Parallel Cross-Linked Pores

It was discussed in Sec. 3.5.d that the most realistic version of this model for catalyst pellets is the communicating pores limiting case. With uncorrelated tortuosity, Eq. 3.5.d-9 gives the diffusion flux  

It will be postulated that the pore-size distribution does not vary with time, and is position independent (i.e., a macroscopically uniform pellet—not always true, see Satterfield [40]). The communicating pore limit, with concentrations only a function of position, 2, as discussed in Sec. 3.S.d, then leads to the mass balance 

the flux Nj is from Eq. 3.9.b-l, and the rate term is also averaged over the pore-size distribution  

For a simple first-order reaction, the usual solution would then be found, 

There is not really any experience with the use of Eq. 3.9.b-8 as yet (althou see Steisel and Butt [139]) and whether it, or less restrictive, versions of the parallel cross-linked pore model are adequate representations of reactions in complex pore systems is not known. 

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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,... 

More general models for the porous structure have also been developed by Johnson and Stewart [60] and by Feng and Stewart [43], called the parallel cross-linked pore model. Here, Eqs. 3.5.b-4 to 6 or Eq. 3.5.b-7 are considered to apply to a single pore of radius r in the solid, and the diffusivities interpreted as fte actual values rather than effective diffusivities corrected for porosity and tortuosity. A pore size and orientation distribution function /(r, Q), similar to Eq. 3.4-2, is defined. Then /(r, Q)dri is the fraction open area of pores with radius r and a direction that forms an angle Q with the pdlet axis. The total porosity is then... 

It is more crucial here to consider specifically the pore-size distribution, since the large molecules will presumably not fit into the smaller pores. The parallel cross-linked pore model can be combined with the above to yield the following steady-state mass balance ... 

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]. 

Pore network models are an example of a discrete model. The earlier pore network models consisted of parallel pores [18] and randomly oriented cross-linked pores [19]. Bethe lattice [20], and regular networks [21] have also been used to represent catalyst structures. Pore network models have been used to analyze the complicated interactions between diffusion and reaction that may occur in catalyst particles, for example Sharatt and Mann [21] used their cubic network... 

Figure 2. ESR spectra of Cu(II) at S-band (2.4 GHz) and 120 K in chemically cross-linked polyacrylamide gels with pore diameter of 3.2 nm. TTie inset shows the two low field lines of the parallel quartet for pore diameter of 3.2 nm (A) and 0.7 nm (B). Solid lines are experimental spectra dotted lines are spectra calculated using the appropriate values of AH, SAn, and Sgn given in Table I, and with g = 2.408, 2.080, A, = 0.0133 cm, A = 0.0005 cm and AHf = 16.5 Gauss.

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