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I'm looking at the radiosity equations for heat transfer

Specifically, I'm hesitant to accept the equation:

$$ \dot Q_i = \frac{ A_i \epsilon_i }{ 1 - \epsilon_i } ( \sigma T_i^4 - J_i )$$

My analysis is as follows: power goes out via Stefan-Boltzmann term, which is given by a term like

$$ - A_i \epsilon_i \sigma T_i^4 $$

and radiation goes in via absorption of all emissions from nearby surfaces, that is:

$$ A_i \epsilon_i H_i = A_i \epsilon_i \sum_j F_{ij} J_j $$

where the radiosity $J_j$ expression is correct on the article, and we are assuming Kirchoff's law of thermal radiation to hold. But the above implies that the actual expression for the net heat transfer is:

$$ \dot Q_i = - A_i \epsilon_i ( \sigma T_i^4 - H_i ) = - A_i \epsilon_i ( \sigma T_i^4 - \sum_j F_{ij} J_j ) $$

Thoughts? Is my derivation correct, or is the one on the article?

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