Macropore model, diffusion coefficients

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]. 

A new mathematical model based on moment techniques to describe micro- and macropore diffusion is used to study the mass-transfer resistances of Ci to C4 saturated hydrocarbons in H and Na mordenites between 127° C and 272° C. The intracrystalline diffusion coefficient decreases as the number of carbon atoms increases while the energy of activation increases with the number of carbons. The contribution from individual mass-transfer resistances to the overall mass-transfer processes is estimated. 

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... 

A single effective diffusion coefficient cannot adequately characterize the mass transfer within a bidisperse-structured catalyst when the influence of the two individual systems is equally important. In a realistic model the separate identity of the macropore and micropore structures must be maintained, and the diffusion must be described in... 

In Equation 6.23 the transport in the pore fluid is modeled as free diffusion in the macropores and mesopores, but the diffusion coefficient Dpo. i is usually smaller than the molecular diffusivity characterizing transport in the liquid mobile phase due to the random orientations and variations in the diameter of the pores (tortuosity) (Section 6.5.8). 

The balances of mass of the chemical species i and the terms for the adsorption kinetics (mass transfer, pore diffusion) are listed in Table 9.5-1 for the three systems with Cj as the concentration in the fluid phase and Xj as the mass loading of the adsorbent. J3 denotes the mass transfer coefficient of a pellet and sj, is its internal porosity. The tortuosity factor will be explained later. The derivation of equations describing instationary diffusion in spheres has already been presented in Sect. 4.3.3. With respect to diffusion in macropores it is important to consider that diffusion can take place in the fluid as well as in the adsorbate phase. In Table 9.5-1 special initial and boimdaty conditions valid for a completely unloaded bed (adsorption) or totally loaded bed (desorption) are given. In this section only the model valid for a thin layer in a fixed bed with the thickness dz and the volmne / dz will be derived, see Fig. 9.5-2. 

The first- and second-order FRFs for the general model (micropore + macropore diffusion -F surface barrier -F film resistance), defining the relation between (q) and p, were derived analytically for the case of constant diffusion coefficients and slab geometry. The following expressions were obtained [61] ... 

Modeling of transient data. The model takes mass transport into account at two different levels Knudsen flow in the interstitial voids of the bed and in the macropores of the matrix, lumped into one diffusion coefficient and an activated diffusion process inside the micropores of the zeolite. The reversible sorption between the gas phase and zeolite sorbate... 

An industrial DMTO fluidized bed catalyst pellet is basically composed of SAPO-34 zeofite particles and catalyst support (or matrix). The pores of zeolite particles and matrix are interconnected as a complex network. The pores inside zeofite particles are typically micropores (less than 2 nm) and the matrix normally has either mesopores (2-50 nm) or macropores (>50 nm), or both (Krishna and Wesselingh, 1997). The bulk diffusion coefficients in the meso- and macropores might be several orders of magnitude larger than surface diffusion coefficients in the micropores. Kortunov et al. (2005) found that the diffusion in macro- and mesopores also plays a crucial part in the transport in catalyst pellets. Therefore, other than a model for SAPO-34 zeofite particles, a modeling approach for diffusion and reaction in MTO catalyst pellets, which are composed of SAPO-34 zeofite particles and catalyst support, is needed. 

Capacitive charging leading to desalination has been discussed in the literature, where both the GCS and mD model are used and both a single monovalent salt solution is considered, as well as mixtures of salts including ions of multiple valencies. Here, we will assume a priori that we only have a monovalent salt with both ions having the same diffusion coefficient, D. Note that this is an effective diffusion coefficient for transport in the macropores, which contains a contribution of pore tortuosity. We will not consider transport outside the electrode (thus, the Biot number is set to infinity), but we will consider the presence of an ion-selective membrane... 

The reactor is divided into three zones two inert zones of quartz beads between which the catalyst is placed. The diffusion in all three zones is described by Knudsen diffusion. In the carbon zone, the flux into the carbon particles is included in the model. Actually, the diffusion into the meso- and macropores is relatively fast and can therefore be lumped into the Knudsen diffusion coefficient according to the following equation [5]... 

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. 

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. 

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. 

Mass transfer through the external fluid film, and macropore, micropore and surface diffusion may all need to be accounted for within the particles in order to represent the mechanisms by which components arrive at and leave adsorption sites. In many cases identification of the rate controlling mechanism(s) allows for simplification of the model. To complicate matters, however, the external film coefficient and the intraparticle diffusivities may each depend on composition, temperature and pressure. In addition the external film coefficient is dependent on the local fluid velocity which may change with position and time in the adsorption bed. 

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