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13.3.6 Discrete Ordinates (DO) Radiation Model Theory

The discrete ordinates (DO) radiation model solves the radiative transfer equation (RTE) for a finite number of discrete solid angles, each associated with a vector direction ${\vec s}$ fixed in the global Cartesian system ( $x,y,z$). The fineness of the angular discretization is controlled by you, analogous to choosing the number of rays for the DTRM. Unlike the DTRM, however, the DO model does not perform ray tracing. Instead, the DO model transforms Equation  13.3-1 into a transport equation for radiation intensity in the spatial coordinates ( $x,y,z$). The DO model solves for as many transport equations as there are directions ${\vec s}$. The solution method is identical to that used for the fluid flow and energy equations.

Two implementations of the DO model are available in FLUENT: uncoupled and (energy) coupled. The uncoupled implementation is sequential in nature and uses a conservative variant of the DO model called the finite-volume scheme [ 61, 294], and its extension to unstructured meshes [ 256]. In the uncoupled case, the equations for the energy and radiation intensities are solved one by one, assuming prevailing values for other variables.

Alternatively, in the coupled ordinates method (or COMET) [ 232], the discrete energy and intensity equations at each cell are solved simultaneously, assuming that spatial neighbors are known. The advantages of using the coupled approach is that it speeds up applications involving high optical thicknesses and/or high scattering coefficients. Such applications slow down convergence drastically when the sequential approach is used.



The DO Model Equations


The DO model considers the radiative transfer equation (RTE) in the direction ${\vec s}$ as a field equation. Thus, Equation  13.3-1 is written as


 \nabla \cdot (I({\vec r},{\vec s}){\vec s}) + (a + \sigma_s)... ...ec s} \; ') \; \Phi({\vec s}\cdot {\vec s} \; ') \; d{\Omega}' (13.3-37)

FLUENT also allows the modeling of non-gray radiation using a gray-band model. The RTE for the spectral intensity $I_{\lambda} ({\vec r},{\vec s})$ can be written as


 \nabla \cdot (I_{\lambda}({\vec r},{\vec s}){\vec s}) + (a_{... ...ec s} \; ') \; \Phi({\vec s}\cdot {\vec s} \; ') \; d{\Omega}' (13.3-38)

Here $\lambda$ is the wavelength, $a_{\lambda}$ is the spectral absorption coefficient, and $I_{b \lambda}$ is the black body intensity given by the Planck function. The scattering coefficient, the scattering phase function, and the refractive index $n$ are assumed independent of wavelength.

The non-gray DO implementation divides the radiation spectrum into $N$ wavelength bands, which need not be contiguous or equal in extent. The wavelength intervals are supplied by you, and correspond to values in vacuum ( $n=1$). The RTE is integrated over each wavelength interval, resulting in transport equations for the quantity $I_{\lambda}\Delta \lambda$, the radiant energy contained in the wavelength band $\Delta \lambda$. The behavior in each band is assumed gray. The black body emission in the wavelength band per unit solid angle is written as


[F(0 \rightarrow n\lambda_2 T) - F(0 \rightarrow n \lambda_1 T)]n^2 \frac{\sigma T^4}{\pi} (13.3-39)

where $F(0 \rightarrow n \lambda T)$ is the fraction of radiant energy emitted by a black body [ 248] in the wavelength interval from 0 to $\lambda$ at temperature $T$ in a medium of refractive index $n$. $\lambda_2$ and $\lambda_1$ are the wavelength boundaries of the band.

The total intensity $I({\vec r},{\vec s})$ in each direction ${\vec s}$ at position ${\vec r}$ is computed using


 I({\vec r},{\vec s}) = \sum_k I_{\lambda_k}({\vec r}, {\vec s}) \Delta \lambda_k (13.3-40)

where the summation is over the wavelength bands.

Boundary conditions for the non-gray DO model are applied on a band basis. The treatment within a band is the same as that for the gray DO model.



Energy Coupling and the DO Model


The coupling between energy and radiation intensities at a cell (which is also known as COMET) [ 232] accelerates the convergence of the finite volume scheme for radiative heat transfer. This method results in significant improvement in the convergence for applications involving optical thicknesses greater than 10. This is typically encountered in glass-melting applications. This feature is advantageous when scattering is significant, resulting in strong coupling between directional radiation intensities. This DO model implementation is utilized in FLUENT by enabling the DO/Energy Coupling option for the DO model in the Radiation Model panel. The discrete energy equations for the coupled method are presented below.

The energy equation when integrated over a control volume $i$, yields the discrete energy equation:


 \sum_{j=1}^{N} \mu_{ij}^{T} T_j - \beta_{i}^{T} T_i - \alpha... ...^{T} \sum_{k=1}^{L} I_{i}^{k} \omega_{k} - S_{i}^{T} S_{i}^{h} (13.3-41)


where $\alpha_{i}^{T}$ = $\kappa \Delta V_{i}$
  $\beta_{i}^{T}$ = $ 16 \kappa \sigma T_{i}^{*3} \Delta V_{i}$
  $S_{i}^{T}$ = $ 12 \kappa \sigma T_{i}^{*4} \Delta V_{i}$
  $\kappa$ = absorption coefficient
  $\Delta V$ = control volume

The coefficient $\mu_{ij}^{T}$ and the source term $S_{i}^{h}$ are due to the discretization of the convection and diffusion terms as well as the non-radiative source terms.

Combining the discretized form of Equation  13.3-37 and the discretized energy equation, Equation  13.3-41, yields [ 232]:


 \vec{P}_{i} \vec{q}_{i} + \vec{r}_{i} = 0 (13.3-42)

where


 \vec{q}_{i} = \left[ \begin{array}{c} I_{i}^{1}\\ I_{i}^{2}\\ :\\ :\\ I_{i}^{L}\\ T_{i} \end{array}\right] (13.3-43)


 \vec{P}_{i} = \left[ \begin{array}{ccc} M_{ii}^{1} + \eta_{i... ...-\alpha_{i}^{T}\omega_{2} & ... M_{ii}^{T} \end{array}\right] (13.3-44)


 \vec{r}_{i} = \left[ \begin{array}{c} \sum_{j=1, i \neq j}^{... ...\mu_{ij}^{T} T_{j} + S_{i}^{T} + S_{i}^{h} \end{array}\right] (13.3-45)

Limitations of DO/Energy Coupling

There are some instances when using DO/Energy coupling is not recommended or is incompatible with certain models:

To find out how to apply DO/Energy coupling, refer to Section  13.3.13.



Angular Discretization and Pixelation


Each octant of the angular space $4\pi$ at any spatial location is discretized into $N_\theta \times N_{\phi}$ solid angles of extent $\omega_i$, called control angles. The angles $\theta$ and $\phi$ are the polar and azimuthal angles respectively, and are measured with respect to the global Cartesian system $(x,y,z)$ as shown in Figure  13.3.3. The $\theta$ and $\phi$ extents of the control angle, $\Delta \theta$ and $\Delta \phi$, are constant. In two-dimensional calculations, only four octants are solved due to symmetry, making a total of $4N_\theta N_{\phi}$ directions in all. In three-dimensional calculations, a total of $8 N_\theta N_{\phi}$ directions are solved. In the case of the non-gray model, $4N_\theta N_{\phi}$ or $8 N_\theta N_{\phi}$ equations are solved for each band.

Figure 13.3.3: Angular Coordinate System
figure

When Cartesian meshes are used, it is possible to align the global angular discretization with the control volume face, as shown in Figure  13.3.4. For generalized unstructured meshes, however, control volume faces do not in general align with the global angular discretization, as shown in Figure  13.3.5, leading to the problem of control angle overhang [ 256].

Figure 13.3.4: Face with No Control Angle Overhang
figure

Figure 13.3.5: Face with Control Angle Overhang
figure

Essentially, control angles can straddle the control volume faces, so that they are partially incoming and partially outgoing to the face. Figure  13.3.6 shows a 3D example of a face with control angle overhang.

Figure 13.3.6: Face with Control Angle Overhang (3D)
figure

The control volume face cuts the sphere representing the angular space at an arbitrary angle. The line of intersection is a great circle. Control angle overhang may also occur as a result of reflection and refraction. It is important in these cases to correctly account for the overhanging fraction. This is done through the use of pixelation [ 256].

Each overhanging control angle is divided into $N_{\theta_p} \times N_{\phi_p}$ pixels, as shown in Figure  13.3.7.

Figure 13.3.7: Pixelation of Control Angle
figure

The energy contained in each pixel is then treated as incoming or outgoing to the face. The influence of overhang can thus be accounted for within the pixel resolution. FLUENT allows you to choose the pixel resolution. For problems involving gray-diffuse radiation, the default pixelation of $1\times1$ is usually sufficient. For problems involving symmetry, periodic, specular, or semi-transparent boundaries, a pixelation of $3\times3$ is recommended. You should be aware, however, that increasing the pixelation adds to the cost of computation.



Anisotropic Scattering


The DO implementation in FLUENT admits a variety of scattering phase functions. You can choose an isotropic phase function, a linear anisotropic phase function, a Delta-Eddington phase function, or a user-defined phase function. The linear anisotropic phase function is described in Equation  13.3-7. The Delta-Eddington function takes the following form:


 \Phi({\vec s} \cdot {\vec s} \; ') = 2f\delta({\vec s} \cdot {\vec s} \; ') + (1-f)(1 + C {\vec s} \cdot {\vec s} \; ') (13.3-46)

Here, $f$ is the forward-scattering factor and $\delta({\vec s} \cdot {\vec s} \; ')$ is the Dirac delta function. The $f$ term essentially cancels a fraction $f$ of the out-scattering; thus, for $f=1$, the Delta-Eddington phase function will cause the intensity to behave as if there is no scattering at all. $C$ is the asymmetry factor. When the Delta-Eddington phase function is used, you will specify values for $f$ and $C$.

When a user-defined function is used to specify the scattering phase function, FLUENT assumes the phase function to be of the form


 \Phi({\vec s} \cdot {\vec s} \; ') = 2f\delta({\vec s} \cdot {\vec s} \; ') + (1-f)\Phi^*({\vec s} \cdot {\vec s} \; ') (13.3-47)

The user-defined function will specify $\Phi^*$ and the forward-scattering factor $f$.

The scattering phase functions available for gray radiation can also be used for non-gray radiation. However, the scattered energy is restricted to stay within the band.



Particulate Effects in the DO Model


The DO model allows you to include the effect of a discrete second phase of particulates on radiation. In this case, FLUENT will neglect all other sources of scattering in the gas phase.

The contribution of the particulate phase appears in the RTE as:


 \nabla \cdot (I{\vec s}) + (a + a_{p} + \sigma_p) I({\vec r}... ...c s} \; ') \; \Phi({\vec s} \cdot {\vec s} \; ') \; d{\Omega}' (13.3-48)

where $a_p$ is the equivalent absorption coefficient due to the presence of particulates, and is given by Equation  13.3-10. The equivalent emission $E_p$ is given by Equation  13.3-9. The equivalent particle scattering factor $\sigma_p$, defined in Equation  13.3-13, is used in the scattering terms.

For non-gray radiation, absorption, emission, and scattering due to the particulate phase are included in each wavelength band for the radiation calculation. Particulate emission and absorption terms are also included in the energy equation.



Boundary Condition Treatment at Opaque Walls


The discrete ordinates radiation model allows the specification of opaque walls that are interior to a domain (with adjacent fluid or solid zones on both sides of the wall), or external to the domain (with an adjacent fluid or solid zone on one side, only). Opaque walls are treated as gray if gray radiation is being computed, or non-gray if the non-gray DO model is being used.

Figure  13.3.8 shows a schematic of radiation on an opaque wall in FLUENT.

Figure 13.3.8: DO Radiation on Opaque Wall
figure

The diagram in Figure  13.3.8 shows incident radiation $q_{\rm in, a}$ on side a of an opaque wall. Some of the radiant energy is reflected diffusely and specularly, depending on the diffuse fraction $f_d$ for side a of the wall that you specify as a boundary condition.

Some of the incident radiation is absorbed at the surface of the wall and some radiation is emitted from the wall surface as shown in Figure  13.3.8. The amount of incident radiation absorbed at the wall surface and the amount emitted back depends on the emissivity of that surface and the diffuse fraction. For non-gray DO models, you must specify internal emissivity for each wavelength band. Radiation is not transmitted through an opaque wall.

Radiant incident energy that impacts an opaque wall can be reflected back to the surrounding medium and absorbed by the wall. The radiation that is reflected can be diffusely reflected and/or specularly reflected, depending on the diffuse fraction $f_d$. If $q_{\rm in}$ is the amount of radiative energy incident on the opaque wall, then the following general quantities are computed by FLUENT for opaque walls:

where $f_d$ is the diffuse fraction, $n$ is the refractive index of the adjacent medium, $\epsilon_w$ is the wall emissivity, $\sigma$ is Boltzmann's Constant, and $T_w$ is the wall temperature.

figure   

There is no emission or absorption in the specular component of reflected energy for an opaque wall surface.

Note that although FLUENT uses emissivity in its computation of radiation quantities, it is not available for postprocessing. Absorption at the wall surface assumes that the absorptivity is equal to the emissivity. For a purely diffused wall, $f_d$ is equal to $1$ and there is no specularly reflected energy. Similarly, for a purely specular wall, $f_d$ is equal to $0$ and there is no diffusely reflected energy. A diffuse fraction between $0$ and $1$ will result in partially diffuse and partially reflected energy.

figure   

Note that in practice, fully specular surfaces (diffuse fraction = $0$) do have some emission and absorption occurring, albeit very low. To define a problem that has a purely specular surface with emissivity, you can set the internal emissivity to $1$ and set the diffuse fraction to the intended emissivity. To define a problem that has a partially-specular surface, set the internal emissivity to suit the net surface emissivity (not necessarily $0$) that you expect to achieve (internal emissivity $=$ net emissivity $/$diffuse fraction).

Gray Diffuse Walls

For gray diffuse radiation, the incident radiative heat flux, $q_{\rm in}$, at the wall is


 q_{\rm in} = \int_{{\vec s} \cdot {\vec n} > 0} I_{\rm in} {\vec s} \cdot {\vec n} d \Omega (13.3-49)

The net radiative flux leaving the surface is given by


 q_{\rm out} = (1 - \epsilon_w)q_{\rm in} + n^2 \epsilon_w \sigma T^{4}_{w} (13.3-50)

where $n$ is the refractive index of the medium next to the wall, $\epsilon_w$ is the wall emissivity, $\sigma$ is Boltzmann's Constant, and $T_w$ is the wall temperature. This equation is also valid for specular radiation with emissivity = $0$.

The boundary intensity for all outgoing directions ${\vec s}$ at the wall is given by


 I_0 = \frac{q_{\rm out}}{\pi} (13.3-51)

Non-Gray Diffuse Walls

There is a special set of equations that apply uniquely to non-gray diffuse opaque walls. These equations assume that the absorptivity is equal to the emissivity for the wall surface. For non-gray diffuse radiation, the incident radiative heat flux $q_{{\rm in},\lambda}$ in the band $\Delta \lambda$ at the wall is


 q_{{\rm in}, \lambda} = \Delta \lambda \int_{{\vec s} \cdot ... ... n} > 0} I_{{\rm in},\lambda} {\vec s} \cdot {\vec n} d \Omega (13.3-52)

The net radiative flux leaving the surface in the band $\Delta \lambda$ is given by


 q_{{\rm out}, \lambda} = (1 - \epsilon_{w \lambda})q_{{\rm i... ... T_w) - F(0 \rightarrow n \lambda_1 T_w)] n^2 \sigma T^{4}_{w} (13.3-53)

where $\epsilon_{w \lambda}$ is the wall emissivity in the band. $F(n,\lambda,T)$ provides the Planck distribution function. The boundary intensity for all outgoing directions ${\vec s}$ in the band $\Delta \lambda$ at the wall is given by


 I_{0 \lambda} = \frac{q_{{\rm out},\lambda}}{\pi \Delta \lambda} (13.3-54)



Boundary Condition Treatment at Semi-Transparent Walls


FLUENT allows the specification of interior and exterior semi-transparent walls for the DO model. In the case of interior semi-transparent walls, incident radiation can pass through the wall and be transmitted to the adjacent medium (and possibly refracted), it can be reflected back into the surrounding medium, and absorbed through the wall thickness. Transmission and reflection can be diffuse and/or specular. You specify the diffuse fraction for all transmitted and reflected radiation; the rest is treated specularly. For exterior semi-transparent walls, there are two possible sources of radiation on the boundary wall: an irradiation beam from outside the computational domain and incident radiation from cells in adjacent fluid or solid zones.

For non-gray radiation, semi-transparent wall boundary conditions are applied on a per-band basis. The radiant energy within a band is transmitted, reflected, and refracted as in the gray case; there is no transmission, reflection, or refraction of radiant energy from one band to another.

By default the DO equations are solved in all fluid zones, but not in any solid zones. Therefore, if you have an adjacent solid zone for your thin wall, you will need to specify the solid zone as participating in radiation in the Solid panel as part of the boundary condition setup.

figure   

If you are interested in the detailed temperature distribution inside your semi-transparent media, then you will need to model a semi-transparent wall as a solid zone with adjacent fluid zone(s), and treat the solid as a semi-transparent medium. This is discussed in a subsequent section.

Semi-Transparent Interior Walls

Figure  13.3.9 shows a schematic of an interior (two-sided) wall that is treated as semi-transparent in FLUENT and has zero thickness. Incident radiant energy depicted by $q_{\rm in,a}$ can pass through the semi-transparent wall if and only if the contiguous fluid or solid cell zones participate in radiation, thereby allowing the radiation to be coupled. Radiation coupling is set when a wall is specified as semi-transparent. Note that by default, radiation is not coupled and you will need to explicitly specify radiation coupling on the interior wall by changing the boundary condition type to semi-transparent in the Wall panel (under the Radiation tab).

Figure 13.3.9: DO Radiation on Interior Semi-Transparent Wall
figure

Incident radiant energy that is transmitted through a semi-transparent wall can be transmitted specularly and diffusely. Radiation can also be reflected at the interior wall back to the surrounding medium if the refractive index $n_a$ for the fluid zone that represents medium $a$ is different than the refractive index $n_b$ for medium $b$. Reflected radiation can be reflected specularly and diffusely. The fraction of diffuse versus specular radiation that is transmitted and reflected depends on the diffuse fraction for the wall. The special cases of purely diffuse and purely specular transmission and reflection on semi-transparent walls is presented in the following sections.

If the semi-transparent wall has thickness, then the thickness and the absorption coefficient determine the absorptivity of the 'thin' wall. If either the wall thickness or absorption coefficient is set to $0$, then the wall has no absorptivity. Although incident radiation can be absorbed in a semi-transparent wall that has thickness, note that the absorbed radiation flux does not affect the energy equation. FLUENT does not consider emission from semi-transparent walls except for the case when a specified temperature boundary condition is defined.

Specular Semi-Transparent Walls

Consider the special case for a semi-transparent wall, when the diffuse fraction $f_d$ is equal to $0$ and all of the transmitted and reflected radiant energy at the semi-transparent wall is purely specular.

Figure  13.3.10 shows a ray traveling from a semi-transparent medium $a$ with refractive index $n_a$ to a semi-transparent medium $b$ with a refractive index $n_b$ in the direction ${\vec s}$. Surface $a$ of the interface is the side that faces medium $a$; similarly, surface $b$ faces medium $b$. The interface normal ${\vec n}$ is assumed to point into side $a$. We distinguish between the intensity $I_{a}({\vec s})$, the intensity in the direction ${\vec s}$ on side $a$ of the interface, and the corresponding quantity on the side $b$, $I_{b}({\vec s})$.

Figure 13.3.10: Reflection and Refraction of Radiation at the Interface Between Two Semi-Transparent Media
figure

A part of the energy incident on the interface is reflected, and the rest is transmitted. The reflection is specular, so that the direction of reflected radiation is given by


 {\vec s}_r = {\vec s} - 2\left({\vec s} \cdot {\vec n}\right){\vec n} (13.3-55)

The radiation transmitted from medium $a$ to medium $b$ undergoes refraction. The direction of the transmitted energy, ${\vec s}_t$, is given by Snell's law:


 \sin \theta_{b} = \frac{n_a}{n_b} \sin \theta_{a} (13.3-56)

where $\theta_{a}$ is the angle of incidence and $\theta_{b}$ is the angle of transmission, as shown in Figure  13.3.10. We also define the direction


 {\vec s}\; ' = {\vec s}_t - 2\left( {\vec s}_t \cdot {\vec n}\right){\vec n} (13.3-57)

shown in Figure  13.3.10.

The interface reflectivity on side $a$ [ 248]


 r_a({\vec s}) = \frac{1}{2} \left(\frac{n_a \cos \theta_b - ... ...\cos \theta_b}{n_a \cos \theta_a + n_b \cos \theta_b}\right)^2 (13.3-58)

represents the fraction of incident energy transferred from ${\vec s}$ to ${\vec s}_r$.

The boundary intensity $I_{w,a}({\vec s}_r)$ in the outgoing direction ${\vec s}_r$ on side $a$ of the interface is determined from the reflected component of the incoming radiation and the transmission from side $b$. Thus


 I_{w,a}({\vec s}_r) = r_a({\vec s}) I_{w,a}({\vec s}) + \tau_b({\vec s} \; ')I_{w,b}({\vec s} \; ') (13.3-59)

where $\tau_b({\vec s} \; ')$ is the transmissivity of side $b$ in direction ${\vec s} '$. Similarly, the outgoing intensity in the direction ${\vec s}_t$ on side $b$ of the interface, $I_{w,b}({\vec s}_t)$, is given by


 I_{w,b}({\vec s}_t) = r_b({\vec s} \; ') I_{w,b}({\vec s} \; ') + \tau_a({\vec s})I_{w,a}({\vec s}) (13.3-60)

For the case $n_a < n_b$, the energy transmitted from medium $a$ to medium $b$ in the incoming solid angle $2\pi$ must be refracted into a cone of apex angle $\theta_c$ (see Figure  13.3.11) where


 \theta_c = \sin^{-1} \frac{n_a}{n_b} (13.3-61)

Figure 13.3.11: Critical Angle $\theta_c$
figure

Similarly, the transmitted component of the radiant energy going from medium $b$ to medium $a$ in the cone of apex angle $\theta_c$ is refracted into the outgoing solid angle $2\pi$. For incident angles greater than $\theta_c$, total internal reflection occurs and all the incoming energy is reflected specularly back into medium $b$. The equations presented above can be applied to the general case of interior semi-transparent walls that is shown in Figure  13.3.9.

When medium $b$ is external to the domain as in the case of an external semi-transparent wall (Figure  13.3.12), $I_{w,b}({\vec s} \; ')$ is given in Equation  13.3-59 as a part of the boundary condition inputs. You supply this incoming irradiation flux in terms of its magnitude, beam direction, and the solid angle over which the radiative flux is to be applied. Note that the refractive index of the external medium is assumed to be $1$.

Diffuse Semi-Transparent Walls

Consider the special case for a semi-transparent wall, when the diffuse fraction $f_d$ is equal to $1$ and all of the transmitted and reflected radiant energy at the semi-transparent wall is purely diffuse.

In many engineering problems, the semi-transparent interface may be a diffuse reflector. For such a case, the interfacial reflectivity $r({\vec s})$ is assumed independent of ${\vec s}$, and equal to the hemispherically averaged value $r_d$. For $n = n_a/n_b > 1$, $r_{d,a}$ and $r_{d,b}$ are given by [ 333]


$\displaystyle r_{d,a}$ $\textstyle =$ $\displaystyle 1 - \frac{(1-r_{d,b})}{n^2}$ (13.3-62)
$\displaystyle r_{d,b}$ $\textstyle =$ $\displaystyle \frac{1}{2} + \frac{( 3n+1)(n-1)}{6(n+1)^2} + \frac{n^2(n^2-1)^2}{(n^2+1)^3} \ln \left(\frac{n-1}{n+1} \right) - \; \;$  
    $\displaystyle \frac{2n^3(n^2 + 2n -1)}{(n^2+1)(n^4-1)} + \frac{8n^4(n^4+1)}{(n^2+1)(n^4-1)^2} \ln(n)$ (13.3-63)

The boundary intensity for all outgoing directions on side $a$ of the interface is given by


 I_{w,a} = \frac{r_{d,a} q_{{\rm in},a} + \tau_{d,b} q_{{\rm in},b}}{\pi} (13.3-64)

Similarly for side $b$,


 I_{w,b} = \frac{r_{d,b} q_{{\rm in},b} + \tau_{d,a} q_{{\rm in},a}}{\pi} (13.3-65)

where


$\displaystyle q_{{\rm in},a}$ $\textstyle =$ $\displaystyle -\int_{4\pi}I_{w,a} {\vec s} \cdot {\vec n} d\Omega, \; \; \; {\vec s}\cdot {\vec n} <0$ (13.3-66)
$\displaystyle q_{{\rm in},b}$ $\textstyle =$ $\displaystyle \int_{4\pi}I_{w,b} {\vec s} \cdot {\vec n} d\Omega, \; \; \; {\vec s}\cdot {\vec n} \ge 0$ (13.3-67)

When medium $b$ is external to the domain as in the case of an external semi-transparent wall (Figure  13.3.12), $q_{\rm in,b}$ is given as a part of the boundary condition inputs. You supply this incoming irradiation flux in terms of its magnitude, beam direction, and the solid angle over which the radiative flux is to be applied. Note that the refractive index of the external medium is assumed to be $1$.

Partially Diffuse Semi-Transparent Walls

When the diffuse fraction $f_d$ that you enter for a semi-transparent wall is between $0$ and $1$, the wall is partially diffuse and partially specular. In this case, FLUENT includes the reflective and transmitted radiative flux contributions from both diffuse and specular components to the defining equations.

Semi-Transparent Exterior Walls

Figure  13.3.12 shows the general case of an irradiation beam $q_{\rm irrad}$ applied to an exterior semi-transparent wall with zero thickness and a non-zero absorption coefficient for the material property. Refer to the previous section for the radiation effects of wall thickness on semi-transparent walls.

Figure 13.3.12: DO Irradiation on External Semi-Transparent Wall
figure

An irradiation flux passes through the semi-transparent wall from outside the computational domain (Figure  13.3.12) into the adjacent fluid or solid medium a. The transmitted radiation can be refracted (bent) and dispersed specularly and diffusely, depending on the refractive index and the diffuse fraction that you provide as a boundary condition input. Note that there is a reflected component of $q_{\rm irrad}$ when the refractive index of the wall ( $n_b$) is not equal to $1$, as shown.

There is an additional flux beyond $q_{\rm irrad}$ that is applied when the Mixed or Radiation boundary conditions are selected in the Thermal tab. This external flux at the semi-transparent wall is computed by FLUENT as


 Q_{ext} = \epsilon_{external} \sigma T_{rad}^4 (13.3-68)

The fraction of the above energy that will enter into the domain depends on the transmissivity of the semi-transparent wall under consideration. Note that this energy is distributed across the solid angles (i.e., similar treatment as diffuse component.)

Incident radiation can also occur on external semi-transparent walls. Refer to the previous discussion on interior walls for details, since the radiation effects are the same.

The irradiation beam is defined by the magnitude, beam direction, and beam width that you supply. The irradiation magnitude is specified in terms of an incident radiant heat flux (W/m $^2$). Beam width is specified as the solid angle over which the irradiation is distributed (i.e., the beam $\theta$ and $\phi$ extents). The default beam width in FLUENT is $1e-6$ degrees which is suitable for collimated beam radiation. Beam direction is defined by the vector of the centroid of the solid angle. If you select the feature Apply Irradiation Parallel to Beam in the Wall boundary condition panel, then you supply $q_{\rm irrad}$ for irradiation (Figure  13.3.12) and FLUENT computes and uses the surface normal flux $q_{\rm irrad,normal}$ in its radiation calculation. If this feature is not checked, then you must supply the surface normal flux $q_{\rm irrad,normal}$ for irradiation.

Figure  13.3.13 shows a schematic of the beam direction and beam width for the irradiation beam. You provide these inputs (in addition to irradiation magnitude) as part of the boundary conditions for a semi-transparent wall.

Figure 13.3.13: Beam Width and Direction for External Irradiation Beam
figure

The irradiation beam can be refracted in medium a depending on the refractive index that is specified for the particular fluid or solid zone material.

Limitations

The thin wall treatment should be used for semi-transparent walls only where absorption or emission by the walls is not significant. In cases with significant emission or absorption of radiation in a participating solid material, such as the absorption of long wavelength radiation in a glass window, the use of semi-transparent thin walls can result in the prediction of unphysical temperatures in the numerical solution. To avoid potentially erroneous results, one or more solid cell zones should be used to represent the solid material in such a case.

Solid Semi-Transparent Media

The discrete ordinates radiation model allows you to model a solid zone that has adjacent fluid or solid zones on either side as a "semi-transparent'' medium. This is done by designating the solid zone to participate in radiation as part of the boundary condition setup. Modeling a solid zone as a semi-transparent medium allows you to obtain a detailed temperature distribution inside the semi-transparent zone since FLUENT solves the energy equation on a per-cell basis for the solid and provides you with the thermal results. By default however, the DO equations are solved in fluid zones, but not in any solid zones. Therefore, you will need to specify the solid zone as participating in radiation in the Solid panel as part of the boundary condition setup.



Boundary Condition Treatment at Specular Walls and Symmetry Boundaries


At specular walls and symmetry boundaries, the direction of the reflected ray ${\vec s}_r$ corresponding to the incoming direction ${\vec s}$ is given by Equation  13.3-55. Furthermore,


 I_w({\vec s}_r) = I_w({\vec s}) (13.3-69)



Boundary Condition Treatment at Periodic Boundaries


When rotationally periodic boundaries are used, it is important to use pixelation in order to ensure that radiant energy is correctly transferred between the periodic and shadow faces. A pixelation between $3\times 3$ and $10\times 10$ is recommended.



Boundary Condition Treatment at Flow Inlets and Exits


The treatment at flow inlets and exits is described in Section  13.3.5.


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