Journal of Chromatography A

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1 Journal of Chromatography A, 7 ) 7 Contents lsts avalable at ScenceDrect Journal of Chromatography A journal homepage: A thermodynamcally consstent explct compettve adsorpton sotherm model based on second-order sngle component behavour Mlca Ilć a,, Detrch Flocerz a, Andreas Sedel-Morgenstern a,b, a Max Planc Insttute for Dynamcs of Complex Techncal Systems, Sandtorstrasse, D-96 Magdeburg, Germany b Otto von Guerce Unversty, Char of Chemcal Process Engneerng, Unverstätsplatz, D-96 Magdeburg, Germany artcle nfo abstract Artcle hstory: Receved 9 November 9 Receved n revsed form 8 January Accepted February Avalable onlne February Keywords: Adsorpton sotherms Bnary mxture Ideal adsorbed soluton theory Langmur sotherm Quadratc sotherm A compettve adsorpton sotherm model s derved for bnary mxtures of components characterzed by sngle component sotherms whch are second-order truncatons of hgher order equlbrum models suggested by mult-layer theory and statstcal thermodynamcs. The compettve sotherms are determned usng the deal adsorbed soluton IAS) theory whch, n case of complex sngle component sotherms, does not generate explct expressons to calculated equlbrum loadngs and causes tme consumng teratons n smulatons of adsorpton processes. The explct model derved n ths wor s based on an analyss of the roots of a cubc polynomal resultng from the set of IAS equatons. The suggested thermodynamcally consstent and wdely applcable compettve sotherm model can be recommended as a flexble tool for effcent smulatons of fxed-bed adsorber dynamcs. Elsever B.V. All rghts reserved.. Introducton The compettve adsorpton sotherms are the most essental nformaton whch should be provded to desgn and optmze separaton processes based on selectve adsorpton [,]. Stll the most relable way to obtan sotherms s to perform experments []. Unfortunately, the expermental determnaton of compettve sotherms s dffcult and tme consumng. Therefore, compettve sotherms are typcally predcted usng thermodynamc models explotng nowledge about the sngle component sotherms, whch can be easer measured. One of the most successful and wdely appled models s the deal adsorbed soluton IAS) theory [4]. Although real systems can devate from dealty [5], the thermodynamcally consstent compettve sotherms predcted by the IAS theory are very useful n smulatng, desgnng and optmzng adsorpton processes. Explct forms of compettve sotherm models based on the IAS theory are avalable only for a few relatvely smple sngle component sotherm models e.g. the Henry and Langmur equatons [4]). For more complex sngle component models explct solutons of the IAS theory are not avalable and, therefore, the compet- Correspondng author at: Otto von Guerce Unversty, Char of Chemcal Process Engneerng, Unverstätsplatz, D-96 Magdeburg, Germany. Tel.: ; fax: E-mal address: sedel-morgenstern@mp-magdeburg.mpg.de A. Sedel-Morgenstern). Present address: Damler AG, D-7 Krchhem unter Tec/Nabern, Germany. tve sotherms have to be calculated numercally usng teratve procedures. In order to extend the lbrary of the avalable thermodynamcally consstent compettve sotherm models, the man tas of ths theoretcal study s to fnd an explct soluton for the case of second-order truncatons of sngle component sotherms quadratc sotherms ), whch are suggested by varous theoretcal concepts and whch are more flexble and, thus, often more accurate. These second-order models are e.g. capable to descrbe frequently observed nflecton ponts n the sotherm courses. In order to valdate the derved explct compettve sotherm model, predcted equlbrum loadngs wll be compared wth results of correspondng numercal solutons. Addtonally, the obtaned sotherm model wll be used n numercal smulatons of fxed-bed dynamcs, agan n comparson wth smulatons based on solvng teratvely the set of IAS equatons.. Sngle component adsorpton sotherms The most often used nonlnear sngle component adsorpton sotherm model s the Langmur model [,,6] whch correlates the flud phase concentratons C of a component and the correspondng equlbrum loadng of the sold phase, Q, as follows: Q C ) = Q sat, b C + b C ) -967/$ see front matter Elsever B.V. All rghts reserved. do:.6/j.chroma...6

2 M. Ilć et al. / J. Chromatogr. A 7 ) 7 Nomenclature A defned n Eq. a) [g/l] b, parameter of the adsorpton sotherm model, Eq. 4) [l/g] b, parameter of the adsorpton sotherm model, Eq. 4) [l /g ] B defned n Eq. b) [g/l] C defned n Eq. c) [g/l] C concentraton of component n the lqud phase [g/l] C sngle solute concentraton [g/l] C,fc hypothetcal sngle solute concentraton [g/l] C nj njecton concentraton [g/l] C mx lqud phase concentraton n mxture [g/l] D defned n Eq. d) [g/l] D app apparent axal dsperson coeffcent [m /s] f * defned n Eq. 5) [g/l] F cubc polynomal, defned n Eq. ) [g/l] l column length [m] N number of components present n a mxture N p number of theoretcal plates, Eq. 8) Q concentraton n the sold phase [g/l] Q sat, saturaton capacty parameter of the adsorpton sotherm model) [g/l] Q tot total loadng [g/l] Q mx loadng n mxture [g/l] R unversal gas constant [J/g K] S ads adsorbent surface [m ] t tme coordnate [s] T temperature [K] u lnear velocty [m/s] V ads adsorbent volume [m ] V nj njecton volume [l] x space coordnate [m] z molar fracton z * defned n Eq. 5) Gree symbols defned n Eq. 4) ˇ defned n Eq. 4), [l/g] defned n Eq. 4), [l/g] ı defned n Eq. 5) runnng varable n Eq. 8), correspondng to sngle solute concentraton [g/l] ε total porosty RTV ads /S ads, Eq. 8) [J m/g] * defned n Eq. 5) defned n Eq. ) spreadng pressure, Eq. 8) [J/m ] In ths equaton Q sat, s the saturaton maxmum) capacty of the sold phase and b s a postve temperature dependent parameter quantfyng the adsorpton energy. Ths classcal model assumes that each adsorpton ste s energetcally equvalent and s avalable only for one molecule. Molecular nteractons between the molecules adsorbed are neglected. Eq. ) accounts only for loadngs up to the formaton of so-called mono-layers. Expermentally observed sotherms are often characterzed by more complex sotherm shapes, whch can not be descrbed accurately by Eq. ) [7]. A typcal feature s the occurrence of nflecton ponts n the sotherm courses, whch can be caused, e.g. by mult-layer formaton, the presence of energetcally heterogeneous adsorbent surfaces, lateral nteractons between adsorbed molecules and capllary condensaton phenomena. In order to descrbe more complex sotherms shapes, varous models have been suggested. A very general and flexble model was suggested by Hll [8]. It results from classcal concepts of statstcal thermodynamcs appled to adsorpton equlbra. The model can be expressed n the followng specal form of a Mth order Padé approxmant [9]: [ ] d Q C ) = Q C sat, C )/dc C ) wth C ) = + M b,j C )j ) j The b,j are postve parameters quantfyng vares types of nteracton energes. For M = Eq. ) reduces to Eq. ). For the second-order approxmaton M = ) holds: C ) = + b,c + b, C ) Ths leads to the followng well nown quadratc sotherm model [,,]. Q O C ) = Q sat, C b + b C ) + b C + b C Eq. 4) s qute flexble and capable of descrbng nflecton ponts n the sotherm courses. The same quadratc sotherm model gven by Eq. 4) can be also derved explotng the extended BET-theory [] or usng a lattce model []. If the lqud phase concentraton approaches nfnty, Eq. 4) predcts that the sold phase concentraton reaches a saturaton value Q sat, = Q sat,.. Dervaton of compettve adsorpton sotherm model for a bnary system A well-nown and wdely used extenson of the Langmur model Eq. )) to the case of compettve adsorpton of N components n a mxture s [,]: Q mx C mx,...,c mx N ) = Q sat, b C mx N + = b C mx 4) =,N 5) Ths equaton predcts the loadngs of a component n the mxture, Q mx, as a functon of the flud phase concentratons C mx.its thermodynamcally consstent only f the saturaton capactes are the same for all components [],.e.: Q sat = Q sat, =,N 6) The most successful and often appled approach to derve compettve sotherms from sngle component data s the deal adsorbed soluton IAS) theory. Ths theory was ntally developed by Myers and Prausntz [4] for gas adsorpton and extended to treat adsorpton from dlute lqud solutons by Rade and Prausntz [4]. An mportant advantage of the IAS theory s that t provdes for any set of sngle component sotherms thermodynamcally consstent compettve sotherms. Usng the framewor of IAS theory, for a N component mxture, characterzed by the N ndependent flud phase concentratons C mx,...,c mx, the followng N + equatons have to be N solved smultaneously to determne the N + dependent varables C,fc,...,C,fc, Q mx,...,q mx N N, Q tot mx [4,4]: C,fc ) = + C,fc ) =,N 7a) +

3 4 M. Ilć et al. / J. Chromatogr. A 7 ) 7 N mx C = 7b) C,fc = N Q mx = 7c) Q = C,fc ) Q mx Q mx tot = Cmx C,fc =,N Because of Eq. 7d), Eq. 7c) can be replaced by: N C mx = Qtot mx C,fc Q = C,fc ) 7d) 7e) An mportant tool of the theory s the Gbbs adsorpton sotherm whch quantfes the two-dmensonal spreadng pressure ) present n the adsorbed phase Eq. 7a)). The C,fc are hypothetcal fctve) concentratons of a component n a hypothetcal sngle component system, whch generate the same spreadng pressure as the mxture does. The Q mx are the equlbrum loadngs of component to be determned and Qtot mx s the total equlbrum loadng. The spreadng pressures of a sngle component for the fctve concentraton C,fc can be determned from the sngle component Gbbs adsorpton equaton as [4,4]: C,fc ) = C,fc Q ) d =,N 8) Hereby substtutes RTV ads /S ads and Q stands for the sngle component sotherm model. Provded Eq. 6) holds, the set of Eqs. 7) and 8) can be solved analytcally for the Langmur sngle component sotherms Eq. )). The result s the expresson gven by Eq. 5). Solvng Eqs. 7) and 8) s more complex for almost all other sngle component sotherm models. If Eq. 6) does not hold, already n case of applyng Eq. ) numercal methods must be used. Below we present an explct compettve sotherm model for the two components of a bnary mxture usng the IAS theory and Eq. 4) for the sngle solute sotherms. In addton t s assumed that Eq. 6) holds. In ths case the spreadng pressure expresson, Eq. 8), provdes: C,fc ) = Q sat ln[ + b, C,fc Usng Eqs. 7a) and 9) leads to: C mx + b, z [ C mx + b, z + b, C,fc ) ] =, 9) ] = + b, C mx z [ ] C mx + b, ) z Hereby, followng a suggeston made recently [5], adsorbed phase molar fractons z have been ntroduced for the hypothetcal concentratons C,fc see Eq. 7d)): z = Cmx C,fc =, wth z = z ) Eq. ) can be reformulated as F = for the followng cubc polynomal F wth x, y) C mx,c mx ) R + and z z : Fx, y, z) = Ax, y)z + Bx, y)z + Cx, y)z + Dx, y) ) wth Ax, y) = x + y> Bx, y) = xˇx ) y + y) a) b) Cx, y) = x ˇx) Dx, y) = ˇx and the parameters c) d) = b, >, ˇ = b,, = b, b, b, b, for b, >, b, =, 4) The zero of F n [,] solvng Eq. ) provdes the z we are loong for. In Appendx A, followng [6], s gven the dervaton of ths unque zero n [,] desgnated as z II. The derved analytcal expresson s: z = z II z II x, y) = z II C mx,c mx ) = z x, y) + ıx, y) cos x, y) + 4 ) [, ] 5, A5b) wth z x, y) = Bx, y) [Ax, y)] ı ıx, y) = [B / x, y) Ax, y)cx, y)] [Ax, y)] x, y) = ) arccos f x, y) Ax, y)ı x, y) f x, y) = Fx, y, z x, y)) [, ] Usng ths expresson for z C mx,c mx ) = z II and Eq. 7e), the equlbrum loadngs Q mx can be determned as follows: [ ] Q mx C mx,c mx ) = Qtot mx z = z Q C,fc ) + z Q C,fc ) =, 6a) wth C,fc = C mx, C,fc = C mx 6b) z z ) Ths soluton allows drect calculaton of compettve IAS sotherms nvolvng two second-order sngle component sotherm models Q C ) and Q C ). It s also vald for bnary mxtures nvolvng a combnaton of frst and second-order truncatons Langmur and quadratc sotherms) and can be effcently appled n smulatons of fxed-bed dynamcs, where typcally a large amount of equlbrum calculatons has to be performed. 4. Applcaton of the derved compettve sotherm model 4.. Equlbrum loadngs For llustraton and also to chec the derved analytcal compettve adsorpton sotherm equaton, the followng parameters were used n the sngle solute sotherm model Eq. 4)): Q sat, = Q sat, = 5 g/l, b, = l/g, b, =l /g, b, = l/g, b, =l /g. In addton to applyng Eqs. ) 6), numercal solutons of Eqs. 7) and 8) were calculated usng three methods mplemented n Matlab [7],.e. the Gauss Newton, the Levenberg Marquardt and Trust-Regon- Dogleg methods. A general survey regardng these well establshed nonlnear least-square methods s gven n [8]. Detals concernng the Levenberg Marquardt method can be found n [9,], whle z

4 M. Ilć et al. / J. Chromatogr. A 7 ) 7 5 As n the frst example no dfferences were found between the compettve sotherms calculated analytcally and numercally. The presence of nflecton ponts s nown to be the reason for unusual pea shapes f effcent chromatographc columns are appled [4]. To nvestgate ths phenomenon the sotherms of ths second example were appled wthn a model capable to predct column dynamcs. 4.. Applcaton of the derved compettve sotherm model n fxed-beds models Fg.. Comparson of the derved analytcal soluton Eqs. ) 6)) and results of numercal calculatons solvng Eqs. 7) and 8). The sngle component sotherms sold lnes) correspond to Eq. 4) and the parameters gven n the text. Blaccomponent, grey-component, squares and crcles derved analytcal soluton, dotted lnes numercal solutons : mxtures). an overvew regardng the Trust-Regon-Dogleg method s gven n []. In Fg. are presented the sngle solute sotherms of the two components as sold lnes. The exstence of the nflecton ponts n the courses of these two sotherms s hardly vsble n ths presentaton. To llustrate ths feature better plots of the sotherm dervatves dq/dc vs. C or of Q/C vs. C are more sutable. For the sae of brevty they are not gven here. Selected compettve sotherms are gven n Fg. for : mxtures n the lqud phase,.e. C mx = C mx. The expected reducton of loadngs n the mxture case can be clearly seen. Of mportance for the purpose of ths study s the fact that the squares and crcles, correspondng to the analytcal soluton, concde perfectly wth the numercal results shown as dotted lnes. Regardng the latter there were no sgnfcant dfferences found for the three methods appled. However, the Gauss Newton method was not able to provde for some pars of C mx and C mx the soluton wth the default parameters of the solver. The derved analytcal soluton can be also reduced to descrbe the case, where one of the sotherms s only of the frst-order type. The combnaton of frst-order Langmur) and second-order quadratc, BET) types of sngle component sotherms was analysed n detal n []. Results of usng Eqs. ) 6) were compared agan wth correspondng numercal solutons for such a stuaton. The sngle component sotherm parameters used are related to the adsorpton of Tröger s base enantomers on mcrocrystallne cellulose tracetate wth ethanol as solvent. In [] ths system was studed theoretcally and expermentally. It was found that the sotherm of the )-enantomer, whch elutes frst, s of the Langmur type Eq. )), whle the sotherm of the +)- enantomer has an nflecton pont and can be modelled wth the quadratc sotherm model Eq. 4)). The sngle solute sotherm model parameters used here were taen from the parameters ftted n [] to expermental data just the saturaton capactes were slghtly modfed to guarantee consstency): Q sat, = Q sat = g/l, b, =.57 l/g, b, =l /g, Q sat, = Q sat / = g/l, b, =.948 l/g, b, =.7 l /g. The obtaned sotherms are presented n Fg.. There s a strong reducton n the equlbrum loadngs of the )- enantomers n : mxtures, whch s due to the relatvely large separaton factor and also the pronounced nflecton pont n the course of the sngle component sotherm of the +)-enantomer. The equlbrum-dspersve model [] as one of the most frequently appled model was used to predct the development of concentraton profles n fxed-beds. Ths model assumes sothermal condtons and that the lqud and sold phases are permanently and at all local postons n equlbrum. Radal gradents are neglected and t s further assumed that all band broadenng contrbutons due to, e.g. axal dsperson and fnte rates of mass transfer processes) can be lumped nto an apparent axal dsperson coeffcent. The correspondng mass balance for a component n a N component mxture s []: C mx í t + ε ε Q mxc mx,...,c mx N ) t + u Cmx x C mx = D app, x,=,,...,n 7) where the C mx are agan the concentratons n the flud phase, the Q mx the correspondng equlbrum concentratons n the sold phase, t the tme coordnate and x s the space coordnate. Further, ε s the total column porosty, u the ntersttal flud phase velocty and D app the apparent axal dsperson coeffcent. For effcent columns small D app, values) and smlar dsperson behavour of all components D app = D app, ) can be assumed []: N p = ul 8) D app In ths equaton N p stand for the number of theoretcal plates, whch s typcally appled to evaluate the effcency of chromatographc columns, and l for the column length. Fg.. Comparson of the derved analytcal soluton Eqs. ) 6)) and results of numercal calculatons solvng Eqs. 7) and 8). Sngle component sotherms sold lnes) follow Eq. ) component, )-enantomer, blac) and Eq. 4) component, +)-enantomer, grey), respectvely, for the parameters gven n the text. Squares and crcles derved analytcal soluton, dotted lnes numercal solutons : mxtures).

5 6 M. Ilć et al. / J. Chromatogr. A 7 ) 7 Sutable boundary condtons vald for fxed-bed adsorpton have been suggested by Danwerts [5]. Analytcal solutons of Eq. 7) can be derved only for lnear adsorpton sotherms. Otherwse numercal methods must be appled. In ths study a fnte dfference method wth forward-n-space and bacward-n-tme approxmatons [6] was used, combned wth both the analytcal soluton for the local equlbrum Eqs. ) 6)) and the numercal soluton of Eqs. 7) and 8) The technque s based on solvng effcently a reduced form of Eq. 6) settng the rght-hand sde to zero) on a coarse grd. The followng expresson for the lqud phase concentraton of a component at the space poston n + and the tme poston j represents the scheme: C j,n+ =Cj,n x u t [Cj,n Cj,n + ε ε Q j,n j Q )] =,N 9),n The x and t are the space and tme ncrements, respectvely, chosen n a way that numercal and physcal dspersons match [6]. The Courant Fredrchs Lewy convergence condton provdes a lmt for the Courant number a cou n order to assure stablty of the scheme [7]: a cou = u t x ) Although the descrbed numercal method as also other alternatves) s relatvely fast usng modern computers, there s stll a sgnfcant amount of computaton tme needed f the equlbrum loadngs of the compounds nvolved are not gven va explct expressons. To demonstrate the potental of the derved analytcal solutons of the IAS theory, eluton profles were calculated for the sotherms dentfed for the Tröger s base enantomers dssolved n ethanol n contact wth mcrocrystallne cellulose tracetate. As n [] the followng parameters were used: a column length of l = 5 cm, an nternal column dameter of 4.6 mm, a total column porosty of ε =.66, number of theoretcal plates N P = 8, volumetrc flow-rate of.5 ml/mn, njecton concentratons C nj = Cnj =.5 g/l. In four calculatons the njecton volume was ncreased stepwse V nj = 6, 8, and ml). Fg. a d llustrate eluton profles for both enantomers calculated wth a cou =. as well as the correspondng total concentraton profles. Only very small n the fgures hardly to detect) dscrepances were found between the profles generated usng the analytcal and numercal solutons of the IAS sotherm model. However, the tmes requred for these smulatons dffered sgnfcantly. On an ordnary PC the calculaton of an eluton profle usng Eqs. ) 6) too less than s, whle the numercal calculaton requred sgnfcantly more tme several mnutes up to hours, dependng on the method used and the startng values provded). The eluton profles shown n Fg. for ths seres of volume overloadng reveal the strong mpact of the nflecton pont n the sotherm of the longer retaned second component the +)- enantomer of Tröger s base) on the band shapes. The courses of the desorpton branches of the second component clearly reveal the dspersed Langmuran) behavour for concentratons above the sotherm nflecton pont followed by a shoc for lower concentratons ant-langmuran behavour). The competton between the two components leads to effluent concentratons of the frst elutng component the )-enantomer) clearly above the njecton concentraton. Fg.. Eluton profles predcted wth the equlbrum-dspersve model usng the sotherm parameters underlyng Fg. and the addtonal parameters gven n the text. Vared parameter was the njecton volume: a) 6 ml, b) 8 ml, c) ml and d) ml. Sold lne total concentraton; dashed and dotted lnes partly hdden) components and. Hardly to dstngush: blac lne numercal soluton of equlbrum model, grey lne analytcal equlbrum model Eqs. ) 6)).

6 M. Ilć et al. / J. Chromatogr. A 7 ) 7 7 It should be fnally mentoned that the course of predcted eluton profles wll be of course also nfluenced by the fxed-bed model used and the netc parameters ncluded. For models more detaled then the equlbrum-dspersve model, the applcaton of explct sotherm expressons can be computatonally even more benefcal than for the case presented here for llustraton. 5. Conclusons An explct adsorpton sotherm model was derved for bnary mxtures usng the IAS theory. The model s applcable for components characterzed by second-order sngle component sotherms. Eqs. ) 6) allow for rapd and thermodynamcally consstent calculaton of the concentratons n the sold phase wthout requrng numercal methods. The soluton derved can be mplemented easly n any fxed-bed model reducng there sgnfcantly the computaton tme. Due to ts flexblty t can be very helpful n desgnng and optmzng varous types of adsorpton processes. Acnowledgements The support of Deutsche Forschungsgemenschaft SFB 578) and Fonds der Chemschen Industre s gratefully acnowledged. Appendx A. Here wll be gven more detals on the dervaton of the obtaned general sotherm model followng [6]. We show that the polynomal Fx,y,z) Eq. )) possesses for x, y) R + three real smple zeros z I x, y) <z II x, y) < z III x, y) If we defne the pont of symmetry of the cubc F as: z x, y),f x, y)) wth z x, y) = Bx, y) [Ax, y)], f x, y) = Fx, y, z x, y)) and ntroduce the postve ı ıx, y):= [B / x, y) Ax, y)cx, y)] [Ax, y)] and the angle x, y):= ) arccos f x, y) Ax, y)ı x, y) then the zeros of F are gven by z I z I x, y) = z x, y) + ıx, y) cos [, ] x, y) + ), A) A) A) A4) A5a) z II z II x, y) = z x, y) + ıx, y) cos x, y) + 4 ) [, ], A5b) z III z III x, y) = z x, y) + ıx, y) cos x, y)) A5c) wth z I < f and only f ˇ > and z III > f and only f >. The zero z I vanshes dentcally for ˇ =, the zero z III s dentcally equal to for =. Proof. Note frst that A s always postve. One has n R + Fx, y, ) =+ˇx, Fx, y, ) = y, A6) d A7a) Fx, y, ) = Cx, y) = x ˇx) dz and d Fx, y, ) = Ax, y) + Bx, y) + Cx, y) = y y) dz A7b) In case ˇ >, > Eq. A6) mples Eq. A) wth strct nequaltes. In case ˇ =, > Eq. A6) and d/dz)fx, y, ) = x > Eq. A7a)) lead to Eq. A) wth z I = <z II < z III. In case ˇ >, > Eq. A6) and d/dz)fx, y, ) = y> Eq. A7b)) yeld Eq. A) wth z I < <z II < = z III. So we arrve n case ˇ =, = ateq. A) wth z I = <z II < = z III. These three smple zeros mply the exstence of a strct local maxmum taen at z = z ı and of a strct local mnmum taen at z = z + ı) wth values: Fx, y, z ıx, y)) = f x, y) + hx, y) >, Fx, y, z + ıx, y)) = f x, y) hx, y) < A8a) A8b) for hx, y):=ax, y)ı x, y) mplyng f x, y)/hx, y) +. Thereby, * s well-defned by Eq. A4) wth: [ ] cos ),, cos + ) [, ] and cos + 4 ) [, ] Thus, Eqs. A5a) A5c) follow from the substtuton of z = z x, y) + ıx, y) cos nto Fx,y,z) = snce Fx, y, z x, y)) + ıx, y) cos) = hx, y)[4 cos cos ] + f x, y) = hx, y) cos) + f x, y)! = A9) can be rewrtten as cos) = f x, y) hx, y) References A) [] G. Guochon, A. Felnger, D.S. Shraz, A. Katt, Fundamentals of Preparatve and Nonlnear Chromatography, Academc Press, New Yor, 6. [] H. Schmdt-Traub, Preparatve Chromatography of Fne Chemcals and Pharmaceutcal Agents, WILEY-VCH Verlag, Wenhem, 5. [] A. Sedel-Morgenstern, J. Chromatogr. A 7 4) 55. [4] A.L. Myers, J.M. Prausntz, AIChE J. 965). [5] A.L. Myers, AIChE J. 9 98) 69. [6] I. Langmur, J. Am. Chem. Soc. 8 96). [7] C.H. Gles, D. Smth, A. Hutson, J. Collod Interface Sc ) 755. [8] T.L. Hll, An Introducton to Statstcal Thermodynamcs, Addson-Welsley, Readng, 96. [9] G.A. Baer Jr., Graves-Morrs, Padé Approxmants, Cambrdge Unversty Press, Cambrdge, 996. [] D.M. Ruthven, Prncples of Adsorpton and Adsorpton Processes, Wley, New Yor, 984. [] F. Grtt, G. Guochon, J. Chromatogr. A 8 4) 97. [] M. Moreau, P. Valentn, C. Vdal-Majdar, B.C. Ln, G. Guochon, J. Collod Interface Sc. 4 99) 7. [] D.B. Brougthon, Ind. Eng. Chem ) 56. [4] C.J. Rade, J.M. Prausntz, AIChE J. 8 97) 76. [5] A. Tarafder, M. Morbdell, M. Mazzott, st Internatonal Symposum on Preparatve and Process Chromatography PREP 8, San Jose, USA, , Poster P--T. [6] R.W.D. Ncalls, Math. Gazette 77 99) 54. [7] D. Redfern, C. Campbell, The Matlab 5 Handboo, Sprnger, New Yor, 998. [8] J.E.J. Denns, n: D. Jacobs Ed.), State of the Art n Numercal Analyss, Academc Press, New Yor, 977. [9] K. Levenberg, Quart. Appl. Math. 944) 64. [] D. Marquardt, Soc. Ind. Appl. Math., J. Appl. Math. 96) 4. [] J.J. Moré, n: G.A. Watson Ed.), Numercal Analyss, Sprnger Verlag, New Yor, 977. [] F. Grtt, G. Guochon, J. Chromatogr. A 8 ). [] A. Sedel-Morgenstern, G. Guochon, Chem. Eng. Sc ) 787. [4] W. Zhang, Y. Shan, A. Sedel-Morgenstern, J. Chromatogr. A 7 6) 6. [5] P.V. Danwerts, Chem. Eng. Sc. 95). [6] P. Rouchon, M. Schonauer, P. Valentn, G. Guochon, Sep. Sc. Technol. 987) 79. [7] R. Courant, K. Fredrchs, H. Lewy, Math. Annalen 98).