An adiabatic two-stage compressor is used to compress a gas, which is cooled to the inlet gas temperature between the stages, the theoretical work can be expressed by the following equation:

\[W=\frac{k p_{1} V_{1}}{k-1}\left[\left(\frac{p_{2}}{p_{1}}\right)^{(k-1) / k}-2+\left(\frac{p_{3}}{p_{2}}\right)^{(k-1) / k}\right]\]

Where, $k=C_{p} / C_{v} ; p_{1}=$ pressure inlet; $p_{2}=$ intermediate stage pressure; $p_{3}=$ outlet pressure; $V_{1}=$ inlet volume

We are interested to optimize the intermediate pressure $p_{2}$ such that the work is a minimum.