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.

Question

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 [latex]η_{th}[/latex] of the cycle.

Known: Turbine inlet pressure [latex]P_2 = P_3[/latex] = 6 MPa, turbine inlet temperature [latex]T_3[/latex] = 400 °C, reheat pressure [latex]P_4 = P_5[/latex] = 0.8 MPa, reheat temperature [latex]T_5[/latex] = 300 °C, condenser pressure [latex]P_1 = P_6[/latex] = 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 [latex]∆S_{34}[/latex] = 0 and [latex]∆S_{56}[/latex] = 0.

Governing Equations:

Work output from turbines [latex]w_t=(h_3-h_4)+(h_5-h_6)[/latex]

Heat input to boiler [latex]q_H=(h_3-h_2)+(h_5-h_4)[/latex]

Thermal efficiency of Rankine cycle [latex]η_{th, Rankine}=\frac{w_t-w_p}{q_H}[/latex]

Accepted Answer

From Example 9.2 the enthalpy at the condenser exit is [latex] h_1 [/latex] = 289.23 kJ / kg.
The enthalpy at the pump exit is [latex]h_2 [/latex] = 295.33 kJ / kg.
For superheated steam at [latex]P_3 [/latex] = 6 MPa and [latex]T_3 [/latex] = 400 °C (Appendix 8c), specific enthalpy [latex]h_3 [/latex] = 3177.2 kJ / kg and specific entropy [latex]s_3 [/latex] = 6.5408 kJ / kgK.
For saturated water at [latex]P_4 [/latex] = 0.8 kPa (Appendix 8b), specific enthalpy of liquid [latex]h_{4,f} [/latex] = 721.11 kJ / kg and vapour [latex]h_{4,g} [/latex] = 2769.1 kJ / kg, and specific entropy of liquid [latex]s_{4,f} [/latex] = 2.0462 kJ / kgK and vapour [latex]s_{4,g} [/latex] = 6.6628 kJ / kgK. If expansion through the turbine is isentropic, [latex]s_4 = s_3 [/latex] = 6.5408 kJ / kgK and the mixture quality in the high pressure turbine is

[latex]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.[/latex]

The enthalpy at the turbine exit is then

[latex]h_4=h_{4,f}+x_4(h_{4,g}-h_{4,f}), \ h_4=721.11 \ kJ/kg+0.973 \ 57 imes (2769.1 \ kJ/kg-721.11 \ kJ/kg)=2715.0 \ kJ/kg.[/latex]

For superheated steam at [latex]P_5 [/latex]= 0.8 MPa and [latex]T_5[/latex] = 300 °C (Appendix 8c), specific enthalpy [latex]h_5[/latex] = 3056.5 kJ / kg and specific entropy [latex]s_5[/latex] = 7.2328 kJ / kgK.
For saturated water at [latex]P_6[/latex] = 30 kPa (Appendix 8b), specific enthalpy of liquid [latex]h_{6,f}[/latex] = 289.23 kJ / kg
and vapour [latex]h_{6,g}[/latex] = 2625.3 kJ / kg, and specific entropy of liquid [latex]s_{6,f}[/latex] = 0.9439 kJ / kgK and vapour [latex]s_{6,g}[/latex] = 7.7686 kJ / kgK. If expansion through the turbine is isentropic, [latex]s_6= s_5[/latex] = 7.2328 kJ / kgK and the mixture quality in the low pressure turbine is

[latex]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.[/latex]

The enthalpy at the turbine exit is then

[latex]h_6=h_{6,f}+x_6(h_{6,g}-h_{6,f}), \ h_6=289.23 \ kJ/kg +0.92149 imes (2625.3 \ kJ/kg-289.23 \ kJ/kg)=2441.9 \ kJ/kg.[/latex]

(a) The work done by both turbines is

[latex]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.[/latex]

(b) The heat input to the boiler is

[latex]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.[/latex]

(c) The work input to the pump is

[latex]w_p=(h_2-h_1) , \ w_p=295.33 \ kJ/kg-289.23 \ kJ/kg=6.1 \ kJ/kg.[/latex]

Then the thermal efficiency of the Rankine cycle is

[latex]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.[/latex]

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%.

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