Adsorption mathematical modeling

Adsorption of macromolecules has been widely investigated both theoretically [9—12] and experimentally [13 -17]. In this paper our purpose was to analyze the probable structures of polymeric stationary phases, so we shall not go into complicated mathematical models but instead consider the main features of the phenomenon. The current state of the art was comprehensively summarized by Fleer and Lyklema [18]. According to them, the reversible adsorption of macromolecules and the structure of adsorbed layers is governed by a subtle balance between energetic and entropic factors. For neutral polymers, the simplest situation, already four contributor factors must be distinguished ... 

Mathematical models have also predicted a low volatility for methyl parathion (Jury et al. 1983 McLean et al. 1988). One study using a laboratory model designed to mimic conditions at soil pit and evaporation pond disposal sites (Sanders and Seiber 1983) did find a high volatility from the soil pit model (75% of the deposited material), but a low volatility for the evaporation pond model (3. 7% of the deposited material). A study of methyl parathion and the structurally similar compound ethyl parathion, which have similar vapor pressures, foimd that methyl parathion underwent less volatilization than ethyl parathion in a review of the data, the reduced level of volatilization for methyl parathion was determined to be due to its adsorption to the soil phase (Alvarez-Benedi et al. 1999). 

A mathematical model for reservoir souring caused by the growth of sulfate-reducing bacteria is available. The model is a one-dimensional numerical transport model based on conservation equations and includes bacterial growth rates and the effect of nutrients, water mixing, transport, and adsorption of H2S in the reservoir formation. The adsorption of H2S by the roek was considered. 

H. Zhang, E. J. Mackay, K. S. Sorbie, and P. Chen. Non-equilibriimi adsorption and precipitation of scale inhibitors corefloods and mathematical modelling. In Proceedings Volume, page 18 PR SPE Oil Gas Int Conf In China (Beijing, China, 11/7-11/10), 2000. 

Alternative mechanisms have been recently proposed [78,79] based on a kinetic investigation of NO reduction by n-octane under isothermal (200°C) and steady-state conditions in the presence of H2. The authors built up a mathematical model based on supposed reaction pathways, which account for molecular adsorption of NO and CO and dissociative ones for H2 and 02. The elementary steps, which have been considered for modelling their results are reported in Table 10.3. Interesting kinetic information can be provided by the examination of this mechanism scheme in particular the fast bimolecular... 

There Is so little adsorption that the material moves with the water front. The mathematical model breaks down under these circumstances. 

Hougen-Watson Models for the Case of Equilibrium Adsorption. This section treats Hougen-Watson mathematical models for cases where the rate limiting step is the chemical reaction rate on the surface. In all cases it is assumed that equilibrium is established with respect to adsorption of all species. 

The dynamics of high-temperature CO adsorption and desorption over Pt-alumina was analyzed in detail using a transient mathematical model. The model combined the mechanism of CO adsorption and desorption (established from ultrahigh-vacuum studies over single-crystal or polycrystalline Pt surfaces) with extra- and intrapellet transport resistances. The numerical values of the parameters which characterize the surface processes were taken from the literature of clean surface studies ... 

In general, there is an array of equilibrium-based mathematical models which have been used to describe adsorption on solid surfaces. These include the widely used Freundlich equation, a purely empirical model, and the Langmuir equation as discussed in the following sections. More detailed modeling approaches of sorption mechanisms are discussed in more detail in Chap. 3 of this volume. 

For all reactions, the mass transport regime is controlled by the diffusion of the reacting ligand only, as the mercury electrode serves as an inexhaustible source for mercury ions. Hence, with respect to the mathematical modeling, reactions (2.205) and (2.206) are identical. This also holds true for reactions (2.210) and (2.211). Furthermore, it is assumed that the electrode surface is covered by a sub-monomolecular film without interactions between the deposited particles. For reactions (2.207) and (2.209) the ligand adsorption obeys a linear adsorption isotherm. Assuming semi-infinite diffusion at a planar electrode, the general mathematical model is defined as follows ... 

Irrespective of the sources of phenolic compounds in soil, adsorption and desorption from soil colloids will determine their solution-phase concentration. Both processes are described by the same mathematical models, but they are not necessarily completely reversible. Complete reversibility refers to singular adsorption-desorption, an equilibrium in which the adsorbate is fully desorbed, with release as easy as retention. In non-singular adsorption-desorption equilibria, the release of the adsorbate may involve a different mechanism requiring a higher activation energy, resulting in different reaction kinetics and desorption coefficients. This phenomenon is commonly observed with pesticides (41, 42). An acute need exists for experimental data on the adsorption, desorption, and equilibria for phenolic compounds to properly assess their environmental chemistry in soil. 

In 1976 he was appointed to Associate Professor for Technical Chemistry at the University Hannover. His research group experimentally investigated the interrelation of adsorption, transfer processes and chemical reaction in bubble columns by means of various model reactions a) the formation of tertiary-butanol from isobutene in the presence of sulphuric acid as a catalyst b) the absorption and interphase mass transfer of CO2 in the presence and absence of the enzyme carboanhydrase c) chlorination of toluene d) Fischer-Tropsch synthesis. Based on these data, the processes were mathematically modelled Fluid dynamic properties in Fischer-Tropsch Slurry Reactors were evaluated and mass transfer limitation of the process was proved. In addition, the solubiHties of oxygen and CO2 in various aqueous solutions and those of chlorine in benzene and toluene were determined. Within the framework of development of a process for reconditioning of nuclear fuel wastes the kinetics of the denitration of efQuents with formic acid was investigated. 

This overview will outline surfactant mixture properties and behavior in selected phenomena. Because of space limitations, not all of the many physical processes involving surfactant mixtures can be considered here, but some which are important and illustrative will be discussed these are micelle formation, monolayer formation, solubilization, surfactant precipitation, surfactant adsorption on solids, and cloud point Mechanisms of surfactant interaction will be as well as mathematical models which have been be useful in describing these systems,... 

This brief review has attempted to discuss some of the important phenomena in which surfactant mixtures can be involved. Mechanistic aspects of surfactant interactions and some mathematical models to describe the processes have been outlined. The application of these principles to practical problems has been considered. For example, enhancement of solubilization or surface tension depression using mixtures has been discussed. However, in many cases, the various processes in which surfactants interact generally cannot be considered by themselves, because they occur simultaneously. The surfactant technologist can use this to advantage to accomplish certain objectives. For example, the enhancement of mixed micelle formation can lead to a reduced tendency for surfactant precipitation, reduced adsorption, and a reduced tendency for coacervate formation. The solution to a particular practical problem involving surfactants is rarely obvious because often the surfactants are involved in multiple steps in a process and optimization of a number of simultaneous properties may be involved. An example of this is detergency, where adsorption, solubilization, foaming, emulsion formation, and other phenomena are all important. In enhanced oil recovery. 

Such limits for Kx are not required. The mathematical model requires only that Kx be positive, a necessity for the constant to have physical meaning. The mechanism of the interaction of the gel with the experimental molecule is unspecified, and may be adsorption, ion exchange, or any other process representable by an "equilibrium constant ... 

In most adsorption processes the adsorbent is contacted with fluid in a packed bed. An understanding of the dynamic behavior of such systems is therefore needed for rational process design and optimization. What is required is a mathematical model which allows the effluent concentration to be predicted for any defined change in the feed concentration or flow rate to the bed. The flow pattern can generally be represented adequately by the axial dispersed plug-flow model, according to which a mass balance for an element of the column yields, for the basic differential equation governing llie dynamic behavior,... 

The objective in this section is to derive a mathematical model that can be used to extract the rate of adsorption from experimentally obtained dynamic surface tension data. Various investigators have speculated that the mechanism of surfactant adsorption involves two subsequent steps ... 

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. 

The studies of Ertl and co-workers showed that the reason for self-oscillations [142, 145, 185-187] and hysteresis effects [143] in CO oxidation over Pt(100) in high vacuum ( 10 4 Torr) is the existence of spatio-temporal waves of the reversible surface phase transition hex - (1 x 1). The mathematical model [188] suggests that in each of the phases an adsorption mechanism with various parameters of CO and 02 adsorption/desorption and their interaction is realized, and the phase transition is modelled by a semi-empirical method via the introduction of discontinuous non-linearity. Later, an imitation model based on the stochastic automat was used [189] to study the qualitative characteristics for the dynamic behaviour of the surface. 

This effect is relatively small until the total magnesium ion concentrations reach about 1000 ppm. o The effect of Mg2+ concentration on limestone dissolution rate can be explained by a surface adsorption model. The adsorption of Mg2+ reduces the limestone dissolution rate because the surface is partially blinded by the adsorbed magnesium ions. The competitive adsorption of calcium and magnesium ions was described by a mathematical model based on the Langmuir adsorption isotherm. The model was used to explain the sensitivity of limestone dissolution rate to magnesium ion concentration under limestone DA operating conditions. 

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