Ind. E n g . Chem. Res. 1988,27, 1212-1217
1212
V = orthogonal matrix, n X n (Ws)i = norm of solution vector w using the first i components of the least-squares solution w = principal components (eq 13) xB = fraction of light component in bottoms zD = fraction of light component in distillate llxllz = Euclidean norm of vector x Greek Symbols ui = singular values, i = 1, ..., r 2 = singular value matrix (eq 7 ) Z+ = pseudoinverse of 2
= approximate pseudoinverse of I: generated by ridge regression approach 2&+ = approximate pseudoinverse of I: generated by component selection approach (AI?), = first change in variable R in response to a step change in set point Au = vector of future control changes 2RRf
Subscripts CS = denotes components selection method RR = denotes ridge regression method Superscript
+ = denotes pseudoinverse
Literature Cited Bristol, E. IEEE Trans. Autom. Control 1966, AC-11, 133. Cutler, C. R. Ph.D. Dissertation, University of Houston, 1983. Cutler, C. R.; Ramaker, B. L. The Joint Automatic Control Conference Preprints; IEEE: New York, 1980; paper WP5-B. Dempster, A. P.; Schatzoff, M.; Wermuth, N. J. Am. Statist. Assoc. 1977, 72, 17.
Garcia, C. E.; Prett, D. M. “Advances in Industrial Model-Predictive Control”. Proceedings of Chemical Process Control-III; Morari, M., McAvoy, T. J., Eds.; Elsevier: New York, 1986; pp 245-293. Golub, G. H.; Heath, M. T.; Wahba, G. Technometrics 1979,21,215. Hoerl, A. E. Chem. Eng. Prog. 1962, 60, 54. Klema, V. C.; Laub, A. J. IEEE Trans. Autom. Control 1980, AC-25, 164. Laub, A. J. IEEE Trans. Autom. Control 1985, AC-30, 97. Marchetti, J. L. Ph.D. Dissertation, University of California, Santa Barbara, 1982. Marchetti, J. L.; Mellichamp, D. A.; Seborg, D. E. Znd. Eng. Chem. Process Des. Dev. 1983, 22, 488. Maurath, P. R. Ph.D. Dissertation, University of California, Santa Barbara, 1985. Maurath, P. R.; Mellichamp, D. A.; Seborg, D. E. Ind. Eng. Chem. Res. 1988, in press. Maurath, P. R.; Seborg, D. E.; Mellichamp, D. A., Paper presented at the AIChE Meeting, Chicago, IL, 1985. Mehra, R. K.; Rouhani, R.; Eterno, J.; Richalet, J.; Rault, J. “Model Algorithmic Control: Review and Recent Developments”. In Chemical Process Control 2; Edgar, T. F., Seborg, D. E., Eds.; Engineering Foundation: New York, 1982. Penrose, R. Proc. Cambridge Philos. SOC.1956,52, 17-19. Richalet, J.; Rault, A.; Testud, J. L.; Papon, J. “Model Predictive Heuristic Control: Applications to Industrial Processes”. Automatica 1978, 14, 413. Rohani, R.; Mehra, R. K. “Model Algorithmic Control (MAC); Basic Theoretical Properties”. Automatica 1982, 18, 401. Stewart, G. W. Introduction to Matrix Computations; Academic: New York, 1973. Weischedel, K.; McAvoy, T. J. Ind. Eng. Chem. Fundam. 1980,19, 379. Wood, R. K.; Berry, M. W. Chem. Eng. Sci. 1973, 28, 1707.
Received for review November 5, 1986 Revised manuscript received October 23, 1987 Accepted November 30, 1987
Multicomponent Adsorption of Continuous Mixtures M. Cristina Annesini, Fausto Gironi, and Luigi Marrelli* Dipartimento di Ingegneria Chimica, Facoltd di Zngegneria, Univeristy of Rome “La Sapienza”, via Eudossiana, 18-00184 Rome, Italy
Continuous thermodynamics is applied t o study adsorption equilibrium from multisolute aqueous mixtures. A theoretical framework is given by the ideal adsorbed solution model, and adsorption equilibrium equations are rewritten in a continuous form. Details of the calculation procedure are reported for linear and Langmuir adsorption isotherms. As a characterizing variable of each component, the slope of the adsorption isotherm a t infinite dilution is used. T h e work shows also the method for obtaining the distribution function from differential adsorption tests. The integral adsorption behavior is then predicted, and the results are compared with the experimental data. In many adsorption problems, the fluid mixture to be processed is composed of a very large number of different components. A typical example is the treatment of industrial and municipal wastewaters. In addition to water, which is the main component, the waste mixture contains many inorganic and organic pollutants thac are difficult to isolate and to identify by ordinary chemical analysis. In this case, the composition of the mixture cannot be described through conventional discrete quantities for each component, such as mole fractions, since their concentrations are not known and, in any case, there would be far too many of them to solve the problem in a simple way. A possible approach for characterizing such mixtures is based on grouping similar components and assuming each group as a pseudocomponent (Ramaswami and Tien, 1986). Another way of describing mixtures of many components is offered by the continuous thermodynamics approach which has been proposed and developed in recent years by several investigators (Ratzsch and Kehlen, 1983; 0888-5885/88/2627-1212$01.50/0
Cotterman et al. 1985; Kehlen and Ratzsch, 1985). The original foundation of continuous thermodynamics consists in replacing the true multicomponent discrete system with a continuous mixture containing an infinite number of components. Each one of these components is identified through one or more continuous characterizing variables (such as molar mass, boiling temperature, etc.). The composition of the continuous mixture is described through a distribution function of the characterizing variables, and chemical thermodynamic equations are converted into a suitable form accounting for this new way of describing the composition. Many of the works on continuous mixtures have considered only one characterizing variable for describing the system. In this way, the mathematical treatment is greatly simplified, but the procedure can be utilized for more complex mixtures which require more than one characterizing variable (Brian0 and Glandt, 1983). In this work, the continuous thermodynamic approach 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 7, 1988 1213 is extended to multisolute adsorption from liquid solutions. The theoretical framework is the ideal adsorbed solution model by Radke and Prausnitz (1972); this model appears to be adequate for representing the adsorption from dilute solutions (like wastewaters) because it is not restricted to a specific form of the adsorption isotherm. Furthermore, the method appears attractive because of its ability to estimate multisolute adsorption using data for single-solute equilibria. In the first part of this work, adsorption equilibrium equations are rewritten. Some comments are devoted to the calculation of the spreading pressure that appears implicitly in the equations. The results obtained for two cases of adsorption isotherms are discussed as examples. In the second part, a procedure is illustrated to calculate the composition of the two phases which are obtained, at equilibrium, by contacting with the adsorbent solid a liquid mixture of known composition (flash calculation). Finally, in the last part of the paper, a procedure is discussed which allows us to obtain the distribution function from experiments of differential adsorption equilibrium. Furthermore, an application of the method is shown with reference to experimental data reported in the literature. Distribution Functions The first step in converting thermodynamic equations into a continuous form is to represent the composition of the system. If we deal with a class of very similar components, only one variable, M , is enough for characterizing the identity of each chemical species. In the case of adsorption, a useful choice of M might be some parameter of the adsorption isotherm, such as the slope a t infinite dilution. Independent of the definition of M , the composition of a multicomponent liquid mixture can be described by the following function of M
cL(M’,M’’) = SM’hL(M) M‘ dM
(1)
where cL(M’,M”) is the molar concentration of those components with M values ranging from M’up to M” and wL(M) is the distribution function in the liquid phase. The overall solute concentration CLis obtained from eq 1 if the integration is extended to the whole domain of the definition of M: CL =
S,WL(M)
dM
(2)
Therefore, the following intensive distribution function can be introduced: WL(M)= WL(M)/CL (3) with the obvious condition (4)
In a similar way, distribution functions can be introduced for the adsorbed phase. For example, the distribution function ma(M) is defined by
1 ma(M) dM M“
ii,(M’,M’’) =
M‘
(5)
where ii, are the moles adsorbed per unit amount of adsorbent solid. Therefore, the distribution function corresponding to the invariant adsorption (see Appendix I) can be defined by wa(M) =
ma(M)
- ( ~ ~ ~ ~ / c L * ) w L ( M ) (6)
where superscript s indicates the solvent. For dilute solution, the following simplified form of eq 6 can be ob-
tained (Radke and Prausnitz, 1972): wa(M) = (V/Q)(WL’(M) - WL(M))
(7)
where V and Q are the volume of solution and the mass of solid, respectively, and oLo(M)is the distribution function of the initial solution. The intensive function, corresponding to wa(M), is
W a ( W = wa(M)/
S w,(M) d~ M
(8)
In this way the adsorbed-phase mole fraction of chemical species with M values ranging between M’ and M“ is obtained by integrating Wa(M)over the pertinent range of
M. Thermodynamics of Multisolute Adsorption Radke and Prausnitz’s (1972) ideal dilute solution theory is based on the assumption that the adsorbed phase, in equilibrium with a multisolute mixture, forms an ideal solution. In order to extend this theory to a continuous mixture system, we define the fugacity in the adsorbed phase, fa, as Gpa(M,T,x;Wa)= R T In fa(M,T,x;Wa) T = const (9)
where x indicates the spreading pressure. The quantity M~ is the chemical potential of a species characterized by
M in the adsorbed phase defined according to Salacuse and Stell (1982). It is a function of T and x , but functional with respect to the distribution function Wa(M). According to the assumption of an ideal adsorbed phase, the fugacity fa(M,T,x;W,) of the species characterized by M a t temperature T and spreading pressure x is given by fa(M,T,x;Wa) = Wa(M)fa*(M,T,x) (11) where f,*(M,T,x) indicates the fugacity of the species characterized by M when it adsorbs singly from dilute solution at the same T and x as those of the multisolute mixture. Radke and Prausnitz (1972) showed that the overall invariant adsorption depends on the composition of the adsorbed phase and on the equilibrium properties of each solute. In a similar way, for a continuous mixture, the overall invariant adsorption is given by
where na* stands for the invariant adsorption of M when it adsorbs singly from a solution at the same temperature and x as in the mixture. Therefore, the total adsorbed amount can be calculated independently of the multisolute mixture properties. In addition to eq 12, the calculation of multisolute equilibria requires the relationship of equal chemical potential in the adsorbed and liquid phase for each M . If we assume the ideality of the adsorbed phase and of the liquid dilute solution, the above equilibrium condition gives the following relationships between the distribution functions in the liquid and adsorbed phases: Wa(M)CL*(M) = WL(M) [~,7’1 (13)
where eq 14 is easily obtained by accounting for eq 8. In eq 13 and 14, cL*(M) is the concentration of solute characterized by M in a single-solute mixture in equilib-
1214 Ind. Eng. Chem. Res., Vol. 27, No. 7, 1988
rium with the adsorbed phase at the same temperature and spreading pressure as those of the mixture. Obviously cL*(M) and n,*(M) are related to each other through the single-soluteadsorption isotherm at the same temperature and spreading pressure as those of the mixture. The dependence of cL*(M) on the spreading pressure (Radke and Prausnitz, 1972) is given by
where A stands for the area of the solution-solid interphase. Fixing a a value independent of M in eq 15 gives, for each component, the corresponding concentration CL*(M). Obviously, the integration in eq 15 requires knowledge of how n,*(M) depends on cL*(M);i.e., a form of adsorption isotherm has to be assumed. Therefore, the procedure for the calculation of T consists of the following steps: 1. An adsorption isotherm is obtained or assumed in order to assign the dependence on cL*(M) of the invariant adsorption n,*(M). 2. The integration in eq 15 is carried out, and the function cL*(M) is obtained for each value of T : CL*(M) = f ( M (16) 3. Equation 16 is introduced into eq 14 together with a distribution function in the liquid phase wL(M), and the integration over M is carried out; the appropriate spreading pressure, a, is that one which satisfies eq 14. This value of the spreading pressure is used in order to calculate cL*(M) from eq 16 and n,*(M) from the adsorption isotherm. In this way, for each solute, we evaluate the equilibrium conditions which correspond, in a singlesolute mixture, to the same spreading pressure as that of the multisolute solution. Equilibrium Calculation Once the spreading pressure has been calculated, eq 12 and 13 allow us to solve the equilibrium problem, that is, the calculation of the total amount of solutes adsorbed and their distribution in the adsorbed phase. The intensive distribution function in the adsorbed phase W,(M) is directly obtained by eq 13, where the distribution function WL(W of the liquid phase in equilibrium has to be used together with the cL*(M) corresponding to the spreading pressure of the mixture. The total adsorbed amount N , can be calculated by an integration over all the components of the distribution function w,(M). Using eq 12 and 13, we obtain
In the right-hand side of eq 17, variables cL*(M) and n,*(M) appear so that the calculation requires only data about adsorption isotherms of single-solute mixtures. The method is not restricted to a specific form of the adsorption isotherm. However, in order to explain the calculation procedure in a simple way, in Appendix I1 two examples are reported for linear and Langmuir adsorption isotherms, respectively. Mass Balance and Flash Calculation For engineering purposes, it is often more useful to calculate the compositions of the two phases in equilibrium
LJ
.L
I,
.n
I,
.n
Figure 1. Distribution function in a single equilibrium stage.
which are obtained by contacting an adsorbent solid with a multisolute liquid mixture of known composition. In this case, together with equilibrium equations, the mass balance equation must be solved for each component or for each value of the characterizing variable M . For dilute solutions, such a balance is given in eq 7 where wL(M) and w,(M) are related through the equilibrium condition. The use of mass balance and equilibrium equations allows one to find the distribution functions of phases in equilibrium, if wo(M) in the feed solution is known. However, the practical solution of this problem can be an arduous task for most of the functional forms of wo(M) and for nonlinear adsorption isotherms. In these cases, numerical procedures are required to find wL(M) and wa(M). Alternatively, the approximate method of moments can be used (Cotterman and Prausnitz, 1985). In Appendix 111, two examples of flash calculations are reported for linear and Langmuir adsorption isotherms, respectively. In these examples, we choose as characterizing variable M the ratio between the slope of the adsorption isotherm a t infinite dilution and the same slope for a reference component. Figure 1 shows the results of the flash calculation procedure. A gamma distribution ( a = 2, p" = 10) is chosen to represent the feed composition; a Langmuir isotherm (A = 10 mol/kg, K , = 1 (mol/m3)-') is assumed for each component. The distribution functions in the liquid and adsorbed phase are reported for different values of cp (amount of solid used per unit volume of solution). As expected, the maximum of the liquid-phase distribution shifts toward lower M values, i.e., toward less adsorbable compounds. Furthermore, the total solute concentration decreases when cp increases.
Evaluation of Distribution Functions from Experimental Data The continuous model of adsorption equilibria requires knowledge of the distribution functions. For example, the distribution function in the feed is required in the flash calculation. However, in many of the cases for which continuous thermodynamics is useful, this information cannot be obtained by ordinary chemical analysis, because of the very high number of components which are often not identifiable. Therefore, some different procedure must be used. In adsorption problems, we can obtain the distribution function from the differential adsorption equilibrium (D.A.E.) curve. This curve is obtained by adding a small amount, SQ, of adsorbent to the solution and measuring the overall concentration in the liquid phase after equi-
Ind. Eng. Chem. Res., Vol. 27, No. 7, 1988 1215 librium is achieved. The solid is then removed and a new SQ of fresh adsorbent is added to the same solution. A new measurement is obtained, and the operation can be repeated, thus obtaining the variation of CL with the total amount of adsorbent used. From the D.A.E. curve, the distribution function of the solution can be obtained by the following procedure. A differential mass balance for each M gives SWL(M) = -wa(M)GV
(18)
where 6cp = SQ/V. The distribution function wa(Mdepends on the cp ratio used and on the initial distribution function, wo(M). Therefore, eq 18 can be written in this form
0
.2
I I
I IO
,
1
2 II
10
0.
1
kg/m
8 .
(1
,
,
50
h0
I
'
Figure 2. Differential adsorption equilibrium curve.
with the boundary condition
An interesting case occurs when the gamma distribution is assumed for wo(M):
Integration of eq 19 gives
where CLoindicates the overall concentration in the initial solution. The most important property of eq 21 is that the lefthand side is expressed in terms of the measurable variable CL(cp). Therefore, if experiments of differential adsorption are carried out in order to measure CL vs cp, eq 21 gives unequivocally the initial distribution function wo(M). However, integration in eq 21 requires knowledge of the dependence of w,(M) on wo(M) and 9. In other words, the form of the adsorption isotherm must be known. From a theoretical point of view, any kind of adsorption isotherm can be used; however, for simplicity, the procedure is illustrated in two simple cases. If a linear adsorption isotherm is assumed, the evaluation of wo(M) is straightforward. In fact, a simple relation holds between wL(M) and w,(M) in equilibrium (see Appendix 11, eq 11-5), and integration of eq 19 with condition 20 gives wa(M)
= MHrWo(M) exp(-MHrcp)
(22)
Substitution of eq 22 into eq 21 shows that CL(cp) is the Laplace transform of wo(M), with the Laplace parameter Hrp:
C L ( =~ l M w o ( M )exp(-MH,cp) dM = 4 w o ( M ) J
(23)
Therefore, wo(M) can be straightforwardly obtained as w0(M) = ~-'(CL(Hr~)l
(24)
Following the same procedure but by the use of the Langmuir isotherm for expressing the equilibrium condition between wa(M) and wL(M) (Appendix 11, eq II.lO), integration of eq 19 and substitution into eq 21 lead to
cL(cp)= S,WO(M) exp{-MK,(ficp + ~ L ( c p )- CLV
d~ (25)
where ii is the maximum adsorption in a Langmuir isotherm assumed to be independent of M . Solution of eq 25 to obtain wo(M) can be difficult because of the implicit form by which this function appears. Some numerical procedure is often required for constructing wo(M) or, even if the form of w'(M) is assumed, for integration of eq 25.
With this assumption, mass balance and equilibrium conditions show that also the distribution function wL(M) is a gamma distribution. Its a parameter is the same as that of the initial distribution function, and parameter p depends on the amount of solid used according to the following equation:
Now, if eq 26 is introduced into eq 25, the following relationship is obtained: CLW = cLo(Po/P)"+l
(28)
Equation 28 gives the dependence of CL(cp)on the variable cp.
A more useful form can be obtained from eq 28 by some rearrangements:
p"((C,(cp)/cLo)-l~(~+l) - 1)CL°Kr{CL(P)/CLo - 1) = fiKrV (29)
From this equation, parameter a, p", and ri can be obtained by fitting the experimental data.
Application to Wastewater Treatment As application of the procedure, we analyzed some adsorption measurements carried out by Okazaki et al. (1981). These authors report data from differential and integral adsorption tests performed on multicomponent industrial wastewaters with activated carbon as adsorbent. The wastewater mainly contains molasses, but the type and concentration of the solutes are not known. The D.A.E. curves were obtained by measuring the total organic carbon (TOC) in the liquid phase vs the total amount of carbon added. We used these data to evaluate the distribution function wo(M). We can observe that the experimental trend of CL(cp)/ CLovs cp reported in Figure 2 depends on the initial solute concentration, CLo. This behavior could not be reproduced by a simple linear adsorption isotherm, so the adoption of a more complex isotherm is required. As a first approximation, we used the Langmuir isotherm with maximum adsorbed amount fi independent of the component. Following the procedure reported above, parameters a,pO,
1216 Ind. Eng. Chem. Res., Vol. 27, No. 7, 1988
nancially supported by Minister0 Pubblica Tstruzione (Italy).
Nomenclature A = area of the solution-solid interface c = molar concentration C = overall molar concentration f = fugacity
H = Henry's constant I
f
1
I
z
,,It
f
,,'7
