Two-stage reciprocating compressor with perfect intercooling Let p1 be the suction pressure to the low-pressure (LP) cylinder and p3 the delivery pressure from the high-pressure (HP) cylinder. For a single-acting, two-stage reciprocating air compressor with complete intercooling and equal piston speed/stroke, what is the correct ratio of cylinder diameters D1/D2 (LP/HP)?

Introduction / Context:
In multi-stage reciprocating compressors with perfect intercooling, equalising the pressure ratio per stage minimises work. This also sets a design relation between LP and HP cylinder sizes because the same mass flow must pass through both stages at different suction pressures.

Given Data / Assumptions:

• Suction to LP: p1; discharge from HP: p3.
• Complete intercooling returns the HP suction temperature approximately to the LP suction temperature.
• Single-acting cylinders, same speed and stroke; volumetric efficiencies taken similar for the sizing relation.

Concept / Approach:
For equal mass flow m through both stages at equal inlet temperature T, we use m ∝ p * V / (R * T) at suction. Thus p1 * V1 = p2 * V2, where p2 is HP suction pressure. With perfect intercooling and optimal staging, p2 = sqrt(p1 * p3). Cylinder displacement V ∝ D^2 (for fixed stroke and speed), so D1^2 / D2^2 = p2 / p1.

Step-by-Step Solution:
p2 = sqrt(p1 * p3).Mass continuity at equal T: p1 * V1 = p2 * V2.With V ∝ D^2: D1^2 / D2^2 = p2 / p1.So D1/D2 = sqrt(p2 / p1) = sqrt( sqrt(p1 * p3) / p1 ) = (p3 / p1)^(1/4).

Verification / Alternative check:
Dimensionally and physically consistent: higher overall pressure ratio (p3/p1) demands a relatively larger LP cylinder to ingest the same mass at the lower pressure.

Why Other Options Are Wrong:

• (p1 p3)^(1/2): ignores the square-root nesting from displacement proportionality.
• (p1/p3)^(1/4) and (p1 p3)^(1/4): invert or misplace the dependence on p3/p1.
• (p3/p1)^(1/2): overpredicts the diameter ratio by omitting the square-root from area–diameter relation.

Common Pitfalls:
Forgetting that cylinder capacity scales with D^2 (at fixed stroke), not with D directly; and overlooking the perfect-intercooling relation p2 = sqrt(p1 p3).

Final Answer:
D1/D2 = (p3/p1)^(1/4)