# Fuel Cell Equivalent Circuit

Polymer-electrolyte-membrane fuel cell using electrical circuit elements and dynamic membrane water content

*Since R2024b*

**Libraries:**

Simscape /
Battery /
Cells

## Description

The Fuel Cell Equivalent Circuit block models a polymer-electrolyte-membrane (PEM) fuel cell by using electrical circuit elements and a dynamic membrane water content model that determines the cell ohmic losses.

The Fuel Cell Equivalent Circuit block models these parts of a fuel cell:

Fuel Cell Potential — Model an ideal PEM fuel cell potential with Tafel equation approximation of losses.

Fuel Cell Dynamic Overpotential — Model the dynamic overpotential contributions of the fuel cell.

Fuel Cell Membrane — Model the water content dynamics.

### Fuel Cell Potential

The block models ideal PEM fuel cell potential by approximating the losses using the Tafel equation:

$$E={E}_{oc}-A\cdot \mathrm{ln}\left(\frac{I}{{I}_{0}}\right).$$

In this equation:

*E*is the nominal potential and is equal to the value of the_{OC}**Open-circuit voltage**parameter.*A*is the value of the**Tafel slope**parameter, in volts.*I*is the value of the_{0}**Nominal exchange current**parameter, in Amperes.*I*is the current drawn from the fuel cell, in Amperes.

### Fuel Cell Dynamic Overpotential

The block models the dynamic overpotential contributions of the fuel cell:

$$\frac{1}{{R}_{d}}\left(\tau \frac{d{v}_{d}}{dt}+{v}_{d}\right)=I.$$

In this equation:

*τ*is the value of the**Overpotential time constant**parameter, in seconds.*R*is the value of the_{d}**Activation and concentration equivalent resistance**parameter, in ohms.*v*is the voltage drop that accounts for the fuel cell dynamics._{d}

### Fuel Cell Membrane

The block models the ohmic resistance dynamics of a fuel cell membrane by discretizing the membrane thickness into slices, from the anode to the cathode. The net molar flow of the water through each slice determines the local water content and, consequently, the local resistance to the proton flow. The block computes the total membrane ohmic resistance by summing all slice resistances.

This figure describes a control volume analysis of a thin element of a fuel cell membrane:

The block applies the law of conservation of mass to generate the governing equation of
the water content inside the *i*th membrane slice,

$$\alpha \frac{d{\lambda}_{i}}{dt}={W}_{in}-{W}_{out,}$$

where:

*W*is the molar flow rate, in mol/s, of the water molecules that flow into the membrane slice._{in}*W*is the molar flow rate, in mol/s of the water molecules that flow out of the slice control volume._{out}*λ*is a nondimensional value that captures the local water concentration relative to the membrane material. The accumulated water is proportional to the net molar flow of the water through the membrane slice._{i}

In this equation, *α* is a constant equal to

$$\alpha =\frac{S\cdot \frac{\delta}{N}\cdot \rho}{{\rm M}},$$

where:

*S*is the value of the**Active surface area**parameter.*δ/N*is the slice thickness.*δ*is the value of the**Total membrane thickness**parameter.*N*is the value of the**Number of membrane discretizations**parameter.*ρ*is the value of the**Dry density of material**parameter.*Μ*is the value of the**Molecular mass of material**parameter.

Diffusion and electro-osmotic drag are the two key mechanisms for water transportation through a fuel cell membrane. A hydrogen fuel cell generates water molecules at the cathode and, through a diffusion mechanism, the molecules of water diffuse over towards the anode side of the membrane. The electro-osmotic drag transfers the water molecules from the anode side to the cathode side of the membrane. While the diffusion mechanisms can be bidirectional between the anode and the cathode, the electro-osmotic drag causes the molecules of water to flow only from the anode to the cathode.

These equations describe the diffusion and electro-osmotic drag mechanisms:

$$\begin{array}{l}{W}_{drag}={\kappa}_{1}GI\lambda \\ {W}_{diff}={\kappa}_{2}D\frac{d\lambda}{dz}\end{array}$$

In these equations:

*dz*refers to a length into the membrane, measured from the anode or membrane interface.*dλ*represents the difference of water content across a membrane length of*dz*.*G*is the electro-osmotic drag coefficient. To specify this value, set the**Interface for electro-osmotic drag parameters**parameter to:`Mask Parameters`

—*G*is a constant and is equal to the value of the**Electro-osmotic drag coefficient**parameter.`Lookup table`

— The block calculates the value of*G*by using the value of the**Drag table data (1-D)**and**Electro-osmotic drag coefficient breakpoints**parameters. The values of the lookup table are based on the average membrane water content.`Literature Heuristic`

— The block controls*G*by using literature heuristic and the value of the**Drag scale factor**parameter.

*D*is the diffusion coefficient. To specify this value, set the**Interface for diffusion parameters**parameter to:`Mask Parameters`

—*D*is a constant and is equal to the value of the**Diffusion coefficient**parameter.`Lookup table`

— The block calculates the value of*D*by using the value of the**Diffusion coefficient table data (1-D)**and**Diffusion coefficient breakpoints**parameters. The values of the lookup table are based on the temperature.`Literature Heuristic`

— The block controls*D*by using an empirical function derived from experiments.

$${\kappa}_{1}=\frac{1}{22F},$$ where

*F*is the Faraday constant in`S*A/mol`

.$${\kappa}_{2}=\frac{S\cdot \rho}{{\rm M}\cdot \frac{\delta}{N}}.$$

The block considers the full membrane thickness as a series of linked slices with the
appropriate and respective boundary conditions. At the *i*th slice, the
block considers the water molar inflow and outflow as contributions from the two water
transport mechanisms. The block assumes that these two mechanisms independently contribute
to the water molar mass flow:

$$\begin{array}{l}{W}_{in}={W}_{drag,in}+{W}_{diff,in}\\ {W}_{out}={W}_{drag,out}+{W}_{diff,out}\end{array}$$

For a generic membrane slice *i* of *δ/N* thickness,
the block assumes that:

The molecules of water that enter the

*i*slice due to electro-osmotic drag are equal to the molecules of water that leave the*i-1*slice due to electro-osmotic drag.The molecules of water that exit the

*i*slice due to electro-osmotic drag are equal to:$$\begin{array}{l}{W}_{drag,out,i}={\kappa}_{1}GI{\lambda}_{i}\\ {W}_{drag,in,i}={W}_{drag,out,i-1}\end{array}$$

The molecules of water that enter and exit the

*i*slice due to diffusion are equal to:$$\begin{array}{l}{W}_{diff,out,i}={\kappa}_{2}D\frac{{\lambda}_{i}-{\lambda}_{i+1}}{\frac{\delta}{N}}\\ {W}_{diff,in,i}={\kappa}_{2}D\frac{{\lambda}_{i-1}-{\lambda}_{i}}{\frac{\delta}{N}}\end{array}$$

The current

*I*is constant for all membrane slices. The proton flow is strictly in plane and is equal in each slice.

By solving the dynamic systems for
*λ _{i}*, the block determines the conductivity

*σ*, in S/m, by evaluating this equation for each slice:

$$\sigma =\left(0.005139\lambda -0.00326\right){e}^{1268\left(\frac{1}{303.15}-\frac{1}{T}\right)},$$

where *T* is the temperature, in Kelvin. You can convert
each slice conductivity into slice resistance. The total membrane resistance is equal to the
sum of the resistance of each slice.

**Boundary Conditions**

The fuel cell membrane model considers the water content dynamics for a discrete slice of the membrane. The block must also consider the environment at the outer edges of the membrane. To establish the boundary conditions of the entire membrane, the block assumes that:

At the anode side, the water molar inflow is due to diffusion only.

At the cathode side, the water molar outflow is due to both the electro-osmotic drag and diffusion.

To set the boundary conditions of the membrane, set the **Interface for lambda
boundary conditions** parameter to one of these options:

`Mask Parameters`

— Specify the values of the boundary conditions directly by using the**Anode water content**and**Cathode water content**parameters.`Physical Signal Inputs`

— Control the boundary conditions externally using the**Anode water content**and**Cathode water content**input ports.

## Examples

## Ports

### Input

### Conserving

## Parameters

## References

[1] Zhou, Daming, Fei Gao, Elena
Breaz, Alexandre Ravey, Abdellatif Miraoui, and Ke Zhang. "Dynamic Phenomena Coupling Analysis
and Modeling of Proton Exchange Membrane Fuel Cells." *IEEE Transactions on Energy
Conversion* 31, no. 4 (December 2016): 1399–1412.
https://doi.org/10.1109/TEC.2016.2587162.

[2] Wu, Hao, Peter Berg, and Xianguo
Li. "Non-Isothermal Transient Modeling of Water Transport in PEM Fuel Cells."
*Journal of Power Sources* 165, no. 1 (2007): 232–43.
https://doi.org/10.1016/j.jpowsour.2006.11.061.

## Extended Capabilities

## Version History

**Introduced in R2024b**

## See Also

Battery Equivalent Circuit | Fuel Cell (Simscape Electrical)