Reheat to maximize work output: For two-stage expansion with ideal reheating in a gas turbine (or steam turbine), the reheating should bring the pressure to which intermediate value (p_i) between maximum p1 and minimum p2 for maximum total work?

Introduction / Context:
In multi-stage expansion with reheating, choosing the optimal intermediate pressure maximizes total work and balances irreversibilities. This is a classic result in gas/steam turbine cycle analysis.

Given Data / Assumptions:

• Two equal-efficiency stages with reheating to approximately the same turbine inlet temperature between stages.
• Idealized isentropic stage analysis for the optimum condition.
• Neglecting pressure losses in reheater and piping for the derivation.

Concept / Approach:

For maximum work, distribute the overall pressure ratio equally across stages. This occurs when the individual stage pressure ratios are equal, so that p1 / p_i = p_i / p2, leading to p_i^2 = p1 * p2 and hence p_i = √(p1 * p2). Equalizing stage ratios tends to equalize temperature drops, improving mean effective temperature of heat addition and reducing exergy destruction.

Step-by-Step Solution:

Verification / Alternative check:

Similar result appears for optimum intercooling pressure in multi-stage compression, reinforcing the symmetry between expansion with reheating and compression with intercooling.

Why Other Options Are Wrong:

Arithmetic means or ad hoc multiples do not equalize pressure ratios. The cube-root form pertains to three-stage problems, not two-stage.

Common Pitfalls:

Confusing arithmetic with geometric mean; ignoring stage equality assumptions and reheater losses when applying the ideal result to real plants.

Final Answer:

p_i = √(p1 * p2)