Binary Metal Sorption on Bone Char Mass Transport Model Using

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Binary Metal Sorption on Bone Char Mass Transport Model Using IAST Chun Wai Cheung, Danny C. K. Ko, John F. Porter, and Gordon McKay* Department of Chemical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR, China Received August 14, 2002. In Final Form: January 10, 2003 The binary sorption of Cu-Cd, Cd-Zn, and Cu-Zn onto bone char has been studied using an equilibrium and batch agitation system. The sorption capacities and the selectivity of metal ions onto bone char follows the order Cu2+ > Cd2+ > Zn2+, which is in the reverse order of the hydrated ionic radii. The binary sorption equilibria were predicted by the ideal adsorbed solution theory (IAST) on the basis of single component isotherm data using a Langmuir or Langmuir-Freundlich isotherm. The overall performance of the IAST provides a reasonable curve fitting to the experimental data. The single component film-pore diffusion model was extended to the multicomponent systems to correlate the batch kinetic data by incorporating the shrinking core model and the IAST. All the diffusivities in the binary systems are similar to or less than the pore diffusivities in single component systems.

1. Introduction Due to their extensive use in industry, metals such as cadmium, copper, and zinc are of great environmental concern. In recent years several new economically priced adsorbents have been tested to access their ability to remove these metal ions. The sorption of cadmium has been tested using agricultural waste,1 waste iron (III)/ chromium(III) hydroxide,2 carbon,3 peat,4,5 and biomass, plants, and algae.6,7 A number of sorbents have been used to remove copper from wastewaters including carbon,8,9 chitosan,10 flyash,11 goethite,12 and peat.13,14 A limited number of studies have been carried out on zinc and include use of carbon,3 sludge,15 and peat.4 In this research an adsorption model has been developed for the multicomponent batch sorption of cadmium, copper, and zinc in three binary component mixtures. A filmpore diffusion model has been developed, incorporating the ideal adsorbed solution theory (IAST) to predict equilibrium surface solute concentrations in multicomponent systems. 2. Theory Adsorption Equilibrium Models. In the study of adsorption equilibrium, the experimental data for single * Author to whom all correspondence should be addressed. E-mail: [email protected]. (1) Periasamy, K.; Namasivayam, C. Ind. Eng. Chem. Res. 1994, 61, 57. (2) Namasivayam, C.; Ranganathan, K. Water Res. 1995, 29, 1737. (3) Gabaldon, C.; Marzal, P.; Ferrer, J.; Seco, A. Water Res. 1996, 30, 3050. (4) McKay, G.; Porter, J. F. J. Chem. Technol. Biotechnol. 1997, 69, 309. (5) McKay, G.; Porter, J. F. Trans. Inst. Chem. Eng. 1997, 75, 171. (6) Trujillo, E. M.; Teffers, T. H.; Feruson, C.; Stevenson, Q. Environ. Sci. Technol. 1991, 25, 1559. (7) Yang, J.; Volesky, B. Environ. Sci. Technol. 1999, 33, 751. (8) Netzer, A.; Hughes, D. E. Water Res. 1984, 18, 927. (9) Namasivayam, C.; Kadirvelu, K. Chemosphere 1997, 34, 377. (10) Findon, A.; McKay, G.; Blair, H. S. J. Environ. Sci. Health, Part A 1993, 28, 173. (11) Panday, K. K.; Prasad, G.; Singh, V. N. Water Res. 1985, 19, 869. (12) Grossl, P. R.; Sparks, D. L.; Alnsworth, C. C. Environ. Sci. Technol. 1994, 28, 1422. (13) Chen, X. H.; Gossett, T.; Thevenot, D. R. Water Res. 1990, 24, 1463. (14) Allen, S. J.; Murray, S.; Brown, P.; Flynn, O. Resour., Conserv. Recycl. 1994, 11, 25. (15) Gao, S.; Walker, W. J.; Dahlgre, R. A.; Bold, J. Water, Air, Soil, Pollut. 1997, 93, 331.

component systems will be correlated by equilibrium equations. The best-fit isotherm equation for single component sorption will be determined and incorporated in the ideal adsorbed solution theory to predict the sorption capacities for multicomponent systems. Adsorption Isotherms for Single Component Systems. In the literature survey, only Gu and co-workers16 used the Langmuir equation to analyze the sorption of uranyl ions on bone char and iron-impregnated bone char from contaminated groundwater. Although references for the sorption of metal ions onto bone char are limited, the sorption of metal ions on CaHAP can be utilized as a reference base to study the adsorption equilibria. Fuierer and co-workers17 used the Langmuir equation to correlate the sorption of zinc and magnesium ions onto CaHAP. Xu and co-workers18 suggested the Langmuir equation to interpret the equilibrium data for the sorption of cadmium and zinc ions onto CaHAP. Kanzaki and co-workers19 used the Langmuir and Temkin equations to study the inhibitory effect of magnesium and zinc on the crystallization kinetics of CaHAP. All authors using the Langmuir equation know that the sorption of metal ions on bone char does not obey the fundamental assumptions of the Langmuir equation. However, the mathematical fit of the experimental data is one of the reasons for using the Langmuir equation. The other reason to use the Langmuir equation is that the Langmuir equation can be readily incorporated into the kinetic equation. Hence, the Langmuir equation applied in these sorption systems should be termed a “Langmuir-type” equation. Ideal Adsorbed Solution Theory (IAST). This method has been applied to predicting the multisolute adsorption capacity using only data for single solute adsorption from dilute liquid solution. The original method was proposed by Myers and Prausnitz20 for gas-solid systems. Later, Radke and Prausnitz21 modified the (16) Gu, B.; Liang, L.; Dickey, M. J.; Yin, X.; Dai, S. Environ. Sci. Technol. 1998, 32 (21), 3366. (17) Fuierer, T. A.; LoRe, M.; Puckett, S. A.; Nancollas, G. H. Langmuir 1994, 10, 4721. (18) Xu, H.; Schwartz, F.; Traina, S. J. Environ. Sci. Technol. 1994, 28, 1472. (19) Kanzaki, N.; Onuma, K.; Treboux, G.; Tsutsumi, S.; Ito, A. J. Phys. Chem. B 2000, 104 (17), 4189. (20) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121. (21) Radke, C. J.; Prausnitz, J. M. AIChE J. 1972, 18 (4), 761.

10.1021/la020719f CCC: $25.00 © 2003 American Chemical Society Published on Web 04/09/2003

Binary Metal Sorption on Bone Char Mass Transport

method for use in liquid-solid adsorption systems. In the IAST, the equilibrium equation for single components can take any form, which can fit the experimental data best. The Langmuir, Freundlich, and Langmuir-Freundlich isotherms are commonly used in the IAST, since these equations can be solved by analytical methods to reduce the complexity of the computer program. The success of the calculation of IAST depends on how well the single component data are fitted, especially in the low concentration region. The IAST has been used for many sorption systems. For instance, McKay and Al-Duri22 used the IAST theory by a graphical method to predict the sorption of dyes onto activated carbon. The fitting of the IAST using the Freundlich isotherm could predict better fitting than the Langmuir isotherm. Sheintuch and Rebhun23 used the IAST for the sorption of unknown compounds. The equilibrium data of single components could be correlated by the Langmuir and Freundlich equations to predict the sorption capacity of the multicomponent system. Siedel and Gelbin24 used the IAST to predict the sorption of p-nitrophenol and p-chloranilin on activated carbon. The authors proposed that the original IAST could not predict the binary systems accurately. After introducing a parameter to account for extrapolation errors in the single solution isotherm at very low concentrations, the IAST could predict the binary data very well. As the fitting of single component isotherm data can directly affect the prediction of multicomponent equilibrium, Porter and coworkers25 used five different error methods to reduce the fitting deviation of single components for dye adsorption on carbon. The authors concluded that the “best-fit” single component isotherm did not necessarily result in the “bestfit” IAST model predictions. Reasons for the apparent behavior were attributed to the liquid-phase interaction. Walker and Weatherley26 used the IAST in the sorption of acid dyes on activated carbon. The authors accurately simulated the experimental data with an average deviation of approximately 3% between modeled and experimental data. The above authors used the IAST in the sorption of organic compounds onto various sorbents. This is because the adsorbed solution theory is an “ideal” state. Organic compounds in dilute aqueous solution often do not have large interaction between the sorbate-sorbate and the sorbate-sorbent. However, the interaction between metal ions may be large. Therefore, literature studies on the sorption of metal ions using IAST are rarely found. Recently, Al-Asheh and co-workers27 compared several correlations, namely, extended Langmuir, extended Freundlich, and extended Langmuir-Freundlich equations, with the IAST for the sorption of metal ions onto pine bark. The fitting of the extended Langmuir-Freundlich equation was the worst, but the other methods including the IAST showed reasonably good agreement with the experimental data. Therefore, the IAST can be used to predict the sorption equilibria of multicomponent metal ion systems under certain conditions. Solution Methodology. This IAST was proposed for dilute liquid adsorption systems by Radke and Prausnitz.21 The major advantages of the IAST are that it has a sound (22) McKay, G.; Al-Duri, B. Chem. Eng. Sci. 1988, 43 (5), 1133. (23) Sheintuch, M.; Rebhun, M. Water Res. 1988, 22 (4), 421. (24) Seidel, A.; Gelbin, D. Chem. Eng. Sci. 1988, 43 (1), 79. (25) Porter, J. F.; McKay, G.; Choy, K. H. Chem. Eng. Sci. 1999, 54, 5863. (26) Walker, G. M.; Weatherley, L. R. Trans. IChemE, Part B 2000, 78, 219. (27) Al-Asheh, S.; Banat, F.; Al-Omari, R.; Duvnjak, Z. Chemosphere 2000, 41, 659.

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thermodynamic basis and requires only single component isotherm parameters and system operating conditions to predict the sorption equilibrium for the multicomponent system. The IAST is based on the assumption that the adsorbed phase can be treated as an ideal solution of the adsorbed components. The IAST uses several equations to calculate the mole fraction of adsorbed-phase concentrations. The uses of these equations are shown and discussed. From the mass balance for each nonvolatile component i, the sum of the material in the solid phase and the liquid phase at equilibrium must be equal to its initial solutionphase concentration,

C0,i ) Ce,i +

W q V e,i

(1)

From the overall mass balance, the sum of the moles in each phase must be exactly equal to unity; that is, for the solid phase n

si ) 1 ∑ i)1

(2)

The IAST dictates that the spreading pressure should be constant for each component in a given system:

π1 ) π2 ) π3 ... ) πn

(3)

The best-fit isotherms for the single component systems are used to calculate the spreading pressure; for instance, the Langmuir and Langmuir-Freundlich isotherms may be used. The spreading pressure, π, and reduced spreading pressure, Π, can be evaluated from eqs 5 or 7:

For the Langmuir isotherm: q°e,i ) Πi )

πiSA ) RgT

qm,iaL,iC°e,i 1 + aL,iC°e,i

q° (C° )

∫0C° e,iC°e,ie,i e

(4)

dC°e,i ) qm,i ln(1 + aL,iC°e,i) (5)

For Langmuir-Freundlich isotherm: q°e,i ) Πi )

πiSA ) RgT

∫0C°

qm,i(aLF,iC°e,i)ni

(6)

1 + (aLF,iC°e,i)ni

q°e,i(C°e,i) dC°e,i ) C°e,i qm,i ln[1 + (aLF,iC°e,i)ni] (7) ni

e,i

( )

Hence, transforming the reduced spreading pressure into those based on single component data gives the following for one component of a multicomponent system:

[

C′0,i ) C°e,i +

( )]

W V

n

sj

∑ j)1 q°

si

(8)

e,i

Since

Ce,i ) C°e,isi

(9)

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( ) n

qe,i ) siqT ) si

sj

∑ i)1q°

Cheung et al.

-1

(10)

e,j

For the Langmuir isotherm, rearranging eqs 4 and 5 allows us to substitute into eq 8 to obtain the equivalent pure component fluid and adsorbed-phase concentrations at the specified spreading pressure:

C°e,i )

exp(Π/qm,i) - 1 aL,i

(11)

Using the Newton-Raphson method will identify values of Π and si that minimize the error between the experimental initial liquid-phase solute concentrations, which are predicted by the IAST. Intraparticle Diffusion Mechanism. The sorption rate for metal ions onto bone char may be controlled by an ion exchange process or a mass transport process. In early studies, the ion exchange process was considered essentially as a chemical reaction best described in terms of rate coefficients and kinetic orders corresponding to the stoichiometry of exchange.28 Boyd and co-workers29 studied the sorption rate of metal ions on ion exchange zeolites and found out that the rate of ion exchange increases with decreasing particle size of the ion exchanger. It is well-known that the liquid film mass transfer limitation is associated with particle size in ion exchange systems; and in the case of ion exchange, ion transport and potential concentration gradients with the exchange solid play an important role in determining kinetics. This leads to the inescapable conclusion that mass transport rather than an actual reaction will be the rate-controlling mechanism. The authors identified the potentially ratecontrolling steps as diffusion in the liquid to the particle surface, diffusion within the particle, and the actual exchange reaction at a fixed site, the slowest of these steps being the bottleneck and therefore limiting the overall rate. These conclusions indicate that ion exchange may also be a mass transport controlled process. When the sorbate is transferred to the sorbent surface, the sorbate will be transferred along the pore to the sorption sites inside the sorbent. There are three possible mass transfer mechanisms to transport the sorbate from the outer surface of the sorbent to its inner surface. Pore diffusion is one mechanism by which adsorbates move within the pores of the adsorbent before being adsorbed onto the surfaces of the pores. The pore diffusivity is basically proportional to the porosity, tortuosity factor, and molecular diffusivity. As a result, the value of pore diffusivity can be predicted from the molecular diffusivity of the adsorbate and the properties of the adsorbent. The dependency of pore diffusivity on temperature can be predicted using a theory similar to that for molecular diffusivity.30,31 It should be noted that the pore diffusivity is independent of adsorbate concentration. Surface diffusion occurs along the pore surface of the adsorbent after initial adsorption has taken place. The driving force for the surface diffusion is the difference in the amount adsorbed on the pore surface (i.e., solid-phase

concentration gradient).32 While pore diffusion is always operating, the importance of surface diffusion is dependent upon the sorptive affinity between the adsorbate and the adsorbent. Experimental results have shown that the value of surface diffusivity may be a function of the solidphase concentration and temperature. At the present time, the numerical value cannot be predicted from thermodynamic theories and must be determined experimentally. Several empirical correlations are available for the calculation of surface diffusivity. These empirical formulas have limited applications and apply only under the experimental conditions used to derive them. The relative importance of pore and surface diffusion in adsorption mass transport is dependent upon the adsorbate-adsorbent system used and must be evaluated on a case-by-case basis.32 Intraparticle diffusion is normally characterized by a lumped parameter, the effective pore diffusivity (if pore diffusion is considered as a primary diffusion mechanism and the surface diffusion is lumped into the pore diffusion coefficient) or the effective surface diffusivity (if surface diffusion is considered as a primary diffusion mechanism and pore diffusion is lumped into the surface diffusion coefficient). For low-affinity solutes in macroporous sorbents, pore diffusion is usually the dominant intraparticle diffusion mechanism.32 For highaffinity solutes, the surface diffusivity can be higher than the pore diffusivity by orders of magnitude.33 But the effective diffusivity should not be larger than the molecular diffusivity, which is physically impossible.34 IAST for Solid-Phase Concentration in Mass Transport Modeling. In the previous section, the IAST was proposed for use in the prediction of solid-phase concentrations at equilibrium. However, the metal ions in the liquid-filled pores may diffuse at different speeds. For the binary kinetic system, there are two reaction fronts moving toward the center of the particle. The solid-phase concentration of metal ions M1 and M2 at radius R - r2 can be calculated by the IAST. The solid-phase concentration for the faster moving fast metal ions, M1, at radius r2 - r1 (as shown in Figure 1) should be calculated by the single component equation. For the metal ions, M1 and M2, the film and pore diffusion relationships can be derived

(28) Liberti, L.; Helfferich, F. G. Mass transfer and kinetics of ion exchange; Kluwer: Boston, 1982. (29) Boyd, G. E.; Adamson, A. W.; Myers, L. S. J. Am. Chem. Soc. 1947, 69, 2836. (30) Lide, D. R.; Frederikse, H. P. R. CRC handbook of chemistry and physics, 74th ed.; CRC Press: London, 1994. (31) Wilke, C. R.; Chang, P. AIChE J. 1955, 1 (2), 264.

(32) Furuya, E. G.; Chang, H. T.; Miura, Y.; Yokomura, H.; Tajima, S.; Yamashita, S.; Noll, K. E. J. Environ. Eng. Div. (Am. Soc. Civ. Eng.) 1996, 122 (10), 909. (33) Ma, Z.; Whitley, R. D.; Wang, N. H. L. AIChE J. 1996, 42 (5), 1244. (34) Robinson, S. M.; Arnold, W. D.; Byers, C. H. AIChE J. 1994, 40 (12), 2045.

Figure 1. Mass transport of metal ions M1 and M2 at different speeds in the sorbent.


from eq 12:

Langmuir, Vol. 19, No. 10, 2003 4147 Table 1. Physical and Chemical Properties of Bone Char (Source: Tate & Lyle Company Limited)

(Ct)i ) ψi(Ce,t)i

(12)

where i is the metal ion Mi. For the M1 ions, the diffusion in the pore water according to the first Fick’s law is

Nt,1 )

4πDp,1(Ce,t)1 1/r1 - 1/R

(13)

The liquid-phase concentration at radius R - r1 is equivalent to (Ce,t)1. The velocity of the concentration front is obtained from the mass balance on a spherical element:35

dr1 Nt,1 ) -4πr12Fp(qe,t)°1 dt

(14)

As M1 is moving faster than M2 in the sorbent, the solidphase concentration (qe,t)°1 from r2-r1 can be calculated by the Langmuir equation. Therefore, the shrinking core radius r1 can be obtained by solving eqs 13 and 14.

Dp,1(Ce,t)1R dr1 )dt r1Fp(R - r1)(qe,t)°1

(15)

For the M2 ions, the diffusion in the pore water according to the first Fick’s law is35

Nt,2 )

4πDp,2(Ce,t)2 1/r2 - 1/R

(16)

The velocity of the concentration front is obtained from the mass balance on a spherical element:35

dr2 Nt,2 ) -4πr2 Fp(qe,t)2 dt

(17)

The shrinking core radius r2 can be obtained by solving eqs 16 and 17.

Dp,2(Ce,t)2R dr2 )dt r2Fp(R - r2)(qe,t)2

(18)

item

limit

item

limit

acid insoluble ash calcium carbonate calcium sulfate carbon content calcium HAP iron as Fe2O3

3 wt % max 7-9 wt % 0.1-0.2 wt % 9-11 wt % 70-76 wt % Cd2+ g Cu2+. The result is not (40) Chen, X.; Wright, J. V.; Conca, J. L.; Peurrung, L. M. Environ. Sci. Technol. 1997, 31, 624. (41) Lee, D. H.; Moon, H. Korean J. Chem. Eng. 2001, 18 (2), 247.

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Figure 6. Comparison of calcium exchange capacity and total sorption capacity of Cu ions onto bone char.

Figure 7. Experimental data and predictive curves for the sorption of Cd-Cu ions at a Cd/Cu ) 7:3 mole ratio.

Table 3. Selected Properties of the Metal Ions30,42 metal ion

ionic radius (Å)

hydrated ionic radius (Å)

Pauling’s electronegativity

Cd2+ Cu2+ Zn2+

0.95 0.73 0.74

4.26 4.19 4.30

1.69 1.90 1.65

matched with ionic radii but coincides reasonably well with the reversed order of hydrated ionic radii as Cs+ (3.30 Å) > Pb2+ (4.01 Å) > Cu2+ (4.19 Å) > Cd2+ (4.26 Å). As zeolite removes the metal ions from solution by both ion adsorption and ion exchange effects, the selectivity properties should be similar to those of CaHAP. The selectivity of metal ions onto bone char follows the order Cu2+ > Cd2+ > Zn2+. The reversed order of hydrated ionic radii is Cu2+ (4.19 Å) > Cd2+ (4.26 Å) > Zn2+ (4.30 Å) (Table 3).42 This result agrees with that of Lee and Moon.41 In the study on the selectivity of zirconium phosphate for univalent cations, it is found that the selectivity also follows a pattern similar to that in zeolite, that is, in a reversed order of hydrated ionic radii, viz. Cs+ (3.29 Å) > Rb+ (3.29 Å) > K+ (3.31 Å) > Na+ (3.58 Å) > Li+ (3.82 Å).43 A more detailed explanation has been given by Hillel44 regarding the ion exchange properties of clay materials. The smaller the ionic radius and the greater the valence, the more closely and strongly is the ion adsorbed. On the other hand, the greater the ion’s hydration, the farther it is from the adsorbing surface and the weaker its adsorption. From Figures 3 and 4, the predictions for the sorption of Cd-Cu and Cu-Zn ions are quite reasonable. Using the Langmuir and the Langmuir-Freundlich isotherms in the IAST predicts theoretical isotherm data which are also close to the experimental points. When one of the Langmuir-Freundlich curves is close to the experimental points, the other Langmuir-Freundlich curve deviates more from the experimental points. A similar effect can be found in the Langmuir curves. However, Figure 5 shows that the IAST does not accurately predict the experimental data; both curves are below the experimental data points. A similar effect can be found in Cd-Cu in 3:7 or 7:3 ratio systems. The sorption capacities for these binary systems may be enhanced. The interaction between metal ions and sorbent may be the main factor affecting the quality (42) Nightingale, E. R. J. Phys. Chem. 1959, 63, 1381. (43) Amphlett, C. B. Inorganic ion exchangers; Elsevier: Amsterdam, 1964. (44) Hillel, D. Environmental soil physics; Academic Press: San Diego, CA, 1998.



of prediction. In addition, the results for unequal mole ratio systems are shown in Figures 7-10. The prediction for the Cd-Cu in the 4:6 or 6:4 ratio systems appears better than that for the Cd-Cu in the 3:7 or 7:3 ratio systems. As illustrated in Figures 7-10, the sorption isotherms of copper ions in the multicomponent Langmuir isotherm appear to produce an increasing trend even in low mole ratios. The overall performance of the IAST using the Langmuir and the Langmuir-Freundlich equations can provide a


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Figure 12. Mass effect for the sorption of Cd ions from Cd-Cu solution onto bone char using the IAST-film-pore diffusion equation.

Figure 11. Concentration effect for the sorption of Cd ions from Cd-Cu solution onto bone char using the IAST-filmpore diffusion equation.

Figure 13. Concentration effect for the sorption of Cu ions from Cd-Cu solution onto bone char using the IAST-filmpore diffusion equation.

reasonable curve fitting to the experimental data. However, the prediction deviates from the experimental data in the nonequimolar ratio systems except Cu-Cd in 6:4 or 4:6 ratio systems. Introducing an interaction factor into the IAST may assist the accuracy of the correlation of experimental data. Multicomponent Mass Transport Modeling. In binary mixtures, there are two reaction fronts moving inside the particle at different speeds, namely, the sorbent’s two species (binary system) in the outer layer; a single species in the middle layer; and finally the unreacted core of sorbent in the inner layer (see Figure 1). As the middle layer of sorbent only contains a single species, the sorption capacity can be predicted from the single component isotherm. The IAST is used to predict the solid-phase concentration of sorbent at specified radii in the outer layer because multicomponent sorption is taking place in this layer. Figure 11 shows the correlation of kinetic data using the IAST-film-pore diffusion model. The model can correlate the concentration effect for the sorption of cadmium ions from the Cd-Cu solution very well. However, Figure 12 shows that the model prediction deviates from the experimental data points for the effect of sorbent mass. The concentration and the sorbent mass effect for the sorption of copper ions from the Cd-Cu solution are shown in Figures 13 and 14.

Figure 14. Mass effect for the sorption of Cu ions from Cd-Cu solution onto bone char using the IAST-film-pore diffusion equation.

Figure 15 shows the sorption of cadmium ions from the Cd-Zn solution at different initial concentrations. The concentrations at 2.7, 3.1, and 4.1 mM can be correlated by the film-pore diffusion model. In high and low concentrations (i.e. C0 ) 2.1 and 4.8 mM), the concentration-decay curves show deviation from the experimental data points. The film-pore diffusion model can also

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Figure 15. Concentration effect for the sorption of Cd ions from Cd-Zn solution onto bone char using the IAST-filmpore diffusion equation.

Figure 16. Mass effect for the sorption of Cd ions from Cd-Zn solution onto bone char using the IAST-film-pore diffusion equation.

correlate the sorbent mass effect in Figure 16. For the sorption of zinc ions from the Cd-Zn solution, the concentration and the sorbent mass effects can also be correlated by the film-pore diffusion model. The pore diffusivities for all three binary sorption systems are summarized in Table 4. All diffusivities in binary systems are similar to or less than the pore diffusivities in single component systems. As the pore diffusivity for cadmium ions in the Cd-Zn system is lower than that of zinc ions, in the numerical calculation, the zinc ions are moving faster than the cadmium ions inside the sorbent. This implies that only zinc ions exist in the middle layer of sorbent. In the outer layer of sorbent, the cadmium and zinc ions coexist in this region. A similar result can be found in the Cu-Zn system. However, in the Cd-Cu system, the shrinking core radius for cadmium ions is moving faster than the copper ions in the numerical solution method. This is because the pore diffusivities for cadmium and copper ions in Cd-Cu are very close. In addition, the sorbent surface is more favorable to the

adsorption of copper ions rather than cadmium ions. Therefore, the unadsorbed cadmium ions can keep moving in the sorbent. A similar explanation was given by Mijangos and Diaz,36 who used ion exchange resins to remove the copper and cobalt ions from aqueous solution. Are´valo et al.35 used ion exchange resin (i.e. resin-Na) to remove cobalt and copper ions from aqueous solution. A shrinking core model was used to correlate the kinetic data for the Co-Cu system. In the calculation of the solidphase concentration, the authors used a mass action lawbased equation to be incorporated into the pore diffusion model. As the copper ions displaced the cobalt ions in the outer layer of resin, the resin was preloaded with the cobalt ions and then the ion exchange experiments with copper ions were performed to calculate the equilibrium constants for the Co-Cu system. In comparison with the Are´valo model, the IAST-film-pore diffusion model only requires the single component isotherm parameters. Therefore, the present IAST-film-pore diffusion model makes it easier to calculate the solid-phase concentration in the binary component layer and is more fundamentally correct than the Are´valo model. The pore diffusivities obtained by Are´valo and co-workers are 1.15 × 10-10 m2/s for copper in a single component solution and 8.2 × 10-11 m2/s for copper in the Co-Cu solution.35 The pore diffusivity for copper ion in the binary component is lower than the value in the single component system. Referring to Table 4, similar phenomena can be found in the Cu-Zn and the Cd-Zn systems. However, the pore diffusivity for cadmium ions is slightly increased in the Cd-Cu system. The increase in pore diffusivity may be due to the fact that the correlation of the binary component system data is poorer than that for the single component systems and the single component equilibrium isotherms in the IAST do not predict the multicomponent equilibrium data exactly. Another consideration is the interference from the counterdiffusion of ions, since the calcium ion is exchanged during the metal ion sorption process within the bone char. The counterdiffusion of Ca has not been taken into account in the current model and could be responsible for a variable “effective” diffusivity; it will be incorporated into the next phase of the diffusion model development. 5. Conclusions The Langmuir and the Langmuir-Freundlich equations were used in IAST and modeling to correlate the experimental data in multicomponent systems. The results show that the sorption of metal ions for multicomponent systems can be predicted reasonably well from the IAST with the Langmuir equation or the Langmuir-Freundlich equation for metal ions. Although the equal molar and unequal molar equilibrium data can generally be predicted by the IAST, the results for the mole ratios 3:7 and 7:3 for the Cd-Cu systems cannot be predicted very well. A possible reason may be interaction between the metal ions affecting the accuracy of calculation without incorporating activity coefficients into the model. A new solution method for the single component filmpore diffusion model was extended to enable the pore diffusivities to be determined for multicomponent systems.

Table 4. Pore Diffusivities and the SSEs of Multicomponent Systems Using the Film-Pore Diffusion Model and IAST Cd-Cu metal ion Cd Cu Zn

single component Dp 1.14 × 10-6 1.59 × 10-6 1.21 × 10-6

(cm2/s)

Dp

(cm2/s)

1.20 × 10-6 1.30 × 10-6

Cd-Zn SSE 5.97 8.00

Dp

(cm2/s)

Cu-Zn SSE

2.25 × 10-7

2.59

1.10 × 10-6

1.73

Dp

(cm2/s)

1.50× 10-6 1.00 × 10-6

SSE 8.87 7.11


The pore diffusivities for multicomponent systems were calculated by incorporating the ideal adsorbed solution theory into the film-pore diffusion model. In a multicomponent metal ion sorption model by Are´valo,35 the diffusion model required the equilibrium constant to calculate the solid-phase capacities for two moving concentration fronts inside the resin. The Are´valo model was restricted by the need to preload one metal ion and then exchange the other metal ion to determine the equilibrium constant for the binary system. Therefore, extra experimental work is required for that model, which has been overcome in this research using a model based on the IAST and the new solution to the film-pore diffusion model. Nomenclature aL ) Langmuir isotherm constant (dm3/mg) aL,i ) binary Langmuir isotherm constant (dm3/mg) aLF ) Langmuir-Freundlich isotherm constant (dm3/mg) C0 ) initial liquid-phase concentration (mg/dm3) Ce ) equilibrium liquid-phase concentration (mg/dm3) Ce,t ) tie line liquid-phase concentration at particle surface at time t (mg/dm3) Ct ) liquid-phase concentration at time t (mg/dm3) Dp ) pore diffusion coefficient (cm2/s) Di ) vessel inside diameter (cm)

Langmuir, Vol. 19, No. 10, 2003 4153 KL ) Langmuir constant (dm3/g) n ) Langmuir-Freundlich isotherm constant Nt ) mass transfer flux (mg/s‚cm2) qe ) equilibrium solid-phase concentration (mg/g) qe,t ) tie line solid-phase concentration at time t (mg/g) qm ) monolayer capacity of Langmuir equation (mg/g) qt ) mean equilibrium solid-phase concentration at time t (mg/g) qT ) total adsorbed-phase concentration (mg/g) r ) radius of concentration front of metal ions penetrating adsorbent (cm) R ) radius of adsorbent particle (cm) Rg ) gas constant (kJ/mol‚K) s ) mole fraction of sorbate in adsorbed phase SA ) surface area of sorbent (m2/g) t ) contact time (h) V ) liquid-phase volume (dm3) W ) weight of sorbent (g) Fp ) particle density (g/cm3) π ) spreading pressure (mg/g) ψ ) relationship between film and intraparticle diffusion, ψ ) 1 + Deffr/(kfR(R - R))

Acknowledgment. The authors are grateful to DAG and RGC, Hong Kong SAR, for the provision of financial support during this research program. LA020719F

Binary Metal Sorption on Bone Char Mass Transport Model Using

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