Average Blade Lift Coefficient Calculator

The average lift coefficient of a rotor blade indicates the workload that each rotor blade has to produce lift. Being a dimensionless number which represents the ratio between the lift produced by a blade and the dynamic pressure acting on its area, it can be used for rotors of any size.

This tool uses a simplified method based on blade element momentum theory to estimate. It converts two input values - the thrust coefficient and rotor solidity - into a representative lift coefficient for the rotating blade.

What is the average lift coefficient?

Rotor blades are not rigid wings. As the speed at which they move through the air constantly increases from the hub to the blade tip, individual wind tunnel measurements cannot represent the entire blade.

The average lift coefficient solves this problem by giving a value that is derived from the area within 75 percent of the radius where most of the lift is generated. Wind turbine blades and helicopter rotor blades are essentially rotating wings, and this value is the lift coefficient for rotating wings, or the coefficient used to describe an aircraft's wing.

As it is a dimensionless number, 0.6 has the same meaning for both camera drones and large transport helicopters. This allows engineers to compare different rotor configurations on drawings before they start machining metal parts.

Instructions for using this calculator:

If you enter the thrust coefficient and rotor solidity, then the average lift coefficient will be shown immediately. You can leave one of the three fields blank and enter the remaining two values to allow the calculator to work out the missing value. So the same tool can be used both to determine the size of the rotor and to recalculate the required strength.

By activating the Performance Limitations options you can add the maximum lift coefficient and advance ratio. This shows estimates for the stall margin, thrust margin and the situation that the rotor blades face on the back side during forward flight.

The two sensor graphs and the one operating condition map below the results show how the lift coefficient reacts to changes in thrust coefficient or rotor solidity.

Formula:

The average lift coefficient is calculated by integrating the total rotor disk lift assuming uniform inflow conditions and based on blade element theory.

Here, $C_T$ is the thrust coefficient, and $\sigma$ is rotor solidity. We consider rotors with $C_T = 0.006$ and $\sigma = 0.06$.

Assuming uniform flow, integration yields a constant of 6, thus limiting the result to the effective working point of the blades at about 75 percent of the radius.

The rotor solidity itself is a geometric ratio, namely the ratio of the blade area to the disc area.

When this relationship is expanded, the individual physical quantities can be identified: Thrust $T$, air density $\rho$, number of blades $N$, chord length $c$, radius $R$ and rotational speed $\Omega$.

This leads to an important point; increasing the blade area will decrease the lift coefficient, and increasing the tip speed of the blade will cause a rapid drop in the lift coefficient. This is because the tip speed of the blade is squared into the calculation. The following table shows the individual symbols and their values for the example above.

Compute any value.

This relationship is a simple proportion so the formula can be simplified directly. A calculation tool will select the appropriate formula based on which fields are filled in.

The first formula is used to calculate the thrust that a rotor can produce given its lift coefficient and solidity. The second formula is used to back-calculate the required rotor solidity, i.e. blade area, and ensure that the lift coefficient is within safe limits in order to achieve the target thrust.

Operating range and blade load

High performance rotors typically operate in the flight range of 0.3 to 0.7. Below 0.3 rotor loading is low, while above 0.7 blade workloads increase and above about 1.0 gradually leads to flow separation.

A closely related value is the blade loading coefficient, which is the ratio of the thrust coefficient to rotor solidity ($C_T/\sigma$). This corresponds to one sixth of the lift coefficient and is a value that many rotorcraft engineers monitor. Typical values in hover are about 0.05 to 0.12 while the actual upper limit before stall is about 0.12 to 0.14.

Lift coefficient and angle of attack

The lift coefficient and the angle of attack are interdependent. If the blade rotation is increased so that the tip of the blade points upwards, then the coefficient increases almost linearly but only up to a certain point.

For most helicopter rotor blades, this limit, the maximum lift coefficient, is about 1.2 to 1.4. If this range is exceeded, flow separation occurs. The air flows away from the upper surface, which causes the lift to suddenly decrease and the drag to increase sharply. The lift coefficient is not a constant property of the blade but changes depending on the angle of attack at which the blade operates during flight.

Flow separation and forward flight

In hover all blades are operating at nearly the same pitch. In forward flight the front blade needs less rotation as it is exposed to faster airflow while the rear blade requires more rotation due to slower airflow in order to maintain the same lift.

This inequality is the reason why all helicopters have a top speed. As forward ratio increases, the rear blade requires ever greater lift coefficients until it stalls. The power limitation option estimates this value for the rear blade and displays a warning when the stall limit is exceeded.

The scope of these values.

The average lift coefficient is encountered in nearly all phases of the rotor design process. In the early design phase it helps to determine the size of the rotor. Once an acceptable lift and desired thrust coefficients are chosen, then the required rotor density can be back-calculated.

Later in the design process it will indicate whether a design is operating within an efficient region or approaching stall at high speed or heavy load. It also affects decisions about rotor blade count, blade width and twist, illustrating trade-offs; more rotor area reduces required lift coefficient but increases weight and drag.

Tips for accurate results:

This formula gives representative values at 75% of the radius and assumes rectangular blades with no twist. Actual rotor blades use a twist and airfoil profile to distribute lift more evenly across the blade area but this simplified model cannot represent these features.

This formula works well for hover and slow flight speeds. When the speed exceeds about 50 knots, errors due to asymmetric flow and blade flapping become significant so a more comprehensive analysis is required. The results are probably most accurate when the thrust loading is below about 0.012 and the rotor density range from 0.05 to 0.12.