Adsorbent phase concentration, equilibrium

Combined Pore and Solid Diffusion In porous adsorbents and ion-exchange resins, intraparticle transport can occur with pore and solid diffusion in parallel. The dominant transport process is the faster one, and this depends on the relative diffusivities and concentrations in the pore fluid and in the adsorbed phase. Often, equilibrium between the pore fluid and the solid phase can be assumed to exist locally at each point within a particle. In this case, the mass-transfer flux is expressed by ... 

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 ... 

Proportionate Pattern Behavior. If the isotherm is unfavorable (as in Fig. 1,111), the stable dynamic situation leading to constant pattern behavior can never be achieved. The equilibrium adsorbed-phase concentration then lies above rather than below the actual adsorbed-phase profile. As the mass transfer zone progresses through the column it broadens, but the limiting situation, which is approached in a long column, is simply local equilibrium at all points (c = c ) and the profile therefore continues to... 

After equilibrium was attained over a period of several days, the carbon was allowed to settle, and the supernatant analyzed to determine the decrease on concentration AC by adsorption. For dilute solutions, a Radke and Prausnitz (11) have shown that the equilibrium adsorbent phase concentration are given by... 

Constant Pattern Behavior. In a real system the finite resistance to mass transfer and axial mixing in the column lead to departures from the idealized response predicted by equilibrium theory. In the case of a favorable isotherm the shock wave solution is replaced by a constant pattern solution. The concentration profile spreads in the initial region until a stable situation is reached in which the mass transfer rate is the same at all points along the wave front and exactly matches the shock velocity. In this situation the fluid-phase and adsorbed-pliase profiles become coincident, as illustrated in Figure 13. This represents a stable situation and the profile propagates without further change in shape—lienee the term constant pattern. The form of the concentration profile under constant pattern conditions may be easily deduced by integrating the mass transfer rate expression subject to the condition c/c0 = q/qQy where qfj is the adsorbed phase concentration in equilibrium with c(y... 

Basic thermodynamic considerations require that, at sufficiently low adsorbed-phase concentrations on a homogeneous surface, the equilibrium isotherm for physical adsorption should always approach linearity (Henry s law). The limiting slope of the isotherm is called the Henry constant ... 

Most of the experimental applications of the ZLC technique have been with gaseous systems, and for these systems the technique may now be regarded as a standard method. Based on our experience it is possible to suggest some guidelines as to how the experiments should be carried out. The key parameter is L, which from its definition (Eq. 17) can be considered the ratio of the diffusional and washout time constants R /D and KVs/F. This parameter is also equal to the dimensionless adsorbed phase concentration gradient at the surface of the solid at time zero. From either of these definitions it is evident that L gives an indication of how far removed the system is from equilibrium control. This parameter is proportional to the flow rate, so it can be easily varied, and to extract a reliable time constant, it is necessary to run the experiment at at least two different flow rates. 

The surface flux equation (7.9-9b) is written in terms of the gradient of adsorbed phase concentration. It can also be written in terms of the gas phase concentration provided that there is a local equilibrium between the gas and adsorbed phases. By local equilibrium here, we mean that at any given point within the particle the gas and solid phases are in equilibrium with each other, despite the gradients of concentration in both phases are present. This is acceptable if the rates of adsorption and desorption at any point are much faster than the rates of diffusion in both phases. If this equilibrium is governed by the Henry law, that is... 

Inversely, for a given set of the adsorbed phase concentrations (C = (C, C 2>., C n ), there will also exist a set of partial pressures such that the two phases are in equilibrium with each other, that is ... 

The mass balance equation involves the gas phase and adsorbed phase concentrations. We assume equilibrium is established between the gas and surface phases hence at any point within the particle, the adsorbed concentration at the patch of sites having an adsorption energy E is related to the gas phase concentration, C, according to eq.( 11.2-6), or written in terms of the fractional loading ... 

For physical adsorption there is no change in molecular state on adsorption (i.e., no association or dissociation). It follows that for adsorption on a uniform surface at sufficiently low concentrations such that all molecules are isolated from their nearest neighbors, the equilibrium relationship between fluid phase and adsorbed phase concentrations will be linear. This linear relationship is commonly referred to as Henry s law by analogy with the limiting behavior of solutions of gases in liquids and the constant of proportionality, which is simply the adsorption equilibrium constant and is referred to as the Henry constant. The Henry constant may be expressed in terms of either pressure or concentration ... 

A somewhat different definition was introduced by Broughton who defined the activity coefficient (y, ) as the ratio of the equilibrium pressure of component i in the mixture to the equilibrium pressure for the pure adsorbed species at the same temperature and at a concentration equal to the total adsorbed phase concentration for the mixture ... 

Despite its industrial importance, adsorption from the liquid phase has been studied much less extensively than adsorption from the vapor phase. There is no difference in principle between adsorption from liquid and vapor phases since, thermodynamically, the adsorbed phase concentration in equilibrium with a liquid must be precisely the same as that which is in equilibrium with the saturated vapor. The differences arise in practice because in adsorption from the liquid phase one is almost invariably concerned with high adsorbed phase concentrations close to the saturation limit. The simple model isotherms, developed primarily to describe adsorption from the vapor phase, are at their best at low sorbate concentrations and become highly unreliable as saturation is approached. Such models are therefore of only very limited applicability for the correlation of liquid phase adsorption data. 

FIGURE 5.6, (a) Equilibrium isotherms and (b) variation of surface diffusivity with adsorb phase concentration. (Reprinted with permission of ref. 40. Copyright 1974 American Chemical Society.)... 

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