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Film and intraparticle mass transfer during the adsorption of metal ions onto bone char
Journal of Colloid and Interface Science 271 (2004) 284–295 www.elsevier.com/locate/jcis
Film and intraparticle mass transfer during the adsorption of metal ions onto bone char Keith K.H. Choy, Danny C.K. Ko, Chun W. Cheung, 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 27 February 2003; accepted 5 December 2003
Abstract The sorption of three metal ions, namely, copper, cadmium, and zinc, onto bone char has been studied in terms of equilibrium and rate studies. Equilibrium studies have been analyzed using the Langmuir isotherm equation and the maximum sorption capacities for the metals were 0.477, 0.709, and 0.505 mmol g−1 bone char for cadmium, copper, and zinc ions, respectively. The kinetic experimental data were used to analyze the effect of external film boundary layer and intraparticle mass transfer resistance on the sorption process and its significance. Four methods of determining the external film transport coefficient were developed and tested; three utilized experimental data to obtain the coefficient and the fourth method was completely empirical. The three experimentally based models give very similar results and consequently similar values of the deviation error values, whereas the error values for the empirical correlation were greater than these three values. The results also demonstrated that the methods for determining the film coefficient could be integrated into more complex diffusion-transport models such as film–intraparticle diffusion processes. 2004 Elsevier Inc. All rights reserved. Keywords: Bone char; Copper; Cadmium; Zinc; Sorption; External mass transfer; Intraparticle diffusion
1. Introduction Bone char has been used extensively as an adsorbent for the decolorization of cane sugar [1] and to a lesser extent for the defluoridation of drinking water. Results for color removal [2] and removal of metal ions from waste waters [3] have also been reported in recent research publications. Bone char is still used extensively in sugar refining as a decolorizing adsorbent. After use it is regenerated by washing and calcining so that the char can pass through many operating cycles before its activity has decreased to an unacceptably low level. Abdel Kader and co-workers [4] reported that the decolorization process removes more than color because colorants interact with color precursors, colloidal material, organic nonsugars, and ash-forming inorganic constituents so that they are taken out with the color. Thus, the value of bone char as a sugar refining aid lies in its simple action in removing both colorants and ash-forming inorganic constituents. * Corresponding author.
E-mail address: [email protected] (G. McKay). 0021-9797/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2003.12.015
Recently, studies into the uses of bone char have increased, though its main use remains as an adsorbent. Shen et al. [2] combined the use of ferrous ions, lime, and bone charcoal to remove the color from the effluent of a local chemical manufacturer. The effluent contained high levels of soluble organic compounds. The intense color of the effluent made common coagulation or adsorption techniques unsuitable for color removal. However, the report showed that a combination of FeCl2 ·4H2 O (as a reducing agent), lime, and bone charcoal removed up to 97% of the color. Lewis [5] used bone char in the treatment of rural water supplies. Bone char was used as a filter and its performance compared with that of a sand filter. The results showed that the pH, color, turbidity, and metal ions in treated water were improved. Abdel Raouf and Daifullah [3] used bone charcoal in the removal of antimony and europium radioisotopes from radioactive wastes. The authors showed that bone char was a good sorbent for 123 SbIII and 123SbV from liquid organic radioactive wastes and for the retention of 152 EuIII from aqueous wastes. Physical sorption was suggested to be the main mechanism operating in the sorption of antimony radioisotopes with appreciably higher fixation on the sorbent when loaded from the liquid organic radioactive waste. In
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contrast, chemisorption was postulated as the main operating mechanism for 152EuIII with a higher and irreversible fixation on bone charcoal from a neutral aqueous medium. Laul et al. [6] used bone charcoal and other sorbents to remove uranium from contaminated groundwater. Due to its high toxicity several researchers have studied the sorption of cadmium onto various materials [7,8]. In adsorption systems, the mass transfer of solute or sorbate onto and within the sorbent particle directly affects the adsorption rate. It is important to study the rate at which the solute is removed from aqueous solution in order to apply adsorption by solid particles to industrial uses. There are essentially four stages in the adsorption process: 1. Transport of sorbate from the bulk of the solution to the exterior film surrounding of the adsorbent. 2. Movement of sorbate across the external liquid film boundary layer to external surface sites. 3. Migration of sorbate within the pores of the sorbent by intraparticle diffusion. 4. Sorption of sorbate at internal surface sites. All these processes may be involved in the control of the sorbate removal rate. However, in a fully mixed agitated tank, mass transport from the bulk solution to the external surface is usually fast. The transport of sorbate from the bulk of the solution to the exterior film surrounding the adsorbent is usually neglected. In addition, the adsorption of sorbate at surface sites (step 4) is usually rapid. Thus, these processes usually are not considered to be the rate-limiting steps in the sorption process. Some metal sorption systems have been analyzed using complexation models [9,10], but in the present system there is strong evidence to support the conclusion that ion exchange is a main mechanism contributing to the sorption process. In fact, the ion exchange of metal ions onto ion exchangers is also considered to be controlled by diffusion [11]. In most cases, steps (2) and (3) may control the sorption mechanisms, because the mass transport (external and internal) of adsorbate to adsorbent is a relatively slow process. In the present paper, a quantitative study of the external mass transport of three metal ions, namely, cadmium, copper, and zinc, onto bone char has been investigated.
2. Materials and methods 2.1. Materials—bone char The production and composition of bone char were described by Lambert and Graham [12]. Bone char is produced from the destructive distillation of dried, crushed cattle bones. After being ground to the appropriate particle size, the bone fragments are calcined at 500–700 ◦ C under controlled conditions for 4–6 h. X-ray diffraction reveals that bone char is a mixed adsorbent composed approxi-
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mately 80–90% of basic tricalcium phosphate and 10% of amorphous carbon. Structurally, calcium phosphate is in the hydroxyapatite form [13]. The amorphous carbon fraction is distributed throughout the whole of the hydroxyapatite structure but most exists as a highly active thin film over about 50% of the porous hydroxyapatite surface. 2.2. Equilibrium isotherm studies A fixed mass of bone charcoal was weighed into 120-ml test bottles. Metal ion solutions were prepared and then pipetted into the test bottles. The initial pH of the solutions was adjusted to 4.9 ± 0.1 by the addition of dilute sulfuric acid. The test bottles were put in the shaker bath for 5 days at 20 ◦ C and were shaken at the maximum shaking rate (200 rpm) to allow the bone charcoal to adsorb the metal ions until the solution reached equilibrium. The initial and final concentrations of the solutions were measured by ICP–AES. These data were used to calculate the adsorption capacity, qe , of the adsorbent. Finally, a diagram of adsorption capacity, qe , against equilibrium concentration, Ce , was plotted. 2.3. Batch kinetic studies A standard tank configuration was used to derive the relative dimensions of the vessel and its components [14]. The following relationships hold with respect to the vessel inside diameter, Di : height of baffles = 0.2 m, baffle width = 0.075Di , height of liquid in the vessel = Di , distance between impeller blade and vessel bottom = 0.5Di , width of impeller blade = 0.1Di , impellerdiameter = 0.5Di . A schematic diagram for the experimental setup is shown in Fig. 1. The adsorber vessel used was a 2-dm3 plastic beaker of internal diameter 0.13 m holding a volume of 1.7 dm3 metal ion solution. Mixing was provided by a sixbladed flat plastic impeller of diameter 0.065 m and blade height 0.013 m. A Heidolph variable motor was used to drive the impeller using a 0.005-m-diameter plastic shaft. Six plastic baffles were evenly spaced around the circumference of the vessel, positioned at 60◦ intervals and held securely in place on top of the vessel. The purpose of the baffles was to prevent the formation of a vortex and the consequential reduction in relative motion between liquid and solid particles and power losses due to air entrainment at the impeller. Polystyrene baffles were 0.2 m long and 0.01 m wide. They were secured in a position slightly away from the vessel wall and the bottom of the tank in order to prevent particle accumulation. Evaporation of liquid was prevented by using a thick polystyrene sheet on top of the vessel. The standard
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carbon in an agitated tank. The experimental data are interpreted by assuming a three-step model: (1) mass transfer of benzene from bulk liquid to particle; (2) intraparticle diffusion; (3) adsorption at internal site. It is assumed that step three is rapid with respect to the first two steps. In fully mixed agitated slurry adsorber, mixing in the liquid phase is rapid. The concentration of sorbate, Ct , with respect to time is related to the fluid–particle mass-transfer coefficient by the equations dCt = −kf SA (Ct − Cs ) ⇒ dt dCt = −kf SA Ct + kf SA Cs dt 1 dCt . or Cs = Ct + kf SA dt
(1)
(2)
Assuming smooth spherical particles, the surface area for mass transfer to the particles can be obtained from ms , which is defined as the concentration of the adsorbent in the liquid phase: W . V Thus the surface area for mass transfer is defined, ms =
Fig. 1. Standard tank configuration for contact time studies.
condition was fixed at the mass of bone char (8.5 g) at a fixed particle size (500–710 µm), a fixed temperature (20 ± 2 ◦ C), a fixed pH (4.9 ± 0.1), and a fixed initial concentration of metal ion solution (3 mM). Therefore, the effect of the initial concentration of metal ion solution on the adsorption rate was studied by varying the initial concentrations of metal ion solution (i.e., 2, 2.5, 3, 4, and 5 mM) and the other conditions were fixed. The effects of sorbent mass and particle size were studied using the same condition as in the previous description.
3. Theory The film diffusion coefficients can be obtained by two methods. The first method is a correlation method and the second method is a dimensional analysis method. In the correlation method, the experimental data of the sorption system is substituted into the film diffusion equation to calculate the film diffusion coefficient. In the dimensional analysis method, the characterization of the sorbent and sorbate and the terminal velocity of sorbent are used for calculating the film diffusion coefficient. The results from the correlation method and dimensional analysis method will be compared to select the most accurate values for the external film masstransfer coefficients.
SA =
6ms . dp ρs (1 − εp )
(3)
(4)
The incremental mass balance on the solid becomes dqt ms (5) = kf SA (Ct − Cs ). dt The differential mass balance of metal ions within the particles, assuming a constant effective intraparticle diffusivity, Deff , is given by 2 ∂qr ∂Cr 2 ∂Cr ∂ Cr = εp , Deff (6) + − ρp 2 ∂r r ∂r ∂t ∂t ∂Cr Deff (7) = kf (Ct − Cs ), ∂r r=R ∂Cr = 0 at r = 0, (8) ∂r Cr = 0 at t = 0 for 0 r R. (9) Boundary and initial conditions corresponding to the experimental conditions are shown in Eqs. (2), (7), (8), and (9). Since equilibrium is assumed for adsorption at an interior site, qr and Cr are related by the instantaneous equilibrium expression ∂qr ∂qr ∂Cr ∂ KL Cr ∂Cr (10) = = . ∂t ∂Cr ∂t ∂Cr 1 + aL Cr ∂t
3.1. Linear adsorption isotherm
For a single resistance model, intraparticle diffusion is neglected, Cr is equal to Cs for any r, and q is uniform throughout the particle. For a linear adsorption isotherm, where 1 aL Cr and constant KL value, Eq. (10) gives
This model was developed by Furusawa and Smith [14]. The authors studied the adsorption of benzene with activated
dqt dCs = KL dt dt
at qt = 0 when t = 0.
(11)
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Combining Eqs. (5) and (11), dCs = kf SA (Ct − Cs ). dt Differentiating Eq. (2) with respect to time,
ms KL
d 2 Ct dCt dCs + kf SA . = −kf SA dt dt dt 2 Substituting Eqs. (12) and (2) into Eq. (13) gives d 2 Ct dCt = −kf SA 2 dt dt kf2 SA2 1 dCt Ct − Ct + . + KL ms kf SA dt
(12)
(13)
3.3. Dimensional analysis (14)
By rearrangement, Eq. (15) becomes the more conventional form ms KL 1 Ct = ln − ln C0 1 + ms KL 1 + ms KL 1 + ms KL + − kf SA t . (16) ms KL As t → 0, surface mass transfer will predominate and the two assumptions of negligible intraparticle diffusion and a linear isotherm (qe = KL Ce ) are valid; consequently a plot of ln(Ct /C0 − 1/(1 + ms KL )) versus t will yield a straight line as t → 0 of intercept ms KL /(1 + ms KL ) and slope −((1 + ms KL )/ms KL )kf SA at t = 0 from which the surface mass-transfer coefficient kf can be obtained. 3.2. Nonlinear adsorption isotherm The equation methodology and assumptions of film diffusion for the nonlinear adsorption isotherm analysis are similar to the linear adsorption isotherm. It is assumed that adsorption at an interior site is assumed to occur rapidly in comparison with mass transfer and to be reversible. For the case of Langmuir isotherm expression applied in the single resistance model, Eq. (10) gives
Substituting Eq. (5) into Eq. (17), kf SA dCs = (1 + aL Cs )2 (Ct − Cs ). dt ms KL
(17)
(18)
Combining Eqs. (13) and (18), kf2 SA2 d 2 Ct dCt = + K S (1 + aL Cs )2 (Ct − Cs ). f A dt 2 dt ms KL
Substituting Eq. (2) into Eq. (19) and rearranging it gives kf SA d 2 Ct + kf SA + (1 + aL Ct ) dt 2 ms KL
aL dCt 2 dCt = 0. + (20) kf SA dt dt Equation (20) can be solved by a numerical method to determine the external mass-transfer coefficient.
Then Eq. (14) can be integrated into Eq. (15) when Ct = C0 at t = 0: Ct 1 = C0 1 + ms KL ms KL 1 + ms KL kf SA t . + exp − (15) 1 + ms KL ms KL
KL dCs dqt = . dt (1 + aL Cs )2 dt
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(19)
For liquid adsorption taking place in batch or continuous flow tanks, the liquid phase is often agitated to increase the interphase mass transfer. According to Harriott [15], the interphase mass-transfer coefficient between liquid and suspended particles in an agitated vessel, kf , and the masstransfer coefficient of the same particles moving at their terminal velocity, uT , through the same liquid, kf∗ , may be approximated by the relationship kf ∼ = 2.0. kf∗
(21)
To estimate kf∗ , Harriott suggested using the equation kf∗ dp dp uT 0.5 υ 0.33 (22) = 2.0 + 0.6 . DM µ DM The terminal velocity, uT , may use the correlation of Nienow [15], given as uT =
0.153g 0.71dp1.14 $ρ 0.77 ρ 0.29 µ0.43
,
(23)
where g is the gravitational acceleration (980 cm s−2 ) and $ρ = ρp + εp ρ is the density difference between the wet particle and the liquid density. Furusawa and Smith [14] compared the correlation of the Harriott method and the single resistance method. The authors concluded that the external mass-transfer coefficients obtained from the single resistance method fell between the ion exchange and dissolution correlations based Harriott’s data.
4. Results and discussion 4.1. Equilibrium studies The equilibrium sorption data are analyzed using the Langmuir equation as shown by Eq. (24). The isotherm constants of the equation are determined by minimizing the difference between the experimental and theoretical data using the SSE error method, and the results are shown in Table 1. Fig. 2 shows the Langmuir isotherm plots of the experimental data for the sorption of copper, cadmium, and zinc ions onto bone char: KL Ce . qe = (24) 1 + aL Ce
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Table 1 Langmuir isotherm constants for copper(II), cadmium(II), and zinc(II) ions on bone char KL (dm3 g−1 ) aL (dm3 mmol−1 ) qmax (mmol g−1 )
Copper
Cadmium
Zinc
49.054 69.199 0.709
75.354 157.812 0.477
31.399 62.176 0.505
4.2. Kinetic studies—external mass transfer For the agitated tank, a method to determine the film diffusion coefficient was proposed by Furusawa and Smith [14] for the sorption of benzene onto activated carbon. Mathews and Weber [16] directly determined the external masstransfer coefficient from Eq. (1). Later, McKay and Allen [17,18] used the Henry, Langmuir, and Freundlich equations to calculate the external film mass-transfer coefficient for the sorption of basic dyes on peat. Allen and co-workers [19] applied the Henry equation to the sorption of metal ions onto peat and concluded that the mass-transfer coefficient was dependent on the initial concentration of the solution in the system. Tien [15] proposed to use the Harriot method to determine the external mass-transfer coefficient in batch agitation system. The external mass-transfer coefficient can be calculated from the Sherwood number, Reynolds number, and Schmidt number. Therefore, the external masstransfer coefficient is concentration-independent. Recently, Carta and co-workers [20–22], based on a chemical potential equation, developed a new approximation to calculate the external mass-transfer coefficients, which is concentrationdependent. The external mass-transfer coefficient can be directly calculated by the film diffusion equation proposed by Mathews and Weber [16] from Eq. (1) and referred to as the M&W model. Equation (1) can be used to calculate the external
mass-transfer coefficient when t → 0 and Cs → 0. Then, Eq. (1) can be integrated to become Ct = −kf SA t. ln (25) C0 An alternative method referred to as the F&S model, which was developed by Furusawa and Smith [14], calculates the external mass-transfer coefficient by incorporating the linear adsorption isotherm into Eq. (1) to become Eq. (16). This equation has been used to calculate the external masstransfer coefficient for many sorption systems [14,18,19]. In this research, the external mass-transfer coefficients using Eqs. (16) and (25) are compared. Initially, Eq. (16) is rearranged to Ct Ct 1 1 ln 1− − 1 + (1/ms KL ) C0 ms KL C0 = −kf SA t.
(26)
By plotting 1 1 Ct Ct − 1− ln 1 + (1/ms KL ) C0 ms KL C0 against t, the external mass-transfer coefficient can be determined from the slope of the straight line. Plotting the linear Eq. (26) is easier than plotting Eq. (16) because the intercept of the straight line must be adjusted to ln(ms KL /(1 + ms KL )), while the intercept of Eq. (26) is equal to zero. This results in easier determination of the initial gradient (as t → 0). In addition, Eq. (26) can be converted to Eq. (25) when V 1 1 = → 0, (27) ms KL W qm aL where qm and aL are the maximum capacity and the affinity, respectively, of sorbate to sorbent. These parameters are the
Fig. 2. Sorption of three metal ions onto bone char using the Langmuir isotherm.
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intrinsic properties of sorbent in relation to sorbate, which cannot be changed. In Eq. (27), the V /m ratio is the only adjustable parameter in this expression. When the mass of sorbent increases, the term 1/ms KL tends to zero. Consider the assumption in Eq. (10) when 1 + aL Cr ≈ 1, Eq. (16) can be obtained. The conditions 1+ aL Cr ≈ 1 and 1/ms KL → 0 apply to very dilute solution systems or the use of a large mass of sorbent in solution. Therefore, the use of Eqs. (25) and (26) for dilute solution is identical. Figs. 3, 4, and 5 show the plots of the external mass transfer equations for the three metal ions. The plots of Eqs. (25) and (26) are very close.
Fig. 3. The sorption of Cd ions onto bone char correlating the Furusawa and Smith equation and the film-diffusion equation of Mathews and Weber.
Fig. 4. The sorption of Cu ions onto bone char correlating the Furusawa and Smith equation and the film-diffusion equation of Mathews and Weber.
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The Harriott equation was developed using dimensional analysis to calculate the external mass-transfer coefficient. Fig. 6 is the plot of Sh/Sc0.33 against particle Reynolds number. Increasing the particle sizes will slightly decrease the external mass-transfer coefficients in Table 2. The smaller sorbent particle travel faster in the agitated solution and experience more shear at the particle surface, therefore reducing the boundary layer film. A similar plot was produced by Harriott who used the equation to calculate the external mass-transfer coefficients of ion-exchange resins [10]. The external mass-transfer coefficients for different bone char particle sizes were calculated using the Harriott equation and are shown in Table 3. The external mass-transfer coefficients
Fig. 5. The sorption of Zn ions onto bone char correlating the Furusawa and Smith equation and the film-diffusion equation of Mathews and Weber.
Fig. 6. Sh/Sc0.33 vs Re for the sorption of Cd ions onto bone char with different particle sizes.
Table 2 External mass-transfer coefficients using the Harriott equation Particle size (µm)
kf dp ν −1/3 Sh Sc−1/3 = D Dm m
ε¯ dp4 1/3 Re = 3 ν
kf (cm s−1 )
250–355
355–500
500–710
710–1000
0.24
0.32
0.43
0.58
0.95
1.51
2.40
3.81
2.79 × 10−3
2.59 × 10−3
2.46 × 10−3
2.38 × 10−3
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Table 3 The external mass-transfer coefficients for different particle size ranges using the Harriott equation Particle size (µm)
Cd2+ (cm s−1 )
Cu2+ (cm s−1 )
Zn2+ (cm s−1 )
250–355 355–500 500–710 710–1000
2.79 × 10−3 2.59 × 10−3 2.46 × 10−3 2.38 × 10−3
2.77 × 10−3 2.57 × 10−3 2.44 × 10−3 2.37 × 10−3
2.74 × 10−3 2.54 × 10−3 2.41 × 10−3 2.34 × 10−3
using different methods have been calculated and are listed in Tables 4–9. The differences between the linear and nonlinear external mass-transfer coefficients are relatively small. The magnitudes of these coefficients, which were calculated by Eqs. (25) and (26), are of order 10−4 cm s−1 . The external mass-transfer coefficients calculated by these methods do vary with the initial concentration of solution and mass of sorbent. The results in Tables 4–6 indicate that kf values show a slow steady decrease with increasing initial metal ion concentration. This would not normally be expected in a single resistance mass transport model and is indicative that intraparticle diffusion is playing a significant role in the mass transport process. When the mass of sorbent is increased, the external mass-transfer coefficients vary only slightly and without trend. In contrast, the external mass-transfer coef-
ficients calculated by the Harriott equation are completely independent of initial concentrations and sorbent masses. The magnitude of the external mass-transfer coefficients by the Harriott equation is 10−3 cm s−1 . Classical treatment of external mass transfer or boundary layer theory [23,24] assumes that it is dependent on the diffusion of molecular species or ions across a stagnant layer of liquid, “the film,” of thickness (δ). Under limiting conditions these concepts are valid and the external mass-transfer coefficient, kf , is a constant value depending on the fixed extent of agitation of the system and the constant diffusivity of the system. The Harriott equation is a solution of these fixed conditions, correlating with kf through the Sherwood number as a function of Reynolds and Schmidt numbers. Using this model, kf will only vary as a function of agitation. However, based on the evaluations of kf in the present paper and other works [25], kf may show changes with liquid and solid phase concentration. In the present systems the diffusion in the boundary layer not only is based on metal ions flowing across the boundary layer from the bulk solution to the bone char surface, but there is a concentration gradient across the boundary due to calcium ions diffusing in a counter direction from the surface to the bulk solution. This rate depends on the concentration of metal ions and the mass concentration of bone char resulting in a variable apparent kf .
Table 4 The parameters of the external mass-transfer equations for the sorption of cadmium ions onto bone char at different initial concentrations of solution C0 (mmol dm−3 )
M&W (cm s−1 )
F&S (linear) (cm s−1 )
F&S (nonlinear) (cm s−1 )
Harriott (cm s−1 )
2.0 2.5 3.0 4.0 5.0
6.23 × 10−4 5.28 × 10−4 4.95 × 10−4 4.12 × 10−4 3.55 × 10−4
7.30 × 10−4 6.14 × 10−4 5.84 × 10−4 4.94 × 10−4 4.36 × 10−4
7.45 × 10−4 6.23 × 10−4 5.94 × 10−4 5.01 × 10−4 4.42 × 10−4
2.46 × 10−3 2.46 × 10−3 2.46 × 10−3 2.46 × 10−3 2.46 × 10−3
Table 5 The parameters of the external mass-transfer equations for the sorption of copper ions onto bone char at different initial concentrations of solution C0 (mmol dm−3 )
M&W (cm s−1 )
F&S (linear) (cm s−1 )
F&S (nonlinear) (cm s−1 )
Harriott (cm s−1 )
2.0 2.5 3.0 4.0 5.0
9.14 × 10−4 6.84 × 10−4 7.02 × 10−4 5.60 × 10−4 4.80 × 10−4
1.06 × 10−3 9.02 × 10−4 8.44 × 10−4 7.01 × 10−4 6.14 × 10−4
1.09 × 10−3 9.38 × 10−4 8.64 × 10−4 7.19 × 10−4 6.29 × 10−4
2.44 × 10−3 2.44 × 10−3 2.44 × 10−3 2.44 × 10−3 2.44 × 10−3
Table 6 The parameters of the external mass-transfer equations for the sorption of zinc ions onto bone char at different initial concentrations of solution C0 (mmol dm−3 )
M&W (cm s−1 )
F&S (linear) (cm s−1 )
F&S (nonlinear) (cm s−1 )
Harriott (cm s−1 )
2.0 2.5 3.0 4.0 5.0
5.97 × 10−4 5.54 × 10−4 4.75 × 10−4 4.65 × 10−4 3.89 × 10−4
7.39 × 10−4 6.34 × 10−4 5.95 × 10−4 5.53 × 10−4 4.75 × 10−4
7.58 × 10−4 6.44 × 10−4 6.09 × 10−4 5.63 × 10−4 4.83 × 10−4
2.41 × 10−3 2.41 × 10−3 2.41 × 10−3 2.41 × 10−3 2.41 × 10−3
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Table 7 The parameters of the external mass-transfer equations for the sorption of cadmium ions onto bone char at different masses of sorbent Mass (g)
M&W (cm s−1 )
F&S (linear) (cm s−1 )
F&S (nonlinear) (cm s−1 )
Harriott (cm s−1 )
6.5 7.5 8.5 9.5 10.5
4.52 × 10−4 4.54 × 10−4 4.95 × 10−4 5.18 × 10−4 5.23 × 10−4
5.47 × 10−4 5.61 × 10−4 5.84 × 10−4 6.28 × 10−4 6.23 × 10−4
5.56 × 10−4 5.69 × 10−4 5.94 × 10−4 5.32 × 10−4 6.29 × 10−4
2.46 × 10−3 2.46 × 10−3 2.46 × 10−3 2.46 × 10−3 2.46 × 10−3
Table 8 The parameters of the external mass-transfer equations for the sorption of copper ions onto bone char at different masses of sorbent Mass (g)
M&W (cm s−1 )
F&S (linear) (cm s−1 )
F&S (nonlinear) (cm s−1 )
Harriott (cm s−1 )
6.5 7.5 8.5 9.5 10.5
6.32 × 10−4 7.24 × 10−4 7.02 × 10−4 6.70 × 10−4 7.74 × 10−4
8.01 × 10−4 8.42 × 10−4 8.44 × 10−4 8.07 × 10−4 8.99 × 10−4
8.11 × 10−4 8.62 × 10−4 8.64 × 10−4 8.34 × 10−4 9.26 × 10−4
2.44 × 10−3 2.44 × 10−3 2.44 × 10−3 2.44 × 10−3 2.44 × 10−3
Table 9 The parameters of the external mass-transfer equations for the sorption of zinc ions onto bone char at different masses of sorbent Mass (g)
M&W (cm s−1 )
F&S (linear) (cm s−1 )
F&S (nonlinear) (cm s−1 )
Harriott (cm s−1 )
6.5 7.5 8.5 9.5 10.5
4.34 × 10−4 5.09 × 10−4 4.75 × 10−4 4.55 × 10−4 5.24 × 10−4
5.34 × 10−4 6.20 × 10−4 5.95 × 10−4 5.71 × 10−4 6.40 × 10−4
5.37 × 10−4 6.34 × 10−4 6.09 × 10−4 5.85 × 10−4 6.59 × 10−4
2.41 × 10−3 2.41 × 10−3 2.41 × 10−3 2.41 × 10−3 2.41 × 10−3
The operating lines of Cu2+ ion systems join (C0 , q0 ) to (Ce , qe ) at equilibrium and the effect of changing C0 with constant mass is to generate a series of parallel operating lines, as shown in Fig. 7. The driving force for mass transfer, based on the difference between the bulk liquid concentration and the particle surface liquid concentration, is timedependent, and this condition is reflected by the tie lines. Thus the external mass-transfer coefficients are apparently increasing as C0 decreases for the three metal ions being adsorbed onto bone char. In Figs. 3–5, the initial adsorption rate in the first 5 min is higher, but considerable error is involved using only one data point to measure the external mass-transfer coefficient for these systems. On this basis it was decided to evaluate the kf values over the first six (5-min interval) data points, that is, the first 30 min of metal ion solution in contact with bone char. Although there was no statistical justification for this selection, it does represent the initial sorption period covering approximately 15% of the bone sorption capacity. Moreover, based on the application of the models to the first 30 min of sorption, the sorption capacities of the metal ions on bone char are only 15%. This means that the (Ct , qt ) points up to this position in time all lie on the steep initial slope of the isotherms. Therefore, the
Fig. 7. The relationship of the adsorption isotherm and the mass-transport model (effect of initial concentration and sorbent mass on the sorption of Cd ions onto bone char).
assumption of a steep initial linear slope is not unreasonable. This assumption could lead to some degree of error in the evaluation of kf with C0 . McKay and Allen [25] served that the external mass-transfer coefficients for the adsorption of dye ions on peat presented a linear relationship between ln(C0 ) and ln(Sh/Sc0.33 ). The operating lines for several carbon masses and a constant initial dye concentration are also shown in Fig. 7. Since the masses are varying, the slopes of the operating lines vary accordingly. The external mass-transfer coefficient may depend on the driving force per unit area, and in this case, since C0 is constant, increasing the mass of carbon increases the surface for adsorption and hence the rate of metal ion removal is increased. Since the particle size range is constant, the surface area will be directly proportional to the mass of bone char in the system. The kf values are shown in Tables 7–9 and demonstrate a dependence on mass. This effect is probably due to the fact that for small masses a small amount of external surface is presented to the metal ion and therefore there is a large driving force from the metal ion per unit surface area of bone char. In these film diffusion studies, the discrepancies in the values of the film mass-transfer coefficients between the experimental methods and the dimensional analysis method can also be explained by the fast external mass transport of sorbates. The first three methods used the experimental data to calculate the film diffusion coefficients. However, the external mass transport of sorbate in these sorption systems may be very fast in comparison with the intraparticle diffusion. The experimental data used to calculate the film masstransfer coefficients may also be in the intraparticle diffusion region. In addition, the single-resistance model based on the mass transport of sorbate at the initial stage (as t → 0) only assumes that mass transfer is controlled by the external mass transport of sorbate only, but if the mass transport at the initial stages also involves film–pore or pore diffusion, the results of Eqs. (20), (25), and (26) will be affected to some extent.
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Fig. 8. The sorption capacity at specified times against the square root of time (concentration effect for the sorption of Cd ions onto bone char).
Fig. 9. The sorption capacity at specified time against the square root of time (particle size effect for the sorption of Cd ions onto bone char).
In order to minimize the extent that intraparticle diffusion may be occurring less during the initial 30 min of the sorption process, one experimental data point at the first 5 min is used to calculate the external mass-transfer coefficient using Eq. (26). A series of gradients were drawn through points on the experimental curve at times ranging from 0 to 30 min. At a time of 5 min, the gradient through the points (0, 0) and (5, qt ) gave the “closest fit” to the best fit kf vales obtained using the two-resistance simulation model. Therefore the time of 5 min was selected for kf determination. Another approach is to use the effectiveness factor method [25], which measures the fractional deviation of the single-resistance kf model from the experimental curve, which was still a tworesistance diffusion model. Therefore both approaches are still subject to the fact that most experimental sorption systems are two- or multiresistance mass transport systems. The external mass-transfer coefficients for the sorption of Cd2+ , Cu2+ , and Zn2+ ions at 3 mM are equivalent to 1.03 × 10−3 , 1.65 × 10−3 , and 1.09 × 10−3 cm s−1 , respectively. The magnitudes of external mass-transfer coefficients from this less statistically accurate approach are higher than the values obtained from the first three methods. Therefore, the choice and number of data points to calculate the external mass-transfer coefficient may be very important. In order to assess the contribution from intraparticle diffusion, Fig. 8 shows the sorption capacity of cadmium ions at specified times against the square root of time. Weber and Morris [26] stated that if intraparticle diffusion is the rate-controlling factor, uptake of the adsorbate varies in proportion to the square of time. When the linear plots of qt versus time0.5 (t 0.5 ) pass through the origin, then intraparticle diffusion may be the predominant rate-controlling step in the removal of Cd2+ ions [27,28]. Approximations of classical diffusion mathematics [29] show that for a diffusion process controlled by intraparticle diffusion only a plot of qt versus t 0.5 will yield a straight line passing through the origin. This has been demonstrated experimentally for the adsorption of surfactants on activated carbons [26]. In
several other reported systems [30,31], plots of qt versus t 0.5 have demonstrated linearity over most of the adsorption period except for a short time at the beginning of the adsorption process. This initial rapid uptake was attributed to an initial rapid surface adsorption, after the external surface loading was completed; however, the rate-controlling process becomes intraparticle diffusion, as demonstrated by the qt versus t 0.5 plots. In the case of Cd2+ , the qt versus t 0.5 plots are shown for four different particle size ranges in Fig. 8. In the two systems, with the large mean particle diameters and therefore relatively small external surface areas, the qt plots are linear and pass through or very close to the origin. This indicates that there is little or no external film control. However, in the two systems for dp , 355–500 and 250–355 µm, the t 0.5 intercepts show considerable deviation from the origin (0,0) demonstrating for an initial time period there is some controlling sorption contribution from external mass transfer (see Fig. 9). In the transition period from t = 0 to when intraparticle diffusion control takes over, there will be a transitional time zone. The sorptions of copper and zinc ions onto bone char have a similar characteristic pattern. Therefore, from Fig. 8, substituting the experimental data into the film diffusion equations to determine the film diffusion coefficient is only valid as t → 0 because intraparticle diffusion is the predominant mechanism throughout the bulk of the sorption process. More interesting results are shown in Fig. 9, showing that when the particle sizes of sorbent decrease, the intercept of the plot of the square root of time for the smaller particle sizes deviates from the point of origin. In the previous discussion for a bone char particle size range of 500–710 µm, the intraparticle diffusion is the main rate-limiting step for most of the metal ion/bone char sorption systems and the film diffusion is not the rate-limiting step except at the early stages of the sorption of metal ions onto bone char process. However, as the radius of the sorbent particle decreases, the mass transport of sorbate inside the sorbent becomes shorter, and the film diffusion effect on the metal ion removal rate
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Table 10 Simulation results from film–pore diffusion model using different kf values M&W
F&S (linear)
F&S (nonlinear)
Harriott
Film–pore best-fit
Cd
Averaged kf (cm s−1 ) Optimized Dp (cm2 s−1 ) SSE
4.69 × 10−4 1.22 × 10−6 1537
5.63 × 10−4 1.22 × 10−6 1969
5.58 × 10−4 1.22 × 10−6 1948
24.6 × 10−4 1.22 × 10−6 4442
4.04 × 10−4 1.22 × 10−6 807
Cu
Averaged kf (cm s−1 ) Optimized Dp (cm2 s−1 ) SSE
6.56 × 10−4 1.41 × 10−6 386
8.04 × 10−4 1.41 × 10−6 413
8.26 × 10−4 1.41 × 10−6 419
24.4 × 10−4 1.41 × 10−6 767
6.38 × 10−4 1.41 × 10−6 385
Zn
Averaged kf (cm s−1 ) Optimized Dp (cm2 s−1 ) SSE
4.89 × 10−4 1.28 × 10−6 532
5.96 × 10−4 1.28 × 10−6 683
6.08 × 10−4 1.28 × 10−6 698
24.1 × 10−4 1.28 × 10−6 1491
2.76 × 10−4 1.28 × 10−6 290
Fig. 10. The effect of kf for four different models on the initial 30 min of the batch kinetic experiment using the film–pore diffusion model (Cu2+ ion, C0 = 2.11 mmol dm−3 , dp = 500–710 µm, Dp = 1.4 × 10−6 cm2 s−1 ).
becomes more and more significant. In contrast, when the particle size increases, the mass transport distance of metal ions inside the sorbent becomes longer, and the film masstransfer effect becomes less significant. These trends are apparent in Fig. 9. As the external mass-transfer coefficients obtained from the four methods are different, it is difficult to decide which approach is correct. Similar findings were obtained by Allen et al. [32]. The authors used both the Furusawa and Smith equation and the homogeneous solid diffusion model (HSDM) [33] to determine the external mass-transfer coefficient. The magnitudes based on the HSDM are 10−3 cm s−1 , but using the Furusawa and Smith equation the value is 10−4 cm s−1 , and the authors selected the best-fit external mass-transfer coefficient from HSDM. Therefore, in the present work it was decided to compare the goodness of data fitting of a two-resistance film–pore intraparticle diffusion model for the sorption systems [34] based on these external mass-transfer coefficients, and the results are shown in Table 10 and Fig. 10. Visually, all four different methods provide a good correlation between the experimental data and the theoretical prediction of the sorption of Cu2+ ion
(C0 = 134 ppm) onto bone char (W = 8.5 g) in the period from 0 to 30 min. By comparing the sum of the square of the errors (SSE) values in Table 10 between the four different methods to obtain kf , it is found that the data-fitting methods (M&W, linear F&S, and nonlinear F&S) based on the experimental data points yield lower SSE values than the empirical correlated data (Harriot method), while the M&W method yields the lowest SSE in the four different methods. Thus, the use of the experimental data to obtain kf does result in an improvement over the kf determined from the empirical correlation. This would be expected because all three methods imply obtaining gradients or data from the initial experimental data points. If intraparticle diffusion control participates in the adsorption-rate-controlling resistance from very short times and onward, then even the early data points are indicative of combined diffusion mass transport. This highlights the major problem of trying to isolate and determine the film mass-transfer coefficient for most conventional adsorption systems. Consequently the effect of particle size was used to study which correlative method provides the most appropriate kf values. Therefore, simulation studies were performed to optimize the correlation between experimental and model data using a film–intraparticle diffusion model. The best-fit values of kf were obtained and compared with the various model values in Table 11. The simulation models confirm that the M&W method provides the best correlation for the external mass-transfer coefficients. The influence of the sensitivity of kf , in terms of the range of values relevant to the present work, is shown in Fig. 11 for the first 30 min of the sorption of Cu2+ ion onto bone char process. Seven theoretical values of kf between 5 × 10−2 and 5 × 10−5 cm s−1 have been selected in the sensitivity analysis, and it can be seen that the lower the kf value, the greater becomes the sensitivity of kf . At high values of kf (>5×10−3 ), the film– pore diffusion model slightly overestimates the rate of Cu2+ ion adsorption onto bone char. Using the range of 5 × 10−4 to 8.5 × 10−4 kf values, obtained from M&W, F&S (linear), and F&S (nonlinear) methods in the film–pore diffusion model does make a very small difference in the prediction of adsorption process decay curves. However, great sensitivity
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Table 11 The parameters of the external mass-transfer equations for the sorption of cadmium ions onto bone char at different particle sizes Particle size (µm)
M&W (cm s−1 )
F&S (linear) (cm s−1 )
F&S (nonlinear) (cm s−1 )
Harriott (cm s−1 )
Best fit (cm s−1 )
250–355 355–500 500–710 710–1000
9.21 × 10−4 6.91 × 10−4 4.69 × 10−4 6.93 × 10−4
9.96 × 10−4 7.83 × 10−4 5.63 × 10−4 7.84 × 10−4
1.07 × 10−3 7.96 × 10−4 5.58 × 10−4 7.99 × 10−4
2.79 × 10−3 2.59 × 10−3 24.6 × 10−4 2.38 × 10−3
8.32 × 10−4 6.09 × 10−4 4.04 × 10−4 7.10 × 10−4
Acknowledgments The authors are grateful to DAG and RGC, Hong Kong SAR, for the provision of financial support during this research program.
Appendix A. Nomenclature aL Bi C0 Ce Fig. 11. Sensitivity analysis of the effect of kf on the initial 30 min of the batch kinetic experiment using the film–pore diffusion model (Cu2+ ion, C0 = 2.11 mmol dm−3 , dp = 500–710 µm, Dp = 1.4 × 10−6 cm2 s−1 ).
was found on the concentration decay curve when the kf value is smaller than 2.5 × 10−4 .
5. Summary The application of the intraparticle diffusion, root time model demonstrates that intraparticle diffusion is by far the dominant mechanism responsible for the sorption of copper, cadmium, and zinc ions onto bone char, except at the very beginning of the sorption process for a bone char particle size of diameter 500–710 µm. However, as the particle size decreases the influence of the external film external mass transfer on the sorption of the metal ions becomes much more significant. Results for bone char particle diameter ranges of 250–355 and 355–500 show film to be increasingly important. The film coefficients are best predicted by the Mathews and Weber model, followed by the linear Furusawa and Smith model and the nonlinear Furusawa and Smith model, with the empirical correlation model giving the poorest prediction. Overall this work highlights the problems of easily defining an absolute kf value that is representative of conventional agitated batch adsorption systems independent of other system parameters. Several experimental data points are required at short contact times. These “short-time” points are subject to errors due to measurement of small changes at high concentrations, establishing uniform experimental conditions, such as agitation mixing, adsorbent wetting, and time of sampling during short contact times.
Cs Cr Ct dp Deff Dm kf KL ms qe qr r R Re SA Sc Sh t W εp ρp ρ
Langmuir isotherm constant (dm3 mmol−1) Biot number, Bi = kf R/Deff initial liquid-phase concentration (mmol dm−3 ) equilibrium liquid-phase concentration (mmol dm−3 ) liquid-phase concentration at external sorbent surface (mmol dm−3 ) liquid-phase concentration at radius r (mmol dm−3 ) liquid-phase concentration at time t (mmol dm−3 ) diameter of sorbent (µm) effective diffusivity (cm2 s−1 ) molecular diffusion coefficient of metal ions (cm2 s−1 ) external mass-transfer coefficient (cm s−1 ) Langmuir constant (dm3 g−1 ) concentration of sorbent in liquid phase (g dm−3 ) equilibrium solid-phase concentration (mmol g−1 ) equilibrium solid-phase concentration at radius r (mmol g−1 ) radius of concentration front of metal ions penetrating adsorbent (cm) radius of adsorbent particle (cm) Reynold number, Re = (¯εdp4 /ν 3 )1/3 surface area of sorbent (m2 g−1 ) Schmidt number, Sc = ν/Dm Sherwood number, Sh = kf dp /Dm contact time (min) mass of sorbent (g) porosity particle density (g cm−3 ) solution density (g cm−3 )
References [1] M.C. Bennett, J.C. Abram, J. Colloid Interface Sci. 23 (1967) 513. [2] X. Shen, A. Bousher, R.G.J. Edyvean, in: IChemE Reseach Event— 1st European Conf. Young Res. Chem. Eng., vol. 1, 1995, p. 469.
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[3] M.W. Abdel Raouf, A.A.M. Daifullah, Adsorpt. Sci. Technol. 15 (8) (1997) 559. [4] A. Abdel Kader, A.N.H. Aly, B.S. Girgis, Int. Sugar J. 98 (1174) (1996) 546. [5] J. Lewis, J. Chartered Inst. Water Environ. 9 (4) (1995) 385. [6] J.C. Laul, M.C. Rupert, M.J. Harris, A. Duran, in: Proc. 6th Annual Int. Conf. High Level Radioactive Waste Management, 1995, p. 231. [7] M.J. Angove, B.B. Johnson, J.D. Wells, J. Colloid Interface Sci. 204 (1998) 93. [8] R. Apak, K. Guclu, M.H. Turgut, J. Colloid Interface Sci. 203 (1998) 122. [9] W.H. Van Riemsdijk, J.C.M. De Wit, L.K. Koopal, G.H. Bolt, J. Colloid Interface Sci. 116 (1987) 511. [10] K.F. Hayes, J.O. Leckie, J. Colloid Interface Sci. 115 (1985) 564. [11] F.G. Helfferich, Ion Exchange, Dover, New York, 1995. [12] S.D. Lambert, N.J.D. Graham, Environ. Technol. Lett. 10 (9) (1989) 785. [13] B.S. Girgis, A. Abdel Kader, A.N.H. Aly, Adsorpt. Sci. Technol. 15 (4) (1997) 277. [14] T. Furusawa, J.M. Smith, Ind. Eng. Chem. Fundam. 12 (1973) 197. [15] C. Tien, Adsorption Calculations and Modeling, Butterworth–Heinemann, Newton, 1994. [16] A.P. Mathews, W.J. Weber Jr., AIChE J. 73 (1976) 91. [17] G. McKay, S.J. Allen, Can. J. Chem. Eng. 58 (1980) 521.
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[18] G. McKay, S.J. Allen, J. Sep. Process Technol. 4 (3) (1983) 1. [19] S. Allen, P. Brown, G. McKay, O. Flynn, J. Chem. Technol. Biotechnol. 54 (1992) 271. [20] G. Carta, A. Cincotti, G. Cao, Sep. Sci. Technol. 34 (1) (1999) 1. [21] G. Carta, R.K. Lewus, Sep. Sci. Technol. 34 (14) (1999) 2685. [22] G. Carta, R.K. Lewus, Adsorption 6 (2000) 5. [23] R. Highie, Trans. AIChE 31 (1935) 365. [24] E.V. Murphree, Ind. Eng. Chem. 24 (1932) 726. [25] G. McKay, S.J. Allen, I.F. McConvey, M.S. Otterburn, J. Colloid Interface Sci. 80 (2) (1980) 323. [26] W.J. Weber Jr., J.C. Morris, J. Sanitary Eng. Div. Proc. ASCE 89 (SA2) (1963) 31. [27] C. Namasivayam, K. Ranganathan, Environ. Pollut. 82 (1993) 255. [28] T. Viraraghavan, M.M. Dronamraju, J. Environ. Sci. Health A 28 (1993) 1261. [29] J. Crank, The Mathematics of Diffusion, second ed., Clarendon, Oxford, 1975. [30] G. McKay, V.J.P. Poots, J. Chem. Technol. Biotechnol. 30 (1980) 279. [31] C.C. Lin, H.S. Liu, Ind. Eng. Chem. Res. 39 (2000) 161. [32] S. Allen, L.J. Whitten, M. Murray, O. Duggan, J. Chem. Technol. Biotechnol. 68 (1997) 442. [33] J.R. Rosen, J. Chem. Phys. 20 (1952) 387. [34] K.K.H. Choy, J.F. Porter, G. McKay, Adsorption 7 (4) (2001) 305.
Film and intraparticle mass transfer during the adsorption of metal ions onto bone char Keith K.H. Choy, Danny C.K. Ko, Chun W. Cheung, 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 27 February 2003; accepted 5 December 2003
Abstract The sorption of three metal ions, namely, copper, cadmium, and zinc, onto bone char has been studied in terms of equilibrium and rate studies. Equilibrium studies have been analyzed using the Langmuir isotherm equation and the maximum sorption capacities for the metals were 0.477, 0.709, and 0.505 mmol g−1 bone char for cadmium, copper, and zinc ions, respectively. The kinetic experimental data were used to analyze the effect of external film boundary layer and intraparticle mass transfer resistance on the sorption process and its significance. Four methods of determining the external film transport coefficient were developed and tested; three utilized experimental data to obtain the coefficient and the fourth method was completely empirical. The three experimentally based models give very similar results and consequently similar values of the deviation error values, whereas the error values for the empirical correlation were greater than these three values. The results also demonstrated that the methods for determining the film coefficient could be integrated into more complex diffusion-transport models such as film–intraparticle diffusion processes. 2004 Elsevier Inc. All rights reserved. Keywords: Bone char; Copper; Cadmium; Zinc; Sorption; External mass transfer; Intraparticle diffusion
1. Introduction Bone char has been used extensively as an adsorbent for the decolorization of cane sugar [1] and to a lesser extent for the defluoridation of drinking water. Results for color removal [2] and removal of metal ions from waste waters [3] have also been reported in recent research publications. Bone char is still used extensively in sugar refining as a decolorizing adsorbent. After use it is regenerated by washing and calcining so that the char can pass through many operating cycles before its activity has decreased to an unacceptably low level. Abdel Kader and co-workers [4] reported that the decolorization process removes more than color because colorants interact with color precursors, colloidal material, organic nonsugars, and ash-forming inorganic constituents so that they are taken out with the color. Thus, the value of bone char as a sugar refining aid lies in its simple action in removing both colorants and ash-forming inorganic constituents. * Corresponding author.
E-mail address: [email protected] (G. McKay). 0021-9797/$ – see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2003.12.015
Recently, studies into the uses of bone char have increased, though its main use remains as an adsorbent. Shen et al. [2] combined the use of ferrous ions, lime, and bone charcoal to remove the color from the effluent of a local chemical manufacturer. The effluent contained high levels of soluble organic compounds. The intense color of the effluent made common coagulation or adsorption techniques unsuitable for color removal. However, the report showed that a combination of FeCl2 ·4H2 O (as a reducing agent), lime, and bone charcoal removed up to 97% of the color. Lewis [5] used bone char in the treatment of rural water supplies. Bone char was used as a filter and its performance compared with that of a sand filter. The results showed that the pH, color, turbidity, and metal ions in treated water were improved. Abdel Raouf and Daifullah [3] used bone charcoal in the removal of antimony and europium radioisotopes from radioactive wastes. The authors showed that bone char was a good sorbent for 123 SbIII and 123SbV from liquid organic radioactive wastes and for the retention of 152 EuIII from aqueous wastes. Physical sorption was suggested to be the main mechanism operating in the sorption of antimony radioisotopes with appreciably higher fixation on the sorbent when loaded from the liquid organic radioactive waste. In
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contrast, chemisorption was postulated as the main operating mechanism for 152EuIII with a higher and irreversible fixation on bone charcoal from a neutral aqueous medium. Laul et al. [6] used bone charcoal and other sorbents to remove uranium from contaminated groundwater. Due to its high toxicity several researchers have studied the sorption of cadmium onto various materials [7,8]. In adsorption systems, the mass transfer of solute or sorbate onto and within the sorbent particle directly affects the adsorption rate. It is important to study the rate at which the solute is removed from aqueous solution in order to apply adsorption by solid particles to industrial uses. There are essentially four stages in the adsorption process: 1. Transport of sorbate from the bulk of the solution to the exterior film surrounding of the adsorbent. 2. Movement of sorbate across the external liquid film boundary layer to external surface sites. 3. Migration of sorbate within the pores of the sorbent by intraparticle diffusion. 4. Sorption of sorbate at internal surface sites. All these processes may be involved in the control of the sorbate removal rate. However, in a fully mixed agitated tank, mass transport from the bulk solution to the external surface is usually fast. The transport of sorbate from the bulk of the solution to the exterior film surrounding the adsorbent is usually neglected. In addition, the adsorption of sorbate at surface sites (step 4) is usually rapid. Thus, these processes usually are not considered to be the rate-limiting steps in the sorption process. Some metal sorption systems have been analyzed using complexation models [9,10], but in the present system there is strong evidence to support the conclusion that ion exchange is a main mechanism contributing to the sorption process. In fact, the ion exchange of metal ions onto ion exchangers is also considered to be controlled by diffusion [11]. In most cases, steps (2) and (3) may control the sorption mechanisms, because the mass transport (external and internal) of adsorbate to adsorbent is a relatively slow process. In the present paper, a quantitative study of the external mass transport of three metal ions, namely, cadmium, copper, and zinc, onto bone char has been investigated.
2. Materials and methods 2.1. Materials—bone char The production and composition of bone char were described by Lambert and Graham [12]. Bone char is produced from the destructive distillation of dried, crushed cattle bones. After being ground to the appropriate particle size, the bone fragments are calcined at 500–700 ◦ C under controlled conditions for 4–6 h. X-ray diffraction reveals that bone char is a mixed adsorbent composed approxi-
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mately 80–90% of basic tricalcium phosphate and 10% of amorphous carbon. Structurally, calcium phosphate is in the hydroxyapatite form [13]. The amorphous carbon fraction is distributed throughout the whole of the hydroxyapatite structure but most exists as a highly active thin film over about 50% of the porous hydroxyapatite surface. 2.2. Equilibrium isotherm studies A fixed mass of bone charcoal was weighed into 120-ml test bottles. Metal ion solutions were prepared and then pipetted into the test bottles. The initial pH of the solutions was adjusted to 4.9 ± 0.1 by the addition of dilute sulfuric acid. The test bottles were put in the shaker bath for 5 days at 20 ◦ C and were shaken at the maximum shaking rate (200 rpm) to allow the bone charcoal to adsorb the metal ions until the solution reached equilibrium. The initial and final concentrations of the solutions were measured by ICP–AES. These data were used to calculate the adsorption capacity, qe , of the adsorbent. Finally, a diagram of adsorption capacity, qe , against equilibrium concentration, Ce , was plotted. 2.3. Batch kinetic studies A standard tank configuration was used to derive the relative dimensions of the vessel and its components [14]. The following relationships hold with respect to the vessel inside diameter, Di : height of baffles = 0.2 m, baffle width = 0.075Di , height of liquid in the vessel = Di , distance between impeller blade and vessel bottom = 0.5Di , width of impeller blade = 0.1Di , impellerdiameter = 0.5Di . A schematic diagram for the experimental setup is shown in Fig. 1. The adsorber vessel used was a 2-dm3 plastic beaker of internal diameter 0.13 m holding a volume of 1.7 dm3 metal ion solution. Mixing was provided by a sixbladed flat plastic impeller of diameter 0.065 m and blade height 0.013 m. A Heidolph variable motor was used to drive the impeller using a 0.005-m-diameter plastic shaft. Six plastic baffles were evenly spaced around the circumference of the vessel, positioned at 60◦ intervals and held securely in place on top of the vessel. The purpose of the baffles was to prevent the formation of a vortex and the consequential reduction in relative motion between liquid and solid particles and power losses due to air entrainment at the impeller. Polystyrene baffles were 0.2 m long and 0.01 m wide. They were secured in a position slightly away from the vessel wall and the bottom of the tank in order to prevent particle accumulation. Evaporation of liquid was prevented by using a thick polystyrene sheet on top of the vessel. The standard
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carbon in an agitated tank. The experimental data are interpreted by assuming a three-step model: (1) mass transfer of benzene from bulk liquid to particle; (2) intraparticle diffusion; (3) adsorption at internal site. It is assumed that step three is rapid with respect to the first two steps. In fully mixed agitated slurry adsorber, mixing in the liquid phase is rapid. The concentration of sorbate, Ct , with respect to time is related to the fluid–particle mass-transfer coefficient by the equations dCt = −kf SA (Ct − Cs ) ⇒ dt dCt = −kf SA Ct + kf SA Cs dt 1 dCt . or Cs = Ct + kf SA dt
(1)
(2)
Assuming smooth spherical particles, the surface area for mass transfer to the particles can be obtained from ms , which is defined as the concentration of the adsorbent in the liquid phase: W . V Thus the surface area for mass transfer is defined, ms =
Fig. 1. Standard tank configuration for contact time studies.
condition was fixed at the mass of bone char (8.5 g) at a fixed particle size (500–710 µm), a fixed temperature (20 ± 2 ◦ C), a fixed pH (4.9 ± 0.1), and a fixed initial concentration of metal ion solution (3 mM). Therefore, the effect of the initial concentration of metal ion solution on the adsorption rate was studied by varying the initial concentrations of metal ion solution (i.e., 2, 2.5, 3, 4, and 5 mM) and the other conditions were fixed. The effects of sorbent mass and particle size were studied using the same condition as in the previous description.
3. Theory The film diffusion coefficients can be obtained by two methods. The first method is a correlation method and the second method is a dimensional analysis method. In the correlation method, the experimental data of the sorption system is substituted into the film diffusion equation to calculate the film diffusion coefficient. In the dimensional analysis method, the characterization of the sorbent and sorbate and the terminal velocity of sorbent are used for calculating the film diffusion coefficient. The results from the correlation method and dimensional analysis method will be compared to select the most accurate values for the external film masstransfer coefficients.
SA =
6ms . dp ρs (1 − εp )
(3)
(4)
The incremental mass balance on the solid becomes dqt ms (5) = kf SA (Ct − Cs ). dt The differential mass balance of metal ions within the particles, assuming a constant effective intraparticle diffusivity, Deff , is given by 2 ∂qr ∂Cr 2 ∂Cr ∂ Cr = εp , Deff (6) + − ρp 2 ∂r r ∂r ∂t ∂t ∂Cr Deff (7) = kf (Ct − Cs ), ∂r r=R ∂Cr = 0 at r = 0, (8) ∂r Cr = 0 at t = 0 for 0 r R. (9) Boundary and initial conditions corresponding to the experimental conditions are shown in Eqs. (2), (7), (8), and (9). Since equilibrium is assumed for adsorption at an interior site, qr and Cr are related by the instantaneous equilibrium expression ∂qr ∂qr ∂Cr ∂ KL Cr ∂Cr (10) = = . ∂t ∂Cr ∂t ∂Cr 1 + aL Cr ∂t
3.1. Linear adsorption isotherm
For a single resistance model, intraparticle diffusion is neglected, Cr is equal to Cs for any r, and q is uniform throughout the particle. For a linear adsorption isotherm, where 1 aL Cr and constant KL value, Eq. (10) gives
This model was developed by Furusawa and Smith [14]. The authors studied the adsorption of benzene with activated
dqt dCs = KL dt dt
at qt = 0 when t = 0.
(11)
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Combining Eqs. (5) and (11), dCs = kf SA (Ct − Cs ). dt Differentiating Eq. (2) with respect to time,
ms KL
d 2 Ct dCt dCs + kf SA . = −kf SA dt dt dt 2 Substituting Eqs. (12) and (2) into Eq. (13) gives d 2 Ct dCt = −kf SA 2 dt dt kf2 SA2 1 dCt Ct − Ct + . + KL ms kf SA dt
(12)
(13)
3.3. Dimensional analysis (14)
By rearrangement, Eq. (15) becomes the more conventional form ms KL 1 Ct = ln − ln C0 1 + ms KL 1 + ms KL 1 + ms KL + − kf SA t . (16) ms KL As t → 0, surface mass transfer will predominate and the two assumptions of negligible intraparticle diffusion and a linear isotherm (qe = KL Ce ) are valid; consequently a plot of ln(Ct /C0 − 1/(1 + ms KL )) versus t will yield a straight line as t → 0 of intercept ms KL /(1 + ms KL ) and slope −((1 + ms KL )/ms KL )kf SA at t = 0 from which the surface mass-transfer coefficient kf can be obtained. 3.2. Nonlinear adsorption isotherm The equation methodology and assumptions of film diffusion for the nonlinear adsorption isotherm analysis are similar to the linear adsorption isotherm. It is assumed that adsorption at an interior site is assumed to occur rapidly in comparison with mass transfer and to be reversible. For the case of Langmuir isotherm expression applied in the single resistance model, Eq. (10) gives
Substituting Eq. (5) into Eq. (17), kf SA dCs = (1 + aL Cs )2 (Ct − Cs ). dt ms KL
(17)
(18)
Combining Eqs. (13) and (18), kf2 SA2 d 2 Ct dCt = + K S (1 + aL Cs )2 (Ct − Cs ). f A dt 2 dt ms KL
Substituting Eq. (2) into Eq. (19) and rearranging it gives kf SA d 2 Ct + kf SA + (1 + aL Ct ) dt 2 ms KL
aL dCt 2 dCt = 0. + (20) kf SA dt dt Equation (20) can be solved by a numerical method to determine the external mass-transfer coefficient.
Then Eq. (14) can be integrated into Eq. (15) when Ct = C0 at t = 0: Ct 1 = C0 1 + ms KL ms KL 1 + ms KL kf SA t . + exp − (15) 1 + ms KL ms KL
KL dCs dqt = . dt (1 + aL Cs )2 dt
287
(19)
For liquid adsorption taking place in batch or continuous flow tanks, the liquid phase is often agitated to increase the interphase mass transfer. According to Harriott [15], the interphase mass-transfer coefficient between liquid and suspended particles in an agitated vessel, kf , and the masstransfer coefficient of the same particles moving at their terminal velocity, uT , through the same liquid, kf∗ , may be approximated by the relationship kf ∼ = 2.0. kf∗
(21)
To estimate kf∗ , Harriott suggested using the equation kf∗ dp dp uT 0.5 υ 0.33 (22) = 2.0 + 0.6 . DM µ DM The terminal velocity, uT , may use the correlation of Nienow [15], given as uT =
0.153g 0.71dp1.14 $ρ 0.77 ρ 0.29 µ0.43
,
(23)
where g is the gravitational acceleration (980 cm s−2 ) and $ρ = ρp + εp ρ is the density difference between the wet particle and the liquid density. Furusawa and Smith [14] compared the correlation of the Harriott method and the single resistance method. The authors concluded that the external mass-transfer coefficients obtained from the single resistance method fell between the ion exchange and dissolution correlations based Harriott’s data.
4. Results and discussion 4.1. Equilibrium studies The equilibrium sorption data are analyzed using the Langmuir equation as shown by Eq. (24). The isotherm constants of the equation are determined by minimizing the difference between the experimental and theoretical data using the SSE error method, and the results are shown in Table 1. Fig. 2 shows the Langmuir isotherm plots of the experimental data for the sorption of copper, cadmium, and zinc ions onto bone char: KL Ce . qe = (24) 1 + aL Ce
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Table 1 Langmuir isotherm constants for copper(II), cadmium(II), and zinc(II) ions on bone char KL (dm3 g−1 ) aL (dm3 mmol−1 ) qmax (mmol g−1 )
Copper
Cadmium
Zinc
49.054 69.199 0.709
75.354 157.812 0.477
31.399 62.176 0.505
4.2. Kinetic studies—external mass transfer For the agitated tank, a method to determine the film diffusion coefficient was proposed by Furusawa and Smith [14] for the sorption of benzene onto activated carbon. Mathews and Weber [16] directly determined the external masstransfer coefficient from Eq. (1). Later, McKay and Allen [17,18] used the Henry, Langmuir, and Freundlich equations to calculate the external film mass-transfer coefficient for the sorption of basic dyes on peat. Allen and co-workers [19] applied the Henry equation to the sorption of metal ions onto peat and concluded that the mass-transfer coefficient was dependent on the initial concentration of the solution in the system. Tien [15] proposed to use the Harriot method to determine the external mass-transfer coefficient in batch agitation system. The external mass-transfer coefficient can be calculated from the Sherwood number, Reynolds number, and Schmidt number. Therefore, the external masstransfer coefficient is concentration-independent. Recently, Carta and co-workers [20–22], based on a chemical potential equation, developed a new approximation to calculate the external mass-transfer coefficients, which is concentrationdependent. The external mass-transfer coefficient can be directly calculated by the film diffusion equation proposed by Mathews and Weber [16] from Eq. (1) and referred to as the M&W model. Equation (1) can be used to calculate the external
mass-transfer coefficient when t → 0 and Cs → 0. Then, Eq. (1) can be integrated to become Ct = −kf SA t. ln (25) C0 An alternative method referred to as the F&S model, which was developed by Furusawa and Smith [14], calculates the external mass-transfer coefficient by incorporating the linear adsorption isotherm into Eq. (1) to become Eq. (16). This equation has been used to calculate the external masstransfer coefficient for many sorption systems [14,18,19]. In this research, the external mass-transfer coefficients using Eqs. (16) and (25) are compared. Initially, Eq. (16) is rearranged to Ct Ct 1 1 ln 1− − 1 + (1/ms KL ) C0 ms KL C0 = −kf SA t.
(26)
By plotting 1 1 Ct Ct − 1− ln 1 + (1/ms KL ) C0 ms KL C0 against t, the external mass-transfer coefficient can be determined from the slope of the straight line. Plotting the linear Eq. (26) is easier than plotting Eq. (16) because the intercept of the straight line must be adjusted to ln(ms KL /(1 + ms KL )), while the intercept of Eq. (26) is equal to zero. This results in easier determination of the initial gradient (as t → 0). In addition, Eq. (26) can be converted to Eq. (25) when V 1 1 = → 0, (27) ms KL W qm aL where qm and aL are the maximum capacity and the affinity, respectively, of sorbate to sorbent. These parameters are the
Fig. 2. Sorption of three metal ions onto bone char using the Langmuir isotherm.
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intrinsic properties of sorbent in relation to sorbate, which cannot be changed. In Eq. (27), the V /m ratio is the only adjustable parameter in this expression. When the mass of sorbent increases, the term 1/ms KL tends to zero. Consider the assumption in Eq. (10) when 1 + aL Cr ≈ 1, Eq. (16) can be obtained. The conditions 1+ aL Cr ≈ 1 and 1/ms KL → 0 apply to very dilute solution systems or the use of a large mass of sorbent in solution. Therefore, the use of Eqs. (25) and (26) for dilute solution is identical. Figs. 3, 4, and 5 show the plots of the external mass transfer equations for the three metal ions. The plots of Eqs. (25) and (26) are very close.
Fig. 3. The sorption of Cd ions onto bone char correlating the Furusawa and Smith equation and the film-diffusion equation of Mathews and Weber.
Fig. 4. The sorption of Cu ions onto bone char correlating the Furusawa and Smith equation and the film-diffusion equation of Mathews and Weber.
289
The Harriott equation was developed using dimensional analysis to calculate the external mass-transfer coefficient. Fig. 6 is the plot of Sh/Sc0.33 against particle Reynolds number. Increasing the particle sizes will slightly decrease the external mass-transfer coefficients in Table 2. The smaller sorbent particle travel faster in the agitated solution and experience more shear at the particle surface, therefore reducing the boundary layer film. A similar plot was produced by Harriott who used the equation to calculate the external mass-transfer coefficients of ion-exchange resins [10]. The external mass-transfer coefficients for different bone char particle sizes were calculated using the Harriott equation and are shown in Table 3. The external mass-transfer coefficients
Fig. 5. The sorption of Zn ions onto bone char correlating the Furusawa and Smith equation and the film-diffusion equation of Mathews and Weber.
Fig. 6. Sh/Sc0.33 vs Re for the sorption of Cd ions onto bone char with different particle sizes.
Table 2 External mass-transfer coefficients using the Harriott equation Particle size (µm)
kf dp ν −1/3 Sh Sc−1/3 = D Dm m
ε¯ dp4 1/3 Re = 3 ν
kf (cm s−1 )
250–355
355–500
500–710
710–1000
0.24
0.32
0.43
0.58
0.95
1.51
2.40
3.81
2.79 × 10−3
2.59 × 10−3
2.46 × 10−3
2.38 × 10−3
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Table 3 The external mass-transfer coefficients for different particle size ranges using the Harriott equation Particle size (µm)
Cd2+ (cm s−1 )
Cu2+ (cm s−1 )
Zn2+ (cm s−1 )
250–355 355–500 500–710 710–1000
2.79 × 10−3 2.59 × 10−3 2.46 × 10−3 2.38 × 10−3
2.77 × 10−3 2.57 × 10−3 2.44 × 10−3 2.37 × 10−3
2.74 × 10−3 2.54 × 10−3 2.41 × 10−3 2.34 × 10−3
using different methods have been calculated and are listed in Tables 4–9. The differences between the linear and nonlinear external mass-transfer coefficients are relatively small. The magnitudes of these coefficients, which were calculated by Eqs. (25) and (26), are of order 10−4 cm s−1 . The external mass-transfer coefficients calculated by these methods do vary with the initial concentration of solution and mass of sorbent. The results in Tables 4–6 indicate that kf values show a slow steady decrease with increasing initial metal ion concentration. This would not normally be expected in a single resistance mass transport model and is indicative that intraparticle diffusion is playing a significant role in the mass transport process. When the mass of sorbent is increased, the external mass-transfer coefficients vary only slightly and without trend. In contrast, the external mass-transfer coef-
ficients calculated by the Harriott equation are completely independent of initial concentrations and sorbent masses. The magnitude of the external mass-transfer coefficients by the Harriott equation is 10−3 cm s−1 . Classical treatment of external mass transfer or boundary layer theory [23,24] assumes that it is dependent on the diffusion of molecular species or ions across a stagnant layer of liquid, “the film,” of thickness (δ). Under limiting conditions these concepts are valid and the external mass-transfer coefficient, kf , is a constant value depending on the fixed extent of agitation of the system and the constant diffusivity of the system. The Harriott equation is a solution of these fixed conditions, correlating with kf through the Sherwood number as a function of Reynolds and Schmidt numbers. Using this model, kf will only vary as a function of agitation. However, based on the evaluations of kf in the present paper and other works [25], kf may show changes with liquid and solid phase concentration. In the present systems the diffusion in the boundary layer not only is based on metal ions flowing across the boundary layer from the bulk solution to the bone char surface, but there is a concentration gradient across the boundary due to calcium ions diffusing in a counter direction from the surface to the bulk solution. This rate depends on the concentration of metal ions and the mass concentration of bone char resulting in a variable apparent kf .
Table 4 The parameters of the external mass-transfer equations for the sorption of cadmium ions onto bone char at different initial concentrations of solution C0 (mmol dm−3 )
M&W (cm s−1 )
F&S (linear) (cm s−1 )
F&S (nonlinear) (cm s−1 )
Harriott (cm s−1 )
2.0 2.5 3.0 4.0 5.0
6.23 × 10−4 5.28 × 10−4 4.95 × 10−4 4.12 × 10−4 3.55 × 10−4
7.30 × 10−4 6.14 × 10−4 5.84 × 10−4 4.94 × 10−4 4.36 × 10−4
7.45 × 10−4 6.23 × 10−4 5.94 × 10−4 5.01 × 10−4 4.42 × 10−4
2.46 × 10−3 2.46 × 10−3 2.46 × 10−3 2.46 × 10−3 2.46 × 10−3
Table 5 The parameters of the external mass-transfer equations for the sorption of copper ions onto bone char at different initial concentrations of solution C0 (mmol dm−3 )
M&W (cm s−1 )
F&S (linear) (cm s−1 )
F&S (nonlinear) (cm s−1 )
Harriott (cm s−1 )
2.0 2.5 3.0 4.0 5.0
9.14 × 10−4 6.84 × 10−4 7.02 × 10−4 5.60 × 10−4 4.80 × 10−4
1.06 × 10−3 9.02 × 10−4 8.44 × 10−4 7.01 × 10−4 6.14 × 10−4
1.09 × 10−3 9.38 × 10−4 8.64 × 10−4 7.19 × 10−4 6.29 × 10−4
2.44 × 10−3 2.44 × 10−3 2.44 × 10−3 2.44 × 10−3 2.44 × 10−3
Table 6 The parameters of the external mass-transfer equations for the sorption of zinc ions onto bone char at different initial concentrations of solution C0 (mmol dm−3 )
M&W (cm s−1 )
F&S (linear) (cm s−1 )
F&S (nonlinear) (cm s−1 )
Harriott (cm s−1 )
2.0 2.5 3.0 4.0 5.0
5.97 × 10−4 5.54 × 10−4 4.75 × 10−4 4.65 × 10−4 3.89 × 10−4
7.39 × 10−4 6.34 × 10−4 5.95 × 10−4 5.53 × 10−4 4.75 × 10−4
7.58 × 10−4 6.44 × 10−4 6.09 × 10−4 5.63 × 10−4 4.83 × 10−4
2.41 × 10−3 2.41 × 10−3 2.41 × 10−3 2.41 × 10−3 2.41 × 10−3
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Table 7 The parameters of the external mass-transfer equations for the sorption of cadmium ions onto bone char at different masses of sorbent Mass (g)
M&W (cm s−1 )
F&S (linear) (cm s−1 )
F&S (nonlinear) (cm s−1 )
Harriott (cm s−1 )
6.5 7.5 8.5 9.5 10.5
4.52 × 10−4 4.54 × 10−4 4.95 × 10−4 5.18 × 10−4 5.23 × 10−4
5.47 × 10−4 5.61 × 10−4 5.84 × 10−4 6.28 × 10−4 6.23 × 10−4
5.56 × 10−4 5.69 × 10−4 5.94 × 10−4 5.32 × 10−4 6.29 × 10−4
2.46 × 10−3 2.46 × 10−3 2.46 × 10−3 2.46 × 10−3 2.46 × 10−3
Table 8 The parameters of the external mass-transfer equations for the sorption of copper ions onto bone char at different masses of sorbent Mass (g)
M&W (cm s−1 )
F&S (linear) (cm s−1 )
F&S (nonlinear) (cm s−1 )
Harriott (cm s−1 )
6.5 7.5 8.5 9.5 10.5
6.32 × 10−4 7.24 × 10−4 7.02 × 10−4 6.70 × 10−4 7.74 × 10−4
8.01 × 10−4 8.42 × 10−4 8.44 × 10−4 8.07 × 10−4 8.99 × 10−4
8.11 × 10−4 8.62 × 10−4 8.64 × 10−4 8.34 × 10−4 9.26 × 10−4
2.44 × 10−3 2.44 × 10−3 2.44 × 10−3 2.44 × 10−3 2.44 × 10−3
Table 9 The parameters of the external mass-transfer equations for the sorption of zinc ions onto bone char at different masses of sorbent Mass (g)
M&W (cm s−1 )
F&S (linear) (cm s−1 )
F&S (nonlinear) (cm s−1 )
Harriott (cm s−1 )
6.5 7.5 8.5 9.5 10.5
4.34 × 10−4 5.09 × 10−4 4.75 × 10−4 4.55 × 10−4 5.24 × 10−4
5.34 × 10−4 6.20 × 10−4 5.95 × 10−4 5.71 × 10−4 6.40 × 10−4
5.37 × 10−4 6.34 × 10−4 6.09 × 10−4 5.85 × 10−4 6.59 × 10−4
2.41 × 10−3 2.41 × 10−3 2.41 × 10−3 2.41 × 10−3 2.41 × 10−3
The operating lines of Cu2+ ion systems join (C0 , q0 ) to (Ce , qe ) at equilibrium and the effect of changing C0 with constant mass is to generate a series of parallel operating lines, as shown in Fig. 7. The driving force for mass transfer, based on the difference between the bulk liquid concentration and the particle surface liquid concentration, is timedependent, and this condition is reflected by the tie lines. Thus the external mass-transfer coefficients are apparently increasing as C0 decreases for the three metal ions being adsorbed onto bone char. In Figs. 3–5, the initial adsorption rate in the first 5 min is higher, but considerable error is involved using only one data point to measure the external mass-transfer coefficient for these systems. On this basis it was decided to evaluate the kf values over the first six (5-min interval) data points, that is, the first 30 min of metal ion solution in contact with bone char. Although there was no statistical justification for this selection, it does represent the initial sorption period covering approximately 15% of the bone sorption capacity. Moreover, based on the application of the models to the first 30 min of sorption, the sorption capacities of the metal ions on bone char are only 15%. This means that the (Ct , qt ) points up to this position in time all lie on the steep initial slope of the isotherms. Therefore, the
Fig. 7. The relationship of the adsorption isotherm and the mass-transport model (effect of initial concentration and sorbent mass on the sorption of Cd ions onto bone char).
assumption of a steep initial linear slope is not unreasonable. This assumption could lead to some degree of error in the evaluation of kf with C0 . McKay and Allen [25] served that the external mass-transfer coefficients for the adsorption of dye ions on peat presented a linear relationship between ln(C0 ) and ln(Sh/Sc0.33 ). The operating lines for several carbon masses and a constant initial dye concentration are also shown in Fig. 7. Since the masses are varying, the slopes of the operating lines vary accordingly. The external mass-transfer coefficient may depend on the driving force per unit area, and in this case, since C0 is constant, increasing the mass of carbon increases the surface for adsorption and hence the rate of metal ion removal is increased. Since the particle size range is constant, the surface area will be directly proportional to the mass of bone char in the system. The kf values are shown in Tables 7–9 and demonstrate a dependence on mass. This effect is probably due to the fact that for small masses a small amount of external surface is presented to the metal ion and therefore there is a large driving force from the metal ion per unit surface area of bone char. In these film diffusion studies, the discrepancies in the values of the film mass-transfer coefficients between the experimental methods and the dimensional analysis method can also be explained by the fast external mass transport of sorbates. The first three methods used the experimental data to calculate the film diffusion coefficients. However, the external mass transport of sorbate in these sorption systems may be very fast in comparison with the intraparticle diffusion. The experimental data used to calculate the film masstransfer coefficients may also be in the intraparticle diffusion region. In addition, the single-resistance model based on the mass transport of sorbate at the initial stage (as t → 0) only assumes that mass transfer is controlled by the external mass transport of sorbate only, but if the mass transport at the initial stages also involves film–pore or pore diffusion, the results of Eqs. (20), (25), and (26) will be affected to some extent.
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Fig. 8. The sorption capacity at specified times against the square root of time (concentration effect for the sorption of Cd ions onto bone char).
Fig. 9. The sorption capacity at specified time against the square root of time (particle size effect for the sorption of Cd ions onto bone char).
In order to minimize the extent that intraparticle diffusion may be occurring less during the initial 30 min of the sorption process, one experimental data point at the first 5 min is used to calculate the external mass-transfer coefficient using Eq. (26). A series of gradients were drawn through points on the experimental curve at times ranging from 0 to 30 min. At a time of 5 min, the gradient through the points (0, 0) and (5, qt ) gave the “closest fit” to the best fit kf vales obtained using the two-resistance simulation model. Therefore the time of 5 min was selected for kf determination. Another approach is to use the effectiveness factor method [25], which measures the fractional deviation of the single-resistance kf model from the experimental curve, which was still a tworesistance diffusion model. Therefore both approaches are still subject to the fact that most experimental sorption systems are two- or multiresistance mass transport systems. The external mass-transfer coefficients for the sorption of Cd2+ , Cu2+ , and Zn2+ ions at 3 mM are equivalent to 1.03 × 10−3 , 1.65 × 10−3 , and 1.09 × 10−3 cm s−1 , respectively. The magnitudes of external mass-transfer coefficients from this less statistically accurate approach are higher than the values obtained from the first three methods. Therefore, the choice and number of data points to calculate the external mass-transfer coefficient may be very important. In order to assess the contribution from intraparticle diffusion, Fig. 8 shows the sorption capacity of cadmium ions at specified times against the square root of time. Weber and Morris [26] stated that if intraparticle diffusion is the rate-controlling factor, uptake of the adsorbate varies in proportion to the square of time. When the linear plots of qt versus time0.5 (t 0.5 ) pass through the origin, then intraparticle diffusion may be the predominant rate-controlling step in the removal of Cd2+ ions [27,28]. Approximations of classical diffusion mathematics [29] show that for a diffusion process controlled by intraparticle diffusion only a plot of qt versus t 0.5 will yield a straight line passing through the origin. This has been demonstrated experimentally for the adsorption of surfactants on activated carbons [26]. In
several other reported systems [30,31], plots of qt versus t 0.5 have demonstrated linearity over most of the adsorption period except for a short time at the beginning of the adsorption process. This initial rapid uptake was attributed to an initial rapid surface adsorption, after the external surface loading was completed; however, the rate-controlling process becomes intraparticle diffusion, as demonstrated by the qt versus t 0.5 plots. In the case of Cd2+ , the qt versus t 0.5 plots are shown for four different particle size ranges in Fig. 8. In the two systems, with the large mean particle diameters and therefore relatively small external surface areas, the qt plots are linear and pass through or very close to the origin. This indicates that there is little or no external film control. However, in the two systems for dp , 355–500 and 250–355 µm, the t 0.5 intercepts show considerable deviation from the origin (0,0) demonstrating for an initial time period there is some controlling sorption contribution from external mass transfer (see Fig. 9). In the transition period from t = 0 to when intraparticle diffusion control takes over, there will be a transitional time zone. The sorptions of copper and zinc ions onto bone char have a similar characteristic pattern. Therefore, from Fig. 8, substituting the experimental data into the film diffusion equations to determine the film diffusion coefficient is only valid as t → 0 because intraparticle diffusion is the predominant mechanism throughout the bulk of the sorption process. More interesting results are shown in Fig. 9, showing that when the particle sizes of sorbent decrease, the intercept of the plot of the square root of time for the smaller particle sizes deviates from the point of origin. In the previous discussion for a bone char particle size range of 500–710 µm, the intraparticle diffusion is the main rate-limiting step for most of the metal ion/bone char sorption systems and the film diffusion is not the rate-limiting step except at the early stages of the sorption of metal ions onto bone char process. However, as the radius of the sorbent particle decreases, the mass transport of sorbate inside the sorbent becomes shorter, and the film diffusion effect on the metal ion removal rate
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293
Table 10 Simulation results from film–pore diffusion model using different kf values M&W
F&S (linear)
F&S (nonlinear)
Harriott
Film–pore best-fit
Cd
Averaged kf (cm s−1 ) Optimized Dp (cm2 s−1 ) SSE
4.69 × 10−4 1.22 × 10−6 1537
5.63 × 10−4 1.22 × 10−6 1969
5.58 × 10−4 1.22 × 10−6 1948
24.6 × 10−4 1.22 × 10−6 4442
4.04 × 10−4 1.22 × 10−6 807
Cu
Averaged kf (cm s−1 ) Optimized Dp (cm2 s−1 ) SSE
6.56 × 10−4 1.41 × 10−6 386
8.04 × 10−4 1.41 × 10−6 413
8.26 × 10−4 1.41 × 10−6 419
24.4 × 10−4 1.41 × 10−6 767
6.38 × 10−4 1.41 × 10−6 385
Zn
Averaged kf (cm s−1 ) Optimized Dp (cm2 s−1 ) SSE
4.89 × 10−4 1.28 × 10−6 532
5.96 × 10−4 1.28 × 10−6 683
6.08 × 10−4 1.28 × 10−6 698
24.1 × 10−4 1.28 × 10−6 1491
2.76 × 10−4 1.28 × 10−6 290
Fig. 10. The effect of kf for four different models on the initial 30 min of the batch kinetic experiment using the film–pore diffusion model (Cu2+ ion, C0 = 2.11 mmol dm−3 , dp = 500–710 µm, Dp = 1.4 × 10−6 cm2 s−1 ).
becomes more and more significant. In contrast, when the particle size increases, the mass transport distance of metal ions inside the sorbent becomes longer, and the film masstransfer effect becomes less significant. These trends are apparent in Fig. 9. As the external mass-transfer coefficients obtained from the four methods are different, it is difficult to decide which approach is correct. Similar findings were obtained by Allen et al. [32]. The authors used both the Furusawa and Smith equation and the homogeneous solid diffusion model (HSDM) [33] to determine the external mass-transfer coefficient. The magnitudes based on the HSDM are 10−3 cm s−1 , but using the Furusawa and Smith equation the value is 10−4 cm s−1 , and the authors selected the best-fit external mass-transfer coefficient from HSDM. Therefore, in the present work it was decided to compare the goodness of data fitting of a two-resistance film–pore intraparticle diffusion model for the sorption systems [34] based on these external mass-transfer coefficients, and the results are shown in Table 10 and Fig. 10. Visually, all four different methods provide a good correlation between the experimental data and the theoretical prediction of the sorption of Cu2+ ion
(C0 = 134 ppm) onto bone char (W = 8.5 g) in the period from 0 to 30 min. By comparing the sum of the square of the errors (SSE) values in Table 10 between the four different methods to obtain kf , it is found that the data-fitting methods (M&W, linear F&S, and nonlinear F&S) based on the experimental data points yield lower SSE values than the empirical correlated data (Harriot method), while the M&W method yields the lowest SSE in the four different methods. Thus, the use of the experimental data to obtain kf does result in an improvement over the kf determined from the empirical correlation. This would be expected because all three methods imply obtaining gradients or data from the initial experimental data points. If intraparticle diffusion control participates in the adsorption-rate-controlling resistance from very short times and onward, then even the early data points are indicative of combined diffusion mass transport. This highlights the major problem of trying to isolate and determine the film mass-transfer coefficient for most conventional adsorption systems. Consequently the effect of particle size was used to study which correlative method provides the most appropriate kf values. Therefore, simulation studies were performed to optimize the correlation between experimental and model data using a film–intraparticle diffusion model. The best-fit values of kf were obtained and compared with the various model values in Table 11. The simulation models confirm that the M&W method provides the best correlation for the external mass-transfer coefficients. The influence of the sensitivity of kf , in terms of the range of values relevant to the present work, is shown in Fig. 11 for the first 30 min of the sorption of Cu2+ ion onto bone char process. Seven theoretical values of kf between 5 × 10−2 and 5 × 10−5 cm s−1 have been selected in the sensitivity analysis, and it can be seen that the lower the kf value, the greater becomes the sensitivity of kf . At high values of kf (>5×10−3 ), the film– pore diffusion model slightly overestimates the rate of Cu2+ ion adsorption onto bone char. Using the range of 5 × 10−4 to 8.5 × 10−4 kf values, obtained from M&W, F&S (linear), and F&S (nonlinear) methods in the film–pore diffusion model does make a very small difference in the prediction of adsorption process decay curves. However, great sensitivity
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Table 11 The parameters of the external mass-transfer equations for the sorption of cadmium ions onto bone char at different particle sizes Particle size (µm)
M&W (cm s−1 )
F&S (linear) (cm s−1 )
F&S (nonlinear) (cm s−1 )
Harriott (cm s−1 )
Best fit (cm s−1 )
250–355 355–500 500–710 710–1000
9.21 × 10−4 6.91 × 10−4 4.69 × 10−4 6.93 × 10−4
9.96 × 10−4 7.83 × 10−4 5.63 × 10−4 7.84 × 10−4
1.07 × 10−3 7.96 × 10−4 5.58 × 10−4 7.99 × 10−4
2.79 × 10−3 2.59 × 10−3 24.6 × 10−4 2.38 × 10−3
8.32 × 10−4 6.09 × 10−4 4.04 × 10−4 7.10 × 10−4
Acknowledgments The authors are grateful to DAG and RGC, Hong Kong SAR, for the provision of financial support during this research program.
Appendix A. Nomenclature aL Bi C0 Ce Fig. 11. Sensitivity analysis of the effect of kf on the initial 30 min of the batch kinetic experiment using the film–pore diffusion model (Cu2+ ion, C0 = 2.11 mmol dm−3 , dp = 500–710 µm, Dp = 1.4 × 10−6 cm2 s−1 ).
was found on the concentration decay curve when the kf value is smaller than 2.5 × 10−4 .
5. Summary The application of the intraparticle diffusion, root time model demonstrates that intraparticle diffusion is by far the dominant mechanism responsible for the sorption of copper, cadmium, and zinc ions onto bone char, except at the very beginning of the sorption process for a bone char particle size of diameter 500–710 µm. However, as the particle size decreases the influence of the external film external mass transfer on the sorption of the metal ions becomes much more significant. Results for bone char particle diameter ranges of 250–355 and 355–500 show film to be increasingly important. The film coefficients are best predicted by the Mathews and Weber model, followed by the linear Furusawa and Smith model and the nonlinear Furusawa and Smith model, with the empirical correlation model giving the poorest prediction. Overall this work highlights the problems of easily defining an absolute kf value that is representative of conventional agitated batch adsorption systems independent of other system parameters. Several experimental data points are required at short contact times. These “short-time” points are subject to errors due to measurement of small changes at high concentrations, establishing uniform experimental conditions, such as agitation mixing, adsorbent wetting, and time of sampling during short contact times.
Cs Cr Ct dp Deff Dm kf KL ms qe qr r R Re SA Sc Sh t W εp ρp ρ
Langmuir isotherm constant (dm3 mmol−1) Biot number, Bi = kf R/Deff initial liquid-phase concentration (mmol dm−3 ) equilibrium liquid-phase concentration (mmol dm−3 ) liquid-phase concentration at external sorbent surface (mmol dm−3 ) liquid-phase concentration at radius r (mmol dm−3 ) liquid-phase concentration at time t (mmol dm−3 ) diameter of sorbent (µm) effective diffusivity (cm2 s−1 ) molecular diffusion coefficient of metal ions (cm2 s−1 ) external mass-transfer coefficient (cm s−1 ) Langmuir constant (dm3 g−1 ) concentration of sorbent in liquid phase (g dm−3 ) equilibrium solid-phase concentration (mmol g−1 ) equilibrium solid-phase concentration at radius r (mmol g−1 ) radius of concentration front of metal ions penetrating adsorbent (cm) radius of adsorbent particle (cm) Reynold number, Re = (¯εdp4 /ν 3 )1/3 surface area of sorbent (m2 g−1 ) Schmidt number, Sc = ν/Dm Sherwood number, Sh = kf dp /Dm contact time (min) mass of sorbent (g) porosity particle density (g cm−3 ) solution density (g cm−3 )
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