Adsorption equilibrium parameters

The model parameters, in addition to the adsorption equilibrium parameters, are ... 

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] ... 

Figure 6.17. Effective double layer adsorption equilibrium isotherms for kj=l and various values of the parameter Ajll.

Another problem which can appear in the search for the minimum is intercorrelation of some model parameters. For example, such a correlation usually exists between the frequency factor (pre-exponential factor) and the activation energy (argument in the exponent) in the Arrhenius equation or between rate constant (appears in the numerator) and adsorption equilibrium constants (appear in the denominator) in Langmuir-Hinshelwood kinetic expressions. 

Change the chemical equilibrium parameters (K and Cmax) to see how they effect the distribution and transport of the solute (see Lyman et al. (1982) for a comprehensive set of data). Experiment by using a different model for the adsorption of the solute, e.g., Freundlich linear adsorption. 

When a specific feed composition is given, the constraints on m1 and m4 as well as the complete separation region in the (m2, m3) plane can be determined,since these depend only on the parameters of the adsorption equilibrium isotherms and the feed composition itself. Based on these values an operating point can be selected, i. e. a set of four values of = 1,..., 4 fulfilling the complete separation requirements. Since the flow rate ratios are dimensionless groups combining column volumes, flow rates and switching intervals, the constraints on the flow rate ratios are independent of the size and productivity of the SMB unit. 

Adsorption equilibrium constant for species i Estimate of a parameter after ith iteration of nonlinear estimation program Vector of estimated rate and adsorption parameters Vector of initial estimates of K for use in nonlinear least squares... 

On the contrary, a more advanced methodology makes use of nonlinear chromatography experiments If the adsorption isotherms are measured under variable temperatures, the corresponding thermodynamic parameters for each site can be obtained in view of the van t Hoff dependency (site-selective thermodynamics measurements) [51,54]. Thus, the adsorption equilibrium constants of the distinct sites bi a = ns, s) are related to the enthalpy (A// ) and entropy (A5j) according to the following equation [54] ... 

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). 

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. 

Figure 7 shows the effects of the equilibrium parameter P on the uptake curve for a given r) and s. Fastest adsorption occurs at p = 0 which corresponds to the isothermal case. 

Measurement of the mean retention time and dispersion of a concentration perturbation passing through a packed adsorption column provides a useful method of determining kinetic and equilibrium parameters. The carrier should be inert, and the magnitude of the concentration change must be kept small to ensure linearity of the system. 

When the relative volumes are known and the diffusion coefficients in the capsule core and capsule membrane can be estimated a priori in single component adsorption, the parameter to work with is the effective diffusivity in the adsorbent pore (Dn). Then, with the above estimated parameter values, the parameters of competitive adsorption are the maximum concentration at the solid phase of the adsorbent (CsmT), and the equilibrium constants of the target product (KS1) and byproduct (KS2)-... 

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.

Here, C( , z, t) is the scaled solute concentration in the fluid phase, Cw the solute concentration at the wall, 6 the normalized adsorbed concentration (O<0< 1), K the adsorption equilibrium constant, p the transverse Peclet number, T represents the adsorption capacity (ratio of adsorption sites per unit tube volume to the reference solute concentration), and Da is the local Damkohler number (ratio of transverse diffusion time to the characteristic adsorption time). We shall assume that p 4Cl while T and Da are order-one parameters. (In physical terms, this implies that transverse molecular diffusion and adsorption processes are much faster compared to the convection.)... 

The adsorption equilibrium constant is K, with K = k Jk a-The dimensionless throughput parameter is defined as... 

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