Rankine cycle 2 MPa: efficiency vs Carnot

Steam table values (read from tables)

Boiler: 2 MPa, 400 °C (superheated): h₃ = 3248.2 kJ/kg, s₃ = 7.1290 kJ/kg·K.

Condenser: saturated at 50 °C: P_sat = 12.35 kPa; h_f = 209.34, h_g = 2591.3 kJ/kg; v_f = 0.001012 m³/kg; s_f = 0.7038, s_g = 8.0749 kJ/kg·K.

The cycle: 1 (saturated liquid at 50 °C) → 2 (pump) → 3 (turbine inlet) → 4 (condenser inlet).

Pump

w_p = v_f (P₂ − P₁) = 0.001012 × (2000 − 12.35) = 2.01 kJ/kg h₂ = h₁ + w_p = 209.34 + 2.01 = 211.35 kJ/kg

Turbine (isentropic, s₄ = s₃)

x₄ = (s₃ − s_f)/(s_g − s_f) = (7.1290 − 0.7038)/(8.0749 − 0.7038) = 0.8717 h₄ = 209.34 + 0.8717 × (2591.3 − 209.34) = 2285.6 kJ/kg

w_t = h₃ − h₄ = 3248.2 − 2285.6 = 962.6 kJ/kg

Heat and efficiency

q_in = h₃ − h₂ = 3248.2 − 211.35 = 3036.9 kJ/kg w_net = w_t − w_p = 962.6 − 2.0 = 960.6 kJ/kg

η_Rankine = 960.6/3036.9 = 0.316 → 31.6 %

Carnot cycle between same limits

T_H = 400 + 273.15 = 673.15 K, T_L = 50 + 273.15 = 323.15 K

η_Carnot = 1 − 323.15/673.15 = 0.520 → 52.0 %

Comparison

Answer: η_Rankine = 31.6 %, η_Carnot = 52.0 %. The Rankine efficiency is about 61 % of the Carnot value. Heat in the Rankine cycle is added over a range of temperatures (liquid heating from about 50 °C up to the boiling point of 212 °C), not all at the highest temperature, so its mean heat addition temperature is lower.