Stack-level modeling

One of the first models to examine transients in polymer-electrolyte fuel cells was a stack-level model by Amphlett et al. Their model is mainly empirical and examines temperature and gas flow rates. They showed that transient behavior lasts for a few minutes in a stack before a new steady state is reached. In a similar stack-level analysis, Yerramalla et al.2 9 used a slightly more complicated single-cell model and examined the shape of the transients. They noticed voltage behavior that had oscillations in it and some leakage current. Their overall analysis was geared to the development of a controller for the stack. 

Detailed CFD models of fuel cells (see Chapters 3 and 4), on the other hand, use continuum assumption to predict the 3-D distributions of the physical quantities inside the fuel cells. These models are more complex and computationally expensive compared to reduced order models especially due to the disparity between the smallest and largest length scales in a fuel cell. The thickness of the electrodes and electrolyte is usually tens of microns whereas the overall dimensions of a fuel cell or stack could be tens of centimeters. Though some authors used detailed 3-D models for cell or stack level modeling, they are mostly confined to component level modeling. In what follows, we present the governing equations for some of these models. 

It should be noted that the analytic solutions to the membrane transport problems above and the scalar formulas for the oxygen transport and electrochemistry below are not necessary for the stack level modeling framework described in this chapter. They do allow a concise description in this expository setting and lead to fast computational methods. However, it is possible to introduce a grid in the through-MEA direction y and compute numerical approximations to more complicated models at each channel point. These more complicated relationships can be combined at the unit cell and stack levels as discussed below just as easily as the simple models presented here. 

To study the stresses in the system, it is first necessary to calculate the temperature distributions of the SOFC stack. Owing to the coupled nature of the SOFC multi-physics, the temperatures in the stack wiU affect both the electrochemical performance and the mechanical stresses of the stack [49]. The electrochemical performance of the SOFC is coupled to the temperature through the Nernst equation [Eq. (26.11)]. Stack-level models are often used to consider the temperature distributions and how the operating conditions and design of the stack affect the temperatures [1, 48, 49]. In these models, the energy conservation equation [Eq. (26.7)] is solved in the gas and sohd phases, and includes the effects of convection in the fuel and air charmels, radiation between the soHd tri-layer and the gas, radiation between the stack and its surroundings, conduction through the tri-layer, and heat sources due to chemical and electrochemical reactions [1, 50]. The balance... 

The stack-level models are not used as frequently as the other two levels (component and cell). However, they are important when the stack design and control are the focus. Validation has been conducted to some extent. In Figures 31.22 and 31.24, experimental data are compared with model predictions, showing good agreement. 

The cell- and stack-level models can improve understanding of the complex interactions between fluid dynamic, thermal, chemical, and electrochemical phenomena. The combined models can therefore help maximise efflciency or power density by optimising PEN element design, cell configuration, and stack architecture for a given set of operating conditions. Most SOFC modelling focuses on cell- and stack-level performance for exactly this purpose. 

For cell- and stack-level modelling it is necessary to have reliable values of the total polarisation of cathode, r)c, and anode, t a, as a function of local bulk gas composition, pressure, and temperature, as well as the local current density. 

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