Two-stage reciprocating compressor with perfect intercooling: find the optimum intermediate pressure Compression from p₁ to p₃ is performed in two stages with perfect intercooling and negligible pressure losses. What is the optimum intermediate pressure p₂ that minimizes work for equal polytropic law in both cylinders? Choose the correct formula.

Given data

• Two-stage reciprocating compressor.
• Perfect intercooling (air cooled back to initial suction temperature between stages).
• Same polytropic law in both cylinders; negligible losses.

Concept/Approach
For minimum total work with equal polytropic index in each stage and perfect intercooling, the pressure ratio should be equally split across the stages. This yields the geometric-mean rule for the intermediate pressure.

Derivation snapshot
Let overall pressure ratio be r = p₃/p₁. For equal stage ratios: r₁ = r₂ = √r.Therefore, p₂ = p₁ r₁ = p₁ √(p₃/p₁) = √(p₁ p₃).

Numerical check
If p₁ = 1 bar and p₃ = 9 bar ⇒ p₂ = √(1×9) = 3 bar (equal ratios 3 and 3).

Final Answer
p₂ = √(p₁ p₃)