Adsorption equilibrium parameters calculation

The constraints on m1 and m4 are explicit. The lower limit of m, however, does not depend on the other flow rate ratios, whereas the upper limit of m4 is an explicit function of the flow rate ratios m2 and m3 and of the feed composition respectively [25]. The constraints on m2 and m3 are implicit (see Eq. 4), but they do not depend on m1 and m4. Therefore, they define a unique region of complete separation in the (m2, m3) plane, which is the triangle-shaped region abw in Fig. 4. The boundaries of this region can be calculated explicitly in terms of the adsorption equilibrium parameters and the feed composition as follows [25] ... 

Because the inverse Debye length is calculated from the ionic surfactant concentration of the continuous phase, the only unknown parameter is the surface potential i/io this can be obtained from a fit of these expressions to the experimental data. The theoretical values of FeQx) are shown by the continuous curves in Eig. 2.5, for the three surfactant concentrations. The agreement between theory and experiment is spectacular, and as expected, the surface potential increases with the bulk surfactant concentration as a result of the adsorption equilibrium. Consequently, a higher surfactant concentration induces a larger repulsion, but is also characterized by a shorter range due to the decrease of the Debye screening length. 

An attempt has been made to summarize the available literature for comparison of adsorption constants and forms of the equations used. Table XV presents a number of parameters reported by different authors for several model compounds on CoMo/A1203 in the temperature range 235-350°C (5,33,104,122,123,125-127). The data presented include the adsorption equilibrium constants at the temperatures employed in the studies and the exponential term (n) of the denominator function of the 0 parameter that was used in the calculation. The numbers shown in parentheses, relating to the value of n, indicate that the hydrogen adsorption term (Xh[H2]) is expressed as the square root of this product in the denominator. Data are presented for both the direct sulfur extraction site (cr) and the hydrogenation site (t). 

Fig. 16. The repulsive force between two plates with adsorption equilibrium at various NaCl concentrations. The following parameters are employed in the calculations ctq = —0.01 C/m2, e1 = 10, e11 = 80, S = 10 A, A eq = 1000. NaCl concentration (1) 0.001 (2) 0.01 M. The solid lines are for the force predicted by the new model and the dashed lines are for the force predicted by the Ciouy Chapman theory.

Adsorption equilibrium and kinetics of Ar on the 4 adsorbents were measured by volumetric method at three temperatures. The constant volume apparatus and the calculation method have been described in detail elsewhere [3]. Small pressure steps ( 0.1 bar) were given to make sure that the measurements were in the linear range (q = Kc, where K is Henry law constant, q and c, in mmol/cc, are concentrations in adsorbed phase and gas phase). As such, constant, limiting kinetic parameters were extracted. 

The relationships should be established between these constants and such parameters as the type of the zeolite lattice, the type and concentration of cations, the degree of decationization, and the structure of the adsorbate molecule. As a result, semiempirical relationships may be obtained which could be used for the practical calculation of the adsorption equilibrium. 

Much more work on the experimental verification and quantification of this physical adsorption process has been conducted in our own laboratory for the chloroplatinic acid (HzPtCL, or CPA)/alumina system [3,4]. The Revised Physical Adsorption (RPA) model [4], with which all known sets of Pt/alumina adsorption data can be satisfactorily simulated with no adjustable parameters, is a result of these efforts. The basis of the RPA model is the a priori calculation of adsorption equilibrium constants, seen as the center regime of Fig. 1. 

In the best strategy of experimentation for model discrimination and parameter estimation, experimental studies are started by determining the Pt dependence of (-B.4)o, which will indicate the functional form of the rate expression. Secondly, pure feed experiments are conducted both at space times within the initial rates region and at longer space times to determine the reaction rate constant and adsorption equilibrium coefficients of reactants. Thirdly, mixed feed experiments carried out in the initial rates region and at longer space times are used in the calculation of adsorption equilibrium coefficients of products [17]. 

The mass transfer in the adsorption column can be deseribed by a set of differential partial equations, but the analytical solution is often difficult to establish. The adsorption parameters can also directly be obtained by the determination of the moments of the beakthrough curve. The fisrt moment, fi, is only related to equilibrium parameters, while the second moment, is related to mass transfer parameters [11], and they can be directly calculated by integration of the response curve. 

Of particular interest has been the study of the polymer configurations at the solid-liquid interface. Beginning with lattice theories, early models of polymer adsorption captured most of the features of adsorption such as the loop, train, and tail structures and the influence of the surface interaction parameter (see Refs. 57, 58, 62 for reviews of older theories). These lattice models have been expanded on in recent years using modem computational methods [63,64] and have allowed the calculation of equilibrium partitioning between a poly-... 

Having estimated the sticking coefficient of nitrogen on the Fe(lll) surface above, we now consider the desorption of nitrogen, for which the kinetic parameters are readily derived from a TPD experiment. Combining adsorption and desorption enables us to calculate the equilibrium constant of dissociative nitrogen adsorption from... 

The slopes of the peaks in the dynamic adsorption experiment is influenced by dispersion. The 1% acidified brine and the surfactant (dissolved in that brine) are miscible. Use of a core sample that is much longer than its diameter is intended to minimize the relative length of the transition zone produced by dispersion because excessive dispersion would make it more difficult to measure peak parameters accurately. Also, the underlying assumption of a simple theory is that adsorption occurs instantly on contact with the rock. The fraction that is classified as "permanent" in the above calculation depends on the flow rate of the experiment. It is the fraction that is not desorbed in the time available. The rest of the adsorption occurs reversibly and equilibrium is effectively maintained with the surfactant in the solution which is in contact with the pore walls. The inlet flow rate is the same as the outlet rate, since the brine and the surfactant are incompressible. Therefore, it can be clearly seen that the dynamic adsorption depends on the concentration, the flow rate, and the rock. The two parameters... 

According to the equilibrium dispersive model and adsorption isotherm models the equilibrium data and isotherm model parameters can be calculated and compared with experimental data. It was found that frontal analysis is an effective technique for the study of multicomponent adsorption equilibria [92], As has been previously mentioned, pure pigments and dyes are generally not necessary, therefore, frontal analysis and preparative RP-HPLC techniques have not been frequently applied in their analysis. 

EXTRACT and O-METHYLATED EXTRACT. The sorption of benzene by the extract and the O-methylated extract is characterized by a rapid, initial uptake followed by a very slow approach to equilibrium. Such sorption behavior is very similar to that of glassy polymers. Thus we have chosen to interpret the sorption curves shown in Figures 2 and 3 in terms of the Berens-Hopfenberg model developed for the sorption of organic vapors into glassy polymers.(lS) By doing so, we attempt to correct the total sorption values for surface adsorption in order to calculate x parameters. 

Figure 9.9 Left BET adsorption isotherms plotted as total number of moles adsorbed, n, divided by the number of moles in a complete monolayer, ri7non, versus the partial pressure, P, divided by the equilibrium vapor pressure, Po. Isotherms were calculated for different values of the parameter C. Right Adsorption isotherms of water on a sample of alumina (Baikowski CR 1) and silica (Aerosil 200) at 20°C (P0 = 2.7 kPa, redrawn from Ref. [379]). The BET curves were plotted using Eq. (9.37) with C = 28 (alumina) and C = 11 (silica). To convert from n/nmo to thickness, the factors 0.194 nm and 0.104 nm were used, which correspond to n-mon = 6.5 and 3.6 water molecules per nm2, respectively.

Another important characteristic of the surface processes is a ratio g of the adspecies migration rate constant to those of the surface reaction, adsorption, and desorption rates. At small coverages the parameter g controls the surface process conditions r 1 in the kinetic and g l in the diffusion mode. A fast surface mobility of the adspecies and their equilibrium distribution on the surface are the most frequently adopted assumptions. At r < 1 the macroscopic concentrations of adspecies 6 cannot be used for calculating the process rates, and a more detailed description of their distribution is essential. 

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