A 3-dimensional mathematical model to study effects of 6 geometrical parameters on performance of solid oxide fuel cell

11 Abstract 13 A 3D mathematical model is developed to study effects of various geometrical parameters 14 such as cathode to anode thickness ratio, rib width, and channel width under various flow 15 conditions, on the performance of solid oxide fuel cell (SOFC). These parameters represent 16 the cathode supported configuration of the solid oxide fuel cell. It is observed from 17 simulation results that performance of SOFC fuel cell is increased at higher cathode to anode 18 thickness. Simulation results also showed that for different volumetric flow rates, the current 19 density and fuel cell performance decrease as rib width increases, what is due to higher 20 contact resistance. It is also shown that by increasing the channel width, the fuel cell 21 performance was increased due to increase in the reaction surface area. Simulation results 22 are compared and validated with literature experimental data, showing well


Introduction 27
In past several years, solid oxide fuel cells (SOFCs) of planar configurations were established as 28 some of the most efficient energy conversion devices, convenient for various industrial applications. Shichuan et al. [4] studied a 3D numerical model of anode and cathode supported SOFC stack to 41 observe effects of cell design on the fuel cell stack performance. The authors postulated that for the 42 optimal rib width, cathode supported fuel cell is more efficient in comparison to anode supported 43 fuel cell. Zaccaria et al. [5,6] studied a 1D transient model to simulate a co-flow parallel anode 44 supported SOFC to study the effect of model characteristics on fuel cell degradation. The authors 45 postulated that the current density, fuel utilization and temperature at inlet are reduced with time, 46 whereas at cell outlet these parameters increase with time. These authors also studied the fuel cell 47 performance for ohmic, activation and diffusion losses on degradation of SOFC. Cunio et al. [ authors showed that with an increase of cell temperature, the overall performance of SOFC 72 increases due to enhanced electrochemical reaction rate and lower concentration loss. The 73 authors also stated that with reduced anode thickness and electrolyte thickness, the fuel cell 74 performance increases as result of reduced ohmic loss and concentration loss, respectively. 75 In this study, a 3-dimensional mathematical model is developed to study the effect of various 76 geometrical parameters such as cathode to anode thickness ratio, and the effect of flow 77 conditions at different rib to channel width ratios on the performance of solid oxide fuel cell 78 (SOFC). To the best of our knowledge, this has never been studied before. 79   In eqns. (3-8), ρ is the density, ua and uc is the inlet velocity vector at anode and cathode, p is 104 the pressure, τi,j is the viscous stress tensor, Ii,j is identity tensor and µ is the dynamic viscosity. 105

Secondary current distribution 106
The electrochemical reactions at the electrodes, and their kinetics responsible for activation 107 over-potential are described by Butler-Volmer equation.
[17,18] 108 where i is the current density, i0 is the exchange current density, and R and 0 are transfer 110 coefficients, T is the operating temperature, F and R are the Faraday and the gas constants, and  111 is the overpotential. 112 When the overpotential is high as for the cathode, Charge transport in the electrode and electrolyte is based on Ohm's law, described by: In eqn. (11), il is the current density, Ql is a source term, l is the conductivity and l is the 122 potential in electrolyte. In eqn, (12), is the current density Qs is a source term, s is the conductivity 123 and s is the potential in electrode.

124
The concentrations of hydrogen and oxygen at the electrode-membrane interface can be 125 determined from Henry's law equation of the forms expressed in the following two equations 126 [17,18]: 127 where, xH and xO are mass fraction of hydrogen and oxygen respectively. KH and KO are Henry's 130 constants and pH, pO pressure for hydrogen and oxygen in fuel cell channel. 131

Brinkman equations (anode) 132
In porous media of the catalyst and diffusion layers, the Navier-Stokes equation is changed into 133 the Brinkman equations and chemical species transport in ideal gas mixtures is described by the 134 Maxwell-Stefan equation [17,18]. 135 where εp is gas diffusion layer porosity, pc and pa are pressure at cathode and anode, uc and ua is 141 inlet velocity vector at cathode and anode respectively,  is permeability of porous media, Ii,j is 142 identity tensor,  is density and Qm is mass source. Transport of concentrated species for anode and cathode 152 The equation is given by 153 where ji is flux density, u is velocity vector,  is density and i is mass fraction.

Results 191
Cathode to anode thickness ratio 192 Solid oxide fuel cell performance is studied under various operating conditions and parameters. 193 Figure 4 shows the current density variation for different SOFC configurations with respect to 194 cathode to anode thickness ratio. It is observed from Figure 4 that as the cathode to anode thickness 195 ratio increased from 1 to 2, the current density increases by 8.5 %, from 4800 to 5250 A/m 2 . 196 Increasing the cathode thickness gives rise to several outcomes. The reactive active sites (RAS) for 197 the evolution of oxygen ions increases with increasing the cathode thickness. The cathode thickness 198 should be optimized in such a way that there is sufficient RAS for the cathode reaction, but the 199 reactant gas should be able to diffuse through to the reaction sites. Increasing the cathode thickness 200 also increases the ohmic resistance across the cell. Thus, optimizing of the cathode thickness is very 201 critical to SOFC performance. The generation of oxygen ions on the cathode is the driving force for 202 the anode reaction as per equation 2. Figure 4 indicates that increasing the cathode to anode 203 thickness ratio from 1 to 1.5 causes an increase in the performance of the SOFC, while increasing the 204 cathode to anode thickness ratio from 1.5 to 2 causes a very limited increase in the performance of 205 the SOFC. Figure 5 shows that as cathode to anode thickness ratio increased from 1 to 2, the average 206 cell power increases by 13.3 %, from 1280 to 1450 W/m 2 . However, it is seen that the increase in 207 power density when the thickness ratio is increased from 1.5 to 2 is lesser than when the thickness 208 ratio is increased from 1 to 1.5. This indicates that increasing the thickness ratio beyond a certain 209 value does not improve SOFC performance. It is highly possible that with increasing cathode 210 thickness, RAS increases, increasing the generation of oxygen ions which drive the power density of 211 SOFC. However, the increasing thickness of the cathode film increases the ohmic resistance of the 212 cell, causing power density to decrease. The optimum value of cathode thickness needs to be 213 identified so that the maximum power density of the SOFC can be obtained. Sun et al.
[2] postulated 214 that for the optimum performance of SOFC, the cathode to anode area should be close to 1. Our 215 simulation results indicate that the cathode/anode thickness of 1.5 would be ideal for the optimum 216 performance of SOFC. Figures 6 and 7 show the fuel utilization and hydrogen mole fraction variation 217 across SOFC channel length. It is observed from these figures that as cathode to anode thickness 218 ratio increases, reactive area increases, and more hydrogen fuel is consumed per unit length of SOFC 219 channel. Figures 6 and 7 show that increasing cathode to anode thickness ratio from 1 to 2 leads to 220 3% increment in hydrogen fuel consumption. Figure 8 shows oxygen mole fraction along the SOFC 221 channel length. It is observed that with increase in cathode to anode thickness ratio, fuel oxidation 222 increases, and more oxygen is consumed with increase in the reaction rate. Sun et al.
[2] have 223 postulated that cathode to anode thickness ratio should be close to 1 for efficient performance of 224 the fuel cell. Although in our simulation we further increased cathode to anode thickness ratio from 225 1 to 2, which increases cathode reaction area, more oxygen ion formation for driving force to anode 226 reaction as defined earlier and leads to increase in more current and power density. under wide ribs. Narrow ribs are required to facilitate more uniform distribution of the reactive 246 gases across the area of the electrode surface and promote electrochemical performance. It is 247 very important to understand this trade-off between rib dimensions and performance of the SOFC. 248 Figure 9 shows polarization curve of SOFC for various rib widths. It is observed from this figure that 249 as rib width of SOFC increases from 1 to 1.25 mm, the current density decreases by 10.11 %, from 250 4450 to 4000 A/m 2 . Further increasing of rib width to 1.5 mm leads to further reduction of current 251 density by 10%, from 4000 to 3600 A/m 2 . Increase of rib width will also lead to larger contact area 252 resistance which will lead to decrease in the fuel cell performance. When the rib width is higher, 253 due to more contact resistance, gas concentration is not uniform across fuel cell and lower 254 compared to the narrow rib width. Hence, lower rib width is better than higher rib width 255 configuration. Figure 10 shows the variation of cell power and current density for different rib 256 widths. It shows that as the rib width increases from 1 to 1.5 mm, peak cell power decreases by 257 13.27 %, from 1311 to 1137 W/m 2 . Figures 11 and 12 show hydrogen fuel consumption across 258 channel length for various rib widths. These figures show that with increase of rib width from 1 to 259 1.5 mm, hydrogen fuel consumption is decreased from 18 to 6 %. Figure 13 shows oxygen mole 260 fraction for various rib widths along flow channel length. It shows that with an increase in rib 261 width, oxygen consumption is reduced from 23.80 to 9.5 %. 262 263 Flow rate to channel width ratio 292 SOFC fuel cell performance is studied for various flow rate to channel width ratios, and 293 polarization curves are presented in Figure 16. From Figure 16 it is observed that as the channel 294 width was increased from 0.5 to 1 mm, the current density increased. Increasing the volumetric 295 flow rate is also seen to increase the current density. Further increase in channel width from 1 to 1.5 mm also increased the fuel cell performance 300 but with less increment in current density values. Similarly, Figure 17 shows average cell power 301 along with current density variation for various flow rate to channel width ratios. For higher flow 302 rate to channel width ratio, current density, as well as average cell power increased. With an 303 increase in channel width, the reaction area increases what leads to transport of more fuel for 304 reaction driving the reaction rate. 305 306 307 Figure 17. Cell power and current density variation of SOFC for various flow rate to channel 308 width ratios 309 Figure 18 shows the comparison between simulation results and experimental data. Sun et al. 310 [2] studied experimental modelling of single chamber solid oxide fuel cell at various temperatures 311 and electrode particles. In Figure 18,  increase in channel width, the current density is increased due to increase of the reaction area and 324 consequently, the average cell power is also increased. Simulation results indicate that the 325 cathode supported SOFC show better performance as the cathode to anode thickness ratio was 326 increased. It is also observed that flow rate plays major role in the fuel cell performance. It is seen 327 that with increase in the volumetric flow rate at various rib and channel widths, the performance 328 of SOFC fuel cell increases. Higher cathode to anode thickness ratio, smaller rib widths, larger 329 channel widths and increasing volumetric flow rate are found to increase the performance of the 330 fuel cell. Model results are compared with experimental data taken from the literature and found 331 to compare well. Some or all data, models, or code that support the findings of this study are available from the 356 corresponding author upon reasonable request. 357