# Fuel Cell

Fuel cell electrical system

**Library:**Simscape / Electrical / Sources

## Description

The Fuel Cell block models a fuel cell that converts the chemical energy of hydrogen into electrical energy.

This chemical reaction defines the electrical conversion:

$${H}_{2}+\frac{1}{2}{O}_{2}\to {H}_{2}O+heat$$

with these catalytic sub-reactions:

$$\begin{array}{l}{H}_{2}\to 2{H}^{+}+2{e}^{-}\\ \frac{1}{2}{O}_{2}+2{e}^{-}\to {O}^{2-}\end{array}$$

A fuel cell stack comprises several series-connected fuel cells. This figure shows the equivalent circuit of a single fuel cell

where:

*V*is the cell voltage._{cell}*R*is the_{i}**Internal resistance**.*R*is the_{d}**Sum of activation and concentration resistances**.*C*is the parallel RC capacitance that accounts for time dynamics in the cell._{dl}

### Equations

You can use the **Model fidelity** parameter to set the
Fuel Cell block to two different levels of fidelity:

`Simplified - nominal conditions`

— The block calculates the Nernst voltage at the nominal condition of temperature and pressure.`Detailed with physical inputs`

— The block calculates the Nernst voltage by considering the pressures and flow rates of fuel and air.

**Simplified Electrical Model**

When **Model fidelity** is set to ```
Simplified - nominal
conditions
```

, the Fuel Cell block
calculates the Nernst voltage, *E*, at the nominal condition of
temperature and pressure, as defined by these equations:

$$\begin{array}{l}E={E}_{oc}-NAln\left(\frac{{i}_{FC}}{{i}_{0}}\right)\\ {v}_{FC}={N}_{unit}E-{R}_{i}{i}_{FC}-{v}_{d}\\ \frac{1}{{R}_{d}}\left(\tau \frac{d{v}_{d}}{dt}+{v}_{d}\right)={i}_{FC}\end{array}$$

where:

*E*is the_{oc}**Open-circuit voltage**.*N*is the**Number of cells per module**.*N*is the_{unit}**Module units (Series)**.*A*is the**Tafel slope**, in volts.*i*is the_{0}**Nominal exchange current**.$$\tau =\frac{{C}_{dl}}{{R}_{d}}$$.

**Detailed Electrical Model**

When **Model fidelity** is set to ```
Detailed with physical
inputs
```

, the Fuel Cell block
calculates the Nernst voltage, *E*, by considering the
pressures and flow rates of fuel and air.

These equations determine the rates of utilization of hydrogen,
*U _{H2}*,
and oxygen,

*U*

_{O2}$$\begin{array}{l}{U}_{{H}_{2}}=\frac{60000N{i}_{FC}{V}_{e}}{{p}_{fuel}{q}_{fuel}{x}_{{H}_{2}}}\\ {U}_{{O}_{2}}=\frac{60000N{i}_{FC}{V}_{e}}{2{p}_{air}{q}_{air}{x}_{{O}_{2}}}\end{array}$$

where:

*V*is the thermal voltage at room temperature._{e}*p*is the supply pressure of the fuel, in_{fuel}`bar`

.*q*is the fuel flow rate._{fuel}*x*is the concentration of hydrogen in the fuel._{H2}*p*is the supply pressure of air, in_{air}`bar`

.*q*is the air flow rate._{air}*x*is the concentration of oxygen in the air._{O2}

These equations define the partial pressures:

$$\begin{array}{l}{p}_{{H}_{2}}={p}_{fuel}{x}_{{H}_{2}}-{U}_{{H}_{2}}\\ {p}_{{O}_{2}}={p}_{air}{x}_{{O}_{2}}-{U}_{{O}_{2}}\\ {p}_{{H}_{2}O}={p}_{air}{x}_{{H}_{2}O}-2{U}_{{O}_{2}}\end{array}$$

where
*x _{H2O}* is
the concentration of vapor in air.

Then, the block computes the Nernst voltage as

where:

$${K}_{z}=\frac{\frac{{E}_{oc\_adm}}{{K}_{c}N}-1.229}{\left({T}_{nom}-298\right)\frac{-44.43}{{z}_{0}F}+\frac{R{T}_{nom}}{{z}_{0}F}\mathrm{ln}\left({p}_{n}{H}_{2}{p}_{n{O}_{2}}^{\frac{1}{2}}\right)}$$.

$${E}_{Tafel}=N{A}_{T}ln\left(\frac{{i}_{FC}}{{i}_{0}}\right)$$ is the electrokinetic term for the activation.

$${E}_{conc}={V}_{e}\frac{T}{298}\mathrm{ln}\left(\frac{{i}_{lim}}{{i}_{lim}-{i}_{FC}}\right)$$ is the electrokinetic term for the concentration.

$${E}_{oc\_adm}=max\left({E}_{oc},N\left(1.229+{\scriptscriptstyle \frac{R{T}_{nom}}{2F}}\mathrm{ln}\left({p}_{n}{H}_{2}{p}_{n{O}_{2}}^{\frac{1}{2}}\right)\right)\right)$$.

*K*is the voltage constant at nominal condition of operation._{c}*T*is the value of the_{nom}**Nominal temperature**parameter.*z*is the number of moving electrons per second.*z*is the number of moving electrons per second at the value of the_{0}**Nominal exchange current**parameter.*F*is the Faraday constant.*R*is the universal gas constant.*p*is the nominal hydrogen pressure, in_{nH2}`bar`

.*p*is the nominal oxygen pressure, in_{nO2}`bar`

.$${A}_{T}=\frac{AT}{\mathrm{ln}(10)\ast 298}$$ is the Tafel slope as a function of the temperature.

*i*is the value of the_{lim}**Collapse current**parameter.The voltage

`1.229`

represents the standard cell potential for the Nernst equation.

The block computes the power dissipated, or the heat released in the fuel cell, by using this equation

$${P}_{dissipated}={Z}_{e\_agg}\left(T\Delta S\right)+{R}_{i}{i}_{FC}+{v}_{d}^{2}\frac{1}{{R}_{d}}$$

where:

$${Z}_{e\_agg}=\frac{\left({N}_{unit}E-{R}_{i}{i}_{FC}-{v}_{d}\right){i}_{FC}}{2\Delta G}$$ is the total electron circulation rate, in

`mol/s`

.*ΔS*is the entropy fuel cell reaction, in`kJ/(mol*K)`

.*ΔG*is the free Gibbs fuel cell reaction, in`kJ/mol`

.

### Variables

Use the **Variables** section of the block
interface to set the priority and initial target values for the block
variables prior to simulation. For more information, see Set Priority and Initial Target for Block Variables.

## Limitations

The Fuel Cell block does not allow electrolysis.

## Ports

### Input

### Conserving

## Parameters

## Model Examples

## References

[1] Do, T.C., et al. “Energy
Management Strategy of a PEM Fuel Cell Excavator with a Supercapacitor/Battery Hybrid
Power Source”. *Energies* 12, no. 22, (November 2019). DOI.org
(Crossref), doi:10.3390/en13010136.

[2] Motapon, Souleman N., O. Tremblay
and L. Dessaint, “A generic fuel cell model for the simulation of fuel cell
vehicles.” *2009 IEEE Vehicle Power and Propulsion
Conference*, Dearborn, MI, 2009, pp. 1722-1729, doi:
10.1109/VPPC.2009.5289692

[3] Hirschenhofer, J. H.,, D.B.
Stauffer, R.R. Engleman, and M.G. Klett. *Fuel Cell Handbook (4th
Ed)*. U.S. Department of Energy Office of Fossil Energy, 1988.

[4] Larminie, James, and Andrew Dicks.
*Fuel Cell Systems Explained*. West Sussex, England: John Wiley
& Sons, Ltd,., 2003. https://doi.org/10.1002/9781118878330.

## Extended Capabilities

## See Also

**Introduced in R2021a**