Assuming that Bartlett's and Walsh's theorems are applicable to a hollow cylindrical black body with perfect diffusive internal surface kept at a uniform temperature, the effective emittance of the cylinder bottom can be calculated approximately by Rossmann's principle.
1. The case of free opening
The effective emittance of the bottom is given by ε
α+ε
α' where ε
α is the original emissivity of the bottom and ε
α' the increment in emissivity resulting from multiple reflections between the bottom and the walls. If the free end is imagined to be closed by a lid of the emissivity 1, and if ε
1" be the increment in emissivity of the bottom by direct radiation from this imaginary lid and ε
1''' that by indirect radiation reflected by the walls, then it would be _??_ The problem is then reduced to the calculation of ε
1" and ε
1'''.
If
l is the length of the cylinder measured by its radius as the unit, ε
1" is given as
_??_
where _??_
In evaluating ε
1''', if only the primary reflection on the walls is considered, we have
_??_
where _??_
2. The case of the opening being closed by a lid with a circular hole in its center
The effect of the lid is to increase the effective emittance of the bottom given in the case 1 by the amount ε
α"+ε
α''' where ε
α" is the increment due to mutual radiations emitted and reflected directly between the lid and the bottom, and ε
α''' to those indirectly through reflections on the walls.
If
p is the length of the radius of the hole in the lid measured by the cylinder radius as the unit, and
k=1-
p2, ε
α" and ε
α''' are given as
_??_
_??_
where _??_
For
l=1, 2, 3, ••••10, and
k=0, 0.75, the calculated results are shown in tables and figures with some other data as reference.
The author's plots are closely in agreement with Nakaji's who used a digital computor for numerical solution of Buckley's equations.
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