Question 4.28: The turbine–generator system for Example Problem 4.27 is ill...

(a) The shaft power delivered to the generator by the turbine, (P_{t})_{out} = ωT_{shaft,out} = (P_{g})_{in} = ωT_{shaft,in} defined for Equation 4.174\eta _{generator}= \frac{electricalpoweroutputfromgenerator}{shaftpowerinputfromturbine} = \frac{\overset{\cdot }{W}_{elect,out} }{(\overset{\cdot }{W}_{shaft,in} )_{e}} = \frac{P_{out of gen}}{P_{in to gen}} = \frac{(P_{g})_{out}}{(P_{g})_{in}} is applied as follows:

w: = 350 \frac{rad}{sec}                                                T_{ shaftout }: = 2200 N.m

P_{tout} : = w. T_{ shaftout } = 770 kW                  P_{gin} : = P_{tout} = 770 kW

which is illustrated in Figure EP 4.28.
(b) The turbine efficiency defined in Equation 4.167 \eta _{turbine}= \frac{shaft power}{hydraulic power} = \frac{\overset{\cdot }{W}_{turbine} }{(\overset{\cdot }{W}_{turbine} )_{e}} = \frac{\overset{\cdot }{W}_{shaft,out} }{(\overset{\cdot }{W}_{turbine} )_{e}} = \frac{wT_shaft,out}{(\overset{\cdot }{W}_{turbine} )_{e}} = \frac{P_{out of turbine}}{P_{in of turbine}} = \frac{(P_{t})_{out}}{(P_{t})_{in}} is applied as follows:

P_{tin} : = 800 kW                            \eta _{ turbine} : = \frac{P_{tout} }{P_{tin}} = 0.963

Thus, the turbine is 96.3% efficient.
(c) The generator efficiency defined in Equation 4.174 \eta _{generator}= \frac{electricalpoweroutputfromgenerator}{shaftpowerinputfromturbine} = \frac{\overset{\cdot }{W}_{elect,out} }{(\overset{\cdot }{W}_{shaft,in} )_{e}} = \frac{P_{out of gen}}{P_{in to gen}} = \frac{(P_{g})_{out}}{(P_{g})_{in}} is applied as follows:

P_{gout}: = 700 kW                              \eta _{ generator} : = \frac{P_{gout} }{P_{gin}} = 0.909

Thus, the generator is 90.9% efficient.
(d) The turbine–generator system efficiency defined in Equation 4.176 \eta _{turbine – generator}= \frac{electricalpower}{hydraulic power} = \frac{\overset{\cdot }{W}_{elect,out} }{(\overset{\cdot }{W}_{turbine} )_{e}} = \frac{(P_{g})_{out}}{(P_{t})_{in}} = \eta _{turbine} \eta _{generator} is applied as follows:

\eta _{turbine generator} : = \frac{P_{gout} }{P_{tin}} = 0.875
\eta _{turbine generator} :\eta _{ turbine} . \eta _{ generator} = 0.875

Thus, the turbine–generator system is 87.5% efficient.