Porosity coefficient, axial

Comparing with the conventional three-phase beds, the axial solid holdup distribution is much more uniform and the radial distribution of gas holdup (sg) is much flatter in circulating beds, due to the relatively high Ul and solid circulation. The values of Eg and bed porosity can be predicted by Eqs. (7) and (8) with a correlation coefficient of 0.94 and 0.95, respectively. 

The general approach for modelling catalyst deactivation is schematically organised in Figure 2. The central part are the mass balances of reactants, intermediates, and metal deposits. In these mass balances, coefficients are present to describe reaction kinetics (reaction rate constant), mass transfer (diffusion coefficient), and catalyst porous texture (accessible porosity and effective transport properties). The mass balances together with the initial and boundary conditions define the catalyst deactivation model. The boundary conditions are determined by the axial position in the reactor. Simulations result in metal deposition profiles in catalyst pellets and catalyst life-time predictions. 

Where Ez is the axial dispersion coefficient, interpaiticle porosity, and Re is Reynolds number. 

The parameters defined in this chapter are divided into model parameters and evaluation parameters. Model parameters are porosity, voidage and axial dispersion coefficient, type and parameters of the isotherm as well as mass transfer and diffusion coefficient. All of them are decisive for the mass transfer and fluid flow within the column. They are needed for process simulation and optimisation. Therefore their values have to be valid over the whole operation range of the chromatographic process. Experimental as well as theoretical methods for determining these parameters are explained and discussed in Chapter 6. 

Based on tracer and solute experiments (Fig. 6.11) the model parameters are determined step-by-step, beginning with the void fraction, total porosity and the axial dispersion coefficient (Section 6.5.6). All experimental data must be corrected for plant effects (Eq. 6.132). 

Assumption 7 We assume that the column is operated under constant conditions, e.g., imder constant temperature, pressure, mobile phase flow rate, so that all the physico-chemical parameters remain constant e.g., diffusion coefficients). In writing Eqs. 2.1b to 2.Id, it was assumed that the porosity remains constant, which is not always true (see later. Section 2.1.6). If the phase ratio, the mobile phase velocity, and /or the axial dispersion coefficient are not constant but depend on the space variable or on the solute concentration, it is easy to modify Eq. 2.2 by leaving the corresponding term under the appropriate differential operator. 

This noninvasive method could allow the differentiation between the various packing materials used in chromatography, a correlation between the chromatographic properties of these materials that are controlled by the mass transfer kinetics e.g., the coliunn efficiency) and the internal tortuosity and pore coimectivity of their particles. It could also provide an original, accurate, and independent method of determination of the mass transfer resistances, especially at high mobile phase velocities, and of the dependence of these properties on the internal and external porosities, on the average pore size and on the parameters of the pore size distributions. It could be possible to determine local fluctuations of the coliunn external porosity, of its external tortuosity, of the mobile phase velocity, of the axial and transverse dispersion coefficients, and of the parameters of the mass transfer kinetics discussed in the present work. Further studies along these lines are certainly warranted. 

In these equations, Mq is the superficial velocity of the mobile phase, e the interparticle void fraction, p and pp the porosity and the density of the packing material, respectively, K the adsorption equilibrium constant, Di the axial dispersion coefficient, Rp the particle radius, kf the external mass transfer coefficient, and Dg the... 

Both diffusion coefficients and mass transfer are important, but they depend on the different solids, drainage (flow), and particles porosity. The effective diffusion involves Knudsen and convective diffusion, which depends on the phase of fluid (gas or liquid) and pore size (large or small). These coefficients are characterized by Peclet number (Pe), which depends on the axial or radial dispersion and diffusivity. Depending on the velocity profile, these coefficients can vary radially or axially. The diffusion and dispersion coefficients can also vary due to its dependence on the radial position. If the coefficients vary along the reactor, as in heterogeneous reactors, for example, the velocity is not constant. Thus, the axial dispersion occurs. 

In Equations (11.122)-(11.124) r and x are the nondimensional time and axial coordinate, respectively, c and q the nondimensional concentrations in the fluid and in the soUd phase, respectively, defined as relative deviations from their steady-state values Cg and (gs, e is the bed porosity, and N the number of theoretical plates. The nonlinear equilibrium relation (adsorption isotherm) 0 is again represented in the Taylor series form (the coefficients a, b,c,... depend on the steady-state concentration). 

Mass transfer rates in gas-flowing solids-fixed bed contactors are expected to be high, according to fluid dynamics and heat transfer behavior. Somewhat lower values of mass transfer coefficients than those expected were reported in the literature [6,35-37]. The reasons for that are the effects of segregation as well as strong influence of axial backmixing. Apart from this, mass transfer rates depend on size and structure (porosity) of flowing solids [36]. 

Ep, Fj, Radial and axial dispersion coefficients kf Interphase (or external) mass transfer coefficient. q Average adsorbed-phase concentration (on a mass basis) t, r, z Independent variables of time, radial distance, and axial distance, respectively Mj Superficial velocity e Fixed-bed porosity Ep Particle porosity Particle density X Tortuosity factor... 

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