## 2 Models for SOFC and their Verification Degree
### Modelling of the Temperature of an SOFC Stack  
In Figure 1, a schematic overview of the SOFC test rig available at the Chair of Mechatronics at the University of Rostock is given. The key component of this test rig is an SOFC stack module with planar fuel cells in electrical series connection. The SOFC stack converts the chemical energy of the fuel gas (supplied at the anode side) into both electricity and process heat by electrochemical reactions with the oxygen contained in the cathode gas. The electrochemical energy conversion starts as soon as the required operating temperature has been reached. For the heating phase, which begins at the ambient temperature and finishes at temperatures above 800 K, two independent electric preheaters are used. During this phase, the anode gas consists of pure nitrogen (N\({}_{2}\)) while air serves as the cathode gas. The time constants of both anode and cathode gas preheaters play a major role in controlling the SOFC temperature at usual operating conditions. Additionally, time constants of the corresponding supply lines (representing parasitic thermal capacities) coupling the preheater outlets with the inlet gas manifold of the SOFC stack need to be taken into account. Variations in the electric power taken from the stack module can cause local overtemperatures in the interior of the SOFC stack. These overtemperatures might lead to an accelerated degradation or even to a destruction of the SOFC stack. Hence, they need to be prevented by manipulating the cathode gas enthalpy flow with the help of a robust control strategy. The efficiency of such controllers significantly depends on the mathematical model for the overall system. A representation of the SOFC stack by a single finite volume element with the assumption of a homogeneous temperature is sufficient for heating the system in a predefined manner. However, a spatial discretization of the temperature distribution in the interior of the stack module by a largernumber volume elements is inevitable if local overtemperatures need to be detected and prevented reliably at high-temperature operating conditions. In this case, the time constants of the preheaters and of the gas supply lines need to be considered to achieve acceptable settling times for the required control procedures. As shown experimentally, linear first-order differential equations are sufficiently accurate as a mathematical model for these components. Finite volume modelling techniques for the temperature distribution in the interior of the SOFC stack rely on an integral formulation of the energy conservation law over finitely large domains. In such a way, the partial differential equation for heat transfer can be replaced by a finite dimensional set of typically nonlinear ordinary differential equations. As pointed out above, the number of volume elements can vary. If we take one volume element for the whole stack, we obtain a single ODE for the temperature, which would be efficient wrt. the computing time, but would give us no information about temperature gradients inside the stack. Using more volume elements allows us to simulate the spatial temperature variation. If we discretise the stack by \(L\) elements in the \(x\)-axis direction, \(M\) elements in the \(y\) direction and \(N\) elements in the \(z\) direction, then the models we consider are referred to their spatial configurations \(L\times M\times N\). As typical representatives, we consider a model with one volume element (configuration \(1\times 1\times 1\)), a model with three volume elements (\(1\times 3\times 1\)), and finally nine volume elements (\(3\times 3\times 1\)). Nonlinearities in the ODEs arise because heat capacities of the anode and cathode gases depend on temperature in a nonlinear way. The same holds for the temperature dependency of the exothermic reaction enthalpy of hydrogen.