Predicting adsorption of water/organic mixtures using molecular simulation

P edic ing Adso p ion o Wa e /O ganic Mix u es Using Molecula Simula ion Miguel Jo ge and Nigel A. Sea on School o Enginee ing and Elec onics, Uni e si y o Edinbu gh, Edinbu gh, EH9 3JL, U.K. The use o Mon e Ca lo simula ion o p edic he adso p ion o mix u es o pola and nonpola species on ac i®a ed ca bon was in®es iga ed using wa e and e hane on BPL ca bon as a p o o ype sys em. The s uc u e o he adso ben was modeled by an a ay o sli -shaped po es, cha ac e ized by a po e-size dis ibu ion. The chemical he e ogene- i y o he ca bon was aken in o accoun by including oxygen-con aining si es on he su ace o he po es. The po e-size dis ibu ion was ob ained om pu e-e hane adso p- ion on he same ca bon sample, while he concen a ion and dis ibu ion o su ace si es we e de e mined by analyzing pu e-wa e adso p ion. Model p edic ions ag ee well wi h expe imen al mul icomponen da a. In oduc ion A subs an ial p opo ion o indus ial adso p ion p ocesses Ž in ol e ca bon adso ben s Bansal e al., 1988; Si ca e al., .Ž 1996 . In many impo an applica ions such as ai pu i ica- . ion and pe sonal p o ec ion , he e is a need o selec i ely adso b o ganic species om gas s eams con aining a ce ain amoun o wa e apo . I is well known ha he p esence o wa e in he eeds eam can signi ican ly a ec he e iciency o he sepa a ion p ocess. Reduced adso p ion capaci ies and ea lie b eak h ough imes ha e been epo ed o adso p- ion o a wide ange o o ganics on ac i a ed ca bon in he Ž p esence o wa e Gong and Keene , 1993; Scameho n, 1979; . Shin e al., 2002 . The de elopmen o a me hod o p edic he adso p ion o mix u es o wa e and o ganic species on ac i a ed ca bon is, he e o e, essen ial o he e icien de- sign o such p ocesses. This in u n equi es a de ailed unde - s anding o he undamen al aspec s ha a ec adso p ion o bo h pola and nonpola componen s. The mos common app oach o p edic ing mul icompo- nen adso p ion is based on classical he modynamic me h- Ž. ods. O hese, he ideal adso bed solu ion heo y IAST Ž. Mye s and P ausni z, 1965 is he s anda d me hod used in indus y. I is based on he concep o sp eading p essu e and assumes ha he adso bed phase is an ideal solu ion in equi- lib ium wi h he bulk gas phase. Fo sys ems such as mix u es o wa e and hyd oca bons adso bed on ac i a ed ca bon, he IAST is likely o pe o m badly, since his sys em is a om ideal. O he he modynamic me hods ha e he e o e been Co espondence conce ning his a icle should be add essed o N. A. Sea on. Ž. p oposed. To ou knowledge, Okazaki e al. 1978 we e he i s o a emp a p edic ion o mul icomponen adso p ion in his ype o sys em. In hei me hod, wa e is adso bed by capilla y condensa ion, while he adso p ion o he o ganic species is he esul o h ee con ibu ions: adso p ion on he d y adso ba e su ace, dissolu ion in o he condensed phase, and adso p ion on o he we su ace. They applied his me hod o bina y adso p ion o wa e and se e al o ganic Ž. species ace one, me hanol, benzene, and oluene on ac i- a ed ca bon. Al hough he p edic ions we e in easonable ag eemen wi h expe imen al da a, he me hod has he signi - ican disad an age o equi ing la ge amoun s o da a as in- Ž pu apa om he pu e-componen adso p ion iso he ms, i also equi es liquid-phase iso he ms o he o ganic species . in wa e and apo ᎐liquid equilib ium da a . A la e me hod, Ž. based on po en ial heo y, is due o Doong and Yang 1987 . Thei me hod uses he concep o maximum a ailable po e olume, oge he wi h he Dubinin-Radushke ich equa ion Žand la e , he Dubinin-As akho equa ion; Doong and Yang, . 1988 wi h pa ame e s ob ained om he pu e-componen iso he ms. A sys em wi h no la e al in e ac ions be ween ad- so bed species is assumed, bu he maximum a ailable mic o- po e olume o a gi en species is educed by he amoun adso bed o all o he componen s. This me hod was seen o pe o m well o he same sys ems s udied by Okazaki e al. Ž. 1978 . A simila app oach was used by La anchy and S oeckli Ž. 1999 o p edic bina y adso p ion o wa e and 2-chlo op o- pane on ac i a ed ca bon. Thei me hod is essen ially a sim- pli ied o m o he Doong-Yang me hod using he Dubinin- Augus 2003 Vol. 49, No. 8AIChE Jou nal 2059 As akho equa ion. La anchy and S oeckli also applied IAST, coupled wi h he Dubinin-As akho equa ion o ep esen he pu e-componen da a, o adso p ion o wa e o ganic mix- u es. They concluded ha IAST pe o ms easonably well when he o ganic componen is soluble in wa e , while hei implemen a ion o he Doong-Yang me hod is equi ed o wa e -insoluble o ganics. Simila conclusions we e d awn Ž. mo e ecen ly by Linde s e al. 2001 . Finally, he e o s o Ž. Appel e al. 1998 o desc ibe mul icomponen adso p ion by using mix u e i ial coe icien s a e no ewo hy. These au- ho s success ully applied his heo y o bina y wa e hexane adso p ion on ac i a ed ca bon, by using he equa ion o Talu Ž. and Meunie 1996 o he pu e-componen wa e iso he ms. Howe e , he calcula ion o he i ial mix u e coe icien s e- qui es he i ing o mul icomponen da a, and, hence, hei me hod was only applied in a co ela i e manne . The he modynamic me hods desc ibed p e iously ha e some impo an disad an ages: hey become inaccu a e as he Ž sys ems mo e away om ideali y in he sense o Raoul ’s . law , due o chemical dissimila i y among he adso p i e species; hey equi e a signi ican amoun o expe imen al da a; and hey gi e us no in o ma ion on he s uc u e o he adso ben . Recen ly, an al e na i e app oach, based on s a is- ical mechanical me hods such as densi y unc ional heo y and molecula simula ion, has begun o be applied o mul i- Ž componen adso p ion p edic ion see Razmus and Hall, 1991; Sega a and Gland , 1994; Guse and O’B ien, 1998; Pan e al., 1999; Vuong and Monson, 1999; Macedonia and . Maginn, 1999; and e e ences he ein . In pa icula , molecu- la simula ion me hods ha e been shown o be ex emely powe ul in p edic ing mul icomponen adso p ion in ac i- a ed ca bon as well as in cha ac e izing he adso ben i sel Ž. Da ies and Sea on, 2000 . Molecula simula ion can be used o ob ain in o ma ion abou he chemical and physical cha - ac e is ics o he adso ben ha a ec he adso p ion o he molecula species o in e es , using a limi ed amoun o ex- pe imen al da a as inpu . Wi h his in o ma ion, a model ad- so ben , ep esen a i e o he eal adso ben , can be con- s uc ed and used o p edic adso p ion in he eal adso ben . The e ec o physical and chemical changes on adso p ion can also be in es iga ed using he model adso ben , he eby using molecula simula ion as an adso ben design ool. In his a icle, we a emp o de elop such a model o he case o wa e o ganic mix u es. Models o Ac i a ed Ca bon Ac i a ed ca bons a e composed o g aphi e pla es s acked Ž. oge he in a mo e o less i egula way McEnaney, 1988 . The in e s ices o hese pla es o m a ne wo k o in e con- nec ed po es, be ween which adso p ion occu s. These ca - bons a e usually mic opo ous and ha e a e y s ong a ini y owa d hyd oca bons. Howe e , in spi e o hei gene ally hy- d ophobic na u e, mos ac i a ed ca bons used in indus ial applica ions a e seen o s ongly adso b wa e . This e ec is due o he p esence o pola g oups ha appea on he edges o he g aphi ic pla es o on de ec si es, as a consequence o he ac i a ion p ocess. The mos common ypes a e oxygen- con aining g oups, such as ca boxyl, ca bonyl, and hyd oxyl Ž. Boehm, 1994 . These g oups o m nuclea ion si es o wa e , inc easing he hyd ophilic p ope ies o he ca bon. Thus, he s uc u e o he ca bon has a s ong e ec on adso p ion o all chemical species, while he chemically he e ogeneous si es a ec mainly he adso p ion o pola species. In he case o wa e o ganic mix u es, bo h ypes o he e ogenei y mus be aken in o accoun when de eloping a model o he adso - ben . The mos widely used model o he s uc u e o ac i a ed ca bon is an a ay o sli -shaped po es, cha ac e ized by a Ž. po e-size dis ibu ion PSD . This model akes ad an age o he shape o he g aphi e pla ele s and assumes ha he en- i e ca bon can be ep esen ed by a bundle o in ini e po es o sli shape, which a e in equilib ium wi h he bulk gas phase. The o al adso p ion on he ca bon can hen be ela ed o he Ž.Ž PSD ia he adso p ion in eg al equa ion AIE Sea on e . al., 1989 ⬁ NT,P,ys w ␳ T,P,y,wdw 1 Ž. Ž. Ž. Ž . H ii 0 Ž. whe e NT,P,yis he o al adso p ion o componen ia i empe a u e T, bulk-phase p essu e P, and composi ion y; i Ž. ␳ T,P,y,wis he mola densi y o he adso ben a he i Ž.Ž . same condi ions in a po e o wid h w; w sdV dw is mic he PSD; and Vis he speci ic mic opo e olume o he mic adso ben . The single-po e iso he ms can be calcula ed in a numbe o di e en ways, anging om mac oscopic models like he Kel in equa ion o he mo e complex molecula sim- ula ion me hods. A emp s o include chemical he e ogenei y in model ca - Ž. bons ha e so a been ew. Sega a and Gland 1994 ha e modeled ac i a ed ca bons as a diso de ed s acking o disk- shaped g aphi ic pla ele s. Su ace pola i y was ep esen ed by dipole momen s smea ed along he edges o he pla es. Simula ed adso p ion iso he ms in his model we e compa ed o expe imen al da a, bu he ag eemen was only quali a i e. Ž. Subsequen wo k Go don and Gland , 1997 sugges s ha he smea ed dipoles a e oo homogeneous o ep esen he pola i y o eal adso ben s, and ha a disc e e dis ibu ion o pola i y would be mo e ealis ic. A la e model, by McCallum Ž. e al. 1999 , combined a disc e e ep esen a ion o he su - ace si es wi h a PSD model o he ca bon s uc u e. The PSD was de e mined by using densi y unc ional heo y o analyze a ni ogen adso p ion iso he m, and su ace si es we e andomly dis ibu ed on he su ace o he po es. Simpli ied molecula models, based on squa e-well po en ials o mimic hyd ogen bonding, we e employed o ep esen wa e molecules and su ace oxygen a oms. The o al concen a ion o pola si es on he ca bon was de e mined expe imen ally by i a ion me hods and he in e ac ion po en ial be ween wa e molecules and su ace si es was ob ained om a i o low-p essu e wa e adso p ion da a. Ag eemen wi h expe i- men was easonable. Howe e , he po en ial models igno e he long- ange na u e o he elec os a ic in e ac ions among wa e molecules, and be ween he wa e molecules and pola su ace g oups. This makes i di icul o ex end he model o Ž mo e gene al cases such as o he empe a u es, mul icompo- . nen adso p ion . A mo e ecen model o ac i a ed ca bon o include chemical he e ogenei y was p oposed by B ennan Ž. e al. 2001 . The s uc u e o he adso ben is ep esen ed by a s acking o g aphi ic pla es o se e al sizes, which is op i- Augus 2003 Vol. 49, No. 8 AIChE Jou nal2060 mized o ep oduce expe imen ally de e mined ca bon᎐ ca bon adial dis ibu ion unc ions by a p ocedu e e med Ž e e se Mon e Ca lo simula ion Thomson and Gubbins, . 2000 . Chemical he e ogenei y was included by andomly placing pola ca bonyl si es on he edges o he pla es. Al- hough his model p o ides us wi h a mo e ealis ic ep esen- a ion o ac i a ed ca bon, i is also ex emely complex, which ca ies disad an ages such as e y compu a ionally demand- ing simula ions and di icul y o applica ion o a wide ange o adso ben s. To ou knowledge, no compa ison o his model wi h an expe imen has ye been p esen ed. The model p oposed he e is based on a PSD o ep esen he physical s uc u e o he ac i a ed ca bon, combined wi h pola su ace si es o desc ibe he chemical he e ogenei y. This dis ibu ion is ob ained om he AIE using expe imen- al da a o pu e-e hane adso p ion oge he wi h g and Ž. canonical Mon e Ca lo GCMC simula ion o calcula e he single-po e iso he ms. We ha e included chemical he e o- genei y in he model by placing pola oxygena ed si es on he su ace o he po es, he concen a ion and dis ibu ion o which a e ob ained om an analysis o expe imen al pu e- wa e adso p ion. This model is expec ed o cap u e all he undamen al cha ac e is ics o he adso ben ha a ec he adso p ion o bo h pola and nonpola species, while e ain- ing a deg ee o simplici y ha will enable i s use in an indus- ial con ex . Molecula Simula ion in Single Po es We ha e used GCMC simula ion o calcula e equilib ium adso p ion iso he ms in model po es. The Mon e Ca lo me hod is a s ochas ic p ocess based on he laws o s a is ical mechanics. In e ec , one samples a la ge numbe o molecu- la con igu a ions ha a e consis en wi h a gi en se o Ž.Ž. mac oscopic he modynamic a iables Hill, 1960 . Fo mos adso p ion calcula ions, he g and canonical ensemble is he mos con enien . In his ensemble he empe a u e, olume, and chemical po en ial a e kep cons an , while he o al numbe o molecules is allowed o luc ua e. This si ua ion is analogous o an adso p ion expe imen , and, hus, adso p ion iso he ms can be ob ained om GCMC simula ions by calcu- la ing he equilib ium adso bed densi y a each se o condi- ions. In p ac ice, ins ead o ixing he chemical po en ial, i is mo e con enien o speci y p ope ies o he bulk phase, Ž. Ž. such as p essu e Pand composi ion x. One mus hen i employ an app op ia e equa ion o s a e o calcula e he chemical po en ial o he bulk phase. In he simula ions e- po ed in his a icle, he Peng-Robinson equa ion o s a e Ž. Sandle , 1989 was used o his pu pose. The GCMC simu- la ion me hod has been widely used and is well documen ed ŽŽ. see, o example, Allen and Tildesley 1989 and F enkel and Ž.. Smi 1996 . De ails o he implemen a ion o his pa icu- Ž la sys em ha e been published elsewhe e Jo ge and Sea on, . 2002a . The simula ion cell is ec angula and is bounded in he z-dimension by he po e walls, he dis ance be ween which Ž. de ines he po e wid h w. In bo h di ec ions pa allel o he Ž. walls xand y, pe iodic bounda y condi ions we e used o ep esen a semi-in ini e po e. The leng h o he cell in hese di ec ions was 3 nm, which is su icien o a oid any ini e-size Ž. e ec s Jo ge and Sea on, 2002b . E hane molecules we e Table 1. Geome ic and Po en ial Pa ame e s U o he Molecula Models Used in This Wo k Pa ame e s G aphi e E hane Wa e Ca bonyl Ž. ␴ nm 0.34 0.3512 0.3166 0.296 y1 Ž. ⑀ Jⴢmol 232.8 1162.3 650.2 879.6 Ž. ⌬nm 0.335 ᎏᎏᎏ y3 Ž. ␳ nm 114 ᎏᎏᎏ sŽ. Bond leng h nm ᎏ0.2353 0.1 0.1233 Ž. Bond angle ⬚ᎏᎏ109.47 ᎏ yŽ. qe ᎏᎏy0.8476 y0.5 qŽ. qe ᎏᎏq0.4238 q0.5 U Ž. G aphi e pa ame e s we e aken om S eele 1974 ; e hane pa ame e s Ž. om C acknell and Nicholson 1994 ; wa e pa ame e s om Be endsen Ž. e al. 1987 ; ca bonyl pa ame e s om Jo gensen and Ti ado-Ri es Ž. 1988 . Ž. ep esen ed by wo Lenna d-Jones L-J si es, one o each CH g oup, wi h pa ame e s aken om he wo k o C ack- 3 Ž. nell and Nicholson 1994 . The poin -cha ge SPC E model o Ž. Be endsen e al. 1987 was chosen o ep esen wa e molecules. The in e ac ion be ween an L-J cen e in an ad- so ba e molecule and each o he po e walls is gi en by he Ž. 10-4-3 po en ial o S eele 1974 o g aphi e. The oxygena ed si es on he su ace o he ca bon we e ep esen ed by ca - bonyl g oups, desc ibed by he OPLS po en ial o C⫽O Ž. g oups in aminoacids Jo gensen and Ti ado-Ri es, 1988 . This model is composed o an L-J cen e and a nega i e poin cha ge o he oxygen a om and a posi i e poin cha ge on he ca bon a om in he basal plane o g aphi e, as desc ibed Ž. in a p e ious publica ion Jo ge e al., 2002 . Geome ic and po en ial pa ame e s o wa e , e hane, g aphi e, and ca - bonyl si es a e gi en in Table 1. The po en ial be ween wo poin cha ges is long- anged and i s calcula ion equi es he use o special echniques. The ull ex en o hese long- ange in e ac ions was accoun ed o using he me hod o Heyes Ž. and an Swol 1981 . C oss-species Lenna d-Jones pa ame- e s we e calcula ed using he Lo en z-Be helo combining ules. When compa ing simula ion esul s wi h expe imen s, he o me needs o be co ec ed o accoun o he di e ence be ween absolu e and excess adso p ion. The esul o a sim- ula ion is he o al adso p ion o each species in a model po eᎏabsolu e adso p ion. On he o he hand, adso p ion expe imen s measu e he di e ence be ween absolu e ad- so p ion and he amoun o adso ba e ha would be p esen unde he same he modynamic condi ions i he e we e no in e ac ions wi h he adso ben ᎏexcess adso p ion. The sim- ula ed absolu e adso p ion iso he ms a e con e ed o excess iso he ms using he me hod p oposed by Da ies and Sea on Ž. 1999 . Po e-Size Dis ibu ion Ž. The PSD is ob ained om he AIE Eq. 1 , using expe i- men al da a o he adso p ion o a pu e componen , which gi es he numbe o moles adso bed, as a unc ion o empe - Ž. a u e and p essu e, NT,P,ys1 . As men ioned p e iously, i he single-po e iso he ms we e calcula ed by GCMC simula- Ž. ions. Equa ion 1 is hen sol ed o he PSD, w. This is an ill-posed p oblem, as in p inciple an in ini e numbe o PSDs can sa is y he expe imen al da a. We ha e used he me hod Augus 2003 Vol. 49, No. 8AIChE Jou nal 2061 Ž. o Da ies e al. 1999 o sol e he AIE. A b ie ou line o his me hod is gi en nex . Fi s , he AIE is disc e ized, using a numbe o quad a u e poin s ha is less han o equal o he numbe o expe imen- al da a poin s. In o de o a oid excessi e spikiness o he PSD, a egula iza ion ac o is inco po a ed in he minimiza- ion p ocedu e. The op imal egula iza ion ac o is de e - mined by analyzing he e o o he i , using bo h gene al- ized c oss- alida ion and L-cu es. This p ocedu e elimina es he need o es ic he PSD o a speci ic unc ional o m, while s ill ensu ing ha a physically meaning ul esul is ob- ained. When es ablishing he quad a u e in e als, one mus ake in o accoun he concep o ‘‘window o eliabili y,’’ i s Ž. in oduced by Guse e al. 1997 . Essen ially, his es ablishes lowe and uppe limi s o he po e sizes ha can be eliably iden i ied in a PSD analysis. The lowe limi is he smalles po e ha he adso ba e o in e es can en e , while he uppe limi is he la ges po e o which he single-po e iso he ms a e s ill linea ly independen . Fo la ge po es, he adso p ion will occu i ually independen ly on each po e wall, and he po es will hen become indis inguishable om each o he . In his me hod, all he po es abo e he uppe limi o eliabili y a e accoun ed o in a single quad a u e in e al. The limi s o eliabili y depend on he adso ba e, on he empe a u e, Ž. and on he p essu e. Da ies and Sea on 2000 ha e ecen ly shown ha he window o eliabili y is ex emely impo an when p edic ing adso p ion, while he esolu ion o he PSD plays only a mino pa . Acco ding o he p eceding p ocedu e, he disc e ized AIE becomes m NT,P,y w ␳ T,P,y,w ␦ w2 Ž. Ž. Ž. Ž . Ý ininn ns1 whe e mis he numbe o quad a u e in e als used in he analysis, wi h m-1 in e als lying wi hin he window o elia- bili y and one in e al accoun ing o he adso p ion in all he po es abo e he uppe limi . Figu e 1. Po e-size dis ibu ion o BPL ac i a ed ca bon de e mined om he analysis o pu e-e hane adso p ion a 273, 298, and 323 K. Figu e 2. Fi o he pu e-e hane expe imen al da a o () Russell and LeVan 1997 using he PSD o Figu e 1. Diamonds a e o da a a 273 K, open squa es a e o 298 K, and iangles a e o 323 K. Ž. Russell and LeVan 1997 ha e epo ed da a o he ad- so p ion o pu e wa e , pu e e hane, and a ious mix u es o hese componen s a empe a u es o 273, 298 and 323 K. We ha e used he p eceding me hod o calcula e a PSD o BPL ca bon based on hei expe imen al da a o e hane a all h ee empe a u es. I was decided o use h ee empe a- u es, a he han a single empe a u e as in ea lie wo k Ž. Da ies and Sea on, 1999 , because i was in ended o p edic adso p ion o wa e e hane mix u es o e his empe a u e ange. A e he single-po e iso he ms o e hane we e ob- ained om GCMC simula ions, he AIE was sol ed using a Ž. nonlinea minimiza ion ou ine MINOS, 1988 in e aced h ough he GAMS ma hema ical p og amming package Ž. GAMS, 1998 . The window o eliabili y was de e mined o lie be ween app oxima ely 0.6 and 2 nm, and he maximum Ž esolu ion possible co esponding o he numbe o expe i- . men al da a poin s was used. The dis ibu ion wi h he op i- Ž mal egula iza ion he gene alized c oss- alida ion me hod and he L-cu e yielded e y simila egula iza ion pa ame- . e s, esul ing in essen ially indis inguishable PSDs is shown in Figu e 1, and he i o he expe imen al da a in Figu e 2. The calcula ed PSD i s he expe imen al da a well o e he en i e p essu e ange. Chemical He e ogenei y The PSD model o he s uc u e o he ca bon has been success ul in p edic ing pu e- and mul icomponen adso p- Ž. ion in ol ing nonpola Da ies and Sea on, 1999 and weakly Ž. pola Heuchel e al., 1999 species. Howe e , as we ha e discussed, a desc ip ion o he chemical he e ogenei y o he ca bon is equi ed when highly pola species, like wa e , a e conce ned. He e we a emp o include his ex a dimension in o ou model ca bon by placing oxygen-con aining si es on he su ace o he sli po es. The beha io o he eal ca bon is likely o be an a e age o he con ibu ions o a wide a ie y o chemical g oups Ž. Ž Boehm, 1994 . Howe e , bo h expe imen al Ba on e al., Augus 2003 Vol. 49, No. 8 AIChE Jou nal2062 .Ž. 1994 and molecula simula ion Jo ge e al., 2002 esul s sugges ha he amoun o wa e adso bed on ac i a ed ca - bon depends p ima ily on he concen a ion o oxygen on he su ace, ega dless o i s unc ionali y. The e o e, we ha e as- sumed ha all he pola si es p esen in he ca bon can be ep esen ed by ca bonyl g oups. As o he loca ion o hese si es, i is known ha hey p edomina e nea he edges o he g aphi e pla es. Since ou model ep esen s he ca bon s uc u e by a collec ion o Ž. semi-in ini e po es wi h no edges , we ha e chosen o place he pola si es on a squa e la ice supe imposed on he po e walls. Fu he mo e, he si es we e egula ly dis ibu ed and sepa a ed by a leas 0.75 nm. These simpli ica ions will ha e an e ec on he s uc u e o he adso bed wa e in he model, which will be di e en om ha in eal ca bons. Howe e , we ha e obse ed om p e ious molecula simula ion s udies ha he dis ibu ion o pola si es on he su ace o he ca - bon has a s ong impac a low p essu e, bu almos no e ec Ž. on he amoun adso bed a high p essu es Jo ge e al., 2002 . Ž An analysis o pu e-wa e adso p ion a low p essu e p e e - . ably in he Hen y’s law limi migh , in p inciple, p o ide us wi h in o ma ion on he si e dis ibu ion. Un o una ely, such low-p essu e expe imen al da a a e ha d o come by and we e una ailable o he pa icula ca bon unde s udy. Howe e , since mos o he a ailable expe imen al da a we e ob ained Ž. a high pa ial p essu es o wa e see Figu e 5 , he assump- ion o a egula si e dis ibu ion in a po e wi h no edges should ha e a negligible e ec on he esul s. Fu he mo e, he si ua ion o echnological in e es , and di icul y, is when wa e adso p ion on he ca bon is subs an ial, which means ha o mos p ac ical cases, he p ecise dis ibu ion o he pola si es on he ca bon su ace will no be an impo an ac o . The las ac o one mus conside is he dis ibu ion o po- la si es be ween po es o di e en wid h. An ob ious i s Ž choice is simply o say ha his dis ibu ion is uni o m ha . is, all po es con ain he same densi y o ca bonyl g oups . This assump ion, oge he wi h he p e ious ones, educes he p ocess o cha ac e izing he chemis y o he ac i a ed ca - bon o he de e mina ion o a single a iableᎏ he concen a- Figu e 3. Po e-size dis ibu ion o BPL ac i a ed ca bon educed o 15 disc e e po e sizes. ion o ca bonyl g oups on he su ace. To es ima e his con- cen a ion, we ha e i s loca ed he in lec ion poin in he Ž. expe imen al ypically S-shaped pu e-wa e adso p ion Ž. iso he m a 298 K see Figu e 5 . Then we ha e de e mined he modal po e wid h o he PSD ob ained om pu e-e hane Ž. adso p ion Figu e 1 , which was calcula ed as 1.24 nm. A se ies o GCMC simula ions in a model po e o ha wid h, o di e en concen a ions o pola si es, was pe o med, aking no e o he a ia ion o he p essu e o spon aneous condensa ion wi h he si e concen a ion. The chosen con- cen a ion was such ha he p essu e o spon aneous con- densa ion in he single-po e iso he m co esponded oughly o he in lexion poin o he expe imen al iso he m. This con- cen a ion, 2.214 ␮ mol m 2 , was used as a i s es ima e in he model. A e choosing he concen a ion o ca bonyl g oups, we a e able o calcula e he o al pu e-wa e adso p ion iso he m co esponding o he model ca bon. This in ol es calcula ing a se ies o single-po e iso he ms by GCMC, one o each po e size o he PSD, wi h he chosen concen a ion o pola si es. Howe e , he e o in ol ed in calcula ing 57 di e en wa e iso he ms is p ohibi i e, e en a e ully op imizing he calcu- la ion o he long- anged po en ial o cha ged molecules. We ha e he e o e educed he esolu ion o he o iginal PSD o only 15 po es. This numbe was su icien o e ain all he cha ac e is ics o he PSD, while signi ican ly educing he compu a ion cos . The educed PSD is shown in Figu e 3 o compa ison. I is wo h no ing ha he i o he pu e-e hane adso p ion iso he ms using his educed PSD is essen ially indis inguishable om ha ob ained wi h he o iginal PSD Ž. Figu e 2 . This is in ag eemen wi h he conclusions o Da ies Ž. and Sea on 2000 , co obo a ing hei s a emen ha he esolu ion o he PSD is o mino impo ance when adso p- ion p edic ion is conce ned. Some o he single-po e iso he ms calcula ed om GCMC simula ion a e shown in Figu e 4. Figu e 4. GCMC adso p ion iso he ms o pu e wa e a 298 K in sli -shaped po es wi h a concen a- ion o su ace si es o 2.214 ␮ mol m 2 and di e en wid hs. Filled squa es ᎏ0.76 nm; open ci clesᎏ0.84 nm; c osses ᎏ0.89 nm; open ianglesᎏ1.04 nm; illed ci cles ᎏ1.24 nm; open squa esᎏ1.44 nm; illed iangles ᎏ 2.04 nm. The p essu e is educed by he sa u a ion p essu e Ž. a hesame empe a u e 3167Pa. Augus 2003 Vol. 49, No. 8AIChE Jou nal 2063 We can see ha a low p essu e he e is ha dly any wa e adso bed in he po es. A a ce ain cha ac e is ic p essu e Ž. speci ic o each po e wid h , he e is a s eplike inc ease in he amoun adso bed, un il he po es a e comple ely illed. This discon inuous ansi ion is e idence o capilla y conden- sa ion, and was obse ed in all he iso he ms a his empe a- u e, down o he smalles po e wid h. I is wo h no ing ha du ing a GCMC simula ion in a semi-in ini e po e a subc i i- cal condi ions, he ue equilib ium apo ᎐liquid ansi ion is ne e a ained in p ac ice. Ins ead, du ing adso p ion calcu- la ions, he apo like phase emains me as able up o p es- Ž su es well abo e equilib ium liquid me as abili y also occu s .Ž . du ing deso p ion uns Jo ge e al., 2002 . Thus, he e ical s ep in he single-po e iso he ms ep esen s a nonequilib ium w spon aneous condensa ion p ocess. The equilib ium ansi- Ž ion can, howe e , be simula ed using special me hods Jo ge .x and Sea on, 2002b . The apo me as abili y ha occu s in po ous solids du ing eal adso p ion expe imen s is no s ic ly compa able o he co esponding phenomenon in a simula- ion, as he dynamics o he simula ed and eal p ocesses a e Ž. dis inc Sa kiso and Monson, 2000; Schoen e al., 1989 . Ž. Howe e , Sa kiso and Monson 2000 ha e obse ed ha in a complex po e s uc u e, he hys e esis a ising om GCMC simula ions seems o co espond well o he expe imen al hys e esis. Fu he mo e, ecen s udies in well-cha ac e ized po ous ma e ials in ol ing mos ly open unconnec ed po es Ž. Ra iko i ch e al., 2001 show good ag eemen be ween sim- ula ion and expe imen and sugges ha he expe imen al ad- so p ion b anch is close o he poin o spon aneous conden- sa ion in GCMC simula ion han o he equilib ium ansi- ion. In ligh o his, we will assume ha he spon aneous condensa ion p essu e ob ained om GCMC is a good ap- p oxima ion o he alue obse ed expe imen ally. The o al adso p ion iso he m o he model ca bon was ob ained by subs i u ing he single-po e iso he ms and he PSD in o Eq. 1, and calcula ing he amoun adso bed, N. This iso he m is compa ed o he expe imen al da a in Figu e 5. As we can see om his igu e, he simula ed iso he m does no cap u e he S-shape o he expe imen al iso he m. Ins ead, i shows wo dis inc ‘‘s eps,’’ e lec ing he bimodal cha ac e o he PSD. Ano he impo an obse a ion is ha he simula ion esul s lie almos en i ely a lowe p essu es han he expe imen , which sugges s ha he concen a ion o su ace si es is oo high. In a mo e encou aging no e, he alue o simula ed adso p ion a he bulk sa u a ion p essu e closely ma ches he expe imen al alue, which lends u he c edibili y o he ep esen a ion o he s uc u e o he ca - bon. We ha e epea ed he p ocedu e desc ibed ea lie o a Ž 2 . lowe concen a ion o ca bonyl si es 1.476 ␮ mol m . The simula ed iso he m is also shown in Figu e 5. The educ ion in he su ace si e concen a ion shi s he iso he m o highe p essu es, bu does no al e i s shape. In ac , he simula ion esul s will e ain he bimodal cha ac e o any si e concen- a ion. Changing he ype o oxygen-con aining g oup used in he model, o changing he in e ac ion pa ame e s o he ca bonyl g oup, would ha e a simila e ec on he simula ion esul s. Choosing a di e en dis ibu ion o he si es on he po e walls should b ing abou a signi ican change in he low-p essu e egion o he single-po e iso he ms. Howe e , he wa e up ake is mainly a esul o capilla y condensa ion, () Figu e 5. Expe imen al da a o Russell and LeVan 1997 () o pu e wa e a 298 K diamonds and iso he ms om he model ca bon a wo di - e en concen a ions o uni o mly dis ibu ed 2 () ca bonyl si es: 2.214 ␮ mol m solid line and 2 () 1.476 ␮ mol m dashed line . The do ed line connec ing he expe imen al da a poin s is a guide o he eye. which means ha he o al adso p ion iso he m would be i - ually unchanged. Since elaxing ou wo assump ions o uni o m si e ype and egula si e dis ibu ion on he po e walls will no im- p o e he compa ison be ween simula ion and expe imen , we a e le wi h elaxing he hi d and las ᎏ he uni o m si e dis ibu ion on po es o di e en wid h. We ha e now al- lowed he densi y o ca bonyl si es in each o he 15 po es o a y, while ying o ma ch he simula ion o he expe imen al Ž iso he m. This means using a si e densi y dis ibu ion a he . han a single o e all densi y alue o ep esen he chemical he e ogenei y o he ca bon. Such a dis ibu ion was calcu- la ed by i ing he simula ed iso he m o he expe imen al Ž pu e-wa e adso p ion da a. The op imal dis ibu ion which . we will e m dis ibu ion I is shown in Figu e 6, and he co esponding i o he expe imen al pu e-wa e iso he m is shown in Figu e 7. Combining a PSD o he s uc u e o he ca bon wi h a dis ibu ion o ca bonyl si es o desc ibe he chemical he e o- Ž genei y p o ided good i s o bo h pu e-e hane he i shown in Figu e 2 is i ually una ec ed by he p esence o pola . si es and pu e-wa e adso p ion iso he ms. The dis ibu ion o pola si es shows wo peaks, one a ound po es o 1.44 nm in wid h and he o he a ound 0.76 nm. The la e peak is somewha ill-de ined, due o he sho age o expe imen al da a a low p essu e, and i is na u al o ask how a dis ibu- ion ha omi ed his peak al oge he would pe o m. To ind ou , we ha e calcula ed iso he ms o he model ca bon using he wo dis ibu ions o su ace si es shown in Figu e 8. Dis- ibu ion II is based on a no mal dis ibu ion cen e ed a ound Ž 1.49 nm wi h a s anda d de ia ion o 0.3 nm he eby, inco - po a ing he main peak o dis ibu ion I, bu neglec ing he . peak a low po e size , while dis ibu ion III is based on an Ž in e se exponen ial and is, he e o e, mono onically inc eas- . ing, unlike bo h dis ibu ions I and II . When calcula ing hese Augus 2003 Vol. 49, No. 8 AIChE Jou nal2064 Figu e 6. Op imal dis ibu ion o pola si es on BPL ca - bon de e mined by analysis o pu e-wa e ad- () so p ion a 298 K dis ibu ion I . dis ibu ions, he densi y o su ace si es was ounded o o allow o an in ege numbe o ca bonyl g oups in he simula- ion cell. In Figu e 9 we compa e he expe imen al iso he m wi h he simula ed one, ob ained using dis ibu ions II and III. As we can see, dis ibu ion II i s he expe imen al da a e y sa - is ac o ily abo e a ela i e p essu e o 0.4. This is expec ed, since he peak o dis ibu ion II closely esembles he second Ž. peak o dis ibu ion I he bes - i dis ibu ion . This peak lies in he egion o la ge mic opo es, which a e esponsible o he wa e up ake a high ela i e p essu es. Howe e , he low-p essu e adso p ion is se e ely unde es ima ed; his can also be ela ed o he pola si e dis ibu ion. The low-p es- su e up ake is mainly dic a ed by adso p ion in he small mi- c opo es. In dis ibu ion II, hese po es a e almos de oid o pola si es, and, hus, condensa ion occu s much la e . The i o expe imen is much wo se o he case o dis ibu ion Figu e 7. Fi o he pu e-wa e expe imen al da a o () Russell and LeVan 1997 a 298 K on BPL ca bon using he PSD o Figu e 1 and he po- la si e dis ibu ion o Figu e 6. The diamonds ep esen he expe imen al da a and he solid line is he simula ed iso he m. () Figu e 8. Two ‘‘smoo h’’pola -si e dis ibu ions: a dis- () ibu ion II; b dis ibu ion III. III. In ac , i is almos as bad as o he uni o m si e dis ibu- ions shown p e iously. The eason o his is ha , because i is mono onically inc easing, dis ibu ion III is unable o com- () Figu e 9. Expe imen al da a o Russell and LeVan 1997 () o pu e wa e a 298 K diamonds and iso he ms om he model ca bon wi h wo di - e en dis ibu ions o ca bonyl si es: dis ibu- ()( ) ion II solid line and III dashed line . Augus 2003 Vol. 49, No. 8AIChE Jou nal 2065 pensa e o he bimodal PSD, wi h he simula ed iso he m e aining he wo dis inc s eps. F om his analysis, we can conclude ha while a sui ably chosen unimodal dis ibu ion o su ace si es can desc ibe he da a abo e a ela i e p es- su e o a ound 0.4, below his p essu e he peak a small po e size shown in dis ibu ion I mus be aken in o accoun . A p ecise desc ip ion o he si e densi y in hese small po es is di icul in he p esen case, due o he educed amoun o expe imen al da a ob ained a low ela i e p essu e. In sum- ma y, he p edic ed adso p ion o wa e is qui e sensi i e o he p ecise o m o he pola si e dis ibu ion as a unc ion o po e size. We a e now able o compa e he physical p ope ies o ou Ž model ca bon composed o he PSD o Figu e 3 and he . pola si e dis ibu ion o Figu e 6 wi h hose o eal BPL ca bon, de e mined expe imen ally. We ha e calcula ed he Ž. mic opo e olume, he speci ic su ace a ea Sand he su - Ž. ace oxygen concen a ion C o he model ca bon. The O o al olume o mic opo es o he model ca bon is gi en by m Vs w ␦ w3 Ž. Ž. Ý mic nn ns1 In o de o compa e his olume o he expe imen ally de e - mined alues on he same basis, he esul o Eq. 3 was modi- ied o accoun o he olume o he po e ha is inaccessible wŽ. x o he adso ba e molecules see Jo ge 2003 , o de ails . The speci ic su ace a ea o he model is simply gi en by m2 w ␦ w Ž. nn Ss4 Ž. Ýw n ns1 Finally, using he pola si e dis ibu ion and aking in o ac- coun ha each ca bonyl g oup con ains one oxygen a om, he o al concen a ion o oxygen on he model ca bon can be ob ained om m2 w ␦ w Ž. nn Cscw 5 Ž. Ž. Ý O n w n ns1 Ž. whe e cw is he mola concen a ion o ca bonyl si es pe uni su ace a ea in a po e o wid h w. Table 2 shows a compa ison be ween he speci ic p ope - ies o he model wi h hose o eal BPL ca bon. B adley and Ž. Rand 1995 ha e ob ained a consis en alue o he o al mic opo e olume o BPL ca bon by applying he Dubinin- Radushke i ch equa ion o iso he ms o se e al nonpola ad- Table 2. Speci ic P ope ies o BPL Ca bon om Expe imen- al Measu emen s s. Those o he Model Ca bon Model Expe imen U 3 Ž. Vcm g 0.417 0.43 mic UU 2 Ž. Sm g 925 1,088y1,120 † Ž. Cmmol g 1.16 1.14y1.40 O U Ž. Taken om B adley and Rand 1995 . UU Ž. Ž. Taken om Ba on e al. 1998 , MacDonald e al. 2000 , and Mac- Ž. Donald and E ans 2002 . † Ž. Ž. Taken om Ba on e al. 1996, 1997 , MacDonald e al. 2000 , and Ž. MacDonald and E ans 2002 . so ba es. Ba on and cowo ke s ha e pe o med ex ensi e ex- pe imen al cha ac e iza ion s udies on BPL ca bon. They ha e ob ained speci ic su ace a eas by applying he s anda d Ž BET me hod o N adso p ion Ba on e al., 1998; MacDon- 2 . ald e al., 2000; MacDonald and E ans, 2002 . They ha e also pe o med a se ies o empe a u e-p og ammed deso p ion s udies o de e mine he oxygen concen a ion on he su ace Ž o BPL ca bon Ba on e al., 1996; Ba on e al., 1997; Mac- . Donald e al., 2000; MacDonald and E ans, 2002 . In Table 2, he ange o su ace a eas and oxygen concen a ions ob- ained on di e en samples is gi en. The mic opo e olume o he model ca bon shows good ag eemen wi h he expe imen al de e mina ion, bu he spe- ci ic su ace a ea is sligh ly below he BET su ace a ea. Howe e , i is likely ha he BET me hod o e es ima es he eal speci ic a ea o he ca bon, since i does no ake in o accoun he enhancemen o he adso p ion ene gy in small po es. As o he oxygen concen a ion o he model, i is wi hin he ange o independen ly de e mined alues. This compa ison should only be aken as a ough guide, since he oxygen a oms included in he model e ec i ely accoun o all he di e en si es o a eal ca bon ha in e ac wi h wa- e , including o he he e oa oms and basic si es. Ne e he- less, he good ag eemen ob ained sugges s ha he app oach p esen ed he e is physically easonable. The e o e, i can be concluded ha bo h he s uc u e and he chemis y o he model ca bon a e consis en wi h expe imen al da a ob ained on eal BPL ca bon. P edic ion o Mul icomponen Adso p ion We ha e assessed he p edic i e capabili ies o he model by compa ing simula ion esul s wi h expe imen al measu e- men s o a mix u e o wa e and e hane on he same sample o BPL. The adso p ion iso he m o he model was ob ained by pe o ming GCMC simula ions in he 15 sli -shaped po es conside ed p e iously, wi h a concen a ion o ca bonyl si es gi en by he dis ibu ion o Figu e 6. These iso he ms we e hen weigh ed by he PSD o Figu e 3, acco ding o Eq. 2, o yield he o al adso p ion iso he m. Figu e 10 shows a compa ison be ween he mul icompo- nen adso p ion iso he ms ob ained om expe imen and om molecula simula ion. The ag eemen is easonable, e en hough he e is a sligh unde es ima ion o wa e up ake a Ž high pa ial p essu es o e hane co esponding o low mole . ac ions o wa e in he bulk mix u e . This is balanced by an o e es ima ion o he e hane up ake in his egion. Wa e up- ake a e y low mole ac ions o wa e is mainly dic a ed by adso p ion in po es o wid h below 0.84 nm, since e hane is p e e en ially adso bed in he la ge po es. This is p ecisely he ange o po e sizes in which he pola si e dis ibu ion is poo ly de ined. As explained p e iously, his is because o he es ic ed amoun o a ailable expe imen al pu e-wa e da a a low ela i e p essu e. I is, he e o e, no su p ising ha he mul icomponen adso p ion p edic ions in his ange, whe e he pola si e densi y has a pa icula ly s ong e ec on adso p ion, a e poo e . The di e en adso p ion beha io in po es o di e en sizes is illus a ed in Figu e 11, which shows snapsho s ob ained om wa e e hane simula ions in wo po es o di e en wid h and a a high e hane pa ial Augus 2003 Vol. 49, No. 8 AIChE Jou nal2066 Figu e 10. Expe imen al da a o Russell and LeVan () ( ) 1997 o bina y wa e solid diamonds and () e hane open diamonds adso p ion a 298 K ( wi h iso he ms om he model ca bon solid ) line using he PSD o Figu e 3 and he po- la -si e dis ibu ion o Figu e 6. p essu e. As we can see, he smalle po e is illed wi h wa e , while he la ge po e con ains mos ly e hane molecules. Again, i would be in e es ing o see wha he beha io o he model ca bon would be, wi h espec o wa e e hane mix u e adso p ion, i a uni o m ca bonyl si e dis ibu ion was Ž. assumed ins ead o dis ibu ion I . We ha e calcula ed he Figu e 11. Molecula simula ion o a mix u e o e hane () pa ial p essu e s73,416 Pa and wa e () pa ial p essu e s1584 Pa a 298 K ad- so bed on ac i a ed ca bon po es o wid h: () () a 0.8 nm; b 1.24 nm. The magen a sphe es ep esen he L-J cen e s o e hane, he anspa en g ay sphe es ep esen he L-J cen e s o wa e , and he g een sphe es ep esen he ca bonyl si es. The poin cha ges o he models a e ep esen ed by solid sphe es, blue o a nega i e cha ge and ed o posi i e. Figu e 12. Expe imen al da a o Russell and LeVan () ( ) 1997 o bina y wa e solid diamonds and () e hane open diamonds adso p ion a 298 K wi h iso he ms om he model ca bon using he PSD o Figu e 3 and wo di e en con- cen a ions o uni o mly dis ibu ed ca bonyl 2 () si es: 2.214 ␮ mol m solid lines and 1.476 2 () ␮ mol m dashed lines . mul icomponen adso p ion iso he ms a he same condi- ions, using wo di e en alues o he uni o m su ace si e densi y in he po es. The esul s a e compa ed wi h he ex- pe imen in Figu e 12. As we can see, he p edic ions a e now ex emely poo , wi h wa e adso p ion being subs an- ially o e es ima ed and e hane adso p ion unde es ima ed. This is due o he la ge si e densi y in he po es a ound 1 nm, which leads o a high wa e up ake in he en i e p essu e ange. The e ec he e is analogous o he o e es ima ion o he pu e-wa e adso p ion a ela i e p essu es below 0.5, is- ible in Figu e 5. F om his analysis, he impo ance o a com- ple e and de ailed desc ip ion o he chemical he e ogenei y o he ac i a ed ca bon is once again emphasized. We ha e also assessed he pe o mance o wo classical he modynamic me hods o his sys em, namely he IAST Ž. Mye s and P ausni z, 1965 and he Doong-Yang me hod Ž. 1988 . The implemen a ion o hese me hods was as de- sc ibed in he o iginal a icles, excep whe e indica ed below. In he case o he IAST, he in eg a ion o he pu e-compo- nen iso he ms was pe o med nume ically by Simpson’s ule. When ex apola ion o he pu e-e hane iso he m o high p essu e was equi ed, his was done using he To h equa ion Ž. To h, 1962 . The nonideali y o he bulk gas phase was aken in o accoun by calcula ing ugaci ies using he Peng-Robin- son equa ion o s a e. In he case o he Doong-Yang me hod, he Dubinin-As akho equa ion was i ed o he pu e-com- ponen iso he ms. Al hough he me hod allows o di e en limi ing po e olumes o each o he componen s, we ha e Ž used he same alue o bo h wa e and e hane using di e - . en olumes yielded no imp o emen o he p edic ions . The p edic ions using IAST and he Doong-Yang me hod a e p esen ed in Figu e 13. As we can see, bo h me hods ail o co ec ly p edic he bina y wa e adso p ion iso he m, subs an ially o e es ima - ing he up ake. The IAST subs an ially o e p edic s he ad- Augus 2003 Vol. 49, No. 8AIChE Jou nal 2067