A distributed charge transfer (DCT) model has been developed to analyze solid oxide fuel cells (SOFCs) and electrolyzers operating in H_{2}–H_{2}O and CO–CO_{2} atmospheres. The model couples mass transport based on the dusty-gas model (DGM), ion and electron transport in terms of charged species electrochemical potentials, and electrochemical reactions defined by Butler–Volmer kinetics. The model is validated by comparison to published experimental data, particularly cell polarization curves for both fuel cell and electrolyzer operation. Parametric studies have been performed to compare the effects of microstructure on the performance of SOFCs and solid oxide electrolysis cells (SOECs) operating in H_{2}–H_{2}O and CO–CO_{2} gas streams. Compared to the H_{2}–H_{2}O system, the power density of the CO–CO_{2} system shows a greater sensitivity to pore microstructure, characterized by the porosity and tortuosity. Analysis of the pore diameter concurs with the porosity and tortuosity parametric studies that CO–CO_{2} systems are more sensitive to microstructural changes than H_{2}–H_{2}O systems. However, the concentration losses of the CO–CO_{2} system are significantly higher than those of the H_{2}–H_{2}O system for the pore sizes analyzed. While both systems can be shown to improve in performance with higher porosity, lower tortuosity, and larger pore sizes, the results of these parametric studies imply that CO–CO_{2} systems would benefit more from such microstructural changes. These results further suggest that objectives for tailoring microstructure in solid oxide cells (SOCs) operating in CO–CO_{2} are distinct from objectives for more common H_{2}-focused systems.

## Introduction

The desire for long-term, sustainable energy conversion and storage has driven the development of SOCs. SOCs are electrochemical devices that convert chemical energy directly to electrical energy while operating as an SOFC and vice versa during SOEC operations. SOCs are named for the solid, ion-conducting electrolyte which is commonly composed of rare earth ceramics such as yttria-stabilized zirconia (YSZ), scandium-stabilized zirconia, and gadolinium-doped ceria. The electrolyte is between two porous composite electrodes consisting of ion conductors and electron conductors or a mixed ionic electronic conductor.

SOCs operate at high temperatures, typically in the range of 600–1000 °C. These high operation temperatures enhance chemical reaction kinetics and allow SOCs to use less expensive catalysts, typically Ni, which are generally resistant to carbon monoxide poisoning. The combination of these characteristics allows SOFCs to have considerable fuel flexibility for the conversion of chemical energy to electrical energy. Hydrogen and carbon monoxide have been the primary focus as fuel sources in the majority of SOC studies since the simplicities of the molecular structure are more predictable than other fuel sources. Studies have been conducted to analyze the utility of hydrocarbons as fuel sources. These studies start with the relatively simple short-chain hydrocarbons, e.g., methane [1,2], and extend to the complex long-chain hydrocarbons, e.g., dodecane [3].

Under applied current, SOCs are able to electrolyze water and carbon dioxide. This electrolysis approach produces hydrogen and carbon monoxide, respectively, as well as oxygen. SOECs have been widely investigated for hydrogen production [4–8], as high-temperature devices SOECs offer several distinct advantages over low-temperature polymer electrolyte cells. Most notably, the use of an inexpensive Ni catalyst that is tolerant of carbon monoxide allows SOEs to reduce carbon dioxide [5,9–11], a capability that enables generation of syngas from steam and CO_{2} for fuel production [5]. High-temperature operation also slightly reduces power requirements [4,5]. In conjunction with the fuel flexibility of SOCs, studies have been conducted of the co-electrolysis of water and carbon dioxide to produce syngas or hydrocarbons via the Fischer–Tropsch process [5,11–13].

The combination of SOFC and SOEC operations enables SOCs to be used as energy storage devices. SOCs can oxidize hydrogen and carbon monoxide to generate power, as well as electrolyze water and carbon dioxide to produce fuel and oxygen [9,10,13–15]. The use of carbon dioxide and carbon monoxide may be beneficial since they may offer simpler storage approaches relative to hydrogen. The operation of SOFCs and electrolysis cells (SOFCs/ECs) in CO–CO_{2} has been studied preliminarily [16–20]. These analyses focus on the performance characterization of SOECs in CO–CO_{2} atmospheres. For the application of SOCs with carbon monoxide as reliable devices for energy conversion and storage, fuel cell and electrolyzer operation requires further investigation.

The composite electrodes of SOCs are inherently multiscale material systems composed of distinct solid and pore phases that fulfill multiple functions [21,22]. The solid and pore phases support the transport of ions, electrons, and gaseous reactants. The solid phases supporting ion and electron transport to electrochemically active sites are typically formed from discrete particles. At nano- and microscales, the discrete particles that comprise solid oxide electrodes form grains, grain boundaries, and mesoscale percolating networks that influence transport of charge [23–26] and gaseous reactants [26]. Assemblies of sintered grains form conducting networks that span from submicron domains to electrode thicknesses on the scale of millimeters. Charge transport in the solid phase is coupled to transport of gaseous reactants through a network of open pores by electrochemical reactions. This interaction connects to device scales where component geometry influences ohmic [27–29] and concentration losses [30,31].

SOC electrodes are commonly formed from random packings of particles that may be assembled by a number of methods including tape casting, screen printing, or pellet pressing. Development of more deterministic microstructures with regularly shaped features assembled in well-defined cell architectures can garner significant improvements in electrode and cell performance [21,23,24,32–34]. Fabrication of deterministic SOCs has been pursued through the application of regularly shaped particles and functionally graded structures to control microstructural features [34–37] and the application of direct write techniques to achieve tailored cell architectures [38–42]. The merger of these fabrication approaches with multiscale, multiphysics design and analysis may enable realization of next generation SOC technologies [21].

The microstructure of SOC electrodes has been analyzed through the application of modeling techniques in an effort to improve SOC performance and longevity [43–48]. In some cases, this work focuses on the macroscopic structure of the cell to determine which component should be the supporting feature [49], though anode supported cells are currently quite common. Other modeling studies have focused on the effects of various electrode degradation mechanisms on the performance of SOCs [50–54], primarily nickel coarsening and carbon deposition.

Further analysis of microstructural effects on the performance of SOCs in H_{2}–H_{2}O and CO–CO_{2} environments can provide insight into favorable microstructural geometry and conditions for operation within these respective systems. It is expected that SOC performance in CO–CO_{2} will benefit more from microstructures that enhance diffusion rates through the electrode since the molecules are significantly larger than H_{2}. This claim is explored by the development of a multiphysics finite element model. The model is used to perform parametric studies of the porosity and pore diameter of SOC electrodes. The effects of microstructure on cell performance are compared for both H_{2}–H_{2}O and CO–CO_{2} reactant systems. A nondimensional analysis is also implemented to study H_{2}–H_{2}O and CO–CO_{2} systems on a broad range of parameters. The nondimensional results are also used to examine how secondary reactions, e.g., the water–gas shift reaction (WGSR), compare with the oxidation of hydrogen and carbon monoxide. The nondimensional analysis provides an estimated range in which the WGSR is competitive with hydrogen and carbon monoxide oxidation. This provides a method to tailor SOC electrodes to be favorable for specific reaction kinetics.

## Methodology

### Governing Equations.

Due to the physical complexity of SOCs, the cell model is formulated from a system of coupled nonlinear differential equations describing mass and charge transport in the presence of an electrochemical reaction. The system of equations is applied to the computational domain to formulate the model, see Fig. 1.

The model presented was developed to predict cell performance based on physical and electrochemical properties. The methodology of the model resembles that of Suwanwarangkul et al. [55] for mass transport and Cannarozzo et al. [56] with respect to charge transfer. However, charge transport and related overpotentials are cast in terms of charged species electrochemical potentials, an approach that has been previously applied toward SOFC electrode microstructural analysis by Shikazono et al. [57,58]. The multiphysics model presented predicts the current density of an SOC subject to applied voltage and gas reactant stream compositions. Transport-related losses, i.e., ohmic and concentration overpotentials, are accounted for through the governing differential equations and effective transport properties. The Butler–Volmer equation is used to account for the production and consumption of all the reactants in the cell as well as activation overpotential. The model is derived under the following assumptions:

- (1)
Steady-state conditions,

- (2)
Uniform temperature and total pressure across the thickness of the cell (

*x*-direction in Fig. 1), - (3)
Binary gas mixtures (CO–CO

_{2}, H_{2}–H_{2}O, and O_{2}–N_{2}), - (4)
Impermeable and electronically insulating electrolyte,

- (5)
One-dimensional transport of mass, ions, and electrons, and

- (6)
Constant and uniform fuel and air stream composition.

*x*-direction indicated in Fig. 1. Under this condition, the DGM on the cathode side takes the form of the following equation:

The mass transport boundary conditions are similar for both electrodes. At the electrode/electrolyte interface, a zero flux condition is imposed since the electrolyte is considered impermeable to gas diffusion. A Dirichlet condition is used to establish the reactant mole fractions at the gas/electrode interface.

*z*= −2) and the electron (

*z*= −1). The ionic and electronic fluxes are related based on current conservation so that the sum of their derivatives is zero.

The DCT boundary conditions are dictated by the assumptions listed above and, again, are similar for both electrodes. A zero ionic flux condition is imposed at the gas channel interfaces since the current collector is considered to be an electronic conductor only. At the fuel-side electrode/electrolyte interface, a Neumann condition is set such that the ionic flux is equal to the current density of the cell. This cell current density is defined as the integral of the Butler–Volmer source term over one of the electrodes, in this case the air-side electrode. The ionic flux through the electrolyte is constant since there is no electrochemical activity in that portion of the cell. For the electronic electrochemical potential, Dirichlet boundary conditions are applied at the current collectors. Specifically, the electron electrochemical potential, $\mu \u0303el$, is set at the anode based on the cell voltage (*V*_{cell} = − $\mu \u0303el$/*F*), and a ground condition is imposed at the cathode. Since the electrolyte is assumed to be electrically insulating, a zero electronic flux condition is applied at that interface for the electrodes. These boundary conditions are summarized in Table 1.

_{2}–H

_{2}O and CO–CO

_{2}atmospheres, respectively [57]. The activation overpotential for the air-side (O

_{2}–N

_{2}) electrode is shown in Eq. (9) [58]. Here, it is important to note that the overpotentials defined in Eqs. (7)–(9) vary locally across the solution domain. The ohmic and concentration overpotentials characteristic of transport losses are accounted for within this local variation. These transport losses are determined by the governing equations of mass and charge transport (Eqs. (1)–(5)), and the physical characteristics of the cell set by the defined microstructural parameters and related effective diffusivities and conductivities. A voltage is applied across the cell, and the model calculates a corresponding current density that accounts for all of the overpotentials

The system of coupled nonlinear differential equations is solved using finite element analysis software (comsol multiphysics). Porosity, tortuosity, and pore diameter can be varied within the model to allow for expedient studies to be performed to assess the effects of microstructure on the operation of SOCs in H_{2}–H_{2}O and CO–CO_{2} atmospheres.

Porosity, tortuosity, and pore diameter are related to the fabrication techniques that are used to create the electrodes, e.g., tape casting. Porosity describes the void fraction in the electrode and can be controlled during fabrication by the addition of a pore former which is not included in the completed cell. Tortuosity is the ratio of the path length that gas particles take through the electrode to the thickness of the electrode. The pore diameter is the average diameter of the pores in the electrode.

### Property Estimates From Percolation Theory.

The microstructure of SOC electrodes is generally disorderly when common fabrication techniques are used, e.g., tape casting from mixtures of powdered materials. This disorder affects the microstructural parameters of the electrode, and a random packing sphere model can be applied to account for this structure. In portions of the studies presented, percolation theory was used to describe the electrode microstructure based on an assemblage of spherical particles. The application of percolation theory primarily affects the charge transfer for a model, but these estimates also contribute to the definition of an average pore radius and the triple-phase boundary (TPB) length. Percolation theory is used to describe clusters of continuous spheres of the same type. Coordination number theory is used to determine the average number of contact points that is created by a sphere and its neighbors. The approach applied for percolation theory property estimates in the present work follows that presented by Zhu and Kee [45].

*Z*

_{tot}= 6 for a binary system of randomly packed spheres. It is then necessary to determine the coordination numbers of spheres for contact with specific particles, see Eq. (15)

*P*, can be used to calculate the effective conductivity of the ion and electron conducting phases, see Eq. (17). Here, the parameter $\gamma $ is the Bruggeman factor. This represents the effects of the solid phase tortuosity and constriction resistances due to the randomness of the particles. If the solid phase was straight with a constant cross section then $\gamma =1$, but since the assumption is that of a network of randomly packed particles $\gamma \u22481.5$

_{α}*n*

_{tot}, is estimated using Eq. (19). The TPB length per contact of the ion and electron conducting particles, $\u2113io\u2212el$, is found according to Eq. (20). The width of a given TPB,

*w*

_{TPB}, and the contact angle

*θ*is treated as constants within the model, but is considered adjustable to allow an estimation of the TPB active area to unit volume. Typical values for the TPB width and contact angle may be estimated from the literature [68,70–72]

_{c}## Results and Discussion

The multiphysics model described above simulates the current density generated by an SOC in response to an imposed voltage and gas reactant concentrations. Activation losses are accounted for by the Butler–Volmer reaction terms (Eqs. (6)–(9)), while ohmic and concentration losses are accounted for through the governing transport equations, see Eqs. (1)–(5). Thus, the cell current density may be predicted without further substitution of field variables into an expression for the cell potential. Imposing a voltage below the open circuit voltage (OCV) simulates fuel cell operation. Conversely, when the voltage exceeds OCV, the model simulates electrolysis. To test the validity of this model, a comparison was made to the experimental data published by Ebbesen and Mogensen [18], see Fig. 2. The solid and dashed lines represent the current density predicted directly by the finite element model. The percentage of carbon monoxide shown in the legend of the figure denotes the carbon monoxide mole fraction applied in the boundary condition at the electrode/fuel channel interface. These values correspond to the CO percentages reported by Ebbesen and Mogensen [18]. Porosity values for the anode and cathode were set at 0.35 and 0.3, respectively, and the tortuosity was adjusted to obtain agreement between the model prediction and the experimental data. Since porosity is typically measured for electrodes, the comparison of the model to experimental data allows for an approximation of the tortuosity. A value of *τ* = 2.1 was applied in a nominal scenario to obtain agreement between the model predictions and the experimental observations. This value is considered representative of tortuosities estimated from data obtained from direct 3D imaging of electrode microstructures [25,26,65,67–69]. The exact microstructure of the electrodes used by Ebbesen and Mogensen is not available, and it is desirable to understand the variability of model predictions given this uncertainty. To this end, a sensitivity study was performed by varying the tortuosity by ±10% of the nominal value. This arrangement essentially modifies the prefactor *ε*/*τ*^{2} applied in estimating an effective diffusion coefficient for each reactant gas. These studies were also run under a pure O_{2} cathode scenario to account for flow conditions applied in Ref. [18]. The ranges of predictions in these sensitivity studies are indicated by the shaded regions that bracket the experimental data and model predictions.

The comparison shows the ability of the model to represent both fuel cell and electrolysis operations. For both cases, the model closely predicts the experimental data, and in general, the range of the sensitivity studies captures the experimental data, although agreement is stronger for predictions of fuel cell operation. There are some differences worth discussing to critically assess the comparison in Fig. 2. These differences arise primarily for electrolytic operation. First, there are lower concentration losses shown in the nominal predictions of electrolytic operation. This discrepancy arises primarily due to the application of air as the gas on the cathode side. Ebbesen and Mogensen supplied a stream of pure O_{2} to the cathode in their electrolysis measurements. Air was applied on the cathode side in the nominal fits shown to demonstrate the seamless simulation of reversible operation. The sensitivity studies were also run under a pure O_{2} cathode gas stream to check the effects of increased O_{2} content on performance. Application of air on the cathode side reduces the overall mixture mass, diffusion coefficient, and the estimated OCV for the cell. These effects combine to shift the prediction of the polarization curve downward. Simulated operation under pure O_{2} demonstrates better agreement with the experimental results at higher cell voltages, but some error manifests closer to OCV. This error near OCV could be a result of assumed reaction kinetics for CO oxidation and CO_{2} electrolysis, specifically the application of exchange current density based on the values for fuel cell operation. Future analysis should focus on further defining the kinetics and catalytic reactions to improve the results of the model. The curve of the model follows the data trend reasonably well and is considered sufficient for an investigative analysis of SOC microstructure.

### Parametric Studies.

The model developed is capable of capturing experimentally observed cell behavior based on reasonable estimates of microstructural parameters. Therefore, it may be used to perform preliminary analyses to study SOC operating conditions and the effects of microstructure, as initially described by lumped parameters, on performance. One such study that was performed was the analysis of the varying effect that microstructure has on CO–CO_{2} atmospheres in comparison to H_{2}–H_{2}O operations. The expectation in this study is that CO–CO_{2} fuel mixtures would show a greater sensitivity to changes of the microstructure of the pore phase in relation to H_{2}–H_{2}O mixtures. The attendant microstructural design of electrodes for CO–CO_{2} and hydrocarbon fueled SOCs would therefore need to address such limitations. Parametric studies were performed by varying the porosity and the pore radius to explore these effects. The tortuosity of the electrode was held constant. However, variation of the porosity cannot be strictly separated from a variation in tortuosity due to the method of calculating effective diffusion coefficients.

To ensure that an equivalent comparison is made between the two different operating atmospheres, all the parameters, other than porosity and pore diameter, were held constant between both systems. The analysis was focused on the anode microstructure, so the cathode remained unchanged during the study. Table 2 outlines the parameters that were held constant throughout the study. The fuel mixture that was used was 90% fuel to 10% product. This would result in a carbon monoxide rich fuel stream which could potentially lead to coking in the electrode [1,16,73]. This possibility was excluded from the study to mitigate variations between the gas systems to be able to solely investigate the effects of nondegraded microstructure on performance. The tortuosity values applied are comparable to pore phase tortuosity values reported for 3D microstructural data from SOC electrodes [65–67]. Electronic and ionic conductivities as well as exchange current densities were set to values comparable to those reported in the literature. Concentration dependence of the exchange current density was not addressed in the present work. However, this dependence may be incorporated in future studies.

The first study that was performed varied the porosity of the anode while holding the pore diameter at a value of 0.2 *μ*m. The resulting voltage–current density curves can be seen in Fig. 3 and show a considerable difference in the losses experienced by the cells operating in the two gaseous systems. The H_{2}–H_{2}O operations only begin to show concentration losses at a porosity of 0.2 and seem to be primarily controlled by ohmic losses. On the other hand, the CO–CO_{2} system is dominated by concentration losses.

Figure 4 shows that H_{2}–H_{2}O systems have considerably higher power densities in comparison to CO–CO_{2}. Upon further analysis, it can be shown that the CO–CO_{2} systems experience greater performance increases from the changes in porosity. Table 3 shows the maximum power densities that were calculated for the different porosity and fuel cases. From these values, it can be determined that the H_{2}–H_{2}O system experiences an increase of 12% in maximum power density from 20% to 40% porosity, while the CO–CO_{2} system has a 52% gain. While the overall power density for hydrogen operations is higher, the carbon monoxide fuel source allows for a greater increase in performance when a diffusion-enhancing microstructure is applied.

The next study that was performed examined how the pore diameters affected SOC performance. Variations in pore diameter produce slightly different results since it only effects the Knudsen diffusion in the electrode. Figure 5 shows that the CO–CO_{2} atmosphere is still primarily dominated by concentration losses while the same losses are only noticeably present for H_{2}–H_{2}O at the smallest pore diameter used.

The power density curves shown in Fig. 6 are similar to those in Fig. 4, as expected. For the chosen range of pore diameters, the power density shows a larger increase in comparison to the results from the porosity study. It is still clear to see that diffusion-enhancing microstructures are beneficial for both operating systems which can be seen tabulated in Table 4. These results show us that the H_{2}–H_{2}O system experienced an increase in power density of 25% from a pore diameter of 0.1 *μ*m to 0.7 *μ*m while over the same range the CO–CO_{2} power density increased by 90%, both determined at the maximum power density calculated.

Again, Figs. 3 and 5 show the effects of concentration polarization on CO–CO_{2} operations. As the microstructure of the electrode is modified to allow for faster diffusion rates, the concentration losses begin to be overcome. On the other hand, since H_{2}–H_{2}O operations have relatively negligible concentration losses within this study the faster transport of gases through the electrode produces significantly smaller increases in performance. These results compliment the original hypothesis for the study that CO–CO_{2} systems are more sensitive to microstructural changes, which leads to the conclusion that to improve those operations focus should be put on improving the diffusion rates through the electrodes. Furthermore, since H_{2}–H_{2}O operations are considerably less affected by microstructure changes it is more important to focus on the improvement of the materials applied in SOC construction to reduce ohmic polarization losses. Ionic conduction would need to be the primary focus of that study since electronic resistivity of the Ni electrocatalyst is negligible relative to the ionic resistivity. One possibility could be the microstructural analysis of the ion-conducting phase rather than the pore phase. As shown in Sec. 3.2, percolation theory provides a method for estimating the effects of microstructure on ionic conductivity. Negative effects to the ionic conductivity translate into large performance effects due to the already relatively small conductivity.

### Performance Assessment Based on Percolation Theory.

Since the microstructural parameters define a specific format for the electrodes, it must be taken into account that the parameters are coupled and changes to one parameter affect every other one. Percolation theory provides a method for determining the effects microstructure has on various transport and reaction parameters. The addition of percolation theory is necessary to avoid arbitrary specifications of parameters which can increase error in the model by defining an electrode microstructure that is improbable. Figure 7 shows that as the porosity of the electrode increases, the effective ionic conductivity decreases. This decrease is to be expected since the volume fraction of the ionic conducting material must decrease as the electrode becomes more porous. It is interesting to note that the decreasing trend in effective conductivity is not strictly linear. The slight nonlinearity of this trend is a result of the ionic conductor volume fraction being a percentage of the electrode that remains from the porosity, being balanced by the electronic conductor.

The active area per unit volume of the electrode was calculated for various values of the ionic conductor total volume fraction, see Fig. 8. The radii of the ion conductor and electron conductor particles were both set at 0.5 *μ*m for calculating effective conductivities. The electronic conductor volume fraction was varied as a function of the ionic conductor total volume fraction and the electrode porosity. For the case shown, an anode porosity of 45% was applied. The resulting curve is parabolic in nature with a maximum active area at an ion conductor total volume fraction of approximately 0.275 for this study. As expected, this maximum TPB area occurs at a 50:50 ratio of ionic conductor to electronic conductor [74,75]. It should be noted that the electrode microstructure must ultimately balance several modes of transport. The effective ionic conductivity is dependent on the volume fraction and size of the ionic conductor. Figure 8 also shows how the effective ionic conductivity changes with respect to the ion conductor volume fraction. The increase with increasing ionic conductor content is expected since the conductivity increases as the conductive path increases across the electrode. This suggests that the maximization of the TPB area may not be the sole indicator of optimal performance for the SOC. Further studies are underway to assess the effects of particle sizes and volume fractions. The remaining studies in the present work focus on the effects of electrode porosity on the performance related to gas transport.

As with the ionic conductivity, the effective diffusion coefficient of the gas species will also depend upon the particle sizes and volume fractions. Since percolation theory couples certain microstructural parameters it produces interesting effects on the effective diffusion coefficients for the gaseous reactants. Figure 9 displays how the effective diffusion coefficients vary as the porosity of the electrode increases. Binary diffusion is only effected by changes in porosity, as far as microstructure is concerned, so it is not surprising that the effective binary diffusion coefficient increases proportionally to the increase of the porosity. The effective Knudsen diffusion coefficient is a function of porosity and the pore radius. This relationship produces an exponential increase of the coefficient as the porosity rises. The effective Knudsen diffusion coefficient experiences an increase in value due to the change in porosity just like the effective binary diffusion coefficient, but the percolation theory definition for the pore radius causes the value to tend toward infinity as the porosity approaches unity. This definition then causes the effective Knudsen diffusion coefficient to tend toward infinity as the porosity increases. This trend is an artifact of the mathematical definition of the pore radius, see Eq. (22). In a more rigorous sense, Knudsen diffusion becomes negligible as pore radii significantly exceed the mean free path of the diffusing gas. The secondary axis of Fig. 9 shows the rapid decrease of the Knudsen number as the Knudsen diffusion coefficient increases. For the present studies, these results suggest that the contribution of Knudsen diffusion becomes negligible as porosity exceeds 0.78.

Figure 10 shows voltage–current density curves that are calculated for H_{2}–H_{2}O and CO–CO_{2} atmospheres based on percolation theory. The curves were determined at porosities of 0.3 and 0.6 for the fuel-side electrodes. These porosity values were chosen to represent how percolation theory affects the modeling results at the edges of the percolation parameters. The results of H_{2}–H_{2}O and CO–CO_{2} for a porosity of 0.3 are as expected since the resulting curves are similar to the ones found in Figs. 3 and 5. The curves for a porosity of 0.6 produce interesting results. For the H_{2}–H_{2}O system, an increase in porosity has little effect on the concentration polarization since the diffusion of hydrogen is relatively fast. On the other hand, the increase in porosity decreases the percentage of ion-conducting material in the electrode. The decrease in ionic conductivity increases the ohmic losses that the system experiences and results in an overall less efficient system. These results are in agreement with the previous statement that the increase of ionic conductivity in SOC electrodes is more important to efficiency than electrodes that have favorable mass diffusion microstructures. The results for the CO–CO_{2} atmosphere with an electrode porosity of 0.6 show an overall increase of efficiency in operation. The increase in efficiency is the result of the mass diffusion favorable microstructure since the concentration losses that are experienced at a porosity of 0.3 are mostly mitigated.

### Electrode Catalytic Effectiveness.

_{2}–H

_{2}O and CO–CO

_{2}systems can be furthered by examining the general capabilities of electrodes to support oxidation of hydrogen and carbon monoxide. This comparison is accomplished by application of a catalytic effectiveness factor, defined general in Eq. (23), and related Thiele moduli (Eq. (24)), that compare Faradaic reaction rates to diffusion within the porous electrode. A baseline comparison can be made using a first-order reaction model [76]. For the boundary conditions imposed on the electrodes in the models presented above, a catalytic effectiveness factor for the first-order reaction may be defined using Eq. (25)

Through the application of the effectiveness factor and the Thiele modulus, a comparison of H_{2}–H_{2}O and CO–CO_{2} systems can be performed. Figure 11 shows how the effectiveness factors for hydrogen and carbon monoxide oxidation differ from one another by determining the Thiele moduli for the oxidations and comparing them to the first-order reaction case. While there is overlap of the two oxidation reactions, hydrogen generally operates at a higher effectiveness factor than carbon monoxide. This is consistent with Fig. 10 in that the operational losses, specifically concentration losses, for CO–CO_{2} systems are typically higher than that of H_{2}–H_{2}O systems which would decrease the effectiveness factor for CO–CO_{2}. Although, the polarization curves for a porosity of 0.7 are similar for the two systems. Not surprisingly this latter scenario coincides with a region in which the effectiveness factors for H_{2} and CO oxidation overlap.

Thiele moduli for the WGSR were also calculated to demonstrate how the dimensionless analysis may be applied to secondary reactions that occur in SOC electrodes, particularly electrodes operating with multicomponent fuel mixtures, e.g., syngas or hydrocarbon reformate. This preliminary assessment illustrates a general domain of microstructures in which such reactions are competitive with, or exceed, the electrochemical oxidation reactions. For this analysis, the WGSR was assumed to be a purely gas phase reaction, that is, the reaction rate term is decoupled from the microstructure of the electrode. The microstructure is accounted for in the calculation of the effective diffusion coefficients. Reaction rates were estimated based on the published rate constants [77,78]. The ranges of Thiele moduli for the WGSR are also shown in Fig. 11. In general, it can be seen that the WGSR tends to exceed CO oxidation for most of the electrode microstructures considered in the present analysis. However, there is a region of overlap between the two reactions, which suggests that for certain microstructural geometries, both reactions may contribute to the consumption of CO fuel.

## Conclusion

Microstructural designs for optimum performance in SOCs are dependent upon the atmosphere in which the SOCs are intended to operate. This dependence has been investigated using a multiphysics model that simulates mass and charge transport in SOCs. Initial studies were performed based solely on variations of the pore phase microstructure. These studies compared how H_{2}–H_{2}O and CO–CO_{2} operation were affected as the porosity and pore size of the electrodes were varied. The results of these initial studies showed that the CO–CO_{2} atmosphere benefited considerably more when a microstructure was applied that was favorable for the diffusion of the gaseous reactant, specifically when the Knudsen diffusion coefficient was increased. Simulated operation in an H_{2}–H_{2}O atmosphere did exhibit an increase in performance as well, but since the changes in the porous microstructure primarily affect the concentration overpotential of the cell, the increase was minor in comparison to the CO–CO_{2} system. This reduced influence is due to the relatively high diffusion coefficients of H_{2}.

The initial studies were performed using parameters that are within typical ranges of current electrode fabrication techniques. The results of the initial study led to the desire to determine how microstructural parameters could be tailored to optimize operations for H_{2}–H_{2}O and CO–CO_{2} atmospheres. At this point, percolation theory was incorporated into the model to better simulate the randomly packed nature of the SOC electrodes that results from common fabrication methods. The use of percolation theory couples the effective transport properties that describe how the electrode microstructural morphology affects mass, ion, and electron transport. With the use of percolation theory, the microstructural parameters of the electrodes were altered to analyze the effects of a microstructure with relatively high porosity, which results in mass and ion transport properties that are not typically seen in SOC electrodes. These studies were compared to results from an analysis that implemented microstructural parameters that are in proximity of values that are commonly found in recent published data. The results showed that an open microstructure is beneficial to CO–CO_{2} operations, which is in agreement with the initial studies. The open microstructure does cause an increase in the activation and ohmic losses, but since CO–CO_{2} systems are primarily limited due to their diffusion rates the decrease in concentration losses outweighs the other losses that are experienced. Conversely, the H_{2}–H_{2}O systems saw a decrease in performance with a more open microstructure. Since the concentration losses for H_{2}–H_{2}O are already relatively low, the decrease in total active area and ionic conductivity imposed activity limits for the electrochemical reaction. The diffusion of hydrogen and steam through the electrode was already at high enough rates to complement the reaction rates that were associated with a less porous microstructure. This analysis demonstrates how the limiting factors of SOC operations associated with electrode design may vary based upon the reactant gas streams. The optimization of H_{2}–H_{2}O operations is strongly linked to the increase of electrochemical activity in the electrode, specifically through increased active surface area and improved ionic conduction. By reducing these latter characteristics, developing a more open microstructure for H_{2}–H_{2}O systems could potentially decrease the efficiency of SOCs without first increasing the rate of reaction. Alternatively, operation in CO–CO_{2} atmospheres presents a limitation from the relatively high concentration losses that are experienced. This limitation can, theoretically, be countered by developing an electrode that is favorable to the diffusion of the carbon monoxide and carbon dioxide through the electrode pores.

The application of nondimensional analysis provides a means through which SOC electrode microstructures can be compared for the purpose of estimating the performance of the cell based on specific reactions. This approach was demonstrated through a first-order assessment of electrode performance for H_{2} and CO oxidation in terms of catalytic effectiveness and Thiele moduli. In general, H_{2}-based operation exhibits a greater catalytic effectiveness. However, the effectiveness of CO-based systems can be comparable with appropriately tuned microstructures. This observation is in agreement with the more detailed assessment of the parametric studies.

The multiphysics model presented offers the ability to analyze electrode structure and calculate detailed information on how a system can be expected to perform, while the nondimensional approach provides a rapid analytic approach to designing SOC electrodes that meet certain performance requirements. These methods can be applied to the development of SOC electrode microstructures that are tailored to operate within prescribed regimes that are more favorable to the desired reaction kinetics for a specific application.

## Acknowledgment

The financial support from a Research Infrastructure Development grant funded by the National Aeronautics and Space Administration through the Alabama Space Grant Consortium (Grant No. NNX13AB09A) is gratefully acknowledged.

## Nomenclature

- $AS/V$ =
active surface area-to-volume ratio, m

^{−1} - $Dijeff$ =
effective binary diffusion of components

*i*and*j*, m^{2}s^{−1} - $Di,keff$ =
effective Knudsen diffusion of component

*i*, m^{2}s^{−1} - $F$ =
Faraday's constant, C m

^{−1} - $hT$ =
Thiele modulus

- $irxn$ =
transfer current density, A m

^{−2} - $i0$ =
exchange current density, A m

^{−2} - $Mi$ =
molecular weight of component

*i*, g mol^{−1} - $p$ =
partial pressure, Pa

- $P$ =
total pressure, Pa

- $P\alpha $ =
percolation probability

- $r$ =
particle radius,

*μ*m - $rp$ =
pore radius, m or cm or

*μ*m - $R$ =
universal gas constant, J mol

^{−1}K^{−1} - $T$ =
temperature, K

- $yi$ =
mole fraction of component

*i* - Z =
coordination number

### Greek Symbols

- $\beta $ =
charge transfer coefficient

- $\epsilon $ =
porosity

- $\zeta $ =
number fraction

- $\eta $ =
effectiveness factor

- $\mu \u0303$ =
electrochemical potential, J mol

^{−1} - $\nu $ =
volume fraction

- $\sigma $ =
conductivity, S m

^{−1} - $\tau $ =
tortuosity

- $\varphi $ =
overpotential, V

- $\psi $ =
volume fraction with respect to the solid phase