Bond Graph Modelling of Engineering Systems pp 355-382 | Cite as

# Bond Graph Modelling of a Solid Oxide Fuel Cell

## Abstract

Fuel cells are environmentally friendly futuristic power sources. They involve multiple energy domains and hence bond graph method is suitable for their modelling. A true bond graph model of a solid oxide fuel cell is presented in this chapter. This model is based on the concepts of network thermodynamics , in which the couplings between the various energy domains are represented in a unified manner. The simulations indicate that the model captures all the essential dynamics of the fuel cell and therefore is useful for control theoretic analysis.

## Keywords

Solid oxide fuel cell Bond graph Network thermodynamics Electrochemical reaction Fuel utilization## Notation

*A*_{c}Effective cell area (m

^{2})*c*_{p},*c*_{v}Specific heat capacity at constant pressure and volume (J kg

^{−1}K^{−1})*E*Activation energy (J mol

^{−1})*F*Faraday’s constant (C mol

^{−1})*G*Gibbs free energy (J)

*h*Specific enthalpy (J kg

^{−1})*H*Enthalpy (J)

*i*Current (A)

*K*Valve coefficient (m s)

*m*Mass (kg)

- \(\dot {m}\)
Mass flow rate (kg s

^{−1})*M*Molar mass (g)

*n*Number of moles (mol)

*n*_{e}Number of electrons participating in the reaction

*p*Pressure (N m

^{−2})*R*Specific gas constant (J kg

^{−1}K^{−1})- R
Universal gas constant (J mol

^{−1}K^{−1})*s*Specific entropy (J kg

^{−1}K^{−1})*S*Entropy (J K

^{−1})- \(\dot {S}\)
Entropy flow rate (J K

^{−1}*s*^{−1})*T*Temperature (K)

*u*Specific internal energy (J kg

^{−1})*U*Internal energy (J)

*v*Specific volume (m

^{3}kg^{−1})*V*Volume (m

^{3})- \(\dot {V}\)
Volume flow rate (m

^{3}s^{−1})*w*Mass fraction

*x*Valve stem position (m)

*ν*Stoichiometric coefficient

*η*Over-voltage (V)

*μ*Chemical potential (J kg

^{−1})*ψ*Pre-exponential coefficient (A m

^{−2})*ξ*Reaction advancement coordinate (mol)

- \(\zeta_\mathrm{f} ,\zeta_\mathrm{o} \)
Fuel and oxygen utilisations

*β*Charge transfer coefficient

*λ*Convection heat trans. coefficient (J m

^{−2}s^{−1}K^{−1})

## Subscripts

- ai
Anode side inlet

- an
Anode

- ao
Anode side outlet

- act
Activation

- AS
Air source

- b
Bulk

- ca
Cathode

- ci
Cathode side inlet

- co
Cathode side outlet

- conc
Concentration

- d
Downstream side

- ENV
Environment

- gen
Generated

- H
Hydrogen gas

- HS
Hydrogen source

- I1
Interconnect on anode side

- I2
Interconnect on cathode side

- L
Limiting

- M
Membrane electrode assembly

- N
Nitrogen gas

- ohm
Ohmic

- O
Oxygen gas

- PL
Polarisation losses

- r
Reaction

- TPB
Triple phase boundary

- u
Upstream side

- W
Water vapour

## Superscripts

- i
Inlet

- o
Outlet

- r
Reaction

- ref
Reference state

- 0
Initial state

## Notes

### Acknowledgments

The first author would like to acknowledge Prof. Moses Tadé, Dean of Engineering, Curtin University of Technology, for kindly permitting him to write this chapter during his stay as a research associate at the University.

## References

- 1.Aguiar P, Adjiman CS, Brandon (2004) Anode-supported intermediate-temperature direct internal reforming solid oxide fuel cell I. Model-based steady-state performance. J Power Sources 138: 120–136.CrossRefGoogle Scholar
- 2.Benson RS (1977) Advanced Engineering Thermodynamics, 2nd ed. Pergamon Press Limited, Oxford.Google Scholar
- 3.Bockris JO’M, Reddy AKN, Gamboa-Aldeco M (1998) Modern Electrochemistry: Fundamentals of Electrodics, 2nd ed. Kluwer/Plenum, Dordrecht.Google Scholar
- 4.Breedveld PC (1984) Physical Systems Theory in Terms of Bond Graphs. Ph.D. Thesis, Twente University, Enschede.Google Scholar
- 5.Callen HB (1985) Thermodynamics and an Introduction to Thermostatistics. Wiley, New York, NY.zbMATHGoogle Scholar
- 6.Feenstra PJ (2000) A Library of Port-Based Thermo-Fluid Submodels. M.Sc.Thesis, University of Twente.Google Scholar
- 7.Karnopp DC, Margolis DL, Rosenberg RC (2006) System Dynamics: Modeling and Simulation of Mechatronic Systems, 4th ed. Wiley, Hoboken, NJ.Google Scholar
- 8.Mukherjee A, Karmakar R, Samantaray AK (2006) Bond Graph in Modeling, Simulation and Fault Identification. CRC Press, Boca Raton, FL.Google Scholar
- 9.Perelson AS (1975) Network thermodynamics, an overview. Biophys J 15: 667–685.CrossRefGoogle Scholar
- 10.Samantaray AK, Mukherjee A (2006) Users Manual of SYMBOLS Shakti. (High-Tech Consultants, STEP, Indian Institute of Technology, Kharagpur, <http://www.htcinfo.com/>)
- 11.Thoma J, Ould Bouamama B (2000) Modelling and Simulation in Thermal and Chemical Engineering. Springer, New York, NY.Google Scholar
- 12.Vijay P (2009) Modelling, Simulation and Control of a Solid Oxide Fuel Cell System: A Bond Graph Approach. Ph.D. Thesis, Indian Institute of Technology, Kharagpur, India.Google Scholar
- 13.Vijay P, Samantaray AK, Mukherjee A (2008) Bond graph model of a solid oxide fuel cell with a C-field for mixture of two gas species. Proc IMechE, Part I: J Syst Control Eng 222(4): 247–259.CrossRefGoogle Scholar
- 14.Vijay P, Samantaray AK, Mukherjee A (2009) On the rationale behind constant fuel utilization control of solid oxide fuel cells. Proc IMechE, Part I: J Syst Control Eng 223(2): 229–252.CrossRefGoogle Scholar
- 15.Vijay P, Samantaray AK, Mukherjee A (2009) A bond graph model-based evaluation of a control scheme to improve the dynamic performance of a solid oxide fuel cell. Mechatronics 19(4): 489–502.CrossRefGoogle Scholar
- 16.Vijay P, Samantaray AK, Mukherjee A (2010) Constant fuel utilization operation of a SOFC system: An efficiency viewpoint. Trans ASME J Fuel Cell Sci Technol 7(4): 041011 (7 pages).CrossRefGoogle Scholar
- 17.Vijay P, Samantaray AK, Mukherjee A (2010) Parameter estimation of chemical reaction mechanisms using thermodynamically consistent kinetic models. Comput Chem Eng 34(6): 866–877.CrossRefGoogle Scholar
- 18.Zemansky MW, Dittman DH (1997) Heat and Thermodynamics. McGraw-Hill, Singapore.Google Scholar