In a two-stage air compressor with an intercooler, perfect intercooling means the temperature after intercooling returns to the initial suction temperature. Under perfect intercooling, what is the optimal intermediate pressure p2 in terms of suction pressure p1 and delivery pressure p3?

Introduction / Context:
Staging a compressor with intercooling reduces work input. The question tests knowledge of the condition for minimum work in a two-stage compressor with ideal (perfect) intercooling and asks for the correct expression for the intermediate pressure p2.

Given Data / Assumptions:

• Two stages with an intercooler that cools the air back to the initial suction temperature (perfect intercooling).
• Same mass flow through both stages.
• Identical polytropic/isentropic behavior in both stages for simplicity.

Concept / Approach:
For minimum total work in two-stage compression with perfect intercooling, the pressure ratio should be split equally between stages. That means the pressure ratio in the first stage equals the pressure ratio in the second stage. This leads directly to the geometric mean relation for p2.

Step-by-Step Solution:

Verification / Alternative check:
If p2 is the geometric mean of p1 and p3, both stages perform equal pressure ratio and roughly equal work, minimizing the sum. This well-known result appears in standard compressor design references and is consistent with energy minimization analyses.

Why Other Options Are Wrong:

• Arithmetic mean (p1 + p3)/2: Does not equalize stage ratios; leads to higher total work.
• p2 = p1 or p2 = p3: Collapses one stage and overloads the other; not minimum work.
• p2 = p1 * p3: Dimensional nonsense for pressure without normalization.

Common Pitfalls:
Confusing geometric mean with arithmetic mean; forgetting the equal-ratio requirement; ignoring that perfect intercooling restores temperature to the initial suction temperature.

Final Answer:
p2 = sqrt(p1 * p3)