Breakthrough Curve Profiles

Favourable Equilibrium If the ion exchange equilibrium is very favourable (. [ 1, 1) the exchange zone maintains a 

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Adsorption Dynamics. An outline of approaches that have been taken to model mass-transfer rates in adsorbents has been given (see Adsorption). Detailed reviews of the extensive Hterature on the interrelated topics of modeling of mass-transfer rate processes in fixed-bed adsorbers, bed concentration profiles, and breakthrough curves include references 16 and 26. The related simple design concepts of WES, WUB, and LUB for constant-pattern adsorption are discussed later. 

The solution to this model for a deep bed indicates an increase in velocity of the fluid-phase concentration wave during breakthrough. This is most dramatic for the rectangular isotherm—the instant the bed becomes saturated, the fluid-phase profile Jumps in velocity from that of the adsorption transition to that of the fluid, and a near shocklike breakthrough curve is obseived [Coppola and LeVan, Chem. Eng. Sci.,36, 967(1981)]. 

FIG. 16-3 Bed profiles (top and middle) and breakthrough curve (bottom). The bed profiles show the mass-transfer zone (MTZ) and equilibrium section at breakthrough. The stoichiometric front divides the MTZ into two parts with contributions to the length of equivalent equilibrium section (LES) and the length of equivalent unused bed (LUB). 

The profile of the tracer recovery curves (breakthrough curves) is best approximated and simplified using a single channel communication between the injection and production wells both for 1,6 and 2,6 NDS tracers. 

Figure /. Breakthrough curves of arsenate and pH profiles of column effluents during adsorption from feeds in t he absence and presence of foreign anions.

The breakthrough curves measured for the monolithic columns with different proteins are very sharp and confirm again the fast mass transport kinetics of the monoliths [133, 134]. The frontal analysis used for the determination of the breakthrough profile can also be used for calculation of the dynamic capacity of the column. For example, the capacity for the 60x16mm i.d. monolith at 1% breakthrough is 324 mg of ovalbumin and represents the specific capacity of 40.0 mg/g of separation medium or 21.6 mg/ml of column volume. 

Fig. 10.1 Effect of different mechanisms on behavior of contaminants advancing through a column of porous material the relative concentration is given by c/c. (a) temporal breakthrough curves at the column outlet, showing effects of diffusion and dispersion (b) spatial concentration profiles along the column, at different times (c) spatial concentration profiles illustrating effects of retardation caused by contaminant absorption.

Figure 1. Case I (fully stable) profiles (a) progression of concentration front, (b) progression of temperature front, (c) breakthrough curve, (d) equilibrium (Y ) and operating (Y) lines for steady-state front, (e) graphical representation of the integral used in the prediction of steady-state LUB...

In many circumstances, however, the assumption of equilibrium sorption is inappropriate. In many laboratory and field studies, contaminant concentration vs time profiles (breakthrough curves) have been observed that exhibited asymmetry. While the assumption of equilibrium sorption results in a prediction of relatively symmetric breakthrough curves, the sharp initial breakthrough and late-tailing curves that are frequently observed are indicative of rate-limited (non-equilibrium) transport [9]. Rate-limited sorption is also a... 

For the case with porous particles, the pore fluid can be treated as a mass transfer medium rather than a separate phase thus enabling it to be combined with the bulk fluid in the overall mass balance. Under plug flow transfer conditions, at the end of each time increment, the pore fluid was assumed to remain stagnant, and only the bulk fluid was transferred to the next section. Based on these assumptions and initial conditions, the concentrations of the polypeptide or protein adsorbate in both liquid and solid phase can be calculated. The liquid phase concentration in the last section C , is the outlet concentration. The concentration-time plot, i.e., the breakthrough curve, can then be constructed. Utilizing this approach, the axial concentration profiles can also be produced for any particular time since the concentrations in each section for each complete time cycle are also derived. 

The analytical solution of this set of differential equations [Eqs. (5) and (6)] was first given by Thomas [38] for the frontal breakthrough curve. The concentration profile is... 

In contrast with the results of Fig. 5. important deviations between the theoretical profile and the experimental one are observed for adsorption studies on the polyclonal antibody [23], even at low HSA feeding concentrations. The frontal elution model given by Eq. (7) with k d = 0 correlates well with the first part of the breakthrough curve, but later a deviation is observed even at very low feeding HSA concentrations. In this case, the simplified model assuming a uniform adsorbent surface is not appropriate. The polyclonal antibody is made of different populations of antibodies of various affinities. With polyclonal immunoadsorbents, the values of ka in Table 2 are to be considered as apparent adsorption rates. 

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