The random pore model

The random pore model, or macro- micro-pore model, of Wakao and Smith [1962, 1964] is intended for application to pellets manufactured by compression of small particles. The void fraction and pore radius distributions are each replaced by two averaged values 8m, I m for the macro and for the micro distribution (often a pore radius of -100 A is used as the dividing point between macro and micro). The particles which contain the micro-pores are randomly positioned in the pellet space. The interstices are the macro-pores of the pellet (see Fig. 3.5.2.1-1). The diffusion flux consists of three parallel contributions the first through the macro-pores, the second through the micro-pores and the third through interconnected macro-micro-pores in which the dominant resistance lies in the latter. The contributions to the diffusivity are added up to yield  

In the second and third terms is based on the microvoid area, so that the value of the ratio of microvoid to total particle area is required. In (3.5.2.1-1), A/and Df, are obtained from (3.4-9), but without correcting for porosity and tortuosity, which are already accounted for in (3.5.2.1-1)  

Diffusion areas in random pore model. Adapted from Smith [1970]. 

Note that the tortuosity factor, t, does not appear in (3.5.2.1-1). For either = 0 or sm= 0, this equation reduces to 

For catalysts without clear distinction between micro- and macro-pores, a different approach is required. 

Wakao and Smith [20] originally developed the random pore model to account for the behaviour of bidisperse systems which contain both micro- and macro-pores. Many industrial catalysts, for example, when prepared in pellet form, contain not only the smaller intraparticle pores, but also larger pores consisting of the voids between compressed particles. Transport within the pellet is assumed to occur through void regions 

The parameters D and Dk whether for macro (denoted by subscript m) or for micro (denoted by subscript ju) regions, are normal bulk and Knudsen diffusion coefficients, respectively, and can be estimated from kinetic theory, provided the mean radii of the diffusion channels are known. Mean radii, of course, are obtainable from pore volume and surface area measurements, as pointed out in Sect. 3.1. For a bidisperse system, two peaks (corresponding to macro and micro) would be expected in a differential pore size distribution curve and this therefore provides the necessary information. Macro and micro voidages can also be determined experimentally. 

For the actual pore-size distribution to be taken into account, the above relations for single pore sizes are usually assumed to remain true, and they are combined with the pore size distribution information. The random pore model, or micromacro pore model, of Wakao and Smith [57,58]) is useful for compressed particle type pellets. The pellet pore-size distribution is, somewhat arbitrarily, broken up into macro (M) and micro (g) values for the pore volume and average pore radius / m and s. (often a pore radius of 1(X) A is used as the dividing point). Based on random placement of the microparticles within the macropellet pores. 

Note again that no tortuosity factor appears in Eq. 3.5.C-1 for either e = 0 or Cl, = 0, it reduces to 

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In the case that the effective diffusion coefficient approach is used for the molar flux, it is given by N = —Da dci/dr), where Dei = (Sp/Tp)Dmi according to the random pore model. Standard boundary conditions are applied to solve the particle model Eq. (8.1). 

Computed from the random pore model of Wakao and Smith (22) and given here at 101.3 kPa and 723 K. 

If data are available on the catalyst pore- structure, a geometrical model can be applied to calculate the effective diffusivity and the tortuosity factor. Wakao and Smith [36] applied a successful model to calculate the effective diffusivity using the concept of the random pore model. According to this, they established that ... 

Fresh Zeolite. The diffusivities within usual porous catalyst (pore radius a few nm) can be estimated by the parallel pore model (18) or the random pore model (19). However, configurational diffusion occurs within the pores of zeolites (pore diameter < 1 nm) and there are only a few reports on the measurement or estimation methods of the diffusivities of zeolites, especially at higher temperature range (20,21). Here we will review the results of ZSM-5, which first explains the diffusivity of fresh ZSM-5, then the results of coke loaded ZSM-5. 

From the characteristics of our reactivity curves (presented later), we selected the random pore model developed by Bhatia and Perlmutter as the model can represent the behaviour of a system that shows a maximum in the reactivity curve as well as that of a system that shows no maximum. The maximum arises from two opposing effects the growth of the reaction surface associated with the growing pores and the loss of surface as pores progressively collapse at their intersections (coalescence). In the kinetically controlled regime, the model equations derived for the reaction surface variation (S/S ) with conversion and conversion-time behaviour are given by ... 

The Random-pore Model This model was originally developed for pellets containing a bidisperse pore system, such as the alumina described in Chap. 8 (Table 8-5 and Fig. 8-10). It is supposed that the pellet consists of an assembly of small particles. When the particles themselves contain pores (micropores), there exists both a macro and a micro void-volume distribu-... 

The random-pore model can also be applied to monodisperse systems. For a pellet containing only macropores, = 0 and Eq. (11-25) becomes... 

Comparison of these last two equations with Eq. (11-24) indicates that 5=1/8. The significance of the random-pore model is that the effective diffusivity is proportional to the square of the porosity. This has also been proposed by Weisz and Schwartz. Johnson and Stewart have developed another method for predicting that utilizes the pore-volume distribution. Evaluation of their model and the random-pore model with extensive experimental data has been carried out by Satterfield and Cadle" and Brown et al. ... 

Thus the random-pore model does not involve an adjustable parameter. 

If we assume that all the diffusion is in the macropores, Eq. (11-28) gives for the random-pore model. Combining Eq. (11-28) with Eq. (11-24) yields... 

Example 11-5 Vycor (porous silica) appears to have a pore system with fewer interconnections than alumina. The pore system is monodisperse, with the somewhat unusual combination of a small mean pore radius (45 A) and a low porosity 0.31. Vycor may be much closer to an assembly of individual voids than to an assembly of particles surrounded by void spaces. Since the random-pore model is based on the assembly-of-particles concept, it is instructive to see how it applies to Vycor. Rao and Smith measured an effective diffusivity for hydrogen of 0.0029 cm /sec in Vycor. The apparatus was similar to that shown in Fig. 11-1, and data were obtained using an H2-N2 system at 25°C and 1 atm. Predict the effective diffusivity by the random-pore model. 

This value is 70% greater than the experimental result—evidence that the random-pore model is not very suitable for Vycor. 

In contrast, the tortuosity predicted by the random-pore model would be 1/e = 3.2.,... 

To obtain r/caic we must estimate the effective dilFusivity. Since the macropores are much larger than the micropores (see and in Table 1T5), it is safe to assume that diffusion is predominantly through the macropores. Then, according to the random-pore model [Eqs. (11-25) and (11-26)],... 

Example 12-2 Using the intrinsic rate equation obtained in Example 12-1, calculate the global rate of the reaction o-Hj p- % at 400 psig and — 196°C, at a location where the mole fraction of ortho hydrogen in the bulk-gas stream is 0.65. The reactor is the same as described in Example 12-1 that is, it is a fixed-bed type with tube of 0.50 in. ID and with x -in. cylindrical catalyst pellets of Ni on AljOj. The superficial mass velocity of gas in the reactor is 15 lb/(hr)(ft ). The effective diffusivity can be estimated from the random-pore model if we assume that diffusion is predominately in the macropores where Knudsen diffusion is insignificant. The macroporosity of the pellets is 0.36. Other properties and conditions are those given in Example 12-1. 

According to the random-pore model, the effective diffusivity for macropore diffusion is given by Eq. (11-28). If the mass transfer is solely by bulk diffusion, Eq. (11-26) shows that so that... 

FIGURE 11.12 The random pore model of gas-solid noncatalytic reactions stages in surface development. 

Perhaps the most realistic model is the random pore model of Bhatia and Perlmutter (1980 1981a, b 1983), which assumes that the actual reaction surface of the reacting solid B is the result of the random overlapping of a set of cylindrical pores. Surface development as envisaged in this model is illustrated in Figure 11.12. The first step in model development is therefore the calculation of the actual reaction surface, based on which the conversion-time relationship is established in terms of the intrinsic structural properties of the solid. In the absence of intraparticle and boundary layer resistances, the following relationship is obtained ... 

As mentioned before, the carbonation of CaO is a typical non-catalytic, gas-solid reaction. As snch, it has been extensively modelled using either random pore or grain models. The random pore model was developed and first applied by Bhatia and Perlmntter [57] to model the sulphation of lime and subsequently extended by Sun et al. [65]. The pores were assumed as an assembly of randomly oriented cylinders of uniform diameter, which initially overlapped (Fig. 6.17). The initial increase in the reaction rate was attributed to the growth of the surface area of the CaO-CaCOs interface, which is, however, overshadowed in later stages by the intersection of the growing surfaces, leading subsequently to a decrease in the reaction rate. 

The two major pore models that have been used extensively over the years for practical purposes are the parallel-pore model proposed by Wheeler in 1955 [5, 9] and the random-pore model proposed by Wakao and Smith in 1962 [34]. Among the more recent advanced models are the parallel cross-linked pore model [35] and pore-network models [36, 37]. 

The random-pore model describes a bidisperse pore-volume distribution and uses separate void fractions in macropores... 

Different theoretical models have been proposed to estimate the effective dif-fusivity Dwj- The most commonly used approximations are the mean transport pore model [18] and the random pore model [19]. 

It should be mentioned that the mean transport pore model is probably not the best option for SCR catalyst layers, as it assumes a uniform pore size distribution. The actual pore size distribution of the SCR layers is highly bimodal with two distinct maxima micropores and mesopores. The random pore model considers a bi-dispersive washcoat material with two characteristic pore sizes with their respective mean pore sizes and void fractions. The total dififusivity is calculated as a combination of the respective Knudsen dififusivities ... 

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