Pore network modelling modelled diffusion

Rieckmann and Keil (1997) introduced a model of a 3D network of interconnected cylindrical pores with predefined distribution of pore radii and connectivity and with a volume fraction of pores equal to the porosity. The pore size distribution can be estimated from experimental characteristics obtained, e.g., from nitrogen sorption or mercury porosimetry measurements. Local heterogeneities, e.g., spatial variation in the mean pore size, or the non-uniform distribution of catalytic active centers may be taken into account in pore-network models. In each individual pore of a cylindrical or general shape, the spatially ID reaction-transport model is formulated, and the continuity equations are formulated at the nodes (i.e., connections of cylindrical capillaries) of the pore space. The transport in each individual pore is governed by the Max-well-Stefan multicomponent diffusion and convection model. Any common type of reaction kinetics taking place at the pore wall can be implemented. 

A similar model has been applied to the modeling of porous media with condensation in the pores. Capillary condensation in the pores of the catalyst in hydroprocessing reactors operated close to the dew point leads to a decrease of conversion at the particle center owing to the loss of surface area available for vapor-phase reaction, and to the liquid-filled pores that contribute less to the flux of reactants (Wood et al., 2002b). Significant changes in catalyst performance thus occur when reactions are accompanied by capillary condensation. A pore-network model incorporates reaction-diffusion processes and the pore filling by capillary condensation. The multicomponent bulk and Knudsen diffusion of vapors in each pore is represented by the Maxwell-Stefan model. 

A great number of studies have been published to deal with relation of transport properties to structural characteristics. Pore network models [12,13,14] are engaged in determination of pore network connectivity that is known to have a crucial influence on the transport properties of a porous material. McGreavy and co-workers [15] developed model based on the equivalent pore network conceptualisation to account for diffusion and reaction processes in catalytic pore structures. Percolation models [16,17] are based on the use of percolation theory to analyse sorption hysteresis also the application of the effective medium approximation (EMA) [18,19,20] is widely used. 

In general it is clear that PFG spin-echo data can provide information on the pore structure at both the gradient length scale, /q, and the diffusion length scale, y (/JA). The NMR data can be analysed to obtain simple structure factors, which may then be related to particular pore network models. ... 

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... 

On initial inspection the results obtained from serial sectioning of LMPA intruded samples appear at odds with the principle theory behind intrusion and retraction as predicted by the Washburn equation. But further inspection shows it is not the Washburn equation, but mercury porosimetry that is at fault. Pore network models have often been used to characterise the behaviour of pore structure in relation to mercury porosimetry. But the model is only as good as the assumptions and the data that it is based iqron. Without artificially shielding the network, the model caimot propa ly detomine the correct psd and cannot derive a more spatially accurate structure that could be used for diffusion and reaction modelling. In order to characterise the pore structure more accurately, we need to introduce some of the elements usually revealed by LMPA intrusion tests. 

Gostick, J.T., loannidis, M.A., Fowler, M.W., and Pritzker, M.D. (2007) Pore network modeling of fibrous gas diffusion layers for polymer electrolyte membrane fuel cells. J. Power Sources, 173, 2TJ 29Q. 

Sinha, P.K. and Wang, C.-Y. (2007) Pore-network modeling of liquid water transport in gas diffusion layer of a polymer electrolyte fuel cell. Bectrochim. Acta, 52, 7936 7945. 

Calculating the transport properties of the calculated structure (e.g., still in the case of PEMFC electrodes, by using pore network modeling of liquid water propagation in the porous electrode and its impact on the effective oxygen diffusion properties). 

Recently, pore network modeling has been applied to simulate the accumulation of liquid water saturation within the porous electrodes of polymer electrolyte membrane fuel cells (PEMFCs). The impetus for this effort is the understanding that liquid water must reside in what would otherwise be reactant diffusion pathways. It therefore becomes important to be able to describe the effect that saturation levels have on reactant diffusion. Equally important is the understanding of how the properties of porous materials affect local saturation levels. This requirement is in contrast to most continuum modeling of the PEMFC, where porous materials are treated with volume-averaged properties. For example, the relationship between bulk liquid saturation and capillary pressures foimd through packed sand and other soil studies are often employed in continuum models. ... 

A pore network model of GDL invasion must include assumptions of the mechanisms that produce liquid water within the GDL. Two mechanisms have been employed thus far in the literature hquid water enters at the GDL/catalyst layer interface due to pressme buildup of condensed hquid water in the catalyst layer,or liquid water enters within the bulk of the GDL due to a condensation mechanism. These mechanisms can be explained by the high humidity levels near the catalyst layer driving the diffusive flux of water vapor to the gas charmel, and by the relatively low temperatures near the ribs of the flow field, respectively. A third mechanism has yet to be applied liquid water entering the GDL at the GDL/gas charmel interface due to upstream accumulation. 

Few pore network models of GDL materials have been applied to calculate the material s relative diffnsivity after an invasion process. This could be due to the fact that most models apply inlet conditions snch as nniform pressure or uniform flnx, where it is assnmed that liquid water is present at each throat at the catalyst layer GDL interface. However, with a nniform pressure boundary condition, Gostick et al modeled diffusion at various stages of saturation and calculated the limiting current density due to reactant transport across the GDL (Fig. 10.13). They were able to do this for two reasons. First, they did not consider the effect of the inlet reservoir on reactant transport. Therefore, reactants could diffuse through inlet throats as long as they had not yet been invaded by water. Secondly, they investigated a scenario where there was a thin film of air in liquid-saturated pores and throats through which reactants could diffuse. 

In this chapter we have shown the scope of multi-scale modeling in PEM fuel cells. The multi-scale modeling consists of pore-scale modeling and macroscopic upscaling. Pore network model and lattice Boltzmann model were specifically discussed in the context of two-phase transport in the PEMFC gas diffusion layer... 

Wu, R., Zhu, X., Liao, Q. A., 2010, Determination of oxygen effective diffusivity in porous gas diffusion layer using a three-dimensional pore network model , Electrochim. Acta, 55 (24) pp. 7394. 

In the past, many mcxlels describing the influence of die porous structure of catalyst bodies on the activity have been presented. Pioneering in diis field were Thiele [1], Zel-dowitsch [2] and Jiittner [3], and some important extensions are by Wheeler [4] and Aris [5]. Ihe models are based on the effective diffusivity of the reacting gases in the pore network. The effective diffusivity is derived from the porous structure. Several approaches are possible, such as the dusty gas model, mixed Knudsen-ocmtinuum models, and others [6,7]. 

It appears that a loose interpretation of this type may be the origin of a discrenancy found by Otanl and Smith [59] in attempting to apply effective diffusivities from Wakao and Smith s [32] isobaric diffusion data to measurements on a chemically reacting system. This was pointed out by Steisel and Butt [60], and further pursued to the point of detailed computer modeling of a particular pore network by Wakao and Nardse [61]. 

Some experimental studies point out that the diffusion rate of pure hydrocarbons decreases with the coke content in the zeolite [6-7]. Theoretical approaches by the percolation theory simulate the accessibility of active sites, and the deactivation as a function of time on stream [8], or coke content [9], for different pore networks. The percolation concepts allow one to take into account the change in the zeolite porous structure by coke. Nevertheless, the kinetics of coke deposition and a good representation of the pore network are required for the development of these models. The knowledge of zeolite structure is not easily acquired for an equilibrium catalyst which contains impurity and structural defects. 

Pore networks in 2-D and 3-D are still being developed as computer-aided representations of real porous materials since the idea was first proposed some 40 years ago (2). Subsequently, 2-D networks were applied to porosimetry (5) and low-temperature gas adsorption (4), and 2-D and 3-D models have been compared (5). More recent work has applied 3-D networks to porosimetry (< ), to flow and transport behaviour (7), as well as to diffusion and reaction in catalysis (S). The equivalence of pore networks to a continuum representation for porosity has lately been established (9) and a review of recent developments and applications is available 10). 

Barzykin and Tachiya have also developed a kinetics modeF to predict the behavior of bimolecular reactions in interconnected pore networks (Fig. 6). This effective model medium, where diffusion between pores is modeled by a potential barrier, shows an acceleration of diffusion-limited reactions on short time scale. However, if the diffusion between pores is small, only a long time scale slowing down becomes apparent, the apparent interpore kinetics constant being strongly time dependent. 

Decimal (not-whole) numbers are normally obiained for d (apparent and overall order of deactivation). This fact occurs in the analysis of all runs. With the work of Corclla ei iil. (13), Airs (6), or Astarita (7,8), ihis fact can be explained. Note that the gas-phase and the catalyst surface are continuous mixtures. Other approaches and models such as diffusion/percolaiioa theory through a pore network of the FCC catalyst, can explain the values of d found hereof 1.4 to 2.8,... 

Note that for the RIPD model, aqueous-phase diffusion in a fixed-pore network is assumed, and that rate-limited behavior is due only to retardation occurring by instantaneous sorption to pore walls. This conceptualization, in essence, means that hindered diffusion is assumed to not be a factor for RIPD. However, it has been suggested, based upon analyses of applications of the RIPD model to experimental observations, that this model can not describe all aspects of the data without calling upon the concept of hindered diffusion (Steinberg et al., 1987 Brusseau and Rao, 1989a Brusseau et al., 1991a). 

Friedman, S.P., L. Zhang, and N. A. Seaton. 1995. Gas and solute diffusion coefficients in pore networks and their description by a simple capillary model. Transp. Por. Media 19 281 301. 

We apply simple effective medium models in an attempt to understand the diffusion process in the complex pore network of a porous SiC sample. There is an analogy between the quantities involved in the electrostatics problem and the steady state diffusion problem for a uniform external diffusion flux impinging on a coated sphere. Kalnin etal. [17] provide the details of such a calculation for the Maxwell Garnett (MG) model [18]. The quantity involved in the averaging is the product of the diffusion constant and the porosity for each component of the composite medium. The effective medium approach does not take into account possible effects due to charges on the molecules and/or pore surfaces, details in the size and shape of the protein molecules, fouling (shown to be negligible in porous SiC), and potentially important features of the microstructure such as bottlenecks. 

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