Mass transfer constant separation factors

FIG. 16-14 Constant separation factor batch adsorption curves for external mass-transfer control with an infinite fluid volume and n j = 0. 

The treatment here is restricted to the Langmuir or constant separation factor isotherm, single-component adsorption, dilute systems, isothermal behavior, and mass-transfer resistances acting alone. References to extensions are given below. Different isotherms have been considered, and the theory is well understood for general isotherms. 

The reaction kinetics approximation is mechanistically correct for systems where the reaction step at pore surfaces or other fluid-solid interfaces is controlhng. This may occur in the case of chemisorption on porous catalysts and in affinity adsorbents that involve very slow binding steps. In these cases, the mass-transfer parameter k is replaced by a second-order reaction rate constant The driving force is written for a constant separation factor isotherm (column 4 in Table 16-12). When diffusion steps control the process, it is still possible to describe the system Iw its apparent second-order kinetic behavior, since it usually provides a good approximation to a more complex exact form for single transition systems (see Fixed Bed Transitions ). 

Model 1. The liquid phase mass transfer with a linear driving force is the controlling element. This model assumes that there are no concentration gradients in the particle, that there is a quasi-stationary state of liquid phase mass transfer, that there is a linear driving force and that there is a constant separation factor at a given solution concentration. 

Expressions for the steady-state concentration profile have also been derived for some more complex countercurrent systems. The extension to a (plug flow) system in which the equilibrium relationship is of Langmuir rather than linear form (constant separation factor) is given by Pratt. The solution for a linear plug flow system in which the mass transfer rate is controlled by intraparticle diffusion rather than by the linear rate law has been derived by Amundson and Kasten while the asymptotic behavior of a dispersed plug flow Langmuir system has been investigated by Rhee and Amundson. ... 

This leads to rate equations with constant mass transfer coefficients, whereas the effect of net transport through the film is reflected separately in thej/gj and Y factors. For unidirectional mass transfer through a stagnant gas the rate equation becomes... 

The growth rate, characterized by the change of the radius with time, is proportional to the driving force for the phase separation, given by the differences between 2 > the chemical composition of the second phase in the continuous phase at any time, and, its equihbrium composition given by the binodal line. The proportionahty factor, given by the quotient of the diffusion constant, D, and the radius, r, is called mass transfer coefficient. Furthermore the difference between the initial amount of solvent, (])o, and c]) must be considered. The growth rate is mathematically expressed by [101]... 

We note, however, the following possible difficulty with the original "hibernation" scenario. If the secondary s atmosphere is isothermal (due to irradiation by the WD), then it can be expected that the mass transfer rate will be reduced by a factor 10-100 depending on Aa/H, where Aa is the change in the separation and H is the (constant) scaleheight (Livio and Shara 1987). If, however, the secondary s atmosphere is con-vective, then M (aR)3, where AR is the distance by which the Roche lobe is overfilled. In such a case, an increase in the separation by Aa/a io-l+ will result in a decrease in I by at most a factor 2 (Edwards and Pringle 1987). Thus, it is not clear at all whether an increased separation can produce a significant decrease in M. 

The application of the z-transform and of the coherence theory to the study of displacement chromatography were initially presented by Helfferich [35] and later described in detail by Helfferich and Klein [9]. These methods were used by Frenz and Horvath [14]. The coherence theory assumes local equilibrium between the mobile and the stationary phase gleets the influence of the mass transfer resistances and of axial dispersion (i.e., it uses the ideal model) and assumes also that the separation factors for all successive pairs of components of the system are constant. With these assumptions and using a nonlinear transform of the variables, the so-called li-transform, it is possible to derive a simple set of algebraic equations through which the displacement process can be described. In these critical publications, Helfferich [9,35] and Frenz and Horvath [14] used a convention that is opposite to ours regarding the definition of the elution order of the feed components. In this section as in the corresponding subsection of Chapter 4, we will assume with them that the most retained solute (i.e., the displacer) is component 1 and that component n is the least retained feed component, so that... 

Consider separation of a binary mixture in a membrane module with the crossflow pattern shown in Figure 9.3. The feed passes across the upstream membrane surface in plug flow with no longitudinal mixing. The pressure ratio and the ideal separation factor are assumed to remain constant. Film mass-transfer resistances external to the membrane are assumed to be negligible. A total mass balance around the differential-volume element gives... 

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