The Parallel Cross-Linked Pore Model

The parallel cross-linked pore model has been developed by Johnson and Stewart [1965] and Feng and Stewart [1973]. Equation (3.4-7) is considered to apply to a single pore of radius r in the solid, and the diffusivities are interpreted as the actual values, rather than effective diffusivities corrected for porosity and tortuosity. A pore size and orientation distribution function f r, Q) is defined. Then, f r, fi)d/dfi is the fraction open area of pores with radius rand a direction that forms an angle Q with the pellet axis. The total porosity is then 

The pellet flux is obtained by integrating the flux in a single pore with orientation iand by accounting for the distribution function  

Perfectly communicating pores. The concentrations are identical at a given position z, that is, Cj z, r, Q) = Cj(.  

Noncommunicating pores. The complete profile C/z r. Si) is first derived for a given pore and then averaged. 

For pure diffusion at steady state, dNJdz = 0 ot Nj = constant, and (3.5.2.2-6) can be directly integrated  

Perfect]) communicating pores, where the concentrations are identical at a given position z—that is, C/z r, 12) = C/z). 

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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. 

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]. 

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,... 

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... 

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