Cylindrical diffusion model particles

For the sake of comparison hydrogenation experiments with large cylindrical catalyst particles were carried out. The increase of the particle size diminished the velocity of catalytic hydrogenation. These experimental results provide a path for the process scale-up, i.e. a prediction of the hydrogenation rate on large catalyst particles starting from crushed particles. The values of the kinetic constants obtained for crushed particles were utilized and the ratio of porosity to tortuosity from the reaction-diffusion model was adjusted (0.167) to fit successfully the experimental data (Figure 10.40). 

Among the dynamical properties the ones most frequently studied are the lateral diffusion coefficient for water motion parallel to the interface, re-orientational motion near the interface, and the residence time of water molecules near the interface. Occasionally the single particle dynamics is further analyzed on the basis of the spectral densities of motion. Benjamin studied the dynamics of ion transfer across liquid/liquid interfaces and calculated the parameters of a kinetic model for these processes [10]. Reaction rate constants for electron transfer reactions were also derived for electron transfer reactions [11-19]. More recently, systematic studies were performed concerning water and ion transport through cylindrical pores [20-24] and water mobility in disordered polymers [25,26]. 

Scanning electron microscopy and other experimental methods indicate that the void spaces in a typical catalyst particle are not uniform in size, shape, or length. Moreover, they are often highly interconnected. Because of the complexities of most common pore structures, detailed mathematical descriptions of the void structure are not available. Moreover, because of other uncertainties involved in the design of catalytic reactors, the use of elaborate quantitative models of catalyst pore structures is not warranted. What is required, however, is a model that allows one to take into account the rates of diffusion of reactant and product species through the void spaces. Many of the models in common use simulate the void regions as cylindrical pores for such models a knowledge of the distribution of pore radii and the volumes associated therewith is required. 

The Samuel-Magee model can be extended to a-particle tracks, considered as cylindrical columns formed by excessive spur overlap due to high LET. To a good approximation, the length I of the cylinder remains constant while its radius grows by diffusion. In this geometry, the normalized radical distribution is given by... 

The performance of a reactor for a gas-solid reaction (A(g) + bB(s) -> products) is to be analyzed based on the following model solids in BMF, uniform gas composition, and no overhead loss of solid as a result of entrainment. Calculate the fractional conversion of B (fB) based on the following information and assumptions T = 800 K, pA = 2 bar the particles are cylindrical with a radius of 0.5 mm from a batch-reactor study, the time for 100% conversion of 2-mm particles is 40 min at 600 K and pA = 1 bar. Compare results for /b assuming (a) gas-film (mass-transfer) control (b) surface-reaction control and (c) ash-layer diffusion control. The solid flow rate is 1000 kg min-1, and the solid holdup (WB) in the reactor is 20,000 kg. Assume also that the SCM is valid, and the surface reaction is first-order with respect to A. 

As shown in Example 22-3, for solid particles of the same size in BMF, the form of the reactor model resulting from equation 22.2-13 depends on the kinetics model used for a single particle. For the SCM, this, in turn, depends on particle shape and the relative magnitudes of gas-film mass transfer resistance, ash-layer diffusion resistance and surface reaction rate. In some cases, as illustrated for cylindrical particles in Example 22-3(a) and (b), the reactor model can be expressed in explicit analytical form additional results are given for spherical particles by Levenspiel(1972, pp. 384-5). In other f l cases, it is convenient or even necessary, as in Example 22-3(c), to use a numerical pro-... 

Let us consider a shallow fluidized bed combustor with multiple coal feeders which are used to reduce the lateral concentration gradient of coal (11). For simplicity, let us assume that the bed can be divided into N similar cylinders of radius R, each with a single feed point in the center. The assumption allows us to use the symmetrical properties of a cylindrical coordinate system and thus greatly reduce the difficulty of computation. The model proposed is based on the two phase theory of fluidization. Both diffusion and reaction resistances in combustion are considered, and the particle size distribution of coal is taken into account also. The assumptions of the model are (a) The bed consists of two phases, namely, the bubble and emulsion phases. The voidage of emulsion phase remains constant and is equal to that at incipient fluidization, and the flow of gas through the bed in excess of minimum fluidization passes through the bed in the form of bubbles (12). (b) The emulsion phase is well mixed in the axial... 

Vanadium molecular size distributions in residual oils are measured by size exclusion chromatography with an inductively coupled plasma detector (SEC-ICP). These distributions are then used as input for a reactor model which incorporates reaction and diffusion in cylindrical particles to calculate catalyst activity, product vanadium size distributions, and catalyst deactivation. Both catalytic and non-catalytic reactions are needed to explain the product size distribution of the vanadium-containing molecules. Metal distribution parameters calculated from the model compare well with experimental values determined by electron microprobe analysis, Modelling with feed molecular size distributions instead of an average molecular size results in predictions of shorter catalyst life at high conversion and longer catalyst life at low conversions. 

In view of evidence such as that in Fig. 8-5, it is unlikely that detailed quantitative descriptions of the void structure of solid catalysts will become available. Therefore, to account quantitatively for the variations in rate of reaction with location within a porous catalyst particle, a simplified model of the pore structure is necessary. The model must be such that diffusion rates of reactants through the void spaces into the interior surface can be evaluated. More is said about these models in Chap. 11. It is sufficient here to note that in all the widely used models the void spaces are simulated as cylindrical pores. Hence the size of the void space is interpreted as a radius 2 of a cylindrical pore, and the distribution of void volume is defined in terms of this variable. However, as the example of the silver, catalyst indicates, this does not mean that the void spaces are well-defined cylindrical pores. 

Investigator Type of correlation Phases involved Model associated Wakao and Kato [40] Particle-to-bed radiative heat transfer coefficient Fluid-solid Assumes that any two hemispheres in contact are circumscribed with a diffusely reflecting cylindrical walk Takes into account the overall view factor... 

BRENNER, H. 6c GAYDOS, L.J. 1977. The constrained Brownian movement of spherical particles in cylindrical pores of comparable radius Models of the diffusive and convective transport of solute molecules in membranes and porous media. J. Colloid Interface Sci. 58, 312-356. 

Ogston et al. [72] developed a model for the diffusion of spherical particles through a randomly oriented array of straight, cylindrical fibers, with radius /-f, occupying a volume fraction 4> ... 

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