Film effectiveness factor

By comparing this relationship with the solution for the effectiveness factor in absence of interphase concentration gradients (cq 51), it becomes obvious that the overall effectiveness factor t] can be expressed as the product of separate pore and external (film) effectiveness factors ... 

The conservation equations for mass and enthalpy for this special situation have already been given with eqs 76 and 62. As there is no diffusional mass transport inside the pellet, the overall catalyst effectiveness factor is identical to the film effectiveness factor i/cxl which is defined as the ratio of the effective reaction rate under surface conditions divided by the intrinsic chemical rate under bulk fluid phase conditions (see eq 61). For an nth order, irreversible reaction we have the following expression ... 

The subscript, G, refers to a global —particle + film—effectiveness factor, which includes both resistances, and which reduces to Eq. 3.6.a-6 for x. Equa-... 

Here, kQ is the mass transfer coefficient describing the diffusion resistance and H represents the membrane permeability. Following [129], an expression can be derived relating the film effectiveness factor to these two dimensionless variables. High values of O indicate high mass transfer resistance of the film compared to the membrane and thus strong concentration polarization. 

Figure 7.8 Film effectiveness factor rj according to Eq. (7.20) as a function of for the two extreme cases zero hydrogen partial pressure in the permeate side (p = 0) and almost identical partial hydrogen pressure on both sides (p = 0.999)...

It is usually possible to investigate very thin films (up to the subnanometer range) by use of infrared wavelengths, which are much greater than the thickness of the film (a factor of 10000) because of the interference optics of the strong oscillator (Berreman effect). 

The quantitative solution to the problem is given in section 11.3. The effectiveness factor T)P (< 1) which expresses the extent to which the promoting ion is fully utilized (qP=l) depends on three dimensionless parameters n, J and P n is the dimensionless dipole moment of the promoting ion, J is a dimensionless current and P, a promotional Thiele modulus, is proportional to the film thickness, L. 

II. The promotional effectiveness factor, t]p, (Chapter 11) must be significant, larger than, at least, 0.1. This requires small promotional Thiele modulus, Op, and significant dimensionless current, J, values. This implies thin (low L) catalyst films and slow kinetics of promoter destmction (low k values, Chapter 11). 

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 pore diffusion is controlUng, we repeat the effectiveness factor calculations in Chapter 10. Equation (10.29) has the form of Equation (11.48), and it includes both film resistance and pore diffusion. 

Actually, it is recognized that two different mechanisms may be involved in the above process. One is related to the reaction of a first deposited metal layer with chalcogen molecules diffusing through the double layer at the interface. The other is related to the precipitation of metal ions on the electrode during the reduction of sulfur. In the first case, after a monolayer of the compound has been plated, the deposition proceeds further according to the second mechanism. However, several factors affect the mechanism of the process, hence the corresponding composition and quality of the produced films. These factors are associated mainly to the com-plexation effect of the metal ions by the solvent, probable adsorption of electrolyte anions on the electrode surface, and solvent electrolysis. 

Use of bioflocs rather than supported film particles will maximize the effectiveness factor for a given particle, but uneven growth of floes can cause severe stratification in the bed. If stratification can be overcome by methods such as the use of a tapered bed to control porosity the removal, breaking up, and recycle of biomass at the bottom of the bed or, ideally, the use of microbial strains or species that will stop growing at a desirable floe size, such as a Zymomonas mobilis strain that stops growing at one millimeter in diameter (Scott, 1983), the use of bioflocs rather than support particles can improve reactor productivity. 

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... 

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. 

It is possible that the pores of wetted catalyst particles eire filled with liquid. Hence, by virtue of the low values of liquid diffusivities (ca. 10 cm s" ), the effectiveness factor will almost certainly be less than unity. A criterion for assessing the importance of mass transfer in the trickling liquid film has been suggested by Satterfield [40] who argued that if liquid film mass transport were important, the rate of reaction could be equated to the rate of mass transfer across the liquid film. For a spherical catalyst particle with diameter dp, the volume of the enveloping liquid fim is 7rdp /6 and the corresponding interfacial area for mass transfer is TTdn. Hence... 

In the preceding expression we include an effectiveness factor r to account for pore diffusion limitations of A. Hi fact, if the catalyst film thickness on the wall of the reactor is small enough that we can assume it planar, then the effectiveness factor becomes... 

The resistance to mass transfer of reactants within catalyst particles results in lower apparent reaction rates, due to a slower supply of reactants to the catalytic reaction sites. Ihe long diffusional paths inside large catalyst particles, often through tortuous pores, result in a high resistance to mass transfer of the reactants and products. The overall effects of these factors involving mass transfer and reaction rates are expressed by the so-called (internal) effectiveness factor f, which is defined by the following equation, excluding the mass transfer resistance of the liquid film on the particle surface [1, 2] ... 

Reinius (R4), 1961 Studies of water flows in open channels at small slopes, Nr, = 50-13,000. Data on film thicknesses, film friction factors, effects of wall roughness. 

In the case of gel entrapped biocatalysts, or where the biocatalyst has been immobilised in the pores of the carrier, then the reaction is unlikely to occur solely at the surface. Similarly, the consumption of substrate by a microbial film or floe would be expected to occur at some depth into the microbial mass. The situation is more complex than in the case of surface immobilisation since, in this case, transport and reaction occur in parallel. By analogy with the case of heterogeneous catalysis, which is discussed in Chapter 3, the flux of substrate is related to the rate of reaction by the use of an effectiveness factor rj. The rate of reaction is itself expressed in terms of the surface substrate concentration which in many instances will be very close to the bulk substrate concentration. In general, the flux of substrate will be given by ... 

Figure 15. (a) Effectiveness factor, t), as a function of c ) for various aspect ratios. (b) Examples of film thickness variations for different values of ( >. 

According to this equation, the effectiveness factor is controlled by two terms, namely the ratio of the rate constants ks/kb, governed by the temperature difference over the external fluid film, and the ratio of the surface concentration versus the bulk concentration cs/cb- Defining equations for both of these terms have already been given with eqs 71 and 79. Substituting these into eq 94 and using eq 81, we obtain the effectiveness factor for arbitrary reaction order as a function of the observable variable rjDa ... 

Ca relates the concentration difference over the film to procurable quantities and is therefore a so-called observable [4]. A criterion for the absence of extra-particle gradients in the rate data can be derived from the definition of an effectiveness factor for a particle. This should not deviate more than 5% from unity ... 

Oxidation-reduction cycle. A sample of freshly electropolished polycrystalline copper with an initial roughness factor of almost unity was treated by an oxidation-reduction cycle in which an oxide film effective thickness of 200 A. was formed and removed. This was done (1) to illustrate the usefulness of the adsorption technique for defining the... 

It plays the same role as the effectiveness factor in heterogeneous catalysis and is a measure of the film thickness uniformity. It represents the ratio of the total reaction rate on each pair of wafers to that we would obtain if the concentration in the cell formed by the two wafers were equal to the bulk concentration everywhere. Thus, if the surface reaction is the rate controlling step, n = 1, whereas if the diffusion between the wafers controls, n < 1. In the limit of strong diffusion resistance the deposition is confined to a narrow outer band of the wafers and a strongly nonuniform film results. 

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