Q$2$

A three$-$stage compressor is used to raise the pressure of an ideal gas initially at $P_1=\frac{1}{b}ar,T_1=300$ K to a certain final pressure $P_2=\frac{10}{b}ar.$ The pressurized gas is cooled in an inter$-$cooling heat exchanger after each stage of compression as shown in the figure.

Ideally, the temperature after each inter$-$cooler drops down back to the initial temperature $T_1.$ Assume that there is no pressure drop in the intercoolers and that the isentrepic efficiency of the compression process is identical in all three stages, ${}_{c,isen}=0\mathrm{.}8.$

a$)$ Develop an expression for the minimum work requirement in this three$-$stage compression process $($by optimizing over the pressure ratio or intermediate pressure levels $P_a$ and $P_b$ after each compressor$).$

Note: The optimal intermediate pressures follow a geometric series between $P_1$ and $P_2.$

b$)$ Develop an expression for the compression work if the ideal gas is compressed isothermally from the initial pressure $P_1$ to the final desired pressure $P_2.$

Formation of NO in hot air:

When heated, air can partially react to yield NO$.$ Assume air is made up of $21\%O2,$ and

$79\%$ N$2.$ Consider air at $1500$ K and $2000$ K at $1$ atm pressure. Determine the

concentration of NO in these two conditions.