Consider the steam power cycle with reheat shown in Figure 3.13. The pres-sures and temperatures at several states are identified in the accompanying table. In addition, the isentropic efficiencies of the turbines and the pump are known. A full energy analysis of this cycle results in values for the energy transfer rates indicated on the diagram. Knowing the energy transfer rates, the thermal efficiency of the cycle can also be found.
To determine the energy transfer rates and thermal efficiency, the follow-ing solution strategy is suggested:
1. Construct a complete table of thermodynamic properties at each state in the cycle.
2. Apply balance and conservation laws to determine the required information.
When constructing the table of properties (step 1), it is important to always be mindful of the State Postulate for a pure fluid, as discussed in Section 3.3.
Condenser
5 1 2
3
HPT LPT
6
1 11.9 MPa 0.80 MPa 0.79 MPa 7 kPa 5 kPa 12.0 MPa
State P T
2 3 4 5 Steam mass flow rate, m·=380kg/s 4 6
Pump Boiler Q˙ b
Q˙ cond W˙ t
ηp = 0.65
ηHPT = ηLPT = 0.85
450˚C
380˚C
30˚C
FIGURe 3.13
Steam power cycle with reheat with known properties in the accompanying table.
Recall that the State Postulate for a pure fluid specifies that two independent, intensive properties are required to fix the thermodynamic state. Therefore, to construct the table of properties, two independent, intensive properties need to be identified at each state. Referring to the cycle shown in Figure 3.13, three of the states (1, 3, and 5) are defined by pressure and temperature. The P,T combination at these three states are independent. This can be verified by a quick check of the saturation temperatures for the given pressures.
Therefore, all of the properties at states 1, 3, and 5 can be found.
At states 2 and 4, only the pressure is known. However, these states are the exhaust of the turbines and a key performance indicator of the turbines is known; the isentropic efficiency. Writing the isentropic efficiency expression for the high-pressure turbine (HPT) gives
η = −
HPT 1− 2
1 2s
h h
h h (3.115)
The unknown in Equation 3.115 is the enthalpy leaving the HPT, h2. The enthalpy leaving the isentropic turbine, h2s can be found because the pres-sure and entropy are known at that state.
( )
The entropy at state 1 is known because the pressure and temperature are known. The three equations summarized in Equations 3.115 and 3.116 allow for the calculation of the enthalpy at state 2. State 2 is now identified with pressure and enthalpy. Therefore, the rest of the properties can be found.
Similar calculations allow for the determination of the properties at state 4, the exhaust of the low-pressure turbine.
State 6 is the exhaust of the pump. The only property known at this state is the pressure. However, the isentropic efficiency of the pump is known.
η = −
The unknown in Equation 3.117 is the enthalpy leaving the pump, h6. The isentropic enthalpy leaving the pump, h6s, is determined from the pump exhaust pressure and inlet entropy.
( )
The entropy at state 5 is known because pressure and temperature are known at that state. Equations 3.117 and 3.118 can then be solved for the enthalpy, h5. Then, state 5 is fixed with pressure and enthalpy.
Table 3.4 shows the property table determined using the procedure out-lined above for the cycle shown in Figure 3.13. The properties in Table 3.4
were determined using EES. The values shown in the bold font represent the independent, intensive properties that fix each state.
Notice that the quality, x, column in Table 3.4 contains values of 100 and –100.
When using EES to calculate properties, a quality value of 100 means that the state is in the vapor or supercritical phase. If the value of x is –100, this implies that the state is in the liquid phase.
Once a table of properties is constructed, it is often helpful to superim-pose the states on a thermodynamic diagram. This helps visualize how the cycle is operating. EES is capable of drawing several different property plots. Furthermore, if the properties are stored in arrays, then the EES arrays table can be overlaid on the property plot. Figure 3.14 shows a temperature–
entropy diagram for the cycle shown in Figure 3.13.
The second step of the solution strategy is to perform the required ther-modynamic analyses. This often includes the application of balance or
5
Temperature–entropy diagram for the cycle shown in Figure 3.13.
Table 3.4
Calculated Properties for the Cycle Shown in Figure 3.13
State P (MPa) T (°C) h (kJ/kg) s (kJ/kg·K) x
1 11.9 450.0 3211.50 6.30846 100
2 0.80 170.4 2701.64 6.51132 0.9674
3 0.79 380.0 3225.57 7.51573 100
4 0.007 39.0 2468.49 7.94374 0.9571
5 0.005 30.0 125.74 0.43676 –100
6 12.0 30.8 139.87 0.44374 –100
conservation laws to the cycle. Figure 3.13 shows four different system boundaries surrounding various components in the cycle.
The power delivered by the turbines can be determined by considering a system boundary surrounding both the high-pressure and low-pressure turbines. If the individual power delivered by each turbine is needed, then two system boundaries need to be analyzed: one around each turbine. For the combined turbines, the conservation of energy equation reduces to
( ) ( )
= + − + = 481.4 MW
t 1 3 2 4
W m h h h h (3.119)
The power required by the pump can be found by applying the conservation of energy equation to a system boundary surrounding the pump.
( )
= − = 5.37 MW
p 6 5
W m h h (3.120)
The net power delivered from the cycle can now be determined by
= − = 476.1 MW
net t p
W W W (3.121)
The heat transfer rate into the boiler can be found by applying the conserva-tion of energy equaconserva-tion to the system boundary surrounding the boiler.
( ) ( )
= + − + = 1167 MW
b 1 3 6 2
Q m h h h h (3.122)
The heat transfer rate from the condenser can be found in one of two ways.
The conservation of energy equation can be applied to a system boundary that surrounds the condenser (Figure 3.13). An alternative way is to apply the conservation of energy equation to a system boundary that encompasses the complete cycle.
The thermal efficiency of the cycle can now be calculated.
η =th net =0.348 34.8%=
Many other parameters can be calculated for this cycle including entropy generation rates for the turbines and pump, exergy destruction rates for the turbines and pump, and exergetic efficiencies of the components in the cycle.
Building the table of properties first has a type of domino effect on the prob-lem. Once the table is fully constructed, the application of the conservation and balance laws is fairly straightforward. As mentioned earlier, some ther-modynamic analysis may be required to complete the property table.