Diffusion constant, spherical particles

This equation gives the change of concentration in a finite volume element with time. In the approach of Barrer and Jost, the diffusivity is assumed to be isotropic throughout the crystal, as Dt is independent of the direction in which the particles diffuse. Assuming spherical particles. Pick s second law can be readily solved in radial coordinates. As a result, all information about the exact shape and connectivity of the pore structure is lost, and only reflected by the value of the diffusion constant. 

FIG. 16-27 Constant pattern solutions for R = 0.5. Ordinant is cfor nfexcept for axial dispersion for which individual curves are labeled a, axial dispersion h, external mass transfer c, pore diffusion (spherical particles) d, surface diffusion (spherical particles) e, linear driving force approximation f, reaction kinetics. [from LeVan in Rodrigues et al. (eds.), Adsorption Science and Technology, Kluwer Academic Publishers, Dor drecht, The Nether lands, 1989 r eprinted with permission.]... 

The relationship between adsorption capacity and surface area under conditions of optimum pore sizes is concentration dependent. It is very important that any evaluation of adsorption capacity be performed under actual concentration conditions. The dimensions and shape of particles affect both the pressure drop through the adsorbent bed and the rate of diffusion into the particles. Pressure drop is lowest when the adsorbent particles are spherical and uniform in size. External mass transfer increases inversely with d (where, d is particle diameter), and the internal adsorption rate varies inversely with d Pressure drop varies with the Reynolds number, and is roughly proportional to the gas velocity through the bed, and inversely proportional to the particle diameter. Assuming all other parameters being constant, adsorbent beds comprised of small particles tend to provide higher adsorption efficiencies, but at the sacrifice of higher pressure drop. This means that sharper and smaller mass-transfer zones will be achieved. 

In this exercise we shall estimate the influence of transport limitations when testing an ammonia catalyst such as that described in Exercise 5.1 by estimating the effectiveness factor e. We are aware that the radius of the catalyst particles is essential so the fused and reduced catalyst is crushed into small particles. A fraction with a narrow distribution of = 0.2 mm is used for the experiment. We shall assume that the particles are ideally spherical. The effective diffusion constant is not easily accessible but we assume that it is approximately a factor of 100 lower than the free diffusion, which is in the proximity of 0.4 cm s . A test is then made with a stoichiometric mixture of N2/H2 at 4 bar under the assumption that the process is far from equilibrium and first order in nitrogen. The reaction is planned to run at 600 K, and from fundamental studies on a single crystal the TOP is roughly 0.05 per iron atom in the surface. From Exercise 5.1 we utilize that 1 g of reduced catalyst has a volume of 0.2 cm g , that the pore volume constitutes 0.1 cm g and that the total surface area, which we will assume is the pore area, is 29 m g , and that of this is the 18 m g- is the pure iron Fe(lOO) surface. Note that there is some dispute as to which are the active sites on iron (a dispute that we disregard here). 

Equation (1) predicts that the rate of release can be constant only if the following parameters are constant (a) surface area, (b) diffusion coefficient, (c) diffusion layer thickness, and (d) concentration difference. These parameters, however, are not easily maintained constant, especially surface area. For spherical particles, the change in surface area can be related to the weight of the particle that is, under the assumption of sink conditions, Eq. (1) can be rewritten as the cube-root dissolution equation ... 

Gas diffusion in the nano-porous hydrophobic material under partial pressure gradient and at constant total pressure is theoretically and experimentally investigated. The dusty-gas model is used in which the porous media is presented as a system of hard spherical particles, uniformly distributed in the space. These particles are accepted as gas molecules with infinitely big mass. In the case of gas transport of two-component gas mixture (i = 1,2) the effective diffusion coefficient (Dj)eff of each of the... 

Size of the particle. Assuming a spherical shape, the value of the rotational diffusion constant Dp can provide an average radius using the following relationship ... 

This result implicitly requires (a) that no strongly attractive/repulsive electrostatic interaction occurs as the particles approach each other, and (b) that a stationary diffusional concentration field surrounds the particle. [Note Einstein s result for spherical particle diffusivity i.e.,D = k Tlfm-qr, where is the Boltzmann constant, T is temperature in kelvin, 17 is the viscosity of the medium, and r is the radius) indicates that RuD will be approximately Ak TKyni].]... 

In this section, the basic theory required for the analysis and interpretation of adsorption and ion-exchange kinetics in batch systems is presented. For this analysis, we consider the transient adsorption of a single solute from a dilute solution in a constant volume, well-mixed batch system, or equivalently, adsorption of a pure gas. Moreover, uniform spherical particles and isothermal conditions are assumed. Finally, diffusion coefficients are considered to be constant. Heat transfer has not been taken into account in the following analysis, since adsorption and ion exchange are not chemical reactions and occur principally with little evolution or uptake of heat. Furthermore, in environmental applications,... 

For computing the diffusion parameter, D /ro, Fick s diffusion equation was assumed to be applicable to the system, with D independent of concentration of the diffusing species. Solving Fick s law for a spherical particle, where the external gas pressure is constant gives ... 

While this equation is thought to overestimate the diffusion-limited rate constant slightly, it is a good approximation. If the diffusing particles are approximately spherical, diffusion constants DA and DB can be calculated from Eq. 9-25, and Eq. 9-28 becomes Eq. 9-29. 

If the catalyst were not to decay but for some other reason, perhaps temperature control, the particles were taken out and recycled, then each might be supposed to be in pristine condition on entering the bed. Each particle would then undergo a transition during which the steady state profile of reactant within the particle would be built up. The analysis of Amundson and Aris (1962 this part is not tainted with the error mentioned above in fn. 13) may be used. We assume spherical particles of radius R, and call the profile of concentration at time a, c r, a). If D is the diffusivity of the reactant and k the rate constant per unit volume of catalyst,... 

A constant reaction kernel is to be expected in the absence of enzyme reaction if the aggregation rate is determined by the Brownian motion of spherical particles which coalesce to form larger spheres. To a first approximation, the increased collisional cross-section is then compensated for by the decrease in diffusion rate (von Smoluchowski,... 

A is the wavelength of the laser in vacuum and q is the magnitude of the so called scattering vector. In turn, for spherical particles, the diffusion constant is related to the particle diameter through the Stokes-Einstein equation ... 

It is possible that the rate-determining process in the kinetics of ion exchange is the film diffusion. Consider a spherical particle encircled by an aqueous solution sphere (see Figure 7.8), in which the zeolite is homoionic at t = 0, the electrolytic solution has a very high volume (i.e., C2A == constant), and the diffusion is stationary and one-dimensional in a direction perpendicular to the zeolite surface [44], Then, under the conditions discussed above, it is possible to calculate the exchange flux of cation B, that is,. 7 , as follows [23] ... 

Table 6.8 presents the details of calculations for spherical particles with an equivalent diameter of 2.4mm. It may be observed that the pore diffusion considerably affects the process rate, particularly at higher temperatures. The external mass transfer plays a minor role. Their combination leads to a global effectiveness that drops from 75% to 35% when the temperature varies from 160 to 220°C. Based on the above elements the apparent reaction constant may be expressed by the following Arrhenius law ... 

A quantitative treatment of the effects of electrolytes on colloid stability has been independently developed by Deryagen and Landau and by Verwey and Over-beek (DLVO), who considered the additive of the interaction forces, mainly electrostatic repulsive and van der Waals attractive forces as the particles approach each other. Repulsive forces between particles arise from the overlapping of the diffuse layer in the electrical double layer of two approaching particles. No simple analytical expression can be given for these repulsive interaction forces. Under certain assumptions, the surface potential is small and remains constant the thickness of the double layer is large and the overlap of the electrical double layer is small. The repulsive energy (VR) between two spherical particles of equal size can be calculated by ... 

H. Ohshima, Diffuse double layer interaction between two spherical particles with constant surface charge density in an electrolyte solution, Colloid Polymer Sci. 263, 158-163 (1975). 

Quantitative measurements of electrokinetic phenomena permit the calculation of the zeta potential by use of the appropriate equations. However, in the deduction of the equations approximations are made this is because in the interfacial region physical properties such as concentration, viscosity, conductivity, and dielectric constant differ from their values in bulk solution, which is not taken into account. Corrections to compensate for these approximations have been introduced, as well as consideration of non-spherical particles and particles of dimensions comparable to the diffuse layer thickness. This should be consulted in the specialized literature. 

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