Macropore transport through

For a macroporous sorbent the situation is slightly more complex. A differential balance on a shell element, assuming diffusivity transport through the macropores with rapid adsorption at the surface (or in the micropores), yields... 

The major design concept of polymer monoliths for separation media is the realization of the hierarchical porous structure of mesopores (2-50 nm in diameter) and macropores (larger than 50 nm in diameter). The mesopores provide retentive sites and macropores flow-through channels for effective mobile-phase transport and solute transfer between the mobile phase and the stationary phase. Preparation methods of such monolithic polymers with bimodal pore sizes were disclosed in a US patent (Frechet and Svec, 1994). The two modes of pore-size distribution were characterized with the smaller sized pores ranging less than 200 nm and the larger sized pores greater than 600 nm. In the case of silica monoliths, the concept of hierarchy of pore structures is more clearly realized in the preparation by sol-gel processes followed by mesopore formation (Minakuchi et al., 1996). 

Compared with microporous and mesoporous materials, the larger, interconnected voids in macroporous materials potentially provide easier molecule transportation through the materials. This is of particular interest for the transport of large biomolecules (e.g., proteins and cells). The pore sizes in macroporous materials are usually from tens to hundreds of nanometers, and the pores are typically... 

Sorption/desorption is the key property for estimating the mobility of organic pollutants in solid phases. There is a real need to predict such mobility at different aqueous-solid phase interfaces. Solid phase sorption influences the extent of pollutant volatilization from the solid phase surface, its lateral or vertical transport, and biotic or abiotic processes (e.g., biodegradation, bioavailability, hydrolysis, and photolysis). For instance, transport through a soil phase includes several processes such as bulk flow, dispersive flow, diffusion through macropores, and molecular diffusion. The transport rate of an organic pollutant depends mainly on the partitioning between the vapor, liquid, and solid phase of an aqueous-solid phase system. 

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]. 

Macroporous membranes - these are membranes containing large pores. The pore size is usually between 0.1 and 1 pm. Convective transport through the pore space is the mechanism of diffusion in this case. 

The case of transport through microporous membranes is different from that of macroporous membranes in that the pore size approaches the size of the diffusing solute. Various theories have been proposed to account for this effect. As reviewed by Peppas and Meadows [141], the earliest treatment of transport in microporous membranes was given by Faxen in 1923. In this analysis, Faxen related a normalized diffusion coefficient to a parameter, X, which was the ratio of the solute radius to the pore radius... 

Adsoiptive molecules transport through macropores to the mesopores and finally enter the micropores. The micropores usually constitute the largest portion of the internal surface and contribute the most to the total pore volume. The attractive forces are stronger and the pores are filled at low relative pressures in the microporosity, and therefore, most of the adsorption of gaseous adsoiptives occurs within that region. Thus, the total pore volume and the pore size distribution determine the adsorption capacity. 

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 transport properties across an MIP membrane are controlled by both a sieving effect due to the membrane pore structure and a selective absorption effect due to the imprinted cavities [199, 200]. Therefore, different selective transport mechanisms across MIP membranes could be distinguished according to the porous structure of the polymeric material. Meso- and microporous imprinted membranes facilitate template transport through the membrane, in that preferential absorption of the template promotes its diffusion, whereas macroporous membranes act rather as membrane absorbers, in which selective template binding causes a diffusion delay. As a consequence, the separation performance depends not only on the efficiency of molecular recognition but also on the membrane morphology, especially on the barrier pore size and the thickness of the membrane. 

Considering the microstructure of membranes, they can be categorized as porous, which allow transport through their pores, or dense, which permit transport through the bulk of the material [19]. Porous membranes are classified as microporous, mesoporous, and macroporous (see Section 6.2). 

The boundary condition of zero accumulation on the interface between macropores and solid phase is imposed. The effective diffusivity of the porous sample G1 with bimodal pore size distribution is summarized in Fig. 16, where the sample macro-porosity macro is varied on the horizontal axis. This effective diffusivity is compared with a situation where the diffusion transport in nanopores is omitted. The contribution of the transport through the nano-porous solid phase to the total diffusion flux is significant. The calculated effective... 

In the domain where the entity that is transported through a membrane is immiscible or not completely soluble in the contacting (exit) phase, such as the case of gas phase air or oxygen in water, the interfacial factor becomes overwhelmingly important over the transport characteristics of the bulk membrane phase, which is empty space. It is important to recognize that the surface of macroporous membrane consists of the solid phase and the gas phase (in the pore diameter exposed to the interface), and the interfacial aspect of the solid surface dominates the behavior of the gas phase that expands out of the pore. 

For porous membranes the mass transport mechanisms that prevail depend mainly on the membrane s mean pore size [1.1, 1.3], and the size and type of the diffusing molecules. For mesoporous and macroporous membranes molecular and Knudsen diffusion, and convective flow are the prevailing means of transport [1.15, 1.16]. The description of transport in such membranes has either utilized a Fickian description of diffusion [1.16] or more elaborate Dusty Gas Model (DGM) approaches [1.17]. For microporous membranes the interaction between the diffusing molecules and the membrane pore surface is of great importance to determine the transport characteristics. The description of transport through such membranes has either utilized the Stefan-Maxwell formulation [1.18, 1.19, 1.20] or more involved molecular dynamics simulation techniques [1.21]. 

The consideration of thermal effects and non-isothermal conditions is particularly important for reactions for which mass transport through the membrane is activated and, therefore, depends strongly on temperature. This is, typically, the case for dense membranes like, for example, solid oxide membranes, where the molecular transport is due to ionic diffusion. A theoretical study of the partial oxidation of CH4 to synthesis gas in a membrane reactor utilizing a dense solid oxide membrane has been reported by Tsai et al. [5.22, 5.36]. These authors considered the catalytic membrane to consist of three layers a macroporous support layer and a dense perovskite film (Lai.xSrxCoi.yFeyOs.s) permeable only to oxygen on the top of which a porous catalytic layer is placed. To model such a reactor Tsai et al. [5.22, 5.36] developed a two-dimensional model considering the appropriate mass balance equations for the three membrane layers and the two reactor compartments. For the tubeside and shellside the equations were similar to equations (5.1) and... 

As noted in Section 1.5, many commercial adsorbents consist of smaU microporous crystals formed into a macroporous pellet Such adsorbents offer two distinct diffusional resistances to mass transfer the micrq>ore resistance of the adsorbent crystals or microparticles and the ihacropore diffdkional resistance of the pellet. When adsorption occurs from a binary (or multicomponent) fluid mixture, there may be an additional resistance to mass transfer associated with transport through the laminar fluid boundary layer surrounding the particle (see Section 6.7). The general situation is as sketched in Figure... 

If even macropore transport is sufficiently quick not to be rate-determining, one may apply a linear driving force model to the overall kinetics, cf., [102]. For this purpose, the molar flux, N, through the pellet interface is correlated with the particular average gas concentration in the pellet that is in equilibrium with the concentration of sorbing species in the zeolite ... 

Liquid and gaseous molecules have been known to exhibit characteristic transport behaviors in each type of porous material. For example, mass transport can be obtained via viscous flow and molecular diffusion in a macroporous material, through surface diffusion and capillary flow in a mesoporous material and by activated diffusion in a microporous material. 

According to Verweij et al. [14], the transport through the membiane pore depends on the pore radius and the interaction of the permeant and the membrane material. According to the lUPAC terminology and the gas sorption analysis the pores are classified as follows. Micorpore (p <2 nm mesopore l<(p <50 nm and macropore ) >50 nm. Verweij et al. also classified the membrane pore transport, depending on the pore size, as shown in Table 8.11. 

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