I am trying to derive the Betz factor from scratch, and I did it but I just assumed the initial part of the standard derivation as given in the textbooks being correct and now I have some questions about this very first step. I have successfully derived the correct value of the Betz factor as 0,593, but then I'd like to understand why the initial assumptions are reasonable. Precisely, why Newton's 2nd law is written in the following form? $$ F = ma = m\frac{\mathrm{d} v}{\mathrm{d} t}=\dot{m}\Delta v=\dot{m}(v_1{}-v_2{})=\rho Sv(v_1{}-v_2{}). $$ I've never seen someone stating that $ m\frac{\mathrm{d} v}{\mathrm{d} t}=\dot{m}\Delta v$ and I don't know what does it mean neither mathematically nor physically. After this assumption, the mathematics involved in the derivation is very simple: just substitute variables from one equation to another, get the final equation, differentiate it by speed ratio, set the differentiated equation equal to zero and find the function maximum which is 0,593. I won't write the whole process here but you can find it in this Wikipedia entry Application of conservation of mass (continuity equation) section. It would be very helpful and a piece of crucial information if someone could explain why the above equation holds.
Kind regards and thank you all in advance,
T.