Macro-pore diffusion

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. 

Diffusivity, m /s / Molar flux inside of the macro-pores, kmol/jm s)... 

In order to utilize the absorption properties or the synthetic zeolite crystals in processes, the commercial materials arc prepared as pelleted aggregates combining a high percentage of the crystalline zeolite with an inert binder. The formation of these aggregates introduces macro pores in the pellet which may result in some capillary condensation at high adsorhate concentrations. In commercial materials, the inacropores contribute diffusion paths. However, the main pan of the adsorption capacity is contained in the voids within the crystals. 

We now establish the coupled clay cluster/macro-pore model at the meso-scale. For the sake of simplicity we adopt a particular form of mesostructure wherein the clay clusters are isolated from each other by the fissure (macropore) system.. Denote Vj. C, D. Jj the velocity, concentration and diffusion coefficient and the overall flux of species (NaCl), the governing equations in Q f reduce to... 

Fio. 16. Effective diffusivity of the examined porous sample G1 calculated for (i) Fick s diffusion both in macro- and nano-pores, and (ii) Fick s diffusion in macro-pores only. The experimental value of ij/ = Dcii/D = 0.199 was determined in Grahams diffusion cell. Only grains larger than 10 pm were used in the reconstruction of macro-porous media (from Salejova et al., 2004). 

The characteristic length of the washcoat section to be simulated is 10 pm thus, we may consider constant temperature profile on this scale. Since the volume diffusion in the macro-pores is much faster than the Knudsen diffusion in the meso-pores of the y-Al203 particles, we may further assume that the CO and 02 gas concentrations in the macro-pores are constant within the simulated washcoat section. For the surface-deposited components CO and O, a zero diffusivity is used, i.e., Df1 — 0. For gaseous CO and 02, the effective diffusivi-ties are based on the Knudsen diffusivity in the meso-pores (with diameter T0nm) of the y-Al203. 

Macro pores on p -Si due to a rate limiting process by diffusion of holes in the space charge layer Lehmann Ronebeck ... 

The internal mass transfer is modeled with Fick s diffusion inside the (macro) pores (Eq. 6.82) as well as surface or micropore diffusion in the solid phase (Eq. 6.83). Note that Eqs. 6.82 and 6.83 no longer include the number of particles (Eq. 6.20) and therefore represent the balance in one particle. 

The equations and plots presented in the foregoing sections largely pertain to the diffusion of a single component followed by reaction. There are several other situations of industrial importance on which considerable information is available. They include biomolecular reactions in which the diffusion-reaction problem must be extended to two molecular species, reactions in the liquid phase, reactions in zeolites, reactions in immobilized catalysts, and extension to complex reactions (see Aris, 1975 Doraiswamy, 2001). Several factors influence the effectiveness factor, such as pore shape and constriction, particle size distribution, micro-macro pore structure, flow regime (bulk or Knudsen), transverse diffusion, gross external surface area of catalyst (as distinct from the total pore area), and volume change upon reaction. Table 11.8 lists the major effects of all these situations and factors. 

In our case these particles possess diameters of 100-200 nm and between them a system of permanent pores (macro pores) is formed which have diameters of 20-50 nm (see Figure 4) It can be assumed that the diffusion within macropores is completely unhindered. At higher magnification (right side of Figure 4 ) the micro-cavities are shown. The black particles represent the templates which are on the order of 1-2 nm and can be situated near the inner surface or inside the denser nuclei. Our polymers were optimized along several lines. 

The diffusion process in porous media is strongly influenced by the pore size distribution. Based on the size the pores are classified as micro pores d < 2nm), macro pores (rfp > 50... 

In a previous paper we have done optimization calculations of pore structure with a micro-macro-pore model [3]. In this work, we have taken more realistic random three-dimensional network models for our optimization calculations [4]. We have investigated the influence of connectivity, diffusion coefficient, outer dimension of pellet and operation time on optimal pore structure. Numerical methods for solving network problems will be discussed. 

The duration and intensity of the transient crystallization ptressure dep>end on three factors M (1) the rate of supply of solute (2) the rate of growth of crystal (3) the rate of diffusion of solute to macro-pores. High evaporation can result in high sup>ersaturation, and increase the growth of crystal and result in a high transient stress IW [20] leading to severe damage by salt crystallization. 

Miscellaneous effects A number of factors can influence the effectiveness factor, some of which are particle size distribution in a mixture of particles/pellets, change in volume upon reaction, pore shape and constriction (such as ink-bottle-type pores), radial and length dispersion of pores, micro-macro pore structure, flow regime (such as bulk or Knudsen), surface diffusion, nonuniform environment around a pellet, dilution of catalyst bed or pellet, distribution of catalyst... 

The random pore model, or macro- micro-pore model, of Wakao and Smith [1962, 1964] is intended for application to pellets manufactured by compression of small particles. The void fraction and pore radius distributions are each replaced by two averaged values 8m, I m for the macro and for the micro distribution (often a pore radius of -100 A is used as the dividing point between macro and micro). The particles which contain the micro-pores are randomly positioned in the pellet space. The interstices are the macro-pores of the pellet (see Fig. 3.5.2.1-1). The diffusion flux consists of three parallel contributions the first through the macro-pores, the second through the micro-pores and the third through interconnected macro-micro-pores in which the dominant resistance lies in the latter. The contributions to the diffusivity are added up to yield ... 

However, a pelletized or extruded catalyst prepared by compacting fine powder typically exhibits a bimodal (macro-micro) pore-size distribution, in which case the mean pore radius is an inappropriate representation of the micropores. There are several analytical approaches and models in the literature which represent pelletized catalysts, but they involve complicated diffusion equations and may require the knowledge of diffusion coefficients and void fractions for both micro- and macro-pores [31]. An easier and more pragmatic approach is to consider the dimensional properties of the fine particles constituting the pellet and use the average pore size of only the micropore system because diffusional resistances will be significantly higher in the micropores than in the macropores. This conservative approach will also tend to underestimate Detr values and provide an upper limit for the W-P criterion. 

In effect, then, each of these clumps is like a virtual catalyst with a bimiodal pore structure that is, a few large pores interconnecting the particles plus the small pore network inherent in the catalyst particle. The effective diffusivity of such a clump will be a combination of the effective diffusivity of the porous particle and these large "macro-pores . But the effective size of the catalyst particle is no longer the size of the particle but instead is the size of the clump. 

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