Effectiveness factor, particle dependence

The catalyst activity depends not only on the chemical composition but also on the diffusion properties of the catalyst material and on the size and shape of the catalyst pellets because transport limitations through the gas boundary layer around the pellets and through the porous material reduce the overall reaction rate. The influence of gas film restrictions, which depends on the pellet size and gas velocity, is usually low in sulphuric acid converters. The effective diffusivity in the catalyst depends on the porosity, the pore size distribution, and the tortuosity of the pore system. It may be improved in the design of the carrier by e.g. increasing the porosity or the pore size, but usually such improvements will also lead to a reduction of mechanical strength. The effect of transport restrictions is normally expressed as an effectiveness factor q defined as the ratio between observed reaction rate for a catalyst pellet and the intrinsic reaction rate, i.e. the hypothetical reaction rate if bulk or surface conditions (temperature, pressure, concentrations) prevailed throughout the pellet [11], For particles with the same intrinsic reaction rate and the same pore system, the surface effectiveness factor only depends on an equivalent particle diameter given by... 

Obviously, the internal effectiveness factor, qi, depends on the effective diffusivity, Dg, and kinetic parameters such as the rate coefficient, fcy,p/ but also on the shape of the catalyst particle. 

However, the interface area liquid/solid, ai, and the effectiveness factor are dependent on the particle diameter. From Figure 21.3, it can be seen that the overall resistance varies linearly with the inverse of the solid mass. This demonstrates the dependence on particle diameter. Thus, the resistance to the rate decreases with the reduction of particle diameter (cases c and b). 

The mass transport influence is easy to diagnose experimentally. One measures the rate at various values of the Thiele modulus the modulus is easily changed by variation of R, the particle size. Cmshing and sieving the particles provide catalyst samples for the experiments. If the rate is independent of the particle size, the effectiveness factor is unity for all of them. If the rate is inversely proportional to particle size, the effectiveness factor is less than unity and

experimental points allow triangulation on the curve of Figure 10 and estimation of Tj and ( ). It is also possible to estimate the effective diffusion coefficient and thereby to estimate Tj and ( ) from a single measurement of the rate (48). 

The effectiveness factor depends, not only on the reaction rate constant and the effective diffusivity, but also on the size and shape of the catalyst pellets. In the following analysis detailed consideration is given to particles of two regular shapes ... 

It will be shown, however, that the effectiveness factor does not critically depend on the shape of the particles, provided that their characteristic length is defined in an appropriate way. Some comparison is made be made between calculated results and experimental measurements with particles of frequently ill-defined shapes. 

The solution of this equation is in the form of a Bessel function 32. Again, the characteristic length of the cylinder may be defined as the ratio of its volume to its surface area in this case, L = rcJ2. It may be seen in Figure 10.13 that, when the effectiveness factor rj is plotted against the normalised Thiele modulus, the curve for the cylinder lies between the curves for the slab and the sphere. Furthermore, for these three particles, the effectiveness factor is not critically dependent on shape. 

The concentration of gas over the active catalyst surface at location / in a pore is ai [). The pore diffusion model of Section 10.4.1 linked concentrations within the pore to the concentration at the pore mouth, a. The film resistance between the external surface of the catalyst (i.e., at the mouths of the pore) and the concentration in the bulk gas phase is frequently small. Thus, a, and the effectiveness factor depends only on diffusion within the particle. However, situations exist where the film resistance also makes a contribution to rj so that Steps 2 and 8 must be considered. This contribution can be determined using the principle of equal rates i.e., the overall reaction rate equals the rate of mass transfer across the stagnant film at the external surface of the particle. Assume A is consumed by a first-order reaction. The results of the previous section give the overall reaction rate as a function of the concentration at the external surface, a. ... 

Inspection of Fig. 15.3 reveals that while for jo 0.1 nAcm , the effectiveness factor is expected to be close to 1, for a faster reaction with Jo 1 p,A cm , it will drop to about 0.2. This is the case of internal diffusion limitation, well known in heterogeneous catalysis, when the reagent concentration at the outer surface of the catalyst grains is equal to its volume concentration, but drops sharply inside the pores of the catalyst. In this context, it should be pointed out that when the pore size is decreased below about 50 nm, the predominant mechanism of mass transport is Knudsen diffusion [Malek and Coppens, 2003], with the diffusion coefficient being less than the Pick diffusion coefficient and dependent on the porosity and pore stmcture. Moreover, the discrete distribution of the catalytic particles in the CL may also affect the measured current owing to overlap of diffusion zones around closely positioned particles [Antoine et ah, 1998]. 

When a solid acts as a catalyst for a reaction, reactant molecules are converted into product molecules at the fluid-solid interface. To use the catalyst efficiently, we must ensure that fresh reactant molecules are supplied and product molecules removed continuously. Otherwise, chemical equilibrium would be established in the fluid adjacent to the surface, and the desired reaction would proceed no further. Ordinarily, supply and removal of the species in question depend on two physical rate processes in series. These processes involve mass transfer between the bulk fluid and the external surface of the catalyst and transport from the external surface to the internal surfaces of the solid. The concept of effectiveness factors developed in Section 12.3 permits one to average the reaction rate over the pore structure to obtain an expression for the rate in terms of the reactant concentrations and temperatures prevailing at the exterior surface of the catalyst. In some instances, the external surface concentrations do not differ appreciably from those prevailing in the bulk fluid. In other cases, a significant concentration difference arises as a consequence of physical limitations on the rate at which reactant molecules can be transported from the bulk fluid to the exterior surface of the catalyst particle. Here, we discuss... 

Since the absolute thickness of the effective hydrodynamic boundary layer is very small, below a particular size range minimum, no hydrodynamic effects are perceived experimentally with varying agitation. This, however, does not mean, that there are no such influences Further, the mechanisms of mass transfer and dissolution may change for very small particles depending on a number of factors, such as the fluid viscosity, the Sherwood number (the ratio of mass diffusivity to molecular diffusivity), and the power input per unit mass of fluid. 

Petersen [12] points out that this criterion is invalid for more complex chemical reactions whose rate is retarded by products. In such cases, the observed kinetic rate expression should be substituted into the material balance equation for the particular geometry of particle concerned. An asymptotic solution to the material balance equation then gives the correct form of the effectiveness factor. The results indicate that the inequality (23) is applicable only at high partial pressures of product. For low partial pressures of product (often the condition in an experimental differential tubular reactor), the criterion will depend on the magnitude of the constants in the kinetic rate equation. 

The importance of the wetting efficiency results mainly from the fact that it is closely related to the reaction yield, and more specifically to the catalyst effectiveness factor (Burghardt et al., 1995). The reaction rate over incompletely covered catalytic particles can be smaller or greater than the rate observed on completely wetted packing, depending on whether the limiting reactant is present only in the liquid-phase or in both gas and liquid-phases. 

The quantity /is a numerical factor that depends on the ratio a/K, where a is the radius of the spherical or cylindrical particle. In other words, / depends on the ratio of the radius of the particle to the effective thickness K 1 of the diffuse layer. When alKX is large, (the particle large in comparison with the diffuse-charge thickness), the numerical factor is always equal to irrespective of the shape of the particle. When the particle is small compared with the thickness of the double layer,/is i for cylindrical particles parallel to the field and for spherical particles (Fig. 6.140). 

When the pore-diffusion is limiting, the effectiveness factor is rj = Thiele modulus, (p = rtJ3 (k/DP) 12, m being the radius of the catalyst particle, and DP the coefficient of pore-diffusion. The overall rate of the process depends then on the reciprocal modulus,

[Pg.77]

Diffusion within the catalyst particle can be accounted for by using an effectiveness factor, tj, but x should no longer be defined to make k(x) linear in x. Rather, it would be sensible to make tjk linear in x. Of course, straightening out the monotone dependence on x of one parameter, distorts the distribution, as in equation (4), and it may be better to think of the Damkohler number as a function of x. We can alway write it as Da.u>(x), where a>(x) has a mean value of 1. 

The design of a gas-liquid-solid reactor is very much dependent upon the size of the solid particles chosen for the reaction and the anticipated value of the effectiveness factor is one of the most important considerations. Generally, the smaller the particle size the closer the effectiveness factor will be to unity. Particles smaller than about 1 mm in diameter cannot, however, be used in the form of a fixed bed. There would be problems in supporting a bed of smaller particles the pressure drop would be too great and perhaps, above all, the possibility of the interstices between the particles becoming blocked too troublesome. There may, however, be other good reasons for choosing a fixed-bed type of reactor. 

As before, we are interested in the effectiveness factor r/ of the pellet and its dependence on T. For a spherical particle the effectiveness factor rj is defined as... 

A final point worth mentioning is the effect of local fields on the optical nonlinearities of strongly QC nanostructures. These arise from embedding QD s in a medium of different dielectric constant (2). One requires to know how the field intensity inside the particle varies at saturation in excitonic absorption. This has been approached theoretically by defining a local field factor f such that Em = f Eout (2). The factor f depends on the shape of the QD and the dielectric constant of the QD e = + E2 relative to that of the surrounding medium. Here... 

The diffusion of small particles depends upon many factors. In addition to Brownian motion, we must consider the effect of gravity and the motion of the fluid in which the particles reside. Ordinary diffusion as understood in colloid chemistry must be modified considerably when we deal with turbulence. However, we still retain the usual definition of diffusion, namely that it is the number or mass of particles passing a unit cross section of the fluid in unit-time and unit-concentration gradient. That is, if dw particles (or mass) move through an area / in time dt and dC/dx is the concentration increase in the jc-directior then... 

Respiratory Effects. The hazard from inhaled uranium aerosols, or from any noxious agent, is the likelihood that the agent will reach the site of its toxic action. Two of the main factors that influence the degree of hazard from toxic airborne particles are 1) the site of deposition in the respiratory tract of the particles and 2) the fate of the particles within the lungs. The deposition site within the lungs depends mainly on the particle size of the inhaled aerosol, while the subsequent fate of the particle depends mainly on the physical and chemical properties of the inhaled particles and the physiological status of the lungs. 

The concentration profile and the effectiveness factor are clearly dependent on the geometry of a catalyst particle. Table 6.3.1 summarizes the results for catalyst particles with three common geometries. 

Recall that k T) is a strong function of temperature. The effectiveness factor for an exothermic reaction can be le than, equal to, or greater than unity, depending on how k T) increases relative to F(Q) within the particle. Thus, there are cases where the increase in k T) can be much larger than the decrease in F Ca), for example,... 

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