c
d
e
R
θ
θ
b a
2
1
D
�
� �
�
�
� �
�
Let X = 1 + D/(2R)
2
F 1−2 =
2
(x − 1) 1/2 + sin −1 1 − X
π
X
2
π
1
2
1/2 +
−1
F 1−2 =
(x − 1)
− cos
− X
π
2
X
246
Analytical Heat Transfer
FIGURE 12.7
View factor between two opposite circular tubes.
Example 12.4
Determine the view factor between two circular tubes with partial blockage as
shown in Figure 12.8. From Hottel’s cross-string method, the view factor can be
determined as follows:
The sum of the length of crossed strings: L A−B−D−G−I + L H−C −D−E −F
The sum of the length of uncrossed strings: L A−F + L H−C −D−G−I
Therefore,
L A−B−D−G−I + L H−C −D−E −F − L A−F + L H−C −D−G−I
L A−B−C −H F 1−2 = 2
2
Table 12.1 shows many useful view factors for 2-D geometries that can be
determined by using Hottel’s cross-string method [2,4].
D
G
I
B
A
R
1
2
E
F
H C
d
d
h
h
α
β
α
FIGURE 12.8
View factor between two circular tubes with partial blockage.
d
e
R
θ
θ
b a
2
1
D
�
� �
�
�
� �
�
Let X = 1 + D/(2R)
2
F 1−2 =
2
(x − 1) 1/2 + sin −1 1 − X
π
X
2
π
1
2
1/2 +
−1
F 1−2 =
(x − 1)
− cos
− X
π
2
X
246
Analytical Heat Transfer
FIGURE 12.7
View factor between two opposite circular tubes.
Example 12.4
Determine the view factor between two circular tubes with partial blockage as
shown in Figure 12.8. From Hottel’s cross-string method, the view factor can be
determined as follows:
The sum of the length of crossed strings: L A−B−D−G−I + L H−C −D−E −F
The sum of the length of uncrossed strings: L A−F + L H−C −D−G−I
Therefore,
L A−B−D−G−I + L H−C −D−E −F − L A−F + L H−C −D−G−I
L A−B−C −H F 1−2 = 2
2
Table 12.1 shows many useful view factors for 2-D geometries that can be
determined by using Hottel’s cross-string method [2,4].
D
G
I
B
A
R
1
2
E
F
H C
d
d
h
h
α
β
α
FIGURE 12.8
View factor between two circular tubes with partial blockage.