# Calculating the change of a Wind Turbine's RPM due to Airflow as f(t)

• Al-Layth
In summary, the conversation discusses the challenges of mathematically modeling a wind turbine's performance and the need to consider factors such as the lift on each blade, the effects of the spinning blades on the incoming airflow, and the turbine's control scheme. Existing codes like FAST can be used for this purpose, but a solid understanding of physics, fluid dynamics, calculus, numerical analysis, and airfoil theory is crucial. The conversation also highlights the importance of considering the turbine's control scheme in the modeling process.
Al-Layth
TL;DR Summary
I have a typical 3 bladed HAWT with a known blade and hub geometry along with full knowledge of the initial airflow (density, temp, pressure, speed). How can I model the motion of the turbine (RPM as a function of time) ?
the tldr covers everything I think. I don't expect there will be an analytic solution here lol. but I don't even know where to even begin formulating this problem mathematically. I assume I should find an expression for the initial lift on the blades, multiply by 3 and then model it as a circular motion problem?

But the problem is, that would only be valid for the instant t=0 and no time after it because:

(1) once the blades start moving, the lift per blade will be different too since the blades are now spinning whereas they were stationary at first.

(2) The spinning of the blades might create additional fluid effects that would distort the incoming airflow before it hits the blades.

So I'm mighty confused on how to deal with these issues in mathematical modelling.

You cannot really.
For any turbine RPM and wind speed, the angle of attack must be computed at each radial point along the blade. The lift and drag of each section along the blade must then be computed. Those forces must be integrated, and the angular acceleration of the turbine = "transducer disc" computed.

The solution will be a numerical solution as described in your quote below:
Al-Layth said:
I expect you will somehow need to calculate the lift on each turbine blade multiply by the number of them and model it as a circular motion problem. but then you also need to take into account the change of lift per blade due to the fact the blade will spin, (whereas it was stationary at first) and also the effects of the spinning blades on the incoming fluid as well. All issues currently beyond my modelling abilities
You start at the hub, and analyze the blade as a series of segments. Each segment has an airspeed, an angle of attack, and a radius. The airspeed and angle of attack allow you to calculate/find the ##C_L## and ##C_D##, from which you calculate the torque. Sum over the length of the blade, multiply by the number of blades, and you have total drive torque.

Total drive torque minus friction minus power generated equals net acceleration torque. Divide that by the total inertia and you have the angular acceleration at that point in time. Start at time 0, HAWT velocity 0, and select an appropriate time step. Integrate accordingly, then proceed to the next time step. Repeat until finished.

If the above is gibberish, you need to go back and study the fundamentals - physics, fluid dynamics, calculus, numerical analysis, and airfoil theory.

MTA: And that assumes a constant wind velocity and direction. The turbine will affect the wind in the vicinity of the turbine. But it's a good place to start.

Al-Layth and berkeman
Your best bet is to use existing code. FAST, from NREL (https://www.nrel.gov/wind/nwtc/fast.html) is an excellent wind turbine modeling code that will do everything you want.

However, the most important factor in this is going to be the wind turbine's control scheme, both for blade pitch and generator torque, and I notice you haven't really mentioned it here. It's an incredibly complicated problem, so even if you want to design your own from scratch, I'd start by looking at what FAST does (luckily it's open source and there are a lot of papers using and describing it) and working from there.

Al-Layth, berkeman and Baluncore

## 1. How is the change in a wind turbine's RPM calculated?

The change in a wind turbine's RPM is calculated by using the formula: RPM = (wind speed x blade length) / (2 x pi). This formula takes into account the wind speed and the length of the turbine's blades to determine the resulting RPM.

## 2. What is the role of airflow in changing a wind turbine's RPM?

Airflow is essential in changing a wind turbine's RPM as it is the force that drives the blades to rotate. The faster the airflow, the higher the RPM, and vice versa.

## 3. How does the change in air density affect a wind turbine's RPM?

The change in air density can affect a wind turbine's RPM as it determines the amount of force that the wind can exert on the blades. Higher air density results in a higher RPM, while lower air density results in a lower RPM.

## 4. Can other factors besides airflow affect a wind turbine's RPM?

Yes, there are other factors that can affect a wind turbine's RPM, such as the angle of the blades, the weight of the blades, and the design of the turbine. These factors can all impact the efficiency and performance of the turbine.

## 5. How can the change in a wind turbine's RPM due to airflow be measured?

The change in a wind turbine's RPM due to airflow can be measured by using an anemometer to measure wind speed and a tachometer to measure the RPM. These measurements can then be used to calculate the change in RPM using the formula mentioned in the first question.

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