Packed beds modulus

Notice that in the region of fast chemical reaction, the effectiveness factor becomes inversely proportional to the modulus h2. Since h2 is proportional to the square root of the external surface concentration, these two fundamental relations require that for second-order kinetics, the fraction of the catalyst surface that is effective will increase as one moves downstream in an isothermal packed bed reactor. 

A certain spherical porous catalyst with a pellet diameter of 1/8 in. has a Thiele modulus of 0.5 for a first-order reaction and gives 90% conversion in a packed bed reactor. It is proposed to... 

In this chapter generalized mathematical models of three dimensional electrodes are developed. The models describe the coupled potential and concentration distributions in porous or packed bed electrodes. Four dimensionless variables that characterize the systems have been derived from modeling a dimensionless conduction modulus ju, a dimensionless diffusion (or lateral dispersion) modulus 5, a dimensionless transfer coefficient a and a dimensionless limiting current density y. The first three are... 

If the catalyst is dispersed throughout the pellet, then internal diffusion of the species within the pores of the pellet, along with simultaneous reaction(s) must be accounted for if the prevailing Thiele modulus > 1. This aspect gives rise to the effectiveness factor" problem, to which a significant amount of effort, summarized by Aris ( ), has been devoted in the literature. It is important to realize that if the catalyst pellet effectiveness factor is different from unity, then the packed-bed reactor model must be a heterogeneous model it cannot be a pseudohomogeneous model. 

The conditions used for pellet forming can have a major influence on several important catalyst properties, including pore size distribution, pellet strength, and abrasion resistance. Both the size and shape of catalyst pellets affect the pressure drop across a packed bed reactor and also, as indicated earlier, affect the Thiele modulus and thus the effectiveness factor. Recently, monolith catalysts have begun to be used in circumstances where low-pressure drop and/or... 

Closure After completing this chapter, the reader should be able to derive differential equations describing diffusion and reaction, discuss the meaning of the effectiveness factor and its relationship to the Thiele modulus, and identify the regions of mass transfer control and reaction rate control. The reader should be able to apply the Weisz-Prater and Mears criteria to identify gradients and diffusion limitations. These principles should be able to be applied to catalyst particles as well as biomaierial tissue engineering. The reader should be able to apply the overall effectiveness factor to a packed bed reactor to calculate the conversion at the exit of the reactor. The reader should be able to describe the reaction and transport steps in slurry reactors, trickle bed reactors, fluidized-besd reactors, and CVD boat reactors and to make calculations for each reactor. 

The solid phase could be a reactant, product, or catalyst. In general the decision on the choice of the particle size rests on an analysis of the extra-and intra-particle transport processes and chemical reaction. For solid-catalyzed reactions, an important consideration in the choice of the particle size is the desire to utilize the catalyst particle most effectively. This would require choosing a particle size such that the generalized Thiele modulus < gen, representing the ratio of characteristic intraparticle diffusion and reaction times, has a value smaller than 0.4 see Fig. 13. Such an effectiveness factor-Thiele modulus analysis may suggest particle sizes too small for use in packed bed operation. The choice is then either to consider fluidized bed operation, or to used shaped catalysts (e.g., spoked wheels, grooved cylinders, star-shaped extrudates, four-leafed clover, etc.). Another commonly used procedure for overcoming the problem of diffu-sional limitations is to have nonuniform distribution of active components (e.g., precious metals) within the catalyst particle. 

The Weisz modulus [11] allows for the estimation of intraparticle diffusion limitations in packed bed microchannel reactors ... 

It should be noted that the elasticity modulus E is not merely a property of the solid material in the bed. In general, is a complex function of the structure of packing, material properties of packing particles, particle size, and particle contact and cohesion forces between particles. 

The relationship between the structure of the disordered heterogeneous material (e.g., composite and porous media) and the effective physical properties (e.g., elastic moduli, thermal expansion coefficient, and failure characteristics) can also be addressed by the concept of the reconstructed porous/multiphase media (Torquato, 2000). For example, it is of great practical interest to understand how spatial variability in the microstructure of composites affects the failure characteristics of heterogeneous materials. The determination of the deformation under the stress of the porous material is important in porous packing of beds, mechanical properties of membranes (where the pressure applied in membrane separations is often large), mechanical properties of foams and gels, etc. Let us restrict our discussion to equilibrium mechanical properties in static deformations, e.g., effective Young s modulus and Poisson s ratio. The calculation of the impact resistance and other dynamic mechanical properties can be addressed by discrete element models (Thornton et al., 1999, 2004). 

In contrast, the Enskog model results in nonmonotonous dependencies for fluidized bed particulate pressure and the bulk elasticity modulus. Furthermore, both quantities fall off to zero as the bed attains the state of close packing. The problem of choosing one of the utilized approximate statistical models therefore assumes a fundamental significance. As has been pointed out, this problem can be successfully resolved by considering the behavior of fluidized beds at concentrations differing little from that corresponding to the close-packed state [25]. 

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