- #1
bobfei
- 30
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Hi,
https://www.physicsforums.com/attachment.php?attachmentid=60066&stc=1&d=1372909704
I would like to ask a question on thermal radiation. In the attached image, if all three surfaces [itex]\{ dA,d{S_1},d{S_2}\} [/itex] are black body surface, and [itex]d{S_2} = d{S_1} = S[/itex], then does [itex]d{S_2}[/itex] receive the same thermal from [itex]dA[/itex] as [itex]d{S_1}[/itex]? In other words, are the view factor [itex]{F_{A \to 1}}[/itex] and [itex]{F_{A \to 2}}[/itex] equal?
I find this problem perplexing:
Bob
https://www.physicsforums.com/attachment.php?attachmentid=60066&stc=1&d=1372909704
I would like to ask a question on thermal radiation. In the attached image, if all three surfaces [itex]\{ dA,d{S_1},d{S_2}\} [/itex] are black body surface, and [itex]d{S_2} = d{S_1} = S[/itex], then does [itex]d{S_2}[/itex] receive the same thermal from [itex]dA[/itex] as [itex]d{S_1}[/itex]? In other words, are the view factor [itex]{F_{A \to 1}}[/itex] and [itex]{F_{A \to 2}}[/itex] equal?
I find this problem perplexing:
- Straightforward intuition suggests that [itex]d{S_1}[/itex] should receive more energy since it is facing [itex]dA[/itex], and [itex]d{S_2}[/itex] might get less because it is at an oblique angle with dA.
- In heat transfer there is a reciprocity relationship which states that for two diffuse emitters: [itex]Are{a_1} \cdot {F_{1 \to 2}} = Are{a_2} \cdot {F_{2 \to 1}}[/itex]. For the pair of [itex]dA[/itex] and [itex]d{S_2}[/itex], if we stand at the position of [itex]d{S_2}[/itex] and look at [itex]dA[/itex], clearly we see only a stretched elliptical projection of [itex]dA[/itex], and the area times view factor is [itex]d{S_2} \cdot {\rm{Solidangle}}(dA\cos \theta ) = S \cdot \frac{{S\cos \theta }}{{4\pi {R^2}}} = \frac{{{S^2}}}{{4\pi {R^2}}}\cos \theta [/itex]; because of the reciprocity relationship this equals [itex]dA \cdot {F_{A \to 2}}[/itex], and we get [itex]S \cdot {F_{A \to 2}} = \frac{{{S^2}}}{{4\pi {R^2}}}\cos \theta \;\; \Rightarrow {F_{A \to 2}} = \frac{{S\cos \theta }}{{4\pi {R^2}}}[/itex], so that the view factor from dA is dependent on [itex]\theta [/itex]. This would suggest that [itex]d{S_2}[/itex], because of larger [itex]\theta [/itex], would receive smaller faction of [itex]dA[/itex]’s energy than [itex]d{S_1}[/itex].
- But from a microscopic perspective, if we consider each molecule of [itex]dA[/itex]’s radiation, we see that for a small molecule, because [itex]d{S_2} = d{S_1} = S[/itex], the two will subtend the same solid angle and hence receive the same fraction of the molecule’s radiation. Then if we sum up the contribution of all molecules on [itex]dA[/itex], this seems suggest that the fraction of energy [itex]d{S_1}[/itex] and [itex]d{S_2}[/itex] received are equal.
- Yet another question is the radiation’s interference. At the position of [itex]d{S_2}[/itex], particularly if we push [itex]d{S_2}[/itex] to the extreme that its center lies on the plane of [itex]dA[/itex], intuitively much of the energy from the left side of [itex]dA[/itex] traveling to [itex]d{S_2}[/itex] will be blocked by molecules on its right side, or cancelled. I have tried to sketch a proof by assuming some distribution of each molecule’s phase, but haven’t yet arrived at any result. Does interference really has a role here?
Bob