Lithium Ion Battery Discharge Curve — Cantera 3.2.0 documentation

Go to the end to download the full example code.
In this example, we illustrate how to calculate the open circuit voltage (voltage when the external applied current is zero) for a lithium ion battery as a function of anode and cathode lithium content. The open circuit voltage here is calculated via two means: kinetically and thermodynamically.
The thermodynamics are based on a graphite anode and a LiCoO₂ cathode (the typical active materials for commercial batteries, as of 2019), and are modeled using the BinarySolutionTabulatedThermo class.
The system here can be thought of as consisting of two particles—anode and cathode—connected by a liquid electrolyte:
For the sake of simplicity, we’re going to assume that the anode and cathode capacities are perfectly balanced. That is, if the cathode lithium content is X percent of its max possible (its capacity), then we will assume that the anode is at 1-X percent. Without loss of generality, we will define the anode composition.
The routine below returns the steady-state cell voltage of a lithium-ion cell for a given cell current, active material lithium stoichiometries, and electrolyte ionic resistance. This functionality is presented in greater detail in the reference (which also describes the derivation of the BinarySolutionTabulatedThermo class):
M. Mayur, S. C. DeCaluwe, B. L. Kee, W. G. Bessler, “Modeling and simulation of the thermodynamics of lithium-ion battery intercalation materials in the open-source software Cantera,” Electrochim. Acta 323, 134797 (2019), https://doi.org/10.1016/j.electacta.2019.134797
In the future, this example may be developed further to demonstrate simulating charge-discharge of the lithium ion battery.
Requires: cantera >= 2.6.0, matplotlib >= 2.0, scipy >= 1.10.0
Tags: Python surface chemistry kinetics electrochemistry battery plotting
In the kinetic equilibrium problem, steady state is achieved when the faradaic current at each electrode (anode and cathode) surface and the ionic current through the electrolyte are all zero. There are, essentially, four relevant electric potentials which must be determined:
\(\phi_{\rm anode}\): electric potential of graphite anode.
\(\phi_{\rm elyte,\, anode}\): electric potential of electrolyte at anode interface.
\(\phi_{\rm elyte,\, cathode}\): electric potential of electrolyte at cathode interface.
\(\phi_{\rm cathode}\): electric potential of LCO cathode.
Setting one of these four to the reference potential of zero (because it is the difference in electric potential which drives currents, the actual potential values are irrelevant. Let’s assume the anode electric potential is zero), there is only one distribution of electric potentials across the cell such that the current is invariant across the cell. I.e. we want the potentials such that:
where \(i_{\rm app}\) is the user input for the applied current. For this example, we assume an applied current of 0, to calculate the equilibrium voltage.
The faradaic current for this model is calculated using Butler-Volmer kinetics. For a Li-ion battery, this is:
where \(S_{\rm elde}\) is the specific surface area of the electrode in question, \(F\) is Faraday’s constant, \(\beta\) is the charge-transfer symmetry parameter, \(R\) the universal gas constant, \(T\) the temperature, and \(\eta\) the overpotential, which is the electric potential difference between the electrode and electrolyte, \(\Delta \phi = \phi_{\rm elde} - \phi_{\rm elyte}\), relative to that at equilibrium, \(\Delta \phi_{\rm eq}\):
\(i_\circ\) is known as the “exchange current density,” which is equal to the rate of the forward and reverse current at equilibrium (which are equal). \(i_\circ\) and \(\beta\) are provided as user inputs in the YAML file. At any particular state, (user-specified electric potentials, pressure, temperature, and chemical compositions), Cantera calculates \(\eta\) as part of the routine to evaluate reaction rates of progress \(\left(\dot{q} = i_{\rm Far}/F \right)\). The user simply sets the state values mentioned above.
The electrolyte is modeled as a resistor with user-defined ionic resistance \(R_{\rm io}\), and hence the ionic current is calculated as:
where positive current is defined as delivering Li\(^+\) to the anode interface. Given \(i_{\rm app}\), this equation can be inverted, to calculate the electric potential of the electrolyte at the cathode interface, relative to that at the anode interface:
Again: in this example, \(i_{\rm app} = 0\) and hence the two electric potential values in the electrolyte are equal.
For the kinetic routine, there are three processes to determine the cell voltage \(\phi_{\rm cathode} - \phi_{\rm anode}\) which corresponds to the user-provided \(i_{\rm app}\):
Determine the \(\phi_{\rm elyte,\,anode}\) value which corresponds to \(i_{\rm app}\), given \(X_{\rm Li, anode}\), the percentage of Li in the anode active material.
Determine \(\phi_{\rm elyte,\,cathode}\), given \(\phi_{\rm elyte,\,anode}\) and \(i_{\rm app}\).
Determine the \(\phi_{\rm cathode}\) which corresponds to \(i_{\rm app}\), given \(\phi_{\rm elyte,\,cathode}\) and \(X_{\rm Li, anode}\), the percentage of Li in the anode active material.
The routines below are written generally such that an interested user may set \(i_{\rm app}\) to any value of interest.
The phase thermodynamics are defined according to experimentally-measured open circuit voltage values, as described in the reference provided above. These values are stored in the input file lithium_ion_battery.yaml.
Stoichiometries X_Li_ca and X_Li_an [-] (can be vectors)
Externally-applied current i_app [A]
Electrolyte resistance R_elyte [Ohm]
Anode total surface area S_an [m^2]
Cathode total surface area S_ca [m^2]
For the \(i_{app} = 0\) case, we can also calculate the voltage using thermodynamics. At equilibrium, the net electrochemical potential change of the reaction must be zero:
where \(\tilde{\mu}_k = \mu_k + z_kF\Phi_k\), where, in turn \(\mu_k = \frac{\partial g_k