Pore Surface Diffusion Model Research Articles

Predicting per‐ and polyfluoroalkyl substances removal in pilot‐scale granular activated carbon adsorbers from rapid small‐scale column tests

AbstractPer‐ and polyfluoroalkyl substances (PFAS) occur widely in drinking water, and consumption of contaminated drinking water is an important human exposure route. Granular activated carbon (GAC) adsorption can effectively remove PFAS from water. To support the design of GAC treatment systems, a rapid bench‐scale testing procedure and scale‐up approach are needed to assess the effects of GAC type, background water matrix, and empty bed contact time (EBCT) on GAC use rates. The overarching goal of this study was to predict PFAS breakthrough curves obtained at the pilot‐scale from rapid small‐scale column test (RSSCT) data. The scale‐up protocol was developed for pilot data obtained with coagulated/settled surface water (TOC = 2.3 mg/L), three GACs, and two EBCTs. Between 7 and 11 PFAS breakthrough curves were available for each pilot column. RSSCT designs were investigated that assumed intraparticle diffusivity is independent of GAC particle size (i.e., constant diffusivity [CD]) or linearly dependent on GAC particle size (i.e., proportional diffusivity [PD]). CD‐RSSCTs effectively predicted the bed volumes of water that could be treated at the pilot‐scale to reach 50% breakthrough (BV50%) of individual PFAS. In contrast, PD‐RSSCTs overpredicted BV50% obtained at the pilot‐scale by a factor of ~2–3. The shape of PFAS breakthrough curves obtained with CD‐RSSCTs deviated from those obtained at the pilot‐scale, indicating that intraparticle diffusivity was dependent on GAC particle diameter (dp). Using the pore surface diffusion model (PSDM), intraparticle diffusivity was found to be proportional to (dp)0.25 when considering data up to about 70% PFAS breakthrough. This proportionality factor can be used to design RSSCTs or scale up existing CD‐RSSCT data using the PSDM. Using pilot‐scale data obtained with groundwater and wastewater‐impacted groundwater as well as with additional GACs, the developed RSSCT scale‐up approach was validated for PFAS breakthrough percentages up to 70%. The presented methodology permits the rapid prediction of GAC use rates for PFAS removal.

Adsorption mechanism and mass transfer modeling of cesium and europium ions adsorption onto hematite-loaded polyacrylamide-gel composite: Pore Volume and Surface Diffusion Model Approach

The overall adsorption rate of cesium (Cs+) and europium (Eu3+) ions onto hematite-loaded polyacrylamide-gel composite (HPAC) in single and binary pollutant systems was explained with diffusional model that takes into consideration both external mass transfer and intra-particle diffusion. The influence of changing system temperature on the overall adsorption rate of Cs+ and Eu3+ onto HPAC was studied. The fitting process of the experimental data with the three studied isotherm models (Langmuir, Freundlich and Prausnitz–Radke (P-R)) showed that Prausnitz–Radke model has the highest correlation coefficients with the smallest average relative error percentage. The maximum P-R adsorption capacity of HPAC decreased from 2.06 and 3.35 L/g in single pollutant solution to 1.02 and 1.51 L/g in binary pollutant solution at 298 K for Cs+ and Eu3+,respectively. The values of surface diffusion coefficient (DS) ranged within 10−8 for Cs+ and 10−8-10−9 cm2/s for Eu3+ in both studied single and binary pollutant systems. The external mass transfer coefficient (KF) was in the range of 10−4 for Cs+ and 10−3 cm/s for Eu3+ in both studied systems. Lastly, the Biot number (Bi) exposed that the diffusion inside the particle controlled the studied cations's adsorption onto HPAC material.

Mechanistic Study of Arsenate Adsorption onto Different Amorphous Grades of Titanium (Hydr)Oxides Impregnated into a Point-of-Use Activated Carbon Block.

Millions of households still rely on drinking water from private wells or municipal systems with arsenic levels approaching or exceeding regulatory limits. Arsenic is a potent carcinogen, and there is no safe level of it in drinking water. Point-of-use (POU) treatment systems are a promising option to mitigate arsenic exposure. However, the most commonly used POU technology, an activated carbon block filter, is ineffective at removing arsenic. Our study aimed to explore the potential of impregnating carbon blocks with amorphous titanium (hydr)oxide (THO) to improve arsenic removal without introducing titanium (Ti) into the treated water. Four synthesis methods achieved 8-16 wt.% Ti loading within the carbon block with 58-97% amorphous THO content. The THO-modified carbon block could adsorb both oxidation states of arsenic (arsenate and arsenite) in batch or column tests. Modified carbon block with higher Ti and amorphous content always led to better arsenate removal, achieving arsenic loadings up to 31 mg As/mg Ti after 70,000 bed volumes in continuous flow tests. Impregnating carbon block with amorphous THO consistently outperformed impregnation using crystalline TiO2. The best-performing system (TTIP-EtOH carbon block) was an amorphous THO derived using titanium isopropoxide, ethanol, and acetic acid via sol-gel technique, aged at 80° for 18 hours and dried overnight at 60°. Comparable pore size distribution and surface area of the impregnated carbon blocks suggested that chemical properties play a more crucial role than physical and textural properties in removing arsenate via amorphous Ti-impregnated carbon block. Freundlich isotherms indicated energetically favorable adsorption for amorphous chemically synthesized adsorbents. The mass transport coefficients for the amorphous TTIP-EtOH carbon block were fitted using a pore surface diffusion model, resulting in Dsurface = 3.1×10-12 cm2/s and Dpore = 3.2×10-6 cm2/s. Impregnating the carbon block with THO enabled effective arsenic removal from water without adversely affecting the pressure drop across the unit or the carbon block's ability to remove polar organic chemical pollutants efficiently.

Develop and validate a mathematical model to estimate the removal of indoor VOCs by carbon filters

Adsorption-based air filters, especially carbon-based ones, are usually employed to remove VOCs from indoor air. Conducting tests at ppm level, which is substantially higher than the actual indoor concentration of VOCs, is recommended for reducing the test duration. Consequently, developing a model for estimating the service life of adsorptive filters under actual conditions is imperative.A comprehensive model that considers axially dispersed plug flow for interpellet mass transfer and pore surface diffusion model (PSDM) with variable surface diffusivity for intrapellet mass transfer was used to predict the performance of two filters exposed to three VOCs. First, the results of experiments conducted at concentrations ranging from 9 to 90 ppm in a bench-scale setup were used to compute the Dubinin-Radushkevich (D-R) isotherm parameters, which are concentration-independent. Then, the surface diffusivities at zero loading for various adsorbate-adsorbent systems were determined by fitting the developed model into the results of the experimental data performed at concentrations of 9 ppm or 30 ppm. Finally, the model was validated using experimental data which were conducted at lower concentrations, a higher velocity and on a full-scale setup.The inter-model comparison was carried out by comparing the comprehensive model with four other models to study the importance of various mass transfer steps. Finally, in sensitivity analysis, five dimensionless parameters (Pe, St, Edp, Eds and Dg) were examined to investigate their impact on the filter's efficiency.

Pore volume and surface diffusion model to characterize batch adsorption of Cu(II) over chemically modified Cucurbita moschata biosorbent: simulation using gPROMS

Abstract This work describes the successful application of the pore volume and surface diffusion (PVSD) model characterizing the batch adsorption of Cu(II) on a chemically modified Cucurbita moschata biosorbent. The PVSD model captures the convective transport of Cu(II) from the bulk solution to the biosorbent surface, followed by its surface and pore diffusion inside the biosorbent. The adsorption of Cu(II) is mimicked using the Langmuir isotherm. The algebraic, ordinary, and partial differential equations, involved in the PVSD model, are solved using the general process modeling system (gPROMS). The model simulation results, depicted by the Cu(II) concentration decay curve, show an excellent match with experimental data. The external mass transfer coefficient (≈10−3 m/s) indicated no restriction on approaching Cu(II) toward the biosorbent surface. Within the biosorbent, surface diffusion was dominant over pore volume diffusion. The statistical analysis of the PVSD model results has been done by calculating R2, Chi-square value, normalized standard deviation, p-value, and root-mean-square error. The PVSD model approach presented in this work could be beneficial to other heavy metal–biosorbent systems.

Construction of metal-organic framework/polymer beads for efficient lead ions removal from water: Experiment studies and full-scale performance prediction.

Metal-organic frameworks (MOFs) show great promise in heavy metal removal; however, their applications are restricted by the poor separability and water instability. Herein, granular Zr-based MOF-polymer composite beads (MPCB(Zr)) (mean diameter∼1.74mm) were synthesized using a facile dropping method, and applied on efficient lead ions (Pb(II)) removal. The as-prepared MPCB(Zr) demonstrated deep Pb(II) removal capability by reducing its concentration to∼0.002mgL-1 after adsorption equilibrium at 360min. The distribution coefficient for Pb(II) reached 8.0×106mLg-1, and the theoretical adsorption capacity for Pb(II) was 144.5mgg-1 (0.70mmolg-1, 30°C). The resulting MPCB(Zr) was highly selective for Pb(II), with the selectivity coefficient up to∼1.0-3.6×103 for the background cations (Na(I), K(I), Ca(II), and Mg(II)). Moreover, the MPCB(Zr) exhibited a broad working pH range (3.0-6.0) and satisfactory anti-interference to dissolved organic matters (humic acid and fuvic acid). Notably, the MPCB(Zr) also demonstrated excellent reusability with the Pb(II) removal efficiency over 99.0% after 20 cycles. Combined physicochemical characterizations unveiled that the thiol and oxygen-containing groups (e.g., hydroxyl, carboxylate) were responsible for the effective Pb(II) removal. To provide guidance for engineering application, the full-scale performance of the MPCB(Zr) under varying operation conditions was systematically evaluated via the validated pore surface diffusion model. This work provides an effective methodology to construct macroscopic MOF-polymer beads for effective Pb(II) removal, and promote the actual application of MOFs in water treatment.

Radix Astragali residue-derived porous amino-laced double-network hydrogel for efficient Pb(II) removal: Performance and modeling

Valorizing solid waste for heavy metal adsorption is highly desirable to avoid global natural resources depletion. In this study, we developed a new protocol to valorize Radix Astragali residue (one of the Chinese medicine residues) into a low-cost, chemically robust, and highly permeable (ca. 90%) amino-laced porous double-network hydrogel (NH2-CNFs/PAA) for efficient Pb(II) adsorption. The NH2-CNFs/PAA showed (i) excellent Pb(II) adsorption capacity (i.e., 994.5 mg g−1, ~4.8 mmol g−1), (ii) fast adsorption kinetics (kf = 2.01 ×10−5 m s−1), (iii) broad working pH range (2.0–6.0), and (iv) excellent regeneration capability (~15 cycles). (v) excellent performance in various real water matrices on Pb(II) removal. Moreover, its high selectivity (distribution coefficient Kd ~2.4 ×106 mL g−1) toward Pb(II) was owing to the present of abundant amino groups (-NH2). Furthermore, the fix-bed column test indicated the NH2-CNFs/PAA can effectively remove 114.6 bed volumes (influent concentration ~5000 μg L−1) with an enrichment factor 10.9. The full-scale system modeling (i.e., pore surface diffusion model (PSDM)) has been applied to predict the NH2-CNFs/PAA performance on Pb(II) removal. Overall, we have provided an alternative “win-win” scenario that can resolve the Chinese medicine residues disposal issue by valorizing it into high performance gel-based adsorbents for efficient heavy metal removal.

Modeling of anthocyanins adsorption onto chitosan films: An approach using the pore volume and surface diffusion model

The use of smart packaging is a global trend, and several works are being reported in this sense. Special attention is given to biodegradable materials like chitosan and natural pigments like anthocyanins. For example, previous studies report that it is possible to produce smart packaging by extracting anthocyanins from red cabbage and adsorbing it into chitosan films. This work elucidates for the first time the mass transfer aspects in the anthocyanins adsorption on chitosan films (CFS). Anthocyanins were extracted from red cabbage and subsequently adsorbed from the leach liquor using chitosan films. The Freundlich model well represented the adsorption isotherms of anthocyanins on CFS. In addition, the PVSDM (pore volume and surface diffusion model) model was implemented to represent the adsorption data and to elucidate the mass transfer. This model was suitable to represent the adsorption data, with a coefficient of determination (R2) higher than 0.95 and an average relative error (ARE) lower than 2.00%. The surface diffusion coefficient values (DS) ranged from 3.70 × 10−10 cm2 s−1 to 1.16 × 10−9 cm2 s−1. The external mass transfer coefficient (kF) was in the range of 10−2 cm s−1. Finally, the Biot number (Bi) revealed that mechanisms inside the particle controlled the anthocyanin's adsorption onto CFS.

A comprehensive investigation of external mass transfer and intraparticle diffusion for batch and continuous adsorption of heavy metals using pore volume and surface diffusion model

Separation of heavy metals from water bodies is a need of hour, to protect human kind, marine life, and the environment. In this context, adsorptive separation of heavy metal over cheap and environmental friendly biosorbent remains promising. Numerous studies have been carried out to characterize the adsorption of heavy metal ions through isotherms, kinetics, and simple mathematical models. However, such studies lacked in detailed mass transfer models evaluating heavy metal adsorption in batch and continuous modes. This work reports the application of pore volume and surface diffusion (PVSD) model investigating mass transfer processes of Cu (II), Cd (II), and Pb (II) adsorptions over alkali-treated Caryotaurens seeds, chemically modified Albizia lebbeck pods, and Schleichera oleosa bark, respectively, in batch and continuous modes. Heavy metal-biosorbent systems are represented by Freundlich, Temkin, and Langmuir isotherms, respectively. PVSD model equations (algebraic, ordinary, and partial differential equations) for batch and continuous modes are solved in gPROMS (General Process Modelling System). For validation, concentration decay and breakthrough curves concerning batch and continuous modes, respectively, are predicted and matched excellently with in-house experimental data and justified with R2 and RMSE values. In batch mode, external mass transfer coefficient suggested no constraint for heavy metal external transport, whereas, surface and pore volume diffusion coefficients were found to control the adsorption process. For continuous mode, breakthrough curves are predicted concerning metal solution flowrate, initial metal concentration, and bed height, and parameters viz. breakthrough time, exhaustion time, adsorption column capacity, and mass transfer zone are calculated. Axial dispersion and mass transfer coefficients signify negligible resistance for metal ions' bulk transport within the column. Surface and pore volume diffusions affected continuous adsorption process, with surface diffusion being dominant. The current PVSD model may represent other metal-biosorbent systems to estimate transport parameters suitable for continuous adsorber operation.

Three-dimensional mass transport modeling of pharmaceuticals adsorption inside ZnAl/biochar composite

Three-dimensional Pore Volume and Surface Diffusion Model (3D-PVSDM) was implemented to predict the adsorption of acetaminophen, ketoprofen, and ibuprofen inside ZnAl/biochar composite. The experimental results revealed that ZnAl/biochar composite presented a remarkable potential as adsorbent in the studied adsorption systems since the maximum adsorption capacity was 1,108.43 mg/g for acetaminophen, 1,081.35 mg/g for ketoprofen, and 1,032.81 mg/g for ibuprofen. The equilibrium adsorption of acetaminophen was well fitted by Freundlich isotherm, while Henry isotherm properly adjusted the equilibrium adsorption of ketoprofen and ibuprofen. The 3D-PVSDM simulations indicated that the surface diffusion flux was the priority transport mechanism during the adsorption of acetaminophen, ketoprofen, and ibuprofen onto ZnAl/biochar composite. Therefore, it was possible to neglect the pore volume diffusion and apply the simplified three-dimensional Surface Diffusion Model (3D-SDM) to accurately simulate the mass transport of pharmaceutical compounds on ZnAl/biochar composite. The surface diffusion coefficients ranged from 2.58 × 10−9 to 9.52 × 10−9 cm²/s for acetaminophen, from 1.87 × 10−9 to 8.21 × 10-9 cm²/s for ibuprofen, and from 1.15 × 10-9 to 1.65 × 10−9 cm²/s for ketoprofen.

Differentiating between adsorption and biodegradation mechanisms while removing trace organic chemicals (TOrCs) in biological activated carbon (BAC) filters

Efficient adsorption of certain trace organic chemicals (TOrCs) present in secondary treated municipal wastewater treatment plant (WWTP) effluents onto granular activated carbon (GAC) has already been demonstrated at lab- and full-scale. Due to high organic matter concentrations in WWTP effluents, GAC filters eventually develop a biofilm and turn into biological activated carbon filters (BAC), where removal of organic compounds is governed by biodegradation as well as by adsorption. However, determining TOrC breakthrough by conducting a long-term BAC column experiment to discern between the removal mechanisms is not possible due to competition for adsorption sites, fluctuating water quality, and other variables. Therefore, a rapid small scale column test (RSSCT) was conducted to determine the contribution of adsorption for select chemicals at 10,000 bed volumes treated (BVT). These results were then used in the pore surface diffusion model (PSDM) to model adsorption behavior at 40,000 BVTs. Pseudo-Freundlich K values obtained from the PSDM model were compared with K values obtained from an integral mass balance calculation. This comparison revealed that the modeling was most accurate for moderately to poorly adsorptive compounds. In comparing RSSCT results to long-term BAC columns, the modeling approach best predicted BAC removal of well adsorbing compounds, such as atenolol, trimethoprim, metoprolol, citalopram, and benzotriazole. However, differences in predicted vs observed BAC removal for the removals of venlafaxine, tramadol and carbamazepine revealed that BAC adsorption capacity was not yet exhausted for these compounds. Therefore, a comparison was not possible. The approach would be improved by operation at longer EBCT and improved calculation of compound fouling indices.

Semiempirical investigation of Hg adsorption process in a continuous mode

ABSTRACT Decreasing the concentration of pollutants in wastewater is a necessary issue particularly in the third-world countries. Mercury is known as one of the most important metal contaminants with harmful impacts on the environment. The present work is based on the lab-scale Hg elimination from synthetic wastewater. For this purpose, adsorption application in continuous mode has been chosen and a modified activated carbon which is sulphurised via dimethyl-disulphide was applied as adsorbent. Thus, the original activated carbon performance in Hg2+ adsorption was compared with the sulphurised sorbent. Investigation of the dynamic behaviour of adsorption fixed bed system which is highly demanded prior to industrialisation, was studied in the present work to predict the mass transfer coefficients and diffusional properties of adsorbent-adsorbate system. Therefore, a diffusional model namely Axial-dispersion Film Pore-Surface Diffusion model including all the main mass transfer mechanisms in porous beds was implemented. Apart from the mathematical model, one of the most common shortcut approaches (Thomas model) was applied to describe the adsorption behaviour in fixed beds for comparison with the mathematical model results. Based on the Thomas model and experimental results, several important properties of both adsorbate/adsorbent system such as equilibrium adsorption capacity and exhaustion rate were calculated and explained. According to the results of mathematical modelling, the factors and mechanisms with the lowest degree of importance were determined in each system, and the more appropriate/simpler mathematical models were suggested.

NUMERICAL ANALYSIS OF LIQUID-SOLID ADSORPTION MODEL

In this paper, the numerical algorithms for solution of pore volume and surface diffusion model of adsorption systems are constructed and investigated. The approximation of PDEs is done by using the finite volume method for space derivatives and ODE15s solvers for numerical integration in time. The analysis of adaptive in time integration algorithms is presented. The main aim of this work is to analyze the sensitivity of the solution with respect to the main parameters of the mathematical model. Such a control analysis is done for a linearized and normalized mathematical model. The obtained results are compared with simulations done for a full nonlinear mathematical model.

Insight into mass transfer during ultrasound-enhanced adsorption/desorption of blueberry anthocyanins on macroporous resins by numerical simulation considering ultrasonic influence on resin properties

The mass transfer mechanism about ultrasonic enhancement of blueberry anthocyanin adsorption and desorption on macroporous resins were investigated. Ultrasound treatment led to the decrease of resin size, whereas average pore diameter, specific surface area, total pore volume were less affected. Then, the pore volume and surface diffusion model considering ultrasound-induced changes of particle size and surface area was employed to model anthocyanin adsorption process numerically. Generally, sonication enhanced external mass transfer coefficient (kL) and surface diffusion coefficient (Ds) during adsorption. The values of kL and Ds under sonication at 279 W/L and 20 °C were 7.578 × 10−2 cm/s and 2.000 × 10−9 cm2/s, which were 2.0% and 140.1% higher than the kL and Ds values under orbital agitation at 20 °C. Accordingly, anthocyanins penetrated into the interior of macroporous resins faster under sonication. The overall intraparticle diffusion for anthocyanin adsorption was dominated by pore volume diffusion with the contribution percentage higher than 50%. Sonication enhanced the contribution of surface diffusion to the overall diffusion, which was consistent with the increase of Ds with ultrasound energy. Moreover, a general diffusion model also incorporating the variations of particle size and surface area was utilized to model anthocyanin desorption process. The overall anthocyanin diffusivity increased with both ultrasound energy level and temperature. Anthocyanins distributed more homogenously inside resins under sonication at 279 W/L than under other treatments. This work could be a reference for the valorization of anthocyanins in food industry, especially juice and wine industries.

Nonlinear Frequency Response Analysis as a Tool for Identification of Adsorption Kinetics: Case Study—Pore-Surface Diffusion Control

In the present paper, the Nonlinear Frequency Response (NFR) analysis is applied for theoretical study of kinetics of adsorption governed by pore-surface diffusion. The concept of higher-order frequency response functions (FRFs) is used. Based on a nonlinear mathematical model for adsorption of pure gas and spherical adsorbent particles, the theoretical first- and second-order FRFs, which relate the adsorbate concentration in the particle to the surrounding pressure (F1(ω) and F2(ω,ω)), have been derived. The obtained FRFs have been simulated for different steady-state pressures and different ratios (between zero and one) of surface to pore diffusion coefficients. The results show that, unlike F1(ω), F2(ω,ω) exhibits features which unambiguously distinguish the pore-surface diffusion model from pure pore diffusion and micropore diffusion. Based on the characteristic features of F1(ω) and F2(ω,ω), a new methodology for direct estimation of the separate values of the pore and surface diffusion coefficients has been established.

Three-dimensional mass transfer modeling of ibuprofen adsorption on activated carbon prepared by sonication

A three-dimensional mass transfer model (pore volume and surface diffusion model, PVSDM 3D) was applied to elucidate the ibuprofen adsorption on standard (SAC) and ultrasound modified (USAC) activated carbons. Experimental isotherms and kinetic curves were constructed at pH 2.0 and 298 K. The results revealed that USAC presented higher values of surface area, pore diameter, pore volume and void fraction than SAC, demonstrating that the ultrasonic modification was efficient to improve the characteristics of activated carbon. The adsorption capacities obtained using USAC carbon were around 25% higher in relation to the obtained using SAC carbon. For both adsorbents, the Redlich–Peterson model was able to predict the isotherm data. At pH of 2.0, dispersive interactions π–π and donor–acceptor interactions between the aromatic ring of ibuprofen and carbonyl groups of the activated carbons occurred. The concentration decay curves obtained with USAC required less time to reach equilibrium due to its modification of textural properties. The application of PVSDM 3D model evidenced that the DS (surface diffusion coefficient) values for USAC were around 1.7 folds greater than those obtained by SAC, besides it was corroborated that the magnitude of total intraparticular flux is a function of time and position inside the particle. It was verified that the ibuprofen molecule diffuses exclusively by surface diffusion and that the diffusion in the pore volume can be neglected. The DS values ranged from 1.8 × 10−9 to 33.6 × 10−9 cm2 s−1.

An improved film–pore–surface diffusion model in the fixed-bed column adsorption for heavy metal ions: Single and multi-component systems

Abstract The present work was investigated the adsorption of heavy metal ions including lead (Pb2+), Cadmium (Cd2+) and Nickel (Ni2+) onto a commercial activated carbon (AC) in single and multi-component aqueous fixed bed column. The model, named film pore-concentration dependant-surface diffusion (FPCDSD), was able adequately to consider the details of inter- and intra-pellet transport mechanisms in column mode and predict film, pore and surface diffusivities of the adsorption process considering a nonlinear isotherm model. A diffusional kinetic model was developed, to investigate the effects of several parameters like initial concentration, flow rate, and bed height, and also to explain the adsorption kinetics and evaluation of the mass transfer coefficients. The detailed calculation methods to determine the contribution of each resistance against mass transfer were addressed in the paper. The model with mass transfer parameters (with the order of magnitude of Dp ≈ 10−10 m2 s−1, Ds ≈ 10−15 m2 s−1, Dz ≈ 10−5 m s−2, and kf ≈ 10−5 m s−1) was found successful in correlating adsorption data in single, and multi-component system under different operating conditions. Sensitivity analysis revealed that the value of Dp and kf dictate the model performance. The Biot number was found to increase with increasing flow rate and initial concentration, indicating an increase in intraparticle diffusion resistance.

Statistical evaluation of pore volume and surface diffusion model in adsorption systems

The Pore Volume and Surface Diffusion Model (PVSDM) was applied to represent the adsorption of malachite green onto bentonite. A detailed analysis of the diffusion parameters was performed and was conclude that the external mass transfer coefficient (kF) was dependent on the operating conditions and ranged from 1.99×10−3 to 2.65×10−3cms−1; the pore volume diffusion coefficient (Dp) was independent on the initial dye concentration and was 6.11×10−7cm2s−1; and the surface diffusion coefficient values (Ds) increased with the initial dye concentration and ranged from 6.83×10−10 to 7.92×10−10cm2s−1. Also, the relative contribution of superficial diffusion mechanism was increased with the initial dye concentration. Was determinate an optimal global parameter Ds=7.73×10−10cm2s−1 was determinate and statically suitable with the Student’s t test, χ2 test and the Fisher’s exact test. The model was validated using the optimal value of Ds which is valid over the entire initial MG concentration range compared.

Detailed numerical solution of pore volume and surface diffusion model in adsorption systems

In this work, the numerical solution of pore volume and surface diffusion model (PVSDM) was developed and presented in detail. The finite difference approximations method was employed to solve the partial differential equations of the diffusional model. The experimental adsorption of Malachite Green dye (MG) on bentonite clay was selected as case study. The equilibrium data were obtained from batch systems and, the Redlich–Peterson isotherm was suitable to represent the results. Due to non-linearity of the isotherm, the non-linear least squares technique was used to estimate the diffusional parameters. The Biot number has shown that the adsorption was simultaneously controlled by external mass transfer and intraparticle diffusion. Thus, both internal resistances (pore volume and surface) must be considered. The MG concentrations as a function of time decreases and the amount of MG mass adsorbed on bentonite clay as a function of the radial position increases until reaching equilibrium. The experimental concentration decay curve of MG was properly represented by PVDSM model. Further, the amount of MG mass adsorbed at higher radial positions was larger than at lower radial positions, within the PVSDM boundaries (particle boundaries), indicating that the method of finite difference approximations was appropriate for the numerical solution of PVSDM model.

Solid-phase extraction of d -glucaric acid from aqueous solution

Abstract Since d -glucaric acid was produced by Escherichia coli and other microorganism, the purification of d -glucaric acid has been an important issue for downstream production. This work studied the adsorption of d -glucaric acid by means of seven ion exchange resins and three non-ionic adsorbents. Ion exchange resins showed superiority in adsorption capacity. Their isotherms could be fitted by Freundlich equation at pH 3.9 and Langmuir equation at pH 6.0, respectively. The highest adsorption capacity is 350 mg/mL at pH 3.9 by using 335 ion exchange resin. Compared to ion exchange resins, non-ionic adsorbents showed less capacity and Freundlich isotherm was suitable for both pH conditions. For adsorption kinetics, pore diffusion model and surface diffusion model were combined with Freundlich isotherm to study the uptake rate of ion exchange resins. 335 resin has the pore diffusion coefficient of 18.3 × 10 −10 m 2 /s and the surface diffusion coefficient of 1.29 × 10 −11 m 2 /s. Generally, 335 resin is the best choice for d -glucaric acid adsorption owing to its superior capacity and diffusivity as well.