$9\mathrm{.}2$ Flow in a converging$-$diverging nozzzle

An experiment is conducted where compressed air from a large reservoir is expanded and accelerated through a nozzle $($figure $9\mathrm{.}11).$ There is no external work transfer to or from the air, and the expansion can satisfactorily be modeled as a one$-$dimensional isentropic process $($isentropic meaning that the process is fully reversible, i$.$e$.$ without losses to friction, and adiabatic, i$.$e$.$ without heat transfer$).$

Figure $9\mathrm{.}11-$ A nozzle in which compressed air in a reservoir $($left$)$ is expanded as it accelerates towards the right. The back pressure $p_b$ is controlled by a valve $($represented on the right$).$ The flow conditions of interest in this exercise are labeled $1,2$ and $3.$

The air $(c_{p\text{a}\text{i}\text{r}}=1005Jk{g}^{-1}{K}^{-1})$ flows through a converging$-$diverging nozzle. The entrance conditions into the nozzle are $A_1=0,15m^2,p_1=620$kPa,$V_1=180m{s}^{-1},$ and $T_1=430$ K $.$

In the expanding section of the nozzle, we wish to know the conditions at a point $2$ of cross$-$sectional area $A_2=A_1=0,15m^2.$

What is the mass flow through the nozzle?

If we wish the flow to become sonic $($i$.$e$.$ reach $[Ma]=1),$ what is approximately the required throat area $A_T?$

Which two flow regimes can be generated at section $2?$ What is approximately the pressure $p_2$ required for each?

The nozzle is now operated so that the flow becomes sonic at the throat, and accelerates to supersonic speeds in the expanding section of the nozzle.

The nozzle cross$-$sectional area increases further downstream of point $2,$ up until it reaches an area $A_3=0,33m^2.$ Downstream of point $3,$ the duct is of constant area and the back pressure $p_b$ is set to $p_b=p_3=p_2.$

$4.$ Describe qualitatively $($i$.$e$.$ without numerical data$)$ the pressure distribution throughout the nozzle.

Finally, we wish to expand the air with a fully$-$isentropic process, expanding the air from the reservoir to point $3$ without a shock wave.

$5.$ Propose and quantify one modification to the experimental set$-$up that would allow an isentropic$,$ supersonic expansion of the flow into the outlet of area $A_3.$ I want the answer on a paper