374
5 Numerical Models for Pebble-Bed Heat Transfer
Fig. 5.111 View factor between two spheres with different distance
Fig. 5.112 View factor case of three spheres (a) and the numerical results of the effect of the third
sphere (b)
• At | P 1 − P 2 | = 4d p without obstruction P 3 , it is f ( P 1 , P 2 ) = 0.162 (Fig. 5.112b).
• P 3 cannot occupy E and F regions because its distance from P 1 or P 2 is less than
2d p .
• If the sphere is at point O 1 (in region A), the rays from P 1 to P 2 are fully obstructed
by P 3 and f ( P 1 , P 2 , P 3 ) = 0.
• If the sphere is at point O 2 (in region D), it is very far away from P 1 and P 2 ; the
rays are not affected by P 3 and f ( P 1 , P 2 , P 3 ) = f ( P 1 , P 2 ).
Figure 5.113 sketches the cases in region B, C and D that do not affect the rays
from P 1 to P 2 or vice versa. Therefore, the view factor from P 1 to P 2 will be affected
only when the sphere P 3 is in region A. When sphere P 3 is on the central line of
P 1 P 2 , the results of the view factor of P 1 and P 2 are shown in Fig. 5.114, where L
is the distance from P 3 to the line P 1 P 2 . For example, it has L ≤ 2d p in the region
5 Numerical Models for Pebble-Bed Heat Transfer
Fig. 5.111 View factor between two spheres with different distance
Fig. 5.112 View factor case of three spheres (a) and the numerical results of the effect of the third
sphere (b)
• At | P 1 − P 2 | = 4d p without obstruction P 3 , it is f ( P 1 , P 2 ) = 0.162 (Fig. 5.112b).
• P 3 cannot occupy E and F regions because its distance from P 1 or P 2 is less than
2d p .
• If the sphere is at point O 1 (in region A), the rays from P 1 to P 2 are fully obstructed
by P 3 and f ( P 1 , P 2 , P 3 ) = 0.
• If the sphere is at point O 2 (in region D), it is very far away from P 1 and P 2 ; the
rays are not affected by P 3 and f ( P 1 , P 2 , P 3 ) = f ( P 1 , P 2 ).
Figure 5.113 sketches the cases in region B, C and D that do not affect the rays
from P 1 to P 2 or vice versa. Therefore, the view factor from P 1 to P 2 will be affected
only when the sphere P 3 is in region A. When sphere P 3 is on the central line of
P 1 P 2 , the results of the view factor of P 1 and P 2 are shown in Fig. 5.114, where L
is the distance from P 3 to the line P 1 P 2 . For example, it has L ≤ 2d p in the region