Ind. Eng. Chem. Res. 1994,33, 2764-2770
2764
Continuous Model for Complex Mixture Adsorption Maria Cristina Annesini,’ Massimiliano Giona,?and Fausto Gironi Dipartimento di Ingegneria Chimica, via Eudossiana 18, 00184 Roma, Italy Adsorption of mixtures containing several ill-identified solutes can be studied by discrete or continuous models. I n the first case, all the solutes are lumped into a few pseudocomponents. More than one pseudocomponent is generally required to obtain a reliable description of the system, and the computational effort increases with the number of pseudocomponents used. In the second, the mixture composition is described by means a continuous distribution function. If the feed distribution is known, the distribution in the liquid phase a t equilibrium depends solely on the degree of total solute removal. Under some hypotheses on the feed distribution function, which are largely borne out in real systems, a limiting distribution is achieved for a high degree of solute removal. The difference between the feed distribution and the limiting distribution is significant when the variance of the former is large. The overall adsorption isotherms can be predicted if the feed distribution function is known. An identification method is proposed for a fairly complete statistical characterization of the feed distribution function extracting from the experimental data.
Introduction Adsorption is a unit operation widely used in water quality control. In recent years great attention has been devoted to the treatment of drinking water that contains a wide spectrum of usually ill-identified organic solutes. In particular, adsorption processes can be used to remove humic and fulvic acids representing from 40 to 99% of the carbon content of dissolved organic matter. These compounds have been recognized not only as colored bodies with an esthetically unpleasant effect but above all as trihalomethane precursors with all the possible implications for the public health. It is therefore necessary to remove these compounds from drinking water in order to reduce the health risks associated with disinfection by-products. Studies performed on natural and synthetic solutions of humic substances reveal a wide distribution of molecular weight in the dissolved organic matter (Admirtharajah et al., 1993), i.e., a variety of thermodynamic characteristics of the components. The design of equipment for the adsorption process requires a knowledge of the thermodynamic equilibrium of such complex systems, but classical procedures for multicomponent adsorption are unsuitable in these cases, mainly because of the impossibility of isolating and identifying the components by ordinary chemical analysis. Furthermore, if the components are known, it would be a complicated problem to manage systems consisting of a large number of equations. A n approximate method is therefore required to describe such adsorption systems. In the classical approach, components are lumped into fractions and each fraction is regarded as an equivalent pure component. Alternatively, if the mixture is assumed t o form a continuum, the solute concentration may be described by means of a continuous probability density function. The theoretical framework of continuous thermodynamics (Kehlen and Ratzsch, 1985) has recently been applied to the study of adsorption equilibrium from multisolute aqueous mixtures (Annesini et al., 1988). The adsorption of
* To whom correspondence
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a multisolute mixture with a rectangular distribution function of the adsorption equilibrium coefficients has been studied by Sheintuch and Rebhun (1988). These authors also consider the coadsorption of a known micropollutant (tracer) and an unknown multisolute mixture (background) described as a continuous mixture. In the present work we generalize all the previous results (Annesini et al., 1988; Sheintuch and Rebhun, 1988) on the modeling of the adsorption phenomena in a continuous mixture referring to an arbitrary feed distribution. In particular, the statistical structure of the distribution function a t different degrees of solute removal is derived and related to the moment hierarchy of the feed distribution. All the main results obtained in the present paper are independent of the particular choice of the feed distribution and of the functional form of the adsorption isotherm. The article is organized as follows. Firstly we briefly compare the lumped vs continuous modeling of adsorption in multicomponent mixtures. We then analyze the functional form of the distribution function at different degrees of solute removal and derive a closed form expression for the resulting moment hierarchy. The limiting case of high solute removal is developed in detail, deriving general criteria for the limit distribution function and applying these results to the case of two particular families (gamma and log-normal) of feed distributions. Finally the behavior of the experimentally measurable quantities (total solute concentration, total adsorbed amount) is related to the functional form of the feed distribution and to the adsorption isotherm. The results obtained represent a general framework for approaching the identification problem in continuous mixtures.
Comparison of Discrete and Continuous Models The simplest approach treats the complex mixture as a single pseudocomponent whose adsorption parameters are determined from the “overall isotherm” referring to an overall organic concentration (COD, TOC, and so on). The main shortcoming of this procedure is that for a multisolute mixture the relation between adsorbed amount and equilibrium liquid phase concentration is not univocally determined but depends on the experi0 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33,No. 11, 1994 2765 mental procedure (Sheintuch and Rebhun, 1988). A single pseudocomponent approach is therefore unsatisfactory, and parameters obtained from a pseudo single solute model cannot be used to represent conditions different from those tested. The natural way to improve the description of the mixture is to group all the solutes into a few classes and to calculate the adsorption amount in terms of standard multisolute adsorption procedure. This procedure has been widely discussed by Calligaris and Thien (1982), Jayari and Thien (1985), and Kage and Thien (1987). The accuracy of the method depends, to a large degree, on the number of pseudospecies used, but unfortunately, the number of pseudospecies required cannot be determined a priori. Moreover, there is no unique way t o group the single components into pseudospecies. It is therefore possible t o observe substantial differences between the fitting parameters obtained from each independent search. On the other hand, increasing the number of pseudocomponents increases the computational effort, both in the fitting of parameters and in the subsequent design calculations. A procedure serving to avoid the introduction of too many empirical parameters is suggested by Moon et al. (1991). In contrast to the use of discrete pseudospecies, a previous paper (Annesini et al., 1988) suggests continuous thermodynamics as a way to describe the adsorption of mixtures with several ill-defined components. In this procedure, a mixture with overall concentration C is described by means of a distribution function w ( n such that the concentration of components with characteristic variable I in the range (I'J") is given by
and the overall concentration C is given by the integral of w ( n over the whole range of I. The distribution function may be characterized by the moment hierarchy {M,}, where
adsorbed solution theory gives the following expression for the total adsorbed amount N:
N = mf(RC9)
(4)
with K = M1/Mo = MdC. The adsorbed amount of the component characterized by K is given by
Kw(K) n(K ) = mf(KC,p)KC
(5)
The distribution function naturally refers t o the liquid mixture in equilibrium with the adsorbed phase and differs from that of the feed.
General Properties of the Distribution Function at Different Degree of Solute Removal The distribution function a t different degrees of total solute removal can be obtained from the material balance equation for each component
w " ( K )- w ( K )= &(K)
(6)
where C$ is the amount of solid per unit amount of solution and the superscript " refers to the feed mixture. By integrating eq 6 over the whole range of K, we obtain c" - C = C$N, Equations 4-6 give
w"(K ) - w(K )
C"- c
Kw(K ) --
(7)
KC
Following a procedure similar to that reported by Marczewski et al. (19901, the distribution function in the liquid phase can be expressed as
Kw"(K 1% w(K)= xK+(1 -x)K-
w"(K 1 1+(1-x-
-
KC" J X w ( X ) dK'
F [w(K),.K 1 (8) The first-order moment M I is related to the mean value of I (I= M1/C),and the second-order moment is related to the variance (a2= MdC - 12). If single-solute adsorption isotherms are known, the ideal adsorbed solution theory (Radke and Prausnitz, 1972) allows us t o predict the total adsorbed amount N and the adsorbed phase composition as reported in the previous work. In a fairly general way, it is possible to assume that the single-solute adsorption isotherm may be written as
where m and K are parameters with dimensions of n and c-l respectively andp is a dimensionless parameter. For example, the Langmuir isotherm may be written in this form with m = nmaxand referring to p as the exponent of Kc (p = 1). With the Freundlich isotherm, m may be regarded as an arbitrary parameter required to adjust the dimensions in eq 2. If the samep value is used for all the components, K itself may be used as the characteristic variable. It can be proved (Appendix A l ) that, with the singlesolute adsorption isotherms given by eq 3, the ideal
where x = C/C"E (0,l) and F is a nonlinear functional of w(K ) depending explicitly on K. The above equation relates the distribution function of the mixture to the overall amount of solute removed from the liquid phase x and t o the feed distribution function wo(K). To simplify the notation, we have not reported explicitly in the functional F its dependence on x. An equation similar t o eq 8 arises in developing the continuous formalism of a simple reaction in a perfectly stirred reactor (Astarita and Nigam, 1989). We shall now present some general mathematical properties of the functional equation (8) which may be useful in modeling the adsorption of a continuous mixture. From eq 7 it follows that
The last equation implies that the moment hierarchy {Ma}can be directly evaluated from { M O n } once x and M1 are known. The first-order moment M I is the solution of the implicit nonlinear equation
1+
Ml
2766 Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994
The last equation can be solved if wo(K1 is explicitly known. Examples are reported in the next section. If w"(K ) is a summable function in [O,m), as it should be for physical reasons, it follows from eq 8 that
w ( K ) Iw"(K ), b' K
E [O,w)
(12)
w(n+lj(K = F w (n)( K ) , K l It can be proved that the sequence { w(,)(K 1) converges to the solution w(K 1, as reported in Appendix A3. It is convenient to normalize the distribution by defining W(K ) as
W ( K ) = w(K )/xCo
w"(K 1 (1 - x)KC"
(14)
M" 1
With this normalization M*1 is equal to the mean value of K , K,which satisfies the nonlinear implicit equation
C
x - + K(1 - Y ) -K
dK
(15)
Limiting Distribution for High Solute Removal
-
It is worth studying the limiting behavior of the distribution W(K) for x 0, i.e., for high solute removal. In other words, we look for the limiting distribution
W "(K ) = lim x-0
and the limiting value of
xco+
(1- .. XIKC"
(16)
K
a
E" = l i m K
(19) This contradicts the above hypothesis that K" > 0. This case has an immediate physical meaning: a feed distribution with a nonzero value for K = 0 corresponds to a mixture containing a finite or an infinitesimal amount of nonadsorbable compounds. Dealing with a mixture containing a finite concentration of a nonadsorbable compound, cna,the limiting distribution attains the form w"(K) = cnad(K) and C = Cna for x 0. Therefore we can determine Cna from the high loading adsorption data and we can refer to a new distribution, w o ( K ) - cnad(K),which does not contain a finite concentration of nonadsorbable compound. (b) If wo(K)/K is a summable function with w " ( K )= 0 only for K = 0, the sequence (14)converges uniformly in any open interval (a,b),b > a > 0, and therefore
(13)
and correspondingly MY;, = Mn/xCo. The normalized distribution satisfies the equation
cox+
which implies that E" = 0, In fact, if we assume that K " > 0 , we have W "(K 1 = K "wo(K)/KC"and therefore
-
w("(K 1 = F [w'"(K ),K 1
W ( K )=
(18)
(11)
and w(K ) is therefore still summable in [O,-). It can be proved that the functional equation ( 8 ) admits only one solution (see Appendix A2). This solution can be obtained by applying an iterative schema where the successive approximation of w(K ), w(,)(K), can be obtained from eq 8, starting from an initial distribution function equal to the feed distribution
ul'"(K) = F [w"(K1,K I
W " ( K )= d ( K )
(17)
x-0
Since W(K is summable for %veryy, (see_eqs 11 and 13), it follows from eq 16 that K < m, i.e., K is bounded and K is finite. Some general properties of the asymptotic distribution W "(K 1 can be stated independently of the explicit form of wo(K1. (a) The feed distribution attains a finite nonzero value for K = 0 ( ~ " ( 0t)0). In this case the normalized limit distribution is given by a Dirac delta:
(20) The proof is given in Appendix A4. It is interesting to observe that all the fundamental features of K- lie in the summability condition of w"(K YK, i.e., in the local behavior of w o ( K )in the neighborhood of K = 0. A similar result can be proved by following the same technique in the case of an initial distribution wo(K) > 0 in a countable union of intervals (in this case the uniform convergence holds true in the union of all the bounded open intervals where w Y K ) t 0). As a consequence, we can also treat mixtures containing different classes of solutes, whose distribution function is expressed as the sum of discrete broadened peaks (multimodal distribution). Once K " is known, the entire moment hierarchy M", of the limiting distribution W "(K ) is obtained according to eq 9
M*,+l = K"(M",/C") for
x
-
0
(21)
Equation 21 is the most general way of representing the relationship existing between w"(K ) and the initial distribution, holding true for arbitrary expressions of wo(K1. The above results can be applied by choosing the explicit form of w"(K ). For example, Marczewski et al. (1990) analyzed the behavior of square and triangleshaped distributions. We consider gamma and lognormal distribution since they represent versatile families of unimodal distributions widely used in continuous thermodynamics. In particular, the log-normalfunction is suitable for describing broad-band distribution with arbitrary large values of the variancelsquare average ratio and regular behavior at K = 0. Without loss of generality, we normalize w"(K1 so that the first-order moment Mol is equal t o C" (Le., that K " = 1).
Gamma Distribution. In this case, the feed distribution is given by
Ind. Eng. Chem. Res., Vol. 33,No. 11, 1994 2767
and depends on /3 > 0. Different cases should be considered separately: /3 > 1: In this case w"(K)/Kis summable and W-YK converges uniformly to K w"(K ) / C"K; therefore K = /3/(/3 - 1)is obtained from eq 20. /3 I 1: I n this case w"(K)/K is not summable and therefore K must be obtained by integration of eq 15 in the limit for x 0. An explicit solution of eq 9 is obtained (Gradshteyn and Ryzhik, 1987):
-
P E (O,m)-{N+) where y = xm(1 Gamma function
x) and
(23)
IYa,d is the incomplete
Since for /3 E N+ we have
in the limit of y
- 0,
eq 23 becomes
adsorbed amount vs the overall solute concentration, given the initial distribution w"(K1. On the other hand, since the overall solute concentration and the overall adsorbed amount are the output variables of a standard adsorption experiment, the analysis proposed in this section can also be used to tackle the inverse problem, i.e., the problem of evaluating the statistical parameters of the initial distribution. The latter problem was clearly stated by Sheintuch and Rebhun (19881, who suggested a qualitative criterion for the estimation of the mixture heterogeneity and, in other words, for discriminating between pseudo single-component mixture and continuous mixture. As previously discussed, for a multisolute mixture, the relation N vs C depends on the feed composition and on the experimental procedure. In this paper we consider the overall isotherms obtained by placing a feed solution of k n o w n composition in contact with an increasing amount of adsorbent solid. Different isotherms are obtained by diluting the feed, i.e., by varying the overall solute concentration of the feed C" while keeping the distribution function W "(K constant. The c y e N vs C can be readly obtained from eq 5 as N = mf(KC,p),where K depends on the degree of solute removal through eq 15. The general properties of the curves N vs C obtained by placing the solution in contact with increasing amounts of adsorbent solid, for fixed c" values, are reported below. (a) All the curves lie between the curves N = m f (K "C,p),pertinent t o the initial distribution function, and N = m f ( K "Cp),pertinent to the limiting distribution function. In particulg, for any C" values, the curves start from N = m f ( K "C",p)for C C_. whereas all the curves collapse in the curve N = mf ( K "C,p)for c/c" 0. (b) In accordance with eq 4, the slope of the curve N vs C is given by
-
-
and therefore K - = O. To sum up, analysis-of the asymptotic behavior of K in the limit of x 0 ( K ") yields
-
PE(O,l)
/3 E [1,-3
K"=O
(27)
K" = (p - l)/B
Log-Normal Distribution. In the case of a lognormal distribution with unit mean
0 " '
where f ' = df(KC)/d(KC) and the term in square brackets represents an enhancing factor with respect i o a single-solute mixture with the value of K equal to K. To evaluate this term, it must be borne in mind that for constant feed concentration
(31)
= d2 ( 2 8 )
the condition of summability of w"(K)/Kis always satisfied. The limiting mean value K" is therefore obtained straightforwardly from eq 20 as
Furthermore, if the hypothesis of uniform convergence of W(K ) holds, the WdX is given by
&
Mz-rnl
dx
XMl
-_ -
(29)
(32)
In both cases, K - is related to the variance of the distribution of the feed. Figure 1 shows the effect of feed variance in the case of a log-normal distribution: go significant variation of the distribution occurs if aoZ/ K RZ 1,whereas a significant shift is observed for high ao2values.
where M I and M Z_are the moments of the fupction w"(K )4(1- x)K + j K 12. In the-limit of C c",M Iand reduce to C"IK and MdK respectively. Conversely, for C 0, f i 1 and M Zredu_e t o the moments of the function w"(K )lK 2, i.e., to c"/K" and to c",respectively. Therefore, for C C" we have
Overall Adsorption Isotherms and Identification of the Initial Distribution The analysis carried out above allows us to predict the "overall isotherms", i.e., the plot of the overall
whereas for C
O2
-
-
-
O2
02,
-
0 we have
2768 Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994 31
a
2.5 C
.-c0
3
P, c L
.-U v)
U 0)
-.-N
2
b z
0.5
0
1
1.5
0
2
1
0.5
Characteristic variable
1.5
2
Characteristic variable
Figure 1. Feed and limiting distribution for different variances of the feed distribution. (a) aazIKO2 = 2.71; (b) ao21K = 1.04. O2
(34) Therefore, all the isotherms start with C = C" and N = m f ( K"c")but with a slope geater than that of a singlesolute mixture with K = KO. Furthermore, all the curves collapse into the c;rve corresponding to a singlesolute mixture with K = K for low C values. Figure 2 shows the curves obtained for multisolute mixtures with the Freundlich isotherm with the same p value for all the solutes. For example, Figure 2 shows the curves obtained for a multisolute mixture where all the components obey the Freundlich isotherm with the same p value and an initial log-normal distributton. The upper dotted line in the plot refers to N-= ( K "CY while the-lower one refers to N = (K -CY'. K is obtained from K " and ao2 according to eq 29. It should be noted, from a qualitative point of view, that the greater the variance of the initial distribution, the greater the difference between KO and K ", the greater the distance between the upper and the lower straight line in the plot and the greater the difference between the curves N vs C obtained for different C" values. Furthermore, the overall isotherms have a nonlinear trend in a log-log scale. In fact, the slope for C c" is given by
0
E
-
-
:E- ( g2)
--
P l+_o,
(35)
while for low C values we have dlnNdlnC-'
(36)
Again, the greater the feed variance, the greater the deviation of the curve from the linear trend. We can now consider the inverse problem, i.e., the problem of identifying the shape and the statistical parameters of the initial distribution from N vs C experimental data. For sake of simplicity we again refer to a Freundlich-type mixture with the samep value for all the components, but the same analysis can be
/ 0.1 L - ' 0.01
'
Km=O. 1054
0.1
1
-
100
10
Overall solute concentration,
EC
Figure 2. Overall adsorption isotherms for a continuo_usmixture with a log-normal distribution function ( p = 113; ao21K = 9.5). O2
developed in the case of Langmuir or other functional form of the gdsorgtion isotherm. The above analysis shows that KO, K " , and p can be obtained from the experimental data in the limiting of C C" and C/c" 0; furthermore, the slope of the isotherms for C c" gives the feed variance, ao2,according to eq 35. On the other hand, if we consider, as usual, a-two-parameter family of initial distribution,the ratio K Y K " is univocally related to the ratio oo2/K Figure 3 shows these curves for some types of distribution: curves for gamma and log-normal distributions are obtained from eqs 27 and 29, respectively, while the curve for rectangular distribution is given by
-
-
-
02.
K
2(31/2)a"/K "
K"
1
-- -
(37)
+ (31'2)a"/K"
In 1 - (31'2)a"/K "
The figure shows that different cjistrjbution functions result in differentbehavipr in the K "/K " vs ao2/K plot. Since K V K " and oo2/K values are obtained from experimental data, a shortcut method to choose the O2
O2
Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994 2769 I = characteristic variable
K = parameter of single-solute adsorption isotherm m = parameter of single-solute adsorption isotherm '
Ik
M,,= nth-order moment of w(K)
8 lY
fin = nth-order moment of the distribution function used
M",,= nth-order moment of W ( K )
in eq 31 n = distribution function in the adsorbed phase N = overall adsorbed amount p = parameter of single-solute adsorption isotherm w = liquid-phase distribution function W = normalized liquid-phase distribution function ~
0
0.5
1
15
2
25
3
35
4
Greek Symbols
~~
4.5
5
Qo2/p2
Figure 3. Ide_ntifjcationof-the shape of the feed distribution function from K V K ' vs aa2IK data. O2
shape of the distribution function is given by comparing these values with the theoretical curves reported in the elot. In other words, the analysis in the K -1K " vs ao21 K plane represents a closure condition for testing the validity of the assumed shape of the distribution. Of course, a complete validation of the assumed distribution can be obtained by comparing the whole trend of the reconstructed N vs C curve with the experimental data. To sum up, the outlined method of analysis allows us to characterize the initial distribution function completely, in its shape and statistical parameters, from a complete set of experimental data. In fact, analysis of experimental data obtained with a single c" value or data obtained in a limited range of CIC" may result in a misleading model. For example, analysis of single C" data in a limited range of CIC" may lead to the system being described as a single pseudo solute mixture with an erroneousp-value. Of course, in this case erroneous prediction of the adsorbed amount would be obtained for the same mixture a t different C" values. O2
Concluding Remarks We have developed a general method of studying the adsorption of multicomponent mixtures in the framework of continuous thermodynamics. The analysis of the distribution function a t different degrees of solute removal permits complete description of the adsorption process. This result can be used in many practical applications, for example in the water treatment process, if it is necessary to monitor some specific pollutants. The initial distribution function can be derived from experimental data of the overall adsorption (N vs C ) . The identification procedure discussed in the last section enables us to obtain not only the mean and the variance of W "(K) but to discriminate between the shapes of the distribution function (see the discussion in connection with Figure 3). The proposed approach can be applied to systems of practical interest such as the removal of humic substances from drinking water. Acknowledgment This work was financially supported by Minister0 dell'universita e della Ricerca Scientifica e Tecnologica MURST (Italy). Nomenclature c = molar concentration in the liquid phase C = overall solute concentration in the liquid phase
,B = parameter of the distribution function n = spreading pressure u2 = variance of the distribution function 4 = amount of solid per unit amount of solution
x = degree of solute removal Superscript
- = mean value " = initial value 00
= limiting value for x
-
0
Appendix Al. In the theoretical framework of the ideal adsorbed solution theory, the total adsorbed amount, N , in equilibrium with a continuous mixture with a distribution function w ( K ) is given by (Annesini et al., 1988):
where c*(K ) and n*(K)stand for the concentration and the adsorbed amount of a single-solute mixture a t the same temperature and spreading pressure, n,of the continuous mixture. The spreading pressure of the continuous mixture is given from the equation
If single-solute adsorption isotherms are written according to eq 3, it is also possible to write KcYK as a function of the spreading pressure, n, and of the parameters m andp: KcYK ) = h(n,m,p). The function h(n,m,p) does not depend on K. Therefore, eq A2 gives
or KcYK = KC. Equation A1 becomes
as reported in eq 4. For example, with the Freudlich isotherms with th_e samep value for all the components, we have N = m(KC)P. A!2. The functional equation
Kw "(K)x w(K)=
-
xK
+ (1 - x)K = F [w(K)B1
(A5)
admits only one solution. Proof: Let us assume that there exist two different functions w1 and w2,solution of eq A5, for a given value
2770 Ind. Eng. Chem. Res., Vol. 33, No. 11, 1994
of x. From eq 10, their first-order moments MIJ and M1,z satisfy the equation
The functiong(x,x), for x belonging to the open set (O,-), is a monotonic increasing function of its argument x with
uniformly in (a,b). The last equation implies
Since where p = K - K and therefore, for every x,there exists an open set (a(d),b(d))such that
g(x,M1) as a function of MI is globally convex. Consequently the solution of MI = g(x,M1)exists for MI > 0 and is unique. If the first-order moments of the two functions coincide, according to eq 9, w1 and w2 have the same hierarchy of moments and therefore coincide almost everywhere. A3. For every bounded, w(")(K),the sequence W(~+''(K ) = F [ w ( ~ ) (),K K 1
(-49)
is bounded. Moreover, i f w'")(K) = wo(K), then the sequence (wini(K))is bounded, monotonically decreasing and therefore admits a unique accumulation point w". Proof: Since w ( ~ ) ()KIw"(K ), V K E [O,-); n = 1 , 2 , ... (A10) the boundedness is proved. For the sake of clarity, let us denote the first-order moment of a generic distribution w as MJwI. If w(")(K)= wo(K1, then M ~ [ w ( ~I) l Ml[w(")land therefore
' M,[W'~']
(All) Applying the same techniques iteratively, one obtains W(n+lj
5 W(n).
A4. Let w"(K)IK be a summable, unimodal function in [O,m) (i.e., with only one local maximum), with w"(K) = 0 only for K = 0, the sequence
-
converges uniformlyLfor x 0, in any open interval (a,b), b a_> 0. Since K is bounded, the sequence w(K) = G(K) KIC" converges also uniformly for x 0. Proof: Since K is bounded for every x, the limit function of the sequence is w"(K)lK. The uniform convergence in a open interval (a,b) implies that for every 6 there exists a value x* such that for x < x*
-
and uniform convergence is proved in (a(d),b(d)),since all the terms on the right side of eq 13 are bounded. Moreover if 61 > 62 it follows that (a(61),b(61))c (a(62),b(62))and therefore the sequence of open intervals (a(d),b(d))forms a nested sequence covering the whole axis for 6 0. The restriction to unimodal function has been added only in order to simplify the proof. A similar result holds true for every summable function w"(K )IK.
-
Literature Cited Admirtharajah, A.; Dennett, K. E.; Studstill, A. Ferric Chloride Coagulation for Removal of Dissolved Organic Matter and Trihalomethane Precursors. Water Sci. Technol. 1993,27,113. Annesini, M. C.; Gironi, F.; Marrelli, L. Multicomponent Adsorption of Continuous Mixtures. Znd. Eng. Chem. Res. 1988,27, 1212. Astarita, G.; Nigam, A. Lumping Nonlinear Kinetics in a CSTR. M C h E J . 1989,35 (121, 1927. Calligaris, M. B.; Tien, C. Species Grouping in Multicomponent Adsorption Calculations. Can. J . Chem. Eng. 1982,60,772. Gradshteyn, I. S.; Ryzhik, I. M. Table of Integrals, Series and Products; Jeffrey, A., Ed.; Academic Press: San Diego, 1987. Jayary, K.; Thien, C. Characterization of Adsorption Affinity of Unknown Substances in Aqueous Solutions. Znd. Eng. Chem Process Des. Dev. 1985,24,1230. Kage, H.; Tien, C. Further Development of the Adsorption Minity Characterization Procedures for Aqueous Solutions with Unknown Compositions. Znd. Eng. Chem. Res. 1987,26,284. Kehlen, H.; Ratzsch, M. T.; Bergmann, J. Continuous thermodynamics of multicomponent systems. AZChE J . 1985,31,1136. Marczewski, A. W.; Derylo-Marczewska, A,; Jaroniec, M. A Simple Method for Describing Multi-Solute Adsorption Equilibria on Activated Carbons. Chem. Eng. Sci. 1990,45,143. Moon, H.; Park, H. C.; Tien, C. Adsorption of Unknown Substances from Aqueous Solutions. Chem. Eng. Sci. 1991,46,23. Radke, C. J.; Prausnitz, J. M. Thermodynamics of Multi-Solute Adsorption from Dilute Liquid Solutions. AIChE J. 1972,18 (4), 761. Sheintuch, M.; Rebhun, M. Adsorption Isotherms for Multisolute Systems with Known and Unknown Composition. Water Res. 1988,22, 421.
Received for review July 12, 1993 Accepted J u n e 21, 1994@ Abstract published in Advance ACS Abstracts, August 1, 1994. @
