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Heat and Mass Transfer in Fuel Cells and Stacks

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50 Years of CFD in Engineering Sciences

Abstract

A review is presented of the evolution of heat and mass transfer in modern fuel cells, and the role of computational fluid dynamics in the prediction of their performance. Both polymer electrolyte and solid oxide fuel cells are considered. The mathematical details of the mass transfer driving force and the transferred substance state, as well as the distributed resistance analogy, are derived. It is shown how the transferred substance state may be used to prescribe generalised convection–diffusion boundary conditions (inlet/outlet/wall). The combination of the mass transfer driving force and the application of the distributed resistance analogy concept to fuel cell stack models are explained in detail. In addition to the governing equations for thermofluids, the mathematical modelling of fuel cells requires additional thermodynamic, electrochemical kinetic and electric considerations to be taken into account (physicochemical hydrodynamics). Moreover, the results of original research conducted over two decades and culminating in very recent results are presented and explained. This work is research-in-motion and some future possibilities are outlined in the conclusion.

Submitted to: 50 Years of CFD in Engineering Sciences. A Commemorative Volume in Honour of D. Brian Spalding. Springer.

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Notes

  1. 1.

    It is perhaps a pity that the first symbol to be introduced should be so elaborate as \(\dot{m}^{{\prime \prime }}\); at any rate, an immediate explanation of the symbol is called for. The dot and the dashes have the significances given to them by Jakob (1949), who introduced them into heat transfer work: the superscribed dot signifies “per unit time”; one dash signifies “per unit length”; two dashes signify “per unit area”; and three dashes signify “per unit volume”. In accordance with this convention, we elsewhere employ the symbol \(\dot{m}^{{\prime }}\) for mass flow rate per unit width of a fluid film, and the symbol \(\dot{m}_{j}^{{{\prime \prime \prime }}}\) for the mass rate of generation of chemical substance j per unit volume due to a chemical transformation; similarly \(\dot{q}^{{\prime \prime }}\) will be used for the rate of heat transfer per unit area.

    D. B. Spalding

  2. 2.

    Sometimes a geometric factor G pre-multiplies C, S = GC(V – yp), depending on the patch type (volume, area, etc.). For convenience, it is excluded here.

  3. 3.

    In the interests of readability indices for the volume fractions, r = rk, and state-variables of each ‘space’ have been excluded.

Abbreviations

A :

Cell area, m2

b :

Blowing parameter

B :

Spalding number

C :

Source term coefficient, [units of solved variable]

c :

Specific heat, J/(kg K)

D h :

Hydraulic diameter, m

E :

Nernst potential, V

E act :

Activation energy, J/mol

F:

Faraday’s constant, 96,485 C/mol

F :

Distributed resistance, kg/(m2 s)

f :

Friction factor

g :

Mass transfer coefficient, kg/(m2 s)

g* :

Zero-flux mass transfer coefficient, kg/(m2 s)

H :

Height, m

H fg :

Heat of evaporation, J/mol

h :

Heat transfer coefficient, W/(m2 K)

\(i^{{\prime \prime }}\) :

Current density, A/m2

\(i_{0}^{{\prime \prime }}\) :

Exchange current density, A/m2

\(j^{{\prime \prime }}\) :

Diffusion flux, kg/(m2 s)

k R :

Reaction rate, m/s

L :

Length, m

M :

Molecular weight, kg/mol

\(\dot{m}^{{\prime \prime }}\) :

Mass flux, kg/(m2 s)

P:

Péclet number

p :

Pressure, N/m2

Q:

Reaction quotient, [arbitrary]

R:

Universal gas constant, 8.31446 J/(mol K)

R :

Resistance, Ohm m2

Re:

Reynolds number

r :

Volume fraction

\(\dot{S}\) :

Source term, [units of solved variable] × kg/s

\(\dot{S}^{{\prime \prime }}\) :

Source term, [units of solved variable] × kg/(m2 s)

s :

Saturation

Sh:

Sherwood number

T :

Temperature, K

t :

Time, s

\({\user2{U}}\) :

Superficial velocity, m/s

\({\user2{u}}\) :

Interstitial velocity, m/s

u :

Velocity, m/s

V :

Cell voltage, V, source term value, kg/s

x :

Displacement, m, mole fraction

y :

Mass fraction

z :

Charge number

α:

Charge transfer coefficient, volumetric heat transfer coefficient W/(m3 K)

Γ:

Exchange coefficient, kg/(m s)

γ:

Reaction order

δ:

Cell half-width, m

η:

Overpotential, V

κ:

Permeability, m2

μ:

Dynamic viscosity, kg/(m s)

ν:

Kinematic viscosity, m2/s

ρ:

Density, kg/m3

τ:

Tortuosity

Φ:

Polarisation

b :

Bulk

e :

Electrode

eff:

Effective

g :

Gas

H2:

Hydrogen

H2O:

Water

int:

Interface

l :

Liquid

O2:

Oxygen

NB:

Neighbour value

t :

Transformed substance state

P :

Nodal value

w :

Wall

0:

Ambient, external

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Author information

Authors and Affiliations

  1. Forschungszentrum Jülich GmbH, Institute of Energy and Climate Research, IEK-3, 52425, Jülich, Germany

    S. B. Beale, S. Zhang, M. Andersson & W. Lehnert

  2. Mechanical and Materials Engineering, Queen’s University, Kingston, ON, K7L 3N6, Canada

    S. B. Beale & J. G. Pharoah

  3. Department of Energy Sciences, Lund University, 22100, Lund, Sweden

    M. Andersson

  4. Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, UK

    R. T. Nishida

  5. Modeling in Electrochemical Process Engineering, RWTH Aachen University, 52425, Aachen, Germany

    W. Lehnert

  6. JARA-HPC, 52425, Jülich, Germany

    W. Lehnert

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Correspondence to S. B. Beale .

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  1. ACRI, The CFD Innovators, Los Angeles, CA, USA

    Akshai Runchal

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Beale, S.B., Zhang, S., Andersson, M., Nishida, R.T., Pharoah, J.G., Lehnert, W. (2020). Heat and Mass Transfer in Fuel Cells and Stacks. In: Runchal, A. (eds) 50 Years of CFD in Engineering Sciences. Springer, Singapore. https://doi.org/10.1007/978-981-15-2670-1_14

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