1456
Ind. Eng. Chem. Res. 1996, 35, 1456-1466
A Test of the Polynomial-Fitting Method of Determining Binary-Gas-Mixture Adsorption Equilibria Mark J. Heslop,* Bryan A. Buffham, and Geoffrey Mason Department of Chemical Engineering, Loughborough University of Technology, Loughborough, Leicestershire LE11 3TU, United Kingdom
The polynomial-fitting method is used to determine the adsorption isotherms of binary gas mixtures. The method involves measuring retention times in a chromatographic column at different compositions; a polynomial is then fitted to a specific function of the retention time versus composition data. An algebraic expression can be obtained for each isotherm, with the coefficients depending on the polynomial coefficients. In this paper we examine the detailed working of the method assuming perfect experimental data is available. We use hypothetical systems that obey the binary-Langmuir theory. Starting with the hypothetical system, the set of retention times is calculated, and these are used in the polynomial-fitting method to estimate the binary isotherms. Comparison of the original and estimated isotherms allows a direct assessment of the precision of the method. The results show that the method is best suited to systems where both binary isotherms are close to linear. Introduction There are many industrial adsorption processes which separate or concentrate gas mixtures. An example is the separation of air into oxygen and nitrogen by using 5A zeolite. Design of adsorption processes requires gasmixture adsorption data in the form of multicomponent isotherms; these show how the amount of each component adsorbed depends on gas-mixture composition. These data may be acquired by using static or dynamic methods. Conventional static methods, in which the adsorbent is allowed to equilibrate with a fixed amount of gas, are best suited to single-component isotherms. Extension of these static methods to mixtures is difficult for two main reasons: the composition of the adsorbed phase is hard to determine, and the adsorption equilibrium can take a long time to establish. An alternative is to use one of the many chromatographic techniques. These are relatively simple and quick. A disturbance is made to the composition of a gas stream entering a chromatographic column, and the effluent composition is measured for a period of time. From the effluent composition history, the retention time can be evaluated. The adsorption isotherms are constructed from a set of retention time measurements. Chromatographic techniques can be divided into two categories. The first category involves adding a pulse of a radioisotope-tagged version of one of the components and is termed the tracer-pulse method. Helfferich (1964) has shown that the tracer-pulse retention time depends only on the equilibrium of the respective tracer component; complete evaluation of the system thus requires a tracer for each component, and so multiple retention times have to be determined. For the other category, the perturbation is an injection of a pulse or step of one of the pure components. A suitable detector measures the effluent composition by monitoring a physical property of the gas mixture. The retention time of the pulse is the time taken for the concentration disturbance to travel through the column. For a binary mixture, it can be shown (Peterson and Helfferich, 1965) that the * Author to whom correspondence should be addressed. Email
[email protected]. Fax: +44 1509 231746. Phone: +44 1509 263171 ext 4094.
0888-5885/96/2635-1456$12.00/0
retention time depends on both binary isotherm gradients. In an experimental situation there is only one measured retention time from which two isotherm gradients have to be determined. This is not enough information. To account for this shortfall in information, van der Vlist and van der Meijden (1973) suggested a clever method in which the isotherms are assumed to be polynomials, the coefficients of which can be found from the measured retention times at several compositions coupled with the amounts of adsorption of both pure components. This technique is termed the polynomial-fitting method. A new chromatographic method (Buffham et al., 1985; Mason and Buffham, 1991) based on the modelindependent theory of Buffham (1978) provides extra information by utilizing the effluent flow rate as well as the effluent composition. Measurement of these two variables allows both binary gradients to be calculated directly from measurements at a single composition. The method has been used by Mason and Buffham (1991, 1995) to investigate the nitrogen-argon-5A zeolite system at ambient conditions. In this paper, the theory of the new method will be used in reverse to test the polynomial-fitting method. The strategy of the test is as follows. First, the isotherms for an original, hypothetical system are defined. The equations of the modelindependent method (Buffham et al., 1985; Mason and Buffham, 1991, 1995) are then used to calculate a corresponding set of retention times for a range of compositions. These retention times are then used in the polynomial-fitting method to estimate the isotherms. The estimates are then compared with the original isotherms to see how well the polynomial-fitting method worked. The Polynomial-Fitting Method Consider a chromatographic column through which flows a binary gas mixture consisting of components A and B; the mole fractions are YA and YB, respectively. The column is at equilibrium, and the effluent composition does not vary with time. The concentrations of A and B in the gas and the solid are cA, cB and qA, qB, respectively. At time t ) 0, a disturbance is made to the equilibrium by adding a gas of different composition. © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1457
This creates a disturbance that moves through the column at velocity v. Assuming negligible axial dispersion and instantaneous mass transfer equilibrium, Peterson and Helfferich (1965) showed that the speed v of the front depends on the interstitial velocity u, the column voidage , the binary-gas composition, and the shapes of the binary equilibrium isotherms. If a binary equilibrium constant is defined as
dqA dqB K ) YB + YA dcA dcB
u 1- 1+ K
(
(2)
)
The binary equilibrium constant K depends on both binary isotherm gradients and the gas-mixture composition. Ruthven and Kumar (1980) showed that eq 2 is still valid in the presence of axial dispersion. It is usual to express the relationship in terms of the concentration retention time τR and the unretained pulse retention time τU by dividing both sides of eq 2 by the column length:
{
τR ) τU 1 +
(1 - )[Y
B
[ [
]}
dqA dqB + YA dcA dcB
B1 B2 + 2 3
(8)
qBM ) cT C0 +
C1 C2 + 2 3
(9)
Using the four A coefficients enables four other equations to be obtained by substituting eqs 5, 6, and 7 into eq 3 and then equating the coefficients for each power of YA. The resulting six simultaneous equations are solved easily because the coefficients are of a convenient form which allows manipulation of the equations. The results are
B0 ) A0
(10)
qAM + qBM - 12A0 - 6A1 - 3A2 - 2A3 cT
(11)
(
B1 ) 6
)
B2 )
( )[ ] τR -1 1 - τU
[
1 6qAM - 6A0 - 2B2 3 cT
]
(12)
(3)
Equation 3 can then be rearranged to obtain the equilibrium constant K explicitly in terms of the measurable quantities τU and τR:
K)
] ]
qAM ) cT B0 +
(1)
then the speed v of the front is given by
v)
coefficients are obtained by integrating eqs 6 and 7 across the composition range and by making the substitutions cA ) cTYA and cB ) cT(1 - YA):
C0 ) A1 + A0 - B1
(13)
C1 ) A2 + B1 - B2
(14)
C2 ) A3 + B2
(15)
(4)
If one of the components, say B, is inert, then dqB/ dcB is zero and eq 3 can be used directly to determine the gradient dqA/dcA of the single-component A isotherm. However, if both components are adsorbed, then there are two unknown gradients, but an experiment yields only one piece of information (K). Van der Vlist and van der Meijden (1973) have suggested a way of overcoming this problem. They represent the equilibrium constant K by a third-order polynomial and the isotherm gradients as second-order polynomials:
K ) A0 + A1YA + A2YA2 + A3YA3
(5)
dqA ) B0 + B1YA + B2YA2 dcA
(6)
dqB ) C0 + C1YA + C2YA2 dcB
(7)
A least-squares polynomial regression fit to experimental values of K determined for several values of YA gives the four A coefficients. However, to find the binary isotherms requires the determination of the six unknown B and C coefficients, and so two extra pieces of information are required. These are the pure-component amounts adsorbed at the total system pressure when the gaseous concentration is cT, namely qAM and qBM. The relations between these and the B and C
The isotherms for each component are obtained by integrating eqs 6 and 7:
[
qA ) B0YA + qB )
]
B1YA2 B2YA3 + cT 2 3
[
(16)
]
C1(1 - YA2) C2(1 - YA3) + C0(1 - YA) + cT (17) 2 3
Obviously after such tortured, although elegant, analysis, the resulting two isotherms may be subject to systematic errors, such as an inadequate fit to the original K values or the use of improper values for qAM and qBM, and to the usual random errors. In this paper we are concerned with the goodness of fit of the polynomial. We define a variance
σK2 )
1
ND
∑
ND i)1
[
]
Ki,fit - Ki Ki
2
(18)
to represent the discrepancy between the true and fitted values of K over the whole composition range. Here ND is the number of data points and Ki,fit is the value of Ki given by the best-fitting polynomial at the ith composition. The value of σK2 allows us to judge how well the polynomial fits the set of K values; the smaller the variance the better the fit. Similarly, we need to
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measure the quality of each estimated binary isotherm and the variances 2
σA )
σB2 )
1
[ ∑[ NV
∑
NV i)1 1
NV
NV i)1
] ]
qA,est,i - qA,i qA,i
qB,est,i - qB,i qB,i
2
(19)
2
(20)
will give measures of how close the estimated isotherm is to the original isotherm. Here, NV is the number of compositions for which there are original (actual) values of qA and qB. Extension of the Polynomial-Fitting Method to Fourth-Order Fitting A third-order polynomial might not give a satisfactory fit to a set of K values, and a fourth-order polynomial might be considered. However, for a fourth-order polynomial, eqs 10-15 are no longer valid, and we require an alternative analysis to use the best-fitting polynomial coefficients (A values) to obtain equations for the recalculated isotherms (B and C values). This analysis can take two forms. For one form, Hyun and Danner (1982) suggested adding an extra term (either B3YA3 or C3YA3) to one of the polynomials representing the isotherm gradients. In this case, five equations are obtained by equating coefficients for each power of YA, and the seven unknown B and C coefficients are evaluated in a similar manner to that above. The fourth-order analysis of van der Vlist and van der Meijden (1973) is to add an extra term to both polynomials representing the mixture isotherms gradients (B3YA3 and C3YA3), but then there are eight unknown B and C coefficients and only seven equations. For this case, an extra piece of independent information is required at a mixture composition YAI (I is for independent). This can be either an amount adsorbed of one of the individual components (qAI or qBI) or a total amount adsorbed (qTI). The latter has the advantage that the total amount can be obtained with the same apparatus used for the single-component isotherms; it is not necessary to evaluate the unknown adsorbed-phase composition. For the total amount case, the eighth equation is written:
[
B1YAI2 B2YAI3 B3YAI4 + + + 2 3 4 C1(1 - YAI2) C2(1 - YAI3) C0(1 - YAI) + + + 2 3 C3(1 - YAI4) cT (21) 4
qTI ) B0YAI +
]
For this case, the eight simultaneous equations are solved by Gaussian elimination. Clearly, by determining the total adsorption at more compositions, even higher order polynomials can be used. The Use of Retention Volumes in the Polynomial-Fitting Method So far, all the discussion has been concerned with the K value, because it is dimensionless and there is no need
to specify any units. However, rather than use the K value, some authors have chosen to use the net retention volume. Before defining the retention volume, it is helpful to make a distinction between standard conditions (subscripted 0) and the column conditions (subscripted T). The total gas concentrations are c0 and cT, respectively, and are obtained from a suitable equation of state. Because the molar flow rate of the carrier is the same everywhere, the volumetric flow rates at standard conditions Q0 and column conditions QT are related by the continuity equation:
QTcT ) Q0c0
(22)
The net retention volume at standard conditions, (VR - VG)0, is defined as the product of the volumetric flow rate at standard conditions, Q0, and the net retention time (τR - τU)
(VR - VG)0 ) Q0(τR - τU)
(23)
If VG is the actual void space in the column and VS is the volume of the column packing, then the net retention volume at standard conditions
(VR - VG)0 ) VS
[
]
cT dqA dqB cT YB + YA ≡ VS K c0 dcA dcB c0
(24)
is obtained by multiplying both sides of eq 3 by QT and making the substitutions VS ) VG(1 - )/, VG ) τUQT, and the continuity relation Q0 ) QTcT/c0. Hyun and Danner (1982) have stated eq 24 using the amounts adsorbed of each component (wA and wB) rather than the concentrations adsorbed, and the gas-phase mole fractions rather than the gas-phase concentrations. This can be achieved by making the substitutions dcA ) cT dYA, dcB ) cT dYB, dwA ) VS dqA, and dwB ) VS dqB:
(VR - VG)0 )
[
]
dwA dwB 1 YB + YA c0 dYA dYB
(25)
To use the polynomial-fitting method for retention volumes, the procedure is similar to that for K values; the product of net retention volume and gas-phase concentration c0 is represented as a third-order polynomial and the isotherm gradients as second-order polynomials:
(VR - VG)0c0 ) A0 + A1YA + A2YA2 + A3YA3
(26)
dwA ) B0 + B1YA + B2YA2 dYA
(27)
dwB ) C0 + C1YA + C2YA2 dYB
(28)
Alternatively, it is possible to fit the net retention volume alone and multiply the fitting coefficients by c0. As distinct from the procedure for K values, the A coefficients are not dimensionless; they have the units of the product of retention volume and gas-phase concentration. The pure-component amounts adsorbed,
Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1459
namely wAM and wBM, are obtained by integrating eqs 27 and 28 across the composition range:
wAM ) B0 +
B1 B2 + 2 3
(29)
wBM ) C0 +
C1 C2 + 2 3
(30)
Using the four A coefficients, and the pure-component amounts adsorbed in eqs 29 and 30, the six unknown B and C coefficients are obtained in a similar manner to that outlined in eqs 10-15 for the K values. It is helpful to define a variance
σRV2 )
1
ND
∑
ND i)1
[
]
(VR - VG)0,i,fit - (VR - VG)0,i (VR - VG)0,i
2
(31)
to represent the discrepancy between the true and fitted retention volumes across the composition range. By using eq 24 to substitute for the net retention volume, it is seen that σRV2 ) σK2 which means that it is possible to make numerical comparisons between retention volume and K-value data.
Figure 1. Experimental net retention volumes at standard conditions and the best-fitting third-order and fourth-order polynomials for the ethane-ethylene-13X zeolite system investigated by Hyun and Danner (1982). The retention volumes were not measured at the two end points of the composition range. The polynomial is shown extended to the end points of the composition range.
Previous Work on the Polynomial-Fitting Method Van der Vlist and van der Meijden (1973) not only suggested the polynomial-fitting method but were the first to use it. They obtained single-component and binary isotherms for nitrogen-oxygen mixtures on 5A zeolite, but they made no comparison with static data. Ruthven and Kumar (1979, 1980) investigated many mixtures of gases on 5A and 4A zeolites but only made comparisons with the static data for the single-component isotherms. The most complete comparison has been made by Hyun and Danner (1982) who studied three systems: carbon monoxide-methane-activated carbon, isobutane-ethylene-13X zeolite, and ethane-ethylene-13X zeolite. For the first and second systems, they calculated net retention volumes from the static data using eq 25. These retention volumes were then used in the polynomial-fitting method outlined in eqs 26-30 to regenerate the static data. The estimated isotherms for the first system were excellent, and those for the second system were good. For the third system, however, the net retention volumes were calculated directly from chromatographic data using eq 23. Figure 1 shows the net retention volumes of Hyun and Danner (1982), along with the best-fitting third-order polynomial. It can be seen that the polynomial cannot cope with the variation in retention volumes. Figure 2 shows the consequence of this and just how badly the estimated isotherms calculated from the polynomial-fitting method compare with the static data of Danner and Choi (1978). Hyun and Danner (1985) later investigated the isobutane-ethylene-13X zeolite system at the same conditions (298 K and 138 kPa). They made several comparisons using this system, and it is important to be clear about what they did and what they concluded. Originally, Hyun (1980) measured the isotherms for this system using a static method. In 1982, Hyun and Danner used these isotherms to predict a set of retention volumes using eq 25. They then used these net retention volumes in the polynomial-fitting method to regenerate their original isotherms and made a com-
Figure 2. Static data of Danner and Choi (1978) and estimations obtained by Hyun and Danner (1982) using the polynomial-fitting method for the ethane-ethylene-13X zeolite system. It can be seen that the ethylene isotherm is underpredicted across the whole composition range. The ethane isotherm is mainly overpredicted, although the agreement is fairly good in the composition range below 40% ethane.
parison. They found good agreement. In 1985, Hyun and Danner reported new chromatographic measurements of the isobutane-ethylene-13X zeolite system and the results of using experimental retention volumes in the polynomial-fitting method. Conversely, they found poor agreement between the isotherms from the static method and those from the polynomial-fitting method. They concluded that the polynomial-fitting method worked well only for hypothetical retention volumes (or K values) but not for experimental data. We will return to this conclusion later. Shah (1988) investigated three binary systems on 5A zeolite: oxygen-nitrogen, ethane-methane, and cyclopropane-propane. Comparisons were not made with static data but with a statistical predictive model. The estimated isotherms for the more-adsorbed component exhibited kinking toward the high mole fraction region and adsorption of the less-adsorbed component was overestimated across the whole composition range. These findings are similar to those of Hyun and Danner (1982). Tezel et al. (1993) obtained single-component and binary isotherms for krypton-nitrogen mixtures on H-mordenite and silicalite zeolites at two different temperatures. For both adsorbents, the third-order polynomial was only able to fit the K data at the higher temperature. Comparisons were not made with static data but with two predictive methods. At the lower temperature, the estimated isotherms for the more-
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adsorbed component exhibited the characteristic kinking toward the high mole fraction region. Model-Independent Chromatographic Method This experimental method (Buffham et al., 1985; Mason and Buffham, 1991, 1995) is novel in that it involves measuring the column outlet flow rate as well as the composition. This extra measurement allows both mixture isotherm gradients to be evaluated directly. The analysis involves a molar balance on a chromatographic column of volume VC and voidage . We will not describe the method in detail, but the end result is an equation for the composition retention time τX involving two correction factors FPVCA and FPVCB:
{
[
τX ) τU FPVCAYB 1 +
(1 - ) dc ] + 1 - dq F Y [1 + ( (32) ) dc ]} dqA A
B
PVCB
A
B
For a negligible column pressure drop, both correction factors FPVCA and FPVCB tend to unity and eq 32 simplifies to the form of eq 3 (Buffham et al., 1985):
{
τX ) τU 1 +
(1 - )[Y
B
]}
dqA dqB + YA dcA dcB
(33)
For the rest of this paper, the column pressure drop will be considered negligible. We will show in a separate paper the effects of a finite column pressure drop. The fact that the form of eq 33 is the same as eq 3 suggests that, for a negligible column pressure drop, τX is equivalent to τR. However, τR is derived assuming negligible axial dispersion and interphase mass-transfer resistance in the column, whereas τX is valid for any column. The unretained pulse time τU is dependent on the column volume VC and on any variation of gas flow rate with the composition (caused by the variation in gas-mixture viscosity). But for any binary system with both binary gradients defined, it is possible to calculate values of τX/τU across the whole composition range. These values are independent of the column volume and the carrier flow rate as well as the mass-transfer resistance and axial dispersion in the column.
Figure 3. Binary-Langmuir isotherms, for the first and second systems defined in Table 1, obtained using eqs 36 and 37. For each system, the total adsorbed-phase concentration is obtained from the sum of eqs 36 and 37. The adsorbed-phase concentration units are moles per unit volume. Table 1. Parameter Values for the Binary-Langmuir Systems mol/unit volume
1st system 2nd system
qA )
a1cA 1 + a2cA
(34)
qB )
b1cB 1 + b2cB
(35)
The two pairs of parameters (a1, a2) and (b1, b2) are usually deduced by nonlinear regression of the singlecomponent experimental data of components A and B, respectively. Markham and Benton (1931) extended the Langmuir theory to the simultaneous adsorption of multicomponent mixtures by considering the competi-
(mol/unit volume)-1
qAM
qBM
a1
b1
a2
b2
800 800
400 400
13.33 40
5 5
0.0067 0.04
0.0025 0.0025
tion between each molecular species for a fixed number of sites. For a binary mixture the adsorbed-phase concentration depends on both gas-phase concentrations as follows:
qA )
a1cA 1 + a2cA + b2cB
(36)
qB )
b1cB 1 + a2cA + b2cB
(37)
For a fixed, low system pressure the gas molar density cT, which is equal to cA + cB, is necessarily constant. We shall be calculating several isotherms, and for the sake of convenience cT will be taken as 100 mol per unit volume. Equations can be written for the isotherms and the isotherm gradients in terms of the respective component concentrations:
qA )
Application of the Polynomial-Fitting Method to Binary-Langmuir Systems The Langmuir isotherm is a commonly used representation for many single-component systems. It is derived by assuming that the adsorbent consists of a fixed number of adsorption sites. It is a two-parameter isotherm. For components A and B, respectively
dimensionless
qB )
a1cA 1 + 100b2 + (a2 - b2)cA b1cB 1 + 100a2 + (b2 - a2)cB
(38)
(39)
dqA a1(1 + 100b2) ) dcA [1 + 100b + (a - b )c ]2 2 2 2 A
(40)
b1(1 + 100a2) dqB ) dcB [1 + 100a + (b - a )c ]2 2 2 2 B
(41)
We also need to choose typical values of the parameters a1, a2, b1, and b2. Values which give realistic isotherms are shown in Table 1. The first system has an almost linear pair of isotherms; the second system is nonlinear. The binary-Langmuir isotherms for both systems and the total amounts adsorbed are shown in Figure 3. For both systems, K values were calculated by differentiating eqs 36 and 37 and by using eq 1 for composition intervals of 5%. Polynomials were then
Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1461
Figure 4. Set of original K values at a composition interval of 5% for the first binary-Langmuir system (coefficients defined in Table 1), along with the best-fitting third-order polynomial. The polynomial coefficients are given in Table 2. It can be seen that the third-order polynomial gives an excellent fit of the K values. Table 2. Best-Fitting Polynomial Coefficients for the Binary-Langmuir Systems order 1st system 2nd system 2nd system
3rd 3rd 4th
A0
A1
A2
A3
A4
13.7 29.7 31.2
-24.7 -119.7 -158.0
20.6 175.4 358.0
-7.3 -85.5 -374.0
144.2
fitted to the K values (of which there were 21 for each system). For the second system, a fourth-order polynomial was fitted as well as a third-order polynomial. The polynomial coefficients are shown in Table 2. For the third-order polynomial cases, the best-fitting A coefficients were then used in eqs 10-15 to obtain the B and C coefficients of the binary isotherm gradients. These B and C coefficients are shown in Table 3. For the second system, the fourth-order coefficients are also given. Details of the first system are shown in Figures 3-5. Figure 3 shows the original binary-Langmuir isotherms, according to the parameters in Table 1. For the binaryLangmuir isotherms, one isotherm is concave and the other is convex to the composition axis. This type of system is reported frequently in the literature. Figure 4 shows the calculated K values at a composition interval of 5% together with the best-fitting third-order polynomial. It can be seen that the third-order polynomial gives a very good fit of the set of K values; the fitting coefficients are shown in Table 2. Figure 5 shows that the estimated isotherms, calculated from the B and C coefficients in Table 3, are virtually identical to the original values. Overall, the polynomial-fitting method works very well for this system. This conclusion confirms that of Hyun and Danner (1982). They used some actual isotherms obtained by Wilson (1980) for carbon monoxide-methane-activated carbon instead of our binary-Langmuir isotherms, calculated the net retention volumes using eq 25, and regenerated the original isotherms using the polynomial-fitting method. The results for the second system are shown in Figures 6-8. Figure 3 shows the binary-Langmuir isotherms given by the parameters in Table 1. Figure 6 shows the calculated K values at a composition interval of 5% together with the best-fitting third- and fourth-order polynomials. It can be seen that the fourth-
Figure 5. Original binary isotherms and estimated lines for the first binary-Langmuir system defined in Table 1 using the coefficients obtained from the third-order polynomial in Figure 4. The coefficients used to plot the estimated lines are given in Table 3. It can be seen that the estimations for both isotherms are excellent. The adsorbed-phase concentration units are moles per unit volume.
Figure 6. Set of original K values at a composition interval of 5% for the second binary-Langmuir system defined in Table 1, along with the best-fitting third-order and fourth-order polynomials. The coefficients of these polynomials are given in Table 2. It can be seen that the fourth-order polynomial gives a much improved fit of the set of K values.
order polynomial gives a better fit of the K values than the third-order polynomial. The estimated lines shown in Figure 7 were obtained using the third-order B and C coefficients in Table 3. They are compared with the original binary-Langmuir isotherms. The estimated lines compare badly with the original isotherms; component B is underestimated for most of the composition range, and component A shows the characteristic underestimation in the high mole fraction region. In order to use the fourth-order polynomial coefficients, we need the total adsorption at an independent mixture point as well as the two single-component amounts adsorbed. A total amount constraint was chosen at a mixture composition of 80% component A and calculated from the binary-Langmuir isotherms, namely eqs 38 and 39. Figure 8 shows lines for the estimated isotherms obtained using the fourth-order B and C coefficients in Table 3. It can be seen that the original and estimated total amounts agree at the 80% A composition (this is a check; they have to). Overall, a substantial improvement in the isotherms is obtained by using the fourthorder polynomial coefficients and the extra “experimental” point at a composition of 80% component A; the
Table 3. Estimated Isotherm Coefficients for the Binary-Langmuir Systems order 1st system 2nd system 2nd system
3rd 3rd 4th
B0
B1
B2
13.7 29.7 31.2
-17.3 -95.6 -136.2
9 78.1 228.6
B3
C0
C1
C2
C3
-125.4
6.3 5.6 9.3
-5.7 1.7 -6.8
1.7 -7.4 -20.0
18.8
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Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 Table 5. Polynomial Coefficients and Variances for Second Binary-Langmuir System
Figure 7. Original binary-Langmuir isotherms and estimated lines for the second binary-Langmuir system defined in Table 1 using the coefficients obtained from the third-order polynomial in Figure 6. The coefficients of this polynomial are given in Table 2. The coefficients required to plot the estimated lines are shown in Table 3. It can be seen that the component B isotherm is overpredicted for most of the composition range and there is a significant kinking in the component A isotherm in the high mole fraction region. The adsorbed-phase concentration units are moles per unit volume.
Figure 8. Original binary isotherms and estimated lines for the second binary-Langmuir system defined in Table 1 using the coefficients obtained from the fourth-order polynomial of Figure 6 and the original, total amount adsorbed at a composition of 80% component A. The coefficients required to plot the estimated lines are given in Table 3. It can be seen that the estimations for both components are much better than those of Figure 7. The original and estimated total amounts adsorbed agree at a composition of 80% component A: this is a constraint that the necessary constraint has been incorporated in the solution of the polynomialfitting equations. The adsorbed-phase concentration units are moles per unit volume. Table 4. Variances and Quality of Predictions for Binary-Langmuir Systems 1st system 2nd system 2nd system
order
ND
σK2
NV
σA2
σB2
3rd 3rd 4th
21 21 21
0.00004 0.102 0.017
21 21 21
0.00003 0.0055 0.0005
0.0002 0.201 0.026
latter fixes agreement at this composition. By the same reasoning, employment of a fifth-order polynomial would require two extra experimental points; the isotherms should be further improved by the enforced agreement at two intermediate compositions. Table 4 gives the polynomial-fitting and estimated isotherm variances for the two binary-Langmuir systems shown in Figures 3-8. Not surprisingly, the smaller the fitting variance σK2, the smaller the isotherm error variances σA2 and σB2 and the better the quality of the estimated isotherms. The variances for component A are smaller than those for component B because qA is generally larger than qB; from eqs 19 and 20 the variances are normalized. Overall, it would
order
A0
A1
A2
A3
ND
σK2
NV
σA2
σB2
3rd 3rd 3rd 3rd 3rd
32.0 31.8 31.6 30.6 29.7
-124.6 -129.0 -129.5 -125.1 -119.7
174.1 186.0 189.0 184.3 175.4
-80.0 -88 -90.4 -89.6 -83.5
4 5 6 11 21
0 0.047 0.092 0.130 0.102
21 21 21 21 21
0.0093 0.0075 0.0073 0.0063 0.0055
0.415 0.321 0.297 0.230 0.201
appear that the estimations for the isotherms are very good when the error variances σA2 and σB2 are less than 0.0001. For the second system using the third-order coefficients, the variance σB2 is clearly very large (0.201). By using the fourth-order coefficients and the constraint at 80% A, there is a clear improvement in the estimations of both isotherms; both error variances are reduced by a factor of 10. Effect of the Number of K Values on the Quality of the Estimated Isotherms For the binary-Langmuir systems defined in Table 1, the estimated isotherms were obtained using 21 K values in the initial regression; this corresponds to a composition interval of 5%. From the literature, it is rare to use as many as 21 experimental K values, and so it is instructive to know the effect of the number (ND) of initial K values on the quality of the estimated isotherms. Heslop et al. (1995) investigated the second (difficult) binary-Langmuir system of Table 1 by obtaining the estimated isotherms for ND values of 21, 11, 6, and 4. This investigation has been extended to include an intermediate value of 5. To prevent ambiguity, the K values were evenly spaced over the composition range and included both end points. For example, for ND ) 6, the K values were obtained at compositions of 0%, 20%, 40%, 60%, 80%, and 100% A. Table 5 shows the best-fitting third-order polynomial coefficients (A values) and the variances for the K values as well as for each estimated isotherm. These values of A are used in eqs 10-15 to obtain the values of B and C and hence the estimated isotherms. It can be seen that the values of A and hence the estimated isotherms depend only slightly on the number of evenly spaced K values used in the initial regression. This is confirmed by the variation in the variances σA2 and σB2. The quality of both estimated isotherms increases steadily as ND is increased from 4 to 21; however, the error variances are only reduced by a factor of 2. Heslop et al. (1995) concluded that for difficult systems, increasing the number of K values (experimental points) does not greatly improve the quality of the estimated isotherms. Making ND ) 4 is hardly worse than making ND ) 21. The variation of σK2 with ND is interesting. As ND is reduced from 21 to 5, σK2 is reduced by a factor of 2, but the overall estimations (sum of σA2 and σB2) actually worsen. For ND ) 4, σK2 ) 0 because a third-order polynomial can be fitted exactly to four points. These cases show the importance of the number of values ND as well as the fitting variance σK2. Effect of Removing End-Point K Values on the Quality of the Estimated Isotherms For the binary-Langmuir systems in the previous two sections, the estimated isotherms were obtained using various numbers (ND) of K values evenly spaced across the composition range starting at 0% A and ending at 100% A. It was shown that the quality of the estimated
Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1463
Figure 9. Set of original K values for the second binary-Langmuir system in the composition range from 10% to 95% A and the bestfitting third-order polynomial. This is similar to Figure 6 except that the polynomial is obtained omitting K values at the compositions of 0%, 5%, and 100% A. The polynomial is shown extended outside its fitting range to the end points of the composition range. The third-order polynomials of Figures 6 and 9 are significantly different; in the former the polynomial intersects the K-axis around 30, and in the latter the intersection is around 25. The coefficients of the third-order polynomials in Figures 6 and 9 are given in Table 6. Table 6. Polynomial Coefficients and Variances for Second Binary-Langmuir System order 3rd 3rd
A0
A1
A2
A3
29.7 -119.7 175.4 -85.5 25.1 -88.6 118.4 -54.8
ND
σK2
NV
21 18
0.102 0.010
21 21
σA2
σB2
0.0055 0.201 0.0074 0.090
isotherms was roughly independent of ND as long as the K value at each end point of the composition range was included in the regression to find the best-fitting polynomial. In this section, we will repeat the polynomial-fitting method for the second (difficult) binaryLangmuir system using ND ) 21 but omitting the K values at 0%, 5%, and 100% A. The reason for this omission is that for the data shown in Figure 1 for the ethane-ethylene-13X system, Hyun and Danner (1982) carried out the polynomial regression without the retention volumes at these compositions. Figure 9 shows the 18 K values and the best-fitting third-order polynomial. The polynomial is shown extended outside its fitting range to the end points of the composition range. In comparison with the third-order polynomial of Figure 6, the polynomial in Figure 9 is different because it intersects the K-axis at a different position. Table 6 shows the best-fitting polynomial coefficients for ND values of 18 and 21. On comparison of Tables 5 and 6, it can be seen that the polynomial coefficients (and hence the estimated isotherms) are much more affected by removing the end-point K values than by reducing ND from 21 to 4. It can be seen that reducing ND from 21 to 18, the value of σK2 is reduced by a factor of 10, showing that the polynomial gives a much better fit of the 18 K values. Figure 10 shows the original and estimated isotherms obtained using 18 K values in the polynomial regression. On comparison of Figure 10 with Figure 7, in the former it can be seen that the original and estimated isotherms for component A do not intersect, while the estimated isotherm for component B shows a distinct improvement. Table 6 shows that the estimated isotherm for component A is slightly worse (σA2 is increased from 0.0055 to 0.0074) and confirms that the estimated isotherm for component B shows a greater improvement (σB2 is reduced from 0.201 to 0.090). Surprisingly, the overall error (sum of σA2 and σB2) is reduced by reducing ND from 21 to 18. So, for the
Figure 10. Original binary-Langmuir isotherms and estimated lines for the second system defined in Table 1 using the coefficients obtained from the polynomial in Figure 9. In a comparison of Figures 10 and 7, in the former the original isotherm and estimated line for component A does not intersect, while the estimated line for component B shows a distinct improvement. The adsorbed-phase concentration units are moles per unit volume.
difficult ethane-ethylene-13X zeolite system investigated by Hyun and Danner (1982) for which there was poor agreement between the isotherms obtained by the static method and the polynomial-fitting method, the omission of the retention volumes at the extremes of the composition range is actually the best procedure. However, even with this modification, there is still poor agreement. Application of the Polynomial-Fitting Method to Experimental Data In the previous three sections, the results for the binary-Langmuir systems were obtained using a computer program which calculated exact binary isotherm gradients and K values from specified isotherms. This set of K values was then used in the polynomial-fitting method to try to regenerate the specified isotherms. Comparison of the original and estimated isotherms provided a direct test on how well the method worked. In this section, however, the program has been modified to allow the input of experimental retention volumes as well the binary isotherms determined by an independent (usually static) method. Similarly to the previous three sections, the retention volumes are used in the polynomial-fitting method to try to regenerate the static data. The problem with using experimental retention volumes and static data is that there is no check on the quality of the data. For example, if the retention volumes are exact and a third-order polynomial gives an exact fit, but there is error in the static data, comparisons between the predicted and “exact” isotherms will indicate that the method does not work for the particular system. The modified program has been applied to the ethaneethylene-13X zeolite system at 298 K and 138 kPa investigated by Hyun and Danner in 1982. Figure 1 shows the experimental retention volumes at standard conditions using the chromatographic method, namely eq 23. Hyun and Danner (1982) did not state the polynomial coefficients, so we have fitted both thirdorder and fourth-order polynomials, the coefficients of which are given in Table 7. To proceed with the method, as indicated in eq 26, a value for the total gas concentration at standard conditions c0 is required; from the ideal gas equation c0 ) 44.6 mol/m3. Finally, from eqs 29 and 30, the pure-component amounts adsorbed at 298 K and 138 kPa are required: wethane ) 2.185 mol/
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error variances have been reduced using the fourthorder procedure; the reduction for ethane is only marginal because the error has been shifted toward the lower values of the isotherm. Any estimations for this system using higher order polynomials are limited because there are no retention volumes above 95% ethane. Although the fourth-order polynomial gives a better fit of the retention volumes, the extrapolated value at 100% ethane is not large enough; from Figure 11 the gradient of the estimated isotherm for ethylene is clearly too small. With this limitation, it is difficult to obtain a significant improvement in the estimated isotherms using a fourth-order polynomial and a fixing constraint.
Figure 11. Static data obtained by Danner and Choi (1978) for the ethane-ethylene-13X zeolite system and estimations of the polynomial-fitting method using the fourth-order coefficients and a fixing constraint. As indicated, the constraint is the agreement of the estimated isotherm for ethane with the static data at a composition of 71.7% ethane. In comparison with Figure 2, both estimated isotherms have been improved.
Discussion The results for the second binary-Langmuir system shown in Figures 6 and 7 are similar in character to those for the ethane-ethylene-13X zeolite system shown in Figures 1 and 2. The sets of retention volumes and K values show the same type of variation, a steep descent followed by a plateau, and for each system, the third-order polynomial cannot cope with this type of variation. For each system, the adsorption of the leastadsorbed component (component B and ethane) is overestimated for most of the composition range. For the more-adsorbed component (component A and ethylene), there is the characteristic underestimated kinking in the high mole fraction region. From Tables 4 and 9, the estimated isotherm for ethane (σethane2 ) 0.0596) is better than that for component B (σB2 ) 0.201). Conversely, the estimated isotherm for ethylene (σethylene2 ) 0.1504) is much worse than that for component A (σA2 ) 0.0055). This paradox is explained by a fundamental difference between the ethylene and component A isotherms in Figures 2 and 7, respectively; the original and estimated isotherms for the latter intersect, but those for the former do not. This is because the net retention volumes at either end of the composition range were not included in the polynomial regression. Indeed, this “nonintersecting” behavior can be repeated for the second binary-Langmuir system in Figure 10 by omitting the K values at the ends of the composition range, as shown in Figure 9. From Table 6, removing the K values at the end points gives a much improved fit of the remaining data (σK2 is reduced by a factor of 10); the quality of the estimated isotherm for component A is slightly reduced, and there is a large improvement in the estimations for component B. Hence, this would suggest that if all the retention volumes had been included for the ethane-ethylene-13X zeolite system, the value of σRV2 would be much higher than 0.0289; the estimations for ethylene would show an improvement (σethylene2 would be smaller than 0.1504), and the estimations for ethane would be worse (σethylene2 would be greater than 0.0596). Overall, the estimations of the polynomial-fitting method for the ethane-ethylene13X zeolite system are worse than those for the second binary-Langmuir system shown in Figure 7 because the former system is more curved. It is possible to obtain a binary-Langmuir system for which the polynomialfitting method gives even poorer estimations by increas-
Table 7. Best-Fitting Polynomial Coefficients for Experimental Data of Hyun and Danner (1982) order
A0
A1
A2
A3
A4
3rd 4th
2.3 19.6
194.6 -147.8
-614.4 978.6
615.3 -1948
1326
kg and wethylene ) 2.765 mol/kg. There is now enough information to calculate the third-order isotherm coefficients for ethane and ethylene; the three B and three C values are given in Table 8. Figure 2 shows the binary isotherms determined independently by Danner and Choi (1978), along with the estimated isotherms from the polynomial-fitting method using these B and C coefficients. It can be seen that the ethylene isotherm is underestimated for the entire composition range; the estimated isotherm for ethane, though, is much better. These trends are confirmed by Table 9, which shows that the error variance for ethylene is about three times that of ethane. Table 9 shows that a fourth-order polynomial gives a much improved fit of the net retention volumes; σRV2 is reduced by a factor of 3, and the polynomial coefficients are given in Table 7. In order to use these fourth-order coefficients in the polynomial-fitting method, it is necessary to specify an extra piece of information, namely a mixture point from the static data of Danner and Choi (1978). This fixing constraint enables us to specify an extra term for both the ethane and ethylene isotherms. Hyun and Danner (1982) also did a fourth-order fitting but without this constraint; they could specify an extra term for only one of the binary isotherms. We have obtained estimated isotherms for the fourth-order coefficients using the following constraint: the amount of ethane adsorbed in a 71.7% ethane mixture (0.688 mol/ kg). Table 8 shows the estimated isotherm coefficients for this constraint. Figure 11 shows the estimated isotherms obtained using the fourth-order polynomial coefficients and the ethane isotherm constraint. This constraint is confirmed because the estimated isotherm for ethane passes through the static data at this composition. Comparing with Figure 2, it can be seen that there is a noticeable improvement in the estimated isotherms for both components. From Table 9, both
Table 8. Estimated Isotherm Coefficients for Experimental Data of Hyun and Danner (1982) order
constraint
B0
B1
B2
3rd 4th
none [ethane]@71.7%
0.103 0.875
1.712 -6.11
3.68 13.8
B3
C0
C1
C2
C3
-0.866
7.08 0.388
-29.4 23.8
31.2 -72.4
58.3
Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996 1465 Table 9. Variances and Quality of Predictions for Experimental Data of Hyun and Danner (1982) order
constraint
ND
σRV2
NV
σethylene2
σethane2
3rd 4th
none [ethane]@71.7%
22 22
0.0289 0.0097
14 14
0.1504 0.1102
0.0596 0.0468
ing the curvature of the isotherms. This can be achieved by increasing the values of a1 and a2 in eqs 36 and 37. Figure 3 shows that the binary isotherms are less curved for the first system, and Figure 4 shows that the third-order polynomial gives an excellent fit of the set of K values. Consequently, Figure 5 shows that the estimated isotherms are virtually identical to the original isotherms. The polynomial method suits systems with both isotherms close to linear since the corresponding set of K values is close to linear and can be easily fitted with a third-order polynomial. Conversely, the method is not well suited to systems with highly curved isotherms. The results from the simulation suggest that the estimations for these difficult systems can be improved by using a fourth-order polynomial and an extra independent mixture point. A point of interest concerns the isobutane-ethylene13X zeolite system investigated by Hyun and Danner (1982, 1985) using two different methods. Their first investigation involved taking the original static data, calculating the binary isotherm gradients graphically and using eq 25 to calculate the retention volumes. A third-order polynomial was then fitted to the retention volume-composition data and the best-fitting polynomial coefficients used in the polynomial-fitting method to regenerate the original isotherms. This is essentially the same method as that used in this paper, although in the present work the method calculates the original isotherms directly and applies irrespective of dispersion and mass-transfer resistance because of the definition of τX. Hyun and Danner’s (1982) estimated isotherms compared well with the static data, and their conclusion was that their particular binary system was well suited to the polynomial-fitting method. The second investigation was chromatographic and involved measuring τU and τR for each gas-mixture composition and using eq 23 to determine a set of retention volumes which could be used with the polynomial-fitting method. However, the experimental set of retention volumes was significantly different from those obtained using the static data, and a third-order polynomial was unable to give a good fit; the subsequent predictions compared badly with the original isotherms. Hyun and Danner (1985) suggest that the reason for this disparity is that the polynomial-fitting method works only for hypothetical retention volumes and not for experimental data. An alternative explanation is that the experimental results for the two methods are inconsistent; one set may be wrong. For example, if the retention volumes taken from the Figure 3 of their 1982 paper are compared with the experimental retention volumes taken from the Figure 4 of their 1985 paper (see our Figure 12), it can be seen that the two sets are incompatible. The inconsistency may be because there is more error in using eq 23 to process the experimental measurements than in using eq 25 to calculate the retention volumes from the static data. For eq 23, there is error due to measuring both retention times τU and τR. These errors are exaggerated because the column length is comparatively short, giving small values of the retention times.
Figure 12. Comparison of the net retention volumes at standard conditions, obtained by Hyun and Danner in 1982 and 1985, for the isobutane-ethylene-13X zeolite system at 298 K and 138 kPa. In the 1982 investigation, the retention volumes were obtained from the static data using eq 25. In the 1985 investigation, the retention volumes were obtained from chromatographic experiments using eq 23.
Conclusions The polynomial-fitting method for determining binarygas-mixture adsorption isotherms gives better estimations the straighter the isotherms. The quality of the estimations depends critically upon how well a thirdorder polynomial fits the set of K values or retention volumes. From our worked examples using 21 points in the initial regression, we conclude that a σK2 (σRV2) variance of less than 0.0001 will give excellent results and a σK2 (σRV2) variance of more than 0.01 will give poor predictions. For systems in which both isotherms are close to linear, the variation in K value (or retention volume) with composition is also close to linear and the third-order polynomial provides a good fit. Problematic binary systems can be characterized as having a highly curved convex and concave isotherm. For this case, the K-value variation consists of a steep ascent followed by a plateau and the third-order polynomial cannot cope with this type of variation. For these problematic systems, it has been shown that by using a fourth-order polynomial the predictions are greatly improved, although an independent mixture point is required. Acknowledgment This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) of the U.K. Nomenclature A0, A1, A2, A3, A4 ) best-fitting polynomial coefficients a1 ) first Langmuir coefficient for component A a2 ) second Langmuir coefficient for component A (mol/ unit volume)-1 B0, B1, B2, B3 ) coefficients for estimated isotherm of component A b1 ) first Langmuir coefficient for component B b2 ) second Langmuir coefficient for component B (mol/ unit volume)-1 C0, C1, C2, C3 ) coefficients for estimated isotherm of component B c0 ) total gas-phase concentration at standard conditions cj ) gas-phase concentration of component j in the column cT ) total gas-phase concentration in the column FPVCj ) pressure and viscosity correction factor for component j in the column Ki ) binary equilibrium constant at composition i Ki,fit ) fitted binary equilibrium constant at composition i
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ND ) number of K values or net retention volumes used in the polynomial regression NV ) number of compositions for which there are original values of qA, qB or wA, wB Q0 ) average volumetric flow rate at standard conditions QT ) average volumetric flow rate through the column qj,est,i ) estimated adsorbed phase concentration of component j in the column at composition i (mol/unit volume) qj,i ) adsorbed-phase concentration of component j in the column at gas-phase composition i (mol/unit volume) qjM ) adsorbed-phase concentration of pure component j at the total system pressure (mol/unit volume) u ) average interstitial gas velocity in the column VC ) volume of column (mL) VG ) volume of the void space in the column (mL) (VR - VG)0,i ) net retention volume at standard conditions at composition i (mL/g) (VR - VG)0,i,fit ) fitted net retention volume at standard conditions at composition i (mL/g) VS ) volume of the solid phase in the column (mL) v ) velocity of the disturbance wj ) amount adsorbed of component j in the column (mol/ kg) Yj ) gas-phase mole fraction of component j in the column Greek Symbols ) voidage of the column σK2 ) variance of the polynomial fitting of the K values σRV2 ) variance of the polynomial fitting of the retention volumes σi2 ) variance in the estimated isotherm of component i τR ) concentration pulse retention time (s) τU ) retention time of unretained pulse (s) τX) retention time of composition front (s)
Literature Cited Buffham, B. A. Model-Independent Aspects of Perturbation Chromatography Theory. Proc. R. Soc. London, Ser. A 1978, 364, 443-455. Buffham, B. A.; G. Mason; Yadav, G. D. Retention Volumes and Retention Times in Binary Chromatography: Determination and Equilibrium Properties. J. Chem. Soc., Faraday Trans. 1 1985, 81, 161-173. Danner, R. P.; Choi, E. C. F. Mixture Adsorption of Ethane and Ethylene on 13X Molecular Sieve. Ind. Eng. Chem. Fundam. 1978, 17, 248-253. Helfferich, F. Travel of Molecules and Disturbances in Chromatographic Columns. J. Chem. Educ. 1964, 41, 410-413.
Heslop, M. J.; Buffham, B. A.; Mason, G. Limitations of the polynomial-fitting method of determining binary gas-mixture adsorption isotherms. Paper presented at The 1995 IChemE Research Event, Edinburgh, 1995. Hyun, S. H. Adsorption of Azeotropic Binary Gas Mixtures on Molecular Sieve Type 13X. M.S. Dissertation, Pennsylvania State University, 1980. Hyun, S. H.; Danner, R. P. Determination of Gas Adsorption Equilibria by the Concentration Pulse Technique. AIChE Symp. Ser. 1982, 78, 19-28. Hyun, S. H.; Danner, R. P. Gas Adsorption Isotherms by Use of Perturbation Chromatography. Ind. Eng. Chem. Fundam. 1985, 24, 95-101. Markham, E. C.; Benton, A. F. The Adsorption of Gas Mixtures by Silica. J. Am. Chem. Soc. 1931, 53, 497-507. Mason, G.; Buffham, B. A. New Chromatographic Method for Determining Binary Gas Adsorption Equilibrium Data. Fundamentals of Adsorption; Mersmann, A. B., Scholl, S. E., Eds.; United Engineering Trustees: New York, 1991; pp 553-562. Mason, G.; Buffham, B. A. Proc. R. Soc. London, Ser. A 1996, in press. Peterson, D. L.; Helfferich, F. Towards a Generalised Theory of Gas Chromatography at High Solute Concentration. J. Phys. Chem. 1965, 69, 1283-1293. Ruthven, D. M.; Kumar, R. A Chromatographic Study of the Diffusion of Nitrogen, Methane and Binary Nitrogen-Methane Mixtures on 4A Molecular Sieve. Can. J. Chem. Eng. 1979, 57, 342-348. 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. Shah, D. B. Binary Sorption Equilibria by Pulse Chromatography. ACS Symp. Ser. 1988, No. 358, 409-420. Tezel, F. H.; Tezel, H. O.; Ruthven, D. M. Determination of Pure and Binary Isotherms for Nitrogen and Krypton. J. Colloid Interface Sci. 1993, 149, 197-207. 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. Wilson, R. J. Adsorption of Synthesis Gas Mixture Components. M.S. Dissertation, Pennsylvania State University, 1980.
Received for review February 6, 1995 Revised manuscript received November 14, 1995 Accepted December 12, 1995X IE950101J
X Abstract published in Advance ACS Abstracts, February 15, 1996.
