The electrochemical models of fuel cells consider the maximum voltage possible (as a consequence of the chemical reaction within the fuel cell) and then subtract various current-dependent losses from this voltage. In particular, ohmic losses can occur due to the electrical resistance of the electrodes and electrolyte layer, activation losses can occur predominantly due to the kinetics at the electrodes, and concentration losses can occur as a consequence of the gas concentrations being reduced at the electrodes when consumed by the reaction. This result is an equation for the voltage across the fuel cell as a function of the current produced by the fuel cell. In this manner, the electrochemical fuel cell model is capable of capturing the behavior of the devices, but require the input of parameters that are experimentally obtained. The models, therefore, do not generally provide insights into how the properties of the fuel cells emerge from the device physics, but rather can be used to predict how the devices might behave in external (and potentially time-dependent) circuits.
We start by considering the open-circuit (or no-load) voltage. This is the maximum voltage that can be obtained in a fuel cell due to the reactions. In particular, the energy difference between the initial states of the reagents (H2 and 102) and the final state of the reagents (H2O) enable this voltage to be calculated through the Nernst expression. The maximum voltage is given by
RT f 1 1
En = Eo + — |ln(Pn2 ) + 2ln(Po2 )j (7.17)
where E0 is the maximum theoretical voltage (1.229 V), T is the temperature in Kelvin, R is the universal gas constant (8314 J K-1 mol-1), and F is the Faraday constant (96487 C). The voltage depends on the hydrogen pressure (in atm), Ph2, and the oxygen pressure (also in atm), Po2 at the anode and cathode, respectively.
The activation losses in voltage arise due to the kinetics at the electrodes. The voltage losses at the anode, however, are much lower than the voltage losses at the cathode and for fuel cells fed with pure hydrogen the voltage losses at the anode can be neglected. The activation voltage losses occur at relatively low currents in the fuel cell. The activation losses can be calculated from the following equation:
where I is the current in the fuel cell, the constants a are phenomenological and, while they may be derived from equations that capture aspects of the
fuel cells kinetic, thermodynamic and electrochemical behavior, they can also be obtained experimentally. The oxygen concentration is also included in the above equation as
where PO2 is the pressure of the oxygen and T is temperature.
As fuel is consumed at an electrode, the concentration of fuel neighboring the electrode will be reduced. It is not possible for the fuel cell concentration to be instantaneously replenished, and this can result in a voltage loss at high currents. In other words, at high currents the fuel is being consumed at the electrodes at a higher rate, and because it cannot be replenished as quickly there is a drop in voltage. The loss of voltage as a consequence of the concentration gradient near the electrodes is of the form
where B is a constant, which can’t be obtained from theoretical considerations, and it is more accurate to allow this constant to be fitted. The current, IL, is the limit current. This represents the point at which the voltage will abruptly decline.
The last mechanism by which voltage can be lost in the fuel cell is through the ohmic resistance of the polymer electrolyte layer (to the ionic flux) and the electrical resistance of the electrodes. The voltage drop as a consequence of ohmic losses is
Vohm — RsI (7.21)
where Rs is the combined membrane and electrodes resistance.
Combining the above equations results in the voltage across the cell as a function of the current through it. In other words, the fuel cell’s voltage is
Vcell — nc (EN Vactiv Vconc Vohm) (7.22)
where nc accounts for the number of cell in series. In other words, the fuel cell’s voltage can now be calculated as a function of the open circuit voltage and the current-dependent voltage drops. We now calculate these voltages using a simple spreadsheet model.
Figure 7.6 depicts the spreadsheet implementation of an electrochemical model of a fuel cell. At the top of the spreadsheet is the constants required for the model (which are taken directly from the literature). The concentration of O2 is calculated from the pressure of the oxygen, PO2, using the equation above. In column B is the current, which the fuel cell’s voltage will depend upon. The maximum voltage, EN, calculated from the Nernst expression, is included in column C and the code in cell C16 is
=$C$3 + (($C$4*$F$3)/(2*$C$5)) * (LN($F$4) + 0.5*LN($F$5))
This does not depend on the current through the fuel cell and is a constant as a function of current, going down the column. In Column D the spreadsheet calculates the voltage drop due to activation losses and the code in cell D16 is
=-($C$7 + $C$8*$F$3 + $C$9*$F$3*LN($F$7) + $C$10*$F$3*LN(B16))
where it is the last term which is current-dependent (referencing cell B16). The voltage drop due the concentration gradient at the electrodes is calculated in column E. The code in cell E16 is
which results in the voltage dropping off at high currents. In column F are the ohmic losses in voltage, which are simply proportional to the current and, therefore, the code in cell F16 is simply
Finally, the voltage across the entire fuel cell is calculated in column G with the code in cell G16 being
and consisting of the open cell voltage minus the current-dependent losses.
A plot of voltage as a function of current for the fuel cell is depicted in Figure 7.7. In particular, the maximum open-circuit voltage is plotted across the top and depicted by the dashed line. Next, the open-circuit voltage minus the voltage drop arising from activation losses is plotted as the more spaced out dotted line. The effects of concentration losses are considered next as the the open-circuit voltage minus the voltage drop arising from activation losses and concentration losses are plotted as the closely spaced dotted line. Finally, the voltage across the entire fuel cell (open circuit voltage minus voltage drops from all losses) is plotted as a solid line. In this manner, the voltage drop due to the different loss mechanisms are clearly observable. In particular, the ohmic losses can be seen to linearly increase with current, while the concentration losses occur predominantly at higher currents. Therefore, the model can be seen to capture the behavior of the fuel cell and predict how the fuel cell will operate under variable conditions.