385
Ind. Eng. Chem. Res. 1991,30, 385-395 tation, University of Wisconsin-Madison, 1986. Cuthrell, J. E.; Biegler, L. T. On the Optimization of Differentiation-Algebraic Process Systems. AIChE J. 1987, 33, 1257-1270. Cuthrell, J. E.; Biegler, L. T. Simultaneous Optimization and Solution Methods for Batch Reactor Control Profiles. Comput. Chem. Eng. 1989, 13 (1/2), 49. Dennis, J. E.; Gay, D. M.; Welsch, R. E. An Adaptive Nonlinear Least-Squares Algorithm. ACM Trans. Math. Softw. 1981, 7 (3), 348-368. Fariss, R. H.; Law, V. H. An Efficient Computational Technique for Generalized Application of Maximum Likelihood to Improve Correlation of Experimental Data. Comput. Chem. Eng. 1979,3, 95-104. Fletcher, R.; Xu, C. Hybrid Methods for Nonlinear Least Squares. IMA J. Numer. Anal. 1979, 7, 371-389. Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design; Wiley: New York, 1979. Fuguitt, R. E.; Hawkins, J. E. Rate of Thermal Isomerization of tu-Pinene in the Liquid Phase. J . Am. Chem. Soc. 1947,69,319. Hertzberg, T.; A s b j m " , 0. A. Parameter Estimation in Nonlinear Differential Equations. Proceedings of the Symposium CHEMDATA.77, 1977, Helsinki.
Jennrich, R. I.; Sampson, P. F. Application of Stepwise Regression to Non-Linear Estimation. Technometrics 1968, 10, 63-72. Murtagh, B. A,; Saunders, M. A. MINOS 5.0 User's Guide; Report SOL 83-20, Department of Operation Research, Stanford University: Stanford, CA, 1983. Nowak, U.; Deuflhard, P. Numerical Identification of Selected Rate Constants in Large Chemical Reaction Systems. Appl. Numer. Math. 1985, I , 59-75. Varah, J. M. A Spline Least Squares Method for Numerical Parameter Estimation in Differential Equations. SIAM J . Sci. Stat. Comput. 1982, 3, 28-46. Varga, K.; Marsi, I.; Tasi, G.; Kiricsi, I.; Fejes, P. Facilitation of Computer Estimation of Kinetics Parameters of a Heterogeneous Catalytic Reaction via the Use of Adsorption Data. Comput. Chem. Eng. 1988, 12 (2/3), 127-133. Vasantharajan, S.; Biegler, L. T. Large-Scale Decomposition for Successive Quadratic Programming. Comput. Chem. Eng. 1988, 12 (ll),1087-1101.
Received for review April 27, 1990 Revised manuscript received August 9, 1990 Accepted August 27, 1990
SEPARATIONS Multicomponent Dye Adsorption onto Carbon Using a Solid Diffusion Mass-Transfer Model Gordon McKay* and Bushra A1 Duri Department of Chemical Engineering, The Queen's University of Belfast, Belfast B T 7 l N N , Northern Ireland
The adsorption of basic dyes onto activated carbon has been studied for single and multicomponent systems. Three single-component systems, one binary dye adsorption system, and one ternary dye adsorption system have been investigated. Equilibria and kinetic studies have been performed and the effects of varying initial dye concentration and carbon mass have been investigated during the kinetic experiments. Prediction of experimental equilibrium data has been performed by use of the ideal adsorbed solute theory, the extended Redlich-Peterson isotherm, and a modified extended Redlich-Peterson isotherm. The predictions from the modified extended Redlich-Peterson isotherm, incorporating an interaction factor, correlated experimental data well and also provided a reasonably simple format that could be incorporated into a kinetic model. T h e kinetic model developed for agitated batch adsorbers is based on the film solid diffusion model coupled with the modified extended Redlich-Peterson isotherm. Kinetic data for the binary and ternary systems are correlated with experimental data for up to 3 h over a wide range of dye concentrations and carbon masses.
Introduction Adsorption from multicomponent solutions plays an important role in a great number of natural and industrial systems such as fundamental biological studies, separation and purification processes, recovery of chemical compounds, catalysis, and waste treatment operations. Adsorption processes using activated carbon have proven efficiency and are economically feasible for the treatment of wastewater (Borowko and Jaroniec, 1983). Ideally the prediction of full scale adsorber performance should be based on experiments carried out rapidly using simple equipment. Unfortunately multicomponent adsorption is frequently complicated by interactions and competition between sorbates and sorbent in the system. However, an efficient, accurate and cost-effective design should account for the effects associated with multicomponent adsorption (Weber and Smith, 1990). A t present, adsorber design is 0888-5885/91/2630-0385$02.50/0
mainly based on extensive pilot plant scale experiments (Yen and Singer, 1984) which supply design parameters for the set of variables studied but are not applicable to different system conditions. Generally, to develop a mathematical model that describes or predicts adsorption dynamics, the following are required: (i) a full description of equilibrium behavior, that is, the maximum level of adsorption attained in a sorbatelsorbent system as a function of the sorbate liquidphase concentration; (ii) a mathematical characterization of the associated rate of adsorption, which is controlled by the resistances within the sorbent particles; (iii) a material balance for each component within the system. Steps i, ii, and iii are combined in a complete model that consists of a system of partial differential equations (PDE's) that describe the continuity of each component in each phase together with equations expressing system 0 1991 American Chemical Society
386 Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991
initial and boundary conditions. Because the system of PDE’s is nonlinear, numerical methods are normally adopted for solution. In the present work a mathematical model, previously applied to single-component dye adsorption (McKay, 1985), has been adapted to predict multicomponent adsorption system equilibria and kinetics for bi- and trisolute dye-carbon adsorption processes. It utilizes the RedlichPeterson equilibrium isotherm previously used by Mathews and Weber (1976) and Jossens et al. (1978). The homogeneous solid diffusion kinetic model is a semianalytical model developed from the numerical approach by Mathews and Weber (1976). The experimental work was carried out in an agitated batch adsorber, and a bisolute system (Basic Red 22 and Basic Yellow 21) and a trisolute system (Basic Red 22, Basic Yellow 21 and Basic Blue 69) were studied. Literature S u r v e y (i) Multicomponent Equilibria. Kolthoff and Van der Groot in 1929 and then Amiot in 1934 reported experimental data of binary mixtures but did not attempt to predict or correlate data. Other early investigations demonstrated there is no “perfect” isotherm. Some approaches are totally predictive (Butler and Ockrent, 1930; Jain and Snoeyink, 1973; Myers and Prausnitz, 1965; Kidnay and Myers, 1966; Jaroniec et al., 1983). These authors have assumed that nonideal or unequal competition for adsorption sites is zero. Weber and Morris (1964) applied the Langmuir model for competitive adsorption, but Young and Crowell (1962) pointed that due to the strong heterogeneity of the carbon surface the Langmuir model is inapplicable and also violates the Gibbs adsorption equation and consequently is thermodynamically inconsistent. The most theoretically sound predictive model is the ideal adsorbed solute theory, IAST (Myers and Prausnitz, 1965). This approach has been developed, extended, and simplified (Radke and Prausnitz, 1972a,b; Baldauf et al., 1977; DiGiano et al., 1978, 1980; Reich et al., 1980; Singer and Yen, 1980; Fritz and Schlunder, 1981; O’Brien and Myers, 1984, 1985; Yonge et al., 1985; Crittenden et al., 1985). Despite all these efforts its predictions are not always close to experimental data and its computational time is extensive. Other correlative approaches (Schay, 1956; Yon and Turnock, 1971; Mathews, 1975; Crittenden, 1976) have attempted to include the competitive and interactive effects in multisolute systems, but all have experienced difficulties in applying the relationships for a wide range of adsorption systems and to quantify interaction terms. Another group of researchers studied surface heterogeneity and energy distribution on the sorbent surface (Jaroniec et al., 1983; Jaroniec, 1982; Rudzinski et al., 1983; Price and Danner, 1987) but again assumed interactive and competitive effects were zero. Rudzinski et al. (1973,1973) and Oscik et al. (1976) made a registry of surface sites incorporating topography and heterogeneity. Several isotherms were tested incorporating heterogeneity term such as the Freundlich-Sips (Dabrowski et al., 1978; Dabrowski and Jaroniec, 1980a,b),the Dubinin-Radushkevich (Goworek et al., 1981), the Freundlich (DiGiano et al., 1978; Sheindorf et al., 1981; Jossens et al., 1978; Fritz et al., 1981; Jaroniec, 1982; Jaroniec and Derylo, 1982), and the Toth isotherm (Jossens et al., 1978). Except for Toth’s equation these equations do not reduce to Henry’s law at low concentrations. This paper highlights the dilemma in selecting an appropriate isotherm for multicomponent equilibria and concludes that it is appropriate to adopt several approaches
to achieve an optimum compromise. (ii) Multicomponent Kinetics. System dynamics describe the rate and progress of the adsorption process. The dynamic behavior of multicomponent adsorption systems provides a fundamental requisite for the design of industrial adsorbers. A typical concentration decay curve shows that the adsorption rate is a maximum initially when the system is furthest from equilibrium, and the rate decreases asymptotically to zero as the system approaches equilibrium (Mathews and Weber, 1969). Early multicomponent adsorption studies were directed toward column work (Weber and Morris, 1964; Klein et al., 1967; Seiler, 1972), and most approaches to mathematical modeling vary in the mode of intraparticle diffusion used (Fritz et al., 1981). The solid-phase interactions have been found to cause diffusional flow or counterdiffusion Crittenden (1976) and Toor (1965) used StefanMaxwell equations for steady-state mass transfer to show four different types of diffusion, Le., barrier, osmotic, reverse, and normal, may occur. However, most models of multicomponent adsorption assume only normal diffusion takes place (Mathews, 1975; Crittenden, 1976; Liapis and Rippin, 1977; Fritz et al., 1981) and that solid-phase solute-solute interactions are accounted for through the isotherm equations only. The first models (Wilson, 1940; De Vault, 1943, Walter, 1945; Gluckauf, 1946; Rhee e t al., 1970) assumed local equilibrium between fluid and sorbent phases for predicting elution curves in chromatographic columns. Klein et al. (1967) and Helfferich and Klein (1970) have developed a multicomponent ion-exchange model for fixed beds. Other studies on binary systems (Weber and Crittenden, 1975; Klaus et al., 1977) assume incomplete equilibrium and that external mass transfer only is rate determining. Fritz et al. (1981) applied this film diffusion model to binary organic liquid mixtures and obtained fair agreement at low concentrations. However, a t high concentrations it is apparent that intraparticle diffusion becomes predominant. The simplest form for providing a quantitative description of diffusion within the sorbent particles is the linear driving force approximation (Cooney and Strusi, 1972; Hsieh et al., 1971). The film pore diffusion model has been solved by Carter and Husain (1974), Ott and Rys (1973), Liapis and Rippin (1977), and Fritz et al. (1981). Liapis and Rippin developed a solution for N-component systems and solved the nonlinear P D E s by orthogonal collocation via a fifth-order Runge-Kutta method. Other film pore diffusion solutions have been applied to gas adsorption studies (Thomas and Lombardi, 1971; Ferrell et al., 1976). A film solid diffusion has been extended to multicomponent systems (Mathews, 1975) by use of the RedlichPeterson isotherm. A numerical solution of Crank-Nicolson’s finite difference method was solved by use of a Newton-Raphson iterative technique. Crittenden (1976) adopted a similar approach for fixed-bed absorption of binary systems. Theory (i) Multicomponent Equilibrium. It is a major advantage if it is possible to determine multicomponent adsorption equilibrium data solely from single-component data. A high level of accuracy is required since equilibrium data are a fundamental prerequisite in mathematical modeling for multicomponent system kinetics. The Redlich-Peterson isotherm offers a compromise between the Langmuir and Freundlich expressions. For a single component i
Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991 387 (c) intraparticle solid diffusion
(7)
For a multicomponent system the Redlich-Peterson equation can be extended:
L
where Kj", bJoand PJ" are the single-component-isotherm constants. Equation 2 includes the competitive effect of all components by extending the denominator of the single-component isotherm. The equation does not take the affinity change into account, nor does it consider dye-dye interaction at the sorbent surface. It is too simplified for the complexity of a multicomponent system and therefore is not recommended for the description of system equilibrium. The empirical equation (2) can be improved. The nature of multicomponent system interactions has not been fully understood yet and therefore cannot be evaluated by exact mathematical equations. However, Schay (1956) has introduced the interaction term, which is specific to each component in the system and is a function of other components present (Yon and Turnock, 1971). For a multicomponent system of n components with mp points of each component, the interaction term, 7, is evaluated by minimizing the variance between experimental and calculated solid-phase concentrations. The objective function F is given by (3)
qj = -3S R q i j r 2 dr R2 0 (e) equilibrium isotherm Qsj
=
1+
6
Equation 4 is a significant improvement to eq 2 because it is based on correlating single-solute and multisolute data. Therefore 7 is a correlation factor that is based on minimizing the deviation between experimental multisolute data and the data based on extending single-solute isotherms (eq 4). It is particularly suitable for systems of similar equilibrium behavior (Fritz et al. 1981) because it does not account for the change in carbon affinity, noticed strongly with systems of dissimilar equilibrium behavior. It can also be used for multisolute systems apart from bisolute systems. (ii) Kinetic Analysis. Mass transfer in the present work is described by external film diffusion followed by intraparticle solid diffusion or the surface-hopping mechanism. This theory has been presented for single-component systems (McKay, 1985). However, in a multicomponent system of n components, the same set of equations is to be solved for each species j . It is stated as follows: (a) material balance (5)
kc:
(9)
qij(r90) = 0
( 104
C,j(O) = COJ
(lob)
The boundary conditions are
qij(R,t) = q s j ( t )
(11)
The dimensionless groups are uj = qijx and r, = tDsj/R2, where x = r / R ; eq 7 becomes auj azuj -=d7j ax2
The coupling equation becomes
j=1
(b) the coupling equation
+ k2= l hkCs,koh
Equation 9 is the general multicomponent empirical isotherm. The well-known isotherms selected to describe the present systems are special cases of eq 9. The initial conditions are
(4)
bjjo(csi/~j)'J~
Kjcsj
=
Introducing 7 into the Redlich-Peterson isotherm: (?si
J
(d) average solid-phase concentration
K J i O csi
qsi =
KJi"(Csi/qi)
.
Also 1 qj
= 3 s ujx dx 0
and
dC,j -V-=mdt
dqj dt
The initial and boundary conditions are Uj(0,7j)
= 0 = u(x,O)
'Jj(R,Tj)
=
usj(7j)
(17) (18)
Solution Method For the solution the set of equations (13)-(18) is solved numerically for t and analytically for x . The same procedure is exactly followed for each component in the multicomponent system. The method has been outlined by McKay (1985). For multicomponent systems of n components, each step is repeated n times before the consecutive step is followed.
388 Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991 Table 1. Physical Properties of Activated Carbon Filtrasorb F400 total surface area (N2 BET method), m2 kg-’ (1.05-1.20) x IO6 solid-phase density, kg m-3 2100 1300-1400 particle density (wetted in water), kg m-3 porosity 0.40 iodine number 1000-1 100 methylene blue number 260-300
This provides data for all components in each step as they are needed simultaneously in eq 19, which represents a semianalytical solution of eq 13. U(X,T)
Table 11. Absorption Coefficients for Basic Blue 69, Basic Red 22, and Basic Yellow 21 Being A, B, and C, with Maximum Absorbance Wavelengths A,. A,. and A, A, = 585 nm A, = 537 nm A3 = 417 nm kA2 = 1.24 X kA3 = 7.88 X k A , = 1.96 X k B 2 = 7.01 x lo-’ kgl = 3.36 x k ~ = 3 3.29 x kcl = 1.29 X kC2 = 1.35 X lo-* kc3 = 6.50 X
Maxillon Red BL-NCI 11055 (Basic Red 22), supplied by Ciba-Geigy Ltd., has the following structure:
= CH3
9
Astrazon Yellow (Basic Yellow 21) has the structure Computerization of Solution Method A computer program, MODMXJ.FOR, using the above equations has been developed to produce multicomponent system kinetics. The program has been supplied by the following input data: (i) mass-transfer and diffusion parameters: kfj (cm/s) and Dsj (cm2/s) for each component j ; (ii) adsorbent properties and specifications: mass (g), solid density (g/cm3),voidage, and mean particle radius (cm); (iii) adsorbate specifications: initial concentration (mg dm-3) and volume of solution (L); (iv) equilibrium data: isotherm constants and interaction terms; (v) mathematical specifications: number of time increments and number of integration steps. The program gives the time, average solid-phase concentration qtj, liquid-phase concentration Ctj, and equilibrium solid- and liquid-phase concentrations qSjand Csj, respectively, for each component j in the system. Generally, the equilibrium data have been solved by a method for roots of more than one transcendental equation. The COBNBF NAG FORTRAN Library Routine Document has been utilized. COBNBF is an easy to use routine to find a zero of a system of N nonlinear functions of N variables by a modification of the Powell hybrid method. Experimental Section Sorbent. Activated carbon Filtrasorb F400 has been selected as the adsorbent for all investigated systems. Carbon, supplied by Chemviron Ltd., is crushed by a hammer mill and screened to a series of particle size ranges by sieve analysis. To remove fines, the sorbent is washed thoroughly with distilled water and then dried in an oven at 110 “C for 24 h. The physical properties of carbon Filtrasorb F400 are given in Table I. Sorbates. Basic dyes in aqueous solution are the subject of this study. Three basic dyes in their single-component, bicomponent, and tricomponent solutions are studied in finite bath or batch adsorption systems. Their names and structures are given as follows. Astrazon Blue FRR (Basic Blue 69) (Supplied by Bayer Ltd). The struture of this dye is still unknown, but the general structure of the group of methine dyes it belongs to is:
3HgQ CH=CH-N CH3
CH3
Systems under Investigation. The systems under investigation fall into two categories. (i) The first is single solute solutions/carbon, obtained by studying the individual dyes on a single-component basis. These are (a) Basic Blue 69, BB69; (b) Basic Red 22, BR22; (c) Basic Yellow 21, BY21. (ii) The second category is multisolute solutions/carbon, obtained by possible combinations of dyes to form binary and ternary mixtures forming the following systems: (a) system I, (Basic Red 22 + Basic Yellow 2l)/carbon; and (b) system 11, (Basic Blue 69 + Basic Red 22 + Basic Yellow 2l)/carbon. Apparatus. Batch experimental work has been carried out in a finite batch system. It consists of an agitated batch adsorber which is a 2.00-dm3cylindrical glass vessel of 0.130-m diameter and 0.200-m height containing 1.70 dm3 of liquid to give a liquid depth of 0.130 m. Mixing is provided by a six flat-bladed impeller extending vertically halfway down the solution. The agitator is driven by a Heidolph Type 5011 variable-speed motor. For complete mixing eight stainless stell baffles are distributed around the circumference of the vessel at 45O angles held firmly in place by a PVC baffle holder. Measurements and Analyses. (i) The wavelength at which maximum absorbance for each dye in aqueous solution was determined by scanning the light absorption curve using a Perkin-Elmer Model 550s spectrophotometer. The maximum wavelength ,A, (nm) for the present dyes are Basic Blue 69 (BB69), 585 nm; Basic Red 22 (BR22), 537 nm; and Basic Yellow 21 (BY21), 417 nm. (ii) For calibration purposes a range of concentrations of each dye was analyzed for its optical density measured at A,, by use of the spectrophotometer, Perkin-Elmer Model 550s. (iii) The optical densities are plotted against the corresponding dye concentrations and an equation, for each dye, of the following form is obtained: C, = k d where C, is the dye concentration, k is the absorption coefficient, and d is the optical density. (iv) For multicomponent systems, given three dyes A, B, and C in their single-component solutions with maximum absorbances A,, A2, and A3 respectively, the optical density of each dye has been measured at AI, A2, and A3 to yield a set of calibration lines with absorption coeffi-
Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991 389 Table 111. Comparison of Redlich-Peterson's Constants of T h r e e Basic Dyes i n Single Components w i t h T h e i r Combined Mixtures single components BR22 BY21 triple mixtures
+
dye BB69 BY21 BR22
KJ" 202 463 597
P'
bJb 0.223 0.771 1.094
" Units in dm3 gl.*Units in dm3 mg-'.
KJ
0.999 0.993 0.999
P
bJ
642 598
KJ 359 161 121
0.956 0.991
P
bJ 0.726 0.861 0.549
0.959 0.967 0.995
'Dimensionless.
Table IV. Molar Volumes, Molar Diameters, Diameters of BR22, BB69, a n d BY21 Dyes molar vol, molar diam, dye cm3 mol-' cm mol-' BR22 200 3.63 BB69 322 4.25 BY21 424 4.66
I
a n d Molecular molecular diam, cm 6.03 x 10-24 7.06 x 10-24 7.74 x 10-24
I
A
I
-
n
P YI
c
500
0
100
2oc
7t-O
.I
200
IO0
C,,
mq dm-'
Figure 1. Adsorption isotherms for three single-component basic dyes.
cients of kAl, kA2,and kA3 for, say, component A. Table I1 shows the absorption coefficients for the dyes under investigation. Concentrations of dyes in single-solute, bisolute, and trisolute solutions are found from their optical densities at their respective wavelengths together with the interference constants (A1 Duri, 1988).
Results and Discussion Multicomponent Adsorption Equilibria. Studying multicomponent adsorption system equilibria must commence with an accurate description of each component in its single (pure) component equilibrium state. Adsorption equilibrium data are conventionally represented mathematically by isotherm equations. The three single-component dye isotherms are shown in Figure 1,and the data were analyzed by use of the Redlich-Peterson equation (1). The constants Kj,bj, and 4, are shown in Table 111. Figure 1 shows that basic dyes have a strong affinity for carbon, with BB69 being adsorbed to a capacity of 871 mg of dye/g of carbon, BY21 having a saturation capacity of 635 mg of dye/g of carbon, and BR22 having a capacity of 543 mg of dye/g of carbon. The Redlich-Peterson equation (1)selected for this work has a Langmuir component for equal energy physisorption and an exponential term for heterogeneity. It therefore provides a compromise between pure Langmuir and Freundlich adsorption. In Figure 2, the experimental sorption capacity is 272 mg of dye/g of carbon for BY21 and also 272 mg of dye/g of carbon for BR22; in the single-component systems the values were 16% different. In an attempt to predict the bisolute equilibria data from single-component data eq 2 was adopted. Equation 2 extends the denominator of pure-component isotherms
Figure 2. (A) Experimental equilibrium isotherms of BR22 in (a) (BR22), (b) (BR22 + BB69), (c) (BR22 + BY211 and (d) (BR22 + BB69 + BY21) systems on carbon. (B) Experimental equilibrium isotherms for BY21 in (a) (BY21), (b) (BY21 + BB69), (c) (BY21 + BR22), and (d) (BY21 + BB69 + BR22) systems on carbon. 300
IC
200
F
=
100
I0 300
7-
230
B
d
100
-' z
500
10
20
30
40
50
60
00.
a
7OC , m&m.3
90
(b)
t/
0
I
I C l
-
0
0.
a
0
0
0. 0
/
0
5
10
15
20
390 Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991
I
[ a
I
Table V. Interaction Terms (11,) for Systems I-IV system I system I1 system I11 system IV BR22+ BB69+ BB69+ BB69 + component BY21 BR22 BY21 BR22 + BY21 BB69 0.712 0.401 0.214 BR22 1.306 13.66 5.144 BY21 0.755 3.952 4.466 Table VI. Variances (2)between Experimental and Calculated Isotherms for Four Dye Mixture Systems on Carbon method (equation) extended improved system component empirical (eq 1) extended (eq 2) I BR22 0.0573 0.00372 BY21 0.0847 0.00430 IV BB69 0.889 0.208 BR22 0.293 0.0464 BY21 0.129 0.0851
’%’
-
“pel,.ted
ix3e-llen-3’
“
Ke-
Figure 4. Application of eq 4 to (a) BY21, (b) BR22, and (c) BB69 in system 11: (BY21 t BR22 t BB69)/carbon.
a
c E
200 c
100
behavior. Furthermore, eq 2 assumes ideal adsorption and independent diffusion of components, two factors that do not occur in reality. Consequently, it was necessary to use a correlative formula that incorporates multisolute experimental data. Since the dyes under investigation in this study have comparable molar volumes and molar diameters (Table IV), it was not possible to identify a molecular sieve effect. The presence of the various effects acting together affects the shape and constants of equilibrium isotherms. The latter change from single-solute to bisolute to trisolute state is demonstrated by Figure 4. Taking these effects into account in the studies of the dynamics of multicomponent adsorption systems makes mathematical modeling almost impossibly difficult. Also, none of these effects have been mathematically described to date. For the previous two reasons, in the multicomponent system kinetics-independent adsorption has been assumed; Le., multicomponent effects are equal to zero. They are accounted for in the description of equilibrium only. This makes the accuracy in isotherm determination and description more essential and adds to its role as a major design parameter. Having applied the extended predictive method unsuccessfully, it was decided to use a correlative approach incorporating an interaction factor to account for the system nonidealities. Equation 3 defines F , the objective function of q for the Redlich-Peterson isotherm. Equation 3 shows that the interaction term is based on the solid-phase concentration of each species in the system; hence its value depends on the equilibrium contribution of each component in the adsorbed phase. Yon and Turnock (1971) applied Schay’s (1956) interaction term using the loading ratio capacity (LRC) with gaseous binary systems of nitrogen + methane, carbon dioxide + methane, and hydrogen sulfide + methane on molecular sieves. These systems, having similar adsorption characteristics yielded only slight deviations. Introducing 7 improved the standard deviation between experiment and theory to +15% /-14%, +7% 1 -15%, and +29%/-23% for the previous three systems, respectively. Applying the IAST gave less satisfactory results due to the sensitivity of 7ri to variation in tem-
II 0
400
-
:heovetical
0
50
100
Exoerinental
153
,
‘t y -- I.
I I
230
100
-T h e o r e t i c a 1 0
Experimental
Figure 5. Application of eq 2 to (a) BY21 and (b) BR22 in system I: (BY21 + BR22)/carbon.
peratures which is vital for gas systems. Mathews (1975) tested q using the Redlich-Peterson isotherm with p chlorophenol + p-cresol and acetone + propionitrile/carbon systems. He obtained satisfactory results for total adsorption. For component adsorption better results were obtained with the IAST, but the mathematical complexity invoked with Redlich-Peterson isotherm made the IAST less favorable. In the present work 7 has been evaluated by developing a computer program in Fortran 77. This program has yielded n values of vi,where n is the number of components in the mixture. EOIUAF NAG FORTRAN Library Routine Document has been utilized to minimize the function F where F = f(ql,qZ,...,qn). A sequential augmented Lagrangian method is used, the minimization subroutines involved being solved by a quasi-Newton method. Table V gives vi values for the present systems. 7 has certainly produced a substantial improvement on the results expressed by the variance u2 given as
where m is the number of points in a given set of data. Table V f presents uz values for the different approaches
Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991 391 Table VII. Mass-Transfer Parameters Obtained by Application of Film Homogeneous Diffusion Model to the Equimolar Binary Mixture of Basic Red 22 and Basic Yellow 21 on Carbon
Vlm," dm3 g" component basic red 22
mass-transfer Param 2.67 i03kf,cm s-l 2.70 109D,, cm2 s-' 0.300 basic yellow 21 103kf,cm s-l 1.80 109D,, cm2 s-l 0.100
For all values CoJ= 57.0 mg d&.
2.00 3.00 0.350 2.00 0.130
1.33 3.20 0.380 2.00 0.070
1.00 3.20 0.380 2.00 0.055
27.0 2.70 0.085 1.80 0.040
Coj,bmg dm-3 42.0 2.70 0.150 1.80 0.100
57.0 2.70 0.300 1.80 0.100
single-component data 2.00 f 0.00 0.085 + 0.005 2.70 f 0.4 0.23 f 0.04
Coj is for each component, j . For all values V / m = 2.67 dm3 g-l,
Table VIII. Mass-Transfer Parameters Obtained by the Application of BY 2 1)/Carbon Vim: dm3 g-' component mass-transfer param 2.67 2.00 1.33 Basic Blue 69 i03kf,cm 2.80 2.50 2.40 i09D,, cm2 s-l 0.030 0.016 0.010 Basic Red 22 i03kf,cm s? 1.50 2.00 1.50 i09D,, cm2 0.055 0.085 0.070 Basic Yellow 21 103kf,cm s? 1.20 1.80 1.20 i09D,, cm2 s-l 0.045 0.070 0.055
MODMXJ.FOR
to System I V (BB69
+ BR22 +
Coj,bmg dm-3
8.13 3.50 0.020 2.00 0.045 1.80 0.030
16.0 2.50 0.022 1.50 0.048 1.20 0.040
32.7 sinde-comDonent data 2.80 2.85 f 0.4 0.030 0.020 f 0.004 1.50 2.00 f 0.00 0.055 0.85 f 0.005 1.20 2.70 f 0.4 0.045 0.23 f 0.04
aFor all V / m values C, = 33.3 mg d d . bFor all C, values V / m = 2.67 dm3 g-'.
adopted to find equilibrium data for the present systems. For the ternary system, %a2 improved from 29.3% to 4.6% for BR22 using the extended Redlich-Peterson isotherm. Figures 3 and 5 illustrate the applications of eq 2 to systems I1 and I, respectively, and the results show the improvement due to incorporating the interaction term. As a dye species is introduced in a multicomponent system, it interacts with other species and undergoes a carbon affinity (or capacity) change, causing a change in its diffusivity and energy of adsorption. The interaction term accounts for some of these effects, although it is apparent from the ternary system that the variance for BB69 is high. The adsorption capacity for BB69 is high compared to the other two dyes, and its diffusive properties are different. Multicomponent Adsorption Kinetics. Most successful models describing multicomponent adsorption kinetics start from single-component diffusion equations and then combine them through mass-transfer parameters and equilibrium equations. The highly nonlinear terms arising from the coupled parabolic PDEs are complex, and in the present model certain assumptions are made: (i) Each component diffuses independently of others with no cross-flow caused by interactions. (ii) One-dimensional radial diffusion occurs in homogeneous, constant-sized spherical particles. (iii) Complete and uniform mixing occurs. However, "idealities in multicomponent systems, which are primarily attributed to the heterogeneous structure of the carbon surface, are accounted for in the production of equilibrium data for these systems, and consequently the assumption of homogeneity in kinetics is valid. A solid diffusion model has been selected because surface diffusion contributes about 20 times as much as pore diffusion in the adsorption process as a whole (Komiyama and Smith, 1974). Previous application of the pore diffusion mechanism to single-component systems has yielded diffusivity values far larger than the theoretically predicted values (Furusawa and Smith, 1973a,b) which is, again, due to the surface migration of dye molecules on the pore walls which adds to the contribution of surface diffusion. On the other hand, Fritz et al. (1981) applying the film pore diffusion model to multicomponent organic solutions on carbon, obtained results that deviated from experimental values. Consequently, it has been decided to take the film solid diffusion into further consideration, to be extended
for the description of the present multicomponent systems. These are BR22 BY21 and the ternary system BR22 BY21 + BB69. Tables VI1 and VI11 present the values of mass-transfer parameters resulting from applying the film solid diffusion model to the binary and ternary systems under investigation, respectively. An overall look at these tables shows that, for each component, a single kfwithin a narrow range of values describes the whole experimental range. In previous literature kfhas been evaluated by the initial slope method (Spahn and Schluender, 1975) whereby kfequals the slope of the concentration decay curve at initial conditions. Furusawa and Smith (1973a,b) have assumed a single external resistance model for their system of benzene/carbon. The so obtained kf is lower than the one obtained experimentally because the model assumes that all resistance is contained in the boundary layer. This is an erroneous assumption because intraparticle diffusion controls the main course of adsorption. Another factor that affects the applicability of a single resistance model to evaluate kf is the solid/liquid equilibrium sorbate distribution. A model that assumes a single resistance ignores even the presence of the internal solid surface and concentrates on the boundary layer only. On the basis of the above, kf in the present work has been evaluated by the initial slope method (Spahn and Schluender, 1975) as an initial estimate which is then readjusted by superposition of experimental and theoretical data. However, the tables show a single diffusivity D, for each set of system conditions, but solid diffusivity is noticed to increase with the sorbate initial concentration. This has been previously obtained (Neretnieks, 1976a,b; Fritz et al., 1981) and attributed to the fact that sorbate-sorbent bonding energy decreases with surface coverage, leading to a higher diffusivity. This confirms the high contribution of surface diffusion to intraparticle diffusion in the process (Furusawa and Smith, 1973a,b; Komiyama and Smith, 1974). The more weakly adsorbed species BR22 (characterized by the higher D, values) appears to be a stronger function of concentration than the other component BY21 in the binary system. In the ternary system the strongly adsorbed BB69 shows a concentration dependence but its value is now affected by two dyes. On the other hand, compared to single-solute results in Table VII, D, for BR22 has increased upon introducing another component while the opposite has occurred to BY21. This indicates that,
+
+
392 Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991
upon mixing two species of different diffusional behavior, the species of higher diffusivity is inhibited by the slower diffuser while the latter is accelerated. The same conclusion has been reached by Fritz et al. (1981)applying the film homogeneous diffusion model to the binary mixture of 2,4-dichlorophenol and dodecylbenzenesulfonic acid on carbon. This effect is confirmed in the ternary system. Different ways to evaluate D, have been followed. One is by matching experimental and theoretical curves and finding D, that makes them congruent (Liapis and Rippin, 1977). Another is to compare experimental and theoretical data at one point of the concentration time curve with the aid of diagrams, say q / q = 0.6 (Fritz et al., 1981). In this work, D, values have been obtained by superposition of theoretical and experimental concentration time curves applied to graphical comparison. In their film homogeneous diffusion model, Liapis and Rippin (1977)have obtained D, values of 128 X and 224 X cmz/s for butanol-2 and a-amyl alcohol, respectively, in their binary mixture on carbon. Fritz et al. (1981)with their five binary mixtures on carbon, namely, p-nitrophenollphenol, p-nitrophenollp-chlorophenol, p-nitrophenol/ benzoic acid, p-chlorophenol/PES, and 2,4-dichlorophenol/dodecylbenzenesulfonatehave obtained values between 50.0 X and 250 X cm2/s. Compared to the above, values in the present system are low. This is attributed to the size of dye molecules, first, and their nature, second. The dyes under investigation have large molecular diameters compared to other organic molecules, and this lowers the speed of their diffusion along the sorbent surface. Also, the strong polar nature of dyes enhances a lot of multidirectional interactions in the sorbent surface the factor that, again, lowers the diffusivities of these dyes. The present kinetic model assumes a structurally homogeneous adsorbent and therefore does not relate the rate of diffusion to the internal structure of the sorbent. Nevertheless, the structural effect appears in the low diffusivity value. Another kinetic model, namely, the branched pore model, previously investigated by the authors (McKay and A1 Duri, 1988) relates the sorbent structure to the diffusion rate. However, it is too complicated and too time-consuming in its computation times to be extended to multisolute systems. Therefore, the solid diffusion is adequate at this stage. This variability of D, with concentration depends to a large extent on (i) the nature of adsorption in the system-chemi- or physisorption; (ii) the sorbent surface properties-heterogeneity and topography; (iii) the sorbate physical and chemical properties-molecular size, polarity, ionic charge, and adsorbing functional groups; and (iv) the relative diffusive and equilibrium properties of the species. In general the increase in D, with initial sorbate concentration is attributed to the decrease in bond energies with surface coverage resulting from the progressive filling of sites with decreasing energy. Sladek et al. (1974)and Neretnieks (1976a,b)developed exponential functions for D, in terms of differential heats of adsorption and surface coverage, respectively. Furthermore, Leyva-Ramos and Geankoplis (1985) found that, for strongly adsorbed species, the contribution of surface diffusion is more effective than for weak or physical adsorbed species. It is noticed that D, does not demonstrate a definite pattern with the change in the mass of carbon and hence the V / m ratio due to a number of competing factors: (i) the hydrodynamic properties of the flow, (ii) the surface area available for adsorption, (iii) the driving force per unit area, and (iv) the slope of the isotherm at the point of intersection of the operating line with the isotherm.
0.4
, BR22
0.1
BY21
1 0
10
20
30
40
50 C
.,
OJ
60
I
70
n~qdrn-~
Figure 6. Dsj versus C , for BR22 and BY21 in system I, using MODMXJ.FOR to predict system kinetics.
I
0.5
0
60
'*'
0
60
lZo Time, m i n
Time, m i n
180
J
Figure 7. (a) Film homogeneous diffusion model applied to several masses of BR22 in (BR22 + BY21) equimolar mixture on carbon. (b) Film homogeneous diffusion model applied to several concentrations of BR22 in (BR22 + BY21) equimolar mixture on carbon.
Table VI1 shows the parameters for (BR22 + BY21)/ carbon, and a single value of k, has succeeded in describing the film diffusion for each component. These values are f 5.3% cm s-l for BR22 2.95 x f 17% and 1.90 X and BY21,respectively. The values of kf.and D are both higher for BR22 than BY21, i.e., kfl > 1, and bs1> Ds2. Table VI1 and Figure 6 show D, for BR22 to be a stronger function of concentration than BY21. On the other hand, equilibrium studies of BR22 and BY21 in the single state and system 1solutions show that the two components have comparable equilibrium behavior, yet carbon has a slightly stronger affinity for BY21 than for BR22 and hence the former is more strongly adsorbed. This argument shows that the less strongly adsorbed species has a higher diffusivity than the more strongly adsorbed species, and this D, is a stronger function of initial sorbate concentration. This, in fact, agrees with previous interpretations of high and low diffusivity values (Sladek et al., 1974;Mathews, 1975;Fritz et al., 1981). Figures 7 and 8 show the concentration decay rate of a few masses and concentrations of BR22 and BY21,respectively, as predicted by the model and compared to the experimental data. Table VI11 presents the parameters obtained by applying the model to the ternary mixture. Results show that surface diffusivities all follow the same trend; that is, they increase with initial sorbate concentration. However, the increase is at a slower rate than in the binary mixtures.
Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991 393
-
I
J
, 33.0 c
0
.0
u
120
-i m e ,
min
=.E
I80
E
20.3
"
1 sn
e
:
C 0: Ingdm-': 34.3
e
40
u
Z
60
>
59
D
30 20
ii
10
rI
0
I 60
1 2 @ . - n e . x;c
130
Figure 8. (a) Application of film homogeneous diffusion model to several masses of BY21 in (BR22 + BY21) equimolar mixture on carbon. (b) Application of film homogeneous diffusion model to several concentrations of (BR22 + BY21) equimolar mixture on carbon.
B
-
I
23.:
I
4-
a
13 I
C
60
IZC
T l V ,
"1"
I90
Figure 10. Application of MODMXLFOR to several concentrations of (a) BB69, (b) BR22, and (c) BY21 in system IV. 1.3
0 3 ,
"2
0.5
I . . . . 60 . * . ' . . . " '
183
I
1
120
1.0
/
0
1
0
'
10
20 C o 3 , m g d n ~ - ~
30
Figure 9. D, versus C , for BR22, BY21, and BB69 in system IV using MODMX.I.FOR to predict system kinetics.
, Y
I 7
0
1 . .
This is expected as the competitive and interactive effects propagate with the number of components leading to more nonidealities in the adsorbed phase. The values of Dsj and k , for the ternary system are given in Table VI11 for the experimental runs. The variation in Dsj with Coj is shown in Figure 9, and the agreement between experimental data and theoretical data is shown in Figures 10 and 11. Figure 10 shows concentration decay curves for the three dyes for various initial dye concentrations. Figure 11shows the effect of varying carbon mass for the ternary system.
Conclusions From the present investigation the following can be concluded: (i) Solid diffusivity is an increasing function of initial sorbate concentration of each component. However, it is more so with the more slowly diffusing component. It does not show a definite trend with V l m ratio. (ii) Obtaining multicomponent equilibrium data by correlative methods is preferred to predictive methods due to the lack of accuracy in the latter. However, it is still associated with mathematical difficulties.
I
60
i2c
7,le. mln
'90
.
00 4-' i
Figure 11. (a) Application of MODMXJ.FOR to a few masses of BB69 in system IV. (b) Application of MODMXJ.FOR to a few masses of BR22 in system IV. (c) Application of MODMXJ.FOR to a few masses of BY21 in system IV.
(iii) Defining an isotherm that predicts experimental data accurately is vital. The isotherm should be thermodynamically consistent. (iv) This model is highly favorable due to the evidently large contribution of surface diffusion in the adsorption process. (v) A single kfhas described each component for a given range of experimental conditions.
394 Ind. Eng. Chem. Res., Vol. 30, No. 2, 1991
Nomenclature b = Redlich-Peterson's constant, dm3 mg-' C, = liquid-phase concentration, mg dm-3 C, = equilibrium liquid-phase concentration, mg dm-3 Co = initial liquid-phase concentration, mg dm-3 D, = solid diffusivity, cm2 D, = pore diffusivity, cm2 s-l d = solid particle diameter, cm fl= objective function in eq 3 K = Redlich-Peterson's constant, dm3 g-' kf = external mass transfer coefficient, cm s-l m = mass of adsorbent, g 4 = average solid-phase concentration, mg g-l qi = point solid-phase concentration, mg g-' q, = equilibrium solid-phase concentration, mg g-I R = carbon particle radius (a constant), cm r = carbon particle radius (a variable from 0 to r ) , cm t = time, min u = dimensionless solid-phase concentration, q i x V = volume of solution, dm3 x = dimensionless distance, r / R Greek Letters d = a constant in the general isotherm equation /3 = Redlich-Peterson's constant, dimensionless p = carbon particle density, g dm-3 cp = carbon particle voidage 7 = T
interaction factor
= dimensionless time, t D , / R 2
Subscripts
calc = calculated data exp = experimental data j = jth component k = kth component where k s = surface
#
j
Subscript O = single solute data
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