Constant pattern condition

Under constant pattern conditions the LUB is independent of column length although, of course, it depends on other process variables. The procedure is therefore to determine the LUB in a small laboratory or pilot-scale column packed with the same adsorbent and operated under the same flow conditions. The length of column needed can then be found simply by adding the LUB to the length calculated from equiUbrium considerations, assuming a shock concentration front. 

Constant pattern condition This condition reduces the mass balance equation (4.128) to the simple relation C/C0=q/qm lx (see the section A look into the constant pattern condition). Practically, the constant pattern assumption holds if the equilibrium is favorable, and at high residence times (Perry and Green, 1999 Wevers, 1959 Michaels, 1952 Hashimoto et al., 1977). However, the constant pattern assumption is weak if the system exhibits very slow kinetics (Wevers, 1959). 

This relationship is the constant pattern condition. According to the literature, a criterion for the constant pattern assumption is the following (Perry and Green, 1999 Wevers, 1959) ... 

The first term is the ratio of maximum loading of solid for a specific inlet concentration to that concentration, whereas the second term is the space velocity (the reciprocal of the residence time), and the third term is the slope of the breakthrough curve. Thus, the constant pattern condition is achieved for... 

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... 

Solution of the Breakthrough Curve under Constant Pattern Condition. . 653... 

According to Eqs. 14.6 and 14.7, the dimensionless plot of x versus — 1) depends on the single parameter R q, which indicates the deviation of the isotherm from linear behavior. Figure 14.1 illustrates this result by showing breakthrough profiles obtained for values of Req between 0 and 0.8, with the liquid film linear driving force model, under constant pattern conditions. 

The most comprehensive study of the combined effects of axial dispersion and mass transfer resistance under constant pattern conditions has been done by Rhee and Amimdson [17,18] using the shock layer theory. These authors assumed a solid film linear driving force model (Eq. 14.3) and wrote the mass balance equation as... 

In Chapter 14, we discussed the case of a single-component band. In practice, there are almost always several components present simultaneously, and they have different mass transfer properties. As seen in Chapter 4, the equilibrium isotherms of the different components of a mixture depend on the concentrations of all the components. Thus, as seen in Chapters 11 to 13, the mass balances of the different components are coupled, which makes more complex the solution of the multicomponent kinetic models. Because of the complexity of these models, approximate analytical solutions can be obtained only under the assumption of constant pattern conditions. In all other cases, only numerical solutions are possible. The problem is further complicated because the diffusion coefficients and the rate constants depend on the concentrations of the corresponding components and of all the other feed components. However, there are still relatively few papers that discuss this second form of coupling between component band profiles in great detail. In most cases, the investigations of mass transfer kinetics and the use of the kinetic models of chromatography in the literature assume that the rate constants and the diffusion coefficients are concentration independent. This seems to be an acceptable first-order approximation in many cases, albeit separation problems in which more sophisticated theoretical approaches are needed begin to appear as the accuracy of measru ments improve and more interest is paid to complex... 

In this case, there are no analytical solutions, even under constant pattern conditions [3], Numerical solutions can be obtained. We can just speculate that, since in the single-component case the two effects are additive imder almost aU cases of practical significance, we may use the same approach in the multicomponent case. This assumption is supported by its agreement with the results of experimental determinations. 

Experimental results [7,8] obtained in the case of the breakthrough curves of binary mixtiues imder constant pattern condition have been compared with the anal5ttical solution. Figure 16.1 compares the experimental breakthrough curves obtained in the case of the vapor phase adsorption of benzene and toluene carried by nitrogen through a bed of activated carbon [8] with the analytical solution calculated from the binary adsorption data and imder the assumption of constant pattern behavior [1,3]. The agreement achieved is excellent. 

The constant pattern approximation is a very useful design-estimation method, since the computation of the adsorption wave under such conditions is straightforward. As shown in Chapter 4, the constant pattern condition is expressed in the material balance as... 

Note Plug flow w ith constant linear velocity is assumed in al cases. Constant-pattern conditions c/cq / o- isotherm q /= bc(l 4- be) or... 

To derive an expression for the asymptotic form of the breakthrough curve for a plug flow system under constant-pattern conditions it is only necessary to integrate the rate expression subject to the constant-pattern condition... 

By comparing the concentration profiles derived from the full solution to the differential model equations with the asymptotic profiles calculated from the constant-pattern condition it is possible to determine the dimensionless distance required to approach constant-pattern behavior. The results of such calculations are summarized in Figure 8.21. 

An equivalent analysis of the combined effects of axial dispersion and mass transfer resistance has been presented by Rhee and Amundson, based on shock layer theory. From the mass balance over the shock layer it may be shown that the propagation velocity [Eq. (8.13)] is not affected by mass transfer resistance or axial dispersion. For an equilibrium system with axial dispersion the differential mass balance [Eq. (8.1)] becomes, under constant pattern conditions. 

Fleck, R.D. Kirwan. D.J., and Hall, K.R., Mixed-resistance diffusion kinetics in fixed-bed adsorption under constant pattern conditions, Ind. Eng. Chem. Fund., 12(1), 95-99 (1973). 

This equation is transformed by using constant pattern condition (Eq. (7-31)) and the dimensionless form becomes... 

For a single component under constant pattern conditions an expression for the asymptotic form of breakthrough curve can be obtained by integrating the appropriate rate expression, subject to the constant pattern condition. Consider the case of plug flow and constant velocity, a linear rate of adsorption based on solid phase conditions (equation (6.48)), a Langmuir isotherm, a simple substitution (equation (6.49)) and the constant pattern condition (equation (6.47)) ... 

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