Isotherm characteristic curve

The curve of Fig. XVII-15 is essentially a characteristic curve of the Polanyi theory, but in the form plotted in might better be called a characteristic isotherm. Furthermore, as would be expected from the Polanyi theory, if the data for a given adsorbate are plotted with RTln P/f ) as the abscissa instead of just ln(P/P ), then a nearly invariant shape is obtained for different temperatures. The plot might then be called the characteristic adsorption curve. 

The BET surface area values are also reported with the distribution of porosity between microporosity (pore diameter <1.8 nm) deduced from N2 adsorption isotherms (t-curves) and mesoporosity (pore diameter > 1.8 nm). The following trend is observed for high atomic M/HPA ratio used for the precipitation, the precipitates exhibited high surface area mainly due to microporosity. However, depending on the nature of the coxmter cation and also of the previous ratio values, the textural characteristics were not similar. In particular, it is interesting to note the presence of mesopores for (NH4)2.4P, CS2.9P, CS2.7P and Cs2.4Si samples. 

With the development of modern computation techniques, more and more numerical simulations occur in the literature to predict the velocity profiles, pressure distribution, and the temperature distribution inside the extruder. Rotem and Shinnar [31] obtained numerical solutions for one-dimensional isothermal power law fluid flows. Griffith [25], Zamodits and Pearson [32], and Fenner [26] derived numerical solutions for two-dimensional fully developed, nonisothermal, and non-Newtonian flow in an infinitely wide rectangular screw channel. Karwe and Jaluria [33] completed a numerical solution for non-Newtonian fluids in a curved channel. The characteristic curves of the screw and residence time distributions were obtained. 

Equation 3 was verified experimentally (3) over wide ranges of temperature and equilibrium pressure for the adsorption of various vapors on active carbons with different parameters for the microporous structure. For adsorption on zeolites, this equation fitted the experimental results well only in the range of high values of 0 (4, 5, 6, 7). Among other equations proposed for the characteristic curve (4, 5, 8, 9, 10) we chose to use the Cohen (4) and Kisarov s (10) equation, which starts from the following adsorption isotherm equation ... 

Dubinin was the pioneer of the concept of micropore filling. His approach was based on the early potential theory of Polanyi, in which the physisorption isotherm data were expressed in the form of a temperature-invariant characteristic curve . 

Various attempts were made by Dubinin and his co-workers to apply the fractional volume filling principle and thereby obtain a characteristic curve for the correlation of a series of physisorption isotherms on a zeolite (Dubinin, 1975). As was noted in Chapter 4, the original Dubinin-Radushkevich (DR) equation (i.e. Equation (4.39)) was found to be inadequate and in its place the more general Dubinin-Astakhov (DA) equation was applied (i.e. Equation (4.45)). 

Although many experimental isotherms appear to obey the DA equation over appreciable ranges of pressure, the theoretical basis of this conformity is highly questionable. However, as Ruthven (1984) points out, even with NaX and other zeolites the temperature invariant characteristic curve can provide a useful empirical means of correlating engineering data. 

Adsorption isotherms of N2 at 77 K and CO2 at 273 K can be fairly compared by plotting them as a characteristic curve. Figures 5 and 6 show the characteristic curves for N2 and CO2 forbothPS and MCM-41. 

Single-component adsorption equilibria on activated carbon of the n-alkanes Q-C4 and of the odorant tert-butyl mercaptan were measured at the operating conditions expected in a large-scale facility for adsorbed natural gas (ANG) storage. The experimental data were correlated successfully with the Adsorption Potential theory and collapsed into a single temperature-independent characteristic curve. The obtained isotherm model should prove to be very useful for predicting the adsorption capacity of an ANG storage tank and to size and optimize the operation of a carbon-based filter for ANG applications. 

Figure 1 displays the experimental characteristic curve obtained and the p values employed to generate it. The existence of very little scatter in the data demonstrates that the isotherms of the various adsorbates were successfully correlated as a single temperature-independent characteristic curve. This fact corroborates the applicability of the Potential theory to the carbon under study. 

FIGURE 24-10 Characteristic curves of rfF against retention temperature for the hydrocarbons ranging from propane (1) to octane (6) obtained from the chromatographic system of Figure 24-4. Solid lines, calculated from isothermal data points, experimental. From Harris and Habgood. )... 

Figure 3. Characteristic curve of CO2 adsorption on LiX points correspond to experimental adsorption isotherms for temperatures from 273° to 363°K (A, cal/mole)...

Let us first consider the most common case. An analysis of many adsorption isotherms on zeolites of various vapors with relatively large molecules has shown that the characteristic curves are expressed by an equation similar to Equation 2, but with power (distribution order, n) higher than 2 ... 

Naturally, for adsorption on cations in zeolite voids, which we have called electrostatic adsorption, the concept of the invariance of characteristic curves is a rather crude approximation. This is directly indicated by characteristic curves constructed from experimental adsorption isotherms for a sufficiently wide temperature range. However, for the purposes of practical description of adsorption equilibria, such an approximate assumption in a controlled temperature range is reasonable. A general evaluation will be made below. 

The potential theory has been widely used in the last decades Such a success is mainly due to the predictive character of this method Once v f, f, and (3 are known, it is possible to calculate any isotherm if the reference characteristic curve of the adsorbent is known. A lot of research teams have tried to improve the initial theory by proposing different correlations for the determination of v> [15,16,17] The calculation of Ps and thus f, for supercritical fluids is another problem which has been much debated [18,19,20] Besides, it is possible to relate the reference characteristic curve to the porous structure of the adsorbent so that the theoretical treatments are very ofen used as characterization methods The way the mathematical expression of the reference characteristic curve can be related to the micropore size distribution function of the adsorbent has also been widely discussed in the literature [4,5,6] In this paper we consider two different methods for the calculation of v, and f ... 

The adsorption isotherms have been treated using the first method. Using equations 1, 2 and 3 to calculate V and Srcf (the reference adsorbate is N2) and equations 4, 5, 6 and the R-K-S eos for the calculation of Va, f and f, we obtained the reference characteristic curve. It is presented in figure 1. 

Using experimental data in such a wide range of experimental conditions allows to obtain a characteristic curve in the whole domain of Crcf Such a domain is larger than the one obtained with only one standard isotherm (77 K N2 isotherm). 

Among the methods available for the calculation of the reference characteristic curve from experimental isotherms, we chose two procedures The first one based on the works by Ozawa, Dubinin and Agarwal, is a classical one easy to use The second one based on the... 

A proper comparison of the N2 and CO2 adsorption experiments as well as the differences between samples is conducted by using the characteristic curves plots [9,10]. The characteristic curves have been calculated by applying the Dubinin-Radushkevich equation [24-25] (eq. 1) to the different adsorption isotherms. 

In order to clarify this deviation some additional experiments with zeolite NaY were completed. Thus, the CO2 adsorption isotherm at 273K was repeated in the same sample after evacuation at 373 K in vacuum (to remove the physisorbed CO2). The isotherm obtained is very similar to the first one and no hysteresis is observed. These results indicate that CO2 is not significantly chemisorbed on the zeolite NaY at the experimental conditions used. Additionally, CO2 adsorption was performed at a higher temperature (i.e., 298 K) in order to discard any diffusional limitations of this adsorptive. The characteristic curves for the isotherm at 273 and 298 K are shown in Figure 7. The shape of the characteristic curves at the two temperatures is similar. Both curves exhibit a downward deviation at high adsorption potentials and the slope at lower potentials as well as the ordinate at the origin are very close (see dashed line). These experiments confirm that the deviation observed in the CO2 characteristic curve of zeolite NaY is due to neither CO2 chemisorption nor to diffusional problems as it was expected because the CO2 can enter into smaller pores like in the case of zeolite NaA (Table 2). Therefore, this deviation at low relative pressures must be related with the surface chemistry of the zeolite. 

Since N2 adsorption is done at 77 K and CO2 at 273 or 298 K, the experiments cannot be directly compared, which introduces strong concerns about the similarities and differences among both adsorptives. Thus, a better way to compare the two experiments is to plot the characteristic curves [33—35, 37], These characteristic curves, obtained applying the Dubinin-Radushkevich (DR) equation [47] to the adsorption isotherms, are the plot of the logarithm of the volume of liquid adsorbed versus the square of the adsorption potential corrected for the affinity coefficient ((3) of the adsorptive ((/l//3) = (RTln(/o//)/[3), T being the temperature, / the fugacity, and/ the saturation fugacity). 

Figure 17.5 Characteristic curves for an activated carbon fiber (ACF) that includes the Nj adsorption data at 77 K (relative pressure from 10 to 1) ( ), the COj adsorption data at 273 K at subatmospheric pressures ( ) and the CO2 adsorption isotherm at high pressures (A).

Figure 17.8 Characteristic curves for samples CFC14, CFC40, and CFS50 in the (A/P) range of 200-600 (y/mol) for the Nj adsorption isotherm at 77 K. (Reproduced with permission from Ref. [33]. Copyright 1996 American Chemical Society.)...

The adsorption capacity, , and maximum loading, for this isotherm are expressed as the volume adsorbed per unit mass or volume of adsorbent. The function = /( ) is called a characteristic curve and can fit data for one component over a range of temperature—and sometimes for a family of components. The adsorption potential, 8, is defined in Table 14.1. p is called an affinity coefficient that can force data for diverse components to fit a common characteristic curve. 

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