Pore effectiveness factor

The first term in parentheses in eq 60, together with the preceding factor 3/, is equivalent to eq 51 this represents the pore effectiveness factor whereas the second expression in parentheses denotes the external effectiveness factor rjtM. 

From this figure, it can be concluded that the reduction of the effectiveness factor at large values of becomes more pronounced as the Biot number is decreased. This arises from the fact that the reactant concentration at the external pellet surface drops significantly at low Biot numbers. However, a clear effect of interphase diffusion is seen only at Biot numbers below 100. In practice, Bim typically ranges from 100 to 200. Hence, the difference between the overall and pore effectiveness factor is usually small. In other words, the influence of intraparticle diffusion is normally by far more crucial than the influence of interphase diffusion. Thus, in many practical situations the overall catalyst efficiency may be replaced by the pore efficiency, as a good approximation. 

The intraparticle concentration and temperature gradients in a porous particle can always be neglected, when the pore effectiveness factor rj is close to 1. Assuming that rj... 

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. 

For this case, the pore effectiveness factor is a function of the so-called Thiele modulus m (Eq. 22) [222] ... 

We introduce another factor for porous bodies, it is called the pore effectiveness factor rjp. This is understood to be the ratio of the actual amount of substance transferred NA0 to the amount which would be transferred NA, if the concentration cao prevailed throughout the porous body. This basically means, if the effective diffusion coefficient DeS, which is used in place of the molecular diffusion coefficient D, was very large DeS — oo, then Ha — 0. Then according to (2.367) the reaction rate of substance A... 

The pore effectiveness factor is valid for pores with constant cross sectional area. It corresponds to the fin efficiency for a straight, rectangular fin, eq. (2.79). The pore effectiveness factor for pores of any cross sectional shape, can also be calculated, as a fairly good approximation using (2.377), as shown by Aris [2.81], if the length L is formed as a characteristic length... 

Volume fraction of i phase in support pores. Effectiveness factor. 

The pore effectiveness factor is given by the ratio of the moles of reactant consumed in the presence of diffusion resistance, to the moles of reactant consumed in the absence of diffusion resistance. The single pore effectiveness is used to calculate the effectiveness factor for the network... 

Up to now, we have only considered particles with one size of pore. In reality, we may have a wide spread of pore sizes or a micro-macroporous material (see Example 4.5.11 for the pore effectiveness factor in this case). Pellets are often prepared by compressing a porous powder, and thus we may get at least two pore sizes, large macropores between the agglomerated particles and small micropores within each particle. The micropores can be considered to be in series with the macropores and only the latter communicate with the external particle surface and the bulk phase of the fluid. For a strong resistance to both macro- and micropore diffusion, the following equation is obtained for the maximum selectivity of the intermediate, if external mass transfer limitations are neglected (Carberry, 1962 Froment and Bischoff, 1990 Levenspiel, 84,83) ... 

To calculate the pore effectiveness factor of pellets that have macro- and micropores, we use the following simple model and respective assumptions ... 

For spherical microporous grains and an irreversible first-order reaction, the pore effectiveness factor is to a good approximation, see Table 4.5.5, given by ... 

For the macroporous system of the spherical pellet, we obtain a pore effectiveness factor of ... 

Figure 6.1.6 Influence of temperature and particle size on pore effectiveness factor (a) 30 bar, particle diameter 2.6 mm [data from Bokhoven and Raayen (1954)] (b) 100 bar, 500°C, differential conversion, feed gas with 4% NH3, H2/N2 = 3 [data from Jennings and Ward (1989)].

In industrial reactors, particle diameters in the range 1-10 mm are used (Appl, 1999). Thus, pore diffusion of the reactants and of ammonia may influence the effective rate as discussed by Akehata et al. (1961), Bokhoven and van Raayen (1954), and by Jennings and Ward (1989) [see also Appl (1999) and Nielsen, (1971)]. The ratio of the effective rate to the intrinsic (maximum) rate in the absence of internal mass transport restrictions is characterized by the pore effectiveness factor >jpore (Section 4.5.4). Figure 6.1.6 shows values of >jpore determined in a laboratory reactor at different temperatures and particle sizes. For technically relevant temperatures of 400-500 °C and particles up to 10 mm, rjpore is in a range of 1 down to 0.2. 

Figure6.1.7 ContentofNHs and pore effectiveness factor at different axial positions in a hypothetical (isothermal) fixed bed reactor [450°C, 214 bar, 12% inerts in feed gas, particle diameter 5.7 mm, space velocity 15 000 m gas (STP) h m catalyst data from Nielsen (1971)].

From the standpoint of a high effective reaction rate, it is desirable to use very fine particles as we then have a pore effectiveness factor approaching unity. However, we have to consider the pressure drop in a technical reactor (Figure 6.1.8). 

FIG U RE 3.28 The impact of pore length and metal charging properties on the pore effectiveness factor Tpore For a reference set parameter with = 0.7 V versus SHE, Rp = 5 nm, and... 

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