ADSORPTION BASIC NOTION

Transcription

1 bruary 8, 005 ChE 505 CHPTER DSORPTION BSIC NOTION dsrptin is a prcss by which a chmical cmpnnt (spcis) frm a fluid phas (gas r liquid) is rmvd by attachmnt t a slid phas. Th spcis bing adsrbd is ftn calld th adsrbat, whn it rsids n th slid, and th slid is calld th adsrbnt. Evry adsrptin at givn cnditins f tmpratur, prssur and cmpsitin f th fluid phas can b charactrizd by its quilibrium stat and by its dynamics cnsisting f th kintics f th adsrptin-dsrptin prcss. Cnsidr fr xampl spcis in a fluid phas (f) that is adsrbd n slid sits ( S ) by a simpl mchanism indicatd by quatin () blw: k + () k a ( f ) S ( s) S () s d If w assum that all slid sits ar idntical in thir affinity fr spcis, and ar all qually accssibl t th fluid cntaining, thn a singl adsrptin rat cnstant, and a singl dsrptin rat cnstant,, ar sufficint t dscrib th adsrptin-dsrptin mchanism f quatin (). k d k a Th rat f adsrptin can nw b writtn as: ( )( S ) ra = ka a) and th rat f dsrptin as ( S ) rd = kd (b)

2 t quilibrium th tw rats ar qual r a = r d (3a) which yilds th quilibrium cnstant (in apprpriat units): K k a = = k d ( S ) ( S )( ) q (3b) Nt that th ttal numbr f activ sits (actually cncntratin f activ sits) availabl fr adsrptin f is cnstant and qual t ( S. Thus, ttal sits ar mad up f ithr unccupid sits ( S ) r sits cntaining th adsrbat ( ) ) S as xprssd by quatin (4): ( S ) ( S ) + ( S ) = (4) Simultanus slutin f quatins (3b) and (4) yilds th fllwing xprssins fr th fractin f th sits cvrd by th adsrbat, θ, and fr th fractin f fr sits, ( ) θ at quilibrium ( S ) ( S ) ( ) ( ) K θ = = (5a) + K = q ( S ) ( S ) = + K ( ) θ (5b) q It is infrmativ t plt th fractin f th surfac sits cvrd by th adsrbat as a functin f fluid phas cncntratin f, ( ) with which th slid is quilibratd.

3 θ ½ /K () IGURE : Langmuir Isthrm This plt f quatin (5a) is th s calld Langmuir isthrm, i.. th plt f th adsrbat cncntratin in quilibrium at cnstant tmpratur cnditins with th surrunding fluid. Th plt rvals that at th fluid phas cncntratin f th adsrbing spcis qual t th rciprcal f th adsrptin quilibrium cnstant (xprssd in apprpriat units) th quilibrium fractinal cvrag f th adsrbnt is xactly n half. s th fluid cncntratin f is incrasd indfinitly (which f curs has practical limitatins) th quilibrium surfac cvrag f th adsrbnt by th adsrbat tnds t unity. Hw fast this apprach t unity ccurs dpnds n th valu f th quilibrium cnstant K. r an irrvrsibl adsrptin plt in igur is a Hvisids stp functin at ( ) =, θ = H (( ) ) K and θ = at all cncntratins f i.. th 0. Lt us xamin nw th ffct f tmpratur n th adsrptin prcss. Th tru thrmdynamic cnstant, K, fr th abv cnsidrd adsrptin prcss is unit-lss and givn in trms f activitis: K = a S = G RT a S a (6) r spcis that ar prfrntially adsrbd n th slids sits (and w clarly spak f chmisrptin hr) th Gibbs fr nrgy fr adsrptin is ngativ as th prcss prcds spntanusly (i.. G < 0 ). This mans 3

4 G = H T S < 0 (7) Hwvr, as th mlculs f gt adsrbd, thy ls a dgr f frdm (as thir mtin n th surfac f th slid is cnstraind in tw dimnsins cmpard t thr dimnsinal mtin in th fluid phas) and thrfr thr is a dcras in ntrpy du t adsrptin, i.. S < 0. Nw fr th lft hand sid f quatin (7) t b ngativ, as it shuld, n must hav H < 0. Th nthalpy chang du t adsrptin ( H H H ) = is ngativ, i.. hat is vlvd (rlasd) ~ S ~ du t adsrptin. This is an accrd with ur cmmn xprincs as adsrptin is a prcss akin t cndnsatin (.g. vapr mlculs cndns n a surfac) which als is accmpanid with hat rlas. Oftn n talks abut th hat f adsrptin dfind by q a H = (8) W rcall th Van t Hffs quatin is: d ln K dt H = (9a) RT r d ln K dt H = (9b) RT Sinc H < 0, K dcrass with incrasd tmpratur, and th fractinal cvrag f th surfac with th adsrbat, θ, dcrass at givn cncntratin f with incrasd tmpratur. (Thr ar fw xcptins t this gnral rat). 4

5 W still nd t cnsidr th rats f adsrptin and th apprpriat units fr varius cnstants. In a liquid stat w can xprss th activitis in trms f th prduct f th activity cfficint and cncntratin α = γ C Thn j j j ( ) () K = γ S S = K γ S S γ K (0) ()γ Th units f K ar ths f cncntratin, say (ml/l) r M, and th units f K thn ar M -. Th γ cncntratin f in th fluid is thn masurd in (ml/l) r M. In gass, it is ftn mr cnvnint t us th partial prssur f P ( atm), as a masur f gas phas cncntratin f th adsrbing spcis. Thn, basd n idal gas quatin n gts ( atm ) = K ( RT ) = K K ( atm ) K p γ () Th quilibrium cnstant ( atm ) (5b) whil K p is thn usd in th adsrptin isthrm f quatin (5a) and P rplacs th cncntratin f, ( ). Th nt rat f adsrptin is th diffrnc btwn th rat f adsrptin and dsrptin which can b writtn with th hlp f (a), (b) : r ad ( S ) ( )( ) k ( S ) θ θ d 0 = k () a Strictly spaking basd n th cncpts intrducs s far w wuld xpct t masur th rat in ( s) mls adsrbd by slid pr unit surfac ara f slid sits and unit tim, i.. ml ( dm) us f dm is ndd sinc ( dm) 3 masur f ( in mls pr unit surfac f th slid r nt. Th L = and cncntratins ar masurd in ( ml L). Th natural S ) 0 i.. ( dm) and dsrptin rat cnstants shuld hav th fllwing units: ( ) ( ) 3 k a ( dm ml s) and ( s ) ml in which cas th adsrptin k d. 5

6 ~ in Hwvr, in practic th nt rat f adsrptin is mst frquntly xprssd as ( ml g slid s) which cas th capacity f th slid absrbnt fr th adsrbat is xprssd as ( ) ( ml g slid ) Th adsrptin and dsrptin rat cnstants rmain unchangd. If, hwvr, n rplacs th cncntratin f in quatin (), (), with partial prssur f P ( atm) ndd adsrptin cnstant K bcm ( atm s ). a p r ad S 0,, thn th units f th. On shuld nt that by subtracting th nt rat f adsrptin at quilibrium (which is zr) frm quatin () n can rprsnt th rat f adsrptin as bing drivn by th fractinal surfac cvrag dpartur frm quilibrium, i.. r ad = [ k a () S 0 ()+ k d S ( θ θ () 0 ] ) (3) whr ( ) ( ) K θ = (5a) + K On shuld nt that th abv simplifid dvlpmnt assums a chmisrptin prcss, i.. a prcss rquiring bnd frmatin btwn and which has an activatin nrgy. It is assumd that nly micr mlcular cvrag ccurs. This dvlpmnt als ignrs pssibl ffct f mass transfr rats via bundary layr diffusin n th rat f adsrptin. It ds nt cnsidr prus slids and th ffct f pr diffusin n th adsrptin prcss. S It ds, hwvr, prsnt a vry basic ntin f adsrptin. dditinal varius typs f dissciativ, dual sit, cmptitiv and inhibitry adsrptin mchanisms will b discussd in th appndix. In catalytic prcsss ftn it is assumd that adsrptin is rapid and quilibratd whil th rat limiting stp is th surfac ractin that cnvrts S stp is thn prprtinal t th quilibrium surfac fractinal cvrag, θ. t a nw spcis. Th rat f that rat limiting 6

7 Instad f discussing htrgnus catalyzd ractins, that always invlv adsrptin as an imprtant stp, w fcus hr n rmval f cmpnnts frm a gas r liquid stram by adsrptin which is a frquntly usd unit prcss in nvirnmntal nginring. PCKED DSORBTION COLUMNS ND BREKTHROUGH CURVES Whn w ar intrstd in rmving a particular cmpnnt frm a fluid phas by adsrptin, w mst frquntly us packd (fixd) bds f slid adsrbnt. This is dn bcaus such an arrangmnt prvids th largst pssibl mass f slids pr unit vlum f th fluid and th flw pattrn f th fluid is th clsst t plug flw. Bth f ths ar cnduciv t achiving a high vlumtric prductivity and thrughput rats. Cnsidr a cylindrical clumn packd with adsrbnt as shwn in igur Q C f Q C f IGURE : Typical Packd Bd dsrbr Th clumn has a diamtr D and lngth H. Cnstant vlumtric flw rat f fluid Q f pumpd thrugh th clumn. Th packing in th clumn is unifrm and has an quivalnt diamtr f. Initially thr is n adsrbat n th packing in th clumn. t tim t = 0 w switch at th inlt frm a flw f pur carrir fluid t a flw (at th sam flw rat Qf) f a stram that cntains C ( ml L) in th fd. Th clumn is kpt isthrmal. t th xit w mnitr cntinuusly th d p cncntratin f, C, in th fflunt i.. w btain th brak thrugh curv. gnral brak thrugh curv will hav th shap indicatd in igur 3. 7

8 C C C C dt IGURE 3: Schmatic f a Brakthrugh Curv W ar intrstd in undrstanding and quantifying th faturs f this curv. mass balanc n th adsrbing spcis fr th whl clumn rquirs: Ttal accumulatin f = Ttal input f Ttal utput f (4) Ttal accumulatin f in clumn = munt f in th clumn at infinit tim munt f in th clumn at tim 0 = ε b VC M + ( ε b )Vρ p W 0 (5) whr ε b = bd vidag 8

9 V(L)= clumn vlum ( π D 4) H C M ( ml L ) = mlar cncntratin f in th fd ( kg ml ) = mlcular wight f ρ p ( kg L) = dnsity f frsh slid adsrbnt w ( kg kg slid ) = saturatin capacity f frsh adsrbnt fr i.. hw many kg pr kg frsh srbnt can n pick up at quilibrium at. C Ttal = Input f Q f C M dt (6a) Ttal = Output f Q f C M dt (6b) whr Q f ( Ls) = vlumtric flw rat thrugh th clumn C ( ml L) = xit cncntratin f as a functin f tim t () s = tim n stram (sinc C intrducd) Equatin (4) nw bcms ε b VC + ε ( b)v ρ p w = Q f ( C C dt M ) (7) This can b rwrittn as: ε b V Q f + ε ( b)v ρ p w = C dt (8) Q f C M C 9

10 Th right hand sid f quatin (8) rprsnts th dashd ara in igur 3. Hnc, a brakthrugh xprimnt, n a mdl fr basis, prvids dirct infrmatin n th adsrptin saturatin capacity f th slid. Th first trm n th lft hand sid is th man rsidnc tim f th flwing fluid xtrnal t th packing. It tlls us hw lng it taks t rplnish th vlum f th fluid in th clumn with incming flw rat. It als tlls us hw lng n th avrag a fluid lmnt stays in th fluid phas f th clumn. Th scnd trm n th lft hand sid is th rati f th capacity f th slid adsrbnt in th bd fr adsrbat t th carrying capacity f th inlt flw fr. It tlls us hw lng a tim th inlt flw rat must b sustaind t prvid nugh t saturat th whl bd. In a way it is a masur f th avrag rsidnc tim f th adsrbat n th adsrbnt. Equatin (8) and th abv discussin mak it clar that frm a brakthrugh xprimnt n can calculat th adsrbnt saturatin capacity w M ( ml g slid) which crrspnds t fluid phas cncntratin C. By rpating th brakthrugh xprimnts at diffrnt lvls f C n can find th adsrptin isthrm f w M vs C. W rcgniz nw that th ara rprsntd by ( C C ) dt is dirctly prprtinal t th saturatin capacity f th bd. Nxt w xamin th shap f th curv. Lt us call th dimnsinlss -curv, r brakthrugh curv, fr th adsrbat. C = t b C ls cnsidr i = Ci C0 as a stp rspns, brakthrugh curv, f a nn-adsrbing inrt spcis i. Thn th fllwing situatins dpictd in igur 4 may aris. 0

11 i i i tim, t tim, t a) b) i tim, t tim, t c) d) IGURE 4: Schmatic f Brakthrugh Curvs f Inrt (Nnadsrbing) and dsrbing Spcis In igur 4a, th brakthrugh, i, f an inrt (nnadsrbing) spcis is shwn tgthr with th brakthrugh f th adsrbat. Th sharp stp-wis ris f th i curv indicats that th flw in th clumn is plug flw, (pistn flw) as all th fluid lmnts that ntr tgthr flw tgthr until th xit. Th ara ( ) i dt = t is th man rsidnc tim f th fluid. Th fact that is als a f sharp rising stp indicats that all adsrptin-dsrptin prcsss ar instantanus n th tim scal f flw. Th ara is givn by quatin (8). In igur 4b, th fact that i is f sigmidal S shap indicats th rspns f a flw prfil thr than plug flw, and pints t pssibilitis f flw maldistributin. Sinc i is th sam in shap as, but just translatd in tim, indicats that th rat prcsss invlvd in adsrptin-dsrptin ar

12 still instantanus n th tim scal f flw. Th ara ( )dt quals th sum f th tw trms n th lft f quatin (8) whil th ara (8). (- i )dt quals th first trm n th lft f quatin In igur 4c, th sharp stp-lik ris f i indicats a prfct plug flw pattrn in th clumn. Th fact that is nw f sigmidal S-shap is indicativ f th prsnc f substantial rsistancs in th adsrptin-dsrptin prcss. In thr wrds, th kintics, diffusin and thr factrs affcting th adsrptin must b dscribd t prdict th shap f th S-curv. igur 4d cmpars tw brakthrughs, and f spcis and, rspctivly. Th slid adsrbnt has th sam adsrptin capacity fr bth and.g. ( ) dt = ( ) dt. Sinc is a sharp rising stp-wis curv this indicats that th flw in th clumn is plug flw and all th stps invlvd in adsrptin f, ar ssntially instantanus n th tim scal cnsidrd. Clarly, majr rsistancs ar prsnt in adsrptin f which rsults in th sigmidal curv. In adsrptin th gal ftn is t rmv a cmpnnt frm th fluid phas and, hnc, whn that cmpnnt braks thrugh at an undsird lvl, th flw must b stppd and switchd t anthr adsrptin clumn. Clarly thn in igur 4d th adsrbnt usd is xcllnt fr rmval f but ds nt yild a gd brakthrugh curv fr adsrptin. Thus runs f small adsrptin clumns culd rapidly dtrmin th suitability f varius adsrbnts fr rmval f diffrnt adsrbats. stp brakthrugh curv with a lng brakthrugh tim is sught. Oftn adsrptin clumns ar usd in pridic pratin. Th adsrptin stp is fllwd by th dsrptin stp in which th adsrbat is rlasd in a nw fluid phas frquntly in cncntratd

13 frm. r xampl, adsrptin at lw tmpratur can b fllwd by dsrptin at lvatd tmpratur upn hating th clumn. In gas phas high prssur adsrptin can b fllwd by lw prssur dsrptin. This swing prssur adsrptin prcss is usd xtnsivly in industrial applicatins. On shuld als nt that th thry f adsrptin clumns is intimatly tid t th thry f chrmatgraphy. r xampl, if w dfin d i Ei = and dt d E = dt (9) Thn E and E rprsnt th impuls rspns f th clumn t a dlta functin (instantanus) i unit injctin f inrt I and spcis. E i E i IGURE 5: Impuls Rspnss fr igur 4b Impuls rspnss drivd frm brakthrugh curvs f igur 4b ar shwn in igur 5. Th cntrid f ths can b shwn t b t E i () t ε b V dt = Q f (0a) te ()dt t = ε V b Q f ( + ε b)vρ p w Q f C M (0b) 3

14 This is prcisly th principl usd in chrmatgraphic clumns t sparat cmpnnts that hav diffrnt adsrptin quilibrium cnstants ( w M C ). W xamin nw nginring mdls fr th brakthrugh curv. ENGINEERING MODELS OR BREKTHROUGH CURVES Th task is t hav a rbust nt vrly cmplx mdl fr prdictin f th brakthrugh tim and th sigmidal shap f th brakthrugh curv and th ability t calculat th mdl paramtrs bth by fitting xprimntal brakthrugh curvs and t prdict thn th thry. It is imprtant that th mdl has th ability t assss th ffct f th chang in scal f th clumn, r with prating cnditins, n th shap f th brakthrugh curv. r many systms, including adsrbrs, it is pssibl t rprsnt th dimsinabl impuls rspns t a unit dlta functin input in th inlt in trms f th incmplt gamma functin E i (θ) = θ ( σ D ) xp( θ /σ D ) Γ( σ D ) whr θ = t /t with t bing givn by quatin (0b) and θ D = σ whr t () σ = (t t ) E i (t)dt (3) whr σ is drivd frm a mdl f adsrptin, flw, transprt in th bd. () r dtails s Link and Dudukvic (98). Th us f such a rprsntatin in dscribing th prfrmanc f a cyclic md f pratin is dscribd fr hat rgnratrs by Dudukvic and Ramachandran (985, 99). Th analgy t adsrbrs shuld b slf vidnt. 4

15 Rfrncs:. Dudukvic, M.P. and P.. Ramachandran, Hat Rgnratrs: Dsign and Evaluatin, Hat Transfr Dsign and Mthds (J.M. McKn, d.), Marcl Dkkr, 99 pp Dudukvic, M.P. and P.. Ramachandran, Quick Dsign and Evaluatin f Hat Rgnratrs, Chmical Enginring, Jun 0, 985, pp Link,. and M.P. Dudukvic, Rprsntatin f Brakthrugh Curvs fr ixd-bd dsrbrs and Ractrs Using Mmnts f th Impuls Rspns, Th Chm. Eng. Jurnal 3, 3-36 (98). 5