Thermodynamics

354 | ThermodynamicsThen the power output of the turbine is determined from the rate form of theenergy balance to beE # in E # (steady)out dE system /dt ¡0 0⎫⎪⎪⎬⎪⎪⎭1444444442444444443Rate of net energy transferby heat, work, and massRate of change in internal,kinetic, potential, etc., energiesE # in E # outm # h 1 W # out m # h 2 1since Q # 0, ke pe 02W # out m # 1h 1 h 2 2 1118.2 kg>s2 1232.3 222.82 kJ>kg 1123 kWFor continuous operation (365 24 8760 h), the amount of power producedper year isAnnual power production W # out ¢t 11123 kW2 18760 h>yr2 0.9837 10 7 kWh>yrAt $0.075/kWh, the amount of money this turbine can save the facility isAnnual power savings 1Annual power production2 1Unit cost of power2 10.9837 10 7 kWh>yr2 1$0.075>kWh2 $737,800/yrThat is, this turbine can save the facility $737,800 a year by simply takingadvantage of the potential that is currently being wasted by a throttlingvalve, and the engineer who made this observation should be rewarded.Discussion This example shows the importance of the property entropy sinceit enabled us to quantify the work potential that is being wasted. In practice,the turbine will not be isentropic, and thus the power produced will be less.The analysis above gave us the upper limit. An actual turbine-generatorassembly can utilize about 80 percent of the potential and produce morethan 900 kW of power while saving the facility more than $600,000 a year.It can also be shown that the temperature of methane drops to 113.9 K (adrop of 1.1 K) during the isentropic expansion process in the turbine insteadof remaining constant at 115 K as would be the case if methane wereassumed to be an incompressible substance. The temperature of methanewould rise to 116.6 K (a rise of 1.6 K) during the throttling process.INTERACTIVETUTORIALSEE TUTORIAL CH. 7, SEC. 9 ON THE DVD.7–9 ■ THE ENTROPY CHANGE OF IDEAL GASESAn expression for the entropy change of an ideal gas can be obtained fromEq. 7–25 or 7–26 by employing the property relations for ideal gases (Fig.7–31). By substituting du c v dT and P RT/v into Eq. 7–25, the differentialentropy change of an ideal gas becomesdTds c v T dvRv(7–30)