Two-stage reciprocating air compressor — optimum intercooler pressure For a two-stage reciprocating air compressor with perfect intercooling, the intermediate (intercooler) pressure that minimizes the total compression work is given by which relation (p1 = intake pressure, p3 = delivery pressure)?

Introduction / Context:
Staging and intercooling in reciprocating compressors reduce compression work and discharge temperature. For two stages with perfect intercooling (the air is cooled back to the initial intake temperature between stages), there is an optimum intermediate pressure that minimizes the total work.

Given Data / Assumptions:

• Two compression stages with equal polytropic/isentropic characteristics.
• Perfect intercooling to the intake temperature between stages.
• Negligible pressure losses in intercooler and piping (idealization).

Concept / Approach:
For minimum total work, the pressure ratio is split equally between stages. That means p2/p1 = p3/p2. Solving this yields p2^2 = p1 * p3, or p2 = (p1 * p3)^(1/2). This balances the work of both stages and achieves the lowest possible sum for a given overall ratio.

Step-by-Step Solution:
Set equal pressure ratios: r1 = p2/p1 and r2 = p3/p2.For minimum work: r1 = r2.Therefore, p2/p1 = p3/p2 → p2^2 = p1 * p3.Hence, p2 = (p1 * p3)^(1/2).

Verification / Alternative check:
Equal temperature rises per stage under perfect intercooling also result when pressure ratios are equal, consistent with the optimum condition.

Why Other Options Are Wrong:
(b) Arithmetic mean ignores thermodynamic work minimization; (c) and (d) have incorrect dimensional/ratio logic and do not balance stage works.

Common Pitfalls:
Forgetting to include the square root; writing p2 = p1 * p3 is a frequent mistake that overpressurizes the first stage.

Final Answer:
p2 = (p1 * p3)^(1/2)