Fuel Cell Modeling

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 DMFCs, 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.


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.

Dr. Colleen Spiegel Posted by Dr. Colleen Spiegel

Dr. Colleen Spiegel is a mathematical modeling and technical writing consultant (President of SEMSCIO) and Professor holding a Ph.D. and an MSc degree in Engineering. She has seventeen years of experience in engineering, statistics, data science, research & technical writing work for many companies as a consultant, employee, and independent business owner. She is the author of ‘Designing and Building Fuel Cells’ (McGraw-Hill, 2007) and ‘PEM Fuel Cell Modeling and Simulation Using MATLAB’ (Elsevier Science, 2008). She previously owned Clean Fuel Cell Energy, LLC, which was a fuel cell organization that served scientists, engineers, and professors world-wide.

Products related to this article

Related Articles

Fuel Cell Operating Conditions

Fuel cell operating conditions depend upon the cell and stack design. The operating parameters that affect fuel cell performance are: Operating Pressure, Operating Temperature, Flow Rates of Reactants, and Humidity of Reactants. Using the correct operating condition for each parameter is...

Fuel Cell Characterization

Different characterization techniques enable the quantitative comparison of every property and part of the fuel cell stack. By characterizing the fuel cell properly, you can understand why the fuel cell is performing well or poorly. These techniques help discriminate between activation, ohmic and concentration losses, fuel crossover, and...

Fuel Cell System Design

Fuel cell system designs range from very simple to very complex depending upon the fuel cell application and the system efficiency desired. A fuel cell system can be very efficient with just the fuel cell stack and a few other balance-of-plant components or may require many outside components to optimize...

Fuel Cell Heat Management

Creating high-efficiency fuel cells requires proper temperature control, and heat management to ensure that the fuel cell system runs consistently. Depending upon the fuel cell type, the optimal temperature can range from room temperature to 1000 ºC, and any deviation from the designed temperature range can result in...

Water Management For PEM Fuel Cells

One of the greatest challenges associated with PEMFCs is the water balance in the fuel cell stack. As the chemical reaction occurs in each cell, water is generated. Depending upon the load and the operating conditions, there is a tendency for the fuel cells to both flood and dry-out. The water content in the...

Techniques for Measuring Fuel Cell Resistance

The fuel cell polarization curve provides useful information on fuel cell performance, however; additional information is needed to study its performance characteristics accurately. Cell resistance provides insightful information about a fuel cell that is not completely captured by polarization curves. Since fuel cell current densities are high in comparison with...

Flow-Field Design

In fuel cells, the flow field plates are designed to provide an adequate amount of the reactants (hydrogen and oxygen) to the gas diffusion layer (GDL) and catalyst surface while minimizing pressure drop. The most popular channel configurations for PEM fuel cells are serpentine, parallel, and...

Biological Fuel Cells (BFCs) and the Bio-production of Hydrogen

A biological fuel cell (BFC) or microbial fuel cell (MFC) is a type of fuel cell that converts biochemical energy into electrical energy. Like other types of fuel cells, a biological fuel cell consists of an anode, a cathode, and a membrane that conducts ions. In the anode compartment, fuel is oxidized by microorganisms, and the result is...

Introduction to Fuel Cell Testing

Those who wish to learn more about fuel cells, and even to build their own, may also want to learn how to test those fuel cells. In this post, we will review some basic terms, and introduce low-cost testing equipment and more sophisticated testing setups. First, however, an understanding of the fuel cell and electrical basics will...

Mathematical Models

Mathematical models are a precise description of a problem, process, or technology in the form of mathematics. These models are built to learn more about a technology, system or method. The models explain why the system or process works the way it does and helps to study the effects and...

How to Estimate Electricity Requirements

When you receive your energy bill each month, you may not understand exactly how the total amount has been calculated. Every device or appliance in your household contributes to the total sum of the bill. To figure out which appliances and devices are using the most energy, you can estimate the...

Metal Hydrides

Fuel cells usually use compressed hydrogen as the fuel, but there are many other types of fuels that can be used. The type of fuel used depends upon the fuel cell application. Fuels are often in their final form before entering the fuel cell; however, certain fuel cell types can be processed on the inside of the fuel cell. Alternative fuel types are...

Fuel Cell Electrolyte Layer Modeling

The electrolyte layer is essential for a fuel cell to work properly. In PEM fuel cells (PEMFCs), the fuel travels to the catalyst layer and gets broken into protons (H+) and electrons. The electrons travel to the external circuit to power the load, and the hydrogen protons travel through the electrolyte until it reaches the cathode to combine with oxygen to form...

Modeling the Catalyst Layers

The fuel cell electrode layer is made up of the catalyst and porous gas diffusion layer. When the fuel in the flow channels meets the electrode layer, it diffuses into the porous electrode. The reactant travels to the catalyst layer where it is broken into protons and electrons. The electrons move to the...

Model Validation Using Residuals

Model validation is the most important step in the model building process; however, it is often neglected. Even when the model is validated, it is often not done adequately. It often consists of taking a few experimental data points and plotting these points on the same graph as the model. There are two different types of models: engineering or...

Fuel Cell Modeling Basics

Fuel cell modeling is helpful for fuel cell developers because it can lead to fuel cell design improvements, as well as cheaper, better, and more efficient fuel cells. The model must be robust and accurate and be able to provide solutions to fuel cell problems quickly. A good model should predict fuel cell performance under a wide range of...

A Review of Mathematical Modeling of Proton Exchange Membrane and Direct Methanol Fuel Cells

There has been a lot of emphasis on the development of long-lasting, efficient and portable, power sources for further technology improvement in commercial electronics devices, medical diagnostic equipment, mobile communication and military applications. These systems all require...

Compact Transient Model for Nafion Membranes

A numerical model was developed to predict the water concentration, temperature, potential and pressure across a Nafion membrane used in proton exchange membrane (PEM) based fuel cells. The numerical model consists of simultaneously calculating the diffusive flux for water and hydrogen, the proton potential and the pressure and temperature at each node...

0 Comments To "Fuel Cell Modeling"

Write a comment

Your Name:

Enter the code in the box below:

Your Comment:
Note: HTML is not translated!