Adsorption isotherm linear isotherm

The values of q are plotted as a function of the equilibrium concentration. For constituents at low or moderate concentrations, the relationship between q and C can be generated. If n = 1, the (q-C) relationship will be linear (Eq. 9), and the slope of the line (i.e.,ITd) defines the adsorption distribution of the pollutant. Kd is generally identified as the distribution or partition coefficient, and is used to describe pollutant partitioning between liquid and solids only if the reactions that cause the partitioning are fast and reversible, and if the isotherm is linear. For cases where the partitioning of the pollutants can be adequately described by the distribution coefficient (i. e.,fast and reversible adsorption, with linear isotherm), the retardation factor (R) of the subsurface environment can be used as follows ... 

According to this equation, 6 depends linearly on lnp therefore, we call (123) logarithmic adsorption isotherm. This isotherm, as it was mentioned in Section IX, describes experimental data well in many cases. 

Example 9.6 Fixed-Bed Adsorption with Linear Isotherm... 

Vlad, M. O. Cerofofini, G. F. Ross, J. Statistical fractal adsorption isotherms, linear energy relations, and power-law trapping-time distributions in porous media. Phys Rev. E 2000, 62, 837-844. 

Very often the fall in heat of adsorption is more nearly linear than logarithmic, and it is this type of behavior that led to the derivation of the Temkin adsorption isotherm. The isotherm is, in fact, derived from the Langmuir adsorption isotherm by inserting the condition that the heat of adsorption decreases linearly with surface coverage. Such an effect can arise from repulsive forces on a uniform surface or from surface heterogeneity of the surface. 

This is the simplest non-linear relation which is exhibited by single solutes under mass-overloaded conditions. The relation in Eq. (3) is the Langmuir Adsorption Isotherm. Other isotherms relating the stationary and mobile phase concentrations are possible, depending upon the individual properties of the solutes, mobile phases and packing materials. Very many solutes follow the Langmuir isotherm, which is one... 

Vlugt T J H, Krishna R and Smit B 1999 Molecular simulations of adsorption isotherms for linear and branched alkanes and their mixtures in silicalite J. Phys. Ohem. B 103 1102-18... 

A rather simpler situation arises when the bulk concentrations are sufficiently small that the adsorption isotherms approach linearity. Then (7,4), for example, shows that... 

The 2eohte and the resin adsorbents show different adsorption isotherm characteristics, particularly at higher concentration (51). The resin adsorbent isotherm is slightly concave upward, whereas the 2eohte isotherm is linear, or even slightly concave downward. Resins, therefore, have an advantage in a UOP Sarex operation that involves high feed-soHds concentration. 

FIG. 16-2 Limiting fixed-bed behavior simple wave for unfavorable isotherm (top), square-root spreading for linear isotherm (middle), and constant pattern for favorable isotherm (bottom). [From LeVan in Rodtigues et al. (eds.), Adsorption Science and Technology, Kluwer Academic Publishers, Dotdtecht, The Nethedands, 1989 reptinted withpeimission.]... 

In either equation, /c is given by Eq. (16-84) for parallel pore and surface diffusion or by Eq. (16-85) for a bidispersed particle. For nearly linear isotherms (0.7 < R < 1.5), the same linear addition of resistance can be used as a good approximation to predict the adsorption behavior of packed beds, since solutions for all mechanisms are nearly identical. With a highly favorable isotherm (R 0), however, the rate at each point is controlled by the resistance that is locally greater, and the principle of additivity of resistances breaks down. For approximate calculations with intermediate values of R, an overall transport parameter for use with the LDF approximation can be calculated from the following relationship for sohd diffusion and film resistance in series... 

Axial Dispersion Effects In adsorption bed calculations, axial dispersion effects are typically accounted for by the axial diffusionhke term in the bed conservation equations [Eqs. (16-51) and (16-52)]. For nearly linear isotherms (0.5 < R < 1.5), the combined effects of axial dispersion and mass-transfer resistances on the adsorption behavior of packed beds can be expressed approximately in terms of an apparent rate coefficient for use with a fluid-phase driving force (column 1, Table 16-12) ... 

For a linear isotherm tij = KjCj), this equation is identical to the con-seiwation equation for sohd diffusion, except that the solid diffusivity D,i is replaced by the equivalent diffusivity = pDj,i/ p + Ppi< ). Thus, Eqs. (16-96) and (16-99) can be used for pore diffusion control with infinite and finite fluid volumes simply by replacing D,j with D. When the adsorption isotherm is nonhnear, a numerical solution is... 

Bidispersed Particles For particles of radius Cp comprising adsorptive subparticles of radius r, that define a macropore network, conservation equations are needed to describe transport both within the macropores and within the subparticles and are given in Table 16-11, item D. Detailed equations and solutions for a hnear isotherm are given in Ruthven (gen. refs., p. 183) and Ruckenstein et al. [Chem. Eng. Sci., 26, 1306 (1971)]. The solution for a linear isotherm with no external resistance and an infinite fluid volume is ... 

The major surfactant in the foam may usually be considered to be present at the bubble surfaces in the form of an adsorbed monolayer with a substantially constant F, often of the order of 3 X 10" (g mol)/ cm", for a molecular weight of several hundred. On the other hand, trace materials follow the linear-adsorption isotherm Tj = KiCj if their concentration is low enough. For a wider range of concentration a Langmuir or other type of isotherm may be applicable (Davies and Rideal, loc. cit.). 

In LC, at very low concentrations of moderator in the mobile phase, the solvent distributes itself between the two phases in much the same way as the solute. However, as the dilution is not infinite, the adsorption isotherm is not linear and takes the form of the Langmuir isotherm. 

The mobile phase was water in which the moderator alcohols were dissolved. It is seen that the linear relationship is completely validated and the data can provide the adsorption isotherms in the manner discussed. The mean surface area was found to be 199 m /g with a standard deviation between the different alcohols of 11 m /g. 

In addition, the concentration of solute in both phases increased almost linearly with the amount of solute added. This would indicate that the system was operating over that part of the adsorption isotherm that was linear. Thus, the solute was interacting... 

A countercurrent moving-bed adsorption column is used to remove benzene from a gaseous emission. Activated carbon is employed as the adsorbent. The flowrate of the gas is 1.2 kg/s and it contains 0.027 wt/wt% of benzene. It is desired to recover 99% of this pollutant. The activated carbon entering the column has 2 X 10 wt/wt% of benzene. Over the operating range, the adsorption isotherm (Yaws et al., 1995) is linearized to... 

Cost estimation and screening external MSAs To determine which external MSA should be used to remove this load, it is necessary to determine the supply and target compositions as well as unit cost data for each MSA. Towards this end, one ought to consider the various processes undergone by each MSA. For instance, activated carbon, S3, has an equilibrium relation (adsorption isotherm) for adsorbing phenol that is linear up to a lean-phase mass fraction of 0.11, after which activated carbon is quickly saturated and the adsorption isotherm levels off. Hence, JC3 is taken as 0.11. It is also necessary to check the thermodynamic feasibility of this composition. Equation (3.5a) can be used to calculate the corresponding... 

The low-coverage parts of the adsorption isotherms evaluated at different temperatures have shown a remarkable feature of linear dependence between the adsorption and the logarithm of gas pressure. This sort of behavior corresponds to the well-known Temkin equation of adsorption... 

For the case of a binary system with linear adsorption isotherms, very simple formulas can be derived to evaluate the better TMB flow rates [19, 20]. For the linear case, the net fluxes constraints are reduced to only four inequalities, which are assumed to be satisfied by the same margin /3 (/3 > 1) and so ... 

Fig. 9-7. Region for complete separation under Equilibrium Theory. Linear adsorption isotherms.

The competitive adsorption isotherms were determined experimentally for the separation of chiral epoxide enantiomers at 25 °C by the adsorption-desorption method [37]. A mass balance allows the knowledge of the concentration of each component retained in the particle, q, in equilibrium with the feed concentration, < In fact includes both the adsorbed phase concentration and the concentration in the fluid inside pores. This overall retained concentration is used to be consistent with the models presented for the SMB simulations based on homogeneous particles. The bed porosity was taken as = 0.4 since the total porosity was measured as Ej = 0.67 and the particle porosity of microcrystalline cellulose triacetate is p = 0.45 [38]. This procedure provides one point of the adsorption isotherm for each component (Cp q. The determination of the complete isotherm will require a set of experiments using different feed concentrations. To support the measured isotherms, a dynamic method of frontal chromatography is implemented based on the analysis of the response curves to a step change in feed concentration (adsorption) followed by the desorption of the column with pure eluent. It is well known that often the selectivity factor decreases with the increase of the concentration of chiral species and therefore the linear -i- Langmuir competitive isotherm was used ... 

Linear case This case is met when the adsorption isotherm is considered linear, which means operation under diluted conditions. Taking into account the saturation capacities of the CSP, this behavior is usually met for concentrations around or below 1 g for separation of enantiomers. 

A more general criterion for linearity can be derived noting that the denominator of the Langmuirian adsorption isotherms must approach 1, and consequently ... 

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