Random-pore models

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. 

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 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]. 

Accordingly, in addition to rate parameters and reaction conditions, the model requires the physicochemical, geometric and morphological characteristics (porosity, pore size distribution) of the monolith catalyst as input data. Effective diffusivities, Deffj, are then evaluated from the morphological data according to a modified Wakao-Smith random pore model, as specifically recommended in ref. [63[. 

Different theoretical models applied to this pore size distribution can give relatively large variations of the calculated effective diffusivity value (DelT). The most commonly used approximations are (i) random-pore model (Wakao and Smith, 1962) using two characteristic transport pores (micropores ji and macropores M)... 

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 ... 

The literature data on the tortuosity factor r show a large spread, with values ranging from 1.5 to 11. Model predictions lead to values of 1/e s (8), of 2 (parallel-path pore model)(9), of 3 (parallel-cross-linked pore model)(IQ), or 4 as recently calculated by Beeckman and Froment (11) for a random pore model. Therefore, it was decided to determine r experimentally through the measurement of the effective diffusivity by means of a dynamic gas chromatographic technique using a column of 163.5 cm length,... 

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. 

IS a modified Damkohler number = A nhsCno ts the dimensionless NH3 adsoiption constant, D, is the molecular diffusivity of species 1 is the effective intraporous diffusivity of species i evaluated according to the Wakao-Smith random pore model [411. Equation (4) is taken from Ref. 39. Equations (6)-(8) provide an approximate analytical solution of the intraporous diffusion-reaction equations under the assumption of large Thiele moduli (i.e., the concentration of the limiting reactant is zero at the centerline of the catalytic wall) the same equations are solved numencally in Ref. 36. 

ABSTRACT The idnetically controlled charcoal reactivity with CO2 at 800°C can very well be described over the entire conversion range when extending Bhatia and Perlmutter s random pore model derivation with two additional parameters only. With untreated charcoal, the extension addresses mainly non-porous phenomena associated which the gradual disintegration of the particle structure at the higher conversions, but the extended kinetic relations are also well suited to describe reactivity effects dominated by metal catalyst accumulation (or re-activation) in the charcoal with progressing conversion. 

For the metal catalysed gasification, several authors report the occurrence of a reactivity maximum around X 0,7 with sodium or potassium eiuiched chars, But, up to date, no efforts have been undertaken to extend the kinetic relations provided by state-of-the-art random pore models, to account for the occurrence of the maximum around X 0,7. We believe that this late reactivity maximum results from catalyst accumulation (but not saturation) in the charcoal. 

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 ... 

Fig. 2 Left plot Gasification reactivity of untreated, respectively, HCl washed charcoal with COj at 800 C as a function of the conversion degree. The dashed lines represent model fits obtained with the original random pore model of Bhatia and Perlmutter. A and B indicate regions of mismatch between experiment and model, Right plot Gasification reactivity of untreated, respectively, Na COj impregnated charcoal versus...

A. Tsetsekou and G. Androutsopoulos, Mercury porosimetry hysteresis and entrapment predictions based on a corrugated random pore model, Chem. Eng. Comm., 110 (1991) 1. 

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 relate and requires a model for the porous structure. The parallel-pore and random-pore models have been applied to surface diffusion by J. H. Krasuk and J. M. Smith Ind. Eng. Chem., Fund. Quart., 4, 102 (1965)] and J. B. Rivarola and J. M. Smith [Ind. Eng. Chem., Fund. Quart., 3, 308 (1964)]. 

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... 

The simplest models are those in which the internal structure of a pellet is not considered, and its behavior as a whole is modeled. These are normally called the macroscopic or basic models. In other models, the behavior of the distinctive elements of a pellet, such as the grain, micrograin, or the pore, constitutes the central feature such models account for structural changes during reaction. The so-called random pore models are the most common. 

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 ... 

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