$2.$ Consider a turbine engine that powers helicopter flight with ambient conditions at $298$ K and $1\mathrm{.}0$ atmospheres of pressure. There are two turbine stages with different rotational speeds; the first one drives the compressor while the other drives the main propeller of the helicopter. The turbine exit pressure is practically $1\mathrm{.}0$ atmospheres. Consider a combustor exit temperature of $1600$ K and a mass mixture ratio $\backslash (\backslash$mu$=30\backslash ).$ Consider that $\backslash (3\backslash \%\backslash )$ of the air is taken from the flow downstream of the compressor but before entry to the combustor to be used for cooling of the turbine. The specific heat of combustion products equals $\backslash (0\mathrm{.}30\backslash$mathrm$\{$cal$\}/\backslash$mathrm$\{$gm$\}^\{\backslash$circ$\}\backslash$mathrm$\{$K$\}\backslash )$ with $\backslash (\backslash$gamma$=1\mathrm{.}3\backslash ),$ and the specific heat of air is $0\mathrm{.}24\backslash (\backslash$mathrm$\{$cal$\}/\backslash$mathrm$\{$gm$\}^\{\backslash$circ$\}\backslash$mathrm$\{$K$\}\backslash )$ with $\backslash (\backslash$gamma$=1\mathrm{.}4\backslash ).$ Assume that flight velocity has negligible effect on the intake flow pressure.

a$.)$ What is the pressure entering the compressor? Assume a polytropic diffuser efficiency equal to $0\mathrm{.}95.$

b$.)$ Assume a pressure ratio of $20$ across the compressor. If the polytropic compressor efficiency is $0\mathrm{.}95,$ what is the power requirement of the compressor $($per unit mass flux$)?$

c$.)$ Neglect pressure change across the combustor. The first turbine stage drives the compressor and has a polytropic efficiency of $0\mathrm{.}92.$ Determine the pressure change and temperature change across that stage.

d$.)$ All of the power from the second stage drives the propeller. The polytropic efficiency of this stage is $0\mathrm{.}95.$ The mechanical efficiency for power transfer to the propeller is $0\mathrm{.}9.$ What is the final exhaust$-$gas temperature? What is the extracted power $($per unit mass flux$)$ delivered to the propeller?