M e m b r a n e P r o c e s s e s — D i a l y s i s

a n d E l e c t r o d i a l y s i s

ELlAS KLEIN*RICHARD A. WARD*Division of NephrologySchool of MedicineUniversity of LouisvilleLouisville, Kentucky

ROBERT E. LACEYS

Southern Research InstituteBirmingham, Alabama

21.1 DIALYSIS

Dialysis first was reported in 1861 by Graham,1 who used parchment paper as a membrane. His experimentswere based on the observations of a school teacher, W. G. Schmidt, that animal membranes were lesspermeable to colloids than to sugar or salt.2 Over the next 100 years, dialysis became widely used as alaboratory technique for the purification of small quantities of solutes but, with minor exceptions, it realizedno large-scale industrial applications. In the last 20 years, development of dialysis for the treatment ofkidney failure has brought about a resurgence of interest in dialysis for a wide range of separations.

Dialysis is a diffusion-based separation process that uses a semipermeable membrane to separate speciesby virtue of their different mobilities in the membrane. A feed solution, containing the solutes to beseparated, flows on one side of the membrane while a solvent stream, the dialysate, flows on the otherside (Fig. 21.1-1). Solute transport across the membrane occurs by diffusion driven by the difference insolute chemical potential between the two membrane-solution interfaces. In practical dialysis devices, anobligatory transmembrane hydraulic pressure may add an additional component of convective transport.Convective transport also may occur if one stream, usually the feed, is highly concentrated, thus givingrise to a transmembrane osmotic gradient down which solvent will flow. In such circumstances, the de-scription of solute transport becomes more complex since it must incorporate some function of the trans-membrane fluid velocity.

The relative transfer of two solutes across a dialysis membrane is a function of both their diffusivitiesin the membrane and their driving forces. Separations will be efficient only for species that differ signifi-cantly in diffusion coefficient. Since diffusion coefficients are a relatively weak function of molecular size,

^Authors of Section 21.1.§Author of Section 21.2.

C H A P T E R 2 1

FIGURE 21.1-1 Schematic representation of dialysis. Small, highly mobile species (O) diffuse across themembrane, from feed to dialysate, while larger molecules ( • ) , to which the membrane is relatively im-permeable, remain in the feed stream.

dialysis is limited in practice to separating species that differ significantly in molecular size, for example,separation of crystalloids from polymers or colloids. In addition to this limitation, dialysis is a usefultechnique only when the soiute(s) to be separated is present in high concentration. This is because solutefluxes in dialysis are directly dependent on the transmembrane concentration gradient, an intrinsic propertyof the feed and dialysate streams. (In other membrane separation processes, such as reverse osmosis andultrafiltration, transmembrane fluxes depend on an applied pressure which is independent of the propertiesof the process stream.) Thus, if the transmembrane concentration gradient is low, practical solute recoveriescan be obtained only by increasing membrane area, which may compromise the economics of the process.Because of these considerations, dialysis is characterized by low flux rates in comparison to other membraneseparation techniques. However, the passive nature of dialysis may be an advantage in those circumstanceswhere the species to be separated are sensitive to mechanical degradation by high pressures or high shearrates. Dialytic separations can be enhanced under certain circumstances by taking advantage of chargerepulsion effects between a solute and the membrane34 (Donnan dialysis), by complexing one or more ofthe species to be separated5'6 (liquid membranes), by chemical conversion of the permeating solute in thedialysate,78 or by staging dialytic cells.9

21.1-1 Applications of Dialysis

Until about 1960, only a few industrial uses of dialysis had reached large-scale application. The mostprominent of these was dialysis of concentrated sodium hydroxide solutions containing hemicellulose, aby-product of viscose rayon production.10 A plate-and-frame dialyzer, containing parchmentized cottoncloth membranes, was used to recover sodium hydroxide from the hemicellulose-contaminated solution. Inthe 1950s, the development of synthetic polymer membranes with greater chemical resistance1' led to theapplication of dialysis for recovering nickel in the electrolytic refining of copper.12 However, the evolutionof other membrane processes, in particular ultrafiltration, together with general advances in chemicaltechnology resulted in dialysis remaining a relatively unimportant industrial separation process.

Beginning in the 1960s, dialysis found what has become its major application, the treatment of end-stage renal disease.l3 In a process known as hemodialysis, small-molecular-weight metabolic waste productsare removed from a patient's blood by dialytic transfer across a membrane that is impermeable to normalblood proteins. The dialysate is formulated to normalize blood electrolyte concentrations and acid-basebalance by dialytic exchange between the blood and dialysate. Fluid balance is restored by superimposinga transmembrane hydrostatic pressure gradient. In 1982, the lives of upward of 57,000 people in the UnitedStates alone were sustained by hemodialysis.

The growth of hemodialysis has resulted in the development of new membranes and more efficientdialysis devices. In turn, the availability of such devices has generated new interest in dialysis for a varietyof applications. One area where dialysis may find a role as a separation process is in the biotechnologyindustry, where products must be separated from fragile (e.g., shear-sensitive) or heat-sensitive solutions.In 1969, Schultz and Gerhardt14 reviewed the use of dialysis as a means of controlling bacterial culturesfor a variety of applications. By circulating the culture solution from a fermenter through a dialyzer, theyield of the fermenter can be enhanced either by the dialytic addition of nutrient or removal of productsthat may inhibit bacterial growth and metabolism. Since then, several groups have conducted laboratory-and pilot-scale studies of a number of microbiological systems. These include the recovery of enzymes andproteins from Staphylococcus aureus cultures15"16 and the growth of yeast cultures using dialytic transferof lactose from whey as substrate.17 Other applications involving sensitive solutions include the desaltingof cheese whey solids18 and reduction of the alcohol content in beer.19

Generally, dialysis has been restricted to use with aqueous solutions, although there is no fundamentalreason why it cannot be used with organic solvents, given the development of suitable membranes. An

FEEOSTREAM

OtALYSATE

TABLE 21.1-1 Dialysis Membrane Materials

Membrane Material Manufacturer

Regenerated cellulose Asahi Medical Co., Tokyo, JapanEnka AG, Wuppertal, West GermanyTerumo Corporation, Tokyo, Japan

Cellulose acetate Asahi Medical Co., Tokyo, JapanCD Medical, Miami, FloridaNipro Medical Industries, Tokyo, Japan

Ethylene-Polyvinyl alcohol Kuraray Co., Osaka, Japan

Poly aery lonitrile Hospal, Lyon, France

Polycarbonate Gambro Ab, Lund, Sweden

Polymethylmethacrylate Toray Industries Inc., Tokyo, Japan

Polyperfluoro (ethylene-co-ethylene E. I. Du Pont de Nemours, Wilmington,sulfonic acid) Delaware

Polysulfone Fresenius AG, Bad Homburg, West Germany

example of dialysis in nonaqueous systems has been described by Benkler and Reineccius,20 who useddialysis to separate volatile flavor components from higher-molecular-weight oils. The membrane used wasa perfluorosulfonic acid copolymer. There also has been a continuing effort to use dialysis in the separationand concentration of ionic species. Using the Donnan effect, a number of potential applications have beenstudied, including the removal of contaminating cations from dilute wastewater solutions,3*21 softening offeedwater to reverse osmosis desalination systems,22 and acid recovery in aluminum anodizing plants.

However, none of these new applications of dialysis, with the possible exception of acid recovery inaluminum anodizing, appears to have progressed beyond the pilot-plant scale, and today the only majorapplication of dialysis remains hemodialysis for the treatment of renal failure.

21.1-2 Membrane Materials and Configurations

The materials commonly used for dialysis membranes are listed in Table 21.1-1. The choice of a membranefor a particular application is dependent on both the separation required and the environment in which themembrane is to operate.

The principal membrane material used in medical applications is regenerated cellulose, derived eitherfrom hydrolysis of cellulose acetate or by the cuprammonium process.24 Cellulose membranes are hydrogelsthat contain up to 50% water when equilibrated with aqueous solutions. The permeability of hydrogelsincreases with the degree of hydration. To maintain the expanded solvation state in the dry membrane,and thus its permeability, during extended storage, cellulose membranes are produced containing 5-40%by weight of glycerol. Once wetted, subsequent drying of the membrane in the absence of a plasticizerresults in an irreversible collapse of the membrane structure and loss of permeability. This property maylimit the usefulness of cellulose membranes in some applications.

Dialysis membranes fabricated from glassy polymers, such as methacrylates, polysulfones, polycar-bonates, and polyacrylonitriles, are not hydrogels and do not undergo irreversible alteration on drying.However, they are generally hydrophobic and their permeability may decrease after drying due to trappingof air in the membrane pores. This can be reversed by first wetting the membrane with a low-surface-tension solvent, such as alcohol, before exposure to aqueous solutions. Fabrication of membranes fromglassy polymers is complicated by the tendency for membranes with adequate diffusive permeabilities tohave high hydraulic permeabilities, which limits their use in applications where volume control is important.Methacrylate, polycarbonate, and polysulfone membranes which circumvent this problem have been de-veloped for hemodialysis. The hydrophobic nature of most glassy polymers causes them to adsorb proteins,if present, from the solutions in contact with the membrane. Such adsorption of protein can cause a decreasein membrane permeability26 and this may limit the role of such membranes for some applications inbiotechnology.

The choice of a membrane for a particular separation is governed also by considerations of temperatureand pH. Membranes of glassy polymers generally are restricted to use at temperatures below 800C, whereascellulosic membranes will function satisfactorily at higher temperatures. Although most membranes fab-ricated from glassy polymers will operate over a wide range of pH, polycarbonate membranes will dete-riorate rapidly at pH values greater than 7.5. Cellulosic membranes hydrolyze slowly at low pH (<2) andswell extensively in highly alkaline solutions (4% sodium hydroxide).

Dialysis membranes are available in three basic configurations: flat sheet, tubular, and hollow fiber(Fig. 21.1-2). Continuously cast flat sheet membranes are used in plate-and-frame devices. Most of theearly industrial applications of dialysis were based on dialyzers of the plate-and-frame type. The availabilityof cellophane tubing, used as sausage casing, led to the development of coil dialyzers for hemodialysis.The dry tubular membrane was laid flat and wound into a coil with a mesh spacer to provide a channel fordialysate flow. Coil configurations provided a relatively large membrane surface area in a compact form;however, their use has been limited entirely to hemodialysis. More recently, hollow-fiber dialysis mem-branes have been developed27 that enable fabrication of compact devices with large surface areas. Inaddition, their structure allows them to be operated without the need for a membrane support structure.These two properties of hollow fibers have made them the dominant membrane form in use today. Anindustrial hollow-fiber dialysis unit is shown in Fig. 21.1-3.

21.1-3 Membrane Transport

For homogeneous membranes, the flux Jx of a solute across the membrane at any point is given by

(21.1-1)

where 3)M is the diffusivity of the solute in the membrane, dC/dx is the solute concentration gradient atthe external surface of the membrane, Jv is the transmembrane solvent velocity, a is the reflection coefficient,and C is the local solute concentration within the membrane.28 The reflection coefficient is a measure of asolute's ability to enter the pores of the membrane. For solutes able to pass the membrane freely, a = 0,while for solutes completely excluded by the membrane, a — I. Integration of Eq. (21.1-1), at steadystate, results in the following expression:

(21.1-2)

where PM is the membrane permeability, AC is the concentration difference across the membrane, andC is the average concentration^ in the membrane. When the transmembrane Peclet number, Pe =J1Xl - O)IPM* is l e s s *nan 3.0, C is given by

(21.1-3)

where Cw is the feed-side concentration at the membrane wall.28 Use of Eqs. (21.1-2) and (21.1-3) requiresknowledge of the solute concentrations at the feed- and dialysate-side membrane surfaces. As a firstapproximation, the solute concentration at the dialysate-side membrane surface can be equated to the bulkdialysate concentration. The concentration at the feed-side membrane surface, CV, can be related to thebulk feed-side concentration, CF, using thin-film boundary-layer theory.29 For dialysis, Villarroel et al.28

have shown that the ratio of wall concentration to bulk concentration is given by

(21.1-4)

where £ = [exp (Pe) - I]"1 and ^ = [exp (0) - I]"1; for hollow-fiber membranes, the parameter 6 isgiven by 0 = 0.109Jy(Xd3NZQpS)2)0333, where Jt is the axial distance from the beginning of the membraneto the point at which Cw is being evaluated.29 In dialytic processes, the transmembrane solvent velocity Jv

is generally less than 3 x 10"3 cm/s and permeability coefficients are of the order of 10~4 cm/s, so thatthe Peclet number is usually less than 0.1. When high transmembrane pressures are obligatory because offlow constraints, the value of Jv may increase, and convection may become an important contribution tooverall mass transfer.

A number of attempts have been made to correlate PM with membrane structure. The concept generallyaccepted by membrane chemists for diffusion in very highly swollen membranes, such as those used indialysis, is that solute diffusion occurs through solvent held in the membrane structure. The membranestructure may be flexible, as in cellulose hydrogels, or it may be glassy, as in porous polysulfone. In eithercase, the diffusion path is thought to consist of channels with a distribution of sizes and whose lengths aregreater than the membrane thickness by virtue of their tortuous paths through the membrane.

Yasuda et al.25 have developed the concept of a homogeneous solvent-swollen membrane in whichthermally induced movement of segments of randomly coiled polymer molecules leaves an interstitial freevolume available for solute transport. They concluded that the permeability characteristics of highly swollensystems cannot be represented by a single coefficient. Values of solute and solvent permeabilities dependon the conditions of measurement, in particular, the magnitude of diffusive flux relative to convective flux.

FIGURE 21.1-2 Basic dialyzer configurations: (A) plate-and-frame, (B) coil and (C) hollow fiber. (Cour-tesy of Travenol Laboratories, Deerfield, IL.)

OiALYSATEOUT

FEEO

OULYSiKrEM

FEEOOUT

NTERLEAVEDFLAT SHEETMEMBRANE ANDSUPPORT PLATES

DIALYSATE M

TUBULARMEMBRANE

MESH SPACERFEED OUT

FECOM

HOLLOW FIBERMEMBRANE BUNDLE

FEED OUTDIALYSATEOUT

DIALYSATEIN

DtACTSATE OUT

FEED M

FIGURE 21.1-3 Industrial hollow-fiber dialysis unit. (Courtesy of Enka AG, Wuppertal, West Germany.)

Yasuda and coworkers found the following relationships between diffusive permeability and membranestructure:

1. The permeability of solutes, whose size is small compared to the membrane pore size, is propor-tional to the degree of membrane hydration.

2. Membrane permeability decreases exponentially with increasing molecular size, where the latter isexpressed in terms of molecular cross-sectional area.

3. Solute reflection coefficients change markedly when solute size approaches the average pore sizeof the membrane.

A different approach was used by Klein et al.30 for a range of dialytic membranes that included notonly the homogeneous, swollen gels studied by Yasuda and coworkers, but also glassy polymers havingporous structures. Their model was based on hypothetical pore structures whose dimensions are hydrody-namically equivalent to cylindrical pores.3132 Since neither the fractional cross-sectional area of the poreopenings AP nor the actual pore path length lP can be measured independently, an experimental method isused to derive their ratio. From the Poiseuille relationship, the hydraulic permeability L& is related to theratio Ap/lp and the hydrodynamic pore radius /> by

(21.1-5)

where n is the viscosity of the solution. The value of L9 is determined by measuring the volume flux ofwater across the membrane in response to an applied pressure gradient. Water also can diffuse through themembrane in response to a concentration gradient. Using a diffusional model,3132 we can relate the diffusivepermeability coefficient of the membrane PM to the ratio Ap/lP by

(21.1-6)

where q is the ratio of solute to pore diameter, SD is the solute diffusion coefficient in the solvent, K1 is apower series in qy and £>w is the effective solute diffusivity in the membrane.

For a solute, such as tritiated water, whose radius is small compared to the pore radius, q is approxi-mately equal to zero, Kx = 1, and Eqs. (21.1-5) and (21.1-6) can be solved simultaneously to yield30

Test Solute Moteculor WeightFIGURE 21.1-4 Ratio of solute diffusivity in the membrane (SDM) to solute diffusivity in solution (3D) asa function of solute molecular weight for three cellulosic dialysis membranes.

(21.1-7)

Once the average pore size of the membrane has been determined using Eq. (21.1-7), the permeabilityof any other solute of known radius through the same membrane can be calculated from simultaneoussolution of Eqs. (21.1-5) and (21.1-6). In the absence of specific solute-membrane interactions, such ascharge or hydrophobic bonding, this model is useful for predicting solute permeability coefficients througha characterized membrane, for values of q less than 0.6.

On the basis of either model, it is clear that dialytic transport will decrease with increasing solutemolecular size, not only because of the smaller solution diffusivity of a large molecule but also because ofincreasing values of q. This effect is seen in Fig. 21.1-4, where the ratio of diffusivity in the membrane(£>w) to diffusivity in solution (£>) is plotted as a function of solute Stokes radii for several dialysismembranes.

The membrane models described above relate solute and solvent fluxes to concentration differences atthe dialyzer's two membrane-solution interfaces, values that cannot be determined experimentally. Since,in practice, dialysis always involves the movement of solute from one bulk phase to another, some meansof expressing the fluxes in terms of bulk concentrations is needed.

As solute is removed from the feed-side membrane-solution interface by dialysis, the layer is depletedand its concentration must be restored from the bulk solution. In laminar flow, which is usual in small-bore hollow-fiber and thin-film plate-and-frame devices, there is no convection and repletion of the inter-facial layer is solely by diffusion from the bulk solution. As such diffusion occurs, the concentration gradientfrom the bulk solution to the interfacial layer decreases. Thus, the rate of restoration of the interfaciaisolute concentration is a function of the solute size, the transmembrane flux, and the rate of solute supply,that is, the axial feed flow rate.

As solute crosses the membrane, it must be transported away from the dialysate-side interfacial layerto maintain the transmembrane concentration difference. The rate at which this occurs is a function of theconcentration gradient between the membrane-dialysate interface and the bulk dialysate. Thus, mass transferon the dialysate side of the membrane is a function of the transmembrane flux, the bulk dialysate concen-tration, which in turn is dependent on the dialysate velocity, and, in the case of laminar flow, the solutediffusivity.

Thus, the resistance to mass transfer from the bulk feed stream to the bulk dialysate stream can beconsidered the sum of three terms: concentration polarization on the feed and dialysate sides of the mem-brane and the resistance to diffusion in the membrane itself. For laminar flow devices, these resistances

Stokes Rodius

are combined to give the following expression for the overall mass transfer coefficient &oV-

(21.1-8)

where kD and kF are the mass transfer coefficients for the concentration polarization layers on the dialysateand feed sides of the membrane, respectively. Both kF and kD can be considered in terms of the ratio ofthe diffiisivity of the solute in a hypothetical stagnant layer to the thickness of the layer. Figure 21.1-4shows that as solute size increases diffusivity in the membrane decreases more rapidly than diffusivity inthe solution. Translating this observation to Eq. (21.1-8), we see that membrane resistance, which is directlyproportional to solute diffusivity in the membrane [Eq. (21.1-6)], will become the dominant factor con-tributing to /fcov a s solute molecular size increases. However, for small solutes, it is clear from Eq.(21.1-8) that membrane resistance will limit mass transfer only when kD and kF are very large relative toPM. Therefore, dialyzer design and fluid dynamics, in the first instance, can limit transport if they permitexcessive concentration polarization layers to develop. When concentration polarization has been mini-mized, further improvements in the rate of mass transfer depend on increased membrane permeability,which in turn will generate further demands on the design of the fluid pathways to minimize concentrationpolarization.

21.1-4 Overall Mass Transfer in Dialyzers

In the previous section, mass transfer was discussed for a single point on the membrane. For practicalapplications, this analysis must be extended to describe mass transfer for the dialyzer as a whole. Neglectingconvective contributions, we can write the following equation for the mass of solute dm transferred acrossan element of membrane of length dx and area dAM per unit time (Fig. 21.1-5):

(21.1-9)

where CF and CD are the solute concentrations in the feed and dialysate, respectively, in element dx.Equation (21.1-9) can be integrated along the length of the membrane to give the following equation foroverall mass transfer:

(21.1-10)

where ACx^0 and ACx m, are the transmembrane concentration differences at either end of the device. Forcountercurrent flow, ACx 0 = CFi — C00 and ACx 1 = CFo — CDi. This leads to the following equationfor overall mass transfer in a dialyzer with this flow configuration:

(21.1-11)

A similar expression can be derived for cocurrent flow by redefining ACx m0 and ACx«/.If ultrafiltradon is negligible, inlet and outlet flows are equal for both the feed and dialysate streams.

Using this assumption, the following overall mass balances may be written for each stream:

(21.1-12)

Finally, the performance of a dialyzer can be described in terms of a "dialysance" D, which is definedas the rate of mass transfer divided by the concentration difference between inlet feed and inlet dialysate;that is,

FIGURE 21.1-5 Diffusive mass transfer across an incremental length of membrane dx.

DIALYSATE

MEMBRANE

FEED

(21.1-13)

Combining Eqs. (21.1-11)-(21.1-13), we arrive at the following expression for the dialysance of acountercurrent dialyzer in terms of membrane properties (kov. AM) and operating conditions (QF. Q0):

(21.1-14)

Note that if the feed and dialysate flows are essentially equal (QF = QD), then

(21.1-15)

Similar expressions can be derived for other flow configurations.33

The foregoing assumes that the solute is distributed freely in the solvent phase. This assumption maynot always be valid; for example, in protein solutions, small solutes may bind to the protein molecules andexist in equilibrium between the free and bound states. In such circumstances, modifications of Eq.(21.1-14) must be used to determine dialysance.34

Equation (21.1-14) can be used to estimate the degree of separation of two solutes by a given dialyzer.For any solute, the fractional extraction into the dialysate, E, of solute from the feed stream is given by

(21.1-16)

For negligible ultrafiltration, this reduces to

(21.1-17)

If the inlet concentration of solute in the dialysate is zero (Q,, = 0), Eq. (21.1-17) further reduces to

(21.1-18)

By combining Eqs. (21.1-14) and (21.1-18), fractional extraction can be obtained as a function ofkoVAM/QF for different ratios of feed to dialysate flow rate. These relationships are shown in Fig. 21.1-6.The separation factor ajk in a dialytic process can be considered to be the ratio of the fractional masses oftwo solutes removed from their common feed stream, under a given set of operating conditions. By applyingEq. (21.1-16) to each solute, we can see that aJk is equal to the ratio of the fractional extractions of thetwo solutes. Thus, Fig. 21.1-6 can be used to estimate the relative separation of two solutes from aknowledge of their overall mass transfer coefficients, membrane area, and feed and dialysate flow rates.

21.1-5 Contributions of Convection to Mass Transfer

The above derivations are based on the assumption that convective mass transfer is negligible, that is,significant ultrafiltration is not occurring. If significant ultrafiltration does occur, mass transfer will beenhanced by the addition of a convective component to the diffusive mass transfer described above. Ul-trafiltration will occur from the feed to the dialysate in response to a pressure gradient which may be eitherapplied, in order to concentrate the feed, or obligatory, as a consequence of the geometry of the deviceand the desired feed flow rate. Less commonly, if the feed is highly concentrated and the dialysate dilute,ultrafiltration may occur from the dialysate to the feed in response to an osmotic gradient. In the first case,ultrafiltration will enhance solute transport while in the second it will impede it.

As discussed earlier, at a steady state the overall solute flux J5 at a point on the membrane can beconsidered to consist of the sum of a diffusive and a convective component:

(21.1-2)

where the first term, PM AC, represents diffusive transfer and the second term, Jv (1 — a) C, convectivetransfer. The relative magnitudes of diffusive and convective transfer are functions of the membrane perme-ability PM and the reflection coefficient cr. For species of small molecular size, the resistance to diffusivetransfer through the membrane is low, that is, PM is large, and diffusive transfer always significantly exceeds

FIGURE 21.1-6 Fractional extraction £ as a function of kowAM/QF for varying feed flow to dialysate flowratios (QF/QD).

convective transfer at low Peclet numbers. However, as solute molecular size increases, membrane perme-ability decreases logarithmically,35 whereas reflection coefficients increase at a much lesser rate and therelative importance of convection to overall mass transfer increases. This effect is illustrated in the followingexample.

Figure 21.1-7 shows membrane permeability and reflection coefficient as a function of molecular weightfor a cellulosic dialysis membrane.36 For two solutes, A and B, having molecular weights of 200 and 2000daltons, respectively, feed-side concentrations at a point on _the membrane surface of 0.1 g/cm3, andnegligible dialysate-side concentrations, AC = 0.1 g/cm3 and C = 0.067 g/cm3. For solute A, PM - 3.6X 10~4 cm/s and 1 - a = 0.8. Substituting these values into Eq. (21.1-2), we obtain

(21.1-19)

As the ultrafiltration rate increases from 0 to 0.5 x 10~4 cm/s, J5 increases from 3.6 x 10"5 g/s-cm2 to3.9 x 10"5 g/s-cm2; that is, convection increases overall solute transfer by 8%. For solute B, PM = 0.40X 10"4 cm/s and 1 - a = 0.6. Under these conditions, Eq. (21.1-2) reduces to

(21.1-20)

Now, for an increase in ultrafiltration rate from 0 to 0.5 x 10"4 cm/s, Jx increases from 0.40 x 10~5 to0.60 x 10"5 g/s-cm2, an increase in overall mass transfer of 50%.

The above considerations deal with a point on the membrane. Expansion of these concepts to thedialyzer as a whole requires that the feed- and dialysate-side solute concentrations at a point on themembrane be expressed in terms of known concentrations. Such an approach has been developed bySchindhelm et al.,37 who, by making the assumptions of a zero inlet dialysate concentration and a lineardialysate concentration profile, developed the following expression for the dialysance of solutes greaterthan 300 daltons molecular weight in a countercurrent dialyzer in the presence of simultaneous diffusionand convection. [Note that the equations for X and Y given in Ref. 37 contain misprints; the correctexpressions are as given in Eqs. (21.1-22) and (21.1-23).]

(21.1-21)

where

(21.1-22)

FRAC

TION

AL

EXTR

ACTI

ON (E

)

SOLUTE MOLECULAR WEIGHT (Daltons)FIGURE 21.1-7 Membrane permeability P ( • ) and reflection coefficient (D) versus solute molecularweight for a regenerated cellulose membrane (Cuprophan 150 PM, Enka AG, Wuppertal, West Germany).Data from Ref. 36.

(21.1-23)

(21.1-24)

A similar analysis, but one mat requires a numerical methods solution, has been described by Jaffrinet al.38

The contribution of convection to overall mass transfer, illustrated previously for a point on the mem-brane, can now be estimated for the dialyzer as a whole. Referring to the previous example, let us considerthe contribution of convection to dialysance for the two solutes A and B in a dialyzer containing 104 cm2

of cellulosic membrane operating in a countercurrent configuration with feed and dialysate flow rates of 8and 16 cm3/s, respectively. For such a membrane, the overall mass transfer coefficients of solutes A andB are on the order of 3.23 x 10~4 and 0.37 x 10"4 cm/s, respectively, while the reflection coefficientsare on the order of 0.2 and 0.4, respectively. Using these data in Eqs. (21.1-21)-(21.1-24), we can calculatethat the dialysance of solute A will increase by 12% (from 2.47 to 2.78 cm3/s) as the ultrafiltration rateincreases from 0 to 0.5 cnrVs, while the same increase in ultrafiltration rate will result in an increase of84% (from 0.36 to 0.66 cnrVs) in the dialysance of solute B. This example further demonstrates theimportant contribution of convection to overall mass transfer for larger-molecular-weight species.

21.1-6 Estimation of Overall Mass Transfer Coefficient

Use of Eqs. (21.1-21)—(21.1 -24) for calculation of dialysance requires the knowledge of a dialyzer's overallmass transfer coefficient kov. An experimentally determined value of kow for the solute, membrane, anddialyzer configuration of interest is to be preferred. Estimates of kow can be made from a knowledge ofthe geometry and fluid dynamics of the dialyzer and the nature of the solute. Such methods are describedbelow; however, they should be used only to provide a rough guide since they involve numerous assump-tions, such as well-defined geometries and Newtonian fluid dynamics. In addition, it can be difficult todetermine membrane area accurately in some dialyzers, such as plate-and-frame devices, where membrane

REFLECTION COEFFICIENT MMEMB

RANE

PER

MEAB

ILIT

Y* P

11Hx

IOW

sec)

masking by the support plates may occur. In such circumstances, experimental determination of the productof mass transfer coefficient and effective membrane area (kovAM)y using Eqs. (21.1-11) and (21.1-12), isthe only practical means of obtaining useful design data. Regardless of whether or not an experimentalmeasurement or a calculated estimate of Ic0V is used, it is important to make the determination at theprojected operating temperature because of the strong temperature dependence of solute diffusivity.

As discussed earlier, the overall mass transfer resistance (/?ov = l/&ov) c a n ^ considered the sum ofthe individual boundary-layer and membrane resistances; that is,

(21.1-25)

If the individual resistances can be estimated, then a value for &Ov can be derived from Eq. (21.1-25).Estimation of the membrane resistance (RM = \IPM) at a point has been discussed already, and this valuecan be assumed to be invariate along the length of the dialyzer. Boundary-layer resistances generally dovary along the length of the dialyzer; however, in practice, average resistances can be used in Eq. (21.1-25). Estimates of these can be obtained for simple geometries such as parallel plates or hollow fibers bymaking use of the analogy between heat and mass transfer.

Colton et al.39 used a Graetz-type solution to calculate a log-mean feed-side Sherwood number (ShF =fc^/3D, where h is the channel height) for laminar flow in a flat duct with permeable walls as a functionof the wall Sherwood number (Sh^ = P^/3)) and the dimensionless length of the dialyzer. This analysisassumes a constant dialysate-side resistance which is combined with the membrane resistance in Sfv; anultrafiltration rate of zero also is assumed. Subsequently, Cooney et al.40 and Jagannathan and Shettigar41

included a variable dialysate-side resistance in reformulating the analysis as a conjugated boundary-valueproblem for parallel-plate and hollow-fiber configurations, respectively. These latter approaches requiresolution by numerical methods. The analysis of Jagannathan and Shettigar41 for hollow fibers, which alsoconsiders the effects of ultrafiltration, predicts that, for constant membrane area, fiber diameter and lengthhave very little effect on overall mass transfer for low-permeability membranes. However, for high-perme-ability membranes, overall mass transfer decreases with increasing membrane diameter.

Unlike the feed side, flow on the dialysate side of the membrane is often turbulent. For turbulent flow,the dialysate-side mass transfer coefficient can be described in terms of a Sherwood number which can berelated to two other dimensionless groups, the Reynolds number (Re) and the Schmidt number (Sc), by acorrelation of the form:

(21.1-26)

where a, b, and c are constants. Using a stirred-batch dialyzer, Kaufmann and Leonard42 found values of0.68 and 0.38 for b and c, respectively, for Re > 20,000, while Marangozis and Johnson43 determinedvalues of 0.65-0.70 and 0.33 for a variety of published data. Values of the remaining constant, a, varyand may depend on the geometry of the system.44

21.1-7 Fluid Dynamics

The contribution of concentration polarization to overall dialytic mass transfer resistance suggests thatdialyzers should be operated with flows in the turbulent flow region to minimize boundary-layer formation.While this is typically the case on the dialysate side of the membrane, dialyzers usually are operated withlaminar flow on the feed side. In hemodialysis, this is partly to avoid undue mechanical stress on the bloodcells which may result in their destruction. However, a consideration of fluid dynamics dictates that feed-side flow be laminar in nearly all applications.

For laminar flow and in the absence of ultrafiltration, the pressure drop (AP) on the feed side of adialyzer is given by the Hagen-Poiseuille equation for hollow fibers,

(21.1-27a)

or parallel plates,

(21.1-27b)

where fx is the viscosity of a fluid flowing at a rate QF through a dialyzer containing N channels of diameter<i, or height h and width w, and length /. In the case of a hollow-fiber dialyzer, the Reynolds number isgiven by

(21.1-28)

where the variables are as defined for Eq. (21.1-27a) and p is the fluid density. Typically, hollow-fiberdialysis membranes have an internal diameter of 0.025 cm, so that a dialyzer of 2.5 X 1& cm2 surfacearea might contain 13,000 fibers, each 20 cm in length. For a solution with a density of 1.0 g/cm3 and aviscosity of 0.05 dyn-s/cm2 flowing through the dialyzer at 200 cm3/s, the Reynolds number is 16 [Eq.(21.1-28)]. well within the laminar flow region, while the pressure drop is 1.61 X 106 dyn/cm2 (Eq.(21.1-27a)]. Such a pressure drop mandates a high feed-side pressure which leads to difficulties in con-trolling ultrafiltration, particularly for highly permeable membranes and, in addition, may place an unduemechanical stress on the membrane. For these reasons, fluid flow on the feed side of dialyzers remainswell inside the laminar flow region.

In an attempt to decrease boundry-layer resistance, while at the same time maintaining laminar flow,Abel et al.*5 have designed a parallel-plate dialyzer that uses vortex mixing to disrupt the boundary layers.Through this mechanism, they were able to reduce overall mass transfer resistance by a factor of 2-3compared with conventional parallel-plate dialyzers containing the same membrane.

21.1-8 Dialysis as a Unit Operation

Dialysis can be used as a unit operation in two basic configurations—as a batch process or in a continuousprocess stream. Figure 21.1-8 depicts the flow diagram for batch dialysis. A well-mixed reservoir of solutionvolume V and solute concentration CF is dialyzed continuously against a dialysate with an inlet soluteconcentration CDi. The feed and dialysate streams enter the dialyzer with flow rates QF2fo& QD, respectively.The feed stream exiting the dialyzer returns to the feed reservoir while the spent dialysate is discarded.Such a configuration could be used for stripping low-molecular-weight contaminants from a high-molecular-weight product. It represents, in essence, the practice of hemodialysis with the reservoir representing thebody solute pool.

A mass balance for the reservoir may be written

(21.1-29)

where the dialysance D is given by Eq. (21.1-14) or (21.1-21). A volume balance also may be written forthe reservoir:

(21.1-30)

where V0 is the initial reservoir volume and QVF is the rate of ultrafiltration from feed to diaiysate.Substitution for V in Eq. (21.1-29) leads to

(21.1-31)

Rearrangement of Eq. (21.1-31) and integration lead to the following expression for the concentration ofsolute in the reservoir as a function of time:

(21.1-32)

BATCHRESERVOIR

DlALYZER

DIALYSATE FIGURE 21.1-8 Schematic representation of batch dialysis.

DIALYSATEFIGURE 21.1-9 Schematic representation of dialysis in a continuous process.

where /3 is given by

(21.1-33)

If ultrafiltration is negligible, that is, QUF = 0, then the volume of the reservoir is constant and Eq.(21.1-31) reduces to

(21.1-34)

In this instance, the solute concentration of the reservoir is given by

(21.1-35)

Dialysis may be operated in a variety of configurations as part of a continuous process. Figure 21.1-9illustrates a typical example. Here, a well-mixed reservoir of volume V represents a reaction vessel wherea high-molecular-weight substrate, introduced into the reservoir in a reagent stream of flow rate QRy isprocessed to yield a small-molecular-weight product with a concentration of CF in the reservoir. The reactionvolume is maintained by removing solvent by ultrafiltration through the dialyzer. An example of such aprocess might be the enzymatic hydrolysis of corn starch to yield low-molecular-weight saccharides. It isassumed that the enzymatic reaction rate G is constant. The reaction mix passes through a dialyzer whereproduct is removed while the substrate returns to the reservoir; the high-molecular-weight enzyme alsowould be retained in the reservoir by virtue of its negligible dialysance.

Again, a mass balance for the reaction product may be written

(21.1-36)

and a volume balance can be written

(21.1-37)

Solution of Eqs. (21.1-36) and (21.1-37) leads to the following equation for CF as a function of time:

(21.1-38)

where 7 is given by

(21.1-39)

REAGENT STREAM

PROCESSREACTOR

OfALYZER

FIGURE 21.1-10 Schematic representation of a multistage dialysis process for continuous fractionationof solutes. The process combines a nonselective solvent stripper (concentrator) with two crosscurrentdialysis cascades (Dl-Dn and Dl-Dn'). (Reprinted by permission of the American Institute of ChemicalEngineers.)

Dialysis also may be used in cascade configurations to give multistage separations. Series staging ofdialyzers serves to increase the area available for dialysis and the equations developed in the precedingsections are applicable. Noda and Gryte9 have described a multistaged system in which two crosscurrentdialyzer cascades are combined with a nonselective solvent stripper, such as a reverse osmosis unit, toincrease the slope of the fractional extraction versus kovAM/QF curves shown in Fig. 21.1-6 and thus therelative separation of two solutes. Their process configuration is shown in Fig. 21.1-10. Assuming identicaldialyzers, equal feed and dialysate flow rates throughout the system, and negligible ultrafiltration, theyhave shown that the fractional extraction of the system as a whole is given by

(21.1-40)

where n is the number of dialyzers in each cascade. Remembering that the separation factor ajk is equal tothe ratio of the fractional extractions of components j and k, we can use Eq. (21.1-40) to calculate theimprovement in separation brought about by using the system shown in Fig. 21.1-10 rather than a singledialyzer. For example, if the values of kovAMIQF for two solutes in a given dialyzer with equal feed anddialysate flow rates are 0.5 and 1.5, respectively, the fractional extractions of the two solutes are 0.33 and0.60 and their separation factor is 1.82. If the same dialyzer then is used in the system shown in Fig.21.1-10, with three dialyzers in each cascade (n = 3), Eq. (21.1-40) predicts fractional extractions of 0.11and 0.77 for the two solutes and a separation factor of 7.00. Although multistaging of dialyzers clearlyimproves separations, fluid flow considerations place a practical limitation on the number of dialytic stageswhich can be incorporated.

21.2 ELECTRODIALYSlS

In electrodialysis, electrolytes are transferred through solutions and membranes by an electrical drivingforce. Electrodialysis is used to change the concentration or composition of solutions, or both. The processusually involves multiple, thin compartments of solutions separated by membranes that allow passage ofeither positive ions (cations) or negative ions (anions) and block the passage of the oppositely charged ions.But the process may be operated with only one membrane that separates two electrode-rinse solutions topurify one of the solutions (e.g., production of salt-free sodium hydroxide).

As in any membrane process, selective membranes are critically important in electrodialysis. Therefore,the nature of the ion-exchange membranes used in electrodialysis will be discussed first.

21.2-1 Ion-Exchange Membranes

Seemingly solid membranes are actually a jumble of intertwined polymer chains. In most ion-exchangemembranes, the polymer chains are attached to each other at points by crosslinking. Transferring ions passthrough spaces between the intertangled polymer chains. Commercially available ion-exchange membranesusually have high crosslinking densities (i.e., the polymer chains are attached to each other at many points).The segments between points of crosslinking are short enough to be relatively inflexible. The short, in-

HIGH MOLECULARWEIGHT PRODUCT

FILTRATE

CONCENTRATOR,1

FEED

LOW MOLECULARWEIGHT PRODUCT

CONCENTRATE

RECY

CLE

flexible polymer segments and small spaces between segments result in high resistances to transfer becauseof interactions between the moving ions and the polymer segments.

Cation-exchange membranes are permeable to cations but not anions. Anion-exchange membranes arepermeable to anions but not cations. The source of this ability to discriminate between cations and anionsis discussed with the aid of Fig. 21.2- L

In Fig. 21.2-1, polymer chains are shown that have negatively charged groups chemically attached tothem. The polymer chains are intertwined and also crosslinked at various points. Positive ions are showndispersed in the voids between polymer chains but near a negative fixed charge. The fixed negative chargeson the chains repulse negative ions that try to enter the membrane and exclude them. Thus, negative ionscannot permeate the membrane, but positive ones can (i.e., the membrane is a cation-exchange membrane).If positive fixed charges are attached to the polymer chains instead of negative fixed charges, positive ionscannot permeate the membrane, but negative ions can; and the membrane is an anion-exchange membrane.This exclusion as a result of electrostatic repulsion is termed Donnan exclusion.

For an ion-exchange membrane to be practical for low-cost processing, it must be low in resistance toion transfer as well as being ion selective. To decrease resistance, a membrane may be made thinner, thecrosslinking density may be decreased, or the fixed-charge density may be increased. Decreases in cross-linking density tend to increase the length and flexibility of polymer segments between points of cross-

FIXED NEGATIVELY CHARGED EXCHANGE SITE; I.E.,SO§MOBILE POSITIVELY CHARGED EXCHANGEABLE CATION; I.E.,Na*POLYSTYRENE CHAINDIVINYLBENZENE CROSSLINK

FIGURE 21.2-1 Representation of an ion-exchange membrane.

linking. This increases the size of the void spaces between segments, which makes it easier for ions tomove through the membrane. Increases in fixed-charge density also tend to increase the size of the voidspaces because of the repulsion of like fixed charges. But, if the void spaces between polymer chainsbecome too large due to either decreased crosslinking density or increased fixed-charge density, there canbe volumes in the center of the voids that are not affected by the fixed charges on the chains (as at pointA in Fig. 21.2-1) because the repulsion of fixed charges falls off rapidly with the distance away from thecharges. Such volumes that are unaffected by fixed charges result in ineffective repulsion of the undesiredions and lowered selectively. Usually, a compromise between selectivity and low resistance must be made.In membranes now available, it has been possible to combine excellent selectivity with low resistance,high physical strength, and long lifetimes.

Heterogeneous ion-exchange membranes have been made by incorporating ion-exchange particles insolutions of film-forming polymers and allowing the solvents to evaporate. They also have been made bydispersing ion-exchange materials in partially polymerized film-forming polymers, casting films of thedispersion, and completing the polymerization.

Homogeneous ion-exchange membranes have been made by graft polymerization of anionic or cationicmoieties into preformed films, by casting films from a solution of a linear film-forming polymer and alinear poly electrolyte and allowing the solvent to evaporate, or by polymerization of mixtures of reactantsthat can undergo polymerization by either addition or condensation reactions. In the last method, one ofthe reactants must be capable of being converted to either anionic or cationic moieties.

The resistances of commercially available ion-exchange membranes range from 2 to 20 ohm-cm2,depending on the types of membrane and solution being treated. The current efficiencies attainable withcommercially available membranes range from 85 to 95% for most operations. Names and addresses ofsuppliers of ion-exchange membranes are given in Table 21.2-1.

With membranes made by the methods described above, the maximum concentrations of the solutionsthat can be processed efficiently are limited because Donnan exclusion decreases with increasing solutionconcentration. With decreases in Donnan exclusion, the current efficiency decreases. Cation-exchangemembranes based on fluorocarbon polymers have been developed with which very concentrated solutionscan be treated with high current efficiencies. The efficiency achievable with these membranes has beenshown to be due to a phenomenon termed "ion clustering" within the membranes.'"5 Because of theinternal structure of such membranes, both high current efficiencies and low resistances can be achievedin concentrated solutions.

These membranes are based on perfluorinated polymers that will withstand oxidative conditions andhigh temperatures that would be destructive to hydrocarbon-based membranes. Sulfonic or carboxylic acidgroups, or both, are affixed chemically to the perfluorinated polymers to impart cation-exchange charac-teristics.

Some of the membranes are composite structures. These have a thin "tight" layer that provides highcurrent efficiencies (i.e., high transport numbers for cations) backed by a thicker layer of "looser" polymerto provide low resistance combined with strength and convenience of handling. Typical resistances rangefrom 2 to 4 ohm-cm2 at room temperature depending on the solutions in which they are used. Typicaltransport numbers are 0.94-0.97, which provide current efficiencies of 94-97%.

The perfluorinated membranes have excellent dimensional stability and long lifetimes. For example,they change in dimensions only 4-6% after being treated in 2% sodium hydroxide solutions and then in24% sodium chloride/1 % sodium hydroxide in preparation for use in chlor-alkali cells. (By assembling themembranes after treatment in such solutions, wrinkle-free surfaces are obtained.)

The major use of perfluorinated membranes, at present, is as separators in chlor-alkali cells. Thecombination of low resistances, high current efficiencies at high solution concentrations, and high temper-atures that can be achieved results in 20-30% lower energy requirements than those achieved with dia-phragm or mercury cells. The long membrane lifetime (typically 2 years) results in low cost for membranereplacement. (Asbestos diaphragms usually last for only a year or less.)

The caustic soda produced contains very low levels of salt (less than 60 ppm at 50% caustic strength).In addition, the environmental problems encountered with mercury cells and the potential for health hazardsfrom handling asbestos are eliminated. The chlorine gas that is coproduced is also a highly pure product.

TABLE 21.2-1 Suppliers of Membranes

Asahi Chemical Industry Co., Ltd., 8345 Yako-cho, Daishigawara, Kawasaki, JapanAsahi Glass Co., Ltd., 14,2-chome, Marunouchi, Chikoda-ku, Tokyo, JapanE. I. Du Pont de Nemours, Wilmington, DelawareIonac Chemical Co., Birmingham, New JerseyIonics, Inc., 65 Grove Street, Watertown, MassachusettsTokuyama Soda Co., Ltd., Tokuyama, JapanZerolit, Ltd., 632-52 London Road, Isleworth, Middlesex, England

For new plants, the capital investment for membrane cells is lower than that for comparable diaphragmor mercury cells. The high conversion of salt reduces the cost of brine handling facilities below that neededfor mercury cells. The high concentrations of caustic from membrane cells reduces the cost of evaporatorsbelow that needed for diaphragm cells. Moreover, existing mercury cells or diaphragm cells can be replacedby membrane cells at lower costs than required for building new plants. Additional information aboutperfluorinated membranes can be obtained from Asahi Chemical Industry Co., Asahi Glass Co., E. I. DuPont Company, and Tokuyama Soda Co. (see Table 21.2-1).

A number of other uses for these membranes have been or are being studied. Some of these uses are:

1. In Donnan dialysis for decontamination of radioactive wastes from nuclear power plants.2. For removal of zinc from textile wastes.3. For recovering copper from industrial wastes or from low-grade ores.4. For recovery of nickel, cadmium, and chromium from electroplating wastes.5. For stripping common pollutants, such as mercury, cadmium, or lead, from industrial wastes to

meet environmental requirements.6. For production of certain high-purity materials, such as potassium stannate.

21.2-2 Electrodialysis Process and Electrodialysis Stacks

In electrodialysis, cation-exchange membranes are alternated with anion-exchange membranes in a parallelarray to form thin solution compartments (0.5-1.0 mm thick). The entire assembly of membranes is heldbetween two electrodes to form an electrodialysis stack as shown in Fig. 21.2-2. A solution to be treatedis circulated through the solution compartments. With the application of a dc electrical potential to theelectrodes, all cations transfer toward the cathode (negative electrode) and all anions move toward theanode (positive electrode). The ions in the even-numbered compartments can transfer through the firstmembranes they encounter (cations through cation-exchange membranes, anions through anion-exchangemembranes), but they are blocked by the next membranes they encounter as indicated by the arrows in thediagram. Ions in the odd-numbered compartments are blocked in both directions. Ions are removed fromthe solution circulating through one set of compartments (even) and transferred to the other set of com-

DESALTED PRODUCT WATER

CONCENTRATED BRINE

FEED WATER

A = ANION-PERMEABLE MEMBRANEC = CATION-PERMEABLE MEMBRANE

FIGURE 21.2-2 Simplified representation of the electrodialysis process.

partments (odd). Ion-depletion is accomplished for one solution, and ion concentration is accomplished forthe second solution.

Figure 21.2-3 is an exploded view of part of an electrodialysis stack that shows the main components.Component 1 in Fig. 21.2-3 is one of the two end frames, each of which has provisions for holding anelectrode and introducing and withdrawing the depleting, the concentrating, and the electrode-rinse solu-tions. The end frames are made thick and rigid to resist bending when pressure is applied to hold the stackcomponents together. The inside surfaces of the electrodes may be recessed, as shown, to form an electrode-rinse compartment when an ion-exchange membrane, component 2, is clamped in place. Components 3and 5 are spacer frames. Spacer frames have gaskets at the edges and ends so that solution compartmentsare formed when ion-exchange membranes and spacer frames are clamped together.

Usually, the supply ducts for the various solutions are formed by matching holes in the spacer frames,membranes, gaskets, and end frames. Each spacer frame is provided with solution channels (at point E inFig. 21.2-3) that connect the solution-supply ducts with the solution compartments. The spacer frameshave mesh spacers or other devices in the compartment spaces to support the ion-exchange membranes andso to prevent collapse when there is a differential pressure between two compartments. These mesh spacersare selected or designed to aid in lateral mixing, which decreases the thickness of boundary layers.

An electromembrane stack usually has many repeated sections, each consisting of components 2, 3, 4,and 5, with a second end frame at the end. These repeating units are termed cell pairs.

There are three basic types of electrodialysis stacks—tortuous-path, sheet-flow, and unit-cell stacks. Inthe tortuous-path stack, the solution flow path is a long, narrow channel which makes several 180° bendsbetween the entrance and exit ports of a compartment as illustrated in Fig. 21.2-4. The bottom half of thespacer gasket in Fig. 21.2-4 shows the individual narrow solution channels and the cross straps used topromote mixing, whereas the cross straps have been omitted in the top half of the figure so that the flowpath could be better depicted. The ratio of channel length to width is high, usually greater than 100:1.Spacer screens to support the membranes may or may not be used in tortuous-path stacks.

In sheet-flow stacks, spacer screens almost always are needed because the width between gasketingdevices is much greater than that in the usual tortuous-path stacks. The solution flow in sheet-flow stacksis in approximately a straight path from one or more entrance ports to an equal number of exit ports asindicated in Fig. 21.2-3. As the solution flows in and around the filaments of the spacer screens, a mixingaction is imparted to the solutions to aid in reducing the thicknesses of diffiisional boundary layers at thesurfaces of the membranes.

Solution velocities in sheet-flow stacks are typically in the range of 5-14 cm/s, whereas the velocitiesin tortuous-path stacks usually are much higher, 30-100 cm/s. The drop in hydraulic pressure through asheet-flow stack, and thus the pumping power, is normally lower than that through a tortuous-path stackbecause of the lower velocities and shorter path lengths.

Unit-cell stacks were developed specifically for concentrating solutions. Readers desiring informationabout unit-cell stacks are referred to Ref. 6.

21.2-3 Concentration Polarization

The detailed nature of ion transfer through a system of solutions and ion-exchange membranes warrantsthorough discussion because it is not only the source of the depletion and concentration that occurs inelectrodialysis but also the source of excessive concentration polarization, which is responsible for mostof the difficulties encountered in electrodialytic processing.

Figure 21.2-5 shows a cation- and an anion-exchange membrane mounted between two electrodes. Asolution of an electrolyte flows through the compartments formed by the membranes and electrodes. Withthe passage of an electrical current through the system of membranes and solutions, anions transfer towardthe anode, and cations transfer toward the cathode. These ion transfers are-the way in which electricalcurrent is carried in an electrolytic medium. The fraction of the current carried by an ionic species is termedits transference number (J+ or t~).

The hydraulic characteristics of an electrodialysis solution compartment and the boundary layers ofnearly static solution at the surfaces of the membranes can have a controlling effect on the current densitiesthat can be used in electrodialysis and therefore a controlling influence on the rate of demineralization.Because of the flow of solution through the center compartment formed by the two membranes in Fig.21.2-5, there is a zone of relatively well-mixed solution near the center of the compartment. The velocityof the solution and thus the degree of mixing diminish as the surfaces of the membranes are approached.In Fig. 21.2-5, an idealization is used such that there is a completely mixed zone in the center of thecompartment and completely static zones of solution in boundary layers adjacent to the membranes. In thestatic boundary layers, ions are transferred only by electrical transfer and diffusion, but in the mixed zoneions are transferred electrically, by diffusion, and by physical mixing.

If the transference number of anions through the anion-exchange membranes (t~) is 1.0 and that ofanions through the solution (t~) is 0.5, only half as many ions will be transferred electrically through thesolution in the static boundary layer on the side of the membrane that anions enter as will be transferredthrough the membranes. The solution at the membrane interface will be depleted of ions. For the same

FIGURE 21.2-3 Main components of an electrodialysis stack.

DILUTINGSOLUTION

CONCENTRATINGSOLUTION

ELECTRODE-RINSESOLUTION

FEED SOLUTION _

ELECTRODE'RINSESOLUTION

CATION-EXCHANGEMEMBRANE

ANION-EXCHANGEMEMBRANE

MANYMOREMEMBRANES,SPACER FRAMES,AND ANOTHEREND FRAME

FIGURE 21.2-4 Diagram of a tortuous-path spacer foran electrodialysis stack.

reasons, the static boundary layer on the other side of the membrane will accumulate ions. The interfacialconcentrations will change until concentration gradients are established in the boundary layers which aresufficiently large to transfer by diffusion the ions not transferred through the boundary layers electrically,as indicated by the dashed lines in Fig. 21.2-5.

If the current density through the system is increased, the rate of electrical transfer of ions increases,and the diffusional transfer through the boundary layers must increase to supply the additional ions trans-ferred electrically. With any given thickness of boundary layer, a current density can be reached at whichthe concentrations of electrolytes at the membrane interfaces on the depleting sides will approach zero. Atthis current density, called the limiting current density, H+ and OH" ions from ionization of water willbegin to be transferred through the membranes. This continuous ionization of water is termed water splitting.The water splitting caused by exceeding the limiting current density has detrimental effects on the operationof electrodialysis, as discussed below.

Scaling of membranes by pH-sensitive electrolytes occurs at anion-exchange membranes because OH"ions transfer through the membranes when water splitting occurs. The OH~ ions increase the pH withinthe membrane and at the interface on the concentrating side so that pH-sensitive substances, such as calciumcarbonate, precipitate.

Operation at or above the limiting current density causes the stack to have resistances higher than normalfor the following reasons. When water splitting occurs, the rate of dissociation of water is increased becausethe H+ and OH" ions are transferred continuously away from the membrane interfaces, which are thelocations of dissociation. An increased voltage is necessary to induce this continuous dissociation of water.Also, a thin film of highly depleted solution forms at the depleting sides of the membranes. These filmshave high specific resistances, which are in series with the other resistances in an electrodialysis stack.Both the increased voltages needed for continuous ionization of water and the high specific resistances ofdepleted films contribute to increased stack resistance.

Fouling of ion-exchange membranes by organic materials also can occur when concentration polarizationoccurs. Fouling of anion-exchange membranes has been shown to be caused by large organic ions becomingattached to charged groups on membranes having the opposite charge.7 9

-STATIC-ZONES%

WELL-MIXEDZONE

CATHODEANODE

S - STATIC BOUNDARY LAYERSC - CATION-EXCHANGE MEMBRANEA = ANION-EXCHANGE MEMBRANE

FIGURE 21.2-5 Idealized representation of concentration gradients in electrodialysis.

Because of these detrimental consequences of excessive concentration polarization, the minimizationof polarization is an important factor in the design of electrodialysis stacks.

21.2-4 Minimizing Concentration Polarization

As was implied in the discussion of Fig. 21.2-5, the changes in electrolyte concentration in the solutionsat the membrane-solution interfaces, which are responsible for the deleterious consequences of polarization,can be minimized by decreasing the thickness of the nearly static boundary layers.

With the idealization of completely static and completely mixed zones used in Fig. 21.2-5, Eq.(21.2-1) has been derived to determine the thickness of boundary layers:1011

(21.2-1)

where 5 is the thickness (cm) of the boundary layers in depleting compartments, D is the diffusivity (cm/s) of the salt being treated, F is the Faraday number, 96,500 C/eq, N is the concentration (eq/cm3) of thesalt in the well-mixed zone, i,lm is the limiting current density (mA/cm2)—that is, the current density whenthe interfacial concentration becomes zero, tm is the transference number of the counterion in the membrane,and r, is the transference number of the counterion in solution.

Equation (21.2-1) has been rearranged to a form in which the polarization parameter ilim/N is expressedin terms of the above variables:

(21.2-2)

The polarization parameter is more convenient to use in design and scale-up of electrodialytic equipmentthan the thickness of the boundary layer. It can be measured easily in small-scale stacks with a given valueof bulk concentration N and used to predict limiting current densities in larger stacks at other values of N.

Although boundary-layer thickness at any point in a cell compartment depends on several factors,12 theaverage boundary-layer thickness in Eq. (21.2-2) is primarily a function of the solution velocity in thedepleting compartments and the mixing characteristics of spacer materials. Because the boundary-layerthickness varies with solution velocity, the polarization parameter does also. The relationships between

solution velocity, the polarization parameter, and the nature of ion-exchange membrane have been studiedby different methods, including determinations of changes in pH and stack resistances which indicate thelimiting current density has been reached.

Values of iym/N may be measured in laboratory stacks assembled with the membranes to be tested. Thesolution of interest is circulated through the depleting compartments, and the stack voltage is adjusted togive a desired stack current. The flow rate and the influent and effluent concentrations of the solution beingdepleted are measured. The current is increased in increments, and, at each increment, measurements aremade of the stack voltage Ex and stack current /v. Values of EJIx at each increment are plotted as a functionof current density. At the limiting current density (/|im), there will be a change in the slope of the curve.If plots of EJIx versus the reciprocal of current density are used, the break in the curve is more pronouncedso that i,lm can be defined more closely. Then the polarization parameter is calculated as iHm divided by thelog-mean concentration M

Because of the ill effects of exceeding the limiting current density and the fact that there can be stagnantor semistagnant areas in even well-designed electrodialysis stacks, most process designers use operatingvalues of UN that are 50-70% of the limiting values.

21.2-5 Electrodes

The most common reaction at the cathode of an electrodialysis stack is the decomposition of water

This reaction gives rise to both gas (H2) and high values of pH. The gas can blanket the electrode andincrease the resistance of the stack. High values of pH can cause precipitation of any pH-sensitive materials(e.g., calcium carbonate) in the electrode-rinse stream, which can raise resistance and even make the stackinoperative.

Accumulation of basic precipitates in the catholyte compartment can be prevented by acidification orsoftening of the electrode-rinse solution. The OH" ions generated at the cathode can transfer into themultiple compartments and cause precipitates to form. Therefore, isolating compartments that are flushedwith acidified or softened catholyte solutions often are used. A high solution velocity is used in the catholytecompartment to sweep out the gas as it is formed.

The choice of cathode materials is not too critical because the cathode is not subjected to severecorrosion. Relatively low-cost materials, such as stainless steel or graphite, are used.

A variety of electrode reactions14 can occur at the anode. Some of them result in the formation of gases,corrosive attacks on most electrode materials, or oxidative attacks on the membrane in contact with theanolyte solution. Platinum is too costly for use as an anode material, but titanium or tantalum coated withextremely thin (microinches) layers of platinum has been used. Lead dioxide deposited on and withingraphite has been used also. Oxidation-resistant membranes (fluorocarbon-based) often are used adjacentto the anolyte solution even though they are expensive.

21.2-6 Design Considerations

In electrodialysis, there are three categories of costs: those that increase with current density (the cost ofenergy); those that decrease with current density (the cost of membrane replacement and amortization ofcapital investment); and those that are essentially invariant with current density (cost of chemicals, main-tenance, and labor). Because two of these categories of costs vary in opposite directions with currentdensity, there is an economically optimum current density. However, for most applications of electrodi-alysis, the limiting current density is lower than the optimum current density. Therefore, determination ofthe limiting current density iUm by the previously described methods is usually the first step in design.

The polarization parameter i\tm/N can be calculated next. During the experiments to determine i,jm, datamay be taken on voltage, current, membrane area, and number of cell pairs in the experimental stack. Thecell-pair voltage Ecp can be determined for various values of concentration N, and current density /, andcurves of Ecp versus i for various values of N can be prepared. The current efficiency to be expected inlarge-scale use can be calculated from the experiments to determine f,im:

(21.2-3)

where r\c is the current efficiency, Q is the volumetric flow rate (cm3/s) through the stack, F is the Faradaynumber, 96,500 C/eq, Nf and Ne are the feed and effluent concentration (eq/cm3), respectively, 5cp is thenumber of cell pairs in the stack (exclusive of isolating compartments), and Ix is the current (A) throughthe stack.

As stated previously, the operating current density io is usually 50-70% of the limiting value. Vendorsof electrodialysis equipment will recommend the percentage to be used in their equipment.

For a given demineralization problem, the feed concentration Nf and the desired final concentrationusually are known. The number of stages of electrodialysis can be calculated by calculating the effluentconcentration N, for each stage:

(21.2-4)

where N, and A are the effluent and feed concentrations (eq/cm3), respectively, io is the operating currentdensity (A/cm2), a is the effective area (cm2) of membrane, TJ(. is the current efficiency, and F is the Faradaynumber, 96,500 C/eq.

The concentration of the effluent from the first stage is the feed concentration for the second stage. Thecalculation of Ne is continued until the effluent concentration from a stage is equal to the desired finalconcentration, and the number of stages needed is found.

The cell-pair voltage Ecp can be read from the curves of Ecp versus / for the average concentrations ineach stage. The cell-pair voltages multiplied by the number of cell pairs per stack will provide an approx-imation of the voltage per stack for each stage.

The operating current densities for each stage multiplied by the effective membrane area gives the stackcurrents for each stage.

The product of stack voltage and stack current will give the dc-power requirements for each stage.The number of lines of stacks to be arranged in parallel to provide a given throughput can be determined

by dividing the desired throughput by the throughput in the depleting compartments of a stack, which canbe obtained from the vendor.

Some water will be transferred out of the depleting compartments by electroosmosis. The rate ofelectroosmotic transfer is determined by the nature of the membranes. Estimates of the decrease in theamount of demineralized product because of this water transfer can be obtained from the supplier of theion-exchange membranes.

With the information described above, predesign estimates of capital and operating costs can be pre-pared. Capital costs include costs for pretreatment equipment (if needed), electrodialysis stacks, storagetanks, and pumps. These costs may be estimated by standard predesign estimating methods.

Operating costs include costs for electrical energy (for stacks, pumping, lighting, and auxiliaries),membrane replacement, chemicals (e.g., acids for electrode-rinse streams), maintenance supplies and labor,operating labor, overhead items, amortization of investment, taxes, and insurance.

NOTATION FOR SECTION 21.1

Nomenclature

constant defined in Eq. (21.1-26) (dimensionless)membrane area (cm2)fractional cross-sectional area of membrane pore openings (dimensionless)constant defined in Eq. (21.1-26) (dimensionless)constant defined in Eq. (21.1-26) (dimensionless)solute concentration (g/cm3)solute concentration difference (g/cm3)diameter (cm)dialysance (cm3/s)diffusivity in bulk solution (cnr/s)diffusivity in the membrane (cm2/s)fractional extraction (dimensionless)reaction rate (g/s)channel height (cm)transmembrane solute flux (g/cm2 • s)transmembrane solvent velocity (cm/s)mass transfer coefficient (cm/s)power series in q defined in Eq. (21.1-6) (dimensionless)length (cm)pore path length (cm)hydraulic permeability (cnrVdyn • s)

mass of solute (g)number of channels or fibers (dimensionless)pressure difference (dyn/cm2)membrane permeability (cm/s)

1 membrane permeability of tritiated water (cm/s)

transmembrane Peclet number (dimensionless)ratio of solute to pore diameter (dimensionless)volumetric flow rate (cm"Vs)hydrodynamic pore radius (cm)mass transfer resistance (s/cm)Reynolds number (dimensionless)Schmidt number (dimensionless)Sherwood number (dimensionless)time (s)volume (cm3)channel width (cm)length (cm)variable defined in Eq. (21.1-22) (cnrVs)variable defined in Eq. (21.1-23) (dimensionless)variable defined in Eq. (21.1-24) (dimensionless)

Greek Symbols

separation factor (dimensionless)variable defined in Eq. (21.1-33) (dimensionless)variable used in Eq. (21.1-38) (dimensionless)variable used in Eq. (21.1-4) (dimensionless)viscosity (dyn • s/cm2)variable used in Eq. (21.1-4) (dimensionless)density (g/cm3)reflection coefficient (dimensionless)variable used in Eq. (21.1-4) (dimensionless)

Subscripts

dialysatefeedinletoutletoverallreagentultrafiltratemembrane wall

Superscripts

initial value (time = 0)value at time = taverage value

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