Macropore diffusivities

Fig. 6. Concentration profiles through an idealized biporous adsorbent particle showing some of the possible regimes. (1) + (a) rapid mass transfer, equihbrium throughout particle (1) + (b) micropore diffusion control with no significant macropore or external resistance (1) + (c) controlling resistance at the surface of the microparticles (2) + (a) macropore diffusion control with some external resistance and no resistance within the microparticle (2) + (b) all three resistances (micropore, macropore, and film) significant (2) + (c) diffusional resistance within the macroparticle and resistance at the surface of the...

Macropore Diffusion. Transport in a macropore can occur by several different mechanisms, the most important of which ate bulk molecular... 

Also shown are the corresponding curves calculated for the same system assuming a diffusion model in place of the linear rate expression. For intracrystalline diffusion k = 15Dq/v, whereas for macropore diffusion k = 15e /R ) Cq/q ), in accordance with the Glueckauf approximation (21). 

Pore diffusion in fluid-filled pores. These pores are sufficiently large that the adsorbing molecule escapes the force field of the adsorbent surface. Thus, this process is often referred to as macropore diffusion. The driving force for such a diffusion process can be approximated by the gradient in mole fraction or, if the molar concentration is constant, by the gradient in concentration of the diffusing species within the pores. 

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. 

Micropore mass transfer resistance of zeoUte crystals is quantified in units of time by r /Dc, where is the crystal radius and Dc is the intracrystalline diffusivity. In addition to micropore resistance, zeolitic catalysts may offer another type of resistance to mass transfer, that is resistance related to transport through the surface barrier at the outer layer of the zeoHte crystal. Finally, there is at least one additional resistance due to mass transfer, this time in mesopores and macropores Rp/Dp. Here Rp is the radius of the catalyst pellet and Dp is the effective mesopore and macropore diffusivity in the catalyst pellet [18]. 

The rate of n-paraffin desorption generally controls the overall production rate (18, 19). The diffusion of n-paraffins in commercial 5A molecular sieves is reported to be controlled by either micropore diffusion or macropore diffusion, or both, depending on the molecular sieve crytal size and macropore size distribution of the adsorbent (20). A 5A molecular sieve adsorbent with smaller crystal size and optimum macropore size distribution would have a faster adsorption-desorption rate and, therefore, a higher effective capacity. 

Purification of Air Prior to Liquefaction. Separation of air by cryogenic fractionation processes requires removal of water vapor and carbon dioxide to avoid heat exchanger freeze-up. Many plants today are using a 13X (Na-X) molecular sieve adsorbent to remove both water vapor and carbon dioxide from air in one adsorption step. Since there is no necessity for size selective adsorption, 13X molecular sieves are generally preferred over type A molecular sieves. The 13X molecular sieves have not only higher adsorptive capacities but also faster rates of C02 adsorption than type A molecular sieves. The rate of C02 adsorption in a commercial 13X molecular sieve seems to be controlled by macropore diffusion 37). The optimum operating temperature for C02 removal by 13X molecular sieve is reported as 160-190°K 38). 

There are three distinct mass-transfer resistances (1) the external resistance of the fluid film surrounding the pellet, (2) the diffusional resistance of the macropores of the pellet, and (3) the diffusional resistance of the zeolite crystals. The external mass-transfer resistance may be estimated from well-established correlations (4, 5) and is generally negligible for molecular sieve adsorbers so that, under practical operating conditions, the rate of mass transfer is controlled by either macropore diffusion or zeolitic diffusion. In the present analysis we consider only systems in which one or other of these resistances is dominant. If both resistances are of comparable importance the analysis becomes more difficult. 

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

Figure 2. Theoretical breakthrough curves for macropore diffusion control at X = 1.0 saturation (----------------------), regeneration (------)...

Figure S. Comparison of asymptotic constant pattern saturation breakthrough curves for X = 0.1 5 (1) zeolitic diffusion control with Dz independent of concentration, (2) zeolitic diffusion control, (8) macropore diffusion control...

Also shown in Table I are the estimated values of the time constant for macropore diffusion based on estimated macropore diffusivities. From the ratio of the time constants for macropore diffusion and zeolitic diffusion, it is clear that the assumption of zeolitic diffusion control is a valid approximation for these systems. 

D limiting zeolitic diffusivity at zero sorbate concentration Dp macropore diffusivity (based on pore sectional area) rrt ratio of bed void space to zeolite crystal volume = e/(l — e ) q local sorbate concentration in a zeolite crystal... 

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. 

The energies of activation for Z)a and A are given in Table III. The low activation energy for macropore diffusion may be explained as follows. The magnitude of the macropores in the pelletized zeolites can usually be assumed about 1 /uneter. From the kinetic theory of gases, the mean free... 

Results of these calculations for H mordenite are presented in Table IV. The macropore diffusion plays a role far from negligible even at high temperature and in some instances (e.g., low temperature and large particles) is the major contribution to the total mass-transfer resistance. No single step controls the overall mass-transfer process as no resistance has a relatively large enough contribution to dominate the process. In every... 

Macropore diffusion Diffusion in macropores —pores that are large compared with the molecular diameter. Several different mechanisms contribute to macropore diffusion, notably ordinary molecular diffusion in larger macropores at higher pressures or in liquids and Knudsen diffusion in smaller macropores at low pressures. Also referred to as intraparticle diffusion. 

FIGURE 4 Schematic diagram of a biporous adsorbent pellet showing the three resistances to mass transfer (external fluid film, macropore diffusion, and micropore diffusion). R9 pellet radius rc crystal radius. 

Since the transport processes within macropores are fairly well understood, it is generally possible to make a reasonable a priori estimate of the effective macropore diffusivity, at least within a factor of 2. 

The customary way of measuring intraparticle macropore diffusivities is the Wicke-Kallenbach method, which depends on measuring the flux through a pellet under steady-state conditions when the two faces are maintained at... 

Alternatively one can in principle derive both micropore and macropore diffusivities from measurements of the transient uptake rate for a particle (or assemblage of crystals) subjected to a step change in ambient sorbate pressure or concentration. The main problem with this approach is that the overall uptake rate may be controlled by several different processes, including both heat and extraparticle mass transfer as well as intraparticle or intracrystalline diffusion. The intrusion of such rate processes is not always obvious from a cursory examination of the experimental data, and the literature of the subject is replete with incorrect diffusivities (usually erroneously low values) obtained as a result of intrusion of such extraneous effects. Nevertheless, provided that intraparticle diffusion is sufficiently slow, the method offers a useful practical alternative to the Wicke-Kallen bach method. 

The foregoing discussion refers solely to intraparticle diffusivity (micropore diffusion) as distinct from interparticle effects (macropore diffusion). Since a practical zeolite catalyst will consist of composite particles, each containing a large number of individual zeolite crystals, it is important to make a clear distinction between these two types of diffusion. In some cases macropore diffusion may be important in determining the overall reaction kinetics but will obviously not introduce or affect shape selectivity in any way. 

The elimination or estimation of the axial dispersion contribution presents a more difficult problem. Established correlations for the axial dispersion coefficient are notoriously unreliable for small particles at low Reynolds number(17,18) and it has recently been shown that dispersion in a column packed with porous particles may be much greater than for inert non-porous particles under similar hydrodynamic conditions(19>20). one method which has proved useful is to make measurements over a range of velocities and plot (cj2/2y ) (L/v) vs l/v2. It follows from eqn. 6 that in the low Reynolds number region where Dj. is essentially constant, such a plot should be linear with slope Dj, and intercept equal to the mass transfer resistance term. Representative data for several systems are shown plotted in this way in figure 2(21). CF4 and iC io molecules are too large to penetrate the 4A zeolite and the intercepts correspond only to the external film and macropore diffusion resistance which varies little with temperature. 

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. 

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