Condition for minimum work in multistage compression State whether the following is true: For minimum total work in multi-stage compression with intercooling, the work done on each stage should be equal.

Introduction / Context:
Optimizing interstage pressures in multi-stage compressors is critical to reduce shaft power. A classic result shows that minimum total work occurs when every stage shares an equal pressure ratio and, consequently, equal work input (with perfect intercooling).

Given Data / Assumptions:

• Multiple stages with intercoolers.
• Intercoolers bring the air back near the initial intake temperature.
• Similar mechanical and volumetric efficiencies stage-to-stage.

Concept / Approach:
Total work is the sum of stage works. By calculus of variations or by symmetry arguments using equal temperature resets, equalising the pressure ratio per stage minimises the sum for a fixed overall ratio. With equal ratios and equal inlet temperatures, the per-stage work becomes equal.

Step-by-Step Solution:
Let overall ratio be r_total; divide into k stages with r_stage = r_total^(1/k).Perfect intercooling → each stage starts at the same temperature.Work per stage becomes equal; total is k * W_stage, and minimised under the equal-split condition.

Verification / Alternative check:
Deriving with polytropic relations shows d(W_total)/d(p2) = 0 gives p2 such that ratios are equal. This generalises to any number of stages.

Why Other Options Are Wrong:

• “No” contradicts the optimal-split principle.
• “Only two stages” is unnecessarily restrictive.
• Dependence on n = 1 is false; the result holds for practical polytropic indices with proper intercooling.

Common Pitfalls:
Ignoring pressure drops and intercooler effectiveness; while these shift exact values, the equal-work principle remains the guiding design rule.

Final Answer:
Yes