Three-stage compressor with perfect intercooling: pressure relationship for minimum work For a three-stage reciprocating air compressor with perfect intercooling and negligible intercooler pressure loss (overall pressures p1 → p4), which relationship ensures minimum total work?

Introduction / Context:
Staging reduces the work of compression, particularly when heat is removed between stages. For a fixed overall pressure ratio, there is an optimal distribution of pressure ratios among stages that minimizes total work. This is a standard design question for multistage reciprocating compressors.

Given Data / Assumptions:

• Three compression stages with intercooling between stages.
• Perfect intercooling: each stage inlet temperature equals the initial intake temperature.
• Negligible pressure drops in intercoolers and piping.

Concept / Approach:
For perfect intercooling and identical polytropic behavior in each stage, the total work is minimized when the pressure ratio is the same across all stages. This equal-ratio condition distributes the enthalpy rise uniformly and produces the smallest sum of stage works for the given overall pressure ratio.

Step-by-Step Solution:

Verification / Alternative check:
Minimization can be shown by differentiating the total work with respect to intermediate pressures and setting the derivative to zero, which leads to equal pressure ratios per stage as the optimum.

Why Other Options Are Wrong:

Products like p1 p2 = p3 p4 or p1 p3 = p2 p4 are not generally satisfied by equal-ratio staging for three stages and do not guarantee minimum work.Statements such as p2^2 = p1 p4 or p3^2 = p2 p4 only apply to specific two-stage relationships, not to three-stage equal ratio design.

Common Pitfalls:
Applying the two-stage geometric mean relation (p2^2 = p1 p3) incorrectly to three-stage problems or overlooking the assumption of perfect intercooling.

Final Answer:

p2 / p1 = p3 / p2 = p4 / p3