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12.2.2 Method 2—Double-Area Integration
Use direct integration to determine the view factor between two adjacent
surface areas i and j as shown in Figure 12.9.
1
cos θ i cos θ j dA i dA j
πR 2
F Ai−Aj = A i
A i A j
where
2
R
2
= (x i − x j )
2
+ (y i − y j )
2
+ (z i − z j )
1 [
]
cos θ i =
l i (x j − x i ) + m i (y j − y i ) + n i (z j − z i )
R
1
cos θ j =
l j (x i − x j ) + m j (y i − y j ) + n j (z i − z j )
R
Therefore, the view factor can be determined by performing the following
integration:
b
b
a
c
1 dx j dx i z j dz j
y i dy i
2
2
π[(x i − x j ) 2 + y i + z j ] 2
(12.10)
F Ai−Aj = bc
0
0
0
0
250
Analytical Heat Transfer
TABLE 12.1 (continued)
View Factors for 2-D Geometries
Geometry
Small area perpendicular to the axis of
a surface of revolution
Relation
F 12 = sin 2 θ
2
θ
1
Area on the inside of a sphere
F 12 =
A 2
4πR 2
A 1
A 2
R
Source: Data from A. Mills, Heat Transfer, Richard D. Irwin, Inc., Boston, MA, 1992; F. Incropera
and D. Dewitt, Fundamentals of Heat and Mass Transfer, John Wiley & Sons, Fifth Edition, 2002.