Single-pore model

The earliest models of fuel-cell catalyst layers are microscopic, single-pore models, because these models are amenable to analytic solutions. The original models were done for phosphoric-acid fuel cells. In these systems, the catalyst layer contains Teflon-coated pores for gas diffusion, with the rest of the electrode being flooded with the liquid electrolyte. The single-pore models, like all microscopic models, require a somewhat detailed microstructure of the layers. Hence, effective values for such parameters as diffusivity and conductivity are not used, since they involve averaging over the microstructure. 

There are two main types of single-pore models. In the first, the approach of Giner and Hunter is taken in which there are straight, cylindrical gas pores of a defined radius. These pores extend the... 

Porous-Electrode Models. The porous-electrode models are based on the single-pore models above, except that, instead of a single pore, the exact geometric details are not considered. Euler and Nonnenmacher and Newman and Tobias were some of the first to describe porous-electrode theory. Newman and Tiedemann review porous-electrode theory for battery applications, wherein they had only solid and solution phases. The equations for when a gas phase also exists have been reviewed by Bockris and Srinivasan and DeVidts and White,and porous-electrode theory is also discussed by New-man in more detail. 

Q. To proceed further at this point one has to specify a pore model for the catalyst, and a model for the active site distribution. Froment and co-workers have examined a variety of cases such as single pore models (single-ended pores and pores open on both sides) with both deterministic and stochastic active site distributions, the bundle of parallel pores model and various tree-like models of the porous structure, which were earlier used by Pismen (40) to describe transport and reaction in porous systems. Such treelike models contain interconnected pores but lack any closed loops and are usually called Bethe networks or lattices. They are completely characterized by their coordination number Z, which is the number of pores connected to the same site of the network. 

One of the simplest ways of extending this single-pore model is to assume that variations in the size of pore spaces can be represented by a variable-diameter assembly of such pores, referred to as a parallel bundle of pores. An example is shown in Fig. 3, for a variation of the model applied to a supported zeolite cracking catalyst. In this example [10] the zeolite pores are simply configured along the pore walls, so that the parallel bundle represents the Si/Alumina-support pore spaces. 

A characteristic feature associated with pore condensation is the occurrence of sorption hysteresis, i.e pore evaporation occurs usually at a lower p/po compared to the condensation process. The details of this hysteresis loop depend on the thermodynamic state of the pore fluid and on the texture of adsorbents, i.e. the presence of a pore network. An empirical classification of common types of sorption hysteresis, which reflects a widely accepted correlation between the shape of the hysteresis loop and the geometry and texture of the mesoporous adsorbent was published by lUPAC [10]. However, detailed effects of these various factors on the hysteresis loop are not fully understood. In the literature mainly two models are discussed, which both contribute to the understanding of sorption hysteresis [8] (i) single pore model. hysteresis is considered as an intrinsic property of the phase transition in a single pore, reflecting the existence of metastable gas-states, (ii) neiM ork model hysteresis is explained as a consequence of the interconnectivity of a real porous network with a wide distribution of pore sizes. 

The random structure of the porous electrode, illustrated in Figure 13.11(a), leads to a distribution of pore diameters and lengths. Nevertheless, the porous electrode is usually represented by the simplified single-pore model shown in Figure 13.11(b) in which pores are assumed to have a cylindrical shape with a length i and a radius r. The impedance of the pore can be represented by the transmission... 

Ramachandran, P. Smith, J. A single-pore model for gas-solid noncatalytic reactions. AIChE J. 1977, 23, 353-361. 

All models discussed in this section are single-pore models describing the morphology of a single pore. The topology of the disordered material. 

The analysis presented in the preceding sections is based on the model of a single pore in an otherwise impermeable membrane. Of course, real membranes deviate in one or more ways from this highly idealized model. It is worthwhile to briefly consider the limitations in applying the single-pore model to the SECM analysis of real membranes. 

Kramer, M. and M. Tomkiewicz, Porous electrodes. I. Numerical simulation using random network and single-pore models. Journal of the Electrochemical Society, 1984. 131 pp. 1283-1288... 

Figure 4.4.45. Ideal single-pore model for the growth of porous magnetite film in acidic chloride solutions at high temperature. (Reprinted with permission from J. R. Park and D, D. Macdonald, Impedance Studies of the Growth of Porous Magnetite Films on Carbon Steel in High Temperature Aqueous Systems, Corros. Sci. 23, 295 [1983], Copyright 1983, Pergamon Journals Ltd.)...

Lane, A.M., Single-pore model A simplified diffusion simulation. AlChE J.. 37(8), 1245-1248 (1991). 

Supercapacitor modeling enables us to predict their behavior in different apphcations, on the basis of a representation of the main physical phenomena occurring in the coirqxment. There are many different models for snpeicapacitors (two-branch model, model based on a transmission line, single-pore model, multi-pore model, etc.) [BEL 01 HAM 06]. These models are in the form of equivalent electrical circuits. Using them, we can describe the supercapacitor s behavior quite accurately. 

Macroscale models of PEM operation that do not include the proper pressure-controlled equilibrium conditions at the single pore level fail in predicting correctly the responses of membrane water sorption, transport properties, and fuel cell operation to changes in external conditions. Single pore models, on the other hand, that do not account for statistical spatial fluctuations in microscopic membrane properties must fail because they cannot predict the dispersion in pore sizes and the evolution of the pore size distribution upon water uptake. 

FIGURE 3.25 Solution of the single pore model in the ID Poisson-Boltzmann limit (a) the electrostatic effectiveness factor, as a function of the metal surface charge density, ctm, for various values of Rp (b) radial variation of the normalized proton concentration in the pore for various values of Rp at gm = —0.05 C (Reprinted from Chan, K. and Eikerling, M. 2011. /. Electrochem. Soc., 158(1), B18-B28, Figures 1,2,3,4,5,6. Copyright (2011), the Electrochemical Society. With permission.)... 

In the considered model, electrostatic interaction between protons and metal surface charge determines the distributions of protons and electrostatic potential in the pore. These phenomena distinguish the present pore model from the gas- and electrolyte-filled single pore models pioneered by Srinivasan et al. (1967), Srinivasan and Hurwitz (1967) and De Levie (Levie, 1967). With the explicit consideration of the pore wall surface charge, the potential of zero charge of the catalyst material... 

The additional parameters that remain to be defined in the single pore model are k cuid Dg. The rate constant can be estimated from initial reaction rate data and the appropriate value of determined by matching the cibove model to actual experimental data. 

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