-power estimate is:

The owner requires a service margin of:

Estimate service delivered power:

Solution

Substitute:

Engineering Comment

Service margin accounts for waves, wind, fouling, aging, loading variation, steering corrections, and machinery degradation. It should be based on the vessel mission, not applied blindly.

Exercise 6: Propeller Advance Ratio

A propeller diameter is:

shaft speed is:

and advance speed into the propeller is:

Compute advance ratio:

Solution

Substitute:

Engineering Comment

Advance ratio connects vessel speed, wake, propeller diameter, and rpm. Propeller efficiency, thrust, torque, and cavitation risk are functions of the operating point, not fixed constants.

Exercise 7: Propeller Thrust Coefficient

A propeller produces thrust:

in seawater:

with:

and:

Compute thrust coefficient:

Solution

Convert:

Compute denominator:

Then:

Engineering Comment

The coefficient is useful only with matching propeller geometry and advance ratio. Compare it with open-water curves or self-propulsion data at the same operating point.

Exercise 8: Propeller Torque Coefficient

Propeller torque is:

Use:

Compute:

Solution

Convert:

Denominator:

Then:

Engineering Comment

Torque coefficient helps connect propeller loading to shaft torque and machinery limits. Shafting, gearbox, bearing, and motor or engine ratings must be checked at transient as well as steady conditions.

Exercise 9: Shaft Power from Torque and Speed

Using propeller torque:

and shaft speed:

estimate shaft power:

where:

Solution

Angular speed:

Power:

Therefore:

Engineering Comment

Compare this with delivered-power estimates and machinery ratings. Differences may reveal boundary errors, efficiency assumptions, or operating points that do not match the resistance estimate.

Exercise 10: Cavitation Pressure Screen

A propeller blade section operates at depth:

below the free surface. Use seawater density:

and atmospheric pressure:

Estimate static absolute pressure:

Solution

Hydrostatic term:

Absolute pressure:

Engineering Comment

This static pressure is only the starting point. Local blade suction can reduce pressure sharply. Cavitation screening must include vapor pressure, blade loading, wake nonuniformity, rpm, water temperature, and maneuvering conditions.

Exercise 11: Fuel Consumption Estimate

A diesel engine delivers brake power:

for:

with specific fuel consumption:

Estimate fuel mass.

Solution

Energy output:

Fuel mass:

Convert:

Engineering Comment

This estimate assumes constant power and SFC. Real fuel use depends on load point, generator or gearbox losses, hotel load, sea state, fouling, fuel temperature, density, and machinery condition.

Exercise 12: Sea-Trial Evidence Checklist

A speed trial reports:

at shaft power:

List the evidence needed before using this as a service-performance claim.

Solution

The trial record should include:

1. displacement, draft, trim, and ballast state;
2. water depth, wind, waves, current, and water temperature;
3. hull and propeller condition;
4. shaft speed, shaft torque or power-meter calibration;
5. fuel flow if endurance or economy is claimed;
6. rudder angle and steering activity;
7. machinery configuration and auxiliary loads;
8. measurement uncertainty and correction method;
9. repeated runs in reciprocal directions where appropriate;
10. comparison with predicted speed-power curve.

Engineering Comment

A single speed-power point is not a universal performance guarantee. Trial data become engineering evidence only when boundary conditions and corrections are clear.