# Thermal modeling of batteries

## By Apoorv Shaligram

Writing a blog after a long time. To be honest, I only thought about this subject because I had the time to think due to a COVID infection quarantine, my second one now (In the first one last year, I learnt the basics of 3D surface modeling in Blender). A friend asked me a question regarding thermal modeling and got me thinking about some aspects of it that people may not be aware of. With the work that I am currently involved in, we could make thermal modeling of batteries pretty much irrelevant in the normal scheme of things. But then, we will count the chickens when they hatch…

To jump into the technicals, thermal modeling of batteries is about predicting the temperature of batteries at different points in its geometry as a function of time. As such, predicting the exact amount of heat being generated in the battery at any point in space at any point of time, is critical to accurately model the battery’s temperature profile. Most people seem to think of heat generated as Ohmic heat (I²*R) and use the formula in thermal modeling. However, the value of the resistance R itself is an indirectly calculated averaged quantity from overpotentials and as such is not an accurate measure of heat. Instead, it is much better to use the overpotentials at given current rate and state-of-charge (SOC) for calculation of heat.

Why do I look at overpotentials when I want heat generated? By definition, the area between the voltage profile at any given charging/load profile and the equilibrium voltage profile under open circuit conditions (w/ both profiles plotted against charge capacity) denotes energy that was wasted or simply, heat. Thus, the overpotential at any point of time can be denoted as dQ/dq.

The next question that arises is that how do we get the equilibrium open-circuit voltage (OCV) curve? By definition, the voltage curve as measured under any charge/load profile will have current flowing at all times and as such, cannot give us an “equilibrium” curve. One way to get this curve is running a test known as a Galvanostatic Intermittent Titration Technique (GITT). In GITT, we charge and discharge the cell at a given charge rate with rest steps at regular intervals. The rest steps allow the cell to “equilibrate” or settle the concentration gradients across the cell. Thus, the longer the rest step is, the closer the observed voltage at its end is to the actual equilibrium OCV.

Thus, to accurately model the thermal behavior of a battery pack, one should have the data from a GITT test on the cell as well as the data from constant current charge-discharge cycles at various rates. If we wish to go a step further in accuracy, the model should also account for the unequal distribution of current that arises from cells being at different temperatures and hence at different voltages despite the same amount of charge having flown through.

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