Electrothermal instability
Encyclopedia
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
.
of the electron gas
in a non-equilibrium
plasma (i.e. where the electron temperature
Te is greatly higher than the overall gas temperature Tg). It arises when a magnetic field
powerful enough is applied in such a plasma, reaching a critical Hall parameter βcr.
Locally, the number of electron
s 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 electrode
s 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".
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 Ωe and the electron-heavy particles collision frequency ν:
where
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:
,
where σ is the electrical conductivity (in siemens
per metre),
is a matrix
, because the electrical conductivity σ is a matrix:
σS is the scalar electrical conductivity:
where ne 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.
We have
where μ is the electron mobility
(in m2/(V
·s
))
and
where Ei 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 βcr greatly varies according to the degree of ionization
α :
where ni is the ion density and nn the neutral density (in particles per cubic metre).
The electron-ion collision frequency νei is much greater than the electron-neutral collision frequency νen.
Therefore with a weak energy degree of ionization α, the electron-ion collision frequency νei can equal the electron-neutral collision frequency νen.
Magnetohydrodynamics
Magnetohydrodynamics is an academic discipline which studies the dynamics of electrically conducting fluids. Examples of such fluids include plasmas, liquid metals, and salt water or electrolytes...
(MHD) instability
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...
appearing in magnetized non-thermal plasmas
Plasma (physics)
In physics and chemistry, plasma is a state of matter similar to gas in which a certain portion of the particles are ionized. Heating a gas may ionize its molecules or atoms , thus turning it into a plasma, which contains charged particles: positive ions and negative electrons or ions...
used in MHD converters. It was first theoretically discovered in 1962 and experimentally measured into a MHD generator
MHD generator
The MHD generator or dynamo transforms thermal energy or kinetic energy directly into electricity. MHD generators are different from traditional electric generators in that they can operate at high temperatures without moving parts...
in 1964 by Evgeny Velikhov
Evgeny Velikhov
Evgeny Pavlovich Velikhov is a physicist and scientific leader in the Russian Federation. His scientific interests include plasma physics, lasers, controlled nuclear fusion, power engineering and magnetohydrodynamics...
.
Physical explanation and characteristics
This instability is a turbulenceTurbulence
In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...
of the electron gas
Free electron model
In solid-state physics, the free electron model is a simple model for the behaviour of valence electrons in a crystal structure of a metallic solid. It was developed principally by Arnold Sommerfeld who combined the classical Drude model with quantum mechanical Fermi-Dirac statistics and hence it...
in a non-equilibrium
Non-equilibrium thermodynamics
Non-equilibrium thermodynamics is a branch of thermodynamics that deals with systems that are not in thermodynamic equilibrium. Most systems found in nature are not in thermodynamic equilibrium; for they are changing or can be triggered to change over time, and are continuously and discontinuously...
plasma (i.e. where the electron temperature
Electron temperature
If the velocities of a group of electrons, e.g., in a plasma, follow a Maxwell-Boltzmann distribution, then the electron temperature is well-defined as the temperature of that distribution...
Te is greatly higher than the overall gas temperature Tg). It arises when a magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
powerful enough is applied in such a plasma, reaching a critical Hall parameter βcr.
Locally, the number of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s and their temperature fluctuate (electron density
Electron density
Electron density is the measure of the probability of an electron being present at a specific location.In molecules, regions of electron density are usually found around the atom, and its bonds...
and thermal velocity
Thermal velocity
The thermal velocity or thermal speed is a typical velocity of the thermal motion of particles which 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 peak in the Maxwell-Boltzmann particle velocity...
) as the electric current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
and the electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
.
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
Standing wave
In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling...
phenomenon just applying a transverse magnetic field with a permanent magnet
Magnet
A magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, and attracts or repels other magnets.A permanent magnet is an object...
on the low-pressure control gauge (Geissler tube
Geissler tube
A Geissler tube is an early gas discharge tube used to demonstrate the principles of electrical glow discharge. The tube was invented by the German physicist and glassblower Heinrich Geissler in 1857...
) provided on vacuum pumps. In this little gas-discharge bulb
Gas-discharge lamp
Gas-discharge lamps are a family of artificial light sources that generate light by sending an electrical discharge through an ionized gas, i.e. a plasma. The character of the gas discharge critically depends on the frequency or modulation of the current: see the entry on a frequency classification...
a high voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
electric potential
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...
is applied between two electrode
Electrode
An electrode is an electrical conductor used to make contact with a nonmetallic part of a circuit...
s which generates an electric glow discharge
Electric glow discharge
An electric glow discharge is a plasma formed by the passage of current at 100 V to several kV through a gas, often argon or another noble gas. It is found in products such as neon lamps and plasma-screen televisions, and is used in plasma physics and analytical chemistry.-Basic operating...
(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 effectHall effect
The Hall effect is the production of a voltage difference across an electrical conductor, transverse to an electric current in the conductor and a magnetic field perpendicular to the current...
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 Ωe and the electron-heavy particles collision frequency ν:
where
- e is the electron chargeElementary chargeThe elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
(1.6 × 10−19 coulomb) - B is the magnetic field (in teslasTesla (unit)The tesla is the SI derived unit of magnetic field B . One tesla is equal to one weber per square meter, and it was defined in 1960 in honour of the inventor, physicist, and electrical engineer Nikola Tesla...
) - me 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
Current density
Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...
vector J is no more colinear with the electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
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 lawOhm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...
,
where σ is the electrical conductivity (in siemens
Siemens (unit)
The siemens is the SI derived unit of electric conductance and electric admittance. Conductance and admittance are the reciprocals of resistance and impedance respectively, hence one siemens is equal to the reciprocal of one ohm, and is sometimes referred to as the mho. In English, the term...
per metre),
is a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
, because the electrical conductivity σ is a matrix:
σS is the scalar electrical conductivity:
where ne 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 (Te > Tg) regime when the Hall parameter is higher that a critical value βcr.We have
where μ is the electron mobility
Electron mobility
In solid-state physics, the electron mobility characterizes how quickly an electron can move through a metal or semiconductor, when pulled by an electric field. In semiconductors, there is an analogous quantity for holes, called hole mobility...
(in m2/(V
Volt
The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...
·s
Second
The second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
))
and
where Ei is the ionization energy
Ionization potential
The ionization energy of a chemical species, i.e. an atom or molecule, is the energy required to remove an electron from the species to a practically infinite distance. Large atoms or molecules have a low ionization energy, while small molecules tend to have higher ionization energies.The property...
(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 βcr greatly varies according to the degree of ionization
Degree of ionization
The degree of ionization refers to the proportion of neutral particles, such as those in a gas or aqueous solution, that are ionized into charged particles...
α :
where ni is the ion density and nn the neutral density (in particles per cubic metre).
The electron-ion collision frequency νei is much greater than the electron-neutral collision frequency νen.
Therefore with a weak energy degree of ionization α, the electron-ion collision frequency νei can equal the electron-neutral collision frequency νen.
- For a weakly ionized gas (non-Coulombian plasma, when νei < νen ):
- For a fully ionized gas (Coulombian plasma, when νei > νen ):
NB: The term "fully ionized gas", introduced by Lyman SpitzerLyman SpitzerLyman Strong Spitzer, Jr. was an American theoretical physicist and astronomer best known for his research in star formation, plasma physics, and in 1946, for conceiving the idea of telescopes operating in outer space...
, 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 (Te >> Tg) is a key feature for practical MHD converters, because it allows the gas to reach sufficient electrical conductivity while protecting materials from thermalHeatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
ablationAblationAblation is removal of material from the surface of an object by vaporization, chipping, or other erosive processes. This occurs in spaceflight during ascent and atmospheric reentry, glaciology, medicine, and passive fire protection.-Spaceflight:...
. 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 densityCurrent densityCurrent density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...
due to the electrothermal instability ruined many MHD projects worldwide, while previous calculation envisaged energy conversion efficienciesEnergy conversion efficiencyEnergy conversion efficiency is the ratio between the useful output of an energy conversion machine and the input, in energy terms. The useful output may be electric power, mechanical work, or heat.-Overview:...
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 plantsPower stationA power station is an industrial facility for the generation of electric energy....
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 Tg 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 laserLaserA laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
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.
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. 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 PetitJean-Pierre PetitJean-Pierre Petit is a French scientist, senior researcher at National Center for Scientific Research as an astrophysicist in Marseille Observatory, now retired...
in 1983 at the 9th MHD International conference in Moscow, made the following comment to the inventor of the magnetic stabilization method:
However it should be noted that 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 flightHypersonic flightHypersonic flight is flight through the atmosphere at speeds above about Mach 5.5, a speed where disassociation of air begins to become significant and high heat loads exist....
).
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 book Partially Ionized Gases, John Wiley & SonsJohn Wiley & SonsJohn Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing and markets its products to professionals and consumers, students and instructors in higher education, and researchers and practitioners in scientific, technical, medical, and...
, 1973 (reprint 1992), Mechanical Engineering Department, Stanford UniversityStanford UniversityThe Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is a private research university on an campus located near Palo Alto, California. It is situated in the northwestern Santa Clara Valley on the San Francisco Peninsula, approximately northwest of San...
, CA, USA. ISBN 0-471-61172-7
- M. Mitchner, C.H. Kruger Jr., Two-temperature ionization instability: Chapter 4 (MHD) – Section 10, pp. 230–241. From the plasma physics course book Partially Ionized Gases, John Wiley & Sons
- For a fully ionized gas (Coulombian plasma, when νei > νen ):