Diffusion adsorption kinetics model

An adsorption kinetic model was developed to evaluate the adsorption rates of five pure gases (Nj, O2, Ar, CO, and CH4) on a Takeda-3A CMS over a wide range of pressures up to ISatm. The kinetic characteristics of adsorption on the CMS were studied by using the adsorption equilibrium of five pure gases measured at three different temperatures and their physical properties. Since the diffiisional time constants of all the components showed much stronger dependence of pressure than those expected by the traditional Darken relation, a structural diffusion model was applied to predict the strong pressure dependence. The proposed model successfully predicted the dif ional time constant up to high pressure on the CMS. 

The first physically sound model for adsorption kinetics, which was derived by Ward and Tordai [18], is based on the assumption that the time dependence of a surface or interfacial tension (which is directly proportional to the surface excess F, in mol m ) is caused by diffusion and transport of surfactant molecules to the interface. This is referred to as diffusion-controlled adsorption kinetics model . The interfacial surfactant concentration at any time t, T(t), is given by the following expression,... 

The adsorption kinetics of interfacial active molecules at liquid interfaces, for example surfactants at the aqueous solution/air or solution/organic solvent interface, can be described by quantitative models. The first physically founded model for interfaces with time invariant area was derived by Ward Tordai (1946). It is based on the assumption that the time dependence of interfacial tension, which is directly correlated to the interfacial concentration T of the adsorbing molecules, is caused by a transport of molecules to the interface. In the absence of any external influences this transport is controlled by diffusion and the result, the so-called diffusion controlled adsorption kinetics model, has the following form... 

The disadvantage of this thermodynamic criterion is that it can be used only in combination with a purification procedure. Thus, if such a procedure is not available, the purity of a surfactant solution with respect to its useful interfacial study caimot be checked. Therefore, a second criterion was developed, which is based on an adsorption kinetics model. If adsorption of both, the main component and the impurity is assumed to be diffusion-controlled, the difference between the surface tension values, measured after a definite time t j after adsorption and desorption, respectively, is a measure of the purity of the solution. 

Appendix 4E Application of the Laplace Transform to solve the diffusion-controlled ADSORPTION KINETICS MODEL... 

As mentioned above many adsorption kinetics models were discussed in the literature. These models consider specific mechanisms of the molecular transfer from the subsurface to the interface, or vice versa in the case of desorption [5, 6, 7, 8, 9, 10, 11, 12, 14, 23], The models, which assume only the transfer mechanism as rate determining step are called kinetic-controlled. More advanced models, the so-called mixed models, consider both the transport by diffusion in the bulk and the transfer mechanism (cf. Fig. 4.1). Such mixed models were first derived by Baret in 1969 [9] who combined Eq. (4.1) with a certain transfer mechanism. 

As it is the case in adsorption kinetics models, besides the diffusion theory, other models exist to describe the exchange of matter. We refer here only to the review by Miller et al. [1] and the book by Dukhin et al. [2]. 

Most spraying processes work under dynamic conditions and improvement of their efficiency requires the use of surfactants that lower the liquid surface tension yLv under these dynamic conditions. The interfaces involved (e.g. droplets formed in a spray or impacting on a surface) are freshly formed and have only a small effective age of some seconds or even less than a millisecond. The most frequently used parameter to characterize the dynamic properties of liquid adsorption layers is the dynamic surface tension (that is a time dependent quantity). Techniques should be available to measure yLv as a function of time (ranging firom a fraction of a millisecond to minutes and hours or days). To optimize the use of surfactants, polymers and mixtures of them specific knowledge of their dynamic adsorption behavior rather than equilibrium properties is of great interest [28]. It is, therefore, necessary to describe the dynamics of surfeictant adsorption at a fundamental level. The first physically sound model for adsorption kinetics was derived by Ward and Tordai [29]. It is based on the assumption that the time dependence of surface or interfacial tension, which is directly proportional to the surface excess F (moles m ), is caused by diffusion and transport of surfeictant molecules to the interface. This is referred to as the diffusion controlled adsorption kinetics model . This diffusion controlled model assumes transport by diffusion of the surface active molecules to be the rate controlled step. The so called kinetic controlled model is based on the transfer mechanism of molecules from solution to the adsorbed state and vice versa [28]. 

The time required for surface tension reduction depends on diffusion processes involved in surfactant adsorption. Kinetic models for surfactant adsorption divide the adsorption process into two steps [67]. The first step is the transport of the surfactant to the subsurface, driven by a concentration gradient or hydrody-... 

Winters and Lee134 describe a physically based model for adsorption kinetics for hydrophobic organic chemicals to and from suspended sediment and soil particles. The model requires determination of a single effective dififusivity parameter, which is predictable from compound solution diffusivity, the octanol-water partition coefficient, and the adsorbent organic content, density, and porosity. 

In any case, exceptions to the FIAM have been pointed out [2,11,38,44,74,76,78]. For example, the uptake has been shown to depend on the Cj M or rMI (e.g. in the case of siderophores [11] or hydrophobic complexes [43,50]), rather than on the free c M. Several authors [11,12,15] showed that a scheme taking into account the kinetics of parallel transfer of M from several solution complexes to the internalisation transporter ( ligand exchange ) can lead to exceptions to the FIAM, even if there is no diffusion limitation. Adsorption equilibrium has been assumed in all the models discussed so far in this chapter, and the consideration of adsorption kinetics is kept for Section 4. Within the framework of the usual hypotheses in this Section 3, we would expect that the FIAM is less likely to apply for larger radii and smaller diffusion coefficients (perhaps arising from D due to the labile complexation of M with a large macromolecule or a colloid particle, see Section 3.3). 

Three kinetic models were applied to adsorption kinetic data in order to investigate the behavior of adsorption process of adsorbates catechol and resorcinol onto ACC. These models are the pseudo-first-order, the pseudo-second-order and the intraparticle diffusion models. Linear form of pseudo-first-order model can be formulated as... 

Adsorption has a significant impact on the movement of allelo-chemical substances in soil. Such movement in soil by water is important from the standpoint of mechanism of phytotoxin activity in the receiving species at a site remote from the donor plant. Adsorption reduces the solute concentration in the soil solution and consequently minimizes redistribution in the environment. Solute transport has been described by Pick s second law of diffusion and the kinetic models for adsorption and degradation of reactive solutes (, 44). The contribution of adsorption is measured and expressed as the retardation factor, R. 

The effects of physical transport processes on the overall adsorption on porous solids are discussed. Quantitative models are presented by which these effects can be taken into account in designing adsorption equipment or in interpreting observed data. Intraparticle processes are often of major importance in adsorption kinetics, particularly for liquid systems. The diffusivities which describe intraparticle transfer are complex, even for gaseous adsorbates. More than a single rate coefficient is commonly necessary to represent correctly the mass transfer in the interior of the adsorbent. 

Adsorption Kinetics Diffusion and Kinetic Controlled Models... 

Active Oxygen Method (AOM), 535, 544 Adsorption, and interfacial properties diffusion and kinetic controlled models, 617-618 (figs.), 620-622 Gibbs adsorption isotherm, 617-619 kinetics of surface-active substances, 639... 

Englezos et al. (1987a,b) generated a kinetic model for methane, ethane, and their mixtures to match hydrate growth data at times less than 200 min in a high pressure stirred reactor. Englezos assumed that hydrate formation is composed of three steps (1) transport of gas from the vapor phase to the liquid bulk, (2) diffusion of gas from the liquid bulk through the boundary layer (laminar diffusion layer) around hydrate particles, and (3) an adsorption reaction whereby gas molecules are incorporated into the structured water framework at the hydrate interface. 

There have been few studies reported in the literature in the area of multi-component adsorption and desorption rate modeling (1, 2,3., 4,5. These have generally employed simplified modeling approaches, and the model predictions have provided qualitative comparisons to the experimental data. The purpose of this study is to develop a comprehensive model for multi-component adsorption kinetics based on the following mechanistic process (1) film diffusion of each species from the fluid phase to the solid surface (2) adsorption on the surface from the solute mixture and (3) diffusion of the individual solute species into the interior of the particle. The model is general in that diffusion rates in both fluid and solid phases are considered, and no restrictions are made regarding adsorption equilibrium relationships. However, diffusional flows due to solute-solute interactions are assumed to be zero in both fluid and solid phases. 

A mathematical model has been developed to describe the kinetics of multicomponent adsorption. The model takes into account diffusional processes in both the solid and fluid phases, and nonlinear adsorption equilibrium. Comparison of model predictions with binary rate data indicates that the model predictions are in excellent for solutes with comparable diffusion rate characteristics. For solutes with markedly different diffusion rate constants, solute-solute interactions appear to affect the diffusional flows. In all cases, the total mixture concentration profiles predicted compares well with experimental data. 

This point can be appreciated more quantitatively after consideration of an important (but simple) model of transport-controlled adsorption kinetics, the film diffusion process.34 35 This process involves the movement of an adsorptive species from a bulk aqueous-solution phase through a quiescent boundary layer ( Nemst film ) to an adsorbent surface. The thickness of the boundary layer, 5, will be largest for adsorbents that adsorb water strongly and smallest for aqueous solution phases that are well stirred. If j is the rate at which an... 

And finally, the coke burning is a heterogeneous process. Its modeling includes description of the processes of molecule adsorption on the surface, surface reactions, desorption of reaction products, diffusion through the pores and diffusion to the particle surface. At present the majority of these processes for coke are relatively poorly known. The key distinction of surface reactions from reactions at the gas phase consists in the necessity to attract for description of their rates such notions as surface active centers and adsorbed particles. And in the kinetic models a different nature of active centers (different energy of dislocations) necessitates consideration of the same particles adsorbed on them as different compounds because of... 

A useful literature relating to polypeptide and protein adsorption kinetics and equilibrium behavior in finite bath systems for both affinity and ion-ex-change HPLC sorbents is now available160,169,171-174,228,234 319 323 402"405 and various mathematical models have been developed, incorporating data on the adsorption behavior of proteins in a finite bath.8,160 167-169 171-174 400 403-405 406 One such model, the so-called combined-batch adsorption model (BAMcomb), initially developed for nonporous particles, takes into account the dynamic adsorption behavior of polypeptides and proteins in a finite bath. Due to the absence of pore diffusion, analytical solutions for nonporous HPLC sorbents can be readily developed using this model and its two simplified cases, and the effects of both surface interaction and film mass transfer can be independently addressed. Based on this knowledge, extension of the BAMcomb approach to porous sorbents in bath systems, and subsequently to packed-, expanded-, and fluidized-bed systems, can then be achieved. 

McCoy and Liapis [36] used two different kinetic models to represent the column affinity process. In both models the transport of the adsorbate in the adsorbent particle is considered to be governed by the diffusion into the pores. In model I the adsorption is assumed to be completely reversible with no interaction between the adsorbed molecules, In model 2, it is assumed that the biomolecule may change conformation after adsorption. Although these two models represent different overall adsorption mechanisms, the differences between the simulated breakthrough curves is very small. 

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