# A Multiscale Approach to the Numerical Simulation of the Solid Oxide Fuel Cell
## 1 Introduction  
Fuel cells are electrochemical devices used to convert the chemical energy of a fuel, usually hydrogen, to the electricity directly. Among the fuel cells, the solid oxide fuel cells (SOFCs) are an attractive solution for the future of electric power generation systems in almost all system scales, from small domestic to large industrial generators. Therefore, SOFCs gain more and more attraction in the scientific community. Solid oxide fuel cells utilize good oxygen anions mobility in the elevated temperature (up to 1000 \({}^{\circ}\)C) [1]. The high temperature enables fast electrochemical kinetics without a need to use expensive precious metal catalysts, which is the case of low-temperature fuel cells [2, 3]. The high quality of the waste heat makes it possible to install SOFCs in combined heat and power systems. The high operating temperature also allows for converting electrochemically the carbon monoxide or even to conduct the in-stack internal reforming process, which leads to wide fuelflexibility--the variety of reformed hydrocarbons may be used as a fuel for the SOFC [4; 5; 6]. Among other advantages there are: low emissions of the greenhouse gases (for the hydrogen-fueled fuel cells virtually there is no emissions) and no emissions of the toxic compounds, relatively high tolerance to fuel impurities [7] and outstanding energy conversion efficiency [2]. However, the high operating temperature creates also several problems, especially with the degradation of the materials, corrosion and sealing [1]. This leads to the concept of intermediate temperature solid oxide fuel cells (IT-SOFCs), which work in the temperature range \(550\,\mathrm{\SIUnitSymbolCelsius}\) to \(800\,\mathrm{\SIUnitSymbolCelsius}\)[8]. A typical solid oxide fuel cell contains two porous electrodes (an anode and a cathode) and a ceramic solid electrolyte, which separates the electrodes. The schematic diagram of a SOFC is shown in Figure 1. On the anode, the fuel is provided and oxidized according to the following reactions:  
* hydrogen oxidation: \(\mathrm{H_{2}}+\mathrm{O^{2-}}\)\(\longrightarrow\)\(\mathrm{H_{2}O}+2\,\mathrm{e^{-}}\);
* carbon monoxide oxidation: \(\mathrm{CO}+\mathrm{O^{2-}}\)\(\longrightarrow\)\(\mathrm{CO_{2}}+2\,\mathrm{e^{-}}\). The produced electrons are transported through the external circuit to the cathode. On the cathode, the electrons take part in the oxygen reduction reaction: \(\frac{1}{2}\,\mathrm{O_{2}}+2\,\mathrm{e^{-}}\)\(\longrightarrow\)\(\mathrm{O^{2-}}\). The oxygen ions are then conducted to the anode through the electrolyte to oxidize the fuel, which closes the process. The electrodes are characterized by the complex microstructure. For example, the electrochemical reactions can occur only on the sites, where the electron conductor, ion conductor and pore (i.e., \"gas conductor\") are in contact. These places are called triple-phase boundaries (TPBs) or--in the case of an electrode material with mixed conductivity (e.g., lanthanum strontium cobalt ferrite (LSCF))--double-phase boundaries (DPBs). The microstructure is a key factor which allows the control of the SOFC performance [9]. Since the construction and testing of novel inventions in the IT-SOFC technology are time-consuming and rather expensive, progress in computational technology and in the methods of mathematical and numerical modeling of solid oxide fuel cells are more and more important. Moreover, the results of numerical models lead to deeper insight into the processes occurring within the SOFC. Figure 1: Principle of the solid oxide fuel cell operation. To analyze the behavior of the SOFCs, usually two classes of the mathematical models are in use. The macroscale models primarily utilize the mass and energy balances or the computational fluid dynamics (CFD) methods to analyze the behavior of the SOFC-based systems [10; 11; 12], stacks [13; 14; 15; 16] or single cells [17; 18; 19; 20; 21].