Pore-surface diffusion model

Isothermal Micropore and Pore-Surface Diffusion Models. 317... 

The FRFs for the pore-surface diffusion model were derived for the case of constant pore and surface diffusion coefficients. The first-order FRF can be derived analytically for all three particle geometries (the solution is analogous to the one obtained for the micropore diffusion model). On the other hand, the second- and higher-order FRFs can be derived analytically only for the slab particle geometry. These are the expressions for the first- and second-order FRFs, for Dp = const, Ds = const, and o- = 0 [56,58] ... 

Deff is the effective or apparent diffusion coefficient, which is often defined for the pore-surface diffusion model. 

An example of the first- and second-order FRFs for the pore-surface diffusion model, corresponding to a favorable adsorption isotherm (l pp < 0), is shown in Figure 11.9. 

Figure 11.10 shows the first- and second-order FRFs for a special case of the pore - surface diffusion model, for 0 = 0 (pure pore diffusion). For this case, the characteristic behavior of the second-order FRFs become even more significant the amplitude of F2,pp(u>, ,—o)) approaches 0 at high frequencies and not to some finite value. For this case, the effective diffusion coefficient Deff, defined in Equation (11.48), reduces to the pore diffusion coefficient Dp. 

For the case of local equilibrium at the micropore mouth, these expressions reduce to the ones corresponding to the pore-surface diffusion model (Equation (11.106) and Equation (11.107)). 

The mass transfer coefficient can be calculated as the ratio of the estimated values of Op and Tp. 2, Isothermal Micropore and Pore-Surface Diffusion Models... 

The expressions for the FRFs for the isothermal micropore and pore-surface diffusion models were obtained for constant diffusion coefficients. If this assumption is not met, that is, if the concentration dependence of the diffusion coefficient has to be taken into account, the value estimated from the maximum of the — Imag(Fi p(diffusion coefficient corresponding to the steady-state concentration. 

Adsorption equilibrium of CPA and 2,4-D onto GAC could be represented by Sips equation. Adsorption equilibrium capacity increased with decreasing pH of the solution. The internal diffusion coefficients were determined by comparing the experimental concentration curves with those predicted from the surface diffusion model (SDM) and pore diffusion model (PDM). The breakthrough curve for packed bed is steeper than that for the fluidized bed and the breakthrough curves obtained from semi-fluidized beds lie between those obtained from the packed and fluidized beds. Desorption rate of 2,4-D was about 90 % using distilled water. 

There are several correlations for estimating the film mass transfer coefficient, kf, in a batch system. In this work, we estimated kf from the initial concentration decay curve when the diffusion resistance does not prevail [3]. The value of kf obtained firom the initial concentration decay curve is given in Table 2. In this study, the pore diffusion coefficient. Dp, and surface diffusion coefficient, are estimated by pore diffusion model (PDM) and surface diffusion model (SDM) [4], The estimated values of kf. Dp, and A for the phenoxyacetic acids are listed in Table 2. 

Extension of the equilibrium model to column or field conditions requires coupling the ion-exchange equations with the transport equations for the 5 aqueous species (Eq. 1). To accomplish this coupling, we have adopted the split-operator approach (e.g., Miller and Rabideau, 1993), which provides considerable flexibility in adjusting the sorption submodel. In addition to the above conceptual model, we are pursuing more complex formulations that couple cation exchange with pore diffusion, surface diffusion, or combined pore/surface diffusion (e.g., Robinson et al., 1994 DePaoli and Perona, 1996 Ma et al., 1996). However, the currently available data are inadequate to parameterize such models, and the need for a kinetic formulation for the low-flow conditions expected for sorbing barriers has not been established. These issues will be addressed in a future publication. 

The above model has been refined based on the dusty gas model [Mason and Malinauskas, 1983] for transport through the gas phase in the pores and the surface diffusion model [Sloot, 1991] for transport due to surface flow. Instead of Equation (10-101), the following equation gives the total molar flux through the membrane pores which are assumed to be cylindrically shaped... 

The major difference between the various GRM models is due to the mechanism of intraparticle diffusion that they propose, namely pore diffusion, siuface diffusion or a combination of both, independent or competitive diffusion. The pore diffusion model assumes that the solute diffuses into the pore of the adsorbent mainly or only in the free mobile phase that impregnates the pores of the particles. The surface diffusion model considers that the intraparticle resistance that slows the mass transfer into and out of the pores proceeds mainly through surface diffusion. In the GRM, diffusion within the mobile phase filling the pores is usually assumed to control intraparticle diffusion (pore diffusion model or PDM). This kind of model often fits the experimental data quite well, so it can be used for the calculation of the effective diffusivity. If this model fails to fit the data satisfactorily, other transport formulations such as the Homogeneous Surface Diffusion Model (HSDM) [27] or a model that allows for simultaneous pore and siuface diffusion may be more successful [28,29]. However, how accurately any transport model can reflect the actual physical events that take place within the porous... 

The difference between the pore diffusion model and the surface diffusion model is that in the pore diffusion model it takes longer to equilibriate the solid with higher adsorption affinity (higher K) while in the surface diffusion model the diffusion time is independent of the Henry constant. This is so because in the case of surface diffusion adsorption occurs at the pore mouth at which point the... 

The half time can be calculated from eq.( 10.4-11a) by setting the fractional uptake to one half. Unlike the parallel pore and surface diffusion model discussed in Chapter 9 where the half time is proportional to the square of the particle radius, the half time of the bimodal diffusion model is proportional to R , where a is equal to 2 when macropore diffusion dominates the transport and a is equal to zero when micropore diffusion controls the uptake. An approximate expression for the half time for a bimodal diffusion model is given by Do (1990) ... 

With this form, the model equations will reduce to the pore and surface diffusion model dealt with extensively in Chapter 9 ... 

Since the void fraction distribution is independently measurable, the only remaining adjustable parameters are the A, so when surface diffusion is negligible equations (8.23) provide a completely predictive flux model. Unfortunately the assumption that (a) is independent of a is unlikely to be realistic, since the proportion of dead end pores will usually increase rapidly with decreasing pore radius. 

Though a porous medium may be described adequately under non-reactive conditions by a smooth field type of diffusion model, such as one of the Feng and Stewart models, it does not necessarily follow that this will still be the case when a chemical reaction is catalysed at the solid surface. In these circumstances the smooth field assumption may not lead to appropriate expressions for concentration gradients, particularly in the smaller pores. Though the reason for this is quite simple, it appears to have been largely overlooked,... 

Reverse osmosis models can be divided into three types irreversible thermodynamics models, such as Kedem-Katchalsky and Spiegler-Kedem models nonporous or homogeneous membrane models, such as the solution—diffusion (SD), solution—diffusion—imperfection, and extended solution—diffusion models and pore models, such as the finely porous, preferential sorption—capillary flow, and surface force—pore flow models. Charged RO membrane theories can be used to describe nanofiltration membranes, which are often negatively charged. Models such as Dorman exclusion and the... 

Step 3. Transport within a catalyst pore is usually modeled as a one-dimensional diffusion process. The pore is assumed to be straight and to have length The concentration inside the pore is ai =ai(l,r,z) where I is the position inside the pore measured from the external surface of the catalyst particle. See Figure 10.2. There is no convection inside the pore, and the diameter of the pore is assumed to be so small that there are no concentration gradients in the radial direction. The governing equation is an ODE. 

The concentration of gas over the active catalyst surface at location / in a pore is ai [). The pore diffusion model of Section 10.4.1 linked concentrations within the pore to the concentration at the pore mouth, a. The film resistance between the external surface of the catalyst (i.e., at the mouths of the pore) and the concentration in the bulk gas phase is frequently small. Thus, a, and the effectiveness factor depends only on diffusion within the particle. However, situations exist where the film resistance also makes a contribution to rj so that Steps 2 and 8 must be considered. This contribution can be determined using the principle of equal rates i.e., the overall reaction rate equals the rate of mass transfer across the stagnant film at the external surface of the particle. Assume A is consumed by a first-order reaction. The results of the previous section give the overall reaction rate as a function of the concentration at the external surface, a. ... 

PALS is based on the injection of positrons into investigated sample and measurement of their lifetimes before annihilation with the electrons in the sample. After entering the sample, positron thermalizes in very short time, approx. 10"12 s, and in process of diffusion it can either directly annihilate with an electron in the sample or form positronium (para-positronium, p-Ps or orto-positronium, o-Ps, with vacuum lifetimes of 125 ps and 142 ns, respectively) if available space permits. In the porous materials, such as zeolites or their gel precursors, ort/zo-positronium can be localized in the pore and have interactions with the electrons on the pore surface leading to annihilation in two gamma rays in pick-off process, with the lifetime which depends on the pore size. In the simple quantum mechanical model of spherical holes, developed by Tao and Eldrup [18,19], these pick-off lifetimes, up to approx. 10 ns, can be connected with the hole size by the relation ... 

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