Hence,

1

A 1 F 1−4 = [A i F i−j − A 1 F 1−3 − A 2 F 2−4 ]
(12.15)
2
where F i−j , F 1−3 , and F 2−4 , are available from Table 12.2 formulas or charts.
And, A 1 F 1−4 = A 4 F 4−1 .
254
Analytical Heat Transfer
TABLE 12.2
View Factors for 3-D Geometries
Geometry
Relation
X = X/L, Y = Y/L
⎧ �
� 1/2
Aligned parallel rectangles
⎨
2
(1 + X 2 )(1 + Y 2 )
F ij =
ln
πXY ⎩
1 + X 2 + Y 2
Y
L
i
j
+X(1 + Y 2 ) 1/2 tan −1
X
(1 + Y 2 ) 1/2
+Y(1 + X 2 ) 1/2 tan −1
Y
(1 + X 2 ) 1/2
⎫
⎬
X
−X tan −1 X − Y tan −1 Y ⎭
Coaxial parallel disks
R i = r i /L, R j = r j /L
r j
1 + R 2
j
j
i
S = 1 +
R 2
i
1
L
F ij = 2
{S − [S 2 − 4(r j /r i ) 2 ] 1/2 }
r i
Perpendicular rectangles with a
H = Z/X, W = Y/X
⎛
1 ⎜
1
1
common edge
F ij =
⎝ W tan −1
+ H tan −1
πW
W
H
X
Z
i
j
−(H 2 + W 2 ) 1/2 tan −1
1
(H 2 + W 2 ) 1/2
⎧

⎪

�
� W 4
⎨
1
(1 + W 2 )(1 + H 2 ) W 2 (1 + W 2 + H 2 )
+ ln
4 ⎪ 1 + W 2 + H 2
(1 + W 2 )(W 2 + H 2 )
⎩
⎫ ⎞
Y
�
� H 2 ⎪ ⎬
H 2 (1 + H 2 + W 2 )
⎟
⎠
⎪
×
(1 + H 2 )(H 2 + W 2 )
⎭
Source: F. Incropera and D. Dewitt, Fundamentals of Heat and Mass Transfer, John Wiley & Sons,
Fifth Edition, New York, NY, 2002.