Binary Langmuir model

Adsorption of Mixtures. The Langmuir model can be easily extended to binary or multicomponent systems ... 

As has become clear adsorption phenomena play an important, if not, decisive role in this behaviour, and good data and modelling of adsorption are mandatory, too, to serve as the input parameters for the permeation description. This should not be l ted to the T.angmnir model, but other theories like the IAS (ideal adsorbed solution) and NIAS (non-ideal) should be considered, since they sometines work well for binary systems where the Langmuir model fails. 

On the other hand, the quantitative prediction of competitive isotherm behavior for the components of binary mixtmes is not possible using the competitive Langmuir isotherm model when the difference between the column satmation capacities for the two components exceeds 5 to 10%. For example, the adsorption isotherms of pure cis- and trans-androsterone on sihca are well accoimted for by the Langmuir model [9]. However, the two column saturation capacities differ by 30%, due to the nearly flat structure of the trans isomer compared to the folded structure of the cis isomer. As a consequence, the competitive Langmuir model accounts poorly for the competitive adsorption data [9,10]. Much improved results are obtained with the more complex LeVan-Vermeulen isotherm (Section 4.1.5). Another approach could use the random adsorption site model, with different exclusion siuface areas for the competing molecules [12],... 

The method has been used for the determination of competitive isotherms in cases in which the deviation from the Langmuir model is moderate [115]. An HPLC chromatograph configured so as to permit injecting into the column a wide rectangular pulse, e.g., by pumping into it either the pure mobile phase or, for a known time, a solution of the compound of interest, was used to make the measurements described [115]. Excellent agreement was observed with other experimental data and with the experimental band profiles recorded in overloaded elution for binary samples of various compositions [48]. 

In this discussion, we assume that the mobile phase contains a single additive or modifier dissolved in a weak solvent, that the sample is a binary mixture, that the competitive isotherms of the additive and the two components are described by the Langmuir model, and that the sample is less retained than the additive, so that ao > 1, as stated above. The presence of an adsorbed modifier in the mobile phase (1) decreases the retention of the two components, (2) causes the elution of an additive system peak and two component system peaks, and (3) may result in considerable departure of one or both component band profiles from their expected Langmuirian behavior. 

A major advantage of the Langmuir model is that it allows a straightforward extension to binary and multicomponent systems ... 

This justifies the use of Henry constants for prehminary screening of selective adsorbents, but in practice a significant loading dependence of the separation factor is generally observed as a result of deviations from the binary (or multicomponent) Langmuir model. 

In contrast to the binary Langmuir or SSTM models, the ideal adsorbed solution theory does not lead to a simple explicit relation for the adsorbed-phase composition and loading in terms of the partial pressures. Calculation of the equilibrium for a particular gas-phase composition therefore requires a trial and error procedure. 

Table 1 shows the differences between nf and n and 9, and for binary adsorption of CO2 (component 1) + CH4 (component 2) mixtures on BPL carbon at soil K [13]. They were estimated at three different total gas pressure levels and at different gas-phase compositions. The table also gives the fractional adsorbate loadings (6,) of the components [0 — nT(P, T, y,)/w]. m is the saturation capacity (surface excess) for both components. The pure and binary gas adsorption isotherms for this system can be described by the Langmuir model [17]. The model parameters are given in Table 1. A value of 0.80cm /g was used as for these calculations. It can be seem from Table 1 that the differences between nf and are relatively small because CO2 (component 1) is more selectively adsorbed than CH4 on the carbon. The differences between n and 2 are, however, much larger. Furthermore, the differences between qi and q, are much larger than the corresponding differences between and. The differences between qi and qi get even bigger as the system pressure P) and the mole fraction of component 1 (y) in the gas phase are increased. These examples demonstrate the weakness of the shortcut method.

The equilibrium factor measures the affinity of the adsorbent for a particular component relative to the same component in the fluid phase whereas the binary separation factor measures the relative preference of the adsorbent for two different competing adsorbates. If the equilibrium obeys the multicomponent Langmuir model ... 

This equation can be readily obtained from the extended Langmuir model, Eq. 3.4, for the binary system. Thus, when the saturated amounts for the pure gases are the same, the IAS is identical to the extended Langmuir model. 

Due to the high pressures, Langmuir is used with fugacities determined from the virial equation of state. It was found that the predictions with the multi-component Langmuir model were better than with lAST for two binary gas mixtures (H2-CO and CO-CH4) at various temperatures, but lAST proved to be superior when modelling the systems CO-CO2 and CH4-CO2 and for aU the ternary and quaternary systems. However, overall both models proved to adequately predict the mixed gas data and the predictions from the two models were very similar. From a mathematical and computational point of view, the explicit Langmuir model is simpler, while lAST needs an iterative solution method ... 

For nonlinear systems, however, the evaluation of the flow rates is not straightforward. Morbidelli and co-workers developed a complete design of the binary separation by SMB chromatography in the frame of Equilibrium Theory for various adsorption equilibrium isotherms the constant selectivity stoichiometric model [21, 22], the constant selectivity Langmuir adsorption isotherm [23], the variable selectivity modified Langmuir isotherm [24], and the bi-Langmuir isotherm [25]. The region for complete separation was defined in terms of the flow rate ratios in the four sections of the equivalent TMB unit ... 

The Sips [38] model (Eq. 10) can easily be extended to binary or multicomponent systems [34, 74]. The resulting expression for the multicomponent Langmuir-Freundlich adsorption model is ... 

The sorption and diffusion behaviour of gas mixtures is of particular interest from the point of view of membrane gas separation, which is steadily gaining in importance by virtue of its low energy requirements. On the basis of the dual mode sorption model, one may reasonably expect that sorption of a binary gas mixture A, B in the polymer matrix will exhibit little gas-gas interaction and hence will tend to occur essentially additively. In the Langmuir-like mode of sorption, on the other hand, there will be competition between A and B for the limited number of available sites. These considerations led 67) to the following reformulation of Eqs. (8) and (9)... 

Langmuir isotherm or model Simple mathematical representation of a favorable (type I) isotherm defined by Eq. (2) for a single component and Eq. (4) for a binary mixture. The separation factor for a Langmuir system is independent of concentration. This makes the expression particularly useful for modeling adsorption column dynamics in multicomponent systems. 

Figure 2 Conversion of heptane and nonane from their binary mixture in iiquid phase plotted against the average conversion (symbols experimental data points, full curves Langmuir adsorption model, dotted curves no selective adsorption).

A linear model predictive control law is retained in both cases because of its attracting characteristics such as its multivariable aspects and the possibility of taking into account hard constraints on inputs and inputs variations as well as soft constraints on outputs (constraint violation is authorized during a short period of time). To practise model predictive control, first a linear model of the process must be obtained off-line before applying the optimization strategy to calculate on-line the manipulated inputs. The model of the SMB is described in [8] with its parameters. It is based on the partial differential equation for the mass balance and a mass transfer equation between the liquid and the solid phase, plus an equilibrium law. The PDE equation is discretized as an equivalent system of mixers in series. A typical SMB is divided in four zones, each zone includes two columns and each column is composed of twenty mixers. A nonlinear Langmuir isotherm describes the binary equilibrium for each component between the adsorbent and the liquid phase. 

Using the IAS theory, LeVan and Vermeulen derived a competitive binary isotherm equation that accounts for differences in the column saturation capacities for the two components, when the single-component adsorption isotherms follow a Langmuir or a Freundlich isotherm model [33]. 

Figure 8.3 illustrates a schematic of the solution of the ideal model in the case of a narrow rectangular pulse of a binary mixture with a competitive Langmuir... 

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