Analytical Description of Fixed-Bed Sorption of Two Langmuir Solutes under Nonequilibrium Conditions David 0. Cooney* and F. Paolo Strusi Department of Chemical Engineering, Clarkson College of Technology, Potsdam, N . Y . 13676
It i s shown that the concentration relationship which exists between two Langmuir solutes during nonequilibrium constant-pattern saturation of a sorbent bed i s essentially the same as that for equilibrium operationwhen the ratio of the solutes’ mass transfer coefficients is near unity. Analytical expressions for column concentration profiles, derived using the equilibrium concentration relationship, are presented.
A t t e m p t s t o obtain analytical solutions to the equations describing multicomponent interactive sorption in fixed beds have so far been largely unsuccessful. I n general terms, t h e problem is simply t o find solutions to the set of equations
dq,*/dcl = dq2*/dc2 under equilibrium nondispersive conditions allows derivation of the analytical c1-c2 relationship asbici
=
aibzc2X
-
(as
- ~i)X/(l
+ X)
(6)
where A is either the positive or negative root of the quadratic X2
aq%iat= f(Cz,cz*, 9%)qt*)
qi*
qz*(ct, CI,
(2)
.)
(3) for each species i, for various selections of the mass transfer rate law (eq 2 ) and the distribution relationship (eq 3). Note that, since the distribution function, qt*, for each component i is riot only a function of cz but of all other solute concentrations as well, then the behavior of component i is intimately linked to the behavior of all of the other species. This is what is meant b y “interactive” sorption. Even for single solute systems, solutions t o t h e above equations have been generally found only for linear equilibria. An important exception is the nonlinear Thomas chemical reaction model. Since equilibria in mulLi-solute interactive systems are necessarily nonlinear, it would seem t h a t analytical two-solute solutions might be very difficult to find. There is, however, one approach which can yield analytical results which are often realistic, although not completely general. This is t o recognize t h a t the behavior of sorption fronts in fixed beds tends toward one of two limiting forms as time goes 011. One limit, for diffuse fronts, consists of equilibrium nondispersire, or proportionatepattern, behavior. The other limit, for self-sharpening fronts, is constant-pattern behavior. For either of these limiting forms many analytical solutions for nonlinear single-solute systems can be derived. A t present, the only available analytical solution for multiple solute cases of the interactive kind is t h a t of Glueckauf (1946), for two solutes following Langmuir isotherms, under equilibrium nondispersive conditions. For solutes having the isotherms q,* = UlCl/(l blCl bzcz) (4) =
’
,
+ + q2* = azcz/(I + blcl +
b2~2)
(5)
where a2 > al (Le,, the more strongly sorbed solute is designated “ 2 ” ) Glueckauf showed t h a t use of the fact that
+ X[l
- azblc,o/albzcz0 -
(a2
- al)/alb2cz01 azb,c10/a,b*c20
= 0
(7)
The negative root applies to saturation conditions and the positive root t o elution conditions. The existence of a n analytical relationship between CI and cz permits rewriting the isotherms in “pseudo-binary” form, in which ql* is expressed in terms of only c1, and similarly for qz*. The effect of this is to divide the original two-solute problem into two single-solute problems, each of which can be solved independently. Because analytical single-solute solutions for nonlinear isotherms do exist for limiting cases such as proportionate and constant-pattern fronts, it is clear t h a t : if pseudo-binary isotherms which are valid for such limiting situations can be found for any multi-solute systems, then analytical descriptions of these systems could possibly be written (for such limiting conditions). Unfortunately, the existence of analytical pseudo-binary relationships seems to encompass only the Langmuir case. Glueckauf mentions t h a t in his searches no solutions were found “for any conceivable mixed isotherm apart from the linear Langmuir isotherm.” Glueckauf’s paper goes on t o describe how the Langmuir pseudo-binary isotherms can be used t o determine column concentratiorl profiles under equilibrium nondispersive conditions. The purpose of the present paper is to indicate the extent t o which these equilibrium analytical pseudo-binary isotherms are also valid for describing nonequilibrium Langmuir sorption. Although Glueckauf’s pseudo-binary isotherms apply strictly only to equilibrium operation, it seemed likely to us that the relationship between c, and cz in a nonequilibrium zone should not be vastly different from that which prevails in a n equilibrium zone, particularly if the two solutes are chemically similar and have roughly the same mass transfer coefficients. If this is true, then use of the equilibrium pseudobinary isotherms for the description of nonequilibrium sorption fronts would be valid. Ind. Eng. Chem. Fundam., Vol. 1 1, No. 1 , 1972
123
c and q in zone 11. Employing the widely used mass transfer
rate law
we may obtain, b y writing this equation for solutes 1 and 2 and taking the ratio of the two resultant expressions A
DI S SA NCE
Figure 1 . Zones generated during saturation of sorbent b e d with two Langmuir solutes
X
XI
Figure 2. xl-xz relationship constant-pattern behavior
for
equilibrium
and
for
Sorption Zones for Saturation with Two Langmuir Solutes
The types of sorption zones which arise during fixed-bed saturation operation, with a feed containing two solutes which follow Langmuir isotherms, are shown in Figure 1. ,411 of the fronts are of the self-sharpening type. Single-solute theories can adequately describe the behavior in zone IV. What we wish to consider here is the description of what occurs in zone 11. Xote that solute 2 goes to zero in this zone, whereas solute 1, by virtue of interaction with solute 2, actually undergoes a n increase in concentration. Prediction of which solute concentration drops to zero in zone I1 can be made using the rules proposed b y Cooney (1966), which indicate that, for saturation, the solute with the higher po/cO ratio is the one whose concentration vanishes. The asymptotic concentration of solute 1 a t the down, stream end of zone I1 may be determined from the requirement that &i/ACi
=
&z/ACz
= qeo/Czo
(8)
The downstream and upstream concentrations must be determined so as to define dimensionless concentration variables. Determination of xI-x2 Relation for Constant-Pattern Conditions
With the assumption that axial dispersion may be neglected relative to the influence of mass transfer effects, eq 1 reduces, as shown by Cooney (1965), to 2: = y, where z and y are dimensionless fluid-phase and fixed-phase concentrations. They are defined as x = ( e - c,)/Ac and y = (q - pm)/Aq, where Ac and Aq are the overall concentration changes in zone 11, and c, and p, are the minimum values attained by 124 Ind. Eng. Chem. Fundam., Vol. 11, No. 1, 1972
where dimensionless concentrations ha#ve been substituted. Now, since z = y for constant-pattern conditions, we may generate from eq 10 the following differential ~ 1 - zrelation ~
where y1* and y2* are each functions of both z1and zz.Here we have denoted the ratio of mass transfer coefficients b y a. We have integrated this relationship numerically, using a standard Runge-Kutta routine, for various choices of a,and for arbitrarily chosen isotherm constants and initial conditions of a1 = 1, bl = 2 , a2 = 3, bp = 4,cIo = 1, and c2O = l/2. The results of these computations are shown in Figure 2. For a = 1 the numerical results indicate that, t o a large number of significant figures, the a-zZ relation is exactly linear. Additionally, the parameters all az, 61, be, elo, and cz0 were all varied independently and in combination by factors of up to four or more, and a linear concentration relation was always obtained for a = 1 (the relations for a + 1 were, however, different for the various cases). A mathematical proof of linearity when a = 1 has since been obtained, and is presented in Appendix 11. Some difficulties encountered in performing the integrations were: (a) since eq 11 is indeterminate at both ends of the sorption zone (zl = 0, xz = 1 and r1 = 1, x2 = 0) L'Hospital's rule was used to obtain initial values for dzl/dz2; (b) the numerical integration was found to be unstable for very high and very low CY values; and (e) the direction (upstream, dolvnstream) of integration and the choice of whether dzl/dzz or dx2/dx1 was integrated both affected the stability of the solution. Figure 2 also includes, for comparison, the z1-z2 relation obtained using Glueckauf's equilibrium solution, eq 6. I n terms of r1and x2 this relation, which is linear, is 21
=
*VMS? -N
(12)
+
+
where Ji' = alb2Xc20/aeblAcl; N = [azblclo (az - al)h/(l h)]/azblAcl. The boundary conditions of z1 = 0 when xz = 1 and 51 = 1 when 22 = 0 imply that -11= N = -1, that is, the z1-xz relation is z1 = 1 - zz. While many selected sets of values for al,az,bl, b2, c10, and c20 have verified that J P and N are indeed numerically equal t o - 1 (to any number of significant figures that one wishes to carry), mathematical proofs of such have not yet been found. It may be seen from Figure 2 that, when the two Langmuir solutes have nearly the same mass transfer coefficients (which is often the case) , the r1-z2 relationship for constant-pattern conditions is essentially the same as that for equilibrium liondispersive conditions. This being so, the following pseudobinary isotherms, obtained using the equilibrium 51-z2 relation, may be used
Y2*
+
+ EzczO) (Dz+ E2~2@22) - a~)A/(l+ A ) ; E1
12
z2(02
I
where D1 = alX (a2 D2 = a? - (a, - a l ) X / ( l
+
I
I
=
0
+
=
+
I
lot
alblA azbl; A); and E2 = a262 alb2A.
I 4 EQUILIBRIUM SOLUTION (COINCIDES WITH a = l CURVE)
Integrated Expressions for Concentration Profiles
z* = A
02
s,. FGj dx
0 0-10 0
+
where A = -eo(l - e)/{K,a[e (1 - e)(Aq/Ac)]} a s shonn by Cooiiey (1965). Here z* is an axial coordinate which travels with the front, defined as
and xc is that value of x for which z* is zero. The position' z , a t which z* = 0 is the position of a perfectly sharp front (step change) a t timet. Numerical computation of the profiles for the two solutes is readily performed once the z1-z2 relationsnip is known. Integrations of eq 15 have been carried out' for various a values, iising K , = 0.0006, a = 144, E = 0.4, and v = 0.1, and the results are shown in Figures 3 and 4. I n these same figures the profiles which result from using Glueckauf's equilibrium z1-z2 relatioil (eq 12) in eq 15 are plotted for comparison. Again it may be noted t h a t agreement is good in the region of a close to unity. The profiles derived from Glueckauf's result can be expressed analytically as Z* =
A[P 111 2 2 - (0
z* = A [ @In (1 -
+ 1) In (1 - ( p + 1) I n
22)
21)
,I
0 6 t
The column concentration profiles for constant-pattern conditions, negligible axial dispersion, and the rate law of cq 9 may be determined from the expression
'
1
I
a=050,
x1
- I] - I]
(17)
-8
4
- 4U
-6
-2
-0
2
41
61
8
z:cm
10
Figure 3. Concentration profiles for solute 1 under equilibrium and under constant-pattern conditions.
I
x
zycm
Figure 4. Concentration profiles for solute 2 equilibrium and under constant-pattern conditions
under
(18)
where P = D 2 / ( E 2 c Z 0 Derivations ). of t'hese equations are given in Appendix I. Conclusion
Inasmuch as analytical expressions for the concentration profiles generated under constant-pattern conditions are obtaiiiable when the equilibrium X I - X T ? relat'ion is used, it is important to know when the equilibrium relation is a n accurate reflection of the constant-pattern relat'ion. It has been shown that, !Then the Laiigmuir solutes have similar mass transfer coefficients, the constant-pattern and equilibrium x1-.r2 relatiom and the corresponding coiiceiitratioii profiles agree very closely. I n fact, when the mass-transfer coefficients are t'lie same, the 51-52 relations for the constantpattern and equilibrium cases are exactly the same and are linear-regardless of the isotherm constants and jeed concentrations. T h a t is, the relation is always 5 1 = 1 - 5 2 when a = 1. Some reasons for the agreement of these two solutions are considered briefly in Appendix 111. \T7e believe this is the first analytical computation of the behavior of a two-solute sorption operat,ion under nonequilibrium conditions. Considering solute 2, we may substitut'e the pseudobinary isotherm, eq 14, into the expression for z * , eq 15, and integrate to obtain
A [ B In x 2 - ( p
+ 1)
111
(1
P ln zo
- z2) -
+ (P + 1) In (1 - x,) 1
Figure origin
(19)
DISTANCE
5. Diagram used to locate position of L* coordinate
The quantity z,may be obtained b y situating a vertical line as shown in Figure 5 , such that the mass of solute represented b y areas A , summed for both phases, equals the mass of solute represented by areas B , summed for both phases. This requirement may be stated mathematically as
1 1
z*(z) dz
Appendix I. Derivation of Eq 17 and 18
z* =
V
+ q@(1- E)
1'
z*(y) dy
=
0
For the constant-pattern, zero dispersion case, where x this reduces to s,'z*(x)
=
0
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
(20) = y,
(21) 125
Application of this requirement to eq 19 gives the following expression for x, 1 - ,8 In x,
+ ( p + 1) In (1 - z,)
=
0
(22)
From this i t is apparent that eq 19 may be simplified to the form given b y eq 17. The solution for solute 1 may be obtained by a similar procedure, but an easier way to obtain the solution for solute 1 is simply t o combine eq 17 with the 21-22 relation, eq 12, to produce the result given in eq 18. Appendix II. Proof that Equilibrium Solution and Constant1 Are Identical Pattern Solution for a
=
Equilibrium Solution
where we have defined f and g as follows: yl*(zl, 22) = f(z1, $2) ; YZ*(XI,$2) = g(z1, 2 2 ) . Glueckauf proved that, for saturation, XI = 1 - z2 is the solution. Hence, it is easily shown that f+g=1. Constant Pattern Solution ( a 1). Equation 11 can be writ.ten
=
dsi - (YI* - XI) - _ f -_21 dzz (YZ*- 2 2 ) g - xz Let us assume a solution of the form z1 = 1 - x2, Then we may use the facts t h a t f g = 1 and XI = 1 - 2 2 to reduce the above equation t o
arbitrary and only an approximation to reality. There is, therefore, no a priori reason why it should match the equilibrium solution as both K , values go to infinity. I n the present situation, the great similarity of the two Langmuir isotherms (note that the denominators of eq 4 and 5 are identical, and both numerators are linear) causes the similarity and linearity of the two solutions cited. The quantities x1 and x2 are interrelated not only through the mass balances, but especially because yl* and y2* are each functions of both z1 and x2.The behaviors of the solutes are coupled, and any variation in the concentration of one has an effect on the concentration of the other. K i t h Langmuir isotherms, the relative strengths of dependence of yl* on $1 and y2* on x2 are essentially the same. Likewise the crossdependencies, of y1* on x2 and y2* on xl, are essentially identical in strength. With similar isotherms having similar major (e.g., y z * on za) and minor (e.g., y r * on z J ) dependencies, it would not be surprising that the two solutions cited above should be close to each other and near-linear. T h a t they are identical and exactly linear would not have been anticipated, but nevertheless seems to be the special situation for which such is the case. Detailed physical arguments can be made in support of the notion that the mathematical solutions should be very close, but these mill be omitted here. They all ultimately rest on the basic fact of the great similarity of the isotherms for the two solutes.
+
dxi - = -1 dx2 Applying the appropriate boundary conditions gives, upon integration, z1 = 1 - x2,which is the same as the assumed solution. Since this satisfies the differential equation and boundary conditions, i t is clearly a correct solution (although not necessarily unique). Appendix 111. Discussion of the Coincidence of the Equilibrium X I - X ~ Relation with That for Constant-Pattern, a 1, Conditions
=
The fact that the 2 1 - 2 2 relations for these two situations are identical [XI = 1 - z2]may seem surprising. Therefore, we wish t o discuss here how this congruence might arise. The equilibrium 21-23 relation [ZI = 1 - z2]is the solution to the equation
and the constant-pattern, a = 1, relation [also 51 = 1 - 2 2 1 is the solution to t h e equation dxl I - -_VI* __X_ dx2 yz* - 2 2 The congruence of these relations, and the fact t h a t they are linear, is apparently unique to Langmuir isotherms. Another two solute system, entailing polynomial isotherms, was recently considered by us. Although the equilibrium and constant-pattern ( a = 1 case) solutions were similar, and not far from being linear, it was nevertheless clear that they were different and nonlinear. This shows, incidentally, that one cannot argue that: because the a = 1 solution also pertains for 01 = w / w , it should therefore agree with the equilibrium solution. The form of the chosen rate equation is 126 Ind. Eng. Chem. Fundam., Vol. 1 1, No. 1, 1972
mass transfer surface area per unit bed volume. cm2/cm3 constants in Langmuir isotherms solute concentration in fluid phase, moles/l. disperqion coefficient, cm2/sec overall fixed-phase ma3s transfer coefficient, cm/sec solute concentration in fixed phase, moles/l. time, see interstitial velocity, cm/sec dimensionless fluid-phase concentration value of z a t z* = 0 dimensionless fixed-phase concentrat ion axial distance from top of bed, cm distance coordinate moving r i t h sorption front, cm GREEKLETTERS = ratio of mass transfer coefficients, dimensiona less = absolute value of t h e change from one end of A zone I1 to the other end of zone I1 = bed void fraction, dimensionless e = roots of the quadratic eq 7 x SUBSCRIPTS AND SUPERSCRIPTS = subscript denoting lowest value attained in m zone I1 = superscript denoting concentrations a t inlet of 0 bed * = superscript denoting equilibrium concentrations W = superscript denoting concentrations a t downstream end of zone I1 literature Cited
Cooney, D. O., IKD. ENG.C H E h f . , FCXDAM. 4, 233 (1965). Cooney, D. O., Ind. Eng. Chem., Process Des. Develop. 5 , 25 (1966). Glueckauf, E., Proc. Boy. SOC.,Ser. A , 186, 35 (1946). RECEIWDfor revieTv March 17, 1971 ACCEPTEDOctober 18. 1971
