Equations 4 to 7 can be solved to give
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(8) |
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(9) |
Equation 9 is the additional indirect heat exchange between  and  as obtained by reflection and re- radiation from surface 
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(10) |
By equation 4, 9 and 10
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(11) |
Where  is a composite geometrical factor for the closed system of two black surfaces  and
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(12) |
It is to be noted that in the above derivation we have identified furnace to consist of source, sink and refractory as independent entities and each one is at uniform temperature. However, in practical situations, it is not very often easy to separate the source, sink and refractory and the temperatures may be far from uniform. Thus a careful engineering judgment is necessary for accurate calculations.
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