(450f) Asymptotically Reduced Zero-Dimensional Model for an All-Vanadium Redox Flow Battery

Recently, mathematical modeling and numerical simulations have come to play an important part in the research and development of all-vanadium redox flow batteries (VRFBs). Detailed mechanistic models consider conservation of mass, momentum, species, energy and charge transfer in the various layers (current collectors, electrodes, and membrane) and thus result in a computationally intensive system of coupled partial differential equations (PDEs). The associated computational cost for a system of PDEs typically decreases as the number of spatial dimensions is decreased. Hence, the lower dimensional models would be more suitable for wide-ranging parametric studies of a single cell, stack or system, real-time optimization and feedback control. There is thus a need to derive reduced dimensional models that can predict the cell behavior at a minimal computational cost while preserving the essential physics.

In this work, we present a reduced zero-dimensional model that both preserves essential physics and reduces computing time. The methodology uses scale analysis and asymptotic reduction. The model reduction from 3D to 2D is straightforward and does not require detailed analysis. The porous nature of the carbon felts and the membrane and the solid nature of the current collector allow for a reduction from three to two dimensions because changes in the dependent variables in the spanwise direction are negligible; hence a 2D model ought to be able to capture the behavior of a 3D cell. However, further reduction in the dimensionality, i.e., 2D to 1D or 0D needs to be carefully analyzed. In this regard, we carry out scale analysis of a 2D model accounting for conservation of species and charge in a VRFB to derive a reduced 0D model. The conditions when a 0D model can capture the behavior of a 2D cell are identified in terms of two non-dimensional numbers and their limits. We verify the reduced model by comparing its charge-discharge curve predictions with that of a full two-dimensional model. The reduced model takes the form of an explicit algebraic equation and can be implemented in a high level programming language such as Fortran or C with a minimum computation overhead, i.e., only few kilobytes of memory and on the order of milliseconds for the complete charge-discharge curve. The proposed analysis leading to reduction in dimensionality is generic and can be employed for other types of redox flow batteries.