## 1.0. Introduction:

On the 25th of August 2020, a Tokyo-based engineering company, SkyDrive Inc., carried out the first ever successful test of a piloted eVTOL (electric vertical take-off and landing vehicle). Not only did this flight represent humanity taking one step closer towards a completely novel era of personal transportation, but it also served to remind us about the significance of a not-so-novel and oft taken for granted – but nonetheless brilliant – feat of engineering: the propeller. The very first propellers came into use on ships in the early 19th century, and now, after 200 years of innovation and advanced scientific understanding, the technology behind them can be found everywhere from planes to wind turbines. Given all of this, a deeper look into the physics behind propellers and turbines – more specifically, ‘the transfer between rotational power and linear motion in air’ – is very topical and is hence the subject of this essay.

## 2.0. Modelling the Efficiency of a Wind Turbine:

### 2.1. Equal and Opposite Forces and the Transfer of Momentum:

The key concepts on which both propellers and turbines work are those described by Newton’s Third Law and the Conservation of Momentum. During any interaction between two bodies with mass, the total momentum within a system will be conserved if no external forces act on the system. Additionally, if either of the bodies exerts a force on the other, by the very nature of physics, an equal force in the opposite direction will be exerted upon it. Therefore, described in the very simplest of terms, a propeller is a device which causes the air around it to move backwards resulting in a forward force on it. A turbine involves the same transfer but occurring in the opposite direction: it gets in the way of moving air and causes it to slow down and the momentum loss of the air allows power to be generated.

Therefore, the fact that turbines work on the same principle as propellers, and also that they are inherently easier to model, means that they provide the perfect basis on which we can begin to mathematically analyse the transfer between the linear motion of air, and rotationalpower. (Note that the reason that turbines are easier to model is because turbines are stationary and so we only need to deal with the change in momentum of the moving body of air. In addition, unlike propellers, the design of turbine blades can more easily be thought of as a simple mechanism for slowing down air without compromising the accuracy of our model.)

### 2.2. The Power of a Moving Body of Air:

As previously stated, a wind turbine generates power by getting in the way of moving air. The kinetic energy of a mass, $m$, of air moving at velocity, $v$, is given by

$E = \frac{1}{2}mv^2$

Power is the rate of change of energy and hence:

$P = \frac{dE}{dt} = \frac{1}{2} \frac{dm}{dt}v^2$

where $\frac{dm}{dt}$ describes the mass flow rate of the air, in other words, the rate at which a certain mass of air travels past the turbine. The mass flow rate can also be written in terms of the density of air, $\rho$, the sweep area of the turbine, $A$, and the velocity of the air, $v$, as follows:

$\frac{dm}{dt} = \frac{d(\rho V)}{dt} = \frac{d(\rho Ax)}{dt} = \rho A \frac{dx}{dt} = \rho Av$

Thus, substituting this into the equation for power, we get that the power of wind travelling through a turbine is:

$P = \frac{1}{2} \rho Av^3$

From this, it is evident that the power increases with the cube of the wind speed, hence why it is so importance to position turbines in windy locations. However, the effective useable wind power is less than indicated by the above equation because the wind speed behind the turbine can never be zero, since no air could follow. Thus, there is a limit on the power that can be generated, and this is called the Betz Limit after Albert Betz, the German physicist who first described it.

### 2.3. The Betz Limit:

The wind speed in front of a wind turbine is larger than the wind speed behind it, and as the mass flow must be continuous, the area $A_2$ must be bigger than the area $A_1$ (See Figure 1).

The power that can be generated by the turbine, $P_{gen}$, is equal to the difference between the wind power in front of the turbine and the wind power behind it. Therefore:

Thus, we can work out the power coefficient, which is defined as the ratio between the generated power and the power of the moving air:

$C_p = \frac{P_{gen}}{P_{wind}} = \frac{(v_1 + v_2)(v_1^2 - v_2^2)}{2v_1^3}$

We can now find the greatest possible power coefficient (the ideal power coefficient) by letting $x$ be the ratio $\frac{v_2}{v_1}$, and then finding the maximum value of the function $f(x) = \frac{(1+x)(1-x^2)}{2}$

Therefore, according to the Betz Limit, the theoretical maximum power efficiency of any design of wind turbine is approximately 59%, and this occurs when the speed of air behind the turbine is $\frac{1}{3}$ of the speed of air in front (see Figure 2). However, the real-world limit is well below this, with power coefficient values typically in the range of 35% – 45% due to the strength and durability requirements of various designs.

## 3.0. The Fundamental Concepts behind Propellers:

### 3.1. Comparing Propellers to Turbines:

Propellers and turbines are described by the same basic theoretical principles (See Figure 3). However, propellers, unlike turbines, have to be able to ‘suck in’ air and speed it up to produce thrust.

This leads to a number of key differences in design. Firstly, turbine blades are designed to rotate in large volumes of slow-moving air and to create as little turbulence as possible that could foul the next blade; aircraft propellers are designed to move in high velocity air.

Secondly, turbines use blade pitch control to keep the rotor speed within operating limits as the wind speed changes. Pitch control is also an important part of the operation of propellers but it is used as key factor in optimising the thrust rather than keeping the rate of rotations low.

Perhaps the most important distinction to make between the design of turbines and propellers has to do with the shape of the blades. In wind turbines, the blade design has the primary purpose of providing the best possible lift-to-drag ratio so that the blade can move as easily as possible through the air, maximising the efficiency of the turbine. Turbine blades are twisted so they can always present an angle that takes advantage of the ideal lift-to-drag force ratio.

A propeller’s blades have an even more distinctive aerofoil profile which is necessary to create a pressure gradient between air in front of the propeller and air behind it. Although flat propellers are still able to produce thrust, they are very inefficient, and so an aerofoil shape is crucial to create high air pressure behind the propeller and therefore generate ‘lift’ in the direction of movement (See Figure 4).

In summary, the most important parameter for wind turbine blades is the lift-to-drag ratio. In addition, an aerofoil profile may be used at a high angle of attack, as is the case with stall-controlled turbines*. In contrast, in aircraft design, the objective of the design process is usually quite different, for example, to decrease the drag for a fixed lift coefficient. It is worth noting that many of the differences described above come about as a result of the Reynolds number** being much lower for wind turbines than they are for aeroplanes’ propellers, which may change the flow behaviour of the air considerably.

### 3.2. The Physics Behind Aerofoils – Bernoulli’s Principle:

We have now established that aerofoils make up a crucial part of propeller design. In order to understand how they have the effects described in 3.1. it is first necessary to understand Bernoulli’s principle (see Figure 5).

Bernoulli’s principle is based on the conservation of energy. As described by Swiss mathematician and physicist, Daniel Bernoulli, in 1738, it states that there is a positive correlation between the pressure and the speed of a fluid at a specific point. Thus, aerofoils work by causing air that travels over the foil to speed up and thus to have a reduced pressure compared to air under the foil. This results in a force, ‘lift’, which acts perpendicular to the direction of airflow (See Figure 6).

### 3.3. Putting it All Together – A Basic Model for the Thrust Produced by a Propeller:

Numerous theories have been developed to allow physicists and engineers to mathematically model the transfer of rotational power into linear thrust. The details of propeller propulsion are very complex because a propeller can be thought of as a rotating wing. The blades are typically long and thin and have an aerofoil profile. In addition, they are usually twisted to increase efficiency. On top of this, the angle of attack (the angle between the oncoming air flow and the orientation of the propeller blades) at the tip is lower than at the hub because the tip is moving at a higher velocity. (This is one of the key concepts in rotational motion. By observing the equation $v = \omega r$, where $\omega$ is the angular velocity and $r$ is the distance from the centre, it is clear that the further out one goes from the hub, the faster the velocity of the blade.) All together, these characteristics make analysing the airflow through the propeller a very complex task.

However, we can attempt to create a very basic model of propeller thrust by using the simplified momentum theory and by assuming, as we did when modelling the turbine, that a spinning propeller acts like a disk through which the surrounding air passes (see Figure 7).

The engine turns the propeller and does work on the airflow. This results in an abrupt change in pressure across the propeller disk. From Bernoulli’s principle, we know that the pressure over the top of an aerofoil wing is lower than the pressure below the wing and therefore, a spinning propeller sets up a pressure lower than free stream in front of the propeller and higher than free stream behind the propeller.

In our model, the thrust generated by a propeller disk, $F$, can be worked out by multiplying the pressure differential across the disk, $\Delta p$, by the area of the disk, $A$:

$F_{thrust} = A \Delta p$

The total pressure in front of the propeller disk is equal to the static pressure, $p$, added to the dynamic pressure. In fact, these are respectively the first two terms in Bernoulli’s equation, contained within Figure 5. Looking at this equation, we can see that the dynamic pressure of a moving body of gas is given by $\frac{1}{2} \rho v^2$, where $\rho$ is the density of air and $v$ is the velocity. Therefore, we can find an expression for the total pressure in front of the propeller disk and as well as behind it. (Note that I have used subscript 0 for quantities in front of the propeller, and subscript $e$ for quantities behind it in order to be in accordance with Figure 7.)

$p_{total_0} = p + \frac{1}{2}\rho v_0^2$

$p_{total_e} = p + \frac{1}{2}\rho v_e^2$

Therefore, we can work out the change in pressure across the disk:

$\Delta p = p_{total_e} - p_{total_0} = p + \frac{1}{2}\rho v_e^2 - (p + \frac{1}{2}\rho v_0^2) = \frac{1}{2}\rho (v_e^2 - v_0^2)$

Finally, we substitute this into our equation for thrust to give:

$F_{thrust} = \frac{1}{2} \rho A(v_e^2 - v_0^2)$

where $v_0$ is the velocity of the oncoming airflow relative to the propeller, and $v_e$ is the exit velocity of the air from the propeller.

It is worth noting that this value is an ideal number and that it does not account for the many losses that occur in practical, high speed propellers, such as tip losses and slip. It is, however, what we set out to achieve: a model of how the thrust produced by a propeller is related to the linear motion of air.

## Extras

* Stall is a sudden reduction in the lift generated by an aerofoil when the critical angle of attack is reached. Around two thirds of the wind turbines currently being installed in the world are stall-controlled machines.

** The Reynolds number of a system is a dimensionless value used to categorise the behaviour of fluids in that system: it is the ratio of inertial forces to viscous forces. It is proportional to the density and flow speed of the fluid, and the linear scale of the system, and inversely proportional to the dynamic viscosity. I highly recommend this link and this link to read more about the effect of the Reynolds number on the thrust produced by a propeller.

## Sources

Abdallah, G. G. (2017). Propeller Force-Constant Modeling for Multirotor UAVs from Experimental Estimation of Inflow Velocity. International Journal of Aerospace Engineering

Engineering, R. A. (Accessed: 2021, January 31). Wind Turbine Power Calculations. Retrieved from https://www.raeng.org.uk/publications/other/23-wind-turbine

Flying Car Gets Off The Ground. (2020, September 1). Retrieved from Born To Engineer: https://www.borntoengineer.com/flying-car-gets-off-the-ground

Layton, J. (Accessed: 2021, January 31). Wind Power. Retrieved from How Stuff Works: https://science.howstuffworks.com/environmental/green-science/wind-power.htm