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Rankine cycle plots and efficiency

Short answer

The Rankine cycle consists of isentropic pump and turbine, constant‑pressure boiler and condenser. Efficiency: η = [(h₃−h₄)−(h₂−h₁)]/(h₃−h₂) ≈ (h₃−h₄)/(h₃−h₁). Superheat raises average heat‑addition temperature, improving efficiency. p‑v and T‑s diagrams show the four processes including evaporation and superheat.

Sketch the Rankine cycle on p-v and T-s plots when the state of steam at the boiler outlet is saturated and superheated vapor respectively. Derive an expression for the efficiency of Rankine cycle.

The Rankine cycle is the ideal cycle of a steam power plant. It has four processes:

  • 1-2: isentropic compression of saturated liquid in the pump
  • 2-3: constant-pressure heat addition in the boiler (and superheater)
  • 3-4: isentropic expansion in the turbine
  • 4-1: constant-pressure heat rejection in the condenser (to saturated liquid)

Diagrams

(a) Saturated vapour at boiler outlet

 T-s diagram                        P-v diagram
 T                                  P
 |           .-'''-.                |  2 _________________ 3 (sat. vapour)
 |         .'       '.              |  |                    \
 |   2'  /____________3             |  |                     '.
 |      /             |\            |  |                       '-.
 |     /              | \           |  1 ______________________ 4
 |  2 /               |  \          +-------------------------------> v
 |  1/________________4   \
 |  /                      \
 +-----------------------------> s
 (2 lies just above 1; 2-2' liquid heating, 2'-3 boiling)

(b) Superheated vapour at boiler outlet

 T-s diagram                        P-v diagram
 T                                  P
 |                       3          |  2 ____________________ 3
 |           .-'''-.   .'|          |  |                       \
 |         .'       '.'  |          |  |                        '.
 |   2'  /____________3' |          |  |                          '-.
 |      /              \ |          |  1 _________________________ 4
 |     /                \|          +----------------------------------> v
 |  2 /                  |\
 |  1/___________________4 \
 |  /                       \
 +-----------------------------> s
 (2 lies just above 1; 2'-3' boiling, 3'-3 superheating)

Efficiency derivation

Per kg of steam (steady flow, kinetic and potential energies neglected):

  • Pump work: w_p = h₂ − h₁ = v_f (P₂ − P₁)
  • Heat added: q_in = h₃ − h₂
  • Turbine work: w_t = h₃ − h₄
  • Heat rejected: q_out = h₄ − h₁

Net work: w_net = w_t − w_p = (h₃ − h₄) − (h₂ − h₁)

η_Rankine = w_net/q_in = [(h₃ − h₄) − (h₂ − h₁)]/(h₃ − h₂) = 1 − (h₄ − h₁)/(h₃ − h₂)

The pump work is very small compared with the turbine work, so approximately η ≈ (h₃ − h₄)/(h₃ − h₁).

The efficiency rises with higher boiler pressure, higher superheat temperature and lower condenser pressure.

✓ Answer checked by Claude Opus 5.5 (high effort)

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