After you understand the basic concepts around designing, building, and testing **fuel cells**, the next step is optimization. Optimization often involves extensive experimentation and testing, however, sometimes experimentation does not yield the expected results. Mathematical modeling is useful when phenomena cannot be visually examined. In fuel cells, many processes cannot be adequately monitored because they occur inside of the fuel cell.

**Essential Model Characteristics**

Fuel cell modeling is helpful because it can lead to fuel cell design improvements, as well as cheaper and more efficient fuel cells. The model must be robust and accurate to provide solutions to fuel cell problems quickly. A good model should predict fuel cell performance under a wide range of fuel cell operating conditions. A very simple fuel cell model can have considerable predictive power. A few examples of simple models are:

• Mass balances

• Energy balances

• Fick’s law of diffusion

• Heat conduction/convection equations

Relevant parameters to include in a fuel cell model are the flow rates, temperatures, pressures, cell potential and the weight fraction of the reactants:

• Flow rates should be included in a mathematical model because the flow rate determines the amount of hydrogen and oxygen that can reach the catalyst sites. This, in turn, determines how much electricity is produced.

• The temperatures and pressures of the reactants also determine the reactant amount that reaches the catalyst sites.

• The load placed on the fuel cell determines the required flow rate, weight fraction, and other operating conditions of the reactants.

An example of some of the parameters that can be included in a mathematical model is shown in Figure 1.

**Figure 1: Parameters that can be included in a mathematical model**

**Review of Fuel Cell Models**

In addition to reactant and load conditions, improvements in fuel cell performance and operation may include superior design and materials. Complex fuel cell performance issues can only be addressed if realistic mathematical process models are available. Many published models for fuel cells exist, and some of the characteristics of these models are as follows:

• **Number of Dimensions:** 1, 2, or 3

• **Dynamic or Steady-State**

• **Anode and Cathode Kinetics:** Tafel type expressions, Butler-Volmer, complex kinetics equations

• **Anode and Cathode Phase:** Gas, liquid, combination of gas and liquid

• **Mass Transport (Anode and Cathode):** Effective Fick’s diffusion, Nerst-Plank, Nerst-Plank + Schlogl, Maxwell-Stefan

• **Mass Transport (Electrolyte):** Nerst-Plank + Schlogl, Nerst-Plank + drag coefficient, Maxwell-Stefan

• **Membrane Swelling:** Empirical or thermodynamic models.

• **Energy Balance:** Isothermal or full energy balance

Fuel cell models usually have one or two dimensions, but in recent years, there have also been many three-dimensional models developed. Although 3D models seem like they would be superior to 1D or 2D models, 3D models are often created by extending the 1D model to 3D, which means that it is just a 1D model in three dimensions. Therefore, this model does not necessarily have the advantages of a 3D model.

A fuel cell model can be dynamic or steady-state. Most models have steady-state voltage characteristics and concentration profiles to keep the models simple. The **electrodes** are usually modeled using simple Tafel-type expressions, although certain models use Butler-Volmer type expressions or complex multi-step reaction kinetics for the electrochemical reactions. The reactant streams usually consist of two phases (liquid and gas) under a variety of operating conditions. On the anode side, there is the production of carbon dioxide in the catalyst layer, especially at elevated temperatures. Inside the cathode structure, water may condense, and block the way for fresh oxygen to reach the catalyst layer.

An essential element of models is the mass transport of the reactants. Simple Fick diffusion models typically use experimentally determined transport coefficients. Certain models use Nernst-Planck mass transport expressions that combine Fick’s diffusion with convective flow. The convective flow is usually calculated using Darcy’s Law using different formulations of the hydraulic permeability coefficient. The Maxwell-Stefan mass transport formulation is also used for multicomponent mixtures. Mass transport models that use effective transport coefficients and drag coefficients usually only yield good approximations to experimental data under a limited range of operating conditions.

There are several characteristics of the ionic membrane that can be included in a mathematical model. The phenomena includes ion and water transport along with the swelling of **polymer membranes**. Membrane swelling is modeled through empirical or thermodynamic models. For **PEM** and **DMFC**s, the water uptake may be described by an empirical correlation, and in other cases, a thermodynamic model based upon the change of Gibbs free energy is used.

Some fuel cell models include energy balances (conservation of energy); however, most models assume an isothermal cell operation, and therefore, no energy balances included. The inclusion of energy balance equations is useful for predicting the “drying out” of the polymer membrane or evaporation of water within the fuel cell stack.

**Model Assumptions**

A model is only as accurate as its assumptions. Each assumption needs to be considered to understand the model’s limitations and accurately interpret its results. Common assumptions used in fuel cell modeling are:

• Ideal gas properties

• Incompressible flow

• Laminar flow

• Isotropic and homogeneous electrolyte, electrode, and bipolar material structures

• A negligible ohmic potential drop in components

• Mass and energy transport modeled from volume-averaged conservation equations

Most equations used for fuel cell modeling can be applied to all fuel cell types and many fuel cell geometries. Even simple fuel cell models will provide tremendous insight into determining why a fuel cell system performs well or poorly.

**Steps for Creating Mathematical Models**

The basic steps for creating a mathematical model are:

1. Model selection

2. Model fitting

3. Model validation.

These three basic steps are used iteratively until an appropriate model has been developed. In the model selection step, plots of the data, process knowledge and assumptions about the process are used to determine the form of the model to fit the data. Then, using the selected model and information about the system, a model-fitting method can be used to estimate the unknown parameters in the model. When the parameter estimates have been made, the model is then carefully assessed to determine if the underlying assumptions of the analysis appear plausible. If the assumptions seem valid, the model can be used to answer the scientific or engineering questions that prompted the modeling effort. If the validation process identifies problems with the model, the modeling process is repeated using information about the model validation step to select and fit an improved model.

*A Variation on the Basic Steps*

The three basic steps of process modeling assume that data has already been collected and can be used to fit the models. Although this is often the case, it is not uncommon to need additional data to fit a new model. In this case, experimental design and data collection, can be added to the basic sequence between model selection and model-fitting. The flow chart below shows the basic model-fitting sequence with the integration of the data collection steps into the model-building process.

*Model Building Sequence*

*Design of Initial Experiment*

Of course, considering the model selection and fitting before collecting data is also a good idea. Without data in hand, a hypothesis about what the data will look like is needed to create an initial model. Hypothesizing the outcome of an experiment is not always possible, of course, but efforts made in early stages of a project often maximize the efficiency of the model-building process and result in the best possible models for the process.

**Conclusion**

While the fuel cell is a unique and fascinating system, accurate system selection, design and modeling for prediction of performance are needed to obtain optimal performance and design. To make strides in performance, cost, and reliability, an understanding of mathematical modeling can aid in improvements in the stack, system and operating conditions.