Question 10.5: The Ideal Regenerative Rankine Cycle Consider a steam power ...

A steam power plant operates on the ideal regenerative Rankine cycle with one open feedwater heater. The fraction of steam extracted from the turbine and the thermal efficiency are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Kinetic and potential
energy changes are negligible. Analysis The schematic of the power plant and the T-s diagram of the cycle are shown in Fig. 10–18. We note that the power plant operates on the ideal regenerative Rankine cycle. Therefore, the pumps and the turbines are isentropic; there are no pressure drops in the boiler, condenser, and feedwater heater; and steam leaves the condenser and the feedwater heater as saturated liquid. First, we determine the enthalpies at various states:

State 1:        \left.\begin{array}{l} P_{1}=10 kPa \\ \text { Sat. liquid } \end{array}\right\} \quad \begin{aligned} &h_{1}=h_{f @ 10 kPa }=191.81 kJ / kg \\ &v_{1}=v_{f @ 10 kPa }=0.00101 m ^{3} / kg \end{aligned}

State 2:    \begin{aligned} P_{2} &=1.2 MPa \\ s_{2} &=s_{1} \end{aligned}

\begin{aligned} w_{\text {pump } I , \text { in }} &=v_{1}\left(P_{2}-P_{1}\right)=\left(0.00101 m ^{3} / kg \right)[(1200-10) kPa ]\left(\frac{1 kJ }{1 kPa \cdot m ^{3}}\right) \\ &=1.20 kJ / kg \\ h_{2} &=h_{1}+w_{\text {pump I,in }}=(191.81+1.20) kJ / kg =193.01 kJ / kg \end{aligned}

State 3:      \left.\begin{array}{l} \begin{array}{l} P_{3}=1.2 MPa \\ \text { Sat. liquid } \end{array} \end{array}\right\} \begin{aligned} &v_{3}=v_{f @ 1.2 MPa }=0.001138 m ^{3} / kg \\ &h_{3}=h_{f @ 1.2 MPa }=798.33 kJ / kg \end{aligned}

State 4:      \begin{aligned} P_{4} &=15 MPa \\ s_{4} &=s_{3} \end{aligned}

\begin{aligned} w_{\text {pump II,in }} &=v_{3}\left(P_{4}-P_{3}\right) \\ &=\left(0.001138 m ^{3} / kg \right)[(15,000-1200) kPa ]\left(\frac{1 kJ }{1 kPa \cdot m ^{3}}\right) \\ &=15.70 kJ / kg \\ h_{4} &=h_{3}+w_{\text {pump II,in }}=(798.33+15.70) kJ / kg =814.03 kJ / kg \end{aligned}

State 5:      \begin{array}{ll} P_{5}=15 MPa & h_{5}=3583.1 kJ / kg \\ T_{5}=600^{\circ} C & \begin{array}{l} s_{5}=6.6796 kJ / kg \cdot K \end{array} \end{array}

State 6:      \left.\begin{array}{rl} P_{6} & =1.2 MPa \\ s_{6} & =s_{5} \end{array}\right\} \begin{aligned} h_{6} &=2860.2 kJ / kg \\ \left(T_{6}\right.&\left.=218.4^{\circ} C \right) \end{aligned}

State 7:    P_{7}=10 kPa

\begin{aligned} &s_{7}=s_{5} \quad x_{7}=\frac{s_{7}-s_{f}}{s_{f g}}=\frac{6.6796-0.6492}{7.4996}=0.8041 \\ &h_{7}=h_{f}+x_{7} h_{f g}=191.81+0.8041(2392.1)=2115.3 kJ / kg \end{aligned}

The energy analysis of open feedwater heaters is identical to the energy analysis of mixing chambers. The feedwater heaters are generally well insulated (\dot{Q}=0) , and they do not involve any work interactions (\dot{W}=0) . By neglecting the kinetic and potential energies of the streams, the energy balance reduces for a feedwater heater to

\dot{E}_{\text {in }}=\dot{E}_{\text {out }} \rightarrow \sum_{\text {in }} \dot{m} h=\sum_{\text {out }} \dot{m} h

or

  y h_{6}+(1-y) h_{2}=1\left(h_{3}\right)

where y is the fraction of steam extracted from the turbine \left(=\dot{m}_{6} / \dot{m}_{5}\right) . Solving for y and substituting the enthalpy values, we find

y=\frac{h_{3}-h_{2}}{h_{6}-h_{2}}=\frac{798.33-193.01}{2860.2-193.01}= 0 . 2 2 7 0

Thus,

\begin{aligned} q_{\text {in }} &=h_{5}-h_{4}=(3583.1-814.03) kJ / kg =2769.1 kJ / kg \\ q_{\text {out }} &=(1-y)\left(h_{7}-h_{1}\right)=(1-0.2270)(2115.3-191.81) kJ / kg \\ &=1486.9 kJ / kg \end{aligned}

and

\eta_{\text {th }}=1-\frac{q_{\text {out }}}{q_{\text {in }}}=1-\frac{1486.9 kJ / kg }{2769.1 kJ / kg }=0.463 \text { or } 46.3 \%

Discussion This problem was worked out in Example 10–3c for the same pressure and temperature limits but without the regeneration process. A comparison of the two results reveals that the thermal efficiency of the cycle has increased from 43.0 to 46.3 percent as a result of regeneration. The net work output decreased by 171 kJ/kg, but the heat input decreased by 607 kJ/kg, which results in a net increase in the thermal efficiency.