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.
Mechanical Engineering
Compressors, Gas Dynamics and Gas Turbines
Difficulty: Medium
Choose an option
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Ap₂ = √(p₁ p₃)
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Bp₂ = (p₁ + p₃) / 2
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Cp₂ = p₁ · p₃
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Dp₂ = p₃ / p₁
Answer
Correct Answer: p₂ = √(p₁ p₃)
Explanation
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/ApproachFor 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 snapshotLet overall pressure ratio be r = p₃/p₁. For equal stage ratios: r₁ = r₂ = √r.Therefore, p₂ = p₁ r₁ = p₁ √(p₃/p₁) = √(p₁ p₃).
Numerical checkIf p₁ = 1 bar and p₃ = 9 bar ⇒ p₂ = √(1×9) = 3 bar (equal ratios 3 and 3).
Final Answerp₂ = √(p₁ p₃)