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Ind. Eng. Chem. Res. 2000, 39, 1514-1519
Tests of the Consistency between Binary Gas Adsorption Isotherms and Binary Chromatographic Retention Volumes Mark J. Heslop,* Bryan A. Buffham, and Geoffrey Mason Department of Chemical Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom
The amounts of two gases simultaneously adsorbed on a solid can be measured directly or may be inferred from chromatographic retention times. Adsorption isotherms obtained by the two methods should agree within experimental error. It is desirable to have a quantitative test of the degree of agreement between experimental results made by both methods. We establish here a new adsorption function and relationship that should exist between the measured retention times and static data if both methods and the experimental results are in perfect agreement. This function gives a test of consistency between static data and retention times that has advantages over alternative methods. Background When two gases, A and B, are in equilibrium with an adsorbing solid at a particular temperature, the adsorption isotherms
wA ) wA(yA,Fg)
(1)
wB ) wB(yB,Fg)
(2)
relate the amount adsorbed per unit mass of the adsorbent (wA and wB) to the mole fractions in the gas phase (yA and yB) and the gas-phase molar density (Fg) The adsorption isotherms can be measured directly by one of the various static methods.1 For example, a binary gas mixture of A and B is admitted to an evacuated chamber of predetermined volume. The equilibrium amounts of A and B in the adsorbed phase are found from gas compositions and pressure measurements, using material balances. This avoids the need to measure adsorbed-phase concentrations. The mass of the adsorbed phase can also be monitored2 and provides an extra check. Static methods are slow and can be prone to cumulative errors. An alternative is the chromatographic method in which the times taken for composition transients to pass through a chromatographic column are measured. This has the advantage of being relatively fast. A binary mixture, mole fraction yA of A, flows through the column at volumetric flow rate Qc. A pulse of one of the pure components is added. The time taken for the composition transient to pass through the column is the retention time tr. In addition, the time tg for a nonadsorbed substance to pass through the column is measured. The retention time tr is related to the slopes of the binary adsorption isotherms by
tr ) tg +
(
)
dwB dwA W yA + yB Qc F g dyB dyA
(3)
This result was originally established3 for ideal chro* To whom correspondence should be addressed. E-mail:
[email protected]. Fax: +44 1509 223923. Phone: +44 1509 263171, ext 4094.
matography but is more generally true.4 The binary equilibrium constant Kw is defined by
Kw ) yA
dwB dwA + yB dyB dyA
(4)
The term QcFg is the molar flow rate of the gas mixture passing through the column. For convenience, it is usual to replace this term with Q0F0 where Q0 and F0 are the equivalent volumetric flow rate and density at standard conditions. Substituting and rearranging, we obtain an explicit expression for Kw in terms of measurable quantities:
[
Kw ) F0
]
(tr - tg)Q0 W
(5)
The term in square brackets is termed the net retention volume at standard conditions. The problem with using eq 3 is that for a single mixture composition yA and measurement of tr, there are two unknown isotherm gradients to determine. van der Vlist and van der Meijden5 tried to surmount this problem by assuming that wA, wB, and Kw could be represented by third-order polynomials in yA. They showed that the wA and wB coefficients could be found from the Kw coefficients and the pure-component amounts adsorbed. This method has been used in other investigations.6-8 Triebe and Tezel9 proposed a different functional form of Kw for use in van der Vlist and van der Meijden’s procedure. Buffham et al.10 suggested that a functional form be assumed for the isotherms rather than the Kw values. Hyun and Danner11 compared the predictions of the polynomial-fitting method with independently obtained (static) data. They examined three systems. The first system was carbon monoxide-methane-BPL activated carbon at 345 kPa and 298 K for which the static data were reported by Wilson.12 For convenience, this will now be referred to as System 1 and the details are summarized in Table 1. The second system (System 2A) was isobutane-ethylene-13X zeolite at 138 kPa and 298 K for which the static data were reported by Hyun and Danner.13 The third system (System 3) was ethane-
10.1021/ie990396r CCC: $19.00 © 2000 American Chemical Society Published on Web 03/31/2000
Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1515 Table 1. Summary of Previous Investigations system 1 2A 2B 3
gases; adsorbent; temp. and total pressure carbon monoxide and methane BPL activated carbon; 298 K and 345 kPa isobutane and ethylene; 13X zeolite; 298 K and 138 kPa isobutane and ethylene; 13X zeolite; 298 K and 138 kPa ethane and ethylene; 13X zeolite; 298 K and 138 kPa
static data reported by
method of Kw determination
Kw values reported by
Wilson (ref 12)
Hyun and Danner (ref 11)
drawing tangents to static data
Hyun and Danner (ref 13)
Hyun and Danner (ref 11)
drawing tangents to static data
Hyun and Danner (ref 13)
Hyun and Danner (ref 15)
chromatographic experiments
Danner and Choi (ref 14)
Hyun and Danner (ref 11)
chromatographic experiments
ethylene-13X zeolite at 138 kPa and 298 K for which the static data were reported by Danner and Choi.14 For Systems 1 and 2A, Hyun and Danner11 calculated Kw values by drawing tangents to curves drawn through the static data points and substituting their slopes into eq 4. In each case, excellent third-order polynomial fits to these “tangent” Kw values were obtained, and the subsequent polynomial predictions agreed with the relevant static data. But this of course was only a test of the van der Vlist procedure because for each system there was only one set of static experimental results and no chromatographic measurements. For System 3, the Kw values were obtained from the chromatographic11 method by measuring the retention times and using eqs 3 and 4. This set of Kw values could only be fitted approximately with a third-order polynomial and the wA and wB predictions11 did not agree with the relevant static data.14 Later, Hyun and Danner15 calculated Kw values from chromatographic experiments for the second system and to avoid confusion this will be termed System 2B. The Kw values calculated from static data (System 2A) are significantly different from the chromatographic values (System 2B). This is worrisome because the static data for Systems 2A and 2B are, of course, the same. The disagreement between the “synthetic” and “actual” Kw values for Systems 2A and 2B (reported by Hyun and Danner11,15), and for System 3 (reported by Buffham et al.10), implies that the static and chromatographic data are inconsistent. But how can this inconsistency be tested? In the next section, we describe existing methods for testing the consistency between chromatographic and static data. Then, in a further section we derive an expression for the relationship between chromatographic Kw values and the amounts adsorbed (wA and wB), at each section along the whole composition range. This relation does not involve the graphical determination of gradients or fitting the curves with functions. Existing Methods of Testing Chromatographic and Static Data It is important to establish consistency between the chromatographic and static data to provide a check on the measurement errors and assumptions of both methods. Essentially, the chromatographic method gives values of yA (dwB/dyB) + yB (dwA/dyA) and the static methods give values of wA and wB directly. One way of establishing consistency is to convert the values of wA and wB into corresponding values of yA (dwB/dyB) + yB (dwA/dyA) and then compare them with the reported chromatographic data. The other way is to convert the
values of yA (dwB/dyB) + yB (dwA/dyA) into corresponding values of wA and wB and similarly compare with the reported static data. These two categories will now be dealt with in turn. Calculating Gradients from Static Data. It is possible, in principle, to test the consistency by using eq 4 directly. This would involve determining the slopes of the wA versus yA and wB versus yB curves and substituting these values as well as the gas-phase composition yA into eq 4. These slopes can be obtained by either of two methods. The first is to draw curves through the static data points by eye and then to construct the tangents. But slopes are notoriously difficult to estimate from discrete points, especially when the data are noisy and the slopes are relatively large. The second, improved, method is to fit suitable isotherms for wA and wB to the static data and then obtain the isotherm slopes by differentiation. For example, the binary-Langmuir equations can be written10 as
w0A yA
wA )
yA +
wB )
w0A RABw0B
(6) yB
w0B yB RABw0B yA + yB w0A
(7)
There are three parameters, namely, the pure-component amounts adsorbed, w0A and w0B, and RAB, the ratio of the two Henry’s constant for components A and B. The corresponding binary-equilibrium constant is
RAByB Kw )
(
w0B RAByA yB + 0 w0A wB
)
2
(8)
Now consider a binary system for which we have chromatographic and static data. If we can obtain a perfect fit of the static data with eqs 6 and 7, then by substituting the corresponding value of RAB into eq 8, we can generate the Kw values that are consistent with the static data. These are then compared with the experimental Kw values to establish the degree of consistency between the chromatographic and static data. The consistency can be quantified by a simple root-
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Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000
mean-square index. If the static data cannot be adequately fitted with eqs 6 and 7, then other isotherms must be selected for wA and wB (and hence Kw) and the procedure repeated. One problem with using extra terms and parameters is that the fitted isotherms can develop strange shapes, especially if the static data points are not at evenly spaced compositions. Another problem is that it is, strictly speaking, only possible to quantify the degree of consistency between the static and chromatographic data if the functions give a perfect fit to the experimental values of wA and wB across the entire composition range. It is not a trivial matter to satisfy this condition. Calculating Isotherms from Chromatographic Measurements. Now, the situation is reversed. The intention is to go from the Kw values to the isotherm values of wA and wB. For example, if eq 8 gives a perfect fit to the Kw values, then the isotherms that are consistent with these Kw values are obtained by substituting the corresponding value of RAB into eqs 6 and 7 because eq 8 is derived from eqs 6 and 7. These isotherms can then be compared with the actual static data and the degree of consistency between the chromatographic and static data can be established. The consistency can again be quantified by a simple rootmean-square index using the differences between the actual static data and the isotherms calculated from the Kw values. If, however, eq 8 does not give a good fit to the reported values, then it is not possible to determine reasonable isotherms for these data using the binaryLangmuir set of equations. Hence, it is not possible to make any conclusions about the consistency between the chromatographic and static data. If we are to find a route from the chromatographic data to the isotherms, we require a function for Kw that will satisfy two criteria. First, the function must give a good fit to the Kw values. Second, it must be possible to determine the parameters for the consistent values of wA and wB from the parameters in the function for Kw.
Figure 1. The new consistency test shown pictorially. For perfect consistency in the composition range from yA1 to yA2, the sum of the two shaded areas in the top picture should be exactly equal to the shaded area in the bottom picture. One advantage of this new test is that any section of the composition range can be examined. Also, if the wA, wB, and Kw data points are at the same composition intervals; only curve fitting across a limited range is required.
New Method of Testing Chromatographic and Static Data Clearly, to test chromatographic data against static data, a method which does not depend on fitting functions to data points is to be preferred. Integration of the expression for the binary equilibrium constant Kw, eq 4, across a section of the composition range from yA ) yA1 to yA ) yA2 yields
∫
yA2
yA1
Kw dyA )
∫
yA2
yA1
yA
dwB dy + dyB A
∫
yA2
yA1
yB
dwA dy (9) dyA A
Because yA + yB ) 1, dyA/dyB ) -1 and eq 9 becomes
∫
yA2
yA1
Kw dyA )
∫
wB2
wB1
yA dwB +
∫
wA2
wA1
yB dwA
(10)
These integrals are represented by the shaded areas in Figure 1, from which it can be seen by letting yA1 f 0 and yA2 f 1 that
∫K 1
0
∫ (w 1
w dyA )
0
A
+ wB) dyA
(11)
So here is the first test given by the new method. The
Figure 2. The simple version of the new consistency test shown pictorially. The top shaded area should be equal to the bottom shaded area. The two areas can be determined by the trapezoidal rule and localized curve fitting.
total area under the Kw versus yA graph should equal the total area under the total amount adsorbed (wA + wB) versus yA curve. This is shown in Figure 2. If the two areas are not the same, there is a problem with the measurement errors or assumptions of one or both of the methods. In other words, the chromatographic measurements and the static data are inconsistent. It is not possible to identify the location of any problem. The total areas can be determined by the trapezoidal rule with localized curve fitting to obtain minor correc-
Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1517 Table 2. Normalized Discrepancies for Systems 1, 2A, 2B, and 3 D dsections
system 1
system 2A
system 2B
system 3
-0.003 0.03
-0.04 0.07
-0.07 0.23
-0.15 0.20
tions. Localized curve fitting has a second-order effect and does not introduce significant errors. For some systems, Kw values may not be available at the ends of the composition range. For these it is necessary to extrapolate in order to estimate the Kw values at yA ) 0 and yA ) 1. It is convenient to define
∫K D) 1
0
∫ (w 1
w
dyA -
∫ (w
0
A
+ wB) dyA
1
0
A
(12)
+ wB) dyA
Figure 3. Comparison of the 10 sectional areas under the Kw graph (termed “Gradients from static data”) and the corresponding areas under the yCO vs wmethane and ymethane vs wCO graphs (termed “Static data”). The Kw data were obtained by drawing tangents to curves drawn through the static data. There is excellent agreement between the two sets of points.
as a measure of the discrepancy between the two sides of eq 11. This dimensionless discrepancy enables comparisons to be made between measurements in different units. Equation 10 provides a test for any section of the composition range. The shaded area in the bottom diagram of Figure 1 should equal the sum of the two shaded areas in the top diagram. Consider now that the composition range is divided into N equal sections. The above procedure can be applied to each section, with eq 10 now giving di for the ith section. The overall discrepancy of all the sections, dsections, is defined as a root-mean-square measure of the agreement for the areas of all N equal sections:
dsections )
x
N
di ∑ i)1
2
N
(13)
For N ) 1, dsections will be the same as D. For the systems studied (see Table 2), as N is increased from 1 to 2, 5, and 10 with equal divisions of gas composition, the value of dsections increases. This is because dividing the composition range into smaller sections causes less canceling out of positive and negative areas. Low values of D and dsections indicate that the chromatographic and static data are consistent. The advantage of the overall discrepancy is that each section is considered individually and this helps identify “good” and “bad” sections of the composition range. Application of New Method to Reported Data In the literature, the chromatographic data for Systems 1, 2A, 2B, and 3 are given in terms of the net retention volumes at standard conditions, rather than the binary equilibrium constant Kw. At standard conditions, the gas-phase molar density F0 is 4.46 × 10-5 mol mL-1. The Kw values are obtained from the net retention volumes by using this value of F0 in eq 5. For Systems 1 and 2A, Hyun and Danner11 have calculated Kw values from the slopes of curves drawn through the static data. We have applied our test using 10 sectional areas (Figures 3 and 4) and found the Kw values to be consistent with the data from which they were calculated. The good consistencies for these two systems are confirmed by the low D and dsections values
Figure 4. Comparison of the 10 sectional areas under the Kw graph (termed “Gradients from static data”) and the corresponding areas under the yisobutane vs wethylene and yethylene vs wisobutane graphs (termed “Static data”). The Kw data were obtained by drawing tangents to curves drawn through the static data. There is a systematic discrepancy between the two sets of points, implying that the calculated isotherm gradients are too low.
in Table 2. However, close inspection of Figure 4 reveals that there is a systematic variation between the two sets of data points: one set is below the other. In a later investigation, Hyun and Danner15 obtained Kw values from chromatographic experiments for the isobutane-ethylene-13X zeolite system, System 2B in Table 1. They did not report Kw values at the two ends of the composition range. However, to calculate D, it is necessary to know Kw across the entire composition range. Hyun and Danner15 fitted the Kw values with a fourth-order polynomial and obtained an excellent fit. We have extrapolated this polynomial to the ends of the composition range. Figure 5 shows that the static and chromatographic data are inconsistent. This is confirmed by the relevant values in Table 2: D ) -0.07 and dsections ) 0.23. Finally, consider System 3, the details of which are given in Table 1. Hyun and Danner15 obtained Kw values from chromatographic measurements, but as with the previous system, they did not report values at the ends of the composition range. They found that a third-order polynomial could not cope with the plateau and steep ascent as yethane increases from 0 to 1. We found10 that a three-parameter binary-Langmuir expression for Kw gave an excellent fit to the reported Kw values. The best 0 ) 4.4024 mol/kg and w0ethylene ) fitting values of wethane
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Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000
Figure 5. Comparison of the 10 sectional areas under the Kw graph (termed “Chromatographic data”) and the corresponding areas under the yisobutane vs wethylene and yethylene vs wisobutane graphs (termed “Static data”). This time, the Kw data were obtained from chromatographic measurements. The static and chromatographic data are clearly inconsistent.
Figure 6. Comparison of the 10 sectional areas under the Kw graph (termed “Chromatographic data”) and the corresponding areas under the yethane vs wethylene and yethylene vs wethane graphs (termed “Static data”). The Kw data were obtained from chromatographic measurements. The static and chromatographic data are clearly inconsistent. 0 wethane
1.7579 mol/kg (compare with the static data: ) 2.185 mol/kg and w0ethylene ) 2.765 mol/kg) are additional pieces of evidence that the chromatographic and static data are inconsistent. We used this fit to extrapolate the function to the ends of the composition range. Figure 6 shows that the static and chromatographic data are inconsistent. This poor agreement is shown by the large discrepancies in Table 2: D ) -0.15 and dsections ) 0.20. Discussion Calculating Kw Values from Drawing Tangents. The good agreement and corresponding low discrepancies D and dsections for Systems 1 and 2A (Figures 3 and 4) are not surprising. None of the data points were obtained from chromatographic experiments. The low values do not necessarily mean that the static data are good. Rather, they are a measure of how well smooth curves can be drawn through sets of discrete points (this depends on the scatter) and how accurately the gradients can be obtained from these curves. With this in mind, we might expect some di values to be positive and others to be negative. This is the case in Figure 3, but in Figure 4 all the di values are negative. This might indicate some systematic error in measuring or processing the static data to calculate the gradients and hence
the Kw values. This set of negative di values fits in with the predictions of the polynomial-fitting method obtained by Hyun and Danner.11 In their Figure 3, they obtained an excellent fit to the Kw values with a thirdorder polynomial. One might expect that the predicted isotherms from the polynomial-fitting method should agree with the static data. However, in their Figure 5, the predicted isotherms have consistently lower values than the static data, apart from the end points of the composition range where the polynomial-fitting method requires agreement with the pure-component amounts adsorbed. These “low” predictions are consistent with the negative value of D in Table 2. Calculating Kw Values from Chromatographic Experiments. For Systems 2B and 3 (Figures 5 and 6) it is clear that the chromatographic and static data are inconsistent. So there is a problem either with the static or with the chromatographic data. Hyun and Danner11,15 obtained both sets of Kw values by injecting “small” pulses of pure gas into gases of different compositions rather than by the addition of small continuous flows. They do not say whether they used just one component for the pulse gas or used each component in turn and averaged the Kw values. Tezel et al.16 investigated the adsorption of nitrogen and krypton on two different adsorbents (dealuminated H-mordenite and silicalite) at two different temperatures (226 and 298 K). For each adsorbent and temperature they found two distinct sets of Kw values, one for the krypton addition and the other for the nitrogen addition. This is a violation of eq 3 because the retention time tr should be independent of the component used as the perturbation gas. The problem probably arises from the injection of pure components which give, at least initially, large changes in composition. One of the assumptions inherent in eq 3 is that changes in composition are small. Conclusions We have described a simple method for establishing the consistency between chromatographic and static data, which only involves calculating the areas under static and chromatographic data. The method is an improvement over alternative methods which require either fitting functions to the static data to obtain the isotherm slopes or fitting a function to the chromatographic data. This is not a trivial matter and ideally requires perfect fitting to quantify the degree of consistency. For the isobutane-ethylene-13X zeolite system at 138 kPa and 298 K and the ethane-ethylene-13X zeolite system at 138 kPa and 298 K, the test shows that the published chromatographic and static data are inconsistent. The discrepancy indicates a fault in either set of experiments and, in the absence of any further information, it is most likely to lie in the way in which the retention times are measured and processed. Acknowledgment This work was supported in part by the United Kingdom Engineering and Physical Sciences Research Council (EPSRC). Nomenclature di ) normalized consistency for ith section of the composition range
Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1519 dsections ) rms normalized consistency for all i sections of the composition range D ) normalized consistency for total areas under the Kw, wA, and wB curves Kw ) binary equilibrium constant (mol M-1) N ) number of sections in composition range Q0 ) volumetric flow rate at standard conditions (L3 T-1) Qc ) volumetric flow rate in column (L3 T-1) RAB ) binary-Langmuir isotherm parameter for both components A and B tg ) retention time for unadsorbed species (T) tr ) retention time (T) W ) mass of adsorbent in column (M) wA, wB ) molar amounts of A and B adsorbed per unit mass of adsorbent (mol M-1) 0 wA,w0B ) pure-component amounts adsorbed per unit mass of adsorbent (mol M-1) yA, yB ) mole fractions of A and B in column gas phase (mol M-1) F0 ) gas-phase molar density at standard conditions (mol L-3) Fg ) gas-phase molar density in column (mol L-3)
Literature Cited (1) Talu, O. Needs, Status, Techniques and Problems with Binary Gas Adosrption Experiments. Adv. Colloid Interface Sci. 1998, 76-77, 227-269. (2) Dreisbach, F.; Staudt, R.; Keller, J. U. High-Pressure Adsorption Data of Methane, Nitrogen, Carbon Dioxide and Their Binary and Ternary Mixtures on Activated Carbon. Adsorp. J. Int. Adsorp. Soc. 1999, 5, 215-227. (3) Peterson, D. L.; Helfferich, F. Towards a Generalized Theory of Gas Chromatography at High Solute Concentration. J. Phys. Chem. 1965, 69, 1283-1293. (4) 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. 1 1985, 81, 161-173.
(5) 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. (6) 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. (7) 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. (8) Shah, D. B. Binary Sorption Equilibria by Pulse Chromatography. ACS Symp. Ser. 1988, 358, 409-420. (9) 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. (10) Buffham, B. A.; Mason, G.; Heslop, M. J. Binary Adsorption Isotherms from Chromatographic Retention Times. Ind. Eng. Chem. Res. 1999, 38, 1114-1124. (11) Hyun, S. H.; Danner, R. P. Determination of Gas Adsorption Equilibria by the Concentration Pulse Technique. AIChE Symp. Ser. 1982, 78, 19-28. (12) Wilson, R. J. Adsorption of Synthesis Gas Mixture Components. M. S. Dissertation, Pennsylvania State University, 1980 (as reported by Hyun and Danner11). (13) Hyun, S. H.; Danner, R. P. Equilibrium Adsorption of Ethane, Ethylene, Isobutane, Carbon Dioxide, and Their Binary Mixtures on 13X Molecular Sieve. J. Chem. Eng. Data 1982, 27, 196-200. (14) 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. (15) Hyun, S. H.; Danner, R. P. Gas Adsorption Isotherms by Use of Perturbation Chromatography. Ind. Eng. Chem. Fundam. 1985, 24, 95-101. (16) 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.
Received for review June 4, 1999 Revised manuscript received February 3, 2000 Accepted February 10, 2000 IE990396R
