# Bond Graph Modelling of a Solid Oxide Fuel Cell

Chapter

## 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

Ac

Effective cell area (m2)

cp, cv

Specific heat capacity at constant pressure and volume (J kg−1 K−1)

E

Activation energy (J mol−1)

F

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)

ne

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 (m3 kg−1)

V

Volume (m3)

$$\dot {V}$$

Volume flow rate (m3 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

$$\zeta_\mathrm{f} ,\zeta_\mathrm{o}$$

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

i

Inlet

o

Outlet

r

Reaction

ref

Reference state

0

Initial state

## References

1. 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.
2. 2.
Benson RS (1977) Advanced Engineering Thermodynamics, 2nd ed. Pergamon Press Limited, Oxford.Google Scholar
3. 3.
Bockris JO’M, Reddy AKN, Gamboa-Aldeco M (1998) Modern Electrochemistry: Fundamentals of Electrodics, 2nd ed. Kluwer/Plenum, Dordrecht.Google Scholar
4. 4.
Breedveld PC (1984) Physical Systems Theory in Terms of Bond Graphs. Ph.D. Thesis, Twente University, Enschede.Google Scholar
5. 5.
Callen HB (1985) Thermodynamics and an Introduction to Thermostatistics. Wiley, New York, NY.
6. 6.
Feenstra PJ (2000) A Library of Port-Based Thermo-Fluid Submodels. M.Sc.Thesis, University of Twente.Google Scholar
7. 7.
Karnopp DC, Margolis DL, Rosenberg RC (2006) System Dynamics: Modeling and Simulation of Mechatronic Systems, 4th ed. Wiley, Hoboken, NJ.Google Scholar
8. 8.
Mukherjee A, Karmakar R, Samantaray AK (2006) Bond Graph in Modeling, Simulation and Fault Identification. CRC Press, Boca Raton, FL.Google Scholar
9. 9.
Perelson AS (1975) Network thermodynamics, an overview. Biophys J 15: 667–685.
10. 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. 11.
Thoma J, Ould Bouamama B (2000) Modelling and Simulation in Thermal and Chemical Engineering. Springer, New York, NY.Google Scholar
12. 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. 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.
14. 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.
15. 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.
16. 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).
17. 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.
18. 18.
Zemansky MW, Dittman DH (1997) Heat and Thermodynamics. McGraw-Hill, Singapore.Google Scholar