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Adsorption of unknown substances for aqueous solutions
Chemical Englneerlng Science. Vol. 46. No. 1, pp. 23-31, Printed in Great Britain.
ADSORPTION
1991.
ooo9-2509/91 93.00 + 0.00 0 1990 Pergamon F%M plc
OF UNKNOWN SUBSTANCES AQUEOUS SOLUTIONS
FROM
HEE MOON and HEUNG CHUL PARK Department of Chemical Technology, Chonnam National University, Kwangju 5ClG757, Korea and
CHI TIEN Department of Chemical Engineering and Materials Science, Syracuse University, Syracuse, NY U.S.A. (First received
6 December 1989; accepted
in revised form
13244,
18 May 1990)
procedure proposed by Jayaraj and Tien for characterizing the adsorption affinity of solutions with unknown compositions was modified by assuming that the solute concentrations may be described by the binomial distribution function. This new method yields results comparable to those of the original Jayaraj-Tien procedure although it requires much less computation. Predictions of batch and fixed-bed adsorption based on the characterization results were found to agree well with experiments.
Abstract-The
an aqueous solution. Kage (1980) and Okazaki
INTRODUCTION In designing adsorption systems for treating water and waste water, one is often handicapped by the fact
et al.
(198 1) argued that the large number of solutes present
in an aqueous solution may be assumed to form a continuum such that the solute concentration may be described by a continuous probability density function of suitable variables represented by a parameter vector, M. This approach was recently justified on theoretical grounds by Annesini et al. (1988). However, the characterization results based on a continuous representation are not consistent with most adsorption algorithms which are formulated for systems with a finite number of solutes. The major difficulty encountered in applying the discrete approach in characterizing solutions with unknown compositions is the large computation demand. If a solution is approximated as one with a number of pseudo species, the accuracy of the approximation, to a large degree, depends upon the number of pseudo species used. Unless the solution to be characterized is simple, one is faced with a search of a relatively large set of parameters. Based on past experiences, such a search is likely to be difficult. The present work is concerned with a further development of the characterization method. To facilitate the subsequent usage of the characterization results, the discrete representation approach is used. The major emphasis of the development is placed on the reduction of the computational demand which is accomplished by assuming that the pseudo species concentrations follow a discrete distribution function (i.e. the binomial distribution function). The main features of this method as well as several examples of its application are given in the following section.
that aqueous systems to be treated often contain a large number of solutes which cannot be completely identified. As the dynamics of any adsorption process depend upon the adsorption equilibrium relationships of the relevant adsorbate-adsorbent system, lacking information on, the solute identities and concentrations makes it difficult, if not impossible, to apply adsorption theories to the design of adsorption systems. In order to circumvent this difficulty, a number of investigators over the past decade have proposed various methods for characterizing aqueous solutions with unknown compositions. Broadly speaking, two different approaches were used in these studies. Frick and Sontheimer (1983) first suggested that solutes present in a solution of unknown composition may be grouped into three classes, i.e. non-adsorbable, weakly adsorbable and strongly adsorbable, and proposed a method for characterizing these three classes of substances and determining their concentrations. This method was subsequently formalized and extended by several investigators (Jayaraj and Tien, 1984, 1985; Kage and Tien, 1987; Crittenden et al., 1985). According to Tien and coworkers, a solution of unknown composition may be represented as one with a number of pseudo species (or theoretical components in the terminology of Crittenden et al.). A pseudo species, in reality, is a group of solutes exhibiting similar adsorption affinity and its use in simplifying multicomponent adsorption calculations has proved successful in previous investigations (Calligaris and Tien, 1982; Mehrotra and Tien, 1984; Ramaswami and Tien, 1986). In contrast to the use of a finite number of pseudo species in representing the unknown solutes present in
PRINCIPLES OF THE PROPOSED METHOD
Characterization of pseudo species concentration discrete distribution function
by a
Similar in concept to the continuous representation 23
24
HEE MOON et al.
approach, we represent the composition of a solution with a large number of solutes by a discrete distribution function and specifically the binomial function of a parameter vector, M. In principle, M may be composed of a number of parameters which characterize the solute adsorption affinity. Since the adsorption affinity of a solute with respect to an adsorbent is most conveniently described by the relevant purecomponent adsorption isotherm, one may choose M to be the constants of the pure-component adsorption isotherm expression. Using this argument and assuming that the purecomponent isotherm data may be described by the Freundlich equation with K and n as the coefficient and the reciprocal of the exponent, M may then be considered to be either K or n or both. Since in the previous work of Kage and Tien (1987), it was shown that an average value of n can be assigned to all the pseudo species as a matter of convenience, M may be assumed to be K. The parametric representation of the pseudo species concentration as a function of the Freundlich coefficient, K, according to the binomial distribution function, is given as K,=
xi=
k,j’
(1)
N! j!(N
-j)!
d(i
- sjN-j
for j = 0, 1, 2, _ _ . , N (2)
where N is the number of pseudo species (note that we also allow the presence of one non-adsorbable species corresponding to j = 0), and K, is the Freundlich coefficient of the jth pseudo species. X, is the total organic carbon (TOC) mass fraction defined as C,
5 Ci, where C, is the concentration of the jth 1 i=1 species expressed in terms of the surrogate quantity, the TOC concentration. k, and s are two constants. The capability of eqs (1) and (2) as a general representation of solutions with a large number of solutes is demonstrated in Fig. 1. In Fig. 1, the histograms of the TOC fractions for N = 8 and different values of s are shown. It is clear that the binomial distribution is indeed capable of approximating compositions of
Fig. 1. Solute concentration distribution according to eq. (2) for different values of s.
various types. The only case for which the use of the binomial distribution function is not satisfactory is when the distribution of the solute concentration is bimodal with the Freundlich coefficients corresponding to the two peak concentrations being significantly different. In the discussions which follow, Xj is assumed to be given by eq. (2) as a matter of convenience and simplicity. According to eqs (1) and (2), characterization of solutions of unknown composition requires the search (or determination) of the values N, n, s and k,_ Before discussing the procedure which may be used to determine these parameters, some general remarks on their orders of magnitude is helpful. For a solution with a large number of solutes of different adsorption affinity, a small value of IV cannot provide a sufficiently accurate representation. On the other hand, there is a practical limit as to the number of the pseudo species one may use. As an example, consider the design of fixed-bed adsorbers for water treatment. Although there exists a number of fixed-bed adsorption algorithms capable of considering an arbitrarily large number of solutes [for example see Hsieh et al. (1977), Wang and Tien (1982) and Moon and Tien (1988a)], the required calculation becomes impractical if N exceeds 10. For most cases, N should be chosen within the range of 5-8. Concerning the magnitude of n, the reciprocal of the exponent of the pseudo species, it is necessary that, for adsorption to be effective, most of the solutes, if not all of them, must be adsorbable and exhibit favourable adsorption behaviour in their pure-component adsorption isotherm data, or l/n c 1.0. The condition that n > 1.0
(3)
should therefore be regarded as one of the constraints in the search of the parameter values. The value of s in eq. (2) determines the skewness of the pseudo species concentration distribution and is within the range 0 -ZZs < 1.0, with s = 0.5 corresponding to a normal distribution type. The parameter k, can be viewed as a quantity which directly determines the adsorption affinity of the pseudo species. For a fixed N, a small value of k, represents a solution with less adsorbable solutes. Regardless of the value of k,, from eq. (l), it is simple to see that the distribution of Kj is not uniform over its range (which is from zero to N2 k,). The spacing of K, between the less adsorbable species is closer than that between the more adsorbable species. This arrangement of K values is made since in multicomponent adsorption, the less adsorbable species often determines the initial breakthrough of adsorption (Hubele and Sontheimer, 1985; Moon and Tien, 1988a). It is also interesting to note the difference between the present method and the method of Jayaraj and Tien (1984, 1985). The original Jayaraj-Tien method requires, for a fixed N, the determination of the Freundlich coefficients, exponents and concentrations for each pseudo species or a total of 3N - 1 quantities.
Adsorption of unknown substances from aqueous solutions
The procedure of Kage and Tien (1987) involves a search to be conducted in successive steps for increasing values of N. For each step, corresponding to specific values of N, K, and n, it is necessary to determine N - 1 concentration values of the N pseudo species. In each case, the number of parameters to be determined increases almost linearly with the value of N. For the present method, only the determination of four parameters is required. The potential advantage of the present method in terms of savings in computation time is therefore rather obvious. N, k,, n and s To obtain these parameter values, certain adsorption data of the solution to be characterized are assumed to be available. Let McXp denote the experimental measurement value and Mprc the calculated value for a particular type of adsorption experiment. The parameter values can be determined from the minimization of the objective function, E, defined as Search of parameters,
where k denotes the kth set of measurement and 1 1 the absolute value. m is the total number of measurements. From the definition of E, it is clear that the objective function is an indication of the fitting error. Most investigators in the past have used batch adsorption data as the basis for characterizing solutions of unknown compositions. Two types of measurements are available; the integral adsorption experiment (IAE) and the differential adsorption experiment (DAE). Detailed experimental procedures for each type have been given elsewhere (Okazaki et al., 1981; Jayaraj and Tien, 1985). For batch adsorption from solutions with N solutes, with the use of the ideal adsorbed solution (IAS) theory for multicomponent adsorption equilibrium and assuming that the pure-component adsorption isotherms of the individual solutes obey the Freundlich equation, the total adsorbate equilibrium concentration can be found from the following equations (Calligaris and Tien, 1982; Tien, 1986):
c, =
$
ci
(5)
i=l
Ci = Zi(l-I/nKi) zi =
,i
(6)
cio /C(n/nKi)” + (“lv)tnln)l
(rl/nK.)n
I
+$,v)(n,n) - l = O-
(7)
(8)
Ci is the equilibrium concentration of the ith component, and n is the reciprocal of the Freundlich exponent (for all the components). A4 and I’ are the mass of the adsorbent and volume of solution. C, is the initial concentration of the ith component. Zi is the mole fraction of the ith component in the adsorbed phase. II is the dimensionless spreading pressure defined as (aA/R T), where R is the spreading pressure. A is the
25
adsorption surface area per unit mass of the adsorbent, R is the gas law constant, and T is the absolute temperature. It is clear that with n, Ki, Ci,. M and Y known, II can be found from eq. (8). Once II is known, Zi can be found from eq. (7). With Zi and II, one can readily determine Ci (i = 1, 2, . . . , N), the sum of which gives Cr. Equations (SH8) can be applied to estimate both the IAE and DAE results. In the former case, the calculation is direct and straightforward. In the latter case, repeated use of these equations is required such that the equilibrium concentrations obtained upon the addition of k th increments of adsorbent are used as the initial concentrations for estimating the equilibrium concentration with the addition of the (k + 1)th increment of adsorbent. In carrying out the parameter search, although it is theoretically possible to search for the optimum values of the four parameters simultaneously, it was found advisable, from the sample calculation results to be given in the following section, to carry out the search in two steps. First, a relatively large value of N is assumed (for example N = 8) and values of n and s are determined from the minimization of E for significantly different values of k,. The value of k, which gives the smallest value of E and also yields a value of n consistent with the criterion of eq. (3) can then be considered as the correct value. Once k, is selected, one can conduct the second step of search, namely, the search for optimum values of n and s for different values of N. SAMPLE
APPLICATIONS
To demonstrate the working procedure of the proposed method, we apply the method to three types of aqueous solutions containing a large number of solutes; namely, solution II prepared by Kage and Tien (1987), the aqueous solution of humic substances used by Hubele and Sontheimer (1985), and the industrial waste examined by Okazaki et al. (1981). The results obtained arc summarized as follows. of solution II of Kage and Tien (1987) Solution II was composed of 10 adsorbates [see Table I of Kage and Tien (1987) for the composition]. Batch adsorption equilibrium concentration data were obtained using solution II with different total initial concentrations (but with the same concentration ratios of the adsorbates) and are shown in Fig. 2. Characterization of solution II was made using the batch adsorption data with CT0 = 216 ppm. The results are given in Table 1. As explained earlier, the search for the optimum values of parameters, N, k,, n and s proceeds in two steps. It was first assumed that N = 8 and the optimum values of n and s were determined for &. = 0.1, 1.0 and 10.0 [see (a) of Table 11. From the results, k, was chosen to be 1.0 since it gave the smallest fitting error (i.e. the smallest value of the objective function) as well as a value of n greater than unity [which satisfies the condition of eq. (3)]. Next, with k, fixed, the search was continued for n Characterization
HEE MOON et al.
26
lob-+--+4 PPM
CT,
0
, I
0
Fig. 2. Batch adsorption data obtained from Kage and Tien (1987).
, 2
I 3
i-W”.
, 4
5
KD/Y3
(4
I
I
‘*I
I
PREDICTION CTD=52.3PPM
-
= 22.2 PPM
O0
n
.
.
0.2
I I.5
1.0 WV
2.0
. KWh3
(b) 01
1
0
1
0.25
0.50
SKEW”ESS
1
0.75
I 1.00
PARAMETER , l
Fig. 3. Contour map of the object function, E, in characterizing solution II of Kage and Tien (1987) with N = 8 and KS = 1.0.
Fig. 4. (a) Experimental and predicted IAE data of Kage and Tien (1987) with C,, = 125 and 216ppm. (b) Experimental and predicted IAE data of Kage and Tien (1987) with C,, = 22.2 and 52.3 ppm.
Table 1. Characterization results of solution II of Kage and Tien
and s with N = 6,8 and 10 [see (b) of Table 11. From the respective values of E of the three cases, it is clear that all the three N values give very similar results. These results also demonstrate the compensating effect between n and s; namely, an increase in n is accompanied by a decrease in s. As a demonstration of the search of the optimum, a contour map of the objective function, E, is shown in Fig. 3 with N = 8 and ks = 1.0. Each closed curve represents a contour with the same value of E and the point denoted by * gives the optimum point at which E is a minimum in the N-s domain. As pointed out by Jayaraj and Tien (1984, 1985) and Kage and Tien (1987) previously, if the characterizing
results
are valid,
one
should
be able
to apply
them, with appropriate adjustment of the concentration values, to both the concentrated and the diluted
versions of the solution ization was based. Fbr by diluting the original the same value of k,, N, tion half of the value of we demonstrate this by
on whose data the characterexample, a solution obtained solution by 50% should have s and II, and a total concentraoriginal solution. In Fig. 4(a), comparing the predicted total
Adsorbate: mixed organic chemicals = 216 ppm TOC
Type: IAE data with C,
(a) First step with N = 8
k 0.1 1.0 10.0
”
l.ClO 1.51 2.64
s
0.507 0.359 0.240
(E) 3.88 3.84 6.45
(b) Second step with k, = 1.0
N
?I
s
6 8 10
1.44 1.51 1.55
0.422 0.359 0.301
3.82 3.84 3.87
adsorbate equilibrium concentration CT from batch adsorption experiments using solution 11 with an initial total concentration C,, = 125 ppm. The predictions were made with N = 8, k, = 1.0, n = 1.51 and
Adsorption
27
of unknown substances from aqueous solutions
Table 2. Search results of solution II based on the modified Jayarai_Tien procedure with N = 8 Search number
Ki 0 1 2 3 4 5 6 7 8 E (%)
0 1 4 9 16 25 36 49 64
Average’
xi [from eq. (2)]
2
3
4
0.0975 0.0000 0.2866 0.3897 0.203 1 0.0023 0.0189 0.0019 O.CK300
0.0925 0.0136 0.2461 0.4773 0.1500 0.0068 0.0083 0.0006 0.0090
0.0872 0.0110 0.3 502 0.2050 0.3162 0.0148 o.ooo2 0.0149 0.0004
0.0919 0.0043 0.3202 0.4058 0.0012 0.0758 0.0089 0.0065 0.0854
0.0923 0.0073 0.3007 0.3684 0.1676 0.0249 0.0091 O.MMO 0.0237
0.0285 0.1277 0.2502 0.2804 0.1963 0.0879 0.0246 0.0039 0.0003
2.75
2.63
3.27
3.52
3.04’
3.84
1
xi,
‘Average TOC fractions (X,-j obtained from the four searches. *Predi&on
error by using &rage
TOC
fractions.
s = 0.359. Also included in Fig. 4(a) are the results for
solution II with CT0 = 216 ppm. This latter comparison, of course, is merely an indication of the degree of fitness of the characterization results to the ex~rimental data on which they are based. Furthermore, Fig. 4(b) shows the characterization results of diluted versions of solution II with C, = 52.3 and 22.2 ppm, respectively. As expected, the predictions are not satisfactory, which means that the applicability of the proposed method is not good to very diluted versions. Kage and Tien (1987) already pointed out that the predictive accuracy for very diluted solutions are not good [see Tables VII and IX of Kage and Tien (1987)]. As a comparison of the present method and the previous method of Jayaraj and Tien based on the value of N = 8, k, = 1.0 and n = 1.51, the TOC mass fractions of pseudo species were determined by a random search. The mass fraction values obtained from a number of independent searches are shown in Table 2. As expected from the Jayaraj-Tien method, different results were obtained from each search but, by and large, they are similar. Also included in Table 2 are values of X, calculated according to eq. (2) and the average values of the four independent searches. It is clear that these two sets of values are approximately although the computation requirement of the present method is much less than the Jayaraj-Tien method. Other characterization results As a further demonstration of the utility of the present method, it was also applied to the characterization of two other aqueous solutions considered by previous investigators, namely the aqueous solution of humic substances (Hubele and Sontheimer, 1985) and the industrial waste examined by Okazaki et al. (1981). Characterization of the solution of humic substances was made based on IAE data and that of the other solution was based on DAE data. The characterization results are summarized in Table 3. For both characterizations, N was chosen to be 8 and the k, value for the first step of search was found
Table 3. Characterization results of two with N = 8
other solutions
(a) Solution of humic substances (Hubele, 1985)
k.
n
s
0.1 1.0 10.0
0.86 1.38 12.33
1.000 0.46 0.312
(Z) 7.63 3.79 (0.97)’ 23.03
(6) Waste water of Okazaki et al. (1981)
k,
n
s
0.001 0.01 0.1
0.88: 1.75 15.00
0.789 0.499 0.283
(6) 4.90 5.01(4.50)’ 9.75
‘Fitting error obtained by using the Jayaraj-Tien procedure. *This violates the condition given by eq. (3).
to be 1.0 and 0.01, respectively. Based on these values, the optimum values of n and s were determined. Also included in Table 3 are values of E corresponding to the respective characterization results as well as the values of E obtained by using the Jayaraj-Tien method. From the comparison of these E values, it is apparent that, in the case of the solution of humic substances, the binomial distribution function gives somewhat less satisfactory results than the Jayaraj-Tien method. Nevertheless, the characterization results were found to give sufficiently accurate prediction of fixed-bed adsorption results to be discussed later. In addition to the DAE data which were used as the basis for characterization, Okazaki et al. provided additional DAE and IAE data using less concentrated solutions. In Figs 5 and 6 the predicted equilibrium concentration results of the DAE and IAE experi-
HEE Moow 20
I
I
\
f5
- --
\
0
\
Table 4. Experimental conditions of fixed-bed experiments
I
BY THIS METHOD
FITlED
-
-“b
r
I
t
et al.
Conditions
PREDtcnoN c+0=13.5
Solution II
YCnlY3
Activated carbon used
I
I
I
10
20
30
Cl 0
HUbeMS solution
40
I
I
30
so
F-400
Particle diameter ( x lo3 m) Bed density (kg/m3) Bed height (m) Bed porosity Superficial velocity ( x 10” m/s) Initial TOC (or DOC) concentration (ppm)
TC
1.000 490.0 0.15 0.367 0.728
F-300 1.320 485.0 2.35 0.420 1.389
94.10
7.970
EWV.KG/M3
Fig. 5. Experimental and predicted DAE data of Okazaki et al. (1981).
2.0
15
ea t
)
I
I
I
I
i“‘--
#I2 a
0
-
PREDICTIGK
0
4
,
0
CTo=15.5MOL/M3
I
I
I
I
5
10
15
20
m/V.
KG/Y3
Fig. 6. Experimental and predicted IAE data of Okazaki et a[. (1981).
ments were compared with experiments. Good ment was observed in both cases.
bulk density of adsorbent. C, and Qi are the solution phase and average adsorbed phase concentrations, respectively. CT is the inlet value of Ci and Qio is the initial value of Qi. Fixed-bed adsorption may be characterized by its breakthrough curve or the effluent concentration history. The solution of eqs (9Hll) together with appropriate rate expressions gives the efIluent histories with various bed lengths. For the simplest case, if local equilibrium is assumed [namely, Ci and Qi are in equilibrium or related by eqs (5x8) if the IAS theory is applied], eqs (9Hll) can then be solved to obtain the effluent concentration history. As shown by Moon and Tien (1988b), breakthrough curves for fixed-bed multicomponent adsorption are composed of a number of plateau and are characterized by the transition times and plateau concentration values if the adsorbate species exhibit favourable adsorption behaviour. These quantities may be determined from the following equations:
agreeOjCi, =
aQij+
$
k=j+l
(Ok-, - O,)C,,
i>j+l (121
PREDKTION
OF FIXED-BED
ADSORPTION
Both Kage and Tien and Hubele and Sontheimer conducted fixed-bed adsorption measurements using their respective solutions. One can therefore compare these experimental data with predictions based on characterization results discussed above. Agreements between predictions and experiments constitute further validation of the characterization method. The conditions used in conducting these experiments are summarized in Table 4. For a solution containing N adsorbable solutes passing through a bed packed with adsorbent, the macroscopic equations of conservation of the individual solutes are
ac,
udz+pbs=O
i=l,2,3
Qi = Qio
z > 0,
ci = cp
z=o,
,...,
oto o>o
N
(9)
(10) (11)
where z and 0 denote the axial distance and corrected time (defined as 0 = t - Z&/U,with t, E and u being the time, bed porosity and superficial velocity). P,, is the
OjCij = Qij c,
= 0
ci, = cp a = L&/U
i=j
(13)
i
(14) (15) (16)
where L is the bed length, 0, is the transition time from the (j + l)th plateau to the jth plateau (note 0, < Oj- 1 for all js), Cij is the solution phase concentration of the ith solute in thejth plateau, and Qij and C, are in equilibrium or, in other words, they are related by eqs (5)-(g). The above results apply to solutions with N adsorbable solutes with the solutes arranged in the descending order of their adsorption affinity. For solutes with the same Freundlich exponent, the solutes should be enumerated in the descending order of their K values. The fixed-bed adsorption calculation results were compared with experiments and are shown in Table 5, Fig. 7 (solution II with CO, = 94.1 ppm TOC of Kage and Tien) and Fig. 8 (aqueous solution of humic substances with CO, = 7.97 ppm dissolved organic carbon). In Table 5 and Figs 7 and 8, predictions based
Adsorption Table
of unknown
29
substances from aqueous solutions
5. Estimated transition times and plateau concentrations of effluent histories of fixed-bed adsorption on different characterization results Solution
Hubele’s solution
II Modified Jayaraj-Tien
This work Time (h)
Time W
Cl+%
0 11.0 33.9 71.5 133.0 234.0 390.0 614.0 918.0
0.029 0.199 0.513 0.784 0.930 0.984 0.998 0.999 1.000
based
Time (h)
G-/G
0 22.3 32.4 67.4 135.0 254.0 431.0 667.0 901.0
Modified Jayaraj-Tien
This work
0.092 0.113 0.522 0.849 0.959 0.974 0.980 0.985 1.000
Time 01)
WC;
0 271.0 772.0 1480.0 2520.0 4060.0 6320.0 9480.0 13,660.O
0.007 0.074 0.280 0.569 0.808 0.939 0.987 0.999 1.000
Cr/G
0 350.0 806.0 1430.0 2420.0 4080.0 6520.0 9870.0 13,840.O
0.020 0.046 0.231 0.585 0.865 0.959 0.983 0.988 1.000
1.0
O#” k 0
0.5
e
O -
= 94.1
THIS
-
-
-
PPM
TOC
WORK
MODIFIED
JAYAAAJ
-
TIEN
PROCEDURE 0
Ii 0
Fig. 7. Predicted
and experimental
effluent history of solution
II with C; = 94.1 ppm TOC.
1.0
0.5
-
THIS WORK -
-
MDDIFIW
I
JAYARAJ
-TIEN
A
OLJJ-
I
I
0
I
I
2 TIME
Fig. 8. Predicted
on
characterization
and experimental
results
from
the
I
4 X 103,
Jayaraj-Tien
I
I
8
HR
effluent history of Hubele’s organic carbon.
method are also included. As shown in Table 5, for both solutions, different predictions of transition times and plateau concentrations were observed.
I
6
solution
with C; = 7.97 ppm dissolved
However, these differences in most cases are not significant. In terms of their comparisons with experimental data, as shown in Figs 7 and 8, the comparison between predicted and experimental results is not
30
HEE MOON et al.
especially good. It is not surprising that the comparisons are not good, bearing in mind that the predictions are based on the assumption of equilibrium control. CONCLUSIONS
We have demonstrated that the relatively simple method developed in this study can be used to characterize solutions with unknown compositions for adsorption calculations. The method has two advantages over the Jayaraj-Tien procedure: first, unlike the previously developed method, it yields a unique representation of the solution to be characterized when the number of adsorbable pseudo species is fixed. Second, the computation requirement is rather modest. In a number of sample cases, characterization results obtained from applying the present method were found useful in estimating adsorption performance in batch and fixed-bed operations. Acknowledgement--Hee Moon and Heung Chul Park wish to acknowledge the support they received from Korean Science and Engineering Foundation under Grant No. 8811005-015-02.
NOTATION
area of the solution-solid interface concentration in solution liquid phase concentration at bed inlet total adsorbate concentration in solution object function or fitting error constant in eq. (I) Freundlich coefficient of the jth pseudo species bed length number of data mass of adsorbent parameter vector experimental value predicted or calculated value reciprocal of Freundlich exponent number of adsorbable pseudo species concentration in the adsorbed phase gas constant constant in eq. (2) or skewness factor temperature superficial velocity volume of solution TOC
mass fraction
Subscripts i, j ith and j th pseudo species experimental exp predicted or calculated pre T total 0 initial Abbreviations DAE differential adsorption equilibrium IAE integral adsorption equilibrium IAS ideal adsorbed solution TOC total organic carbon
REFERENCES
Annesini, M. C., Gironi, F. and Marrclli. L., 1988, Multicomponent adsorption of continuous mixtures. 2nd. Engng Chem. Res. 27, 1212-1217. CaIIigaris. M. B. and Tien, C.. 1982, Species grouping in multicomponent adsorption calculations. Can. J. &em. Engng60, 772-780. Crittenden,J. C., Luft, P. J., Hand, D. W. and Friedman, G., 1985, Prediction of multicomponent adsorption cquilibrium in backnround mixtures of unknown comnosition. J. Wat. Res. 17:312-319. Dobbs, R. A. and Cohen, j. M., 1980, Carbon adsorption isotherms for toxic organics. EPA-60018-80-02. Frick, B. R. and Sontheimer, H., 1983, Adsorption equilibrium in multisolute mixtures of known and unknown composition, in Treatment of Water by Granular Activated Carbon (Edited by M. J. Mcguire and I. Suffet), pp. 247-268. Advances in Chemistry Series 202, American Chemical Society, Washington, DC. Hiseh. J. S. C., Turian. R. M. and Tien, C., 1977, Multicomponent liquid phase adsorption in fixed bed. A.I.Ch.E.
J. 23, 263-275. Hub&, C., 1985, Adsorption and biologischen abbau van huminstoffen in aktivkohelefiltetn. Dr-Ind dissertation, Karlsruhe University, Karlsruhe. Hubele, C. and Sontheimer, H., 1985, Isotherm evaluation and filter breakthrough calculations for waters with a multicomponent mixture of organ& using the simplified competitive adsorption model (SCAM). Unpublished work. Jayaraj, K. and Tien, C., 1984, Characterization of adsorp tion affinity of solutions, in Proceedings of National Environmental Enaineerina Conference (Edited by M. Pirbazari and J. S-Deviniy), pp: 394-399. ASCE. Jayaraj. K. and Ticn, C., 1985, Characterization of adsorption affinity of unknown substances in aqueous solutions. Ind. Enana - _ Chem. Process Des. Dev. 24, 1230-1239. Kage, H. and Tien, C., 1987, Further development of the adsorption affinity characterization procedure for aqueous solutions with unknown compositions. Ind. Engng
Chem. Res. 26, 284292. of the jth pseudo
species in
solution axial distance TOC mass fraction of the jth pseudo species in the adsorbed phase
Kage. H.. 1980. Characteristics of multi-solute adsorution in $1&e iqme&s solution. DEng thesis, Kyoto, Japk. Mehrotra. A. K. and Tien. C.. 1984, Further work in species grouping in multicomp&e& adsorption calculation. Can. J. them. Engng 62, 632443. Moon, H. and Tien, C., 1988a, Fixed-bed multicomponent adsorption under local equilibrium. A.I.Ch.E. Symp. Ser.
No. 264 84,94-108.
Greek fetters K
defined
s
differenti& m ’ -’ spreading pressure dimensionless spreading pressure bulk density of adsorbent
; Pb
bv ea. f161
Moon, H. and Tien, C., 1988b, Prediction of fixed-bed of solutions of unknown composition. adsorption
A.I.C%E. Symp. Ser. No. 264 84, 23-35. Okazaki, M., Kage. H., Iijima, F. and Toei, R., 1981, Approximate description of multi-solute adsorption equilibrium in organic aqueous solution. J. &em. Engng Japan 14, 26-31.
Adsorption of unknown substances from aqueous solutions Radke, C. J. and Prausnitz, J. M., 1972, Adsorption of organic solutes from dilute aqueous solution on activated carbon. Znd. Engng Chem Fundam. 4,445-451. Ramaswami, S. and Tien, C., 1986, Simplification of multicomponent fixed-bed adsorption calculations by species grouping. Znd. Engng Chem. Process Des. Dev. 25,133-139.
31
Tien, C., 1986. Incorporation of IAS theory in multicomponent adsorption calculations. Cheza. Engng Cummun. 40, 265-279. Wang, S.-C. and Tien, C., 1982, Further work on multicomponent liquid phase adsorption in fixed-beds. A.Z.Ch.E. J. 28, 565-572.
ADSORPTION
1991.
ooo9-2509/91 93.00 + 0.00 0 1990 Pergamon F%M plc
OF UNKNOWN SUBSTANCES AQUEOUS SOLUTIONS
FROM
HEE MOON and HEUNG CHUL PARK Department of Chemical Technology, Chonnam National University, Kwangju 5ClG757, Korea and
CHI TIEN Department of Chemical Engineering and Materials Science, Syracuse University, Syracuse, NY U.S.A. (First received
6 December 1989; accepted
in revised form
13244,
18 May 1990)
procedure proposed by Jayaraj and Tien for characterizing the adsorption affinity of solutions with unknown compositions was modified by assuming that the solute concentrations may be described by the binomial distribution function. This new method yields results comparable to those of the original Jayaraj-Tien procedure although it requires much less computation. Predictions of batch and fixed-bed adsorption based on the characterization results were found to agree well with experiments.
Abstract-The
an aqueous solution. Kage (1980) and Okazaki
INTRODUCTION In designing adsorption systems for treating water and waste water, one is often handicapped by the fact
et al.
(198 1) argued that the large number of solutes present
in an aqueous solution may be assumed to form a continuum such that the solute concentration may be described by a continuous probability density function of suitable variables represented by a parameter vector, M. This approach was recently justified on theoretical grounds by Annesini et al. (1988). However, the characterization results based on a continuous representation are not consistent with most adsorption algorithms which are formulated for systems with a finite number of solutes. The major difficulty encountered in applying the discrete approach in characterizing solutions with unknown compositions is the large computation demand. If a solution is approximated as one with a number of pseudo species, the accuracy of the approximation, to a large degree, depends upon the number of pseudo species used. Unless the solution to be characterized is simple, one is faced with a search of a relatively large set of parameters. Based on past experiences, such a search is likely to be difficult. The present work is concerned with a further development of the characterization method. To facilitate the subsequent usage of the characterization results, the discrete representation approach is used. The major emphasis of the development is placed on the reduction of the computational demand which is accomplished by assuming that the pseudo species concentrations follow a discrete distribution function (i.e. the binomial distribution function). The main features of this method as well as several examples of its application are given in the following section.
that aqueous systems to be treated often contain a large number of solutes which cannot be completely identified. As the dynamics of any adsorption process depend upon the adsorption equilibrium relationships of the relevant adsorbate-adsorbent system, lacking information on, the solute identities and concentrations makes it difficult, if not impossible, to apply adsorption theories to the design of adsorption systems. In order to circumvent this difficulty, a number of investigators over the past decade have proposed various methods for characterizing aqueous solutions with unknown compositions. Broadly speaking, two different approaches were used in these studies. Frick and Sontheimer (1983) first suggested that solutes present in a solution of unknown composition may be grouped into three classes, i.e. non-adsorbable, weakly adsorbable and strongly adsorbable, and proposed a method for characterizing these three classes of substances and determining their concentrations. This method was subsequently formalized and extended by several investigators (Jayaraj and Tien, 1984, 1985; Kage and Tien, 1987; Crittenden et al., 1985). According to Tien and coworkers, a solution of unknown composition may be represented as one with a number of pseudo species (or theoretical components in the terminology of Crittenden et al.). A pseudo species, in reality, is a group of solutes exhibiting similar adsorption affinity and its use in simplifying multicomponent adsorption calculations has proved successful in previous investigations (Calligaris and Tien, 1982; Mehrotra and Tien, 1984; Ramaswami and Tien, 1986). In contrast to the use of a finite number of pseudo species in representing the unknown solutes present in
PRINCIPLES OF THE PROPOSED METHOD
Characterization of pseudo species concentration discrete distribution function
by a
Similar in concept to the continuous representation 23
24
HEE MOON et al.
approach, we represent the composition of a solution with a large number of solutes by a discrete distribution function and specifically the binomial function of a parameter vector, M. In principle, M may be composed of a number of parameters which characterize the solute adsorption affinity. Since the adsorption affinity of a solute with respect to an adsorbent is most conveniently described by the relevant purecomponent adsorption isotherm, one may choose M to be the constants of the pure-component adsorption isotherm expression. Using this argument and assuming that the purecomponent isotherm data may be described by the Freundlich equation with K and n as the coefficient and the reciprocal of the exponent, M may then be considered to be either K or n or both. Since in the previous work of Kage and Tien (1987), it was shown that an average value of n can be assigned to all the pseudo species as a matter of convenience, M may be assumed to be K. The parametric representation of the pseudo species concentration as a function of the Freundlich coefficient, K, according to the binomial distribution function, is given as K,=
xi=
k,j’
(1)
N! j!(N
-j)!
d(i
- sjN-j
for j = 0, 1, 2, _ _ . , N (2)
where N is the number of pseudo species (note that we also allow the presence of one non-adsorbable species corresponding to j = 0), and K, is the Freundlich coefficient of the jth pseudo species. X, is the total organic carbon (TOC) mass fraction defined as C,
5 Ci, where C, is the concentration of the jth 1 i=1 species expressed in terms of the surrogate quantity, the TOC concentration. k, and s are two constants. The capability of eqs (1) and (2) as a general representation of solutions with a large number of solutes is demonstrated in Fig. 1. In Fig. 1, the histograms of the TOC fractions for N = 8 and different values of s are shown. It is clear that the binomial distribution is indeed capable of approximating compositions of
Fig. 1. Solute concentration distribution according to eq. (2) for different values of s.
various types. The only case for which the use of the binomial distribution function is not satisfactory is when the distribution of the solute concentration is bimodal with the Freundlich coefficients corresponding to the two peak concentrations being significantly different. In the discussions which follow, Xj is assumed to be given by eq. (2) as a matter of convenience and simplicity. According to eqs (1) and (2), characterization of solutions of unknown composition requires the search (or determination) of the values N, n, s and k,_ Before discussing the procedure which may be used to determine these parameters, some general remarks on their orders of magnitude is helpful. For a solution with a large number of solutes of different adsorption affinity, a small value of IV cannot provide a sufficiently accurate representation. On the other hand, there is a practical limit as to the number of the pseudo species one may use. As an example, consider the design of fixed-bed adsorbers for water treatment. Although there exists a number of fixed-bed adsorption algorithms capable of considering an arbitrarily large number of solutes [for example see Hsieh et al. (1977), Wang and Tien (1982) and Moon and Tien (1988a)], the required calculation becomes impractical if N exceeds 10. For most cases, N should be chosen within the range of 5-8. Concerning the magnitude of n, the reciprocal of the exponent of the pseudo species, it is necessary that, for adsorption to be effective, most of the solutes, if not all of them, must be adsorbable and exhibit favourable adsorption behaviour in their pure-component adsorption isotherm data, or l/n c 1.0. The condition that n > 1.0
(3)
should therefore be regarded as one of the constraints in the search of the parameter values. The value of s in eq. (2) determines the skewness of the pseudo species concentration distribution and is within the range 0 -ZZs < 1.0, with s = 0.5 corresponding to a normal distribution type. The parameter k, can be viewed as a quantity which directly determines the adsorption affinity of the pseudo species. For a fixed N, a small value of k, represents a solution with less adsorbable solutes. Regardless of the value of k,, from eq. (l), it is simple to see that the distribution of Kj is not uniform over its range (which is from zero to N2 k,). The spacing of K, between the less adsorbable species is closer than that between the more adsorbable species. This arrangement of K values is made since in multicomponent adsorption, the less adsorbable species often determines the initial breakthrough of adsorption (Hubele and Sontheimer, 1985; Moon and Tien, 1988a). It is also interesting to note the difference between the present method and the method of Jayaraj and Tien (1984, 1985). The original Jayaraj-Tien method requires, for a fixed N, the determination of the Freundlich coefficients, exponents and concentrations for each pseudo species or a total of 3N - 1 quantities.
Adsorption of unknown substances from aqueous solutions
The procedure of Kage and Tien (1987) involves a search to be conducted in successive steps for increasing values of N. For each step, corresponding to specific values of N, K, and n, it is necessary to determine N - 1 concentration values of the N pseudo species. In each case, the number of parameters to be determined increases almost linearly with the value of N. For the present method, only the determination of four parameters is required. The potential advantage of the present method in terms of savings in computation time is therefore rather obvious. N, k,, n and s To obtain these parameter values, certain adsorption data of the solution to be characterized are assumed to be available. Let McXp denote the experimental measurement value and Mprc the calculated value for a particular type of adsorption experiment. The parameter values can be determined from the minimization of the objective function, E, defined as Search of parameters,
where k denotes the kth set of measurement and 1 1 the absolute value. m is the total number of measurements. From the definition of E, it is clear that the objective function is an indication of the fitting error. Most investigators in the past have used batch adsorption data as the basis for characterizing solutions of unknown compositions. Two types of measurements are available; the integral adsorption experiment (IAE) and the differential adsorption experiment (DAE). Detailed experimental procedures for each type have been given elsewhere (Okazaki et al., 1981; Jayaraj and Tien, 1985). For batch adsorption from solutions with N solutes, with the use of the ideal adsorbed solution (IAS) theory for multicomponent adsorption equilibrium and assuming that the pure-component adsorption isotherms of the individual solutes obey the Freundlich equation, the total adsorbate equilibrium concentration can be found from the following equations (Calligaris and Tien, 1982; Tien, 1986):
c, =
$
ci
(5)
i=l
Ci = Zi(l-I/nKi) zi =
,i
(6)
cio /C(n/nKi)” + (“lv)tnln)l
(rl/nK.)n
I
+$,v)(n,n) - l = O-
(7)
(8)
Ci is the equilibrium concentration of the ith component, and n is the reciprocal of the Freundlich exponent (for all the components). A4 and I’ are the mass of the adsorbent and volume of solution. C, is the initial concentration of the ith component. Zi is the mole fraction of the ith component in the adsorbed phase. II is the dimensionless spreading pressure defined as (aA/R T), where R is the spreading pressure. A is the
25
adsorption surface area per unit mass of the adsorbent, R is the gas law constant, and T is the absolute temperature. It is clear that with n, Ki, Ci,. M and Y known, II can be found from eq. (8). Once II is known, Zi can be found from eq. (7). With Zi and II, one can readily determine Ci (i = 1, 2, . . . , N), the sum of which gives Cr. Equations (SH8) can be applied to estimate both the IAE and DAE results. In the former case, the calculation is direct and straightforward. In the latter case, repeated use of these equations is required such that the equilibrium concentrations obtained upon the addition of k th increments of adsorbent are used as the initial concentrations for estimating the equilibrium concentration with the addition of the (k + 1)th increment of adsorbent. In carrying out the parameter search, although it is theoretically possible to search for the optimum values of the four parameters simultaneously, it was found advisable, from the sample calculation results to be given in the following section, to carry out the search in two steps. First, a relatively large value of N is assumed (for example N = 8) and values of n and s are determined from the minimization of E for significantly different values of k,. The value of k, which gives the smallest value of E and also yields a value of n consistent with the criterion of eq. (3) can then be considered as the correct value. Once k, is selected, one can conduct the second step of search, namely, the search for optimum values of n and s for different values of N. SAMPLE
APPLICATIONS
To demonstrate the working procedure of the proposed method, we apply the method to three types of aqueous solutions containing a large number of solutes; namely, solution II prepared by Kage and Tien (1987), the aqueous solution of humic substances used by Hubele and Sontheimer (1985), and the industrial waste examined by Okazaki et al. (1981). The results obtained arc summarized as follows. of solution II of Kage and Tien (1987) Solution II was composed of 10 adsorbates [see Table I of Kage and Tien (1987) for the composition]. Batch adsorption equilibrium concentration data were obtained using solution II with different total initial concentrations (but with the same concentration ratios of the adsorbates) and are shown in Fig. 2. Characterization of solution II was made using the batch adsorption data with CT0 = 216 ppm. The results are given in Table 1. As explained earlier, the search for the optimum values of parameters, N, k,, n and s proceeds in two steps. It was first assumed that N = 8 and the optimum values of n and s were determined for &. = 0.1, 1.0 and 10.0 [see (a) of Table 11. From the results, k, was chosen to be 1.0 since it gave the smallest fitting error (i.e. the smallest value of the objective function) as well as a value of n greater than unity [which satisfies the condition of eq. (3)]. Next, with k, fixed, the search was continued for n Characterization
HEE MOON et al.
26
lob-+--+4 PPM
CT,
0
, I
0
Fig. 2. Batch adsorption data obtained from Kage and Tien (1987).
, 2
I 3
i-W”.
, 4
5
KD/Y3
(4
I
I
‘*I
I
PREDICTION CTD=52.3PPM
-
= 22.2 PPM
O0
n
.
.
0.2
I I.5
1.0 WV
2.0
. KWh3
(b) 01
1
0
1
0.25
0.50
SKEW”ESS
1
0.75
I 1.00
PARAMETER , l
Fig. 3. Contour map of the object function, E, in characterizing solution II of Kage and Tien (1987) with N = 8 and KS = 1.0.
Fig. 4. (a) Experimental and predicted IAE data of Kage and Tien (1987) with C,, = 125 and 216ppm. (b) Experimental and predicted IAE data of Kage and Tien (1987) with C,, = 22.2 and 52.3 ppm.
Table 1. Characterization results of solution II of Kage and Tien
and s with N = 6,8 and 10 [see (b) of Table 11. From the respective values of E of the three cases, it is clear that all the three N values give very similar results. These results also demonstrate the compensating effect between n and s; namely, an increase in n is accompanied by a decrease in s. As a demonstration of the search of the optimum, a contour map of the objective function, E, is shown in Fig. 3 with N = 8 and ks = 1.0. Each closed curve represents a contour with the same value of E and the point denoted by * gives the optimum point at which E is a minimum in the N-s domain. As pointed out by Jayaraj and Tien (1984, 1985) and Kage and Tien (1987) previously, if the characterizing
results
are valid,
one
should
be able
to apply
them, with appropriate adjustment of the concentration values, to both the concentrated and the diluted
versions of the solution ization was based. Fbr by diluting the original the same value of k,, N, tion half of the value of we demonstrate this by
on whose data the characterexample, a solution obtained solution by 50% should have s and II, and a total concentraoriginal solution. In Fig. 4(a), comparing the predicted total
Adsorbate: mixed organic chemicals = 216 ppm TOC
Type: IAE data with C,
(a) First step with N = 8
k 0.1 1.0 10.0
”
l.ClO 1.51 2.64
s
0.507 0.359 0.240
(E) 3.88 3.84 6.45
(b) Second step with k, = 1.0
N
?I
s
6 8 10
1.44 1.51 1.55
0.422 0.359 0.301
3.82 3.84 3.87
adsorbate equilibrium concentration CT from batch adsorption experiments using solution 11 with an initial total concentration C,, = 125 ppm. The predictions were made with N = 8, k, = 1.0, n = 1.51 and
Adsorption
27
of unknown substances from aqueous solutions
Table 2. Search results of solution II based on the modified Jayarai_Tien procedure with N = 8 Search number
Ki 0 1 2 3 4 5 6 7 8 E (%)
0 1 4 9 16 25 36 49 64
Average’
xi [from eq. (2)]
2
3
4
0.0975 0.0000 0.2866 0.3897 0.203 1 0.0023 0.0189 0.0019 O.CK300
0.0925 0.0136 0.2461 0.4773 0.1500 0.0068 0.0083 0.0006 0.0090
0.0872 0.0110 0.3 502 0.2050 0.3162 0.0148 o.ooo2 0.0149 0.0004
0.0919 0.0043 0.3202 0.4058 0.0012 0.0758 0.0089 0.0065 0.0854
0.0923 0.0073 0.3007 0.3684 0.1676 0.0249 0.0091 O.MMO 0.0237
0.0285 0.1277 0.2502 0.2804 0.1963 0.0879 0.0246 0.0039 0.0003
2.75
2.63
3.27
3.52
3.04’
3.84
1
xi,
‘Average TOC fractions (X,-j obtained from the four searches. *Predi&on
error by using &rage
TOC
fractions.
s = 0.359. Also included in Fig. 4(a) are the results for
solution II with CT0 = 216 ppm. This latter comparison, of course, is merely an indication of the degree of fitness of the characterization results to the ex~rimental data on which they are based. Furthermore, Fig. 4(b) shows the characterization results of diluted versions of solution II with C, = 52.3 and 22.2 ppm, respectively. As expected, the predictions are not satisfactory, which means that the applicability of the proposed method is not good to very diluted versions. Kage and Tien (1987) already pointed out that the predictive accuracy for very diluted solutions are not good [see Tables VII and IX of Kage and Tien (1987)]. As a comparison of the present method and the previous method of Jayaraj and Tien based on the value of N = 8, k, = 1.0 and n = 1.51, the TOC mass fractions of pseudo species were determined by a random search. The mass fraction values obtained from a number of independent searches are shown in Table 2. As expected from the Jayaraj-Tien method, different results were obtained from each search but, by and large, they are similar. Also included in Table 2 are values of X, calculated according to eq. (2) and the average values of the four independent searches. It is clear that these two sets of values are approximately although the computation requirement of the present method is much less than the Jayaraj-Tien method. Other characterization results As a further demonstration of the utility of the present method, it was also applied to the characterization of two other aqueous solutions considered by previous investigators, namely the aqueous solution of humic substances (Hubele and Sontheimer, 1985) and the industrial waste examined by Okazaki et al. (1981). Characterization of the solution of humic substances was made based on IAE data and that of the other solution was based on DAE data. The characterization results are summarized in Table 3. For both characterizations, N was chosen to be 8 and the k, value for the first step of search was found
Table 3. Characterization results of two with N = 8
other solutions
(a) Solution of humic substances (Hubele, 1985)
k.
n
s
0.1 1.0 10.0
0.86 1.38 12.33
1.000 0.46 0.312
(Z) 7.63 3.79 (0.97)’ 23.03
(6) Waste water of Okazaki et al. (1981)
k,
n
s
0.001 0.01 0.1
0.88: 1.75 15.00
0.789 0.499 0.283
(6) 4.90 5.01(4.50)’ 9.75
‘Fitting error obtained by using the Jayaraj-Tien procedure. *This violates the condition given by eq. (3).
to be 1.0 and 0.01, respectively. Based on these values, the optimum values of n and s were determined. Also included in Table 3 are values of E corresponding to the respective characterization results as well as the values of E obtained by using the Jayaraj-Tien method. From the comparison of these E values, it is apparent that, in the case of the solution of humic substances, the binomial distribution function gives somewhat less satisfactory results than the Jayaraj-Tien method. Nevertheless, the characterization results were found to give sufficiently accurate prediction of fixed-bed adsorption results to be discussed later. In addition to the DAE data which were used as the basis for characterization, Okazaki et al. provided additional DAE and IAE data using less concentrated solutions. In Figs 5 and 6 the predicted equilibrium concentration results of the DAE and IAE experi-
HEE Moow 20
I
I
\
f5
- --
\
0
\
Table 4. Experimental conditions of fixed-bed experiments
I
BY THIS METHOD
FITlED
-
-“b
r
I
t
et al.
Conditions
PREDtcnoN c+0=13.5
Solution II
YCnlY3
Activated carbon used
I
I
I
10
20
30
Cl 0
HUbeMS solution
40
I
I
30
so
F-400
Particle diameter ( x lo3 m) Bed density (kg/m3) Bed height (m) Bed porosity Superficial velocity ( x 10” m/s) Initial TOC (or DOC) concentration (ppm)
TC
1.000 490.0 0.15 0.367 0.728
F-300 1.320 485.0 2.35 0.420 1.389
94.10
7.970
EWV.KG/M3
Fig. 5. Experimental and predicted DAE data of Okazaki et al. (1981).
2.0
15
ea t
)
I
I
I
I
i“‘--
#I2 a
0
-
PREDICTIGK
0
4
,
0
CTo=15.5MOL/M3
I
I
I
I
5
10
15
20
m/V.
KG/Y3
Fig. 6. Experimental and predicted IAE data of Okazaki et a[. (1981).
ments were compared with experiments. Good ment was observed in both cases.
bulk density of adsorbent. C, and Qi are the solution phase and average adsorbed phase concentrations, respectively. CT is the inlet value of Ci and Qio is the initial value of Qi. Fixed-bed adsorption may be characterized by its breakthrough curve or the effluent concentration history. The solution of eqs (9Hll) together with appropriate rate expressions gives the efIluent histories with various bed lengths. For the simplest case, if local equilibrium is assumed [namely, Ci and Qi are in equilibrium or related by eqs (5x8) if the IAS theory is applied], eqs (9Hll) can then be solved to obtain the effluent concentration history. As shown by Moon and Tien (1988b), breakthrough curves for fixed-bed multicomponent adsorption are composed of a number of plateau and are characterized by the transition times and plateau concentration values if the adsorbate species exhibit favourable adsorption behaviour. These quantities may be determined from the following equations:
agreeOjCi, =
aQij+
$
k=j+l
(Ok-, - O,)C,,
i>j+l (121
PREDKTION
OF FIXED-BED
ADSORPTION
Both Kage and Tien and Hubele and Sontheimer conducted fixed-bed adsorption measurements using their respective solutions. One can therefore compare these experimental data with predictions based on characterization results discussed above. Agreements between predictions and experiments constitute further validation of the characterization method. The conditions used in conducting these experiments are summarized in Table 4. For a solution containing N adsorbable solutes passing through a bed packed with adsorbent, the macroscopic equations of conservation of the individual solutes are
ac,
udz+pbs=O
i=l,2,3
Qi = Qio
z > 0,
ci = cp
z=o,
,...,
oto o>o
N
(9)
(10) (11)
where z and 0 denote the axial distance and corrected time (defined as 0 = t - Z&/U,with t, E and u being the time, bed porosity and superficial velocity). P,, is the
OjCij = Qij c,
= 0
ci, = cp a = L&/U
i=j
(13)
i
(14) (15) (16)
where L is the bed length, 0, is the transition time from the (j + l)th plateau to the jth plateau (note 0, < Oj- 1 for all js), Cij is the solution phase concentration of the ith solute in thejth plateau, and Qij and C, are in equilibrium or, in other words, they are related by eqs (5)-(g). The above results apply to solutions with N adsorbable solutes with the solutes arranged in the descending order of their adsorption affinity. For solutes with the same Freundlich exponent, the solutes should be enumerated in the descending order of their K values. The fixed-bed adsorption calculation results were compared with experiments and are shown in Table 5, Fig. 7 (solution II with CO, = 94.1 ppm TOC of Kage and Tien) and Fig. 8 (aqueous solution of humic substances with CO, = 7.97 ppm dissolved organic carbon). In Table 5 and Figs 7 and 8, predictions based
Adsorption Table
of unknown
29
substances from aqueous solutions
5. Estimated transition times and plateau concentrations of effluent histories of fixed-bed adsorption on different characterization results Solution
Hubele’s solution
II Modified Jayaraj-Tien
This work Time (h)
Time W
Cl+%
0 11.0 33.9 71.5 133.0 234.0 390.0 614.0 918.0
0.029 0.199 0.513 0.784 0.930 0.984 0.998 0.999 1.000
based
Time (h)
G-/G
0 22.3 32.4 67.4 135.0 254.0 431.0 667.0 901.0
Modified Jayaraj-Tien
This work
0.092 0.113 0.522 0.849 0.959 0.974 0.980 0.985 1.000
Time 01)
WC;
0 271.0 772.0 1480.0 2520.0 4060.0 6320.0 9480.0 13,660.O
0.007 0.074 0.280 0.569 0.808 0.939 0.987 0.999 1.000
Cr/G
0 350.0 806.0 1430.0 2420.0 4080.0 6520.0 9870.0 13,840.O
0.020 0.046 0.231 0.585 0.865 0.959 0.983 0.988 1.000
1.0
O#” k 0
0.5
e
O -
= 94.1
THIS
-
-
-
PPM
TOC
WORK
MODIFIED
JAYAAAJ
-
TIEN
PROCEDURE 0
Ii 0
Fig. 7. Predicted
and experimental
effluent history of solution
II with C; = 94.1 ppm TOC.
1.0
0.5
-
THIS WORK -
-
MDDIFIW
I
JAYARAJ
-TIEN
A
OLJJ-
I
I
0
I
I
2 TIME
Fig. 8. Predicted
on
characterization
and experimental
results
from
the
I
4 X 103,
Jayaraj-Tien
I
I
8
HR
effluent history of Hubele’s organic carbon.
method are also included. As shown in Table 5, for both solutions, different predictions of transition times and plateau concentrations were observed.
I
6
solution
with C; = 7.97 ppm dissolved
However, these differences in most cases are not significant. In terms of their comparisons with experimental data, as shown in Figs 7 and 8, the comparison between predicted and experimental results is not
30
HEE MOON et al.
especially good. It is not surprising that the comparisons are not good, bearing in mind that the predictions are based on the assumption of equilibrium control. CONCLUSIONS
We have demonstrated that the relatively simple method developed in this study can be used to characterize solutions with unknown compositions for adsorption calculations. The method has two advantages over the Jayaraj-Tien procedure: first, unlike the previously developed method, it yields a unique representation of the solution to be characterized when the number of adsorbable pseudo species is fixed. Second, the computation requirement is rather modest. In a number of sample cases, characterization results obtained from applying the present method were found useful in estimating adsorption performance in batch and fixed-bed operations. Acknowledgement--Hee Moon and Heung Chul Park wish to acknowledge the support they received from Korean Science and Engineering Foundation under Grant No. 8811005-015-02.
NOTATION
area of the solution-solid interface concentration in solution liquid phase concentration at bed inlet total adsorbate concentration in solution object function or fitting error constant in eq. (I) Freundlich coefficient of the jth pseudo species bed length number of data mass of adsorbent parameter vector experimental value predicted or calculated value reciprocal of Freundlich exponent number of adsorbable pseudo species concentration in the adsorbed phase gas constant constant in eq. (2) or skewness factor temperature superficial velocity volume of solution TOC
mass fraction
Subscripts i, j ith and j th pseudo species experimental exp predicted or calculated pre T total 0 initial Abbreviations DAE differential adsorption equilibrium IAE integral adsorption equilibrium IAS ideal adsorbed solution TOC total organic carbon
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Annesini, M. C., Gironi, F. and Marrclli. L., 1988, Multicomponent adsorption of continuous mixtures. 2nd. Engng Chem. Res. 27, 1212-1217. CaIIigaris. M. B. and Tien, C.. 1982, Species grouping in multicomponent adsorption calculations. Can. J. &em. Engng60, 772-780. Crittenden,J. C., Luft, P. J., Hand, D. W. and Friedman, G., 1985, Prediction of multicomponent adsorption cquilibrium in backnround mixtures of unknown comnosition. J. Wat. Res. 17:312-319. Dobbs, R. A. and Cohen, j. M., 1980, Carbon adsorption isotherms for toxic organics. EPA-60018-80-02. Frick, B. R. and Sontheimer, H., 1983, Adsorption equilibrium in multisolute mixtures of known and unknown composition, in Treatment of Water by Granular Activated Carbon (Edited by M. J. Mcguire and I. Suffet), pp. 247-268. Advances in Chemistry Series 202, American Chemical Society, Washington, DC. Hiseh. J. S. C., Turian. R. M. and Tien, C., 1977, Multicomponent liquid phase adsorption in fixed bed. A.I.Ch.E.
J. 23, 263-275. Hub&, C., 1985, Adsorption and biologischen abbau van huminstoffen in aktivkohelefiltetn. Dr-Ind dissertation, Karlsruhe University, Karlsruhe. Hubele, C. and Sontheimer, H., 1985, Isotherm evaluation and filter breakthrough calculations for waters with a multicomponent mixture of organ& using the simplified competitive adsorption model (SCAM). Unpublished work. Jayaraj, K. and Tien, C., 1984, Characterization of adsorp tion affinity of solutions, in Proceedings of National Environmental Enaineerina Conference (Edited by M. Pirbazari and J. S-Deviniy), pp: 394-399. ASCE. Jayaraj. K. and Ticn, C., 1985, Characterization of adsorption affinity of unknown substances in aqueous solutions. Ind. Enana - _ Chem. Process Des. Dev. 24, 1230-1239. Kage, H. and Tien, C., 1987, Further development of the adsorption affinity characterization procedure for aqueous solutions with unknown compositions. Ind. Engng
Chem. Res. 26, 284292. of the jth pseudo
species in
solution axial distance TOC mass fraction of the jth pseudo species in the adsorbed phase
Kage. H.. 1980. Characteristics of multi-solute adsorution in $1&e iqme&s solution. DEng thesis, Kyoto, Japk. Mehrotra. A. K. and Tien. C.. 1984, Further work in species grouping in multicomp&e& adsorption calculation. Can. J. them. Engng 62, 632443. Moon, H. and Tien, C., 1988a, Fixed-bed multicomponent adsorption under local equilibrium. A.I.Ch.E. Symp. Ser.
No. 264 84,94-108.
Greek fetters K
defined
s
differenti& m ’ -’ spreading pressure dimensionless spreading pressure bulk density of adsorbent
; Pb
bv ea. f161
Moon, H. and Tien, C., 1988b, Prediction of fixed-bed of solutions of unknown composition. adsorption
A.I.C%E. Symp. Ser. No. 264 84, 23-35. Okazaki, M., Kage. H., Iijima, F. and Toei, R., 1981, Approximate description of multi-solute adsorption equilibrium in organic aqueous solution. J. &em. Engng Japan 14, 26-31.
Adsorption of unknown substances from aqueous solutions Radke, C. J. and Prausnitz, J. M., 1972, Adsorption of organic solutes from dilute aqueous solution on activated carbon. Znd. Engng Chem Fundam. 4,445-451. Ramaswami, S. and Tien, C., 1986, Simplification of multicomponent fixed-bed adsorption calculations by species grouping. Znd. Engng Chem. Process Des. Dev. 25,133-139.
31
Tien, C., 1986. Incorporation of IAS theory in multicomponent adsorption calculations. Cheza. Engng Cummun. 40, 265-279. Wang, S.-C. and Tien, C., 1982, Further work on multicomponent liquid phase adsorption in fixed-beds. A.Z.Ch.E. J. 28, 565-572.