Stochastic pore networks

Figure 6. Regular cubic 3-D stochastic pore network (10x10x10) (Reproduced with permission. Copyright 1993 Institution of Chemical Engineers.)...

However, from Figure 12 it is evident that even though an interpretation of the porosimeter curve via a 3-D stochastic pore network gives rise to much larger estimates... 

In conclusion it would appear that the combination of 3-D stochastic pore networks with mercury porosimetry and low melting point alloy impregnation offers a new framework for the description and calculation of the role of pore spaces in typical porous catalyst particles. 

Khalaf, K., 1988, "Application of 3-D stochastic pore networks to FCC particles", Ph D. Thesis, UMIST... 

If a large number of open-ended cylindrical pore segments (like the one in Fig. 2) are interconnected such that the diameter of any pore is independent of the size of neighbor pores, a so-called randomized, or stochastic, pore network is formed. Such a set can be assembled from a cohort obeying any stipulated pore diameter distribution function. If all the pore segments are of equal length with a connectivity of 4, a square network... 

The stochastic pore network visualized in Fig. 5 can now be examined with respect to the widely used characterization techniques of mercury porosimetry and low-temperature capillary gas adsorption. Both these equilibrium processes can be computed on a pore-by-pore basis. 

Mercury porosimetry is governed in each pore by an equilibrium force/surface tension balance (the Washburn equation) that relates the diameter of a cylindrical pore to the pressure needed to force mercury into it. The pressured step-by-step invasion of a pore network is then controlled by a pattern of pone accessibly at each given pressure. Systematic penetration, starting from an empty network surrounded by mercury, can be readily performed. Results for the network in Fig. 5 are given in Fig. 6, showing both the penetration curve and the retraction curve. Stochastic pore networks implicitly predict hysteresis between penetration and retraction as well as a residual final entrapment of mercury. In Fig. 6, the final entrapment is about 45%, with much of the retained mercury entrapped in the larger pores [11]. More details of the pore-by-pore calculation have been published [4]. 

The parallel bundle, by its nature, cannot give hysteresis or entrapment in porosimetry, and the stochastic pore network indicates that both these phenomena arise from randomness and connections amongst pores. Applications of this simple model to oil-... 

The rate processes of diffusion and catalytic reaction in simple square stochastic pore networks have also been subject to analysis. The usual second-order diffusion and reaction equation within individual pore segments (as in Fig. 2) is combined with a balance for each node in the network, to yield a square matrix of individual node concentrations. Inversion of this 2A matrix gives (subject to the limitation of equimolar counterdiffusion) the concentration profiles throughout the entire network [14]. Figure 8 shows an illustrative result for a 20 X 20 network at an intermediate value of the Thiele modulus. The same approach has been applied to diffusion (without reaction) in a Wicke-Kallenbach configuration. As a result of large and small pores being randomly juxtaposed inside a network, there is a 2-D distribution of the frequency of pore fluxes with pore diameter. 

Figure 8 Concentration profile in a 2-D stochastic pore network. (From Ref. 11.)...

Figure 12 Variations on a regular 2-D stochastic pore network, (a) Random configuration (b) subordering of corrugations between nodes (c) random node displacement with subordering. (From Ref. 11.)...

In this section, two examples are presented for the application of a technique of low-melting-point alloy (LMPA) impregnation that provides for a visualization of the invasion of a nonwetting fluid into the pore spaces in a typical porous article. The visualization can be linked to the modeling of mercury porosimeter curves using 3-D stochastic pore networks. This makes the quantification of pore structure more direct. Quantified structures can be visually examined against sample particle sections. The visual comparison can be made more precise by image analysis of the accessible porosity made visible by metal penetration over a series of pressures. 

Figure 26 Stochastic pore network for reforming catalyst (From Ref. 20.)...

R. Mann, G.P. Androutsopoulos, and H. Golshan, Application of stochastic pore networks to oil bearing rock with observations relevant to oil recovery, Chem. Eng. Sci. 36 337 (1981). R. Mann and H. Golshan, Application of stochastic pore networks to a catalyst pellet, Chem, Eng. Commun. 12 371 (1981). 

G. Thomson and R. Mann, Application of stochastic pore networks to interpret adsorption isotherms, in Adsorption Science and Technology, E.A. Rodrigues, ed., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989, p. 63. 

R. Mann, H.N.S. Yousef, D.K. Friday, and J.J. Mahle, Interpretation of water isotherm hysteresis for an activated charcoal using stochastic pore networks. Adsorption 7 256 (1995). 

A.N. Patwardhan and R. Mann, Effective diffusivity and tortuosity in Wicke-Kallenbach experiments Direct interpretation using stochastic pore networks, Trans. /. Chem, E. 691A),205 (1991). 

R. Mann, P.N. Sharratt, and G. Thomson, Catalyst deactivation by fouling Diffusion, reaction and coke deposition in stochastic pore networks, Chem, Eng. Sci, 47 711 (1986). 

A. Al-Lamy, Characterization of catalyst pore structure by image reconstruction from 3-D stochastic pore networks, Ph.D. dissertation, UMIST (1995). 

Over a period of time, particularly the last twenty years, researchers have attempted to improve and create models oqrable of describing the influence of porous media in catalytic reaction processes, and they have been aided by the development of computing power and computer modelling techniques. Hence a continual progression has been made from the simple parallel bundle models, which have been the basis of most textbook treatments [1], to stochastic pore network models [2-3] and chamber and throat pore models [4], and more recently fiactal-based models, first introduced by Mann and Wasilewski [5], and subsequently expanded upon by other workers [6-8]. 

Previous work had demonstrated the use of a Woods Metal, or LMPA, in the production of photographed SEM images of serial sections of impregnated porous media [13]. This sq)proach was adapted and used in a new process, exactly similar in principle to mercury porosimetry. Mercury was replaced by LMPA, which had similar surface tension and contact angle when in liquid form. Serial sections from LMPA intruded samples could then be compared with theoretical visualised sections constructed from data from stochastic pore networks. 

Curve fitting using a random stochastic pore network... 

J.H.Petropoulos, J.K.Petrou and N.K.Kanellopoulos, Explicit relation between relative permeability and structural parameters in stochastic pore networks, Chem.Eng. Sci.,44 (1989) 2967... 

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