Diffusion random pore

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). 

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,... 

Therefore, the section shown in Figure 15, and the pore network in 3-D from which it arises, are both absolutely defined in a quantitative way. Inasmuch as the 3-D networks are felt to be a realistic representation of random pore spaces, it is feasible to compute directly several important macroscopic properties for the FCC powder particles. Amongst these properties are permeability and effective difflisivity, so that diffusion and reaction calculations relevant to gas-oil cracking in the FCC particles can be directly undertaken. Also important in this respect are calculations of deactivation due to coke laydown within the particles. It is also possible that the pore networks could be used to deduce strength and abrasion resistance of the particles. 

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. 

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. ... 

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. 

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 definition of tortuosity factor in Eq. 3.5.b-2 includes both the effect of altered diffusion path length as well as changing cross-sectional areas in constrictions for some applications, especially with two-phase fluids in porous media, it may be better to keep the two separate (e.g., Van Brakel and Heertjes [39]). This tortuosity factor should have a value of approximately 3 for loose random pore structures, but measured values of 1.5 up to 10 or more have been reported. Satterfield [40] states that many common catalyst materials have a t 2 3 to 4 he also gives further data. 

Figure 3.5.C-1 Diffusion areas in random pore model. (Adapted from Smith [24].)... 

Finally, Sun et al. [63] calculated the activation energy for the slow stage of the carbonation reaction by modelling the carbonation reaction with a random pore model. They reported a value of 215 kJ/mol for tanperatures between 500 and 850 °C indicative of solid-state diffusion. 

Bhatia SK, Perlmntter DD (1981) A random pore model for fluid-solid reactions II. Diffusion and transport effects. AIChE J 27 247-254... 

In the future network models will be completed by random pore models in which diffusion and reaction problems can be treated by Monte Carlo methods only. 

In Section 3.4 already the homogeneous medium diffusivity was corrected for the ratio of surface holes to total surface area of the catalyst. For a random pore network this ratio is, according to Dupuit s law, equal to the internal void fraction, es. Another adaptation is required because of the tortuous nature of the pores and eventual pore constrictions. The diffusion path length along the... 

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 ... 

The porosity of the particle, Sso, is 0.45. Neglect the change caused by the oxidation. Calculate the effective diffusivity of ojg gen for a random pore model with only macropores. 

Attempts have been made to predict gasification rates using mathematical models. This area has been briefly reviewed by Rafsanjani et al. (2002) who discuss the use of (what are termed) the grain model, the random pore model, the simple particle model and the volume reaction model. They report that differential mass conservation equations are required for the oxidant gas and char particle. These authors use a simplified mathematical model (the quantise method (QM)) for activation of coal chars when both diffusion and kinetic effects have to be considered. Results are compared with other methods when it is found that QM predictions of rate are more accurate than predictions by the random pore model and the simple particle model. 

When the catalyst is made from a precipitated powder, a random pore structure will result. The pores are not straight channels, but rather the spaces between the powder particles. When these are approximately cubic or spherical, the pores are interconnected in all directions, and the pore structure may be quite isotropic. Diffusion rates will then be independent of spatial directions. 

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