Analytical Solutions of the Nernst-Planck-Poisson Equations and Its Application for Protonic Ceramic Fuel and Electrolysis Cells

Proton conducting ceramics have emerged as a significant alternative for the sustainable production of chemicals and electrical energy due to their versatility and low precious metal content in their construction. One of the major advantages of this technology is its intermediate operating temperature range (400 – 700 °C), which allows the material to be used for several applications, including fuel cells, electrolyzers, separation, and electrochemical compression, under various gas compositions. These benefits, however, come at a cost. To achieve optimal performance, these materials must transport several point mobile defects, such as protons, polarons (or electron holes), and oxygen vacancies, whose behavior can vary considerably depending on the operating conditions. Consequently, modeling the transport and electrochemical behavior of this class of materials can be challenging. This work presents a novel method for modeling electrochemical cells using barium zirconate-based ceramics, leveraging analytical solutions of the Nernst-Planck-Poisson equations in addition to empirical equations published in the open literature for the cell activation overpotentials. These solutions enable the quick and efficient calculation of defect and gas fluxes under diverse operating conditions. Such analytical solutions can be used to study the transport of mobile point defects (protons, vacancies, and polarons). They can also be extended to include more charge carriers if the material is well-characterized. Furthermore, these solutions illustrate the impact of defect transport properties on the voltage drop across the electrolyte. To ensure the robustness of the analytical solutions, the derived equations were compared with more computationally intensive numerical solutions available in the literature, demonstrating excellent agreement. Figure 1 shows the impact of operating conditions on key cell characteristics, including polarization, faradaic efficiency, and energy efficiency as elucidated by our model. The figure shows the cell energy efficiency of a BaCe₀.₇Zr₀.₁Y₀.₁Yb₀.₁O₃₋δ (BCZYYb711) electrolyte-based cell as a function of temperature and current density. The cell uses a Ni-BCZYYb711 negatrode and a positrode consisting of a mixture of 80% BCFZY and 20% BCZYYb711. The fuel gas at the negatrode is a 50/50 mixture of hydrogen and steam, while the oxidant at the positrode is a 50/50 mixture of steam and oxygen in molar basis. The cell operates at 1 atm. This figure illustrates the complex interplay between defect transport properties, cell geometry, and operating conditions. An optimal temperature region exists for low current densities. As current density increases, the combined activation and ohmic overpotential rises, requiring a higher power density for electrolysis operation and thus decreasing cell efficiency. For fuel cell operation, the increase of ohmic and activation overpotential decreases the available power from the cell also resulting in an efficiency decrease. Beyond cell polarization characteristics, faradaic efficiency also significantly impacts cell energy efficiency. As temperature increases, the polaron flux across the membrane increases due to the exponential relationship between temperature and the diffusivity of charged species. This increased polaron flux reduces the proton flux, consequently lowering the faradaic efficiency. Finally, the figure shows negative energy efficiencies near zero current density in electrolysis mode. This occurs because, at low current densities, charge carrier transport within the electrolyte is dominated by diffusion, and under these conditions, the proton flux can be in the opposite direction to the applied bias. This simplified analytical approach provides valuable insights into the operation of protonic ceramic cells in both fuel cell and electrolysis modes, identifying optimal operating points as a function of these conditions. While demonstrated using barium zirconate-based materials, our model is readily adaptable to other ceramic materials, provided their transport and thermodynamic properties are adequately characterized. This versatility makes our model a powerful tool which can be easily implemented, for example using a spreadsheet software such as excel, for the development and optimization of proton-conducting ceramics across a broad spectrum of applications. Figure 1