Discrepancy Between “Total heating [W]” by PyBaMM and Bernardi Formula in Lumped Thermal Models

General
1

Hello,

I have some questions regarding the calculation of heat generation in the lumped thermal model. In PyBaMM, the heat generation is calculated using information such as the exchange current density and electrode potentials, while the commonly used Bernardi formula estimates heat generation based on battery current, terminal voltage, and open circuit voltage, i.e., Q=I(U-Uocv) (neglecting the reversible heat term for now).

Theoretically, these two approaches differ primarily in their level of detail, and the results should be generally comparable. However, when I simulate using PyBaMM, I notice that the value of “Total heating [W]” output by the model differs significantly from the value calculated using the Bernardi formula based on simulated current and voltage.

import pybamm
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from matplotlib import rcParams

DFN = pybamm.lithium_ion.DFN({"thermal": "lumped"})

VAR_PTS = {"x_n": 15, "x_s": 10, "x_p": 15, "r_n": 10, "r_p": 10}
SOLVER = pybamm.CasadiSolver(dt_max=100, atol=1e-6, rtol=1e-6, return_solution_if_failed_early=True)

PARAM = pybamm.ParameterValues('Chen2020')

experiment_1C = pybamm.Experiment(
        [
            ("Charge at 1C until 4.2V",
             "Rest for 10 min"),
        ],
        period="1 seconds")

sim = pybamm.Simulation(DFN, solver=SOLVER, experiment=experiment_1C, parameter_values=PARAM, var_pts=VAR_PTS)
    # 仿真
solution_ = sim.solve(initial_soc=0)

t_sim = solution_["Time [s]"].entries
V_sim = solution_["Terminal voltage [V]"].entries
ocv_sim = solution_["Battery open-circuit voltage [V]"].entries
I_sim = solution_["Current [A]"].entries
Total_heat_sim= solution_["Total heating [W]"].entries

def calculate_total_heat(I, Ut, Uocv):
    P = np.zeros_like(I)
    for i in range(len(I)):
        P[i] = I[i]*(Ut[i]-Uocv[i])
    return P

Total_heating_calculate = calculate_total_heat(-I_sim, V_sim, ocv_sim)

plt.rcParams['font.family'] = 'Times New Roman'  ## 将图中字体设置为新罗马字体

fig, ax = plt.subplots(figsize=(7.5, 6))

ax.plot(t_sim, Total_heating_calculate, label='sim_self_cal')
ax.plot(t_sim, Total_heat_sim, label='sim_model_output')
ax.set_ylabel('I*(Ut-Uocv) [W]', fontsize=20)
ax.set_xlabel('Time [s]', fontsize=20)
ax.tick_params(labelsize=18)
ax.legend(fontsize=15)
ax.set_title('Total heating [W]', fontsize=20)

plt.show()

image
image1039×853 85.9 KB

I would like to ask:

  • Is there something I may have misunderstood about the way PyBaMM calculates heat generation?
  • Or could there be an inconsistency in the heat generation calculation within PyBaMM?
  • Additionally, is there an existing interface or recommended approach to directly calculate battery heat generation using the Bernardi formula within PyBaMM?

Thank you very much for your support!

2

Hi,

In PyBaMM, "Total heating [W]" is the sum of multiple heat-generation mechanisms, namely ohmic heating Qohmic​= solution_["Ohmic heating [W]"], irreversible electrochemical (reaction/overpotential) heating Qrxn=solution_["Irreversible electrochemical heating [W]"], reversible (entropic) heating Qrev​= solution_["Reversible heating [W]"], and heat of mixing due to concentration gradients Qmixing​= solution_["Heat of mixing [W]"] (with hysteresis heating included if enabled), so that Qtotal=Qohmic+Qrxn+Qrev+Qmixing +Qhyst ; in contrast, I(Ut−Uocv) represents a lumped electrical loss and does not account for how energy is temporarily stored or redistributed by diffusion and entropy effects, which is why the two do not generally match at moderate–high C-rates, but when the C-rate is reduced to very low values (e.g. ∼0.01C) the two curves come very close, clearly indicating that polarization, concentration gradients, and transport limitations have become negligible and the system is effectively quasi-equilibrium.

Sunil