105-2019201-p2 Solutions

Question 1

(a)

$P_{i}={\frac {1}{2}}\rho {\frac {\Delta V}{\Delta t}}v_{0}^{2}={\frac {\rho A}{2}}v_{0}^{3}$

(b)

Immediately before and immediately after the blade

$\rho {\frac {\Delta V}{\Delta t}}=A{\frac {(v_{0}+v_{1})}{2}}$

Thus

$P_{e}ff=P_{i}-P_{f}={\frac {1}{2}}\rho {\frac {\Delta V}{\Delta t}}(v_{0}^{2}-v_{1}^{2})$ $={\frac {\rho A}{4}}(v_{0}+v_{1})(v_{0}^{2}-v_{1}^{2})$

(c)

$C_{p}={\frac {(v_{0}+v_{1})(v_{0}^{2}-v_{1}^{2})}{2v_{0}^{3}}}={\frac {(1+x)(1-x^{2})}{2}}$

${\frac {d}{dx}}C_{p}=0\rightarrow x={\frac {1}{3}}$

$C_{p}=\%59$