Calculations with context

Propeller open-water coefficients

Interpolate a supplied J–KT–actual KQ table without extrapolation; inspect thrust, torque, shaft power and unclipped open-water efficiency.

Propeller open-water coefficient workspace

Interpolate a declared J–KT–KQ table and inspect thrust, torque and open-water efficiency. The default curve is deliberately synthetic and does not represent a manufacturer, vessel or validated propeller.

J = VA/(nD) · T = ρn²D⁴KT · Q = ρn²D⁵KQ · PD = 2πnQ · η₀ = TVA/PD = JKT/(2πKQ)

Enter a teaching case

Use a decimal point, no thousands separators. Scientific notation is accepted. Bounds are numerical teaching controls, not evidence that a vessel or propeller is validated.

Meaning of speed input

Declared speed chosen unitAdvance mode: VA. Ship/wake mode: VS. Converted speed must be 0–50 m/s.

Speed unit

Propeller rotational speed rev/min0.1–6,000 rev/min. n = rpm/60; n must be positive.

Propeller diameter D m0.01–20 m. Fixed for this table calculation.

Water density ρ kg/m³100–2,000 kg/m³. Does not establish model/full-scale similarity.

User table: J, KT, actual KQ2–30 rows, no header. One J,KT,KQ triplet per line. J: 0–3, strictly increasing by at least 0.000001; KT: 0–2; actual KQ: 0.000001–1. Use decimal points and comma separators. Do not enter 10KQ as KQ. No automatic coefficient rescaling.

Default values are synthetic teaching data. Replace the entire table with a documented compatible dataset before any interpretation of real equipment. This workspace does not verify that dataset.

The initial result below is the synthetic/default teaching case.

Calculated example
Quantity	Value	Unit
VS	Not defined	m/s
VA	6	m/s
n	2	rev/s
J	0.6	1
KT	0.25	1
Actual KQ	0.045	1
Thrust T	640.625	kN
Torque Q	576.5625	kN·m
Shaft power at propeller PD	7245.2981	kW
Useful open-water power TVA	3843.75	kW
Open-water efficiency η₀ (unclipped)	0.53051648	1

Physical interpretation

Input shaft power PD = 2πnQ is related to propeller thrust T and useful TVA power. No ship resistance balance is imposed. Schematic only; arrow lengths and drawing dimensions are not to scale.

Substitution using these inputs

VA = 6 m/s
n = 120 / 60 = 2 rev/s
J = 6 / (2 × 5) = 0.6
α = (0.6 − 0.4) / (0.8 − 0.4) = 0.5
KT = 0.3 + 0.5 × (0.2 − 0.3) = 0.25
KQ = 0.05 + 0.5 × (0.04 − 0.05) = 0.045
T = 1025 × 2² × 5⁴ × 0.25 = 640625 N
Q = 1025 × 2² × 5⁵ × 0.045 = 576562.5 N·m
PD = 2π × 2 × 576562.5 = 7245298.1 W
TVA = 640625 × 6 = 3843750 W
η₀ = 0.6 × 0.25 / (2π × 0.045) = 0.53051648
PD − 2πnQ = 0 W; η₀ − TVA/PD = 0.0000000000000001110223

Curve and numerical values

Solid line and circles show KT; dashed line and squares show actual KQ. Both axes are dimensionless. Piecewise straight segments join only the entered J range. Vertical dotted line marks the current advance ratio. The complete supplied rows and computed efficiencies are in the following table.

Input rows; KQ is the actual torque coefficient. Efficiencies are computed from each row, not entered.
Row	J	KT	Actual KQ	η₀
1	0	0.4	0.06	0
2	0.4	0.3	0.05	0.38197186
3	0.8	0.2	0.04	0.63661977
4	1.2	0.08	0.025	0.61115498

Speed sensitivity

Only the declared input speed is multiplied by 0.5, 0.75, 1, 1.25 and 1.5. rpm, D, ρ, the supplied table, speed mode and any explicit w are held fixed. J, KT and KQ are recomputed. A point outside the curve or teaching bounds is unavailable; there is no extrapolation.

Only the declared input speed is multiplied by 0.5, 0.75, 1, 1.25 and 1.5. rpm, D, ρ, the supplied table, speed mode and any explicit w are held fixed. J, KT and KQ are recomputed. A point outside the curve or teaching bounds is unavailable; there is no extrapolation.
V ×	V · m/s	VA · m/s	J	KT	Actual KQ	T · kN	Q · kN·m	PD · kW	η₀	Status
0.5	3	3	0.3	0.325	0.0525	832.8125	672.65625	8452.8477	0.29557347	Calculated from the current inputs.
0.75	4.5	4.5	0.45	0.2875	0.04875	736.71875	624.60938	7849.0729	0.42237273	Calculated from the current inputs.
1	6	6	0.6	0.25	0.045	640.625	576.5625	7245.2981	0.53051648	Calculated from the current inputs.
1.25	7.5	7.5	0.75	0.2125	0.04125	544.53125	528.51563	6641.5232	0.61491683	Calculated from the current inputs.
1.5	9	9	0.9	0.17	0.03625	435.625	464.45312	5836.4901	0.67174362	Calculated from the current inputs.

The CSV is generated locally and includes inputs, units, assumptions, model version, sources, interpolation details, every supplied curve row and sensitivity rows where applicable. Extra digits describe arithmetic, not experimental accuracy.

Model and equations

A coefficient table is data, not a propeller design. This workspace accepts only ahead rotation with positive n, nonnegative advance speed and thrust coefficient, and positive torque coefficient. Negative-thrust, braking, reverse and windmilling quadrants require a different model and are rejected here.

ITTC defines n in revolutions per second. Convert rpm by dividing by 60 before applying n² or n³. The actual torque coefficient is KQ = Q/(ρn²D⁵). Some datasets may label an ordinate 10KQ: that ordinate is ten times KQ, so confirm the source and convert it yourself before entry. The program never guesses which convention a source intended.

VA is the undisturbed axial advance speed used for the open-water representation. It is not automatically ship speed VS. Direct mode takes VA as supplied and leaves VS unknown. Ship/wake mode explicitly assumes the scalar relation VA = VS(1−w). The entered w is an assumption, not a prediction of hull interaction or an effective-wake measurement.

For adjacent rows (J₁, KT₁, KQ₁) and (J₂, KT₂, KQ₂), let α = (J−J₁)/(J₂−J₁). Interpolate KT = KT₁ + α(KT₂−KT₁), and KQ in the same way. Exact row matches use that row. Efficiency is calculated from these coefficients; it is not linearly interpolated from endpoint efficiencies. Outside the table, no answer is supplied.

The dimensions provide useful checks: ρn²D⁴ has units N and ρn²D⁵ has units N·m. Shaft angular speed is 2πn radians per second, so PD = 2πnQ has units W. Useful open-water power is TVA, yielding η₀ = TVA/PD. PD is power delivered at the represented propeller; it is not engine brake power.

With positive input torque, η₀ > 1 violates the declared motoring energy interpretation. Such raw results remain visible and flagged, including inconsistencies at other table rows or inside interpolation intervals. The check tests the quadratic JKT−2πKQ throughout each linear segment. Passing this necessary energy check does not validate blade geometry, test quality, similarity or performance.

Symbols and units

Symbols and units
Symbol	Quantity	Unit
VA / VS	Advance speed / explicitly declared ship speed	m/s
w	User-assumed scalar wake fraction	1
n / rpm	Rate of rotation / revolutions per minute	rev/s / rev/min
D / ρ	Diameter / fluid density	m / kg/m³
J	Advance ratio VA/(nD)	1
KT / KQ	Thrust / actual torque coefficient	1
T / Q	Thrust / torque	N / N·m
PD / TVA	Propeller shaft power / useful open-water power	W
η₀	Open-water efficiency; no clipping	1

Worked example

The synthetic table contains (J,KT,KQ) = (0,0.4,0.06), (0.4,0.3,0.05), (0.8,0.2,0.04), (1.2,0.08,0.025). Load the example with VA = 6 m/s, rpm = 120, D = 5 m and ρ = 1025 kg/m³. No measured curve or named propeller is implied.

n = 120/60 = 2 rev/s. J = 6/(2×5) = 0.6. Between J = 0.4 and 0.8, α = (0.6−0.4)/(0.8−0.4) = 0.5. Therefore KT = 0.3 + 0.5(0.2−0.3) = 0.25, and KQ = 0.05 + 0.5(0.04−0.05) = 0.045.

T = 1025 × 2² × 5⁴ × 0.25 = 640625 N. Q = 1025 × 2² × 5⁵ × 0.045 = 576562.5 N·m. PD = 2π × 2 × 576562.5 ≈ 7245298.057 W. TVA = 640625 × 6 = 3843750 W; η₀ ≈ 0.530516477.

The explicit wake example gives the same VA by taking VS = 7.5 m/s and assumed w = 0.2: VA = 7.5(1−0.2) = 6 m/s. The identical arithmetic demonstrates the assumed transformation only. It does not prove that w is correct or determine thrust deduction, hull efficiency, relative rotative efficiency, cavitation or the vessel’s attainable speed.

At VA = 0, J = 0 only if the entered table includes that endpoint. A positive KT and KQ can then give finite thrust, torque and shaft power but TVA = η₀ = 0. A stationary propeller n = 0 is outside this model because J and the coefficients cannot be used through division by zero.

Assumptions and limits

The table is user data with a bounded numerical format, not a supplied test certificate. The starting values are synthetic. Data may depend on propeller geometry, Reynolds number, immersion and experimental conditions that this workspace does not model.
Piecewise linear interpolation is an explicit teaching approximation, not the ITTC test procedure or a fitted physical law. It cannot recover unresolved humps or changes between sparse rows. Two rows are permitted but provide only one straight segment; no extra information is invented. The minimum-J-gap check allows only a scale-aware four-machine-epsilon rounding margin. Strict order and the no-extrapolation boundary are unchanged.
The model accepts J ≥ 0, KT ≥ 0, KQ > 0 and n > 0 only. Negative/reversed, zero-torque, four-quadrant and windmilling operation are rejected. η₀ > 1 is flagged, not silently made equal to 1. Even η₀ within [0,1] is not sufficient evidence of physically consistent data.
No cavitation onset, ventilation, strength, noise, hull interaction, scale extrapolation, wake prediction, propeller–engine matching, sea-trial performance or attainable ship speed is inferred. The cited ITTC open-water test definitions concern model-scale testing and do not themselves provide full-scale prediction.
Changing D or rpm while reusing the same coefficients does not establish similarity. This workspace does not couple thrust to a ship-resistance curve or apply a thrust-deduction factor. Shaft power here cannot be read as engine power without additional losses and propulsion relationships.

Primary sources

Public ITTC equations checked on 9 October 2026. This simplified workspace does not implement the complete test or extrapolation procedures.

propulsion-teaching-1.0.1

Related context

The method explanation and worked example are on this page. The articles below provide additional context.

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