Diffusion macro

Radical polymerizations of macromonomers are greatly influenced by the diffusion control effect [44]. Segmental diffusivity and translational diffusivity of the growing chains of macromonomers are strongly affected by the feed concentration and the molecular weight of the macromonomers. Furthermore, there is little difference in the degree of polymerization between macro-... 

The efficacy of polymers when used to protect metals from corrosive environments is influenced by their efficiency as barrier materials. When applied to metals by some techniques, such as fluidised bed coating, there is always the danger of macro-diffusion through pinholes which are gross imperfections in the surface and which do not have to be visible to be very much greater than the dimension of penetrating molecules. 

Decreasing with chain transfer to monomer or transfer agents. (This process is not as limited by diffusion as macro-radical-macroradical termination reactions.)... 

The term numerical diffusion describes the effect of artificial diffusive fluxes which are induced by discretization errors. This effect becomes visible when the transport of quantities with small diffusivities [with the exact meaning of small yet to be specified in Eq. (42)] is considered. In macroscopic systems such small diffusivities are rarely found, at least when being looked at from a phenomenological point of view. The reason for the reduced importance of numerical diffusion in many macroscopic systems lies in the turbulent nature of most macro flows. The turbulent velocity fluctuations induce an effective diffusivity of comparatively large magnitude which includes transport effects due to turbulent eddies [1]. The effective diffusivity often dominates the numerical diffusivity. In contrast, micro flows are often laminar, and especially for liquid flows numerical diffusion can become the major effect limiting the accuracy of the model predictions. 

The convective diffusion equations presented above have been used to model tablet dissolution in flowing fluids and the penetration of targeted macro-molecular drugs into solid tumors [5], In comparison with the nonequilibrium thermodynamics approach described below, the convective diffusion equations have the advantage of theoretical rigor. However, their mathematical complexity dictates a numerical solution in all but the simplest cases. 

Diffusion of electroactive species to the surface of conventional disk (macro-) electrodes is mainly planar. When the electrode diameter is decreased the edge effects of hemi-spherical diffusion become significant. In 1964 Lingane derived the corrective term bearing in mind the edge effects for the Cotrell equation [129, 130], confirmed later on analytically and by numerical calculation [131,132], In the case of ultramicroelectrodes this term becomes dominant, which makes steady-state current proportional to the electrode radius [133-135], Since capacitive and other diffusion-unrelated currents are proportional to the square of electrode radius, the signal-to-noise ratio is increased as the electrode radius is decreased. 

Let us consider the uptake of a given species, either a nutrient or a pollutant heavy metal or an organic (macro)molecule, etc., which will be referred to as M. M is present in the bulk of the medium at a concentration, c, ar d we assume that the only relevant mode of transport from the medium to the organism s surface is diffusion. The internalisation sites are taken to be located on the spherical surface of the microorganism or in a semi-spherical surface of a specialised region of the organism with radius ro (see Figure 1). Thus, diffusion prescribes ... 

Macro pores on p -Sidue to hole diffusion in space charge layer39... 

Equation 9.15, when solved for the case of macro-pore diffusion gives us, in the low loading Hmit, the famihar relationship that mass uptake is proportional to the square root of time. The same relationship can be derived for micro-pore diffusion as well. The solution to this equation can be used with the appropriate particle sizes to estimate diffusivity from uptake rate measurements. 

Equation (9.15) was written for macro-pore diffusion. Recognize that the diffusion of sorbates in the zeoHte crystals has a similar or even identical form. The substitution of an appropriate diffusion model can be made at either the macropore, the micro-pore or at both scales. The analytical solutions that can be derived can become so complex that they yield Httle understanding of the underlying phenomena. In a seminal work that sought to bridge the gap between tractabUity and clarity, the work of Haynes and Sarma [10] stands out They took the approach of formulating the equations of continuity for the column, the macro-pores of the sorbent and the specific sorption sites in the sorbent. Each formulation was a pde with its appropriate initial and boundary conditions. They used the method of moments to derive the contributions of the three distinct mass transfer mechanisms to the overall mass transfer coefficient. The method of moments employs the solutions to all relevant pde s by use of a Laplace transform. While the solutions in Laplace domain are actually easy to obtain, those same solutions cannot be readily inverted to obtain a complete description of the system. The moments of the solutions in the Laplace domain can however be derived with relative ease. 

Diffusion in the macro-pores of a formed parhcle is generally speaking a very important mechanism. If we speak in terms of resistances to mass transfer macropore resistance is often the largest of the resistances to mass transfer. For transport in the macro-pores we must introduce two parameters that influence the transport. 

Reid, Sherwood and Prausnitz [11] provide a wide variety of models for calculation of molecular diffusion. Dr is the Knudsen diffusion coefficient. It has been given in several articles as 9700r(T/MW). Once we have both diffusion coefficients we can obtain an expression for the macro-pore diffusion coefficient 1/D = 1/Dk -i-1/Dm- We next obtain the pore diffusivity by inclusion of the tortuosity Dp = D/t, and finally the local molar flux J in the macro-pores is described by the famiUar relationship J = —e D dcjdz. Thus flux in the macro-pores of the adsorbent product is related to the term CpD/r. This last quantity may be thought of as the effective macro-pore diffusivity. The resistance to mass transfer that develops due to macropore diffusion has a length dependence of R]. 

It may not be obvious but driving selectivity to a high value is best done by driving N2 adsorption to some acceptably high value and then driving O2 to a minimum. This dramatically changes the volume of gas that must pass in and out of the macro-pore structure of the adsorbent In aU PSA separations it is the macropore diffusion that is the dominant resistance to mass transfer. 

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