Blade Element Momentum (BEM) for propellers and turbines: part 1 linear and angular momentum

blade element momentum theory. mostly from perspective of a propeller in this particular derivation (see my notes for turbine version). linear momentum balance, angular momentum balance, efficiency.
For more detail see the following two references:
Ning, A., “Using Blade Element Momentum Methods with Gradient-Based Design Optimization,” Structural and Multidisciplinary Optimization, Vol. 64, No. 2, pp. 994-1014, May 2021. doi:10.1007/s00158-021-02883-6
Ning, A., “A Simple Solution Method for the Blade Element Momentum Equations with Guaranteed Convergence,” Wind Energy, Vol. 17, No. 9, pp. 1327-1345, Sep. 2014. doi:10.1002/we.1636

[@dormg22]

amazing channel, thank you so much for uploading these lessons professor

[@apocalypt0723]

Thank you so much for the lectures. Learning so much from them

[@samdougherty5559]

So from my understanding, for most of the calculations in this video the units are in Torque and Thurst. It is only when you make A_d = 2*pi*r*(dr) and then divide LHS and RHS by (dr) do the equations become Torque per unit length and Thrust per unit length?

[@grahamflowers]

There is something wrong with Betz limit if kinetic energy in the wind is 1/2 mass or 50% and Betz limit is 0.5926% then where can the 0.926% come from not the wind ? regards Graham Flowers MEng

[@casper6349]

At 35:56, you are missing a V_\infty in the equation for torque. Furthermore, in case you would insert the velocity to arrive at the same equation as 34:08, you would end up at torque in terms of units, not torque per unit length. Won't this mess up the integration over the span later on in the calculations?

[@BYUFLOWLab]

Good catch, I indeed made a typo on the slide and the V_\infty should be there. With the Vinf include you do get the correct units (torque per unit length, N*m/m or just N). Fortunately, when I combine the momentum and blade element parts in the next video I have the V_\infty term back in so everything else is correct - just a typo on that slide.