The strategy for inter-particle radiation involves adapting Ray-tracing for evaluating view fac- tors and then solving for radiation exchange between the interacting surfaces using a novel algorithm. View factors in radiation exchange problems are purely geometrical quantities that indicate how surfaces view each other and consequently determining the radiation exchange between them. Besides few simple cases, deriving analytical formula for view factor calcula- tion can be extremely challenging or impossible. Statistical Ray-tracing approach is a suitable alternative for cases involving complex surfaces.
A1
A2
A3
F12 = N2/P N
F13= N3/P N
Figure 3.10 Schematic for evaluating view factor between interacting surfaces
Ray-tracing method to calculate view factor is explained here for a sample case shown in Fig. 3.10, where N2 and N3 are number of rays intercepted by the surfaces 2 and 3 respectively and
ΣN is total number of rays leaving the surface 1. By definition the view factor is the fraction of total emitted radiation from one surface that is intercepted by another surface. If we represent the radiation leaving a surface by sufficiently large number of discrete rays and register the total number of such rays intercepted by other surfaces, then the ratio of intercepted rays to total rays fired can be used as view factor.
The method is very simple to apply and nearly independent of geometrical complexities. However, it is important that average behaviour of the rays is representative of actual radiation leaving the surface, by ensuring uniform ray density and following directional or spectral de- pendence (if any). This is achieved by dividing the emitting surface into a number of elemental areas, where each area acts as an origin for multiple rays which are fired in different directions at random while following the corresponding probability distribution of emitting radiation (e.g. ray direction for a diffuse surface is given by ϕ = 2πΥϕ and θ = sin−1
√
Υθ, where Υθ and Υϕ
are random numbers between 0 and 1) [64].
In order to obtain uniformly distributed points on a spherical surface, the following method is used. In spherical coordinate system a point in three-dimensional space is defined by (r, θ, ϕ) as shown in Fig. 3.11, where θ and ϕ are zenith and azimuth angles, respectively, such that 0 6 θ 6 π and 0 < ϕ 6 2π. For obtaining evenly distributed points on a sphere of radius r centred at origin, the standard practice of incremental addition by a fixed value (e.g. θi = θi−1+ ∆θ or
ϕi = ϕi−1+ ∆ϕ, where ∆θ = Nπ
θ, ∆ϕ =
2π
Nϕ) yields a non-uniform distribution of points with clustering near the poles( see Fig. 3.12(a)). Even the random selection of θ and ϕ values as per equations
t θ
ϕ r
Figure 3.11 Spherical coordinate system
where Υθ and Υϕare random numbers between 0 and 1 results in an equally biased distribution
(see Fig. 3.12(b)).
To avoid clustering, the whole surface area of sphere is divided in such a way that solid angle subtended by each elemental area is equal. The geometric centres of these areas will have a uniform distribution over the sphere. Mathematically solid angle is given by
dΩ = sin θdθdϕ · (3.9)
When dθ and dϕ are assumed to be constant, variation of dΩ with θ leads to the clustering of points near the poles. To avoid this clustering, a new variable u is introduced so that
Clustering Clustering N o clustering
a b c
Figure 3.12 Distribution of ray origin points on spherical surface (a) uniform increment of θ
and ϕ, (b) random selection of θ and ϕ, (c) uniform solid angle.
du = sin θdθ (3.10)
that yields
By substituting u in Eqn. (3.9) for solid angle, we obtain
dΩ = dudϕ · (3.12)
Now a random selection of u and ϕ from a given range (u ∈ [−1, 1], ϕ ∈ [0, 2π)) satisfies the condition dΩ = dudϕ ≈ constant, since the randomly selected points are assumed to be sufficiently uniform over the total range [131]. Fig. 3.12(c) shows the generated distribution of points with no clustering observed.
A B C D Intersecting N on − intersecting O ~ R(ψ) ~ R0(ψ) ϕ ϕ β β θ 6 AOB = cos−1 ~ OA · ~OB | ~OA|| ~OB| ! 6 ϕ = cos−1 ~ OA · ˆeR | ~OA| ! 6 β = cos−1 ~ OB · ˆeR | ~OB| ! x y
Figure 3.13 Ray-tracing to calculate view factor between two dimensional surfaces.
Ray-surface intersection is a key component of the view factor calculation using ray tracing. Just qualifying the rays as intersecting or non-intersecting with a given surface is sufficient to evaluate view factor, while determination of exact position of ray-surface intersection is redun- dant. Consider a sample case involving two straight lines as shown in Fig. 3.13. Segment AB is selected as interceptor and CD as shooting segment. Now an arbitrary ray ~R(ψ) from segment CD will intersect line AB if the angles which the ray make with OA and OB are both smaller then6 AOB. Necessary conditions for intersection are given in form of logical truth Table 3.2.
Similarly for three-dimensional case, general condition for a ray intersecting with plane poly- gon ABCD (see Fig. 3.14) is obtained by ensuring that any point on the ray should be on the same side of the plane (containing ray origin and two vertices) as that of next vertex of polygon following a given order. Considering a set of three vertices A, B, C as shown in Fig. 3.14, a point on the ray ~R(ψ)is on the same side of plane AOB as that of vertex C if
ˆ
where ˆ n = ~ OA × ~OB ~ OA × ~OB ,
and ˆeOC and ˆeRare unit vectors in the direction of vectors ~OC and ray ~R(ψ), respectively. When
the condition (3.13) is true for all the sets of ordered pairs of three vertices of interceptor poly- gon, the ray is termed intersecting. A ray-sphere intersection algorithm [45] has been already discussed in the previous section. Important steps of algorithm for view factor calculation are presented in the form of a flow chart in Fig. 3.15.
Table 3.2 Truth table
ϕ ≤ θ β ≤ θ Intersecting
(ϕ ≤ θ ∧ β ≤ θ)
True True True
True False False
False True False
False False False