External effectiveness factors

As the reaction rate increases in the region of low effectiveness factors, external mass transfer becomes important and eventually controls the overall rate. Since bulk diffusivity in gases depends on the apparent activation energy will show a further decrease, and the plot of rate vs. jT will be as shown in Figure 4.9. Sometimes the transition from the pore diffusion region to the external mass transfer region occurs over a small... 

V Ilex- Overall effectiveness factor, external, of porous catalyst —... 

Intraparticle mass transport resistance can lead to disguises in selectivity. If a series reaction A — B — C takes place in a porous catalyst particle with a small effectiveness factor, the observed conversion to the intermediate B is less than what would be observed in the absence of a significant mass transport influence. This happens because as the resistance to transport of B in the pores increases, B is more likely to be converted to C rather than to be transported from the catalyst interior to the external surface. This result has important consequences in processes such as selective oxidations, in which the desired product is an intermediate and not the total oxidation product CO2. 

Rates and selectivities of soHd catalyzed reactions can also be influenced by mass transport resistance in the external fluid phase. Most reactions are not influenced by external-phase transport, but the rates of some very fast reactions, eg, ammonia oxidation, are deterrnined solely by the resistance to this transport. As the resistance to mass transport within the catalyst pores is larger than that in the external fluid phase, the effectiveness factor of a porous catalyst is expected to be less than unity whenever the external-phase mass transport resistance is significant, A practical catalyst that is used under such circumstances is the ammonia oxidation catalyst. It is a nonporous metal and consists of layers of wire woven into a mesh. 

An immobilized enzyme-carrier complex is a special case that can employ the methodology developed for evaluation of a heterogeneous cat ytic system. The enzyme complex also has external diffusional effects, pore diffusional effects, and an effectiveness factor. When carried out in aqueous solutions, heat transfer is usually good, and it is safe to assume that isothermal conditions prevail for an immobihzed enzyme complex. 

A hydrocarbon is cracked using a silica-alumina catalyst in the form of spherical pellets of mean diameter 2.0 mm. When the reactant concentration is 0.011 kmol/m3, the reaction rate is 8.2 x 10"2 kmol/(m3 catalyst) s. If the reaction is of first-order and the effective diffusivity De is 7.5 x 10 s m2/s, calculate the value of the effectiveness factor r). It may be assumed that the effect of mass transfer resistance in the. fluid external Lo the particles may be neglected. 

Few fixed-bed reactors operate in a region where the intrinsic kinetics are applicable. The particles are usually large to minimize pressure drop, and this means that diffusion within the pores. Steps 3 and 7, can limit the reaction rate. Also, the superficial fluid velocity may be low enough that the external film resistances of Steps 2 and 8 become important. A method is needed to estimate actual reaction rates given the intrinsic kinetics and operating conditions within the reactor. The usual approach is to define the effectiveness factor as... 

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. ... 

Consider a nonporous catalyst particle where the active surface is all external. There is obviously no pore resistance, but a film resistance to mass transfer can still exist. Determine the isothermal effectiveness factor for first-order kinetics. 

If diffusion of reactants to the active sites in pores is slower than the chemical reaction, internal mass transfer is at least partly limiting and the reactant concentration decreases along the pores. This reduces the reaction rate compared to the rate at external surface conditions. A measure of the reaction rate decrease is the effectiveness factor, r, which has been defined as ... 

C. Holmstrom and S. Kjelleberg, The effect of external biological factors on settlement of marine invertebrate and new antifouling technology. Biofouling, 1994, 8, 147. 

All of these steps are rate processes and are temperature dependent. It is important to realize that very large temperature gradients may exist between active sites and the bulk gas phase. Usually, one step is slower than the others, and it is this rate-controlling step. The effectiveness factor is the ratio of the observed rate to that which would be obtained if the whole of the internal surface of the pellet were available to the reagents at the same concentrations as they have at the external surface. Generally, the higher the effectiveness factor, the higher the rate of reaction. 

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. 

In the limit of low effectiveness factors where tj becomes inversely proportional to the Thiele modulus, the apparent order of the reaction may differ from the true order. In this case, since the rate is proportional to the product of the effectiveness factor and the external concentration... 

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... 

At this point it is instructive to consider the possible presence of intraparticle and external mass and heat transfer limitations using the methods developed in Chapter 12. In order to evaluate the catalyst effectiveness factor we first need to know the combined diffusivity for use... 

The activity calculated from (7) comprises both film and pore diffusion resistance, but also the positive effect of increased temperature of the catalyst particle due to the exothermic reaction. From the observed reaction rates and mass- and heat transfer coefficients, it is found that the effect of external transport restrictions on the reaction rate is less than 5% in both laboratory and industrial plants. Thus, Table 2 shows that smaller catalyst particles are more active due to less diffusion restriction in the porous particle. For the dilute S02 gas, this effect can be analyzed by an approximate model assuming 1st order reversible and isothermal reaction. In this case, the surface effectiveness factor is calculated from... 

For a more detailed analysis of measured transport restrictions and reaction kinetics, a more complex reactor simulation tool developed at Haldor Topsoe was used. The model used for sulphuric acid catalyst assumes plug flow and integrates differential mass and heat balances through the reactor length [16], The bulk effectiveness factor for the catalyst pellets is determined by solution of differential equations for catalytic reaction coupled with mass and heat transport through the porous catalyst pellet and with a film model for external transport restrictions. The model was used both for optimization of particle size and development of intrinsic rate expressions. Even more complex models including radial profiles or dynamic terms may also be used when appropriate. 

The expression for the effectiveness factor q in the case of zero-order kinetics, described by the Michaelis-Menten equation (Eq. 8) at high substrate concentration, can also be analytically solved. Two solutions were combined by Kobayashi et al. to give an approximate empirical expression for the effectiveness factor q [9]. A more detailed discussion on the effects of internal and external mass transfer resistance on the enzyme kinetics of a Michaelis-Menten type can be found elsewhere [10,11]. 

The value of the extractable lipid measurement is two-fold. First, it indicates how much lipid has to be removed in the subsequent clean-up process and second, it allows the levels of organic pollutants in the matrix to be expressed on a lipid basis. This normalization reduces the differences among samples purely as a result of the lipid in the sample and the effect of external factors that affect lipid levels. 

Zone II combustion proceeds with partial penetration of oxygen, resulting in simultaneous variations in particle density and diameter as the pores closest to the particle surface undergo oxidation, in addition to the external surface of the particle. The ratio of the actual burning rate to the maximum possible rate if the entire particle was subject to the oxygen partial pressure at the external particle surface is known as the effectiveness factor. 

The study of the intra-phase mass transfer in SCR reactors has been addressed by combining the equations for the external field with the differential equations for diffusion and reaction of NO and N H 3 in the intra-porous region and by adopting the Wakao-Smith random pore model to describe the diffusion of NO and NH3 inside the pores [30, 44]. The solution of the model equations confirmed that steep reactant concentration gradients are present near the external catalyst surface under typical industrial conditions so that the internal catalyst effectiveness factor is low [27]. 

There are a number of examples of tube waU reactors, the most important being the automotive catalytic converter (ACC), which was described in the previous section. These reactors are made by coating an extruded ceramic monolith with noble metals supported on a thin wash coat of y-alumina. This reactor is used to oxidize hydrocarbons and CO to CO2 and H2O and also reduce NO to N2. The rates of these reactions are very fast after warmup, and the effectiveness factor within the porous wash coat is therefore very smaU. The reactions are also eternal mass transfer limited within the monohth after warmup. We wUl consider three limiting cases of this reactor, surface reaction limiting, external mass transfer limiting, and wash coat diffusion limiting. In each case we wiU assume a first-order irreversible reaction. 

Besides the impressive difference in the chemiexcitation efficiency, also the fluorescence yield of the meta-pattemed emitter m-17 is by more than an order of magnitude ( ) higher than that of the para regioisomer p-17 . Evidently, crossed-conjugated emitters are advantageous for the design of efficient intramolecular CIEEL systems. In Sections V.A-V.C we shall consider additional internal (substrate structural effects) and external (medium influence) factors, which play an essential role in the development of efficient dioxetane-based analytical probes. 

The external mass-transfer effects on the activity of an itmnobilized biocatalyst can be expressed quantitatively by the external effectiveness factor defined as the ratio of... 

Figure 11.19 shows the dependence of the external effectiveness factor on and Da. Similar plots have been obtained by other authors (Mosbach, 1976). 

Figure 11.19 Plots of the external effectiveness factor as a function of the substrate modulus Da for different values of the dimensionless bulk substrate concentration is the limiting first-order effectiveness factor attained at sufficiently low concentrations. Adapted from C.Horvath and J.M.Engasser. Biotechnol.Bioeng., 16, 909 (1974).

For the first order reaction, the external effectiveness factor has an analytical solution, which is given by ... 

It is useful to introduce the external effectiveness factor i]c . as the ratio of the observed overall rate r to the chemical reaction rate r without diffusion resistance (C = C,) ... 

A high Damkohler number means that the global rate is controlled by mass transfer phenomena. So, the process rate can be rewritten in terms of the Damkohler number and the external effectiveness factor for each reaction order can be deduced, as shown in Table 5.5. In Figure 5.3, the external effectiveness factor versus the Damkohler number is depicted for various reaction orders. It is clear that the higher the reaction order, the more obvious the external mass transfer limitation. For Damkohler numbers higher than 0.10, external mass transfer phenomena control the global rate. In the case of n = 1, the external effec-... 

The global rate expression and the external effectiveness factor for an isothermal catalytic reaction... 

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