The model that is outlined in Chapter 4 can be extended to a more complete three-dimensional description of the PEMFC. Since the validity of the model ap-proach is shown, the model concept can be transferred to the three-dimensional case. However, time-dependent three-dimensional models that contain all coupled transport phenomena require large computational resources. In this context, the scaling analysis of the model equations can predict important characteristic quan-tities, such as current density and temperature profiles, prior to numerical compu-tations. Moreover, scaling analysis indicates which phenomena can be considered decoupled without a significant loss of accuracy.
The mathematical description of the liquid water transport has been transferred from the field of ground water flow. The porous media investigated in this field have much larger typical length scales than the GDLs and the electrodes of a PEMFC. In particular, the two-phase flow properties of the fuel cell components require detailed investigation.
The capillary pressure-saturation relation for the gas diffusion layers and cata-lyst layers commonly used in PEMFCs have to be determined. The functional relationship that describes the dependence of the capillary pressure on the liquid water saturation is specific to a certain material. Hence, the use of standard litera-ture approaches reduces the predictive power of fuel cell models. Furthermore, the drainage and imbibition of a structure with irregular pore geometry is expected to cause a hysteresis in the capillary pressure-saturation curve that is not captured by current model approaches. The gas diffusion layer can be considered as a random fiber material that is surface-treated with Teflon. In the mathematical description of the capillary pressure the contact angle of the layer is used. The contact angle
is defined by the Laplace equation that is valid for a regular pore geometry and usually measured on smooth surfaces. Since the pores of the GDL have irregular shape and the coating with Teflon is inhomogeneous, the use of a single contact angle cannot describe the surface properties inside the porous structure accurately.
The catalyst layer contains ionomer and platinum catalyst particles, usually on car-bon support. Thus the catalyst layer may have the property of mixed wettability, i.e. it can neither be considered to be simply hydrophilic nor hydrophobic. One could argue that the two-phase flow properties of the catalyst layer are of minor importance compared to the gas diffusion layer which is usually up to 30 times thicker than the catalyst layer. This is not the case, since the state of the catalyst layer determines the properties of the membrane. The membrane in turn is ex-tremly important for the water distribution within the fuel cell. The catalyst layer is commonly modeled as a separate layer adjacent to the membrane on one side and to the gas diffusion layer on the other side. However, the polymer conducting phase of the membrane extends into the catalyst layer, resulting in an irregular interface rather than a well-defined boundary. Moreover, the polymer phase tends to swell due to the uptake of water, which gives rises to dynamic behavior on the part of the interface. Refined fuel cell models should consider this fact.
Recently developed gas diffusion media contain a so-called microporous layer that is inserted between the GDL and the catalyst layer. While experimental re-sults indicate that the microporous layer can improve the durability and the per-formance of PEMFCs, the dynamic water transport properties are changed signif-icantly. Hence, current empirical control approaches that have been developed for fuel cells without a microporous layer are insufficient and require major modifi-cations. Accordingly, future models should integrate the microporous layer into the mathematical description.
When modeling the phase transition of water, evaporation and condensation rate constants are commonly used. Rather than using constants, more sophisticated models should contain a functional dependence of the evaporation and condensa-tion rates. Condensacondensa-tion, for example, occurs on pre-existing liquid surfaces and, hence, the surface energy influences the condensation rate. Furthermore, the im-pact of the temperature, the average gas velocity, and the relative humidity should be studied in detail.
Furthermore, the physical phenomena at the interface between the GDL and the open gas channel require clarification. During operation, liquid water is generated in the fuel cell and transported through the GDL. During experiments, it has been observed in transparent fuel cells that droplets emerge and grow at the surface of the GDL. These droplets are eventually removed by the convective gas flux in the channels. Fluid dynamic calculations clarify some of the properties associ-ated with the transport of droplets through the channels. For example, the moving droplets tend to accumulate at the bends of the flowfield due to the inhomogeneous
velocity and pressure field of the gas flux. However, it is an unresolved question how to couple the liquid water transport equations in the GDL with the convective transport processes in the open gas channel without improper simplification.
6.3 Dynamic fuel cell stack modeling
Modeling results and validation
An analysis of the dynamic behavior of a PEMFC stack is given based on a novel stack model. The mathematical formulation of the stack model is a coupled differ-ential algebraic equation system. Ordinary differdiffer-ential equations in time describe the transport phenomena, and the oxygen reduction at the cathode is modeled by an algebraic relation. The model is physically detailed and considers the impor-tant couplings among the transport phenomena and the electrochemical reaction.
At the same time, the model is computationally efficient.
The realistic simulation results show that the modeling approach is appropriate for the dynamic description of a PEMFC stack. The model can predict the stack voltage and the molar fluxes of hydrogen, oxygen, and water vapor, given an arbi-trary load profile. Moreover, the temperature of the off-gas and, most important, of the stack itself is calculated. The stack temperature is coupled with all the other phenomena that occur in a fuel cell stack. For example, the protonic conductivity is treated as a function of the temperature and its evolution is calculated.
A thorough understanding of the dynamic response of a PEMFC stack is crucial for the operational control of integrated fuel cell systems. The stack model is suitable for control applications, since the convergence behavior is excellent, as shown by the simulation of a current-step-profile. This model property allows one to examine a wide parameter range and to simulate different operating scenarios.
Moreover, the model is computationally efficient. In a computing time of less than one second the dynamic response of the stack upon varying load can be predicted for an operation time of more than one hour.
The stack temperature is one of the most important parameters during operation.
Accordingly, the first step of the validation was to compare the simulated with the measured stack temperature. A PEMFC stack constructed for the use in portable applications was operated under constant load. The input parameters of the model were controlled by the software of the test stand. Agreement between the simula-tion results and the measurement is excellent. For the validasimula-tion of the electrical model a comparison of simulated and measured stack voltage is given. Deviations between the predictions of the model and the experimental results are explained by the generation and accumulation of liquid water in the stack.
Model applications
The stack model is able to identify the dominant time-dependent physical pro-cesses under changing load conditions and for different operating regimes. For example, the influence of the humidity of the inlet gases, or the heat flux that should be removed via the surface of the stack, can be simulated. By integrating the stack model into a system simulation environment, it can be utilized to study the dynamic interaction between the fuel cell stack and its peripheral components like pumps, fans, and valves. From the results of such a study, guidelines for the optimization of fuel cell systems can be drawn up. The most important applica-tion is the model-based development of a control algorithm. Currently, the control of fuel cell systems is usually managed by algorithms that are based on empiri-cal observations and practiempiri-cal experience. By developing a control scheme that is based on a physical fuel cell stack model the performance of fuel cell systems can be improved.