Approximate explicit analytic solution of the ElenbaasHeller equation
Abstract
The ElenbaasHeller equation describing the temperature field of a cylindrically symmetrical nonradiative electric arc has been solved, and approximate explicit analytic solutions are obtained. The radial distributions of the heatflux potential and the electrical conductivity have been figured out briefly by using some special simplification techniques. The relations between both the core heatflux potential and the electric field with the total arc current have also been given in several easy explicit formulas. Besides, the special voltageampere characteristic of electric arcs is explained intuitionally by a simple expression involving the Lambert Wfunction. The analyses also provide a preliminary estimation of the Joule heating per unit length, which has been verified in previous investigations. Helium arc is used to examine the theories, and the results agree well with the numerical computations.
 Authors:
 Department of Thermal Science and Energy Engineering, University of Science and Technology of China, Hefei 230026 (China)
 Publication Date:
 OSTI Identifier:
 22597668
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Applied Physics; Journal Volume: 120; Journal Issue: 6; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ANALYTICAL SOLUTION; APPROXIMATIONS; CYLINDRICAL CONFIGURATION; ELECTRIC ARCS; ELECTRIC CONDUCTIVITY; ELECTRIC FIELDS; ELECTRIC POTENTIAL; EQUATIONS; HEAT; HEAT FLUX; HELIUM; JOULE HEATING; LENGTH; SPATIAL DISTRIBUTION
Citation Formats
Liao, MengRan, Li, Hui, and Xia, WeiDong, Email: xiawd@ustc.edu.cn. Approximate explicit analytic solution of the ElenbaasHeller equation. United States: N. p., 2016.
Web. doi:10.1063/1.4960777.
Liao, MengRan, Li, Hui, & Xia, WeiDong, Email: xiawd@ustc.edu.cn. Approximate explicit analytic solution of the ElenbaasHeller equation. United States. doi:10.1063/1.4960777.
Liao, MengRan, Li, Hui, and Xia, WeiDong, Email: xiawd@ustc.edu.cn. 2016.
"Approximate explicit analytic solution of the ElenbaasHeller equation". United States.
doi:10.1063/1.4960777.
@article{osti_22597668,
title = {Approximate explicit analytic solution of the ElenbaasHeller equation},
author = {Liao, MengRan and Li, Hui and Xia, WeiDong, Email: xiawd@ustc.edu.cn},
abstractNote = {The ElenbaasHeller equation describing the temperature field of a cylindrically symmetrical nonradiative electric arc has been solved, and approximate explicit analytic solutions are obtained. The radial distributions of the heatflux potential and the electrical conductivity have been figured out briefly by using some special simplification techniques. The relations between both the core heatflux potential and the electric field with the total arc current have also been given in several easy explicit formulas. Besides, the special voltageampere characteristic of electric arcs is explained intuitionally by a simple expression involving the Lambert Wfunction. The analyses also provide a preliminary estimation of the Joule heating per unit length, which has been verified in previous investigations. Helium arc is used to examine the theories, and the results agree well with the numerical computations.},
doi = {10.1063/1.4960777},
journal = {Journal of Applied Physics},
number = 6,
volume = 120,
place = {United States},
year = 2016,
month = 8
}

The ElenbaasHeller equation is nondimensionalized and solved using regular perturbation theory to provide closedform analytical solutions to describe structures of cylindrically symmetrical steady electric arc discharges with negligible radiant heat transfer. Based on available data, it is assumed that the electrical conductivity varies with the heatflux potential in an Arrhenius fashion. The leadingorder solution is equivalent to an asymptotic solution proposed by Kuiken [J. Appl. Phys. 58, 1833 (1991)]. Higherorder terms are also derived in the present paper, and it is shown that quantitatively accurate analytical solutions can be developed when higherorder terms are included. Analysis shows that appreciable Joulemore »

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