Lecture 12. Transport in Membranes (2)

Transcription

1 Lecture 12. Transport n embranes (2) odule Flow Patterns - Perfect mxng - Countercurrent flow - Cocurrent flow - Crossflow embrane Cascades External ass-transfer Resstances Concentraton Polarzaton and Foulng

2 odule Flow Patterns Perfect mxng Countercurrent flow Cocurrent flow Crossflow The flow pattern can sgnfcantly affect the degree of separaton and the membrane area For countercurrent and cocurrent flow, permeate flud at a gven locaton on the downstream sde conssts of flud that has just passed through the membrane plus the permeate flud flowng to that locaton For crossflow, there s no flow of permeate flud along the membrane

3 Crossflow Pattern (1) Expresson for the local permeate composton y * é x - ry ù = a, B ê ú 1 - y ë ( 1- x) - r( 1 - y) û The pressure rato, r=p P /P F, and the deal separaton factor, a *,B, are assumed constant t the dfferental element, local mole fractons n the retentate and permeate are x and y æ n ö P q: cut, fracton of feed permeated ç è nf ø Penetrant molar flux: dn/d ateral balance for around the dfferental-volume element ydn = d( nx) = xdn + ndx dn dx = n y - x é 1 + ( a - 1) x ù = ê ú dx ë x( a - 1)( 1- x) û a = * a, B

4 Crossflow Pattern (2) Integraton from the ntermedate locaton of the dfferental element to the fnal retentate (from n to n R ; from x to x R ) é êæ x ö - n = nr ê ç x ç R 1 - x êè ø è ø ë æ 1 ö æ 1 ö ç è a -1 ø ç æ 1 x è a -1 ø R ö ole fracton of n the fnal permeate ù ú ú ú û æ a ö é 1 a ç ù æ ö æ ö x a 1 a R 1 a 1 q x è - ø æ ö ç ç a 1 F ç 1 a yp = y q ò x dn n è - ø æ - ö ê æ ö è - ø è - ø ú F = xr ( 1 x ) F R x R ç q ê x ú è ø ç - ê è F ø ú ë û Dfferental rate of mass transfer of across the membrane P d ydn = ( xpf - ypp ) l Total membrane surface area = ò x x F R l ydn P ( x P - y P ) F P

5 Cascades (1) Cascades: aggregates of stages - ccomplsh separatons that cannot be acheved n a sngle stage - Reduce the amounts of separatng agent requred - ake effcent use of raw materals Two or more streams are ntmately contacted Promote rapd mass and heat transfer The separated phases leavng the stage approach equlbrum

6 Cascades (2) Sngle secton of stages - Streams enterng and leavng are only from the ends - Used to recover components from a feed stream Lnear countercurrent Two sectons of stages - Consst of one secton above the feed and one below - Used to make a sharp separaton between two selected feed components, key components Two-secton, countercurrent Lnear crosscurrent

7 embrane Cascades (1) embrane-separaton systems often consst of multplemembrane modules because a sngle module may not be large enough to handle the requred feed rate - number of modules of dentcal sze n parallel wth retentates and permeates from each module combned - The parallel unts functon as a sngle stage - In multple stages, the combned retentate from each stage becomes the feed for the next stage One stage ultple stage

8 embrane Cascades (2) Sngle-stage membrane-separaton process : a sngle membrane module or a number of such modules arranged n parallel or n seres wthout recycle - The extent to whch a feed mxture can be separated s lmted and determned by the separaton factor, a - The separaton factor depends on module flow patterns, the permeablty rato (deal separaton factor), the cut (q), and the drvng force for membrane mass transfer To mprove purty and recovery, membrane stages are cascaded wth recycle

9 embrane Cascades (3) ultple-stage countercurrent recycle cascades - Permeate s enrched n components of hgh permeablty n an enrchng secton - The retentate s enrched n components of low permeablty n a strppng secton - For a cascade, addtonal factors that affect the degree of separaton are the number of stages and the recycle rato (permeate recycle rate/permeate product rate)

10 embrane Cascades (4) It s best to manpulate the cut and reflux at each stage so as to force compostons of the two streams enterng each stage to be dentcal (deal) : Ths corresponds to the least amount of entropy producton for the cascade and, thus, the hghest second-law effcency In the case of gas permeaton, compresson costs are hgh often lmted to just two or three stages Two-stage strppng cascade Two-stage enrchng cascade - Desgned to obtan a purer retentate - Desgned to obtan a purer permeate

11 embrane Cascades (5) Two-stage enrchng cascade wth addtonal premembrane stage - ddton of a premembrane stage may be attractve when (1) feed concentraton s low n the component to be passed preferentally through the membrane (2) desred permeate purty s hgh (3) separaton factor s low (4) a hgh recovery of the more permeable component s desred

12 External ass-transfer Resstances (1) When mass-transfer resstances external to the membrane are not neglgble, - Gradents exst n the boundary layers (or flms) adjacent to the membrane surfaces - Reduces the drvng force for mass transfer across the membrane and, therefore, the flux of penetrant Gas permeaton by soluton-dffuson s slow compared to dffuson n gas boundary layers or flms external mass-transfer resstances are neglgble embrane processes nvolvng lqud (dalyss, reverse osmoss, pervaporaton): dffuson n lqud boundary layers and flms s slow concentraton polarzaton (accumulaton of non-permeable speces on the upstream surface of the membrane) cannot be neglected

13 External ass-transfer Resstances (2) ass transfer of lquds wth a porous membrane (at steady state) De N = k ( c ) ( ) ( ) F - c F = c 0 - c 0 = k L c P - c L P l N = c F - c 1 l k D k P e F P ass transfer of lquds wth a nonporous membrane (at steady state) K D N = k ( c ) ( ) ( ) F - c F = c 0 - c 0 = k L c P - c L P l N = F 1 l k K D k F c - c P P k F & k P : mass-transfer coeffcents

14 External ass-transfer Resstances (3) 1 l æ l ö 1 k De è K D ø k Resstances to mass flux:, ç or, F P ass transfer coeffcents depend on flud propertes, flowchannel geometry, and flow regme l D embrane resstances and can be replaced by or e l K D The emprcal flm-model correlaton of mass-transfer coeffcents for channel flow l P 1 P NSh = kd H D b = an N ( d L) Re Sc H d N N Re Sc = d vr m H = m rd d H : hydraulc dameter v : velocty

15 Concentraton Polarzaton and Foulng (1) Concentraton polarzaton occurs n membrane separators when the membrane s permeable to, but relatvely mpermeable to B - olecules of B are carred by bulk flow to the upstream surface of the membrane, where they accumulate, causng ther concentraton at the surface of the membrane to ncrease n a polarzaton layer - The equlbrum concentraton of B n ths layer s reached when ts back-dffuson to the bulk flud on the feed-retentate sde equals ts bulk flow toward the membrane, B B

16 Concentraton Polarzaton and Foulng (2) Concentraton polarzaton s most common n pressure-drven membrane separatons nvolvng lquds, such as reverse osmoss and ultrafltraton, where t reduces the flux of - The polarzaton effect can be serous f the concentraton of B reaches ts solublty lmt on the membrane surface precptate of gel may form, the result beng foulng on the membrane surface or wthn membrane pores, wth a further reducton n the flux of B The most straghtforward way of mnmzng concentraton polarzaton s to reduce the flm thckness by ncreasng turbulent mxng at the membrane surface