Two Adsorbable Components

As a result of the continuity condition a binary isothermal system with two adsorbable components (no carrier) has a single mass transfer zone which propagates with velocity given by Eq. (8.22). This provides the basis of a simple chromatographic method for the experimental determination of binary equilibrium isotherms. The propagation velocity for a small perturbation is measured over a range of compositions for the binary mixture. The variation with composition of the apparent equilibrium constant defined by 

The isotherms for the individual components are then expressed in polynomial form with unknown coefficients  

The equilibrium adsorbed phase concentrations for the individual components are found by integration of Eq. (8.29) (from K = 0 to T for component 1 and from Y = 1 to yi for component 2)  

Comparing coefficients between Eqs. (8.28) and (8.30) provides four relations between the six unknown coefficients Bq, B,, Cq, C,, Cj. Two further equa- 

FINITE MASS TRANSFER RESISTANCE (LINEAR EQUILIBRIUM) 

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When no analyhcal soluhon can describe the process satisfactorily it may be possible, working from Eq. (9.18) (which describes the length of the wave) and either Eq. (9.11) or (9.13) (the expression for the velocity of the adsorption wave), to assemble a simple wave mechanics solution that approximates the length and movement of the mass transfer front in the bed. As with analytical solutions this method can deliver useful results that may approximate the wave shape inside the bed and thus can be used to describe the shape and duration of the breakthrough curve that occurs as the wave intercepts and crosses the end of the bed. Such methods are generally only applicable for one or at most two adsorbable components. 

Such systems can be of any of the following types (1) isothermal, two adsorbable components plus inert carrier ... 

Butler and Ockrent1 have obtained data with solutions of several organic substances, sometimes containing two adsorbed components in such cases naturally the more strongly adsorbed component predominates at the surface. 

A similar equation can be described for component 2. The ratio of the coverages of the two adsorbed components is... 

In the case of solid-liquid systems, a similar equation can be written for two adsorbed components using K = I/A and K2 = 1 /b2 (Equation 1.65). In this case, the form of the competitive Langmuir isotherm analogous to Equation 1.64 for component 1 would be... 

In this problem we will simulate a batch adsorption process that takes place with two adsorbate components. The simulation will allow us to do computational experiments with the aim of learning how the adsorption and desorption parameters affect the behavior of this process. Building the simulation will provide new experience in developing the model equations, utilizing more complex constitutive relationships, finding numerical solutions to these equations, and displaying the results graphically. 

If the system contains two adsorbable components, rather than one adsorb-able component in an inert carrier, mass balance equations analogous to Eqs. (8.1) and (8.2) may be written for both species. However, since the contintiity condition must also be satisfied (C -f Cj constant) these equations are not independent and there is still only one mass transfer zone. The behavior of a system which contains an inert carrier in addition to the two adsorbable species is entirely different since two distinct mass transfer zones are then formed. The discussion of such systems is deferred uritU Chapter 9. 

To illustrate the detailed application of equilibrium theory we consider Glueckaufs treatment of a system with two adsorbable components (1 and 2)... 

The development outlined here may be generalized for systems with more than two adsorbable components although the algebra becomes tedious. ... 

The effect of mass transfer resistance is to broaden the mass transfer zone relative to the profile deduced from equilibrium theory. Where equilibrium theory predicts a shock transition the actual profile will approach constant-pattern form. Since the location of the mass transfer zone and the concentration change over which the transition occurs are not affected by mass transfer resistance, the extension of equilibrium theory is in this case straightforward and requires only the integration of the rate expression, subject to the constant-pattern approximation, to determine the form of the concentration profile. This is in essence the approach of Cooney and Strusi who show that for a Langmuir system with two adsorbable components a simple analytic expression for the concentration profile may be obtained when both mass transfer zones are of constant-pattern form. 

Most PSA processes depend on equilibrium selectivity and the simplest approach to the modeling of these systems is through equilibrium theory. Such an analysis was first developed by Shendalman and Mitchell for the case of a single adsorbable species in a nonadsorbing carrier. The theory was extended by Chan, Hill, and Wong to a system with two adsorbable components subject to the restriction that the equilibrium relationships for both species are linear and the more strongly adsorbed species is present only at low concentration. The system is described by the following equations ... 

As shown briefly here, equilibrium theory can give a range of operating conditions for providing a pure product. This treatment has been further extended to apply to two adsorbable component systems (Chan, Hill and Wong, 1982 and Knaebel and Hill, 1982). Also Kawazoe and Kawai (1973) used equilibrium assumption to estimate a concentration profile during the blowdown step. 

Fractionation of a vapor mixture For the separation of a vapor mixture consisting of. for example, two adsorbable components for which the adsorbent exhibits a relative adsorptivity, calculations are made in the manner for moving beds to locate operating lines and equilibrium curves [Eqs. (11,53) to (11.65) and Fig. 11.33], Theoretical stages can then be determined by the usual step construction in the lower part of such a figure. 

The dynamic response of the bed is given by the simultaneous solution of equations (6.19) and (6.20), subject to the imposed initial and boundary conditions. If the fluid comprises more than one adsorbate then the conservation equation (6.19) and the rate equation (6.20) must be written for each component. In addition, the continuity equation must be satisfied. For example, for a system which contains two adsorbable components, rather than one adsorbate in a non-adsorbing carrier fluid, equations (6.19) and (6.20), which must be written for both components, are not independent and there is only one MTZ. The continuity equation must be satisfied ... 

This simple analysis for an isothermal and equilibrium controlled process can be extended to concentrated systems in which u must remain within the differential of the second term in equation (6.19). The analysis can also be extended to systems which include more than a single adsorbable component. Consider the case of a feed stream which contains only two adsorbable components, i.e. a system which does not include a non-adsorbing carrier fluid. In this case both components can be ejqiected to be concentrated in the fluid and hence the variation in fluid velocity over the MTZ must be taken into account. Two differential fluid phase mass balance equations must be written, one for each component. Equation (6.31) is shown for component 1. The axial dispersion term is retained to create a general equation. 

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