An aircraft flies with $290($m$)/($s$)$ at an altitude of $20$ km $,$ where the ambient pressure and

temperature are $26\mathrm{.}5$ kPa and $223\mathrm{.}16$ K $,$ respectively. The jet speed, pressure, and temperature

are $600($m$)/($s$),750$K$,$ and $30$ kPa $,$ respectively. The propulsion system of the aircraft consumes $205$

k$($g$)/($s$)$ of air and burns $6$k$($g$)/($s$)$ of fuel having a heating value of Q$\_(\backslash$Omega $)=42,600$k$($J$)/($k$)$g$($T$)/($m$)$a k$=1\mathrm{.}4,$R$=287($J$)/($k$)$g$.$K

Air enters a compressor operating at steady state at $1$ atm with a specific enthalpy of $290$k$($J$)/($k$)$g

and exits at a higher pressure with a specific enthalpy of $1023$k$($J$)/($k$)$g$.$ The mass flow rate is $0\mathrm{.}1$

k$($g$)/($s$).$ If the compressor power input is $77$ kW $,$ determine the rate of heat transfer between the

compressor and its surroundings, in kW$.$ Neglect kinetic and potential energy effects and

assume the ideal gas model.

A rocket has three nozzles canted at $30\backslash$deg with respect

to the rocket's centerline. The gases exit at $2200($m$)/($s$)$

through the exit area of $1\mathrm{.}2$m$^(2).$ The density of the

exhaust gases is $0\mathrm{.}35$k$($g$)/($m$^(3)),$ and the pressure is $55$

kPa $.$ The atmospheric pressure is $12$ kPa $.$ Determine

the thrust on the rocket.

Figure$\mathrm{.}1$ A rocket with three nozzles.

Show that the specific thrust and the TSFC for a turbofan engine can be expressed as follows,

$($T$)/($m$\_($a$)^())=(1+$f$)/(1+\backslash$beta $)$u$\_($e$\_($k$))+(\backslash$beta $)/(1+\backslash$beta $)$u$\_($e$\_($c$))-$u$,$TSFC$=($f$)/((1+\backslash$beta $)(($T$)/($m$\_($a$)^())))$