Rectangular isotherm

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)]. 

The rectangular isotherm has received special attention. For this, many of the constant patterns are developed fuUy at the bed inlet, as shown for external mass transfer [Klotz, Chem. Rev.s., 39, 241 (1946)], pore diffusion [Vermeulen, Adv. Chem. Eng., 2, 147 (1958) Hall et al., Jnd. Eng. Chem. Fundam., 5, 212 (1966)], the linear driving force approximation [Cooper, Jnd. Eng. Chem. Fundam., 4, 308 (1965)], reaction kinetics [Hiester and Vermeulen, Chem. Eng. Progre.s.s, 48, 505 (1952) Bohart and Adams, J. Amei Chem. Soc., 42, 523 (1920)], and axial dispersion [Coppola and LeVan, Chem. Eng. ScL, 38, 991 (1983)]. 

Isotherm contours in Figure 5 show how large K must be to give rectangular isotherms like those usually observed for water in zeolites and also show some values of 0 at x = 1. With Equation 7, Equation 6 integrates to... 

In a particular limiting case where the formation of the stable RB complex, on an absolute basis, is formally consistent with the rectangular isotherm of B ion invasion of the B-RA exchange, the sharp boundary of this exchange advancing toward the bead center. The existence of such a boundary has been demonstrated elsewhere [32-34]. In this chapter it is presented in the microphotographs (Fig. 1) obtained for the bead of the... 

An approximate analytical solution of the system with Eq. (8) was obtained [26,27] for the rectangular isotherm of the invading B ion by using the integral relations method [61,62]. In this case the conversion (F) versus time (t) dependence is too close to what it is for conventional ion exchange [2,16,30,41] however, the effective diffusion coefficient D, is determined by the somewhat different relation [26,27] presented below ... 

At the initial stage the dependence of fractional conversion F on (Fo) is linear. The exchange rate is dependent on the value of ag/Cg and independent of Dy. This is in accordance with the analytical solution (11) for the rectangular isotherm of the B counterion. The greatest spread of concentration profiles appears in situations where both factors act together to spread the concentration distribution of the B ion i.e., at Df/ Dg < 1 and Kg /Kgg < 1 (Fig. 3, Vg, curve Il.e). In this instance the diffusion of the A ion is slower than the diffusion of the B ion (D, < Dg) and this results in accumulation of the A ion (Fig. 3, curve Il.e). The accumulation is partially attenuated due to the effect of the selectivity factor when Kg /Kgs < 1 continues to prevail (compare concentration profiles in variants II and Il.e). Co-ion Y also enters the bead (Fig. 3, Cy, curve Il.e) although not as vigorously as in the case with variant II. 

Figure 14 Simulated Vg concentration radial distributions obtained experimentally with the cone cell filled by VPC resin beads for various IE systems forward exchange Ag-RNa (x, rectangular isotherm) forward exchange H-RAg, (A, solid line) and reverse exchange Ag-RT (O, dashed line) at fractional conversion F = 0.1.

The satisfactory s eement between the experimental and the calculated (theoretical) kinetic curves and concentration profiles suggests that the values of D = 10" cmVs, = 10 cmVs, and = 1.5 10 cm /s adopted for the calculations do indeed characterize the mobility of and Ag+ ions in ampholyte VPC. By using these values and the above ratio, Dji/D = 7, it was possible to evaluate the rate of the Ag/Na exchange. A rectangular isotherm characterizes the sorption of Ag" " ions by the VPC ampholyte and this warrants the use of the approximate analytical solution (11, Sec. II) [26,27] for the above computations. 

A study was done to compare the isotherm plots of Nahon 117 and lonac MC 3470. The comparison in Figure 34.20b shows that Nahon 117 has a more rectangular isotherm than lonac MC 3470, indicating that the homogeneous membrane has a higher selectivity for Al ions. In fact, the average selectivity coefficient was determined to be 3.21 for Nahon 117 and 0.11 (over 30 times lower) for lonac MC 3470. 

Nahon 117 from DuPont Chemical Co. was found to be the most effective membrane in terms of kinetics of ion transfer. The equilibrium isotherm showed that the membrane had a high selectivity for trivalent aluminum ions. This resulted in a rectangular isotherm for the Nahon 117 membrane. 

Rectangular Isothermal Fins on Vertical Surfaces. Vertical rectangular fins, such as shown in Fig. 4.23a, are often used as heat sinks. If WIS > 5, Aihara [1] has shown that the heat transfer coefficient is essentially the same as for the parallel-plate channel (see the section on parallel isothermal plates). Also, as WIS - 0, the heat transfer should approach that for a vertical flat plate. Van De Pol and Tierney [270] proposed the following modification to the Elenbaas equation [88, 89] to fit the data of Welling and Wooldridge [283] in the range 0.6 < Ra < 100, Pr = 0.71,0.33 < WIS < 4.0, and 42 < HIS < 10.6 ... 

It was found experimentally that water adsorption facilitates the H2S removal by activated carbons, when its value exceeds 10-20 %wt. of the H2S adsorption practically does not depend on H2O content. Taking this into account, and also the absence of any VOCs at the laboratory conditions (lab), the process of HiS adsorption can be described by mass balance and kinetics equations for one component (H2S) with corresponding boundary conditions. In this type of model the rectangular isotherm Eq.(39) is commonly used to describe the equilibrium in a reaction system. 

For zero axial dispersion, the differential mass balance equation was solved using tlie rate expression and rectangular isotherm. The solution can be expressed as [127] ... 

A variant of the zero-length column (ZLC) method has also been developed to permit rapid measurement of both Henry constants and complete isotherms [4]. This method works well provided the curvature of the isotherm is moderate but it breaks down for highly favorable (rectangular) isotherms. 

The pressure response to the voliune perturbation was recorded with a high-accuracy differential Baratron pressure transducer (MKS 698A11TRC) (Eq. 10) at each frequency over three to five square-wave cycles (256 pressure readings per cycle) after the periodic steady-state had been established. The isotherm describing the equilibrium sorption conditions can be linear or curved. However, the horizontal region of a rectangular isotherm cannot be used as there is no sorption/desorption following the square-wave modulation of the equilibrium volume. 

We see from Figure 3.2-1 that the larger is the value of n, the more nonlinear is the adsorption isotherm, and as n is getting larger than about 10 the adsorption isotherm is approaching a so-called rectangular isotherm (or irreversible isotherm). The term "irreversible isotherm" is normally used because the pressure (or concentration) needs to go down to an extremely low value before adsorbate molecules would desorb from the surface. 

We first discuss the various simple models, and start with linear models, favoured for the possibility of analytical solution which allows us to study the system behaviour in a more explicit way. Next we will discuss nonlinear models, and under special conditions such as the case of rectangular isotherm with pore diffusion analytical solution is also possible. Nonisothermal conditions are also dealt with by simply adding an energy balance equation to mass balance equations. We then discuss adsorption behaviour of multicomponent systems. 

By splitting the particle domain into two, for which the domain close to the particle exterior surface is saturated with the adsorbed species while the inner domain is free from any sorbates, the problem can be solved to yield analytical solution (Suzuki and Kawazoe, 1974). Using a rigorous perturbation method. Do (1986) has proved that the penetration of the adsorption front in the case of rectangular isotherm can be derived from the full pore diffusion and adsorption... 

From the solutions for the fractional uptake in Table 9.2-4, the half time can be obtained by setting the fractional uptake F to one half, and they are tabulated in the third column of Table 9.2-4. For this case of rectangular isotherm, the half time is proportional to the square of particle radius, the maximum adsorptive capacity, and inversely proportional to the bulk concentration. The time it takes to reach equilibrium is half when the bulk concentration is doubled. This is because when the bulk concentration is doubled the driving force for mass transfer is doubled while the adsorptive capacity is remained constant (that is saturation concentration) hence the time to reach saturation will be half. Recall that when the isotherm is linear, the time scale for adsorption is independent of bulk concentration. Hence, for moderately nonlinear isotherm, the time scale would take the following form ... 

Wakao suggests as a limiting expression for a system with a rectangular isotherm... 

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. 

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