C. For a small convex surface, A1, completely enclosed by a very large concave surface, A2, A1 < < A2 and F12 = 1, then Eq. (2.66) becomes
The last equation also applies for a flat-plate collector cover radiating to the surroundings, whereas case B applies in the analysis of a parabolic trough collector receiver where the receiver pipe is enclosed in a glass cylinder.
As can be seen from Eqs. (2.67)-(2.69), the rate of radiative heat transfer between surfaces depends on the difference of the fourth power of the surface temperatures. In many engineering calculations, however, the heat transfer equations are linearized in terms of the differences of temperatures to the first power. For this purpose, the following mathematical identity is used:
(T2 - T22)(T2 + T22) = (T - T2XT1 + T2XT2 + T22) (2.70)
Therefore, Eq. (2.66) can be written as
Q12 = Alhr(T - T2) with the radiation heat transfer coefficient, hr, defined as
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