Applied Adsorption Kinetics Model for Removal of Hazards from Aqueous Solution : More Informational Parameters for Industrial Design

Adsorption is one of the widely applied techniques for removal of hazards from aqueous solution. Its kinetics studies are of great significance to evaluate the adsorption performance and reveal the underlying mechanisms. There are several mathematic models to describe the adsorption process and diffusion mode. In recently years, more and more adsorption kinetics models with more informational parameters are investigated, which could give guidance for industrial conversion through supplying paramount importance parameters. Here, we reviewed several widely used adsorption kinetics models and paid more attention to their potential values of industrial application guiding. We believe that the review is of certain significance and improvement of applied adsorption kinetics modeling.

models, like pseudo first order, Langmuir and Freundlich models, which are widely used in the treatment study of wastewater, are used to try to investigate the mechanism of adsorption process [9][10][11][12][13][14][15].In recent years, the study of adsorption models, which could indicate the applied parameters, such as adsorption equilibrium time and optimal adsorbent pore size, is being researched extensively [16][17][18].These models are expected to be applied prosperously since the supplied parameters are very vital to optimize the design of the industrial adsorption system.
As scientists begin to identify the adsorption mechanisms that contribute to separation efficiency, more informed and personalized removal strategy of hazard from aqueous solution can be developed.This increased insight has led to the directed design of adsorption kinetics models that better exploit the physical and chemical properties of adsorbent.For instance, in industrial scale objects, the adsorbent and adsorbate are in contact with each other for a short time period, therefore, predicating the equilibrium time and the rate of adsorption is of paramount importance [19][20][21][22].Meanwhile, differences in particle size and the pore nature of the adsorbent plays a very important role in the efficiency of the adsorption process [23][24][25][26].The better description of adsorbent size contribution could supply a better guidance for the adsorbent size selection in big-scale application of water treatment.
Many hypotheses have been considered to describe the adsorption kinetic and various theoretical and empirical models have been presented for modeling of adsorption kinetics.The classical adsorption kinetics model, Langmuir kinetics model (equation 1): assumes the surface of adsorbate is energetically homogeneous [27][28].
(p A is adsorbate's partial pressure; V is the volume of adsorbate; θ A is the fractional occupancy of the adsorption sites; V m is the volume of the monolayer; K eq is the associated equilibrium constant).By contrast, the Langmuir-Freundlich model (equation 2) is another equation, which can't be solved analytically and considers the effect of surface heterogeneity [29][30][31].However, the rate constant of this model can be obtained by an approximation method.
(θ A is the fractional occupancy of the adsorption sites; α F and C F are fitting parameters) The other two important empirical rate models: pseudo first order (equation 3) and pseudo second order (equation 4): could supply a better validity under certain conditions [32][33][34].
(k 1 is the pseudo first order rate coefficient; q e and q are the amount of the adsorbed species per unit mass of adsorbent at the equilibrium state and any time of adsorption) (k 2 is the pseudo second order rate coefficient) All these classical adsorption kinetics models are used generally in the study of wastewater treatment to investigate the adsorption mechanism [21,[35][36][37][38][39].However beside the traditional adsorption theory, they cannot give any specific guidance parameter for the industry-scale application.As more and more lab research of wastewater began to be transformed via industrial conversion, more informative process parameters needs to be developed from kinetics model to guide industrial production.In 2017, Azizian et al. reported that the shape of the adsorption kinetic curve could be described by the mathematical model of hyperbolic tangent properly (equation 5) [16] Followed by the traditional model deduction process, the y and x were defined as  =    and x=kt.When the equilibrium state is set as q/q e ≥0.995, the adsorption kinetics model could be described as ).In addition to consider the different degrees of curvature of kinetics curves, which are derived from the adsorbent surface heterogeneity, different functional groups on the ad-species, possibility of binding to the surface with different geometries, and effect of diffusion on the adsorption overall rate, the n power law was introduced for the curvature degree: (t e is the required time for adsorption to reach the equilibrium state; π is the setting of q/q e ≥0.995; n is power constant) This hyperbolic tangent model was evaluated through three different data series on two different testing conditions: adsorption same concentration of adsorbate onto the different amount of adsorbent; different concentration of adsorbate onto the same concentration of adsorbent.For the different amount of adsorbent, the values of the equilibrium time derived from hyperbolic tangent model decreased by increasing the adsorbent dosage.For the different amount of adsorbate, the equilibrium time, t e , increased by increasing the adsorbate concentration in solution.
The predicted equilibrium times of adsorption by the hyperbolic tangent kinetic model are very similar to the experimental ones.This model is simple and can be used easily for modeling of adsorption kinetics data in industrial scales to give the equilibrium time of adsorption.This predicted equilibrium time is much useful for controlling the industrial wastewater treatment process, which could benefit the industrial design and production cost.It is a perfect example of how basic science of model and engineering go hand in hand.This work investigated the model with the restriction setting of equilibrium state as q/q e >0.995.Actually, this model cannot only be used under this scale, but also any designed equilibrium degree.So it could accurately predict the equilibrium time in any setting scale of q/q e according to the experiment needed.By leveraging our ever-growing knowledge of wastewater adsorption and its kinetics model, it is possible to engineer a progressive hyperbolic tangent kinetics models to give more industrialguiding parameters under more flexible equilibrium definition.
Recently, rates of adsorption have been observed to exhibit biphasic behavior: a rapid adsorption in the order of minutes followed by a slower phase with a timescale in the order of hours [40][41].The latter phase, which may result in a significant fraction being sequestered, [42][43][44] is frequently rate limiting for abstraction and separation [45][46][47].Mechanism proposed as potentially responsible for the commonly observed slow adsorption of hazards on resins includes intraorganic matter diffusion [48,49] and hindered pore diffusion [50][51][52].Diffusion through hazardous materials is analogous to diffusion in polymers; i.e, diffusing molecules must penetrate and migrate through a polymeric matrix [53].Pore diffusion can be either sterically hindered or retarded by sorption to organic phases associated with pore wells.Adsorption on high-energy sites may also be an important rate-limiting step.In condensed polymeric organic matter, active compounds may adsorb strongly in molecule-sized voids or "holes" [54] while during pore diffusion they are subject to high-energy sorption in molecule-sized pores [55][56].Now, adsorption has been modeled both as a diffusive and as a first-order rate process.Several different rate models have been used to describe the adsorption of active compounds onto adsorbent.
(ϕ s is fast adsorption rate coefficient; k s is fast adsorption rate; k r is slow adsorption rate; t is adsorption time, q(t) is adsorption amount (mg/g) at the time of t; q 0 is initial adsorption amount(mg/g).
(ϕ s is fast adsorption rate coefficient; k s is fast adsorption rate; k r is slow adsorption rate; t is adsorption time, q(t) is adsorption amount (mg/g) at the time of t; q 0 is initial adsorption amount (mg/g)) Shuler studied a new two-parameter model (equation 8) to express the kinetics of adsorption based on the gamma function, which assumes the adsorption and desorption from heterogeneous sites occurs over a continuum of energies and rates [60].This model theoretically describes a continuous distribution of adsorptive compartments with only two parameters.An advantage of this model is that it theoretically describes a continuous distribution of sorptive compartments with only two parameters.However, the gamma model underpredicted adsorption at late stages as it "flattened out" too much, indicating that the model does not mechanistically describe the observed adsorption process; the two calculated parameters do not have obvious physical significance and do not logically correlate to observed trends in slow adsorption rates and resistant fractions.
(D aq is the diffusion coefficient; ρ g is the density of the adsorbate; ε i is the intrapariticle porosity; τ f is the tortuosity factor) Roberts and Gschwend used a one-parameter spherical pore-diffusion model (equation 9) to describe the experimental data of adsorption, which only has one fitting parameter, and it can theoretically estimate adsorption rates a priori based on adsorbate and adsorbent physical characteristics [61].Advantages of this model are that it has only one fitting parameter, and it can theoretically estimate adsorption rates a priori based on sorbate and sorbent physical characteristics.However, the simple spherical pore-diffusion model often does not fit entire adsorption rate profiles, and it essentially has no rapidly adsorbing fraction.
(C r is the immobile aqueous-phase concentration, r is the radial coordinate; D a is the apparent diffusion coefficient) Although there are several reports about the adsorption kinetics of resins, all of them attribute the adsorption to a simple process without considering the effect of sphere size.Corroll and Berens suggested a three-parameter, twocompartment model (equation 10) accounting for the influence of sphere diameter, which assumed biphasic polymer diffusion processes [62].They studied rates of adsorption of solutes to poly-(vinyl chloride) polymer powders of non-uniform particle size and found that a simple modification of the Fickian model satisfactorily accounts for the effect of particle non-uniformity upon sorption kinetics for conditions under which transport is diffusion-controlled.
This monophasic model always gives a good fit of data in each case, however the adsorption is not considered as rapid, slow and even slower procedures, but only a global process instead.
(ϕ r is fast adsorption rate coefficient; a r and a s are the diameters of the spheres; D r and D s are diffusion coefficient) My study was performed in order to determine the adsorption profiles of adsorbate on resins functionalized with chloromethy and amino groups [63][64].A new adsorption isotherm model, the multi-layer inductive effect model (equation 11): in which the inductive effect passed to a higher layer, and a new adsorption kinetics model (equation 12): the multi-compartment kinetics model, in which the adsorption process contained more compartments, were created according to the multiparameter theory and Karickhoff's theory by investigating the regression of the experimental results [65].
(K 1 , K 2, …, K i are the equilibrium adsorption constants for the first layer, second layer,…,i th layer, respectively; θ 1 , θ 1 ,…, θ i are the fracitions of surface overage at the first layer, second layer,…, i th layer, respectively; C e is the equilibrium adsorbate concentration; Q m is the surface site density ) are the rate constants of every adsorption level; k 1' , k' 2 ,…, k i' are the rate constants of every desorption level, respectively; M 0 is mass present at t=0) These new models explained the adsorption process better than the other kinetic models, and provided a good fit of adsorption data, which is not surprising in that they had at least two more fitting parameters than any of the other models.In order to guide the industry application, a sphere size parameter was considered to be included to explain adsorption kinetics more explicitly.Previous studies have demonstrated that due to the repulsive or attractive action between adsorbate and adsorbent, adsorption of multiple components can be rationally expected to be either competitive or synergistic.In some studies, [41,45,56,66] weeks or months of incubation for the adsorption systems were tested in order to make sure the sorption equilibrium was achieved.In fact, some previous studies have shown that aging a long time has negligible additional effect on adsorption-resistant fractions or adsorption rates.
The equilibration time is based on several factors, such as driving force, sphere diameter, and the concentration of the solution.The effects of driving force, solution concentration and sphere size on the adsorption process have been discussed in our previous work [67].
Comparison of experimental data with Fick's Law requires a solution of the diffusion equation appropriate for the sample geometry and experimental boundary conditions.The simplest geometry for powder is a collection of spherical particles of uniform size.This condition can, in fact, be closely approached in specific powder prepared by emulsion polymerization.Consequently, to approximate adsorption of adsorbate by synthetic resins, we made a model for the adsorption on nonuniform resins in the case of uniform initial concentration throughout the sphere, a constant concentration at the surface, and constant D r .According to the theory of the first-order, two-component four-parameter model, a new model [68] incorporating sphere size can be described by the following equitation 13: (M t is the solid-phase sorbate concentration at a given time, M 0 is the initial solid-phase adsorbate concentration, F 1 and F 2 are different compartments' fractions, D r is the diffusion coefficient, and r 1 and r 2 are the diameters of the spheres, on which the different compartments of adsorption process are mainly carried out) It should be noted, however, that this first-order, two-compartment, four-parameter model does not necessarily reflect the reasonable procedures of adsorption.This model contains just two steps, and does not take into account the differences be-tween fast adsorption, slow adsorption, and even slower adsorption [69].Accordingly, a modified model with sphere size was created according to Karickhoof's theory and the multiparameter adsorption kinetics model.A sphere-size model in which the adsorption process contains more compartments was proposed as equation 14.
Through studying the adsorption processes of same adsorbate on different resins and comparing the fitting results of different models, the values of R 2 from the sphere-size kinetics were identical with our previous multiparameter kinetics model.Meanwhile, this model supplied very useful parameters, which reflects the effect of sphere-size distribution on adsorption kinetics process.This sphere-size model was created to present the first experimental evidence that sphere size affects the adsorption process.The new sphere-size model has more detailed parameters to more accurately describe the adsorption process, and will assist in the development of an efficient, industrial wastewater purification process.However, this model still lacks several important parameters, which could guide the industrial-scale production better, such as equilibrium time.
Overall, the progress toward adsorption kinetics models with more detailed parameter is encouraging, and this is hope that further development of new adsorption theory will help to bring forth better industrial outcomes.
Different groups have carried out adsorption kinetics studies.Norris et al. presented the first experimental evidence to produce a three-parameter model (equation 6) to describe adsorption rates for both a slowly adsorbing fraction and a rapidly desorbing fraction[57][58].The model's parameters are useful for distinguishing the rapidly and slowly adsorbing compartments.An advantage of this model is that its parameters are useful for distinguishing between rapidly and slowly adsorbing fractions.()  0 =   exp(−  ) + (1 −   )exp (−  )

Furthermore, model results
are intuitive given the biphasic nature of the adsorption data.Noort et al. stated a technique using Tenax TA beads as a "sink" for the solute was employed to measure the kinetics of adsorption and desorption of chlorobenzenes, polychlorinated biphenyles from laboratory-contaminated sediment[59].They suggested a three-compartment model (equation 7) in which the slow fraction is divided into two compartments, slowly and very slowly adsorbing.An advantage of this model is that it theoretically describes a continuous distribution of sorptive compartments with only two parameters.However, convergence on a five-parameter solution was laborious and uncertain, with final fitted parameters strongly depending on initial guesses; researchers rarely recommend the triphasic model over others.exp(−  ) +   exp(−  ) + (1 −   −   )exp (−  )