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13.3.5 Discrete Transfer Radiation Model (DTRM) Theory

The main assumption of the DTRM is that the radiation leaving the surface element in a certain range of solid angles can be approximated by a single ray. This section provides details about the equations used in the DTRM.



The DTRM Equations


The equation for the change of radiant intensity, $dI$, along a path, $ds$, can be written as


 \frac{dI}{ds} + a I = \frac{a \sigma T^4} {\pi} (13.3-30)


where $a$ = gas absorption coefficient
  $I$ = intensity
  $T$ = gas local temperature
  $\sigma$ = Stefan-Boltzmann constant (5.672 $\times$ 10 $^{-8}$ W/m $^2$-K $^4$)

Here, the refractive index is assumed to be unity. The DTRM integrates Equation  13.3-30 along a series of rays emanating from boundary faces. If $a$ is constant along the ray, then $I(s)$ can be estimated as


 I(s) = \frac{\sigma T^{4}}{\pi} (1- e^{-as}) + I_0 e^{-as} (13.3-31)

where $I_0$ is the radiant intensity at the start of the incremental path, which is determined by the appropriate boundary condition (see the description of boundary conditions, below). The energy source in the fluid due to radiation is then computed by summing the change in intensity along the path of each ray that is traced through the fluid control volume.

The "ray tracing'' technique used in the DTRM can provide a prediction of radiative heat transfer between surfaces without explicit view-factor calculations. The accuracy of the model is limited mainly by the number of rays traced and the computational grid.



Ray Tracing


The ray paths are calculated and stored prior to the fluid flow calculation. At each radiating face, rays are fired at discrete values of the polar and azimuthal angles (see Figure  13.3.2). To cover the radiating hemisphere, $\theta$ is varied from $0$ to $\frac{\pi}{2}$ and $\phi$ from $0$ to $2 \pi$. Each ray is then traced to determine the control volumes it intercepts as well as its length within each control volume. This information is then stored in the radiation file, which must be read in before the fluid flow calculations begin.

Figure 13.3.2: Angles $\theta$ and $\phi$ Defining the Hemispherical Solid Angle About a Point $P$
figure



Clustering


DTRM is computationally very expensive when there are too many surfaces to trace rays from and too many volumes crossed by the rays. To reduce the computational time, the number of radiating surfaces and absorbing cells is reduced by clustering surfaces and cells into surface and volume "clusters''. The volume clusters are formed by starting from a cell and simply adding its neighbors and their neighbors until a specified number of cells per volume cluster is collected. Similarly, surface clusters are made by starting from a face and adding its neighbors and their neighbors until a specified number of faces per surface cluster is collected.

The incident radiation flux, $q_{\rm in}$, and the volume sources are calculated for the surface and volume clusters respectively. These values are then distributed to the faces and cells in the clusters to calculate the wall and cell temperatures. Since the radiation source terms are highly non-linear (proportional to the fourth power of temperature), care must be taken to calculate the average temperatures of surface and volume clusters and distribute the flux and source terms appropriately among the faces and cells forming the clusters.

The surface and volume cluster temperatures are obtained by area and volume averaging as shown in the following equations:


 T_{sc} = \left(\frac{\sum_f A_f T_f^4}{\sum A_f} \right) ^{1/4} (13.3-32)


 T_{vc} = \left(\frac{\sum_c V_c T_c^4}{\sum V_c} \right) ^{1/4} (13.3-33)

where $T_{sc}$ and $T_{vc}$ are the temperatures of the surface and volume clusters respectively, $A_f$ and $T_f$ are the area and temperature of face $f$, and $V_c$ and $T_c$ are the volume and temperature of cell $c$. The summations are carried over all faces of a surface cluster and all cells of a volume cluster.



Boundary Condition Treatment for the DTRM at Walls


The radiation intensity approaching a point on a wall surface is integrated to yield the incident radiative heat flux, $q_{\rm in}$, as


 q_{\rm in} = \int_{{\vec s} \cdot {\vec n} > 0} I_{\rm in} {\vec s} \cdot {\vec n} d \Omega (13.3-34)

where $\Omega$ is the hemispherical solid angle, $I_{\rm in}$ is the intensity of the incoming ray, ${\vec s}$ is the ray direction vector, and ${\vec n}$ is the normal pointing out of the domain. The net radiative heat flux from the surface, $q_{\rm out}$, is then computed as a sum of the reflected portion of $q_{\rm in}$ and the emissive power of the surface:


 q_{\rm out} = (1 - \epsilon_w)q_{\rm in} + \epsilon_w \sigma T^{4}_{w} (13.3-35)

where $T_{w}$ is the surface temperature of the point $P$ on the surface and $\epsilon_w$ is the wall emissivity which you input as a boundary condition. FLUENT incorporates the radiative heat flux (Equation  13.3-35) in the prediction of the wall surface temperature. Equation  13.3-35 also provides the surface boundary condition for the radiation intensity $I_0$ of a ray emanating from the point $P$, as


 I_0 = \frac{q_{\rm out}}{\pi} (13.3-36)



Boundary Condition Treatment for the DTRM at Flow Inlets and Exits


The net radiative heat flux at flow inlets and outlets is computed in the same manner as at walls, as described above. FLUENT assumes that the emissivity of all flow inlets and outlets is 1.0 (black body absorption) unless you choose to redefine this boundary treatment.

FLUENT includes an option that allows you to use different temperatures for radiation and convection at inlets and outlets. This can be useful when the temperature outside the inlet or outlet differs considerably from the temperature in the enclosure. See Section  13.3.15 for details.


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