Blade element momentum theory (BEM) for wind turbines and tidal turbines cuts the blade into rings and solves two coupled equations for each: the balance of forces on the airfoil and the momentum balance of the fluid. It provides the axial and tangential induced velocities, hence the thrust and torque of each element. This page establishes it for energy-capturing rotors, with the two classic corrections of Prandtl and Glauert.

In this page:

• Blade element theory coupled to Froude
• Momentum balance on a blade ring
• Velocities, angle of attack and forces on the airfoil
• Axial and tangential induction factors
• Prandtl correction: tip losses
• Glauert correction: high induction
• BEM calculation algorithm

1: Blade element theory coupled to Froude (BEM method)

Froude theory ("momentum theory") links the axial induced velocity to the thrust force. It assumes the flow does not rotate, and says nothing about the shape to give the blade. Yet conservation of angular momentum requires the fluid to rotate, in the direction opposite to the rotor, so that the rotor can extract a useful torque.

BEM builds on the results of Froude to estimate the axial force of an element, and adds the performance of the airfoils distributed along the blade. It introduces the rotation of the air and provides the tangential and axial induced velocities that make up the induced velocity of Froude. Using the lift and drag coefficients Cl and Cd of the airfoils determines the blade geometry, so the blade can actually be built.

We thus define an axial induction factor (a) and a tangential induction factor (a'). A system of equations, solved by converging iterations with the Cl and Cd performance of the blade elements, gives the induced velocities and the performance of each element. We will also see two corrections: the first compensates for the infinite number of blades assumed by Froude (Prandtl tip loss factor), the second corrects high axial induction, where Froude's theory stops being valid (Glauert correction, beyond an axial induction factor of about 0.4).

The model rests on four assumptions:

• the upstream flow, far from the rotor plane, is entirely axial;
• at the rotor plane the air rotates; this rotation decreases a lot far downstream, so that the static pressure there equals atmospheric pressure;
• there is no interference between adjacent blade elements;
• the air flow around a blade element is two-dimensional.

2: Momentum balance on a blade ring

In Froude's method, the control volume is global. Here, we use rings as control volumes to find the elementary momentum change of each ring of the rotor. The sum of the performance of these rings gives that of the rotor.

Figure 1: a ring of thickness dr, at radius r, cuts one element out of the blade.

The change of momentum of a ring of thickness dr at radius r gives the axial traction force T on the element:

• u₁ is the velocity far downstream, V₀(1 − 2a); u is the velocity at the disc, V₀(1 − a);
• a is the axial induction factor.

It also gives the torque MR on the element, by introducing the tangential induced velocity:

• C_θ is the downstream tangential velocity;
• C_θ = 2a'ωr, considering that the upstream tangential velocity is zero;
• a' is the tangential induction factor;
• r is the distance to the axis, in metres, and ω the angular speed of the rotor, in rad/s.

Introducing a and a' into equations 6.1 and 6.2, we obtain:

and

3: Velocities, angle of attack and lift and drag forces on the airfoil

Let us recall the quantities of figure 2:

• the velocity Ve (m/s) seen by an element combines the tangential velocity Ω r (1 + a') and the axial velocity V (1 − a);
• the tangential induced velocity is Wt (m/s) = Ω r a';
• the axial induced velocity is Wa (m/s) = −a V;
• β (radians) is the pitch angle, between the rotation plane and the airfoil chord;
• φ (radians) is the apparent angle, that the resultant of the rotation speed Ω r and the upstream velocity V makes with the rotation plane;
• φ + αi (radians) is the angle between the rotation plane and the relative velocity Ve;
• α (radians) is the angle of attack seen by the airfoil: α = (φ + αi) − β.

We note that:

Figure 2: velocities, angles and forces on a wind turbine airfoil. Wa and Wt add to the fluid velocity to form the velocity Ve seen by the airfoil.

The lift dL is perpendicular to the velocity Ve, and the drag dD is parallel to it. If the lift and drag coefficients are known, they are written:

1. lift dL (N) = 0.5 ρ Ve² c Cl dr;
2. drag dD (N) = 0.5 ρ Ve² c Cd dr.

• Cl is the lift coefficient (Cz in French literature), taken from the airfoil database;
• Cd is the drag coefficient (Cx), taken from the same database;
• c (m) is the airfoil chord, between leading edge and trailing edge;
• ρ is the fluid density, in kg/m³.

We look for the force perpendicular to the rotation plane (the thrust) and the force parallel to the rotation plane (the one that makes the rotor turn). Lift and drag are projected in these two directions, with the angles of figure 2:

• force normal to the rotation plane (N/m) Pn = dL cos(φ + αi) + dD sin(φ + αi);
• tangential force, in the rotation plane (N/m) Pt = dL sin(φ + αi) − dD cos(φ + αi).

Normalised by 0.5 ρ Ve² c, they give the coefficients:

• Cn = Cl cos(φ + αi) + Cd sin(φ + αi), that is Cn = Pn / (0.5 ρ Ve² c) (6.14);
• Ct = Cl sin(φ + αi) − Cd cos(φ + αi), that is Ct = Pt / (0.5 ρ Ve² c) (6.15).

Figure 2 also gives:

• Ve sin(φ + αi) = V (1 − a) (6.16);
• Ve cos(φ + αi) = Ω r (1 + a') (6.17).

The solidity σ is the share of the annular area of radius r that is covered by the blades:

• σ(r) = c(r) B / (2 π r) (6.18),

with B the number of blades, c(r) the local chord and r the radial position of the ring. Since Pn and Pt are forces per unit length, the thrust T and the torque MR on the ring of thickness dr are:

• thrust on the element (N): T = B Pn dr (6.19);
• torque on the element (N·m): MR = r B Pt dr (6.20).

Using (6.14) for Pn and (6.16) for Ve, equation (6.19) becomes:

Using (6.15) for Pt, and (6.16) and (6.17) for Ve, equation (6.20) becomes:

4: Axial and tangential induction factors (formulas)

If we equate equations (6.21) and (6.4) and apply the definition of solidity, we obtain the expression of the axial induction factor a:

If we equate equations (6.22) and (6.5), we obtain the tangential induction factor a':

5: Prandtl correction, tip losses

Froude's method assumes an infinite number of blades on the disc. The tip loss factor derived by Prandtl corrects the performance for the number of blades. This factor F corrects equations (6.4) and (6.5), which become:

and

Prandtl's loss factor F is computed as follows:

where

B is the number of blades, R the tip radius, r the local radius of the element and φ + αi the flow angle seen by the element (figure 2). Using (6.31) and (6.32) in place of (6.4) and (6.5) to establish a and a', we obtain:

and

6: Glauert correction for high axial induction

When the axial induction factor exceeds about 0.4, Froude's momentum theory is no longer valid. Glauert proposes corrections tied to the thrust coefficient. Several empirical relations between the thrust coefficient C_T and the axial induction factor have been established to match measurements. We retain the one from Spera (1994), where a_c is approximately 0.2:

The tip loss factor F corrects the assumption of an infinite number of blades. The thrust on an element is given by:

The thrust coefficient C_T is, by definition:

Introducing (6.21) into (6.39), we obtain for C_T:

We can now introduce the correction (6.38). If a < a_c:

which gives:

This is equation (6.35), unchanged. If on the other hand a > a_c:

which gives:

with:

A note on notation: in this page, Ct denotes the tangential coefficient of the airfoil (equation 6.15), whereas C_T denotes the thrust coefficient of the ring (equation 6.39).

7: BEM calculation algorithm for a wind turbine

These relations allow an algorithm that converges by iterations. It computes the induced velocities of each blade element separately, with the Prandtl and Glauert corrections, from the performance of the airfoils. It gives the thrust and torque of each element; summing them, we obtain the performance of each blade, then of the whole rotor. The steps are:

1. initialise a and a' to zero;
2. compute the angle φ + αi with equation (6.7);
3. compute the angle of attack α = (φ + αi) − β;
4. read from the database the Cl and Cd performance of the element's airfoil for this angle α;
5. compute Cn and Ct from equations (6.14) and (6.15);
6. compute a and a' from equations (6.44) or (6.42), and (6.36);
7. if a and a' changed by more than a given tolerance, return to step 2, otherwise stop;
8. compute the forces on the elements.

Reminder of figure 2, which serves as a reference for all the formulas.

To go further, we can review Froude theory and the Betz limit, see how this method applies to propulsion propellers, or consult the bibliographic references.

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