XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX''"> 8.3 Mean Beam Length
For the enclosure with an isothermal participating medium (in equilibrium with itself), we may define a quantity analogous to the viewfactor above, called the exchange factor, by the equation:
(8‑2)
subject to the restrictions that
does not intersect any view obstructing surfaces and the product
is negative, and
e | = | Surface e ID |
f | = | Surface f ID |
i | = | Node ID on surface e |
j | = | Node ID on surface f |
Ae | = | Surface e area |
Af | = | Surface f area |
k | | media extinction coefficient |
| = | Primary variable finite element interpolation function associated with the ith node on surface e |
| = | Vector from a point on surface e to a point on surface f |
| = | Unit normal to surface e |
| = | Magnitude of a vector |
For the optically thin media, the exponential function in the above equation may be approximated by:
(8‑3)
Using this approximation and assuming that the extinction coefficient is constant (or close enough to constant that it may be taken outside the integration), define the mean beam length by the equation:
(8‑4)
Then in Patran Thermal the exchange factor is approximated by the equation:
(8‑5)
In this way the geometry of the model is isolated in the viewfactor and mean beam length quantities. The material properties, such as the extinction coefficient, may be time and temperature dependent functions in Patran Thermal without requiring recalculation of the viewfactor and mean beam length. This results in great saving of computer processing time.
The assumptions made for the participating media model in Viewfactor and Patran Thermal are as follows:
1. The media is isothermal;
2. The media is in equilibrium with its internal energy;
3. The media extinction coefficient is not a function of position;
4. The optical thickness is small enough that the linear approximation of the exponential function is reasonable and the media is only weakly interacting with itself.
