Desorption curve equation

A numerical solution of this equation for a constant surface concentration (infinite fluid volume) is given by Garg and Ruthven [Chem. Eng. ScL, 27, 417 (1972)]. The solution depends on the value of A. = n i — n )/ n — n ). Because of the effect of adsorbate concentration on the effective diffusivity, for large concentration steps adsorption is faster than desorption, while for small concentration steps, when D, can be taken to he essentially constant, adsorption and desorption curves are mirror images of each other as predicted by Eq. (16-96) see Ruthven, gen. refs., p. 175. 

In most cases the authors prefer the second way of treatment of the desorption data, which is analytic in its nature the Arrhenius equation, whose parameters are assumed to be constant, is solved either in a closed form or numerically. The resulting quantities determining the location, height, and shape of a maximum on the desorption curve are analyzed and expressed whenever possible, in at least approximately linear form, and then compared with the experimental results. A simple analytical expression of the time-temperature function is essential for this kind of treatment. 

The required distribution of initial populations ntu can be obtained in the following manner (32). Let us consider a system with Ed mi = 20 kcal/ mole and Ed max = 45 kcal/mole. Assuming that kd = 1013 sec-1 and x = 1, we can calculate theoretical desorption rates dnai/dt for Ed = 20, 21, 22,..., 45 kcal/mole as a function of nBOi. With increasing temperature, 25 values of dnjdt are measured at temperatures corresponding to Ed of 20, 21, 22,. . ., 45 kcal/mole. Since the total desorption rate at any moment must be equal to the sum of the individual desorption processes, we obtain 25 linear equations. Their solution permits the computation of the initial populations of the surface sites in the energy spectrum considered, i.e. the function n,oi(Edi). From the form of this function, desorption processes can be determined which exhibit a substantial effect on the experimental desorption curve. 

In the Dynamic method a flow of He is passed over the sample at 77 K. A small amount of N2 is introduced into the He stream. The gas stream coming from the sample is monitored using mass spectroscopy. N2 is only detected after a monolayer is formed. The N2 supply is then switched off and the desorption curve plotted. Integration of this curve gives the information required for the BET equation. 

This solution is valid for the initially linear portion of the sorption (or desorption) curve when MtIM is plotted against the square root of time. These equations also demonstrate that for Fickian processes the sorption time scales with the square of the dimension. Thus, to confirm Fickian diffusion rigorously, a plot of MJM vs. Vt/T should be made for samples of different thicknesses a single master curve should be obtained. If the data for samples of different thicknesses do not overlap despite transport exponents of 0.5, the transport is designated pseudo-Fickian. ... 

Fig. 3.36 (a) Desorption curves for the (MgHj + Xwt%Al) composites with the content of A1 additive equivalent to the content of free A1 formed after decomposition of Awt%LiAlH in (MgH + Xwt%LiAlH ) composites (300°C 0.1 MPa Hj). The (MgH + Xwt%Al) composites were baU milled for 20 h. (b) Dependence of an effective kinetic parameter, k, in the JMAK equation on the equivalent content of A1 metal additive... 

The analysis of the ZLC desorption curve involves solving Fickian diffusion equation with appropriate initial and boundary conditions (9). The solution of the desorption curve for spherical geometry is given as ... 

Fortunately, these complex expressions reduce to two approximations, reliable to better than 1 %, valid for different parts of the desorption curve. The early time approximation, which holds for the initial portion of the curve, derived from Equation (12.4), is... 

The late time approximation, which holds for the final portion of the desorption curve, derived from Equation (12.3), is... 

In the case of SiC12T sample and benzene as a wetting liquid the desorption curve is smooth without characteristic step. It may be explained by not satisfactory wetting of this silica by benzene and restricted penetration of narrow pores. Observed effect is confirmed by small pore volume for the same sample, derived from benzene desorption data. The localization of desorption steps on temperature axis corresponds to emptying of pores with dominant share in total pore volume. Converting the temperature into pore radius, by using the Kelvin equation, the dimensions of pores and pore size distributions PSD, AV Affp vs. R, may be calculated in the manner described earher [9]. 

This equation has the same form as that obtained for solid diffusion control with D replaced by the equivalent concentration-dependent diffusivity Dp = epDp,/[ppn)K,(l - n,/nif. Numerical results for the case of adsorption on an initially clean particle are given in Fig. 16-18 for different values of X = n1/n = 1 - R. The upt e curves become increasingly steeper, as the nonlinearity of the isotherm, measured by the parameter X, increases. The desorption curve shown for a particle with nf/n) = 0.9 shows that for the same step in concentration, adsorption occurs much more quickly than desorption. This difference, however, becomes smaller as the value of X is reduced, and in the linear region of the adsorption isotherm (X —> 0), adsorption and desorption curves are mirror images. The solution in Fig. 16-18 is applicable to a nonzero initial adsorbent loading by redefining X as (n, - - ng)... 

A comparison of the desorption rates at pH 7, shown in Figure 7 for the plutonium sorbed from fresh and aged solutions, indicates that the total desorption curve may be interpreted in terms of two different sorbed species. This is expressed in Equations 2, 3, and 4 as two first order processes. For both the fresh and aged systems, the relative quantities of the Ao(d or loosely-held species were almost identical, as were their desorption rate constants. It is likely that the A0<2 or tightly-held species were colloidal in size, since irreversibility is a widely known characteristic of colloid sorption. This was found to apply, for example, in the case of the sorption of colloidal americium on quartz (27). 

Bmr tracer desorption curve as determined from Figure ia. lc Determination of T,nl from the desorption curve (= shaded area) by Equation 0 (Reproduced with permission from Ref. 7. Copyright 1952, American Institute of Chemical Engineers)... 

It is seen from Fig. 2.31a and b [109] that the sublimation rate of VCI (G-2) diminishes both in the bulk and in the surface layer of PE films. The kinetic curve of the initial VCI sublimation has a linear character. At the same time, all desorption curves of G-2 from the extruded PE films (Fig. 2.31a) display parabolic dependencies of the mjmo = art kind and obey Boltzmann s solution of the diffusion equation in a semi-infinite medium [110]. Therefore, it is possible to anticipate that the VCI desorption rate from the film carrier is limited by its diffusion. At the initial moment of diffusion, the surface concentration of the diffusant in the film is equal to that in the volume, although a concentration gradient is formed with time. So, diffusion of VCI in the films within a wide time and temperature range is described by the relation... 

Finally Figure 8 shows the measured change with time of dissolved CO2 concentration, fermentor overhead pressure, and CO2 outlet fow rate at 14° C in the pilot plant. As demonstrated by the good agreement between the experimental points and the calculated curves, equations 14 to 16 provide an adequate modeling of CO2 production, dissolution and desorption. 

Equations (37.23) illustrate the typical temperature dependence of the ion current, and the second-order desorption curve being nearly symmetrical, the first-order curve decreasing faster on the high-temperature side and... 

The numerical integration of the desorption curve (part B of Figure 3) using equation (2) and considering the curve in the absence of solid, permitted quantification of loosely adsorbed fraction (niads) of 370 pimol.g-i released during isothermal desorption. 

Numerical solutions were applied to the dual-mode sorption and transport model for gas permeation, sorption, and desorption rate curves allowing for mobility of the Langmuir component. These rate curves were almost consistent with the curves predicted by Fickian-type diffusion equation, except the desorption curve in which a slight sigmoidal deviation from the Fickian model line was apparent. The sorption... 

For estimation of desorption curves from an exhausted adsorbent bed, the basic equations for mass and heat balances in an adiabatic column are as follows. 

The solution to the diffusion equation yields a series of exponentials and it is difficult from a single ZLC experiment to distinguish different mass transfer mechanisms, i.e. surface barriers vs internal diffusion. For linear systems the shape of the initial part of the desorption curves should be distinctive [3] and the analysis of the moments of the desorption curves can also provide a means to distinguish the two mechanisms [4]. 

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