Thermodynamics

702 | Thermodynamicsentropy generation during an adiabatic mixing process that forms an idealsolution becomesS gen S out S in ¢S mixing R u aiN i ln y i 1ideal solution2(13–51a)orsgen s out s in ¢s mixing R u aiy i ln y i 1per unit mole of mixture2(13–51b)Also noting that X destroyed T 0 S gen , the exergy destroyed during this (and anyother) process is obtained by multiplying the entropy generation by the temperatureof the environment T 0 . It givesX destroyed T 0 S gen R u T 0 aiN i ln y i 1ideal soluton2(13–52a)orxdestroyed T 0 s gen R u T 0 aiy i ln y i 1per unit mole of mixture2(13–52b)ABMixingchamberT 0W rev X destruction T 0 S genA + BmixtureFIGURE 13–21For a naturally occurring processduring which no work is produced orconsumed, the reversible work is equalto the exergy destruction.Exergy destroyed represents the wasted work potential—the work that wouldbe produced if the mixing process occurred reversibly. For a reversible or“thermodynamically perfect” process, the entropy generation and thus theexergy destroyed is zero. Also, for reversible processes, the work output is amaximum (or, the work input is a minimum if the process does not occurnaturally and requires input). The difference between the reversible work andthe actual useful work is due to irreversibilities and is equal to the exergydestruction. Therefore, X destroyed W rev W actual . Then it follows that for anaturally occurring process during which no work is produced, the reversiblework is equal to the exergy destruction (Fig. 13–21). Therefore, for the adiabaticmixing process that forms an ideal solution, the reversible work (totaland per unit mole of mixture) is, from Eq. 13–52,W rev R u T 0 aiN i ln y i andw rev R u T 0 aiy i ln y i(13–53)A reversible process, by definition, is a process that can be reversed withoutleaving a net effect on the surroundings. This requires that the direction of allinteractions be reversed while their magnitudes remain the same when theprocess is reversed. Therefore, the work input during a reversible separationprocess must be equal to the work output during the reverse process of mixing.A violation of this requirement will be a violation of the second law ofthermodynamics. The required work input for a reversible separation processis the minimum work input required to accomplish that separation since thework input for reversible processes is always less than the work input of correspondingirreversible processes. Then the minimum work input required forthe separation process can be expressed asW min,in R u T 0 aiN i ln y i andw min,in R u T 0 aiy i ln y i(13–54)