Page 518 - Bird R.B. Transport phenomena

498 Chapter 16 Energy Transport by Radiation
Body 2 Fig. 16.4-2. Radiant interchange
between two black bodies.

Bodyl

Solid angle
01 sin Oi dd x йф

Of the energy leaving dA x at an angle 0 U only the fraction given by the following ratio
will be intercepted by dA : 2
f area of dA 2 projected onto a \
\plane perpendicular to r 12 / dA 2 cos 0 2
/area formed by the intersection \ Ai s m #i d&\ & (16.4-4)
I of the solid angle sin 6 ] d9 } d$^ j
1 with a sphere of radius r with I
12
\center at dA x /
Multiplication of these last two expressions then gives
_ crT\ cos 0! cos 0
2 dAdA (16.4-5)
12 * A ? l 2
This is the radiant energy emitted by dA x and intercepted by dA 2 per unit time. In a simi-
lar way we can write

vT\ cos 0 cos 0 . . . .
rfQ^ = -= 1 z 2 dA dA 2 (16.4-6)
}
21 rf 2
which is the radiant energy emitted by dA 2 that is intercepted by dA x per unit time. The
net rate of energy transport from dA^ to dA is then
2
dQi2 = dQ_+ - i
12
COS 0i COS : dA dA (16.4-7)
2
r u } 2
Therefore, the net rate of energy transfer from an isothermal black body 1 to another
isothermal black body 2 is
COS 0] COS 0 2
(16.4-8)

Here it is understood that the integration is restricted to those pairs of areas dA A and dA 2
that are in full view of each other. This result is conventionally written in the form
Q 12 = \ ~ T\) (16.4-9)