Multi-stage air compression — conditions for minimum total work For compressing and delivering a given mass of air using multi-stage compression with intercooling, which set of conditions minimizes the total work input?

Introduction / Context:
Staging and intercooling are used to reduce work input and discharge temperature in air compression. There is a classic optimal design point that distributes the compression across stages to minimize total power for a given overall pressure ratio.

Given Data / Assumptions:

• N-stage compression with identical polytropic behavior per stage.
• Intercooling available between stages with negligible pressure loss.
• Objective: minimize total compressor work for a fixed overall ratio.

Concept / Approach:
With perfect intercooling (air returned to intake temperature), the minimum work condition occurs when the pressure ratio is the same in every stage. At this point, each stage performs the same work, which naturally follows from equal ratios and identical thermodynamics per stage.

Step-by-Step Solution:
Let overall ratio be R = p_out/p_in.For N stages, set stage ratio r_s so that r_s^N = R → r_s = R^(1/N).Under perfect intercooling, the temperature entering each stage is equal, making equal work per stage and minimizing the total.

Verification / Alternative check:
Calculus-based minimization of total polytropic work with respect to intermediate pressures yields equal stage ratios, confirming the condition.

Why Other Options Are Wrong:
Each of (a), (b), and (c) is correct individually, so selecting anything other than “All of the above” would omit valid requirements for minimum work.

Common Pitfalls:
Forgetting to account for pressure losses in real intercoolers, which slightly shift the true optimum in practice.

Final Answer:
All of the above