Battery Thermal Runaway: Cell Spacing and Barrier Margin

A derived symmetric two-node thermal network represents one donor and one identical neighbor. Each has $C=mc_p$ and cooling conductance $G_s$. Exponential release is $P(t)=Q\exp(-t/t_r)/t_r$; the instantaneous option deposits the allocated energies at $t=0$. The donor receives $Q_b=f_bQ$ and the neighbor directly receives $Q_j=f_jQ$. The rest leaves the model.

For equal long cylinders, $X=1+g/D$ and $F_{12}=[\sqrt{X^2-1}+\arcsin(1/X)-X]/\pi$ (Zhang et al. 2023, Eq. 5, attached p. 6; also F-Chart EES). With $A_s=\pi DL$ and $A_p=DL$, gas conductance is $k_gA_p/g$. The donor adiabatic bound is $T_b=T_a+Q_b/C$ in kelvin. Since $f_b\ge f_j$, neither node exceeds this bound. The mean-value theorem gives $\sigma(T_d^4-T_n^4)\le4\sigma T_b^3(T_d-T_n)$. Thus $G_r=4\sigma T_b^3 A_sF_{12}$ bounds direct blackbody exchange. Finite-end losses and radiation to ambient are not added.

The selected pair conductance is $G=\eta(G_g+G_r)+G_{bridge}$. The barrier is a measured effective transmission ratio, including reradiation. Let $S=\theta_d+\theta_n$ and $D_\theta=\theta_d-\theta_n$. Their decay rates are $a=G_s/C$ and $b=(G_s+2G)/C$. For exponential release, $H(r,t)=[e^{-t/t_r}-e^{-rt}]/(rt_r-1)$, with its continuous limit when $rt_r=1$. Then $S=(Q_b+Q_j)H(a,t)/C$, $D_\theta=(Q_b-Q_j)H(b,t)/C$, and $\theta_n=(S-D_\theta)/2$. Instantaneous release uses $H=e^{-rt}$.

Integrating the full event gives $Q_{in}=Q_j+G(Q_b-Q_j)/(G_s+2G)$. The heat margin is $Q_{crit}-Q_{in}$; the peak temperature is found by a scalar search of the analytic transient. Maximum transmission is the inverted heat-budget equation, clipped to 0–1; the separate bypass test identifies infeasibility. Energy conservation is checked over the full event. These network equations and the radiation envelope are derived here, not a reproduction of the cited papers’ chemical or FEM models.

For stable numerical evaluation, the finite-release response uses $H=(t/t_r)e^{-\min(r,1/t_r)t}E(|r-1/t_r|t)$, where $E(z)=-\operatorname{expm1}(-z)/z$ and $E(0)=1$. This retains unequal rates even arbitrarily close to equality. The integral follows the exact identity $r\int_0^t H\,dt=1-e^{-t/t_r}-H$, with the released fraction also evaluated using expm1.

Overview charts use a positive logarithmic time grid spanning the fastest release/redistribution timescale to the cooling tail. Physical timescales and calculated donor and neighbor peak points are inserted explicitly. The separate linear onset chart includes $t=0$, including the instantaneous-deposition initial state. Plot resolution is therefore independent of the ratio between release and ambient-cooling timescales.

Two identical lumped cells begin at ambient; constant mass, heat capacity and equal cooling conductance. The neighbor has no chemical heat generation. Long, parallel cylinders exchange direct radiation as black surfaces; gas conduction uses a planar projected-area approximation. The barrier changes only the covered gas/radiation path, while cooling, bridges and direct deposition stay fixed. The source fractions sum to at most one and donor-body deposition is no smaller than direct neighbor deposition.

Published defaults: NASA 2020 attached p. 25 reports a 59.7 kJ mean for Samsung 18650-30Q; Tang et al. Table 1 gives HG2 mass 46.9 g; Zhang et al. Table 1 gives representative specific heat 1000 J/(kg·K). Tang et al. report about 3 kJ external heat and 124 °C voltage-failure temperature for HG2. These are different experiments and cell types. The default fractions 0.30 and 0.01, 0.05 W/(m·K) gap conductivity, 0.01 W/K bridge, 0.2 W/K cooling and 0.15 transmission are analyst assumptions, not measured catalogue data. Replace them with applicable measurements.

A comparative external-heating screen, not a prediction that propagation will or will not occur. Positive margins refer only to the entered heat budget and temperature. The radiation envelope is conservative for this symmetric network with the stated unchanged cooling and direct-exchange assumptions; it does not bound omitted chemistry, flames, enclosure reflection, electrical heating or damage to a barrier.

Tang et al.’s voltage-failure heat and temperature were measured under slow, nearly uniform ARC heating at 100% SOC. Their Table 2 distinguishes voltage failure from later runaway: HG2 failure was 124 °C, while its peak runaway temperature was 805 °C. Do not transfer these criteria unchanged to a rapid local jet, another cell chemistry, SOC, age or format. The animation shows the thermal network response; its time is not an escape or intervention window.

Effective barrier transmission must include hot-face reradiation, assembly details and the event duration of interest. A value of zero for maximum permitted transmission can mean that bypass exposure is already too large; read the budget comparison. Retest cooling after adding insulation, since a real barrier can also reduce normal heat rejection. The reference cases validate the cylinder view factor and an independent cooling limit, not actual runaway propagation.