ENERGY 201C
Spring ‘25
Supplemental Note for HW#3: Problem 2
These notes are adapted from the book “Control of Fuel Cell Power Systems: Principles, Modeling, Analysis and Feedback Design by J. T. Pukrushpan, A. G. Stefanopolou and H. Peng” [1]. The notation used in the equations is updated to match the notation in the lecture slides.
The irreversible fuel cell voltage can be determined with the following relation:
where the individual terms are provided as follows.
1. Nernst potential, E
As shown in the lecture notes, the reversible voltage of the hydrogen fuel cell is:
This is known as “Nernst Potential” of a hydrogen fuel cell. The Gibbs free energy changes at standard temperature and pressure (STP) (25oC and 1 bar) gives a reference potential of 1.229 V.
Using thermodynamic values of the standard-state entropy change, the expression can be rewritten as:
where
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E
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Nernst potential [V]
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Tfc
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Stack temperature [K]
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PH2
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Hydrogen partial pressure [atm] (1 atm = 101325 Pa)
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Po2
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Oxygen partial pressure [atm]
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2. Activation overvoltage, ηact
This term maybe be estimated by using the Tafel equation, which relates the current density to the activation overvoltage. However, better results can be obtained by using the following correlation obtained from empirical data on a Nafion 117 membrane:
where η0 is the voltage drop at zero current density in [V], and ηa in [V] and c1 are constants. The parameters in the expression above are:
where
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ηact
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Activation overvoltage [V]
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i
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Current density [A/cm2]
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Tfc
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Stack temperature [K]
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pca
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Cathode pressure [bar]
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poz
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Oxygen partial pressure [bar]
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psat
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Water saturation pressure [bar]
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The saturation pressure of water as a function of temperature [2] is given by:
In the formula above, psat is in [kPa].
3. Ohmic overvoltage, ηohm
The Ohmic overvoltage is proportional to the membrane resistance:
where Rohm is the internal specific resistance [Ω .cm2]. The value for this parameter can be determined as follows:
where tm is the membrane thickness [cm]. The membrane conductivity σm is a function of the membrane water content λm and the temperature Tfc as in the following:
Here, we can assume that the water content of the membrane is equal to the relative humidity of the cathode and it varies in the range of 0 to 14 corresponding (proportionally) to 0 to 100% RH respectively.
4. Concentration overvoltage, ηcon
An approximation of the concentration overvoltage [V] to match experimental data is:
where c2, c3 and imax are constants that depend on the temperature and reactant partial pressure and can be determined empirically.
where i is the current density in [A/cm2] and Tfc is the temperature in [K].
References
[1] J. T. Pukrushpan, A. G. Stefanopoulou, and H. Peng, “Control of fuel cell power systems: principles, modeling, analysis and feedback design”, Springer Science & Business Media, 2004.
[2] J.C. Amphlett, R.M. Baumert, R.F. Mann, B.A. Peppley, and P.R. Roberge, “Performance modeling of the Ballard Mark IV solid polymer electrolyte fuel cell”, Journal of Electrochemical Society, 142(1):9-15, 1995.