## 2 Materials and Methods
where the radiosity has been expressed as the sum of diffusively reflected and emitted radiation from Cell 2. Consequently, the net radiation heat flux on Cell 2 can be written as:  
\[q=F_{12}G-J \tag{4}\]  
In these equations, \(\varepsilon\) is the emissivity, \(\sigma\) is the Stefan-Boltzmann constant, T\({}_{1}\) is the temperature of Cell 1, T\({}_{2}\) is the temperature of Cell 2, T\({}_{\text{amb}}\) is ambient temperature, and F\({}_{12}\) is the view factor from Cell 1 to Cell 2. In order to validate the thermal radiation modelling, a quasi-steady state radiative heat transfer problem between two equal cylinders of radius R = 9 mm is considered. With the given separation distance and cylinder temperatures, we can numerically simulate the radiation heat flux received by Cell 2 among the total radiation emitted from Cell 1, that is, \(GF_{12}\). Meanwhile, in classical heat transfer theory, the radiation heat flux between two cylinders is analytically available with a known view factor, which can be used to validate the simulation results. The view factor between two equal cylinders has been derived from the view factor for general parallel cylinders as follows [37]:  
\[F_{12}=\frac{\sqrt{\left(\frac{2R+d}{R}\right)^{2}-4}-\frac{2R+d}{R}+2 arcsin \left(\frac{2R}{2R+d}\right)}{2\pi} \tag{5}\]  
As shown in Figure 4a,b, the numerical radiation heat flux is compared with the analytical theory between two equal blackbody cylinders (i) with fixed temperature but varying distance and (ii) with fixed distance but varying temperature of the Cell 1. The results suggest that as the distance of two cells increases, the radiation heat flux decreases. Also, as the temperature of Cell 1 increases, the radiation heat flux received by Cell 2 increases. The numerical calculation of the radiation heat flux is in excellent agreement with analytical results, further substantiating the simulation tool. In the thermal runaway simulation, we fix the temperature T\({}_{1}\) of Cell 1 and the ambient temperature (293 K) in each case. The initial temperature of Cell 2 is also taken to be 293 K. The evolution of T\({}_{2}\) is numerically solved by accounting for radiation, conduction, and thermal runaway chemical heat release based on Equation (1). The surface-to-surface radiation is simulated using the hemicube method. A time-dependent Multifrontal Massively Parallel Sparse (MUMPS) solver is used to solve for the thermal energy conservation equation. The backward differentiation formula (BDF) method is used for time stepping. The relative tolerance for the solution is \(10^{-3}\). Based on this general methodology, a number  
Figure 4: Validation of simulation results against analytical model for surface-to-surface radiation modeling: (**a**) Radiation heat flux vs. distance d between two cells with fixed cell temperatures. (**b**) Radiation heat flux vs. temperature T\({}_{1}\) of Cell 1 with fixed distance d and temperature T\({}_{2}\) of Cell 2.  
of investigations are carried out to understand the nature of radiation-induced thermal runaway, as well as the coupling between radiation and chemical heat generation. These results are discussed in the next section. In the following (Sections 3.1-3.3), we have adopted a cell spacing d of 1 mm to investigate the feature of radiation-induced thermal runaway. There are two primary considerations behind the choice of cell separation d: energy density and safety. For energy considerations, the larger the cell spacing, the fewer number of cells can be accommodated in a given space, hence reducing the energy density of the pack. For safety, however, larger spacing is preferred, as thermal runaway propagation is less likely. As such, there is eventually a trade-off effect between these two considerations. Cell spacing of 1 mm in a battery pack has been adopted previously in [38; 39; 40]. Note that concepts and techniques developed in this work can also easily be applied to investigate radiative heat transfer for other cell spacing configurations.