Breakthrough curve irreversible case

The asymptotic solutions given in Eqs. 14.6 and 14.7 were derived assuming that axial dispersion is negligible. Acrivos [15] has discussed the influence on the shape of the constant pattern breakthrough curve of the combination of axial dispersion and mass transfer resistance. An exact analytical solution can be derived only in the case of an irreversible adsorption isotherm (Req = 1/(1 - - bCo) = 0, or b infinite), and assuming a liquid film linear driving force model [15]. 

Under linear-equilibrium conditions, the combined effect of two diffusional resistances in series is treated by simple addition of the resistances. In the irreversible case, either the external or the internal resistance alone controls, with the latter over-taking the former part-way along the breakthrough curve (Section III, B, 2d). However, when the resistances are in relatively equal balance, with 3NP > Nf > 0.3NP, it is possible throughout the range of r values to calculate the breakthrough behavior by assuming an equivalent reaction-kinetic resistance. 

In Vermeulen s notation, the breakthrough curve for the irreversible case can be presented in dimensionless form... 

IRREVERSIBLE ADSORPTION. Irreversible adsorption with a constant mass-transfer coefficient is the simplest case to consider, since the rate of mass transfer is then just proportional to the fluid concentration. A truly constant coefficient is obtained only when all resistance is in the external film, but a moderate internal resistance does not change the breakthrough curve very much. Strongly favorable adsorption gives almost the same results as irreversible adsorption, because the equilibrium concentration in the fluid is practically zero until the solid concentration is over half the saturation value. If the accumulation term for the fluid is neglected, Eqs. (25.6) and (25.7) are combined to give... 

If pore diffusion controls the rate of adsorption, the breakthrough curve has the opposite shape from that for external-film control. The corresponding line in Fig. 25.10 was taken from the work of Hall et al., who presented breakthrough curves for several cases of irreversible adsorption. For pore diffusion control the initial slope of the curve is high, because the solid near the front of the mass-transfer zone has almost no adsorbate, and the average diffusion distance is a very small fraction of the particle radius. The curve has a long tail because the final molecules adsorbed have to diffuse almost to the center of the particle. 

When the equilibrium relationship is nonlinear it is generally not possible to determine a general analytic solution for the breakthrough curve. Such solutions have been obtained, however, for a number of special cases of which the irreversible or rectangular isotherm is the simplest. The irreversible isotherm, sketched in Figure 8.14, may be considered as the extreme limit of a favorable type 1 isotherm for which /8 0 and, as such, it represents an important limiting case. 

Except for the limiting case of the irreversible isotherm discussed above the prediction of the temperature and concentration profiles requires the simultaneous solution of the coupled differential heat and mass balance equations which describe the system. The earliest general numerical solutions for a nonisothermal adsorption column appear to have been given almost simultaneously by Carter and by Meyer and Weber. These studies all deal with binary adiabatic or near adiabatic systems with a small concentration of an adsorbable species in an inert carrier. Except for a difference in the form of the equilibrium relationship and the inclusion of intraparticle heat conduction and finite heat loss from the column wall in the work of Meyer and Weber, the mathematical models are similar. In both studies the predictive value of the mathematical model was confirmed by comparing experimental nonisothermal temperature and concentration breakthrough curves with the theoretical curves calculated from the model using the experimental equilibrium... 

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