After the mathematical model was formulated, a series of 3D numerical simulations were performed using the finite-volume method. For the finite-volume method, a computational mesh is required. To build the mesh of each of the designs described in
Section 2, the 3D models were modified to include elements that represent the paths of the gases. To reduce the number of mesh elements, the ceramic covers were virtually removed and replaced with the proper thermal boundary conditions, representing the behavior of the furnace that maintains a constant temperature (Dirichlet boundary conditions). Each mesh was created as a compromise between quality and the number of elements, which determines the resolution, accuracy, and computational time. The difference in scale between the electrodes/electrolyte and gas channels is significant. A smooth transition of element size was utilized to properly model the fluxes between both domains. Each of the meshes was built using hexahedral elements. The size of the basic cubical mesh element is 1 mm × 1 mm × 1 mm. In the electrochemically active area, the elements are flattened to 0.01 mm, to retain a minimum of three layers per element. In
Figure 3, a computational mesh of the multistack design (see
Figure 2c) is illustrated, including the detailed view, which shows the inflation of elements’ thickness in the area of contact of solid and fluid zones. Other designs’ computational meshes are consistent with the illustrated one. Boundary conditions applied in this section are presented in
Table 1.
4.1. Gas Flow Rates Influence Study
This subsection presents a numerical analysis of gas flow rates’ influence on the performance of the initial design (design (a) in
Figure 2). As shown in Part 1 of this article, the model has proved its validity for two different temperatures (
and
) and three gas flow rates (
,
, and
;
,
, and
;
,
, and
) by comparing model results to the experimental study. As the model proved its accuracy, it is assumed to simulate the SOFC behavior for other operating parameters relatively close to the ones used in the validation. The previous study, presented in Part 1 of this paper, showed the lack of oxygen and hydrogen as the performance limiting factor. To clarify the model validation and present the performance difference for different gas flow rates,
Figure 4 is presented. In the upper left graph, a comparison of the electric characteristics of the model and experimental study is shown to present the model validation. The operational parameters are
,
, and
. The upper right graph in
Figure 4 presents the performance comparison of two modeled cases for
with different gas flow rates; solid lines represent
and
, while dashed lines represent
and
. At the bottom, a comparison of potential (power) distribution along the stack cells is shown, with an appropriate indication, also located at the electric characteristics comparison for both gas flow rates, to visualize the analyzed points. The graph marked “(a)” shows the decreasing trend in potential values along the stack, which indicates the lack of fuel and air as the performance-limiting factor. To study the performance, while the feed of gases is in surplus, the hydrogen gas flow rate was increased ten times, and the air gas flow rate was increased twenty times. As shown in the upper right graph in
Figure 4, the increase in gas flow rates caused the distinct performance change and, as shown in the graph marked “(b)”, a uniform potential distribution along the stack. The power increased from
at the current of
for the “(a)” case, up to
at
for the ”(b)” case, which gives a
% performance increase. To better visualize the influence of gases adjustment, the investigation is extended to the numerical analysis, as below. All further figure marks “(a)” and “(b)”, shown in this subsection, are consistent with “(a)” and “(b)” marked in
Figure 4. The investigation in this subsection studies the design marked as “(a)” in
Figure 2, which was also studied experimentally. The full experimental study was presented in Part 1 of this article.
The distribution of the hydrogen mass fraction is shown in
Figure 5 to confirm the influence of hydrogen on performance and provide a clear visualization of hydrogen behavior along the stack. The distribution is located 5
m above anodes’ surfaces, viewed from the side of the anode. Flow direction is from left to right. The values were fitted to the local range. Two different color sets are proposed to make the value range more visible. The distribution shows a lack of hydrogen in case “(a)” and a surplus in case “(b)”, which confirms the potential distribution difference shown in
Figure 4.
To analogously study the fraction of oxygen mass, the distribution located 5
m above the cathodes’ surfaces is shown in
Figure 6. The view is from the cathode side. Flow direction, again, is from left to right. Values are adjusted to the local range. Oxygen remains similar to hydrogen; in the “(a)” case, the oxygen mass fraction declines even earlier than hydrogen on the anode side; in the “(b)” case, the surplus of oxygen is exhibited. This behavior of oxygen utilization has a decent influence on performance and potential distribution along the stack, which is visible in
Figure 4.
To prove the preservation of the electrochemical reaction of hydrogen and oxygen and study the ventilation inside the stack, the distribution of the water vapor mass fraction on the anode side is investigated. In
Figure 7, the distribution is shown in the plane located 5
m above the anode surface. The flow direction is consistent with the
x-axis. Values are determined by the local range. The “(a)” case distribution shows large steam fraction values in most of the stack area. Despite the steam being produced in higher amounts in the “(b)” case, the mass fraction of steam is lower, which indicates a higher velocity to insufflate the channel. Both cases can suggest providing better steam drainage in future prototype designs.
To investigate the velocity dispersing from the inlet tubes to the flow channels and the resultant supply of gases to the electrodes, the distribution of both discussed cases “(a)” and “(b)”, shown in
Figure 8, is provided. Since hydrogen supply and steam ventilation are dependent on the velocity distribution on the anode side, velocity investigation is crucial. The distribution is visualized in the symmetry plane of the flow channels, and the
x-axis determines the flow direction. Maximum values are specified for each case to make the distribution more accurate. The maximum global values in much smaller inlet tubes reach much higher values than in the area of interest—air and fuel channels. To make changes of lower velocity values in the area of wider channels more visible, fixing the range is required. In this case, the maximum represents a greater or equal value of velocity. The distribution exhibits poor velocity dispersing in the electrodes’ area, and the eddy is observed in the “(b)” case. The analysis of a velocity distribution has shown low-velocity values in the electrode area compared to the values in the height of the inlet and outlet tubes. Large velocity differences within the height of the channels indicate that the flow is concentrated in the area between the inlet and outlet rather than spreading over the entire channels. These phenomena cause a limitation in the fuel and air supply as a result of insufflation mainly between the inlet and outlet tubes.
The temperature distribution inside the stack is strictly dependent on the electric load and flow rates. To inspect such an influence, the temperature distribution, located in the flow channel symmetry plane, is shown in
Figure 9. Flow direction is determined by the
x-axis (from left to right). Values are presented in the global range, so the minimum and maximum values represent the extremes for the whole domain for both cases together. Despite the lower current flux in the “(a)” case, the stack temperature is higher than in the “(b)” case, where the temperature is effectively decreased by a higher flow rate, especially at the cathode side. The temperature difference shown in “(c)” of
Figure 9 points to a major difference in the flow channel area on the cathode side and a minor difference on the anode side, but overall, the increase in flow rates caused a decrease in the stack temperature, which has a positive influence on mechanical issues by potentially decreasing thermal stress in the stack. As shown, due to cooling down by higher flow rates, stack temperature is more uniform, and hotspots in the areas of electrodes are partially eliminated. The air channel outlet area also shows the difference; due to the insufflation of hot air, the heat is collected and directed into the outlet.
4.2. Geometric Improvements Study
To make a comparison of each investigated design, the boundary conditions, electrochemical, material, and operating parameters were the same in all numerical cases (see
Table 1). To simulate the behavior of the furnace interior and substitute the ceramic covers, as discussed above, constant temperature and zero species flux boundary conditions were established at the outer surfaces of fluid domains. No current leakage was allowed on the outer walls of the solid parts, except for the current collectors’ taps. At the current collectors’ taps, on the anodic side, a zero potential is assumed, whereas on the cathodic side, a current flux is given. The constant current load simulates the series connection of the stack to multiply the electric potential value. The boundary conditions for constant velocity, temperature, and gas mixture composition were given at the inlets of the flow channels. Constant atmospheric pressure and furnace temperature were applied to the channels’ outlets. The boundary conditions applied for all cases are presented in
Table 1. Pressure–velocity coupling was performed using the SIMPLE scheme with the Rhie–Chow distance-based flux type. For spatial discretization of the convection terms, the first-order upwind scheme was used. The choice of such a scheme was dictated by numerical stability. The Green–Gauss node-based method was used to calculate the gradient, and the pressure was computed using the second-order scheme.
To make a comparison of each design shown in
Figure 2, a series of simulations was performed. Each design studied in this section consists of the same SOFC stack supported on a 105
× 30
electrolyte. To obtain a current–power characteristic, an electric load of constant current was applied to the current collectors of the cathodes with a value ranging from zero to the maximal (unknown) value, at which the power of the stack drops to zero. Four different plots are shown in
Figure 10. For reference, the marks (a), (b1), (b2), and (c) are consistent with the geometries shown in
Figure 2. The upper left graph of
Figure 10 illustrates the current–voltage and current–power characteristics of cases (a) and (b1), for the comparison of half-tubular covers and rectangular ones, with the electrolyte of 0.3 mm of thickness. The difference between the (a) and (b1) cases is almost unrecognizable. Flattening the channels by changing the shape to rectangular resulted in a power drop from
(a) to
(b1). The graph on the upper right compares the characteristics of the initial case (a) and the design with a rectangular shape of covers with a reduced electrolyte thickness (0.1 mm) (b2). As expected, reducing the thickness of the electrolyte resulted in a performance increase because of the reduced losses associated with ionic conductivity. The power of the (b2) case reaches
. The lower left graph compares the performance of the initial case (a) and the multistack design (b), which consists of 12 cells. The multistack design (c) is the most powerful arrangement option. The power reaches the value of
, but the power density is slightly lower than in the case (b2). Furthermore, in the (c) case, a concentration loss is visible in the characteristics, which could mean insufficient hydrogen supply due to the doubling of the cell number, which could be a reason for the drop in power density. Moreover, it is worth underlining that the multistack design (c) volume is still reduced compared to the initial design (a), so the volumetric power density, which considers the whole system volume, would increase. To better illustrate the performance comparison of the four cases (a), (b1), (b2), and (c), a bar plot of the maximal power density, located in the lower right part of
Figure 10, is presented.
The comparison of the temperature distribution, shown in
Figure 11, is presented to analyze the possibilities to improve heat transfer. Design marks (a), (b1), (b2), and (c) are consistent with the geometries shown in
Figure 2. The initial design (a), as well as designs (b1) and (b2), is arranged in co-flow, with a flow direction from left to right. The multistack design cross-flow is shown in
Figure 3. The distributions illustrated in
Figure 11 are located on the channels’ symmetry planes. Each distribution was presented for the maximum power point, determined during the study of the electric characteristics (see
Figure 10). The temperature distribution does not change significantly between (a) and (b1). For both cases, the maximum power retains the same current value. After implementing a thinner electrolyte (b2), the increase in current allowed the maximum power to increase. Due to the higher load, there is a significant temperature increase. The analysis of the multistack design reveals a significant temperature increase in the fuel channel between two sets of anodes, which is caused by heat generation. From the outside, the constant temperature boundary condition cools the neighboring areas. The distribution analysis suggests that a multistack prototype could remain at operating temperature with heating lower than the remaining designs.
The hydrogen distribution is presented in
Figure 12 to inspect the hydrogen consumption in the anodes. The distribution is shown in the channels’ symmetry planes. Design marks (a), (b1), (b2), and (c) are consistent with the geometries shown in
Figure 2. In the (a), (b1), and (b2) cases, fuel channels are located at the top. Case (c) has a fuel channel enclosed by two stacks and is located in the middle of the geometry. Fuel flows along the
x-axis in all cases. Each case was presented for its maximum power point in the power–voltage characteristics (
Figure 10). The operating parameters for each case were the same and are shown in
Table 1. Cases (a) and (b1) present similar hydrogen usage, because the generated power is almost the same. Case (b2), with a higher power output, due to a thinner electrolyte and a lower ionic resistance, presents a higher hydrogen usage throughout the channel, compared to (b1). In case (c), which consists of two stacks and 12 electrodes (instead of 6, as in reference design (a)), it is clearly visible that the amount of hydrogen is insufficient. This confirms the high concentration loss, visible in
Figure 10 in the high-current region.
A study of the oxygen distribution on the cathode side was conducted; this was consistent with the analysis of the hydrogen mass fraction. Results are shown in
Figure 13. The airflow in cases (a), (b1), and (b2) is along the
x-axis. The geometry of case (c) is a cross-flow configuration, so the airflow direction is determined by the opposite of the
z-axis (see
Figure 3). Due to the different direction of airflow in the air channels of case (c), the oxygen mass fraction distribution presents an entirely different behavior, as in the other cases (a), (b1), and (b2).
To study the production of water vapor due to the electrochemical reaction, a distribution similar to the distributions of the hydrogen and oxygen mass fractions is shown in
Figure 14. The consumption of oxygen and hydrogen should produce water vapor in the fuel channels, which is clearly visible in the water vapor mass fraction distribution presented and confirms the proper operation of the model. As the hydrogen and oxygen mass fractions decrease along the stacks, steam fills the fuel channel.