Lithium-ion battery has been widely used in many industrial applications. For better understanding of the performance of lithium-ion batteries, various numerical models have been developed. These models can be categorized into two groups: empirical models and the physics-based models. In empirical models, electrochemical physics is not resolved. Instead, correlations are established by curve-fitting experimental data of a battery. In physics-based models, the electrochemical kinetics and transport phenomena are considered and the models can produce more accurate prediction than empirical models. All these electrochemical models are proposed by electrochemical engineers and are used to predict a battery's electric performance such as charging/discharging characteristics. These models are usually zero dimensional (e.g., equivalent circuit models) or one dimensional (e.g., Newman's P2D model).

In many battery applications such as electric vehicles, batteries are connected in parallel and/or in series to form large format of modules or packs. Thermal management becomes a critical issue in the design of such systems. To perform thermal analysis, electrochemical models developed by electrochemical engineers like the ones mentioned in the previous paragraph need to be coupled in the simulation since heat released during battery's operation relies on the prediction from these models. However, thermal analysis, even for a single battery cell, is three dimensional by nature. A multi-physics model has to be used to handle the complex interplay of different physics. A modeling methodology, called multi-scale multi-dimensional method (MSMD), has been proposed by Kim et al. for such applications. The method has been proven to be very effective in simulating single battery or battery packs. It has been implemented in a general computational fluid dynamics commercial code, Ansys FLUENT.

In the MSMD method, different physics are resolved at different scales and in different domains. Then they are intimately coupled by exchanging information from one domain to another. One key idea in the MSMD method is that a battery's electric behavior can be represented by two co-located potential fields. The dual potential fields are solved on the same mesh as other flow variables like temperature. In this method every computational cell (control volume) in the active zone of a battery after discretization is treated as a mini-battery. The electrochemical model is applied to each mini-battery, which is at a subscale level. Since different physics is resolved in different domain and at different scale, in theory, any electrochemical submodel can be used in the MSMD method. Both empirical models and physics-based models have been used in the past under this methodology.

Although the MSMD method is very effective, there are some challenges preventing the method from being used for large problems in a battery or pack simulation. One challenge is to couple a physics-based electrochemical submodel such as Newman's P2D model in a three dimensional thermal simulation. Several partial differential equations already need to be solved in the electrochemical submodel. If the submodel is called by every CFD computational cell, the high computational cost makes the method practically unfeasible even in a single battery simulation, let alone battery packs. Guo and White noticed that, under a battery's normal operation, electrochemical reactions progress pretty evenly within a battery and they proposed a linear approximation model. By using that assumption, the computational cost can be reduced by orders of magnitude, thus makes the physics-based model feasible in a real three dimensional electrochemical-thermal coupled simulation (ECT).

The MSMD method needs two potential equations to be solved. Although the equations do not have transient terms, they need to be solved repeatedly at each time step in a transient fashion due to the time varying electric load condition and the change of state of charge of a battery. In this presentation, a linear scaling procedure is found to reconstruct the two potential fields from reference fields that can be calculated in advance. The procedure completely removes the need of solving two partial differential potential equations repeatedly. It can save the computational time significantly in an ECT simulation. The procedure is valid on the same basis as the linear approximation model. With this procedure, very large problems in battery application can be tackled. The cost of an ECT simulation is reduced to about the same as of a pure thermal simulation, which is the best as one can expect.

In this presentation, the MSMD methodology is first reviewed. Then the scaling procedure, called the reduced order method (compared to solving two potential equations directly), is proposed in detail. The method is illustrated by simulating a single battery and a battery pack.