# An Equivalent Electrical Circuit Model of Proton Exchange Membrane Fuel Cells Based on Mathematical Modelling

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{®}for the electrical circuit model, and COMSOL

^{®}for the other. Finally, the steady state and transient responses will be compared in order to make sure of the equivalence of the models.

## 2. PEM Fuel Cell Mathematical Model

_{3}O

^{+}(here simply quoted as H

^{+}) and electrons:

**Figure 3.**Model regions ($\text{\Omega}$) and model boundaries ($\partial \text{\Omega}$) definition.

- (1)
- water exists only in the gas phase within the electrodes, and as solute water in the membrane,
- (2)
- the cell temperature remains constant and homogeneous all over the cell,
- (3)
- the gas diffusion layers (GDLs) and the membrane are isotropic and homogeneous,
- (4)
- the catalyst layers are very thin and are considered as reactive surfaces,
- (5)
- the membrane is gas-tight,
- (6)
- all electrical contact resistances are neglected,
- (7)
- the current density is homogeneous at collectors.

Parameter | Symbol | Value |
---|---|---|

GDL thicknesses | L_{d} | 400 μm |

Membrane thickness | L_{m} | 15 μm |

Cell active area | A_{cell} | 100 cm^{2} |

Membrane type | Gore Primea 5761 | |

Open circuit voltage | V_{OC} | ~1 V |

Nominal voltage | V_{N} | 0.6 V |

Rated power | P_{N} | 30 W |

#### 2.1. Gas Diffusion Layers

#### 2.1.1. Charge Transport Equations

_{s}is the electronic conductivity of the electrodes. Charge conservation is given by:

#### 2.1.2. Mass Transport Equations

_{d}begins at x = 0 on the anode side (respectively at x = 2 L

_{d}+ L

_{m}on the cathode side) and comes to an end at the point x = L

_{d}(respectively at x = L

_{d}+ L

_{m}). The interactions between a pair of species (i, j) are characterized by the binary diffusion coefficient D

_{ij}[18]:

Diffusivity name | Ref. temperature T_{0} [K] | Diffusivity value [m^{2}.s^{−1}] |
---|---|---|

${\text{D}}_{{\text{H}}_{2},{\text{H}}_{2}\text{O}}^{0}$ | 307.1 | 9.15 × 10^{−5} |

${\text{D}}_{{\text{O}}_{2},{\text{H}}_{2}\text{O}}^{0}$ | 308.1 | 2.82 × 10^{−5} |

${\text{D}}_{{\text{O}}_{2},{\text{N}}_{2}}^{0}$ | 293.2 | 2.20 × 10^{−5} |

${\text{D}}_{{\text{N}}_{2},{\text{H}}_{2}\text{O}}^{0}$ | 307.5 | 2.56 × 10^{−5} |

_{p}is the pore diameter, and M

_{i}the molar mass of the specie i.

_{a}and P

_{c}are evaluated assuming the gas to be ideal and adiabatic.

#### 2.1.3. Boundary Conditions for Charge Transport

_{cell}, calculated from the whole cell current I

_{cell}, and from the actual active area of the membrane (100 cm

^{2}):

_{a}and j

_{c}being given by usual electrochemical kinetics relations (Butler-Volmer equations).

#### 2.1.4. Boundary Conditions for Mass Transport

_{sat}in atm unit [21]:

_{a}and λ

_{c}versus water activity a = P

_{H2O}/P

_{sat}. At 30 °C and 80 °C, these curves are defined as follows:

#### 2.1.5. Equivalent Circuit Modeling of Mass Transport

_{H2}corresponds to the partial pressure of H

_{2}, and the current density N

_{H2}is the molar flow density of H

_{2}.

Transport model | Electrical distributed model |
---|---|

pressure: P [Pa] | voltage: U [V] |

molar flow density: N [mol·m^{−2}·s^{−1}] | current density: J [A·m^{−2}] |

$\frac{\text{RT}}{\text{D}}$ | electrical resistivity: ρ_{el} [Ω·m] |

$\frac{\partial \text{P}}{\partial \text{x}}=-\frac{\text{RT}}{\text{D}}\cdot \text{N}$ | $\frac{\partial U}{\partial \text{x}}=-{\text{\rho}}_{\text{el}}\cdot \text{J}$ |

$\frac{\text{\epsilon}}{\text{RT}}$ | specific capacitance: c_{v} [F·m^{−3}] |

$\frac{\partial \text{N}}{\partial \text{x}}=-\frac{\text{\epsilon}}{\text{RT}}\cdot \frac{\partial \text{P}}{\partial \text{t}}$ | $\frac{\partial \text{J}}{\partial \text{x}}=-{\text{c}}_{\mathrm{v}}\cdot \frac{\partial \text{U}}{\partial \text{t}}$ |

^{2}) is composed of a constant-value resistance R

_{H2}and a voltage-dependent resistance R

_{H2,H2O}, which depends on water molar fraction:

_{H2,H2O}depends on water molar flow density, and on hydrogen molar fraction:

^{−2}) represents the mass balance expressed in Equation (12):

_{d}/10).

#### 2.2. Membrane

_{H2O}, and electrolyte potential φ

_{m}.

#### 2.2.1. Charge Transport Equations

_{m}is the ionic conductivity of the membrane that greatly depends on the membrane hydration state. Charge conservation is given by:

#### 2.2.2. Water Transport Equations

_{d}is the electro-osmotic drag coefficient (number of water molecules dragged per proton), and ${\text{D}}_{{\text{H}}_{2}\text{O}}^{\text{m}}$ is the water diffusion coefficient. These coefficients depend on the membrane water content [3]. The conservation of water quantity in the membrane can be written as:

#### 2.2.3. Parametric Laws

_{3}

^{¯}available in the polymer. It can also be stated versus membrane equivalent weight EW, membrane density ρ

_{m}, and water concentration as follows:

^{®}membrane [26], and propose for water diffusion coefficient versus water content and temperature:

#### 2.2.4. Boundary Conditions for Charge Transport

#### 2.2.5. Boundary Conditions for Water Transport

_{a}and λ

_{c}are calculated versus water activity, through a linear interpolation based on the empirical sorption curves defined by Equation (19).

#### 2.2.6. Equivalent Circuit Modeling of Water Transport

_{m}/10).

_{w}[Ω·m

^{2}], the current-controlled voltage source V

_{w}[V] and the capacitance C [F·m

^{−2}] can be established as follows, using Equations (29–31):

#### 2.3. Catalyst Layers

_{a}and j

_{c}, which contain a faradic component governed by Butler-Volmer equations [28]:

_{dla}and c

_{dlc}are double layer capacitances. Therefore, the current densities in the anode and cathode active layer are the sum of these two currents:

^{2}Gore-type single cell, the double layer capacitance has been evaluated to 2 F, by using an impedance spectroscopy method. In Equation (41), η

_{a}and η

_{c}are electrode electrochemical overvoltages. They are calculated versus electrode potential and membrane potential at membrane-electrode interfaces as follows:

#### 2.4. Membrane Electrodes Assembly (MEA) Model

^{®}. Boundary conditions have not been represented, in order to make it clear. Figure 8 depicts the circuit model as it has been implemented in SABER

^{®}.

_{BVa}and j

_{BVc}, and electrochemical overvoltages η

_{a}and η

_{c}, respectively. This representation has been chosen because it respects the electrical engineering rules of source connection.

## 3. Numerical Simulations

^{®}(version 3.5). Parameters required for the description of the electrochemical reaction and the water transport across the membrane are based on literature values, and are given in Table 4. Geometric parameters and operating conditions are detailed in Table 5 and Table 6, respectively. First of all, polarization curves are compared. Then dynamic voltage responses to different current waveforms are plotted versus time. All initial conditions correspond to the steady state.

Parameter | Symbol | Value | Unit | Ref. |
---|---|---|---|---|

dry GDL porosity | ε_{s} | 0.6 | - | [29] |

dry membrane density | ρ_{m} | 2020 | kg·m^{−3} | [30] |

equiv. membrane weight | EW | 0.95 | kg·mol^{−1} | [31] |

anod. exch. curr. density | ${\text{j}}_{\mathrm{a}}^{0}$ | 5000 | A·m^{−2} | [32] |

cath. exch. curr. density | ${\text{j}}_{\mathrm{c}}^{0}$ | 20 | A·m^{−2} | estimated |

anod. transfer coefficient | α_{a} | 2 | - | [32] |

cath. transfer coefficient | α_{c} | 0.5 | - | [32] |

double layer capacitor | c_{dla}, c_{dlc} | 2 | F | measured |

Parameter | Symbol | Value | Unit |
---|---|---|---|

inlet channel height | l_{ch} | 0.5 | mm |

outlet channel height | l_{ch} | 0.5 | mm |

current collector height | l_{sh} | 1 | mm |

GDL thickness | L_{d} | 400 | μm |

membrane thickness | L_{m} | 15 | μm |

Parameter | Symbol | Value | Unit |
---|---|---|---|

cell temperature | T | 333 | K |

anode relative humidity | HR_{a} | 0% | - |

cath. relative humidity | HR_{c} | 79% | - |

anode outlet pressure | ${\text{p}}_{\mathrm{a}}^{\text{out}}$ | 1.013 × 10^{5} | Pa |

cathode outlet pressure | ${\text{p}}_{\mathrm{c}}^{\text{out}}$ | 1.013 × 10^{5} | Pa |

#### 3.1. In Steady State

^{®}plotted curve in a large current density scale. Thus, the circuit model seems to be in good agreement with the PDE model.

#### 3.2. In Transient State

#### 3.3. Practical Application

^{2}active area and the boost switching frequency is 10 kHz. We assume that the stack model is the same as the cell model; therefore we just have to multiply the results by the cell number of the stack.

_{ref}= 40 A (where I

_{ref}is the setpoint used to compute the gas flow rates). The second way is called adapted gas flows. In this case, the fuel cell current is used to compute the gas flow rates [I

_{ref}= I

_{cell}(t)].

**Figure 12.**Simulated responses to a 10 A/30 A current step of a fuel cell stack connected to a regulated boost converter.

## 4. Conclusions

## Nomenclature:

A | surface area [m ^{2}] |

a | water activity [-] |

C_{i} | concentration of species i [mol·m ^{−3}] |

${\text{D}}_{\text{ij}}^{\text{eff}}$ | binary diffusion coefficient of species i and j [m ^{2}·s^{−1}] |

D_{ik} | Knudsen diffusion coefficient of species i [m ^{2}·s^{−1}] |

${\text{D}}_{{\text{H}}_{2}\text{O}}^{\text{m}}$ | water diffusion coefficient in membrane [m ^{2}·s^{−1}] |

d_{p} | pore diameter of GDLs [m] |

EW | equivalent weight of the membrane [kg·mol ^{−1}] |

F | Faraday’s constant, 96472 [C·mol ^{−1}] |

I | current [A] |

J_{cell} | current density [A·m ^{−2}] |

L_{d} | GDLs thickness [m] |

L_{m} | membrane thickness [m] |

M_{i} | molar mass of species i [kg·mol ^{−1}] |

N | molar flow density of species i [mol·m ^{−2}·s^{−1}] |

n_{d} | electro-osmotic drag coefficient [-] |

P_{i} | partial pressure of species i [Pa] |

${\text{P}}_{\mathrm{i}}^{*}$ | partial pressure of species i at the cell entry [Pa] |

P_{sat} | saturated pressure [Pa] |

P_{a,c} | total pressure at anode or cathode side [Pa] |

R | gas constant, 8.314 [J·mol ^{−1}·K^{−1}] |

RH | relative humidity [-] |

R_{m} | membrane specific resistance [Ω·m] |

t | time [s] |

T | temperature [K] |

t | time [s] |

x_{i} | molar fraction of species i [-] |

z | lenght [m] |

## Greek Letters

α | transfer coefficient [-] |

γ | water transfer coefficient [m·s ^{−1}] |

ε | porosity [-] |

η^{a,c} | anode or cathode overvoltage [V] |

λ | membrane water content [-] |

ζ | stoichiometry [-] |

ρ_{m} | density of dry membrane [kg·m ^{−3}] |

σ_{m} | membrane ionic conductivity [S·m ^{−1}] |

φ | electrical potential [V] |

## Superscripts and Subscripts

a | anode |

c | cathode |

e | entry |

eff | effective value |

GDL | gas diffusion layer |

i | species (H _{2} and H_{2}O for the anode, O_{2}, H_{2}O, and N_{2} for the cathode) |

l | liquid water |

m | membrane |

prod | produced |

s | electronic phase |

v | vapor water |

w | water |

## References

- Dannenberg, K.; Ekdunge, P.; Lindbergh, G. Mathematical model of the PEMFC. J. Appl. Electrochem.
**2000**, 30, 1377–1387. [Google Scholar] [CrossRef] - Rowe, A.; Li, X. Mathematical modeling of proton exchange membrane fuel cells. J. Power Sources
**2001**, 102, 82–96. [Google Scholar] [CrossRef] - Ge, S.H.; Yi, B.L. A mathematical model for PEMFC in different flow modes. J. Power Sources
**2003**, 124, 1–11. [Google Scholar] [CrossRef] - Baschuk, J.J.; Li, X. A general formulation for a mathematical PEM fuel cell model. J. Power Sources
**2005**, 142, 134–153. [Google Scholar] [CrossRef] - Wang, C.; Nehrir, M.H.; Shaw, S.R. Dynamic models and model validation for PEM fuel cells using electrical circuits. IEEE Trans. Energy Convers.
**2005**, 20, 442–451. [Google Scholar] [CrossRef] - Pathapati, P.R.; Xue, X.; Tang, J. A new dynamic model for predicting transient phenomena in a PEM fuel cell system. Renew. Energy
**2005**, 30, 1–22. [Google Scholar] [CrossRef] - Yuan, W.; Tang, Y.; Pan, M.; Li, Z.; Tang, B. Model prediction of effects of operating parameters on proton exchange membrane fuel cell performance. Renew. Energy
**2010**, 35, 656–666. [Google Scholar] [CrossRef] - Ceraolo, M.; Miulli, C.; Pozio, A. Modelling static and dynamic behaviour of proton exchange membrane fuel cells on the basis of electro-chemical description. J. Power Sources
**2003**, 113, 131–144. [Google Scholar] [CrossRef] - Gao, F.; Blunier, B.; Miraoui, A.; El-Moudni, A. Cell layer level generalized dynamic modeling of a PEMFC stack using VHDL-AMS language. Int. J. Hydrogen Energy
**2009**, 34, 5498–5521. [Google Scholar] [CrossRef] - Kim, H.; Cho, C.Y.; Nam, J.H.; Shin, D.; Chung, T. A simple dynamic model for polymer electrolyte membrane fuel cell (PEMFC) power modules: Parameter estimation and model prediction. Int. J. Hydrogen Energy
**2010**, 35, 3656–3663. [Google Scholar] [CrossRef] - Yu, D.; Yuvarajan, S. Electronic circuit model for proton exchange membrane fuel cells. J. Power Sources
**2005**, 142, 238–242. [Google Scholar] [CrossRef] - Andujar, J.M.; Segura, F.; Vasallo, M.J. A suitable model plant for control of the set fuel cell DC/DC converter. Renew. Energy
**2008**, 33, 813–826. [Google Scholar] [CrossRef] - Lazarou, S.; Pyrgioti, E.; Alexandridis, A.T. A simple electric circuit model for proton exchange membrane fuel cells. J. Power Sources
**2009**, 190, 380–386. [Google Scholar] [CrossRef] - Reggiani, U.; Sandrolini, L.; Giuliattini Burbui, G.L. Modelling a PEM fuel cell stack with a nonlinear equivalent circuit. J. Power Sources
**2007**, 165, 224–231. [Google Scholar] [CrossRef] - Alexander, C.; Sadiku, M. Fundamentals of Electric Circuits, McGraw-Hill: New York, NY, USA, 2004.
- Paul, C. Fundamentals of Electric Circuit Analysis; Wiley: Hoboken, NJ, USA, 2001. [Google Scholar]
- Larminie, J.; Dicks, A. Fuel Cell Systems Explained, 2nd ed.; John Wiley & Sons: Hoboken, NJ, USA, 2000; ISBN 0-470-84857-X. pp. 69–81. [Google Scholar]
- Serican, M.F.; Yesilyurt, S. Transient analysis of proton electrolyte membrane fuel cells (PEMFC) at start-up and failure. Fuel Cells
**2006**, 7, 118–127. [Google Scholar] [CrossRef][Green Version] - Daneshpajooh, M.H.; Mason, E.A.; Bresler, E.H.; Wendt, R.P. Equations for membrane transport. Experimental and theoretical tests of the frictional model. Biophys. J.
**1975**, 15, 591–613. [Google Scholar] [CrossRef] [PubMed] - Monroe, C.W.; Newman, J. Onsager reciprocal relations for Maxwell-Stefan diffusion. Ind. Eng. Chem. Res.
**2006**, 45, 5361–5367. [Google Scholar] [CrossRef] - Chiang, M.-S.; Chu, H.-S. Numerical investigation of transport component design effect on a proton exchange membrane fuel cell. J. Power Sources
**2006**, 160, 340–352. [Google Scholar] [CrossRef] - Hinatsu, J.T.; Mizuhata, M.; Takenaka, H. Water uptake of perfluorosulfonic acid membranes from liquid water and water vapour. J. Electrochem. Soc.
**1994**, 141, 1493–1498. [Google Scholar] [CrossRef] - Berg, P.; Promislow, K.; Pierre, J., St.; Stumper, J.; Wetton, B. Water management in PEM fuel cells. J. Electrochem. Soc.
**2004**, 151, A341–A353. [Google Scholar] [CrossRef] - Nguyen, T.V.; White, R.E. A water and thermal management model for proton exchange membrane fuel cells. J. Electrochem. Soc.
**1993**, 140, 2178–2186. [Google Scholar] [CrossRef] - Springer, T.E.; Zawodzinski, T.A.; Gottesfeld, S. Polymer electrolyte fuel cell model. J. Electrochem. Soc.
**1991**, 138, 2334–2342. [Google Scholar] [CrossRef] - Ye, X.; Wang, C.-Y. Measurement of water transport properties through membrane-electrode assemblies. J. Electrochem. Soc.
**2007**, 154, B676–B682. [Google Scholar] [CrossRef] - Siegel, N.P.; Ellis, M.W.; Nelson, D.J.; von Spakovsky, M.R. Single domain PEMFC model based on agglomerate catalyst geometry. J. Power Sources
**2003**, 115, 81–89. [Google Scholar] [CrossRef] - Berning, T.; Lu, D.M.; Djilali, N. Three-dimensional computational analysis of transport phenomena in a PEM fuel cell. J. Power Sources
**2002**, 106, 284–294. [Google Scholar] [CrossRef] - Wang, Y.; Wang, C.-Y. Dynamics of polymer electrolyte fuel cells undergoing load changes. Electrochim. Acta
**2006**, 51, 3924–3933. [Google Scholar] [CrossRef] - Ramousse, J.; Deseure, J.; Lottin, O.; Didierjean, S.; Maillet, D. Modeling of heat, mass and charge transfer in a PEMFC sigle cell. J. Power Sources
**2005**, 145, 416–427. [Google Scholar] [CrossRef] - Liu, F.; Lu, G.; Wang, C.-Y. Water transport coefficient distribution though the membrane in polymer electrolyte fuel cell. J. Membr. Sci.
**2007**, 287, 126–131. [Google Scholar] [CrossRef] - Shimpalee, S.; Ohashi, M.; van Zee, J.W.; Ziegler, C.; Stoeckmann, C.; Sadeler, C.; Hebling, C. Experimental and numerical studies of portable PEMFC stack. Electrochim. Acta
**2009**, 54, 2899–2911. [Google Scholar] [CrossRef]

© 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Hinaje, M.; Raël, S.; Noiying, P.; Nguyen, D.A.; Davat, B. An Equivalent Electrical Circuit Model of Proton Exchange Membrane Fuel Cells Based on Mathematical Modelling. *Energies* **2012**, *5*, 2724-2744.
https://doi.org/10.3390/en5082724

**AMA Style**

Hinaje M, Raël S, Noiying P, Nguyen DA, Davat B. An Equivalent Electrical Circuit Model of Proton Exchange Membrane Fuel Cells Based on Mathematical Modelling. *Energies*. 2012; 5(8):2724-2744.
https://doi.org/10.3390/en5082724

**Chicago/Turabian Style**

Hinaje, Melika, Stéphane Raël, Panee Noiying, Dinh An Nguyen, and Bernard Davat. 2012. "An Equivalent Electrical Circuit Model of Proton Exchange Membrane Fuel Cells Based on Mathematical Modelling" *Energies* 5, no. 8: 2724-2744.
https://doi.org/10.3390/en5082724