Binary Adsorption Isotherms from Chromatographic Retention Times

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Ind. Eng. Chem. Res. 1999, 38, 1114-1124

Binary Adsorption Isotherms from Chromatographic Retention Times Bryan A. Buffham,* Geoffrey Mason, and Mark J. Heslop Department of Chemical Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, U.K.

A new method to determine binary adsorption isotherms from a set of binary chromatographic retention-time experiments is proposed. It differs from existing methods in two main ways. The first is that a functional form is assumed for the isotherms. The second is that it does not require supplementary measurements to be made. The isotherm parameters are found by a computer search. Any other available data may be incorporated into the method. The method is tested on retention-time data for the coadsorption of nitrogen and argon on 5A molecular sieves and of ethane and ethylene on 13X molecular sieves. Very precise experimental data are required if the isotherms are to be reconstructed solely from retention data. The Chromatographic Method In the chromatographic method of determining binary adsorption isotherms, a mixture of the two components is passed through a chromatographic column, and then a chromatographic transient is provoked by the injection of a pulse or the addition (or removal) of a small continuous flow of some material. The material added is usually one of the pure components. The transient is observed by a simple, uncalibrated detector, such as a katharometer, that senses the change of a physical property of the effluent gas and not the concentrations of the individual components. For a binary mixture, one sees a single chromatographic peak (or front).1 Its retention time tr is the time the peak takes to traverse the column. There are several variants of the theory that connects the retention time to the isotherms and the rule by which different authors find the retention time from the chromatogram is not always the same. Nevertheless, the various forms of the theory agree2 in the way that the retention time depends on the volumetric flow rate, Q, the volumes of the mobile and stationary phases, the mobile-phase composition, and the adsorption equilibrium. Suppose a binary gas mixture is at equilibrium with a solid adsorbent. For a constant temperature, the two equilibrium relations between the molar concentrations qA and qB of the two species, A and B, in the solid phase and those, cA and cB, in the gas phase are called the adsorption isotherms. Alternatively, the relations may be between qA and qB and the gas-phase mole fraction, e.g., yA, of one of the components and the gas-phase molar density Fg:

qA ) qA(yA,Fg)

(1)

qB ) qB(yA,Fg)

(2)

This formulation is convenient when the pressure is low and constant, for then the gas-phase molar density is independent of composition and there is just one independent variable, yA. * Author to whom correspondence should be addressed. E-mail: [email protected]. Fax: +44 1509 223923. Telephone: +44 1509 222503.

When the gas density Fg ) cA + cB is constant and the concentration perturbations are very small, the result that the various theories give for the retention time in a column containing a volume Vg of gas and a volume Vs of solid is2

tr )

(

[

)]

dqA dqB 1 Vg + Vs yB + yA Q dcA dcB

(3)

The dimensionless quantity in the parentheses in eq 3 is given the symbol K and is called the binary equilibrium constant. All the theories agree that the retention time is independent of which component is added and is the same for pulse- and step-response experiments. Equation 3 contains the volume Vg of gas in the column. One often-used way of estimating this gas volume is to measure the retention time of a pulse of a nonretained substance injected into another, perhaps retained, substance. Suppose that A is not retaineds meaning that qA is zero whatever the value of cA. Then both dqA/dcA and yA will be zero, the term in parentheses is zero, and Vg can be found directly from the measurement of the appropriate tr and the application of eq 3. In experiments, both this measurement and the measured retention time include the effects of the volumes of the connecting piping between the injection point and the column, of the volume between the column and the detector, and also of any measurement lag. Subtracting the retention time for the pulse of nonretained substance tg from the measured retention time tr compensates for both the piping volume and the measurement lag and gives the net retention time:

t r - tg )

(

)

Vs dqA dqB yB + yA Q dcA dcB

(4)

The gas volume Qtr that flows from the column in the retention time is called the retention volume. Similarly, the net retention volume Q(tr - tg) corresponds to the net retention time tr - tg. Frequently helium is used as the non-retained substance, but a correction must be made because it penetrates the pore space. In practice, it is often more convenient to work in terms of the amounts of the species adsorbed per unit

10.1021/ie980425i CCC: $18.00 © 1999 American Chemical Society Published on Web 02/06/1999

Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999 1115

mass of adsorbent than in terms of solid-phase concentrations. The amounts (moles) of A and B adsorbed per unit mass are

wA )

qA Fs

(5)

wB )

qB Fs

(6)

where Fs is the density (mass per unit volume) of the adsorbent. When the solid-phase concentration is expressed in terms of w and the gas-phase concentration in terms of y, the net-retention-time equation, eq 4, becomes

(

)

VsFs dwA dwB t r - tg ) y + yA QFg B dyA dyB

(7)

where, it must be remembered, the restriction of constant gas-phase density still applies. The term in parentheses in eq 7

yB

dwA dwB Fg + yA ) K ) Kw dyA dyB Fs

(8)

is a new form of the (constant-gas-density) binary equilibrium constant. It has the same dimensions as w, moles per unit mass. As VsFs is the mass, W, of the solid, the net retention time is given in terms of measurable quantities and Kw by

t r - tg )

(

)

WKw dwA dwB W yB + yA ) QFg dyA dyB QFg

(9)

The major difficulty with the chromatographic method is clear from eq 9. The mass of solid, the gas-phase mole fractions, the flow rate, and the gas density are at the experimenter’s choice; the net retention time can be measured. Thus Kw can be found. However, we wish to find wA and wB. There are two dependent variables, dwA/ dyA and dwB/dyB, to be found from one equation. If this problem could be surmounted and a way could be found to determine dwA/dyA and dwB/dyB, then the isotherms, wA ) wA(yA) and wB ) wB(yA), could be constructed by numerical integration of values of dwA/dyA and dwB/dyB determined at several compositions. The integration constants would be established from the fact that the solid-phase concentration of a component is zero when the gas-phase concentration of that component is zero. Implicit in this approach is the idea that the isotherms are smoothssmooth enough for the derivatives to exist and be integrable. With the measurements of tr at several values of the composition considered individually, the problem of finding how wA and wB vary with composition is insoluble. However, the differentials dwA/dyA and dwB/ dyB are likely to vary smoothly (in the above sense) with yA and so adjacent values are related to each other. Consequently, it may be possible to circumvent the problem of insufficient information by considering the results of several experiments together. That the experiments are relatively quick to do means that it is not impractical to make measurements of tr at many different values of yA.

Figure 1. Experimental values (b) of the binary equilibrium constant Kw for the adsorption of ethane-ethylene mixtures on 13X molecular sieves at 298 K and 138 kPa calculated from the retention volumes reported by Hyun and Danner.5 Also shown is the best-fitting third-order polynomial.5,6 Note that the fit has an inflexion that is not present in the experimental data.

Van der Vlist and van der Meijden suggested3 an ingenious way of interrelating the results of several experiments by parametrization. They assume that Kw may be represented by a third-order polynomial in yA and that dwA/dyA and dwB/dyB may be represented by second-order polynomials in yA. [They actually used w’s and partial pressures. We translate other authors’ variables into w’s and y’s.] This assumption makes the differentials dwA/dyA and dwB/dyB smooth functions. The four coefficients of the Kw polynomial are estimated by polynomial regression of a set of net-retention-time measurements at different compositions. Expressions for the six parameters in the isotherms can then be deduced in terms of the four Kw coefficients and the two values of the pure-component amounts adsorbed.3-6 [Note that in the paper6 by Heslop et al. the left-hand side of eq 11 should be B2 and the left-hand side of eq 12 should be B1.] The logical sequence of van der Vlist and van der Meijden’s method is to fit a function to the values of Kw and then work back to the isotherms. We call this van der Vlist and van der Meijden’s solution procedure. The original method outlined above works well in some cases and less well in others.4-6 It is unsatisfactory when a third-order polynomial cannot be fitted satisfactorily to the Kw against composition data. Sometimes the experimental data form an obvious smooth curve but the polynomial fits it with a wiggle.5-7 This can be seen in Figure 1, for example. Triebe and Tezel7 have proposed that van der Vlist and van der Meijden’s procedure be applied to a different functional form of Kw. They suggest that Kw be represented by the sum (the Triebe and Tezel expansion) of four terms of the form Aj(yA + β)-j, with the integer j taking the values from -1 to +2, and each of the differentials dwA/dyA and dwB/dyB by the sum of three terms of the forms Bj(yA + β)-j and Cj(yA + β)-j with j taking the values from 0 to 2. Van der Vlist and van der Meijden’s third-order polynomial for Kw is thus replaced by the ratio of a general third-order polynomial to a perfect square, (yA + β)2. Ideally, one would wish to make an additional measurement at each gas composition and derive an additional relation to complement eq 9. We,2,8 and others,9 have advocated using outlet flow-rate measurements. When this is done, the binary isotherm gradients at a

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Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999

single composition may be determined2,8 from outlet composition and flow-rate measurements. Nevertheless, because retention times are relatively easy to measure, it would be useful if the chromatographic method could be modified to allow any isotherm to be used. Then we could exploit generic background information such as “guessing” that a particular isotherm type would be suitable because that type is known to fit similar systems, and use an algorithm to optimize the fit. Van der Vlist and van der Meijden’s solution procedure as applied by them and by Triebe and Tezel involves “guessing” a function to fit the Kw. This choice limits the algebraic form of the isotherm functions. Using an isotherm of established type accepts not only the smoothness of the variations of dwA/dyA and dwB/dyB with composition but also that they vary in typical ways and are related to each other. Therefore, in practice, we require more than that the isotherm expressions be mathematically smooth (i.e. be continuous and have continuous derivatives). They also have to be smooth overall, be monotonic, and have few, if any, points of inflexion. It would also be useful if fewer parameters could be used to give an estimate, however approximate, of the isotherms. It would be even more useful for preliminary studies if the method could dispense with the need for pure-component data, which are often measured by a different technique. The purpose of this paper is to show that all these objectives can be reached and to highlight potential difficulties. Fitting a Model Isotherm We propose parametrizing the problem just as van der Vlist and van der Meijden3 as well as Triebe and Tezel7 did, but we reverse the van der Vlist and van der Meijden solution procedure. We assume that the isotherms (instead of the Kw function) belong to a family having a particular algebraic form and involving some unspecified parameters. This makes the functions smooth and, perhaps, monotonic. The family may be purely empirical (such as polynomials or the Triebe and Tezel isotherms), but the choice can be informed by theoretical considerations such as the need for thermodynamical consistency or a proposed mechanism or theory for the adsorption process. It is sensible to pick an established form for the isotherms. For any model isotherm equations, the binary equilibrium constant can be derived by using eq 8. This step is nonspecific: any form may be chosen for the isotherms. It is also straightforward: it involves only differentiation and multiplication. The resulting expression can be fitted to the experimental Kw data by using an appropriate algorithm to choose the values of the parameters that meet a “best-fit” criterion. For present purposes, we adopt the view that the objective is to obtain equilibrium compositions to use in the design of adsorbers. Computational convenience may help guide the choice of the model isotherm. In other circumstances, the objective could be to study a theory of adsorption. Then there would be less, perhaps no, freedom of choice. In this paper we consider three isotherms. We use the van der Vlist and van der Meijden polynomial and the Triebe and Tezel isotherms because they have been used before, and we examine the benefits of using the binary Langmuir isotherm because of its simplicity.

Table 1. Retention Times8 for the Passage of a Nitrogen-Argon Composition Front through a Column of 5A Molecular Sieves at 323 K and 105 kPa and the Corresponding Binary Equilibrium Constantsa binary equilibrium constant, mole fraction flow rate, perturbation retention of nitrogen Q/(cm3/s) gas time tr/sa Kw/(mol/kg) 0.000

0.516

0.300

0.548

0.531

0.579

0.737

0.616

1.000

0.660

nitrogen argon nitrogen argon nitrogen argon nitrogen argon nitrogen argon

200.1

0.2748

156.6 158.6 124.2 123.4 98.8 98.6

0.2248 0.2280 0.1853 0.1840 0.1537 0.1533

73.2

0.1177

a

Note that the retention time depends on the gas used to generate the front.

The Data To Be Fitted We are looking to test the analysis of data produced by the chromatographic method. To do this we need gravimetric data and chromatographic data for the same system. Remarkably few data sets have been found using both methods simply because of the effort involved. We examine two. One is for mixtures of argon and nitrogen on 5A molecular sieves (by us) and the other is for ethane-ethylene on 13X molecular sieves (by Danner et al.). For the argon-nitrogen-5A system we have measured retention times8 and calculated isotherms10 by using a combination of retention times and mass balances. These results rank on a par with gravimetric results. Our retention times8 for the passage of nitrogenargon concentration fronts through a column of 5A molecular sieves at 323 K and 105 kPa are given in Table 1. The fronts were generated by adding a small extra stream (perturbation gas) of one component to a steady stream of mixed gas. Results were obtained by using each component as perturbation gas at several compositions. Slightly different retention times were obtained by using the two perturbation gases. This is an important point to which we shall return later. Values of Kw calculated, eq 9, from the retention times, flow rates, and column data8 are also given in Table 1. The Langmuir isotherm is known11 to fit results for the adsorption of nitrogen-oxygen mixtures on 5A sieves and the adsorptive behavior of argon is similar to that of oxygen. Hyun and Danner have reported5 net retention volumes for concentration-pulse measurements for the ethane-ethylene-13X system at 298 K and 138 kPa. In addition to their own results they also quote the results of similar experiments performed by Nicoletti.12 Both sets of results are consistent.5 [There is a scale error in Hyun and Danner’s5a Figure 6: the net retention volumes should be multiplied by 10.5b] Previously, Danner and Choi had reported13 equilibrium data for the same system measured by a static method. Values of Kw calculated from the net retention volumes are given in Figure 1. Calculation Method Suppose that we have retention times (and hence Kw’s) measured at each of several different compositions. We may have other data such as the pure-component


amounts adsorbed, w0A and w0B. Our procedure is as follows. Choose a suitable functional form as the adsorption isotherm model. Then derive an expression for Kw by using eq 8. If other information is available, use it to reduce the number of parameters in the expression for Kw. For example, when w0A and w0B are known, set yA and yB equal to one in the model isotherms to obtain two equations and use them to eliminate two parameters. [Some search algorithms allow supplementary information to be used in the form of constraints.] Next, find the values of the parameters that minimize a suitable objective function such as the sum of the squares of the differences between the n experimental values, Kexp,i, of Kw calculated from the retention times and the values, Kmodel,i, given by the model at the experimental compositions: n

S2 )

(Kexp,i - Kmodel,i)2 ∑ i)1

(10)

The minimum value of S2 gives a joint measure of how well the model fits the experimental results and experimental scatter, and so allows quantitative comparison of the ability of different isotherms to fit a given set of experimental data. The dimensionality of S2 is equal to the number of parameters in the adsorption model less the number of equality constraints provided by the supplementary data. We used proprietary implementations (Mathcad) of Powell’s algorithm14 in multidimensional searches and Brent’s algorithm14 in one-dimensional searches to minimize S2. The values of the parameters in the original isotherm are thus determined and, finally, the isotherms can be calculated. The normalized version

xS2/n

s) 1

)

n

xnS2

(11)

n

Kexp,i ∑Kexp,i ∑ i)1

Kw ) (A1 + 2A2yA + 3A3yA2)yB + (B1 + 2B2yB + 3B3yB2)yA (14) Kw ) (w0A - A2 - A3 + 2A2yA + 3A3yA2)yB + (w0B - B2 - B3 + 2B2yB + 3B3yB2)yA (15) Because yB ) 1 - yA, these expressions for Kw are thirdorder polynomials in yA, as in van der Vlist and van der Meijden’s original method. It is not, therefore, possible to find unique values of the six parameters by curve fitting experimental values of Kw. When w0A and w0B are known, it may be possible to find unique values of the four remaining parameters (two A’s and two B’s) in eq 15. General Polynomial Isotherms. The above procedure is not limited to third-order polynomials, nor is it necessary that the two polynomials be of the same order, but changing the order of the polynomials does not remove the need for extra information such as w0A and w0B. Triebe and Tezel Isotherms. The Triebe and Tezel isotherms are

wA ) B0yA + B1 ln wB ) C0(1 - yA) - C1 ln

of the root-mean-square discrepancy between the experimental and fitted values of Kw (obtained by dividing the rms discrepancy by the average of the experimental values) is appropriate to compare between different data sets.

(16)

yA + β C2(1 - yA) + 1+β (1 + β)(yA + β) (17)

where the B’s, C’s, and β stand for constants. Note that there are seven parameters, β is shared between the two isotherms, and the B’s and C’s are not exchangeable. Differentiating these equations and substituting the differentials into eq 8 gives an expression for the binary equilibrium constant Kw:

[

Kw ) yB B0 +

n i)1

yA + β B2yA + β β(yA + β)

]

B2 B1 + + yA + β (y + β)2 A C1 C2 (18) yA C0 + + yA + β (y + β)2 A

[

]

The Isotherm Forms and Their Kw’s

Setting yA ) 1 in eq 16 and yA ) 0 in eq 17 gives relations between w0A and w0B and the model parameters that allow w0A and w0B to be introduced into eq 18 and B0 and C0 to be eliminated:

Van der Vlist and van der Meijden Polynomial Isotherms. Van der Vlist and van der Meijden’s isotherms may be written as

Kw ) yB w0A - B1 ln

wA ) A1yA + A2yA2 + A3yA3 ) (w0A - A2 - A3)yA + A2yA2 + A3yA3 (12) 2

3

wB ) B1yB + B2yB + B3yB )

(w0B

- B2 - B3)yB + 2

3

B2yB + B3yB (13) where the A’s and B’s are constants and w0A and w0B are the pure-component amounts adsorbed. Using eqs 12 and 13 to substitute for wA and wB in eq 8 gives

[

B2

] [

(yA + β)2

B2 B1 1+β + + β β(1 + β) yA + β

+ yA w0B - C1 ln

C2 1+β + β β(1 + β)

]

C2 C1 (19) + yA + β (y + β)2 A

There are seven parameters in eq 18 and also in eq 19. These expressions for Kw are equivalent to the (fiveparameter) Triebe and Tezel expansion as may be confirmed by replacing yB by 1 - yA and then writing the result in terms of yA + β. Thus only five parameters (the A’s) may be determined uniquely by curve fitting

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and extra information, such as w0A and w0B, is needed. [We have used A’s and B’s both for Triebe and Tezel parameters (the A’s for the Triebe and Tezel expansion coefficients) and for van der Vlist and van der Meijden polynomial parameters. There is no danger of confusion, because the two sets are not used in the same context.] Binary Langmuir Isotherms. When expressed in terms of the amounts adsorbed and mole fractions, the Langmuir isotherm equations are

wA )

HAyA 1 + LAyA + LByB

(20)

wB )

HByB 1 + LAyA + LByB

(21)

Note that LA and LB are shared between the two isotherms. HA and HB are the w-y forms of the Henry’s law constants and are appropriate for constant gas density. They have dimensions of moles per unit mass. In contrast to the true (q-c) Henry’s law constants, they vary with gas density and hence with pressure. We shall call the dimensionless L constants Langmuir constants. Differentiating eqs 20 and 21 and substituting the differentials into eq 8 gives

Kw )

HA(1 + LB)yB + HB(1 + LA)yA (1 + LAyA + LByB)2

(22)

Relations between the pure-component amounts adsorbed and the H’s and L’s are found from eqs 20 and 21. These allow the Langmuir constants, LA and LB, to be eliminated from eq 22 and the binary equilibrium constant to be expressed in terms of the composition, the pure-component amounts adsorbed, and the ratio

RAB )

HA HB

(23)

of the Henry’s law constants. The result is

RAB Kw )

(

(

yA

w0A yA

+

) )

yB

w0B yB 2

RAB 0 + 0 wA wB

(24)

Note that Kw depends on just three parameters, not four, and also that this expression belongs to the Triebe and Tezel7 class. RAB is equal to the ratio of the true Henry’s law constants as well as the H’s and is a true constant, but the amounts adsorbed, w0A and w0B, depend on the gas density. At low pressure, the gas density is constant when the pressure is constant. In the derivation1 of the Langmuir isotherm it is assumed that the adsorbent has a fixed possible molar capacity for the adsorbate: a saturation value of the total amount adsorbed qA + qB is attained at very large values of cA or cB or both. This provides a relation between the isotherm parameters. This constraint has not been used in deriving eq 24. Replacing the ones in the denominators of eqs 20 and 21 by yA + yB, dividing the numerator and denominator of eq 20 by HA and those of eq 21 by HB, and then using eq 23 and the expressions for w0A and w0B given by eqs

Figure 2. Fitted values of the binary equilibrium constant Kw for the adsorption of nitrogen-argon mixtures on 5A molecular sieves at 323 K and 105 kPa, obtained by using the binary Langmuir model isotherm and retention times only, compared with the experimental values of Table 1. The fitted values are shown by the curves; experimental results obtained by using argon perturbation are shown by O and those obtained by using nitrogen perturbation by b.

20 and 21 shows that

wA )

w0AyA yA +

wB )

w0A RABw0B

(25) yB

w0ByB RABw0B yA + yB w0A

(26)

There are three parameters in all in eqs 25 and 26, namely w0A, w0B, and RAB. However, when these three are grouped as w0A/RABw0B, there are two parameters per component isotherm. We have seen that it is not possible to find unique values of the van der Vlist and van der Meijden or Triebe and Tezel isotherm parameters by curve fitting experimental values of Kw. The Langmuir isotherm is different because it implies a relationship between dwA/ dyA and dwB/dyB. Fitting experimental Kw values yields three parameters. The isotherms can be calculated from these parameters with no need for extra information. Fitting the Isotherms to the Experimental Results for Argon-Nitrogen-5A Binary Langmuir. We consider only Langmuir adsorption because it requires fewer parameters and because the isotherms are so nearly linear. We use different amounts of information in our procedure to find the parameters that give the “best” fit of the data. i. Retention Times Only. Figure 2 compares the fitted values of Kw, calculated by minimizing the sum of squares in the way just outlined, with the experimental values given in Table 1. The two sets were processed separately to give the two branches in Figure 2. The minimum normalized values of the rms discrepancy, s, between the experimental and fitted values of Kw were 2.867 × 10-3 for nitrogen perturbation and 3.503 × 10-3 for argon. Figure 3 compares the calculated isotherms with results obtained10 from mass balances


Figure 3. Binary Langmuir isotherms (shown by the dotted line for argon perturbation and the broken line for nitrogen perturbation) for the adsorption of nitrogen-argon mixtures on 5A molecular sieves at 323 K and 105 kPa calculated by least-squares fitting of the binary Langmuir model isotherm to the measured the binary equilibrium constants (Kw’s) of Table 1. Estimated equilibrium compositions are compared with those (shown by the solid line) calculated from mass balances.10

Figure 5. Binary Langmuir isotherms (shown by O for argon perturbation and by b for nitrogen perturbation) calculated for the adsorption of nitrogen-argon mixtures on 5A molecular sieves at 323 K and 105 kPa by least-squares fitting of experimental binary equilibrium constants with the pure component amounts adsorbed prescribed. Estimated equilibrium compositions are compared with those calculated from mass balances10 and shown by a full line. Note that O and b are markers for calculated curves and not experimental points.

3. Using the pure-component amounts adsorbed has forced the isotherms through the same end points and this has improved the agreement. Fitting the Isotherms to the Experimental Results for Ethane-Ethylene-13X

Figure 4. Fitted values of the binary equilibrium constant Kw for the adsorption of nitrogen-argon mixtures on 5A molecular sieves at 323 K and 105 kPa obtained by using the binary Langmuir model isotherm and experimental binary equilibrium constants, with the pure-component amounts adsorbed prescribed, compared with the experimental values of Table 1. Experimental results obtained by using argon perturbation are shown by O and those obtained by using nitrogen perturbation by b. The pure0 component amounts adsorbed are wnitrogen ) 0.2565 mol/kg and 0 wargon ) 0.1243 mol/kg. If the data and fit were perfect, the points would fall on the diagonal.

on adsorption transients. Because two retention times are known for each composition (one for the nitrogen perturbation and one for the argon perturbation) there are four isotherms calculated directly from retention times. From the curves on Figure 3 we can conclude that when retention times are the only data available, very precise experimental data are needed to obtain satisfactory isotherms. ii. Retention Times plus Pure Component Data. Figure 4 shows the quality of the fit between the experimental and fitted values of Kw when the retention times are supplemented by pure-component data. The normalized rms discrepancies are somewhat greater than before (4.927 × 10-3 for nitrogen perturbation and 15.78 × 10-3 for argon). Even so, the two pairs of isotherms calculated from retention times and the isotherms calculated from mass balances are indistinguishable at this scale, Figure 5. Compare with Figure

Van der Vlist and van der Meijden Polynomial Isotherms. i. Retention Times Only. Although only four parameters are unique, we tried searching for six parameters to see what would happen. The information used was the Kw data of Figure 1. Perhaps the nonunique values would combine to give the same isotherms. We were able to find several sets of parameters that gave the same value (s ) 0.1298) to the objective function, eq 10. In each case the Kw data were fitted with a wiggle rather like that in Figure 1. However, in every one of these cases one or other of the isotherms indicated negative amounts adsorbed, which is a physical impossibility. ii. Retention Times plus Pure Component Data. The information used was the Kw data of Figure 1 and the pure-component amounts adsorbed of Danner and 0 ) 2.185 mol/kg and w0ethylene Choi13 (converted to wethane 0 ) 2.765 mol/kg). When wA and w0B are known, there are only four parameters to search for in eq 15, namely A2, A3, B2, and B3. The isotherms, calculated using eqs 12 and 13, are shown in Figure 6. The results we obtained in this way are exactly the same as those obtained5,6 using the original van der Vlist and van der Meijden method. Considering that the ends of the isotherms are fixed, the fit is extremely poor. Triebe and Tezel Isotherms. i. Retention Times Only. Although only five parameters are unique, we tried searching for seven. We were able to obtain several parameter sets that minimize the objective function, eq 10, and give a good fit (s ) 0.0395) to the Kw data. Some of the calculated isotherms looked reasonable, but there is no way of knowing which is best. The value of s (0.0395) probably indicates the amount of random scatter. Interestingly, the same value (-1.2302) was found for β in several cases.

1120


Figure 6. Van der Vlist and van der Meijden isotherms for the adsorption of ethane-ethylene mixtures on 13X molecular sieves at 298 K and 138 kPa calculated by least-squares fitting of experimental binary equilibrium constants (Figure 1) with the pure component amounts adsorbed prescribed. Estimated equilibrium compositions are compared with the static data13 (O for ethane and b for ethylene).

Figure 7. Triebe and Tezel isotherms for the adsorption of ethane-ethylene mixtures on 13X molecular sieves at 298 K and 138 kPa calculated by least-squares fitting of experimental binary equilibrium constants (Figure 1) with the pure component amounts adsorbed prescribed. Five parameters are sought. Estimated equilibrium compositions are compared with the static data13 (O for ethane and b for ethylene).

ii. Retention Times plus Pure Component Data. When w0A and w0B are known, our method proceeds as before except that now only the five parameters of eq 19 are sought. The calculated isotherms are given in Figure 7. The fit is better than that given by van der Vlist and van der Meijden’s method. The same values were found for the objective function (s ) 0.0395) and for β (-1.2302) as were found when the pure-component amounts adsorbed were not prescribed. We found multiple minima of the objective function but only one with s ) 0.0395. (This does not mean that there may not be others nor does it mean that there is not a set of parameters with a lower value of s.) The results obtained using our procedure are the same as those we found using van der Vlist and van der Meijden’s procedure with Triebe and Tezel isotherms. Binary Langmuir Isotherms. i. Retention Times Only. Figure 8 compares the fitted values of Kw for the adsorption of ethane-ethylene mixtures on 13X sieves calculated using Langmuir isotherms with the experimental values given in Figure 1. Calculated isotherms are displayed together with the static data of Danner

Figure 8. Fitted values of the binary equilibrium constant Kw for the adsorption of ethane-ethylene mixtures on 13X molecular sieves at 298 K and 138 kPa obtained by using the binary Langmuir model isotherm and retention times only. Also shown are the experimental Kw values (from Figure 1) that have been fitted. The fit is excellent.

Figure 9. Binary Langmuir isotherms for the adsorption of ethane-ethylene mixtures on 13X molecular sieves at 298 K and 138 kPa calculated by least-squares fitting of the binary Langmuir model isotherm to the measured binary equilibrium constants Kw of Figure 1. Estimated equilibrium compositions are compared with the static data13 (O for ethane and b for ethylene). The agreement is poor.

and Choi13 in Figure 9. The lack of agreement between the calculated isotherms and the static data is surprising. In practice, one would not be aware of the discrepancy. Although the agreement between the fitted and experimental Kw values, Figure 8, is good (s ) 0.0411), the resulting isotherms in Figure 9 are very wide of the mark. ii. Retention Times plus Pure Component Data. The pure-component amounts adsorbed being known, a single observation of the retention time would be sufficient information to find the sole remaining unknown in eq 24, the ratio, RAB, of the Henry’s law constants. This could be done analytically because eq 24 is quadratic in RAB. However, we have supposed that we have retention times measured at several compositions, so it would be preferable to find the value of RAB that minimizes an objective function such as that in eq 10. The advantage of this approach is that the value of the sum of squares (s ) 0.2445) gives an idea of how well (or poorly) the model fits the experimental results. Figure 10 compares the fitted values of Kw for the adsorption of ethane-ethylene mixtures found for the binary Langmuir isotherm and the experimental re-


Figure 10. Fitted values of the binary equilibrium constant Kw for the adsorption of ethane-ethylene mixtures on 13X molecular sieves at 298 K and 138 kPa obtained by using the binary Langmuir model isotherm and experimental binary equilibrium constants, with the pure-component amounts adsorbed13 prescribed, compared with the experimental values of Figure 1. If the data and fit were perfect, the points would fall on the diagonal.

Figure 12. Binary Langmuir fit of Danner and Choi’s static data13 for the coadsorption of ethane (O) and ethylene (b) on 13X molecular sieves at 298 K and 138 kPa.

method did not work for this system because not only are data from the same series of experiments used for all the calculations, so there can be no discrepancy caused by experimental inconsistency, but also the isotherms are nearly straight. For the ethane-ethylene system we have seen that reconstructing the isotherms from fitted Kw values does not give even an approximate fit to the static data, no matter which isotherm or technique is used. How can this be? We wondered whether our objective function might be laying too much stress on one part of the Kw against yA curve. In the case of ethane-ethylene mixtures, the range of Kw values is large and there is a large change of slope; see Figure 1. Replacing eq 10 by n

Σ2 ) Figure 11. Binary Langmuir isotherms for the adsorption of ethane-ethylene mixtures on 13X molecular sieves at 298 K and 138 kPa calculated by least-squares fitting of experimental binary equilibrium constants Kw together with the pure component amounts adsorbed prescribed. Estimated equilibrium compositions are compared with the static data13 (O for ethane and b for ethylene).

sults. Figure 11 compares the calculated equilibrium compositions with the static data. The agreement between the new results and the static data is not good but it is better than was obtained by using the van der Vlist and van der Meijden procedure and isotherms, Figure 6. The greatest discrepancy occurs at intermediate compositions, simply because the method has forced agreement with the static data at the ends of the composition range. Comparison of Figure 11 (Langmuir isotherm) with Figure 7 (Triebe and Tezel isotherm) shows that the fit to the static data is not improved by increasing the number of parameters from one to five. Discussion For the nitrogen-argon system, we have seen, Figure 3, that reconstructing the isotherms from fitted Kw values gives a reasonable fit to the data calculated from mass balances. The fit would be better if the results obtained by using the two perturbation gases were averaged. When the pure component amounts adsorbed are used in addition to the Kw values, the agreement is excellent, Figure 5. It would be very worrying if the new

∑ i)1

(

)

Kexp,i - Kmodel,i Kexp,i

(27)

did not make a great difference to the fits obtained. The results obtained by using eq 27 as the objective function agreed with the static data a little better, judged subjectively, than those obtained by using eq 10. However, we have no way of judging from Hyun and Danner’s report5 whether it would be better to use the objective function defined in eq 27 than to use that in eq 10. Moreover, in practice one would not be aware of the static data and so would not be able to make the subjective judgment. When eq 27 is used as the objective function, an appropriate root-mean-square index of the quality of fit is

σ ) xΣ2/n

(28)

Values of σ from eq 28 obtained when using eq 27 for the objective function were not much different from those obtained previously for s by using eqs 10 and 11. We also wondered if the binary Langmuir isotherm was a poor choice for the ethane-ethylene system and so we fitted Danner and Choi’s static data13 with a pair of binary Langmuir isotherms, eqs 25 and 26. All of the data points were weighted equally in the objective function. The fit is excellent, Figure 12, particularly when it is realized that there were only three param0 ) 2.187 eters. The values of the parameters are wethane 0 mol/kg, wethylene ) 2.777 mol/kg, and R ) Hethane/

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Table 2. Langmuir Isotherms Calculated from the Static Data of Danner and Choi13 for the Adsorption of Ethane-Ethylene Mixturesa Triebe and Tezel isotherms calculated by van der Vlist and Langmuir fit van der Meijden’s mole to static data procedure our procedure fraction of ethane wethane wethylene wethane wethylene wethane wethylene 0.0 0.2 0.4 0.6 0.8 0.9 1.0

0.000 0.078 0.197 0.399 0.816 1.252 2.187

2.777 2.678 2.527 2.271 1.741 1.187 0.000

0.000 0.079 0.200 0.404 0.825 1.260 2.185

2.765 2.670 2.523 2.270 1.742 1.188 0.000

0.000 0.079 0.200 0.404 0.824 1.260 2.185

2.765 2.670 2.523 2.270 1.742 1.188 0.000

a A three-parameter Langmuir fit was obtained by fitting the static data to eqs 25 and 26. Kw’s were calculated from the Langmuir parameters and then Triebe and Tezel isotherms were calculated from these Kw’s and the pure-component amounts adsorbed by van der Vlist and van der Meijden’s procedure and by our procedure.

Hethylene ) 0.1171. When the pure-component amounts 0 ) adsorbed are known (in the present case wethane 0 2.185 mol/kg and wethylene ) 2.765 mol/kg), there is only a single remaining parameter, R ) Hethane/Hethylene. We have calculated this fit as well (for which R ) 0.1165) and it is virtually identical to Figure 12. We tested whether our method could reconstruct Langmuir isotherms from Kw values calculated from Langmuir isotherms. To do this we calculated a set of 21 Kw values at evenly-spaced mole fractions by using eq 24 and the three parameters obtained by fitting Danner and Choi’s static data.13 Then we searched to find the values of the parameters that gave the best fit to the Kw values. The parameters were almost identical 0 ) 2.185 mol/kg, w0ethylene ) 2.778 mol/kg, and R (wethane 0 ) 2.187 mol/ ) 0.1171) to the original values (wethane 0 kg, wethylene ) 2.777 mol/kg, and R ) 0.1171). We also tried both van der Vlist and van der Meijden’s procedure and our procedure to estimate Triebe and Tezel isotherms from these calculated Kw values together with 0 , and w0ethylene. The isotherms so estimated were wethane in very good agreement with the original data, Table 2. Poor fits to the Kw values and unrealistic isotherms were obtained when van der Vlist and van der Meijden isotherms were used instead of Triebe and Tezel isotherms. The success of the Triebe and Tezel isotherms could be because of the relationship between the Triebe and Tezel and Langmuir Kw expressions. The Langmuir isotherm parameters obtained by fitting the static data can be used in eq 24 to calculate Kw values which may be compared with the Kw values calculated from the retention volumes. This is shown in Figure 13 and it seems that Hyun and Danner’s retention volumes5 are incompatible with Danner and Choi’s static data.13 It is possible to make the two sets agree by subtracting 0.25 mol/kg from the Kw values calculated from retention volumes and multiplying the corresponding ethane mole fractions by 0.95. Hyun and Danner5 realized that van der Vlist and van der Meijden’s original procedure does not work when the polynomial does not fit the Kw data well. Later16 they concluded that the procedure would not work for experimental retention data, although they thought5 that the method would work if they calculated retention times from static data and then applied the

Figure 13. Binary equilibrium constants calculated from the parameters of the fitted isotherms in Figure 12 compared with those calculated from the retention volumes reported by Hyun and Danner.5 The curves can only be made to agree if one is displaced relative to the other.

method to this “synthetic” data. However, there may be a more fundamental problem. In the experiments the requirements of the theory behind eq 3 must be met. In our experiments8 on the nitrogen-argon system, we added a small perturbation stream and recorded a frontal retention time. Different experimental retention times were obtained by using argon and nitrogen perturbations. This is almost certainly because the increase in flow rate caused by adding the perturbation stream increases the mean pressure in the column slightly and so the constant density requirement of eq 3 is not precisely met. The concentration perturbations were, however, small and of known size. Some idea of the size of this effect can be seen from the differences in retention times in Table 1. In Hyun and Danner’s experiments,5 a pulse injection of one of the pure components was made using a sampling valve and so the constant density condition would be closely met (the total flow does not increase). However, the other requirement of eq 3, that the concentration perturbations be small, is probably not met, particularly when the pure-component pulse first reaches the column. Peak spreading dilutes the concentration perturbation once the pulse is in the column, but the extent is unknown. An estimate of the scale of this effect can be seen in the work of Tezel et al.17 on the nitrogen-krypton system. Working with pulses, they obtained retention times that differed with the component injected. Sometimes the difference was as much as ten per cent. We have already seen (for the nitrogen-argon system, refer to Figure 3) that isotherms calculated from front retention times are very dependent on the precision of the retention-time measurements. So the explanation of why the chromatographic and static data do not correspond could be that Hyun and Danner’s injections (with a sampling valve) were too large, thus giving slightly incorrect retention times, and that this error was magnified by the method. This explanation is reinforced by the radioactive-tracer studies by Danner et al.,18 which replicated the static measurements. The theory of the tracer-pulse technique does not require small concentration changes. If this explanation for the discrepancy between the chromatographic and static data for the same system is correct, it raises a dilemma for experimentalists. They can use frontal chromatography and keep the concentration step under control but violate the constant


pressure condition, or they can use pulses of pure components and maintain the constant pressure constraint but then the small-concentration-perturbation condition is not met. A possible solution would be to design a frontal experiment in which the input composition is changed but the input molar flow rate remains constant. This was done in our first work2 on measuring effluent flowrate changes in frontal chromatography. We changed to experiments in which both the input molar flow rate and composition are changed8 because substantial advantages are gained in calculating the isotherm slopes. Even with constant input molar flow rate, there could still be pressure effects caused by viscosity changes.10 Another way forward would be to extend the theory of retention times in frontal chromatography to incorporate the effect of pressure changes. We know that making the perturbation in a different way gives a different retention time. If this effect were properly understood, then extra information could be extracted from a series of experiments that use different composition perturbations with the same carrier composition. We have already used this general idea to obtain partial differential coefficients of binary adsorption isotherms when the flow transients are also measured.19 It may be that the same idea would allow unambiguous fitting of two isotherm gradients from the two different retention times measured for two different perturbations. The requirements for highly precise data and meeting the assumptions of the theory may be relaxed by using additional information. It is possible to find the Henry’s law constants separately using the same apparatus as is used in measuring the retention times. Injection of a retained substance into an unretained carrier gives the Henry’s law constant for the retained substance.15 Finally, we must emphasize that attention must be paid to determining the retention time correctly. It is known2,8,15,20 that retention times can be unambiguously related to equilibrium (thermodynamic) properties, and are independent of rate properties, provided that the proper material-balance integral is used to determine the retention time. Hence, the method used to calculate the retention time from the chromatographic response should be based on a material balance. Conclusions The new chromatographic procedure to construct binary gas adsorption isotherms from chromatographic retention times gives the same results as van der Vlist and van der Meijden’s3 procedure with either their or Triebe and Tezel’s7 isotherms. The new method has the advantage over that procedure that it can be used with any isotherm equation and can incorporate other available information. Some isotherm equations may be used either with or without knowledge of the pure-component amounts adsorbed. When the new method is applied to retention times by using the Langmuir isotherm far fewer parameters are needed and the method works reasonably well. Precise experimental data are required if the isotherms are to be constructed with any degree of certainty. Acknowledgment This work was supported in part by the United Kingdom Engineering and Physical Sciences Research Council (EPSRC).

Nomenclature A: B: cA, cB: C: HA, HB: K: Kexp,i: Kmodel,i: Kw: LA, LB: n: qA, qB: Q: RAB: S 2:

parameter, Triebe and Tezel expansion or van der Vlist and van der Meijden isotherm (mol M-1) parameter, Triebe and Tezel isotherm or van der Vlist and van der Meijden isotherm (mol M-1) molar concentration of A, B in the gas phase (mol L-3) parameter, Triebe and Tezel isotherm (mol M-1) w-y form of the Henry’s law constants for the adsorption of A, B (mol M-1) binary equilibrium constant, eq 3 ith experimental value of Kw (mol M-1) value of Kw given by the model at the conditions of ith experiment (mol M-1) binary equilibrium constant, eq 8 (mol M-1) w-y form of the Langmuir constants for the adsorption of A, B number of determinations of the binary equilibrium constant molar concentration of A, B in the solid phase (mol L-3) volumetric flow rate (L3 T-1) ratio of Henry’s law constants, RAB ) HA/HB

n (Kexp,i - Kmodel,i)2 (mol2 M-2) sum of squares, ∑i)1 s: normalized root-mean-square discrepancy between the experimental and fitted values of Kw, eq 28 tg: retention time for unretained species (T) tr: retention time (T) Vg: volume of gas phase (L3) Vg: volume of solid phase (L3) wA, wB: molar amounts of A and B adsorbed per unit mass of solid (mol M-1) 0 0 molar amounts of pure A and pure B adsorbed wA, wB: per unit mass of solid at a given pressure (mol M-1) W: mass of adsorbent (M) yA, yB: mole fraction of A, B in the gas phase

Greek Symbols β: Fg: Fs: σ:

Σ2:

parameter, Triebe and Tezel expansion and isotherm molar density of the gas phase (mol L-3) density (mass per unit volume) of the adsorbent (M L-3) normalized root-mean-square discrepancy between the experimental and fitted values of Kw, eq 28 n [(Kexp,i - Kmodel,i)/Kexp,i]2 sum of squares, ∑i)1

Literature Cited (1) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley-Interscience: New York, 1984. (2) Buffham, B. A.; Mason, G.; Yadav, G. D. Retention Volumes and Retention Times in Binary Chromatography: Determination of Equilibrium Properties. J. Chem. Soc., Faraday Trans. I 1985, 81, 161-173. (3) Van der Vlist, E.; van der Meijden, J. Determination of Adsorption Isotherms of the Components of Binary Gas Mixtures by Gas Chromatography. J. Chromatogr. 1973, 79, 1-13.

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(4) Ruthven, D. M.; Kumar, R. An Experimental Study of Single Component and Binary Adsorption Equilibria by a Chromatographic Method. Ind. Eng. Chem. Fundam. 1980, 19, 27-32. (5) (a) Hyun. S. H.; Danner, R. P. Determination of Binary Gas Adsorption Equilibria by the Concentration-Pulse Technique. AIChE Symp. Ser. 1982, 78, 19-28. (b) Danner, R. P. Private communication, 1998. (6) Heslop, M. J.; Buffham, B. A.; Mason, G. A Test of the Polynomial-Fitting Method of Determining Binary-Gas-Mixture Adsorption Equilibria. Ind. Eng. Chem. Res. 1996, 35, 1456-1466. (7) Triebe, R. W.; Tezel, F. H. Adsorption of Nitrogen and Carbon Monoxide on Clinoptilite: Determination and Prediction of Pure and Binary Isotherms. Can. J. Chem. Eng. 1995, 73, 717724. (8) Mason, G.; Buffham, B. A. Gas Adsorption Isotherms from Composition and Flow-Rate Transient Times in Chromatographic Columns I. Basic Theory and Binary Experimental Test. Proc. R. Soc. London A 1996, 452, 1263-1285. (9) Kluge, G.; Franke, Th.; Scho¨llner, R. Estimation of Component Loadings in Fixed Bed Adsorption from Breakthrough Curves of Binary Gas Mixtures in Nontrace Systems. Chem. Eng. Sci. 1991, 46, 368-371. (10) Mason, G.; Buffham, B. A.; Heslop, M. J. Gas Adsorption Isotherms from Composition and Flow-Rate Transient Times in Chromatographic Columns III. Effect of Gas Viscosity Changes. Proc. R. Soc. London A 1997, 453, 1569-1592. (11) Verelst, H.; Baron, G. V. Adsorption of Oxygen, Nitrogen, and Argon on 5A Molecular Sieve. J. Chem. Eng. Data 1985, 30, 66-70. (12) Nicolletti, M. P. M.S. Thesis, Pennsylvania State University, 1976. Reported by Hyun and Danner.5

(13) Danner, R. P.; Choi, E. C. F. Mixture Adsorption Equilibria of Ethane and Ethylene on 13X Molecular Sieves. Ind. Eng. Chem. Fundam. 1978, 17, 248-253. (14) Press, W. H.; Vetterling, W. T.; Teukolsky, S. A.; Flannery, B. P. Numerical Recipes in Fortran: the Art of Scientific Computing; Cambridge University Press: Cambridge, U.K., 1992. (15) Buffham, B. A. Model-Independent Aspects of Tracer Chromatography Theory. Proc. R. Soc. London A 1973, 333, 8998. (16) Hyun. S. H.; Danner, R. P. Gas Adsorption Isotherms by Use of Perturbation Chromatography. Ind. Eng. Chem. Fundam. 1985, 24, 95-101. (17) Tezel F. H.; Tezel H. O.; Ruthven, D. M. Determination of Pure and Binary Isotherms for Nitrogen and Krypton. J. Colloid Interface Sci. 1992, 149, 197-207. (18) Danner, R. P.; Nicolletti, M. P.; Al-Ameeri, R. S. Determination of Gas Mixture Adsorption Equilibria by the Tracer Pulse Technique. Chem. Eng. Sci. 1980, 35, 2129-2133. (19) Heslop, M. J.; Mason, G.; Buffham, B. A. Measurement of the Partial Differential Coefficients of Binary Gas Adsorption Isotherms. In Fundamentals of Adsorption 6; Meunier, F., Ed.; Elsevier: Paris, France, 1998; pp 231-236. (20) Buffham, B. A. Model-Independent Aspects of Perturbation Chromatography Theory. Proc. R. Soc. London A 1978, 364, 443455.

Received for review July 1, 1998 Revised manuscript received November 30, 1998 Accepted December 18, 1998 IE980425I

Binary Adsorption Isotherms from Chromatographic Retention Times

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