Thermodynamics

Chapter 13 | 703It can also be expressed in the rate form asW # min,in R u T 0 aiN # i ln y i N # mR u T 0 aiy i ln y i 1kW2(13–55)where Ẇ min,in is the minimum power input required to separate a solution thatapproaches at a rate of N # (or m # m N # m kmol>smM m kg>s) into its components.The work of separation per unit mass of mixture can be determinedfrom w min,in w min,in>M m , where M m is the apparent molar mass of themixture.The minimum work relations above are for complete separation of thecomponents in the mixture. The required work input will be less if the exitingstreams are not pure. The reversible work for incomplete separation canbe determined by calculating the minimum separation work for the incomingmixture and the minimum separation works for the outgoing mixtures, andthen taking their difference.Reversible Mixing ProcessesThe mixing processes that occur naturally are irreversible, and all the workpotential is wasted during such processes. For example, when the fresh waterfrom a river mixes with the saline water in an ocean, an opportunity to producework is lost. If this mixing is done reversibly (through the use of semipermeablemembranes, for example) some work can be produced. Themaximum amount of work that can be produced during a mixing process isequal to the minimum amount of work input needed for the correspondingseparation process (Fig. 13–22). That is,Ay AB(a) MixingW max,out = 5 kJ/kg mixtureMixingchamberA + Bmixturey BAW max,out,mixing W min,in,separation(13–56)Therefore, the minimum work input relations given above for separation canalso be used to determine the maximum work output for mixing.The minimum work input relations are independent of any hardware orprocess. Therefore, the relations developed above are applicable to any separationprocess regardless of actual hardware, system, or process, and can beused for a wide range of separation processes including the desalination ofsea or brackish water.Second-Law EfficiencyThe second-law efficiency is a measure of how closely a process approximatesa corresponding reversible process, and it indicates the range availablefor potential improvements. Noting that the second-law efficiency rangesfrom 0 for a totally irreversible process to 100 percent for a totally reversibleprocess, the second-law efficiency for separation and mixing processes canbe defined asy ABy B(b) SeparationW max,in = 5 kJ/kg mixtureSeparationunitA + BmixtureFIGURE 13–22Under reversible conditions, the workconsumed during separation is equalto the work produced during thereverse process of mixing.h II,separation W# min,inW # act,in w min,inandhw II,mixing W# act,outact,in W # w act,outwmax,out max,out(13–57)where Ẇ act,in is the actual power input (or exergy consumption) of the separationplant and Ẇ act,out is the actual power produced during mixing. Note that