Electrothermal instability

__NOTOC__The electrothermal instability (also known as the ionization instability or Velikhov instability in the literature) is a magnetohydrodynamic (MHD) instability appearing in magnetized non-thermal plasmas used in MHD converters. It was first theoretically discovered in 1962 and experimentally measured into a MHD generator in 1964 by Evgeny Velikhov. [cite conference
author = E.P. Velikhov
year = 1962
title = Hall instability of current-carrying slightly-ionized plasmas
conference = 1^st International Conference on MHD Electrical Power Generation, Paper 47
booktitle = Newcastle-upon-Tyne, England] [cite conference
author = E.P. Velikhov
coauthors = A.M. Dykhne
date = 13-18 July 1963
title = Plasma turbulence due to the ionization instability in a strong magnetic field
conference = 6^th International Conference on Ionization Phenomena in Gases
booktitle = Proceedings
editor = Paris, France
volume = 4] [cite conference
author = E.P. Velikhov
coauthors = A.M. Dykhne, I.Ya Shipuk
year = 1965
title = Ionization instability of a plasma with hot electrons
conference = 7^th International Conference on Ionization Phenomena in Gases
booktitle = Belgrade, Yugoslavia]

Physical explanation and characteristics

This instability is a turbulence of the electron gas in a non-equilibrium plasma (i.e. where the electron temperature T_e is greatly higher than the overall gas temperature T_g). It arises when a magnetic field powerful enough is applied in such a plasma, reaching a critical Hall parameter β_cr.

Locally, the number of electrons and their temperature fluctuate (electron density and thermal velocity) as the electric current and the electric field.

The Velikhov instability is a kind of ionization wave system, almost frozen in the two temperature gas. The reader can evidence such a stationary wave phenomenon just applying a transverse magnetic field with a permanent magnet on the low-pressure control gauge (Geissler tube) provided on vacuum pumps. In this little gas-discharge bulb a high voltage electric potential is applied between two electrodes which generates an electric glow discharge (pinkish for air) when the pressure has become low enough. When the transverse magnetic field is applied on the bulb, some oblique grooves appear in the plasma, typical of the electrothermal instability.

The electrothermal instability occurs extremely quickly, in a few microseconds. The plasma becomes non-homogeneous, transformed into alternating layers of high free electron and poor free electron densities. Visually the plasma appears stratified, as a "pile of plates".

Hall effect in plasmas

The Hall effect in ionized gases has nothing to do with the Hall effect in solids (where the Hall parameter is always very inferior to unity). In a plasma, the Hall parameter can take any value.

The Hall parameter β in a plasma is the ratio between the electron gyrofrequency Ω_e and the electron-heavy particles collision frequency ν:

:

where: "e" is the electron charge (1.6 × 10^-19 coulomb): "B" is the magnetic field (in teslas): m_e is the electron mass (0.9 × 10^-30 kg)

The Hall parameter value increases with the magnetic field strength.

Physically, when the Hall parameter is low, the trajectories of electrons between two encounters with heavy particles (neutral or ion) are almost linear. But if the Hall parameter is high, the electron movements are highly curved. The current density vector J is no more colinear with the electric field vector E. The two vectors J and E make the Hall angle θ which also gives the Hall parameter:

:

Plasma conductivity and magnetic fields

In a non-equilibrium ionized gas with high Hall parameter, Ohm's law,

:$mathbf\{J\}\; =\; sigmamathbf\{E\}$where "σ" is the electrical conductivity (in siemens per metre),

is a matrix, because the electrical conductivity σ is a matrix:

:

σ_S is the scalar electrical conductivity:

:$sigma\_s\; =\; frac\; \{n\_e\; e^2\}\{m\_e\; u\}$

where n_e is the electron density (number of electrons per cubic meter).

The current density J has two components:

:

Therefore

:

The Hall effect makes electrons "crabwalk".

When the magnetic field B is high, the Hall parameter β is also high, and

Thus both conductivities

become weak, therefore the electric current cannot flow in these areas. This explains why the electron current density is weak where the magnetic field is the stongest.

Critical Hall parameter

The electrothermal instability occurs in a plasma at a (T_e > T_g) regime when the Hall parameter is higher that a critical value β_cr.

We have

:$f\; =\; frac\{left(\; frac\{delta\; mu\}\{mu\}\; ight)\}\{left(\; frac\{delta\; n\_e\}\{n\_e\}\; ight)\}$

where μ is the electron mobility (in m²/(V·s))

and

:$s\; =\; frac\{2\; k\; T\_e^2\}\{E\_i;\; (T\_e\; -\; T\_g)\}\; imes\; frac\; \{1\}\{1\; +\; dfrac\{3\}\{2\}\; dfrac\{k;\; T\_e\}\{E\_i$

where "E_i" is the ionization energy (in electron volts) and "k" the Boltzmann constant.

The growth rate of the instability is

:

And the critical Hall parameter is

:

The critical Hall parameter β_cr greatly varies according to the degree of ionization α :

:$alpha\; =\; frac\{n\_i\}\{n\_n\}$

where n_i is the ion density and n_n the neutral density (in particles per cubic metre).

The electron-ion collision frequency ν_ei is much greater than the electron-neutral collision frequency ν_en.

Therefore with a weak energy degree of ionization α, the electron-ion collision frequency ν_ei can equal the electron-neutral collision frequency ν_en.

* For a weakly ionized gas (non-Coulombian plasma, when &nu;_ei < &nu;_en ): :

* For a fully ionized gas (Coulombian plasma, when &nu;_ei > &nu;_en ): :

NB: The term "fully ionized gas", introduced by Lyman Spitzer, does not mean the degree of ionization is unity, but only that the plasma is Coulomb-collision dominated, which can correspond to a degree of ionization as low as 0.01%.

Technical problems and solutions

A two-temperature gas, globally cool but with hot electrons (T_e >> T_g) is a key feature for practical MHD converters, because it allows the gas to reach sufficient electrical conductivity while protecting materials from thermal ablation. This idea was first introduced for MHD generators in the early 1960s by Jack L. Kerrebrock [Cite journal
author = J.L. Kerrebrock
date = November 1, 1960
title = Non-equilibrium effects on conductivity and electrode heat transfer in ionized gases
journal = Technical Note #4
publisher = AFOSR-165
location = Guggenheim Jet Propulsion Center, Caltech, Pasadena, California.OSTI|4843920] and Alexander E. Sheindlin [cite conference
author = A.E. Sheindlin
coauthors = V.A. Batenin, E.I. Asinovsky
date = July 6, 1964
title = investigation of non-equilibrium ionization in a mixture of argon and potassium
conference = International symposium on magnetohydrodynamic electric power generation
booktitle = CONF-640701-102
editor = Paris, FranceOSTI|5024025] .

But the unexpected large and quick drop of current density due to the electrothermal instability ruined many MHD projects worldwide, while previous calculation envisaged energy conversion efficiencies over 60% with these devices. Whereas some studies were made about the instability by various researchers, [cite conference
author = A. Solbes
date = 24–30 July 1968
title = A quasi linear plane wave study of electrothermal instabilities
conference = 8^th International Conference on MHD Electrical Power Generation
booktitle = SM/107/26
editor = International Atomic Energy Agency, Warsaw, Poland] [cite conference
author = A.H. Nelson
coauthors = M.G. Haines
date = 26–28 March 1969
title = Analysis of the nature and growth of electrothermal waves
conference = 10^th Symposium in Engineering Aspects of MHD
booktitle = Proceedings
editor = MIT, Cambridge, MA, USA
doi = 10.1088/0032-1028/11/10/003] no real solution was found at that time. This prevented further developments of non-equilibrium MHD generators and compelled most engaged countries to cancel their MHD power plants programs and to retire completely from this research field in the early 1970s, because this technical problem was considered as an impassable stumbling block in these days.

Nevertheless experimental studies about the growth rate of the electrothermal instability and the critical conditions showed that a stability region still exists for high electron temperatures. [cite conference
author = J.P. Petit
coauthors = J. Valensi, J.P. Caressa
date = 24–30 July 1968
title = Theoretical and experimental study in shock tube of non-equilibrium phenomena in a closed-cycle MHD generator
conference = 8^th International Conference on MHD Electrical Power Generation
booktitle = Proceedings
editor = International Atomic Energy Agency, Warsaw, Poland
volume = 2
pages = 745–750] [cite journal
author = J.P. Petit
coauthors = J. Valensi
date = September 1, 1969
title = Growth rate of electrothermal instability and critical Hall parameter in closed-cycle MHD generators when the electron mobility is variable
journal = Comptes rendus de l'Académie des sciences
publisher = French Academy of Sciences
location = Paris
issue = 269
pages = 365&ndash;367] The stability is given by a quick transition to "fully ionized" conditions (fast enough to overtake the growth rate of the electrothermal instability) where the Hall parameter decreases cause of the collision frequency rising, below its critical value which is then about 2. Stable operation with several megawatts in power output had been experimentally achieved as from 1967 with high electron temperature. [cite conference
author = J.P. Petit
coauthors = J. Valensi, J.P. Caressa
date = 24–30 July 1968
title = Electrical characteristics of a converter using as a conversion fluid a binary mix of rare gases with non-equilibrium ionization
conference = 8^th International Conference on MHD Electrical Power Generation
booktitle = Proceedings
editor = International Atomic Energy Agency, Warsaw, Poland
volume = 3] [cite journal
author = J.P. Petit
coauthors = J. Valensi, D. Dufresnes, J.P. Caressa
date = January 27, 1969
title = Electrical characteristics of a Faraday linear generator using a binary mix of rare gases, with non-equilibrium ionization
journal = Comptes rendus de l'Académie des sciences
publisher = French Academy of Sciences
location = Paris
volume = 268
issue = A
pages = 245&ndash;247] [cite journal
author = S. Hatori
coauthors = S. Shioda
year = 1974
month = March
title = Stabilization of Ionization Instability in an MHD Generator
journal = Journal of the Physical Society of Japan
volume = 36
issue = 3
publisher = Tokyo Institute of Technology, Yokohama, Japan
pages = 920–920
doi = 10.1143/JPSJ.36.920] But this electrothermal control does not allow to decrease T_g low enough for long duration conditions (thermal ablation) so such a solution is not practical for any industrial energy conversion.

Another idea to control the instability would be to increase non-thermal ionisation rate thanks to a laser which would act like a guidance system for streamers between electrodes, increasing the electron density and the conductivity, therefore lowering the Hall parameter under its critical value along these paths. But this concept has never been tested experimentally.

In the 1970s and more recently, some researchers tried to master the instability thanks to oscillating fields. Oscillations of the electric field or of an additional RF electromagnetic field locally modify the Hall parameter. [cite journal
author = G.I. Shapiro
coauthors = A.H. Nelsone
date = April 12, 1978
title = Stabilization of ionization instability in a variable electric field
journal = Pis'ma v Zhurnal Tekhnicheskoi Fiziki
volume = 4
issue = 12
publisher = Akademiia Nauk SSSR, Institut Problem Mekhaniki, Moscow, USSR
pages = 393–396] [cite journal
author = T. Murakami
coauthors = Y. Okuno, H. Yamasaki
year = 2005
month = May
month = December
title = Dynamic stabilization of the electrothermal instability
journal = Applied Physics Letters
volume = 86
issue = 19
publisher = Tokyo Institute of Technology, Yokohama, Japan
pages = 191502–191502.3
doi = 10.1063/1.1926410]

Finally, a solution has been found in the early 1980s to annihilate completely the electrothermal instability within MHD converters, thanks to non-homogeneous magnetic fields. A strong magnetic field implies a high Hall parameter, therefore a low electrical conductivity in the medium. So the idea is to make some "paths" linking an electrode to the other, "where the magnetic field is locally attenuated". Then the electric current tends to flow in these low B-field paths as thin plasma cords or "streamers", where the electron density and temperature increase. The plasma becomes locally Coulombian, and the local Hall parameter value falls, while its critical threshold is risen. Experiments where streamers do not present any inhomogeneity has been obtained with this method. [cite journal
author = J.P. Petit
coauthors = M. Billiotte
date = April 27, 1981
title = Method for eliminating the Velikhov instability
journal = Comptes-rendus de l'Académie des Sciences
publisher = French Academy of Sciences
location = Paris
pages = 158–161] [cite conference
author = J.P. Petit
year = 1983
month = September
title = Cancellation of the Velikhov instability by magnetic confinment
conference = 8^th International Conference on MHD Electrical Power Generation
booktitle = Moscow, Russia
url = http://www.mhdprospects.com/pdf/cancellation_of_the_velikhov_instability_by_magnetic_confinment.pdf
format = PDF] This effect, strongly nonlinear, was unexpected but led to a very effective system for streamer guidance.

But this last working solution was discovered too late, 10 years after all the international effort about MHD power generation had been abandoned in most nations. Vladimir S. Golubev, coworker of Evgeny Velikhov, who met Jean-Pierre Petit in 1983 at the 9^th MHD International conference in Moscow, made the following comment to the inventor of the magnetic stabilization method:

cquote
"You bring the cure, but the patient already died..."

However it should be noted that this electrothermal stablization by magnetic confinment, if found too late for the development of MHD power plants, might be of interest for future applications of MHD to aerodynamics (magnetoplasma-aerodynamics for hypersonic flight).

See also

* Magnetohydrodynamics
* MHD generator
* Evgeny Velikhov

External links

* M. Mitchner, C.H. Kruger Jr., [http://navier.stanford.edu/PIG/C4_S10.pdf Two-temperature ionization instability] : Chapter 4 (MHD) - Section 10, pp. 230–241. From the plasma physics course book [http://navier.stanford.edu/PIG/PIGdefault.html Partially Ionized Gases] , John Wiley & Sons, 1973 (reprint 1992), Mechanical Engineering Department, Stanford University, CA, USA. ISBN 0-471-61172-7

References