Two-stage compression with perfect intercooling: If n1 and n2 are the polytropic indices in stage-1 and stage-2, then the work ratio with equal pressure ratios is W1/W2 = [n1/(n1 − 1)] / [n2/(n2 − 1)].

Introduction / Context:In multi-stage air compression with perfect intercooling and equal pressure ratios per stage, each stage can follow a different polytropic index n. Comparing stage works helps in assessing valve timing, cooling, and stage matching.

Given Data / Assumptions:

• Two stages with polytropic indices n1 (stage-1) and n2 (stage-2).
• Perfect intercooling restores the interstage temperature to the initial suction temperature.
• Equal pressure ratios per stage.
• Ideal gas behavior with constant specific heats.

Concept / Approach:For polytropic compression p * v^n = constant, the specific work (per unit mass) is proportional to n/(n − 1) times a function of the pressure ratio raised to (n − 1)/n. With equal pressure ratios and identical inlet temperature to each stage (due to perfect intercooling), the ratio of works reduces to the ratio of n/(n − 1) factors.

Step-by-Step Solution:

Verification / Alternative check:If n1 = n2, the ratio becomes 1, matching the expectation that both stages do equal work under perfect intercooling with equal pressure ratios.

Why Other Options Are Wrong:

• Inverse ratio: Swaps the stage roles incorrectly.
• (n1 − 1)/(n2 − 1): Omits the n terms; not dimensionally consistent with polytropic work dependence.
• n2/n1: Also incorrect; polytropic work does not scale simply as n.

Common Pitfalls:Assuming stage work equality without checking n values; perfect intercooling equalizes temperatures, but differing n values still change the polytropic factor.

Final Answer:W1/W2 = [n1/(n1 − 1)] / [n2/(n2 − 1)]