Understanding Power Enhancement of SOFC by Built-in Chemical Iron Bed: A Computational Approach

Solid oxide fuel cells (SOFCs) with enhanced fast ramping power capability and overload tolerance can ﬁnd important applications in grid stability management and critical data center overload protection. Recently, we have demonstrated a new SOFC conﬁguration featuring a built-in chemical Fe-bed in the anode chamber of a tubular SOFC with exceptional fast power ramping capability and overload tolerance. In the present study, we showed our theoretical understanding of the enhanced performance through a two-dimensional axial symmetrical numerical model. The model couples the charge and mass transport in the tubular SOFC with chemical reaction kinetics in the Fe-bed, producing longitudinal distributions of Nernst potential, H 2 O/H 2 molar ratio, local current density and fuel utilization under various operating conditions. The crucial role of Fe-bed in providing instant H 2 to support fast ramping and overload currents has been explicitly explained by this computational model.

Solid Oxide Fuel Cell (SOFC) has long been deemed an ideal efficient and clean power generator for uninterrupted steady-state baseload and distributed generation. [1][2][3] In reality, however, the load is not always constant, fluctuating periodically from peak to valley throughout a day. Typically, SOFC can follow these load variations provided that the interval of load change is long enough to allow SOFC system to adjust fuel flow to meet the power demand. 4,5 However, if the load increase is too sudden that the fuel supply system could not have sufficient time to respond and provide the necessary fuel for the overload, it would result in oxidation of Ni-based anode and cause mechanical failure of SOFC stacks. [6][7][8] Recently, we presented a new tubular SOFC concept featuring a chemical Fe-bed loaded in the anode chamber and demonstrated that such Fe-bed loaded SOFC can produce fast ramping power and operate at high overload currents. 9 The built-in Fe-bed reacts with H 2 O in the fuel stream to produce extra H 2 for supporting fast power and overload operation. The demonstrated new functionalities suggest that the Fe-bed SOFCs are well suited for applications in grid stability management to provide timely fast power for grid power management and critical overload protection for data center. However, the fundamental aspects including distributions of Nernst potential (E N ) and local current density of the newly configured SOFC are still lacking at this point. It is worth mentioning that the Fe-bed concept presented in this study is fundamentally different from early studies that used Fe to alloy with Ni in the anode to boost its coking resistance so as to operate on methane fuel. 10,11 In the present study, we report our theoretical analysis of the enhanced dynamic and electrochemical performance of an anodesupported tubular SOFC with a built-in Fe-bed through a twodimensional axial symmetrical multi-physics numerical model. The physical-chemical processes considered by the model include the charge/mass transport in a tubular SOFC and chemical reaction kinetics in the Fe-bed. For high-fidelity prediction, the model parameters were validated by experimental V-I data obtained from a pilot-scale anode-supported tubular SOFC. Longitudinal distributions of Nernst potential, composition, local current density and fuel utilization are simulated under various load conditions.

The Numerical Model
The numerical model is built from the configuration of a tubular solid oxide fuel cell loaded with a Fe-bed in the anode chamber shown in Fig. 1. The operating temperature for the model is set to 750 • C to reflect the practical operating temperature of such SOFC. 9 At this temperature, FeO is the prevalent product of Fe oxidation by H 2 O through the reaction Fe + H 2 O = FeO +H 2 . The cell operates at ambient pressure with the cathode constantly open to air.
The physical and chemical processes occurring in the tubular SOFC with a Fe-bed include: electron/oxygen-ion transport in the SOFC, electrochemical reactions at triple phase boundaries of SOFC electrodes, and gas diffusion through the porous electrode, open-space chamber and porous Fe-bed. [12][13][14][15][16][17] In addition, the kinetics of steam oxidation in the Fe-bed depends on temperature and Fe/FeO molar ratio. The fuel velocity in the chamber's open space is assumed to be constant, while that in the porous electrode and porous Fe-bed is deemed zero for simplicity. The governing equations and corresponding variables for each domain are given in Table I, while the source terms and  associated parameters are listed in Table II.

Construction of Numerical Model with Boundary/Initial Conditions
According to the schematic shown in Fig. 2a, there are three components considered in the model: SOFC, free space (chamber) filled with H 2 and H 2 O and Fe-bed. Figure 2b shows a 2D axial symmetric computational domain corresponding to the area highlighted by the dashed line in Fig. 2a. The SOFC is an anode-supported tubular design with a 22 mm OD, 18 mm ID and 323 mm length. The effective surface area is 220 cm 2 . The makeup of the cell consists of a ∼2 mm thick Ni-Zr 0.84 Y 0.16 O 2-δ (Ni-YSZ) anode substrate, a ∼10 μm thick YSZ    9 The whole system is operated at 750 • C. The boundary conditions for the model are given in Table III. The initial gas composition in the chamber was taken as mole ratio x H 2 :x H 2 O = 0.97:0.03.

Numerical Solutions and Model Validation
There are a total of 8 dependent variables in the model, including the electronic potential φ e , ionic potential φ i , mass fraction of the gas species, ω j (ω 1 ,ω 2 ,ω 3 in the porous air electrode, ω 4 and ω 5 in the fuel electrode, chamber and Fe-bed), the reacted fraction ratio of Fe (or Fe utilization) x Fe . By combining governing equations in Tables  I-II with the boundary conditions in Table III and initial conditions,   Table II. Associated parameters used in the model.

Source terms/Parameters
Mathematical expressions   those variables can be solved simultaneously as a function of time and positions.

Binary diffusion
The computation was performed with a commercial software package COMSOL Multiphysics 4.3 using a workstation equipped with an Intel Core i7-4700MQ processor @2.4GHz and 2.39 GHz and 16 GB of RAM. The computational domain was discretized by swept mesh and refined until a converged solution was reached. The Direct Solver was applied to solve the transient problem. Electric Currents module, Transport of Concentrated Species module, and General form of PDEs were utilized as the governing equations.
To ensure the accuracy of the numerical analysis, the model parameters were first validated by the experimental V-I curve under 750 • C and fuel utilization of 75% without Fe-bed. The reaction rate constant of Fe-oxidization in the Fe-bed is determined by an early study of solid oxide iron-air redox battery under the same temperature. 13 Since some of the parameters could not be obtained directly from experiments, a best fitting approach is used to extract these parameters, which has been a common practice in the SOFC community. 22,23 Figure 3 shows that the fitted results agree well with the experimental V-I curve (the adjustable parameters marked with ' * " in Table IV). Overall, the obtained parameters are well within a reasonable range as reported in the open literature. 22 Note that the use of obtained exchange current density for Fe-bed SOFC represents a conservative action since higher H 2 concentration provided by the Fe-bed could also increase the effective exchange current density. The model primarily considers the effect of high H 2 -concentration enabled by the Fe-bed on local Nernst potential. Therefore, the model should produce meaningful results for understanding the Fe-bed effect.

Results and Discussion
A focus of the present work is to investigate the dynamic responses of the power generation of a tubular SOFC loaded with and without a Fe-bed. In the following section, the performance improvement of the tubular SOFC by the Fe-bed will be presented under different cell voltages and different Fe loadings. To understand the mechanism of the performance enhancement, the longitudinal distributions of Nernst potential, local current density, fuel utilization and molar ratio of H 2 O and H 2 at a predefined stabilizing time step of 180 seconds are computed.
Transient average current density.-For a tubular SOFC, the current is typically collected by two bus bars attached longitudinally to the anode and cathode, respectively. Under this circumstance, the electric potential at the electrode/current collector interface along the longitudinal direction is uniform, equalling the potential applied. The overall performance of the tubular SOFC evaluated by the average current density along the longitudinal axis under four operating ) unless CC License in place (see abstract   voltages is shown in Fig. 4a. A transient process is observed at the very first few seconds, followed by a stabilized current density. Therefore, in the following, we choose a step of 180 seconds as a stabilization time to study the longitudinal distributions under various operating voltages. A replot of cell voltage (V cell ) vs stabilized average current density (J avg ) in Fig. 4a is shown in Fig. 4b. The increase of J avg by the Fe-bed is evident, i.e. at V cell = 0.904, 0.858, 0.805 and 0.755 V, J avg is increased by 5.6%, 12.5%, 19.3% and 26.4%, respectively, compared to that without Fe-bed. The lower the operating voltage, the greater the improvement. Therefore, the Fe-bed is more beneficial to high current density operation. [24][25][26] the fuel is introduced through a fuel injector to the closeend and then flowed back toward the open-end over the inner surface of the anode for a pilot scale anode supported tubular SOFC, where it meets with air that flows over the outer cathode surface of the cell. 9 As both fuel and air co-flow along the longitudinal direction, the electrochemical cathodic and anodic reactions take place, resulting in a depleted fuel (or increased oxide) distribution along the inner anode surface of the flow direction, thus lowered Nernst potential (E N (z)). In other words, instead of a constant E N for the button cell, there will be an E N (z) distribution along the longitudinal direction of the cell during operation.

Distribution of nernst potential (E N (z)) along the longitudinal direction (z).-Different from the button cell previously studied in our lab,
In this section, we compute the E N (z) profiles along the longitudinal direction for both the baseline and Fe-bed SOFC under different voltages. The results are shown in Fig. 5, where it is fairly obvious that for the baseline cell, Fig. 5a, the E N (z) drops rather quickly in the first 50-mm, followed by a slower linear decrease with cell length, especially under high operating voltages. The lower the cell voltage or the higher the cell current, the more pronounced the E N decrease along the cell length. In comparison, for a tubular Fe-SOFC, the E N distribution, Fig. 5b, is less sensitive to the cell voltage than that of the baseline cell. It is also noticed that the distribution of Nernst potentials are similar in magnitude for both cases for the first ∼100 mm. The major difference is seen toward the open-end of the cell: the E N (z) of Fe-bed SOFC stabilizes at a certain value for each operating voltage, whereas it falls for the baseline cell. Such a "stabilized Nernst potential" phenomenon toward the open-end region infers that H 2 produced from the reaction of Fe-H 2 O stabilizes the H 2 /H 2 O that exclusively determines the E N (z) at a fixed temperature. Table  II   the baseline cell observed in Fig. 5. It is also worth mentioning that the nature of nonlinear i-η relationship (Butler-Volmer equation in Table II) and the logarithm relationship between E N (z) and r(z) can yield a more pronounced variation of E N (z) along the cell length in a regular baseline cell under a load condition. However, if an extra H 2 -source such as Fe-bed is present very next to the anode, the E N (z) variation can be effectively mitigated. This is extremely important to applications such as fast ramping power and overload protection since conventional SOFCs may suffer Ni-oxidation due to slow response to fuel demand, ultimately threatening the lifetime of stack.

Distribution of local current density (J loc (z)) along the longitudinal direction z.-In
Distribution of nernst potential (E N (z)) along the longitudinal direction (z) section, we show that E N varies along the length of the tubular SOFC. Since the operating cell voltage is fixed along the length by the bus bars, the local current density will vary along the cell. In this section, we here compute the local current density (J loc (z)) profiles along the cell length under different cell voltages for both baseline and Fe-bed SOFC.
For the baseline cell, Fig. 7a, the highest J loc (z) is located at the close-end (z=0 mm), where fresh fuel is introduced. This is understandable in that E N is the highest at the closed-end. Above the close-end, J loc (z) monotonically decreases with cell length z. It is also noticed that the distribution of J loc (z) is deeper at lower cell voltage than at higher voltage. In comparison, for the Fe-bed SOFC, Fig. 7b, J loc (z) also decreases with cell length z, but exhibits a much flatter profile than the baseline cell. It appears that the lower the cell voltage, the lower the z-value at which the local current density is stabilized. For example, for V cell = 0.904 V, the stabilizing J loc (z) occurs at z = 200 mm; however, it is 150 mm for V cell = 0.755 V. This observation implies that there are more Fe taking part in the reaction due to higher concentration of H 2 O at lower cell voltage (or higher current density).

Distribution of fuel utilization (U f (z)) along the longitudinal direction z.-
The fuel utilization U f (z) is calculated by the ratio of hydrogen molar fraction x H 2 (z) at a certain z location vs x H 2 (0) at the close-end of the cell, i.e. U f (z) = x H 2 (z)/x H 2 (0). Therefore, it is a direct indication of the H 2 fuel molar fraction in the anode stream. Because of the 2D axial symmetric configuration, we used the radial integration to calculate the average fuel molar fraction x H 2 (z) at each z. The results are shown in Fig. 8 for both baseline and Fe-bed SOFCs.
For the baseline cell, Fig. 8a, the local U f (z) increases monotonously with the cell length z. The lower the operating voltage, the steeper the profile. As the fuel approaches the open-end of the cell, the slope of the fuel utilization tends to be smaller than that at the close-end. Overall, U f (z) at the open-end can be very high, e.g. U f (z=324 mm)=95% at 0.755 V. This is a dangerous gas composition since excessive H 2 O could potentially oxidize the Ni-anode. In   comparison, the Fe-bed SOFC, Fig. 8b, shows a much flatter profile than the baseline cell at all operating voltage simulated. The highest U f (z=324 mm) is 55% at 0.755 V. The additional H 2 produced in situ by the Fe-bed is clearly the fundamental reason for the low local U f (z), demonstrating the tolerance of Fe-bed SOFC to overload operation.

Longitudinal distribution of Fe-utilization in the chemical bed.-
One observation from the above simulation is that the lower the cell voltage or the higher the cell current, the closer the active Fe-bed to the close-end, inferring that more Fe are involved in the H 2 O-Fe reaction. To confirm this prediction, we also computed the Fe utilization (x Fe (z)) profiles (integration along the radial direction at each z location) under different cell voltages; the results are shown in Fig. 9. It is evident that the Fe-bed begins to be oxidized at different locations for different cell voltages, i.e. z = 180, 124, 91.5 and 75.4 mm for V cell =0.904, 0.858, 0.805 and 0.755 V, respectively. The lower the operating voltage the closer to the close-end the Fe-bed to produce H 2 . More Fe is consumed at lower cell voltage due to higher concentration of H 2 O resulted from higher cell current.
The effect of Fe -loading on cell performance.-The effect of Feloading on cell performance is also investigated. Figure 10 summarizes the results, which compares the performance of the baseline cell with 34 g Fe-loading with that of a cell with half of the loading, i.e. 17g. Clearly, there is more impact from Fe-loading at a lower operating voltage, 0.755 V. The average current density J avg is increased by 170 A/m 2 by doubling Fe-loading, see Fig. 10a. In comparison, at higher operating voltage, 0.904 V, the average current density J avg is only increased by 10 A/m 2 by doubling Fe-loading.
Correspondingly, at lower operating voltage of 0.755 V, by doubling Fe loading E N at the open-end is increased by 0.03 V  (see Fig. 10b), while the H 2 O/H 2 molar fraction ratio at the openend is decreased by 1.5 times (see Fig. 10c). Such an enhancement can also be seen from the local current density (J loc ) and fuel utilization (U f ) profiles shown in Figs. 10d and 10e, where J loc increases and U f decreases from z=100 mm to the open-end, respectively. To achieve such performance improvement, the Fe is utilized to generate extra H 2 to reach a new dynamic equilibrium in the anode chamber. With higher Fe-loading, i.e. 34 g, the Fe utilization is relatively small, which results in a higher Fe-bed reaction rate according to the last rate equation listed in Table I. Therefore, the H 2 O/H 2 molar fraction ratio is only slightly deviated from its equilibrium, which means more hydrogen exists in the anode chamber and is beneficial to achieve a higher cell performance.
The average current density J avg @ 180 s as a function of Feloading is shown in Fig. 11. From the slope change of the curve with Fe-loading, it is inferred that the improvement resulted from Feloading is more significant at low Fe-loading than at high Fe-loading. In other words, as Fe loading increases the corresponding increase of J ave becomes gradually limited. On the other hand, more Fe-loading occupies more space of the anode chamber, resulting in an increased General comments on Fe-bed tubular SOFC.-Based on the above simulation results, one can conclude that the inclusion of a Fe-bed in the anode chamber of a tubular SOFC is beneficial when the cell is operated under overload or fast ramping conditions. The mechanism leading to the cell performance enhancement is due to the extra H 2 generated by the Fe-bed, which increases the local E N and the current density. The minimum Fe-loading depends on the operating current density and operation duration. There is a trade-off between the Fe-loading and fuel flow pressure drop in the anode chamber, since the increased amount of Fe occupies more space in the anode chamber and leads to an increased pumping power. In addition, beyond the minimum Fe-loading content, the improvement of cell performance becomes less significant.

Conclusions
In summary, a 2D axial symmetrical multi-physics model has been constructed for an anode-supported tubular solid oxide fuel cell loaded with and without a Fe-bed. The model parameters are validated by experimental data to ensure a high-fidelity computation. The model features the coupling of charge and mass transport in a tubular SOFC with chemical reaction kinetics in the Fe-bed, producing longitudinal distributions of Nernst potential, H 2 O/H 2 molar ratio, local current density, fuel utilization and Fe-utilization under different load conditions. The simulations explicitly show that the Fe-bed loaded SOFC exhibit a higher and flatter distribution of E N (z), xH 2 /xH 2 O(z), local J loc (z) and local fuel utilization along the cell length z. The lower the cell voltage (or higher cell current) the more Fe is used. The fundamental reason for the enhanced performance is the production of H 2 through in situ H 2 O-Fe reaction. The ability of Fe-SOFC to produce extra H 2 next to the Ni-anode enables the cell to tolerate fast transient overload and provide fast ramping power for practical power generations.