Heat Transfer: Radiation View Factors

Radiation View Factors 
The above equations for blackbodies and
graybodies assumed that the small body could see only the large
enclosing body and nothing else. Hence, all radiation leaving the
small body would reach the large body. For the case where two
objects can see more than just each other, then one must introduce a
view
factor F and the heat transfer calculations become
significantly more involved.
The view factor F_{12} is used to parameterize the fraction of thermal power leaving object 1 and reaching object 2. Specifically, this quantity is equal to, 
Likewise, the fraction of thermal power leaving object 2 and reaching object 1 is given by, 
The case of two blackbodies in thermal equilibrium can be used to derive the following reciprocity relationship for view factors, 
Thus, once one knows F_{12},
F_{21} can be calculated immediately.
Radiation view factors can be analytically derived for simple geometries and are tabulated in several references on heat transfer (e.g. Holman, 1986). They range from zero (e.g. two small bodies spaced very far apart) to 1 (e.g. one body is enclosed by the other). 
Heat Transfer Between Two Finite Graybodies 
The heat flow transfered from Object 1 to Object 2 where the two objects see only a fraction of each other and nothing else is given by, 
This equation demonstrates the usage of
F_{12}, but it represents a nonphysical case since
it would be impossible to position two finite objects such that they
can see only a portion of each other and "nothing" else. On the
contrary, the complementary view factor (1  F_{12})
cannot be neglected as radiation energy sent in those directions
must be accounted for in the thermal bottom line.
A more realistic problem would consider the same two objects surrounded by a third surface that can absorb and readmit thermal radiation yet is nonconducting. In this manner, all thermal energy that is absorbed by this third surface will be readmitted; no energy can be removed from the system through this surface. The equation describing the heat flow from Object 1 to Object 2 for this arrangement is, 
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