In order to estimate accurately the allowable plate dissipiation of vacuuI_??_ tubes, some analyses are performed on the temperature distribution of a thin-walled hollow cylinder at a high temperature where the heat conduction through cylinder wall is negligibly small compared with the radiation from cylinder surfaces.
The fundamental equation is deduced under the following assumptions.
1) Heat radiation follows Lambert's law.
2) Heat conduction through cylinder wall is negligible.
3) Total emissivity of the cylinder surfaces is independent of temperature.
The equation is given in following formula.
_??_
where
A: total emissivity,
a: radius of the cylinder,
l: length of the cylinder,
P(
Z1): input energy
Q(
Z1): heat energy coming away from inner surface of the cylinder.
This is a Fredholm's integral equation of the second kind.
Two methods of numerical solutions of the equation are shown, and the results for some values of A and
l/2
a are tabulated. For example, T
O/T
5, the ratio of the maximum and minimum temperatures, is 0.990730 for
A=0.2 &
l/2
a=1, and 0.900059 for
A=0.2 &
l/2
a=8.
It may be said that the temperature distribution is chiefly dependent on the geometrical dimensions of the cylinder but little dependent on the total emissivity of its surface.
Short discussion is given on the method of obtaining the cylinder temperature availing so-called “Winkelverhältniss” which is deduced under the assumption that the cylinder has an uniform temperature distribution.
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