Journal of Membrane Sczence, 73 (1992) 103-118 103 Elsevler Science Publishers B V , Amsterdam
Review
Diffusional phenomena in membrane separation processes*
G B van den Berg” and C A Smoldersb “Unzlever Research, Process Engzneerzng Sectzon, Colworth House, Sharnbrook, Bedford, MK44 1LQ (UK) bDept of Chemzcal Engzneerzng, Unzverszty of Twente, P 0 Box 217, 7500 AE Enschede (Netherlands) (Received June 25, 1991, accepted m revised form April 6,1992)
Abstract
Nowadays membrane filtration processes are used mdustrlally as an alternative to conventional sep- aration methods Membrane separation methods can be divided into classes according to their separation characterlstlcs (1) separation by sieving action, (11) separation due to a difference m affinity and dlf- fuslvlty, (111) separation due to a difference m charge of molecules, (m) carrier-fachtated transport, and (v) the process of (time-) controlled release by diffusion In all these cases dlffuslon processes play an important role m the transport mechanism of the solutes Various mechanisms have been distinguished to describe the transport m membranes transport through bulk material (dense membranes), Knudsen diffusion m narrow pores, viscous flow m wide pores or surface diffusion along pore walls In practice, the transport can be a result of more than only one of these mechanisms For all of these mechamsms models have been derived The characterlstlcs of a membrane, e g its crystalhmty or its charge, can also have major consequences for the rate of diffusion m the membrane, and hence for the flux obtained Apart from the diffusion transport processes zn membranes mentloned above, other important diffusion processes are related to membrane processes, viz diffusion zn the boundary layer near the membrane (concentration polanzatlon phenomena) and diffusion durzng membrane formatzon The degree of con- centration polarlzatlon IS related to the magnitude of the mass transfer coefficient which, m turn, 1s influenced by the diffusion coefficient The effect of concentration polarlzatlon can be rather different for the various membrane processes The phase mverslon membrane formation mechanism 1s determined to a large extent by the kinetic aspects during membrane formation, which are diffusion of solvent and of non-solvent and the kinetics of the phase separation itself
Keywords diffusion, theory, review
Introduction used mdustrlally as an alternative to conven- tional separation methods such as dlstlllatlon, Membrane filtration processes nowadays are centrlfugatlon and extraction. Since the first asymmetric reverse osmosis membranes be- Correspondence to G B van den Berg, Umlever Research, came avallable m the early slxtles membrane Process Engineering Sectlon, Colworth House, Sharn- technology has developed enormously This is brook, Bedford MK44 lLQ, UK expressed m the vast amount of research that *Paper presented at the Int Symp “Progress m Membrane Science and Technology”, Enschede, Netherlands, June 25- has gone mto developmg the right membrane 28,199l type and module for different kinds of separa-
0376-7388/92/$05 00 0 1992 Elsevler Science Publishers B V All rights reserved 104 G B van den Berg and CA Smolders/J Membrane Scr 73 (1992) 103-118 tlon processes, developing new processes and rier membranes Carrier-facilitated transport the best possible circumstances for separation can also be determined by diffusion only, which These efforts have resulted m the present day depends for example on the kmd of membrane commerclahzation of ultrafiltration (UF), ml- applied Furthermore, the technique of con- crofiltratlon (MF), reverse osmosis or hyper- trolled release of drugs from (biodegradable) filtration (RO), gas separation (GS), perva- reservoirs is a process that is mainly deter- poratlon (PV), (kidney-) dialysis and mined by diffusion electrodlalysls (ED ) Applications of already commercialized A membrane separation process can be gen- techniques include erally represented as m Fig 1 - Food industry: whey processmg (RO and UF ) , The various membrane separation methods concentration of milk for cheese production can be divided mto three classes according to (UF), clarification and/or sterilization of their separation characteristics various fluids such as wine, vmegar and apple (1) UF and MF use the size of the solutes to Juice (MF ) and whey desalting (ED ) separate particles by sieving action, with a - Water treatment production of high reslstiv- pressure difference as the driving force, the ity ( > 18 M&cm) water for the electronics membranes used m UF can have pores from 1 industry (MF and RO) and production of to 50 nm, while for MF the pore range is from clean boiler feedwater, potable water and 005tolOpm clean waste water (RO and ED ) (u) RO, gas separation and pervaporatlon, - Others oil-water separation (UF and MF), which are associated with (partly) dense mem- recovery of pamt and latices from waste water brane structures (pores < 1 nm), make use of effluents (UF), hemodlalysls, membrane a difference m affimty between several feed electrodlalysis and recovery of gases (GS) components and the membrane, and of a dlf- The membranes used m the various mem- ference m dlffuslvlty through the membrane, brane processes can be very different, both the the drlvmg force is a pressure difference m case material and the configuration (modules) offer of RO and gas separation, and m case of dl- several posslblhtles A membrane can be made alysls and pervaporatlon the driving force is a out of a polymeric or inorganic material Well concentration gradient known polymer materials are polysulfone, cel- (111) Electrodlalysls uses anion- and cation se- lulose-acetate, polycarbonate, polypropylene lective membranes to separate charged mole- and polyacrylomtrlle A large variety of poly- cules from uncharged ones The ions are trans- meric membranes are produced to optimize ported by a diffusional mechanism, as a result their permeability and separation charactens- of an applied potential difference tics. Inorganic membranes (usually MF-type) Some other membrane processes are under can be made from e g a-Al,O, or silica ( S102) development at the moment, for instance, fa- eventually supphed with a less porous ~-A1~0~ cilitated transport by hqmd and fixed site car- top layer Membranes can be subdivided m symmetric and asymmetric membranes Symmetric mem-
dnvmg force branes can be dense, or can have straight or feed AP AC etc permeate sponge-like pores Polymeric asymmetric cross-flow membranes, usually made by the phase-mver- velocity flux slon method, consist of a thm dense skin layer t membrane (0 l-l ,um thickness, contaunng pores or not) on top of a much more porous sublayer The Fig 1 Schematx representation of a membrane process thinness of the skm layer results m a low resls- G B van den Berg and CA Smolders/J Membrane SCL 73 (1992) 103-118 105 tance for transport through the membrane gulshed m relation to membrane structure Another example of an asymmetric membrane -transport through bulk material (dense 1s a composite membrane, which 1s usually made membranes), of a very permeable UF membrane mto which -Knudsen diffusion, m narrow pores, a thm dense layer, often of another polymer -viscous flow, m wide pores, type, 1s applied and which 1s chosen for its high -surface dlffuslon, along pore walls selectivity In practice, the overall transport can be the re- Apart from the diffusion processes men- sult of more than only one of these mecha- tioned above, that determine the actual mem- nisms, this 1s for example the case m an asym- brane separation, other important diffusion metric membrane, as shown m Fig 2. processes are related to membrane processes, It 1s not possible to define fixed pore-dlmen- viz diffusion m the boundary layer near the slons coupled to the various transport mecha- membrane (concentration polarlzatlon phe- nisms that occur Apart from the case m which nomena) and diffusion during membrane for- there are no pores at all (dense membranes m mation: the inflow and outflow of solvents and which sorption and diffusion of the permeants non-solvents during the phase mverslon pro- occurs ) , transport will be a combmatlon of the cess determine the structure of the membrane various mechanisms mentloned. In general, the to be formed transport can be considered to be mainly of the The various diffusion processes that can oc- Knudsen diffusion type when the pore radius r cur during processes m which membranes are 1s smaller than lo-’ m (=lO nm) at ambient involved are summarized m Table 1. pressures, and it will be mainly viscous (Pol- The aim of this paper 1s to show the lmpor- seullle) flow when r 1s larger than 10m5 m ( = 10 tance of diffusion related to membrane sepa- ,um) These values also depend on the applied ration processes A description of the different pressure and temperature In between these diffusion processes will be given m the para- pore sizes the flow 1s a combmatlon of Knudsen graphs to come and Polseullle flow [ 1 ]
1. Diffusion in membranes 1 1 Transport In dense membranes
When describing transport m membranes In the literature usually one of the followmg various possible mechanisms can be dlstm- three theories/models are used to describe the
TABLE I
Survey of the various areas where diffusion occurs
1 In membranes (a) m dense membranes (gas separation, pervaporatlon) (b) m porous membranes -gases -1lqulds -faclhtated transport (liquid membranes ) -solutes (controlled release) -electrodlalysls 2 In the boundary layer near the membrane 3 During membrane formation 106 G B van den Berg and CA Smolders/J Membrane SCL 73 (1992) 103-118
- skm - bulk dlffwon small subsystems m which a local equihbrmm narrow pores - Knudsen exists or is approached. )_ dlffusron wide pores + “lSCO”Sflow Equations for transport rates (fluxes), re- I- tention and other membrane related parame- Fig 2 Transport m an asymmetric membrane can be the ters can be derived usmg general thermody- result of various mechamsms namic equations For example, the flux for each component, J,, is described as [ 41
transport in dense membranes [ 2,3] : I -thermodynamics of irreversible processes (TIP), II -preferential sorption capillary flow model where the phenomenologlcal coefficients, L,, (PSCF), are often complex functions of composition and III -solution-&ffusion model (SD ) concentration The driving forces, F,, which are Smce diffusion is the only possible transport chemical potential gradients, are given by mechanism m dense membranes a diffusion coefficient appears, directly or mdirectly, m the three models mentioned above (TIP ) : diffusion coefficients are mcorpo- rated via the frictional coefficients, (PSCF) after (preferential) sorption, surface m which the first term indicates the gradient diffusion occurs at the walls of the due to a pressure difference and the second term ‘pores’, expresses the concentration related gradients (SD): the transport m the membrane is de- According to the friction model, the general- termined by both the sorption and ized force actmg on solute z m the membrane is the diffusion of the permeants balanced by the frictional force between solute z and the membrane, and that between solute I and solute J 1 1 1 Thermodynamxs of rrreverstble processes Smce transport phenomena are due to non- Ft = CfJK -u,) +funU, eqmhbrmm conditions and thermodynamics deal with equihbrmm systems, the classical theory of thermodynamics is m prmclple less where U,and u, are the linear velocities of solute appropriate to describe transport processes We z and/ m the membrane In practice the friction assume firstly that simultaneously occurrmg coefficients, fCm,are always larger than the ‘free processes can be clearly separated into non-m- friction coefficients’, f,, which are related to teractmg mechanisms of change m the system, diffusion coefficients by and secondly that the steady state irreversible flows affect only each of the processes sepa- f, = RTI W,,D, 1 rately At near eqmhbrmm conditions of the ir- reversible processes, the thermodynamic treat- ment may be applied by dlvldmg the system m where D, 1s the ‘free’ diffusion coefficient, m a G B van den Berg and CA Smolders/J Membrane Scr 73 (1992) 103-118 107 binary solution f, can be described by, e g N Stokes’ relation* S= -C’DSOlew( -qlm’RT) (dG,/dp) v.. From experiments it appeared that the surface f, = 67vRS diffusion coefflclents can be described very m which R, 1s the Stokes’ radius of the solute generally without taking the influence of the Of course it 1s quite a problem to get quantlta- surface concentration mto conslderatlon as [ 61: tlve values for the various frlctlon coefficients, D 6x10-‘exp( -0 45q/m’RT) especially fLmvalues sL=l Though the existence of pores seems to be 1 1 2 The preferentral sorption capdlary flow contradictory to the tightness of dense mem- model [5,6] branes, this surface diffusion model can be used If the pores m a dense membrane are small to describe transport m dense membranes. the enough one should consider the transport phe- mtermolecular voids are considered to be the nomena taking place at the pore walls. Surface ‘pores’ For the description of transport m RO diffusion, or surface flow, can take place when membranes the sltuatlon 1sthought to be as de- the residence time of a molecule at the pore wall scribed m Fig 3 [ 71 1slarger than zero Depending on the migration Furthermore, the surface &ffuslon approach energy and the surface dlffusivlty a molecule can also be used m combmatlon with other can move along a pore wall Because of dlffer- models For instance, Jonsson et al [8] de- ences m energies and dlffuslvltles a certain se- scribed the flux as a result of a combmatlon of lectlvlty m transport rate can be the result. dlffuslonal surface flow and viscous flow For component I the flux, N,, due to surface through the ‘pores’ of a RO membrane diffusion can be described as 1 1 3 The sol&on-dlffuston model Ns, = - C’& (dG,ldp ) r’p This model, which m practice 1smost widely where C’ 1s a geometrical parameter that de- used for dense membranes, describes the trans- pends on the pore system, and (dC,,/dp) can port by permeablhty = sorption x drffuslon [ 91. be determined from the adsorption isotherm This can be seen m the general expression for The surface diffusion coefficient DsL1s a func- the flux J, tion of the temperature as 1s given by J,= -D,c?dWlRT)ldx DsL= &Olexp ( - E/ET) while Dsol depends on the mean surface free- t PreSSUU? demmerallzed wafel N&l Hz0 N.&l H20 al the mterface path length, the Jump frequency and the sur- Na+Cl H 0 Na+Cl HO _-___-----_-__*__---2 face concentration (especially at low surface rzzrz=z=== concentrations) The activation energy for sur- N&Cl H&l N&l H20 N&l H,O NatCl Hz0 face diffusion (migration energy) E 1sassumed to be a fraction of the differential heat of ad- sorption q porous - film cm~calpore dnmeter = 21 E=q/m’ Fig 3 Schematical representation of the preferential sorp- where m’ = 1, 2 or 3, which depends on the tlon capillary flow model The dimension of the cntical pore type of surface bmdmg. Now it can be derived diameter 1s two times the thickness of the adsorbed water that layer 108 G B van den Berg and CA Smolders/J Membrane SCL 73 (1992) 103-118 where the gradient m the chemical potential mlxmg of a solvent and a polymer is [ 21 across the membrane d(pT)/dx is the driving F y=RT[ln@,+ (l-l/m)!DP+#] force The concentration cp m the membrane can be calculated usmg, e g , the Flory-Huggms where x is the Huggms’ parameter, m is the equation molar volume ratio of polymer and solvent, and In the absence of strong interaction effects QP and Cp,are the volume fractions of the poly- between the permeating components, the mer and the solvent, respectively, from which permeabihty constants P of the pure compo- the concentration CT can be calculated. Hug- nents can be used to describe the selectivity gins parameter x depends on the solubihty pa- rameters of solvent and polymer, among other ~A/I~=PAIPB=(SA/SB) X (DAIDB) things where the preferential sorption is expressed by In case of gas separation a different model is SA/Sn and the ratio of the mobihties by DA/ frequently used the dual-mode sorption model Dr+ This model combines Henry’s law with a Lang- Both the sorption and the diffusion are de- mmr isotherm The latter reflecting the sorp- pendent on the characteristics of the mem- tion of a gas m voids that are assumed to exist brane material and the permeants In the lit- m a glassy polymer membrane For compari- erature many examples can be found son, the solubihtles of gases m rubbery poly- attempting to describe the permeation selectiv- mers above the 7’, often are described by Hen- ity as a function of the sorption and/or the dd- ry’s law only The resultmg descriptions of the fusion coefficient. Both in experimental and sorption equihbria are [ 111 theoretical papers the solution-diffusion model T> T; C=lz,p has been used For instance, Lee et al [lo] showed that, during pervaporation of orgamcs Y
The MM-S equation uses the coefficients of self diffusion, *D,,, of a species 2 in a mixture In all the equations mentioned above the mem- Fig 4 Possible effect of crystalhmty on diffusion resls- brane can be considered to be one of the com- tance Taken from Ref [ 121 ponents of the mixture The self diffusion coef- ficients, *D, in a multicomponent mixture can be described by a Vigne equation, which read originally In (D,) = x,ln (“D,,) + x,ln (“D,,), modified to
n natural rubber In (*D,,) = x,ln (*D,,) + C 5 In ( O”D, ) J=l
where 03D, is the diffusivity of z in] at mfimte 10 ioo 1000 10000 100000 dilution of z. Molecular weight (D) The crystalhmty of a membrane can also have Fig 5 Diffusion coefficients, m water, m natural rubber major consequences for the rate of diffusion m and m glassy polystyrene, as a function of the molecular weight of the diffusing particle the membrane, and hence for the flux The dif- fusion coefficient as a function of the crystal- these different media is shown to be several or- hmty can be described as follows: ders of magnitude for an average size molecule When selectively permeable materials are mcorporated m a dense membrane, both the se- lectivity and the flux can be positively changed. where !J’=is the fraction crystalline polymer, B Cussler [ 141 showed that m flake-filled mem- is a constant and the exponent n< ( << ) 1 The branes the selectivity and the relative flux can effect can be a hundredfold increase m diffu- be greatly enhanced An optimum ratio of the sion resistance, Fig 4 [ 121 diffusion coefficient m the polymer and the dif- The influence of the molecular weight (hence fusion coefficient m the flakes can be calcu- size) of the components on the diffusion is lated using variable flake fractions m the mem- shown m Fig 5 To emphasize the influence of brane and variable flake-width/thickness the rubbery/glassy state of the polymer the dif- ratios fusion m water, natural rubber (rubbery state) and polystyrene (glassy state) are given [ 131 1.2. Transport in porous membranes The drawn lines have been obtained from a large number of experimental data. The differ- As stated m the mtroduction transport ence m value for the diffusion coefficient m through bulk material (dense membranes), will 110 G B van den Berg and CA SmoldersfJ Membrane Set 73 (1992) 103-118 be different from the transport phenomena m (m which f 1s the fraction lffuse reflection more porous membranes Knudsen dlffusron m which the pore wall); this is roughly equal to narrow pores and viscous flow m wide pores are $Gr when f equals about unity more likely to occur m porous membranes The The importance of Knudsen dlffuslon rela- various posslblhtles to consider are transport tive to molecular drffuslon, whrch 1s given by phenomena of gases, solutes and hqulds, facll- the number of colhslons with the wall com- ltated transport, solutes m controlled release pared to the number of mutual colhslons, can and electrodlalysls These subJects will be de- be descrrbed for hard spheres of pure compo- scribed m the next paragraphs nent A as [15]-
D d 1 2 1 Knudsen dtffutxon tn narrow pores -=PKA Knudsen dlffusron occurs when the pore dr- D AA ;1 ameter 1smuch smaller than the mean free path For example, for Nz at a pressure of 0 1 MPa lengths, 1, of the (gas) molecules to be sepa- and pores of d,,= 5 nm, more than 90% of the rated. 3,IS defined as: colhslons will be colhsions with the wall and so Knudsen dlffuslon mainly determines the transport Pg 8RT c The importance of surface drffuslon relative where the vrscoslty q and the den&y p are to Knudsen dlffuslon has been described in lit- functions of T and LIP The flux due to Knud- erature as well- Kelzer et al. [ 1] found that the sen drffuslon m straight pores can be described observed permeability of CO, was larger than by could be calculated from the permeability of Nz and ascribed the additional transport to sur- face dlffusron (Fig. 6). JA=gE&E Uhlhorn et al [ 161, using silica modified y- or by alumina membranes, found much higher selec- - tlvltles for condensable gases than Knudsen AP VA J A=: ltnk1’3-- dlffuslon could account for The modlficatlon S RT of the y-alumma membranes wrth srhca con- G is the average molecular velocity; rt 1s nor- mally represented by. z co* 3-J4 uA=JW E N2 ,z 3 ---_-___-_--_- CO2 calculated (Knudsen) From the first flux equation the selectlvlty cy E can be derrved, as aA,B=Jm The second equation shows the relation to dlf- fusion. The Knudsen dlffuslon coefficient m a 0 100 200 y~.sure (kPa) pore with diameter dp, or radius r, can be cal- culated as Fig 6 The permeablhtles of CO, and N, as a function of pressure for the separation (top-) layer of ceramic mem- D branes [ 11 The dashed hne IS the expected value for CO, if only Knudsen dlffuslon occurs G B van den Berg and CA Smolders/J Membrane Scr 73 (1992) 103-118 111 sisted of a silica layer of approximately 30 nm the flux m the Knudsen-Polseuille transition thickness, with pores smaller than 1 nm, on top range by of the membrane and probably some filling of J= u(P/P,,,)~,~P the pores with silica as well. From permeability experiments it could be concluded that trans- where a is the membrane constant and g is a port of inert gases, e.g hehum and nitrogen, was measure of the extent of the Polsemlle flow The determined by Knudsen diffusion only How- parameter g increases with mcreasmg molecu- ever, for condensable gases the transport was lar weight of the gas and with increasing mem- greatly enhanced, the permeabihties were de- brane pore size pendent on the applied pressure and surpns- Until now the examples on the performance mgly high permselectivities could be observed of the membrane were mainly determined by These phenomena were attributed to surface the pore size of the membrane Obviously, the diffusion. nature of the membrane also determines the In a further study a y-alumma membrane was separation characteristics An example is the impregnated with magnesia, introducing more description of a diffusion-reaction m a multr- strong-base sites and fewer weak-base sites membrane contaunng urease The equihbrmm This resulted m stronger bonding of COZ, at reaction NH&!ONH2 + 2H,O4-+2NH,+ + about equal amounts, and less surface diffusion CO:- is strongly affected when one of the (the surface diffusion accounts for approxi- membranes m the cell is switched from a neu- mately 30% of the CO, transport on unmodi- tral celluloslc membrane to an anion-exchange fied y-alumma membranes, see Fig 6) [ 171 membrane. The outward transport of the re- action products and so the conversion are 12 2 VWOW flow In wide pores greatly reduced because of the nature of the The average velocity of a liquid medium m a membrane [ 19 ] pore 1s described by the Hagen-Poiseullle relation: 1 2 3 Fachtated transport m lrqurdand fixed sate carrier membranes (U) A$$ Liquid membranes (LM) were developed be- cause of the relatively small transmembrane from which the flux J can be calculated. flux of polymer membranes A liquid mem- brane is a fluid or quasi-fluid phase which sep- nkApore J=A (7~) =E(u) =constantXr4 arates two other phases that are immiscible with membrane the hquid membrane Various types of liquid In this equation no molecule-related quantities membranes exist, e g bulk-, emulsion- and are left, from which it can be easily concluded supported liquid membranes. that Poiseuille flow will not result in any selec- Because of the nature of the facilitated tivity. Furthermore, it can be observed that now transport, m which a carrier is used for the se- the flux is proportional to r4, which was r3 for lective transport of e g. an ion, the transport m Knudsen flow. So in a matrix conslstmg of very the hquld membrane is mamly determined by small pores few large pores can have great im- the diffusion of the carrier m the membrane pact on o! phase Transport of the ion can even take place The afore-mentioned transport of combined against its own concentration gradient. This is Knudsen and Poiseuille flow has been de- possible because of the phenomenon of coupled scribed by Schofield et al [ 181 They described transport, 1.e a different ion of equal charge is 112 G B van den Berg and CA Smolders/J Membrane Set 73 (1992) 103-118
feed LM StrIpping phase fusion coefficient of the carrier-anion complex is usually calculated using the Stokes-Emstem equation. D = kT/ (67cry,r). Other diffusivities can play a role as well; e g the diffusion of the anion in the stagnant film at both sides of the membrane or the back-diffusion of the carrier However, m case of nitrate removal the simple
Fig 7 The (supported liquid membrane) separation with Fick’s law can be applied a countertransport mechanism for the removal of mtrate Noble [21] developed a model for transport ions m fixed-site carrier membranes (the complex- mg agents are cast directly into the polymer feed LM stnpplng phase films) This model yields a dual-mode sorption kmd of description m the case of diffusion-hm- ited transport The diffusion m this case is de- termined by the diffusion coefficient of the sol- ute-carrier complex This implies that the transport is morphology dependent Further- more, it was shown that the change m mobility Fig 8 Ideal concentration profile for transport of per- caused by changes m morphology may result m meatmg NO,-ions through a ltqutd membrane NO3 stands for the carrier-Ion complex a percolation limit transported back This ion can be provided with 1 2 4 Solute transport m controlled release a large chemical potential gradient Mostly cat- (CR) [221 ions are transported although also amon trans- The release of a drug (e g the contraceptive port has been realized, e g the removal of m- steroid levenorgestrel) from a CR-membrane, trates from ground water [20] (Fig. 7) In this which is usually a microporous hollow fibre process the followmg steps of the permeate ion membrane, can be described m two ways de- can be observed pendmg on the tightness of the skin (a) diffusion of the ion to the membrane (a) One way of release is described as a so- surface, lution-diffusion mechamsm, 1 e the steroid (b) chemical reaction (complexation) at the dissolves into the (dense) polymeric skm, dif- surface with a carrier molecule, fuses through the polymer and finally partition (c > diffusion of the carrier bounded ion occurs into the receiving fluid surroundmg the through the LM-phase, hollow fibre On the assumption that drug dis- (d) chemical reaction at the surface with the solution is not rate-hmitmg for such a mecha- stripping phase (decomplexation) , nism two extreme conditions may exist. (1) the (e) diffusion of the ion to the bulk of the rate-controlhng step is formed by diffusion of stripping phase the steroid through the hollow fibre skm The expected concentration profile of the ion (membrane-controlled permeation process ) , or is shown M Fig 8 In the most simple represen- (11) by diffusion of the steroid from the hollow tation4f the process the flux of the amon can fibre mto the receivmg fluid (diffusion layer- be represented by F&s law hke J=dCD/G and hmitmg partition-controlled process) so it will be a linear function of the diffusion In case (1) the transport is proportional to coefficient of the complex m the LM The dif- the diffusion coefficient of the steroid m the G B van den Berg and CA Smolders/J Membrane Scr 73 (1992) 103-118 113 polymeric skm DSP,proportional to the concen- ions. The diffusion mvolved is diffusion of ions tration difference between the concentration m m swollen polymers, and and outside of the fibre and inversely propor- (b) the transport through the mevltable con- tional to the thickness of the skm layer: centration polarization boundary layer In this case the diffusion is diffusion of an ion m the dM,/dt = k1D,&/d, solvent In case (11) (diffusion layer-limiting partl- Usually concentration polarization phenom- tlon-controlled process) the transport is pro- ena influence the electrodlalysls process to a portional to the &ffuslon coefficient D,, and the large extent, which results m the observation solublhty C, m the recelvmg fluid, and m- that the transport is mainly determined by the versely proportional to the thickness of the dif- diffusion process of the ions in the boundary fusion boundary layer &. layers near the membrane For instance, Tan- aka [ 241 has shown that ion-exchange electro- di%ldt = kJLC,ldi, dialysis of NaCl results m concentration polar- (b) When the asymmetric (hollow fibre) ization boundary layers on both the amon and membrane contains a microporous skin a pore- the cation exchange membranes Due to the diffusion mechanism can be imagined The diffusion coefficient of Cl--ions being larger steroid release rate, into the liquid-filled pores, than the diffusion coefficient of the Na+-ions, is then described as follows the concentration near the cation exchange membrane will rise considerably As can be ex- m,ldt= keR, (E/T) G/d, pected the concentration near the anion ex- where d, 1s the thickness of the microporous change membrane will drop compared to the skin, and E and r are the porosity and the tor- bulk concentration However, this is only the tuoslty of the membrane wall, respectively case when the convective flow rate towards the membrane stays below a critical value, if not, a 1 2 5 Electrodtalysu [23] concentration equal to the bulk concentration The separation of ions by electrically charged can be expected membranes under the influence of the driving In practice, the concentration polarization force AE 1s schematically represented in Fig 9 problem can be reduced considerably by mtro- The transport during electrodlalysls is deter- ducmg the electrodlalysls reversal mode m mined by two processes m which &ffuslon plays which the polarity of the electrodes is reversed an important role periodically, typically every 15 mm [ 251 (a) the transport through the membranes that causes the actual separation of anions and cat- 2. Diffusion in the boundary layer near the membrane (concentration feed solution polarization related phenomena) [26]
The description of concentration polarlza- tlon phenomena during membrane separation processes can be generally divided mto two groups (Fig 10). -cake filtration, which is used for membrane Fig 9 Schematic representation of the electrodlalysls processes that mvolve large particles, like m MF membrane separation process and sometimes UF, and 114 G B van den Berg and CA Smolders/J Membrane Scr 73 (1992) 103-118
TABLE 2
The influence of concentration polarlzatlon on various membrane separation processes
Process Influence of Reason concentration a) the cake hltratlontype b) the concentration profile polarlzatlon of descrlptlon accordmg tothe film theory
I I Reverse osmosis moderate k large Ultrafiltration strong k small Fig 10 The concentration profiles due to concentration Mlcroflltratlon strong k small polarization phenomena according to the cake-filtration Gas separation (very) low k very large/J small type of description (a) and the film theory (b) Pervaporation low k large/J small Dialysis low J small -film theory descrlptlon, which 1s used for all Electrodlalysls strong processes that separate hqulds, Ions and small particles. Examples are pervaporatlon (L-L), ED (Ions ) , RO (small molecules ) and often UF (macromolecules) 400 In most concentration polarrzation equa- tions a flux expressron hke J= lzxf( Cb, C,, ) 1sused, e g the well known equation
300 where K 1s the mass transfer coefficient, and "E iz=D/6, m which 6 1s the thickness of the 2 boundary layer The dlffuslon coefficient of the 0 200 solute m the solution that has to be used m thus equation 1s usually taken to be independent of concentration However, some researchers have shown that a concentratron dependent dlffu- 100 slon coefficient can describe the process much better [ 271 The effect of concentratron polar- lzatron can be rather drfferent for the various 0 processes (see Table 2 ) 0 100 200 300 For ultrafiltration a number of different models have been described m literature, both X(F) for dead-end and for crossflow filtration A Fig 11 Simulated concentration profiles near the mem- brane mterface for ultraflltratlon of the protein BSA, as a model that appeared to be able to predict fluxes fun&on of time and distance from the membrane, using accurately during dead-end ultrafiltration of D=69XlO-“m’/sec proteins IS the Boundary Layer Resistance model [ 28,291 Mamly using characterlstrcs of tratlon of a BSA solution are reproduced m Fig the proteins, like sedlmentatlon coefficrent and 11 [29] specrfic volume, the flux behavlour could be The influence of the value of the dlffuslon calculated numermally For instance, the con- coefficient on the build-up of the concentratron centratron profiles after various periods of fil- profile was also calculated (Fig 12 ) . The figure G B van den Berg and CA Smolders/J Membrane Scr 73 (1992) 103-118 115
. D=6 10 I2 nl*/s mixture of solvent and non-solvent can also be
x D=3 10 ” m*/s used as the coagulation bath
q D=2 10 lo m*ls The membrane formation mechamsm by phase separation is determined by. (a) ther- o- 600 E modynamic eqmhbria m the polymer/solvent/ ii non-solvent system, and (b) the kinetic as- pects during membrane formation, which are (bl) diffusion of solvent (out) and non-sol- 200 vent (into the polymer film) and (b2) kinetics of the phase separation itself Usually the phase 0 0 10 20 30 40 50 behavlour of the three component system poly- x (elm) mer/solvent/non-solvent is represented m a
Fig 12 The calculated concentration profiles after 1000 phase diagram The changing composition dur- set ultrafiltration of BSA using various values of the dlf- mg membrane formation will determine which fusion coefficient type of membrane will be developed. Two pos- sible composition paths (leading to either ag- clearly shows the increasing concentration near gregate formation or L-L demixmg) during co- the membrane with decreasing diffusion coef- agulation determine the type of membrane that ficient, this phenomenon will result m an m- will be formed These diffusion determined creased resistance of the concentrated bound- processes are represented m the phase hagram ary layer near the membrane, while the (Fig. 13). separation characteristics are also affected The calculation of the composition profile is strongly done using (a) ternary diffusion equations, (b) the appropriate boundary conditions, and (c) 3. Diffusion during membrane formation concentration dependent diffusion coeffi- 1301 cients. The fluxes of the various components are depicted m Fig 14 The most important preparation technique The ternary diffusion equations are for the for polymeric membranes is the phase mver- polymer solution: sion method. This method results m various types of membranes by precipitation from a a(~,/~~)lat=alat[u,~‘CL,(d~~,lam) 1 polymer solution using a non-solvent that dif- (El, 2,3) fuses into a film of polymer solution. Both sym- metric and asymmetric membranes can be P formed, with porous or dense top layers, de- Possible composltlon paths pending on the membrane formation mecha- dung coagulation nism. Frequently used polymers are polysul- 1 gwngaggregate formatlon fone, polyacrylonitrile, polypropylene, 2 glvmg llquld lkquld demwng cellulose-acetate and polyamide. The preparation technique is based on pre- cipitation after immersing the polymer solu- tion m a non-solvent bath, resulting m the name Fig 13 Phase diagram of the three component system sol- ‘immersion precipitation’ Instead of immers- vent (S) /non-solvent (NS) /polymer (P) and the possible mg the polymer solution m pure non-solvent, a composltlon paths dunng coagulation 116 G B van den Berg and CA Smolders/J Membrane SCL 73 (1992) 103-118
mtelface diffusional transport processes m membranes,
J, = c L,J (4,>9,) *$l diffusion m the boundary layer near the mem- J brane (concentration polarization phenom-
(1) non SOlvent ena) and diffusion durmg membrane forma- (2) Solvent tion are of high interest The different diffusion (3) polymer processes are described by one of the numerous models Fig 14 The situation during membrane formatlon Various mechanisms have been distm- gulshed to describe the transport m mem- branes transport through bulkmaterial (dense 6 1 membranes ) , Knudsen diffusion in narrow po- res, viscous flow m wide pores, or surface dif- fusion along pore walls In practice, the trans- port can be a result of more than only one of these mechanisms. / \, The characteristics of a membrane, e.g its acetone 6 4 water crystalhmty or its charge, can also have maJor Fig 15 Approximated change of the composltlon path at consequences for the rate of diffusion m the different lmmerslon times for a 10 vol % cellulose acetate membrane, and hence for the flux obtamed. (CA) solution Immersed mto a pure water bath The effect of concentration polarization can be rather different for the various processes, and m the coagulation bath which is mainly determined by the mass trans- a~,lat=alat[o(~,)a~,/axl (z=L 2) fer coefficient, and thus by the diffusion coefficient. The phenomenologlcal coefficients L, can be The phase mverslon membrane formation obtamed from solvent-non-solvent diffusion mechanism is determined to a large extent by coefficients and the sedimentation coefficient the diffusion of solvent and of non-solvent of polymer in solvent. Different experimental circumstances can be List of symbols simulated also. m Fig 15 the influence of the time of immersion m the coagulation bath on a membrane permeability constant the composition path is represented [ 301. From (kg/m2-set-Pa) the figure it will be clear that m this case a cer- A surface area (m”) tam time is needed before liquid-liquid demlx- b affinity constant (l/atm) mg will occur During this delay-time a dense B constant (-) surface layer is formed by loss of solvent from CL concentration of z ( kg/m3 ) the polymer solution, resulting m an asymmet- C amount of sorbed gas per amount of ric membrane structure Consequently the ki- polymer [ m3 ( STP ) /m” ] netlcs of demixmg determme the type of mem- C’ geometrical parameter (-) brane that is formed Cb concentration m the bulk ( kg/m31 G-I Langmuir capacity constant 4. Conclusions (m3(STP)/m3) CIJ concentration m the permeate (kg/ Diffusion processes play an important role m m3) membrane separation processes apart from the C* solubihty m receiving fluid ( kg/m3 ) G B van den Berg and CA Smolders/J Membrane Scl 73 (1992) 103-118 117
concentratron of z at the surface (kg/ Ns, molar flux due to surface drffuslon m3) ( mol/sec-m2) thickness of the drffusron boundary r pore radius (m) layer (m) P relative pressure (Pa) pore diameter (m ) P (hydrauhc ) pressure (Pa)
drffuslon coefficient of z in J (m’/sec ) PA permeablhty constant of the pure drffuslvlty of z ml at mfimte dllutlon component A ( m3/m2-set-Pa-m) of z (m”/sec ) P/P,,, pressure wlthm the membrane/refer- coefficient of self diffusron of z m a ence pressure (- ) mixture (m”/sec ) differential heat of adsorption (J/ surface diffusion coefficient ( m2/sec) mol ) diffusion coefficient of steroid n-rthe gas constant (J/mol-K) polymeric skin (m”/sec ) Stokes’ radius (-) diffusion coefficient of steroid m re- temperature (K) celvmg liquid (m”/sec) glass-rubber transition temperature actlvatlon energy for surface dlffuslon (K) (J/mol) velocity of z m a membrane (m/set ) potential difference (V) average molecular velocity (m/set ) fraction diffuse reflection at the pore partial specific volume ( m3/kg) wall (-) (molar) fraction (-) friction coefficient ( J-sec/m2) driving force (N) selectivity (-) partial molar free energy of mixing thrckness (m ) (Jbol) porosity (-) exponent m Schofield’s equation (-) viscosity (Pa-set ) flux of component z (m/set) chemical potential, p: 1s concentra- volume flux (m/set) tion dependent part of ,uC(J/kg) mass transfer coefficient (m/set) mean free path length of (gas) mole- Henry’s law solubllity coefficient cules (m) [ m3 (STP) /m3-atm] density ( kg/m3) constant ( m2 ) tortuosity (-) phenomenologlcal coefficient (kg- volume fraction of component z (-) set/m ) volume fraction of the polymer (- ) molar volume ratio of polymer and volume fraction of the solvent (-) solvent (m) Huggms’ parameter (- ) factor depending on type of binding fraction crystalline polymer (-) (-) molecular weight (kg/kmol) References mass (kg) exponent (-) number of pores (-) 1 K Kelzer, R J R Uhlhorn, R J van Vuren and A J Avogadro’s number (l/mol) Burggraaf, Gas separation mechamsms m mxropo- rous modified y-Al,03 membranes, J Membrane Scl , molar flux ( mol/sec-m2 ) 39 (1988)285-300 118 G B van den Berg and CA Smolders/J Membrane SCL 73 (1992) 103-118
2 J G A Bitter, Transport mechamsms m membrane Knudsen-Polseullle transition, J Membrane Scl (53 separation processes, Plenum Pubhshmg Comp , New (1990) 159-171 York, NY, 1991 19 M Yonese, H Murabayashl and H Klshlmoto, DL 3 M Soltameh and W N Gill, Reverse osmosis, Chem fusion-reaction of urea through multi-membrane con- Eng Comm ,12 (1981) 279 taming urease Effects of micro-environments around 4 K S Splegler and 0 Kedem, Thermodynamics of hy- urease and asymmetry of the membrane, J Mem- perfiltration (reverse osmosis) criteria for efficient brane Scl ,54 (1990) 145-162 membranes, Desalination, 1 (1966) 311-326 20 T Neplenbroek, Nitrate removal with supported hq- 5 E R Gllhland, R F Badour, G P Perkmson and K J uld membranes, transport mechanism, Ph D Thesis, Sladek, Diffusion on surfaces I Effect of concentra- Umv of Twente, Enschede, 1989, Chap 2 tion on the dlffuslvlty of physically adsorbed gases, 21 R D Noble, Analysis of ion transport with fixed site Ind Eng Chem , Fundam ,13 (1974) 95-99 carrier membranes, J Membrane Scl ,56 (1991) 229- 6 K J Sladek, E R G&land and R F Badour, Dlffu- 234 slon on surfaces II Correlation of dlffuslvltles of 22 M J D Eemnk, Biodegradable hollow fibres for the physically and chemically adsorbed species, Ind Eng controlled release of drugs II In vctro release of the Chem , Fundam, 13 (1974) loo-105 contraceptive steroid levenorgestrel, Ph D Thesis, 7 S SounraJan, Reverse Osmosis, Academic Press, New Umv of Twente, Enschede, 1987, Chap 7 York, NY, 1970 23 H Strathmann, Electrodlalysls, m PM Bungay 8 G Jonsson and C E Boesen, Water and solute trans- (Ed ), Synthetic Membranes Science, Engmeermg port through cellulose acetate reverse osmosis mem- and Apphcatlons, Reldel, Dordrecht, 1986, pp 197- branes, Desahnatlon, 17 (1975) 145-165 223 9 H K Lonsdale, U Merten and R L Riley, Transport 24 Y Tanaka, Concentration polarlzatlon m ion ex- properties of cellulose acetate osmotic membranes, J change membrane electrodlalysls, J Membrane Scl , Appl Polym Scl ,9 (1965) 1341 57 (1991)217-235 10 Y M Lee, D Bourgeois and G Belfort, Sorption, dlf- 25 W E Katz, The electrodlalysls reversal (“EDR”) fusion and pervaporatlon of orgamcs m polymer process, Paper presented at the International Con- membranes, J Membrane Scl ,44 (1989) 161-181 gress on Desalination and Water Re-use, Tokyo, Ja- 11 W J Koros, A H Chan and D R Paul, Sorption and pan (1977) transport of various gases m polycarbonate, J Mem- 26 G B van den Berg and C A Smolders, Flux decline m brane Scl ,2 (1977) 165-190, and references therem membrane processes, Filtration and Separation, 12 J G A Bitter, Effect of crystalhmty and swelhng on March (1988) 115-121 the permeability and selectlvlty of polymer mem- 27 S Nakao, T Tanabe and S Klmura, Mass transfer branes, Desalination, 51 (1984) 19-35 coefficient in ultrafiltration of macromolecular solu- 13 H Strathmann, Trennung von molekularen MIS- tions, Poster presented at the 5th Int Symp on Syn- chungen mlt Hllfe synthetlscher Membranen, Stem- thetic Membranes m Science and Industry, Tubm- kopf Verlag, Darmstadt, 1979 gen, Germany, September 2-5,1986 14 E L Cussler, Membranes contammg selective flakes, 28 J G WlJmans, S Nakao, J W A van den Berg, F R J Membrane Scl ,52 (1990) 275-288 Troelstra and C A Smolders, Hydrodynamic rests- 15 N Wakao, J Otam and J M Smith, AIChE J , 11 tance of concentration polanzatlon boundary layers (1965) 435-438 m ultraflltratlon, J Membrane Scl , 22 (1985) 117- 16 R J R Uhlhorn, M H B J Huls m ‘t Veld, K Kelzer 135 and A J Burggraaf, High permselectmltles of mlcro- 29 G B van den Berg and C A Smolders, The boundary porous slhca modified y-alumina membranes, J Mat layer resistance model for unstirred ultraflltratlon A Scl Lett ,8(10) (1989) 1135-1138 new approach, J Membrane Scl ,40 (1989) 149-172 17 R J R Uhlhorn, K Kelzer and A J Burggraaf, Gas 30 A J Reuvers, Membrane formation diffusion m- and surface diffusion m modified y-alumma systems, duced demlxmg processes m ternary polymeric sys- J Membrane SC], 46 (1989) 225-241 tems, Ph D Thesis, Umv of Twente, Enschede, 1987 18 R W Schofield, A G Fane and C J D Fell, Gas and See also J Membrane Scl ,34 (1987) 45-65 and 67- vapour transport through mlcroporous membranes I 86