Fractal-like Vermeulen Kinetic Equation for the Description of Diffusion

Article pubs.acs.org/JPCC

Fractal-like Vermeulen Kinetic Equation for the Description of Diffusion-Controlled Adsorption Dynamics Marco Balsamo† and Fabio Montagnaro*,‡ †

Dipartimento di Ingegneria Chimica, dei Materiali e della Produzione Industriale, Università degli Studi di Napoli Federico II, Piazzale Vincenzo Tecchio 80, 80125 Napoli, Italy ‡ Dipartimento di Scienze Chimiche, Università degli Studi di Napoli Federico II, Complesso Universitario di Monte Sant’Angelo, 80126 Napoli, Italy ABSTRACT: In this paper, we validated the applicability of a new fractallike Vermeulen equation for the analysis of Cd2+ kinetic adsorption data on fly ashes both as received and as treated by means of mechanical sieving and CO2/steam gasification. The modeling results showed a better agreement between theoretical and experimental data in the whole time range for the fractal-like equation, with respect to its classic counterpart, the latter generally underestimating the capture outcomes for short adsorption times. In the case of the raw ash, the nonfractal adsorption pattern was linked to its more homogeneous pore structure while the widening of the pore size distribution induced by the mechanical sieving determined a time-dependent intraparticle diffusivity. In the case of the gasified samples, the occurrence of a significant fraction of smaller pores ( 500 min, the classic formulation of the Vermeulen model underestimates the adsorption data for shorter times. As an example, Θ(t) at t = 250 min equals 0.41 from experiments while it is 0.39 and 0.35 for the FVER and VER models, respectively. Differences in the interpretation of kinetic adsorption data between the FVER and VER equations become more marked when both the gasified samples (DG10 and SG10) are taken into account. As a matter of fact, the fractal-like Vermeulen model is able to provide a very good match with the experimental data in the whole time range and for both DG10 and SG10 samples. On the other hand, the VER model completely fails in predicting the kinetic data for t < 1500 min, and discrepancies tend to increase at shorter adsorption times. For the three treated ashes (F25, DG10, and SG10), the fractallike approach gives values for + 0P in the 10−16−10−15 m2 min−1 range. The reported analysis clearly demonstrates that in the case of diffusion-controlled adsorption (cf. also Balsamo et al.11,13) a more reliable description of the process dynamics can be achieved by considering intraparticle diffusivities exhibiting temporal “memories”. 3.2. Intertwining between Fractal-like Diffusivities and Fly Ashes Textural Properties. Figure 2 depicts the time evolution of the intraparticle diffusivity obtained for the

2. MATHEMATICAL MODELING OF KINETIC DATA The homogeneous solid diffusion model is widely applied for the description of adsorption dynamics. The pollutant material balance in a differential element of an adsorbent particle considered as spherical can be expressed as (see Nomenclature for the symbols meaning)2 + ∂ ⎛ ∂Θ ⎞ ∂Θ = 2P ⎜r 2 ⎟ ∂t r ∂r ⎝ ∂r ⎠

(1)

Different approximate solutions of eq 1 have been proposed, and most of them generally provide an accurate description of the kinetic data only in specific Θ(t) ranges.2 The Vermeulen equation for the adsorption kinetics is a diffusion-based model, derived as an approximate solution of eq 1, that can be used to predict experimental data in the whole t-range. Its integral form, here named VER, is15 Θ(t ) =

⎛ 4π 2 + ⎞ P 1 − exp⎜ − t⎟ dS 2 ⎠ ⎝

(2)

In order to account for a nonuniform distribution of the pollutant in the adsorbent pores system during the adsorption, due to diffusion limitations, we propose the following fractal-like time dependence for the intraparticle diffusivity: +P = + 0Pt −h

0≤ h ≤ 1

(t ≥ 1)

(3) 16

where h represents a heterogeneity parameter. By introducing eq 3 into eq 2, one obtains the following fractal-like form of the Vermeulen model (named FVER): Θ(t ) =

⎞ ⎛ 4π 2 + 0 P (1 − h) ⎟ 1 − exp⎜ − t dS 2 ⎠ ⎝

(4)

In this work, the dynamic data modeling for cadmium adsorption onto fly ashes both raw (sample named CCA, coal combustion ash) and beneficiated by means of mechanical sieving (fraction finer than 25 μm), 10 min CO2 (dry) and steam gasification (samples denoted as F25, DG10, and SG10, respectively) was performed via nonlinear regression of experimental 8782

DOI: 10.1021/acs.jpcc.5b01783 J. Phys. Chem. C 2015, 119, 8781−8785

Article

The Journal of Physical Chemistry C

Figure 1. Cadmium adsorption onto (a) CCA, (b) F25, (c) DG10, and (d) SG10; C0 = 50 mg L−1, m = 1 g, V = 0.1 L: comparison between experimental data (black circles) and by fitting with classical (VER; gray lines) and fractal-like (FVER; black dashed lines) Vermeulen kinetic models. The inserts show the behavior for short adsorption times.

t = 1000 min DG10, SG10, and the raw sample exhibit similar values for +P ((5−6.6) × 10−16 m2 min−1), which is still 5-fold the value determined for the mechanically sieved sample. Finally, the intraparticle diffusivity is greater for CCA when compared to the other samples for adsorption times longer than 1000 min. In order to better compare the different kinetic behavior in the adsorption process for the investigated sorbents, it is useful to recall that the characteristic diffusion time is proportional to the ratio between the square of the particle diameter and the intraparticle diffusivity.11 Accordingly, for the gasified samples the larger value of +P (see Figure 2) and the lower value of dS with respect to the data derived for CCA (cf. Table 1, where some sorbents microstructural parameters have been reported) contribute in determining faster capture kinetics. In the case of the F25 sorbent, the reduction of the particle diameter induced by the mechanical sieving (dS is 3 and 10 μm for F25 and CCA, respectively) appears to be a more crucial factor in determining shorter distances for the adsorbate to be covered to reach the adsorption sites. Additionally, the porosity increase (see Φ values in Table 1) observed for the beneficiated ashes, and consequently the greater number of available diffusive paths for the adsorbate, should further produce faster cadmium removal rates with respect to CCA. Peculiar observations can be drawn when comparing the time dependence of the intraparticle diffusivity according to the proposed eq 3 with respect to other literature mathematical analyses derived in the case of pure diffusion into porous systems

Figure 2. Time evolution of intraparticle diffusivities for cadmium adsorption onto the investigated sorbents (semilog scale).

investigated sorbents according to eq 3. As already outlined in section 3.1, the intraparticle diffusivity is constant (h = 0) for CCA. +P vs t profiles show values of the intraparticle diffusivity significantly greater for the gasified samples with respect to both CCA and F25 in almost the entire temporal range analyzed. More specifically, the initial value (t = 1 min) of the intraparticle diffusivity is 1.8, 14.5, and 31.9 times greater for the CO2-gasified sample DG10 with respect to SG10, CCA, and F25, respectively. For longer adsorption times, the differences diminish, and for 8783

DOI: 10.1021/acs.jpcc.5b01783 J. Phys. Chem. C 2015, 119, 8781−8785

Article

The Journal of Physical Chemistry C

intraparticle diffusivity. The applicability of the fractal-like Vermeulen equation was verified for kinetic uptake data obtained from cadmium adsorption on combustion fly ash both raw and treated via mechanical sieving and CO2/steam gasification. Theoretical kinetic profiles clearly demonstrated the superior fitting accuracy of the fractal-like equation with respect to its canonical formulation for all the beneficiated ashes, while for the parent sorbent a time-independent intraparticle diffusivity was derived. A very interesting relationship was observed between the fractal-like adsorption behavior and the sorbents microstructural properties. In fact, we related the greater degree of porosimetric heterogeneity determined by the beneficiation treatments to a more marked time dependence of the intraparticle diffusivity. Finally, we physically explained the limiting zero value of the intraparticle diffusion coefficient for long adsorption times, predicted by the proposed fractal-like expression, by considering that under equilibrium conditions the diffusion of the adsorbate in the porous system is strongly hindered by interaction forces exerted by adsorption sites.

(i.e., in the absence of adsorption phenomena). In fact, according to Sen,6 the intraparticle diffusivity approaches a nonzero value for t → ∞ for diffusion in a well-connected system because the walker displacement is hindered only by collisions with pore walls. On the contrary, the proposed fractal-like dependence produces the limiting case +P = 0 for long adsorption times. From a physical point of view, this could be justified considering that when approaching saturation conditions, the force fields determined by pore walls strongly confine the adsorbate molecules, thus avoiding the exploration of the pore space. Interesting links between the modeling analysis and the sorbents textural parameters can be built by comparing the values of the fractal exponent h listed in Table 1 with the fly ashes absolute porosimetric distributions depicted in Figure 3. It can be

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +39 081 674029. Fax: +39 081 674090. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The contribution of Giorgio Totarella (UniNa) in performing some modeling evaluation is gratefully acknowledged. Figure 3. Absolute porosimetric distributions for CCA, F25, DG10, and SG10 (semilog scale).

C0 dS +P + 0P h m r Rp t V Vp

generally argued that the greater degree of porosimetric heterogeneity induced by the beneficiation treatments should determine a more marked fractal behavior in the adsorption process (i.e., larger values for the h parameter). As a matter of fact, both CCA and F25 display a substantially unimodal distribution, which is wider for the mechanically sieved sample because of an increase of the pore fraction in the 300−1000 nm size range. The observed differences should be responsible for the fractal pattern observed for the cadmium uptake onto F25, whereas more similar pore sizes for the raw ash should reduce the pollutant intraparticle concentration gradients, which in turn translates to time-independent transport properties. The gasified samples generally display very similar and bimodal porosimetric distributions. When compared to CCA, it is observed a slight reduction of the fraction of pores wider than 500 nm, but both gasification treatments end up in the occurrence of a relevant fraction of pores smaller than 50 nm. Consequently, the more pronounced heterogeneity degree highlighted for the pore size distributions of the gasified samples can be linked to the larger values of the fractal exponent. Moreover, it is reasonable to hypothesize that very small pores act as “bottlenecks” for the diffusion, hindering the establishment of uniform composition profiles along the particle radial coordinate.

NOMENCLATURE cadmium initial concentration in the liquid phase [mg L−1] mean Sauter particle diameter [μm] intraparticle diffusivity [m2 min−1] fractal diffusion kinetic constant [m2 min−(1−h)] fractal exponent [−] sorbent dose [g] particle radial coordinate [m] pore radius [nm] adsorption time [min] volume of liquid solution [L] specific pore volume [mm3 g−1]

Greek Symbols

Θ surface coverage degree with respect to equilibrium conditions [−] Φ specific cumulative pore volume [mm3 g−1]

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REFERENCES

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4. CONCLUSIONS In this work, we extended the recently proposed fractal concepts for the description of the adsorption dynamics to diffusioncontrolled capture processes by proposing a new Vermeulenbased kinetic equation including a fractal-like expression for the 8784

DOI: 10.1021/acs.jpcc.5b01783 J. Phys. Chem. C 2015, 119, 8781−8785

Article

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DOI: 10.1021/acs.jpcc.5b01783 J. Phys. Chem. C 2015, 119, 8781−8785