Patterned behavior

Fig. 13. Schematic diagram showing (a) approach to constant pattern behavior for a system with a favorable isotherm and (b) approach to proportionate pattern behavior for a system with an unfavorable isotherm, jy axis cj qlj q,----------------------- < q,-- From ref. 7.

Favorable and unfavorable equihbrium isotherms are normally defined, as in Figure 11, with respect to an increase in sorbate concentration. This is, of course, appropriate for an adsorption process, but if one is considering regeneration of a saturated column (desorption), the situation is reversed. An isotherm which is favorable for adsorption is unfavorable for desorption and vice versa. In most adsorption processes the adsorbent is selected to provide a favorable adsorption isotherm, so the adsorption step shows constant pattern behavior and proportionate pattern behavior is encountered in the desorption step. 

Treatments of constant pattern behavior have been carried out for multicomponent adsorption [Vermeulen, Adv. in Chem. Eng., 2, 147 (1958) Vermeulen et., Ruthven, gen. refs. Rhee and Amundson, Chem. Eng. ScL, 29, 2049 (1974) Cooney and Lightfoot, Jnd. Eng. Chem. Fundam., 5, 25 (1966) Cooney and Strusi, Jnd. Eng. Chem. Fundam., 11, 123 (1972) Bradley and Sweed, AJChE Symp. Ser. No. 152, 71, 59 (1975)]. The behavior is such that coexisting compositions advance through the bed together at a uniform rate this is the coherence concept of Helfferich and coworkers [gen. refs.]. 

Sequences of aggressive behavior that are composed of characteristic acts and postures following each other rapidly are disrupted. These disorganizing effects parallel the analysis of amphetamine effects on other intricately patterned behaviors such as feeding, maternal care, play behavior, or reproductive interactions. For example, amphetamine suppresses play... 

However, reinforcement does not necessarily happen all the time or even regularly. For example, even though many of us work every day, we do not necessarily get paid at the end of that day for what we did. And in some instances reinforcement is unpredictable, like when you receive an unexpected phone call that is rewarding to you from a close friend. When reinforcement doesn t occur in a predictable way, it is referred to as being on a variable or intermittent (random or unpredictable) schedule (or pattern). Behavioral researchers have found that a variable reinforcement schedule produces behavior patterns that are much more difficult to change than behavior patterns reinforced regularly. 

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 transferrate is the same at all points along the wave front and exactly matches the shock velocity. In this situation the fluid-phase and adsorbed-phase profiles become coincident. This represents a stable situation and the profile propagates without further change in shape—hence the term constant pattern. 

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... 

Pratt, J., and Keil, H. The focusing effect as patterned behavior... 

FIGURE 7 Schematic diagram showing (a) approach to constant-pattern behavior for a system with favorable equilibrium and (b) approach to proportionate-pattern limit for a system... 

In the case of an unfavorable isotherm (or equally for desorption with a favorable isotherm) a different type of behavior is observed. The concentration front or mass transfer zone, as it is sometimes called, broadens continuously as it progresses through the column, and in a sufficiently long column the spread of the profile becomes directly proportional to column length (proportionate pattern behavior). The difference between these two limiting types of behavior can be understood in terms of the relative positions of the gas, solid, and equilibrium profiles for favorable and unfavorable isotherms (Fig. 7). 

Fig.l shows several types of breakthrough curves obtained for IPA-TCE -Y-type zeolite system. For this system, reversal of the order of breakthrough (turn over) occurred twice at concentrations of 0.25 and 0.75 mole fractions of IP A, respectively. When the mole fractions were 0.25 and 0.75, the mixture of two components behaved as if it was a single component system as shown in Fig.2 (B) and (D). For other azeotropic mixture systems, the turnover occurred only once. The breakthrough curves for other systems always showed so-called constant pattern behavior for the whole concentration range.

A very detailed study of the combined effects of axial dispersion and mass-transfer resistance under a constant pattern behavior has been conducted by Rhee and Amundson [10]. They used the shock-layer theory. The shock layer is defined as a zone of a breakthrough curve where a specific concentration change occurs (i.e., a concentration change from 10% to 90%). The study of the shock-layer thickness is a new approach to the study of column performance in nonlinear chromatography. The optimum velocity for minimum shock-layer thickness (SLT) can be quite different from the optimum velocity for the height equivalent to a theoretical plate (HETP) [9]. 

Numerical Solution of the Breakthrough Curve under Constant Pattern Behavior 657... 

Bohart and Adams [4] were the first to recognize the existence of this constant pattern behavior. They assumed irreversible adsorption and the control of the mass transfer kinetics by the rate of adsorption. Later Wicke [5] gave an analytical solution in the case of irreversible adsorption, assuming that the mass transfer kinetics are controlled by the diffusion rate inside the particles. The asymptotic nature of constant pattern behavior has been discussed by Cooney and Lightfoot [6], who demonstrated its existence for aU convex-upward isotherms. 

Figure 14.1 Constant-pattern behavior of a breakthrough profile in the case of fluid-resistance-controlled kinetics. Plot of x versus at different values of R (Eqs. 14.7 and 14.8a). Reproduced from T. Ver-meulen, M.D. LeVan, N.K. Hiester and G. Klein, in Handbook of Chemical Engineering," Perry Ed., 6th ed., 1984, Chapter 16 (Fig. 16.13), with permission from McCraw- Hill, 1984.

The solutions given in Eqs. 14.6 and 14.7 are not rigorous solutions of the problem, but make the following two assumptions. First, the derivation of these equations neglects the contribution of the axial dispersion. Second, these equations are asymptotic solutions, as obtained under constant pattern behavior. 

Equation 14.45 applies in linear chromatography. The correct HETP equation in frontal analysis imder constant pattern behavior, and with the same solid film linear driving force model is Eq. 14.36b. Comparison of these two equations shows that an error is made when the latter is used to replace the former, in the equilibrium-dispersive model. We should replace in Eq. 14.44 and 14.45 Atq by k — FAq)/AC. In the case of the Langmuir isotherm, this would give k = fc )/(l + bCo) = X. 

Analytical Solution for Binary Mixture Constant Pattern Behavior.736... 

Analytical Solution for a Binary Mixture under Constant Pattern Behavior... 

A more comprehensive analysis of constant pattern behavior for a binary system has been given by Rhee and Amundson [3]. In this work, these authors have extended to binary systems the analysis of the combined effects of mass transfer resistance and axial dispersion that they had previously made in the case of single-component bands [4]. Rhee and Amundson [3] assumed the solid film linear driving force model, finite axial dispersion, and no particular isotherm model. The system of equations becomes... 

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. 

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