Problem $1$: A nozzle with circular cross$-$section is connected to a large reservoir of air at $p_0$ and $T_0.$ At the nozzle inlet the Mach number is $0\mathrm{.}5$ and the inlet radius is $20$ cm $.$ The nozzle outlet velcity is sonic. The edges of the nozzle are straight, having an angle of $10$ with the horizontal $($e$.$g$.$ the radius decreases linearly with $x).$

a$.)$ Determine the area distribution for the nozzle. e$.$g$.,$ find $A(x).$

b$.)$ Determine the Mach number variation in the nozzle. e$.$g$.$ compute $M(x).$ In order to do that, you will have to solve

$(\frac{A}{A^{**}}{)}^2=\frac{1}{M^2}[\frac{2}{+1}(1+\frac{-1}{2}{M}^2){]}^{+\frac{1}{}-1}$

Using a simple Newton$-$Raphson technique, express Equation $(1)$ as a function of Mach number, $f(M).$ Then take the derivative, $d\frac{f}{d}M.$ The Newton$-$Raphson technique makes a new Mach number estimate based on an old estimate as

$M_{\text{n}\text{e}\text{w}}=M_{\text{o}\text{l}\text{d}}-\frac{f(M_{\text{o}\text{l}\text{d}})}{\frac{df}{dM}(M_{\text{o}\text{l}\text{d}})}$

when $M_{\text{n}\text{e}\text{w}}$ and $M_{\text{o}\text{l}\text{d}}$ are sufficently close, you have determined the Mach number. Include sufficient details of your solution procedure and clearly state your acceptance threshold.

c$.)$ Plot $M(x),T(x)$ and $p(x)$ vs$.$ the length of the nozzle for a reservoir temperature of $400$ K and reservoir pressures of $250$kPa,$500$kPa and $750$ kPa $.$