Surface diffusion Models

The spacing, between successive ledges emanating from the dislocation is given by (31) 

Information about the actual dissolution mechanism based on surface adsorption layers may be obtained from analysis of kinetic data on the undersaturation dependence of dissolution rate and observation of surface morphology of etched surfaces (36). 

The formation of etch pits at the sites of disiocations is possibie when the rate of nucieation of pits of unit depth aiong the disiocation iine, 

In Sec. 2 the overall surface dissolution was considered. Except in the case of surface diffusion, we assumed that a dissolving surface is free from defects. Since real crystals usually contain dislocations and other defects, it Is necessary to know their effect on dissolution rates and the mechanism of formation of etch pits at their emergence points. 

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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. 

The surface diffusion model (Eq. 60) is usually approximated by the linear-driving-force relation [124] ... 

Choy and McKay used a homogeneous surface diffusion model (HSDM) taking into account both external and internal transport, and found that the mean value of the solid diffusion coefficient is 3.72 X 10-9 cm2/s while kf = 6.06 X 10- 4 cm/s. 

Based upon the results shown in Figure 6 and the surface diffusion model given by equation (21), the values of ( Dos/is) which best fit the three systems were computed, both from the standpoint of assuming complete macropore control and from that of assuming the existence of micropore resistance as well, in the amount predicted from Figure 6. These values are given in Table III. 

Using the computer programs discussed above, it is possible to extract from these breakthrough curves the effective local mass transfer coefficients as a function of CO2 concentration within the stable portion of the wave. These mass transfer coefficients are shown in Figure 15, along with the predicted values with and without the inclusion of the surface diffusion model. It is seen that without the surface diffusion model, very little change in the local mass transfer coefficient is predicted, whereas with surface diffusion effects included, a more than six-fold increase in diffusion rates is predicted over the concentrations measured and the predictions correspond very closely to those actually encountered in the breakthrough runs. Further, the experimentally derived results indicate that, for these runs, the assumption that micropore (intracrystalline) resistances are small relative to overall mass transfer resistance is justified, since the effective mass transfer coefficients for the two (1/8" and 1/4" pellets) runs scale approximately to the inverse of the square of the particle diameter, as would be expected when diffusive resistances in the particle macropores predominate. 

Smith, E.H. 1991. Modified solution of homogeneous surface diffusion model for adsorption. J Environ. Eng-ASCE 117(3) 320-338. 

The second model extends the surface diffusion model to include the importance of the atomic placement of atoms in the randomly packed alloy. The model considers that a continuous connected cluster of the less noble atoms must exist to maintain the selective dissolution process for more than just the few monolayers of the alloy. This percolating cluster of atoms provides a continuous active pathway for the corrosion process as well as a pathway for the electrolyte to penetrate the solid. This is expected to depend on a sharp critical composition of the less noble element, below which dealloying does not occur.54, (Corcoran)5... 

The BCF surface diffusion model, eqn. (85), may be simplified under the conditions... 

Miller, C. T., and J. A. Pedit. 1992. Use of reactive surface-diffusion model to describe apparent sorption-desorption hysteresis and abiotic degradation of lindane in subsurface material. Environ. Sci. Technol. 26 1417-1427. 

Abuzaid, N. S. and G. Nakhla (1997). Predictability of the homogeneous surface diffusion model for activated carbon adsorption kinetics formulation of a new mathematical model, J. Environment. Science Health, Part A Environment. Eng. Toxic Hazardous Substance Control. 32, 7, 1945-1961. 

A multicomponent HSDM for acid cfye/carbon adsorption has been developed based on the ideal adsorbed solution theory (lAST) and the homogeneous surface diffusion model (H SDM) to predict the concentration versus time decay curves. The lAST with the Redlich-P eterson equation is used to determine the pair of liquid phase concentrations, Q and Qj, from the corresponding pair of solid phase concentrations, q j and q jy at fha surface of the carbon particle in the binary component. 

McKay, G., Application of Surface Diffusion Model to the Adsorption of Dyes on Bagasse Pith, Adsorption 4 (1998) pp. 361-372. 

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 GRM Formulated with the Maxwell-Stefan Surface Diffusion Model. 765... 

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... 

Hand, D.W., J.C. Crittenden, and W.E. Thacker. 1981. User oriented solutions to the homogeneous surface diffusion model Batch reactor solutions. Proc. 54th Annual Conf, Water Pollution Control Federation, Detroit, ML 4-9 Oct. 1981. Water Pollution Control Federation, Alexandria, VA. 

Figure 5 Influence of the surface diffusion model on the ethene hydrogenation rate 5 CONCLUSIONS...

The model we have just described is called the BCF surface diffusion model because diffusion on the crystal surface is considered to be the rate-controlling step. While this is true in vapor growth, it is often not true in solution growth where diffusion from the bulk solution to the crystal-liquid interface can often be the limiting rate step. 

Eq. (2.45) shows that at low relative supersaturations, the growth rate follows a parabolic relation with supersaturation. This is the same result as the BCF surface diffusion model. In addition, however, the growth rate, G, decreases with an increase in 6, the boundary layer thickness. This is an important result since the boundary layer thickness is directly related to hydrodynamic conditions and stirring rates. If (S — I) is much greater than Scr but much <1, Eq. (2.44) results in a relationship in which G increases linearly with supersaturation and declines with increasing S. 

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