A straightforward Rankine cycle utilises water as the working fluid. The boiler runs at a pressure of 7000 kPa and the condenser at a pressure of 50 kPa. The temperature at the turbine's entry is 450 degrees celcius. The turbine is isentropically efficient at 90%, pressure and pump losses are low, and water exiting the condenser is subcooled to 6.32 degrees celcius. The boiler has been designed to operate at a mass flow rate of 20 kg/s. Determine:
a) The rate of heat addition in the boiler b) The amount of energy needed to run the pumps
c) The cycle's net power output and thermal efficiency
$Given:\phantom{\rule{0ex}{0ex}}{P}_{1}=7000KPa\phantom{\rule{0ex}{0ex}}{P}_{2}=50KPa\phantom{\rule{0ex}{0ex}}{T}_{1}=450C\phantom{\rule{0ex}{0ex}}\eta =0.9\phantom{\rule{0ex}{0ex}}{T}_{3}=6.32\xb0C\phantom{\rule{0ex}{0ex}}m=20Kg/s$
$Propertiesareobtainedfromsteamtable:\phantom{\rule{0ex}{0ex}}AT{P}_{1}=7000KPaand{T}_{1}=450\xb0C\phantom{\rule{0ex}{0ex}}{h}_{1}=3287.78KJ/Kg\phantom{\rule{0ex}{0ex}}{s}_{1}=6.6354KJ/Kg.K\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}At{P}_{2}=50KPa:\phantom{\rule{0ex}{0ex}}{T}_{s}=81.317\xb0C{v}_{f}=0.00102993{m}^{3}/Kg{v}_{g}=3.2400{m}^{3}/Kg\phantom{\rule{0ex}{0ex}}{h}_{f}=340.54KJ/Kg{h}_{g}=2645.2KJ/Kg{h}_{fg}=2304.7KJ/Kg\phantom{\rule{0ex}{0ex}}{s}_{f}=1.0912KJ/Kg.K{s}_{g}=7.5930KJ/Kg.K{s}_{fg}=6.5018KJ/Kg\phantom{\rule{0ex}{0ex}}{s}_{2}={s}_{1}=6.6354KJ/Kg(process1-2isentropicexpansion)\phantom{\rule{0ex}{0ex}}{s}_{2}={s}_{f}+x*{s}_{fg}\phantom{\rule{0ex}{0ex}}6.6354=1.0912=x*6.5018\phantom{\rule{0ex}{0ex}}x=0.8527\phantom{\rule{0ex}{0ex}}{h}_{2}={h}_{f}+x*{h}_{fg}\phantom{\rule{0ex}{0ex}}{h}_{2}=340.54+0.8527*2304.7\phantom{\rule{0ex}{0ex}}{h}_{2}=2305.8KJ/Kg\phantom{\rule{0ex}{0ex}}Isentropicefficiencyofturbine:\phantom{\rule{0ex}{0ex}}{\eta}_{T}=\frac{{h}_{1}-{h}_{2\text{'}}}{{h}_{1}-{h}_{2}}\phantom{\rule{0ex}{0ex}}0.9=\frac{3287.78-{h}_{2\text{'}}}{3287.78-2305.8}\phantom{\rule{0ex}{0ex}}{h}_{2\text{'}}=2403.99KJ/Kg\phantom{\rule{0ex}{0ex}}{h}_{3}={h}_{f}-C*({T}_{s}-6.32)\phantom{\rule{0ex}{0ex}}{h}_{3}=340.54-4.186*(81.317-6.32)\phantom{\rule{0ex}{0ex}}{h}_{3}=26.60KJ/Kg\phantom{\rule{0ex}{0ex}}{s}_{3}={s}_{f}-4.186*\mathrm{ln}\frac{(81.317+273)}{(6.32+273)}\phantom{\rule{0ex}{0ex}}{s}_{3}=1.0912-4.186*\mathrm{ln}\frac{354.317}{279.32}\phantom{\rule{0ex}{0ex}}{s}_{3}=0.09562KJ/Kg.K\phantom{\rule{0ex}{0ex}}Forprocess3-4:isentropiccompression\phantom{\rule{0ex}{0ex}}{s}_{4}={s}_{3}=0.09562\phantom{\rule{0ex}{0ex}}At{P}_{4}=7000KPa\phantom{\rule{0ex}{0ex}}{T}_{s4}=285.829\xb0C\phantom{\rule{0ex}{0ex}}{h}_{f}=1267.7KJ/Kg{h}_{g}=2772.6KJ/Kg{h}_{g}=1505.0KJ/Kg\phantom{\rule{0ex}{0ex}}{s}_{f}=3.1224KJ/Kg.K{s}_{g}=5.8148KJ/Kg.K{s}_{fg}=2.6924KJ/Kg.K\phantom{\rule{0ex}{0ex}}{s}_{4}={s}_{f}-C*\mathrm{ln}\frac{{T}_{s4}}{{T}_{4}}\phantom{\rule{0ex}{0ex}}0.09562=3.1224-4.186*\mathrm{ln}\frac{(285.829+273)}{{T}_{4}}\phantom{\rule{0ex}{0ex}}{T}_{4}=271.177K=-1.82\xb0C\phantom{\rule{0ex}{0ex}}{h}_{4}={h}_{f}-C*({T}_{s4}-{T}_{4})\phantom{\rule{0ex}{0ex}}{h}_{4}=1267.7-4.186*(285.829+1.82)\phantom{\rule{0ex}{0ex}}{h}_{4}=63.59KJ/Kg\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}a)Rateofheatadditiontoboiler:\phantom{\rule{0ex}{0ex}}{Q}_{a}=m*({h}_{1}-{h}_{4})\phantom{\rule{0ex}{0ex}}{Q}_{a}=20*(3287.78-63.59)\phantom{\rule{0ex}{0ex}}{Q}_{a}=64483.8KW\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}b)Energyneededtorunpumps:\phantom{\rule{0ex}{0ex}}{W}_{p}=m*({h}_{4}-{h}_{3})\phantom{\rule{0ex}{0ex}}{W}_{p}=20*(63.59-26.60)\phantom{\rule{0ex}{0ex}}{W}_{p}=739.8KW\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}c)Netpoweroutput:\phantom{\rule{0ex}{0ex}}{W}_{net=}m*[({h}_{1}-{h}_{2}\text{'})-{W}_{p}]\phantom{\rule{0ex}{0ex}}{W}_{net}=20*[(3287.78-2403.99)-739.8]\phantom{\rule{0ex}{0ex}}{W}_{net}=2879.8KW\phantom{\rule{0ex}{0ex}}Thermalefficiency:\phantom{\rule{0ex}{0ex}}\eta =\frac{{W}_{net}}{{Q}_{a}}=\frac{2879.8}{64483.8}\phantom{\rule{0ex}{0ex}}\eta =0.044=4.4\%\phantom{\rule{0ex}{0ex}}$