Effectiveness factor, particle overall

Where, tIc is overall catal54ic effectiveness factor. The overall catalytic effectiveness factor for a spherical catalyst particle can be expressed 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. ... 

The effect of catalyst particle size was investigated by two different catalyst particle size fractions 63-93 pm and 150-250 pm, respectively. The effect of the particle size is very clear as demonstrated by Figure 47.2. The overall hydrogenation rate was for smaller particles 0.17 mol/min/gNi while it was 0.06 mol/min/gNi, for the larger particles, showing the presence of diffusion limitation. This kind of studies can be used to determine the effectiveness factors. The conversion levels after 70 min time-on-stream were 21% and 3%, respectively, for these two cases. 

We consider the effects of cA and Tseparately, deferring the latter to Section 8.5.5. In focusing on the particle effectiveness factor, we also ignore the effect of any difference in concentration between bulk gas and exterior surface (cAg and cAs.) in Section 8.5.6, we introduce the overall effectiveness factor to take this into account. 

The particle effectiveness factor 17 defined by equation 8.5-5 takes into account concentration and temperature gradients within the particle, but neglects any gradients from bulk fluid to the exterior surface of the particle. The overall effectiveness factor y)0 takes both into account, and is defined by reference to bulk gas conditions (cA, Tg) rather than conditions at the exterior of the particle (cAj, Ts) ... 

The starting points for the continuity and energy equations are again 21.5-1 and 21.5-6 (adiabatic operation), respectively, but the rate quantity7 (—rA) must be properly interpreted. In 21.5-1 and 21.5-6, the implication is that the rate is the intrinsic surface reaction rate, ( rA)int. For a heterogeneous model, we interpret it as an overall observed rate, (—rA)obs, incorporating the transport effects responsible for the gradients in concentration and temperature. As developed in Section 8.5, these effects are lumped into a particle effectiveness factor, 77, or an overall effectiveness factor, r]0. Thus, equations 21.5-1 and 21.5-6 are rewritten as... 

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... 

Overall, the most effective factor in Equation 5.20 is the particle size. The smaller the particle size, the higher the column efficiency. Equations 5.13, 5.15, and 5.18 are depicted in Figure 5.6 against flow velocity as A, B, and C, respectively. The band spreading is thus influenced by Equation 5.15 at a low flow rate. The band spreading is influenced by Equations 5.18 and 5.19 at a high flow rate. For gas chromatography curve D is obtained. 

As mentioned earlier, if the rate of a catalytic reaction is proportional to the surface area, then a catalyst with the highest possible area is most desirable and that is generally achieved by its porous structure. However, the reactants have to diffuse into the pores within the catalyst particle, and as a result a concentration gradient appears between the pore mouth and the interior of the catalyst. Consequently, the concentration at the exterior surface of the catalyst particle does not apply to die whole surface area and the pore diffusion limits the overall rate of reaction. The effectiveness factor tjs is used to account for diffusion and reaction in porous catalysts and is defined as... 

It is possible to combine the resistances of internal and external mass transfer through an overall effectiveness factor, for isothermal particles and first-order reaction. Two approaches can be applied. The general idea is that the catalyst can be divided into two parts its exterior surface and its interior surface. Therefore, the global reaction rates used here are per unit surface area of catalyst. 

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] ... 

In the case that the mass transfer effects are not negligible, the required height of the packed bed is greater than that without mass transfer effects. In some practices, the ratio of the packed-bed heights without and with the mass transfer effects is defined as the overall effectiveness factor (-), the maximum value of which is unity. However, if the right-hand side of Equation 7.47 is multiplied it cannot be integrated simply, since is a function of C, U, and other factors. If the reaction is extremely rapid and the liquid phase mass transfer on the particle surface controls the overall rate, then the rate can be estimated by... 

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]

The overall rate of reaction calculated for the three-phase fluidised-bed reactor above is approximately one tenth of the rate calculated for the agitated tank slurry reactor in Example 4.6. The main reasons are the very poor effectiveness factor and the relatively smaller external surface area for mass transfer caused by using the larger particles. Even the gas-liquid transfer resistance is greater for the three-phase fluidised-bed, in spite of the larger particles being able to produce relatively small bubbles these bubbles are not however as small as can be produced... 

For the particle sizes used in industrial reactors (> 1.5 mm), intraparticle transport of the reactants and ammonia to and from the active inner catalyst surface may be slower than the intrinsic reaction rate and therefore cannot be neglected. The overall reaction can in this way be considerably limited by ammonia diffusion through the pores within the catalysts [211]. The ratio of the actual reaction rate to the intrinsic reaction rate (absence of mass transport restriction) has been termed as pore effectiveness factor E. This is often used as a correction factor for the rate equation constants in the engineering design of ammonia converters. 

To calculate the effectiveness factor, the actual reaction rate throughout the catalyst is divided by the rate determined at the conditions of the external surface, that is, Rate(X,5, X25). The overall reaction rate throughout the particle is equivalent to the flux X, evaluated at the external surface of the catalyst. Thus, the final solution is ... 

It is not easy to compare the activity of the V-W-Ti catalysts here tested with the lot of chromia, Pt and Pd based catalysts previously used because they have different shapes (monoliths and spheres) and because very different particle sizes arc involved (having thus very different effectiveness factors). For conqiarison purposes, all X-T curves were adjusted to a simple fust order kinetic model (with rate based on overall volume of catalyst, both for monoliths and for fixed beds). From the kinetic constants so obtained (see details of the method in ref 7), the preexponential factors (ko) of the Arrhenius law and the apparent energies of activation (E, p) were calculated for all catalysts. One example is shown in Figure 17. By the well Imown compensation effect between ko and E,pp, the kg values so obtained were recalculated for a given E.pp value of 44 kJ/mol. Such new ko value was used [7] as an activity index of the catalyst. 

Intraparticle Diffusion and External Mass-Transfer Resistance For typical industrial conditions, external mass transfer is important only if there is substantial intraparticle diffusion resistance. This subject has been discussed by Luss, Diffusion-Reaction Interactions in Catalyst Pellets, in Carberry and Varma (eds.), Chemical Reaction and Reactor Engineering, Dekker, 1987. This, however, may not be the case for laboratory conditions, and care must be exerted in including the proper data interpretation. For instance, for a spherical particle with both external and internal mass-transfer limitations and first-order reaction, an overall effectiveness factor r, can be derived, indicating the series-of-resistances nature of external mass transfer followed by intraparticle diffusion-reaction ... 

Closure After completing this chapter, the reader should be able to derive differential equations describing diffusion and reaction, discuss the meaning of the effectiveness factor and its relationship to the Thiele modulus, and identify the regions of mass transfer control and reaction rate control. The reader should be able to apply the Weisz-Prater and Mears criteria to identify gradients and diffusion limitations. These principles should be able to be applied to catalyst particles as well as biomaierial tissue engineering. The reader should be able to apply the overall effectiveness factor to a packed bed reactor to calculate the conversion at the exit of the reactor. The reader should be able to describe the reaction and transport steps in slurry reactors, trickle bed reactors, fluidized-besd reactors, and CVD boat reactors and to make calculations for each reactor. 

Wang and Yang (1991b) have proposed a more realistic model for triphase catalysis in a batch reactor, where they consider mass transfer of reactants in the bulk aqueous and organic phases, diffusion of reactants within the pores of the solid catalyst particle, and intrinsic reactivities of the ion-ex-change and organic reactions at the active sites within the solid catalyst. An apparent overall effectiveness factor of the catalyst is obtained in this case by applying the pseudo-... 

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