Numerous books on SOFC modeling present equivalent circuits that are useful for incorpo- rating the charge double layer into fuel cell models. O’Hayre, et al., for instance, [43] discuss an equivalent circuit model based (qualitatively) on electrochemical impedance spectroscopy data. In this circuit, the charge double layer is represented as a capacitor, the ohmic resis- tance and reaction kinetics are both represented as resistors, and species diffusion through the electrodes is represented as a Warburg element. The properties of each of these ele- ments are obtained from Nyquist plots. A Nyquist plot displays the fuel cell’s impedance in the complex plane. Each equivalent circuit element (or polarization) produces a unique pattern on the Nyquist plot. Ohmic resistance, for instance, appears as a single point on the Nyquist plot. Hence, its value is simply read off of the real axis. Activation kinetics and the charge double layer, on the other hand, appear together as semi-circles on the Nyquist plot. The Warburg element may appear linear on the Nyquist plot, if the diffusion thickness is sufficiently large, or it may appear circular, if the diffusion thickness is relatively small.
Nehrir and Wang [64], Gemmen [65], and Larminie and Dicks [26] present similar equivalent circuit models. These references, however, do not provide as extensive of a discussion on electrochemical impedance spectroscopy data as that found in O’Hayre, et al. [43].
Prior modeling studies have investigated the charge double layer. Qi, et al. [66] developed a state-space SOFC model that incorporated the charge double layer via an equivalent cir- cuit. The model tracked changes in the current, operating voltage, reaction rates, and species diffusion rates on the millisecond timescale in response to step changes in the load resistance and species partial pressures. The authors found that the ohmic resistance responded in- stantaneously to a step change in the load resistance. The charge double layer, on the other hand, exhibited a slower (but still millisecond-scale) response. The authors also investigated the influence of diffusion resistance through the electrodes and boundary layers on SOFC
performance. The authors found that increasing the diffusion layer thickness from 1 mm to 3 mm significantly increased the diffusion rate’s settling time due to increased concentration loss. As expected, increasing the diffusion layer thickness also decreased the SOFC’s output voltage. Qi, et al. [67] used the same equivalent circuit in a tubularSOFCmodel (discretized axially and radially) to investigate the responses of the fuel cell’s operating voltage, current, and exit gas properties (i.e, exit temperature, pressure, composition, and flow velocities) to step changes in the load resistance and inlet gas properties. The authors found that the charge double layer, again, diminished within milliseconds. The authors also found that the inlet flow velocity minimally influenced the fuel cell’s dynamic response, whereas the inlet pressure and temperature had a larger influence.
Wang and Nehrir [68] developed a lumped-parameter SOFCmodel that tracked changes in the operating voltage on millisecond, second, and minute timescales in response to step changes in the current. These authors used an equivalent circuit to combine the charge double layer with the ohmic, activation, and concentration polarizations. The authors varied the charge double layer capacitance value between 0.4 F and 4 F. Similar to Qi, et al. [66,67], these authors found that the double layer polarization settled in a span of milliseconds following a step change in load. The mass flow dynamics, on the other hand, settled in a span of seconds following a step change in load. The thermal dynamics lasted the longest, settling on the minute timescale. Wang and Nehrir [69] furthermore experimentally verified
a similar equivalent circuit model. The authors compared their results to those obtained from an SR-12 Avista Labs PEMfuel cell. The authors imposed step changes in current and tracked the operating voltage on short (electrochemical) and long (thermal) timescales. The simulation results agreed reasonably well with the experimental results in terms of settling times.
The foregoing studies indicate that electrochemical dynamics tend to diminish within milliseconds following a load change. Few (if any) studies, however, have investigated this assumption across a wide range of operating conditions. Many studies have investigated the dynamic response of an SOFC on second and minute timescales, but these fuel cell models do not include the charge double layer [70–81]. The present study investigates electrochemi- cal dynamics under various operating conditions to determine if the electrochemical settling time could possibly last seconds (or longer). It bears mentioning that the present model is a macroscale model that incorporates the charge double layer (a microscale phenomenon) via an equivalent circuit (a macroscale representation). As such, the present model does not capture the same level of detail as a microscale model, particularly in terms of elementary reaction chemistry, mass transfer through the PEN structure, and electric potential distri- butions. (Further information on these phenomena may be found in Refs. [82–86], among others). Nevertheless, the use of an equivalent circuit permits (computationally) investiga- tion into the dynamic behavior of the charge double layer under a wide range of operating conditions, involving not only dynamic electrochemistry but also dynamic mass flow, energy, and momentum balances, and on longer-than-usual timescales (greater than milliseconds, which is the charge double layer’s characteristic time). Thus, the present model is consid- ered suitable for the task at hand.