Nonlinear Equilibrium Isotherm Research Articles

Assessment of activated carbon/alginate for the concurrent removal efficiency of paracetamol and caffeine from wastewater in their binary solutions

The increasing amount of pharmaceutical chemicals in wastewater is a concern for both public health and the environment. This research investigates the effectiveness of a new composite material made of alginate-activated carbon in removing caffeine and paracetamol from wastewater simultaneously. The composite material, produced using a simple technique, demonstrates excellent removal rates and exceptional adsorption properties for both medicinal chemicals. The structural stability and integrity of the composite are confirmed through comprehensive characterization methods including scanning electron microscopy (SEM), Fourier-transform infrared spectroscopy (FTIR), X-ray diffraction, energy dispersive X-ray (EDX), and BET surface area analysis. The study uses batch adsorption experiments to assess the impact of various factors, such as pH, initial concentration, and contact time, on the efficiency of caffeine and paracetamol removal. The highest values for the adsorption process were 89.63% for Paracetamol and 97.32% for Caffeine, achieved under optimal experimental conditions (pH 7, 0.1 g dose of adsorbent at elevated temperature). Additionally, Kinetic studies and thermodynamic parameters such as ΔS°, ΔGº, and ΔH° were calculated, indicating that the adsorption process was endothermic and spontaneous for paracetamol, and exothermic and non-spontaneous for caffeine. Seven non-linear equilibrium isotherm models were utilized to fit the experimental data for paracetamol and caffeine at pH 7 showing maximum adsorption capacities (qmax) of 606.80 and 725.05 mg/g for Paracetamol and Caffeine, respectively with a high regression coefficient (R2) of 0.99 using alginate-activated carbon as an adsorbent, The results demonstrate the potential of alginate-activated carbon as an effective adsorbent for the simultaneous removal of various pharmaceutical pollutants, highlighting the synergistic adsorption effects of the composite material. The study also examines the safety of the composite material, particularly in the context of potential medical applications. The study emphasizes the potential for sustainable and repeated use of the activated carbon/alginate composite, demonstrating that it maintains both its structural integrity and adsorption effectiveness after multiple cycles of use. This study evaluated the suitability of the suggested analytical method using BAGI, a metric with a unique formula. The BAGI is a complementary tool to GAPI, Complex GAPI, AGREE, AGREE prep, and ESA. A major focus is on the practical elements of white analytical chemistry centered around “blue”.

Highly efficient remediation of Cu (II) from water by novel poly(o-phenylenediamine)/zinc oxide nanohybrids: Kinetic, equilibrium, and thermodynamic studies

Herein, novel nanohybrids of poly(o-phenylenediamine)/zinc oxide (POPD/ZnO) were synthesized using an ultrasound-assisted technique via a facile in situ polymerization method for the removal of Cu (II) ions from water. These nanohybrids were characterized using infrared spectra (FTIR), UV–Visible spectra, scanning electron microscopy (SEM), X-ray scattering (XRD), thermogravimetric analysis (TGA), and BET (Brunauer-Emmett-Teller surface area). Numerous parameters were optimized to see the performance of the adsorption process like effect of POPD loading, adsorbent dose, initial Cu (II) ion concentration, pH, contact time, and temperature. POPD/ZnO-13/87 was found the best adsorbent based on above analysis. Non-linear equilibrium isotherm (Langmuir and Freundlich), kinetics (pseudo-first and second order, intraparticle diffusion model), and thermodynamic studies were performed to analyze the adsorption process. Langmuir isotherm and pseudo-second-order kinetic model were found appropriate while thermodynamic parameters Gibbs free energy change (ΔG°), entropy change (ΔS°) (166.32 J mol−1 K−1), and enthalpy change (ΔH°) (34.03 kJ mol−1) were calculated showing an endothermic and spontaneous process for adsorption of Cu (II) ions. According to Langmuir isotherm, the maximum adsorption capacity was found at 2485 mg g−1 while experimental adsorption capacity was found 2360 mg g−1 at adsorbent doses = 0.4 g L−1, Cu (II) ion concentration = 1000 mg L−1, pH = 6.5, temperature = 25 °C in 90 min that was higher than that of other reported materials till date to the best of our knowledge. Mechanism studies suggested that the electrostatic interaction and cation-π interaction between nanohybrids and Cu (II) ions was the main driving force for the adsorption of Cu (II) ions.

Linear and Non-Linear of 2-Parameters Adsorption Equilibrium Isotherm Models of Synthetic Arsenic Wastewaters

Contamination of groundwater and surface water with arsenic (Asic) has become emerging health and environmental problem around the world. This problem has received significant attention amongst scientists for the development of new adsorbents to remediate Asic -contaminated water. The ability of the immobilized powdered eggshell (Poes), as adsorbent, to remove Asic was studied under batch conditions. Equilibrium data were analysed using non-linear and linearized two-parameter adsorption isotherms models (Langmuir, Freundlich, Elovich, Flory–Huggins, Temkin, Frenkel- Hasley- Hill; Langmuir- Vageler, Hill-de Boer, Kiselev, Fowler- Guggenheim, Dubinin –Radushkevich, Jovanovic, Harkins–Jura and Halsey). The performance of adsorption equilibrium isotherm models was evaluated statistically using the following analysis of variance (ANOVA), model of' selection criterion (MSC), Coefficient of Determination (CD), Correlation coefficient (R) and Akaike Information Criterion (AIC). The study revealed that for non-linear equilibrium isotherm models, Freundlich (0.986 and 3.906) > Fowler—Guggenheim (0.996 and 5.176) and Hasley (0.986 and 3.906) performed well in predicting experimental data-based on the magnitudes of R and MSC. The linearized adsorption equilibrium isotherm models, Dubinin – Radushkevich (0.993 and 4.621) <Temkin (0.994 and 4.701) < Kiselev (0.9999 and 8.856). These three models are the best isotherm models for Asic adsorption onto Poes. It was concluded that Poes particles contain numerous materials that aid Asic adsorption. Based on the performance indicators and to ensure reliable results of adsorption equilibrium data analysis through the adsorption isotherm models, it is necessary that these data sets should be evaluated by both non-linear and linear regression analyses.

Experimental and theoretical studies of Rhodamine B direct dye sorption onto clay-cellulose composite

Dyes are one of the main water pollutants and many biological and environmental problems are associated with them. Rhodamine B (RhB) is one of the most commonly used dye in the textile, printing, paints, and paper industry. The present work reports the sorptive removal of Rhodamine B direct dye from wastewater onto developed cellulose and clay composites. Sorbent material cellulose (48 g) was extracted from 80 g of bagasse. Then cellulose and two types of acid-activated clays were used to make efficient sorbent namely, composite I and II. Various characterization techniques were used to study the physiochemical properties of the synthesized composites. Different sorption affecting parameters were optimized such as initial dye concentration, time, temperature, pH, and composite dose for the efficient sorption of RhB onto composites. Equilibrium time was 60 min for composite-I and 80 min for composite-II Non-linear equilibrium isotherm and kinetic models demonstrated the fitness of Pseudo-second order and Redlich-Peterson isotherm. Composite-I and II removed 85.9% and 95.6% of RhB at pH 2 in 120 min, respectively. The sorption efficiency was checked, and sorbents were applied to real textile effluent which showed promising removal efficiency of over 90%. To confirm the experimental results, computational optimization and vibrational calculations were carried out using the Gaussian 09 program package with 3–21 G, 6–311 G, and 6–311+G basic sets. Geometric parameters showed the planar geometry. In the case of FTIR spectra, fundamental ring vibrations were observed with C-H and C-C. This study suggests that the developed composites have exceptional sorption ability to remove the dye contents from aqueous media.

Development of new organic-inorganic, hybrid bionanocomposite from cellulose and clay for enhanced removal of Drimarine Yellow HF-3GL dye

Cellulose/clay composites were prepared and applied for the adsorption of Drimarine Yellow HF-3GL direct dye. The prepared composites were characterized by FTIR, TGA, EDX, SEM and XRD techniques. Bagasse was used as a cellulose source, while clay was obtained from local source, which was modified chemically before composite preparation. Adsorption efficiencies were compared of composite I and II as a function of contact time, temperature, pH, initial dye concentration and composite dose. Non-linear kinetic and equilibrium isotherm employed and dye adsorption data fitted well to pseudo-second order kinetics model. Among isotherms, the Redlich-Peterson isotherm well defined the sorption process of dye on to composites. Thermodynamic factors (ΔS°, ΔH° and ΔG°) revealed that the sorption process was spontaneous, exothermic and feasible. Cellulose/clay composite I and II removed 88.64% and 89.95% dye with 60 min at pH 2 and 30 °C, respectively. For reusability, desorption was performed using different eluting agents and NaOH showed higher desorption efficiency. For the treatment of wastewater, the developed composites were applied to textile effluents and color removal of (90–96.07%) and (97–98.23%) was achieved using cellulose/clay composite I and II, respectively. The results showed that cellulose/clay composite are efficient for the removal dyes and could possibly be used for the treatment of textile effluents.

Response to “Some remarks on a critical review of the estimation of the thermodynamic parameters on adsorption equilibria. Wrong use of equilibrium constant in the van't Hoff equation for calculation of thermodynamic parameters of adsorption - Journal of Molecular Liquids 273 (2019) 425–434.”

• The response to the letter to the editor • Determination of thermodynamic parameters by using the van't Hoff equation. • Use of nonlinear equilibrium isotherms at several temperatures to obtain the K.

Silica rich MIL-101(Cr) for enhanced hydrogen uptake

Hybrid composite of MIL-101 with silica rich rice husk ash (RHA) was fabricated to explore such materials for improved low pressure hydrogen storage applications, compared to the well explored carbon based composites of MIL-101. RHA–MIL-101 was prepared by in situ incorporation of RHA in MIL-101 during the synthesis, under hydrothermal conditions. The incorporation of RHA in MIL-101 was confirmed by P-XRD, FTIR, TGA, SEM, EDS, and N2 adsorption and desorption isotherms studies. The as-synthesized RHA–MIL-101 composite displayed enhanced BET surface area (8.6% compared to bare MIL-101), whereas AC–MIL-101 showed an enhancement of 12.7% in BET surface area compared to bare MIL-101. Hydrogen uptake properties of these materials were evaluated at 77 K and 1 bar. Despite that RHA–MIL-101 exhibited lower surface area as compared to AC–MIL-101, the hydrogen uptake capacities of RHA–MIL-101 reached an enhanced value of 1.54 wt%, which is obviously higher than the bare MIL-101 (1.40 wt%) and AC–MIL-101 (1.48 wt%) by 9.1% and 5.7%, respectively, as well as comparable to most of the reported carbon incorporated MOFs. It has been observed that improved hydrogen uptake properties was due to the bifunctional properties of the synthesized RHA–MIL-101, abundance of silanol bonds of RHA (which shows high affinity towards H2 molecules), and tuned porous properties of RHA–MIL-101. Moreover, hydrogen adsorption isotherms data of these materials were best fitted with three parameter non-linear adsorption equilibrium isotherm equations.

Transport of Cadmium and Phosphate in Soils

This study quantified and modeled cadmium (Cd) transport in the presence to phosphate (P) in three different soils, Mahan (acidic, kaolinitic), Webster (neutral, mixed mineralogy), and Windsor (acidic, sandy). Two sets of miscible displacement experiments were carried out: (1) sequential pulses of 200 mg L−1 Cd as Cd(NO3)2 then 200 mg L−1 P as KH2PO4, both in 0.01 M KNO3 background; and (2) two consecutive pulses of mixed 200 mg L−1 Cd and P in background solution. Isotherms for Cd and P (1-d equilibration) were also determined. Breakthrough curves (BTCs) of Cd applied without P were generally consistent with the (Freundlich) isotherms and soil properties, Cd retention decreasing: Webster, 72% > Mahan, 61% > Windsor, 37%. Retention of Cd increased when applied with P in Webster and Windsor but decreased in Mahan. Phosphate BTCs indicated high mobility (≥92% recovery) and little effect of Cd, thus limited Cd-P precipitation. Residual distribution of Cd with depth also showed no clear evidence for precipitation. Cadmium BTCs could not be described solely by either linear or nonlinear equilibrium isotherms. Rather, the data were best described by nonlinear, reversible kinetic sorption with an irreversible phase. However, P sorption during transport was well described without irreversible sorption, thus indirect evidence of limited Cd-P precipitation. Although transport of Cd applied without P was well described (r2 = 0.95, Webster and Windsor), description of BTCs was more qualitative when applied with P, accurately locating effluent peak but approximating the sorption and desorption sides.

Simulated moving bed chromatography for the separation of ethyl esters of eicosapentaenoic acid and docosahexaenoic acid under nonlinear conditions

In this study, ethyl esters of eicosapentaenoic acid and docosahexaenoic acid were separated with simulated moving bed (SMB) chromatography, where the stationary phase was C18 silica gel with particle size of 10μm packed in eight columns, and the mobile phase was pure methanol. The Henry constants, transport parameters and total porosity were measured from pulse response chromatographic experiments using a single column. The Henry constants were obtained from the first moment analysis. The transport parameters including axial dispersion coefficients and effective mass transfer coefficients were obtained from the second moment analysis. Nonlinear adsorption equilibrium isotherms for the pure components and their mixture were determined from adsorption-desorption method. The Langmuir model was used to fit the experimental data, and the corresponding parameters were further used to predict the competitive adsorption equilibria of the mixture. The validity of mathematical model parameters was checked by a frontal chromatography experiment. The simulated results of the SMB process using these parameters agreed well with the experimental results. At the feed concentration of 100g/L, the SMB separation was able to produce both solutes with relative purity above 99%, productivity of 13.11g/L adsorbent/h, and solvent consumption of 0.46L/g.

Analysis and numerical investigation of two dynamic models for liquid chromatography

This paper presents the analytical and numerical investigations of two established models for simulating liquid chromatographic processes namely the equilibrium dispersive and lumped kinetic models. The models are analyzed using Dirichlet and Robin boundary conditions. The Laplace transformation is applied to solve these models analytically for single component adsorption under linear conditions. Statistical moments of step responses are calculated and compared with the numerical predictions for both types of boundary conditions. The discontinuous Galerkin finite element method is proposed to numerically approximate the more general lumped kinetic model. The scheme achieves high order accuracy on coarse grids, resolves sharp discontinuities, and avoids numerical diffusion and dispersion. For validation, the results of the suggested method are compared with some flux-limiting finite volume schemes available in the literature. A good agreement of the numerical and analytical solutions for simplified cases verifies the robustness and accuracy of the proposed method. The method is also capable to solve chromatographic models also for non-linear and competitive adsorption equilibrium isotherms.

Application of wave propagation theory to adsorption breakthrough studies of toluene on activated carbon fiber beds

Based on the wave propagation theory, a dynamics model that combines the nonlinear equilibrium isotherm and the linear mass-transfer equation has been developed to predict the breakthrough behaviour of toluene adsorption in a fixed bed packed with activated carbon fibers. The experimental results showed that the constant-pattern wave model using the Langmuir isotherm equation could capture the dynamic behaviour of the adsorption column. Two important parameters, the half breakthrough time (t1/2) and the volumetric mass-transfer coefficients (kGα) in the model were obtained from linear fitting of the model to experimental breakthrough data. kGα was found to be insensitive to the initial concentration and increased with the increasing the superficial velocity. It was also observed that t1/2 decreases with increasing the superficial velocity and the initial concentration, and increases with increasing the bed height. A sensitivity analysis showed that external mass-transfer had a much stronger influence on the breakthrough curve than internal mass-transfer, confirming that the overall mass-transfer for toluene adsorption onto activated carbon fibers in fixed bed is controlled by external mass-transfer.

Predicting emission characteristics of volatile organic compounds from wet surface coatings

A mathematical model is developed to describe the emission characteristics of VOCs from homogeneous wet coating materials deposited on impermeable substrates. The model considers mass transfer in the material and in air, boundary layer resistance and the change in the coating thickness with time due to emission of VOC. Key features of the model are incorporation of concentration dependent VOC diffusion coefficients predicted from a physical model and nonlinear equilibrium isotherm at the coating/air interface. The model is applied to predict emission characteristics of MMA from acrylic based surface coatings. In an attempt to investigate the influences of equilibrium isotherm type and diffusion formalism on the predictions, simulations are performed with either constant or concentration dependent diffusivities and linear or nonlinear equilibrium isotherms. The lowest MMA concentration in air is predicted by incorporating the concentration dependent diffusivity and nonlinear equilibrium isotherm. The results suggest that assuming the diffusivity of MMA constant or equilibrium isotherm linear may lead to wrong conclusions about the emission rates from wet coatings. The model is general, fully predictive and can be used to predict emission rates of different VOCs from different coating materials if diffusion and thermodynamic parameters are available.

Separation Region and Strategies for Proteins Separation by Salt Gradient Ion‐Exchange SMB

A gradient SMB model is used to analyze the separation performance of proteins in a step‐wise salt gradient ion‐exchange simulated moving bed (IE‐SMB) for the nonlinear adsorption equilibrium isotherms. The strategy of the selection of salt gradient in IE‐SMB chromatography is discussed based on the adsorption equilibrium isotherms of both BSA and myoglobin on Q‐Sepharose FF anion exchanger in the pH 8 Tris buffer. The separation region is used to select the flow rates in the salt gradient IE‐SMB chromatography for the separation of proteins. The separation region is calculated by the gradient SMB model where both the mass transfer resistance in the ion exchangers and axial liquid dispersion in columns are taken into account. Some strategies are suggested for the selections of salt gradient and flow rates in the IE‐SMB chromatography with open loop configuration for two cases: (a) binary separation of proteins with the requirement of the complete recovery from extract stream and raffinate stream; and (b) protein purification from a stream with some impurities.

Improved performance for continuous separation of 1,1′-bi-2-naphthol racemate based on simulated moving bed technology

The success of simulated moving bed (SMB) chromatography for separating difficult or nearly impossible mixture has drawn strong interest in improving its performance in recent years. In this work, experimentally verified mathematical model was used to find the best possible operating conditions for continuous separation of 1,1′-bi-2-naphtol racemate under specified product quality requirement. Equilibrium dispersive model for SMB systems coupled with non-linear equilibrium isotherm and lumped kinetic approximation constitute the mathematical model in simulating the system behavior. Single objective as well as multi-objective optimization with diverse objectives was carried out using a state-of-the-art AI-based non-traditional optimization technique, elitist non-dominated sorting genetic algorithm with jumping genes. The optimization results showed that significant improvement could be achieved depending on desired objectives. Equilibrium triangle theory was used to explain the optimization results.

Adsorption-enhanced steam-methane reforming with intraparticle-diffusion limitations

The adsorption-enhanced reaction (AER) process is theoretically analyzed for hydrogen production by steam-methane reforming (SMR). It uses a fixed-bed packed column of an admixture of a SMR catalyst and an adsorbent for selective removal of CO 2 from the reaction zone. A mathematical model taking into account multicomponent (six species) mass balances, overall mass balance, Ergun relation for pressure drop, energy balance for bed-volume element including the heat-transfer to the column wall, and nonlinear adsorption equilibrium isotherm coupled with three main reactions was derived to describe AER process with the intraparticle-diffusion limitations. The numerical solution of the model equations for this process was obtained by using the method of orthogonal collocation. The validity of the model prediction was checked by comparing the simulated results with experimental data from literature. The mechanism of the adsorption-enhanced SMR is studied by analysis of the profiles of the bed concentrations, temperature, velocity, pressure, and reaction and adsorption rates. The intraparticle-diffusion limitations on the adsorption-enhanced SMR are evaluated by the effectiveness factors. The effect of the operating conditions (reaction temperature, pressure and length of adsorptive reactor) on the hydrogen purity, hydrogen productivity and methane conversion is studied by numerical simulation; a high purity of hydrogen product gas (90–98%) with methane as the prime impurity and traces of CO 2 (below 400 ppm) and CO (below 30 ppm) can be produced directly from the adsorptive reactor under conditions of 450–490 °C and 222.9–891.4 kPa.

Simulation of five‐step one‐bed sorption‐enhanced reaction process

AbstractA five‐step, one‐bed, sorption‐enhanced reaction process proposed by Carvill et al. in 1996 for hydrogen production by steam–methane reforming was analyzed. For the simulated results of the first step, data from Hufton et al. (1999) and Ding and Alpay (2000) for a fixed‐bed column of an admixture of a catalyst and a sorbent that selectively removes CO2 from the reaction zone were used. The sorbent is periodically regenerated by using the principles of pressure‐swing adsorption. The process steps allow direct production of hydrogen with high purity and high methane conversion. A model considering multicomponent and overall mass balance, Ergun relation for pressure drop, energy balance for the bed‐volume element, and nonlinear adsorption equilibrium isotherm coupled with three main reactions was derived to describe the sorption‐enhanced reaction cyclic process. Two different isotherms were used under wet and dry conditions. The LDF model adopted describes the mass‐transfer rate of CO2 in the adsorbent. Numerical solution of model equations for the cyclic process was obtained by the orthogonal collocation method. The operating conditions allow the combination of a sufficiently high purity of hydrogen (average purity over 80%) with traces of CO and CO2, high methane conversion, fast adsorbent regeneration, and cyclic steady‐state operation. The model predictions agree reasonably with the literature data. The package is suitable for the design and analysis of sorption‐enhanced reaction process.

Sorption-enhanced reaction process with reactive regeneration

Sorption-enhanced reaction process with reactive regeneration of adsorbent was proposed where the purge step is performed at a low temperature of 400°C (compared with the reaction temperature of 450°C) with 10% H 2 in nitrogen under atmospheric pressure. Hydrogen-enriched stream with traces of CO 2 and CO can be produced by steam–methane reforming. A mathematical model taking into account multicomponent (six species) mass balances, overall mass balance, Ergun relation for the pressure drop, energy balance for the bed-volume element including the heat transfer to the column wall, and nonlinear adsorption equilibrium isotherm coupled with three main reactions was derived to describe this new cyclic process. The feasibility and effectiveness of this cyclic process is analyzed by numerical simulation. The results show that using either a 4 m or a 6 m long adsorptive reactor (depending on the operation time of the first step), a product gas with above 88% hydrogen purity and traces of CO 2 and CO (CO less than 30 ppm ) can be continuously produced and directly used in the fuel cell applications. The validity of the model prediction was checked by comparing the simulated results with experimental data from the literature where the regeneration of adsorbent is carried out by steam purge at subatmospheric pressure. The model results qualitatively agree with experimental data. The package is needed to improve the design and analysis of sorption-enhanced reaction process.

Measurements of Effective Self-diffusion Coefficients in a Gel-Type Cation Exchanger by the Zero-Length-Column Method

The zero-length column (ZLC), conventionally developed for measuring the diffusivity of gases and liquids in zeolite crystals, has been extended to the measurement of intraparticle diffusivities in ion exchangers. A theoretical model which considers the ZLC cell as a continuous stirred tank reactor is developed. The model allows the use of nonlinear ion-exchange equilibrium isotherms, describes ion fluxes with Nernst-Planck equations, and includes the resistances to mass transfer both in the particle and in the film. Simulations have been carried out to analyze the effect of model parameters on the system response and to choose experimental operating conditions. The model validity has been tested by comparison with the experimental results of the ZLC for a step perturbation in the binary systems Na + -H + , K + -H + , and Na + -K + .

Heat Effects in ZLC Experiments

The problem of nonisothermal desorption in a zero length column (ZLC) experiment is considered theoretically. Simple analytical expressions for the ZLC desorption curve are derived for certain limiting situations in which the governing equations reduce to a linear form. More general numerical solutions are calculated for a wide range of experimental conditions assuming both negligible mass transfer resistance and finite mass transfer resistance controlled by intraparticle diffusion. A simple criterion for negligible thermal effects is developed. It is shown that when the ZLC technique is applied to the measurement of diffusion in unaggregated zeolite crystals, as originally intended, heat effects are generally insignificant. However, when applied to the measurement of macropore diffusion in relatively large adsorbent particles heat effects can become important and may cause major modification of both the desorption rate and the shape of the desorption curve. A recent experimental ZLC study carried out with commercial adsorbent particles, under conditions of macropore diffusion control, showed an anomalous dependence of the desorption rate on both temperature and particle size. These effects can be qualitatively explained by the nonisothermal model. A more precise quantitative representation of these experiments will require a more refined model incorporating a nonlinear equilibrium isotherm as well as intraparticle diffusional resistance.

Intraparticle Diffusivity of Cd Ions in a New Biosorbent Material

Cadmium equilibrium sorption isotherms were determined for formaldehyde crosslinked Sargassum fluitans, establishing that an effective regeneration of the new biosorbent material is possible by an acid wash. Batch desorption kinetics were investigated at pH values of 1.0 and 2.0. By incorporating the linear and non-linear Langmuir equilibrium isotherm relationships into the rate equations, a mathematical model was proposed for modeling the metal desorption process. The model was solved numerically and a MATLAB computer program was used to curve-fit the experimental data. The model successfully predicted the Cd 2+ elution concentration profile in a batch reactor. The average values of the intraparticle diffusivity of Cd 2+ in the algal biosorbent calculated from the model were 3.40 x 10 -6 and 1.65 x 10 -6 cm 2 s -1 , 0.473 and 0229 times the molecular diffusivity of Cd 2+ in water, at pH values of 1.0 and 2.0, respectively. These values agreed well with theoretical predictions.