Question 4.26: The turbine–generator system for Example Problem 4.25 is ill...

(a) The shaft power delivered to the generator by the turbine, (P_{t})_{out}= wT_{shaft,out} = (P_{g})_{in}= wT_{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: = 320 \frac{rad}{sec}                     T_{shaftout}: = 2100 N.m                    P_{tout}: = w . T_{shaftout} = 672 kW                    P_{gin}: = P_{tout} = 672 kW

(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}: = 700 kW                            \eta _{turbine}: = \frac{P_{tout}}{P_{tin}} = 0.96

Thus, the turbine is 96% 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}: = 600 kW                          \eta _{ generator}: = \frac{P_{gout}}{P_{gin}}

Thus, the generator is 89.3% 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 _{turbinegenerator }: = \frac{P_{gout}}{P_{tin}} = 0.857
\eta _{turbinegenerator }: = \eta _{turbine}. \eta _{ generator} = 0.857

Thus, the turbine–generator system is 85.7% efficient.