Description of binary liquid mixtures transport through …uchp.icpf.cas.cz/Membrane-Group/docs/Description of...Journal of Membrane Science 214 (2003) 293–309 Description of binary

Journal of Membrane Science 214 (2003) 293–309

Description of binary liquid mixtures transport through non-porousmembrane by modified Maxwell–Stefan equations

P. Izáka,∗, L. Bartovskáb, K. Friessb, M. Šıpekb, P. Uchytilaa Institute of Chemical Process Fundamentals, Rozvojova 135, 16502 Prague 6, Czech Republicb The Institute of Chemical Technology in Prague, Technicka 5, 16628 Prague 6, Czech Republic

Received 22 March 2002; received in revised form 25 November 2002; accepted 25 November 2002

Abstract

This paper deals with a description of transport parameters during pervaporation of binary liquid mixture through non-porouspolymer membrane. The main task of this work is the application of the model that would be able to describe the transport ofthe penetrant in the membrane. The influence of the C-number of aliphatic alcohols on transport parameters is also under study.

The concentration dependent density of the membrane, vapor sorption isotherms, the dependence of pure componentdiffusion coefficients on their concentration in the membrane and the dependence of the solubility of liquid binary mixture ina membrane was necessary to determine experimentally. All mention dependencies were mathematically described and thenintroduced into modified Maxwell–Stefan equations. The model was used for the liquid binary mixtures (ethanol, propan-1-ol,butan-1-ol with toluene) and the low-density polyethylene membrane at 25C.

Pervaporation data for all the three systems were measured in order to calculate coupled diffusion coefficient and to comparethe model with the experiment. The dependencies of pervaporation fluxes and weight fraction of preferable penetrant inpermeate on concentration of preferable penetrant in the feed are compared with the calculations based on the first Fick’s lawwith constant diffusion coefficient and modified Maxwell–Stefan equations.© 2002 Elsevier Science B.V. All rights reserved.

Keywords:Pervaporation; Sorption; Coupled diffusion; Modified Maxwell–Stefan equation

1. Introduction and theory

The transport of the permeate through a perm-selectivemembrane involves according to the solution-diffusionmodel three successive steps[1]:

1. Upstream partitioning of the feed components be-tween the liquid mixture and the swollen upstreamlayer of the membrane, i.e.selective sorption.

2. Selective diffusionof the penetrants through theunevenly swollen perm-selective membrane.

∗ Corresponding author. Tel.:+420-2-96780268;fax: +420-2-20920661.E-mail address:[email protected] (P. Izak).

3. Permeatedesorption, which takes place on thedownstream surface of the membrane. This multi-stage process is evidently much more complex thana single evaporation step, and it is understandablethat the composition of the permeate may widelydiffer from that of mixed vapor evolved after theestablishment of a free liquid–vapor equilibrium.

This is in fact the main advantage of pervaporationseparation technique[2]. The pervaporation transportof a binary liquid mixture is not additive, comparedto pervaporation transport of components investigatedseparately. The permeability of the operating mem-brane is directly governed by the extent to which it

0376-7388/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.doi:10.1016/S0376-7388(02)00580-X

294 P. Izak et al. / Journal of Membrane Science 214 (2003) 293–309

Nomenclature

ai activity of componentiA membrane area (m2)c molar concentration (mol m−3)D diffusion coefficient (m2 s−1)D0 diffusion coefficient at zero

concentration (m2 s−1)DiM averaged diffusion coefficient of

a pure componenti in the activelayer of the membrane (m2 s−1)

h change in length of the quartzspring (m)

J pervaporation flux (g m−2 h−1)k constant of the spiral quartz

balance (g m−1)l membrane thickness (m)m mass of the swelled foil (kg)mo mass of the dry polymer foil (kg)mt mass uptake at timet (kg)m∞ mass uptake at equilibrium (kg)Mi molar mass of the component

i (g mol−1)Qe equilibrium swelling degree

=(m−mo)/moR universal gas constant

(Pa m3 mol−1 K−1)T absolute temperature (K)w1F weight fraction of the faster

permeant 1 in the feedw2P weight fraction of the slower

permeant 2 in the permeatew′

1 weight fractions of component1 in a membrane

xoi mole fraction of the component

i before sorptionxi mole fraction of the component

i in the equilibrium bulk liquidxi = experimentally determined changexoi − xi of solution concentration caused

by the sorption

Greek symbolsα separation factorδM thickness of the membrane (m)µi chemical potential of componenti

νj local velocities of thecomponents (m s−1)

ξ plasticising exponential constantρM mean density of the swollen

membrane (kg m−3)φ crystallinity of the membraneΦj volume fraction of componentjΩi(xi) preferential sorption as a function

of the bulk liquid composition

swells in the processed liquid mixture. The main diffi-culty encountered when attempting to account for thepervaporation of a binary mixture lies in the fact thatthe transport of each permeant is not independent ofeach other.

In the literature on pervaporation, the term“coupling” is used to designate either the thermody-namical constraint which correlates the flows of thedifferent mobile species, or the mutual drag betweendifferent permeates, originating from short-rangemolecular interactions between them and making itpossible for a given penetrant to move against its ownconcentration gradient in certain regions of the system.

1.1. A generalized Maxwell–Stefan model fortransport processes of mixtures through densemembrane

The generalized transport equations for mixtureshave been derived by Mason and Viehland[3] frombasic principles of statistical mechanics and pre-sented in the form of the generalized Maxwell–Stefanequations. Starting from the concept of irreversiblethermodynamics, the following expression is obtainedfor the relation between driving forces and frictionresistances in a multicomponent mixture[4]:

dµidz

=n∑

j=1

xj (νj − νi)RT

D0ji

(1)

whereµi is the chemical potential of componenti,dµi/dz the gradient of the chemical potential of com-ponent i, xj the mole fractions of the componentsj = 1,2, . . . , n, νj are the local velocities of thecomponents,RT/D0

ji has the meaning of a frictioncoefficient accounting for the frictional effect exertedby componentj on componenti [5]. Eq. (1) is well

P. Izak et al. / Journal of Membrane Science 214 (2003) 293–309 295

known as the Maxwell–Stefan equation. ModifiedMaxwell–Stefan approach in pervaporation processis well described by Heintz and Stephan[4]. Thediffusive mass fluxesJi = ρiνi and weight fractionsof componenti in a membranew′

i instead of mo-lar fractions,xj are introduced and also is assumedDij = Dji . Then the following equations for twopermeating components are obtained fromEq. (1):

−J1 = D1M

(w′

1D2M +D12

D12 + w′1D2M + w′

2D1M

)ρM

dw′1

dz

+D1M

(w′

1D2M

D12 + w′1D2M + w′

2D1M

)ρM

dw′2

dz

(2a)

−J2 = D2M

(w′

2D1M +D12

D12 + w′2D1M + w′

1D2M

)ρM

dw′2

dz

+D2M

(w′

2D1M

D12 + w′2D1M + w′

1D2M

)ρM

dw′1

dz

(2b)

wherew′1 andw′

2 are the weight fractions of compo-nents 1 and 2 in a membrane, respectively,ρM is themean density of the swollen membrane,D2M andD1Mare the diffusion coefficients of components 2 and 1in membraneM, if the Hittorf frame of reference isused (νM = 0).

The equations of coupled diffusion (2a) and (2b)can be rearranged by dividingEq. (2a)by Eq. (2b):

dw′2

dw′1=(J2

J1

) (w′1D2M +D12)−D2Mw

′2

(D2M/D1M)(D12 + w′2D1M)

−(J2/J1)w′1

(3)

This differential equation can be solved for given per-meate pressure, values ofw′

iF (denotes weight fractionof componenti in feed in the membrane) and ratioof J2/J1 providing values ofw′

iP (weight fraction ofcomponenti in permeate in the membrane) and finallywiP (weight fraction of componenti in permeate).

The integration has to be performed numerically bythe following procedure. At a given compositionwiF,w′iF can be calculated according to the UNIQUAC

model[6,7] or can be measured by the sorption of thecomponents in the membrane from the liquid mixture[8–12].

At a given permeate pressure, the unknown weightfraction of componenti in permeatewiP has to be es-timated in the first attempt. Using sorption data fromthe liquid mixture in a membrane, the correspondingequilibrium weight fractions in the membranew′

1F andw′

2F are obtained. Permeate weight fraction of com-ponent 1w1P is defined by:

w1P = J1

J1 + J2= 1

1 + (J2/J1)(4)

wherew1P defines the ratioJ2/J1 to be substitutedinto Eq. (3). Numerical integration ofEq. (3)providesvalues ofw′

1P andw′2P which will differ from the first

estimate and, consequently, from these results newvalues forw1P andJ2/J1 are obtained. SolvingEq. (3)again gives a third set of solution forw1P. The itera-tive procedure is repeated until two following sets ofsolution forw1P differ by less than 0.01%.

Eq. (3)has to be solved in each iteration step. Thiscalculation method is tedious and time consuming.Eqs. (2a) and (2b)can be also solved by an approx-imation which deals with difference equations ratherthan differential equations[4]. Therefore,Eqs. (2a)and (2b)can be rewritten as:

J1 = D1M

(D2Mw

′1 +D12

D12 + w′1D2M + w′

2D1M

)ρM

w′1

δM

+D1M

(D2Mw

′1

D12 + w′1D2M + w′

2D1M

)ρM

w′2

δM

(5a)

J2 = D2M

(D1Mw

′2 +D12

D12 + w′2D1M + w′

1D2M

)ρM

w′2

δM

+D2M

(D1Mw

′2

D12 + w′2D1M + w′

1D2M

)ρM

w′1

δM

(5b)

where

w′i = 1

2(w′iF + w′

iP), w′i = w′

iF − w′iP

DiM is the average diffusion coefficient of a pure com-ponenti in the active layer of the membrane and isdefined by:

DiM =∫ w′

iFw′iPDiM(w

′i )dw′

i

w′iF − w′

iP(6)


ρM is the averaged density of the polymer membraneand is defined by:

ρM =∫ w′

iFw′iPρM(w

′i )dw′

i

w′iF − w′

iP(7)

The concentration polarization is neglected in case ofpervaporation process[1] andD12 has to be adjustedto the experimental partial fluxes.

Eqs. (5a) and (5b)represent nonlinear equations forw′

1P andw′2P at given values ofw′

1F andw′2F. Solv-

ing Eqs. (5a) and (5b)is much more convenient thanEq. (3). In order to obtain consistent results for thepermeate composition (w1P, w2P) at a given permeatepressure, the iteration process has to be repeated untilthew1P values fulfill Eq. (4).

To calculate partial fluxes of the permeate com-ponents through dense membrane by modifiedMaxwell–Stefan equations, we need these input data:concentration dependence of the density of a mem-brane, concentration dependence of the diffusioncoefficients of pure componenti in the membrane,weight fraction of componenti in feed in the mem-branew′

iF, coupled diffusion coefficientDij , and thethickness of the membraneδM .

2. Experimental

Three liquid binary mixtures (ethanol, propan-1-ol,butan-1-ol with toluene) and polyethylene membranewere measured at 25C.

2.1. Chemicals and the membrane

All chemicals are of analytical grade and syntheticZeolite type A4 is used for their dehydration. Thepurity of all chemicals are checked by measuringtheir density (densimeter Paar DMA 45) and refrac-tive indexes at 25C, and their values are comparedwith the literature ones[13]. Binary mixtures usedfor experiments are prepared by weighting.

The membrane used for experiment is a high-pressure, low-density,Polyethylene–Bralen FB2-30,from Slovnaft Bratislava, Slovakia (denoted PE in thetext) in the form of foil (thickness 50m). AntistaticPE foil has a melting index ITT= (1.7–2.3)/600 s.Its density (ρ = 919± 2 kg m−3) is determined with

mercury pycnometer. This density value correspondsto the volume fraction of crystalline phase,φ = 0.455and glass temperatureTg = −120C. Prior to ex-periments, the foil is washed in distilled water, driedin the drying box for 12 h at 60C and then kept invacuum desiccators over magnesium perchlorate.

2.2. Swelling measurements of polymer membranein liquid organic solutions

A device for measuring the extension of the poly-mer membrane swelling in liquid organic solutions[14,15] was developed at the Department of PhysicalChemistry of the Institute of Chemical Technologyin Prague. The device (Fig. 1) is able to measure theextension of the polymer membrane during swellingcontinuously, while the membrane is immersed in liq-uid. The device has the photoelectric sensors that arerolling above the membrane in two vertical directions(important especially for anisotropic materials). Thesensors are driven by stepping motor with movablescrew. The rise of the screw and radian step deter-mines the basic period of the measurement, which inour case is 12.5m.

The measuring cuvette is made of glass and Teflon.The chain pigeonhole prevents the membrane fromwringing during swelling. The transparent membranesare marked with stain steel end-marks, for measuringthe extension of the membranes. The results of exten-sion measurements of anisotropic polyethylene mem-brane in binary liquid mixture butan-1-ol+toluene areshown inFigs. 2 and 3. The experimental error of theextension measurements does not exceed 2%.

It is obvious from swelling measurements that thedensity of polyethylene membrane is concentrationdependent. It is necessary to know this dependencefor successful solving ofmodified Maxwell–StefanEqs. (5a) and (5b)that are used for prediction ofpervaporation characteristics.

The polyethylene membrane extends in three di-mensions during its swelling in the solvents. Thethickness (Z-axis) of the membrane is usually verydifficult to measure because of its micron’s width.The low-density Polyethylene–Bralen FB2-30 haspolyethylene bonds oriented inX-direction. There-fore, we can assume that the flat polyethylene mem-brane will extend in Z-direction similarly as inY-direction during swelling. We can then calculate


Fig. 1. The sectional view of the measuring cuvette and system of sensors.

Fig. 2. An example of a record written by the device for measuring of an extension of the PE membrane.

concentration dependence of density of the polyethy-lene membrane from experimentally known depen-dence of relative extension in the solvent mixtures(Fig. 4).

It is possible to describe the density dependence ofthe PE membrane on concentration of solvent withaccuracy up to 2% by the polynomial equation ofsecond order(ρM = ρ0

M − bρM + aρ2M). The density

of the dry PE membrane (ρ0M ) was experimentally

measured by mercury pycnometer. It was found outfrom another swelling experiment that the mixturesethanol, propan-1-ol with toluene behaved identi-cally like mixture of butan-1-ol+ toluene, which was

Fig. 3. The results of the extension measurement of the PE mem-brane in butan-1-ol+ toluene mixture at 25C.


Fig. 4. Density of the PE membrane vs. mole fraction of toluenein butan-1-ol+ toluene mixture at 25C.

measured, because of much higher (more than 10times) toluene sorption in PE membrane.

2.3. Selective sorption of single components inpolyethylene membrane from binary liquid mixtures

In order to establish the composition of the sorbedphase in the polyethylene membrane, which differsfrom that of the bulk equilibrium solution, two dif-ferent experiments are performed: determining thepreferential sorption and determining the equilibriumswelling degree.

2.3.1. Preferential sorption experimentThe measurements of the preferential sorption from

binary liquid solutions over the whole concentrationrange at constant temperature, yield the isotherm ofconcentration change (composite isotherm):

Ωi(xi) = No

moxi (8)

whereΩi(xi ) denotes the preferential sorption as afunction of the bulk liquid composition,mo the massof the polymer brought in contact withNo moles ofbinary solution,xi = xo

i − xi the experimentallydetermined change of solution concentration causedby the sorption,xo

i the mole fraction of the componenti before sorption,xi the mole fraction of the componenti in the equilibrium bulk liquid.

The composition of the sorbed liquid is obtained bycombining the data about preferential sorption with theinformation about the total amount of liquid, sorbedby the polymer, that are both gained from the liquid

sorption experiments:

xsi = Qexi +ΩiMj

Qe +Ωi(Mj −Mi)(9)

whereQe = (m−mo)/mo is the equilibrium swellingdegree,mo the mass of the dry polymer foil,m themass of the swelled foil,Mi and Mj are the molarmass of the componentsi and j, respectively.

Polyethylene foil is cut to pieces 2 cm×3 cm. About1.3 g of these pieces are weighed into a dry 25 cm3

Erlenmeyer flask and then about 3 g of the solution ofcertain concentration (mole fractionxo

T ) is weightedin. Tightly closed flask is kept at a constant temper-ature of 25C, its contents being often stirred. After24 h, when the sorption equilibrium is established (ex-periments after 2, 3 and 5 days gave the same results)the solution surrounding the foil is analyzed by inter-ferometry measurements[8,9].

Standard Zeiss differential interferometer with0.5 cm double cell is used. The equilibrium solution(mole fractionxT ) is here compared with the initialsolution (mole fractionxo

T ). The difference in thecomposition of these two solutions are determinedby comparing the reading of the interferometer withthe calibration curve, constructed in the vicinity ofeach measuring point. The interferometer is very con-venient for this type of measurements: it is able todetect very small concentration differences and thechange in solution concentration is obtained directlyin a single measurement.

Moreover, being a differential instrument, the in-terferometer has an important advantage: the require-ments of temperature control are not so exacting as inthe case of other methods.

2.3.2. Swelling experimentsThe total sorption is determined by gravimet-

ric method [12]. A piece of PE membrane (about0.2 g) is immersed into a thermostatted Erlenmeyerflask containing about 40 g of binary liquid mixturealcohol+ toluene. After about 24 h1 the swelled foilis weighed.

The amount of sorbed liquid is very small. There-fore, weighting the swelled sample after drying

1 Other experiments (measurement of the dimensions of a mem-brane immersed into the liquid[14]) confirmed that this time issufficient for establishing the equilibrium.


between two sheets of filter paper gave entirely unre-liable results.

For that reason a special procedure must be used.The foil is transferred into a tightly closed weightingbottle inlaid by filter paper. The foil must be fixed in aposition, in which only its edges touched the paper (thedimensions of the weighting bottle must correspondto the dimensions of the foil sample). After 3 h, whenthe paper is drained off the excess liquid from the foiland the equilibrium in the bottle is established, the fullbottle is weighted. The foil is then taken out quicklyand the bottle with the wet paper is weighted again.The difference represents the weight of the swollenmembrane. This procedure is found to be reproduciblewithin 2%.

2.4. Diffusion coefficients of pure components

The method that is very useful for describingand understanding the transport of small moleculesthrough polymers isfree volume theory.

The free volume approach also gives satisfactoryresults for interacting systems. Deviation may causeclustering of the penetrant molecules, i.e. the com-ponent diffuses not as a single molecule, but in itsdimeric or trimetric form.

Fig. 5. A scheme of the sorption apparatus: A, glass tube with McBain spiral sorption balance; B, pressure gauge with mercury reservoir;C, vapor reservoir; D1 and D2, vessels with liquid sample; E, cooling trap; F, vacuum pump; 1–6, valves with Teflon core (3 and 6 areair-inlet valves).

The clustering ability may be described by theZimm–Lundberg theory[16]; this analysis tool hasbeen applied to various models and experimental data[17,18]. Its use showed that Flory–Huggins theorysystematically underestimates solvent clustering, evenfor non-polar mixtures[18].

It is striking, however, that although cluster forma-tion seems to be proved by experimental or theoreticalstudies, no simple equilibrium theory has yet beenproposed that would take such a particularity intoaccount.

2.4.1. Vapor sorption measurementThe measured sample solution is introduced by

syringe into the vessel D2 (seeFig. 5). The vessel isimmersed into the liquid medium of the temperaturecorresponding to the selected relative pressure of themeasured sample. Mercury level in the left branchesof U-manometer B is set up in a way that the vaporcannot penetrate into the right branch. The sorp-tion kinetic measurements start with opening valve2 (start stop-watch). The changes in length of thequartz spring balance in specific time interval can beread with cathetometer. The relative pressure of themeasured sample is determined from the differenceof mercury level in U-manometer B.


2.4.2. Determination of sorption isotherms anddiffusion coefficients

Relative pressureprel, i of solvent vapori is givenby prel,i = pi/p

0i , wherepi is the partial pressure of

the solventi in the sorption apparatus andp0i is the

saturated vapor pressure of the solvent.Vapor sorption experiments also allow the de-

termination of the solvent diffusion coefficients inthe membrane material. Assuming thermodynamicequilibrium between liquid and vapor phase of thesolvent at any time, the change of the vapor pressureas a function of time is monitored by U-manometerbuilt inside the apparatus. Due to diffusion processes,the response of the balance will be a time depen-dent increase of the membrane weight, which can bemeasured as a function of time.

Diffusion coefficients can be determined from thisfunction, provided that the following assumptions aremade[19]:

1. According to the basic ideas of the solution-diffu-sion model, thermodynamic equilibrium exists atany time between the actual vapor pressure andthe solvent concentration in the membrane at thephase boundary between the vapor and the swollenmembrane.

Fig. 6. Scheme of the pervaporation apparatus: 1, membrane; 2, support layer; 3 and 4, cell from stainless; 5, electric heater; 6, water cooler;7, thermometer; 8 and 10, slots for thermometer and stirrer; 9, stirrer; 11 and 13, glass cold trap; 12, channel; 14, spiral quartz balance; 15,absorbent (active carbon); 16, glass coat; 17, oil rotating pump; 18, cold trap for oil vapor; 19, vacuometer; 20–24, valves with Teflon core.

2. During sorption step the diffusion coefficient of thesolvent in the membrane is a constant.

3. The thickness of the membrane sample is negligiblecompared to its length and width.

The time dependent mass uptake of the solvent va-por in the membrane sample can be described quanti-tatively by solving second Fick’s law of diffusion. Wecan solve second Fick’s law[20,21]for polymer mem-branes of finite dimensions bounded by planesz = l/2andz = −l/2 (wherel is the membrane thickness):

∂ci

∂t= Di

(∂2ci

∂z2

)(10)

under the initial and boundary conditions

−l/2 < z < l/2, t = 0, c = 0z = l/2, t ≥ 0, c = c1z = −l/2, t ≥ 0, c = c1

leads to infinite series given byEq. (11):

mt

m∞= 1 − 8

π2

n=∞∑n=0

[1

(2n+ 1)2

exp

(−Di

(2n+ 1)2π2t

l2

)](11)


Fig. 7. Preferential sorption isotherms of all measured systems inPE membrane at 25C.

wheremt is the mass uptake at timet, m∞ the massuptake at equilibrium.

If the ratio mt /m∞ > 0.5 with sufficient accuracywe can applyEq. (12) [6]:

ln

(1 − mt

m∞

)= −π2Di

l2t + ln

(8

π2

)(12)

The corresponding value of diffusion coefficientDi

can be obtained from the slope of the dependenceln(1 −mt /m∞) versus time.

Fig. 9. Sorption isotherms of solvents in the polyethylene membrane at 25C.

Fig. 8. Total sorption isotherms of all measured systems in PEmembrane at 25C.

2.5. Pervaporation technique

Pervaporation device is designed to achieve rapiddesorption of the permeates at the downstream surfaceof the membrane in order to reduce their activitiesto a virtually zero value in the subjacent layer of theoperating barrier. One may assume that this require-ment is fulfilled, if the perm-selective film is kept un-der very low pressure, since the transported permeateis then immediately removed by continuously pump-ing system. At the Department of Physical Chemistryof the Institute of Chemical Technology in Prague thepervaporation apparatus was constructed for sensitive


Fig. 10. Experimental data of the diffusion coefficientDi (Hittorf frame of reference) as a function of the weight fraction of pure componenti in the PE membrane at 25C.

measurement of very small pervaporation fluxes bysorption method[22]. The comparison of the classicdirect gravimetric method (whereby the permeate iscondensed and then weighted) and the sorption method(where the permeate is absorbed into active carbonand the mass is determined from the change in lengthof the quartz spring balance, on which active carbon issuspended) is described and their results are comparedon the system butan-1-ol+ toluene+ low-density PEmembrane at 25C in our previous study[23].

It was found in comparison with the direct method,that the sorption method gives more accurate results[23]. The measured values are pervaporation fluxand concentration of the permeants in feed and inpermeate.

2.5.1. Measurement of pervaporation fluxThe pervaporation apparatus (Fig. 6) is evacuated

and it takes about 1 h. Valves 24, 22, 21 are opened andvalves 23, 20 are closed. The supply of cooling wateris switched on at this point (refer to scheme part 6).The electric stirrer and the electric heater, parts 9 and 5on the scheme, are also switched on. The thermometerbegins reading at a specific temperature and remainsconstant throughout the entire experiment. A constantvalue of 25C is chosen in each case. The permeateis condensed in the cold trap (11), until steady-state isreached within the apparatus.

Steady-state is achieved when the value of the per-vaporation flux from the initial run is identical with thevalue obtained in any of the successive flux runs. The

values above are measured to two decimal places. Themeasurement of the pervaporation flux is carried outin duplicate. It is necessary to regenerate active carbonand this is carried out by heating it to 200C for 10 minin a reheating furnace. The degree of regeneration ofthe active carbon is measured using a spiral quartzbalance. The quartz spring length is measured usinga cathetometer and noted. This value should remainconstant upon attaching the regenerated carbon to thequartz spring in the vacuum. The constant value isused as the reference point for the proceeding measure-ments. Any deviation from the reference point priorto permeate introduction indicates incomplete carbonregeneration. Valve 24 is closed and valves 21–23 are

Fig. 11. Dependence of separation factor on the weight fractionof toluene in feed at 25C.


opened, which allows the pervaporation flux to enterinto the evacuated apparatus.

The permeate then travels along the apparatus tothe cold trap 13, where it will be condensed. The timerequired for this to occur will depend on the feed con-centration of the initial binary system. A 0.05 ml ofthe permeate is required for accurate measurement ofthe pervaporation flux. This amount is condensed inthe cold trap, until the calibration mark indicates soon the trap. At this point, valves 23, 22, 21 are closedand valve 24 is opened to maintain steady-state. Theliquid nitrogen source surrounding the cold trap 13 isremoved, which allowed the cold trap to be warmed bysurrounding atmosphere and the condensed permeateto be returned to the vapor state. Valve 22 is openedand the permeate proceeds into the evacuated appara-tus where it is absorbed by the active carbon, whichis suspended on the quartz spring. A cathetometer isused to read the change in length of the quartz spring.When the reading has a constant value (i.e. to seconddecimal place), absorption is completed.

The previous procedure is repeated up to the point,where liquid permeate is obtained. At this point a sam-ple is extracted for analysis. Valve 22 is removed andthe sample is extracted using a syringe. A 0.5 ml iscollected for further analysis. Measurement of weightfraction of toluene in the retentate could then proceed.

Pervaporation flux is given by the relationship:

J = m

At(13)

Here

m = k h (14)

Fig. 12. Separation diagram for the system ethan-1-ol+ toluene+ PE membrane at 25C.

wherem is weight of the permeate passing throughthe membrane areaA during the timet, k is constantof the spiral quartz balance,h the change in length ofthe quartz spring.

Partial pervaporation flux is given by the relations:

J1 = Jw1P (15)

J2 = J − J1 (16)

whereJ1 is partial pervaporation flux of component1, J2 the partial pervaporation flux of component 2,J the total pervaporation flux, andw1P is the weightfraction of component 1 in the permeate.

Analysis of the retentate and the permeate is per-formed by the refractometric method. The dependen-cies of refractive index on weight fraction of toluenein the binary mixtures are obtained at 25C. Weightfraction values are substituted intoEqs. (15) and (17)for the determination of partial pervaporation flux andseparation factor of the membrane:

α12 = w1P/w2P

w1F/w2F(17)

The selectivity of separation of measured binarymixtures are represented by the dependence of weightfraction of toluene in permeate, and separation factoron weight fraction of toluene in feed (Figs. 11, 12, 15and 18).

3. Discussion of the results

There is a basic difference between a ternary sys-tem (a binary liquid mixture and a membrane) and


Fig. 13. Partial fluxes of toluene for the system ethan-1-ol+ toluene+ PE membrane at 25C.

Fig. 14. Partial fluxes of ethan-1-ol for the system ethan-1-ol+ toluene+ PE membrane at 25C.

Fig. 15. Separation diagram for the system propan-1-ol+ toluene+ PE membrane at 25C.


Fig. 16. Partial fluxes of toluene for the system propan-1-ol+ toluene+ PE membrane at 25C.

Fig. 17. Partial fluxes of propan-1-ol for the system propan-1-ol+ toluene+ PE membrane at 25C.

Fig. 18. Separation diagram for the system butan-1-ol+ toluene+ PE membrane at 25C.


Fig. 19. Partial fluxes of toluene for the system butan-1-ol+ toluene+ PE membrane at 25C.

a binary system (a membrane and a pure liquid). Inthe former case, not only the amount of liquid insidethe membrane (overall sorption) is an important para-meter. Even more important is the composition of theliquid mixture inside the membrane. This latter value,the preferential sorption, represents the sorption selec-tivity. It can be seen fromFig. 7 that the experimentalpreferential sorption isotherms obtained for the liquidbinary systems of aliphatic alcohol+ toluene+PE areof common U-shape. The position of the maximumshifts slightly to the higher concentration of toluenewith increasing C-number in the alcohol molecule andhas lower values. The total sorption isotherms increasewith mole fraction of toluene more linearly with in-creasing C-number in the alcohol molecule (Fig. 8).The sorption equilibrium (full diamonds) is comparedto the vapor–liquid equilibrium[24] (broken lines) and

Fig. 20. Partial fluxes of butan-1-ol for the system butan-1-ol+ toluene+ PE membrane at 25C.

to the pervaporation experiments (asterisks) (Figs. 12,15 and 18).

The sorption apparatus (Fig. 5) is constructed ina way that the pressure in the apparatus (where themembrane is placed) rises immediately after thevalve is opened. This procedure provides data ofthe diffusion coefficients related to Hittorf frameof reference as function of relative pressure of thesolvent.

All measured sorption isotherms increased expo-nentially with increasing relative pressure of the sol-vent (Fig. 9). It is obvious that the sorption of toluenein PE membrane is one order of magnitude higher thanthe sorption of aliphatic alcohols. It was found fromour sorption measurements an influence of C-numberof aliphatic alcohols on their sorption in PE mem-brane. It was observed with increasing C-number an


exponential increase of mass fraction of the solvent inmembrane (seeFig. 9).

The diffusion coefficients of aliphatic alcohols in-crease with increase of their concentration in the mem-brane (Fig. 10). Values of diffusion coefficientDi

obtained for toluene pass the maximum in their de-pendence on concentration in the membrane. A sim-ilar behavior was observed by Lützow in the sys-tem toluene+ linear low-density polyethylene (φ =0.45, ρ70C = 0.9097) at 70C [25]. The decreaseof the diffusion coefficient of toluene with its relativepressure is probably caused by forming of clusters athigher relative pressures[26].

No unique relationship exists for the concentrationdependence of the diffusion coefficient. The empiricalexponential relationship[27] is often used:

Di = D0i exp(ξΦi) (18)

whereD0i is the diffusion coefficient of penetranti at

zero concentration,Φi the volume fraction of pene-trant i, ξ the plasticising constant indicating the plas-ticising action of the penetrant on segmental motion.

However, in our systems, the dependence ofDi

onw′i , which is needed for calculation of pervapora-

tion fluxesEqs. (5a) and (5b), is more complicated.Therefore, it is not possible to use this exponential re-lationship (18). The best fit for our experimental datais represented by polynomial equations of the thirdorder.2 All diffusions coefficients are evaluated fromsorption kinetics with relative error approximately 5%.

Vacuum pervaporation of the binary liquid mix-tures ethanol, propan-1-ol, butan-1-ol with toluenethrough a PE membrane is described by the experi-mental dependencies of the weight fraction of toluenein the permeate (Figs. 12, 15 and 18), partial fluxes(Figs. 13, 14, 16, 19 and 20), and separation fac-tor (Fig. 11) on the weight fraction of toluene inthe feed. The azeotropic mixtures were separated bypervaporation in the whole concentration range.

As the toluene concentration in the feed becamehigh, the total pervaporation flux through the polyethy-lene membrane increased exponentially. The experi-mental error in pervaporation flux does not exceed 5%of the total value.

2 Personal discussion with Marcel Mulder, Ph.D. on the 41stMicro-Symposium of Polymer Membranes, Prague, Czech Repub-lic, 16–19 July 2001.

Table 1Calculated diffusion coupling coefficients

Measured system Diffusion coupling coefficientD12 × 1013 (m2 s−1)

Ethan-1-ol+ toluene 5.63Propan-1-ol+ toluene 1.07Butan-1-ol+ toluene 2.71

We did not find any influence of the C-number inthe alcohol molecule on pervaporation flux throughthe membrane. The curves of separation factor weredecreasing with increasing concentration of toluene inthe feed. The separation process in all measured binarymixtures is most successful in cases where there is asmall amount of toluene in alcohol.

The mixture with the highest separation factor(Fig. 11), i.e. for which the separation is most effi-cient, is ethanol with toluene.

4. Conclusions

Pervaporation of liquid binary mixtures (ethanol,propan-1-ol, butan-1-ol with toluene) through thepolyethylene membrane Bralen is a complex process,which is influenced by many factors.

The most simple calculation procedure for descrip-tion of pervaporation process is based on the firstFick’s law with constant diffusion coefficientDi

3

(Figs. 12–20). It turns out that the strong non-idealsolubility and diffusivity behavior of measured liquidbinary mixture components in polyethylene mem-brane and diffusive coupling effects play the dom-inant roles in the pervaporation process. Therefore,the results based on the first Fick’s law with constantdiffusion coefficient are completely unsatisfactory.

An essential improvement of the model descriptioncan be achieved by introduction of the concentrationdependence of all diffusion coefficients, density ofthe membrane and diffusion coupling using modifiedMaxwell–Stefan equations with one additional ad-justable parameterDij (seeTable 1). These diffusioncoupling coefficients are calculated using Newton

3 Di is an arithmetic average from all experimentally determinedvalues over the whole concentration range.


iterative method by correlation of experimentallyobtained and calculated partial pervaporation fluxes.

The separation diagrams (Figs. 12, 15 and 18) rep-resent the selectivity of the pervaporation process.The velocity of the separation is represented by plot ofpartial fluxes of the components versus feed concen-tration (Figs. 13, 14, 16, 17, 19 and 20). These figurescompare model calculations based on the first Fick’slaw with constant diffusion coefficient and modifiedMaxwell–Stefan model with experimental data.

The modified Maxwell–Stefan model takes into ac-count the non-ideal multicomponent solubility effect,non-ideal diffusivity of all permeating components,concentration dependent density of the membrane anddiffusion coupling. Therefore, this model is able todescribe semi-quantitatively the performance of a per-vaporation membrane for given separation problems.

Acknowledgements

The financial support of the Grant Agency ofthe Czech Republic (Grants No. 23400008, and104010945) are gratefully acknowledged. Specialthanks to MSc. Jaroslava Machkova for collabora-tion in pervaporation experiments and to Mgr. IvanaDvorakova for support and big help.

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