Solar Lift Coefficient Calculator

A solar thermal chimney converts sunlight indirectly into electricity. A large glass roof heats the air below it, and this heated air quickly rises in a tall tower. An on-ground turbine captures that airflow. The lift coefficient is a number used to determine whether or not the blades of the turbine can actually capture that airflow. This tool calculates the fifth value if you enter four of the following five values.

What is the lift coefficient?

The lift coefficient is denoted by $C_L$ and is the ratio of the lift to the dynamic pressure force acting on the projected area of the rotor blades. Since it has no units, it can be used for both paper models in a wind tunnel and rotors as large as 40 meters, so long as they have the same $C_L$. This is why it is often used to compare designs of different scales.

For a flat plate at small angles of attack the value is about 0.8. A well-designed airfoil can have values between 1.2 and 1.5 under ideal conditions. When the blade stalls, this value drops dramatically.

How to use this calculator:

When you input lift, air density, wind speed and wing area, the tool calculates the lift coefficient. This is similar to how it's done in a wind tunnel. You set up an airflow, measure the force, then divide to get the coefficient.

To calculate values, leave the desired element blank and fill in the remaining four fields. This tool will then show you either the lift, the wind speed required to produce a known amount of lift or the area corresponding to a given coefficient. You can mix SI units and yard-pound units for each element. The conversion is done automatically in the background. By default, air density is set to sea level ($1.225\ \text{kg}/\text{m}^3$), but if your device is located at higher altitude where the air is thinner, adjust this value accordingly.

Two optional features can also be enabled. One estimates the air density and lift lost due to altitude of location. The other takes a drag coefficient and returns the ratio of lift to drag, a key metric for measuring rotor efficiency.

Lift coefficient equation:

The lift coefficient is directly derived from the lift equation. When solving for $C_L$, we get:

Here, $L$ is the lift in Newtons, $\rho$(rho) is the air density, $v$ is the wind speed and $A$ is the rotor area (projected area). In its full form the equation reads:

Assume that the rotor produces a lift of 147 N at a wind speed of 10 m/s, an air density at sea level of $1.225\ \text{kg}/\text{m}^3$ and a rotor area of $2\ \text{m}^2$. First calculate the dynamic pressure: $\tfrac{1}{2}\times 1.225\times 10^{2}=61.25\ \text{Pa}$. Then divide the lift by the product of the dynamic pressure and the area.

$C_L$ is 1.2 and falls within the range of optimized airfoils with high performance. The following table shows the symbols used and their respective values for this example.

Why is wind speed squared?

It is important to note that $v^2$ applies. If the wind speed doubles, lift quadruples. This is why rotor blades must be sized for the strongest winds possible at a given installation site and not just average breezes. The lift coefficient itself changes little with speed but the force associated with it increases rapidly.

Typical lift coefficients

The lift coefficient is mainly determined by the shape of the wing and its angle of attack. These are general reference values that can be used to estimate or compare results.

Air density and altitude

Lift is proportional to air density, so a solar chimney built at high altitude will produce proportionally less lift than one on sea level for the same wind speed. At 1500 m elevation, air density is about 12 percent lower and lift produced by the same wind speed is also 12 percent lower. If the size of the blades is not adjusted to these conditions, performance will drop off directly.

The standard atmosphere model provides a relationship between altitude and density in the troposphere.

Here $h$ is the altitude of the location, in meters. By checking the Altitude option and entering an altitude you can simultaneously check both density and lift loss compared to sea level.

Lift, air resistance and turbine efficiency

The goal of a solar thermal chimney is to keep the turbine spinning from sunrise until the afternoon. Engineers want the blades to have high $C_L$ at low wind speeds so that rotation can begin earlier in the day, but they need to be able to control $C_L$ at higher winds to ensure that the rotor does not get destroyed by strong gusts.

The lift coefficient is used in combination with the drag coefficient $C_D$ and relates to the lift-to-drag ratio $L/D = C_L / C_D$. This ratio measures how much useful force the wings can generate per unit of air resistance. This ratio determines the tip speed ratio, which is the ratio of the wingtip's speed to the wind speed, and allows you to maximize energy generation from the updraft. You can check this ratio by enabling the "Lift-to-drag ratio" option and entering the drag coefficient.

Tips to improve accuracy of results:

Use the actual air density at your installation site instead of the standard sea level value. The blade area is measured as a projected area from above and not as the wetted area of the blade. A single value $C_L$ represents only one operating point. Since lift depends on the angle of attack, for comprehensive design studies both wind speed and angle of attack should be varied simultaneously to recalculate $C_L$ at each point.