Broad Pore-Size Distribution

Broad Pore-Size Distribution Several models have been developed to estimate the effective diffusivity in catalysts with broad pore-size distributions. One of the simplest is the parallel pore model of Johnson and Stewart. In their approach, the value ofDp ir) is calculated by averaging over the whole range of pore sizes  

For values of r where diffusion is in the Knudsen regime, DA,t(r) does not depend on composition. However, if r is such that bulk diffusion is important, a concentration-average value of DA,t(.r) should be calculated for each r before the integration is performed. 

The effectiveness factor can be used to account for internal transport effects in the sizing and analysis of heterogeneous catalytic reactors. At this point, it is no longer necessary to assume that internal concentration gradients are negligible. 

The differential form of the design equation for an ideal plug flow reactor in which a heterogeneous catalytic reaction takes place is 

The reaction rate —rA now can be written as i x (—r ), where —r is shorthand for the rate that would exist if there were no gradients inside the catalyst particle. 

---

The broad pore size distribution of Sepharose makes it well apt for the analysis of broad molecular mass distributions of large molecules. One example is given by the method for determination of MWD of clinical dextran suggested in the Nordic Pharmacopea (Nilsson and Nilsson, 1974). Because Superose 6 has the same type of pore size distributions as Sepharose 6, many analytical applications performed earlier on Sepharose have been transformed to Superose in order to decrease analysis time. However, Sepharose is suitable as a first try out when no information about the composition of the sample, in terms of size, is available. 

The mesopores are with regular and well-defined shapes and have a broad pore size distribution. N2 adsorption analysis revealed also a broad pore size distribution centered around 28 nm. The... 

The reason for the high selectivity of zeohte catalysts is the fact that the catalytic reaction typically takes place inside the pore systems of the zeohtes. The selectivity in zeohte catalysis is therefore closely associated to the unique pore properties of zeohtes. Their micropores have a defined pore diameter, which is different from all other porous materials showing generally a more or less broad pore size distribution. Therefore, minute differences in the sizes of molecules are sufficient to exclude one molecule and allow access of another one that is just a little smaller to the pore system. The high selectivity of zeolite catalysts can be explained by three major effects [14] reactant selectivity, product selectivity, and selectivity owing to restricted size of a transition state (see Figure 4.11). 

Ultrafiltration (UF) is an important component in wastewater treatment and in food industry [109,110]. With increasing concerns and regulations in environment as well as in food safety, the process of ultrafiltration has become more critical, whereby new technology development to provide faster and more efficient water treatment is not only necessary but also urgent. Currently, conventional polymeric UF membranes are prepared mainly by the phase immersion process, typically generating an asymmetric porous structure with two major limitations (1) relatively low porosity and (2) fairly broad pore-size distribution [111,112],... 

The polymerization temperature, through its effects on the kinetics of polymerization, is a particularly effective means of control, allowing the preparation of macroporous polymers with different pore size distributions from a single composition of the polymerization mixture. The effect of the temperature can be readily explained in terms of the nucleation rates, and the shift in pore size distribution induced by changes in the polymerization temperature can be accounted for by the difference in the number of nuclei that result from these changes [61,62]. For example, while the sharp maximum of the pore size distribution profile for monoliths prepared at a temperature of 70 °C is close to 1000 nm, a very broad pore size distribution curve spanning from 10 to 1000 nm with no distinct maximum is typical for monolith prepared from the same mixture at 130°C [63]. 

Drying without the occurrence of large capillary stresses was obtained with supercritical drying in an autoclave. In this case a mean pore size was obtained which was twice that obtained under normal drying conditions and with a broad pore size distribution in accordance with the expectation for a noncompressed, random packing of particles. 

The shape of the hysteresis loop in the adsorption/desorption isotherms provides information about the nature of the pores. The loops have been classified according to shape as A, B and E (De Boer, 1958) or as HI - H4 by lUPAC (Sing et al, 1985). Ideally, the different loop shapes correspond to cylindrical, slit shaped and ink-bottle pores the loops in the isotherm IV and V of Figure 5.3 correspond to cylindrical pores. Wide loops indicate a broad pore size distribution (for an example see Fig. 14.9). The absence of such a loop may mean that the sample is either nonporous or microporous. These generalizations provide some initial assistance in assessing the porosity of a sample. In fact the adsorption/desorption isotherms are often more complicated than those shown in Figure 5.3 owing to a mixture of pore types and/or to a wide pore size distribution. 

As indicated by XRD patterns, there exist just 2-3 broad peaks in the calcined acid-made materials (Fig. 3A). Moreover, the N2 adsorption/desorption isotherm shown in Fig. 3B, the calcined acid-made mesoporous silica indeed possesses a broad capillary condensation at the partial pressure p/p0 of ca. 0.2-0.4, indicating a broad pore size distribution with a FWHM ca. 1.0 nm calculated from the BJH method. This is attributed to the occurrence of partial collapse of the mesostructure during the high temperature calcination. The hexagonal structure completely collapsed when subjected to further hydrothermal treatment in water at 100 °C for 3 h. Mesoporous silica materials synthesized from the acid route are commonly believed to be less stable than those from the alkaline route [6,7]. 

The pore size distributions of Ti-MCM-41 synthesized in this work are shown in Fig. 2. All of the samples showed a sharp distribution without addition of TMB and the use of methanol solvent resulted in the expansion of pore channel size. The average pore sizes determined by N, adsorption were 4.0nm and 2.8nm when the added solvents were methanol and ethanol, respectively. In this case, the used surfactant was C22TMAC1. In addition, the expansion of BJH pore size of Ti-MCM-41 was observed by the addition of TMB. A broad pore size distribution was investigated by using TMB as an auxiliary chemical. The mean pore size was ca. 7.5nm in methanol solvent. 

Micro and mesoporous broad pore size distribution (5A - 100A) +... 

Experimentally determined effective transport properties of porous bodies, e.g., effective diffusivity and permeability, can be compared with the respective effective transport properties of reconstructed porous media. Such a comparison was found to be satisfactory in the case of sandstones or other materials with relatively narrow pore size distribution (Bekri et al., 1995 Liang et al., 2000b Yeong and Torquato, 1998b). Critical verification studies of effective transport properties estimated by the concept of reconstructed porous media for porous catalysts with a broad pore size distribution and similar materials are scarce (Mourzenko et al., 2001). Let us employ the sample of the porous... 

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. 

Porosity within a particle is a manifestation of the shape of a particle. Fractal particles will have internal porosity as a result of their shapes. Fractal particles with low fractal dimensions (i.e., <2.0) will have a broad pore size distribution, where the largest pore approaches the size of the aggregate. Fractal particles with large fractal dimensions (i.e., >2.0) will have narrower pore size distributions with most of the porosity occurring at a size much smaller than that of the aggregate. Calcination of metal salt particles or metal hydroxides to produce oxides is another common method to produce internal porosity. In the gas evolution that takes place in transformation to the oxide, pores are opened up in the particle structure. The opening of pores in a hydrous alumina powder can increase its surface area from 0.5 m /gm (its external area) to 450 mVgm (its internal pore area). 

Many commercial ceramic membranes have two three or even four layers in structure and their pore size distributions are similar to that shown in Figure 4.12. It should be noted, however, that determination of those multi-layered, broad pore size distributions is not suaightforward. The major reason for this is the overwhelmingly small pore volume of the thin, fine pore membrane layer compared to those of the support layer(s) of the structure. It is possible, although very tedious, to remove most of the bulk support layer to increase the relative percentage of the pore volumes of the membrane and other thin support layers. Provided the amount of bulk support layer removed is known and the mercury porosimeu data of the "shaved" membrane/support sample is determined, it is feasible to construct a composite pore size distribution such as the one shown in Figure 4.12. 

Macromolecular fractionation, which involves high-resolution separation of solutes having comparable molecular weights using ultrafiltration, is challenging primarily due to the broad pore-size distribution of ultrafiltration membranes. This implies that purely size-based fractionation is not feasible using membranes currently available. The development of advanced membranes with narrow pore-size distributions could make fractionation more feasible. With currently available membranes... 

It is known that the effect of the surface area in the gasification of charcoal is intimately related to the very broad pore size distribution of this material. Random pore structure models accounting for the effects of pore growth and coalescence have been proposed by various authors and have often shown satisfactory agreement between theory and experiment, but none of the proposed kinetic relations describes the charcoal reactivity in the conversion range beyond X 0.7 satisfactorily. For the latter conversion... 

Fig 4 shows the /-plots of the four mesoporous silica samples These plots consist of three linear stages According to Branton et al the initial stage represents multilayer adsorption on the pore wall the following steep one, capillary condensation in mesopores, and the last one, adsorption on the external surface It is noted that the second stage for the two commercial silicas covers a broader / range (over 3 A), compared to the range for the two MCM-41 samples (less than 0 5 A). The capillary condensation over a wide / values implies a broad pore size distribution in the samples. 

The importance of diffusional limitations on catalyst deactivation has been recognized for more than four decades (ref.1-2). Voluminous studies have been reported on the deactivation of resid hydrodesulfurization (RHDS) catalysts (ref. 3-4). However, most of the experiments were carried out over the catalysts which have a broad pore size distribution This would cause the difficulty in determining the effect of pore size on the deactivation. 

Generally, high selectivity is related to membrane properties, such as small pores and high hydraulic resistance or low permeability. It can be compromised by a broad pore size distribution. The permeability increases with increasing density of pores, and the overall membrane resistance is directly proportional to its thickness. Therefore, a good membrane must have a narrow range of pore sizes, a high porosity, and a thin layer of material. 

Fig. 1 compares the pore size distributions of major commercial adsorbents discussed in this section. Activated carbons have a broad pore size distribution like activated alumina and silica gel. Although activated carbon is thought to be hydrophobic, it does adsorb... 

Ordered mesoporous carbons (OMCs) are new carbon materials that were developed over the last ten years. Their mesopores have a defined width with a very narrow pore size distribution. This sets them aside from older nanoporous carbons, such as activated carbons or activated carbon fibers. The last two classes of carbons are produced from various carbon-containing materials by carbonization followed by partial oxidation (activation). To a certain degree, the pore structure of these materials can be controlled by the carbonization and activation conditions. However, it is not possible to produce purely mesoporous activated carbons or activated carbon fibers. Furthermore, these materials generally exhibit a broad pore size distribution [1, 2]. 

---