External effectiveness factor reaction

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... 

Reaction Overall rate (rov) External effectiveness factor (f/ex)... 

Figure 5.3 The external effectiveness factor for reaction order n.

For positive reaction orders the existence of significant concentration differences between the bulk of the fluid phase and the external surface of the catalyst pellet leads to lower reaction rates. This can be expressed by the external effectiveness factor, r e ... 

Figure 7.9 shows the temperature and concentration profiles caused by transfer limitations in the case of an endothermic reaction. For an exothermic reaction the temperature at the external surface, T,-, increases with increasing transfer limitations. Hence the external effectiveness factor, Tie, becomes greater than 1 as soon as the decrease of the reactant concentration is compensated by the increase of the temperature. 

The external effectiveness factor is expressed as the ratio of the observed rate to the reaction rate at bulk fluid conditions... 

The first step in heterogeneous catalytic processes is the transfer of the reactant from the bulk phase to the external surface of the catalyst pellet. If a nonporous catalyst is used, only external mass and heat transfer can influence the effective rate of reaction. The same situation will occur for very fast reactions, where the reactants are completely exhausted at the external catalyst surface. As no internal mass and heat transfer resistances are considered, the overall catalyst effectiveness factor corresponds to the external effectiveness factor,... 

Figure 11.3 Isothermal external effectiveness factor as a function of the second Damkohler number and different reaction orders, n.

The external effectiveness factors as function of the second Damkohler number are obtained by solving Equation 2.141. This is done for reaction orders = 1, 2, i/2> and -1 and displayed in Figure 2.20 [27]. 

Whereas Figure 2.20 is quite instructive, it is not of practical use for estimating the importance of the mass transfer influence from experimental data, as the intrinsic rate constant is normally unknown. Replotting the effectiveness factor as function of the ratio between observed reaction rate to the maximum mass transfer rate called as Carberry number Ca) allows estimating the external effectiveness factor plotted in Figure 2.21. 

The isothermal external effectiveness factor stated for the case of negligible (Ts Ty) carries only the ACa effect and is based on i-RA)p=JiCAs> Tb) and (-RA)b=f(,CAb, Ty). Keeping the first-order surface reaction assumption, the following simple expression is found for y ... 

Topic 4.5.5 External effectiveness factor for an exothermic heterogeneously catalyzed gas reaction... 

Figure 4.5.15 External effectiveness factor as a function ofthe ratio ofthe (measurable) effective reaction rate to the maximum rate (complete control by external mass transfer) fora constant Arrhenius number of 20. For a Prater number jSex < 0, the reaction is endothermic, for /Sex > 0 exothermic, and for /Sex = 0 we have isothermal conditions. Arrows and dashed line indicate ignition, as explained in the text.

The influence of the internal effectiveness factor, t, on global rate thus has similarities to that of the external effectiveness factor, fj, in that a) the higher the reaction order, the greater the diffusional effect b) t unity for small values of the Thiele modulus, (/>, and similarly, fj unity for small values of the Damkohler number, Dao and c) at large values of these two moduli, T = l/(/)(for 0 > 3) and fj = 1/Dao. Assuming that external mass transfer limitations have been removed (Cg = Co), the effect of internal (pore) diffusion on the observed kinetics can be determined i.e., for cf) > 3, i] = l/4> and... 

Figure 4.6 Nonisothermal external effectiveness factors for first-order reactions. (Carberry and Kulkarini 1973. Reprinted with permission from Journal of Catalysis. Copyright by Academic Press.)...

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. 

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. ... 

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 ... 

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... 

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 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. 

---