Numerical methods adsorption

Recent polymer adsorption theories, such as those of Roe (3) and of Scheutjens and Fleer (h) allow the calculation of displacement isotherms, so that we could study the dependency of these isotherms on various parameters by numerical methods. However, all the essential features of displacement can also be demonstrated by means of a simple analytical expression for the critical point, which can be derived in a straightforward way. 

As seen in equations (32)-(34), the forward adsorptive flux depends upon the concentration of free cell surface carriers. Unfortunately, there is only limited information in the literature on determinations of carrier concentrations for the uptake of trace metals. In principle, graphical and numerical methods can be used to determine carrier numbers and the equilibrium constant, As, corresponding to the formation of M — Rcen following measurement of [M] and (M —Rceii. For example, a (Scatchard) plot of (M — RCeii /[M] versus (M — RCeii should yield a straight line with a slope equal to the reciprocal of the dissociation constant and abscissa-intercept equal to the total carrier numbers (e.g. [186]). 

These models describe the development of surface charge and potential together with ion adsorption in a quantitative manner. They have in common a set of simultaneous equations that can be solved by numerical methods using the appropriate... 

The numerical optimization methods do not require additional assumptions of the temporal constancy, or even neglect some physical constants, for example surface potential. Used for the optimization of the edl parameters (surface hydroxyl group reaction constants, capacity and density of adsorption sites) the numerical methods allow us to find the closest values to the experimentally available data (surface charge density, adsorption of ions, zeta potential, colorimetric measurements). Usually one aims to find the parameters, accepted from physical point of view, where a function, that expresses square of the deviation between calculated and measured values will be the smallest. 

The determination of the ion reaction constants with hydroxyl groups on the surface of metal oxide, may be done in the similar way, as the determination of the surface hydroxyl group ionization constants by the extrapolation or numerical methods. The example of the first one is a method proposed for the oxide surface by Schindler [16]. According to the previous remarks the surface adsorption of the simple cations may take place on two hydroxyl groups at most. Then it may be described as follows ... 

Horstmann and Chase [35] have used the mass transfer parameters determined in stirred tank experiments to simulate the breakthrough curves of affinity chromatography experiments. Numerical methods using different computer packages were carried out to solve the differential equations of the stirred tank adsorption and to predict the performances of a packed bed chromatographic column. 

For the solution of sophisticated mathematical models of adsorption cycles including complex multicomponent equilibrium and rate expressions, two numerical methods are popular. These are finite difference methods and orthogonal collocation. The former vary in the manner in which distance variables are discretized, ranging from simple backward difference stage models (akin to the plate theory of chromatography) to more involved schemes exhibiting little numerical dispersion. Collocation methods are often thought to be faster computationally, but oscillations in the polynomial trial function can be a problem. The choice of best method is often the preference of the user. 

All parameters used in equations (1) and (2) were calculated from single gas adsorption and membrane permeation experiment. The numerical method for solving above equations were MOL(Method Of Line). These calculations were executed using LSODE solver(FORTRAN code). 

Direct determination of the column saturation capacity requires measurement of the adsorption isotherm. Use of methods such as frontal analysis, elution by characteristic point are classical techniques. Frontal analysis and elution by characteri.stic point require mg or gram quantities of pure product component. It is also possible to estimate the column saturation capacity from single-component overloaded elution profiles using the retention time method or using an iterative numerical method from a binary mixture [66J. 

Regarding adsorption and diffusion without reaction, Jordi and Do (49) simulated the expected results for the frequency response by completely numerical methods, with no need for linearization. In a later study, they used a linearized model coupled with analytic solutions for the diffusion inside the particles, which also took into account transport in both macropores and micropores (50). The mathematical details are clearly presented in these papers. 

AEDs derived for six different compoimds, rmder different experimental conditions, on the same Kromasil Cig allowed the selection of the isotherm model that, in each case, best accoimts for the adsorption data of these compotmds. Knowing the isotherm model made it possible to use the rapid numerical method of derivation of the isotherm coefficients that is based on the solution of the in-... 

NUMERICAL METHODS OF SOLVING THE INTEGRAL EQUATION OF ADSORPTION... 

A consequence of the ill-posed nature of Eq. (14), therefore, is that different PSD results can be obtained for the same material if different methods are applied to solve the adsorption integral equation, even if the same experimental data and adsorption model are used in both cases. A standard protocol has not yet been agreed upon for the use of regularization in pore size characterization. To avoid confusion in comparing PSD results, therefore, the numerical method employed to solve for the PSD and the type of regularization, if any, implemented to smooth the PSD should both be clearly identihed. 

For common adsorbates the equilibrium constants of reactions involving only solution species are available from literature for less common adsorbates they can be determined in separate experiments that do not involve the adsorbent. The equilibrium constants of (hypothetical) surface reactions are the adjustable parameters of the model, and they are determined from the adsorption data by means of appropriate fitting procedure. With simple models (e.g. the model leading to Langmuir equation which has two adjustable parameters) the analytical equations exist for least-square best-fit model parameters as the function of directly measured quantities, but more complicated models require numerical methods to calculate their parameters. 

It is a common feature of the numerous methods of separating globin from heme or hematin that one component is removed from the mixture by adsorption, as on kieselguhr (hematin) or precipitation, as by acetone (globin). It is, in principle, impossible to determine the amounts of split products present in labile equilibrium by removal of one component, since removal of one product of the dissociation is bound to result in a shift of the equilibrium point in the direction of increased dissociation. In addition, when the medium is changed, as by addition of acetone, profound changes must occur in the acidic dissociations in the protein which affect the reaction (see Section II of this review). Both of these effects would... 

While DFT allows us to calculate values for q(p, e), it of course provides no analytic form for the function, and in general the form of f(e) is also unknown. However, by using carefully designed numerical methods, model isotherms calculated by MNLDFT can be used in carrying out the inversion of the discrete form of the integral equation of adsorption. In this way one can determine the effective adsorptive potential distribution of the adsorbent from the experimental adsorption isotherm. The method used can be expressed by... 

Mamleev, V. Sh. and Bekturov, E.A. (1996). Numerical method for analysis of surface heterogeneity in a case of finite diversity of adsorption sites. Langmuir, 12, 441-9. 

Weronski, P. Eumelech, M. 2008 Novel numerical method for calculating initial flux of colloid particle adsorption through an energy barrier. Journal of Colloid and Interface Science 319,406 15. 

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