Concept of Effectiveness Factor

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... 

The concept of an effectiveness factor is useful in estimating the reaction rate per catalyst pellet (volume or mass). It is, however, mainly useful for simple reactions and simple kinetics. When there are complex reaction pathways, the concept of effectiveness factor is no longer easily applicable, and species and energy balance equations inside the particle may have to be solved to obtain the reaction rates per unit volume of... 

The concept of effectiveness factor has been developed to calculate the overall reaction rate in terms of the concentration at the external surface C , an nth-order irreversible reaction Vp / (n + 1) kCZ (7-103)... 

Garside, J. (1971) The concept of effectiveness factors in crystal growth. Chemical Engineering Science, 26, 1425-1431. 

Electrode response Substrate addition Concept of effectivity factors Transport limitation Transport enhancement... 

The factor (j) is, after Carman, the surface factor, which is calculated from the particle thickness, grain diameter, and specific particle surface area. With this, an approximate value for /cl2 can be estimated using Equ. 4.1a. For biological films, the dependence of /cl2 cn the flow velocity has been found to be 0.7 (La Motta, 1976a). A preliminary estimate can be made of the relative importance of the individual transport steps in G L S process, using the concept of effectiveness factors tj (cf. Sect. 4.5 and Equ. 2.50). 

Figure 4.36. Graphical representation of the concept of effectiveness factors (a) The effectiveness factor of reaction, rj, as a function of the Damkoehler number of the second kind, Dan, or of the Thiele modulus, (j) (cf. Equ. 4.74). (b) The effective reaction rate, r ff, as a function of the diameter of the particle, d. Part (b) can be used to obtain the value of for a given average diameter, of a population of floes with a distribution d. The range of validity of kinetic control (r js) and of diffusion control (Wfds) is indicated in part a.

Define, evaluate, and use the concepts of effectiveness factor, Thiele modulus, and Weisz modulus. 

First-Order Reactions. The Concept of Effectiveness Factor... 

After the input has been read and sorted, heat transfer coefficients and other thermodunamic data are calculated at the beginning of each catalyst zone. Temperature and conversion profile in the catalyst bed is then calculated by an axial integration. The mathematical model used in the integrations is described in. This model allows in principle the determination of diffusion restrictions and calculation of effectiveness factors for each reaction in cases where several reactions take place simultaneously. In such cases the concept of effectiveness factor may become rather dubious as shown below for the methanol synthesis, and this may be reflected in difficulties in the calculations. 

There is a potential for the forward shift reaction. This means that the shift reaction would run in one direction at bulk conditions but in the other direction inside the particles. Negative effect-ivences factor is therefore calculated. In the lower part of the catalyst bed the equilibrium temperature is in all points above the actual temperature. This means that the direction of the shift reaction is the same in bulk and inside particles, and therefore positive effectiveness factors are calculated. It is clear that in cases as this the concept of "effectiveness factor" has become rather meaningless due to the coupling between the reactions. Mathematically, the asympotic singular point is difficult to handle, and in many cases problems are encountered in the calculations. Therefore, it may be more convenient in cases with strong coupling between reactions as described above to develop kinetic equations valid for whole particles with possible diffusion effects built into the equations. 

Several authors " introduced the concept of effectiveness factor tj, which is the ratio of observed rate to the rate which would result if diflFusion where so rapid as to have no influence at all. 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

This relation is plotted as curve Bin Figure 12.11. Smith (66) has shown that the same limiting forms for are observed using the concept of effective dififusivities and spherical catalyst pellets. Curve B indicates that, for fast reactions on catalyst surfaces where the poisoned sites are uniformly distributed over the pore surface, the apparent activity of the catalyst declines much less rapidly than for the case where catalyst effectiveness factors approach unity. Under these circumstances, the catalyst effectiveness factors are considerably less than unity, and the effects of the portion of the poison adsorbed near the closed end of the pore are not as apparent as in the earlier case for small hr. With poisoning, the Thiele modulus hp decreases, and the reaction merely penetrates deeper into the pore. 

The International Commission on Radiological Protection (1977) introduced the concept of Effective Dose, which takes into account the probability of mortality from cancers in various tissues receiving irradiation. For irradiation of the thyroid, a weighting factor, wT, equal to 0.03... 

As a consequence of diffusion there is a reduction in the reaction rate as we progress inside the catalyst with a result that the overall rate is much less than would be achieved if the reactant were at a concentration as supplied at the outer surface. Thus the catalyst regions are not effectively used and the concept of effectiveness is introduced. Effectiveness is defined as the average reaction rate, i.e., with diffusion, divided by the reaction rate if the rate of reaction is evaluated at the boundary condition value at X = 1. The effectiveness factor can be generally given by... 

Here it is assumed that it is possible to use the concept of an effective diffusion coefficient without making too large an error. Hence the effect of micro properties will not be studied here and it is assumed the value of De is known. The discussion is restricted to the impact of the macro properties and reaction properties on the effectiveness factor. Furthermore only simple reactions are discussed. Generalized formulae are provided that enable calculation of effectiveness factor for varying properties of the catalyst or the reacting system. 

Even the concept of certainty factors is not derived from a formal mathematical basis it is the most commonly used method to describe uncertainty in expert systems. This is mainly due to the fact that certainty factors are easy to compute and can be used to effectively reduce search by eliminating branches with low certainty. However, it is difficult to produce a consistent and accurate set of certainty factors. In addition, they are not consistently reliable a certainty factor may produce results opposite to probability theory. 

In Chapter 11, we explained the concept of winddull factors. We said that the heat transfer rates from your body to the surroundings increase on a cold, windy day. Simply stated, you lose more body heat on the cold, windy day than you do on a calm day. The windchill index accounts for the combined effect ofwind speed and the air temperature. It accoimts for the additional body heat loss that occurs on a cold, windy day. The windchill values are determined empirically, and a common correlation used to determine the windchill index is... 

Finally, hychogen evolution becomes crucial factor determining the shape of powder particles of the group of the inert metals, and the concept of effective overpotential is applicable to explain the formation of these powder particles. Due to vigorous hychogen evolution, dendritic growth is almost completely inhibited. Analysis of the polarization curves for Co and Ni [12, 26, 30] showed that their... 

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