Model distributed electrode

The preceding sections have discussed what occurs inside the membrane. The potential has been that in the ionomer. Our goal here is to develop an analytic model of the potential from the metal on each side into and through the ionomer. As this chapter is concerned with membrane modeling, we will only consider a hydrogen pump cell with hydrogen electrodes on the anode and cathode sides. We will treat point electrodes on the boundaries of the one-dimensional membrane. The point electrode model will supply the necessary interface equations to use in a distributed electrode model, which one would need in order to expect the model to match experimental data. However, this chapter focuses on the membrane, not the electrode. We will assume we can maintain equal H2 gas pressure on each side so that the gas contribution to any Nemst potential will be zero. 

The distributed resistor model neglects the effect of mobile electrolyte ions. Much of our following discussion of the electrolyte s influence neglects, for simplicity, the distributed resistance. In a real dye cell, both effects operate simultaneously. Both tend toward the same result An applied potential will be more or less confined near the substrate electrode, depending on the relative rates of charge transport and interfacial charge transfer and on the concentration of electrolyte. 

Fig. 42. Schematic of regions considered in PEFC air electrode modeling, including (from left to right) gas flow channel, gas-diffusion backing, and cathode catalyst layer. Oxygen is transported in the backing through the gas-phase component of a porous/tortuous medium and through the catalyst layer by diffusion through a condensed medium. The catalyst layer also transports protons and is assumed to have evenly distributed catalyst particles within its volume [100]. (Reprinted by permission of the Electrochemic Society).

The theory of gas-diffusion electrodes has a long history [11, 12, 105, 106]. Specifically, catalyst utilization and specific effective surface area in composite electrodes were always in the focus of attention (see, e.g., Ref. 13 and the articles quoted therein). A comprehensive review of the theory and models of ionic-into-electronic current transformation in two-and three-phase distributed electrodes can be found in Ref. 14. 

A subsequent description by Bockris and associates drew attention to further complexities as shown in Figure 15. The metal surface now is covered by combinations of oriented structured water dipoles, specifically adsorbed anions, followed by secondary water dipoles along with the hydrated cation structures. This model serves to bring attention to the dynamic situation in which changes in potential involve sequential as well as simultaneous responses of molecular and atomic systems at and near an electrode surface. Changes in potential distribution involve interactions extending from atom polarizability, through dipole orientation, to ion movements. The electrical field effects are complex in this ideal polarized electrode model. 

Gas diffusion electrodes have been characterized with the objective to Imk structural parameters (such as permeability, fraction of hydrophobic pores, pore size distribution and volume, and catalyst layer thickness and composition [26,27,28]) to cell performance. Although this information is valuable to validate existing gas diffusion electrode models [29, 30, 31], the Imk between... 

In the industrial applications of electrochemistiy, the use of smooth surfaces is impractical and the electrodes must possess a large real surface area in order to increase the total current per unit of geometric surface area. For that reason porous electrodes are usually used, for example, in industrial electrolysis, fuel cells, batteries, and supercapacitors [400]. Porous siufaces are different from rough surfaces in the depth, /, and diameter, r, of pores for porous electrodes the ratio Hr is very important. Characterization of porous electrodes can supply information about their real surface area and electrochemical utilization. These factors are important in their design, and it makes no sense to design pores that are too long and that are impenetrable by a current. Impedance studies provide simple tools to characterize such materials. Initially, an electrode model was developed by several authors for dc response of porous electrodes [401-406]. Such solutions must be known first to be able to develop the ac response. In what follows, porous electrode response for ideally polarizable electrodes will be presented, followed by a response in the presence of redox processes. Finally, more elaborate models involving pore size distribution and continuous porous models will be presented. 

Figure 4. 45. One-dimensional network distributed homogeneous cylindrical porous electrode model after Gbhr [1997].

The model presented here is a comprehensive full three-dimensional, non-isothermal, singlephase, steady-state model that resolves coupled transport processes in the membrane, eatalyst layer, gas diffusion eleetrodes and reactant flow channels of a PEM fuel cell. This model accounts for a distributed over potential at the catalyst layer as well as in the membrane and gas diffusion electrodes. The model features an algorithm that allows for a more realistie representation of the loeal activation overpotentials which leads to improved prediction of the local current density distribution. This model also takes into aeeount convection and diffusion of different species in the channels as well as in the porous gas diffusion layer, heat transfer in the solids as well as in the gases, electrochemical reactions and the transport of water through the membrane. 

Depending on the resolution of the mathematical model, different forms of the species conservation equations may be considered in the porous electrodes. For instance, in the multi-scale modehng of Khaleel et al. [18], a mesoscale lattice Boltzmarm model of the electrodes resolves the species transport in the gas, on the surface of the electrode, and through the bulk solid of the electrode. In this model, Eq. (26.1) is solved in three separate domains with corresponding transport properties and source terms. In contrast, in the macroscale distributed electrochemistry model of Ryan et al. [19], the porous medium of the SOFC electrodes is not explicitly resolved but is included in the model via effective properties. In the effective properties model, the diffusion coefficient of Eq. (26.1) is replaced with an effective diffusion coefficient, which is discussed in Section 26.3.3. 

In these equations, i represents the current density distribution through the current collectors at the anode and cathode and i and ic are the current densities obtained from the electrode. They depend on the local potential differences at the respective electrode and are obtained from the electrode model ... 

The equations presented in this chapter are a common basis for many spatially distributed MCFC models. The full diversity of existing models cannot be reflected here Table 28.1 and the references therein give a good overview of the spectrum of these models. A focus in MCFC modeling is on the description of the processes in the electrodes, which are illustrated in a separate section of this chapter. Also, the combination of electrode models and MCFC models has been shown in detail. 

Continuum electrode models Multi-particle models Local current density distribution models Micro-kinetic models... 

Going from planar to porous electrode introduces another length scale, the electrode thickness. In the case of a PEM fuel cell catalyst layer, the thickness lies in the range of IcL — 5-10 pm. The objective of porous electrode theory is to describe distributions of electrostatic potentials, concentrations of reactant and product species, and rates of electrochemical reactions at this scale. An accurate description of a potential distribution that accounts explicitly for the potential drop at the metal/electrolyte interface would require spatial resolution in the order of 1 A. This resolution is hardly feasible (and in most cases not necessary) in electrode modeling because of the huge disparity of length scales. The simplified description of a porous electrode as an effective medium with two continuous potential distributions for the metal and electrolyte phases appears to be a consistent and practicable option for modeling these stmctures. 

In one-dimensional electrode modeling, (x) denotes the metal phase potential and (x) the electrolyte phase potential. The gradient of the metal (carbon) phase potential drives the electron flux, while protons move along the potential gradient of the electrolyte (ionomer) phase. At equilibrium, these gradients are zero and the potentials in the distinct phases are constant, (p (x) = and O (x) = 4) . The potential distribution of a working PEFC with porous electrodes of finite thickness is shown in Figure 1.9. For illustrative purposes, a simple assembly of anode catalyst layer, PEM and cathode catalyst layer is displayed. 

Fig, 6 Equivalent circuits of the IPMC. (a) Lumped capacitance and resistance model, (b) Distributed circuit model that takes into account the roughness of the plated electrodes, (c) A possible distributed circuit model that takes into account the surface resistance of the plated electrodes, (d) A possible circuit model that takes into account the ion transport in the porous electrodes... 

The distribution line, the pole, and the home appliances in the house shown in Figure 6.16 can be represented by horizontal and vertical distribution line models and lumped parameter circuits [27,28]. The grounding electrodes of the pole, the telephone line SPD, and the home appliances, if grounded, are modeled by a combination of a distributed line and a lumped parameter circuit to simulate the transient characteristic [29]. However, this section adopts a simple resistance model with a resistance value taken from the experiments discussed in References 29 and 30, as the vertical grounding electrode used for a home appliance is short and the transient period is much shorter in the phenomenon investigated in this chapter. A protection device (PD) is installed in a home appliance, and the NTT SPD is represented by a time-controlled switch prepared in the EMTP [31]. 

Distribution line model of a grounding electrode, (a) Representation of a horizontal electrode with length x = n Ax and (b) model circuit. 

Garcia etal. [41] developed a two-dimensional porous electrode model and accounted for potential and charge distributions in the electrolyte. They employed transport equations derived from dilute solution theory, which is generally not adequate for LIB systems. The stress generation effect is built into the 2D DNS modeling framework with a simplified, sphere-packed electrode microstmcture description. 

The distribution line, the pole, and home appliances in the house in Figure 6.16 can be readily represented by horizontal and vertical distributed line models and lumped-parameter circuits [27,28]. The grounding electrodes of the pole, the telephone line SPD, and the home appliances if those are grounded... 

The objective of an electrode model is to analyse the point-to-point distribution of the reaction in an SOFC electrode, leading to current, potential, and species concentration distributions. The result of the analysis is a prediction of the polarisation of the electrode due to (i) kinetic resistance, (ii) mass transfer resistance, and (iii) ohmic resistance. 

One-Dimensional Porous Electrode Models Based on Complete Concentration, Potential, and Current Distributions... 

General impedance models for distributed electrode processes... 

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