Froude theory for propulsion and traction propellers

Froude theory for propulsion propellers shows that creating thrust means accelerating a flow of fluid. It puts a number on it: thrust equals the mass flow rate crossing the propeller times the acceleration it gives the fluid, T = ṁ (V2 − V0), and the velocity at the disc is the mean of the upstream and downstream velocities. This page establishes these results with all their formulas. It relies on the momentum balance.

In this page:

1: Froude theory, the propeller seen as a disc

Froude's approach evaluates the axial induced velocity with a control volume around the stream tube crossing the propeller. The propeller accelerates the fluid, the tube contracts, and fluid must therefore enter the control volume sideways to compensate. We can compute that volume.

Following Froude, we regard the propeller as a uniformly loaded disc with an infinite number of blades (figure 1). Four assumptions simplify the problem:

  1. we ignore the rotation of the flow;
  2. outside the stream tube, the flow is undisturbed;
  3. the pressure far upstream and far downstream equals the static pressure of the undisturbed flow;
  4. the disturbances are small enough for the fluid density to be constant.

Froude theory for a propulsion propeller: stream tube contracted by the propeller in a control volume, with V0 and p0 upstream, V1 and pressures p, p + Delta p at the disc, V2 downstream

Figure 1: the accelerated stream tube contracts. On the left, V0 and p0 upstream; at the disc, the pressure jumps from p to p + Δp; on the right, V2 and the tube radius r downstream, inside a volume of radius R.

2: Momentum balance of a propulsive propeller

We write V0 for the velocity far upstream, V1 for the velocity at the disc, V2 for the velocity far downstream, r for the radius of the stream tube far downstream and R for the radius of the control volume. The continuity equation gives the volume flow rate Q crossing the outer surface of the control volume:

Volume flow rate Q = V2 pi r2 + V0 pi (R2 - r2) - V0 pi R2 = pi r2 (V2 - V0)

The momentum theorem gives the traction (or propulsion) force T of the propeller, that is the change of momentum between sections 2 and 0, projected on the horizontal axis:

T = M2 - M0, change of axial momentum

The time derivative of the momentum is:

Momentum derivative = rho V Q

The momentum M2 leaving the control volume at section 2 is:

M2 = rho pi r2 V2 squared + rho pi (R2 - r2) V0 squared

The total momentum M1 entering the control volume is the momentum at the volume inlet, plus the momentum entering sideways:

M1 = rho pi R2 V0 squared + rho (V2 - V0) pi r2 V0

The axial force (T = M2 − M1) therefore becomes, with the simplification on the second line:

T = rho pi r2 V2 (V2 - V0) after simplifying the balance

3: Pressure and induced velocity at the propeller disc

The traction force can also be written with the resultant of the static pressure on the disc:

T = A Delta p

where A is the area of the disc swept by the propeller and Δp the pressure difference across the disc. The Bernoulli equation reads, from 0 to 1 upstream:

p0 + 1/2 rho V0 squared = p + 1/2 rho V1 squared

and from 2 to 1 downstream:

p0 + 1/2 rho V2 squared = p' + 1/2 rho V1 squared

The static pressure difference between the downstream and upstream faces of the disc is:

Delta p = 1/2 rho (V2 squared - V0 squared) = 1/2 rho (V2 - V0)(V2 + V0)

The continuity equation between the disc and section 2 reads pi r2 V2 = A V1, which allows us to write:

T = rho pi r2 V2 (V2 - V0) = A Delta p = A 1/2 rho (V2 - V0)(V2 + V0)

We conclude that the flow velocity at the disc is the arithmetic mean of the upstream and downstream velocities:

V1 = (V2 + V0) / 2

The induced velocity, written w, is the increase of velocity at the disc, and a is the axial interference factor:

w = V1 - V0 = a V1

w = V2 - V1 = V2 - 2 a V0

Reading note: these two images mix two conventions for a. Since V1 is the mean of V0 and V2, the induced velocity is w = V1 − V0 = V2 − V1 = (V2 − V0)/2. With the usual definition a = w/V0, we have V1 = (1 + a) V0 and V2 = (1 + 2a) V0. This is the convention of the page on induced velocity, where the axial velocity seen by the blade is V(1 + a).

With the mean V1 = (V2 + V0) / 2, the traction force acting on the disc is:

T = rho pi r2 V2 (V2 - V0) = rho A (V0 + w) 2 w

4: Propeller thrust formula, mass flow and acceleration

This expression of the thrust brings out the mass flow rate. The mass of fluid crossing the propeller per second is, in kg/s:

ṁ = ρ A V1 = ρ π r2 V2

with ρ the density in kg/m3, A the disc area in m2 and r the radius of the stream tube far downstream (section 2), in metres. The thrust of the propeller is then:

T = ṁ (V2 − V0)

Creating thrust therefore means increasing the velocity of a mass flow of fluid. We have two levers, the mass moved or the acceleration given, and the choice between them is decisive for the propulsive efficiency.

5: Worked example of thrust, induced velocity and Froude efficiency

The values below are an example chosen to illustrate the formulas; they are not measurements. Take a propeller 2 m in diameter advancing at V0 = 20 m/s in air (ρ = 1.225 kg/m3), and assume the propeller imposes an induced velocity w = 3 m/s at the disc.

The useful power, the one that moves the vehicle, is T × V0 ≈ 531 × 20 ≈ 10,620 W. The power given to the fluid is T × V1 ≈ 531 × 23 ≈ 12,215 W, which ½ ṁ (V22 − V02) confirms. The difference, about 1,590 W, is the kinetic energy left in the wake. The ratio of the two is the Froude efficiency (ideal efficiency): η = V0 / V1 = 20 / 23 ≈ 0.87.

This efficiency shows the lever mentioned above. For the same 531 N of thrust, a propeller twice as large would move almost twice the flow and need almost half the acceleration: V1 would come closer to V0 and the Froude efficiency would rise. This is why slow, large-diameter propellers are more efficient than small, fast ones at equal thrust. At zero forward speed (static thrust), η falls to zero: the thrust is real, but the useful power is zero, because nothing moves forward.

6: Limits of Froude theory for a propulsive propeller

This efficiency is a ceiling, not a forecast. The theory contains neither airfoil drag nor wake rotation: a real propeller spins the fluid, and the energy of that rotation is lost on top of the axial acceleration. It also assumes an infinite number of blades, hence no tip loss, and a uniform flow over the disc. A real propeller therefore stays below the Froude efficiency, all the more so as the rotation is strong, that is at low forward speed and high torque.

To know which blade shape actually produces this thrust, we need blade element theory for a propulsive propeller, which brings in the airfoils and their Cl and Cd. For its wind turbine counterpart, see BEM for energy-capturing rotors.

Froude theory gives the axial induced velocity. Coupled to blade element theory, it splits this induced velocity into axial and tangential components, which are needed to predict angles of attack and blade geometry.

To go further, we can look at the blade element calculation for propulsion propellers, the energy-capturing case with the Betz limit, or the bibliographic references.