Question $4[20$ marks$]$

Air enters a convergent$-$divergent nozzle, as schematically shown in Figure Q$4,$ at stagnation pressure of $1\mathrm{.}0$ MPa and stagnation temperature of $800$ K with negligible velocity $($i$.$e$.$ the air velocity at the nozzle inlet is $\backslash ($ V$\_\{\backslash$text $\{$inlet $\}\}\backslash$approx $0\backslash )).$ The throat area of the nozzle is $\backslash (20\backslash$mathrm$\{$~cm$\}^\{2\}\backslash ).$ The flow Mach number is $2\mathrm{.}0$ at the exit from the nozzle. Assume the air flow in the nozzle is a one$-$dimensional, steady, isentropic and adiabatic compressible flow problem. For air, the specific heat ratio can be taken as $1\mathrm{.}4$ and the gas constant as $287\backslash (\backslash$mathrm$\{$J$\}/\backslash$mathrm$\{$kg$\}/\backslash$mathrm$\{$K$\}\backslash ).$ Air can be assumed to be a perfect gas.

$($a$)$ Calculate the stagnation density of air at the inlet to the nozzle.

$($b$)$ Determine the flow Mach number at the throat, the critical cross$-$sectional area of the nozzle, and critical temperature of air in the nozzle.

$($c$)$ Determine the cross$-$sectional area of the exit from the nozzle.

$($d$)$ Determine the pressure of air at the exit from the nozzle.

$($Note: When you attempt Question $4($d$),$ you may need to use the table on Page $15$ of the exam paper.$)$

$($e$)$ Explain if it is possible to increase the mass flow rate of air in the nozzle $($relative to the setup of this question$)$ by lower the back pressure of air beyond the exit from the nozzle.