Consider a two connected steam turbines with the state properties as shown in the Figure 5.19. Assume that the changes in kinetic and potential energies are negligible. (a) Write mass, energy, entropy, and exergy balance equations for each turbine separately, (b) calculate the work rates produced

Question

Consider a two connected steam turbines with the state properties as shown in the Figure 5.19. Assume that the changes in kinetic and potential energies are negligible. (a) Write mass, energy, entropy, and exergy balance equations for each turbine separately, (b) calculate the work rates produced by both turbines, (c) find the temperature of the water leaving turbine 2, and (d) calculate the total entropy generation rate and total exergy destruction rate.

Accepted Answer

a) Write mass, energy and entropy balance equations for each turbine separately.

[latex]MBE : \dot{m}_{1}=\dot{m}_{2}=\dot{m}_{3}=\dot{m}_{w}[/latex]

[latex] ext { EBE (turbine 1) }: \dot{m}_{w} h_{1}=\dot{m}_{w} h_{2}+\dot{W}_{ ext {out }, 1}+\dot{Q}_{ ext {out }, 1}[/latex]

[latex] ext { EBE (turbine 2) }: \dot{m}_{w} h_{2}=\dot{m}_{w} h_{3}+\dot{W}_{ ext {out }, 2}+\dot{Q}_{ ext {out }, 2}[/latex]

[latex]\operatorname{EBE}( ext { overall }): \dot{m}_{w} h_{1}=\dot{m}_{w} h_{3}+\dot{W}_{ ext {out }, t}+\dot{Q}_{ ext {out }, 1}+\dot{Q}_{ ext {out }, 2}[/latex]

[latex] ext { EnBE (turbine 1): } \dot{m}_{w} S_{1}+\dot{S}_{ ext {gen }, 1}=\dot{m}_{w} S_{2}+\dot{Q}_{ ext {out }, 1} / T_{a v, 1}[/latex]

[latex] ext { EnBE (turbine 2) : } \dot{m}_{w^{S}}+\dot{S}_{ ext {gen, } 2}=\dot{m}_{w^{S} 3}+\dot{Q}_{ ext {out }, 2} / T_{a v, 2}[/latex]

[latex] ext { EnBE (overall) }: \dot{m}_{w} S_{1}+\dot{S}_{g e n}=\dot{m}_{w} S_{3}+\dot{Q}_{o u t, 1} / T_{a v, 1}+\dot{Q}_{o u t, 2} / T_{a v, 2}[/latex]

[latex]\operatorname{ExBE}( ext { turbine } 1): \dot{m}_{w} e x_{1}=\dot{m}_{w} e x_{2}+\dot{W}_{ ext {out }, 1}+\dot{E} x_{\dot{Q}_{ ext {out }, 1}}+\dot{E} x_{d}[/latex]

[latex] ext { ExBE (turbine 2): } \dot{m}_{w} e x_{2}=\dot{m}_{w} e x_{3}+\dot{W}_{ ext {out }, 2}+\dot{E} x_{\dot{Q}_{ ext {out }, 2}}+\dot{E} x_{d}[/latex]

[latex]\operatorname{ExBE}( ext { overall }): \dot{m}_{w} e x_{1}=\dot{m}_{w} e x_{3}+\dot{W}_{ ext {out }, t}+\dot{E} x_{\dot{Q}_{ ext {ou }, 1}}+\dot{E} x_{\dot{Q}_{ ext {out }, 2}}+\dot{E} x_{d}[/latex]

where t stands for total and w for water.

b) Calculate the work rates produced by both turbines.

From a properties based software (such as EES) or properties tables (such as Appendix B-1c), one obtains:

[latex]\left.\begin{array}{c} P_{1}=8000 kPa \ T_{1}=600{ }^{\circ} C \end{array} ight\} \begin{array}{c}h_{1}=3642.4 kJ / kg \ s_{1}=7.0221 kJ / kg K \end{array}[/latex]

[latex]\left.\begin{array}{c} P_{2}=300 kPa \ x_{2}=1 \end{array} ight\} h_{2}=2724.9 kJ / kg[/latex]

[latex]\dot{m}_{w} h_{1}=\dot{m}_{w} h_{2}+\dot{W}_{ ext {out }, 1}+\dot{Q}_{ ext {out }, 1}[/latex]

[latex]\left(13 \frac{ kg }{ s } imes 3642.4 \frac{ kJ }{ kg } ight)=\left(13 \frac{ kg }{ s } imes 2724.9 \frac{ kJ }{ kg } ight)+\left(800 \frac{ kJ }{ s } ight)+\dot{W}_{ out , 1}[/latex]

[latex]\dot{W}_{ ext {out }, 1}=11128 k W[/latex]

[latex]h_{3}=h_{f}+x\left(h_{f g} ight)[/latex]

[latex]h_{3}=191.81+0.85 imes(2392.1)=2225.095 kJ / kg[/latex]

[latex]\dot{m}_{w} h_{2}=\dot{m}_{w} h_{3}+\dot{W}_{ ext {out }, 2}+\dot{Q}_{ ext {out }, 2}[/latex]

[latex]\left(13 \frac{ kg }{ s } imes 2724.9 \frac{ kJ }{ kg } ight)=\left(13 \frac{ kg }{ s } imes 2225.095 \frac{ kJ }{ kg } ight)+\left(200 \frac{ kJ }{ s } ight)+\dot{W}_{ out , 2}[/latex]

[latex]\dot{W}_{ ext {out }, 2}=6297 kW[/latex]

The total turbine work is calculated as follows:

[latex]\dot{W}_{ ext {out }, t}=\dot{W}_{ ext {out }, 1}+\dot{W}_{ ext {out }, 2}=17425 k W[/latex]

c) Find the temperature of the water leaving turbine 2.

By using a properties based software or properties tables (such as Appendix B-1b):

[latex]\left.\begin{array}{c} P_{3}=10 kPa \ x_{3}=0.85 \end{array} ight\} T _{3}= 4 5 . 8 2 { }^{\circ} C[/latex]

d) Calculate the total entropy generation rate and the total exergy destruction rate.

[latex]s_{3}=s_{f}+x\left(s_{f g} ight)[/latex]

[latex]s_{3}=0.6492+0.85 imes 8.1488=7.57568 kJ / kg[/latex]

[latex]\dot{m}_{w} S_{1}+\dot{S}_{g e n}=\dot{m}_{w} S_{3}+\dot{Q}_{o u t, 1} / T_{a v g, 1}+\dot{Q}_{o u t, 2} / T_{a v g, 2}[/latex]

[latex]\dot{S}_{g e n}=13 \frac{ kg }{ s } imes(7.57568-7.0221) \frac{ kJ }{ kgK }+\frac{800 kW }{(873+406.6) / 2}+\frac{200 kW }{(318.8+406.6) / 2}[/latex]

[latex]\dot{S}_{ ext {gen }}=8.998 k W / K[/latex]

The total exergy destruction rate then becomes

[latex]\dot{E} x _{ d }=T_{o} \dot{S}_{ ext {gen }}=298 K imes 8.998 kW / K=2681.4 k W[/latex]

Question 5.17: Consider a two connected steam turbines with the state prope...

Related Answered Questions

Air compressor, as shown in Figure 5.15, takes in air at atmospheric pressure and at a temperature of 280 K, and the air exiting the compressor leaves at a pressure and a temperature of 600 kPa and 400 K. The air mass flow rate of the air being compressed is 1 kg/s; the compressor loses heat ...

Verified Answer:

Verified Answer:

A balloon with a 12 kg of refrigerant R134a in the saturated vapor state at a pressure of 240 kPa, as shown in Figure 5.9, is expanded in a process, during which a total of 300 kJ of heat is transferred to the refrigerant. Take the source temperature of heat as 300 ºC. (a) Write the mass, ...

Verified Answer:

Hot water at a temperature of 95 °C and a pressure of 100 kPa flows in a pipe at a flow rate of 1 kg/s, as shown in Figure 5.10. The pipe loses heat at a rate of 1 kW per meter along the pipe length. (a) Write the mass, energy, entropy, and exergy balance equations, (b) find the temperature ...

Verified Answer:

Consider a diffuser as shown in Figure 5.11a with inlet and exit pressures of 90 and 94 kPa, respectively. The inlet air temperature, the entrance area and the inlet air velocity are 10 °C, 6.4 m2 and 94 m/s. Treat air as an ideal gas and use its properties. (a) Write the mass, energy, entropy ...

Verified Answer:

Verified Answer:

The hot exhaust gases leaving an internal combustion engine at 400 C, 150 kPa, and at a rate of 0.8 kg/s are used to produce saturated steam at 200 C in an insulated heat exchanger, as shown in Figure 5.14. Water enters the heat exchanger at the ambient temperature of 20 C and the exhaust gases ...

Verified Answer:

Verified Answer:

In a vapor-compression refrigeration cycle, consider refrigerant R134a at a pressure of 0.8 MPa and in the saturated liquid state flowing through an expansion valve, as shown in Figure 5.17. The refrigerant is then throttled down to a pressure of 0.12 MPa. (a) Write the mass, energy, entropy, ...

Verified Answer:

Verified Answer: