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

[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.

 

[Baluncore]

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.

 

[jrmichler]

The solution will be a numerical solution as described in your quote below:

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.

 

[cjl]

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.