# Fuel Cell

Proton Exchange Membrane (PEM) Fuel Cell component in Schematic Editor

component | component dialog window | component parameters |
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## PEM Fuel Cell modeling

The hydrogen fuel cell component is implemented using a signal controlled voltage source, a current measurement, and a fuel cell signal processing model as seen in Figure 1. The proton exchange membrane fuel cell in particular is known to have slower reaction kinetics at the cathode side (air) than at the anode side (hydrogen). As consequence, the cathode activation loss is an order of magnitude higher than the anode activation loss. In this PEM fuel cell model, the anode side activation loss effects are neglected for the sake of simplification purposes.

Considering this, the PEM fuel cell stack output terminal voltage can be expressed by the following equations:

${V}_{stack}={{N}_{cell}\bullet V}_{cell}$

${V}_{cell}={E}_{cell}-{V}_{ohm}-{V}_{act}$

where ${E}_{cell}$ is the thermodynamic voltage of the fuel cell, ${V}_{ohm}$ is the voltage drop on the electrolyte ionic resistance and ${V}_{act}$ is the cathode activation losses voltage.

**Thermodynamic voltage of the PEM fuel cell**can be modeled from the Nernst equation:

${E}_{cell}=1.229-0.85\bullet {10}^{-3}\left({T}_{cell}-298.15\right)+\frac{R\bullet {T}_{cell}}{2F}\mathrm{ln}\left({P}_{{H}_{2}cata}\bullet \sqrt{{P}_{{O}_{2}cata}}\right)$

where
${T}_{cell}$
is the temperature of the cell in Kelvins,
${P}_{{O}_{2}cata}$
is the oxygen pressure at the cathode catalytic interface in atm,
${P}_{{H}_{2}cata}$
is the hydrogen pressure at the anode catalytic interface in atm, *R=8.314* is
the universal gas constant in J/(mol·K), and *F=96485.3* is the Faraday constant in C/mol.

The differences between the gas supply pressures and the catalyst interface pressures (effective gas pressure for electrochemical reactions) are due to the gas diffusion phenomenon through the porous electrodes in a PEM fuel cell, which can be modeled by Fick's law:

${P}_{xcata}={P}_{x}-\frac{{N}_{x}\bullet {\delta}_{GDL}\bullet R\bullet {T}_{cell}}{{D}_{x-{H}_{2}Oeff}\bullet {A}_{\mathrm{M}\mathrm{E}\mathrm{A}}}\mathrm{x}\in (\mathrm{H}2,\mathrm{O}2)$

where
${\delta}_{GDL}$
is the gas diffusion layer thickness,
${D}_{x-{H}_{2}Oeff}$
is the effective gas diffusion coefficient, and the reactant gas (hydrogen and oxygen)
molar flow rates are directly related to the fuel cell current *i*:

${N}_{{H}_{2}}=\frac{i}{2\bullet F}$

${N}_{{O}_{2}}=\frac{i}{4\bullet F}$

The effective gas diffusion coefficients ${D}_{x-{H}_{2}Oeff}$ , can be obtained from the binary gas diffusion coefficient and the Bruggemann correction term for the porous electrode of the fuel cell:

${D}_{x-{H}_{2}Oeff}={D}_{x-{H}_{2}O}\bullet {\epsilon}^{\tau}\mathrm{x}\in (\mathrm{H}2,\mathrm{O}2)$

where
${D}_{x-{H}_{2}O}$
is the binary gas diffusion coefficient in m^{2}/s, between the reactants
${H}_{2}$
,
${O}_{2}$
and product
${H}_{2}O$
, *ε *is the corresponding gas diffusion layer porosity, and
$\tau $
is the corresponding gas diffusion layer tortuosity.

**Cathode activation
losses voltage** can be modeled using the following differential equation:

$\frac{d}{dt}{V}_{act}=\frac{i}{{C}_{dl}}\left(1-\frac{1}{{\eta}_{act}}{V}_{act}\right)$

where
${C}_{dl}$
is the single fuel cell cathode double layer capacitance in Farads, *i* is the
fuel cell current in Amperes, and
${\eta}_{act}$
is the cathode steady-state activation loss value, calculated using the well-known
Tafel equation:

${\eta}_{act}=\frac{R\bullet {T}_{cell}}{2\bullet \alpha \bullet F}\mathrm{ln}\left(\frac{i}{{j}_{\mathrm{0,C}}\bullet {A}_{\mathrm{M}\mathrm{E}\mathrm{A}}}\right)$

where
$\alpha $
is the electrochemical reaction symmetry factor (usually between 0.2 and 0.5),
${A}_{MEA}$
is the Membrane Electrode Assembly geometric surface area of a single cell in
m^{2}, and
${j}_{\mathrm{0,C}}$
is the cathode exchange current density in A/m^{2}, calculated by the
following empirical equation:

${j}_{\mathrm{0,C}}=\gamma \bullet {P}_{{O}_{2}cata}^{\beta}\bullet {e}^{-\frac{{E}_{c}}{R\bullet {T}_{cell}}\left(1-\frac{{T}_{cell}}{298.15}\right)}$

where $\gamma $ and $\beta $ are two empirical parameters that need to be identified through fuel cell experimental tests, and ${E}_{c}$ is the constant representing oxygen activation energy at electrode platinum interface (66000 J/mol).

*Double layer capacitance dynamics*, as seen in the Tab: Electrochemical characteristic.

**Electrolyte ionic resistance voltage drop** can be calculated from the
membrane resistance:

${V}_{ohm}=i\bullet {R}_{mem}$ and ${R}_{mem}=\frac{{\delta}_{mem}}{{\sigma}_{mem}\bullet {A}_{MEA}}$

where ${\delta}_{mem}$ is the PEM thickness, and ${\sigma}_{mem}$ (S/m) is the proton conductivity of a Nafion type polymer membrane, which is highly related to the water content of the membrane. Its value can be calculated by:

${\sigma}_{mem}=\mathrm{(0.5139}\bullet {\lambda}_{mem}-\mathrm{0.326)}\bullet {e}^{\left[1268\left(\frac{1}{303}-\frac{1}{{T}_{cell}}\right)\right]}$

**Thermal
calculation:**

The fuel cell's energy balance can be calculated by considering the various energy terms (sources, sinks, etc.) during fuel cell operation:

${Q}_{cooling}=hA\bullet \left({T}_{cell}-{T}_{cooling}\right)$

${Q}_{theo}=\frac{i}{2\bullet F}\mathrm{\Delta}H$

${Q}_{elec}=i\bullet {V}_{cell}$

where ${Q}_{cooling}$ is the heat removal rate from a single fuel cell by the cooling circuit (either forced cooling or natural cooling) in J/s, ${Q}_{theo}$ is the total theoretical thermodynamic power produced during the electrochemical reaction in J/s, $\mathrm{\Delta}H$ is the hydrogen-oxygen reaction enthalpy change in the fuel cell at a higher heating value (286 kJ/mol with liquid water as product), and ${Q}_{elec}$ is the fuel cell total output electrical power (W).

Using the heat transfer formula, cell temperature can be calculated with:

$M\bullet {C}_{p}\bullet \left(\frac{d{T}_{cell}}{dt}\right)={Q}_{theo}-{Q}_{elec}-{Q}_{cooling}$

where *hA* is the cooling coefficient in W/K, *Cp* is the equivalent thermal
capacity of the fuel cell in J/(kg*K), *M* is the mass of the fuel cell in kg, and
${T}_{cooling}$
is the ambient temperature in Kelvins.

*Temperature model type*, as seen in the Tab: Thermal model.

## Tab: General

*N cell* represents the number of fuel cells per fuel cell stack (connected in series).
In this component, this parameter scales the output voltage by the number of fuel cells without
any additional effects.

*Execution rate* represents the signal processing execution rate of this component.

*Signal outputs*is a checkbox that allows you to enable or to disable signal processing outputs. When enabled, a vectorized signal output port appears on the component in Schematic Editor. Output signals are in the following order:

- Temperature of the fuel cell(s)
- Terminal output voltage of the fuel cell stack
- Terminal fuel cell current
- Output voltage of a single fuel cell

## Tab: Electrochemical characteristic

*Double layer capacitance dynamics*checkbox allows you to enable or to disable the differential equation connecting the cathode steady-state activation loss ${\eta}_{act}$ , to the output voltage ${V}_{act}$ .

- When the property checkbox is disabled, ${V}_{act}$ will equal ${\eta}_{act}$ .
- When the property checkbox is enabled, ${V}_{act}$ relates to ${\eta}_{act}$ according to $\frac{d}{dt}{V}_{act}=\frac{i}{{C}_{dl}}\left(1-\frac{1}{{\eta}_{act}}{V}_{act}\right)$ .

*C dl* is a parameter which represents the single fuel cell double layer capacitance.
This capacitance is used only when the property *Double layer capacitance dynamics* is
checked.

*Alpha constant* represents the electrochemical reaction symmetry factor (usually between
0.2 and 0.5).

*Beta constant* and *Gamma constant* are empirical parameters that need to be
identified through fuel cell experimental tests. They affect exchange current density of cathode
electrode and thus the value of cathode activation loss.

## Tab: Fluidic coefficients

*D O2-H2O* represents the binary oxygen gas to vapor diffusion coefficient.

*D H2-H2O* represents the binary hydrogen gas to vapor diffusion coefficient.

## Tab: Fuel cell geometry

*Eta GDL* represents gas diffusion layer porosity, which is used to calculate the
effective gas diffusion coefficient in the fuel cell porous electrode.

*Tau GDL* represents gas diffusion layer tortuosity, which is used to calculate the
effective gas diffusion coefficient in the fuel cell porous electrode.

*Delta GDL* represents gas diffusion layer thickness.

*A MEA* is used to represent the Membrane Electrode Assembly (MEA) surface area of a
single fuel cell (this property represents the geometric surface area and not the effective
surface area of the catalyst).

*Delta MEM* represents proton exchange membrane thickness.

## Tab: Thermal model

*Temperature model type*combo box allows you to model the heat transfer or to set the temperature to a constant value.

*Static model*forces the temperature of the fuel cell to be constant and equal to*Tcell*.*Dynamic model*enables heat transfer equations and calculates temperature based on cooling inlet/ambient temperature*Tcooling*, thermal model parameters, and fuel cell conditions.

*Tcell* represents the fuel cell temperature. The entered value is kept constant during
the simulation. This parameter is disabled when *Temperature model type* is set to
*Dynamic model*.

*Tcooling* represents the external cooling temperature. It can be the cooling inlet
temperature (for forced cooling) or the ambient temperature (for natural cooling). This
parameter is disabled when *Temperature model type* is set to *Static model*.

*Cp* represents the equivalent thermal capacity of a single fuel cell's materials. This
parameter is disabled when *Temperature model type* is set to *Static model*.

*hA* represents the cooling coefficient (forced or natural) of the fuel cell (a product
of the heat transfer coefficient in W/(m^{2}*K) and the cell’s effective cooling surface
in m^{2}).

- In the case of forced cooling the heat transfer coefficient is the fluid-solid surface's forced heat transfer coefficient of the coolant (water or air in most cases). The effective cooling surface is the total contact surface area of cooling channels.
- In the case of natural cooling the heat transfer coefficient is the surface's natural heat transfer coefficient of air and the effective cooling surface is the external surface area of fuel cell stack exposed to ambient air.

This parameter is disabled when *Temperature model type* is set to *Static
model*.

*M* represents the mass of a single fuel cell. This parameter is disabled when
*Temperature model type* is set to *Static model*.

*Cp*,

*hA*, and

*M*can be entered for the fuel cell stack as well as for a single fuel cell. It is important however that all three parameters together reference only one of these two options.

## Tab: Operating conditions

*P H2* represents the hydrogen gas supply pressure of the fuel cell in atm.

*P O2* represents the oxygen gas supply pressure of the fuel cell in atm. If supplied by
air, a coefficient of 21% should be used, representing the O_{2} molar fraction in
air.

*Lambda mem* represents the membrane water content.

## Tab: Polarization curve

*Imax* represents the maximum current of the polarization curve. If during the
simulation, the terminal current of the fuel cell component increases beyond this value an
*overcurrent flag* will be raised to 1. This signal measurement is located inside the
specific fuel cell component and it is always enabled. If the terminal fuel cell current drops
below the *Imax* value, *overcurrent flag* will again return to 0.

The fuel cell's static characteristics are represented by its polarization curve. This curve
shows the output voltage of a single fuel cell as function of its current when the *Preview
Polarization Curve* button is clicked.