Fuel Cell

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 2. 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.

Figure 2. Schematic diagram of fuel cell model

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

$V_{s t a c k} = {N_{c e l l} ∙ V}_{c e l l}$

$V_{c e l l} = E_{c e l l} - V_{o h m} - V_{a c t}$

where $E_{c e l l}$ is the thermodynamic voltage of the fuel cell, $V_{o h m}$ is the voltage drop on the electrolyte ionic resistance and $V_{a c t}$ is the cathode activation losses voltage.

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

$E_{c e l l} = 1.229 - 0.85 ∙ 10^{- 3} (T_{c e l l} - 298.15) + \frac{R ∙ T_{c e l l}}{2 F} \ln (P_{H_{2} c a t a} ∙ \sqrt{P_{O_{2} c a t a}})$

where $T_{c e l l}$ is the temperature of the cell in Kelvins, $P_{O_{2} c a t a}$ is the oxygen pressure at the cathode catalytic interface in atm, $P_{H_{2} c a t a}$ 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:

$P_{x c a t a} = P_{x} - \frac{N_{x} ∙ δ_{G D L} ∙ R ∙ T_{c e l l}}{D_{x - H_{2} O e f f} ∙ A_{M E A}} x \in (H 2, O 2)$

where $δ_{G D L}$ is the gas diffusion layer thickness, $D_{x - H_{2} O e f f}$ 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:

$N_{H_{2}} = \frac{i}{2 ∙ F}$

$N_{O_{2}} = \frac{i}{4 ∙ F}$

The effective gas diffusion coefficients $D_{x - H_{2} O e f f}$ , can be obtained from the binary gas diffusion coefficient and the Bruggemann correction term for the porous electrode of the fuel cell:

$D_{x - H_{2} O e f f} = D_{x - H_{2} O} ∙ ε^{τ} x \in (H 2, O 2)$

where $D_{x - H_{2} O}$ is the binary gas diffusion coefficient in m²/s, between the reactants $H_{2}$ , $O_{2}$ and product $H_{2} O$ , ε is the corresponding gas diffusion layer porosity, and $τ$ is the corresponding gas diffusion layer tortuosity.

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

$\frac{d}{d t} V_{a c t} = \frac{i}{C_{d l}} (1 - \frac{1}{η_{a c t}} V_{a c t})$

where $C_{d l}$ is the single fuel cell cathode double layer capacitance in Farads, i is the fuel cell current in Amperes, and $η_{a c t}$ is the cathode steady-state activation loss value, calculated using the well-known Tafel equation:

$η_{a c t} = \frac{R ∙ T_{c e l l}}{2 ∙ α ∙ F} \ln (\frac{i}{j_{0,C} ∙ A_{M E A}})$

where $α$ is the electrochemical reaction symmetry factor (usually between 0.2 and 0.5), $A_{M E A}$ is the Membrane Electrode Assembly geometric surface area of a single cell in m², and $j_{0,C}$ is the cathode exchange current density in A/m², calculated by the following empirical equation:

$j_{0,C} = γ ∙ P_{O_{2} c a t a}^{β} ∙ e^{- \frac{E_{c}}{R ∙ T_{c e l l}} (1 - \frac{T_{c e l l}}{298.15})}$

where $γ$ and $β$ 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).

Note: The Fuel Cell component can be switched between dynamic cathode activation losses voltage calculation as

V_{a c t}

, or steady-state activation losses voltage as

η_{a c t}

by changing the combo box Double layer capacitance dynamics, as seen in the Electrochemical characteristic (Tab).

Note: Because the cathode activation loss of the fuel cell is modeled by the explicit Tafel equation, the model results might be less accurate than experimental ones when electrical current value is very close to zero. To accurately calculate cathode activation loss for extremely small current values, the full Butler-Volmer equation in its implicit form should be used.

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

$V_{o h m} = i ∙ R_{m e m}$ and $R_{m e m} = \frac{δ_{m e m}}{σ_{m e m} ∙ A_{M E A}}$

where $δ_{m e m}$ is the PEM thickness, and $σ_{m e m}$ (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:

$σ_{m e m} = (0.5139 ∙ λ_{m e m} - 0.326) ∙ e^{[1268 (\frac{1}{303} - \frac{1}{T_{c e l l}})]}$

where

λ_{m e m}

is the membrane water content:

λ_{m e m} \approx 7

for a dry membrane,

λ_{m e m} \approx 14

for a well-humidified membrane and

λ_{m e m} \approx 22

for a flooded membrane.

Note: The polymer membrane proton conductivity model is applied for Nafion-type materials only.

Thermal calculation:

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

$Q_{c o o l i n g} = h A ∙ (T_{c e l l} - T_{c o o l i n g})$

$Q_{t h e o} = \frac{i}{2 ∙ F} Δ H$

$Q_{e l e c} = i ∙ V_{c e l l}$

where $Q_{c o o l i n g}$ is the heat removal rate from a single fuel cell by the cooling circuit (either forced cooling or natural cooling) in J/s, $Q_{t h e o}$ is the total theoretical thermodynamic power produced during the electrochemical reaction in J/s, $Δ H$ 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 $Q_{e l e c}$ is the fuel cell total output electrical power (W).

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

$M ∙ C_{p} ∙ (\frac{d T_{c e l l}}{d t}) = Q_{t h e o} - Q_{e l e c} - Q_{c o o l i n g}$

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 $T_{c o o l i n g}$ is the ambient temperature in Kelvins.

Note: The Fuel Cell component has can be switched between temperature calculation and constant temperature by changing the checkbox Temperature model type, as seen in the Thermal model (Tab).

Note: The thermal calculation of Fuel Cell component is enabled only in case Scientific is selected as Parameter definition.

General (Tab)

Parameter definition
- Specifies the set of parameters by choosing between Simplified, Detailed and Scientific.
If Simplified is selected as Parameter definition, the following properties can be used:
- Preset Fuel Cell
  - Provides a set of predetermined Fuel Cell staks found on the market.
- Voltage at 0 A and 1 A
  - 2-dimensional vector representing fuel cell voltages at 0 A and 1 A.
- Nominal operating point
  - 2-dimensional vector representing nominal fuel cell curent (A) and voltage (V).
- Maximum operating point
  - 2-dimensional vector representing fuel cell curent (A) and voltage (V) at maximum power.

If Detailed is selected as Parameter definition, the following properties can be used:
- Preset Fuel Cell
  - Provides a set of predetermined Fuel Cell staks found on the market.
- Voltage at 0 A and 1 A
  - 2-dimensional vector representing fuel cell voltages at 0 A and 1 A.
- Nominal operating point
  - 2-dimensional vector representing nominal fuel cell curent (A) and voltage (V).
- Maximum operating point
  - 2-dimensional vector representing fuel cell curent (A) and voltage (V) at maximum power.
- Number of cells
  - Number of fuel cells per fuel cell stack (connected in series).
- Nominal stack efficiency
  - Fuel cell efficiency at nominal operating point.
- Operating temperature
  - Constant temperature of fuel cell.
  - The property is disabled if temperature input is enabled.
- Nominal fuel flow rate
  - Constant fuel flow rate of fuel cell.
  - The property is disabled if fuel flow rate input is enabled.
- Nominal air flow rate
  - Constant air flow rate of fuel cell.
  - The property is disabled if air flow rate input is enabled.
- Nominal supply presure
  - 2-dimesional vector representing constant supply pressure of fuel and air.
  - The property is disabled if both input representing supply pressure of fuel and supply pressure of air are enabled. If one input is enabled, and other is disabled, the nominal value corresponing to enabled input will be ignored.
- Nominal composition
  - 3-dimesional vector representing constant percentage of hydrogen in the fuel, oxygen and water in the oxidant.
  - If one of the relevant inputs is enabled (Fuel compoistion, Air composition), the nominal value corresponing to enabled input will be ignored.
If Scientific is selected as Parameter definition, the following properties can be used:
- Number of cells
  - Number of fuel cells per fuel cell stack (connected in series)
  - The parameter scales the output voltage by the number of fuel cells without any additional effects.
- Signal outputs
  - Enables/disables signal processing outputs. When enabled, a signal output port appears on the component.
  - Available outputs are in the following order:
    1. Temperature of the fuel cell(s)
    2. Terminal output voltage of the fuel cell stack
    3. Terminal fuel cell current
    4. Output voltage of a single fuel cell
Execution rate
- Signal processing execution rate of the fuel cell component

Input (Tab)

This Tab is only available if Detailed is selected as Parameter definition.

Temperature
- Enables/disables signal processing input for controlling the fuel cell temperature
Fuel flow rate
- Enables/disables signal processing input for controlling the fuel flow rate
Air flow rate
- Enables/disables signal processing input for controlling the air flow rate
Fuel supply pressure
- Enables/disables signal processing input for controlling the fuel supply pressure
Air supply pressure
- Enables/disables signal processing input for controlling the air supply pressure
Fuel composition
- Enables/disables signal processing input for controlling the percentage of hydrogen in the fuel
Air composition
- Enables/disables signal processing input for controlling the percentage of oxygen in the oxidant