Prove that the optimum pressure ratio for intercooling of gas turbines with heat exchange is given below:

\[\frac{p_{2}}{p_{1}}=\frac{p_{i}}{p_{1}}=\left(\frac{p_{2^{\prime}}}{p_{1}}\right)^{1 / 2}\]

Assume that the processes in the turbine and compressor are both isentropic, and that the heat exchanger effectiveness, \(\varepsilon=1.0\).