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Pore-Surface Diffusion Modeling for Dyes from Effluent on Pith Buning Chen, Chi Wai Hui, and Gordon McKay* Department of Chemical Engineering, The Hong Kong University of Science & Technology, Clear Water Bay, Kowloon, Hong Kong, SAR China Received May 12, 2000. In Final Form: October 11, 2000 A new pore-surface diffusion model (PSDM) for batch adsorption systems based on shrinking core theory has been developed. The new PSDM incorporates pore and surface diffusion into a variable effective diffusion coefficient, but the contributions from both pore and surface diffusion are evaluated independently. A further advantage of the new model is that it incorporates a time dependent surface diffusivity based on surface adsorbate concentration. The predicted results have been compared using four sets of experimental data for the batch adsorption of dyes on pith.
Introduction Adsorption now plays a key role in modern industries, especially in the field of environmental protection engineering, with the increasing environmental awareness of people all over the world. Adsorption processes are being employed widely for large-scale biochemical, chemical, and environmental recovery and purification applications.1-4 Liquid-solid adsorption operations are concerned with the ability of certain solids to preferentially concentrate specific substances from solution onto their surfaces, such as the removal of moisture dissolved in gasoline, the decolorization of petroleum products, and the removal of pollutants from aqueous or gaseous effluents. The requirements of a model are the solid-liquid equilibrium relationship, the mass balance equation, and the diffusional mass-transport relationships. Analysis of equilibrium studies gives the best model to describe the equilibrium relationship of dyestuff between liquid phase and adsorbed phase. Batch kinetic adsorption studies enable dynamic models, which incorporate these equilibrium models, to be developed and tested. Single resistance diffusion models based on external film mass transfer only5 or intraparticle diffusion only6,7 have been developed. The film-pore diffusion model is the batch kinetic analysis method which is a two-resistance diffusion model.8,9 An analytical solution of the pore diffusion model was proposed by McKay10 using the Langmuir isotherm; this model also has been applied to several sorption * Author for correspondence. (1) Dechow, F. J. Separation and Purification Techniques in Biotechnology: Noyes: Park Ridge, NJ, 1989. (2) Asenjo, J. A., Ed. Separation Processes in Biotechnology; Marcel Dekker: New York, 1983. (3) Wase, D. A. J.; Forster, C. F. Bio-sorbents for Metal Ions; Taylor and Francis: U.K., 1997. (4) McKay, G. Use of Adsorbents for the Removal of Pollutants from Wastewater; CRC Press Inc.: Boca Raton, FL, 1995. (5) Furusawa, T.; Smith, J. M. Fluid Particle and Intraparticle Mass Transfer Rates in Slurries. Ind. Eng. Chem. Fundam. 1973, 12, 197. (6) Crank, J. The Mathematics of Diffusion, 2nd ed.; Clarendon Press: Oxford, 1975. (7) McKay, G.; Allen, S. J.; McConvey, I. F.; Otterburn, M. S., Adsorption of dyes onto wood and peat surfaces. J. Colloid Interface Sci. 1981, 80, 323. (8) Cooper, R. S. Slow Particle Diffusion in Ion Exchange Columns. Ind. Eng. Chem. Fundam. 1965, 4, 308-318. (9) Cooper, R. S.; Liberman, D. A. Fixed-bed Adsorption Kinetics with Pore Diffusion Control. Ind. Eng. Chem. Fundam. 1970, 9, 620629.
systems: basic dyes on activated carbon11 and the sorption of Omega Chrome Red ME on fly ash.12 In 1988, McKay further tested the film-pore diffusion model: the effective diffusivity was found to have values much larger than those of pore diffusivities calculated from liquid diffusivities, and its value decreased with increasing initial dye concentration.11 This was attributed to the effect of surface diffusivity in the model. The film-surface diffusion model is another tworesistance kinetic analysis method. The homogeneous surface diffusion model (HSDM) has been successfully used to examine the dynamics of the adsorption process for various organic compounds on granular activated carbon,13-15 and for propionic acid on activated carbon.16 Hand et al.17 developed a procedure for determining the surface diffusion coefficient by eliminating liquid film mass-transfer resistance and comparing batch adsorption data to empirical solutions with the HSDM. The mechanism of sorption of adsorbate molecules on the surface of adsorbents has been discussed.18,19 Miyahara and Okazaki14,15 showed a significant concentration dependence of surface diffusivity in an aqueous adsorption by batch kinetic experiments. These single-resistance mass-transport models usually only have limited success in predicting experimental (10) McKay, G. The Adsorption of Basic Dye onto Silica from Aqueous Solution-Solid Diffusion Model. Chem. Eng. Sci. 1984, 39, 129. (11) McKay, G.; Al Duri, B. Branched-Pore Model Applied to the Adsorption of Basic Dyes on Carbon. Chem. Eng. Process 1988, 24, 1. (12) Gupta, G. S.; Prasad, G.; Singh, V. N. Removal of Chrome Dye from Aqueous Solutions by Mixed Adsorbents: Fly Ash and Coal. Water Res. 1990, 24, 45. (13) Suidan, M. T.; Traegner, U. K. Evaluation of Surface and Film Diffusion Coefficients for Carbon Adsorption. Water Res. 1988, 23 (3), 267. (14) Miyahara, M.; Okazaki, M. Concentration Dependence of Surface Diffusivity of Nitrobenzene and Benzonitrile in Liquid Phase Adsorption onto an Activated Carbon. J. Chem. Eng. Jpn. 1992, 25 (4), 407. (15) Miyahara, M.; Okazaki, M. Correlation of ConcentrationDependent Surface Diffusivity in Liquid Phase Adsorption. J. Chem. Eng. Jpn. 1993, 26 (5), 510. (16) Suzuki, M.; Tatao, F. Concentration Dependence of Surface Diffusion Coefficient of Propionic Acid in Activated Carbon Particles. AIChE J. 1982, 28, 380. (17) Hand, D. W.; Crittenden, J. C.; Thacker, W. E. User Oriented Batch Reactor Solutions to the Homogeneous Surface Diffusion Model. J. Environ, Eng. Div. (Am. Soc. Civ. Eng.) 1983, 109, 82. (18) Kapoor, A.; Yang, R. T.; Wong, A. C. Surface Diffusion on Energetically Heterogeneous SurfacessAn Effective Medium Approximation Approach. Catal. Rev. Sci. Eng. 1989, 31, 129. (19) Suzuki, M.; Talbot, J.; Jin, X.; Wang, N. H. L. New Equations for Multicomponent Adsorption Kinetics: Adsorption Engineering; Elsevier: Amsterdam, 1990; Vol. 10, p 1663.
10.1021/la000668r CCC: $20.00 © 2001 American Chemical Society Published on Web 01/05/2001
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Figure 1. Proposed concentration profiles inside a particle.
adsorption data, and therefore two-resistance models have been developed. A linear driving force (LDF) model for approximating the uptake rates by spherical pellets was first proposed by Glueckauf and Coates.20a Several investigators have attempted to improve the LDF model and have met with various degrees of success.20b-23 More recently, Yao and Tien24 analyzed the relationship between the LDF model and the approximate solutions of the intrapellet diffusion equation. The predicted results of the film-pore diffusion model do not agree well with experimental data; sometimes large errors appeared even if individual Deff values were used for each initial concentration. The main reason is that the Deff value used in the film-pore diffusion model cannot represent the actual diffusivity completely. Deff should reflect the joint effect of pore and surface diffusions, and their relationships with both solid and liquid concentrations while the contact time is changing. In the present paper, a new pore-surface diffusion model has been developed on the basis of the shrinking core model (SCM), and this model has been tested and compared with a previous form of the SCM25,26 using four experimental adsorption systems. The experimental systems studied are the adsorption of Acid Blue 25, Acid Red 114, Basic Blue 69, and Basic Red 22 dyes onto pith. Theory. The development of a kinetic mass-transport model needs a number of simplifying assumptions. In terms of the sorption process, there are likely to be many, making a predictive kinetic model based on the chemical reactions and complexes formed both difficult and complex. (20) (a) Glueckauf, F.; Coates, J. I. Theory of chromatography. IV. The influence of incomplete equilibrium on the front boundary of chromatograms and on the effectiveness of separation. J. Chem. Soc. 1947, 41, 1315. (b) Vermeulen, T. Kinetic Relationships for Ion Exchange Processes in Adsorption, Dialysis, and Ion Exchange. Chem. Eng. Prog. Symp. Ser. 1959, 55-71. (21) Hall, K. R.; Eagleton, L. C.; Acrivos, A.; Vermeulen, T. Pore-and Solid-Diffusion Kinetics in Fixed-bed Adsorption under Constantpattern Conditions. Ind. Eng. Chem. Fundam. 1966, 5, 212-221. (22) Rice, R. G. Approximate solutions for Batch, Packed Tube and Radical Flwo AdsorberssComparisons with Experiment. Chem. Eng. Sci. 1982, 37, 83-92. (23) Do, D.; Rice, R. G. On the Relative Importance of Pore and Surface Diffusion in Nonequilibrium Adsorption Rate Processes. Chem. Eng. Sci. 1987, 42, 2269. (24) Yao, C.; Tien, C. Approximations of uptake rate of Spherical Adsorbent Pellets and Their Application to Batch Adsorption Calculations. Chem. Eng. Sci. 1993, 48, 187. (25) Spahn, H.; Schlunder, E. U. The Scale-up of Activated Carbon Columns for Water Purification, I: Based on Results from Batch Tests. Chem. Eng. Sci. 1975, 30, 529. (26) McKay, G. Analytical Solution Using a Pore Diffusion Model for a Pseudo-irreversible Isotherm for the Adsorption of Basic Dye on Silica. AIChE J. 1984, 30, 692.
Pith comprises thousands of different chemical species and several species groups; the main ones are celluloses, pentosans, lignin, fulvic acid, and ash. Since the four dyes in the study also ionize to a certain degree in water, basic dyes producing positive dye ions and acidic dyes producing negative ions, then ion-exchange sorption as well as physical sorption on surface sites can also take place. On the basis of this complex myriad of chemistry, it was decided to see if a model based on generic physical rate controlling steps could be developed and used to describe the system and its variables. This model would be used to generate theoretical concentration versus time decay curves, which are compared to the experimentally obtained data points. The effect of any film resistance due to the external liquid film boundary layer surrounding the particle has been assumed negligible in this well-agitated system. Therefore, the basic assumptions in establishing this mass-transfer-based sorption model are as follows: (i) The mass-transport resistance in the particle external boundary layer is negligible. (ii) The internal mass transport rate within the particle is due to two rate-controlling mechanisms. The first ratecontrolling mechanism is pore diffusion, in which the pith particle pores are filled with dye solution and the dye molecules/ions diffuse through the liquid-filled pores to the particle surface where they undergo adsorption; the dye solution in the pores is constantly being replenished and is maintained at the bulk liquid-phase concentration. The second rate-controlling mechanism is surface diffusion, by migration of the adsorbed surface species by a surface hopping mechanism; the speed of this will depend on the bond strength of the attached sorption site and the affinity of the recipient site. This will account for the different types of chemical/physical interactions due to the wide ranging number of sites. (iii) For pore diffusion to the surface sites the adsorption at the site will be rapid and not be involved in the ratecontrolling mass-transfer processswhich will now be accounted for pore diffusion and surface diffusion masstransport resistance. The mass-transfer analysis is based on the adsorption of dyes on the particle within a shrinking core or spherical shell model (SCM). The mass-transfer analysis is based on the adsorption of dyes on the pith particle within a shrinking core or sperical shell model (SCM). In the SCM, it is assumed that the adsorbate molecule enters the spherical particle from the outer-layer to the inner or core of the particle. The particle radius changes from r ) R to r ) 0 during the process and from r to r - dr in time dt. The proposed
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simultaneously based on the following derivation:
δC δY Nt ) DP + FsDs δr δr
(3)
Equation 3 incorporates both pore and surface diffusion terms and can be rearranged to
δY δC Nt ) DP + FsDs δC δr
(
)
(4)
However, the rate of internal mass transfer from “filmpore diffusion” is given by
δC Nt ) Deff δr
Figure 2. Langmuir isotherm.
concentration profile inside the particle is represented in Figure 1. In Figure 1, at time t, the volume of particle bounded between r ) R and r ) r′ + dr′ has a solid-phase concentration defined by the tie line as Ye,t, with a corresponding liquid-phase concentration, Ce,t. The adsorption front, represented by an element thickness, dr, is at a radius r and average solid-phase concentration, Ye,i. On the basis of the assumption in this model, namely, that equilibrium is instantaneous and there is no film resistance, only pore and surface diffusion, then the tie lines are vertical and Ct ) Ce,t and Ye,t is readily found from the isotherm equation as represented in Figure 2. The model assumes that the concentration front moves toward the center of the particle at a pseudo-steady-state, that is, that the adsorbate diffusion rate is the same and is a constant in the process. The driving force at the beginning of the diffusion process is greater than that at the end of it, the former being much bigger than the latter. So, this assumption could result in an error. Another assumption of the shrinking-core model is that the effective diffusivity is assumed to be a constant throughout the whole process. The PSDM model for batch adsorption is based on the solid and liquid diffusion adsorption rate equations. The shrinking core model (SCM)27,28 is used to describe the mechanism, and equilibrium is correlated by the Langmuir isotherm equation. The fundamental equations in the SCM are represented: (i) The velocity of the concentration-front is obtained from the mass balance on a spherical element and is shown by eq 1:
dr Nt ) -4πr Ye,tFs dt 2
(1)
(5)
So, Deff is represented by
δY Deff ) DP + FsDs δC
(6)
It has been shown11,16 that the estimated effective diffusion coefficient can be greater than the pore diffusivity Dp, due to surface migration on the pore wall. The authors also found that surface diffusion is concentration dependent. Komiyama and Smith31 found that surface diffusion can contribute 20 times as much as pore diffusion. Therefore, surface diffusion has an inevitable effect on the effective diffusivity defined by the “film-pore diffusion” model. (iii) Equilibrium. The Langmuir isotherm is used to represent the dye pith-dye solution solid-liquid equilibrium relationship because this form provides a good fit for all four experimental systems.
Ye,t )
KLCe,t 1 + aLCe,t
(7)
(iv) The adsorbate mass balance at time t is
R3 - r3 WYe,t ) V(C0 - Ce,t) R3
(8)
for the solid-phase shrinking core (R - r) in the pith particles with the liquid-phase volume V of solution. From eqs 1 and 2, we can get
dr DeffR(1 + aLCe,t) ) dt rFsKL(r - R)
(9)
From eqs 7 and 8, we obtain where Ye,t is the solid-phase equilibrium dye concentration on the element dr at contact time t. (ii) The adsorption rate for pore liquid diffusion according to Fick’s law16,29,30 is
Nt )
4πDeffCe,t 1 1 r R
(
)
(2)
where Deff is the effective diffusion coefficient in the pore liquid, which can be expanded to incorporate Dp and Ds, when pore and surface diffusion are taking place (27) Yagi, S.; Kunii, D. Fluidised Solids Reactors with Continuous Solids Feeds. Chem. Eng. Sci. 1961, 16, 364. (28) Levenspiel, O. Chemical Reaction Engineering; Wiley: New York, 1962.
Ce,t ) f(r)
(10)
Equation 9 can be written as follows:
dr )
DeffR[1 + aLf(r)] dt rFsKL(r - R)
(11)
Equation 11 can be solved by a numerical method. The results of eq 11 will be more precise when the step distances (29) Aris, R. Interpretation of Sorption and Diffusion Data in Porous Solids. Ind. Eng. Chem. Fundam. 1983, 22, 150. (30) Riekert, L. The Relative Contribution of Pore Volume Diffusion and Surface Diffusion to Mass Transfer in Capillaries and Porous Media. AIChE J. 1985, 31, 863. (31) Komiyama, H.; Smith, J. M. Surface Diffusion in Liquid-Filled Pores. AIChE J. 1974, 20, 1110.
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of particle radius (dr) and time (dt) are very small. Then we use eq 12, which defines the tie lines as demonstrated in Figure 2 (the second dotted line from the left):
Ct ) Ce,t
(12)
Ye ) Ye,t
(13)
KL δY δYe ) ) δC δCe (1 + a C )2 L e,t
(14)
KL Deff ) DP + FsDs (1 + aLCe,t)2
(15)
Usually, the best fit agreement between experimental data and prediction results of the pore-surface diffusion model is when dt is set and dr is evaluated; values of dt are varied until convergence of the solution is achieved. Usually, dt is set at values around 1-60 s, and this results in dr < 10-5 cm. Several workers11,23,32-35 have reported Ds as a function of the fractional surface coverage, Ye,t/Ymax. Therefore, the expression for surface diffusivity, Ds, is
Ds )
Ds,0 exp(-E/RT) Ye,t 1Ymax
(
)
(16)
A detailed discussion on the correlation between Ds and Ds,0 has been given.18 In the earlier versions of the pore diffusion model,25,36,37 the diffusion coefficient, Deff, was based on combining constant values of Dp and Ds. The cases in which this assumption is justified have been discussed in detail.38-42 We solved eq 11 by using Excel and also developed a GAMS file to optimize the parameters over the range of experimental conditions and system variables studied in the work. GAMS is a general algebraic modeling system and enables us to solve eq 11 simply and fast.43 Since the present work has been performed isothermally, the relationship between Ds and Ds,0 is primarily dependent on surface coverage. The relationship proposed (32) Thacker, W. E.; Crittenden, J. C.; Snoeyink, V. L. Modeling of Adsorber Performance: Variable Influent Concentration and Comparison of Adsorbents. J. Water Pollut. Control Fed. 1984, 56, 243. (33) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley-Interscience: New York, 1984. (34) Hu, X.; Do, D. D. Experimental Concentration Dependence of Surface Diffusivity of Hydrocarbon in Activated Carbon. Chem. Eng. Sci. 1994, 49, 2145. (35) Yoshida, H.; Kataoka, T.; Fujikawa, S. Kinetics in a Chelate Ion-ExchangersII Experimental. Chem. Eng. Sci. 1986, 41, 2525. (36) McKay, G. The Adsorption of Dyestuffs from Aqueous Solutions Using Activated Carbon: An External Mass Transfer and Homogeneous Surface Diffusion Model. AIChE J. 1985, 31, 335. (37) Neretnieks, I. Adsorption of Components Having a Saturation Isotherm. Chem. Ing. Technol. 1974, 46, 781. (38) Crittenden, J. C.; Wong, B. W. C.; Thacker, W. E.; Snoeyink, V. O.; Heinrichs, R. L. Mathematical Model of Sequential Loading in Fixedbed Adsorber. J. Water Pollut. Control Fed. 1980, 52, 2780. (39) Yeroshenkova, G. V.; Volkov, S. A.; Sakodynskii, K. I. Effect of Packing Irregularities along the Bed Length. J. Chromatogr. 1983, 262, 19. (40) Neogi, P.; Ruchenstein, E. Transport Phenomena in Solids with Bidispersed Pores. AIChE J. 1980, 26, 787. (41) Whitaker, S. Diffusion in Packed Beds of Porous Particles. AIChE J. 1988, 34, 679. (42) Hui, C. W.; Chen, B.; McKay, G. Contact Time Optimization of 2-Stage Batch Adsorber Systems Using the Modified Film-Pore Diffusion Model. Submitted. (43) Brooke, A.; Kendrick, D.; Meeraus, A. GAMS: A Users Guide; Scientific Press: Palo Alto, CA, 1992.
by Higashi et al.44 is
Ds )
Ds,0 exp(-E/RT) 1-θ
(17)
where θ ) Ye/Ymax. There is an anomaly at the monolayer, because as the surface coverage θ f 1, then Ds f ∞. Other authors45,46 have considered the relationship between the diffusivities in terms of chemical potential. The surface diffusion flux, Js, is related to the chemical potential, µ, gradient by
Js ) -LYe
dµ dc
(18)
At equilibrium in an ideal solution, the chemical potential is related to the reference chemical potential, µ0, as follows:
µ ) µ0 + RT ln Ce
(19)
Combining eqs 18 and 19, the flux is
d ln Ce Js ) -Ds,0 exp(-E/RT)Ye dc
(20)
However, from eq 18
Ds,0 exp(-E/RT) ) LRT
(21)
Then, we obtain eq 22:
∂ ln Ce ∂ ln Ye
Ds ) Ds,0 exp(-E/RT)
(22)
Using the Langmuir isotherm from eq 7, ∂ ln Ce/∂ ln Ye is obtained, and in terms of fractional coverage θ ()Ye/Ymax); then eq 22 becomes
Ds )
Ds,0 exp(-E/RT) 1-θ
(17)
that is, the same as the Higashi model. Yang et al.47 and Chen and Yang48 have developed modifications to eq 17. These papers incorporated expressions to represent activation energies for surface diffusion on first and second layers, respectively. The early work by Yang et al.47 enables eq 17 to be modified and represented by
Ds )
Ds,0 exp(-E/RT) 1 - fθ
(23)
This equation has been used to modify eq 17. The use of eq 23 is still only an approximation because the activation energy is not constant and the surface mobility49 has been assumed to be unity. Experimental Section The design of the standard agitated batch adsorber has been described in previous papers.5,7,50,51,52 The methodology for the kinetic mass transport studies in the agitated batch adsorber (44) Higashi, K.; Ito, H.; Oishi, J. Surface Diffusivity. Nihon Genshiryoku Gakkaishi 1963, 5, 24-32. (45) Darken, L. S. Diffusion, Mobility and Their Interrelation through Free Energy in Binary Metallic Systems. Trans. AIME 1948, 174, 184. (46) Aharoni, Ch.; Evans, M. J. B. Fundamentals of Adsorption; Proc. IVth Int. Conf., Kyoto, May, Elsevier: Tokyo, 1992. (47) Yang, R. T.; Fenn, J. B.; Haller, G. L. Modification to the Higashi Model for Surface Diffusion. AIChE J. 1973, 19, 1052. (48) Chen, Y. D.; Yang, R. T. Concentration Dependence of Surface Diffusion and Zeolitic Diffusion. AIChE J. 1991, 37, 1579. (49) Miyabe, K.; Suzuki, M. Chromatography of Liquid-Phase Adsorption on Octadecylsilyl-Silica Gel. AIChE J. 1992, 38, 901.
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Table 1. Langmuir Isotherm Constants adsorbent name
dye name
KL (dm3/g)
aL (dm3/mg)
Ymax (mg/g)
cfs
pith pith pith pith
Acid Blue 25 Acid Red 114 Basic Blue 69 Basic Red 22
0.50 1.10 15.9 18.0
0.023 0.048 0.101 0.235
21.74 22.92 157.43 76.60
0.996 0.989 0.995 0.993
and the methodology for measuring equilibrium isotherms have also been presented in these papers. Four systems have been studied, namely, the adsorption of Acid Blue 25, Acid Red 114, Basic Blue 69, and Basic Red 22 dyes on pith. These four systems offer a wide range of characteristics over which to test the adsorption models, since colored acid dye molecules exhibit a negative charge in solution and basic dyes a positive charge. The dye concentrations in solution were all measured using UV/VIS spectrophotometry. The general composition of pith has been determined as R-cellulose, 53.7%; pentosan, 27.9%; lignin, 20.2%; alcohol/benzene soluble components, 7.5%; and ash, 6.6%.
Figure 3. Langmuir isotherms for Acid Blue 25 and Acid Red 114 on pith.
Results and Discussion Equilibrium Isotherms. The equilibrium isotherms were measured by contacting the adsorbents with dye solutions for 21 days, which was sufficient to achieve equilibrium. The data were analyzed by eq 7, and the Langmuir constants KL and aL and the saturation monolayer capacity Ymax ()KL/aL) are shown in Table 1. The best fit values of KL and aL were determined by linear regression, and the correlation coefficients, cfs, are also shown in Table 1. Since the correlation factors are so high, we considered the Langmuir model sufficiently accurate to be used in the testing of the mass-transport model. The maximum adsorption capacities, Ymax, are shown in Table 1. The capacities of the two basic dyes are significantly higher than those of the acidic dyes. In solution, basic dyes lose negatively charged anions, such as hydroxyl and chloride groups. The residual charge on the colored molecule is positive. In solution, certain groups in the pith, such as lignin, cellulose, and pentosan, lose hydrogen ions and form a particle surface with a negative ζ-potential. Consequently, the affinity and capacity for the positively charged colored basic dye ions are high due to Coulombic attraction. The colored acidic dye ions are negatively charged, and the net effect will be repulsion
from the negative ζ-potential at the particle surface. However, there are a significant number of species and groups in the acid dye molecules that enable a limited amount of physical adsorption of negatively charge acid dye ions to take place. Pore-Surface Diffusion Model. The theoretical model represented by eq 23 has been applied to the adsorption of the four dyes onto bagasse pith. Table 2 shows the values of the parameters DP, Ds,0 exp(-E/RT), and f for the effect of pith mass and initial dye concentration for the adsorption of the two acid dyes, AB25 and AR114, onto pith. Figures 5-8 show the plots of experimental points with the theoretical curves predicted by the pore-surface diffusion model. The average relative error has been determined for each experimental system and is shown and is based on eq 24.
average % error )
summation of absolute errors number of data points (24)
Table 3 shows the parameters and error values for the adsorption of the two basic dyes onto pith for different
Table 2. Prediction Results of Acid Blue 25 and Acid Red 114 on Pith by Using the Pore-Surface Diffusion Model system adsorbate adsorbent Acid Blue 25
pith
Acid Blue 25
pith
Acid Red 114
pith
Acid Red 114
pith
S (g)
C0 (mg/dm3)
f
Dp (×10-7 cm2/s)
Ds,0e(-E/RT) (×10-9 cm2/s)
3.4 2.55 2.125 1.70 1.275 0.85 3.4 3.4 3.4 3.4 3.4 3.4 3.4 2.55 2.125 1.70 1.275 0.85 3.4 3.4 3.4 3.4 3.4 3.4
100 100 100 100 100 100 26 53 79 100 129 166 100 100 100 100 100 100 25 50 75 100 125 150
0.2
2.15
3.0
0.8
4.0
1.04
average error (%) this model ref 53 1.22 0.63 0.40 0.25 0.16 0.17 2.38 1.25 1.08 0.68 0.42 0.22 1.17 0.66 0.49 0.32 0.27 0.31 4.28 2.06 1.93 1.22 0.69 0.66
2.25 1.47 1.08 0.85 0.45 0.23 6.61 2.67 0.77 0.69 1.35 1.71 1.38 0.96 0.56 0.45 0.22 0.18 7.54 4.36 1.84 0.46 0.67 1.07
Pore-Surface Diffusion Modeling
Figure 4. Langmuir isotherms of Basic Blue 69 and Basic Red 22 on pith.
Figure 5. Effect of pith mass for Acid Blue 25 on pith by using the pore-surface diffusion model.
Langmuir, Vol. 17, No. 3, 2001 745
Figure 6. Effect of initial concentration for Acid Blue 25 on pith by using the pore-surface diffusion model.
Figure 7. Effect of pith mass for Acid Red 114 on pith by using the pore-surface diffusion model.
initial dye concentrations and pith masses. The best fit value of f for the basic dyes is zero, implying that the adsorption energy is constant. This could be explained if the dominant mechanism is a chemisorption process between positively charged basic dye ions and negatively charged ζ-potential sites on the surface of the pith. The isotherm studies indicated that the adsorption capacity is much higher for basic dyes, and this is confirmed by these observations. Figures 9-12 show the plots of experimental points with the theroretically predicted curves for the effect of initial dye concentration and pith mass on the adsorption of the two basic dyes. The values of f, Dp, and Ds for the curves in Figures 5-12 are obtained by optimizing the three parameters using the GAMS software. Film-Pore Diffusion Model. This model was developed previously53 and has been tested for the present (50) McKay, G. The Adsorption of Dyestuffs from Aqueous Solution Using Activated CarbonsAnalytical Solution for Batch Adsorption Based on External Mass Transfer and Pore Diffusion. Chem. Eng. J. 1983, 27, 187. (51) Arevalo, E.; Rendueles, M.; Fernandez, A.; Rodrigues, A.; Diaz, M. Uptake of Copper and Cobalt in a Complexing Resin: ShrinkingCore Model with Two Reaction Fronts. Sep. Purif. Technol. 1998, 13, 37. (52) McKay, G.; Allen, S. J. Surface Mass Transfer Processes Using Peat as an Adsorbent for Dyestuffs. Can. J. Chem. Eng. 1980, 58, 521. (53) McKay, G. Two Solutions to Adsorption Equations for Pore Diffusion. Water, Air, Soil Pollut. 1991, 60, 117.
Figure 8. Effect of initial concentration for Acid Red 114 on pith by using the pore-surface diffusion model.
experimental data. The correlation of experimental and predicted data is shown in the final columns of Tables 2 and 3. In all cases for AB25 adsorption, the new poresurface model gives a better correlation of the experimental data than the film-pore model. In the case of Acid Red 114 adsorption, the pore-surface model provides the better fit except for two low-mass systems and one initial dye concentration. Certainly, it would be expected that high
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Table 3. Prediction Results of Basic Blue 69 and Basic Red 22 on Pith by Using the Pore-Surface Diffusion Model system adsorbate adsorbent Basic Blue 69
pith
Basic Blue 69
pith
Baisc Red 22
pith
Basic Red 22
pith
S (g)
C0 (mg/dm3)
f
Dp (×10-6 cm2/s)
Ds,0e(-E/RT) (×10-9 cm2/s)
2.55 2.125 1.70 1.275 0.85 0.425 1.7 1.7 1.7 1.7 1.7 1.7 2.55 2.125 1.70 1.275 0.85 0.425 1.7 1.7 1.7 1.7 1.7 1.7
200 200 200 200 200 200 300 250 200 150 100 50 200 200 200 200 200 200 300 250 200 150 100 50
0
5.0
4.0
0
2.3
4.6
Figure 9. Effect of pith mass for Basic Blue 69 on pith by using the pore-surface diffusion model.
average error (%) this model ref 53 9.98 6.09 3.75 1.40 0.86 0.83 2.11 2.76 2.65 2.73 2.37 4.87 3.96 2.22 0.80 0.48 0.55 0.84 0.95 1.07 1.91 1.64 1.24 6.58
6.12 2.17 1.48 2.50 1.36 1.07 3.71 3.53 3.03 1.39 12.56 31.71 4.65 2.83 1.13 0.54 0.39 0.79 0.65 0.85 1.77 1.99 3.54 20.89
Figure 10. Effect of initial concentration for Basic Blue 69 on pith by using the pore-surface diffusion model.
pith mass concentrations would favor a surface diffusion mechanism. In Table 3 for the adsorption of Basic Blue 69 dye, the new pore-surface diffusion model provides a significantly better fit to the experimental data than the film-pore model. In the case of Basic Red 22 adsorption, the average errors for the two mass transport model are much closer together, and at low pith masses and high initial dye concentrations the film-pore model shows slightly better correlations with the experimental results. The extent of film or pore diffusion is often assessed using the Biot number, Bi:
Bi )
KfR Deff
(25)
The values of the external mass-transfer coefficients are 4.3 × 10-3, 3.46 × 10-3, 1.75 × 10-3, and 7.5 × 10-3 cm/s, and the effective “best fit single values” are the coefficients 7.2 × 10-6, 3.1 × 10-6, 3.5 × 10-7, and 6.2 × 10-7 cm2/s for BB69, BR22, AB25, and AR114, respectively. The
Figure 11. Effect of pith mass for Acid Red 22 on pith by using the pore-surface diffusion model.
resulting Biot numbers are 36.1, 67.5, 30.3, and 73.2, respectively. For systems in which film diffusion only is rate controlling, the Biot number is of the order of unity or less, and for Biot numbers greater than 10, pore diffusion
Pore-Surface Diffusion Modeling
Langmuir, Vol. 17, No. 3, 2001 747
Figure 14. Sensitivity analysis for the effect of surface diffusivity.
Figure 12. Effect of initial concentration for Basic Red 22 on pith by using the pore-surface diffusion model.
results of increasing Dp by a factor of 5 and also decreasing Dp by a factor of 5. The predictions are compared with the best fit experimental model data for Acid Blue 15 dye adsorption using a pith mass of 1.7 g and an initial dye concentration of 100 mg/cm3. The theoretical model predictions demonstrate the sensitivity of the model to the pore diffusion coefficient, Dp. The results of the sensitivity analysis for the surface diffusion coefficient, Ds,o, are shown in Figure 14. The best fit experimental model data from Figure 13 have again been used as the baseline data for the sensitivity analysis. The effect of changing Ds,o is very apparent, and Figure 14 shows that this new model is also sensitive to the surface diffusivity. Conclusions
Figure 13. Sensitivity analysis for the effect of pore diffusivity.
becomes the predominant mechanism with Biot numbers greater than 50 signifying almost complete internal diffusion control. The four Biot numbers indicate a major dominance for internal diffusion control, with the BB69 and AB25 systems indicating some contribution from external film resistance. The contributions from film control are responsible for the few cases in which the average percent errors are lower for the film-pore model. We have not yet completed the development of this threeresistance mass-transport model and the incorporation of the external mass-transfer coefficient into the poresurface diffusion model due to mathematical complexities. Significantly large average percent errors are observed for the film-pore model results for all four dye-pith systems at large pith masses and also for low initial dye concentrations. These are conditions which favor surface diffusion/adsorption rather than pore diffusion. Comparison of the average relative errors of the two models shows that the pore-surface diffusion model produces significantly lower error values than the filmpore diffusion model in most systems. However, some of the error values are still significant and indicate that it is still necessary to incorporate an external film masstransfer coefficient into the model under certain conditions. It is important to assess the sensitivity of any adsorption mass-transfer model in which mass-transport diffusion coefficients represent and control the rate of sorption of a range of chemical and physical surface reactions. Therefore, a sensitivity analysis has been carried out on Dp and Ds,o to ensure the theoretical model predictions are sensitive to these parameters. Figure 13 shows the
A pore-surface diffusion model, that can be used to predict concentration decay curves, has been developed. The predicted results of four different adsorption systems, Acid Blue 25, Acid Red 114, Basic Blue 69, and Basic Red 22 dyes on pith, prove that the pore-surface diffusion model is a universal model applicable for most batch adsorption processes. The model only contains parameters related to the nature of the system, that is, pore diffusivity and surface diffusivity, adsorbent weight and density, and liquid volume and initial solute concentration. The model overcomes the disadvantage of assuming Ye and Deff as constants throughout the whole concentration region. This results in more precise driving forces and concentration versus time prediction results. The model can predict theoretical concentration versus time decay curves very accurately in the medium and low concentration regions, which cannot be done accurately by the filmpore diffusion model. Due to its simple and fast solution, GAMS makes the optimization of large amounts of experimental data easy to process and the best values of Ds,0 and Dp readily obtainable. There are still problems at very low concentrations for certain systems possessing a high adsorption capacity and extremely steep initial isotherm slopes. Notation a aL b Bi cfs C C0
correlative coefficient in eq 29 Langmuir constant, dm3/mg correlative coefficient in eq 29 Biot number, unitless correlation coefficient liquid-phase concentration, mg/dm3 initial liquid concentration, mg/dm3
748 Ce Ce,t Ct Cexptl Ccalc Deff DL Dp Ds Ds,0 E f f(r) kf KL Nt n r R
Langmuir, Vol. 17, No. 3, 2001 equilibrium liquid-phase concentration, mg/dm3 equilibrium liquid-phase concentration at time t, mg/dm3 liquid concentration at time t, mg/dm3 experimental value of liquid concentration, mg/ dm3 prediction value of liquid concentration, mg/dm3 effective diffusion coefficient, cm2/s liquid-phase diffusivity, cm2/s pore diffusivity, cm2/s surface diffusivity, cm2/s surface diffusivity, cm2/s potential energy, kJ coefficient of area coverage function of r, mg/dm3 external film mass-transfer coefficient, cm/s Langmuir constant, dm3/g adsorption rate, mg/s number of experimental data points particle radius, cm particle diameter, cm; gas constant, 8.31 kJ/mol‚ K
Chen et al. ARE SES T t V W Ye Ye,i Ye,t Ye,0 Ymax
average relative error, % sum of error square temperature, K time, s liquid volume, dm3 adsorbent weight, g equilibrium solid-phase concentration, mg/g mean instantaneous solid-phase concentration in the core element, mg/g equilibrium solid-phase concentration at time t, mg/g initial equilibrium solid-phase concentration, mg/g maximum equilibrium solid-phase concentration, mg/g
Greek Symbols Fs µ θ
adsorbent particle density, g/cm3 tortuosity factor or chemical potential surface coverage LA000668R
