If n is the polytropic index of the compression and \(\frac{{{P_4}}}{{{P_1}}}\) is the pressure ratio for a three-stage compressor with ideal intercooling, the expression for total work of three stage reciprocating air compressor will be

1
\(\frac{{3n}}{{\left( {n - 1} \right)}}{P_1}{V_1}\left[ {{{\left( {\frac{{{P_4}}}{{{P_1}}}} \right)}^{\frac{{n - 1}}{n}}} - 1} \right]\)
2
\(\frac{n}{{\left( {n - 1} \right)}}{P_1}{V_1}\left[ {3{{\left( {\frac{{{P_4}}}{{{P_1}}}} \right)}^{\frac{{n - 1}}{{3n}}}} - 1} \right]\)
3
\(\frac{n}{{\left( {n - 1} \right)}}{P_1}{V_1}\left[ {{{\left( {\frac{{{P_4}}}{{{P_1}}}} \right)}^{\frac{{n - 1}}{{3n}}}} - 1} \right]\)
4
\(\frac{{3n}}{{\left( {n - 1} \right)}}{P_1}{V_1}\left[ {{{\left( {\frac{{{P_4}}}{{{P_1}}}} \right)}^{\frac{{n - 1}}{{3n}}}} - 1} \right]\)

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