Bidispersed macropore diffusion

The described treatment of mass transport presumes a simple, relatively uniform (monomodal) pore size distribution. As previously mentioned, many catalyst particles are formed by tableting or extruding finely powdered microporous materials and have a bidisperse porous structure. Mass transport in such catalysts is usually described in terms of two coefficients, a effective macropore diffusivity and an effective micropore diffusivity. 

The FR measurements have been carried out for several systems using bidispersed structured sorbents [64,76,77]. All the spectra, however, indicate that either micropore diffusion or macropore diffusion, with or without a surface resistance, was the rate-controlUng step for these systems. 

Chapter 10 deals with zeolite type particle, where the particle is usually in bidisperse form, that is small pores (channels inside zeolite crystal) are grouped together within a crystal, and the intercrystal void would form a network of larger pores. In other words, there are two diffusion processes in the particle, namely micropore diffusion and macropore diffusion. In the micropore network, only one phase is possible the adsorbed phase. Depending on the relative time scales between these two diffusion processes, a system can be either controlled by the macropore diffusion, or by micropore diffusion, or by a combination of both. Isothermal as well as nonisothermal conditions will be addressed in this chapter. 

Model assumptions include the following (i) The adsorbent has an uniform bidisperse pore structure, (ii) The pellets have spherical geometry, (iii) The reactor behaves like a CSTR, (iv) Ideal pulse input, (v) Macropore diffusion is Fickian, (vi) No external film resistance, (vii) Linear equilibrium, (viii) First order irreversible reaction, (ix) The crystals are small (<0.2 pm) agglomerates and diffusion resistance in these can be neglected. The following differential mass balances for species i result ... 

In bidisperse porous adsorbents such as zeolite pellets there are two diffusion mechanisms the macropore diffusion with time constant Rp /Dp and the micropore diffusion with time constant rc /Dc. Bidisperse porous models for ZLC desorption curves have been recently developed by Brandani [28] and Silva and Rodrigues [29]. In bidisperse porous adsorbents, it is important to carry out experiments in pellets with different sizes but with the same crystal size (different Rp, same rc) or pellets with the same size but with different crystals (same Rp, different rc). If macropore diffusion is controlling, time constants for diffusion should depend directly on pellet size and should be insensitive to crystal size changes. If micropore diffusion controls the reverse is true. The influence of temperature is also important when macropore diffusion is dominant the apparent time constant of diffusion defined by Rp2(H-K)/Dp is temperature dependent in the same order of K (directly related to the heat of adsorption) which is determined independently from the isotherm. The type of purge gas is... 

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. 

The available transport models are not reliable enough for porous material with a complex pore structure and broad pore size distribution. As a result the values of the model par ameters may depend on the operating conditions. Many authors believe that the value of the effective diffusivity D, as determined in a Wicke-Kallenbach steady-state experiment, need not be equal to the value which characterizes the diffusive flux under reaction conditions. It is generally assumed that transient experiments provide more relevant data. One of the arguments is that dead-end pores, which do not influence steady state transport but which contribute under reaction conditions, are accounted for in dynamic experiments. Experimental data confirming or rejecting this opinion are scarce and contradictory [2]. Nevertheless, transient experiments provide important supplementary information and they are definitely required for bidisperse porous material where diffusion in micro- and macropores is described separately with different effective diffusivities. 

The method can be applied to investigate the bidisperse pore structures, which consist of small microporous particles formed into macroporous pellets with a clay binder. In such a structure there are three distinct resistances to mass transfer, associated with diffusion through the external fluid film, the pellet macropores, and the micropores. Haynes and Sarma [24] developed a suitable mathematical model for such a system. 

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... 

Conventional procedures can be used to find the reaction effectiveness factors of the mono- and bidisperse pellets. If diffusion limitations in the macropores are significant the average reaction rates are... 

The porous structure of most catalysts is polydisperse. Therefore, capillary condensate fills only part of the pore-space — mostly small pores. The bidisperse globular structure (Figure 23.1) is convenient to consider as a model for rough estimations of the influence of external mass transfer and intraparticle diffusion on the total reaction rate. Such an analysis was made by Ostrovskii and Bukhavtsova [8]. According to this model, the only pores inside globules (micropores) will fill with liquid, and space between globules (macropores) fill with gas. Then the total porosity can be written as... 

The small and uniform fiber diameter has a direct and important implication on the mass-transfer rates for both adsorption and desorption. The uptake and desorption rates are directly related to the diffusion time constant, D/R (where D is diffusivity and R is radius). The value of R /D is approximately the time for accomplishing 99% diffusion upon a step change for a spherical particle with clean initial condition. The value decreases rapidly with decreasing R. ACF has only micropores of a diffusion length less than R (i.e., fiber radius), whereas GAC has both micropores and meso/macropores. Hence two diffusion time constants are involved. The interplay of these two diffusion time constants has been delineated from the analysis of diffusion in a bidisperse porous structme by Ruck-enstein et al. (1971). In all commercial sorbents, the resistance by the macropoie (in pellet) diffusion is as important as the micropore diffusion because of the much larger diffusion distance in the macropore. 

Adsorbents can have bidisperse pore structures when they are produced by combining primary particles which themselves are porous (Fig 6.5). The resulting particle has two pore systems micropores within the small particles and macropores corresponding to the space between primary particles Generally, inadequate diffusivities may result if diffusion data in bidisperse adsorbents are not analyzed by modek which account for both micro- and macropores. 

The random pore model of Wakao and Smith (1962) for a bidisperse pore structure may also be applied in order to estimate De. It was supposed that the porous solid is composed of stacked layers of microporous particles with voids between the particles forming a macroporous network. The magnitude of the micropores and macropores becomes evident from an experimental pore size distribution analysis. If Dm and Dp are the macropore and micropore diffusivities calculated from equations (4.9) and (4.10), respectively, the random pore model gives the effective diffusivity as... 

An alternative model for diffusion in porous media has been proposed by Wakao and Smith [39], who noted that many solids of interest have a bidisperse pore structure. That is, the solids consist of compacts of solid particles that are themselves porous. The solid therefore contains micropores (pores within the particles) and macropores (interstices between particles) and diffusion occurs through macropores, through micropores, and through micropores and macropores in series. For diffusion at constant pressure, we have... 

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