Library / Dimensional analysis and similarity / DIM-22
Dimensional analysis and similarity
Propeller thrust from the thrust coefficient
Thrust of a propeller from its non-dimensional thrust coefficient, the fluid density, the rotational speed in revolutions per second and the diameter.
- Ref
- DIM-22
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- T [N]
T = CT ρ n² D⁴ gives the thrust of a propeller from its thrust coefficient CT, which comes from propeller data at the required advance ratio, blade geometry and pitch. n is the speed in revolutions per second and D the diameter. At matched coefficients, with s, d and z the speed, diameter and density ratios, thrust scales as z s² d⁴. Allow for changes in blade Reynolds number, tip Mach number and, in water, cavitation.
Thrust of a propeller from its non-dimensional thrust coefficient, the fluid density, the rotational speed in revolutions per second and the diameter. The coefficient depends on the advance ratio and the blade geometry.
Calculator
- Rotational speed1,200 rpm
- Thrust with the speed doubled, 4T12 N
- Thrust with the diameter doubled, 16T48 N
- Thrust per square metre of disc area15.28 N/m²
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With CT = 0.1 at 1,200 rpm, a 0.5 m propeller in a fluid of 1.2 kg/m³ gives 3 N of thrust.
- Thrust rises with the square of the speed and the fourth power of the diameter: doubling the speed quadruples the thrust, while doubling the diameter multiplies it by sixteen (at the same coefficient and advance ratio).
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| D Propeller diameter | 0.05 to 10 m | 3e-4 | 480,000 | 480,000 | +10% | +46.4 % |
| ρ Fluid density | 0.1 to 2,000 kg/m³ | 0.25 | 5,000 | 5,000 | +10% | +10 % |
| n Rotational speed | 1 to 200 rev/s | 0.0075 | 300 | 300 | +10% | +21 % |
| CT Thrust coefficient (from propeller data at the advance ratio) | 0.001 to 1 | 0.03 | 30 | 29.97 | +10% | +10 % |
Sweep of Rotational speed
| Rotational speed (rev/s) | Thrust (N) |
|---|---|
| 1 | 0.0075 |
| 25.88 | 5.021 |
| 50.75 | 19.32 |
| 75.63 | 42.89 |
| 100.5 | 75.75 |
| 125.4 | 117.9 |
| 150.3 | 169.3 |
| 175.1 | 230 |
| 200 | 300 |
Sweep one variable
Variables and how each one contributes
Thrust
What the equation gives you. Everything else feeds it.
Thrust coefficient (from propeller data at the advance ratio)
The thrust coefficient at the operating advance ratio, a plain number from propeller data. Raises the thrust in direct proportion. Raising it by 10 % (to 0.11) moves the result by +10 %, from 3 to 3.3 N.
Fluid density
The fluid density, in kg/m³ (air about 1.2, seawater about 1025). Raises the thrust in direct proportion. Raising it by 10 % (to 1.32) moves the result by +10 %, from 3 to 3.3 N.
Rotational speed
The rotational speed, in revolutions per second (rpm divided by 60). Raises the thrust with the square of the speed. Raising it by 10 % (to 22) moves the result by +21 %, from 3 to 3.63 N.
Propeller diameter
The propeller diameter, in m. Raises the thrust with the fourth power of the diameter, so diameter is the strongest lever. Raising it by 10 % (to 0.55) moves the result by +46.4 %, from 3 to 4.392 N.
Worked example with the values above
- Rotational and size terms: n²D⁴ = 20² × 0.5⁴ = 25 m²/s².
- Multiply by the density and coefficient: 0.1 × 1.2 × 25.
- T = 3 N.
Assumptions and limits
- CT is taken at the operating advance ratio for the actual blade geometry and pitch.
- Axial inflow, with the fluid density at the operating state and n in revolutions per second.
Common mistakes
- Putting the speed in rpm instead of revolutions per second.
- Using a coefficient defined with a different convention (for example based on angular speed or the radius).
- Using a static coefficient at a nonzero advance speed, or the reverse.
Statistics and uncertainty
The result is a product of powers, so to first order its relative error is the root sum of squares of each input’s relative error multiplied by its exponent (CT × 1, ρ × 1, n × 2, D × 4). The coefficient is the main uncertainty, since it depends on the advance ratio and the blade details, and the diameter error counts four times.
Where it is used
- Estimating the thrust of a drone, aircraft, fan or marine propeller from tabulated coefficients.
- Scaling thrust from a test propeller to another speed, size or fluid.
Maths you need
- Advance ratio: axial speed relative to the blade-tip speed
- Dimensionless coefficients: thrust, torque and power written without units