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Figure 6. Profile (I): Temperature profile for the mean cell temperature T: (a) Teny fixed, h varied; (b) h fixed, Teny varied.

Figure 6 Profile (I): Temperature profile for the mean cell temperature T: (a) Teny fixed, h varied; (b) h fixed, Teny varied.

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Figure 1. Scheme of the several domain levels and their corresponding coordinate systems. (a) Particle domain; (b) electrode domain; (c) cell domain.
Figure 1. Scheme of the several domain levels and their corresponding coordinate systems. (a) Particle domain; (b) electrode domain; (c) cell domain.
Table 1. Effective thermal parameter used in simulation in comparison with effective thermal parameter given in literature.
Table 1. Effective thermal parameter used in simulation in comparison with effective thermal parameter given in literature.
The simulations were performed with the two current load profiles with a constant current densi! i = +80 A/m? as seen in Figure 2, in which there are no relaxation periods between the chargin d discharging pulses with duration of 250s each in the first profile Profile I . In the second profi ‘ofile II) relaxation periods of 250s between charging and discharging pulses are used. A comple rrent cycle lasts 500s and 1000s. The maximal simulation time is tsim = 5 x 104s and fim = 10° r the time integration in COMSOL Multiphysics, a BDF integration scheme is chosen with a minimus Jer of 1 and a maximum order of 5 using a variable step size with a maximum time step of 1s anda solute tolerance of 0.001. Since the spatial discretisation in COMSOL Multiphysics is based on the FE] -thod, we have used an adaptive spatial discretisation in the three models of the particle domai > electrode domain and the cell domain respectively. The model of the particle domain is solve tomatically in the battery module of COMSOL Multiphysics. Therefore only a spatial discretisatio ‘ the electrode domain and the cell domain is required. In the electrode domain, the maximu1 ment size in the discretisation is chosen as 1 pm. In total the discretisation contains 168 element 1adratic basis functions were chosen for the one-dimensional finite element discretisation in tt ctrode domain. In the cell domain the spatial discretisation is performed in the r — z-plane usin 56 triangular elements with the element size in the interval [3.9 x 10~4,8.45 x 10-2] m and quadrat sis functions. The FEM discretisation is shown in Figure 3. Figure 2. (a) First 1000s of the current load profile I without relaxation times and (b) of the current load profile II with relaxation times of 250 s between charging and discharging pulses.
The simulations were performed with the two current load profiles with a constant current densi! i = +80 A/m? as seen in Figure 2, in which there are no relaxation periods between the chargin d discharging pulses with duration of 250s each in the first profile Profile I . In the second profi ‘ofile II) relaxation periods of 250s between charging and discharging pulses are used. A comple rrent cycle lasts 500s and 1000s. The maximal simulation time is tsim = 5 x 104s and fim = 10° r the time integration in COMSOL Multiphysics, a BDF integration scheme is chosen with a minimus Jer of 1 and a maximum order of 5 using a variable step size with a maximum time step of 1s anda solute tolerance of 0.001. Since the spatial discretisation in COMSOL Multiphysics is based on the FE] -thod, we have used an adaptive spatial discretisation in the three models of the particle domai > electrode domain and the cell domain respectively. The model of the particle domain is solve tomatically in the battery module of COMSOL Multiphysics. Therefore only a spatial discretisatio ‘ the electrode domain and the cell domain is required. In the electrode domain, the maximu1 ment size in the discretisation is chosen as 1 pm. In total the discretisation contains 168 element 1adratic basis functions were chosen for the one-dimensional finite element discretisation in tt ctrode domain. In the cell domain the spatial discretisation is performed in the r — z-plane usin 56 triangular elements with the element size in the interval [3.9 x 10~4,8.45 x 10-2] m and quadrat sis functions. The FEM discretisation is shown in Figure 3. Figure 2. (a) First 1000s of the current load profile I without relaxation times and (b) of the current load profile II with relaxation times of 250 s between charging and discharging pulses.
The main purpose of this section is to compute the overall mean cell temperature, below which the LIB shows normal behavior and above which the cell shows a thermal runaway with respect to some system parameters like the environmental temperature or the convective heat transfer coefficient. Furthermore, critical values for such parameters will be computed from the simulations and a first coarse classification of the thermal runaway will be given.
The main purpose of this section is to compute the overall mean cell temperature, below which the LIB shows normal behavior and above which the cell shows a thermal runaway with respect to some system parameters like the environmental temperature or the convective heat transfer coefficient. Furthermore, critical values for such parameters will be computed from the simulations and a first coarse classification of the thermal runaway will be given.
Table 2. Classification of the time evolution of the temperature during thermal runaway. In summarize one can see the difference between Model A and Model B due to the exothermic heat sources in the temperature development over the time and corresponding heating rate. Furthermore Model B gives a good agreement with experimental results from literature [26,32] and a coarse classification of the time evolution of a thermal runaway event has been made. This topic will be considered in more detail in a subsequent publication.
Table 2. Classification of the time evolution of the temperature during thermal runaway. In summarize one can see the difference between Model A and Model B due to the exothermic heat sources in the temperature development over the time and corresponding heating rate. Furthermore Model B gives a good agreement with experimental results from literature [26,32] and a coarse classification of the time evolution of a thermal runaway event has been made. This topic will be considered in more detail in a subsequent publication.
Figure 5. Cont. Table 3. Thermal runaway time trr/|s] with respect to heat transfer coefficient h and the environment al temperature Teny. (a) Profile (I), (b) Profile (II).
Figure 5. Cont. Table 3. Thermal runaway time trr/|s] with respect to heat transfer coefficient h and the environment al temperature Teny. (a) Profile (I), (b) Profile (II).
Figure 7. Profile (I): Temperature profile for the mean cell temperature T: (a) Teny fixed, h varied; (b) h fixed, Teny varied.
Figure 7. Profile (I): Temperature profile for the mean cell temperature T: (a) Teny fixed, h varied; (b) h fixed, Teny varied.
Table 4. Critical parameter intervals: Profile (I) (a) Teny fixed, (b) h fixed, Profile (II) (c) Teny fixed, (d) h fixed. Additional from the temperature curves of the mean cell temperature T in the Figures 6 and 7 the critical mean cell temperature is Torit & 400 K. Since the development of a complete mathematical theory is far beyond the scope of this work an extension of the theories of Semenov [52] and Frank-Kamenetskii [38] and the corresponding critical parameter will be considered in future work.. In summarize the following general statement can be given from the observation of the simulation results:
Table 4. Critical parameter intervals: Profile (I) (a) Teny fixed, (b) h fixed, Profile (II) (c) Teny fixed, (d) h fixed. Additional from the temperature curves of the mean cell temperature T in the Figures 6 and 7 the critical mean cell temperature is Torit & 400 K. Since the development of a complete mathematical theory is far beyond the scope of this work an extension of the theories of Semenov [52] and Frank-Kamenetskii [38] and the corresponding critical parameter will be considered in future work.. In summarize the following general statement can be given from the observation of the simulation results:
Table A1. Simulation parameters.
Table A1. Simulation parameters.
Table A1. Cont. Table A2. Exothermic simulation parameters.
Table A1. Cont. Table A2. Exothermic simulation parameters.
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