Rectangular Isotherm Research Articles

Modelling of porous core–shell adsorbent particles with various morphologies suspended in batch adsorber from analytical solutions of diffusion equations

AbstractConsidering both intra‐particle diffusion and film resistance for mass transfer, analytical solutions of transient concentration of adsorbate inside adsorbents with spherical, cylindrical, or slab‐type particles were derived for batch adsorbers by solving governing equations using the Laplace transform. Assuming Henry's or rectangular isotherm, the average concentration inside adsorbents as well as transient bulk concentration were also obtained for the particles with or without the inert core. Computations were performed to compare the results according to the shape of adsorbents by adjusting adsorbent loading, Biot number (Bi), and inert core thickness. Regardless of particle morphologies, steady‐state bulk concentration was only affected by adsorbent loading and inert core thickness, whereas the effect of Bi was confirmed from the decreasing reduction rate of adsorbate concentration with decreasing Bi. When diffusivity was dependent on time, time‐decaying diffusivity caused the increase in steady‐state concentration that was predicted by eigenfunction expansion. Experimental results using porous fibres by electrospinning were compared with the mathematical solution of a cylindrical adsorbent for the estimation of intra‐particle diffusivity. Using the solutions of the diffusion equation model, novel core–shell cylindrical adsorbents can be designed and synthesized as core–shell fibres by electrospinning with a coaxial nozzle to save the cost of the active shell layer. Such core–shell structured fibres can be adopted as adsorbents for novel batch adsorption processes and the present modelling results can be extended to other processes like fixed bed adsorbers.

Ion-exchanged ETS-10 adsorbents for CO2/CH4 separation: IAST assisted comparison of performance with other zeolites

ETS-10, the well-known member of titanosilicates, was synthesized, characterized, and then employed to study CO2 and CH4 adsorption on its cation exchanged forms. Equilibrium isotherms of as-synthesized, K-, Mg-, and Sr-ETS-10 were analyzed at 298 K and pressures up to 100 kPa. High selectivities of CO2/CH4 were observed for all the adsorbents. By a closer look at equilibrium isotherms, adsorption selectivity was found to be in the order of K- > as-synthesized- > Sr- > Mg-ETS-10. Analyses of the data revealed that ETS-10 adsorbents containing monovalent cations have more rectangular isotherms and higher CO2/CH4 selectivity. On the other hand, ETS-10 adsorbents exchanged by divalent cations showed lower selectivity but more curvature isotherms which makes the regeneration easier in the pressure swing adsorption process (PSA). High-pressure adsorption experiments at different temperatures up to 348 K were also studied for the most selective adsorbent, i.e., K-ETS-10. Finally, the ETS-10 adsorbents performance in different binary mixtures of CO2/CH4 was investigated by Ideal Adsorption Solution Theory (IAST), and the results were evaluated and compared with the performance of other aluminosilicates and high silica zeolites in CO2/CH4 separation. The comparison led to a new trade-off plot, i.e., adsorption capacity versus CO2/CH4 selectivity, which can be helpful for adsorbent selection.

Modeling and simulation of the adsorption of biogas hydrogen sulfide on treated sewage–sludge

• The dynamic behavior of a fixed bed with treated sewage–sludge for H 2 S removal is modeled. • The Bohart and Adams model does not predict suitably the complete breakthrough curves. • The Klinkenberg solution implies a linear isotherm, and the prediction is not good enough. • The LDF model with a non-linear isotherm provides a good approach of the adsorption process. • As an application, a scale-up of the lab scale to a larger one was performed and assessed. For the design of a fixed-bed adsorber, it is essential to know the prediction of the column breakthrough. This paper compares two modeling approaches of the dynamic behavior of a fixed bed that contains treated sewage–sludge for biogas desulfurization. The first approach is based on the simple model by Bohart and Adams, which has been widely used to describe the adsorption dynamics when chemical reaction takes place, and the second one uses the linear driving force model (LDF), solving it analytically by Klinkenberg equation and numerically with the aid of Aspen Adsorption. While the Bohart–Adams model uses rectangular isotherm to describe the equilibrium, the LDF model is solved with a more realistic and suitable isotherm. The Klinkenberg solution implies a linear isotherm, and the prediction is not good enough, although better than that of the approach of Bohart–Adams. The trouble is that the linear isotherm used in the analytical solution is not realistic and a nonlinear isotherm is required. This paper presents a methodology to estimate the overall mass transfer coefficient dynamically as well as the isotherm that best fits. Thus, the LDF model with the Freundlich isotherm provide a breakthrough curve that match the experimental results much better than the Bohart–Adams approach. In order to apply the prediction obtained, a scaling up of the lab scale was performed, discussed and assessed.

Predicting protein dynamic binding capacity from batch adsorption tests

The dynamic binding capacity (DBC) and its dependence on residence time influence the design and productivity of adsorption columns used in protein capture applications. This paper offers a very simple approach to predict the DBC of an adsorption column based on a measurement of the equilibrium binding capacity (EBC) and of the time needed to achieve one-half of the EBC in a batch adsorption test. The approach is based on a mass transfer kinetics model that assumes pore diffusion with a rectangular isotherm; however, the same approach is also shown to work for other systems where solute transport inside the particle occurs through other transport mechanisms.

ChemInform Abstract: Chromatographic Purification and Superpurification of Biologically Active Compounds Using Heteroreticular and Composite Ion Exchange Resins at Low Pressure

The proposed methods of isolation, separation, and fine p urification of biologically active compounds (BAC) are based on a theoretical analysis of sorption dynamics and chromatography and the u se o f a new type of biosorbents (heteroreticular polymer ion exchangers and composite sorbents o f the Cellosorb type). FSne separation is carried out by a directed parallel shift in the thermodynamic and kinetic parameters in the sorbent-sorbate system with a stepwise change in the properties of t he eluting solutions. FUNDAMENTAL PRINCIPLES OF SHARP BOUNDARY FORMATION The analysis of selective column sorption and selective elution by our methodology is based on the formation of eharp boundaries of chromatographic zones of the compounds (Refs, 1,2,3). In order to determine the conditions of the formation of sharp boundaries, it is convenient to use the dimensionless criteria h , , A,, and A,, which predict the experimentally confirmed conditions of complete saturation of the column or its part and complete elution when h ( A, A,, A ) - 1 + 4. This range depends on the type of the ieotherms of interpkaae substance distribution: highly selective sorption linear or intemediate type isotherme, A dimensionless parameter is introduced for sorbent grains that sorb substances throughout the mass at linear or rectangular isotherms under the c onditions of gel (internal) limitation due to the kinetics of heterogeneous mass exchange:

Fixed bed sorption: Setting the record straight on the Bohart–Adams and Thomas models

The Bohart–Adams model and the so-called Thomas model are commonly used in the modeling of fixed bed breakthrough curves in environmental sorption and biosorption research. Some authors fit the two equations to the same set of experimental breakthrough data and compare their modeling capabilities, implying that the two expressions are separate and independent models. This article clarifies unambiguously and demonstrates conclusively that the two models are in reality mathematically equivalent to each other and their parameters interchangeable. The salient features of the Bohart–Adams model and the genuine Thomas model are outlined here and their relationship delineated. The main difference between the Bohart–Adams and actual Thomas models lies in the form of the sorption isotherm assumed. The former assumes a rectangular (irreversible) isotherm while the latter assumes a Langmuir (favorable) isotherm. It is shown that, when the sorption isotherm is highly favorable, the actual Thomas model reduces to the Bohart–Adams model.

Analysis of mass transport models based on Maxwell–Stefan theory and Fick's law for protein uptake to porous anion exchanger

Parallel mass transport of protein in the pore fluid and on the pore wall of porous adsorbent is modeled based on the Maxwell–Stefan theory that uses chemical potential gradient as the diffusive driving force. The uptake kinetics of proteins (BSA and γ-globulin) to anion exchanger is studied by batch adsorption. The parallel diffusion model based on the MS approach (MS-ParD model) and its simplified form, the surface diffusion model (MS-SD model), are analyzed and compared with those based on the Fick's law (F-ParD and F-SD models). It is found that the models from the MS equation and the Fick's law are quite different from each other. For the MS-ParD model, both the pore and surface diffusion coefficients are constant, while the surface or pore diffusion coefficient for the F-ParD model varies significantly with initial protein concentration. Moreover, the MS-SD model can be used to describe the uptake of a protein that shows nearly a rectangular isotherm, in which the surface diffusion contributes very small to the mass transport. In addition, an equation that describes the relationship between the surface diffusion coefficients for the MS-SD and F-SD models is proposed. It is observed that the ratio of the surface diffusion coefficients for the MS-ParD and F-ParD models for less favorably adsorbed protein can be qualitatively described by a theoretical equation. It is concluded that the MS-ParD model is superior to the F-ParD model for describing the non-ideality of adsorbed protein.

Prediction of protein breakthrough behavior using simplified analytical solutions

The predictability of the breakthrough behavior of lysozyme on the strong cation exchanger SP Sepharose FF was evaluated using two different but related analytical solutions. The first solution, known as the Thomas equation, assumes that adsorption can be represented by a pseudo second-order reaction rate law which translates into a Langmuir isotherm at equilibrium. The second solution, known as the Bohart–Adams equation, assumes that adsorption can be described by a quasi-chemical rate law which reduces to a rectangular isotherm at equilibrium. The Bohart–Adams solution can be regarded as a limiting form of the Thomas solution. The equilibrium and rate parameters embedded in the two solutions were extracted from batch experiments. Predictions using the two simplified solutions agreed with experimental breakthrough data. When the isotherm is highly favorable the simpler Bohart–Adams solution can be used as a practically useful modeling tool in lieu of the mathematically complicated Thomas solution.

Kinetic Study of the Mass Transfer of Bovine Serum Albumin on Cibacron Blue Cellulose Membranes by Using the Multi-Plate and Transport Models

The mass transfer kinetics of bovine serum albumin on Cibacron blue F3GA cellulose affinity membranes has been investigated. It was found that the multi-plate (MP) and transport models successfully described experimental breakthrough curves obtained by single-step frontal analysis. The correlation between the two models was used to estimate the rate coefficients of mass transfer from experimental data. The flow rate was found to have little effect on the performance of affinity membrane separations. The improvement of the performance by increasing the thickness of the membranes was limited. The transport model was simplified by approximating the very sharp Langmuir isotherm determined by a rectangular isotherm and an analytical solution obtained although this was found to be unsuited to describe the experimental data.

Breakthrough and Elution Curves for Adsorption of Phosphates on an OH-Type Strongly Basic Ion Exchanger

The breakthrough and elution curves for adsorption of H3PO4, H2PO4- and HPO42-, PO43- ions on an OH-type strongly basic ion exchanger were investigated in a packed-bed column. The results appeared to be technically feasible. In all phosphate systems, sharp breakthrough curves were obtained, and they were affected by inlet concentrations. The breakthrough times decreased with an increase in the inlet concentration. The ion-exchange capacities of phosphates in the column decreased in the following order: H3PO4 aqueous solution > NaH2PO4 aqueous solution > Na2HPO4 aqueous solution > Na3PO4 aqueous solution. The values of the ratio between the ion-exchange capacity and the total ion-exchange capacity of the resin were in the range of 100−30%. The experimental breakthrough data were correlated by neglecting the effect of axial dispersion and by using a linear driving force assumption in the intraparticle diffusion. At a separation factor of R ≥ 0.0594, the theoretical breakthrough lines were calculated numerically for the Langmuir isotherm, while at 0.000 42 ≤ R ≥ 0.035 27, the theoretical breakthrough lines were all calculated using an analytical solution for the rectangular isotherm. The values of the intraparticle effective diffusivity increased with an increase in the influent concentration. In H3PO4 and NaH2PO4 systems, the contribution of surface and pore diffusions was significant, while in Na2HPO4 and Na3PO4 systems, the contribution of the pore diffusion was significant. The phosphate ions adsorbed in the column bed were completely eluted by using a NaOH aqueous solution.

Modeling separation of proteins by inert core adsorbent in a batch adsorber

Adsorption/desorption kinetics of protein on the binding ligand of inert core adsorbent in a batch adsorber is analyzed theoretically for Langmuir isotherm coupled with the intraparticle diffusion and film mass transfer resistances. For the two limiting cases of Langmuir isotherm, there are analytical solutions. New analytical solutions are derived for Henry isotherm, and the analytical solution of shrinking core model is recommended for rectangular isotherm. The effects of the inert core radius, equilibrium constant, intraparticle diffusion and film mass transfer resistances on the time evolution of bulk concentration and particle radial profiles were investigated. The applicable range of the analytical solution with rectangular isotherm is given. A new method to estimate both film mass transfer coefficient, k f , and effective pore diffusivity, D pe , from a single bulk concentration–time curve in batch adsorber is given and tested with literature data for the adsorption of BSA on CB-6AS inert core adsorbent.

Comparison between mass transfer properties of weak-anion-exchange resins with graft-functionalized polymer layers and traditional ungrafted resins

Glycidylmethacrylate was grafted to Toyopearl HW-65M and subsequently modified with diethylamine to obtain a weak anion exchanger. The degree of grafting was varied from 11 to 50%. The binding capacity for bovine serum albumin was 11 mg/ml for the lowest degree of grafting and 97 mg/ml for the highest degree of grafting. The maximum binding capacity was observed at 27% degree of grafting. The mass transfer properties of the grafted resins and an ungrafted resin (Toyopearl DEAE 650M) were investigated assuming rectangular isotherms. Simple models for reaction kinetics, pore- and surface diffusion and film diffusion were used to describe the concentration–time data in batch mode. The data were best fitted by a pore diffusion model. The estimated pore diffusion coefficients ( D P) for bovine serum albumin were fitted by a polynome to the degree of grafting with an maximum value at 27% of D P=1.95·10 −11 m 2/s. Compared to published data of other ungrafted resins and to the molecular diffusion coefficient of bovine serum albumin in free solution of D P=5.6·10 −11 m 2/s, the diffusion in grafted layers seems to be accelerated. The breakthrough curves for columns packed with various resins showed a decrease in sharpness with increasing degree of grafting which could not be described by a simple pore diffusion model using the calculated transport coefficients from batch mode. The shape of the breakthrough curves could be well described by a combined film and pore diffusion model. For the ungrafted Toyopearl DEAE 650M resin the breakthrough curve is more favorable and the influence of film diffusion to the mass transfer is reduced. It can be concluded that grafting will increase the capacity and the pore diffusion in batch mode but in column operation the grafting layer has a film resistance which plays an important role in the overall mass transfer.

Model for parallel surface and pore diffusion of an adsorbate in a spherical adsorbent particle

A model is presented to describe the rate of uptake of a species by a spherical adsorbent particle when intraparticle transport can occur by parallel diffusion through the pore space and along the surfaces of pore walls. Although the conventional shrinking core model (SCM) has been applied to such systems previously, its use is valid only when adsorption onto the pore walls is described by a rectangular isotherm. As with the SCM, the new model differs from homogeneous models by envisaging the advance of the adsorbate to be marked by a distinct inward-moving interface, but it generalizes the SCM to allow for an incompletely saturated adsorbed shell behind the front governed by any Langmuir isotherm. The model has been applied to previously published experimental data for the uptake of bovine serum albumin by chitosan beads and compared to the results obtained when the same data are analysed using the corresponding homogeneous model. The fit of the two models to the convension-time data yields comparable results, but significant differences in the predicted absorbate concentration profiles within adsorbent particles are observed, particularly at high conversions.

ADSORPTION AND DESORPTION KINETICS FOR NONLINEAR ADSORPTION ISOTHERMS

The theoretical calculations of the mass transfer rate are based on a model which considers a nonlinear adsorption isotherms and simultaneous resistance to mass transfer in The pore adsorbent and the convective mass transfer. Numerical solutions to the diffusive-convective-controlled adsorption and desorption processes are calculated for the Langmuir and rectangular adsorption isotherms. It is shown that for the rectangular adsorption isotherm the adsorption kinetics is governed by the diffusive-convective mass transfer over a wide range of times and the desorption kinetics is irreversible process. The equations are derived to calculate from the experimental kinetics data: (a) the coefficient diffusion in the pore adsorbent, (b) the relaxation times characterizing the adsorption and desorption processes, and (c) the times needed to reach the quasiequilibrium state for the adsorption and desorption processes.

The Rectangular Isotherm Model for Adsorption Kinetics

The transient uptake response of an adsorbent particle, subjected to a step change in surface concentration, is considered. It is shown that, when the isotherm is highly favorable, the theoretical curves derived for a Langmuirian system reduce asymptotically to the much simpler form for a rectangular isotherm. The simple rectangular model provides a useful approximation even when the form of the actual isotherm is quite far from the rectangular limit.

Mass transfer mechanisms in Hyper D media for chromatographic protein separation

In order to elucidate the mass transfer mechanisms that allow improved performance for the separation of proteins of the chromatographic packing material Q Hyper D (BioSepra, Villeneuve la Garenne, France), when compared with traditional packing materials, several experiments were carried out using a commercial HPLC column, with BSA and myoglobin as test proteins. First, elution chromatographic runs under unretained conditions at several flowrates and protein concentrations were made. The HETP vs. superficial velocity plot is almost a plateau, except for the low flowrate region. Elution experiments were also carried out under weakly retained conditions (with salt concentration 0.3 M NaCl) in the linear region of the adsorption equilibrium isotherm. Using Rodrigues' equation, the experimental initial slope and plateau of a Van Deemter plot enable the determination of the effective diffusivity and particle permeability respectively. Then, breakthrough experiments under retaining conditions were run at several feed concentrations and flowrates. The experimental adsorption equilibrium isotherm is rectangular; using a simplified fixed-bed adsorber model for rectangular isotherms, an apparent (augmented) diffusivity is calculated as a function of flowrate. The determined augmented diffusivities are shown to follow the dependence on velocity established by Rodrigues et al.: this clearly shows that intraparticle convection is present as a mass transfer mechanism, since the real intraparticle diffusivity is independent of flowrate.

Nonlinear adsorption and hydrodynamic dispersion in self-similar networks

Self-similar models proposed by Villermaux and Schweich (1992, J. Phys. II France 2, 1023–1043) are modified and applied to simulate convection and nonlinear adsorption of a solute in a disordered porous medium. We explore the dispersive behaviour of three self-similar patterns. Conservation equations are integrated numerically for the Langmuir adsorption isotherm, and an analytical procedure is developed for the rectangular isotherm. Breakthrough curves rapidly approach an asymptote with a nonzero breadth as the isotherm becomes increasingly concave-downward, in contrast to results of conventional dispersion models. The networks are simple to use, and we discuss how they can be applied in practice.

Adsorption of direct dye on cross-linked chitosan fiber: breakthrough curve

The recovery of direct dye by adsorption on cross-linked fiber (ChF-B), which we have developed, appeared technically feasible. The concentration of amino group fixed in the adsorbent phase was 3.30 mol/kg dry fiber. A typical direct dye, Brilliant Yellow (M.W.=625, divalent anion) was used in this experimental study. The breakthrough curves for adsorption of the dye were measured for different flow rates, bed heights, influent concentrations of the dye, and temperatures. The experimental breakthrough curves showed that ChF-B recovered the dye efficiently even from the hot dye solution. Axial dispersion coefficient Ez (m2/s) in fibrous bed was measured and the following correlation for the dispersion coefficient was obtained: Ez = 5.4 × 10−6Re′1.1 (ɛ = 0.698) Ez = 4.7 × 10−6Re′1.2 (ɛ = 0.813) where ɛ denotes void fraction of the bed. The theoretical breakthrough curve obtained by considering the effect of axial dispersion was close to that obtained from the analytical solution for rectangular isotherm system without considering the axial dispersion in Re′ > 0.46. The experimental breakthrough curves in pH ≤ 4 agreed well with the analytical solution. The intrafiber effective phase diffusivity of the dye at 323 K was about two times larger than that at 298 K and the bed capacity at 323 K was almost the same as that at 298 K.

Potassium removal from water-methanol-polyol mixtures by ion exchange on Amberlite 252

Abstract Equilibrium isotherms for the uptake of potassium from polyol-water-methanol mixtures on Amberlite 252, a strong acid ion exchange resin, have been obtained in order to assess the possibility of using such a system to purify industrial polyether-polyols. The selectivity was very high. A two-step reaction scheme was proposed and equilibrium data were correlated by an equation derived according to the mass action law. Effective intraparticle diffusivities were obtained using a homogeneous model and the effects of the viscosity of the bulk solution and resin swelling on the exchange rate are discussed. A simple empirical equation correlating diffusivity with viscosity and resin swelling is suggested. Breakthrough curves were obtained under a wide range of conditions, being well correlated by the expression for breakthrough curves with rectangular isotherm. The proposed purification procedure is generally applicable to different polyol types, being economically and technically feasible.

Adsorption of direct dye on cross-linked chitosan fiber: Breakthrough curve

The recovery of direct dye by adsorption on cross-linked fiber (ChF-B), which we have developed, appeared technically feasible. The concentration of amino group fixed in the adsorbent phase was 3.30 mol/kg dry fiber. A typical direct dye, Brilliant Yellow (M.W.=625, divalent anion) was used in this experimental study. The breakthrough curves for adsorption of the dye were measured for different flow rates, bed heights, influent concentrations of the dye, and temperatures. The experimental breakthrough curves showed that ChF-B recovered the dye efficiently even from the hot dye solution. Axial dispersion coefficient E z (m 2 /s) in fibrous bed was measured and the following correlation for the dispersion coefficient was obtained: Ez = 5.4 × 10 −6 Re ′ 1.1 (ɛ = 0.698) Ez = 4.7 × 10 −6 Re ′ 1.2 (ɛ = 0.813) where ɛ denotes void fraction of the bed. The theoretical breakthrough curve obtained by considering the effect of axial dispersion was close to that obtained from the analytical solution for rectangular isotherm system without considering the axial dispersion in Re ′ > 0.46. The experimental breakthrough curves in pH ≤ 4 agreed well with the analytical solution. The intrafiber effective phase diffusivity of the dye at 323 K was about two times larger than that at 298 K and the bed capacity at 323 K was almost the same as that at 298 K.