284
I n d . Eng. C h e m . Res. 1987, 26, 284-292
Valov, P. I.; Blyumberg, E. A.; Emanuel', N. M. Izu. Akad. Nauk SSSR, Ser. Khim. 1969, 791. Walling, C. J . Am. Chem. SOC.1969, 91, 7590. Wang, C . K. M.S. of National Cheng Kung University, Tainan, Taiwan, R.O.C., 1984. Wohlers, H. C.; Sack, M.; Leven, H. P. Ind. Eng. Chem. 1958,50(11), 1685.
Wisniak, J.; Cancino, A.; Vega, J. C. Ind. Eng. Chem. Prod. Res. Deu. 1964, 3(4), 306. Wisniak, J.; Navarrete, E. Ind. Eng. Chem. Prod. Res. Deu. 1970, 9(1), 33.
Received for review November I, 1985 Accepted August 18, 1986
Further Development of the Adsorption Affinity Characterization Procedure for Aqueous Solutions with Unknown Compositions H. Kage* and Chi Tien Department of Chemical Engineering and Materials Science, Syracuse University, Syracuse, New York 13210
A systematic application of the Jayaraj-Tien procedure which describes a solution containing a large number of unknown adsorbates with a finite number of pseudospecies is made, thus demonstrating the practical utility of the Jayaraj-Tien procedure for characterizing the adsorption affinity of solutions of unknown compositions. In two previous publications (Jayaraj and Tien, 1984, 1985) proposed a procedure to characterize the adsorption affinity of aqueous solutions which contain a large number of unknown adsorbates. By application of this procedure to relavant adsorption data, a solution of unknown composition can be described as one of a number of pseudospecies identified by isotherm constants and with specified concentrations. The need for such a procedure is obvious. Many liquid wastes and natural water systems which require adsorption treatment for purification or removal of specific constituents often contain a large number of dissolved organic substances whose identities are not completely known. The lack of information on adsorbate identity and composition had made it difficult, if not impossible, to apply adsorption theories to the rational design of treatment processes for these systems. The results of the earlier works have shown the utility of the characterization procedure. It was found that predictions of adsorption equilibrium concentrations based on the characterization results were much more accurate than those obtained from current practices of lumping all the constitutes into a single group. The present study represents a continuing effort to develop further the characterization procedure. The purpose is to devise a specific scheme for a straightforward and systematic application of the Jayaraj-Tien procedure. This scheme is established through case studies of adsorption data of solutions with a large number of disparate adsorbates. The work reported in this paper consists of three parts: (I) Carbon adsorption experiments of solutions containing a fairly large number of adsorbates at different total concentration levels were performed to obtain adsorption equilibrium data; these data were then used in the case studies. (11) A systematic scheme of applying the characterization procedure to the data was then developed in which the number of pseudospecies used to represent the solution was increased step by step in order to improve the accuracy of representation. (111)Predictions of equilibrium concentrations (based on characterization results of data which were obtained using solutions at one concentration) *Present address: Department of Industrial Chemistry, Kyushu Institute of Technology, Kitakyushu, Japan.
involving solution at more dilute concentration levels were made and compared with experiments. The comparisons provided empirical evidence of the validity and accuracy of the procedure as well as guidelines to properly apply the characterization procedures.
Principles of Characterization Procedures Detailed information on the characterization procedure has been given elsewhere (Jayaraj and Tien, 1984, 1985), and only a brief description will be given below as background for presenting the results of the present study. Consider a solution with a large number of adsorbates ranging from highly adsorbable to only slightly adsorbable. If one assumes that the adsorption affinity of compounds can be described by a parameter vector, a, and furthermore for the sake of discussion a is assumed to have two components, cy1 and cy2, each adsorbate is represented by a point on the cy1-cy2 domain as shown in Figure 1. Tien and co-workers (Tien, 1984; Calligaris and Tien, 1982; Mehrotra and Tien, 1984) have shown that multicomponent adsorption calculations can be significantly simplified by combining adsorbates with similar affinities into a group and considering the group as a single entity (species grouping). By applying the species grouping principle, one may combine adsorbates represented by various points in the cy1-cy2 plane near a given grid point of the cy1-cy2 network (as shown in Figure 1) to form a pseudospecies identified by the coordinates of that grid point. In this manner, a solution with a large number of adsorbates may be approximated by one with a fewer number of pseudospecies identified by the coordinates of the point of the grid network. The characterization of the solution is completed by assigning concentration values to the pseudospecies. The assignment of concentration values to various species may proceed in the following manner. The total adsorbate equilibrium concentration achieved when a given volume of the solution is brought into contact with a specified quantity of adsorbents indicates the adsorption affinity of the adsorbates involved. On the other hand, assuming that a solution is properly characterized as one with a number of pseudospecies with specified concentrations, it would be possible to calculate the equilibrium concentrations achieved under given conditions with the
0888-SSS5/87/2626-0284~01.50/0 0 1987 American Chemical Society
Ind. Eng. Chem. Res. Vol. 26, No. 2, 1987 285 Table I. Compositions of Solutions Used
"mox
0 .
C
i
P
g
I
W X
1
01
I "3
I?
dz J
E
"2
B d
u
0
E w r
"1
0
"min
Amin A I
0
42 A3 FREUMLICH COEFFICENT. A
Figure 1. Representation of a solution of large numbers of adsorbates by pseudospecies, each point corresponding to a given adsorbate.
use of appropriate multicomponent adsorption theories. Accordingly, with equilibrium concentration data available, the pseudospecies concentrations can be determined on the basis that they give the best estimates of the equilibrium concentrations achieved under experimental conditions. Let u denote certain concentration values achieved under adsorption equilibrium, which can be predicted as well as experimentally determined. An objective function 4 is defined as
where the superscript k denotes the kth value of u and the subscripts exptl and calcd denote the experimental and calculated values, respectively. The pseudospecies concentrations are determined by minimizing the objective function subject to the constraint that the sum of the pseudospecies concentration is specified and equal to the total adsorbate concentration of the solution cT0 or Thus, the problems involved in the characterization procedure can be described to be (a) the selection of the parameter vector (a)to identify the pseudospecies, (b) the development of a method of calculation which can be used to predict the equilibrium concentrations achieved between the solution to be characterized and adsorbents from the knowledge about the pseudospecies (i.e., their parameter vector values and concentrations), and (c) the development of efficient algorithms to search the conditions which give a minimum value to the objective function. Jayaraj and Tien (1984,1985) suggested the use of the Freundlich constants ( A ,n) of single adsorbate adsorption isotherms to be the parameter vector. With this selection, the second problem (namely, the selection of an appropriate theory for predicting adsorption equilibrium) is readily solved with the use of the ideal adsorbed solution (IAS) theory. However, it should be noted that the characterization procedure is not limited to the use of the IAS theory. To carry out the optimization search, the complex method was found to be most effective. To implement the characterization procedure, both the ranges of the parameter vector (a) and the number of the pseudospecies must be specified. The parameter vector (a) describes the adsorption affinity of a compound. To
substance adipic acid benzoic acid cyclohexanone a-cresol 2,4-dichlorophenol dichloromethane nitrobenzene p-nitrophenol phenol propionic acid s-tetrachloroethane 1,1,2-trichloroethane
concn ratio (percentage of the adsorbate of the total adsorbates in organic C) solution I solution I1 11.54 9.09 4.62 2.45 3.85 0.91 24.55 11.54 3.64 7.69 4.55 7.69 3.85 15.38 26.36 15.38 21.27 15.38 0.45 1.54 1.54 0.73
provide a correct characterization, the pseudospecies must cover a range of adsorption affinities comparable to that of the adsorbates present in the solution. Although this information can be obtained by using the tracer compound technique of Frick and Sontheimer (1983))the technique is tedious and often impractical. As for the number of pseudospecies to be used in characterization, one may argue intuitively that the accuracy of representation improves with the number of pseudospecies used. No evidence, theoretical or empirical, however, is available to support this argument.
Experimental Work Equilibrium concentration data were obtained by contacting aqueous solutions containing multiple adsorbates at various concentration levels with granular activated carbon (Filtrasorb 400, Calgon Corporation, Pittsburgh, PA) corresponding to different adsorbent-solution ratios. Two kinds of aqueous solutions containing 12 and 10 adsorbates, respectively, at different concentration levels were used. These adsorbates are listed in Table I. The experimental measurements were carried out by introducing test solutions and carbon particles into 250-mL flasks with different volume-to-mass ratios and placing these flasks in an oscillating water bath maintained at 20 OC. Samples were taken and their concentrations analyzed until equilibrium was reached. Generally speaking, it took 10-14 days to reach equilibrium. The total adsorbate concentrations were determined by using a total carbon analyzer (Beckman Model 915 TOCAMASTER). The experimental results can be expressed in the form of q T vs. CT where q T and CT denote the total adsorbate concentrations in the adsorbed and solution phases, respectively, or in the form of cT vs. M I V where M and V are the mass of carbon and the volume of the solutions used in the measurement. Figures 2 and 3 show the results obtained in the form of q T vs. cT. Application of Characterization Procedure The characterization of each of the two solutions was based on the total adsorbate equilibrium concentration data obtained by using that solution at the highest concentration level. On the basis of the characterization results, predictions of CT values involving the same type of solution but at lower concentration levels were made and compared with experiments. The rationale for doing this is obvious. If a solution is properly characterized as one with a number of pseudospecies with specific concentration values, diluted versions of that solution can be simply represented by the same pseudospecies with appropriate reductions of their concentration values. Thus, agreement
286 Ind. Eng. Chem. Res. Vol. 26, No. 2 , 1987
where N
When eq 6 is substituted into eq 5, ci0 is found to be
1
1+-Vl+D
The total adsorbate equilibrium concentration, cT., can be found to be I
KXX,
=I -
I
1
1
I l l /
o
523ppm 4 1 . 5 ~
0
222ppm
"
' 0 0 7
"
0 I)
i
l
-
"
A
and A
A .
n 0
A
n
. 0
0
A A
1+-Vl+D
-
A
U
0
l
A A
01
!i
I
I , I I V
~ ~ - 2 1 6 ~ ~ 125ppm
A A
c--
L-
I
I
0 .
.
.
bicio
N
.*
D = C
--
i=1~
0
n
.
M
1+--
-
Figure 3. Equilibrium concentration data of solution 11.
between the predictions and experiments constitutes a validation of the characterization results. As mentioned earlier, the characterization procedure requires the use of adsorption theories which enable the prediction of the multicomponent adsorption equilibrium from the single species isotherm data of the individual components. The two theories used in this work are the Langmuir theory and the ideal adsorbed solution theory. (1) Langmuir Theory. For a solution containing N species with concentrations cl, c2, ..., cN, the adsorbed phase concentration, qi, in equilibrium with the solution is given as
Thus, by knowing the composition of the solution, D can be found from eq 10 (by trial and error). Once D is known, cT, can be obtained directly from eq 9. (2) Ideal Adsorbed Solution Theory. Several investigators (Wang and Tien, 1982; Calligaris and Tien, 1982; Tien, 1986) have incorporated the ideal adsorbed solution theory in multicomponent adsorption calculations. The counterparts of eq 9 and 10, based on the assumption that the single species isotherm data of the individual component obey the Freundlich expression, are (Calligaris and Tien, 1982) N CT,
= Cci,
(11)
i=l
N
where ai and bi are the Langmuir constants of the ith species in its single species state. In other words, for a solution with the ith species alone and with concentration c;O,the adsorbed phase concentration in equilibrium with the solution, q:, is given as
ai V l + D
1-C
S
- - cN
Ci0
nFl0
=0
(14)
=0
(15)
P
(4) A given quantity (M)of fresh adsorbent is added to a specified volume (V) of solution with concentration clo, czo, ..., cNo. A t equilibrium, the concentrations of the solution and adsorbed phases are cl, CZ,, ..., C N , and qi=,qz., ...,qN; By mass balance, one has (5) WC,,- c,J = Mqlm Since the Langmuir theory is assumed to be applied, by eq 3, one has
where A , and n, are the Freundlich coefficient and the reciprocal of the exponent, respectively. The concentrations (elm and q,,) are expressed as the organic carbon mass of the species per unit volume of the solution or unit mass of the adsorbent. M I and X I denote, respectively, the molecular weight and the mass of adsorbate per unit mass of the organic carbon of the ith species, which may be assumed to be the same for all the pseudospecies. By knowing the solution composition and the values of M and
Ind. Eng. Chem. Res. Vol. 26, No. 2, 1987 287 Table SI. Characterization Results of Solution I (IAS Theory) (A) representation by a single species search no. A n 1 2 3 4 5
22.30 21.77 21.86 22.24 22.15
E , 90
2.718 2.674 2.678 2.711 2.711
5.7692 5.8716 5.9628 5.8442 5.6403
(B) representation by two pseudospecies concn, mg of organic C / L
search 1 2 3 4 5
search no. 1
2 3 4 5
search no. 1 2 3 4 5
search no.
(10.0, 2.7)
(30, 2.7)
E, %
1 2 3 4 5
34.6 34.7 34.6 34.9 34.5
230.4 230.3 230.4 230.1 230.5
12.33 12.35 12.33 12.39 12.31
(C) representation by four pseudospecies concn, mg of organic C / L no. (10, 21 (10, 4) 130, 21 5.66 6.20 74.40 26.20 71.40 12.80 4.07 73.40 14.80 3.62 72.50 25.80 69.80 (D)reoresentation bv six oseudospecies concn, mg of organic C/L (10, 21 w,41 (30, 21 130, 41 8.3 55.6 156.0 9.2 35.7 134.6 15.5 3.7 24.5 113.1 10.2 7.8 101.6 9.9 105.9 (E) representation by nine pseudospecies concn, mg of organic C/L (10, 2) (10, 4) (10, 6) (30, 2) (30, 4) (30, 8) 11.0 5.41 0.16 93.3 12.9 1.61 1.74 40.2 79.3 14.4 10.8 8.52 10.1 0.03 0.04 14.6 3.22 0.13 13.4 7.61 7.36 0.02
V, the two quantities, S and ?r (or more correctly, SXi/Mi and TMJX~),can be found from eq 14 and 15. Once S and T are known, the value of ziand, subsequently, ci, and cT, can be calculated from eq 13, 12, and 11. With the availability of eq 9 and 10 or eq 11and 15, the characterization procedure, in essence, involves the assignment of ci0 values to the various pseudospecies such that the value of the objective function (4) is a minimum. The computation can be made with the algorithm developed earlier (Jayaraj and Tien, 1984, 1985). Before the characterization procedure can be carried out, the number of pseudospecies as well as their values has to be selected first. In the following examples, a systematic scheme for selecting these values which was found useful in this work will be described. (A) Characterization of Solution Type I with the IAS Theory. In this case, the solution will be represented by a number of pseudospecies with the assumption that the IAS theory can be used to predict the multicomponent adsorption equilibrium. The data base was the equilibrium concentration data obtained by using the concentration at the highest concentration level (cT0= 265 mg of organic carbon/L of solution) as showin in Figure 2. In the absence of any information concerning the identities of the adsorbates in a solution, the logical starting point is to group all the adsorbates into a single species. A random search of the Freundlich constants (A, n) of the
150, 21 3.1 5.6
130~41
E, 70
178.70 167.40 174.80 174.10 169.40
6.20 6.27 6.22 6.06 6.05
(50, 4)
E, %
45.1 82.4 108.2 145.5 143.5
5.82 5.19 6.00 5.85 5.84
(50, 2)
(50, 4)
(50, 6)
E , 70
0.04
153.0 92.3 59.7 70.8 70.6
2.08 22.4 175.9 176.3 166.0
5.57 6.11 2.10 1.73 2.27
pseudospecies was made, and the results are given in Table 11. As an indication of the degree of fitting, an average error, E , is defined as
where the subscripts exptl and pred denote, respectively, the experimental and predicted values. The index i denotes the ith data value and the n such values. The search results shown in Table I demonstrate two points worth mentioning. All five independent searches yield essentially the same results. Note that it is more direct to obtain the A and n values from regression analysis rather than from a search base on the complex method. In an indirect way, the results do demonstrate the efficacy of the search algorithms developed earlier. More importantly, one should conclude that a good fit (as indicated by a relatively small value of E ) does not necessarily imply valid representation. The inadequacy of using a single pseudospecies representative is obvious as shown in Figure 2 . Should it indeed be possible to combine all the adsorbates to form a single group, all the equilibrium concentration data obtained from using solution I of different concentrations should follow the same trend. Instead, as shown in Figure 2, the q T vs. cT data
288 Ind. Eng. Chem. Res. Vol. 26, No. 2, 1987 Table 111. Errors in Predicting c V . Solution I (IAS Theorv) E ( % ) for no. of pseudospecies used in characterization 1 cT:
138 67.1 31.4
max 22.18 45.12 66.16
min 21.66 44.75 64.00
av 21.95 44.92 64.99
max 16.15 30.29 53.73
2 min 15.98 30.25 53.41
av 16.08 30.27 53.54
max 22.42 38.72 58.49
4 min 21.01 37.77 51.62
av 21.63 38.27 54.90
max 22.97 37.99 47.97
6 min 21.74 33.87 37.48
av 22.55 35.40 42.37
max 25.37 35.04 45.92
9 min 21.78 28.38 28.85
av 23.81 31.55 35.46
Initial solution chemistry in milligrams of organic carbon/liter.
obtained from solution I of different concentrations form separate and significantly different curves. It is obvious that an improved characterization of the solution requires that the solution be represented as one with more than a single species of adsorbate. If so, what is the number of pseudospecies and what are their identities? Since there is an obvious trade-off between the complexity of the required search and the number of pseudospecies used, it is reasonable to increase the number of the pseudospecies in stages in order to give some insight about the relationship between the accuracy of representation and the number of the pseudospecies used in the representation. On this basis, it was decided to represent the solution as one with two pseudospecies as well as one with four pseudospecies. If the results based on the single pseudospecies may claim some degree of validity, one may argue that pseudospecies should be identified by A and n values around A = 22 and n = 2.7. Accordingly, for the two-pseudospecies representation, the pseudospecies were assumed to be (10, 2.7) and (30, 2.71, where the first number in the braces denotes the A value and the second number the n value. For the four-species representation, the pseudospecies were assumed to be (10, 21, (30, 21, (10, 41, and 130, 4). The characterization results are given as parts B and C of Table 11. In both types of representation, the results indicate high concentrations of the species with the highest A value and n value (for the six-pseudospecies case). Thus, it was natural to inquire whether more species should be considered in order to expand the ranges of the A and n values. It is also of some importance to note that the average error of fitting increased from case A to case B. The values of E for the case where four pseudospecies were used, however, were comparable to those obtained with the single pseudospecies case. To further expand the number of species of pseudospecies, two versions of representation, in which the pseudospecies were assumed to be six and nine, respectively, were considered. For the first case, the pseudospecies were (10, 21, (10, 41, (30, 21, (30, 41, (50, 21, and (50, 4). Note that the first four species were the same as the four pseudospecies case. The last two were additional ones in order to expand the range of the A value. For the nine-pseudospecies case, the pseudospecies were (10, 2), (10, 4), 110, 61, (30, 21, (30, 41, (30, 61, 150, 21, (50,4),and (50,6). Comparing it to the six-pseudospecies case, the additional species were {lo,61, (30, 61, and (50, 6). In other words, as the number of pseudospecies increased from six to nine, the maximum n value included extended from four to six. The results of five independent searches for these two cases are given in parts D and E of Table 11. From these results, two significant inferences can be drawn. For the use of a given number of pseudospecies, the five independent searches yielded different results. The differences were rather small when the number of the pseudospecies was small but increased as the number of pseudospecies increased. This lack of uniqueness results directly from the fact that species grouping which forms the basis of the
characterization, is not unique. The degree of freedom increases with the increase in the number of pseudospecies used, giving rise to the situation that in the nine-pseudospecies cases, there were substantial differences among the concentration values obtained from each independent search. Furthermore, the average error, E, improved with the use of a greater number of pseadospecies. Because of these considerations, it does not appear useful to consider a representation which uses more than nine pseudospecies. Intuitively, one may argue that the inclusion of more than 10 adsorbates would render most adsorption calculations impractical. The validity of the various characterization results was tested by comparing predictons based on all the results (five sets of results for a given number of pseudospecies used) against experimental data obtained by using solution I but at lower concentration levels (in other words, all the data points shown in Figure 2 except those corresponding to cT, = 265 mg of organic carbon/L, which were used in the characterization). The maximum, minimum, and average values of E are tabulated in Table 111. The results of Table I11 indicate that the major advantage of employing a large number of pseudospecies in the solution representation is to provide a better and, equally important, more consistent accuracy in prediction. It is true that when cT, is predicted involving the solution of cTo = 138 mg of organic C/L, the average error in prediction based on a single-pseudospecies representation is marginally better than that based on a nine-pseudospecies representation (namely, 21.95% vs. 23.81 %). However, consider the solutions collectively; the predictive accuracy achieved by the nine-species representation is far superior to that achieved by the single-species representation (prediction errors of 23.81%, 31.55%, and 35.40% vs. 21.95%, 44.92%, and 64.99%). In other words, the ninepseudospecies representation not only gives much less error in predicting cT, for more diluted solutions but also provides a predictive accuracy relatively independent of the initial concentrations of the solutions. This same kind of result was also observed in the case of solution I1 to be discussed later. (B) Characterization of Solution I with the Langmuir Theory. This case study was based on the same data as A [namely, cT, (or qT,) data obtained by using solution I with initial concentration cT0 = 265 mg of organic C/L] with the exception that the solution was assumed to obey the Langmuir theory. Accordingly, the pseudospecies were identified by the Langmuir constants a and b, and the calculation of cT, (in both search and prediction) was made according to eq 6-10. The scheme used to carry out the characterization was the same as before. Initially, the solution was assumed to be of a single species. Subsequently, the number of species considered increased to two (by expanding the ranges of a ) and four (by expanding both of the ranges of a and b). The values of a and b were further expanded, and the corresponding number of the pseudospecies were increased to six and finally to nine. The results corresponding to five independent searches for each case are summarized
Ind. Eng. Chem. Res. Vol. 26, No. 2, 1987 289 Table IV. Characterization Results of Solution I (Langmuir Theory) (A) representation bv a single sDecies a b search no.
E, 70
0.0424 0.0427 0.0425 0.0426 0.0423
6.132 6.146 6.124 6.143 6.119
3.25 3.26 3.24 3.25 3.24
(B)representation by two pseudospecies concn, mg of organic C / L (10, 0.04) E , 90 1 190.8 74.2 3.11 74.2 3.11 190.8 74.6 3.15 190.4 74.4 3.13 190.6 74.3 3.12 190.7 (C) representation by four pseudospecies concn, mg of organic C/L search no. 15, 0011 15, 0.1) (10, 0.01) (10, 0.1) E,% 1 94.9 57.7 60.7 51.7 3.18 2 107 51.7 46.9 59.4 3.23 3 94.1 59.1 62.1 49.7 3.16 4 68.3 69.2 87.9 39.6 3.14 5 70.6 67.9 85.1 41.4 3.15 (D)representation by six pseudospecies concn, mg of organic C/L 11, 0.011 11, 0.11 (7, 0.011 17, 0.1) (15, 0.01) (15, 0.1) E, % search no. 1 14.5 41.4 108.6 56.3 44.2 3.00 2 8.8 2.7 99.0 79.7 74.8 3.27 3 17.6 69.6 73.6 1.9 102.2 3.18 4 7.6 4.7 79.7 65.1 107.7 3.33 5 15.6 38.4 102.6 50.9 57.5 3.02 (E)remesentation of nine DseudosDecies concn, mg of organic C/L searchno. 11, 0.0051 (15, 0.05) (1,0.5) (7, 0.005) (7, 0.5) (15, 0.005) (1,0.05) (7, 0.05) (15, 0.5) E, % 1 10.0 2.6 1.2 57.5 59.6 1.8 53.4 52.8 26.0 3.42 2 20.6 35.6 53.4 12.7 56.9 71.1 14.7 3.08 3 8.3 9.7 46.7 15.3 11.3 42.3 60.2 21.2 3.22 4 21.1 29.7 13.8 37.0 17.2 138.1 8.15 3.04 5 20.8 46.7 34.3 13.8 84.5 49.6 15.3 3.02 (5, 0.041
search no.
Table V. Errors in Predicting
CT..,
Solution I (Langmuir Theory) E (%) for no. of pseudospecies used in characterization 2
1
cT; 138 67.1 31.4 a
max 17.12 48.29 115.4
min 17.03 48.03 115.1
av 17.06 48.16 115.3
max 15.08 49.17 116.8
min 15.07 49.03 116.6
4
av 15.08 44.12 116.7
max 17.03 45.65 110.6
min 16.87 42.69 105.5
6
9
av
max
min
av
max
min
av
16.96 44.16 108.0
18.03 38.39 96.85
17.14 34.94 90.66
17.51 36.79 92.81
18.36 35.41 89.96
17.34 29.94 81.12
17.97 33.10 86.41
Initial concentration in milligrams of organic carbon/liter.
in Table IV. Table V, similar to Table 111, gives the errors between the predicted values of cT, based on the characterization result of Table IV with experimental data obtained from solution I1 a t lower concentration levels (cT0 = 138, 67.1, and 31.4 mg of organic C/L). The results of these calculations are qualitatively similar to those given in Tables I1 and 111. An increase in the number of pseudospecies does not change the degree of fitting. The values of E for all five cases (see Table IV) are very comparable. On the other hand, the predictive accuracy increases with the increase of the number of pseudospecies used especially at low concentration levels. Comparing the characterization results based on the Langmuir theory and those based on the IAS theory, the latter results consistently give much better predictions. These conclusions, however, could not be expected if the closeness in data fitting was considered as the sole criteria
to determine the quality of representation. (C) Characterization of Solution of Type 11. The equilibrium concentration data shown in Figure 3 were obtained by using solutions of type I1 a t five different concentration levels (cT0= 216, 125, 525,41.5, and 22.2 mg of organic C/L). The equilibrium concentration data corresponding to the highest value of cT0 = 216 mg of organic C/L were used as the basis of characterization. Both the Langmuir and the IAS theories were used to predict the multicomponent adsorption equilibria behavior. The characterization results based on the IAS theory are summarized in Table VI. On the basis of the results, predictions of cTAnfusing solutions of lower concentration levels were made and compared with experimental data. The error of comparisons for all the cases are given in Table VII. The characterization results based on the Langmuir theory are given in Table VIII. Table IX gives
290 Ind. Eng. Chem. Res. Vol. 26, No. 2, 1987 Table VI. Characterization Results of Solution I1 (IAS Theory) (A) representation by a single species search no. A n 1.544 1.047 0.741 0.881 0.730 0.879 0.722 0.877 0.733 0.88 (B)representation by two pseudospecies concn. me of oreanic C / L search no. E.5, 0.9) (10, 0.9) 159.1 56.9 158.8 57.2 159.5 56.5 56.4 159.6 158.8 57.2 (C) representation by four pseudospecies concn. me of oreanic C / L
E, % 5.79 4.05 4.03 4.01 4.03
E, % 6.70 6.66 6.76 6.77 6.66
112.0 111.8 111.9 111.0 112.2
69.3 68.9 65.5 71.4 66.7
34.7 35.6 38.6 33.6 37.1
8.68 8.71 3.60 9.00 8.62
(D) representation by six pseudospecies search no. 1 2 3 4 5
{0.5, 13) 11.0 15.5 32.5 15.2 13.7
P.5, 51
concn, mg of organic C/L 110, 11 iw51 44.7 38.3 43.2 34.4 43.7 21.5 37.6 35.2 45.5 36.1
150, 11
4.26
E , 9'0
150, 51 122.0 122.9 118.2 123.8 120.7
3.14 3.06 3.12 3.31 3.13
(E) representation by nine pseudospecies concn, mg of organic C/L search no. 1
2 3 4 5
(0.5, 0.5) 49.7 1.3 0.1
(0.5, 11 23.2 15.2 38.4 18.3 6.0
(0.5, 5)
110, 0.51
2.0 1.3
(10, 11 1.1 43.3 35.7 40.2 35.0
110, 5) 29.2 34.9 16.7 32.7 42.6
150, 0.51
150, 1)
0.1
0.4 0.8 7.1
(50, 51 112.8 121.2 122.7 122.7 125.4
E , 74 5.93 3.15 3.28 3.24 3.52
Table VII. E r r o r s in Predicting cT-,Solution I1 (IAS Theory) E I% ) for no. of DseudosDecies used in characterization
'"
125 21.59 17.41 20.65 20.11 19.74 19.91 14.56 13.42 13.83 3.12 2.77 2.97 5.85 2.59 3.83 31.74 33.16 37.34 32.02 33.63 92.99 92.29 92.61 62.40 59.82 60.79 34.16 52.3 107.7 93.05 104.7 79.48 74.39 76.79 82.33 71.75 77.0 41.5 159.5 149.6 157.4 133.6 132.9 133.2 104.3 103.0 103.5 263.5 158.4 132.4 140.1 179.8 120.2 147.4 22.2 407.5 356.7 396.9 357.9 356.3 357.0 268.0 260.2 Initial concentrtion in milligrams of organic carbon/liter.
the comparison errors between experimental data and predictions based on the characterization results of Table VIII. The scheme used to carry out the characterization was identical with that used to characterize solution I, beginning with a single-species representation and gradually increasing the number to nine. The results of Table VI1 are qualitatively similar to those of Table 111, and those of Table IX are similar to those of Table V. Generally speaking, if the IAS theory is used for predicting cT., the use of a large number of pseudospecies yields a more consistent predictive accuracy for different diluted solutions. An improvement in predictive accuracy of more diluted solutions with the increase in the number of pseudospecies is observed. The improvement, however, does not go on indefinitely. As shown in Table VII, as the
number of the pseudospecies increased from six to nine, the predictive accuracy actually declined slightly. It should also be noted that for both types I and 11 solutions, the IAS theory always gave better accuracy than the Langmuir theory.
Conclusions A simple scheme of applying the characterization method of Jayaraj and Tien was established in this study. The empirical evidence obtained from the case studied indicate the following. (a) Adequate characterization of the adsorption affinity of solutions containing a disparate group of adsorbates requires the use of more than a single pseudospecies. While the number of pseudospecies required cannot be determined a priori, a systematic application of the
Ind. Eng. Chem. Res. Vol. 26, No. 2, 1987 291 Table VIII. Characterization Results of Solution I1 (Langmuir Theory) _____ (A) representation by a single species search no. a b ~~
~
E , 90 1.623 X 5.06 1.285 6.351 X 5.09 1.282 1.575 X 5.08 1.282 3.06 X lo6 5.06 1.285 1.281 4.54 x 10-5 5.09 (Bj representation by two pseudospecies concn, mg of organic C/L lO.001. 0.002l 15, 0.002\ E, % search no. 176.9 18.83 1 39.1 2 39.2 176.8 18.83 3 39.5 176.5 18.82 176.7 18.82 4 39.3 5 39.4 176.6 18.82 (Cj representation by four pseudospecies concn, mg of organic C/L no. (0.001, 0.OOll 10.001, 0.11 15.0, 0.OOll 15.0, 0.011 31.2 184.8 1.9 29.6 184.5 31.2 184.8 30.9 185.1 4.73 27.0 184.3 (D)representation by six pseudospecies concn, mg of organic C / L (0.001,0.OOOll (0.001, 0.011 (5, 0.OOll (5, 0.011 (10, 0.001l (10, 0.011 31.2 181.8 3 31.3 183.4 1.4 30.9 185.1 30.8 184.4 1.8 2 31.2 0.1 182.7 (E) representation by nine pseudospecies 1 2 3 4 5
search 1 2 3 4 5
search no. 1 2 3 4 5
E, % 7.78 7.91 7.78 7.77 8.25
E, % 8.15 7.90 7.97 8.07 8.02
concn, mg of organic C/L search no. (0.001, 0.0001) 1 1.1 2 3 4 0.6 5
(0.001, 0.0051
(0.001, 0.05) 21.5 22.4 24.6 21.5 22.8
(5, 0.001) 18.1 37.9 18.4 50.6 33.3
(5, 0.005) 42.6 13.9 1.1 7.5 13.7
(5, 0.05) 52.9 56.2 41.9 63.8 53.0
110, O.OOOl] 38.5 13.2 40.4 43.9 53.5
(10, 0.005) 22.1 56.7 28.4 24.8 12.0
(10, 0.05) 19.3 15.7 61.2 3.2 27.7
E , 70 2.21 2.17 2.27 2.22 2.17
Table IX. Errors in Predicting cTm, Solution I1 (Langmuir Theory) E ( % ) for no. of pseudospecies used in characterization 1 2 4 6 9 max min av max min av max min av max min av max min av 6.98 125 18.72 18.49 18.62 15.07 14.84 14.97 6.52 6.79 6.82 6.50 6.69 3.57 2.55 2.73 46.89 47.32 47.18 46.68 46.89 35.41 31.16 33.78 52.5 97.12 96.83 96.99 48.98 48.41 48.74 47.57 41.5 152.4 152.2 152.3 90.03 89.65 84.87 94.81 94.42 94.66 94.42 93.92 94.23 85.28 79.36 83.14 190.2 175.0 184.6 22.2 370.5 369.9 370.2 239.2 238.1 238.7 229.7 228.1 229.0 228.5 227.6 228 cT,.O
Initial concentration in milligrams of organic carbon/liter.
characterization method in which the number of pseudospecies increases incremently with the identities of the pseudospecies, determined on the basis of the previous characterization results, would lead eventually to a successful characterization. (b) The closeness in data fitting alone does not guarantee that the results can be used to obtain good predictions. (c) A good characterization should be the one which provides both a good and consistent predictive accuracy for different diluted solutions. (d) Comparing the results based on the Langmuir theory and those based on the IAS theory, the latter results yielded much better predictions than the former. This conclusion is also consistent with all the available evidence which initiates that the IAS theory predicts carbon adsorption equilibria of multicomponent systems with much
greater accuracy than the Langmuir theory.
Acknowledgment This study was performed under Grant CPE 83 09508 from the National Science Foundation.
Nomenclature A, = Freundlich coefficient a,, b, = Langmuir constants c, = concentration of the ith species in solution phase c,, = initial value of c, cLm= equilibrium value of c, cT = total adsorbate concentration, C,c, cT0 = initial value of CT cT, = equilibrium value of CT D = quantity defined by eq I
Ind. Eng. Chem. Res. 1987, 26, 292-296
292
E = average error defined by eq 16 M = mass of adsorbent M , = molecular weight of the ith species n, = reciprocal of Freundlich exponent q, = concentration of the ith species in the adsorbed phase qL,= equilibrium value of q1 q T = total adsorbate concentration in the adsorbed phase, xq, S = parameter appearing in eq 15 V = volume of solution v = concentration values used in optimization search X,= mass of the adsorbate per unit mass of the organic carbon of the adsorbate z, = mole fraction of the ith species in the adsorbed phase Greek Symbols a*,a2 = parameters characterizing adsorption affinity 4 = objective functions
Literature Cited Calligaris, M.; Tien, C. Can. J. Chem. Eng. 1982, 60, 772. Frick, F. B.; Sontheimer, H. In Treatment of Water by Granular Activated Carbon; McGuire, M. J., Suffet, I. H., Eds.; Advances in Chemistry Series 202; American Chemical Society: Washington, DC, 1983; p 247. Jayaraj, K.; Tien, C. Proc. Enuiron. Eng. Conf. 1984, p 394. Jayaraj, K.; Tien, C. Ind. Eng. Chem. Process Des. Deu. 1985, 24, 1230. Mehrotra, A. K.; Tien, C. Can. J . Chem. Eng. 1984, 62, 632. Tien, C. In Fundamentals of Adsorption; Myers, A. L., Belfort, G., Eds.; United Engineering Trustees: New York, 1984; p 647. Tien, C. Chem. Eng. Commun. 1986, 40, 265. Wang, S.-C.; Tien, C. AIChE J . 1982, 28, 575.
Received for review December 5 , 1985 Accepted August 20, 1986
Cinematic Modeling of Dynamics of Solids Mixing in Fluidized Beds Chandrasekharan C. L a k s h m a n a n t and O w e n E. P o t t e r * Department of Chemical Engineering, Monash University, Clayton 3168, Victoria, Australia
T h e dynamic response of a gas-solid fluidized bed to a pulse input of the tracer at the top of the bed is obtained. The tracer distribution between the phases is represented by a convective model. The model includes a fast downflow of dense phase at the wall parallel with the tubes. But for this, the model is very similar to the countercurrent backmixing model. T h e simulation is performed by using a numerical model (“cinematic” model), and the results are in good agreement with available results. Solids mixing in fluidized beds plays an important role in the scaling up of fluidized beds to suit commercial requirements. Studies on solids mixing are usually conducted by using stimulus-response techniques. Bart (1950) injected solids impregnated with sodium chloride halfway up the bed and found that the data fitted the diffusion equation approximately. Radioisotopes were used to study the mixing pattern of solids in catalytic cracking units by Singer et al. (1957). Hull and von Rosenberg (1960) reported tests performed on a pilot-scale catalytic cracking unit using radioactive zirconium-niobium-95. Gilliland and Mason (1952) used thermal inputs to the bed and represented the heat flow from top to the bottom of the bed by an apparent axial thermal diffusion coefficient. Reman (1955) reported a similar study of solids mixing in a fluidized bed of cracking catalyst. May.(1959) employed fluid beds up to 5 f t in diameter in his study of solids mixing using tagged solids. These solids were injected near the top of the bed, and the radioactivity a t various locations below the injection point was monitored. May (1959) suggested that the gross circulation pattern was superimposed by a smaller scale mixing. van Deemter (1967) and Bailie (1967) presented two phase models for the dynamics of solids mixing in fluidized beds. The models suggested by them are very similar in mathematical representations, but the techniques used by them to solve the model equations are bascially different. Ishida and Wen (1973) used the bubble assemblage model to study the steadystate solids mixing behavior and used it to predict the reactor performance for noncatalytic gas-solid reactions. Avidan and Yerushalmi (1985) investigated solids mixing of a group A fine powder in a 0.15-m-internal-diameter *Author to whom all correspondence should be addressed.
Present address: BHP, Melbourne Research Laboratories, Clayton 3168, Victoria, Australia.
expanded top fluidized bed with ferromagnetic tracer. They found that the countercurrent flow model of solids mixing described the tracer distribution well in the bubbling fluidization regime, and at higher gas velocities, a one-dimensional Taylor dispersion model fitted the data well. They also reported that the axial dispersion coefficient increased with the square root of the bed diameter. Gwyn et al. (1970) considered a three-phase model for gas and solids mixing in gas-solid fluidized beds. Their model was essentially a countercurrent backmixing model. The model equations were solved by using hybrid and digital computer techniques. The equations were continuously integrated with respect to the space variable, and finite difference methods were used to integrate with respect to time. Solids mixing in batch-operated tapered and nontapered gas fluidized beds was studied by impulse-response experiments. The countercurrent backmixing model was used to represent the dynamic response of the bed, and the resulting model equations were solved by using Laplace transforms. During the simulation, the bed was assumed to extend to infinity, and the solution obtained until the tracer front reached the distributor. To conform with the solution, experiments were conducted to collect samples before the tracer front reached the distributor. Potter (1971) reviewed the earlier works on solids mixing and the experimental studies using tracer solids. We developed a numerical model (“cinematic” model) to simulate the dynamic response of countercurrent systems. In this work, the cinematic model is applied to simulate the dynamic response of a fluidized bed with horizontal tubes to a pulse input of tracer at the top of the bed. The simulation is performed to describe the experimental tracer distribution published by Sitnai (1981). For details about the tube layout and other experimental aspects, the published work of Sitnai (1981) may be con-
0888-5885/87 / 2626-0292$01.50/0 b 1987 American Chemical Society
