Question 9.3: Problem: Superheated steam at a pressure of 6 MPa and temper...
Problem: Superheated steam at a pressure of 6 MPa and temperature of 400 °C enters the first stage of a turbine in a Rankine cycle where it expands to 0.8 MPa. The steam is then reheated to 300 °C and expanded in the second stage of the turbine which it leaves at a condenser pressure of 30 kPa. Find (a) the work done by the turbine per unit mass of fluid, (b) the heat added per unit mass of fluid and (c) the thermal efficiency of the cycle.
Find: (a) Work done w per unit mass of fluid, (b) heat added q per unit mass of fluid, (c) thermal efficiency η_{th} of the cycle.
Known: Turbine inlet pressure P_2 = P_3 = 6 MPa, turbine inlet temperature T_3 = 400 °C, reheat pressure P_4 = P_5 = 0.8 MPa, reheat temperature T_5 = 300 °C, condenser pressure P_1 = P_6 = 30 kPa.
Process Diagram: The Rankine cycle with reheat on a T‐s diagram (Figure E9.3).
Assumptions: Expansion through the turbines is isentropic so ∆S_{34} = 0 and ∆S_{56} = 0.
Governing Equations:
Work output from turbines w_t=(h_3-h_4)+(h_5-h_6)
Heat input to boiler q_H=(h_3-h_2)+(h_5-h_4)
Thermal efficiency of Rankine cycle η_{th, Rankine}=\frac{w_t-w_p}{q_H}
From Example 9.2 the enthalpy at the condenser exit is h_1 = 289.23 kJ / kg.
The enthalpy at the pump exit is h_2 = 295.33 kJ / kg.
For superheated steam at P_3 = 6 MPa and T_3 = 400 °C (Appendix 8c), specific enthalpy h_3 = 3177.2 kJ / kg and specific entropy s_3 = 6.5408 kJ / kgK.
For saturated water at P_4 = 0.8 kPa (Appendix 8b), specific enthalpy of liquid h_{4,f} = 721.11 kJ / kg and vapour h_{4,g} = 2769.1 kJ / kg, and specific entropy of liquid s_{4,f} = 2.0462 kJ / kgK and vapour s_{4,g} = 6.6628 kJ / kgK. If expansion through the turbine is isentropic, s_4 = s_3 = 6.5408 kJ / kgK and the mixture quality in the high pressure turbine is
x_4=\frac{s_4-s_{4,f}}{s_{4,g}-s_{4,f}}=\frac{6.5408 \ kJ/kgK-2.0462 \ kJ/kgK}{6.6628 \ kJ/kgK-2.0462 \ kJ/kgK}=0.973 \ 57.
The enthalpy at the turbine exit is then
h_4=h_{4,f}+x_4(h_{4,g}-h_{4,f}), \\ h_4=721.11 \ kJ/kg+0.973 \ 57 \times (2769.1 \ kJ/kg-721.11 \ kJ/kg)=2715.0 \ kJ/kg.
For superheated steam at P_5 = 0.8 MPa and T_5 = 300 °C (Appendix 8c), specific enthalpy h_5 = 3056.5 kJ / kg and specific entropy s_5 = 7.2328 kJ / kgK.
For saturated water at P_6 = 30 kPa (Appendix 8b), specific enthalpy of liquid h_{6,f} = 289.23 kJ / kg
and vapour h_{6,g} = 2625.3 kJ / kg, and specific entropy of liquid s_{6,f} = 0.9439 kJ / kgK and vapour s_{6,g} = 7.7686 kJ / kgK. If expansion through the turbine is isentropic, s_6= s_5 = 7.2328 kJ / kgK and the mixture quality in the low pressure turbine is
x_6=\frac{s_6-s_{6,f}}{s_{6,g}-s_{6,f}}=\frac{7.2328 \ kJ/kgK-0.9439 \ kJ/kgK}{7.7686 \ kJ/kgK-0.9439 \ kJ/kgK}=0.92 \ 149.
The enthalpy at the turbine exit is then
h_6=h_{6,f}+x_6(h_{6,g}-h_{6,f}), \\ h_6=289.23 \ kJ/kg +0.92149 \times (2625.3 \ kJ/kg-289.23 \ kJ/kg)=2441.9 \ kJ/kg.
(a) The work done by both turbines is
w_t=(h_3-h_4)+(h_5-h_6), \\ w_t=(3177.2 \ kJ/kg -2715.0 \ kJ/kg)+(3056.5 \ kJ/kg-2441.9 \ kJ/kg)=1076.8 \ kJ/kg.
(b) The heat input to the boiler is
q_H=(h_3-h_2)+(h_5-h_4), \\ q_H=(3177.2 \ kJ/kg-295.33 \ kJ/kg)+(3056.5 \ kJ /kg-2715.0 \ kJ/kg)=3223.4 \ kJ/kg.
(c) The work input to the pump is
w_p=(h_2-h_1) , \\ w_p=295.33 \ kJ/kg-289.23 \ kJ/kg=6.1 \ kJ/kg.
Then the thermal efficiency of the Rankine cycle is
η_{th,Rankine}=\frac{w_t-w_p}{q_H} , \\ η_{th,Rankine}= \frac{1076.8 \ kJ/kg-6.1 \ kJ/kg}{3223.4 \ kJ/kg}=0.332 16.
Answer: (a) The work done by the turbine is 1076.8 kJ / kg, (b) with heat input of 3223.3 kJ / kg of working fluid, (c) the thermal efficiency of the Rankine cycle is 33.2%.