Minimum work in three-stage reciprocating compression Under optimum staging and perfect intercooling, how does the total minimum work required by a three-stage reciprocating air compressor compare to the work required by each individual stage?

Introduction / Context:
Multi-stage compression with intercooling reduces work input. When pressure ratios are optimally split, each stage does the same amount of work, which leads to a simple relation between per-stage work and the total work.

Given Data / Assumptions:

• Three stages with perfect intercooling between stages.
• Equal pressure ratio per stage for minimum work.
• Similar efficiencies and speeds across stages.

Concept / Approach:
For minimum work, the interstage pressures are chosen such that the per-stage work is equal. The total work is then the sum of the three equal contributions, hence three times the work of any single stage.

Step-by-Step Solution:
Choose interstage pressures so that p2 = (p1 * p3)^(1/2) and p3 = (p2 * p4)^(1/2), etc., equalising pressure ratio per stage.With perfect intercooling, inlet temperature to each stage returns to the initial value, equalising stage work.Total work = W1 + W2 + W3 = 3 * W_stage.

Verification / Alternative check:
Indicator diagrams and analytical formulas for polytropic or isentropic compression confirm equal areas (works) for equal pressure ratios with perfect intercooling.

Why Other Options Are Wrong:

• Equal to or double: underestimate the sum of three equal stage works.
• Six times: overestimates and has no basis.
• Half: contradicts additive nature of work.

Common Pitfalls:
Confusing “minimum total work” with “minimum per-stage work”; both are satisfied simultaneously when work is equalised per stage.

Final Answer:
three times