Simulation of multicomponent gas separation in a hollow fiber ...

Journal of Membrane Science 173 (2000) 61–71Simulation of multicomponent gas separation in a hollow fiber membraneby orthogonal collocation — hydrogen recovery from refinery gasesS.P. Kaldis, G.C. Kapantaidakis, G.P. Sakellaropoulos ∗Aristotle University of Thessaloniki and Chemical Process Engineering Research Institute, P.O. Box 1520, Thessaloniki 54006, GreeceReceived 26 August 1998; received in revised form 7 February 2000; accepted 8 February 2000AbstractModeling of hollow fiber asymmetric membranes can provide useful guidelines to achieve desirable separations of multicomponentgas mixtures. Especially in cases of high commercial interest, such as hydrogen recovery from refinery streams,the accurate prediction of membrane separation performance is important. In this work, the appropriate model equationsare solved by orthogonal collocation to approximate differential equations, and to solve the resulting system of non-linearalgebraic equations by the Brown method. This technique is applied for the first time in a multicomponent gas separation byhollow fiber membranes and offers minimum computational time and effort, and improved solution stability. The predictionsof the mathematical model are compared with experimental results for the separation and recovery of hydrogen from a typicalgas oil desulfurization unit for various feed pressures, temperatures and stage cuts. In general, there is a very good agreementbetween simulation and experimental results. Further application of the developed mathematical model to various refinerygas streams of interest reveals that high permeate purity (99.95+), and high recovery (0.6–0.9), can be achieved even in aone-stage membrane unit. The reported experimental results and the theoretical analysis demonstrate the potential whichpolymer membrane technology has for the separation of hydrogen from refinery gas streams. © 2000 Elsevier Science B.V.All rights reserved.Keywords: Gas separation; Orthogonal collocation; Simulation; Hydrogen recovery; Refinery1. IntroductionPolymer membrane technology appears as an attractivealternative for the separation of gases inindustrial processes. Compared with conventionalseparation technologies, membranes exhibit simplicityof operation and maintenance, small size, andeffective and reliable performance. These appealingfeatures enabled membrane technology to penetrate awide variety of markets and applications. After almost∗ Corresponding author. Tel.: +30-31-996271;fax: +30-31-996168.E-mail address: sakel@vergina.eng.auth.gr (G.P. Sakellaropoulos)two decades of continuous research development andefforts, polymer membranes have found applicationsinto a variety of industrial processes, such as air dehumidification,nitrogen production, refinery hydrogenrecovery, ammonia purge gas separation, landfill gasupgrading, acid and sour gas treatment, helium separation,and volatile organic compound recovery [1].Most of these applications refer to the separation ofmulticomponent gas streams. As in the case of binarygaseous mixtures, the prediction of the separationperformance and the evaluation of the membrane processare important. Successful membrane modelingand simulation can provide valuable information forthe design, optimization and economics of the overall0376-7388/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved.PII: S0376-7388(00)00353-7

62 S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71separation process with minimum cost and effort. Forthese reasons, the multicomponent gas permeation hasattracted considerable attention. Shindo et al. [2,3]developed approximate calculation methods for multicomponentseparations in five flow patterns:one-sidemixing, complete mixing, cross flow, countercurrentand cocurrent flow. The main assumptions in thesestudies is that concentration gradients and pressuredrop on both sides of the permeator are negligible. Inorder to deal with concentration and pressure gradientsalong the permeator, Chen et al. [4] supplementedthe previous model by introducing an average drivingforce approximation. Sengupta and Sirkar [5,6] studiedexperimentally and theoretically the separation ofa ternary gas mixture in a cocurrent and countercurrentpermeator. Their model offers a good representationof the operation of a hollow fiber permeatoroperation by taking into account the concentrationand pressure gradients. However, the model is limitedto only three-component mixtures and is based on afinite difference algorithm; its extension to a highernumber of components would require considerablecalculation effort. The most detailed representation ofmulticomponent separation in hollow fiber asymmetricmembranes has been presented thus far by Pan [7].However, the solution method of the nonlinear differentialequations is very complicated and requiresinitial estimates of pressure and concentration profilesalong the fiber. Trying to deal with this mathematicalcomplexity, Kovvali et al. [8] presented a computationalmethod which is based on a linear approximationof permeate and feed compositions at certainintervals along the fiber length. With this technique,solution accuracy depends strongly on the number ofintervals across the fiber. The error, compared with theexact solution of Pan, is between 5 and 30% for high(30) and low (10) number of intervals, respectively.To overcome solution difficulties with multicomponentmixtures, Coker et al. [9] used a staged approach(100–1000 stages) to convert the pertinent differentialequations to a set of coupled, non-linear differenceequations. This corresponds to a first order finite differencemethodology, with an iterative solution of thederived tridiagonal matrices. This method was usedto simulate the separation of a refinery stream and ofimpurity containing air streams, without further experimentalvalidation. The sensitivity of the techniqueto initial estimates has not been discussed.In the present work, the orthogonal collocationtechnique was adopted to solve the mathematicalmodel which describes the multicomponent gas separationin hollow fiber asymmetric membrane permeators.This technique has been already used effectivelyin binary systems [10], while its application in multicomponentsystems has been announced [11,12].Orthogonal collocation is a widely applied methodin numerous chemical engineering problems similarto or more complicated than the one examined here.In all applications to boundary value, nonlinear differentialequations, this method offers more accurate,and stable solutions, with less computation time, ascompared to other conventional techniques, such asRunge-Kutta or finite differences [13,14].Among the numerous industrial areas where multicompomentgas separation occurs, the recovery ofhydrogen from refinery gases has been chosen hereas a case study for the application of the developedmathematical model. Hydrogen is a valuable gaswhich is produced and consumed in large quantitiesand in a wide variety of oil refining processes.Over the next decade, the demand for hydrogen inrefineries is expected to increase rapidly due to thegrowing demand for environmentally friendly fuels,and the deterioration of crude oil quality. Excess hydrogencan be either produced on site or recoveredfrom a number of available processes which generatehydrogen as a by-product. The developed mathematicaltechnique for multicomponent gas separations bymembranes has been applied here to evaluate hydrogenseparation from various refinery streams in termsof permeate purity, hydrogen recovery, residue compositionand stage cut. In order to validate the resultsof the mathematical model, experimentation with amulticomponent gaseous mixture has been carriedout on a bench scale separation unit equipped with apolyimide hollow fiber membrane module.2. Mathematical modelingThe developed model is based on Pan’s initialtheoretical formulation [7], modified and augmentedto account for the dependence of gas viscosity andpermeability on temperature and pressure. The modelconsists of a system of differential and algebraicequations with the appropriate boundary conditionsat the fibers entrance and exit.

S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71 63For the mathematical formulation, the followingmain assumptions are made [7]:1. The porous supporting layer of the membrane hasnegligible resistance to the gas flow, and diffusionalong the pore path is insignificant due to high permeateflux.2. Feed gas pressure drop is negligible.3. Permeate flow is governed by the Hagen–Poiseuilleequation.4. The deformation of the fiber under pressure is negligible.5. The membrane permeabilities depend on temperatureby an Arrhenius type equation; pressure andconcentration dependence can also be incorporatedeasily, when needed (e.g. a dual sorption or equivalentmodel [10] can be used for CO 2 ).6. Gas viscosities are assumed to vary linearly withconcentration, and are calculated by Thodos’smethod [15].7. The effect of back-diffusion is negligible, thus, theconcentration of the permeate leaving the membraneskin surface, is identical to that of the bulkpermeate stream outside the porous layer. This assumptionis valid for membranes with ultrathin skinand a highly porous supporting layer. The use ofthis assumption in the present work was dictatedby the complexity which back-diffusion would introducein the multicomponent model. The requirementthat the sum of concentrations at every pointof the membrane must be equal to unity results, as itwill be shown further, in the development of highlynonlinear equations. By assuming that the permeatebulk concentration is equal to the porous supportconcentration (i.e. no back-diffusion present)solution is simplified considerably.The permeation and transport of a multicomponent gasmixture containing (n) gases through a differential elementof the membrane are described by the followingset of differential and algebraic equations:2.1. Residue concentrationsdx idS = 1 ( )u α xii(x i − γy i ) − 1y ifor i = 1, 2,...n− 1 (1)n∑x i = 1 (2)i=12.2. Permeate concentrationsy i =α i x i y o1 − γ + γα i y ofor i = 1, 2,... ,n− 1 (3)n∑y i = 1 (4)i=12.3. Feed side flowratedudS=−(1 − γ)y o(5)2.4. Permeate pressure build-updγ 2dS = A µ (u c − u) (6)µ 1whereu = U U f(7)γ = p P( )( )Q1 PS = πD o zN Fd U f(8)(9)256µ 1 RTU 2 fA =π 2 (Q 1 /d)P 3 D o Di 4N F2 (10)The three simultaneous differential Eqs. (1), (5) and(6), together with Eqs. (2)–(4), describe the permeateand residue mole fractions, the feed side flowrate, andpermeate pressure drop along the fiber. The permeateflow rate can be easily obtained from the materialbalances.The boundary conditions depend strongly on theoperational mode of the fiber. In countercurrent modethe area coordinate starts at the fiber’s exit (z=0, S=0)and the following conditions apply:2.5. B.C. for countercurrent operation(fiber exit) S = 0:x i = x fi u = 1 γ = γ o(fiber closed end) S= S o : dx idS = 0 u = u dγcdS =0(11)

64 S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71In the case of cocurrent operation only slight modificationsare necessary, i.e.:1. u c in Eq. (6) should be replaced by 1, which is thedimensionless feed flow rate.2. The area coordinate should start now at the fiber’sclosed end, and the boundary conditions should be:2.6. B.C. for cocurrent operation(fiber closed end) S= 0: dx idS =0 u = u dγcdS =0 .(fiber exit) S = S o : x i = x fi u = 1 γ = γ o(12)Solution of the multicomponent model is more complicatedthan for the binary mixture case [9] becauseof the requirement that the local sum of all molefractions of the components must be equal to unityat every point along the membrane. This results inthe introduction of the additional nonlinear equations(Eqs. (3) and (4)), with a nonlinearity order (n) equalto the number of the mixture components. In eachiteration, the value of y o is obtained from the solutionof Eqs. (3) and (4) and introduced into Eq. (5).Therefore, the solution algorithm should include aniterative Newton–Raphson (or equivalent) method inorder to deal with these additional equations.To solve this set of differential and non-linear algebraicequations, orthogonal collocation [13] wasemployed along with the Brown method [16], as in thecase of a binary mixture model reported previously[10]. With this technique, the boundary value natureof this problem is retained and the basic non-lineardifferential equations for hollow fiber permeators areexpressed along the fiber. The accuracy of the techniquedepends on the number of collocation points.As discussed previously for a binary mixture separation[9], a small number of collocation points givespoor solution accuracy, while a high number of pointsprolongs the solution convergence time with an insignificanteffect on accuracy. Hence, six points wereused here, because they offer satisfactory accuracywith modest computational time. Results obtainedwith six collocation points for multicomponent separation,using Pan’s data [7], show a difference of lessthan 5% compared to the solution obtained by Pan.The proposed technique does not require initialguessing of the concentration profile inside the fiber,as in Pan’s finite difference solution [7]. It is also notsensitive to starting values of residue concentrationand permeate pressure. In this work, a starting valueof unity was used for the residue concentration andfor the outlet value of the permeate pressure. Theorthogonal collocation technique permits easy evaluationof the performance of hollow fiber asymmetricmembrane permeators, for a variety of gases, membranematerials and operating conditions, with highaccuracy, minimum computational time and effort,thus having improved stability compared to othermethods used.3. ExperimentalIn order to validate the predictions of the developedmathematical model, experimental data with amulticomponent mixture were used, obtained on amembrane bench scale unit, equipped with a modulesupplied by UBE Industries Ltd., Japan. The modulecontains polyimide hollow fibers with 400 m outerand 200 m inner diameter, and its effective surfacearea is 10 cm 2 . The membrane module can withstandpressures and temperatures of 35 bars and 100 ◦ C,respectively, while it can handle gas stream flows ofup to 30 Nl/h. Fig. 1, shows a flow diagram of theexperimental set up. The membrane unit contains allnecessary ancillary instrumentation for pressure, temperature,flow rate measurement and control. The feedpressure is maintained by a back-pressure regulatorinstalled in the residue gas stream. Both permeateand residue flowrates are measured after expansionto atmospheric pressure. The hollow fiber module isplaced into a water bath for controlled temperatureexperiments. The feed gas enters the shell side at highpressure and flows inside the fibers in a counter currentmode to the permeate flow. The composition ofeach gas stream is analyzed by a Perkin Elmer 8500Gas Chromatograph equipped with a Porapak Q anda Molecular Sieve 5A columns.The polyimide membrane module was initiallycharacterized by studying the permeability of H 2 ,CH 4 ,C 2 H 6 ,CO 2 , pure gases. Then, tests were carriedout with a multicomponent gas mixture simulatingthe composition of a gas oil desulfurization refinerystream. Such streams usually contain H 2 , light hydrocarbons(C 1 ,C 2 ) and H 2 S. Taking into account

S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71 65Fig. 1. Flow diagram of the experimental set up.that polyimide membranes exhibit similar CO 2 andH 2 S permeability values and plasticization behavior,H 2 S was replaced in the test mixture by CO 2 mainlyfor safety and handling reasons. The latter exist inconcentrations between 5–10%. Thus the gaseousmixture which was fed to the hollow fiber membraneunit contained 67.5% H 2 , 16.7% CH 4 , 4.3% C 2 H 6and 11.5% CO 2 .4. Results and discussionTable 1, summarises some potential sources for hydrogenrecovery in a typical oil refinery, as well asstream compositions, flowrates, pressures and temperatures[13]. In all cases, the concentration of hydrogenvaries from 30 to 90%, the stream temperaturesare relatively low (30–40 ◦ C) and the pressure variesfrom 12 to 40 bar. Undoubtedly, these values of majorprocess variables are all compatible with polymericmembrane separation technology.The permeation rates of pure H 2 and CH 4 gaseswere measured through the polyimide membrane, at40 ◦ C and 10 bar pressure, and of CO 2 and C 2 H 6 at40 ◦ C and 2 bar pressure. Activation energies were estimatedfrom permeation rate measurements of puregases in the temperature range 20–80 ◦ C, by applyingan Arrhenius type equation. Table 2 gives the calcu-Table 1Refinery gas streams available for hydrogen recoveryNaptha Light Naptha Mild Mild Gas Gas OilHydrotreater (a) Isomerization (b) Hydrocracking (c) Hydrocracking (d) Oil (e) Desulfurization (f)Gas composition (%)H 2 90.0 85.0 32.4 35.7 74.0 79.9CH 4 5.2 5.2 23.1 25.4 13.3 14.4C 2 H 6 1.8 2.0 17.3 19.1 3.6 3.9C 3 ,C 4+ 3.0 7.8 18.0 19.8 1.7 1.8H 2 S – – 9.2 – 7.4 –Flowrate (Nm 3 /h) a 500 400 8000 7264 2500 2315Temperature ( ◦ C) a 35 30 40 40 40 40Pressure (bar g) a 40 19 12 12 25 25a Typical values for small-medium refineries.

66 S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71Table 2Permeabilities and activation energies of pure gases in polyimide membraneComponent Permeability, P a Activation Energy,(cm 3 (STP)/cm 2 /s/cm Hg)E p (kJ/mol)H 2 2.9×10 −4 13.9CO 2 0.93×10 −4 12.3CH 4 0.037×10 −4 9.3C 2 H 6 0.0064×10 −4 7.7a Measured at a pressure of 10 bar for H 2 and CH 4 , and 2 bar for CO 2 and C 2 H 6 ; temperature was 40 ◦ C for all gases.lated permeation rates and activation energies for eachpure gas component. These values were then used insimulating the hollow fiber module performance withthe investigated gas mixtures. All input data used inthis work are experimental, and no trial and errormatch is included. No extrapolation of permeabilitieshas been done using activation energies.The gaseous mixture was fed to the hollow fibermembrane unit at three different temperatures (22,40, 60 ◦ C) and five different pressures (5, 10, 15, 20,25 bar), while the permeate pressure was in all cases1 bar. The feed flow rate varied from 5 to 30 Nl/h.Gaseous concentrations in the permeate and residuestreams were measured as a function of stage cut. Inall cases, flowrates and concentrations were measuredwith an accuracy of ±5 and ±3%, respectively.Initially, the gaseous mixture was fed to the membraneunit at T=40 ◦ C and P=20 bar. Figs. 2 and 3,show the effect of stage cut on the composition ofthe residue and permeate stream, respectively. In allcases, experimental results (symbols) compare verywell with the predictions of the mathematical model(lines), for the same operating conditions. The concentrationof hydrogen in the residue decreases from60 to 40%, while the concentrations of the less permeablecomponents increase slightly with stage cut. Thepermeate contains mainly H 2 and CO 2 , and only smallamounts of hydrocarbons. Their concentration variesonly slightly with stage cut. Hydrogen is enriched inthe permeate stream up to about 93%. Carbon dioxideconcentrations in the permeate vary from 6.5 to 9.5%,while the total concentration of methane and ethaneis lower than 0.5%. Hydrogen recovery of the processcan be estimated as the product of hydrogen concentrationin the permeate and of stage cut; for the experimentalresults of this work it increases linearly withstage cut from 20 to 65%.The effect of pressure on residue and permeatestream compositions, are shown in Figs. 4 and 5, respectively.Results are presented for a constant stageFig. 2. Effect of stage cut on residue composition, T=40 ◦ C,P=20 bar, points: experimental, lines: simulations.Fig. 3. Effect of stage cut on permeate composition, T=40 ◦ C,P=20 bar, points: experimental, lines: simulations.

S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71 67Fig. 4. Effect of feed pressure on residue composition at T=40 ◦ C,stage cut=0.5, points: experimental, lines: simulations.Fig. 6. Effect of feed temperature on residue composition atP=16 bar, stage cut=0.5, points: experimental, lines: simulations.cut (0.5) at 40 ◦ C and pressures varying from 5 to25 bar. Again good agreement between theoretical predictionsand experimental results is obtained. With theexception of a slight increase of hydrogen permeateconcentration, the other gases in both streams remainalmost unaffected by feed pressure. This behaviourcan be attributed to the fact that the permeabilities ofconstant gases, such as H 2 , do not depend on pressure.The latter is not valid for non-ideal gases suchas CO 2 and C x H y which show pressure dependentpermeabilities [1]. Nevertheless, their low feed partialpressures (about 0.06– 0.25 bar) in the experiments ofthis work, result in a negligible change of permeabilitywith feed pressure and, therefore, in no pressuredependence of the exit stream concentration. Residueand permeate concentrations were also examined forthree different temperatures (22, 40, 60 ◦ C), at a feedpressure of 16 bar and a stage cut of 0.5 (Figs. 6 and7). The gases which have the highest activation energiesof permeation [1], i.e. H 2 and CO 2 , show someFig. 5. Effect of feed pressure on permeate composition at T=40 ◦ C,stage cut=0.5, points: experimental, lines: simulations.Fig. 7. Effect of feed temperature on permeate composition atP=16 bar, stage cut=0.5, points: experimental, lines: simulations.

68 S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71Fig. 8. Effect of feed pressure and temperature on feed flow rate,stage cut=0.5, points: experimental, lines: simulations.Fig. 9. Effect of stage cut on H 2 theoretical permeate mole fraction,(a–f: Refinery processes, cf. Table 1).concentration change with temperature, especially inthe residue stream. Such change is small, because ofthe low activation energies (10–25 kJ/mole) associatedwith the permeation process.Although the role of pressure and concentration oncomponent concentrations seems to be minor, theseparameters have a pronounced effect on the modulefeed flow rate. The latter is directly proportional tothe flow capacity of a hollow fiber module with certaindimensions and characteristic properties. Fig. 8shows that both variables, and especially pressure atelevated temperatures, result in significant increaseof the throughput of the polyimide hollow fiber unit.Since flow rate is inversely proportional to the requiredmembrane surface area, at given operating conditions(T, P concentration, etc.), process economics shouldimprove appreciably at elevated pressures and temperatures.Simulation and experimental results here,show that for a five fold pressure increase (5–25 bar)at 22 ◦ C the membrane area requirements decrease bya factor of five. A further area decrease, to about 10%of the original one, should be possible by operating atelevated temperatures (e.g. 60 ◦ C).Following validation, the mathematical model wasused to predict the performance of a membrane unitfor the separation of hydrogen from other refineryoff-gases. The input values of pressure and temperatureto the model were, in each case, those of therespective operating conditions at the exit of the refin-ery process (c.f. Table 1) while the permeate pressurewas 1 bar in all cases. Fig. 9 summarizes the effect ofstage cut on the concentration of hydrogen in the permeatestream for a number of feed gas streams. Whenthe refinery stream contains significant amounts of H 2(85–90%), hydrogen purity in the permeate stream isquite high (99.9%, curves a, b and f) and relatively insensitiveto stage cut. For low hydrogen feed concentrations,as in the case of mild hydrocracking process(curves c, d) permeate purity declines rapidly withstage cut. Hydrogen recovery in the permeate streamcan reach values up to 90% at high stage cuts, even inthe cases where the corresponding H 2 purity is relativelylow (Figs. 9 and 10, respectively, curves e andc). Permeate streams rich in hydrogen can be recycledin the refinery, depending on the refining processused. For example, streams with 90% H 2 can be usedeffectively in hydrotreatment processes for heavy oilupgrading and for oil desulfurization or denitrogenation.Apart from the H 2 -rich permeate, the membraneresidue stream is also important as an additional gasfuel source. Fig. 11 demonstrates the effect of stagecut on hydrogen residue concentration. In all cases,high stage cut values lead to hydrogen depleted residuestreams. The depletion is more intense for low hydrogenconcentration in the feed (curves c, d), wherelow to moderate (0.1–0.3) stage cuts can be achieved.Similarly, the effect of stage cut on the concentration

S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71 69Fig. 10. Effect of stage cut on H 2 theoretical recovery in permeatestream, (a–f: Refinery processes, cf. Table 1).Fig. 12. Effect of stage cut on CH 4 theoretical residue molefraction, (a–f: Refinery processes, cf. Table 1).of the light hydrocarbons (methane, ethane) for variousprocesses is shown in Figs. 12 and 13. In general,the residue mole fraction of the least permeable componentincreases most with stage cut. For instance,ethane concentration in the residue becomes 2.5–3times higher than that in the feed of mild hydrocrackingprocess (curves c, d; Fig. 13).From the above discussion it is evident that high purityhydrogen can be recovered from various refinerygas streams even in a one stage membrane unit and forworking conditions as imposed by the off-gas com-Fig. 13. Effect of stage cut on C 2+ theoretical residue molefraction, (a–f: Refinery processes, cf. Table 1).position, pressure and temperature. With the reportedhere procedure of orthogonal collocation, and its goodagreement with experiment, a variety of operating conditionscan be simulated easily, rapidly and accurately,for further process evaluation and optimization.5. ConclusionsFig. 11. Effect of stage cut on H 2 theoretical residue mole fraction,(a–f: Refinery processes, cf. Table 1).The majority of gas separations encountered inindustrial level refer to multicomponent mixtures.

70 S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71Among them the separation of hydrogen from lighthydrocarbons in oil refineries is an issue of majorimportance. Polymer membranes can be viewed asan attractive alternative of conventional competingmethods. In that direction, process simulation andoptimization by modeling of membrane multicomponentseparations, can be a powerful theoretical tool.In this work, membrane modeling is approached forfirst time by the orthogonal collocation method, toensure solution stability with low computational timeand effort. Model predictions have been validatedby experimental results obtained from the separationof a multicomponent mixture in a polyimide membraneunit and the general comparison is very good.The mathematical model has been applied furtherto various other refinery streams. Results show thateven for a one-stage membrane unit, high hydrogenpermeate purity (up to 99+%) and significant totalrecovery (up to 90%) can be achieved for moderateto high (0.2–0.6) stage cuts. The residue stream, richin hydrocarbons and containing 10% of the initialhydrogen can be further separated in a second stage,or used as a gas fuel.u dimensionless feed side gas flow rateu c dimensionless final residue gas flow rate(u c =U c /U f )x fi feed concentration of component i(mole fraction)x i residue concentration of component i,mole fractiony i permeate concentration of component i,mole fractiony o sum of permeate concentrations definedby Eqs. (3) and (4)z fibers length variableGreek symbolsα i membrane permselectivity (permeabilityof component i to permeability of lesspermeable component Q 1 )γ ratio of permeate pressure to feed pressureγ o ratio of permeate pressure to feed pressureat the permeate outletµ viscosity of gas mixture (Pa s)µ 1 viscosity of the less permeablecomponent (Pa s)6. NomenclatureA dimensionless constant defined by Eq. (10)d effective skin thickness of asymmetricmembrane (m)D i hollow fiber inside diameter (m)D o hollow fiber outside diameter (m)L total fiber length (m)NF total number of active fibers in the hollowfiber moduleP feed-side pressure (Pa)p permeate-side pressure (Pa)Q 1 permeability of the less permeablecomponent (mol/m s Pa)R ideal gas constantS dimensionless membrane area defined byEq. (9)S o dimensionless membrane area, with totalfibers lengthT temperature (K)U feed side gas flow rate (mol/s)U c final residue gas flow rate (mol/s)feed gas flow rate (mol/s)U fAcknowledgementsThe authors wish to thank the European Union andthe General Secretariat of Research and Technologyof Greece for financial support of this work throughprojects ECSC 7220-EC/703, 7220-ED/082 and HY-PER 8991. Mr Kapantaidakis thanks also the HellenicAspropyrgos Refinery, Greece, for a research scholarship.Part of this work was presented at the 10th NorthAmerican Membrane Society Meeting held in Cleveland,Ohio, USA, 16–21 May 1998.References[1] R.D. Noble, S.A. Stern, Membrane Separations Technology,Elsevier, Amsterdam, 1995.[2] Y. Shindo, T. Hakuta, H. Yoshitome, Calculation methodsfor multicomponent gas separation by permeation, SeparationSci. Tech. 20 (1985) 445.[3] Y. Shindo, N. Itoh, K. Haraga, A theoretical analysis ofmulticomponent gas separation by means of a membrane withperfect mixing, Separation Sci. Tech. 24 (1989) 599.

S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71 71[4] H. Chen, G. Jiang, R. Xu, An approximate solution forcountercurrent gas permeation separating multicomponentmixtures, J. Memb. Sci. 95 (1994) 11.[5] A. Sengupta, K.K. Sirkar, Ternary gas mixture separation intwo-membrane permeators, AIChE J. 33 (1987) 529.[6] A. Sengupta, K.K. Sirkar, Ternary gas mixture separationusing two different membranes, J. Memb. Sci. 39 (1988) 61.[7] C.Y. Pan, Gas separation by high-flux, asymmetrichollow-fiber membrane, AIChE J. 32 (1986) 2020.[8] A.S. Kovvali, S. Vemury, W. Admassu, Modeling ofmulticomponent countercurrent gas permeators, Ind. Eng.Chem. Res. 33 (1994) 896.[9] D.T. Coker, B.D. Freeman, G.K. Fleming, Modelingmulticomponent gas separation using hollow-fiber membranecontactors, AIChE J. 44 (1998) 1289.[10] S.P. Kaldis, G.C. Kapantaidakis, T.I. Papadopoulos, G.P.Sakellaropoulos, Simulation of binary gas separationin hollow-fiber asymmetric membranes by orthogonalcollocation, J. Memb. Sci. 142 (1998) 43.[11] G.C. Kapantaidakis, S.P. Kaldis, T.I. Papadopoulos, G.P.Sakellaropoulos, Hydrogen separation in gasification gasstreams by asymmetric polymer hollow fiber membranes, in:T.N. Vesiroglu, C.J. Winter, J.P. Baselt, G. Kreysa (Eds.),Hydrogen Energy Progress XI, IHAs, Frankfurt am Main,Germany, 1996, pp. 855–859.[12] S. Tessendorf, R. Gani, M.L. Michelsen, Design andAnalysis of Membrane-Based Gas Separation Processes,AIChE Annual Meeting, Paper 201i, Los Angeles, CA, 19November 1997.[13] J. Villadsen, M.L. Michelsen, Solution of DifferentialEquation Models by Polynomial Equations, Prentice-Hall,Englewood Cliffs, NJ, 1978.[14] B.A. Finlayson, Nonlinear Analysis in Chemical Engineering,McGraw-Hill, New York, 1980.[15] R.C. Reid, J.M. Prausnitz, T.K. Sherwood, The Properties ofGases and Liquids, McGraw-Hill, New York, 1977.[16] K.M. Brown, Solution of simultaneous non-linear equations,Comm. ACM 10 (1967) 728.