Problem $3(15$ points$)$

Consider a rocket nozzle which has been designed for optimal operation at an ambient

pressure of $p_a=25$kPa to produce an ideal thrust of approximately $80\mathrm{.}5$ kN with an ideal

chamber pressure of $3\mathrm{.}95$ MPa by employing a nozzle with a throat diameter of $12\mathrm{.}5$ cm and

exit diameter of $48$ cm $.$ This nozzle will be attached to a previously built thrust chamber

which operates at $T_0=3300K$ using combustion gases with $=1\mathrm{.}25$ and $R=370\frac{(J)}{kgK}.$

However, test measurements at the stated pressure conditions yield a thrust of only $72\mathrm{.}5$ kN

when the measured flow rate is $32\mathrm{.}0k\frac{g}{s}.$ Find the applicable correction factors $,$

and ${}_{C_F})$ and the actual specific impulse assuming frozen flow throughout the nozzle.