Effectiveness factor behavior

Thus we have expressions for the effectiveness factor for different catalyst geometries. The Thiele modulus can be computed from catalyst geometry and surface area parameters. The characteristic size is 21 for a porous slab and 2R for a cylinder or sphere. While the expressions for r)(0) appear quite different, they are in fact very similar when scaled appropriately, and they have the same asymptotic behavior,... 

The conclusion was reached that the mechanical properties of glass-reinforced unsaturated polyester are influenced by the chemical structure of the spacer groups in the methacrylate functional silane. Effective factors include hydro-phobicity, reactivity of the double bond, chain flexibility of the backbone, and adsorption behavior. 

The effectiveness factor versus the Weisz modulus according to Kao and Satterfield [61] is shown in Fig. 21 for C = 0.5 and different values of B. From this diagram, a similar behavior is seen as in the case of a simple, first order, reversible reaction (see Fig. 18) with decreasing value of B, the effectiveness factor is reduced. A decline of the effectiveness factor is also observed for a rise of the parameter C, which corresponds to a shift towards the chemical equilibrium, and hence to a reduction of the net reaction rate [91]. 

Shiriyama Y, Chen ACH, Nakagawa S, Russell DS, Duman RS. Brain-derived neurotrophic factor produces antidepressant effects in behavioral models of depression. J. Neurosci. 2002 22 3251-3261. 

Effectiveness factors for a first-order reaction in a spherical, nonisothermal catalysts pellet. (Reprinted from R B. Weisz and J. S. Hicks, The Behavior of Porous Catalyst Particles in View of Internal Mass and Heat Diffusion Effects, Chem. Eng. Sci., 17 (1962) 265, copyright 1962, with permission from Elsevier Science.)... 

Exemplary results of modeling processes inside the catalytic layer are presented in Fig. 9. The solid lines show the dependency of the overall effectiveness factor on the relative distribution of the catalyst between the comers and the side regions. The two cases represent two levels of the first-order rate constants, with the faster reaction in case (b). As expected, the effectiveness factor of the first reaction drops as more catalyst is deposited in the comers. The effectiveness factor for the second reaction increases in case (a) but decreases in case (b). The latter behavior is caused by depletion of B deep inside the catalytic layer. What might be surprising is the rather modest dependency of the effectiveness factor on the washcoat distribution. The explanation is that internal diffusion is not important for slow reactions, while for fast reactions the available external surface area becomes the key quantity, and this depends only slightly on the washcoat distribution for thin layers. The dependence of the effectiveness factor on the distribution becomes more pronounced for consecutive reactions described by Langmuir-Hinshelwood-Hougen-Watson kinetics [26]. 

The ideal Thiele-Damkohler theory assumes isothermal behavior of the catalyst particle. However, if the particle is large and the reaction is highly exothermic, heat transfer may not be fast enough to remove the heat of reaction from the interior. The particle center then heats up, causing the reaction rate to be higher than under isothermal conditions. The effectiveness factor can be larger than unity This may occur if the structural material of the particle and the imbibed fluid are poor heat conductors, as might be the case, for example, in gas reactions in silica-ceous particles. 

A set of simultaneous PDEs (ODEs if the pellets are spherical) must be solved to estimate the extent of reaction and conversion occurring within a single pellet. These local values are then substituted into Equations 9.1 and 9.3 so that we need to solve a set of PDEs that are embedded within a set of PDEs. The resulting system truly reflects the complexity of heterogeneous reactors, but practical solutions rarely go to this complexity. Most industrial reactors are designed on the basis of pseudohomogeneous models as in Equations 9.1 and 9.3, and the local catalyst behavior is described by the effectiveness factor defined in Chapter 10. 

Lidsky et al. (2003) point out that cognitive development of children and effects of behavior as a consequence of long-term exposure on low levels of blood in the environment is known, but there are evidences that certain genetic and ecological factors can increase the adverse effects of lead to neurological development, so certain part of children are more vulnerable to lead s neurotoxicity. 

Equation 9 combined with Equation 5 gives the asymptotic behavior of the effectiveness factor for large values of the Thiele modulus. Other values for the effectiveness factor are obtained by numerical integration of Equation 7 to yield the desired concentration profile ... 

Figure 6 Effectiveness factor vs the Thiele modulus for a distribution of reacting centres on the one-dimensional line localized on a Cantor middle-third set. Curve d) is the theoretical behavior, eqs. (18),(23).

Figure 6 shows the behavior of the effectiveness factor vs the Thiele modulus for this structure. It can be proved analytictJly (although the proof is not reported here for the sake of brevity) that in a difliision-controlled regime the exponent appearing in eq. (18) is given by... 

In the previous examples, we have exploited the idea of an effectiveness factor to reduce fixed-bed reactor models to the same form as plug-flow reactor models. This approach is useful and solves several important cases, but this approach is also limited and can take us only So far. In the general case, we must contend with multiple reactions that are not first order, nonconstant thermochemical properties, and nonisothermal behavior in the pellet and the fluid. For these cases, we have no alternative but to solve numerically for the temperature and species concentrations profiles in both the pellet and the bed. As a final example, we compute the numerical solution to a problem of this type. 

Asymptotic behavior of the effectiveness factor versus Thiele modulus first-order reaction in spherical pellet. . . 389... 

---