simplify the calculations, the integral method may again be substituted for the differential method to yield a n 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 i n 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 i n first, second, etc., plateau zone; levels 1, 2, etc., i n 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 a t 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 ) . T h e 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. T h e forms of the VOL. 6
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351
A general analytical solution Is provided for the simultaneous material balance and constant-separationfactor equilibrium relations pertaining to zones of varying composition in multicomponent, fixed-bed ion exchange columns. Uniform presaturation and a step change in feed composition are considered. Algebraic and numerical methods are presented for determining the constants occurring in the analytical solution, and for obtaining the over-all concentration profiles of three- and four-component systems. Criteria for classifying ternary pattern types are developed to permit the rapid construction of schematic profiles. Numerical examples illustrate the calculation methods.
equations for two-component and three-component cases, for instance, are different, and the parameters involved are not susceptible of a simple common interpretation. T h e basic concepts underlying the following discussion have been developed in (4); here the pertinent equilibrium relations are summarized. Ion exchange equilibria involving only ions of the same valence (homovalent ions) can often be characterized by constant separation factors a 7 t , where i and j designate any two of the ion species present: a I' - -Y i- x1
x i Y1
factor systems ( 4 ) . For constant-separation-factor systems, simple analytical solutions are developed here. Basic Equations. Walter's equations (7), applicable to uniform presaturation and constant feed composition, can be generalized to apply to any number of components in either phase, and to any transition (gradual or abrupt). T h e key to this generalization lies in extrapolating the concentration profiles, as illustrated for a ternary system in Figure 2. Here the solid-phase profiles in a gradual transition are shown, the
(1)
(Even for ions of different valences, the assumption of constant a ' s may be a good approximation.) If the total number of exchangeable ion species is n, there are n - 1 independent equations of this type. These, together with the relations n
EXi =
1
(2)
n
C Y t = l
(3)
I
+
permit calculation of n 1 concentration variables if the remaining n - 1 variables are known. T o find the liquid-phase concentrations from given solidphase concentrations, Equations 1, 2, and 3 can be combined to yield
Y,= 1
YA'1
where the summation is carried out over all components, and k designates a n arbitrarily chosen component. aii, a j i , and a k k are regarded as equal to one. An analogous relation is obtained for y t in terms of the x ' s :
Figure 1. Graphical representation of ternary constant-separation-factor isotherm CYCA =
4,
OlCB
= 2
If x I is constant, y r becomes a linear function of the other y's. Similarly, if y i is constant, x i becomes a linear function of the other x's. This property has been used in constructing Figure 1, which shows a triangular composition diagram for components A , B , and C, with equilibria characterized by CYBA = 2 and c i C B = 2. Here, the basic grid represents the solid-phase composition. A contour line for xA = 0.1, for example, is constructed by connecting its intercepts with the lines ye = 0 and yc = 0 with a straight line. T h e intercepts, in turn, are obtained from the binary A-Cand A-B equilibrium relations. Transition Profiles
Numerical methods are generally needed for calculating the transition-zone profiles in multicomponent variable-separation352
l&EC FUNDAMENTALS
Figure 2. Ternary transition profile showing extrapolated concentrations
extrapolated portions being represented by dashed lines. T h e abscissa is a dimensionless bed volume, given by the reciprocal of the throughput parameter, T : 1 .._ T
cessively to components i and j, but written for the same reference component, k , yields the following relation, after elimination of d T / T k = obetween the two resulting equations
UQP
C, ( V
- UC)
T h e profiles intersect the 1 / T axis a t l/TA=o, 1/TB=0, and l/T,=o, Lvith each T,=a designating the value of T a t lvhich y j (and x,) is zero. The component selected to go to zero is termed the "reference component." T h e values of y a , y e , ,and y c a t T = TA-0 can be designated, respectively, as ( y J a = o , lvhich is zero; (yB)a=o;and (yc)a=o. Analogous definitions are used for T = TB,o and T = Tc-0. ( Y ~ ) + values, ~ being extrapolated concentrations, can fall outside the normal range of the y's from 0 to 1. TVhere a plateau zone begins or ends a t l/Tj=o, the ( y i ) j = ~ ' s have the same values as the plateau y's, as illustrated f o r i = A in Figure 2. T h e y ' s in the upstream plateau zone have been designated by y i l ; thoz,e in the dobvnstream plateau zone, by yi2. An analogous graph can be drawn in terms of liquidphase concentrations ( x ) against T . Extrapolated concenmations obey Equations 1 through 5. I n conjunction ivith Equation 31 of ( 4 )Equations 4 and 5 make it possible to express Tj,o in terms of a's and the ( x Z ) ~ , ~ or ( Y $ ) ? =values: ~
T,=o is thus seen to be the average a i ) weighted by the y+o values; and 1 /TI+,, the average aIk,weighted by the xj=o values. Tt'ith the constants just defined, the generalized Walter equations become :
These equations can br shown to satisfy the differential material balance equation. [Equations 19 of ( 4 ) ] the , equilibrium relations (Equation 1). m d the stoichiometric relations (Equations 2 and 3). For z = k , the expressions become indeterminate, but xi, and yi. can be obtained from the known xa's and yz's through Equation 2 or 3. I n Equations 8 and 9. x i is linear in and y i is linear Since xt and y a must be independent of the choice in of the reference component, the following three relations may thus be obtained by w ~ i t i n gthese equations successively for j and X as reference components and equating the constant terms and the coefficients of t/l/T and
dy
dF
Equations 8 and 9 can be used to interrelate any two concentration variables. I:q uation 9, for example, applied suc-
or, in combination with Equation 11,
(14)
If n is the total number of components present in the system, then n - 2 independent equations of this type define a linear composition "path" (compositions arising with varying T in a transition) in a n (n - 1) dimensional composition space. [The method of deriving rules from the topology of composition paths was first used by Helfferich ( Z ) . ] The detailed properties of such composition paths in three-component systems are considered below. Calculation of Constants in Basic Equations. T h e starting data from which transition profiles are calculated are the a's, the concentrations in a plateau zone adjacent to the transition, and the components present in the transition. If the concentration of any component k becomes zero a t one of the bounds of the transition zone, the plateau-zone concentrations x, and y , a t this bound become x , = (x,)k=O and ya = (y,)r=o, and Tk=o can be computed immediately from Equation 7. If no concentration becomes zero a t either of the bounds of the transition, the extrapolated concentrations may be obtained by the following procedure, illustrated here for solid-phase profiles. Equation 13 can be written in the form
Introducing this expression for each component into Equation 3 yields
When y a and y 3 are the concentrations in an adjacent plateau zone, Equations 15 and 16 permit determination of the extrapolated concentrations. The n - 1 real roots in (y3)k=0of Equation 16 are found algebraically or by direct search. Each of these roots, introduced into Equation 15, written successively for each component, leads to a set of extrapolated concentrations. The manner of selecting the proper set for a given transition is described below. As shown ( 4 ) ,a given composition leads to n - 1 roots of the T equation [Equation 29 of ( 4 ) ]such that each corresponds to a different transition (root rule), Similarly, here, the various roots of Equation 16 must allow the composition in question to occur in n - 1 transitions. Therefore, each root corresponds to a different transition, and, with Equations 15, must lead to a set of extrapolated concentrations which satisfies the slope criterion in the transition of interest. The slope of the concentration profile is related to the extrapolated concentrations by the following equation, obtained by differentiating Equation 9.
If the component for which the resin has the smallest selectivity is chosen as reference component and designated k , then the ski's are all larger than 1 and the denominators in EquaVOL. 6
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353
tion 17 are negative. T h e slope of the i profile therefore has the same sign as (yt)k=o. Since the slope rule indicates that the slope of the A profile is positive in all transitions, all (ya)k=o’s must be positive. (Throughout the present paper, the capital letters designating components are so chosen that the selectivity of the exchanger for the components decreases in alphabetical order.) T h e slope of the B profile is negative in the first transition counted from the inlet, and positive in all others; hence there must be one negative (yB)k=0and n - 2 positive ones. Application of this reasoning to all components determines which root of Equation 16 applies to a given transition. Extrapolated liquid-phase concentrations can be obtained through similar operations; or the yk=;s may be calculated first, as indicated, and the xk=O)s obtained by Equation 4. From the known yk=0’s or xk=0’s, Tk=Ois now calculated from Equation 7. Thus all the constants in Equations 8 and 9 can be obtained from the composition of a plateau zone adjacent to the transition of interest. The concentration profiles in the first and last transition zones, which are adjacent to the end plateau zones of known compositions, can then be determined directly. The problem of calculating concentration profiles in intermediate zones is considered in the section on plateau-zone compositions and bounds. Integral Material Balance
For abrupt transitions, the following integral material balance applies for any type of isotherm (4):
where is the value of the throughput parameter a t the concentration discontinuity, and subscripts 1 and 2 refer to the two sides of the discontinuity. For constant-separation-factor systems, any two composition points within a gradual transition zone, including the bounds, also satisfy Equation 18. the stoichiometric average value of T, is the geometric mean of the T values a t the two compositions considered. These relations are shown in the following way: Let x t l and yI1 be the concentrations of component i a t TI within the transition, and x , and ~ yt2 be the concentrations a t Ts. Introduction of the expressions of Equations 8 and 9 for x i and y i into Equation 18 yields:
Composition Paths in Ternary Systems
The general properties of composition paths in the ternarycomposition diagram will now be surveyed. A path corresponding to the profile of Figure 2 is shown in Figure 3, together with the various extrapolated concentrations. Because of the alphabet rule ( 4 ) a composition path for a n upstream 1-2 transition in a ternary diagram intersects the y~ = 0 and Y B = 0 borders of the triangle; and a path for a 2-3 transition, the y B = 0 and yc = 0 borders. A path can be constructed from its intercepts with the borders. For a 1-2 path, for example, after a value of ( y c ) a = ois picked, or calculated from given data, is found immediately from Equation 11. The analogous procedure is used to draw 2-3 paths, using (ya)B=c and (ya)c=o intercepts. I n this manner, a grid of composition paths corresponding to given a’s may be constructed readily; thus Figures 4 and 5 show, respectively, 1-2 and 2-3 composition paths in the ternary system with aCA = 4 and olcB = 2. Also shown in these diagrams, as curves, are lines of constant T calculated from Equation 9. When both rectilinear grids are drawn on the same diagram, the composition of an intermediate plateau zone can be determined readily from the composition of the two terminal plateau zones. A 1-2 path is drawn through the point representing plateau 1, by interpolation between the lines of the grid. Similarly, a 2-3 path is drawn through plateau 3. The intersection of these two paths represents the composition of plateau 2. Transition Parameters. I t is of interest to examine the availability of normalized concentration variables for each particular transition, whose values within the transition range from 0 to 1. By means of such variables, Vermeulen and Hiester ( 6 ) were able to extend their generalized treatment of dynamic binary fixed-bed systems to cases in which feed, presaturant, or both, contain both components. The respective liquid and solid-phase “transition parameters” are
where superscripts ’ and I f refer to the bounds of the transition. When the x’s and y’s are replaced by the expressions of Equations 8 and 9, written for the bounds of the transition and for a point within the transition, the transition parameters become:
fi-dF wi =
ddT; Tff-
A,
dT” dT
= wi-
(21)
C
T h e constant before the square root is unity by virtue of Equations 5 and 7. Ti2 given by this expression is identical for all components. Since Equations 8 and 9 are solutions of the differential material balance applicable to gradual transitions [Equation 19 of (4)], the integral and the differential material balancesfor constant-a systems only-lead to the same composition path in a composition space, whether the transition is abrupt or gradual. Although not of direct physical significance there, Equations 8 and 9 can be applied to abrupt transitions in order to define “fictitious” concentration profiles. Fictitious profiles have been discussed a t some length ( 4 ) . When used in calculations, they will satisfy the integral material-balance (Equation 18) for the actual transition, follow the slope and alphabet rules, and yield the correct adjacent plateau-zone compositions. 354
l&EC FUNDAMENTALS
Figure 3. Extrapolated solid-phase concentrations for first transition in ternary system czcA = 1.75, = 1.25 Corresponding diagram for second transition obtained by interchanging A ond C
Figure 4.
1-2 transition paths and lines of constant TU ffCA
= 4;
2
“Alphabet Rule“
At a given time and point in the column, these parameters thus have the same value: for all components. The breadth of a transition, in binary systems, depends upon the parameter,,, ( 3 ) . By analogy, an cGeffective breadth parameter” can be defined here from the range of the T values bracketing the transition:
reff.= ~ / T ’ / T ”
ffcB =
A rule is derived here that predicts the levels in the column a t which the concentrations of various components can become ‘ero. I n an n-component system with plateau ‘Ones and n - 1 real or fictitious transition zones, there are 2(n 1) zone bounds. If these are numbered successively in the direction of increasing 1/T, the profile of component A can go through zero a t bound 1, that of B a t bound 2 or 3, that of C a t bound 4 or 5, etc. T h e component for which the exchanger has the lowest selectivity can go to zero only a t bound 2(n 1).
-
(22)
If T” > T’, in a fictitious profile, then r,ff < 1, matching the criterion for abrupt transitions in binary systems.
-
T.0.25
k0.5
A YA
Figure 5.
2-3 transition paths and lines of constant T23 ffCA
= 4;ffCB = 2
V O L 6 NO. 3
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355
Because the alphabetical symbols for the various components are again so chosen that the selectivity of the exchanger for the latter decreases in alphabetical order, this relation is referred to as the alphabet rule. I n the proof of this rule for constant-separation-factor systems, it is convenient to picture all components as being present throughout the column-Le., to replace zero concentrations by theoretical trace concentrations. For any one transition, Equation 12 establishes that the zeros of the concentration profiles of the various components (extrapolated if necessary) occur in alphabetical order along the l / T axis. If the exchanger prefers component j over component k (ak’ > l ) , the j-profile can thus be zero only a t a point (l/T+o) upstream of the point a t which k can vanish ( l/Tkso). The extrapolated profiles of components A, B, C, etc., of a particular transition will thus cross the 1/T axis in the order indicated. This situation is illustrated schematically in Figure 6 for a (k, k 1) transition with extrapolated profiles. Here, the 1, selectivity of the exchanger for components k - 1, k , k and k 2, decreases in the order given. According to the slope rule, the profiles of the components up to and including k must have positive slopes; the profiles of the subsequent components, negative slopes. We have just established that the 1, and k 2 along zero points for components k - 1, k , k the l/T axis are arranged in the order given. Only componentsk and k 1 can go to zero in this transition; for, a t the values of l / T a t which any of the other components could go to zero, the concentrations of k and k 1 would have to have negative values. These considerations, applied to real as well as to fictitious profiles, establish the alphabet rule.
+ +
Figure 6. Schematic transition profiles illustrating derivation of alphabet rule
+
+
.I
+
1
.
Basic. Pattern a a
+
+
Over-all Concentration Patterns
Classification. Various patterns can develop in a system of a given number of components. They are obtained by considering all possible combinations of gradual and abrupt transitions, and all possible combinations indicated by the alphabet rule. Thus, if a gradual transition is denoted by g and an abrupt transition by a, the order of the transitions in a three-component system can be aa, ug, gg, or ga. I n a n n-component system, there are 2”-l such “basic” combinations, each of which introduces the various possibilities for appearance or disappearance of components that are indicated by the alphabet rule. The number of these possibilities is obtained by considering that all components but the most strongly and the most weakly held can go to zero in either of two transitions, or not go to zero, introducing for each such component a factor of 3 in the total number of cases. Component A and the last component in alphabetical order can either not go to zero, or go to zero in only one transition, introducing for each of these a factor of 2. The total number of possibilities, or “subcombinations,” for each of the 2”-l basic combinations is thus 4(3)n-2. Figures illustrating the 48 patterns possible in ternary systems, and the basic patterns of four-component systems, are presented elsewhere ( 5 ) . Here the four basic patterns that can arise in ternary systems in which all three components are present in both the presaturated resin and the feed solution are shown in Figure 7. Abrupt transitions are represented by fictitious profiles reflecting application of the slope rule. From given feed-concentration and presaturation levels, to arrive a t the unique concentration profile representing the mathematical solution, it may be necessary to calculate partially several concentration profiles, each in accordance with the slope rule and the alphabet rule. If discrepancies arise, 356
l&EC FUNDAMENTALS
Y
0
I/T
Figure 7. Basic profile-pattern types possible in ternary systems in which all three components are present throughout crcA = 1.50; acB = 1.25
alternative profile-pattern types are tried. This procedure is illustrated below for a four-component system. For ternary systems, criteria are presented here which have been developed through direct calculation of the pertinent extrapolated concentrations. These criteria, listed in Tables I through IV, are explained below. Table I is used where all three components are present in the feed and the presaturated exchanger (and therefore in all intermediate zones). Table I1 applies where a single component is absent from either feed or presaturated exchanger, and Table 111, where one component is absent from the feed and another from the presaturated exchanger. Finally, Table I V applies to single-component feed or single-component presaturant. For all rows of Tables I, 11, and I V having no entries in column 4, and of Table I11 having entries in neither column 3 nor 4, the slope and alphabet rules establish the profile type. The most general criterion employed in the remaining cases involves a concept derivable from the slope rule, or from Figures 4 and 5. A feed-composition point to the right (on the ”A-lean” side) of the 2-3 composition path going through the presaturation composition implies an increase of yA in the 1-2 transition, which must thus be gradual; and a feed-composition point above (on the “A-lean” side of) the 1-2 composition path going through the presaturation composition implies a gradual 2-3 transition for the same reason. Feed compositions on the other sides of these paths correspond to abrupt transitions. T h e converse holds for the location of the presaturation-composition point relative to the composition
Table 1.
Profile Types in Ternary Systems
[All components present in feed and presaturated exchanger (and in intermediate plateau zone)] Basic Pattern
Feed and Presaturation Criteria
YAl
Yci
> YAa
YCl
{
I
YCl YCI YCl
YCl
Sa aa
< Yca
>
Ye3
YC3 < Ycs > Ye3
Table II.
Profile Types in Ternary Systems
(Single component absent from either feed or presaturated exchanger) Additional Criterion
Feed and Presaturation Criteria
i YE1
= 0
'jYCl
3
IYCl
YB3
Basic Pattern
Components Present in Second Plateau Zone
YB3
Y
YCl
< Ye3
YCl
3
Yc3
Table 111.
Profile Types in Ternary Systems
(Single component absent from feed and another single component absent from presaturated exchanger) Components Present in Second Plateau Zone
Feed and Piesaturation Criteria (Yc3
= 0
iyc3> I >< Ye3
Yc3
Ye3
6
A , B, C A. B. C
A,c
Y
A, C
Y Y
VOL. 6
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AUGUST 1967
357
path through the feed composition : a presaturation-composition point to the right (A-lean side) of the 2-3 path going through the feed composition implies that ya decreases in the 1-2 transition, which must thus be abrupt; and a presaturation point above (A-lean side of) the 1-2 path through the feed composition implies an abrupt 2-3 transition, for the same reason. T o express this criterion formally, one may designate the left member of Equation 13 by Zk:
Table IV. Profile Types in Ternary Systems (Single-component feed or single-component presaturant)
Free and Presaturation Criteria
yAl
Basic Pattern
1
YA3
= 1
yc1
=
1
Additional Criteria 0 iYA3 > 0
yA3
gg gg
Components Present in Second Plateau Zone
B A, B A, C A, C
with (Ydk-0
+ (Yr)rc=o
=
1
(24)
so that z k = 0 is the equation of a composition path which divides the triangular diagram into a zone in which coordinates y i and y j of any point render Z k negative, and a zone in which ZI, is positive. The sign of the Zk function thus determines whether the pertinent composition lies on the A-rich or the A-lean side of the path Z k = 0. Since Zk is negative for the corner point yk = 1 ( y i = 0, y 3 = 0), Zk is negative for all points on the same side of the path as this corner point. I n Tables I and 11, (Zkp,)t represents the value of z k obtained by letting y z = y z l , y 3 = y J l (corresponding to the feed composition), and by letting ( y z ) k = oand (y3)k=0be the extrapolated concentrations based on the 1-2 path going through the presaturation composition. Similarly, (ZkI2)3is based on y z = yi3, y j = y33, and extrapolated concentrations based on the 1-2 path going through the feed composition. The definitions of (Zkz,), and (Zk23)3are analogous. I n Table 11, component k has been chosen so that the extrapolated concentrations used coincide with actual feed or presaturation concentrations. When the coordinates of a point make z k equal to zero, this point lies on the composition path defined by Z k = 0. This condition is thus the criterion for the feed composition (plateau 1) and the presaturation composition (plateau 3) to lie on the same composition path. T h e common path is a 2-3 path if ( Z k z s ) ~= 0, in which case the 1-2 transition vanishes, plateaus 1 and 2 having the same composition. If ( z k 1 , ) 3 = 0, the common path is a 1-2 path, and the 2-3 transition vanishes, plateaus 2 and 3 having the same compositiqn. Where = 0 , or yB3 = 0 , simple criteria involving the constant
may be used, y being equal to the ratios in Equation 11 with i = C, j = B , k = A . T h e first of these criteria consists in comparing ycl to y (if y B 1 = 0), or yc3 to y (if yB3 = 0), and allows one to find in which transition B appears or disappears. This information then leads to further conclusions about the abrupt or gradual character of the transitions, and the components present, as illustrated below :
yel = 0
~
abrupt ycl 3 y. B appears in the 2-3 transition, which is
y . B disappears in the 2-3 transition, which is abrupt
I
The second of these criteria can be considered as a limiting 358
l&EC FUNDAMENTALS
case of the Z criterion, and applies to certain cases of Table I11 where B is one of the missing components. Since both feed and presaturation compositions lie on a border of the triangular diagram, we need determine only the position of the point lying on the y B = 0 border, say, with respect to the intercept on this border of a composition path going through the other composition point. For example, if Y A I = 0, the feed-composition point lies on the y A = 0 border, and is determined by y c l . T h e intercept on the Y B = 0 border of a 1-2 transition path going through the feed composition is given The posiby Equation 11 : (ycIp)B=o = y(yclz)A=o = yycl. tion of the presaturation composition defined by yc3, with respect to the intercept, will then determine whether the 2-3 transition is abrupt or gradual. Here, if yc3 > y , we note that the second y criterion is not necessary. The special case where yc3/yc1 = y, as for the Z criterion, corresponds to the feed and presaturation compositions lying on the same path, here a 1-2 path, so that the 2-3 transition vanishes. Similarly Y A S / Y A l = 1 - y corresponds to a vanishing 1-2 transition, feed and presaturation lying on the same 2-3 composition path. The criteria of Tables I through I V are given in terms of solid-phase concentrations. Analogous criteria using liquidphase concentrations are obtained by replacing y by x, y by y ‘ = yaBA, and Z k by
Plateau-Zone Compositions and Bounds. For constantseparation-factor systems, as shown above, the integrated differential material balance leads to Equation 18 whether the transition under consideration is abrupt or gradual. With known feed and presaturation compositions, Equations 1 (equilibrium), 2, 3, and 18, written for each transition, suffice to solve for the unknown concentrations in each intermediate plateau zone, and for the n - 1 T’s in a n n-component system. I n practice, the method outlined below is preferable to simultaneous solution of the equations just indicated. This method utilizes the analytic expressions for real or fictitious concentration-profile curves and composition paths. Simultaneous solution of the equations available in this case gives physically meaningful bounds for the gradual transitions; with abrupt transitions, fictitious zone bounds are obtained, so that the T marking the composition discontinuity must still be calculated from Equation 18, or as the geometric mean of
the fictitious zone bounds (Equation 19). For ternary systems, the rapid graphical method described in the discussion of composition paths may be used as a n alternative for determining the composition of the intermediate plateau zone. First, a relation is established between the concentrations in any intermediate plateau zone of a multicomponent system, and the extrapolated concentrations corresponding to the two transition zones adjacent to this plateau zone. This is done by rewriting Equation 14 for two successive transitions, identil ) , and for concenfied by subscripts (m - 1, m ) and ( m , rn trations yfmand yjm in the intermediate plateau:
+
Yim (Yd.m-1,m)l-o
-+
Yjm
= 1
(27)
(yj,m-l,m)t-o
Y*m - $, YW (Yi,m ,m+d,=o (Y,,m,m+JI-o
= 1
(28)
Solving this linear system for ytm,one obtains
y,, = [(YJ,m--l,m) t=o - (Y3,m,m+l)i-o1 (Yi,m-l,m)l=O ( Y t , m , m + d j = 0 (Y3,m-l.m) 1-0 - (Yt,m-l,m)j-o
(yt,m,m+l)j-o
(Yj,m,m+l)i-o
(29)
T h e expression for yjm is obtained by interchanging i and j . Equation 29 permits calculation of the intermediate plateau compositions once the extrapolated concentrations are known, as illustrated in a n example for a four-component system. For a three-component system, Equation 29, with Equation 11 and the relation (yt12)*=0 = 1, reduces to
+
y t z = (ytlz)r=o (Yt23)k=O
# j # k # i)
(2
(30)
T h e extrapolated concentrations in Equation 30 are obtained directly from the feed and presaturation compositions through Equation 16. T h e particularly simple result of Equation 30 has been used to determine the profiles in Figure 7 . Once the extrapolated concentrations and the intermediate-plateau compositions are known, the profile curves and the bounds are easily constructed from Equations 8 and 9, and if applicable, 18. T h e general procedure to calculate the extrapolated concentrations in intermediate zones will now be outlined in terms of a n n-component system. T h e concentration yi of component i in a given intermediate plateau zone is equated to the value of yt obtained from Equation 9, written for a transition adjacent to the plateau, where T is replaced by its value a t the bound between the transition and the plateau. Using the two transitions adjacent to the plateau considered, two equations are thus obtained, from which y, is eliminated. From the set of equations obtained by doing this for all components present in the plateau, except the reference component, and for each intermediate plateau, the T/T,=o’s, corresponding to the various bounds, are eliminated. The result is the following set of relations between the extrapolated concentrations: t =A,B,p j =B , p J k =p,A,B
[
(yt ,m ,m+l)ref=O (yj,m .m+l)ref=O
(yt,m-l,m)ref-O
+
(yj,m-l,m)ref=o
1
(yl~ ,m, m + J r e i = o = 0 (YX,m--l,m)ref=o
(31)
valid in conjunction with igref
(Yt,m-1,m)rei=o = 1
(2
=
A , B, C
*
. .)
equation has three terms. I n the first, i = A , j = B, k = p ; in the second, i = B, j = p, k = A ; and in the third, i = p, j = A , k = B. Equation 32 is written for each of the n - 3 intermediate transitions. (It need not be written for the first and last transition, in which the extrapolated concentrations can be determined directly from the feed and presaturation compositions.) A computer program (AD1 Baer) for solving such a system is available. However, no examples involving more than four components have been treated. A simple numerical solution for a four-component system is presented below. Numerical Examples
Several three- and four-component examples have been presented ( 5 ) . The first of the four-component examples given here was selected to illustrate the method of determining the transition types where criteria such as those of Tables I to IV are not available, and where the alphabet and slope rules (used with feed and presaturation conditions) d o not allow a n unequivocal assignment of gradual and abrupt transitions. I n some cases, one or more components of intermediate selectivity coefficients (components other than A and D in a four-component system) are absent either from the feed solution or from the presaturated column, and their profiles, according to the alphabet rule, must go through zero in either of two transitions. T h e general method of solution then consists in establishing all profile-type subcombinations corresponding to the feed and presaturation compositions, and calculating successively the transitions and plateaus pertaining to the various profile types until the “consistent” profile is found. I n such a profile, the plateau zones must have finite or zero length; the smallest value of T in a gradual transition (or of a n abrupt transition) is larger than the largest value of T in the next gradual transition downstream (or than T of a n abrupt transition). T h e first system considered has the following separation factors and feed and presaturation compositions : P
ffD‘
A R C
1.728
0
0.900
1.440
0
D
1.000
0.600 0 0.400
i ,200
Yi4
Xi1
0.100 0
Application of the slope and alphabet rules shows that A, being absent from the feed, must appear in transition 1-2, which is thus gradual; D, being absent from the presaturant, must disappear in transition 3-4, which is thus gradual. No conclusions can be drawn as yet as to the appearance of C and disappearance of B, and as to the gradual or abrupt character of transition 2-3. T h e solid-phase concentrations in equilibrium with the feed obtained from Equation 5 and the xtl values are yAl = 0, ye1 = 0.6835, ycl = 0, and yD1 = 0 . 3 1 6 5 . We assume first that component B disappears in the 1-2 transition; and that component C appears in the 3-4 transition. The transitions are described by Equation 9. I t is convenient to take A as reference component in the first transition, and D in the last, because then the extrapolated concentrations to be used coincide with the actual feed or presaturation concentrations. T h e equations for transition 1-2 thus become
(32)
For each pair of suc:cessive transitions, identified by subl), Equation 30 is written n - 3 scripts (m - 1, m) and ( m , m times-namely, w i t h p := C, D . . . (p # ref). Each resulting
+
VOL. 6
NO. 3
AUGUST 1967
359
where TA-o= asAy ~ fl agAy o 1 = 1.3671, from Equation 7. For the 3-4 transition, yA34
= YA4
yC34
= yC4
(dT/TD-O
- aDA)/(l
(m -
aDc)/(l
=
yD34
1
- aDA)
(36)
- aDc)
(37)
- YA34 - yC34
(34)’
1.0
Y
(38)
+
with T D - 0 = a~~ Y,44 OcD y c 4 = 0.6040. The values of T a t the zone bounds (12)” and (34)’ are obtained by letting = 0 in Equation 32, and yc34 = 0 in Equation 37, to yield
T(lz)”=
(adB)’
TA=O= 0.9494;
T(34)’=
Figure 8.
- a A D ) / ( l - aAD) = (adB - aAD)/(l - a A D ) = 0.1913
(40)
- a D A ) / ( l - a n A ) = 0.6527
(41)
(dT(12)”ITA-0 YO1
y A 4 (UDc
and, because only A and D are present in the two intermediate plateau zones, Y A ~ = 1 - YDZ = 0.8087, and Y D 3 = 1 - y A 3 = 0.3473. Since y A 3 < Y A ~ ,the slope rule indicates the 2-3 transition to be abrupt. With x A 2 = 0.7099 and x A 3 = 0.5210, calculated from Equation 5, Equation 18 yields T23 = ( y A 2 Y A 3 ) / ( X A 2 - XAp) = (0.8087 - 0.6527)/(0.7099 - 0.5210) = 0.8258, so that i i i 2 3 < T(34)’< T ( I ~ ) ~Since J . 1/T(I2)” (where component B disappears) is upstream of l/T23, the assumption that B disappears in the 1-2 transition is justified. The Cprofile, however, cannot go through zero in the 3-4 transition, as assumed, because of the impossible result that T(34)t> F23. I n repeating the calculation on the basis of the appearance of C in the 2-3 transition, the previous calculations for transition 1-2 and plateau zone 2 are not affected. Downstream of the 1-2 transition, the system contains only components A, C, and D, and the relations for ternary systems apply. From Equation 30, Y A ~= (Yaza)c=o ( Y A ~ ~ ) D = o . For c as reference component in the 2-3 transition, and D in the 3-4 transition, (yA23)C=o = y A 2 and (YA34)D=O = y A 4 , SO that y A 3 = YAZ y.44 = 0.8087 x 0.9000 = 0.7278. T(34)’ = 0.7838 is obtained by substituting the last value as YA34 into Equation 36. This result, substituted into Equation 37, yields y c 3 = 0.0304. From Equation 38, y ~ = 3 0.2418. T o complete the profile, we again compute T23 from Equation 18, using the new value of Y A 3 , and x A 3 = 0.6121, obtained as before: T 2 3 = (0.8087 - 0.7278)/(0.7099 - 0.6121) = 0.8272, SO that now, T(34)~< T23 < T(12)”. The calculated profile is shown in Figure 8. The second four-component example considered has all components present in both feed and presaturant, and illustrates the application of the general method presented in the section on plateau-zone compositions and bounds. The data for this example are :
i
QDi
A B C D 360
1.728 1.440 1.200 1 .ooo
I&EC FUNDAMENTALS
Xi1
Yir
0.2500 0.2500 0.2500 0.2500
0.4592 0.0380 0.4005 0.1023
t.5
1.0
2.0
I /f Profile of first numerical example
(39)
T h e compositions in plateau zones 2 and 3 are calculated by substituting these values into Equations 34 and 36: = yDl
0.5
0
(a~’)’ TD-0 =
0.8697
yD2
0
T h e extrapolated concentrations in transitions 1-2 and 3-4 are first calculated, using Equation 16, which here can be written as a cubic equation. With j = A , k = D, and yc replaced by yil for all i, Equation 16 becomes :
having the roots 1.7218, 0.4509, and 0.2806, obtained analytically. The largest value, ( y A l z ) D = o = 1.7218, is the correct root to employ for transition 1-2, as explained above. Equation 15 yields ( Y B ~ Z ) D = O= -0.6034. Finally (YCIZ)D-O = 1 - ( Y A ~ ~ ) D = O- ( y B l z ) ~ = o = -0.1184. For transition 3-4, letting y. = y 1 4 in Equation 16 yields (yA)DI.03
-
+ 2.7419 (YA)D-o
3.0035 (YA)D-O’
- 0.7348
= 0
with the smallest root, (YA34)D=O = 0.4806, assigned to transition 3-4. Then, from Equation 15, ( y B a ) D = o = 0.0410 and (YC34)D=0 = 1 - (yA34)D=O - (yB34)D=O = 0.4783. Using the values found for the various extrapolated concentrations ==2.088 ~ and 1/(T34)~=0= in Equation 7 gives ~ / ( T I z ) D 1.418. Equation 9 can now be written for each component in the first and last transition, all the constants being known. The values of T a t bounds (12)’ and (34)” may be obtained by letting Y A = y A 1 and YA = Y A ~ ,for instance, in the respective equations. T o determine the extrapolated concentrations for transition 2-3, the system of Equations 31 and 32 can be written as aDB)
(ODA-
(YA23
f yC23) f
yB23
___
YAl2 YBl2 y B 2 3 yC23 y B l 2 YAl2
+- + yA23)
(OD”
- OD“) x
YClZ
-
(OD‘
ODA) YClZ Y A l 2
yAl2
YE12
(42) (aDA
-
OD”)
( y 1 2 3 yB23
+ --
yC23)
(YB23
yC23
YE34 y C 3 4
+
(aDB
- ODc) x
(Ye23
YA23
~
YA34 yW34
yC34
yA23)
f - f
(aDc
-
ODA)
~
yC34 yA34
yA34
+YE23)
=
0
YE34
(43) yA23
f
YE23
f
yC23
=
(44)
The subscript D = 0 pertaining to the y’s has been dropped for simplicity. Direct application of Equation 32 leads to Equation 43 with yiz3and y , 3 4 interchanged, but the form used here is equivalent. y e 2 3 is eliminated successively between Equation 44, and Equations 42 and 43, to yield two quadratic
forms in the unknowns ~ 4 2 3and y B 2 . 3 . Introducing the numerical values of the separation factors and of the extrapolated concentrations y,lz and y , 3 4 calculated earlier, one obtains: 2.5900
+ 3.3593
2.2966
~
~
+ 4.6711 2
3
~
~
~y A 2 3 y s 2 3
2.3994
y.423
y ~ 2 3 ~
+ 2.4324 = 0
-2 6.6668 3 ye23 - 12.2256
-
1.2416
-
yA23
=
-
+
-
+ 0.6021 = 0
~ B 2 3
- 0.3071 ~ ~ 2 0.1022 3 0.0559 y e 2 3 - 0.1432
yB!~3' -
I
I
1
1
-
Y -
~ B 2 3 ~
The numerical solution of such a system is readily found by constructing the two conics represented by these equations and determining their intersections. Since here both y A 2 3 and y B 2 3 must be positive, the construction may be limited to the first quadrant. By taking the geometric mean of Y A l z and Y A 3 4 as a preliminary guess for y.423, the search for the intersection of interest is, moreover, limited to a small zone. T o construct the intersection, it is convenient to eliminate square terms by transforming the equations into linear combinations. O n multiplying the first equation by 2.2966, the second by 2.5900, and subtracting, a n equation is obtained which does not contain ~ , 4 2 3 ~ .y s 2 i 2 may be eliminated similarly. T h e two resulting equations can then be made explicit for one variable in terms of the other:
I
A
-
-
y ~ 2 f 3 ~6.2265 y ~ 2 y3 ~ 2 3
5.1618
0.5
I
I
I
D C \-
B
0
I
Figure 9.
B I
I
I
I
I
I
I
Profile of second numerical example
Y D = '
Plateau
YA
YB
2 3
0.5231 0.4282
0.0916 0.0336
YC
0.1816 0.2886
- Y A -
Y e - Yc
0,2037 0.2496
T h e zone bounds may now be calculated from Equation 9. The calculated profile is shown in Figure 9. Acknowledgment
T h e authors are indebted to the Office of Saline Water,
U.S. Department of the Interior, for financial support of these A numerical search for the intersection of the two curves represented by these equations can now be undertaken in the neighborhood of y A 2 3 = d y ~ l YzA 3 4 0.91. Examining the sign of Y E 2 3 reduces the interval further. T h e numerator of the second equation is positive for values of y A 2 3 between about 0.807 and 1.00, while the denominator is positive for y A 2 3 larger than 0.9331. T h e interval of search must thus be 0.9331 > ~ ~ >2 1.00. 3 The most efficient procedure is to substitute the ~ ~ calculated 2 3 from the second equation for a n assumed 2 the 3 first equation. Similarity of the value of value of ~ ~ into ~ ~ thus 2 found 3 to the value assumed then serves as criterion of convergence. Intermediate results are tabulated below. YA23
yw23
YAls
Assumed
Calculated
Calculated
0.9400 1.0000
0,5650 0.11044 0.l991 0 ,4200 0 . :3084 0 . :3667 0 . :3474 0 . :3492 0 . :3487 0 . :34867
2.2214 0.7059 0.6113 1 ,2497 0.8148 1.0113 0.9390 0.9456 0.9436 0.9433
0,9500
0.9420 0.9450 0.9430 0,9435 0.94345 0.943465 0.943466
(
YA YA
Calculated) Assumed
2.36 0.706 0,644 1.326 0.862 1.072 0.995 1.0022 1.00014 0.99987
T h e result may be talcen as ( y & a ) D = O = 0.94346, ( Y B S S ) U = O 0.34865, and ( ~ ~ 2 3 ) ~ = = 0 1 - 0.94346 - 0.34865 = -0.29211. Substituting these values into Equation 29 and using Equation 11 yields the following concentrations for plateaus 2 and 3. =
studies, and to Theodore Vermeulen for helpful collaboration and guidance. Nomenclature
In addition to nomenclature defined in ( 4 ) : effective breath parameter, dimensionless solid-phase pattern-criterion function for ternary systems, based on concentrations extrapolated to y k = 0 (cf. Equation 23) solid-phase pattern-criterion function in 1-2 transition, applied to composition of plateau 1 liquid-phase pattern-criterion function, based on concentrations extrapolated to xk = 0; cf. Equation 26 pattern-criterion constant, dimensionless liquid-phase transition parameter, dimensionless; cf. Equation 20 solid-phase transition parameter, dimensionless; cf. Equation 21 Cited
(1) Glueckauf, E., Proc. Roy. Soc. (London) A186, 35-57 (1946). (2) Helfferich, F., private communication, 1964. (3) Hiester, N. K., Vermeulen, T., Chem. Erg. Progr. 48, 505-16 11052). --, I - -
(4) Klein, G., Tondeur, D., Vermeulen, T., IND. ENG. CHEM. FUNDAMENTALS 6 , 339 (1967). (5) Tondeur, D., Klein, G., Vermeulen, T., University of California Sea TVater Conversion Laboratory, Rept. 65-4 (July 1965). ( 6 ) Veimeulen, T., Hiester, N. K., J . Chem. Phys. 22, 96-101 ( 1954). (7) TValter, J. E., Zbid.,13, 299-34 (1945).
RECEIVED for review August 31, 1966 ACCEPTED February 27, 1967
VOL. 6
NO. 3
AUGUST
1967
361
