Macropore diffusion equation

If the intracrystalline diffusional resistance for certain components could be neglected compared to resistances of other types, then the solution to the system of transport equations for a biporous pellet reduces to the macropore diffusion equation (8), in conjunction with the equilibrium relationship, eqs. (15)-(16). For an effective macropore diffusion coefficient, it holds ... 

If zeolitic diffusion is sufficiently rapid so that the sorbate concentration through any particular crystal is essentially constant and in equilibrium with the macropore fluid just outside the crystal, the rate of mass transfer will be controlled by transport through the macropores of the pellet. Transport through the macropores may be assumed to occur by a diffusional process characterized by a constant pore diffusion coefficient Z)p. The relevant form of the diffusion equation, neglecting accumulation in the fluid phase within the macropores which is generally small in comparison with accumulation within the zeolite crystals, is... 

The breakthrough curve for the case of macropore diffusion control may thus be obtained from the solution of Equations 2-4 and 13-17. 

If the formerly discussed conditions for viscous diffusion are satisfied for a cylindrical macropore, that is, a pore of diameter larger than 50 nm, as soon as the pore diameter is large relative to the mean free path, collisions between diffusing molecules will take place considerably more often than collisions between molecules and the pore surface [2,20], Under these circumstances, the pore surface effect is negligible, and, consequently, diffusion will take place by basically the same mechanism as in the bulk gas. Therefore, the pore diffusivity is equal to the molecular gaseous diffusivity (Equation 5.93). 

This technique may also be used to measure effective macropore diffusivities in biporous adsorbent pellets [13,14]. For such a system with a linear equilibrium isotherm and assuming rapid intracrystalhne diffusion, the governing diffusion equation is of the same form as for micropore control. The solution is identical to Eq. 1 except that R now refers to the particle radius and the diffusivity D is replaced by the effective diffusivity De = Dp p/(ep + (1 - p)fC). Since the equilibrium constant (K) is generally large and varies with temperature according to the van t Hoff equation (K = it is clear that a macropore-controlled system will gener-... 

This type of diffusion takes place when the pore diameter is large compared to the mean free path of the gas molecules. In this case collisions between the molecules represent the main barrier to diffusion. In contrast to micropore diffusion, macropore diffusion, which is largely determined by this mechanism, is not an activated process. The order of magnitude of the diffusion coefficient for a gas A can be estimated by means of the simple kinetic gas theory (Equation 2.1-22)... 

As a rule a pore size distribution in adsoibents can be expected. Some authors apply a mean pore size width which is valid for a cumulative volume distribution 0.5 of the pores. The macropore diffusion coefficient present in the balance equations is composed of the molecular diffusion coefficient and the Knudsen diffusion coefficient according to... 

If the time scale of diffusion in the micropore is very short compared to that in the macropore, we will have a macropore diffusion model with the characteristic length being the particle dimension. This case is called the macropore diffusion control. The model equations of this macropore diffusion case are similar to those obtained in Chapter 9 for homogeneous-type solids. The only difference is that in the case of macropore diffusion control for zeolite particles, there is no contribution of the surface diffusion. 

The macropore diffusion of nc adsorbates is described by the Maxwell-Stefan equation as learnt in Chapter 8 (Section 8.8). The micropore diffusion in crystal is activated and is described by eq. (10.6-11), and the adsorption process at the micropore mouth is assumed to be very fast compared to diffusion so that local equilibrium is established at the mouth. Adsorption and desorption of adsorbates are associated with heat release which in turn causes a rise or drop in temperature of the pellet. We shall assume that the thermal conductivity of the pellet is large such that the pellet temperature is uniform and all the heat transfer resistance is located at the thin film surrounding the pellet. How large the pellet temperature will change during the course of adsorption depends on the interplay between the rate of adsorption, the heat of adsorption and the rate of heat dissipation to the surrounding. But the rate of adsorption at any given time depends on the temperature. Thus the mass and heat balances are coupled and therefore their balance equations must be solved simultaneously for the proper description of concentration and temperature evolution. 

The appropriate form of the diffusion equation for a macropore-controlled system may be obtained from a differential mass balance on a spherical shell element ... 

If macropore diffusion within the particle is controlling, the equations are... 

The analysis of macropore diffusion in binary or multicomponent systenis presents no particular problems since the transport properties of one compos nent are not directly affected by changes ini the concentration of the bther components. In an adsorbed phase the situation is more complex since ih addition to any possible direct effect on thei mobility, the driving force for each component (chemical potential gradient is modified, through the multi-component equilibrium isotherm, by the coiicentration levels of all components in the system. The diffusion equations for each component are therefore directly coupled through the equilibrium relationship. Because of the complexity of the problem, diffusion in a mixed adscjrbed phase has been studied tjs only a limited extent. 

Multiscale models can be based only on continuum mathematical descriptions, e.g., as in the zeolite-based chemical reactor illustrated in Fig. 5, where the conservation equation describing the reactant transport along the reactor has a source term being calculated from the boundary condition of a macropore diffusion model with a source term calculated in turn from the boundary condition of a micropore diffusion model with a reaction kinetics source term [44]. 

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. 

The condition that the electrochemical processes occur largely in the region of the meniscus is only met if the thin film of electrolyte is absent or if the thickness is very small. The simple-pore model [64] is an example of the first case. The meniscus is assumed to form at the intersection of micropores with macropores. While the micropores are filled with electrolyte up to the intersection, the macropores are filled with gas. The meniscus may be treated as flat in a first approximation. The walls of micropores are the seat of the electrode reaction. The simple-pore model was suggested [64] as applying to non-wetted systems like the Teflon-bonded platinum black electrodes. The limiting current due to the diffusion of species into a micropore was derived [64] as the steady-state solution of the two-dimensional diffusion equation in cylindrical coordinates. The summation of the currents of the individual pores leads to ... 

A model of a composite zeolite pellet must thus be represented by a combination of coupled equations for intracrystalline diffusion and macropore diffusion. The diffusion of adsorbate within crystals was discussed in the previous section and intracrystalline diffusion is given by equation (4.17). Macropore diffusion for a spherical pellet of radius Rp, macropore diffusivity Dp and porosity Cp is described by... 

The first thing to notice about these results is that the influence of the micropores reduces the effective diffusion coefficient below the value of the bulk diffusion coefficient for the macropore system. This is also clear in general from the forms of equations (10.44) and (10.48). As increases from zero, corresponding to the introduction of micropores, the variance of the response pulse Increases, and this corresponds to a reduction in the effective diffusion coefficient. The second important point is that the influence of the micropores on the results is quite small-Indeed it seems unlikely that measurements of this type will be able to realize their promise to provide information about diffusion in dead-end pores. 

In a particle having a bidispersed pore structure comprising spherical adsorptive subparticles of radius forming a macroporous aggregate, separate flux equations can be written for the macroporous network in terms of Eq. (16-64) and for the subparticles themselves in terms of Eq. (16-70) if solid diffusion occurs. 

Table 16-4 shows the IUPAC classification of pores by size. Micropores are small enough that a molecule is attracted to both of the opposing walls forming the pore. The potential energy functions for these walls superimpose to create a deep well, and strong adsorption results. Hysteresis is generally not observed. (However, water vapor adsorbed in the micropores of activated carbon shows a large hysteresis loop, and the desorption branch is sometimes used with the Kelvin equation to determine the pore size distribution.) Capillary condensation occurs in mesopores and a hysteresis loop is typically found. Macropores form important paths for molecules to diffuse into a par-... 

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. 

To quantify such transport, the advection-dispersion equation, which requires a narrow pore-size distribution, often is used in a modified framework. Van Genuchten and Wierenga (1976) discuss a conceptualization of preferential solute transport throngh mobile and immobile regions. In this framework, contaminants advance mostly through macropores containing mobile water and diffuse into and out of relatively immobile water resident in micropores. The mobile-immobile model involves two coupled equations (in one-dimensional form) ... 

Thus, two mass balance equations are written in the lumped pore diffusion model for the two different fractions of the mobile phase, the one that percolates through the network of macropores between the particles of the packing material and the one that is stagnant inside the pores of the particles ... 

For gel-type exchange resins and for macropore resins with a low degree of cross-linking, the following equation is proposed for the solid diffusion coefficient (Helfferich, 1962) ... 

The computer-reconstructed catalyst is represented by a discrete volume phase function in the form of 3D matrix containing information about the phase in each volume element. Another 3D matrix defines the distribution of active catalytic sites. Macroporosity, sizes of supporting articles and the correlation function describing the macropore size distribution are evaluated from the SEM images of porous catalyst (Koci et al., 2006 Kosek et al., 2005). Spatially 3D reaction-diffusion system with low concentrations of reactants and products can be described by mass balances in the form of the following partial differential equations (Koci et al., 2006, 2007a). For gaseous components ... 

Prediction of the breakthrough performance of molecular sieve adsorption columns requires solution of the appropriate mass-transfer rate equation with boundary conditions imposed by the differential fluid phase mass balance. For systems which obey a Langmuir isotherm and for which the controlling resistance to mass transfer is macropore or zeolitic diffusion, the set of nonlinear equations must be solved numerically. Solutions have been obtained for saturation and regeneration of molecular sieve adsorption columns. Predicted breakthrough curves are compared with experimental data for sorption of ethane and ethylene on type A zeolite, and the model satisfactorily describes column performance. Under comparable conditions, column regeneration is slower than saturation. This is a consequence of non-linearities of the system and does not imply any difference in intrinsic rate constants. 

Incidentally, these features cannot be accounted for by assuming different values of macropore radius or tortosity factor in the predictive equations. Even with the assumptions of negligible Knudsen resistance (rpore- ) and no tortuosity (rw = D, the predicted macropore resistances (excluding surface effects) would be lowered by only forty percent, which is still insufficient to account for the low LUB values, at least in the 5A and 13X systems. There appears, therefore, to be a fairly strong case for the presence of a surface diffusion effect in these systems, with the possibility of such an effect in the CO2/ air/4A system as well. 

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. 

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... 

The viscous diffusion mechanism is also valid for transport process in the liquid phase. Then, if we have a liquid filtration process through a porous (i.e., macroporous or mesoporous) membrane, the following form of the Carman-Kozeny equation can be used [9]... 

In Equation 8 the first two terms (x = 1 - E ) give the conversion for the unreacted core model. The remaining terms in this equation give the conversion in the inner core for the two stage model. The data at 1000 °C and 1040 °C showed that fractional conversions of the samples are approximately the same as the values predicted by the first two terms of Equation 8. This shows that at high temperatures unreacted core model becomes the controlling mechanism due to the increased concentration of CO2 in the pores and diffusion limitations. Experiments carried out at different temperatures also showed that the ratio of macropore... 

The characteristics of pore structure in polymers is a key parameter in the study of diffusion in polymers. Pore sizes ranging from 0.1 to 1.0 pm (macroporous) are much larger than the pore sizes of diffusing solute molecules, and thus the diffusant molecules do not face a significant hurdle to diffuse through polymers comprising the solvent-filled pores. Thus, a minor modification of the values determined by the hydrodynamic theory or its empirical equations can be made to take into account the fraction of void volume in polymers (i.e., porosity, e), the crookedness of pores (i.e., tortuosity, x), and the affinity of solutes to polymers (i.e., partition coefficient, K). The effective diffusion coefficient, De, in the solvent-filled polymer pores is expressed by ... 

The principles and basic equations of continuous models have already been introduced in Section 6.2.2. These are based on the well known conservation laws for mass and energy. The diffusion inside the pores is usually described in these models by the Fickian laws or by the theory of multicomponent diffusion (Stefan-Maxwell). However, these approaches basically apply to the mass transport inside the macropores, where the necessary assumption of a continuous fluid phase essentially holds. In contrast, in the microporous case, where the pore size is close to the range of molecular dimensions, only a few molecules will be present within the cross-section of a pore, a fact which poses some doubt on whether the assumption of a continuous phase will be valid. 

Models with varying degrees of complexity have been employed to analyze the experimental results by a variety of techniques. The most comprehensive models include terms to account for axial dispersion in the packed bed, external mass transfer, intraparticle diffusion in both macropore and micropore regions of the pellet and a finite rate of adsorption. Of the several methods of analysis, the most popular ones are based on the moments of the response curve. The first moment of the chromatogram is defined by Equation 5.25 in which the concentration now is taken at the outlet of the column. The second central moment is calculated from equation... 

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