__NOTOC__
The electrothermal instability (also known as
ionization instability, non-equilibrium instability or Velikhov instability in the literature) is a
magnetohydrodynamic
Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydromagnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magnetofluids include plasmas, liquid metals, ...
(MHD)
instability
In numerous fields of study, the component of instability within a system is generally characterized by some of the outputs or internal states growing without bounds. Not all systems that are not stable are unstable; systems can also be mar ...
appearing in
magnetized non-thermal plasmas used in
MHD converter __NOTOC__
A magnetohydrodynamic converter (MHD converter) is an electromagnetic machine with no moving parts involving magnetohydrodynamics, the study of the kinetics of electrically conductive fluids (liquid or ionized gas) in the presence of el ...
s. It was first theoretically discovered in 1962 and experimentally measured into a
MHD generator in 1963 by
Evgeny Velikhov.
Physical explanation and characteristics
![electrothermal instability evolution](https://upload.wikimedia.org/wikipedia/commons/2/28/Electrothermal_instability_evolution.gif)
This instability is a
turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
of the
electron gas
An ideal Fermi gas is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. T ...
in a
non-equilibrium plasma (i.e. where the
electron temperature
Plasma parameters define various characteristics of a plasma, an electrically conductive collection of charged particles that responds ''collectively'' to electromagnetic forces. Plasma typically takes the form of neutral gas-like clouds or char ...
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
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no ...
s and their temperature fluctuate (
electron density
In quantum chemistry, electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial va ...
and
thermal velocity Thermal velocity or thermal speed is a typical velocity of the thermal motion of particles that make up a gas, liquid, etc. Thus, indirectly, thermal velocity is a measure of temperature. Technically speaking, it is a measure of the width of the pea ...
) 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
A Geissler tube is an early gas discharge tube used to demonstrate the principles of electrical glow discharge, similar to modern neon lighting. The tube was invented by the German physicist and glassblower Heinrich Geissler in 1857. It cons ...
) provided on vacuum pumps. In this little
gas-discharge bulb a high
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
electric potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
is applied between two
electrode
An electrode is an electrical conductor used to make contact with a nonmetallic part of a circuit (e.g. a semiconductor, an electrolyte, a vacuum or air). Electrodes are essential parts of batteries that can consist of a variety of materials d ...
s which generates an
electric glow discharge
A glow discharge is a plasma formed by the passage of electric current through a gas. It is often created by applying a voltage between two electrodes in a glass tube containing a low-pressure gas. When the voltage exceeds a value called the st ...
(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
The Hall effect is the production of a voltage difference (the Hall voltage) across an electrical conductor that is transverse to an electric current in the conductor and to an applied magnetic field perpendicular to the current. It was dis ...
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
Cyclotron resonance describes the interaction of external forces with charged particles experiencing a magnetic field, thus already moving on a circular path. It is named after the cyclotron, a cyclic particle accelerator that utilizes an oscillati ...
Ω
e and the electron-heavy particles collision frequency ν:
:
where
: ''e'' is the
electron charge
The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...
(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,
:
where ''σ'' is the
electrical conductivity (in
siemens per metre),
is a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, because the electrical conductivity σ is a matrix:
:
σ
S is the scalar electrical conductivity:
:
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 strongest.
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
:
where μ is the
electron mobility (in m
2/(
V·
s))
and
:
where ''E
i'' is the
ionization energy (in
electron volt
In physics, an electronvolt (symbol eV, also written electron-volt and electron volt) is the measure of an amount of kinetic energy gained by a single electron accelerating from rest through an electric potential difference of one volt in vacuum ...
s) and ''k'' the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas 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 :
:
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 ν
ei < ν
en ):
:
* For a fully ionized gas (Coulombian plasma, when ν
ei > ν
en ):
:
NB: The term "fully ionized gas", introduced by
Lyman Spitzer
Lyman Spitzer Jr. (June 26, 1914 – March 31, 1997) was an American theoretical physicist, astronomer and mountaineer. As a scientist, he carried out research into star formation, plasma physics, and in 1946, conceived the idea of telesco ...
, 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
A thermal column (or thermal) is a rising mass of buoyant air, a convective current in the atmosphere, that transfers heat energy vertically. Thermals are created by the uneven heating of Earth's surface from solar radiation, and are an example ...
ablation
Ablation ( la, ablatio – removal) is removal or destruction of something from an object by vaporization, chipping, erosive processes or by other means. Examples of ablative materials are described below, and include spacecraft material for a ...
. This idea was first introduced for MHD generators in the early 1960s by
Jack L. Kerrebrock and
Alexander E. Sheindlin.
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, 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
A power station, also referred to as a power plant and sometimes generating station or generating plant, is an industrial facility for the generation of electric power. Power stations are generally connected to an electrical grid.
Many pow ...
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.
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.
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
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fi ...
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 with oscillating fields. Oscillations of the electric field or of an additional RF electromagnetic field locally modify the Hall parameter.
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 rises. Experiments where streamers do not present any inhomogeneity have been obtained with this method.
This effect, strongly
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
, 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
Jean-Pierre Petit is a French engineer.
Education
Jean-Pierre Petit obtained his engineer's degree in 1961 at the Institut supérieur de l'aéronautique et de l'espace (Supaéro). Petit defended his doctoral thesis, ''Applications de la thé ...
in 1983 at the 9th MHD International conference in Moscow, made the following comment to the inventor of the magnetic stabilization method:
However, this electrothermal stabilization by magnetic confinement, 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
Hypersonic flight is flight through the atmosphere below altitudes of about 90 km at speeds greater than Mach 5, a speed where dissociation of air begins to become significant and high heat loads exist. Speeds of Mach 25+ have been achiev ...
).
See also
*
Magnetohydrodynamics
Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydromagnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magnetofluids include plasmas, liquid metals, ...
*
MHD generator
*
Evgeny Velikhov
External links
* M. Mitchner, C.H. Kruger Jr.
Two-temperature ionization instability Chapter 4 (MHD) – Section 10, pp. 230–241. From the plasma physics course boo
John Wiley & Sons
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, ...
, 1973 (reprint 1992), Mechanical Engineering Department,
Stanford University, CA, USA.
References
{{reflist
Plasma instabilities