Editing Solid oxide fuel cell (section)

==Polarizations==

Polarizations, or overpotentials, are losses in voltage due to imperfections in materials, microstructure, and design of the fuel cell. Polarizations result from ohmic resistance of oxygen ions conducting through the electrolyte (iRΩ), electrochemical activation barriers at the anode and cathode, and finally concentration polarizations due to inability of gases to diffuse at high rates through the porous anode and cathode (shown as ηA for the anode and ηC for cathode).<ref>{{cite journal |last1=Shimada |first1=Hiroyuki |last2=Suzuki |first2=Toshio |last3=Yamaguchi |first3=Toshiaki |last4=Sumi |first4=Hirofumi |last5=Hamamoto |first5=Koichi |last6=Fujishiro |first6=Yoshinobu |title=Challenge for lowering concentration polarization in solid oxide fuel cells |journal=Journal of Power Sources |date=January 2016 |volume=302 |pages=53–60 |doi=10.1016/j.jpowsour.2015.10.024|bibcode=2016JPS...302...53S }}</ref> The cell voltage can be calculated using the following equation:

:<math> {V} = {E}_0 - {iR}_\omega - {\eta}_{cathode} - {\eta}_{anode} </math>

where:
*<math>{E}_0</math> = [[Nernst potential]] of the reactants 
*<math>R</math> = [[Thévenin's theorem|Thévenin equivalent]] resistance value of the electrically conducting portions of the cell
*<math>{\eta}_{cathode}</math> = polarization losses in the cathode
*<math>{\eta}_{anode}</math> = polarization losses in the anode

In SOFCs, it is often important to focus on the ohmic and concentration polarizations since high operating temperatures experience little activation polarization. However, as the lower limit of SOFC operating temperature is approached (~600&nbsp;°C), these polarizations do become important.<ref>{{cite journal |author1=Hai-Bo Huo |author2=Xin-Jian Zhu |author3=Guang-Yi Cao | title=Nonlinear modeling of a SOFC stack based on a least squares support vector machine | journal=Journal of Power Sources | year=2006 | volume=162 | issue=2 | pages=1220–1225 | doi=10.1016/j.jpowsour.2006.07.031|bibcode=2006JPS...162.1220H }}</ref>

Above mentioned equation is used for determining the SOFC voltage (in fact for fuel cell voltage in general). This approach results in good agreement with particular experimental data (for which
adequate factors were obtained) and poor agreement for other than original experimental working parameters. Moreover, most of the equations used require the addition of numerous factors which are difficult or impossible to determine. It makes very difficult any optimizing process of the SOFC working parameters as well as design architecture configuration selection. Because of those circumstances a few other equations were proposed:<ref name="Milewski J, Miller A. 2006 396-402">{{cite journal | doi=10.1115/1.2349519 |vauthors=Milewski J, Miller A | title=Influences of the Type and Thickness of Electrolyte on Solid Oxide Fuel Cell Hybrid System Performance | journal=Journal of Fuel Cell Science and Technology | year=2006 | volume=3 | issue=4 | pages=396–402}}</ref>

:<math>E_{SOFC} = \frac{E_{max}-i_{max}\cdot\eta_f\cdot r_1}{\frac{r_1}{r_2}\cdot\left( 1-\eta_f \right) + 1}
</math>

where: 
*<math>E_{SOFC}</math> = cell voltage
*<math>E_{max}</math> = maximum voltage given by the Nernst equation
*<math>i_{max}</math> = maximum current density (for given fuel flow)
*<math>\eta_f</math> = fuel utilization factor<ref name="Milewski J, Miller A. 2006 396-402"/><ref>{{cite journal |author1=M. Santarelli |author2=P. Leone |author3=M. Calì |author4=G. Orsello | title=Experimental evaluation of the sensitivity to fuel utilization and air management on a 100 kW SOFC system | journal=Journal of Power Sources | year=2007 | volume=171 | issue=2 | pages=155–168 | doi=10.1016/j.jpowsour.2006.12.032|bibcode=2007JPS...171..155S }}</ref>
*<math>r_1</math> = ionic specific resistance of the electrolyte
*<math>r_2</math> = electric specific resistance of the electrolyte.

This method was validated and found to be suitable for optimization and sensitivity studies in plant-level modelling of various systems with solid oxide fuel cells.<ref name="Kupecki J, Milewski J, Jewulski J">{{cite journal | doi=10.2478/s11532-013-0211-x |author1=Kupecki J. |author2=Milewski J. |author3=Jewulski J. | title=Investigation of SOFC material properties for plant-level modeling | journal=Central European Journal of Chemistry | year=2013 | volume=11 | issue=5 | pages=664–671| doi-access=free }}</ref> With this mathematical description it is possible to account for different properties of the SOFC. There are many parameters which impact cell working conditions, e.g. electrolyte material, electrolyte thickness, cell temperature, inlet and outlet gas compositions at anode and cathode, and electrode porosity, just to name some. The flow in these systems is often calculated using the [[Navier–Stokes equations]].

===Ohmic polarization===

Ohmic losses in an SOFC result from ionic conductivity through the electrolyte and electrical resistance offered to the flow of electrons in the external electrical circuit. This is inherently a materials property of the crystal structure and atoms involved. However, to maximize the ionic conductivity, several methods can be done. Firstly, operating at higher temperatures can significantly decrease these ohmic losses. Substitutional doping methods to further refine the crystal structure and control defect concentrations can also play a significant role in increasing the conductivity. Another way to decrease ohmic resistance is to decrease the thickness of the electrolyte layer.

====Ionic conductivity====
An ionic specific resistance of the electrolyte as a function of temperature can be described by the following relationship:<ref name="Milewski J, Miller A. 2006 396-402"/>

:<math>r_1 = \frac{\delta}{\sigma}</math>

where: <math>\delta</math> – electrolyte thickness, and <math>\sigma</math> – ionic conductivity.

The ionic conductivity of the solid oxide is defined as follows:<ref name="Milewski J, Miller A. 2006 396-402"/>

:<math>\sigma = \sigma_0\cdot e^\frac{-E}{R\cdot T}</math>

where: <math>\sigma_0</math> and <math>E</math> – factors depended on electrolyte materials, <math>T</math> – electrolyte temperature, and <math>R</math> – ideal gas constant.

===Concentration polarization===

The concentration polarization is the result of practical limitations on mass transport within the cell and represents the voltage loss due to spatial variations in reactant concentration at the chemically active sites. This situation can be caused when the reactants are consumed by the electrochemical reaction faster than they can diffuse into the porous electrode, and can also be caused by variation in bulk flow composition. The latter is due to the fact that the consumption of reacting species in the reactant flows causes a drop in reactant concentration as it travels along the cell, which causes a drop in the local potential near the tail end of the cell.

The concentration polarization occurs in both the anode and cathode. The anode can be particularly problematic, as the oxidation of the hydrogen produces steam, which further dilutes the fuel stream as it travels along the length of the cell. This polarization can be mitigated by reducing the reactant utilization fraction or increasing the electrode porosity, but these approaches each have significant design trade-offs.

===Activation polarization===
The activation polarization is the result of the kinetics involved with the electrochemical reactions. Each reaction has a certain activation barrier that must be overcome in order to proceed and this barrier leads to the polarization. The activation barrier is the result of many complex electrochemical reaction steps where typically the rate limiting step is responsible for the polarization. The polarization equation shown below is found by solving the [[Butler–Volmer equation]] in the high current density regime (where the cell typically operates), and can be used to estimate the activation polarization:

:<math> {\eta}_{act} = \frac {RT} {{\beta}zF} \times ln \left(\frac {i} {{i}_0} \right) </math>

where:
*<math>R</math> = gas constant
*<math>{T}_0</math> = operating temperature
*<math>{\beta}</math> = electron transfer coefficient
*<math>z</math> = electrons associated with the electrochemical reaction
*<math>F</math> = Faraday's constant
*<math>i</math> = operating current
*<math>i_0</math> = exchange current density

The polarization can be modified by microstructural optimization. The Triple Phase Boundary (TPB) length, which is the length where porous, ionic and electronically conducting pathways all meet, directly relates to the electrochemically active length in the cell. The larger the length, the more reactions can occur and thus the less the activation polarization. Optimization of TPB length can be done by processing conditions to affect microstructure or by materials selection to use a mixed ionic/electronic conductor to further increase TPB length.

==Mechanical Properties==
Current SOFC research focuses heavily on optimizing cell performance while maintaining acceptable mechanical properties because optimized performance often compromises mechanical properties. Nevertheless, mechanical failure represents a significant problem to SOFC operation. The presence of various kinds of load and [[Thermal stress]] during operation requires high mechanical strength. Additional stresses associated with changes in gas atmosphere, leading to reduction or oxidation also cannot be avoided in prolonged operation.<ref name="Progress in Material Selection">{{cite journal |last1=Mahato |first1=N |last2=Banerjee |first2=A |last3=Gupta |first3=A |last4=Omar |first4=S |last5=Balani |first5=K |title=Progress in material selection for solid oxide fuel cell technology: A review |journal=Progress in Materials Science |date=2015-07-01 |volume=72 |pages=141–337 |doi=10.1016/j.pmatsci.2015.01.001}}</ref> When electrode layers delaminate or crack, conduction pathways are lost, leading to a redistribution of current density and local changes in temperature. These local temperature deviations, in turn, lead to increased thermal strains, which propagate cracks and [[Delamination]]. Additionally, when electrolytes crack, separation of fuel and air is no longer guaranteed, which further endangers the continuous operation of the cell.<ref>{{cite journal |last1=Nakajo |first1=Arata |last2=Kuebler |first2=Jakob |last3=Faes |first3=Antonin |last4=Vogt |first4=Ulrich |last5=Schindler |first5=Hansjürgen |last6=Chiang |first6=Lieh-Kwang |last7=Modena |first7=Stefano |last8=Van Herle |first8=Jan |title=Compilation of mechanical properties for the structural analysis of solid oxide fuel cell stacks. Part I. Constitutive materials of anode-supported cells. |journal=Ceramics International |date=2012-01-25 |volume=38 |pages=3907–3927 |doi=10.1016/j.ceramint.2012.01.043}}</ref>

Since SOFCs require materials with high oxygen conductivity, thermal stresses provide a significant problem. The [[Coefficient of thermal expansion]] in mixed ionic-electronic perovskites can be directly related to oxygen vacancy concentration, which is also related to ionic conductivity.<ref>{{cite journal |last1=Ullmann |first1=H. |last2=Trofimenko |first2=N. |last3=Tietz |first3=F. |last4=Stöver |first4=D. |last5=Ahmad-Khanlou |first5=A. |title=Correlation between thermal expansion and oxide ion transport in mixed conducting perovskite-type oxides for SOFC cathodes |journal=Solid State Ionics |date=1 December 2000 |volume=138 |issue=1–2 |pages=79–90 |doi=10.1016/S0167-2738(00)00770-0}}</ref> Thus, thermal stresses increase in direct correlation with improved cell performance. Additionally, however, the temperature dependence of oxygen vacancy concentration means that the CTE is not a linear property, which further complicates measurements and predictions.

Just as thermal stresses increase as cell performance improves through improved ionic conductivity, the fracture toughness of the material also decreases as cell performance increases. This is because, to increase reaction sites, porous ceramics are preferable. However, as shown in the equation below, fracture toughness decreases as porosity increases.<ref>{{cite journal |last1=Radovic |first1=M. |last2=Lara-Curzio |first2=E. |title=Mechanical properties of tape cast nickel-based anode materials for solid oxide fuel cells before and after reduction in hydrogen |journal=Acta Materialia |date=December 2004 |volume=52 |issue=20 |pages=5747–5756 |doi=10.1016/j.actamat.2004.08.023|bibcode=2004AcMat..52.5747R }}</ref>

<math> K_{IC} = K_{IC,0}\exp{(-b_{k}p')} </math>

Where:

<math>K_{IC}</math> = fracture toughness

<math>K_{IC,0}</math> = fracture toughness of the non-porous structure

<math>b_k</math> = experimentally determined constant

<math>p' </math> = porosity

Thus, porosity must be carefully engineered to maximize reaction kinetics while maintaining an acceptable fracture toughness. Since fracture toughness represents the ability of pre-existing cracks or pores to propagate, a potentially more useful metric is the failure stress of a material, as this depends on sample dimensions instead of crack diameter. Failure stresses in SOFCs can also be evaluated using a ring-on ring biaxial stress test. This type of test is generally preferred, as sample edge quality does not significantly impact measurements. The determination of the sample's failure stress is shown in the equation below.<ref>{{cite web |last1=ASTM |title=Standard Test Method for Monotonic Equibiaxial Flexural Strength of Advanced Ceramics at Ambient Temperature, ASTM Standard C1499-04 |url=https://www.astm.org/c1499-19.html}}</ref>

<math> \sigma_{cr}= \frac{3F_{cr}}{2\pi h_{s}^{2}}+
\Biggl((1-\nu)\frac{D_{sup}^{2}-D_{load}^{2}}{2D_{s}^{2}}+(1+\nu)\ln\left ( \frac{D_{sup}}{D_{load}} \right )\Biggr) </math>

Where:

<math> \sigma_{cr} </math> = failure stress of the small deformation

<math> F_{cr} </math> = critical applied force

<math> h_s </math> = height of the sample

<math> \nu </math> = Poisson's ratio

<math> D </math> = diameter (sup = support ring, load = loading ring, s = sample)

However, this equation is not valid for deflections exceeding 1/2h,<ref>{{Cite journal |last1=Kao |first1=Robert |last2=Perrone |first2=Nicholas |last3=Capps |first3=Webster |date=1971 |title=Large-Deflection Solution of the Coaxial-Ring-Circular-Glass-Plate Flexure Problem |url=https://onlinelibrary.wiley.com/doi/10.1111/j.1151-2916.1971.tb12209.x |journal=Journal of the American Ceramic Society |language=en |volume=54 |issue=11 |pages=566–571 |doi=10.1111/j.1151-2916.1971.tb12209.x |issn=0002-7820|url-access=subscription }}</ref> making it less applicable for thin samples, which are of great interest in SOFCs. Therefore, while this method does not require knowledge of crack or pore size, it must be used with great caution and is more applicable to support layers in SOFCs than active layers. In addition to failure stresses and fracture toughness, modern fuel cell designs that favor mixed ionic electronic conductors (MIECs), [[Creep (deformation)]] pose another great problem, as MIEC electrodes often operate at temperatures exceeding half of the melting temperature. As a result, diffusion creep must also be considered.<ref>{{Cite journal |last1=Nakajo |first1=Arata |last2=Kuebler |first2=Jakob |last3=Faes |first3=Antonin |last4=Vogt |first4=Ulrich F. |last5=Schindler |first5=Hans Jürgen |last6=Chiang |first6=Lieh-Kwang |last7=Modena |first7=Stefano |last8=Van herle |first8=Jan |last9=Hocker |first9=Thomas |date=2012-01-25 |title=Compilation of mechanical properties for the structural analysis of solid oxide fuel cell stacks. Constitutive materials of anode-supported cells |url=https://linkinghub.elsevier.com/retrieve/pii/S0272884212000466 |journal=Ceramics International |language=en |volume=38 |issue=5 |pages=3907–3927 |doi=10.1016/j.ceramint.2012.01.043|url-access=subscription }}</ref>

<math> \dot{\epsilon}_{eq}^{creep}=\frac{\tilde{k}_0D}{T}\frac{\sigma_{eq}^{m}}{d_{grain}^{n}} </math>

Where:

<math> \dot{\epsilon}_{eq}^{creep} </math> = equivalent creep strain

<math> D </math> = [[Diffusion]] coefficient

<math> T </math> = temperature

<math> \tilde{k}_0 </math> = kinetic constant

<math> \sigma_{eq} </math> = equivalent stress (e.g. von Mises)

<math> m </math> = creep stress exponential factor

<math> n </math> = particle size exponent (2 for [[Nabarro–Herring creep]], 3 for [[Coble creep]])

To properly model creep strain rates, knowledge of [[Microstructure]] is therefore of significant importance. Due to the difficulty in mechanically testing SOFCs at high temperatures, and due to the microstructural evolution of SOFCs over the lifetime of operation resulting from [[Grain growth]] and coarsening, the actual creep behavior of SOFCs is currently not completely understood