Chemrml Engmeerrttg Saenre Vol 35 pp 1545-1556 Pergamon Press Ltd 1980 Prmted nn Great Bntam

COMPOSITE SEPARATION UNITS AND THEIR APPLICATION IN DIALYSIS FOR THE ISOLATION

OF INTERMEDIATE-SIZED MOLECULES

ISA0 NODA and CARL C GRYTE* Department of Chemical Engmeermg and Apphed Chemistry,

Columbia Umverslty, New York, NY 10027, U S A

(Recerved 15 September 1978)

Abstract-Composite separation units (Le assemblages of simtlar elementary separation umts) are studied Dialysis 1s used as an example of a basic umt The overall mput-output response of a composite group made up of a system of interconnected dlalyzers IS developed m relation to the system configuration In certam flow configurations, a composite &alyzcr has the permeability charactenstlcs of a band-pass filter Smce the selectivity of composite dlalyzer systems can be ampltied by repeating the units m a cascading manner, a spenfic solute of intermediate permeability can, m prmclple, be extracted from a complex multicomponent s01ut10n

INTRODUCTION Separation of chemical substances from a mixture 1s possibly one of the most important steps 111 a chemical process Great amounts of tune and effort are spent on the estimation of the performance of various separation units which are interconnected with each other and with chemical reactors to form complex process networks A number of slmllar separation umts are often employed to construct an assemblage of separators which can accomplish a desrred overall separation result This type of assemblage, or the composite separation unit, can be treated as a single mdependent separation unit conslstmg of numerous elementary units Tbe performance of a composite separation unit IS usually qmte different from the performance of the mdlvldual elementary units which compose it The countercurrent cascade used In dlstlllatlon IS a classical example of a composite separation unit The assemblage of numerous smgle stage vapor-liquid contactors, or bubble cap trays m a vertical countercurrent cascade configuration, IS usually treated as a single separation unit a dlstlllatton column The separation efficiency ot a composite unit whose elementary parts are assembled m a counter- current configuration is often, but not always, higher than that of the elementary component separators

The type of configuration of a composite system IS not lunlted to that of a sunple countercurrent cascade Much more complicated configurations are possible for composite units For a given set of elementary units, there are many possible composite separation umt configurations whose separation performance differs considerably from one another The novel performance of qome composite separation units which can be estimated from that of therr component units is the subject of this investigation

Dialysis IS chosen as the simple separation process to which the concept of composite separation 1s to be

*To whom correspondence should be addressed

applied for analysis The input-output response of an ideal countercurrent dlalyzer can be expressed mathematically m a very sunple form, so that the performance of an assemblage of such dlalyzers can be easily estunated Although the analysis m the report IS restncted to that of dialysis separation, the apphcatlon of the basic concept developed here could be extended to other separation processes

The separation of substances m dialysis results from the differences m the rate of dlffuslonal transport of each solute across a permeable membrane Dialysis IS traditionally used for the separation of solutes which are highly permeable to the membrane barrier from those partlculates and colloidal solutes which are unpermeable Cl] The particular apphcatlon of dialysis used most successfully IS that of artlficlal kidney hemodlalysls [21 -I-he permeability differentiation of solutes by currently available membrane materials 1s poor As a result, it IS difficult to separate solutes by conventional single stage dialysis unless their membrane permeablhtles are widely different Various authors [3-51 have measured the diffusive permeablhtles of membranes to various solutes Using conventional membranes, the input-output response of simple dlalyzers IS thus not sufficiently sensitive to solute permeability to permit practical separations of solutes having sumlar molecular structure However, by cascading dlalyzers m conJunction with ultrafiltration units m a specific configuration, it IS possible to amplify the selectlvlty of a membrane system and overcome the mablhty of smgle membranes to differentiate between similar solutes [6]

Since the dlalysls-ultrafiItratlon system can amplify the system selectlvlty, it IS useful to investigate the configuratlon of the multunembered dlalyzer component-the composite dlalyzer Even a minute change m the input-utput response, introduced by an evolution of new flow geometries m the dlalyzer network, becomes important if it can be amplified to a

1545

1546 ISAO NODA and CARL C GRYTE

larger magmtude Any pecuhantles m the performance of a dlalyzer or an assembly of dlalyzers can be amplified Just as electrical signals are amphfied m a complex electrical clrcult Analogies between electrical and dialysis systems are not lmuted to the ability to amplify the slgnaIs As electrical elements such as resistors and capacitors are assembled to produce an electrical clrcmt to perform a specific task, it IS possible to construct an mtrlcate network of dlalyzers or a diffusion cncult, which yields a umque mput-output response The concept of composite separation units can be applied to the systematic development of such clrcults Using dlalyzers as elements, several examples of composite separation units are developed m this study Although the slgmficance of these composite units m an Immediate apphcatlon IS not yet known, the performance of composite dlalyzer units of various topological configurations will give us a new insight mto the development of complex separation processes

ANALYSIS

(1) Szngle dzalyzer Consider a simple counter-current mass exchange

dlalyzer (Fig 1) The Aows of the two solutions are separated by a membrane of surface area S, and solutes can be transferred by &ffuslon from one stream to the other The net volumetric flux across the membrane due to hydraulic and osmotic pressure IS assumed to be very small so that the volumetric flow rates Qr and Qrr of the two solution streams are practically constant The bulk concentrations C, and C,, of a solute are, on the other hand, functions of the distance from the entry points of the solutions to the dlalyzer The solute molar flow rates of the entering and exiting streams J g, t(, and v are defined by

f = (CrQ,),, (1)

g = (c,,Qr~),n (2)

u = (C,Q,L, (3)

21 = (C~IQIIL~ (4)

The mput+output response of a dlalyzer IS expressed m the form

u=(l --E,)f+E,,g (5)

u = E,f + (1 - E,,)g (6)

INPUT

OUTPUT

where the fractzonai extractzons E, and E,, are the fraction of a solute transferred from one solution to the other during dialysis The expressions for the fractional extractions of dlalyzers are gven by Mlchaels [7] for various flow geometries

When the volumetric flow rates of the two solution streams are identical m magnitude but opposite m the flow configuration, the fractional extractlons E, and E,, become

E, = E,, = E (7)

In the present discussion, the term szngle dzalyzer refers to this counter-current dialysis unit m which the two streams have volumetric flow rates of the same magnitude Such a single dlalyzer IS represented schematically m Fig 2 One of the unportant charactenstlcs of the smgle &alyzer IS the symmetric nature of the mass transfer of a solute Equation (7) IS a direct result of this symmetry The fraction of a solute transferred from the upper section to the lower section during dlalysls IS identical to the fraction of the same solute transferred from the lower section to the upper section Thus, the mput-output response remains the same when the positions of the two sections are reversed The symmetry holds even if the two sectlons, for mstance, the inside and the outside regions of hollow fibers, are geometrically quite different

The fractional extraction E of a single dlalyzer can be expressed m a very simple form

e EC--

i+e

The dralyszs coeflczent 0 IS a dunenslonless number charactenzmg the ease with which a solute IS transferred from one solution stream to the other and IS given by

where h IS the permeability, or the overall mass transfer coefficient, of the solute m the dlalyzer This overall mass transfer coefficient IS calculated dmectly from the fiber spacmg and the membrane permeability for the case of a uniform array of hollow fibers under conditions of equal volumetric counter-current flow [S] Substltutmg eqns (7) and (8) mto eqns (5) and (6),

OUTPUT

QI CI Y --/ Ql CL REGION I ___------------ QII $1 REGION II INPUT

\ l-PERMEABLE WMBRANE

Fig 1 Counter-current mass exchanger dlalyzer

Composite separation umts and their apphcatlon m dmlysls 1547

INPUT f

1 F'

d - OUTPUT u INPUT g

Fig 2 SchematIc diagram for the single dialyzer

the mput-output response of the smgle dlalyzer 1s obtamed

(10)

(L1)

The fractlonal extraction E of the smgle dlalyzer IS plotted m Fig 3 as a function of the dlalysls coefficient 19 As the value of 0 increases, the fractional extraction asymptotically approaches unity The validity of eqns (10 and 11) has been demonstrated expernnentally by Noda and Gryte [9]

(2) Two-unit composite dlalyzers

In order to study the systems of dlalyzers, It IS convenient to develop a concept of composite dlalyzers

A composite dlalyzer IS an assemblage of smgle dlalyzers which are mterconnected m a complex manner such that the assemblage as a whole will behave as if It were an independent dialysis umt A composite dalyzer, like a single &alyzer, has two pairs of entering and exiting solution streams and has a unique mass transfer input-output response The only separating agent, or the drlvmg force of separation, which a composite dlalyzer can exploit IS the concentration difference between the two input solutions If the volumetric flow rates of all the entering

V Hz - A a SERIES DIALYZER

_f, l---b+-

08 -

00’ I I I

01 10 10

DlALYSIS COEFFICIENT e

Fig 3 E vs 0 curves of the smgle dlalyzer, the series dlalyzer, and the parallel dlalyzer

and exiting solutions are the same, the symmetry m the extraction of solute IS observed as m the case of the single dlaIyzer

The simplest type of composite dlalyzer 1s the two- umt composite dlalyzer which has only two single dlalyzers as elements Four nontrivial and topologl- tally Independent configurations (Fig 4) of two-unit composite dlalyzers are found For the sake of convenience, simple names are arbitrarily assigned to each of the composite dlalyzers the series dlalyzer, the parallel dlalyzers, the cross dlalyzer, and the knot dlalyzer Each two-umt composite dlalyzer IS studied m detail

(a) Series dlalyzer The first type of two-umt composite dlalyzer IS the serzes dlalyzer shown m Fig 4(a) Two units of single dlalyzers are cascaded m the counter-current series configuration In this composite dlalyzer, one stream of solution enters the upper section of the first single dlalyzer with a solute molar flow rate f, comes out of the first single dlalyzer, and enters the upper section of the second smgle dlalyzer, and finally leaves the second single dlalyzer with solute molar flow rate u Slmllarly, the other solution stream enters the lower section of the second single dlalyzer

u w2 0 CROSS DIALYZER

mm u w2 3

b PARALLEL DIALYZER V d KNOT DIALYZEIt 9

Fig 4 Two-unit composite dlalyzers

1548 ISAONODA and CARL C GRYTE

with solute molar flow rate g, and comes out of the lower sectlon of the first smgle dmlyzer with solute molar flow rate u

Suppose S, and S, are the membrane surface areas of the two smgle dlalyzers bulldmg this series dlalyzer In order to obtain the overall mput-output response of a composite dlalyzer, the mput-output response developed m eqns (10) and (11) 1s applied to the mdlvldual single dlalyzer element of the composite dlalyzer For the first single dlalyzer, the mput-output response becomes

W- l-&+&W2

v-*f +&p

where

(12)

(13)

The solute molar flow rates wI and wZ are assigned to the intermediate stream leaving the first single Qalyzer and to the stream leaving the second single dlalyzer respectively The input-output response of the second single dlalyzer IS given by

1 02 u=-wi +----- 1 + 0, 1 +ezg

(15)

*2 1 w -p

2-1+8, w’+l+ (16)

where

(17)

Solving eqns (12), (13), (15) and (16) simultaneously, the overall mput-output response of the senes &alyzer can be expressed m terms of two dlalysls coefficients defined m eqns (14) and (17)

U=1+Rf+B,f+l~;~02g (18)

v=l?;:B,f+l+Bf+B,g (19)

The fractional extraction E for the series dlalyzer becomes

E= Q, + 02

1 +e,+e, (20)

Introducmg the overall dialysis coefficient C#I of the series dlalyzer,

c#J = 8, + 8, (21)

Equation (20) can be rewritten as

4 E=l+C#l

which 1s snnllar to eqn (8) for a single dlalyzer Equation (20) shows that the fractional extraction of this composite dlalyzer 1s invariant even if the values of the two dlalysls coefficients @I and 0, are exchanged The elements m a senes dlalyzer, therefore, are commutative, me the overall performance of the composite dlalyzer IS not altered by rearranging the posltlon of the first and the second single dlalyzers, so long as the senes configuration of the composite dlalyzer 1s mamtamed

Suppose the two single dlalyzers have membranes of identical surface area S

s, = s, = s (23)

From eqn (21), the overall dlalysls coefficient # of the series dlalyzer IS gven by

From eqn (9), eqn (24) becomes

Cp = 28 (25)

The input-output response of this particular series dlalyzer 1s

“‘&f t&g

v=&f -c&-g The above equations are identical to the input-output response of a single dlalyzer having the membrane surface area of 2S, so that cascading two units of single &alyzers m a series configuration produces the same effect as doublmg the membrane surface area of an ordinary single dlaiyzer This result 1s not surprlsmg since twice as much membrane 1s used m the series dlalyzer to exchange solutes from one stream to the other The fractional extraction E of the series dlalyzer conslstmg of two identical single dlalyzers 1s plotted m Fig 3 as a function of dlalysls coefficient 0

The configuration of a series dlalyzer resembles the configuration of counter-current cascades of equlll- brmrn stages which are used extensively m separation processes In general, this counter-current cascade configuration enhances the separation components If the separation IS based on the equlhbrmm &strlbutlon of components between two phases This enhancement

Composite separation umts and their apphcatlon In dlalysrs 1549

1s not expected for dlalysls, which 1s a rate controlled separation process The separation of solutes m dlalysls IS achieved by the selective &ffuslon of solutes from one solution to the other In order to have the dlffuslon of solutes, there must be concentration gradients across the membrane, the equlhbrlum between the two solutions separated by a membrane 1s usually not observed The increase of the membrane surface area will increase the rate of exchange of solutes from one stream to the other, but the selectivity of this cascaded dlalyzer will not be unproved since the increase of the rate IS umforrn for all the solutes

(b) Parallel dlalyzer The second typ of two-unit composite dlalyzer 1s the parallel dralyzer shown m Fig 4(b) This composite dlalyzer also has the counter- current cascade configuration sunllar to that of a series dlalyzer Unlike any of the other two-umt composite dialyzers m Fig 4, the parallel dlalyzer has three independent streams the left stream whlLh enters the left section of the first single dlalyzer with a solute molar flow rate S and comes out with u, the right stream which enters the right section of the second single dlalyzer with a solute molar flow rate g and comes out with v, and the middle stream which circulates between the right section of the first single dlalyzer and the left section of the second single dlalyzer The volumetric flow rate of each stream can be arbltranly adJusted without violating the mass balance of the system If the flow rate of the middle stream IS Infinitely large, the bulk concentration of a solute m this stream becomes everywhere uniform The counter-current charactenstlcs of the single dlalyzers, 1 e the fractional extraction approaches unity as the dlalysls coefficient becomes infinitely large, will be lost On the other hand, If the volumetric flow rate of the middle stream 1s too small, there will be no exchange of solutes between the left and the right streams

If the volumetric tlow rate of each stream has the same value Q. then the mput-output responses for the first and the second single dlalyzers become

and

1 W-

0, -WI +- 2-l+82 1 +Ozg

0, I “=~%W 1+8/

(31)

where wl and wZ are the solute molar flow rates m the mrddle stream leaving the first single dlalyzer and m the stream leaving the second single dlalyzer respectively The definitions of the dialysis coefficients 8, and 19, are the same as m eqns (14) and (17) Solvmg

eqns (2gH31) snnultaneously, the overall mput-output response of a parallel dlalyzer IS obtained

’ = 8, + e2 + e1e2 f

wb + e1 + 8, + elezg

“=e, + e, + e,e, f

4 + e2 + el + e2 + elezg (33)

The fractional extraction E for the parallel dlalyzer 1s

(34)

Equation (34) shows that the two single dlalyzers m the parallel dlalyzer are also commutatlve. so that the overall performance ofthls composite dlalyzer remains the same even if the posltlons of the two units are exchanged By defining the overall dialysis coefficient q5 of the parallel dlalyzer as

cp= 1 l/e, + I/e,

(35)

Equation (34) can be reduced to eqn (22) Suppose the first and the second dlalyzers m the

parallel dlalyzer have membranes of Identical surface area S From eqns (14), (17), (23) and (35), the overall dlalysls coefficient 4 of the parallel dlalyzer 1s given by

+=Q l Q

hS+hS 1 2

1 hS _- =2 Q (36)

From eqn (9), 4 can be expressed m terms of 0 as

4 = )e (37)

The input-utput response of this parallel dlalyzer becomes

(39)

This result 1s ldentlcal to that of a single dmlyzer having a membrane whose permeablhty for a given solute IS h/2 Cascading two units of single dlalyzers m

15.50 ISAO NODA and CARL C GRYTE

a parallel configuration produces the same effect as redunng the membrane permeablhty mto one-half The reduction of apparent permeablllty IS expected since, unlike the single dlalyzer, the solutes must diffuse through membranes twice to be extracted from one stream to the other The fractlonal extractlon E of the parallel dlalyzer conslstmg of two identical single dlalyzers IS plotted m Fig 3 as a function of dlalysls coefficient 8

(c) Cross d~alyzer The configuration of the third type of two-unit composlt dlalyzer, the cross dralyzer shown m Fig 4(c), IS somewhat more complicated than the configurations of the semes and the parallel dlalyzers As m the case of the parallel dlalyzer, one stream of solution enters the left section of the first single dlalyzer with a solute molar flow rate f and comes out with u The other solution stream enters the lower section of the second single dlalyzer with a solute molar flow rate g, comes out of the second single dlalyzer, and enters the right section of the first smgle dlalyzer, after leaving the first smgle dalyzer, this solution stream returns to the upper section of the second dlalyzer and comes out with a solute molar flow rate II Both the upper and lower sections of the second dlalyzer are used by the same solution stream, le the solution stream 1s dialyzed against itself

The configuration of the cross dlalyzer IS possible only for the separation units which use some sort of mechanical barriers to segregate two streams of mixtures The existence of a barrier, m this study the permeable membrane, IS essential for the cross configuration due to the “self-dlalysls” of the solution stream m the second single dlalyzer Other separation processes, such as dlstlllatlon and liquid-liquid extraction, do not use a mechanical segregation of solution streams but thoroughly depend on phase separation These processes will fall to achieve an effective separation with this configuration since the separation umt m the posltlon of the second single dlalyzer will behave as a simple mixer of the feed and the product solutions

The mput&output responses for the first and the second single dlalyzers are

u=&j+$+* 0

W- Af+Lw i-l+8, l-CO, *

and

w- *-&02++q

v=j+g+&wl

(40)

(41)

The conventions for the solute molar flow rates w1 and w2 and for the dialysis coefficients O1 and 8, are the

same as m the senes and the parallel &alyzers Solvmg eqns (4oH43) simultaneously, the overall mput-output response of the cross dlalyzer IS obtained

1 + e,e, l4 = 1 + e1 + 8,8, s

+ *I 1 + 8, + elezg

” 6 = 1 + 0, + 0,8,

f

1 + 818, + 1 + 8, + e1e2g (45)

The fractional extraction E of the cross dlalyzer IS

E= @I

1 + 8, + 818, (46)

Equation (46) can be reduced to eqn (22) by defining the overall dialysis coefficient 4 of the cross dlalyzer as

&= 8l 1 + BIB2 (47)

Unlike other two-unit composite dlalyzers, the overall mput-output response of the cross dlalyzer depends on the position m which the two single dtalyzers are placed The elements m the composite dlalyzer are not commutative m this particular configuration From eqn (46), the fractional extraction E of the cross dlalyzer increases with the increase of the dlalysls coefficient 8, of the first single dlalyzer On the other hand, if the value of the other dlalysls coefficient e2 of the second single dlalyzer IS increased, the extraction m this composite dlalyzer actually de- creases The two single dlalyzers constructing the cross dlalyzer, therefore, contribute to the overall input-utput response m an opposing manner If the order of arrangement of the single dlalyzer elements m Fig 4(c) IS reversed so that the second single dlalyzer IS placed on the left side of the first single dlalyzer, then the fractional extraction E becomes

E= 82 1 + e* + 818, (48)

which IS different from eqn (46) The noncommutatlve nature of the cross configuration by no means distorts the symmetry of the input-output response of the composite dlalyzer The value of the fractional extraction may be varied by switching the two single dlalyzer elements wlthm the cross dlalyzer, but It will not be varied by switching the feed points of the two entenng solution streams The fraction of a solute which enters the first single dlalyzer and leaves the second single dlalyzer IS identical to the fraction of the

Composite separation umts and their apphcatlon m dialysis 1551

solute which enters the second smgle dlalyzer and leaves the first smgle dlalyzer

It IS convement to define the reference surface area S of a two-unit composite dlalyzer to be the geometric mean of the surface area of membranes m the single dlalyzer elements

In addition, the membrane surface area devlatlon factor b of the single dlalyzers 1s defined

b= Sl J- s, From eqns (9), (49) and (50), the dlalysls coefficients of single dlalyzers given m eqns (14) and (17) can be rewritten m the form

Q1 = 68 (51)

Cl2 =$O

Then the extraction E and the overall dlalysls coefficient 9 are expressed m terms of the reference dlalysls coefficient 0 and the membrane surface area deviation factor b of the composite dlalyzer

6

E = 6 + (e + e-l)

The fractIona extraction E of the cross dlalyzer 1s plotted m Fig 5 as d function of 8 and 6 Notice that the shape of the extraction curves for the cross dlalyzer IS quite different from that in Fig 3 While the fractlonal extractions for the single dlalyzer, the series dlalyzer, and the parallel dmlyzer increase asymptotically with the mcrease of the dialysis coefficient 0, the fractional extraction for the cross dlalyzer has a maxunum at a particular value of 8 and decreases with the further mcrease of 8 The maxunum extraction IS gven by

6 E max =- at O=l

a+2

The peak-shape of the extraction curve in Fig 5 nnphes the followmg m the cross dlalyzer not only the nnpermeable solutes but also the highly permeable solutes, instead of being extracted to the other solution, remam m the same solution stream after dialysis, only the solutes having intermediate perme- ablhtles are extracted to the other stream In short, the band-pass behavior on the permeability spectrum is observed

The peculiar mput-output response of the cross dlalyzer comes from the followmg reasons Most of the

08 -

,,I- 01 10 10

DIALYSIS COEFFICIENT e

Fig 5 E vs 0 curve of the cross dlalyzer for various b

less permeable solutes which give values of tr much less than one will be retained m the same solution as the solution under-going dialysis m either single dlalyzer element In fact, the input-output response of the cross dlalyzer approaches asymptotically that of the single dlalyzer as 8 becomes very small The highly permeable solutes which can easily diffuse across the membranes of dlalyzers, on the other hand, behave m a somewhat different way The large fraction of highly permeable solutes m the feed solution whtch enters the left section of the first single dlalyzer will diffuse across the membrane mto the solution stream m the nght section entering the upper section of the second single dlalyzer with the extracted solutes The solutes m the upper sectlon of the second single dlalyzer, however, will be extracted by the solution stream m the lower section of the same dlalyzer which returns to the right section of the first single dlalyzer Only a small fraction of the highly permeable solute from the first feed solution, therefore, reaches the exit pomt of the upper section of the second single dlalyzer

Meanwhile, the highly permeable solutes m the second feed stream which enter the lower section ofthe second single dlalyzer diffuse across the membrane mto the solution stream m the upper section of the same smgle dlalyzer Most of the highly permeable solutes m the second feed stream do not even reach the first single dlalyzer, where the exchange of solutes between the two solution streams entering the cross dlalyzer occurs The highly permeable solutes m the feed solutions, therefore, are kept m the same streams by two somewhat different mechanisms the recycle of extracted solutes and the bypass of the solutes The apparent exchange of highly permeable solutes between the two solution streams flowing through the cross dlalyzer S, m both cases, kept to the mmnnum

The mput-output response of the cross dlalyzer depends on the membrane surface area devlatlon factor 6 as well as the dlalysls coefficient 0 Figure 5 shows that the extraction of the cross dmlyzer increases with the increase of 6 Unfortunately, the

1552 ISAO NODA and CARL C GRYTE

selectivity of the cross dlalyzer does not increase with the Increase of 6, so that the extractIon curve for the dlalyzer becomes rather flat as S becomes large The posltlon of the maxlmum point 1s independent of S and 1s found always at 0 = 1 From eqn (53), the maxnnum extractlon of the cross dlalyzer becomes 50% at S = 2 and 20% at 6 = 5 This difference m the fractlonal extraction 1s another mdlcatlon that the two single dlalyzer elements m the cross dlalyzer are not commutative The mput-output response of the cross dlalyzer depends on whether the membrane surface area of the first single dlalyzer 1s larger or smaller than the membrane surface area of the second dlalyzer

The value of the dlalysls coefficient 0 defined m eqn (9) 1s a function not only of the pcrmeablhty h, which 1s more or less the property of the type of solute for a gwen membrane, and the membrane surface area S, which 1s specified during the construction of dlalyzers, but also of the volumetric flow rate Q Even if the size of the dlalyzer 1s fixed, for a wide range of solutes the dlalysls coefficient 0 can take a value of one merely by ddJUStlng the volumetric flow rate of the two streams flowing through the dlalyzer The maxlmum extrac- tion m the cross dlalyzer, therefore, 1s achieved for any solute If the volumetric flow rate IS appropriately tuned The capability of the cross dlalyzer to set the maxnnum extraction pomt on a specified target solute is extremely nnportant

Even though the selectivity of the cross dlalyzer 1s not very high, the peak-shaped mput-output response of the dlalyzer suggests an nnportant posslblhty the construction of a &alysls system which extracts a specific solute from a solution mixture by settmg the maxunum extraction condltlon on the solute Using the smgle dlalyzer, all the solutes m a multi-component feed solution are separated mto two simple groups, the permeable solutes and the less permeable solutes Using the cross dlalyzer, on the other hand, the solutes are separated mto three groups the solutes the permeabllltles of which give the value of tl around unity, the highly permeable (0 >> 1) solutes, and the less permeable (0 c< 1) solutes A smgle dlalyzer can only separate the feed solution mto two solution mixtures, even If a method to amplify the dlalysls selectlvlty (Noda and Gryte [6]) 1s developed, it cannot extract a specific solute from a solution mixture unless the tnrger solute 1s either the most or the least permeable among the solutes m the feed If, on the other hand, the selectlvlty of the cross dlalyzer IS amplified, it IS possible to extract from among the solutes m the feed a specific solute which has an Intermediate permeability

(d) Knot dlalyzer The last type of two-unit dlalyzer 1s the knot dmlyzer shown m Fig 4(d) Two solution streams enter the first smgle dlalyzer with solute molar flow ratesf and g, then both solutions enter the second single dlalyzer, and come out with solute molar flow rates u and 2) Compared to the counter-current cascade configuration seen m the series and the parallel dlalyzers, the knot dlalyzer has the co-current

configuratlon as an overall structure, even though the mdlvrdual single dlalyzer element m the knot dlalyzer is a counter-current separation umt

The mput-output responses for the first and the second single dlalyzers are

(561

(57)

and

1 U=L+l+

Q2 T-T-igw2 (58)

02 1

v=j-$-yqw8,+-w 1+8, 2

(59)

where w1 and wp are the solute molar flow rates of the intermediate streams leaving the first single dlalyzer Substrtutmg eqns (56) and (57) into eqns (58) and (59), the overall Input-output response of the knot dlalyzer 1s obtained

1 + 818,

u = i + 8, + 8, + o,e, f

e, + e2 + i + o1 + e, + elezg

e, + e2 Y=i+e,+e,+e,e, f

1 + e,f?,

+ 1 + 8, + e2 + elezg (611

Equations (60) and (61) show that the two smgle dlalyzer elements m the knot dlalyzer are commuta- tive The Input-output response of this composite dlalyzer does not change by swltchmg the posltlon of the single dlalyzers Using the reference area S and the membrane surface area devlatlon factor S defined m eqns (49) and (SO), eqns (60) and (61) can be rewritten as

b + se2 u = s + e + s2e + se2 J

8 -t s2e + b + 8 + bze + sezg

8 + b2e ’ = b + 8 + s2e + se2 f

s + be2 + s + 8 + 6% + sezg

(62)

(63)

Composite separation umts and their appllcatlon m dlalysls 1553

The fractlonal extractlon E and the overall dlalysls coeffiaent 4 are

E= 6 +6--z

6+6-‘+8+8--’

and

(64)

The fractlonal extractlon E of the knot dlalyzer IS plotted m Fig 6 as a function of 8 and 6 Again the peak-shaped extraction curves are obtained The maxnnum extraction IS given by

E 1 +P

¶llBX = 1 + 26 + 62 at 8=1 (66)

The mput-output responses of a knot dtalyzer and the cross dlalyzer are very similar They are, m fact, identical to each other when the value ofthe membrane surface area deviation factor 6 for the cross &alyzer IS equal to the value of (b + 6-l) for the knot dlalyzer Thus, the extraction of the cross dlalyzer having 6 = 2 and that of the knot dlalyzer having b = 1 are both 50% Given any knot dralyzer, there IS a cross dlalyzer which has the mput-output response identical to that of the knot dlalyzer The opposite IS, however, not always true, the single dlalyzer elements are commuta- tive m the knot &alyzer but not In the cross dlalyzer The fractional extraction E of a knot dlalyzer given zn eqn (64) does not change if the value of 6-l IS substituted for S Unlike the extraction of the cross dlalyzer, that of the knot dlalyzer has a lower lunlt at 6 = 1 The mput-output response given by the cross dlalyzer which has the value of 6 less than two cannot be obtamed by the knot dlalyzer

T----l

00 1 1 J 01 10 10

DIAL"SIS CXFFlC'EdT 9

Fig 6 E vs 0 curve of the knot dlalyzer for various d

(3) Two-step separatzon of zntermedzate solutes The band pass behavior found for the cross and knot

dlalyzers IS relatively rare m chemical separation processes The fractlonatlon of intermediate sub- stances usually requzres two separatzon steps which produce three products low, mtermedlate, and high separation factor substances When fractionation of only intermediate substances IS required, any separat- mg agents consumed by the unnecessary separation of high and low separation factor substances are wasted

Figure 7 shows a two-step dlalysls scheme for extracting the intermediate permeablhty solutes The feed solution enters the left sectlon of the first dlalyzer with a solute molar flow rate f Two dlalysate streams which do not contam the solute enter the right sections of the first and second dlalyzers (broken lines) and produce the product streams with solute molar flow rates o and w The product stream solute molar flow rates normalized by the feed solute molar flow rate are given by

1 + a-‘8

ulf = (1 + %I)(1 + a-18)

68 u/f- = (1 + s&(1 + X18)

e2 w/f = (1 + se)(i + b-9)

(67)

(69)

The definition of the surface area devlatlon factor b IS the same as gven m eqn (50) The normahzed solute molar flow rates u/J, u/f, and w/f, correspondmg respectively to the product streams rich m the less, intermediate, and highly permeable solutes, are plotted m Fig 8 as functions of 0 at 6 = 1 The curve for v/f has a maximum point at 8 = 1

(df),., = & ( 1 2

at e=i (70)

The shape of this curve 1s similar to but somewhat flatter than that for the cross and knot dlalyzers Note that twice the amount of dlalysate, compared with two-unit composite dlalyzers (Fzg 4), IS required for the two-step dlalysls (l-lg 7), yet a less selective input-output response IS obtained

H

I L 2

c-D-v--

u a $0 V

Fig. 7 Two-step dlalysls to extract the mtermedlate solute The broken lines indicate the absence of the solute

ISAO NODA and CARL C GRYTE

I I I (5) Selectxxty ampllfrcatlon of composrte dralyzers

10 10

DIALYSIS COEFFICIENT 9

Fig 8 Normahzed solute molar flow rates of the product streams from the two-step dlalysls

(4) Composite dlalyzers wrth three or more elements

The number of topologlcally dlstmgmshable con- figurations of composite dralyzers becomes very large when more than two single dlalyzers are used Figure 9 shows twenty examples of three-unit composite dlalyzers len of them (Fig 9(k-t)) are known to have extraction curves with at least one local maxlmum point The shape of the extraction curves for the composite dlalyzers with many dlalyzer elements becomes very intricate The possible composltlons of the product from a given multlcomponent feed IS greatly widened

The multi-staged dlalysls system shown m Figure 10 IS developed by Noda and Gryte [6] to amphfy the selectlvlty of smgle dralyzers By cascading angle dlalyzer elements mth a nonselective solvent stripper, the single dialyzer extraction curve becomes sign?_ ficantly steeper (Fig 11). so that a high degree of solute separation 1s achieved The fractional extractlon E of the multi-staged smgle dlalyzer system IS given by

8” E=-

I + en (71)

where n 1s the order of cascade The amphficatlon of dialysis selectnMy IS not restncted to the single dialyzers, each dialyzer element in Fig 10 could be a composite dlaIyzer for which the input-output response IS quite different from that of a single dlalyzer The fractional extraction of the multi-staged com- posite dialyzers are easily obtamed by using the overall dialysis coefficient

4 E=l+&” (72)

Suppose the knot dlalyzers (Fig 4(d)) are used as the repeating dlalyzer elements m Fig 10 From eqns (65) and (72), the fractlonal extractlon E becomes

E= (6 + cs-ly

(6 + s-1~ + (e + e-l)n (731

Ftg 9 Three-unit composite dlalyzers

Composite separation units and then application m dlalysls 1555

”

CONCENTRATE W I .I. I A

7_*- __-_ -_A_*

t___________Y_j__ _ FILlRATE L- -5-J

"1 * un

I *

Fig 10 Scheme to amphfy the dlalysls selecttwty

01 10 10

DIALYSIS COEFFICIENT 8

vs 0 curve of the cascaded smgle dlalyzers for vanous n

The fractlonal extractlon IS plotted m Fig 12 as a function of the dlalysls coefficient 8 for various orders of cascade n The extraction curves having the sharp peak-shape are obtamed for the multi-staged knot

dlalyzers The steepness 1s increased by increasing the order of cascade n At 0 = b and 0 = 6- ‘, the value for E IS SO%, regardless of the order of cascade The membrane surface area devlatlon factor b, therefore, determines the width of the peak

It 1s Important to notice that a dlalysls unit which has the extractlon curve with a sharp and narrow peak can be a solute selectwe separation unit which extracts a specific solute from a solution mixture The mixture may contain solutes the permeablhtles of which are either higher or lower than the permeabhty of the target solute The operating condltlons of the multi- staged composite dlalyzer system are adjusted so that the value for the dlaiysls coefficient 0 becomes unity around the target solute Thus, a selective separation unit which 1s “tunable” to a wide range of soIutes 1s obtamed

CONCLUSIONS

The smgle dlalyzers can be assembled to create d composite dlalyzer which has a unique input-output

10 10

DIIL,S S CCE==‘CIEYT ?

Fig 12 E vs 0 curve of the cascaded knot dlalyzers for various n at h = 1 4

response Gwen a fixed number of smgle dlalyzer elements, it 1s possible to construct several composite dlalyzers the configurations of which are topologlcnlly independent Four examples of two-umt composite dlalyzers dre considered, the characterlstlcs of which are studied m detail The band-pass behavior 1s observed for the Input-utput responses of the knot and cross dlalyzers By cascading the composite dlalyzers with d nonselective concentrator, the selectlvlty of the band-pass behavior IS amphfied A dialysis unit which has a very sharp and narrow band- pass input-output response can extract a specific solute from the feed of solution mixture The selection of the desired target solute 1s accomplished by adJustmg the volumetric flow rate

NOTATION bulk concentration of a solute, mole/cm3 fractional extraction molar flow rate of a solute m a feed, mole/set molar flow rate of a solute m feed, moie/sec permeability of a solute, cm/set order of cascade volumetnc flow rate, cm3/sec membrane surface area. cm’

1556 ISAO No~~and CARL C GRYTE

U molar Aow rate of a solute m product, mole/set V molar flow rate of a solute m product, mole/set [II

W molar flow rate of a solute, mole/set PI

Greek symbols

6 surface area devlatlon factor defined m eqn (50) E31

0 dialysis coefficient defined m eqn (9) c41 4 overall dialysis coefficient of a composite dlalyzer

[51 Subscripts

Z upper (or left) sectlon of a dlalyzer [61

ZZ lower (or right)) sectlon of a dlalyzer 1 first single dlalyzer element :;; 2 second smgle dlalyzer element PI

REFERENCES Graham T, Roy Sot London Phd 7Ton.s 1861 151 183-224 Klein E, Holland E F , Lebouf A, Donnaud A and Smith J K J Memb Scz 1976 1 371 Colton C K , Smith K A, Merrill E W and Farrell P C J Boomed Mater Res 1971 5 459488 Frltzmger B K, Brauman S K and Lyman D J J Boomed Mater Res 1971 5 3-16 Lonsdale H K , Cross B P , Graber F M and Mllstead C E J Macromol Scr Phys 1971 5 167-188 Noda 1 and Gryte C C A Z Ch E J , accepted for Dubhcatlon (19791 kchaels A ‘b, ASAl frans 1966 12 387 Noda I and Gryte C C AZChE J 1979 25(l) 113 Noda I, Brown-West D G and Gryte C C J Memb Set 1979 5 209