# Generic Dynamical Model of PEM Electrolyser under Intermittent Sources

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## Abstract

**:**

## 1. Introduction

## 2. BG Technique for Model Building

#### 2.1. BG Elements and BG Variables

#### 2.2. Different Levels of Modelling Abstraction

#### 2.3. Modular Building (Capsules)

#### Grammar and Connectivity Rules

## 3. PEM Electrolyser Modelling

- The cells constituting the stack are identical in nature and connected in series. Thus, the stack with N cells can be modelled as an equivalent single cell that has the same dynamics of the stack.
- Uniform fluid flows and current distribution are considered between cells.
- Overpotential due to mass transport or diffusion is negligible with the assumption that PEM system usually operates at low current density.
- Electrolysis reaction kinetics is assumed firmly as a Faradic process and considers that there is no mass limitation problem in the system.
- Gases produced are assumed to have similar properties as that of an ideal gas and the partial pressures of these gases are governed by Dalton’s law.
- Temperature is homogenous throughout the stack.
- Cell is operated below the boiling temperature of the water.
- The system parameters are considered as lumped parameters. Pumps and fans are assumed as perfect mass flow sources.

#### 3.1. Technological Representation

#### 3.2. Modular Representation

#### 3.2.1. Stack Model

#### Electrochemical Sub-Model of the Stack

#### Chemical-Fluidic Sub-Model of the Stack

#### Thermal Sub-Model of the Stack

#### Fluidic and Mass Transfer Sub-Model

#### 3.2.2. Converter Sub-Model

#### 3.2.3. Separator Sub-Models

#### 3.2.4. Cooling and Recirculation Circuits

#### 3.2.5. Hydrogen Purification Subsystem

#### 3.2.6. System Enclosure

#### 3.3. Efficiency of the PEM Electrolysis System

#### 3.3.1. Efficiency of Cell/Stack

#### 3.3.2. Efficiency of System Including the Auxiliaries

## 4. Experimental Validation and Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AEM | Anion Exchange Membrane |

BG | Bond Graph |

DC | Direct Current |

HHV | Higher Heating Value |

HSV | Hydrogen Separator Vessel |

LHV | Lower Heating Value |

MEA | Membrane Electrode Assembly |

NTU | Number of Transfer Unit |

OSV | Oxygen Separator Vessel |

PEM | Proton Exchange Membrane |

RES | Renewable Energy Sources |

Nomenclature | |

${\alpha}_{k}$ | Charge transfer or symmetry factor coefficients for ${k}^{th}$ electrode |

$\beta $ | Transformer coefficient of the converter |

$\Delta {G}_{R}$ | Gibb’s free energy of water dissociation reaction, J·mol${}^{-1}$ |

$\Delta {H}_{R}$ | Enthalpy Change of water dissociation reaction, J·mol${}^{-1}$ |

$\Delta {S}_{R}$ | Entropy change of water dissociation reaction, J·mol${}^{-1}$·K${}^{-1}$ |

$\dot{\xi}$ | Rate of reaction flow, mol·s${}^{-1}$ |

$\dot{H}$ | Enthalpy rate, J·s${}^{-1}$ |

${\dot{m}}_{i}$ | Mass flow rate of the ${i}^{th}$ species, kg·s${}^{-1}$ |

${\dot{n}}_{i}$ | Gas mass flow rate for ${i}^{th}$ species, kg·s${}^{-1}$ |

$\dot{Q}$ | Rate of heat flow, J·s${}^{-1}$ |

$\gamma $ | Hydration of the membrane |

${\nu}_{i}$ | Coefficient of stoichiometry for ${i}^{th}$ species |

${\rho}_{w}$ | Water density, kg·m${}^{-3}$ |

${\sigma}_{M}$ | Conductivity of the membrane, S·m${}^{-1}$ |

$\epsilon $ | Efficiency |

${A}_{M}$ | Cross-sectional area of the membrane, m${}^{2}$ |

${A}_{sep,i}$ | Cross-sectional area of the ${i}^{th}$ separator, m${}^{2}$ |

${a}_{{H}_{2}O}$ | Chemical activity of water |

${C}_{i}$ | Matter storage capacity of the ${i}^{th}$ species ($i={H}_{2},{O}_{2},{H}_{2}O$), kg${}^{2}$·J${}^{-1}$ |

${C}_{p}$ | Specific heat at constant pressure, J·kg${}^{-1}$·K${}^{-1}$ |

${C}_{cold}^{th}$ | Thermal capacitance of cooling tank, J·K${}^{-1}$ |

${C}_{cool}^{th}$ | Thermal capacitance of cooling circuit, J·K${}^{-1}$ |

${C}_{dry,i}^{ch}$ | Dryer’s chemical capacitance of the purification unit for the ${i}^{th}$ species, mol·Pa${}^{-1}$ |

${C}_{dry}^{th}$ | Dryer’s thermal capacitance of the purification unit, J·K${}^{-1}$ |

${C}_{enc}^{th}$ | Thermal capacity of the enclosure, J·K${}^{-1}$ |

${C}_{Hsep,i}^{ch}$ | Chemical capacitance of hydrogen separator for the ${i}^{th}$ species, mol·Pa${}^{-1}$ |

${C}_{Osep,i}^{ch}$ | Chemical capacitance of oxygen separator for the ${i}^{th}$ species, mol·Pa${}^{-1}$ |

${C}_{rec,k}^{th}$ | Thermal capacitance of recirculation circuit (anode/cathode side), J·K${}^{-1}$ |

${C}_{sep,{H}_{2}}^{fl-th}$ | Field capacitance element representing fluidic capacitance and thermal capacitance of hydrogen separator |

${C}_{sep,{O}_{2}}^{fl-th}$ | Field capacitance element representing fluidic capacitance and thermal capacitance of oxygen separator |

${C}_{stack}$ | Thermal capacitance of the stack, J·K${}^{-1}$ |

${D}_{i}$ | Parameter for diffusion, m${}^{2}$·s${}^{-1}$ |

${d}_{M}$ | Ratio of length to the cross-sectional area of the membrane, m${}^{-1}$ |

${E}_{act,k}$ | Activation overvoltages for ${k}^{th}$ electrode, V |

${E}_{cell}$ | Cell voltage, V |

${E}_{ohm}$ | Ohmic overvoltage, V |

${E}_{rev}$ | Reversible voltage, V |

${E}_{rev}^{0}$ | Standard reversible cell voltage at STP, V |

g | Acceleration due to gravity, m·s${}^{-2}$ |

${H}_{i}$ | Henry’s Parameter, Pa·m${}^{3}$·mol${}^{-1}$ |

${I}_{cell}$ | Cell current, A |

J | Current density, A·m${}^{-2}$ |

${J}_{0,k}$ | Standard current exchange density for ${k}^{th}$ electrode, A·m${}^{-2}$ |

${L}_{M}$ | Length of the membrane, m |

${L}_{sep,i}$ | Water level in Separators (HSV and OSV), m |

${L}_{sep,i}$ | Water level of the ${i}^{th}$ separator, m |

${M}_{i}$ | Molar mass for ${i}^{th}$ species, kg mol${}^{-1}$ |

${n}_{eo}$ | Electro-osmosis coefficient |

P | Pressure, Pa |

${p}_{i}$ | Partial pressure of ${i}^{th}$ species, Pa |

${R}_{c}$ | Coupling element for fluidic flow to thermal flow |

${R}_{act,k}$ | Non linear activation resistance for ${k}^{th}$ electrode |

${R}_{ads}$ | Coupling resistance of adsorbed water molar flow and the enthalpy flow towards dryer |

${R}_{diff,i}$ | Diffusion resistance of the ${i}^{th}$ species, Pa·s·kg${}^{-1}$ |

${R}_{dry,enc}$ | Thermal resistance between purification unit and enclosure, K·s·J${}^{-1}$ |

${R}_{dry}$ | Internal pneumatic resistance of the dryer, Pa·s·kg${}^{-1}$ |

${R}_{enc}$ | Thermal resistance between the enclosure and the atmosphere, K·s·J${}^{-1}$ |

${R}_{exhaust}$ | Pneumatic resistance of the exhaust valve, Pa·s·kg${}^{-1}$ |

${R}_{Hrec,enc}$ | Thermal resistance between hydrogen recirculation circuit and enclosure, K·s·J${}^{-1}$ |

${R}_{Hsep,enc}$ | Thermal resistance between hydrogen separator and enclosure, K·s·J${}^{-1}$ |

${R}_{htex}$ | Thermal resistance of heat exchanger, K·s·J${}^{-1}$ |

${R}_{hyst,k}$ | Internal fluidic resistance of the stack at th ${k}_{th}$ electrode side, Pa·s·kg${}^{-1}$ |

${R}_{Lrec,k}$ | Hydraulic resistance representing leakage in recirculation circuit (anode/cathode side), Pa·s·kg${}^{-1}$ |

${R}_{ohm}$ | Total ohmic resistance of the cell, $\Omega $ |

${R}_{Orec,enc}$ | Thermal resistance between oxygen recirculation circuit and enclosure, K·s·J${}^{-1}$ |

${R}_{Osep,enc}$ | Thermal resistance between oxygen separator and enclosure, K·s·J${}^{-1}$ |

${R}_{others}$ | Ohmic resistance of the cell except membrane, $\Omega $ |

${R}_{sep,hc}$ | Pneumatic resistance between hydrogen separator and hydrogen circuit, Pa·s·kg${}^{-1}$ |

${R}_{sep,oc}$ | Pneumatic resistance between oxygen separator and oxygen circuit, Pa·s·kg${}^{-1}$ |

${R}_{sepv}$ | Hydraulic resistance of the separator valve, Pa·s·kg${}^{-1}$ |

${R}_{stack}$ | Thermal resistance of the stack, K·s·J${}^{-1}$ |

${R}_{tank}$ | Hydraulic resistance between tank and oxygen Separator, Pa·s·kg${}^{-1}$ |

T | Temperature, K |

${V}_{Hsep,i}$ | Volume of the ${i}^{th}$ species in HSV, m${}^{3}$ |

${V}_{Osep,i}$ | Volume of the ${i}^{th}$ species in OSV, m${}^{3}$ |

${x}_{i}$ | Mass fraction of the ${i}^{th}$ species |

${\mu}_{i}$ | Chemical potential of the ${i}^{th}$ species, J·kg${}^{-1}$ |

${A}_{i}$ | Chemical affinity of the ${i}^{th}$ species, J·mol${}^{-1}$ |

${C}_{ads}^{ch}$ | Water adsorption capacity of the purification unit, mol·Pa${}^{-1}$ |

F | Faraday’s constant, C·mol${}^{-1}$ |

R | Ideal gas constant, J·mol${}^{-1}$·K${}^{-1}$ |

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**Figure 2.**(

**a**) General schematic of a Proton Exchange Membrane (PEM) electrolyser and (

**b**) Multi-physics phenomena in PEM electrolyser cell/stack (inspired by Reference [2]).

**Figure 4.**Representation of vector bond (

**a**) with a small ring around the power bond, (

**b**) with multi-bonds and (

**c**) with separate power bond for different energy domains.

**Figure 16.**BG recirculation sub-models (

**a**) anode side including cooling circuit and (

**b**) cathode side.

**Figure 22.**(

**a**) Contribution of overvoltages, (

**b**) efficiency of electrolysis cell and (

**c**) temperature evolution of cell.

**Figure 23.**(

**a**) Powers consumed by the cell for 12 h run and (

**b**) Input voltage for the cell running on intermittent sources.

**Figure 24.**(

**a**) Current drawn by the cell during 12 h of operation, (

**b**) hydrogen production and (

**c**) corresponding cell efficiency for the cell.

Energy Domain | Flow (f) | Effort (e) |
---|---|---|

Electrical | Current intensity (A) | Voltage (V) |

Fluidic | Volume flow rate (m${}^{3}$·s${}^{-1}$) | Pressure (Pa) |

Fluidic (Pseudo BG) | Mass flow (kg·s${}^{-1}$) | Pressure (Pa) |

Thermal | Entropy flow (J·K${}^{-1}$·s${}^{-1}$) | Temperature (K) |

Thermal (Pseudo BG) | Thermal flow (J·s${}^{-1}$) | Temperature (K) |

Chemical (Transformation) | Molar flow (mol·s${}^{-1}$) | Chemical potential (J·mol${}^{-1}$) |

Chemical (Kinetic) | Reaction flow rate (mol·s${}^{-1}$) | Chemical affinity (J·mol${}^{-1}$) |

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## Share and Cite

**MDPI and ACS Style**

Sood, S.; Prakash, O.; Boukerdja, M.; Dieulot, J.-Y.; Ould-Bouamama, B.; Bressel, M.; Gehin, A.-L.
Generic Dynamical Model of PEM Electrolyser under Intermittent Sources. *Energies* **2020**, *13*, 6556.
https://doi.org/10.3390/en13246556

**AMA Style**

Sood S, Prakash O, Boukerdja M, Dieulot J-Y, Ould-Bouamama B, Bressel M, Gehin A-L.
Generic Dynamical Model of PEM Electrolyser under Intermittent Sources. *Energies*. 2020; 13(24):6556.
https://doi.org/10.3390/en13246556

**Chicago/Turabian Style**

Sood, Sumit, Om Prakash, Mahdi Boukerdja, Jean-Yves Dieulot, Belkacem Ould-Bouamama, Mathieu Bressel, and Anne-Lise Gehin.
2020. "Generic Dynamical Model of PEM Electrolyser under Intermittent Sources" *Energies* 13, no. 24: 6556.
https://doi.org/10.3390/en13246556