1230
Ind. Eng. Chem. Process Des. Dev. 1985, 24,1230-1239
temperatures. The agreement between the calculated and experimental results is reasonable. Similar results for the Illinois coal liquid in mixtures with hydrogen are reported in Table VIII. The calculations with the CCOR equation for this coal liquid are not as good as those for the Wyoming coal liquid. The calculated K value of hydrogen deviates from the experimental by 8.4% on the average, which is about twice that for the Wyoming coal liquid. The calculated flash vaporization temperature deviates from the experimental by 1.6% on the average, corresponding to about 10 K. The largest deviation is observed for datum no. 2 for which the calculated temperature is low by 22 K. The relatively large deviations appear to be due to the large amount of residue (62.78%) of the Illinois coal liquid which remained unvaporized at the end of the TBP fractionation. The calculation should be improved if the residue can be better characterized. Acknowledgment Funds for this research were provided by the Electric Power Research Institute through research Project RP-367. Catalytic, Inc., and Hydrocarbon Research, Inc., supplied coal liquids. T. M. Guo performed the VLE calculations. Literature Cited
Grayson, H. G.; Streed, C. W. Paper presented at the 6th World Petroleum Conference, Frankfurt am Main, Qermany, June 19-26, 1963; Paper 20, Sec. VII. Henry, R. M. "Vapor-Liquid Equilibrium Measurements for the SCR-I I Process"; DOE/ET/ 1004-1, Gulf Science and Technology Co.: Pittsburgh, PA, Oct 1980. Kim. H.; Lin, H. M.; Chao K. C. Paper presented at the Proceedings of the 3rd Pacific Chemical Engineering Congress, Seoui, Korea, May 8-1 1, 1983; Vol. 11, p 321. Kim, H.; Lin, H. M.; Chao, K. C. Ind. Eng. Chem. Fundam., in press. Lewis, H. E. Plant Manager, Quarterly Technical Progress Report, Jan.March, 1982, Catalytic, Inc., Wilsonville, AL. Dist. Category UC-Sod, DOE/ET/10154-122. Lin, H. M.; Chao, K. C. AIChE J. 1984,30, 981. Lin, H. M.; Kim, H.; Guo, T. M.; Chao, K. C. Ind. Eng. Chem. Process Des. Dev. in press. Lin, H. M.; Sebastian, H. M.; Simnick, J. J.; Chao, K. C. Ind. Eng. Chem. Process Des. Dev. 1981,2 0 , 253. Merdenger, M. Final Report to US. Department of Energy (DOE Contract DE-AC05-77ET-10152) and the Electric Power Research Institute (AP2623, Research Project 238-3). Hydrocarbon Research, Inc., Lawrenceville, NJ, Oct 1962. Radosz, M.; Lin, H. M.; Chao, K. C. Ind. €no. Chem. Process Des. Dev. 1982,2 1 , 653. Sebastian, H. M.: Lin, H. M.; Chao, K. C. AIChE J. 198la,2 7 , 138. Sebastian, H. M.; Lln, H. M.; Chao, K. C. Ind. Eng. Chem. Fundam. 1981b, 20 - , 346 - . ..
Sebastian, H. M.; Lin. H. M.; Chao, K. C. Ind. Eng. Chem. Process Des. Dev. 1981c,2 0 , 508. Soave, G. Chem. Eng. Sci. 1972. 2 7 , 1197. Sung, C. Ph.D. Thesis, University of Ptttsburgh, PA, 1981. Watanasiri, S.; Brule, M. R.; Starling, K. E. AIChE J. 1982,28, 626. Wilson, G. M.; Johnston, R. H.; Hwang, S. C.; Tsonopoulos, C. Ind. Eng. Chem. Process Des. Dev. 1981,2 0 , 94.
El-Twaty. A. I.; Prausnitz, J. M. Chem. Eng Sci. 1980,35, 1765. Gray, R. D.; Heidman, J. L.; Hwang, S. C.; Tsonopoulos, C. Nuid Phase Equilib. 1983, 13, 59.
Received for review October 9, 1984 Accepted M a r c h 11, 1985
Characterization of Adsorption Affinity of Unknown Substances In Aqueous Solutions Kumaraswamy JayaraJtand Chi Tlen' Department of Chemical Englneering and Materials Science, Syracuse University, Syracuse. New York 132 10
An aqueous solution with a large number of unknown adsorbates may be approximated by one with a fewer number of pseudospecies identified by their Freundlich constants. A procedure is proposed which calculates the pseudospecies composition from total adsorbate equilibrium concentration data obtained from batch contacting measurements. The method is applied to a number of wastewater systems to demonstrate its utility and validity.
The development of rational design methods in adsorption is handicapped by the fact that adsorption-treated aqueous solutions frequently contain a number adsorbates which cannot be completely identified. The dynamics of any adsorption process is defined by its stoichiometry, the equilibrium relationship between the solution and adsorbed phases, and the rate of the adsorption process, with the equilibrium relationship and rate parameters attributed to the adsorbate-adsorbent system. Because of a lack of information on the adsorbate identity and concentration, a rational design development based on adsorption theories becomes impractical. In recent years, a few investigators have suggested procedures to characterize the adsorption affinity of such solutions. Frick (1980) proposed that organics present in 'Present address: Honeywell Inc., Bloomington, MN 55420. 0196-4305/85/1124-1230$01.50/0
river water may be grouped into three categories: nonadsorbable, moderately adsorbable, and highly adsorbable. Each of these can be identified by their Freundlich constants, A and n. On the basis that the calculated concentration values which correspond to certain specified conditions must agree with experimental values, the Freundlich constants were determined, and the total organic carbon concentration was apportioned into the three categories. A more rigorous procedure was adopted by Kage (1980) and Okazaki et ai. (1980,1981). These investigators contended that for a solution containing a large number of adsorbates, the parameters characterizing the adsorption affinity of the individual adsorbates are described by their Langmuir constants, a and b. These constants may range from zero to infinity. Such a solution is, therefore, described by its adsorbate concentration frequency function m(a,b),defined as such that the concentration of adsor0 1985 American Chemical
Society
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985 1231
bates with Langmuir constants a and b within the range a. 5 a 5 a. da and bo Ib Ibo db is m(ao,bo)da db. The total adsorbate concentration, cT, becomes
+
+
The application of the Langmuir theory enables one to determine a corresponding frequency function, w(a,b), in the adsorbed phase, which is in equilibrium with the solution with a concentration frequency function m(a,b). By definition, w(ao,bo)da db represents the concentration of adsorbates, with a. 5 a Ia. da and bo 5 b 5 bo + db in the adsorbed phase. When the solution and adsorbed phases are in equilibrium, w(a,b) and m(a,b) are related to one another by the expression
"ox
I
I
l
1
e .
+
. .
u
$
"min Am,, AI
1+
J,
Jim(a,b) da db
A solution with unknown adsorbates can then be characterized by its adsorbate concentration frequency function; this function can be used in adsorption calculations. In batch contacting processes, for example, the problem is to determine the time dependence of m(a,b) and w(a,b) and, ultimately, the concentration functions at equilibrium, m,(a,b) and w,(a,b). In the case of fixed-bed adsorption, the problem is to predict, from an influent with a given concentration frequency function, the history of the concentration frequency function of the effluent. The scheme of characterization, while conceptually simple, is difficult to implement. Both Kage and Okazaki et al. found it necessary to assume that the adsorbate concentration frequency function is a function of only one of the two Langmuir constants (either a or b). The selection of the Langmuir constants was left arbitrary. Frick, Kage, and Okazaki et al. all reported success with their respective methods. Because of the extreme simplifications introduced in their respective methods and the limited testa conducted, their utility as general procedures for characterizations, however, cannot be assured. The present study establishes a more systematic and complete procedure which characterizes the adsorption affinity of aqueous solutions according to appropriate adsorption data obtained from different types of measurements. Like Frick's work, the present study assumes the validity of the IAS theory in describing multicomponent adsorption equilibrium. Unlike Fricks study, however, there is no limit to the number of pseudospecies, and their Freundlich constants may vary over any range. An outline of the prinicple of characterization, the types of experimental data on which the characterization is based, and the computation procedures of the proposed method are given below, followed by a detailed discussion of the results obtained by applying the method to a number of real wastewater systems. Principles of Characterization The proposed characterization method is based on the prinicple of species grouping. In a number of recent studies (Calligaris and Tien, 1982; Tien, 1983; Mehrotra and Tien, 1984), it was shown that multicomponent adsorption calculations can be significantly simplified by grouping similar adsorbates into pseudospecies. For a solution with a large number of adsorbates, and assuming that one may characterize adsorbates by their Freundlich constants (on the basis that their single species isotherm data obey the Freundlich expression), the various adsorbates in the solution are represented by various points as shown in Figure 1. By applying the species grouping
A2 "3 FREUNDLICH COEFFICIENT, A
-
Amox
Figure 1. Representation of a solution of large numbers of adsorbates by pseudospecies. Each point corresponding to a given adsorbate.
principle, one may combine the adsorbates represented by the various point near a given grid point of the A-n network (see Figure 1)to form a pseudospecies with A and n values corresponding to those of the grid point. In this manner, a solution with a large number of adsorbates may be approximated by one with a fewer number of pseudospecies with specified A and n values. The solution is then characterized by assigning concentration values to the pseudospecies. The principle of characterization used in this study is as follows. Once a solution is properly characterized as described above, and assuming that the IAS theory is valid, one can estimate the equilibrium concentrations achieved by contacting the solution with known adsorbents under a variety of conditions. The requirement that the calculated value agrees with the corresponding experimental value, therefore, provides a basis for determining the concentrations of the pseudospecies. Let u denote certain concentration values which can be predicted as well as experimentally determined. An objective function, 4, is defined as (3)
where the subscripts calcd and exptl denote the calculated and experimental values, respectively. The determination of the pseudospecies concentration is made by minimizing the objective function, 4, subject to the constraint that the sum of the concentrations of the pseudospecies, cio, is specified and is equal to the total adsorbate concentration, cT0, or cTo =
ccio i
(4)
Experimental Data Base for Characterization The method of characterization as outlined above is based on minimizing the objective function (eq 3). Thus, in principle, any experimental adsorption data of the solution in question can be considered a basis for characterization. The choice is, in most cases, largely a mater of convenience. The following discussion outlines three kinds of measurements commonly used to assess adsorption affinity. (A) Integral Batch Contacting Measurement. The simplest and most common test of assessing adsorption affinity is to contact a fixed volume of the solution in question with various amounts of adsorbents and to allow sufficient contact time until equilibrium between the so-
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Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
lution phase and the adsorbed phase is reached. For a given solution, the data can be expressed as a function of c T , vs. M / V where M and V denote the amount of the adsorbent and the volume of the solution, respectively. Alternatively, cT, can be expressed as a function of the total adsorbate concentration in the adsorbed phase, which is often erroneously referred to as the adsorption isotherm. Different kinds of the integral batch contacting measurements can also be made by using either concentrated or diluted solutions. The total adsorbate concentration of the tested solution changes as a result of the dilution or concentration. However, the relative amount of the various species remains unaltered. Measurements of these kinds are often intended to reveal the interaction differences among adsorbates at different concentration levels. (B) Differential Batch Contacting Measurements. Kage (1980) suggested this type of measurement to determine the concentration frequency function mentioned earlier. A small amount of adsorbent ( A M ) is added to a given volume of solution. After equilibrium is reached and the total adsorbate concentration in the solution phase is determined, the saturated adsorbent is removed and a new increment of fresh adsorbent is added. This process is then repeated. Let cy! be the total adsorbate concentration in the solution phase after the addition of the kth increment of adsorbent. A relationship for c[ymlvs. kAM/V can be obtained. This relationship, in general, will be different from that for cT, vs. M / V, as described in (A). (C) Integral Batch Contacting Measurements with Tracer Compound. Frick suggested measurement of this kind as a means of qualitatively estimating the maximum (or minimum) adsorption affinity of adsorbates present in a solution of unknown composition. A known compound is added to the tested solution before carrying out measurements such as described under (A). One can then determine experimentally the total adsorbate concentration (excluding the tracer compound) and the added tracer compound concentration in the solution phases for different dosages of adsorbents applied ( M / V). While normally the trace compound is not among the original adsorbates of the solution, this condition cannot be guaranteed if the identities of the solution's original adsorbates are unknown. From a practical standpoint, this lack of certainty presents no significant problem if (a) the amount of the tracer to be added is substantial, compared with the original total adsorbate concentration, and if (b) among the large number of adsorbates in the solution, no single species dominates.
Numerical Procedures Used in Characterization The numerical computations involved in carrying out the characterization are of two types: (a) the calculation of the particular kind of concentration values used as the basis of characterization and (b) the optimization-search procedure to determine the concentration values of the various pseudospecies. The methods used for these calculations are described below. (A) Calculation of Equilibrium Concentrations of Adsorbates in the Solution Phase. The calculations of three types of equilibrium concentrations, cT,, cy:, and c?:~, that correspond to the three kinds of adsorption measurements are as follows: (a) cT, and qT, from Integral Batch Contacting Measurements. For a solution with N adsorbates and initial concentrations cT, and assuming that the ideal adsorbed solution theory applies, the total adsorbate equilibrium concentration in the solution phase achieved when a given volume of the solution (V) is brought into contact with a fixed mass of fresh adsorbents (M)can be found
from the following equations (Calligaris and Tien, 1982):
[Aini] + v s where the concentrations (cT,, qT,) are expressed as the mass of adsorbates (in organic carbon) per unit volume of solution or mass of adsorbent. The use of this particular concentration unit is necessitated by the fact that the adsorbate concentration of a solution with unknown composition can be determined only in this manner. M , and x , are the molecular weight and the mass of adsorbate per unit mass of organic carbon for the ith species. The value of M J x , is assumed to be the same for all pseudo species. Accordingly, the above set of equations can be used to calculate cT, (or qT,) if M and V are known and the solution is completely characterized (Le., a solution with N species of adsorbates with A,, n,, and c,, given). (b) cy! for Differential Batch Contacting Measurements. If the experimental data base used for characterization is obtained from the differential contacting measurements, the value of &) can be obtained from eq 5-10, with c[fi,')replacing c,,, c:,k-O replacin cc0,and AM replacing M. In this manner, the value of c(,=) (17 can be found from the value of c ciz) from c!:), etc. Incrementally, the relationship of c@ vs. kAM/ V can be predicted. (c) cTmand cTr, from Tracer Compound Measurement. With the addition of a tracer compound, the solution may be considered to have ( N + l) species of adsorbates, with N pseudospecies (i = 1, 2, ..., N)and the tracer compound as the ( N + 1) th adsorbate. Both cT, = c , ~(total adsorbate concentration excluding the tracer compound) and cn, = cN+1. can be readily calculated from eq 5-10. (B) Optimization-SearchProcedure. On the basis of the type of experimental cata available for characterization, the optimization-search procedure assigns concentration values to the various pseudospecies so the objective function, 4, has a minimum value. A number of methods were considered and tried; the complex method was found to be the most suitable. Generally speaking, a two-level application of the complex method was necessary, the essential features of which are described below. (a) An initial complex with K vortices where K 2 N is selected. Here, a vortex is used to denote a set of concentration values c,,, i = 1, 2, ..., N , for the assumed N pseudospecies. A set of random concentration values for each vortex can be determined in the following manner.
xi"=,
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985 1233
A set of random values, [bi]i = 1, ...,N with 0 Ib I1, is first generated by using a random number generator. The concentration value, cio,is then calculated according to the expression
bi
where cT is the total adsorbate concentration of the solution an$ can be determined, for example, by using a TOC (total organic carbon) analyzer. Since there are K complexes, it is necessary to generate K sets of random numbers, [b,]. In a sense, each vortex corresponds to a point randomly located in the N-dimensional space with the coordinates of [cl,lk = 1, 2, ..., K. The centroid of the complex, by definition, is located at [e,], defined as CCl,
e, = -
(12)
[c(C,,- c,)2]”2
(13)
N The distance between the kth vortex and the centroid, dk, is given as dk =
1
(b) With the type of experimental data base for characterization specified and the values of A,, n,, and c,, assigned, the objective function, $, at each vortex can be evaluated by using the appropriate procedure described earlier. The vortex which gives the largest value of the objective function can then be identified. Let that vortex be the kth vortex, with coordinates (c,,)k. (c) To replace the worst vortex which gives the largest value of the objective function, the reflection point of that vortex is found. The coordinates of the reflection point are ( C L , ) ~=
e,, - [(c,,)k - EL,]
(14)
The reflection point becomes the new vortex if the following two conditions are met. (i) The coordinates of the reflection point must be physically meaningful. In other words, (cJR should always be positive (or zero) since concentrations cannot be negative. (ii) The value of the objective function at the reflection point cannot be the largest as compared with the other k - 1 vortexes. In the event that (i) is not satisfied, a point midway between the centroid of the complex and the vortex which gives its maximum value is chosen as the vortex. The process may be repeated until the condition is met. In the event that condition (ii) is not met, the midpoint between the reflection point and the centroid will be used to replace the kth vortex. (d) Once a replacement for the kth vortex is found, steps (b) and (c) can be repeated until an optimum is found. Ideally, the optimum is when all the vortexes collapse into a single point. In practice, a situation develops whereby the coordinates of atl the vortexes become very similar but not necessarily identical. Then, the vortex with the smallest value of the objective function is taken to be the optimum or, more likely, the starting value for the second-level search. The second-level search, if necessary, is conducted in the neighborhood of the results obtained in the first-level search. A new complex with K vortexes is chosen first. The initial concentration values assigned to each vortex, (Cl,,)k
k = 1, 2, ..., N, i = 1, 2, ..., N , are chosen randomly such that these values will be within 10% of the results of the first-level search. This random selection can be done as follows: A sequence of random numbers, [b,],is generated for each of the K vortexes. These initial concentration values can then be calculated according to the expressions
Si =
(~iJ*[l
+ O.l(bi - 0.5)]
(16)
where (ciJ* are the values obtained from the first-level search. Once the ci ’s are assigned for the K vortexes, the procedures outlined in (a)-(d) can be used to conduct the search. In assigning values to the various species, it is also stipulated that the minimum value of ci0should not be less than 1% of the maximum value of cio. Any cia's failing to meet this requirement are set to zero and omitted from subsequent calculations. The optimization-search procedure can be simplified if all the pseudospecies have the same n values. Accordingly, one has s = n (17) and eq 9 and 10 become identical and reduce to =0
(18)
+ -One can, therefore, calculate “Milxi from eq 17 through trial and error, assuming that M j / x jis the same for all the pseudospecies. Once M j r / x j is known, eq 5-8 can then be used to obtain CT, (or qT.). For the optimization-search calculation, concentration values are assigned to the pseudospecies, and the Freundlich exponent, l l n , is selected on the basis of the minimization of the objective function. Assigning values to ciofollows the procedure described earlier. For the value n, one extra random number, bNfl, is generated in the random number sequence [bi]. n is calculated according to the expression (19) n = nmin + (nmax - nmin)bN+1 Applications In order to demonstrate the present method and its validity, a number of aqueous solutions containing different adsorbates (both known and unknown) were characterized by using this method. Generally speaking, a perfect characterization scheme should identify the adsorbates in the solution and their concentrations. One may, therefore, argue that the validity of the characterization method depends upon its ability to yield correct information about the composition of the solution. The present method, however, will not withstand such a test. Species grouping, which forms the basis for the present method, gives only approximate representations. A pseudospecies generally includes more than one adsorbate. Furthermore, the species grouping selected is not the only configuration available. Rather, variety of plausible combinations can be used to form the pseudospecies. Consequently, a multiplicity of results is obtained from the optimization-search calculations. The fact that the characterization method fails to give unique representation does not necessarily negate its utility if the different results of characterization yield essentially the same answers in adsorption calculations. This re-
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Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
yielded the total adsorbate equilibrium concentration, cT,, as a function of V I M (i.e., integral batch contacting measurements). Diluted leachate was used in the “type B” and “type C” measurements. In the sample calculations discussed here, the data from the type A measurements were used as the basis for leachate characterization. Once the leachate was characterized, type B and C measurements could be predicted and compared against experiments. The degree of agreement provided a measure of the method’s validity. As discussed earlier, the present method permits a solution to be characterized in two ways. It can be represented either as a solution with a number of pseudospecies with the same n value but different A values or as a solution with varying A and n values. In addition, as an extreme simplification, the system may be considered as a single-species solution so that all the adsorbates are grouped to form a pseudospecies, as was customarilydone. Then, by regresssion analysis, an isotherm expression can be obtained from type A measurement data. The characterization results according to these three types of representation for two kinds of leachate are shown in Table I. The results are the average values of a number of independent searches. On the basis of these results, the type B and C measurements were predicted and compared against experiments as shown in Table 11. It is clear that the predictions based on either type of representation give better agreement with experiments than the conventional single-species representation. The error in predictions increases as the adsorbates in the leachate becomes depleted, especially if the system is represented as a singlespecies solution. It was pointed out above that the present characterization method does not yield unique results. There are many choices in applying the species-groupingprinciple, with each choice leading to one set of results. Accordingly, it may be instructive to examine in detail the difference in results obtained from independent searches. For the present characterization example, a total of 23 independent searches was made, and the results, as expected, are not identical. The value of n ranges from nearly unity to almost five. On the other hand, the overwhelming majority of the n values is between 2.5 and 3.5. The average per-
Table I. Characterization Results of Leachate Case A:
Filtered and Lime-Treated EPA No. 1
(I) single-spec represent = 4.836
x 10-7~6.217
g of organic carbon per g of adsorbent g of organic carbon per L of soln (11) mult-spec represent with const n u n = 2.895 Ai 0 0.01 0.1 0.5 1.0 q: c:
5.0
ci, 2.093 3.57 1.42 0.12 0.042 0.11 (111) mult-spec representa ni/Ai 0.05 0.1 1.0 10.0 1.5 2.45 0.098 0.153 3.5 1.44 0.14 concn of nonadsorbable spec 3.89 Case B: Filtered, Lime-Treated, Air-Stripped, and H,SO,-Added
( I ) single-spec represent 4 = 6.426 x 1 0 - 4 ~ 5 4 q : g of organic carbon per g of adsorbent c: g of organic carbon per L of soln (11) mult-species representation with const n u n = 2.189 Ai 0 0.01 0.1 0.5 1.0 cj 0.80 4.178 3.05 0.02 0.031 (111) mult-spec representa ni/Ai 2.0
4.0
0.01 1.16 2.2
0.1 2.35 1.0
5.0 0.081
1.0 0.061 0.267
concn of nonadsorbable spec 1.0039 a Concentration u n i t in grams of organic carbon per liter of solution.
quirement is indeed satisfied by the present method as shown in the examples given below. (A) Characterization Leachate. Ahlert and Gorgol (1983) measured to determine the treatability of landfill leachate by carbon adsorption. Three types of adsorption measurements were taken. Their “type A measurements” were made by contacting treated leachate with carbon and
Table 11. Comparisons between Experimental D a t a of Type B Measurement a n d Predictions Based on Results of Table I Case A (A) a n d Case B ( B ) a n d of Type C Measurements and Predictions Based on Results of Table I Case A (C) a n d Case B (D) mult-spec represent mult-spec represent single-spec represent with const n “70 error MI V cT.. exptl CT% error CT% error CTA 5.40 7.24 7.2 7.52 0.56 -3.79 7.53 -4.05 -1.51 2.875 3.74 3.99 3.80 -6.66 3.83 -2.27 1.02 1.56 1.63 1.55 -4.57 1.56 0.84 -0.18 1.20 2.34 0.51 0.798 0.82 0.78 0.79 -2.36 B
5.0 2.50 1.658 0.5
7.42 3.22 2.12 0.735
7.50 4.07 2.72 0.816
-1.02 -26.48 -28.15 -11.02
1.0 0.6 0.3 0.1
7.94 4.62 2.18 0.695
7.72 4.79 2.43 0.816
2.74 -3.65 -11.7 -17.45
1.0 0.3 0.01 0.002
7.76 1.7 0.047 0.003 65
7.81 2.45 0.0816 0.0163
1.68 -44.0 -73.6 -347.1
7.35 3.77 2.54 0.78
0.92 -17.0 -19.81 -6.33
7.36 3.76 2.52 0.77
0.76 -16.65 -19.0 -5.03
7.82 4.65 2.28 0.724
1.54 -0.62 -4.64 -4.18
7.83 4.67 2.31 0.74
1.41 -1.06 -5.75 -6.52
C
D 7.78 2.25 0.058 0.01
-0.21 -32.4 -22.96 -169.5
7.73 2.24 0.058 0.008
0.33 -31.6 -23.3 -131.2
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
1235
Table 111. Detailed Results of Independent Searches in Characterizing Filtered and Lime-Treated EPA No. 1 Leacheate ci, Ai
search no.
n
1, 0
2, 0.01
3, 0.1
4, 0.5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
3.720 3.610 0.936 2.260 3.080 4.940 3.050 2.510 2.490 3.470 1.020 3.230 3.610 1.070 2.800 2.570 3.370 3.230 2.680 2.600 3.630 3.650 3.050
1.560 1.140 3.750 2.610 2.750 4.500 5.460 2.970 2.900 5.650 0.046 1.070 0.670 2.230 1.822 2.370 3.760 5.690 0.712 5.460 2.380 1.410 5.460
4.460 5.280 3.150 3.770 3.640 3.260 0.921 1.550 3.460 0.750 7.580 5.330 5.740 4.870 4.560 4.000 2.640 0.715 5.670 0.843 4.040 5.010 0.910
1.470 1.480 1.180 1.620 1.530 0.411 1.550 2.210 1.620 1.510 0.410 1.520 1.480 0.825 1.560 1.610 1.510 1.520 1.570 1.610 1.470 1.470 1.550
0.270 0.276
5, 1.0
6, 5.0
0.095 0.176 0.116
0.132 0.085 0.213
0.051
0.192
0.072
0.117 0.273 0.245
0.134
0.105 1.230 0.193 0.138
0.220 0.197 0.172 0.139
0.276 0.278 0.085
0.091 0.113 0.214 0.038
0.166
0.051
0.105
% error”
1046
2.03 2.04 3.35 2.17 2.07 4.80 2.08 2.20 2.15 2.04 3.20 2.05 2.04 3.07 2.12 2.13 2.05 2.07 2.13 2.12 2.03 2.04 2.08
0.990 0.990 1.370 1.020 0.998 1.500 1.000 1.040 1.010 0.990 1.030 0.995 0.988 1.340 1.000 1.010 0.990 0.996 1.000 1.060 0.990 0.990 1.000
“Average 70 error between type B and type C data and predictions. Table IV. Characterization Results of Kage’s Wastewater V
Symbol 0 A
(A) single-spec represent q = 0.049 x c’.OS3 of organic carbon per kg of adsorbent mol q: c: mol of organic carbon per m 3 of soln (B) mult-spec represent with const n n = 1.088 ai 0 0.005 0.01 0.1 1.0 5.0 ci 3.33 0.07 0.318 1 4 . 8 6 0.002 0.003 (C) mult-spec represent ni/Ai 0.0005 0.01 0.1 1.0 1.0 0.39 3.38 13.2 3.5 0.607 0.261 0.65 concn of nonadsorbable spec 0.087
Search NO 5
IO
‘ O t
4
I I , 01”r
I I 2 3 FRNNOLKH COEFFICIENT, A
1
1
4
Figure 2. Cumulative concentration distribution function of leachate A.
centage errors of the predicted type B and C measurements based on the results of each independent search are quite comparable, between 2% and 3% (see Table 111). A further test of this method’s validity can be made from the following considerations. If a cumulative adsorbate distributive function, M(A,n), is defined as M(A,n) = lAJ’m(A,n) dA dn 0
0
then M(Ao,no)represents the concentration of the adsorbates with A IA. at n Ino. In Figure 2, the values of M are plotted against A (since in this case, all species have the same n) according to the results of the independent searches summarized in Table 111. Comparable results on the cumulative adsorbate distribution function add credence to the characterization results. (B) Characterization of Industrial Waste. In connection with his earlier characterization studies, Kage
obtained both integral and differential batch contacting measurement data of an industrial waste consisting mainly of molasses. These integral adsorption data were used as the basis for characterization. In turn, based on the characterization results, differential measurements were predicted and compared with experiments. The characterization results are given in Table IV. As before, the waste was characterized in two ways: multiple species with constant n and multiple species with varying A and n. A single-species representation of the waste was also made based on the integral adsorption data. The comparison of the differential measurements is given in Table V. All three methods of representation give very good predictions. In fact, the single-species representation has the smallest percentage error. The significance of this fact will be discussed later. (C) Characterization of Aqueous Solutions of Multiple Organic Adsorbates. In addition to the industrial waste discussed earlier, Kage obtained both integral and differential batch adsorption data for aqueous solutions containing up to 11 organics. Solution characterizations were based on the integral adsorption data. As before, the solution was represented in two ways as well as single-species solution. The results for three of the solutions (designated by Kage as wastewater I, 111, and IV) are quite similar (see
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Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
Table V. Comparison between Differential Batch Contacting Measurement Data of Kage's Wastewater V and Predictions Based on Results of Table IV single-spec represent
CAM/ V
cT..
2.06 8.61 19.5 36.0 57.5
exptl
16.19 11.5 7.37 3.55 1.78
CT-
90 error
CT..
16.28 11.57 6.67 3.01 1.14
-0.54 -0.61 9.46 15.35 36.1
16.07 11.18 6.78 4.35 3.62
Table VI. Characterization Results of Kage's
Wastewater I, 111, and IV
(A)single-spec represent wastewater I q = 6.6 x c " * ~ ' wastewater I11 q = 6.456 x c0.4w wastewater IV q = 5 . 7 2 X c " . ~ q : mol of organic carbon per kg of adsorbent c : mol of organic carbon per m3 of soln
(B) mult-spec represent with const n wastewater I n = 3.225 Ai 0 0.1 1.0 5.0 10.0 ci
0.915
0.335
wastewater I11 ai ci
0
1.0
0.007
0.408
wastewater IV Ai
0
ci
0.002
5 9.74
0.177
0.304
15.05
100
-
n = 3.00 5.0 4.39
10.0 12.95
25.0 0.318
100 0.241
n = 3.57 10 50 12.11
(C) mult-spec represent wastewater I ni/Ai 1.0 5.0 2.0 0.54 5.0
1.77
0.684
4.4
10.0
50.0
7.71 8.06
0.249
concn of nonadsorbable spec 0 . 7 1 wastewater I11 ni/Ai 2.0 5.0
0.1 0.192
1.0 0.141 0.181
5.0 0.196 5.73
10.0 2.65 4.0
50.0 4.65 0.567
concn of nonadsorbable spec 0.0015 wastewater IV ni/Ai 0.01 1.0 10.0 50.0 2.5 5.0
0.94
0.413 0.423
0.007 24.3
mult-spec represent with const n
0.133
concn of nonadsorbable spec 0.046
Tables VI and VIII). Generally speaking, independent searches for each solution led to different but essentially similar results. Predictions of differential adsorption based on the characterization results give better agreement with experiments than do those based on single-species representation. Between the two kinds of representations (constant n vs. variable n),the latter was found to be more accurate. In particular, the best agreement was found in the case with the largest number of adsorbates. The differences between prediction and experiment become significant as the solution is depleted of adsorbates (in other words, large values of k A M / V or low concentration) for all three kinds of representations. This deviation occurs, however, at a much higher value of k A M / V for the more elaborate representation (varying n and A ) than for the single-species representation. For one of the solutions studied by Kage (designated as wastewater 11),that significantly different characterization results were obtained implied the possibility for error in the optimization scheme as outlined above. In all six cases
% error 0.75 2.83 7.93 -22.5 -103.6
mult-spec represent % error
CT-
16.17 10.89 6.56 4.05 3.02
0.14 5.4 10.9 -14.1 -69.9
discussed above, independent searches gave quite similar results. For wastewater 11, the results of the 21 independent searches (see Table VIII) can be grouped into two categories, with significantly different average values as shown in Table IX. Which one of these two sets of results should be used to represent wastewater II? As table IX illustrates, in terms of their respective accuracy in predicting the differential batch contacting measurements, there is little to choose from (22.4% vs. 25.3%). The objective function had the value of 8.526 X lod and 6.906 X lo4, respectively, corresponding to the two sets of results. Since the basis of solution characterization is the minimization of the objective function, one may conclude tentatively that the first set of results should be selected. The optimization-searchcalculations which yielded two sets of results as shown in Table VI11 are based on the integral batch contacting measurement data. In order to ascertain that this duality of results is not caused by using this particular type of data, optimization calculations were also made based on the differential batch contacting measurements data. For this purpose, a total of 23 independent searches was made, and similar behavior was observed as shown in Table X. The average values of the two sets of results obtained (see Table XI) are very similar to those given in Table IX although they are not identical. The set of results with a larger value of n now has better predictive accuracy as well as smaller values of the objective function. The results summarized in Tables VI11 and X make it difficult, if not impossible, to determine which set of characterization results should be selected. The adsorbates present in wastewater I1 may possibly be grouped into two categories, with their respective n values approximately equal to two and five. If this is the case, the systems should be represented as a solution of adsorbates with different n and A values. The results of such a representation are given in Table XII. In this representation, wastewater I1 is characterized as a solution with six adsorbable species, identified as [1.0,2.0], [5.0,2.0], [10.0,20], [1.0,5.0], [5.0,5.0], and [ 10.0,5.0], in addition to a nonadsorbable species. The amount of the species with n = 5.0, however, is rather insignificant and represents less than 2% of the total adsorbate concentration. On this basis, the set of results with n = 2.0 should be considered a better representation of wastewater I1 than the set of results with n = 5.0. A comparison between predictions of the differential batch contacting measurements and experiments is given in Table XIII. The predictions used the characterization results of Table XI1 as well as the results from the best search results of Table VIII. (Search no. 2 had the minimum value of 4.) Also included are predictions based on the single-species assumption with the isotherm expression obtained from the integral batch contacting measurement data. The percent error of the predictions based on the single-species assumption is approximately twice as large as that of the other two predictions. The improvement introduced by assuming the presence of adsorbates with
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
1237
Table VII. Comparison between Differential Batch Contacting Measurement Data in Kage’s Wastewaters I (A), I11 (B), and IV (C) and Predictions Based on Results Given in Table VI mult-spec represent with const n
single-spec represent
.EM1V
exptl
cT..
% error
CT-
mult-spec represent
CT-
% error
CT..
% error
16.46 12.36 6.80 3.16 1.59 1.34 1.27
-3.4 -7.0 7.8 23.97 16.4 -64.4 -124.8
16.44 12.38 6.80 2.96 1.18 0.82 0.73
15.75 10.16 4.3 0.99 0.35 0.22
-7.55 2.89 21.4 40.6 48.7 53.1
13.49 8.86 4.86 1.56 0.56 0.39
22.01 16.25 9.85 3.5 0.9 0.04
11.3 15.1 30.2 43.5 59.8 96.1
24.4 19.0 11.8 2.63 1.60 1.39
A 0.1 0.3 0.63 0.96 1.32 1.68 2.12
15.91 11.55 7.38 4.13 1.94 0.81 0.56
16.59 12.73 7.37 3.34 0.68 0.0006 0.0
-4.25 -10.22 0.08 19.23 64.41 99.93 100.0
0.1 0.4 0.82 1.34 1.66 1.87
14.64 10.46 5.47 1.66 0.68 0.46
16.18 11.07 5.09 0.69 0.002 0
-10.5 -5.8 6.9 58.2 99.7 100
0.1 0.4 0.835 1.55 2.0 2.30
22.64 17.58 13.09 5.88 2.18 1.04
21.59 14.84 7.33 0.77 0.0 0.0
4.6 15.6 44.0 86.8 100 100
-3.3 -7.2 7.8 28.2 38.0 -0.4 -28.9
B 7.9 15.3 1.11 5.8 17.7 16.1
C
Table VIII. Detailed Results of Independent Searches in Characterizing Kaae’s Wastewater I1 A search no. 1 2 3 4 5 6 7 8 9 10 11 12
n 1.940 1.911 2.279 1.910 1.940 1.960 2.960 2.160 2.120 2.000 1.930 1.960
1 2 3 4 5 6 7 8 9
5.00 5.00 5.05 5.00 5.00 5.00 5.01 5.11 5.00
0.568 0.003 0.180 0.003 0.005
5 16.20 15.45 11.90 15.90 15.70 15.40 15.33 14.60 14.50 15.50 15.83 15.30
1 1.820 2.350 2.230 2.350 2.310 2.320 2.340 1.393 2.210 2.020 2.340 2.330
0.703
0.420 0.029 1.650 0.990 1.630 0.017 0.117 0.110
4.05 4.07 4.05 4.08 4.07 4.08 4.06 3.95 4.07
2.268 1.840 2.230 0.608 1.270 0.622 2.250 2.150 2.150
8.63 8.54 8.63 8.56 8.59 8.53 8.56 8.74 8.57
0 0.313 0.002 0.003 0.002 0.013 0.008
Table IX. Average Characterization Results of Two Grows of Kaae’s Wastewater I1 from Table VI11 ____ group I group I1 2.0058 0.0916 15.1758 2.1678 0.5304 0.0621 0.3038 22.43 0.8527
5.0187 0.5514 4.0533 1.7089 8.5944 1.4081 2.0177 25.31 6.9089
n = 5.0 in addition to those with n = 2.0 is only marginal, since adsorbates with n = 2.0 are the dominant species. Conclusions The method developed in this study characterizes the adsorption affinity of solutions containing unknown multiple adsorbates. Based on experimental data which
10
100
4.096 0.050 0.287 0.570 0.640 0.353 0.003
0.132 0.115 0.161 0.230 0.911 1.911 3.390 3.080 0.074 2.202 0.139 0.918
50 0.003 0.030 0.123 0.007 0.067 0.020 1.400 1.500 0.496
3.160 2.540 1.480 0.043 0.336 3.380 1.422 3.230 2.510
1.7 0.68 16.4 57.5 28.2 -33.5
~~
% error
23.9 23.2 19.3 23.2 22.6 22.3 22.3 24.1 20.5 22.4 23.0 22.3
1046 1.050 0.134 1.870 0.144 0.257 0.216 0.200 3.350 1.950 0.867 0.192 0.202
23.5 25.5 23.7 32.2 19.5 31.6 23.8 24.0 24.0
6.98 6.95 6.80 7.07 7.02 7.07 6.93 6.43 6.93
can be gathered easily and by applying the method, a solution with unknown adsorbates is described as one with a number of adsorbates identified by their Freundlich constants and with specified concentrations. Two versions of the characterization method were developed. For the general case, the pseudospecies are allowed to have different values of both the Freundlich coefficients and Freundlich exponents. The determined quantities are the concentrations of the various pseudospecies identified by specified Freundlich constants. For the simplifed case, the Freundlich exponent of all the pseudospecies is assumed to be constant. In addition to determining the pseudospecies concentrations, the value of the exponent also needs to be searched. In either case, there is no limit as to the number of included pseudospecies nor to the magnitude of the values of A and n. The practical advantages of the proposed method can be seen by comparing it against the current practice and other competitive methods. In dealing with solutions of
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
1238
Table X. Detailed Results of Independent Searches in Characterizing Kage’s Wastewater I1 Based on Differential Batch Contacting Measurement Data concn (A =) search no. n 0 1.0 5.0 10.0 50.0 100 1O”d % error 1 2 3 4 5 6 7 8 9 10
2.170 2.190 2.160 2.170 2.820 2.970 2.150 2.190 2.76 1.255
3.080 3.230 1.500 1.430 0.286 0.212 0.070 2.337 3.070 0.008
0.173
1 2
5.114 5.320 5.200 5.000 5.040 5.000 5.340 5.000 5.000 5.200 5.000 5.010 5.085
2.190 2.890 1.320 1.990 1.820 0.310 2.980 0.145 2.630 3.830 0.655 1.530 0.741
1.040
3 4 5 6 7 8 9 10 11 12 13
1.960 2.050 2.760 3.550 3.570 0.985 0.181 9.850
1.850 1.190 1.480 2.940 0.330 3.100 0.584 2.570 1.690 2.500
15.80 15.64 15.60 15.50 9.96 15.40 15.48 15.80 9.18 3.45 3.68 3.31 3.56 3.29 3.44 3.00 3.45 3.48 3.52 3.47 3.50 3.50
Table XI. Average Characterization Results of Two GrouDs of Kage’s Wastewater I1 from Table X group I1 group I 2.2251 1.5523 2.5059 12.8360 1.8502 0.0243 0.0138 20.42 31.20
n C1
e2
c3 c4 c5 c6
% error 1034
5.0953 1.7178 1.4826 3.4269 12.4000 0.0160 0.0078 12.59 7.59
Table XII. Characterization Results of Kage’s Wastewater I1 ~
~~
(A) single-spec represent q = 2.818
x
q : mol of organic carbon per kg of adsorbent
c : mol of organic carbon per m3 of soln (B) mult-spec represent with const n n = 1.91 Ai 0 5.0 10.0 50.0 0.0019
c,
15.95
(C) mult-species represent nilAi 1.0 2.0 5.0
1.84 0.317
2.35 5.0 14.83 0.082
0.0268
10.0 1.23
concn of nonadsorbable spec 0.0213
unkown composition, the common practice assumes (either explicitly or implicitly) that all the adsorbates may be
0.003 0.175
6.000 12.320
0.002 0.001 0.020 0.060 0.050
25.60 25.70 22.70 22.40 14.60 11.90 20.00 23.70 25.50
0.005
12.12
10.60 10.85 10.90 11.00 25.30 8.00 11.20 11.50 10.60 210.00
0.03 0.019 0.009 0.002
12.50 9.34 11.21 11.90 12.45 12.30 14.80 18.90 12.11 11.70 12.10 12.21 12.12
7.5 7.03 6.83 8.05 8.70 8.00 5.52 8.00 8.00 7.11 8.00 7.99 8.00
0.245 0.004 12.37 12.47 12.57 12.31 12.25 12.37 12.72 12.36 12.34 12.36 12.35 12.33 12.40
0.208 0.020 0.002 0.040 0.001 0.006
grouped together (for example, in terms of the total organic carbon concentration) and obtains the so-called adsorption isotherm data from integral batch contacting measurements. Since the adsorption equilibrium relationship involving multiple adsorbates is determined not only by the total adsorbate concentration but also by the relative amount of individual adsorbates, application of the socalled single-species adsorption isotherm data to conditions different from those used to determine the isotherm data may lead to serious errors, as the various comparisons given in this work show. With the use of the proposed method, this limitation is, to a large extent, eliminated. The proposed method is also more advantageous and flexible than either Frick’s method or the method suggested by Kage. Frick’s method requires determining 3N - 1 quantities for a solution with N pseudospecies, while the present method requires the determining only N - 1 quantities. Furthermore, the complex method of optimization used in the present study offers a more efficient and systematic search method than the ad hoc procedure used by Frick. In contrast to Kage’s method, the present method is more complete in the sense that it allows the identification of pseudospecies by two parameters (A,n)instead of only one parameter (a or b ) , as in Kage’s method. The use of the IAS theory to predict multicomponent adsorption equilibrum is also preferable to the use of the Langmuir expression because of the higher accuracy of the IAS theory, at least in carbon adsorption. It should be emphasized that the proposed method, however, is not limited to the use of the IAS theory. It can be readily extended to any multicomponent adsorption theory, although the
Table XIII. Comparison between Differential Batch Contacting Measurements Data of Kage’s Wastewater I1 with Predictions Based on Results of Table XI1 mult-spec represent mult-spec represent single-spec represent with const n 70 error D M 1v CT. exptl I CT% error CT.. % error CT0.1 0.4 0.8 1.2 1.6 1.8
17.28 13.53 7.91 5.05 3.28 2.70
16.9 11.52 6.44 3.23 1.39 0.84
2.16 14.89 18.61 35.95 57.59 68.78
16.75 11.53 6.47 3.4 2.00 1.67
3.0 14.8 18.2 32.6 39.1 38.2
16.74 11.58 6.59 3.48 2.03 1.71
3.1 14.4
16.7 31.1 38.1 36.7
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
complexity of the required computation is directly related to the complexity of the adsorption theory used. The present method is not without its problem. It is based on the species-grouping assumption, which at best is only an approximation. The use of the Freundlich constant to characterize the adsorption behavior of substances may not always be correct. Furthermore, the application of the species-grouping principle is not unique. Consequently, a solution with a given composition may be characterized in a number of ways. The optimum obtained from each search may be viewed as the optimum corresponding to a particular grouping. For this reason, one must do a number of independent searches to ensure the validity of the characterization, thus increasing the required computation effort. Similar to any optimization calculation involving the determination of a large number of quantities, the optimiztaion computations may become stuck. In carrying out calculations for the seven application examples, this situation occurred once. As shown in the case of Kage's wastewater 11, when the optimization calculations become stuck, one is confrontred with two sets of characterization results. Additional computation is required to make the proper selection. Two observations can be made from the examples examined in this study. Firstly, the manner with which a solution may be adequately characterized depends upon the complexity of the solution. A simple solution, which has few adsorbates or a dominant species, can best be represented as one with few or even a single pseudospecies. If a solution with a single adsorbate is to be represented as a solution of multiple species, the characterization is, in fact, made by applying species grouping in reverse namely, by dividing that adsorbate into several pseudospecies. A more complex characterization in certain cases may actually introduce greater error than a simple characterization. Secondly, the method is best suited to characterize solutions with a large number of adsorbates with comparable concentrations, as shown in the case of characterizing Kage's wastewater 111. A potential application of this method would be to characterize the so-called background substance in natural water systems. A logical question a t this stage is to what degree of accuracy the proposed method is capable of representing a solution of unknown composition. It is clear that the pseudospecies representation is more than an attempt of curve fitting. This is shown in the various examples in which the characterization results were found to give accurate prediction under much diluted conditions. More important, is the utility of the method in practical application such as in the calculation of fixed-bed adsorption of solutions with unknown adsorbates. As mentioned earlier, the dynamics of fixed-bed adsorption depends on the equilibrium relationship as well as on the rate expression. A complete characterization of the solution, therefore, requires identifying the pseudospecies by their Freundlich constants as well as by their rate parameters. While it is possible to formulate an optimization-search procedure to determine the additional rate parameters for the pseudospecies, the optimization will be difficult to implement because of the large quantities to be deter-
1239
mined. A more practical way of solving the fixed-bed adsorption problem would be, first, to obtain breakthrough curves of fixed-beds based on the local equilibrium assumption (namely, to ignore the mass-transfer effect) and, then, to correct the mass-transfer effect empirically. Such an investigation constitutes a logical continuation of the present work and is strongly recommended. Acknowledgment This was performed under Contract R809482, US Environmental Protection Agency. Nomenclature A = Freundlich coefficient Ai = Freundlich coefficient of the ith species a, b = Langmuir constants [bi] = random number ci = concentration of the ith species in solution c . = initial value of ci cy = concentration of the ith species of the single state in the solution phase ci. = equilibrium value of ci cT = total adsorbate concentration in solution cT0 = initial value of CT cTr = tracer substance concentrations in solution c = equilibrium values of cTr cgY= value of CT achieved the kth addition of adsorbents Eio = coordinates of centroid (eq 12) d k = distance between centroid and vortex K = number of vortexes M = mass of adsorbent M . = molecular weight of the ith species Mi = equal to k(AA4) M ( A ) = cumulative concentration distribution function of solution m = frequency concentration distribution function of solution N = number of species n = reciprocal of Freundlich coefficient ni = value of n for the ith species q . = concentration of the ith species adsorbent phase qp = value of q i in the single species state q T = total adsorbate concentration in the adsorbed phase qT, = equilibrium value of qT S = quantity appearing in eq 9 and 10 Si = number defined by eq 16 v = volume of solution vk = concentration value (see eq 4) w = frequency distribution function in the adsorbed phase x i = mass of the adsorbate per unit mass of organic carbon zi = mole fraction of the ith species in the adsorbed phase Greek Letters AM = incremental amount of adsorbent C$ = objective function ?r = quantity appearing in eq 9 and 10 Literature Cited Ahiert, R. C.; Gorgoi, J. F. Environ. Prog. 1983, 2 , 54. Caiiigaris, M.: Tien, C. Can. J. Chem. Eng. 1982, 60, 772. Frick, F. B., Dr.-Ing. Dissertation, Karisruhe University, 1980. Kage, H. Dr. Eng. Dissertation, Kyoto University, 1980. Mehrotra, A. K.; Tien, C. Can. J . Chem. Eng. 1984, 62, 632. Okazaki, M.; Kage, H.; Toei, R. J . Chem. Eng. Jpn. 1981, 13, 286. Okazaki, M.; lijma, F.; Toei. R. J. Chem. Eng. Jpn. 1981, 74, 26. Tien, C. In "Fundamentals of Adsorption", Myers, A. L., Belfort, G., Eds.; United Engineering Trustees: New York, 1984; p 647.
Receiued for review October 22, 1984 Revised manuscript received March 5 , 1985
