The effect of a number of open feedwater heaters on the thermal efficiency of an ideal cycle is to be studied. Steam leaves the steam generator at 20 MPa, 600°C, and the cycle has a condenser pressure of 10 kPa. Determine the thermal efficiency for each of the following cases. A: No feedwater

Question

The effect of a number of open feedwater heaters on the thermal efficiency of an ideal cycle is to be studied. Steam leaves the steam generator at 20 MPa, 600°C, and the cycle has a condenser pressure of 10 kPa. Determine the thermal efficiency for each of the following cases. A: No feedwater heater. B: One feedwater heater operating at 1 MPa. C: Two feedwater heaters, one operating at 3 MPa and the other at 0.2 MPa.

Accepted Answer

a) no feed water heater

[latex]\begin{aligned}w _{ P } & =\int_1^2 vdP \& \approx 0.00101(20000-10) \& =20.2\, kJ / kg\end{aligned}[/latex]

[latex]\begin{aligned}h _2 & = h _1+ w _{ p }=191.8+20.2=212.0 \s _4 & = s _3=6.5048 \& =0.6493+ x _4 imes 7.5009 \x _4 & =0.78064\end{aligned}[/latex]

[latex]\begin{aligned}& h _4=191.83+0.78064 imes 2392.8 \& \quad=2059.7 \& w _{ T }= h _3- h _4=3537.6-2059.7 \& \quad=1477.9 \,kJ / kg \& w _{ N }= w _{ T }- w _{ P }=1477.9-20.2=1457.7 \& q _{ H }= h _3- h _2=3537.6-212.0=3325.6\end{aligned}[/latex]

[latex]\eta_{ TH }=\frac{ w _{ N }}{ q _{ H }}=\frac{1457.7}{3325.6}= 0 . 4 3 8[/latex]

b) one feedwater heater

[latex]\begin{aligned}& w _{ P 12}=0.00101(1000-10) \& \quad=1.0 \,kJ / kg \& h _2= h _1+ w _{ P 12}=191.8+1.0=192.8 \& w _{ P 34}=0.001127(20000-1000) \& \quad=21.4\, kJ / kg \& h _4= h _3+ w _{ P 34}=762.8+21.4=784.2 \& s _6= s _5=6.5048 \& \quad=2.1387+ x _6 imes 4.4478\end{aligned}[/latex]

[latex]\begin{aligned}& x _6=0.9816 \& h _6=762.8+0.9816 imes 2015.3=2741.1\end{aligned}[/latex]

CV: heater
const: [latex]m _3= m _6+ m _2=1.0 \,kg[/latex]

1st law: [latex]m _6 h _6+ m _2 h _2= m _3 h _3[/latex]

[latex]m _6=\frac{762.8-192.8}{2741.1-192.8}=0.2237[/latex]

[latex]\left. m _2=0.7763, h _7=2059.7\left(= h _4 ext { of part } a ight) ight)[/latex]

CV: turbine

[latex]\begin{aligned}& w _{ T }=\left( h _5- h _6 ight)+ m _2\left( h _6- h _7 ight) \&=(3537.6-2741.1)+0.7763(2741.1-2059.7)=1325.5 \,kJ / kg\end{aligned}[/latex]

CV: pumps

[latex]\begin{aligned}& w _{ P }= m _1 w _{ P 12}+ m _3 w _{ P 34}=0.7763(1.0)+1(21.4)=22.2 \,kJ / kg \& w _{ N }=1325.5-22.2=1303.3 \,kJ / kg\end{aligned}[/latex]

CV: steam generator

[latex]\begin{aligned}& q _{ H }= h _5- h _4=3537.6-784.2=2753.4 \,kJ / kg \& \eta_{ TH }= w _{ N } / q _{ H }=1303.3 / 2753.4= 0 . 4 7 3\end{aligned}[/latex]

c) two feedwater heaters

[latex]\begin{aligned}& W _{ P 12}=0.00101 imes \& (200-10) \& =0.2 \,kJ / kg \& h _2= w _{ p 12}+ h _1 \& =191.8+0.2 \& =192.0 \& w _{ P 34}=0.001061 imes \& ext { (3000 - 200) } \& =3.0 \,kJ / kg \& h _4= h _3+ w _{ P 34} \& =504.7+3.0 \& =507.7 \end{aligned}[/latex]

[latex]\begin{aligned}w _{ P 56} & =0.001217(20000-3000) \& =20.7 \,kJ / kg \h _6= & h _5+ w _{ P 56}=1008.4+20.7=1029.1\end{aligned}[/latex]

[latex]\left.\begin{array}{l}s _8= s _7=6.5048 \ ext { at } P _8=3 \,MPa\end{array} ight\} \begin{aligned}& T _8=293.2{ }^{\circ} C \& h _8=2974.8\end{aligned}[/latex]

[latex] \quad \quad s_9 = s_8 = 6.5048 = 1.5301 + x_9 imes 5.5970[/latex]

[latex]x _9=0.8888 \Rightarrow h _9=504.7+0.888 imes 2201.9=2461.8 \,kJ / kg[/latex]

CV: high pressure heater

[latex]\begin{array}{cl} ext { cont: } \quad m _5= m _4+ m _8=1.0 \,kg ; \quad 1 st ext { law: } m _5 h _5= m _4 h _4+ m _8 h _8 \m _8=\frac{1008.4-507.7}{2974.8-507.7}=0.2030 \quad m _4=0.7970\end{array}[/latex]

CV: low pressure heater

[latex]\begin{aligned}& ext { cont: } \quad m _9+ m _2= m _3= m _4 ; \quad 1 ext { st law: } m _9 h _9+ m _2 h _2= m _3 h _3 \& m _9= \frac{0.7970(504.7-192.0)}{2461.8-192.0}=0.1098 \& m _2=0.7970-0.1098=0.6872\end{aligned}[/latex]

CV: turbine

[latex]\begin{aligned}w _{ T }= & \left( h _7- h _8 ight)+\left(1- m _8 ight)\left( h _8- h _9 ight)+\left(1- m _8- m _9 ight)\left( h _9- h _{10} ight) \= & (3537.6-2974.8)+0.797(2974.8-2461.8) \& +0.6872(2461.8-2059.7)=1248.0\,kJ / kg\end{aligned}[/latex]

CV: pumps

[latex]\begin{aligned}& w _{ P }= m _1 w _{ P 12}+ m _3 w _{ P 34}+ m _5 w _{ P 56} \& =0.6872(0.2)+0.797(3.0)+1(20.7)=23.2 \,kJ / kg \& w _{ N }=1248.0-23.2=1224.8 \,kJ / kg\end{aligned}[/latex]

CV: steam generator

[latex]\begin{aligned}& q _{ H }= h _7- h _6=3537.6-1029.1=2508.5 \,kJ / kg \& \eta_{ TH }= w _{ N } / q _{ H }=1224.8 / 2508.5= 0 . 4 8 8\end{aligned}[/latex]

Question 11.CSGP.132: The effect of a number of open feedwater heaters on the ther......

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