1+1D Non-Isothermal and Two-Phase Transient Model of PEM Fuel Cells for Real-Time Estimation and Control
Authors: Goshtasbi Alireza , Pence Benjamin , Ersal Tulga 
Published: February 2020

In this work we present a dynamic model for PEM fuel cells with the overarching goal of enhancing their performance and durability by enabling real-time control applications such as online estimation and prediction of system states. The need for a dynamic model stems from the fact that PEM fuel cells undergo transient behaviors on multiple time scales, and these transients have significant influence on the fuel cell system performance and health. Some of the most important phenomena that affect fuel cell performance include: 1) Water crossover through, absorption into and desorption out of the membrane; 2) two-phase flow and flooding in the gas diffusion layers (GDLs); 3) two-phase flow and flooding in flow channels; and 4) transient spatially distributed temperature effects. Although there are several models in the literature that address one or some of these issues [1], taking all of these transient effects into account in a computationally efficient manner is still an open research question. Specifically, most of the models reported to date are developed for design purposes with no constraints on computational cost [2, 3]. Hence, such models do not lend themselves to real-time control applications. On the other hand, control-oriented models are also available in the literature, but they ignore most of the abovementioned physical phenomena to arrive at a computationally efficient formulation [4, 5]. Thus, these models suffer from insufficient predictive capability for water and thermal management and are applicable only in a narrow range of operating conditions.
To fill this gap, this work develops a 1+1 dimensional model that accounts for transport in both down-the-channel and through-the-membrane directions in a decoupled manner, thereby assuming weak coupling to achieve computational efficiency while maintaining the predictive capability of a 2-dimensional model.
The following phenomena are included in the model:
a) Transport of gas species in the flow fields is described through a set of ordinary differential equations.
b) Total pressure drop along the flow channels is modeled via a separated flow approach [6].
c) Transport of reactant gases inside the gas diffusion layers is modeled by Fickian diffusion.
d) Water exchange between the diffusion layers and flow channels in both liquid and vapor form is considered.
e) The membrane water content is approximated by a linear distribution based on the observation that the nonlinear diffusion model yields a linear distribution for membranes thinner than 30 microns.
f) The Butler-Volmer equation is used to describe the kinetics of reactions on the catalyst sites. The effects of diffusion and catalyst layer flooding are included in the voltage submodel.
g) The anode and cathode catalyst layers are modeled as thin interfaces on the membrane boundary and Henry’s law is used to estimate the gas species concentration reaching the active catalyst sites.
The model takes advantage of spatial decoupling, disparate time scales, and efficient numerical schemes to provide a faster-than-real-time solution. Specifically, the current Matlab implementation of the model is about twice faster than real-time. Our preliminary results indicate qualitative agreement with published experimental studies. Quantitative agreement will be investigated after a rigorous parameter identification study is carried out. Ongoing efforts also include experimental validation as well as further complexity reduction of the model.
In conclusion, the contribution of this work is twofold: 1) It provides a non-isothermal and two-phase transient model for PEM fuel cells that incorporates all of the phenomena that have been previously shown to affect the overall fuel cell performance; 2) through spatiotemporal decoupling it provides a computationally efficient solution that can be used for online applications.
References:
1. Promislow, K., et al., Two-phase unit cell model for slow transients in polymer electrolyte membrane fuel cells. Journal of The Electrochemical Society, 2008. 155(7): p. A494-A504.
2. Cao, T.-F., et al., Numerical investigation of the coupled water and thermal management in PEM fuel cell. Applied energy, 2013. 112: p. 1115-1125.
3. Shah, A., et al., Transient non-isothermal model of a polymer electrolyte fuel cell. Journal of Power Sources, 2007. 163(2): p. 793-806.
4. Pukrushpan, J.T., H. Peng, and A.G. Stefanopoulou, Control-oriented modeling and analysis for automotive fuel cell systems. Journal of dynamic systems, measurement, and control, 2004. 126(1): p. 14-25.
5. Philipps, S. and C. Ziegler, Computationally efficient modeling of the dynamic behavior of a portable PEM fuel cell stack. Journal of Power Sources, 2008. 180(1): p. 309-321.
6. Kandlikar, S.G., E.J. See, and R. Banerjee, Modeling Two-Phase Pressure Drop along PEM Fuel Cell Reactant Channels. Journal of The Electrochemical Society, 2015. 162(7): p. F772-F782.