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 $10\backslash$deg 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

A A$2=1$ M$22$ $+11+$ $12$ M$2$ $+1/$$1$

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

M$\_$new $=$ M$\_$old $$ f$($M$\_$old$)/($df$/$dM $($M$\_$old$))$

when Mnew and Mold 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.