Adsorption rate parameters

K Langmuir constant (m moL ) k reaction rate constant (m moL s ) k overall adsorption rate parameter (s ) k intra-particle mass transfer coefficient (s )... 

From the experimental point of view there is one more experiment that can be done to obtain adsorption rate parameters and this is the use of a semicontinuous approach. Here the adsorbate is fed at a mass flow rate that is equal to the rate of adsorption, and a very sensitive pressure transducer is slaved to a mass flow controller for the adsorbate. As this gas is adsorbed, and were there no flow into the system, the pressure would drop. This would lead to the complexities we have just analyzed. If, however, the mass flow controller is slaved in such a way that it opens whenever there is a slight 6P of pressure drop below the fixed experimental pressure set point, then the pressure can be maintained as a constant. The mass flow rate into the system is the same as the mass rate of adsorption "out" of the gas phase and "onto" the adsorbent phase II. 

Na is the adsorption capacity, is the adsorption rate parameter and F is the flow rate. The critical bed depth, Zq, is the theoretical depth of carbon sufficient to prevent the solute concentration from exceeding the Q value at f = 0. By letting / = 0, Zo is obtained by solving Eq. (15.31) for Z. The final result is ... 

The relationship between adsorption capacity and surface area under conditions of optimum pore sizes is concentration dependent. It is very important that any evaluation of adsorption capacity be performed under actual concentration conditions. The dimensions and shape of particles affect both the pressure drop through the adsorbent bed and the rate of diffusion into the particles. Pressure drop is lowest when the adsorbent particles are spherical and uniform in size. External mass transfer increases inversely with d (where, d is particle diameter), and the internal adsorption rate varies inversely with d Pressure drop varies with the Reynolds number, and is roughly proportional to the gas velocity through the bed, and inversely proportional to the particle diameter. Assuming all other parameters being constant, adsorbent beds comprised of small particles tend to provide higher adsorption efficiencies, but at the sacrifice of higher pressure drop. This means that sharper and smaller mass-transfer zones will be achieved. 

We now study the consequences of these BEP choices to the dependence of predicted rate of methane production on Eads- Making the additional simplifying assumption that the adsorption energy parameters in Eqs. (1.9b) and (1.9c) are the same, one finds for the rate of methane production an expression... 

In connection with practical situations where CO oxidation is important, we must also consider the perennial question of how to connect the low pressure results onto those at high pressure. Qualitatively this has been done for the CO oxidation reaction but it would still be worthwhile to attempt a numerical prediction of high pressure results based on low-pressure rate parameters. A very nice paper modeling steady-state CO oxidation data over a supported Pt catalyst at CO and O2 pressures of several torr has very recently appeared (.25). Extension of this work to other systems in warranted and, even though unresolved questions continue to exist, every indication is that the high and low pressure data can be reliably modeled with the same rate parameters if no adsorption - desorption equilibria are assumed. 

The rate parameters presented in Table I were used together with the parameter values listed in Table II to predict the product responses during the adsorption of NO on a hydrogen covered Rh surface and the subsequent reduction of the adsorbed... 

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] ... 

For the simulation of SMB-separations efficient software packages,based on the Triangle-Theory, are commercially available. The number of columns, the column dimensions, the theoretical number of plates in the columns, the feed concentration, the bi-Langmuir adsorption isotherm parameters and the number of cycles need to be defined by the user. Then the separation is simulated and values for the flow rate ratios, the flow rates, the switching time and the quality of the separation, purity and yield, are calculated. Based on these values an actual separation can be performed. However, some optimization/further development is usually necessary, since the simulations are based on an ideal model and the derived parameters and results therefore can only be taken as indications for the test runs. 

In reality, it is believed that the oxidation of carbonaceous surfaces occurs through adsorption of oxygen, either immediately releasing a carbon monoxide or carbon dioxide molecule or forming a stable surface oxygen complex that may later desorb as CO or C02. Various multi-step reaction schemes have been formulated to describe this process, but the experimental and theoretical information available to-date has been insufficient to specify any surface oxidation mechanism and associated set of rate parameters with any degree of confidence. As an example, Mitchell [50] has proposed the following surface reaction mechanism ... 

This empirical rate expression considers the active sites of the catalyst as only a fraction of the total adsorption sites for ammonia and is consistent vfith the presence of a reservoir of ammonia adsorbed species which can take part in the reaction. The ammonia reservoir is likely associated vfith poorly active but abundant W and Ti surface sites, which can strongly adsorb ammonia in fact, nhs roughly corresponds to the surface coverage of V. Once the ammonia gas-phase concentration is decreased, the desorption of ammonia species originally adsorbed at the W and Ti sites can occur followed by fast readsorption. When readsorption occurs at the reactive V sites, ammonia takes part in the reaction. Also, the analysis of the rate parameter estimates indicates that at steady state the rate of ammonia adsorption is comparable to the rate of its surface reaction with NO, whereas NH3 desorption is much slower. Accordingly, the assumption of equilibrated ammonia adsorption, which is customarily assumed in steady-state kinetics, may be incorrect, as also suggested by other authors [55]. 

Ba(NO3)2, decomposition of BaOa to BaO and oxygen and the reversible spillover of NO2 between Pt sites and BaO sites. Essentially the model assumes that the adsorption of NO proceeds through the nitrate route and does not consider the nitrite route. Olsson et al. [76] estimated part of the rate parameters in their model from theoretical considerations, part were taken from the literature or calculated from thermodynamic constraints and part were estimated by fitting a set of experimental data. 

Figure 1.4 gives an example of the adsorption of a compound to suspended sediment, modeled as two resistances in series. At first, the compound is dissolved in water. For successful adsorption, the compound must be transported to the sorption sites on the surface of the sediment. The inverse of this transport rate can also be considered as a resistance to transport, Ri. Then, the compound, upon reaching the surface of the suspended sediment, must find a sorption site. This second rate parameter is more related to surface chemistry than to diffusive transport and is considered a second resistance, R2, that acts in series to the first resistance. The second resistance cannot... 

Table 13 Parameter values in the case of an infinite adsorption rate Ic — +oo...

According to their analysis, if is zero (practically much lower than 1), then the liquid-film diffusion controls the process rate, while if tfis infinite (practically much higher than 1), then the solid diffusion controls the process rate. Essentially, the so-called mechanical parameter represents the ratio of the diffusion resistances (solid and liquid film). The authors did not refer to any assumption concerning the type of isotherm for the derivation of the above-mentioned criterion it is sufficient to be favorable (not only rectangular). They noted that for >1.6, the particle diffusion is more significant, whereas if < 0.14, the external mass transfer controls the adsorption rate. 

The rate expressions Rj — Rj(T,ck,6m x) typically contain functional dependencies on reaction conditions (temperature, gas-phase and surface concentrations of reactants and products) as well as on adaptive parameters x (i.e., selected pre-exponential factors k0j, activation energies Ej, inhibition constants K, effective storage capacities i//ec and adsorption capacities T03 1 and Q). Such rate parameters are estimated by multiresponse non-linear regression according to the integral method of kinetic analysis based on classical least-squares principles (Froment and Bischoff, 1979). The objective function to be minimized in the weighted least squares method is... 

The rate parameters of importance in the multicomponent rate model are the mass transfer coefficients and surface diffusion coefficients for each solute species. For accurate description of the multicomponent rate kinetics, it is necessary that accurate values are used for these parameters. It was shown by Mathews and Weber (14), that a deviation of 20% in mass transfer coefficients can have significant effects on the predicted adsorption rate profiles. Several mass transfer correlation studies were examined for estimating the mass transfer coefficients (15, jL6,17,18,19). The correlation of Calderbank and Moo-Young (16) based on Kolmogaroff s theory of local isotropic turbulence has a standard deviation of 66%. The slip velocity method of Harriott (17) provides correlation with an average deviation of 39%. Brian and Hales (15) could not obtain super-imposable curves from heat and mass transfer studies, and the mass transfer data was not in agreement with that of Harriott for high Schmidt number values. 

For this study, mass transfer and surface diffusions coefficients were estimated for each species from single solute batch reactor data by utilizing the multicomponent rate equations for each solute. A numerical procedure was employed to solve the single solute rate equations, and this was coupled with a parameter estimation procedure to estimate the mass transfer and surface diffusion coefficients (20). The program uses the principal axis method of Brent (21) for finding the minimum of a function, and searches for parameter values of mass transfer and surface diffusion coefficients that will minimize the sum of the square of the difference between experimental and computed values of adsorption rates. The mass transfer and surface coefficients estimated for each solute are shown in Table 2. These estimated coefficients were tested with other single solute rate experiments with different initial concentrations and different amounts of adsorbent and were found to predict... 

The equations (20) to (30) provide the basis for predicting the adsorption rate profiles for the binary system. The input parameters required for the model are the single-solute film transfer and surface diffusion coefficients, the single-solute isotherm constants and the mixture equilibria coefficients. The rate parameters were obtained from single solute rate data (20), and the equilibrium parameters were obtained from single and multi-solute equilibrium data. 

The proposed mathematical model for encapsulated adsorbents can describe various diffusion characteristics in addition to the intrinsic binding characteristics of the encapsulated adsorbents. The performance of encapsulated adsorbent in an in situ product separation process can be evaluated using the proposed model for the adsorption rate of a target product, berberine. The performance of the encapsulated adsorbents is influenced by design parameters such as the adsorbent content in the capsule (Ns), the capsule size ( R ), the number of capsules (n), the membrane thickness ( Rm), and the ratio of the single capsule volume to the total capsule volume (Nc). 

Chase [32] used the adsorption rate-limited model [Eqs. (7 —(11)[ to analyze the experimental breakthrough curves in affinity chromatography. This empirical approach assumes that all the rate-limiting processes can be represented by an apparent single second-order Langmuir adsorption rate equation in which k is considered a lumped" parameter. 

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