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Pore-Surface Diffusion Model for Batch Adsorption Processes Chi-Wai Hui, Buning Chen, and Gordon McKay* Department of Chemical Engineering, The Hong Kong University of Science & Technology, Clear Water Bay, Kowloon, Hong Kong, SAR China Received September 30, 2002. In Final Form: January 24, 2003 A new pore-surface diffusion model (PSDM) for batch adsorption systems 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. This model is compared with two previous models, which were based on a “lumped” effective diffusion coefficient, for which rapid analytical solutions have been developed. A further advantage of the new model is that it incorporates a time dependent surface diffusivity based on surface adsorbate concentration. Previous rapid analytical solutions to the effective pore diffusion models have assumed a constant time independent surface adsorbate concentration. The three models have been compared using four sets of experimental data for the batch adsorption of dyes onto wood and peat.
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.1,2 Liquid-solid adsorption operations concern 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 effluents. Most mass transfer models for batch adsorption systems assume the adsorbent particle is a sphere or a cylinder. The next requirement of a model is to define the solidliquid equilibrium relationships, the mass balance equation, and the diffusional mass transport relationships. In terms of adsorption capacity, the equilibrium data are usually expressed in the form of an isotherm equation. Literature shows that equilibrium data may be assumed to follow a linear isotherm,3 a nonlinear isotherm,4 or an irreversible isotherm.5 Most mechanisms have been based on two resistance mass transport models, namely, film or external mass transfer and internal diffusional mass transport. The internal diffusion transport process has been assumed to occur by pore diffusion6,7 or homogeneous surface/solid-phase diffusion.8,9 A theoretical solution for the case of three-resistance controlled mass transfer was * Author for correspondence. (1) Wase, D. A. J.; Forster, C. F. Bio-sorbents for Metal Ions; Taylor and Francis: U.K., 1997. (2) McKay, G. Use of Adsorbents for the Removal of Pollutants from Wastewater; CRC Press Inc.: Boca Raton, FL, 1995. (3) Dryden, C. E.; Kay, W. B. Kinetics of Batch Adsorption and Desorption. Ind. Eng. Chem. 1954, 46, 2294. (4) Tien, C. Adsorption Kinetics of a Nonflow System with Nonlinear Equilibrium Relationship, AIChE J. 1961, 7, 410. (5) Liapis, A. I.; Rippin, D. W. T. The Simulation of Binary Adsorption in Activated Carbon Columns Using Estimates of Diffusional Resistance within the Carbon Particles Drived from Batch Experiments. Chem. Eng. Sci. 1978, 33, 593. (6) Furusawa, T.; Smith, J. M. Fluid Particle and Intraparticle Mass Transfer Rates in Slurries. Ind. Eng. Chem. Fundam. 1973, 12, 197. (7) Fritz, W.; Schlunder, E. U. Competitive Adsorption of Two Dissolved Organics onto Activated Carbon I: Adsorption Equilibria. Chem. Eng. Sci. 1981, 36, 721.
developed,10 and other three-resistance models based on the macro/meso/micropore structure of adsorbents have been developed and applied to various experimental systems.11,12 In reality, the true mechanism involves external film, pore and surface diffusion apart from sorbate-sorbent interactions. The mathematical models implement assumptions to reduce mathematical complexity and data processing time while optimizing the accuracy of theoretical predictions when compared with experimental data. Early developments of pore-surface diffusion models (PSDMs) inferred diffusivities by matching model predictions with breakthrough curves.10,13 Other reasonably successful attempts to incorporate pore and surface diffusion processes into a combined model are based on film-pore diffusion models.14-17 In these models a “lumped” effective diffusion coefficient, Deff, was adopted incorporating the effects of surface diffusion and pore diffusion. Consequently, a single Deff value was used for each solute concentration. However, the predicted results of the film-pore diffusion model do not agree well with experimental data. The main reason is that Deff used in the film-pore diffusion model (8) Miller, C. O. M.; Clump, C. W. A Liquid-Phase Adsorption Study of the Rate of Diffusion of Phenol from Aqueous Solution into Activated Carbon. AIChE J. 1970, 16, 169. (9) Hashimoto, K.; Miura, K.; Nagata, S. Intraparticle Diffusivities in Liquid-Phase Adsorption with Non-Linear Isotherms. J. Chem. Eng. Jpn. 1975, 8, 367. (10) Masamune, S.; Smith, J. M. Adsorption Rate Studiess Significance of Pore Diffusion. Ind. Eng. Chem. Fundam. 1964, 10, 246. (11) Peel, R. G.; Benedek, A.; Crowe, C. M. A Branched Pore Kinetics Model for Activated Carbon Adsorption. AIChE J. 1981, 27, 26. (12) McKay, G.; Al-Duri, B. Study of the Mechanism of Pore Diffusion in Batch Adsorption Systems. J. Chem. Technol. Biotechnol. 1990, 48, 269. (13) Schneider, P.; Smith, J. Adsorption Rate Constant from Chromatography. AIChE J. 1986, 14, 762. (14) 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. (15) Fritz, W.; Merk, W.; Schlunder, E. U. Competitive Adsorption of Two Dissolved Organics onto Activated Carbon II: Adsorption Kinetics. Chem. Eng. Sci. 1981, 36, 731. (16) 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. (17) McKay, G. Two Solutions to Adsorption Equations for Pore Diffusion. Water, Air, Soil Pollut. 1991, 60, 117.
10.1021/la026624v CCC: $25.00 © 2003 American Chemical Society Published on Web 04/16/2003
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Figure 1. Proposed concentration profiles inside the particle.
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 two previous forms of the SCM14,18 using four experimental adsorption systems. The experimental systems studied are the adsorption of Basic Blue 69 dye onto peat and wood and the adsorption of Acid Blue 25 dye onto peat and wood. Theory 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 to r during the process and from r to dr at any time t. The proposed concentration profile inside the particle is represented in Figure 1. The model assumes that the concentration front moves toward the center of the particle at a pseudo-steady-state with a constant diffusivity. The driving force at the beginning of the diffusion process is greater than that at the end of it. The PSDM model for batch adsorption is based on the solid and liquid diffusion adsorption rate equations. The shrinking core model (SCM)19 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 obtained from the mass balance on a spherical element is shown by eq 1:
Nt ) -4πr2 Ye,tFs
dr dt
(1)
(ii) The adsorption rate for pore liquid diffusion according to Fick’s law20,21 is (18) McKay, G. The Adsorption of Basic Dye onto Silica from Aqueous Solution - Solid Diffusion Model. Chem. Eng. Sci. 1984, 39, 129. (19) Yagi, S.; Kunii, D. Fluidised Solids Reactors with Continuous Solids Feeds. Chem. Eng. Sci. 1961, 16, 364. (20) Aris, R. Interpretation of Sorption and Diffusion Data in Porous Solids. Ind. Eng. Chem. Fundam. 1983, 22, 150. (21) 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.
4πDeffCe,t 1 1 r R
Nt )
(
(2)
)
where Deff is the effective diffusion coefficient in the pore liquid, which can be expanded to incorporate Dp and Ds on the basis of the following derivation:
Nt ) Dp
δC δY + FsDs δr δr
(3)
Equation 3 incorporates both pore and surface diffusion terms:
(
Nt ) Dp + FsDs
δY δC δC δr
)
(4)
However, the rate of internal mass transfer from “filmpore diffusion” is given by
Nt ) Deff
δC δr
(5)
So, Deff is represented by
Deff ) Dp + FsDs
δY δC
(6)
It has been shown22 that the estimated effective diffusion coefficient was 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 Smith23 found that surface diffusion contributed 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 solid-liquid equilibrium relationship because this form provides a good fit for all four experimental systems.
Ye,t )
KLCe,t 1 + aLCe,t
(7)
(22) McKay, G.; Al Duri, B. Branched-Pore Model Applied to the Adsorption of Basic Dyes on Carbon. Chem. Eng. Process. 1988, 24, 1. (23) Komiyama, H.; Smith, J. M. Surface Diffusion in Liquid-Filled Pores. AIChE J. 1974, 20, 1110.
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the expression for surface diffusivity, Ds, is
Ds )
Ds,0 exp(-E/RT) Ye,t 1Ymax
(
)
(16)
A detail discussion on the correlation between Ds and Ds,0 has been given.26 This equation can also be derived by chemical potential analysis, and Yang et al.27 and Chen and Yang28 have developed modifications to eq 16. These papers incorporated expressions to represent activation energies for surface diffusion on first and second layers, respectively. The work by Yang et al.27 enables eq 16 to be modified and represented by
Ds ) Figure 2. Langmuir isotherm.
(8)
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
Ce,t ) f(r)
(10)
Equation 9 can be written as follows
DeffR[1 + aLf(r)] rFsKL(r - R)
(17)
Experimental Section
3
R -r WYe,t ) V(C0 - Ce,t) R3
dr )
(1 - fθ)
This equation has been used to modify model 3.
(iv) The adsorbate mass balance at time t is 3
Ds,0 exp(-E/RT)
dt
(11)
Equation 11 can be solved by a numerical method. The results of eq 11 will be more precise when the step distances 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 predictions of the pore-surface diffusion model results 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 workers22,24,25 have reported Ds as a function of the fractional surface coverage, θ ) Ye,t/Ymax. Therefore, (24) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Wiley-Interscience: New York, 1984.
The design of the standard agitated batch adsorber has been described in previous work.29,30 The methodology for the kinetic mass transport studies in the agitated batch adsorber and the methodology for measuring equilibrium isotherms have also been presented in these papers. Four systems have been studied, namely, the adsorption of Basic Blue 69 dye onto peat and wood and that of Acid Blue 25 dye onto peat and wood. Basic dyes were selected because they have a high sorption capacity and high affinity for peat and wood, whereas acid dyes have a low capacity and low affinity for the adsorbents. Peat and wood were selected as widely available low-cost adsorbents, with peat containing a rich array of potential exchange sorption groups (carboxylic acids, phenolics, ketones) whereas wood has a more uniform unreactive cellulosic structure. These four systems offer a wide range of characteristics over which to test the adsorption models. The dye concentrations in solution were all measured using UV/vis spectrophotometry.
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 are 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’s, are also shown in Table 1. The saturation capacities for Basic Blue on peat and wood are 624.3 and 81.93 mg/g, respectively. The high capacity of the Basic Blue on peat reflects the greater exchange sorption capacity between the positively charged dye species and the hydrogen ions from the humic and fulvic acids than that on the more unreactive sites of wood. The capacity of both adsorbents for Acid Blue dye is lower than the adsorption capacity of each adsorbent for (25) Hu, X.; Do, D. D. Experimental Concentration Dependence of Surface Diffusivity of Hydrocarbon in Activated Carbon. Chem. Eng. Sci. 1994, 49, 2145. (26) Kapoor, A.; Yang, R. T.; Wong, C. Surface Diffusion. Catal. Rev. Sci. Eng. 1989, 31, 129. (27) Yang, R. T.; Fenn, J. B.; Haller, G. L. Modification to the Higashi Model for Surface Diffusion. AIChE J. 1973, 19, 1052. (28) Chen, Y. D.; Yang, R. T. Concentration Dependence of Surface Diffusion and Zeolitic Diffusion. AIChE J. 1991, 37, 1579. (29) Furusawa, T.; Smith, J. M. Diffusivities from Dynamic Data. AIChE J. 1974, 19, 40. (30) 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.
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Langmuir, Vol. 19, No. 10, 2003 4191 Table 2. Prediction Results of Model 1 (Ye Changes with Initial Liquid Concentration C0) Using the Searched Best Individual Deff Values system adsorbate
adsorbent C0 (mg/dm3) Deff (cm2/s) avg err (%)
Acid Blue 25
wood
Basic Blue 69
wood
Acid Blue 25
peat
Basic Blue 69
peat
20 50 100 200 50 100 200 500 20 50 100 200 50 100 200 500
Figure 3. Concentration curves for the adsorption of Acid Blue 25 on wood.
4.90 × 10-8 5.70 × 10-8 5.30 × 10-8 6.50 × 10-8 4.80 × 10-7 9.40 × 10-7 1.16 × 10-6 3.00 × 10-7 1.02 × 10-5 1.03 × 10-5 5.00 × 10-8 4.10 × 10-8 6.00 × 10-7 8.60 × 10-7 1.30 × 10-7 2.50 × 10-7
2.85 2.47 1.10 0.60 30.21 11.23 1.31 0.35 67.39 58.37 3.93 1.67 63.21 54.19 38.68 15.14
the experimental data (symbols) for Acid Blue 25 on wood and Basic Blue 69 dye on wood. The values of the effective diffusion coefficient for each system are shown in Table 2. This solution assumes that the contribution from surface diffusion is negligible or constant and independent of initial dye concentration. The Deff values are obtained by minimizing the error between theoretical and experimental data points over all the four initial dye concentration studies. The error definition used was the average relative error (ARE), as shown by eq 19 and the error values given in Table 2.
ARE ) Figure 4. Concentration curves for the adsorption of Basic Blue 69 on wood. Table 1. Langmuir Isotherm Constants adsorbent wood wood peat peat
dye
KL (dm3/g)
aL (dm3/mg)
Ymax (mg/g)
cfs
Acid Blue 25 Basic Blue 69 Acid Blue 25 Basic Blue 69
0.257 11.47 0.621 22.47
0.0159 0.140 0.0130 0.036
16.06 81.93 47.77 624.3
0.992 0.983 0.985 0.975
Basic Blue dye. Wood has a lower capacity for Acid Blue dye than does peat. Since the Acid Blue dye ions are negatively charged, the higher sorption capacity of peat may be due to exchange sorption with negatively charged hydroxyls on peat. Adsorption on peat is not purely a function of the exchange capacity of the end groups, since after alkali extraction of the acidic compounds peat still has a very significant adsorption capacity for positively charged species. DiffusionsModel 1.14 This model applies the SCM to the pore diffusion model. In the first application of this model a single effective diffusivity is applied to all the different values of initial dye concentrations for each of the four experimental systems. To enable rapid analytical solution of the problem, it is necessary to input a single value of Ye,t. Spahn and Schlunder15 used a value
Ye,t ) 0.5Ye,0
(18)
Figures 3 and 4 show concentration decay curves for the theoretically predicted data (solid lines) compared with
1
n
∑ ni ) 1
(
)
|Cexp - Ccalc| Ccalc
(19)
Visual examination of Figure 4 shows that there is good agreement between theoretical and experimental data for Acid Blue on wood at all four initial dye concentrations. In the case of Basic Blue 69 on wood, the correlation at high initial dye concentrations is good but at low dye concentrations the agreement between theoretically predicted data (solid lines) and experimental data (points) is poor. Table 2 confirms the error is high, namely 30%. A similar trend is observed in Table 2 for the adsorption of Acid Blue 25 on peat. Since Deff is changed for each initial dye concentration, then the reasons for the error must be (i) that Deff may vary throughout each experiment and (ii) that a single value of Ye,t ()0.5Ye,0) is a poor assumption. At low concentrations the slope ∂Y/∂C is large, and this will both affect the surface diffusion flux contribution to Deff and cause Ye,t to vary significantly. For the adsorption of Basic Blue dye on peat, the error for all four initial dye concentrations is large, from 15% to 63%. The reason for this is that the isotherm rises very steeply and the values of the slope, ∂Y/∂C, are extremely high; therefore, the contribution from surface diffusion must be significant and both time and concentration dependent. DiffusionsModel 2.30,31 The fundamentals of this model are the same as those for model 1, except that a different constant value for Ye,t was used on the basis of the operating line equilibrium adsorption capacity, that is (31) McKay, G.; El-Geundi, M.; Nassar, M. M. Pore Diffusion During the Adsorption of Dyes onto Bagasse Pith. Trans IChemE Part B: Process Safety Environ. Protection 1996, 74, 277.
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Ye,t ) Ye
Hui et al.
(20)
At the corresponding value of Ct,
Ct ) Ce,t ) Ce
(21)
Two approaches for Deff were tested in this model. In the first case, Deff was maintained constant for each system but also for every initial dye concentration; in other words, Deff was assumed to be constant and independent of dye concentration in all cases. The theoretical data (solid lines) and experimental data (symbols) are shown in Figures 5 and 6 for the adsorption of Acid Blue 25 and Basic Blue 69 dye on wood, respectively. The diffusivities are shown in Table 3. The error values are greater than those for model 1, but a constant system Deff was used compared with a constant Deff for each individual experiment, as shown in Table 3. McKay and Bino32 and McKay30 used this model successfully for the adsorption of phenol onto carbon and Basic Blue dye 69 onto silica. However, in those two previous studies, the model was restricted to systems in which all the operating lines terminated on the monolayer, that is
Ye,t ) Ymax
(22)
In the second case, eq 22 was used in eq 13 and a Deff was selected for each individual experiment for the adsorption of the dyes. The data are compared in Figures 7 and 8 for the adsorption of Acid Blue 25 dye on wood and peat, respectively. Table 4 shows the diffusivities and the error values. This model provides a significantly better fit to experimental data than the previous model on the basis of the error values in Table 3. At low concentrations, the Acid Blue 25-peat system is still not well correlated and the three lower concentrations of the Basic Blue 69-peat system show poor correlation, with an avg err (%) > 25 in all three cases. In the present work, the heterogeneous nature of the peat surface must be significant in influencing the adsorption rate throughout each individual experiment. Therefore, a variable contribution to surface diffusion is required. The data in Table 4 can be correlated, as the Deff values vary as a function of C0. The data are plotted as Deff versus C0 in Figure 9 and may be correlated by an equation of the general form
Deff ) aC0b
Figure 5. Concentration curves for the adsorption of Acid Blue 25 on wood.
Figure 6. Concentration curves for the adsorption of Basic Blue 69 on wood. Table 3. Prediction Results of Model 2 (Ye Changes with Initial Liquid Concentration C0) Using the Searched Best System Deff Values system adsorbate
adsorbent C0 (mg/dm3) Deff (cm2/s) avg err (%)
Acid Blue 25
wood
Basic Blue 69
wood
Acid Blue 25
peat
Basic Blue 69
peat
(23)
The values of a and b and the relative errors (%) are shown in Table 5. The results in Table 5 demonstrate that there is a definite trend in the values of Deff, and although a reasonable correlation between Deff and C0 can obtained, this model is still based on a generic effective diffusivity. A more accurate representation of the diffusion mechanism should be obtained by dividing Deff into its representative components of pore and surface diffusivities. DiffusionsModel 3: Pore-Surface Diffusion Model. The theoretical development of this new model was presented in the Theory section of this paper. In this case, Deff is a function of the pore diffusion coefficient and a combination of the variable surface diffusivity and the mass transfer concentration gradient, ∂Y/∂C. There is only (32) McKay, G.; Bino, M. J. Application of Two Resistance Mass Transfer Model to Adsorption Systems. Chem. Eng. Res. Des. 1985, 63, 168.
20 50 100 200 50 100 200 500 20 50 100 200 50 100 200 500
9.50 × 10-8
2.65 × 10-6
2.60 × 10-7
3.00 × 10-6
3.24 2.28 1.25 1.61 33.50 3.91 14.42 13.95 47.82 34.10 2.72 6.70 66.89 55.94 28.21 16.57
limited information in the literature on surface diffusivities32 in liquid-phase systems. Most studies have employed indirect methods to infer surface diffusivities by matching breakthrough curves with model predictions.23,34 Recently, surface diffusivities have been determined more directly (33) Ma, Z.; Whitley, R. D.; Wang, N. H. L. Pore and Surface Diffusion in Multicomponent Adsorption and Liquid Chromatography Systems. AIChE J. 1996, 42, 1244. (34) Mathews, A. P.; Weber, J. Effects of External Mass Transfer and Intraparticle Diffusion on Adsorption Rates in Slurry Reactors. AIChE Symp. Ser. 1976, 73, 166.
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Figure 7. Concentration curves for the adsorption of Acid Blue 25 on wood. Figure 9. Concentration decay curves of Acid Blue 25 on peat.
participating in the adsorption process. Theoretical concentration decay curves were determined from a constant Dp and the best-fit Ds,0 as defined in eq 17. The values of the diffusivities, Dp and Ds,0, are shown in Table 6. The experimental data points and theoretical curves are shown in Figures 9 and 10 for the adsorption of Acid Blue 25 on peat and Basic Blue 69 on peat, respectively. The percentage errors are also shown in Table 6, and the values are seen to be a significant improvement on the data for model 3 shown in Table 3. Pore diffusivity is related to molecular diffusivity by eq 24:
Dp ) Figure 8. Concentration curves for the adsorption of Acid Blue 25 on peat. Table 4. Prediction Results of Model 2 (Ye Changes with Initial Liquid Concentration C0) Using the Searched Best Individual Deff Values system adsorbate
adsorbent C0 (mg/dm3) Deff (cm2/s) avg err (%)
Acid Blue 25
wood
Basic Blue 69
wood
Acid Blue 25
peat
Basic Blue 69
peat
20 50 100 200 50 100 200 500 20 50 100 200 50 100 200 500
1.23 × 10-7 1.08 × 10-7 8.10 × 10-8 7.60 × 10-8 1.10 × 10-5 3.11 × 10-6 1.25 × 10-6 3.00 × 10-7 3.13 × 10-4 7.70 × 10-6 2.60 × 10-7 9.20 × 10-8 9.30 × 10-6 1.39 × 10-5 5.30 × 10-6 1.70 × 10-6
1.84 1.86 1.00 0.53 3.42 2.16 1.14 0.34 40.60 24.93 2.72 1.28 51.82 40.65 25.61 6.88
using pulsed data35 and laser fluorescence.36,37 In the present work, Ds is inferred by matching theoretical and experimental breakthrough curves, and then Ds,0 is evaluated as a function of the fractional surface area (35) Miyabe, K.; Suzuki, M. Chromatography of Liquid-Phase Adsorption on Octadecylsilyl-Silica Gel. AIChE J. 1992, 38, 901. (36) Tilton, R. D.; Robertson, C. R.; Gast, A. P. Lateral Diffusion of Bovine Serum Albumin Adsorbed at the Solid-Liquid Interface. J. Colloid Interface Sci. 1990, 137, 192. (37) Tilton, R. D.; Gast, A. P.; Robertson, C. R. Surface Diffusion of Interacting Proteins. Biophys. J. 1990, 58, 1321.
DL µ
(24)
The tortuosity or labyrinth factor is strongly dependent on the pore structure, pore size, and distribution of pores throughout the adsorbent particle. In addition, the molecular size, mobility, and affinity of the adsorbate in solution will influence the way the adsorbate molecules move through this labyrinth of pores. The tortuosity factor is assumed to be a constant value for a particular adsorbate-adsorbent system, which is reasonable, providing the pore size distribution is homogeneous throughout the entire adsorbent particle. However, should the ratio of micropores/macropores change with particle penetration, this would not be the case. We have assumed a constant tortuosity in the present work. The values of DL and µ are given in Table 7. Analysis of the data in Table 6 provides information relating to the adsorption mechanism. In the case of Acid Blue 25 on wood, the process is independent of the value of f and obviously independent of Ds,0 too. The mechanism is pore diffusion only, because wood has only limited active surface sites in its predominantly cellulosic structure. These active sites are due to weak carboxylic acid groups present in the fulvic acids. The acid dye ions are negatively charged and will be relatively inactive to ion exchange and bonding with the carboxylic hydrogens in fulvic acids. The adsorption of Basic Blue 69 on wood is characterized by an f value which is almost unity. This fact implies equal energies of adsorption on the surface binding sites of the chemisorption processes as opposed to physical sorption via pore diffusion on the low energy cellulosic sites. Since basic dyes ionize to form positively charged colored ions, these cations exchange with the hydrogen ions of the carboxylic groups in the fulvic acids. It would appear that the activation energy for each carboxylate-
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Table 5. Correlation Parameters for Effective Diffusivity and Initial Dye Concentration system adsorbate
adsorbent
C0 (mg/dm3)
a
b
err (%)
Acid Blue 25
wood
8.70 × 10-2
-2.633
Basic Blue 69
wood
2.53 × 10-1
-2.306
Acid Blue 25
peat
3.75 × 10-2
-1.598
Basic Blue 69
peat
20 50 100 200 50 100 200 500 20 50 100 200 50 100 200 500
2.37 × 10-2
-1.586
99.6 96.3 82.8 0 64.0 49.7 0 98.6 0 89.4 98.9 98.8 70.9 41.6 0 37.2
Table 6. Prediction Results of the Pore-Surface Diffusion Model system adsorbate
C0 (mg/dm3)
adsorbent
Acid Blue 25
wood
Basic Blue 69
wood
Acid Blue 25
peat
Basic Blue 69
peat
20 50 100 200 50 100 200 500 20 50 100 200 50 100 200 500
f
DP (cm2/s) 10-8
Ds,0 exp(-E/RT) (cm2/s)
avg err (%)
2.0 × 10-14
1.52 1.04 1.64 0.71 12.49 11.08 4.08 0.46 21.98 18.30 7.54 3.32 28.23 19.44 14.63 11.47
0-1
5.78 ×
0.992
8.0 × 10-8
3.6 × 10-9
0.001
1.0 × 10-9
9.27 × 10-10
0.465
6.07 × 10-7
1.11 × 10-9
Table 7. Molecular Diffusivities and Tortuosity Factors for the Adsorption Systems adsordate
adsorbent
DL (×10-6 cm2/s)
µ (-)
temp (°C)
Acid Blue 25 Basic Blue 69 Acid Blue 25 Basic Blue 69
wood wood peat peat
6.7 8.2 6.7 8.2
116 103 6700 13.5
20 20 20 20
dye bond is the same and therefore Ds varies exactly as a function of surface coverage only as in eq 16. In the case of Acid Blue dye on peat, we observed a different trend from that for the previous system. The area coverage factor due to energy band levels is very low; hence, fθ ≈ 0, and therefore, Ds is constant. In this system, Ds is independent of surface coverage and the capacity of Acid Blue 25 for peat (47.8 mg of dye/g of peat) is significantly higher than that for wood (16.1 mg of dye /g of wood). Since the same dependence on surface coverage as that of Basic Blue 69 on wood attributed to ion exchange is not observed, it appears that a different mechanism is occurring. As Basic Blue 69/wood and Basic Blue 69/peat systems have higher sorption capacities, it seems likely that this is a low energy process. The complex chemistry of peat containing alcohols, aldehydes, phenolic hydroxides, ketones, carboxylic acids, ethers, and a rich array of ring compounds makes an exact prediction of the mechanism difficult.38,39 It would (38) Zhipei, Z.; Junlu, Y.; Zengnui, W.; Piya, C. A preliminary study of the removal of Pb2+, Cd2+, Zn2+, Ni2+ and Cr2+, from wastewaters with several Chinese peats. Proc. 7th Int. Peat Congr. (Dublin, Ireland) 1984, 3, 147. (39) Gosset, T.; Trancart, J. L.; Thevenot, D. R. Batch metal removal by peat kinetics and thermodynamics. Water Res. 1996, 20, 21.
Figure 10. Concentration decay curves of Basic Blue 69 on peat.
appear that the negatively charged Acid Blue dye ions are sorbed at positive sites which have been produced by some of the strongly electronegative phenolic hydroxyls in peat. The ability of peat to adsorb positively charged ions, particularly metal ions, has been known for many years. Much of the capacity of peat is provided by the formation of dye-carboxylate ring compounds whereby the dye displaces the acidic hydrogen ions. There are several carboxylic acid forms in peat, such as humic and fulvic. Each will have an associated energy barrier level; thus, we obtain an optimum fit energy level surface factor, f, of 0.465. Many other researchers have implicated carboxylic acid
Model for Batch Adsorption Processes
(COOH) groups in the reaction of divalent metals with humic acids.40-42 They support the general view that the reaction of metal ions, such as Cu and Fe, with humic acids is one of chelate ring formation involving adjacent aromatic carboxylate COOH and phenolic OH groups or, less predominantly, two adjacent COOH groups which participate in ion exchange reactions by binding metal ions with the release of H+ ions. In more recent work, Stevenson and Chen43 determined stability constants for copper(II)-humate complexes on the basis of binding at two sites. These authors proposed that binding of Cu(II) may occur through the formation of a coordinate link with a single carboxylate (COO-), a link between two carboxylates on the same molecule, a chelate ring structure with a COOH phenolic OH site, or 2:1 complexes with Cu(II) serving as a bridge between two macromolecules. Spark et al.44 proposed that copper(II) has a tendency to form cross-linked spherocolloids involving the carboxylate functional groups on the humic acid macromolecules. Furthermore, some of the colloid components and, hence, the sorbate-colloid complex are soluble. A more recent study by Francioso et al.,45 using IR, Raman, and NMR spectroscopy evidence, again supported a metal ion complex involving the carboxylic acid group. It appears that the presence of a greater number of carboxylate and phenolic groups can explain the high affinity of peat for positively charged dye ions. Other recent work, involving NMR and FT-IR structural studies, was performed by Averett et al.46 and Leenheer et al.47 on copper binding with carboxylate groups on humic and fulvic acid. These authors conclude that an innersphere complex of the metal binding fraction occurred based on chemisorption in which the copper ions were not irreversibly bound and an outer-sphere binding complex was also formed. Other mechanisms, such as chelation, exchange sorption, and polar organic bonding compete in the sorption process, since pH calculations on metal ion sorption systems48-51 fail to balance. Other mechanistic factors which should be studied in (40) Schnitzer, M. In Soil Organic Matter; Schnitzer, M., Khan, S. U., Eds.; Elsevier: New York, 1978; p 47. (41) Vinkler, B. L.; Meisel, J. Infrared spectroscopic investigations of humic substances and their metal complexes. Geoderma 1976, 15, 231. (42) Boyd, S. A.; Sommers, L. E.; Nelson, D. W. Copper(II) and iron(III) complexation by the carboxylate group of humic acid. J. Soil Sci. Soc. Am. 1981, 45, 1241. (43) Stevenson, F. J.; Chen, Y. Stability constants of Copper (II)humate complexes determined by modified potentiometric titration. J. Soil Sci. Soc. Am. 1991, 55, 1586. (44) Spark, K. M.; Wells, J. D.; Johnson, B. B. The interaction of a humic acid with heavy metals. Aust. J. Soil Res. 1997, 35, 89. (45) Francioso, O.; Sanchez-Cortes, S.; Tugnoli, V.; Ciavatta, C.; Sitti, L.; Gessa, C. Infrared, Raman and nuclear magnetic resonance (1H, 13C and 31P) spectroscopy in the study of fractions of peat humic acids. Appl. Spectrosc. 1996, 50, 1165. (46) Averett, R. C.; Leenheer, J. A.; McKnight, D. M.; Thron, K. A. Humic sunstances in the Sumanne River, Georgia: Interactions, Properties and properties and proposed structure; US Governemnt Printing Office: Washington, DC, 1994; p 224. (47) Leenheer, J. A.; Brown, G. K.; McCarthym, P.; Cabaniss, S. E. Models of metal binding structures in fulvic acid from the Suwannee River, Georgia. Environ. Sci. Technol. 1998, 32, 2410. (48) Shallcross, D. C.; Herrmann, C. C.; McCoy, B. J. An improved model for the prediction of multicomponent ion exchange equilibria. Chem. Eng. Sci. 1998, 43, 279. (49) Dzombak, D. A.; Morel, F. M. M. Surface Complexation Modelling; John Wiley and Sons: New York, 1990. (50) Ho, Y. S.; Wase, D. A. J.; Forster, C. F. The adsorption of divalent copper ions from aqueous solution by sphagnum moss peat. Trans. I. Chem. Part B: Proc. Safety Environ. Prot. 1994, 17, 185. (51) Nicolas-Simmonnot, J. O.; Fernandez, M. A.; Cheneviere, P.; Bailly, M.; Grevillot, G. Model for capacity variation of a weak-acid and weak-base ion exchangers as a function of the ionic environment. React. Polym. 1992, 17, 39.
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more detail are those based on molecular weight distribution, size, and functional group content of humic substances, and specific values for various peat-dye ion complexes.52-54 The peat in the present experimental studies was washed extensively, dried, milled, and sieved prior to use. Therefore, much of the soluble colloidal component of peat will have been removed. A tortuosity of 1 was used for the adsorption of phenol, p-nitrophenol, and p-chlorophenol15 because this represents the limiting value of tortuosity. The authors, however, did not consider this substitution to be appropriate for large organic molecules, and these authors used a tortuosity value of 3 for adsorption of dodecyl benzenesulfonate and Alizarin Blue dye onto activated carbon with a porosity of 0.79. A value of 4 was selected for the adsorption of benzene on carbon,29 and a value of 6,12 for the adsorption of Basic Blue 69 dye onto activated carbon. In the present work initially DL was used as a first estimate for Dp and µ was allowed to vary between 1 and 10, to optimize Dp and Ds,0 by minimizing the error function, ARE. However, this was completely unsatisfactory and the tortuosity values were eventually calculated directly from eq 24, using the best-fit values of Dp and calculated values of DL for the two dyes. The µ values shown in Table 7 are substantially higher than most literature values, which are less than 10. One high value of tortuosity was a value of 65 for activated carbon55 while another value for Darco carbon was 10.56 Consequently, the high values obtained in the present paper are unusual and represent the complexity, chemically and physically, of the structure of the wood and peat particles and also the complex nature of dyestuffs which are organic chemicals but ionic. Furthermore, dye molecules are prone to micelle formation57 and these agglomerates could be responsible for increasing the molecular volume of the dyes in the calculation of DL, thus leading to high tortuosity values. Conclusions A new pore-surface diffusion model that can be used to predict sorption concentration decay curves has been developed. The predicted results of four different adsorption systems 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 (52) Breuer, K.; Melzer, A. Heavy metal accumulation (lead and cadmium) and ion exchange in three species of Sphagnacese I. Main Principles of heavy metal accumulation in Sphagnacese. Oecologia 1990, 82, 461. (53) Breuer, K.; Melzer, A. Heavy metal accumulation (lead and cadmium) and ion exchange in three species of Sphagnacese II. Chemical Equilibrium of ion exchange and the selectivity of single ions. Oecologia 1990, 82, 468. (54) Bartschat, B. M. Oligoelectrolyte model for cation binding by humic substances. Environ. Sci. Technol. 1992, 26, 284. (55) Yang, R. T. Gas Separation by Adsorption Processes; Bulterworths: Boston, 1987. (56) Larson, A. C.; Tien, C. Multicomponent liquid Phase Adsorption in Batch. Part II: Experiment on Carbon Adsorption from Solutions of Phenol, o-Cresol and 2,4-Dichlorophenol. Chem. Eng. Commun. 1984, 27, 359. (57) Kirk, R. E.; Othmer, D. F.; Kroschwitz, J. I.; Howe-Grant, M. Encyclopedia of Chemical Technology, 4th ed.; Wiley: New York, 1991; Vol. 8, p 683.
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curves very accurately in the medium and low concentration regions that 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 an extremely steep initial isotherm slope. Acknowledgment. The authors acknowledge the support of the RGC, Hong Kong, for this research project. Notation a ) correlative coefficient in eq 23 aL ) Langmuir constant, dm3/mg b ) correlative coefficient in eq 23 cfs ) correlation coefficient C ) liquid-phase concentration, mg/dm3 C0 ) initial liquid concentration, mg/dm3 Ce,t ) equilibrium liquid-phase concentration at time t, mg/dm3 Ct ) liquid concentration at time t, mg/dm3 Cexp ) experimental value of liquid concentration, mg/ dm3 Ccalc ) prediction value of liquid concentration, mg/dm3 Deff ) effective diffusion coefficient, cm2/s DL ) liquid-phase diffusivity, cm2/s
Hui et al. Dp ) pore diffusivity, cm2/s Ds ) surface diffusivity, cm2/s Ds,0 ) surface diffusivity, cm2/s E ) potential energy, kJ f ) coefficient of area coverage f(r) ) function of r, mg/dm3 KL ) Langmuir constant, dm3/g Nt ) adsorption rate, mg/s n ) number of experimental data points r ) particle radius, cm R ) particle diameter, cm; gas constant, 8.31 kJ/mol‚K ARE ) average relative error, % SES ) sum of error square T ) temperature, K t ) time, s V ) liquid volume, dm3 W ) adsorbent weight, g Ye,t ) equilibrium solid-phase concentration at time t, mg/g Ye,0 ) initial equilibrium solid-phase concentration, mg/g Ymax ) maximum equilibrium solid-phase concentration, mg/g Greek Symbols Fs ) adsorbent particle density, g/cm3 µ ) tortuosity factor, or chemical potential θ ) surface coverage LA026624V
