*2.2. The Radiation Models*

Thermal radiation makes the temperature distribution more uniform in an enclosed space by transferring thermal energy from a hot surface to a cold one. Generally, thermal radiation accounts for 30–70% of the total heat transfer rate [45]. In this study, radiation heat transfer was taken into consideration for modeling the cabin thermal environment.

The P-1 radiation model is the simplest case of the P-N model, which is based on the expansion of the radiation intensity into an orthogonal series of spherical harmonics functions. The directional dependence in radiative transfer equation (RTE) is integrated out, resulting in a diffusion equation for incident radiation. Equation (1) is obtained for the radiation flux *qr*, only considering scattering and absorption when modeling gray radiation.

$$q\_r = -\frac{1}{3(a + \sigma\_s) - \mathbb{C}\sigma\_s} \nabla G \tag{1}$$

An advection-diffusion equation is solved to determine the local radiation intensity *G* in the P-1 model.

$$-\nabla \cdot q\_{\mathbb{T}} = \kappa G - 4\sigma T^4 \tag{2}$$

where *qr* is the radiation heat flux, *a* is the absorption coefficient, σ*<sup>s</sup>* is the scattering coefficient, *G* is the incident radiation, *C* is the linear-anisotropic phase function coefficient, σ is the Stefan-Boltzmann constant.

The Surface to Surface (S2S) radiation model is applicable for modeling radiation in situations where there are no participating media. All surfaces involved in radiation are assumed to be gray and diffuse, ignoring absorption, emission and scattering and preserving only "surface-to-surface" radiation. The energy flux leaving a given surface is composed of directly emitted and reflected energy. The reflected energy flux is dependent on the incident energy flux from the surroundings, which then can be expressed in terms of the energy flux leaving all other surfaces. The energy leaving from the kth adiabatic surface can be expressed as Equation (3).

$$
\eta\_{\rm out,k} = \varepsilon\_k \sigma T\_k^4 + \rho\_k \eta\_{\rm in,k} \tag{3}
$$

where *qout*,*<sup>k</sup>* is the energy flux leaving the surface, ε*<sup>k</sup>* is the emissivity, ρ*<sup>k</sup>* is the reflectivity of surface k, *qin*,*<sup>k</sup>* is the energy flux incident on the surface from the surroundings.

The Discrete Ordinates (DO) radiation model is regarded as the most comprehensive radiation model, which solves the RTE for a discrete number of finite solid angles, as shown in Equation (4). Each associated with a vector direction <sup>→</sup> *s* is fixed in the global Cartesian system (x, y, z). Accuracy can be increased by using a better discretization, while it may be CPU-intensive with many ordinates.

$$\nabla \cdot \left( l \stackrel{\textstyle \neg}{r}, \stackrel{\textstyle \neg}{s} \stackrel{\textstyle \neg}{\mathbf{s}} \right) + (a + \sigma\_{\sf s}) l \stackrel{\textstyle \neg}{\mathbf{r}}, \stackrel{\textstyle \neg}{\mathbf{s}} \right) = a \mathbf{n}^2 \frac{\sigma T^4}{4 \pi} + \frac{\sigma\_{\sf s}}{4 \pi} \int\_0^{4 \pi} l \stackrel{\textstyle \neg}{\mathbf{r}} \stackrel{\neg}{\mathbf{s}} \stackrel{\textstyle \neg}{\mathbf{s}} \Phi \Big( \stackrel{\textstyle \neg}{s} \stackrel{\neg}{\mathbf{s}} \Big) d\Omega' \tag{4}$$

where <sup>→</sup> *r* is the position vector, <sup>→</sup> *<sup>s</sup>* is the direction vector, <sup>→</sup> *s* is the scattering direction vector, α is the absorption coefficient, *n* is the refractive index, *I* is the radiation intensity, which depends on the position -→ *r* and direction -→ *s* , *T* is the local temperature, Φ is the phase function, Ω is the solid angle.

For the Discrete Transfer Radiation Model (DTRM), the main assumption is that radiation leaving a surface element within a specified range of solid angles can be approximated by a single ray. The energy source in the fluid due to radiation is computed by summing the change in intensity *dI* along the path of each ray *ds* that is traced through the fluid control volume.

$$\frac{dI}{ds} + aI = \frac{a\sigma T^4}{\pi} \tag{5}$$
