Convex isotherm

The three solutions corresponding to the three boundary conditions can be used to obtain approximate solutions for other convex bodies, such as a cube, for which there are no analytical solutions available. The dimensionless parameters Bi and Fo are defined with respect to the equivalent sphere radius, which is obtained by setting the surface area of the sphere equal to the surface area of the given body, i.e., a = Vi4/(4tt). This will be considered in the following section, which covers transient external conduction from isothermal convex bodies. 

Transient External Conduction From Isothermal Convex Bodies... 

External transient conduction from an isothermal convex body into a surrounding space has been solved numerically (Yovanovich et al. [149]) for several axisymmetric bodies circular disks, oblate and prolate spheroids, and cuboids such as square disks, cubes, and tall square cuboids (Fig. 3.10). The sphere has a complete analytical solution [11] that is applicable for all dimensionless times Fovr = all A. The dimensionless instantaneous heat transfer rate is QVa = Q AI(kAQn), where k is the thermal conductivity of the surrounding space, A is the total area of the convex body, and 0O = T0 - T, is the temperature excess of the body relative to the initial temperature of the surrounding space. The analytical solution for the sphere is given by... 

M. M. Yovanovich, P. Teertstra, and J. R. Culham, Modeling Transient Conduction From Isothermal Convex Bodies of Arbitrary Shape, Journal of Thermophysics and Heat Transfer (9/3) 385-390,1995. 

It will be clear that all systems having a negative heat of absorption (generally those having a smaller affinity than the ideal system), will also yield sorption isotherms convex to the pressure axis. Curves concave to the pressure axis hence indicate an affinity sensibly larger than that of the ideal system. 

When studying the isotherms of water vapour adsorption one may note the isotherm convexity for SCN carbons at the low P/Pj values. This is probably associated with the chemically-bound nitrogen atoms present in the sorbent. 

For a second active carbon, AG, the DR plot was convex to the logio(p7p) This carbon was believed from X-ray results to have a wider distribution of pores. It was found that the isotherms of both benzene and cyclohexane could be interpreted by postulating that the micropore system consisted of two sub-systems each with its own Wq and and with m = 2 ... 

Both Type III and Type V isotherms are characterized by convexity towards the relative pressure axis, commencing at the origin. In Ty )e III isotherms the convexity persists throughout their course (Fig. 5.1(a), whereas in Type V isotherms there is a point of inflection at fairly high relative pressure, often 0-5 or even higher, so that the isotherm bends over and reaches a plateau DE in the multilayer region of the isotherm (cf. Fig. 5.1 (b)) sometimes there is a final upward sweep near saturation pressure (see DE in Fig. 5.1(b)) attributable to adsorption in coarse mesopores and macropores. 

The weakness of the adsorbent-adsorbate forces will cause the uptake at low relative pressures to be small but once a molecule has become adsorbed, the adsorbate-adsorbate forces will promote the adsorption of further molecules—a cooperative process—so that the isotherms will become convex to the pressure axis. 

I (curve D). Thus the micropores had been able to enhance the adsorbent-adsorbate interaction sufficiently to replace monolayer-multilayer formation by micropore filling and thereby change the isotherm from being convex to being concave to the pressure axis. 

McCabe-Thiele diagrams for nonlinear and more practical systems with pertinent inequaUty constraints are illustrated in Figures 11 and 12. The convex isotherms are generally observed for 2eohtic adsorbents, particularly in hydrocarbon separation systems, whereas the concave isotherms are observed for ion-exchange resins used in sugar separations. 

So far, it has been assumed that elution is independent of analyte load or the presence of multiple components in a mixture. If this condition holds, then the analyte concentration in the mobile phase is directly proportional to the concentration in the stationary phase, no matter what the concentration is. Experimentally, this could be determined by incubating various concentrations of an analyte with a fixed amount of stationary phase and measuring the amount adsorbed. A plot of the concentration of analyte in the mobile phase on the x-axis vs. that in the stationary phase on the y-axis would be linear, and such a plot is called a "linear isotherm". A convex isotherm implies that tailing would be expected, and a concave isotherm implies that fronting is expected. 

In some systems, three stages of adsorption may be discerned. In the activated alumina-air-water vapour system at normal temperature, the isotherm is found to be of Type IV. This consists of two regions which are concave to the gas concentration axis separated by a region which is convex. The concave region that occurs at low gas concentrations is usually associated with the formation of a single layer of adsorbate molecules over the... 

An isotherm which is convex to the fluid concentration axis is termed unfavourable. This leads to an adsorption zone which gradually increases in length as it moves through the bed. For the case of a linear isotherm, the zone goes through the bed unchanged. Figure 17.19 illustrates the development of the zone for these three conditions. 

These isotherms are sometimes referred to as anti-Langmuir models because their initial curvature is convex down. The fact that the curvature of these isotherms at the origin and at low concentrations... 

The empirical constants La and Fr are related to the particular system under investigation and are obtained from laboratory experiments (Chen el al., 1968 Chern and Chien, 2002). Generally, an isotherm is favorable if its shape is convex upward, and unfavorable... 

Let us assume that an experimental isotherm is perfectly described by the Cohen-Kisarov equation. When plotting the experimental points with the previous coordinates, three different cases may occur (1) if cmlA < 2, a case which was not yet found ((4) and Table I), the curve exhibits a constant convex curvature towards the ordinate axis (2) if cm /A > 2, the curve exhibits two distinct inflection points (Figure 3) where the experimental curve may easily be confused with the tangent to the inflection point, thus explaining the previous observations (3) if cm /A decreases to a value of 2, these inflection points are unified to give a large linear section, and the Dubinin-Radushkevich equation behaves as a limiting case of the Cohen-Kisarov equation. 

Frontal analysis brings with it the requirement of the system to have convex isotherms (see Section 1.2.6). This results in the peaks having sharp fronts and well-formed steps. An inspection of Figure 1.3 reflects the problem of analytical frontal analysis— it is difficult to calculate initial concentrations in the sample. One can, however, determine the number of components present in the sample. If the isotherms are linear, the zones may be diffuse. This may be caused by three important processes inhomogeneity of the packing, large diffusion effects, and nonattainment of sorption equilibrium. 

As with frontal analysis, displacement analysis requires convex isotherms. Once equilibrium conditions have been attained, an increase in column length serves no useful purpose in this technique because the separation is more dependent on equilibrium conditions than on the size of column. 

The convex isotherm demonstrates that the K value is changing... 

The shape of the adsorption front, the width of the MTZ, and the profile of the effluent concentration depend on the nature of the adsorption isotherm and the rate of mass transfer. Practical bed depths may be expressed as multiples of MTZ, values of 5-10 multiples being economically feasible. Systems that have linear adsorption isotherms develop constant MTZs whereas MTZs of convex ones (such as Type I of Figure 15.1) become narrower, and those of concave systems become wider as they progress through... 

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