MULTICOMPONENT ION EXCHANGE
IN FIXED BEDS General Properties of Equilibrium Systems GERHARD KLEIN, DANIEL TONDEUR, AND THEODORE VERMEULEN Sea Water Conversion Laboratory, University of California, Richmond, Calif.
A theoretical analysis of multicomponent ion exchange in fixed beds is presented. Equilibrium operation, uniform prescituration, and constant feed composition are assumed. Rules for outlining over-all concentration profiles, formulated in terms of dimensionless parameters, predict the number of composition changes (transitions) between zones of constant composition (plateau zones); whether these transitions exhibit constant or proportionate patterns; the order in which the concentrations of various components can become zero; and the proper selection of roots in calculating throughput-parameter values within transitions. Differences are discussed between the integral material-balance equation, applicable to abrupt composition changes, and the integrated differential material-balance equation applicable to gradual composition changes. Methods based on these material balances are developed for variable-separation-factor systems to calculate concentration-profile curves, the location of concentration discontinuities, and the composition of plateau zones Where all of the components present in a transition zone (boundary of chromatography) are also present in the adjacent plateau zones, approximate ideal extrapolation of the profile curves of one component to zero concentration facilitates calculation. Application of these methods is illustrated. The profiles obtaihed are readily convertible into effluent-concentration histories.
ORPTION operations
(adsorption, ion exchange, ion retardation, ion exclusion, etc.) are commonly carried out in fixed-bed columns. Exhaustion (or saturation) and regeneration constitute a cycle The performance of a fixed sorption bed is governed by a combination of stoichiometric, equilibrium, and rate relationships. Consideration of only the first two of these factors leads to the equilibrium theory of sorption operations, developed here for ion exchange but adaptable also to multicomponent adsorption. Application of this theory is useful in the following I-espects: From equilibrium data, feed composition, and initial composition of the exchanger alone, equilibrium theory readily predicts ideal process performance, on the basis of which technical and economic feasibility may be estimated or the qualitative effects of any change in operating conditions readily ascertained. 'This qualitative validity is especially useful where a process must pass through a number of cycles before it operates reproducibly. In uniformly presaturated exchanger beds receiving a feed of constant composition, zones of constant composition (plateau zones) and of varying composition (transition zones) arise. The effluent concentrations arising from these zones are, respectively, constant or transitory. When such beds are operated under practical conditions, if the detention period is sufficient, plateau zones still develop. The equilibrium theory then predicts the number of transitions, identifies each transition as being of the constant-pattern or proportionate-pattern type, and yields the exact composition in each plateau zone and therefore also the stoichiometric-average solution volume around which concentration changes center. For multicomponent systems, this information is important, both directly and as a basis for calculations accounting also for mass-transfer inefficiencies. The basic equations for fixed-bed sorption of several solutes under equilibrium conditions were set forth in 1943 by De Vault (2). Two years later, Walter provided explicit solutions for certain ion-exchange systems in which the equilibria between the exchanging ion species are governed by constant
separation factors (74). This treatment was largely paralleled in a later article by Glueckauf, giving a theory of chromatography based on Langmuir isotherms and involving up to three adsorbate species ( 4 ) . Silltn used a somewhat different approach to obtain similar results, but in a simpler way owing to the use of normalized variables (72). More recently, multicomponent equilibrium sorption was critically reviewed by BaylC and Klinkenberg (7, 9 ) . Helfferich has treated the phenomena occurring in sorption columns on the basis of the relative rate of migration of the components and the concept of composition paths ( 5 ) . Methods are developed here for generalizing the previously published theory and reducing it to practice. Applications include the difficult case of variable-separation-factor isotherms. Analytical solutions for constant-separation-factor systems are presented by Tondeur and Klein (73). As contributions to the fixed-bed equilibrium theory, this work includes general rules for determining the number of zones of constant composition, the sign of the slope of concentration-profile curves, and the order of points a t which the concentrations of various components can become zero. The employment of fictitious concentration profiles to replace concentration discontinuities, and to extrapolate concentration-profile curves to zero concentration of one component, has been found valuable in simplifying the calculations. The dimensionless parameters introduced by Hiester and Vermeulen (6) have been used throughout. While the computation methods as presented in specific numerical examples extend to systems of three exchangeable ion species only, the other findings are applicable to any number of components. A broader coverage of introductory material than possible here has been given by the present authors (7). Stagewise computation methods have been developed by Pandya, Klein, and Vermeulen (70). VOL. 6
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Basic Concepts
Assumptions. The “equilibrium” theory is based on the assumptions of local equilibrium a t any time and at all points of the exchanger bed; absence of hydrodynamic mixing and diffusion in the flow direction; homogeneity of the bed; and absence of secondary processes, such as neutralization, weakacid or weak-base formation in the fluid phase, complex formation, or precipitation, reported on elsewhere ( 7 7 ) . Characterization of Fixed-Bed Performance. The course of an ion-exchange column is usually described by an effluentconcentration history-Le., by the concentrations of the exchangeable ion species as functions of time or of effluent volume. Under equilibrium conditions with uniform presaturation and constant feed composition, it can be shown that one obtains a unique concentration history for a given ion exchange material, feed, and initial exchanger composition, by plotting the effluent concentrations against the ratio of the equivalents of solution to equivalents of exchanger. This dimensionless ratio, called the “throughput parameter,” is defined in terms of bed volume u, void-fraction E, feed-solution volume V , exchange capacity Q, bulk density of the exchanger p , and total solution normality C,, by
C,(V
T =
- UE)
1-
l/T Figure 1. Typical concentration Drofile for four-component system. exhibiting plateau zones and both gradual and abrupt transitions
D goes to zero while B rises to its presaturation level. The presaturated zone is counted as the fourth (in general, the nth) plateau zone. Equilibrium Relations. A general expression for the equilibrium relationships of k exchangeable ion species (the “components”) between solution and exchanger may be written in the form
~ Q P f l (XI,XZ,
T o the normalized effluent-concentration history, there corresponds a unique normalized concentration profile within a column of sufficient length, obtained by representing the concentrations as functions of 1/T. Either the solution-phase or exchanger-phase concentrations may logically be selected to give a complete picture of the course of the operation; the latter is generally used in this discussion. Normalized liquid- and solid-phase concentrations (equivalent-fractions) of the ith component of a system are defined, respectively, as
(3) where c t is the normality of the species in solution, and qr is the exchange capacity occupied by this species. In most cases changes in total solution concentration, due to physical adsorption, volume changes of the exchanger, or secondary reactions, can be neglected for practical purposes. The total capacity is also approximately constant, leading to k
c x i = l i=l
(4)
f2 (XI,X2,
. . . . ., XA-1,Yl) . . . . ., X L l , Y 2 )
= 0 = 0
.......................... ..........................
fE
(Xl,XZ,
. . , . ., X k - 1 , y O
= 0
.......................... fk-I(Xl,X%
. . . . ., Xk-1,YL-I)
= 0
Because it applies strictly a t only one specified temperature, this set of equations constitutes an equilibrium “isotherm.” Equation 4 indicates that it is sufficient to specify only k 1 x’s in each equation; Equation 5 implies that there are only k 1 independent equations of the type of Equation 6. Thus, 1 concentration variables (either x’s or y’s) if any set of k is specified, these, with Equations 4, 5 , and 6 will define the values of all the remaining concentrations in the system. Where the presence of other components does not affect the equilibrium relationships between any two of the components significantly, the isotherm can be expressed approximately in terms of k - 1 independent equations of the type
-
-
-
k
CY$=1 i=l
(7) (5)
where k designates the number of ion species locally present. A typical four-component profile, as determined from the ensuing analysis, is shown in Figure 1 for a column fully presaturated with component B, and receiving a feed containing the exchangeable species A, C, and D. The first (upstream) plateau zone contains the feed components. (Throughout this paper, the various zones and transitions are numbered in the direction of flow.) I n the first transition (abrupt here), the concentration of component A goes to zero, with a compensating increase in the concentrations of C and D. The second plateau zone is followed by a gradual transition in which B rises from zero and C goes to zero. B and D are present in the third, short plateau zone. I n the last transition, 340
l&EC FUNDAMENTALS
1
2
where i and j designate different components. A binary equilibrium relation frequently employed is that of massaction type without activity corrections,
Here K B is~ the selectivity coefficient, and a and b are the absolute values of the respective valences of ion species A and B entering into the ion exchange reaction
bA’,
+ a BRb = OBtb f bAR,
(9)
R denotes the ion exchanger, usually of the resinous type, and AR, and BRb are the resinates.
O n combining Equations 2 and 3 with Equation 8, one obtains an expression for the “dimensionless selectivity coefficient” :
The separation factor, a g A , is the ratio of distribution coefficients for components A and B at a specific set of concentrations:
If a # b , Equation 11 implies that aBAvaries with composition; is constant. while for a = b , a B A = (.KBA)’/‘ = (KBA)’lU I n the event that k -- 1 x’s are specified, the remaining x is obtained by differen.ce from unity, utilizing Equation 4. Where selectivity coefficients are known, solution for the y’s involves the equilibrium1 relations
where Y ; is the absolute value of the valence of component i. For any component i, E,quation 5 then yields a polynomial in which only y t is unknown:
-
Likewise, if k 1 yI)s are specified, the xI)s can be determined by solving the respective relations:
Figure 2.
The prablem of finding the remaining concentrations if k - 2 x$s (or y 2 s ) and one y i (or xi) are known, is illustrated here for a three-component system in which X A and y A are known. The selectivity-coefficient relations then yield y B / x B (= kB) and y C / x c (= kc). Elimination of XC, y e , and y c yields 1- k B X B = (1 - X A - X B ) k C (15) and hence :
Three-component isotherms can be represented on a triangular diagram. The regular grid of the diagram is used to represent the composition of one of the phases; the composition in the other phase is given by a set of contour lines, each corresponding to a constant concentration of one of the components. For constant-separation-factor isotherms, the preparation of such a diagram is facilitated by the rectilinearity of these contour lines, derivable from the ternary isotherm equations (73). Figure 2 shows such a diagram for variable-separationfactor equilibria as characterized by Equation 10 and used in the examples below. The solid-phase ( y ) concentrations can be read off the original grid, and the solution ( x ) concentrations are given by the contours. For constant-separation-factor isotherms, the contour lines are rectilinear (73), so that variable-separation-factor systems may be viewed as distorted constant-separation-factor systems and expected to exhibit properties identical to those of constant-separation-factor systems over infinitesimal composition changes, and qualitatively similar behavior over finite composition changes.
Graphical representation of ternary isotherm
(YA/XA)/(XC/YC)’
= 8.06 (YB/XB)/(XC/YC)~ = 3.87
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Material Balances. GRADUAL TRANSITIONS. The difference in the amounts of the ion species i that enter and leave a column “slice” perpendicular to the direction of flow must be balanced by changes in the liquid- and solid-phase concentrations in the slice. For an infinitesimal slice, through which an infinitesimal amount of solution is being passed, this relationship may be expressed by the system of partial differential equations, one for each component,
J
2
z Glueckauf ( 3 )has derived a slightly different material balance which includes axial mass transfer by molecular diffusion and hydraulic dispersion. For continuous concentration changes, Equation 17 applies to any geometry, such as radial or axial flow in cylindrical or conical beds; it is not restricted to equilibrium operation. I t is evident that Equation 17 is satisfied by solutions of the form
xi
=
constant; y i = constant
(1 8)
corresponding to a plateau-zone composition. For the case of uniform presaturation and constant feed composition, Silltn (72) has transformed Equation 17 into the ordinary differential equation
where T is the throughput parameter defined in Equation 1, and (dyi/dxi)pathis evaluated along a “composition path” (5) followed throughout a transition. For set compositions of the plateau zones bounding the transition, this path, which must satisfy the isotherm, is unique. Methods for evaluating the path derivative are developed below. For any composition, their application leads to a specific set of values of (dyJ dxt)psth,and thus, according to Equations 1 and 19, of V/v. For a uniformly presaturated column receiving a feed of constant composition, the rate of movement of this composition, relative to the rate of advance of the liquid front in the column, is thus
This relation is useful in calculating the concentration profiles arising in columns with step changes in presaturation composition, that receive a feed undergoing stepwise composition changes. Equation 17 is not valid in abrupt transitions, involving concentration discontinuities. Here formal application of Equation 19 leads to concentration profiles which are physically meaningless in the sense that the plateau zones on either side of the concentration change are calculated to overlap during the transition. However, such “fictitious” profiles are useful in determining the qualitative course of the over-all concentration profiles and as conceptual aid in calculating them. Equation 19, applicable to gradual transitions between plateau zones, readily lends itself to integration:
0;
Figure 3. equation
7’
T2
Integration of
Ti
differential material-balance
Shaded area = yil
-
yiz
T h e physical significance of this equation is illustrated in Figure 3, which represents a portion of a variable effluentconcentration history for a component i. The equilibrium solid-phase concentration has been plotted for comparison. Methods of constructing such curves are given below. The integral of Equation 21 represents the shaded area. Solid-phase concentration profiles can be integrated similarity:
where the symbols denoting concentrations have the same meaning as in Equation 21. ABRUPTTRANSITIONS. Criteria for the occurrence of gradual or abrupt concentration changes in multicomponent systems are developed below. For an abrupt transition, a material balance establishes the following relation between the concentrations of any two components i and j in the plateau zones adjacent to the transition: Yi2
- Yil
xi2
- xi1
-
Yj2
xj2
- Yj1 - XI1
-
T,,
(23)
Here T is the stoichiometric value of the throughput parameter corresponding to the transition, and subscripts 1 and 2 refer to the upstream and downstream plateau zones, respectively. The equality of T of one transition for all components is due to the interrelation of all concentrations through equilibrium ; a change in the value of any one of these variables must entail a change in all the others. For abrupt transition in a constant-separation-factor system, the fictitious profiles given by Equation 19 satisfy Equation 23. For a variable-separation-factor system, this is no longer true, but the fictitious profile gives an approximate guide to abrupttransition behavior. Conversely, Equation 23 applies to gradual transitions only in the case of a constant-separation-factor system. For a gradual transition to which a variable separation factor applies, a T ican be defined (as shown) for each component, but the values for the different components are no longer exactly equal. The correspondence between Equations 19 and 23 in the constant-separation-factor case is demonstrated in another paper (73). Plateaus and Transitions
where xil, y i ~ ,and xiz, yiz are equilibrium concentrations corresponding to the bounding values T I and Tz of a T interval. 342
I&EC FUNDAMENTALS
T h e present section shows how the principles just set forth can be utilized to predict equilibrium-column performance. The calculation methods require certain algebraic relations stated only implicitly, or without proof, in the previous litera-
ture. I n addition to clarifying these relations, the present section introduces several previously unidentified properties of over-all concentration profiles. Specifically: Methods are developed for obtaining the composition path on which the compositions of two plateau zones separated by an abrupt transition must lie. An analogous method is presented for obtaining the composition path for gradual transitions. The relations governing the two types of path are not necessarily the same. A simplified, approximate procedure is developed for computing composition paths in gradual transitions. An equation for calculating concentration profiles from composition paths is derived for mass-action-type equilibria. The number of plateau zones is shown to equal the number of components in the system, in general. The equation giving the throughput parameter for a transition is shown to have n -- 1 positive roots in T, n being the total number of components occurring in the system. The assignment of these roots to the various transition zones according to their magnitude has been termed the “root rule.” The “slope rule” establishes the sign of the concentrationprofile slopes of the various components in each transition. The “alphabet rule” defines the points a t which the concentrations of various components can become zero. Individual Transitions. ABRUPT TRANSITIONS. If the point set (xzl,xJ1, . , ytl,yjl, ) in Equation 23 designates the known composition of a plateau zone (an “anchor point”), this equation imposes restrictions on the allowable composi) in an adjoining plateau zone tions (xt2,xj2, , yz2, y32, separated from the “anchor” plateau by an abrupt transition. If the total number of exchangeable components in the two zones is X, the concentration variables must satisfy k - 1 independent equilibrium relationships and k - 2 independent expressions of the type of Equation 23, in addition to Equations 4 and 5. There are 2k concentrations in the unknown plateau zone. If one of these concentrations is arbitrarily set, the 2k 1 others can be calculated from the 2k - 1 equations just mentioned. The locus of all points having the concentrations that satisfy the relations mentioned as coordinates may be termed a composition path T h e solid curves of Figure 4 show such paths on triangular coordinates for a representative three-component variable-
-
/&
separation-factor system. The exchanger-phase concentrations are shown. The equilibria used for this plot, as well as the detailed calculation method, are given in example 2 below. Point Pi corresponds to the given composition of a plateau zone. Using the coordinates of this point in the equations indicated, three composition paths can be generated, only one of which is shown in the figure (as path 1). This path corresponds to the transition between plateau zones 1 and 2. Another path corresponds to the transition between plateau zones 2 and 3. A third, spurious, path arises because of the complexity of the equilibrium relations used. The second solid curve on the diagram (path 1’) belongs to the same transition as path 1, but was generated using the concentrations corresponding to the intersection of the first curve with the line y B = 0 as the given “anchor point.” The two curves are seen to be different. I n general, let path 1 be the path generated from a point 1, and let point 1’ be some other point on path 1. The finding just described then indicates that a path l‘, based on point l’, may be different from path 1. I n our example, both paths are seen to go through both reference points. T h a t this must be so in general is evident from the interchangeability of concentrations l and 2 in Equation 23. This behavior of the two composition paths may be viewed as a type of hysteresis. As shown (73), the composition paths in constant-separation-factor systems are linear and therefore identical, whether generated from point 1 or 1I . (Path 1” is discussed below.) I n the particular example illustrated in Figure 4, calculating a point on path 1 involves solution of a cubic equation, while obtaining a point on path 1 ’, starting from point 1 I , on the y B = 0 line, requires only solving a linear and a quadratic equation successively. One may therefore utilize the relative similarity of the two paths by calculating point 1’ first, then computing points on path 1‘, and considering path 1‘ as a n approximation to path 1. Although composition paths calculated by these methods may be useful over their entire range, only part of them has real meaning. For the type of curves shown in Figure 4, only the portion starting from the anchor point and going in the direction of decreasing concentration of component A corresponds to an abrupt transition. GRADUALTRANSITIONS. Here, the differential materialbalance equation (Equation 19) applies instead of the integral material balance. To construct a composition path for this case, it is convenient to have expressions for derivatives of the type dyc/dyj(= dx,/dxj). Differentiating Equations 6 with respect to xi gives
fi’(xi, XZ, . . . . xk-1, yi, dxzldxi, d ~ / d x i ,. . . . dXk-l/dXl, dylldxl) fz’(x1, ~
=
0,
=
0,
. . . . Xk-1, yz, dxildxz, dxaldxz, . . . .
2 ,
dxk-i/dxz, dyzldxz)
................................................. fi’(X1, X Z ,
. . . . xk-1, yi, dxlldxi, dxzldxi,
,
(24)
....
dxk-lldxi, dyildxi)
=
0,
.................................................. fic-l’(Xi, XZ,
. . . . Xk-1, yk-1, dXildXk-1, dxzldxli-1, . . . . dXk-a/d%-i,
Figure 4. Composition paths for first transition in ternary variable-separation-factor system Data of example 2 P I corresponds to plateau zone composition
--
-____
dyt-1ldxk-d
=
0
Also, it is necessary to differentiate Equation 4 with respect to xk,where the kth component is selected as the reference:
Derived from integral material balance Derived from differential material balance
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If dyi/dxiin Equations 24 is replaced by T,and each derivative dxi/dxj in Equations 25 is replaced by (dxi/dxk)/(dxj/dxk), a system of k equations is obtained in the k unknowns dxl/dxk, dxn/dxk, . . ., dxk-l/dxx, and T,which in principle can be solved simultaneously. For mass-action-type isotherms, Equation 10 may be written in the form
for the ith and kth ion species, where U~ and vk are usually the absolute values of the valences. (They may be purely empirical coefficients.) Implicit differentiation with respect to xk, after rearrangement, leads to
h’ow Kxi is replaced by the expression of Equation 26, and
dyk/dxk and dyi/dx, each by T (cf. Equation 19), to yield dxi - dyt - u l xiyi TXk - yk dxk dyk vlc XkYk T x i - yi
(28)
The identity of dxi/dxk and dyi/dyk is again derived from Equation 19. If Equation 4 is next differentiated with respect to xk, and the dxi/dxi;terms in the resulting equation are replaced by the expression of Equation 28, one obtains
For an n-component system, Equation 29 has n - 1 positive roots in T. A4sshown below, the largest of these roots corresponds to compositions in the first transition zone; the nextto-the-largest root, to compositions in the second transition zone, etc. Where the concentration of one of the components is zero, Equation 29 becomes indeterminate. The value of T for this point, however, can be obtained from Equation 19 as
(T),=,
=
lim Yl: xi-0 yi+o
xi
I n the case of mass-action-type equilibria characterized by Equation 10, this limit is easily computed as
Here, 2, and y j are the concentrations of any component other than i present in the adjacent plateau zone. T o get the value of dxi/dxk or dyt/dykfor a given composition, the appropriate root of Equation 29 may now be substituted into Equation 28. The assignment of the reference component k is arbitrary, but k cannot be i. A composition path may now be computed by replacing the differentials in dyi/dy, by finite increments. After selecting a small increment for the y of any one component, the corresponding increment in the y’s of the other components (and hence the successive values of all these y’s) can be computed, starting from a known plateau-zone composition. I n each repetition of this calculation step, the composition last calculated is used to obtain new values for the derivatives dyildyj. 344
I&EC FUNDAMENTALS
Using the method indicated, a composition path (path 1”) going through point PI in Figure 4 was constructed, which conforms closely to path 1 except in the dashed region. This path (as a gradual transition) has real meaning only in the direction of increasing concentration of component A. Path l’, which is more readily constructed, can be used as an approximation to path 1” also. Application of the finite-increment method to the construction of concentration profiles is illustrated in example 1 ; to obtain the profile, the reciprocal of the T values corresponding to concentrations along the composition path is simply plotted against these concentrations. APPROXIMATELY INTEGRAL RULEFOR GRADUAL TRASSITIONS. Equation 21 may be used to define a stoichiometric or mean value of the throughput parameter in a transition zone:
where subscripts 1 and 2 refer to two levels in the transition zone. For constant-separation-factor (and for binary) systems, the equality of (T,)12 for all components leads to an equation formally identical with Equation 23. The approximate validity of this equation for transition zones in variableseparation-factor systems can be used for the approximate calculation of concentration paths and profiles in such systems. When paths 1 and 1’ as defined in the preceding section are different, the paths generated using the differential and integral material balance must be different also. I n a region of gradual composition change, since the differential material balance must apply, the integral material balance cannot apply also. The explanation is as follows. For any given transition, Equation 28 defines a direction field through which one and only one differential concentration path can be laid for a given set of boundary conditionsLe., for a given set of concentration coordinates of a point on the path. Because Equation 19 is a limiting expression of Equation 23, the differential path (path 1”) must be tangent to path 1 (at point 1 in Figure 4). If the hypothesis were correct that path 1 ” coincides with path 1 over its entire range, both paths would have to go through any point P’ on either path. As established before, the path based on the integral material balance and generated from this point is not the same as path 1. Since the differentialmaterial-balance path through this point would have to have the slope of both path 1 and the new path generated from this point, which is manifestly impossible, the hypothesis must be false. Over-all Concentration Profiles. In the present section, rules and procedures are presented which make it possible to calculate over-all concentration profiles in multicomponent systems. NUMBEROF PLATEAUZONES. A relation highly useful in obtaining a qualitative picture of column performance is now developed: that the maximum number of plateau zones is equal to the number of exchangeable components present in the system. Let the exchanger bed initially have presaturation levels yip, . . ., y i p ’ , . . ., ynP, and let the feed concentrations be X I I , X P I , . . ., xil, . . ,, xnl, the first subscripts referring to exchangeable ion species, and the second to the plateau-zone number. I f p is the maximum number of plateau regions after all of the interstitial solution present initially has been pushed out of the column,p - 1 transitions will occur in the exchanger bed. The equations available will include 2p stoichiometric
relationships of the type of Equations 4 and 5, (n - 1)p independent equilibrium relations of the type of Equations 6, and ( n - 2)(p - 1) independent differential or integral material-balance equations (Equations 28 or 31)-that is, 2np p - n 2 equations in all. There are 2 np concentration 1) quantities x,1, X Q , . . ; and yi1, yi2, . . ., of which 2 ( n are given in the form of independent feed concentrations and n 1) presaturation levels. ’There are therefore 2(np unknowns. ‘4n exchange system involving a bed in a known initial state and a feed of known composition, with specified equilibrium relations, can be expected to be fully defined. I t follows that the number of independent equations and unknown concentrations must be equal-Le., 2np - p - n 2 = 2np 2n 2 , or p = n. This means that the number of plateau zones must equal the number of components. If some of the presaturant components are absent from the feed, or some of the feed components are not present in the exchanger initially, the adjusted numbers of equations and unknowns still lead to the general statement that the maximum possible number of plateau regions is equal to the total number n of distinct exchangeable ion species. Three cases arise in which the number of plateau zones is less than the number of components: As operation of the bed is continued, the downstream plateau zones will disappear from the bed ; two separate plateau compositions will coincide so that the intermediate transition (or transitions) vanishes (Figure 6 b ) ; and the length of a plateau zone will shrink to zero without losing the transitions adjacent to it, which can result in a sharp peak of the concentration profile of at least one component. PLATEAU-ZOXE COhfl~OSITION. Where a plateau zone of known composition adjoins a transition zone, a t whose other boundary the concentration profile of one of the components goes to zero (see Figure 5), the composition of the next plateau zone can be found by constructing the concentration profiles (or the composition path) for the transition zone, using the procedure given below. T h e concentrations sought are given by the ordinates of the profiles a t the value of 1/T a t which the mentioned component vanishes. The procedure is illustrated, for a three-component system, in the first transition zone of example 2. If the plateau zones of known and unknown compositions are separated by a n abrupt transition, Equation 23 are used in conjunction with Equations 4 and 5 and the appropriate equilibrium relations. The procedure is illustrated in example 2 for the relatively difficult case of a ternary feed entering a column presaturated with the same three components. T h e exact solution for the case considered in the example requires a
+
-
-
+
+
+
numerical solution by use of a computer, with numerical or graphical construction of the intersection of compositions in the two transition zones bracketing the unknown plateau zone. An intrinsically approximate method, involving the extrapolation of concentration profiles to the zero of one component, leads readily to a similar solution by hand computation alone. The application of Equation 23 to gradual transitions in this way also yields useful approximate solutions. THE“ROOTRULE.” Equation 29 enables one to find T for any given composition within a transition. Hoizever, since this equation has multiple roots in T for systems of more than two exchangeable ion species, a criterion (termed the “root rule”) is needed for selection of the proper root. Equation 29 can be transformed so that its left member becomes a polynomial of degree k - 1. For a given composition, the equation thus has k 1 positive roots in T. Experience with numerous fully calculated constant-separation-factor examples, for which analytical solutions are available, has shown that the largest of these roots applies in the first upstream transition zone, the next-to-largest in the second transition zone, etc. This assignment of the roots is in harmony with the fact that, as we approach the limiting case in which all n plateau zones have the same composition, from the case of distinct compositions, we would expect the adjacent downstream and upstream boundaries of neighboring plateau zones to approach each other. One would thus approach n - 1 values of T for a single composition. The largest value of T (corresponding to the smallest value of 1/T) must therefore be assigned to the transition zone nearest the column inlet, and the other values, successively, to the other zones. THE“SLOPERULE.” Two rules applicable to concentration changes in multicomponent systems are useful for predicting whether a concentration change is abrupt or gradual, for establishing the qualitative composition of various zones, and for reducing the number of qualitatively possible over-all profile patterns, thus facilitating subsequent detailed computations. T h e rules, termed the “slope rule” and the “alphabet rule,” are applicable to any system in which the order of selectivity is not altered by changes in the mixture composition. Symbols A, B, C, etc., in alphabetical order, are considered to refer to ion species whose separation factor decreases, relative to any given component, with A held by the exchanger the most strongly of all. We consider gradual transitions first. For these, the slope rule states that, in the first upstream transition zone, the profile of A, if A exists in this zone, has a positive slope ; in the second transition zone, the profiles of A and B ; in the third, those of A, B, and C, etc. The slopes of all the other profiles are negative. These relations are summarized in Table I, where the transition zones are designated by the numbers of the adjacent plateau zones. We can extend the rule to abrupt transitions if we assign a positive slope to any component having a higher concentration in the upstream than in the downstream plateau zone adjacent to the transition.
-
Com-
0
ponent A
l/T
Figure 5. Ternary-system transition zone with two profile curves going to zero
B C D
..
Table 1. Sign of Transition-Profile Slopes Zone Zone Zone Zone 23 34 45 12
+-
-
..
+-+ -
++ +-
..
..
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.
I
.
... ... ... ...
...
... 345
a
1 Y 0
1
Y
Figure 6. Schematic concentration profiles for bed presaturated with components A and C, receiving feed containing components B and D
_-__
Fictitious profiles
The slope rule is illustrated in all the diagrams of Figure 6, where the six qualitatively distinct profile patterns are shown which are possible in a column presaturated with components A and C, receiving a feed containing components B and D. For the sake of simplicity, concentration-profile curves are represented by sloping straight lines. Full lines represent real profiles; dashed lines, fictitious profiles. The mathematical zone boundaries are shown as thin vertical lines and labeled (12)’, (12)”, etc., the numbers in parentheses indicating the adjacent plateau zones, and the primes, the position of the boundary within the transition zone. We can prove the slope rule for gradual transitions in systems containing n components throughout, provided that the affinity series A, B, C, . . ., Z holds throughout the given system. This condition implies that the inequalities
KZA> KzB > KZC . . .
>1
(33)
are sufficiently great to ensure the validity of the relation
y - A > y e > y c , , , > - Yz XA
XB
XC
over the entire concentration range of interest, a condition which is frequently met. Equation 29, applied to n components, has n 1 positive roots TI,TP,. . . , Tn-1, in T, such that
-
G > T l > P > T 2 > - Yc . . . XA XB xc
(35)
If T I is used, the first term of Equation 29 is negative, and all others are positive; if Tz is used, the first two terms only are negative, etc. Since, as was mentioned above, Ti applies to the ith transition zone, this means that the term corresponding to component A is negative in the first transition zone, the terms corresponding to components A and B are negative in the second transition zone, etc., all other terms being positive. T h e sign of each term can be shown to determine the slope of the concentration profiles of the components whose concentrations the term contains. 346
l&EC F U N D A M E N T A L S
T = -dYt < dxi
Yi XI
(36)
applicable to a negative term in Equation 29, ensures convexity upward of the y rus. x t curve, equivalent to * (y/x) means that x and y decrease with 1/T and conversely, if the inequality sign is reversed, that x and y increase with 1/T. Therefore the condition
d2yt - dT dx? dxi
(38)
Hence dT/dxr and dxi/dT are negative if the term for component i is negative, and the slope of the concentration profile, dx,/d(l/T), must be positive. Since all T values are positive, it follows from Equation 19 that the slope of the yi profile has the same sign as the corresponding term in Equation 29. The slope rule can be applied also where some of the components are absent from part of the system. T o prove this, it is merely necessary to substitute trace concentrations, and to let these vary toward zero, for the absent components. The proof of the slope rule for abrupt transitions follows from the mutual tangency of the differential-material-balance and integral-material-balance compo;ition paths when both are generated from the same composition point. THE“ALPHABET RULE.”This rule has been derived by the authors for constant-separation-factor systems (73), but its applicability to other systems has been demonstrated by calculations and experiments. The alphabet rule states that
the mathematical profile curves which rise from zero, or go to zero, do so a t distinct column levels; and that these levels, when labeled with the symbols of the corresponding components, are arranged in alphabetical order. Specifically, if we designate the successive zone bounds by (12)’, (12)”, (23) ’, (23) ”, etc., as before, the various components can vanish a t the column levels indicated below: Zone bound Zero of component
(12)’ (12)’’ (23)’ (23)” (34)’ (34)” A
B
B
C
C
D
,
.,
...
I n Figure 6, a, b , c, for instance, A rises from zero a t level (12) I , B goes to zero a t (12)”, C rises from zero at (34’), and D goes to zero at (34)”. I n Figure 6d, A, B, and D behave similarly, but the fictitious profile of C rises from zero at The (23)”; the actual profile starts discontinuously a t (B). case depicted in Figure 6f is analogous; here, however, the fictitious profile of B goes to zero a t (23)’, while actually, B disappears discontinuously at (5).I n Figure 6e, B dis; the alphabet appears, and C appears discontinuously a t rule is again obeyed when fictitious profiles are considered. APPLICATION O F SLOPEA N D ALPHABET RULES. T h e diagrams of Figure 6 were obtained in the following manner, which illustrates also the general manner of applying the two rules just discussed. Since the system involves four exchangeable ion species, we expect four plateau zones to develop. From the feed and presaturation compositions, it is known that the first plateau zone (counted from the upstream end) must contain components B and D ; and the last, components A and C. Arbitrary levels of these components can be marked in these zones. Since components A and C are absent from the feed, and B and D from the exchanger in its initial state, the profiles of each of the four components must intersect zero a t some point. I n accordance with the alphabet rule, the profile of A can rise from zero only at level (12)’, causing the first transition to be gradual. Similarly, the least strongly held component, D, must go to zero a t (34)” in a profile of negative slope, and the third transition is gradual also. Component B can vanish only a t (12)” or (23)’. If B vanishes a t (12)”, C can begin to rise a t (34)‘, leading to the qualitatively symmetrica.1 patterns a, b , and c; or a t (23)”, leading to profile d. If, on the other hand, the fictitious profile of B goes to zero a t (23)’, C must begin to rise a t (23)” (diagram e), or a t (34) ’ (diagramf). Which of the patterns shown develops depends on the feed composition and presaturation conditions. T h e latter are the same for patterns a through d but the proportion of B in the feed is rising from pattern a to pattern d. With a relatively low concentration of B in th,? feed, the concentration of A in the second plateau zone is lower than in the third, and the intermediate transition zone is real ( a ) , while, with a higher con-
(z)
Table II.
centration of B, the concentration of A in the second plateau zone is higher than in the third, and the intermediate transition zone is therefore fictitious, the actual discontinuity being at (%) (diagram c). A singular intermediate case is shown as pattern b , where the t\vo intermediate plateau zones merge into one. When the proportion of B in the feed becomes still higher than in example c, pattern d results, again with a discontinuity a t ( E ) . Case e and f correspond to reversed proportions of A and C in the presaturated exchanger; e with the same feed composition as in d, and b with the same feed composition as in a . Examples
I n the present section, calculations involving ternary variable-separation-factor systems illustrate the construction of concentration profiles corresponding to gradual and to abrupt transitions. I n each case, both an exact and an approximate method are given. I n practice, the exact method requires the use of a high-speed computer, while the approximate method is amenable to slide-rule or desk-calculator computation. For constant-separation-factor systems, closedform solutions are available (73). T h e equilibria used in the present illustrative examples correspond to those measured for the principal cation species in sea water and sea-water concentrates in contact with Duolite C-25 ( 8 ) . The order of selectivity of this resin for calcium, magnesium, and sodium ions is expressed by designating these species, respectively, as A, B, and C . T h e numerical values of the normalized selectivity coefficients a t the total concentration of normal sea watcr (0.62” are
Example 1, Gradual Transitions. The concentration profiles are to be calculated in a column containing exchanger in A-form and receiving a feed of composition x A ~=
0.007353; x
B ~=
0.08560;
Xc1
0.90705
T h e equilibrium exchanger-phase composition is readily computed from Equation 13, which takes the form
Inserting the numerical values of xAl,xB1, xcl, Kc*‘, and K c B into this quadratic equation, and solving it for yc1, one obtains only one positive root. Y A 1 and yBl are then computed from Equation 12. T h e resultant values, as well as other intermediate and final results, are summarized in Table 11. Calculations starting with the fully known composition of the first plateau zone can proceed by either of two methods. First, however, the over-all profile must be known qualitatively.
Selected Data Obtained for First Transition Zone of Example 1 by Differential Method
Start of Interval
‘YO. 1 2 3 10 20 30 40 49 50
XA
0.0073534 0.0095228 0 . 011747 0.029002 0.059921 0.10095 0.15672 0.22600 0.23527
XB
0.085595 0.084503 0.083389 0.07491 1 0.060449 0.042686 0.021281 0.0000914 -0.0020953
YB
YA
0.039470 0.050479 0.061488 0.13855 0.24864 0.35873 0.46881 0.56789 0.57890
0.22053 0.21501 0.20951 0.17177 0,12040 0.072810 0,030556 0,0001103 -0,0024747
VOL. 6
NO. 3
T
5.138 5.012 4.888 4.084 3.111 2.319 1.681 1.207 1.088
AUGUST 1967
347
QUALITATIVE PROFILE.The general course of the concentration profiles is obtained by application of the slope and alphabet rules. Three plateau zones should arise in a threecomponent system. These, together with the intermediate (real or fictitious) transition zones, are drawn schematically as in Figure 6, and the zone bounds marked. Plateau zone 3 has the presaturation composition, hence contains only component A. The concentrations of A must therefore rise in the second transition zone. They d o so with a positive slope, making the second transition zone real (gradual). One may then assume, tentatively, that the concentration changes between the first and second plateau zones correspond to a fictitious transition zone. This would imply that the concentrations of B and C would increase in this zone, following the slope rule, and therefore, that B and C would be present in the second plateau zone. However, this is impossible, because then both of these components would have to go to zero a t (23)", which the alphabet rule prohibits. The hypothesis of the 12 transition zones' being fictitious is therefore false; this zone is a real transition zone. The only way in which B and C can go through zero and satisfy the alphabet rule is for B to disappear at (12)", and for C to disappear a t (23)". The result is a profile of the type shown by the solid lines in Figure 7 . DIRECT(DIFFERENTIAL) METHOD. The known concentrations of the first plateau zone are substituted into Equation 29, which is quadratic in T for a three-component system. T o obtain solutions pertaining to the first transition, the larger of the T values calculated is placed in Equation 28, from which d y B / d y A is calculated. A y B is next computed for an arbitrarily chosen small interval AYA,using the approximate relation
Calculations with intervals of different magnitude may be made to demonstrate that intervals smaller than the interval used would lead to practically identical profiles. The new composition is YA = y A 1 $- AYA,YB = y ~ -k i AYE, and yc = 1 YA - YB. The xc corresponding to the y values found is next calculated using Equation 14; XA is then obtained from the equilibrium relations, and xB by difference. The process is now repeated, starting with the concentration values just reached. Selected compositions and the corresponding T values calculated by this method are presented in Table 11, and plotted as solid profile curves in the first transition zone of Figure 7 . The second plateau zone begins where component B vanishes. From this point on, the system is binary and only components A and C exchange in the second
-
Figure 7.
Concentration profile of example 1
-Differential method - - - -Approximate extrapolation method 348
l&EC FUNDAMENTALS
transition zone. The profile for this transition is constructed simply by assuming a few suitable values for yA, computing xA from the equilibrium relation, and calculating T from Equation 19. EXTRAPOLATION METHOD (APPROXIMATE). I n using this hand-computation method, we first extrapolate the composition path for the first transition to the zero of component A. From the extrapolated composition, we next calculate various approximate compositions along the path, compute the corresponding T values, and plot the profile. Approximation occurs in using the integral material balance from an anchor point other than a point representing an actual plateau composition; and also in applying this material balance to computing compositions in the direction of increasing A, in which the differential material balance should be used for an exact calculation. Let the concentrations extrapolated from the first plateau Zone be ( X A A=O, ~ ( Y A d ~ = 0 > (XBI)A=O, ( Y B A=O, ~ (XCl)A=O, and ( Y C ~ ) A = O . With ( X A ~ ) A = O = 0 and ( y A l ) A = o = 0, the concentrations of B corresponding to XA = 0 and YA = 0 are obtained from Equation 23: E l = YBl XAl
xB1
-
(yBl)A=O (XBl)A=O
I n the imaginary plateau zone having the extrapolated composition, only components B and C are present. The linear relation just shown, combined with the binary equilibrium expression, yields
This relation becomes a cubic equation in either (xB~)A-o or ( y B l ) A = o , with three real roots; these, as extrapolated concentrations, can fall outside the normal range from 0 to 1. I n consequence of the slope rule, the proper root must lead to values of (xB~)A-o, ( ~ B I ) A = o , (XCl)A=O, and (YCl)A=O that satisfy the conditions (xBi)A=O > x B 1 , ( Y B ~ ) A = O > y B 1 , (XCl)A=O > x c 1 , and ( y c l ) A = o > ycl. I n the present case, the proper root has the value indicated in Table 111. T o obtain approximate compositions within the transition zone, Equation 23 is used again, together with the equilibrium relation involving components A and B; this time, with (XBl)A-O, ( ~ B ~ ) A = o(XCl)A-O, , and (YC1)A'O as coordinates of the anchor point:
yB can thus be calculated for set values of xB. The other concentrations are computed from the equilibrium relations. The T values corresponding to the calculated compositions are found by Equation 29. The results of such calculations are listed in Table I11 and plotted as the dashed lines in Figure 7 . The T values for XA = 0 and for XB = 0 were obtained from Equation 31. The approximate result is seen to be similar to the profiles computed by the differential method. I n the present case, however, even the slight difference suffices to cause the apparent disappearance of the second plateau zone, which was very short as computed by the differential method. Example 2, Abrupt Transitions. The calculation of a plateau-zone composition is illustrated here for a three-component system in which all components are present in the feed
Table 111.
Data Obtained for First Transition of Example 1 by Integral Extrapolation Method (Approximate)
XA
0 =
XB
0.08931 = 0.06 0.04 0.02
(XA1)A-O
0.06063 0,10500 0.15140 0.19964 a
Y A
0 =
(XBI)A=O
Y B
0.24047 = 0.11918 0.06739 0.02926
(YAI)A-O
0.25089 0.36854 0.46139 0.53753
0.00
(YBI)A-O
0.00000
T 5 . 612a 3.093 2.257 1.716 1.292
Extrapolated values.
Table IV. XA
0.034000 0,028000 0.015000 0 . 0 12000 0,011000 0.010000 0.000000
0.007353 0 007712 0 010659 0 010750 0.01073
Compositions Calculated for Integral (Exact) Method
Example 2 by
Xt' YA YB FIRST(12) TRANSITION 0,176000 0.130542 0,324360. 0.109656 0,338464 0.18Cl052 0.188902 0.061423 0.371296 0.049664 0,379353 0.190959 0.045689 0.382082 0.191646 0.041685 0.384833 0.192333 0.000000 0.413604 0.199244 SECOND (23) TRANSITION 0,085595 0.039470 0.220530* 0 040135 0 239866 0 096019 0 044542 0 379458 0 189174 0 044654 0 383346 0 192259 POIUTOF INTERSECTION 0.19SO 0.04462 0.38278
a Anchor point; corresponds to feed composition. correspotds to presaturation lecel i n column.
b
Anchor point;
as well as initially in the exchanger. Because a simultaneous solution of the equations involved is required in systems in which no concentration becomes zero, they are more difficult to calculate than systems in which one or more components are absent from either the feed or the presaturated column. T h e problem illustrated was encountered in a study of softening sea water by ion exchange. I n the proposed process, an impure regenerant is used, containing sodium and magnesium ions in major proportions, together with a low concentration of calcium ion. The question arose as to the maximum possible amount of seal water which could be treated in a column equilibrated with this regenerant solution. The applicable equilibria are those given ahead of Example 1. Feed concentrations x , 1 and presaturation levels y 2 3 were as given in Table IV. T h e normalized softening profile can be expected to exhibit three plateau zones, as shown in Figure 8. From the profile, the number of equivalents of sea water that can be softened by one equivalent of resin per cycle is obtained as the reciprocal of the abscissa of the concentration discontinuity between the first and second plateau zones. I n the present example, two methods of solution will be used, as in example 1 ; but because abrupt transitions are involved, the integral material balance is used for the direct method. DIRECT(INTEGRAL) ~ [ E T H O D . The unknown concentrations y i l and x,3 in the end plateau zones are first found from the equilibrium relations, and then the appropriate composition paths are constructed through the points representing the compositions of the first and third plateau zones. The composition of the intermediate plateau zone is obtained as the point of intersection of these paths. The composition path through the point representing plateau
0.5 l/T Figure 8.
Concentration profile of example 2
zone 1 was obtained in the following manner, using an electronic digital computer: Step 1. From a new value of X A (different from X A l by an arbitarily chosen increment), and from x B 1 , compute a trial set of concentration variables y A , y B ,etc., from the equilibrium relations. Step 2. Compute
Step 3.
Compute
' Step 4. Using the equilibrium relations, compute y ~ and yB' based on X A and x B ' . Step 5. Compare
-I
If the absolute value of TA' - T B ' is larger than a predetermined small number (10-6 in the present calculations), repeat the iterative cycle between steps 3 and 5 until the difference between TAand TBlast calculated meets the criterion just mentioned. Step 6. Repeat the calculations for other selected values of x A , using the nearest available value of X B to launch the computations in steps 1 and 3. I n this manner, two paths can be generated. As x i approaches x i l , the ratio ( y r - y r l ) / ( x r - x n ) approaches d y f / d x i = T , with two different values of T, for the point representing the first plateau-zone composition. Here, the larger value of T corresponds to a first transition path, and the smaller, to a second transition. The calculated concentrations in the first transition are listed in Table IV, together with those in the second transition, obtained similarly, but with the composition of the third plateau zone as the anchor point. T h e composition of the VOL. 6
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AUGUST 1 9 6 7
349
Table V. Compositions Calculated for a 23-Transition with Feed Composition of Example 2 as Anchor Point (Integral Method) XA
0.034000 0,030003 0.023596 0.018267
XB
Y A
YB
0.176000 0.136772 0.078390 0.034395
0.130542 0.125466 0.115622 0.105053
0.324360 0.274534 0.184378 0,094947
intermediate plateau zone is given by the intersection of the two concentration paths, as shown in Figure 9. For comparison, data for a 23-transition path through the feed-composition point are presented in Table V. These data were also calculated by the integral method. EXTRAPOLATION METHOD(APPROXIMATE). Here, as in the corresponding method in example 1, the compositions of the first and third plateau zones are first extrapolated to zero concentration of one of the components along composition paths based on the integral material balance and on the compositions of these plateau zones as anchor points. Again, the extrapolated compositions can be calculated without constructing the composition paths. In the present example, although points 1 and 3 in Figure 9 (corresponding to the given plateau-zone compositions) are closer to the line Y A = 0 than to the other borders of the diagram, component B has been chosen for convenience as the component absent from either of the extrapolated compositions; this choice leads to a simple linear expression for the final result. I n analogy to example 1, the expressions
sions apply for (XA3)B=O and ( y A s ) B = o , based on the composition of the third plateau zone. As before, combination of each of these pairs of equations yields a cubic equation with three real roots, corresponding to the following values of ( x A ) B - O : Calculated Values of
StartTransiing tion Plateau Zone Zone 12 1
23
3
(XA)B-O
1
2
3
0.013642 0.004058
0.319301 0.201581
0,609957 0.492419
According to the slope rule as applied to the qualitative concentration profile, the proper value of ( x ~ 1 ) B - o must be greater than X A l , and the proper value of (XA3)B=O must be smaller than x A 3 ; hence, the first value of ( x A ~ ) B = o , and the second and third values of ( X A ~ ) B - O can be eliminated from further consideration. The second-plateau-zone concentrations x A 2 and y A 2 are computed from Equations 10 and 23: YA2
- (yAl)B-0
xA2
-
-YA -
(XAl)B=O
-K
A ~ E 2
xB2
xA2
and
Here Y B 2 / X B 2 can be eliminated immediately, leaving two equations in x A 2 and Y A 2 . Solved simultaneously, these equations yield =
(XA1)B-0
( Y A 3 ) B-0
[(yAl)B=O
-
-
( x A 3 ) B=O
(YA3)B=OI
( Y A 1 ) B=O
( K s A-
and
and
are solved simultaneously for ( x A 1 ) B - O and ( Y A ~ ) B = o , based on the composition of the first plateau zone. Analogous expres-
( Y A ~ ) B - O and ( Y A 3 ) B - O are readily obtained from the expression utilizing K C A , given above. T h e numerical results, listed together with the corresponding other concentrations, calculated from the equilibrium relations, are :
Calculated Concentrations Based on First Value of ( X A ~ ) B = Oand Third value of Second value of
..
0.5
Figure 9. Composition paths for first and second transitions in example 2 PI,
Pz, Pa.
350
Respective compositions of lst, 2nd, and 3rd plateau zones
I&EC FUNDAMENTALS
( x ~) Bi - o
( X A I )B=O
0,010670 0.044095 0.195348 0.387519 0.793982 0.568386
0.011178 0.044622 0.214136 0.410327 0.774687 0.545051
The third value of ( X A I ) B = O is seen to lead to an xc which violates the slope rule in that xc2 is smaller than x c 1 . T h e solution is therefore represented by the set of values listed in the first column. I t is seen to approximate closely the solution obtained by the direct method and listed in Table IV. I n order to obtain the exact solution for the composition of the intermediate plateau zone in ternary systems with one gradual and one abrupt transition, the differential calculation method must, of course, be used to generate the transition path for the gradual transition, and the integral method for the abrupt-transition path. The point of intersection of these paths then represents the plateau-zone composition. TO
simplify the calculations, the integral method may again be substituted for the differential method to yield an approximate gradual-transition path. As the difficulty of the numerical calculation increases with the number of components and the complexity of the equilibrium relationship, it will \be preferable, in most cases, to simulate the equilibrium constants of variable-separation-factor systems of more than three components by suitable constant separation factors. A method for solving constant-separationfactor systems u p to four components numerically was presented by BaylC and Klinltenberg ( 7 ) . Acknowledgment
T h e authors are indebted to the Office of Saline Water,
U.S. Department of the Interior, for financial support of these studies and to Everett D. Howe, Director of the Sea Water Conversion Laboratory, for his over-all supervision and encouragement. They a:lso gratefully acknowledge discussions with F. G. Helfferich., Shell Development Co.; A. L . Gram, California Institute of Technology; and K . S.Spiegler, Seymour Cherney, Ronald Clazie, Alan Glueck, Klara Shiloh, and B. TY. Tleimat, Sea Water Conversion Laboratory. Thanks are due to Vera Lee Yoakum for preparation of the final manuscript.
n
Q Pi
exchangeable ion species absolute .value of valences of ion species A, B, . . . normality of ion species i, equivalents per liter total solution normality, equivalents per liter exchangeable components selectivity coefficient, (ml. per gram).-*; cf. Equation 8 dimensionless selectivity coefficient; cf. Equation 10 number of exchangeable ion species present locally = total number of components in system = total exchange capacity, meq. per gram of dry exchanger in specified ionic form = concentration of ion species i in exchanger, meq. per gram of dry exchanger in specified ionic form
symbol for (usually resinous) ion exchanger throughput parameter, dimensionless; cf. Equation 1 = stoichiometric value of throughput parameter; cf. Equations 23 and 32 = feed-solution volume, ml. = bed volume, ml. = equivalent fraction of ion species i in solution, dimensionless or a t , . . = xi in first, second, etc., plateau zone; levels 1, 2, etc., in a transition = actual or extrapolated value of xi a t x j = 0 = equivalent fraction of ion species i in exchanger, dimensionless = y i in first, second, etc., plateau zone; or at levels 1, 2, etc., in a transition = actual or extrapolated value of yi a t y j = 0 = separation factor, dimensionless; cf. Equation 11 = void fraction, dimensionless = absolute value of valence of species i = bulk density of column packing, grams per milliliter Units shown are illustrative; any consistent set can be used. = =
literature Cited
(1) Baylt, G. G., Klinkenberg, A., Rec. Trav. Chim. 73, 1037-57 (1954). ( 2 ) DeVault, Don, J . Am. Chem. SOC.6 5 , 532-50 (1943). (3) Glueckauf, E., “Principles of Operation of Ion-Exchange Columns,” Ion Exchange and Its Applications,” pp. 34-46, Society of Chemical Industry, London, 1958. (4) Glueckauf, E., Proc. Roy. SOC. (London) A186, 35-57 (1946). (5) Helfferich, Friedrich, private communication, 1964. (6) Hiester, N. K., Vermeulen, T., Chem. Eng. Progr. 48, 505-16 (1952). (7) Klein, G., Tondeur, D., Vermeulen, T., Sea Water Conversion Laboratorv. Universitv of California. ReDt. 65-3 (June 1965). (8) Klein, G:, Villena-Blanco, M., ’Veimeulen,‘ T., Znd. Eng. Chem. Process Design Develop. 3, 280-7 (1964). (9) Klinkenberg, A , , BaylC, G. G., Rec. Trav. Chim. 76, 607-21 (1057’1 ,.
(10) Pandya, P., Klein, G., Vermeulen, T., Sea Water Conversion
Laboratory, University of California, Rept. 65-2 (June 1965). i l l ) Shiloh. K.. M.S. thesis in chemical engineering. Universitv ” ‘ of California, Berkeley, 1965. ( 1 2 ) Sillen, G. L., Arkiv Kemi 2,477-98 (1950). (13) Tondeur, D., Klein,. G.,. IND.ENC. CHEM.FUNDAMENTALS 6, 351 (1967). (14) Walter, J. E., J . Chem. Phys. 13, 299-34 (1945). U
I
RECEIVED for review August 31, 1966 ACCEPTEDFebruary 27, 1967
MULTICOMPONENT ION EXCHANGE IN FIXED BEDS Constant-Separation-FactorEquilibrium DANIEL TONDEUR AND GERHARD K L E I N Sea Water Conversion Laboratory, University of California, Richmond, Calif.
THE principal earlier contributions to the multicomponent equilibrium theory for fixed-bed sorption columns have been discussed ( 4 ) . The present treatment is limited to constant-separation-factor ion exchange systems. For such systems Walter has provided analytic solutions for the complete solid-phase concentration profiles in a few simple ternary cases; and for the profiles in the first transition of a column initially saturated with the component most strongly
held by the exchanger, receiving a feed of any number of components but not including the presaturant component (7). Walter’s work does not show how the complete concentration profile can be determined in the general case. A parallel, more complete theory has been developed for adsorption by Glueckauf (7), who presented examples for u p to three adsorbate species (analogous to four exchangeable ion species). His solutions also are not general. The forms of the VOL. 6
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