# Approximate explicit analytic solution of the Elenbaas-Heller equation

## Abstract

The Elenbaas-Heller equation describing the temperature field of a cylindrically symmetrical non-radiative electric arc has been solved, and approximate explicit analytic solutions are obtained. The radial distributions of the heat-flux potential and the electrical conductivity have been figured out briefly by using some special simplification techniques. The relations between both the core heat-flux potential and the electric field with the total arc current have also been given in several easy explicit formulas. Besides, the special voltage-ampere characteristic of electric arcs is explained intuitionally by a simple expression involving the Lambert W-function. 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, Meng-Ran, Li, Hui, and Xia, Wei-Dong, E-mail: xiawd@ustc.edu.cn.
```*Approximate explicit analytic solution of the Elenbaas-Heller equation*. United States: N. p., 2016.
Web. doi:10.1063/1.4960777.

```
Liao, Meng-Ran, Li, Hui, & Xia, Wei-Dong, E-mail: xiawd@ustc.edu.cn.
```*Approximate explicit analytic solution of the Elenbaas-Heller equation*. United States. doi:10.1063/1.4960777.

```
Liao, Meng-Ran, Li, Hui, and Xia, Wei-Dong, E-mail: xiawd@ustc.edu.cn. Sun .
"Approximate explicit analytic solution of the Elenbaas-Heller equation". United States.
doi:10.1063/1.4960777.
```

```
@article{osti_22597668,
```

title = {Approximate explicit analytic solution of the Elenbaas-Heller equation},

author = {Liao, Meng-Ran and Li, Hui and Xia, Wei-Dong, E-mail: xiawd@ustc.edu.cn},

abstractNote = {The Elenbaas-Heller equation describing the temperature field of a cylindrically symmetrical non-radiative electric arc has been solved, and approximate explicit analytic solutions are obtained. The radial distributions of the heat-flux potential and the electrical conductivity have been figured out briefly by using some special simplification techniques. The relations between both the core heat-flux potential and the electric field with the total arc current have also been given in several easy explicit formulas. Besides, the special voltage-ampere characteristic of electric arcs is explained intuitionally by a simple expression involving the Lambert W-function. 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 = {Sun Aug 14 00:00:00 EDT 2016},

month = {Sun Aug 14 00:00:00 EDT 2016}

}