Surface Photon Monte Carlo (SPMC) Radiation

The Surface Photon Monte Carlo (SPMC) radiation model applies Monte Carlo statistical methods to randomized bursts of photons to calculate close approximate solutions for radiative energy exchange between surfaces.

In the SPMC method, the key processes of radiation (such as emission, absorption, scattering, and boundary interaction) are modeled explicitly by stochastic methods, as opposed to obtaining a numerical solution of the governing equations. The method entails simulating the radiative processes by a ray trace procedure for a large number of photon bundles (representative samples of radiative energy). The generation of these photon bundles mimics the emission process, and the interaction of these photon bundles with the surfaces as they travel in the computational domain represent the radiation boundary treatment: absorption, reflection, transmission, or refraction. During the ray trace procedure, statistics are collected for the mesh faces to obtain the radiative quantities of interest.

A surface-to-surface radiation simulation with SPMC has the following modeling components:

• Emission modeling
• Movement of photon bundles
• Boundary treatment modeling

Emission Modeling

Emission of radiative energy from surfaces in a SPMC simulation is represented by the generation of photon bundles. This requires decisions on the number of photon bundles to be emitted at a given patch, the direction vector of the bundles, and their starting location on the patch.

In Simcenter STAR-CCM+, boundary surfaces are discretized into smaller elements called patches. These patches are sets of contiguous, non-overlapping boundary cell faces. The SPMC model solves for radiative exchange between the patches. The emissive power and radiation properties are assumed to be uniform over the surface of each patch, and are computed by surface averaging, as shown in Eqn. (1707).

Number of photon bundles from a given patch: For a SPMC simulation, the total number of photon bundles in a problem (i.e., user specified average number of rays per patch times the total patch count in the simulation) is distributed among the patches so that patches with higher emissive power emit more photon bundles. It has been shown in the literature that such sampling of photon bundles leads to lower statistical noise in the results for a given number of the photon bundles.

Emission direction: For diffuse emission from a patch, random direction is sampled according to this equation: [412]:

$$\psi =2\pi R_{\psi };\theta ={\sin }^{-1}\sqrt{R_{\theta }}$$

(1714)

where:

• $\psi$ and $\theta$ the azimuthal and polar angles, respectively, for the emission direction.
• $R_{\psi }$ and $R_{\theta }$ are uniform random numbers ( $\in [0,1]$ ) used in the sampling.

Emission location on a patch: The entire patch surface area emits energy, and therefore the starting location for a photon bundle is randomly chosen on the patch. It is important for the emission locations to be randomly sampled from a patch surface, for an accurate representation of radiative energy exchange between surfaces in a simulation. For an arbitrary patch, the emission location can be sampled according to these equations:

$$\begin{array}{c}{R}_x=\frac{1}{E_p}{\int }_{A_p\forall x^{\prime }<x}^{}ϵ\sigma T^{4}dA\\ R_y=\frac{1}{E_p}{\int }_{A_p\forall y^{\prime }<y}^{}ϵ\sigma T^{4}dA\\ R_z=\frac{1}{E_p}{\int }_{A_p\forall z^{\prime }<z}^{}ϵ\sigma T^{4}dA\end{array}$$

(1715)

where:

• $R_x$ , $R_y$ , and $R_z$ are the uniform random numbers between 0 and 1.
• $A_p$ is the patch area.
• $E_p$ is the emissive power of the patch. ( $E_p={\int }_{A_p}^{}ϵ\sigma T^{4}dA$ )
• $ϵ$ is the emissivity.
• $\sigma$ is the Stefan-Boltzmann constant.
• $T$ is the temperature.

The above expressions are inverted to get the (x,y,z) emission location.

Multi-band modeling: For multi-band simulations, spectral energy content is not sampled randomly on photon bundles and they carry emissive powers for all bands.

External collimated and diffuse sources: External directional or diffuse sources (such as solar loads or external diffuse flux at boundaries), if present, are included in SPMC. The incident fluxes on patches due to the specular component of the solar and environment loads are computed separately, and are used in SPMC to provide the effective diffuse emissive power from patches, using the following equation:

$$E_{\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{,}\mathrm{p}}=E_p+q_{\text{p,src}}+{\rho }_d(q_{\mathrm{d}\mathrm{i}\mathrm{r}\mathrm{,}\mathrm{e}\mathrm{x}\mathrm{t}}+q_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{,}\mathrm{e}\mathrm{x}\mathrm{t}})$$

(1716)

where:

• $E_{\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{,}\mathrm{p}}$ is the effective diffuse emissive power from a patch.
• $E_p$ is the patch local emissive power due to temperature-based emission.
• ${\rho }_d$ is the diffuse reflectivity for the patch.
• $q_{\text{p,src}}$ is the specified external diffuse energy source from the patch.
• $q_{\mathrm{d}\mathrm{i}\mathrm{r}\mathrm{,}\mathrm{e}\mathrm{x}\mathrm{t}}$ and $q_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{,}\mathrm{e}\mathrm{x}\mathrm{t}}$ are the incident fluxes on the patch due to the specular components of the directional and diffuse external sources, respectively.

The effective diffuse emissive power is used in the SPMC formulation for photon bundle emission modeling.

Movement of Photon Bundles

The photon bundles are traced in the direction of their travel till they encounter a boundary, and then the boundary treatment is performed.

Boundary Treatment Modeling

The radiative processes of absorption, reflection and transmission are modeled by a stochastic interaction scheme of photon bundles with the boundary mesh faces. In this scheme, photon bundles deposit some energy on the boundary proportional to its absorptivity, and rest of the radiative processes are modeled in a statistical sense, where a process is randomly chosen (weighted by their energy importance) for the boundary interaction of the incident photon bundle. The photon bundle goes through the randomly chosen radiative process (transmission or diffuse reflection or specular reflection) at the boundary.

The key idea is that for any boundary surface, the radiative interactions for a large number of photon bundles are selected randomly in such a way that they provide the desired statistics as dictated by the boundary properties. Instead of a stochastic approach, a deterministic approach (in which a photon bundle is split at the boundary to account for all possible interactions) can be used for the boundary treatment, and that might provide lower statistical noise in the solution than does the stochastic approach for the same number of the initial photon bundles. However, the subsequent generation of split photon bundles during deterministic boundary treatment can lead to a very large number of photon bundles in the simulation, several orders of magnitude larger than the original number of bundles. Generally, it is far more computationally efficient to employ the stochastic approach, albeit with a relatively higher number of initial/primary photon bundles to have the same level of statistical noise as the deterministic approach.

In this approach, for an arbitrary patch with diffuse reflectivity ${\rho }_d$ , specular reflectivity ${\rho }_s$ , and transmissivity $\tau$ , the boundary interaction for an incident photon bundle can be randomly sampled by drawing a uniform random number ( $R_{{\rho }_d,{\rho }_s,\tau }$ ) between 0 and 1, using the following equations:

$$\begin{array}{l}\text{Diffuse reflection:}0<R_{{\rho }_d,{\rho }_s,\tau }\le \frac{{\rho }_d}{{\rho }_d+{\rho }_s+\tau }\\ \text{Specular reflection:}\frac{{\rho }_d}{{\rho }_d+{\rho }_s+\tau }<R_{{\rho }_d,{\rho }_s,\tau }\le \frac{{\rho }_d+{\rho }_s}{{\rho }_d+{\rho }_s+\tau }\\ \text{Transmission:}\frac{{\rho }_d+{\rho }_s}{{\rho }_d+{\rho }_s+\tau }<R_{{\rho }_d,{\rho }_s,\tau }\le 1\end{array}$$

(1717)

The direction vector after the boundary interaction for the incident photon bundle is determined according to the boundary treatment. For diffuse reflection, expressions given in Eqn. (1714) are used whereas for specular reflection the outgoing direction can be determined from the law of optics. For transmission, refractive effects are considered while determining the direction.

The emission and boundary treatment for SPMC provides the solution for the radiative energy exchange between boundaries. The incident radiative heat fluxes are recorded on the patches during the SPMC ray trace procedure, and these fluxes are treated to account for the statistical nature of SPMC solution. This treatment is performed according to the statistical sampling factor specification in the user-interface for the SPMC Solver, and is discussed below.

The incident radiative heat fluxes, that is, irradiation, on patches after statistical treatment are then used to get the irradiation on faces, which is subsequently used in the computation of the radiation source term for the emission equation. The methodology followed here for the SPMC model is identical to the S2S model. See Eqn. (1708)—Eqn. (1712) for more details.