Application of diffusive transport model for better

Chinese Journal of Chemical Engineering 24 (2016) 1161–1165

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Chinese Journal of Chemical Engineering journal homepage: www.elsevier.com/locate/CJChE

Separation Science and Engineering

Application of diffusive transport model for better insight into retardation mechanisms involved in ion-imprinted membrane transport Ehsan Salehi ⁎ Department of Chemical Engineering, Faculty of Engineering, Arak University, Arak 38156-8-8349, Iran

a r t i c l e

i n f o

Article history: Received 7 September 2015 Received in revised form 15 March 2016 Accepted 22 March 2016 Available online 29 April 2016 Keywords: Ion-imprinted membrane Mathematical modeling Chemisorption Retardation mechanism

a b s t r a c t Heavy metal removal from water is a great concern for environmentalists and engineers. Ion-imprinted membranes are among the state of the art technologies for selective adsorption of heavy metals from aqueous environment. Dialysis permeation of nickel ions through Ni(II)-imprinted membranes has been thermodynamically studied in our prior work. In current study, the diffusive transport model was developed and then applied for better insight into the retardation mechanisms involved in the ion-imprinted membrane transport. The Sips isotherm model was coupled with the transport model to obtain the governing equation. Chemisorption and physical interactions (bulk diffusion and pore-clogging) were the most probable retardation mechanisms according to the modeling results. Relative retardation factor (η) was also defined as; transport-rate controlled by chemical adsorption to that controlled by physical interactions. With the help of the retardation factor, it was understood that the membrane behavior gradually changes from chemisorption to facilitated transport during permeation time. Effect of important operating parameters such as time, temperature and concentration on transport behavior was also investigated. Results indicated that chemisorption rate is rather higher at lower concentrations, early permeation times and reduced temperatures. In addition, η tabulated greater values for Ni(II) compared to Co(II) due to the imprinting effect. © 2016 The Chemical Industry and Engineering Society of China, and Chemical Industry Press. All rights reserved.

1. Introduction Mathematical modeling is an interesting and useful approach to upgrade researchers' knowledge about membrane transport mechanisms. A great deal of researches has been aimed at modeling of transport phenomena in membranes. For example, the effect of concentration gradient is included in the Stephan–Maxwell model [1], to extend the model for describing transport of charged particles through surfacecharged porous membranes [1–3]. Membrane adsorption is a promising technology for the removal of macromolecules and heavy metals from effluents [4–6]. Adsorptive membranes, prepared from reactive and functional polymers through complex physico-chemical synthesizing steps, carry reactive groups as adsorption sites to chelate macromolecules and heavy metals from aqueous phase [6–8]. Mathematical modeling of membrane adsorbents, due to its novelty, is of great importance from both microscopic and macroscopic aspects. From macroscopic outlook, rapid prototyping and presenting efficient scale-up protocols are some of the major advantages of transport modeling. In microscopic view however, useful knowledge about the interactions/mechanisms involved in the process can be obtained in the light of the modeling results. Common adsorption/transport models ⁎ Tel.: +98 918 8523253. E-mail address: [email protected].

are derived based on mass balance principles. These types of mathematical models take advantage of convection, diffusion and adsorption mechanisms, in combination, for describing transport behavior of adsorptive membranes [9,10]. Adsorption isotherms are sometimes combined with the mathematical transport models to characterize the equilibrium or quasi-equilibrium adsorption during transport. Heavy metal removal from water resources is a great concern for environmentalists. Ion-imprinted membrane adsorbents are potential solutions for selective removal of heavy metals. Recently, the ionimprinting technique has been employed to enhance the selectivity of the adsorptive membranes [11]. Moreover, specific recognition sites (artificial receptors) are contrived in an appropriate polymeric background via insertion and subsequent extraction of template ions [12,13]. Shape, orientation and functionality of the introduced cavities/ sites are matching with the track-etched template ions. In sum, ionimprinting process results in adsorptive membranes with specific affinity and selectivity toward the imprinted ions. There is a strong motivation for better understanding of the transport mechanisms involved in imprinted membrane transport. Previous researches have shown that the principle mechanisms for the removal of heavy metal ions by the ion-imprinted adsorbents are adsorption (surface adsorption, complexation and chelation) and ion-exchange [14]. Transport mechanisms however, are not fully understood for the ion-imprinted membranes. A novel mathematical model for describing Ni(II) ions permeation through Ni(II)-imprinted membrane was

http://dx.doi.org/10.1016/j.cjche.2016.04.034 1004-9541/© 2016 The Chemical Industry and Engineering Society of China, and Chemical Industry Press. All rights reserved.

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E. Salehi / Chinese Journal of Chemical Engineering 24 (2016) 1161–1165

developed in our previous study [15]. The proposed model could successfully simulate the transport of Ni(II) and Co(II) ions across the imprinted membrane for a routine dialysis permeation process. In the current work, transport mechanism of nickel and cobalt ions through Ni(II)-imprinted membrane was investigated with more focus on the retardation mechanisms affecting the diffusive transport. The mathematical model was developed and then applied to elucidate the controlling mechanisms/interactions involved in diffusive transport of ions through the membranes i.e., physical interactions and chemical adsorption. As a result, relative retardation factor (η) was defined to illustrate the relative importance of these retardation mechanisms during permeation. Effect of important parameters such as temperature, ion concentration and time on the retardation factor was also examined.

independent variables (x and t). The following boundary and initial conditions were applied for the mathematical analysis of Eq. (2): x ¼ 0; t N0

ð3Þ

∂c ¼ 0 x ¼ δ; t N0 ∂x

ð4Þ

c¼0

ð5Þ

c ¼ c0

xN0; t ¼ 0 :

One can ignore the chemical adsorption term to obtain the pure effect of physical interactions on the diffusive transport of the ions. Accordingly, the reduced Partial Differential Equation (PDE) is obtained as: 2

∂c ∂ c ¼ εDi 2 ∂t ∂x

2. Mathematical Model Extension

ε

Unsteady state diffusive transport model (Eq. (1)), can satisfactorily describe transport of ions/molecules through an adsorptive membrane [15]:

On the other hand, physical interactions may be overlooked to obtain a partial differential equation indicating the pure effect of chemical adsorption on transport mechanism: "

2

∂c ∂Q ∂ c þ ð1−ε Þρs ¼ εDi 2 ∂t ∂t ∂x

ε

ð1Þ

ð1−ε Þ

where ε is the membrane bulk porosity, c is the ion concentration in aqueous phase, Q is the adsorption on recognition sites (solid phase), Di is the ions diffusivity across the membrane, ρs is the dry bulk density, and x indicates the transport direction perpendicular to the membrane surface. Right-hand side of Eq. (1) demonstrates the role of diffusion dialysis in permeation transport. On the left-hand side of Eq. (1), the unsteadystate term indicates couple of the possible ways for the accumulation of ions in adsorptive membrane i.e., physical accumulation (first term) and chemical adsorption (second term). Moreover, adsorption includes diffusion of solute from the bulk fluid to the solid surface in which the chemical surface interactions occur. At the solid surface, the solute (ions) interacts with the reactive sites through either physical interactions (physisorption) or chemical interactions (chemisorption). Chemical interactions with the surface result in chemical bonding, complexation and chelation. Physical step however, includes bulk diffusion, film diffusion, pore clogging, surface diffusion and sieving mechanisms. The final governing equation has been obtained by coupling Eq. (1) with the Sips isotherm model ðQ e ¼

ε þ ð1−ε Þ  ρs 

#

abmcm−1 2

ð1 þ bcm Þ

2

ð1 þ bcmÞ

2

∂c ∂ c ¼ εDi 2 : ∂t ∂x

ð7Þ

" # Chemically‐controlled‐transport 1−ε abmcm−1 : ¼ ρs η¼ 2 Physically‐controlled‐transport ε ð1 þ bcm Þ

ð8Þ

The term "retardation" (Eq. (8)) is an attempt to evaluate the relative importance of the chemical retardation mechanisms (i.e. surface chemical interactions) to that of the physical ones. In other words, intensity and weakness of chemical to physical retardation mechanisms can be monitored during ion transport with the help of Eq. (8). 3. Materials and Methods

abC e m Þ. It is worthy 1þbC e m

Nickel and cobalt nitrates were obtained from Merck. All other chemicals were of analytical grade and used as received. Atomic absorption spectrometry (AA-6300 Shimadzu) was applied for analyzing the nickel and cobalt ions concentration. Membrane fabrication method was perfectly elucidated in another work from the current authors [13]. Briefly, methacrylic acid was polymerized on the surface of PVDF microfiltration membrane in the presence of Ni-dithizone as complex agent and ethylene glycol dimethacrylate as crosslinker. Hydrochloric acid (1 mol·L− 1) was employed for selective dissolution of template ions from the synthesized membrane. Subsequently, the membranes were abundantly rinsed with distilled water and air-dried.

2

∂c ∂ c ¼ εDi 2 : ∂t ∂x

#

abmcm−1

Considering Eqs. (6) and (7), relative retardation factor (η) can be defined as ion transport-rate controlled by chemisorption to that controlled by the physical attachment mechanisms. Accordingly, a straightforward definition for η is obtained by dividing Eq. (7) to Eq. (6), as follows:

of mention that the Sips model has been verified among several tested isotherms according to its higher coefficient of determination [15]. The resultant governing equation is obtained as follows: "

ð6Þ

ð2Þ

a, b and m are the equilibrium parameters of the Sips isotherm which have been obtained in another work from the current authors [15]. These parameters for adsorption at different temperatures are given in Table 1. Eq. (2) includes one dependent variable (c), and couple of

Table 1 Physical and thermodynamic parameters for Ni(II)-imprinted membrane adsorption [15] Temp./°C

Isotherm constants a

30 35 40

Physical properties b

ΔH°/kJ·mol−1

m

Di/cm2·h−1

ρs/g·cm−3

Ni(II)

Co(II)

Ni(II)

Co(II)

Ni(II)

Co(II)

Ni(II)

Co(II)

Ni(II)

Co(II)

Ni(II)

Co(II)

51.6 46.7 43.5

49.4 44.2 41.8

0.08 0.06 0.04

0.05 0.04 0.03

1.68 1.80 1.80

1.70 1.80 1.90

32.41

19.90

0.0022

0.0018

8.91

8.90

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Static (batch) adsorption tests were performed to obtain the equilibrium parameters of the adsorption isotherms. The procedure was fully described in our prior paper [15]. Diffusive transport of ions through the imprinted membrane was investigated according to straightforward dialysis permeation procedure. The experimental setup and applied procedure were explained in detail elsewhere [13,15]. In brief, a two-section dialysis cell was employed for dialysis permeation tests. Membrane samples were placed and sealed between the two half-cells of the setup. Feed- and receive-side half-cells were filled with 100 ml of nickel (or cobalt) nitrate solution (with concentration range of 1 to 25 mg·L−1) and distilled water, respectively. Three hours once, a few milliliter sample solution was taken out from the receive solution and then, immediately compensated with the same amount of distilled water.

permeation of the ions through the membrane. Therefore, onedimensional analysis approach could be employed without any meaningful error. As inferred from Fig. 1, the concentration of the ions inside the membrane matrix increases with time. This fact demonstrates the potential of the imprinted membrane for the ion uptake. Moreover, the ions can easily access the vacant recognition sites via facilitated transport mechanism [13–15]. Gradual extension of the concentration gradient inside the membrane matrix can facilitate the adsorption/transport mechanism. It is obvious from Fig. 1, that the concentration of the ions is higher in regions closer to the membrane inlet presumably due to superior ion uptake rate as well as minimum possible mass-transfer resistance against mass transfer in these regions.

4. Results and Discussion

4.1. Retardation mechanism

The governing equation (Eq. (2)) has been numerically solved to obtain the concentration of the ions as a function of permeation time and location throughout the imprinted membrane. Fig. 1, indicates the results for T = 35 °C, pH = 8 and c0 = 25 mg·L−1. The results were comparable for different initial concentrations (not shown). Model predictions were compared with the experimental data in Fig. 2. Results indicate that the mathematical model can satisfactorily simulate the permeation of Ni(II) ions through the Ni(II)-imprinted membrane. It is obvious from Fig. 1, that the dispersion has negligible effect on the

Based on Eq. (8), retardation factor (η) is the function of equilibrium concentration and temperature. Moreover, isotherm parameters (a, b and m) were employed to calculate η based on Eq. (8). These constants are clearly temperature dependent. In addition, equilibrium concentration is another independent variable of retardation factor. Results for Ni(II) and Co(II) ions are shown in Figs. 3 and 4, respectively. Initial concentration of the ions in the feed phase is in the range of 1 to 25 mg·L−1. According to the modeling results, the chemisorption-induced transport rate is around one thousand order of magnitude greater than the physical-interaction-controlled rate. This fact reveals that the imprinted membrane is originally an ‘adsorptive’ rather than a ‘size-exclusive’

Fig. 1. Concentration distribution of Ni(II) ions versus time and position in the Ni(II)imprinted membrane at T = 35 °C, pH = 8 and c0 = 25 mg·L−1.

Fig. 3. Retarding factor versus initial concentration at different temperatures for Ni(II) ions.

Fig. 2. Permeate concentration versus time: model prediction (dash-line) versus experimental data (triangles).

Fig. 4. Retarding factor versus initial concentration at different temperatures for Co(II) ions.

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membrane [11,15]. Generation of artificial receptor-like adsorption sites as a result of the imprinting process provides the condition for the imprinted membrane to act as a selective-adsorption carrier. In an ion-imprinted membrane, reactive reception sites play an important role in transport of the template ions. These sites can memorize, recognize and interact with the imprinted ions through different mechanisms such as chemical bonding, chelation and complexation. In other words, accumulation of ions in aqueous solution filled in the membrane pores is in lesser importance compared to the selective-chemisorption mechanism in ion-imprinted membrane transport. It is obvious from Figs. 3 and 4 that the retardation factor is higher for the nickel compared to the cobalt ions. This fact is originated from the imprinting process. Moreover, the artificial reception sites are better matching with Ni(II) (the imprinted ion) than with Co(II) ions [14]. Accordingly, the Ni(II)imprinted membrane offers higher affinity toward Ni(II) ions. 4.2. Effect of equilibrium concentration Retardation factor increases sharply with increasing concentration, meets a maximum and then, decreases and switches to a plateau at higher concentrations. Increasing η indicates the impact of chemisorption as controlling mechanism at initial concentration less than 3 mg·L− 1 (low concentrations). This may be attributed to the large volume of vacant reactive sites at lower concentrations due to lower ion uptake rate. In this condition, the imprinted membrane is a better candidate for adsorption compared to physical retardation. Physical interactions are strengthened with increasing ion concentration. Severe uptake of ions at higher concentrations impels the ions to accumulate in the pores due to limited number of the recognition sites. Therefore, the relative retardation factor decreases. This means gradual weakness of chemical adsorption mechanism. At higher concentrations (more than 15 mg·L−1), saturation condition brings about as a result of the concentration-gradient development. Unchanged value of retardation factor confirms this conclusion. At saturation condition; whereas, all the adsorption sites are occupied, there is a balance in the ion exchange rate between the recognition sites (solid phase) and the fluid filled the pores (liquid phase). Accordingly, the relative retardation factor remains constant. This reveals that the relative impact of chemisorption to physical interactions in control of mass transport does not further change. In this condition, imprinted membrane acts as a facilitatedtransport carrier. Fig. 5 schematically shows the change in the

imprinted-membrane transport behavior from chemically adsorptive to facilitated-exchange transport. This conclusion is in tune with the results obtained by other researchers [11–14]. 4.3. Effect of temperature Retardation factor was calculated at different temperatures. As previously mentioned, isotherm constants which play a pivotal role in determination of η are temperature dependent. It is clear from Figs. 3 and 4 that adsorption is superior controlling mechanism at lower temperatures. It is in agreement with the exothermic nature of the adsorption according to the positive amount of enthalpy (Table 1) [15]. In other word, reducing temperature provides more favorable condition for adsorption of ions from thermodynamic viewpoint. The effect of temperature is not significant at higher concentrations as obvious from Figs. 3 and 4. This may be attributed to the dominating effect of exchange rather than adsorption mechanism. In this condition, the kinetic of the adsorption/desorption cycle is not highly affected by raising the temperature. 4.4. Time dependency Variation of retardation factor versus time for Ni(II) permeation at two different initial concentrations is shown in Fig. 6. At first, η increases with time and then switches to a constant. Concentration gradient (or more exactly mass transfer zone) is extended inside the membrane matrix by gradual occupation of adsorption sites located on or near the surface. Adsorption capacity is fulfilled after full development of the concentration gradient. In this condition, the ion adsorption rate is in tune with the desorption rate. So, the concentration of the ions in the aqueous phase and that on the solid phase remains unchanged i.e., η switches to a constant.

Fig. 6. Retarding factor versus time at different initial concentrations for Ni(II) ions.

As inferred from Fig. 6, adsorption is dominating retardation mechanism at lower concentrations. This is attributed to the limited adsorption capacity of the imprinted membrane. At higher concentrations, larger number of the ions can accumulate in the fluid filled in the membrane pores. Accordingly, saturation occurs sooner and transport mechanism rapidly changes from adsorption-conducted to physical ion-exchange.

5. Conclusions Fig. 5. Ion-imprinted membrane transport mechanism; before (A) and after (B) saturation of recognition sites. (Dash lines: ion-exchange paths).

Diffusive transport model was utilized to derive a key-factor indicating relative importance of chemical adsorption to physical interactions

E. Salehi / Chinese Journal of Chemical Engineering 24 (2016) 1161–1165

during ion-imprinted membrane transport. Some bullet results inferred from the current study: 1. Chemisorption is much important controlling mechanism for ionimprinted membrane permeation compared to physical retardation. 2. Retardation factor increases with reducing temperature due to exothermal nature of adsorption. 3. Retardation factor is also greater for the permeation of Ni(II) ions compared to Co(II) ions as a result of the imprinting effect. 4. Separation mechanism of the imprinted membrane changes from adsorption to facilitated transport. 5. Adsorption to physical-attachment ratio meets a constant after saturation of the membrane. Nomenclature a, b, m isotherm constants c equilibrium concentration of ion in the solution in contact with the membrane, mg·L−1 c0 initial concentration of ion in the solution, mg·L−1 Di apparent diffusivity of ion through the membrane, cm2·h−1 Q ion adsorbed amount at time t, mg·g−1 t time, h x direction of ion diffusion through the membrane, cm adsorption enthalpy, kJ·mol−1 ΔH∘ δ membrane thickness, μm ε membrane bulk porosity (dimensionless) η retardation factor (dimensionless) dry bulk density, g·cm−3 ρs Acknowledgements The author acknowledges Arak University for supporting during this study.

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