Analytical Treatment of Radiation and Conduction Losses in Thermal Induction Plasmas

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Abstract
Approximate solutions have been obtained of the Elenbaas‐Heller equation for a two‐zone model of thermal induction plasmas. Most of the heat production takes place in the external zone, where it is kept in balance by conduction losses. The major portion is lost to the wall, the minor transferred to the internal zone. This heat supply accounts for most of the radiation losses, which are confined to the internal zone. A small fraction of radiation is balanced by direct dissipation. Under these conditions, the distribution of the heat conduction potential assumes a parabolic shape for the inner zone, whereas it is described by Bessel functions in the external zone. A coordinate transformation in this zone accounts for the nonuniformity of the induced electric field and explicitly shows the effect of skin depth upon the profile shape. Matching of the zonewise solutions yields temperature distributions that are continuous to the first derivative and display the characteristic minimum at the axis known from experimental work. Generalized charts are presented by which a particular profile can be calculated from the gas transport properties and the time rate of change of primary magnetic flux. The method is applied to argon plasmas at atmospheric pressure with radiation properties calculated from atomic theory. Simple expressions are derived for the total radiated power and the radiation efficiency of the discharge by a further idealization of the model. They indicate that, at most, 50% of the input power can be converted into radiation as long as the discharge remains attached to the tube wall. It is also shown that large field penetration depth improves the efficiency. The results are in qualitative agreement with available experimental data. A quantitative check will require more complete experiments.