Simulation of multicomponent gas separation in a hollow fiber membrane by orthogonal collocation — hydrogen recovery from refinery gases

Journal of Membrane Science 173 (2000) 61–71

Simulation of multicomponent gas separation in a hollow fiber membrane by orthogonal collocation — hydrogen recovery from refinery gases S.P. Kaldis, G.C. Kapantaidakis, G.P. Sakellaropoulos∗ Aristotle University of Thessaloniki and Chemical Process Engineering Research Institute, P.O. Box 1520, Thessaloniki 54006, Greece Received 26 August 1998; received in revised form 7 February 2000; accepted 8 February 2000

Abstract Modeling of hollow fiber asymmetric membranes can provide useful guidelines to achieve desirable separations of multicomponent gas 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 equations are solved by orthogonal collocation to approximate differential equations, and to solve the resulting system of non-linear algebraic equations by the Brown method. This technique is applied for the first time in a multicomponent gas separation by hollow fiber membranes and offers minimum computational time and effort, and improved solution stability. The predictions of the mathematical model are compared with experimental results for the separation and recovery of hydrogen from a typical gas oil desulfurization unit for various feed pressures, temperatures and stage cuts. In general, there is a very good agreement between simulation and experimental results. Further application of the developed mathematical model to various refinery gas streams of interest reveals that high permeate purity (99.95+), and high recovery (0.6–0.9), can be achieved even in a one-stage membrane unit. The reported experimental results and the theoretical analysis demonstrate the potential which polymer 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; Refinery

1. Introduction Polymer membrane technology appears as an attractive alternative for the separation of gases in industrial processes. Compared with conventional separation technologies, membranes exhibit simplicity of operation and maintenance, small size, and effective and reliable performance. These appealing features enabled membrane technology to penetrate a wide variety of markets and applications. After almost ∗ Corresponding author. Tel.: +30-31-996271; fax: +30-31-996168. E-mail address: [email protected] (G.P. Sakellaropoulos)

two decades of continuous research development and efforts, polymer membranes have found applications into a variety of industrial processes, such as air dehumidification, nitrogen production, refinery hydrogen recovery, ammonia purge gas separation, landfill gas upgrading, acid and sour gas treatment, helium separation, and volatile organic compound recovery [1]. Most of these applications refer to the separation of multicomponent gas streams. As in the case of binary gaseous mixtures, the prediction of the separation performance and the evaluation of the membrane process are important. Successful membrane modeling and simulation can provide valuable information for the design, optimization and economics of the overall

0376-7388/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 6 - 7 3 8 8 ( 0 0 ) 0 0 3 5 3 - 7

62

S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71

separation process with minimum cost and effort. For these reasons, the multicomponent gas permeation has attracted considerable attention. Shindo et al. [2,3] developed approximate calculation methods for multicomponent separations in five flow patterns:one-side mixing, complete mixing, cross flow, countercurrent and cocurrent flow. The main assumptions in these studies is that concentration gradients and pressure drop on both sides of the permeator are negligible. In order to deal with concentration and pressure gradients along the permeator, Chen et al. [4] supplemented the previous model by introducing an average driving force approximation. Sengupta and Sirkar [5,6] studied experimentally and theoretically the separation of a ternary gas mixture in a cocurrent and countercurrent permeator. Their model offers a good representation of the operation of a hollow fiber permeator operation by taking into account the concentration and pressure gradients. However, the model is limited to only three-component mixtures and is based on a finite difference algorithm; its extension to a higher number of components would require considerable calculation effort. The most detailed representation of multicomponent separation in hollow fiber asymmetric membranes has been presented thus far by Pan [7]. However, the solution method of the nonlinear differential equations is very complicated and requires initial estimates of pressure and concentration profiles along the fiber. Trying to deal with this mathematical complexity, Kovvali et al. [8] presented a computational method which is based on a linear approximation of permeate and feed compositions at certain intervals along the fiber length. With this technique, solution accuracy depends strongly on the number of intervals across the fiber. The error, compared with the exact solution of Pan, is between 5 and 30% for high (30) and low (10) number of intervals, respectively. To overcome solution difficulties with multicomponent mixtures, Coker et al. [9] used a staged approach (100–1000 stages) to convert the pertinent differential equations to a set of coupled, non-linear difference equations. This corresponds to a first order finite difference methodology, with an iterative solution of the derived tridiagonal matrices. This method was used to simulate the separation of a refinery stream and of impurity containing air streams, without further experimental validation. The sensitivity of the technique to initial estimates has not been discussed.

In the present work, the orthogonal collocation technique was adopted to solve the mathematical model which describes the multicomponent gas separation in hollow fiber asymmetric membrane permeators. This technique has been already used effectively in binary systems [10], while its application in multicomponent systems has been announced [11,12]. Orthogonal collocation is a widely applied method in numerous chemical engineering problems similar to or more complicated than the one examined here. In all applications to boundary value, nonlinear differential equations, this method offers more accurate, and stable solutions, with less computation time, as compared to other conventional techniques, such as Runge-Kutta or finite differences [13,14]. Among the numerous industrial areas where multicompoment gas separation occurs, the recovery of hydrogen from refinery gases has been chosen here as a case study for the application of the developed mathematical model. Hydrogen is a valuable gas which is produced and consumed in large quantities and in a wide variety of oil refining processes. Over the next decade, the demand for hydrogen in refineries is expected to increase rapidly due to the growing demand for environmentally friendly fuels, and the deterioration of crude oil quality. Excess hydrogen can be either produced on site or recovered from a number of available processes which generate hydrogen as a by-product. The developed mathematical technique for multicomponent gas separations by membranes has been applied here to evaluate hydrogen separation from various refinery streams in terms of permeate purity, hydrogen recovery, residue composition and stage cut. In order to validate the results of the mathematical model, experimentation with a multicomponent gaseous mixture has been carried out on a bench scale separation unit equipped with a polyimide hollow fiber membrane module. 2. Mathematical modeling The developed model is based on Pan’s initial theoretical formulation [7], modified and augmented to account for the dependence of gas viscosity and permeability on temperature and pressure. The model consists of a system of differential and algebraic equations with the appropriate boundary conditions at the fibers entrance and exit.

S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71

For the mathematical formulation, the following main assumptions are made [7]: 1. The porous supporting layer of the membrane has negligible resistance to the gas flow, and diffusion along the pore path is insignificant due to high permeate flux. 2. Feed gas pressure drop is negligible. 3. Permeate flow is governed by the Hagen–Poiseuille equation. 4. The deformation of the fiber under pressure is negligible. 5. The membrane permeabilities depend on temperature by an Arrhenius type equation; pressure and concentration dependence can also be incorporated easily, when needed (e.g. a dual sorption or equivalent model [10] can be used for CO2 ). 6. Gas viscosities are assumed to vary linearly with concentration, and are calculated by Thodos’s method [15]. 7. The effect of back-diffusion is negligible, thus, the concentration of the permeate leaving the membrane skin surface, is identical to that of the bulk permeate stream outside the porous layer. This assumption is valid for membranes with ultrathin skin and a highly porous supporting layer. The use of this assumption in the present work was dictated by the complexity which back-diffusion would introduce in the multicomponent model. The requirement that the sum of concentrations at every point of the membrane must be equal to unity results, as it will be shown further, in the development of highly nonlinear equations. By assuming that the permeate bulk concentration is equal to the porous support concentration (i.e. no back-diffusion present) solution is simplified considerably. The permeation and transport of a multicomponent gas mixture containing (n) gases through a differential element of the membrane are described by the following set of differential and algebraic equations: 2.1. Residue concentrations   xi 1 dxi −1 = αi (xi − γ yi ) dS u yi for i = 1, 2, . . . n − 1 n X

xi = 1

i=1

63

2.2. Permeate concentrations yi =

αi xi yo 1 − γ + γ αi yo

for i = 1, 2, . . . , n − 1

n X yi = 1

(3)

(4)

i=1

2.3. Feed side flowrate −(1 − γ ) du = dS yo

(5)

2.4. Permeate pressure build-up µ dγ 2 = A (uc − u) dS µ1

(6)

where u=

U Uf

(7)

γ =

p P

(8) 

S = π Do zNF A=

Q1 d



P Uf



256µ1 RTU2f π 2 (Q1 /d)P 3 Do Di4 NF2

(9)

(10)

The three simultaneous differential Eqs. (1), (5) and (6), together with Eqs. (2)–(4), describe the permeate and residue mole fractions, the feed side flowrate, and permeate pressure drop along the fiber. The permeate flow rate can be easily obtained from the material balances. The boundary conditions depend strongly on the operational mode of the fiber. In countercurrent mode the area coordinate starts at the fiber’s exit (z=0, S=0) and the following conditions apply: 2.5. B.C. for countercurrent operation

(1) (2)

S = 0 : xi = xfi u = 1 γ = γo dxi dγ (fiber closed end) S = So : = 0 u = uc =0 dS dS (11)

(fiber exit)

64

S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71

In the case of cocurrent operation only slight modifications are necessary, i.e.: 1. uc in Eq. (6) should be replaced by 1, which is the dimensionless feed flow rate. 2. The area coordinate should start now at the fiber’s closed end, and the boundary conditions should be: 2.6. B.C. for cocurrent operation dγ dxi =0 u = uc =0 . dS dS S = So : xi = xfi u = 1 γ = γo (12)

(fiber closed end) S = 0 : (fiber exit)

Solution of the multicomponent model is more complicated than for the binary mixture case [9] because of the requirement that the local sum of all mole fractions of the components must be equal to unity at every point along the membrane. This results in the introduction of the additional nonlinear equations (Eqs. (3) and (4)), with a nonlinearity order (n) equal to the number of the mixture components. In each iteration, the value of yo is obtained from the solution of Eqs. (3) and (4) and introduced into Eq. (5). Therefore, the solution algorithm should include an iterative Newton–Raphson (or equivalent) method in order to deal with these additional equations. To solve this set of differential and non-linear algebraic equations, orthogonal collocation [13] was employed along with the Brown method [16], as in the case of a binary mixture model reported previously [10]. With this technique, the boundary value nature of this problem is retained and the basic non-linear differential equations for hollow fiber permeators are expressed along the fiber. The accuracy of the technique depends on the number of collocation points. As discussed previously for a binary mixture separation [9], a small number of collocation points gives poor solution accuracy, while a high number of points prolongs the solution convergence time with an insignificant effect on accuracy. Hence, six points were used here, because they offer satisfactory accuracy with modest computational time. Results obtained with six collocation points for multicomponent separation, using Pan’s data [7], show a difference of less than 5% compared to the solution obtained by Pan. The proposed technique does not require initial guessing of the concentration profile inside the fiber,

as in Pan’s finite difference solution [7]. It is also not sensitive to starting values of residue concentration and permeate pressure. In this work, a starting value of unity was used for the residue concentration and for the outlet value of the permeate pressure. The orthogonal collocation technique permits easy evaluation of the performance of hollow fiber asymmetric membrane permeators, for a variety of gases, membrane materials and operating conditions, with high accuracy, minimum computational time and effort, thus having improved stability compared to other methods used.

3. Experimental In order to validate the predictions of the developed mathematical model, experimental data with a multicomponent mixture were used, obtained on a membrane bench scale unit, equipped with a module supplied by UBE Industries Ltd., Japan. The module contains polyimide hollow fibers with 400 ␮m outer and 200 ␮m inner diameter, and its effective surface area is 10 cm2 . The membrane module can withstand pressures and temperatures of 35 bars and 100◦ C, respectively, while it can handle gas stream flows of up to 30 Nl/h. Fig. 1, shows a flow diagram of the experimental set up. The membrane unit contains all necessary ancillary instrumentation for pressure, temperature, flow rate measurement and control. The feed pressure is maintained by a back-pressure regulator installed in the residue gas stream. Both permeate and residue flowrates are measured after expansion to atmospheric pressure. The hollow fiber module is placed into a water bath for controlled temperature experiments. The feed gas enters the shell side at high pressure and flows inside the fibers in a counter current mode to the permeate flow. The composition of each gas stream is analyzed by a Perkin Elmer 8500 Gas Chromatograph equipped with a Porapak Q and a Molecular Sieve 5A columns. The polyimide membrane module was initially characterized by studying the permeability of H2 , CH4 , C2 H6 , CO2 , pure gases. Then, tests were carried out with a multicomponent gas mixture simulating the composition of a gas oil desulfurization refinery stream. Such streams usually contain H2 , light hydrocarbons (C1 , C2 ) and H2 S. Taking into account

S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71

65

Fig. 1. Flow diagram of the experimental set up.

that polyimide membranes exhibit similar CO2 and H2 S permeability values and plasticization behavior, H2 S was replaced in the test mixture by CO2 mainly for safety and handling reasons. The latter exist in concentrations between 5–10%. Thus the gaseous mixture which was fed to the hollow fiber membrane unit contained 67.5% H2 , 16.7% CH4 , 4.3% C2 H6 and 11.5% CO2 .

stream compositions, flowrates, pressures and temperatures [13]. In all cases, the concentration of hydrogen varies from 30 to 90%, the stream temperatures are relatively low (30–40◦ C) and the pressure varies from 12 to 40 bar. Undoubtedly, these values of major process variables are all compatible with polymeric membrane separation technology. The permeation rates of pure H2 and CH4 gases were measured through the polyimide membrane, at 40◦ C and 10 bar pressure, and of CO2 and C2 H6 at 40◦ C and 2 bar pressure. Activation energies were estimated from permeation rate measurements of pure gases in the temperature range 20–80◦ C, by applying an Arrhenius type equation. Table 2 gives the calcu-

4. Results and discussion Table 1, summarises some potential sources for hydrogen recovery in a typical oil refinery, as well as Table 1 Refinery gas streams available for hydrogen recovery Naptha Hydrotreater (a) Gas composition (%) H2 CH4 C2 H6 C3 , C4+ H2 S

90.0 5.2 1.8 3.0 –

Light Naptha Isomerization (b) 85.0 5.2 2.0 7.8 –

Mild Hydrocracking (c) 32.4 23.1 17.3 18.0 9.2

Mild Hydrocracking (d) 35.7 25.4 19.1 19.8 –

Gas Oil (e) 74.0 13.3 3.6 1.7 7.4

Gas Oil Desulfurization (f) 79.9 14.4 3.9 1.8 –

Flowrate (Nm3 /h)a

500

400

8000

7264

2500

2315

(◦ C)a

35

30

40

40

40

40

40

19

12

12

25

25

Temperature

Pressure (bar g)a a

Typical values for small-medium refineries.

66

S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71

Table 2 Permeabilities and activation energies of pure gases in polyimide membrane Component

Permeability, Pa (cm3 (STP)/cm2 /s/cm Hg)

Activation Energy, Ep (kJ/mol)

H2 CO2 CH4 C2 H6

2.9×10−4 0.93×10−4 0.037×10−4 0.0064×10−4

13.9 12.3 9.3 7.7

a

Measured at a pressure of 10 bar for H2 and CH4 , and 2 bar for CO2 and C2 H6 ; temperature was 40◦ C for all gases.

lated permeation rates and activation energies for each pure gas component. These values were then used in simulating the hollow fiber module performance with the investigated gas mixtures. All input data used in this work are experimental, and no trial and error match is included. No extrapolation of permeabilities has been done using activation energies. The gaseous mixture was fed to the hollow fiber membrane 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 cases 1 bar. The feed flow rate varied from 5 to 30 Nl/h. Gaseous concentrations in the permeate and residue streams were measured as a function of stage cut. In all cases, flowrates and concentrations were measured with an accuracy of ±5 and ±3%, respectively. Initially, the gaseous mixture was fed to the membrane unit at T=40◦ C and P=20 bar. Figs. 2 and 3, show the effect of stage cut on the composition of the residue and permeate stream, respectively. In all

cases, experimental results (symbols) compare very well with the predictions of the mathematical model (lines), for the same operating conditions. The concentration of hydrogen in the residue decreases from 60 to 40%, while the concentrations of the less permeable components increase slightly with stage cut. The permeate contains mainly H2 and CO2 , and only small amounts of hydrocarbons. Their concentration varies only slightly with stage cut. Hydrogen is enriched in the permeate stream up to about 93%. Carbon dioxide concentrations in the permeate vary from 6.5 to 9.5%, while the total concentration of methane and ethane is lower than 0.5%. Hydrogen recovery of the process can be estimated as the product of hydrogen concentration in the permeate and of stage cut; for the experimental results of this work it increases linearly with stage cut from 20 to 65%. The effect of pressure on residue and permeate stream compositions, are shown in Figs. 4 and 5, respectively. Results are presented for a constant stage

Fig. 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

67

Fig. 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 at P=16 bar, stage cut=0.5, points: experimental, lines: simulations.

cut (0.5) at 40◦ C and pressures varying from 5 to 25 bar. Again good agreement between theoretical predictions and experimental results is obtained. With the exception of a slight increase of hydrogen permeate concentration, the other gases in both streams remain almost unaffected by feed pressure. This behaviour can be attributed to the fact that the permeabilities of constant gases, such as H2 , do not depend on pressure. The latter is not valid for non-ideal gases such as CO2 and Cx Hy which show pressure dependent

permeabilities [1]. Nevertheless, their low feed partial pressures (about 0.06– 0.25 bar) in the experiments of this work, result in a negligible change of permeability with feed pressure and, therefore, in no pressure dependence of the exit stream concentration. Residue and permeate concentrations were also examined for three different temperatures (22, 40, 60◦ C), at a feed pressure of 16 bar and a stage cut of 0.5 (Figs. 6 and 7). The gases which have the highest activation energies of permeation [1], i.e. H2 and CO2 , show some

Fig. 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 at P=16 bar, stage cut=0.5, points: experimental, lines: simulations.

68

S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71

Fig. 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 H2 theoretical permeate mole fraction, (a–f: Refinery processes, cf. Table 1).

concentration change with temperature, especially in the residue stream. Such change is small, because of the low activation energies (10–25 kJ/mole) associated with the permeation process. Although the role of pressure and concentration on component concentrations seems to be minor, these parameters have a pronounced effect on the module feed flow rate. The latter is directly proportional to the flow capacity of a hollow fiber module with certain dimensions and characteristic properties. Fig. 8 shows that both variables, and especially pressure at elevated temperatures, result in significant increase of the throughput of the polyimide hollow fiber unit. Since flow rate is inversely proportional to the required membrane surface area, at given operating conditions (T, P concentration, etc.), process economics should improve 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 by a factor of five. A further area decrease, to about 10% of the original one, should be possible by operating at elevated temperatures (e.g. 60◦ C). Following validation, the mathematical model was used to predict the performance of a membrane unit for the separation of hydrogen from other refinery off-gases. The input values of pressure and temperature to the model were, in each case, those of the respective operating conditions at the exit of the refin-

ery process (c.f. Table 1) while the permeate pressure was 1 bar in all cases. Fig. 9 summarizes the effect of stage cut on the concentration of hydrogen in the permeate stream for a number of feed gas streams. When the refinery stream contains significant amounts of H2 (85–90%), hydrogen purity in the permeate stream is quite high (99.9%, curves a, b and f) and relatively insensitive to stage cut. For low hydrogen feed concentrations, as in the case of mild hydrocracking process (curves c, d) permeate purity declines rapidly with stage cut. Hydrogen recovery in the permeate stream can reach values up to 90% at high stage cuts, even in the cases where the corresponding H2 purity is relatively low (Figs. 9 and 10, respectively, curves e and c). Permeate streams rich in hydrogen can be recycled in the refinery, depending on the refining process used. For example, streams with 90% H2 can be used effectively in hydrotreatment processes for heavy oil upgrading and for oil desulfurization or denitrogenation. Apart from the H2 -rich permeate, the membrane residue stream is also important as an additional gas fuel source. Fig. 11 demonstrates the effect of stage cut on hydrogen residue concentration. In all cases, high stage cut values lead to hydrogen depleted residue streams. The depletion is more intense for low hydrogen concentration in the feed (curves c, d), where low 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

Fig. 10. Effect of stage cut on H2 theoretical recovery in permeate stream, (a–f: Refinery processes, cf. Table 1).

69

Fig. 12. Effect of stage cut on CH4 theoretical residue mole fraction, (a–f: Refinery processes, cf. Table 1).

of the light hydrocarbons (methane, ethane) for various processes is shown in Figs. 12 and 13. In general, the residue mole fraction of the least permeable component increases most with stage cut. For instance, ethane concentration in the residue becomes 2.5–3 times higher than that in the feed of mild hydrocracking process (curves c, d; Fig. 13). From the above discussion it is evident that high purity hydrogen can be recovered from various refinery gas streams even in a one stage membrane unit and for working conditions as imposed by the off-gas com-

Fig. 13. Effect of stage cut on C2+ theoretical residue mole fraction, (a–f: Refinery processes, cf. Table 1).

position, pressure and temperature. With the reported here procedure of orthogonal collocation, and its good agreement with experiment, a variety of operating conditions can be simulated easily, rapidly and accurately, for further process evaluation and optimization.

5. Conclusions Fig. 11. Effect of stage cut on H2 theoretical residue mole fraction, (a–f: Refinery processes, cf. Table 1).

The majority of gas separations encountered in industrial level refer to multicomponent mixtures.

70

S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71

Among them the separation of hydrogen from light hydrocarbons in oil refineries is an issue of major importance. Polymer membranes can be viewed as an attractive alternative of conventional competing methods. In that direction, process simulation and optimization by modeling of membrane multicomponent separations, can be a powerful theoretical tool. In this work, membrane modeling is approached for first time by the orthogonal collocation method, to ensure solution stability with low computational time and effort. Model predictions have been validated by experimental results obtained from the separation of a multicomponent mixture in a polyimide membrane unit and the general comparison is very good. The mathematical model has been applied further to various other refinery streams. Results show that even for a one-stage membrane unit, high hydrogen permeate purity (up to 99+%) and significant total recovery (up to 90%) can be achieved for moderate to high (0.2–0.6) stage cuts. The residue stream, rich in hydrocarbons and containing 10% of the initial hydrogen can be further separated in a second stage, or used as a gas fuel.

u uc xfi xi yi yo z

dimensionless feed side gas flow rate dimensionless final residue gas flow rate (uc =Uc /Uf ) feed concentration of component i (mole fraction) residue concentration of component i, mole fraction permeate concentration of component i, mole fraction sum of permeate concentrations defined by Eqs. (3) and (4) fibers length variable

Greek symbols membrane permselectivity (permeability αi of component i to permeability of less permeable component Q1 ) γ ratio of permeate pressure to feed pressure ratio of permeate pressure to feed pressure γo at the permeate outlet µ viscosity of gas mixture (Pa s) viscosity of the less permeable µ1 component (Pa s)

6. Nomenclature Acknowledgements A d Di Do L NF P p Q1 R S So T U Uc Uf

dimensionless constant defined by Eq. (10) effective skin thickness of asymmetric membrane (m) hollow fiber inside diameter (m) hollow fiber outside diameter (m) total fiber length (m) total number of active fibers in the hollow fiber module feed-side pressure (Pa) permeate-side pressure (Pa) permeability of the less permeable component (mol/m s Pa) ideal gas constant dimensionless membrane area defined by Eq. (9) dimensionless membrane area, with total fibers length temperature (K) feed side gas flow rate (mol/s) final residue gas flow rate (mol/s) feed gas flow rate (mol/s)

The authors wish to thank the European Union and the General Secretariat of Research and Technology of Greece for financial support of this work through projects ECSC 7220-EC/703, 7220-ED/082 and HYPER 8991. Mr Kapantaidakis thanks also the Hellenic Aspropyrgos Refinery, Greece, for a research scholarship. Part of this work was presented at the 10th North American 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 methods for multicomponent gas separation by permeation, Separation Sci. Tech. 20 (1985) 445. [3] Y. Shindo, N. Itoh, K. Haraga, A theoretical analysis of multicomponent gas separation by means of a membrane with perfect mixing, Separation Sci. Tech. 24 (1989) 599.

S.P. Kaldis et al. / Journal of Membrane Science 173 (2000) 61–71 [4] H. Chen, G. Jiang, R. Xu, An approximate solution for countercurrent gas permeation separating multicomponent mixtures, J. Memb. Sci. 95 (1994) 11. [5] A. Sengupta, K.K. Sirkar, Ternary gas mixture separation in two-membrane permeators, AIChE J. 33 (1987) 529. [6] A. Sengupta, K.K. Sirkar, Ternary gas mixture separation using two different membranes, J. Memb. Sci. 39 (1988) 61. [7] C.Y. Pan, Gas separation by high-flux, asymmetric hollow-fiber membrane, AIChE J. 32 (1986) 2020. [8] A.S. Kovvali, S. Vemury, W. Admassu, Modeling of multicomponent countercurrent gas permeators, Ind. Eng. Chem. Res. 33 (1994) 896. [9] D.T. Coker, B.D. Freeman, G.K. Fleming, Modeling multicomponent gas separation using hollow-fiber membrane contactors, AIChE J. 44 (1998) 1289. [10] S.P. Kaldis, G.C. Kapantaidakis, T.I. Papadopoulos, G.P. Sakellaropoulos, Simulation of binary gas separation in hollow-fiber asymmetric membranes by orthogonal collocation, J. Memb. Sci. 142 (1998) 43.

71

[11] G.C. Kapantaidakis, S.P. Kaldis, T.I. Papadopoulos, G.P. Sakellaropoulos, Hydrogen separation in gasification gas streams 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 and Analysis of Membrane-Based Gas Separation Processes, AIChE Annual Meeting, Paper 201i, Los Angeles, CA, 19 November 1997. [13] J. Villadsen, M.L. Michelsen, Solution of Differential Equation 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 of Gases and Liquids, McGraw-Hill, New York, 1977. [16] K.M. Brown, Solution of simultaneous non-linear equations, Comm. ACM 10 (1967) 728.