S2S (Surface-to-Surface) Radiation

The S2S model is based on enclosure theory [397], [400] and uses view factors between patches to perform radiative energy exchange.

The calculation approach uses the following main elements:

The boundaries enclosing the computational domain and internal boundaries are spatially subdivided into contiguous, non-overlapping patches.
From the center of each patch, a specified number of beams are emitted over the enclosing hemisphere with solid angles that are discretized using an angular quadrature. Each beam is traced through the computational domain until it intercepts an opposing patch, thus defining a pair of patches that exchange radiative energy.
The radiation energy transfer to or from each patch is then calculated from the radiation transport equation and the boundary conditions.

Surface Patches and View (Interaction) Factors

The surfaces in the model can have either diffuse or specular surface properties. The surface emissivity is assumed diffuse, but the reflectivity can be either diffuse or specular, and all transmissivity is specular. Specular properties are considered in the calculation of the view factors. Hence, the view factors are not merely geometric properties, but also include the influence of the specular surface properties.

The amount of radiation that is exchanged between surfaces depends on the (diffuse) emission from the surfaces and the position and orientation of the surfaces relative to each other. The amount of radiation also depends on the presence of any specularly transmissive or reflective surfaces that can alter the transfer between the surfaces. The latter two influences—geometry and specular properties—are accounted for by the view (or interaction) factors which, by definition, represent the “fraction of uniform diffuse radiation leaving a surface that directly reaches another surface” [397].

The following Eqn. (1688) to Eqn. (1693) define the view factor relationships for gray and diffuse surfaces.

The following diagram shows the radiation exchange between two elemental surfaces:

The total amount of radiation power that $d S_{1}$ emits and $d S_{2}$ receives is:

Figure 1. EQUATION_DISPLAY

P_{1 - 2} = {i′}_{1} d S_{1} \cos (β_{1}) (\frac{d S_{2} \cos (β_{2})}{L^{2}})

(1688)

where:

$P_{1 - 2}$ is the total power leaving surface 1 and incident on surface 2 [W]
${i′}_{1}$ is the total intensity leaving surface 1, where intensity is defined as the “radiative energy passing through an area per unit solid angle, per unit of the area projected normal to the direction of passage, and per unit time” [397] [ $\frac{W}{m^{2} s r}$ ]
$β$ is the angle between the surface normal and a line joining two surfaces [rad]
$L$ is the distance

By taking the fraction of the power that $d S_{2}$ receives over the total radiation power that is emitted from $d S_{1}$ $(P_{1} = {π i'}_{1} d S_{1})$ , the view factor for the two surfaces is obtained as:

Figure 2. EQUATION_DISPLAY

{d F}_{1 - 2} = \frac{P_{1 - 2}}{{π i'}_{1} d S_{1}}

(1689)

Applying definition Eqn. (1689) to two black surfaces i and j in thermal equilibrium at the same temperature (no net energy exchange) yields the requirement that view factors must satisfy the reciprocity relation:

Figure 3. EQUATION_DISPLAY

{d F}_{i - j} d S_{i} = {d F}_{j - i} d S_{j}

(1690)

For diffuse radiation, relationship Eqn. (1690) applies to any two surfaces, whether black or not.

These definitions can be extended to surfaces of finite size, for which the view factors are obtained using integration over constituent elemental surfaces. The following diagram shows the radiation exchange between two finite surfaces:

For the (S1, S2) pair:

Figure 4. EQUATION_DISPLAY

F_{1 - 2} = \frac{\int_{S_{1}} \int_{S_{2}} {i^{'}}_{1} \frac{\cos (β_{1}) \cos (β_{2})}{π L^{2}} d S_{1} d S_{2}}{\int_{S_{1}} {i'}_{1} d S_{1}}

(1691)

and (assuming $i'$ is constant over element 1):

Figure 5. EQUATION_DISPLAY

F_{1 - 2} = \frac{1}{S_{1}} \int_{S_{1}} \int_{S_{2}} \frac{\cos (β_{1}) \cos (β_{2})}{π L^{2}} d S_{1} d S_{2}

(1692)

Likewise:

Figure 6. EQUATION_DISPLAY

F_{2 - 1} = \frac{1}{S_{2}} \int_{S_{2}} \int_{S_{1}} \frac{\cos (β_{1}) \cos (β_{2})}{π L^{2}} d S_{1} d S_{2} or F_{2 - 1} = F_{1 - 2} \frac{S_{1}}{S_{2}}

(1693)

In Simcenter STAR-CCM+, the boundary surfaces are discretized into smaller elements called patches. These patches are sets of contiguous boundary cell faces, and view factors are calculated for each patch pair. By definition, patches do not straddle boundaries and are therefore at most as large as an entire boundary or as small as a boundary cell face. The emissive power (and therefore the radiation intensity) and radiation properties are assumed to be uniform over the surface of each patch.

The view factors as in Eqns. Eqn. (1692) and Eqn. (1693) above are purely topological quantities, dependent only on the geometry of the two surfaces. However, the presence of partially transmissive and/or specularly reflective surfaces in Simcenter STAR-CCM+ can introduce a further dependence on other surfaces and their respective surface properties. As a result, the collection of view factors is, in general, dependent on the surface geometry and the respective transmissivity and specular reflectivity of those surfaces. For a fixed geometry and fixed surface transmissivity and specular reflectivity, the view factors must be obtained only once, usually at the start of a simulation.

The double integral is approximated using a ray tracing approach. Collections of polygons represent surfaces, and the solid angle is discretized using an angular quadrature. For each surface or patch, a predefined number of rays is traced through the computational domain starting at the patch centroid. The directions and weights are based on discretization of an ideal hemisphere over the patch.

A ray shot from one patch can do any of the following:

Hit an opaque patch.
Pass through a semi-transparent patch (with reduced strength proportional to the transmissivity of the boundary to which the hit patch belongs).
Reflect from a specularly reflective patch.
Pass through a transparent boundary or an opening and be dispersed in the environment.

In all these cases a view factor is generated between the shooting patch and, respectively, the opaque patch, the semi-transparent patch, the specularly reflective patch, or the environment.

This method is fairly simple and cost-effective compared to other methods (hemicube, hemisphere, or Monte Carlo). There are some analogies to the Unit Sphere Method [398]. After the beam tracking has finished, the array of view factors obtained satisfies conservation but not reciprocity, so a post-correction procedure is necessary to enforce reciprocity. Conservation is also enforced, since as soon as reciprocity is imposed conservation is lost.

The general definitions above describe the direct, unimpeded interaction between two isolated surfaces. In reality, when more surfaces are involved, radiation energy issuing from a patch can reach another patch after having gone through one or more non-opaque surfaces. The radiation energy can also undergo reflection by one or more specularly reflective surfaces.

The S2S radiation model makes it possible to model specular transmission, where radiation emitted from a surface is transmitted through another surface along its original direction. This situation is schematized below for boundaries $B_{1}$ to $B_{4}$ and selected patches:

Computing, for example, the irradiation from patch 2 on $B_{1}$ onto patch 8 on $B_{4}$ , involves summing up the following: the energy which reaches 8 through 5 directly, the energy which reaches 8 through 6 directly, and the energy which reaches 8 through both 6 and 7. Each time a radiation ray is transmitted through a surface, the transmissivity of that surface attenuates its intensity. Therefore, the only way to account properly for all the possible paths is to account for them and the transmissivities at the time when the view factors are calculated. The calculation uses integration over small, discretized solid angles.

The influence of specularly reflective surfaces is similarly considered during the ray tracing process. Specularly reflective surfaces change the direction of rays and potentially the ray strength when the surface is less than fully reflective. Furthermore, surfaces can be both specularly reflective and transmissive. In this special case, each incident ray is split into two child rays -- one transmitted through the surface and one reflected from the surface. The strengths of the resulting two child rays are determined from the strength of the parent ray and by the respective specular transmissivity and reflectivity.

When transmissive and/or specularly reflective surfaces are present, the view factors for a patch pair are global view factors. These global view factors already account for all possible radiation interactions in the space between the two patches. The global view factors no longer are purely topological quantities, and must be recalculated every time the transmissivity or specular reflectivity of a surface in the domain is changed.

Environmental Patch

In addition to the patches on physical boundaries, a virtual patch is created when the view factors calculator detects non-opaque external surfaces. The environmental patch is internal to the code and makes it possible to collect and account for all the energy radiated to and from the environment around the regions. By definition, the environmental patch temperature is the radiation temperature of the thermal environment.

Radiation Equilibrium

After the surfaces have been decomposed into patches and view factors have been calculated for the patch pairs, radiant fluxes on each patch are obtained. The radiant fluxes are obtained by enforcing radiation equilibrium on the entire closed set of surfaces for each radiation spectrum or spectral band independently.

For a given radiation spectrum or spectral band, $λ$ , radiation contributions on each patch can be expressed in terms of the incident, emitted, and transmitted radiation, as illustrated below for a generic patch $i$ . The following diagram shows a schematic representation of radiation fluxes on patch i:

For radiation to balance over the entire closed set of surfaces, all radiation received by each patch must come from the other patches. Summing over all the patches, the total radiation incident on patch $i$ is then:

Figure 7. EQUATION_DISPLAY

Q_{I, i, λ} = S_{i} I_{i, λ} = \sum_{j = 1}^{N_{p}} \frac{F_{j-i, λ}}{(1 - τ_{i, λ} - ρ_{s, i, λ})} J_{d, j, λ} S_{j} + \frac{F_{e-i, λ}}{(1 - τ_{i, λ} - ρ_{s, i, λ})} J_{e, λ} S_{e}

(1694)

where:

$S_{i}, S_{j}$ are the surface areas of patches $i$ and $j$ , respectively
$I_{i}$ is the irradiation or incident radiant flux on patch $i$ [ $\frac{W}{m^{2}}$ ]
$N_{p}$ is the total number of patches
$F_{j-i, λ}$ is the view factor from patch $j$ to $i$ resulting from the approximation of Eqns. Eqn. (1688) to Eqn. (1693) using ray tracing
$J_{d, j, λ}$ is the diffuse component of the radiant flux leaving patch $j$ , sometimes termed the radiosity
$F_{e -i, λ}$ is the view factor from the environmental patch to patch $i$
$τ_{i, λ}$ is the transmissivity
$ρ_{s, i, λ}$ is the specular component of reflectivity

The view factors include the influence of any transmissive or specularly reflective surfaces. Correspondingly, the view factors are effectively normalized using the diffuse portion of the surface properties at patch $i$ : $(1 - τ_{i, λ} - ρ_{s, i, λ})$ .

The general reciprocity relationship for view factors between surfaces that can be transmissive and/or specularly reflective is:

Figure 8. EQUATION_DISPLAY

F_{j-i, λ} = F_{i-j, λ} [\frac{S_{i} (1 - τ_{i, λ} - ρ_{s, i, λ})}{S_{j} (1 - τ_{j, λ} - ρ_{s, j, λ})}]

(1695)

By introducing the reciprocity relationship into Eqn. (1694), the incoming radiation flux becomes:

Figure 9. EQUATION_DISPLAY

I_{i, λ} = \sum_{j = 1}^{N_{p}} \frac{F_{i-j, λ}}{1 - τ_{j, λ} - ρ_{s, j, λ}} J_{d, j, λ} + \frac{F_{i-e, λ}}{1 - τ_{e, λ} - ρ_{s, e, λ}} J_{e, λ}

(1696)

where $J_{d, j, λ}$ is the diffuse radiosity:

Figure 10. EQUATION_DISPLAY

J_{d, j, λ} = {ρ_{d, j, λ} I}_{j, λ} + E_{j, λ} + E_{d, j, λ}

(1697)

where:

$E_{j, λ}$ is the emission based on Planck’s Law
$E_{d, j, λ}$ is the user-specified non-Planck emission

After introducing the effective radiosity as:

Figure 11. EQUATION_DISPLAY

J_{i, λ}^{e f f} = \frac{J_{d, i, λ}}{1 - τ_{i, λ} - ρ_{s, i, λ}}

(1698)

the irradiation can be written as:

Figure 12. EQUATION_DISPLAY

I_{i, λ} = \sum_{j = 1}^{N_{p}} F_{i-j, λ} J_{j, λ}^{e f f} + F_{i-e, λ} J_{e, λ}^{e f f}

(1699)

The total diffuse radiation that patch $i$ emits is the sum of the diffusely reflected and emitted components:

Figure 13. EQUATION_DISPLAY

J_{d, i, λ} = ρ_{d, i, λ} I_{i, λ} + E_{i, λ} + E_{d, i, λ}

(1700)

Combining Eqns. Eqn. (1699) and Eqn. (1700) for each patch yields an equation for $J_{i, λ}^{e f f}$ :

Figure 14. EQUATION_DISPLAY

\begin{matrix} (1 - τ_{i, λ} - ρ_{i, λ}^{s}) (1 - F_{i-i, λ}) {J^{eff}}_{i, λ} - ρ_{d, i, λ} \sum_{(j = 1), (j \neq i)}^{N_{p}} [F_{i-j, λ} {J^{eff}}_{j, λ}] \\ = E_{i, λ} + E_{d, i, λ} + ρ_{d, i, λ} [F_{i-e, λ} (E_{env, λ} + J_{d, source, λ}) + J_{dir, source, λ}] \end{matrix}

(1701)

where:

$J_{d, source, e}$ is a specified external diffuse radiative flux source, for example solar radiation
$J_{dir, source, i, λ}$ is a specified external collimated flux source

To close the system of equations, the emitted energy and environment radiosity on the right-hand side of Eqn. (1701) can be further specified. These terms are defined below for an arbitrary spectrum or band. As such, the definitions are applicable to either the gray or multiband spectrum models. Simplifications for the gray spectrum model (gray thermal radiation) are noted where appropriate.

The surface emissive power for a given spectrum or band is the product of the emissivity and the blackbody emissive power for that spectrum or band:

Figure 15. EQUATION_DISPLAY

E_{i, λ} = ε_{i, λ} Δ_{(λ_{1} \to λ_{2}, T_{i})} σ T_{i}^{4}

(1702)

where:

$Δ_{(λ_{1} \to λ_{2}, T_{i})}$ is the fraction of total blackbody emission for the spectrum or band
$λ_{1}$ and $λ_{2}$ represent the lower and upper wavelength bounds of the spectrum or band
$ε_{i, λ}$ is the emissivity
$σ$ is the Stefan-Boltzmann constant [ $\frac{W}{m^{2} K^{4}}$ ]

The following graph shows the fractional blackbody emission for a band:

For gray radiation, the wavelength bounds are effectively zero and infinity and this factor is simply unity. For multiband radiation, this fraction is a function of the surface temperature and wavelength bounds:

Figure 16. EQUATION_DISPLAY

Δ_{(λ_{1} \to λ_{2}, T)} = Δ_{(0 \to λ_{2}, T)} - Δ_{(0 \to λ_{1}, T)} = f (λ_{2} T) - f (λ_{1} T)

(1703)

The fractional blackbody emission function can be evaluated by either integrating the Planck distribution or by interpolating from a table that is based on this integration [397].

The environment is modeled as a black body with unity emissivity and zero reflectivity and transmissivity. Thus, the effective radiosity of the environment is:

Figure 17. EQUATION_DISPLAY

J_{e, λ}^{e f f} = Δ_{(λ_{1} \to λ_{2}, T_{e})} σ T_{e}^{4}

(1704)

As before, the fraction of total blackbody emission for a given spectrum or band is dependent on the extent of the spectrum or band and the environment temperature. For Gray Thermal Radiation, the fraction reduces to unity.

If solar loads are considered, two more sources can be present—direct solar radiation and diffuse solar radiation. For purposes of apportioning these loads by spectrum or band, the spectral distribution of the solar emissions is assumed to be represented with a blackbody at the average surface temperature of the sun. The direct and diffuse solar fluxes augment the effective radiosity of the environment:

Figure 18. EQUATION_DISPLAY

{F_{i-e, λ} J}_{e, λ}^{e f f} = {F_{i-e, λ} [Δ_{(λ_{1} \to λ_{2}, T_{e})} σ T_{e}^{4} + Δ_{(λ_{1} \to λ_{2}, T_{s u n})} J_{d, solar, e}] + J}_{dir, solar, i, λ}

(1705)

This patch-based quantity is dependent on the specified sun position, and the specified direct solar flux, and the bounds of the given spectral waveband (used for apportioning the total direct flux). This quantity is also dependent on the specified spectrum/band transmissivities and the model geometry.

Substituting Eqns. Eqn. (1702) and Eqn. (1705) into Eqn. (1701) and expanding the $i$ to $i$ terms out of the sum yields the general system of equations for the effective radiosity:

Figure 19. EQUATION_DISPLAY

\begin{matrix} (1 - {τ_{i, λ} - ρ}_{s, i, λ}) (1 - F_{i-i, λ}) J_{i, λ}^{e f f} - ρ_{d, i, λ} \sum_{j = 1, j \neq i}^{N_{p}} F_{i-j, λ} J_{j, λ}^{e f f} \\ = ε_{i, λ} Δ_{(λ_{1} \to λ_{2}, T_{i})} σ T_{i}^{4} + E_{d, i, λ} \\ + ρ_{d, i, λ} {F_{i-e, λ} [Δ_{(λ_{1} \to λ_{2}, T_{e})} σ T_{e}^{4} + Δ_{(λ_{1} \to λ_{2}, T_{s u n})} J_{d, solar, e}] + J_{dir, solar, i, λ}} \end{matrix}

(1706)

For the gray spectrum model, only one system of equations exists, and that system represents the full thermal spectrum. For the multiband spectrum model, a separate system exists for each spectral band. In the absence of solar radiation, the solar load terms are omitted from Eqn. (1706).

Numerical Solution

The patch values for emissive power, reflectivity, and transmissivity are obtained by surface-averaging the cell-face values for all boundary faces that constitute each patch:

Figure 20. EQUATION-DISPLAY

\begin{matrix} E_{i, λ} & = & ε_{i, λ} Δ_{(λ_{1} \to λ_{2}, T_{i})} σ T_{i}^{4} = \frac{\sum_{f_{1} (i)}^{f_{N} (i)} ϵ_{f_{k}, λ} Δ_{(λ_{1} \to λ_{2}, T_{f_{k}})} T_{f_{k}}^{4} S_{f_{k}}}{\sum_{f_{1} (i)}^{f_{N} (i)} S_{f_{k}}} \\ ρ_{i, λ} & = & \frac{\sum_{f_{1} (i)}^{f_{N} (i)} {ρ_{f_{k}, λ} S}_{f_{k}}}{\sum_{f_{1} (i)}^{f_{N} (i)} S_{f_{k}}} \\ τ_{i, λ} & = & \frac{\sum_{f_{1} (i)}^{f_{N} (i)} {τ_{f_{k}, λ} S}_{f_{k}}}{\sum_{f_{1} (i)}^{f_{N} (i)} S_{f_{k}}} \end{matrix}

(1707)

Surface-averaging the face emissive power rather than averaging the face emissivity and temperature independently is required to conserve energy. Given the emissive powers and properties for the entire set of patches, the set of radiation balance equations, Eqn. (1706), is solved in matrix form by the S2S solver and its AMG Linear Solver for each spectrum or band.

Based on the updated values of the effective radiosities, the patch irradiation fluxes are obtained directly from Eqn. (1699). The contribution to the irradiation from the environment in Eqn. (1699) is based on either Eqn. (1704) or Eqn. (1705) depending on whether solar loads are considered. Radiosities are then extracted from the effective radiosities using Eqn. (1698). In the presence of solar radiation, these fluxes include the influence of the solar loads.

The radiative fluxes on the faces are then derived from the patch fluxes. The irradiation (incident flux) at each face f is equal to the irradiation on the associated patch i. The face radiosity is a combination of local face emissive power and the reflected irradiation:

Figure 21. EQUATION_DISPLAY

\begin{matrix} I_{f, λ} = I_{i, λ} \end{matrix}

(1708)

Figure 22. EQUATION_DISPLAY

\begin{matrix} J_{f, λ} = E_{f, λ} + ρ_{f, λ} I_{i, λ} \end{matrix}

(1709)

The reflectivity in the above equation is the full reflectivity (diffuse + specular). The use of the local face emissive power rather than the patch emissive power is critical to ensuring local energy balances when the net radiative flux is later derived from the radiosity.

The full-thermal-spectrum fluxes are calculated by summing the fluxes from the individual spectra or bands:

Figure 23. EQUATION_DISPLAY

\begin{matrix} I_{f} = \sum_{λ \in s p e c t a}^{} I_{f, λ} \end{matrix}

(1710)

Figure 24. EQUATION_DISPLAY

\begin{matrix} J_{f} = \sum_{λ \in s p e c t r a}^{} J_{f, λ} \end{matrix}

(1711)

Figure 25. EQUATION_DISPLAY

{(q_{r})}_{f} = \sum_{λ \in s p e c t r a}^{} {(q_{r})}_{f, λ} = \sum_{λ \in s p e c t a}^{} [(1 - τ_{f, λ}) I_{f, λ} - J_{f, λ}]

(1712)

For the gray spectrum model, the summation operation is trivial; the gray fluxes are the total fluxes. For the multiband spectrum model, the summation extends over all spectral bands. Eqn. (1712) defines the total or full-spectrum net radiative heat flux to the boundary (total absorption minus total emission). A positive value indicates net absorption by the boundary.

Mean Radiant Temperature (MRT)

The MRT is a useful variable for thermal comfort analysis as it provides “the uniform temperature of an imaginary enclosure in which radiant heat transfer from the human body equals the radiant heat transfer in the actual nonuniform enclosure.” See [401].

The MRT is derived by equating the actual heat flux with the idealized heat flux (that from an imaginary room at a uniform temperature and with unity emissivity). By substituting the expressions for the equated heat fluxes and rearranging, the MRT can be expressed in terms of the irradiation:

Figure 26. EQUATION_DISPLAY

{(T_{M R T})}_{f} = {(\frac{I_{f}}{σ})}^{\frac{1}{4}}

(1713)

The irradiation or incident heat flux in the above expression is the result of the surface-to-surface solution. This formulation considers the potential influences of solar radiation, radiation from the thermal environment, transmission through non-opaque surfaces, and emission from non-black surfaces.