Cell-and Stack-Level Modelling

The coupled continuum-level electrochemical, flow, and thermal models are usually discretised in a finite element mesh [23,24]. When the necessary material properties, geometrical parameters, operating parameters, and boundary conditions are supplied, cell- and stack-level performance can be analysed. The combined models can determine the cell/stack voltage, the total current output, temperature distribution, species concentration, etc. 

An important outcome of the combined models at the cell level is the cell efficiency, and at the stack level, the stack efficiency as well. The electrical efficiency of a cell and of the stack is defined as 

The electrical efficiency therefore depends to some extent on the definition of the fuel energy input and on whether power for gas pumping and the like is subtracted from the generated power, egi may be expressed in terms of current, voltage, etc., as follows  

This can be considered the product of three additional fundamental efficiencies, namely the ideal or thermodynamic efficiency AG/AH, the voltage efficiency (V/Eo), and the fuel utilisation (Uf)  

The fuel utilisation, Uf, is the ratio of the delivered current to the stoichiometric current equivalent to the fuel flow rate  

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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. 

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. 

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. 

Traditionally, the majority of SOFC modeling efforts have taken place at the macroscale and have provided valuable insight into the operation of SOFC cells and stacks [15, 36-38], In macroscale modeling, simplified models of the SOFC multi-physics are used to simulate the operating conditions and performance of the SOFC. Macroscale models are used to investigate the performance of experimental cells and stacks [35, 39], SOFC system-level operation and controls [40, 41], and thermal stresses and strains in the cells and stacks [42-44]. 

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. 

Despite the availability of quite sophisticated fuel cell models with well-written code and convenient user interfaces, the fuel cell developer or engineer must be a critical user. As mentioned above, obtaining experimental data on the behavior of fuel cells (especially internally and at the micro-level) can be difficult, time-consuming, and expensive. Unfortunately this has lead to a dearth of accurate and detailed data of sufficient quality and quantity to allow thorough validation of the mathematical models. Much of the data on fuel cell performance reported in the literature is, while phenomenologically often interesting, insufficiently accurate and accompanied by far too little detail on the test conditions to be usable for model validation. In particular, with much of the cell and stack taken at modest utilization, it is almost impossible to infer kinetic data without spatially resolved data on current density, temperature and species concentrations. As a consequence, the validity of fuel cell models must be critically considered for each use. The user of the model must be thoroughly familiar with the assumptions and limitations embedded in the models. 

Electrode-level models describe the performance of SOFC electrodes in detail. They take into account the distribution of species concentrations, electric potential, current, and even temperature in the electrode. Their purpose is to (i) interpret the performance (polarisation curve) of electrodes in terms of rate-limiting resistances such as kinetic (activation), mass transfer, and ohmic resistance and (ii) predict the local polarisation in full-scale cell and stack models. 

Modelling of the IP-SOFC tube involves different levels of detail, such as the electrode, the single cell, the stack, and the system level. At the electrode level, the main... 

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